Atlas › Modular forms › Weight 4 › Level 135 › 135.4.a

Newspace 135.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(135))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
135.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 4q^{4} + 5q^{5} + O(q^{8})$	−
135.4.a.b	1	$\mathbb{Q}$	$q - q^{2} - 7q^{4} - 5q^{5} - 6q^{7} + O(q^{8})$	+
135.4.a.c	1	$\mathbb{Q}$	$q + q^{2} - 7q^{4} + 5q^{5} - 6q^{7} + O(q^{8})$	−
135.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 4q^{4} - 5q^{5} + O(q^{8})$	+
135.4.a.e	3	degree 3	$3q - 5q^{2} + 17q^{4} - 15q^{5} - 4q^{7} + O(q^{8})$	+
135.4.a.f	3	degree 3	$3q - q^{2} + 23q^{4} + 15q^{5} + 44q^{7} + O(q^{8})$	−
135.4.a.g	3	degree 3	$3q + q^{2} + 23q^{4} - 15q^{5} + 44q^{7} + O(q^{8})$	+
135.4.a.h	3	degree 3	$3q + 5q^{2} + 17q^{4} + 15q^{5} - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
135.4.a.a	$x + 2$
135.4.a.b	$x + 19$
135.4.a.c	$x + 17$
135.4.a.d	$x - 2$
135.4.a.e	$x^{3} + 17x^{2} - 6923x - 243099$
135.4.a.f	$x^{3} - 131x^{2} + 5137x - 58008$
135.4.a.g	$x^{3} - 133x^{2} + 4753x - 37044$
135.4.a.h	$x^{3} + 7x^{2} - 7403x - 201189$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(135))$	16
Old forms	32
Cusp forms $S_{4}(\Gamma_0(135))$	48
Eisenstein series $E_{4}(\Gamma_0(135))$	12
Modular forms $M_{4}(\Gamma_0(135))$	60

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(135)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.129 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, 5, 9, 15, 27, 45, 135]: [1, 2, 3, 4, 8, 10, 20, 56] (certified at ell = 2147483629)
T = [(2, 1), (7, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 4 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(135, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 135:4 1000

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