Atlas › Modular forms › Weight 2 › Level 291 › 291.2.a

Newspace 291.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(291))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
291.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$	291a1 rank 0
291.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$	291b1 rank 0
291.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	291c1 rank 1
291.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	291d1 rank 0
291.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - 2q^{3} + 3q^{4} - 6q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
291.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 4q^{5} - q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
291.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 2q^{3} + 3q^{4} + 6q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
291.2.a.h	7	degree 7	$7q + 7q^{3} + 8q^{4} + 4q^{5} + 9q^{7} + O(q^{8})$	− +	$+1$

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 $T$
291.2.a.a	$x - 7$
291.2.a.b	$x + 7$
291.2.a.c	$x + 1$
291.2.a.d	$x - 5$
291.2.a.e	$x^{2} + 19x + 87$
291.2.a.f	$x^{2} + 13x + 11$
291.2.a.g	$x^{2} - 21x + 109$
291.2.a.h	$x^{7} - 12x^{6} - 140x^{5} + 1507x^{4} + 8023x^{3} - 51956x^{2} - 205788x + 161693$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(291))$	17
Old forms	14
Cusp forms $S_{2}(\Gamma_0(291))$	31
Eisenstein series $E_{2}(\Gamma_0(291))$	3
Modular forms $M_{2}(\Gamma_0(291))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(291)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 97, 291]: [0, 1, 8, 34] (certified at ell = 2147483137)
T = [(2, 1), (5, 3)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 3 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(291, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 291:2 1000

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