Atlas › Modular forms › Weight 2 › Level 970 › 970.2.a

Newspace 970.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(970))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
970.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + O(q^{8})$	+ + −	$+1$	970a1 rank 0
970.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− − −	$+1$	970b1 rank 0
970.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} + 2q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
970.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} + 2q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
970.2.a.e	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
970.2.a.f	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + O(q^{8})$	+ − +	$+1$
970.2.a.g	4	degree 4	$4q - 4q^{2} + 4q^{4} - 4q^{5} + 2q^{7} + O(q^{8})$	+ + +	$-1$
970.2.a.h	5	degree 5	$5q - 5q^{2} - 6q^{3} + 5q^{4} + 5q^{5} + 6q^{6} - 6q^{7} + O(q^{8})$	+ − −	$-1$
970.2.a.i	5	degree 5	$5q + 5q^{2} + 5q^{4} - 5q^{5} + 2q^{7} + O(q^{8})$	− + +	$+1$
970.2.a.j	5	degree 5	$5q + 5q^{2} + 4q^{3} + 5q^{4} + 5q^{5} + 4q^{6} + 8q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
970.2.a.a	$x + 2$
970.2.a.b	$x + 6$
970.2.a.c	$x^{2} + 7x + 11$
970.2.a.d	$x^{2} + 3x + 1$
970.2.a.e	$x^{3} + x^{2} - 31x - 2$
970.2.a.f	$x^{3} - 3x^{2} - x + 2$
970.2.a.g	$x^{4} - 2x^{3} - 5x^{2} + 12x - 5$
970.2.a.h	$x^{5} + 12x^{4} + 25x^{3} - 156x^{2} - 715x - 794$
970.2.a.i	$x^{5} - 2x^{4} - 33x^{3} + 90x^{2} + 161x - 482$
970.2.a.j	$x^{5} - 12x^{4} + 45x^{3} - 58x^{2} + 27x - 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(970))$	31
Old forms	112
Cusp forms $S_{2}(\Gamma_0(970))$	143
Eisenstein series $E_{2}(\Gamma_0(970))$	7
Modular forms $M_{2}(\Gamma_0(970))$	150

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(970)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(194))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(485))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.347 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, 5, 10, 97, 194, 485, 970]: [0, 1, 1, 3, 8, 26, 50, 150] (certified at ell = 2147483137)
T = [(3, 1), (7, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 4 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(970, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 970:2 1000

← 969.2.a · 971.2.a →