Atlas › Modular forms › Weight 2 › Level 773 › 773.2.a

Newspace 773.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(773))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
773.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + 2q^{3} + 4q^{4} - q^{5} - 4q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
773.2.a.b	24	degree 24	$24q - 3q^{2} - 15q^{3} + 11q^{4} - 3q^{5} + q^{6} - 26q^{7} + O(q^{8})$	+	$-1$
773.2.a.c	38	degree 38	$38q + 7q^{2} + 13q^{3} + 47q^{4} + 2q^{5} + 3q^{6} + 31q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
773.2.a.a	$x^{2} + 24x - 36$
773.2.a.b	$x^{24} + 245x^{23} + 24737x^{22} + 1241011x^{21} + 23112114x^{20} - 747328611x^{19} - 55499973884x^{18} - 1295148898021x^{17} - 5706038649147x^{16} + 363951979642581x^{15} + 7718769500352685x^{14} + 24404654075696169x^{13} - 1095782759499365803x^{12} - 13916740324078385976x^{11} + 11987736286624284089x^{10} + 1316048672488956685099x^{9} + 6423312557746343874084x^{8} - 42974702677054936690657x^{7} - 436577669395101105586993x^{6} + 33005353018738192029961x^{5} + 10516619806018666060299789x^{4} + 20367700931150119202703024x^{3} - 86228980579801024488026788x^{2} - 215842876222358007504207959x + 144766832133517776639104591$
773.2.a.c	$x^{38} - 273x^{37} + 24527x^{36} - 43031x^{35} - 135418364x^{34} + 7458258841x^{33} + 118236088598x^{32} - 24276714698577x^{31} + 527237248687023x^{30} + 28917583198457157x^{29} - 1511338516714563694x^{28} - 3903993281186300096x^{27} + 1718964557708717010313x^{26} - 26038806269253029794262x^{25} - 977359999036290250191090x^{24} + 31541417401471284483774325x^{23} + 196693884488117656977592818x^{22} - 18629460569625279807947781937x^{21} + 96675450403103865095712540244x^{20} + 6464273739105764490231039516559x^{19} - 84839062358268252734080936249173x^{18} - 1303733267604666398569496919745814x^{17} + 30002834203807757862174738646535010x^{16} + 116869123111844004604138424282206266x^{15} - 6232689776538360823113904982636037531x^{14} + 8299489932770326052304313650198346943x^{13} + 813641997362244301164816511421842813079x^{12} - 3497125820998165559009338377853656846654x^{11} - 67576081274197722167024569826661506841215x^{10} + 406723104914326561591808010870941860693852x^{9} + 3592940299948764210378797434837229330920638x^{8} - 23748608996041177813987395767943813155230763x^{7} - 123832853145113120967552512381736790901326669x^{6} + 707655105478009750723877295571146942124056581x^{5} + 2615693596020483698695637366828978098132224251x^{4} - 9017686881782041376154274771176799286023793755x^{3} - 23183726753761948214324562943782648609155379961x^{2} + 41031466720428474967536967951873889624133433960x + 42658788046281151520920882464148305345922300236$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(773))$	64
Old forms	0
Cusp forms $S_{2}(\Gamma_0(773))$	64
Eisenstein series $E_{2}(\Gamma_0(773))$	1
Modular forms $M_{2}(\Gamma_0(773))$	65

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.297 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, 773]: [0, 65] (certified at ell = 2147480893)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 64 = 1 x 64 (phi(ord eps) x dim S_k^new); 14 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(773, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 773:2 1000

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