Atlas › Modular forms › Weight 2 › Level 867 › 867.2.a

Newspace 867.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(867))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
867.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{6} - 4q^{7} + O(q^{8})$	+	867b1 rank 1
867.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} + O(q^{8})$	−	867d1 rank 0
867.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	+	867a1 rank 1
867.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+	867c1 rank 1
867.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	−	867e1 rank 0
867.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 2q^{3} + 5q^{4} - 3q^{5} - q^{6} + O(q^{8})$	−
867.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} + O(q^{8})$	+
867.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	−
867.2.a.i	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	+
867.2.a.j	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	−
867.2.a.k	4	degree 4	$4q - 2q^{2} - 4q^{3} + 6q^{4} - 6q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+
867.2.a.l	4	degree 4	$4q - 2q^{2} + 4q^{3} + 6q^{4} + 6q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	−
867.2.a.m	4	degree 4	$4q - 4q^{3} + 4q^{5} + 8q^{7} + O(q^{8})$	+
867.2.a.n	4	degree 4	$4q + 4q^{3} - 4q^{5} - 8q^{7} + O(q^{8})$	−
867.2.a.o	6	degree 6	$6q + 3q^{2} - 6q^{3} + 9q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+
867.2.a.p	6	degree 6	$6q + 3q^{2} + 6q^{3} + 9q^{4} - 3q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
867.2.a.a	$x - 9$
867.2.a.b	$x - 57$
867.2.a.c	$x + 104$
867.2.a.d	$x + 108$
867.2.a.e	$x - 42$
867.2.a.f	$x^{2} - 103x + 2546$
867.2.a.g	$x^{2} - 38x - 1561$
867.2.a.h	$x^{2} + 86x + 1607$
867.2.a.i	$x^{3} + 66x^{2} - 729x - 75367$
867.2.a.j	$x^{3} + 126x^{2} + 933x - 200269$
867.2.a.k	$x^{4} + 66x^{3} - 1123x^{2} - 62700x + 265852$
867.2.a.l	$x^{4} - 202x^{3} + 13689x^{2} - 352928x + 3053824$
867.2.a.m	$x^{4} - 104x^{3} - 4052x^{2} + 45280x - 68612$
867.2.a.n	$x^{4} + 288x^{3} + 28916x^{2} + 1192752x + 17107196$
867.2.a.o	$x^{6} - 246x^{5} + 16722x^{4} + 154174x^{3} - 48360213x^{2} + 1238291820x - 8883517501$
867.2.a.p	$x^{6} + 42x^{5} - 4398x^{4} - 294274x^{3} - 5604357x^{2} - 37088124x - 68681953$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(867))$	45
Old forms	40
Cusp forms $S_{2}(\Gamma_0(867))$	85
Eisenstein series $E_{2}(\Gamma_0(867))$	35
Modular forms $M_{2}(\Gamma_0(867))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(867)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(289))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.671 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, 3, 17, 51, 289, 867]: [0, 1, 2, 8, 26, 104] (certified at ell = 2147483249)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (29, 8)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 13 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(867, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 867:2 1000

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