Atlas › Modular forms › Weight 4 › Level 126 › 126.4.a

Newspace 126.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(126))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
126.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 18q^{5} + 7q^{7} + O(q^{8})$	+ −
126.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 6q^{5} + 7q^{7} + O(q^{8})$	+ −
126.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} - 2q^{5} - 7q^{7} + O(q^{8})$	+ +
126.4.a.d	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 12q^{5} + 7q^{7} + O(q^{8})$	+ −
126.4.a.e	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{4} + 22q^{5} - 7q^{7} + O(q^{8})$	+ +
126.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} - 22q^{5} - 7q^{7} + O(q^{8})$	− +
126.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} + 6q^{5} + 7q^{7} + O(q^{8})$	− −
126.4.a.h	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{4} + 14q^{5} - 7q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11}$ 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 $T$
126.4.a.a	$x - 198$
126.4.a.b	$x + 96$
126.4.a.c	$x - 22$
126.4.a.d	$x + 132$
126.4.a.e	$x - 100$
126.4.a.f	$x + 100$
126.4.a.g	$x - 96$
126.4.a.h	$x - 98$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(126))$	8
Old forms	56
Cusp forms $S_{4}(\Gamma_0(126))$	64
Eisenstein series $E_{4}(\Gamma_0(126))$	16
Modular forms $M_{4}(\Gamma_0(126))$	80

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(126)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(9))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(18))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.152 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]: [1, 2, 2, 5, 3, 4, 8, 11, 10, 28, 26, 76] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 126:4 1000

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