Properties

Label 804.2.o.d.365.7
Level $804$
Weight $2$
Character 804.365
Analytic conductor $6.420$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [804,2,Mod(365,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.365");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.o (of order \(6\), degree \(2\), 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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 365.7
Character \(\chi\) \(=\) 804.365
Dual form 804.2.o.d.641.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.869231 + 1.49814i) q^{3} +2.66956 q^{5} +(0.767031 + 0.442845i) q^{7} +(-1.48888 - 2.60447i) q^{9} +O(q^{10})\) \(q+(-0.869231 + 1.49814i) q^{3} +2.66956 q^{5} +(0.767031 + 0.442845i) q^{7} +(-1.48888 - 2.60447i) q^{9} +(-2.39703 + 4.15178i) q^{11} +(-5.90147 + 3.40722i) q^{13} +(-2.32046 + 3.99938i) q^{15} +(5.55864 - 3.20928i) q^{17} +(2.50179 + 4.33322i) q^{19} +(-1.33017 + 0.764188i) q^{21} +(-1.46659 + 0.846736i) q^{23} +2.12654 q^{25} +(5.19605 + 0.0333319i) q^{27} +(5.11405 + 2.95260i) q^{29} +(-5.28723 - 3.05259i) q^{31} +(-4.13639 - 7.19995i) q^{33} +(2.04763 + 1.18220i) q^{35} +(0.639895 + 1.10833i) q^{37} +(0.0252378 - 11.8029i) q^{39} +(-1.45058 + 2.51248i) q^{41} +12.1986i q^{43} +(-3.97464 - 6.95278i) q^{45} +(3.58366 + 2.06903i) q^{47} +(-3.10778 - 5.38283i) q^{49} +(-0.0237716 + 11.1173i) q^{51} -5.25036 q^{53} +(-6.39901 + 11.0834i) q^{55} +(-8.66643 - 0.0185311i) q^{57} -0.587414i q^{59} +(-5.85867 + 3.38250i) q^{61} +(0.0113630 - 2.65705i) q^{63} +(-15.7543 + 9.09576i) q^{65} +(2.91744 - 7.64778i) q^{67} +(0.00627189 - 2.93317i) q^{69} +(1.38128 + 0.797483i) q^{71} +(3.71930 + 6.44202i) q^{73} +(-1.84845 + 3.18586i) q^{75} +(-3.67719 + 2.12303i) q^{77} +(10.0799 + 5.81963i) q^{79} +(-4.56650 + 7.75546i) q^{81} +(-7.12573 + 4.11404i) q^{83} +(14.8391 - 8.56736i) q^{85} +(-8.86870 + 5.09509i) q^{87} -9.09197i q^{89} -6.03548 q^{91} +(9.16904 - 5.26764i) q^{93} +(6.67867 + 11.5678i) q^{95} +(7.83565 - 4.52391i) q^{97} +(14.3820 + 0.0615055i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 4 q^{9} - 36 q^{13} + 18 q^{15} + 16 q^{21} + 76 q^{25} + 6 q^{31} + 4 q^{33} + 42 q^{37} - 21 q^{39} + 2 q^{49} + 18 q^{51} + 20 q^{55} + 18 q^{57} - 24 q^{61} - 12 q^{63} - 8 q^{67} + 3 q^{69} + 14 q^{73} + 72 q^{79} - 12 q^{81} - 18 q^{85} - 21 q^{87} - 68 q^{91} + 9 q^{93} - 48 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −0.869231 + 1.49814i −0.501851 + 0.864954i
\(4\) 0 0
\(5\) 2.66956 1.19386 0.596931 0.802292i \(-0.296386\pi\)
0.596931 + 0.802292i \(0.296386\pi\)
\(6\) 0 0
\(7\) 0.767031 + 0.442845i 0.289910 + 0.167380i 0.637901 0.770118i \(-0.279803\pi\)
−0.347991 + 0.937498i \(0.613136\pi\)
\(8\) 0 0
\(9\) −1.48888 2.60447i −0.496292 0.868156i
\(10\) 0 0
\(11\) −2.39703 + 4.15178i −0.722732 + 1.25181i 0.237169 + 0.971468i \(0.423780\pi\)
−0.959901 + 0.280340i \(0.909553\pi\)
\(12\) 0 0
\(13\) −5.90147 + 3.40722i −1.63677 + 0.944992i −0.654840 + 0.755767i \(0.727264\pi\)
−0.981934 + 0.189225i \(0.939402\pi\)
\(14\) 0 0
\(15\) −2.32046 + 3.99938i −0.599141 + 1.03264i
\(16\) 0 0
\(17\) 5.55864 3.20928i 1.34817 0.778365i 0.360178 0.932884i \(-0.382716\pi\)
0.987990 + 0.154519i \(0.0493827\pi\)
\(18\) 0 0
\(19\) 2.50179 + 4.33322i 0.573950 + 0.994110i 0.996155 + 0.0876097i \(0.0279228\pi\)
−0.422205 + 0.906500i \(0.638744\pi\)
\(20\) 0 0
\(21\) −1.33017 + 0.764188i −0.290268 + 0.166760i
\(22\) 0 0
\(23\) −1.46659 + 0.846736i −0.305805 + 0.176557i −0.645048 0.764142i \(-0.723163\pi\)
0.339243 + 0.940699i \(0.389829\pi\)
\(24\) 0 0
\(25\) 2.12654 0.425308
\(26\) 0 0
\(27\) 5.19605 + 0.0333319i 0.999979 + 0.00641474i
\(28\) 0 0
\(29\) 5.11405 + 2.95260i 0.949654 + 0.548283i 0.892974 0.450109i \(-0.148615\pi\)
0.0566808 + 0.998392i \(0.481948\pi\)
\(30\) 0 0
\(31\) −5.28723 3.05259i −0.949615 0.548261i −0.0566539 0.998394i \(-0.518043\pi\)
−0.892961 + 0.450133i \(0.851376\pi\)
\(32\) 0 0
\(33\) −4.13639 7.19995i −0.720053 1.25335i
\(34\) 0 0
\(35\) 2.04763 + 1.18220i 0.346113 + 0.199829i
\(36\) 0 0
\(37\) 0.639895 + 1.10833i 0.105198 + 0.182208i 0.913819 0.406121i \(-0.133119\pi\)
−0.808621 + 0.588330i \(0.799786\pi\)
\(38\) 0 0
\(39\) 0.0252378 11.8029i 0.00404127 1.88998i
\(40\) 0 0
\(41\) −1.45058 + 2.51248i −0.226543 + 0.392384i −0.956781 0.290809i \(-0.906076\pi\)
0.730238 + 0.683192i \(0.239409\pi\)
\(42\) 0 0
\(43\) 12.1986i 1.86027i 0.367218 + 0.930135i \(0.380310\pi\)
−0.367218 + 0.930135i \(0.619690\pi\)
\(44\) 0 0
\(45\) −3.97464 6.95278i −0.592504 1.03646i
\(46\) 0 0
\(47\) 3.58366 + 2.06903i 0.522730 + 0.301798i 0.738051 0.674745i \(-0.235746\pi\)
−0.215321 + 0.976543i \(0.569080\pi\)
\(48\) 0 0
\(49\) −3.10778 5.38283i −0.443968 0.768975i
\(50\) 0 0
\(51\) −0.0237716 + 11.1173i −0.00332869 + 1.55673i
\(52\) 0 0
\(53\) −5.25036 −0.721192 −0.360596 0.932722i \(-0.617427\pi\)
−0.360596 + 0.932722i \(0.617427\pi\)
\(54\) 0 0
\(55\) −6.39901 + 11.0834i −0.862842 + 1.49449i
\(56\) 0 0
\(57\) −8.66643 0.0185311i −1.14790 0.00245451i
\(58\) 0 0
\(59\) 0.587414i 0.0764748i −0.999269 0.0382374i \(-0.987826\pi\)
0.999269 0.0382374i \(-0.0121743\pi\)
\(60\) 0 0
\(61\) −5.85867 + 3.38250i −0.750126 + 0.433085i −0.825739 0.564052i \(-0.809242\pi\)
0.0756136 + 0.997137i \(0.475908\pi\)
\(62\) 0 0
\(63\) 0.0113630 2.65705i 0.00143160 0.334757i
\(64\) 0 0
\(65\) −15.7543 + 9.09576i −1.95408 + 1.12819i
\(66\) 0 0
\(67\) 2.91744 7.64778i 0.356422 0.934325i
\(68\) 0 0
\(69\) 0.00627189 2.93317i 0.000755047 0.353112i
\(70\) 0 0
\(71\) 1.38128 + 0.797483i 0.163928 + 0.0946439i 0.579720 0.814816i \(-0.303162\pi\)
−0.415792 + 0.909460i \(0.636495\pi\)
\(72\) 0 0
\(73\) 3.71930 + 6.44202i 0.435312 + 0.753982i 0.997321 0.0731491i \(-0.0233049\pi\)
−0.562009 + 0.827131i \(0.689972\pi\)
\(74\) 0 0
\(75\) −1.84845 + 3.18586i −0.213441 + 0.367872i
\(76\) 0 0
\(77\) −3.67719 + 2.12303i −0.419055 + 0.241941i
\(78\) 0 0
\(79\) 10.0799 + 5.81963i 1.13408 + 0.654760i 0.944957 0.327194i \(-0.106103\pi\)
0.189120 + 0.981954i \(0.439437\pi\)
\(80\) 0 0
\(81\) −4.56650 + 7.75546i −0.507389 + 0.861717i
\(82\) 0 0
\(83\) −7.12573 + 4.11404i −0.782150 + 0.451575i −0.837192 0.546909i \(-0.815804\pi\)
0.0550415 + 0.998484i \(0.482471\pi\)
\(84\) 0 0
\(85\) 14.8391 8.56736i 1.60953 0.929261i
\(86\) 0 0
\(87\) −8.86870 + 5.09509i −0.950825 + 0.546251i
\(88\) 0 0
\(89\) 9.09197i 0.963747i −0.876241 0.481874i \(-0.839956\pi\)
0.876241 0.481874i \(-0.160044\pi\)
\(90\) 0 0
\(91\) −6.03548 −0.632691
\(92\) 0 0
\(93\) 9.16904 5.26764i 0.950785 0.546229i
\(94\) 0 0
\(95\) 6.67867 + 11.5678i 0.685217 + 1.18683i
\(96\) 0 0
\(97\) 7.83565 4.52391i 0.795589 0.459334i −0.0463372 0.998926i \(-0.514755\pi\)
0.841927 + 0.539592i \(0.181422\pi\)
\(98\) 0 0
\(99\) 14.3820 + 0.0615055i 1.44545 + 0.00618154i
\(100\) 0 0
\(101\) 6.83885 11.8452i 0.680491 1.17865i −0.294340 0.955701i \(-0.595100\pi\)
0.974831 0.222945i \(-0.0715669\pi\)
\(102\) 0 0
\(103\) 0.0432333 0.0748822i 0.00425990 0.00737836i −0.863888 0.503685i \(-0.831977\pi\)
0.868147 + 0.496306i \(0.165311\pi\)
\(104\) 0 0
\(105\) −3.55098 + 2.04005i −0.346540 + 0.199088i
\(106\) 0 0
\(107\) 16.7666i 1.62089i −0.585816 0.810444i \(-0.699226\pi\)
0.585816 0.810444i \(-0.300774\pi\)
\(108\) 0 0
\(109\) 3.22325i 0.308732i 0.988014 + 0.154366i \(0.0493334\pi\)
−0.988014 + 0.154366i \(0.950667\pi\)
\(110\) 0 0
\(111\) −2.21666 0.00473979i −0.210396 0.000449881i
\(112\) 0 0
\(113\) 7.19242 12.4576i 0.676606 1.17192i −0.299391 0.954130i \(-0.596784\pi\)
0.975997 0.217785i \(-0.0698831\pi\)
\(114\) 0 0
\(115\) −3.91515 + 2.26041i −0.365089 + 0.210784i
\(116\) 0 0
\(117\) 17.6605 + 10.2973i 1.63272 + 0.951983i
\(118\) 0 0
\(119\) 5.68486 0.521131
\(120\) 0 0
\(121\) −5.99151 10.3776i −0.544683 0.943418i
\(122\) 0 0
\(123\) −2.50317 4.35711i −0.225703 0.392867i
\(124\) 0 0
\(125\) −7.67087 −0.686103
\(126\) 0 0
\(127\) 1.64965 2.85728i 0.146383 0.253542i −0.783505 0.621385i \(-0.786570\pi\)
0.929888 + 0.367843i \(0.119904\pi\)
\(128\) 0 0
\(129\) −18.2753 10.6034i −1.60905 0.933578i
\(130\) 0 0
\(131\) 7.00155i 0.611729i −0.952075 0.305864i \(-0.901055\pi\)
0.952075 0.305864i \(-0.0989454\pi\)
\(132\) 0 0
\(133\) 4.43162i 0.384270i
\(134\) 0 0
\(135\) 13.8711 + 0.0889816i 1.19384 + 0.00765831i
\(136\) 0 0
\(137\) 1.93779 0.165557 0.0827785 0.996568i \(-0.473621\pi\)
0.0827785 + 0.996568i \(0.473621\pi\)
\(138\) 0 0
\(139\) 13.8960i 1.17865i 0.807898 + 0.589323i \(0.200605\pi\)
−0.807898 + 0.589323i \(0.799395\pi\)
\(140\) 0 0
\(141\) −6.21473 + 3.57038i −0.523374 + 0.300680i
\(142\) 0 0
\(143\) 32.6688i 2.73190i
\(144\) 0 0
\(145\) 13.6522 + 7.88212i 1.13376 + 0.654575i
\(146\) 0 0
\(147\) 10.7656 + 0.0230197i 0.887934 + 0.00189864i
\(148\) 0 0
\(149\) 16.0183i 1.31227i 0.754644 + 0.656134i \(0.227809\pi\)
−0.754644 + 0.656134i \(0.772191\pi\)
\(150\) 0 0
\(151\) 3.32600 + 5.76080i 0.270666 + 0.468807i 0.969033 0.246933i \(-0.0794228\pi\)
−0.698367 + 0.715740i \(0.746089\pi\)
\(152\) 0 0
\(153\) −16.6346 9.69907i −1.34483 0.784123i
\(154\) 0 0
\(155\) −14.1146 8.14905i −1.13371 0.654548i
\(156\) 0 0
\(157\) 10.0978 + 17.4899i 0.805893 + 1.39585i 0.915686 + 0.401894i \(0.131648\pi\)
−0.109793 + 0.993954i \(0.535019\pi\)
\(158\) 0 0
\(159\) 4.56377 7.86579i 0.361931 0.623798i
\(160\) 0 0
\(161\) −1.49989 −0.118208
\(162\) 0 0
\(163\) 4.14521 7.17971i 0.324678 0.562358i −0.656769 0.754091i \(-0.728078\pi\)
0.981447 + 0.191733i \(0.0614109\pi\)
\(164\) 0 0
\(165\) −11.0423 19.2207i −0.859645 1.49633i
\(166\) 0 0
\(167\) 9.82527 + 5.67262i 0.760302 + 0.438961i 0.829404 0.558649i \(-0.188680\pi\)
−0.0691020 + 0.997610i \(0.522013\pi\)
\(168\) 0 0
\(169\) 16.7183 28.9569i 1.28602 2.22745i
\(170\) 0 0
\(171\) 7.56089 12.9675i 0.578196 0.991646i
\(172\) 0 0
\(173\) 13.7279 7.92578i 1.04371 0.602586i 0.122828 0.992428i \(-0.460804\pi\)
0.920882 + 0.389842i \(0.127470\pi\)
\(174\) 0 0
\(175\) 1.63112 + 0.941728i 0.123301 + 0.0711880i
\(176\) 0 0
\(177\) 0.880031 + 0.510598i 0.0661472 + 0.0383789i
\(178\) 0 0
\(179\) −10.8008 −0.807289 −0.403645 0.914916i \(-0.632257\pi\)
−0.403645 + 0.914916i \(0.632257\pi\)
\(180\) 0 0
\(181\) 4.46568 7.73479i 0.331932 0.574922i −0.650959 0.759113i \(-0.725633\pi\)
0.982891 + 0.184191i \(0.0589664\pi\)
\(182\) 0 0
\(183\) 0.0250547 11.7173i 0.00185210 0.866169i
\(184\) 0 0
\(185\) 1.70824 + 2.95875i 0.125592 + 0.217532i
\(186\) 0 0
\(187\) 30.7710i 2.25020i
\(188\) 0 0
\(189\) 3.97077 + 2.32661i 0.288831 + 0.169236i
\(190\) 0 0
\(191\) 8.44762 + 14.6317i 0.611249 + 1.05871i 0.991030 + 0.133637i \(0.0426658\pi\)
−0.379782 + 0.925076i \(0.624001\pi\)
\(192\) 0 0
\(193\) 24.5120 1.76441 0.882207 0.470862i \(-0.156057\pi\)
0.882207 + 0.470862i \(0.156057\pi\)
\(194\) 0 0
\(195\) 0.0673736 31.5086i 0.00482473 2.25638i
\(196\) 0 0
\(197\) 11.1465 19.3064i 0.794159 1.37552i −0.129213 0.991617i \(-0.541245\pi\)
0.923372 0.383906i \(-0.125421\pi\)
\(198\) 0 0
\(199\) −3.86657 6.69709i −0.274094 0.474744i 0.695812 0.718224i \(-0.255045\pi\)
−0.969906 + 0.243479i \(0.921711\pi\)
\(200\) 0 0
\(201\) 8.92155 + 11.0184i 0.629278 + 0.777181i
\(202\) 0 0
\(203\) 2.61509 + 4.52946i 0.183543 + 0.317906i
\(204\) 0 0
\(205\) −3.87241 + 6.70721i −0.270461 + 0.468452i
\(206\) 0 0
\(207\) 4.38886 + 2.55900i 0.305047 + 0.177863i
\(208\) 0 0
\(209\) −23.9874 −1.65925
\(210\) 0 0
\(211\) −5.91693 10.2484i −0.407338 0.705531i 0.587252 0.809404i \(-0.300210\pi\)
−0.994591 + 0.103873i \(0.966876\pi\)
\(212\) 0 0
\(213\) −2.39540 + 1.37616i −0.164130 + 0.0942931i
\(214\) 0 0
\(215\) 32.5649i 2.22091i
\(216\) 0 0
\(217\) −2.70365 4.68286i −0.183536 0.317893i
\(218\) 0 0
\(219\) −12.8840 0.0275494i −0.870621 0.00186162i
\(220\) 0 0
\(221\) −21.8694 + 37.8790i −1.47110 + 2.54802i
\(222\) 0 0
\(223\) 7.01373 0.469674 0.234837 0.972035i \(-0.424544\pi\)
0.234837 + 0.972035i \(0.424544\pi\)
\(224\) 0 0
\(225\) −3.16615 5.53850i −0.211077 0.369234i
\(226\) 0 0
\(227\) 3.74588 + 2.16268i 0.248623 + 0.143542i 0.619133 0.785286i \(-0.287484\pi\)
−0.370511 + 0.928828i \(0.620817\pi\)
\(228\) 0 0
\(229\) 0.261357 0.150895i 0.0172710 0.00997139i −0.491340 0.870968i \(-0.663493\pi\)
0.508611 + 0.860997i \(0.330159\pi\)
\(230\) 0 0
\(231\) 0.0157256 7.35437i 0.00103467 0.483882i
\(232\) 0 0
\(233\) −12.7168 + 22.0262i −0.833106 + 1.44298i 0.0624566 + 0.998048i \(0.480107\pi\)
−0.895563 + 0.444935i \(0.853227\pi\)
\(234\) 0 0
\(235\) 9.56678 + 5.52338i 0.624068 + 0.360306i
\(236\) 0 0
\(237\) −17.4804 + 10.0425i −1.13547 + 0.652333i
\(238\) 0 0
\(239\) 0.837087 1.44988i 0.0541466 0.0937847i −0.837682 0.546159i \(-0.816089\pi\)
0.891828 + 0.452374i \(0.149423\pi\)
\(240\) 0 0
\(241\) −12.1313 −0.781444 −0.390722 0.920509i \(-0.627775\pi\)
−0.390722 + 0.920509i \(0.627775\pi\)
\(242\) 0 0
\(243\) −7.64945 13.5826i −0.490713 0.871321i
\(244\) 0 0
\(245\) −8.29639 14.3698i −0.530037 0.918051i
\(246\) 0 0
\(247\) −29.5285 17.0483i −1.87885 1.08476i
\(248\) 0 0
\(249\) 0.0304733 14.2514i 0.00193117 0.903147i
\(250\) 0 0
\(251\) −6.13618 10.6282i −0.387312 0.670845i 0.604775 0.796397i \(-0.293263\pi\)
−0.992087 + 0.125552i \(0.959930\pi\)
\(252\) 0 0
\(253\) 8.11861i 0.510412i
\(254\) 0 0
\(255\) −0.0634597 + 29.6781i −0.00397400 + 1.85852i
\(256\) 0 0
\(257\) 23.4922 + 13.5632i 1.46540 + 0.846050i 0.999253 0.0386567i \(-0.0123079\pi\)
0.466149 + 0.884706i \(0.345641\pi\)
\(258\) 0 0
\(259\) 1.13350i 0.0704321i
\(260\) 0 0
\(261\) 0.0757608 17.7154i 0.00468947 1.09656i
\(262\) 0 0
\(263\) 16.0693i 0.990875i −0.868644 0.495437i \(-0.835008\pi\)
0.868644 0.495437i \(-0.164992\pi\)
\(264\) 0 0
\(265\) −14.0161 −0.861004
\(266\) 0 0
\(267\) 13.6211 + 7.90302i 0.833597 + 0.483657i
\(268\) 0 0
\(269\) 7.74125i 0.471992i 0.971754 + 0.235996i \(0.0758353\pi\)
−0.971754 + 0.235996i \(0.924165\pi\)
\(270\) 0 0
\(271\) 17.0339i 1.03474i −0.855762 0.517369i \(-0.826911\pi\)
0.855762 0.517369i \(-0.173089\pi\)
\(272\) 0 0
\(273\) 5.24623 9.04203i 0.317516 0.547248i
\(274\) 0 0
\(275\) −5.09738 + 8.82892i −0.307384 + 0.532404i
\(276\) 0 0
\(277\) 4.64989 0.279385 0.139693 0.990195i \(-0.455389\pi\)
0.139693 + 0.990195i \(0.455389\pi\)
\(278\) 0 0
\(279\) −0.0783264 + 18.3153i −0.00468928 + 1.09651i
\(280\) 0 0
\(281\) −9.56232 16.5624i −0.570440 0.988031i −0.996521 0.0833459i \(-0.973439\pi\)
0.426081 0.904685i \(-0.359894\pi\)
\(282\) 0 0
\(283\) 20.5705 1.22279 0.611396 0.791325i \(-0.290608\pi\)
0.611396 + 0.791325i \(0.290608\pi\)
\(284\) 0 0
\(285\) −23.1355 0.0494699i −1.37043 0.00293034i
\(286\) 0 0
\(287\) −2.22528 + 1.28477i −0.131354 + 0.0758374i
\(288\) 0 0
\(289\) 12.0990 20.9560i 0.711704 1.23271i
\(290\) 0 0
\(291\) −0.0335093 + 15.6713i −0.00196435 + 0.918665i
\(292\) 0 0
\(293\) 27.9889i 1.63513i −0.575837 0.817564i \(-0.695324\pi\)
0.575837 0.817564i \(-0.304676\pi\)
\(294\) 0 0
\(295\) 1.56814i 0.0913004i
\(296\) 0 0
\(297\) −12.5935 + 21.4929i −0.730747 + 1.24715i
\(298\) 0 0
\(299\) 5.77003 9.99398i 0.333689 0.577967i
\(300\) 0 0
\(301\) −5.40210 + 9.35671i −0.311372 + 0.539312i
\(302\) 0 0
\(303\) 11.8013 + 20.5418i 0.677969 + 1.18010i
\(304\) 0 0
\(305\) −15.6401 + 9.02979i −0.895547 + 0.517044i
\(306\) 0 0
\(307\) 0.231235 + 0.400510i 0.0131973 + 0.0228583i 0.872549 0.488527i \(-0.162466\pi\)
−0.859351 + 0.511386i \(0.829132\pi\)
\(308\) 0 0
\(309\) 0.0746047 + 0.129860i 0.00424411 + 0.00738746i
\(310\) 0 0
\(311\) −9.65840 −0.547678 −0.273839 0.961776i \(-0.588293\pi\)
−0.273839 + 0.961776i \(0.588293\pi\)
\(312\) 0 0
\(313\) 13.1989i 0.746047i 0.927822 + 0.373024i \(0.121679\pi\)
−0.927822 + 0.373024i \(0.878321\pi\)
\(314\) 0 0
\(315\) 0.0303342 7.09315i 0.00170914 0.399653i
\(316\) 0 0
\(317\) 12.0664 6.96656i 0.677718 0.391281i −0.121276 0.992619i \(-0.538699\pi\)
0.798995 + 0.601338i \(0.205365\pi\)
\(318\) 0 0
\(319\) −24.5170 + 14.1549i −1.37269 + 0.792523i
\(320\) 0 0
\(321\) 25.1188 + 14.5740i 1.40199 + 0.813443i
\(322\) 0 0
\(323\) 27.8131 + 16.0579i 1.54756 + 0.893485i
\(324\) 0 0
\(325\) −12.5497 + 7.24558i −0.696133 + 0.401913i
\(326\) 0 0
\(327\) −4.82890 2.80175i −0.267039 0.154937i
\(328\) 0 0
\(329\) 1.83252 + 3.17401i 0.101030 + 0.174989i
\(330\) 0 0
\(331\) −2.55961 1.47779i −0.140689 0.0812269i 0.428003 0.903777i \(-0.359217\pi\)
−0.568692 + 0.822550i \(0.692551\pi\)
\(332\) 0 0
\(333\) 1.93389 3.31675i 0.105976 0.181757i
\(334\) 0 0
\(335\) 7.78828 20.4162i 0.425519 1.11546i
\(336\) 0 0
\(337\) 10.9976 6.34948i 0.599078 0.345878i −0.169601 0.985513i \(-0.554248\pi\)
0.768679 + 0.639635i \(0.220914\pi\)
\(338\) 0 0
\(339\) 12.4115 + 21.6038i 0.674098 + 1.17336i
\(340\) 0 0
\(341\) 25.3473 14.6343i 1.37263 0.792491i
\(342\) 0 0
\(343\) 11.7049i 0.632005i
\(344\) 0 0
\(345\) 0.0167432 7.83027i 0.000901423 0.421568i
\(346\) 0 0
\(347\) 17.3954 30.1298i 0.933836 1.61745i 0.157141 0.987576i \(-0.449772\pi\)
0.776696 0.629876i \(-0.216894\pi\)
\(348\) 0 0
\(349\) −34.3154 −1.83686 −0.918430 0.395582i \(-0.870543\pi\)
−0.918430 + 0.395582i \(0.870543\pi\)
\(350\) 0 0
\(351\) −30.7779 + 17.5073i −1.64280 + 0.934473i
\(352\) 0 0
\(353\) 11.2119 + 19.4195i 0.596747 + 1.03360i 0.993298 + 0.115583i \(0.0368737\pi\)
−0.396551 + 0.918013i \(0.629793\pi\)
\(354\) 0 0
\(355\) 3.68741 + 2.12893i 0.195707 + 0.112992i
\(356\) 0 0
\(357\) −4.94146 + 8.51675i −0.261530 + 0.450754i
\(358\) 0 0
\(359\) 24.3510i 1.28520i 0.766203 + 0.642599i \(0.222144\pi\)
−0.766203 + 0.642599i \(0.777856\pi\)
\(360\) 0 0
\(361\) −3.01789 + 5.22714i −0.158836 + 0.275113i
\(362\) 0 0
\(363\) 20.7551 + 0.0443800i 1.08936 + 0.00232934i
\(364\) 0 0
\(365\) 9.92890 + 17.1974i 0.519702 + 0.900151i
\(366\) 0 0
\(367\) −17.9252 10.3491i −0.935687 0.540219i −0.0470815 0.998891i \(-0.514992\pi\)
−0.888606 + 0.458672i \(0.848325\pi\)
\(368\) 0 0
\(369\) 8.70341 + 0.0372205i 0.453081 + 0.00193762i
\(370\) 0 0
\(371\) −4.02718 2.32510i −0.209081 0.120713i
\(372\) 0 0
\(373\) −14.8627 8.58099i −0.769562 0.444307i 0.0631565 0.998004i \(-0.479883\pi\)
−0.832718 + 0.553697i \(0.813217\pi\)
\(374\) 0 0
\(375\) 6.66776 11.4921i 0.344321 0.593448i
\(376\) 0 0
\(377\) −40.2405 −2.07249
\(378\) 0 0
\(379\) −8.38111 + 4.83883i −0.430508 + 0.248554i −0.699563 0.714571i \(-0.746622\pi\)
0.269055 + 0.963125i \(0.413289\pi\)
\(380\) 0 0
\(381\) 2.84669 + 4.95505i 0.145840 + 0.253855i
\(382\) 0 0
\(383\) 6.59191 + 11.4175i 0.336831 + 0.583408i 0.983835 0.179079i \(-0.0573117\pi\)
−0.647004 + 0.762487i \(0.723978\pi\)
\(384\) 0 0
\(385\) −9.81648 + 5.66755i −0.500294 + 0.288845i
\(386\) 0 0
\(387\) 31.7709 18.1622i 1.61500 0.923237i
\(388\) 0 0
\(389\) 4.79644 2.76923i 0.243189 0.140405i −0.373452 0.927649i \(-0.621826\pi\)
0.616642 + 0.787244i \(0.288493\pi\)
\(390\) 0 0
\(391\) −5.43483 + 9.41340i −0.274851 + 0.476056i
\(392\) 0 0
\(393\) 10.4893 + 6.08597i 0.529117 + 0.306996i
\(394\) 0 0
\(395\) 26.9089 + 15.5358i 1.35393 + 0.781693i
\(396\) 0 0
\(397\) −11.8894 −0.596714 −0.298357 0.954454i \(-0.596439\pi\)
−0.298357 + 0.954454i \(0.596439\pi\)
\(398\) 0 0
\(399\) −6.63921 3.85210i −0.332376 0.192846i
\(400\) 0 0
\(401\) 23.8611 1.19157 0.595783 0.803146i \(-0.296842\pi\)
0.595783 + 0.803146i \(0.296842\pi\)
\(402\) 0 0
\(403\) 41.6033 2.07241
\(404\) 0 0
\(405\) −12.1905 + 20.7036i −0.605752 + 1.02877i
\(406\) 0 0
\(407\) −6.13539 −0.304120
\(408\) 0 0
\(409\) 9.72849 + 5.61675i 0.481043 + 0.277730i 0.720851 0.693090i \(-0.243751\pi\)
−0.239808 + 0.970820i \(0.577084\pi\)
\(410\) 0 0
\(411\) −1.68439 + 2.90310i −0.0830849 + 0.143199i
\(412\) 0 0
\(413\) 0.260134 0.450565i 0.0128003 0.0221708i
\(414\) 0 0
\(415\) −19.0225 + 10.9827i −0.933780 + 0.539118i
\(416\) 0 0
\(417\) −20.8183 12.0789i −1.01947 0.591504i
\(418\) 0 0
\(419\) 13.0009 7.50607i 0.635135 0.366695i −0.147603 0.989047i \(-0.547156\pi\)
0.782738 + 0.622351i \(0.213822\pi\)
\(420\) 0 0
\(421\) 12.3082 + 21.3184i 0.599863 + 1.03899i 0.992841 + 0.119445i \(0.0381116\pi\)
−0.392978 + 0.919548i \(0.628555\pi\)
\(422\) 0 0
\(423\) 0.0530892 12.4140i 0.00258129 0.603591i
\(424\) 0 0
\(425\) 11.8207 6.82466i 0.573386 0.331045i
\(426\) 0 0
\(427\) −5.99171 −0.289959
\(428\) 0 0
\(429\) 48.9426 + 28.3967i 2.36297 + 1.37101i
\(430\) 0 0
\(431\) −28.4860 16.4464i −1.37212 0.792195i −0.380927 0.924605i \(-0.624395\pi\)
−0.991195 + 0.132410i \(0.957728\pi\)
\(432\) 0 0
\(433\) −5.46719 3.15648i −0.262736 0.151691i 0.362846 0.931849i \(-0.381805\pi\)
−0.625582 + 0.780158i \(0.715138\pi\)
\(434\) 0 0
\(435\) −23.6755 + 13.6016i −1.13515 + 0.652149i
\(436\) 0 0
\(437\) −7.33819 4.23671i −0.351033 0.202669i
\(438\) 0 0
\(439\) 7.84323 + 13.5849i 0.374337 + 0.648371i 0.990228 0.139461i \(-0.0445370\pi\)
−0.615891 + 0.787832i \(0.711204\pi\)
\(440\) 0 0
\(441\) −9.39230 + 16.1085i −0.447252 + 0.767069i
\(442\) 0 0
\(443\) −4.22784 + 7.32284i −0.200871 + 0.347918i −0.948809 0.315850i \(-0.897710\pi\)
0.747938 + 0.663768i \(0.231044\pi\)
\(444\) 0 0
\(445\) 24.2716i 1.15058i
\(446\) 0 0
\(447\) −23.9977 13.9236i −1.13505 0.658563i
\(448\) 0 0
\(449\) −8.74757 5.05041i −0.412823 0.238344i 0.279179 0.960239i \(-0.409938\pi\)
−0.692002 + 0.721896i \(0.743271\pi\)
\(450\) 0 0
\(451\) −6.95417 12.0450i −0.327459 0.567176i
\(452\) 0 0
\(453\) −11.5216 0.0246362i −0.541331 0.00115751i
\(454\) 0 0
\(455\) −16.1121 −0.755346
\(456\) 0 0
\(457\) 1.61348 2.79462i 0.0754753 0.130727i −0.825818 0.563937i \(-0.809286\pi\)
0.901293 + 0.433210i \(0.142619\pi\)
\(458\) 0 0
\(459\) 28.9899 16.4903i 1.35313 0.769701i
\(460\) 0 0
\(461\) 27.3169i 1.27227i 0.771576 + 0.636137i \(0.219469\pi\)
−0.771576 + 0.636137i \(0.780531\pi\)
\(462\) 0 0
\(463\) −19.1655 + 11.0652i −0.890695 + 0.514243i −0.874170 0.485621i \(-0.838594\pi\)
−0.0165250 + 0.999863i \(0.505260\pi\)
\(464\) 0 0
\(465\) 24.4773 14.0623i 1.13511 0.652122i
\(466\) 0 0
\(467\) −16.1193 + 9.30648i −0.745912 + 0.430653i −0.824215 0.566277i \(-0.808383\pi\)
0.0783028 + 0.996930i \(0.475050\pi\)
\(468\) 0 0
\(469\) 5.62455 4.57411i 0.259718 0.211213i
\(470\) 0 0
\(471\) −34.9798 0.0747960i −1.61178 0.00344642i
\(472\) 0 0
\(473\) −50.6459 29.2404i −2.32870 1.34448i
\(474\) 0 0
\(475\) 5.32015 + 9.21477i 0.244105 + 0.422803i
\(476\) 0 0
\(477\) 7.81713 + 13.6744i 0.357922 + 0.626107i
\(478\) 0 0
\(479\) −9.27106 + 5.35265i −0.423605 + 0.244569i −0.696619 0.717442i \(-0.745313\pi\)
0.273013 + 0.962010i \(0.411980\pi\)
\(480\) 0 0
\(481\) −7.55264 4.36052i −0.344371 0.198823i
\(482\) 0 0
\(483\) 1.30375 2.24706i 0.0593228 0.102245i
\(484\) 0 0
\(485\) 20.9177 12.0768i 0.949824 0.548381i
\(486\) 0 0
\(487\) −37.0488 + 21.3902i −1.67884 + 0.969280i −0.716442 + 0.697647i \(0.754231\pi\)
−0.962401 + 0.271634i \(0.912436\pi\)
\(488\) 0 0
\(489\) 7.15310 + 12.4509i 0.323474 + 0.563051i
\(490\) 0 0
\(491\) 4.04352i 0.182481i 0.995829 + 0.0912407i \(0.0290833\pi\)
−0.995829 + 0.0912407i \(0.970917\pi\)
\(492\) 0 0
\(493\) 37.9028 1.70706
\(494\) 0 0
\(495\) 38.3937 + 0.164192i 1.72567 + 0.00737990i
\(496\) 0 0
\(497\) 0.706324 + 1.22339i 0.0316830 + 0.0548765i
\(498\) 0 0
\(499\) −30.1152 + 17.3870i −1.34814 + 0.778349i −0.987986 0.154544i \(-0.950609\pi\)
−0.360154 + 0.932893i \(0.617276\pi\)
\(500\) 0 0
\(501\) −17.0388 + 9.78886i −0.761239 + 0.437334i
\(502\) 0 0
\(503\) −9.58680 + 16.6048i −0.427454 + 0.740373i −0.996646 0.0818321i \(-0.973923\pi\)
0.569192 + 0.822205i \(0.307256\pi\)
\(504\) 0 0
\(505\) 18.2567 31.6216i 0.812413 1.40714i
\(506\) 0 0
\(507\) 28.8496 + 50.2166i 1.28125 + 2.23020i
\(508\) 0 0
\(509\) 17.0124i 0.754061i 0.926201 + 0.377031i \(0.123055\pi\)
−0.926201 + 0.377031i \(0.876945\pi\)
\(510\) 0 0
\(511\) 6.58831i 0.291450i
\(512\) 0 0
\(513\) 12.8550 + 22.5990i 0.567561 + 0.997771i
\(514\) 0 0
\(515\) 0.115414 0.199902i 0.00508574 0.00880875i
\(516\) 0 0
\(517\) −17.1803 + 9.91903i −0.755587 + 0.436239i
\(518\) 0 0
\(519\) −0.0587074 + 27.4557i −0.00257697 + 1.20517i
\(520\) 0 0
\(521\) 4.28481 0.187721 0.0938605 0.995585i \(-0.470079\pi\)
0.0938605 + 0.995585i \(0.470079\pi\)
\(522\) 0 0
\(523\) −14.2199 24.6296i −0.621793 1.07698i −0.989152 0.146897i \(-0.953071\pi\)
0.367359 0.930079i \(-0.380262\pi\)
\(524\) 0 0
\(525\) −2.82867 + 1.62508i −0.123453 + 0.0709242i
\(526\) 0 0
\(527\) −39.1864 −1.70699
\(528\) 0 0
\(529\) −10.0661 + 17.4350i −0.437656 + 0.758042i
\(530\) 0 0
\(531\) −1.52990 + 0.874586i −0.0663920 + 0.0379538i
\(532\) 0 0
\(533\) 19.7698i 0.856324i
\(534\) 0 0
\(535\) 44.7594i 1.93512i
\(536\) 0 0
\(537\) 9.38838 16.1811i 0.405139 0.698268i
\(538\) 0 0
\(539\) 29.7977 1.28348
\(540\) 0 0
\(541\) 19.5488i 0.840468i 0.907416 + 0.420234i \(0.138052\pi\)
−0.907416 + 0.420234i \(0.861948\pi\)
\(542\) 0 0
\(543\) 7.70612 + 13.4136i 0.330701 + 0.575631i
\(544\) 0 0
\(545\) 8.60466i 0.368583i
\(546\) 0 0
\(547\) −28.8062 16.6313i −1.23167 0.711103i −0.264289 0.964444i \(-0.585137\pi\)
−0.967377 + 0.253341i \(0.918471\pi\)
\(548\) 0 0
\(549\) 17.5325 + 10.2226i 0.748267 + 0.436289i
\(550\) 0 0
\(551\) 29.5471i 1.25875i
\(552\) 0 0
\(553\) 5.15439 + 8.92767i 0.219187 + 0.379643i
\(554\) 0 0
\(555\) −5.91749 0.0126532i −0.251183 0.000537096i
\(556\) 0 0
\(557\) 10.6629 + 6.15624i 0.451802 + 0.260848i 0.708591 0.705619i \(-0.249331\pi\)
−0.256789 + 0.966468i \(0.582664\pi\)
\(558\) 0 0
\(559\) −41.5633 71.9898i −1.75794 3.04484i
\(560\) 0 0
\(561\) −46.0994 26.7471i −1.94632 1.12926i
\(562\) 0 0
\(563\) 44.2623 1.86543 0.932717 0.360610i \(-0.117431\pi\)
0.932717 + 0.360610i \(0.117431\pi\)
\(564\) 0 0
\(565\) 19.2006 33.2564i 0.807774 1.39911i
\(566\) 0 0
\(567\) −6.93711 + 3.92642i −0.291331 + 0.164894i
\(568\) 0 0
\(569\) 27.8744 + 16.0933i 1.16856 + 0.674667i 0.953340 0.301899i \(-0.0976206\pi\)
0.215218 + 0.976566i \(0.430954\pi\)
\(570\) 0 0
\(571\) 2.23640 3.87355i 0.0935903 0.162103i −0.815429 0.578857i \(-0.803499\pi\)
0.909019 + 0.416754i \(0.136832\pi\)
\(572\) 0 0
\(573\) −29.2634 0.0625728i −1.22249 0.00261401i
\(574\) 0 0
\(575\) −3.11876 + 1.80062i −0.130061 + 0.0750909i
\(576\) 0 0
\(577\) 19.6185 + 11.3267i 0.816729 + 0.471538i 0.849287 0.527931i \(-0.177032\pi\)
−0.0325585 + 0.999470i \(0.510366\pi\)
\(578\) 0 0
\(579\) −21.3066 + 36.7226i −0.885472 + 1.52614i
\(580\) 0 0
\(581\) −7.28754 −0.302338
\(582\) 0 0
\(583\) 12.5853 21.7983i 0.521228 0.902794i
\(584\) 0 0
\(585\) 47.1458 + 27.4892i 1.94924 + 1.13654i
\(586\) 0 0
\(587\) −5.41622 9.38117i −0.223551 0.387202i 0.732333 0.680947i \(-0.238432\pi\)
−0.955884 + 0.293745i \(0.905098\pi\)
\(588\) 0 0
\(589\) 30.5477i 1.25870i
\(590\) 0 0
\(591\) 19.2348 + 33.4808i 0.791216 + 1.37722i
\(592\) 0 0
\(593\) −1.57009 2.71947i −0.0644758 0.111675i 0.831986 0.554797i \(-0.187204\pi\)
−0.896461 + 0.443122i \(0.853871\pi\)
\(594\) 0 0
\(595\) 15.1761 0.622158
\(596\) 0 0
\(597\) 13.3942 + 0.0286402i 0.548186 + 0.00117217i
\(598\) 0 0
\(599\) 13.9261 24.1207i 0.569006 0.985547i −0.427659 0.903940i \(-0.640662\pi\)
0.996665 0.0816064i \(-0.0260051\pi\)
\(600\) 0 0
\(601\) 2.16618 + 3.75193i 0.0883601 + 0.153044i 0.906818 0.421522i \(-0.138504\pi\)
−0.818458 + 0.574566i \(0.805171\pi\)
\(602\) 0 0
\(603\) −24.2621 + 3.78821i −0.988029 + 0.154268i
\(604\) 0 0
\(605\) −15.9947 27.7036i −0.650276 1.12631i
\(606\) 0 0
\(607\) 18.4717 31.9940i 0.749744 1.29860i −0.198201 0.980161i \(-0.563510\pi\)
0.947945 0.318434i \(-0.103157\pi\)
\(608\) 0 0
\(609\) −9.05891 0.0193703i −0.367085 0.000784925i
\(610\) 0 0
\(611\) −28.1985 −1.14079
\(612\) 0 0
\(613\) −2.55105 4.41856i −0.103036 0.178464i 0.809898 0.586571i \(-0.199522\pi\)
−0.912934 + 0.408107i \(0.866189\pi\)
\(614\) 0 0
\(615\) −6.68236 11.6315i −0.269459 0.469029i
\(616\) 0 0
\(617\) 15.7934i 0.635819i 0.948121 + 0.317909i \(0.102981\pi\)
−0.948121 + 0.317909i \(0.897019\pi\)
\(618\) 0 0
\(619\) 22.6888 + 39.2981i 0.911940 + 1.57953i 0.811321 + 0.584601i \(0.198749\pi\)
0.100618 + 0.994925i \(0.467918\pi\)
\(620\) 0 0
\(621\) −7.64869 + 4.35079i −0.306931 + 0.174591i
\(622\) 0 0
\(623\) 4.02634 6.97382i 0.161312 0.279400i
\(624\) 0 0
\(625\) −31.1105 −1.24442
\(626\) 0 0
\(627\) 20.8506 35.9367i 0.832694 1.43517i
\(628\) 0 0
\(629\) 7.11389 + 4.10720i 0.283649 + 0.163765i
\(630\) 0 0
\(631\) −32.9816 + 19.0419i −1.31297 + 0.758046i −0.982588 0.185799i \(-0.940513\pi\)
−0.330387 + 0.943846i \(0.607179\pi\)
\(632\) 0 0
\(633\) 20.4968 + 0.0438276i 0.814675 + 0.00174199i
\(634\) 0 0
\(635\) 4.40384 7.62767i 0.174761 0.302695i
\(636\) 0 0
\(637\) 36.6809 + 21.1777i 1.45335 + 0.839092i
\(638\) 0 0
\(639\) 0.0204627 4.78486i 0.000809490 0.189286i
\(640\) 0 0
\(641\) 3.27214 5.66752i 0.129242 0.223854i −0.794141 0.607733i \(-0.792079\pi\)
0.923383 + 0.383880i \(0.125412\pi\)
\(642\) 0 0
\(643\) 3.04449 0.120063 0.0600314 0.998196i \(-0.480880\pi\)
0.0600314 + 0.998196i \(0.480880\pi\)
\(644\) 0 0
\(645\) −48.7869 28.3064i −1.92098 1.11456i
\(646\) 0 0
\(647\) −0.189531 0.328277i −0.00745122 0.0129059i 0.862276 0.506439i \(-0.169038\pi\)
−0.869727 + 0.493533i \(0.835705\pi\)
\(648\) 0 0
\(649\) 2.43881 + 1.40805i 0.0957317 + 0.0552707i
\(650\) 0 0
\(651\) 9.36569 + 0.0200263i 0.367070 + 0.000784893i
\(652\) 0 0
\(653\) −21.6786 37.5485i −0.848350 1.46938i −0.882680 0.469974i \(-0.844263\pi\)
0.0343306 0.999411i \(-0.489070\pi\)
\(654\) 0 0
\(655\) 18.6911i 0.730320i
\(656\) 0 0
\(657\) 11.2405 19.2782i 0.438532 0.752113i
\(658\) 0 0
\(659\) −31.4183 18.1394i −1.22388 0.706609i −0.258140 0.966108i \(-0.583109\pi\)
−0.965744 + 0.259498i \(0.916443\pi\)
\(660\) 0 0
\(661\) 4.46606i 0.173710i −0.996221 0.0868549i \(-0.972318\pi\)
0.996221 0.0868549i \(-0.0276816\pi\)
\(662\) 0 0
\(663\) −37.7386 65.6892i −1.46565 2.55116i
\(664\) 0 0
\(665\) 11.8305i 0.458766i
\(666\) 0 0
\(667\) −10.0003 −0.387212
\(668\) 0 0
\(669\) −6.09655 + 10.5076i −0.235706 + 0.406247i
\(670\) 0 0
\(671\) 32.4319i 1.25202i
\(672\) 0 0
\(673\) 4.22151i 0.162727i −0.996684 0.0813636i \(-0.974072\pi\)
0.996684 0.0813636i \(-0.0259275\pi\)
\(674\) 0 0
\(675\) 11.0496 + 0.0708817i 0.425299 + 0.00272824i
\(676\) 0 0
\(677\) −2.63905 + 4.57097i −0.101427 + 0.175677i −0.912273 0.409583i \(-0.865674\pi\)
0.810846 + 0.585260i \(0.199007\pi\)
\(678\) 0 0
\(679\) 8.01358 0.307533
\(680\) 0 0
\(681\) −6.49604 + 3.73200i −0.248929 + 0.143010i
\(682\) 0 0
\(683\) −20.1597 34.9177i −0.771391 1.33609i −0.936801 0.349864i \(-0.886228\pi\)
0.165410 0.986225i \(-0.447105\pi\)
\(684\) 0 0
\(685\) 5.17306 0.197652
\(686\) 0 0
\(687\) −0.00111770 + 0.522713i −4.26428e−5 + 0.0199427i
\(688\) 0 0
\(689\) 30.9848 17.8891i 1.18043 0.681521i
\(690\) 0 0
\(691\) 1.47918 2.56202i 0.0562708 0.0974639i −0.836518 0.547940i \(-0.815412\pi\)
0.892789 + 0.450476i \(0.148746\pi\)
\(692\) 0 0
\(693\) 11.0042 + 6.41620i 0.418016 + 0.243731i
\(694\) 0 0
\(695\) 37.0962i 1.40714i
\(696\) 0 0
\(697\) 18.6213i 0.705332i
\(698\) 0 0
\(699\) −21.9445 38.1975i −0.830019 1.44476i
\(700\) 0 0
\(701\) −20.1270 + 34.8610i −0.760186 + 1.31668i 0.182569 + 0.983193i \(0.441559\pi\)
−0.942755 + 0.333487i \(0.891775\pi\)
\(702\) 0 0
\(703\) −3.20176 + 5.54561i −0.120757 + 0.209157i
\(704\) 0 0
\(705\) −16.5906 + 9.53133i −0.624837 + 0.358971i
\(706\) 0 0
\(707\) 10.4912 6.05711i 0.394563 0.227801i
\(708\) 0 0
\(709\) 9.12206 + 15.7999i 0.342586 + 0.593377i 0.984912 0.173055i \(-0.0553638\pi\)
−0.642326 + 0.766432i \(0.722030\pi\)
\(710\) 0 0
\(711\) 0.149326 34.9175i 0.00560017 1.30951i
\(712\) 0 0
\(713\) 10.3389 0.387196
\(714\) 0 0
\(715\) 87.2113i 3.26152i
\(716\) 0 0
\(717\) 1.44450 + 2.51435i 0.0539460 + 0.0939003i
\(718\) 0 0
\(719\) 12.8737 7.43265i 0.480109 0.277191i −0.240353 0.970686i \(-0.577263\pi\)
0.720462 + 0.693494i \(0.243930\pi\)
\(720\) 0 0
\(721\) 0.0663225 0.0382913i 0.00246998 0.00142604i
\(722\) 0 0
\(723\) 10.5449 18.1744i 0.392168 0.675913i
\(724\) 0 0
\(725\) 10.8752 + 6.27881i 0.403896 + 0.233189i
\(726\) 0 0
\(727\) 1.33011 0.767938i 0.0493309 0.0284812i −0.475132 0.879915i \(-0.657600\pi\)
0.524463 + 0.851433i \(0.324266\pi\)
\(728\) 0 0
\(729\) 26.9978 + 0.346389i 0.999918 + 0.0128292i
\(730\) 0 0
\(731\) 39.1488 + 67.8076i 1.44797 + 2.50796i
\(732\) 0 0
\(733\) 43.9650 + 25.3832i 1.62388 + 0.937550i 0.985868 + 0.167527i \(0.0535780\pi\)
0.638016 + 0.770023i \(0.279755\pi\)
\(734\) 0 0
\(735\) 28.7395 + 0.0614525i 1.06007 + 0.00226671i
\(736\) 0 0
\(737\) 24.7587 + 30.4445i 0.911998 + 1.12144i
\(738\) 0 0
\(739\) 8.00593 4.62223i 0.294503 0.170031i −0.345468 0.938431i \(-0.612280\pi\)
0.639971 + 0.768399i \(0.278946\pi\)
\(740\) 0 0
\(741\) 51.2078 29.4190i 1.88117 1.08074i
\(742\) 0 0
\(743\) −12.0681 + 6.96754i −0.442737 + 0.255614i −0.704758 0.709448i \(-0.748944\pi\)
0.262021 + 0.965062i \(0.415611\pi\)
\(744\) 0 0
\(745\) 42.7617i 1.56667i
\(746\) 0 0
\(747\) 21.3242 + 12.4334i 0.780212 + 0.454915i
\(748\) 0 0
\(749\) 7.42501 12.8605i 0.271304 0.469912i
\(750\) 0 0
\(751\) −5.81892 −0.212335 −0.106168 0.994348i \(-0.533858\pi\)
−0.106168 + 0.994348i \(0.533858\pi\)
\(752\) 0 0
\(753\) 21.2563 + 0.0454516i 0.774623 + 0.00165635i
\(754\) 0 0
\(755\) 8.87895 + 15.3788i 0.323138 + 0.559691i
\(756\) 0 0
\(757\) 39.7347 + 22.9409i 1.44418 + 0.833800i 0.998125 0.0612044i \(-0.0194942\pi\)
0.446058 + 0.895004i \(0.352827\pi\)
\(758\) 0 0
\(759\) 12.1628 + 7.05694i 0.441483 + 0.256151i
\(760\) 0 0
\(761\) 19.1770i 0.695167i −0.937649 0.347584i \(-0.887002\pi\)
0.937649 0.347584i \(-0.112998\pi\)
\(762\) 0 0
\(763\) −1.42740 + 2.47233i −0.0516755 + 0.0895045i
\(764\) 0 0
\(765\) −44.4070 25.8922i −1.60554 0.936136i
\(766\) 0 0
\(767\) 2.00145 + 3.46661i 0.0722680 + 0.125172i
\(768\) 0 0
\(769\) −36.6831 21.1790i −1.32283 0.763734i −0.338647 0.940913i \(-0.609969\pi\)
−0.984179 + 0.177179i \(0.943303\pi\)
\(770\) 0 0
\(771\) −40.7398 + 23.4051i −1.46721 + 0.842914i
\(772\) 0 0
\(773\) −19.1042 11.0298i −0.687129 0.396714i 0.115407 0.993318i \(-0.463183\pi\)
−0.802536 + 0.596604i \(0.796516\pi\)
\(774\) 0 0
\(775\) −11.2435 6.49144i −0.403879 0.233180i
\(776\) 0 0
\(777\) −1.69814 0.985271i −0.0609206 0.0353464i
\(778\) 0 0
\(779\) −14.5162 −0.520097
\(780\) 0 0
\(781\) −6.62195 + 3.82318i −0.236952 + 0.136804i
\(782\) 0 0
\(783\) 26.4744 + 15.5123i 0.946118 + 0.554364i
\(784\) 0 0
\(785\) 26.9567 + 46.6904i 0.962126 + 1.66645i
\(786\) 0 0
\(787\) 18.2496 10.5364i 0.650528 0.375583i −0.138130 0.990414i \(-0.544109\pi\)
0.788659 + 0.614831i \(0.210776\pi\)
\(788\) 0 0
\(789\) 24.0741 + 13.9679i 0.857061 + 0.497271i
\(790\) 0 0
\(791\) 11.0336 6.37026i 0.392310 0.226500i
\(792\) 0 0
\(793\) 23.0499 39.9235i 0.818524 1.41773i
\(794\) 0 0
\(795\) 12.1833 20.9982i 0.432095 0.744729i
\(796\) 0 0
\(797\) −6.94721 4.01097i −0.246083 0.142076i 0.371886 0.928278i \(-0.378711\pi\)
−0.617969 + 0.786202i \(0.712044\pi\)
\(798\) 0 0
\(799\) 26.5603 0.939637
\(800\) 0 0
\(801\) −23.6797 + 13.5368i −0.836683 + 0.478300i
\(802\) 0 0
\(803\) −35.6611 −1.25845
\(804\) 0 0
\(805\) −4.00405 −0.141124
\(806\) 0 0
\(807\) −11.5975 6.72893i −0.408252 0.236870i
\(808\) 0 0
\(809\) −19.8709 −0.698623 −0.349311 0.937007i \(-0.613584\pi\)
−0.349311 + 0.937007i \(0.613584\pi\)
\(810\) 0 0
\(811\) −0.712665 0.411458i −0.0250251 0.0144482i 0.487435 0.873159i \(-0.337933\pi\)
−0.512460 + 0.858711i \(0.671266\pi\)
\(812\) 0 0
\(813\) 25.5193 + 14.8064i 0.895002 + 0.519284i
\(814\) 0 0
\(815\) 11.0659 19.1666i 0.387621 0.671378i
\(816\) 0 0
\(817\) −52.8593 + 30.5183i −1.84931 + 1.06770i
\(818\) 0 0
\(819\) 8.98608 + 15.7192i 0.313999 + 0.549274i
\(820\) 0 0
\(821\) −9.55155 + 5.51459i −0.333351 + 0.192461i −0.657328 0.753605i \(-0.728313\pi\)
0.323977 + 0.946065i \(0.394980\pi\)
\(822\) 0 0
\(823\) −2.64571 4.58250i −0.0922235 0.159736i 0.816223 0.577737i \(-0.196064\pi\)
−0.908446 + 0.418001i \(0.862731\pi\)
\(824\) 0 0
\(825\) −8.79620 15.3110i −0.306244 0.533060i
\(826\) 0 0
\(827\) 31.4356 18.1494i 1.09312 0.631115i 0.158718 0.987324i \(-0.449264\pi\)
0.934406 + 0.356209i \(0.115931\pi\)
\(828\) 0 0
\(829\) 5.65056 0.196252 0.0981261 0.995174i \(-0.468715\pi\)
0.0981261 + 0.995174i \(0.468715\pi\)
\(830\) 0 0
\(831\) −4.04183 + 6.96621i −0.140210 + 0.241655i
\(832\) 0 0
\(833\) −34.5500 19.9475i −1.19709 0.691138i
\(834\) 0 0
\(835\) 26.2291 + 15.1434i 0.907696 + 0.524059i
\(836\) 0 0
\(837\) −27.3710 16.0376i −0.946079 0.554341i
\(838\) 0 0
\(839\) −47.8662 27.6356i −1.65253 0.954086i −0.976029 0.217639i \(-0.930164\pi\)
−0.676496 0.736446i \(-0.736502\pi\)
\(840\) 0 0
\(841\) 2.93564 + 5.08467i 0.101229 + 0.175334i
\(842\) 0 0
\(843\) 33.1248 + 0.0708295i 1.14088 + 0.00243950i
\(844\) 0 0
\(845\) 44.6304 77.3021i 1.53533 2.65927i
\(846\) 0 0
\(847\) 10.6132i 0.364676i
\(848\) 0 0
\(849\) −17.8806 + 30.8177i −0.613659 + 1.05766i
\(850\) 0 0
\(851\) −1.87693 1.08364i −0.0643402 0.0371468i
\(852\) 0 0
\(853\) 21.5827 + 37.3823i 0.738976 + 1.27994i 0.952957 + 0.303106i \(0.0980236\pi\)
−0.213981 + 0.976838i \(0.568643\pi\)
\(854\) 0 0
\(855\) 20.1842 34.6174i 0.690286 1.18389i
\(856\) 0 0
\(857\) 28.9434 0.988687 0.494344 0.869267i \(-0.335409\pi\)
0.494344 + 0.869267i \(0.335409\pi\)
\(858\) 0 0
\(859\) 9.53921 16.5224i 0.325474 0.563737i −0.656135 0.754644i \(-0.727810\pi\)
0.981608 + 0.190907i \(0.0611430\pi\)
\(860\) 0 0
\(861\) 0.00951645 4.45055i 0.000324320 0.151674i
\(862\) 0 0
\(863\) 18.0028i 0.612824i 0.951899 + 0.306412i \(0.0991285\pi\)
−0.951899 + 0.306412i \(0.900871\pi\)
\(864\) 0 0
\(865\) 36.6473 21.1583i 1.24605 0.719405i
\(866\) 0 0
\(867\) 20.8784 + 36.3416i 0.709067 + 1.23423i
\(868\) 0 0
\(869\) −48.3236 + 27.8997i −1.63927 + 0.946431i
\(870\) 0 0
\(871\) 8.84043 + 55.0735i 0.299547 + 1.86610i
\(872\) 0 0
\(873\) −23.4487 13.6721i −0.793618 0.462732i
\(874\) 0 0
\(875\) −5.88379 3.39701i −0.198909 0.114840i
\(876\) 0 0
\(877\) −2.51276 4.35223i −0.0848499 0.146964i 0.820477 0.571679i \(-0.193708\pi\)
−0.905327 + 0.424715i \(0.860374\pi\)
\(878\) 0 0
\(879\) 41.9314 + 24.3288i 1.41431 + 0.820590i
\(880\) 0 0
\(881\) 6.59259 3.80623i 0.222110 0.128235i −0.384817 0.922993i \(-0.625735\pi\)
0.606927 + 0.794758i \(0.292402\pi\)
\(882\) 0 0
\(883\) −15.7960 9.11982i −0.531577 0.306906i 0.210081 0.977684i \(-0.432627\pi\)
−0.741659 + 0.670778i \(0.765961\pi\)
\(884\) 0 0
\(885\) 2.34929 + 1.36307i 0.0789706 + 0.0458191i
\(886\) 0 0
\(887\) −40.0349 + 23.1142i −1.34424 + 0.776098i −0.987427 0.158077i \(-0.949471\pi\)
−0.356815 + 0.934175i \(0.616137\pi\)
\(888\) 0 0
\(889\) 2.53066 1.46108i 0.0848758 0.0490030i
\(890\) 0 0
\(891\) −21.2529 37.5492i −0.711999 1.25794i
\(892\) 0 0
\(893\) 20.7051i 0.692868i
\(894\) 0 0
\(895\) −28.8333 −0.963792
\(896\) 0 0
\(897\) 9.95694 + 17.3314i 0.332453 + 0.578679i
\(898\) 0 0
\(899\) −18.0261 31.2221i −0.601204 1.04132i
\(900\) 0 0
\(901\) −29.1848 + 16.8499i −0.972288 + 0.561350i
\(902\) 0 0
\(903\) −9.32203 16.2263i −0.310218 0.539976i
\(904\) 0 0
\(905\) 11.9214 20.6485i 0.396281 0.686378i
\(906\) 0 0
\(907\) 12.4865 21.6273i 0.414609 0.718124i −0.580778 0.814062i \(-0.697252\pi\)
0.995387 + 0.0959381i \(0.0305851\pi\)
\(908\) 0 0
\(909\) −41.0327 0.175478i −1.36097 0.00582025i
\(910\) 0 0
\(911\) 7.82418i 0.259227i 0.991565 + 0.129613i \(0.0413736\pi\)
−0.991565 + 0.129613i \(0.958626\pi\)
\(912\) 0 0
\(913\) 39.4459i 1.30547i
\(914\) 0 0
\(915\) 0.0668850 31.2800i 0.00221115 1.03409i
\(916\) 0 0
\(917\) 3.10061 5.37041i 0.102391 0.177346i
\(918\) 0 0
\(919\) 19.5487 11.2864i 0.644851 0.372305i −0.141630 0.989920i \(-0.545234\pi\)
0.786481 + 0.617615i \(0.211901\pi\)
\(920\) 0 0
\(921\) −0.801019 0.00171279i −0.0263945 5.64383e-5i
\(922\) 0 0
\(923\) −10.8688 −0.357751
\(924\) 0 0
\(925\) 1.36076 + 2.35691i 0.0447416 + 0.0774947i
\(926\) 0 0
\(927\) −0.259397 0.00110932i −0.00851972 3.64350e-5i
\(928\) 0 0
\(929\) 22.2867 0.731202 0.365601 0.930772i \(-0.380863\pi\)
0.365601 + 0.930772i \(0.380863\pi\)
\(930\) 0 0
\(931\) 15.5500 26.9334i 0.509630 0.882706i
\(932\) 0 0
\(933\) 8.39538 14.4697i 0.274852 0.473716i
\(934\) 0 0
\(935\) 82.1449i 2.68643i
\(936\) 0 0
\(937\) 15.4389i 0.504366i 0.967680 + 0.252183i \(0.0811484\pi\)
−0.967680 + 0.252183i \(0.918852\pi\)
\(938\) 0 0
\(939\) −19.7739 11.4729i −0.645297 0.374404i
\(940\) 0 0
\(941\) 29.8924 0.974466 0.487233 0.873272i \(-0.338006\pi\)
0.487233 + 0.873272i \(0.338006\pi\)
\(942\) 0 0
\(943\) 4.91304i 0.159990i
\(944\) 0 0
\(945\) 10.6002 + 6.21103i 0.344824 + 0.202045i
\(946\) 0 0
\(947\) 37.2442i 1.21027i −0.796122 0.605136i \(-0.793119\pi\)
0.796122 0.605136i \(-0.206881\pi\)
\(948\) 0 0
\(949\) −43.8987 25.3449i −1.42501 0.822732i
\(950\) 0 0
\(951\) −0.0516023 + 24.1328i −0.00167332 + 0.782560i
\(952\) 0 0
\(953\) 40.6494i 1.31676i −0.752684 0.658382i \(-0.771241\pi\)
0.752684 0.658382i \(-0.228759\pi\)
\(954\) 0 0
\(955\) 22.5514 + 39.0602i 0.729747 + 1.26396i
\(956\) 0 0
\(957\) 0.104848 49.0340i 0.00338924 1.58504i
\(958\) 0 0
\(959\) 1.48635 + 0.858144i 0.0479967 + 0.0277109i
\(960\) 0 0
\(961\) 3.13656 + 5.43268i 0.101179 + 0.175248i
\(962\) 0 0
\(963\) −43.6680 + 24.9634i −1.40718 + 0.804433i
\(964\) 0 0
\(965\) 65.4363 2.10647
\(966\) 0 0
\(967\) 3.93507 6.81574i 0.126543 0.219180i −0.795792 0.605570i \(-0.792945\pi\)
0.922335 + 0.386391i \(0.126278\pi\)
\(968\) 0 0
\(969\) −48.2330 + 27.7100i −1.54947 + 0.890173i
\(970\) 0 0
\(971\) 33.4335 + 19.3028i 1.07293 + 0.619457i 0.928981 0.370128i \(-0.120686\pi\)
0.143950 + 0.989585i \(0.454020\pi\)
\(972\) 0 0
\(973\) −6.15379 + 10.6587i −0.197282 + 0.341702i
\(974\) 0 0
\(975\) 0.0536691 25.0994i 0.00171879 0.803823i
\(976\) 0 0
\(977\) −21.1843 + 12.2308i −0.677745 + 0.391296i −0.799005 0.601324i \(-0.794640\pi\)
0.121260 + 0.992621i \(0.461307\pi\)
\(978\) 0 0
\(979\) 37.7479 + 21.7937i 1.20643 + 0.696531i
\(980\) 0 0
\(981\) 8.39486 4.79902i 0.268027 0.153221i
\(982\) 0 0
\(983\) −32.0043 −1.02078 −0.510390 0.859943i \(-0.670499\pi\)
−0.510390 + 0.859943i \(0.670499\pi\)
\(984\) 0 0
\(985\) 29.7564 51.5395i 0.948116 1.64219i
\(986\) 0 0
\(987\) −6.34801 0.0135737i −0.202059 0.000432056i
\(988\) 0 0
\(989\) −10.3290 17.8903i −0.328443 0.568880i
\(990\) 0 0
\(991\) 36.6991i 1.16578i 0.812550 + 0.582892i \(0.198079\pi\)
−0.812550 + 0.582892i \(0.801921\pi\)
\(992\) 0 0
\(993\) 4.43885 2.55013i 0.140863 0.0809259i
\(994\) 0 0
\(995\) −10.3220 17.8783i −0.327230 0.566779i
\(996\) 0 0
\(997\) −14.4622 −0.458022 −0.229011 0.973424i \(-0.573549\pi\)
−0.229011 + 0.973424i \(0.573549\pi\)
\(998\) 0 0
\(999\) 3.28798 + 5.78026i 0.104027 + 0.182879i
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 804.2.o.d.365.7 36
3.2 odd 2 inner 804.2.o.d.365.12 yes 36
67.38 odd 6 inner 804.2.o.d.641.12 yes 36
201.38 even 6 inner 804.2.o.d.641.7 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
804.2.o.d.365.7 36 1.1 even 1 trivial
804.2.o.d.365.12 yes 36 3.2 odd 2 inner
804.2.o.d.641.7 yes 36 201.38 even 6 inner
804.2.o.d.641.12 yes 36 67.38 odd 6 inner