Atlas › Modular forms › Weight 2 › Level 242 › 242.2.a

Newspace 242.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(242))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
242.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 3q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+	242b1 rank 0
242.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 3q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	−	242a1 rank 1
242.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{7} + O(q^{8})$	+
242.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 3q^{3} + 2q^{4} + 2q^{5} - 3q^{6} + 4q^{7} + O(q^{8})$	+
242.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + O(q^{8})$	−
242.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 3q^{3} + 2q^{4} + 2q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
242.2.a.a	$x - 140$
242.2.a.b	$x + 176$
242.2.a.c	$x^{2} + 72x + 864$
242.2.a.d	$x^{2} + x - 1711$
242.2.a.e	$x^{2} + 276x + 18072$
242.2.a.f	$x^{2} - 179x - 10901$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(242))$	10
Old forms	12
Cusp forms $S_{2}(\Gamma_0(242))$	22
Eisenstein series $E_{2}(\Gamma_0(242))$	23
Modular forms $M_{2}(\Gamma_0(242))$	45

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(242)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 11, 22, 121, 242]: [0, 1, 2, 5, 12, 35] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 4 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(242, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 242:2 1000

← 241.2.a · 243.2.a →