[N,k,chi] = [74,4,Mod(47,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.47");
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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).
\(n\)
\(39\)
\(\chi(n)\)
\(-\beta_{2}\)
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
\( T_{3}^{8} + 5T_{3}^{7} + 63T_{3}^{6} - 126T_{3}^{5} + 1384T_{3}^{4} - 984T_{3}^{3} + 9384T_{3}^{2} - 7040T_{3} + 48400 \)
T3^8 + 5*T3^7 + 63*T3^6 - 126*T3^5 + 1384*T3^4 - 984*T3^3 + 9384*T3^2 - 7040*T3 + 48400
acting on \(S_{4}^{\mathrm{new}}(74, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 4)^{4} \)
(T^2 - 2*T + 4)^4
$3$
\( T^{8} + 5 T^{7} + 63 T^{6} + \cdots + 48400 \)
T^8 + 5*T^7 + 63*T^6 - 126*T^5 + 1384*T^4 - 984*T^3 + 9384*T^2 - 7040*T + 48400
$5$
\( T^{8} + 10 T^{7} + 276 T^{6} + \cdots + 729 \)
T^8 + 10*T^7 + 276*T^6 + 556*T^5 + 42529*T^4 + 203268*T^3 + 1345716*T^2 - 31266*T + 729
$7$
\( T^{8} - 3 T^{7} + \cdots + 8611097616 \)
T^8 - 3*T^7 + 647*T^6 - 830*T^5 + 318364*T^4 - 318560*T^3 + 61086232*T^2 + 127316112*T + 8611097616
$11$
\( (T^{4} - 32 T^{3} - 2880 T^{2} + \cdots - 460800)^{2} \)
(T^4 - 32*T^3 - 2880*T^2 + 77440*T - 460800)^2
$13$
\( T^{8} + 61 T^{7} + \cdots + 60535681600 \)
T^8 + 61*T^7 + 4879*T^6 + 50906*T^5 + 5294096*T^4 + 100390856*T^3 + 3408321664*T^2 + 14952342880*T + 60535681600
$17$
\( T^{8} - 12 T^{7} + \cdots + 1531116688689 \)
T^8 - 12*T^7 + 7586*T^6 - 358512*T^5 + 59307643*T^4 - 1696020528*T^3 + 40926188178*T^2 - 277059952764*T + 1531116688689
$19$
\( T^{8} + 71 T^{7} + \cdots + 6992428262400 \)
T^8 + 71*T^7 + 7377*T^6 + 97504*T^5 + 12161856*T^4 - 67888960*T^3 + 23516753920*T^2 - 348204057600*T + 6992428262400
$23$
\( (T^{4} + 26 T^{3} - 31000 T^{2} + \cdots - 15115776)^{2} \)
(T^4 + 26*T^3 - 31000*T^2 - 2143124*T - 15115776)^2
$29$
\( (T^{4} - 161 T^{3} - 40343 T^{2} + \cdots - 72087162)^{2} \)
(T^4 - 161*T^3 - 40343*T^2 + 5980165*T - 72087162)^2
$31$
\( (T^{4} + 56 T^{3} - 99216 T^{2} + \cdots + 712674048)^{2} \)
(T^4 + 56*T^3 - 99216*T^2 - 8437024*T + 712674048)^2
$37$
\( T^{8} - 557 T^{7} + \cdots + 65\!\cdots\!81 \)
T^8 - 557*T^7 + 71521*T^6 + 28620314*T^5 - 12160873546*T^4 + 1449704765042*T^3 + 183503318498089*T^2 - 72388689065857889*T + 6582952005840035281
$41$
\( T^{8} - 92 T^{7} + \cdots + 24\!\cdots\!25 \)
T^8 - 92*T^7 + 118138*T^6 + 11151088*T^5 + 10424771171*T^4 + 344270642160*T^3 + 170803202873050*T^2 - 824886470179500*T + 2417419909609430625
$43$
\( (T^{4} - 266 T^{3} - 32132 T^{2} + \cdots - 712153104)^{2} \)
(T^4 - 266*T^3 - 32132*T^2 + 12592876*T - 712153104)^2
$47$
\( (T^{4} - 140 T^{3} - 300064 T^{2} + \cdots + 6878004480)^{2} \)
(T^4 - 140*T^3 - 300064*T^2 + 44250720*T + 6878004480)^2
$53$
\( T^{8} - 159 T^{7} + \cdots + 32\!\cdots\!00 \)
T^8 - 159*T^7 + 304151*T^6 - 5733230*T^5 + 63684370720*T^4 - 1237351403000*T^3 + 5664614130522400*T^2 + 452288284015476000*T + 326342570248097640000
$59$
\( T^{8} - 263 T^{7} + \cdots + 86\!\cdots\!00 \)
T^8 - 263*T^7 + 850913*T^6 - 204461968*T^5 + 674321904496*T^4 - 165160003346880*T^3 + 34786562021395200*T^2 - 1901707229543616000*T + 86030240301901440000
$61$
\( T^{8} + 206 T^{7} + \cdots + 48\!\cdots\!29 \)
T^8 + 206*T^7 + 470364*T^6 + 159504404*T^5 + 215571709073*T^4 + 55849334396284*T^3 + 12363488609419852*T^2 + 859446811740938778*T + 48172011272443276729
$67$
\( T^{8} - 245 T^{7} + \cdots + 58\!\cdots\!24 \)
T^8 - 245*T^7 + 743189*T^6 + 639851628*T^5 + 432935571584*T^4 + 149580016132416*T^3 + 39343641090497024*T^2 + 5693550403411551232*T + 580852685161830682624
$71$
\( T^{8} + 957 T^{7} + \cdots + 13\!\cdots\!04 \)
T^8 + 957*T^7 + 1415831*T^6 + 470196402*T^5 + 715695255988*T^4 + 259689552567888*T^3 + 219113121634868808*T^2 + 5583104997783101424*T + 138539254648791922704
$73$
\( (T^{4} + 136 T^{3} - 581912 T^{2} + \cdots - 6461408784)^{2} \)
(T^4 + 136*T^3 - 581912*T^2 - 125038736*T - 6461408784)^2
$79$
\( T^{8} - 173 T^{7} + \cdots + 15\!\cdots\!00 \)
T^8 - 173*T^7 + 631329*T^6 + 454099192*T^5 + 343674114832*T^4 + 101015996208960*T^3 + 23254542709526016*T^2 + 2147964517323901440*T + 150604061411323929600
$83$
\( T^{8} - 1217 T^{7} + \cdots + 16\!\cdots\!16 \)
T^8 - 1217*T^7 + 1968827*T^6 - 891685270*T^5 + 1182001307584*T^4 - 460374556264440*T^3 + 531830261605137912*T^2 - 29950889041745651232*T + 1626571073644778806416
$89$
\( T^{8} + 2136 T^{7} + \cdots + 21\!\cdots\!25 \)
T^8 + 2136*T^7 + 3615614*T^6 + 2276658096*T^5 + 1214308772191*T^4 + 77590761399696*T^3 + 60008385052774134*T^2 + 5886020719006243200*T + 2146012339814002625625
$97$
\( (T^{4} - 2631 T^{3} + \cdots - 492264212614)^{2} \)
(T^4 - 2631*T^3 + 1604321*T^2 + 524168339*T - 492264212614)^2
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