Atlas › Modular forms › Weight 2 › Level 168 › 168.2.a

Newspace 168.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(168))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
168.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + q^{7} + O(q^{8})$	+ −			168b1 rank 0
168.2.a.b	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} - q^{7} + O(q^{8})$	− +			168a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19} + 11T_{23}$ 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 $T$
168.2.a.a	$x - 10$
168.2.a.b	$x + 46$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(168))$	2
Old forms	23
Cusp forms $S_{2}(\Gamma_0(168))$	25
Eisenstein series $E_{2}(\Gamma_0(168))$	15
Modular forms $M_{2}(\Gamma_0(168))$	40

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(168)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(42))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(84))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.129 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, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168]: [0, 1, 1, 2, 3, 1, 3, 5, 4, 4, 8, 7, 12, 12, 22, 40] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (19, 9), (23, 11)]; 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(168, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 168:2 1000

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