Atlas › Modular forms › Weight 2 › Level 563 › 563.2.a

Newspace 563.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(563))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
563.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 4q^{5} + q^{6} - 5q^{7} + O(q^{8})$	−	$+1$	563a1 rank 2
563.2.a.b	3	degree 3	$3q - q^{2} - q^{3} + q^{4} - 6q^{5} + 5q^{6} - q^{7} + O(q^{8})$	+	$-1$
563.2.a.c	3	degree 3	$3q - q^{2} + q^{3} + q^{4} + 2q^{5} - 7q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
563.2.a.d	9	degree 9	$9q - 2q^{2} - 7q^{3} + 2q^{4} - 3q^{5} - q^{6} + O(q^{8})$	+	$-1$
563.2.a.e	31	degree 31	$31q + 5q^{2} + 8q^{3} + 43q^{4} + 13q^{5} - 4q^{6} + 17q^{7} + O(q^{8})$	−	$+1$

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 $T$
563.2.a.a	$x + 4$
563.2.a.b	$x^{3} + 4x^{2} - 60x + 50$
563.2.a.c	$x^{3} - 2x^{2} - 12x + 8$
563.2.a.d	$x^{9} + 23x^{8} + 136x^{7} - 534x^{6} - 8397x^{5} - 26006x^{4} + 4518x^{3} + 132740x^{2} + 151351x - 2503$
563.2.a.e	$x^{31} - 29x^{30} - 232x^{29} + 13940x^{28} - 29071x^{27} - 2854522x^{26} + 17436564x^{25} + 323072810x^{24} - 2947680439x^{23} - 21641390352x^{22} + 277811464813x^{21} + 811522697989x^{20} - 16679883834991x^{19} - 9432300449469x^{18} + 667675215353705x^{17} - 603005215280691x^{16} - 17961687979930377x^{15} + 34631974898035001x^{14} + 318074977860994150x^{13} - 869108179103888194x^{12} - 3502162576795864405x^{11} + 12224028134824904082x^{10} + 21134994752699241653x^{9} - 95471435144729741208x^{8} - 49876001792401340687x^{7} + 373450074946241515600x^{6} - 17215836078819647498x^{5} - 671219715827607257056x^{4} + 179329507139615961430x^{3} + 491100370305546682708x^{2} - 137018612980874512408x - 96271559248689353032$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(563))$	47
Old forms	0
Cusp forms $S_{2}(\Gamma_0(563))$	47
Eisenstein series $E_{2}(\Gamma_0(563))$	1
Modular forms $M_{2}(\Gamma_0(563))$	48

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 563]: [0, 48] (certified at ell = 2147482367)
T = [(2, 1), (3, 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(563, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 563:2 1000

← 562.2.a · 564.2.a →