[N,k,chi] = [175,3,Mod(26,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.26");
S:= CuspForms(chi, 3);
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 \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
\(n\)
\(101\)
\(127\)
\(\chi(n)\)
\(1 + \beta_{3}\)
\(1\)
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
\( T_{2}^{12} - 2 T_{2}^{11} + 19 T_{2}^{10} - 26 T_{2}^{9} + 244 T_{2}^{8} - 338 T_{2}^{7} + 1249 T_{2}^{6} - 986 T_{2}^{5} + 3532 T_{2}^{4} - 3618 T_{2}^{3} + 3579 T_{2}^{2} - 1386 T_{2} + 441 \)
T2^12 - 2*T2^11 + 19*T2^10 - 26*T2^9 + 244*T2^8 - 338*T2^7 + 1249*T2^6 - 986*T2^5 + 3532*T2^4 - 3618*T2^3 + 3579*T2^2 - 1386*T2 + 441
acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} - 2 T^{11} + 19 T^{10} - 26 T^{9} + \cdots + 441 \)
T^12 - 2*T^11 + 19*T^10 - 26*T^9 + 244*T^8 - 338*T^7 + 1249*T^6 - 986*T^5 + 3532*T^4 - 3618*T^3 + 3579*T^2 - 1386*T + 441
$3$
\( T^{12} - 6 T^{11} - 16 T^{10} + \cdots + 5184 \)
T^12 - 6*T^11 - 16*T^10 + 168*T^9 + 435*T^8 - 4284*T^7 + 7688*T^6 + 3618*T^5 - 14999*T^4 - 6300*T^3 + 19368*T^2 + 18144*T + 5184
$5$
\( T^{12} \)
T^12
$7$
\( T^{12} - 2 T^{11} + \cdots + 13841287201 \)
T^12 - 2*T^11 + 14*T^10 - 348*T^9 + 1588*T^8 - 14434*T^7 + 136318*T^6 - 707266*T^5 + 3812788*T^4 - 40941852*T^3 + 80707214*T^2 - 564950498*T + 13841287201
$11$
\( T^{12} + 14 T^{11} + \cdots + 4134617957376 \)
T^12 + 14*T^11 + 547*T^10 + 3398*T^9 + 140377*T^8 + 590852*T^7 + 23674912*T^6 - 2111872*T^5 + 2132951008*T^4 - 5406058752*T^3 + 161963985408*T^2 - 569995960320*T + 4134617957376
$13$
\( T^{12} + 546 T^{10} + \cdots + 77792400 \)
T^12 + 546*T^10 + 79881*T^8 + 3303936*T^6 + 42864696*T^4 + 177884640*T^2 + 77792400
$17$
\( T^{12} + 48 T^{11} + \cdots + 8707129344 \)
T^12 + 48*T^11 + 648*T^10 - 5760*T^9 - 87636*T^8 + 647136*T^7 + 11076768*T^6 - 13874112*T^5 - 200358000*T^4 + 284259456*T^3 + 2789282304*T^2 - 8948994048*T + 8707129344
$19$
\( T^{12} + 30 T^{11} + \cdots + 1590595171344 \)
T^12 + 30*T^11 - 765*T^10 - 31950*T^9 + 682305*T^8 + 25288920*T^7 - 36561816*T^6 - 5052679560*T^5 + 43549675740*T^4 + 19685376000*T^3 - 266558601600*T^2 - 116231086080*T + 1590595171344
$23$
\( T^{12} - 14 T^{11} + \cdots + 44219321357121 \)
T^12 - 14*T^11 + 1543*T^10 - 26030*T^9 + 2008060*T^8 - 29553974*T^7 + 641665549*T^6 - 4654809638*T^5 + 84040909636*T^4 - 623745806958*T^3 + 6475997136111*T^2 - 17939233143486*T + 44219321357121
$29$
\( (T^{6} - 32 T^{5} - 1494 T^{4} + \cdots + 486804)^{2} \)
(T^6 - 32*T^5 - 1494*T^4 + 12304*T^3 + 333061*T^2 - 1354800*T + 486804)^2
$31$
\( T^{12} - 132 T^{11} + \cdots + 76\!\cdots\!04 \)
T^12 - 132*T^11 + 7128*T^10 - 174240*T^9 + 615948*T^8 + 46242144*T^7 + 899582400*T^6 - 109893459456*T^5 + 3376076775696*T^4 - 56190191121792*T^3 + 554869606608576*T^2 - 3092977274429952*T + 7676274676379904
$37$
\( T^{12} + \cdots + 110961448062976 \)
T^12 + 44*T^11 + 3923*T^10 - 180*T^9 + 5324905*T^8 + 42813304*T^7 + 2789132224*T^6 - 8673187712*T^5 + 443389126336*T^4 + 1193089594368*T^3 + 20862837395456*T^2 - 40995283431424*T + 110961448062976
$41$
\( T^{12} + 13194 T^{10} + \cdots + 77\!\cdots\!25 \)
T^12 + 13194*T^10 + 64686087*T^8 + 147580502796*T^6 + 157331965881231*T^4 + 66986328667983210*T^2 + 7714430452211186025
$43$
\( (T^{6} - 2 T^{5} - 3556 T^{4} + \cdots - 9258464)^{2} \)
(T^6 - 2*T^5 - 3556*T^4 + 16894*T^3 + 2026169*T^2 - 14028392*T - 9258464)^2
$47$
\( T^{12} + 204 T^{11} + \cdots + 11\!\cdots\!04 \)
T^12 + 204*T^11 + 16143*T^10 + 463284*T^9 - 5404251*T^8 - 462288816*T^7 + 3563699148*T^6 + 323896522608*T^5 - 151113871044*T^4 - 89804273963328*T^3 + 1133755303189968*T^2 - 5718382930746432*T + 11420780538274704
$53$
\( T^{12} + 196 T^{11} + \cdots + 18\!\cdots\!00 \)
T^12 + 196*T^11 + 30067*T^10 + 2734084*T^9 + 232858381*T^8 + 15660085720*T^7 + 1040997364060*T^6 + 53648620154800*T^5 + 2354233969145500*T^4 + 71006479149576000*T^3 + 1634370363491706000*T^2 + 20920710418536960000*T + 183359739907728810000
$59$
\( T^{12} - 72 T^{11} + \cdots + 92\!\cdots\!84 \)
T^12 - 72*T^11 - 3204*T^10 + 355104*T^9 + 17593260*T^8 - 466060608*T^7 - 17236557648*T^6 + 430665682464*T^5 + 11697950662416*T^4 - 345744787893120*T^3 + 1198460557631232*T^2 + 26029175189133312*T + 92863132555382784
$61$
\( T^{12} - 72 T^{11} + \cdots + 23\!\cdots\!24 \)
T^12 - 72*T^11 - 3150*T^10 + 351216*T^9 + 18349503*T^8 - 211927824*T^7 - 22404607038*T^6 + 84839055696*T^5 + 30054034565217*T^4 + 859577292425520*T^3 + 11561045288973228*T^2 + 79080264157124160*T + 230080557521105424
$67$
\( T^{12} - 138 T^{11} + \cdots + 28\!\cdots\!56 \)
T^12 - 138*T^11 + 17664*T^10 - 1040380*T^9 + 70596915*T^8 - 2482614048*T^7 + 164251304184*T^6 - 4187735372262*T^5 + 219452544787065*T^4 - 1555443324703420*T^3 + 141292966343239944*T^2 - 1579086693160818912*T + 28421955565045973056
$71$
\( (T^{6} + 4 T^{5} - 15420 T^{4} + \cdots + 46762703904)^{2} \)
(T^6 + 4*T^5 - 15420*T^4 - 543512*T^3 + 51908596*T^2 + 3246772368*T + 46762703904)^2
$73$
\( T^{12} - 528 T^{11} + \cdots + 15\!\cdots\!64 \)
T^12 - 528*T^11 + 115344*T^10 - 11835648*T^9 + 334795164*T^8 + 34240726656*T^7 - 324128790720*T^6 - 582877279779264*T^5 + 67739133309222672*T^4 - 3752939867276920320*T^3 + 117296427373600776192*T^2 - 2007788828856569561088*T + 15079580207403723915264
$79$
\( T^{12} + 12 T^{11} + \cdots + 560965048576 \)
T^12 + 12*T^11 + 10872*T^10 - 67280*T^9 + 113056620*T^8 + 263196912*T^7 + 26818072896*T^6 - 262690846944*T^5 + 5491560890256*T^4 - 21190927480256*T^3 + 79286149589184*T^2 + 6592990063872*T + 560965048576
$83$
\( T^{12} + 39468 T^{10} + \cdots + 26\!\cdots\!00 \)
T^12 + 39468*T^10 + 573758538*T^8 + 3888167046948*T^6 + 12449477606526801*T^4 + 15728476620981481560*T^2 + 2640007882389289280400
$89$
\( T^{12} - 204 T^{11} + \cdots + 81\!\cdots\!44 \)
T^12 - 204*T^11 + 1578*T^10 + 2507976*T^9 - 42657933*T^8 - 21749962968*T^7 + 674880773130*T^6 + 97846183456092*T^5 - 1210893398226207*T^4 - 219073101146669520*T^3 + 5406563315005687152*T^2 + 128967743626202780928*T + 816171631707942514944
$97$
\( T^{12} + 48624 T^{10} + \cdots + 11\!\cdots\!00 \)
T^12 + 48624*T^10 + 783676032*T^8 + 4513801706496*T^6 + 4678831632208896*T^4 + 893569269968732160*T^2 + 117070399866470400
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