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Properties

Label 187.4.g.a
Level $187$
Weight $4$
Character orbit 187.g
Analytic conductor $11.033$
Analytic rank $0$
Dimension $88$
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] = [187,4,Mod(69,187)]
 
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([8, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.69");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 187.g (of order \(5\), 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: \(11.0333571711\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 2 q^{2} + 14 q^{3} - 82 q^{4} + 42 q^{5} + 12 q^{6} + 16 q^{7} + 104 q^{8} - 178 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 2 q^{2} + 14 q^{3} - 82 q^{4} + 42 q^{5} + 12 q^{6} + 16 q^{7} + 104 q^{8} - 178 q^{9} - 130 q^{10} - 20 q^{11} - 674 q^{12} + 24 q^{13} + 173 q^{14} + 280 q^{15} + 242 q^{16} - 374 q^{17} - 265 q^{18} - 92 q^{19} + 347 q^{20} - 480 q^{21} - 248 q^{22} - 1312 q^{23} + 512 q^{24} + 146 q^{25} + 1045 q^{26} + 314 q^{27} + 51 q^{28} - 526 q^{29} - 712 q^{30} + 1424 q^{31} - 166 q^{32} - 1036 q^{33} + 34 q^{34} - 446 q^{35} - 260 q^{36} + 1344 q^{37} - 1006 q^{38} - 92 q^{39} - 272 q^{40} - 1002 q^{41} + 649 q^{42} + 764 q^{43} + 1894 q^{44} - 2452 q^{45} + 1154 q^{46} + 462 q^{47} + 1399 q^{48} - 1872 q^{49} - 2336 q^{50} + 238 q^{51} - 2266 q^{52} + 1606 q^{53} + 4800 q^{54} + 3492 q^{55} + 54 q^{56} + 2012 q^{57} + 603 q^{58} + 286 q^{59} - 5684 q^{60} - 1348 q^{61} + 508 q^{62} - 2936 q^{63} - 1576 q^{64} - 304 q^{65} + 6572 q^{66} - 6884 q^{67} - 969 q^{68} - 878 q^{69} + 1370 q^{70} - 1432 q^{71} + 753 q^{72} + 3480 q^{73} - 6737 q^{74} - 694 q^{75} + 2522 q^{76} + 3870 q^{77} + 2240 q^{78} + 2118 q^{79} + 1640 q^{80} - 1950 q^{81} - 3285 q^{82} + 5758 q^{83} + 7968 q^{84} - 306 q^{85} - 5066 q^{86} + 3936 q^{87} - 97 q^{88} - 5984 q^{89} + 1819 q^{90} + 2240 q^{91} + 2111 q^{92} - 4076 q^{93} - 1788 q^{94} + 10810 q^{95} - 15349 q^{96} + 3358 q^{97} + 4604 q^{98} + 5224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −4.16200 + 3.02387i −2.07129 − 6.37477i 5.70631 − 17.5622i 14.1522 + 10.2822i 27.8972 + 20.2685i −1.01675 + 3.12924i 16.6382 + 51.2072i −14.5040 + 10.5378i −89.9932
69.2 −4.09432 + 2.97470i 0.319974 + 0.984778i 5.44248 − 16.7502i −5.01624 − 3.64451i −4.23949 − 3.08017i −8.53295 + 26.2617i 15.0325 + 46.2652i 20.9761 − 15.2400i 31.3794
69.3 −3.45303 + 2.50877i −1.63031 − 5.01757i 3.15733 − 9.71725i 1.56107 + 1.13418i 18.2174 + 13.2357i 10.3128 − 31.7394i 2.92451 + 9.00072i −0.674633 + 0.490150i −8.23580
69.4 −3.12144 + 2.26786i −0.686753 − 2.11361i 2.12808 − 6.54956i −9.25221 − 6.72212i 6.93704 + 5.04005i −3.90604 + 12.0215i −1.32747 − 4.08553i 17.8477 − 12.9671i 44.1251
69.5 −2.73512 + 1.98718i 1.73282 + 5.33306i 1.05986 − 3.26192i −5.73256 − 4.16495i −15.3372 − 11.1431i 1.54020 − 4.74024i −4.77461 − 14.6947i −3.59539 + 2.61221i 23.9557
69.6 −2.58153 + 1.87559i 2.85418 + 8.78427i 0.674329 − 2.07537i 15.4264 + 11.2079i −23.8439 − 17.3236i 4.99270 − 15.3660i −5.73671 − 17.6558i −47.1736 + 34.2736i −60.8454
69.7 −1.32131 + 0.959989i −1.99477 − 6.13926i −1.64785 + 5.07156i −3.20546 − 2.32890i 8.52933 + 6.19692i −6.24735 + 19.2274i −6.72889 − 20.7094i −11.8680 + 8.62258i 6.47113
69.8 −1.21822 + 0.885088i 1.94689 + 5.99190i −1.77146 + 5.45199i −3.23546 − 2.35070i −7.67508 − 5.57627i −3.44584 + 10.6052i −6.39001 − 19.6664i −10.2690 + 7.46086i 6.02208
69.9 −1.20982 + 0.878984i −1.36481 − 4.20044i −1.78109 + 5.48163i 12.6241 + 9.17197i 5.34328 + 3.88212i −9.84983 + 30.3147i −6.36035 − 19.5751i 6.06247 − 4.40464i −23.3349
69.10 −1.17223 + 0.851678i 0.272867 + 0.839800i −1.82336 + 5.61172i 12.1365 + 8.81772i −1.03510 − 0.752046i 8.75701 − 26.9513i −6.22400 − 19.1555i 21.2127 − 15.4119i −21.7367
69.11 −1.14481 + 0.831753i −2.81299 − 8.65749i −1.85336 + 5.70405i −15.9701 − 11.6030i 10.4212 + 7.57147i 7.59675 − 23.3804i −6.12085 − 18.8380i −45.1957 + 32.8366i 27.9336
69.12 0.417015 − 0.302979i −1.10292 − 3.39443i −2.39003 + 7.35576i −1.69147 − 1.22892i −1.48837 − 1.08137i 5.45767 − 16.7970i 2.50624 + 7.71343i 11.5377 − 8.38263i −1.07771
69.13 1.04043 − 0.755917i 1.65485 + 5.09310i −1.96105 + 6.03550i 14.1218 + 10.2601i 5.57171 + 4.04808i −3.72872 + 11.4758i 5.70127 + 17.5467i −1.35766 + 0.986394i 22.4485
69.14 1.51197 − 1.09851i 1.07478 + 3.30783i −1.39280 + 4.28661i −13.7098 − 9.96076i 5.25874 + 3.82070i 8.40424 − 25.8656i 7.22319 + 22.2307i 12.0569 − 8.75981i −31.6709
69.15 1.55773 − 1.13176i 2.24441 + 6.90757i −1.32648 + 4.08249i −6.97938 − 5.07082i 11.3139 + 8.22003i −2.69096 + 8.28193i 7.31410 + 22.5105i −20.8337 + 15.1366i −16.6110
69.16 2.11621 − 1.53752i −3.14805 − 9.68871i −0.357740 + 1.10101i 1.59764 + 1.16075i −21.5585 − 15.6632i −8.96511 + 27.5918i 7.40235 + 22.7821i −62.1175 + 45.1310i 5.16562
69.17 2.29922 − 1.67048i −1.65295 − 5.08726i 0.0237610 − 0.0731288i 13.8623 + 10.0715i −12.2986 − 8.93548i 2.42144 − 7.45244i 6.95825 + 21.4153i −1.30447 + 0.947753i 48.6966
69.18 3.00388 − 2.18244i 0.333901 + 1.02764i 1.78807 − 5.50312i 1.55572 + 1.13030i 3.24577 + 2.35819i −8.36761 + 25.7529i 2.53993 + 7.81709i 20.8989 − 15.1839i 7.14001
69.19 3.36401 − 2.44410i −0.858514 − 2.64224i 2.87082 − 8.83549i −13.9525 − 10.1371i −9.34593 − 6.79022i −1.17874 + 3.62778i −1.65780 − 5.10219i 15.5991 − 11.3334i −71.7123
69.20 3.37987 − 2.45562i 2.59397 + 7.98341i 2.92132 − 8.99090i 6.72000 + 4.88237i 28.3715 + 20.6131i 3.09987 − 9.54043i −1.87658 − 5.77553i −35.1627 + 25.5472i 34.7020
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.4.g.a ✓ 88
11.c even 5 1 inner 187.4.g.a ✓ 88
11.c even 5 1 2057.4.a.t 44
11.d odd 10 1 2057.4.a.s 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.g.a ✓ 88 1.a even 1 1 trivial
187.4.g.a ✓ 88 11.c even 5 1 inner
2057.4.a.s 44 11.d odd 10 1
2057.4.a.t 44 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} - 2 T_{2}^{87} + 131 T_{2}^{86} - 278 T_{2}^{85} + 9602 T_{2}^{84} - 20518 T_{2}^{83} + \cdots + 24\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(187, [\chi])\). Copy content Toggle raw display