Atlas › Modular forms › Weight 2 › Level 935 › 935.2.a

Newspace 935.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(935))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
935.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} - q^{5} + 3q^{7} + O(q^{8})$	+ − −	$-1$	935a1 rank 1
935.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} + q^{5} + 5q^{7} + O(q^{8})$	− + +	$+1$	935b1 rank 0
935.2.a.c	3	degree 3	$3q - 2q^{2} - q^{3} + 3q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$
935.2.a.d	3	degree 3	$3q - 2q^{2} - q^{3} + 4q^{4} - 3q^{5} + 5q^{6} - 6q^{7} + O(q^{8})$	+ − −	$-1$
935.2.a.e	3	degree 3	$3q - 3q^{3} + 3q^{5} - 3q^{6} + O(q^{8})$	− + −	$-1$
935.2.a.f	3	degree 3	$3q + q^{2} + q^{3} + 3q^{4} - 3q^{5} + 9q^{6} + O(q^{8})$	+ + −	$+1$
935.2.a.g	4	degree 4	$4q - q^{3} + 2q^{4} - 4q^{5} - q^{6} + q^{7} + O(q^{8})$	+ + +	$-1$
935.2.a.h	6	degree 6	$6q - 2q^{2} + 4q^{3} + 8q^{4} - 6q^{5} - 10q^{6} + 6q^{7} + O(q^{8})$	+ + −	$+1$
935.2.a.i	9	degree 9	$9q + 3q^{2} + 3q^{3} + 11q^{4} - 9q^{5} + 9q^{6} + 8q^{7} + O(q^{8})$	+ − +	$+1$
935.2.a.j	11	degree 11	$11q - q^{2} + 5q^{3} + 21q^{4} + 11q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
935.2.a.k	11	degree 11	$11q + q^{3} + 12q^{4} + 11q^{5} + 7q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{19}$ 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 $T$
935.2.a.a	$x - 9$
935.2.a.b	$x + 45$
935.2.a.c	$x^{3} + 162x^{2} + 7271x + 55859$
935.2.a.d	$x^{3} + 56x^{2} + 79x - 14951$
935.2.a.e	$x^{3} + 72x^{2} + 405x - 36369$
935.2.a.f	$x^{3} - 33x^{2} - 4206x + 21037$
935.2.a.g	$x^{4} + 107x^{3} + 1375x^{2} - 124346x - 2298391$
935.2.a.h	$x^{6} - 92x^{5} - 8752x^{4} + 1449588x^{3} - 68605567x^{2} + 1358399674x - 9522159621$
935.2.a.i	$x^{9} - 321x^{8} + 37662x^{7} - 1673175x^{6} - 18029415x^{5} + 4064262457x^{4} - 110483585239x^{3} - 18830643442x^{2} + 33594374545822x - 272608762193991$
935.2.a.j	$x^{11} - 131x^{10} - 15269x^{9} + 1917632x^{8} + 93800341x^{7} - 9233602520x^{6} - 278628761014x^{5} + 14507295901499x^{4} + 311990848742617x^{3} - 1096680867619979x^{2} - 11646089767863942x + 21030195703171617$
935.2.a.k	$x^{11} - 163x^{10} - 3821x^{9} + 1536080x^{8} - 17584987x^{7} - 4613611584x^{6} + 59660689512x^{5} + 5854973561291x^{4} - 27825393485427x^{3} - 2529653329841501x^{2} + 1360396429539412x + 272613537858153363$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(935))$	55
Old forms	50
Cusp forms $S_{2}(\Gamma_0(935))$	105
Eisenstein series $E_{2}(\Gamma_0(935))$	7
Modular forms $M_{2}(\Gamma_0(935))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(935)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(85))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(187))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.48 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, 5, 11, 17, 55, 85, 187, 935]: [0, 1, 2, 2, 8, 10, 20, 112] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (13, 7), (19, 9)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 14 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(935, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 935:2 1000

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