Properties

Label 3640.2.a.x.1.3
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,2,Mod(1,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, 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("3640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.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: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.344151\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.34415 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.19326 q^{9} +O(q^{10})\) \(q+1.34415 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.19326 q^{9} +1.02414 q^{11} +1.00000 q^{13} -1.34415 q^{15} -0.881560 q^{17} +3.90570 q^{19} -1.34415 q^{21} +0.975857 q^{23} +1.00000 q^{25} -5.63637 q^{27} +0.320007 q^{29} -4.46726 q^{31} +1.37660 q^{33} +1.00000 q^{35} +6.56622 q^{37} +1.34415 q^{39} +12.5044 q^{41} +3.71244 q^{43} +1.19326 q^{45} +5.12311 q^{47} +1.00000 q^{49} -1.18495 q^{51} -1.12311 q^{53} -1.02414 q^{55} +5.24985 q^{57} +1.91926 q^{59} -8.24621 q^{61} +1.19326 q^{63} -1.00000 q^{65} +0.430138 q^{67} +1.31170 q^{69} -13.6569 q^{71} +8.49971 q^{73} +1.34415 q^{75} -1.02414 q^{77} +14.3912 q^{79} -3.99636 q^{81} +16.7459 q^{83} +0.881560 q^{85} +0.430138 q^{87} -1.55630 q^{89} -1.00000 q^{91} -6.00467 q^{93} -3.90570 q^{95} -16.5090 q^{97} -1.22207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) 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
\(3\) 1.34415 0.776046 0.388023 0.921650i \(-0.373158\pi\)
0.388023 + 0.921650i \(0.373158\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.19326 −0.397753
\(10\) 0 0
\(11\) 1.02414 0.308791 0.154395 0.988009i \(-0.450657\pi\)
0.154395 + 0.988009i \(0.450657\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.34415 −0.347058
\(16\) 0 0
\(17\) −0.881560 −0.213810 −0.106905 0.994269i \(-0.534094\pi\)
−0.106905 + 0.994269i \(0.534094\pi\)
\(18\) 0 0
\(19\) 3.90570 0.896030 0.448015 0.894026i \(-0.352131\pi\)
0.448015 + 0.894026i \(0.352131\pi\)
\(20\) 0 0
\(21\) −1.34415 −0.293318
\(22\) 0 0
\(23\) 0.975857 0.203480 0.101740 0.994811i \(-0.467559\pi\)
0.101740 + 0.994811i \(0.467559\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.63637 −1.08472
\(28\) 0 0
\(29\) 0.320007 0.0594239 0.0297119 0.999559i \(-0.490541\pi\)
0.0297119 + 0.999559i \(0.490541\pi\)
\(30\) 0 0
\(31\) −4.46726 −0.802343 −0.401171 0.916003i \(-0.631397\pi\)
−0.401171 + 0.916003i \(0.631397\pi\)
\(32\) 0 0
\(33\) 1.37660 0.239636
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 6.56622 1.07948 0.539740 0.841832i \(-0.318523\pi\)
0.539740 + 0.841832i \(0.318523\pi\)
\(38\) 0 0
\(39\) 1.34415 0.215236
\(40\) 0 0
\(41\) 12.5044 1.95286 0.976428 0.215845i \(-0.0692507\pi\)
0.976428 + 0.215845i \(0.0692507\pi\)
\(42\) 0 0
\(43\) 3.71244 0.566143 0.283071 0.959099i \(-0.408647\pi\)
0.283071 + 0.959099i \(0.408647\pi\)
\(44\) 0 0
\(45\) 1.19326 0.177881
\(46\) 0 0
\(47\) 5.12311 0.747282 0.373641 0.927573i \(-0.378109\pi\)
0.373641 + 0.927573i \(0.378109\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.18495 −0.165926
\(52\) 0 0
\(53\) −1.12311 −0.154270 −0.0771352 0.997021i \(-0.524577\pi\)
−0.0771352 + 0.997021i \(0.524577\pi\)
\(54\) 0 0
\(55\) −1.02414 −0.138095
\(56\) 0 0
\(57\) 5.24985 0.695360
\(58\) 0 0
\(59\) 1.91926 0.249867 0.124933 0.992165i \(-0.460128\pi\)
0.124933 + 0.992165i \(0.460128\pi\)
\(60\) 0 0
\(61\) −8.24621 −1.05582 −0.527910 0.849301i \(-0.677024\pi\)
−0.527910 + 0.849301i \(0.677024\pi\)
\(62\) 0 0
\(63\) 1.19326 0.150336
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 0.430138 0.0525498 0.0262749 0.999655i \(-0.491635\pi\)
0.0262749 + 0.999655i \(0.491635\pi\)
\(68\) 0 0
\(69\) 1.31170 0.157910
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810386 0.585896i \(-0.800743\pi\)
−0.810386 + 0.585896i \(0.800743\pi\)
\(72\) 0 0
\(73\) 8.49971 0.994816 0.497408 0.867517i \(-0.334285\pi\)
0.497408 + 0.867517i \(0.334285\pi\)
\(74\) 0 0
\(75\) 1.34415 0.155209
\(76\) 0 0
\(77\) −1.02414 −0.116712
\(78\) 0 0
\(79\) 14.3912 1.61913 0.809567 0.587027i \(-0.199702\pi\)
0.809567 + 0.587027i \(0.199702\pi\)
\(80\) 0 0
\(81\) −3.99636 −0.444040
\(82\) 0 0
\(83\) 16.7459 1.83810 0.919052 0.394137i \(-0.128956\pi\)
0.919052 + 0.394137i \(0.128956\pi\)
\(84\) 0 0
\(85\) 0.881560 0.0956186
\(86\) 0 0
\(87\) 0.430138 0.0461157
\(88\) 0 0
\(89\) −1.55630 −0.164968 −0.0824839 0.996592i \(-0.526285\pi\)
−0.0824839 + 0.996592i \(0.526285\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −6.00467 −0.622655
\(94\) 0 0
\(95\) −3.90570 −0.400717
\(96\) 0 0
\(97\) −16.5090 −1.67624 −0.838120 0.545487i \(-0.816345\pi\)
−0.838120 + 0.545487i \(0.816345\pi\)
\(98\) 0 0
\(99\) −1.22207 −0.122822
\(100\) 0 0
\(101\) 19.3210 1.92251 0.961257 0.275653i \(-0.0888941\pi\)
0.961257 + 0.275653i \(0.0888941\pi\)
\(102\) 0 0
\(103\) −4.40541 −0.434078 −0.217039 0.976163i \(-0.569640\pi\)
−0.217039 + 0.976163i \(0.569640\pi\)
\(104\) 0 0
\(105\) 1.34415 0.131176
\(106\) 0 0
\(107\) 7.36237 0.711748 0.355874 0.934534i \(-0.384183\pi\)
0.355874 + 0.934534i \(0.384183\pi\)
\(108\) 0 0
\(109\) 14.1225 1.35269 0.676346 0.736584i \(-0.263562\pi\)
0.676346 + 0.736584i \(0.263562\pi\)
\(110\) 0 0
\(111\) 8.82599 0.837726
\(112\) 0 0
\(113\) 10.7366 1.01001 0.505007 0.863115i \(-0.331490\pi\)
0.505007 + 0.863115i \(0.331490\pi\)
\(114\) 0 0
\(115\) −0.975857 −0.0909991
\(116\) 0 0
\(117\) −1.19326 −0.110317
\(118\) 0 0
\(119\) 0.881560 0.0808125
\(120\) 0 0
\(121\) −9.95113 −0.904648
\(122\) 0 0
\(123\) 16.8078 1.51551
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.6470 1.29971 0.649854 0.760059i \(-0.274830\pi\)
0.649854 + 0.760059i \(0.274830\pi\)
\(128\) 0 0
\(129\) 4.99009 0.439353
\(130\) 0 0
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 0 0
\(133\) −3.90570 −0.338667
\(134\) 0 0
\(135\) 5.63637 0.485102
\(136\) 0 0
\(137\) 1.63171 0.139406 0.0697030 0.997568i \(-0.477795\pi\)
0.0697030 + 0.997568i \(0.477795\pi\)
\(138\) 0 0
\(139\) 7.23630 0.613775 0.306887 0.951746i \(-0.400713\pi\)
0.306887 + 0.951746i \(0.400713\pi\)
\(140\) 0 0
\(141\) 6.88623 0.579925
\(142\) 0 0
\(143\) 1.02414 0.0856432
\(144\) 0 0
\(145\) −0.320007 −0.0265752
\(146\) 0 0
\(147\) 1.34415 0.110864
\(148\) 0 0
\(149\) 1.12311 0.0920084 0.0460042 0.998941i \(-0.485351\pi\)
0.0460042 + 0.998941i \(0.485351\pi\)
\(150\) 0 0
\(151\) 2.03406 0.165529 0.0827646 0.996569i \(-0.473625\pi\)
0.0827646 + 0.996569i \(0.473625\pi\)
\(152\) 0 0
\(153\) 1.05193 0.0850435
\(154\) 0 0
\(155\) 4.46726 0.358819
\(156\) 0 0
\(157\) 17.3194 1.38224 0.691120 0.722740i \(-0.257117\pi\)
0.691120 + 0.722740i \(0.257117\pi\)
\(158\) 0 0
\(159\) −1.50962 −0.119721
\(160\) 0 0
\(161\) −0.975857 −0.0769083
\(162\) 0 0
\(163\) −14.4908 −1.13501 −0.567504 0.823371i \(-0.692091\pi\)
−0.567504 + 0.823371i \(0.692091\pi\)
\(164\) 0 0
\(165\) −1.37660 −0.107168
\(166\) 0 0
\(167\) −20.4608 −1.58330 −0.791650 0.610974i \(-0.790778\pi\)
−0.791650 + 0.610974i \(0.790778\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −4.66052 −0.356399
\(172\) 0 0
\(173\) 2.43320 0.184993 0.0924963 0.995713i \(-0.470515\pi\)
0.0924963 + 0.995713i \(0.470515\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 2.57978 0.193908
\(178\) 0 0
\(179\) 1.51224 0.113030 0.0565152 0.998402i \(-0.482001\pi\)
0.0565152 + 0.998402i \(0.482001\pi\)
\(180\) 0 0
\(181\) −16.8862 −1.25514 −0.627572 0.778559i \(-0.715951\pi\)
−0.627572 + 0.778559i \(0.715951\pi\)
\(182\) 0 0
\(183\) −11.0842 −0.819364
\(184\) 0 0
\(185\) −6.56622 −0.482758
\(186\) 0 0
\(187\) −0.902844 −0.0660225
\(188\) 0 0
\(189\) 5.63637 0.409986
\(190\) 0 0
\(191\) −8.75481 −0.633476 −0.316738 0.948513i \(-0.602588\pi\)
−0.316738 + 0.948513i \(0.602588\pi\)
\(192\) 0 0
\(193\) 1.95638 0.140823 0.0704116 0.997518i \(-0.477569\pi\)
0.0704116 + 0.997518i \(0.477569\pi\)
\(194\) 0 0
\(195\) −1.34415 −0.0962566
\(196\) 0 0
\(197\) −10.6628 −0.759692 −0.379846 0.925050i \(-0.624023\pi\)
−0.379846 + 0.925050i \(0.624023\pi\)
\(198\) 0 0
\(199\) 18.0847 1.28199 0.640996 0.767544i \(-0.278521\pi\)
0.640996 + 0.767544i \(0.278521\pi\)
\(200\) 0 0
\(201\) 0.578171 0.0407810
\(202\) 0 0
\(203\) −0.320007 −0.0224601
\(204\) 0 0
\(205\) −12.5044 −0.873343
\(206\) 0 0
\(207\) −1.16445 −0.0809348
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 22.8155 1.57068 0.785342 0.619063i \(-0.212487\pi\)
0.785342 + 0.619063i \(0.212487\pi\)
\(212\) 0 0
\(213\) −18.3569 −1.25779
\(214\) 0 0
\(215\) −3.71244 −0.253187
\(216\) 0 0
\(217\) 4.46726 0.303257
\(218\) 0 0
\(219\) 11.4249 0.772022
\(220\) 0 0
\(221\) −0.881560 −0.0593002
\(222\) 0 0
\(223\) −25.5839 −1.71322 −0.856611 0.515963i \(-0.827434\pi\)
−0.856611 + 0.515963i \(0.827434\pi\)
\(224\) 0 0
\(225\) −1.19326 −0.0795506
\(226\) 0 0
\(227\) 25.5190 1.69375 0.846876 0.531790i \(-0.178480\pi\)
0.846876 + 0.531790i \(0.178480\pi\)
\(228\) 0 0
\(229\) −19.5216 −1.29002 −0.645011 0.764173i \(-0.723147\pi\)
−0.645011 + 0.764173i \(0.723147\pi\)
\(230\) 0 0
\(231\) −1.37660 −0.0905738
\(232\) 0 0
\(233\) 10.7459 0.703989 0.351994 0.936002i \(-0.385504\pi\)
0.351994 + 0.936002i \(0.385504\pi\)
\(234\) 0 0
\(235\) −5.12311 −0.334195
\(236\) 0 0
\(237\) 19.3439 1.25652
\(238\) 0 0
\(239\) 8.40075 0.543399 0.271700 0.962382i \(-0.412414\pi\)
0.271700 + 0.962382i \(0.412414\pi\)
\(240\) 0 0
\(241\) −6.12835 −0.394762 −0.197381 0.980327i \(-0.563244\pi\)
−0.197381 + 0.980327i \(0.563244\pi\)
\(242\) 0 0
\(243\) 11.5374 0.740125
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 3.90570 0.248514
\(248\) 0 0
\(249\) 22.5090 1.42645
\(250\) 0 0
\(251\) −27.3105 −1.72383 −0.861913 0.507057i \(-0.830734\pi\)
−0.861913 + 0.507057i \(0.830734\pi\)
\(252\) 0 0
\(253\) 0.999417 0.0628328
\(254\) 0 0
\(255\) 1.18495 0.0742044
\(256\) 0 0
\(257\) 29.6155 1.84737 0.923683 0.383158i \(-0.125163\pi\)
0.923683 + 0.383158i \(0.125163\pi\)
\(258\) 0 0
\(259\) −6.56622 −0.408005
\(260\) 0 0
\(261\) −0.381852 −0.0236360
\(262\) 0 0
\(263\) 23.3234 1.43818 0.719092 0.694915i \(-0.244558\pi\)
0.719092 + 0.694915i \(0.244558\pi\)
\(264\) 0 0
\(265\) 1.12311 0.0689918
\(266\) 0 0
\(267\) −2.09191 −0.128023
\(268\) 0 0
\(269\) 13.0748 0.797186 0.398593 0.917128i \(-0.369499\pi\)
0.398593 + 0.917128i \(0.369499\pi\)
\(270\) 0 0
\(271\) −18.5021 −1.12392 −0.561961 0.827164i \(-0.689953\pi\)
−0.561961 + 0.827164i \(0.689953\pi\)
\(272\) 0 0
\(273\) −1.34415 −0.0813517
\(274\) 0 0
\(275\) 1.02414 0.0617582
\(276\) 0 0
\(277\) −18.3760 −1.10411 −0.552054 0.833808i \(-0.686156\pi\)
−0.552054 + 0.833808i \(0.686156\pi\)
\(278\) 0 0
\(279\) 5.33059 0.319134
\(280\) 0 0
\(281\) −28.8591 −1.72159 −0.860795 0.508952i \(-0.830033\pi\)
−0.860795 + 0.508952i \(0.830033\pi\)
\(282\) 0 0
\(283\) 20.0071 1.18930 0.594648 0.803986i \(-0.297291\pi\)
0.594648 + 0.803986i \(0.297291\pi\)
\(284\) 0 0
\(285\) −5.24985 −0.310975
\(286\) 0 0
\(287\) −12.5044 −0.738110
\(288\) 0 0
\(289\) −16.2229 −0.954285
\(290\) 0 0
\(291\) −22.1906 −1.30084
\(292\) 0 0
\(293\) −15.8645 −0.926812 −0.463406 0.886146i \(-0.653373\pi\)
−0.463406 + 0.886146i \(0.653373\pi\)
\(294\) 0 0
\(295\) −1.91926 −0.111744
\(296\) 0 0
\(297\) −5.77245 −0.334952
\(298\) 0 0
\(299\) 0.975857 0.0564353
\(300\) 0 0
\(301\) −3.71244 −0.213982
\(302\) 0 0
\(303\) 25.9704 1.49196
\(304\) 0 0
\(305\) 8.24621 0.472177
\(306\) 0 0
\(307\) 5.69822 0.325214 0.162607 0.986691i \(-0.448010\pi\)
0.162607 + 0.986691i \(0.448010\pi\)
\(308\) 0 0
\(309\) −5.92154 −0.336865
\(310\) 0 0
\(311\) 32.5018 1.84301 0.921503 0.388372i \(-0.126962\pi\)
0.921503 + 0.388372i \(0.126962\pi\)
\(312\) 0 0
\(313\) 3.34654 0.189158 0.0945788 0.995517i \(-0.469850\pi\)
0.0945788 + 0.995517i \(0.469850\pi\)
\(314\) 0 0
\(315\) −1.19326 −0.0672325
\(316\) 0 0
\(317\) −30.7188 −1.72534 −0.862670 0.505767i \(-0.831209\pi\)
−0.862670 + 0.505767i \(0.831209\pi\)
\(318\) 0 0
\(319\) 0.327733 0.0183496
\(320\) 0 0
\(321\) 9.89614 0.552349
\(322\) 0 0
\(323\) −3.44311 −0.191580
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 18.9828 1.04975
\(328\) 0 0
\(329\) −5.12311 −0.282446
\(330\) 0 0
\(331\) 17.2221 0.946610 0.473305 0.880898i \(-0.343061\pi\)
0.473305 + 0.880898i \(0.343061\pi\)
\(332\) 0 0
\(333\) −7.83520 −0.429366
\(334\) 0 0
\(335\) −0.430138 −0.0235010
\(336\) 0 0
\(337\) −29.1491 −1.58785 −0.793925 0.608016i \(-0.791966\pi\)
−0.793925 + 0.608016i \(0.791966\pi\)
\(338\) 0 0
\(339\) 14.4316 0.783817
\(340\) 0 0
\(341\) −4.57511 −0.247756
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.31170 −0.0706195
\(346\) 0 0
\(347\) −8.77065 −0.470833 −0.235416 0.971895i \(-0.575645\pi\)
−0.235416 + 0.971895i \(0.575645\pi\)
\(348\) 0 0
\(349\) −25.0566 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(350\) 0 0
\(351\) −5.63637 −0.300847
\(352\) 0 0
\(353\) 15.3766 0.818414 0.409207 0.912442i \(-0.365805\pi\)
0.409207 + 0.912442i \(0.365805\pi\)
\(354\) 0 0
\(355\) 13.6569 0.724832
\(356\) 0 0
\(357\) 1.18495 0.0627142
\(358\) 0 0
\(359\) 10.0896 0.532510 0.266255 0.963903i \(-0.414214\pi\)
0.266255 + 0.963903i \(0.414214\pi\)
\(360\) 0 0
\(361\) −3.74548 −0.197131
\(362\) 0 0
\(363\) −13.3758 −0.702048
\(364\) 0 0
\(365\) −8.49971 −0.444895
\(366\) 0 0
\(367\) −30.0422 −1.56819 −0.784096 0.620640i \(-0.786873\pi\)
−0.784096 + 0.620640i \(0.786873\pi\)
\(368\) 0 0
\(369\) −14.9210 −0.776754
\(370\) 0 0
\(371\) 1.12311 0.0583087
\(372\) 0 0
\(373\) −3.51691 −0.182099 −0.0910493 0.995846i \(-0.529022\pi\)
−0.0910493 + 0.995846i \(0.529022\pi\)
\(374\) 0 0
\(375\) −1.34415 −0.0694116
\(376\) 0 0
\(377\) 0.320007 0.0164812
\(378\) 0 0
\(379\) 29.6735 1.52422 0.762112 0.647445i \(-0.224162\pi\)
0.762112 + 0.647445i \(0.224162\pi\)
\(380\) 0 0
\(381\) 19.6877 1.00863
\(382\) 0 0
\(383\) −14.1874 −0.724944 −0.362472 0.931995i \(-0.618067\pi\)
−0.362472 + 0.931995i \(0.618067\pi\)
\(384\) 0 0
\(385\) 1.02414 0.0521952
\(386\) 0 0
\(387\) −4.42991 −0.225185
\(388\) 0 0
\(389\) −32.3039 −1.63787 −0.818937 0.573883i \(-0.805436\pi\)
−0.818937 + 0.573883i \(0.805436\pi\)
\(390\) 0 0
\(391\) −0.860277 −0.0435061
\(392\) 0 0
\(393\) 12.2628 0.618578
\(394\) 0 0
\(395\) −14.3912 −0.724099
\(396\) 0 0
\(397\) 1.68160 0.0843970 0.0421985 0.999109i \(-0.486564\pi\)
0.0421985 + 0.999109i \(0.486564\pi\)
\(398\) 0 0
\(399\) −5.24985 −0.262821
\(400\) 0 0
\(401\) −27.2894 −1.36277 −0.681383 0.731927i \(-0.738621\pi\)
−0.681383 + 0.731927i \(0.738621\pi\)
\(402\) 0 0
\(403\) −4.46726 −0.222530
\(404\) 0 0
\(405\) 3.99636 0.198581
\(406\) 0 0
\(407\) 6.72475 0.333333
\(408\) 0 0
\(409\) −1.34949 −0.0667279 −0.0333639 0.999443i \(-0.510622\pi\)
−0.0333639 + 0.999443i \(0.510622\pi\)
\(410\) 0 0
\(411\) 2.19326 0.108185
\(412\) 0 0
\(413\) −1.91926 −0.0944407
\(414\) 0 0
\(415\) −16.7459 −0.822025
\(416\) 0 0
\(417\) 9.72667 0.476317
\(418\) 0 0
\(419\) −15.6321 −0.763680 −0.381840 0.924228i \(-0.624710\pi\)
−0.381840 + 0.924228i \(0.624710\pi\)
\(420\) 0 0
\(421\) 8.23893 0.401541 0.200770 0.979638i \(-0.435656\pi\)
0.200770 + 0.979638i \(0.435656\pi\)
\(422\) 0 0
\(423\) −6.11319 −0.297234
\(424\) 0 0
\(425\) −0.881560 −0.0427620
\(426\) 0 0
\(427\) 8.24621 0.399062
\(428\) 0 0
\(429\) 1.37660 0.0664630
\(430\) 0 0
\(431\) −3.56759 −0.171845 −0.0859223 0.996302i \(-0.527384\pi\)
−0.0859223 + 0.996302i \(0.527384\pi\)
\(432\) 0 0
\(433\) 21.6639 1.04110 0.520551 0.853831i \(-0.325727\pi\)
0.520551 + 0.853831i \(0.325727\pi\)
\(434\) 0 0
\(435\) −0.430138 −0.0206236
\(436\) 0 0
\(437\) 3.81141 0.182324
\(438\) 0 0
\(439\) 17.6711 0.843396 0.421698 0.906736i \(-0.361434\pi\)
0.421698 + 0.906736i \(0.361434\pi\)
\(440\) 0 0
\(441\) −1.19326 −0.0568218
\(442\) 0 0
\(443\) 7.01686 0.333381 0.166690 0.986009i \(-0.446692\pi\)
0.166690 + 0.986009i \(0.446692\pi\)
\(444\) 0 0
\(445\) 1.55630 0.0737759
\(446\) 0 0
\(447\) 1.50962 0.0714027
\(448\) 0 0
\(449\) 7.13244 0.336601 0.168300 0.985736i \(-0.446172\pi\)
0.168300 + 0.985736i \(0.446172\pi\)
\(450\) 0 0
\(451\) 12.8063 0.603024
\(452\) 0 0
\(453\) 2.73408 0.128458
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) 14.5346 0.679898 0.339949 0.940444i \(-0.389590\pi\)
0.339949 + 0.940444i \(0.389590\pi\)
\(458\) 0 0
\(459\) 4.96880 0.231924
\(460\) 0 0
\(461\) −1.79318 −0.0835169 −0.0417584 0.999128i \(-0.513296\pi\)
−0.0417584 + 0.999128i \(0.513296\pi\)
\(462\) 0 0
\(463\) 23.0041 1.06909 0.534545 0.845140i \(-0.320483\pi\)
0.534545 + 0.845140i \(0.320483\pi\)
\(464\) 0 0
\(465\) 6.00467 0.278460
\(466\) 0 0
\(467\) −5.47090 −0.253163 −0.126582 0.991956i \(-0.540401\pi\)
−0.126582 + 0.991956i \(0.540401\pi\)
\(468\) 0 0
\(469\) −0.430138 −0.0198619
\(470\) 0 0
\(471\) 23.2799 1.07268
\(472\) 0 0
\(473\) 3.80208 0.174820
\(474\) 0 0
\(475\) 3.90570 0.179206
\(476\) 0 0
\(477\) 1.34016 0.0613615
\(478\) 0 0
\(479\) 18.6818 0.853593 0.426797 0.904348i \(-0.359642\pi\)
0.426797 + 0.904348i \(0.359642\pi\)
\(480\) 0 0
\(481\) 6.56622 0.299394
\(482\) 0 0
\(483\) −1.31170 −0.0596844
\(484\) 0 0
\(485\) 16.5090 0.749637
\(486\) 0 0
\(487\) −11.4431 −0.518537 −0.259268 0.965805i \(-0.583481\pi\)
−0.259268 + 0.965805i \(0.583481\pi\)
\(488\) 0 0
\(489\) −19.4778 −0.880818
\(490\) 0 0
\(491\) −0.869608 −0.0392449 −0.0196224 0.999807i \(-0.506246\pi\)
−0.0196224 + 0.999807i \(0.506246\pi\)
\(492\) 0 0
\(493\) −0.282106 −0.0127054
\(494\) 0 0
\(495\) 1.22207 0.0549279
\(496\) 0 0
\(497\) 13.6569 0.612594
\(498\) 0 0
\(499\) 9.15395 0.409787 0.204894 0.978784i \(-0.434315\pi\)
0.204894 + 0.978784i \(0.434315\pi\)
\(500\) 0 0
\(501\) −27.5023 −1.22871
\(502\) 0 0
\(503\) 16.5504 0.737945 0.368973 0.929440i \(-0.379710\pi\)
0.368973 + 0.929440i \(0.379710\pi\)
\(504\) 0 0
\(505\) −19.3210 −0.859775
\(506\) 0 0
\(507\) 1.34415 0.0596958
\(508\) 0 0
\(509\) −12.6176 −0.559264 −0.279632 0.960107i \(-0.590212\pi\)
−0.279632 + 0.960107i \(0.590212\pi\)
\(510\) 0 0
\(511\) −8.49971 −0.376005
\(512\) 0 0
\(513\) −22.0140 −0.971942
\(514\) 0 0
\(515\) 4.40541 0.194126
\(516\) 0 0
\(517\) 5.24679 0.230754
\(518\) 0 0
\(519\) 3.27059 0.143563
\(520\) 0 0
\(521\) 7.87748 0.345119 0.172559 0.984999i \(-0.444796\pi\)
0.172559 + 0.984999i \(0.444796\pi\)
\(522\) 0 0
\(523\) 27.6735 1.21008 0.605039 0.796196i \(-0.293158\pi\)
0.605039 + 0.796196i \(0.293158\pi\)
\(524\) 0 0
\(525\) −1.34415 −0.0586635
\(526\) 0 0
\(527\) 3.93816 0.171549
\(528\) 0 0
\(529\) −22.0477 −0.958596
\(530\) 0 0
\(531\) −2.29018 −0.0993851
\(532\) 0 0
\(533\) 12.5044 0.541625
\(534\) 0 0
\(535\) −7.36237 −0.318303
\(536\) 0 0
\(537\) 2.03268 0.0877167
\(538\) 0 0
\(539\) 1.02414 0.0441130
\(540\) 0 0
\(541\) −43.7891 −1.88264 −0.941320 0.337517i \(-0.890413\pi\)
−0.941320 + 0.337517i \(0.890413\pi\)
\(542\) 0 0
\(543\) −22.6976 −0.974049
\(544\) 0 0
\(545\) −14.1225 −0.604942
\(546\) 0 0
\(547\) 3.72178 0.159132 0.0795658 0.996830i \(-0.474647\pi\)
0.0795658 + 0.996830i \(0.474647\pi\)
\(548\) 0 0
\(549\) 9.83986 0.419955
\(550\) 0 0
\(551\) 1.24985 0.0532456
\(552\) 0 0
\(553\) −14.3912 −0.611975
\(554\) 0 0
\(555\) −8.82599 −0.374642
\(556\) 0 0
\(557\) 18.9510 0.802980 0.401490 0.915863i \(-0.368492\pi\)
0.401490 + 0.915863i \(0.368492\pi\)
\(558\) 0 0
\(559\) 3.71244 0.157020
\(560\) 0 0
\(561\) −1.21356 −0.0512365
\(562\) 0 0
\(563\) −37.9774 −1.60056 −0.800279 0.599628i \(-0.795315\pi\)
−0.800279 + 0.599628i \(0.795315\pi\)
\(564\) 0 0
\(565\) −10.7366 −0.451692
\(566\) 0 0
\(567\) 3.99636 0.167831
\(568\) 0 0
\(569\) 13.2545 0.555658 0.277829 0.960630i \(-0.410385\pi\)
0.277829 + 0.960630i \(0.410385\pi\)
\(570\) 0 0
\(571\) −31.8390 −1.33242 −0.666210 0.745765i \(-0.732084\pi\)
−0.666210 + 0.745765i \(0.732084\pi\)
\(572\) 0 0
\(573\) −11.7678 −0.491606
\(574\) 0 0
\(575\) 0.975857 0.0406960
\(576\) 0 0
\(577\) 43.1418 1.79602 0.898008 0.439980i \(-0.145014\pi\)
0.898008 + 0.439980i \(0.145014\pi\)
\(578\) 0 0
\(579\) 2.62967 0.109285
\(580\) 0 0
\(581\) −16.7459 −0.694738
\(582\) 0 0
\(583\) −1.15022 −0.0476373
\(584\) 0 0
\(585\) 1.19326 0.0493352
\(586\) 0 0
\(587\) 32.5350 1.34286 0.671431 0.741067i \(-0.265680\pi\)
0.671431 + 0.741067i \(0.265680\pi\)
\(588\) 0 0
\(589\) −17.4478 −0.718923
\(590\) 0 0
\(591\) −14.3324 −0.589556
\(592\) 0 0
\(593\) 4.27333 0.175484 0.0877422 0.996143i \(-0.472035\pi\)
0.0877422 + 0.996143i \(0.472035\pi\)
\(594\) 0 0
\(595\) −0.881560 −0.0361404
\(596\) 0 0
\(597\) 24.3086 0.994885
\(598\) 0 0
\(599\) 19.3964 0.792517 0.396258 0.918139i \(-0.370308\pi\)
0.396258 + 0.918139i \(0.370308\pi\)
\(600\) 0 0
\(601\) −1.73717 −0.0708607 −0.0354303 0.999372i \(-0.511280\pi\)
−0.0354303 + 0.999372i \(0.511280\pi\)
\(602\) 0 0
\(603\) −0.513266 −0.0209018
\(604\) 0 0
\(605\) 9.95113 0.404571
\(606\) 0 0
\(607\) −16.3197 −0.662395 −0.331197 0.943562i \(-0.607453\pi\)
−0.331197 + 0.943562i \(0.607453\pi\)
\(608\) 0 0
\(609\) −0.430138 −0.0174301
\(610\) 0 0
\(611\) 5.12311 0.207259
\(612\) 0 0
\(613\) −6.47376 −0.261473 −0.130736 0.991417i \(-0.541734\pi\)
−0.130736 + 0.991417i \(0.541734\pi\)
\(614\) 0 0
\(615\) −16.8078 −0.677754
\(616\) 0 0
\(617\) 28.7823 1.15873 0.579365 0.815068i \(-0.303301\pi\)
0.579365 + 0.815068i \(0.303301\pi\)
\(618\) 0 0
\(619\) 34.6713 1.39356 0.696779 0.717286i \(-0.254616\pi\)
0.696779 + 0.717286i \(0.254616\pi\)
\(620\) 0 0
\(621\) −5.50029 −0.220719
\(622\) 0 0
\(623\) 1.55630 0.0623520
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.37660 0.214721
\(628\) 0 0
\(629\) −5.78852 −0.230803
\(630\) 0 0
\(631\) −44.9767 −1.79050 −0.895248 0.445569i \(-0.853002\pi\)
−0.895248 + 0.445569i \(0.853002\pi\)
\(632\) 0 0
\(633\) 30.6675 1.21892
\(634\) 0 0
\(635\) −14.6470 −0.581247
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 16.2962 0.644667
\(640\) 0 0
\(641\) 14.0228 0.553866 0.276933 0.960889i \(-0.410682\pi\)
0.276933 + 0.960889i \(0.410682\pi\)
\(642\) 0 0
\(643\) −5.79957 −0.228713 −0.114356 0.993440i \(-0.536481\pi\)
−0.114356 + 0.993440i \(0.536481\pi\)
\(644\) 0 0
\(645\) −4.99009 −0.196484
\(646\) 0 0
\(647\) 23.2205 0.912891 0.456445 0.889751i \(-0.349122\pi\)
0.456445 + 0.889751i \(0.349122\pi\)
\(648\) 0 0
\(649\) 1.96560 0.0771565
\(650\) 0 0
\(651\) 6.00467 0.235341
\(652\) 0 0
\(653\) −6.50904 −0.254718 −0.127359 0.991857i \(-0.540650\pi\)
−0.127359 + 0.991857i \(0.540650\pi\)
\(654\) 0 0
\(655\) −9.12311 −0.356469
\(656\) 0 0
\(657\) −10.1424 −0.395691
\(658\) 0 0
\(659\) −0.825988 −0.0321759 −0.0160880 0.999871i \(-0.505121\pi\)
−0.0160880 + 0.999871i \(0.505121\pi\)
\(660\) 0 0
\(661\) −30.6071 −1.19048 −0.595238 0.803549i \(-0.702942\pi\)
−0.595238 + 0.803549i \(0.702942\pi\)
\(662\) 0 0
\(663\) −1.18495 −0.0460196
\(664\) 0 0
\(665\) 3.90570 0.151457
\(666\) 0 0
\(667\) 0.312281 0.0120916
\(668\) 0 0
\(669\) −34.3886 −1.32954
\(670\) 0 0
\(671\) −8.44530 −0.326027
\(672\) 0 0
\(673\) −25.0887 −0.967098 −0.483549 0.875317i \(-0.660653\pi\)
−0.483549 + 0.875317i \(0.660653\pi\)
\(674\) 0 0
\(675\) −5.63637 −0.216944
\(676\) 0 0
\(677\) −38.5111 −1.48010 −0.740051 0.672551i \(-0.765198\pi\)
−0.740051 + 0.672551i \(0.765198\pi\)
\(678\) 0 0
\(679\) 16.5090 0.633559
\(680\) 0 0
\(681\) 34.3013 1.31443
\(682\) 0 0
\(683\) 7.10066 0.271699 0.135850 0.990729i \(-0.456624\pi\)
0.135850 + 0.990729i \(0.456624\pi\)
\(684\) 0 0
\(685\) −1.63171 −0.0623443
\(686\) 0 0
\(687\) −26.2399 −1.00112
\(688\) 0 0
\(689\) −1.12311 −0.0427869
\(690\) 0 0
\(691\) −14.3103 −0.544391 −0.272195 0.962242i \(-0.587750\pi\)
−0.272195 + 0.962242i \(0.587750\pi\)
\(692\) 0 0
\(693\) 1.22207 0.0464225
\(694\) 0 0
\(695\) −7.23630 −0.274488
\(696\) 0 0
\(697\) −11.0234 −0.417540
\(698\) 0 0
\(699\) 14.4441 0.546327
\(700\) 0 0
\(701\) 8.62690 0.325833 0.162917 0.986640i \(-0.447910\pi\)
0.162917 + 0.986640i \(0.447910\pi\)
\(702\) 0 0
\(703\) 25.6457 0.967246
\(704\) 0 0
\(705\) −6.88623 −0.259350
\(706\) 0 0
\(707\) −19.3210 −0.726642
\(708\) 0 0
\(709\) −40.2993 −1.51347 −0.756736 0.653721i \(-0.773207\pi\)
−0.756736 + 0.653721i \(0.773207\pi\)
\(710\) 0 0
\(711\) −17.1724 −0.644016
\(712\) 0 0
\(713\) −4.35940 −0.163261
\(714\) 0 0
\(715\) −1.02414 −0.0383008
\(716\) 0 0
\(717\) 11.2919 0.421703
\(718\) 0 0
\(719\) −3.61348 −0.134760 −0.0673801 0.997727i \(-0.521464\pi\)
−0.0673801 + 0.997727i \(0.521464\pi\)
\(720\) 0 0
\(721\) 4.40541 0.164066
\(722\) 0 0
\(723\) −8.23743 −0.306353
\(724\) 0 0
\(725\) 0.320007 0.0118848
\(726\) 0 0
\(727\) 17.5031 0.649155 0.324577 0.945859i \(-0.394778\pi\)
0.324577 + 0.945859i \(0.394778\pi\)
\(728\) 0 0
\(729\) 27.4971 1.01841
\(730\) 0 0
\(731\) −3.27274 −0.121047
\(732\) 0 0
\(733\) −12.0198 −0.443963 −0.221981 0.975051i \(-0.571252\pi\)
−0.221981 + 0.975051i \(0.571252\pi\)
\(734\) 0 0
\(735\) −1.34415 −0.0495797
\(736\) 0 0
\(737\) 0.440523 0.0162269
\(738\) 0 0
\(739\) −7.58398 −0.278981 −0.139491 0.990223i \(-0.544546\pi\)
−0.139491 + 0.990223i \(0.544546\pi\)
\(740\) 0 0
\(741\) 5.24985 0.192858
\(742\) 0 0
\(743\) 29.9552 1.09895 0.549475 0.835510i \(-0.314828\pi\)
0.549475 + 0.835510i \(0.314828\pi\)
\(744\) 0 0
\(745\) −1.12311 −0.0411474
\(746\) 0 0
\(747\) −19.9822 −0.731111
\(748\) 0 0
\(749\) −7.36237 −0.269015
\(750\) 0 0
\(751\) 10.1995 0.372186 0.186093 0.982532i \(-0.440417\pi\)
0.186093 + 0.982532i \(0.440417\pi\)
\(752\) 0 0
\(753\) −36.7095 −1.33777
\(754\) 0 0
\(755\) −2.03406 −0.0740269
\(756\) 0 0
\(757\) −33.0452 −1.20105 −0.600524 0.799607i \(-0.705041\pi\)
−0.600524 + 0.799607i \(0.705041\pi\)
\(758\) 0 0
\(759\) 1.34337 0.0487611
\(760\) 0 0
\(761\) 16.3460 0.592540 0.296270 0.955104i \(-0.404257\pi\)
0.296270 + 0.955104i \(0.404257\pi\)
\(762\) 0 0
\(763\) −14.1225 −0.511270
\(764\) 0 0
\(765\) −1.05193 −0.0380326
\(766\) 0 0
\(767\) 1.91926 0.0693005
\(768\) 0 0
\(769\) −31.4991 −1.13589 −0.567944 0.823067i \(-0.692261\pi\)
−0.567944 + 0.823067i \(0.692261\pi\)
\(770\) 0 0
\(771\) 39.8077 1.43364
\(772\) 0 0
\(773\) 9.42489 0.338990 0.169495 0.985531i \(-0.445786\pi\)
0.169495 + 0.985531i \(0.445786\pi\)
\(774\) 0 0
\(775\) −4.46726 −0.160469
\(776\) 0 0
\(777\) −8.82599 −0.316631
\(778\) 0 0
\(779\) 48.8384 1.74982
\(780\) 0 0
\(781\) −13.9866 −0.500480
\(782\) 0 0
\(783\) −1.80368 −0.0644583
\(784\) 0 0
\(785\) −17.3194 −0.618157
\(786\) 0 0
\(787\) 15.3137 0.545876 0.272938 0.962032i \(-0.412005\pi\)
0.272938 + 0.962032i \(0.412005\pi\)
\(788\) 0 0
\(789\) 31.3502 1.11610
\(790\) 0 0
\(791\) −10.7366 −0.381749
\(792\) 0 0
\(793\) −8.24621 −0.292832
\(794\) 0 0
\(795\) 1.50962 0.0535408
\(796\) 0 0
\(797\) 30.8323 1.09213 0.546067 0.837741i \(-0.316124\pi\)
0.546067 + 0.837741i \(0.316124\pi\)
\(798\) 0 0
\(799\) −4.51633 −0.159776
\(800\) 0 0
\(801\) 1.85707 0.0656165
\(802\) 0 0
\(803\) 8.70492 0.307190
\(804\) 0 0
\(805\) 0.975857 0.0343944
\(806\) 0 0
\(807\) 17.5745 0.618653
\(808\) 0 0
\(809\) 34.0530 1.19724 0.598619 0.801034i \(-0.295716\pi\)
0.598619 + 0.801034i \(0.295716\pi\)
\(810\) 0 0
\(811\) −38.1128 −1.33832 −0.669162 0.743117i \(-0.733347\pi\)
−0.669162 + 0.743117i \(0.733347\pi\)
\(812\) 0 0
\(813\) −24.8696 −0.872215
\(814\) 0 0
\(815\) 14.4908 0.507591
\(816\) 0 0
\(817\) 14.4997 0.507281
\(818\) 0 0
\(819\) 1.19326 0.0416958
\(820\) 0 0
\(821\) −14.9828 −0.522903 −0.261452 0.965217i \(-0.584201\pi\)
−0.261452 + 0.965217i \(0.584201\pi\)
\(822\) 0 0
\(823\) −29.1799 −1.01715 −0.508574 0.861018i \(-0.669827\pi\)
−0.508574 + 0.861018i \(0.669827\pi\)
\(824\) 0 0
\(825\) 1.37660 0.0479272
\(826\) 0 0
\(827\) −17.1926 −0.597844 −0.298922 0.954278i \(-0.596627\pi\)
−0.298922 + 0.954278i \(0.596627\pi\)
\(828\) 0 0
\(829\) −19.1085 −0.663667 −0.331833 0.943338i \(-0.607667\pi\)
−0.331833 + 0.943338i \(0.607667\pi\)
\(830\) 0 0
\(831\) −24.7001 −0.856838
\(832\) 0 0
\(833\) −0.881560 −0.0305443
\(834\) 0 0
\(835\) 20.4608 0.708074
\(836\) 0 0
\(837\) 25.1791 0.870318
\(838\) 0 0
\(839\) −35.7960 −1.23582 −0.617908 0.786251i \(-0.712020\pi\)
−0.617908 + 0.786251i \(0.712020\pi\)
\(840\) 0 0
\(841\) −28.8976 −0.996469
\(842\) 0 0
\(843\) −38.7910 −1.33603
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 9.95113 0.341925
\(848\) 0 0
\(849\) 26.8925 0.922948
\(850\) 0 0
\(851\) 6.40769 0.219653
\(852\) 0 0
\(853\) −7.97883 −0.273190 −0.136595 0.990627i \(-0.543616\pi\)
−0.136595 + 0.990627i \(0.543616\pi\)
\(854\) 0 0
\(855\) 4.66052 0.159386
\(856\) 0 0
\(857\) −13.1791 −0.450190 −0.225095 0.974337i \(-0.572269\pi\)
−0.225095 + 0.974337i \(0.572269\pi\)
\(858\) 0 0
\(859\) −41.0040 −1.39904 −0.699519 0.714614i \(-0.746602\pi\)
−0.699519 + 0.714614i \(0.746602\pi\)
\(860\) 0 0
\(861\) −16.8078 −0.572807
\(862\) 0 0
\(863\) −14.3609 −0.488849 −0.244425 0.969668i \(-0.578599\pi\)
−0.244425 + 0.969668i \(0.578599\pi\)
\(864\) 0 0
\(865\) −2.43320 −0.0827312
\(866\) 0 0
\(867\) −21.8060 −0.740569
\(868\) 0 0
\(869\) 14.7386 0.499974
\(870\) 0 0
\(871\) 0.430138 0.0145747
\(872\) 0 0
\(873\) 19.6996 0.666729
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 29.7678 1.00519 0.502593 0.864523i \(-0.332379\pi\)
0.502593 + 0.864523i \(0.332379\pi\)
\(878\) 0 0
\(879\) −21.3242 −0.719249
\(880\) 0 0
\(881\) −45.3973 −1.52947 −0.764737 0.644342i \(-0.777131\pi\)
−0.764737 + 0.644342i \(0.777131\pi\)
\(882\) 0 0
\(883\) 3.57942 0.120457 0.0602286 0.998185i \(-0.480817\pi\)
0.0602286 + 0.998185i \(0.480817\pi\)
\(884\) 0 0
\(885\) −2.57978 −0.0867182
\(886\) 0 0
\(887\) 58.7606 1.97299 0.986494 0.163799i \(-0.0523749\pi\)
0.986494 + 0.163799i \(0.0523749\pi\)
\(888\) 0 0
\(889\) −14.6470 −0.491243
\(890\) 0 0
\(891\) −4.09284 −0.137115
\(892\) 0 0
\(893\) 20.0093 0.669587
\(894\) 0 0
\(895\) −1.51224 −0.0505487
\(896\) 0 0
\(897\) 1.31170 0.0437963
\(898\) 0 0
\(899\) −1.42956 −0.0476783
\(900\) 0 0
\(901\) 0.990085 0.0329845
\(902\) 0 0
\(903\) −4.99009 −0.166060
\(904\) 0 0
\(905\) 16.8862 0.561317
\(906\) 0 0
\(907\) 8.00373 0.265760 0.132880 0.991132i \(-0.457578\pi\)
0.132880 + 0.991132i \(0.457578\pi\)
\(908\) 0 0
\(909\) −23.0550 −0.764686
\(910\) 0 0
\(911\) −16.0484 −0.531707 −0.265854 0.964013i \(-0.585654\pi\)
−0.265854 + 0.964013i \(0.585654\pi\)
\(912\) 0 0
\(913\) 17.1502 0.567589
\(914\) 0 0
\(915\) 11.0842 0.366431
\(916\) 0 0
\(917\) −9.12311 −0.301271
\(918\) 0 0
\(919\) 18.6416 0.614930 0.307465 0.951559i \(-0.400519\pi\)
0.307465 + 0.951559i \(0.400519\pi\)
\(920\) 0 0
\(921\) 7.65926 0.252381
\(922\) 0 0
\(923\) −13.6569 −0.449521
\(924\) 0 0
\(925\) 6.56622 0.215896
\(926\) 0 0
\(927\) 5.25680 0.172656
\(928\) 0 0
\(929\) 19.6882 0.645948 0.322974 0.946408i \(-0.395317\pi\)
0.322974 + 0.946408i \(0.395317\pi\)
\(930\) 0 0
\(931\) 3.90570 0.128004
\(932\) 0 0
\(933\) 43.6873 1.43026
\(934\) 0 0
\(935\) 0.902844 0.0295262
\(936\) 0 0
\(937\) 48.8909 1.59720 0.798598 0.601865i \(-0.205575\pi\)
0.798598 + 0.601865i \(0.205575\pi\)
\(938\) 0 0
\(939\) 4.49825 0.146795
\(940\) 0 0
\(941\) −36.7870 −1.19922 −0.599612 0.800291i \(-0.704678\pi\)
−0.599612 + 0.800291i \(0.704678\pi\)
\(942\) 0 0
\(943\) 12.2025 0.397367
\(944\) 0 0
\(945\) −5.63637 −0.183351
\(946\) 0 0
\(947\) −45.3232 −1.47281 −0.736403 0.676543i \(-0.763477\pi\)
−0.736403 + 0.676543i \(0.763477\pi\)
\(948\) 0 0
\(949\) 8.49971 0.275912
\(950\) 0 0
\(951\) −41.2907 −1.33894
\(952\) 0 0
\(953\) −28.5184 −0.923801 −0.461900 0.886932i \(-0.652832\pi\)
−0.461900 + 0.886932i \(0.652832\pi\)
\(954\) 0 0
\(955\) 8.75481 0.283299
\(956\) 0 0
\(957\) 0.440523 0.0142401
\(958\) 0 0
\(959\) −1.63171 −0.0526905
\(960\) 0 0
\(961\) −11.0436 −0.356246
\(962\) 0 0
\(963\) −8.78522 −0.283100
\(964\) 0 0
\(965\) −1.95638 −0.0629781
\(966\) 0 0
\(967\) −11.0837 −0.356428 −0.178214 0.983992i \(-0.557032\pi\)
−0.178214 + 0.983992i \(0.557032\pi\)
\(968\) 0 0
\(969\) −4.62806 −0.148675
\(970\) 0 0
\(971\) 50.5926 1.62359 0.811797 0.583940i \(-0.198490\pi\)
0.811797 + 0.583940i \(0.198490\pi\)
\(972\) 0 0
\(973\) −7.23630 −0.231985
\(974\) 0 0
\(975\) 1.34415 0.0430473
\(976\) 0 0
\(977\) 9.36261 0.299536 0.149768 0.988721i \(-0.452147\pi\)
0.149768 + 0.988721i \(0.452147\pi\)
\(978\) 0 0
\(979\) −1.59388 −0.0509406
\(980\) 0 0
\(981\) −16.8518 −0.538037
\(982\) 0 0
\(983\) −36.5853 −1.16689 −0.583445 0.812152i \(-0.698296\pi\)
−0.583445 + 0.812152i \(0.698296\pi\)
\(984\) 0 0
\(985\) 10.6628 0.339745
\(986\) 0 0
\(987\) −6.88623 −0.219191
\(988\) 0 0
\(989\) 3.62281 0.115199
\(990\) 0 0
\(991\) −0.908676 −0.0288650 −0.0144325 0.999896i \(-0.504594\pi\)
−0.0144325 + 0.999896i \(0.504594\pi\)
\(992\) 0 0
\(993\) 23.1491 0.734613
\(994\) 0 0
\(995\) −18.0847 −0.573325
\(996\) 0 0
\(997\) 0.638561 0.0202234 0.0101117 0.999949i \(-0.496781\pi\)
0.0101117 + 0.999949i \(0.496781\pi\)
\(998\) 0 0
\(999\) −37.0097 −1.17093
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 3640.2.a.x.1.3 4
4.3 odd 2 7280.2.a.br.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.x.1.3 4 1.1 even 1 trivial
7280.2.a.br.1.2 4 4.3 odd 2