Atlas › Modular forms › Weight 2 › Level 338 › 338.2.a

Newspace 338.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(338))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
338.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+	338f1 rank 1
338.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+	338d1 rank 0
338.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 4q^{7} + O(q^{8})$	+	338a1 rank 1
338.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	−	338e1 rank 1
338.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} + 4q^{7} + O(q^{8})$	−	338b1 rank 0
338.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} + q^{7} + O(q^{8})$	−	338c1 rank 0
338.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 2q^{5} - 3q^{6} + 4q^{7} + O(q^{8})$	+
338.2.a.h	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} + 2q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
338.2.a.a	$x$
338.2.a.b	$x - 8$
338.2.a.c	$x - 3$
338.2.a.d	$x + 10$
338.2.a.e	$x + 3$
338.2.a.f	$x - 10$
338.2.a.g	$x^{3} + 3x^{2} - 130x - 41$
338.2.a.h	$x^{3} - 9x^{2} - 22x + 211$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(338))$	12
Old forms	20
Cusp forms $S_{2}(\Gamma_0(338))$	32
Eisenstein series $E_{2}(\Gamma_0(338))$	27
Modular forms $M_{2}(\Gamma_0(338))$	59

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(338)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(169))^{\oplus 2}$

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, 2, 13, 26, 169, 338]: [0, 1, 1, 5, 15, 47] (certified at ell = 2147483053)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (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(338, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 338:2 1000

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