Atlas › Modular forms › Weight 6 › Level 97 › 97.6.a

Newspace 97.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(97))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
97.6.a.a	19	degree 19	$19q - 21q^{2} - 36q^{3} + 271q^{4} - 89q^{5} - 217q^{6} - 451q^{7} + O(q^{8})$	+	$-1$
97.6.a.b	21	degree 21	$21q + 19q^{2} + 36q^{3} + 367q^{4} + 161q^{5} + 71q^{6} + 431q^{7} + O(q^{8})$	−	$+1$

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 $T$
97.6.a.a	$x^{19} + 57x^{18} - 1419x^{17} - 114616x^{16} + 530703x^{15} + 89842506x^{14} + 41322237x^{13} - 35779833235x^{12} - 50154130409x^{11} + 7973134290642x^{10} + 6458003318898x^{9} - 998874734142033x^{8} + 111681197774926x^{7} + 66509663868178071x^{6} - 44694438735091999x^{5} - 2156553233205876047x^{4} + 1704702184994107324x^{3} + 26701814162382725930x^{2} - 17126810926953462663x - 8843110132446779050$
97.6.a.b	$x^{21} - 55x^{20} - 2547x^{19} + 168860x^{18} + 2504403x^{17} - 215916290x^{16} - 1259627675x^{15} + 151709284357x^{14} + 399225634775x^{13} - 64340187943430x^{12} - 119898480991918x^{11} + 16816624245301463x^{10} + 38856972476800042x^{9} - 2609970037080355949x^{8} - 8633908487208140123x^{7} + 213249827629529255929x^{6} + 928608538993680509432x^{5} - 6617738129308778421454x^{4} - 32082235446548653441595x^{3} + 14884766154031266322350x^{2} + 29190294826633520852028x - 2173801845288971483640$

Orbit

Characteristic polynomial of

T

97.6.a.a

x^{19} + 57x^{18} - 1419x^{17} - 114616x^{16} + 530703x^{15} + 89842506x^{14} + 41322237x^{13} - 35779833235x^{12} - 50154130409x^{11} + 7973134290642x^{10} + 6458003318898x^{9} - 998874734142033x^{8} + 111681197774926x^{7} + 66509663868178071x^{6} - 44694438735091999x^{5} - 2156553233205876047x^{4} + 1704702184994107324x^{3} + 26701814162382725930x^{2} - 17126810926953462663x - 8843110132446779050

97.6.a.b

x^{21} - 55x^{20} - 2547x^{19} + 168860x^{18} + 2504403x^{17} - 215916290x^{16} - 1259627675x^{15} + 151709284357x^{14} + 399225634775x^{13} - 64340187943430x^{12} - 119898480991918x^{11} + 16816624245301463x^{10} + 38856972476800042x^{9} - 2609970037080355949x^{8} - 8633908487208140123x^{7} + 213249827629529255929x^{6} + 928608538993680509432x^{5} - 6617738129308778421454x^{4} - 32082235446548653441595x^{3} + 14884766154031266322350x^{2} + 29190294826633520852028x - 2173801845288971483640

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(97))$	40
Old forms	0
Cusp forms $S_{6}(\Gamma_0(97))$	40
Eisenstein series $E_{6}(\Gamma_0(97))$	2
Modular forms $M_{6}(\Gamma_0(97))$	42

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 97]: [1, 42] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(97, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 97:6 1000

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