Atlas › Modular forms › Weight 2 › Level 154 › 154.2.a

Newspace 154.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(154))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
154.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 4q^{5} - q^{7} + O(q^{8})$	+ + +	$-1$	154a1 rank 1
154.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	154c1 rank 0
154.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} - q^{7} + O(q^{8})$	− + +	$+1$	154b1 rank 0
154.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
154.2.a.a	$x + 12$
154.2.a.b	$x - 8$
154.2.a.c	$x - 6$
154.2.a.d	$x^{2} - 4x - 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(154))$	5
Old forms	16
Cusp forms $S_{2}(\Gamma_0(154))$	21
Eisenstein series $E_{2}(\Gamma_0(154))$	7
Modular forms $M_{2}(\Gamma_0(154))$	28

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(154)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(77))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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, 7, 11, 14, 22, 77, 154]: [0, 1, 1, 2, 4, 5, 10, 28] (certified at ell = 2147482951)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (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(154, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 154:2 1000

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