Atlas › Modular forms › Weight 2 › Level 744 › 744.2.a

Newspace 744.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(744))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
744.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	+ +	744e1 rank 0
744.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	+ +	744a1 rank 1
744.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 3q^{7} + O(q^{8})$	+ −	744f1 rank 1
744.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + 5q^{7} + O(q^{8})$	+ +	744d1 rank 0
744.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} + q^{7} + O(q^{8})$	− −	744c1 rank 1
744.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + O(q^{8})$	− +	744b1 rank 0
744.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 3q^{7} + O(q^{8})$	− +	744g1 rank 1
744.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{3} + 2q^{5} + 3q^{7} + O(q^{8})$	− −
744.2.a.i	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{3} + 6q^{5} - q^{7} + O(q^{8})$	− +
744.2.a.j	3	degree 3	$3q - 3q^{3} + 2q^{5} - 5q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ 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 $T$
744.2.a.a	$x - 71$
744.2.a.b	$x + 73$
744.2.a.c	$x + 103$
744.2.a.d	$x + 37$
744.2.a.e	$x + 99$
744.2.a.f	$x - 102$
744.2.a.g	$x + 33$
744.2.a.h	$x^{2} - 46x + 461$
744.2.a.i	$x^{2} - 64x - 8201$
744.2.a.j	$x^{3} - 134x^{2} + 5455x - 68338$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(744))$	14
Old forms	107
Cusp forms $S_{2}(\Gamma_0(744))$	121
Eisenstein series $E_{2}(\Gamma_0(744))$	15
Modular forms $M_{2}(\Gamma_0(744))$	136

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(744)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(93))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(186))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(248))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(372))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.718 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, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744]: [0, 1, 1, 2, 3, 3, 5, 8, 3, 10, 12, 19, 36, 36, 70, 136] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(744, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 744:2 1000

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