Atlas › Modular forms › Weight 2 › Level 743 › 743.2.a

Newspace 743.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(743))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
743.2.a.a	21	degree 21	$21q - 4q^{2} - 9q^{3} + 12q^{4} - 10q^{5} - 7q^{6} - 18q^{7} + O(q^{8})$	+	$-1$
743.2.a.b	41	degree 41	$41q + 3q^{2} + 11q^{3} + 47q^{4} + 12q^{5} + 5q^{6} + 22q^{7} + O(q^{8})$	−	$+1$

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 $T$
743.2.a.a	$x^{21} + 13x^{20} + 29x^{19} - 269x^{18} - 1196x^{17} + 1678x^{16} + 14297x^{15} + 738x^{14} - 83873x^{13} - 51957x^{12} + 271112x^{11} + 236280x^{10} - 495394x^{9} - 463919x^{8} + 509773x^{7} + 421036x^{6} - 298445x^{5} - 159852x^{4} + 100402x^{3} + 17053x^{2} - 14743x + 1727$
743.2.a.b	$x^{41} - 14x^{40} - 62x^{39} + 1680x^{38} - 1174x^{37} - 86844x^{36} + 230944x^{35} + 2545175x^{34} - 10178588x^{33} - 46493528x^{32} + 250346361x^{31} + 539909484x^{30} - 4021791553x^{29} - 3707722816x^{28} + 44930048043x^{27} + 8579288415x^{26} - 359769189093x^{25} + 98924943648x^{24} + 2094162757011x^{23} - 1214086681000x^{22} - 8898514245448x^{21} + 6877993014323x^{20} + 27530982862938x^{19} - 24149143133837x^{18} - 61579147928303x^{17} + 55365553207990x^{16} + 98610032779687x^{15} - 82444137487698x^{14} - 111661269368849x^{13} + 76568827025919x^{12} + 87476628213712x^{11} - 40431278158507x^{10} - 45088099660554x^{9} + 9352318132162x^{8} + 13642610937882x^{7} + 340754120220x^{6} - 1851961089579x^{5} - 369249587015x^{4} + 34284952976x^{3} + 11987932992x^{2} - 58099941x - 105718473$

Orbit

Characteristic polynomial of

T

743.2.a.a

x^{21} + 13x^{20} + 29x^{19} - 269x^{18} - 1196x^{17} + 1678x^{16} + 14297x^{15} + 738x^{14} - 83873x^{13} - 51957x^{12} + 271112x^{11} + 236280x^{10} - 495394x^{9} - 463919x^{8} + 509773x^{7} + 421036x^{6} - 298445x^{5} - 159852x^{4} + 100402x^{3} + 17053x^{2} - 14743x + 1727

743.2.a.b

x^{41} - 14x^{40} - 62x^{39} + 1680x^{38} - 1174x^{37} - 86844x^{36} + 230944x^{35} + 2545175x^{34} - 10178588x^{33} - 46493528x^{32} + 250346361x^{31} + 539909484x^{30} - 4021791553x^{29} - 3707722816x^{28} + 44930048043x^{27} + 8579288415x^{26} - 359769189093x^{25} + 98924943648x^{24} + 2094162757011x^{23} - 1214086681000x^{22} - 8898514245448x^{21} + 6877993014323x^{20} + 27530982862938x^{19} - 24149143133837x^{18} - 61579147928303x^{17} + 55365553207990x^{16} + 98610032779687x^{15} - 82444137487698x^{14} - 111661269368849x^{13} + 76568827025919x^{12} + 87476628213712x^{11} - 40431278158507x^{10} - 45088099660554x^{9} + 9352318132162x^{8} + 13642610937882x^{7} + 340754120220x^{6} - 1851961089579x^{5} - 369249587015x^{4} + 34284952976x^{3} + 11987932992x^{2} - 58099941x - 105718473

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(743))$	62
Old forms	0
Cusp forms $S_{2}(\Gamma_0(743))$	62
Eisenstein series $E_{2}(\Gamma_0(743))$	1
Modular forms $M_{2}(\Gamma_0(743))$	63

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 743]: [0, 63] (certified at ell = 2147477851)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (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(743, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 743:2 1000

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