Properties

Label 1764.2.i.d.373.3
Level $1764$
Weight $2$
Character 1764.373
Analytic conductor $14.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(373,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.373");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.3
Root \(0.500000 - 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 1764.373
Dual form 1764.2.i.d.1537.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.349814 + 1.69636i) q^{3} +(-0.849814 - 1.47192i) q^{5} +(-2.75526 + 1.18682i) q^{9} +O(q^{10})\) \(q+(0.349814 + 1.69636i) q^{3} +(-0.849814 - 1.47192i) q^{5} +(-2.75526 + 1.18682i) q^{9} +(-1.23855 + 2.14523i) q^{11} +(0.388736 - 0.673310i) q^{13} +(2.19963 - 1.95649i) q^{15} +(1.40545 + 2.43430i) q^{17} +(-2.49381 + 4.31941i) q^{19} +(-0.356004 - 0.616617i) q^{23} +(1.05563 - 1.82841i) q^{25} +(-2.97710 - 4.25874i) q^{27} +(-2.25526 - 3.90623i) q^{29} -5.09888 q^{31} +(-4.07234 - 1.35059i) q^{33} +(3.43818 - 5.95510i) q^{37} +(1.27816 + 0.423902i) q^{39} +(-2.93818 + 5.08907i) q^{41} +(2.32691 + 4.03033i) q^{43} +(4.08836 + 3.04695i) q^{45} -12.9876 q^{47} +(-3.63781 + 3.23569i) q^{51} +(-0.944368 - 1.63569i) q^{53} +4.21015 q^{55} +(-8.19963 - 2.71941i) q^{57} -14.2880 q^{59} -14.3090 q^{61} -1.32141 q^{65} +7.98762 q^{67} +(0.921468 - 0.819611i) q^{69} -10.2632 q^{71} +(2.49381 + 4.31941i) q^{73} +(3.47091 + 1.15113i) q^{75} -9.21015 q^{79} +(6.18292 - 6.53999i) q^{81} +(4.40545 + 7.63046i) q^{83} +(2.38874 - 4.13741i) q^{85} +(5.83743 - 5.19218i) q^{87} +(4.82691 - 8.36046i) q^{89} +(-1.78366 - 8.64953i) q^{93} +8.47710 q^{95} +(4.32072 + 7.48371i) q^{97} +(0.866524 - 7.38061i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} + q^{5} - 4 q^{9} - 2 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} - 14 q^{23} + 6 q^{25} - 7 q^{27} - q^{29} + 6 q^{31} - 17 q^{33} + 3 q^{37} + 6 q^{39} - 3 q^{43} + 13 q^{45} - 42 q^{47} + 8 q^{51} - 6 q^{53} - 12 q^{55} - 37 q^{57} - 62 q^{59} - 12 q^{61} + 30 q^{65} + 12 q^{67} - 5 q^{69} + 34 q^{71} - 3 q^{73} - 8 q^{75} - 18 q^{79} + 32 q^{81} + 20 q^{83} + 15 q^{85} - 7 q^{87} + 12 q^{89} - 30 q^{93} + 40 q^{95} - 9 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See 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
\(3\) 0.349814 + 1.69636i 0.201965 + 0.979393i
\(4\) 0 0
\(5\) −0.849814 1.47192i −0.380048 0.658263i 0.611020 0.791615i \(-0.290759\pi\)
−0.991069 + 0.133352i \(0.957426\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.75526 + 1.18682i −0.918420 + 0.395607i
\(10\) 0 0
\(11\) −1.23855 + 2.14523i −0.373437 + 0.646812i −0.990092 0.140422i \(-0.955154\pi\)
0.616655 + 0.787234i \(0.288487\pi\)
\(12\) 0 0
\(13\) 0.388736 0.673310i 0.107816 0.186743i −0.807069 0.590457i \(-0.798948\pi\)
0.914885 + 0.403714i \(0.132281\pi\)
\(14\) 0 0
\(15\) 2.19963 1.95649i 0.567942 0.505163i
\(16\) 0 0
\(17\) 1.40545 + 2.43430i 0.340871 + 0.590405i 0.984595 0.174852i \(-0.0559448\pi\)
−0.643724 + 0.765258i \(0.722611\pi\)
\(18\) 0 0
\(19\) −2.49381 + 4.31941i −0.572119 + 0.990940i 0.424229 + 0.905555i \(0.360545\pi\)
−0.996348 + 0.0853846i \(0.972788\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.356004 0.616617i −0.0742320 0.128574i 0.826520 0.562907i \(-0.190317\pi\)
−0.900752 + 0.434334i \(0.856984\pi\)
\(24\) 0 0
\(25\) 1.05563 1.82841i 0.211126 0.365682i
\(26\) 0 0
\(27\) −2.97710 4.25874i −0.572943 0.819595i
\(28\) 0 0
\(29\) −2.25526 3.90623i −0.418791 0.725368i 0.577027 0.816725i \(-0.304213\pi\)
−0.995818 + 0.0913573i \(0.970879\pi\)
\(30\) 0 0
\(31\) −5.09888 −0.915787 −0.457893 0.889007i \(-0.651396\pi\)
−0.457893 + 0.889007i \(0.651396\pi\)
\(32\) 0 0
\(33\) −4.07234 1.35059i −0.708904 0.235108i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.43818 5.95510i 0.565233 0.979012i −0.431795 0.901972i \(-0.642120\pi\)
0.997028 0.0770405i \(-0.0245471\pi\)
\(38\) 0 0
\(39\) 1.27816 + 0.423902i 0.204669 + 0.0678786i
\(40\) 0 0
\(41\) −2.93818 + 5.08907i −0.458866 + 0.794780i −0.998901 0.0468628i \(-0.985078\pi\)
0.540035 + 0.841643i \(0.318411\pi\)
\(42\) 0 0
\(43\) 2.32691 + 4.03033i 0.354851 + 0.614620i 0.987092 0.160151i \(-0.0511982\pi\)
−0.632241 + 0.774771i \(0.717865\pi\)
\(44\) 0 0
\(45\) 4.08836 + 3.04695i 0.609457 + 0.454212i
\(46\) 0 0
\(47\) −12.9876 −1.89444 −0.947220 0.320586i \(-0.896120\pi\)
−0.947220 + 0.320586i \(0.896120\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.63781 + 3.23569i −0.509395 + 0.453088i
\(52\) 0 0
\(53\) −0.944368 1.63569i −0.129719 0.224680i 0.793849 0.608115i \(-0.208074\pi\)
−0.923568 + 0.383436i \(0.874741\pi\)
\(54\) 0 0
\(55\) 4.21015 0.567696
\(56\) 0 0
\(57\) −8.19963 2.71941i −1.08607 0.360194i
\(58\) 0 0
\(59\) −14.2880 −1.86014 −0.930069 0.367385i \(-0.880253\pi\)
−0.930069 + 0.367385i \(0.880253\pi\)
\(60\) 0 0
\(61\) −14.3090 −1.83208 −0.916042 0.401082i \(-0.868634\pi\)
−0.916042 + 0.401082i \(0.868634\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.32141 −0.163901
\(66\) 0 0
\(67\) 7.98762 0.975843 0.487922 0.872887i \(-0.337755\pi\)
0.487922 + 0.872887i \(0.337755\pi\)
\(68\) 0 0
\(69\) 0.921468 0.819611i 0.110932 0.0986696i
\(70\) 0 0
\(71\) −10.2632 −1.21802 −0.609011 0.793162i \(-0.708433\pi\)
−0.609011 + 0.793162i \(0.708433\pi\)
\(72\) 0 0
\(73\) 2.49381 + 4.31941i 0.291878 + 0.505548i 0.974254 0.225454i \(-0.0723864\pi\)
−0.682376 + 0.731002i \(0.739053\pi\)
\(74\) 0 0
\(75\) 3.47091 + 1.15113i 0.400786 + 0.132921i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.21015 −1.03622 −0.518111 0.855313i \(-0.673365\pi\)
−0.518111 + 0.855313i \(0.673365\pi\)
\(80\) 0 0
\(81\) 6.18292 6.53999i 0.686991 0.726666i
\(82\) 0 0
\(83\) 4.40545 + 7.63046i 0.483561 + 0.837551i 0.999822 0.0188798i \(-0.00600997\pi\)
−0.516261 + 0.856431i \(0.672677\pi\)
\(84\) 0 0
\(85\) 2.38874 4.13741i 0.259095 0.448765i
\(86\) 0 0
\(87\) 5.83743 5.19218i 0.625839 0.556660i
\(88\) 0 0
\(89\) 4.82691 8.36046i 0.511652 0.886207i −0.488257 0.872700i \(-0.662367\pi\)
0.999909 0.0135071i \(-0.00429956\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.78366 8.64953i −0.184957 0.896915i
\(94\) 0 0
\(95\) 8.47710 0.869732
\(96\) 0 0
\(97\) 4.32072 + 7.48371i 0.438703 + 0.759856i 0.997590 0.0693880i \(-0.0221047\pi\)
−0.558887 + 0.829244i \(0.688771\pi\)
\(98\) 0 0
\(99\) 0.866524 7.38061i 0.0870890 0.741779i
\(100\) 0 0
\(101\) 1.20582 2.08854i 0.119983 0.207817i −0.799777 0.600297i \(-0.795049\pi\)
0.919761 + 0.392479i \(0.128383\pi\)
\(102\) 0 0
\(103\) −2.16690 3.75317i −0.213511 0.369811i 0.739300 0.673376i \(-0.235156\pi\)
−0.952811 + 0.303565i \(0.901823\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.59888 + 16.6258i −0.927959 + 1.60727i −0.141228 + 0.989977i \(0.545105\pi\)
−0.786732 + 0.617295i \(0.788228\pi\)
\(108\) 0 0
\(109\) −9.48143 16.4223i −0.908156 1.57297i −0.816623 0.577171i \(-0.804157\pi\)
−0.0915329 0.995802i \(-0.529177\pi\)
\(110\) 0 0
\(111\) 11.3047 + 3.74920i 1.07299 + 0.355859i
\(112\) 0 0
\(113\) −6.46472 + 11.1972i −0.608150 + 1.05335i 0.383395 + 0.923584i \(0.374755\pi\)
−0.991545 + 0.129762i \(0.958579\pi\)
\(114\) 0 0
\(115\) −0.605074 + 1.04802i −0.0564235 + 0.0977283i
\(116\) 0 0
\(117\) −0.271971 + 2.31650i −0.0251437 + 0.214161i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.43199 + 4.21233i 0.221090 + 0.382939i
\(122\) 0 0
\(123\) −9.66071 3.20397i −0.871077 0.288892i
\(124\) 0 0
\(125\) −12.0865 −1.08105
\(126\) 0 0
\(127\) 17.6291 1.56433 0.782163 0.623073i \(-0.214116\pi\)
0.782163 + 0.623073i \(0.214116\pi\)
\(128\) 0 0
\(129\) −6.02290 + 5.35715i −0.530287 + 0.471670i
\(130\) 0 0
\(131\) −2.84362 4.92530i −0.248449 0.430326i 0.714647 0.699485i \(-0.246587\pi\)
−0.963096 + 0.269160i \(0.913254\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.73855 + 8.00119i −0.321763 + 0.688633i
\(136\) 0 0
\(137\) 9.72617 16.8462i 0.830963 1.43927i −0.0663128 0.997799i \(-0.521124\pi\)
0.897276 0.441471i \(-0.145543\pi\)
\(138\) 0 0
\(139\) −1.49381 + 2.58736i −0.126703 + 0.219457i −0.922397 0.386242i \(-0.873773\pi\)
0.795694 + 0.605699i \(0.207106\pi\)
\(140\) 0 0
\(141\) −4.54325 22.0317i −0.382611 1.85540i
\(142\) 0 0
\(143\) 0.962937 + 1.66786i 0.0805249 + 0.139473i
\(144\) 0 0
\(145\) −3.83310 + 6.63913i −0.318322 + 0.551350i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.04944 7.01384i −0.331743 0.574596i 0.651111 0.758983i \(-0.274303\pi\)
−0.982854 + 0.184387i \(0.940970\pi\)
\(150\) 0 0
\(151\) 4.43199 7.67643i 0.360670 0.624699i −0.627401 0.778696i \(-0.715881\pi\)
0.988071 + 0.153997i \(0.0492147\pi\)
\(152\) 0 0
\(153\) −6.76145 5.03913i −0.546631 0.407390i
\(154\) 0 0
\(155\) 4.33310 + 7.50516i 0.348043 + 0.602829i
\(156\) 0 0
\(157\) 8.76509 0.699530 0.349765 0.936837i \(-0.386261\pi\)
0.349765 + 0.936837i \(0.386261\pi\)
\(158\) 0 0
\(159\) 2.44437 2.17417i 0.193851 0.172423i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.993810 1.72133i 0.0778412 0.134825i −0.824477 0.565895i \(-0.808531\pi\)
0.902318 + 0.431070i \(0.141864\pi\)
\(164\) 0 0
\(165\) 1.47277 + 7.14192i 0.114655 + 0.555998i
\(166\) 0 0
\(167\) 1.31089 2.27053i 0.101440 0.175699i −0.810838 0.585270i \(-0.800988\pi\)
0.912278 + 0.409571i \(0.134322\pi\)
\(168\) 0 0
\(169\) 6.19777 + 10.7349i 0.476751 + 0.825758i
\(170\) 0 0
\(171\) 1.74474 14.8608i 0.133424 1.13643i
\(172\) 0 0
\(173\) 5.22981 0.397615 0.198808 0.980039i \(-0.436293\pi\)
0.198808 + 0.980039i \(0.436293\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.99814 24.2375i −0.375683 1.82181i
\(178\) 0 0
\(179\) 2.38255 + 4.12669i 0.178080 + 0.308443i 0.941223 0.337786i \(-0.109678\pi\)
−0.763143 + 0.646230i \(0.776345\pi\)
\(180\) 0 0
\(181\) 10.4313 0.775352 0.387676 0.921796i \(-0.373278\pi\)
0.387676 + 0.921796i \(0.373278\pi\)
\(182\) 0 0
\(183\) −5.00550 24.2732i −0.370017 1.79433i
\(184\) 0 0
\(185\) −11.6872 −0.859264
\(186\) 0 0
\(187\) −6.96286 −0.509175
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.3214 0.963904 0.481952 0.876198i \(-0.339928\pi\)
0.481952 + 0.876198i \(0.339928\pi\)
\(192\) 0 0
\(193\) −14.6414 −1.05391 −0.526957 0.849892i \(-0.676667\pi\)
−0.526957 + 0.849892i \(0.676667\pi\)
\(194\) 0 0
\(195\) −0.462249 2.24159i −0.0331023 0.160524i
\(196\) 0 0
\(197\) −18.4858 −1.31706 −0.658528 0.752556i \(-0.728821\pi\)
−0.658528 + 0.752556i \(0.728821\pi\)
\(198\) 0 0
\(199\) 11.8083 + 20.4527i 0.837071 + 1.44985i 0.892333 + 0.451378i \(0.149067\pi\)
−0.0552614 + 0.998472i \(0.517599\pi\)
\(200\) 0 0
\(201\) 2.79418 + 13.5499i 0.197086 + 0.955734i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.98762 0.697566
\(206\) 0 0
\(207\) 1.71270 + 1.27643i 0.119041 + 0.0887178i
\(208\) 0 0
\(209\) −6.17742 10.6996i −0.427301 0.740107i
\(210\) 0 0
\(211\) 7.27747 12.6050i 0.501002 0.867761i −0.498998 0.866603i \(-0.666298\pi\)
0.999999 0.00115718i \(-0.000368342\pi\)
\(212\) 0 0
\(213\) −3.59022 17.4101i −0.245998 1.19292i
\(214\) 0 0
\(215\) 3.95489 6.85007i 0.269721 0.467171i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −6.45489 + 5.74138i −0.436181 + 0.387967i
\(220\) 0 0
\(221\) 2.18539 0.147005
\(222\) 0 0
\(223\) 4.72253 + 8.17966i 0.316244 + 0.547750i 0.979701 0.200464i \(-0.0642449\pi\)
−0.663457 + 0.748214i \(0.730912\pi\)
\(224\) 0 0
\(225\) −0.738550 + 6.29059i −0.0492367 + 0.419372i
\(226\) 0 0
\(227\) 9.55563 16.5508i 0.634230 1.09852i −0.352448 0.935831i \(-0.614651\pi\)
0.986678 0.162687i \(-0.0520159\pi\)
\(228\) 0 0
\(229\) 5.72253 + 9.91171i 0.378155 + 0.654984i 0.990794 0.135379i \(-0.0432252\pi\)
−0.612639 + 0.790363i \(0.709892\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.595243 1.03099i 0.0389956 0.0675424i −0.845869 0.533391i \(-0.820917\pi\)
0.884865 + 0.465848i \(0.154251\pi\)
\(234\) 0 0
\(235\) 11.0371 + 19.1168i 0.719979 + 1.24704i
\(236\) 0 0
\(237\) −3.22184 15.6237i −0.209281 1.01487i
\(238\) 0 0
\(239\) −12.1414 + 21.0296i −0.785365 + 1.36029i 0.143416 + 0.989663i \(0.454191\pi\)
−0.928781 + 0.370630i \(0.879142\pi\)
\(240\) 0 0
\(241\) −10.7095 + 18.5493i −0.689857 + 1.19487i 0.282027 + 0.959406i \(0.408993\pi\)
−0.971884 + 0.235461i \(0.924340\pi\)
\(242\) 0 0
\(243\) 13.2570 + 8.20066i 0.850440 + 0.526073i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.93887 + 3.35822i 0.123367 + 0.213678i
\(248\) 0 0
\(249\) −11.4029 + 10.1425i −0.722629 + 0.642752i
\(250\) 0 0
\(251\) 2.67996 0.169158 0.0845789 0.996417i \(-0.473045\pi\)
0.0845789 + 0.996417i \(0.473045\pi\)
\(252\) 0 0
\(253\) 1.76371 0.110884
\(254\) 0 0
\(255\) 7.85414 + 2.60483i 0.491846 + 0.163121i
\(256\) 0 0
\(257\) −5.54256 9.60000i −0.345736 0.598832i 0.639752 0.768582i \(-0.279037\pi\)
−0.985487 + 0.169750i \(0.945704\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 10.8498 + 8.08608i 0.671587 + 0.500516i
\(262\) 0 0
\(263\) −6.70396 + 11.6116i −0.413384 + 0.716002i −0.995257 0.0972776i \(-0.968987\pi\)
0.581873 + 0.813279i \(0.302320\pi\)
\(264\) 0 0
\(265\) −1.60507 + 2.78007i −0.0985989 + 0.170778i
\(266\) 0 0
\(267\) 15.8709 + 5.26357i 0.971281 + 0.322125i
\(268\) 0 0
\(269\) 2.04511 + 3.54224i 0.124693 + 0.215974i 0.921613 0.388111i \(-0.126872\pi\)
−0.796920 + 0.604085i \(0.793539\pi\)
\(270\) 0 0
\(271\) 3.06182 5.30323i 0.185992 0.322148i −0.757918 0.652350i \(-0.773783\pi\)
0.943910 + 0.330201i \(0.107117\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.61491 + 4.52915i 0.157685 + 0.273118i
\(276\) 0 0
\(277\) −7.88255 + 13.6530i −0.473616 + 0.820327i −0.999544 0.0302019i \(-0.990385\pi\)
0.525928 + 0.850529i \(0.323718\pi\)
\(278\) 0 0
\(279\) 14.0488 6.05146i 0.841077 0.362291i
\(280\) 0 0
\(281\) 10.5946 + 18.3503i 0.632018 + 1.09469i 0.987139 + 0.159867i \(0.0511065\pi\)
−0.355120 + 0.934821i \(0.615560\pi\)
\(282\) 0 0
\(283\) −6.87636 −0.408757 −0.204378 0.978892i \(-0.565517\pi\)
−0.204378 + 0.978892i \(0.565517\pi\)
\(284\) 0 0
\(285\) 2.96541 + 14.3802i 0.175656 + 0.851809i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.54944 7.87987i 0.267614 0.463521i
\(290\) 0 0
\(291\) −11.1836 + 9.94740i −0.655595 + 0.583127i
\(292\) 0 0
\(293\) −13.7534 + 23.8216i −0.803482 + 1.39167i 0.113829 + 0.993500i \(0.463689\pi\)
−0.917311 + 0.398172i \(0.869645\pi\)
\(294\) 0 0
\(295\) 12.1421 + 21.0308i 0.706943 + 1.22446i
\(296\) 0 0
\(297\) 12.8233 1.11190i 0.744082 0.0645192i
\(298\) 0 0
\(299\) −0.553566 −0.0320135
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.96472 + 1.31490i 0.227767 + 0.0755390i
\(304\) 0 0
\(305\) 12.1600 + 21.0618i 0.696281 + 1.20599i
\(306\) 0 0
\(307\) 21.5178 1.22809 0.614043 0.789273i \(-0.289542\pi\)
0.614043 + 0.789273i \(0.289542\pi\)
\(308\) 0 0
\(309\) 5.60872 4.98874i 0.319069 0.283800i
\(310\) 0 0
\(311\) −18.3855 −1.04255 −0.521273 0.853390i \(-0.674543\pi\)
−0.521273 + 0.853390i \(0.674543\pi\)
\(312\) 0 0
\(313\) 0.00137742 7.78563e−5 3.89281e−5 1.00000i \(-0.499988\pi\)
3.89281e−5 1.00000i \(0.499988\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0989 0.791872 0.395936 0.918278i \(-0.370420\pi\)
0.395936 + 0.918278i \(0.370420\pi\)
\(318\) 0 0
\(319\) 11.1730 0.625568
\(320\) 0 0
\(321\) −31.5611 10.4672i −1.76157 0.584223i
\(322\) 0 0
\(323\) −14.0197 −0.780075
\(324\) 0 0
\(325\) −0.820724 1.42154i −0.0455256 0.0788526i
\(326\) 0 0
\(327\) 24.5414 21.8287i 1.35714 1.20713i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.9629 0.767468 0.383734 0.923444i \(-0.374638\pi\)
0.383734 + 0.923444i \(0.374638\pi\)
\(332\) 0 0
\(333\) −2.40545 + 20.4883i −0.131818 + 1.12275i
\(334\) 0 0
\(335\) −6.78799 11.7571i −0.370868 0.642362i
\(336\) 0 0
\(337\) −12.0982 + 20.9547i −0.659031 + 1.14147i 0.321836 + 0.946795i \(0.395700\pi\)
−0.980867 + 0.194679i \(0.937633\pi\)
\(338\) 0 0
\(339\) −21.2559 7.04953i −1.15446 0.382878i
\(340\) 0 0
\(341\) 6.31522 10.9383i 0.341988 0.592341i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.98948 0.659811i −0.107110 0.0355230i
\(346\) 0 0
\(347\) −32.7156 −1.75626 −0.878132 0.478418i \(-0.841210\pi\)
−0.878132 + 0.478418i \(0.841210\pi\)
\(348\) 0 0
\(349\) −11.8887 20.5919i −0.636389 1.10226i −0.986219 0.165445i \(-0.947094\pi\)
0.349830 0.936813i \(-0.386240\pi\)
\(350\) 0 0
\(351\) −4.02476 + 0.348986i −0.214826 + 0.0186275i
\(352\) 0 0
\(353\) 10.0309 17.3740i 0.533889 0.924724i −0.465327 0.885139i \(-0.654063\pi\)
0.999216 0.0395847i \(-0.0126035\pi\)
\(354\) 0 0
\(355\) 8.72184 + 15.1067i 0.462907 + 0.801779i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.15087 + 7.18953i −0.219075 + 0.379449i −0.954525 0.298130i \(-0.903637\pi\)
0.735451 + 0.677578i \(0.236971\pi\)
\(360\) 0 0
\(361\) −2.93818 5.08907i −0.154641 0.267846i
\(362\) 0 0
\(363\) −6.29487 + 5.59905i −0.330395 + 0.293874i
\(364\) 0 0
\(365\) 4.23855 7.34138i 0.221856 0.384266i
\(366\) 0 0
\(367\) 5.77197 9.99735i 0.301294 0.521857i −0.675135 0.737694i \(-0.735915\pi\)
0.976429 + 0.215837i \(0.0692480\pi\)
\(368\) 0 0
\(369\) 2.05563 17.5088i 0.107012 0.911472i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.42580 2.46956i −0.0738250 0.127869i 0.826750 0.562570i \(-0.190187\pi\)
−0.900575 + 0.434701i \(0.856854\pi\)
\(374\) 0 0
\(375\) −4.22803 20.5030i −0.218335 1.05877i
\(376\) 0 0
\(377\) −3.50680 −0.180609
\(378\) 0 0
\(379\) 35.9519 1.84672 0.923361 0.383932i \(-0.125430\pi\)
0.923361 + 0.383932i \(0.125430\pi\)
\(380\) 0 0
\(381\) 6.16690 + 29.9052i 0.315940 + 1.53209i
\(382\) 0 0
\(383\) −0.915278 1.58531i −0.0467685 0.0810054i 0.841694 0.539956i \(-0.181559\pi\)
−0.888462 + 0.458950i \(0.848226\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.1945 8.34299i −0.569050 0.424098i
\(388\) 0 0
\(389\) 5.69530 9.86454i 0.288763 0.500152i −0.684752 0.728776i \(-0.740089\pi\)
0.973515 + 0.228624i \(0.0734227\pi\)
\(390\) 0 0
\(391\) 1.00069 1.73324i 0.0506070 0.0876539i
\(392\) 0 0
\(393\) 7.36033 6.54674i 0.371280 0.330240i
\(394\) 0 0
\(395\) 7.82691 + 13.5566i 0.393815 + 0.682107i
\(396\) 0 0
\(397\) −5.21565 + 9.03377i −0.261766 + 0.453392i −0.966711 0.255870i \(-0.917638\pi\)
0.704945 + 0.709262i \(0.250972\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.0371 + 29.5091i 0.850790 + 1.47361i 0.880496 + 0.474053i \(0.157210\pi\)
−0.0297058 + 0.999559i \(0.509457\pi\)
\(402\) 0 0
\(403\) −1.98212 + 3.43313i −0.0987364 + 0.171016i
\(404\) 0 0
\(405\) −14.8807 3.54299i −0.739427 0.176053i
\(406\) 0 0
\(407\) 8.51671 + 14.7514i 0.422158 + 0.731199i
\(408\) 0 0
\(409\) −3.97524 −0.196563 −0.0982815 0.995159i \(-0.531335\pi\)
−0.0982815 + 0.995159i \(0.531335\pi\)
\(410\) 0 0
\(411\) 31.9796 + 10.6060i 1.57744 + 0.523156i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.48762 12.9689i 0.367553 0.636620i
\(416\) 0 0
\(417\) −4.91164 1.62894i −0.240524 0.0797697i
\(418\) 0 0
\(419\) −4.72184 + 8.17847i −0.230677 + 0.399544i −0.958008 0.286743i \(-0.907427\pi\)
0.727331 + 0.686287i \(0.240761\pi\)
\(420\) 0 0
\(421\) 3.16002 + 5.47331i 0.154010 + 0.266753i 0.932698 0.360658i \(-0.117448\pi\)
−0.778688 + 0.627411i \(0.784115\pi\)
\(422\) 0 0
\(423\) 35.7843 15.4140i 1.73989 0.749453i
\(424\) 0 0
\(425\) 5.93454 0.287867
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.49243 + 2.21693i −0.120336 + 0.107034i
\(430\) 0 0
\(431\) 13.8770 + 24.0357i 0.668434 + 1.15776i 0.978342 + 0.206995i \(0.0663683\pi\)
−0.309908 + 0.950766i \(0.600298\pi\)
\(432\) 0 0
\(433\) −11.2473 −0.540510 −0.270255 0.962789i \(-0.587108\pi\)
−0.270255 + 0.962789i \(0.587108\pi\)
\(434\) 0 0
\(435\) −12.6032 4.17985i −0.604278 0.200409i
\(436\) 0 0
\(437\) 3.55122 0.169878
\(438\) 0 0
\(439\) 15.0865 0.720040 0.360020 0.932945i \(-0.382770\pi\)
0.360020 + 0.932945i \(0.382770\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.93316 0.376916 0.188458 0.982081i \(-0.439651\pi\)
0.188458 + 0.982081i \(0.439651\pi\)
\(444\) 0 0
\(445\) −16.4079 −0.777810
\(446\) 0 0
\(447\) 10.4814 9.32284i 0.495755 0.440955i
\(448\) 0 0
\(449\) −32.5636 −1.53677 −0.768386 0.639987i \(-0.778940\pi\)
−0.768386 + 0.639987i \(0.778940\pi\)
\(450\) 0 0
\(451\) −7.27816 12.6061i −0.342715 0.593600i
\(452\) 0 0
\(453\) 14.5723 + 4.83292i 0.684668 + 0.227070i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.4079 0.533640 0.266820 0.963746i \(-0.414027\pi\)
0.266820 + 0.963746i \(0.414027\pi\)
\(458\) 0 0
\(459\) 6.18292 13.2326i 0.288594 0.617645i
\(460\) 0 0
\(461\) 2.45853 + 4.25830i 0.114505 + 0.198329i 0.917582 0.397547i \(-0.130138\pi\)
−0.803077 + 0.595876i \(0.796805\pi\)
\(462\) 0 0
\(463\) −7.59957 + 13.1628i −0.353182 + 0.611729i −0.986805 0.161913i \(-0.948234\pi\)
0.633623 + 0.773642i \(0.281567\pi\)
\(464\) 0 0
\(465\) −11.2156 + 9.97590i −0.520113 + 0.462621i
\(466\) 0 0
\(467\) 11.8905 20.5950i 0.550228 0.953022i −0.448030 0.894018i \(-0.647874\pi\)
0.998258 0.0590037i \(-0.0187924\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.06615 + 14.8687i 0.141281 + 0.685115i
\(472\) 0 0
\(473\) −11.5280 −0.530058
\(474\) 0 0
\(475\) 5.26509 + 9.11941i 0.241579 + 0.418427i
\(476\) 0 0
\(477\) 4.54325 + 3.38597i 0.208021 + 0.155033i
\(478\) 0 0
\(479\) 3.02909 5.24654i 0.138403 0.239720i −0.788489 0.615048i \(-0.789137\pi\)
0.926892 + 0.375328i \(0.122470\pi\)
\(480\) 0 0
\(481\) −2.67309 4.62992i −0.121882 0.211106i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.34362 12.7195i 0.333457 0.577564i
\(486\) 0 0
\(487\) −0.568012 0.983825i −0.0257391 0.0445814i 0.852869 0.522125i \(-0.174861\pi\)
−0.878608 + 0.477544i \(0.841527\pi\)
\(488\) 0 0
\(489\) 3.26764 + 1.08371i 0.147768 + 0.0490072i
\(490\) 0 0
\(491\) 16.4382 28.4718i 0.741845 1.28491i −0.209810 0.977742i \(-0.567285\pi\)
0.951655 0.307170i \(-0.0993821\pi\)
\(492\) 0 0
\(493\) 6.33929 10.9800i 0.285507 0.494513i
\(494\) 0 0
\(495\) −11.6001 + 4.99669i −0.521384 + 0.224584i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.0989 22.6879i −0.586387 1.01565i −0.994701 0.102810i \(-0.967217\pi\)
0.408314 0.912841i \(-0.366117\pi\)
\(500\) 0 0
\(501\) 4.31020 + 1.42948i 0.192566 + 0.0638644i
\(502\) 0 0
\(503\) 25.8516 1.15267 0.576333 0.817215i \(-0.304483\pi\)
0.576333 + 0.817215i \(0.304483\pi\)
\(504\) 0 0
\(505\) −4.09888 −0.182398
\(506\) 0 0
\(507\) −16.0421 + 14.2688i −0.712454 + 0.633701i
\(508\) 0 0
\(509\) 17.5858 + 30.4595i 0.779478 + 1.35009i 0.932243 + 0.361832i \(0.117849\pi\)
−0.152766 + 0.988262i \(0.548818\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 25.8196 2.23881i 1.13996 0.0988459i
\(514\) 0 0
\(515\) −3.68292 + 6.37900i −0.162289 + 0.281092i
\(516\) 0 0
\(517\) 16.0858 27.8615i 0.707453 1.22535i
\(518\) 0 0
\(519\) 1.82946 + 8.87163i 0.0803045 + 0.389421i
\(520\) 0 0
\(521\) −8.93130 15.4695i −0.391287 0.677730i 0.601332 0.798999i \(-0.294637\pi\)
−0.992620 + 0.121270i \(0.961303\pi\)
\(522\) 0 0
\(523\) −11.4320 + 19.8008i −0.499886 + 0.865828i −1.00000 0.000131698i \(-0.999958\pi\)
0.500114 + 0.865960i \(0.333291\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.16621 12.4122i −0.312165 0.540685i
\(528\) 0 0
\(529\) 11.2465 19.4795i 0.488979 0.846937i
\(530\) 0 0
\(531\) 39.3671 16.9573i 1.70839 0.735883i
\(532\) 0 0
\(533\) 2.28435 + 3.95661i 0.0989462 + 0.171380i
\(534\) 0 0
\(535\) 32.6291 1.41068
\(536\) 0 0
\(537\) −6.16690 + 5.48523i −0.266121 + 0.236705i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.1538 + 19.3190i −0.479541 + 0.830589i −0.999725 0.0234656i \(-0.992530\pi\)
0.520184 + 0.854054i \(0.325863\pi\)
\(542\) 0 0
\(543\) 3.64902 + 17.6952i 0.156594 + 0.759374i
\(544\) 0 0
\(545\) −16.1149 + 27.9118i −0.690287 + 1.19561i
\(546\) 0 0
\(547\) −10.8083 18.7206i −0.462131 0.800435i 0.536936 0.843623i \(-0.319582\pi\)
−0.999067 + 0.0431882i \(0.986249\pi\)
\(548\) 0 0
\(549\) 39.4251 16.9822i 1.68262 0.724784i
\(550\) 0 0
\(551\) 22.4968 0.958394
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.08836 19.8258i −0.173541 0.841556i
\(556\) 0 0
\(557\) −1.58768 2.74993i −0.0672720 0.116518i 0.830428 0.557127i \(-0.188096\pi\)
−0.897700 + 0.440608i \(0.854763\pi\)
\(558\) 0 0
\(559\) 3.61822 0.153034
\(560\) 0 0
\(561\) −2.43571 11.8115i −0.102836 0.498682i
\(562\) 0 0
\(563\) −43.7628 −1.84438 −0.922190 0.386737i \(-0.873602\pi\)
−0.922190 + 0.386737i \(0.873602\pi\)
\(564\) 0 0
\(565\) 21.9752 0.924505
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −23.8626 −1.00037 −0.500186 0.865918i \(-0.666735\pi\)
−0.500186 + 0.865918i \(0.666735\pi\)
\(570\) 0 0
\(571\) 10.2212 0.427742 0.213871 0.976862i \(-0.431393\pi\)
0.213871 + 0.976862i \(0.431393\pi\)
\(572\) 0 0
\(573\) 4.66002 + 22.5979i 0.194675 + 0.944040i
\(574\) 0 0
\(575\) −1.50324 −0.0626893
\(576\) 0 0
\(577\) −18.0185 31.2089i −0.750120 1.29925i −0.947764 0.318972i \(-0.896663\pi\)
0.197645 0.980274i \(-0.436671\pi\)
\(578\) 0 0
\(579\) −5.12178 24.8371i −0.212854 1.03220i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.67859 0.193767
\(584\) 0 0
\(585\) 3.64084 1.56828i 0.150530 0.0648403i
\(586\) 0 0
\(587\) −10.5142 18.2111i −0.433966 0.751651i 0.563245 0.826290i \(-0.309553\pi\)
−0.997211 + 0.0746391i \(0.976220\pi\)
\(588\) 0 0
\(589\) 12.7156 22.0242i 0.523939 0.907489i
\(590\) 0 0
\(591\) −6.46658 31.3585i −0.266000 1.28991i
\(592\) 0 0
\(593\) −12.5803 + 21.7897i −0.516612 + 0.894798i 0.483202 + 0.875509i \(0.339474\pi\)
−0.999814 + 0.0192889i \(0.993860\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.5643 + 27.1858i −1.25091 + 1.11264i
\(598\) 0 0
\(599\) −2.22253 −0.0908100 −0.0454050 0.998969i \(-0.514458\pi\)
−0.0454050 + 0.998969i \(0.514458\pi\)
\(600\) 0 0
\(601\) 14.0494 + 24.3343i 0.573089 + 0.992619i 0.996246 + 0.0865627i \(0.0275883\pi\)
−0.423158 + 0.906056i \(0.639078\pi\)
\(602\) 0 0
\(603\) −22.0080 + 9.47987i −0.896234 + 0.386050i
\(604\) 0 0
\(605\) 4.13348 7.15939i 0.168050 0.291071i
\(606\) 0 0
\(607\) −3.26509 5.65531i −0.132526 0.229542i 0.792124 0.610361i \(-0.208975\pi\)
−0.924650 + 0.380819i \(0.875642\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.04875 + 8.74470i −0.204251 + 0.353773i
\(612\) 0 0
\(613\) −5.36398 9.29068i −0.216649 0.375247i 0.737132 0.675748i \(-0.236179\pi\)
−0.953781 + 0.300501i \(0.902846\pi\)
\(614\) 0 0
\(615\) 3.49381 + 16.9426i 0.140884 + 0.683191i
\(616\) 0 0
\(617\) 15.5265 26.8928i 0.625075 1.08266i −0.363451 0.931613i \(-0.618402\pi\)
0.988526 0.151049i \(-0.0482650\pi\)
\(618\) 0 0
\(619\) −0.723217 + 1.25265i −0.0290685 + 0.0503482i −0.880194 0.474615i \(-0.842587\pi\)
0.851125 + 0.524963i \(0.175921\pi\)
\(620\) 0 0
\(621\) −1.56615 + 3.35186i −0.0628475 + 0.134505i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.99312 + 8.64834i 0.199725 + 0.345934i
\(626\) 0 0
\(627\) 15.9894 14.2220i 0.638555 0.567971i
\(628\) 0 0
\(629\) 19.3287 0.770686
\(630\) 0 0
\(631\) 0.0741250 0.00295087 0.00147544 0.999999i \(-0.499530\pi\)
0.00147544 + 0.999999i \(0.499530\pi\)
\(632\) 0 0
\(633\) 23.9283 + 7.93581i 0.951063 + 0.315420i
\(634\) 0 0
\(635\) −14.9814 25.9486i −0.594520 1.02974i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 28.2779 12.1806i 1.11866 0.481857i
\(640\) 0 0
\(641\) 23.5204 40.7384i 0.928998 1.60907i 0.143996 0.989578i \(-0.454005\pi\)
0.785002 0.619494i \(-0.212662\pi\)
\(642\) 0 0
\(643\) −16.8647 + 29.2105i −0.665077 + 1.15195i 0.314187 + 0.949361i \(0.398268\pi\)
−0.979264 + 0.202587i \(0.935065\pi\)
\(644\) 0 0
\(645\) 13.0036 + 4.31266i 0.512018 + 0.169811i
\(646\) 0 0
\(647\) 22.4814 + 38.9390i 0.883836 + 1.53085i 0.847042 + 0.531526i \(0.178381\pi\)
0.0367945 + 0.999323i \(0.488285\pi\)
\(648\) 0 0
\(649\) 17.6964 30.6510i 0.694644 1.20316i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.8578 36.1267i −0.816228 1.41375i −0.908443 0.418010i \(-0.862728\pi\)
0.0922143 0.995739i \(-0.470606\pi\)
\(654\) 0 0
\(655\) −4.83310 + 8.37118i −0.188845 + 0.327089i
\(656\) 0 0
\(657\) −11.9975 8.94138i −0.468065 0.348837i
\(658\) 0 0
\(659\) 10.5259 + 18.2313i 0.410029 + 0.710191i 0.994892 0.100941i \(-0.0321852\pi\)
−0.584863 + 0.811132i \(0.698852\pi\)
\(660\) 0 0
\(661\) −22.4437 −0.872958 −0.436479 0.899714i \(-0.643775\pi\)
−0.436479 + 0.899714i \(0.643775\pi\)
\(662\) 0 0
\(663\) 0.764480 + 3.70720i 0.0296899 + 0.143976i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.60576 + 2.78126i −0.0621754 + 0.107691i
\(668\) 0 0
\(669\) −12.2236 + 10.8725i −0.472593 + 0.420354i
\(670\) 0 0
\(671\) 17.7225 30.6962i 0.684168 1.18501i
\(672\) 0 0
\(673\) 5.83929 + 10.1140i 0.225088 + 0.389864i 0.956346 0.292237i \(-0.0943996\pi\)
−0.731258 + 0.682101i \(0.761066\pi\)
\(674\) 0 0
\(675\) −10.9294 + 0.947691i −0.420674 + 0.0364766i
\(676\) 0 0
\(677\) −10.4684 −0.402335 −0.201167 0.979557i \(-0.564474\pi\)
−0.201167 + 0.979557i \(0.564474\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 31.4188 + 10.4201i 1.20397 + 0.399297i
\(682\) 0 0
\(683\) −16.4079 28.4193i −0.627832 1.08744i −0.987986 0.154543i \(-0.950609\pi\)
0.360154 0.932893i \(-0.382724\pi\)
\(684\) 0 0
\(685\) −33.0617 −1.26322
\(686\) 0 0
\(687\) −14.8120 + 13.1747i −0.565113 + 0.502647i
\(688\) 0 0
\(689\) −1.46844 −0.0559431
\(690\) 0 0
\(691\) −5.90112 −0.224489 −0.112245 0.993681i \(-0.535804\pi\)
−0.112245 + 0.993681i \(0.535804\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.07784 0.192614
\(696\) 0 0
\(697\) −16.5178 −0.625656
\(698\) 0 0
\(699\) 1.95715 + 0.649089i 0.0740263 + 0.0245508i
\(700\) 0 0
\(701\) −12.3782 −0.467519 −0.233759 0.972294i \(-0.575103\pi\)
−0.233759 + 0.972294i \(0.575103\pi\)
\(702\) 0 0
\(703\) 17.1483 + 29.7018i 0.646761 + 1.12022i
\(704\) 0 0
\(705\) −28.5679 + 25.4101i −1.07593 + 0.957000i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13.2829 −0.498850 −0.249425 0.968394i \(-0.580242\pi\)
−0.249425 + 0.968394i \(0.580242\pi\)
\(710\) 0 0
\(711\) 25.3764 10.9308i 0.951688 0.409937i
\(712\) 0 0
\(713\) 1.81522 + 3.14406i 0.0679806 + 0.117746i
\(714\) 0 0
\(715\) 1.63664 2.83474i 0.0612067 0.106013i
\(716\) 0 0
\(717\) −39.9210 13.2398i −1.49088 0.494449i
\(718\) 0 0
\(719\) −12.1847 + 21.1045i −0.454413 + 0.787066i −0.998654 0.0518628i \(-0.983484\pi\)
0.544242 + 0.838929i \(0.316817\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −35.2126 11.6783i −1.30957 0.434319i
\(724\) 0 0
\(725\) −9.52290 −0.353672
\(726\) 0 0
\(727\) −7.99450 13.8469i −0.296500 0.513552i 0.678833 0.734293i \(-0.262486\pi\)
−0.975333 + 0.220740i \(0.929153\pi\)
\(728\) 0 0
\(729\) −9.27375 + 25.3574i −0.343472 + 0.939163i
\(730\) 0 0
\(731\) −6.54070 + 11.3288i −0.241917 + 0.419012i
\(732\) 0 0
\(733\) −21.1414 36.6181i −0.780877 1.35252i −0.931431 0.363917i \(-0.881439\pi\)
0.150554 0.988602i \(-0.451894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.89307 + 17.1353i −0.364416 + 0.631187i
\(738\) 0 0
\(739\) 1.54325 + 2.67299i 0.0567695 + 0.0983276i 0.893014 0.450030i \(-0.148587\pi\)
−0.836244 + 0.548357i \(0.815253\pi\)
\(740\) 0 0
\(741\) −5.01849 + 4.46376i −0.184359 + 0.163980i
\(742\) 0 0
\(743\) −3.31522 + 5.74213i −0.121624 + 0.210658i −0.920408 0.390959i \(-0.872143\pi\)
0.798784 + 0.601617i \(0.205477\pi\)
\(744\) 0 0
\(745\) −6.88255 + 11.9209i −0.252157 + 0.436749i
\(746\) 0 0
\(747\) −21.1941 15.7954i −0.775453 0.577924i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 21.3702 + 37.0142i 0.779808 + 1.35067i 0.932052 + 0.362325i \(0.118017\pi\)
−0.152243 + 0.988343i \(0.548650\pi\)
\(752\) 0 0
\(753\) 0.937489 + 4.54618i 0.0341640 + 0.165672i
\(754\) 0 0
\(755\) −15.0655 −0.548288
\(756\) 0 0
\(757\) −31.0232 −1.12756 −0.563779 0.825926i \(-0.690653\pi\)
−0.563779 + 0.825926i \(0.690653\pi\)
\(758\) 0 0
\(759\) 0.616972 + 2.99189i 0.0223947 + 0.108599i
\(760\) 0 0
\(761\) 11.8182 + 20.4697i 0.428409 + 0.742025i 0.996732 0.0807799i \(-0.0257411\pi\)
−0.568323 + 0.822805i \(0.692408\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.67123 + 14.2346i −0.0604233 + 0.514655i
\(766\) 0 0
\(767\) −5.55425 + 9.62025i −0.200553 + 0.347367i
\(768\) 0 0
\(769\) −1.73422 + 3.00376i −0.0625375 + 0.108318i −0.895599 0.444862i \(-0.853253\pi\)
0.833061 + 0.553180i \(0.186586\pi\)
\(770\) 0 0
\(771\) 14.3462 12.7604i 0.516665 0.459554i
\(772\) 0 0
\(773\) 17.2985 + 29.9619i 0.622184 + 1.07765i 0.989078 + 0.147392i \(0.0470879\pi\)
−0.366894 + 0.930263i \(0.619579\pi\)
\(774\) 0 0
\(775\) −5.38255 + 9.32284i −0.193347 + 0.334886i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.6545 25.3824i −0.525053 0.909418i
\(780\) 0 0
\(781\) 12.7115 22.0170i 0.454854 0.787831i
\(782\) 0 0
\(783\) −9.92147 + 21.2338i −0.354564 + 0.758834i
\(784\) 0 0
\(785\) −7.44870 12.9015i −0.265855 0.460475i
\(786\) 0 0
\(787\) 12.1593 0.433431 0.216715 0.976235i \(-0.430466\pi\)
0.216715 + 0.976235i \(0.430466\pi\)
\(788\) 0 0
\(789\) −22.0426 7.31041i −0.784736 0.260258i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.56243 + 9.63442i −0.197528 + 0.342128i
\(794\) 0 0
\(795\) −5.27747 1.75027i −0.187173 0.0620758i
\(796\) 0 0
\(797\) 2.89493 5.01416i 0.102544 0.177611i −0.810188 0.586170i \(-0.800635\pi\)
0.912732 + 0.408559i \(0.133969\pi\)
\(798\) 0 0
\(799\) −18.2534 31.6158i −0.645759 1.11849i
\(800\) 0 0
\(801\) −3.37704 + 28.7639i −0.119322 + 1.01632i
\(802\) 0 0
\(803\) −12.3548 −0.435993
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.29349 + 4.70837i −0.186340 + 0.165742i
\(808\) 0 0
\(809\) 24.5908 + 42.5926i 0.864568 + 1.49748i 0.867476 + 0.497479i \(0.165741\pi\)
−0.00290803 + 0.999996i \(0.500926\pi\)
\(810\) 0 0
\(811\) −40.7266 −1.43010 −0.715052 0.699072i \(-0.753597\pi\)
−0.715052 + 0.699072i \(0.753597\pi\)
\(812\) 0 0
\(813\) 10.0672 + 3.33880i 0.353074 + 0.117097i
\(814\) 0 0
\(815\) −3.37822 −0.118334
\(816\) 0 0
\(817\) −23.2115 −0.812069
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0938 0.526777 0.263388 0.964690i \(-0.415160\pi\)
0.263388 + 0.964690i \(0.415160\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −6.76833 + 6.02018i −0.235643 + 0.209596i
\(826\) 0 0
\(827\) 35.2348 1.22523 0.612616 0.790381i \(-0.290117\pi\)
0.612616 + 0.790381i \(0.290117\pi\)
\(828\) 0 0
\(829\) −1.61745 2.80151i −0.0561765 0.0973006i 0.836570 0.547861i \(-0.184558\pi\)
−0.892746 + 0.450560i \(0.851224\pi\)
\(830\) 0 0
\(831\) −25.9177 8.59562i −0.899077 0.298179i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.45606 −0.154208
\(836\) 0 0
\(837\) 15.1799 + 21.7148i 0.524694 + 0.750574i
\(838\) 0 0
\(839\) −15.5197 26.8808i −0.535798 0.928030i −0.999124 0.0418419i \(-0.986677\pi\)
0.463326 0.886188i \(-0.346656\pi\)
\(840\) 0 0
\(841\) 4.32760 7.49563i 0.149228 0.258470i
\(842\) 0 0
\(843\) −27.4226 + 24.3914i −0.944483 + 0.840083i
\(844\) 0 0
\(845\) 10.5339 18.2453i 0.362377 0.627656i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.40545 11.6648i −0.0825547 0.400334i
\(850\) 0 0
\(851\) −4.89602 −0.167833
\(852\) 0 0
\(853\) −8.03637 13.9194i −0.275160 0.476591i 0.695015 0.718995i \(-0.255398\pi\)
−0.970176 + 0.242403i \(0.922064\pi\)
\(854\) 0 0
\(855\) −23.3566 + 10.0608i −0.798779 + 0.344072i
\(856\) 0 0
\(857\) 9.61058 16.6460i 0.328291 0.568617i −0.653882 0.756597i \(-0.726861\pi\)
0.982173 + 0.187980i \(0.0601940\pi\)
\(858\) 0 0
\(859\) −7.40112 12.8191i −0.252523 0.437382i 0.711697 0.702487i \(-0.247927\pi\)
−0.964220 + 0.265104i \(0.914594\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.38441 + 12.7902i −0.251368 + 0.435382i −0.963903 0.266255i \(-0.914214\pi\)
0.712535 + 0.701637i \(0.247547\pi\)
\(864\) 0 0
\(865\) −4.44437 7.69787i −0.151113 0.261735i
\(866\) 0 0
\(867\) 14.9585 + 4.96099i 0.508018 + 0.168484i
\(868\) 0 0
\(869\) 11.4072 19.7579i 0.386964 0.670241i
\(870\) 0 0
\(871\) 3.10507 5.37815i 0.105211 0.182232i
\(872\) 0 0
\(873\) −20.7865 15.4917i −0.703518 0.524313i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −26.1916 45.3651i −0.884427 1.53187i −0.846369 0.532597i \(-0.821216\pi\)
−0.0380575 0.999276i \(-0.512117\pi\)
\(878\) 0 0
\(879\) −45.2211 14.9976i −1.52527 0.505855i
\(880\) 0 0
\(881\) 31.3214 1.05525 0.527623 0.849479i \(-0.323084\pi\)
0.527623 + 0.849479i \(0.323084\pi\)
\(882\) 0 0
\(883\) −43.0494 −1.44873 −0.724363 0.689419i \(-0.757866\pi\)
−0.724363 + 0.689419i \(0.757866\pi\)
\(884\) 0 0
\(885\) −31.4283 + 27.9543i −1.05645 + 0.939673i
\(886\) 0 0
\(887\) −7.48831 12.9701i −0.251433 0.435494i 0.712488 0.701685i \(-0.247568\pi\)
−0.963921 + 0.266190i \(0.914235\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.37195 + 21.3639i 0.213468 + 0.715718i
\(892\) 0 0
\(893\) 32.3887 56.0988i 1.08385 1.87727i
\(894\) 0 0
\(895\) 4.04944 7.01384i 0.135358 0.234447i
\(896\) 0 0
\(897\) −0.193645 0.939046i −0.00646562 0.0313538i
\(898\) 0 0
\(899\) 11.4993 + 19.9174i 0.383524 + 0.664282i
\(900\) 0 0
\(901\) 2.65452 4.59776i 0.0884348 0.153174i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.86467 15.3541i −0.294671 0.510386i
\(906\) 0 0
\(907\) −15.2280 + 26.3756i −0.505636 + 0.875787i 0.494343 + 0.869267i \(0.335409\pi\)
−0.999979 + 0.00652002i \(0.997925\pi\)
\(908\) 0 0
\(909\) −0.843624 + 7.18555i −0.0279812 + 0.238330i
\(910\) 0 0
\(911\) 9.97593 + 17.2788i 0.330517 + 0.572473i 0.982613 0.185664i \(-0.0594435\pi\)
−0.652096 + 0.758136i \(0.726110\pi\)
\(912\) 0 0
\(913\) −21.8255 −0.722317
\(914\) 0 0
\(915\) −31.4746 + 27.9954i −1.04052 + 0.925501i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −22.8145 + 39.5159i −0.752582 + 1.30351i 0.193985 + 0.981004i \(0.437859\pi\)
−0.946567 + 0.322506i \(0.895475\pi\)
\(920\) 0 0
\(921\) 7.52723 + 36.5019i 0.248031 + 1.20278i
\(922\) 0 0
\(923\) −3.98969 + 6.91034i −0.131322 + 0.227457i
\(924\) 0 0
\(925\) −7.25890 12.5728i −0.238671 0.413391i
\(926\) 0 0
\(927\) 10.4247 + 7.76926i 0.342392 + 0.255176i
\(928\) 0 0
\(929\) −56.3722 −1.84951 −0.924755 0.380562i \(-0.875730\pi\)
−0.924755 + 0.380562i \(0.875730\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −6.43151 31.1884i −0.210558 1.02106i
\(934\) 0 0
\(935\) 5.91714 + 10.2488i 0.193511 + 0.335171i
\(936\) 0 0
\(937\) 36.8530 1.20393 0.601967 0.798521i \(-0.294384\pi\)
0.601967 + 0.798521i \(0.294384\pi\)
\(938\) 0 0
\(939\) 0.000481840 0.00233659i 1.57243e−5 7.62519e-5i
\(940\) 0 0
\(941\) 8.76000 0.285568 0.142784 0.989754i \(-0.454395\pi\)
0.142784 + 0.989754i \(0.454395\pi\)
\(942\) 0 0
\(943\) 4.18401 0.136250
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.6452 0.865852 0.432926 0.901430i \(-0.357481\pi\)
0.432926 + 0.901430i \(0.357481\pi\)
\(948\) 0 0
\(949\) 3.87773 0.125877
\(950\) 0 0
\(951\) 4.93199 + 23.9168i 0.159931 + 0.775554i
\(952\) 0 0
\(953\) 24.3039 0.787282 0.393641 0.919264i \(-0.371215\pi\)
0.393641 + 0.919264i \(0.371215\pi\)
\(954\) 0 0
\(955\) −11.3207 19.6081i −0.366330 0.634502i
\(956\) 0 0
\(957\) 3.90848 + 18.9534i 0.126343 + 0.612677i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −5.00138 −0.161335
\(962\) 0 0
\(963\) 6.71565 57.2004i 0.216409 1.84326i
\(964\) 0 0
\(965\) 12.4425 + 21.5511i 0.400539 + 0.693753i
\(966\) 0 0
\(967\) 5.22872 9.05641i 0.168144 0.291234i −0.769623 0.638498i \(-0.779556\pi\)
0.937767 + 0.347264i \(0.112889\pi\)
\(968\) 0 0
\(969\) −4.90428 23.7824i −0.157548 0.764000i
\(970\) 0 0
\(971\) −20.8578 + 36.1267i −0.669358 + 1.15936i 0.308726 + 0.951151i \(0.400098\pi\)
−0.978084 + 0.208211i \(0.933236\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.12433 1.88951i 0.0680331 0.0605129i
\(976\) 0 0
\(977\) 5.89011 0.188441 0.0942207 0.995551i \(-0.469964\pi\)
0.0942207 + 0.995551i \(0.469964\pi\)
\(978\) 0 0
\(979\) 11.9567 + 20.7097i 0.382139 + 0.661885i
\(980\) 0 0
\(981\) 45.6141 + 33.9950i 1.45635 + 1.08538i
\(982\) 0 0
\(983\) 20.9196 36.2338i 0.667232 1.15568i −0.311443 0.950265i \(-0.600812\pi\)
0.978675 0.205415i \(-0.0658543\pi\)
\(984\) 0 0
\(985\) 15.7095 + 27.2096i 0.500545 + 0.866969i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.65678 2.86963i 0.0526826 0.0912489i
\(990\) 0 0
\(991\) 27.3578 + 47.3851i 0.869049 + 1.50524i 0.862970 + 0.505255i \(0.168602\pi\)
0.00607865 + 0.999982i \(0.498065\pi\)
\(992\) 0 0
\(993\) 4.88441 + 23.6860i 0.155002 + 0.751653i
\(994\) 0 0
\(995\) 20.0698 34.7619i 0.636255 1.10203i
\(996\) 0 0
\(997\) −9.02476 + 15.6313i −0.285817 + 0.495050i −0.972807 0.231617i \(-0.925598\pi\)
0.686990 + 0.726667i \(0.258932\pi\)
\(998\) 0 0
\(999\) −35.5970 + 3.08661i −1.12624 + 0.0976561i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1764.2.i.d.373.3 6
3.2 odd 2 5292.2.i.e.1549.3 6
7.2 even 3 1764.2.j.e.589.1 6
7.3 odd 6 1764.2.l.e.949.1 6
7.4 even 3 1764.2.l.f.949.3 6
7.5 odd 6 252.2.j.a.85.3 6
7.6 odd 2 1764.2.i.g.373.1 6
9.2 odd 6 5292.2.l.f.3313.1 6
9.7 even 3 1764.2.l.f.961.3 6
21.2 odd 6 5292.2.j.d.1765.3 6
21.5 even 6 756.2.j.b.253.1 6
21.11 odd 6 5292.2.l.f.361.1 6
21.17 even 6 5292.2.l.e.361.3 6
21.20 even 2 5292.2.i.f.1549.1 6
28.19 even 6 1008.2.r.j.337.1 6
63.2 odd 6 5292.2.j.d.3529.3 6
63.5 even 6 2268.2.a.h.1.3 3
63.11 odd 6 5292.2.i.e.2125.3 6
63.16 even 3 1764.2.j.e.1177.1 6
63.20 even 6 5292.2.l.e.3313.3 6
63.25 even 3 inner 1764.2.i.d.1537.3 6
63.34 odd 6 1764.2.l.e.961.1 6
63.38 even 6 5292.2.i.f.2125.1 6
63.40 odd 6 2268.2.a.i.1.1 3
63.47 even 6 756.2.j.b.505.1 6
63.52 odd 6 1764.2.i.g.1537.1 6
63.61 odd 6 252.2.j.a.169.3 yes 6
84.47 odd 6 3024.2.r.j.1009.1 6
252.47 odd 6 3024.2.r.j.2017.1 6
252.103 even 6 9072.2.a.by.1.1 3
252.131 odd 6 9072.2.a.bv.1.3 3
252.187 even 6 1008.2.r.j.673.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.3 6 7.5 odd 6
252.2.j.a.169.3 yes 6 63.61 odd 6
756.2.j.b.253.1 6 21.5 even 6
756.2.j.b.505.1 6 63.47 even 6
1008.2.r.j.337.1 6 28.19 even 6
1008.2.r.j.673.1 6 252.187 even 6
1764.2.i.d.373.3 6 1.1 even 1 trivial
1764.2.i.d.1537.3 6 63.25 even 3 inner
1764.2.i.g.373.1 6 7.6 odd 2
1764.2.i.g.1537.1 6 63.52 odd 6
1764.2.j.e.589.1 6 7.2 even 3
1764.2.j.e.1177.1 6 63.16 even 3
1764.2.l.e.949.1 6 7.3 odd 6
1764.2.l.e.961.1 6 63.34 odd 6
1764.2.l.f.949.3 6 7.4 even 3
1764.2.l.f.961.3 6 9.7 even 3
2268.2.a.h.1.3 3 63.5 even 6
2268.2.a.i.1.1 3 63.40 odd 6
3024.2.r.j.1009.1 6 84.47 odd 6
3024.2.r.j.2017.1 6 252.47 odd 6
5292.2.i.e.1549.3 6 3.2 odd 2
5292.2.i.e.2125.3 6 63.11 odd 6
5292.2.i.f.1549.1 6 21.20 even 2
5292.2.i.f.2125.1 6 63.38 even 6
5292.2.j.d.1765.3 6 21.2 odd 6
5292.2.j.d.3529.3 6 63.2 odd 6
5292.2.l.e.361.3 6 21.17 even 6
5292.2.l.e.3313.3 6 63.20 even 6
5292.2.l.f.361.1 6 21.11 odd 6
5292.2.l.f.3313.1 6 9.2 odd 6
9072.2.a.bv.1.3 3 252.131 odd 6
9072.2.a.by.1.1 3 252.103 even 6