Properties

Label 171.3.p.f
Level $171$
Weight $3$
Character orbit 171.p
Analytic conductor $4.659$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19163381760000.1
Defining polynomial: \( x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (2 \beta_{6} + \beta_{4} + 3 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{6} - \beta_{4} - 5) q^{7} + (2 \beta_{7} - \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (2 \beta_{6} + \beta_{4} + 3 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{6} - \beta_{4} - 5) q^{7} + (2 \beta_{7} - \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{8} + ( - \beta_{4} + 4 \beta_{2} - 8) q^{10} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{11} + (2 \beta_{4} + 5 \beta_{2} - 10) q^{13} + (\beta_{7} + \beta_{5}) q^{14} + (2 \beta_{6} + 4 \beta_{4} + 11 \beta_{2} - 11) q^{16} + (4 \beta_{3} - 2 \beta_1) q^{17} + ( - 3 \beta_{6} - 5 \beta_{4} - 6 \beta_{2} - 9) q^{19} + ( - 3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{20} + (7 \beta_{6} + 10 \beta_{2} + 10) q^{22} + ( - 7 \beta_{7} + \beta_{3} - 9 \beta_1) q^{23} + (12 \beta_{6} + 6 \beta_{4} - 9 \beta_{2}) q^{25} + ( - 2 \beta_{5} - 7 \beta_{3} - 9 \beta_1) q^{26} + ( - 4 \beta_{6} - 2 \beta_{4} + 15 \beta_{2}) q^{28} + 10 \beta_{3} q^{29} + ( - \beta_{6} - \beta_{4} + 22 \beta_{2} - 11) q^{31} + ( - 2 \beta_{7} + 4 \beta_{5} - 9 \beta_{3} + 2 \beta_1) q^{32} + ( - 6 \beta_{4} - 14 \beta_{2} + 28) q^{34} + (11 \beta_{7} - 11 \beta_{5} + 18 \beta_{3} - 9 \beta_1) q^{35} + (6 \beta_{6} + 6 \beta_{4} - 50 \beta_{2} + 25) q^{37} + ( - 3 \beta_{7} + 5 \beta_{5} + 14 \beta_{3} - 14 \beta_1) q^{38} + ( - 13 \beta_{6} + 14 \beta_{2} + 14) q^{40} + (4 \beta_{7} + 4 \beta_{5}) q^{41} + (7 \beta_{6} + 14 \beta_{4} + 25 \beta_{2} - 25) q^{43} + (3 \beta_{7} - 13 \beta_{3} + 29 \beta_1) q^{44} + ( - 31 \beta_{6} - 31 \beta_{4} - 56 \beta_{2} + 28) q^{46} + ( - 8 \beta_{7} - 8 \beta_1) q^{47} + ( - 10 \beta_{6} + 10 \beta_{4} + 6) q^{49} + (12 \beta_{7} - 6 \beta_{5} - 9 \beta_{3} + 15 \beta_1) q^{50} + ( - 21 \beta_{6} - 35 \beta_{2} - 35) q^{52} + ( - 5 \beta_{7} + 10 \beta_{5} - 21 \beta_{3} + 5 \beta_1) q^{53} + (4 \beta_{6} + 8 \beta_{4} - 10 \beta_{2} + 10) q^{55} + ( - 12 \beta_{7} + 6 \beta_{5} - 17 \beta_{3} + 11 \beta_1) q^{56} + (10 \beta_{6} - 10 \beta_{4} + 70) q^{58} + ( - 5 \beta_{7} - 5 \beta_{5} + 25 \beta_1) q^{59} + ( - 20 \beta_{6} - 10 \beta_{4} - 17 \beta_{2}) q^{61} + ( - \beta_{7} + \beta_{5} - 20 \beta_{3} + 10 \beta_1) q^{62} + (7 \beta_{6} - 7 \beta_{4} - 1) q^{64} + (2 \beta_{7} - \beta_{5} + 7 \beta_{3} - 6 \beta_1) q^{65} + ( - 11 \beta_{4} + 15 \beta_{2} - 30) q^{67} + (6 \beta_{5} + 12 \beta_{3} + 18 \beta_1) q^{68} + (17 \beta_{4} - 30 \beta_{2} + 60) q^{70} + (8 \beta_{7} + 8 \beta_{5} + 10 \beta_1) q^{71} + (18 \beta_{6} + 36 \beta_{4} + 15 \beta_{2} - 15) q^{73} + (6 \beta_{7} - 6 \beta_{5} + 38 \beta_{3} - 19 \beta_1) q^{74} + ( - 18 \beta_{6} - 30 \beta_{4} - 55 \beta_{2} + 98) q^{76} + ( - 5 \beta_{5} + 5 \beta_{3}) q^{77} + (19 \beta_{6} + 35 \beta_{2} + 35) q^{79} + ( - \beta_{7} + 3 \beta_{3} - 7 \beta_1) q^{80} + (16 \beta_{6} + 8 \beta_{4} - 20 \beta_{2}) q^{82} + (4 \beta_{5} - 14 \beta_{3} - 10 \beta_1) q^{83} + ( - 4 \beta_{6} - 2 \beta_{4} + 24 \beta_{2}) q^{85} + (7 \beta_{7} - 14 \beta_{5} - 46 \beta_{3} - 7 \beta_1) q^{86} + (23 \beta_{6} + 23 \beta_{4} + 120 \beta_{2} - 60) q^{88} + ( - 3 \beta_{7} + 6 \beta_{5} - 51 \beta_{3} + 3 \beta_1) q^{89} + (5 \beta_{4} - 5 \beta_{2} + 10) q^{91} + ( - 3 \beta_{7} + 3 \beta_{5} + 110 \beta_{3} - 55 \beta_1) q^{92} + ( - 32 \beta_{6} - 32 \beta_{4} - 48 \beta_{2} + 24) q^{94} + (23 \beta_{7} - 13 \beta_{5} + 32 \beta_{3} - 13 \beta_1) q^{95} + (8 \beta_{6} - 70 \beta_{2} - 70) q^{97} + ( - 10 \beta_{7} - 10 \beta_{5} - 44 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} - 40 q^{7} - 48 q^{10} - 60 q^{13} - 44 q^{16} - 96 q^{19} + 120 q^{22} - 36 q^{25} + 60 q^{28} + 168 q^{34} + 168 q^{40} - 100 q^{43} + 48 q^{49} - 420 q^{52} + 40 q^{55} + 560 q^{58} - 68 q^{61} - 8 q^{64} - 180 q^{67} + 360 q^{70} - 60 q^{73} + 564 q^{76} + 420 q^{79} - 80 q^{82} + 96 q^{85} + 60 q^{91} - 840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\nu^{6} - 177\nu^{4} + 2478\nu^{2} - 361 ) / 3363 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -14\nu^{7} + 177\nu^{5} - 2478\nu^{3} + 3724\nu ) / 3363 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{6} + 413\nu^{4} - 4661\nu^{2} + 13167 ) / 3363 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34\nu^{7} - 590\nu^{5} + 7139\nu^{3} - 23617\nu ) / 3363 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -39\nu^{6} + 413\nu^{4} - 4661\nu^{2} - 5320 ) / 3363 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -53\nu^{7} + 590\nu^{5} - 7139\nu^{3} - 11685\nu ) / 3363 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + \beta_{4} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - \beta_{5} - 10\beta_{3} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{6} + 28\beta_{4} + 79\beta_{2} - 79 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{7} - 28\beta_{5} - 121\beta_{3} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -177\beta_{6} + 177\beta_{4} - 973 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -177\beta_{7} - 177\beta_{5} - 1858\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−3.05907 1.76616i
−1.06868 0.617004i
1.06868 + 0.617004i
3.05907 + 1.76616i
−3.05907 + 1.76616i
−1.06868 + 0.617004i
1.06868 0.617004i
3.05907 1.76616i
−3.05907 1.76616i 0 4.23861 + 7.34149i 0.533068 0.923301i 0 0.477226 15.8150i 0 −3.26139 + 1.88296i
46.2 −1.06868 0.617004i 0 −1.23861 2.14534i 4.08850 7.08149i 0 −10.4772 7.99294i 0 −8.73861 + 5.04524i
46.3 1.06868 + 0.617004i 0 −1.23861 2.14534i −4.08850 + 7.08149i 0 −10.4772 7.99294i 0 −8.73861 + 5.04524i
46.4 3.05907 + 1.76616i 0 4.23861 + 7.34149i −0.533068 + 0.923301i 0 0.477226 15.8150i 0 −3.26139 + 1.88296i
145.1 −3.05907 + 1.76616i 0 4.23861 7.34149i 0.533068 + 0.923301i 0 0.477226 15.8150i 0 −3.26139 1.88296i
145.2 −1.06868 + 0.617004i 0 −1.23861 + 2.14534i 4.08850 + 7.08149i 0 −10.4772 7.99294i 0 −8.73861 5.04524i
145.3 1.06868 0.617004i 0 −1.23861 + 2.14534i −4.08850 7.08149i 0 −10.4772 7.99294i 0 −8.73861 5.04524i
145.4 3.05907 1.76616i 0 4.23861 7.34149i −0.533068 0.923301i 0 0.477226 15.8150i 0 −3.26139 1.88296i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 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.f 8
3.b odd 2 1 inner 171.3.p.f 8
19.d odd 6 1 inner 171.3.p.f 8
57.f even 6 1 inner 171.3.p.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.p.f 8 1.a even 1 1 trivial
171.3.p.f 8 3.b odd 2 1 inner
171.3.p.f 8 19.d odd 6 1 inner
171.3.p.f 8 57.f even 6 1 inner

Hecke kernels

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

\( T_{2}^{8} - 14T_{2}^{6} + 177T_{2}^{4} - 266T_{2}^{2} + 361 \) Copy content Toggle raw display
\( T_{5}^{8} + 68T_{5}^{6} + 4548T_{5}^{4} + 5168T_{5}^{2} + 5776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 14 T^{6} + 177 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 68 T^{6} + 4548 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$7$ \( (T^{2} + 10 T - 5)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 140 T^{2} + 1900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 30 T^{3} + 335 T^{2} + 1050 T + 1225)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 168 T^{6} + 25488 T^{4} + \cdots + 7485696 \) Copy content Toggle raw display
$19$ \( (T^{4} + 48 T^{3} + 1178 T^{2} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 3332 T^{6} + \cdots + 7687066773136 \) Copy content Toggle raw display
$29$ \( T^{8} - 1400 T^{6} + \cdots + 36100000000 \) Copy content Toggle raw display
$31$ \( (T^{4} + 746 T^{2} + 124609)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4470 T^{2} + 2295225)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 2720 T^{6} + \cdots + 14786560000 \) Copy content Toggle raw display
$43$ \( (T^{4} + 50 T^{3} + 3345 T^{2} + \cdots + 714025)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 3968 T^{6} + \cdots + 14541836517376 \) Copy content Toggle raw display
$53$ \( T^{8} - 7044 T^{6} + \cdots + 32083206953616 \) Copy content Toggle raw display
$59$ \( T^{8} - 15500 T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + 34 T^{3} + 3867 T^{2} + \cdots + 7349521)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 90 T^{3} + 2165 T^{2} + \cdots + 286225)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 390758009760000 \) Copy content Toggle raw display
$73$ \( (T^{4} + 30 T^{3} + 10395 T^{2} + \cdots + 90155025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 210 T^{3} + 14765 T^{2} + \cdots + 4225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 7208 T^{2} + 12527536)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 32580 T^{6} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + 420 T^{3} + 72860 T^{2} + \cdots + 197683600)^{2} \) Copy content Toggle raw display
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