Atlas › Modular forms › Weight 2 › Level 550 › 550.2.a

Newspace 550.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(550))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
550.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	550d1 rank 0
550.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	+ −	550c1 rank 0
550.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	+ −	550g1 rank 1
550.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} + O(q^{8})$	+ +	550a1 rank 1
550.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 3q^{7} + O(q^{8})$	+ −	550f1 rank 1
550.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 3q^{7} + O(q^{8})$	+ −	550b1 rank 0
550.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} + q^{7} + O(q^{8})$	+ +	550e1 rank 0
550.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - 3q^{6} - q^{7} + O(q^{8})$	− +	550j1 rank 1
550.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - 5q^{7} + O(q^{8})$	− −	550i1 rank 1
550.2.a.j	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + 3q^{7} + O(q^{8})$	− −	550k1 rank 0
550.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	− −	550l1 rank 0
550.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	− −	550m1 rank 0
550.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	550h1 rank 0
550.2.a.n	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
550.2.a.a	$x - 188$
550.2.a.b	$x - 332$
550.2.a.c	$x + 238$
550.2.a.d	$x + 322$
550.2.a.e	$x + 34$
550.2.a.f	$x - 86$
550.2.a.g	$x + 162$
550.2.a.h	$x + 106$
550.2.a.i	$x + 142$
550.2.a.j	$x - 6$
550.2.a.k	$x + 30$
550.2.a.l	$x + 160$
550.2.a.m	$x - 104$
550.2.a.n	$x^{2} - 100x - 50300$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(550))$	15
Old forms	64
Cusp forms $S_{2}(\Gamma_0(550))$	79
Eisenstein series $E_{2}(\Gamma_0(550))$	23
Modular forms $M_{2}(\Gamma_0(550))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(550)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(55))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(110))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(275))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.796 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, 11, 22, 25, 50, 55, 110, 275, 550]: [0, 1, 1, 3, 2, 5, 3, 9, 8, 22, 32, 94] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 15 = 1 x 15 (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(550, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 550:2 1000

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