Properties

Label 136.4.s.a.11.1
Level $136$
Weight $4$
Character 136.11
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,4,Mod(3,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.s (of order \(16\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label 11.1
Root \(-0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 136.11
Dual form 136.4.s.a.99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.61313 - 1.08239i) q^{2} +(-1.54267 - 7.75551i) q^{3} +(5.65685 + 5.65685i) q^{4} +(-4.36332 + 21.9359i) q^{6} +(-8.65914 - 20.9050i) q^{8} +(-32.8234 + 13.5959i) q^{9} +O(q^{10})\) \(q+(-2.61313 - 1.08239i) q^{2} +(-1.54267 - 7.75551i) q^{3} +(5.65685 + 5.65685i) q^{4} +(-4.36332 + 21.9359i) q^{6} +(-8.65914 - 20.9050i) q^{8} +(-32.8234 + 13.5959i) q^{9} +(-71.4636 - 14.2150i) q^{11} +(35.1452 - 52.5985i) q^{12} +64.0000i q^{16} +(21.0091 + 66.8701i) q^{17} +100.488 q^{18} +(13.9013 + 5.75812i) q^{19} +(171.357 + 114.497i) q^{22} +(-148.771 + 99.4055i) q^{24} +(47.8354 + 115.485i) q^{25} +(37.4640 + 56.0688i) q^{27} +(69.2731 - 167.240i) q^{32} +576.166i q^{33} +(17.4802 - 197.480i) q^{34} +(-262.587 - 108.767i) q^{36} +(-30.0934 - 30.0934i) q^{38} +(-416.511 + 278.304i) q^{41} +(-76.5305 + 31.7000i) q^{43} +(-323.847 - 484.671i) q^{44} +(496.353 - 98.7307i) q^{48} +(131.260 - 316.891i) q^{49} -353.553i q^{50} +(486.202 - 266.095i) q^{51} +(-37.2096 - 187.065i) q^{54} +(23.2120 - 116.695i) q^{57} +(-251.472 - 607.106i) q^{59} +(-362.039 + 362.039i) q^{64} +(623.637 - 1505.59i) q^{66} +984.492i q^{67} +(-259.429 + 497.120i) q^{68} +(568.445 + 568.445i) q^{72} +(-1024.67 - 684.661i) q^{73} +(821.851 - 549.143i) q^{75} +(46.0649 + 111.211i) q^{76} +(-301.245 + 301.245i) q^{81} +(1389.63 - 276.415i) q^{82} +(43.2971 - 104.528i) q^{83} +234.296 q^{86} +(321.648 + 1617.04i) q^{88} +(-1029.99 - 1029.99i) q^{89} +(-1403.90 - 279.253i) q^{96} +(-866.591 + 1296.94i) q^{97} +(-686.000 + 686.000i) q^{98} +(2538.94 - 505.028i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{7}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61313 1.08239i −0.923880 0.382683i
\(3\) −1.54267 7.75551i −0.296887 1.49255i −0.784850 0.619685i \(-0.787260\pi\)
0.487964 0.872864i \(-0.337740\pi\)
\(4\) 5.65685 + 5.65685i 0.707107 + 0.707107i
\(5\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(6\) −4.36332 + 21.9359i −0.296887 + 1.49255i
\(7\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(8\) −8.65914 20.9050i −0.382683 0.923880i
\(9\) −32.8234 + 13.5959i −1.21568 + 0.503552i
\(10\) 0 0
\(11\) −71.4636 14.2150i −1.95883 0.389635i −0.989731 0.142943i \(-0.954343\pi\)
−0.969094 0.246691i \(-0.920657\pi\)
\(12\) 35.1452 52.5985i 0.845461 1.26532i
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000i 1.00000i
\(17\) 21.0091 + 66.8701i 0.299733 + 0.954023i
\(18\) 100.488 1.31585
\(19\) 13.9013 + 5.75812i 0.167852 + 0.0695264i 0.465027 0.885296i \(-0.346045\pi\)
−0.297175 + 0.954823i \(0.596045\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 171.357 + 114.497i 1.66061 + 1.10959i
\(23\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(24\) −148.771 + 99.4055i −1.26532 + 0.845461i
\(25\) 47.8354 + 115.485i 0.382683 + 0.923880i
\(26\) 0 0
\(27\) 37.4640 + 56.0688i 0.267035 + 0.399646i
\(28\) 0 0
\(29\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(30\) 0 0
\(31\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(32\) 69.2731 167.240i 0.382683 0.923880i
\(33\) 576.166i 3.03932i
\(34\) 17.4802 197.480i 0.0881716 0.996105i
\(35\) 0 0
\(36\) −262.587 108.767i −1.21568 0.503552i
\(37\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(38\) −30.0934 30.0934i −0.128468 0.128468i
\(39\) 0 0
\(40\) 0 0
\(41\) −416.511 + 278.304i −1.58654 + 1.06009i −0.626809 + 0.779173i \(0.715639\pi\)
−0.959731 + 0.280919i \(0.909361\pi\)
\(42\) 0 0
\(43\) −76.5305 + 31.7000i −0.271414 + 0.112423i −0.514239 0.857647i \(-0.671926\pi\)
0.242826 + 0.970070i \(0.421926\pi\)
\(44\) −323.847 484.671i −1.10959 1.66061i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 496.353 98.7307i 1.49255 0.296887i
\(49\) 131.260 316.891i 0.382683 0.923880i
\(50\) 353.553i 1.00000i
\(51\) 486.202 266.095i 1.33494 0.730603i
\(52\) 0 0
\(53\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(54\) −37.2096 187.065i −0.0937702 0.471415i
\(55\) 0 0
\(56\) 0 0
\(57\) 23.2120 116.695i 0.0539387 0.271168i
\(58\) 0 0
\(59\) −251.472 607.106i −0.554895 1.33964i −0.913763 0.406247i \(-0.866837\pi\)
0.358868 0.933388i \(-0.383163\pi\)
\(60\) 0 0
\(61\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −362.039 + 362.039i −0.707107 + 0.707107i
\(65\) 0 0
\(66\) 623.637 1505.59i 1.16310 2.80797i
\(67\) 984.492i 1.79515i 0.440865 + 0.897573i \(0.354672\pi\)
−0.440865 + 0.897573i \(0.645328\pi\)
\(68\) −259.429 + 497.120i −0.462653 + 0.886539i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(72\) 568.445 + 568.445i 0.930443 + 0.930443i
\(73\) −1024.67 684.661i −1.64285 1.09772i −0.907515 0.420021i \(-0.862023\pi\)
−0.735339 0.677699i \(-0.762977\pi\)
\(74\) 0 0
\(75\) 821.851 549.143i 1.26532 0.845461i
\(76\) 46.0649 + 111.211i 0.0695264 + 0.167852i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(80\) 0 0
\(81\) −301.245 + 301.245i −0.413231 + 0.413231i
\(82\) 1389.63 276.415i 1.87145 0.372255i
\(83\) 43.2971 104.528i 0.0572587 0.138235i −0.892661 0.450728i \(-0.851164\pi\)
0.949920 + 0.312494i \(0.101164\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 234.296 0.293776
\(87\) 0 0
\(88\) 321.648 + 1617.04i 0.389635 + 1.95883i
\(89\) −1029.99 1029.99i −1.22673 1.22673i −0.965195 0.261531i \(-0.915773\pi\)
−0.261531 0.965195i \(-0.584227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1403.90 279.253i −1.49255 0.296887i
\(97\) −866.591 + 1296.94i −0.907103 + 1.35757i 0.0266459 + 0.999645i \(0.491517\pi\)
−0.933748 + 0.357930i \(0.883483\pi\)
\(98\) −686.000 + 686.000i −0.707107 + 0.707107i
\(99\) 2538.94 505.028i 2.57751 0.512699i
\(100\) −382.683 + 923.880i −0.382683 + 0.923880i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −1558.53 + 169.078i −1.51291 + 0.164130i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1831.56 1223.81i −1.65480 1.10570i −0.882311 0.470667i \(-0.844013\pi\)
−0.772486 0.635032i \(-0.780987\pi\)
\(108\) −105.245 + 529.101i −0.0937702 + 0.471415i
\(109\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1080.59 214.942i −0.899584 0.178938i −0.276427 0.961035i \(-0.589150\pi\)
−0.623157 + 0.782097i \(0.714150\pi\)
\(114\) −186.965 + 279.814i −0.153605 + 0.229885i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1858.64i 1.45001i
\(119\) 0 0
\(120\) 0 0
\(121\) 3675.29 + 1522.36i 2.76130 + 1.14377i
\(122\) 0 0
\(123\) 2800.93 + 2800.93i 2.05326 + 2.05326i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(128\) 1337.92 554.185i 0.923880 0.382683i
\(129\) 363.911 + 544.631i 0.248376 + 0.371721i
\(130\) 0 0
\(131\) 380.926 570.097i 0.254059 0.380226i −0.682414 0.730966i \(-0.739070\pi\)
0.936473 + 0.350740i \(0.114070\pi\)
\(132\) −3259.29 + 3259.29i −2.14912 + 2.14912i
\(133\) 0 0
\(134\) 1065.61 2572.60i 0.686973 1.65850i
\(135\) 0 0
\(136\) 1216.00 1018.23i 0.766700 0.642006i
\(137\) 3206.99 1.99994 0.999970 0.00779827i \(-0.00248229\pi\)
0.999970 + 0.00779827i \(0.00248229\pi\)
\(138\) 0 0
\(139\) −556.393 2797.17i −0.339515 1.70686i −0.653078 0.757291i \(-0.726523\pi\)
0.313563 0.949567i \(-0.398477\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −870.138 2100.70i −0.503552 1.21568i
\(145\) 0 0
\(146\) 1936.51 + 2898.20i 1.09772 + 1.64285i
\(147\) −2660.14 529.135i −1.49255 0.296887i
\(148\) 0 0
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) −2741.99 + 545.415i −1.49255 + 0.296887i
\(151\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(152\) 340.468i 0.181681i
\(153\) −1598.75 1909.27i −0.844781 1.00886i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 1113.26 461.126i 0.539912 0.223639i
\(163\) −801.206 1199.09i −0.385002 0.576196i 0.587461 0.809252i \(-0.300127\pi\)
−0.972463 + 0.233056i \(0.925127\pi\)
\(164\) −3930.47 781.819i −1.87145 0.372255i
\(165\) 0 0
\(166\) −226.281 + 226.281i −0.105800 + 0.105800i
\(167\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(168\) 0 0
\(169\) 2197.00i 1.00000i
\(170\) 0 0
\(171\) −534.576 −0.239065
\(172\) −612.244 253.600i −0.271414 0.112423i
\(173\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 909.759 4573.67i 0.389635 1.95883i
\(177\) −4320.48 + 2886.85i −1.83473 + 1.22593i
\(178\) 1576.64 + 3806.34i 0.663899 + 1.60280i
\(179\) −629.206 + 260.626i −0.262732 + 0.108827i −0.510161 0.860079i \(-0.670414\pi\)
0.247429 + 0.968906i \(0.420414\pi\)
\(180\) 0 0
\(181\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −550.829 5077.42i −0.215404 1.98555i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 3366.30 + 2249.29i 1.26532 + 0.845461i
\(193\) −574.544 + 2888.43i −0.214283 + 1.07727i 0.712499 + 0.701673i \(0.247563\pi\)
−0.926782 + 0.375600i \(0.877437\pi\)
\(194\) 3668.31 2451.09i 1.35757 0.907103i
\(195\) 0 0
\(196\) 2535.13 1050.08i 0.923880 0.382683i
\(197\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(198\) −7181.22 1428.43i −2.57751 0.512699i
\(199\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(200\) 2000.00 2000.00i 0.707107 0.707107i
\(201\) 7635.24 1518.74i 2.67935 0.532955i
\(202\) 0 0
\(203\) 0 0
\(204\) 4255.64 + 1245.11i 1.46056 + 0.427331i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −911.586 609.103i −0.301702 0.201591i
\(210\) 0 0
\(211\) −681.025 + 455.046i −0.222198 + 0.148468i −0.661687 0.749780i \(-0.730159\pi\)
0.439490 + 0.898248i \(0.355159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 3461.45 + 5180.42i 1.10570 + 1.65480i
\(215\) 0 0
\(216\) 847.713 1268.69i 0.267035 0.399646i
\(217\) 0 0
\(218\) 0 0
\(219\) −3729.18 + 9003.03i −1.15066 + 2.77794i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(224\) 0 0
\(225\) −3140.25 3140.25i −0.930443 0.930443i
\(226\) 2591.06 + 1731.29i 0.762631 + 0.509574i
\(227\) −1272.05 + 6395.02i −0.371933 + 1.86983i 0.110464 + 0.993880i \(0.464766\pi\)
−0.482397 + 0.875953i \(0.660234\pi\)
\(228\) 791.432 528.818i 0.229885 0.153605i
\(229\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3604.16 5394.01i 1.01338 1.51662i 0.165654 0.986184i \(-0.447026\pi\)
0.847722 0.530441i \(-0.177974\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2011.77 4856.85i 0.554895 1.33964i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −514.788 2588.01i −0.137595 0.691737i −0.986575 0.163311i \(-0.947783\pi\)
0.848980 0.528426i \(-0.177217\pi\)
\(242\) −7956.21 7956.21i −2.11341 2.11341i
\(243\) 4314.89 + 2883.12i 1.13910 + 0.761120i
\(244\) 0 0
\(245\) 0 0
\(246\) −4287.48 10350.9i −1.11122 2.68272i
\(247\) 0 0
\(248\) 0 0
\(249\) −877.464 174.539i −0.223321 0.0444214i
\(250\) 0 0
\(251\) −5534.06 + 5534.06i −1.39166 + 1.39166i −0.570054 + 0.821607i \(0.693078\pi\)
−0.821607 + 0.570054i \(0.806922\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −4096.00 −1.00000
\(257\) 7577.61 + 3138.75i 1.83922 + 0.761828i 0.956366 + 0.292170i \(0.0943774\pi\)
0.882849 + 0.469658i \(0.155623\pi\)
\(258\) −361.440 1817.08i −0.0872182 0.438475i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1612.48 + 1077.42i −0.380226 + 0.254059i
\(263\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(264\) 12044.7 4989.10i 2.80797 1.16310i
\(265\) 0 0
\(266\) 0 0
\(267\) −6399.16 + 9577.03i −1.46675 + 2.19515i
\(268\) −5569.13 + 5569.13i −1.26936 + 1.26936i
\(269\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −4279.69 + 1344.58i −0.954023 + 0.299733i
\(273\) 0 0
\(274\) −8380.27 3471.22i −1.84770 0.765344i
\(275\) −1776.87 8932.95i −0.389635 1.95883i
\(276\) 0 0
\(277\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(278\) −1573.72 + 7911.61i −0.339515 + 1.70686i
\(279\) 0 0
\(280\) 0 0
\(281\) −6897.49 + 2857.03i −1.46431 + 0.606535i −0.965552 0.260209i \(-0.916209\pi\)
−0.498753 + 0.866744i \(0.666209\pi\)
\(282\) 0 0
\(283\) 8703.40 + 1731.21i 1.82814 + 0.363640i 0.984793 0.173731i \(-0.0555824\pi\)
0.843346 + 0.537371i \(0.180582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 6431.22i 1.31585i
\(289\) −4030.23 + 2809.77i −0.820320 + 0.571905i
\(290\) 0 0
\(291\) 11395.3 + 4720.10i 2.29555 + 0.950850i
\(292\) −1923.37 9669.43i −0.385468 1.93788i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 6378.55 + 4262.01i 1.26532 + 0.845461i
\(295\) 0 0
\(296\) 0 0
\(297\) −1880.29 4539.42i −0.367359 0.886883i
\(298\) 0 0
\(299\) 0 0
\(300\) 7755.51 + 1542.67i 1.49255 + 0.296887i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −368.519 + 889.685i −0.0695264 + 0.167852i
\(305\) 0 0
\(306\) 2111.16 + 6719.64i 0.394402 + 1.25535i
\(307\) −7204.00 −1.33926 −0.669632 0.742693i \(-0.733548\pi\)
−0.669632 + 0.742693i \(0.733548\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(312\) 0 0
\(313\) 8873.70 5929.22i 1.60246 1.07073i 0.652769 0.757557i \(-0.273607\pi\)
0.949695 0.313175i \(-0.101393\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −6665.77 + 16092.6i −1.15902 + 2.79813i
\(322\) 0 0
\(323\) −92.9914 + 1050.56i −0.0160191 + 0.180974i
\(324\) −3408.20 −0.584397
\(325\) 0 0
\(326\) 795.767 + 4000.59i 0.135195 + 0.679670i
\(327\) 0 0
\(328\) 9424.58 + 6297.30i 1.58654 + 1.06009i
\(329\) 0 0
\(330\) 0 0
\(331\) −4312.01 10410.1i −0.716041 1.72868i −0.684322 0.729180i \(-0.739902\pi\)
−0.0317191 0.999497i \(-0.510098\pi\)
\(332\) 836.227 346.377i 0.138235 0.0572587i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6697.07 1332.13i 1.08253 0.215329i 0.378574 0.925571i \(-0.376415\pi\)
0.703957 + 0.710242i \(0.251415\pi\)
\(338\) −2378.02 + 5741.04i −0.382683 + 0.923880i
\(339\) 8712.09i 1.39580i
\(340\) 0 0
\(341\) 0 0
\(342\) 1396.91 + 578.621i 0.220867 + 0.0914860i
\(343\) 0 0
\(344\) 1325.38 + 1325.38i 0.207731 + 0.207731i
\(345\) 0 0
\(346\) 0 0
\(347\) −6772.37 + 4525.15i −1.04772 + 0.700066i −0.955295 0.295654i \(-0.904463\pi\)
−0.0924274 + 0.995719i \(0.529463\pi\)
\(348\) 0 0
\(349\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7327.82 + 10966.9i −1.10959 + 1.66061i
\(353\) 6465.40 6465.40i 0.974840 0.974840i −0.0248514 0.999691i \(-0.507911\pi\)
0.999691 + 0.0248514i \(0.00791128\pi\)
\(354\) 14414.7 2867.26i 2.16421 0.430489i
\(355\) 0 0
\(356\) 11653.0i 1.73485i
\(357\) 0 0
\(358\) 1926.29 0.284379
\(359\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(360\) 0 0
\(361\) −4689.95 4689.95i −0.683767 0.683767i
\(362\) 0 0
\(363\) 6136.90 30852.3i 0.887338 4.46095i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(368\) 0 0
\(369\) 9887.53 14797.7i 1.39492 2.08764i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) −4056.38 + 13864.2i −0.560830 + 1.91684i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10979.4 7336.21i −1.48806 0.994290i −0.992032 0.125984i \(-0.959791\pi\)
−0.496028 0.868306i \(-0.665209\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) −6361.95 9521.34i −0.845461 1.26532i
\(385\) 0 0
\(386\) 4627.77 6925.94i 0.610226 0.913268i
\(387\) 2081.00 2081.00i 0.273342 0.273342i
\(388\) −12238.8 + 2434.45i −1.60137 + 0.318532i
\(389\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −7761.20 −1.00000
\(393\) −5009.04 2074.81i −0.642932 0.266311i
\(394\) 0 0
\(395\) 0 0
\(396\) 17219.3 + 11505.6i 2.18511 + 1.46004i
\(397\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −7391.04 + 3061.47i −0.923880 + 0.382683i
\(401\) 8837.76 + 13226.6i 1.10059 + 1.64715i 0.664601 + 0.747198i \(0.268601\pi\)
0.435989 + 0.899952i \(0.356399\pi\)
\(402\) −21595.7 4295.66i −2.67935 0.532955i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −9772.81 7859.91i −1.18585 0.953734i
\(409\) 3903.53 0.471925 0.235962 0.971762i \(-0.424176\pi\)
0.235962 + 0.971762i \(0.424176\pi\)
\(410\) 0 0
\(411\) −4947.32 24871.9i −0.593755 2.98501i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −20835.2 + 8630.22i −2.44677 + 1.01349i
\(418\) 1722.80 + 2578.36i 0.201591 + 0.301702i
\(419\) 11792.0 + 2345.58i 1.37489 + 0.273482i 0.826591 0.562803i \(-0.190277\pi\)
0.548296 + 0.836285i \(0.315277\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 2272.14 451.957i 0.262100 0.0521349i
\(423\) 0 0
\(424\) 0 0
\(425\) −6717.51 + 5625.00i −0.766700 + 0.642006i
\(426\) 0 0
\(427\) 0 0
\(428\) −3437.95 17283.7i −0.388270 1.95197i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(432\) −3588.40 + 2397.69i −0.399646 + 0.267035i
\(433\) −3151.72 7608.93i −0.349797 0.844484i −0.996644 0.0818641i \(-0.973913\pi\)
0.646847 0.762620i \(-0.276087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 19489.6 19489.6i 2.12614 2.12614i
\(439\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(440\) 0 0
\(441\) 12186.0i 1.31585i
\(442\) 0 0
\(443\) 15449.4 1.65694 0.828471 0.560032i \(-0.189211\pi\)
0.828471 + 0.560032i \(0.189211\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8671.90 5794.38i 0.911475 0.609028i −0.00894477 0.999960i \(-0.502847\pi\)
0.920420 + 0.390932i \(0.127847\pi\)
\(450\) 4806.88 + 11604.8i 0.503552 + 1.21568i
\(451\) 33721.5 13967.9i 3.52080 1.45836i
\(452\) −4896.82 7328.62i −0.509574 0.762631i
\(453\) 0 0
\(454\) 10245.9 15334.1i 1.05918 1.58517i
\(455\) 0 0
\(456\) −2640.50 + 525.228i −0.271168 + 0.0539387i
\(457\) −2878.32 + 6948.89i −0.294622 + 0.711280i 0.705375 + 0.708834i \(0.250779\pi\)
−0.999997 + 0.00244611i \(0.999221\pi\)
\(458\) 0 0
\(459\) −2962.24 + 3683.18i −0.301232 + 0.374545i
\(460\) 0 0
\(461\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −15256.6 + 10194.1i −1.51662 + 1.01338i
\(467\) 7712.53 + 18619.7i 0.764226 + 1.84500i 0.432546 + 0.901612i \(0.357615\pi\)
0.331679 + 0.943392i \(0.392385\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −10514.0 + 10514.0i −1.02531 + 1.02531i
\(473\) 5919.76 1177.51i 0.575456 0.114465i
\(474\) 0 0
\(475\) 1880.84i 0.181681i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1456.04 + 7320.00i −0.137595 + 0.691737i
\(483\) 0 0
\(484\) 12178.8 + 29402.3i 1.14377 + 2.76130i
\(485\) 0 0
\(486\) −8154.69 12204.4i −0.761120 1.13910i
\(487\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(488\) 0 0
\(489\) −8063.57 + 8063.57i −0.745699 + 0.745699i
\(490\) 0 0
\(491\) −8178.81 + 19745.4i −0.751740 + 1.81486i −0.202311 + 0.979321i \(0.564845\pi\)
−0.549429 + 0.835540i \(0.685155\pi\)
\(492\) 31688.9i 2.90375i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 2104.01 + 1405.85i 0.189323 + 0.126501i
\(499\) 2995.83 15061.0i 0.268761 1.35115i −0.576631 0.817005i \(-0.695633\pi\)
0.845392 0.534147i \(-0.179367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20451.2 8471.18i 1.81829 0.753162i
\(503\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −17038.9 + 3389.24i −1.49255 + 0.296887i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 10703.4 + 4433.48i 0.923880 + 0.382683i
\(513\) 197.948 + 995.152i 0.0170363 + 0.0856472i
\(514\) −16403.9 16403.9i −1.40767 1.40767i
\(515\) 0 0
\(516\) −1022.31 + 5139.49i −0.0872182 + 0.438475i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −19360.2 3850.97i −1.62799 0.323828i −0.705165 0.709043i \(-0.749127\pi\)
−0.922827 + 0.385215i \(0.874127\pi\)
\(522\) 0 0
\(523\) −9447.21 + 9447.21i −0.789862 + 0.789862i −0.981471 0.191609i \(-0.938629\pi\)
0.191609 + 0.981471i \(0.438629\pi\)
\(524\) 5379.80 1070.11i 0.448507 0.0892136i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −36874.6 −3.03932
\(529\) −11240.8 4656.11i −0.923880 0.382683i
\(530\) 0 0
\(531\) 16508.3 + 16508.3i 1.34915 + 1.34915i
\(532\) 0 0
\(533\) 0 0
\(534\) 27087.9 18099.6i 2.19515 1.46675i
\(535\) 0 0
\(536\) 20580.8 8524.85i 1.65850 0.686973i
\(537\) 2991.94 + 4477.76i 0.240432 + 0.359831i
\(538\) 0 0
\(539\) −13884.9 + 20780.3i −1.10959 + 1.66061i
\(540\) 0 0
\(541\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 12638.7 + 1118.73i 0.996105 + 0.0881716i
\(545\) 0 0
\(546\) 0 0
\(547\) 2101.33 + 10564.1i 0.164253 + 0.825757i 0.971773 + 0.235917i \(0.0758092\pi\)
−0.807520 + 0.589840i \(0.799191\pi\)
\(548\) 18141.5 + 18141.5i 1.41417 + 1.41417i
\(549\) 0 0
\(550\) −5025.76 + 25266.2i −0.389635 + 1.95883i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 12675.8 18970.6i 0.966857 1.44700i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −38528.3 + 12104.7i −2.89958 + 0.910985i
\(562\) 21116.4 1.58495
\(563\) −18945.3 7847.39i −1.41820 0.587439i −0.463795 0.885943i \(-0.653512\pi\)
−0.954408 + 0.298504i \(0.903512\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20869.2 13944.4i −1.54982 1.03556i
\(567\) 0 0
\(568\) 0 0
\(569\) 9209.41 + 22233.5i 0.678521 + 1.63809i 0.766712 + 0.641991i \(0.221891\pi\)
−0.0881913 + 0.996104i \(0.528109\pi\)
\(570\) 0 0
\(571\) −4941.81 7395.94i −0.362186 0.542050i 0.604965 0.796252i \(-0.293187\pi\)
−0.967151 + 0.254202i \(0.918187\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 6961.11 16805.6i 0.503552 1.21568i
\(577\) 19651.9i 1.41789i −0.705266 0.708943i \(-0.749173\pi\)
0.705266 0.708943i \(-0.250827\pi\)
\(578\) 13572.8 2979.98i 0.976735 0.214448i
\(579\) 23287.6 1.67150
\(580\) 0 0
\(581\) 0 0
\(582\) −24668.4 24668.4i −1.75694 1.75694i
\(583\) 0 0
\(584\) −5440.11 + 27349.3i −0.385468 + 1.93788i
\(585\) 0 0
\(586\) 0 0
\(587\) 9562.15 3960.77i 0.672355 0.278498i −0.0202721 0.999795i \(-0.506453\pi\)
0.692627 + 0.721296i \(0.256453\pi\)
\(588\) −12054.8 18041.3i −0.845461 1.26532i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8139.44 19650.3i 0.563654 1.36078i −0.343171 0.939273i \(-0.611501\pi\)
0.906825 0.421507i \(-0.138499\pi\)
\(594\) 13897.3i 0.959955i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) −18596.4 12425.7i −1.26532 0.845461i
\(601\) −2609.98 + 13121.3i −0.177144 + 0.890561i 0.785307 + 0.619106i \(0.212505\pi\)
−0.962451 + 0.271455i \(0.912495\pi\)
\(602\) 0 0
\(603\) −13385.1 32314.4i −0.903950 2.18233i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(608\) 1925.98 1925.98i 0.128468 0.128468i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1756.55 19844.4i 0.116020 1.31072i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 18825.0 + 7797.56i 1.23732 + 0.512514i
\(615\) 0 0
\(616\) 0 0
\(617\) −21448.2 14331.3i −1.39947 0.935097i −0.999830 0.0184247i \(-0.994135\pi\)
−0.399641 0.916672i \(-0.630865\pi\)
\(618\) 0 0
\(619\) 22345.5 14930.8i 1.45095 0.969495i 0.454039 0.890982i \(-0.349983\pi\)
0.996913 0.0785136i \(-0.0250174\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11048.5 + 11048.5i −0.707107 + 0.707107i
\(626\) −29605.8 + 5888.97i −1.89024 + 0.375991i
\(627\) −3317.63 + 8009.46i −0.211313 + 0.510155i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(632\) 0 0
\(633\) 4579.71 + 4579.71i 0.287563 + 0.287563i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18031.4 26986.0i 1.11107 1.66284i 0.546495 0.837462i \(-0.315962\pi\)
0.564580 0.825378i \(-0.309038\pi\)
\(642\) 34837.0 34837.0i 2.14160 2.14160i
\(643\) 9694.64 1928.38i 0.594587 0.118271i 0.111381 0.993778i \(-0.464472\pi\)
0.483205 + 0.875507i \(0.339472\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1380.11 2644.58i 0.0840554 0.161068i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 8906.06 + 3689.01i 0.539912 + 0.223639i
\(649\) 9341.05 + 46960.6i 0.564974 + 2.84032i
\(650\) 0 0
\(651\) 0 0
\(652\) 2250.77 11315.4i 0.135195 0.679670i
\(653\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −17811.5 26656.7i −1.06009 1.58654i
\(657\) 42941.7 + 8541.64i 2.54995 + 0.507216i
\(658\) 0 0
\(659\) −21039.3 + 21039.3i −1.24366 + 1.24366i −0.285191 + 0.958471i \(0.592057\pi\)
−0.958471 + 0.285191i \(0.907943\pi\)
\(660\) 0 0
\(661\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(662\) 31870.2i 1.87111i
\(663\) 0 0
\(664\) −2560.08 −0.149624
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7294.65 + 10917.2i 0.417813 + 0.625301i 0.979355 0.202147i \(-0.0647919\pi\)
−0.561542 + 0.827448i \(0.689792\pi\)
\(674\) −18942.2 3767.83i −1.08253 0.215329i
\(675\) −4683.00 + 7008.60i −0.267035 + 0.399646i
\(676\) 12428.1 12428.1i 0.707107 0.707107i
\(677\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(678\) 9429.89 22765.8i 0.534149 1.28955i
\(679\) 0 0
\(680\) 0 0
\(681\) 51559.0 2.90124
\(682\) 0 0
\(683\) 3570.43 + 17949.8i 0.200028 + 1.00561i 0.942112 + 0.335300i \(0.108838\pi\)
−0.742084 + 0.670307i \(0.766162\pi\)
\(684\) −3024.02 3024.02i −0.169044 0.169044i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2028.80 4897.95i −0.112423 0.271414i
\(689\) 0 0
\(690\) 0 0
\(691\) 8843.56 + 1759.09i 0.486867 + 0.0968438i 0.432419 0.901673i \(-0.357660\pi\)
0.0544477 + 0.998517i \(0.482660\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 22595.0 4494.43i 1.23587 0.245830i
\(695\) 0 0
\(696\) 0 0
\(697\) −27360.8 22005.3i −1.48689 1.19585i
\(698\) 0 0
\(699\) −47393.4 19631.0i −2.56450 1.06225i
\(700\) 0 0
\(701\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 31018.9 20726.2i 1.66061 1.10959i
\(705\) 0 0
\(706\) −23893.0 + 9896.80i −1.27369 + 0.527579i
\(707\) 0 0
\(708\) −40770.9 8109.83i −2.16421 0.430489i
\(709\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12613.1 + 30450.8i −0.663899 + 1.60280i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −5033.65 2085.00i −0.262732 0.108827i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7179.07 + 17331.8i 0.370052 + 0.893384i
\(723\) −19277.2 + 7984.89i −0.991601 + 0.410735i
\(724\) 0 0
\(725\) 0 0
\(726\) −49430.7 + 73978.3i −2.52692 + 3.78181i
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 11301.7 27284.8i 0.574187 1.38621i
\(730\) 0 0
\(731\) −3727.62 4451.62i −0.188606 0.225238i
\(732\) 0 0
\(733\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13994.5 70355.3i 0.699451 3.51638i
\(738\) −41854.3 + 27966.2i −2.08764 + 1.39492i
\(739\) −6770.36 16345.1i −0.337012 0.813619i −0.997999 0.0632220i \(-0.979862\pi\)
0.660988 0.750397i \(-0.270138\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4019.64i 0.196882i
\(748\) 25606.3 31838.2i 1.25168 1.55631i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(752\) 0 0
\(753\) 51456.7 + 34382.3i 2.49029 + 1.66396i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(758\) 20749.9 + 31054.5i 0.994290 + 1.48806i
\(759\) 0 0
\(760\) 0 0
\(761\) −29281.5 + 29281.5i −1.39482 + 1.39482i −0.580691 + 0.814124i \(0.697218\pi\)
−0.814124 + 0.580691i \(0.802782\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 6318.77 + 31766.6i 0.296887 + 1.49255i
\(769\) −27307.9 27307.9i −1.28056 1.28056i −0.940351 0.340207i \(-0.889503\pi\)
−0.340207 0.940351i \(-0.610497\pi\)
\(770\) 0 0
\(771\) 12652.9 63610.3i 0.591027 2.97130i
\(772\) −19589.5 + 13089.3i −0.913268 + 0.610226i
\(773\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(774\) −7690.38 + 3185.46i −0.357138 + 0.147932i
\(775\) 0 0
\(776\) 34616.6 + 6885.66i 1.60137 + 0.318532i
\(777\) 0 0
\(778\) 0 0
\(779\) −7392.56 + 1470.47i −0.340008 + 0.0676318i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 20281.0 + 8400.67i 0.923880 + 0.382683i
\(785\) 0 0
\(786\) 10843.5 + 10843.5i 0.492079 + 0.492079i
\(787\) −33357.0 22288.5i −1.51086 1.00953i −0.987536 0.157396i \(-0.949690\pi\)
−0.523328 0.852131i \(-0.675310\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −32542.7 48703.6i −1.46004 2.18511i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 22627.4 1.00000
\(801\) 47811.4 + 19804.1i 2.10903 + 0.873589i
\(802\) −8777.77 44128.8i −0.386476 1.94295i
\(803\) 63494.0 + 63494.0i 2.79035 + 2.79035i
\(804\) 51782.8 + 34600.1i 2.27144 + 1.51773i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15771.5 23603.6i −0.685408 1.02578i −0.997137 0.0756140i \(-0.975908\pi\)
0.311730 0.950171i \(-0.399092\pi\)
\(810\) 0 0
\(811\) −805.820 + 1205.99i −0.0348905 + 0.0522173i −0.848505 0.529188i \(-0.822497\pi\)
0.813614 + 0.581405i \(0.197497\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 17030.1 + 31116.9i 0.730603 + 1.33494i
\(817\) −1246.41 −0.0533736
\(818\) −10200.4 4225.15i −0.436001 0.180598i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(822\) −13993.1 + 70348.2i −0.593755 + 2.98501i
\(823\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(824\) 0 0
\(825\) −66538.5 + 27561.1i −2.80797 + 1.16310i
\(826\) 0 0
\(827\) −46528.5 9255.09i −1.95641 0.389155i −0.992345 0.123496i \(-0.960589\pi\)
−0.964067 0.265658i \(-0.914411\pi\)
\(828\) 0 0
\(829\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 23948.2 + 2119.81i 0.996105 + 0.0881716i
\(834\) 63786.3 2.64837
\(835\) 0 0
\(836\) −1711.11 8602.32i −0.0707893 0.355882i
\(837\) 0 0
\(838\) −28275.2 18892.9i −1.16557 0.778811i
\(839\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(840\) 0 0
\(841\) −9333.27 22532.5i −0.382683 0.923880i
\(842\) 0 0
\(843\) 32798.3 + 49086.1i 1.34002 + 2.00548i
\(844\) −6426.59 1278.33i −0.262100 0.0521349i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 70170.0i 2.83655i
\(850\) 23642.2 7427.85i 0.954023 0.299733i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9723.99 + 48885.8i −0.388270 + 1.95197i
\(857\) 34971.7 23367.3i 1.39394 0.931404i 0.394020 0.919102i \(-0.371084\pi\)
0.999924 0.0123023i \(-0.00391605\pi\)
\(858\) 0 0
\(859\) −31446.3 + 13025.5i −1.24905 + 0.517373i −0.906532 0.422138i \(-0.861280\pi\)
−0.342518 + 0.939511i \(0.611280\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 11972.2 2381.42i 0.471415 0.0937702i
\(865\) 0 0
\(866\) 23294.5i 0.914063i
\(867\) 28008.5 + 26922.0i 1.09714 + 1.05458i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10811.3 54352.3i 0.419139 2.10715i
\(874\) 0 0
\(875\) 0 0
\(876\) −72024.3 + 29833.4i −2.77794 + 1.15066i
\(877\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −38296.4 + 7617.63i −1.46452 + 0.291311i −0.862046 0.506830i \(-0.830817\pi\)
−0.602471 + 0.798141i \(0.705817\pi\)
\(882\) 13190.1 31843.7i 0.503552 1.21568i
\(883\) 24867.2i 0.947733i 0.880597 + 0.473866i \(0.157142\pi\)
−0.880597 + 0.473866i \(0.842858\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −40371.3 16722.4i −1.53081 0.634084i
\(887\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 25810.3 17245.9i 0.970456 0.648438i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −28932.6 + 5755.04i −1.07516 + 0.213862i
\(899\) 0 0
\(900\) 35527.8i 1.31585i
\(901\) 0 0
\(902\) −103237. −3.81089
\(903\) 0 0
\(904\) 4863.58 + 24450.9i 0.178938 + 0.899584i
\(905\) 0 0
\(906\) 0 0
\(907\) −8174.09 + 41093.9i −0.299246 + 1.50441i 0.479762 + 0.877399i \(0.340723\pi\)
−0.779008 + 0.627013i \(0.784277\pi\)
\(908\) −43371.5 + 28979.9i −1.58517 + 1.05918i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(912\) 7468.46 + 1485.57i 0.271168 + 0.0539387i
\(913\) −4580.03 + 6854.50i −0.166021 + 0.248468i
\(914\) 15042.8 15042.8i 0.544390 0.544390i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 11727.4 6418.30i 0.421634 0.230757i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 11113.4 + 55870.8i 0.397610 + 1.99892i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42035.7 + 8361.41i 1.48455 + 0.295295i 0.869791 0.493421i \(-0.164254\pi\)
0.614758 + 0.788716i \(0.289254\pi\)
\(930\) 0 0
\(931\) 3649.39 3649.39i 0.128468 0.128468i
\(932\) 50901.4 10124.9i 1.78898 0.355850i
\(933\) 0 0
\(934\) 57003.6i 1.99702i
\(935\) 0 0
\(936\) 0 0
\(937\) 10300.9 + 4266.79i 0.359143 + 0.148762i 0.554957 0.831879i \(-0.312735\pi\)
−0.195814 + 0.980641i \(0.562735\pi\)
\(938\) 0 0
\(939\) −59673.3 59673.3i −2.07387 2.07387i
\(940\) 0 0
\(941\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 38854.8 16094.2i 1.33964 0.554895i
\(945\) 0 0
\(946\) −16743.6 3330.51i −0.575456 0.114465i
\(947\) −19139.9 + 28644.9i −0.656773 + 0.982931i 0.342287 + 0.939595i \(0.388798\pi\)
−0.999061 + 0.0433353i \(0.986202\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 2035.80 4914.86i 0.0695264 0.167852i
\(951\) 0 0
\(952\) 0 0
\(953\) −28444.1 −0.966836 −0.483418 0.875390i \(-0.660605\pi\)
−0.483418 + 0.875390i \(0.660605\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 27523.3 11400.5i 0.923880 0.382683i
\(962\) 0 0
\(963\) 76756.7 + 15267.9i 2.56848 + 0.510903i
\(964\) 11727.9 17552.1i 0.391837 0.586426i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(968\) 90014.3i 2.98881i
\(969\) 8291.06 899.463i 0.274868 0.0298193i
\(970\) 0 0
\(971\) −21256.7 8804.82i −0.702534 0.290999i 0.00267705 0.999996i \(-0.499148\pi\)
−0.705211 + 0.708997i \(0.749148\pi\)
\(972\) 8099.33 + 40718.1i 0.267270 + 1.34366i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2433.83 + 5875.78i 0.0796981 + 0.192408i 0.958706 0.284399i \(-0.0917940\pi\)
−0.879008 + 0.476807i \(0.841794\pi\)
\(978\) 29799.1 12343.2i 0.974303 0.403570i
\(979\) 58965.4 + 88248.0i 1.92497 + 2.88092i
\(980\) 0 0
\(981\) 0 0
\(982\) 42744.5 42744.5i 1.38904 1.38904i
\(983\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(984\) 34299.8 82807.1i 1.11122 2.68272i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(992\) 0 0
\(993\) −74083.8 + 49501.2i −2.36755 + 1.58195i
\(994\) 0 0
\(995\) 0 0
\(996\) −3976.35 5951.03i −0.126501 0.189323i
\(997\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(998\) −24130.4 + 36113.7i −0.765366 + 1.14545i
\(999\) 0 0
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.s.a.11.1 8
8.3 odd 2 CM 136.4.s.a.11.1 8
17.14 odd 16 inner 136.4.s.a.99.1 yes 8
136.99 even 16 inner 136.4.s.a.99.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.s.a.11.1 8 1.1 even 1 trivial
136.4.s.a.11.1 8 8.3 odd 2 CM
136.4.s.a.99.1 yes 8 17.14 odd 16 inner
136.4.s.a.99.1 yes 8 136.99 even 16 inner