Atlas › Modular forms › Weight 2 › Level 784 › 784.2.a

Newspace 784.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(784))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
784.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{5} + O(q^{8})$		784b1 rank 1
784.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + O(q^{8})$		784j1 rank 1
784.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$		784d1 rank 0
784.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + O(q^{8})$		784g1 rank 0
784.2.a.e	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$		784c1 rank 0
784.2.a.f	1	$\mathbb{Q}$	$q + O(q^{8})$	$\mathbb{Q}(\sqrt{-7})$	784h1 rank 1
784.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} + O(q^{8})$		784i1 rank 1
784.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - q^{5} + O(q^{8})$		784a1 rank 1
784.2.a.i	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{5} + O(q^{8})$		784e1 rank 0
784.2.a.j	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + O(q^{8})$		784f1 rank 0
784.2.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
784.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
784.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
784.2.a.n	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
784.2.a.a	$x + 139$
784.2.a.b	$x + 230$
784.2.a.c	$x - 167$
784.2.a.d	$x - 141$
784.2.a.e	$x - 204$
784.2.a.f	$x - 32$
784.2.a.g	$x + 185$
784.2.a.h	$x - 101$
784.2.a.i	$x + 130$
784.2.a.j	$x - 87$
784.2.a.k	$x^{2} + 388x + 36914$
784.2.a.l	$x^{2} - 252x - 4526$
784.2.a.m	$x^{2} - 96x - 1928$
784.2.a.n	$x^{2} - 352x + 22264$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(784))$	18
Old forms	71
Cusp forms $S_{2}(\Gamma_0(784))$	89
Eisenstein series $E_{2}(\Gamma_0(784))$	47
Modular forms $M_{2}(\Gamma_0(784))$	136

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(784)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(112))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(196))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(392))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.01 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, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784]: [0, 1, 2, 1, 3, 4, 4, 7, 5, 12, 16, 20, 31, 60, 116] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 6 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(784, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 784:2 1000

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