Atlas › Modular forms › Weight 2 › Level 959 › 959.2.a

Newspace 959.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(959))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
959.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - q^{4} + 6q^{5} - 5q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
959.2.a.b	7	degree 7	$7q - 2q^{2} - 3q^{3} - 3q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− −	$-1$
959.2.a.c	10	degree 10	$10q + q^{2} - 3q^{3} + 5q^{4} - 3q^{5} - 7q^{6} - 10q^{7} + O(q^{8})$	+ +	$-1$
959.2.a.d	24	degree 24	$24q - q^{2} + 7q^{3} + 29q^{4} + 9q^{5} + 3q^{6} - 24q^{7} + O(q^{8})$	+ −	$+1$
959.2.a.e	26	degree 26	$26q + 4q^{2} + 3q^{3} + 42q^{4} - 3q^{5} + 6q^{6} + 26q^{7} + O(q^{8})$	− +	$+1$

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 $T$
959.2.a.a	$x^{2} - x - 1$
959.2.a.b	$x^{7} + 5x^{6} - 5x^{5} - 44x^{4} - 18x^{3} + 79x^{2} + 30x - 49$
959.2.a.c	$x^{10} + 2x^{9} - 17x^{8} - 17x^{7} + 110x^{6} - 10x^{5} - 247x^{4} + 258x^{3} - 102x^{2} + 17x - 1$
959.2.a.d	$x^{24} - 6x^{23} - 79x^{22} + 503x^{21} + 2397x^{20} - 16452x^{19} - 37986x^{18} + 282573x^{17} + 354387x^{16} - 2829031x^{15} - 2048831x^{14} + 17035625x^{13} + 7306617x^{12} - 60759939x^{11} - 14349947x^{10} + 120887315x^{9} + 8862223x^{8} - 118856063x^{7} + 9660010x^{6} + 42915013x^{5} - 7847278x^{4} - 3323492x^{3} + 603222x^{2} + 40115x - 4231$
959.2.a.e	$x^{26} - 7x^{25} - 89x^{24} + 689x^{23} + 3072x^{22} - 27945x^{21} - 50042x^{20} + 606737x^{19} + 333527x^{18} - 7714351x^{17} + 675991x^{16} + 59702004x^{15} - 23399959x^{14} - 287591753x^{13} + 140549782x^{12} + 874695699x^{11} - 404927025x^{10} - 1677628214x^{9} + 640484108x^{8} + 1998594054x^{7} - 559786313x^{6} - 1419016552x^{5} + 239038013x^{4} + 543485091x^{3} - 23491387x^{2} - 85123544x - 9413891$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(959))$	69
Old forms	22
Cusp forms $S_{2}(\Gamma_0(959))$	91
Eisenstein series $E_{2}(\Gamma_0(959))$	3
Modular forms $M_{2}(\Gamma_0(959))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(959)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(137))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 7, 137, 959]: [0, 1, 12, 94] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(959, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 959:2 1000

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