Atlas › Modular forms › Weight 4 › Level 197 › 197.4.a

Newspace 197.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(197))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
197.4.a.a	22	degree 22	$22q - 6q^{2} - 34q^{3} + 68q^{4} - 31q^{5} - 24q^{6} - 102q^{7} + O(q^{8})$	−	$-1$
197.4.a.b	27	degree 27	$27q + 4q^{2} + 32q^{3} + 128q^{4} + 29q^{5} + 36q^{6} + 122q^{7} + O(q^{8})$	+	$+1$

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 $T$
197.4.a.a	$x^{22} + 40x^{21} + 329x^{20} - 6072x^{19} - 100431x^{18} + 144531x^{17} + 9083658x^{16} + 19067268x^{15} - 392460045x^{14} - 1438341113x^{13} + 9079515260x^{12} + 42490536383x^{11} - 114097738749x^{10} - 645713381556x^{9} + 735908238004x^{8} + 5231463344257x^{7} - 2185179691603x^{6} - 21701865697784x^{5} + 4032659396506x^{4} + 43218324786288x^{3} - 9787100279456x^{2} - 31803440477952x + 11283007562240$
197.4.a.b	$x^{27} - 36x^{26} - 68x^{25} + 16252x^{24} - 112139x^{23} - 2851397x^{22} + 32647270x^{21} + 244213597x^{20} - 4189266744x^{19} - 9464930368x^{18} + 307196007987x^{17} - 23617252088x^{16} - 13999868912662x^{15} + 17525045280228x^{14} + 409380157020981x^{13} - 723530049661428x^{12} - 7743074206694232x^{11} + 13751046737341387x^{10} + 94245991494291925x^{9} - 124410009485050507x^{8} - 719251601102723307x^{7} + 372588494689392154x^{6} + 3048359668868492862x^{5} + 1011598092283840628x^{4} - 4593769899760536280x^{3} - 3914396346055392752x^{2} + 315610957355591072x + 768305593258318400$

Orbit

Characteristic polynomial of

T

197.4.a.a

x^{22} + 40x^{21} + 329x^{20} - 6072x^{19} - 100431x^{18} + 144531x^{17} + 9083658x^{16} + 19067268x^{15} - 392460045x^{14} - 1438341113x^{13} + 9079515260x^{12} + 42490536383x^{11} - 114097738749x^{10} - 645713381556x^{9} + 735908238004x^{8} + 5231463344257x^{7} - 2185179691603x^{6} - 21701865697784x^{5} + 4032659396506x^{4} + 43218324786288x^{3} - 9787100279456x^{2} - 31803440477952x + 11283007562240

197.4.a.b

x^{27} - 36x^{26} - 68x^{25} + 16252x^{24} - 112139x^{23} - 2851397x^{22} + 32647270x^{21} + 244213597x^{20} - 4189266744x^{19} - 9464930368x^{18} + 307196007987x^{17} - 23617252088x^{16} - 13999868912662x^{15} + 17525045280228x^{14} + 409380157020981x^{13} - 723530049661428x^{12} - 7743074206694232x^{11} + 13751046737341387x^{10} + 94245991494291925x^{9} - 124410009485050507x^{8} - 719251601102723307x^{7} + 372588494689392154x^{6} + 3048359668868492862x^{5} + 1011598092283840628x^{4} - 4593769899760536280x^{3} - 3914396346055392752x^{2} + 315610957355591072x + 768305593258318400

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(197))$	49
Old forms	0
Cusp forms $S_{4}(\Gamma_0(197))$	49
Eisenstein series $E_{4}(\Gamma_0(197))$	2
Modular forms $M_{4}(\Gamma_0(197))$	51

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 197]: [1, 51] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(197, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 197:4 1000

← 196.4.a · 198.4.a →