Atlas › Modular forms › Weight 2 › Level 551 › 551.2.a

Newspace 551.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(551))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
551.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 4q^{6} - q^{7} + O(q^{8})$	− −	$-1$	551d1 rank 1
551.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$	551b1 rank 1
551.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	551a1 rank 1
551.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 2q^{3} + 2q^{4} - q^{5} - 4q^{6} - q^{7} + O(q^{8})$	− −	$-1$	551c1 rank 1
551.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +	$-1$
551.2.a.f	3	degree 3	$3q + 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
551.2.a.g	16	degree 16	$16q + 3q^{2} + 4q^{3} + 21q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	− +	$+1$
551.2.a.h	18	degree 18	$18q + 2q^{2} - 2q^{3} + 26q^{4} + 7q^{5} + 12q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
551.2.a.a	$x + 8$
551.2.a.b	$x - 2$
551.2.a.c	$x - 4$
551.2.a.d	$x + 4$
551.2.a.e	$x^{2} + 2x - 44$
551.2.a.f	$x^{3} - 4x + 2$
551.2.a.g	$x^{16} - 15x^{15} - 238x^{14} + 4307x^{13} + 17710x^{12} - 466979x^{11} - 345637x^{10} + 25536507x^{9} - 16692344x^{8} - 762940298x^{7} + 976014183x^{6} + 12268004321x^{5} - 18939270821x^{4} - 95472218438x^{3} + 158205848978x^{2} + 276459290012x - 483739053720$
551.2.a.h	$x^{18} + 4x^{17} - 455x^{16} - 1651x^{15} + 82815x^{14} + 264175x^{13} - 7822476x^{12} - 20991598x^{11} + 416972019x^{10} + 886440154x^{9} - 12730474715x^{8} - 20153860580x^{7} + 212629826642x^{6} + 251190935871x^{5} - 1710560127212x^{4} - 1901247698194x^{3} + 5371040369594x^{2} + 6113521093148x - 1982484758400$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(551))$	43
Old forms	6
Cusp forms $S_{2}(\Gamma_0(551))$	49
Eisenstein series $E_{2}(\Gamma_0(551))$	3
Modular forms $M_{2}(\Gamma_0(551))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(551)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.312 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, 19, 29, 551]: [0, 2, 3, 52] (certified at ell = 2147483269)
T = [(2, 1), (3, 3)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 6 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(551, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 551:2 1000

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