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Properties

Label 187.4.a.d
Level $187$
Weight $4$
Character orbit 187.a
Self dual yes
Analytic conductor $11.033$
Analytic rank $1$
Dimension $10$
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 = [10,-8] 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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 55 x^{8} + 72 x^{7} + 1037 x^{6} - 812 x^{5} - 7851 x^{4} + 2526 x^{3} + 20108 x^{2} + \cdots - 5760 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{7} + \beta_{4} - 4) q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 3) q^{6} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \cdots - 5) q^{7}+ \cdots + (11 \beta_{9} - 22 \beta_{7} + \cdots + 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{2} - 9 q^{3} + 40 q^{4} - 41 q^{5} + 31 q^{6} - 63 q^{7} - 96 q^{8} + 61 q^{9} - 47 q^{10} + 110 q^{11} - 171 q^{12} - 99 q^{13} - 95 q^{14} - 150 q^{15} + 180 q^{16} - 170 q^{17} - 207 q^{18}+ \cdots + 671 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 55 x^{8} + 72 x^{7} + 1037 x^{6} - 812 x^{5} - 7851 x^{4} + 2526 x^{3} + 20108 x^{2} + \cdots - 5760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8581 \nu^{9} + 87924 \nu^{8} - 477211 \nu^{7} - 4373978 \nu^{6} + 3699365 \nu^{5} + \cdots - 66046272 ) / 38091008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3389 \nu^{9} - 51454 \nu^{8} + 53385 \nu^{7} + 1596748 \nu^{6} - 4145275 \nu^{5} - 10681936 \nu^{4} + \cdots + 85931328 ) / 9522752 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65857 \nu^{9} - 19492 \nu^{8} - 3591727 \nu^{7} - 1423226 \nu^{6} + 66918273 \nu^{5} + \cdots + 447588544 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 100245 \nu^{9} + 308020 \nu^{8} + 3973787 \nu^{7} - 9256382 \nu^{6} - 41124309 \nu^{5} + \cdots - 190803392 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 114545 \nu^{9} + 219204 \nu^{8} + 6247231 \nu^{7} - 8101958 \nu^{6} - 112129585 \nu^{5} + \cdots - 77756096 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 131631 \nu^{9} - 771020 \nu^{8} - 6038849 \nu^{7} + 33447594 \nu^{6} + 97500335 \nu^{5} + \cdots - 829705408 ) / 76182016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 107255 \nu^{9} + 351052 \nu^{8} + 5474489 \nu^{7} - 14597242 \nu^{6} - 95190391 \nu^{5} + \cdots + 724671168 ) / 38091008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{3} + \beta_{2} + 20\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} + 7\beta_{8} - 2\beta_{6} + 7\beta_{5} - 3\beta_{4} + 26\beta_{2} + 45\beta _1 + 210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42 \beta_{9} + 74 \beta_{8} + 42 \beta_{7} + 22 \beta_{6} + 86 \beta_{5} - 10 \beta_{4} + 52 \beta_{3} + \cdots + 418 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 296 \beta_{9} + 342 \beta_{8} + 86 \beta_{7} - 82 \beta_{6} + 358 \beta_{5} - 152 \beta_{4} + \cdots + 4999 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1531 \beta_{9} + 2488 \beta_{8} + 1517 \beta_{7} + 313 \beta_{6} + 3028 \beta_{5} - 588 \beta_{4} + \cdots + 16115 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10275 \beta_{9} + 12859 \beta_{8} + 5048 \beta_{7} - 2678 \beta_{6} + 14211 \beta_{5} - 5725 \beta_{4} + \cdots + 137170 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53272 \beta_{9} + 82310 \beta_{8} + 51012 \beta_{7} + 1064 \beta_{6} + 101470 \beta_{5} - 24920 \beta_{4} + \cdots + 576936 \) 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
−4.38145
−3.88370
−3.27779
−1.54722
−0.813593
0.447769
2.44876
3.25015
3.99555
5.76153
−5.38145 −8.82835 20.9600 −12.9495 47.5093 −22.6868 −69.7435 50.9398 69.6868
1.2 −4.88370 2.22066 15.8506 17.8151 −10.8451 −27.4352 −38.3399 −22.0687 −87.0038
1.3 −4.27779 2.36616 10.2995 −19.2230 −10.1219 31.3553 −9.83664 −21.4013 82.2318
1.4 −2.54722 −5.06531 −1.51166 −0.212294 12.9025 8.35551 24.2283 −1.34267 0.540760
1.5 −1.81359 4.59414 −4.71088 2.19824 −8.33191 −8.16262 23.0524 −5.89384 −3.98672
1.6 −0.552231 −9.39468 −7.69504 9.49824 5.18803 15.9512 8.66728 61.2599 −5.24522
1.7 1.44876 9.43069 −5.90109 −19.3257 13.6628 −33.2851 −20.1394 61.9379 −27.9984
1.8 2.25015 1.42575 −2.93684 −11.7202 3.20814 23.4792 −24.6095 −24.9672 −26.3722
1.9 2.99555 −2.94594 0.973314 8.57150 −8.82472 −19.5556 −21.0488 −18.3214 25.6764
1.10 4.76153 −2.80312 14.6722 −15.6524 −13.3472 −31.0159 31.7697 −19.1425 −74.5294
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.d ✓ 10
3.b odd 2 1 1683.4.a.m 10
11.b odd 2 1 2057.4.a.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.a.d ✓ 10 1.a even 1 1 trivial
1683.4.a.m 10 3.b odd 2 1
2057.4.a.i 10 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 8 T_{2}^{9} - 28 T_{2}^{8} - 320 T_{2}^{7} + 43 T_{2}^{6} + 3842 T_{2}^{5} + 2272 T_{2}^{4} + \cdots + 13336 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(187))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 8 T^{9} + \cdots + 13336 \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{9} + \cdots - 1126032 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 597348544 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 10064173681152 \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T + 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 55\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 88\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 41\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 90\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 74\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 35\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
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