[N,k,chi] = [603,2,Mod(37,603)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(603, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("603.37");
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 \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).
\(n\)
\(136\)
\(470\)
\(\chi(n)\)
\(-\beta_{5}\)
\(1\)
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
\( T_{2}^{10} + 8T_{2}^{8} + 57T_{2}^{6} - T_{2}^{5} + 56T_{2}^{4} - 16T_{2}^{3} + 49T_{2}^{2} - 7T_{2} + 1 \)
T2^10 + 8*T2^8 + 57*T2^6 - T2^5 + 56*T2^4 - 16*T2^3 + 49*T2^2 - 7*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} + 8 T^{8} + 57 T^{6} - T^{5} + \cdots + 1 \)
T^10 + 8*T^8 + 57*T^6 - T^5 + 56*T^4 - 16*T^3 + 49*T^2 - 7*T + 1
$3$
\( T^{10} \)
T^10
$5$
\( (T^{5} + 3 T^{4} - 6 T^{3} - 11 T^{2} + 16 T - 4)^{2} \)
(T^5 + 3*T^4 - 6*T^3 - 11*T^2 + 16*T - 4)^2
$7$
\( T^{10} + T^{9} + 10 T^{8} - 9 T^{7} + \cdots + 1 \)
T^10 + T^9 + 10*T^8 - 9*T^7 + 77*T^6 - 9*T^5 + 37*T^4 - 18*T^3 + 16*T^2 - 4*T + 1
$11$
\( T^{10} + 6 T^{9} + 53 T^{8} + 230 T^{7} + \cdots + 64 \)
T^10 + 6*T^9 + 53*T^8 + 230*T^7 + 1553*T^6 + 6030*T^5 + 23048*T^4 + 44216*T^3 + 70496*T^2 + 2144*T + 64
$13$
\( T^{10} + T^{9} + 26 T^{8} + 35 T^{7} + \cdots + 1 \)
T^10 + T^9 + 26*T^8 + 35*T^7 + 665*T^6 + 769*T^5 + 651*T^4 + 250*T^3 + 70*T^2 + 10*T + 1
$17$
\( T^{10} + 8 T^{9} + 80 T^{8} + \cdots + 208849 \)
T^10 + 8*T^9 + 80*T^8 + 370*T^7 + 2863*T^6 + 13367*T^5 + 55817*T^4 + 138511*T^3 + 264432*T^2 + 281055*T + 208849
$19$
\( T^{10} + 5 T^{9} + 104 T^{8} + \cdots + 37613689 \)
T^10 + 5*T^9 + 104*T^8 + 305*T^7 + 6449*T^6 + 18363*T^5 + 213653*T^4 + 429314*T^3 + 4524314*T^2 + 9457086*T + 37613689
$23$
\( T^{10} - 7 T^{9} + 87 T^{8} + \cdots + 7198489 \)
T^10 - 7*T^9 + 87*T^8 - 314*T^7 + 3181*T^6 - 9601*T^5 + 76453*T^4 - 118938*T^3 + 863919*T^2 - 786119*T + 7198489
$29$
\( T^{10} - 12 T^{9} + 194 T^{8} + \cdots + 3690241 \)
T^10 - 12*T^9 + 194*T^8 - 346*T^7 + 6077*T^6 + 28647*T^5 + 351731*T^4 + 1184927*T^3 + 3497168*T^2 + 4032179*T + 3690241
$31$
\( T^{10} + 12 T^{9} + 181 T^{8} + \cdots + 2166784 \)
T^10 + 12*T^9 + 181*T^8 + 508*T^7 + 6105*T^6 - 4340*T^5 + 245024*T^4 - 355648*T^3 + 1653248*T^2 + 1436672*T + 2166784
$37$
\( T^{10} + 17 T^{9} + 264 T^{8} + \cdots + 47761921 \)
T^10 + 17*T^9 + 264*T^8 + 1637*T^7 + 11979*T^6 + 27529*T^5 + 276049*T^4 + 291962*T^3 + 5294770*T^2 - 7270372*T + 47761921
$41$
\( T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 26122321 \)
T^10 + 13*T^9 + 202*T^8 + 1071*T^7 + 11549*T^6 + 48321*T^5 + 472627*T^4 + 869826*T^3 + 4337350*T^2 - 3628810*T + 26122321
$43$
\( (T^{5} + 2 T^{4} - 79 T^{3} - 131 T^{2} + \cdots + 2032)^{2} \)
(T^5 + 2*T^4 - 79*T^3 - 131*T^2 + 1452*T + 2032)^2
$47$
\( T^{10} - 25 T^{9} + \cdots + 560884489 \)
T^10 - 25*T^9 + 464*T^8 - 5125*T^7 + 48629*T^6 - 335667*T^5 + 2336813*T^4 - 12552826*T^3 + 67220114*T^2 - 212152314*T + 560884489
$53$
\( (T^{5} - 6 T^{4} - 109 T^{3} + 391 T^{2} + \cdots - 1964)^{2} \)
(T^5 - 6*T^4 - 109*T^3 + 391*T^2 + 2340*T - 1964)^2
$59$
\( (T^{5} - 6 T^{4} - 125 T^{3} + 707 T^{2} + \cdots - 20672)^{2} \)
(T^5 - 6*T^4 - 125*T^3 + 707*T^2 + 3872*T - 20672)^2
$61$
\( T^{10} - 9 T^{9} + 162 T^{8} + \cdots + 3916441 \)
T^10 - 9*T^9 + 162*T^8 - 927*T^7 + 16267*T^6 - 105661*T^5 + 520821*T^4 - 1545714*T^3 + 3441904*T^2 - 4460666*T + 3916441
$67$
\( T^{10} - 2 T^{9} + \cdots + 1350125107 \)
T^10 - 2*T^9 + 108*T^8 - 663*T^7 - 609*T^6 - 70902*T^5 - 40803*T^4 - 2976207*T^3 + 32482404*T^2 - 40302242*T + 1350125107
$71$
\( T^{10} + 29 T^{9} + 527 T^{8} + \cdots + 9591409 \)
T^10 + 29*T^9 + 527*T^8 + 5950*T^7 + 49181*T^6 + 286715*T^5 + 1253229*T^4 + 3819518*T^3 + 8457343*T^2 + 11313341*T + 9591409
$73$
\( T^{10} - 12 T^{9} + 122 T^{8} + \cdots + 100489 \)
T^10 - 12*T^9 + 122*T^8 - 462*T^7 + 1833*T^6 - 2003*T^5 + 9539*T^4 - 1991*T^3 + 57304*T^2 + 51037*T + 100489
$79$
\( T^{10} + T^{9} + 232 T^{8} + \cdots + 7678441 \)
T^10 + T^9 + 232*T^8 - 2195*T^7 + 52189*T^6 - 229993*T^5 + 1010985*T^4 - 1093622*T^3 + 2757222*T^2 - 526490*T + 7678441
$83$
\( T^{10} - 6 T^{9} + 82 T^{8} + \cdots + 375769 \)
T^10 - 6*T^9 + 82*T^8 - 22*T^7 + 2277*T^6 + 2555*T^5 + 59597*T^4 + 165613*T^3 + 445952*T^2 + 449329*T + 375769
$89$
\( (T^{5} - 2 T^{4} - 407 T^{3} + \cdots - 130084)^{2} \)
(T^5 - 2*T^4 - 407*T^3 + 1559*T^2 + 34336*T - 130084)^2
$97$
\( T^{10} - 11 T^{9} + 108 T^{8} + \cdots + 3481 \)
T^10 - 11*T^9 + 108*T^8 - 463*T^7 + 2305*T^6 - 6251*T^5 + 29839*T^4 - 58626*T^3 + 150816*T^2 + 22184*T + 3481
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