Properties

Label 189.2.s.b
Level $189$
Weight $2$
Character orbit 189.s
Analytic conductor $1.509$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{9} + \beta_{8} + \beta_{2}) q^{5} + (2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{7} + ( - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (\beta_{9} + \beta_{8} + \beta_{2}) q^{5} + (2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{7} + ( - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + (4 \beta_{8} + 5 \beta_{7} - 6 \beta_{6} + 3 \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + \cdots - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{4} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{4} + 3 q^{7} - 15 q^{10} + 6 q^{13} + 6 q^{14} - 6 q^{16} + 12 q^{17} + 3 q^{19} + 3 q^{20} + 5 q^{22} - 14 q^{25} - 3 q^{26} + 2 q^{28} + 15 q^{29} - 9 q^{31} - 48 q^{32} + 3 q^{34} + 15 q^{35} + 6 q^{37} - 36 q^{38} + 9 q^{41} + 3 q^{43} - 24 q^{44} - 13 q^{46} - 15 q^{47} - 23 q^{49} - 3 q^{50} + 9 q^{53} + 51 q^{56} - 16 q^{58} + 18 q^{59} + 12 q^{61} - 12 q^{62} + 6 q^{64} + 3 q^{65} - 10 q^{67} + 54 q^{68} + 9 q^{70} + 3 q^{73} + 9 q^{76} - 45 q^{77} + 20 q^{79} + 30 q^{80} + 9 q^{82} + 15 q^{83} + 18 q^{85} + 16 q^{88} - 24 q^{89} - 24 q^{91} - 39 q^{92} - 3 q^{94} + 6 q^{97} - 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232 ) / 72795 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583 ) / 218385 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546 ) / 218385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 ) / 14559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856 ) / 43677 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101 ) / 218385 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + 3\beta_{6} - \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + 5\beta_{8} + \beta_{7} - 12\beta_{6} - 5\beta_{5} - \beta_{3} - 5\beta_{2} + 5\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{9} - \beta_{8} - \beta_{7} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} + 11\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{9} - 8\beta_{8} - 14\beta_{7} + 16\beta_{5} + 9\beta_{4} - 7\beta_{3} + 22\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} - 31\beta_{8} - 8\beta_{7} + 31\beta_{5} - 30\beta_{4} + 8\beta_{3} + \beta_{2} + 43\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + 76 \beta_{3} + 8 \beta_{2} - 112 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 95\beta_{9} + 189\beta_{8} + 94\beta_{7} + 37\beta_{5} + 84\beta_{4} + 47\beta_{3} - 194\beta_{2} - 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.827154 1.43267i
−1.04536 + 1.81062i
−0.539982 + 0.935277i
0.187540 0.324828i
1.07065 1.85442i
0.827154 + 1.43267i
−1.04536 1.81062i
−0.539982 0.935277i
0.187540 + 0.324828i
1.07065 + 1.85442i
−1.81474 1.04774i 0 1.19552 + 2.07070i 2.08983 0 −0.879217 + 2.49539i 0.819421i 0 −3.79250 2.18960i
17.2 −1.30778 0.755047i 0 0.140193 + 0.242822i 0.775876 0 2.05881 1.66171i 2.59678i 0 −1.01468 0.585823i
17.3 0.254498 + 0.146935i 0 −0.956820 1.65726i −3.06027 0 −1.22581 2.34465i 1.15010i 0 −0.778834 0.449660i
17.4 0.621951 + 0.359083i 0 −0.742118 1.28539i 1.44755 0 2.19442 + 1.47801i 2.50226i 0 0.900304 + 0.519791i
17.5 2.24607 + 1.29677i 0 2.36322 + 4.09323i −1.25299 0 −0.648211 2.56512i 7.07116i 0 −2.81429 1.62483i
89.1 −1.81474 + 1.04774i 0 1.19552 2.07070i 2.08983 0 −0.879217 2.49539i 0.819421i 0 −3.79250 + 2.18960i
89.2 −1.30778 + 0.755047i 0 0.140193 0.242822i 0.775876 0 2.05881 + 1.66171i 2.59678i 0 −1.01468 + 0.585823i
89.3 0.254498 0.146935i 0 −0.956820 + 1.65726i −3.06027 0 −1.22581 + 2.34465i 1.15010i 0 −0.778834 + 0.449660i
89.4 0.621951 0.359083i 0 −0.742118 + 1.28539i 1.44755 0 2.19442 1.47801i 2.50226i 0 0.900304 0.519791i
89.5 2.24607 1.29677i 0 2.36322 4.09323i −1.25299 0 −0.648211 + 2.56512i 7.07116i 0 −2.81429 + 1.62483i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s 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.s.b 10
3.b odd 2 1 63.2.s.b yes 10
4.b odd 2 1 3024.2.df.b 10
7.b odd 2 1 1323.2.s.b 10
7.c even 3 1 1323.2.i.b 10
7.c even 3 1 1323.2.o.d 10
7.d odd 6 1 189.2.i.b 10
7.d odd 6 1 1323.2.o.c 10
9.c even 3 1 63.2.i.b 10
9.c even 3 1 567.2.p.d 10
9.d odd 6 1 189.2.i.b 10
9.d odd 6 1 567.2.p.c 10
12.b even 2 1 1008.2.df.b 10
21.c even 2 1 441.2.s.b 10
21.g even 6 1 63.2.i.b 10
21.g even 6 1 441.2.o.d 10
21.h odd 6 1 441.2.i.b 10
21.h odd 6 1 441.2.o.c 10
28.f even 6 1 3024.2.ca.b 10
36.f odd 6 1 1008.2.ca.b 10
36.h even 6 1 3024.2.ca.b 10
63.g even 3 1 441.2.s.b 10
63.h even 3 1 441.2.o.d 10
63.i even 6 1 567.2.p.d 10
63.i even 6 1 1323.2.o.d 10
63.j odd 6 1 1323.2.o.c 10
63.k odd 6 1 63.2.s.b yes 10
63.l odd 6 1 441.2.i.b 10
63.n odd 6 1 1323.2.s.b 10
63.o even 6 1 1323.2.i.b 10
63.s even 6 1 inner 189.2.s.b 10
63.t odd 6 1 441.2.o.c 10
63.t odd 6 1 567.2.p.c 10
84.j odd 6 1 1008.2.ca.b 10
252.n even 6 1 1008.2.df.b 10
252.bn odd 6 1 3024.2.df.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b 10 9.c even 3 1
63.2.i.b 10 21.g even 6 1
63.2.s.b yes 10 3.b odd 2 1
63.2.s.b yes 10 63.k odd 6 1
189.2.i.b 10 7.d odd 6 1
189.2.i.b 10 9.d odd 6 1
189.2.s.b 10 1.a even 1 1 trivial
189.2.s.b 10 63.s even 6 1 inner
441.2.i.b 10 21.h odd 6 1
441.2.i.b 10 63.l odd 6 1
441.2.o.c 10 21.h odd 6 1
441.2.o.c 10 63.t odd 6 1
441.2.o.d 10 21.g even 6 1
441.2.o.d 10 63.h even 3 1
441.2.s.b 10 21.c even 2 1
441.2.s.b 10 63.g even 3 1
567.2.p.c 10 9.d odd 6 1
567.2.p.c 10 63.t odd 6 1
567.2.p.d 10 9.c even 3 1
567.2.p.d 10 63.i even 6 1
1008.2.ca.b 10 36.f odd 6 1
1008.2.ca.b 10 84.j odd 6 1
1008.2.df.b 10 12.b even 2 1
1008.2.df.b 10 252.n even 6 1
1323.2.i.b 10 7.c even 3 1
1323.2.i.b 10 63.o even 6 1
1323.2.o.c 10 7.d odd 6 1
1323.2.o.c 10 63.j odd 6 1
1323.2.o.d 10 7.c even 3 1
1323.2.o.d 10 63.i even 6 1
1323.2.s.b 10 7.b odd 2 1
1323.2.s.b 10 63.n odd 6 1
3024.2.ca.b 10 28.f even 6 1
3024.2.ca.b 10 36.h even 6 1
3024.2.df.b 10 4.b odd 2 1
3024.2.df.b 10 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 7T_{2}^{8} + 42T_{2}^{6} + 24T_{2}^{5} - 46T_{2}^{4} - 21T_{2}^{3} + 52T_{2}^{2} - 21T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 7 T^{8} + 42 T^{6} + 24 T^{5} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - 9 T^{3} + 6 T^{2} + 12 T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + 16 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 44 T^{8} + 684 T^{6} + \cdots + 2883 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + 3 T^{8} + 54 T^{7} + \cdots + 3267 \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + 111 T^{8} + \cdots + 263169 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} - 24 T^{8} + \cdots + 2187 \) Copy content Toggle raw display
$23$ \( T^{10} + 65 T^{8} + 1299 T^{6} + \cdots + 27 \) Copy content Toggle raw display
$29$ \( T^{10} - 15 T^{9} + 5 T^{8} + \cdots + 186003 \) Copy content Toggle raw display
$31$ \( T^{10} + 9 T^{9} - 6 T^{8} + \cdots + 16875 \) Copy content Toggle raw display
$37$ \( T^{10} - 6 T^{9} + 88 T^{8} + \cdots + 369664 \) Copy content Toggle raw display
$41$ \( T^{10} - 9 T^{9} + 171 T^{8} + \cdots + 40487769 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + 136 T^{8} + \cdots + 12243001 \) Copy content Toggle raw display
$47$ \( T^{10} + 15 T^{9} + 186 T^{8} + \cdots + 321489 \) Copy content Toggle raw display
$53$ \( T^{10} - 9 T^{9} - 28 T^{8} + \cdots + 871563 \) Copy content Toggle raw display
$59$ \( T^{10} - 18 T^{9} + 240 T^{8} + \cdots + 4100625 \) Copy content Toggle raw display
$61$ \( T^{10} - 12 T^{9} - 54 T^{8} + \cdots + 826875 \) Copy content Toggle raw display
$67$ \( T^{10} + 10 T^{9} + 153 T^{8} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{10} + 359 T^{8} + \cdots + 46216875 \) Copy content Toggle raw display
$73$ \( T^{10} - 3 T^{9} - 105 T^{8} + \cdots + 789507 \) Copy content Toggle raw display
$79$ \( T^{10} - 20 T^{9} + \cdots + 1067655625 \) Copy content Toggle raw display
$83$ \( T^{10} - 15 T^{9} + \cdots + 340734681 \) Copy content Toggle raw display
$89$ \( T^{10} + 24 T^{9} + 489 T^{8} + \cdots + 32455809 \) Copy content Toggle raw display
$97$ \( T^{10} - 6 T^{9} - 174 T^{8} + \cdots + 9687627 \) Copy content Toggle raw display
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