Properties

Label 171.3.p.d
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + 1) q^{2} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{4} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1 + 3) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 1) q^{2} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{4} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1 + 3) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{8} + ( - \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 12) q^{10} + ( - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} - 6) q^{11} + ( - 4 \beta_{4} + 2 \beta_{3} + 4) q^{13} + (4 \beta_{5} + 7 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{14} + (4 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} + 4 \beta_1 + 7) q^{16} + (\beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 16) q^{17} + (5 \beta_{5} + 2 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 14) q^{19} + ( - 5 \beta_{5} + 5 \beta_{4} + 3 \beta_{2} - 17) q^{20} + (\beta_{5} + 5 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{22} + (4 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{23} + (2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 8 \beta_1 + 2) q^{25} + (2 \beta_{5} - 2 \beta_{4} - 8 \beta_{2} + 22) q^{26} + ( - 13 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{28} + ( - 13 \beta_{4} + 9 \beta_{2} - 9 \beta_1) q^{29} + ( - 4 \beta_{5} - 4 \beta_{4} + 42 \beta_{3} - \beta_{2} - 2 \beta_1 + 17) q^{31} + ( - 3 \beta_{4} - 8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 16) q^{32} + ( - 15 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 10) q^{34} + (9 \beta_{5} + 18 \beta_{4} + 17 \beta_{3} + 14 \beta_1 + 26) q^{35} + (9 \beta_{5} + 9 \beta_{4} - 24 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{37} + ( - \beta_{5} - 8 \beta_{4} - 33 \beta_{3} - 7 \beta_{2} - 18 \beta_1 - 18) q^{38} + ( - 19 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + \beta_1 - 23) q^{40} + (\beta_{5} + 4 \beta_{3} + 20 \beta_{2} + 10 \beta_1 - 3) q^{41} + ( - 4 \beta_{5} - 8 \beta_{4} - 18 \beta_{3} + 8 \beta_1 - 22) q^{43} + (8 \beta_{5} + 4 \beta_{4} - 24 \beta_{3} + 15 \beta_{2} + 15 \beta_1 + 8) q^{44} + ( - 4 \beta_{5} - 4 \beta_{4} - 22 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 15) q^{46} + ( - 12 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} + \beta_{2} + \beta_1 - 12) q^{47} + ( - 7 \beta_{5} + 7 \beta_{4} + 6 \beta_{2} + 5) q^{49} + ( - 12 \beta_{5} - 12 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 12 \beta_1 - 9) q^{50} + (12 \beta_{5} - 10 \beta_{3} - 24 \beta_{2} - 12 \beta_1 + 22) q^{52} + (12 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 8) q^{53} + (8 \beta_{5} + 16 \beta_{4} - 7 \beta_{3} + 17 \beta_1 + 1) q^{55} + ( - 10 \beta_{5} - 10 \beta_{4} - 42 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 31) q^{56} + ( - 9 \beta_{5} + 9 \beta_{4} + \beta_{2} + 29) q^{58} + ( - 22 \beta_{5} - 20 \beta_{3} - 22 \beta_{2} - 11 \beta_1 - 2) q^{59} + ( - 30 \beta_{5} - 15 \beta_{4} - 30 \beta_{3} - 13 \beta_{2} - 13 \beta_1 - 30) q^{61} + ( - 20 \beta_{5} - 40 \beta_{4} + 17 \beta_{3} + 5 \beta_1 - 3) q^{62} + (12 \beta_{5} - 12 \beta_{4} - 2 \beta_{2} + 27) q^{64} + (2 \beta_{5} + 2 \beta_{4} - 40 \beta_{3} - 14 \beta_{2} - 28 \beta_1 - 18) q^{65} + ( - 8 \beta_{4} + 24 \beta_{3} + 13 \beta_{2} - 13 \beta_1 + 48) q^{67} + ( - 3 \beta_{5} + 3 \beta_{4} - 24 \beta_{2} + 6) q^{68} + ( - 31 \beta_{4} - 31 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 62) q^{70} + ( - 10 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 10) q^{71} + (11 \beta_{5} + 22 \beta_{4} + 14 \beta_{3} - 2 \beta_1 + 25) q^{73} + (11 \beta_{5} + 22 \beta_{4} - 42 \beta_{3} - 15 \beta_1 - 31) q^{74} + (11 \beta_{5} + 31 \beta_{4} - 36 \beta_{3} - 18 \beta_{2} - 11 \beta_1 - 30) q^{76} + (5 \beta_{5} - 5 \beta_{4} + 31 \beta_{2} + 73) q^{77} + (34 \beta_{5} + 8 \beta_{3} + 20 \beta_{2} + 10 \beta_1 + 26) q^{79} + (30 \beta_{5} + 15 \beta_{4} + 50 \beta_{3} + 29 \beta_{2} + 29 \beta_1 + 30) q^{80} + ( - 28 \beta_{5} - 14 \beta_{4} + 25 \beta_{3} + 28 \beta_{2} + 28 \beta_1 - 28) q^{82} + (13 \beta_{5} - 13 \beta_{4} + 37 \beta_{2} - 12) q^{83} + (32 \beta_{5} + 16 \beta_{4} + 48 \beta_{3} - 6 \beta_{2} - 6 \beta_1 + 32) q^{85} + (10 \beta_{4} + 28 \beta_{3} - 16 \beta_{2} + 16 \beta_1 + 56) q^{86} + (5 \beta_{5} + 5 \beta_{4} - 50 \beta_{3} + 11 \beta_{2} + 22 \beta_1 - 20) q^{88} + ( - 9 \beta_{4} - 3 \beta_{3} - 24 \beta_{2} + 24 \beta_1 - 6) q^{89} + ( - 4 \beta_{4} - 26 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 52) q^{91} + (8 \beta_{5} + 16 \beta_{4} - 15 \beta_{3} - 3 \beta_1 - 7) q^{92} + ( - 14 \beta_{5} - 14 \beta_{4} + 62 \beta_{3} + 13 \beta_{2} + 26 \beta_1 + 17) q^{94} + (10 \beta_{5} + 4 \beta_{4} + 26 \beta_{3} + 13 \beta_{2} - 10 \beta_1 - 67) q^{95} + ( - 46 \beta_{5} - 17 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 29) q^{97} + (6 \beta_{5} + 41 \beta_{3} + 40 \beta_{2} + 20 \beta_1 - 35) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 5 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 5 q^{4} + 2 q^{5} - 60 q^{10} - 26 q^{11} + 30 q^{13} - 54 q^{14} + q^{16} + 42 q^{17} + 25 q^{19} - 108 q^{20} - 39 q^{22} - 8 q^{23} - 17 q^{25} + 148 q^{26} + 32 q^{28} + 12 q^{29} - 51 q^{32} - 6 q^{34} + 38 q^{35} + 14 q^{38} - 96 q^{40} - 63 q^{41} - 34 q^{43} + 69 q^{44} - 58 q^{47} + 18 q^{49} + 162 q^{52} + 12 q^{53} - 28 q^{55} + 172 q^{58} + 147 q^{59} + 58 q^{61} + 116 q^{62} + 166 q^{64} + 201 q^{67} + 84 q^{68} - 198 q^{70} + 102 q^{71} + 7 q^{73} - 174 q^{74} - 173 q^{76} + 376 q^{77} - 134 q^{80} - 145 q^{82} - 146 q^{83} - 90 q^{85} + 270 q^{86} + 72 q^{89} - 216 q^{91} - 72 q^{92} - 558 q^{95} + 21 q^{97} - 411 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 8\nu^{4} + 64\nu^{3} - 50\nu^{2} + 7\nu - 56 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu - 385 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 534 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\nu^{5} - 92\nu^{4} + 605\nu^{3} + 211\nu^{2} + 3683\nu - 1430 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} - \beta_{4} + 4\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 7\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} + 16\beta_{4} - 31\beta_{3} - 6\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28\beta_{5} + 14\beta_{4} - 48\beta_{3} - 55\beta_{2} - 55\beta _1 + 28 \) Copy content Toggle raw display

Character values

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

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

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
0.0702177 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 2.91992 5.05745i 0 9.38186 6.24860i 0 −11.6221 + 6.71002i
46.2 0.583430 + 0.336844i 0 −1.77307 3.07105i 1.55311 2.69006i 0 −8.15294 5.08374i 0 1.81226 1.04631i
46.3 2.90671 + 1.67819i 0 3.63264 + 6.29191i −3.47303 + 6.01546i 0 −1.22892 10.9595i 0 −20.1902 + 11.6568i
145.1 −1.99014 + 1.14901i 0 0.640435 1.10927i 2.91992 + 5.05745i 0 9.38186 6.24860i 0 −11.6221 6.71002i
145.2 0.583430 0.336844i 0 −1.77307 + 3.07105i 1.55311 + 2.69006i 0 −8.15294 5.08374i 0 1.81226 + 1.04631i
145.3 2.90671 1.67819i 0 3.63264 6.29191i −3.47303 6.01546i 0 −1.22892 10.9595i 0 −20.1902 11.6568i
\(n\): 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.d 6
3.b odd 2 1 19.3.d.a 6
12.b even 2 1 304.3.r.b 6
19.d odd 6 1 inner 171.3.p.d 6
57.d even 2 1 361.3.d.c 6
57.f even 6 1 19.3.d.a 6
57.f even 6 1 361.3.b.b 6
57.h odd 6 1 361.3.b.b 6
57.h odd 6 1 361.3.d.c 6
57.j even 18 3 361.3.f.h 18
57.j even 18 3 361.3.f.i 18
57.l odd 18 3 361.3.f.h 18
57.l odd 18 3 361.3.f.i 18
228.n odd 6 1 304.3.r.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.d.a 6 3.b odd 2 1
19.3.d.a 6 57.f even 6 1
171.3.p.d 6 1.a even 1 1 trivial
171.3.p.d 6 19.d odd 6 1 inner
304.3.r.b 6 12.b even 2 1
304.3.r.b 6 228.n odd 6 1
361.3.b.b 6 57.f even 6 1
361.3.b.b 6 57.h odd 6 1
361.3.d.c 6 57.d even 2 1
361.3.d.c 6 57.h odd 6 1
361.3.f.h 18 57.j even 18 3
361.3.f.h 18 57.l odd 18 3
361.3.f.i 18 57.j even 18 3
361.3.f.i 18 57.l odd 18 3

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
\( T_{5}^{6} - 2T_{5}^{5} + 48T_{5}^{4} - 164T_{5}^{3} + 2188T_{5}^{2} - 5544T_{5} + 15876 \) 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
$5$ \( T^{6} - 2 T^{5} + 48 T^{4} + \cdots + 15876 \) Copy content Toggle raw display
$7$ \( (T^{3} - 78 T - 94)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 13 T^{2} - 83 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 30 T^{5} + 272 T^{4} + \cdots + 355008 \) Copy content Toggle raw display
$17$ \( T^{6} - 42 T^{5} + 1200 T^{4} + \cdots + 5817744 \) Copy content Toggle raw display
$19$ \( T^{6} - 25 T^{5} + 1026 T^{4} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + 174 T^{4} + \cdots + 381924 \) Copy content Toggle raw display
$29$ \( T^{6} - 12 T^{5} - 1432 T^{4} + \cdots + 54289548 \) Copy content Toggle raw display
$31$ \( T^{6} + 4544 T^{4} + \cdots + 2642351052 \) Copy content Toggle raw display
$37$ \( T^{6} + 3024 T^{4} + 1967760 T^{2} + \cdots + 38988 \) Copy content Toggle raw display
$41$ \( T^{6} + 63 T^{5} + \cdots + 498533643 \) Copy content Toggle raw display
$43$ \( T^{6} + 34 T^{5} + \cdots + 981944896 \) Copy content Toggle raw display
$47$ \( T^{6} + 58 T^{5} + 3198 T^{4} + \cdots + 94361796 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} - 768 T^{4} + \cdots + 48771072 \) Copy content Toggle raw display
$59$ \( T^{6} - 147 T^{5} + \cdots + 1787690763 \) Copy content Toggle raw display
$61$ \( T^{6} - 58 T^{5} + \cdots + 661415524 \) Copy content Toggle raw display
$67$ \( T^{6} - 201 T^{5} + \cdots + 17294403 \) Copy content Toggle raw display
$71$ \( T^{6} - 102 T^{5} + 4272 T^{4} + \cdots + 1259712 \) Copy content Toggle raw display
$73$ \( T^{6} - 7 T^{5} + 3274 T^{4} + \cdots + 340734681 \) Copy content Toggle raw display
$79$ \( T^{6} - 6688 T^{4} + \cdots + 44231363328 \) Copy content Toggle raw display
$83$ \( (T^{3} + 73 T^{2} - 5585 T - 397611)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 72 T^{5} + \cdots + 84829321008 \) Copy content Toggle raw display
$97$ \( T^{6} - 21 T^{5} + \cdots + 21248211843 \) Copy content Toggle raw display
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