Atlas › Modular forms › Weight 2 › Level 749 › 749.2.a

Newspace 749.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(749))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
749.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$	749a1 rank 1
749.2.a.b	6	degree 6	$6q - q^{2} - 4q^{3} + q^{4} - 5q^{6} + 6q^{7} + O(q^{8})$	− −	$-1$
749.2.a.c	12	degree 12	$12q - 5q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 12q^{7} + O(q^{8})$	+ +	$-1$
749.2.a.d	14	degree 14	$14q + q^{2} + 5q^{3} + 17q^{4} + 2q^{5} + 12q^{6} - 14q^{7} + O(q^{8})$	+ −	$+1$
749.2.a.e	20	degree 20	$20q + 2q^{2} + 3q^{3} + 32q^{4} - 2q^{5} + 20q^{7} + O(q^{8})$	− +	$+1$

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 $T$
749.2.a.a	$x$
749.2.a.b	$x^{6} + 5x^{5} + x^{4} - 16x^{3} + x^{2} + 11x - 4$
749.2.a.c	$x^{12} + 5x^{11} - 20x^{10} - 100x^{9} + 125x^{8} + 608x^{7} - 215x^{6} - 1466x^{5} - 105x^{4} + 1466x^{3} + 440x^{2} - 512x - 226$
749.2.a.d	$x^{14} - 6x^{13} - 45x^{12} + 301x^{11} + 612x^{10} - 5183x^{9} - 2229x^{8} + 37649x^{7} - 6392x^{6} - 116657x^{5} + 42842x^{4} + 138850x^{3} - 52673x^{2} - 32348x + 6376$
749.2.a.e	$x^{20} - 5x^{19} - 68x^{18} + 370x^{17} + 1495x^{16} - 9130x^{15} - 14047x^{14} + 100304x^{13} + 70742x^{12} - 559175x^{11} - 232136x^{10} + 1601906x^{9} + 476205x^{8} - 2190936x^{7} - 296383x^{6} + 1392038x^{5} - 74673x^{4} - 325270x^{3} + 61456x^{2} + 9592x - 1186$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(749))$	53
Old forms	18
Cusp forms $S_{2}(\Gamma_0(749))$	71
Eisenstein series $E_{2}(\Gamma_0(749))$	3
Modular forms $M_{2}(\Gamma_0(749))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(749)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 7, 107, 749]: [0, 1, 10, 74] (certified at ell = 2147482621)
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(749, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 749:2 1000

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