Atlas › Modular forms › Weight 10 › Level 39 › 39.10.a

Newspace 39.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(39))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
39.10.a.a	3	degree 3	$3q - 18q^{2} + 243q^{3} + 980q^{4} - 888q^{5} - 1458q^{6} - 14406q^{7} + O(q^{8})$	− −	$-1$
39.10.a.b	4	degree 4	$4q + 36q^{2} - 324q^{3} + 472q^{4} - 1044q^{5} - 2916q^{6} + 5868q^{7} + O(q^{8})$	+ +	$-1$
39.10.a.c	5	degree 5	$5q + 52q^{2} - 405q^{3} + 1646q^{4} + 2624q^{5} - 4212q^{6} - 4802q^{7} + O(q^{8})$	+ −	$+1$
39.10.a.d	6	degree 6	$6q - 2q^{2} + 486q^{3} + 1854q^{4} + 444q^{5} - 162q^{6} + 5868q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
39.10.a.a	$x^{3} + 906x^{2} - 973872x + 186071040$
39.10.a.b	$x^{4} + 1008x^{3} - 4726908x^{2} - 120239568x + 2799291354304$
39.10.a.c	$x^{5} - 2676x^{4} - 4842903x^{3} + 15269148882x^{2} - 6763232044440x + 156120546332736$
39.10.a.d	$x^{6} - 442x^{5} - 10154077x^{4} + 6108636384x^{3} + 25505657329180x^{2} - 20638379351215696x - 1121855873084983360$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(39))$	18
Old forms	22
Cusp forms $S_{10}(\Gamma_0(39))$	40
Eisenstein series $E_{10}(\Gamma_0(39))$	4
Modular forms $M_{10}(\Gamma_0(39))$	44

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(39)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 3, 13, 39]: [1, 4, 11, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(39, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 39:10 1000

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