Atlas › Modular forms › Weight 2 › Level 398 › 398.2.a

Newspace 398.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(398))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
398.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$	398a1 rank 0
398.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - q^{3} + 2q^{4} + q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
398.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
398.2.a.d	6	degree 6	$6q - 6q^{2} + q^{3} + 6q^{4} + 4q^{5} - q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$
398.2.a.e	6	degree 6	$6q + 6q^{2} + 3q^{3} + 6q^{4} + 2q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
398.2.a.a	$x$
398.2.a.b	$x^{2} + x - 1$
398.2.a.c	$x^{2} + 5x + 5$
398.2.a.d	$x^{6} - 5x^{5} - 40x^{4} + 215x^{3} + 230x^{2} - 2257x + 2651$
398.2.a.e	$x^{6} - 5x^{5} - 12x^{4} + 93x^{3} - 90x^{2} - 141x + 185$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(398))$	17
Old forms	32
Cusp forms $S_{2}(\Gamma_0(398))$	49
Eisenstein series $E_{2}(\Gamma_0(398))$	3
Modular forms $M_{2}(\Gamma_0(398))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(398)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(199))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.289 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, 199, 398]: [0, 1, 17, 52] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(398, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 398:2 1000

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