Properties

Label 352.2.m.f.257.3
Level $352$
Weight $2$
Character 352.257
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [352,2,Mod(97,352)] 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("352.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 257.3
Root \(-1.36475 + 0.991547i\) of defining polynomial
Character \(\chi\) \(=\) 352.257
Dual form 352.2.m.f.289.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.212270 - 0.653300i) q^{3} +(-0.607036 + 0.441038i) q^{5} +(0.944137 + 2.90576i) q^{7} +(2.04531 + 1.48600i) q^{9} +(-3.27718 - 0.510019i) q^{11} +(5.02345 + 3.64975i) q^{13} +(0.159274 + 0.490196i) q^{15} +(2.49320 - 1.81142i) q^{17} +(1.30894 - 4.02849i) q^{19} +2.09874 q^{21} -0.264608 q^{23} +(-1.37111 + 4.21983i) q^{25} +(3.07215 - 2.23205i) q^{27} +(-0.513105 - 1.57918i) q^{29} +(3.39534 + 2.46686i) q^{31} +(-1.02884 + 2.03272i) q^{33} +(-1.85467 - 1.34750i) q^{35} +(-0.730823 - 2.24924i) q^{37} +(3.45071 - 2.50709i) q^{39} +(1.75476 - 5.40060i) q^{41} -9.90575 q^{43} -1.89696 q^{45} +(-3.56217 + 10.9632i) q^{47} +(-1.88890 + 1.37237i) q^{49} +(-0.654167 - 2.01332i) q^{51} +(-3.56561 - 2.59057i) q^{53} +(2.21430 - 1.13576i) q^{55} +(-2.35396 - 1.71025i) q^{57} +(-0.258400 - 0.795272i) q^{59} +(-5.06629 + 3.68088i) q^{61} +(-2.38691 + 7.34616i) q^{63} -4.65909 q^{65} +7.76323 q^{67} +(-0.0561683 + 0.172868i) q^{69} +(8.38148 - 6.08950i) q^{71} +(-2.61683 - 8.05376i) q^{73} +(2.46577 + 1.79149i) q^{75} +(-1.61211 - 10.0042i) q^{77} +(-11.9096 - 8.65285i) q^{79} +(1.53764 + 4.73238i) q^{81} +(8.89570 - 6.46311i) q^{83} +(-0.714561 + 2.19919i) q^{85} -1.14059 q^{87} -15.9492 q^{89} +(-5.86246 + 18.0428i) q^{91} +(2.33233 - 1.69454i) q^{93} +(0.982144 + 3.02273i) q^{95} +(0.712979 + 0.518009i) q^{97} +(-5.94495 - 5.91304i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7} - q^{9} - 11 q^{11} - 2 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} + 24 q^{21} - 12 q^{23} + 13 q^{25} + 3 q^{27} + 16 q^{31} - 7 q^{33} - 28 q^{35} - 4 q^{37} + 46 q^{39} - 4 q^{41} - 22 q^{43}+ \cdots - 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/352\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(287\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\)

Coefficient data

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



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
\(3\) 0.212270 0.653300i 0.122554 0.377183i −0.870893 0.491472i \(-0.836459\pi\)
0.993448 + 0.114289i \(0.0364590\pi\)
\(4\) 0 0
\(5\) −0.607036 + 0.441038i −0.271475 + 0.197238i −0.715190 0.698930i \(-0.753660\pi\)
0.443716 + 0.896168i \(0.353660\pi\)
\(6\) 0 0
\(7\) 0.944137 + 2.90576i 0.356850 + 1.09827i 0.954929 + 0.296835i \(0.0959311\pi\)
−0.598079 + 0.801438i \(0.704069\pi\)
\(8\) 0 0
\(9\) 2.04531 + 1.48600i 0.681770 + 0.495335i
\(10\) 0 0
\(11\) −3.27718 0.510019i −0.988106 0.153777i
\(12\) 0 0
\(13\) 5.02345 + 3.64975i 1.39325 + 1.01226i 0.995499 + 0.0947671i \(0.0302106\pi\)
0.397755 + 0.917492i \(0.369789\pi\)
\(14\) 0 0
\(15\) 0.159274 + 0.490196i 0.0411244 + 0.126568i
\(16\) 0 0
\(17\) 2.49320 1.81142i 0.604691 0.439333i −0.242850 0.970064i \(-0.578082\pi\)
0.847541 + 0.530730i \(0.178082\pi\)
\(18\) 0 0
\(19\) 1.30894 4.02849i 0.300290 0.924199i −0.681103 0.732188i \(-0.738499\pi\)
0.981393 0.192011i \(-0.0615008\pi\)
\(20\) 0 0
\(21\) 2.09874 0.457983
\(22\) 0 0
\(23\) −0.264608 −0.0551745 −0.0275873 0.999619i \(-0.508782\pi\)
−0.0275873 + 0.999619i \(0.508782\pi\)
\(24\) 0 0
\(25\) −1.37111 + 4.21983i −0.274221 + 0.843966i
\(26\) 0 0
\(27\) 3.07215 2.23205i 0.591236 0.429558i
\(28\) 0 0
\(29\) −0.513105 1.57918i −0.0952813 0.293246i 0.892046 0.451945i \(-0.149270\pi\)
−0.987327 + 0.158700i \(0.949270\pi\)
\(30\) 0 0
\(31\) 3.39534 + 2.46686i 0.609821 + 0.443061i 0.849351 0.527828i \(-0.176993\pi\)
−0.239530 + 0.970889i \(0.576993\pi\)
\(32\) 0 0
\(33\) −1.02884 + 2.03272i −0.179098 + 0.353851i
\(34\) 0 0
\(35\) −1.85467 1.34750i −0.313497 0.227769i
\(36\) 0 0
\(37\) −0.730823 2.24924i −0.120147 0.369773i 0.872839 0.488008i \(-0.162276\pi\)
−0.992986 + 0.118235i \(0.962276\pi\)
\(38\) 0 0
\(39\) 3.45071 2.50709i 0.552556 0.401455i
\(40\) 0 0
\(41\) 1.75476 5.40060i 0.274048 0.843432i −0.715422 0.698693i \(-0.753765\pi\)
0.989470 0.144740i \(-0.0462345\pi\)
\(42\) 0 0
\(43\) −9.90575 −1.51061 −0.755306 0.655372i \(-0.772512\pi\)
−0.755306 + 0.655372i \(0.772512\pi\)
\(44\) 0 0
\(45\) −1.89696 −0.282782
\(46\) 0 0
\(47\) −3.56217 + 10.9632i −0.519596 + 1.59915i 0.255165 + 0.966898i \(0.417870\pi\)
−0.774761 + 0.632255i \(0.782130\pi\)
\(48\) 0 0
\(49\) −1.88890 + 1.37237i −0.269843 + 0.196052i
\(50\) 0 0
\(51\) −0.654167 2.01332i −0.0916017 0.281921i
\(52\) 0 0
\(53\) −3.56561 2.59057i −0.489775 0.355842i 0.315323 0.948984i \(-0.397887\pi\)
−0.805098 + 0.593142i \(0.797887\pi\)
\(54\) 0 0
\(55\) 2.21430 1.13576i 0.298576 0.153146i
\(56\) 0 0
\(57\) −2.35396 1.71025i −0.311790 0.226529i
\(58\) 0 0
\(59\) −0.258400 0.795272i −0.0336407 0.103536i 0.932826 0.360327i \(-0.117335\pi\)
−0.966467 + 0.256791i \(0.917335\pi\)
\(60\) 0 0
\(61\) −5.06629 + 3.68088i −0.648672 + 0.471288i −0.862819 0.505514i \(-0.831303\pi\)
0.214147 + 0.976802i \(0.431303\pi\)
\(62\) 0 0
\(63\) −2.38691 + 7.34616i −0.300723 + 0.925529i
\(64\) 0 0
\(65\) −4.65909 −0.577889
\(66\) 0 0
\(67\) 7.76323 0.948430 0.474215 0.880409i \(-0.342732\pi\)
0.474215 + 0.880409i \(0.342732\pi\)
\(68\) 0 0
\(69\) −0.0561683 + 0.172868i −0.00676187 + 0.0208109i
\(70\) 0 0
\(71\) 8.38148 6.08950i 0.994699 0.722691i 0.0337539 0.999430i \(-0.489254\pi\)
0.960945 + 0.276739i \(0.0892538\pi\)
\(72\) 0 0
\(73\) −2.61683 8.05376i −0.306276 0.942621i −0.979198 0.202908i \(-0.934961\pi\)
0.672922 0.739714i \(-0.265039\pi\)
\(74\) 0 0
\(75\) 2.46577 + 1.79149i 0.284723 + 0.206863i
\(76\) 0 0
\(77\) −1.61211 10.0042i −0.183717 1.14008i
\(78\) 0 0
\(79\) −11.9096 8.65285i −1.33994 0.973521i −0.999446 0.0332737i \(-0.989407\pi\)
−0.340491 0.940248i \(-0.610593\pi\)
\(80\) 0 0
\(81\) 1.53764 + 4.73238i 0.170849 + 0.525820i
\(82\) 0 0
\(83\) 8.89570 6.46311i 0.976430 0.709418i 0.0195224 0.999809i \(-0.493785\pi\)
0.956908 + 0.290391i \(0.0937854\pi\)
\(84\) 0 0
\(85\) −0.714561 + 2.19919i −0.0775050 + 0.238536i
\(86\) 0 0
\(87\) −1.14059 −0.122284
\(88\) 0 0
\(89\) −15.9492 −1.69061 −0.845306 0.534282i \(-0.820582\pi\)
−0.845306 + 0.534282i \(0.820582\pi\)
\(90\) 0 0
\(91\) −5.86246 + 18.0428i −0.614552 + 1.89140i
\(92\) 0 0
\(93\) 2.33233 1.69454i 0.241851 0.175715i
\(94\) 0 0
\(95\) 0.982144 + 3.02273i 0.100766 + 0.310125i
\(96\) 0 0
\(97\) 0.712979 + 0.518009i 0.0723920 + 0.0525959i 0.623393 0.781909i \(-0.285754\pi\)
−0.551001 + 0.834505i \(0.685754\pi\)
\(98\) 0 0
\(99\) −5.94495 5.91304i −0.597489 0.594283i
\(100\) 0 0
\(101\) −9.30209 6.75837i −0.925593 0.672483i 0.0193169 0.999813i \(-0.493851\pi\)
−0.944910 + 0.327331i \(0.893851\pi\)
\(102\) 0 0
\(103\) −3.37708 10.3936i −0.332753 1.02411i −0.967818 0.251651i \(-0.919027\pi\)
0.635065 0.772459i \(-0.280973\pi\)
\(104\) 0 0
\(105\) −1.27401 + 0.925624i −0.124331 + 0.0903317i
\(106\) 0 0
\(107\) −0.645212 + 1.98576i −0.0623750 + 0.191970i −0.977388 0.211454i \(-0.932180\pi\)
0.915013 + 0.403424i \(0.132180\pi\)
\(108\) 0 0
\(109\) −3.88713 −0.372320 −0.186160 0.982519i \(-0.559604\pi\)
−0.186160 + 0.982519i \(0.559604\pi\)
\(110\) 0 0
\(111\) −1.62456 −0.154197
\(112\) 0 0
\(113\) 4.20708 12.9481i 0.395769 1.21805i −0.532592 0.846372i \(-0.678782\pi\)
0.928361 0.371679i \(-0.121218\pi\)
\(114\) 0 0
\(115\) 0.160626 0.116702i 0.0149785 0.0108825i
\(116\) 0 0
\(117\) 4.85096 + 14.9297i 0.448472 + 1.38025i
\(118\) 0 0
\(119\) 7.61746 + 5.53441i 0.698292 + 0.507339i
\(120\) 0 0
\(121\) 10.4798 + 3.34284i 0.952706 + 0.303895i
\(122\) 0 0
\(123\) −3.15573 2.29277i −0.284543 0.206732i
\(124\) 0 0
\(125\) −2.18813 6.73437i −0.195712 0.602340i
\(126\) 0 0
\(127\) 6.61306 4.80467i 0.586814 0.426345i −0.254360 0.967110i \(-0.581865\pi\)
0.841174 + 0.540764i \(0.181865\pi\)
\(128\) 0 0
\(129\) −2.10269 + 6.47143i −0.185132 + 0.569777i
\(130\) 0 0
\(131\) 5.90996 0.516356 0.258178 0.966097i \(-0.416878\pi\)
0.258178 + 0.966097i \(0.416878\pi\)
\(132\) 0 0
\(133\) 12.9416 1.12218
\(134\) 0 0
\(135\) −0.880490 + 2.70987i −0.0757806 + 0.233229i
\(136\) 0 0
\(137\) 6.71184 4.87644i 0.573431 0.416622i −0.262919 0.964818i \(-0.584685\pi\)
0.836350 + 0.548196i \(0.184685\pi\)
\(138\) 0 0
\(139\) 5.62320 + 17.3064i 0.476953 + 1.46791i 0.843305 + 0.537435i \(0.180607\pi\)
−0.366352 + 0.930476i \(0.619393\pi\)
\(140\) 0 0
\(141\) 6.40614 + 4.65433i 0.539494 + 0.391966i
\(142\) 0 0
\(143\) −14.6013 14.5229i −1.22102 1.21447i
\(144\) 0 0
\(145\) 1.00795 + 0.732318i 0.0837057 + 0.0608157i
\(146\) 0 0
\(147\) 0.495610 + 1.52533i 0.0408773 + 0.125807i
\(148\) 0 0
\(149\) −8.89158 + 6.46011i −0.728426 + 0.529233i −0.889065 0.457781i \(-0.848645\pi\)
0.160639 + 0.987013i \(0.448645\pi\)
\(150\) 0 0
\(151\) 5.42468 16.6955i 0.441454 1.35866i −0.444871 0.895594i \(-0.646751\pi\)
0.886326 0.463062i \(-0.153249\pi\)
\(152\) 0 0
\(153\) 7.79114 0.629877
\(154\) 0 0
\(155\) −3.14907 −0.252940
\(156\) 0 0
\(157\) −4.09117 + 12.5913i −0.326511 + 1.00490i 0.644243 + 0.764821i \(0.277173\pi\)
−0.970754 + 0.240077i \(0.922827\pi\)
\(158\) 0 0
\(159\) −2.44929 + 1.77952i −0.194242 + 0.141125i
\(160\) 0 0
\(161\) −0.249826 0.768885i −0.0196890 0.0605967i
\(162\) 0 0
\(163\) 18.2756 + 13.2780i 1.43146 + 1.04001i 0.989743 + 0.142859i \(0.0456295\pi\)
0.441714 + 0.897156i \(0.354371\pi\)
\(164\) 0 0
\(165\) −0.271961 1.68769i −0.0211721 0.131387i
\(166\) 0 0
\(167\) −13.0270 9.46466i −1.00806 0.732397i −0.0442572 0.999020i \(-0.514092\pi\)
−0.963801 + 0.266623i \(0.914092\pi\)
\(168\) 0 0
\(169\) 7.89716 + 24.3050i 0.607474 + 1.86961i
\(170\) 0 0
\(171\) 8.66352 6.29442i 0.662516 0.481346i
\(172\) 0 0
\(173\) 2.66741 8.20943i 0.202799 0.624151i −0.796998 0.603983i \(-0.793580\pi\)
0.999797 0.0201688i \(-0.00642035\pi\)
\(174\) 0 0
\(175\) −13.5563 −1.02476
\(176\) 0 0
\(177\) −0.574402 −0.0431747
\(178\) 0 0
\(179\) −4.17100 + 12.8370i −0.311755 + 0.959483i 0.665315 + 0.746563i \(0.268297\pi\)
−0.977070 + 0.212920i \(0.931703\pi\)
\(180\) 0 0
\(181\) 6.62903 4.81627i 0.492732 0.357991i −0.313502 0.949588i \(-0.601502\pi\)
0.806234 + 0.591597i \(0.201502\pi\)
\(182\) 0 0
\(183\) 1.32929 + 4.09115i 0.0982643 + 0.302426i
\(184\) 0 0
\(185\) 1.43564 + 1.04305i 0.105550 + 0.0766866i
\(186\) 0 0
\(187\) −9.09452 + 4.66475i −0.665057 + 0.341121i
\(188\) 0 0
\(189\) 9.38633 + 6.81957i 0.682755 + 0.496051i
\(190\) 0 0
\(191\) 6.08452 + 18.7262i 0.440260 + 1.35498i 0.887599 + 0.460617i \(0.152372\pi\)
−0.447339 + 0.894364i \(0.647628\pi\)
\(192\) 0 0
\(193\) 19.3032 14.0246i 1.38948 1.00951i 0.393556 0.919301i \(-0.371245\pi\)
0.995922 0.0902141i \(-0.0287551\pi\)
\(194\) 0 0
\(195\) −0.988986 + 3.04379i −0.0708228 + 0.217970i
\(196\) 0 0
\(197\) 18.9000 1.34657 0.673283 0.739385i \(-0.264884\pi\)
0.673283 + 0.739385i \(0.264884\pi\)
\(198\) 0 0
\(199\) 9.81826 0.695998 0.347999 0.937495i \(-0.386861\pi\)
0.347999 + 0.937495i \(0.386861\pi\)
\(200\) 0 0
\(201\) 1.64790 5.07172i 0.116234 0.357732i
\(202\) 0 0
\(203\) 4.10426 2.98192i 0.288062 0.209290i
\(204\) 0 0
\(205\) 1.31666 + 4.05228i 0.0919598 + 0.283023i
\(206\) 0 0
\(207\) −0.541204 0.393208i −0.0376163 0.0273298i
\(208\) 0 0
\(209\) −6.34422 + 12.5345i −0.438839 + 0.867028i
\(210\) 0 0
\(211\) −10.5325 7.65227i −0.725084 0.526804i 0.162921 0.986639i \(-0.447909\pi\)
−0.888004 + 0.459835i \(0.847909\pi\)
\(212\) 0 0
\(213\) −2.19914 6.76824i −0.150682 0.463752i
\(214\) 0 0
\(215\) 6.01315 4.36881i 0.410093 0.297950i
\(216\) 0 0
\(217\) −3.96242 + 12.1951i −0.268987 + 0.827857i
\(218\) 0 0
\(219\) −5.81700 −0.393076
\(220\) 0 0
\(221\) 19.1357 1.28721
\(222\) 0 0
\(223\) −1.08063 + 3.32583i −0.0723642 + 0.222714i −0.980697 0.195534i \(-0.937356\pi\)
0.908333 + 0.418248i \(0.137356\pi\)
\(224\) 0 0
\(225\) −9.07502 + 6.59339i −0.605001 + 0.439559i
\(226\) 0 0
\(227\) −3.37711 10.3937i −0.224147 0.689852i −0.998377 0.0569486i \(-0.981863\pi\)
0.774231 0.632904i \(-0.218137\pi\)
\(228\) 0 0
\(229\) −9.45596 6.87016i −0.624868 0.453993i 0.229751 0.973250i \(-0.426209\pi\)
−0.854618 + 0.519257i \(0.826209\pi\)
\(230\) 0 0
\(231\) −6.87795 1.07040i −0.452536 0.0704271i
\(232\) 0 0
\(233\) −3.81746 2.77354i −0.250090 0.181701i 0.455677 0.890145i \(-0.349397\pi\)
−0.705767 + 0.708444i \(0.749397\pi\)
\(234\) 0 0
\(235\) −2.67283 8.22613i −0.174356 0.536614i
\(236\) 0 0
\(237\) −8.18096 + 5.94382i −0.531411 + 0.386092i
\(238\) 0 0
\(239\) −0.374310 + 1.15201i −0.0242121 + 0.0745172i −0.962432 0.271521i \(-0.912473\pi\)
0.938220 + 0.346038i \(0.112473\pi\)
\(240\) 0 0
\(241\) −17.6151 −1.13469 −0.567344 0.823481i \(-0.692029\pi\)
−0.567344 + 0.823481i \(0.692029\pi\)
\(242\) 0 0
\(243\) 14.8102 0.950077
\(244\) 0 0
\(245\) 0.541366 1.66615i 0.0345866 0.106447i
\(246\) 0 0
\(247\) 21.2783 15.4596i 1.35391 0.983672i
\(248\) 0 0
\(249\) −2.33406 7.18349i −0.147915 0.455235i
\(250\) 0 0
\(251\) 0.644281 + 0.468097i 0.0406666 + 0.0295460i 0.607933 0.793988i \(-0.291999\pi\)
−0.567266 + 0.823534i \(0.691999\pi\)
\(252\) 0 0
\(253\) 0.867166 + 0.134955i 0.0545183 + 0.00848455i
\(254\) 0 0
\(255\) 1.28505 + 0.933645i 0.0804731 + 0.0584671i
\(256\) 0 0
\(257\) −4.76394 14.6619i −0.297166 0.914584i −0.982485 0.186341i \(-0.940337\pi\)
0.685319 0.728243i \(-0.259663\pi\)
\(258\) 0 0
\(259\) 5.84575 4.24718i 0.363237 0.263907i
\(260\) 0 0
\(261\) 1.29720 3.99238i 0.0802948 0.247122i
\(262\) 0 0
\(263\) −23.1291 −1.42620 −0.713101 0.701062i \(-0.752710\pi\)
−0.713101 + 0.701062i \(0.752710\pi\)
\(264\) 0 0
\(265\) 3.30700 0.203147
\(266\) 0 0
\(267\) −3.38554 + 10.4196i −0.207192 + 0.637670i
\(268\) 0 0
\(269\) −19.5180 + 14.1807i −1.19004 + 0.864612i −0.993268 0.115838i \(-0.963045\pi\)
−0.196768 + 0.980450i \(0.563045\pi\)
\(270\) 0 0
\(271\) −1.56539 4.81777i −0.0950905 0.292658i 0.892187 0.451667i \(-0.149170\pi\)
−0.987277 + 0.159008i \(0.949170\pi\)
\(272\) 0 0
\(273\) 10.5429 + 7.65989i 0.638087 + 0.463597i
\(274\) 0 0
\(275\) 6.64555 13.1298i 0.400742 0.791759i
\(276\) 0 0
\(277\) 4.74659 + 3.44860i 0.285195 + 0.207206i 0.721180 0.692748i \(-0.243600\pi\)
−0.435985 + 0.899954i \(0.643600\pi\)
\(278\) 0 0
\(279\) 3.27876 + 10.0910i 0.196294 + 0.604131i
\(280\) 0 0
\(281\) 22.6389 16.4481i 1.35052 0.981212i 0.351538 0.936174i \(-0.385659\pi\)
0.998985 0.0450385i \(-0.0143411\pi\)
\(282\) 0 0
\(283\) −2.34921 + 7.23013i −0.139646 + 0.429786i −0.996284 0.0861321i \(-0.972549\pi\)
0.856638 + 0.515919i \(0.172549\pi\)
\(284\) 0 0
\(285\) 2.18323 0.129323
\(286\) 0 0
\(287\) 17.3496 1.02411
\(288\) 0 0
\(289\) −2.31846 + 7.13550i −0.136380 + 0.419735i
\(290\) 0 0
\(291\) 0.489760 0.355831i 0.0287102 0.0208592i
\(292\) 0 0
\(293\) 1.21793 + 3.74840i 0.0711522 + 0.218984i 0.980309 0.197470i \(-0.0632726\pi\)
−0.909157 + 0.416454i \(0.863273\pi\)
\(294\) 0 0
\(295\) 0.507603 + 0.368795i 0.0295538 + 0.0214721i
\(296\) 0 0
\(297\) −11.2064 + 5.74797i −0.650260 + 0.333531i
\(298\) 0 0
\(299\) −1.32924 0.965752i −0.0768722 0.0558509i
\(300\) 0 0
\(301\) −9.35239 28.7837i −0.539063 1.65906i
\(302\) 0 0
\(303\) −6.38980 + 4.64246i −0.367084 + 0.266702i
\(304\) 0 0
\(305\) 1.45202 4.46885i 0.0831422 0.255885i
\(306\) 0 0
\(307\) −10.7552 −0.613829 −0.306915 0.951737i \(-0.599297\pi\)
−0.306915 + 0.951737i \(0.599297\pi\)
\(308\) 0 0
\(309\) −7.50698 −0.427057
\(310\) 0 0
\(311\) −8.06019 + 24.8067i −0.457051 + 1.40666i 0.411658 + 0.911338i \(0.364950\pi\)
−0.868710 + 0.495322i \(0.835050\pi\)
\(312\) 0 0
\(313\) 2.43382 1.76827i 0.137568 0.0999487i −0.516873 0.856062i \(-0.672904\pi\)
0.654441 + 0.756113i \(0.272904\pi\)
\(314\) 0 0
\(315\) −1.79099 5.51210i −0.100911 0.310572i
\(316\) 0 0
\(317\) −26.2108 19.0433i −1.47214 1.06958i −0.979985 0.199071i \(-0.936208\pi\)
−0.492160 0.870505i \(-0.663792\pi\)
\(318\) 0 0
\(319\) 0.876127 + 5.43693i 0.0490537 + 0.304410i
\(320\) 0 0
\(321\) 1.16034 + 0.843034i 0.0647637 + 0.0470536i
\(322\) 0 0
\(323\) −4.03383 12.4149i −0.224449 0.690782i
\(324\) 0 0
\(325\) −22.2890 + 16.1939i −1.23637 + 0.898277i
\(326\) 0 0
\(327\) −0.825122 + 2.53946i −0.0456294 + 0.140433i
\(328\) 0 0
\(329\) −35.2197 −1.94172
\(330\) 0 0
\(331\) −19.2715 −1.05926 −0.529629 0.848229i \(-0.677669\pi\)
−0.529629 + 0.848229i \(0.677669\pi\)
\(332\) 0 0
\(333\) 1.84762 5.68640i 0.101249 0.311613i
\(334\) 0 0
\(335\) −4.71256 + 3.42388i −0.257475 + 0.187066i
\(336\) 0 0
\(337\) −6.07494 18.6967i −0.330923 1.01848i −0.968696 0.248252i \(-0.920144\pi\)
0.637772 0.770225i \(-0.279856\pi\)
\(338\) 0 0
\(339\) −7.56593 5.49697i −0.410925 0.298555i
\(340\) 0 0
\(341\) −9.86899 9.81602i −0.534436 0.531568i
\(342\) 0 0
\(343\) 11.5313 + 8.37801i 0.622634 + 0.452370i
\(344\) 0 0
\(345\) −0.0421452 0.129710i −0.00226902 0.00698333i
\(346\) 0 0
\(347\) 1.04704 0.760720i 0.0562081 0.0408376i −0.559326 0.828947i \(-0.688940\pi\)
0.615535 + 0.788110i \(0.288940\pi\)
\(348\) 0 0
\(349\) −3.54495 + 10.9102i −0.189757 + 0.584011i −0.999998 0.00208547i \(-0.999336\pi\)
0.810241 + 0.586097i \(0.199336\pi\)
\(350\) 0 0
\(351\) 23.5792 1.25857
\(352\) 0 0
\(353\) −15.5104 −0.825534 −0.412767 0.910837i \(-0.635438\pi\)
−0.412767 + 0.910837i \(0.635438\pi\)
\(354\) 0 0
\(355\) −2.40216 + 7.39310i −0.127494 + 0.392385i
\(356\) 0 0
\(357\) 5.23259 3.80170i 0.276938 0.201207i
\(358\) 0 0
\(359\) 4.31460 + 13.2790i 0.227716 + 0.700837i 0.998005 + 0.0631419i \(0.0201121\pi\)
−0.770289 + 0.637695i \(0.779888\pi\)
\(360\) 0 0
\(361\) 0.855919 + 0.621861i 0.0450484 + 0.0327295i
\(362\) 0 0
\(363\) 4.40842 6.13684i 0.231382 0.322101i
\(364\) 0 0
\(365\) 5.14052 + 3.73481i 0.269067 + 0.195489i
\(366\) 0 0
\(367\) −4.77524 14.6967i −0.249266 0.767161i −0.994906 0.100812i \(-0.967856\pi\)
0.745640 0.666349i \(-0.232144\pi\)
\(368\) 0 0
\(369\) 11.6143 8.43832i 0.604619 0.439281i
\(370\) 0 0
\(371\) 4.16114 12.8067i 0.216035 0.664889i
\(372\) 0 0
\(373\) 6.75664 0.349846 0.174923 0.984582i \(-0.444032\pi\)
0.174923 + 0.984582i \(0.444032\pi\)
\(374\) 0 0
\(375\) −4.86404 −0.251178
\(376\) 0 0
\(377\) 3.18604 9.80562i 0.164089 0.505015i
\(378\) 0 0
\(379\) 18.7117 13.5948i 0.961153 0.698319i 0.00773497 0.999970i \(-0.497538\pi\)
0.953418 + 0.301651i \(0.0975379\pi\)
\(380\) 0 0
\(381\) −1.73514 5.34020i −0.0888937 0.273587i
\(382\) 0 0
\(383\) 19.3862 + 14.0849i 0.990591 + 0.719706i 0.960050 0.279827i \(-0.0902772\pi\)
0.0305402 + 0.999534i \(0.490277\pi\)
\(384\) 0 0
\(385\) 5.39084 + 5.36191i 0.274743 + 0.273268i
\(386\) 0 0
\(387\) −20.2603 14.7200i −1.02989 0.748259i
\(388\) 0 0
\(389\) 4.12049 + 12.6815i 0.208917 + 0.642980i 0.999530 + 0.0306637i \(0.00976210\pi\)
−0.790613 + 0.612316i \(0.790238\pi\)
\(390\) 0 0
\(391\) −0.659721 + 0.479315i −0.0333635 + 0.0242400i
\(392\) 0 0
\(393\) 1.25451 3.86098i 0.0632816 0.194761i
\(394\) 0 0
\(395\) 11.0458 0.555775
\(396\) 0 0
\(397\) 23.8585 1.19742 0.598712 0.800964i \(-0.295679\pi\)
0.598712 + 0.800964i \(0.295679\pi\)
\(398\) 0 0
\(399\) 2.74712 8.45476i 0.137528 0.423267i
\(400\) 0 0
\(401\) −10.9262 + 7.93833i −0.545627 + 0.396421i −0.826171 0.563420i \(-0.809485\pi\)
0.280543 + 0.959841i \(0.409485\pi\)
\(402\) 0 0
\(403\) 8.05291 + 24.7843i 0.401144 + 1.23459i
\(404\) 0 0
\(405\) −3.02056 2.19457i −0.150093 0.109049i
\(406\) 0 0
\(407\) 1.24788 + 7.74389i 0.0618551 + 0.383850i
\(408\) 0 0
\(409\) −30.4632 22.1328i −1.50631 1.09440i −0.967782 0.251790i \(-0.918981\pi\)
−0.538528 0.842608i \(-0.681019\pi\)
\(410\) 0 0
\(411\) −1.76105 5.41997i −0.0868664 0.267347i
\(412\) 0 0
\(413\) 2.06690 1.50169i 0.101706 0.0738934i
\(414\) 0 0
\(415\) −2.54954 + 7.84668i −0.125152 + 0.385178i
\(416\) 0 0
\(417\) 12.4999 0.612124
\(418\) 0 0
\(419\) −13.2409 −0.646862 −0.323431 0.946252i \(-0.604836\pi\)
−0.323431 + 0.946252i \(0.604836\pi\)
\(420\) 0 0
\(421\) −0.716385 + 2.20481i −0.0349145 + 0.107456i −0.966995 0.254796i \(-0.917992\pi\)
0.932080 + 0.362251i \(0.117992\pi\)
\(422\) 0 0
\(423\) −23.5771 + 17.1298i −1.14636 + 0.832879i
\(424\) 0 0
\(425\) 4.22543 + 13.0045i 0.204964 + 0.630813i
\(426\) 0 0
\(427\) −15.4790 11.2461i −0.749081 0.544239i
\(428\) 0 0
\(429\) −12.5872 + 6.45624i −0.607718 + 0.311710i
\(430\) 0 0
\(431\) 3.16135 + 2.29685i 0.152277 + 0.110636i 0.661315 0.750108i \(-0.269999\pi\)
−0.509038 + 0.860744i \(0.669999\pi\)
\(432\) 0 0
\(433\) 6.06021 + 18.6514i 0.291235 + 0.896329i 0.984460 + 0.175608i \(0.0561891\pi\)
−0.693225 + 0.720721i \(0.743811\pi\)
\(434\) 0 0
\(435\) 0.692381 0.503044i 0.0331971 0.0241191i
\(436\) 0 0
\(437\) −0.346354 + 1.06597i −0.0165684 + 0.0509922i
\(438\) 0 0
\(439\) 4.69615 0.224135 0.112068 0.993701i \(-0.464253\pi\)
0.112068 + 0.993701i \(0.464253\pi\)
\(440\) 0 0
\(441\) −5.90273 −0.281082
\(442\) 0 0
\(443\) 12.9938 39.9908i 0.617354 1.90002i 0.264061 0.964506i \(-0.414938\pi\)
0.353292 0.935513i \(-0.385062\pi\)
\(444\) 0 0
\(445\) 9.68175 7.03420i 0.458959 0.333453i
\(446\) 0 0
\(447\) 2.33297 + 7.18015i 0.110346 + 0.339610i
\(448\) 0 0
\(449\) 19.9789 + 14.5156i 0.942865 + 0.685031i 0.949109 0.314949i \(-0.101987\pi\)
−0.00624385 + 0.999981i \(0.501987\pi\)
\(450\) 0 0
\(451\) −8.50507 + 16.8038i −0.400488 + 0.791258i
\(452\) 0 0
\(453\) −9.75564 7.08789i −0.458360 0.333018i
\(454\) 0 0
\(455\) −4.39882 13.5382i −0.206220 0.634680i
\(456\) 0 0
\(457\) −14.4508 + 10.4991i −0.675978 + 0.491127i −0.872021 0.489468i \(-0.837191\pi\)
0.196043 + 0.980595i \(0.437191\pi\)
\(458\) 0 0
\(459\) 3.61633 11.1299i 0.168796 0.519500i
\(460\) 0 0
\(461\) 29.9746 1.39606 0.698028 0.716071i \(-0.254061\pi\)
0.698028 + 0.716071i \(0.254061\pi\)
\(462\) 0 0
\(463\) −29.0359 −1.34941 −0.674705 0.738087i \(-0.735729\pi\)
−0.674705 + 0.738087i \(0.735729\pi\)
\(464\) 0 0
\(465\) −0.668454 + 2.05729i −0.0309988 + 0.0954045i
\(466\) 0 0
\(467\) −22.7821 + 16.5521i −1.05423 + 0.765942i −0.973012 0.230754i \(-0.925881\pi\)
−0.0812167 + 0.996696i \(0.525881\pi\)
\(468\) 0 0
\(469\) 7.32956 + 22.5581i 0.338447 + 1.04163i
\(470\) 0 0
\(471\) 7.35748 + 5.34552i 0.339015 + 0.246309i
\(472\) 0 0
\(473\) 32.4629 + 5.05212i 1.49264 + 0.232297i
\(474\) 0 0
\(475\) 15.2049 + 11.0470i 0.697646 + 0.506870i
\(476\) 0 0
\(477\) −3.44318 10.5970i −0.157653 0.485205i
\(478\) 0 0
\(479\) 17.9735 13.0585i 0.821230 0.596659i −0.0958345 0.995397i \(-0.530552\pi\)
0.917065 + 0.398739i \(0.130552\pi\)
\(480\) 0 0
\(481\) 4.53792 13.9663i 0.206911 0.636807i
\(482\) 0 0
\(483\) −0.555343 −0.0252690
\(484\) 0 0
\(485\) −0.661265 −0.0300265
\(486\) 0 0
\(487\) 4.04875 12.4608i 0.183466 0.564652i −0.816452 0.577413i \(-0.804062\pi\)
0.999919 + 0.0127614i \(0.00406220\pi\)
\(488\) 0 0
\(489\) 12.5539 9.12094i 0.567707 0.412463i
\(490\) 0 0
\(491\) −4.19446 12.9092i −0.189293 0.582585i 0.810703 0.585458i \(-0.199085\pi\)
−0.999996 + 0.00287325i \(0.999085\pi\)
\(492\) 0 0
\(493\) −4.13982 3.00776i −0.186448 0.135463i
\(494\) 0 0
\(495\) 6.21667 + 0.967485i 0.279419 + 0.0434852i
\(496\) 0 0
\(497\) 25.6079 + 18.6052i 1.14867 + 0.834558i
\(498\) 0 0
\(499\) 6.16899 + 18.9862i 0.276162 + 0.849940i 0.988910 + 0.148519i \(0.0474506\pi\)
−0.712747 + 0.701421i \(0.752549\pi\)
\(500\) 0 0
\(501\) −8.94850 + 6.50147i −0.399789 + 0.290464i
\(502\) 0 0
\(503\) −2.50003 + 7.69431i −0.111471 + 0.343072i −0.991195 0.132413i \(-0.957728\pi\)
0.879724 + 0.475485i \(0.157728\pi\)
\(504\) 0 0
\(505\) 8.62740 0.383914
\(506\) 0 0
\(507\) 17.5548 0.779634
\(508\) 0 0
\(509\) −9.74481 + 29.9914i −0.431931 + 1.32935i 0.464268 + 0.885695i \(0.346317\pi\)
−0.896199 + 0.443652i \(0.853683\pi\)
\(510\) 0 0
\(511\) 20.9316 15.2077i 0.925960 0.672750i
\(512\) 0 0
\(513\) −4.97054 15.2977i −0.219455 0.675412i
\(514\) 0 0
\(515\) 6.63397 + 4.81986i 0.292328 + 0.212388i
\(516\) 0 0
\(517\) 17.2653 34.1117i 0.759328 1.50023i
\(518\) 0 0
\(519\) −4.79701 3.48523i −0.210565 0.152985i
\(520\) 0 0
\(521\) 11.4287 + 35.1740i 0.500701 + 1.54100i 0.807880 + 0.589347i \(0.200615\pi\)
−0.307179 + 0.951652i \(0.599385\pi\)
\(522\) 0 0
\(523\) −5.99740 + 4.35736i −0.262248 + 0.190534i −0.711137 0.703053i \(-0.751820\pi\)
0.448890 + 0.893587i \(0.351820\pi\)
\(524\) 0 0
\(525\) −2.87760 + 8.85634i −0.125589 + 0.386522i
\(526\) 0 0
\(527\) 12.9338 0.563405
\(528\) 0 0
\(529\) −22.9300 −0.996956
\(530\) 0 0
\(531\) 0.653270 2.01056i 0.0283495 0.0872508i
\(532\) 0 0
\(533\) 28.5258 20.7252i 1.23559 0.897709i
\(534\) 0 0
\(535\) −0.484127 1.48999i −0.0209306 0.0644179i
\(536\) 0 0
\(537\) 7.50104 + 5.44983i 0.323694 + 0.235177i
\(538\) 0 0
\(539\) 6.89019 3.53411i 0.296782 0.152225i
\(540\) 0 0
\(541\) 9.36829 + 6.80646i 0.402774 + 0.292633i 0.770670 0.637234i \(-0.219922\pi\)
−0.367896 + 0.929867i \(0.619922\pi\)
\(542\) 0 0
\(543\) −1.73933 5.35310i −0.0746417 0.229723i
\(544\) 0 0
\(545\) 2.35963 1.71437i 0.101075 0.0734356i
\(546\) 0 0
\(547\) −3.10133 + 9.54491i −0.132603 + 0.408111i −0.995210 0.0977649i \(-0.968831\pi\)
0.862606 + 0.505876i \(0.168831\pi\)
\(548\) 0 0
\(549\) −15.8319 −0.675690
\(550\) 0 0
\(551\) −7.03331 −0.299629
\(552\) 0 0
\(553\) 13.8987 42.7759i 0.591035 1.81902i
\(554\) 0 0
\(555\) 0.986167 0.716492i 0.0418605 0.0304134i
\(556\) 0 0
\(557\) 9.61618 + 29.5955i 0.407450 + 1.25400i 0.918832 + 0.394650i \(0.129134\pi\)
−0.511381 + 0.859354i \(0.670866\pi\)
\(558\) 0 0
\(559\) −49.7610 36.1535i −2.10467 1.52913i
\(560\) 0 0
\(561\) 1.11699 + 6.93164i 0.0471593 + 0.292654i
\(562\) 0 0
\(563\) 18.0268 + 13.0973i 0.759741 + 0.551984i 0.898831 0.438296i \(-0.144418\pi\)
−0.139090 + 0.990280i \(0.544418\pi\)
\(564\) 0 0
\(565\) 3.15673 + 9.71542i 0.132805 + 0.408731i
\(566\) 0 0
\(567\) −12.2994 + 8.93603i −0.516526 + 0.375278i
\(568\) 0 0
\(569\) −9.66296 + 29.7395i −0.405092 + 1.24675i 0.515726 + 0.856754i \(0.327522\pi\)
−0.920818 + 0.389992i \(0.872478\pi\)
\(570\) 0 0
\(571\) −5.93719 −0.248464 −0.124232 0.992253i \(-0.539647\pi\)
−0.124232 + 0.992253i \(0.539647\pi\)
\(572\) 0 0
\(573\) 13.5254 0.565032
\(574\) 0 0
\(575\) 0.362805 1.11660i 0.0151300 0.0465654i
\(576\) 0 0
\(577\) −5.40311 + 3.92559i −0.224934 + 0.163424i −0.694545 0.719449i \(-0.744394\pi\)
0.469611 + 0.882874i \(0.344394\pi\)
\(578\) 0 0
\(579\) −5.06479 15.5878i −0.210485 0.647808i
\(580\) 0 0
\(581\) 27.1790 + 19.7467i 1.12757 + 0.819230i
\(582\) 0 0
\(583\) 10.3639 + 10.3083i 0.429229 + 0.426926i
\(584\) 0 0
\(585\) −9.52928 6.92343i −0.393987 0.286249i
\(586\) 0 0
\(587\) 12.6950 + 39.0713i 0.523980 + 1.61264i 0.766324 + 0.642454i \(0.222084\pi\)
−0.242345 + 0.970190i \(0.577916\pi\)
\(588\) 0 0
\(589\) 14.3820 10.4491i 0.592600 0.430549i
\(590\) 0 0
\(591\) 4.01189 12.3473i 0.165027 0.507902i
\(592\) 0 0
\(593\) −0.283994 −0.0116622 −0.00583111 0.999983i \(-0.501856\pi\)
−0.00583111 + 0.999983i \(0.501856\pi\)
\(594\) 0 0
\(595\) −7.06496 −0.289635
\(596\) 0 0
\(597\) 2.08412 6.41427i 0.0852974 0.262519i
\(598\) 0 0
\(599\) −33.6857 + 24.4741i −1.37636 + 0.999985i −0.379152 + 0.925334i \(0.623784\pi\)
−0.997210 + 0.0746511i \(0.976216\pi\)
\(600\) 0 0
\(601\) 1.90750 + 5.87068i 0.0778085 + 0.239470i 0.982394 0.186823i \(-0.0598192\pi\)
−0.904585 + 0.426293i \(0.859819\pi\)
\(602\) 0 0
\(603\) 15.8782 + 11.5362i 0.646610 + 0.469790i
\(604\) 0 0
\(605\) −7.83591 + 2.59274i −0.318575 + 0.105410i
\(606\) 0 0
\(607\) 32.9010 + 23.9040i 1.33541 + 0.970232i 0.999599 + 0.0283021i \(0.00901005\pi\)
0.335810 + 0.941930i \(0.390990\pi\)
\(608\) 0 0
\(609\) −1.07688 3.31428i −0.0436372 0.134302i
\(610\) 0 0
\(611\) −57.9075 + 42.0722i −2.34269 + 1.70206i
\(612\) 0 0
\(613\) 5.33390 16.4160i 0.215434 0.663038i −0.783688 0.621154i \(-0.786664\pi\)
0.999122 0.0418838i \(-0.0133359\pi\)
\(614\) 0 0
\(615\) 2.92684 0.118022
\(616\) 0 0
\(617\) 13.2729 0.534346 0.267173 0.963649i \(-0.413911\pi\)
0.267173 + 0.963649i \(0.413911\pi\)
\(618\) 0 0
\(619\) 3.96107 12.1909i 0.159209 0.489994i −0.839354 0.543585i \(-0.817067\pi\)
0.998563 + 0.0535908i \(0.0170667\pi\)
\(620\) 0 0
\(621\) −0.812916 + 0.590618i −0.0326212 + 0.0237007i
\(622\) 0 0
\(623\) −15.0582 46.3445i −0.603296 1.85675i
\(624\) 0 0
\(625\) −13.6496 9.91704i −0.545985 0.396682i
\(626\) 0 0
\(627\) 6.84209 + 6.80537i 0.273247 + 0.271780i
\(628\) 0 0
\(629\) −5.89640 4.28399i −0.235105 0.170814i
\(630\) 0 0
\(631\) −1.99094 6.12748i −0.0792581 0.243931i 0.903574 0.428431i \(-0.140934\pi\)
−0.982833 + 0.184500i \(0.940934\pi\)
\(632\) 0 0
\(633\) −7.23496 + 5.25650i −0.287564 + 0.208927i
\(634\) 0 0
\(635\) −1.89533 + 5.83321i −0.0752137 + 0.231484i
\(636\) 0 0
\(637\) −14.4976 −0.574416
\(638\) 0 0
\(639\) 26.1917 1.03613
\(640\) 0 0
\(641\) −6.20170 + 19.0869i −0.244953 + 0.753887i 0.750692 + 0.660653i \(0.229720\pi\)
−0.995644 + 0.0932338i \(0.970280\pi\)
\(642\) 0 0
\(643\) 26.2589 19.0782i 1.03555 0.752371i 0.0661375 0.997811i \(-0.478932\pi\)
0.969412 + 0.245440i \(0.0789324\pi\)
\(644\) 0 0
\(645\) −1.57773 4.85576i −0.0621231 0.191195i
\(646\) 0 0
\(647\) −12.7925 9.29429i −0.502925 0.365396i 0.307208 0.951642i \(-0.400605\pi\)
−0.810133 + 0.586246i \(0.800605\pi\)
\(648\) 0 0
\(649\) 0.441217 + 2.73803i 0.0173193 + 0.107477i
\(650\) 0 0
\(651\) 7.12595 + 5.17730i 0.279288 + 0.202915i
\(652\) 0 0
\(653\) −0.927899 2.85578i −0.0363115 0.111755i 0.931258 0.364361i \(-0.118713\pi\)
−0.967569 + 0.252606i \(0.918713\pi\)
\(654\) 0 0
\(655\) −3.58756 + 2.60652i −0.140178 + 0.101845i
\(656\) 0 0
\(657\) 6.61570 20.3610i 0.258103 0.794360i
\(658\) 0 0
\(659\) −3.17515 −0.123686 −0.0618431 0.998086i \(-0.519698\pi\)
−0.0618431 + 0.998086i \(0.519698\pi\)
\(660\) 0 0
\(661\) −30.6169 −1.19086 −0.595430 0.803408i \(-0.703018\pi\)
−0.595430 + 0.803408i \(0.703018\pi\)
\(662\) 0 0
\(663\) 4.06194 12.5014i 0.157753 0.485513i
\(664\) 0 0
\(665\) −7.85603 + 5.70774i −0.304644 + 0.221337i
\(666\) 0 0
\(667\) 0.135772 + 0.417862i 0.00525710 + 0.0161797i
\(668\) 0 0
\(669\) 1.94338 + 1.41195i 0.0751354 + 0.0545891i
\(670\) 0 0
\(671\) 18.4804 9.47897i 0.713429 0.365932i
\(672\) 0 0
\(673\) 14.1071 + 10.2494i 0.543787 + 0.395084i 0.825489 0.564417i \(-0.190899\pi\)
−0.281703 + 0.959502i \(0.590899\pi\)
\(674\) 0 0
\(675\) 5.20663 + 16.0244i 0.200403 + 0.616778i
\(676\) 0 0
\(677\) −32.2671 + 23.4434i −1.24012 + 0.901003i −0.997607 0.0691445i \(-0.977973\pi\)
−0.242517 + 0.970147i \(0.577973\pi\)
\(678\) 0 0
\(679\) −0.832059 + 2.56081i −0.0319315 + 0.0982750i
\(680\) 0 0
\(681\) −7.50705 −0.287671
\(682\) 0 0
\(683\) 37.9702 1.45289 0.726445 0.687225i \(-0.241171\pi\)
0.726445 + 0.687225i \(0.241171\pi\)
\(684\) 0 0
\(685\) −1.92364 + 5.92034i −0.0734984 + 0.226205i
\(686\) 0 0
\(687\) −6.49549 + 4.71925i −0.247819 + 0.180051i
\(688\) 0 0
\(689\) −8.45675 26.0272i −0.322177 0.991558i
\(690\) 0 0
\(691\) −24.5820 17.8599i −0.935145 0.679423i 0.0121022 0.999927i \(-0.496148\pi\)
−0.947247 + 0.320504i \(0.896148\pi\)
\(692\) 0 0
\(693\) 11.5690 22.8573i 0.439470 0.868276i
\(694\) 0 0
\(695\) −11.0463 8.02558i −0.419009 0.304428i
\(696\) 0 0
\(697\) −5.40777 16.6434i −0.204834 0.630414i
\(698\) 0 0
\(699\) −2.62229 + 1.90520i −0.0991840 + 0.0720614i
\(700\) 0 0
\(701\) 4.72041 14.5279i 0.178287 0.548712i −0.821481 0.570236i \(-0.806852\pi\)
0.999768 + 0.0215239i \(0.00685180\pi\)
\(702\) 0 0
\(703\) −10.0176 −0.377822
\(704\) 0 0
\(705\) −5.94149 −0.223770
\(706\) 0 0
\(707\) 10.8557 33.4104i 0.408271 1.25653i
\(708\) 0 0
\(709\) −28.2924 + 20.5556i −1.06254 + 0.771983i −0.974557 0.224139i \(-0.928043\pi\)
−0.0879861 + 0.996122i \(0.528043\pi\)
\(710\) 0 0
\(711\) −11.5007 35.3955i −0.431310 1.32743i
\(712\) 0 0
\(713\) −0.898434 0.652750i −0.0336466 0.0244457i
\(714\) 0 0
\(715\) 15.2687 + 2.37623i 0.571016 + 0.0888658i
\(716\) 0 0
\(717\) 0.673152 + 0.489074i 0.0251393 + 0.0182648i
\(718\) 0 0
\(719\) −3.25274 10.0109i −0.121307 0.373344i 0.871903 0.489678i \(-0.162886\pi\)
−0.993210 + 0.116334i \(0.962886\pi\)
\(720\) 0 0
\(721\) 27.0128 19.6259i 1.00601 0.730908i
\(722\) 0 0
\(723\) −3.73916 + 11.5079i −0.139061 + 0.427985i
\(724\) 0 0
\(725\) 7.36738 0.273618
\(726\) 0 0
\(727\) −8.67301 −0.321664 −0.160832 0.986982i \(-0.551418\pi\)
−0.160832 + 0.986982i \(0.551418\pi\)
\(728\) 0 0
\(729\) −1.46916 + 4.52160i −0.0544132 + 0.167467i
\(730\) 0 0
\(731\) −24.6970 + 17.9435i −0.913453 + 0.663663i
\(732\) 0 0
\(733\) −13.1926 40.6027i −0.487280 1.49969i −0.828651 0.559766i \(-0.810891\pi\)
0.341370 0.939929i \(-0.389109\pi\)
\(734\) 0 0
\(735\) −0.973582 0.707349i −0.0359111 0.0260909i
\(736\) 0 0
\(737\) −25.4415 3.95940i −0.937149 0.145846i
\(738\) 0 0
\(739\) 33.3721 + 24.2463i 1.22761 + 0.891913i 0.996709 0.0810632i \(-0.0258316\pi\)
0.230904 + 0.972976i \(0.425832\pi\)
\(740\) 0 0
\(741\) −5.58302 17.1828i −0.205097 0.631225i
\(742\) 0 0
\(743\) −28.1799 + 20.4739i −1.03382 + 0.751115i −0.969070 0.246786i \(-0.920626\pi\)
−0.0647518 + 0.997901i \(0.520626\pi\)
\(744\) 0 0
\(745\) 2.54836 7.84304i 0.0933646 0.287347i
\(746\) 0 0
\(747\) 27.7987 1.01710
\(748\) 0 0
\(749\) −6.37930 −0.233094
\(750\) 0 0
\(751\) 11.0813 34.1047i 0.404362 1.24450i −0.517064 0.855947i \(-0.672975\pi\)
0.921427 0.388553i \(-0.127025\pi\)
\(752\) 0 0
\(753\) 0.442570 0.321546i 0.0161281 0.0117178i
\(754\) 0 0
\(755\) 4.07034 + 12.5272i 0.148135 + 0.455913i
\(756\) 0 0
\(757\) −8.92416 6.48378i −0.324354 0.235657i 0.413677 0.910424i \(-0.364244\pi\)
−0.738031 + 0.674767i \(0.764244\pi\)
\(758\) 0 0
\(759\) 0.272239 0.537873i 0.00988167 0.0195235i
\(760\) 0 0
\(761\) −7.07652 5.14139i −0.256524 0.186375i 0.452089 0.891973i \(-0.350679\pi\)
−0.708613 + 0.705597i \(0.750679\pi\)
\(762\) 0 0
\(763\) −3.66999 11.2951i −0.132862 0.408909i
\(764\) 0 0
\(765\) −4.72950 + 3.43619i −0.170996 + 0.124236i
\(766\) 0 0
\(767\) 1.60449 4.93810i 0.0579347 0.178305i
\(768\) 0 0
\(769\) 9.61551 0.346744 0.173372 0.984856i \(-0.444534\pi\)
0.173372 + 0.984856i \(0.444534\pi\)
\(770\) 0 0
\(771\) −10.5899 −0.381384
\(772\) 0 0
\(773\) −2.03122 + 6.25146i −0.0730580 + 0.224849i −0.980917 0.194426i \(-0.937716\pi\)
0.907859 + 0.419275i \(0.137716\pi\)
\(774\) 0 0
\(775\) −15.0651 + 10.9454i −0.541155 + 0.393172i
\(776\) 0 0
\(777\) −1.53381 4.72058i −0.0550251 0.169350i
\(778\) 0 0
\(779\) −19.4594 14.1381i −0.697205 0.506549i
\(780\) 0 0
\(781\) −30.5734 + 15.6817i −1.09400 + 0.561134i
\(782\) 0 0
\(783\) −5.10114 3.70620i −0.182300 0.132449i
\(784\) 0 0
\(785\) −3.06976 9.44775i −0.109564 0.337205i
\(786\) 0 0
\(787\) 1.30052 0.944881i 0.0463584 0.0336814i −0.564365 0.825526i \(-0.690879\pi\)
0.610723 + 0.791844i \(0.290879\pi\)
\(788\) 0 0
\(789\) −4.90962 + 15.1102i −0.174787 + 0.537939i
\(790\) 0 0
\(791\) 41.5960 1.47898
\(792\) 0 0
\(793\) −38.8845 −1.38083
\(794\) 0 0
\(795\) 0.701976 2.16046i 0.0248965 0.0766236i
\(796\) 0 0
\(797\) −15.9525 + 11.5902i −0.565068 + 0.410546i −0.833310 0.552805i \(-0.813557\pi\)
0.268242 + 0.963352i \(0.413557\pi\)
\(798\) 0 0
\(799\) 10.9778 + 33.7862i 0.388366 + 1.19527i
\(800\) 0 0
\(801\) −32.6211 23.7006i −1.15261 0.837419i
\(802\) 0 0
\(803\) 4.46823 + 27.7282i 0.157680 + 0.978508i
\(804\) 0 0
\(805\) 0.490761 + 0.356558i 0.0172970 + 0.0125670i
\(806\) 0 0
\(807\) 5.12115 + 15.7613i 0.180273 + 0.554823i
\(808\) 0 0
\(809\) 3.70085 2.68882i 0.130115 0.0945340i −0.520824 0.853664i \(-0.674375\pi\)
0.650939 + 0.759130i \(0.274375\pi\)
\(810\) 0 0
\(811\) 1.08998 3.35461i 0.0382743 0.117796i −0.930094 0.367322i \(-0.880275\pi\)
0.968368 + 0.249526i \(0.0802747\pi\)
\(812\) 0 0
\(813\) −3.47973 −0.122040
\(814\) 0 0
\(815\) −16.9501 −0.593735
\(816\) 0 0
\(817\) −12.9660 + 39.9052i −0.453622 + 1.39611i
\(818\) 0 0
\(819\) −38.8022 + 28.1914i −1.35586 + 0.985088i
\(820\) 0 0
\(821\) −0.762466 2.34663i −0.0266102 0.0818979i 0.936869 0.349679i \(-0.113709\pi\)
−0.963480 + 0.267781i \(0.913709\pi\)
\(822\) 0 0
\(823\) −6.02028 4.37399i −0.209854 0.152468i 0.477894 0.878418i \(-0.341400\pi\)
−0.687748 + 0.725950i \(0.741400\pi\)
\(824\) 0 0
\(825\) −7.16707 7.12861i −0.249525 0.248186i
\(826\) 0 0
\(827\) −8.74279 6.35201i −0.304016 0.220881i 0.425308 0.905049i \(-0.360166\pi\)
−0.729325 + 0.684168i \(0.760166\pi\)
\(828\) 0 0
\(829\) −6.89204 21.2115i −0.239371 0.736707i −0.996512 0.0834553i \(-0.973404\pi\)
0.757141 0.653252i \(-0.226596\pi\)
\(830\) 0 0
\(831\) 3.26053 2.36891i 0.113106 0.0821767i
\(832\) 0 0
\(833\) −2.22348 + 6.84318i −0.0770391 + 0.237102i
\(834\) 0 0
\(835\) 12.0821 0.418119
\(836\) 0 0
\(837\) 15.9372 0.550869
\(838\) 0 0
\(839\) 9.86396 30.3581i 0.340542 1.04808i −0.623386 0.781914i \(-0.714243\pi\)
0.963927 0.266165i \(-0.0857566\pi\)
\(840\) 0 0
\(841\) 21.2310 15.4252i 0.732102 0.531904i
\(842\) 0 0
\(843\) −5.94000 18.2814i −0.204584 0.629646i
\(844\) 0 0
\(845\) −15.5133 11.2710i −0.533672 0.387736i
\(846\) 0 0
\(847\) 0.180846 + 33.6077i 0.00621394 + 1.15478i
\(848\) 0 0
\(849\) 4.22478 + 3.06948i 0.144994 + 0.105344i
\(850\) 0 0
\(851\) 0.193381 + 0.595166i 0.00662903 + 0.0204020i
\(852\) 0 0
\(853\) 13.5827 9.86844i 0.465064 0.337889i −0.330450 0.943823i \(-0.607201\pi\)
0.795515 + 0.605934i \(0.207201\pi\)
\(854\) 0 0
\(855\) −2.48300 + 7.64188i −0.0849167 + 0.261347i
\(856\) 0 0
\(857\) −11.2278 −0.383534 −0.191767 0.981440i \(-0.561422\pi\)
−0.191767 + 0.981440i \(0.561422\pi\)
\(858\) 0 0
\(859\) −50.2349 −1.71399 −0.856996 0.515324i \(-0.827672\pi\)
−0.856996 + 0.515324i \(0.827672\pi\)
\(860\) 0 0
\(861\) 3.68279 11.3345i 0.125509 0.386278i
\(862\) 0 0
\(863\) 5.32058 3.86562i 0.181115 0.131587i −0.493534 0.869726i \(-0.664295\pi\)
0.674649 + 0.738139i \(0.264295\pi\)
\(864\) 0 0
\(865\) 2.00146 + 6.15985i 0.0680515 + 0.209441i
\(866\) 0 0
\(867\) 4.16948 + 3.02930i 0.141603 + 0.102881i
\(868\) 0 0
\(869\) 34.6168 + 34.4310i 1.17429 + 1.16799i
\(870\) 0 0
\(871\) 38.9982 + 28.3339i 1.32140 + 0.960056i
\(872\) 0 0
\(873\) 0.688498 + 2.11898i 0.0233021 + 0.0717166i
\(874\) 0 0
\(875\) 17.5025 12.7163i 0.591694 0.429891i
\(876\) 0 0
\(877\) 13.6684 42.0671i 0.461550 1.42050i −0.401720 0.915762i \(-0.631588\pi\)
0.863270 0.504742i \(-0.168412\pi\)
\(878\) 0 0
\(879\) 2.70736 0.0913170
\(880\) 0 0
\(881\) 37.0273 1.24748 0.623741 0.781631i \(-0.285612\pi\)
0.623741 + 0.781631i \(0.285612\pi\)
\(882\) 0 0
\(883\) 8.59827 26.4627i 0.289355 0.890542i −0.695705 0.718328i \(-0.744908\pi\)
0.985059 0.172214i \(-0.0550922\pi\)
\(884\) 0 0
\(885\) 0.348683 0.253333i 0.0117208 0.00851569i
\(886\) 0 0
\(887\) −3.66741 11.2871i −0.123140 0.378985i 0.870418 0.492314i \(-0.163849\pi\)
−0.993558 + 0.113329i \(0.963849\pi\)
\(888\) 0 0
\(889\) 20.2048 + 14.6797i 0.677648 + 0.492340i
\(890\) 0 0
\(891\) −2.62552 16.2931i −0.0879583 0.545838i
\(892\) 0 0
\(893\) 39.5026 + 28.7003i 1.32190 + 0.960420i
\(894\) 0 0
\(895\) −3.12966 9.63210i −0.104613 0.321965i
\(896\) 0 0
\(897\) −0.913085 + 0.663395i −0.0304870 + 0.0221501i
\(898\) 0 0
\(899\) 2.15344 6.62760i 0.0718212 0.221043i
\(900\) 0 0
\(901\) −13.5824 −0.452496
\(902\) 0 0
\(903\) −20.7896 −0.691835
\(904\) 0 0
\(905\) −1.89990 + 5.84730i −0.0631549 + 0.194371i
\(906\) 0 0
\(907\) −8.23782 + 5.98513i −0.273532 + 0.198733i −0.716092 0.698006i \(-0.754071\pi\)
0.442559 + 0.896739i \(0.354071\pi\)
\(908\) 0 0
\(909\) −8.98269 27.6459i −0.297937 0.916956i
\(910\) 0 0
\(911\) 13.0498 + 9.48123i 0.432359 + 0.314127i 0.782591 0.622536i \(-0.213897\pi\)
−0.350233 + 0.936663i \(0.613897\pi\)
\(912\) 0 0
\(913\) −32.4491 + 16.6438i −1.07391 + 0.550828i
\(914\) 0 0
\(915\) −2.61128 1.89721i −0.0863262 0.0627197i
\(916\) 0 0
\(917\) 5.57982 + 17.1729i 0.184262 + 0.567099i
\(918\) 0 0
\(919\) −15.2546 + 11.0831i −0.503203 + 0.365598i −0.810239 0.586100i \(-0.800663\pi\)
0.307036 + 0.951698i \(0.400663\pi\)
\(920\) 0 0
\(921\) −2.28300 + 7.02635i −0.0752273 + 0.231526i
\(922\) 0 0
\(923\) 64.3291 2.11742
\(924\) 0 0
\(925\) 10.4935 0.345023
\(926\) 0 0
\(927\) 8.53773 26.2764i 0.280416 0.863031i
\(928\) 0 0
\(929\) −10.2972 + 7.48135i −0.337840 + 0.245455i −0.743750 0.668458i \(-0.766955\pi\)
0.405910 + 0.913913i \(0.366955\pi\)
\(930\) 0 0
\(931\) 3.05612 + 9.40576i 0.100160 + 0.308261i
\(932\) 0 0
\(933\) 14.4953 + 10.5314i 0.474555 + 0.344784i
\(934\) 0 0
\(935\) 3.46337 6.84270i 0.113264 0.223780i
\(936\) 0 0
\(937\) −7.21762 5.24391i −0.235789 0.171311i 0.463616 0.886036i \(-0.346552\pi\)
−0.699405 + 0.714725i \(0.746552\pi\)
\(938\) 0 0
\(939\) −0.638586 1.96537i −0.0208395 0.0641373i
\(940\) 0 0
\(941\) 11.1981 8.13587i 0.365047 0.265222i −0.390107 0.920769i \(-0.627562\pi\)
0.755154 + 0.655548i \(0.227562\pi\)
\(942\) 0 0
\(943\) −0.464324 + 1.42904i −0.0151205 + 0.0465360i
\(944\) 0 0
\(945\) −8.70553 −0.283191
\(946\) 0 0
\(947\) −34.5104 −1.12144 −0.560719 0.828006i \(-0.689475\pi\)
−0.560719 + 0.828006i \(0.689475\pi\)
\(948\) 0 0
\(949\) 16.2487 50.0084i 0.527456 1.62334i
\(950\) 0 0
\(951\) −18.0047 + 13.0812i −0.583843 + 0.424187i
\(952\) 0 0
\(953\) 8.13241 + 25.0290i 0.263435 + 0.810769i 0.992050 + 0.125845i \(0.0401643\pi\)
−0.728615 + 0.684923i \(0.759836\pi\)
\(954\) 0 0
\(955\) −11.9525 8.68399i −0.386773 0.281007i
\(956\) 0 0
\(957\) 3.73792 + 0.581724i 0.120830 + 0.0188045i
\(958\) 0 0
\(959\) 20.5066 + 14.8989i 0.662193 + 0.481112i
\(960\) 0 0
\(961\) −4.13658 12.7311i −0.133438 0.410680i
\(962\) 0 0
\(963\) −4.27050 + 3.10270i −0.137615 + 0.0999831i
\(964\) 0 0
\(965\) −5.53238 + 17.0269i −0.178094 + 0.548116i
\(966\) 0 0
\(967\) −4.18054 −0.134437 −0.0672185 0.997738i \(-0.521412\pi\)
−0.0672185 + 0.997738i \(0.521412\pi\)
\(968\) 0 0
\(969\) −8.96690 −0.288058
\(970\) 0 0
\(971\) 5.41506 16.6658i 0.173777 0.534832i −0.825798 0.563966i \(-0.809275\pi\)
0.999576 + 0.0291339i \(0.00927491\pi\)
\(972\) 0 0
\(973\) −44.9792 + 32.6793i −1.44197 + 1.04765i
\(974\) 0 0
\(975\) 5.84820 + 17.9989i 0.187292 + 0.576426i
\(976\) 0 0
\(977\) 17.7304 + 12.8819i 0.567246 + 0.412128i 0.834104 0.551608i \(-0.185985\pi\)
−0.266858 + 0.963736i \(0.585985\pi\)
\(978\) 0 0
\(979\) 52.2684 + 8.13440i 1.67050 + 0.259977i
\(980\) 0 0
\(981\) −7.95039 5.77629i −0.253836 0.184423i
\(982\) 0 0
\(983\) 10.3243 + 31.7749i 0.329293 + 1.01346i 0.969465 + 0.245229i \(0.0788632\pi\)
−0.640172 + 0.768232i \(0.721137\pi\)
\(984\) 0 0
\(985\) −11.4730 + 8.33559i −0.365559 + 0.265594i
\(986\) 0 0
\(987\) −7.47608 + 23.0090i −0.237966 + 0.732385i
\(988\) 0 0
\(989\) 2.62114 0.0833473
\(990\) 0 0
\(991\) −17.4811 −0.555307 −0.277653 0.960681i \(-0.589557\pi\)
−0.277653 + 0.960681i \(0.589557\pi\)
\(992\) 0 0
\(993\) −4.09077 + 12.5901i −0.129817 + 0.399534i
\(994\) 0 0
\(995\) −5.96004 + 4.33022i −0.188946 + 0.137277i
\(996\) 0 0
\(997\) 15.7737 + 48.5464i 0.499558 + 1.53748i 0.809732 + 0.586800i \(0.199613\pi\)
−0.310174 + 0.950680i \(0.600387\pi\)
\(998\) 0 0
\(999\) −7.26562 5.27878i −0.229874 0.167013i
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 352.2.m.f.257.3 yes 12
4.3 odd 2 352.2.m.e.257.1 12
8.3 odd 2 704.2.m.m.257.3 12
8.5 even 2 704.2.m.n.257.1 12
11.3 even 5 inner 352.2.m.f.289.3 yes 12
11.5 even 5 3872.2.a.bn.1.5 6
11.6 odd 10 3872.2.a.bo.1.5 6
44.3 odd 10 352.2.m.e.289.1 yes 12
44.27 odd 10 3872.2.a.bq.1.2 6
44.39 even 10 3872.2.a.bp.1.2 6
88.3 odd 10 704.2.m.m.641.3 12
88.5 even 10 7744.2.a.dv.1.2 6
88.27 odd 10 7744.2.a.du.1.5 6
88.61 odd 10 7744.2.a.dw.1.2 6
88.69 even 10 704.2.m.n.641.1 12
88.83 even 10 7744.2.a.dt.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.m.e.257.1 12 4.3 odd 2
352.2.m.e.289.1 yes 12 44.3 odd 10
352.2.m.f.257.3 yes 12 1.1 even 1 trivial
352.2.m.f.289.3 yes 12 11.3 even 5 inner
704.2.m.m.257.3 12 8.3 odd 2
704.2.m.m.641.3 12 88.3 odd 10
704.2.m.n.257.1 12 8.5 even 2
704.2.m.n.641.1 12 88.69 even 10
3872.2.a.bn.1.5 6 11.5 even 5
3872.2.a.bo.1.5 6 11.6 odd 10
3872.2.a.bp.1.2 6 44.39 even 10
3872.2.a.bq.1.2 6 44.27 odd 10
7744.2.a.dt.1.5 6 88.83 even 10
7744.2.a.du.1.5 6 88.27 odd 10
7744.2.a.dv.1.2 6 88.5 even 10
7744.2.a.dw.1.2 6 88.61 odd 10