Atlas › Modular forms › Weight 6 › Level 83 › 83.6.a

Newspace 83.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(83))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
83.6.a.a	14	degree 14	$14q - 13q^{2} - 17q^{3} + 133q^{4} - 162q^{5} - 15q^{6} - 247q^{7} + O(q^{8})$	+	$-1$
83.6.a.b	20	degree 20	$20q + 7q^{2} + 37q^{3} + 373q^{4} + 88q^{5} + 129q^{6} + 439q^{7} + O(q^{8})$	−	$+1$

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 $T$
83.6.a.a	$x^{14} + 30x^{13} - 1866x^{12} - 45889x^{11} + 1322678x^{10} + 21384849x^{9} - 479646070x^{8} - 3675520827x^{7} + 88586418009x^{6} + 152523286405x^{5} - 7712884204560x^{4} + 16266519583240x^{3} + 236655664683472x^{2} - 1168728914301808x + 1206108837762336$
83.6.a.b	$x^{20} - 44x^{19} - 3166x^{18} + 147918x^{17} + 4167415x^{16} - 207251872x^{15} - 3004775505x^{14} + 157725819186x^{13} + 1321448502777x^{12} - 71055482224630x^{11} - 376148208590796x^{10} + 19333515828475298x^{9} + 71975712093366637x^{8} - 3099869138360500064x^{7} - 9289698811777097141x^{6} + 269978724552071367110x^{5} + 742441695991257810002x^{4} - 10241142817009646582134x^{3} - 27466260911857785565328x^{2} + 52924821601381361181808x + 24826081189085403897680$

Orbit

Characteristic polynomial of

T

83.6.a.a

x^{14} + 30x^{13} - 1866x^{12} - 45889x^{11} + 1322678x^{10} + 21384849x^{9} - 479646070x^{8} - 3675520827x^{7} + 88586418009x^{6} + 152523286405x^{5} - 7712884204560x^{4} + 16266519583240x^{3} + 236655664683472x^{2} - 1168728914301808x + 1206108837762336

83.6.a.b

x^{20} - 44x^{19} - 3166x^{18} + 147918x^{17} + 4167415x^{16} - 207251872x^{15} - 3004775505x^{14} + 157725819186x^{13} + 1321448502777x^{12} - 71055482224630x^{11} - 376148208590796x^{10} + 19333515828475298x^{9} + 71975712093366637x^{8} - 3099869138360500064x^{7} - 9289698811777097141x^{6} + 269978724552071367110x^{5} + 742441695991257810002x^{4} - 10241142817009646582134x^{3} - 27466260911857785565328x^{2} + 52924821601381361181808x + 24826081189085403897680

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(83))$	34
Old forms	0
Cusp forms $S_{6}(\Gamma_0(83))$	34
Eisenstein series $E_{6}(\Gamma_0(83))$	2
Modular forms $M_{6}(\Gamma_0(83))$	36

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 83]: [1, 36] (certified at ell = 2147483323)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(83, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 83:6 1000

← 82.6.a · 84.6.a →