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Properties

Label 187.3.k.a
Level $187$
Weight $3$
Character orbit 187.k
Analytic conductor $5.095$
Analytic rank $0$
Dimension $128$
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] = [187,3,Mod(18,187)]
 
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([7, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.18");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 187.k (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: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 2 q^{3} + 76 q^{4} + 6 q^{5} - 40 q^{8} - 86 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 2 q^{3} + 76 q^{4} + 6 q^{5} - 40 q^{8} - 86 q^{9} - 8 q^{11} + 16 q^{12} + 6 q^{14} - 42 q^{15} - 196 q^{16} - 150 q^{18} - 78 q^{20} - 10 q^{22} + 24 q^{23} - 244 q^{25} + 126 q^{26} - 58 q^{27} + 110 q^{28} - 90 q^{29} + 240 q^{30} + 126 q^{31} + 48 q^{33} + 150 q^{35} - 24 q^{36} - 150 q^{37} + 244 q^{38} + 210 q^{39} + 480 q^{40} - 120 q^{41} + 74 q^{42} - 112 q^{44} + 228 q^{45} + 210 q^{46} + 8 q^{47} - 18 q^{48} + 686 q^{49} - 160 q^{50} - 600 q^{52} - 212 q^{53} + 74 q^{55} - 1004 q^{56} - 640 q^{57} + 62 q^{58} + 278 q^{59} - 540 q^{60} - 160 q^{61} + 320 q^{62} - 160 q^{63} + 360 q^{64} - 432 q^{66} + 528 q^{67} - 322 q^{69} - 268 q^{70} + 330 q^{71} - 990 q^{72} - 610 q^{73} + 1180 q^{74} - 140 q^{75} + 702 q^{77} + 136 q^{78} - 800 q^{79} + 880 q^{80} - 92 q^{81} + 220 q^{82} - 640 q^{83} + 480 q^{84} + 756 q^{86} - 634 q^{88} - 96 q^{89} + 800 q^{90} + 98 q^{91} + 242 q^{92} - 420 q^{93} - 400 q^{94} - 1240 q^{96} - 414 q^{97} + 374 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −2.31249 + 3.18287i 1.20371 − 3.70463i −3.54698 − 10.9165i 4.77177 − 3.46690i 9.00780 + 12.3982i −9.90029 + 3.21680i 27.9813 + 9.09168i −4.99424 − 3.62853i 23.2051i
18.2 −2.16650 + 2.98193i 0.0133565 − 0.0411072i −2.96211 − 9.11644i −2.50609 + 1.82078i 0.0936418 + 0.128887i 11.2776 − 3.66433i 19.5801 + 6.36197i 7.27964 + 5.28897i − 11.4177i
18.3 −2.11389 + 2.90952i −1.15588 + 3.55742i −2.76070 − 8.49657i −3.23959 + 2.35370i −7.90698 − 10.8830i −7.36274 + 2.39230i 16.8754 + 5.48314i −4.03803 − 2.93380i − 14.4011i
18.4 −1.96583 + 2.70574i −0.899736 + 2.76910i −2.22045 − 6.83384i 5.28889 − 3.84260i −5.72374 − 7.87805i 4.61733 − 1.50026i 10.1325 + 3.29224i 0.422747 + 0.307144i 21.8643i
18.5 −1.45361 + 2.00072i −0.264767 + 0.814868i −0.653844 − 2.01232i 5.18447 − 3.76674i −1.24546 − 1.71423i −6.62161 + 2.15149i −4.43143 − 1.43986i 6.68725 + 4.85857i 15.8481i
18.6 −1.42516 + 1.96157i −1.57804 + 4.85672i −0.580594 − 1.78688i 0.773670 − 0.562104i −7.27782 − 10.0171i 2.00036 − 0.649955i −4.89131 − 1.58928i −13.8163 − 10.0382i 2.31870i
18.7 −1.32269 + 1.82052i 0.602902 − 1.85554i −0.328735 − 1.01174i 3.47030 − 2.52132i 2.58061 + 3.55190i −0.277688 + 0.0902262i −6.28390 − 2.04176i 4.20161 + 3.05265i 9.65267i
18.8 −1.30889 + 1.80153i 1.69599 − 5.21973i −0.296254 − 0.911776i 4.80885 − 3.49383i 7.18364 + 9.88743i 11.8455 − 3.84883i −6.44095 − 2.09279i −17.0881 − 12.4152i 13.2363i
18.9 −1.28144 + 1.76375i 0.175313 − 0.539557i −0.232667 − 0.716075i −5.98933 + 4.35151i 0.726993 + 1.00062i 0.967141 − 0.314243i −6.73254 − 2.18754i 7.02077 + 5.10088i − 16.1399i
18.10 −0.900091 + 1.23887i −1.08430 + 3.33712i 0.511435 + 1.57404i −6.24768 + 4.53920i −3.15829 − 4.34701i 0.156244 − 0.0507669i −8.23587 − 2.67600i −2.67951 − 1.94677i − 11.8258i
18.11 −0.709458 + 0.976486i 1.58905 − 4.89059i 0.785875 + 2.41867i −3.08779 + 2.24341i 3.64822 + 5.02135i −10.3811 + 3.37303i −7.51106 − 2.44049i −14.1116 − 10.2527i − 4.60679i
18.12 −0.613699 + 0.844685i −0.425831 + 1.31057i 0.899203 + 2.76746i 0.218666 − 0.158870i −0.845690 − 1.16399i 11.5604 − 3.75620i −6.86142 − 2.22941i 5.74488 + 4.17390i 0.282203i
18.13 −0.357008 + 0.491380i −0.692618 + 2.13166i 1.12207 + 3.45337i 0.882293 − 0.641023i −0.800183 − 1.10136i −13.1946 + 4.28718i −4.40811 − 1.43228i 3.21691 + 2.33722i 0.662391i
18.14 −0.284918 + 0.392156i 0.470795 − 1.44896i 1.16346 + 3.58076i 1.94576 − 1.41368i 0.434080 + 0.597460i −0.490287 + 0.159304i −3.57974 − 1.16313i 5.40332 + 3.92574i 1.16582i
18.15 −0.0486696 + 0.0669879i −0.995158 + 3.06278i 1.23395 + 3.79771i 6.75865 − 4.91045i −0.156735 − 0.215728i 9.72270 − 3.15910i −0.629452 − 0.204521i −1.10913 − 0.805833i 0.691737i
18.16 0.112326 − 0.154603i 1.42515 − 4.38616i 1.22478 + 3.76949i −5.67167 + 4.12071i −0.518033 − 0.713012i 12.5096 − 4.06462i 1.44734 + 0.470269i −9.92621 − 7.21182i 1.33972i
18.17 0.134829 − 0.185576i 0.446425 − 1.37395i 1.21981 + 3.75418i −4.26161 + 3.09624i −0.194782 − 0.268094i 0.534433 − 0.173648i 1.73378 + 0.563340i 5.59270 + 4.06333i 1.20832i
18.18 0.246168 − 0.338822i −1.16131 + 3.57413i 1.18187 + 3.63741i −0.982722 + 0.713989i 0.925117 + 1.27331i −1.09549 + 0.355945i 3.11661 + 1.01265i −4.14462 − 3.01125i 0.508729i
18.19 0.267728 − 0.368496i 1.23150 − 3.79017i 1.17196 + 3.60691i 7.38726 − 5.36716i −1.06695 − 1.46854i −4.11806 + 1.33804i 3.37567 + 1.09682i −5.56764 − 4.04512i − 4.15911i
18.20 0.728638 − 1.00288i −1.81440 + 5.58415i 0.761204 + 2.34275i −3.45324 + 2.50893i 4.27821 + 5.88846i 3.12784 − 1.01630i 7.61999 + 2.47589i −20.6095 − 14.9737i 5.29130i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d 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.k.a ✓ 128
11.d odd 10 1 inner 187.3.k.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.3.k.a ✓ 128 1.a even 1 1 trivial
187.3.k.a ✓ 128 11.d odd 10 1 inner

Hecke kernels

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