Atlas › Modular forms › Weight 2 › Level 491 › 491.2.a

Newspace 491.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(491))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
491.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{3} - q^{4} - q^{5} - 3q^{6} + q^{7} + O(q^{8})$	+	$-1$
491.2.a.b	10	degree 10	$10q - 3q^{2} - 4q^{3} + 3q^{4} - 9q^{5} - q^{6} - 6q^{7} + O(q^{8})$	+	$-1$
491.2.a.c	29	degree 29	$29q + 5q^{3} + 40q^{4} + 12q^{5} + 3q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
491.2.a.a	$x^{2} + 2x - 4$
491.2.a.b	$x^{10} + 15x^{9} - 40x^{8} - 1294x^{7} - 1592x^{6} + 32607x^{5} + 71934x^{4} - 254211x^{3} - 590513x^{2} + 504838x + 1084676$
491.2.a.c	$x^{29} - 15x^{28} - 508x^{27} + 8470x^{26} + 105900x^{25} - 2068203x^{24} - 11395058x^{23} + 288092335x^{22} + 607199795x^{21} - 25447159059x^{20} - 3549960431x^{19} + 1499369059275x^{18} - 1671699297919x^{17} - 60372858562804x^{16} + 118547509096850x^{15} + 1674103404634478x^{14} - 4280470940484677x^{13} - 31813217870552647x^{12} + 94465133527623865x^{11} + 407769720150392060x^{10} - 1314788566114377001x^{9} - 3436528966123750655x^{8} + 11289582404991676004x^{7} + 18473951319027690155x^{6} - 55694246841114426726x^{5} - 62015635093403660324x^{4} + 134806387664166876684x^{3} + 125685460115986926358x^{2} - 107914741621532953472x - 97617544757250161560$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(491))$	41
Old forms	0
Cusp forms $S_{2}(\Gamma_0(491))$	41
Eisenstein series $E_{2}(\Gamma_0(491))$	1
Modular forms $M_{2}(\Gamma_0(491))$	42

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 491]: [0, 42] (certified at ell = 2147474491)
T = [(2, 1), (3, 3)]; h of degree 41 = 1 x 41 (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(491, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 491:2 1000

← 490.2.a · 492.2.a →