Atlas › Modular forms › Weight 2 › Level 819 › 819.2.a

Newspace 819.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(819))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
819.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - q^{5} - q^{7} + O(q^{8})$	+ +	819d1 rank 0
819.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{7} + O(q^{8})$	+ +	819b1 rank 1
819.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− −	819e1 rank 0
819.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{7} + O(q^{8})$	+ +	819a1 rank 1
819.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + q^{5} + q^{7} + O(q^{8})$	− −	819f1 rank 0
819.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{5} - q^{7} + O(q^{8})$	+ +	819c1 rank 0
819.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{7} + O(q^{8})$	− +
819.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{5} + 2q^{7} + O(q^{8})$	− +
819.2.a.i	3	degree 3	$3q - q^{2} + 3q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ −
819.2.a.j	3	degree 3	$3q + 2q^{2} + 4q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	+ +
819.2.a.k	4	degree 4	$4q - q^{2} + 7q^{4} + 3q^{5} + 4q^{7} + O(q^{8})$	− −
819.2.a.l	4	degree 4	$4q + 6q^{4} + 4q^{7} + O(q^{8})$	− +
819.2.a.m	6	degree 6	$6q + 12q^{4} - 6q^{7} + O(q^{8})$	+ −

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
819.2.a.a	$x - 96$
819.2.a.b	$x + 5$
819.2.a.c	$x - 80$
819.2.a.d	$x + 35$
819.2.a.e	$x - 62$
819.2.a.f	$x - 28$
819.2.a.g	$x^{2} + 170x + 263$
819.2.a.h	$x^{2} + 64x - 226$
819.2.a.i	$x^{3} + 307x^{2} + 30996x + 1030418$
819.2.a.j	$x^{3} - 16x^{2} - 8095x - 217576$
819.2.a.k	$x^{4} - 53x^{3} - 11155x^{2} - 395619x - 3981122$
819.2.a.l	$x^{4} - 202x^{3} + 14915x^{2} - 473738x + 5395366$
819.2.a.m	$x^{6} - 70x^{5} - 15938x^{4} + 777616x^{3} + 45104371x^{2} - 2991711330x + 41624936002$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(819))$	30
Old forms	75
Cusp forms $S_{2}(\Gamma_0(819))$	105
Eisenstein series $E_{2}(\Gamma_0(819))$	15
Modular forms $M_{2}(\Gamma_0(819))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(819)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(63))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(273))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.699 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, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819]: [0, 1, 1, 2, 1, 4, 6, 10, 10, 16, 40, 116] (certified at ell = 2147483629)
T = [(2, 1), (5, 2), (11, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 9 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(819, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 819:2 1000

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