Atlas › Modular forms › Weight 2 › Level 785 › 785.2.a

Newspace 785.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(785))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
785.2.a.a	4	degree 4	$4q - 2q^{2} - 4q^{3} + 2q^{4} - 4q^{5} - 4q^{6} + O(q^{8})$	+ +	$-1$
785.2.a.b	9	degree 9	$9q - q^{2} - 2q^{3} + 11q^{4} - 9q^{5} - 2q^{6} - 22q^{7} + O(q^{8})$	+ +	$-1$
785.2.a.c	10	degree 10	$10q - 2q^{2} - 4q^{3} + 4q^{4} + 10q^{5} - 10q^{6} - 14q^{7} + O(q^{8})$	− −	$-1$
785.2.a.d	14	degree 14	$14q + 2q^{2} + 6q^{3} + 16q^{4} - 14q^{5} + 4q^{6} + 20q^{7} + O(q^{8})$	+ −	$+1$
785.2.a.e	16	degree 16	$16q + 4q^{3} + 18q^{4} + 16q^{5} + 20q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
785.2.a.a	$x^{4} + 6x^{3} + 7x^{2} - 6x - 7$
785.2.a.b	$x^{9} + 3x^{8} - 26x^{7} - 35x^{6} + 228x^{5} - 65x^{4} - 359x^{3} + 136x^{2} + 174x - 29$
785.2.a.c	$x^{10} + 6x^{9} - 47x^{7} - 26x^{6} + 136x^{5} + 50x^{4} - 164x^{3} + 9x^{2} + 45x - 11$
785.2.a.d	$x^{14} - 8x^{13} - 22x^{12} + 277x^{11} + 21x^{10} - 3300x^{9} + 1694x^{8} + 17021x^{7} - 8535x^{6} - 38913x^{5} + 11519x^{4} + 31468x^{3} - 7953x^{2} - 7107x + 1731$
785.2.a.e	$x^{16} - 4x^{15} - 49x^{14} + 211x^{13} + 769x^{12} - 3719x^{11} - 4229x^{10} + 26171x^{9} + 4581x^{8} - 67822x^{7} + 5394x^{6} + 48243x^{5} - 1042x^{4} - 9157x^{3} + 632x^{2} + 509x - 61$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(785))$	53
Old forms	24
Cusp forms $S_{2}(\Gamma_0(785))$	77
Eisenstein series $E_{2}(\Gamma_0(785))$	3
Modular forms $M_{2}(\Gamma_0(785))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(785)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(157))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 157, 785]: [0, 1, 13, 80] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(785, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 785:2 1000

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