Properties

Label 171.3.p.e
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: 6.0.6967728.1
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{8} + (\beta_{5} + \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 5) q^{10} + ( - 3 \beta_{4} + 3 \beta_{3} + 2) q^{11} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1 - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} - 8 \beta_1 + 15) q^{14} + (2 \beta_{5} - 2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 6) q^{16} + ( - 4 \beta_{5} - 4 \beta_{2} + 2 \beta_1 + 4) q^{17} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 16 \beta_1 - 1) q^{19} + (\beta_{4} - 5 \beta_{3} - 4 \beta_{2} + 20) q^{20} + (3 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 9) q^{22} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 10 \beta_1 + 11) q^{23} + (8 \beta_{3} - 4 \beta_{2} - 9 \beta_1 + 13) q^{25} + ( - 4 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - 7) q^{26} + ( - 8 \beta_{5} + 8 \beta_{4} - 20 \beta_{3} + 10 \beta_{2} + \beta_1 - 11) q^{28} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 6 \beta_1 - 12) q^{29} + (6 \beta_{5} - 3 \beta_{4} + 11 \beta_{3} - 8 \beta_{2} - 10 \beta_1 + 13) q^{31} + (2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} - 8 \beta_1 - 3) q^{32} + (6 \beta_{2} + 12 \beta_1 + 6) q^{34} + (9 \beta_{5} + 9 \beta_{3} - 9 \beta_{2} - 4 \beta_1 + 9) q^{35} + (18 \beta_{3} - 18 \beta_{2} + 6 \beta_1 + 15) q^{37} + ( - \beta_{5} - 8 \beta_{3} + 21 \beta_{2} + 10 \beta_1 - 33) q^{38} + ( - \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + \beta_{2} - 14 \beta_1 + 27) q^{40} + (6 \beta_{5} - 12 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 2) q^{41} + (\beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{43} + (\beta_{5} - \beta_{4} + 26 \beta_{3} - 13 \beta_{2} + 46 \beta_1 - 33) q^{44} + (2 \beta_{5} - \beta_{4} - 13 \beta_{3} + 14 \beta_{2} + 28 \beta_1 - 28) q^{46} + ( - 4 \beta_{5} + 4 \beta_{4} - 32 \beta_{3} + 16 \beta_{2} + 6 \beta_1 - 22) q^{47} + ( - 6 \beta_{4} - 2 \beta_{3} - 8 \beta_{2} + 20) q^{49} + (8 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + \beta_{2} + 40 \beta_1 - 21) q^{50} + (\beta_{5} - 2 \beta_{4} + 21 \beta_{3} - \beta_{2} + 29 \beta_1 - 57) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} - 21 \beta_{2} - 26 \beta_1 - 5) q^{53} + (2 \beta_{5} - 10 \beta_{3} + 22 \beta_{2} - 48 \beta_1 - 22) q^{55} + ( - 12 \beta_{5} + 6 \beta_{4} - 29 \beta_{3} + 23 \beta_{2} + 12 \beta_1 - 29) q^{56} + ( - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 30) q^{58} + ( - 5 \beta_{5} + 10 \beta_{4} + 13 \beta_{3} + 5 \beta_{2} + 12 \beta_1 - 29) q^{59} + ( - 12 \beta_{5} + 12 \beta_{4} - 44 \beta_{3} + 22 \beta_{2} + 65 \beta_1 - 87) q^{61} + (11 \beta_{5} + 12 \beta_{3} - 13 \beta_{2} + 28 \beta_1 + 13) q^{62} + (\beta_{4} - 13 \beta_{3} - 12 \beta_{2} + 27) q^{64} + (2 \beta_{5} - \beta_{4} - 17 \beta_{3} + 18 \beta_{2} - 56 \beta_1 + 10) q^{65} + (\beta_{5} + \beta_{4} - 17 \beta_{2} - 13 \beta_1 + 4) q^{67} + (10 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} - 42) q^{68} + (9 \beta_{5} + 9 \beta_{4} - 23 \beta_{2} + 18 \beta_1 + 41) q^{70} + ( - 2 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 44 \beta_1 + 86) q^{71} + (20 \beta_{5} + 20 \beta_{2} + 41 \beta_1 - 20) q^{73} + (18 \beta_{5} + 21 \beta_{3} - 24 \beta_{2} + 90 \beta_1 + 24) q^{74} + (4 \beta_{5} - 18 \beta_{4} + 18 \beta_{3} + 4 \beta_{2} - 33 \beta_1 - 11) q^{76} + ( - 29 \beta_{4} + 25 \beta_{3} - 4 \beta_{2} + 22) q^{77} + ( - \beta_{5} + 2 \beta_{4} - 17 \beta_{3} + \beta_{2} + 19 \beta_1 - 39) q^{79} + ( - 9 \beta_{5} + 9 \beta_{4} - 18 \beta_{3} + 9 \beta_{2} + 8 \beta_1 - 17) q^{80} + (2 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 18 \beta_{2} - 44 \beta_1 + 62) q^{82} + ( - 4 \beta_{3} - 4 \beta_{2} + 42) q^{83} + (14 \beta_{5} - 14 \beta_{4} + 12 \beta_{3} - 6 \beta_{2} - 40 \beta_1 + 46) q^{85} + (\beta_{5} + \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 8) q^{86} + (2 \beta_{5} - \beta_{4} + 35 \beta_{3} - 34 \beta_{2} + 76 \beta_1 - 4) q^{88} + (\beta_{5} + \beta_{4} - \beta_{2} - 72 \beta_1 - 71) q^{89} + ( - 15 \beta_{5} - 15 \beta_{4} + 19 \beta_{2} + 21 \beta_1 + 2) q^{91} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 34 \beta_1 - 1) q^{92} + ( - 32 \beta_{5} + 16 \beta_{4} - 50 \beta_{3} + 34 \beta_{2} - 136 \beta_1 + 34) q^{94} + ( - 19 \beta_{5} + 12 \beta_{4} - 35 \beta_{3} + 11 \beta_{2} + 8 \beta_1 + 13) q^{95} + ( - 8 \beta_{5} + 16 \beta_{4} + 44 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 24) q^{97} + ( - 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 28 \beta_1 + 54) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 5 q^{4} + 2 q^{5} + 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 5 q^{4} + 2 q^{5} + 26 q^{7} + 30 q^{10} - 15 q^{13} + 81 q^{14} + q^{16} + 10 q^{17} - 46 q^{19} + 124 q^{20} - 84 q^{22} + 24 q^{23} + 15 q^{25} - 58 q^{26} + 19 q^{28} - 66 q^{29} - 51 q^{32} + 90 q^{34} + 6 q^{35} - 83 q^{38} + 162 q^{40} - 24 q^{41} + 11 q^{43} - 176 q^{44} + 26 q^{47} + 96 q^{49} - 321 q^{52} - 180 q^{53} - 176 q^{55} - 188 q^{58} - 162 q^{59} - 141 q^{61} + 109 q^{62} + 166 q^{64} - 63 q^{67} - 212 q^{68} + 258 q^{70} + 372 q^{71} + 103 q^{73} + 315 q^{74} - 217 q^{76} + 16 q^{77} - 123 q^{79} - 6 q^{80} + 80 q^{82} + 252 q^{83} + 116 q^{85} + 39 q^{86} - 642 q^{89} + 87 q^{91} - 104 q^{92} + 214 q^{95} - 12 q^{97} + 264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu + 8 ) / 393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 927 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -77\nu^{5} + 92\nu^{4} - 605\nu^{3} - 211\nu^{2} - 3683\nu + 1037 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -79\nu^{5} + 108\nu^{4} - 733\nu^{3} - 111\nu^{2} - 3697\nu + 1149 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 147\nu^{5} - 128\nu^{4} + 1155\nu^{3} + 772\nu^{2} + 8103\nu - 503 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta_{2} + 4\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{4} + 9\beta_{3} + 2\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{5} - 11\beta_{3} + 19\beta_{2} - 31\beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{5} + 55\beta_{4} - 166\beta_{3} + 83\beta_{2} - 96\beta _1 + 13 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.0702177 0.121621i
−1.13654 + 1.96854i
1.56632 2.71294i
0.0702177 + 0.121621i
−1.13654 1.96854i
1.56632 + 2.71294i
−1.99014 1.14901i 0 0.640435 + 1.10927i 0.140435 0.243241i 0 −5.24143 6.24860i 0 −0.558972 + 0.322723i
46.2 0.583430 + 0.336844i 0 −1.77307 3.07105i −2.27307 + 3.93708i 0 9.87987 5.08374i 0 −2.65236 + 1.53134i
46.3 2.90671 + 1.67819i 0 3.63264 + 6.29191i 3.13264 5.42589i 0 8.36156 10.9595i 0 18.2113 10.5143i
145.1 −1.99014 + 1.14901i 0 0.640435 1.10927i 0.140435 + 0.243241i 0 −5.24143 6.24860i 0 −0.558972 0.322723i
145.2 0.583430 0.336844i 0 −1.77307 + 3.07105i −2.27307 3.93708i 0 9.87987 5.08374i 0 −2.65236 1.53134i
145.3 2.90671 1.67819i 0 3.63264 6.29191i 3.13264 + 5.42589i 0 8.36156 10.9595i 0 18.2113 + 10.5143i
\(n\): 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.e 6
3.b odd 2 1 57.3.g.a 6
12.b even 2 1 912.3.be.d 6
19.d odd 6 1 inner 171.3.p.e 6
57.f even 6 1 57.3.g.a 6
228.n odd 6 1 912.3.be.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.a 6 3.b odd 2 1
57.3.g.a 6 57.f even 6 1
171.3.p.e 6 1.a even 1 1 trivial
171.3.p.e 6 19.d odd 6 1 inner
912.3.be.d 6 12.b even 2 1
912.3.be.d 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 \(S_{3}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 21T_{2}^{3} + 40T_{2}^{2} - 63T_{2} + 27 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} + 32T_{5}^{4} + 40T_{5}^{3} + 800T_{5}^{2} - 224T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} - 4 T^{4} + 21 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 32 T^{4} + 40 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{3} - 13 T^{2} - 13 T + 433)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 264 T + 304)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 15 T^{5} - 172 T^{4} + \cdots + 3518667 \) Copy content Toggle raw display
$17$ \( T^{6} - 10 T^{5} + 472 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{6} + 46 T^{5} + 1383 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} - 24 T^{5} + 696 T^{4} + \cdots + 1517824 \) Copy content Toggle raw display
$29$ \( T^{6} + 66 T^{5} + 1520 T^{4} + \cdots + 19293888 \) Copy content Toggle raw display
$31$ \( T^{6} + 2033 T^{4} + \cdots + 124768803 \) Copy content Toggle raw display
$37$ \( T^{6} + 5913 T^{4} + \cdots + 1689765867 \) Copy content Toggle raw display
$41$ \( T^{6} + 24 T^{5} + \cdots + 1907539968 \) Copy content Toggle raw display
$43$ \( T^{6} - 11 T^{5} + 110 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$47$ \( T^{6} - 26 T^{5} + \cdots + 2266521664 \) Copy content Toggle raw display
$53$ \( T^{6} + 180 T^{5} + 11016 T^{4} + \cdots + 2495232 \) Copy content Toggle raw display
$59$ \( T^{6} + 162 T^{5} + \cdots + 48350430912 \) Copy content Toggle raw display
$61$ \( T^{6} + 141 T^{5} + \cdots + 558907255201 \) Copy content Toggle raw display
$67$ \( T^{6} + 63 T^{5} + \cdots + 2349816507 \) Copy content Toggle raw display
$71$ \( T^{6} - 372 T^{5} + \cdots + 100019515392 \) Copy content Toggle raw display
$73$ \( T^{6} - 103 T^{5} + \cdots + 214090364601 \) Copy content Toggle raw display
$79$ \( T^{6} + 123 T^{5} + \cdots + 2549342403 \) Copy content Toggle raw display
$83$ \( (T^{3} - 126 T^{2} + 4908 T - 57704)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 642 T^{5} + \cdots + 3516172210368 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots + 3000768049152 \) Copy content Toggle raw display
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