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Properties

Label 189.4.e.e
Level $189$
Weight $4$
Character orbit 189.e
Analytic conductor $11.151$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 40x^{10} + 1147x^{8} + 15564x^{6} + 154089x^{4} + 578934x^{2} + 1633284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{4} + \cdots - 6) q^{4}+ \cdots + (\beta_{10} - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{11} + \beta_{7} + \beta_{4} + \cdots - 6) q^{4}+ \cdots + ( - 9 \beta_{10} - 67 \beta_{9} + \cdots + 256 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 32 q^{4} - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 32 q^{4} - 26 q^{7} + 20 q^{10} + 104 q^{13} + 148 q^{16} + 62 q^{19} - 712 q^{22} + 46 q^{25} - 348 q^{28} + 82 q^{31} + 840 q^{34} + 1132 q^{37} + 444 q^{40} - 3132 q^{43} + 888 q^{46} + 366 q^{49} + 72 q^{52} + 448 q^{55} - 4 q^{58} - 886 q^{61} - 1848 q^{64} + 2084 q^{67} - 4460 q^{70} + 2398 q^{73} + 6408 q^{76} - 984 q^{79} + 3892 q^{82} - 7200 q^{85} + 5796 q^{88} - 6492 q^{91} - 2772 q^{94} - 1364 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 40x^{10} + 1147x^{8} + 15564x^{6} + 154089x^{4} + 578934x^{2} + 1633284 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 173197 \nu^{10} + 6246280 \nu^{8} + 179112079 \nu^{6} + 2135188674 \nu^{4} + \cdots + 24978356640 ) / 65426239998 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 173197 \nu^{11} - 6246280 \nu^{9} - 179112079 \nu^{7} - 2135188674 \nu^{5} + \cdots - 90404596638 \nu ) / 65426239998 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108403 \nu^{10} + 7516849 \nu^{8} + 245835571 \nu^{6} + 4262422452 \nu^{4} + \cdots + 91892207808 ) / 14539164444 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12280 \nu^{11} - 352129 \nu^{9} - 6257731 \nu^{7} - 47305323 \nu^{5} + \cdots - 10924950600 \nu ) / 1842992676 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1600\nu^{10} - 45880\nu^{8} - 1315609\nu^{6} - 6163560\nu^{4} - 23157360\nu^{2} + 1332538101 ) / 153582723 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2443778 \nu^{10} - 113590439 \nu^{8} - 2984596505 \nu^{6} - 37517061303 \nu^{4} + \cdots + 35784531648 ) / 130852479996 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2193901 \nu^{11} - 105145360 \nu^{9} - 3015043198 \nu^{7} - 45990857349 \nu^{5} + \cdots - 1521805596156 \nu ) / 130852479996 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31480 \nu^{11} - 902689 \nu^{9} - 22045039 \nu^{7} - 121268043 \nu^{5} + \cdots + 12437477316 \nu ) / 1842992676 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 777199 \nu^{11} + 22712800 \nu^{9} + 651289540 \nu^{7} + 7118068149 \nu^{5} + \cdots + 328730303880 \nu ) / 43617493332 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 235927 \nu^{10} - 8654518 \nu^{8} - 222205510 \nu^{6} - 2556628047 \nu^{4} + \cdots - 52512120288 ) / 6231070476 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{7} + \beta_{6} - \beta_{4} + 14\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -27\beta_{11} + 31\beta_{7} - 4\beta_{6} + 23\beta_{4} - 248\beta_{2} - 252 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 31\beta_{10} - 19\beta_{8} + 329\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 160\beta_{11} - 467\beta_{6} + 160\beta_{4} + 4389 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -307\beta_{9} + 787\beta_{5} + 6737\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9539\beta_{11} - 18715\beta_{7} + 14127\beta_{6} - 14127\beta_{4} + 99788\beta_{2} + 14127 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -18715\beta_{10} + 4951\beta_{9} + 4951\beta_{8} - 18715\beta_{5} - 142169\beta_{3} - 142169\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -315555\beta_{11} + 431707\beta_{7} - 116152\beta_{6} + 199403\beta_{4} - 2108696\beta_{2} - 2224848 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 431707\beta_{10} - 83251\beta_{8} + 3055361\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.35377 − 4.07684i
1.84379 − 3.19353i
1.02968 − 1.78345i
−1.02968 + 1.78345i
−1.84379 + 3.19353i
−2.35377 + 4.07684i
2.35377 + 4.07684i
1.84379 + 3.19353i
1.02968 + 1.78345i
−1.02968 − 1.78345i
−1.84379 − 3.19353i
−2.35377 − 4.07684i
−2.35377 − 4.07684i 0 −7.08042 + 12.2637i −0.415227 − 0.719194i 0 16.4314 + 8.54463i 29.0024 0 −1.95469 + 3.38563i
109.2 −1.84379 − 3.19353i 0 −2.79910 + 4.84819i 5.93594 + 10.2814i 0 −15.1628 − 10.6344i −8.85681 0 21.8892 − 37.9133i
109.3 −1.02968 − 1.78345i 0 1.87953 − 3.25543i −7.25204 − 12.5609i 0 −7.76857 + 16.8122i −24.2161 0 −14.9345 + 25.8674i
109.4 1.02968 + 1.78345i 0 1.87953 − 3.25543i 7.25204 + 12.5609i 0 −7.76857 + 16.8122i 24.2161 0 −14.9345 + 25.8674i
109.5 1.84379 + 3.19353i 0 −2.79910 + 4.84819i −5.93594 − 10.2814i 0 −15.1628 − 10.6344i 8.85681 0 21.8892 − 37.9133i
109.6 2.35377 + 4.07684i 0 −7.08042 + 12.2637i 0.415227 + 0.719194i 0 16.4314 + 8.54463i −29.0024 0 −1.95469 + 3.38563i
163.1 −2.35377 + 4.07684i 0 −7.08042 − 12.2637i −0.415227 + 0.719194i 0 16.4314 − 8.54463i 29.0024 0 −1.95469 − 3.38563i
163.2 −1.84379 + 3.19353i 0 −2.79910 − 4.84819i 5.93594 − 10.2814i 0 −15.1628 + 10.6344i −8.85681 0 21.8892 + 37.9133i
163.3 −1.02968 + 1.78345i 0 1.87953 + 3.25543i −7.25204 + 12.5609i 0 −7.76857 − 16.8122i −24.2161 0 −14.9345 − 25.8674i
163.4 1.02968 − 1.78345i 0 1.87953 + 3.25543i 7.25204 − 12.5609i 0 −7.76857 − 16.8122i 24.2161 0 −14.9345 − 25.8674i
163.5 1.84379 − 3.19353i 0 −2.79910 − 4.84819i −5.93594 + 10.2814i 0 −15.1628 + 10.6344i 8.85681 0 21.8892 + 37.9133i
163.6 2.35377 − 4.07684i 0 −7.08042 − 12.2637i 0.415227 − 0.719194i 0 16.4314 − 8.54463i −29.0024 0 −1.95469 − 3.38563i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
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.e ✓ 12
3.b odd 2 1 inner 189.4.e.e ✓ 12
7.c even 3 1 inner 189.4.e.e ✓ 12
7.c even 3 1 1323.4.a.bd 6
7.d odd 6 1 1323.4.a.be 6
21.g even 6 1 1323.4.a.be 6
21.h odd 6 1 inner 189.4.e.e ✓ 12
21.h odd 6 1 1323.4.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.e ✓ 12 1.a even 1 1 trivial
189.4.e.e ✓ 12 3.b odd 2 1 inner
189.4.e.e ✓ 12 7.c even 3 1 inner
189.4.e.e ✓ 12 21.h odd 6 1 inner
1323.4.a.bd 6 7.c even 3 1
1323.4.a.bd 6 21.h odd 6 1
1323.4.a.be 6 7.d odd 6 1
1323.4.a.be 6 21.g even 6 1

Hecke kernels

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

\( T_{2}^{12} + 40T_{2}^{10} + 1147T_{2}^{8} + 15564T_{2}^{6} + 154089T_{2}^{4} + 578934T_{2}^{2} + 1633284 \) Copy content Toggle raw display
\( T_{13}^{3} - 26T_{13}^{2} - 3211T_{13} - 35818 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 40 T^{10} + \cdots + 1633284 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 418120704 \) Copy content Toggle raw display
$7$ \( (T^{6} + 13 T^{5} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{3} - 26 T^{2} + \cdots - 35818)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{6} - 31 T^{5} + \cdots + 5047249936)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 1224041183712)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 4861932390441)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 18026291164644)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 69555350983392)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 783 T^{2} + \cdots + 15519697)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 72\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 48\!\cdots\!69)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 19\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 37\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 28\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{3} + 341 T^{2} + \cdots - 1582590381)^{4} \) Copy content Toggle raw display
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