[N,k,chi] = [300,5,Mod(157,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.157");
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 \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
\(n\)
\(101\)
\(151\)
\(277\)
\(\chi(n)\)
\(1\)
\(1\)
\(\beta_{1}\)
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
\( T_{7}^{8} - 140 T_{7}^{7} + 9800 T_{7}^{6} - 245480 T_{7}^{5} + 3113188 T_{7}^{4} - 69856960 T_{7}^{3} + 9400947200 T_{7}^{2} - 172367518720 T_{7} + 1580189787136 \)
T7^8 - 140*T7^7 + 9800*T7^6 - 245480*T7^5 + 3113188*T7^4 - 69856960*T7^3 + 9400947200*T7^2 - 172367518720*T7 + 1580189787136
acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} \)
T^8
$3$
\( (T^{4} + 729)^{2} \)
(T^4 + 729)^2
$5$
\( T^{8} \)
T^8
$7$
\( T^{8} - 140 T^{7} + \cdots + 1580189787136 \)
T^8 - 140*T^7 + 9800*T^6 - 245480*T^5 + 3113188*T^4 - 69856960*T^3 + 9400947200*T^2 - 172367518720*T + 1580189787136
$11$
\( (T^{4} - 144 T^{3} - 30374 T^{2} + \cdots + 311125216)^{2} \)
(T^4 - 144*T^3 - 30374*T^2 + 2512176*T + 311125216)^2
$13$
\( T^{8} + 300 T^{7} + \cdots + 67\!\cdots\!00 \)
T^8 + 300*T^7 + 45000*T^6 + 1215000*T^5 + 7332412500*T^4 + 2045439000000*T^3 + 284411250000000*T^2 + 19553541750000000*T + 672162221025000000
$17$
\( T^{8} - 1020 T^{7} + \cdots + 35\!\cdots\!96 \)
T^8 - 1020*T^7 + 520200*T^6 - 151983240*T^5 + 26901465428*T^4 - 2462232259680*T^3 + 66787209676800*T^2 + 6902530053603840*T + 356691957722668096
$19$
\( T^{8} + 235000 T^{6} + \cdots + 56\!\cdots\!00 \)
T^8 + 235000*T^6 + 18265050000*T^4 + 553343536000000*T^2 + 5658689440000000000
$23$
\( T^{8} + 1320 T^{7} + \cdots + 69\!\cdots\!56 \)
T^8 + 1320*T^7 + 871200*T^6 + 246118560*T^5 + 54945885968*T^4 + 26459061377280*T^3 + 17344277950924800*T^2 + 4925226014150922240*T + 699304155500331667456
$29$
\( T^{8} + 3665716 T^{6} + \cdots + 33\!\cdots\!56 \)
T^8 + 3665716*T^6 + 4621594592196*T^4 + 2281077411268531456*T^2 + 336655065031937929056256
$31$
\( (T^{4} - 736 T^{3} + \cdots - 1071194996864)^{2} \)
(T^4 - 736*T^3 - 2785464*T^2 + 3533902784*T - 1071194996864)^2
$37$
\( T^{8} - 300 T^{7} + \cdots + 28\!\cdots\!96 \)
T^8 - 300*T^7 + 45000*T^6 - 3152007000*T^5 + 32639542595028*T^4 - 29215074875745600*T^3 + 12263317109971920000*T^2 + 8363842992355174300800*T + 2852159369828484093880896
$41$
\( (T^{4} + 1740 T^{3} + \cdots - 3221711600000)^{2} \)
(T^4 + 1740*T^3 - 9134900*T^2 - 16008792000*T - 3221711600000)^2
$43$
\( T^{8} - 6360 T^{7} + \cdots + 17\!\cdots\!36 \)
T^8 - 6360*T^7 + 20224800*T^6 - 28671180480*T^5 + 105215920711488*T^4 - 535202802013271040*T^3 + 1686937162657067827200*T^2 - 2434545992742057436446720*T + 1756738283434595878062145536
$47$
\( T^{8} + 4800 T^{7} + \cdots + 24\!\cdots\!76 \)
T^8 + 4800*T^7 + 11520000*T^6 + 12965788800*T^5 + 10695518268048*T^4 + 16498566355622400*T^3 + 40036587662177280000*T^2 + 44719479738307607961600*T + 24975053879955828808421376
$53$
\( T^{8} + 3900 T^{7} + \cdots + 43\!\cdots\!36 \)
T^8 + 3900*T^7 + 7605000*T^6 - 50409958200*T^5 + 157106201387588*T^4 - 16727644443098400*T^3 + 10551467982183120000*T^2 - 96017729575512193900800*T + 436877807353620747753031936
$59$
\( T^{8} + 34454956 T^{6} + \cdots + 17\!\cdots\!16 \)
T^8 + 34454956*T^6 + 262928039331876*T^4 + 414975152985534645376*T^2 + 178099549083968066939708416
$61$
\( (T^{4} + 5772 T^{3} + \cdots - 162789583002624)^{2} \)
(T^4 + 5772*T^3 - 23280756*T^2 - 160960390272*T - 162789583002624)^2
$67$
\( T^{8} - 920 T^{7} + \cdots + 41\!\cdots\!96 \)
T^8 - 920*T^7 + 423200*T^6 - 99690050240*T^5 + 1272768375531328*T^4 - 4482326578732180480*T^3 + 8554157934335610060800*T^2 - 8462302025126405191106560*T + 4185716239644127750739869696
$71$
\( (T^{4} + 1800 T^{3} + \cdots - 301327179200000)^{2} \)
(T^4 + 1800*T^3 - 58485200*T^2 - 275496864000*T - 301327179200000)^2
$73$
\( T^{8} + 2960 T^{7} + \cdots + 66\!\cdots\!36 \)
T^8 + 2960*T^7 + 4380800*T^6 - 144735141280*T^5 + 1128904875279688*T^4 - 1510300402711578560*T^3 + 1058134891019250291200*T^2 - 375317133615183877089920*T + 66561906227962723543461136
$79$
\( T^{8} + 156443664 T^{6} + \cdots + 71\!\cdots\!36 \)
T^8 + 156443664*T^6 + 7957202957833536*T^4 + 148208390456504405059584*T^2 + 717586080257978299162383630336
$83$
\( T^{8} + 12720 T^{7} + \cdots + 14\!\cdots\!96 \)
T^8 + 12720*T^7 + 80899200*T^6 + 152003655360*T^5 + 150847816022928*T^4 + 51175335116436480*T^3 + 38246079843532800*T^2 - 1040717811311044853760*T + 14159536967070321518186496
$89$
\( T^{8} + 112438600 T^{6} + \cdots + 36\!\cdots\!00 \)
T^8 + 112438600*T^6 + 4440915658410000*T^4 + 71127492316056400000000*T^2 + 366433549440002500000000000000
$97$
\( T^{8} - 15600 T^{7} + \cdots + 19\!\cdots\!56 \)
T^8 - 15600*T^7 + 121680000*T^6 - 343976407200*T^5 + 3866571291108168*T^4 - 50038563220316568000*T^3 + 369277075890006684480000*T^2 - 1192391606029346287904937600*T + 1925109673681371656856860714256
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