Atlas › Modular forms › Weight 2 › Level 441 › 441.2.a

Newspace 441.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(441))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
441.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 2q^{5} + O(q^{8})$		441f1 rank 1
441.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + 2q^{5} + O(q^{8})$		441e1 rank 0
441.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$	441d1 rank 1
441.2.a.d	1	$\mathbb{Q}$	$q - 2q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	441b1 rank 1
441.2.a.e	1	$\mathbb{Q}$	$q - 2q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	441a1 rank 0
441.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} + O(q^{8})$		441c1 rank 1
441.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} + O(q^{8})$
441.2.a.h	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$
441.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 4q^{5} + O(q^{8})$
441.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 4q^{5} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
441.2.a.a	$x + 42$
441.2.a.b	$x + 14$
441.2.a.c	$x + 85$
441.2.a.d	$x + 70$
441.2.a.e	$x - 70$
441.2.a.f	$x + 45$
441.2.a.g	$x^{2} - 32x - 251$
441.2.a.h	$x^{2} - 3703$
441.2.a.i	$x^{2} - 110x - 175$
441.2.a.j	$x^{2} - 102x - 1631$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(441))$	14
Old forms	27
Cusp forms $S_{2}(\Gamma_0(441))$	41
Eisenstein series $E_{2}(\Gamma_0(441))$	31
Modular forms $M_{2}(\Gamma_0(441))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(441)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(147))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 7, 9, 21, 49, 63, 147, 441]: [0, 1, 1, 2, 4, 5, 10, 20, 58] (certified at ell = 2147483647)
T = [(2, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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(441, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 441:2 1000

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