Atlas › Modular forms › Weight 6 › Level 85 › 85.6.a

Newspace 85.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(85))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
85.6.a.a	5	degree 5	$5q - 7q^{2} - 36q^{3} + 43q^{4} + 125q^{5} - 105q^{6} - 204q^{7} + O(q^{8})$	− −	$-1$
85.6.a.b	7	degree 7	$7q + 9q^{2} + 18q^{3} + 139q^{4} - 175q^{5} - 41q^{6} + 50q^{7} + O(q^{8})$	+ −	$+1$
85.6.a.c	8	degree 8	$8q - 3q^{2} - 36q^{3} + 187q^{4} - 200q^{5} - 113q^{6} - 184q^{7} + O(q^{8})$	+ +	$-1$
85.6.a.d	8	degree 8	$8q + 5q^{2} + 18q^{3} + 187q^{4} + 200q^{5} + 111q^{6} - 46q^{7} + O(q^{8})$	− +	$+1$

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 $T$
85.6.a.a	$x^{5} + 43x^{4} + 55x^{3} - 9555x^{2} - 24544x + 552560$
85.6.a.b	$x^{7} - 27x^{6} - 1039x^{5} + 20883x^{4} + 349806x^{3} - 4638840x^{2} - 34543584x + 335086560$
85.6.a.c	$x^{8} + 39x^{7} - 595x^{6} - 26011x^{5} + 89430x^{4} + 2647392x^{3} - 18038204x^{2} + 28736532x + 8865000$
85.6.a.d	$x^{8} - 23x^{7} - 1573x^{6} + 32607x^{5} + 751032x^{4} - 11660860x^{3} - 125115228x^{2} + 599357148x + 5114156640$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(85))$	28
Old forms	14
Cusp forms $S_{6}(\Gamma_0(85))$	42
Eisenstein series $E_{6}(\Gamma_0(85))$	4
Modular forms $M_{6}(\Gamma_0(85))$	46

Decomposition of the old subspace

$S_{6}^{\mathrm{old}}(\Gamma_0(85)) =$ $S_{6}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{6}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.255 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, 5, 17, 85]: [1, 3, 8, 46] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 5 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(85, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 85:6 1000

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