Properties

Label 6960.2.a.co.1.2
Level $6960$
Weight $2$
Character 6960.1
Self dual yes
Analytic conductor $55.576$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.a (trivial)

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: yes
Analytic conductor: \(55.5758798068\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 6960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.393832 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.393832 q^{7} +1.00000 q^{9} +0.393832 q^{11} -2.56511 q^{13} -1.00000 q^{15} +2.07830 q^{17} +0.958939 q^{19} -0.393832 q^{21} -6.15661 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} +10.1566 q^{31} +0.393832 q^{33} +0.393832 q^{35} -7.34192 q^{37} -2.56511 q^{39} -1.65745 q^{41} -10.3279 q^{43} -1.00000 q^{45} +11.5915 q^{47} -6.84490 q^{49} +2.07830 q^{51} -12.3279 q^{53} -0.393832 q^{55} +0.958939 q^{57} +9.54022 q^{59} -6.25340 q^{61} -0.393832 q^{63} +2.56511 q^{65} +7.42023 q^{67} -6.15661 q^{69} -5.98533 q^{71} +3.34192 q^{73} +1.00000 q^{75} -0.155104 q^{77} +2.06745 q^{79} +1.00000 q^{81} -6.41000 q^{83} -2.07830 q^{85} -1.00000 q^{87} +15.8302 q^{89} +1.01022 q^{91} +10.1566 q^{93} -0.958939 q^{95} -18.4575 q^{97} +0.393832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.393832 −0.148855 −0.0744273 0.997226i \(-0.523713\pi\)
−0.0744273 + 0.997226i \(0.523713\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.393832 0.118745 0.0593725 0.998236i \(-0.481090\pi\)
0.0593725 + 0.998236i \(0.481090\pi\)
\(12\) 0 0
\(13\) −2.56511 −0.711433 −0.355716 0.934594i \(-0.615763\pi\)
−0.355716 + 0.934594i \(0.615763\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.07830 0.504063 0.252031 0.967719i \(-0.418901\pi\)
0.252031 + 0.967719i \(0.418901\pi\)
\(18\) 0 0
\(19\) 0.958939 0.219996 0.109998 0.993932i \(-0.464916\pi\)
0.109998 + 0.993932i \(0.464916\pi\)
\(20\) 0 0
\(21\) −0.393832 −0.0859413
\(22\) 0 0
\(23\) −6.15661 −1.28374 −0.641871 0.766813i \(-0.721841\pi\)
−0.641871 + 0.766813i \(0.721841\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.1566 1.82418 0.912090 0.409989i \(-0.134468\pi\)
0.912090 + 0.409989i \(0.134468\pi\)
\(32\) 0 0
\(33\) 0.393832 0.0685574
\(34\) 0 0
\(35\) 0.393832 0.0665698
\(36\) 0 0
\(37\) −7.34192 −1.20700 −0.603502 0.797361i \(-0.706229\pi\)
−0.603502 + 0.797361i \(0.706229\pi\)
\(38\) 0 0
\(39\) −2.56511 −0.410746
\(40\) 0 0
\(41\) −1.65745 −0.258850 −0.129425 0.991589i \(-0.541313\pi\)
−0.129425 + 0.991589i \(0.541313\pi\)
\(42\) 0 0
\(43\) −10.3279 −1.57499 −0.787494 0.616323i \(-0.788622\pi\)
−0.787494 + 0.616323i \(0.788622\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 11.5915 1.69079 0.845397 0.534138i \(-0.179364\pi\)
0.845397 + 0.534138i \(0.179364\pi\)
\(48\) 0 0
\(49\) −6.84490 −0.977842
\(50\) 0 0
\(51\) 2.07830 0.291021
\(52\) 0 0
\(53\) −12.3279 −1.69336 −0.846682 0.532099i \(-0.821404\pi\)
−0.846682 + 0.532099i \(0.821404\pi\)
\(54\) 0 0
\(55\) −0.393832 −0.0531043
\(56\) 0 0
\(57\) 0.958939 0.127015
\(58\) 0 0
\(59\) 9.54022 1.24203 0.621015 0.783798i \(-0.286720\pi\)
0.621015 + 0.783798i \(0.286720\pi\)
\(60\) 0 0
\(61\) −6.25340 −0.800665 −0.400333 0.916370i \(-0.631105\pi\)
−0.400333 + 0.916370i \(0.631105\pi\)
\(62\) 0 0
\(63\) −0.393832 −0.0496182
\(64\) 0 0
\(65\) 2.56511 0.318162
\(66\) 0 0
\(67\) 7.42023 0.906525 0.453262 0.891377i \(-0.350260\pi\)
0.453262 + 0.891377i \(0.350260\pi\)
\(68\) 0 0
\(69\) −6.15661 −0.741168
\(70\) 0 0
\(71\) −5.98533 −0.710328 −0.355164 0.934804i \(-0.615575\pi\)
−0.355164 + 0.934804i \(0.615575\pi\)
\(72\) 0 0
\(73\) 3.34192 0.391142 0.195571 0.980690i \(-0.437344\pi\)
0.195571 + 0.980690i \(0.437344\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.155104 −0.0176757
\(78\) 0 0
\(79\) 2.06745 0.232607 0.116303 0.993214i \(-0.462896\pi\)
0.116303 + 0.993214i \(0.462896\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.41000 −0.703589 −0.351795 0.936077i \(-0.614428\pi\)
−0.351795 + 0.936077i \(0.614428\pi\)
\(84\) 0 0
\(85\) −2.07830 −0.225424
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 15.8302 1.67800 0.839000 0.544131i \(-0.183140\pi\)
0.839000 + 0.544131i \(0.183140\pi\)
\(90\) 0 0
\(91\) 1.01022 0.105900
\(92\) 0 0
\(93\) 10.1566 1.05319
\(94\) 0 0
\(95\) −0.958939 −0.0983851
\(96\) 0 0
\(97\) −18.4575 −1.87407 −0.937036 0.349233i \(-0.886442\pi\)
−0.937036 + 0.349233i \(0.886442\pi\)
\(98\) 0 0
\(99\) 0.393832 0.0395816
\(100\) 0 0
\(101\) −12.8038 −1.27403 −0.637015 0.770852i \(-0.719831\pi\)
−0.637015 + 0.770852i \(0.719831\pi\)
\(102\) 0 0
\(103\) −4.86979 −0.479834 −0.239917 0.970793i \(-0.577120\pi\)
−0.239917 + 0.970793i \(0.577120\pi\)
\(104\) 0 0
\(105\) 0.393832 0.0384341
\(106\) 0 0
\(107\) 2.34255 0.226463 0.113231 0.993569i \(-0.463880\pi\)
0.113231 + 0.993569i \(0.463880\pi\)
\(108\) 0 0
\(109\) 8.55044 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(110\) 0 0
\(111\) −7.34192 −0.696864
\(112\) 0 0
\(113\) −11.2085 −1.05441 −0.527204 0.849739i \(-0.676760\pi\)
−0.527204 + 0.849739i \(0.676760\pi\)
\(114\) 0 0
\(115\) 6.15661 0.574107
\(116\) 0 0
\(117\) −2.56511 −0.237144
\(118\) 0 0
\(119\) −0.818503 −0.0750321
\(120\) 0 0
\(121\) −10.8449 −0.985900
\(122\) 0 0
\(123\) −1.65745 −0.149447
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −20.1566 −1.78861 −0.894305 0.447458i \(-0.852329\pi\)
−0.894305 + 0.447458i \(0.852329\pi\)
\(128\) 0 0
\(129\) −10.3279 −0.909319
\(130\) 0 0
\(131\) 5.50235 0.480742 0.240371 0.970681i \(-0.422731\pi\)
0.240371 + 0.970681i \(0.422731\pi\)
\(132\) 0 0
\(133\) −0.377661 −0.0327474
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 7.49853 0.640643 0.320321 0.947309i \(-0.396209\pi\)
0.320321 + 0.947309i \(0.396209\pi\)
\(138\) 0 0
\(139\) 9.35277 0.793292 0.396646 0.917972i \(-0.370174\pi\)
0.396646 + 0.917972i \(0.370174\pi\)
\(140\) 0 0
\(141\) 11.5915 0.976180
\(142\) 0 0
\(143\) −1.01022 −0.0844790
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) −6.84490 −0.564558
\(148\) 0 0
\(149\) −11.5402 −0.945411 −0.472706 0.881220i \(-0.656723\pi\)
−0.472706 + 0.881220i \(0.656723\pi\)
\(150\) 0 0
\(151\) −6.08212 −0.494956 −0.247478 0.968894i \(-0.579602\pi\)
−0.247478 + 0.968894i \(0.579602\pi\)
\(152\) 0 0
\(153\) 2.07830 0.168021
\(154\) 0 0
\(155\) −10.1566 −0.815798
\(156\) 0 0
\(157\) −11.5953 −0.925407 −0.462704 0.886513i \(-0.653121\pi\)
−0.462704 + 0.886513i \(0.653121\pi\)
\(158\) 0 0
\(159\) −12.3279 −0.977665
\(160\) 0 0
\(161\) 2.42467 0.191091
\(162\) 0 0
\(163\) 0.855118 0.0669780 0.0334890 0.999439i \(-0.489338\pi\)
0.0334890 + 0.999439i \(0.489338\pi\)
\(164\) 0 0
\(165\) −0.393832 −0.0306598
\(166\) 0 0
\(167\) −7.73194 −0.598315 −0.299158 0.954204i \(-0.596706\pi\)
−0.299158 + 0.954204i \(0.596706\pi\)
\(168\) 0 0
\(169\) −6.42023 −0.493863
\(170\) 0 0
\(171\) 0.958939 0.0733319
\(172\) 0 0
\(173\) 8.07449 0.613892 0.306946 0.951727i \(-0.400693\pi\)
0.306946 + 0.951727i \(0.400693\pi\)
\(174\) 0 0
\(175\) −0.393832 −0.0297709
\(176\) 0 0
\(177\) 9.54022 0.717087
\(178\) 0 0
\(179\) −12.4100 −0.927567 −0.463784 0.885949i \(-0.653508\pi\)
−0.463784 + 0.885949i \(0.653508\pi\)
\(180\) 0 0
\(181\) −2.49765 −0.185649 −0.0928246 0.995682i \(-0.529590\pi\)
−0.0928246 + 0.995682i \(0.529590\pi\)
\(182\) 0 0
\(183\) −6.25340 −0.462264
\(184\) 0 0
\(185\) 7.34192 0.539789
\(186\) 0 0
\(187\) 0.818503 0.0598549
\(188\) 0 0
\(189\) −0.393832 −0.0286471
\(190\) 0 0
\(191\) 6.32085 0.457361 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(192\) 0 0
\(193\) 25.2921 1.82057 0.910284 0.413984i \(-0.135863\pi\)
0.910284 + 0.413984i \(0.135863\pi\)
\(194\) 0 0
\(195\) 2.56511 0.183691
\(196\) 0 0
\(197\) −21.2047 −1.51077 −0.755386 0.655280i \(-0.772551\pi\)
−0.755386 + 0.655280i \(0.772551\pi\)
\(198\) 0 0
\(199\) 2.64723 0.187657 0.0938285 0.995588i \(-0.470089\pi\)
0.0938285 + 0.995588i \(0.470089\pi\)
\(200\) 0 0
\(201\) 7.42023 0.523382
\(202\) 0 0
\(203\) 0.393832 0.0276416
\(204\) 0 0
\(205\) 1.65745 0.115761
\(206\) 0 0
\(207\) −6.15661 −0.427914
\(208\) 0 0
\(209\) 0.377661 0.0261234
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) −5.98533 −0.410108
\(214\) 0 0
\(215\) 10.3279 0.704356
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 3.34192 0.225826
\(220\) 0 0
\(221\) −5.33107 −0.358607
\(222\) 0 0
\(223\) −12.5504 −0.840440 −0.420220 0.907422i \(-0.638047\pi\)
−0.420220 + 0.907422i \(0.638047\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −1.30149 −0.0863829 −0.0431914 0.999067i \(-0.513753\pi\)
−0.0431914 + 0.999067i \(0.513753\pi\)
\(228\) 0 0
\(229\) 3.28682 0.217199 0.108600 0.994086i \(-0.465363\pi\)
0.108600 + 0.994086i \(0.465363\pi\)
\(230\) 0 0
\(231\) −0.155104 −0.0102051
\(232\) 0 0
\(233\) 17.7115 1.16032 0.580159 0.814503i \(-0.302990\pi\)
0.580159 + 0.814503i \(0.302990\pi\)
\(234\) 0 0
\(235\) −11.5915 −0.756146
\(236\) 0 0
\(237\) 2.06745 0.134296
\(238\) 0 0
\(239\) −21.2651 −1.37553 −0.687763 0.725935i \(-0.741407\pi\)
−0.687763 + 0.725935i \(0.741407\pi\)
\(240\) 0 0
\(241\) 15.3177 0.986697 0.493349 0.869832i \(-0.335773\pi\)
0.493349 + 0.869832i \(0.335773\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.84490 0.437304
\(246\) 0 0
\(247\) −2.45978 −0.156512
\(248\) 0 0
\(249\) −6.41000 −0.406217
\(250\) 0 0
\(251\) −26.8917 −1.69739 −0.848696 0.528882i \(-0.822612\pi\)
−0.848696 + 0.528882i \(0.822612\pi\)
\(252\) 0 0
\(253\) −2.42467 −0.152438
\(254\) 0 0
\(255\) −2.07830 −0.130148
\(256\) 0 0
\(257\) −6.85512 −0.427611 −0.213805 0.976876i \(-0.568586\pi\)
−0.213805 + 0.976876i \(0.568586\pi\)
\(258\) 0 0
\(259\) 2.89149 0.179668
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 13.6294 0.840423 0.420212 0.907426i \(-0.361956\pi\)
0.420212 + 0.907426i \(0.361956\pi\)
\(264\) 0 0
\(265\) 12.3279 0.757296
\(266\) 0 0
\(267\) 15.8302 0.968794
\(268\) 0 0
\(269\) −8.46129 −0.515894 −0.257947 0.966159i \(-0.583046\pi\)
−0.257947 + 0.966159i \(0.583046\pi\)
\(270\) 0 0
\(271\) −6.34255 −0.385282 −0.192641 0.981269i \(-0.561705\pi\)
−0.192641 + 0.981269i \(0.561705\pi\)
\(272\) 0 0
\(273\) 1.01022 0.0611414
\(274\) 0 0
\(275\) 0.393832 0.0237490
\(276\) 0 0
\(277\) −17.1464 −1.03023 −0.515113 0.857122i \(-0.672250\pi\)
−0.515113 + 0.857122i \(0.672250\pi\)
\(278\) 0 0
\(279\) 10.1566 0.608060
\(280\) 0 0
\(281\) −12.2985 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(282\) 0 0
\(283\) −25.1830 −1.49697 −0.748487 0.663149i \(-0.769219\pi\)
−0.748487 + 0.663149i \(0.769219\pi\)
\(284\) 0 0
\(285\) −0.958939 −0.0568027
\(286\) 0 0
\(287\) 0.652757 0.0385311
\(288\) 0 0
\(289\) −12.6807 −0.745921
\(290\) 0 0
\(291\) −18.4575 −1.08200
\(292\) 0 0
\(293\) 12.3170 0.719569 0.359784 0.933035i \(-0.382850\pi\)
0.359784 + 0.933035i \(0.382850\pi\)
\(294\) 0 0
\(295\) −9.54022 −0.555453
\(296\) 0 0
\(297\) 0.393832 0.0228525
\(298\) 0 0
\(299\) 15.7924 0.913296
\(300\) 0 0
\(301\) 4.06745 0.234444
\(302\) 0 0
\(303\) −12.8038 −0.735561
\(304\) 0 0
\(305\) 6.25340 0.358068
\(306\) 0 0
\(307\) 22.7379 1.29772 0.648860 0.760908i \(-0.275246\pi\)
0.648860 + 0.760908i \(0.275246\pi\)
\(308\) 0 0
\(309\) −4.86979 −0.277032
\(310\) 0 0
\(311\) 5.33810 0.302696 0.151348 0.988481i \(-0.451639\pi\)
0.151348 + 0.988481i \(0.451639\pi\)
\(312\) 0 0
\(313\) −1.96338 −0.110977 −0.0554885 0.998459i \(-0.517672\pi\)
−0.0554885 + 0.998459i \(0.517672\pi\)
\(314\) 0 0
\(315\) 0.393832 0.0221899
\(316\) 0 0
\(317\) 1.29064 0.0724895 0.0362448 0.999343i \(-0.488460\pi\)
0.0362448 + 0.999343i \(0.488460\pi\)
\(318\) 0 0
\(319\) −0.393832 −0.0220504
\(320\) 0 0
\(321\) 2.34255 0.130748
\(322\) 0 0
\(323\) 1.99297 0.110892
\(324\) 0 0
\(325\) −2.56511 −0.142287
\(326\) 0 0
\(327\) 8.55044 0.472840
\(328\) 0 0
\(329\) −4.56511 −0.251683
\(330\) 0 0
\(331\) −28.9971 −1.59382 −0.796911 0.604096i \(-0.793534\pi\)
−0.796911 + 0.604096i \(0.793534\pi\)
\(332\) 0 0
\(333\) −7.34192 −0.402335
\(334\) 0 0
\(335\) −7.42023 −0.405410
\(336\) 0 0
\(337\) −16.7854 −0.914356 −0.457178 0.889375i \(-0.651140\pi\)
−0.457178 + 0.889375i \(0.651140\pi\)
\(338\) 0 0
\(339\) −11.2085 −0.608763
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 5.45257 0.294411
\(344\) 0 0
\(345\) 6.15661 0.331461
\(346\) 0 0
\(347\) 17.8681 0.959210 0.479605 0.877485i \(-0.340780\pi\)
0.479605 + 0.877485i \(0.340780\pi\)
\(348\) 0 0
\(349\) 8.73789 0.467728 0.233864 0.972269i \(-0.424863\pi\)
0.233864 + 0.972269i \(0.424863\pi\)
\(350\) 0 0
\(351\) −2.56511 −0.136915
\(352\) 0 0
\(353\) −22.6017 −1.20297 −0.601484 0.798885i \(-0.705424\pi\)
−0.601484 + 0.798885i \(0.705424\pi\)
\(354\) 0 0
\(355\) 5.98533 0.317668
\(356\) 0 0
\(357\) −0.818503 −0.0433198
\(358\) 0 0
\(359\) 13.1830 0.695772 0.347886 0.937537i \(-0.386900\pi\)
0.347886 + 0.937537i \(0.386900\pi\)
\(360\) 0 0
\(361\) −18.0804 −0.951602
\(362\) 0 0
\(363\) −10.8449 −0.569209
\(364\) 0 0
\(365\) −3.34192 −0.174924
\(366\) 0 0
\(367\) −17.4511 −0.910938 −0.455469 0.890252i \(-0.650528\pi\)
−0.455469 + 0.890252i \(0.650528\pi\)
\(368\) 0 0
\(369\) −1.65745 −0.0862834
\(370\) 0 0
\(371\) 4.85512 0.252065
\(372\) 0 0
\(373\) −32.2018 −1.66734 −0.833672 0.552260i \(-0.813766\pi\)
−0.833672 + 0.552260i \(0.813766\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 2.56511 0.132110
\(378\) 0 0
\(379\) −32.3660 −1.66253 −0.831265 0.555876i \(-0.812383\pi\)
−0.831265 + 0.555876i \(0.812383\pi\)
\(380\) 0 0
\(381\) −20.1566 −1.03265
\(382\) 0 0
\(383\) 25.8739 1.32209 0.661047 0.750345i \(-0.270113\pi\)
0.661047 + 0.750345i \(0.270113\pi\)
\(384\) 0 0
\(385\) 0.155104 0.00790483
\(386\) 0 0
\(387\) −10.3279 −0.524996
\(388\) 0 0
\(389\) 32.6103 1.65341 0.826703 0.562639i \(-0.190214\pi\)
0.826703 + 0.562639i \(0.190214\pi\)
\(390\) 0 0
\(391\) −12.7953 −0.647086
\(392\) 0 0
\(393\) 5.50235 0.277557
\(394\) 0 0
\(395\) −2.06745 −0.104025
\(396\) 0 0
\(397\) 4.39534 0.220596 0.110298 0.993899i \(-0.464820\pi\)
0.110298 + 0.993899i \(0.464820\pi\)
\(398\) 0 0
\(399\) −0.377661 −0.0189067
\(400\) 0 0
\(401\) −19.0658 −0.952099 −0.476049 0.879418i \(-0.657932\pi\)
−0.476049 + 0.879418i \(0.657932\pi\)
\(402\) 0 0
\(403\) −26.0528 −1.29778
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −2.89149 −0.143326
\(408\) 0 0
\(409\) −1.38235 −0.0683530 −0.0341765 0.999416i \(-0.510881\pi\)
−0.0341765 + 0.999416i \(0.510881\pi\)
\(410\) 0 0
\(411\) 7.49853 0.369875
\(412\) 0 0
\(413\) −3.75725 −0.184882
\(414\) 0 0
\(415\) 6.41000 0.314655
\(416\) 0 0
\(417\) 9.35277 0.458007
\(418\) 0 0
\(419\) −2.86979 −0.140198 −0.0700991 0.997540i \(-0.522332\pi\)
−0.0700991 + 0.997540i \(0.522332\pi\)
\(420\) 0 0
\(421\) 13.6440 0.664970 0.332485 0.943109i \(-0.392113\pi\)
0.332485 + 0.943109i \(0.392113\pi\)
\(422\) 0 0
\(423\) 11.5915 0.563598
\(424\) 0 0
\(425\) 2.07830 0.100813
\(426\) 0 0
\(427\) 2.46279 0.119183
\(428\) 0 0
\(429\) −1.01022 −0.0487740
\(430\) 0 0
\(431\) 11.9707 0.576607 0.288303 0.957539i \(-0.406909\pi\)
0.288303 + 0.957539i \(0.406909\pi\)
\(432\) 0 0
\(433\) 17.9907 0.864576 0.432288 0.901736i \(-0.357706\pi\)
0.432288 + 0.901736i \(0.357706\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) −5.90381 −0.282418
\(438\) 0 0
\(439\) 16.2226 0.774260 0.387130 0.922025i \(-0.373466\pi\)
0.387130 + 0.922025i \(0.373466\pi\)
\(440\) 0 0
\(441\) −6.84490 −0.325947
\(442\) 0 0
\(443\) 35.3762 1.68078 0.840388 0.541986i \(-0.182327\pi\)
0.840388 + 0.541986i \(0.182327\pi\)
\(444\) 0 0
\(445\) −15.8302 −0.750425
\(446\) 0 0
\(447\) −11.5402 −0.545834
\(448\) 0 0
\(449\) 41.6971 1.96781 0.983903 0.178702i \(-0.0571898\pi\)
0.983903 + 0.178702i \(0.0571898\pi\)
\(450\) 0 0
\(451\) −0.652757 −0.0307371
\(452\) 0 0
\(453\) −6.08212 −0.285763
\(454\) 0 0
\(455\) −1.01022 −0.0473599
\(456\) 0 0
\(457\) −16.1111 −0.753646 −0.376823 0.926285i \(-0.622983\pi\)
−0.376823 + 0.926285i \(0.622983\pi\)
\(458\) 0 0
\(459\) 2.07830 0.0970069
\(460\) 0 0
\(461\) 17.3396 0.807586 0.403793 0.914850i \(-0.367692\pi\)
0.403793 + 0.914850i \(0.367692\pi\)
\(462\) 0 0
\(463\) −19.3177 −0.897768 −0.448884 0.893590i \(-0.648178\pi\)
−0.448884 + 0.893590i \(0.648178\pi\)
\(464\) 0 0
\(465\) −10.1566 −0.471001
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −2.92232 −0.134940
\(470\) 0 0
\(471\) −11.5953 −0.534284
\(472\) 0 0
\(473\) −4.06745 −0.187022
\(474\) 0 0
\(475\) 0.958939 0.0439992
\(476\) 0 0
\(477\) −12.3279 −0.564455
\(478\) 0 0
\(479\) 40.3877 1.84536 0.922681 0.385565i \(-0.125994\pi\)
0.922681 + 0.385565i \(0.125994\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) 0 0
\(483\) 2.42467 0.110326
\(484\) 0 0
\(485\) 18.4575 0.838110
\(486\) 0 0
\(487\) −2.84171 −0.128770 −0.0643850 0.997925i \(-0.520509\pi\)
−0.0643850 + 0.997925i \(0.520509\pi\)
\(488\) 0 0
\(489\) 0.855118 0.0386698
\(490\) 0 0
\(491\) −0.157863 −0.00712428 −0.00356214 0.999994i \(-0.501134\pi\)
−0.00356214 + 0.999994i \(0.501134\pi\)
\(492\) 0 0
\(493\) −2.07830 −0.0936021
\(494\) 0 0
\(495\) −0.393832 −0.0177014
\(496\) 0 0
\(497\) 2.35722 0.105736
\(498\) 0 0
\(499\) −2.64723 −0.118506 −0.0592531 0.998243i \(-0.518872\pi\)
−0.0592531 + 0.998243i \(0.518872\pi\)
\(500\) 0 0
\(501\) −7.73194 −0.345437
\(502\) 0 0
\(503\) −13.9545 −0.622200 −0.311100 0.950377i \(-0.600697\pi\)
−0.311100 + 0.950377i \(0.600697\pi\)
\(504\) 0 0
\(505\) 12.8038 0.569763
\(506\) 0 0
\(507\) −6.42023 −0.285132
\(508\) 0 0
\(509\) −16.4921 −0.731001 −0.365500 0.930811i \(-0.619102\pi\)
−0.365500 + 0.930811i \(0.619102\pi\)
\(510\) 0 0
\(511\) −1.31616 −0.0582233
\(512\) 0 0
\(513\) 0.958939 0.0423382
\(514\) 0 0
\(515\) 4.86979 0.214588
\(516\) 0 0
\(517\) 4.56511 0.200773
\(518\) 0 0
\(519\) 8.07449 0.354431
\(520\) 0 0
\(521\) −12.2253 −0.535601 −0.267800 0.963474i \(-0.586297\pi\)
−0.267800 + 0.963474i \(0.586297\pi\)
\(522\) 0 0
\(523\) −1.66659 −0.0728748 −0.0364374 0.999336i \(-0.511601\pi\)
−0.0364374 + 0.999336i \(0.511601\pi\)
\(524\) 0 0
\(525\) −0.393832 −0.0171883
\(526\) 0 0
\(527\) 21.1085 0.919501
\(528\) 0 0
\(529\) 14.9038 0.647992
\(530\) 0 0
\(531\) 9.54022 0.414010
\(532\) 0 0
\(533\) 4.25154 0.184155
\(534\) 0 0
\(535\) −2.34255 −0.101277
\(536\) 0 0
\(537\) −12.4100 −0.535531
\(538\) 0 0
\(539\) −2.69574 −0.116114
\(540\) 0 0
\(541\) −34.6558 −1.48997 −0.744984 0.667083i \(-0.767543\pi\)
−0.744984 + 0.667083i \(0.767543\pi\)
\(542\) 0 0
\(543\) −2.49765 −0.107185
\(544\) 0 0
\(545\) −8.55044 −0.366261
\(546\) 0 0
\(547\) −38.2530 −1.63558 −0.817791 0.575515i \(-0.804801\pi\)
−0.817791 + 0.575515i \(0.804801\pi\)
\(548\) 0 0
\(549\) −6.25340 −0.266888
\(550\) 0 0
\(551\) −0.958939 −0.0408522
\(552\) 0 0
\(553\) −0.814230 −0.0346246
\(554\) 0 0
\(555\) 7.34192 0.311647
\(556\) 0 0
\(557\) −28.7760 −1.21928 −0.609639 0.792679i \(-0.708686\pi\)
−0.609639 + 0.792679i \(0.708686\pi\)
\(558\) 0 0
\(559\) 26.4921 1.12050
\(560\) 0 0
\(561\) 0.818503 0.0345572
\(562\) 0 0
\(563\) 32.9809 1.38998 0.694989 0.719020i \(-0.255409\pi\)
0.694989 + 0.719020i \(0.255409\pi\)
\(564\) 0 0
\(565\) 11.2085 0.471546
\(566\) 0 0
\(567\) −0.393832 −0.0165394
\(568\) 0 0
\(569\) 16.8038 0.704453 0.352227 0.935915i \(-0.385425\pi\)
0.352227 + 0.935915i \(0.385425\pi\)
\(570\) 0 0
\(571\) 6.21703 0.260175 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(572\) 0 0
\(573\) 6.32085 0.264057
\(574\) 0 0
\(575\) −6.15661 −0.256748
\(576\) 0 0
\(577\) 11.9977 0.499470 0.249735 0.968314i \(-0.419656\pi\)
0.249735 + 0.968314i \(0.419656\pi\)
\(578\) 0 0
\(579\) 25.2921 1.05111
\(580\) 0 0
\(581\) 2.52447 0.104733
\(582\) 0 0
\(583\) −4.85512 −0.201078
\(584\) 0 0
\(585\) 2.56511 0.106054
\(586\) 0 0
\(587\) 11.2194 0.463073 0.231536 0.972826i \(-0.425625\pi\)
0.231536 + 0.972826i \(0.425625\pi\)
\(588\) 0 0
\(589\) 9.73957 0.401312
\(590\) 0 0
\(591\) −21.2047 −0.872245
\(592\) 0 0
\(593\) −7.69682 −0.316071 −0.158035 0.987433i \(-0.550516\pi\)
−0.158035 + 0.987433i \(0.550516\pi\)
\(594\) 0 0
\(595\) 0.818503 0.0335554
\(596\) 0 0
\(597\) 2.64723 0.108344
\(598\) 0 0
\(599\) −20.0543 −0.819396 −0.409698 0.912221i \(-0.634366\pi\)
−0.409698 + 0.912221i \(0.634366\pi\)
\(600\) 0 0
\(601\) 3.10851 0.126799 0.0633995 0.997988i \(-0.479806\pi\)
0.0633995 + 0.997988i \(0.479806\pi\)
\(602\) 0 0
\(603\) 7.42023 0.302175
\(604\) 0 0
\(605\) 10.8449 0.440908
\(606\) 0 0
\(607\) −37.0441 −1.50357 −0.751786 0.659407i \(-0.770807\pi\)
−0.751786 + 0.659407i \(0.770807\pi\)
\(608\) 0 0
\(609\) 0.393832 0.0159589
\(610\) 0 0
\(611\) −29.7334 −1.20289
\(612\) 0 0
\(613\) −13.3839 −0.540569 −0.270284 0.962781i \(-0.587118\pi\)
−0.270284 + 0.962781i \(0.587118\pi\)
\(614\) 0 0
\(615\) 1.65745 0.0668348
\(616\) 0 0
\(617\) −12.6364 −0.508721 −0.254361 0.967109i \(-0.581865\pi\)
−0.254361 + 0.967109i \(0.581865\pi\)
\(618\) 0 0
\(619\) 41.2447 1.65776 0.828882 0.559424i \(-0.188978\pi\)
0.828882 + 0.559424i \(0.188978\pi\)
\(620\) 0 0
\(621\) −6.15661 −0.247056
\(622\) 0 0
\(623\) −6.23446 −0.249778
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.377661 0.0150823
\(628\) 0 0
\(629\) −15.2587 −0.608406
\(630\) 0 0
\(631\) −29.8009 −1.18635 −0.593177 0.805072i \(-0.702127\pi\)
−0.593177 + 0.805072i \(0.702127\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 20.1566 0.799891
\(636\) 0 0
\(637\) 17.5579 0.695669
\(638\) 0 0
\(639\) −5.98533 −0.236776
\(640\) 0 0
\(641\) 17.7774 0.702167 0.351083 0.936344i \(-0.385813\pi\)
0.351083 + 0.936344i \(0.385813\pi\)
\(642\) 0 0
\(643\) 44.5534 1.75702 0.878508 0.477727i \(-0.158539\pi\)
0.878508 + 0.477727i \(0.158539\pi\)
\(644\) 0 0
\(645\) 10.3279 0.406660
\(646\) 0 0
\(647\) −42.9339 −1.68790 −0.843952 0.536418i \(-0.819777\pi\)
−0.843952 + 0.536418i \(0.819777\pi\)
\(648\) 0 0
\(649\) 3.75725 0.147485
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) −19.8426 −0.776500 −0.388250 0.921554i \(-0.626920\pi\)
−0.388250 + 0.921554i \(0.626920\pi\)
\(654\) 0 0
\(655\) −5.50235 −0.214994
\(656\) 0 0
\(657\) 3.34192 0.130381
\(658\) 0 0
\(659\) −23.0483 −0.897836 −0.448918 0.893573i \(-0.648190\pi\)
−0.448918 + 0.893573i \(0.648190\pi\)
\(660\) 0 0
\(661\) −36.2079 −1.40832 −0.704162 0.710040i \(-0.748677\pi\)
−0.704162 + 0.710040i \(0.748677\pi\)
\(662\) 0 0
\(663\) −5.33107 −0.207042
\(664\) 0 0
\(665\) 0.377661 0.0146451
\(666\) 0 0
\(667\) 6.15661 0.238385
\(668\) 0 0
\(669\) −12.5504 −0.485228
\(670\) 0 0
\(671\) −2.46279 −0.0950749
\(672\) 0 0
\(673\) 22.9000 0.882731 0.441365 0.897327i \(-0.354494\pi\)
0.441365 + 0.897327i \(0.354494\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 36.0068 1.38385 0.691927 0.721967i \(-0.256762\pi\)
0.691927 + 0.721967i \(0.256762\pi\)
\(678\) 0 0
\(679\) 7.26915 0.278964
\(680\) 0 0
\(681\) −1.30149 −0.0498732
\(682\) 0 0
\(683\) −9.12896 −0.349310 −0.174655 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(684\) 0 0
\(685\) −7.49853 −0.286504
\(686\) 0 0
\(687\) 3.28682 0.125400
\(688\) 0 0
\(689\) 31.6223 1.20472
\(690\) 0 0
\(691\) −5.85956 −0.222908 −0.111454 0.993770i \(-0.535551\pi\)
−0.111454 + 0.993770i \(0.535551\pi\)
\(692\) 0 0
\(693\) −0.155104 −0.00589191
\(694\) 0 0
\(695\) −9.35277 −0.354771
\(696\) 0 0
\(697\) −3.44469 −0.130477
\(698\) 0 0
\(699\) 17.7115 0.669910
\(700\) 0 0
\(701\) −7.56955 −0.285898 −0.142949 0.989730i \(-0.545658\pi\)
−0.142949 + 0.989730i \(0.545658\pi\)
\(702\) 0 0
\(703\) −7.04046 −0.265536
\(704\) 0 0
\(705\) −11.5915 −0.436561
\(706\) 0 0
\(707\) 5.04256 0.189645
\(708\) 0 0
\(709\) 14.5209 0.545342 0.272671 0.962107i \(-0.412093\pi\)
0.272671 + 0.962107i \(0.412093\pi\)
\(710\) 0 0
\(711\) 2.06745 0.0775356
\(712\) 0 0
\(713\) −62.5302 −2.34178
\(714\) 0 0
\(715\) 1.01022 0.0377802
\(716\) 0 0
\(717\) −21.2651 −0.794161
\(718\) 0 0
\(719\) −12.2164 −0.455596 −0.227798 0.973708i \(-0.573153\pi\)
−0.227798 + 0.973708i \(0.573153\pi\)
\(720\) 0 0
\(721\) 1.91788 0.0714255
\(722\) 0 0
\(723\) 15.3177 0.569670
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 40.8856 1.51636 0.758182 0.652044i \(-0.226088\pi\)
0.758182 + 0.652044i \(0.226088\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.4645 −0.793892
\(732\) 0 0
\(733\) 25.4915 0.941550 0.470775 0.882253i \(-0.343974\pi\)
0.470775 + 0.882253i \(0.343974\pi\)
\(734\) 0 0
\(735\) 6.84490 0.252478
\(736\) 0 0
\(737\) 2.92232 0.107645
\(738\) 0 0
\(739\) −9.56830 −0.351975 −0.175988 0.984392i \(-0.556312\pi\)
−0.175988 + 0.984392i \(0.556312\pi\)
\(740\) 0 0
\(741\) −2.45978 −0.0903624
\(742\) 0 0
\(743\) 18.0455 0.662025 0.331013 0.943626i \(-0.392610\pi\)
0.331013 + 0.943626i \(0.392610\pi\)
\(744\) 0 0
\(745\) 11.5402 0.422801
\(746\) 0 0
\(747\) −6.41000 −0.234530
\(748\) 0 0
\(749\) −0.922572 −0.0337100
\(750\) 0 0
\(751\) −11.9255 −0.435168 −0.217584 0.976042i \(-0.569818\pi\)
−0.217584 + 0.976042i \(0.569818\pi\)
\(752\) 0 0
\(753\) −26.8917 −0.979989
\(754\) 0 0
\(755\) 6.08212 0.221351
\(756\) 0 0
\(757\) 5.86152 0.213041 0.106520 0.994311i \(-0.466029\pi\)
0.106520 + 0.994311i \(0.466029\pi\)
\(758\) 0 0
\(759\) −2.42467 −0.0880100
\(760\) 0 0
\(761\) 49.8739 1.80793 0.903963 0.427610i \(-0.140644\pi\)
0.903963 + 0.427610i \(0.140644\pi\)
\(762\) 0 0
\(763\) −3.36744 −0.121909
\(764\) 0 0
\(765\) −2.07830 −0.0751412
\(766\) 0 0
\(767\) −24.4717 −0.883621
\(768\) 0 0
\(769\) 29.5121 1.06423 0.532117 0.846671i \(-0.321396\pi\)
0.532117 + 0.846671i \(0.321396\pi\)
\(770\) 0 0
\(771\) −6.85512 −0.246881
\(772\) 0 0
\(773\) −2.41935 −0.0870180 −0.0435090 0.999053i \(-0.513854\pi\)
−0.0435090 + 0.999053i \(0.513854\pi\)
\(774\) 0 0
\(775\) 10.1566 0.364836
\(776\) 0 0
\(777\) 2.89149 0.103731
\(778\) 0 0
\(779\) −1.58939 −0.0569460
\(780\) 0 0
\(781\) −2.35722 −0.0843479
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 11.5953 0.413855
\(786\) 0 0
\(787\) 51.1549 1.82348 0.911738 0.410772i \(-0.134741\pi\)
0.911738 + 0.410772i \(0.134741\pi\)
\(788\) 0 0
\(789\) 13.6294 0.485218
\(790\) 0 0
\(791\) 4.41428 0.156954
\(792\) 0 0
\(793\) 16.0406 0.569619
\(794\) 0 0
\(795\) 12.3279 0.437225
\(796\) 0 0
\(797\) 28.8662 1.02249 0.511247 0.859434i \(-0.329184\pi\)
0.511247 + 0.859434i \(0.329184\pi\)
\(798\) 0 0
\(799\) 24.0907 0.852266
\(800\) 0 0
\(801\) 15.8302 0.559334
\(802\) 0 0
\(803\) 1.31616 0.0464462
\(804\) 0 0
\(805\) −2.42467 −0.0854584
\(806\) 0 0
\(807\) −8.46129 −0.297851
\(808\) 0 0
\(809\) 19.0426 0.669501 0.334750 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(810\) 0 0
\(811\) 20.8783 0.733137 0.366569 0.930391i \(-0.380533\pi\)
0.366569 + 0.930391i \(0.380533\pi\)
\(812\) 0 0
\(813\) −6.34255 −0.222443
\(814\) 0 0
\(815\) −0.855118 −0.0299535
\(816\) 0 0
\(817\) −9.90381 −0.346491
\(818\) 0 0
\(819\) 1.01022 0.0353000
\(820\) 0 0
\(821\) 3.60466 0.125804 0.0629018 0.998020i \(-0.479965\pi\)
0.0629018 + 0.998020i \(0.479965\pi\)
\(822\) 0 0
\(823\) −27.2288 −0.949135 −0.474567 0.880219i \(-0.657395\pi\)
−0.474567 + 0.880219i \(0.657395\pi\)
\(824\) 0 0
\(825\) 0.393832 0.0137115
\(826\) 0 0
\(827\) −47.3936 −1.64804 −0.824019 0.566562i \(-0.808273\pi\)
−0.824019 + 0.566562i \(0.808273\pi\)
\(828\) 0 0
\(829\) −41.8388 −1.45312 −0.726560 0.687103i \(-0.758882\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(830\) 0 0
\(831\) −17.1464 −0.594802
\(832\) 0 0
\(833\) −14.2258 −0.492894
\(834\) 0 0
\(835\) 7.73194 0.267575
\(836\) 0 0
\(837\) 10.1566 0.351064
\(838\) 0 0
\(839\) 31.6385 1.09228 0.546141 0.837693i \(-0.316096\pi\)
0.546141 + 0.837693i \(0.316096\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −12.2985 −0.423584
\(844\) 0 0
\(845\) 6.42023 0.220862
\(846\) 0 0
\(847\) 4.27107 0.146756
\(848\) 0 0
\(849\) −25.1830 −0.864278
\(850\) 0 0
\(851\) 45.2013 1.54948
\(852\) 0 0
\(853\) 44.2000 1.51338 0.756690 0.653773i \(-0.226815\pi\)
0.756690 + 0.653773i \(0.226815\pi\)
\(854\) 0 0
\(855\) −0.958939 −0.0327950
\(856\) 0 0
\(857\) 14.4775 0.494541 0.247270 0.968947i \(-0.420466\pi\)
0.247270 + 0.968947i \(0.420466\pi\)
\(858\) 0 0
\(859\) −22.5883 −0.770703 −0.385352 0.922770i \(-0.625920\pi\)
−0.385352 + 0.922770i \(0.625920\pi\)
\(860\) 0 0
\(861\) 0.652757 0.0222459
\(862\) 0 0
\(863\) 22.9900 0.782590 0.391295 0.920265i \(-0.372027\pi\)
0.391295 + 0.920265i \(0.372027\pi\)
\(864\) 0 0
\(865\) −8.07449 −0.274541
\(866\) 0 0
\(867\) −12.6807 −0.430658
\(868\) 0 0
\(869\) 0.814230 0.0276209
\(870\) 0 0
\(871\) −19.0337 −0.644931
\(872\) 0 0
\(873\) −18.4575 −0.624691
\(874\) 0 0
\(875\) 0.393832 0.0133140
\(876\) 0 0
\(877\) 13.5741 0.458364 0.229182 0.973384i \(-0.426395\pi\)
0.229182 + 0.973384i \(0.426395\pi\)
\(878\) 0 0
\(879\) 12.3170 0.415443
\(880\) 0 0
\(881\) 9.91235 0.333956 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(882\) 0 0
\(883\) −39.3192 −1.32320 −0.661598 0.749859i \(-0.730121\pi\)
−0.661598 + 0.749859i \(0.730121\pi\)
\(884\) 0 0
\(885\) −9.54022 −0.320691
\(886\) 0 0
\(887\) 59.2520 1.98949 0.994743 0.102403i \(-0.0326531\pi\)
0.994743 + 0.102403i \(0.0326531\pi\)
\(888\) 0 0
\(889\) 7.93832 0.266243
\(890\) 0 0
\(891\) 0.393832 0.0131939
\(892\) 0 0
\(893\) 11.1155 0.371968
\(894\) 0 0
\(895\) 12.4100 0.414821
\(896\) 0 0
\(897\) 15.7924 0.527291
\(898\) 0 0
\(899\) −10.1566 −0.338742
\(900\) 0 0
\(901\) −25.6211 −0.853562
\(902\) 0 0
\(903\) 4.06745 0.135356
\(904\) 0 0
\(905\) 2.49765 0.0830248
\(906\) 0 0
\(907\) 5.91210 0.196308 0.0981541 0.995171i \(-0.468706\pi\)
0.0981541 + 0.995171i \(0.468706\pi\)
\(908\) 0 0
\(909\) −12.8038 −0.424676
\(910\) 0 0
\(911\) 12.4185 0.411445 0.205722 0.978610i \(-0.434046\pi\)
0.205722 + 0.978610i \(0.434046\pi\)
\(912\) 0 0
\(913\) −2.52447 −0.0835476
\(914\) 0 0
\(915\) 6.25340 0.206731
\(916\) 0 0
\(917\) −2.16700 −0.0715607
\(918\) 0 0
\(919\) −16.4332 −0.542081 −0.271041 0.962568i \(-0.587368\pi\)
−0.271041 + 0.962568i \(0.587368\pi\)
\(920\) 0 0
\(921\) 22.7379 0.749239
\(922\) 0 0
\(923\) 15.3530 0.505351
\(924\) 0 0
\(925\) −7.34192 −0.241401
\(926\) 0 0
\(927\) −4.86979 −0.159945
\(928\) 0 0
\(929\) 13.8358 0.453936 0.226968 0.973902i \(-0.427119\pi\)
0.226968 + 0.973902i \(0.427119\pi\)
\(930\) 0 0
\(931\) −6.56384 −0.215121
\(932\) 0 0
\(933\) 5.33810 0.174762
\(934\) 0 0
\(935\) −0.818503 −0.0267679
\(936\) 0 0
\(937\) 20.7528 0.677964 0.338982 0.940793i \(-0.389917\pi\)
0.338982 + 0.940793i \(0.389917\pi\)
\(938\) 0 0
\(939\) −1.96338 −0.0640726
\(940\) 0 0
\(941\) 40.1666 1.30940 0.654698 0.755891i \(-0.272796\pi\)
0.654698 + 0.755891i \(0.272796\pi\)
\(942\) 0 0
\(943\) 10.2043 0.332297
\(944\) 0 0
\(945\) 0.393832 0.0128114
\(946\) 0 0
\(947\) 17.4144 0.565894 0.282947 0.959136i \(-0.408688\pi\)
0.282947 + 0.959136i \(0.408688\pi\)
\(948\) 0 0
\(949\) −8.57239 −0.278271
\(950\) 0 0
\(951\) 1.29064 0.0418518
\(952\) 0 0
\(953\) 13.5121 0.437701 0.218851 0.975758i \(-0.429769\pi\)
0.218851 + 0.975758i \(0.429769\pi\)
\(954\) 0 0
\(955\) −6.32085 −0.204538
\(956\) 0 0
\(957\) −0.393832 −0.0127308
\(958\) 0 0
\(959\) −2.95316 −0.0953626
\(960\) 0 0
\(961\) 72.1567 2.32763
\(962\) 0 0
\(963\) 2.34255 0.0754876
\(964\) 0 0
\(965\) −25.2921 −0.814183
\(966\) 0 0
\(967\) 24.0598 0.773712 0.386856 0.922140i \(-0.373561\pi\)
0.386856 + 0.922140i \(0.373561\pi\)
\(968\) 0 0
\(969\) 1.99297 0.0640233
\(970\) 0 0
\(971\) −34.7877 −1.11639 −0.558195 0.829710i \(-0.688506\pi\)
−0.558195 + 0.829710i \(0.688506\pi\)
\(972\) 0 0
\(973\) −3.68342 −0.118085
\(974\) 0 0
\(975\) −2.56511 −0.0821492
\(976\) 0 0
\(977\) 35.7566 1.14396 0.571978 0.820269i \(-0.306176\pi\)
0.571978 + 0.820269i \(0.306176\pi\)
\(978\) 0 0
\(979\) 6.23446 0.199254
\(980\) 0 0
\(981\) 8.55044 0.272995
\(982\) 0 0
\(983\) 7.19064 0.229346 0.114673 0.993403i \(-0.463418\pi\)
0.114673 + 0.993403i \(0.463418\pi\)
\(984\) 0 0
\(985\) 21.2047 0.675638
\(986\) 0 0
\(987\) −4.56511 −0.145309
\(988\) 0 0
\(989\) 63.5847 2.02188
\(990\) 0 0
\(991\) 19.9123 0.632537 0.316268 0.948670i \(-0.397570\pi\)
0.316268 + 0.948670i \(0.397570\pi\)
\(992\) 0 0
\(993\) −28.9971 −0.920194
\(994\) 0 0
\(995\) −2.64723 −0.0839228
\(996\) 0 0
\(997\) −1.57302 −0.0498179 −0.0249089 0.999690i \(-0.507930\pi\)
−0.0249089 + 0.999690i \(0.507930\pi\)
\(998\) 0 0
\(999\) −7.34192 −0.232288
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 6960.2.a.co.1.2 4
4.3 odd 2 435.2.a.j.1.1 4
12.11 even 2 1305.2.a.r.1.4 4
20.3 even 4 2175.2.c.n.349.8 8
20.7 even 4 2175.2.c.n.349.1 8
20.19 odd 2 2175.2.a.v.1.4 4
60.59 even 2 6525.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 4.3 odd 2
1305.2.a.r.1.4 4 12.11 even 2
2175.2.a.v.1.4 4 20.19 odd 2
2175.2.c.n.349.1 8 20.7 even 4
2175.2.c.n.349.8 8 20.3 even 4
6525.2.a.bi.1.1 4 60.59 even 2
6960.2.a.co.1.2 4 1.1 even 1 trivial