Atlas › Modular forms › Weight 2 › Level 969 › 969.2.a

Newspace 969.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(969))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
969.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$	969a1 rank 0
969.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
969.2.a.c	4	degree 4	$4q - 3q^{2} + 4q^{3} + 3q^{4} - 6q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$
969.2.a.d	4	degree 4	$4q - q^{2} - 4q^{3} - q^{4} + 4q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$
969.2.a.e	4	degree 4	$4q + q^{2} + 4q^{3} + 11q^{4} + 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$
969.2.a.f	5	degree 5	$5q - 2q^{2} + 5q^{3} + 6q^{4} - 8q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$
969.2.a.g	5	degree 5	$5q + 2q^{2} - 5q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	+ + +	$-1$
969.2.a.h	7	degree 7	$7q - 7q^{3} + 8q^{4} - 6q^{5} + 9q^{7} + O(q^{8})$	+ + −	$+1$
969.2.a.i	7	degree 7	$7q + 4q^{2} + 7q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
969.2.a.j	8	degree 8	$8q + q^{2} - 8q^{3} + 11q^{4} - 4q^{5} - q^{6} + 9q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
969.2.a.a	$x - 7$
969.2.a.b	$x^{2} - 8x + 14$
969.2.a.c	$x^{4} + 21x^{3} + 98x^{2} - 26x - 25$
969.2.a.d	$x^{4} - 11x^{3} - 24x^{2} + 14x + 1$
969.2.a.e	$x^{4} - 13x^{3} - 90x^{2} + 698x + 3487$
969.2.a.f	$x^{5} + 26x^{4} + 189x^{3} + 26x^{2} - 3253x - 6989$
969.2.a.g	$x^{5} - 8x^{4} - 55x^{3} + 288x^{2} + 1213x + 107$
969.2.a.h	$x^{7} + 18x^{6} - 41x^{5} - 1794x^{4} - 1855x^{3} + 47807x^{2} + 63102x - 310742$
969.2.a.i	$x^{7} - 16x^{6} - 39x^{5} + 1574x^{4} - 4435x^{3} - 22177x^{2} + 56870x + 145050$
969.2.a.j	$x^{8} + 11x^{7} - 241x^{6} - 2455x^{5} + 18615x^{4} + 175758x^{3} - 383943x^{2} - 4142656x - 4330966$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(969))$	47
Old forms	70
Cusp forms $S_{2}(\Gamma_0(969))$	117
Eisenstein series $E_{2}(\Gamma_0(969))$	7
Modular forms $M_{2}(\Gamma_0(969))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(969)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(51))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(323))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.341 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, 19, 51, 57, 323, 969]: [0, 1, 2, 2, 8, 8, 32, 124] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 7 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(969, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 969:2 1000

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