Atlas › Modular forms › Weight 2 › Level 767 › 767.2.a

Newspace 767.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(767))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
767.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
767.2.a.b	3	degree 3	$3q - 3q^{2} - q^{3} - 3q^{4} - q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
767.2.a.c	4	degree 4	$4q + q^{2} - 2q^{3} + 3q^{4} - q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
767.2.a.d	10	degree 10	$10q - 3q^{3} + 6q^{4} - 6q^{5} - 8q^{6} - 11q^{7} + O(q^{8})$	+ +	$-1$
767.2.a.e	17	degree 17	$17q + 3q^{2} + 5q^{3} + 15q^{4} + 4q^{5} + 6q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
767.2.a.f	23	degree 23	$23q + 7q^{3} + 30q^{4} + 4q^{5} - 6q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$

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 $T$
767.2.a.a	$x^{2} + 4x - 4$
767.2.a.b	$x^{3} + 4x^{2} - 2$
767.2.a.c	$x^{4} + x^{3} - 5x^{2} - 7x - 1$
767.2.a.d	$x^{10} + 3x^{9} - 20x^{8} - 49x^{7} + 123x^{6} + 168x^{5} - 350x^{4} - 41x^{3} + 214x^{2} - 52x + 2$
767.2.a.e	$x^{17} - 8x^{16} - 36x^{15} + 399x^{14} + 208x^{13} - 6817x^{12} + 2145x^{11} + 58409x^{10} - 26020x^{9} - 283010x^{8} + 80825x^{7} + 789364x^{6} + 9914x^{5} - 1172712x^{4} - 424769x^{3} + 676118x^{2} + 506904x + 87206$
767.2.a.f	$x^{23} - 7x^{22} - 59x^{21} + 473x^{20} + 1270x^{19} - 12934x^{18} - 10682x^{17} + 184572x^{16} - 11016x^{15} - 1471000x^{14} + 806348x^{13} + 6465044x^{12} - 5337305x^{11} - 14395758x^{10} + 13936888x^{9} + 13585663x^{8} - 14180819x^{7} - 4409444x^{6} + 5151475x^{5} + 689090x^{4} - 758448x^{3} - 77818x^{2} + 38320x + 5328$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(767))$	59
Old forms	10
Cusp forms $S_{2}(\Gamma_0(767))$	69
Eisenstein series $E_{2}(\Gamma_0(767))$	3
Modular forms $M_{2}(\Gamma_0(767))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(767)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 13, 59, 767]: [0, 1, 6, 72] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (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(767, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 767:2 1000

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