Atlas › Modular forms › Weight 2 › Level 761 › 761.2.a

Newspace 761.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(761))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
761.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + q^{3} - 2q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
761.2.a.b	20	degree 20	$20q - 2q^{2} - 5q^{3} + 8q^{4} - 8q^{5} - 5q^{6} - 20q^{7} + O(q^{8})$	+	$-1$
761.2.a.c	41	degree 41	$41q + 3q^{2} + 4q^{3} + 59q^{4} + 3q^{5} + 27q^{7} + O(q^{8})$	−	$+1$

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 $T$
761.2.a.a	$x^{2} + x - 1$
761.2.a.b	$x^{20} + 7x^{19} - 29x^{18} - 264x^{17} + 275x^{16} + 3850x^{15} - 1352x^{14} - 29147x^{13} + 8295x^{12} + 124580x^{11} - 56102x^{10} - 294495x^{9} + 202116x^{8} + 348759x^{7} - 340541x^{6} - 142767x^{5} + 233484x^{4} - 31191x^{3} - 32372x^{2} + 5136x + 1753$
761.2.a.c	$x^{41} - 7x^{40} - 134x^{39} + 1024x^{38} + 7615x^{37} - 65518x^{36} - 239484x^{35} + 2432883x^{34} + 4526946x^{33} - 58730097x^{32} - 51016422x^{31} + 979718856x^{30} + 280110902x^{29} - 11730892179x^{28} + 649741894x^{27} + 103374052167x^{26} - 24745829475x^{25} - 681132004250x^{24} + 213977973926x^{23} + 3382996941624x^{22} - 1053334721297x^{21} - 12673756488247x^{20} + 3234428292052x^{19} + 35566081848218x^{18} - 5908617440484x^{17} - 73656837641666x^{16} + 4545570935290x^{15} + 109936746229324x^{14} + 4805548616920x^{13} - 114308337205673x^{12} - 15936996253454x^{11} + 79017323572689x^{10} + 17216394676817x^{9} - 34120280319948x^{8} - 9338976785016x^{7} + 8514106590742x^{6} + 2580814418699x^{5} - 1113489866850x^{4} - 337776327640x^{3} + 63879820267x^{2} + 16502153740x - 848766755$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(761))$	63
Old forms	0
Cusp forms $S_{2}(\Gamma_0(761))$	63
Eisenstein series $E_{2}(\Gamma_0(761))$	1
Modular forms $M_{2}(\Gamma_0(761))$	64

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 761]: [0, 64] (certified at ell = 2147466641)
T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 7 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(761, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 761:2 1000

← 760.2.a · 762.2.a →