Atlas › Modular forms › Weight 2 › Level 709 › 709.2.a

Newspace 709.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(709))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
709.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	−	$+1$	709a1 rank 2
709.2.a.b	27	degree 27	$27q - 8q^{2} - 13q^{3} + 22q^{4} - 18q^{5} - 14q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
709.2.a.c	30	degree 30	$30q + 8q^{2} + 12q^{3} + 28q^{4} + 15q^{5} + 10q^{6} + 10q^{7} + O(q^{8})$	−	$+1$

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 $T$
709.2.a.a	$x + 3$
709.2.a.b	$x^{27} + 21x^{26} + 146x^{25} + 108x^{24} - 3333x^{23} - 12768x^{22} + 16544x^{21} + 175211x^{20} + 123715x^{19} - 1085217x^{18} - 1771973x^{17} + 3594852x^{16} + 8839012x^{15} - 6529854x^{14} - 24021457x^{13} + 6050457x^{12} + 39024539x^{11} - 2251415x^{10} - 38677045x^{9} + 497428x^{8} + 23029157x^{7} - 1145411x^{6} - 7793692x^{5} + 927808x^{4} + 1310538x^{3} - 245745x^{2} - 75090x + 17011$
709.2.a.c	$x^{30} - 20x^{29} + 84x^{28} + 874x^{27} - 8512x^{26} + 1883x^{25} + 228233x^{24} - 655190x^{23} - 2375382x^{22} + 13468071x^{21} + 4058712x^{20} - 126881813x^{19} + 122418142x^{18} + 647574067x^{17} - 1187800057x^{16} - 1794616340x^{15} + 5217068967x^{14} + 2208355738x^{13} - 12957906012x^{12} + 722606799x^{11} + 19156888135x^{10} - 5296357619x^{9} - 17122782188x^{8} + 5927310772x^{7} + 9174733969x^{6} - 2833237027x^{5} - 2790789719x^{4} + 561393106x^{3} + 414016036x^{2} - 29365645x - 19751209$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(709))$	58
Old forms	0
Cusp forms $S_{2}(\Gamma_0(709))$	58
Eisenstein series $E_{2}(\Gamma_0(709))$	1
Modular forms $M_{2}(\Gamma_0(709))$	59

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.231 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, 709]: [0, 59] (certified at ell = 2147482237)
T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (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(709, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 709:2 1000

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