Atlas › Modular forms › Weight 2 › Level 745 › 745.2.a

Newspace 745.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(745))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
745.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + q^{3} - q^{4} - 2q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
745.2.a.b	10	degree 10	$10q - 2q^{2} + 6q^{4} - 10q^{5} - 9q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
745.2.a.c	11	degree 11	$11q - 5q^{2} - 12q^{3} + 9q^{4} + 11q^{5} + 3q^{6} - 13q^{7} + O(q^{8})$	− −	$-1$
745.2.a.d	12	degree 12	$12q + 5q^{2} - 3q^{3} + 19q^{4} - 12q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
745.2.a.e	14	degree 14	$14q + 4q^{2} + 10q^{3} + 14q^{4} + 14q^{5} - q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
745.2.a.a	$x^{2} - 2x - 4$
745.2.a.b	$x^{10} + 2x^{9} - 110x^{8} - 74x^{7} + 3916x^{6} + 1170x^{5} - 55901x^{4} - 29482x^{3} + 306371x^{2} + 281090x - 205524$
745.2.a.c	$x^{11} + 41x^{10} + 618x^{9} + 3570x^{8} - 4180x^{7} - 135206x^{6} - 449891x^{5} + 194207x^{4} + 2938649x^{3} + 4336765x^{2} + 1813658x - 123316$
745.2.a.d	$x^{12} + 4x^{11} - 276x^{10} - 1064x^{9} + 27555x^{8} + 108010x^{7} - 1262801x^{6} - 5292888x^{5} + 25816053x^{4} + 126114402x^{3} - 140478393x^{2} - 1175519072x - 1255695271$
745.2.a.e	$x^{14} - 34x^{13} + 290x^{12} + 2172x^{11} - 43039x^{10} + 69594x^{9} + 1778759x^{8} - 7308428x^{7} - 28357831x^{6} + 169048842x^{5} + 177581081x^{4} - 1509242984x^{3} - 582496869x^{2} + 4772186886x + 2481836260$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(745))$	49
Old forms	24
Cusp forms $S_{2}(\Gamma_0(745))$	73
Eisenstein series $E_{2}(\Gamma_0(745))$	3
Modular forms $M_{2}(\Gamma_0(745))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(745)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(149))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 149, 745]: [0, 1, 13, 76] (certified at ell = 2147481629)
T = [(2, 1), (3, 3)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(745, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 745:2 1000

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