Atlas › Modular forms › Weight 6 › Level 61 › 61.6.a

Newspace 61.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(61))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
61.6.a.a	11	degree 11	$11q - 9q^{2} - 46q^{3} + 117q^{4} - 129q^{5} - 11q^{6} - 87q^{7} + O(q^{8})$	+	$-1$
61.6.a.b	14	degree 14	$14q + 11q^{2} + 26q^{3} + 261q^{4} + 71q^{5} + 205q^{6} + 109q^{7} + O(q^{8})$	−	$+1$

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 $T$
61.6.a.a	$x^{11} + 55x^{10} - 304x^{9} - 44770x^{8} - 52246x^{7} + 11693356x^{6} + 17185157x^{5} - 1119071457x^{4} - 1870384936x^{3} + 32738322800x^{2} + 98548688400x + 17474904000$
61.6.a.b	$x^{14} - 37x^{13} - 2440x^{12} + 79492x^{11} + 2599692x^{10} - 64837100x^{9} - 1517512423x^{8} + 24774504295x^{7} + 485659259986x^{6} - 4462759655616x^{5} - 76215923769224x^{4} + 366066156099648x^{3} + 4744562847617552x^{2} - 13957028681264016x - 70580968683671904$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(61))$	25
Old forms	0
Cusp forms $S_{6}(\Gamma_0(61))$	25
Eisenstein series $E_{6}(\Gamma_0(61))$	2
Modular forms $M_{6}(\Gamma_0(61))$	27

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 61]: [1, 27] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 5 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(61, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 61:6 1000

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