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Properties

Label 164.3.n.c.39.12
Level $164$
Weight $3$
Character 164.39
Analytic conductor $4.469$
Analytic rank $0$
Dimension $304$
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] = [164,3,Mod(39,164)] 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("164.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.n (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [304,-10,0,-22,-20,-6,0,-10,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(38\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 39.12
Character \(\chi\) \(=\) 164.39
Dual form 164.3.n.c.143.12

$q$-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+(-1.11080 - 1.66316i) q^{2} +(1.70364 - 1.70364i) q^{3} +(-1.53223 + 3.69490i) q^{4} +(0.728137 + 1.00219i) q^{5} +(-4.72585 - 0.941024i) q^{6} +(5.82261 + 11.4275i) q^{7} +(7.84723 - 1.55596i) q^{8} +3.19522i q^{9} +(0.857997 - 2.32425i) q^{10} +(10.5514 - 1.67118i) q^{11} +(3.68441 + 8.90515i) q^{12} +(1.13307 - 2.22378i) q^{13} +(12.5380 - 22.3777i) q^{14} +(2.94786 + 0.466896i) q^{15} +(-11.3045 - 11.3229i) q^{16} +(-7.81450 + 1.23769i) q^{17} +(5.31417 - 3.54926i) q^{18} +(-29.5474 + 15.0552i) q^{19} +(-4.81868 + 1.15480i) q^{20} +(29.3880 + 9.54874i) q^{21} +(-14.5000 - 15.6924i) q^{22} +(38.6093 - 12.5449i) q^{23} +(10.7181 - 16.0196i) q^{24} +(7.25121 - 22.3169i) q^{25} +(-4.95713 + 0.585697i) q^{26} +(20.7763 + 20.7763i) q^{27} +(-51.1450 + 4.00436i) q^{28} +(-45.6684 - 7.23316i) q^{29} +(-2.49797 - 5.42141i) q^{30} +(29.3796 - 40.4375i) q^{31} +(-6.27467 + 31.3788i) q^{32} +(15.1287 - 20.8229i) q^{33} +(10.7389 + 11.6220i) q^{34} +(-7.21293 + 14.1562i) q^{35} +(-11.8060 - 4.89581i) q^{36} +(-3.05713 + 2.22113i) q^{37} +(57.8606 + 32.4189i) q^{38} +(-1.85817 - 5.71887i) q^{39} +(7.27323 + 6.73150i) q^{40} +(39.8611 + 9.59631i) q^{41} +(-16.7632 - 59.4838i) q^{42} +(11.3414 + 34.9052i) q^{43} +(-9.99234 + 41.5469i) q^{44} +(-3.20223 + 2.32655i) q^{45} +(-63.7517 - 50.2787i) q^{46} +(10.1700 - 19.9598i) q^{47} +(-38.5490 - 0.0312268i) q^{48} +(-67.8837 + 93.4339i) q^{49} +(-45.1714 + 12.7298i) q^{50} +(-11.2045 + 15.4217i) q^{51} +(6.48051 + 7.59393i) q^{52} +(-35.7322 - 5.65943i) q^{53} +(11.4760 - 57.6327i) q^{54} +(9.35770 + 9.35770i) q^{55} +(63.4720 + 80.6146i) q^{56} +(-24.6896 + 75.9868i) q^{57} +(38.6987 + 83.9886i) q^{58} +(35.9555 - 11.6827i) q^{59} +(-6.24194 + 10.1767i) q^{60} +(-46.0133 - 14.9506i) q^{61} +(-99.8892 - 3.94492i) q^{62} +(-36.5133 + 18.6045i) q^{63} +(59.1580 - 24.4199i) q^{64} +(3.05369 - 0.483657i) q^{65} +(-51.4369 - 2.03139i) q^{66} +(-107.593 - 17.0411i) q^{67} +(7.40045 - 30.7702i) q^{68} +(44.4044 - 87.1485i) q^{69} +(31.5562 - 3.72844i) q^{70} +(-17.7282 + 2.80786i) q^{71} +(4.97161 + 25.0736i) q^{72} -26.2287i q^{73} +(7.08997 + 2.61726i) q^{74} +(-25.6666 - 50.3735i) q^{75} +(-10.3538 - 132.243i) q^{76} +(80.5340 + 110.846i) q^{77} +(-7.44736 + 9.44299i) q^{78} +(44.1460 - 44.1460i) q^{79} +(3.11646 - 19.5739i) q^{80} +42.0337 q^{81} +(-28.3177 - 76.9552i) q^{82} -25.2030i q^{83} +(-80.3108 + 93.9548i) q^{84} +(-6.93043 - 6.93043i) q^{85} +(45.4550 - 57.6354i) q^{86} +(-90.1252 + 65.4798i) q^{87} +(80.1989 - 29.5316i) q^{88} +(45.9109 - 23.3928i) q^{89} +(7.42649 + 2.74149i) q^{90} +32.0097 q^{91} +(-12.8062 + 161.879i) q^{92} +(-18.8388 - 118.943i) q^{93} +(-44.4934 + 5.25701i) q^{94} +(-36.6028 - 18.6500i) q^{95} +(42.7684 + 64.1480i) q^{96} +(17.4369 - 110.092i) q^{97} +(230.801 + 9.11502i) q^{98} +(5.33977 + 33.7140i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 304 q - 10 q^{2} - 22 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 46 q^{10} - 20 q^{12} - 48 q^{13} - 30 q^{14} + 58 q^{16} - 80 q^{17} + 18 q^{18} - 220 q^{20} - 20 q^{21} - 68 q^{22} - 26 q^{24} + 384 q^{25}+ \cdots + 124 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11080 − 1.66316i −0.555402 − 0.831582i
\(3\) 1.70364 − 1.70364i 0.567880 − 0.567880i −0.363654 − 0.931534i \(-0.618471\pi\)
0.931534 + 0.363654i \(0.118471\pi\)
\(4\) −1.53223 + 3.69490i −0.383058 + 0.923724i
\(5\) 0.728137 + 1.00219i 0.145627 + 0.200439i 0.875599 − 0.483038i \(-0.160467\pi\)
−0.729972 + 0.683477i \(0.760467\pi\)
\(6\) −4.72585 − 0.941024i −0.787641 − 0.156837i
\(7\) 5.82261 + 11.4275i 0.831801 + 1.63250i 0.773151 + 0.634222i \(0.218680\pi\)
0.0586501 + 0.998279i \(0.481320\pi\)
\(8\) 7.84723 − 1.55596i 0.980904 − 0.194494i
\(9\) 3.19522i 0.355024i
\(10\) 0.857997 − 2.32425i 0.0857997 − 0.232425i
\(11\) 10.5514 − 1.67118i 0.959218 − 0.151925i 0.342856 − 0.939388i \(-0.388606\pi\)
0.616362 + 0.787463i \(0.288606\pi\)
\(12\) 3.68441 + 8.90515i 0.307034 + 0.742096i
\(13\) 1.13307 − 2.22378i 0.0871594 − 0.171060i −0.843312 − 0.537424i \(-0.819397\pi\)
0.930472 + 0.366364i \(0.119397\pi\)
\(14\) 12.5380 − 22.3777i 0.895575 − 1.59840i
\(15\) 2.94786 + 0.466896i 0.196524 + 0.0311264i
\(16\) −11.3045 − 11.3229i −0.706534 − 0.707679i
\(17\) −7.81450 + 1.23769i −0.459676 + 0.0728056i −0.381978 − 0.924171i \(-0.624757\pi\)
−0.0776981 + 0.996977i \(0.524757\pi\)
\(18\) 5.31417 − 3.54926i 0.295232 − 0.197181i
\(19\) −29.5474 + 15.0552i −1.55513 + 0.792377i −0.999244 − 0.0388834i \(-0.987620\pi\)
−0.555883 + 0.831260i \(0.687620\pi\)
\(20\) −4.81868 + 1.15480i −0.240934 + 0.0577399i
\(21\) 29.3880 + 9.54874i 1.39943 + 0.454702i
\(22\) −14.5000 − 15.6924i −0.659089 − 0.713289i
\(23\) 38.6093 − 12.5449i 1.67867 − 0.545432i 0.694015 − 0.719961i \(-0.255840\pi\)
0.984652 + 0.174529i \(0.0558401\pi\)
\(24\) 10.7181 − 16.0196i 0.446586 − 0.667485i
\(25\) 7.25121 − 22.3169i 0.290049 − 0.892678i
\(26\) −4.95713 + 0.585697i −0.190659 + 0.0225268i
\(27\) 20.7763 + 20.7763i 0.769491 + 0.769491i
\(28\) −51.1450 + 4.00436i −1.82661 + 0.143013i
\(29\) −45.6684 − 7.23316i −1.57477 − 0.249419i −0.692944 − 0.720991i \(-0.743687\pi\)
−0.881828 + 0.471572i \(0.843687\pi\)
\(30\) −2.49797 − 5.42141i −0.0832658 − 0.180714i
\(31\) 29.3796 − 40.4375i 0.947729 − 1.30444i −0.00480040 − 0.999988i \(-0.501528\pi\)
0.952529 − 0.304448i \(-0.0984720\pi\)
\(32\) −6.27467 + 31.3788i −0.196083 + 0.980587i
\(33\) 15.1287 − 20.8229i 0.458445 − 0.630996i
\(34\) 10.7389 + 11.6220i 0.315849 + 0.341822i
\(35\) −7.21293 + 14.1562i −0.206084 + 0.404462i
\(36\) −11.8060 − 4.89581i −0.327944 − 0.135995i
\(37\) −3.05713 + 2.22113i −0.0826250 + 0.0600306i −0.628331 − 0.777946i \(-0.716262\pi\)
0.545706 + 0.837977i \(0.316262\pi\)
\(38\) 57.8606 + 32.4189i 1.52265 + 0.853128i
\(39\) −1.85817 − 5.71887i −0.0476455 − 0.146638i
\(40\) 7.27323 + 6.73150i 0.181831 + 0.168288i
\(41\) 39.8611 + 9.59631i 0.972223 + 0.234056i
\(42\) −16.7632 − 59.4838i −0.399123 − 1.41628i
\(43\) 11.3414 + 34.9052i 0.263753 + 0.811749i 0.991978 + 0.126411i \(0.0403457\pi\)
−0.728225 + 0.685338i \(0.759654\pi\)
\(44\) −9.99234 + 41.5469i −0.227099 + 0.944249i
\(45\) −3.20223 + 2.32655i −0.0711606 + 0.0517012i
\(46\) −63.7517 − 50.2787i −1.38591 − 1.09302i
\(47\) 10.1700 − 19.9598i 0.216384 − 0.424677i −0.757143 − 0.653249i \(-0.773405\pi\)
0.973527 + 0.228572i \(0.0734055\pi\)
\(48\) −38.5490 − 0.0312268i −0.803104 − 0.000650559i
\(49\) −67.8837 + 93.4339i −1.38538 + 1.90681i
\(50\) −45.1714 + 12.7298i −0.903428 + 0.254596i
\(51\) −11.2045 + 15.4217i −0.219696 + 0.302386i
\(52\) 6.48051 + 7.59393i 0.124625 + 0.146037i
\(53\) −35.7322 − 5.65943i −0.674193 − 0.106782i −0.190055 − 0.981773i \(-0.560867\pi\)
−0.484138 + 0.874992i \(0.660867\pi\)
\(54\) 11.4760 − 57.6327i 0.212518 − 1.06727i
\(55\) 9.35770 + 9.35770i 0.170140 + 0.170140i
\(56\) 63.4720 + 80.6146i 1.13343 + 1.43955i
\(57\) −24.6896 + 75.9868i −0.433151 + 1.33310i
\(58\) 38.6987 + 83.9886i 0.667219 + 1.44808i
\(59\) 35.9555 − 11.6827i 0.609416 − 0.198011i 0.0119802 − 0.999928i \(-0.496186\pi\)
0.597435 + 0.801917i \(0.296186\pi\)
\(60\) −6.24194 + 10.1767i −0.104032 + 0.169611i
\(61\) −46.0133 − 14.9506i −0.754316 − 0.245092i −0.0934786 − 0.995621i \(-0.529799\pi\)
−0.660837 + 0.750529i \(0.729799\pi\)
\(62\) −99.8892 − 3.94492i −1.61112 − 0.0636277i
\(63\) −36.5133 + 18.6045i −0.579577 + 0.295309i
\(64\) 59.1580 − 24.4199i 0.924344 − 0.381561i
\(65\) 3.05369 − 0.483657i 0.0469799 − 0.00744088i
\(66\) −51.4369 − 2.03139i −0.779347 − 0.0307787i
\(67\) −107.593 − 17.0411i −1.60587 − 0.254345i −0.711835 − 0.702346i \(-0.752136\pi\)
−0.894033 + 0.448002i \(0.852136\pi\)
\(68\) 7.40045 − 30.7702i 0.108830 − 0.452503i
\(69\) 44.4044 − 87.1485i 0.643542 − 1.26302i
\(70\) 31.5562 − 3.72844i 0.450803 − 0.0532635i
\(71\) −17.7282 + 2.80786i −0.249692 + 0.0395474i −0.280026 − 0.959992i \(-0.590343\pi\)
0.0303336 + 0.999540i \(0.490343\pi\)
\(72\) 4.97161 + 25.0736i 0.0690502 + 0.348244i
\(73\) − 26.2287i − 0.359297i −0.983731 − 0.179649i \(-0.942504\pi\)
0.983731 − 0.179649i \(-0.0574960\pi\)
\(74\) 7.08997 + 2.61726i 0.0958104 + 0.0353684i
\(75\) −25.6666 − 50.3735i −0.342221 − 0.671647i
\(76\) −10.3538 − 132.243i −0.136235 − 1.74003i
\(77\) 80.5340 + 110.846i 1.04590 + 1.43955i
\(78\) −7.44736 + 9.44299i −0.0954789 + 0.121064i
\(79\) 44.1460 − 44.1460i 0.558810 − 0.558810i −0.370158 − 0.928969i \(-0.620697\pi\)
0.928969 + 0.370158i \(0.120697\pi\)
\(80\) 3.11646 − 19.5739i 0.0389558 − 0.244674i
\(81\) 42.0337 0.518934
\(82\) −28.3177 − 76.9552i −0.345337 − 0.938479i
\(83\) − 25.2030i − 0.303650i −0.988407 − 0.151825i \(-0.951485\pi\)
0.988407 − 0.151825i \(-0.0485151\pi\)
\(84\) −80.3108 + 93.9548i −0.956081 + 1.11851i
\(85\) −6.93043 − 6.93043i −0.0815345 − 0.0815345i
\(86\) 45.4550 − 57.6354i 0.528547 − 0.670180i
\(87\) −90.1252 + 65.4798i −1.03592 + 0.752642i
\(88\) 80.1989 − 29.5316i 0.911351 − 0.335586i
\(89\) 45.9109 − 23.3928i 0.515852 − 0.262840i −0.176627 − 0.984278i \(-0.556519\pi\)
0.692479 + 0.721438i \(0.256519\pi\)
\(90\) 7.42649 + 2.74149i 0.0825165 + 0.0304609i
\(91\) 32.0097 0.351755
\(92\) −12.8062 + 161.879i −0.139197 + 1.75956i
\(93\) −18.8388 − 118.943i −0.202567 − 1.27896i
\(94\) −44.4934 + 5.25701i −0.473334 + 0.0559256i
\(95\) −36.6028 − 18.6500i −0.385292 − 0.196316i
\(96\) 42.7684 + 64.1480i 0.445504 + 0.668208i
\(97\) 17.4369 − 110.092i 0.179761 − 1.13497i −0.718506 − 0.695520i \(-0.755174\pi\)
0.898268 − 0.439449i \(-0.144826\pi\)
\(98\) 230.801 + 9.11502i 2.35512 + 0.0930105i
\(99\) 5.33977 + 33.7140i 0.0539371 + 0.340545i
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 164.3.n.c.39.12 ✓ 304
4.3 odd 2 inner 164.3.n.c.39.20 yes 304
41.20 even 20 inner 164.3.n.c.143.20 yes 304
164.143 odd 20 inner 164.3.n.c.143.12 yes 304
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.c.39.12 ✓ 304 1.1 even 1 trivial
164.3.n.c.39.20 yes 304 4.3 odd 2 inner
164.3.n.c.143.12 yes 304 164.143 odd 20 inner
164.3.n.c.143.20 yes 304 41.20 even 20 inner