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Properties

Label 164.4.g.a.37.5
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 37.5
Character \(\chi\) \(=\) 164.37
Dual form 164.4.g.a.133.5

$q$-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)
 
\(f(q)\) \(=\) \(q-0.00279865 q^{3} +(-5.78452 + 17.8029i) q^{5} +(-6.36253 - 4.62265i) q^{7} -27.0000 q^{9} +(-21.0346 - 64.7378i) q^{11} +(66.1409 - 48.0542i) q^{13} +(0.0161888 - 0.0498242i) q^{15} +(14.3381 + 44.1282i) q^{17} +(-103.585 - 75.2593i) q^{19} +(0.0178065 + 0.0129372i) q^{21} +(-59.1942 + 43.0071i) q^{23} +(-182.356 - 132.490i) q^{25} +0.151127 q^{27} +(-7.59075 + 23.3619i) q^{29} +(-25.9613 - 79.9008i) q^{31} +(0.0588685 + 0.181179i) q^{33} +(119.101 - 86.5319i) q^{35} +(-115.861 + 356.583i) q^{37} +(-0.185105 + 0.134487i) q^{39} +(-258.732 + 44.4858i) q^{41} +(238.755 - 173.465i) q^{43} +(156.182 - 480.679i) q^{45} +(-136.938 + 99.4913i) q^{47} +(-86.8799 - 267.389i) q^{49} +(-0.0401273 - 0.123499i) q^{51} +(-28.5480 + 87.8618i) q^{53} +1274.20 q^{55} +(0.289900 + 0.210624i) q^{57} +(-489.922 + 355.949i) q^{59} +(-340.038 - 247.052i) q^{61} +(171.788 + 124.812i) q^{63} +(472.912 + 1455.47i) q^{65} +(-244.857 + 753.594i) q^{67} +(0.165664 - 0.120362i) q^{69} +(-164.575 - 506.509i) q^{71} -301.305 q^{73} +(0.510352 + 0.370792i) q^{75} +(-165.427 + 509.132i) q^{77} +1149.23 q^{79} +728.999 q^{81} +636.944 q^{83} -868.549 q^{85} +(0.0212439 - 0.0653819i) q^{87} +(-291.676 - 211.915i) q^{89} -642.961 q^{91} +(0.0726567 + 0.223614i) q^{93} +(1939.03 - 1408.79i) q^{95} +(358.940 - 1104.70i) q^{97} +(567.934 + 1747.92i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 2 q^{3} + 270 q^{9} + 83 q^{11} - 94 q^{13} - 48 q^{15} - 144 q^{17} - 179 q^{19} - 176 q^{21} + 180 q^{23} - 542 q^{25} - 236 q^{27} + 68 q^{29} - 298 q^{31} + 823 q^{33} + 178 q^{35} - 253 q^{37}+ \cdots + 3203 q^{99}+O(q^{100}) \) 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
\(3\) −0.00279865 −0.000538600 −0.000269300 − 1.00000i \(-0.500086\pi\)
−0.000269300 1.00000i \(0.500086\pi\)
\(4\) 0 0
\(5\) −5.78452 + 17.8029i −0.517383 + 1.59234i 0.261520 + 0.965198i \(0.415776\pi\)
−0.778903 + 0.627144i \(0.784224\pi\)
\(6\) 0 0
\(7\) −6.36253 − 4.62265i −0.343544 − 0.249600i 0.402611 − 0.915371i \(-0.368103\pi\)
−0.746156 + 0.665771i \(0.768103\pi\)
\(8\) 0 0
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) −21.0346 − 64.7378i −0.576561 − 1.77447i −0.630802 − 0.775944i \(-0.717274\pi\)
0.0542410 − 0.998528i \(-0.482726\pi\)
\(12\) 0 0
\(13\) 66.1409 − 48.0542i 1.41109 − 1.02522i 0.417929 − 0.908480i \(-0.362756\pi\)
0.993163 − 0.116738i \(-0.0372438\pi\)
\(14\) 0 0
\(15\) 0.0161888 − 0.0498242i 0.000278663 − 0.000857636i
\(16\) 0 0
\(17\) 14.3381 + 44.1282i 0.204559 + 0.629568i 0.999731 + 0.0231841i \(0.00738039\pi\)
−0.795172 + 0.606384i \(0.792620\pi\)
\(18\) 0 0
\(19\) −103.585 − 75.2593i −1.25074 − 0.908719i −0.252479 − 0.967602i \(-0.581246\pi\)
−0.998265 + 0.0588833i \(0.981246\pi\)
\(20\) 0 0
\(21\) 0.0178065 + 0.0129372i 0.000185033 + 0.000134434i
\(22\) 0 0
\(23\) −59.1942 + 43.0071i −0.536646 + 0.389896i −0.822838 − 0.568276i \(-0.807610\pi\)
0.286192 + 0.958172i \(0.407610\pi\)
\(24\) 0 0
\(25\) −182.356 − 132.490i −1.45885 − 1.05992i
\(26\) 0 0
\(27\) 0.151127 0.00107720
\(28\) 0 0
\(29\) −7.59075 + 23.3619i −0.0486058 + 0.149593i −0.972414 − 0.233263i \(-0.925060\pi\)
0.923808 + 0.382856i \(0.125060\pi\)
\(30\) 0 0
\(31\) −25.9613 − 79.9008i −0.150413 − 0.462923i 0.847255 − 0.531187i \(-0.178254\pi\)
−0.997667 + 0.0682643i \(0.978254\pi\)
\(32\) 0 0
\(33\) 0.0588685 + 0.181179i 0.000310536 + 0.000955731i
\(34\) 0 0
\(35\) 119.101 − 86.5319i 0.575192 − 0.417902i
\(36\) 0 0
\(37\) −115.861 + 356.583i −0.514794 + 1.58437i 0.268861 + 0.963179i \(0.413353\pi\)
−0.783656 + 0.621196i \(0.786647\pi\)
\(38\) 0 0
\(39\) −0.185105 + 0.134487i −0.000760014 + 0.000552183i
\(40\) 0 0
\(41\) −258.732 + 44.4858i −0.985539 + 0.169452i
\(42\) 0 0
\(43\) 238.755 − 173.465i 0.846738 − 0.615191i −0.0775069 − 0.996992i \(-0.524696\pi\)
0.924245 + 0.381801i \(0.124696\pi\)
\(44\) 0 0
\(45\) 156.182 − 480.679i 0.517383 − 1.59234i
\(46\) 0 0
\(47\) −136.938 + 99.4913i −0.424989 + 0.308772i −0.779642 − 0.626226i \(-0.784599\pi\)
0.354653 + 0.934998i \(0.384599\pi\)
\(48\) 0 0
\(49\) −86.8799 − 267.389i −0.253294 − 0.779559i
\(50\) 0 0
\(51\) −0.0401273 − 0.123499i −0.000110176 − 0.000339085i
\(52\) 0 0
\(53\) −28.5480 + 87.8618i −0.0739881 + 0.227712i −0.981211 − 0.192939i \(-0.938198\pi\)
0.907223 + 0.420651i \(0.138198\pi\)
\(54\) 0 0
\(55\) 1274.20 3.12387
\(56\) 0 0
\(57\) 0.289900 + 0.210624i 0.000673651 + 0.000489436i
\(58\) 0 0
\(59\) −489.922 + 355.949i −1.08106 + 0.785434i −0.977867 − 0.209228i \(-0.932905\pi\)
−0.103190 + 0.994662i \(0.532905\pi\)
\(60\) 0 0
\(61\) −340.038 − 247.052i −0.713729 − 0.518554i 0.170646 − 0.985332i \(-0.445415\pi\)
−0.884374 + 0.466778i \(0.845415\pi\)
\(62\) 0 0
\(63\) 171.788 + 124.812i 0.343544 + 0.249600i
\(64\) 0 0
\(65\) 472.912 + 1455.47i 0.902423 + 2.77737i
\(66\) 0 0
\(67\) −244.857 + 753.594i −0.446479 + 1.37412i 0.434375 + 0.900732i \(0.356969\pi\)
−0.880854 + 0.473389i \(0.843031\pi\)
\(68\) 0 0
\(69\) 0.165664 − 0.120362i 0.000289038 − 0.000209998i
\(70\) 0 0
\(71\) −164.575 − 506.509i −0.275091 − 0.846642i −0.989195 − 0.146603i \(-0.953166\pi\)
0.714105 − 0.700039i \(-0.246834\pi\)
\(72\) 0 0
\(73\) −301.305 −0.483083 −0.241541 − 0.970391i \(-0.577653\pi\)
−0.241541 + 0.970391i \(0.577653\pi\)
\(74\) 0 0
\(75\) 0.510352 + 0.370792i 0.000785738 + 0.000570872i
\(76\) 0 0
\(77\) −165.427 + 509.132i −0.244833 + 0.753519i
\(78\) 0 0
\(79\) 1149.23 1.63669 0.818343 − 0.574730i \(-0.194893\pi\)
0.818343 + 0.574730i \(0.194893\pi\)
\(80\) 0 0
\(81\) 728.999 0.999999
\(82\) 0 0
\(83\) 636.944 0.842333 0.421167 − 0.906983i \(-0.361621\pi\)
0.421167 + 0.906983i \(0.361621\pi\)
\(84\) 0 0
\(85\) −868.549 −1.10832
\(86\) 0 0
\(87\) 0.0212439 − 0.0653819i 2.61791e−5 − 8.05709e-5i
\(88\) 0 0
\(89\) −291.676 − 211.915i −0.347389 − 0.252393i 0.400384 − 0.916348i \(-0.368877\pi\)
−0.747773 + 0.663955i \(0.768877\pi\)
\(90\) 0 0
\(91\) −642.961 −0.740667
\(92\) 0 0
\(93\) 0.0726567 + 0.223614i 8.10124e−5 + 0.000249330i
\(94\) 0 0
\(95\) 1939.03 − 1408.79i 2.09411 − 1.52146i
\(96\) 0 0
\(97\) 358.940 − 1104.70i 0.375720 − 1.15635i −0.567271 − 0.823531i \(-0.692001\pi\)
0.942992 − 0.332817i \(-0.107999\pi\)
\(98\) 0 0
\(99\) 567.934 + 1747.92i 0.576561 + 1.77447i
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.37.5 ✓ 40
41.10 even 5 inner 164.4.g.a.133.5 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.4.g.a.37.5 ✓ 40 1.1 even 1 trivial
164.4.g.a.133.5 yes 40 41.10 even 5 inner