Atlas › Modular forms › Weight 4 › Level 211 › 211.4.a

Newspace 211.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(211))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
211.4.a.a	23	degree 23	$23q - 4q^{2} - 17q^{3} + 72q^{4} - 91q^{5} - 58q^{6} - 32q^{7} + O(q^{8})$	−	$-1$
211.4.a.b	29	degree 29	$29q + 6q^{2} + 13q^{3} + 132q^{4} + 89q^{5} + 38q^{6} + 24q^{7} + O(q^{8})$	+	$+1$

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 $T$
211.4.a.a	$x^{23} + 21x^{22} - 220x^{21} - 6516x^{20} + 13650x^{19} + 812988x^{18} - 30099x^{17} - 55005279x^{16} - 25235650x^{15} + 2221970164x^{14} + 956563588x^{13} - 54939911292x^{12} - 18503593424x^{11} + 837728434420x^{10} + 295352160861x^{9} - 7833843353861x^{8} - 3988848802914x^{7} + 43290763993476x^{6} + 33763135873330x^{5} - 125564892854584x^{4} - 144429469821041x^{3} + 124878229948177x^{2} + 236806934882750x + 86160174317950$
211.4.a.b	$x^{29} - 19x^{28} - 626x^{27} + 13238x^{26} + 157184x^{25} - 3873236x^{24} - 20628323x^{23} + 632487411x^{22} + 1547883114x^{21} - 64802629246x^{20} - 66792573890x^{19} + 4420530320548x^{18} + 1491022737532x^{17} - 206866592992442x^{16} - 9241598212129x^{15} + 6716206509169649x^{14} - 100154652379144x^{13} - 150802480195194600x^{12} - 9489855164796064x^{11} + 2300775891506133952x^{10} + 512360471655410173x^{9} - 22981497716222486607x^{8} - 9950476010729638608x^{7} + 140032421976899095324x^{6} + 94803787397370023880x^{5} - 450490457794146027272x^{4} - 426441598746287398256x^{3} + 517122724552261027664x^{2} + 649115010203497077360x + 97184115874110549968$

Orbit

Characteristic polynomial of

T

211.4.a.a

x^{23} + 21x^{22} - 220x^{21} - 6516x^{20} + 13650x^{19} + 812988x^{18} - 30099x^{17} - 55005279x^{16} - 25235650x^{15} + 2221970164x^{14} + 956563588x^{13} - 54939911292x^{12} - 18503593424x^{11} + 837728434420x^{10} + 295352160861x^{9} - 7833843353861x^{8} - 3988848802914x^{7} + 43290763993476x^{6} + 33763135873330x^{5} - 125564892854584x^{4} - 144429469821041x^{3} + 124878229948177x^{2} + 236806934882750x + 86160174317950

211.4.a.b

x^{29} - 19x^{28} - 626x^{27} + 13238x^{26} + 157184x^{25} - 3873236x^{24} - 20628323x^{23} + 632487411x^{22} + 1547883114x^{21} - 64802629246x^{20} - 66792573890x^{19} + 4420530320548x^{18} + 1491022737532x^{17} - 206866592992442x^{16} - 9241598212129x^{15} + 6716206509169649x^{14} - 100154652379144x^{13} - 150802480195194600x^{12} - 9489855164796064x^{11} + 2300775891506133952x^{10} + 512360471655410173x^{9} - 22981497716222486607x^{8} - 9950476010729638608x^{7} + 140032421976899095324x^{6} + 94803787397370023880x^{5} - 450490457794146027272x^{4} - 426441598746287398256x^{3} + 517122724552261027664x^{2} + 649115010203497077360x + 97184115874110549968

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(211))$	52
Old forms	0
Cusp forms $S_{4}(\Gamma_0(211))$	52
Eisenstein series $E_{4}(\Gamma_0(211))$	2
Modular forms $M_{4}(\Gamma_0(211))$	54

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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, 211]: [1, 54] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 52 = 1 x 52 (phi(ord eps) x dim S_k^new); 7 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(211, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 211:4 1000

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