Atlas › Modular forms › Weight 2 › Level 384 › 384.2.a

Newspace 384.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(384))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
384.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} + 2q^{7} + O(q^{8})$	+	384g1 rank 0
384.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{7} + O(q^{8})$	+	384d1 rank 1
384.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 2q^{7} + O(q^{8})$	+	384b1 rank 0
384.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} - 2q^{7} + O(q^{8})$	+	384f1 rank 0
384.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} - 2q^{7} + O(q^{8})$	−	384h1 rank 1
384.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 2q^{7} + O(q^{8})$	−	384c1 rank 0
384.2.a.g	1	$\mathbb{Q}$	$q + q^{3} + 2q^{7} + O(q^{8})$	−	384a1 rank 0
384.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} + 2q^{7} + O(q^{8})$	−	384e1 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 $T$
384.2.a.a	$x - 90$
384.2.a.b	$x + 82$
384.2.a.c	$x - 82$
384.2.a.d	$x - 22$
384.2.a.e	$x + 102$
384.2.a.f	$x - 66$
384.2.a.g	$x - 30$
384.2.a.h	$x + 42$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(384))$	8
Old forms	41
Cusp forms $S_{2}(\Gamma_0(384))$	49
Eisenstein series $E_{2}(\Gamma_0(384))$	31
Modular forms $M_{2}(\Gamma_0(384))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(384)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(48))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(96))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(128))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(192))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.336 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, 128, 192, 384]: [0, 1, 1, 2, 3, 3, 5, 4, 8, 6, 12, 10, 20, 18, 36, 68] (certified at ell = 2147483489)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(384, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 384:2 1000

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