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Properties

Label 639.2.f.c
Level $639$
Weight $2$
Character orbit 639.f
Analytic conductor $5.102$
Analytic rank $0$
Dimension $20$
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] = [639,2,Mod(199,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([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("639.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.f (of order \(5\), degree \(4\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + 28 x^{18} - 91 x^{17} + 268 x^{16} - 604 x^{15} + 1278 x^{14} - 1990 x^{13} + \cdots + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 71)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + (\beta_{10} + \beta_{7} + \cdots - \beta_{2}) q^{4}+ \cdots + (\beta_{18} - \beta_{13} - \beta_{10} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + (\beta_{10} + \beta_{7} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - 2 \beta_{18} - 2 \beta_{17} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{2} - 10 q^{4} + q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{2} - 10 q^{4} + q^{5} - q^{7} + q^{8} + 15 q^{10} + q^{11} + 3 q^{13} - 5 q^{14} - 16 q^{16} + 2 q^{17} + 3 q^{19} + 72 q^{20} + 10 q^{22} + 22 q^{23} + 14 q^{25} + 42 q^{26} + 4 q^{28} + q^{29} + 6 q^{31} - 52 q^{32} - 34 q^{34} - 3 q^{35} - 6 q^{37} - 25 q^{38} + 65 q^{40} + 60 q^{41} + 23 q^{43} - 37 q^{44} - 19 q^{46} - 29 q^{47} + 22 q^{49} - 36 q^{50} + 43 q^{52} - 2 q^{53} + 10 q^{55} - 31 q^{56} - 33 q^{58} - 31 q^{59} - 2 q^{61} - 5 q^{62} - 65 q^{64} - 54 q^{65} - 38 q^{67} + 3 q^{68} - 34 q^{70} - 45 q^{71} - 21 q^{73} - 21 q^{74} - q^{76} + 12 q^{77} - 59 q^{79} + 16 q^{80} + 53 q^{82} + 15 q^{83} + 13 q^{85} + 19 q^{86} - 64 q^{88} - 16 q^{89} - 18 q^{91} - 86 q^{92} - 62 q^{94} - 15 q^{95} - 58 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 28 x^{18} - 91 x^{17} + 268 x^{16} - 604 x^{15} + 1278 x^{14} - 1990 x^{13} + \cdots + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!52 \nu^{19} + \cdots + 79\!\cdots\!35 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!24 \nu^{19} + \cdots + 20\!\cdots\!43 ) / 14\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!85 \nu^{19} + \cdots - 55\!\cdots\!85 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!17 \nu^{19} + \cdots - 46\!\cdots\!47 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!14 \nu^{19} + \cdots - 58\!\cdots\!91 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 52\!\cdots\!06 \nu^{19} + \cdots - 11\!\cdots\!37 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 59\!\cdots\!02 \nu^{19} + \cdots - 94\!\cdots\!39 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!87 \nu^{19} + \cdots - 12\!\cdots\!35 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!85 \nu^{19} + \cdots + 40\!\cdots\!67 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 34\!\cdots\!65 \nu^{19} + \cdots - 16\!\cdots\!86 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!50 \nu^{19} + \cdots + 15\!\cdots\!37 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45\!\cdots\!47 \nu^{19} + \cdots + 30\!\cdots\!24 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 46\!\cdots\!36 \nu^{19} + \cdots + 83\!\cdots\!27 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!17 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 93\!\cdots\!90 \nu^{19} + \cdots - 12\!\cdots\!43 ) / 92\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 29\!\cdots\!95 \nu^{19} + \cdots - 40\!\cdots\!25 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 33\!\cdots\!15 \nu^{19} + \cdots + 34\!\cdots\!80 ) / 28\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 40\!\cdots\!73 \nu^{19} + \cdots + 32\!\cdots\!68 ) / 28\!\cdots\!78 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - \beta_{15} - 7 \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} + 14 \beta_{7} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} - 8 \beta_{18} - 2 \beta_{17} - 2 \beta_{14} - \beta_{13} + 8 \beta_{12} - 19 \beta_{11} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{19} - 10 \beta_{18} - \beta_{16} + 10 \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + 10 \beta_{12} + \cdots - 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40 \beta_{18} + 20 \beta_{17} - 20 \beta_{16} + 14 \beta_{15} - 100 \beta_{14} - 54 \beta_{13} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 73 \beta_{19} + 88 \beta_{17} - 73 \beta_{16} + 80 \beta_{15} + 44 \beta_{14} + 79 \beta_{12} + \cdots - 73 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 153 \beta_{19} - 213 \beta_{18} + 174 \beta_{17} + 352 \beta_{15} + 1083 \beta_{14} + 352 \beta_{13} + \cdots - 404 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 160 \beta_{19} + 585 \beta_{18} + 160 \beta_{17} + 484 \beta_{16} + 1596 \beta_{14} + 605 \beta_{13} + \cdots + 1480 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 261 \beta_{19} + 2292 \beta_{18} + 1089 \beta_{16} - 2292 \beta_{15} + 1101 \beta_{14} + 1215 \beta_{13} + \cdots + 3404 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 272 \beta_{18} - 1476 \beta_{17} + 1476 \beta_{16} - 4505 \beta_{15} + 4445 \beta_{14} + 4233 \beta_{13} + \cdots - 4890 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2589 \beta_{19} - 10214 \beta_{17} + 2589 \beta_{16} - 9977 \beta_{15} - 15811 \beta_{14} + \cdots + 2589 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 12566 \beta_{19} - 3081 \beta_{18} - 32671 \beta_{17} - 30332 \beta_{15} - 112847 \beta_{14} + \cdots + 36735 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 53518 \beta_{19} - 100358 \beta_{18} - 53518 \beta_{17} - 22793 \beta_{16} - 183741 \beta_{14} + \cdots - 117737 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 131083 \beta_{19} - 216409 \beta_{18} - 101835 \beta_{16} + 216409 \beta_{15} - 264879 \beta_{14} + \cdots - 343121 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 65070 \beta_{18} + 378645 \beta_{17} - 378645 \beta_{16} + 612180 \beta_{15} - 697043 \beta_{14} + \cdots + 127814 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 868734 \beta_{19} + 1668075 \beta_{17} - 868734 \beta_{16} + 1833024 \beta_{15} + 2037321 \beta_{14} + \cdots - 868734 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 2701758 \beta_{19} + 42838 \beta_{18} + 4173987 \beta_{17} + 4626848 \beta_{15} + 15374951 \beta_{14} + \cdots - 2890770 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(569\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.758630 − 2.33482i
−0.420063 − 1.29282i
0.300331 + 0.924322i
0.561200 + 1.72720i
0.699129 + 2.15170i
−0.758630 + 2.33482i
−0.420063 + 1.29282i
0.300331 − 0.924322i
0.561200 − 1.72720i
0.699129 − 2.15170i
−1.16276 − 0.844794i
−0.363618 − 0.264184i
0.685174 + 0.497808i
1.27286 + 0.924789i
2.18637 + 1.58849i
−1.16276 + 0.844794i
−0.363618 + 0.264184i
0.685174 − 0.497808i
1.27286 − 0.924789i
2.18637 − 1.58849i
−0.758630 + 2.33482i 0 −3.25785 − 2.36696i 0.0372909 − 0.0270934i 0 0.331946 − 1.02163i 4.02570 − 2.92484i 0 0.0349684 + 0.107622i
199.2 −0.420063 + 1.29282i 0 0.123097 + 0.0894354i 2.53185 − 1.83950i 0 −1.31457 + 4.04584i −2.36681 + 1.71959i 0 1.31461 + 4.04595i
199.3 0.300331 − 0.924322i 0 0.853860 + 0.620366i 1.37660 − 1.00016i 0 0.656494 − 2.02048i 2.40241 − 1.74545i 0 −0.511035 − 1.57280i
199.4 0.561200 − 1.72720i 0 −1.05022 − 0.763032i −1.21045 + 0.879441i 0 −0.388444 + 1.19551i 1.03119 − 0.749203i 0 0.839664 + 2.58422i
199.5 0.699129 − 2.15170i 0 −2.52299 − 1.83306i −3.04432 + 2.21183i 0 −0.0944386 + 0.290652i −2.04740 + 1.48752i 0 2.63081 + 8.09681i
289.1 −0.758630 − 2.33482i 0 −3.25785 + 2.36696i 0.0372909 + 0.0270934i 0 0.331946 + 1.02163i 4.02570 + 2.92484i 0 0.0349684 − 0.107622i
289.2 −0.420063 − 1.29282i 0 0.123097 − 0.0894354i 2.53185 + 1.83950i 0 −1.31457 − 4.04584i −2.36681 − 1.71959i 0 1.31461 − 4.04595i
289.3 0.300331 + 0.924322i 0 0.853860 − 0.620366i 1.37660 + 1.00016i 0 0.656494 + 2.02048i 2.40241 + 1.74545i 0 −0.511035 + 1.57280i
289.4 0.561200 + 1.72720i 0 −1.05022 + 0.763032i −1.21045 − 0.879441i 0 −0.388444 − 1.19551i 1.03119 + 0.749203i 0 0.839664 − 2.58422i
289.5 0.699129 + 2.15170i 0 −2.52299 + 1.83306i −3.04432 − 2.21183i 0 −0.0944386 − 0.290652i −2.04740 − 1.48752i 0 2.63081 − 8.09681i
451.1 −1.16276 + 0.844794i 0 0.0202979 − 0.0624704i 0.655415 + 2.01716i 0 −0.917632 + 0.666699i −0.859096 − 2.64402i 0 −2.46618 − 1.79178i
451.2 −0.363618 + 0.264184i 0 −0.555609 + 1.70999i −0.225736 − 0.694745i 0 1.83483 − 1.33308i −0.527502 − 1.62348i 0 0.265622 + 0.192986i
451.3 0.685174 − 0.497808i 0 −0.396383 + 1.21994i −0.467113 − 1.43763i 0 2.86367 − 2.08058i 0.859132 + 2.64414i 0 −1.03572 − 0.752493i
451.4 1.27286 − 0.924789i 0 0.146911 − 0.452144i −0.148297 − 0.456410i 0 −4.12180 + 2.99466i 0.741239 + 2.28130i 0 −0.610844 − 0.443804i
451.5 2.18637 − 1.58849i 0 1.63889 − 5.04397i 0.994748 + 3.06152i 0 0.649950 − 0.472216i −2.75886 − 8.49090i 0 7.03810 + 5.11348i
622.1 −1.16276 − 0.844794i 0 0.0202979 + 0.0624704i 0.655415 − 2.01716i 0 −0.917632 − 0.666699i −0.859096 + 2.64402i 0 −2.46618 + 1.79178i
622.2 −0.363618 − 0.264184i 0 −0.555609 − 1.70999i −0.225736 + 0.694745i 0 1.83483 + 1.33308i −0.527502 + 1.62348i 0 0.265622 − 0.192986i
622.3 0.685174 + 0.497808i 0 −0.396383 − 1.21994i −0.467113 + 1.43763i 0 2.86367 + 2.08058i 0.859132 − 2.64414i 0 −1.03572 + 0.752493i
622.4 1.27286 + 0.924789i 0 0.146911 + 0.452144i −0.148297 + 0.456410i 0 −4.12180 − 2.99466i 0.741239 − 2.28130i 0 −0.610844 + 0.443804i
622.5 2.18637 + 1.58849i 0 1.63889 + 5.04397i 0.994748 − 3.06152i 0 0.649950 + 0.472216i −2.75886 + 8.49090i 0 7.03810 − 5.11348i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.f.c 20
3.b odd 2 1 71.2.c.a ✓ 20
71.c even 5 1 inner 639.2.f.c 20
213.h odd 10 1 71.2.c.a ✓ 20
213.h odd 10 1 5041.2.a.i 10
213.i even 10 1 5041.2.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.c.a ✓ 20 3.b odd 2 1
71.2.c.a ✓ 20 213.h odd 10 1
639.2.f.c 20 1.a even 1 1 trivial
639.2.f.c 20 71.c even 5 1 inner
5041.2.a.i 10 213.h odd 10 1
5041.2.a.j 10 213.i even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 6 T_{2}^{19} + 28 T_{2}^{18} - 91 T_{2}^{17} + 268 T_{2}^{16} - 604 T_{2}^{15} + 1278 T_{2}^{14} + \cdots + 961 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 6 T^{19} + \cdots + 961 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - T^{19} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 19321 \) Copy content Toggle raw display
$11$ \( T^{20} - T^{19} + \cdots + 72914521 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 27626096521 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 13374691201 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 375390625 \) Copy content Toggle raw display
$23$ \( (T^{10} - 11 T^{9} + \cdots + 2981269)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} - T^{19} + \cdots + 84842521 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 195132961 \) Copy content Toggle raw display
$37$ \( (T^{10} + 3 T^{9} + \cdots + 1974269)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 30 T^{9} + \cdots + 52111)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 285441025 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2910063025 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 255064391521 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18611898850801 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 422508025 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 373215840252025 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 3714697296025 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11944136448841 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{10} + 29 T^{9} + \cdots + 3911261945)^{2} \) Copy content Toggle raw display
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