[N,k,chi] = [600,2,Mod(121,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.121");
S:= CuspForms(chi, 2);
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 \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
\(n\)
\(151\)
\(301\)
\(401\)
\(577\)
\(\chi(n)\)
\(1\)
\(1\)
\(1\)
\(-\beta_{5}\)
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
\( T_{7}^{6} + 2T_{7}^{5} - 16T_{7}^{4} - 37T_{7}^{3} + 41T_{7}^{2} + 140T_{7} + 80 \)
T7^6 + 2*T7^5 - 16*T7^4 - 37*T7^3 + 41*T7^2 + 140*T7 + 80
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \)
(T^4 - T^3 + T^2 - T + 1)^3
$5$
\( T^{12} + 15 T^{9} - 20 T^{8} + \cdots + 15625 \)
T^12 + 15*T^9 - 20*T^8 - 50*T^7 + 75*T^6 - 250*T^5 - 500*T^4 + 1875*T^3 + 15625
$7$
\( (T^{6} + 2 T^{5} - 16 T^{4} - 37 T^{3} + \cdots + 80)^{2} \)
(T^6 + 2*T^5 - 16*T^4 - 37*T^3 + 41*T^2 + 140*T + 80)^2
$11$
\( T^{12} - 10 T^{11} + 87 T^{10} + \cdots + 256 \)
T^12 - 10*T^11 + 87*T^10 - 411*T^9 + 1874*T^8 - 5119*T^7 + 23559*T^6 + 9073*T^5 + 23757*T^4 - 1508*T^3 - 1296*T^2 + 192*T + 256
$13$
\( T^{12} - 10 T^{11} + 47 T^{10} + \cdots + 361 \)
T^12 - 10*T^11 + 47*T^10 - 104*T^9 + 169*T^8 - 346*T^7 + 2339*T^6 - 8998*T^5 + 24072*T^4 - 33142*T^3 + 35554*T^2 + 1938*T + 361
$17$
\( T^{12} + 9 T^{11} + 107 T^{10} + \cdots + 4116841 \)
T^12 + 9*T^11 + 107*T^10 + 828*T^9 + 5728*T^8 + 31543*T^7 + 141183*T^6 + 521793*T^5 + 1673308*T^4 + 4257148*T^3 + 7646247*T^2 + 8138319*T + 4116841
$19$
\( T^{12} - 13 T^{11} + 69 T^{10} + \cdots + 256 \)
T^12 - 13*T^11 + 69*T^10 - 93*T^9 + 32*T^8 - 127*T^7 + 1029*T^6 + 2101*T^5 + 3429*T^4 + 2164*T^3 + 1632*T^2 + 256
$23$
\( T^{12} + 13 T^{11} + 158 T^{10} + \cdots + 160000 \)
T^12 + 13*T^11 + 158*T^10 + 1041*T^9 + 5605*T^8 + 26351*T^7 + 99388*T^6 + 272933*T^5 + 536341*T^4 + 749180*T^3 + 736400*T^2 + 472000*T + 160000
$29$
\( T^{12} - T^{11} + 44 T^{10} - 50 T^{9} + \cdots + 36481 \)
T^12 - T^11 + 44*T^10 - 50*T^9 + 1575*T^8 + 3514*T^7 + 54711*T^6 + 207091*T^5 + 1439275*T^4 + 3744025*T^3 + 13534549*T^2 + 1133776*T + 36481
$31$
\( T^{12} - 3 T^{11} + 34 T^{10} + \cdots + 160000 \)
T^12 - 3*T^11 + 34*T^10 + 3*T^9 + 301*T^8 - 995*T^7 + 10670*T^6 - 41525*T^5 + 96525*T^4 - 138500*T^3 + 208000*T^2 - 240000*T + 160000
$37$
\( T^{12} + 2 T^{11} - 6 T^{10} + \cdots + 990025 \)
T^12 + 2*T^11 - 6*T^10 + 28*T^9 + 2246*T^8 + 18410*T^7 + 107285*T^6 + 381570*T^5 + 1087885*T^4 + 2050000*T^3 + 2988125*T^2 + 2457650*T + 990025
$41$
\( T^{12} - 12 T^{11} + 49 T^{10} + \cdots + 44342281 \)
T^12 - 12*T^11 + 49*T^10 + 143*T^9 + 3802*T^8 - 88843*T^7 + 945784*T^6 - 5047961*T^5 + 15311454*T^4 - 12169729*T^3 + 58692847*T^2 - 29066535*T + 44342281
$43$
\( (T^{6} - 4 T^{5} - 109 T^{4} + 682 T^{3} + \cdots - 16)^{2} \)
(T^6 - 4*T^5 - 109*T^4 + 682*T^3 - 601*T^2 - 924*T - 16)^2
$47$
\( T^{12} - 23 T^{11} + \cdots + 11337990400 \)
T^12 - 23*T^11 + 264*T^10 - 1197*T^9 + 10211*T^8 - 123405*T^7 + 2789000*T^6 - 26969085*T^5 + 193404585*T^4 - 786621000*T^3 + 2155155200*T^2 - 1288408000*T + 11337990400
$53$
\( T^{12} + 14 T^{11} + \cdots + 2663076025 \)
T^12 + 14*T^11 + 26*T^10 - 726*T^9 + 20786*T^8 + 185870*T^7 + 1182575*T^6 + 10065760*T^5 + 169531735*T^4 - 131289650*T^3 + 2236752425*T^2 + 1196719950*T + 2663076025
$59$
\( T^{12} - 30 T^{11} + \cdots + 1097994496 \)
T^12 - 30*T^11 + 613*T^10 - 9328*T^9 + 120649*T^8 - 1233588*T^7 + 10150931*T^6 - 57780396*T^5 + 217805377*T^4 - 471358356*T^3 + 2010144816*T^2 - 2299770944*T + 1097994496
$61$
\( T^{12} - 12 T^{11} + 189 T^{10} + \cdots + 58997761 \)
T^12 - 12*T^11 + 189*T^10 - 657*T^9 + 4582*T^8 - 100983*T^7 + 1585804*T^6 - 9279021*T^5 + 74317174*T^4 - 513875949*T^3 + 2220365707*T^2 + 212648485*T + 58997761
$67$
\( T^{12} + 2 T^{11} + \cdots + 15332382976 \)
T^12 + 2*T^11 - 11*T^10 + 332*T^9 + 22762*T^8 + 275618*T^7 + 3196584*T^6 + 22713116*T^5 + 157571049*T^4 + 782039984*T^3 + 3240285152*T^2 + 8798933440*T + 15332382976
$71$
\( T^{12} - 25 T^{11} + 343 T^{10} + \cdots + 90935296 \)
T^12 - 25*T^11 + 343*T^10 - 3457*T^9 + 33694*T^8 - 248117*T^7 + 1307151*T^6 - 4962134*T^5 + 20513937*T^4 - 52729464*T^3 + 114226496*T^2 - 142429696*T + 90935296
$73$
\( T^{12} - 12 T^{11} + \cdots + 3713805481 \)
T^12 - 12*T^11 + 51*T^10 + 145*T^9 + 20595*T^8 - 181707*T^7 + 2865169*T^6 - 20790282*T^5 + 177585855*T^4 - 914392470*T^3 + 3781018956*T^2 + 1552350093*T + 3713805481
$79$
\( T^{12} - 34 T^{11} + \cdots + 130627307776 \)
T^12 - 34*T^11 + 569*T^10 - 4945*T^9 + 35390*T^8 - 308699*T^7 + 6082451*T^6 - 84312001*T^5 + 840742765*T^4 - 5326865180*T^3 + 25138070944*T^2 - 61002588416*T + 130627307776
$83$
\( T^{12} - 16 T^{11} + 461 T^{10} + \cdots + 65536 \)
T^12 - 16*T^11 + 461*T^10 - 5644*T^9 + 82897*T^8 - 790884*T^7 + 4927621*T^6 - 5736592*T^5 + 19598849*T^4 - 33357792*T^3 + 27583488*T^2 - 2150400*T + 65536
$89$
\( T^{12} - 13 T^{11} + 263 T^{10} + \cdots + 1575025 \)
T^12 - 13*T^11 + 263*T^10 - 3236*T^9 + 36190*T^8 - 282741*T^7 + 1741013*T^6 - 7009213*T^5 + 20663786*T^4 - 19141490*T^3 + 3739065*T^2 + 5390225*T + 1575025
$97$
\( T^{12} - 28 T^{11} + \cdots + 264355081 \)
T^12 - 28*T^11 + 923*T^10 - 12319*T^9 + 169468*T^8 - 877371*T^7 + 3737782*T^6 - 5396029*T^5 - 3078562*T^4 - 727341*T^3 + 155874693*T^2 - 333195687*T + 264355081
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