Atlas › Modular forms › Weight 2 › Level 401 › 401.2.a

Newspace 401.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(401))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
401.2.a.a	12	degree 12	$12q - 3q^{2} - 5q^{3} + 5q^{4} - 7q^{5} - 9q^{6} - 20q^{7} + O(q^{8})$	+	$-1$
401.2.a.b	21	degree 21	$21q + 3q^{3} + 28q^{4} + 3q^{5} + 9q^{6} + 24q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
401.2.a.a	$x^{12} + 8x^{11} + 7x^{10} - 70x^{9} - 117x^{8} + 161x^{7} + 318x^{6} - 114x^{5} - 274x^{4} + 27x^{3} + 78x^{2} - 2x - 7$
401.2.a.b	$x^{21} - 3x^{20} - 81x^{19} + 255x^{18} + 2530x^{17} - 8171x^{16} - 40338x^{15} + 131197x^{14} + 364234x^{13} - 1158726x^{12} - 1977901x^{11} + 5779265x^{10} + 6846660x^{9} - 16084028x^{8} - 15566480x^{7} + 22964538x^{6} + 21959243x^{5} - 12194854x^{4} - 15013101x^{3} - 1862556x^{2} + 1329605x + 299663$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(401))$	33
Old forms	0
Cusp forms $S_{2}(\Gamma_0(401))$	33
Eisenstein series $E_{2}(\Gamma_0(401))$	1
Modular forms $M_{2}(\Gamma_0(401))$	34

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 401]: [0, 34] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (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(401, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 401:2 1000

← 400.2.a · 402.2.a →