Atlas › Modular forms › Weight 2 › Level 570 › 570.2.a

Newspace 570.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(570))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
570.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + + −	$+1$	570b1 rank 0
570.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + + +	$-1$	570a1 rank 1
570.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + − −	$-1$	570c1 rank 1
570.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − + +	$+1$	570d1 rank 0
570.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ − − +	$-1$	570e1 rank 1
570.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$	570f1 rank 0
570.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	− + + +	$+1$	570g1 rank 0
570.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + − −	$+1$	570h1 rank 0
570.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	− + − −	$+1$	570i1 rank 0
570.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − + −	$+1$	570j1 rank 0
570.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − + −	$+1$	570k1 rank 0
570.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − − +	$+1$	570l1 rank 0
570.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} + O(q^{8})$	− − − +	$+1$	570m1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{23} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ 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 $T$
570.2.a.a	$x - 178$
570.2.a.b	$x + 274$
570.2.a.c	$x + 362$
570.2.a.d	$x + 16$
570.2.a.e	$x - 128$
570.2.a.f	$x - 154$
570.2.a.g	$x - 48$
570.2.a.h	$x - 194$
570.2.a.i	$x - 20$
570.2.a.j	$x + 142$
570.2.a.k	$x + 62$
570.2.a.l	$x + 302$
570.2.a.m	$x - 100$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(570))$	13
Old forms	100
Cusp forms $S_{2}(\Gamma_0(570))$	113
Eisenstein series $E_{2}(\Gamma_0(570))$	15
Modular forms $M_{2}(\Gamma_0(570))$	128

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.147 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, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570]: [0, 1, 1, 1, 3, 3, 4, 2, 10, 7, 8, 12, 24, 34, 44, 128] (certified at ell = 2147483629)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (23, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 13 = 1 x 13 (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(570, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 570:2 1000

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