[N,k,chi] = [825,6,Mod(1,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.1");
S:= CuspForms(chi, 6);
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
\(3\)
\(1\)
\(5\)
\(1\)
\(11\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 9T_{2}^{7} - 141T_{2}^{6} + 1251T_{2}^{5} + 5160T_{2}^{4} - 45922T_{2}^{3} - 22268T_{2}^{2} + 305880T_{2} - 277200 \)
T2^8 - 9*T2^7 - 141*T2^6 + 1251*T2^5 + 5160*T2^4 - 45922*T2^3 - 22268*T2^2 + 305880*T2 - 277200
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
$p$
$F_p(T)$
$2$
\( T^{8} - 9 T^{7} - 141 T^{6} + \cdots - 277200 \)
T^8 - 9*T^7 - 141*T^6 + 1251*T^5 + 5160*T^4 - 45922*T^3 - 22268*T^2 + 305880*T - 277200
$3$
\( (T + 9)^{8} \)
(T + 9)^8
$5$
\( T^{8} \)
T^8
$7$
\( T^{8} + 66 T^{7} + \cdots - 16208419192764 \)
T^8 + 66*T^7 - 43317*T^6 - 2993728*T^5 + 229947387*T^4 + 21428297834*T^3 + 447350231285*T^2 + 956749065620*T - 16208419192764
$11$
\( (T + 121)^{8} \)
(T + 121)^8
$13$
\( T^{8} - 382 T^{7} + \cdots + 29\!\cdots\!92 \)
T^8 - 382*T^7 - 1233660*T^6 + 209409990*T^5 + 409444035723*T^4 + 19399964975608*T^3 - 13279307638628560*T^2 + 269346234065601280*T + 29768594901532270592
$17$
\( T^{8} - 288 T^{7} + \cdots - 29\!\cdots\!00 \)
T^8 - 288*T^7 - 5036646*T^6 - 756711112*T^5 + 7453158971889*T^4 + 3322137594471736*T^3 - 2770652410257892908*T^2 - 1988650484532328777200*T - 292649235990719260953600
$19$
\( T^{8} + 988 T^{7} + \cdots - 27\!\cdots\!23 \)
T^8 + 988*T^7 - 13849084*T^6 - 11869788124*T^5 + 58138788158518*T^4 + 54859898442344996*T^3 - 67612185614305884284*T^2 - 93730456380197130991140*T - 27332013204042933152281023
$23$
\( T^{8} - 5972 T^{7} + \cdots - 31\!\cdots\!28 \)
T^8 - 5972*T^7 - 9368739*T^6 + 109438953460*T^5 - 173962094878713*T^4 + 31761263884922556*T^3 + 69967844531909239279*T^2 - 15281666798746891572444*T - 3113163287280747459049428
$29$
\( T^{8} - 1032 T^{7} + \cdots + 72\!\cdots\!84 \)
T^8 - 1032*T^7 - 80275817*T^6 + 136385794628*T^5 + 1164954080577988*T^4 - 1171904833692208288*T^3 - 3900637172314241359728*T^2 - 701056658308509683974080*T + 721019268277435490735381184
$31$
\( T^{8} + 4682 T^{7} + \cdots - 95\!\cdots\!00 \)
T^8 + 4682*T^7 - 78081908*T^6 - 281234671730*T^5 + 1670428542873099*T^4 + 3873373443108252792*T^3 - 5352600958391634074264*T^2 - 16686681599728745996899200*T - 9548811089779667728137570000
$37$
\( T^{8} + 17200 T^{7} + \cdots - 29\!\cdots\!64 \)
T^8 + 17200*T^7 - 230184946*T^6 - 3983713586852*T^5 + 19959697548846457*T^4 + 280435296785225505916*T^3 - 701894273609355809601168*T^2 - 4912958443181432923706281776*T - 2979488468179736121867155552464
$41$
\( T^{8} + 13220 T^{7} + \cdots - 33\!\cdots\!96 \)
T^8 + 13220*T^7 - 370964634*T^6 - 5533488078480*T^5 + 11528132540235433*T^4 + 335176746239433359676*T^3 + 132384930015974154504480*T^2 - 5767152566317059591942235632*T - 3387960619781355463160179959696
$43$
\( T^{8} - 22872 T^{7} + \cdots + 68\!\cdots\!72 \)
T^8 - 22872*T^7 - 496401418*T^6 + 10754530573240*T^5 + 61204830520082089*T^4 - 845947348005770357728*T^3 - 5231402269785562644416800*T^2 - 3030980410046186134722525696*T + 6887036086166804667936632047872
$47$
\( T^{8} - 6700 T^{7} + \cdots + 21\!\cdots\!00 \)
T^8 - 6700*T^7 - 1101305122*T^6 + 3349876481676*T^5 + 327679652914896601*T^4 - 543270367207572356888*T^3 - 25597009613620819483028592*T^2 + 19018659575461717435612723200*T + 21882747586496394614956372224000
$53$
\( T^{8} - 6224 T^{7} + \cdots - 55\!\cdots\!44 \)
T^8 - 6224*T^7 - 1790258940*T^6 + 27648711321968*T^5 + 766189953687893440*T^4 - 20127833720973047417600*T^3 + 89231236716795802152845312*T^2 + 835679046409407538885553356800*T - 5565713576458488972322419291193344
$59$
\( T^{8} + 77556 T^{7} + \cdots + 27\!\cdots\!00 \)
T^8 + 77556*T^7 + 384019458*T^6 - 120710773688212*T^5 - 4173064390571672647*T^4 - 50001908149771677262728*T^3 - 121990767575788156464951188*T^2 + 1201987195469857775907153442560*T + 2743366557991731049312886402035200
$61$
\( T^{8} - 11554 T^{7} + \cdots - 59\!\cdots\!00 \)
T^8 - 11554*T^7 - 4464951563*T^6 + 40297627546416*T^5 + 5386676365093523372*T^4 - 34244146190891554634016*T^3 - 1001275706637601500104153152*T^2 - 4569623846238468125786049745920*T - 5953365423227504215502221643468800
$67$
\( T^{8} - 20894 T^{7} + \cdots + 45\!\cdots\!00 \)
T^8 - 20894*T^7 - 8412920251*T^6 + 99364436136032*T^5 + 22755249451230039724*T^4 - 121789067973570269886688*T^3 - 20183939111853728106147580992*T^2 + 85509077318569733895162778255360*T + 4519088189120266954217039132912844800
$71$
\( T^{8} + 21648 T^{7} + \cdots + 39\!\cdots\!88 \)
T^8 + 21648*T^7 - 7915443863*T^6 - 52475445464896*T^5 + 20414056284410490259*T^4 - 84148821849415474130392*T^3 - 16069398508580500083112614893*T^2 + 64871194129720108579538376654024*T + 3948948409880827551191314442077559088
$73$
\( T^{8} - 64660 T^{7} + \cdots + 59\!\cdots\!56 \)
T^8 - 64660*T^7 - 5948944640*T^6 + 302465422876720*T^5 + 14271378369496915760*T^4 - 440256258967588567465344*T^3 - 15219584061782581039710435584*T^2 + 202778762153810349231365351985152*T + 5948872230043400057717821823170297856
$79$
\( T^{8} + 22660 T^{7} + \cdots + 21\!\cdots\!12 \)
T^8 + 22660*T^7 - 13891355874*T^6 - 209974290040624*T^5 + 51467418121802758577*T^4 + 862923930759189981514796*T^3 - 46479903432030673144809840272*T^2 - 877639895206039790104616385085376*T + 2168301608717830410466240741593408512
$83$
\( T^{8} - 100390 T^{7} + \cdots + 27\!\cdots\!00 \)
T^8 - 100390*T^7 - 4972035075*T^6 + 460773963541612*T^5 + 9024689880809608652*T^4 - 625589214241151298190656*T^3 - 7282880488687068724280229072*T^2 + 223370555808740738247352179656640*T + 2792046883843657358798194707919156800
$89$
\( T^{8} + 25578 T^{7} + \cdots + 21\!\cdots\!32 \)
T^8 + 25578*T^7 - 21705015635*T^6 - 1192707941416588*T^5 + 71583238270923338260*T^4 + 4017676642241720337104864*T^3 - 61986895828297739631189121984*T^2 - 1760997321138284670668345213564928*T + 21230202926167962665082065539949740032
$97$
\( T^{8} - 142828 T^{7} + \cdots - 13\!\cdots\!75 \)
T^8 - 142828*T^7 - 21162252612*T^6 + 3003974779671580*T^5 + 47547144985686806894*T^4 - 13869675173250767116553844*T^3 + 303156441158629197296100587756*T^2 + 4897025978787858395861019111565380*T - 134234489650421206365056584305842794775
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