Atlas › Modular forms › Weight 2 › Level 741 › 741.2.a

Newspace 741.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(741))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
741.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	− − +	$-1$	741e1 rank 1
741.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	741b1 rank 0
741.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$	741a1 rank 0
741.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + 3q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$	741c1 rank 0
741.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	741d1 rank 0
741.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 2q^{5} - 6q^{7} + O(q^{8})$	− + −	$-1$
741.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} - 2q^{3} + 5q^{4} + 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
741.2.a.h	3	degree 3	$3q + 3q^{3} + 6q^{4} - 3q^{5} + 3q^{7} + O(q^{8})$	− + +	$+1$
741.2.a.i	4	degree 4	$4q - q^{2} - 4q^{3} + 3q^{4} - 4q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$
741.2.a.j	5	degree 5	$5q - q^{2} - 5q^{3} + 3q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + +	$-1$
741.2.a.k	5	degree 5	$5q + q^{2} - 5q^{3} + 9q^{4} + 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ + −	$+1$
741.2.a.l	9	degree 9	$9q + 9q^{3} + 14q^{4} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
741.2.a.a	$x - 3$
741.2.a.b	$x + 8$
741.2.a.c	$x - 4$
741.2.a.d	$x - 10$
741.2.a.e	$x - 5$
741.2.a.f	$x^{2} + 6x + 7$
741.2.a.g	$x^{2} - 7x + 8$
741.2.a.h	$x^{3} + 9x^{2} - 69x - 138$
741.2.a.i	$x^{4} + 13x^{3} - 14x^{2} - 103x + 137$
741.2.a.j	$x^{5} + 4x^{4} - 159x^{3} - 591x^{2} + 6004x + 19757$
741.2.a.k	$x^{5} - 10x^{4} - 96x^{3} + 578x^{2} + 2151x - 7900$
741.2.a.l	$x^{9} - 328x^{7} + 391x^{6} + 35399x^{5} - 70852x^{4} - 1321404x^{3} + 2690593x^{2} + 11787340x - 7974732$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(741))$	35
Old forms	54
Cusp forms $S_{2}(\Gamma_0(741))$	89
Eisenstein series $E_{2}(\Gamma_0(741))$	7
Modular forms $M_{2}(\Gamma_0(741))$	96

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(741)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(247))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 3, 13, 19, 39, 57, 247, 741]: [0, 1, 1, 2, 6, 8, 24, 96] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 5 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(741, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 741:2 1000

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