Properties

Label 189.2.p.b
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} + ( 2 + \beta_{2} ) q^{7} -2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} + ( 2 + \beta_{2} ) q^{7} -2 \beta_{3} q^{8} + ( 4 - 2 \beta_{2} ) q^{10} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{11} + ( -2 + 4 \beta_{2} ) q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{14} + ( 4 - 4 \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 1 + \beta_{2} ) q^{19} -8 q^{22} -5 \beta_{1} q^{23} -\beta_{2} q^{25} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{26} -\beta_{3} q^{29} + ( -6 + 3 \beta_{2} ) q^{31} + ( -2 + 4 \beta_{2} ) q^{34} + ( 4 \beta_{1} - 5 \beta_{3} ) q^{35} + ( 4 - 4 \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{3} ) q^{38} + ( -4 - 4 \beta_{2} ) q^{40} + ( 2 \beta_{1} - \beta_{3} ) q^{41} -7 q^{43} -10 \beta_{2} q^{46} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{47} + ( 3 + 5 \beta_{2} ) q^{49} -\beta_{3} q^{50} + ( -\beta_{1} + \beta_{3} ) q^{53} + ( -8 + 16 \beta_{2} ) q^{55} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{56} + ( 2 - 2 \beta_{2} ) q^{58} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{59} + ( -3 - 3 \beta_{2} ) q^{61} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{62} -8 q^{64} + 6 \beta_{1} q^{65} -2 \beta_{2} q^{67} + ( 10 - 2 \beta_{2} ) q^{70} + 2 \beta_{3} q^{71} + ( 14 - 7 \beta_{2} ) q^{73} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{74} + ( -12 \beta_{1} + 8 \beta_{3} ) q^{77} + ( 4 - 4 \beta_{2} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{80} + ( 2 + 2 \beta_{2} ) q^{82} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{83} + 6 q^{85} -7 \beta_{1} q^{86} + 16 \beta_{2} q^{88} + ( -\beta_{1} + 2 \beta_{3} ) q^{89} + ( -8 + 10 \beta_{2} ) q^{91} + ( 12 - 6 \beta_{2} ) q^{94} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{95} + ( 5 - 10 \beta_{2} ) q^{97} + ( 3 \beta_{1} + 5 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{7} + O(q^{10}) \) \( 4q + 10q^{7} + 12q^{10} + 8q^{16} + 6q^{19} - 32q^{22} - 2q^{25} - 18q^{31} + 8q^{37} - 24q^{40} - 28q^{43} - 20q^{46} + 22q^{49} + 4q^{58} - 18q^{61} - 32q^{64} - 4q^{67} + 36q^{70} + 42q^{73} + 8q^{79} + 12q^{82} + 24q^{85} + 32q^{88} - 12q^{91} + 36q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

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
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 0 −1.22474 + 2.12132i 0 2.50000 + 0.866025i 2.82843i 0 3.00000 1.73205i
26.2 1.22474 + 0.707107i 0 0 1.22474 2.12132i 0 2.50000 + 0.866025i 2.82843i 0 3.00000 1.73205i
80.1 −1.22474 + 0.707107i 0 0 −1.22474 2.12132i 0 2.50000 0.866025i 2.82843i 0 3.00000 + 1.73205i
80.2 1.22474 0.707107i 0 0 1.22474 + 2.12132i 0 2.50000 0.866025i 2.82843i 0 3.00000 + 1.73205i
\(n\): 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.b 4
3.b odd 2 1 inner 189.2.p.b 4
7.c even 3 1 1323.2.c.c 4
7.d odd 6 1 inner 189.2.p.b 4
7.d odd 6 1 1323.2.c.c 4
9.c even 3 1 567.2.i.e 4
9.c even 3 1 567.2.s.c 4
9.d odd 6 1 567.2.i.e 4
9.d odd 6 1 567.2.s.c 4
21.g even 6 1 inner 189.2.p.b 4
21.g even 6 1 1323.2.c.c 4
21.h odd 6 1 1323.2.c.c 4
63.i even 6 1 567.2.s.c 4
63.k odd 6 1 567.2.i.e 4
63.s even 6 1 567.2.i.e 4
63.t odd 6 1 567.2.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.b 4 1.a even 1 1 trivial
189.2.p.b 4 3.b odd 2 1 inner
189.2.p.b 4 7.d odd 6 1 inner
189.2.p.b 4 21.g even 6 1 inner
567.2.i.e 4 9.c even 3 1
567.2.i.e 4 9.d odd 6 1
567.2.i.e 4 63.k odd 6 1
567.2.i.e 4 63.s even 6 1
567.2.s.c 4 9.c even 3 1
567.2.s.c 4 9.d odd 6 1
567.2.s.c 4 63.i even 6 1
567.2.s.c 4 63.t odd 6 1
1323.2.c.c 4 7.c even 3 1
1323.2.c.c 4 7.d odd 6 1
1323.2.c.c 4 21.g even 6 1
1323.2.c.c 4 21.h odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} ) \)
$3$ 1
$5$ \( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 7 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 104 T^{2} + 8007 T^{4} + 292136 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 98 T^{2} + 6123 T^{4} + 341138 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 70 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \)
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