Atlas › Modular forms › Weight 4 › Level 239 › 239.4.a

Newspace 239.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(239))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
239.4.a.a	22	degree 22	$22q - 4q^{2} - 13q^{3} + 50q^{4} - 37q^{5} - 42q^{6} - 52q^{7} + O(q^{8})$	−	$-1$
239.4.a.b	37	degree 37	$37q + 4q^{2} + 11q^{3} + 170q^{4} + 43q^{5} + 54q^{6} + 60q^{7} + O(q^{8})$	+	$+1$

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 $T$
239.4.a.a	$x^{22} + 17x^{21} - 258x^{20} - 5009x^{19} + 28888x^{18} + 617005x^{17} - 2034190x^{16} - 41390734x^{15} + 107841588x^{14} + 1632029584x^{13} - 4275106893x^{12} - 37744643110x^{11} + 112010842410x^{10} + 478614868545x^{9} - 1712417124428x^{8} - 2738309177111x^{7} + 13487828332902x^{6} + 2049227391934x^{5} - 45589459911950x^{4} + 25835353515257x^{3} + 46139351873401x^{2} - 42755616102699x + 4702888740782$
239.4.a.b	$x^{37} - 15x^{36} - 911x^{35} + 13954x^{34} + 361252x^{33} - 5623618x^{32} - 83598787x^{31} + 1308227667x^{30} + 12789770254x^{29} - 197188739887x^{28} - 1388738050889x^{27} + 20435358594063x^{26} + 111901778676198x^{25} - 1503403749091393x^{24} - 6868838597256262x^{23} + 79705297346477658x^{22} + 324328692142421862x^{21} - 3051664537457412835x^{20} - 11714371133621109479x^{19} + 83544342792013661520x^{18} + 317622452426152120757x^{17} - 1596902163126461165915x^{16} - 6272974866942373109781x^{15} + 20393864442433711774337x^{14} + 86528086681031236620165x^{13} - 160383905934810266592252x^{12} - 785785384241772868166524x^{11} + 650314448473378554110644x^{10} + 4305959333510109484347035x^{9} - 682066587236809925042415x^{8} - 12589646313186080405360039x^{7} - 1668160561031770001005467x^{6} + 18357050744681692346284812x^{5} + 2392852785754127948057032x^{4} - 12788079774779435332595217x^{3} + 280844051098231129530930x^{2} + 3125349476742560883195715x - 500496391304187684881350$

Orbit

Characteristic polynomial of

T

239.4.a.a

x^{22} + 17x^{21} - 258x^{20} - 5009x^{19} + 28888x^{18} + 617005x^{17} - 2034190x^{16} - 41390734x^{15} + 107841588x^{14} + 1632029584x^{13} - 4275106893x^{12} - 37744643110x^{11} + 112010842410x^{10} + 478614868545x^{9} - 1712417124428x^{8} - 2738309177111x^{7} + 13487828332902x^{6} + 2049227391934x^{5} - 45589459911950x^{4} + 25835353515257x^{3} + 46139351873401x^{2} - 42755616102699x + 4702888740782

239.4.a.b

x^{37} - 15x^{36} - 911x^{35} + 13954x^{34} + 361252x^{33} - 5623618x^{32} - 83598787x^{31} + 1308227667x^{30} + 12789770254x^{29} - 197188739887x^{28} - 1388738050889x^{27} + 20435358594063x^{26} + 111901778676198x^{25} - 1503403749091393x^{24} - 6868838597256262x^{23} + 79705297346477658x^{22} + 324328692142421862x^{21} - 3051664537457412835x^{20} - 11714371133621109479x^{19} + 83544342792013661520x^{18} + 317622452426152120757x^{17} - 1596902163126461165915x^{16} - 6272974866942373109781x^{15} + 20393864442433711774337x^{14} + 86528086681031236620165x^{13} - 160383905934810266592252x^{12} - 785785384241772868166524x^{11} + 650314448473378554110644x^{10} + 4305959333510109484347035x^{9} - 682066587236809925042415x^{8} - 12589646313186080405360039x^{7} - 1668160561031770001005467x^{6} + 18357050744681692346284812x^{5} + 2392852785754127948057032x^{4} - 12788079774779435332595217x^{3} + 280844051098231129530930x^{2} + 3125349476742560883195715x - 500496391304187684881350

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(239))$	59
Old forms	0
Cusp forms $S_{4}(\Gamma_0(239))$	59
Eisenstein series $E_{4}(\Gamma_0(239))$	2
Modular forms $M_{4}(\Gamma_0(239))$	61

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 239]: [1, 61] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (phi(ord eps) x dim S_k^new); 8 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(239, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 239:4 1000

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