Properties

Label 136.4.k.b.81.4
Level $136$
Weight $4$
Character 136.81
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,4,Mod(81,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.k (of order \(4\), 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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 81.4
Root \(-0.0399724i\) of defining polynomial
Character \(\chi\) \(=\) 136.81
Dual form 136.4.k.b.89.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0399724 + 0.0399724i) q^{3} +(-1.97903 - 1.97903i) q^{5} +(-4.30175 + 4.30175i) q^{7} -26.9968i q^{9} +O(q^{10})\) \(q+(0.0399724 + 0.0399724i) q^{3} +(-1.97903 - 1.97903i) q^{5} +(-4.30175 + 4.30175i) q^{7} -26.9968i q^{9} +(36.5044 - 36.5044i) q^{11} +19.2351 q^{13} -0.158213i q^{15} +(-57.9167 - 39.4798i) q^{17} -25.0096i q^{19} -0.343902 q^{21} +(105.841 - 105.841i) q^{23} -117.167i q^{25} +(2.15838 - 2.15838i) q^{27} +(-14.6850 - 14.6850i) q^{29} +(161.487 + 161.487i) q^{31} +2.91833 q^{33} +17.0266 q^{35} +(-249.300 - 249.300i) q^{37} +(0.768872 + 0.768872i) q^{39} +(-155.469 + 155.469i) q^{41} +51.0771i q^{43} +(-53.4276 + 53.4276i) q^{45} +261.156 q^{47} +305.990i q^{49} +(-0.736967 - 3.89317i) q^{51} +102.139i q^{53} -144.487 q^{55} +(0.999693 - 0.999693i) q^{57} +694.356i q^{59} +(-154.454 + 154.454i) q^{61} +(116.133 + 116.133i) q^{63} +(-38.0669 - 38.0669i) q^{65} -442.817 q^{67} +8.46145 q^{69} +(382.069 + 382.069i) q^{71} +(-190.365 - 190.365i) q^{73} +(4.68344 - 4.68344i) q^{75} +314.065i q^{77} +(-22.3403 + 22.3403i) q^{79} -728.741 q^{81} -669.427i q^{83} +(36.4873 + 192.751i) q^{85} -1.17399i q^{87} +1184.49 q^{89} +(-82.7445 + 82.7445i) q^{91} +12.9100i q^{93} +(-49.4948 + 49.4948i) q^{95} +(218.714 + 218.714i) q^{97} +(-985.502 - 985.502i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{3} + 6 q^{5} + 10 q^{7} - 66 q^{11} - 124 q^{13} + 130 q^{17} - 148 q^{21} - 162 q^{23} - 204 q^{27} + 158 q^{29} - 350 q^{31} + 116 q^{33} + 236 q^{35} + 582 q^{37} - 320 q^{39} + 878 q^{41} - 26 q^{45} - 448 q^{47} - 590 q^{51} + 1460 q^{55} + 292 q^{57} - 1130 q^{61} + 114 q^{63} + 20 q^{65} - 64 q^{67} + 2076 q^{69} - 3274 q^{71} - 1426 q^{73} + 946 q^{75} + 1718 q^{79} + 166 q^{81} - 2018 q^{85} + 476 q^{89} + 2128 q^{91} - 2444 q^{95} + 1926 q^{97} + 3870 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.0399724 + 0.0399724i 0.00769269 + 0.00769269i 0.710943 0.703250i \(-0.248269\pi\)
−0.703250 + 0.710943i \(0.748269\pi\)
\(4\) 0 0
\(5\) −1.97903 1.97903i −0.177010 0.177010i 0.613041 0.790051i \(-0.289946\pi\)
−0.790051 + 0.613041i \(0.789946\pi\)
\(6\) 0 0
\(7\) −4.30175 + 4.30175i −0.232273 + 0.232273i −0.813641 0.581368i \(-0.802518\pi\)
0.581368 + 0.813641i \(0.302518\pi\)
\(8\) 0 0
\(9\) 26.9968i 0.999882i
\(10\) 0 0
\(11\) 36.5044 36.5044i 1.00059 1.00059i 0.000590210 1.00000i \(-0.499812\pi\)
1.00000 0.000590210i \(-0.000187870\pi\)
\(12\) 0 0
\(13\) 19.2351 0.410373 0.205187 0.978723i \(-0.434220\pi\)
0.205187 + 0.978723i \(0.434220\pi\)
\(14\) 0 0
\(15\) 0.158213i 0.00272337i
\(16\) 0 0
\(17\) −57.9167 39.4798i −0.826286 0.563250i
\(18\) 0 0
\(19\) 25.0096i 0.301979i −0.988535 0.150989i \(-0.951754\pi\)
0.988535 0.150989i \(-0.0482459\pi\)
\(20\) 0 0
\(21\) −0.343902 −0.00357360
\(22\) 0 0
\(23\) 105.841 105.841i 0.959540 0.959540i −0.0396723 0.999213i \(-0.512631\pi\)
0.999213 + 0.0396723i \(0.0126314\pi\)
\(24\) 0 0
\(25\) 117.167i 0.937335i
\(26\) 0 0
\(27\) 2.15838 2.15838i 0.0153845 0.0153845i
\(28\) 0 0
\(29\) −14.6850 14.6850i −0.0940322 0.0940322i 0.658526 0.752558i \(-0.271180\pi\)
−0.752558 + 0.658526i \(0.771180\pi\)
\(30\) 0 0
\(31\) 161.487 + 161.487i 0.935608 + 0.935608i 0.998049 0.0624410i \(-0.0198885\pi\)
−0.0624410 + 0.998049i \(0.519889\pi\)
\(32\) 0 0
\(33\) 2.91833 0.0153944
\(34\) 0 0
\(35\) 17.0266 0.0822292
\(36\) 0 0
\(37\) −249.300 249.300i −1.10769 1.10769i −0.993453 0.114240i \(-0.963557\pi\)
−0.114240 0.993453i \(-0.536443\pi\)
\(38\) 0 0
\(39\) 0.768872 + 0.768872i 0.00315687 + 0.00315687i
\(40\) 0 0
\(41\) −155.469 + 155.469i −0.592199 + 0.592199i −0.938225 0.346026i \(-0.887531\pi\)
0.346026 + 0.938225i \(0.387531\pi\)
\(42\) 0 0
\(43\) 51.0771i 0.181144i 0.995890 + 0.0905719i \(0.0288695\pi\)
−0.995890 + 0.0905719i \(0.971131\pi\)
\(44\) 0 0
\(45\) −53.4276 + 53.4276i −0.176989 + 0.176989i
\(46\) 0 0
\(47\) 261.156 0.810501 0.405250 0.914206i \(-0.367184\pi\)
0.405250 + 0.914206i \(0.367184\pi\)
\(48\) 0 0
\(49\) 305.990i 0.892099i
\(50\) 0 0
\(51\) −0.736967 3.89317i −0.00202345 0.0106893i
\(52\) 0 0
\(53\) 102.139i 0.264713i 0.991202 + 0.132357i \(0.0422544\pi\)
−0.991202 + 0.132357i \(0.957746\pi\)
\(54\) 0 0
\(55\) −144.487 −0.354229
\(56\) 0 0
\(57\) 0.999693 0.999693i 0.00232303 0.00232303i
\(58\) 0 0
\(59\) 694.356i 1.53216i 0.642745 + 0.766080i \(0.277796\pi\)
−0.642745 + 0.766080i \(0.722204\pi\)
\(60\) 0 0
\(61\) −154.454 + 154.454i −0.324194 + 0.324194i −0.850374 0.526179i \(-0.823624\pi\)
0.526179 + 0.850374i \(0.323624\pi\)
\(62\) 0 0
\(63\) 116.133 + 116.133i 0.232245 + 0.232245i
\(64\) 0 0
\(65\) −38.0669 38.0669i −0.0726403 0.0726403i
\(66\) 0 0
\(67\) −442.817 −0.807444 −0.403722 0.914882i \(-0.632284\pi\)
−0.403722 + 0.914882i \(0.632284\pi\)
\(68\) 0 0
\(69\) 8.46145 0.0147629
\(70\) 0 0
\(71\) 382.069 + 382.069i 0.638638 + 0.638638i 0.950219 0.311582i \(-0.100859\pi\)
−0.311582 + 0.950219i \(0.600859\pi\)
\(72\) 0 0
\(73\) −190.365 190.365i −0.305212 0.305212i 0.537837 0.843049i \(-0.319242\pi\)
−0.843049 + 0.537837i \(0.819242\pi\)
\(74\) 0 0
\(75\) 4.68344 4.68344i 0.00721062 0.00721062i
\(76\) 0 0
\(77\) 314.065i 0.464819i
\(78\) 0 0
\(79\) −22.3403 + 22.3403i −0.0318162 + 0.0318162i −0.722836 0.691020i \(-0.757162\pi\)
0.691020 + 0.722836i \(0.257162\pi\)
\(80\) 0 0
\(81\) −728.741 −0.999645
\(82\) 0 0
\(83\) 669.427i 0.885291i −0.896697 0.442645i \(-0.854040\pi\)
0.896697 0.442645i \(-0.145960\pi\)
\(84\) 0 0
\(85\) 36.4873 + 192.751i 0.0465600 + 0.245962i
\(86\) 0 0
\(87\) 1.17399i 0.00144672i
\(88\) 0 0
\(89\) 1184.49 1.41074 0.705371 0.708839i \(-0.250781\pi\)
0.705371 + 0.708839i \(0.250781\pi\)
\(90\) 0 0
\(91\) −82.7445 + 82.7445i −0.0953185 + 0.0953185i
\(92\) 0 0
\(93\) 12.9100i 0.0143947i
\(94\) 0 0
\(95\) −49.4948 + 49.4948i −0.0534533 + 0.0534533i
\(96\) 0 0
\(97\) 218.714 + 218.714i 0.228939 + 0.228939i 0.812249 0.583310i \(-0.198243\pi\)
−0.583310 + 0.812249i \(0.698243\pi\)
\(98\) 0 0
\(99\) −985.502 985.502i −1.00047 1.00047i
\(100\) 0 0
\(101\) 805.995 0.794054 0.397027 0.917807i \(-0.370042\pi\)
0.397027 + 0.917807i \(0.370042\pi\)
\(102\) 0 0
\(103\) 531.840 0.508774 0.254387 0.967102i \(-0.418126\pi\)
0.254387 + 0.967102i \(0.418126\pi\)
\(104\) 0 0
\(105\) 0.680594 + 0.680594i 0.000632563 + 0.000632563i
\(106\) 0 0
\(107\) −849.213 849.213i −0.767257 0.767257i 0.210366 0.977623i \(-0.432535\pi\)
−0.977623 + 0.210366i \(0.932535\pi\)
\(108\) 0 0
\(109\) 952.659 952.659i 0.837139 0.837139i −0.151342 0.988481i \(-0.548360\pi\)
0.988481 + 0.151342i \(0.0483596\pi\)
\(110\) 0 0
\(111\) 19.9302i 0.0170423i
\(112\) 0 0
\(113\) 161.682 161.682i 0.134600 0.134600i −0.636597 0.771197i \(-0.719658\pi\)
0.771197 + 0.636597i \(0.219658\pi\)
\(114\) 0 0
\(115\) −418.927 −0.339697
\(116\) 0 0
\(117\) 519.286i 0.410325i
\(118\) 0 0
\(119\) 418.975 79.3110i 0.322751 0.0610960i
\(120\) 0 0
\(121\) 1334.14i 1.00236i
\(122\) 0 0
\(123\) −12.4289 −0.00911119
\(124\) 0 0
\(125\) −479.256 + 479.256i −0.342928 + 0.342928i
\(126\) 0 0
\(127\) 140.286i 0.0980189i 0.998798 + 0.0490095i \(0.0156064\pi\)
−0.998798 + 0.0490095i \(0.984394\pi\)
\(128\) 0 0
\(129\) −2.04167 + 2.04167i −0.00139348 + 0.00139348i
\(130\) 0 0
\(131\) −288.058 288.058i −0.192120 0.192120i 0.604491 0.796612i \(-0.293376\pi\)
−0.796612 + 0.604491i \(0.793376\pi\)
\(132\) 0 0
\(133\) 107.585 + 107.585i 0.0701414 + 0.0701414i
\(134\) 0 0
\(135\) −8.54301 −0.00544641
\(136\) 0 0
\(137\) 2753.80 1.71732 0.858661 0.512545i \(-0.171297\pi\)
0.858661 + 0.512545i \(0.171297\pi\)
\(138\) 0 0
\(139\) −1595.40 1595.40i −0.973523 0.973523i 0.0261350 0.999658i \(-0.491680\pi\)
−0.999658 + 0.0261350i \(0.991680\pi\)
\(140\) 0 0
\(141\) 10.4390 + 10.4390i 0.00623493 + 0.00623493i
\(142\) 0 0
\(143\) 702.166 702.166i 0.410616 0.410616i
\(144\) 0 0
\(145\) 58.1241i 0.0332893i
\(146\) 0 0
\(147\) −12.2311 + 12.2311i −0.00686264 + 0.00686264i
\(148\) 0 0
\(149\) 1689.36 0.928843 0.464422 0.885614i \(-0.346262\pi\)
0.464422 + 0.885614i \(0.346262\pi\)
\(150\) 0 0
\(151\) 658.964i 0.355137i −0.984108 0.177569i \(-0.943177\pi\)
0.984108 0.177569i \(-0.0568232\pi\)
\(152\) 0 0
\(153\) −1065.83 + 1563.57i −0.563184 + 0.826188i
\(154\) 0 0
\(155\) 639.175i 0.331224i
\(156\) 0 0
\(157\) −1725.60 −0.877186 −0.438593 0.898686i \(-0.644523\pi\)
−0.438593 + 0.898686i \(0.644523\pi\)
\(158\) 0 0
\(159\) −4.08272 + 4.08272i −0.00203636 + 0.00203636i
\(160\) 0 0
\(161\) 910.605i 0.445750i
\(162\) 0 0
\(163\) −1603.92 + 1603.92i −0.770728 + 0.770728i −0.978234 0.207505i \(-0.933466\pi\)
0.207505 + 0.978234i \(0.433466\pi\)
\(164\) 0 0
\(165\) −5.77548 5.77548i −0.00272497 0.00272497i
\(166\) 0 0
\(167\) 1736.33 + 1736.33i 0.804559 + 0.804559i 0.983804 0.179246i \(-0.0573657\pi\)
−0.179246 + 0.983804i \(0.557366\pi\)
\(168\) 0 0
\(169\) −1827.01 −0.831594
\(170\) 0 0
\(171\) −675.179 −0.301943
\(172\) 0 0
\(173\) 216.659 + 216.659i 0.0952153 + 0.0952153i 0.753110 0.657895i \(-0.228553\pi\)
−0.657895 + 0.753110i \(0.728553\pi\)
\(174\) 0 0
\(175\) 504.022 + 504.022i 0.217717 + 0.217717i
\(176\) 0 0
\(177\) −27.7551 + 27.7551i −0.0117864 + 0.0117864i
\(178\) 0 0
\(179\) 2741.86i 1.14490i 0.819941 + 0.572448i \(0.194006\pi\)
−0.819941 + 0.572448i \(0.805994\pi\)
\(180\) 0 0
\(181\) 1097.33 1097.33i 0.450628 0.450628i −0.444935 0.895563i \(-0.646773\pi\)
0.895563 + 0.444935i \(0.146773\pi\)
\(182\) 0 0
\(183\) −12.3478 −0.00498785
\(184\) 0 0
\(185\) 986.746i 0.392146i
\(186\) 0 0
\(187\) −3555.40 + 673.029i −1.39036 + 0.263191i
\(188\) 0 0
\(189\) 18.5696i 0.00714678i
\(190\) 0 0
\(191\) 3861.44 1.46285 0.731424 0.681923i \(-0.238856\pi\)
0.731424 + 0.681923i \(0.238856\pi\)
\(192\) 0 0
\(193\) 1043.68 1043.68i 0.389251 0.389251i −0.485169 0.874420i \(-0.661242\pi\)
0.874420 + 0.485169i \(0.161242\pi\)
\(194\) 0 0
\(195\) 3.04325i 0.00111760i
\(196\) 0 0
\(197\) −3782.22 + 3782.22i −1.36788 + 1.36788i −0.504416 + 0.863460i \(0.668292\pi\)
−0.863460 + 0.504416i \(0.831708\pi\)
\(198\) 0 0
\(199\) 2619.61 + 2619.61i 0.933162 + 0.933162i 0.997902 0.0647402i \(-0.0206219\pi\)
−0.0647402 + 0.997902i \(0.520622\pi\)
\(200\) 0 0
\(201\) −17.7005 17.7005i −0.00621141 0.00621141i
\(202\) 0 0
\(203\) 126.342 0.0436822
\(204\) 0 0
\(205\) 615.356 0.209650
\(206\) 0 0
\(207\) −2857.38 2857.38i −0.959427 0.959427i
\(208\) 0 0
\(209\) −912.960 912.960i −0.302157 0.302157i
\(210\) 0 0
\(211\) −708.196 + 708.196i −0.231063 + 0.231063i −0.813136 0.582074i \(-0.802242\pi\)
0.582074 + 0.813136i \(0.302242\pi\)
\(212\) 0 0
\(213\) 30.5444i 0.00982568i
\(214\) 0 0
\(215\) 101.083 101.083i 0.0320643 0.0320643i
\(216\) 0 0
\(217\) −1389.35 −0.434632
\(218\) 0 0
\(219\) 15.2187i 0.00469581i
\(220\) 0 0
\(221\) −1114.03 759.398i −0.339086 0.231143i
\(222\) 0 0
\(223\) 4456.87i 1.33836i 0.743101 + 0.669180i \(0.233354\pi\)
−0.743101 + 0.669180i \(0.766646\pi\)
\(224\) 0 0
\(225\) −3163.13 −0.937224
\(226\) 0 0
\(227\) 2485.09 2485.09i 0.726613 0.726613i −0.243331 0.969943i \(-0.578240\pi\)
0.969943 + 0.243331i \(0.0782401\pi\)
\(228\) 0 0
\(229\) 4609.76i 1.33023i 0.746743 + 0.665113i \(0.231617\pi\)
−0.746743 + 0.665113i \(0.768383\pi\)
\(230\) 0 0
\(231\) −12.5539 + 12.5539i −0.00357571 + 0.00357571i
\(232\) 0 0
\(233\) 2019.71 + 2019.71i 0.567877 + 0.567877i 0.931533 0.363656i \(-0.118472\pi\)
−0.363656 + 0.931533i \(0.618472\pi\)
\(234\) 0 0
\(235\) −516.837 516.837i −0.143467 0.143467i
\(236\) 0 0
\(237\) −1.78599 −0.000489504
\(238\) 0 0
\(239\) 6534.60 1.76857 0.884285 0.466947i \(-0.154646\pi\)
0.884285 + 0.466947i \(0.154646\pi\)
\(240\) 0 0
\(241\) −2360.27 2360.27i −0.630863 0.630863i 0.317421 0.948285i \(-0.397183\pi\)
−0.948285 + 0.317421i \(0.897183\pi\)
\(242\) 0 0
\(243\) −87.4058 87.4058i −0.0230744 0.0230744i
\(244\) 0 0
\(245\) 605.564 605.564i 0.157911 0.157911i
\(246\) 0 0
\(247\) 481.062i 0.123924i
\(248\) 0 0
\(249\) 26.7586 26.7586i 0.00681026 0.00681026i
\(250\) 0 0
\(251\) 2225.36 0.559616 0.279808 0.960056i \(-0.409729\pi\)
0.279808 + 0.960056i \(0.409729\pi\)
\(252\) 0 0
\(253\) 7727.34i 1.92021i
\(254\) 0 0
\(255\) −6.24623 + 9.16320i −0.00153394 + 0.00225028i
\(256\) 0 0
\(257\) 4927.60i 1.19601i −0.801491 0.598006i \(-0.795960\pi\)
0.801491 0.598006i \(-0.204040\pi\)
\(258\) 0 0
\(259\) 2144.85 0.514574
\(260\) 0 0
\(261\) −396.448 + 396.448i −0.0940210 + 0.0940210i
\(262\) 0 0
\(263\) 5423.64i 1.27162i 0.771846 + 0.635810i \(0.219334\pi\)
−0.771846 + 0.635810i \(0.780666\pi\)
\(264\) 0 0
\(265\) 202.136 202.136i 0.0468570 0.0468570i
\(266\) 0 0
\(267\) 47.3470 + 47.3470i 0.0108524 + 0.0108524i
\(268\) 0 0
\(269\) −1401.41 1401.41i −0.317641 0.317641i 0.530220 0.847860i \(-0.322109\pi\)
−0.847860 + 0.530220i \(0.822109\pi\)
\(270\) 0 0
\(271\) −3510.15 −0.786812 −0.393406 0.919365i \(-0.628703\pi\)
−0.393406 + 0.919365i \(0.628703\pi\)
\(272\) 0 0
\(273\) −6.61499 −0.00146651
\(274\) 0 0
\(275\) −4277.11 4277.11i −0.937888 0.937888i
\(276\) 0 0
\(277\) −5410.58 5410.58i −1.17361 1.17361i −0.981342 0.192270i \(-0.938415\pi\)
−0.192270 0.981342i \(-0.561585\pi\)
\(278\) 0 0
\(279\) 4359.62 4359.62i 0.935497 0.935497i
\(280\) 0 0
\(281\) 4519.42i 0.959451i 0.877419 + 0.479726i \(0.159264\pi\)
−0.877419 + 0.479726i \(0.840736\pi\)
\(282\) 0 0
\(283\) 3295.00 3295.00i 0.692111 0.692111i −0.270585 0.962696i \(-0.587217\pi\)
0.962696 + 0.270585i \(0.0872171\pi\)
\(284\) 0 0
\(285\) −3.95685 −0.000822399
\(286\) 0 0
\(287\) 1337.57i 0.275103i
\(288\) 0 0
\(289\) 1795.69 + 4573.08i 0.365498 + 0.930812i
\(290\) 0 0
\(291\) 17.4851i 0.00352231i
\(292\) 0 0
\(293\) 6222.10 1.24061 0.620305 0.784360i \(-0.287009\pi\)
0.620305 + 0.784360i \(0.287009\pi\)
\(294\) 0 0
\(295\) 1374.15 1374.15i 0.271208 0.271208i
\(296\) 0 0
\(297\) 157.581i 0.0307871i
\(298\) 0 0
\(299\) 2035.87 2035.87i 0.393770 0.393770i
\(300\) 0 0
\(301\) −219.721 219.721i −0.0420747 0.0420747i
\(302\) 0 0
\(303\) 32.2175 + 32.2175i 0.00610841 + 0.00610841i
\(304\) 0 0
\(305\) 611.341 0.114771
\(306\) 0 0
\(307\) 478.173 0.0888951 0.0444475 0.999012i \(-0.485847\pi\)
0.0444475 + 0.999012i \(0.485847\pi\)
\(308\) 0 0
\(309\) 21.2589 + 21.2589i 0.00391384 + 0.00391384i
\(310\) 0 0
\(311\) −5088.61 5088.61i −0.927809 0.927809i 0.0697553 0.997564i \(-0.477778\pi\)
−0.997564 + 0.0697553i \(0.977778\pi\)
\(312\) 0 0
\(313\) 6588.67 6588.67i 1.18982 1.18982i 0.212703 0.977117i \(-0.431773\pi\)
0.977117 0.212703i \(-0.0682268\pi\)
\(314\) 0 0
\(315\) 459.664i 0.0822195i
\(316\) 0 0
\(317\) −939.884 + 939.884i −0.166527 + 0.166527i −0.785451 0.618924i \(-0.787569\pi\)
0.618924 + 0.785451i \(0.287569\pi\)
\(318\) 0 0
\(319\) −1072.13 −0.188175
\(320\) 0 0
\(321\) 67.8901i 0.0118045i
\(322\) 0 0
\(323\) −987.374 + 1448.47i −0.170090 + 0.249521i
\(324\) 0 0
\(325\) 2253.72i 0.384657i
\(326\) 0 0
\(327\) 76.1600 0.0128797
\(328\) 0 0
\(329\) −1123.43 + 1123.43i −0.188257 + 0.188257i
\(330\) 0 0
\(331\) 10372.7i 1.72246i 0.508218 + 0.861229i \(0.330305\pi\)
−0.508218 + 0.861229i \(0.669695\pi\)
\(332\) 0 0
\(333\) −6730.30 + 6730.30i −1.10756 + 1.10756i
\(334\) 0 0
\(335\) 876.350 + 876.350i 0.142926 + 0.142926i
\(336\) 0 0
\(337\) −5162.49 5162.49i −0.834476 0.834476i 0.153649 0.988125i \(-0.450897\pi\)
−0.988125 + 0.153649i \(0.950897\pi\)
\(338\) 0 0
\(339\) 12.9256 0.00207087
\(340\) 0 0
\(341\) 11789.9 1.87232
\(342\) 0 0
\(343\) −2791.79 2791.79i −0.439483 0.439483i
\(344\) 0 0
\(345\) −16.7455 16.7455i −0.00261318 0.00261318i
\(346\) 0 0
\(347\) −3136.83 + 3136.83i −0.485285 + 0.485285i −0.906815 0.421529i \(-0.861493\pi\)
0.421529 + 0.906815i \(0.361493\pi\)
\(348\) 0 0
\(349\) 3795.57i 0.582155i 0.956699 + 0.291078i \(0.0940138\pi\)
−0.956699 + 0.291078i \(0.905986\pi\)
\(350\) 0 0
\(351\) 41.5166 41.5166i 0.00631337 0.00631337i
\(352\) 0 0
\(353\) −6298.80 −0.949721 −0.474860 0.880061i \(-0.657501\pi\)
−0.474860 + 0.880061i \(0.657501\pi\)
\(354\) 0 0
\(355\) 1512.26i 0.226091i
\(356\) 0 0
\(357\) 19.9177 + 13.5772i 0.00295282 + 0.00201283i
\(358\) 0 0
\(359\) 4063.54i 0.597397i −0.954348 0.298699i \(-0.903447\pi\)
0.954348 0.298699i \(-0.0965525\pi\)
\(360\) 0 0
\(361\) 6233.52 0.908809
\(362\) 0 0
\(363\) 53.3288 53.3288i 0.00771085 0.00771085i
\(364\) 0 0
\(365\) 753.476i 0.108051i
\(366\) 0 0
\(367\) −6960.90 + 6960.90i −0.990071 + 0.990071i −0.999951 0.00987975i \(-0.996855\pi\)
0.00987975 + 0.999951i \(0.496855\pi\)
\(368\) 0 0
\(369\) 4197.16 + 4197.16i 0.592128 + 0.592128i
\(370\) 0 0
\(371\) −439.374 439.374i −0.0614856 0.0614856i
\(372\) 0 0
\(373\) 10995.7 1.52638 0.763188 0.646177i \(-0.223633\pi\)
0.763188 + 0.646177i \(0.223633\pi\)
\(374\) 0 0
\(375\) −38.3140 −0.00527607
\(376\) 0 0
\(377\) −282.467 282.467i −0.0385883 0.0385883i
\(378\) 0 0
\(379\) 57.7292 + 57.7292i 0.00782415 + 0.00782415i 0.711008 0.703184i \(-0.248239\pi\)
−0.703184 + 0.711008i \(0.748239\pi\)
\(380\) 0 0
\(381\) −5.60758 + 5.60758i −0.000754029 + 0.000754029i
\(382\) 0 0
\(383\) 1528.32i 0.203899i 0.994790 + 0.101949i \(0.0325080\pi\)
−0.994790 + 0.101949i \(0.967492\pi\)
\(384\) 0 0
\(385\) 621.546 621.546i 0.0822777 0.0822777i
\(386\) 0 0
\(387\) 1378.92 0.181122
\(388\) 0 0
\(389\) 7868.90i 1.02563i 0.858500 + 0.512813i \(0.171397\pi\)
−0.858500 + 0.512813i \(0.828603\pi\)
\(390\) 0 0
\(391\) −10308.6 + 1951.39i −1.33332 + 0.252394i
\(392\) 0 0
\(393\) 23.0287i 0.00295584i
\(394\) 0 0
\(395\) 88.4244 0.0112636
\(396\) 0 0
\(397\) −7541.37 + 7541.37i −0.953376 + 0.953376i −0.998961 0.0455841i \(-0.985485\pi\)
0.0455841 + 0.998961i \(0.485485\pi\)
\(398\) 0 0
\(399\) 8.60085i 0.00107915i
\(400\) 0 0
\(401\) −9284.96 + 9284.96i −1.15628 + 1.15628i −0.171013 + 0.985269i \(0.554704\pi\)
−0.985269 + 0.171013i \(0.945296\pi\)
\(402\) 0 0
\(403\) 3106.21 + 3106.21i 0.383949 + 0.383949i
\(404\) 0 0
\(405\) 1442.20 + 1442.20i 0.176947 + 0.176947i
\(406\) 0 0
\(407\) −18201.1 −2.21669
\(408\) 0 0
\(409\) 12487.9 1.50975 0.754877 0.655867i \(-0.227697\pi\)
0.754877 + 0.655867i \(0.227697\pi\)
\(410\) 0 0
\(411\) 110.076 + 110.076i 0.0132108 + 0.0132108i
\(412\) 0 0
\(413\) −2986.95 2986.95i −0.355879 0.355879i
\(414\) 0 0
\(415\) −1324.82 + 1324.82i −0.156705 + 0.156705i
\(416\) 0 0
\(417\) 127.544i 0.0149780i
\(418\) 0 0
\(419\) 1686.42 1686.42i 0.196628 0.196628i −0.601925 0.798553i \(-0.705599\pi\)
0.798553 + 0.601925i \(0.205599\pi\)
\(420\) 0 0
\(421\) 4179.68 0.483860 0.241930 0.970294i \(-0.422220\pi\)
0.241930 + 0.970294i \(0.422220\pi\)
\(422\) 0 0
\(423\) 7050.38i 0.810405i
\(424\) 0 0
\(425\) −4625.72 + 6785.92i −0.527954 + 0.774507i
\(426\) 0 0
\(427\) 1328.85i 0.150603i
\(428\) 0 0
\(429\) 56.1344 0.00631747
\(430\) 0 0
\(431\) 5546.33 5546.33i 0.619854 0.619854i −0.325640 0.945494i \(-0.605580\pi\)
0.945494 + 0.325640i \(0.105580\pi\)
\(432\) 0 0
\(433\) 1323.79i 0.146922i 0.997298 + 0.0734610i \(0.0234044\pi\)
−0.997298 + 0.0734610i \(0.976596\pi\)
\(434\) 0 0
\(435\) −2.32336 + 2.32336i −0.000256084 + 0.000256084i
\(436\) 0 0
\(437\) −2647.05 2647.05i −0.289761 0.289761i
\(438\) 0 0
\(439\) 10146.7 + 10146.7i 1.10313 + 1.10313i 0.994030 + 0.109103i \(0.0347979\pi\)
0.109103 + 0.994030i \(0.465202\pi\)
\(440\) 0 0
\(441\) 8260.75 0.891993
\(442\) 0 0
\(443\) 5281.03 0.566387 0.283193 0.959063i \(-0.408606\pi\)
0.283193 + 0.959063i \(0.408606\pi\)
\(444\) 0 0
\(445\) −2344.15 2344.15i −0.249716 0.249716i
\(446\) 0 0
\(447\) 67.5277 + 67.5277i 0.00714530 + 0.00714530i
\(448\) 0 0
\(449\) 2107.89 2107.89i 0.221553 0.221553i −0.587599 0.809152i \(-0.699927\pi\)
0.809152 + 0.587599i \(0.199927\pi\)
\(450\) 0 0
\(451\) 11350.6i 1.18510i
\(452\) 0 0
\(453\) 26.3404 26.3404i 0.00273196 0.00273196i
\(454\) 0 0
\(455\) 327.508 0.0337447
\(456\) 0 0
\(457\) 1746.31i 0.178750i −0.995998 0.0893750i \(-0.971513\pi\)
0.995998 0.0893750i \(-0.0284870\pi\)
\(458\) 0 0
\(459\) −210.219 + 39.7939i −0.0213773 + 0.00404666i
\(460\) 0 0
\(461\) 12493.9i 1.26226i −0.775679 0.631128i \(-0.782592\pi\)
0.775679 0.631128i \(-0.217408\pi\)
\(462\) 0 0
\(463\) −9171.67 −0.920612 −0.460306 0.887760i \(-0.652260\pi\)
−0.460306 + 0.887760i \(0.652260\pi\)
\(464\) 0 0
\(465\) 25.5493 25.5493i 0.00254800 0.00254800i
\(466\) 0 0
\(467\) 2173.91i 0.215410i 0.994183 + 0.107705i \(0.0343502\pi\)
−0.994183 + 0.107705i \(0.965650\pi\)
\(468\) 0 0
\(469\) 1904.89 1904.89i 0.187547 0.187547i
\(470\) 0 0
\(471\) −68.9765 68.9765i −0.00674792 0.00674792i
\(472\) 0 0
\(473\) 1864.54 + 1864.54i 0.181251 + 0.181251i
\(474\) 0 0
\(475\) −2930.30 −0.283055
\(476\) 0 0
\(477\) 2757.41 0.264682
\(478\) 0 0
\(479\) −11222.9 11222.9i −1.07054 1.07054i −0.997316 0.0732221i \(-0.976672\pi\)
−0.0732221 0.997316i \(-0.523328\pi\)
\(480\) 0 0
\(481\) −4795.31 4795.31i −0.454568 0.454568i
\(482\) 0 0
\(483\) −36.3990 + 36.3990i −0.00342901 + 0.00342901i
\(484\) 0 0
\(485\) 865.686i 0.0810491i
\(486\) 0 0
\(487\) 210.075 210.075i 0.0195470 0.0195470i −0.697266 0.716813i \(-0.745600\pi\)
0.716813 + 0.697266i \(0.245600\pi\)
\(488\) 0 0
\(489\) −128.225 −0.0118579
\(490\) 0 0
\(491\) 8812.02i 0.809941i −0.914330 0.404970i \(-0.867282\pi\)
0.914330 0.404970i \(-0.132718\pi\)
\(492\) 0 0
\(493\) 270.746 + 1430.27i 0.0247338 + 0.130661i
\(494\) 0 0
\(495\) 3900.68i 0.354187i
\(496\) 0 0
\(497\) −3287.13 −0.296676
\(498\) 0 0
\(499\) 13173.4 13173.4i 1.18180 1.18180i 0.202529 0.979276i \(-0.435084\pi\)
0.979276 0.202529i \(-0.0649159\pi\)
\(500\) 0 0
\(501\) 138.810i 0.0123784i
\(502\) 0 0
\(503\) 239.890 239.890i 0.0212647 0.0212647i −0.696394 0.717659i \(-0.745214\pi\)
0.717659 + 0.696394i \(0.245214\pi\)
\(504\) 0 0
\(505\) −1595.09 1595.09i −0.140556 0.140556i
\(506\) 0 0
\(507\) −73.0300 73.0300i −0.00639719 0.00639719i
\(508\) 0 0
\(509\) −9045.53 −0.787693 −0.393847 0.919176i \(-0.628856\pi\)
−0.393847 + 0.919176i \(0.628856\pi\)
\(510\) 0 0
\(511\) 1637.80 0.141785
\(512\) 0 0
\(513\) −53.9802 53.9802i −0.00464578 0.00464578i
\(514\) 0 0
\(515\) −1052.53 1052.53i −0.0900583 0.0900583i
\(516\) 0 0
\(517\) 9533.35 9533.35i 0.810979 0.810979i
\(518\) 0 0
\(519\) 17.3207i 0.00146492i
\(520\) 0 0
\(521\) 2882.23 2882.23i 0.242366 0.242366i −0.575462 0.817828i \(-0.695178\pi\)
0.817828 + 0.575462i \(0.195178\pi\)
\(522\) 0 0
\(523\) 22649.5 1.89368 0.946839 0.321707i \(-0.104257\pi\)
0.946839 + 0.321707i \(0.104257\pi\)
\(524\) 0 0
\(525\) 40.2939i 0.00334966i
\(526\) 0 0
\(527\) −2977.31 15728.2i −0.246098 1.30006i
\(528\) 0 0
\(529\) 10237.7i 0.841436i
\(530\) 0 0
\(531\) 18745.4 1.53198
\(532\) 0 0
\(533\) −2990.46 + 2990.46i −0.243023 + 0.243023i
\(534\) 0 0
\(535\) 3361.24i 0.271625i
\(536\) 0 0
\(537\) −109.599 + 109.599i −0.00880733 + 0.00880733i
\(538\) 0 0
\(539\) 11170.0 + 11170.0i 0.892625 + 0.892625i
\(540\) 0 0
\(541\) −8794.59 8794.59i −0.698908 0.698908i 0.265267 0.964175i \(-0.414540\pi\)
−0.964175 + 0.265267i \(0.914540\pi\)
\(542\) 0 0
\(543\) 87.7256 0.00693308
\(544\) 0 0
\(545\) −3770.69 −0.296364
\(546\) 0 0
\(547\) −9867.73 9867.73i −0.771323 0.771323i 0.207015 0.978338i \(-0.433625\pi\)
−0.978338 + 0.207015i \(0.933625\pi\)
\(548\) 0 0
\(549\) 4169.78 + 4169.78i 0.324156 + 0.324156i
\(550\) 0 0
\(551\) −367.265 + 367.265i −0.0283957 + 0.0283957i
\(552\) 0 0
\(553\) 192.205i 0.0147801i
\(554\) 0 0
\(555\) −39.4426 + 39.4426i −0.00301666 + 0.00301666i
\(556\) 0 0
\(557\) −17027.0 −1.29526 −0.647629 0.761956i \(-0.724239\pi\)
−0.647629 + 0.761956i \(0.724239\pi\)
\(558\) 0 0
\(559\) 982.473i 0.0743366i
\(560\) 0 0
\(561\) −169.020 115.215i −0.0127202 0.00867093i
\(562\) 0 0
\(563\) 23996.7i 1.79634i 0.439649 + 0.898170i \(0.355103\pi\)
−0.439649 + 0.898170i \(0.644897\pi\)
\(564\) 0 0
\(565\) −639.949 −0.0476510
\(566\) 0 0
\(567\) 3134.86 3134.86i 0.232190 0.232190i
\(568\) 0 0
\(569\) 442.338i 0.0325901i 0.999867 + 0.0162951i \(0.00518711\pi\)
−0.999867 + 0.0162951i \(0.994813\pi\)
\(570\) 0 0
\(571\) 4060.36 4060.36i 0.297584 0.297584i −0.542483 0.840067i \(-0.682516\pi\)
0.840067 + 0.542483i \(0.182516\pi\)
\(572\) 0 0
\(573\) 154.351 + 154.351i 0.0112532 + 0.0112532i
\(574\) 0 0
\(575\) −12401.1 12401.1i −0.899411 0.899411i
\(576\) 0 0
\(577\) −6432.83 −0.464129 −0.232064 0.972700i \(-0.574548\pi\)
−0.232064 + 0.972700i \(0.574548\pi\)
\(578\) 0 0
\(579\) 83.4363 0.00598877
\(580\) 0 0
\(581\) 2879.71 + 2879.71i 0.205629 + 0.205629i
\(582\) 0 0
\(583\) 3728.51 + 3728.51i 0.264870 + 0.264870i
\(584\) 0 0
\(585\) −1027.68 + 1027.68i −0.0726317 + 0.0726317i
\(586\) 0 0
\(587\) 5080.96i 0.357264i −0.983916 0.178632i \(-0.942833\pi\)
0.983916 0.178632i \(-0.0571671\pi\)
\(588\) 0 0
\(589\) 4038.71 4038.71i 0.282534 0.282534i
\(590\) 0 0
\(591\) −302.368 −0.0210453
\(592\) 0 0
\(593\) 15267.6i 1.05727i −0.848848 0.528637i \(-0.822703\pi\)
0.848848 0.528637i \(-0.177297\pi\)
\(594\) 0 0
\(595\) −986.125 672.207i −0.0679449 0.0463156i
\(596\) 0 0
\(597\) 209.424i 0.0143570i
\(598\) 0 0
\(599\) 14852.3 1.01310 0.506551 0.862210i \(-0.330920\pi\)
0.506551 + 0.862210i \(0.330920\pi\)
\(600\) 0 0
\(601\) −6958.91 + 6958.91i −0.472313 + 0.472313i −0.902662 0.430349i \(-0.858390\pi\)
0.430349 + 0.902662i \(0.358390\pi\)
\(602\) 0 0
\(603\) 11954.7i 0.807348i
\(604\) 0 0
\(605\) −2640.31 + 2640.31i −0.177428 + 0.177428i
\(606\) 0 0
\(607\) 2781.06 + 2781.06i 0.185964 + 0.185964i 0.793949 0.607985i \(-0.208022\pi\)
−0.607985 + 0.793949i \(0.708022\pi\)
\(608\) 0 0
\(609\) 5.05020 + 5.05020i 0.000336033 + 0.000336033i
\(610\) 0 0
\(611\) 5023.36 0.332608
\(612\) 0 0
\(613\) 3388.18 0.223242 0.111621 0.993751i \(-0.464396\pi\)
0.111621 + 0.993751i \(0.464396\pi\)
\(614\) 0 0
\(615\) 24.5972 + 24.5972i 0.00161277 + 0.00161277i
\(616\) 0 0
\(617\) −8426.48 8426.48i −0.549818 0.549818i 0.376570 0.926388i \(-0.377103\pi\)
−0.926388 + 0.376570i \(0.877103\pi\)
\(618\) 0 0
\(619\) −5634.61 + 5634.61i −0.365871 + 0.365871i −0.865969 0.500098i \(-0.833297\pi\)
0.500098 + 0.865969i \(0.333297\pi\)
\(620\) 0 0
\(621\) 456.891i 0.0295240i
\(622\) 0 0
\(623\) −5095.39 + 5095.39i −0.327677 + 0.327677i
\(624\) 0 0
\(625\) −12748.9 −0.815931
\(626\) 0 0
\(627\) 72.9864i 0.00464879i
\(628\) 0 0
\(629\) 4596.32 + 24280.9i 0.291363 + 1.53918i
\(630\) 0 0
\(631\) 15781.0i 0.995612i 0.867288 + 0.497806i \(0.165861\pi\)
−0.867288 + 0.497806i \(0.834139\pi\)
\(632\) 0 0
\(633\) −56.6165 −0.00355498
\(634\) 0 0
\(635\) 277.631 277.631i 0.0173503 0.0173503i
\(636\) 0 0
\(637\) 5885.75i 0.366094i
\(638\) 0 0
\(639\) 10314.6 10314.6i 0.638562 0.638562i
\(640\) 0 0
\(641\) −13468.8 13468.8i −0.829930 0.829930i 0.157577 0.987507i \(-0.449632\pi\)
−0.987507 + 0.157577i \(0.949632\pi\)
\(642\) 0 0
\(643\) 18993.3 + 18993.3i 1.16489 + 1.16489i 0.983391 + 0.181499i \(0.0580949\pi\)
0.181499 + 0.983391i \(0.441905\pi\)
\(644\) 0 0
\(645\) 8.08108 0.000493321
\(646\) 0 0
\(647\) −22573.9 −1.37167 −0.685835 0.727757i \(-0.740563\pi\)
−0.685835 + 0.727757i \(0.740563\pi\)
\(648\) 0 0
\(649\) 25347.1 + 25347.1i 1.53306 + 1.53306i
\(650\) 0 0
\(651\) −55.5356 55.5356i −0.00334349 0.00334349i
\(652\) 0 0
\(653\) 3162.80 3162.80i 0.189540 0.189540i −0.605957 0.795497i \(-0.707210\pi\)
0.795497 + 0.605957i \(0.207210\pi\)
\(654\) 0 0
\(655\) 1140.15i 0.0680145i
\(656\) 0 0
\(657\) −5139.24 + 5139.24i −0.305176 + 0.305176i
\(658\) 0 0
\(659\) −11439.0 −0.676174 −0.338087 0.941115i \(-0.609780\pi\)
−0.338087 + 0.941115i \(0.609780\pi\)
\(660\) 0 0
\(661\) 11381.6i 0.669730i 0.942266 + 0.334865i \(0.108691\pi\)
−0.942266 + 0.334865i \(0.891309\pi\)
\(662\) 0 0
\(663\) −14.1756 74.8855i −0.000830371 0.00438659i
\(664\) 0 0
\(665\) 425.829i 0.0248315i
\(666\) 0 0
\(667\) −3108.55 −0.180455
\(668\) 0 0
\(669\) −178.152 + 178.152i −0.0102956 + 0.0102956i
\(670\) 0 0
\(671\) 11276.5i 0.648771i
\(672\) 0 0
\(673\) −8821.52 + 8821.52i −0.505267 + 0.505267i −0.913070 0.407803i \(-0.866295\pi\)
0.407803 + 0.913070i \(0.366295\pi\)
\(674\) 0 0
\(675\) −252.891 252.891i −0.0144204 0.0144204i
\(676\) 0 0
\(677\) −8725.07 8725.07i −0.495320 0.495320i 0.414657 0.909978i \(-0.363901\pi\)
−0.909978 + 0.414657i \(0.863901\pi\)
\(678\) 0 0
\(679\) −1881.71 −0.106352
\(680\) 0 0
\(681\) 198.670 0.0111792
\(682\) 0 0
\(683\) −12158.3 12158.3i −0.681146 0.681146i 0.279113 0.960258i \(-0.409960\pi\)
−0.960258 + 0.279113i \(0.909960\pi\)
\(684\) 0 0
\(685\) −5449.86 5449.86i −0.303983 0.303983i
\(686\) 0 0
\(687\) −184.263 + 184.263i −0.0102330 + 0.0102330i
\(688\) 0 0
\(689\) 1964.64i 0.108631i
\(690\) 0 0
\(691\) 2132.44 2132.44i 0.117398 0.117398i −0.645967 0.763365i \(-0.723546\pi\)
0.763365 + 0.645967i \(0.223546\pi\)
\(692\) 0 0
\(693\) 8478.76 0.464764
\(694\) 0 0
\(695\) 6314.69i 0.344647i
\(696\) 0 0
\(697\) 15142.1 2866.36i 0.822882 0.155769i
\(698\) 0 0
\(699\) 161.465i 0.00873700i
\(700\) 0 0
\(701\) −31729.0 −1.70954 −0.854770 0.519007i \(-0.826302\pi\)
−0.854770 + 0.519007i \(0.826302\pi\)
\(702\) 0 0
\(703\) −6234.89 + 6234.89i −0.334500 + 0.334500i
\(704\) 0 0
\(705\) 41.3184i 0.00220729i
\(706\) 0 0
\(707\) −3467.19 + 3467.19i −0.184437 + 0.184437i
\(708\) 0 0
\(709\) 18575.2 + 18575.2i 0.983930 + 0.983930i 0.999873 0.0159433i \(-0.00507512\pi\)
−0.0159433 + 0.999873i \(0.505075\pi\)
\(710\) 0 0
\(711\) 603.117 + 603.117i 0.0318124 + 0.0318124i
\(712\) 0 0
\(713\) 34183.9 1.79551
\(714\) 0 0
\(715\) −2779.22 −0.145366
\(716\) 0 0
\(717\) 261.204 + 261.204i 0.0136051 + 0.0136051i
\(718\) 0 0
\(719\) 17278.6 + 17278.6i 0.896222 + 0.896222i 0.995100 0.0988778i \(-0.0315253\pi\)
−0.0988778 + 0.995100i \(0.531525\pi\)
\(720\) 0 0
\(721\) −2287.84 + 2287.84i −0.118174 + 0.118174i
\(722\) 0 0
\(723\) 188.691i 0.00970607i
\(724\) 0 0
\(725\) −1720.59 + 1720.59i −0.0881396 + 0.0881396i
\(726\) 0 0
\(727\) −7209.98 −0.367818 −0.183909 0.982943i \(-0.558875\pi\)
−0.183909 + 0.982943i \(0.558875\pi\)
\(728\) 0 0
\(729\) 19669.0i 0.999290i
\(730\) 0 0
\(731\) 2016.51 2958.22i 0.102029 0.149677i
\(732\) 0 0
\(733\) 21640.2i 1.09045i −0.838290 0.545225i \(-0.816444\pi\)
0.838290 0.545225i \(-0.183556\pi\)
\(734\) 0 0
\(735\) 48.4117 0.00242951
\(736\) 0 0
\(737\) −16164.8 + 16164.8i −0.807920 + 0.807920i
\(738\) 0 0
\(739\) 34704.0i 1.72748i −0.503940 0.863738i \(-0.668117\pi\)
0.503940 0.863738i \(-0.331883\pi\)
\(740\) 0 0
\(741\) 19.2292 19.2292i 0.000953309 0.000953309i
\(742\) 0 0
\(743\) −23978.2 23978.2i −1.18395 1.18395i −0.978712 0.205241i \(-0.934202\pi\)
−0.205241 0.978712i \(-0.565798\pi\)
\(744\) 0 0
\(745\) −3343.30 3343.30i −0.164415 0.164415i
\(746\) 0 0
\(747\) −18072.4 −0.885186
\(748\) 0 0
\(749\) 7306.20 0.356426
\(750\) 0 0
\(751\) 431.344 + 431.344i 0.0209587 + 0.0209587i 0.717508 0.696550i \(-0.245282\pi\)
−0.696550 + 0.717508i \(0.745282\pi\)
\(752\) 0 0
\(753\) 88.9530 + 88.9530i 0.00430495 + 0.00430495i
\(754\) 0 0
\(755\) −1304.11 + 1304.11i −0.0628629 + 0.0628629i
\(756\) 0 0
\(757\) 11527.3i 0.553457i −0.960948 0.276729i \(-0.910750\pi\)
0.960948 0.276729i \(-0.0892503\pi\)
\(758\) 0 0
\(759\) 308.880 308.880i 0.0147716 0.0147716i
\(760\) 0 0
\(761\) 35908.5 1.71049 0.855246 0.518223i \(-0.173406\pi\)
0.855246 + 0.518223i \(0.173406\pi\)
\(762\) 0 0
\(763\) 8196.19i 0.388889i
\(764\) 0 0
\(765\) 5203.66 985.040i 0.245933 0.0465545i
\(766\) 0 0
\(767\) 13356.0i 0.628758i
\(768\) 0 0
\(769\) −4159.74 −0.195064 −0.0975320 0.995232i \(-0.531095\pi\)
−0.0975320 + 0.995232i \(0.531095\pi\)
\(770\) 0 0
\(771\) 196.968 196.968i 0.00920055 0.00920055i
\(772\) 0 0
\(773\) 38339.5i 1.78393i −0.452106 0.891964i \(-0.649327\pi\)
0.452106 0.891964i \(-0.350673\pi\)
\(774\) 0 0
\(775\) 18920.9 18920.9i 0.876978 0.876978i
\(776\) 0 0
\(777\) 85.7348 + 85.7348i 0.00395845 + 0.00395845i
\(778\) 0 0
\(779\) 3888.21 + 3888.21i 0.178831 + 0.178831i
\(780\) 0 0
\(781\) 27894.4 1.27803
\(782\) 0 0
\(783\) −63.3915 −0.00289327
\(784\) 0 0
\(785\) 3415.03 + 3415.03i 0.155271 + 0.155271i
\(786\) 0 0
\(787\) −10975.5 10975.5i −0.497123 0.497123i 0.413418 0.910541i \(-0.364335\pi\)
−0.910541 + 0.413418i \(0.864335\pi\)
\(788\) 0 0
\(789\) −216.796 + 216.796i −0.00978217 + 0.00978217i
\(790\) 0 0
\(791\) 1391.03i 0.0625276i
\(792\) 0 0
\(793\) −2970.95 + 2970.95i −0.133041 + 0.133041i
\(794\) 0 0
\(795\) 16.1597 0.000720912
\(796\) 0 0
\(797\) 483.686i 0.0214969i 0.999942 + 0.0107484i \(0.00342140\pi\)
−0.999942 + 0.0107484i \(0.996579\pi\)
\(798\) 0 0
\(799\) −15125.3 10310.4i −0.669706 0.456515i
\(800\) 0 0
\(801\) 31977.5i 1.41057i
\(802\) 0 0
\(803\) −13898.3 −0.610785
\(804\) 0 0
\(805\) 1802.12 1802.12i 0.0789023 0.0789023i
\(806\) 0 0
\(807\) 112.035i 0.00488702i
\(808\) 0 0
\(809\) −11349.9 + 11349.9i −0.493252 + 0.493252i −0.909329 0.416077i \(-0.863405\pi\)
0.416077 + 0.909329i \(0.363405\pi\)
\(810\) 0 0
\(811\) −7111.36 7111.36i −0.307908 0.307908i 0.536189 0.844098i \(-0.319863\pi\)
−0.844098 + 0.536189i \(0.819863\pi\)
\(812\) 0 0
\(813\) −140.309 140.309i −0.00605270 0.00605270i
\(814\) 0 0
\(815\) 6348.42 0.272854
\(816\) 0 0
\(817\) 1277.42 0.0547016
\(818\) 0 0
\(819\) 2233.84 + 2233.84i 0.0953072 + 0.0953072i
\(820\) 0 0
\(821\) 29059.7 + 29059.7i 1.23531 + 1.23531i 0.961896 + 0.273415i \(0.0881532\pi\)
0.273415 + 0.961896i \(0.411847\pi\)
\(822\) 0 0
\(823\) −3018.11 + 3018.11i −0.127831 + 0.127831i −0.768128 0.640297i \(-0.778811\pi\)
0.640297 + 0.768128i \(0.278811\pi\)
\(824\) 0 0
\(825\) 341.932i 0.0144298i
\(826\) 0 0
\(827\) 9959.68 9959.68i 0.418781 0.418781i −0.466002 0.884783i \(-0.654306\pi\)
0.884783 + 0.466002i \(0.154306\pi\)
\(828\) 0 0
\(829\) −42024.2 −1.76063 −0.880314 0.474391i \(-0.842668\pi\)
−0.880314 + 0.474391i \(0.842668\pi\)
\(830\) 0 0
\(831\) 432.548i 0.0180565i
\(832\) 0 0
\(833\) 12080.4 17721.9i 0.502475 0.737129i
\(834\) 0 0
\(835\) 6872.51i 0.284830i
\(836\) 0 0
\(837\) 697.099 0.0287876
\(838\) 0 0
\(839\) 32650.7 32650.7i 1.34354 1.34354i 0.451024 0.892512i \(-0.351059\pi\)
0.892512 0.451024i \(-0.148941\pi\)
\(840\) 0 0
\(841\) 23957.7i 0.982316i
\(842\) 0 0
\(843\) −180.652 + 180.652i −0.00738076 + 0.00738076i
\(844\) 0 0
\(845\) 3615.72 + 3615.72i 0.147201 + 0.147201i
\(846\) 0 0
\(847\) 5739.14 + 5739.14i 0.232821 + 0.232821i
\(848\) 0 0
\(849\) 263.418 0.0106484
\(850\) 0 0
\(851\) −52772.4 −2.12575
\(852\) 0 0
\(853\) 33363.7 + 33363.7i 1.33922 + 1.33922i 0.896819 + 0.442397i \(0.145872\pi\)
0.442397 + 0.896819i \(0.354128\pi\)
\(854\) 0 0
\(855\) 1336.20 + 1336.20i 0.0534470 + 0.0534470i
\(856\) 0 0
\(857\) 11067.4 11067.4i 0.441139 0.441139i −0.451255 0.892395i \(-0.649024\pi\)
0.892395 + 0.451255i \(0.149024\pi\)
\(858\) 0 0
\(859\) 34645.2i 1.37611i −0.725658 0.688056i \(-0.758464\pi\)
0.725658 0.688056i \(-0.241536\pi\)
\(860\) 0 0
\(861\) 53.4660 53.4660i 0.00211628 0.00211628i
\(862\) 0 0
\(863\) 28318.9 1.11702 0.558508 0.829499i \(-0.311374\pi\)
0.558508 + 0.829499i \(0.311374\pi\)
\(864\) 0 0
\(865\) 857.550i 0.0337082i
\(866\) 0 0
\(867\) −111.019 + 254.575i −0.00434878 + 0.00997211i
\(868\) 0 0
\(869\) 1631.04i 0.0636700i
\(870\) 0 0
\(871\) −8517.63 −0.331354
\(872\) 0 0
\(873\) 5904.59 5904.59i 0.228912 0.228912i
\(874\) 0 0
\(875\) 4123.28i 0.159306i
\(876\) 0 0
\(877\) 9654.78 9654.78i 0.371743 0.371743i −0.496369 0.868112i \(-0.665334\pi\)
0.868112 + 0.496369i \(0.165334\pi\)
\(878\) 0 0
\(879\) 248.712 + 248.712i 0.00954363 + 0.00954363i
\(880\) 0 0
\(881\) 4554.04 + 4554.04i 0.174154 + 0.174154i 0.788802 0.614648i \(-0.210702\pi\)
−0.614648 + 0.788802i \(0.710702\pi\)
\(882\) 0 0
\(883\) −35594.1 −1.35655 −0.678277 0.734807i \(-0.737273\pi\)
−0.678277 + 0.734807i \(0.737273\pi\)
\(884\) 0 0
\(885\) 109.856 0.00417264
\(886\) 0 0
\(887\) 22055.1 + 22055.1i 0.834881 + 0.834881i 0.988180 0.153299i \(-0.0489898\pi\)
−0.153299 + 0.988180i \(0.548990\pi\)
\(888\) 0 0
\(889\) −603.477 603.477i −0.0227671 0.0227671i
\(890\) 0 0
\(891\) −26602.3 + 26602.3i −1.00023 + 1.00023i
\(892\) 0 0
\(893\) 6531.41i 0.244754i
\(894\) 0 0
\(895\) 5426.24 5426.24i 0.202658 0.202658i
\(896\) 0 0
\(897\) 162.757 0.00605830
\(898\) 0 0
\(899\) 4742.85i 0.175954i
\(900\) 0 0
\(901\) 4032.41 5915.53i 0.149100 0.218729i
\(902\) 0 0
\(903\) 17.5655i 0.000647335i
\(904\) 0 0
\(905\) −4343.30 −0.159532
\(906\) 0 0
\(907\) −13912.2 + 13912.2i −0.509313 + 0.509313i −0.914315 0.405003i \(-0.867271\pi\)
0.405003 + 0.914315i \(0.367271\pi\)
\(908\) 0 0
\(909\) 21759.3i 0.793960i
\(910\) 0 0
\(911\) 25127.5 25127.5i 0.913843 0.913843i −0.0827295 0.996572i \(-0.526364\pi\)
0.996572 + 0.0827295i \(0.0263638\pi\)
\(912\) 0 0
\(913\) −24437.0 24437.0i −0.885813 0.885813i
\(914\) 0 0
\(915\) 24.4367 + 24.4367i 0.000882901 + 0.000882901i
\(916\) 0 0
\(917\) 2478.31 0.0892485
\(918\) 0 0
\(919\) 378.693 0.0135930 0.00679648 0.999977i \(-0.497837\pi\)
0.00679648 + 0.999977i \(0.497837\pi\)
\(920\) 0 0
\(921\) 19.1137 + 19.1137i 0.000683842 + 0.000683842i
\(922\) 0 0
\(923\) 7349.14 + 7349.14i 0.262080 + 0.262080i
\(924\) 0 0
\(925\) −29209.7 + 29209.7i −1.03828 + 1.03828i
\(926\) 0 0
\(927\) 14358.0i 0.508714i
\(928\) 0 0
\(929\) −9344.29 + 9344.29i −0.330007 + 0.330007i −0.852589 0.522582i \(-0.824969\pi\)
0.522582 + 0.852589i \(0.324969\pi\)
\(930\) 0 0
\(931\) 7652.68 0.269395
\(932\) 0 0
\(933\) 406.808i 0.0142747i
\(934\) 0 0
\(935\) 8368.21 + 5704.31i 0.292695 + 0.199520i
\(936\) 0 0
\(937\) 34950.5i 1.21855i 0.792959 + 0.609275i \(0.208540\pi\)
−0.792959 + 0.609275i \(0.791460\pi\)
\(938\) 0 0
\(939\) 526.730 0.0183058
\(940\) 0 0
\(941\) −9450.53 + 9450.53i −0.327395 + 0.327395i −0.851595 0.524200i \(-0.824364\pi\)
0.524200 + 0.851595i \(0.324364\pi\)
\(942\) 0 0
\(943\) 32910.0i 1.13648i
\(944\) 0 0
\(945\) 36.7499 36.7499i 0.00126505 0.00126505i
\(946\) 0 0
\(947\) 18549.3 + 18549.3i 0.636506 + 0.636506i 0.949692 0.313186i \(-0.101396\pi\)
−0.313186 + 0.949692i \(0.601396\pi\)
\(948\) 0 0
\(949\) −3661.68 3661.68i −0.125251 0.125251i
\(950\) 0 0
\(951\) −75.1388 −0.00256208
\(952\) 0 0
\(953\) −19074.4 −0.648352 −0.324176 0.945997i \(-0.605087\pi\)
−0.324176 + 0.945997i \(0.605087\pi\)
\(954\) 0 0
\(955\) −7641.92 7641.92i −0.258939 0.258939i
\(956\) 0 0
\(957\) −42.8557 42.8557i −0.00144757 0.00144757i
\(958\) 0 0
\(959\) −11846.2 + 11846.2i −0.398887 + 0.398887i
\(960\) 0 0
\(961\) 22364.8i 0.750724i
\(962\) 0 0
\(963\) −22926.0 + 22926.0i −0.767166 + 0.767166i
\(964\) 0 0
\(965\) −4130.94 −0.137803
\(966\) 0 0
\(967\) 6142.31i 0.204264i 0.994771 + 0.102132i \(0.0325664\pi\)
−0.994771 + 0.102132i \(0.967434\pi\)
\(968\) 0 0
\(969\) −97.3666 + 18.4312i −0.00322793 + 0.000611039i
\(970\) 0 0
\(971\) 56958.3i 1.88247i 0.337750 + 0.941236i \(0.390334\pi\)
−0.337750 + 0.941236i \(0.609666\pi\)
\(972\) 0 0
\(973\) 13726.0 0.452246
\(974\) 0 0
\(975\) 90.0863 90.0863i 0.00295905 0.00295905i
\(976\) 0 0
\(977\) 11185.4i 0.366277i −0.983087 0.183138i \(-0.941374\pi\)
0.983087 0.183138i \(-0.0586257\pi\)
\(978\) 0 0
\(979\) 43239.2 43239.2i 1.41157 1.41157i
\(980\) 0 0
\(981\) −25718.7 25718.7i −0.837040 0.837040i
\(982\) 0 0
\(983\) 20875.7 + 20875.7i 0.677348 + 0.677348i 0.959399 0.282052i \(-0.0910150\pi\)
−0.282052 + 0.959399i \(0.591015\pi\)
\(984\) 0 0
\(985\) 14970.3 0.484256
\(986\) 0 0
\(987\) −89.8121 −0.00289641
\(988\) 0 0
\(989\) 5406.06 + 5406.06i 0.173815 + 0.173815i
\(990\) 0 0
\(991\) 30805.3 + 30805.3i 0.987450 + 0.987450i 0.999922 0.0124723i \(-0.00397017\pi\)
−0.0124723 + 0.999922i \(0.503970\pi\)
\(992\) 0 0
\(993\) −414.620 + 414.620i −0.0132503 + 0.0132503i
\(994\) 0 0
\(995\) 10368.6i 0.330358i
\(996\) 0 0
\(997\) −28690.5 + 28690.5i −0.911370 + 0.911370i −0.996380 0.0850098i \(-0.972908\pi\)
0.0850098 + 0.996380i \(0.472908\pi\)
\(998\) 0 0
\(999\) −1076.17 −0.0340825
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 136.4.k.b.81.4 14
4.3 odd 2 272.4.o.f.81.4 14
17.2 even 8 2312.4.a.l.1.8 14
17.4 even 4 inner 136.4.k.b.89.4 yes 14
17.15 even 8 2312.4.a.l.1.7 14
68.55 odd 4 272.4.o.f.225.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.k.b.81.4 14 1.1 even 1 trivial
136.4.k.b.89.4 yes 14 17.4 even 4 inner
272.4.o.f.81.4 14 4.3 odd 2
272.4.o.f.225.4 14 68.55 odd 4
2312.4.a.l.1.7 14 17.15 even 8
2312.4.a.l.1.8 14 17.2 even 8