Atlas › Modular forms › Weight 2 › Level 781 › 781.2.a

Newspace 781.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(781))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
781.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+ −	$+1$	781a1 rank 0
781.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} + 3q^{7} + O(q^{8})$	− −	$-1$	781b1 rank 1
781.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} - q^{3} + 3q^{4} - 3q^{5} - 7q^{6} + q^{7} + O(q^{8})$	− −	$-1$
781.2.a.d	6	degree 6	$6q - 2q^{2} - 6q^{3} + 2q^{4} - 5q^{5} + 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
781.2.a.e	12	degree 12	$12q + 3q^{2} + 5q^{3} + 13q^{4} + 2q^{5} + 8q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
781.2.a.f	15	degree 15	$15q - 4q^{2} - 7q^{3} + 10q^{4} - 7q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
781.2.a.g	20	degree 20	$20q + 3q^{2} + 9q^{3} + 25q^{4} + 9q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{5} + 7T_{19} + 8T_{23}$ 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 $T$
781.2.a.a	$x - 78$
781.2.a.b	$x + 54$
781.2.a.c	$x^{2} + 165x + 5373$
781.2.a.d	$x^{6} - 7x^{5} - 4975x^{4} - 27764x^{3} + 623055x^{2} - 601100x - 933057$
781.2.a.e	$x^{12} - 206x^{11} - 1337x^{10} + 2514160x^{9} - 101870406x^{8} - 6291915075x^{7} + 247788360860x^{6} + 10030526810919x^{5} - 124533672550345x^{4} - 7969071095686252x^{3} - 101040196915424540x^{2} - 471509813007124448x - 705956546338392112$
781.2.a.f	$x^{15} + 467x^{14} + 86858x^{13} + 7730366x^{12} + 250623026x^{11} - 10561791359x^{10} - 1095895827115x^{9} - 18762402919569x^{8} + 985523732436690x^{7} + 39114892962908147x^{6} - 119872182328528979x^{5} - 22846311448414289767x^{4} - 172085209976339816458x^{3} + 4549924156066327476604x^{2} + 51743084785940226190703x - 77082284889909335004602$
781.2.a.g	$x^{20} - 286x^{19} - 841x^{18} + 8268674x^{17} - 858344114x^{16} - 21535270027x^{15} + 7932331480382x^{14} - 276012214986854x^{13} - 22433541322644245x^{12} + 1653624868075358377x^{11} + 10271743984685063612x^{10} - 3477118764652482941162x^{9} + 52126473809360611813307x^{8} + 3266970533935251974735960x^{7} - 92723778363307267628068485x^{6} - 1181770569739347831832309813x^{5} + 59098309356910258790112115442x^{4} - 50837096490116755330623684696x^{3} - 14061647363371277044144983106464x^{2} + 79014950725796244201690710991792x + 639296340536105681830117724166048$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(781))$	57
Old forms	14
Cusp forms $S_{2}(\Gamma_0(781))$	71
Eisenstein series $E_{2}(\Gamma_0(781))$	3
Modular forms $M_{2}(\Gamma_0(781))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(781)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.407 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, 11, 71, 781]: [0, 2, 7, 74] (certified at ell = 2147483171)
T = [(2, 1), (3, 2), (7, 3), (13, 4), (17, 5), (5, 6), (19, 7), (23, 8)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 16 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(781, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 781:2 1000

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