Atlas › Modular forms › Weight 12 › Level 15 › 15.12.a

Newspace 15.12.a: $S_{12}^{\mathrm{new}}(\Gamma_0(15))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
15.12.a.a	1	$\mathbb{Q}$	$q - 56q^{2} - 243q^{3} + 1088q^{4} + 3125q^{5} + 13608q^{6} + 27984q^{7} + O(q^{8})$	+ −	$-1$
15.12.a.b	2	$\mathbb{Q}(\sqrt{1609})$	$2q - 22q^{2} + 486q^{3} - 636q^{4} - 6250q^{5} - 5346q^{6} - 10864q^{7} + O(q^{8})$	− +	$-1$
15.12.a.c	2	$\mathbb{Q}(\sqrt{1801})$	$2q - 13q^{2} - 486q^{3} - 3111q^{4} - 6250q^{5} + 3159q^{6} + 7784q^{7} + O(q^{8})$	+ +	$+1$
15.12.a.d	3	degree 3	$3q - q^{2} + 729q^{3} + 4757q^{4} + 9375q^{5} - 243q^{6} - 14608q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{7}$ 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 $T$
15.12.a.a	$x - 27928$
15.12.a.b	$x^{2} + 10886x - 2586004376$
15.12.a.c	$x^{2} - 7771x - 5771617196$
15.12.a.d	$x^{3} + 14609x^{2} - 2706663298x + 22551554333152$

Dimensions

Newforms $S_{12}^{\mathrm{new}}(\Gamma_0(15))$	8
Old forms	12
Cusp forms $S_{12}(\Gamma_0(15))$	20
Eisenstein series $E_{12}(\Gamma_0(15))$	4
Modular forms $M_{12}(\Gamma_0(15))$	24

Decomposition of the old subspace

$S_{12}^{\mathrm{old}}(\Gamma_0(15)) =$ $S_{12}^{\mathrm{new}}(\Gamma_0(1))^{\oplus 4}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(3))^{\oplus 2}$ $\oplus$ $S_{12}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.335 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, 5, 15]: [2, 5, 7, 24] (certified at ell = 2147483629)
T = [(2, 1), (7, 1)]; h of degree 8 = 1 x 8 (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(15, 12)

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

cargo run --release -p sagebrush-web --example atlas -- 15:12 1000

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