Properties

Label 171.2.u.c
Level $171$
Weight $2$
Character orbit 171.u
Analytic conductor $1.365$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{7} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{2} - \zeta_{18} + 1) q^{2} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{7} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{8} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} + 1) q^{10} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}) q^{11} + (3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 2) q^{13} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{14} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{2} - 3) q^{16} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 2 \zeta_{18} - 1) q^{17} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{19} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{20} + 3 \zeta_{18}^{4} q^{22} + ( - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{23} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{25} + (5 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 5) q^{26} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2}) q^{28} + (5 \zeta_{18}^{5} - \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} + 1) q^{29} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} + 3) q^{31} + (3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3) q^{32} + ( - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 4 \zeta_{18} - 2) q^{34} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3}) q^{35} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18}) q^{37} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{38} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 6 \zeta_{18} + 1) q^{40} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4) q^{41} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 2) q^{43} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18} - 1) q^{44} + ( - 2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{47} + ( - \zeta_{18}^{5} + 5 \zeta_{18}^{3} - \zeta_{18}) q^{49} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 2 \zeta_{18} + 5) q^{50} + ( - \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 6 \zeta_{18} + 1) q^{52} + ( - \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{53} + (3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{55} + (5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{56} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} + 6) q^{58} + ( - 2 \zeta_{18}^{2} - 7 \zeta_{18} - 2) q^{59} + ( - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} - 4) q^{61} + ( - 3 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 7 \zeta_{18} + 3) q^{62} + (4 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} - 4) q^{64} + (5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 5 \zeta_{18}^{2}) q^{65} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 4 \zeta_{18} - 4) q^{67} + (4 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 4 \zeta_{18}) q^{68} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18} - 1) q^{70} + ( - 2 \zeta_{18}^{5} - 10 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 10 \zeta_{18} + 2) q^{71} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{3}) q^{73} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + \zeta_{18} - 5) q^{74} + (9 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} + 5) q^{76} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{77} + ( - 7 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 6) q^{79} + (\zeta_{18}^{4} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{80} + ( - 6 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 7 \zeta_{18} - 11) q^{82} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 9 \zeta_{18}^{2} - 6 \zeta_{18}) q^{83} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{85} + ( - 7 \zeta_{18}^{5} + 8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + 7 \zeta_{18}^{2}) q^{86} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 6 \zeta_{18} + 3) q^{88} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18} + 1) q^{89} + (2 \zeta_{18}^{4} + \zeta_{18}^{3} + 5 \zeta_{18}^{2} + \zeta_{18} + 2) q^{91} + ( - 8 \zeta_{18}^{5} + 8 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 6) q^{92} + ( - 2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{94} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 4 \zeta_{18}^{2} - \zeta_{18} - 6) q^{95} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 5 \zeta_{18} + 2) q^{97} + (4 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{5} - 6 q^{8} + 9 q^{10} - 3 q^{13} + 3 q^{14} - 18 q^{16} - 3 q^{17} - 12 q^{19} + 6 q^{20} - 6 q^{23} - 15 q^{26} + 6 q^{28} + 3 q^{29} + 9 q^{31} - 9 q^{32} - 6 q^{35} + 15 q^{38} - 21 q^{41} - 3 q^{43} - 9 q^{44} - 18 q^{46} + 3 q^{47} + 15 q^{49} + 15 q^{50} + 15 q^{52} + 3 q^{53} + 18 q^{55} + 6 q^{56} + 36 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} - 12 q^{64} + 12 q^{65} - 30 q^{67} + 15 q^{68} - 9 q^{70} + 6 q^{71} - 12 q^{73} - 15 q^{74} + 36 q^{76} + 18 q^{77} - 39 q^{79} - 3 q^{80} - 54 q^{82} - 24 q^{86} + 9 q^{88} + 12 q^{89} + 15 q^{91} - 42 q^{92} + 18 q^{94} - 39 q^{95} + 18 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
0.233956 + 1.32683i 0 0.173648 0.0632028i 0.826352 + 0.300767i 0 −0.173648 + 0.300767i 1.47178 + 2.54920i 0 −0.205737 + 1.16679i
55.1 0.233956 1.32683i 0 0.173648 + 0.0632028i 0.826352 0.300767i 0 −0.173648 0.300767i 1.47178 2.54920i 0 −0.205737 1.16679i
73.1 1.93969 + 1.62760i 0 0.766044 + 4.34445i 0.233956 1.32683i 0 −0.766044 1.32683i −3.05303 + 5.28801i 0 2.61334 2.19285i
82.1 1.93969 1.62760i 0 0.766044 4.34445i 0.233956 + 1.32683i 0 −0.766044 + 1.32683i −3.05303 5.28801i 0 2.61334 + 2.19285i
100.1 0.826352 0.300767i 0 −0.939693 + 0.788496i 1.93969 + 1.62760i 0 0.939693 + 1.62760i −1.41875 + 2.45734i 0 2.09240 + 0.761570i
118.1 0.826352 + 0.300767i 0 −0.939693 0.788496i 1.93969 1.62760i 0 0.939693 1.62760i −1.41875 2.45734i 0 2.09240 0.761570i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.1
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 171.2.u.c 6
3.b odd 2 1 19.2.e.a 6
12.b even 2 1 304.2.u.b 6
15.d odd 2 1 475.2.l.a 6
15.e even 4 2 475.2.u.a 12
19.e even 9 1 inner 171.2.u.c 6
19.e even 9 1 3249.2.a.z 3
19.f odd 18 1 3249.2.a.s 3
21.c even 2 1 931.2.w.a 6
21.g even 6 1 931.2.v.a 6
21.g even 6 1 931.2.x.b 6
21.h odd 6 1 931.2.v.b 6
21.h odd 6 1 931.2.x.a 6
57.d even 2 1 361.2.e.h 6
57.f even 6 1 361.2.e.a 6
57.f even 6 1 361.2.e.b 6
57.h odd 6 1 361.2.e.f 6
57.h odd 6 1 361.2.e.g 6
57.j even 18 1 361.2.a.h 3
57.j even 18 2 361.2.c.h 6
57.j even 18 1 361.2.e.a 6
57.j even 18 1 361.2.e.b 6
57.j even 18 1 361.2.e.h 6
57.l odd 18 1 19.2.e.a 6
57.l odd 18 1 361.2.a.g 3
57.l odd 18 2 361.2.c.i 6
57.l odd 18 1 361.2.e.f 6
57.l odd 18 1 361.2.e.g 6
228.u odd 18 1 5776.2.a.bi 3
228.v even 18 1 304.2.u.b 6
228.v even 18 1 5776.2.a.br 3
285.bd odd 18 1 475.2.l.a 6
285.bd odd 18 1 9025.2.a.bd 3
285.bf even 18 1 9025.2.a.x 3
285.bi even 36 2 475.2.u.a 12
399.ca odd 18 1 931.2.v.b 6
399.cb even 18 1 931.2.v.a 6
399.ch odd 18 1 931.2.x.a 6
399.ci even 18 1 931.2.x.b 6
399.cj even 18 1 931.2.w.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 3.b odd 2 1
19.2.e.a 6 57.l odd 18 1
171.2.u.c 6 1.a even 1 1 trivial
171.2.u.c 6 19.e even 9 1 inner
304.2.u.b 6 12.b even 2 1
304.2.u.b 6 228.v even 18 1
361.2.a.g 3 57.l odd 18 1
361.2.a.h 3 57.j even 18 1
361.2.c.h 6 57.j even 18 2
361.2.c.i 6 57.l odd 18 2
361.2.e.a 6 57.f even 6 1
361.2.e.a 6 57.j even 18 1
361.2.e.b 6 57.f even 6 1
361.2.e.b 6 57.j even 18 1
361.2.e.f 6 57.h odd 6 1
361.2.e.f 6 57.l odd 18 1
361.2.e.g 6 57.h odd 6 1
361.2.e.g 6 57.l odd 18 1
361.2.e.h 6 57.d even 2 1
361.2.e.h 6 57.j even 18 1
475.2.l.a 6 15.d odd 2 1
475.2.l.a 6 285.bd odd 18 1
475.2.u.a 12 15.e even 4 2
475.2.u.a 12 285.bi even 36 2
931.2.v.a 6 21.g even 6 1
931.2.v.a 6 399.cb even 18 1
931.2.v.b 6 21.h odd 6 1
931.2.v.b 6 399.ca odd 18 1
931.2.w.a 6 21.c even 2 1
931.2.w.a 6 399.cj even 18 1
931.2.x.a 6 21.h odd 6 1
931.2.x.a 6 399.ch odd 18 1
931.2.x.b 6 21.g even 6 1
931.2.x.b 6 399.ci even 18 1
3249.2.a.s 3 19.f odd 18 1
3249.2.a.z 3 19.e even 9 1
5776.2.a.bi 3 228.u odd 18 1
5776.2.a.br 3 228.v even 18 1
9025.2.a.x 3 285.bf even 18 1
9025.2.a.bd 3 285.bd odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} - 30 T^{3} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$37$ \( (T^{3} - 21 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 21 T^{5} + 162 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 60 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + 84 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{6} + 30 T^{5} + 348 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{6} + 39 T^{5} + 708 T^{4} + \cdots + 654481 \) Copy content Toggle raw display
$83$ \( T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681 \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
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