Atlas › Modular forms › Weight 2 › Level 569 › 569.2.a

Newspace 569.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(569))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
569.2.a.a	16	degree 16	$16q - 3q^{2} - 8q^{3} + 3q^{4} - 6q^{5} - 5q^{6} - 16q^{7} + O(q^{8})$	+	$-1$
569.2.a.b	31	degree 31	$31q + 8q^{3} + 44q^{4} + 4q^{5} + 5q^{6} + 20q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
569.2.a.a	$x^{16} + 11x^{15} + 23x^{14} - 126x^{13} - 477x^{12} + 349x^{11} + 2681x^{10} + 555x^{9} - 5924x^{8} - 3667x^{7} + 5121x^{6} + 5026x^{5} - 798x^{4} - 2141x^{3} - 618x^{2} + 32x + 23$
569.2.a.b	$x^{31} - 8x^{30} - 91x^{29} + 859x^{28} + 3042x^{27} - 38285x^{26} - 41280x^{25} + 947677x^{24} - 76041x^{23} - 14575046x^{22} + 10619419x^{21} + 146936075x^{20} - 165342105x^{19} - 996616116x^{18} + 1373964196x^{17} + 4594601782x^{16} - 7085342868x^{15} - 14386678131x^{14} + 23657773884x^{13} + 30297928436x^{12} - 51499857359x^{11} - 42116783966x^{10} + 71978930511x^{9} + 37436276086x^{8} - 62043100066x^{7} - 20039488429x^{6} + 30310941768x^{5} + 5632082696x^{4} - 6933119138x^{3} - 536973731x^{2} + 406887583x - 5327417$

Orbit

Characteristic polynomial of

T

569.2.a.a

x^{16} + 11x^{15} + 23x^{14} - 126x^{13} - 477x^{12} + 349x^{11} + 2681x^{10} + 555x^{9} - 5924x^{8} - 3667x^{7} + 5121x^{6} + 5026x^{5} - 798x^{4} - 2141x^{3} - 618x^{2} + 32x + 23

569.2.a.b

x^{31} - 8x^{30} - 91x^{29} + 859x^{28} + 3042x^{27} - 38285x^{26} - 41280x^{25} + 947677x^{24} - 76041x^{23} - 14575046x^{22} + 10619419x^{21} + 146936075x^{20} - 165342105x^{19} - 996616116x^{18} + 1373964196x^{17} + 4594601782x^{16} - 7085342868x^{15} - 14386678131x^{14} + 23657773884x^{13} + 30297928436x^{12} - 51499857359x^{11} - 42116783966x^{10} + 71978930511x^{9} + 37436276086x^{8} - 62043100066x^{7} - 20039488429x^{6} + 30310941768x^{5} + 5632082696x^{4} - 6933119138x^{3} - 536973731x^{2} + 406887583x - 5327417

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(569))$	47
Old forms	0
Cusp forms $S_{2}(\Gamma_0(569))$	47
Eisenstein series $E_{2}(\Gamma_0(569))$	1
Modular forms $M_{2}(\Gamma_0(569))$	48

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.207 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, 569]: [0, 48] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(569, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 569:2 1000

← 568.2.a · 570.2.a →