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Properties

Label 189.4.f.b
Level $189$
Weight $4$
Character orbit 189.f
Analytic conductor $11.151$
Analytic rank $0$
Dimension $16$
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] = [189,4,Mod(64,189)]
 
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([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), 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: \(11.1513609911\)
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: \( x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + \cdots + 21307456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{14} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} - 5 \beta_{3}) q^{4} + (\beta_{12} + 4 \beta_{3}) q^{5} + ( - 7 \beta_{3} + 7) q^{7} + ( - \beta_{5} + \beta_{4} - 4 \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{8} - 5 \beta_{3}) q^{4} + (\beta_{12} + 4 \beta_{3}) q^{5} + ( - 7 \beta_{3} + 7) q^{7} + ( - \beta_{5} + \beta_{4} - 4 \beta_{2} + 1) q^{8} + (\beta_{9} + 2 \beta_{7} + \beta_{5} + \cdots - 3) q^{10}+ \cdots - 49 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - 43 q^{4} + 30 q^{5} + 56 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} - 43 q^{4} + 30 q^{5} + 56 q^{7} - 12 q^{8} - 28 q^{10} + 24 q^{11} - 68 q^{13} - 21 q^{14} - 103 q^{16} - 336 q^{17} + 352 q^{19} + 330 q^{20} - 151 q^{22} + 228 q^{23} - 244 q^{25} - 1590 q^{26} - 602 q^{28} + 618 q^{29} - 72 q^{31} + 786 q^{32} + 261 q^{34} + 420 q^{35} + 420 q^{37} + 1032 q^{38} + 375 q^{40} + 420 q^{41} + 2 q^{43} - 774 q^{44} + 804 q^{46} + 570 q^{47} - 392 q^{49} + 1110 q^{50} + 431 q^{52} - 1056 q^{53} - 1676 q^{55} - 42 q^{56} - 37 q^{58} - 150 q^{59} - 578 q^{61} - 2340 q^{62} - 224 q^{64} - 366 q^{65} + 898 q^{67} + 2526 q^{68} - 98 q^{70} - 1764 q^{71} + 1944 q^{73} - 222 q^{74} - 1423 q^{76} - 168 q^{77} + 158 q^{79} - 4950 q^{80} - 422 q^{82} + 2958 q^{83} + 774 q^{85} - 114 q^{86} - 1317 q^{88} - 8760 q^{89} - 952 q^{91} + 4629 q^{92} + 3234 q^{94} + 930 q^{95} + 60 q^{97} - 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + \cdots + 21307456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!31 \nu^{15} + \cdots + 20\!\cdots\!60 ) / 35\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54\!\cdots\!70 \nu^{15} + \cdots + 25\!\cdots\!60 ) / 20\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\!\cdots\!63 \nu^{15} + \cdots + 57\!\cdots\!34 ) / 44\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 87\!\cdots\!81 \nu^{15} + \cdots + 23\!\cdots\!72 ) / 89\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\!\cdots\!80 \nu^{15} + \cdots - 53\!\cdots\!96 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!23 \nu^{15} + \cdots + 24\!\cdots\!52 ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!10 \nu^{15} + \cdots + 33\!\cdots\!80 ) / 20\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!39 \nu^{15} + \cdots - 22\!\cdots\!52 ) / 55\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!41 \nu^{15} + \cdots - 20\!\cdots\!20 ) / 55\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!13 \nu^{15} + \cdots - 92\!\cdots\!60 ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 54\!\cdots\!89 \nu^{15} + \cdots + 18\!\cdots\!48 ) / 64\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 73\!\cdots\!84 \nu^{15} + \cdots - 23\!\cdots\!52 ) / 45\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!67 \nu^{15} + \cdots - 68\!\cdots\!48 ) / 12\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 74\!\cdots\!47 \nu^{15} + \cdots + 15\!\cdots\!76 ) / 32\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 13\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - 20\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 2 \beta_{13} + 2 \beta_{12} - \beta_{10} + 2 \beta_{9} - 27 \beta_{8} - 2 \beta_{7} + \cdots - 257 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} - 4 \beta_{12} - 32 \beta_{8} - 35 \beta_{6} + \cdots - 453 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{11} + 37\beta_{10} - 86\beta_{9} + 46\beta_{7} + \beta_{5} - 687\beta_{4} + 342\beta_{2} + 5775 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 185 \beta_{15} + 106 \beta_{14} + 214 \beta_{13} + 250 \beta_{12} - 106 \beta_{11} + 185 \beta_{10} + \cdots + 3250 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1108 \beta_{15} - 154 \beta_{14} + 2888 \beta_{13} - 640 \beta_{12} + 17494 \beta_{8} + \cdots + 11319 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3868 \beta_{11} - 6392 \beta_{10} + 8332 \beta_{9} + 9964 \beta_{7} - 26952 \beta_{5} + 28969 \beta_{4} + \cdots - 115437 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32012 \beta_{15} + 3856 \beta_{14} - 88864 \beta_{13} + 648 \beta_{12} - 3856 \beta_{11} + \cdots - 3377929 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 198609 \beta_{15} - 121416 \beta_{14} - 284570 \beta_{13} - 335534 \beta_{12} - 851561 \beta_{8} + \cdots - 6725628 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 70910 \beta_{11} + 919910 \beta_{10} - 2619280 \beta_{9} - 379312 \beta_{7} + 411887 \beta_{5} + \cdots + 84708632 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5863177 \beta_{15} + 3540560 \beta_{14} + 9060926 \beta_{13} + 10411970 \beta_{12} - 3540560 \beta_{11} + \cdots + 111743867 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 26357375 \beta_{15} - 685622 \beta_{14} + 75309154 \beta_{13} + 19353238 \beta_{12} + \cdots + 301101285 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 99274802 \beta_{11} - 168292714 \beta_{10} + 276515780 \beta_{9} + 308708012 \beta_{7} + \cdots - 3331785418 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−2.47591 − 4.28840i
−1.96709 − 3.40709i
−1.46974 − 2.54566i
0.403686 + 0.699204i
0.797492 + 1.38130i
1.30789 + 2.26533i
2.28179 + 3.95218i
2.62188 + 4.54123i
−2.47591 + 4.28840i
−1.96709 + 3.40709i
−1.46974 + 2.54566i
0.403686 − 0.699204i
0.797492 − 1.38130i
1.30789 − 2.26533i
2.28179 − 3.95218i
2.62188 − 4.54123i
−2.47591 − 4.28840i 0 −8.26023 + 14.3071i 8.06998 − 13.9776i 0 3.50000 + 6.06218i 42.1917 0 −79.9221
64.2 −1.96709 − 3.40709i 0 −3.73885 + 6.47588i −1.21571 + 2.10567i 0 3.50000 + 6.06218i −2.05480 0 9.56561
64.3 −1.46974 − 2.54566i 0 −0.320267 + 0.554718i −1.28443 + 2.22469i 0 3.50000 + 6.06218i −21.6330 0 7.55109
64.4 0.403686 + 0.699204i 0 3.67408 − 6.36369i 9.11444 − 15.7867i 0 3.50000 + 6.06218i 12.3917 0 14.7175
64.5 0.797492 + 1.38130i 0 2.72801 − 4.72505i −1.27816 + 2.21384i 0 3.50000 + 6.06218i 21.4622 0 −4.07730
64.6 1.30789 + 2.26533i 0 0.578868 − 1.00263i −6.77153 + 11.7286i 0 3.50000 + 6.06218i 23.9546 0 −35.4255
64.7 2.28179 + 3.95218i 0 −6.41313 + 11.1079i 10.3955 − 18.0055i 0 3.50000 + 6.06218i −22.0250 0 94.8813
64.8 2.62188 + 4.54123i 0 −9.74848 + 16.8849i −2.03009 + 3.51621i 0 3.50000 + 6.06218i −60.2873 0 −21.2906
127.1 −2.47591 + 4.28840i 0 −8.26023 − 14.3071i 8.06998 + 13.9776i 0 3.50000 − 6.06218i 42.1917 0 −79.9221
127.2 −1.96709 + 3.40709i 0 −3.73885 − 6.47588i −1.21571 − 2.10567i 0 3.50000 − 6.06218i −2.05480 0 9.56561
127.3 −1.46974 + 2.54566i 0 −0.320267 − 0.554718i −1.28443 − 2.22469i 0 3.50000 − 6.06218i −21.6330 0 7.55109
127.4 0.403686 − 0.699204i 0 3.67408 + 6.36369i 9.11444 + 15.7867i 0 3.50000 − 6.06218i 12.3917 0 14.7175
127.5 0.797492 − 1.38130i 0 2.72801 + 4.72505i −1.27816 − 2.21384i 0 3.50000 − 6.06218i 21.4622 0 −4.07730
127.6 1.30789 − 2.26533i 0 0.578868 + 1.00263i −6.77153 − 11.7286i 0 3.50000 − 6.06218i 23.9546 0 −35.4255
127.7 2.28179 − 3.95218i 0 −6.41313 − 11.1079i 10.3955 + 18.0055i 0 3.50000 − 6.06218i −22.0250 0 94.8813
127.8 2.62188 − 4.54123i 0 −9.74848 − 16.8849i −2.03009 − 3.51621i 0 3.50000 − 6.06218i −60.2873 0 −21.2906
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.f.b 16
3.b odd 2 1 63.4.f.b ✓ 16
9.c even 3 1 inner 189.4.f.b 16
9.c even 3 1 567.4.a.g 8
9.d odd 6 1 63.4.f.b ✓ 16
9.d odd 6 1 567.4.a.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.b ✓ 16 3.b odd 2 1
63.4.f.b ✓ 16 9.d odd 6 1
189.4.f.b 16 1.a even 1 1 trivial
189.4.f.b 16 9.c even 3 1 inner
567.4.a.g 8 9.c even 3 1
567.4.a.i 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 3 T_{2}^{15} + 58 T_{2}^{14} - 129 T_{2}^{13} + 2107 T_{2}^{12} - 4455 T_{2}^{11} + \cdots + 21307456 \) acting on \(S_{4}^{\mathrm{new}}(189, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + \cdots + 21307456 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 28841980243729 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 2073513126204)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 604815137888177)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 71\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 86\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 36\!\cdots\!25)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 44\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
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