Atlas › Modular forms › Weight 2 › Level 147 › 147.2.a

Newspace 147.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(147))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
147.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+	147a1 rank 0
147.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + O(q^{8})$	+	147c1 rank 0
147.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + O(q^{8})$	−	147b1 rank 0
147.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + O(q^{8})$	+
147.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	−

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 $T$
147.2.a.a	$x - 5$
147.2.a.b	$x - 8$
147.2.a.c	$x + 4$
147.2.a.d	$x^{2} + 14x + 41$
147.2.a.e	$x^{2} - 10x - 7$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(147))$	7
Old forms	4
Cusp forms $S_{2}(\Gamma_0(147))$	11
Eisenstein series $E_{2}(\Gamma_0(147))$	15
Modular forms $M_{2}(\Gamma_0(147))$	26

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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]: [0, 1, 1, 4, 5, 20] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 7 = 1 x 7 (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(147, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 147:2 1000

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