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Properties

Label 164.4.g.a.37.9
Level $164$
Weight $4$
Character 164.37
Analytic conductor $9.676$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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] = [164,4,Mod(37,164)] 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("164.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.g (of order \(5\), degree \(4\), 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: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 37.9
Character \(\chi\) \(=\) 164.37
Dual form 164.4.g.a.133.9

$q$-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)
 
\(f(q)\) \(=\) \(q+7.79300 q^{3} +(2.83592 - 8.72806i) q^{5} +(7.21035 + 5.23862i) q^{7} +33.7308 q^{9} +(0.222917 + 0.686068i) q^{11} +(35.9062 - 26.0874i) q^{13} +(22.1003 - 68.0177i) q^{15} +(2.96044 + 9.11131i) q^{17} +(-39.0163 - 28.3470i) q^{19} +(56.1902 + 40.8246i) q^{21} +(-22.5448 + 16.3798i) q^{23} +(32.9906 + 23.9691i) q^{25} +52.4529 q^{27} +(-31.7620 + 97.7534i) q^{29} +(-16.0154 - 49.2903i) q^{31} +(1.73719 + 5.34653i) q^{33} +(66.1709 - 48.0760i) q^{35} +(17.6181 - 54.2231i) q^{37} +(279.817 - 203.299i) q^{39} +(144.626 + 219.099i) q^{41} +(-11.7089 + 8.50699i) q^{43} +(95.6577 - 294.404i) q^{45} +(61.9486 - 45.0083i) q^{47} +(-81.4469 - 250.668i) q^{49} +(23.0707 + 71.0044i) q^{51} +(-50.0447 + 154.022i) q^{53} +6.62022 q^{55} +(-304.054 - 220.908i) q^{57} +(-568.612 + 413.121i) q^{59} +(73.7375 + 53.5735i) q^{61} +(243.211 + 176.703i) q^{63} +(-125.865 - 387.373i) q^{65} +(-217.984 + 670.886i) q^{67} +(-175.692 + 127.648i) q^{69} +(181.773 + 559.440i) q^{71} -849.796 q^{73} +(257.096 + 186.791i) q^{75} +(-1.98674 + 6.11457i) q^{77} -718.930 q^{79} -501.966 q^{81} -1094.21 q^{83} +87.9196 q^{85} +(-247.521 + 761.792i) q^{87} +(-865.645 - 628.928i) q^{89} +395.558 q^{91} +(-124.808 - 384.119i) q^{93} +(-358.062 + 260.147i) q^{95} +(248.658 - 765.290i) q^{97} +(7.51917 + 23.1416i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 2 q^{3} + 270 q^{9} + 83 q^{11} - 94 q^{13} - 48 q^{15} - 144 q^{17} - 179 q^{19} - 176 q^{21} + 180 q^{23} - 542 q^{25} - 236 q^{27} + 68 q^{29} - 298 q^{31} + 823 q^{33} + 178 q^{35} - 253 q^{37}+ \cdots + 3203 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 7.79300 1.49976 0.749881 − 0.661572i \(-0.230111\pi\)
0.749881 + 0.661572i \(0.230111\pi\)
\(4\) 0 0
\(5\) 2.83592 − 8.72806i 0.253652 − 0.780661i −0.740440 − 0.672122i \(-0.765383\pi\)
0.994092 − 0.108539i \(-0.0346171\pi\)
\(6\) 0 0
\(7\) 7.21035 + 5.23862i 0.389322 + 0.282859i 0.765178 − 0.643819i \(-0.222651\pi\)
−0.375856 + 0.926678i \(0.622651\pi\)
\(8\) 0 0
\(9\) 33.7308 1.24929
\(10\) 0 0
\(11\) 0.222917 + 0.686068i 0.00611018 + 0.0188052i 0.954065 − 0.299599i \(-0.0968530\pi\)
−0.947955 + 0.318404i \(0.896853\pi\)
\(12\) 0 0
\(13\) 35.9062 − 26.0874i 0.766046 − 0.556565i −0.134713 − 0.990885i \(-0.543011\pi\)
0.900759 + 0.434320i \(0.143011\pi\)
\(14\) 0 0
\(15\) 22.1003 − 68.0177i 0.380418 − 1.17081i
\(16\) 0 0
\(17\) 2.96044 + 9.11131i 0.0422361 + 0.129989i 0.969951 − 0.243300i \(-0.0782299\pi\)
−0.927715 + 0.373289i \(0.878230\pi\)
\(18\) 0 0
\(19\) −39.0163 − 28.3470i −0.471103 − 0.342277i 0.326768 − 0.945105i \(-0.394040\pi\)
−0.797871 + 0.602828i \(0.794040\pi\)
\(20\) 0 0
\(21\) 56.1902 + 40.8246i 0.583891 + 0.424222i
\(22\) 0 0
\(23\) −22.5448 + 16.3798i −0.204388 + 0.148497i −0.685271 − 0.728288i \(-0.740316\pi\)
0.480883 + 0.876785i \(0.340316\pi\)
\(24\) 0 0
\(25\) 32.9906 + 23.9691i 0.263925 + 0.191753i
\(26\) 0 0
\(27\) 52.4529 0.373873
\(28\) 0 0
\(29\) −31.7620 + 97.7534i −0.203381 + 0.625943i 0.796395 + 0.604777i \(0.206738\pi\)
−0.999776 + 0.0211659i \(0.993262\pi\)
\(30\) 0 0
\(31\) −16.0154 − 49.2903i −0.0927886 − 0.285574i 0.893882 − 0.448301i \(-0.147971\pi\)
−0.986671 + 0.162727i \(0.947971\pi\)
\(32\) 0 0
\(33\) 1.73719 + 5.34653i 0.00916383 + 0.0282034i
\(34\) 0 0
\(35\) 66.1709 − 48.0760i 0.319569 − 0.232181i
\(36\) 0 0
\(37\) 17.6181 − 54.2231i 0.0782812 − 0.240925i −0.904256 − 0.426991i \(-0.859574\pi\)
0.982537 + 0.186066i \(0.0595737\pi\)
\(38\) 0 0
\(39\) 279.817 − 203.299i 1.14889 − 0.834715i
\(40\) 0 0
\(41\) 144.626 + 219.099i 0.550899 + 0.834572i
\(42\) 0 0
\(43\) −11.7089 + 8.50699i −0.0415252 + 0.0301698i −0.608354 − 0.793666i \(-0.708170\pi\)
0.566829 + 0.823835i \(0.308170\pi\)
\(44\) 0 0
\(45\) 95.6577 − 294.404i 0.316885 − 0.975271i
\(46\) 0 0
\(47\) 61.9486 − 45.0083i 0.192258 − 0.139684i −0.487492 − 0.873127i \(-0.662088\pi\)
0.679750 + 0.733444i \(0.262088\pi\)
\(48\) 0 0
\(49\) −81.4469 − 250.668i −0.237455 − 0.730810i
\(50\) 0 0
\(51\) 23.0707 + 71.0044i 0.0633441 + 0.194953i
\(52\) 0 0
\(53\) −50.0447 + 154.022i −0.129701 + 0.399180i −0.994728 − 0.102546i \(-0.967301\pi\)
0.865027 + 0.501725i \(0.167301\pi\)
\(54\) 0 0
\(55\) 6.62022 0.0162304
\(56\) 0 0
\(57\) −304.054 − 220.908i −0.706543 − 0.513334i
\(58\) 0 0
\(59\) −568.612 + 413.121i −1.25469 + 0.911589i −0.998485 − 0.0550327i \(-0.982474\pi\)
−0.256209 + 0.966621i \(0.582474\pi\)
\(60\) 0 0
\(61\) 73.7375 + 53.5735i 0.154773 + 0.112449i 0.662476 − 0.749083i \(-0.269506\pi\)
−0.507704 + 0.861532i \(0.669506\pi\)
\(62\) 0 0
\(63\) 243.211 + 176.703i 0.486376 + 0.353373i
\(64\) 0 0
\(65\) −125.865 − 387.373i −0.240179 − 0.739196i
\(66\) 0 0
\(67\) −217.984 + 670.886i −0.397477 + 1.22331i 0.529538 + 0.848286i \(0.322365\pi\)
−0.927015 + 0.375024i \(0.877635\pi\)
\(68\) 0 0
\(69\) −175.692 + 127.648i −0.306534 + 0.222710i
\(70\) 0 0
\(71\) 181.773 + 559.440i 0.303838 + 0.935118i 0.980108 + 0.198464i \(0.0635952\pi\)
−0.676270 + 0.736654i \(0.736405\pi\)
\(72\) 0 0
\(73\) −849.796 −1.36248 −0.681240 − 0.732060i \(-0.738559\pi\)
−0.681240 + 0.732060i \(0.738559\pi\)
\(74\) 0 0
\(75\) 257.096 + 186.791i 0.395825 + 0.287583i
\(76\) 0 0
\(77\) −1.98674 + 6.11457i −0.00294040 + 0.00904961i
\(78\) 0 0
\(79\) −718.930 −1.02387 −0.511937 − 0.859023i \(-0.671072\pi\)
−0.511937 + 0.859023i \(0.671072\pi\)
\(80\) 0 0
\(81\) −501.966 −0.688567
\(82\) 0 0
\(83\) −1094.21 −1.44704 −0.723522 − 0.690301i \(-0.757478\pi\)
−0.723522 + 0.690301i \(0.757478\pi\)
\(84\) 0 0
\(85\) 87.9196 0.112191
\(86\) 0 0
\(87\) −247.521 + 761.792i −0.305024 + 0.938766i
\(88\) 0 0
\(89\) −865.645 − 628.928i −1.03099 − 0.749059i −0.0624845 − 0.998046i \(-0.519902\pi\)
−0.968507 + 0.248987i \(0.919902\pi\)
\(90\) 0 0
\(91\) 395.558 0.455668
\(92\) 0 0
\(93\) −124.808 − 384.119i −0.139161 − 0.428293i
\(94\) 0 0
\(95\) −358.062 + 260.147i −0.386698 + 0.280953i
\(96\) 0 0
\(97\) 248.658 − 765.290i 0.260282 − 0.801067i −0.732460 − 0.680810i \(-0.761628\pi\)
0.992743 − 0.120257i \(-0.0383719\pi\)
\(98\) 0 0
\(99\) 7.51917 + 23.1416i 0.00763338 + 0.0234931i
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 164.4.g.a.37.9 ✓ 40
41.10 even 5 inner 164.4.g.a.133.9 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.4.g.a.37.9 ✓ 40 1.1 even 1 trivial
164.4.g.a.133.9 yes 40 41.10 even 5 inner