Atlas › Modular forms › Weight 2 › Level 902 › 902.2.a

Newspace 902.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(902))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
902.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 3q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ + +	$-1$	902a1 rank 1
902.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + 3q^{5} - 2q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$	902b1 rank 0
902.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
902.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
902.2.a.e	2	$\mathbb{Q}(\sqrt{21})$	$2q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$
902.2.a.f	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$
902.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 2q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
902.2.a.h	5	degree 5	$5q - 5q^{2} + 5q^{4} + 5q^{7} + O(q^{8})$	+ + −	$+1$
902.2.a.i	5	degree 5	$5q + 5q^{2} + 3q^{3} + 5q^{4} - q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	− − −	$+1$
902.2.a.j	5	degree 5	$5q + 5q^{2} + 4q^{3} + 5q^{4} - 6q^{5} + 4q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$
902.2.a.k	6	degree 6	$6q - 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ 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 $T$
902.2.a.a	$x + 93$
902.2.a.b	$x - 19$
902.2.a.c	$x^{2} + 83x + 1511$
902.2.a.d	$x^{2} + 109x + 869$
902.2.a.e	$x^{2} + 19x + 43$
902.2.a.f	$x^{3} + 44x^{2} + 28x - 2591$
902.2.a.g	$x^{3} + 43x^{2} - 1615x - 60206$
902.2.a.h	$x^{5} - 104x^{4} - 459x^{3} + 348065x^{2} - 11909598x + 112733879$
902.2.a.i	$x^{5} - 8x^{4} - 3684x^{3} - 28482x^{2} + 822699x + 6809938$
902.2.a.j	$x^{5} - 116x^{4} + 2087x^{3} + 107987x^{2} - 1599824x - 27829311$
902.2.a.k	$x^{6} - 106x^{5} - 465x^{4} + 382143x^{3} - 11214138x^{2} - 76563205x + 4496993506$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(902))$	35
Old forms	88
Cusp forms $S_{2}(\Gamma_0(902))$	123
Eisenstein series $E_{2}(\Gamma_0(902))$	7
Modular forms $M_{2}(\Gamma_0(902))$	130

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(902)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(451))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.468 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, 11, 22, 41, 82, 451, 902]: [0, 1, 2, 5, 4, 12, 44, 130] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 9 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(902, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 902:2 1000

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