Atlas › Modular forms › Weight 2 › Level 126 › 126.2.a

Newspace 126.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(126))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
126.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} - q^{7} + O(q^{8})$	+ +			126b1 rank 0
126.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + q^{7} + O(q^{8})$	− −			126a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
126.2.a.a	$x + 34$
126.2.a.b	$x + 16$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(126))$	2
Old forms	15
Cusp forms $S_{2}(\Gamma_0(126))$	17
Eisenstein series $E_{2}(\Gamma_0(126))$	15
Modular forms $M_{2}(\Gamma_0(126))$	32

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(126)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126]: [0, 1, 1, 3, 1, 2, 4, 5, 4, 12, 10, 28] (certified at ell = 2147483647)
T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 2 = 1 x 2 (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(126, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 126:2 1000

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