Atlas › Modular forms › Weight 2 › Level 919 › 919.2.a

Newspace 919.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(919))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
919.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{4} - 3q^{5} + q^{7} + O(q^{8})$	+	$-1$
919.2.a.b	27	degree 27	$27q - 6q^{2} - 9q^{3} + 24q^{4} - 17q^{5} - 9q^{6} - 7q^{7} + O(q^{8})$	+	$-1$
919.2.a.c	47	degree 47	$47q + 5q^{2} + 9q^{3} + 51q^{4} + 22q^{5} + 9q^{6} - 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
919.2.a.a	$x^{2} - 5$
919.2.a.b	$x^{27} + 15x^{26} + 45x^{25} - 364x^{24} - 2299x^{23} + 1543x^{22} + 35787x^{21} + 33699x^{20} - 276422x^{19} - 491502x^{18} + 1188876x^{17} + 2958932x^{16} - 2889930x^{15} - 9919066x^{14} + 3643777x^{13} + 19763270x^{12} - 1470956x^{11} - 23343722x^{10} - 1203764x^{9} + 15601855x^{8} + 1103279x^{7} - 5414747x^{6} - 191953x^{5} + 845279x^{4} - 6010x^{3} - 44508x^{2} + 423x + 421$
919.2.a.c	$x^{47} - 14x^{46} - 94x^{45} + 2131x^{44} + 901x^{43} - 142636x^{42} + 264778x^{41} + 5547552x^{40} - 17949039x^{39} - 139228746x^{38} + 613676418x^{37} + 2357005260x^{36} - 13539700330x^{35} - 27038178582x^{34} + 209688242399x^{33} + 198964425399x^{32} - 2376053216862x^{31} - 694439753070x^{30} + 20171939610946x^{29} - 2825318829408x^{28} - 130146287779632x^{27} + 55808852315424x^{26} + 643425902609305x^{25} - 396850769674481x^{24} - 2446845543022355x^{23} + 1781287739558486x^{22} + 7156435274622321x^{21} - 5557637902796709x^{20} - 16029439289600923x^{19} + 12366526974203900x^{18} + 27247057315711126x^{17} - 19599237348236324x^{16} - 34615771752654148x^{15} + 21689487780174367x^{14} + 32100340707856563x^{13} - 16094632059867808x^{12} - 20960547634786396x^{11} + 7439194816633124x^{10} + 9111443162087001x^{9} - 1845492449419684x^{8} - 2396763376053722x^{7} + 154453946397152x^{6} + 312771039030282x^{5} + 10008189929331x^{4} - 10668133371154x^{3} + 392821967083x^{2} + 21411518790x - 340532271$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(919))$	76
Old forms	0
Cusp forms $S_{2}(\Gamma_0(919))$	76
Eisenstein series $E_{2}(\Gamma_0(919))$	1
Modular forms $M_{2}(\Gamma_0(919))$	77

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.384 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, 919]: [0, 77] (certified at ell = 2147470057)
T = [(2, 1), (3, 1)]; h of degree 76 = 1 x 76 (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(919, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 919:2 1000

← 918.2.a · 920.2.a →