Atlas › Modular forms › Weight 2 › Level 417 › 417.2.a

Newspace 417.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(417))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
417.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + O(q^{8})$	+ −	$+1$	417a1 rank 0
417.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 2q^{5} - q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
417.2.a.c	3	degree 3	$3q - 3q^{3} + 2q^{4} + 2q^{5} + 5q^{7} + O(q^{8})$	+ −	$+1$
417.2.a.d	3	degree 3	$3q + 3q^{3} + 2q^{4} + 4q^{5} + 5q^{7} + O(q^{8})$	− +	$+1$
417.2.a.e	7	degree 7	$7q - 3q^{2} - 7q^{3} + 7q^{4} - 4q^{5} + 3q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
417.2.a.f	7	degree 7	$7q + 7q^{3} + 14q^{4} - 4q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
417.2.a.a	$x - 3$
417.2.a.b	$x^{2} + 3x + 1$
417.2.a.c	$x^{3} - 2x^{2} - 4x + 7$
417.2.a.d	$x^{3} - 4x^{2} + 1$
417.2.a.e	$x^{7} + 7x^{6} - 2x^{5} - 70x^{4} - 94x^{3} + 59x^{2} + 154x + 58$
417.2.a.f	$x^{7} + 4x^{6} - 38x^{5} - 117x^{4} + 479x^{3} + 869x^{2} - 1864x - 778$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(417))$	23
Old forms	22
Cusp forms $S_{2}(\Gamma_0(417))$	45
Eisenstein series $E_{2}(\Gamma_0(417))$	3
Modular forms $M_{2}(\Gamma_0(417))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(417)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(139))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.194 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, 139, 417]: [0, 1, 12, 48] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 23 = 1 x 23 (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(417, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 417:2 1000

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