Atlas › Modular forms › Weight 2 › Level 329 › 329.2.a

Newspace 329.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(329))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
329.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + 3q^{5} + q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	329a1 rank 0
329.2.a.b	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + q^{3} - 2q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
329.2.a.c	3	degree 3	$3q - q^{2} - 4q^{3} - q^{4} - q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$
329.2.a.d	3	degree 3	$3q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + 3q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
329.2.a.e	3	degree 3	$3q + q^{2} + q^{3} + 5q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
329.2.a.f	5	degree 5	$5q + q^{2} + 2q^{3} + 13q^{4} - 4q^{5} + 7q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
329.2.a.g	6	degree 6	$6q + 3q^{3} + 12q^{4} + 5q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
329.2.a.a	$x + 2$
329.2.a.b	$x^{2} + x - 4$
329.2.a.c	$x^{3} + 5x^{2} + 6x + 1$
329.2.a.d	$x^{3} + 3x^{2} - 4x + 1$
329.2.a.e	$x^{3} - 2x^{2} - 12x + 8$
329.2.a.f	$x^{5} - 3x^{4} - 25x^{3} + 76x^{2} + 38x - 83$
329.2.a.g	$x^{6} - 3x^{5} - 18x^{4} + 65x^{3} - 4x^{2} - 96x + 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(329))$	23
Old forms	8
Cusp forms $S_{2}(\Gamma_0(329))$	31
Eisenstein series $E_{2}(\Gamma_0(329))$	3
Modular forms $M_{2}(\Gamma_0(329))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(329)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 7, 47, 329]: [0, 1, 5, 34] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 3 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(329, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 329:2 1000

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