Atlas › Modular forms › Weight 4 › Level 147 › 147.4.a

Newspace 147.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(147))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
147.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} - 3q^{3} + q^{4} + 3q^{5} + 9q^{6} + O(q^{8})$	+
147.4.a.b	1	$\mathbb{Q}$	$q - 3q^{2} + 3q^{3} + q^{4} - 3q^{5} - 9q^{6} + O(q^{8})$	−
147.4.a.c	1	$\mathbb{Q}$	$q - 3q^{2} + 3q^{3} + q^{4} + 18q^{5} - 9q^{6} + O(q^{8})$	−
147.4.a.d	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - 7q^{4} + 12q^{5} + 3q^{6} + O(q^{8})$	+
147.4.a.e	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} - 7q^{4} - 12q^{5} - 3q^{6} + O(q^{8})$	−
147.4.a.f	1	$\mathbb{Q}$	$q + 4q^{2} - 3q^{3} + 8q^{4} - 18q^{5} - 12q^{6} + O(q^{8})$	+
147.4.a.g	1	$\mathbb{Q}$	$q + 4q^{2} + 3q^{3} + 8q^{4} + 4q^{5} + 12q^{6} + O(q^{8})$	−
147.4.a.h	1	$\mathbb{Q}$	$q + 4q^{2} + 3q^{3} + 8q^{4} + 18q^{5} + 12q^{6} + O(q^{8})$	−
147.4.a.i	2	$\mathbb{Q}(\sqrt{57})$	$2q - 3q^{2} - 6q^{3} + 17q^{4} - 6q^{5} + 9q^{6} + O(q^{8})$	+
147.4.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} + O(q^{8})$	+
147.4.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} + O(q^{8})$	−
147.4.a.l	3	degree 3	$3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + O(q^{8})$	+
147.4.a.m	3	degree 3	$3q + q^{2} + 9q^{3} + 25q^{4} - 11q^{5} + 3q^{6} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
147.4.a.a	$x - 6$
147.4.a.b	$x + 12$
147.4.a.c	$x - 51$
147.4.a.d	$x - 35$
147.4.a.e	$x + 37$
147.4.a.f	$x + 50$
147.4.a.g	$x - 16$
147.4.a.h	$x - 58$
147.4.a.i	$x^{2} + 21x - 588$
147.4.a.j	$x^{2} - 62x - 7$
147.4.a.k	$x^{2} + 58x + 41$
147.4.a.l	$x^{3} - 34x^{2} - 1554x + 39984$
147.4.a.m	$x^{3} + 32x^{2} - 1950x - 24852$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(147))$	20
Old forms	28
Cusp forms $S_{4}(\Gamma_0(147))$	48
Eisenstein series $E_{4}(\Gamma_0(147))$	16
Modular forms $M_{4}(\Gamma_0(147))$	64

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(147)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 7, 21, 49, 147]: [1, 2, 3, 10, 15, 58] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(147, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 147:4 1000

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