Properties

Label 171.3.p.c
Level $171$
Weight $3$
Character orbit 171.p
Analytic conductor $4.659$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.92607408.1
Defining polynomial: \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{5} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + ( - 2 \beta_{5} - 10 \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{4} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{5} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + ( - 2 \beta_{5} - 10 \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{8} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{10} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 8) q^{11} + ( - 2 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 1) q^{13} + (\beta_{5} - 8 \beta_{4} + 4 \beta_{3} - \beta_{2} + 8) q^{14} + ( - 2 \beta_{5} + 4 \beta_{4} + 11 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{16} + ( - 2 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{17} + ( - 2 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 5) q^{19} + (10 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 2) q^{20} + (\beta_{5} + 11 \beta_{4} + 8 \beta_{3} - \beta_{2} + 16) q^{22} + (3 \beta_{5} + 3 \beta_{4} - 17 \beta_{3} - 3 \beta_1 - 14) q^{23} + ( - 2 \beta_{4} - 7 \beta_{3} + 2 \beta_1 - 9) q^{25} + (14 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 7 \beta_1 - 14) q^{26} + ( - 4 \beta_{5} + 10 \beta_{4} - 35 \beta_{3} - 10 \beta_1 - 25) q^{28} + ( - 4 \beta_{5} - 2 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} - 2 \beta_1 + 12) q^{29} + (2 \beta_{5} + 31 \beta_{3} + \beta_{2} + 5 \beta_1 + 13) q^{31} + ( - 3 \beta_{4} - 9 \beta_{3} - 3 \beta_1 + 6) q^{32} + (4 \beta_{5} + 18 \beta_{3} + 8 \beta_{2} - 18) q^{34} + ( - 3 \beta_{5} - 7 \beta_{4} - 16 \beta_{3} - 3 \beta_{2} - 14 \beta_1) q^{35} + (8 \beta_{5} - 14 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 1) q^{37} + (3 \beta_{5} - 2 \beta_{4} + 22 \beta_{3} + 5 \beta_{2} + 14 \beta_1 - 18) q^{38} + (\beta_{5} - 23 \beta_{4} - 4 \beta_{3} - \beta_{2} - 8) q^{40} + ( - 2 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} - 24) q^{41} + ( - \beta_{5} - 3 \beta_{4} - 39 \beta_{3} - \beta_{2} - 6 \beta_1) q^{43} + ( - \beta_{5} - 5 \beta_{4} + 31 \beta_{3} + 5 \beta_1 + 26) q^{44} + (6 \beta_{5} + 53 \beta_{3} + 3 \beta_{2} + 17 \beta_1 + 18) q^{46} + (4 \beta_{5} + 18 \beta_{3} + 18) q^{47} + ( - 24 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} - 12 \beta_1 - 18) q^{49} + ( - 4 \beta_{5} - 17 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 10) q^{50} + ( - \beta_{5} - 27 \beta_{4} + 35 \beta_{3} + \beta_{2} + 70) q^{52} + (9 \beta_{5} - 9 \beta_{4} - 13 \beta_{3} + 18 \beta_{2} - 9 \beta_1 + 4) q^{53} + ( - 2 \beta_{5} + 16 \beta_{4} - 34 \beta_{3} - 2 \beta_{2} + 32 \beta_1) q^{55} + (12 \beta_{5} + 91 \beta_{3} + 6 \beta_{2} + 31 \beta_1 + 30) q^{56} + (44 \beta_{4} + 22 \beta_{3} + 2 \beta_{2} + 22 \beta_1 + 2) q^{58} + (\beta_{5} + 11 \beta_{4} + 4 \beta_{3} - \beta_{2} + 8) q^{59} + (10 \beta_{4} + 9 \beta_{3} - 10 \beta_1 + 19) q^{61} + ( - 5 \beta_{5} - 20 \beta_{4} - 62 \beta_{3} - 5 \beta_{2} - 40 \beta_1) q^{62} + ( - 14 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} - 7 \beta_1 - 3) q^{64} + (22 \beta_{5} + 3 \beta_{3} + 11 \beta_{2} + 19 \beta_1 - 8) q^{65} + ( - 9 \beta_{5} + 13 \beta_{4} + 10 \beta_{3} - 18 \beta_{2} + 13 \beta_1 + 3) q^{67} + ( - 20 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} - 10 \beta_1 + 52) q^{68} + (7 \beta_{5} + 29 \beta_{4} + 61 \beta_{3} + 14 \beta_{2} + 29 \beta_1 - 32) q^{70} + (4 \beta_{5} - 30 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} - 36) q^{71} + ( - 8 \beta_{5} - 20 \beta_{4} + 3 \beta_{3} - 8 \beta_{2} - 40 \beta_1) q^{73} + (12 \beta_{5} + 5 \beta_{4} + 82 \beta_{3} + 12 \beta_{2} + 10 \beta_1) q^{74} + (4 \beta_{5} - 28 \beta_{4} - 91 \beta_{3} - 6 \beta_{2} - 32 \beta_1 - 5) q^{76} + (6 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 42) q^{77} + ( - 11 \beta_{5} - \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 6) q^{79} + ( - 3 \beta_{5} + \beta_{4} - 105 \beta_{3} - \beta_1 - 104) q^{80} + ( - 2 \beta_{5} - 14 \beta_{4} - 2 \beta_{3} + 14 \beta_1 - 16) q^{82} + (48 \beta_{4} + 24 \beta_{3} + 6 \beta_{2} + 24 \beta_1 + 80) q^{83} + (34 \beta_{5} + 6 \beta_{4} + 34 \beta_{3} - 6 \beta_1 + 40) q^{85} + (3 \beta_{5} + 44 \beta_{4} + 58 \beta_{3} + 6 \beta_{2} + 44 \beta_1 - 14) q^{86} + ( - 18 \beta_{5} - 111 \beta_{3} - 9 \beta_{2} + 5 \beta_1 - 58) q^{88} + ( - 15 \beta_{5} - 9 \beta_{4} - 19 \beta_{3} - 30 \beta_{2} - 9 \beta_1 + 10) q^{89} + (3 \beta_{5} + 35 \beta_{4} + 68 \beta_{3} + 6 \beta_{2} + 35 \beta_1 - 33) q^{91} + ( - 29 \beta_{5} - 29 \beta_{4} - 78 \beta_{3} - 29 \beta_{2} - 58 \beta_1) q^{92} + ( - 18 \beta_{3} - 26 \beta_1 + 4) q^{94} + (\beta_{5} - 31 \beta_{4} + 60 \beta_{3} - 11 \beta_{2} - 30 \beta_1 + 16) q^{95} + ( - 14 \beta_{4} - 30 \beta_{3} - 60) q^{97} + ( - 12 \beta_{5} + 30 \beta_{4} - 54 \beta_{3} + 12 \beta_{2} - 108) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 22 q^{7} + 54 q^{10} + 36 q^{11} - 3 q^{13} + 57 q^{14} - 23 q^{16} - 38 q^{17} - 10 q^{19} - 32 q^{20} + 36 q^{22} - 54 q^{23} - 21 q^{25} - 118 q^{26} - 101 q^{28} + 102 q^{29} + 63 q^{32} - 150 q^{34} + 24 q^{35} - 119 q^{38} + 30 q^{40} - 96 q^{41} + 107 q^{43} + 94 q^{44} + 50 q^{47} - 48 q^{49} + 399 q^{52} + 90 q^{53} + 148 q^{55} - 116 q^{58} + 27 q^{61} + 121 q^{62} + 46 q^{64} - 39 q^{67} + 388 q^{68} - 354 q^{70} - 84 q^{71} - 77 q^{73} - 219 q^{74} + 215 q^{76} - 260 q^{77} + 9 q^{79} - 312 q^{80} - 4 q^{82} + 348 q^{83} + 68 q^{85} - 249 q^{86} + 72 q^{89} - 393 q^{91} + 118 q^{92} - 104 q^{95} - 228 q^{97} - 540 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 9\nu^{4} - 10\nu^{3} + 98\nu^{2} - 87\nu + 52 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{3} - 2\nu^{2} + 9\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{2} - 7\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + \beta_{3} - 6\beta_{2} - 14\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 5\beta_{4} - 9\beta_{3} - 24\beta_{2} + 45\beta _1 + 81 \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.500000 + 2.69511i
0.500000 0.630453i
0.500000 2.93068i
0.500000 2.69511i
0.500000 + 0.630453i
0.500000 + 2.93068i
−3.08403 1.78057i 0 4.34085 + 7.51857i −2.32722 + 4.03087i 0 −10.6817 16.6722i 0 14.3545 8.28756i
46.2 −0.204011 0.117786i 0 −1.97225 3.41604i −2.88028 + 4.98878i 0 1.94451 1.87150i 0 1.17522 0.678513i
46.3 1.78805 + 1.03233i 0 0.131406 + 0.227602i 3.20750 5.55555i 0 −2.26281 7.71601i 0 11.4703 6.62239i
145.1 −3.08403 + 1.78057i 0 4.34085 7.51857i −2.32722 4.03087i 0 −10.6817 16.6722i 0 14.3545 + 8.28756i
145.2 −0.204011 + 0.117786i 0 −1.97225 + 3.41604i −2.88028 4.98878i 0 1.94451 1.87150i 0 1.17522 + 0.678513i
145.3 1.78805 1.03233i 0 0.131406 0.227602i 3.20750 + 5.55555i 0 −2.26281 7.71601i 0 11.4703 + 6.62239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.p.c 6
3.b odd 2 1 57.3.g.b 6
12.b even 2 1 912.3.be.f 6
19.d odd 6 1 inner 171.3.p.c 6
57.f even 6 1 57.3.g.b 6
228.n odd 6 1 912.3.be.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.b 6 3.b odd 2 1
57.3.g.b 6 57.f even 6 1
171.3.p.c 6 1.a even 1 1 trivial
171.3.p.c 6 19.d odd 6 1 inner
912.3.be.f 6 12.b even 2 1
912.3.be.f 6 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{6} + 3T_{2}^{5} - 4T_{2}^{4} - 21T_{2}^{3} + 46T_{2}^{2} + 21T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} + 56T_{5}^{4} + 184T_{5}^{3} + 2288T_{5}^{2} + 6880T_{5} + 29584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} - 4 T^{4} - 21 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + 56 T^{4} + \cdots + 29584 \) Copy content Toggle raw display
$7$ \( (T^{3} + 11 T^{2} - T - 47)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 18 T^{2} - 96 T + 1084)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} - 340 T^{4} + \cdots + 2883 \) Copy content Toggle raw display
$17$ \( T^{6} + 38 T^{5} + 1408 T^{4} + \cdots + 52186176 \) Copy content Toggle raw display
$19$ \( T^{6} + 10 T^{5} - 249 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} + 54 T^{5} + 2352 T^{4} + \cdots + 21049744 \) Copy content Toggle raw display
$29$ \( T^{6} - 102 T^{5} + \cdots + 487228608 \) Copy content Toggle raw display
$31$ \( T^{6} + 2345 T^{4} + \cdots + 112326483 \) Copy content Toggle raw display
$37$ \( T^{6} + 4713 T^{4} + \cdots + 3652564347 \) Copy content Toggle raw display
$41$ \( T^{6} + 96 T^{5} + 3824 T^{4} + \cdots + 1354752 \) Copy content Toggle raw display
$43$ \( T^{6} - 107 T^{5} + \cdots + 1477402969 \) Copy content Toggle raw display
$47$ \( T^{6} - 50 T^{5} + 1976 T^{4} + \cdots + 5798464 \) Copy content Toggle raw display
$53$ \( T^{6} - 90 T^{5} + \cdots + 6327041328 \) Copy content Toggle raw display
$59$ \( T^{6} - 1048 T^{4} + \cdots + 25509168 \) Copy content Toggle raw display
$61$ \( T^{6} - 27 T^{5} + \cdots + 1215986641 \) Copy content Toggle raw display
$67$ \( T^{6} + 39 T^{5} + \cdots + 11787475467 \) Copy content Toggle raw display
$71$ \( T^{6} + 84 T^{5} + \cdots + 149905029888 \) Copy content Toggle raw display
$73$ \( T^{6} + 77 T^{5} + \cdots + 434693631969 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + \cdots + 32898206883 \) Copy content Toggle raw display
$83$ \( (T^{3} - 174 T^{2} - 4140 T + 1174072)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 72 T^{5} + \cdots + 591631797168 \) Copy content Toggle raw display
$97$ \( T^{6} + 228 T^{5} + \cdots + 68659968 \) Copy content Toggle raw display
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