Atlas › Modular forms › Weight 2 › Level 704 › 704.2.a

Newspace 704.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(704))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
704.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{5} + O(q^{8})$	−	704l1 rank 1
704.2.a.b	1	$\mathbb{Q}$	$q - 3q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	+	704i1 rank 0
704.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + 2q^{7} + O(q^{8})$	−	704k1 rank 1
704.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + 4q^{7} + O(q^{8})$	+	704g1 rank 0
704.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} - 4q^{7} + O(q^{8})$	+	704b1 rank 1
704.2.a.f	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + 2q^{7} + O(q^{8})$	−	704d1 rank 0
704.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 4q^{7} + O(q^{8})$	−	704j1 rank 1
704.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+	704a1 rank 1
704.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	+	704f1 rank 0
704.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	−	704c1 rank 0
704.2.a.k	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} + O(q^{8})$	+	704h1 rank 0
704.2.a.l	1	$\mathbb{Q}$	$q + 3q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	−	704e1 rank 0
704.2.a.m	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	−
704.2.a.n	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 3q^{5} + O(q^{8})$	+
704.2.a.o	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 3q^{5} + O(q^{8})$	−
704.2.a.p	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
704.2.a.a	$x + 364$
704.2.a.b	$x - 246$
704.2.a.c	$x + 406$
704.2.a.d	$x - 396$
704.2.a.e	$x + 204$
704.2.a.f	$x - 266$
704.2.a.g	$x + 56$
704.2.a.h	$x + 102$
704.2.a.i	$x - 154$
704.2.a.j	$x - 64$
704.2.a.k	$x + 64$
704.2.a.l	$x + 242$
704.2.a.m	$x^{2} - 594x + 7272$
704.2.a.n	$x^{2} + 128x - 152576$
704.2.a.o	$x^{2} - 220x - 80992$
704.2.a.p	$x^{2} + 198x + 8424$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(704))$	20
Old forms	65
Cusp forms $S_{2}(\Gamma_0(704))$	85
Eisenstein series $E_{2}(\Gamma_0(704))$	23
Modular forms $M_{2}(\Gamma_0(704))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(704)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 7}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(88))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(176))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(352))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.815 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, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704]: [0, 1, 2, 3, 2, 4, 5, 6, 9, 10, 16, 28, 52, 100] (certified at ell = 2147482801)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 8 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(704, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 704:2 1000

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