Atlas › Modular forms › Weight 6 › Level 89 › 89.6.a

Newspace 89.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(89))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
89.6.a.a	15	degree 15	$15q - 9q^{2} - 9q^{3} + 127q^{4} - 39q^{5} - 177q^{6} - 540q^{7} + O(q^{8})$	+	$-1$
89.6.a.b	21	degree 21	$21q + 7q^{2} + 27q^{3} + 415q^{4} + 61q^{5} + 183q^{6} + 538q^{7} + O(q^{8})$	−	$+1$

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 $T$
89.6.a.a	$x^{15} + 18x^{14} - 2166x^{13} - 24858x^{12} + 1783246x^{11} + 7679362x^{10} - 694004752x^{9} + 958301962x^{8} + 122270879657x^{7} - 571483618796x^{6} - 9075967223946x^{5} + 57345279796360x^{4} + 274859616415624x^{3} - 1951942201184432x^{2} - 3053797921584448x + 20565700276547968$
89.6.a.b	$x^{21} - 34x^{20} - 3808x^{19} + 126724x^{18} + 6066167x^{17} - 193405642x^{16} - 5404488644x^{15} + 159435841626x^{14} + 3009782939429x^{13} - 78024719409244x^{12} - 1100195730707100x^{11} + 23272450677839552x^{10} + 266908972726115013x^{9} - 4139201325112110578x^{8} - 41765366628345153816x^{7} + 402089799120410376858x^{6} + 3836861955173377184818x^{5} - 16350901293463902120726x^{4} - 159421954024603309843140x^{3} - 1085778519425386137184x^{2} + 459773424308937175299840x - 68658727515398122215168$

Orbit

Characteristic polynomial of

T

89.6.a.a

x^{15} + 18x^{14} - 2166x^{13} - 24858x^{12} + 1783246x^{11} + 7679362x^{10} - 694004752x^{9} + 958301962x^{8} + 122270879657x^{7} - 571483618796x^{6} - 9075967223946x^{5} + 57345279796360x^{4} + 274859616415624x^{3} - 1951942201184432x^{2} - 3053797921584448x + 20565700276547968

89.6.a.b

x^{21} - 34x^{20} - 3808x^{19} + 126724x^{18} + 6066167x^{17} - 193405642x^{16} - 5404488644x^{15} + 159435841626x^{14} + 3009782939429x^{13} - 78024719409244x^{12} - 1100195730707100x^{11} + 23272450677839552x^{10} + 266908972726115013x^{9} - 4139201325112110578x^{8} - 41765366628345153816x^{7} + 402089799120410376858x^{6} + 3836861955173377184818x^{5} - 16350901293463902120726x^{4} - 159421954024603309843140x^{3} - 1085778519425386137184x^{2} + 459773424308937175299840x - 68658727515398122215168

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(89))$	36
Old forms	0
Cusp forms $S_{6}(\Gamma_0(89))$	36
Eisenstein series $E_{6}(\Gamma_0(89))$	2
Modular forms $M_{6}(\Gamma_0(89))$	38

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 89]: [1, 38] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (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(89, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 89:6 1000

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