[N,k,chi] = [22,4,Mod(3,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).
\(n\)
\(13\)
\(\chi(n)\)
\(\beta_{2}\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3T_{3}^{7} + 42T_{3}^{6} + 185T_{3}^{5} + 1931T_{3}^{4} - 11455T_{3}^{3} + 224168T_{3}^{2} + 368251T_{3} + 4748041 \)
T3^8 - 3*T3^7 + 42*T3^6 + 185*T3^5 + 1931*T3^4 - 11455*T3^3 + 224168*T3^2 + 368251*T3 + 4748041
acting on \(S_{4}^{\mathrm{new}}(22, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \)
(T^4 - 2*T^3 + 4*T^2 - 8*T + 16)^2
$3$
\( T^{8} - 3 T^{7} + 42 T^{6} + \cdots + 4748041 \)
T^8 - 3*T^7 + 42*T^6 + 185*T^5 + 1931*T^4 - 11455*T^3 + 224168*T^2 + 368251*T + 4748041
$5$
\( T^{8} - 5 T^{7} + 146 T^{6} + \cdots + 68624656 \)
T^8 - 5*T^7 + 146*T^6 + 870*T^5 + 7381*T^4 - 260490*T^3 + 3495696*T^2 - 15739600*T + 68624656
$7$
\( T^{8} + T^{7} + 698 T^{6} + \cdots + 87027360016 \)
T^8 + T^7 + 698*T^6 - 1790*T^5 + 311021*T^4 + 736690*T^3 + 119944612*T^2 - 118591608*T + 87027360016
$11$
\( T^{8} + 155 T^{7} + \cdots + 3138428376721 \)
T^8 + 155*T^7 + 13111*T^6 + 755095*T^5 + 31999176*T^4 + 1005031445*T^3 + 23226936271*T^2 + 365481892105*T + 3138428376721
$13$
\( T^{8} - 7 T^{7} + \cdots + 16022496400 \)
T^8 - 7*T^7 + 6324*T^6 + 154082*T^5 + 12652281*T^4 + 151584980*T^3 + 480153440*T^2 - 5585975400*T + 16022496400
$17$
\( T^{8} - 161 T^{7} + \cdots + 4078198497025 \)
T^8 - 161*T^7 + 14846*T^6 - 835281*T^5 + 33388091*T^4 - 766431885*T^3 + 18801687590*T^2 - 358241219725*T + 4078198497025
$19$
\( T^{8} + 272 T^{7} + \cdots + 3649418019025 \)
T^8 + 272*T^7 + 46919*T^6 + 4795488*T^5 + 296722606*T^4 + 8103481680*T^3 + 113555229660*T^2 + 659909576800*T + 3649418019025
$23$
\( (T^{4} - 314 T^{3} + 13148 T^{2} + \cdots - 257882816)^{2} \)
(T^4 - 314*T^3 + 13148*T^2 + 3418592*T - 257882816)^2
$29$
\( T^{8} - 33 T^{7} + \cdots + 10\!\cdots\!00 \)
T^8 - 33*T^7 + 62954*T^6 - 12395922*T^5 + 1734695821*T^4 - 141070735410*T^3 + 28792899761660*T^2 - 870628334274600*T + 10504778151552400
$31$
\( T^{8} - 323 T^{7} + \cdots + 85\!\cdots\!00 \)
T^8 - 323*T^7 + 55484*T^6 + 2105108*T^5 + 920458121*T^4 - 72151234330*T^3 + 158959422041460*T^2 - 7332152413905000*T + 851731083376128400
$37$
\( T^{8} - 49 T^{7} + \cdots + 10\!\cdots\!56 \)
T^8 - 49*T^7 + 93438*T^6 + 19861870*T^5 + 1569396301*T^4 - 655485872790*T^3 + 99706404336192*T^2 + 1746155227792752*T + 1021563173977639056
$41$
\( T^{8} - 361 T^{7} + \cdots + 60\!\cdots\!41 \)
T^8 - 361*T^7 + 118702*T^6 - 14223373*T^5 + 1716329655*T^4 + 91549373403*T^3 + 75132305104722*T^2 + 9333933513254451*T + 604267458241509241
$43$
\( (T^{4} - 721 T^{3} + 102089 T^{2} + \cdots - 2713875120)^{2} \)
(T^4 - 721*T^3 + 102089*T^2 + 17514420*T - 2713875120)^2
$47$
\( T^{8} + 1069 T^{7} + \cdots + 99\!\cdots\!00 \)
T^8 + 1069*T^7 + 508716*T^6 + 109459414*T^5 + 16617033681*T^4 + 1592442938800*T^3 + 100628143564640*T^2 + 3731587331947200*T + 99091938239238400
$53$
\( T^{8} + 281 T^{7} + \cdots + 39\!\cdots\!96 \)
T^8 + 281*T^7 - 46048*T^6 - 38831162*T^5 + 32160384905*T^4 + 4307888731972*T^3 + 2328714801794112*T^2 - 18110915532571776*T + 395283225207767296
$59$
\( T^{8} + 128 T^{7} + \cdots + 33\!\cdots\!25 \)
T^8 + 128*T^7 + 323639*T^6 + 184278592*T^5 + 46247757326*T^4 + 1785415366240*T^3 + 43227672694140*T^2 + 895729328095800*T + 33947069399763025
$61$
\( T^{8} + 617 T^{7} + \cdots + 23\!\cdots\!00 \)
T^8 + 617*T^7 + 393164*T^6 + 235647198*T^5 + 103722319201*T^4 + 25302515417920*T^3 + 4062760345065760*T^2 + 402170511502232000*T + 23283698433429510400
$67$
\( (T^{4} - 289 T^{3} + \cdots + 35027256944)^{2} \)
(T^4 - 289*T^3 - 516317*T^2 + 17452812*T + 35027256944)^2
$71$
\( T^{8} - 115 T^{7} + \cdots + 23\!\cdots\!96 \)
T^8 - 115*T^7 + 577394*T^6 - 429482520*T^5 + 136584831361*T^4 + 29045324199880*T^3 + 2692276885382584*T^2 + 44195457894499160*T + 23313041323805777296
$73$
\( T^{8} + 1487 T^{7} + \cdots + 32\!\cdots\!41 \)
T^8 + 1487*T^7 + 920762*T^6 + 13427985*T^5 + 315344977721*T^4 - 28472243930925*T^3 + 13526831433558878*T^2 - 2907392007869339999*T + 320582100421882343641
$79$
\( T^{8} - 71 T^{7} + \cdots + 68\!\cdots\!00 \)
T^8 - 71*T^7 + 485856*T^6 + 189283124*T^5 + 59245661401*T^4 - 9552661844550*T^3 + 2988886503577500*T^2 - 500782095988875000*T + 687070385660756250000
$83$
\( T^{8} - 1942 T^{7} + \cdots + 16\!\cdots\!25 \)
T^8 - 1942*T^7 + 2018969*T^6 - 1119059988*T^5 + 1469248478626*T^4 - 294496905242870*T^3 + 264090792772182160*T^2 + 34088647974029230700*T + 1642277606267847558025
$89$
\( (T^{4} + 1101 T^{3} + \cdots - 64943655580)^{2} \)
(T^4 + 1101*T^3 - 412981*T^2 - 413964860*T - 64943655580)^2
$97$
\( T^{8} + 5128 T^{7} + \cdots + 46\!\cdots\!81 \)
T^8 + 5128*T^7 + 12168523*T^6 + 16112162976*T^5 + 13004295996730*T^4 + 6509528139573096*T^3 + 2344091468308582608*T^2 + 316880162854933405068*T + 46941791994123734982681
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