Atlas › Modular forms › Weight 2 › Level 136 › 136.2.a

Newspace 136.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(136))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
136.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−	136a1 rank 1
136.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + O(q^{8})$	+	136b1 rank 0
136.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 4q^{5} + 2q^{7} + O(q^{8})$	−

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 $T$
136.2.a.a	$x + 4$
136.2.a.b	$x - 2$
136.2.a.c	$x^{2} - 2x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(136))$	4
Old forms	11
Cusp forms $S_{2}(\Gamma_0(136))$	15
Eisenstein series $E_{2}(\Gamma_0(136))$	7
Modular forms $M_{2}(\Gamma_0(136))$	22

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(136)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 8, 17, 34, 68, 136]: [0, 1, 2, 3, 2, 6, 12, 22] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(136, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 136:2 1000

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