Properties

Label 189.3.d.e
Level $189$
Weight $3$
Character orbit 189.d
Analytic conductor $5.150$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2355463701504.14
Defining polynomial: \(x^{8} + 9 x^{6} + 79 x^{4} + 18 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( 5 - \beta_{1} ) q^{4} + \beta_{2} q^{5} + ( 2 + \beta_{1} - \beta_{3} ) q^{7} + ( -3 \beta_{4} + \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( 5 - \beta_{1} ) q^{4} + \beta_{2} q^{5} + ( 2 + \beta_{1} - \beta_{3} ) q^{7} + ( -3 \beta_{4} + \beta_{7} ) q^{8} + ( 2 \beta_{3} + \beta_{6} ) q^{10} -\beta_{7} q^{11} -\beta_{3} q^{13} + ( -\beta_{2} + \beta_{5} - \beta_{7} ) q^{14} + ( 7 - 7 \beta_{1} ) q^{16} + \beta_{5} q^{17} + ( 2 \beta_{3} - \beta_{6} ) q^{19} + ( 6 \beta_{2} - 2 \beta_{5} ) q^{20} + 8 \beta_{1} q^{22} + ( -5 \beta_{4} + \beta_{7} ) q^{23} + ( -29 + 3 \beta_{1} ) q^{25} + ( -\beta_{2} + \beta_{5} ) q^{26} + ( -8 + 4 \beta_{1} - 7 \beta_{3} - \beta_{6} ) q^{28} + ( -5 \beta_{4} - 2 \beta_{7} ) q^{29} + ( 3 \beta_{3} - 2 \beta_{6} ) q^{31} + ( -9 \beta_{4} + 3 \beta_{7} ) q^{32} -9 \beta_{3} q^{34} + ( \beta_{2} - 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{35} + ( -38 - \beta_{1} ) q^{37} + ( -6 \beta_{2} - 2 \beta_{5} ) q^{38} + ( 22 \beta_{3} + 2 \beta_{6} ) q^{40} + ( \beta_{2} - 2 \beta_{5} ) q^{41} + ( -23 - 4 \beta_{1} ) q^{43} + ( 16 \beta_{4} - 4 \beta_{7} ) q^{44} + ( 45 - 13 \beta_{1} ) q^{46} + 5 \beta_{2} q^{47} + ( -5 + 6 \beta_{1} + 2 \beta_{6} ) q^{49} + ( 35 \beta_{4} - 3 \beta_{7} ) q^{50} + ( -7 \beta_{3} - \beta_{6} ) q^{52} + ( 9 \beta_{4} - \beta_{7} ) q^{53} + ( -16 \beta_{3} + \beta_{6} ) q^{55} + ( -11 \beta_{2} + 16 \beta_{4} + 3 \beta_{5} ) q^{56} + ( 45 + 11 \beta_{1} ) q^{58} + ( -2 \beta_{2} - 3 \beta_{5} ) q^{59} + ( 14 \beta_{3} + \beta_{6} ) q^{61} + ( -13 \beta_{2} - 3 \beta_{5} ) q^{62} + ( 53 - 5 \beta_{1} ) q^{64} + ( -6 \beta_{4} + 3 \beta_{7} ) q^{65} + ( 35 + 15 \beta_{1} ) q^{67} + ( -9 \beta_{2} + 5 \beta_{5} ) q^{68} + ( 54 - 30 \beta_{1} - 16 \beta_{3} + \beta_{6} ) q^{70} + ( -13 \beta_{4} + 3 \beta_{7} ) q^{71} + 6 \beta_{3} q^{73} + ( 36 \beta_{4} + \beta_{7} ) q^{74} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{76} + ( 8 \beta_{2} - 16 \beta_{4} - 3 \beta_{7} ) q^{77} + ( 86 - 3 \beta_{1} ) q^{79} + ( 14 \beta_{2} - 14 \beta_{5} ) q^{80} + ( 20 \beta_{3} + \beta_{6} ) q^{82} + ( 5 \beta_{2} + 6 \beta_{5} ) q^{83} -27 \beta_{1} q^{85} + ( 15 \beta_{4} + 4 \beta_{7} ) q^{86} + ( -144 + 16 \beta_{1} ) q^{88} + ( -3 \beta_{2} + 3 \beta_{5} ) q^{89} + ( -27 + 3 \beta_{1} + \beta_{6} ) q^{91} + ( -51 \beta_{4} + 9 \beta_{7} ) q^{92} + ( 10 \beta_{3} + 5 \beta_{6} ) q^{94} + ( -36 \beta_{4} - 9 \beta_{7} ) q^{95} + ( -18 \beta_{3} - \beta_{6} ) q^{97} + ( 16 \beta_{2} + 17 \beta_{4} - 6 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 44q^{4} + 12q^{7} + O(q^{10}) \) \( 8q + 44q^{4} + 12q^{7} + 84q^{16} - 32q^{22} - 244q^{25} - 80q^{28} - 300q^{37} - 168q^{43} + 412q^{46} - 64q^{49} + 316q^{58} + 444q^{64} + 220q^{67} + 552q^{70} + 700q^{79} + 108q^{85} - 1216q^{88} - 228q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 79 x^{4} + 18 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 377 \)\()/79\)
\(\beta_{2}\)\(=\)\((\)\( 44 \nu^{6} + 395 \nu^{4} + 3397 \nu^{2} + 397 \)\()/79\)
\(\beta_{3}\)\(=\)\((\)\( 34 \nu^{7} + 316 \nu^{5} + 2765 \nu^{3} + 1244 \nu \)\()/79\)
\(\beta_{4}\)\(=\)\((\)\( 36 \nu^{7} + 316 \nu^{5} + 2765 \nu^{3} + 16 \nu \)\()/79\)
\(\beta_{5}\)\(=\)\((\)\( -81 \nu^{6} - 711 \nu^{4} - 6399 \nu^{2} - 747 \)\()/79\)
\(\beta_{6}\)\(=\)\((\)\( 95 \nu^{7} + 869 \nu^{5} + 7663 \nu^{3} + 3448 \nu \)\()/79\)
\(\beta_{7}\)\(=\)\((\)\( -99 \nu^{7} - 869 \nu^{5} - 7663 \nu^{3} - 44 \nu \)\()/79\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{5} - 9 \beta_{2} - 9 \beta_{1} - 45\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{7} - 11 \beta_{4}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(43 \beta_{5} + 81 \beta_{2} - 81 \beta_{1} - 387\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(35 \beta_{7} - 35 \beta_{6} + 97 \beta_{4} + 97 \beta_{3}\)\()/6\)
\(\nu^{6}\)\(=\)\(79 \beta_{1} + 377\)
\(\nu^{7}\)\(=\)\((\)\(307 \beta_{7} + 307 \beta_{6} + 851 \beta_{4} - 851 \beta_{3}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.238746 + 0.413520i
−0.238746 0.413520i
1.48088 + 2.56496i
1.48088 2.56496i
−1.48088 2.56496i
−1.48088 + 2.56496i
0.238746 0.413520i
0.238746 + 0.413520i
−3.71106 0 9.77200 8.26535i 0 −2.77200 6.42775i −21.4203 0 30.6732i
55.2 −3.71106 0 9.77200 8.26535i 0 −2.77200 + 6.42775i −21.4203 0 30.6732i
55.3 −2.28648 0 1.22800 6.53330i 0 5.77200 + 3.96030i 6.33813 0 14.9383i
55.4 −2.28648 0 1.22800 6.53330i 0 5.77200 3.96030i 6.33813 0 14.9383i
55.5 2.28648 0 1.22800 6.53330i 0 5.77200 3.96030i −6.33813 0 14.9383i
55.6 2.28648 0 1.22800 6.53330i 0 5.77200 + 3.96030i −6.33813 0 14.9383i
55.7 3.71106 0 9.77200 8.26535i 0 −2.77200 + 6.42775i 21.4203 0 30.6732i
55.8 3.71106 0 9.77200 8.26535i 0 −2.77200 6.42775i 21.4203 0 30.6732i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.d.e 8
3.b odd 2 1 inner 189.3.d.e 8
7.b odd 2 1 inner 189.3.d.e 8
21.c even 2 1 inner 189.3.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.d.e 8 1.a even 1 1 trivial
189.3.d.e 8 3.b odd 2 1 inner
189.3.d.e 8 7.b odd 2 1 inner
189.3.d.e 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 19 T_{2}^{2} + 72 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 72 - 19 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 2916 + 111 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2401 - 294 T + 34 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$11$ \( ( 18432 - 280 T^{2} + T^{4} )^{2} \)
$13$ \( ( 648 + 57 T^{2} + T^{4} )^{2} \)
$17$ \( ( 243 + T^{2} )^{4} \)
$19$ \( ( 23328 + 972 T^{2} + T^{4} )^{2} \)
$23$ \( ( 2592 - 835 T^{2} + T^{4} )^{2} \)
$29$ \( ( 5832 - 1435 T^{2} + T^{4} )^{2} \)
$31$ \( ( 887112 + 3585 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1388 + 75 T + T^{2} )^{4} \)
$41$ \( ( 944784 + 2163 T^{2} + T^{4} )^{2} \)
$43$ \( ( 149 + 42 T + T^{2} )^{4} \)
$47$ \( ( 1822500 + 2775 T^{2} + T^{4} )^{2} \)
$53$ \( ( 103968 - 1963 T^{2} + T^{4} )^{2} \)
$59$ \( ( 3272481 + 4494 T^{2} + T^{4} )^{2} \)
$61$ \( ( 12340512 + 12684 T^{2} + T^{4} )^{2} \)
$67$ \( ( -3350 - 55 T + T^{2} )^{4} \)
$71$ \( ( 609408 - 6355 T^{2} + T^{4} )^{2} \)
$73$ \( ( 839808 + 2052 T^{2} + T^{4} )^{2} \)
$79$ \( ( 7492 - 175 T + T^{2} )^{4} \)
$83$ \( ( 43401744 + 18651 T^{2} + T^{4} )^{2} \)
$89$ \( ( 3779136 + 5859 T^{2} + T^{4} )^{2} \)
$97$ \( ( 41806368 + 20172 T^{2} + T^{4} )^{2} \)
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