Atlas › Modular forms › Weight 2 › Level 854 › 854.2.a

Newspace 854.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(854))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
854.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	854a1 rank 1
854.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	854b1 rank 1
854.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	854d1 rank 1
854.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	854c1 rank 1
854.2.a.e	6	degree 6	$6q - 6q^{2} - 2q^{3} + 6q^{4} + 3q^{5} + 2q^{6} - 6q^{7} + O(q^{8})$	+ + −	$+1$
854.2.a.f	6	degree 6	$6q - 6q^{2} + 4q^{3} + 6q^{4} + 3q^{5} - 4q^{6} + 6q^{7} + O(q^{8})$	+ − +	$+1$
854.2.a.g	6	degree 6	$6q + 6q^{2} + 4q^{3} + 6q^{4} + 3q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$
854.2.a.h	7	degree 7	$7q + 7q^{2} + 2q^{3} + 7q^{4} - 3q^{5} + 2q^{6} - 7q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
854.2.a.a	$x - 37$
854.2.a.b	$x + 69$
854.2.a.c	$x + 39$
854.2.a.d	$x + 53$
854.2.a.e	$x^{6} - 52x^{5} - 10162x^{4} + 426903x^{3} + 19674943x^{2} - 520514664x - 7451753413$
854.2.a.f	$x^{6} - 54x^{5} - 5462x^{4} + 94509x^{3} + 9949621x^{2} + 133067276x - 58935069$
854.2.a.g	$x^{6} - 140x^{5} + 4398x^{4} + 92283x^{3} - 3000601x^{2} - 62435568x - 290548281$
854.2.a.h	$x^{7} - 40x^{6} - 7270x^{5} + 77673x^{4} + 17827847x^{3} + 246699214x^{2} - 5827736813x - 102585801686$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(854))$	29
Old forms	92
Cusp forms $S_{2}(\Gamma_0(854))$	121
Eisenstein series $E_{2}(\Gamma_0(854))$	7
Modular forms $M_{2}(\Gamma_0(854))$	128

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(854)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(122))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(427))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.436 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, 61, 122, 427, 854]: [0, 1, 1, 4, 5, 17, 42, 128] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 8 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(854, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 854:2 1000

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