Atlas › Modular forms › Weight 2 › Level 619 › 619.2.a

Newspace 619.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(619))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
619.2.a.a	21	degree 21	$21q - 9q^{2} - 5q^{3} + 15q^{4} - 21q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
619.2.a.b	30	degree 30	$30q + 9q^{2} + q^{3} + 33q^{4} + 21q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
619.2.a.a	$x^{21} + 14x^{20} + 41x^{19} - 251x^{18} - 1477x^{17} + 654x^{16} + 15951x^{15} + 11966x^{14} - 82845x^{13} - 105277x^{12} + 231127x^{11} + 366363x^{10} - 346120x^{9} - 642344x^{8} + 252294x^{7} + 568419x^{6} - 67086x^{5} - 229352x^{4} + 1596x^{3} + 35853x^{2} + 558x - 1089$
619.2.a.b	$x^{30} - 10x^{29} - 69x^{28} + 969x^{27} + 1100x^{26} - 38638x^{25} + 29988x^{24} + 837065x^{23} - 1477892x^{22} - 10941129x^{21} + 26634966x^{20} + 90458051x^{19} - 272129747x^{18} - 480732003x^{17} + 1745601319x^{16} + 1622836338x^{15} - 7271970336x^{14} - 3298093858x^{13} + 19720599786x^{12} + 3414171340x^{11} - 34108263246x^{10} - 419730055x^{9} + 36108603168x^{8} - 2501947163x^{7} - 21997160380x^{6} + 2136206797x^{5} + 6905383230x^{4} - 624395688x^{3} - 839637808x^{2} + 55995696x + 1117888$

Orbit

Characteristic polynomial of

T

619.2.a.a

x^{21} + 14x^{20} + 41x^{19} - 251x^{18} - 1477x^{17} + 654x^{16} + 15951x^{15} + 11966x^{14} - 82845x^{13} - 105277x^{12} + 231127x^{11} + 366363x^{10} - 346120x^{9} - 642344x^{8} + 252294x^{7} + 568419x^{6} - 67086x^{5} - 229352x^{4} + 1596x^{3} + 35853x^{2} + 558x - 1089

619.2.a.b

x^{30} - 10x^{29} - 69x^{28} + 969x^{27} + 1100x^{26} - 38638x^{25} + 29988x^{24} + 837065x^{23} - 1477892x^{22} - 10941129x^{21} + 26634966x^{20} + 90458051x^{19} - 272129747x^{18} - 480732003x^{17} + 1745601319x^{16} + 1622836338x^{15} - 7271970336x^{14} - 3298093858x^{13} + 19720599786x^{12} + 3414171340x^{11} - 34108263246x^{10} - 419730055x^{9} + 36108603168x^{8} - 2501947163x^{7} - 21997160380x^{6} + 2136206797x^{5} + 6905383230x^{4} - 624395688x^{3} - 839637808x^{2} + 55995696x + 1117888

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(619))$	51
Old forms	0
Cusp forms $S_{2}(\Gamma_0(619))$	51
Eisenstein series $E_{2}(\Gamma_0(619))$	1
Modular forms $M_{2}(\Gamma_0(619))$	52

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 619]: [0, 52] (certified at ell = 2147482021)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(619, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 619:2 1000

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