Atlas › Modular forms › Weight 2 › Level 442 › 442.2.a

Newspace 442.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(442))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
442.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	442c1 rank 0
442.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	− + −	$-1$	442b1 rank 1
442.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− + +	$+1$	442a1 rank 0
442.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$	442d1 rank 0
442.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$	442e1 rank 0
442.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 4q^{3} + 2q^{4} + 4q^{6} + O(q^{8})$	+ − −	$-1$
442.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ − +	$+1$
442.2.a.h	3	degree 3	$3q - 3q^{2} - 2q^{3} + 3q^{4} - 4q^{5} + 2q^{6} + O(q^{8})$	+ + +	$-1$
442.2.a.i	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 4q^{5} + 2q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
442.2.a.a	$x - 32$
442.2.a.b	$x + 36$
442.2.a.c	$x - 12$
442.2.a.d	$x - 34$
442.2.a.e	$x + 20$
442.2.a.f	$x^{2} + 32x + 184$
442.2.a.g	$x^{2} - 18x - 524$
442.2.a.h	$x^{3} + 28x^{2} - 1552x - 24128$
442.2.a.i	$x^{3} + 14x^{2} - 944x + 2752$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(442))$	15
Old forms	44
Cusp forms $S_{2}(\Gamma_0(442))$	59
Eisenstein series $E_{2}(\Gamma_0(442))$	7
Modular forms $M_{2}(\Gamma_0(442))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(442)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(221))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 13, 17, 26, 34, 221, 442]: [0, 1, 1, 2, 5, 6, 22, 66] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(442, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 442:2 1000

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