Atlas › Modular forms › Weight 2 › Level 742 › 742.2.a

Newspace 742.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(742))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
742.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	+ + +	$-1$	742a1 rank 1
742.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} + q^{7} + O(q^{8})$	+ − +	$+1$	742c1 rank 0
742.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 4q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	742e1 rank 1
742.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	742b1 rank 0
742.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	742d1 rank 0
742.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	742g1 rank 1
742.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} - 2q^{5} + 3q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	742f1 rank 0
742.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
742.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$
742.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	− + +	$+1$
742.2.a.k	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 2q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
742.2.a.l	3	degree 3	$3q + 3q^{2} - 4q^{3} + 3q^{4} - 5q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
742.2.a.m	6	degree 6	$6q + 6q^{2} + 2q^{3} + 6q^{4} - q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ 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 $T$
742.2.a.a	$x + 47$
742.2.a.b	$x - 33$
742.2.a.c	$x + 100$
742.2.a.d	$x - 74$
742.2.a.e	$x - 102$
742.2.a.f	$x + 166$
742.2.a.g	$x + 34$
742.2.a.h	$x^{2} - 156x - 414$
742.2.a.i	$x^{2} + 76x + 386$
742.2.a.j	$x^{2} - 56x - 1016$
742.2.a.k	$x^{3} + 122x^{2} + 1920x - 20480$
742.2.a.l	$x^{3} + 171x^{2} + 7316x + 91834$
742.2.a.m	$x^{6} - 233x^{5} + 10040x^{4} + 825298x^{3} - 53835280x^{2} - 492790000x + 48112804000$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(742))$	25
Old forms	80
Cusp forms $S_{2}(\Gamma_0(742))$	105
Eisenstein series $E_{2}(\Gamma_0(742))$	7
Modular forms $M_{2}(\Gamma_0(742))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(742)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(371))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.62 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, 7, 14, 53, 106, 371, 742]: [0, 1, 1, 4, 5, 15, 38, 112] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 25 = 1 x 25 (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(742, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 742:2 1000

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