[N,k,chi] = [324,5,Mod(163,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.163");
S:= CuspForms(chi, 5);
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 \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
\(n\)
\(163\)
\(245\)
\(\chi(n)\)
\(-1\)
\(1\)
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
\( T_{5}^{5} - T_{5}^{4} - 1466T_{5}^{3} - 3718T_{5}^{2} + 144841T_{5} - 59017 \)
T5^5 - T5^4 - 1466*T5^3 - 3718*T5^2 + 144841*T5 - 59017
acting on \(S_{5}^{\mathrm{new}}(324, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} + 11 T^{9} + 78 T^{8} + \cdots + 1048576 \)
T^10 + 11*T^9 + 78*T^8 + 412*T^7 + 1832*T^6 + 7440*T^5 + 29312*T^4 + 105472*T^3 + 319488*T^2 + 720896*T + 1048576
$3$
\( T^{10} \)
T^10
$5$
\( (T^{5} - T^{4} - 1466 T^{3} - 3718 T^{2} + \cdots - 59017)^{2} \)
(T^5 - T^4 - 1466*T^3 - 3718*T^2 + 144841*T - 59017)^2
$7$
\( T^{10} + 15720 T^{8} + \cdots + 61\!\cdots\!88 \)
T^10 + 15720*T^8 + 86788272*T^6 + 210231388800*T^4 + 213802428195072*T^2 + 61503314905202688
$11$
\( T^{10} + 91848 T^{8} + \cdots + 19\!\cdots\!68 \)
T^10 + 91848*T^8 + 2993020464*T^6 + 41595645727872*T^4 + 209672238131032320*T^2 + 1959985078880698368
$13$
\( (T^{5} + 89 T^{4} - 78122 T^{3} + \cdots + 18547259857)^{2} \)
(T^5 + 89*T^4 - 78122*T^3 - 5292314*T^2 + 277970489*T + 18547259857)^2
$17$
\( (T^{5} + 11 T^{4} + \cdots - 667567300561)^{2} \)
(T^5 + 11*T^4 - 186710*T^3 + 4768694*T^2 + 8188994917*T - 667567300561)^2
$19$
\( T^{10} + 737352 T^{8} + \cdots + 42\!\cdots\!52 \)
T^10 + 737352*T^8 + 177212603952*T^6 + 16383790665401472*T^4 + 477043043748783558912*T^2 + 4250869004388942912356352
$23$
\( T^{10} + 1100520 T^{8} + \cdots + 59\!\cdots\!52 \)
T^10 + 1100520*T^8 + 390178463664*T^6 + 56127052983404160*T^4 + 3306521803333765296384*T^2 + 59100765417135394444541952
$29$
\( (T^{5} - 1033 T^{4} + \cdots - 189507124642945)^{2} \)
(T^5 - 1033*T^4 - 1537130*T^3 + 1268190842*T^2 + 467924100217*T - 189507124642945)^2
$31$
\( T^{10} + 2953344 T^{8} + \cdots + 71\!\cdots\!32 \)
T^10 + 2953344*T^8 + 2430566658048*T^6 + 699039234277244928*T^4 + 56298601624764558606336*T^2 + 713434729838908124096888832
$37$
\( (T^{5} + 41 T^{4} + \cdots - 852313726745807)^{2} \)
(T^5 + 41*T^4 - 4885178*T^3 + 106825654*T^2 + 3343649210825*T - 852313726745807)^2
$41$
\( (T^{5} + 1826 T^{4} + \cdots + 920805364605728)^{2} \)
(T^5 + 1826*T^4 - 4170728*T^3 - 7278664912*T^2 + 1213110882448*T + 920805364605728)^2
$43$
\( T^{10} + 18102408 T^{8} + \cdots + 73\!\cdots\!48 \)
T^10 + 18102408*T^8 + 115179866201136*T^6 + 306300559331103456384*T^4 + 307157615607302342078996736*T^2 + 73305603365828766231977254453248
$47$
\( T^{10} + 25575072 T^{8} + \cdots + 15\!\cdots\!28 \)
T^10 + 25575072*T^8 + 245963438641920*T^6 + 1090863061983975776256*T^4 + 2184194145967408621349830656*T^2 + 1594884227306307308507570192252928
$53$
\( (T^{5} - 3094 T^{4} + \cdots + 302749170293792)^{2} \)
(T^5 - 3094*T^4 - 7331096*T^3 + 14374955984*T^2 + 17252093040976*T + 302749170293792)^2
$59$
\( T^{10} + 112922016 T^{8} + \cdots + 24\!\cdots\!00 \)
T^10 + 112922016*T^8 + 4637535611038464*T^6 + 86588931490980486488064*T^4 + 746610314053356443407965290496*T^2 + 2416348160279644850186842079546572800
$61$
\( (T^{5} + 4841 T^{4} + \cdots + 815431223481217)^{2} \)
(T^5 + 4841*T^4 - 6779018*T^3 - 22031492570*T^2 - 8992592338663*T + 815431223481217)^2
$67$
\( T^{10} + 125329992 T^{8} + \cdots + 21\!\cdots\!12 \)
T^10 + 125329992*T^8 + 5801265112181808*T^6 + 120396249654546070518912*T^4 + 1036617399095442370031463821568*T^2 + 2146211614055495226642490478924070912
$71$
\( T^{10} + 132091560 T^{8} + \cdots + 95\!\cdots\!00 \)
T^10 + 132091560*T^8 + 6167814567891120*T^6 + 121834118956595551410816*T^4 + 902197527142583606257023439104*T^2 + 950968293442753094648534369053900800
$73$
\( (T^{5} - 5011 T^{4} + \cdots - 29\!\cdots\!47)^{2} \)
(T^5 - 5011*T^4 - 75577958*T^3 + 353589755338*T^2 + 611650594585973*T - 2940816558450161447)^2
$79$
\( T^{10} + 257560488 T^{8} + \cdots + 67\!\cdots\!12 \)
T^10 + 257560488*T^8 + 19757967998173104*T^6 + 442127922032751816652416*T^4 + 3407294832528137239426150990080*T^2 + 6754672868298694038991103746860122112
$83$
\( T^{10} + 335564832 T^{8} + \cdots + 33\!\cdots\!72 \)
T^10 + 335564832*T^8 + 35647724813654784*T^6 + 1324164802936819403759616*T^4 + 15898467377269624638078277189632*T^2 + 3355016856387263998773013273116672
$89$
\( (T^{5} - 9637 T^{4} + \cdots + 21\!\cdots\!15)^{2} \)
(T^5 - 9637*T^4 - 78217910*T^3 + 413213882870*T^2 + 2607156675410629*T + 2161820565841523615)^2
$97$
\( (T^{5} + 7886 T^{4} + \cdots - 43\!\cdots\!16)^{2} \)
(T^5 + 7886*T^4 - 269142632*T^3 - 815426146736*T^2 + 21471225600443024*T - 43369752793313040416)^2
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