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Properties

Label 187.3.l.a
Level $187$
Weight $3$
Character orbit 187.l
Analytic conductor $5.095$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [187,3,Mod(50,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.50");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 187.l (of order \(10\), degree \(4\), 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: \(5.09538094354\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 10 q^{2} + 58 q^{4} - 10 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 10 q^{2} + 58 q^{4} - 10 q^{8} + 48 q^{9} - 40 q^{13} - 90 q^{15} - 110 q^{16} + 70 q^{17} + 70 q^{18} - 70 q^{19} + 142 q^{25} - 38 q^{26} - 100 q^{30} + 174 q^{33} - 98 q^{34} - 330 q^{35} - 280 q^{36} + 228 q^{38} + 510 q^{42} + 36 q^{47} + 44 q^{49} - 750 q^{50} - 455 q^{51} + 520 q^{52} - 144 q^{53} - 614 q^{55} + 424 q^{59} - 296 q^{60} + 494 q^{64} - 392 q^{66} + 12 q^{67} + 690 q^{68} + 246 q^{69} - 536 q^{70} + 1640 q^{72} + 630 q^{77} - 916 q^{81} - 190 q^{83} - 1320 q^{84} + 260 q^{85} - 638 q^{86} - 452 q^{89} + 200 q^{93} + 560 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −3.57739 + 1.16237i −1.20420 + 1.65745i 8.21057 − 5.96533i −3.20007 − 1.03977i 2.38136 − 7.32906i 6.45922 − 4.69290i −13.5948 + 18.7116i 1.48414 + 4.56770i 12.6565
50.2 −3.57739 + 1.16237i 1.20420 − 1.65745i 8.21057 − 5.96533i 3.20007 + 1.03977i −2.38136 + 7.32906i −6.45922 + 4.69290i −13.5948 + 18.7116i 1.48414 + 4.56770i −12.6565
50.3 −3.29469 + 1.07051i −3.37419 + 4.64418i 6.47292 − 4.70285i 6.66799 + 2.16656i 6.14528 − 18.9132i −8.03329 + 5.83652i −8.14689 + 11.2132i −7.40206 − 22.7812i −24.2883
50.4 −3.29469 + 1.07051i 3.37419 − 4.64418i 6.47292 − 4.70285i −6.66799 − 2.16656i −6.14528 + 18.9132i 8.03329 − 5.83652i −8.14689 + 11.2132i −7.40206 − 22.7812i 24.2883
50.5 −2.78817 + 0.905930i −0.947453 + 1.30406i 3.71709 − 2.70063i 1.02892 + 0.334315i 1.46027 − 4.49425i −1.46814 + 1.06667i −1.02457 + 1.41020i 1.97826 + 6.08844i −3.17165
50.6 −2.78817 + 0.905930i 0.947453 − 1.30406i 3.71709 − 2.70063i −1.02892 − 0.334315i −1.46027 + 4.49425i 1.46814 − 1.06667i −1.02457 + 1.41020i 1.97826 + 6.08844i 3.17165
50.7 −2.65582 + 0.862929i −2.13688 + 2.94116i 3.07268 − 2.23243i −9.01183 − 2.92812i 3.13716 − 9.65519i −6.35361 + 4.61617i 0.331499 − 0.456269i −1.30303 − 4.01033i 26.4606
50.8 −2.65582 + 0.862929i 2.13688 − 2.94116i 3.07268 − 2.23243i 9.01183 + 2.92812i −3.13716 + 9.65519i 6.35361 − 4.61617i 0.331499 − 0.456269i −1.30303 − 4.01033i −26.4606
50.9 −2.18981 + 0.711513i −2.79261 + 3.84370i 1.05296 − 0.765020i 1.00132 + 0.325350i 3.38045 − 10.4040i 9.33260 − 6.78053i 3.65205 − 5.02662i −4.19419 − 12.9084i −2.42420
50.10 −2.18981 + 0.711513i 2.79261 − 3.84370i 1.05296 − 0.765020i −1.00132 − 0.325350i −3.38045 + 10.4040i −9.33260 + 6.78053i 3.65205 − 5.02662i −4.19419 − 12.9084i 2.42420
50.11 −1.65779 + 0.538650i −0.956348 + 1.31630i −0.777931 + 0.565200i 7.82445 + 2.54232i 0.876403 − 2.69729i 2.83541 − 2.06005i 5.08350 − 6.99683i 1.96311 + 6.04183i −14.3407
50.12 −1.65779 + 0.538650i 0.956348 − 1.31630i −0.777931 + 0.565200i −7.82445 − 2.54232i −0.876403 + 2.69729i −2.83541 + 2.06005i 5.08350 − 6.99683i 1.96311 + 6.04183i 14.3407
50.13 −0.919654 + 0.298814i −0.836462 + 1.15129i −2.47959 + 1.80153i 3.40132 + 1.10516i 0.425234 − 1.30874i −5.31470 + 3.86135i 4.01555 − 5.52694i 2.15535 + 6.63349i −3.45827
50.14 −0.919654 + 0.298814i 0.836462 − 1.15129i −2.47959 + 1.80153i −3.40132 − 1.10516i −0.425234 + 1.30874i 5.31470 − 3.86135i 4.01555 − 5.52694i 2.15535 + 6.63349i 3.45827
50.15 −0.790508 + 0.256852i −2.01126 + 2.76826i −2.67714 + 1.94505i −2.75445 − 0.894974i 0.878884 − 2.70493i −1.92980 + 1.40208i 3.57095 − 4.91499i −0.836945 − 2.57585i 2.40729
50.16 −0.790508 + 0.256852i 2.01126 − 2.76826i −2.67714 + 1.94505i 2.75445 + 0.894974i −0.878884 + 2.70493i 1.92980 − 1.40208i 3.57095 − 4.91499i −0.836945 − 2.57585i −2.40729
50.17 −0.0158839 + 0.00516100i −3.11381 + 4.28579i −3.23584 + 2.35098i −0.870268 − 0.282767i 0.0273405 − 0.0841455i −6.84191 + 4.97094i 0.0785317 − 0.108090i −5.89104 − 18.1308i 0.0152826
50.18 −0.0158839 + 0.00516100i 3.11381 − 4.28579i −3.23584 + 2.35098i 0.870268 + 0.282767i −0.0273405 + 0.0841455i 6.84191 − 4.97094i 0.0785317 − 0.108090i −5.89104 − 18.1308i −0.0152826
50.19 0.309010 − 0.100404i −1.32381 + 1.82206i −3.15066 + 2.28909i −7.38403 − 2.39922i −0.226128 + 0.695951i 11.0969 − 8.06236i −1.50767 + 2.07513i 1.21370 + 3.73539i −2.52263
50.20 0.309010 − 0.100404i 1.32381 − 1.82206i −3.15066 + 2.28909i 7.38403 + 2.39922i 0.226128 − 0.695951i −11.0969 + 8.06236i −1.50767 + 2.07513i 1.21370 + 3.73539i 2.52263
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
17.b even 2 1 inner
187.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.3.l.a ✓ 136
11.d odd 10 1 inner 187.3.l.a ✓ 136
17.b even 2 1 inner 187.3.l.a ✓ 136
187.l odd 10 1 inner 187.3.l.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.3.l.a ✓ 136 1.a even 1 1 trivial
187.3.l.a ✓ 136 11.d odd 10 1 inner
187.3.l.a ✓ 136 17.b even 2 1 inner
187.3.l.a ✓ 136 187.l odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(187, [\chi])\).