Atlas › Modular forms › Weight 2 › Level 521 › 521.2.a

Newspace 521.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(521))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
521.2.a.a	14	degree 14	$14q - 2q^{2} - 5q^{3} + 2q^{4} - 3q^{5} - 7q^{6} - 20q^{7} + O(q^{8})$	+	$-1$
521.2.a.b	29	degree 29	$29q + q^{2} + 3q^{3} + 43q^{4} - q^{5} + 7q^{6} + 26q^{7} + O(q^{8})$	−	$+1$

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 $T$
521.2.a.a	$x^{14} + 7x^{13} - 5x^{12} - 112x^{11} - 78x^{10} + 644x^{9} + 700x^{8} - 1646x^{7} - 1979x^{6} + 1919x^{5} + 2177x^{4} - 1000x^{3} - 744x^{2} + 250x - 9$
521.2.a.b	$x^{29} - 4x^{28} - 111x^{27} + 461x^{26} + 5095x^{25} - 21773x^{24} - 128913x^{23} + 561875x^{22} + 2021212x^{21} - 8867281x^{20} - 20951966x^{19} + 90454484x^{18} + 149612772x^{17} - 612623050x^{16} - 754460763x^{15} + 2775715805x^{14} + 2708607364x^{13} - 8341358989x^{12} - 6844085520x^{11} + 16199747646x^{10} + 11784432836x^{9} - 19309001614x^{8} - 13046598374x^{7} + 12762977266x^{6} + 8373324855x^{5} - 3697702544x^{4} - 2500068931x^{3} + 155360803x^{2} + 150580769x + 582871$

Orbit

Characteristic polynomial of

T

521.2.a.a

x^{14} + 7x^{13} - 5x^{12} - 112x^{11} - 78x^{10} + 644x^{9} + 700x^{8} - 1646x^{7} - 1979x^{6} + 1919x^{5} + 2177x^{4} - 1000x^{3} - 744x^{2} + 250x - 9

521.2.a.b

x^{29} - 4x^{28} - 111x^{27} + 461x^{26} + 5095x^{25} - 21773x^{24} - 128913x^{23} + 561875x^{22} + 2021212x^{21} - 8867281x^{20} - 20951966x^{19} + 90454484x^{18} + 149612772x^{17} - 612623050x^{16} - 754460763x^{15} + 2775715805x^{14} + 2708607364x^{13} - 8341358989x^{12} - 6844085520x^{11} + 16199747646x^{10} + 11784432836x^{9} - 19309001614x^{8} - 13046598374x^{7} + 12762977266x^{6} + 8373324855x^{5} - 3697702544x^{4} - 2500068931x^{3} + 155360803x^{2} + 150580769x + 582871

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(521))$	43
Old forms	0
Cusp forms $S_{2}(\Gamma_0(521))$	43
Eisenstein series $E_{2}(\Gamma_0(521))$	1
Modular forms $M_{2}(\Gamma_0(521))$	44

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.2 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, 521]: [0, 44] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 5 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(521, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 521:2 1000

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