[N,k,chi] = [76,8,Mod(1,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.1");
S:= CuspForms(chi, 8);
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
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
\( p \)
Sign
\(2\)
\(-1\)
\(19\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 40T_{3}^{5} - 7711T_{3}^{4} + 93190T_{3}^{3} + 16686996T_{3}^{2} + 43655400T_{3} - 7412317344 \)
T3^6 - 40*T3^5 - 7711*T3^4 + 93190*T3^3 + 16686996*T3^2 + 43655400*T3 - 7412317344
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(76))\).
$p$
$F_p(T)$
$2$
\( T^{6} \)
T^6
$3$
\( T^{6} - 40 T^{5} + \cdots - 7412317344 \)
T^6 - 40*T^5 - 7711*T^4 + 93190*T^3 + 16686996*T^2 + 43655400*T - 7412317344
$5$
\( T^{6} + \cdots + 174360864384000 \)
T^6 - 279*T^5 - 382107*T^4 + 89339211*T^3 + 31455689670*T^2 - 5181395099400*T + 174360864384000
$7$
\( T^{6} + 1565 T^{5} + \cdots + 15\!\cdots\!84 \)
T^6 + 1565*T^5 - 2268958*T^4 - 3630360070*T^3 + 1278964317269*T^2 + 2093958739148105*T + 15086336025908584
$11$
\( T^{6} - 7983 T^{5} + \cdots - 47\!\cdots\!56 \)
T^6 - 7983*T^5 - 37767781*T^4 + 432916267239*T^3 - 880155543756408*T^2 + 385525041126730164*T - 47929083669332301456
$13$
\( T^{6} - 1250 T^{5} + \cdots - 15\!\cdots\!56 \)
T^6 - 1250*T^5 - 122286729*T^4 - 6544577750*T^3 + 3312548747774008*T^2 + 589363572143005920*T - 15319634805440895689856
$17$
\( T^{6} - 21735 T^{5} + \cdots + 81\!\cdots\!86 \)
T^6 - 21735*T^5 - 1164308584*T^4 + 36745972939470*T^3 - 231907545730575987*T^2 - 858871018484235580455*T + 8152361263409668732149786
$19$
\( (T - 6859)^{6} \)
(T - 6859)^6
$23$
\( T^{6} - 100920 T^{5} + \cdots - 10\!\cdots\!64 \)
T^6 - 100920*T^5 - 284311201*T^4 + 151244309224680*T^3 + 1502913205656366528*T^2 - 7015145233308404129280*T - 10131803990133889037451264
$29$
\( T^{6} + 58656 T^{5} + \cdots - 19\!\cdots\!56 \)
T^6 + 58656*T^5 - 95711902329*T^4 - 4494973587859440*T^3 + 2578255904089960750008*T^2 + 76523107161988774318618944*T - 19309171134099958491730368909456
$31$
\( T^{6} - 403808 T^{5} + \cdots - 59\!\cdots\!24 \)
T^6 - 403808*T^5 + 18356065296*T^4 + 4120055684069632*T^3 - 319372289042744119808*T^2 + 7665008493910929005383680*T - 59469561500940745594498842624
$37$
\( T^{6} - 808780 T^{5} + \cdots + 12\!\cdots\!96 \)
T^6 - 808780*T^5 + 176256645884*T^4 - 9069720444468640*T^3 - 347190860463975948304*T^2 + 5191054539189353523141440*T + 124536595731119467187730296896
$41$
\( T^{6} - 556944 T^{5} + \cdots + 23\!\cdots\!84 \)
T^6 - 556944*T^5 - 326056713208*T^4 + 174434267004322080*T^3 + 2157162144421067749584*T^2 - 1067514808032839309216904576*T + 23338992175248254326181410627584
$43$
\( T^{6} - 1220735 T^{5} + \cdots - 15\!\cdots\!44 \)
T^6 - 1220735*T^5 + 158935767027*T^4 + 142748983288435795*T^3 - 46051523057753836668440*T^2 + 4661658345451186652836838160*T - 154097329310339035579788080098944
$47$
\( T^{6} - 1915305 T^{5} + \cdots + 10\!\cdots\!56 \)
T^6 - 1915305*T^5 - 384082342837*T^4 + 2150474405962335525*T^3 - 661005833538587861055696*T^2 - 324720105707311102197786553920*T + 104713598761424722953743469567814656
$53$
\( T^{6} - 511650 T^{5} + \cdots - 69\!\cdots\!76 \)
T^6 - 511650*T^5 - 3740637014161*T^4 + 1036250365853981250*T^3 + 3965837149172783092356840*T^2 - 189404486376422674701976467840*T - 695595985358363257740317072458258176
$59$
\( T^{6} - 1300572 T^{5} + \cdots - 86\!\cdots\!56 \)
T^6 - 1300572*T^5 - 1547972074327*T^4 + 1965619922082623202*T^3 + 574513744255956073498212*T^2 - 619638885049121808670044671976*T - 86009270406166492856689005529297056
$61$
\( T^{6} - 565335 T^{5} + \cdots - 23\!\cdots\!64 \)
T^6 - 565335*T^5 - 8735718697011*T^4 + 3565037032223823767*T^3 + 12147376867741010024327214*T^2 - 3269585231287627013776097234316*T - 2301352032237925633469153645158227464
$67$
\( T^{6} + 45010 T^{5} + \cdots - 13\!\cdots\!76 \)
T^6 + 45010*T^5 - 16470947063367*T^4 + 1501881992060699320*T^3 + 85557155517434897219425360*T^2 - 12206470804539541818915606522240*T - 132726823572930625244662477069019198976
$71$
\( T^{6} + 1424106 T^{5} + \cdots + 13\!\cdots\!04 \)
T^6 + 1424106*T^5 - 14152150868392*T^4 - 18728657102643413232*T^3 + 41117713101441495009540432*T^2 + 50907169907215768219104435032352*T + 13535981091827314004436212385699151104
$73$
\( T^{6} + 11153825 T^{5} + \cdots + 96\!\cdots\!26 \)
T^6 + 11153825*T^5 + 46136474610476*T^4 + 92710992204432255410*T^3 + 96226770293526768419688329*T^2 + 49054691402281892204791620009725*T + 9624457063719626672182251795785919226
$79$
\( T^{6} + 6392144 T^{5} + \cdots + 29\!\cdots\!76 \)
T^6 + 6392144*T^5 + 2824710229340*T^4 - 43513478257211180320*T^3 - 79446304014369974498532160*T^2 - 35993795123493685348231918829056*T + 2956440985861855561798670399576498176
$83$
\( T^{6} + 3164160 T^{5} + \cdots - 65\!\cdots\!64 \)
T^6 + 3164160*T^5 - 103202687694196*T^4 - 121042182434330223840*T^3 + 2896490929079852479940429568*T^2 - 1700840388442765245769356686922240*T - 6545218979767270128105855563739588824064
$89$
\( T^{6} + 14502678 T^{5} + \cdots + 18\!\cdots\!76 \)
T^6 + 14502678*T^5 - 112082030411632*T^4 - 2225414419024761179040*T^3 - 1452412221116096437145650176*T^2 + 69701367951459345963376571477455872*T + 188797913343229671059662130763365927755776
$97$
\( T^{6} + 21377010 T^{5} + \cdots + 48\!\cdots\!44 \)
T^6 + 21377010*T^5 - 45190604396916*T^4 - 2330897223876266942920*T^3 - 1588463970838532305718360640*T^2 + 28421264378688452504439526273536000*T + 48176140375807984342995060572820534943744
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