Atlas › Modular forms › Weight 2 › Level 607 › 607.2.a

Newspace 607.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(607))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
607.2.a.a	5	degree 5	$5q - 3q^{2} + q^{4} - 2q^{5} + 5q^{6} + 3q^{7} + O(q^{8})$	+	$-1$
607.2.a.b	7	degree 7	$7q - 4q^{2} - 5q^{3} + 8q^{4} + q^{5} - 5q^{7} + O(q^{8})$	+	$-1$
607.2.a.c	7	degree 7	$7q - q^{2} - 2q^{3} + 7q^{4} - 13q^{5} - 9q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
607.2.a.d	31	degree 31	$31q + 9q^{2} + 7q^{3} + 35q^{4} + 16q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	−	$+1$

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 $T$
607.2.a.a	$x^{5} + 3x^{4} - 11x^{3} - 33x^{2} + 28x + 85$
607.2.a.b	$x^{7} + 9x^{6} + 17x^{5} - 24x^{4} - 41x^{3} + 53x^{2} - 13x - 1$
607.2.a.c	$x^{7} + 3x^{6} - 7x^{5} - 10x^{4} + 7x^{3} + 7x^{2} - x - 1$
607.2.a.d	$x^{31} - 16x^{30} + 10x^{29} + 1098x^{28} - 4650x^{27} - 26695x^{26} + 193656x^{25} + 217924x^{24} - 3776843x^{23} + 1882247x^{22} + 42407349x^{21} - 57905357x^{20} - 296394143x^{19} + 581595415x^{18} + 1334506490x^{17} - 3285297437x^{16} - 3931184788x^{15} + 11525531694x^{14} + 7673061049x^{13} - 25648581939x^{12} - 10184042893x^{11} + 35494149042x^{10} + 9557862606x^{9} - 28675844912x^{8} - 6112499392x^{7} + 11986257558x^{6} + 1967440155x^{5} - 2277034921x^{4} - 154343408x^{3} + 158954469x^{2} - 15427601x + 393127$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(607))$	50
Old forms	0
Cusp forms $S_{2}(\Gamma_0(607))$	50
Eisenstein series $E_{2}(\Gamma_0(607))$	1
Modular forms $M_{2}(\Gamma_0(607))$	51

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.21 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, 607]: [0, 51] (certified at ell = 2147480989)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (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(607, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 607:2 1000

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