Atlas › Modular forms › Weight 2 › Level 650 › 650.2.a

Newspace 650.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(650))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
650.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + 3q^{6} + O(q^{8})$	+ +	650c1 rank 1
650.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - q^{7} + O(q^{8})$	+ −	650g1 rank 1
650.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	+ −	650e1 rank 0
650.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + O(q^{8})$	+ +	650a1 rank 1
650.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + 4q^{7} + O(q^{8})$	+ −	650d1 rank 0
650.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 5q^{7} + O(q^{8})$	+ +	650b1 rank 1
650.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} - q^{7} + O(q^{8})$	+ −	650f1 rank 0
650.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + 5q^{7} + O(q^{8})$	− −	650l1 rank 0
650.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	− +	650k1 rank 1
650.2.a.j	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	− +	650h1 rank 0
650.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + q^{7} + O(q^{8})$	− +	650i1 rank 0
650.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	650j1 rank 0
650.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} + 3q^{6} + O(q^{8})$	− −	650m1 rank 0
650.2.a.n	3	degree 3	$3q - 3q^{2} + 3q^{4} + 2q^{7} + O(q^{8})$	+ +
650.2.a.o	3	degree 3	$3q + 3q^{2} + 3q^{4} - 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
650.2.a.a	$x + 7$
650.2.a.b	$x + 68$
650.2.a.c	$x + 48$
650.2.a.d	$x + 70$
650.2.a.e	$x - 45$
650.2.a.f	$x - 18$
650.2.a.g	$x - 107$
650.2.a.h	$x - 130$
650.2.a.i	$x + 29$
650.2.a.j	$x - 51$
650.2.a.k	$x + 12$
650.2.a.l	$x + 36$
650.2.a.m	$x + 105$
650.2.a.n	$x^{3} - 62x^{2} - 4535x + 240686$
650.2.a.o	$x^{3} - 158x^{2} + 7065x - 80530$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(650))$	19
Old forms	74
Cusp forms $S_{2}(\Gamma_0(650))$	93
Eisenstein series $E_{2}(\Gamma_0(650))$	23
Modular forms $M_{2}(\Gamma_0(650))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(650)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(65))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(130))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(325))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.748 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, 13, 25, 26, 50, 65, 130, 325, 650]: [0, 1, 1, 3, 1, 3, 5, 9, 8, 24, 36, 108] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 6 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(650, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 650:2 1000

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