Atlas › Modular forms › Weight 2 › Level 896 › 896.2.a

Newspace 896.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(896))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
896.2.a.a	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	+	896b1 rank 1
896.2.a.b	1	$\mathbb{Q}$	$q - q^{7} + O(q^{8})$	+	896a1 rank 1
896.2.a.c	1	$\mathbb{Q}$	$q + q^{7} + O(q^{8})$	−	896d1 rank 1
896.2.a.d	1	$\mathbb{Q}$	$q + q^{7} + O(q^{8})$	−	896c1 rank 0
896.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−
896.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	+
896.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+
896.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{5} + 2q^{7} + O(q^{8})$	−
896.2.a.i	3	degree 3	$3q - 2q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+
896.2.a.j	3	degree 3	$3q - 2q^{3} + 2q^{5} + 3q^{7} + O(q^{8})$	−
896.2.a.k	3	degree 3	$3q + 2q^{3} - 2q^{5} + 3q^{7} + O(q^{8})$	−
896.2.a.l	3	degree 3	$3q + 2q^{3} + 2q^{5} - 3q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
896.2.a.a	$x - 38$
896.2.a.b	$x + 302$
896.2.a.c	$x + 162$
896.2.a.d	$x - 298$
896.2.a.e	$x^{2} + 720x + 109428$
896.2.a.f	$x^{2} - 400x - 46700$
896.2.a.g	$x^{2} - 56x - 38204$
896.2.a.h	$x^{2} - 424x + 28516$
896.2.a.i	$x^{3} + 670x^{2} + 128932x + 5660024$
896.2.a.j	$x^{3} - 426x^{2} - 66652x + 8752216$
896.2.a.k	$x^{3} - 38x^{2} - 62556x - 891736$
896.2.a.l	$x^{3} - 782x^{2} + 58820x + 33808072$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(896))$	24
Old forms	89
Cusp forms $S_{2}(\Gamma_0(896))$	113
Eisenstein series $E_{2}(\Gamma_0(896))$	31
Modular forms $M_{2}(\Gamma_0(896))$	144

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(896)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 7}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(112))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(128))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(224))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(448))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.498 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, 32, 56, 64, 112, 128, 224, 448, 896]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 10, 20, 18, 36, 68, 132] (certified at ell = 2147483137)
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), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 9 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(896, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 896:2 1000

← 895.2.a · 897.2.a →