Properties

Label 189.3.n.a
Level $189$
Weight $3$
Character orbit 189.n
Analytic conductor $5.150$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,3,Mod(170,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.170");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.63369648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} - 3) q^{5} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{7} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 1) q^{2} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} - 3) q^{5} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{7} + ( - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{8} + ( - 2 \beta_{5} - 4 \beta_{4} - \beta_{3} - 8 \beta_{2} - 10) q^{10} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_1 + 4) q^{11} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 6) q^{13} + (4 \beta_{5} - 2 \beta_{3} - 9 \beta_{2} + \beta_1 - 11) q^{14} + (4 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 11 \beta_{2} - 7) q^{16} + ( - 6 \beta_{5} - 2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 1) q^{17} + ( - 2 \beta_{3} + 7 \beta_{2} + 2 \beta_1) q^{19} + ( - 7 \beta_{4} - \beta_{3} - 7 \beta_{2} - \beta_1 - 14) q^{20} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 19 \beta_{2} + 18) q^{22} + ( - 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{23} + (4 \beta_{5} - 4 \beta_{4} - \beta_1 + 9) q^{25} + (2 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 2 \beta_1 + 18) q^{26} + ( - \beta_{5} - 11 \beta_{4} + 3 \beta_{3} + 31 \beta_{2} + 2) q^{28} + (\beta_{4} + 6 \beta_{2} + 12) q^{29} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 4) q^{31} + ( - 5 \beta_{4} + 2 \beta_{3} + 30 \beta_{2} + 2 \beta_1 + 60) q^{32} + ( - 22 \beta_{5} - 11 \beta_{4} - 42 \beta_{2} - 22) q^{34} + (10 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 10 \beta_{2} - 3 \beta_1 + 3) q^{35} + (14 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 14) q^{37} + (\beta_{5} + \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 3) q^{38} + (2 \beta_{5} - 2 \beta_{4} - 19) q^{40} + ( - 10 \beta_{5} - 4 \beta_{3} - 5 \beta_{2} + 8 \beta_1 - 5) q^{41} + (8 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 38 \beta_{2} + 3 \beta_1 + 8) q^{43} + (26 \beta_{4} - 2 \beta_{3} - 21 \beta_{2} - 2 \beta_1 - 42) q^{44} + ( - 2 \beta_{5} - 4 \beta_{4} - 21 \beta_{2} - 23) q^{46} + (18 \beta_{5} + 2 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 30) q^{47} + (12 \beta_{5} + 12 \beta_{4} - 3 \beta_{3} + 18 \beta_{2} - 3 \beta_1 + 34) q^{49} + ( - 2 \beta_{5} + 3 \beta_{3} + 31 \beta_{2} - 6 \beta_1 - 33) q^{50} + ( - 11 \beta_{5} + 11 \beta_{4} - 21) q^{52} + (15 \beta_{5} + 3 \beta_{3} - \beta_{2} - 6 \beta_1 + 16) q^{53} + ( - 3 \beta_{5} + 3 \beta_{4} - \beta_1 + 28) q^{55} + (12 \beta_{5} + 24 \beta_{4} - 35 \beta_{2} - 9 \beta_1 - 46) q^{56} + ( - 6 \beta_{5} + 6 \beta_{4} + \beta_1 + 2) q^{58} + ( - 6 \beta_{4} - 3 \beta_{3} - 17 \beta_{2} - 3 \beta_1 - 34) q^{59} + ( - 6 \beta_{5} - 12 \beta_{4} + 8 \beta_{3} - 16 \beta_{2} - 22) q^{61} + ( - 24 \beta_{5} - 24 \beta_{4} + 12 \beta_{3} + 40 \beta_{2} - 6 \beta_1 - 4) q^{62} + ( - 20 \beta_{5} + 20 \beta_{4} - 3 \beta_1 - 22) q^{64} + ( - 8 \beta_{5} - 19 \beta_{2} + 11) q^{65} + ( - 12 \beta_{5} - 6 \beta_{4} + 7 \beta_{3} - 41 \beta_{2} - 7 \beta_1 - 12) q^{67} + ( - 18 \beta_{5} - 18 \beta_{4} - 6 \beta_{3} - 136 \beta_{2} + 3 \beta_1 - 86) q^{68} + (26 \beta_{5} + 19 \beta_{4} + 4 \beta_{3} + 74 \beta_{2} - 3 \beta_1 + 55) q^{70} + (6 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 32 \beta_{2} + 2 \beta_1 - 10) q^{71} + (18 \beta_{5} + 36 \beta_{4} + 7 \beta_{3} - 18 \beta_{2}) q^{73} + (18 \beta_{5} + 18 \beta_{4} + 4 \beta_{3} + 102 \beta_{2} - 2 \beta_1 + 69) q^{74} + (8 \beta_{5} + 16 \beta_{4} + 3 \beta_{3} - 26 \beta_{2} - 18) q^{76} + ( - 17 \beta_{5} - 20 \beta_{4} - 7 \beta_{3} + 35 \beta_{2} + 11 \beta_1 + 29) q^{77} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 29 \beta_{2} - 27) q^{79} + ( - 7 \beta_{5} - 2 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 5) q^{80} + ( - 34 \beta_{5} - 17 \beta_{4} + 2 \beta_{3} - 68 \beta_{2} - 2 \beta_1 - 34) q^{82} + ( - 9 \beta_{3} - 8 \beta_{2} - 9 \beta_1 - 16) q^{83} + ( - 23 \beta_{5} - 46 \beta_{4} - 12 \beta_{3} - 72 \beta_{2} - 95) q^{85} + (29 \beta_{5} + 29 \beta_{4} + 14 \beta_{3} + 70 \beta_{2} - 7 \beta_1 + 64) q^{86} + (23 \beta_{5} - 23 \beta_{4} + 12 \beta_1 + 161) q^{88} + ( - 8 \beta_{4} - 5 \beta_{3} - 28 \beta_{2} - 5 \beta_1 - 56) q^{89} + ( - 12 \beta_{5} - 18 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 13) q^{91} + ( - 9 \beta_{4} + 2 \beta_{3} - 20 \beta_{2} + 2 \beta_1 - 40) q^{92} + ( - 12 \beta_{5} - 6 \beta_{4} + 12 \beta_{3} + 138 \beta_{2} - 12 \beta_1 - 12) q^{94} + (\beta_{4} - 2 \beta_{3} + 16 \beta_{2} - 2 \beta_1 + 32) q^{95} + ( - 24 \beta_{5} - 12 \beta_{4} - \beta_{3} + 42 \beta_{2} + \beta_1 - 24) q^{97} + (5 \beta_{5} + 9 \beta_{4} + 12 \beta_{3} + 96 \beta_{2} + 9 \beta_1 + 110) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 13 q^{4} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 13 q^{4} - 5 q^{7} - 19 q^{10} + 11 q^{13} - 51 q^{14} - 47 q^{16} + 33 q^{17} - 19 q^{19} - 45 q^{20} + 65 q^{22} + 52 q^{25} + 81 q^{26} - 42 q^{28} + 51 q^{29} + 46 q^{31} + 291 q^{32} + 93 q^{34} - 57 q^{35} + 7 q^{37} - 114 q^{40} + 27 q^{41} - 99 q^{43} - 273 q^{44} - 57 q^{46} + 156 q^{47} + 69 q^{49} - 294 q^{50} - 126 q^{52} + 45 q^{53} + 166 q^{55} - 297 q^{56} + 14 q^{58} - 144 q^{59} - 22 q^{61} - 138 q^{64} + 147 q^{65} + 98 q^{67} - 29 q^{70} - 101 q^{73} - 99 q^{76} + 195 q^{77} - 90 q^{79} + 93 q^{80} + 151 q^{82} - 99 q^{83} - 159 q^{85} + 990 q^{88} - 243 q^{89} + 177 q^{91} - 147 q^{92} - 444 q^{94} + 135 q^{95} - 161 q^{97} + 360 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
−1.34153 + 2.32360i
1.98253 3.43384i
−0.140998 + 0.244215i
−1.34153 2.32360i
1.98253 + 3.43384i
−0.140998 0.244215i
−3.28247 + 1.89513i 0 5.18306 8.97732i 0.326165i 0 −4.08365 + 5.68540i 24.1293i 0 −0.618126 1.07063i
170.2 −0.895777 + 0.517177i 0 −1.46506 + 2.53755i 2.42975i 0 6.82589 + 1.55153i 7.16819i 0 1.25661 + 2.17651i
170.3 2.67824 1.54629i 0 2.78200 4.81856i 6.55667i 0 −5.24223 4.63886i 4.83675i 0 −10.1385 17.5604i
179.1 −3.28247 1.89513i 0 5.18306 + 8.97732i 0.326165i 0 −4.08365 5.68540i 24.1293i 0 −0.618126 + 1.07063i
179.2 −0.895777 0.517177i 0 −1.46506 2.53755i 2.42975i 0 6.82589 1.55153i 7.16819i 0 1.25661 2.17651i
179.3 2.67824 + 1.54629i 0 2.78200 + 4.81856i 6.55667i 0 −5.24223 + 4.63886i 4.83675i 0 −10.1385 + 17.5604i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 170.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.n.a 6
3.b odd 2 1 63.3.n.a yes 6
7.c even 3 1 189.3.j.a 6
9.c even 3 1 63.3.j.a 6
9.d odd 6 1 189.3.j.a 6
21.c even 2 1 441.3.n.c 6
21.g even 6 1 441.3.j.c 6
21.g even 6 1 441.3.r.c 6
21.h odd 6 1 63.3.j.a 6
21.h odd 6 1 441.3.r.b 6
63.g even 3 1 63.3.n.a yes 6
63.h even 3 1 441.3.r.b 6
63.k odd 6 1 441.3.n.c 6
63.l odd 6 1 441.3.j.c 6
63.n odd 6 1 inner 189.3.n.a 6
63.t odd 6 1 441.3.r.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.a 6 9.c even 3 1
63.3.j.a 6 21.h odd 6 1
63.3.n.a yes 6 3.b odd 2 1
63.3.n.a yes 6 63.g even 3 1
189.3.j.a 6 7.c even 3 1
189.3.j.a 6 9.d odd 6 1
189.3.n.a 6 1.a even 1 1 trivial
189.3.n.a 6 63.n odd 6 1 inner
441.3.j.c 6 21.g even 6 1
441.3.j.c 6 63.l odd 6 1
441.3.n.c 6 21.c even 2 1
441.3.n.c 6 63.k odd 6 1
441.3.r.b 6 21.h odd 6 1
441.3.r.b 6 63.h even 3 1
441.3.r.c 6 21.g even 6 1
441.3.r.c 6 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} - 8T_{2}^{4} - 33T_{2}^{3} + 100T_{2}^{2} + 231T_{2} + 147 \) acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} - 8 T^{4} - 33 T^{3} + \cdots + 147 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 49 T^{4} + 259 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} - 22 T^{4} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{6} + 457 T^{4} + 60859 T^{2} + \cdots + 1982907 \) Copy content Toggle raw display
$13$ \( T^{6} - 11 T^{5} + 182 T^{4} + \cdots + 499849 \) Copy content Toggle raw display
$17$ \( T^{6} - 33 T^{5} - 252 T^{4} + \cdots + 31434507 \) Copy content Toggle raw display
$19$ \( T^{6} + 19 T^{5} + 422 T^{4} + \cdots + 214369 \) Copy content Toggle raw display
$23$ \( T^{6} + 393 T^{4} + 31491 T^{2} + \cdots + 128547 \) Copy content Toggle raw display
$29$ \( T^{6} - 51 T^{5} + 1144 T^{4} + \cdots + 694083 \) Copy content Toggle raw display
$31$ \( T^{6} - 46 T^{5} + 2440 T^{4} + \cdots + 21455424 \) Copy content Toggle raw display
$37$ \( T^{6} - 7 T^{5} + 2230 T^{4} + \cdots + 564110001 \) Copy content Toggle raw display
$41$ \( T^{6} - 27 T^{5} + \cdots + 1387137027 \) Copy content Toggle raw display
$43$ \( T^{6} + 99 T^{5} + \cdots + 432265681 \) Copy content Toggle raw display
$47$ \( T^{6} - 156 T^{5} + \cdots + 29274835968 \) Copy content Toggle raw display
$53$ \( T^{6} - 45 T^{5} + \cdots + 5141134827 \) Copy content Toggle raw display
$59$ \( T^{6} + 144 T^{5} + \cdots + 813189888 \) Copy content Toggle raw display
$61$ \( T^{6} + 22 T^{5} + \cdots + 25823204416 \) Copy content Toggle raw display
$67$ \( T^{6} - 98 T^{5} + \cdots + 44264793664 \) Copy content Toggle raw display
$71$ \( T^{6} + 5568 T^{4} + \cdots + 109734912 \) Copy content Toggle raw display
$73$ \( T^{6} + 101 T^{5} + \cdots + 653628591729 \) Copy content Toggle raw display
$79$ \( T^{6} + 90 T^{5} + \cdots + 279290944 \) Copy content Toggle raw display
$83$ \( T^{6} + 99 T^{5} + \cdots + 165842832483 \) Copy content Toggle raw display
$89$ \( T^{6} + 243 T^{5} + \cdots + 15857178627 \) Copy content Toggle raw display
$97$ \( T^{6} + 161 T^{5} + \cdots + 36908941689 \) Copy content Toggle raw display
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