Atlas › Modular forms › Weight 2 › Level 444 › 444.2.a

Newspace 444.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(444))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
444.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+ +	444a1 rank 0
444.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	− +	444b1 rank 1
444.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ −
444.2.a.d	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{3} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
444.2.a.a	$x$
444.2.a.b	$x + 6$
444.2.a.c	$x^{2} + 2x - 2$
444.2.a.d	$x^{2} - 4x - 2$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(444))$	6
Old forms	65
Cusp forms $S_{2}(\Gamma_0(444))$	71
Eisenstein series $E_{2}(\Gamma_0(444))$	11
Modular forms $M_{2}(\Gamma_0(444))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(444)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(148))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(222))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.361 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, 12, 37, 74, 111, 148, 222, 444]: [0, 1, 1, 2, 3, 5, 3, 11, 14, 22, 42, 82] (certified at ell = 2147483629)
T = [(5, 1), (7, 1)]; 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(444, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 444:2 1000

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