Atlas › Modular forms › Weight 6 › Level 71 › 71.6.a

Newspace 71.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(71))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
71.6.a.a	11	degree 11	$11q - 4q^{2} - 20q^{3} + 122q^{4} - 72q^{5} - 306q^{6} - 344q^{7} + O(q^{8})$	+	$-1$
71.6.a.b	18	degree 18	$18q + 4q^{2} + 16q^{3} + 346q^{4} + 128q^{5} + 126q^{6} + 244q^{7} + O(q^{8})$	−	$+1$

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 $T$
71.6.a.a	$x^{11} + 24x^{10} - 1030x^{9} - 16930x^{8} + 383751x^{7} + 2709140x^{6} - 42552857x^{5} - 157493648x^{4} + 1534157675x^{3} + 1869051958x^{2} - 15245941404x + 13837652904$
71.6.a.b	$x^{18} - 20x^{17} - 3475x^{16} + 55689x^{15} + 5077140x^{14} - 59891592x^{13} - 4094666944x^{12} + 30287285719x^{11} + 1995247417901x^{10} - 6113372814109x^{9} - 594106892145206x^{8} - 572969120151493x^{7} + 101827025468568239x^{6} + 469839682394350254x^{5} - 8263202611846021008x^{4} - 68766056795622286584x^{3} + 114794575258393730880x^{2} + 2510100582181768929600x + 6366234786642469224000$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(71))$	29
Old forms	0
Cusp forms $S_{6}(\Gamma_0(71))$	29
Eisenstein series $E_{6}(\Gamma_0(71))$	2
Modular forms $M_{6}(\Gamma_0(71))$	31

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 71]: [1, 31] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 6 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(71, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 71:6 1000

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