Atlas › Modular forms › Weight 4 › Level 217 › 217.4.a

Newspace 217.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(217))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
217.4.a.a	1	$\mathbb{Q}$	$q - q^{2} - 8q^{3} - 7q^{4} + 4q^{5} + 8q^{6} + 7q^{7} + O(q^{8})$	− +	$-1$
217.4.a.b	10	degree 10	$10q - 5q^{2} - 12q^{3} + 37q^{4} - 38q^{5} - 48q^{6} + 70q^{7} + O(q^{8})$	− +	$-1$
217.4.a.c	10	degree 10	$10q - 5q^{2} - 6q^{3} + 37q^{4} - 14q^{5} - 4q^{6} - 70q^{7} + O(q^{8})$	+ −	$-1$
217.4.a.d	12	degree 12	$12q + 7q^{2} + 12q^{3} + 61q^{4} - 14q^{5} - 16q^{6} - 84q^{7} + O(q^{8})$	+ +	$+1$
217.4.a.e	13	degree 13	$13q + 6q^{2} + 10q^{3} + 54q^{4} + 46q^{5} + 20q^{6} + 91q^{7} + O(q^{8})$	− −	$+1$

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 $T$
217.4.a.a	$x + 9$
217.4.a.b	$x^{10} + 17x^{9} - 42x^{8} - 1762x^{7} - 2668x^{6} + 52354x^{5} + 76718x^{4} - 719145x^{3} - 197825x^{2} + 4013190x - 3951342$
217.4.a.c	$x^{10} + 11x^{9} - 160x^{8} - 1250x^{7} + 10078x^{6} + 35662x^{5} - 181704x^{4} - 548381x^{3} + 852035x^{2} + 3501666x + 2491058$
217.4.a.d	$x^{12} - 19x^{11} - 164x^{10} + 3874x^{9} + 9934x^{8} - 279902x^{7} - 357780x^{6} + 8524961x^{5} + 8689299x^{4} - 97236644x^{3} - 57215508x^{2} + 369366624x + 26332800$
217.4.a.e	$x^{13} - 16x^{12} - 207x^{11} + 4168x^{10} + 5878x^{9} - 300482x^{8} + 217240x^{7} + 9325847x^{6} - 12143188x^{5} - 133394361x^{4} + 190762892x^{3} + 765926436x^{2} - 1022569344x - 732729088$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(217))$	46
Old forms	16
Cusp forms $S_{4}(\Gamma_0(217))$	62
Eisenstein series $E_{4}(\Gamma_0(217))$	4
Modular forms $M_{4}(\Gamma_0(217))$	66

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(217)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.144 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, 7, 31, 217]: [1, 3, 9, 66] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 46 = 1 x 46 (phi(ord eps) x dim S_k^new); 7 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(217, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 217:4 1000

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