Atlas › Modular forms › Weight 2 › Level 640 › 640.2.a

Newspace 640.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(640))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
640.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{5} + O(q^{8})$	+	640f1 rank 0
640.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + O(q^{8})$	−	640h1 rank 1
640.2.a.c	1	$\mathbb{Q}$	$q - q^{5} - 2q^{7} + O(q^{8})$	+	640b1 rank 1
640.2.a.d	1	$\mathbb{Q}$	$q - q^{5} + 2q^{7} + O(q^{8})$	+	640a1 rank 1
640.2.a.e	1	$\mathbb{Q}$	$q + q^{5} - 2q^{7} + O(q^{8})$	−	640g1 rank 1
640.2.a.f	1	$\mathbb{Q}$	$q + q^{5} + 2q^{7} + O(q^{8})$	−	640c1 rank 0
640.2.a.g	1	$\mathbb{Q}$	$q + 2q^{3} - q^{5} + O(q^{8})$	+	640e1 rank 0
640.2.a.h	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + O(q^{8})$	−	640d1 rank 0
640.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+
640.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−
640.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	+
640.2.a.l	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
640.2.a.a	$x - 414$
640.2.a.b	$x + 346$
640.2.a.c	$x - 216$
640.2.a.d	$x + 288$
640.2.a.e	$x + 224$
640.2.a.f	$x - 88$
640.2.a.g	$x + 182$
640.2.a.h	$x + 30$
640.2.a.i	$x^{2} + 554x + 73604$
640.2.a.j	$x^{2} - 590x + 20900$
640.2.a.k	$x^{2} - 154x - 120476$
640.2.a.l	$x^{2} - 514x + 48644$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(640))$	16
Old forms	65
Cusp forms $S_{2}(\Gamma_0(640))$	81
Eisenstein series $E_{2}(\Gamma_0(640))$	31
Modular forms $M_{2}(\Gamma_0(640))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(640)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(40))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(80))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(128))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(160))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(320))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.68 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, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640]: [0, 1, 2, 1, 3, 3, 4, 6, 6, 10, 10, 16, 18, 28, 52, 100] (certified at ell = 2147483489)
T = [(3, 1), (7, 2), (11, 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 16 = 1 x 16 (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(640, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 640:2 1000

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