Atlas › Modular forms › Weight 2 › Level 888 › 888.2.a

Newspace 888.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(888))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
888.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} - q^{7} + O(q^{8})$	+ −	888a1 rank 0
888.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+ −	888c1 rank 1
888.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + O(q^{8})$	− −	888b1 rank 1
888.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} + O(q^{8})$	− −	888d1 rank 0
888.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	+ +
888.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 2q^{5} + 8q^{7} + O(q^{8})$	+ −
888.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 4q^{7} + O(q^{8})$	− +
888.2.a.h	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{3} + 3q^{7} + O(q^{8})$	− −
888.2.a.i	3	degree 3	$3q - 3q^{3} + 2q^{5} - 5q^{7} + O(q^{8})$	+ +
888.2.a.j	3	degree 3	$3q + 3q^{3} - q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ 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 $T$
888.2.a.a	$x - 61$
888.2.a.b	$x + 34$
888.2.a.c	$x + 90$
888.2.a.d	$x - 18$
888.2.a.e	$x^{2} + 92x + 1816$
888.2.a.f	$x^{2} + 8x - 164$
888.2.a.g	$x^{2} + 80x + 248$
888.2.a.h	$x^{2} - 43x - 3238$
888.2.a.i	$x^{3} - 59x^{2} - 204x + 27368$
888.2.a.j	$x^{3} - 143x^{2} + 6584x - 98360$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(888))$	18
Old forms	127
Cusp forms $S_{2}(\Gamma_0(888))$	145
Eisenstein series $E_{2}(\Gamma_0(888))$	15
Modular forms $M_{2}(\Gamma_0(888))$	160

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(888)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(148))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(222))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(296))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(444))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.633 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, 24, 37, 74, 111, 148, 222, 296, 444, 888]: [0, 1, 1, 2, 3, 3, 5, 8, 3, 11, 14, 22, 42, 42, 82, 160] (certified at ell = 2147483629)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(888, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 888:2 1000

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