[N,k,chi] = [206,2,Mod(9,206)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(206, base_ring=CyclotomicField(34))
chi = DirichletCharacter(H, H._module([26]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("206.9");
S:= CuspForms(chi, 2);
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{34}\).
We also show the integral \(q\)-expansion of the trace form .
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/206\mathbb{Z}\right)^\times\).
\(n\)
\(5\)
\(\chi(n)\)
\(\zeta_{34}^{2}\)
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
\( T_{3}^{16} + 3 T_{3}^{15} - 8 T_{3}^{14} - 24 T_{3}^{13} + 30 T_{3}^{12} + 90 T_{3}^{11} + 32 T_{3}^{10} + 28 T_{3}^{9} + 169 T_{3}^{8} - 3 T_{3}^{7} + 297 T_{3}^{6} + 92 T_{3}^{5} + 140 T_{3}^{4} + 63 T_{3}^{3} + 36 T_{3}^{2} + 6 T_{3} + 1 \)
T3^16 + 3*T3^15 - 8*T3^14 - 24*T3^13 + 30*T3^12 + 90*T3^11 + 32*T3^10 + 28*T3^9 + 169*T3^8 - 3*T3^7 + 297*T3^6 + 92*T3^5 + 140*T3^4 + 63*T3^3 + 36*T3^2 + 6*T3 + 1
acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{16} - T^{15} + T^{14} - T^{13} + T^{12} - T^{11} + \cdots + 1 \)
T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$
\( T^{16} + 3 T^{15} - 8 T^{14} - 24 T^{13} + \cdots + 1 \)
T^16 + 3*T^15 - 8*T^14 - 24*T^13 + 30*T^12 + 90*T^11 + 32*T^10 + 28*T^9 + 169*T^8 - 3*T^7 + 297*T^6 + 92*T^5 + 140*T^4 + 63*T^3 + 36*T^2 + 6*T + 1
$5$
\( T^{16} - 15 T^{15} + 123 T^{14} + \cdots + 57121 \)
T^16 - 15*T^15 + 123*T^14 - 706*T^13 + 3127*T^12 - 11171*T^11 + 32840*T^10 - 80605*T^9 + 170239*T^8 - 326925*T^7 + 589904*T^6 - 948558*T^5 + 1232737*T^4 - 1208719*T^3 + 849741*T^2 - 339141*T + 57121
$7$
\( T^{16} + 7 T^{15} - 2 T^{14} - 184 T^{13} + \cdots + 256 \)
T^16 + 7*T^15 - 2*T^14 - 184*T^13 - 336*T^12 + 1405*T^11 + 5789*T^10 - 8284*T^9 - 38115*T^8 + 64338*T^7 + 292504*T^6 + 188680*T^5 + 119744*T^4 + 32544*T^3 + 10752*T^2 + 1280*T + 256
$11$
\( T^{16} + 17 T^{14} - 102 T^{13} + \cdots + 289 \)
T^16 + 17*T^14 - 102*T^13 + 340*T^12 - 2040*T^11 + 5372*T^10 - 4675*T^9 + 22508*T^8 - 65603*T^7 - 59245*T^6 + 173978*T^5 + 272527*T^4 + 139587*T^3 + 34969*T^2 + 2890*T + 289
$13$
\( T^{16} - 22 T^{15} + 263 T^{14} + \cdots + 5031049 \)
T^16 - 22*T^15 + 263*T^14 - 2046*T^13 + 11386*T^12 - 49484*T^11 + 189467*T^10 - 671646*T^9 + 2286091*T^8 - 7833204*T^7 + 25373189*T^6 - 66729601*T^5 + 138918887*T^4 - 196711695*T^3 + 193396197*T^2 - 56265655*T + 5031049
$17$
\( T^{16} + 15 T^{15} + 157 T^{14} + \cdots + 21077281 \)
T^16 + 15*T^15 + 157*T^14 + 723*T^13 + 1512*T^12 - 10487*T^11 - 40974*T^10 + 24998*T^9 + 1545811*T^8 + 2865671*T^7 - 12364266*T^6 - 38024503*T^5 + 14800913*T^4 + 118383922*T^3 + 192342603*T^2 + 25480050*T + 21077281
$19$
\( T^{16} - 32 T^{15} + 531 T^{14} + \cdots + 33790969 \)
T^16 - 32*T^15 + 531*T^14 - 5840*T^13 + 46970*T^12 - 290464*T^11 + 1426058*T^10 - 5681867*T^9 + 18635758*T^8 - 50624213*T^7 + 113746409*T^6 - 209096320*T^5 + 305777282*T^4 - 338898555*T^3 + 266728092*T^2 - 134547698*T + 33790969
$23$
\( T^{16} - 5 T^{15} + 42 T^{14} + \cdots + 5987809 \)
T^16 - 5*T^15 + 42*T^14 - 142*T^13 + 1169*T^12 - 15722*T^11 + 183432*T^10 - 1542828*T^9 + 9436648*T^8 - 42950206*T^7 + 143773480*T^6 - 349582180*T^5 + 603699550*T^4 - 690703543*T^3 + 455880158*T^2 - 90127904*T + 5987809
$29$
\( T^{16} + 6 T^{15} + 87 T^{14} + \cdots + 329676649 \)
T^16 + 6*T^15 + 87*T^14 + 216*T^13 + 7943*T^12 + 75436*T^11 + 668380*T^10 + 3274350*T^9 + 19991319*T^8 + 42304681*T^7 + 180617365*T^6 - 106076476*T^5 + 1675020808*T^4 - 4542966430*T^3 + 4534330364*T^2 - 1994873276*T + 329676649
$31$
\( T^{16} - 4 T^{15} + 16 T^{14} + \cdots + 47623801 \)
T^16 - 4*T^15 + 16*T^14 + 293*T^13 - 985*T^12 + 6915*T^11 + 89657*T^10 - 1681653*T^9 + 23605385*T^8 - 105311978*T^7 + 241090871*T^6 - 300116296*T^5 + 231311045*T^4 - 162580150*T^3 + 70127600*T^2 - 11172719*T + 47623801
$37$
\( T^{16} + 4 T^{15} - 35 T^{14} + \cdots + 26020201 \)
T^16 + 4*T^15 - 35*T^14 + 13*T^13 + 1769*T^12 + 8181*T^11 + 46732*T^10 + 251987*T^9 + 912000*T^8 + 2840245*T^7 + 8536855*T^6 + 20595530*T^5 + 41294752*T^4 + 59082178*T^3 + 78658285*T^2 + 60594779*T + 26020201
$41$
\( T^{16} + 35 T^{15} + \cdots + 49807973329 \)
T^16 + 35*T^15 + 630*T^14 + 8008*T^13 + 71996*T^12 + 410262*T^11 + 1409369*T^10 + 4181542*T^9 + 16493554*T^8 + 47333390*T^7 + 325193885*T^6 - 828742009*T^5 - 1433088677*T^4 - 1099170665*T^3 + 20657195058*T^2 + 42629708301*T + 49807973329
$43$
\( T^{16} + T^{15} + \cdots + 1928284945129 \)
T^16 + T^15 - 84*T^14 + 137*T^13 + 1344*T^12 + 4846*T^11 + 411622*T^10 + 6444089*T^9 + 66258504*T^8 + 233374046*T^7 + 1685313038*T^6 + 23387219074*T^5 + 120887756790*T^4 - 358768415938*T^3 + 1120964110866*T^2 - 1554570703754*T + 1928284945129
$47$
\( (T^{8} + 9 T^{7} - 227 T^{6} + \cdots + 806957)^{2} \)
(T^8 + 9*T^7 - 227*T^6 - 1790*T^5 + 15387*T^4 + 87528*T^3 - 319232*T^2 - 353404*T + 806957)^2
$53$
\( T^{16} + \cdots + 132986631912289 \)
T^16 + 84*T^15 + 3248*T^14 + 77519*T^13 + 1312911*T^12 + 17549881*T^11 + 201350755*T^10 + 2038292757*T^9 + 17523001159*T^8 + 120067115244*T^7 + 622580491638*T^6 + 2396580401902*T^5 + 7258620452510*T^4 + 19777066094393*T^3 + 45902363452609*T^2 + 50805523412477*T + 132986631912289
$59$
\( T^{16} - 13 T^{15} + \cdots + 298232855449 \)
T^16 - 13*T^15 + 33*T^14 + 319*T^13 + 1429*T^12 - 17778*T^11 - 93382*T^10 + 855487*T^9 + 3818881*T^8 + 2960537*T^7 - 127599876*T^6 - 512781425*T^5 + 2495890735*T^4 + 13987392094*T^3 + 91757791651*T^2 + 86889446449*T + 298232855449
$61$
\( T^{16} - 28 T^{15} + \cdots + 15851565409 \)
T^16 - 28*T^15 + 342*T^14 - 3252*T^13 + 54370*T^12 - 577687*T^11 + 4885638*T^10 - 57899997*T^9 + 587813538*T^8 - 3890476050*T^7 + 17016352689*T^6 - 50038396032*T^5 + 98666009568*T^4 - 125308793524*T^3 + 88950763119*T^2 - 18136956665*T + 15851565409
$67$
\( T^{16} + 18 T^{15} + \cdots + 1248835305169 \)
T^16 + 18*T^15 + 222*T^14 - 322*T^13 - 9349*T^12 - 61947*T^11 + 1473034*T^10 - 2828629*T^9 - 38825160*T^8 + 259902189*T^7 + 74387870*T^6 - 9195146727*T^5 + 64116761826*T^4 - 260216660663*T^3 + 481555028937*T^2 - 199645932476*T + 1248835305169
$71$
\( T^{16} + \cdots + 844940426843881 \)
T^16 + 23*T^15 + 291*T^14 + 5231*T^13 + 71132*T^12 + 446886*T^11 + 3021197*T^10 + 43927402*T^9 + 217564333*T^8 - 4191339*T^7 + 9034664380*T^6 + 32446839120*T^5 - 65261980395*T^4 - 1928442038412*T^3 + 28758452521844*T^2 - 208541133873674*T + 844940426843881
$73$
\( T^{16} - 16 T^{15} + \cdots + 82029049746121 \)
T^16 - 16*T^15 - 33*T^14 - 1206*T^13 + 27779*T^12 + 156503*T^11 + 1244197*T^10 - 25825787*T^9 - 233862012*T^8 - 1419968995*T^7 + 10787629855*T^6 + 199068137906*T^5 + 1814478406131*T^4 + 9750909573089*T^3 + 35439757517136*T^2 + 76575012483222*T + 82029049746121
$79$
\( T^{16} - 44 T^{15} + \cdots + 108222409 \)
T^16 - 44*T^15 + 950*T^14 - 12492*T^13 + 112408*T^12 - 598899*T^11 + 1602565*T^10 - 1112638*T^9 - 4259096*T^8 + 20967232*T^7 + 10212395*T^6 + 35631810*T^5 + 100037023*T^4 + 1140757*T^3 + 26509655*T^2 - 8894565*T + 108222409
$83$
\( T^{16} + 77 T^{15} + \cdots + 13\!\cdots\!41 \)
T^16 + 77*T^15 + 3175*T^14 + 89146*T^13 + 1891096*T^12 + 32006061*T^11 + 444677861*T^10 + 5140136424*T^9 + 49678807278*T^8 + 401949272323*T^7 + 2737708448595*T^6 + 15784652340670*T^5 + 76024647044904*T^4 + 292112176655450*T^3 + 826060611798888*T^2 + 1509281575243951*T + 1339302275197441
$89$
\( T^{16} + \cdots + 365135192637121 \)
T^16 + 16*T^15 + 69*T^14 - 1820*T^13 - 63256*T^12 + 121243*T^11 + 15302007*T^10 + 245029652*T^9 + 3858910122*T^8 + 42035635105*T^7 + 528351517681*T^6 + 5614154415952*T^5 + 49760043173619*T^4 + 294111475320092*T^3 + 1105533063987738*T^2 + 928294999454535*T + 365135192637121
$97$
\( T^{16} + \cdots + 205995892438249 \)
T^16 - 37*T^15 + 842*T^14 - 9938*T^13 + 73657*T^12 + 180790*T^11 - 5167084*T^10 + 58949495*T^9 + 94638370*T^8 - 791512279*T^7 + 16396759479*T^6 + 43132836332*T^5 + 6560050223*T^4 + 255904614137*T^3 - 3743826906412*T^2 - 35161984932932*T + 205995892438249
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