Atlas › Modular forms › Weight 2 › Level 182 › 182.2.a

Newspace 182.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(182))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
182.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	182c1 rank 0
182.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	182e1 rank 0
182.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - q^{7} + O(q^{8})$	− + +	$+1$	182a1 rank 0
182.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	182b1 rank 0
182.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} - 4q^{5} + 3q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	182d1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17}$ 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 $T$
182.2.a.a	$x - 36$
182.2.a.b	$x + 50$
182.2.a.c	$x + 16$
182.2.a.d	$x + 14$
182.2.a.e	$x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(182))$	5
Old forms	20
Cusp forms $S_{2}(\Gamma_0(182))$	25
Eisenstein series $E_{2}(\Gamma_0(182))$	7
Modular forms $M_{2}(\Gamma_0(182))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(182)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 13, 14, 26, 91, 182]: [0, 1, 1, 1, 4, 5, 10, 32] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (11, 5), (17, 7)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(182, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 182:2 1000

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