Atlas › Modular forms › Weight 2 › Level 659 › 659.2.a

Newspace 659.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(659))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
659.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} - 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+	$-1$	659a1 rank 1
659.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	−	$+1$	659b1 rank 0
659.2.a.c	16	degree 16	$16q - 3q^{2} - 9q^{3} + 5q^{4} - 8q^{5} - 7q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
659.2.a.d	37	degree 37	$37q - 2q^{2} + 6q^{3} + 50q^{4} + 13q^{5} + 5q^{6} + 6q^{7} + O(q^{8})$	−	$+1$

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 $T$
659.2.a.a	$x - 7$
659.2.a.b	$x - 5$
659.2.a.c	$x^{16} + 30x^{15} + 241x^{14} - 1031x^{13} - 22460x^{12} - 54560x^{11} + 491119x^{10} + 2168607x^{9} - 4283795x^{8} - 27837029x^{7} + 18529997x^{6} + 166096257x^{5} - 71678991x^{4} - 484067682x^{3} + 268763001x^{2} + 554152139x - 430265413$
659.2.a.d	$x^{37} - 16x^{36} - 696x^{35} + 12031x^{34} + 211097x^{33} - 4023875x^{32} - 36825451x^{31} + 795226127x^{30} + 4103295892x^{29} - 104039379128x^{28} - 306092843132x^{27} + 9562694867589x^{26} + 15547917030430x^{25} - 639100160361504x^{24} - 536807545082192x^{23} + 31682552142268125x^{22} + 12636120700201193x^{21} - 1176637299922349712x^{20} - 232065810744634873x^{19} + 32788659486195473586x^{18} + 5470966147797074895x^{17} - 680996412566936964756x^{16} - 174938589700252184827x^{15} + 10370747432214903941973x^{14} + 4403048825496625796846x^{13} - 112507316374664557977116x^{12} - 70977720906289598593693x^{11} + 829691847299743899669485x^{10} + 695283073396755435701797x^{9} - 3859431642828552834153877x^{8} - 3877555790426439552468916x^{7} + 10042617746321277632384433x^{6} + 10748829939430553931546634x^{5} - 12320357703923855355449723x^{4} - 11219221090794304360003384x^{3} + 7026971358981196439757964x^{2} + 3219979591494491808522298x - 1073425846578906609736581$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(659))$	55
Old forms	0
Cusp forms $S_{2}(\Gamma_0(659))$	55
Eisenstein series $E_{2}(\Gamma_0(659))$	1
Modular forms $M_{2}(\Gamma_0(659))$	56

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 659]: [0, 56] (certified at ell = 2147471831)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(659, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 659:2 1000

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