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Properties

Label 187.4.a.f
Level $187$
Weight $4$
Character orbit 187.a
Self dual yes
Analytic conductor $11.033$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [187,4,Mod(1,187)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("187.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(187, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 187.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.0333571711\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 72 x^{10} + 90 x^{9} + 1924 x^{8} - 1096 x^{7} - 22484 x^{6} + 1486 x^{5} + \cdots + 75328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_{5} q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{4} + 2) q^{5} + (\beta_{7} - 2 \beta_{5} - \beta_1 - 4) q^{6} + (\beta_{10} - \beta_{5} + 3) q^{7} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 10) q^{8}+ \cdots + ( - 11 \beta_{9} - 11 \beta_{8} + \cdots + 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{2} + 3 q^{3} + 60 q^{4} + 29 q^{5} - 43 q^{6} + 39 q^{7} + 120 q^{8} + 119 q^{9} + 53 q^{10} + 132 q^{11} - 27 q^{12} + 171 q^{13} - 3 q^{14} + 6 q^{15} + 356 q^{16} + 204 q^{17} + 365 q^{18}+ \cdots + 1309 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 72 x^{10} + 90 x^{9} + 1924 x^{8} - 1096 x^{7} - 22484 x^{6} + 1486 x^{5} + \cdots + 75328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 133541 \nu^{11} - 1719716 \nu^{10} - 5711968 \nu^{9} + 101156802 \nu^{8} + 104507376 \nu^{7} + \cdots + 42607780512 ) / 1259512320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25463 \nu^{11} - 23258 \nu^{10} - 1979564 \nu^{9} + 941646 \nu^{8} + 55417988 \nu^{7} + \cdots + 185225216 ) / 157439040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 251857 \nu^{11} + 1418932 \nu^{10} + 20300256 \nu^{9} - 109146234 \nu^{8} + \cdots - 82523776544 ) / 1259512320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53131 \nu^{11} - 169486 \nu^{10} - 3329588 \nu^{9} + 7368822 \nu^{8} + 78510636 \nu^{7} + \cdots - 1103095488 ) / 157439040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21861 \nu^{11} + 81816 \nu^{10} + 1400488 \nu^{9} - 4137522 \nu^{8} - 33269576 \nu^{7} + \cdots - 2364204512 ) / 28625280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 964573 \nu^{11} + 4189428 \nu^{10} + 61953024 \nu^{9} - 240675826 \nu^{8} + \cdots - 79083552416 ) / 1259512320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1211857 \nu^{11} + 4083732 \nu^{10} + 82870176 \nu^{9} - 216144634 \nu^{8} + \cdots - 46705439264 ) / 1259512320 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 724677 \nu^{11} + 2105472 \nu^{10} + 49336216 \nu^{9} - 108458034 \nu^{8} + \cdots - 38713874144 ) / 314878080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 21\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{8} + 6\beta_{5} + 27\beta_{2} + 45\beta _1 + 249 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 8 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} + \cdots + 484 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54 \beta_{11} + 40 \beta_{10} - 52 \beta_{9} - 88 \beta_{8} + 30 \beta_{7} - 8 \beta_{6} + 280 \beta_{5} + \cdots + 5930 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 236 \beta_{11} + 146 \beta_{10} - 146 \beta_{9} - 344 \beta_{8} + 434 \beta_{7} - 196 \beta_{6} + \cdots + 16767 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2169 \beta_{11} + 1451 \beta_{10} - 2051 \beta_{9} - 3222 \beta_{8} + 1902 \beta_{7} - 596 \beta_{6} + \cdots + 151391 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10202 \beta_{11} + 6978 \beta_{10} - 7326 \beta_{9} - 14638 \beta_{8} + 17690 \beta_{7} - 7364 \beta_{6} + \cdots + 538208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 78132 \beta_{11} + 52232 \beta_{10} - 73376 \beta_{9} - 111396 \beta_{8} + 85256 \beta_{7} + \cdots + 4066652 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 388528 \beta_{11} + 282324 \beta_{10} - 308772 \beta_{9} - 552192 \beta_{8} + 650660 \beta_{7} + \cdots + 16805831 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.64667
4.91619
4.71127
2.06516
1.47062
0.744598
−1.00243
−1.28193
−2.76170
−3.72673
−4.24119
−4.54055
−4.64667 −4.62923 13.5916 1.63140 21.5105 13.4459 −25.9823 −5.57022 −7.58059
1.2 −3.91619 6.35856 7.33657 −7.95223 −24.9013 −25.4745 2.59810 13.4312 31.1425
1.3 −3.71127 6.19465 5.77353 15.0518 −22.9900 31.3904 8.26304 11.3736 −55.8611
1.4 −1.06516 −1.29693 −6.86543 −15.6676 1.38144 −10.5402 15.8341 −25.3180 16.6886
1.5 −0.470624 −2.60876 −7.77851 21.7227 1.22775 −11.5058 7.42575 −20.1944 −10.2232
1.6 0.255402 8.17844 −7.93477 4.15484 2.08879 20.4156 −4.06976 39.8869 1.06115
1.7 2.00243 −9.34368 −3.99028 −12.7089 −18.7100 −10.0991 −24.0097 60.3043 −25.4486
1.8 2.28193 −3.55867 −2.79279 0.141464 −8.12064 31.0601 −24.6284 −14.3359 0.322812
1.9 3.76170 7.49421 6.15038 8.81271 28.1910 −10.8225 −6.95770 29.1632 33.1508
1.10 4.72673 0.151093 14.3419 13.7842 0.714177 15.3969 29.9766 −26.9772 65.1542
1.11 5.24119 5.22307 19.4700 −14.7961 27.3751 15.7778 60.1166 0.280412 −77.5491
1.12 5.54055 −9.16274 22.6977 14.8257 −50.7667 −20.0444 81.4336 56.9559 82.1427
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(17\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.4.a.f ✓ 12
3.b odd 2 1 1683.4.a.n 12
11.b odd 2 1 2057.4.a.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.a.f ✓ 12 1.a even 1 1 trivial
1683.4.a.n 12 3.b odd 2 1
2057.4.a.j 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 10 T_{2}^{11} - 28 T_{2}^{10} + 520 T_{2}^{9} - 341 T_{2}^{8} - 9028 T_{2}^{7} + \cdots - 20400 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(187))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 10 T^{11} + \cdots - 20400 \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 9091456 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13228722912 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 440051897483264 \) Copy content Toggle raw display
$11$ \( (T - 11)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T - 17)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 18\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 29\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 79\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 83\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 31\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 11\!\cdots\!06 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 53\!\cdots\!48 \) Copy content Toggle raw display
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