Atlas › Modular forms › Weight 2 › Level 737 › 737.2.a

Newspace 737.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(737))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
737.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 4q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	737a1 rank 1
737.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} - 4q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
737.2.a.c	8	degree 8	$8q + 2q^{2} - 6q^{3} + 4q^{4} - 6q^{5} - 2q^{6} - 10q^{7} + O(q^{8})$	− −	$-1$
737.2.a.d	12	degree 12	$12q - 2q^{2} - 4q^{3} + 8q^{4} + 4q^{5} - 4q^{6} - 20q^{7} + O(q^{8})$	+ +	$-1$
737.2.a.e	15	degree 15	$15q + 2q^{2} + 6q^{3} + 18q^{4} - 4q^{5} + 4q^{6} + 24q^{7} + O(q^{8})$	+ −	$+1$
737.2.a.f	17	degree 17	$17q + q^{2} + 10q^{3} + 19q^{4} + 10q^{5} - 6q^{6} + 20q^{7} + O(q^{8})$	− +	$+1$

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 $T$
737.2.a.a	$x - 4$
737.2.a.b	$x^{2} + 12x + 4$
737.2.a.c	$x^{8} + 16x^{7} - 11x^{6} - 1065x^{5} - 2463x^{4} + 15092x^{3} + 42611x^{2} - 10029x - 61865$
737.2.a.d	$x^{12} + 14x^{11} - 113x^{10} - 1915x^{9} + 3477x^{8} + 78892x^{7} - 73317x^{6} - 1158127x^{5} + 1372719x^{4} + 2741282x^{3} - 236656x^{2} - 449796x + 54420$
737.2.a.e	$x^{15} - 20x^{14} - 106x^{13} + 4207x^{12} - 7555x^{11} - 297401x^{10} + 1154057x^{9} + 9638078x^{8} - 46565611x^{7} - 159971067x^{6} + 813310990x^{5} + 1495510897x^{4} - 6241732493x^{3} - 8315944528x^{2} + 15508607612x + 20025303888$
737.2.a.f	$x^{17} - 31x^{16} + 130x^{15} + 5263x^{14} - 61600x^{13} - 111232x^{12} + 4862140x^{11} - 15540389x^{10} - 125227417x^{9} + 824489544x^{8} + 317875805x^{7} - 12882535945x^{6} + 21924665530x^{5} + 61000116027x^{4} - 207753488630x^{3} + 59844796152x^{2} + 352210238192x - 315401306708$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(737))$	55
Old forms	12
Cusp forms $S_{2}(\Gamma_0(737))$	67
Eisenstein series $E_{2}(\Gamma_0(737))$	3
Modular forms $M_{2}(\Gamma_0(737))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(737)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 11, 67, 737]: [0, 2, 6, 70] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(737, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 737:2 1000

← 736.2.a · 738.2.a →