Atlas › Modular forms › Weight 2 › Level 909 › 909.2.a

Newspace 909.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(909))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
909.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - 2q^{7} + O(q^{8})$	−	909c1 rank 1
909.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} + 3q^{5} + O(q^{8})$	+	909a1 rank 0
909.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + q^{5} - 2q^{7} + O(q^{8})$	+	909b1 rank 0
909.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{5} - 4q^{7} + O(q^{8})$	−
909.2.a.e	4	degree 4	$4q - 3q^{2} + 3q^{4} - 2q^{7} + O(q^{8})$	+
909.2.a.f	4	degree 4	$4q + 4q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	+
909.2.a.g	4	degree 4	$4q + 4q^{4} + 4q^{5} + 2q^{7} + O(q^{8})$	−
909.2.a.h	4	degree 4	$4q + 3q^{2} + 3q^{4} - 2q^{7} + O(q^{8})$	−
909.2.a.i	6	degree 6	$6q - q^{2} + 3q^{4} - 6q^{5} + O(q^{8})$	−
909.2.a.j	7	degree 7	$7q + 10q^{4} - 6q^{5} + 6q^{7} + O(q^{8})$	+
909.2.a.k	7	degree 7	$7q + 12q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
909.2.a.a	$x - 3$
909.2.a.b	$x - 9$
909.2.a.c	$x - 5$
909.2.a.d	$x^{2} - 6x + 1$
909.2.a.e	$x^{4} + 3x^{3} - 169x^{2} - 397x + 3711$
909.2.a.f	$x^{4} + 12x^{3} + 30x^{2} - 52x - 151$
909.2.a.g	$x^{4} - 12x^{3} + 30x^{2} + 52x - 151$
909.2.a.h	$x^{4} - 3x^{3} - 169x^{2} + 397x + 3711$
909.2.a.i	$x^{6} + 19x^{5} + 32x^{4} - 962x^{3} - 3944x^{2} + 409x + 11241$
909.2.a.j	$x^{7} + 18x^{6} - 171x^{5} - 4302x^{4} - 5210x^{3} + 231371x^{2} + 1102212x + 738193$
909.2.a.k	$x^{7} - 9x^{6} - 118x^{5} + 1138x^{4} + 3053x^{3} - 39485x^{2} + 19961x + 226183$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(909))$	41
Old forms	58
Cusp forms $S_{2}(\Gamma_0(909))$	99
Eisenstein series $E_{2}(\Gamma_0(909))$	7
Modular forms $M_{2}(\Gamma_0(909))$	106

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(909)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(303))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.288 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, 9, 101, 303, 909]: [0, 1, 2, 9, 36, 104] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 6 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(909, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 909:2 1000

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