Atlas › Modular forms › Weight 2 › Level 192 › 192.2.a

Newspace 192.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(192))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
192.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+	192a1 rank 1
192.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+	192d1 rank 0
192.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + 4q^{7} + O(q^{8})$	−	192b1 rank 0
192.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	−	192c1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ 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 $T$
192.2.a.a	$x + 48$
192.2.a.b	$x - 28$
192.2.a.c	$x + 24$
192.2.a.d	$x + 92$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(192))$	4
Old forms	17
Cusp forms $S_{2}(\Gamma_0(192))$	21
Eisenstein series $E_{2}(\Gamma_0(192))$	23
Modular forms $M_{2}(\Gamma_0(192))$	44

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(192)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(96))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.267 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, 8, 12, 16, 24, 32, 48, 64, 96, 192]: [0, 1, 1, 2, 3, 3, 5, 4, 8, 6, 12, 10, 20, 36] (certified at ell = 2147483489)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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(192, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 192:2 1000

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