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Properties

Label 164.3.n.c.39.20
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 39.20
Character \(\chi\) \(=\) 164.39
Dual form 164.3.n.c.143.20

$q$-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+(0.0789243 + 1.99844i) q^{2} +(-1.70364 + 1.70364i) q^{3} +(-3.98754 + 0.315451i) q^{4} +(0.728137 + 1.00219i) q^{5} +(-3.53909 - 3.27017i) q^{6} +(-5.82261 - 11.4275i) q^{7} +(-0.945126 - 7.94397i) q^{8} +3.19522i q^{9} +(-1.94536 + 1.53424i) q^{10} +(-10.5514 + 1.67118i) q^{11} +(6.25592 - 7.33076i) q^{12} +(1.13307 - 2.22378i) q^{13} +(22.3777 - 12.5380i) q^{14} +(-2.94786 - 0.466896i) q^{15} +(15.8010 - 2.51575i) q^{16} +(-7.81450 + 1.23769i) q^{17} +(-6.38545 + 0.252180i) q^{18} +(29.5474 - 15.0552i) q^{19} +(-3.21962 - 3.76660i) q^{20} +(29.3880 + 9.54874i) q^{21} +(-4.17251 - 20.9545i) q^{22} +(-38.6093 + 12.5449i) q^{23} +(15.1438 + 11.9235i) q^{24} +(7.25121 - 22.3169i) q^{25} +(4.53352 + 2.08887i) q^{26} +(-20.7763 - 20.7763i) q^{27} +(26.8227 + 43.7309i) q^{28} +(-45.6684 - 7.23316i) q^{29} +(0.700406 - 5.92798i) q^{30} +(-29.3796 + 40.4375i) q^{31} +(6.27467 + 31.3788i) q^{32} +(15.1287 - 20.8229i) q^{33} +(-3.09022 - 15.5191i) q^{34} +(7.21293 - 14.1562i) q^{35} +(-1.00794 - 12.7411i) q^{36} +(-3.05713 + 2.22113i) q^{37} +(32.4189 + 57.8606i) 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} +(41.5469 - 9.99234i) q^{44} +(-3.20223 + 2.32655i) q^{45} +(-28.1175 - 76.1684i) q^{46} +(-10.1700 + 19.9598i) q^{47} +(-22.6333 + 31.2051i) q^{48} +(-67.8837 + 93.4339i) q^{49} +(45.1714 + 12.7298i) q^{50} +(11.2045 - 15.4217i) q^{51} +(-3.81668 + 9.22485i) q^{52} +(-35.7322 - 5.65943i) q^{53} +(39.8804 - 43.1599i) q^{54} +(-9.35770 - 9.35770i) q^{55} +(-85.2767 + 57.0551i) q^{56} +(-24.6896 + 75.9868i) q^{57} +(10.8507 - 91.8365i) q^{58} +(-35.9555 + 11.6827i) q^{59} +(11.9020 + 0.931858i) q^{60} +(-46.0133 - 14.9506i) q^{61} +(-83.1308 - 55.5219i) q^{62} +(36.5133 - 18.6045i) q^{63} +(-62.2135 + 15.0161i) q^{64} +(3.05369 - 0.483657i) q^{65} +(42.8073 + 28.5904i) q^{66} +(107.593 + 17.0411i) q^{67} +(30.7702 - 7.40045i) q^{68} +(44.4044 - 87.1485i) q^{69} +(28.8596 + 13.2974i) q^{70} +(17.7282 - 2.80786i) q^{71} +(25.3827 - 3.01988i) q^{72} -26.2287i q^{73} +(-4.68008 - 5.93419i) q^{74} +(25.6666 + 50.3735i) q^{75} +(-113.072 + 69.3539i) q^{76} +(80.5340 + 110.846i) q^{77} +(-11.2822 + 4.16481i) q^{78} +(-44.1460 + 44.1460i) q^{79} +(14.0266 + 14.0038i) q^{80} +42.0337 q^{81} +(-16.0317 + 80.4176i) q^{82} +25.2030i q^{83} +(-120.198 - 28.8055i) q^{84} +(-6.93043 - 6.93043i) q^{85} +(68.8609 - 25.4200i) q^{86} +(90.1252 - 65.4798i) q^{87} +(23.2482 + 82.2405i) q^{88} +(45.9109 - 23.3928i) q^{89} +(-4.90222 - 6.21585i) q^{90} -32.0097 q^{91} +(149.999 - 62.2028i) q^{92} +(-18.8388 - 118.943i) q^{93} +(-40.6912 - 18.7489i) q^{94} +(36.6028 + 18.6500i) q^{95} +(-64.1480 - 42.7684i) q^{96} +(17.4369 - 110.092i) q^{97} +(-192.080 - 128.287i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0789243 + 1.99844i 0.0394622 + 0.999221i
\(3\) −1.70364 + 1.70364i −0.567880 + 0.567880i −0.931534 − 0.363654i \(-0.881529\pi\)
0.363654 + 0.931534i \(0.381529\pi\)
\(4\) −3.98754 + 0.315451i −0.996885 + 0.0788629i
\(5\) 0.728137 + 1.00219i 0.145627 + 0.200439i 0.875599 − 0.483038i \(-0.160467\pi\)
−0.729972 + 0.683477i \(0.760467\pi\)
\(6\) −3.53909 − 3.27017i −0.589848 − 0.545028i
\(7\) −5.82261 − 11.4275i −0.831801 − 1.63250i −0.773151 − 0.634222i \(-0.781320\pi\)
−0.0586501 − 0.998279i \(-0.518680\pi\)
\(8\) −0.945126 − 7.94397i −0.118141 − 0.992997i
\(9\) 3.19522i 0.355024i
\(10\) −1.94536 + 1.53424i −0.194536 + 0.153424i
\(11\) −10.5514 + 1.67118i −0.959218 + 0.151925i −0.616362 − 0.787463i \(-0.711394\pi\)
−0.342856 + 0.939388i \(0.611394\pi\)
\(12\) 6.25592 − 7.33076i 0.521327 − 0.610896i
\(13\) 1.13307 − 2.22378i 0.0871594 − 0.171060i −0.843312 − 0.537424i \(-0.819397\pi\)
0.930472 + 0.366364i \(0.119397\pi\)
\(14\) 22.3777 − 12.5380i 1.59840 − 0.895575i
\(15\) −2.94786 − 0.466896i −0.196524 − 0.0311264i
\(16\) 15.8010 − 2.51575i 0.987561 − 0.157234i
\(17\) −7.81450 + 1.23769i −0.459676 + 0.0728056i −0.381978 − 0.924171i \(-0.624757\pi\)
−0.0776981 + 0.996977i \(0.524757\pi\)
\(18\) −6.38545 + 0.252180i −0.354747 + 0.0140100i
\(19\) 29.5474 − 15.0552i 1.55513 − 0.792377i 0.555883 − 0.831260i \(-0.312380\pi\)
0.999244 + 0.0388834i \(0.0123801\pi\)
\(20\) −3.21962 − 3.76660i −0.160981 − 0.188330i
\(21\) 29.3880 + 9.54874i 1.39943 + 0.454702i
\(22\) −4.17251 − 20.9545i −0.189660 − 0.952475i
\(23\) −38.6093 + 12.5449i −1.67867 + 0.545432i −0.984652 − 0.174529i \(-0.944160\pi\)
−0.694015 + 0.719961i \(0.744160\pi\)
\(24\) 15.1438 + 11.9235i 0.630993 + 0.496814i
\(25\) 7.25121 − 22.3169i 0.290049 − 0.892678i
\(26\) 4.53352 + 2.08887i 0.174366 + 0.0803411i
\(27\) −20.7763 − 20.7763i −0.769491 − 0.769491i
\(28\) 26.8227 + 43.7309i 0.957954 + 1.56182i
\(29\) −45.6684 − 7.23316i −1.57477 − 0.249419i −0.692944 − 0.720991i \(-0.743687\pi\)
−0.881828 + 0.471572i \(0.843687\pi\)
\(30\) 0.700406 − 5.92798i 0.0233469 − 0.197599i
\(31\) −29.3796 + 40.4375i −0.947729 + 1.30444i 0.00480040 + 0.999988i \(0.498472\pi\)
−0.952529 + 0.304448i \(0.901528\pi\)
\(32\) 6.27467 + 31.3788i 0.196083 + 0.980587i
\(33\) 15.1287 − 20.8229i 0.458445 − 0.630996i
\(34\) −3.09022 − 15.5191i −0.0908887 − 0.456445i
\(35\) 7.21293 − 14.1562i 0.206084 − 0.404462i
\(36\) −1.00794 − 12.7411i −0.0279982 − 0.353918i
\(37\) −3.05713 + 2.22113i −0.0826250 + 0.0600306i −0.628331 − 0.777946i \(-0.716262\pi\)
0.545706 + 0.837977i \(0.316262\pi\)
\(38\) 32.4189 + 57.8606i 0.853128 + 1.52265i
\(39\) 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 \(-0.959654\pi\)
0.728225 − 0.685338i \(-0.240346\pi\)
\(44\) 41.5469 − 9.99234i 0.944249 − 0.227099i
\(45\) −3.20223 + 2.32655i −0.0711606 + 0.0517012i
\(46\) −28.1175 − 76.1684i −0.611251 − 1.65584i
\(47\) −10.1700 + 19.9598i −0.216384 + 0.424677i −0.973527 − 0.228572i \(-0.926595\pi\)
0.757143 + 0.653249i \(0.226595\pi\)
\(48\) −22.6333 + 31.2051i −0.471526 + 0.650107i
\(49\) −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
\(52\) −3.81668 + 9.22485i −0.0733977 + 0.177401i
\(53\) −35.7322 − 5.65943i −0.674193 − 0.106782i −0.190055 − 0.981773i \(-0.560867\pi\)
−0.484138 + 0.874992i \(0.660867\pi\)
\(54\) 39.8804 − 43.1599i 0.738526 − 0.799258i
\(55\) −9.35770 − 9.35770i −0.170140 − 0.170140i
\(56\) −85.2767 + 57.0551i −1.52280 + 1.01884i
\(57\) −24.6896 + 75.9868i −0.433151 + 1.33310i
\(58\) 10.8507 − 91.8365i 0.187081 − 1.58339i
\(59\) −35.9555 + 11.6827i −0.609416 + 0.198011i −0.597435 − 0.801917i \(-0.703814\pi\)
−0.0119802 + 0.999928i \(0.503814\pi\)
\(60\) 11.9020 + 0.931858i 0.198367 + 0.0155310i
\(61\) −46.0133 − 14.9506i −0.754316 − 0.245092i −0.0934786 − 0.995621i \(-0.529799\pi\)
−0.660837 + 0.750529i \(0.729799\pi\)
\(62\) −83.1308 − 55.5219i −1.34082 − 0.895514i
\(63\) 36.5133 − 18.6045i 0.579577 − 0.295309i
\(64\) −62.2135 + 15.0161i −0.972086 + 0.234627i
\(65\) 3.05369 − 0.483657i 0.0469799 − 0.00744088i
\(66\) 42.8073 + 28.5904i 0.648596 + 0.433188i
\(67\) 107.593 + 17.0411i 1.60587 + 0.254345i 0.894033 − 0.448002i \(-0.147864\pi\)
0.711835 + 0.702346i \(0.247864\pi\)
\(68\) 30.7702 − 7.40045i 0.452503 − 0.108830i
\(69\) 44.4044 − 87.1485i 0.643542 − 1.26302i
\(70\) 28.8596 + 13.2974i 0.412280 + 0.189962i
\(71\) 17.7282 − 2.80786i 0.249692 − 0.0395474i −0.0303336 − 0.999540i \(-0.509657\pi\)
0.280026 + 0.959992i \(0.409657\pi\)
\(72\) 25.3827 − 3.01988i 0.352538 − 0.0419428i
\(73\) − 26.2287i − 0.359297i −0.983731 − 0.179649i \(-0.942504\pi\)
0.983731 − 0.179649i \(-0.0574960\pi\)
\(74\) −4.68008 − 5.93419i −0.0632444 − 0.0801917i
\(75\) 25.6666 + 50.3735i 0.342221 + 0.671647i
\(76\) −113.072 + 69.3539i −1.48779 + 0.912551i
\(77\) 80.5340 + 110.846i 1.04590 + 1.43955i
\(78\) −11.2822 + 4.16481i −0.144643 + 0.0533950i
\(79\) −44.1460 + 44.1460i −0.558810 + 0.558810i −0.928969 − 0.370158i \(-0.879303\pi\)
0.370158 + 0.928969i \(0.379303\pi\)
\(80\) 14.0266 + 14.0038i 0.175332 + 0.175048i
\(81\) 42.0337 0.518934
\(82\) −16.0317 + 80.4176i −0.195508 + 0.980702i
\(83\) 25.2030i 0.303650i 0.988407 + 0.151825i \(0.0485151\pi\)
−0.988407 + 0.151825i \(0.951485\pi\)
\(84\) −120.198 − 28.8055i −1.43093 − 0.342923i
\(85\) −6.93043 − 6.93043i −0.0815345 − 0.0815345i
\(86\) 68.8609 − 25.4200i 0.800709 − 0.295581i
\(87\) 90.1252 − 65.4798i 1.03592 − 0.752642i
\(88\) 23.2482 + 82.2405i 0.264184 + 0.934552i
\(89\) 45.9109 − 23.3928i 0.515852 − 0.262840i −0.176627 − 0.984278i \(-0.556519\pi\)
0.692479 + 0.721438i \(0.256519\pi\)
\(90\) −4.90222 − 6.21585i −0.0544691 − 0.0690649i
\(91\) −32.0097 −0.351755
\(92\) 149.999 − 62.2028i 1.63042 − 0.676118i
\(93\) −18.8388 − 118.943i −0.202567 − 1.27896i
\(94\) −40.6912 − 18.7489i −0.432886 − 0.199457i
\(95\) 36.6028 + 18.6500i 0.385292 + 0.196316i
\(96\) −64.1480 − 42.7684i −0.668208 − 0.445504i
\(97\) 17.4369 − 110.092i 0.179761 − 1.13497i −0.718506 − 0.695520i \(-0.755174\pi\)
0.898268 − 0.439449i \(-0.144826\pi\)
\(98\) −192.080 − 128.287i −1.96000 − 1.30906i
\(99\) −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.39.20 yes 304
4.3 odd 2 inner 164.3.n.c.39.12 ✓ 304
41.20 even 20 inner 164.3.n.c.143.12 yes 304
164.143 odd 20 inner 164.3.n.c.143.20 yes 304
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.c.39.12 ✓ 304 4.3 odd 2 inner
164.3.n.c.39.20 yes 304 1.1 even 1 trivial
164.3.n.c.143.12 yes 304 41.20 even 20 inner
164.3.n.c.143.20 yes 304 164.143 odd 20 inner