↑

Properties

Label 171.2.e.a
Level $171$
Weight $2$
Character orbit 171.e
Analytic conductor $1.365$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,2,Mod(58,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.58");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.e (of order \(3\), degree \(2\), 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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 150 x^{14} - 237 x^{13} + 816 x^{12} - 966 x^{11} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_1) q^{2} + (\beta_{10} + \beta_{2}) q^{3} + ( - \beta_{17} + \beta_{11} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - \beta_{16} + \beta_{11} + \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_1) q^{2} + (\beta_{10} + \beta_{2}) q^{3} + ( - \beta_{17} + \beta_{11} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - \beta_{16} + \beta_{14} + \beta_{12} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} - q^{3} - 9 q^{4} - 9 q^{5} - 3 q^{6} + 12 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} - q^{3} - 9 q^{4} - 9 q^{5} - 3 q^{6} + 12 q^{8} + 5 q^{9} - 12 q^{11} - 16 q^{12} - 3 q^{14} - 9 q^{15} + 3 q^{16} + 24 q^{17} + 18 q^{18} + 18 q^{19} - 18 q^{20} - 14 q^{21} - 15 q^{23} + 12 q^{24} - 18 q^{25} + 36 q^{26} + 2 q^{27} - 24 q^{29} - 24 q^{30} + 3 q^{31} - 15 q^{32} + 12 q^{33} + 18 q^{35} + 35 q^{36} - 24 q^{37} - 3 q^{38} - 11 q^{39} - 24 q^{41} + 3 q^{42} + 6 q^{43} + 72 q^{44} + 6 q^{45} - 15 q^{47} - 46 q^{48} - 3 q^{49} - 15 q^{50} + 12 q^{51} - 15 q^{52} + 42 q^{53} + 54 q^{54} + 6 q^{55} + 15 q^{56} - q^{57} + 3 q^{58} - 9 q^{59} - 9 q^{60} + 6 q^{61} - 48 q^{62} + 10 q^{63} - 48 q^{64} - 6 q^{65} - 15 q^{66} + 3 q^{67} - 18 q^{68} + 33 q^{69} + 39 q^{70} + 12 q^{72} + 24 q^{73} - 33 q^{74} - 37 q^{75} - 9 q^{76} - 15 q^{77} + 42 q^{78} - 6 q^{79} + 18 q^{80} - 7 q^{81} + 6 q^{82} - 6 q^{83} - 14 q^{84} - 6 q^{85} - 27 q^{86} - 12 q^{88} + 24 q^{89} + 51 q^{90} - 66 q^{91} + 9 q^{92} - 35 q^{93} + 12 q^{94} - 9 q^{95} - 30 q^{96} + 6 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 18 x^{16} - 33 x^{15} + 150 x^{14} - 237 x^{13} + 816 x^{12} - 966 x^{11} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 46\!\cdots\!09 \nu^{17} + \cdots - 18\!\cdots\!17 ) / 94\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!46 \nu^{17} + \cdots + 59\!\cdots\!01 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!83 \nu^{17} + \cdots + 58\!\cdots\!89 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 42\!\cdots\!75 \nu^{17} + \cdots + 32\!\cdots\!53 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!59 \nu^{17} + \cdots + 76\!\cdots\!53 ) / 94\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{17} + \cdots + 19\!\cdots\!35 ) / 94\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 70\!\cdots\!07 \nu^{17} + \cdots + 22\!\cdots\!37 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!93 \nu^{17} + \cdots + 10\!\cdots\!09 ) / 94\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!81 \nu^{17} + \cdots + 10\!\cdots\!27 ) / 94\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 72\!\cdots\!69 \nu^{17} + \cdots - 16\!\cdots\!33 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!91 \nu^{17} + \cdots - 72\!\cdots\!90 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!21 \nu^{17} + \cdots - 51\!\cdots\!06 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!37 \nu^{17} + \cdots + 33\!\cdots\!89 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!89 \nu^{17} + \cdots + 39\!\cdots\!10 ) / 47\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!81 \nu^{17} + \cdots + 64\!\cdots\!03 ) / 31\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 72\!\cdots\!69 \nu^{17} + \cdots + 15\!\cdots\!33 ) / 10\!\cdots\!22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - 3\beta_{11} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{10} + \beta_{7} + 4\beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{17} + \beta_{12} + 14\beta_{11} - 8\beta_{10} + \beta_{9} - \beta_{8} + 6\beta_{7} - 7\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{17} - \beta_{16} + \beta_{15} + 7 \beta_{14} + \beta_{13} + 11 \beta_{11} - 7 \beta_{10} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} - 2 \beta_{16} + \beta_{15} + 10 \beta_{14} + \beta_{12} + 2 \beta_{11} + 42 \beta_{10} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55 \beta_{17} + \beta_{14} - 9 \beta_{13} - \beta_{12} - 78 \beta_{11} + 107 \beta_{10} + \cdots + 96 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 207 \beta_{17} + 21 \beta_{16} - 9 \beta_{15} - 66 \beta_{14} - 9 \beta_{13} - 42 \beta_{12} + \cdots - 411 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 75 \beta_{17} + 63 \beta_{16} - 57 \beta_{15} - 273 \beta_{14} - 15 \beta_{12} - 63 \beta_{11} + \cdots - 612 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1137 \beta_{17} - 102 \beta_{14} + 54 \beta_{13} + 147 \beta_{12} + 2187 \beta_{11} + \cdots - 609 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2808 \beta_{17} - 408 \beta_{16} + 306 \beta_{15} + 1545 \beta_{14} + 306 \beta_{13} + 252 \beta_{12} + \cdots + 4074 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 813 \beta_{17} - 1068 \beta_{16} + 252 \beta_{15} + 3723 \beta_{14} + 765 \beta_{12} + 1068 \beta_{11} + \cdots + 13644 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14454 \beta_{17} + 1323 \beta_{14} - 1443 \beta_{13} - 699 \beta_{12} - 23751 \beta_{11} + \cdots + 15063 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 46032 \beta_{17} + 6795 \beta_{16} - 810 \beta_{15} - 19038 \beta_{14} - 810 \beta_{13} - 9150 \beta_{12} + \cdots - 80460 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 21366 \beta_{17} + 15867 \beta_{16} - 5751 \beta_{15} - 66510 \beta_{14} - 10440 \beta_{12} + \cdots - 166923 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 241398 \beta_{17} - 37305 \beta_{14} - 378 \beta_{13} + 19395 \beta_{12} + 437373 \beta_{11} + \cdots - 158697 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 694449 \beta_{17} - 97614 \beta_{16} + 15462 \beta_{15} + 340200 \beta_{14} + 15462 \beta_{13} + \cdots + 1047150 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
1.24673 − 2.15939i
1.20534 − 2.08772i
0.909577 − 1.57543i
0.541887 − 0.938575i
0.361114 − 0.625468i
−0.102535 + 0.177596i
−0.608496 + 1.05395i
−0.995207 + 1.72375i
−1.05841 + 1.83322i
1.24673 + 2.15939i
1.20534 + 2.08772i
0.909577 + 1.57543i
0.541887 + 0.938575i
0.361114 + 0.625468i
−0.102535 − 0.177596i
−0.608496 − 1.05395i
−0.995207 − 1.72375i
−1.05841 − 1.83322i
−1.24673 − 2.15939i 1.65054 + 0.525095i −2.10865 + 3.65229i −1.98711 + 3.44177i −0.923881 − 4.21881i 1.27190 + 2.20299i 5.52873 2.44855 + 1.73338i 9.90951
58.2 −1.20534 − 2.08772i 0.722909 − 1.57398i −1.90570 + 3.30078i 1.03888 − 1.79940i −4.15737 + 0.387954i −0.352395 − 0.610366i 4.36674 −1.95481 − 2.27568i −5.00884
58.3 −0.909577 − 1.57543i −0.542417 + 1.64493i −0.654662 + 1.13391i −0.845876 + 1.46510i 3.08484 − 0.641647i −2.49887 − 4.32816i −1.25645 −2.41157 − 1.78447i 3.07756
58.4 −0.541887 − 0.938575i 1.23779 + 1.21156i 0.412718 − 0.714848i 1.14985 − 1.99159i 0.466398 − 1.81829i 0.839351 + 1.45380i −3.06213 0.0642459 + 2.99931i −2.49234
58.5 −0.361114 − 0.625468i −1.68928 + 0.382518i 0.739193 − 1.28032i −1.83658 + 3.18105i 0.849277 + 0.918461i 1.04195 + 1.80471i −2.51219 2.70736 − 1.29236i 2.65286
58.6 0.102535 + 0.177596i −0.685888 − 1.59046i 0.978973 − 1.69563i −0.487677 + 0.844681i 0.212132 − 0.284889i −1.50630 − 2.60898i 0.811659 −2.05911 + 2.18175i −0.200016
58.7 0.608496 + 1.05395i −1.69921 − 0.335699i 0.259465 − 0.449407i 0.854539 − 1.48010i −0.680153 − 1.99514i 1.28695 + 2.22907i 3.06552 2.77461 + 1.14084i 2.07993
58.8 0.995207 + 1.72375i −1.03903 + 1.38579i −0.980875 + 1.69892i −0.327903 + 0.567945i −3.42281 − 0.411887i 1.14086 + 1.97603i 0.0761342 −0.840818 − 2.87976i −1.30533
58.9 1.05841 + 1.83322i 1.54459 − 0.783729i −1.24046 + 2.14854i −2.05812 + 3.56477i 3.07156 + 2.00207i −1.22345 − 2.11909i −1.01801 1.77154 − 2.42109i −8.71334
115.1 −1.24673 + 2.15939i 1.65054 − 0.525095i −2.10865 − 3.65229i −1.98711 − 3.44177i −0.923881 + 4.21881i 1.27190 − 2.20299i 5.52873 2.44855 − 1.73338i 9.90951
115.2 −1.20534 + 2.08772i 0.722909 + 1.57398i −1.90570 − 3.30078i 1.03888 + 1.79940i −4.15737 − 0.387954i −0.352395 + 0.610366i 4.36674 −1.95481 + 2.27568i −5.00884
115.3 −0.909577 + 1.57543i −0.542417 − 1.64493i −0.654662 − 1.13391i −0.845876 − 1.46510i 3.08484 + 0.641647i −2.49887 + 4.32816i −1.25645 −2.41157 + 1.78447i 3.07756
115.4 −0.541887 + 0.938575i 1.23779 − 1.21156i 0.412718 + 0.714848i 1.14985 + 1.99159i 0.466398 + 1.81829i 0.839351 − 1.45380i −3.06213 0.0642459 − 2.99931i −2.49234
115.5 −0.361114 + 0.625468i −1.68928 − 0.382518i 0.739193 + 1.28032i −1.83658 − 3.18105i 0.849277 − 0.918461i 1.04195 − 1.80471i −2.51219 2.70736 + 1.29236i 2.65286
115.6 0.102535 − 0.177596i −0.685888 + 1.59046i 0.978973 + 1.69563i −0.487677 − 0.844681i 0.212132 + 0.284889i −1.50630 + 2.60898i 0.811659 −2.05911 − 2.18175i −0.200016
115.7 0.608496 − 1.05395i −1.69921 + 0.335699i 0.259465 + 0.449407i 0.854539 + 1.48010i −0.680153 + 1.99514i 1.28695 − 2.22907i 3.06552 2.77461 − 1.14084i 2.07993
115.8 0.995207 − 1.72375i −1.03903 − 1.38579i −0.980875 − 1.69892i −0.327903 − 0.567945i −3.42281 + 0.411887i 1.14086 − 1.97603i 0.0761342 −0.840818 + 2.87976i −1.30533
115.9 1.05841 − 1.83322i 1.54459 + 0.783729i −1.24046 − 2.14854i −2.05812 − 3.56477i 3.07156 − 2.00207i −1.22345 + 2.11909i −1.01801 1.77154 + 2.42109i −8.71334
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.e.a ✓ 18
3.b odd 2 1 513.2.e.b 18
9.c even 3 1 inner 171.2.e.a ✓ 18
9.c even 3 1 1539.2.a.q 9
9.d odd 6 1 513.2.e.b 18
9.d odd 6 1 1539.2.a.n 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.e.a ✓ 18 1.a even 1 1 trivial
171.2.e.a ✓ 18 9.c even 3 1 inner
513.2.e.b 18 3.b odd 2 1
513.2.e.b 18 9.d odd 6 1
1539.2.a.n 9 9.d odd 6 1
1539.2.a.q 9 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} + 18 T_{2}^{16} + 33 T_{2}^{15} + 150 T_{2}^{14} + 237 T_{2}^{13} + 816 T_{2}^{12} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{18} + T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} + 9 T^{17} + \cdots + 281961 \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + \cdots + 1841449 \) Copy content Toggle raw display
$11$ \( T^{18} + 12 T^{17} + \cdots + 164025 \) Copy content Toggle raw display
$13$ \( T^{18} + 54 T^{16} + \cdots + 600625 \) Copy content Toggle raw display
$17$ \( (T^{9} - 12 T^{8} + \cdots + 75492)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 28333978929 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 16119603369 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 84129582601 \) Copy content Toggle raw display
$37$ \( (T^{9} + 12 T^{8} + \cdots + 189172)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 55471825139481 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 403559079169 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2830133601 \) Copy content Toggle raw display
$53$ \( (T^{9} - 21 T^{8} + \cdots - 19116)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 7410627225 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 46262728392241 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 19\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{9} - 267 T^{7} + \cdots + 5854032)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 12 T^{8} + \cdots + 2073724)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 998270748225 \) Copy content Toggle raw display
$89$ \( (T^{9} - 12 T^{8} + \cdots + 14801940)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
show more
show less