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Properties

Label 190.2.k.d
Level $190$
Weight $2$
Character orbit 190.k
Analytic conductor $1.517$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [190,2,Mod(61,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(18))
 
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("190.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 190.k (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 24 x^{16} - 12 x^{15} + 393 x^{14} - 222 x^{13} + 3518 x^{12} - 2478 x^{11} + 22809 x^{10} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} - \beta_{7} q^{3} - \beta_{11} q^{4} + \beta_{10} q^{5} + \beta_{5} q^{6} + \beta_{17} q^{7} + \beta_{14} q^{8} + (\beta_{15} - \beta_{14} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} - \beta_{7} q^{3} - \beta_{11} q^{4} + \beta_{10} q^{5} + \beta_{5} q^{6} + \beta_{17} q^{7} + \beta_{14} q^{8} + (\beta_{15} - \beta_{14} + \cdots - \beta_{3}) q^{9}+ \cdots + ( - \beta_{17} - 3 \beta_{16} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{8} - 18 q^{9} - 12 q^{11} - 6 q^{13} + 6 q^{14} - 42 q^{18} + 18 q^{20} + 12 q^{21} - 3 q^{22} + 9 q^{23} - 9 q^{26} - 18 q^{27} + 3 q^{28} - 6 q^{29} - 6 q^{31} + 66 q^{33} + 18 q^{34} + 3 q^{35} - 18 q^{36} - 12 q^{37} - 6 q^{38} + 48 q^{39} - 21 q^{41} + 42 q^{42} + 18 q^{43} + 3 q^{44} - 21 q^{45} + 18 q^{46} - 54 q^{47} - 39 q^{49} + 9 q^{50} + 42 q^{51} + 12 q^{52} - 24 q^{53} - 54 q^{54} - 6 q^{55} - 18 q^{57} - 30 q^{59} + 48 q^{61} - 30 q^{62} - 57 q^{63} - 9 q^{64} + 9 q^{65} + 24 q^{66} - 6 q^{67} - 6 q^{68} - 30 q^{69} - 3 q^{70} + 30 q^{71} + 6 q^{73} - 3 q^{74} - 21 q^{76} + 30 q^{77} - 24 q^{78} + 30 q^{79} + 18 q^{81} + 21 q^{82} + 6 q^{83} + 6 q^{84} + 36 q^{86} + 24 q^{87} + 12 q^{88} + 30 q^{89} - 60 q^{91} - 18 q^{92} - 12 q^{93} + 6 q^{94} - 12 q^{95} - 12 q^{97} - 18 q^{98} + 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 24 x^{16} - 12 x^{15} + 393 x^{14} - 222 x^{13} + 3518 x^{12} - 2478 x^{11} + 22809 x^{10} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61\!\cdots\!55 \nu^{17} + \cdots - 40\!\cdots\!72 ) / 96\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!17 \nu^{17} + \cdots - 33\!\cdots\!04 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!97 \nu^{17} + \cdots + 27\!\cdots\!28 ) / 96\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 56\!\cdots\!89 \nu^{17} + \cdots - 58\!\cdots\!72 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!08 \nu^{17} + \cdots - 35\!\cdots\!36 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71\!\cdots\!45 \nu^{17} + \cdots + 27\!\cdots\!92 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!98 \nu^{17} + \cdots + 22\!\cdots\!20 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!28 \nu^{17} + \cdots - 14\!\cdots\!12 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 86\!\cdots\!79 \nu^{17} + \cdots + 22\!\cdots\!24 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91\!\cdots\!23 \nu^{17} + \cdots - 72\!\cdots\!92 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!53 \nu^{17} + \cdots - 51\!\cdots\!08 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19\!\cdots\!21 \nu^{17} + \cdots - 81\!\cdots\!08 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!13 \nu^{17} + \cdots - 20\!\cdots\!12 ) / 75\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!29 \nu^{17} + \cdots + 41\!\cdots\!68 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!70 \nu^{17} + \cdots + 13\!\cdots\!24 ) / 19\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!41 \nu^{17} + \cdots - 36\!\cdots\!28 ) / 19\!\cdots\!96 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{16} + \beta_{15} + 5\beta_{14} - \beta_{13} - \beta_{11} + \beta_{9} - \beta_{4} - \beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 8 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{17} + 10 \beta_{16} - 9 \beta_{15} - 43 \beta_{14} + \beta_{13} - 10 \beta_{12} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{17} - 2 \beta_{15} + 38 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 24 \beta_{11} + 34 \beta_{10} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{16} - 16 \beta_{15} + 86 \beta_{13} + 86 \beta_{12} + 147 \beta_{11} - 147 \beta_{10} + \cdots + 420 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32 \beta_{17} - 36 \beta_{16} - 4 \beta_{15} - 534 \beta_{14} - 40 \beta_{13} + 36 \beta_{12} + \cdots + 783 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1194 \beta_{17} - 856 \beta_{16} + 1037 \beta_{15} + 4366 \beta_{14} - 1037 \beta_{13} + 181 \beta_{12} + \cdots - 4366 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 576 \beta_{16} + 576 \beta_{15} + 78 \beta_{13} + 78 \beta_{12} + 8696 \beta_{11} - 8696 \beta_{10} + \cdots + 6786 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12424 \beta_{17} + 10484 \beta_{16} - 8697 \beta_{15} - 46925 \beta_{14} + 1787 \beta_{13} + \cdots + 8649 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4752 \beta_{17} - 1088 \beta_{16} - 6320 \beta_{15} + 82676 \beta_{14} + 6320 \beta_{13} + \cdots - 82676 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16145 \beta_{16} - 16145 \beta_{15} + 89314 \beta_{13} + 89314 \beta_{12} + 271532 \beta_{11} + \cdots + 513982 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 55406 \beta_{17} - 77236 \beta_{16} - 13592 \beta_{15} - 988598 \beta_{14} - 90828 \beta_{13} + \cdots + 1013533 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1359838 \beta_{17} - 922973 \beta_{16} + 1055883 \beta_{15} + 5695827 \beta_{14} - 1055883 \beta_{13} + \cdots - 5695827 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1089148 \beta_{16} + 1089148 \beta_{15} + 163696 \beta_{13} + 163696 \beta_{12} + 14312704 \beta_{11} + \cdots + 11723112 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 14303403 \beta_{17} + 10521571 \beta_{16} - 9578681 \beta_{15} - 63623749 \beta_{14} + \cdots + 17714382 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7586424 \beta_{17} - 1957914 \beta_{16} - 10959612 \beta_{15} + 138514320 \beta_{14} + \cdots - 138514320 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-\beta_{11}\) \(1\)

Embeddings

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
1.60792 + 2.78500i
−0.0180720 − 0.0313015i
−1.58985 − 2.75370i
1.60792 − 2.78500i
−0.0180720 + 0.0313015i
−1.58985 + 2.75370i
−0.897451 − 1.55443i
−0.288205 − 0.499186i
1.18566 + 2.05362i
−0.897451 + 1.55443i
−0.288205 + 0.499186i
1.18566 − 2.05362i
1.15990 − 2.00901i
0.554587 − 0.960572i
−1.71449 + 2.96958i
1.15990 + 2.00901i
0.554587 + 0.960572i
−1.71449 − 2.96958i
0.939693 + 0.342020i −0.558424 − 3.16698i 0.766044 + 0.642788i 0.766044 − 0.642788i 0.558424 − 3.16698i 0.0116976 − 0.0202608i 0.500000 + 0.866025i −6.89886 + 2.51098i 0.939693 − 0.342020i
61.2 0.939693 + 0.342020i 0.00627632 + 0.0355948i 0.766044 + 0.642788i 0.766044 − 0.642788i −0.00627632 + 0.0355948i −0.918706 + 1.59124i 0.500000 + 0.866025i 2.81785 − 1.02561i 0.939693 − 0.342020i
61.3 0.939693 + 0.342020i 0.552148 + 3.13139i 0.766044 + 0.642788i 0.766044 − 0.642788i −0.552148 + 3.13139i 1.67305 − 2.89781i 0.500000 + 0.866025i −6.68164 + 2.43192i 0.939693 − 0.342020i
81.1 0.939693 − 0.342020i −0.558424 + 3.16698i 0.766044 − 0.642788i 0.766044 + 0.642788i 0.558424 + 3.16698i 0.0116976 + 0.0202608i 0.500000 − 0.866025i −6.89886 − 2.51098i 0.939693 + 0.342020i
81.2 0.939693 − 0.342020i 0.00627632 − 0.0355948i 0.766044 − 0.642788i 0.766044 + 0.642788i −0.00627632 − 0.0355948i −0.918706 − 1.59124i 0.500000 − 0.866025i 2.81785 + 1.02561i 0.939693 + 0.342020i
81.3 0.939693 − 0.342020i 0.552148 − 3.13139i 0.766044 − 0.642788i 0.766044 + 0.642788i −0.552148 − 3.13139i 1.67305 + 2.89781i 0.500000 − 0.866025i −6.68164 − 2.43192i 0.939693 + 0.342020i
101.1 −0.766044 + 0.642788i −1.68666 − 0.613893i 0.173648 − 0.984808i 0.173648 + 0.984808i 1.68666 − 0.613893i −0.680736 + 1.17907i 0.500000 + 0.866025i 0.169813 + 0.142490i −0.766044 − 0.642788i
101.2 −0.766044 + 0.642788i −0.541649 − 0.197144i 0.173648 − 0.984808i 0.173648 + 0.984808i 0.541649 − 0.197144i 2.43209 − 4.21251i 0.500000 + 0.866025i −2.04362 − 1.71480i −0.766044 − 0.642788i
101.3 −0.766044 + 0.642788i 2.22831 + 0.811037i 0.173648 − 0.984808i 0.173648 + 0.984808i −2.22831 + 0.811037i −1.57771 + 2.73267i 0.500000 + 0.866025i 2.00943 + 1.68611i −0.766044 − 0.642788i
111.1 −0.766044 − 0.642788i −1.68666 + 0.613893i 0.173648 + 0.984808i 0.173648 − 0.984808i 1.68666 + 0.613893i −0.680736 − 1.17907i 0.500000 − 0.866025i 0.169813 − 0.142490i −0.766044 + 0.642788i
111.2 −0.766044 − 0.642788i −0.541649 + 0.197144i 0.173648 + 0.984808i 0.173648 − 0.984808i 0.541649 + 0.197144i 2.43209 + 4.21251i 0.500000 − 0.866025i −2.04362 + 1.71480i −0.766044 + 0.642788i
111.3 −0.766044 − 0.642788i 2.22831 − 0.811037i 0.173648 + 0.984808i 0.173648 − 0.984808i −2.22831 − 0.811037i −1.57771 − 2.73267i 0.500000 − 0.866025i 2.00943 − 1.68611i −0.766044 + 0.642788i
131.1 −0.173648 + 0.984808i −1.77707 − 1.49114i −0.939693 − 0.342020i −0.939693 + 0.342020i 1.77707 − 1.49114i 2.45837 + 4.25802i 0.500000 − 0.866025i 0.413538 + 2.34529i −0.173648 − 0.984808i
131.2 −0.173648 + 0.984808i −0.849676 − 0.712963i −0.939693 − 0.342020i −0.939693 + 0.342020i 0.849676 − 0.712963i −2.46456 − 4.26875i 0.500000 − 0.866025i −0.307311 − 1.74285i −0.173648 − 0.984808i
131.3 −0.173648 + 0.984808i 2.62675 + 2.20410i −0.939693 − 0.342020i −0.939693 + 0.342020i −2.62675 + 2.20410i −0.933500 − 1.61687i 0.500000 − 0.866025i 1.52078 + 8.62480i −0.173648 − 0.984808i
161.1 −0.173648 − 0.984808i −1.77707 + 1.49114i −0.939693 + 0.342020i −0.939693 − 0.342020i 1.77707 + 1.49114i 2.45837 − 4.25802i 0.500000 + 0.866025i 0.413538 − 2.34529i −0.173648 + 0.984808i
161.2 −0.173648 − 0.984808i −0.849676 + 0.712963i −0.939693 + 0.342020i −0.939693 − 0.342020i 0.849676 + 0.712963i −2.46456 + 4.26875i 0.500000 + 0.866025i −0.307311 + 1.74285i −0.173648 + 0.984808i
161.3 −0.173648 − 0.984808i 2.62675 − 2.20410i −0.939693 + 0.342020i −0.939693 − 0.342020i −2.62675 − 2.20410i −0.933500 + 1.61687i 0.500000 + 0.866025i 1.52078 − 8.62480i −0.173648 + 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.k.d ✓ 18
5.b even 2 1 950.2.l.i 18
5.c odd 4 2 950.2.u.g 36
19.e even 9 1 inner 190.2.k.d ✓ 18
19.e even 9 1 3610.2.a.bi 9
19.f odd 18 1 3610.2.a.bj 9
95.p even 18 1 950.2.l.i 18
95.q odd 36 2 950.2.u.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.d ✓ 18 1.a even 1 1 trivial
190.2.k.d ✓ 18 19.e even 9 1 inner
950.2.l.i 18 5.b even 2 1
950.2.l.i 18 95.p even 18 1
950.2.u.g 36 5.c odd 4 2
950.2.u.g 36 95.q odd 36 2
3610.2.a.bi 9 19.e even 9 1
3610.2.a.bj 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 9 T_{3}^{16} + 6 T_{3}^{15} + 36 T_{3}^{14} + 216 T_{3}^{13} + 788 T_{3}^{12} + 1422 T_{3}^{11} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} + 9 T^{16} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 51 T^{16} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 5079555441 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 183439936 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 205291584 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 121352256 \) Copy content Toggle raw display
$29$ \( T^{18} + 6 T^{17} + \cdots + 95883264 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 533794816 \) Copy content Toggle raw display
$37$ \( (T^{9} + 6 T^{8} + \cdots + 25992)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 14058877733289 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 448422976 \) Copy content Toggle raw display
$47$ \( T^{18} + 54 T^{17} + \cdots + 1871424 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 24208870464 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 11195428245849 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 261685448704 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 170103363047424 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 35070801984 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 33597696507904 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 57361719357504 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 156263975312481 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 3392829504 \) Copy content Toggle raw display
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