Atlas › Modular forms › Weight 2 › Level 541 › 541.2.a

Newspace 541.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(541))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
541.2.a.a	20	degree 20	$20q - 5q^{2} - 12q^{3} + 15q^{4} - 17q^{5} - 11q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
541.2.a.b	24	degree 24	$24q + 3q^{2} + 12q^{3} + 23q^{4} + 13q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
541.2.a.a	$x^{20} + 17x^{19} + 91x^{18} + 12x^{17} - 1441x^{16} - 3371x^{15} + 6564x^{14} + 29408x^{13} - 2280x^{12} - 104894x^{11} - 58511x^{10} + 170965x^{9} + 159486x^{8} - 104570x^{7} - 156707x^{6} - 9880x^{5} + 51031x^{4} + 22880x^{3} + 1971x^{2} - 668x - 113$
541.2.a.b	$x^{24} - 15x^{23} + 25x^{22} + 670x^{21} - 3356x^{20} - 7598x^{19} + 81291x^{18} - 40040x^{17} - 848951x^{16} + 1363815x^{15} + 4602215x^{14} - 11030071x^{13} - 14041686x^{12} + 45651880x^{11} + 25650428x^{10} - 110928997x^{9} - 31893401x^{8} + 163803892x^{7} + 34820968x^{6} - 142985306x^{5} - 34004996x^{4} + 66259308x^{3} + 20057265x^{2} - 12208382x - 4548403$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(541))$	44
Old forms	0
Cusp forms $S_{2}(\Gamma_0(541))$	44
Eisenstein series $E_{2}(\Gamma_0(541))$	1
Modular forms $M_{2}(\Gamma_0(541))$	45

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 541]: [0, 45] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (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(541, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 541:2 1000

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