Properties

Label 3192.2.a.s.1.2
Level $3192$
Weight $2$
Character 3192.1
Self dual yes
Analytic conductor $25.488$
Analytic rank $1$
Dimension $2$
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] = [3192,2,Mod(1,3192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3192.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3192.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: \(25.4882483252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.46410 q^{13} +0.732051 q^{15} -3.26795 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.92820 q^{23} -4.46410 q^{25} +1.00000 q^{27} -3.26795 q^{29} -2.00000 q^{31} -4.00000 q^{33} +0.732051 q^{35} -4.92820 q^{37} -1.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +0.732051 q^{45} +0.196152 q^{47} +1.00000 q^{49} -3.26795 q^{51} +7.66025 q^{53} -2.92820 q^{55} +1.00000 q^{57} -10.9282 q^{59} +0.928203 q^{61} +1.00000 q^{63} -1.07180 q^{65} +5.46410 q^{67} -6.92820 q^{69} +4.19615 q^{71} -10.3923 q^{73} -4.46410 q^{75} -4.00000 q^{77} +2.92820 q^{79} +1.00000 q^{81} +8.19615 q^{83} -2.39230 q^{85} -3.26795 q^{87} -6.00000 q^{89} -1.46410 q^{91} -2.00000 q^{93} +0.732051 q^{95} +11.8564 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 4 q^{13} - 2 q^{15} - 10 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 10 q^{51} - 2 q^{53} + 8 q^{55} + 2 q^{57} - 8 q^{59} - 12 q^{61} + 2 q^{63} - 16 q^{65} + 4 q^{67} - 2 q^{71} - 2 q^{75} - 8 q^{77} - 8 q^{79} + 2 q^{81} + 6 q^{83} + 16 q^{85} - 10 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} - 2 q^{95} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.732051 0.327383 0.163692 0.986512i \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 0 0
\(15\) 0.732051 0.189015
\(16\) 0 0
\(17\) −3.26795 −0.792594 −0.396297 0.918122i \(-0.629705\pi\)
−0.396297 + 0.918122i \(0.629705\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.26795 −0.606843 −0.303421 0.952856i \(-0.598129\pi\)
−0.303421 + 0.952856i \(0.598129\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0.732051 0.123739
\(36\) 0 0
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) 0 0
\(39\) −1.46410 −0.234444
\(40\) 0 0
\(41\) −8.92820 −1.39435 −0.697176 0.716900i \(-0.745560\pi\)
−0.697176 + 0.716900i \(0.745560\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 0.732051 0.109128
\(46\) 0 0
\(47\) 0.196152 0.0286118 0.0143059 0.999898i \(-0.495446\pi\)
0.0143059 + 0.999898i \(0.495446\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.26795 −0.457604
\(52\) 0 0
\(53\) 7.66025 1.05222 0.526108 0.850418i \(-0.323651\pi\)
0.526108 + 0.850418i \(0.323651\pi\)
\(54\) 0 0
\(55\) −2.92820 −0.394839
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −10.9282 −1.42273 −0.711365 0.702822i \(-0.751923\pi\)
−0.711365 + 0.702822i \(0.751923\pi\)
\(60\) 0 0
\(61\) 0.928203 0.118844 0.0594221 0.998233i \(-0.481074\pi\)
0.0594221 + 0.998233i \(0.481074\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.07180 −0.132940
\(66\) 0 0
\(67\) 5.46410 0.667546 0.333773 0.942653i \(-0.391678\pi\)
0.333773 + 0.942653i \(0.391678\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) 4.19615 0.497992 0.248996 0.968505i \(-0.419899\pi\)
0.248996 + 0.968505i \(0.419899\pi\)
\(72\) 0 0
\(73\) −10.3923 −1.21633 −0.608164 0.793812i \(-0.708094\pi\)
−0.608164 + 0.793812i \(0.708094\pi\)
\(74\) 0 0
\(75\) −4.46410 −0.515470
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 2.92820 0.329449 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.19615 0.899645 0.449822 0.893118i \(-0.351487\pi\)
0.449822 + 0.893118i \(0.351487\pi\)
\(84\) 0 0
\(85\) −2.39230 −0.259482
\(86\) 0 0
\(87\) −3.26795 −0.350361
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 0.732051 0.0751068
\(96\) 0 0
\(97\) 11.8564 1.20384 0.601918 0.798558i \(-0.294403\pi\)
0.601918 + 0.798558i \(0.294403\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −4.73205 −0.470857 −0.235428 0.971892i \(-0.575649\pi\)
−0.235428 + 0.971892i \(0.575649\pi\)
\(102\) 0 0
\(103\) −2.92820 −0.288524 −0.144262 0.989539i \(-0.546081\pi\)
−0.144262 + 0.989539i \(0.546081\pi\)
\(104\) 0 0
\(105\) 0.732051 0.0714408
\(106\) 0 0
\(107\) −12.1962 −1.17905 −0.589523 0.807751i \(-0.700684\pi\)
−0.589523 + 0.807751i \(0.700684\pi\)
\(108\) 0 0
\(109\) −4.53590 −0.434460 −0.217230 0.976120i \(-0.569702\pi\)
−0.217230 + 0.976120i \(0.569702\pi\)
\(110\) 0 0
\(111\) −4.92820 −0.467764
\(112\) 0 0
\(113\) 16.7321 1.57402 0.787009 0.616941i \(-0.211628\pi\)
0.787009 + 0.616941i \(0.211628\pi\)
\(114\) 0 0
\(115\) −5.07180 −0.472947
\(116\) 0 0
\(117\) −1.46410 −0.135356
\(118\) 0 0
\(119\) −3.26795 −0.299572
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −8.92820 −0.805029
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) −6.53590 −0.579967 −0.289984 0.957032i \(-0.593650\pi\)
−0.289984 + 0.957032i \(0.593650\pi\)
\(128\) 0 0
\(129\) 4.92820 0.433904
\(130\) 0 0
\(131\) 10.7321 0.937664 0.468832 0.883287i \(-0.344675\pi\)
0.468832 + 0.883287i \(0.344675\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 0.732051 0.0630049
\(136\) 0 0
\(137\) 4.92820 0.421045 0.210522 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(138\) 0 0
\(139\) 19.3205 1.63874 0.819372 0.573262i \(-0.194322\pi\)
0.819372 + 0.573262i \(0.194322\pi\)
\(140\) 0 0
\(141\) 0.196152 0.0165190
\(142\) 0 0
\(143\) 5.85641 0.489737
\(144\) 0 0
\(145\) −2.39230 −0.198670
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −3.46410 −0.283790 −0.141895 0.989882i \(-0.545320\pi\)
−0.141895 + 0.989882i \(0.545320\pi\)
\(150\) 0 0
\(151\) 13.4641 1.09569 0.547847 0.836579i \(-0.315448\pi\)
0.547847 + 0.836579i \(0.315448\pi\)
\(152\) 0 0
\(153\) −3.26795 −0.264198
\(154\) 0 0
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 0 0
\(159\) 7.66025 0.607498
\(160\) 0 0
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) −12.9282 −1.01262 −0.506308 0.862353i \(-0.668990\pi\)
−0.506308 + 0.862353i \(0.668990\pi\)
\(164\) 0 0
\(165\) −2.92820 −0.227960
\(166\) 0 0
\(167\) 9.46410 0.732354 0.366177 0.930545i \(-0.380666\pi\)
0.366177 + 0.930545i \(0.380666\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −11.8564 −0.901426 −0.450713 0.892669i \(-0.648830\pi\)
−0.450713 + 0.892669i \(0.648830\pi\)
\(174\) 0 0
\(175\) −4.46410 −0.337454
\(176\) 0 0
\(177\) −10.9282 −0.821414
\(178\) 0 0
\(179\) −14.7321 −1.10113 −0.550563 0.834794i \(-0.685587\pi\)
−0.550563 + 0.834794i \(0.685587\pi\)
\(180\) 0 0
\(181\) −7.07180 −0.525643 −0.262821 0.964845i \(-0.584653\pi\)
−0.262821 + 0.964845i \(0.584653\pi\)
\(182\) 0 0
\(183\) 0.928203 0.0686148
\(184\) 0 0
\(185\) −3.60770 −0.265243
\(186\) 0 0
\(187\) 13.0718 0.955904
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) 8.92820 0.642666 0.321333 0.946966i \(-0.395869\pi\)
0.321333 + 0.946966i \(0.395869\pi\)
\(194\) 0 0
\(195\) −1.07180 −0.0767530
\(196\) 0 0
\(197\) 0.928203 0.0661317 0.0330659 0.999453i \(-0.489473\pi\)
0.0330659 + 0.999453i \(0.489473\pi\)
\(198\) 0 0
\(199\) 19.3205 1.36959 0.684797 0.728734i \(-0.259891\pi\)
0.684797 + 0.728734i \(0.259891\pi\)
\(200\) 0 0
\(201\) 5.46410 0.385408
\(202\) 0 0
\(203\) −3.26795 −0.229365
\(204\) 0 0
\(205\) −6.53590 −0.456487
\(206\) 0 0
\(207\) −6.92820 −0.481543
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 4.19615 0.287516
\(214\) 0 0
\(215\) 3.60770 0.246043
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) −10.3923 −0.702247
\(220\) 0 0
\(221\) 4.78461 0.321848
\(222\) 0 0
\(223\) −22.7846 −1.52577 −0.762885 0.646534i \(-0.776218\pi\)
−0.762885 + 0.646534i \(0.776218\pi\)
\(224\) 0 0
\(225\) −4.46410 −0.297607
\(226\) 0 0
\(227\) −7.32051 −0.485879 −0.242940 0.970041i \(-0.578112\pi\)
−0.242940 + 0.970041i \(0.578112\pi\)
\(228\) 0 0
\(229\) 2.39230 0.158088 0.0790440 0.996871i \(-0.474813\pi\)
0.0790440 + 0.996871i \(0.474813\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) −17.3205 −1.13470 −0.567352 0.823475i \(-0.692032\pi\)
−0.567352 + 0.823475i \(0.692032\pi\)
\(234\) 0 0
\(235\) 0.143594 0.00936701
\(236\) 0 0
\(237\) 2.92820 0.190207
\(238\) 0 0
\(239\) −9.46410 −0.612182 −0.306091 0.952002i \(-0.599021\pi\)
−0.306091 + 0.952002i \(0.599021\pi\)
\(240\) 0 0
\(241\) 10.7846 0.694698 0.347349 0.937736i \(-0.387082\pi\)
0.347349 + 0.937736i \(0.387082\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.732051 0.0467690
\(246\) 0 0
\(247\) −1.46410 −0.0931586
\(248\) 0 0
\(249\) 8.19615 0.519410
\(250\) 0 0
\(251\) −1.66025 −0.104794 −0.0523972 0.998626i \(-0.516686\pi\)
−0.0523972 + 0.998626i \(0.516686\pi\)
\(252\) 0 0
\(253\) 27.7128 1.74229
\(254\) 0 0
\(255\) −2.39230 −0.149812
\(256\) 0 0
\(257\) 2.39230 0.149228 0.0746139 0.997212i \(-0.476228\pi\)
0.0746139 + 0.997212i \(0.476228\pi\)
\(258\) 0 0
\(259\) −4.92820 −0.306224
\(260\) 0 0
\(261\) −3.26795 −0.202281
\(262\) 0 0
\(263\) 2.53590 0.156370 0.0781851 0.996939i \(-0.475087\pi\)
0.0781851 + 0.996939i \(0.475087\pi\)
\(264\) 0 0
\(265\) 5.60770 0.344478
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −27.4641 −1.67452 −0.837258 0.546808i \(-0.815843\pi\)
−0.837258 + 0.546808i \(0.815843\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −1.46410 −0.0886115
\(274\) 0 0
\(275\) 17.8564 1.07678
\(276\) 0 0
\(277\) −24.3923 −1.46559 −0.732796 0.680449i \(-0.761785\pi\)
−0.732796 + 0.680449i \(0.761785\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −4.73205 −0.282290 −0.141145 0.989989i \(-0.545078\pi\)
−0.141145 + 0.989989i \(0.545078\pi\)
\(282\) 0 0
\(283\) −8.39230 −0.498871 −0.249435 0.968391i \(-0.580245\pi\)
−0.249435 + 0.968391i \(0.580245\pi\)
\(284\) 0 0
\(285\) 0.732051 0.0433629
\(286\) 0 0
\(287\) −8.92820 −0.527015
\(288\) 0 0
\(289\) −6.32051 −0.371795
\(290\) 0 0
\(291\) 11.8564 0.695035
\(292\) 0 0
\(293\) −27.8564 −1.62739 −0.813694 0.581293i \(-0.802547\pi\)
−0.813694 + 0.581293i \(0.802547\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 10.1436 0.586619
\(300\) 0 0
\(301\) 4.92820 0.284057
\(302\) 0 0
\(303\) −4.73205 −0.271849
\(304\) 0 0
\(305\) 0.679492 0.0389076
\(306\) 0 0
\(307\) 16.9282 0.966144 0.483072 0.875581i \(-0.339521\pi\)
0.483072 + 0.875581i \(0.339521\pi\)
\(308\) 0 0
\(309\) −2.92820 −0.166580
\(310\) 0 0
\(311\) −7.12436 −0.403985 −0.201993 0.979387i \(-0.564742\pi\)
−0.201993 + 0.979387i \(0.564742\pi\)
\(312\) 0 0
\(313\) 19.0718 1.07800 0.539001 0.842305i \(-0.318802\pi\)
0.539001 + 0.842305i \(0.318802\pi\)
\(314\) 0 0
\(315\) 0.732051 0.0412464
\(316\) 0 0
\(317\) −19.2679 −1.08220 −0.541098 0.840960i \(-0.681991\pi\)
−0.541098 + 0.840960i \(0.681991\pi\)
\(318\) 0 0
\(319\) 13.0718 0.731880
\(320\) 0 0
\(321\) −12.1962 −0.680723
\(322\) 0 0
\(323\) −3.26795 −0.181834
\(324\) 0 0
\(325\) 6.53590 0.362546
\(326\) 0 0
\(327\) −4.53590 −0.250836
\(328\) 0 0
\(329\) 0.196152 0.0108142
\(330\) 0 0
\(331\) 8.39230 0.461283 0.230641 0.973039i \(-0.425918\pi\)
0.230641 + 0.973039i \(0.425918\pi\)
\(332\) 0 0
\(333\) −4.92820 −0.270064
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 1.60770 0.0875767 0.0437884 0.999041i \(-0.486057\pi\)
0.0437884 + 0.999041i \(0.486057\pi\)
\(338\) 0 0
\(339\) 16.7321 0.908760
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −5.07180 −0.273056
\(346\) 0 0
\(347\) 19.3205 1.03718 0.518590 0.855023i \(-0.326457\pi\)
0.518590 + 0.855023i \(0.326457\pi\)
\(348\) 0 0
\(349\) −22.7846 −1.21963 −0.609816 0.792543i \(-0.708757\pi\)
−0.609816 + 0.792543i \(0.708757\pi\)
\(350\) 0 0
\(351\) −1.46410 −0.0781480
\(352\) 0 0
\(353\) −25.5167 −1.35811 −0.679057 0.734085i \(-0.737611\pi\)
−0.679057 + 0.734085i \(0.737611\pi\)
\(354\) 0 0
\(355\) 3.07180 0.163034
\(356\) 0 0
\(357\) −3.26795 −0.172958
\(358\) 0 0
\(359\) 29.8564 1.57576 0.787880 0.615828i \(-0.211178\pi\)
0.787880 + 0.615828i \(0.211178\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −7.60770 −0.398205
\(366\) 0 0
\(367\) −15.3205 −0.799724 −0.399862 0.916575i \(-0.630942\pi\)
−0.399862 + 0.916575i \(0.630942\pi\)
\(368\) 0 0
\(369\) −8.92820 −0.464784
\(370\) 0 0
\(371\) 7.66025 0.397701
\(372\) 0 0
\(373\) 30.7846 1.59397 0.796983 0.604001i \(-0.206428\pi\)
0.796983 + 0.604001i \(0.206428\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 4.78461 0.246420
\(378\) 0 0
\(379\) 20.3923 1.04748 0.523741 0.851877i \(-0.324536\pi\)
0.523741 + 0.851877i \(0.324536\pi\)
\(380\) 0 0
\(381\) −6.53590 −0.334844
\(382\) 0 0
\(383\) 0.679492 0.0347204 0.0173602 0.999849i \(-0.494474\pi\)
0.0173602 + 0.999849i \(0.494474\pi\)
\(384\) 0 0
\(385\) −2.92820 −0.149235
\(386\) 0 0
\(387\) 4.92820 0.250515
\(388\) 0 0
\(389\) −13.6077 −0.689938 −0.344969 0.938614i \(-0.612110\pi\)
−0.344969 + 0.938614i \(0.612110\pi\)
\(390\) 0 0
\(391\) 22.6410 1.14501
\(392\) 0 0
\(393\) 10.7321 0.541360
\(394\) 0 0
\(395\) 2.14359 0.107856
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 6.58846 0.329012 0.164506 0.986376i \(-0.447397\pi\)
0.164506 + 0.986376i \(0.447397\pi\)
\(402\) 0 0
\(403\) 2.92820 0.145864
\(404\) 0 0
\(405\) 0.732051 0.0363759
\(406\) 0 0
\(407\) 19.7128 0.977128
\(408\) 0 0
\(409\) −2.53590 −0.125392 −0.0626961 0.998033i \(-0.519970\pi\)
−0.0626961 + 0.998033i \(0.519970\pi\)
\(410\) 0 0
\(411\) 4.92820 0.243090
\(412\) 0 0
\(413\) −10.9282 −0.537742
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 19.3205 0.946129
\(418\) 0 0
\(419\) 8.87564 0.433604 0.216802 0.976216i \(-0.430437\pi\)
0.216802 + 0.976216i \(0.430437\pi\)
\(420\) 0 0
\(421\) −16.5359 −0.805910 −0.402955 0.915220i \(-0.632017\pi\)
−0.402955 + 0.915220i \(0.632017\pi\)
\(422\) 0 0
\(423\) 0.196152 0.00953726
\(424\) 0 0
\(425\) 14.5885 0.707644
\(426\) 0 0
\(427\) 0.928203 0.0449189
\(428\) 0 0
\(429\) 5.85641 0.282750
\(430\) 0 0
\(431\) −40.9808 −1.97397 −0.986987 0.160801i \(-0.948592\pi\)
−0.986987 + 0.160801i \(0.948592\pi\)
\(432\) 0 0
\(433\) 23.8564 1.14647 0.573233 0.819393i \(-0.305689\pi\)
0.573233 + 0.819393i \(0.305689\pi\)
\(434\) 0 0
\(435\) −2.39230 −0.114702
\(436\) 0 0
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) −3.85641 −0.184056 −0.0920281 0.995756i \(-0.529335\pi\)
−0.0920281 + 0.995756i \(0.529335\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 16.3923 0.778822 0.389411 0.921064i \(-0.372679\pi\)
0.389411 + 0.921064i \(0.372679\pi\)
\(444\) 0 0
\(445\) −4.39230 −0.208215
\(446\) 0 0
\(447\) −3.46410 −0.163846
\(448\) 0 0
\(449\) 19.6603 0.927825 0.463912 0.885881i \(-0.346445\pi\)
0.463912 + 0.885881i \(0.346445\pi\)
\(450\) 0 0
\(451\) 35.7128 1.68165
\(452\) 0 0
\(453\) 13.4641 0.632599
\(454\) 0 0
\(455\) −1.07180 −0.0502466
\(456\) 0 0
\(457\) −11.3205 −0.529551 −0.264776 0.964310i \(-0.585298\pi\)
−0.264776 + 0.964310i \(0.585298\pi\)
\(458\) 0 0
\(459\) −3.26795 −0.152535
\(460\) 0 0
\(461\) −9.80385 −0.456611 −0.228305 0.973590i \(-0.573318\pi\)
−0.228305 + 0.973590i \(0.573318\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −1.46410 −0.0678961
\(466\) 0 0
\(467\) −8.19615 −0.379273 −0.189636 0.981854i \(-0.560731\pi\)
−0.189636 + 0.981854i \(0.560731\pi\)
\(468\) 0 0
\(469\) 5.46410 0.252309
\(470\) 0 0
\(471\) −3.07180 −0.141541
\(472\) 0 0
\(473\) −19.7128 −0.906396
\(474\) 0 0
\(475\) −4.46410 −0.204827
\(476\) 0 0
\(477\) 7.66025 0.350739
\(478\) 0 0
\(479\) 19.5167 0.891739 0.445869 0.895098i \(-0.352895\pi\)
0.445869 + 0.895098i \(0.352895\pi\)
\(480\) 0 0
\(481\) 7.21539 0.328993
\(482\) 0 0
\(483\) −6.92820 −0.315244
\(484\) 0 0
\(485\) 8.67949 0.394115
\(486\) 0 0
\(487\) 5.85641 0.265379 0.132690 0.991158i \(-0.457639\pi\)
0.132690 + 0.991158i \(0.457639\pi\)
\(488\) 0 0
\(489\) −12.9282 −0.584634
\(490\) 0 0
\(491\) −6.24871 −0.282000 −0.141000 0.990010i \(-0.545032\pi\)
−0.141000 + 0.990010i \(0.545032\pi\)
\(492\) 0 0
\(493\) 10.6795 0.480980
\(494\) 0 0
\(495\) −2.92820 −0.131613
\(496\) 0 0
\(497\) 4.19615 0.188223
\(498\) 0 0
\(499\) −1.85641 −0.0831042 −0.0415521 0.999136i \(-0.513230\pi\)
−0.0415521 + 0.999136i \(0.513230\pi\)
\(500\) 0 0
\(501\) 9.46410 0.422825
\(502\) 0 0
\(503\) −25.2679 −1.12664 −0.563321 0.826238i \(-0.690477\pi\)
−0.563321 + 0.826238i \(0.690477\pi\)
\(504\) 0 0
\(505\) −3.46410 −0.154150
\(506\) 0 0
\(507\) −10.8564 −0.482150
\(508\) 0 0
\(509\) 5.32051 0.235827 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(510\) 0 0
\(511\) −10.3923 −0.459728
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −2.14359 −0.0944580
\(516\) 0 0
\(517\) −0.784610 −0.0345071
\(518\) 0 0
\(519\) −11.8564 −0.520438
\(520\) 0 0
\(521\) −36.2487 −1.58808 −0.794042 0.607862i \(-0.792027\pi\)
−0.794042 + 0.607862i \(0.792027\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 0 0
\(525\) −4.46410 −0.194829
\(526\) 0 0
\(527\) 6.53590 0.284708
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) −10.9282 −0.474244
\(532\) 0 0
\(533\) 13.0718 0.566202
\(534\) 0 0
\(535\) −8.92820 −0.386000
\(536\) 0 0
\(537\) −14.7321 −0.635735
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) −7.07180 −0.303480
\(544\) 0 0
\(545\) −3.32051 −0.142235
\(546\) 0 0
\(547\) 30.9282 1.32239 0.661197 0.750212i \(-0.270049\pi\)
0.661197 + 0.750212i \(0.270049\pi\)
\(548\) 0 0
\(549\) 0.928203 0.0396147
\(550\) 0 0
\(551\) −3.26795 −0.139219
\(552\) 0 0
\(553\) 2.92820 0.124520
\(554\) 0 0
\(555\) −3.60770 −0.153138
\(556\) 0 0
\(557\) −38.1051 −1.61457 −0.807283 0.590165i \(-0.799063\pi\)
−0.807283 + 0.590165i \(0.799063\pi\)
\(558\) 0 0
\(559\) −7.21539 −0.305178
\(560\) 0 0
\(561\) 13.0718 0.551892
\(562\) 0 0
\(563\) 35.3205 1.48858 0.744291 0.667855i \(-0.232788\pi\)
0.744291 + 0.667855i \(0.232788\pi\)
\(564\) 0 0
\(565\) 12.2487 0.515307
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −29.5167 −1.23740 −0.618701 0.785626i \(-0.712341\pi\)
−0.618701 + 0.785626i \(0.712341\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 18.9282 0.790737
\(574\) 0 0
\(575\) 30.9282 1.28980
\(576\) 0 0
\(577\) 14.7846 0.615491 0.307746 0.951469i \(-0.400425\pi\)
0.307746 + 0.951469i \(0.400425\pi\)
\(578\) 0 0
\(579\) 8.92820 0.371043
\(580\) 0 0
\(581\) 8.19615 0.340034
\(582\) 0 0
\(583\) −30.6410 −1.26902
\(584\) 0 0
\(585\) −1.07180 −0.0443133
\(586\) 0 0
\(587\) 20.1962 0.833584 0.416792 0.909002i \(-0.363154\pi\)
0.416792 + 0.909002i \(0.363154\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0.928203 0.0381812
\(592\) 0 0
\(593\) 9.80385 0.402596 0.201298 0.979530i \(-0.435484\pi\)
0.201298 + 0.979530i \(0.435484\pi\)
\(594\) 0 0
\(595\) −2.39230 −0.0980749
\(596\) 0 0
\(597\) 19.3205 0.790736
\(598\) 0 0
\(599\) −7.80385 −0.318857 −0.159428 0.987210i \(-0.550965\pi\)
−0.159428 + 0.987210i \(0.550965\pi\)
\(600\) 0 0
\(601\) 16.1436 0.658511 0.329255 0.944241i \(-0.393202\pi\)
0.329255 + 0.944241i \(0.393202\pi\)
\(602\) 0 0
\(603\) 5.46410 0.222515
\(604\) 0 0
\(605\) 3.66025 0.148810
\(606\) 0 0
\(607\) 16.7846 0.681266 0.340633 0.940196i \(-0.389359\pi\)
0.340633 + 0.940196i \(0.389359\pi\)
\(608\) 0 0
\(609\) −3.26795 −0.132424
\(610\) 0 0
\(611\) −0.287187 −0.0116183
\(612\) 0 0
\(613\) −23.6077 −0.953506 −0.476753 0.879037i \(-0.658186\pi\)
−0.476753 + 0.879037i \(0.658186\pi\)
\(614\) 0 0
\(615\) −6.53590 −0.263553
\(616\) 0 0
\(617\) 22.7846 0.917274 0.458637 0.888624i \(-0.348338\pi\)
0.458637 + 0.888624i \(0.348338\pi\)
\(618\) 0 0
\(619\) −18.5359 −0.745021 −0.372510 0.928028i \(-0.621503\pi\)
−0.372510 + 0.928028i \(0.621503\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) 16.1051 0.642153
\(630\) 0 0
\(631\) −34.7846 −1.38475 −0.692377 0.721536i \(-0.743436\pi\)
−0.692377 + 0.721536i \(0.743436\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) −4.78461 −0.189871
\(636\) 0 0
\(637\) −1.46410 −0.0580098
\(638\) 0 0
\(639\) 4.19615 0.165997
\(640\) 0 0
\(641\) 3.26795 0.129076 0.0645381 0.997915i \(-0.479443\pi\)
0.0645381 + 0.997915i \(0.479443\pi\)
\(642\) 0 0
\(643\) 0.392305 0.0154710 0.00773550 0.999970i \(-0.497538\pi\)
0.00773550 + 0.999970i \(0.497538\pi\)
\(644\) 0 0
\(645\) 3.60770 0.142053
\(646\) 0 0
\(647\) 10.3397 0.406497 0.203249 0.979127i \(-0.434850\pi\)
0.203249 + 0.979127i \(0.434850\pi\)
\(648\) 0 0
\(649\) 43.7128 1.71588
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) −45.7128 −1.78888 −0.894440 0.447187i \(-0.852426\pi\)
−0.894440 + 0.447187i \(0.852426\pi\)
\(654\) 0 0
\(655\) 7.85641 0.306975
\(656\) 0 0
\(657\) −10.3923 −0.405442
\(658\) 0 0
\(659\) −11.1244 −0.433343 −0.216672 0.976245i \(-0.569520\pi\)
−0.216672 + 0.976245i \(0.569520\pi\)
\(660\) 0 0
\(661\) 5.46410 0.212529 0.106264 0.994338i \(-0.466111\pi\)
0.106264 + 0.994338i \(0.466111\pi\)
\(662\) 0 0
\(663\) 4.78461 0.185819
\(664\) 0 0
\(665\) 0.732051 0.0283877
\(666\) 0 0
\(667\) 22.6410 0.876664
\(668\) 0 0
\(669\) −22.7846 −0.880904
\(670\) 0 0
\(671\) −3.71281 −0.143332
\(672\) 0 0
\(673\) 43.8564 1.69054 0.845270 0.534339i \(-0.179440\pi\)
0.845270 + 0.534339i \(0.179440\pi\)
\(674\) 0 0
\(675\) −4.46410 −0.171823
\(676\) 0 0
\(677\) 0.535898 0.0205962 0.0102981 0.999947i \(-0.496722\pi\)
0.0102981 + 0.999947i \(0.496722\pi\)
\(678\) 0 0
\(679\) 11.8564 0.455007
\(680\) 0 0
\(681\) −7.32051 −0.280522
\(682\) 0 0
\(683\) −27.8038 −1.06388 −0.531942 0.846781i \(-0.678538\pi\)
−0.531942 + 0.846781i \(0.678538\pi\)
\(684\) 0 0
\(685\) 3.60770 0.137843
\(686\) 0 0
\(687\) 2.39230 0.0912721
\(688\) 0 0
\(689\) −11.2154 −0.427272
\(690\) 0 0
\(691\) 29.8564 1.13579 0.567896 0.823101i \(-0.307758\pi\)
0.567896 + 0.823101i \(0.307758\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) 14.1436 0.536497
\(696\) 0 0
\(697\) 29.1769 1.10515
\(698\) 0 0
\(699\) −17.3205 −0.655122
\(700\) 0 0
\(701\) −5.21539 −0.196983 −0.0984913 0.995138i \(-0.531402\pi\)
−0.0984913 + 0.995138i \(0.531402\pi\)
\(702\) 0 0
\(703\) −4.92820 −0.185871
\(704\) 0 0
\(705\) 0.143594 0.00540805
\(706\) 0 0
\(707\) −4.73205 −0.177967
\(708\) 0 0
\(709\) 39.3205 1.47671 0.738356 0.674411i \(-0.235602\pi\)
0.738356 + 0.674411i \(0.235602\pi\)
\(710\) 0 0
\(711\) 2.92820 0.109816
\(712\) 0 0
\(713\) 13.8564 0.518927
\(714\) 0 0
\(715\) 4.28719 0.160332
\(716\) 0 0
\(717\) −9.46410 −0.353443
\(718\) 0 0
\(719\) 11.4115 0.425579 0.212789 0.977098i \(-0.431745\pi\)
0.212789 + 0.977098i \(0.431745\pi\)
\(720\) 0 0
\(721\) −2.92820 −0.109052
\(722\) 0 0
\(723\) 10.7846 0.401084
\(724\) 0 0
\(725\) 14.5885 0.541802
\(726\) 0 0
\(727\) −35.7128 −1.32451 −0.662257 0.749276i \(-0.730401\pi\)
−0.662257 + 0.749276i \(0.730401\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.1051 −0.595669
\(732\) 0 0
\(733\) 37.3205 1.37846 0.689232 0.724541i \(-0.257948\pi\)
0.689232 + 0.724541i \(0.257948\pi\)
\(734\) 0 0
\(735\) 0.732051 0.0270021
\(736\) 0 0
\(737\) −21.8564 −0.805091
\(738\) 0 0
\(739\) −47.8564 −1.76043 −0.880213 0.474578i \(-0.842601\pi\)
−0.880213 + 0.474578i \(0.842601\pi\)
\(740\) 0 0
\(741\) −1.46410 −0.0537851
\(742\) 0 0
\(743\) 5.66025 0.207655 0.103827 0.994595i \(-0.466891\pi\)
0.103827 + 0.994595i \(0.466891\pi\)
\(744\) 0 0
\(745\) −2.53590 −0.0929081
\(746\) 0 0
\(747\) 8.19615 0.299882
\(748\) 0 0
\(749\) −12.1962 −0.445638
\(750\) 0 0
\(751\) 24.3923 0.890088 0.445044 0.895509i \(-0.353188\pi\)
0.445044 + 0.895509i \(0.353188\pi\)
\(752\) 0 0
\(753\) −1.66025 −0.0605030
\(754\) 0 0
\(755\) 9.85641 0.358711
\(756\) 0 0
\(757\) −14.7846 −0.537356 −0.268678 0.963230i \(-0.586587\pi\)
−0.268678 + 0.963230i \(0.586587\pi\)
\(758\) 0 0
\(759\) 27.7128 1.00591
\(760\) 0 0
\(761\) −28.0526 −1.01690 −0.508452 0.861090i \(-0.669782\pi\)
−0.508452 + 0.861090i \(0.669782\pi\)
\(762\) 0 0
\(763\) −4.53590 −0.164211
\(764\) 0 0
\(765\) −2.39230 −0.0864940
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) 8.92820 0.321959 0.160980 0.986958i \(-0.448535\pi\)
0.160980 + 0.986958i \(0.448535\pi\)
\(770\) 0 0
\(771\) 2.39230 0.0861568
\(772\) 0 0
\(773\) 42.7846 1.53886 0.769428 0.638734i \(-0.220542\pi\)
0.769428 + 0.638734i \(0.220542\pi\)
\(774\) 0 0
\(775\) 8.92820 0.320711
\(776\) 0 0
\(777\) −4.92820 −0.176798
\(778\) 0 0
\(779\) −8.92820 −0.319886
\(780\) 0 0
\(781\) −16.7846 −0.600601
\(782\) 0 0
\(783\) −3.26795 −0.116787
\(784\) 0 0
\(785\) −2.24871 −0.0802599
\(786\) 0 0
\(787\) 4.14359 0.147703 0.0738516 0.997269i \(-0.476471\pi\)
0.0738516 + 0.997269i \(0.476471\pi\)
\(788\) 0 0
\(789\) 2.53590 0.0902804
\(790\) 0 0
\(791\) 16.7321 0.594923
\(792\) 0 0
\(793\) −1.35898 −0.0482589
\(794\) 0 0
\(795\) 5.60770 0.198884
\(796\) 0 0
\(797\) 14.7846 0.523698 0.261849 0.965109i \(-0.415668\pi\)
0.261849 + 0.965109i \(0.415668\pi\)
\(798\) 0 0
\(799\) −0.641016 −0.0226775
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 41.5692 1.46695
\(804\) 0 0
\(805\) −5.07180 −0.178757
\(806\) 0 0
\(807\) −27.4641 −0.966782
\(808\) 0 0
\(809\) 49.7128 1.74781 0.873905 0.486097i \(-0.161580\pi\)
0.873905 + 0.486097i \(0.161580\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −9.46410 −0.331513
\(816\) 0 0
\(817\) 4.92820 0.172416
\(818\) 0 0
\(819\) −1.46410 −0.0511599
\(820\) 0 0
\(821\) −41.3205 −1.44210 −0.721048 0.692885i \(-0.756339\pi\)
−0.721048 + 0.692885i \(0.756339\pi\)
\(822\) 0 0
\(823\) 22.7846 0.794222 0.397111 0.917771i \(-0.370013\pi\)
0.397111 + 0.917771i \(0.370013\pi\)
\(824\) 0 0
\(825\) 17.8564 0.621680
\(826\) 0 0
\(827\) 16.9808 0.590479 0.295239 0.955423i \(-0.404601\pi\)
0.295239 + 0.955423i \(0.404601\pi\)
\(828\) 0 0
\(829\) −9.71281 −0.337340 −0.168670 0.985673i \(-0.553947\pi\)
−0.168670 + 0.985673i \(0.553947\pi\)
\(830\) 0 0
\(831\) −24.3923 −0.846160
\(832\) 0 0
\(833\) −3.26795 −0.113228
\(834\) 0 0
\(835\) 6.92820 0.239760
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 42.6410 1.47213 0.736066 0.676910i \(-0.236681\pi\)
0.736066 + 0.676910i \(0.236681\pi\)
\(840\) 0 0
\(841\) −18.3205 −0.631742
\(842\) 0 0
\(843\) −4.73205 −0.162980
\(844\) 0 0
\(845\) −7.94744 −0.273400
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) −8.39230 −0.288023
\(850\) 0 0
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 0.732051 0.0250356
\(856\) 0 0
\(857\) −7.85641 −0.268370 −0.134185 0.990956i \(-0.542842\pi\)
−0.134185 + 0.990956i \(0.542842\pi\)
\(858\) 0 0
\(859\) 37.8564 1.29164 0.645822 0.763488i \(-0.276515\pi\)
0.645822 + 0.763488i \(0.276515\pi\)
\(860\) 0 0
\(861\) −8.92820 −0.304272
\(862\) 0 0
\(863\) −41.3731 −1.40836 −0.704178 0.710024i \(-0.748684\pi\)
−0.704178 + 0.710024i \(0.748684\pi\)
\(864\) 0 0
\(865\) −8.67949 −0.295112
\(866\) 0 0
\(867\) −6.32051 −0.214656
\(868\) 0 0
\(869\) −11.7128 −0.397330
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 11.8564 0.401279
\(874\) 0 0
\(875\) −6.92820 −0.234216
\(876\) 0 0
\(877\) −35.5692 −1.20109 −0.600544 0.799592i \(-0.705049\pi\)
−0.600544 + 0.799592i \(0.705049\pi\)
\(878\) 0 0
\(879\) −27.8564 −0.939573
\(880\) 0 0
\(881\) 23.2679 0.783917 0.391959 0.919983i \(-0.371798\pi\)
0.391959 + 0.919983i \(0.371798\pi\)
\(882\) 0 0
\(883\) −28.7846 −0.968679 −0.484340 0.874880i \(-0.660940\pi\)
−0.484340 + 0.874880i \(0.660940\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 41.1769 1.38259 0.691293 0.722575i \(-0.257042\pi\)
0.691293 + 0.722575i \(0.257042\pi\)
\(888\) 0 0
\(889\) −6.53590 −0.219207
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 0.196152 0.00656399
\(894\) 0 0
\(895\) −10.7846 −0.360490
\(896\) 0 0
\(897\) 10.1436 0.338685
\(898\) 0 0
\(899\) 6.53590 0.217984
\(900\) 0 0
\(901\) −25.0333 −0.833981
\(902\) 0 0
\(903\) 4.92820 0.164000
\(904\) 0 0
\(905\) −5.17691 −0.172086
\(906\) 0 0
\(907\) −34.6410 −1.15024 −0.575118 0.818070i \(-0.695044\pi\)
−0.575118 + 0.818070i \(0.695044\pi\)
\(908\) 0 0
\(909\) −4.73205 −0.156952
\(910\) 0 0
\(911\) 47.5167 1.57430 0.787149 0.616763i \(-0.211556\pi\)
0.787149 + 0.616763i \(0.211556\pi\)
\(912\) 0 0
\(913\) −32.7846 −1.08501
\(914\) 0 0
\(915\) 0.679492 0.0224633
\(916\) 0 0
\(917\) 10.7321 0.354404
\(918\) 0 0
\(919\) −3.71281 −0.122474 −0.0612372 0.998123i \(-0.519505\pi\)
−0.0612372 + 0.998123i \(0.519505\pi\)
\(920\) 0 0
\(921\) 16.9282 0.557803
\(922\) 0 0
\(923\) −6.14359 −0.202219
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) 0 0
\(927\) −2.92820 −0.0961748
\(928\) 0 0
\(929\) 26.5885 0.872339 0.436169 0.899865i \(-0.356335\pi\)
0.436169 + 0.899865i \(0.356335\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −7.12436 −0.233241
\(934\) 0 0
\(935\) 9.56922 0.312947
\(936\) 0 0
\(937\) 52.9282 1.72909 0.864545 0.502556i \(-0.167607\pi\)
0.864545 + 0.502556i \(0.167607\pi\)
\(938\) 0 0
\(939\) 19.0718 0.622385
\(940\) 0 0
\(941\) 30.3923 0.990761 0.495380 0.868676i \(-0.335029\pi\)
0.495380 + 0.868676i \(0.335029\pi\)
\(942\) 0 0
\(943\) 61.8564 2.01432
\(944\) 0 0
\(945\) 0.732051 0.0238136
\(946\) 0 0
\(947\) 10.1436 0.329622 0.164811 0.986325i \(-0.447299\pi\)
0.164811 + 0.986325i \(0.447299\pi\)
\(948\) 0 0
\(949\) 15.2154 0.493912
\(950\) 0 0
\(951\) −19.2679 −0.624806
\(952\) 0 0
\(953\) −1.41154 −0.0457244 −0.0228622 0.999739i \(-0.507278\pi\)
−0.0228622 + 0.999739i \(0.507278\pi\)
\(954\) 0 0
\(955\) 13.8564 0.448383
\(956\) 0 0
\(957\) 13.0718 0.422551
\(958\) 0 0
\(959\) 4.92820 0.159140
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −12.1962 −0.393016
\(964\) 0 0
\(965\) 6.53590 0.210398
\(966\) 0 0
\(967\) −30.7846 −0.989966 −0.494983 0.868903i \(-0.664826\pi\)
−0.494983 + 0.868903i \(0.664826\pi\)
\(968\) 0 0
\(969\) −3.26795 −0.104982
\(970\) 0 0
\(971\) −15.2154 −0.488285 −0.244143 0.969739i \(-0.578506\pi\)
−0.244143 + 0.969739i \(0.578506\pi\)
\(972\) 0 0
\(973\) 19.3205 0.619387
\(974\) 0 0
\(975\) 6.53590 0.209316
\(976\) 0 0
\(977\) 25.5167 0.816350 0.408175 0.912904i \(-0.366165\pi\)
0.408175 + 0.912904i \(0.366165\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −4.53590 −0.144820
\(982\) 0 0
\(983\) 8.78461 0.280186 0.140093 0.990138i \(-0.455260\pi\)
0.140093 + 0.990138i \(0.455260\pi\)
\(984\) 0 0
\(985\) 0.679492 0.0216504
\(986\) 0 0
\(987\) 0.196152 0.00624360
\(988\) 0 0
\(989\) −34.1436 −1.08570
\(990\) 0 0
\(991\) −30.6410 −0.973344 −0.486672 0.873585i \(-0.661789\pi\)
−0.486672 + 0.873585i \(0.661789\pi\)
\(992\) 0 0
\(993\) 8.39230 0.266322
\(994\) 0 0
\(995\) 14.1436 0.448382
\(996\) 0 0
\(997\) −43.0718 −1.36410 −0.682049 0.731307i \(-0.738911\pi\)
−0.682049 + 0.731307i \(0.738911\pi\)
\(998\) 0 0
\(999\) −4.92820 −0.155921
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 3192.2.a.s.1.2 2
3.2 odd 2 9576.2.a.bw.1.1 2
4.3 odd 2 6384.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.s.1.2 2 1.1 even 1 trivial
6384.2.a.bi.1.2 2 4.3 odd 2
9576.2.a.bw.1.1 2 3.2 odd 2