Properties

Label 4925.2.a.k.1.5
Level $4925$
Weight $2$
Character 4925.1
Self dual yes
Analytic conductor $39.326$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4925,2,Mod(1,4925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4925 = 5^{2} \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4925.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.3263229955\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - x^{12} + 66 x^{11} - 83 x^{10} - 218 x^{9} + 446 x^{8} + 178 x^{7} - 782 x^{6} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 985)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.236045\) of defining polynomial
Character \(\chi\) \(=\) 4925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.236045 q^{2} +2.87112 q^{3} -1.94428 q^{4} -0.677712 q^{6} -1.02221 q^{7} +0.931027 q^{8} +5.24331 q^{9} +0.916844 q^{11} -5.58226 q^{12} -4.99719 q^{13} +0.241286 q^{14} +3.66880 q^{16} -0.345734 q^{17} -1.23766 q^{18} -5.30778 q^{19} -2.93487 q^{21} -0.216416 q^{22} +7.21308 q^{23} +2.67309 q^{24} +1.17956 q^{26} +6.44081 q^{27} +1.98746 q^{28} +4.17022 q^{29} +9.99162 q^{31} -2.72806 q^{32} +2.63237 q^{33} +0.0816088 q^{34} -10.1945 q^{36} -2.64110 q^{37} +1.25287 q^{38} -14.3475 q^{39} -9.80072 q^{41} +0.692761 q^{42} +12.7545 q^{43} -1.78260 q^{44} -1.70261 q^{46} +8.43174 q^{47} +10.5336 q^{48} -5.95510 q^{49} -0.992644 q^{51} +9.71595 q^{52} +3.66822 q^{53} -1.52032 q^{54} -0.951701 q^{56} -15.2393 q^{57} -0.984359 q^{58} +4.19018 q^{59} +2.70353 q^{61} -2.35847 q^{62} -5.35974 q^{63} -6.69366 q^{64} -0.621356 q^{66} -9.72899 q^{67} +0.672205 q^{68} +20.7096 q^{69} +0.435154 q^{71} +4.88167 q^{72} +6.14822 q^{73} +0.623419 q^{74} +10.3198 q^{76} -0.937203 q^{77} +3.38665 q^{78} -4.36466 q^{79} +2.76239 q^{81} +2.31341 q^{82} +9.26666 q^{83} +5.70622 q^{84} -3.01064 q^{86} +11.9732 q^{87} +0.853607 q^{88} +14.2366 q^{89} +5.10815 q^{91} -14.0243 q^{92} +28.6871 q^{93} -1.99027 q^{94} -7.83257 q^{96} +11.0151 q^{97} +1.40567 q^{98} +4.80730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 5 q^{6} + 4 q^{7} + 12 q^{8} + 13 q^{9} - 5 q^{11} + 10 q^{12} + 15 q^{13} - 15 q^{14} + 18 q^{16} + 18 q^{17} + 29 q^{18} + 20 q^{19} - 5 q^{21} - 12 q^{22} + 42 q^{23}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.236045 −0.166909 −0.0834544 0.996512i \(-0.526595\pi\)
−0.0834544 + 0.996512i \(0.526595\pi\)
\(3\) 2.87112 1.65764 0.828820 0.559515i \(-0.189013\pi\)
0.828820 + 0.559515i \(0.189013\pi\)
\(4\) −1.94428 −0.972141
\(5\) 0 0
\(6\) −0.677712 −0.276675
\(7\) −1.02221 −0.386357 −0.193179 0.981164i \(-0.561880\pi\)
−0.193179 + 0.981164i \(0.561880\pi\)
\(8\) 0.931027 0.329168
\(9\) 5.24331 1.74777
\(10\) 0 0
\(11\) 0.916844 0.276439 0.138219 0.990402i \(-0.455862\pi\)
0.138219 + 0.990402i \(0.455862\pi\)
\(12\) −5.58226 −1.61146
\(13\) −4.99719 −1.38597 −0.692985 0.720952i \(-0.743705\pi\)
−0.692985 + 0.720952i \(0.743705\pi\)
\(14\) 0.241286 0.0644864
\(15\) 0 0
\(16\) 3.66880 0.917200
\(17\) −0.345734 −0.0838529 −0.0419264 0.999121i \(-0.513350\pi\)
−0.0419264 + 0.999121i \(0.513350\pi\)
\(18\) −1.23766 −0.291718
\(19\) −5.30778 −1.21769 −0.608845 0.793290i \(-0.708367\pi\)
−0.608845 + 0.793290i \(0.708367\pi\)
\(20\) 0 0
\(21\) −2.93487 −0.640441
\(22\) −0.216416 −0.0461401
\(23\) 7.21308 1.50403 0.752015 0.659146i \(-0.229082\pi\)
0.752015 + 0.659146i \(0.229082\pi\)
\(24\) 2.67309 0.545642
\(25\) 0 0
\(26\) 1.17956 0.231331
\(27\) 6.44081 1.23953
\(28\) 1.98746 0.375594
\(29\) 4.17022 0.774391 0.387195 0.921998i \(-0.373444\pi\)
0.387195 + 0.921998i \(0.373444\pi\)
\(30\) 0 0
\(31\) 9.99162 1.79455 0.897274 0.441474i \(-0.145544\pi\)
0.897274 + 0.441474i \(0.145544\pi\)
\(32\) −2.72806 −0.482257
\(33\) 2.63237 0.458236
\(34\) 0.0816088 0.0139958
\(35\) 0 0
\(36\) −10.1945 −1.69908
\(37\) −2.64110 −0.434195 −0.217097 0.976150i \(-0.569659\pi\)
−0.217097 + 0.976150i \(0.569659\pi\)
\(38\) 1.25287 0.203243
\(39\) −14.3475 −2.29744
\(40\) 0 0
\(41\) −9.80072 −1.53061 −0.765307 0.643665i \(-0.777413\pi\)
−0.765307 + 0.643665i \(0.777413\pi\)
\(42\) 0.692761 0.106895
\(43\) 12.7545 1.94505 0.972524 0.232804i \(-0.0747899\pi\)
0.972524 + 0.232804i \(0.0747899\pi\)
\(44\) −1.78260 −0.268738
\(45\) 0 0
\(46\) −1.70261 −0.251036
\(47\) 8.43174 1.22990 0.614948 0.788568i \(-0.289177\pi\)
0.614948 + 0.788568i \(0.289177\pi\)
\(48\) 10.5336 1.52039
\(49\) −5.95510 −0.850728
\(50\) 0 0
\(51\) −0.992644 −0.138998
\(52\) 9.71595 1.34736
\(53\) 3.66822 0.503869 0.251935 0.967744i \(-0.418933\pi\)
0.251935 + 0.967744i \(0.418933\pi\)
\(54\) −1.52032 −0.206889
\(55\) 0 0
\(56\) −0.951701 −0.127176
\(57\) −15.2393 −2.01849
\(58\) −0.984359 −0.129253
\(59\) 4.19018 0.545515 0.272757 0.962083i \(-0.412064\pi\)
0.272757 + 0.962083i \(0.412064\pi\)
\(60\) 0 0
\(61\) 2.70353 0.346152 0.173076 0.984908i \(-0.444629\pi\)
0.173076 + 0.984908i \(0.444629\pi\)
\(62\) −2.35847 −0.299526
\(63\) −5.35974 −0.675264
\(64\) −6.69366 −0.836708
\(65\) 0 0
\(66\) −0.621356 −0.0764837
\(67\) −9.72899 −1.18859 −0.594293 0.804249i \(-0.702568\pi\)
−0.594293 + 0.804249i \(0.702568\pi\)
\(68\) 0.672205 0.0815169
\(69\) 20.7096 2.49314
\(70\) 0 0
\(71\) 0.435154 0.0516432 0.0258216 0.999667i \(-0.491780\pi\)
0.0258216 + 0.999667i \(0.491780\pi\)
\(72\) 4.88167 0.575310
\(73\) 6.14822 0.719595 0.359797 0.933030i \(-0.382846\pi\)
0.359797 + 0.933030i \(0.382846\pi\)
\(74\) 0.623419 0.0724709
\(75\) 0 0
\(76\) 10.3198 1.18377
\(77\) −0.937203 −0.106804
\(78\) 3.38665 0.383463
\(79\) −4.36466 −0.491063 −0.245531 0.969389i \(-0.578962\pi\)
−0.245531 + 0.969389i \(0.578962\pi\)
\(80\) 0 0
\(81\) 2.76239 0.306932
\(82\) 2.31341 0.255473
\(83\) 9.26666 1.01715 0.508574 0.861018i \(-0.330173\pi\)
0.508574 + 0.861018i \(0.330173\pi\)
\(84\) 5.70622 0.622600
\(85\) 0 0
\(86\) −3.01064 −0.324646
\(87\) 11.9732 1.28366
\(88\) 0.853607 0.0909948
\(89\) 14.2366 1.50908 0.754538 0.656256i \(-0.227861\pi\)
0.754538 + 0.656256i \(0.227861\pi\)
\(90\) 0 0
\(91\) 5.10815 0.535480
\(92\) −14.0243 −1.46213
\(93\) 28.6871 2.97471
\(94\) −1.99027 −0.205281
\(95\) 0 0
\(96\) −7.83257 −0.799408
\(97\) 11.0151 1.11842 0.559209 0.829027i \(-0.311105\pi\)
0.559209 + 0.829027i \(0.311105\pi\)
\(98\) 1.40567 0.141994
\(99\) 4.80730 0.483152
\(100\) 0 0
\(101\) 11.5252 1.14680 0.573400 0.819276i \(-0.305624\pi\)
0.573400 + 0.819276i \(0.305624\pi\)
\(102\) 0.234308 0.0232000
\(103\) −4.12363 −0.406313 −0.203157 0.979146i \(-0.565120\pi\)
−0.203157 + 0.979146i \(0.565120\pi\)
\(104\) −4.65252 −0.456217
\(105\) 0 0
\(106\) −0.865864 −0.0841002
\(107\) 10.2697 0.992805 0.496402 0.868092i \(-0.334654\pi\)
0.496402 + 0.868092i \(0.334654\pi\)
\(108\) −12.5228 −1.20500
\(109\) 19.6212 1.87937 0.939685 0.342041i \(-0.111118\pi\)
0.939685 + 0.342041i \(0.111118\pi\)
\(110\) 0 0
\(111\) −7.58292 −0.719739
\(112\) −3.75027 −0.354367
\(113\) 14.9685 1.40812 0.704058 0.710142i \(-0.251369\pi\)
0.704058 + 0.710142i \(0.251369\pi\)
\(114\) 3.59715 0.336904
\(115\) 0 0
\(116\) −8.10809 −0.752817
\(117\) −26.2018 −2.42236
\(118\) −0.989070 −0.0910512
\(119\) 0.353411 0.0323972
\(120\) 0 0
\(121\) −10.1594 −0.923582
\(122\) −0.638154 −0.0577758
\(123\) −28.1390 −2.53721
\(124\) −19.4265 −1.74455
\(125\) 0 0
\(126\) 1.26514 0.112708
\(127\) −5.21953 −0.463159 −0.231579 0.972816i \(-0.574389\pi\)
−0.231579 + 0.972816i \(0.574389\pi\)
\(128\) 7.03611 0.621910
\(129\) 36.6197 3.22419
\(130\) 0 0
\(131\) −1.76360 −0.154087 −0.0770433 0.997028i \(-0.524548\pi\)
−0.0770433 + 0.997028i \(0.524548\pi\)
\(132\) −5.11807 −0.445470
\(133\) 5.42564 0.470463
\(134\) 2.29648 0.198385
\(135\) 0 0
\(136\) −0.321888 −0.0276017
\(137\) 14.9252 1.27514 0.637571 0.770392i \(-0.279939\pi\)
0.637571 + 0.770392i \(0.279939\pi\)
\(138\) −4.88839 −0.416127
\(139\) −10.6367 −0.902195 −0.451097 0.892475i \(-0.648967\pi\)
−0.451097 + 0.892475i \(0.648967\pi\)
\(140\) 0 0
\(141\) 24.2085 2.03873
\(142\) −0.102716 −0.00861971
\(143\) −4.58164 −0.383136
\(144\) 19.2367 1.60306
\(145\) 0 0
\(146\) −1.45125 −0.120107
\(147\) −17.0978 −1.41020
\(148\) 5.13505 0.422099
\(149\) −7.93007 −0.649656 −0.324828 0.945773i \(-0.605306\pi\)
−0.324828 + 0.945773i \(0.605306\pi\)
\(150\) 0 0
\(151\) −23.5423 −1.91584 −0.957921 0.287033i \(-0.907331\pi\)
−0.957921 + 0.287033i \(0.907331\pi\)
\(152\) −4.94169 −0.400824
\(153\) −1.81279 −0.146556
\(154\) 0.221222 0.0178266
\(155\) 0 0
\(156\) 27.8956 2.23344
\(157\) −6.13158 −0.489353 −0.244677 0.969605i \(-0.578682\pi\)
−0.244677 + 0.969605i \(0.578682\pi\)
\(158\) 1.03025 0.0819627
\(159\) 10.5319 0.835234
\(160\) 0 0
\(161\) −7.37325 −0.581093
\(162\) −0.652047 −0.0512297
\(163\) 7.33018 0.574144 0.287072 0.957909i \(-0.407318\pi\)
0.287072 + 0.957909i \(0.407318\pi\)
\(164\) 19.0554 1.48797
\(165\) 0 0
\(166\) −2.18735 −0.169771
\(167\) 10.1434 0.784921 0.392461 0.919769i \(-0.371624\pi\)
0.392461 + 0.919769i \(0.371624\pi\)
\(168\) −2.73244 −0.210813
\(169\) 11.9719 0.920915
\(170\) 0 0
\(171\) −27.8304 −2.12824
\(172\) −24.7984 −1.89086
\(173\) −2.03837 −0.154975 −0.0774873 0.996993i \(-0.524690\pi\)
−0.0774873 + 0.996993i \(0.524690\pi\)
\(174\) −2.82621 −0.214254
\(175\) 0 0
\(176\) 3.36372 0.253550
\(177\) 12.0305 0.904267
\(178\) −3.36047 −0.251878
\(179\) 0.681587 0.0509442 0.0254721 0.999676i \(-0.491891\pi\)
0.0254721 + 0.999676i \(0.491891\pi\)
\(180\) 0 0
\(181\) 3.58428 0.266417 0.133209 0.991088i \(-0.457472\pi\)
0.133209 + 0.991088i \(0.457472\pi\)
\(182\) −1.20575 −0.0893763
\(183\) 7.76216 0.573795
\(184\) 6.71557 0.495078
\(185\) 0 0
\(186\) −6.77144 −0.496506
\(187\) −0.316985 −0.0231802
\(188\) −16.3937 −1.19563
\(189\) −6.58383 −0.478903
\(190\) 0 0
\(191\) 15.4485 1.11782 0.558908 0.829229i \(-0.311220\pi\)
0.558908 + 0.829229i \(0.311220\pi\)
\(192\) −19.2183 −1.38696
\(193\) 4.87672 0.351034 0.175517 0.984476i \(-0.443840\pi\)
0.175517 + 0.984476i \(0.443840\pi\)
\(194\) −2.60006 −0.186674
\(195\) 0 0
\(196\) 11.5784 0.827028
\(197\) 1.00000 0.0712470
\(198\) −1.13474 −0.0806423
\(199\) −20.0659 −1.42244 −0.711218 0.702971i \(-0.751856\pi\)
−0.711218 + 0.702971i \(0.751856\pi\)
\(200\) 0 0
\(201\) −27.9331 −1.97025
\(202\) −2.72046 −0.191411
\(203\) −4.26282 −0.299192
\(204\) 1.92998 0.135126
\(205\) 0 0
\(206\) 0.973361 0.0678173
\(207\) 37.8204 2.62870
\(208\) −18.3337 −1.27121
\(209\) −4.86641 −0.336617
\(210\) 0 0
\(211\) −4.29538 −0.295706 −0.147853 0.989009i \(-0.547236\pi\)
−0.147853 + 0.989009i \(0.547236\pi\)
\(212\) −7.13206 −0.489832
\(213\) 1.24938 0.0856059
\(214\) −2.42410 −0.165708
\(215\) 0 0
\(216\) 5.99657 0.408015
\(217\) −10.2135 −0.693337
\(218\) −4.63148 −0.313683
\(219\) 17.6523 1.19283
\(220\) 0 0
\(221\) 1.72770 0.116218
\(222\) 1.78991 0.120131
\(223\) −13.6428 −0.913589 −0.456795 0.889572i \(-0.651003\pi\)
−0.456795 + 0.889572i \(0.651003\pi\)
\(224\) 2.78863 0.186323
\(225\) 0 0
\(226\) −3.53323 −0.235027
\(227\) 8.32757 0.552720 0.276360 0.961054i \(-0.410872\pi\)
0.276360 + 0.961054i \(0.410872\pi\)
\(228\) 29.6294 1.96226
\(229\) −27.7853 −1.83611 −0.918053 0.396458i \(-0.870239\pi\)
−0.918053 + 0.396458i \(0.870239\pi\)
\(230\) 0 0
\(231\) −2.69082 −0.177043
\(232\) 3.88259 0.254904
\(233\) 15.7592 1.03242 0.516211 0.856462i \(-0.327342\pi\)
0.516211 + 0.856462i \(0.327342\pi\)
\(234\) 6.18480 0.404313
\(235\) 0 0
\(236\) −8.14689 −0.530318
\(237\) −12.5314 −0.814005
\(238\) −0.0834209 −0.00540737
\(239\) −26.6777 −1.72564 −0.862818 0.505515i \(-0.831303\pi\)
−0.862818 + 0.505515i \(0.831303\pi\)
\(240\) 0 0
\(241\) 7.59582 0.489290 0.244645 0.969613i \(-0.421329\pi\)
0.244645 + 0.969613i \(0.421329\pi\)
\(242\) 2.39807 0.154154
\(243\) −11.3913 −0.730752
\(244\) −5.25643 −0.336509
\(245\) 0 0
\(246\) 6.64206 0.423482
\(247\) 26.5240 1.68768
\(248\) 9.30247 0.590707
\(249\) 26.6057 1.68607
\(250\) 0 0
\(251\) −10.2993 −0.650088 −0.325044 0.945699i \(-0.605379\pi\)
−0.325044 + 0.945699i \(0.605379\pi\)
\(252\) 10.4209 0.656452
\(253\) 6.61327 0.415773
\(254\) 1.23204 0.0773053
\(255\) 0 0
\(256\) 11.7265 0.732905
\(257\) 16.8749 1.05263 0.526315 0.850290i \(-0.323573\pi\)
0.526315 + 0.850290i \(0.323573\pi\)
\(258\) −8.64390 −0.538146
\(259\) 2.69975 0.167754
\(260\) 0 0
\(261\) 21.8658 1.35346
\(262\) 0.416289 0.0257184
\(263\) −8.57005 −0.528452 −0.264226 0.964461i \(-0.585116\pi\)
−0.264226 + 0.964461i \(0.585116\pi\)
\(264\) 2.45080 0.150837
\(265\) 0 0
\(266\) −1.28069 −0.0785244
\(267\) 40.8749 2.50151
\(268\) 18.9159 1.15547
\(269\) 26.3927 1.60919 0.804596 0.593822i \(-0.202382\pi\)
0.804596 + 0.593822i \(0.202382\pi\)
\(270\) 0 0
\(271\) 27.5722 1.67489 0.837447 0.546518i \(-0.184047\pi\)
0.837447 + 0.546518i \(0.184047\pi\)
\(272\) −1.26843 −0.0769099
\(273\) 14.6661 0.887633
\(274\) −3.52300 −0.212832
\(275\) 0 0
\(276\) −40.2653 −2.42369
\(277\) −9.89475 −0.594518 −0.297259 0.954797i \(-0.596072\pi\)
−0.297259 + 0.954797i \(0.596072\pi\)
\(278\) 2.51074 0.150584
\(279\) 52.3892 3.13646
\(280\) 0 0
\(281\) 27.3790 1.63330 0.816648 0.577135i \(-0.195830\pi\)
0.816648 + 0.577135i \(0.195830\pi\)
\(282\) −5.71429 −0.340281
\(283\) 7.11007 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(284\) −0.846062 −0.0502045
\(285\) 0 0
\(286\) 1.08147 0.0639488
\(287\) 10.0183 0.591364
\(288\) −14.3040 −0.842874
\(289\) −16.8805 −0.992969
\(290\) 0 0
\(291\) 31.6257 1.85393
\(292\) −11.9539 −0.699548
\(293\) −15.7053 −0.917512 −0.458756 0.888562i \(-0.651705\pi\)
−0.458756 + 0.888562i \(0.651705\pi\)
\(294\) 4.03584 0.235375
\(295\) 0 0
\(296\) −2.45894 −0.142923
\(297\) 5.90522 0.342656
\(298\) 1.87185 0.108433
\(299\) −36.0451 −2.08454
\(300\) 0 0
\(301\) −13.0377 −0.751483
\(302\) 5.55702 0.319771
\(303\) 33.0902 1.90098
\(304\) −19.4732 −1.11686
\(305\) 0 0
\(306\) 0.427900 0.0244614
\(307\) 8.97401 0.512174 0.256087 0.966654i \(-0.417567\pi\)
0.256087 + 0.966654i \(0.417567\pi\)
\(308\) 1.82219 0.103829
\(309\) −11.8394 −0.673521
\(310\) 0 0
\(311\) −9.76960 −0.553983 −0.276992 0.960872i \(-0.589337\pi\)
−0.276992 + 0.960872i \(0.589337\pi\)
\(312\) −13.3579 −0.756243
\(313\) 0.449986 0.0254347 0.0127174 0.999919i \(-0.495952\pi\)
0.0127174 + 0.999919i \(0.495952\pi\)
\(314\) 1.44733 0.0816773
\(315\) 0 0
\(316\) 8.48613 0.477382
\(317\) 27.2882 1.53266 0.766330 0.642447i \(-0.222081\pi\)
0.766330 + 0.642447i \(0.222081\pi\)
\(318\) −2.48600 −0.139408
\(319\) 3.82344 0.214072
\(320\) 0 0
\(321\) 29.4854 1.64571
\(322\) 1.74042 0.0969896
\(323\) 1.83508 0.102107
\(324\) −5.37087 −0.298381
\(325\) 0 0
\(326\) −1.73025 −0.0958297
\(327\) 56.3348 3.11532
\(328\) −9.12473 −0.503829
\(329\) −8.61897 −0.475179
\(330\) 0 0
\(331\) 4.69448 0.258032 0.129016 0.991643i \(-0.458818\pi\)
0.129016 + 0.991643i \(0.458818\pi\)
\(332\) −18.0170 −0.988812
\(333\) −13.8481 −0.758873
\(334\) −2.39430 −0.131010
\(335\) 0 0
\(336\) −10.7675 −0.587413
\(337\) 1.18729 0.0646760 0.0323380 0.999477i \(-0.489705\pi\)
0.0323380 + 0.999477i \(0.489705\pi\)
\(338\) −2.82590 −0.153709
\(339\) 42.9763 2.33415
\(340\) 0 0
\(341\) 9.16076 0.496083
\(342\) 6.56921 0.355222
\(343\) 13.2428 0.715042
\(344\) 11.8748 0.640247
\(345\) 0 0
\(346\) 0.481147 0.0258666
\(347\) −2.21480 −0.118896 −0.0594482 0.998231i \(-0.518934\pi\)
−0.0594482 + 0.998231i \(0.518934\pi\)
\(348\) −23.2793 −1.24790
\(349\) 22.6828 1.21418 0.607092 0.794632i \(-0.292336\pi\)
0.607092 + 0.794632i \(0.292336\pi\)
\(350\) 0 0
\(351\) −32.1860 −1.71796
\(352\) −2.50120 −0.133315
\(353\) 12.2691 0.653016 0.326508 0.945194i \(-0.394128\pi\)
0.326508 + 0.945194i \(0.394128\pi\)
\(354\) −2.83973 −0.150930
\(355\) 0 0
\(356\) −27.6800 −1.46704
\(357\) 1.01469 0.0537029
\(358\) −0.160885 −0.00850304
\(359\) −18.9950 −1.00252 −0.501258 0.865298i \(-0.667129\pi\)
−0.501258 + 0.865298i \(0.667129\pi\)
\(360\) 0 0
\(361\) 9.17256 0.482767
\(362\) −0.846050 −0.0444674
\(363\) −29.1688 −1.53097
\(364\) −9.93169 −0.520562
\(365\) 0 0
\(366\) −1.83222 −0.0957715
\(367\) −35.4991 −1.85304 −0.926519 0.376249i \(-0.877214\pi\)
−0.926519 + 0.376249i \(0.877214\pi\)
\(368\) 26.4633 1.37950
\(369\) −51.3882 −2.67516
\(370\) 0 0
\(371\) −3.74968 −0.194673
\(372\) −55.7759 −2.89184
\(373\) −22.1497 −1.14687 −0.573434 0.819252i \(-0.694389\pi\)
−0.573434 + 0.819252i \(0.694389\pi\)
\(374\) 0.0748225 0.00386898
\(375\) 0 0
\(376\) 7.85018 0.404842
\(377\) −20.8394 −1.07328
\(378\) 1.55408 0.0799332
\(379\) −0.0471576 −0.00242232 −0.00121116 0.999999i \(-0.500386\pi\)
−0.00121116 + 0.999999i \(0.500386\pi\)
\(380\) 0 0
\(381\) −14.9859 −0.767750
\(382\) −3.64654 −0.186573
\(383\) 27.8084 1.42094 0.710470 0.703727i \(-0.248482\pi\)
0.710470 + 0.703727i \(0.248482\pi\)
\(384\) 20.2015 1.03090
\(385\) 0 0
\(386\) −1.15112 −0.0585907
\(387\) 66.8760 3.39950
\(388\) −21.4165 −1.08726
\(389\) −5.91034 −0.299666 −0.149833 0.988711i \(-0.547874\pi\)
−0.149833 + 0.988711i \(0.547874\pi\)
\(390\) 0 0
\(391\) −2.49381 −0.126117
\(392\) −5.54436 −0.280032
\(393\) −5.06351 −0.255420
\(394\) −0.236045 −0.0118918
\(395\) 0 0
\(396\) −9.34675 −0.469692
\(397\) 10.7928 0.541673 0.270836 0.962625i \(-0.412700\pi\)
0.270836 + 0.962625i \(0.412700\pi\)
\(398\) 4.73646 0.237417
\(399\) 15.5777 0.779858
\(400\) 0 0
\(401\) −25.3129 −1.26407 −0.632034 0.774941i \(-0.717780\pi\)
−0.632034 + 0.774941i \(0.717780\pi\)
\(402\) 6.59345 0.328851
\(403\) −49.9300 −2.48719
\(404\) −22.4082 −1.11485
\(405\) 0 0
\(406\) 1.00622 0.0499377
\(407\) −2.42148 −0.120028
\(408\) −0.924178 −0.0457536
\(409\) 11.0910 0.548415 0.274207 0.961671i \(-0.411585\pi\)
0.274207 + 0.961671i \(0.411585\pi\)
\(410\) 0 0
\(411\) 42.8519 2.11373
\(412\) 8.01750 0.394994
\(413\) −4.28322 −0.210764
\(414\) −8.92731 −0.438753
\(415\) 0 0
\(416\) 13.6326 0.668394
\(417\) −30.5393 −1.49551
\(418\) 1.14869 0.0561843
\(419\) −29.8386 −1.45771 −0.728856 0.684667i \(-0.759947\pi\)
−0.728856 + 0.684667i \(0.759947\pi\)
\(420\) 0 0
\(421\) −31.3685 −1.52881 −0.764403 0.644739i \(-0.776966\pi\)
−0.764403 + 0.644739i \(0.776966\pi\)
\(422\) 1.01390 0.0493560
\(423\) 44.2103 2.14958
\(424\) 3.41521 0.165857
\(425\) 0 0
\(426\) −0.294909 −0.0142884
\(427\) −2.76356 −0.133738
\(428\) −19.9671 −0.965147
\(429\) −13.1544 −0.635102
\(430\) 0 0
\(431\) −28.4126 −1.36859 −0.684293 0.729207i \(-0.739889\pi\)
−0.684293 + 0.729207i \(0.739889\pi\)
\(432\) 23.6301 1.13690
\(433\) −29.6797 −1.42631 −0.713157 0.701005i \(-0.752735\pi\)
−0.713157 + 0.701005i \(0.752735\pi\)
\(434\) 2.41084 0.115724
\(435\) 0 0
\(436\) −38.1492 −1.82701
\(437\) −38.2855 −1.83144
\(438\) −4.16672 −0.199094
\(439\) 19.0177 0.907664 0.453832 0.891087i \(-0.350057\pi\)
0.453832 + 0.891087i \(0.350057\pi\)
\(440\) 0 0
\(441\) −31.2244 −1.48688
\(442\) −0.407814 −0.0193977
\(443\) 12.5365 0.595628 0.297814 0.954624i \(-0.403742\pi\)
0.297814 + 0.954624i \(0.403742\pi\)
\(444\) 14.7433 0.699688
\(445\) 0 0
\(446\) 3.22031 0.152486
\(447\) −22.7681 −1.07690
\(448\) 6.84230 0.323268
\(449\) 17.9879 0.848900 0.424450 0.905451i \(-0.360467\pi\)
0.424450 + 0.905451i \(0.360467\pi\)
\(450\) 0 0
\(451\) −8.98573 −0.423122
\(452\) −29.1030 −1.36889
\(453\) −67.5926 −3.17578
\(454\) −1.96568 −0.0922539
\(455\) 0 0
\(456\) −14.1882 −0.664422
\(457\) 27.3085 1.27744 0.638718 0.769441i \(-0.279465\pi\)
0.638718 + 0.769441i \(0.279465\pi\)
\(458\) 6.55858 0.306462
\(459\) −2.22681 −0.103939
\(460\) 0 0
\(461\) −8.85519 −0.412427 −0.206214 0.978507i \(-0.566114\pi\)
−0.206214 + 0.978507i \(0.566114\pi\)
\(462\) 0.635154 0.0295500
\(463\) 17.7027 0.822714 0.411357 0.911474i \(-0.365055\pi\)
0.411357 + 0.911474i \(0.365055\pi\)
\(464\) 15.2997 0.710271
\(465\) 0 0
\(466\) −3.71988 −0.172320
\(467\) 12.5779 0.582035 0.291018 0.956718i \(-0.406006\pi\)
0.291018 + 0.956718i \(0.406006\pi\)
\(468\) 50.9437 2.35488
\(469\) 9.94502 0.459219
\(470\) 0 0
\(471\) −17.6045 −0.811171
\(472\) 3.90117 0.179566
\(473\) 11.6939 0.537687
\(474\) 2.95798 0.135865
\(475\) 0 0
\(476\) −0.687132 −0.0314946
\(477\) 19.2336 0.880648
\(478\) 6.29713 0.288024
\(479\) 21.5408 0.984224 0.492112 0.870532i \(-0.336225\pi\)
0.492112 + 0.870532i \(0.336225\pi\)
\(480\) 0 0
\(481\) 13.1981 0.601781
\(482\) −1.79295 −0.0816668
\(483\) −21.1695 −0.963243
\(484\) 19.7527 0.897852
\(485\) 0 0
\(486\) 2.68886 0.121969
\(487\) −4.30185 −0.194935 −0.0974677 0.995239i \(-0.531074\pi\)
−0.0974677 + 0.995239i \(0.531074\pi\)
\(488\) 2.51706 0.113942
\(489\) 21.0458 0.951724
\(490\) 0 0
\(491\) −7.55139 −0.340789 −0.170395 0.985376i \(-0.554504\pi\)
−0.170395 + 0.985376i \(0.554504\pi\)
\(492\) 54.7102 2.46653
\(493\) −1.44179 −0.0649349
\(494\) −6.26085 −0.281689
\(495\) 0 0
\(496\) 36.6573 1.64596
\(497\) −0.444816 −0.0199527
\(498\) −6.28013 −0.281419
\(499\) 37.8523 1.69450 0.847250 0.531194i \(-0.178256\pi\)
0.847250 + 0.531194i \(0.178256\pi\)
\(500\) 0 0
\(501\) 29.1229 1.30112
\(502\) 2.43110 0.108505
\(503\) −20.1705 −0.899360 −0.449680 0.893190i \(-0.648462\pi\)
−0.449680 + 0.893190i \(0.648462\pi\)
\(504\) −4.99007 −0.222275
\(505\) 0 0
\(506\) −1.56103 −0.0693961
\(507\) 34.3727 1.52654
\(508\) 10.1483 0.450256
\(509\) 19.5293 0.865623 0.432811 0.901484i \(-0.357522\pi\)
0.432811 + 0.901484i \(0.357522\pi\)
\(510\) 0 0
\(511\) −6.28474 −0.278021
\(512\) −16.8402 −0.744239
\(513\) −34.1864 −1.50937
\(514\) −3.98324 −0.175693
\(515\) 0 0
\(516\) −71.1991 −3.13437
\(517\) 7.73060 0.339991
\(518\) −0.637262 −0.0279997
\(519\) −5.85241 −0.256892
\(520\) 0 0
\(521\) 19.9256 0.872958 0.436479 0.899714i \(-0.356225\pi\)
0.436479 + 0.899714i \(0.356225\pi\)
\(522\) −5.16130 −0.225904
\(523\) −0.645222 −0.0282136 −0.0141068 0.999900i \(-0.504490\pi\)
−0.0141068 + 0.999900i \(0.504490\pi\)
\(524\) 3.42894 0.149794
\(525\) 0 0
\(526\) 2.02291 0.0882033
\(527\) −3.45445 −0.150478
\(528\) 9.65763 0.420295
\(529\) 29.0285 1.26211
\(530\) 0 0
\(531\) 21.9704 0.953435
\(532\) −10.5490 −0.457357
\(533\) 48.9760 2.12139
\(534\) −9.64831 −0.417523
\(535\) 0 0
\(536\) −9.05795 −0.391244
\(537\) 1.95692 0.0844472
\(538\) −6.22986 −0.268588
\(539\) −5.45990 −0.235174
\(540\) 0 0
\(541\) 0.776828 0.0333984 0.0166992 0.999861i \(-0.494684\pi\)
0.0166992 + 0.999861i \(0.494684\pi\)
\(542\) −6.50828 −0.279555
\(543\) 10.2909 0.441624
\(544\) 0.943182 0.0404386
\(545\) 0 0
\(546\) −3.46186 −0.148154
\(547\) −8.27780 −0.353933 −0.176967 0.984217i \(-0.556628\pi\)
−0.176967 + 0.984217i \(0.556628\pi\)
\(548\) −29.0187 −1.23962
\(549\) 14.1755 0.604994
\(550\) 0 0
\(551\) −22.1346 −0.942967
\(552\) 19.2812 0.820662
\(553\) 4.46158 0.189726
\(554\) 2.33560 0.0992303
\(555\) 0 0
\(556\) 20.6808 0.877061
\(557\) 11.5911 0.491130 0.245565 0.969380i \(-0.421026\pi\)
0.245565 + 0.969380i \(0.421026\pi\)
\(558\) −12.3662 −0.523503
\(559\) −63.7368 −2.69578
\(560\) 0 0
\(561\) −0.910100 −0.0384244
\(562\) −6.46268 −0.272612
\(563\) −13.9955 −0.589840 −0.294920 0.955522i \(-0.595293\pi\)
−0.294920 + 0.955522i \(0.595293\pi\)
\(564\) −47.0682 −1.98193
\(565\) 0 0
\(566\) −1.67829 −0.0705440
\(567\) −2.82373 −0.118585
\(568\) 0.405140 0.0169993
\(569\) −40.7482 −1.70825 −0.854127 0.520065i \(-0.825908\pi\)
−0.854127 + 0.520065i \(0.825908\pi\)
\(570\) 0 0
\(571\) −17.0553 −0.713742 −0.356871 0.934154i \(-0.616156\pi\)
−0.356871 + 0.934154i \(0.616156\pi\)
\(572\) 8.90801 0.372463
\(573\) 44.3545 1.85294
\(574\) −2.36478 −0.0987039
\(575\) 0 0
\(576\) −35.0970 −1.46237
\(577\) −23.5705 −0.981255 −0.490627 0.871369i \(-0.663232\pi\)
−0.490627 + 0.871369i \(0.663232\pi\)
\(578\) 3.98455 0.165735
\(579\) 14.0016 0.581888
\(580\) 0 0
\(581\) −9.47243 −0.392983
\(582\) −7.46509 −0.309438
\(583\) 3.36319 0.139289
\(584\) 5.72416 0.236867
\(585\) 0 0
\(586\) 3.70715 0.153141
\(587\) −0.0140745 −0.000580917 0 −0.000290459 1.00000i \(-0.500092\pi\)
−0.000290459 1.00000i \(0.500092\pi\)
\(588\) 33.2429 1.37091
\(589\) −53.0334 −2.18520
\(590\) 0 0
\(591\) 2.87112 0.118102
\(592\) −9.68969 −0.398244
\(593\) 7.67621 0.315224 0.157612 0.987501i \(-0.449620\pi\)
0.157612 + 0.987501i \(0.449620\pi\)
\(594\) −1.39390 −0.0571923
\(595\) 0 0
\(596\) 15.4183 0.631558
\(597\) −57.6117 −2.35789
\(598\) 8.50826 0.347928
\(599\) −9.42711 −0.385181 −0.192591 0.981279i \(-0.561689\pi\)
−0.192591 + 0.981279i \(0.561689\pi\)
\(600\) 0 0
\(601\) 3.49753 0.142667 0.0713337 0.997453i \(-0.477274\pi\)
0.0713337 + 0.997453i \(0.477274\pi\)
\(602\) 3.07749 0.125429
\(603\) −51.0121 −2.07737
\(604\) 45.7728 1.86247
\(605\) 0 0
\(606\) −7.81076 −0.317290
\(607\) −11.1635 −0.453112 −0.226556 0.973998i \(-0.572747\pi\)
−0.226556 + 0.973998i \(0.572747\pi\)
\(608\) 14.4799 0.587239
\(609\) −12.2391 −0.495952
\(610\) 0 0
\(611\) −42.1350 −1.70460
\(612\) 3.52458 0.142473
\(613\) −44.5755 −1.80039 −0.900193 0.435490i \(-0.856575\pi\)
−0.900193 + 0.435490i \(0.856575\pi\)
\(614\) −2.11827 −0.0854864
\(615\) 0 0
\(616\) −0.872561 −0.0351565
\(617\) −25.3603 −1.02097 −0.510484 0.859887i \(-0.670534\pi\)
−0.510484 + 0.859887i \(0.670534\pi\)
\(618\) 2.79463 0.112417
\(619\) 17.2831 0.694667 0.347333 0.937742i \(-0.387087\pi\)
0.347333 + 0.937742i \(0.387087\pi\)
\(620\) 0 0
\(621\) 46.4581 1.86430
\(622\) 2.30606 0.0924647
\(623\) −14.5527 −0.583043
\(624\) −52.6382 −2.10721
\(625\) 0 0
\(626\) −0.106217 −0.00424528
\(627\) −13.9720 −0.557989
\(628\) 11.9215 0.475720
\(629\) 0.913120 0.0364085
\(630\) 0 0
\(631\) 13.6827 0.544700 0.272350 0.962198i \(-0.412199\pi\)
0.272350 + 0.962198i \(0.412199\pi\)
\(632\) −4.06362 −0.161642
\(633\) −12.3325 −0.490175
\(634\) −6.44124 −0.255814
\(635\) 0 0
\(636\) −20.4770 −0.811965
\(637\) 29.7587 1.17908
\(638\) −0.902504 −0.0357305
\(639\) 2.28165 0.0902606
\(640\) 0 0
\(641\) 4.80495 0.189784 0.0948921 0.995488i \(-0.469749\pi\)
0.0948921 + 0.995488i \(0.469749\pi\)
\(642\) −6.95987 −0.274684
\(643\) 11.5218 0.454376 0.227188 0.973851i \(-0.427047\pi\)
0.227188 + 0.973851i \(0.427047\pi\)
\(644\) 14.3357 0.564905
\(645\) 0 0
\(646\) −0.433162 −0.0170425
\(647\) 19.8337 0.779742 0.389871 0.920870i \(-0.372520\pi\)
0.389871 + 0.920870i \(0.372520\pi\)
\(648\) 2.57186 0.101032
\(649\) 3.84174 0.150802
\(650\) 0 0
\(651\) −29.3241 −1.14930
\(652\) −14.2519 −0.558149
\(653\) −37.8833 −1.48249 −0.741244 0.671235i \(-0.765764\pi\)
−0.741244 + 0.671235i \(0.765764\pi\)
\(654\) −13.2975 −0.519974
\(655\) 0 0
\(656\) −35.9569 −1.40388
\(657\) 32.2370 1.25769
\(658\) 2.03446 0.0793116
\(659\) −18.2539 −0.711072 −0.355536 0.934663i \(-0.615702\pi\)
−0.355536 + 0.934663i \(0.615702\pi\)
\(660\) 0 0
\(661\) 39.5364 1.53779 0.768895 0.639375i \(-0.220807\pi\)
0.768895 + 0.639375i \(0.220807\pi\)
\(662\) −1.10811 −0.0430678
\(663\) 4.96043 0.192647
\(664\) 8.62751 0.334813
\(665\) 0 0
\(666\) 3.26878 0.126663
\(667\) 30.0801 1.16471
\(668\) −19.7217 −0.763055
\(669\) −39.1701 −1.51440
\(670\) 0 0
\(671\) 2.47872 0.0956898
\(672\) 8.00649 0.308857
\(673\) −14.7965 −0.570364 −0.285182 0.958473i \(-0.592054\pi\)
−0.285182 + 0.958473i \(0.592054\pi\)
\(674\) −0.280254 −0.0107950
\(675\) 0 0
\(676\) −23.2767 −0.895259
\(677\) −37.1563 −1.42803 −0.714017 0.700128i \(-0.753126\pi\)
−0.714017 + 0.700128i \(0.753126\pi\)
\(678\) −10.1443 −0.389590
\(679\) −11.2597 −0.432109
\(680\) 0 0
\(681\) 23.9094 0.916212
\(682\) −2.16235 −0.0828006
\(683\) 29.5420 1.13039 0.565196 0.824957i \(-0.308801\pi\)
0.565196 + 0.824957i \(0.308801\pi\)
\(684\) 54.1101 2.06895
\(685\) 0 0
\(686\) −3.12589 −0.119347
\(687\) −79.7749 −3.04360
\(688\) 46.7938 1.78400
\(689\) −18.3308 −0.698348
\(690\) 0 0
\(691\) 24.4783 0.931198 0.465599 0.884996i \(-0.345839\pi\)
0.465599 + 0.884996i \(0.345839\pi\)
\(692\) 3.96317 0.150657
\(693\) −4.91405 −0.186669
\(694\) 0.522791 0.0198449
\(695\) 0 0
\(696\) 11.1474 0.422540
\(697\) 3.38844 0.128346
\(698\) −5.35416 −0.202658
\(699\) 45.2466 1.71138
\(700\) 0 0
\(701\) 21.2361 0.802077 0.401039 0.916061i \(-0.368649\pi\)
0.401039 + 0.916061i \(0.368649\pi\)
\(702\) 7.59732 0.286742
\(703\) 14.0184 0.528714
\(704\) −6.13704 −0.231299
\(705\) 0 0
\(706\) −2.89605 −0.108994
\(707\) −11.7811 −0.443074
\(708\) −23.3907 −0.879076
\(709\) −10.4005 −0.390599 −0.195300 0.980744i \(-0.562568\pi\)
−0.195300 + 0.980744i \(0.562568\pi\)
\(710\) 0 0
\(711\) −22.8853 −0.858265
\(712\) 13.2547 0.496739
\(713\) 72.0703 2.69905
\(714\) −0.239511 −0.00896348
\(715\) 0 0
\(716\) −1.32520 −0.0495250
\(717\) −76.5948 −2.86048
\(718\) 4.48366 0.167329
\(719\) −35.5394 −1.32540 −0.662698 0.748887i \(-0.730589\pi\)
−0.662698 + 0.748887i \(0.730589\pi\)
\(720\) 0 0
\(721\) 4.21520 0.156982
\(722\) −2.16514 −0.0805780
\(723\) 21.8085 0.811067
\(724\) −6.96885 −0.258995
\(725\) 0 0
\(726\) 6.88514 0.255532
\(727\) 20.0589 0.743944 0.371972 0.928244i \(-0.378682\pi\)
0.371972 + 0.928244i \(0.378682\pi\)
\(728\) 4.75583 0.176263
\(729\) −40.9929 −1.51826
\(730\) 0 0
\(731\) −4.40968 −0.163098
\(732\) −15.0918 −0.557810
\(733\) 7.54137 0.278547 0.139273 0.990254i \(-0.455523\pi\)
0.139273 + 0.990254i \(0.455523\pi\)
\(734\) 8.37937 0.309288
\(735\) 0 0
\(736\) −19.6777 −0.725329
\(737\) −8.91997 −0.328571
\(738\) 12.1299 0.446508
\(739\) −2.97975 −0.109612 −0.0548059 0.998497i \(-0.517454\pi\)
−0.0548059 + 0.998497i \(0.517454\pi\)
\(740\) 0 0
\(741\) 76.1535 2.79757
\(742\) 0.885091 0.0324927
\(743\) −50.0156 −1.83490 −0.917448 0.397856i \(-0.869754\pi\)
−0.917448 + 0.397856i \(0.869754\pi\)
\(744\) 26.7085 0.979180
\(745\) 0 0
\(746\) 5.22832 0.191422
\(747\) 48.5880 1.77774
\(748\) 0.616308 0.0225344
\(749\) −10.4977 −0.383577
\(750\) 0 0
\(751\) 25.8694 0.943989 0.471995 0.881601i \(-0.343534\pi\)
0.471995 + 0.881601i \(0.343534\pi\)
\(752\) 30.9344 1.12806
\(753\) −29.5706 −1.07761
\(754\) 4.91903 0.179140
\(755\) 0 0
\(756\) 12.8008 0.465562
\(757\) 52.5795 1.91103 0.955517 0.294936i \(-0.0952983\pi\)
0.955517 + 0.294936i \(0.0952983\pi\)
\(758\) 0.0111313 0.000404307 0
\(759\) 18.9875 0.689201
\(760\) 0 0
\(761\) −47.2393 −1.71242 −0.856211 0.516626i \(-0.827188\pi\)
−0.856211 + 0.516626i \(0.827188\pi\)
\(762\) 3.53734 0.128144
\(763\) −20.0569 −0.726109
\(764\) −30.0363 −1.08668
\(765\) 0 0
\(766\) −6.56402 −0.237168
\(767\) −20.9391 −0.756068
\(768\) 33.6681 1.21489
\(769\) 19.1183 0.689425 0.344712 0.938708i \(-0.387976\pi\)
0.344712 + 0.938708i \(0.387976\pi\)
\(770\) 0 0
\(771\) 48.4499 1.74488
\(772\) −9.48172 −0.341255
\(773\) 35.0337 1.26007 0.630037 0.776565i \(-0.283040\pi\)
0.630037 + 0.776565i \(0.283040\pi\)
\(774\) −15.7857 −0.567406
\(775\) 0 0
\(776\) 10.2554 0.368147
\(777\) 7.75130 0.278076
\(778\) 1.39510 0.0500169
\(779\) 52.0201 1.86381
\(780\) 0 0
\(781\) 0.398968 0.0142762
\(782\) 0.588650 0.0210501
\(783\) 26.8596 0.959884
\(784\) −21.8481 −0.780288
\(785\) 0 0
\(786\) 1.19521 0.0426319
\(787\) −31.2968 −1.11561 −0.557806 0.829971i \(-0.688357\pi\)
−0.557806 + 0.829971i \(0.688357\pi\)
\(788\) −1.94428 −0.0692622
\(789\) −24.6056 −0.875983
\(790\) 0 0
\(791\) −15.3009 −0.544036
\(792\) 4.47573 0.159038
\(793\) −13.5101 −0.479756
\(794\) −2.54757 −0.0904099
\(795\) 0 0
\(796\) 39.0139 1.38281
\(797\) −31.3216 −1.10947 −0.554734 0.832027i \(-0.687180\pi\)
−0.554734 + 0.832027i \(0.687180\pi\)
\(798\) −3.67702 −0.130165
\(799\) −2.91514 −0.103130
\(800\) 0 0
\(801\) 74.6469 2.63752
\(802\) 5.97498 0.210984
\(803\) 5.63696 0.198924
\(804\) 54.3098 1.91536
\(805\) 0 0
\(806\) 11.7857 0.415134
\(807\) 75.7766 2.66746
\(808\) 10.7303 0.377489
\(809\) 4.76429 0.167504 0.0837518 0.996487i \(-0.473310\pi\)
0.0837518 + 0.996487i \(0.473310\pi\)
\(810\) 0 0
\(811\) −0.713408 −0.0250511 −0.0125256 0.999922i \(-0.503987\pi\)
−0.0125256 + 0.999922i \(0.503987\pi\)
\(812\) 8.28813 0.290856
\(813\) 79.1631 2.77637
\(814\) 0.571578 0.0200338
\(815\) 0 0
\(816\) −3.64181 −0.127489
\(817\) −67.6983 −2.36846
\(818\) −2.61797 −0.0915352
\(819\) 26.7836 0.935896
\(820\) 0 0
\(821\) 1.05247 0.0367315 0.0183658 0.999831i \(-0.494154\pi\)
0.0183658 + 0.999831i \(0.494154\pi\)
\(822\) −10.1150 −0.352800
\(823\) −36.9290 −1.28726 −0.643631 0.765336i \(-0.722573\pi\)
−0.643631 + 0.765336i \(0.722573\pi\)
\(824\) −3.83921 −0.133745
\(825\) 0 0
\(826\) 1.01103 0.0351783
\(827\) 10.4375 0.362948 0.181474 0.983396i \(-0.441913\pi\)
0.181474 + 0.983396i \(0.441913\pi\)
\(828\) −73.5336 −2.55547
\(829\) −17.5141 −0.608291 −0.304145 0.952626i \(-0.598371\pi\)
−0.304145 + 0.952626i \(0.598371\pi\)
\(830\) 0 0
\(831\) −28.4090 −0.985497
\(832\) 33.4495 1.15965
\(833\) 2.05888 0.0713360
\(834\) 7.20863 0.249614
\(835\) 0 0
\(836\) 9.46168 0.327239
\(837\) 64.3541 2.22440
\(838\) 7.04325 0.243305
\(839\) 23.0283 0.795026 0.397513 0.917597i \(-0.369873\pi\)
0.397513 + 0.917597i \(0.369873\pi\)
\(840\) 0 0
\(841\) −11.6093 −0.400319
\(842\) 7.40436 0.255171
\(843\) 78.6084 2.70742
\(844\) 8.35144 0.287468
\(845\) 0 0
\(846\) −10.4356 −0.358783
\(847\) 10.3850 0.356832
\(848\) 13.4580 0.462149
\(849\) 20.4138 0.700601
\(850\) 0 0
\(851\) −19.0505 −0.653042
\(852\) −2.42914 −0.0832211
\(853\) −41.3312 −1.41515 −0.707577 0.706637i \(-0.750212\pi\)
−0.707577 + 0.706637i \(0.750212\pi\)
\(854\) 0.652325 0.0223221
\(855\) 0 0
\(856\) 9.56133 0.326799
\(857\) 14.1899 0.484718 0.242359 0.970187i \(-0.422079\pi\)
0.242359 + 0.970187i \(0.422079\pi\)
\(858\) 3.10503 0.106004
\(859\) −15.4345 −0.526618 −0.263309 0.964712i \(-0.584814\pi\)
−0.263309 + 0.964712i \(0.584814\pi\)
\(860\) 0 0
\(861\) 28.7638 0.980269
\(862\) 6.70664 0.228429
\(863\) 3.73755 0.127228 0.0636138 0.997975i \(-0.479737\pi\)
0.0636138 + 0.997975i \(0.479737\pi\)
\(864\) −17.5709 −0.597774
\(865\) 0 0
\(866\) 7.00573 0.238064
\(867\) −48.4658 −1.64598
\(868\) 19.8579 0.674021
\(869\) −4.00171 −0.135749
\(870\) 0 0
\(871\) 48.6176 1.64734
\(872\) 18.2679 0.618628
\(873\) 57.7558 1.95474
\(874\) 9.03708 0.305684
\(875\) 0 0
\(876\) −34.3210 −1.15960
\(877\) 37.4630 1.26504 0.632518 0.774546i \(-0.282022\pi\)
0.632518 + 0.774546i \(0.282022\pi\)
\(878\) −4.48902 −0.151497
\(879\) −45.0917 −1.52091
\(880\) 0 0
\(881\) −21.1306 −0.711907 −0.355953 0.934504i \(-0.615844\pi\)
−0.355953 + 0.934504i \(0.615844\pi\)
\(882\) 7.37036 0.248173
\(883\) 21.3153 0.717317 0.358658 0.933469i \(-0.383234\pi\)
0.358658 + 0.933469i \(0.383234\pi\)
\(884\) −3.35914 −0.112980
\(885\) 0 0
\(886\) −2.95918 −0.0994156
\(887\) −28.5267 −0.957834 −0.478917 0.877860i \(-0.658971\pi\)
−0.478917 + 0.877860i \(0.658971\pi\)
\(888\) −7.05990 −0.236915
\(889\) 5.33544 0.178945
\(890\) 0 0
\(891\) 2.53268 0.0848480
\(892\) 26.5255 0.888138
\(893\) −44.7539 −1.49763
\(894\) 5.37430 0.179743
\(895\) 0 0
\(896\) −7.19235 −0.240280
\(897\) −103.490 −3.45542
\(898\) −4.24594 −0.141689
\(899\) 41.6673 1.38968
\(900\) 0 0
\(901\) −1.26823 −0.0422509
\(902\) 2.12103 0.0706227
\(903\) −37.4329 −1.24569
\(904\) 13.9361 0.463507
\(905\) 0 0
\(906\) 15.9549 0.530065
\(907\) 7.24122 0.240441 0.120220 0.992747i \(-0.461640\pi\)
0.120220 + 0.992747i \(0.461640\pi\)
\(908\) −16.1912 −0.537322
\(909\) 60.4302 2.00434
\(910\) 0 0
\(911\) −30.4699 −1.00951 −0.504756 0.863262i \(-0.668418\pi\)
−0.504756 + 0.863262i \(0.668418\pi\)
\(912\) −55.9098 −1.85136
\(913\) 8.49609 0.281179
\(914\) −6.44602 −0.213215
\(915\) 0 0
\(916\) 54.0225 1.78495
\(917\) 1.80276 0.0595325
\(918\) 0.525627 0.0173483
\(919\) −56.0316 −1.84831 −0.924157 0.382013i \(-0.875231\pi\)
−0.924157 + 0.382013i \(0.875231\pi\)
\(920\) 0 0
\(921\) 25.7654 0.849000
\(922\) 2.09022 0.0688378
\(923\) −2.17454 −0.0715760
\(924\) 5.23171 0.172111
\(925\) 0 0
\(926\) −4.17863 −0.137318
\(927\) −21.6215 −0.710142
\(928\) −11.3766 −0.373455
\(929\) −17.8294 −0.584963 −0.292481 0.956271i \(-0.594481\pi\)
−0.292481 + 0.956271i \(0.594481\pi\)
\(930\) 0 0
\(931\) 31.6084 1.03592
\(932\) −30.6404 −1.00366
\(933\) −28.0497 −0.918305
\(934\) −2.96894 −0.0971468
\(935\) 0 0
\(936\) −24.3946 −0.797363
\(937\) −10.3366 −0.337681 −0.168841 0.985643i \(-0.554002\pi\)
−0.168841 + 0.985643i \(0.554002\pi\)
\(938\) −2.34747 −0.0766476
\(939\) 1.29196 0.0421616
\(940\) 0 0
\(941\) 4.39565 0.143294 0.0716470 0.997430i \(-0.477174\pi\)
0.0716470 + 0.997430i \(0.477174\pi\)
\(942\) 4.15544 0.135392
\(943\) −70.6933 −2.30209
\(944\) 15.3729 0.500347
\(945\) 0 0
\(946\) −2.76029 −0.0897447
\(947\) 13.1530 0.427415 0.213708 0.976898i \(-0.431446\pi\)
0.213708 + 0.976898i \(0.431446\pi\)
\(948\) 24.3647 0.791328
\(949\) −30.7238 −0.997337
\(950\) 0 0
\(951\) 78.3477 2.54060
\(952\) 0.329036 0.0106641
\(953\) −52.7707 −1.70941 −0.854706 0.519113i \(-0.826262\pi\)
−0.854706 + 0.519113i \(0.826262\pi\)
\(954\) −4.54000 −0.146988
\(955\) 0 0
\(956\) 51.8690 1.67756
\(957\) 10.9776 0.354854
\(958\) −5.08459 −0.164276
\(959\) −15.2566 −0.492661
\(960\) 0 0
\(961\) 68.8325 2.22040
\(962\) −3.11534 −0.100443
\(963\) 53.8470 1.73520
\(964\) −14.7684 −0.475659
\(965\) 0 0
\(966\) 4.99694 0.160774
\(967\) −40.7983 −1.31198 −0.655992 0.754768i \(-0.727750\pi\)
−0.655992 + 0.754768i \(0.727750\pi\)
\(968\) −9.45867 −0.304013
\(969\) 5.26874 0.169256
\(970\) 0 0
\(971\) 59.3602 1.90496 0.952479 0.304603i \(-0.0985239\pi\)
0.952479 + 0.304603i \(0.0985239\pi\)
\(972\) 22.1479 0.710394
\(973\) 10.8729 0.348569
\(974\) 1.01543 0.0325364
\(975\) 0 0
\(976\) 9.91872 0.317491
\(977\) 21.8738 0.699806 0.349903 0.936786i \(-0.386215\pi\)
0.349903 + 0.936786i \(0.386215\pi\)
\(978\) −4.96775 −0.158851
\(979\) 13.0527 0.417167
\(980\) 0 0
\(981\) 102.880 3.28471
\(982\) 1.78247 0.0568808
\(983\) 32.5706 1.03884 0.519421 0.854519i \(-0.326148\pi\)
0.519421 + 0.854519i \(0.326148\pi\)
\(984\) −26.1982 −0.835167
\(985\) 0 0
\(986\) 0.340327 0.0108382
\(987\) −24.7461 −0.787677
\(988\) −51.5701 −1.64066
\(989\) 91.9994 2.92541
\(990\) 0 0
\(991\) −0.493203 −0.0156671 −0.00783356 0.999969i \(-0.502494\pi\)
−0.00783356 + 0.999969i \(0.502494\pi\)
\(992\) −27.2577 −0.865433
\(993\) 13.4784 0.427724
\(994\) 0.104997 0.00333029
\(995\) 0 0
\(996\) −51.7290 −1.63909
\(997\) −19.0675 −0.603872 −0.301936 0.953328i \(-0.597633\pi\)
−0.301936 + 0.953328i \(0.597633\pi\)
\(998\) −8.93483 −0.282827
\(999\) −17.0109 −0.538200
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 4925.2.a.k.1.5 14
5.4 even 2 985.2.a.f.1.10 14
15.14 odd 2 8865.2.a.v.1.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
985.2.a.f.1.10 14 5.4 even 2
4925.2.a.k.1.5 14 1.1 even 1 trivial
8865.2.a.v.1.5 14 15.14 odd 2