Atlas › Modular forms › Weight 2 › Level 719 › 719.2.a

Newspace 719.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(719))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
719.2.a.a	5	degree 5	$5q - 2q^{2} + 2q^{3} + 2q^{4} - 5q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
719.2.a.b	10	degree 10	$10q - 6q^{3} + 2q^{4} - 5q^{5} + q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
719.2.a.c	45	degree 45	$45q + 3q^{2} + 4q^{3} + 59q^{4} + 5q^{5} + 4q^{6} + 16q^{7} + O(q^{8})$	−	$+1$

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 $T$
719.2.a.a	$x^{5} - 8x^{3} + 5x^{2} + 2x - 1$
719.2.a.b	$x^{10} + 6x^{9} - 6x^{8} - 73x^{7} - 5x^{6} + 279x^{5} + 14x^{4} - 399x^{3} + 45x^{2} + 152x - 13$
719.2.a.c	$x^{45} - 7x^{44} - 158x^{43} + 1192x^{42} + 10865x^{41} - 90277x^{40} - 427631x^{39} + 4036666x^{38} + 10628598x^{37} - 119364816x^{36} - 172455321x^{35} + 2479191644x^{34} + 1788398460x^{33} - 37515582867x^{32} - 10175224299x^{31} + 423571172312x^{30} - 272606805x^{29} - 3625112151947x^{28} + 535461862599x^{27} + 23750400601017x^{26} - 4859051239860x^{25} - 119653214491497x^{24} + 23644525228574x^{23} + 463122789342508x^{22} - 69270334887326x^{21} - 1368546079216980x^{20} + 111141800742874x^{19} + 3052980129206047x^{18} - 32802858717122x^{17} - 5063091918094853x^{16} - 239786001458841x^{15} + 6126472997505910x^{14} + 511666893415950x^{13} - 5286826476722800x^{12} - 494165077784072x^{11} + 3154626716420761x^{10} + 248289121961980x^{9} - 1239962798789582x^{8} - 59215485672406x^{7} + 294883072902390x^{6} + 5635820653270x^{5} - 36406604023323x^{4} - 923010494805x^{3} + 1893323660667x^{2} + 110886594211x - 11852305957$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(719))$	60
Old forms	0
Cusp forms $S_{2}(\Gamma_0(719))$	60
Eisenstein series $E_{2}(\Gamma_0(719))$	1
Modular forms $M_{2}(\Gamma_0(719))$	61

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 719]: [0, 61] (certified at ell = 2147481997)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 6 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(719, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 719:2 1000

← 718.2.a · 720.2.a →