Atlas › Modular forms › Weight 8 › Level 59 › 59.8.a

Newspace 59.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(59))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
59.8.a.a	14	degree 14	$14q - 9q^{2} - 54q^{3} + 575q^{4} - 430q^{5} - 346q^{6} - 2390q^{7} + O(q^{8})$	−	$-1$
59.8.a.b	20	degree 20	$20q + 15q^{2} + 1535q^{4} + 570q^{5} + 1382q^{6} + 1040q^{7} + O(q^{8})$	+	$+1$

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 $T$
59.8.a.a	$x^{14} + 63x^{13} - 16414x^{12} - 778234x^{11} + 108969268x^{10} + 2953857592x^{9} - 372659036057x^{8} - 1931712077059x^{7} + 641697130751251x^{6} - 9049902065411305x^{5} - 365543974817926331x^{4} + 12758132007812346643x^{3} - 137286780125869890294x^{2} + 365779297591004785340x + 1516024177011042689408$
59.8.a.b	$x^{20} - 15x^{19} - 34000x^{18} + 396025x^{17} + 472052408x^{16} - 2911059070x^{15} - 3552754121672x^{14} - 1374826293785x^{13} + 15991781079996949x^{12} + 115705562436722720x^{11} - 44164935677746449009x^{10} - 615808226138426530080x^{9} + 72749535295874045358451x^{8} + 1493636838466489814921375x^{7} - 64171656327934873892797505x^{6} - 1795155010460998798297911080x^{5} + 20861775449565486940802034739x^{4} + 908048904433515345397319811290x^{3} + 3058406798694838602198105689478x^{2} - 68021086190713764162698972034980x - 119796481022401687978995917552304$

Orbit

Characteristic polynomial of

T

59.8.a.a

x^{14} + 63x^{13} - 16414x^{12} - 778234x^{11} + 108969268x^{10} + 2953857592x^{9} - 372659036057x^{8} - 1931712077059x^{7} + 641697130751251x^{6} - 9049902065411305x^{5} - 365543974817926331x^{4} + 12758132007812346643x^{3} - 137286780125869890294x^{2} + 365779297591004785340x + 1516024177011042689408

59.8.a.b

x^{20} - 15x^{19} - 34000x^{18} + 396025x^{17} + 472052408x^{16} - 2911059070x^{15} - 3552754121672x^{14} - 1374826293785x^{13} + 15991781079996949x^{12} + 115705562436722720x^{11} - 44164935677746449009x^{10} - 615808226138426530080x^{9} + 72749535295874045358451x^{8} + 1493636838466489814921375x^{7} - 64171656327934873892797505x^{6} - 1795155010460998798297911080x^{5} + 20861775449565486940802034739x^{4} + 908048904433515345397319811290x^{3} + 3058406798694838602198105689478x^{2} - 68021086190713764162698972034980x - 119796481022401687978995917552304

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(59))$	34
Old forms	0
Cusp forms $S_{8}(\Gamma_0(59))$	34
Eisenstein series $E_{8}(\Gamma_0(59))$	2
Modular forms $M_{8}(\Gamma_0(59))$	36

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 59]: [1, 36] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(59, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 59:8 1000

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