Atlas › Modular forms › Weight 2 › Level 215 › 215.2.a

Newspace 215.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(215))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
215.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$	215a1 rank 1
215.2.a.b	3	degree 3	$3q - 2q^{2} + q^{3} + 4q^{4} + 3q^{5} + 8q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
215.2.a.c	5	degree 5	$5q + 2q^{2} - q^{3} + 8q^{4} + 5q^{5} - 12q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$
215.2.a.d	6	degree 6	$6q + 3q^{2} + 4q^{3} + 7q^{4} - 6q^{5} + 8q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
215.2.a.a	$x$
215.2.a.b	$x^{3} - x^{2} - 69x - 3$
215.2.a.c	$x^{5} + x^{4} - 121x^{3} + 124x^{2} + 3641x - 9088$
215.2.a.d	$x^{6} - 15x^{5} - 14x^{4} + 1001x^{3} - 4008x^{2} + 4642x - 519$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(215))$	15
Old forms	6
Cusp forms $S_{2}(\Gamma_0(215))$	21
Eisenstein series $E_{2}(\Gamma_0(215))$	3
Modular forms $M_{2}(\Gamma_0(215))$	24

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(215)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.178 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, 43, 215]: [0, 1, 4, 24] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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(215, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 215:2 1000

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