Properties

Label 475.2.b.c.324.6
Level $475$
Weight $2$
Character 475.324
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
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] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.6
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 475.324
Dual form 475.2.b.c.324.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.80194i q^{2} -0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} -3.04892i q^{7} -10.7899i q^{8} +2.69202 q^{9} +O(q^{10})\) \(q+2.80194i q^{2} -0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} -3.04892i q^{7} -10.7899i q^{8} +2.69202 q^{9} -2.93900 q^{11} +3.24698i q^{12} -3.24698i q^{13} +8.54288 q^{14} +18.5308 q^{16} -2.15883i q^{17} +7.54288i q^{18} +1.00000 q^{19} -1.69202 q^{21} -8.23490i q^{22} -1.19806i q^{23} -5.98792 q^{24} +9.09783 q^{26} -3.15883i q^{27} +17.8388i q^{28} +1.77479 q^{29} -9.34481 q^{31} +30.3424i q^{32} +1.63102i q^{33} +6.04892 q^{34} -15.7506 q^{36} -1.15883i q^{37} +2.80194i q^{38} -1.80194 q^{39} +8.57002 q^{41} -4.74094i q^{42} -5.27413i q^{43} +17.1957 q^{44} +3.35690 q^{46} +2.35690i q^{47} -10.2838i q^{48} -2.29590 q^{49} -1.19806 q^{51} +18.9976i q^{52} -8.82371i q^{53} +8.85086 q^{54} -32.8974 q^{56} -0.554958i q^{57} +4.97285i q^{58} +5.70171 q^{59} -9.96077 q^{61} -26.1836i q^{62} -8.20775i q^{63} -47.9560 q^{64} -4.57002 q^{66} +4.98254i q^{67} +12.6310i q^{68} -0.664874 q^{69} +2.70171 q^{71} -29.0465i q^{72} -13.7778i q^{73} +3.24698 q^{74} -5.85086 q^{76} +8.96077i q^{77} -5.04892i q^{78} -5.66487 q^{79} +6.32304 q^{81} +24.0127i q^{82} +3.00969i q^{83} +9.89977 q^{84} +14.7778 q^{86} -0.984935i q^{87} +31.7114i q^{88} +10.2838 q^{89} -9.89977 q^{91} +7.00969i q^{92} +5.18598i q^{93} -6.60388 q^{94} +16.8388 q^{96} +3.24698i q^{97} -6.43296i q^{98} -7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.80194i 1.98127i 0.136540 + 0.990635i \(0.456402\pi\)
−0.136540 + 0.990635i \(0.543598\pi\)
\(3\) − 0.554958i − 0.320405i −0.987084 0.160203i \(-0.948785\pi\)
0.987084 0.160203i \(-0.0512148\pi\)
\(4\) −5.85086 −2.92543
\(5\) 0 0
\(6\) 1.55496 0.634809
\(7\) − 3.04892i − 1.15238i −0.817315 0.576191i \(-0.804538\pi\)
0.817315 0.576191i \(-0.195462\pi\)
\(8\) − 10.7899i − 3.81479i
\(9\) 2.69202 0.897340
\(10\) 0 0
\(11\) −2.93900 −0.886142 −0.443071 0.896486i \(-0.646111\pi\)
−0.443071 + 0.896486i \(0.646111\pi\)
\(12\) 3.24698i 0.937322i
\(13\) − 3.24698i − 0.900550i −0.892890 0.450275i \(-0.851326\pi\)
0.892890 0.450275i \(-0.148674\pi\)
\(14\) 8.54288 2.28318
\(15\) 0 0
\(16\) 18.5308 4.63270
\(17\) − 2.15883i − 0.523594i −0.965123 0.261797i \(-0.915685\pi\)
0.965123 0.261797i \(-0.0843151\pi\)
\(18\) 7.54288i 1.77787i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.69202 −0.369229
\(22\) − 8.23490i − 1.75569i
\(23\) − 1.19806i − 0.249813i −0.992169 0.124907i \(-0.960137\pi\)
0.992169 0.124907i \(-0.0398631\pi\)
\(24\) −5.98792 −1.22228
\(25\) 0 0
\(26\) 9.09783 1.78423
\(27\) − 3.15883i − 0.607918i
\(28\) 17.8388i 3.37121i
\(29\) 1.77479 0.329570 0.164785 0.986329i \(-0.447307\pi\)
0.164785 + 0.986329i \(0.447307\pi\)
\(30\) 0 0
\(31\) −9.34481 −1.67838 −0.839189 0.543840i \(-0.816970\pi\)
−0.839189 + 0.543840i \(0.816970\pi\)
\(32\) 30.3424i 5.36383i
\(33\) 1.63102i 0.283925i
\(34\) 6.04892 1.03738
\(35\) 0 0
\(36\) −15.7506 −2.62510
\(37\) − 1.15883i − 0.190511i −0.995453 0.0952555i \(-0.969633\pi\)
0.995453 0.0952555i \(-0.0303668\pi\)
\(38\) 2.80194i 0.454534i
\(39\) −1.80194 −0.288541
\(40\) 0 0
\(41\) 8.57002 1.33841 0.669206 0.743077i \(-0.266634\pi\)
0.669206 + 0.743077i \(0.266634\pi\)
\(42\) − 4.74094i − 0.731543i
\(43\) − 5.27413i − 0.804297i −0.915575 0.402148i \(-0.868264\pi\)
0.915575 0.402148i \(-0.131736\pi\)
\(44\) 17.1957 2.59234
\(45\) 0 0
\(46\) 3.35690 0.494947
\(47\) 2.35690i 0.343789i 0.985115 + 0.171894i \(0.0549888\pi\)
−0.985115 + 0.171894i \(0.945011\pi\)
\(48\) − 10.2838i − 1.48434i
\(49\) −2.29590 −0.327985
\(50\) 0 0
\(51\) −1.19806 −0.167762
\(52\) 18.9976i 2.63449i
\(53\) − 8.82371i − 1.21203i −0.795453 0.606015i \(-0.792767\pi\)
0.795453 0.606015i \(-0.207233\pi\)
\(54\) 8.85086 1.20445
\(55\) 0 0
\(56\) −32.8974 −4.39610
\(57\) − 0.554958i − 0.0735060i
\(58\) 4.97285i 0.652968i
\(59\) 5.70171 0.742299 0.371150 0.928573i \(-0.378964\pi\)
0.371150 + 0.928573i \(0.378964\pi\)
\(60\) 0 0
\(61\) −9.96077 −1.27535 −0.637673 0.770307i \(-0.720103\pi\)
−0.637673 + 0.770307i \(0.720103\pi\)
\(62\) − 26.1836i − 3.32532i
\(63\) − 8.20775i − 1.03408i
\(64\) −47.9560 −5.99450
\(65\) 0 0
\(66\) −4.57002 −0.562531
\(67\) 4.98254i 0.608714i 0.952558 + 0.304357i \(0.0984416\pi\)
−0.952558 + 0.304357i \(0.901558\pi\)
\(68\) 12.6310i 1.53174i
\(69\) −0.664874 −0.0800415
\(70\) 0 0
\(71\) 2.70171 0.320634 0.160317 0.987066i \(-0.448748\pi\)
0.160317 + 0.987066i \(0.448748\pi\)
\(72\) − 29.0465i − 3.42317i
\(73\) − 13.7778i − 1.61257i −0.591530 0.806283i \(-0.701476\pi\)
0.591530 0.806283i \(-0.298524\pi\)
\(74\) 3.24698 0.377454
\(75\) 0 0
\(76\) −5.85086 −0.671139
\(77\) 8.96077i 1.02117i
\(78\) − 5.04892i − 0.571677i
\(79\) −5.66487 −0.637348 −0.318674 0.947864i \(-0.603238\pi\)
−0.318674 + 0.947864i \(0.603238\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) 24.0127i 2.65176i
\(83\) 3.00969i 0.330356i 0.986264 + 0.165178i \(0.0528199\pi\)
−0.986264 + 0.165178i \(0.947180\pi\)
\(84\) 9.89977 1.08015
\(85\) 0 0
\(86\) 14.7778 1.59353
\(87\) − 0.984935i − 0.105596i
\(88\) 31.7114i 3.38045i
\(89\) 10.2838 1.09008 0.545041 0.838409i \(-0.316514\pi\)
0.545041 + 0.838409i \(0.316514\pi\)
\(90\) 0 0
\(91\) −9.89977 −1.03778
\(92\) 7.00969i 0.730811i
\(93\) 5.18598i 0.537761i
\(94\) −6.60388 −0.681138
\(95\) 0 0
\(96\) 16.8388 1.71860
\(97\) 3.24698i 0.329681i 0.986320 + 0.164840i \(0.0527109\pi\)
−0.986320 + 0.164840i \(0.947289\pi\)
\(98\) − 6.43296i − 0.649827i
\(99\) −7.91185 −0.795171
\(100\) 0 0
\(101\) 5.09246 0.506718 0.253359 0.967372i \(-0.418465\pi\)
0.253359 + 0.967372i \(0.418465\pi\)
\(102\) − 3.35690i − 0.332382i
\(103\) 14.3110i 1.41010i 0.709157 + 0.705051i \(0.249076\pi\)
−0.709157 + 0.705051i \(0.750924\pi\)
\(104\) −35.0344 −3.43541
\(105\) 0 0
\(106\) 24.7235 2.40136
\(107\) − 8.18598i − 0.791369i −0.918387 0.395684i \(-0.870507\pi\)
0.918387 0.395684i \(-0.129493\pi\)
\(108\) 18.4819i 1.77842i
\(109\) 11.3274 1.08496 0.542482 0.840067i \(-0.317485\pi\)
0.542482 + 0.840067i \(0.317485\pi\)
\(110\) 0 0
\(111\) −0.643104 −0.0610407
\(112\) − 56.4989i − 5.33864i
\(113\) − 3.43535i − 0.323171i −0.986859 0.161585i \(-0.948339\pi\)
0.986859 0.161585i \(-0.0516607\pi\)
\(114\) 1.55496 0.145635
\(115\) 0 0
\(116\) −10.3840 −0.964134
\(117\) − 8.74094i − 0.808100i
\(118\) 15.9758i 1.47069i
\(119\) −6.58211 −0.603381
\(120\) 0 0
\(121\) −2.36227 −0.214752
\(122\) − 27.9095i − 2.52680i
\(123\) − 4.75600i − 0.428834i
\(124\) 54.6752 4.90997
\(125\) 0 0
\(126\) 22.9976 2.04879
\(127\) 1.46144i 0.129681i 0.997896 + 0.0648407i \(0.0206539\pi\)
−0.997896 + 0.0648407i \(0.979346\pi\)
\(128\) − 73.6848i − 6.51288i
\(129\) −2.92692 −0.257701
\(130\) 0 0
\(131\) −13.2295 −1.15587 −0.577934 0.816083i \(-0.696141\pi\)
−0.577934 + 0.816083i \(0.696141\pi\)
\(132\) − 9.54288i − 0.830601i
\(133\) − 3.04892i − 0.264375i
\(134\) −13.9608 −1.20603
\(135\) 0 0
\(136\) −23.2935 −1.99740
\(137\) 16.1739i 1.38183i 0.722936 + 0.690915i \(0.242792\pi\)
−0.722936 + 0.690915i \(0.757208\pi\)
\(138\) − 1.86294i − 0.158584i
\(139\) −13.0978 −1.11094 −0.555472 0.831535i \(-0.687462\pi\)
−0.555472 + 0.831535i \(0.687462\pi\)
\(140\) 0 0
\(141\) 1.30798 0.110152
\(142\) 7.57002i 0.635262i
\(143\) 9.54288i 0.798015i
\(144\) 49.8853 4.15711
\(145\) 0 0
\(146\) 38.6045 3.19493
\(147\) 1.27413i 0.105088i
\(148\) 6.78017i 0.557326i
\(149\) −11.0640 −0.906397 −0.453198 0.891410i \(-0.649717\pi\)
−0.453198 + 0.891410i \(0.649717\pi\)
\(150\) 0 0
\(151\) 10.4426 0.849811 0.424905 0.905238i \(-0.360307\pi\)
0.424905 + 0.905238i \(0.360307\pi\)
\(152\) − 10.7899i − 0.875173i
\(153\) − 5.81163i − 0.469842i
\(154\) −25.1075 −2.02322
\(155\) 0 0
\(156\) 10.5429 0.844106
\(157\) 2.48427i 0.198266i 0.995074 + 0.0991332i \(0.0316070\pi\)
−0.995074 + 0.0991332i \(0.968393\pi\)
\(158\) − 15.8726i − 1.26276i
\(159\) −4.89679 −0.388341
\(160\) 0 0
\(161\) −3.65279 −0.287880
\(162\) 17.7168i 1.39196i
\(163\) − 3.16852i − 0.248178i −0.992271 0.124089i \(-0.960399\pi\)
0.992271 0.124089i \(-0.0396008\pi\)
\(164\) −50.1420 −3.91543
\(165\) 0 0
\(166\) −8.43296 −0.654525
\(167\) 4.74632i 0.367281i 0.982993 + 0.183640i \(0.0587882\pi\)
−0.982993 + 0.183640i \(0.941212\pi\)
\(168\) 18.2567i 1.40853i
\(169\) 2.45712 0.189009
\(170\) 0 0
\(171\) 2.69202 0.205864
\(172\) 30.8582i 2.35291i
\(173\) − 3.96316i − 0.301314i −0.988586 0.150657i \(-0.951861\pi\)
0.988586 0.150657i \(-0.0481389\pi\)
\(174\) 2.75973 0.209214
\(175\) 0 0
\(176\) −54.4620 −4.10523
\(177\) − 3.16421i − 0.237837i
\(178\) 28.8146i 2.15975i
\(179\) 14.0858 1.05282 0.526409 0.850231i \(-0.323538\pi\)
0.526409 + 0.850231i \(0.323538\pi\)
\(180\) 0 0
\(181\) −12.2513 −0.910631 −0.455316 0.890330i \(-0.650474\pi\)
−0.455316 + 0.890330i \(0.650474\pi\)
\(182\) − 27.7385i − 2.05612i
\(183\) 5.52781i 0.408628i
\(184\) −12.9269 −0.952985
\(185\) 0 0
\(186\) −14.5308 −1.06545
\(187\) 6.34481i 0.463979i
\(188\) − 13.7899i − 1.00573i
\(189\) −9.63102 −0.700554
\(190\) 0 0
\(191\) 20.1468 1.45777 0.728884 0.684637i \(-0.240039\pi\)
0.728884 + 0.684637i \(0.240039\pi\)
\(192\) 26.6136i 1.92067i
\(193\) 9.52781i 0.685827i 0.939367 + 0.342913i \(0.111414\pi\)
−0.939367 + 0.342913i \(0.888586\pi\)
\(194\) −9.09783 −0.653186
\(195\) 0 0
\(196\) 13.4330 0.959497
\(197\) 4.96316i 0.353611i 0.984246 + 0.176805i \(0.0565763\pi\)
−0.984246 + 0.176805i \(0.943424\pi\)
\(198\) − 22.1685i − 1.57545i
\(199\) −5.42221 −0.384370 −0.192185 0.981359i \(-0.561557\pi\)
−0.192185 + 0.981359i \(0.561557\pi\)
\(200\) 0 0
\(201\) 2.76510 0.195035
\(202\) 14.2687i 1.00395i
\(203\) − 5.41119i − 0.379791i
\(204\) 7.00969 0.490776
\(205\) 0 0
\(206\) −40.0984 −2.79379
\(207\) − 3.22521i − 0.224168i
\(208\) − 60.1691i − 4.17198i
\(209\) −2.93900 −0.203295
\(210\) 0 0
\(211\) 15.9638 1.09899 0.549495 0.835497i \(-0.314820\pi\)
0.549495 + 0.835497i \(0.314820\pi\)
\(212\) 51.6262i 3.54570i
\(213\) − 1.49934i − 0.102733i
\(214\) 22.9366 1.56791
\(215\) 0 0
\(216\) −34.0834 −2.31908
\(217\) 28.4916i 1.93413i
\(218\) 31.7385i 2.14961i
\(219\) −7.64609 −0.516675
\(220\) 0 0
\(221\) −7.00969 −0.471523
\(222\) − 1.80194i − 0.120938i
\(223\) 23.2150i 1.55459i 0.629134 + 0.777297i \(0.283410\pi\)
−0.629134 + 0.777297i \(0.716590\pi\)
\(224\) 92.5115 6.18119
\(225\) 0 0
\(226\) 9.62565 0.640288
\(227\) − 17.4795i − 1.16015i −0.814562 0.580077i \(-0.803022\pi\)
0.814562 0.580077i \(-0.196978\pi\)
\(228\) 3.24698i 0.215036i
\(229\) 2.32544 0.153669 0.0768346 0.997044i \(-0.475519\pi\)
0.0768346 + 0.997044i \(0.475519\pi\)
\(230\) 0 0
\(231\) 4.97285 0.327190
\(232\) − 19.1497i − 1.25724i
\(233\) 18.9342i 1.24042i 0.784435 + 0.620211i \(0.212953\pi\)
−0.784435 + 0.620211i \(0.787047\pi\)
\(234\) 24.4916 1.60106
\(235\) 0 0
\(236\) −33.3599 −2.17154
\(237\) 3.14377i 0.204210i
\(238\) − 18.4426i − 1.19546i
\(239\) −11.1685 −0.722432 −0.361216 0.932482i \(-0.617638\pi\)
−0.361216 + 0.932482i \(0.617638\pi\)
\(240\) 0 0
\(241\) 7.42998 0.478607 0.239303 0.970945i \(-0.423081\pi\)
0.239303 + 0.970945i \(0.423081\pi\)
\(242\) − 6.61894i − 0.425482i
\(243\) − 12.9855i − 0.833022i
\(244\) 58.2790 3.73093
\(245\) 0 0
\(246\) 13.3260 0.849637
\(247\) − 3.24698i − 0.206600i
\(248\) 100.829i 6.40266i
\(249\) 1.67025 0.105848
\(250\) 0 0
\(251\) −14.7409 −0.930440 −0.465220 0.885195i \(-0.654025\pi\)
−0.465220 + 0.885195i \(0.654025\pi\)
\(252\) 48.0224i 3.02512i
\(253\) 3.52111i 0.221370i
\(254\) −4.09485 −0.256934
\(255\) 0 0
\(256\) 110.548 6.90927
\(257\) 3.34721i 0.208793i 0.994536 + 0.104397i \(0.0332911\pi\)
−0.994536 + 0.104397i \(0.966709\pi\)
\(258\) − 8.20105i − 0.510575i
\(259\) −3.53319 −0.219542
\(260\) 0 0
\(261\) 4.77777 0.295737
\(262\) − 37.0683i − 2.29009i
\(263\) 16.6853i 1.02886i 0.857532 + 0.514430i \(0.171997\pi\)
−0.857532 + 0.514430i \(0.828003\pi\)
\(264\) 17.5985 1.08311
\(265\) 0 0
\(266\) 8.54288 0.523797
\(267\) − 5.70709i − 0.349268i
\(268\) − 29.1521i − 1.78075i
\(269\) −15.5961 −0.950911 −0.475456 0.879740i \(-0.657717\pi\)
−0.475456 + 0.879740i \(0.657717\pi\)
\(270\) 0 0
\(271\) −13.2131 −0.802640 −0.401320 0.915938i \(-0.631449\pi\)
−0.401320 + 0.915938i \(0.631449\pi\)
\(272\) − 40.0049i − 2.42565i
\(273\) 5.49396i 0.332510i
\(274\) −45.3183 −2.73778
\(275\) 0 0
\(276\) 3.89008 0.234156
\(277\) 12.9758i 0.779642i 0.920891 + 0.389821i \(0.127463\pi\)
−0.920891 + 0.389821i \(0.872537\pi\)
\(278\) − 36.6993i − 2.20108i
\(279\) −25.1564 −1.50608
\(280\) 0 0
\(281\) 24.8901 1.48482 0.742409 0.669947i \(-0.233683\pi\)
0.742409 + 0.669947i \(0.233683\pi\)
\(282\) 3.66487i 0.218240i
\(283\) 13.5821i 0.807372i 0.914898 + 0.403686i \(0.132271\pi\)
−0.914898 + 0.403686i \(0.867729\pi\)
\(284\) −15.8073 −0.937992
\(285\) 0 0
\(286\) −26.7385 −1.58108
\(287\) − 26.1293i − 1.54236i
\(288\) 81.6824i 4.81318i
\(289\) 12.3394 0.725849
\(290\) 0 0
\(291\) 1.80194 0.105631
\(292\) 80.6118i 4.71745i
\(293\) − 1.27652i − 0.0745751i −0.999305 0.0372875i \(-0.988128\pi\)
0.999305 0.0372875i \(-0.0118717\pi\)
\(294\) −3.57002 −0.208208
\(295\) 0 0
\(296\) −12.5036 −0.726760
\(297\) 9.28382i 0.538702i
\(298\) − 31.0006i − 1.79582i
\(299\) −3.89008 −0.224969
\(300\) 0 0
\(301\) −16.0804 −0.926857
\(302\) 29.2597i 1.68370i
\(303\) − 2.82610i − 0.162355i
\(304\) 18.5308 1.06281
\(305\) 0 0
\(306\) 16.2838 0.930884
\(307\) − 32.2295i − 1.83944i −0.392580 0.919718i \(-0.628417\pi\)
0.392580 0.919718i \(-0.371583\pi\)
\(308\) − 52.4282i − 2.98737i
\(309\) 7.94198 0.451804
\(310\) 0 0
\(311\) 12.4983 0.708712 0.354356 0.935111i \(-0.384700\pi\)
0.354356 + 0.935111i \(0.384700\pi\)
\(312\) 19.4426i 1.10072i
\(313\) 20.9390i 1.18354i 0.806106 + 0.591771i \(0.201571\pi\)
−0.806106 + 0.591771i \(0.798429\pi\)
\(314\) −6.96077 −0.392819
\(315\) 0 0
\(316\) 33.1444 1.86452
\(317\) 15.8562i 0.890575i 0.895388 + 0.445287i \(0.146898\pi\)
−0.895388 + 0.445287i \(0.853102\pi\)
\(318\) − 13.7205i − 0.769407i
\(319\) −5.21611 −0.292046
\(320\) 0 0
\(321\) −4.54288 −0.253559
\(322\) − 10.2349i − 0.570369i
\(323\) − 2.15883i − 0.120121i
\(324\) −36.9952 −2.05529
\(325\) 0 0
\(326\) 8.87800 0.491707
\(327\) − 6.28621i − 0.347628i
\(328\) − 92.4693i − 5.10576i
\(329\) 7.18598 0.396176
\(330\) 0 0
\(331\) −13.2349 −0.727456 −0.363728 0.931505i \(-0.618496\pi\)
−0.363728 + 0.931505i \(0.618496\pi\)
\(332\) − 17.6093i − 0.966433i
\(333\) − 3.11960i − 0.170953i
\(334\) −13.2989 −0.727682
\(335\) 0 0
\(336\) −31.3545 −1.71053
\(337\) − 26.0780i − 1.42056i −0.703920 0.710279i \(-0.748569\pi\)
0.703920 0.710279i \(-0.251431\pi\)
\(338\) 6.88471i 0.374479i
\(339\) −1.90648 −0.103546
\(340\) 0 0
\(341\) 27.4644 1.48728
\(342\) 7.54288i 0.407872i
\(343\) − 14.3424i − 0.774418i
\(344\) −56.9071 −3.06822
\(345\) 0 0
\(346\) 11.1045 0.596984
\(347\) 8.98254i 0.482208i 0.970499 + 0.241104i \(0.0775095\pi\)
−0.970499 + 0.241104i \(0.922490\pi\)
\(348\) 5.76271i 0.308914i
\(349\) −0.599564 −0.0320939 −0.0160470 0.999871i \(-0.505108\pi\)
−0.0160470 + 0.999871i \(0.505108\pi\)
\(350\) 0 0
\(351\) −10.2567 −0.547460
\(352\) − 89.1764i − 4.75312i
\(353\) − 31.8896i − 1.69731i −0.528945 0.848656i \(-0.677412\pi\)
0.528945 0.848656i \(-0.322588\pi\)
\(354\) 8.86592 0.471218
\(355\) 0 0
\(356\) −60.1691 −3.18896
\(357\) 3.65279i 0.193326i
\(358\) 39.4674i 2.08592i
\(359\) 18.3763 0.969863 0.484931 0.874552i \(-0.338845\pi\)
0.484931 + 0.874552i \(0.338845\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 34.3274i − 1.80421i
\(363\) 1.31096i 0.0688077i
\(364\) 57.9221 3.03594
\(365\) 0 0
\(366\) −15.4886 −0.809601
\(367\) − 14.6377i − 0.764083i −0.924145 0.382042i \(-0.875221\pi\)
0.924145 0.382042i \(-0.124779\pi\)
\(368\) − 22.2010i − 1.15731i
\(369\) 23.0707 1.20101
\(370\) 0 0
\(371\) −26.9028 −1.39672
\(372\) − 30.3424i − 1.57318i
\(373\) 1.02715i 0.0531837i 0.999646 + 0.0265918i \(0.00846544\pi\)
−0.999646 + 0.0265918i \(0.991535\pi\)
\(374\) −17.7778 −0.919267
\(375\) 0 0
\(376\) 25.4306 1.31148
\(377\) − 5.76271i − 0.296795i
\(378\) − 26.9855i − 1.38799i
\(379\) 22.5284 1.15721 0.578603 0.815609i \(-0.303598\pi\)
0.578603 + 0.815609i \(0.303598\pi\)
\(380\) 0 0
\(381\) 0.811035 0.0415506
\(382\) 56.4499i 2.88823i
\(383\) − 32.6698i − 1.66935i −0.550745 0.834674i \(-0.685656\pi\)
0.550745 0.834674i \(-0.314344\pi\)
\(384\) −40.8920 −2.08676
\(385\) 0 0
\(386\) −26.6963 −1.35881
\(387\) − 14.1981i − 0.721728i
\(388\) − 18.9976i − 0.964457i
\(389\) 8.16421 0.413942 0.206971 0.978347i \(-0.433639\pi\)
0.206971 + 0.978347i \(0.433639\pi\)
\(390\) 0 0
\(391\) −2.58642 −0.130801
\(392\) 24.7724i 1.25120i
\(393\) 7.34183i 0.370346i
\(394\) −13.9065 −0.700598
\(395\) 0 0
\(396\) 46.2911 2.32622
\(397\) 37.5478i 1.88447i 0.334954 + 0.942235i \(0.391279\pi\)
−0.334954 + 0.942235i \(0.608721\pi\)
\(398\) − 15.1927i − 0.761541i
\(399\) −1.69202 −0.0847070
\(400\) 0 0
\(401\) 32.2519 1.61058 0.805291 0.592880i \(-0.202009\pi\)
0.805291 + 0.592880i \(0.202009\pi\)
\(402\) 7.74764i 0.386417i
\(403\) 30.3424i 1.51146i
\(404\) −29.7952 −1.48237
\(405\) 0 0
\(406\) 15.1618 0.752468
\(407\) 3.40581i 0.168820i
\(408\) 12.9269i 0.639978i
\(409\) 31.5459 1.55984 0.779921 0.625878i \(-0.215259\pi\)
0.779921 + 0.625878i \(0.215259\pi\)
\(410\) 0 0
\(411\) 8.97584 0.442745
\(412\) − 83.7314i − 4.12515i
\(413\) − 17.3840i − 0.855413i
\(414\) 9.03684 0.444136
\(415\) 0 0
\(416\) 98.5212 4.83040
\(417\) 7.26875i 0.355952i
\(418\) − 8.23490i − 0.402782i
\(419\) 14.6866 0.717490 0.358745 0.933436i \(-0.383205\pi\)
0.358745 + 0.933436i \(0.383205\pi\)
\(420\) 0 0
\(421\) 15.1142 0.736622 0.368311 0.929703i \(-0.379936\pi\)
0.368311 + 0.929703i \(0.379936\pi\)
\(422\) 44.7294i 2.17740i
\(423\) 6.34481i 0.308495i
\(424\) −95.2065 −4.62364
\(425\) 0 0
\(426\) 4.20105 0.203541
\(427\) 30.3696i 1.46969i
\(428\) 47.8950i 2.31509i
\(429\) 5.29590 0.255688
\(430\) 0 0
\(431\) 1.67696 0.0807761 0.0403881 0.999184i \(-0.487141\pi\)
0.0403881 + 0.999184i \(0.487141\pi\)
\(432\) − 58.5357i − 2.81630i
\(433\) 13.1545i 0.632166i 0.948732 + 0.316083i \(0.102368\pi\)
−0.948732 + 0.316083i \(0.897632\pi\)
\(434\) −79.8316 −3.83204
\(435\) 0 0
\(436\) −66.2747 −3.17398
\(437\) − 1.19806i − 0.0573111i
\(438\) − 21.4239i − 1.02367i
\(439\) −6.45580 −0.308118 −0.154059 0.988062i \(-0.549235\pi\)
−0.154059 + 0.988062i \(0.549235\pi\)
\(440\) 0 0
\(441\) −6.18060 −0.294314
\(442\) − 19.6407i − 0.934213i
\(443\) 25.3709i 1.20541i 0.797965 + 0.602704i \(0.205910\pi\)
−0.797965 + 0.602704i \(0.794090\pi\)
\(444\) 3.76271 0.178570
\(445\) 0 0
\(446\) −65.0471 −3.08007
\(447\) 6.14005i 0.290414i
\(448\) 146.214i 6.90795i
\(449\) 3.59956 0.169874 0.0849370 0.996386i \(-0.472931\pi\)
0.0849370 + 0.996386i \(0.472931\pi\)
\(450\) 0 0
\(451\) −25.1873 −1.18602
\(452\) 20.0998i 0.945413i
\(453\) − 5.79523i − 0.272284i
\(454\) 48.9764 2.29858
\(455\) 0 0
\(456\) −5.98792 −0.280410
\(457\) − 12.2784i − 0.574361i −0.957876 0.287181i \(-0.907282\pi\)
0.957876 0.287181i \(-0.0927180\pi\)
\(458\) 6.51573i 0.304460i
\(459\) −6.81940 −0.318302
\(460\) 0 0
\(461\) 39.4935 1.83939 0.919697 0.392628i \(-0.128434\pi\)
0.919697 + 0.392628i \(0.128434\pi\)
\(462\) 13.9336i 0.648251i
\(463\) − 18.3991i − 0.855079i −0.903997 0.427540i \(-0.859380\pi\)
0.903997 0.427540i \(-0.140620\pi\)
\(464\) 32.8883 1.52680
\(465\) 0 0
\(466\) −53.0525 −2.45761
\(467\) − 36.2282i − 1.67644i −0.545332 0.838220i \(-0.683596\pi\)
0.545332 0.838220i \(-0.316404\pi\)
\(468\) 51.1420i 2.36404i
\(469\) 15.1914 0.701472
\(470\) 0 0
\(471\) 1.37867 0.0635256
\(472\) − 61.5206i − 2.83172i
\(473\) 15.5007i 0.712721i
\(474\) −8.80864 −0.404594
\(475\) 0 0
\(476\) 38.5109 1.76515
\(477\) − 23.7536i − 1.08760i
\(478\) − 31.2935i − 1.43133i
\(479\) 34.3129 1.56780 0.783898 0.620890i \(-0.213229\pi\)
0.783898 + 0.620890i \(0.213229\pi\)
\(480\) 0 0
\(481\) −3.76271 −0.171565
\(482\) 20.8183i 0.948249i
\(483\) 2.02715i 0.0922384i
\(484\) 13.8213 0.628242
\(485\) 0 0
\(486\) 36.3846 1.65044
\(487\) − 24.4722i − 1.10894i −0.832203 0.554470i \(-0.812921\pi\)
0.832203 0.554470i \(-0.187079\pi\)
\(488\) 107.475i 4.86518i
\(489\) −1.75840 −0.0795175
\(490\) 0 0
\(491\) 18.3067 0.826168 0.413084 0.910693i \(-0.364452\pi\)
0.413084 + 0.910693i \(0.364452\pi\)
\(492\) 27.8267i 1.25452i
\(493\) − 3.83148i − 0.172561i
\(494\) 9.09783 0.409331
\(495\) 0 0
\(496\) −173.167 −7.77542
\(497\) − 8.23729i − 0.369493i
\(498\) 4.67994i 0.209713i
\(499\) 25.3424 1.13448 0.567241 0.823552i \(-0.308011\pi\)
0.567241 + 0.823552i \(0.308011\pi\)
\(500\) 0 0
\(501\) 2.63401 0.117679
\(502\) − 41.3032i − 1.84345i
\(503\) − 14.1105i − 0.629156i −0.949232 0.314578i \(-0.898137\pi\)
0.949232 0.314578i \(-0.101863\pi\)
\(504\) −88.5605 −3.94480
\(505\) 0 0
\(506\) −9.86592 −0.438594
\(507\) − 1.36360i − 0.0605596i
\(508\) − 8.55065i − 0.379374i
\(509\) 17.8019 0.789057 0.394529 0.918884i \(-0.370908\pi\)
0.394529 + 0.918884i \(0.370908\pi\)
\(510\) 0 0
\(511\) −42.0073 −1.85829
\(512\) 162.380i 7.17625i
\(513\) − 3.15883i − 0.139466i
\(514\) −9.37867 −0.413675
\(515\) 0 0
\(516\) 17.1250 0.753885
\(517\) − 6.92692i − 0.304646i
\(518\) − 9.89977i − 0.434971i
\(519\) −2.19939 −0.0965425
\(520\) 0 0
\(521\) 13.3948 0.586837 0.293418 0.955984i \(-0.405207\pi\)
0.293418 + 0.955984i \(0.405207\pi\)
\(522\) 13.3870i 0.585934i
\(523\) 1.46921i 0.0642439i 0.999484 + 0.0321219i \(0.0102265\pi\)
−0.999484 + 0.0321219i \(0.989774\pi\)
\(524\) 77.4040 3.38141
\(525\) 0 0
\(526\) −46.7512 −2.03845
\(527\) 20.1739i 0.878789i
\(528\) 30.2241i 1.31534i
\(529\) 21.5646 0.937593
\(530\) 0 0
\(531\) 15.3491 0.666095
\(532\) 17.8388i 0.773409i
\(533\) − 27.8267i − 1.20531i
\(534\) 15.9909 0.691994
\(535\) 0 0
\(536\) 53.7609 2.32212
\(537\) − 7.81700i − 0.337329i
\(538\) − 43.6993i − 1.88401i
\(539\) 6.74764 0.290642
\(540\) 0 0
\(541\) −42.7144 −1.83643 −0.918217 0.396077i \(-0.870371\pi\)
−0.918217 + 0.396077i \(0.870371\pi\)
\(542\) − 37.0224i − 1.59025i
\(543\) 6.79895i 0.291771i
\(544\) 65.5042 2.80847
\(545\) 0 0
\(546\) −15.3937 −0.658791
\(547\) 10.6866i 0.456928i 0.973552 + 0.228464i \(0.0733703\pi\)
−0.973552 + 0.228464i \(0.926630\pi\)
\(548\) − 94.6311i − 4.04244i
\(549\) −26.8146 −1.14442
\(550\) 0 0
\(551\) 1.77479 0.0756086
\(552\) 7.17390i 0.305341i
\(553\) 17.2717i 0.734469i
\(554\) −36.3575 −1.54468
\(555\) 0 0
\(556\) 76.6335 3.24999
\(557\) − 34.2083i − 1.44945i −0.689036 0.724727i \(-0.741966\pi\)
0.689036 0.724727i \(-0.258034\pi\)
\(558\) − 70.4868i − 2.98394i
\(559\) −17.1250 −0.724310
\(560\) 0 0
\(561\) 3.52111 0.148661
\(562\) 69.7405i 2.94182i
\(563\) 5.46788i 0.230444i 0.993340 + 0.115222i \(0.0367579\pi\)
−0.993340 + 0.115222i \(0.963242\pi\)
\(564\) −7.65279 −0.322241
\(565\) 0 0
\(566\) −38.0562 −1.59962
\(567\) − 19.2784i − 0.809618i
\(568\) − 29.1511i − 1.22315i
\(569\) 17.4819 0.732878 0.366439 0.930442i \(-0.380577\pi\)
0.366439 + 0.930442i \(0.380577\pi\)
\(570\) 0 0
\(571\) −7.21877 −0.302096 −0.151048 0.988526i \(-0.548265\pi\)
−0.151048 + 0.988526i \(0.548265\pi\)
\(572\) − 55.8340i − 2.33454i
\(573\) − 11.1806i − 0.467076i
\(574\) 73.2127 3.05584
\(575\) 0 0
\(576\) −129.099 −5.37911
\(577\) 1.80625i 0.0751952i 0.999293 + 0.0375976i \(0.0119705\pi\)
−0.999293 + 0.0375976i \(0.988029\pi\)
\(578\) 34.5743i 1.43810i
\(579\) 5.28754 0.219743
\(580\) 0 0
\(581\) 9.17629 0.380697
\(582\) 5.04892i 0.209284i
\(583\) 25.9329i 1.07403i
\(584\) −148.660 −6.15160
\(585\) 0 0
\(586\) 3.57673 0.147753
\(587\) 2.39075i 0.0986767i 0.998782 + 0.0493384i \(0.0157113\pi\)
−0.998782 + 0.0493384i \(0.984289\pi\)
\(588\) − 7.45473i − 0.307428i
\(589\) −9.34481 −0.385046
\(590\) 0 0
\(591\) 2.75435 0.113299
\(592\) − 21.4741i − 0.882580i
\(593\) 42.2650i 1.73562i 0.496899 + 0.867808i \(0.334472\pi\)
−0.496899 + 0.867808i \(0.665528\pi\)
\(594\) −26.0127 −1.06731
\(595\) 0 0
\(596\) 64.7338 2.65160
\(597\) 3.00910i 0.123154i
\(598\) − 10.8998i − 0.445725i
\(599\) −8.95838 −0.366029 −0.183015 0.983110i \(-0.558586\pi\)
−0.183015 + 0.983110i \(0.558586\pi\)
\(600\) 0 0
\(601\) −19.8998 −0.811729 −0.405864 0.913933i \(-0.633029\pi\)
−0.405864 + 0.913933i \(0.633029\pi\)
\(602\) − 45.0562i − 1.83635i
\(603\) 13.4131i 0.546224i
\(604\) −61.0984 −2.48606
\(605\) 0 0
\(606\) 7.91856 0.321669
\(607\) 26.0084i 1.05565i 0.849354 + 0.527823i \(0.176992\pi\)
−0.849354 + 0.527823i \(0.823008\pi\)
\(608\) 30.3424i 1.23055i
\(609\) −3.00298 −0.121687
\(610\) 0 0
\(611\) 7.65279 0.309599
\(612\) 34.0030i 1.37449i
\(613\) − 41.3763i − 1.67117i −0.549360 0.835586i \(-0.685128\pi\)
0.549360 0.835586i \(-0.314872\pi\)
\(614\) 90.3051 3.64442
\(615\) 0 0
\(616\) 96.6854 3.89557
\(617\) 43.9845i 1.77075i 0.464880 + 0.885374i \(0.346098\pi\)
−0.464880 + 0.885374i \(0.653902\pi\)
\(618\) 22.2529i 0.895145i
\(619\) −23.4553 −0.942749 −0.471374 0.881933i \(-0.656242\pi\)
−0.471374 + 0.881933i \(0.656242\pi\)
\(620\) 0 0
\(621\) −3.78448 −0.151866
\(622\) 35.0194i 1.40415i
\(623\) − 31.3545i − 1.25619i
\(624\) −33.3913 −1.33672
\(625\) 0 0
\(626\) −58.6698 −2.34492
\(627\) 1.63102i 0.0651368i
\(628\) − 14.5351i − 0.580014i
\(629\) −2.50173 −0.0997505
\(630\) 0 0
\(631\) −34.8877 −1.38886 −0.694429 0.719562i \(-0.744343\pi\)
−0.694429 + 0.719562i \(0.744343\pi\)
\(632\) 61.1232i 2.43135i
\(633\) − 8.85922i − 0.352122i
\(634\) −44.4282 −1.76447
\(635\) 0 0
\(636\) 28.6504 1.13606
\(637\) 7.45473i 0.295367i
\(638\) − 14.6152i − 0.578622i
\(639\) 7.27306 0.287718
\(640\) 0 0
\(641\) −38.5314 −1.52190 −0.760949 0.648812i \(-0.775266\pi\)
−0.760949 + 0.648812i \(0.775266\pi\)
\(642\) − 12.7289i − 0.502368i
\(643\) 18.6353i 0.734906i 0.930042 + 0.367453i \(0.119770\pi\)
−0.930042 + 0.367453i \(0.880230\pi\)
\(644\) 21.3720 0.842173
\(645\) 0 0
\(646\) 6.04892 0.237991
\(647\) 19.2282i 0.755938i 0.925818 + 0.377969i \(0.123377\pi\)
−0.925818 + 0.377969i \(0.876623\pi\)
\(648\) − 68.2247i − 2.68012i
\(649\) −16.7573 −0.657783
\(650\) 0 0
\(651\) 15.8116 0.619706
\(652\) 18.5386i 0.726026i
\(653\) 3.98553i 0.155966i 0.996955 + 0.0779828i \(0.0248479\pi\)
−0.996955 + 0.0779828i \(0.975152\pi\)
\(654\) 17.6136 0.688745
\(655\) 0 0
\(656\) 158.809 6.20046
\(657\) − 37.0901i − 1.44702i
\(658\) 20.1347i 0.784931i
\(659\) −33.5719 −1.30778 −0.653889 0.756591i \(-0.726864\pi\)
−0.653889 + 0.756591i \(0.726864\pi\)
\(660\) 0 0
\(661\) 36.9909 1.43878 0.719390 0.694607i \(-0.244422\pi\)
0.719390 + 0.694607i \(0.244422\pi\)
\(662\) − 37.0834i − 1.44129i
\(663\) 3.89008i 0.151078i
\(664\) 32.4741 1.26024
\(665\) 0 0
\(666\) 8.74094 0.338704
\(667\) − 2.12631i − 0.0823310i
\(668\) − 27.7700i − 1.07445i
\(669\) 12.8834 0.498100
\(670\) 0 0
\(671\) 29.2747 1.13014
\(672\) − 51.3400i − 1.98048i
\(673\) − 12.0747i − 0.465447i −0.972543 0.232723i \(-0.925236\pi\)
0.972543 0.232723i \(-0.0747637\pi\)
\(674\) 73.0689 2.81451
\(675\) 0 0
\(676\) −14.3763 −0.552934
\(677\) 41.9135i 1.61087i 0.592686 + 0.805434i \(0.298067\pi\)
−0.592686 + 0.805434i \(0.701933\pi\)
\(678\) − 5.34183i − 0.205152i
\(679\) 9.89977 0.379918
\(680\) 0 0
\(681\) −9.70038 −0.371719
\(682\) 76.9536i 2.94671i
\(683\) − 9.17092i − 0.350915i −0.984487 0.175458i \(-0.943859\pi\)
0.984487 0.175458i \(-0.0561405\pi\)
\(684\) −15.7506 −0.602240
\(685\) 0 0
\(686\) 40.1866 1.53433
\(687\) − 1.29052i − 0.0492364i
\(688\) − 97.7338i − 3.72606i
\(689\) −28.6504 −1.09149
\(690\) 0 0
\(691\) −0.111244 −0.00423193 −0.00211596 0.999998i \(-0.500674\pi\)
−0.00211596 + 0.999998i \(0.500674\pi\)
\(692\) 23.1879i 0.881472i
\(693\) 24.1226i 0.916341i
\(694\) −25.1685 −0.955384
\(695\) 0 0
\(696\) −10.6273 −0.402827
\(697\) − 18.5013i − 0.700785i
\(698\) − 1.67994i − 0.0635867i
\(699\) 10.5077 0.397438
\(700\) 0 0
\(701\) −27.7832 −1.04936 −0.524678 0.851301i \(-0.675814\pi\)
−0.524678 + 0.851301i \(0.675814\pi\)
\(702\) − 28.7385i − 1.08467i
\(703\) − 1.15883i − 0.0437062i
\(704\) 140.943 5.31198
\(705\) 0 0
\(706\) 89.3527 3.36283
\(707\) − 15.5265i − 0.583933i
\(708\) 18.5133i 0.695774i
\(709\) −37.4668 −1.40710 −0.703548 0.710648i \(-0.748402\pi\)
−0.703548 + 0.710648i \(0.748402\pi\)
\(710\) 0 0
\(711\) −15.2500 −0.571918
\(712\) − 110.961i − 4.15844i
\(713\) 11.1957i 0.419281i
\(714\) −10.2349 −0.383031
\(715\) 0 0
\(716\) −82.4137 −3.07994
\(717\) 6.19806i 0.231471i
\(718\) 51.4892i 1.92156i
\(719\) 21.0513 0.785081 0.392541 0.919735i \(-0.371596\pi\)
0.392541 + 0.919735i \(0.371596\pi\)
\(720\) 0 0
\(721\) 43.6329 1.62498
\(722\) 2.80194i 0.104277i
\(723\) − 4.12333i − 0.153348i
\(724\) 71.6805 2.66399
\(725\) 0 0
\(726\) −3.67324 −0.136327
\(727\) − 44.7023i − 1.65792i −0.559310 0.828958i \(-0.688934\pi\)
0.559310 0.828958i \(-0.311066\pi\)
\(728\) 106.817i 3.95891i
\(729\) 11.7627 0.435656
\(730\) 0 0
\(731\) −11.3860 −0.421125
\(732\) − 32.3424i − 1.19541i
\(733\) − 25.0301i − 0.924509i −0.886747 0.462254i \(-0.847041\pi\)
0.886747 0.462254i \(-0.152959\pi\)
\(734\) 41.0140 1.51385
\(735\) 0 0
\(736\) 36.3521 1.33996
\(737\) − 14.6437i − 0.539407i
\(738\) 64.6426i 2.37953i
\(739\) 23.9982 0.882788 0.441394 0.897313i \(-0.354484\pi\)
0.441394 + 0.897313i \(0.354484\pi\)
\(740\) 0 0
\(741\) −1.80194 −0.0661958
\(742\) − 75.3798i − 2.76728i
\(743\) − 10.4601i − 0.383744i −0.981420 0.191872i \(-0.938544\pi\)
0.981420 0.191872i \(-0.0614559\pi\)
\(744\) 55.9560 2.05145
\(745\) 0 0
\(746\) −2.87800 −0.105371
\(747\) 8.10215i 0.296442i
\(748\) − 37.1226i − 1.35734i
\(749\) −24.9584 −0.911959
\(750\) 0 0
\(751\) −29.9420 −1.09260 −0.546299 0.837590i \(-0.683964\pi\)
−0.546299 + 0.837590i \(0.683964\pi\)
\(752\) 43.6752i 1.59267i
\(753\) 8.18060i 0.298118i
\(754\) 16.1468 0.588030
\(755\) 0 0
\(756\) 56.3497 2.04942
\(757\) 27.2174i 0.989235i 0.869111 + 0.494617i \(0.164692\pi\)
−0.869111 + 0.494617i \(0.835308\pi\)
\(758\) 63.1232i 2.29274i
\(759\) 1.95407 0.0709281
\(760\) 0 0
\(761\) 18.6813 0.677195 0.338598 0.940931i \(-0.390047\pi\)
0.338598 + 0.940931i \(0.390047\pi\)
\(762\) 2.27247i 0.0823229i
\(763\) − 34.5362i − 1.25029i
\(764\) −117.876 −4.26459
\(765\) 0 0
\(766\) 91.5387 3.30743
\(767\) − 18.5133i − 0.668478i
\(768\) − 61.3497i − 2.21377i
\(769\) −6.34780 −0.228907 −0.114454 0.993429i \(-0.536512\pi\)
−0.114454 + 0.993429i \(0.536512\pi\)
\(770\) 0 0
\(771\) 1.85756 0.0668984
\(772\) − 55.7458i − 2.00634i
\(773\) − 49.7851i − 1.79064i −0.445419 0.895322i \(-0.646945\pi\)
0.445419 0.895322i \(-0.353055\pi\)
\(774\) 39.7821 1.42994
\(775\) 0 0
\(776\) 35.0344 1.25766
\(777\) 1.96077i 0.0703423i
\(778\) 22.8756i 0.820130i
\(779\) 8.57002 0.307053
\(780\) 0 0
\(781\) −7.94033 −0.284127
\(782\) − 7.24698i − 0.259151i
\(783\) − 5.60627i − 0.200352i
\(784\) −42.5448 −1.51946
\(785\) 0 0
\(786\) −20.5714 −0.733756
\(787\) − 9.19865i − 0.327897i −0.986469 0.163948i \(-0.947577\pi\)
0.986469 0.163948i \(-0.0524230\pi\)
\(788\) − 29.0388i − 1.03446i
\(789\) 9.25965 0.329652
\(790\) 0 0
\(791\) −10.4741 −0.372416
\(792\) 85.3678i 3.03341i
\(793\) 32.3424i 1.14851i
\(794\) −105.207 −3.73364
\(795\) 0 0
\(796\) 31.7245 1.12445
\(797\) 13.5084i 0.478493i 0.970959 + 0.239247i \(0.0769004\pi\)
−0.970959 + 0.239247i \(0.923100\pi\)
\(798\) − 4.74094i − 0.167827i
\(799\) 5.08815 0.180006
\(800\) 0 0
\(801\) 27.6843 0.978175
\(802\) 90.3678i 3.19100i
\(803\) 40.4929i 1.42896i
\(804\) −16.1782 −0.570562
\(805\) 0 0
\(806\) −85.0176 −2.99462
\(807\) 8.65519i 0.304677i
\(808\) − 54.9469i − 1.93302i
\(809\) −31.9560 −1.12351 −0.561756 0.827303i \(-0.689874\pi\)
−0.561756 + 0.827303i \(0.689874\pi\)
\(810\) 0 0
\(811\) 40.7012 1.42921 0.714607 0.699526i \(-0.246606\pi\)
0.714607 + 0.699526i \(0.246606\pi\)
\(812\) 31.6601i 1.11105i
\(813\) 7.33273i 0.257170i
\(814\) −9.54288 −0.334478
\(815\) 0 0
\(816\) −22.2010 −0.777192
\(817\) − 5.27413i − 0.184518i
\(818\) 88.3895i 3.09047i
\(819\) −26.6504 −0.931240
\(820\) 0 0
\(821\) −13.4300 −0.468709 −0.234355 0.972151i \(-0.575298\pi\)
−0.234355 + 0.972151i \(0.575298\pi\)
\(822\) 25.1497i 0.877198i
\(823\) 26.7275i 0.931663i 0.884873 + 0.465832i \(0.154245\pi\)
−0.884873 + 0.465832i \(0.845755\pi\)
\(824\) 154.413 5.37924
\(825\) 0 0
\(826\) 48.7090 1.69480
\(827\) 33.3435i 1.15947i 0.814806 + 0.579733i \(0.196843\pi\)
−0.814806 + 0.579733i \(0.803157\pi\)
\(828\) 18.8702i 0.655786i
\(829\) −12.2849 −0.426672 −0.213336 0.976979i \(-0.568433\pi\)
−0.213336 + 0.976979i \(0.568433\pi\)
\(830\) 0 0
\(831\) 7.20105 0.249802
\(832\) 155.712i 5.39835i
\(833\) 4.95646i 0.171731i
\(834\) −20.3666 −0.705237
\(835\) 0 0
\(836\) 17.1957 0.594725
\(837\) 29.5187i 1.02032i
\(838\) 41.1511i 1.42154i
\(839\) −36.9922 −1.27711 −0.638557 0.769575i \(-0.720468\pi\)
−0.638557 + 0.769575i \(0.720468\pi\)
\(840\) 0 0
\(841\) −25.8501 −0.891383
\(842\) 42.3491i 1.45945i
\(843\) − 13.8130i − 0.475743i
\(844\) −93.4016 −3.21502
\(845\) 0 0
\(846\) −17.7778 −0.611212
\(847\) 7.20237i 0.247477i
\(848\) − 163.510i − 5.61497i
\(849\) 7.53750 0.258686
\(850\) 0 0
\(851\) −1.38835 −0.0475922
\(852\) 8.77240i 0.300537i
\(853\) − 7.16959i − 0.245482i −0.992439 0.122741i \(-0.960832\pi\)
0.992439 0.122741i \(-0.0391684\pi\)
\(854\) −85.0936 −2.91184
\(855\) 0 0
\(856\) −88.3256 −3.01891
\(857\) 2.55124i 0.0871486i 0.999050 + 0.0435743i \(0.0138745\pi\)
−0.999050 + 0.0435743i \(0.986125\pi\)
\(858\) 14.8388i 0.506587i
\(859\) −53.9493 −1.84073 −0.920363 0.391065i \(-0.872107\pi\)
−0.920363 + 0.391065i \(0.872107\pi\)
\(860\) 0 0
\(861\) −14.5007 −0.494181
\(862\) 4.69873i 0.160039i
\(863\) − 21.4698i − 0.730840i −0.930843 0.365420i \(-0.880925\pi\)
0.930843 0.365420i \(-0.119075\pi\)
\(864\) 95.8467 3.26077
\(865\) 0 0
\(866\) −36.8582 −1.25249
\(867\) − 6.84787i − 0.232566i
\(868\) − 166.700i − 5.65817i
\(869\) 16.6491 0.564781
\(870\) 0 0
\(871\) 16.1782 0.548178
\(872\) − 122.221i − 4.13891i
\(873\) 8.74094i 0.295836i
\(874\) 3.35690 0.113549
\(875\) 0 0
\(876\) 44.7362 1.51149
\(877\) − 38.9638i − 1.31571i −0.753144 0.657856i \(-0.771463\pi\)
0.753144 0.657856i \(-0.228537\pi\)
\(878\) − 18.0887i − 0.610465i
\(879\) −0.708415 −0.0238942
\(880\) 0 0
\(881\) −13.4373 −0.452713 −0.226357 0.974045i \(-0.572681\pi\)
−0.226357 + 0.974045i \(0.572681\pi\)
\(882\) − 17.3177i − 0.583116i
\(883\) 27.8799i 0.938234i 0.883136 + 0.469117i \(0.155428\pi\)
−0.883136 + 0.469117i \(0.844572\pi\)
\(884\) 41.0127 1.37941
\(885\) 0 0
\(886\) −71.0877 −2.38824
\(887\) − 28.9135i − 0.970821i −0.874286 0.485410i \(-0.838670\pi\)
0.874286 0.485410i \(-0.161330\pi\)
\(888\) 6.93900i 0.232858i
\(889\) 4.45580 0.149443
\(890\) 0 0
\(891\) −18.5834 −0.622568
\(892\) − 135.828i − 4.54785i
\(893\) 2.35690i 0.0788705i
\(894\) −17.2040 −0.575389
\(895\) 0 0
\(896\) −224.659 −7.50533
\(897\) 2.15883i 0.0720814i
\(898\) 10.0858i 0.336566i
\(899\) −16.5851 −0.553144
\(900\) 0 0
\(901\) −19.0489 −0.634611
\(902\) − 70.5733i − 2.34983i
\(903\) 8.92394i 0.296970i
\(904\) −37.0670 −1.23283
\(905\) 0 0
\(906\) 16.2379 0.539467
\(907\) 30.7127i 1.01980i 0.860234 + 0.509900i \(0.170317\pi\)
−0.860234 + 0.509900i \(0.829683\pi\)
\(908\) 102.270i 3.39395i
\(909\) 13.7090 0.454699
\(910\) 0 0
\(911\) 39.3690 1.30435 0.652176 0.758067i \(-0.273856\pi\)
0.652176 + 0.758067i \(0.273856\pi\)
\(912\) − 10.2838i − 0.340531i
\(913\) − 8.84548i − 0.292743i
\(914\) 34.4034 1.13796
\(915\) 0 0
\(916\) −13.6058 −0.449548
\(917\) 40.3357i 1.33200i
\(918\) − 19.1075i − 0.630642i
\(919\) 7.87023 0.259615 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(920\) 0 0
\(921\) −17.8860 −0.589365
\(922\) 110.658i 3.64434i
\(923\) − 8.77240i − 0.288747i
\(924\) −29.0954 −0.957170
\(925\) 0 0
\(926\) 51.5532 1.69414
\(927\) 38.5254i 1.26534i
\(928\) 53.8514i 1.76776i
\(929\) −39.2924 −1.28914 −0.644572 0.764544i \(-0.722964\pi\)
−0.644572 + 0.764544i \(0.722964\pi\)
\(930\) 0 0
\(931\) −2.29590 −0.0752450
\(932\) − 110.781i − 3.62876i
\(933\) − 6.93602i − 0.227075i
\(934\) 101.509 3.32148
\(935\) 0 0
\(936\) −94.3135 −3.08273
\(937\) − 26.7554i − 0.874061i −0.899447 0.437031i \(-0.856030\pi\)
0.899447 0.437031i \(-0.143970\pi\)
\(938\) 42.5652i 1.38980i
\(939\) 11.6203 0.379213
\(940\) 0 0
\(941\) −41.9191 −1.36653 −0.683263 0.730173i \(-0.739440\pi\)
−0.683263 + 0.730173i \(0.739440\pi\)
\(942\) 3.86294i 0.125861i
\(943\) − 10.2674i − 0.334353i
\(944\) 105.657 3.43885
\(945\) 0 0
\(946\) −43.4319 −1.41209
\(947\) 55.4868i 1.80308i 0.432698 + 0.901539i \(0.357562\pi\)
−0.432698 + 0.901539i \(0.642438\pi\)
\(948\) − 18.3937i − 0.597401i
\(949\) −44.7362 −1.45220
\(950\) 0 0
\(951\) 8.79954 0.285345
\(952\) 71.0200i 2.30177i
\(953\) − 8.48081i − 0.274720i −0.990521 0.137360i \(-0.956138\pi\)
0.990521 0.137360i \(-0.0438618\pi\)
\(954\) 66.5561 2.15483
\(955\) 0 0
\(956\) 65.3454 2.11342
\(957\) 2.89472i 0.0935731i
\(958\) 96.1426i 3.10623i
\(959\) 49.3129 1.59240
\(960\) 0 0
\(961\) 56.3256 1.81695
\(962\) − 10.5429i − 0.339916i
\(963\) − 22.0368i − 0.710127i
\(964\) −43.4717 −1.40013
\(965\) 0 0
\(966\) −5.67994 −0.182749
\(967\) − 7.63102i − 0.245397i −0.992444 0.122699i \(-0.960845\pi\)
0.992444 0.122699i \(-0.0391548\pi\)
\(968\) 25.4886i 0.819234i
\(969\) −1.19806 −0.0384873
\(970\) 0 0
\(971\) −19.3599 −0.621288 −0.310644 0.950526i \(-0.600545\pi\)
−0.310644 + 0.950526i \(0.600545\pi\)
\(972\) 75.9764i 2.43695i
\(973\) 39.9342i 1.28023i
\(974\) 68.5695 2.19711
\(975\) 0 0
\(976\) −184.581 −5.90829
\(977\) 55.8883i 1.78802i 0.448042 + 0.894012i \(0.352121\pi\)
−0.448042 + 0.894012i \(0.647879\pi\)
\(978\) − 4.92692i − 0.157546i
\(979\) −30.2241 −0.965968
\(980\) 0 0
\(981\) 30.4935 0.973582
\(982\) 51.2941i 1.63686i
\(983\) 4.58450i 0.146223i 0.997324 + 0.0731114i \(0.0232929\pi\)
−0.997324 + 0.0731114i \(0.976707\pi\)
\(984\) −51.3166 −1.63591
\(985\) 0 0
\(986\) 10.7356 0.341890
\(987\) − 3.98792i − 0.126937i
\(988\) 18.9976i 0.604394i
\(989\) −6.31873 −0.200924
\(990\) 0 0
\(991\) 27.5666 0.875681 0.437840 0.899053i \(-0.355743\pi\)
0.437840 + 0.899053i \(0.355743\pi\)
\(992\) − 283.544i − 9.00254i
\(993\) 7.34481i 0.233081i
\(994\) 23.0804 0.732065
\(995\) 0 0
\(996\) −9.77240 −0.309650
\(997\) 15.9119i 0.503933i 0.967736 + 0.251967i \(0.0810774\pi\)
−0.967736 + 0.251967i \(0.918923\pi\)
\(998\) 71.0079i 2.24772i
\(999\) −3.66056 −0.115815
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 475.2.b.c.324.6 6
5.2 odd 4 475.2.a.d.1.1 3
5.3 odd 4 475.2.a.h.1.3 yes 3
5.4 even 2 inner 475.2.b.c.324.1 6
15.2 even 4 4275.2.a.bn.1.3 3
15.8 even 4 4275.2.a.z.1.1 3
20.3 even 4 7600.2.a.bn.1.2 3
20.7 even 4 7600.2.a.bw.1.2 3
95.18 even 4 9025.2.a.w.1.1 3
95.37 even 4 9025.2.a.be.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.1 3 5.2 odd 4
475.2.a.h.1.3 yes 3 5.3 odd 4
475.2.b.c.324.1 6 5.4 even 2 inner
475.2.b.c.324.6 6 1.1 even 1 trivial
4275.2.a.z.1.1 3 15.8 even 4
4275.2.a.bn.1.3 3 15.2 even 4
7600.2.a.bn.1.2 3 20.3 even 4
7600.2.a.bw.1.2 3 20.7 even 4
9025.2.a.w.1.1 3 95.18 even 4
9025.2.a.be.1.3 3 95.37 even 4