Atlas › Modular forms › Weight 2 › Level 768 › 768.2.a

Newspace 768.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(768))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
768.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+	768b1 rank 1
768.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 4q^{7} + O(q^{8})$	+	768a1 rank 1
768.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 4q^{7} + O(q^{8})$	+	768e1 rank 0
768.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	+	768f1 rank 0
768.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	−	768h1 rank 1
768.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 4q^{7} + O(q^{8})$	−	768g1 rank 1
768.2.a.g	1	$\mathbb{Q}$	$q + q^{3} + 4q^{7} + O(q^{8})$	−	768c1 rank 0
768.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−	768d1 rank 0
768.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + O(q^{8})$	+
768.2.a.j	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + O(q^{8})$	+
768.2.a.k	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + O(q^{8})$	−
768.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = 1904T_{5} + 4595T_{7} + 5697T_{13} + 9850T_{17} + 5931T_{19} + 8769T_{11} + 1705T_{23} + 9694T_{29}$ 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 $T$
768.2.a.a	$x + 78366$
768.2.a.b	$x + 95132$
768.2.a.c	$x - 78436$
768.2.a.d	$x - 86414$
768.2.a.e	$x + 130554$
768.2.a.f	$x + 8308$
768.2.a.g	$x + 53796$
768.2.a.h	$x - 43706$
768.2.a.i	$x^{2} + 157000x + 2843080712$
768.2.a.j	$x^{2} - 70752x + 727949076$
768.2.a.k	$x^{2} - 165648x + 5893207476$
768.2.a.l	$x^{2} - 78200x - 1062902008$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(768))$	16
Old forms	89
Cusp forms $S_{2}(\Gamma_0(768))$	105
Eisenstein series $E_{2}(\Gamma_0(768))$	47
Modular forms $M_{2}(\Gamma_0(768))$	152

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.327 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, 256, 384, 768]: [0, 1, 1, 2, 3, 3, 5, 4, 8, 6, 12, 10, 20, 18, 36, 34, 68, 132] (certified at ell = 2147483137)
T = [(5, 1904), (7, 4595), (13, 5697), (17, 9850), (19, 5931), (11, 8769), (23, 1705), (29, 9694)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(768, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 768:2 1000

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