Atlas › Modular forms › Weight 4 › Level 164 › 164.4.a

Newspace 164.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(164))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
164.4.a.a	3	degree 3	$3q - 2q^{3} - 10q^{5} + O(q^{8})$	+
164.4.a.b	7	degree 7	$7q + 4q^{3} + 10q^{5} + O(q^{8})$	−

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 $T$
164.4.a.a	$x^{3} + 12x^{2} + 6x - 176$
164.4.a.b	$x^{7} - 14x^{6} - 900x^{5} + 14760x^{4} + 146444x^{3} - 3175120x^{2} + 11251500x + 8619000$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(164))$	10
Old forms	50
Cusp forms $S_{4}(\Gamma_0(164))$	60
Eisenstein series $E_{4}(\Gamma_0(164))$	6
Modular forms $M_{4}(\Gamma_0(164))$	66

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(164)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 4, 41, 82, 164]: [1, 2, 3, 12, 34, 66] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 10 = 1 x 10 (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(164, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 164:4 1000

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