Atlas › Modular forms › Weight 4 › Level 247 › 247.4.a

Newspace 247.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(247))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
247.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} - 5q^{3} + q^{4} + 7q^{5} + 15q^{6} - 27q^{7} + O(q^{8})$	− +	$-1$
247.4.a.b	10	degree 10	$10q + q^{3} + 8q^{4} - 27q^{5} - 23q^{6} + 36q^{7} + O(q^{8})$	− +	$-1$
247.4.a.c	13	degree 13	$13q - 9q^{2} - 10q^{3} + 65q^{4} - 34q^{5} - 8q^{6} - 27q^{7} + O(q^{8})$	+ −	$-1$
247.4.a.d	14	degree 14	$14q + 5q^{2} + 2q^{3} + 69q^{4} + 40q^{5} + 28q^{6} + 75q^{7} + O(q^{8})$	+ +	$+1$
247.4.a.e	16	degree 16	$16q + 11q^{2} + 20q^{3} + 53q^{4} + 26q^{5} + 28q^{6} - 9q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ 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 $T$
247.4.a.a	$x - 368$
247.4.a.b	$x^{10} + 609x^{9} - 323545x^{8} - 177045823x^{7} + 49245438137x^{6} + 16601738012593x^{5} - 3848700487429473x^{4} - 491870812798123510x^{3} + 110447284194011549472x^{2} + 3400776052650849323384x - 800986363256914897614976$
247.4.a.c	$x^{13} + 911x^{12} - 284122x^{11} - 411432318x^{10} + 229086630x^{9} + 62997138545708x^{8} + 4560156468926060x^{7} - 4029962463763040747x^{6} - 345017849934175879783x^{5} + 96148434209073470625342x^{4} + 7568869550264821480243628x^{3} - 485135386060116188128953232x^{2} - 24819543134599804990824062320x + 349517582900590961362437518400$
247.4.a.d	$x^{14} - 1375x^{13} - 43789x^{12} + 756013830x^{11} - 254463228756x^{10} - 79142103564950x^{9} + 51936769711595704x^{8} - 4542886577165929679x^{7} - 1513344642277320403777x^{6} + 242731638146137632427453x^{5} + 11886134258606401384529574x^{4} - 3033421295568098397907121064x^{3} + 6284998450732412875539409248x^{2} + 9076117581499035347773694929296x - 211673689764339133998361577218464$
247.4.a.e	$x^{16} - 381x^{15} - 1350902x^{14} + 609568057x^{13} + 659371258801x^{12} - 346881738888904x^{11} - 141213161393034812x^{10} + 90819620311054790555x^{9} + 11965943912922076289611x^{8} - 11608003124128288310508962x^{7} + 15880909578969616236605021x^{6} + 687462299400174944413608196317x^{5} - 48007670325942859443861552038806x^{4} - 14663090652520479849746101506280584x^{3} + 1516567140281532248486518319064457920x^{2} - 4245152422339531144367575939655071744x - 19000049925303081321666855546697805824$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(247))$	54
Old forms	14
Cusp forms $S_{4}(\Gamma_0(247))$	68
Eisenstein series $E_{4}(\Gamma_0(247))$	4
Modular forms $M_{4}(\Gamma_0(247))$	72

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(247)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.254 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, 13, 19, 247]: [1, 5, 6, 72] (certified at ell = 2147483629)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 54 = 1 x 54 (phi(ord eps) x dim S_k^new); 18 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(247, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 247:4 1000

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