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Properties

Label 87.3.h.a.5.2
Level $87$
Weight $3$
Character 87.5
Analytic conductor $2.371$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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] = [87,3,Mod(5,87)] 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("87.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 11])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.h (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.37057829993\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 5.2
Character \(\chi\) \(=\) 87.5
Dual form 87.3.h.a.35.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.17865 - 2.73194i) q^{2} +(-2.77437 + 1.14144i) q^{3} +(-1.82689 + 8.00414i) q^{4} +(0.834882 - 0.665796i) q^{5} +(9.16270 + 5.09261i) q^{6} +(0.965090 + 4.22833i) q^{7} +(13.2540 - 6.38280i) q^{8} +(6.39424 - 6.33353i) q^{9} +(-3.63783 - 0.830310i) q^{10} +(5.61335 + 2.70325i) q^{11} +(-4.06775 - 24.2917i) q^{12} +(6.63886 + 3.19711i) q^{13} +(9.44895 - 11.8486i) q^{14} +(-1.55631 + 2.80013i) q^{15} +(-16.7255 - 8.05458i) q^{16} -19.4140 q^{17} +(-31.2336 - 3.67014i) q^{18} +(32.9793 + 7.52731i) q^{19} +(3.80389 + 7.89886i) q^{20} +(-7.50389 - 10.6294i) q^{21} +(-4.84441 - 21.2247i) q^{22} +(33.5611 + 26.7641i) q^{23} +(-29.4860 + 32.8369i) q^{24} +(-5.30928 + 23.2615i) q^{25} +(-5.72944 - 25.1023i) q^{26} +(-10.5107 + 24.8702i) q^{27} -35.6073 q^{28} +(-15.4325 + 24.5528i) q^{29} +(11.0404 - 1.84876i) q^{30} +(-0.293838 + 0.234328i) q^{31} +(1.34048 + 5.87301i) q^{32} +(-18.6591 - 1.09252i) q^{33} +(42.2963 + 53.0379i) q^{34} +(3.62095 + 2.88761i) q^{35} +(39.0129 + 62.7511i) q^{36} +(-18.2915 - 37.9827i) q^{37} +(-51.2862 - 106.497i) q^{38} +(-22.0680 - 1.29211i) q^{39} +(6.81591 - 14.1534i) q^{40} +33.8611 q^{41} +(-12.6904 + 43.6578i) q^{42} +(-29.1693 - 23.2617i) q^{43} +(-31.8922 + 39.9915i) q^{44} +(1.12160 - 9.54502i) q^{45} -149.996i q^{46} +(-19.1438 - 9.21915i) q^{47} +(55.5965 + 3.25526i) q^{48} +(27.2001 - 13.0989i) q^{49} +(75.1159 - 36.1739i) q^{50} +(53.8617 - 22.1599i) q^{51} +(-37.7186 + 47.2976i) q^{52} +(-40.4947 + 32.2935i) q^{53} +(90.8428 - 25.4689i) q^{54} +(6.48630 - 1.48045i) q^{55} +(39.7800 + 49.8825i) q^{56} +(-100.089 + 16.7603i) q^{57} +(100.699 - 11.3313i) q^{58} +53.5167i q^{59} +(-19.5694 - 17.5724i) q^{60} +(32.1571 - 7.33965i) q^{61} +(1.28034 + 0.292229i) q^{62} +(32.9513 + 20.9246i) q^{63} +(-33.1734 + 41.5981i) q^{64} +(7.67129 - 1.75092i) q^{65} +(37.6669 + 53.3557i) q^{66} +(-31.0774 + 14.9661i) q^{67} +(35.4674 - 155.393i) q^{68} +(-123.660 - 35.9455i) q^{69} -16.1833i q^{70} +(43.2839 - 89.8799i) q^{71} +(44.3238 - 124.758i) q^{72} +(19.2642 + 15.3627i) q^{73} +(-63.9156 + 132.722i) q^{74} +(-11.8216 - 70.5961i) q^{75} +(-120.499 + 250.220i) q^{76} +(-6.01284 + 26.3440i) q^{77} +(44.5483 + 63.1033i) q^{78} +(29.7848 + 61.8488i) q^{79} +(-19.3265 + 4.41116i) q^{80} +(0.772696 - 80.9963i) q^{81} +(-73.7715 - 92.5065i) q^{82} +(99.6349 + 22.7410i) q^{83} +(98.7878 - 40.6435i) q^{84} +(-16.2084 + 12.9258i) q^{85} +130.368i q^{86} +(14.7899 - 85.7337i) q^{87} +91.6538 q^{88} +(-34.8722 - 43.7283i) q^{89} +(-28.5200 + 17.7311i) q^{90} +(-7.11134 + 31.1568i) q^{91} +(-275.536 + 219.732i) q^{92} +(0.547743 - 0.985508i) q^{93} +(16.5214 + 72.3848i) q^{94} +(32.5455 - 15.6731i) q^{95} +(-10.4227 - 14.7638i) q^{96} +(78.9642 + 18.0231i) q^{97} +(-95.0446 - 45.7711i) q^{98} +(53.0142 - 18.2671i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 7 q^{3} - 42 q^{4} - 23 q^{6} - 18 q^{7} - 17 q^{9} - 14 q^{10} - 52 q^{13} - 7 q^{15} - 58 q^{16} + 42 q^{18} - 14 q^{19} - 217 q^{21} + 124 q^{22} - 49 q^{24} + 64 q^{25} + 119 q^{27} + 284 q^{28}+ \cdots + 728 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{11}{14}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17865 − 2.73194i −1.08932 − 1.36597i −0.925178 − 0.379532i \(-0.876085\pi\)
−0.164145 − 0.986436i \(-0.552487\pi\)
\(3\) −2.77437 + 1.14144i −0.924790 + 0.380479i
\(4\) −1.82689 + 8.00414i −0.456723 + 2.00104i
\(5\) 0.834882 − 0.665796i 0.166976 − 0.133159i −0.536437 − 0.843941i \(-0.680230\pi\)
0.703413 + 0.710781i \(0.251658\pi\)
\(6\) 9.16270 + 5.09261i 1.52712 + 0.848769i
\(7\) 0.965090 + 4.22833i 0.137870 + 0.604048i 0.995901 + 0.0904500i \(0.0288305\pi\)
−0.858031 + 0.513598i \(0.828312\pi\)
\(8\) 13.2540 − 6.38280i 1.65675 − 0.797850i
\(9\) 6.39424 − 6.33353i 0.710471 − 0.703726i
\(10\) −3.63783 − 0.830310i −0.363783 − 0.0830310i
\(11\) 5.61335 + 2.70325i 0.510304 + 0.245750i 0.671273 − 0.741210i \(-0.265748\pi\)
−0.160969 + 0.986959i \(0.551462\pi\)
\(12\) −4.06775 − 24.2917i −0.338979 − 2.02431i
\(13\) 6.63886 + 3.19711i 0.510682 + 0.245931i 0.671435 − 0.741063i \(-0.265678\pi\)
−0.160753 + 0.986995i \(0.551392\pi\)
\(14\) 9.44895 − 11.8486i 0.674925 − 0.846330i
\(15\) −1.55631 + 2.80013i −0.103754 + 0.186675i
\(16\) −16.7255 − 8.05458i −1.04534 − 0.503411i
\(17\) −19.4140 −1.14200 −0.571001 − 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(18\) −31.2336 − 3.67014i −1.73520 − 0.203896i
\(19\) 32.9793 + 7.52731i 1.73575 + 0.396174i 0.969252 − 0.246068i \(-0.0791388\pi\)
0.766501 + 0.642243i \(0.221996\pi\)
\(20\) 3.80389 + 7.89886i 0.190194 + 0.394943i
\(21\) −7.50389 − 10.6294i −0.357328 − 0.506160i
\(22\) −4.84441 − 21.2247i −0.220200 − 0.964761i
\(23\) 33.5611 + 26.7641i 1.45918 + 1.16365i 0.953679 + 0.300827i \(0.0972625\pi\)
0.505498 + 0.862828i \(0.331309\pi\)
\(24\) −29.4860 + 32.8369i −1.22858 + 1.36820i
\(25\) −5.30928 + 23.2615i −0.212371 + 0.930459i
\(26\) −5.72944 − 25.1023i −0.220363 − 0.965474i
\(27\) −10.5107 + 24.8702i −0.389284 + 0.921118i
\(28\) −35.6073 −1.27169
\(29\) −15.4325 + 24.5528i −0.532154 + 0.846648i
\(30\) 11.0404 − 1.84876i 0.368014 − 0.0616255i
\(31\) −0.293838 + 0.234328i −0.00947863 + 0.00755896i −0.628218 − 0.778038i \(-0.716215\pi\)
0.618739 + 0.785597i \(0.287644\pi\)
\(32\) 1.34048 + 5.87301i 0.0418899 + 0.183532i
\(33\) −18.6591 − 1.09252i −0.565427 − 0.0331066i
\(34\) 42.2963 + 53.0379i 1.24401 + 1.55994i
\(35\) 3.62095 + 2.88761i 0.103456 + 0.0825031i
\(36\) 39.0129 + 62.7511i 1.08369 + 1.74309i
\(37\) −18.2915 − 37.9827i −0.494365 − 1.02656i −0.987648 − 0.156690i \(-0.949918\pi\)
0.493283 − 0.869869i \(-0.335797\pi\)
\(38\) −51.2862 − 106.497i −1.34964 − 2.80255i
\(39\) −22.0680 − 1.29211i −0.565845 − 0.0331311i
\(40\) 6.81591 − 14.1534i 0.170398 − 0.353834i
\(41\) 33.8611 0.825881 0.412941 − 0.910758i \(-0.364502\pi\)
0.412941 + 0.910758i \(0.364502\pi\)
\(42\) −12.6904 + 43.6578i −0.302153 + 1.03947i
\(43\) −29.1693 − 23.2617i −0.678356 − 0.540971i 0.222585 − 0.974913i \(-0.428551\pi\)
−0.900941 + 0.433943i \(0.857122\pi\)
\(44\) −31.8922 + 39.9915i −0.724822 + 0.908898i
\(45\) 1.12160 − 9.54502i 0.0249244 − 0.212112i
\(46\) − 149.996i − 3.26079i
\(47\) −19.1438 − 9.21915i −0.407314 − 0.196152i 0.218997 − 0.975726i \(-0.429722\pi\)
−0.626311 + 0.779574i \(0.715436\pi\)
\(48\) 55.5965 + 3.25526i 1.15826 + 0.0678180i
\(49\) 27.2001 − 13.0989i 0.555103 − 0.267324i
\(50\) 75.1159 − 36.1739i 1.50232 − 0.723479i
\(51\) 53.8617 − 22.1599i 1.05611 − 0.434508i
\(52\) −37.7186 + 47.2976i −0.725358 + 0.909570i
\(53\) −40.4947 + 32.2935i −0.764051 + 0.609310i −0.926015 − 0.377486i \(-0.876789\pi\)
0.161964 + 0.986797i \(0.448217\pi\)
\(54\) 90.8428 − 25.4689i 1.68227 − 0.471646i
\(55\) 6.48630 − 1.48045i 0.117933 − 0.0269174i
\(56\) 39.7800 + 49.8825i 0.710356 + 0.890759i
\(57\) −100.089 + 16.7603i −1.75594 + 0.294040i
\(58\) 100.699 − 11.3313i 1.73618 − 0.195368i
\(59\) 53.5167i 0.907062i 0.891240 + 0.453531i \(0.149836\pi\)
−0.891240 + 0.453531i \(0.850164\pi\)
\(60\) −19.5694 − 17.5724i −0.326157 − 0.292874i
\(61\) 32.1571 − 7.33965i 0.527165 − 0.120322i 0.0493487 − 0.998782i \(-0.484285\pi\)
0.477817 + 0.878460i \(0.341428\pi\)
\(62\) 1.28034 + 0.292229i 0.0206506 + 0.00471336i
\(63\) 32.9513 + 20.9246i 0.523037 + 0.332136i
\(64\) −33.1734 + 41.5981i −0.518334 + 0.649970i
\(65\) 7.67129 − 1.75092i 0.118020 − 0.0269373i
\(66\) 37.6669 + 53.3557i 0.570710 + 0.808419i
\(67\) −31.0774 + 14.9661i −0.463842 + 0.223375i −0.651184 − 0.758920i \(-0.725727\pi\)
0.187342 + 0.982295i \(0.440013\pi\)
\(68\) 35.4674 − 155.393i 0.521579 − 2.28519i
\(69\) −123.660 − 35.9455i −1.79218 − 0.520950i
\(70\) − 16.1833i − 0.231190i
\(71\) 43.2839 − 89.8799i 0.609632 − 1.26591i −0.336363 − 0.941732i \(-0.609197\pi\)
0.945995 − 0.324181i \(-0.105089\pi\)
\(72\) 44.3238 − 124.758i 0.615608 − 1.73275i
\(73\) 19.2642 + 15.3627i 0.263894 + 0.210448i 0.746493 − 0.665393i \(-0.231736\pi\)
−0.482600 + 0.875841i \(0.660307\pi\)
\(74\) −63.9156 + 132.722i −0.863724 + 1.79354i
\(75\) −11.8216 − 70.5961i −0.157621 − 0.941282i
\(76\) −120.499 + 250.220i −1.58552 + 3.29236i
\(77\) −6.01284 + 26.3440i −0.0780889 + 0.342130i
\(78\) 44.5483 + 63.1033i 0.571132 + 0.809017i
\(79\) 29.7848 + 61.8488i 0.377023 + 0.782896i 1.00000 0.000655610i \(0.000208687\pi\)
−0.622977 + 0.782240i \(0.714077\pi\)
\(80\) −19.3265 + 4.41116i −0.241582 + 0.0551395i
\(81\) 0.772696 − 80.9963i 0.00953946 − 0.999954i
\(82\) −73.7715 − 92.5065i −0.899652 − 1.12813i
\(83\) 99.6349 + 22.7410i 1.20042 + 0.273988i 0.775562 − 0.631271i \(-0.217467\pi\)
0.424859 + 0.905260i \(0.360324\pi\)
\(84\) 98.7878 − 40.6435i 1.17605 − 0.483851i
\(85\) −16.2084 + 12.9258i −0.190688 + 0.152068i
\(86\) 130.368i 1.51591i
\(87\) 14.7899 − 85.7337i 0.169999 − 0.985444i
\(88\) 91.6538 1.04152
\(89\) −34.8722 − 43.7283i −0.391822 − 0.491329i 0.546322 − 0.837575i \(-0.316028\pi\)
−0.938144 + 0.346246i \(0.887456\pi\)
\(90\) −28.5200 + 17.7311i −0.316888 + 0.197012i
\(91\) −7.11134 + 31.1568i −0.0781466 + 0.342383i
\(92\) −275.536 + 219.732i −2.99495 + 2.38840i
\(93\) 0.547743 − 0.985508i 0.00588971 − 0.0105969i
\(94\) 16.5214 + 72.3848i 0.175759 + 0.770051i
\(95\) 32.5455 − 15.6731i 0.342584 − 0.164980i
\(96\) −10.4227 − 14.7638i −0.108569 − 0.153790i
\(97\) 78.9642 + 18.0231i 0.814064 + 0.185805i 0.609232 − 0.792992i \(-0.291478\pi\)
0.204832 + 0.978797i \(0.434335\pi\)
\(98\) −95.0446 − 45.7711i −0.969843 − 0.467052i
\(99\) 53.0142 − 18.2671i 0.535497 − 0.184516i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 87.3.h.a.5.2 ✓ 108
3.2 odd 2 inner 87.3.h.a.5.17 yes 108
29.6 even 14 inner 87.3.h.a.35.17 yes 108
87.35 odd 14 inner 87.3.h.a.35.2 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.3.h.a.5.2 ✓ 108 1.1 even 1 trivial
87.3.h.a.5.17 yes 108 3.2 odd 2 inner
87.3.h.a.35.2 yes 108 87.35 odd 14 inner
87.3.h.a.35.17 yes 108 29.6 even 14 inner