Properties

Label 189.2.p.c
Level $189$
Weight $2$
Character orbit 189.p
Analytic conductor $1.509$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (2 \beta_{2} - 3) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (2 \beta_{2} - 3) q^{7} + \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (2 \beta_{2} - 1) q^{13} + (2 \beta_{3} - 3 \beta_1) q^{14} + ( - \beta_{2} + 1) q^{16} + ( - 2 \beta_{3} - 2 \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + 10 q^{22} - 2 \beta_1 q^{23} + 5 \beta_{2} q^{25} + (2 \beta_{3} - \beta_1) q^{26} + ( - 3 \beta_{2} - 6) q^{28} + 2 \beta_{3} q^{29} + (\beta_{2} - 2) q^{31} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + ( - 20 \beta_{2} + 10) q^{34} + (5 \beta_{2} - 5) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} + ( - 2 \beta_{3} + 4 \beta_1) q^{41} - 7 q^{43} + 6 \beta_1 q^{44} - 10 \beta_{2} q^{46} + (4 \beta_{3} - 2 \beta_1) q^{47} + ( - 8 \beta_{2} + 5) q^{49} + 5 \beta_{3} q^{50} + (3 \beta_{2} - 6) q^{52} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{3} - 2 \beta_1) q^{56} + (10 \beta_{2} - 10) q^{58} + (2 \beta_{3} + 2 \beta_1) q^{59} + (5 \beta_{2} + 5) q^{61} + (\beta_{3} - 2 \beta_1) q^{62} + 13 q^{64} + \beta_{2} q^{67} + ( - 12 \beta_{3} + 6 \beta_1) q^{68} - 4 \beta_{3} q^{71} + ( - 4 \beta_{2} + 8) q^{73} + (5 \beta_{3} - 5 \beta_1) q^{74} + ( - 12 \beta_{2} + 6) q^{76} + (6 \beta_{3} - 2 \beta_1) q^{77} + (11 \beta_{2} - 11) q^{79} + (10 \beta_{2} + 10) q^{82} + (2 \beta_{3} - 4 \beta_1) q^{83} - 7 \beta_1 q^{86} + 10 \beta_{2} q^{88} + (8 \beta_{3} - 4 \beta_1) q^{89} + ( - 4 \beta_{2} - 1) q^{91} - 6 \beta_{3} q^{92} + (10 \beta_{2} - 20) q^{94} + ( - 2 \beta_{2} + 1) q^{97} + ( - 8 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 8 q^{7} + 2 q^{16} - 12 q^{19} + 40 q^{22} + 10 q^{25} - 30 q^{28} - 6 q^{31} - 10 q^{37} - 28 q^{43} - 20 q^{46} + 4 q^{49} - 18 q^{52} - 20 q^{58} + 30 q^{61} + 52 q^{64} + 2 q^{67} + 24 q^{73} - 22 q^{79} + 60 q^{82} + 20 q^{88} - 12 q^{91} - 60 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
26.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
80.1 −1.93649 + 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
80.2 1.93649 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.p.c 4
3.b odd 2 1 inner 189.2.p.c 4
7.c even 3 1 1323.2.c.b 4
7.d odd 6 1 inner 189.2.p.c 4
7.d odd 6 1 1323.2.c.b 4
9.c even 3 1 567.2.i.c 4
9.c even 3 1 567.2.s.e 4
9.d odd 6 1 567.2.i.c 4
9.d odd 6 1 567.2.s.e 4
21.g even 6 1 inner 189.2.p.c 4
21.g even 6 1 1323.2.c.b 4
21.h odd 6 1 1323.2.c.b 4
63.i even 6 1 567.2.s.e 4
63.k odd 6 1 567.2.i.c 4
63.s even 6 1 567.2.i.c 4
63.t odd 6 1 567.2.s.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c 4 1.a even 1 1 trivial
189.2.p.c 4 3.b odd 2 1 inner
189.2.p.c 4 7.d odd 6 1 inner
189.2.p.c 4 21.g even 6 1 inner
567.2.i.c 4 9.c even 3 1
567.2.i.c 4 9.d odd 6 1
567.2.i.c 4 63.k odd 6 1
567.2.i.c 4 63.s even 6 1
567.2.s.e 4 9.c even 3 1
567.2.s.e 4 9.d odd 6 1
567.2.s.e 4 63.i even 6 1
567.2.s.e 4 63.t odd 6 1
1323.2.c.b 4 7.c even 3 1
1323.2.c.b 4 7.d odd 6 1
1323.2.c.b 4 21.g even 6 1
1323.2.c.b 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$53$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$59$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$61$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 240 T^{2} + 57600 \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
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