Atlas › Modular forms › Weight 2 › Level 618 › 618.2.a

Newspace 618.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(618))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
618.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$	618a1 rank 1
618.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$	618b1 rank 1
618.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$	618d1 rank 1
618.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	+ − −	$-1$	618c1 rank 1
618.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	618e1 rank 1
618.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$	618f1 rank 1
618.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$	618g1 rank 0
618.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	+ + −	$+1$
618.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$
618.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
618.2.a.k	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} + 5q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
618.2.a.a	$x + 36$
618.2.a.b	$x + 24$
618.2.a.c	$x - 2$
618.2.a.d	$x + 130$
618.2.a.e	$x + 94$
618.2.a.f	$x - 24$
618.2.a.g	$x + 30$
618.2.a.h	$x^{2} - 38x - 2$
618.2.a.i	$x^{2} - 80x - 3602$
618.2.a.j	$x^{2} - 8x - 34$
618.2.a.k	$x^{4} - 66x^{3} - 4902x^{2} + 182488x + 7744256$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(618))$	17
Old forms	84
Cusp forms $S_{2}(\Gamma_0(618))$	101
Eisenstein series $E_{2}(\Gamma_0(618))$	7
Modular forms $M_{2}(\Gamma_0(618))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(618)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(103))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(206))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(309))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.471 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, 6, 103, 206, 309, 618]: [0, 1, 1, 3, 9, 28, 36, 108] (certified at ell = 2147482501)
T = [(5, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; h of degree 17 = 1 x 17 (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(618, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 618:2 1000

← 617.2.a · 619.2.a →