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Properties

Label 189.4.e.f
Level $189$
Weight $4$
Character orbit 189.e
Analytic conductor $11.151$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 44 x^{12} + 23 x^{11} + 1346 x^{10} + 854 x^{9} + 20545 x^{8} + 27750 x^{7} + \cdots + 254016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{9} - 4 \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{4} + \beta_{3} - 4 \beta_{2} + 11) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{9} - 4 \beta_{5} + \cdots + \beta_1) q^{4}+ \cdots + (6 \beta_{13} + 3 \beta_{12} + \cdots + 807) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 31 q^{4} + q^{5} + 20 q^{7} + 168 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} - 31 q^{4} + q^{5} + 20 q^{7} + 168 q^{8} - 12 q^{10} - 98 q^{11} - 248 q^{13} - 134 q^{14} - 139 q^{16} - 30 q^{17} - 182 q^{19} - 220 q^{20} + 552 q^{22} + 6 q^{23} - 388 q^{25} + 245 q^{26} + 425 q^{28} + 646 q^{29} - 26 q^{31} - 398 q^{32} + 228 q^{34} + 1025 q^{35} + 112 q^{37} - 1015 q^{38} + 147 q^{40} - 1048 q^{41} + 16 q^{43} - 937 q^{44} + 339 q^{46} - 288 q^{47} + 446 q^{49} + 5152 q^{50} + 1075 q^{52} - 1353 q^{53} + 312 q^{55} + 1980 q^{56} + 81 q^{58} - 165 q^{59} - 56 q^{61} - 2430 q^{62} - 3412 q^{64} - 1694 q^{65} + 988 q^{67} - 2625 q^{68} - 4941 q^{70} + 1584 q^{71} - 1487 q^{73} - 2736 q^{74} + 3904 q^{76} - 34 q^{77} + 1273 q^{79} + 2501 q^{80} + 2049 q^{82} - 2340 q^{83} + 432 q^{85} + 160 q^{86} - 9 q^{88} - 1058 q^{89} + 3538 q^{91} + 7668 q^{92} - 1653 q^{94} - 3260 q^{95} - 7460 q^{97} + 10160 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 44 x^{12} + 23 x^{11} + 1346 x^{10} + 854 x^{9} + 20545 x^{8} + 27750 x^{7} + \cdots + 254016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!97 \nu^{13} + \cdots + 40\!\cdots\!44 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!19 \nu^{13} + \cdots + 84\!\cdots\!08 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\!\cdots\!21 \nu^{13} + \cdots + 83\!\cdots\!48 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!13 \nu^{13} + \cdots + 44\!\cdots\!04 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!09 \nu^{13} + \cdots - 15\!\cdots\!88 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!51 \nu^{13} + \cdots - 70\!\cdots\!68 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!79 \nu^{13} + \cdots - 24\!\cdots\!68 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26\!\cdots\!87 \nu^{13} + \cdots + 10\!\cdots\!88 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36\!\cdots\!97 \nu^{13} + \cdots + 19\!\cdots\!36 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93\!\cdots\!41 \nu^{13} + \cdots - 74\!\cdots\!12 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!27 \nu^{13} + \cdots - 20\!\cdots\!36 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 45\!\cdots\!73 \nu^{13} + \cdots - 18\!\cdots\!64 ) / 10\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 12\beta_{5} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + 20\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} + 2\beta_{11} - 23\beta_{9} + \beta_{7} + 242\beta_{5} + 23\beta_{4} - 36\beta _1 - 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{13} + 7 \beta_{12} + 11 \beta_{11} + 3 \beta_{10} - 38 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 24 \beta_{13} + 16 \beta_{12} - 24 \beta_{11} - 24 \beta_{10} + 40 \beta_{8} - 118 \beta_{7} + \cdots + 5331 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 99 \beta_{13} - 99 \beta_{12} - 230 \beta_{11} + 188 \beta_{10} + 1149 \beta_{9} + 188 \beta_{8} + \cdots + 12373 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 836 \beta_{13} + 302 \beta_{12} - 2216 \beta_{11} + 1138 \beta_{10} + 11836 \beta_{9} - 302 \beta_{8} + \cdots + 302 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8946 \beta_{13} - 6708 \beta_{12} - 8946 \beta_{11} - 8946 \beta_{10} + 2238 \beta_{8} + \cdots - 353276 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 28876 \beta_{13} - 28876 \beta_{12} + 75860 \beta_{11} - 31536 \beta_{10} - 277073 \beta_{9} + \cdots - 2878871 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 214792 \beta_{13} + 255367 \beta_{12} + 562265 \beta_{11} + 40575 \beta_{10} - 863393 \beta_{9} + \cdots + 255367 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 342231 \beta_{13} + 1043380 \beta_{12} + 342231 \beta_{11} + 342231 \beta_{10} + 701149 \beta_{8} + \cdots + 68968512 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 550350 \beta_{13} - 550350 \beta_{12} - 9597410 \beta_{11} + 6491216 \beta_{10} + 22762674 \beta_{9} + \cdots + 242906239 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\)

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
2.54592 + 4.40966i
2.38108 + 4.12414i
0.720983 + 1.24878i
−0.270836 − 0.469102i
−0.815927 − 1.41323i
−1.81482 − 3.14336i
−2.24639 − 3.89087i
2.54592 − 4.40966i
2.38108 − 4.12414i
0.720983 − 1.24878i
−0.270836 + 0.469102i
−0.815927 + 1.41323i
−1.81482 + 3.14336i
−2.24639 + 3.89087i
−2.54592 − 4.40966i 0 −8.96343 + 15.5251i 9.29927 + 16.1068i 0 −4.02294 − 18.0781i 50.5460 0 47.3504 − 82.0134i
109.2 −2.38108 − 4.12414i 0 −7.33904 + 12.7116i −9.35067 − 16.1958i 0 14.2072 + 11.8809i 31.8020 0 −44.5293 + 77.1270i
109.3 −0.720983 − 1.24878i 0 2.96037 − 5.12751i −3.30182 − 5.71892i 0 −10.3191 − 15.3791i −20.0732 0 −4.76111 + 8.24648i
109.4 0.270836 + 0.469102i 0 3.85330 − 6.67410i 8.36445 + 14.4876i 0 18.5069 − 0.702629i 8.50782 0 −4.53079 + 7.84756i
109.5 0.815927 + 1.41323i 0 2.66853 − 4.62202i −6.18949 − 10.7205i 0 −6.28171 + 17.4224i 21.7641 0 10.1003 − 17.4943i
109.6 1.81482 + 3.14336i 0 −2.58715 + 4.48108i −2.42081 − 4.19297i 0 16.3234 − 8.74903i 10.2563 0 8.78669 − 15.2190i
109.7 2.24639 + 3.89087i 0 −6.09257 + 10.5526i 4.09907 + 7.09980i 0 −18.4138 − 1.98297i −18.8030 0 −18.4163 + 31.8979i
163.1 −2.54592 + 4.40966i 0 −8.96343 − 15.5251i 9.29927 − 16.1068i 0 −4.02294 + 18.0781i 50.5460 0 47.3504 + 82.0134i
163.2 −2.38108 + 4.12414i 0 −7.33904 − 12.7116i −9.35067 + 16.1958i 0 14.2072 − 11.8809i 31.8020 0 −44.5293 − 77.1270i
163.3 −0.720983 + 1.24878i 0 2.96037 + 5.12751i −3.30182 + 5.71892i 0 −10.3191 + 15.3791i −20.0732 0 −4.76111 − 8.24648i
163.4 0.270836 − 0.469102i 0 3.85330 + 6.67410i 8.36445 − 14.4876i 0 18.5069 + 0.702629i 8.50782 0 −4.53079 − 7.84756i
163.5 0.815927 − 1.41323i 0 2.66853 + 4.62202i −6.18949 + 10.7205i 0 −6.28171 − 17.4224i 21.7641 0 10.1003 + 17.4943i
163.6 1.81482 − 3.14336i 0 −2.58715 − 4.48108i −2.42081 + 4.19297i 0 16.3234 + 8.74903i 10.2563 0 8.78669 + 15.2190i
163.7 2.24639 − 3.89087i 0 −6.09257 − 10.5526i 4.09907 − 7.09980i 0 −18.4138 + 1.98297i −18.8030 0 −18.4163 − 31.8979i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.e.f ✓ 14
3.b odd 2 1 189.4.e.g yes 14
7.c even 3 1 inner 189.4.e.f ✓ 14
7.c even 3 1 1323.4.a.bj 7
7.d odd 6 1 1323.4.a.bk 7
21.g even 6 1 1323.4.a.bh 7
21.h odd 6 1 189.4.e.g yes 14
21.h odd 6 1 1323.4.a.bi 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.f ✓ 14 1.a even 1 1 trivial
189.4.e.f ✓ 14 7.c even 3 1 inner
189.4.e.g yes 14 3.b odd 2 1
189.4.e.g yes 14 21.h odd 6 1
1323.4.a.bh 7 21.g even 6 1
1323.4.a.bi 7 21.h odd 6 1
1323.4.a.bj 7 7.c even 3 1
1323.4.a.bk 7 7.d odd 6 1

Hecke kernels

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

\( T_{2}^{14} + T_{2}^{13} + 44 T_{2}^{12} - 23 T_{2}^{11} + 1346 T_{2}^{10} - 854 T_{2}^{9} + \cdots + 254016 \) Copy content Toggle raw display
\( T_{13}^{7} + 124 T_{13}^{6} - 354 T_{13}^{5} - 620994 T_{13}^{4} - 26760399 T_{13}^{3} + \cdots + 86896171316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots + 254016 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 356440542214689 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 55\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 87\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{7} + 124 T^{6} + \cdots + 86896171316)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 495493458583509)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 6874920774882)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 765727707363728)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 23\!\cdots\!58)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 64\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 86\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
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