Atlas › Modular forms › Weight 2 › Level 850 › 850.2.a

Newspace 850.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(850))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
850.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + 3q^{6} - q^{7} + O(q^{8})$	+ −	850e1 rank 1
850.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	+ −	850a1 rank 0
850.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 5q^{7} + O(q^{8})$	+ −	850d1 rank 1
850.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + O(q^{8})$	+ −	850c1 rank 1
850.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + 4q^{7} + O(q^{8})$	+ −	850b1 rank 0
850.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 3q^{6} - 2q^{7} + O(q^{8})$	− +	850j1 rank 0
850.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 2q^{7} + O(q^{8})$	− −	850k1 rank 1
850.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + O(q^{8})$	− +	850l1 rank 1
850.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + 5q^{7} + O(q^{8})$	− +	850f1 rank 0
850.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	− +	850h1 rank 0
850.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	850g1 rank 0
850.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} + 3q^{6} + q^{7} + O(q^{8})$	− +	850i1 rank 0
850.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +
850.2.a.n	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} - 2q^{7} + O(q^{8})$	+ +
850.2.a.o	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	− −
850.2.a.p	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} + O(q^{8})$	+ +
850.2.a.q	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} + q^{6} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
850.2.a.a	$x - 61$
850.2.a.b	$x - 183$
850.2.a.c	$x + 359$
850.2.a.d	$x + 19$
850.2.a.e	$x + 24$
850.2.a.f	$x - 185$
850.2.a.g	$x + 75$
850.2.a.h	$x + 77$
850.2.a.i	$x - 83$
850.2.a.j	$x + 120$
850.2.a.k	$x + 16$
850.2.a.l	$x + 49$
850.2.a.m	$x^{2} + 158x - 6881$
850.2.a.n	$x^{2} + 305x + 16792$
850.2.a.o	$x^{2} - 206x + 10591$
850.2.a.p	$x^{3} - 457x^{2} + 65424x - 2914400$
850.2.a.q	$x^{3} - 195x^{2} - 20400x + 1560376$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(850))$	24
Old forms	99
Cusp forms $S_{2}(\Gamma_0(850))$	123
Eisenstein series $E_{2}(\Gamma_0(850))$	23
Modular forms $M_{2}(\Gamma_0(850))$	146

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(850)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(170))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(425))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.428 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 5, 10, 17, 25, 34, 50, 85, 170, 425, 850]: [0, 1, 1, 3, 2, 3, 6, 9, 10, 30, 46, 138] (certified at ell = 2147482801)
T = [(3, 1), (7, 2), (11, 3), (13, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(850, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 850:2 1000

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