Atlas › Modular forms › Weight 2 › Level 419 › 419.2.a

Newspace 419.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(419))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
419.2.a.a	9	degree 9	$9q - 2q^{2} - 4q^{3} - 5q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	+	$-1$
419.2.a.b	26	degree 26	$26q + 2q^{2} + 2q^{3} + 38q^{4} + 5q^{5} + 5q^{6} + 7q^{7} + O(q^{8})$	−	$+1$

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 $T$
419.2.a.a	$x^{9} + 6x^{8} - 43x^{6} - 36x^{5} + 37x^{4} + 50x^{3} + 12x^{2} - 3x - 1$
419.2.a.b	$x^{26} - 4x^{25} - 100x^{24} + 409x^{23} + 4086x^{22} - 16885x^{21} - 91646x^{20} + 378396x^{19} + 1267391x^{18} - 5148302x^{17} - 11461636x^{16} + 44663935x^{15} + 70031581x^{14} - 250890860x^{13} - 293751703x^{12} + 902018479x^{11} + 843646917x^{10} - 1986056002x^{9} - 1601257468x^{8} + 2417405024x^{7} + 1815612916x^{6} - 1249649476x^{5} - 949746305x^{4} + 48045096x^{3} + 75306711x^{2} - 1042780x - 1364582$

Orbit

Characteristic polynomial of

T

419.2.a.a

x^{9} + 6x^{8} - 43x^{6} - 36x^{5} + 37x^{4} + 50x^{3} + 12x^{2} - 3x - 1

419.2.a.b

x^{26} - 4x^{25} - 100x^{24} + 409x^{23} + 4086x^{22} - 16885x^{21} - 91646x^{20} + 378396x^{19} + 1267391x^{18} - 5148302x^{17} - 11461636x^{16} + 44663935x^{15} + 70031581x^{14} - 250890860x^{13} - 293751703x^{12} + 902018479x^{11} + 843646917x^{10} - 1986056002x^{9} - 1601257468x^{8} + 2417405024x^{7} + 1815612916x^{6} - 1249649476x^{5} - 949746305x^{4} + 48045096x^{3} + 75306711x^{2} - 1042780x - 1364582

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(419))$	35
Old forms	0
Cusp forms $S_{2}(\Gamma_0(419))$	35
Eisenstein series $E_{2}(\Gamma_0(419))$	1
Modular forms $M_{2}(\Gamma_0(419))$	36

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 419]: [0, 36] (certified at ell = 2147482943)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 4 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(419, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 419:2 1000

← 418.2.a · 420.2.a →