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Properties

Label 189.4.e.h
Level $189$
Weight $4$
Character orbit 189.e
Analytic conductor $11.151$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,4,Mod(109,189)]
 
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([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.e (of order \(3\), degree \(2\), 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: \(11.1513609911\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{9}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2}) q^{2} + (\beta_{4} - \beta_{3} + 6 \beta_1) q^{4} + ( - \beta_{12} - \beta_{11}) q^{5} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + ( - \beta_{9} + 5 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{2}) q^{2} + (\beta_{4} - \beta_{3} + 6 \beta_1) q^{4} + ( - \beta_{12} - \beta_{11}) q^{5} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + (62 \beta_{12} + 64 \beta_{11} + \cdots - 251 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 48 q^{4} - 60 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 48 q^{4} - 60 q^{7} - 44 q^{10} - 168 q^{13} - 156 q^{16} - 12 q^{19} + 448 q^{22} - 408 q^{25} - 152 q^{28} - 800 q^{31} - 1896 q^{34} - 692 q^{37} - 96 q^{40} + 2912 q^{43} - 1524 q^{46} + 2020 q^{49} - 1972 q^{52} - 2560 q^{55} - 2372 q^{58} - 216 q^{61} + 9928 q^{64} - 684 q^{67} + 4316 q^{70} + 4564 q^{73} - 760 q^{76} - 556 q^{79} - 3340 q^{82} + 2592 q^{85} - 6696 q^{88} - 184 q^{91} + 492 q^{94} + 1168 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 51\!\cdots\!86 \nu^{15} + \cdots - 15\!\cdots\!87 ) / 34\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!79 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95\!\cdots\!24 \nu^{15} + \cdots + 13\!\cdots\!43 ) / 29\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 78\!\cdots\!07 \nu^{15} + \cdots + 45\!\cdots\!11 ) / 17\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66\!\cdots\!74 \nu^{15} + \cdots - 22\!\cdots\!33 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 73\!\cdots\!09 \nu^{15} + \cdots + 89\!\cdots\!53 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!03 \nu^{15} + \cdots + 17\!\cdots\!42 ) / 29\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46\!\cdots\!69 \nu^{15} + \cdots - 84\!\cdots\!78 ) / 41\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!57 \nu^{15} + \cdots + 10\!\cdots\!43 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!34 \nu^{15} + \cdots - 45\!\cdots\!93 ) / 29\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!31 \nu^{15} + \cdots + 33\!\cdots\!79 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 49\!\cdots\!31 \nu^{15} + \cdots - 16\!\cdots\!07 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!02 \nu^{15} + \cdots - 33\!\cdots\!25 ) / 58\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!04 \nu^{15} + \cdots - 12\!\cdots\!68 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{15} + \cdots + 86\!\cdots\!95 ) / 14\!\cdots\!92 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 3 \beta_{15} - 2 \beta_{14} - \beta_{13} - 5 \beta_{12} + 6 \beta_{10} + \beta_{8} + 5 \beta_{7} + \cdots + 2 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{15} + 9 \beta_{14} - \beta_{13} + 12 \beta_{12} + 12 \beta_{11} + 20 \beta_{10} + 9 \beta_{9} + \cdots - 565 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 65 \beta_{15} - 144 \beta_{13} - 449 \beta_{11} - 144 \beta_{10} - 83 \beta_{9} - 115 \beta_{8} + \cdots - 13 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 52 \beta_{15} - 312 \beta_{14} + 218 \beta_{13} - 339 \beta_{12} - 706 \beta_{10} - 285 \beta_{8} + \cdots - 436 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5672 \beta_{15} + 2377 \beta_{14} + 8249 \beta_{13} + 17713 \beta_{12} + 17713 \beta_{11} + \cdots + 19531 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1577 \beta_{15} - 983 \beta_{13} - 1614 \beta_{11} - 983 \beta_{10} - 1823 \beta_{9} + 2046 \beta_{8} + \cdots + 122121 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 165693 \beta_{15} - 37672 \beta_{14} - 53663 \beta_{13} - 349141 \beta_{12} + 326682 \beta_{10} + \cdots + 107326 ) / 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 198134 \beta_{15} + 400050 \beta_{14} - 205994 \beta_{13} + 246687 \beta_{12} + 246687 \beta_{11} + \cdots - 28093496 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4545349 \beta_{15} - 8840700 \beta_{13} - 27656533 \beta_{11} - 8840700 \beta_{10} - 2499019 \beta_{9} + \cdots - 101048477 ) / 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10805111 \beta_{15} - 15065301 \beta_{14} + 17122465 \beta_{13} - 4441368 \beta_{12} - 62172506 \beta_{10} + \cdots - 34244930 ) / 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 349102204 \beta_{15} + 84438449 \beta_{14} + 542822449 \beta_{13} + 1100326265 \beta_{12} + \cdots + 4651037963 ) / 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 120521474 \beta_{15} - 31201500 \beta_{13} + 5945004 \beta_{11} - 31201500 \beta_{10} + \cdots + 7846479013 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11044886505 \beta_{15} - 1428712508 \beta_{14} - 4137003643 \beta_{13} - 21978952367 \beta_{12} + \cdots + 8274007286 ) / 42 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 4861786192 \beta_{15} + 22350858507 \beta_{14} - 25647216547 \beta_{13} - 9895364610 \beta_{12} + \cdots - 1860633895699 ) / 42 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 353535630725 \beta_{15} - 548381305872 \beta_{13} - 1762964503613 \beta_{11} - 548381305872 \beta_{10} + \cdots - 11080741238257 ) / 84 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−2.17666 + 3.77009i
1.03883 − 1.79930i
−0.0640761 + 0.110983i
0.128480 − 0.222533i
−0.178435 + 0.309059i
−3.06544 + 5.30949i
3.25730 − 5.64181i
1.06000 − 1.83598i
−2.17666 − 3.77009i
1.03883 + 1.79930i
−0.0640761 − 0.110983i
0.128480 + 0.222533i
−0.178435 − 0.309059i
−3.06544 − 5.30949i
3.25730 + 5.64181i
1.06000 + 1.83598i
−2.73089 − 4.73004i 0 −10.9155 + 18.9063i −0.0995681 − 0.172457i 0 −16.0061 + 9.31685i 75.5424 0 −0.543819 + 0.941923i
109.2 −1.74618 − 3.02447i 0 −2.09829 + 3.63435i 3.93765 + 6.82022i 0 18.4135 − 1.98614i −13.2829 0 13.7517 − 23.8187i
109.3 −1.64637 − 2.85159i 0 −1.42105 + 2.46133i −10.8582 − 18.8070i 0 1.08769 − 18.4883i −16.9836 0 −35.7533 + 61.9265i
109.4 −0.884622 − 1.53221i 0 2.43489 − 4.21735i 6.52561 + 11.3027i 0 −18.4950 − 0.966772i −22.7698 0 11.5454 − 19.9972i
109.5 0.884622 + 1.53221i 0 2.43489 − 4.21735i −6.52561 − 11.3027i 0 −18.4950 − 0.966772i 22.7698 0 11.5454 − 19.9972i
109.6 1.64637 + 2.85159i 0 −1.42105 + 2.46133i 10.8582 + 18.8070i 0 1.08769 − 18.4883i 16.9836 0 −35.7533 + 61.9265i
109.7 1.74618 + 3.02447i 0 −2.09829 + 3.63435i −3.93765 − 6.82022i 0 18.4135 − 1.98614i 13.2829 0 13.7517 − 23.8187i
109.8 2.73089 + 4.73004i 0 −10.9155 + 18.9063i 0.0995681 + 0.172457i 0 −16.0061 + 9.31685i −75.5424 0 −0.543819 + 0.941923i
163.1 −2.73089 + 4.73004i 0 −10.9155 − 18.9063i −0.0995681 + 0.172457i 0 −16.0061 − 9.31685i 75.5424 0 −0.543819 − 0.941923i
163.2 −1.74618 + 3.02447i 0 −2.09829 − 3.63435i 3.93765 − 6.82022i 0 18.4135 + 1.98614i −13.2829 0 13.7517 + 23.8187i
163.3 −1.64637 + 2.85159i 0 −1.42105 − 2.46133i −10.8582 + 18.8070i 0 1.08769 + 18.4883i −16.9836 0 −35.7533 − 61.9265i
163.4 −0.884622 + 1.53221i 0 2.43489 + 4.21735i 6.52561 − 11.3027i 0 −18.4950 + 0.966772i −22.7698 0 11.5454 + 19.9972i
163.5 0.884622 − 1.53221i 0 2.43489 + 4.21735i −6.52561 + 11.3027i 0 −18.4950 + 0.966772i 22.7698 0 11.5454 + 19.9972i
163.6 1.64637 − 2.85159i 0 −1.42105 − 2.46133i 10.8582 − 18.8070i 0 1.08769 + 18.4883i 16.9836 0 −35.7533 − 61.9265i
163.7 1.74618 − 3.02447i 0 −2.09829 − 3.63435i −3.93765 + 6.82022i 0 18.4135 + 1.98614i 13.2829 0 13.7517 + 23.8187i
163.8 2.73089 − 4.73004i 0 −10.9155 − 18.9063i 0.0995681 − 0.172457i 0 −16.0061 − 9.31685i −75.5424 0 −0.543819 − 0.941923i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.e.h ✓ 16
3.b odd 2 1 inner 189.4.e.h ✓ 16
7.c even 3 1 inner 189.4.e.h ✓ 16
7.c even 3 1 1323.4.a.bo 8
7.d odd 6 1 1323.4.a.bn 8
21.g even 6 1 1323.4.a.bn 8
21.h odd 6 1 inner 189.4.e.h ✓ 16
21.h odd 6 1 1323.4.a.bo 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.h ✓ 16 1.a even 1 1 trivial
189.4.e.h ✓ 16 3.b odd 2 1 inner
189.4.e.h ✓ 16 7.c even 3 1 inner
189.4.e.h ✓ 16 21.h odd 6 1 inner
1323.4.a.bn 8 7.d odd 6 1
1323.4.a.bn 8 21.g even 6 1
1323.4.a.bo 8 7.c even 3 1
1323.4.a.bo 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(189, [\chi])\):

\( T_{2}^{16} + 56 T_{2}^{14} + 2151 T_{2}^{12} + 42140 T_{2}^{10} + 593317 T_{2}^{8} + 5029374 T_{2}^{6} + \cdots + 152473104 \) Copy content Toggle raw display
\( T_{13}^{4} + 42T_{13}^{3} - 4475T_{13}^{2} - 94812T_{13} + 5259952 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 152473104 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 39033114624 \) Copy content Toggle raw display
$7$ \( (T^{8} + 30 T^{7} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{4} + 42 T^{3} + \cdots + 5259952)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 141394168701184)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 18\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 11\!\cdots\!81)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 728 T^{3} + \cdots - 12025459739)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 17\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 28\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 87\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{4} - 292 T^{3} + \cdots + 100316149821)^{4} \) Copy content Toggle raw display
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