Properties

Label 171.3.p.b
Level $171$
Weight $3$
Character orbit 171.p
Analytic conductor $4.659$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + 5 q^{7} + ( - 10 \zeta_{6} + 5) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + 5 q^{7} + ( - 10 \zeta_{6} + 5) q^{8} + ( - 4 \zeta_{6} + 8) q^{10} - 4 q^{11} + ( - 7 \zeta_{6} + 14) q^{13} + (5 \zeta_{6} + 5) q^{14} + ( - 11 \zeta_{6} + 11) q^{16} + (8 \zeta_{6} - 8) q^{17} + 19 q^{19} - 4 q^{20} + ( - 4 \zeta_{6} - 4) q^{22} + 4 \zeta_{6} q^{23} + 9 \zeta_{6} q^{25} + 21 q^{26} - 5 \zeta_{6} q^{28} + (32 \zeta_{6} - 64) q^{29} + (6 \zeta_{6} - 3) q^{31} + (9 \zeta_{6} - 18) q^{32} + (8 \zeta_{6} - 16) q^{34} + ( - 20 \zeta_{6} + 20) q^{35} + (18 \zeta_{6} - 9) q^{37} + (19 \zeta_{6} + 19) q^{38} + ( - 20 \zeta_{6} - 20) q^{40} + ( - 8 \zeta_{6} - 8) q^{41} + (11 \zeta_{6} - 11) q^{43} + 4 \zeta_{6} q^{44} + (8 \zeta_{6} - 4) q^{46} - 56 \zeta_{6} q^{47} - 24 q^{49} + (18 \zeta_{6} - 9) q^{50} + ( - 7 \zeta_{6} - 7) q^{52} + ( - 28 \zeta_{6} + 56) q^{53} + (16 \zeta_{6} - 16) q^{55} + ( - 50 \zeta_{6} + 25) q^{56} - 96 q^{58} + (28 \zeta_{6} + 28) q^{59} + 79 \zeta_{6} q^{61} + (9 \zeta_{6} - 9) q^{62} - 71 q^{64} + ( - 56 \zeta_{6} + 28) q^{65} + (11 \zeta_{6} - 22) q^{67} + 8 q^{68} + ( - 20 \zeta_{6} + 40) q^{70} + (64 \zeta_{6} + 64) q^{71} + ( - \zeta_{6} + 1) q^{73} + (27 \zeta_{6} - 27) q^{74} - 19 \zeta_{6} q^{76} - 20 q^{77} + (19 \zeta_{6} + 19) q^{79} - 44 \zeta_{6} q^{80} - 24 \zeta_{6} q^{82} - 112 q^{83} + 32 \zeta_{6} q^{85} + (11 \zeta_{6} - 22) q^{86} + (40 \zeta_{6} - 20) q^{88} + (44 \zeta_{6} - 88) q^{89} + ( - 35 \zeta_{6} + 70) q^{91} + ( - 4 \zeta_{6} + 4) q^{92} + ( - 112 \zeta_{6} + 56) q^{94} + ( - 76 \zeta_{6} + 76) q^{95} + (64 \zeta_{6} + 64) q^{97} + ( - 24 \zeta_{6} - 24) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - q^{4} + 4 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - q^{4} + 4 q^{5} + 10 q^{7} + 12 q^{10} - 8 q^{11} + 21 q^{13} + 15 q^{14} + 11 q^{16} - 8 q^{17} + 38 q^{19} - 8 q^{20} - 12 q^{22} + 4 q^{23} + 9 q^{25} + 42 q^{26} - 5 q^{28} - 96 q^{29} - 27 q^{32} - 24 q^{34} + 20 q^{35} + 57 q^{38} - 60 q^{40} - 24 q^{41} - 11 q^{43} + 4 q^{44} - 56 q^{47} - 48 q^{49} - 21 q^{52} + 84 q^{53} - 16 q^{55} - 192 q^{58} + 84 q^{59} + 79 q^{61} - 9 q^{62} - 142 q^{64} - 33 q^{67} + 16 q^{68} + 60 q^{70} + 192 q^{71} + q^{73} - 27 q^{74} - 19 q^{76} - 40 q^{77} + 57 q^{79} - 44 q^{80} - 24 q^{82} - 224 q^{83} + 32 q^{85} - 33 q^{86} - 132 q^{89} + 105 q^{91} + 4 q^{92} + 76 q^{95} + 192 q^{97} - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i 0 5.00000 8.66025i 0 6.00000 3.46410i
145.1 1.50000 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i 0 5.00000 8.66025i 0 6.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
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.b yes 2
3.b odd 2 1 171.3.p.a 2
19.d odd 6 1 inner 171.3.p.b yes 2
57.f even 6 1 171.3.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.p.a 2 3.b odd 2 1
171.3.p.a 2 57.f even 6 1
171.3.p.b yes 2 1.a even 1 1 trivial
171.3.p.b yes 2 19.d odd 6 1 inner

Hecke kernels

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

\( T_{2}^{2} - 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 4T_{5} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$7$ \( (T - 5)^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 96T + 3072 \) Copy content Toggle raw display
$31$ \( T^{2} + 27 \) Copy content Toggle raw display
$37$ \( T^{2} + 243 \) Copy content Toggle raw display
$41$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$53$ \( T^{2} - 84T + 2352 \) Copy content Toggle raw display
$59$ \( T^{2} - 84T + 2352 \) Copy content Toggle raw display
$61$ \( T^{2} - 79T + 6241 \) Copy content Toggle raw display
$67$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
$71$ \( T^{2} - 192T + 12288 \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 57T + 1083 \) Copy content Toggle raw display
$83$ \( (T + 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 132T + 5808 \) Copy content Toggle raw display
$97$ \( T^{2} - 192T + 12288 \) Copy content Toggle raw display
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