Atlas › Modular forms › Weight 2 › Level 612 › 612.2.a

Newspace 612.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(612))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
612.2.a.a	1	$\mathbb{Q}$	$q - 3q^{5} + 2q^{7} + O(q^{8})$	−	612b1 rank 1
612.2.a.b	1	$\mathbb{Q}$	$q - q^{5} + O(q^{8})$	+	612c1 rank 1
612.2.a.c	1	$\mathbb{Q}$	$q + q^{5} + 4q^{7} + O(q^{8})$	−	612d1 rank 0
612.2.a.d	1	$\mathbb{Q}$	$q + 3q^{5} + 2q^{7} + O(q^{8})$	+	612a1 rank 0
612.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
612.2.a.a	$x - 3$
612.2.a.b	$x + 1$
612.2.a.c	$x - 13$
612.2.a.d	$x - 9$
612.2.a.e	$x^{2} + 6x + 6$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(612))$	6
Old forms	91
Cusp forms $S_{2}(\Gamma_0(612))$	97
Eisenstein series $E_{2}(\Gamma_0(612))$	23
Modular forms $M_{2}(\Gamma_0(612))$	120

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.412 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, 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612]: [0, 1, 1, 2, 3, 2, 5, 2, 5, 6, 9, 8, 12, 22, 20, 42, 58, 114] (certified at ell = 2147483137)
T = [(5, 1), (7, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 1 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(612, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 612:2 1000

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