Properties

Label 4368.2.h.t.337.4
Level $4368$
Weight $2$
Character 4368.337
Analytic conductor $34.879$
Analytic rank $0$
Dimension $10$
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] = [4368,2,Mod(337,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4368.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(-0.535829 + 0.535829i\) of defining polynomial
Character \(\chi\) \(=\) 4368.337
Dual form 4368.2.h.t.337.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.660872i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.660872i q^{5} -1.00000i q^{7} +1.00000 q^{9} -0.393402i q^{11} +(3.36543 + 1.29378i) q^{13} -0.660872i q^{15} -7.29578 q^{17} +6.32008i q^{19} -1.00000i q^{21} -5.56325 q^{23} +4.56325 q^{25} +1.00000 q^{27} -2.97570 q^{29} +8.46340i q^{31} -0.393402i q^{33} -0.660872 q^{35} +10.7608i q^{37} +(3.36543 + 1.29378i) q^{39} +6.81167i q^{41} -0.998337 q^{43} -0.660872i q^{45} -6.04736i q^{47} -1.00000 q^{49} -7.29578 q^{51} +11.6159 q^{53} -0.259989 q^{55} +6.32008i q^{57} +1.07166i q^{59} +6.73087 q^{61} -1.00000i q^{63} +(0.855019 - 2.22412i) q^{65} +12.4709i q^{67} -5.56325 q^{69} +8.39340i q^{71} -7.39397i q^{73} +4.56325 q^{75} -0.393402 q^{77} +5.41993 q^{79} +1.00000 q^{81} -2.75158i q^{83} +4.82157i q^{85} -2.97570 q^{87} -5.84810i q^{89} +(1.29378 - 3.36543i) q^{91} +8.46340i q^{93} +4.17676 q^{95} -6.22937i q^{97} -0.393402i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 10 q^{9} + 2 q^{13} - 2 q^{23} - 8 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{35} + 2 q^{39} + 34 q^{43} - 10 q^{49} - 2 q^{53} - 64 q^{55} + 4 q^{61} + 2 q^{65} - 2 q^{69} - 8 q^{75} + 16 q^{77} + 38 q^{79} + 10 q^{81} - 2 q^{87} + 34 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.660872i 0.295551i −0.989021 0.147775i \(-0.952789\pi\)
0.989021 0.147775i \(-0.0472113\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.393402i 0.118615i −0.998240 0.0593077i \(-0.981111\pi\)
0.998240 0.0593077i \(-0.0188893\pi\)
\(12\) 0 0
\(13\) 3.36543 + 1.29378i 0.933403 + 0.358829i
\(14\) 0 0
\(15\) 0.660872i 0.170636i
\(16\) 0 0
\(17\) −7.29578 −1.76949 −0.884743 0.466079i \(-0.845666\pi\)
−0.884743 + 0.466079i \(0.845666\pi\)
\(18\) 0 0
\(19\) 6.32008i 1.44993i 0.688788 + 0.724963i \(0.258143\pi\)
−0.688788 + 0.724963i \(0.741857\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) −5.56325 −1.16002 −0.580009 0.814610i \(-0.696951\pi\)
−0.580009 + 0.814610i \(0.696951\pi\)
\(24\) 0 0
\(25\) 4.56325 0.912650
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.97570 −0.552573 −0.276287 0.961075i \(-0.589104\pi\)
−0.276287 + 0.961075i \(0.589104\pi\)
\(30\) 0 0
\(31\) 8.46340i 1.52007i 0.649881 + 0.760036i \(0.274819\pi\)
−0.649881 + 0.760036i \(0.725181\pi\)
\(32\) 0 0
\(33\) 0.393402i 0.0684826i
\(34\) 0 0
\(35\) −0.660872 −0.111708
\(36\) 0 0
\(37\) 10.7608i 1.76907i 0.466473 + 0.884536i \(0.345525\pi\)
−0.466473 + 0.884536i \(0.654475\pi\)
\(38\) 0 0
\(39\) 3.36543 + 1.29378i 0.538901 + 0.207170i
\(40\) 0 0
\(41\) 6.81167i 1.06380i 0.846806 + 0.531902i \(0.178523\pi\)
−0.846806 + 0.531902i \(0.821477\pi\)
\(42\) 0 0
\(43\) −0.998337 −0.152245 −0.0761225 0.997098i \(-0.524254\pi\)
−0.0761225 + 0.997098i \(0.524254\pi\)
\(44\) 0 0
\(45\) 0.660872i 0.0985169i
\(46\) 0 0
\(47\) 6.04736i 0.882098i −0.897483 0.441049i \(-0.854607\pi\)
0.897483 0.441049i \(-0.145393\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −7.29578 −1.02161
\(52\) 0 0
\(53\) 11.6159 1.59556 0.797781 0.602948i \(-0.206007\pi\)
0.797781 + 0.602948i \(0.206007\pi\)
\(54\) 0 0
\(55\) −0.259989 −0.0350569
\(56\) 0 0
\(57\) 6.32008i 0.837115i
\(58\) 0 0
\(59\) 1.07166i 0.139518i 0.997564 + 0.0697591i \(0.0222230\pi\)
−0.997564 + 0.0697591i \(0.977777\pi\)
\(60\) 0 0
\(61\) 6.73087 0.861799 0.430900 0.902400i \(-0.358196\pi\)
0.430900 + 0.902400i \(0.358196\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0.855019 2.22412i 0.106052 0.275868i
\(66\) 0 0
\(67\) 12.4709i 1.52356i 0.647836 + 0.761780i \(0.275674\pi\)
−0.647836 + 0.761780i \(0.724326\pi\)
\(68\) 0 0
\(69\) −5.56325 −0.669736
\(70\) 0 0
\(71\) 8.39340i 0.996114i 0.867144 + 0.498057i \(0.165953\pi\)
−0.867144 + 0.498057i \(0.834047\pi\)
\(72\) 0 0
\(73\) 7.39397i 0.865398i −0.901538 0.432699i \(-0.857561\pi\)
0.901538 0.432699i \(-0.142439\pi\)
\(74\) 0 0
\(75\) 4.56325 0.526919
\(76\) 0 0
\(77\) −0.393402 −0.0448324
\(78\) 0 0
\(79\) 5.41993 0.609790 0.304895 0.952386i \(-0.401379\pi\)
0.304895 + 0.952386i \(0.401379\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.75158i 0.302025i −0.988532 0.151012i \(-0.951747\pi\)
0.988532 0.151012i \(-0.0482534\pi\)
\(84\) 0 0
\(85\) 4.82157i 0.522973i
\(86\) 0 0
\(87\) −2.97570 −0.319028
\(88\) 0 0
\(89\) 5.84810i 0.619898i −0.950753 0.309949i \(-0.899688\pi\)
0.950753 0.309949i \(-0.100312\pi\)
\(90\) 0 0
\(91\) 1.29378 3.36543i 0.135624 0.352793i
\(92\) 0 0
\(93\) 8.46340i 0.877614i
\(94\) 0 0
\(95\) 4.17676 0.428527
\(96\) 0 0
\(97\) 6.22937i 0.632497i −0.948676 0.316249i \(-0.897577\pi\)
0.948676 0.316249i \(-0.102423\pi\)
\(98\) 0 0
\(99\) 0.393402i 0.0395384i
\(100\) 0 0
\(101\) 16.1174 1.60374 0.801868 0.597501i \(-0.203840\pi\)
0.801868 + 0.597501i \(0.203840\pi\)
\(102\) 0 0
\(103\) −18.3392 −1.80702 −0.903510 0.428567i \(-0.859018\pi\)
−0.903510 + 0.428567i \(0.859018\pi\)
\(104\) 0 0
\(105\) −0.660872 −0.0644945
\(106\) 0 0
\(107\) −4.58755 −0.443495 −0.221748 0.975104i \(-0.571176\pi\)
−0.221748 + 0.975104i \(0.571176\pi\)
\(108\) 0 0
\(109\) 8.47088i 0.811363i −0.914014 0.405682i \(-0.867034\pi\)
0.914014 0.405682i \(-0.132966\pi\)
\(110\) 0 0
\(111\) 10.7608i 1.02137i
\(112\) 0 0
\(113\) 6.46340 0.608025 0.304013 0.952668i \(-0.401673\pi\)
0.304013 + 0.952668i \(0.401673\pi\)
\(114\) 0 0
\(115\) 3.67659i 0.342844i
\(116\) 0 0
\(117\) 3.36543 + 1.29378i 0.311134 + 0.119610i
\(118\) 0 0
\(119\) 7.29578i 0.668803i
\(120\) 0 0
\(121\) 10.8452 0.985930
\(122\) 0 0
\(123\) 6.81167i 0.614188i
\(124\) 0 0
\(125\) 6.32008i 0.565285i
\(126\) 0 0
\(127\) −6.22189 −0.552104 −0.276052 0.961143i \(-0.589026\pi\)
−0.276052 + 0.961143i \(0.589026\pi\)
\(128\) 0 0
\(129\) −0.998337 −0.0878987
\(130\) 0 0
\(131\) −3.12650 −0.273163 −0.136582 0.990629i \(-0.543612\pi\)
−0.136582 + 0.990629i \(0.543612\pi\)
\(132\) 0 0
\(133\) 6.32008 0.548021
\(134\) 0 0
\(135\) 0.660872i 0.0568788i
\(136\) 0 0
\(137\) 9.51990i 0.813340i 0.913575 + 0.406670i \(0.133310\pi\)
−0.913575 + 0.406670i \(0.866690\pi\)
\(138\) 0 0
\(139\) −16.0266 −1.35936 −0.679681 0.733508i \(-0.737882\pi\)
−0.679681 + 0.733508i \(0.737882\pi\)
\(140\) 0 0
\(141\) 6.04736i 0.509279i
\(142\) 0 0
\(143\) 0.508974 1.32397i 0.0425626 0.110716i
\(144\) 0 0
\(145\) 1.96656i 0.163314i
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 1.57183i 0.128769i −0.997925 0.0643846i \(-0.979492\pi\)
0.997925 0.0643846i \(-0.0205084\pi\)
\(150\) 0 0
\(151\) 5.12982i 0.417459i −0.977973 0.208730i \(-0.933067\pi\)
0.977973 0.208730i \(-0.0669328\pi\)
\(152\) 0 0
\(153\) −7.29578 −0.589829
\(154\) 0 0
\(155\) 5.59322 0.449258
\(156\) 0 0
\(157\) 10.6628 0.850984 0.425492 0.904962i \(-0.360101\pi\)
0.425492 + 0.904962i \(0.360101\pi\)
\(158\) 0 0
\(159\) 11.6159 0.921198
\(160\) 0 0
\(161\) 5.56325i 0.438445i
\(162\) 0 0
\(163\) 4.99086i 0.390914i −0.980712 0.195457i \(-0.937381\pi\)
0.980712 0.195457i \(-0.0626190\pi\)
\(164\) 0 0
\(165\) −0.259989 −0.0202401
\(166\) 0 0
\(167\) 14.7223i 1.13924i 0.821906 + 0.569622i \(0.192911\pi\)
−0.821906 + 0.569622i \(0.807089\pi\)
\(168\) 0 0
\(169\) 9.65229 + 8.70823i 0.742484 + 0.669864i
\(170\) 0 0
\(171\) 6.32008i 0.483309i
\(172\) 0 0
\(173\) −1.52580 −0.116004 −0.0580020 0.998316i \(-0.518473\pi\)
−0.0580020 + 0.998316i \(0.518473\pi\)
\(174\) 0 0
\(175\) 4.56325i 0.344949i
\(176\) 0 0
\(177\) 1.07166i 0.0805508i
\(178\) 0 0
\(179\) 8.41079 0.628652 0.314326 0.949315i \(-0.398222\pi\)
0.314326 + 0.949315i \(0.398222\pi\)
\(180\) 0 0
\(181\) 2.02264 0.150342 0.0751708 0.997171i \(-0.476050\pi\)
0.0751708 + 0.997171i \(0.476050\pi\)
\(182\) 0 0
\(183\) 6.73087 0.497560
\(184\) 0 0
\(185\) 7.11154 0.522850
\(186\) 0 0
\(187\) 2.87018i 0.209888i
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 16.1352 1.16750 0.583751 0.811933i \(-0.301584\pi\)
0.583751 + 0.811933i \(0.301584\pi\)
\(192\) 0 0
\(193\) 7.85336i 0.565297i −0.959224 0.282648i \(-0.908787\pi\)
0.959224 0.282648i \(-0.0912130\pi\)
\(194\) 0 0
\(195\) 0.855019 2.22412i 0.0612292 0.159273i
\(196\) 0 0
\(197\) 17.7013i 1.26117i 0.776122 + 0.630583i \(0.217184\pi\)
−0.776122 + 0.630583i \(0.782816\pi\)
\(198\) 0 0
\(199\) 23.9012 1.69431 0.847155 0.531347i \(-0.178314\pi\)
0.847155 + 0.531347i \(0.178314\pi\)
\(200\) 0 0
\(201\) 12.4709i 0.879628i
\(202\) 0 0
\(203\) 2.97570i 0.208853i
\(204\) 0 0
\(205\) 4.50164 0.314408
\(206\) 0 0
\(207\) −5.56325 −0.386673
\(208\) 0 0
\(209\) 2.48634 0.171983
\(210\) 0 0
\(211\) −25.7215 −1.77074 −0.885372 0.464884i \(-0.846096\pi\)
−0.885372 + 0.464884i \(0.846096\pi\)
\(212\) 0 0
\(213\) 8.39340i 0.575106i
\(214\) 0 0
\(215\) 0.659773i 0.0449961i
\(216\) 0 0
\(217\) 8.46340 0.574533
\(218\) 0 0
\(219\) 7.39397i 0.499638i
\(220\) 0 0
\(221\) −24.5535 9.43910i −1.65164 0.634942i
\(222\) 0 0
\(223\) 21.7851i 1.45884i 0.684066 + 0.729421i \(0.260210\pi\)
−0.684066 + 0.729421i \(0.739790\pi\)
\(224\) 0 0
\(225\) 4.56325 0.304217
\(226\) 0 0
\(227\) 14.1948i 0.942144i −0.882095 0.471072i \(-0.843867\pi\)
0.882095 0.471072i \(-0.156133\pi\)
\(228\) 0 0
\(229\) 23.1410i 1.52920i 0.644504 + 0.764601i \(0.277064\pi\)
−0.644504 + 0.764601i \(0.722936\pi\)
\(230\) 0 0
\(231\) −0.393402 −0.0258840
\(232\) 0 0
\(233\) −2.84119 −0.186132 −0.0930661 0.995660i \(-0.529667\pi\)
−0.0930661 + 0.995660i \(0.529667\pi\)
\(234\) 0 0
\(235\) −3.99653 −0.260705
\(236\) 0 0
\(237\) 5.41993 0.352062
\(238\) 0 0
\(239\) 6.34480i 0.410411i 0.978719 + 0.205205i \(0.0657863\pi\)
−0.978719 + 0.205205i \(0.934214\pi\)
\(240\) 0 0
\(241\) 4.21554i 0.271547i 0.990740 + 0.135773i \(0.0433519\pi\)
−0.990740 + 0.135773i \(0.956648\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.660872i 0.0422215i
\(246\) 0 0
\(247\) −8.17676 + 21.2698i −0.520275 + 1.35337i
\(248\) 0 0
\(249\) 2.75158i 0.174374i
\(250\) 0 0
\(251\) 13.1751 0.831605 0.415802 0.909455i \(-0.363501\pi\)
0.415802 + 0.909455i \(0.363501\pi\)
\(252\) 0 0
\(253\) 2.18860i 0.137596i
\(254\) 0 0
\(255\) 4.82157i 0.301939i
\(256\) 0 0
\(257\) −8.45592 −0.527466 −0.263733 0.964596i \(-0.584954\pi\)
−0.263733 + 0.964596i \(0.584954\pi\)
\(258\) 0 0
\(259\) 10.7608 0.668646
\(260\) 0 0
\(261\) −2.97570 −0.184191
\(262\) 0 0
\(263\) −29.6685 −1.82944 −0.914718 0.404092i \(-0.867587\pi\)
−0.914718 + 0.404092i \(0.867587\pi\)
\(264\) 0 0
\(265\) 7.67659i 0.471569i
\(266\) 0 0
\(267\) 5.84810i 0.357898i
\(268\) 0 0
\(269\) −25.8274 −1.57472 −0.787362 0.616491i \(-0.788554\pi\)
−0.787362 + 0.616491i \(0.788554\pi\)
\(270\) 0 0
\(271\) 28.5015i 1.73134i 0.500611 + 0.865672i \(0.333109\pi\)
−0.500611 + 0.865672i \(0.666891\pi\)
\(272\) 0 0
\(273\) 1.29378 3.36543i 0.0783028 0.203685i
\(274\) 0 0
\(275\) 1.79519i 0.108254i
\(276\) 0 0
\(277\) 6.52334 0.391950 0.195975 0.980609i \(-0.437213\pi\)
0.195975 + 0.980609i \(0.437213\pi\)
\(278\) 0 0
\(279\) 8.46340i 0.506690i
\(280\) 0 0
\(281\) 9.66654i 0.576658i −0.957531 0.288329i \(-0.906900\pi\)
0.957531 0.288329i \(-0.0930996\pi\)
\(282\) 0 0
\(283\) −16.1052 −0.957356 −0.478678 0.877990i \(-0.658884\pi\)
−0.478678 + 0.877990i \(0.658884\pi\)
\(284\) 0 0
\(285\) 4.17676 0.247410
\(286\) 0 0
\(287\) 6.81167 0.402080
\(288\) 0 0
\(289\) 36.2284 2.13108
\(290\) 0 0
\(291\) 6.22937i 0.365172i
\(292\) 0 0
\(293\) 14.5004i 0.847122i −0.905868 0.423561i \(-0.860780\pi\)
0.905868 0.423561i \(-0.139220\pi\)
\(294\) 0 0
\(295\) 0.708229 0.0412347
\(296\) 0 0
\(297\) 0.393402i 0.0228275i
\(298\) 0 0
\(299\) −18.7227 7.19759i −1.08276 0.416248i
\(300\) 0 0
\(301\) 0.998337i 0.0575432i
\(302\) 0 0
\(303\) 16.1174 0.925918
\(304\) 0 0
\(305\) 4.44824i 0.254706i
\(306\) 0 0
\(307\) 0.124834i 0.00712467i 0.999994 + 0.00356234i \(0.00113393\pi\)
−0.999994 + 0.00356234i \(0.998866\pi\)
\(308\) 0 0
\(309\) −18.3392 −1.04328
\(310\) 0 0
\(311\) 4.92680 0.279373 0.139687 0.990196i \(-0.455391\pi\)
0.139687 + 0.990196i \(0.455391\pi\)
\(312\) 0 0
\(313\) −32.8701 −1.85793 −0.928963 0.370172i \(-0.879299\pi\)
−0.928963 + 0.370172i \(0.879299\pi\)
\(314\) 0 0
\(315\) −0.660872 −0.0372359
\(316\) 0 0
\(317\) 25.0302i 1.40584i 0.711270 + 0.702919i \(0.248120\pi\)
−0.711270 + 0.702919i \(0.751880\pi\)
\(318\) 0 0
\(319\) 1.17065i 0.0655437i
\(320\) 0 0
\(321\) −4.58755 −0.256052
\(322\) 0 0
\(323\) 46.1099i 2.56562i
\(324\) 0 0
\(325\) 15.3573 + 5.90382i 0.851870 + 0.327485i
\(326\) 0 0
\(327\) 8.47088i 0.468441i
\(328\) 0 0
\(329\) −6.04736 −0.333402
\(330\) 0 0
\(331\) 26.0793i 1.43345i −0.697358 0.716723i \(-0.745641\pi\)
0.697358 0.716723i \(-0.254359\pi\)
\(332\) 0 0
\(333\) 10.7608i 0.589690i
\(334\) 0 0
\(335\) 8.24165 0.450290
\(336\) 0 0
\(337\) 9.02831 0.491803 0.245902 0.969295i \(-0.420916\pi\)
0.245902 + 0.969295i \(0.420916\pi\)
\(338\) 0 0
\(339\) 6.46340 0.351044
\(340\) 0 0
\(341\) 3.32952 0.180304
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 3.67659i 0.197941i
\(346\) 0 0
\(347\) 27.1272 1.45626 0.728132 0.685437i \(-0.240389\pi\)
0.728132 + 0.685437i \(0.240389\pi\)
\(348\) 0 0
\(349\) 22.5205i 1.20549i 0.797933 + 0.602746i \(0.205927\pi\)
−0.797933 + 0.602746i \(0.794073\pi\)
\(350\) 0 0
\(351\) 3.36543 + 1.29378i 0.179634 + 0.0690566i
\(352\) 0 0
\(353\) 30.4767i 1.62211i −0.584970 0.811055i \(-0.698894\pi\)
0.584970 0.811055i \(-0.301106\pi\)
\(354\) 0 0
\(355\) 5.54696 0.294402
\(356\) 0 0
\(357\) 7.29578i 0.386134i
\(358\) 0 0
\(359\) 22.0515i 1.16383i −0.813248 0.581917i \(-0.802303\pi\)
0.813248 0.581917i \(-0.197697\pi\)
\(360\) 0 0
\(361\) −20.9434 −1.10229
\(362\) 0 0
\(363\) 10.8452 0.569227
\(364\) 0 0
\(365\) −4.88646 −0.255769
\(366\) 0 0
\(367\) 21.3363 1.11374 0.556872 0.830598i \(-0.312001\pi\)
0.556872 + 0.830598i \(0.312001\pi\)
\(368\) 0 0
\(369\) 6.81167i 0.354601i
\(370\) 0 0
\(371\) 11.6159i 0.603065i
\(372\) 0 0
\(373\) −19.6003 −1.01486 −0.507431 0.861692i \(-0.669405\pi\)
−0.507431 + 0.861692i \(0.669405\pi\)
\(374\) 0 0
\(375\) 6.32008i 0.326368i
\(376\) 0 0
\(377\) −10.0145 3.84988i −0.515774 0.198279i
\(378\) 0 0
\(379\) 20.5656i 1.05638i −0.849125 0.528192i \(-0.822870\pi\)
0.849125 0.528192i \(-0.177130\pi\)
\(380\) 0 0
\(381\) −6.22189 −0.318757
\(382\) 0 0
\(383\) 23.0259i 1.17657i 0.808654 + 0.588284i \(0.200196\pi\)
−0.808654 + 0.588284i \(0.799804\pi\)
\(384\) 0 0
\(385\) 0.259989i 0.0132502i
\(386\) 0 0
\(387\) −0.998337 −0.0507483
\(388\) 0 0
\(389\) −0.967723 −0.0490655 −0.0245328 0.999699i \(-0.507810\pi\)
−0.0245328 + 0.999699i \(0.507810\pi\)
\(390\) 0 0
\(391\) 40.5882 2.05264
\(392\) 0 0
\(393\) −3.12650 −0.157711
\(394\) 0 0
\(395\) 3.58188i 0.180224i
\(396\) 0 0
\(397\) 32.5274i 1.63250i −0.577698 0.816251i \(-0.696049\pi\)
0.577698 0.816251i \(-0.303951\pi\)
\(398\) 0 0
\(399\) 6.32008 0.316400
\(400\) 0 0
\(401\) 28.1220i 1.40434i 0.712007 + 0.702172i \(0.247786\pi\)
−0.712007 + 0.702172i \(0.752214\pi\)
\(402\) 0 0
\(403\) −10.9497 + 28.4830i −0.545445 + 1.41884i
\(404\) 0 0
\(405\) 0.660872i 0.0328390i
\(406\) 0 0
\(407\) 4.23334 0.209839
\(408\) 0 0
\(409\) 7.87252i 0.389271i −0.980876 0.194636i \(-0.937648\pi\)
0.980876 0.194636i \(-0.0623524\pi\)
\(410\) 0 0
\(411\) 9.51990i 0.469582i
\(412\) 0 0
\(413\) 1.07166 0.0527329
\(414\) 0 0
\(415\) −1.81844 −0.0892637
\(416\) 0 0
\(417\) −16.0266 −0.784828
\(418\) 0 0
\(419\) 6.81825 0.333093 0.166547 0.986034i \(-0.446738\pi\)
0.166547 + 0.986034i \(0.446738\pi\)
\(420\) 0 0
\(421\) 18.7713i 0.914860i 0.889246 + 0.457430i \(0.151230\pi\)
−0.889246 + 0.457430i \(0.848770\pi\)
\(422\) 0 0
\(423\) 6.04736i 0.294033i
\(424\) 0 0
\(425\) −33.2925 −1.61492
\(426\) 0 0
\(427\) 6.73087i 0.325730i
\(428\) 0 0
\(429\) 0.508974 1.32397i 0.0245735 0.0639219i
\(430\) 0 0
\(431\) 13.4668i 0.648675i 0.945941 + 0.324338i \(0.105141\pi\)
−0.945941 + 0.324338i \(0.894859\pi\)
\(432\) 0 0
\(433\) −14.3097 −0.687681 −0.343841 0.939028i \(-0.611728\pi\)
−0.343841 + 0.939028i \(0.611728\pi\)
\(434\) 0 0
\(435\) 1.96656i 0.0942891i
\(436\) 0 0
\(437\) 35.1602i 1.68194i
\(438\) 0 0
\(439\) −18.1126 −0.864465 −0.432233 0.901762i \(-0.642274\pi\)
−0.432233 + 0.901762i \(0.642274\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) −18.8662 −0.896361 −0.448180 0.893943i \(-0.647928\pi\)
−0.448180 + 0.893943i \(0.647928\pi\)
\(444\) 0 0
\(445\) −3.86485 −0.183211
\(446\) 0 0
\(447\) 1.57183i 0.0743449i
\(448\) 0 0
\(449\) 34.0029i 1.60470i −0.596856 0.802348i \(-0.703584\pi\)
0.596856 0.802348i \(-0.296416\pi\)
\(450\) 0 0
\(451\) 2.67973 0.126183
\(452\) 0 0
\(453\) 5.12982i 0.241020i
\(454\) 0 0
\(455\) −2.22412 0.855019i −0.104268 0.0400839i
\(456\) 0 0
\(457\) 2.83541i 0.132635i −0.997799 0.0663174i \(-0.978875\pi\)
0.997799 0.0663174i \(-0.0211250\pi\)
\(458\) 0 0
\(459\) −7.29578 −0.340538
\(460\) 0 0
\(461\) 2.81935i 0.131310i −0.997842 0.0656551i \(-0.979086\pi\)
0.997842 0.0656551i \(-0.0209137\pi\)
\(462\) 0 0
\(463\) 22.8434i 1.06162i −0.847490 0.530812i \(-0.821887\pi\)
0.847490 0.530812i \(-0.178113\pi\)
\(464\) 0 0
\(465\) 5.59322 0.259379
\(466\) 0 0
\(467\) −31.0146 −1.43519 −0.717593 0.696463i \(-0.754756\pi\)
−0.717593 + 0.696463i \(0.754756\pi\)
\(468\) 0 0
\(469\) 12.4709 0.575852
\(470\) 0 0
\(471\) 10.6628 0.491316
\(472\) 0 0
\(473\) 0.392748i 0.0180586i
\(474\) 0 0
\(475\) 28.8401i 1.32327i
\(476\) 0 0
\(477\) 11.6159 0.531854
\(478\) 0 0
\(479\) 3.40387i 0.155527i 0.996972 + 0.0777634i \(0.0247779\pi\)
−0.996972 + 0.0777634i \(0.975222\pi\)
\(480\) 0 0
\(481\) −13.9221 + 36.2149i −0.634793 + 1.65126i
\(482\) 0 0
\(483\) 5.56325i 0.253137i
\(484\) 0 0
\(485\) −4.11682 −0.186935
\(486\) 0 0
\(487\) 29.7484i 1.34803i 0.738719 + 0.674014i \(0.235431\pi\)
−0.738719 + 0.674014i \(0.764569\pi\)
\(488\) 0 0
\(489\) 4.99086i 0.225694i
\(490\) 0 0
\(491\) 25.3180 1.14258 0.571292 0.820747i \(-0.306442\pi\)
0.571292 + 0.820747i \(0.306442\pi\)
\(492\) 0 0
\(493\) 21.7100 0.977771
\(494\) 0 0
\(495\) −0.259989 −0.0116856
\(496\) 0 0
\(497\) 8.39340 0.376496
\(498\) 0 0
\(499\) 4.52726i 0.202668i 0.994852 + 0.101334i \(0.0323111\pi\)
−0.994852 + 0.101334i \(0.967689\pi\)
\(500\) 0 0
\(501\) 14.7223i 0.657743i
\(502\) 0 0
\(503\) 2.98205 0.132963 0.0664815 0.997788i \(-0.478823\pi\)
0.0664815 + 0.997788i \(0.478823\pi\)
\(504\) 0 0
\(505\) 10.6515i 0.473986i
\(506\) 0 0
\(507\) 9.65229 + 8.70823i 0.428673 + 0.386746i
\(508\) 0 0
\(509\) 32.2809i 1.43083i 0.698702 + 0.715413i \(0.253761\pi\)
−0.698702 + 0.715413i \(0.746239\pi\)
\(510\) 0 0
\(511\) −7.39397 −0.327090
\(512\) 0 0
\(513\) 6.32008i 0.279038i
\(514\) 0 0
\(515\) 12.1199i 0.534066i
\(516\) 0 0
\(517\) −2.37905 −0.104630
\(518\) 0 0
\(519\) −1.52580 −0.0669750
\(520\) 0 0
\(521\) −39.8145 −1.74430 −0.872152 0.489235i \(-0.837276\pi\)
−0.872152 + 0.489235i \(0.837276\pi\)
\(522\) 0 0
\(523\) −1.77365 −0.0775564 −0.0387782 0.999248i \(-0.512347\pi\)
−0.0387782 + 0.999248i \(0.512347\pi\)
\(524\) 0 0
\(525\) 4.56325i 0.199156i
\(526\) 0 0
\(527\) 61.7471i 2.68975i
\(528\) 0 0
\(529\) 7.94973 0.345641
\(530\) 0 0
\(531\) 1.07166i 0.0465060i
\(532\) 0 0
\(533\) −8.81277 + 22.9242i −0.381723 + 0.992959i
\(534\) 0 0
\(535\) 3.03178i 0.131075i
\(536\) 0 0
\(537\) 8.41079 0.362952
\(538\) 0 0
\(539\) 0.393402i 0.0169450i
\(540\) 0 0
\(541\) 23.0972i 0.993027i 0.868029 + 0.496513i \(0.165387\pi\)
−0.868029 + 0.496513i \(0.834613\pi\)
\(542\) 0 0
\(543\) 2.02264 0.0867997
\(544\) 0 0
\(545\) −5.59816 −0.239799
\(546\) 0 0
\(547\) 12.5995 0.538717 0.269359 0.963040i \(-0.413188\pi\)
0.269359 + 0.963040i \(0.413188\pi\)
\(548\) 0 0
\(549\) 6.73087 0.287266
\(550\) 0 0
\(551\) 18.8067i 0.801190i
\(552\) 0 0
\(553\) 5.41993i 0.230479i
\(554\) 0 0
\(555\) 7.11154 0.301868
\(556\) 0 0
\(557\) 21.0197i 0.890635i −0.895373 0.445317i \(-0.853091\pi\)
0.895373 0.445317i \(-0.146909\pi\)
\(558\) 0 0
\(559\) −3.35984 1.29162i −0.142106 0.0546299i
\(560\) 0 0
\(561\) 2.87018i 0.121179i
\(562\) 0 0
\(563\) 18.6818 0.787345 0.393672 0.919251i \(-0.371204\pi\)
0.393672 + 0.919251i \(0.371204\pi\)
\(564\) 0 0
\(565\) 4.27148i 0.179702i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 13.0469 0.546956 0.273478 0.961878i \(-0.411826\pi\)
0.273478 + 0.961878i \(0.411826\pi\)
\(570\) 0 0
\(571\) 19.9203 0.833640 0.416820 0.908989i \(-0.363145\pi\)
0.416820 + 0.908989i \(0.363145\pi\)
\(572\) 0 0
\(573\) 16.1352 0.674057
\(574\) 0 0
\(575\) −25.3865 −1.05869
\(576\) 0 0
\(577\) 23.0394i 0.959141i 0.877503 + 0.479570i \(0.159208\pi\)
−0.877503 + 0.479570i \(0.840792\pi\)
\(578\) 0 0
\(579\) 7.85336i 0.326374i
\(580\) 0 0
\(581\) −2.75158 −0.114155
\(582\) 0 0
\(583\) 4.56971i 0.189258i
\(584\) 0 0
\(585\) 0.855019 2.22412i 0.0353507 0.0919561i
\(586\) 0 0
\(587\) 11.9372i 0.492700i 0.969181 + 0.246350i \(0.0792313\pi\)
−0.969181 + 0.246350i \(0.920769\pi\)
\(588\) 0 0
\(589\) −53.4894 −2.20399
\(590\) 0 0
\(591\) 17.7013i 0.728135i
\(592\) 0 0
\(593\) 43.4795i 1.78549i 0.450565 + 0.892744i \(0.351223\pi\)
−0.450565 + 0.892744i \(0.648777\pi\)
\(594\) 0 0
\(595\) 4.82157 0.197665
\(596\) 0 0
\(597\) 23.9012 0.978210
\(598\) 0 0
\(599\) 30.2417 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(600\) 0 0
\(601\) 18.8654 0.769535 0.384767 0.923014i \(-0.374282\pi\)
0.384767 + 0.923014i \(0.374282\pi\)
\(602\) 0 0
\(603\) 12.4709i 0.507853i
\(604\) 0 0
\(605\) 7.16731i 0.291393i
\(606\) 0 0
\(607\) −2.67781 −0.108689 −0.0543445 0.998522i \(-0.517307\pi\)
−0.0543445 + 0.998522i \(0.517307\pi\)
\(608\) 0 0
\(609\) 2.97570i 0.120581i
\(610\) 0 0
\(611\) 7.82392 20.3520i 0.316522 0.823353i
\(612\) 0 0
\(613\) 23.7973i 0.961164i −0.876950 0.480582i \(-0.840425\pi\)
0.876950 0.480582i \(-0.159575\pi\)
\(614\) 0 0
\(615\) 4.50164 0.181524
\(616\) 0 0
\(617\) 25.1250i 1.01149i 0.862682 + 0.505746i \(0.168783\pi\)
−0.862682 + 0.505746i \(0.831217\pi\)
\(618\) 0 0
\(619\) 10.1886i 0.409514i −0.978813 0.204757i \(-0.934360\pi\)
0.978813 0.204757i \(-0.0656405\pi\)
\(620\) 0 0
\(621\) −5.56325 −0.223245
\(622\) 0 0
\(623\) −5.84810 −0.234299
\(624\) 0 0
\(625\) 18.6395 0.745579
\(626\) 0 0
\(627\) 2.48634 0.0992947
\(628\) 0 0
\(629\) 78.5087i 3.13035i
\(630\) 0 0
\(631\) 5.80475i 0.231084i 0.993303 + 0.115542i \(0.0368604\pi\)
−0.993303 + 0.115542i \(0.963140\pi\)
\(632\) 0 0
\(633\) −25.7215 −1.02234
\(634\) 0 0
\(635\) 4.11187i 0.163175i
\(636\) 0 0
\(637\) −3.36543 1.29378i −0.133343 0.0512612i
\(638\) 0 0
\(639\) 8.39340i 0.332038i
\(640\) 0 0
\(641\) −27.9398 −1.10356 −0.551778 0.833991i \(-0.686050\pi\)
−0.551778 + 0.833991i \(0.686050\pi\)
\(642\) 0 0
\(643\) 26.3582i 1.03947i −0.854329 0.519733i \(-0.826031\pi\)
0.854329 0.519733i \(-0.173969\pi\)
\(644\) 0 0
\(645\) 0.659773i 0.0259785i
\(646\) 0 0
\(647\) 41.8995 1.64724 0.823619 0.567143i \(-0.191951\pi\)
0.823619 + 0.567143i \(0.191951\pi\)
\(648\) 0 0
\(649\) 0.421593 0.0165490
\(650\) 0 0
\(651\) 8.46340 0.331707
\(652\) 0 0
\(653\) −27.9578 −1.09407 −0.547036 0.837109i \(-0.684244\pi\)
−0.547036 + 0.837109i \(0.684244\pi\)
\(654\) 0 0
\(655\) 2.06621i 0.0807336i
\(656\) 0 0
\(657\) 7.39397i 0.288466i
\(658\) 0 0
\(659\) −36.6362 −1.42714 −0.713572 0.700582i \(-0.752924\pi\)
−0.713572 + 0.700582i \(0.752924\pi\)
\(660\) 0 0
\(661\) 17.5721i 0.683474i 0.939796 + 0.341737i \(0.111015\pi\)
−0.939796 + 0.341737i \(0.888985\pi\)
\(662\) 0 0
\(663\) −24.5535 9.43910i −0.953578 0.366584i
\(664\) 0 0
\(665\) 4.17676i 0.161968i
\(666\) 0 0
\(667\) 16.5545 0.640995
\(668\) 0 0
\(669\) 21.7851i 0.842262i
\(670\) 0 0
\(671\) 2.64794i 0.102223i
\(672\) 0 0
\(673\) 18.5167 0.713766 0.356883 0.934149i \(-0.383839\pi\)
0.356883 + 0.934149i \(0.383839\pi\)
\(674\) 0 0
\(675\) 4.56325 0.175640
\(676\) 0 0
\(677\) −4.36566 −0.167786 −0.0838929 0.996475i \(-0.526735\pi\)
−0.0838929 + 0.996475i \(0.526735\pi\)
\(678\) 0 0
\(679\) −6.22937 −0.239061
\(680\) 0 0
\(681\) 14.1948i 0.543947i
\(682\) 0 0
\(683\) 38.9088i 1.48880i −0.667732 0.744401i \(-0.732735\pi\)
0.667732 0.744401i \(-0.267265\pi\)
\(684\) 0 0
\(685\) 6.29143 0.240383
\(686\) 0 0
\(687\) 23.1410i 0.882885i
\(688\) 0 0
\(689\) 39.0924 + 15.0283i 1.48930 + 0.572533i
\(690\) 0 0
\(691\) 48.6783i 1.85181i −0.377757 0.925905i \(-0.623304\pi\)
0.377757 0.925905i \(-0.376696\pi\)
\(692\) 0 0
\(693\) −0.393402 −0.0149441
\(694\) 0 0
\(695\) 10.5916i 0.401761i
\(696\) 0 0
\(697\) 49.6964i 1.88239i
\(698\) 0 0
\(699\) −2.84119 −0.107464
\(700\) 0 0
\(701\) 42.7573 1.61492 0.807461 0.589921i \(-0.200841\pi\)
0.807461 + 0.589921i \(0.200841\pi\)
\(702\) 0 0
\(703\) −68.0094 −2.56502
\(704\) 0 0
\(705\) −3.99653 −0.150518
\(706\) 0 0
\(707\) 16.1174i 0.606155i
\(708\) 0 0
\(709\) 34.7608i 1.30547i −0.757586 0.652735i \(-0.773621\pi\)
0.757586 0.652735i \(-0.226379\pi\)
\(710\) 0 0
\(711\) 5.41993 0.203263
\(712\) 0 0
\(713\) 47.0840i 1.76331i
\(714\) 0 0
\(715\) −0.874974 0.336367i −0.0327222 0.0125794i
\(716\) 0 0
\(717\) 6.34480i 0.236951i
\(718\) 0 0
\(719\) 0.151096 0.00563494 0.00281747 0.999996i \(-0.499103\pi\)
0.00281747 + 0.999996i \(0.499103\pi\)
\(720\) 0 0
\(721\) 18.3392i 0.682989i
\(722\) 0 0
\(723\) 4.21554i 0.156778i
\(724\) 0 0
\(725\) −13.5789 −0.504306
\(726\) 0 0
\(727\) −25.3419 −0.939879 −0.469939 0.882699i \(-0.655724\pi\)
−0.469939 + 0.882699i \(0.655724\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.28365 0.269395
\(732\) 0 0
\(733\) 1.92558i 0.0711229i 0.999367 + 0.0355615i \(0.0113219\pi\)
−0.999367 + 0.0355615i \(0.988678\pi\)
\(734\) 0 0
\(735\) 0.660872i 0.0243766i
\(736\) 0 0
\(737\) 4.90607 0.180718
\(738\) 0 0
\(739\) 6.02953i 0.221800i 0.993832 + 0.110900i \(0.0353733\pi\)
−0.993832 + 0.110900i \(0.964627\pi\)
\(740\) 0 0
\(741\) −8.17676 + 21.2698i −0.300381 + 0.781366i
\(742\) 0 0
\(743\) 33.4818i 1.22833i −0.789178 0.614164i \(-0.789493\pi\)
0.789178 0.614164i \(-0.210507\pi\)
\(744\) 0 0
\(745\) −1.03878 −0.0380578
\(746\) 0 0
\(747\) 2.75158i 0.100675i
\(748\) 0 0
\(749\) 4.58755i 0.167625i
\(750\) 0 0
\(751\) 26.5503 0.968835 0.484418 0.874837i \(-0.339031\pi\)
0.484418 + 0.874837i \(0.339031\pi\)
\(752\) 0 0
\(753\) 13.1751 0.480127
\(754\) 0 0
\(755\) −3.39015 −0.123380
\(756\) 0 0
\(757\) −6.33289 −0.230173 −0.115086 0.993355i \(-0.536715\pi\)
−0.115086 + 0.993355i \(0.536715\pi\)
\(758\) 0 0
\(759\) 2.18860i 0.0794410i
\(760\) 0 0
\(761\) 13.7719i 0.499231i 0.968345 + 0.249616i \(0.0803043\pi\)
−0.968345 + 0.249616i \(0.919696\pi\)
\(762\) 0 0
\(763\) −8.47088 −0.306666
\(764\) 0 0
\(765\) 4.82157i 0.174324i
\(766\) 0 0
\(767\) −1.38649 + 3.60660i −0.0500631 + 0.130227i
\(768\) 0 0
\(769\) 19.1849i 0.691825i −0.938267 0.345912i \(-0.887569\pi\)
0.938267 0.345912i \(-0.112431\pi\)
\(770\) 0 0
\(771\) −8.45592 −0.304532
\(772\) 0 0
\(773\) 9.89641i 0.355949i −0.984035 0.177975i \(-0.943046\pi\)
0.984035 0.177975i \(-0.0569545\pi\)
\(774\) 0 0
\(775\) 38.6206i 1.38729i
\(776\) 0 0
\(777\) 10.7608 0.386043
\(778\) 0 0
\(779\) −43.0503 −1.54244
\(780\) 0 0
\(781\) 3.30199 0.118154
\(782\) 0 0
\(783\) −2.97570 −0.106343
\(784\) 0 0
\(785\) 7.04674i 0.251509i
\(786\) 0 0
\(787\) 21.1383i 0.753500i −0.926315 0.376750i \(-0.877042\pi\)
0.926315 0.376750i \(-0.122958\pi\)
\(788\) 0 0
\(789\) −29.6685 −1.05623
\(790\) 0 0
\(791\) 6.46340i 0.229812i
\(792\) 0 0
\(793\) 22.6523 + 8.70823i 0.804406 + 0.309238i
\(794\) 0 0
\(795\) 7.67659i 0.272261i
\(796\) 0 0
\(797\) −21.9845 −0.778733 −0.389366 0.921083i \(-0.627306\pi\)
−0.389366 + 0.921083i \(0.627306\pi\)
\(798\) 0 0
\(799\) 44.1202i 1.56086i
\(800\) 0 0
\(801\) 5.84810i 0.206633i
\(802\) 0 0
\(803\) −2.90880 −0.102649
\(804\) 0 0
\(805\) 3.67659 0.129583
\(806\) 0 0
\(807\) −25.8274 −0.909167
\(808\) 0 0
\(809\) 47.1785 1.65871 0.829353 0.558725i \(-0.188709\pi\)
0.829353 + 0.558725i \(0.188709\pi\)
\(810\) 0 0
\(811\) 3.00631i 0.105566i 0.998606 + 0.0527830i \(0.0168091\pi\)
−0.998606 + 0.0527830i \(0.983191\pi\)
\(812\) 0 0
\(813\) 28.5015i 0.999592i
\(814\) 0 0
\(815\) −3.29832 −0.115535
\(816\) 0 0
\(817\) 6.30957i 0.220744i
\(818\) 0 0
\(819\) 1.29378 3.36543i 0.0452082 0.117598i
\(820\) 0 0
\(821\) 54.3428i 1.89658i −0.317408 0.948289i \(-0.602812\pi\)
0.317408 0.948289i \(-0.397188\pi\)
\(822\) 0 0
\(823\) 53.1589 1.85300 0.926501 0.376293i \(-0.122801\pi\)
0.926501 + 0.376293i \(0.122801\pi\)
\(824\) 0 0
\(825\) 1.79519i 0.0625006i
\(826\) 0 0
\(827\) 34.5064i 1.19991i 0.800036 + 0.599953i \(0.204814\pi\)
−0.800036 + 0.599953i \(0.795186\pi\)
\(828\) 0 0
\(829\) −4.68892 −0.162853 −0.0814264 0.996679i \(-0.525948\pi\)
−0.0814264 + 0.996679i \(0.525948\pi\)
\(830\) 0 0
\(831\) 6.52334 0.226292
\(832\) 0 0
\(833\) 7.29578 0.252784
\(834\) 0 0
\(835\) 9.72954 0.336705
\(836\) 0 0
\(837\) 8.46340i 0.292538i
\(838\) 0 0
\(839\) 5.21448i 0.180024i 0.995941 + 0.0900120i \(0.0286905\pi\)
−0.995941 + 0.0900120i \(0.971309\pi\)
\(840\) 0 0
\(841\) −20.1452 −0.694663
\(842\) 0 0
\(843\) 9.66654i 0.332933i
\(844\) 0 0
\(845\) 5.75502 6.37893i 0.197979 0.219442i
\(846\) 0 0
\(847\) 10.8452i 0.372647i
\(848\) 0 0
\(849\) −16.1052 −0.552730
\(850\) 0 0
\(851\) 59.8652i 2.05215i
\(852\) 0 0
\(853\) 36.5276i 1.25068i −0.780351 0.625341i \(-0.784960\pi\)
0.780351 0.625341i \(-0.215040\pi\)
\(854\) 0 0
\(855\) 4.17676 0.142842
\(856\) 0 0
\(857\) 54.6394 1.86645 0.933223 0.359298i \(-0.116984\pi\)
0.933223 + 0.359298i \(0.116984\pi\)
\(858\) 0 0
\(859\) −21.8848 −0.746700 −0.373350 0.927691i \(-0.621791\pi\)
−0.373350 + 0.927691i \(0.621791\pi\)
\(860\) 0 0
\(861\) 6.81167 0.232141
\(862\) 0 0
\(863\) 17.4426i 0.593752i 0.954916 + 0.296876i \(0.0959449\pi\)
−0.954916 + 0.296876i \(0.904055\pi\)
\(864\) 0 0
\(865\) 1.00835i 0.0342851i
\(866\) 0 0
\(867\) 36.2284 1.23038
\(868\) 0 0
\(869\) 2.13221i 0.0723304i
\(870\) 0 0
\(871\) −16.1345 + 41.9699i −0.546697 + 1.42210i
\(872\) 0 0
\(873\) 6.22937i 0.210832i
\(874\) 0 0
\(875\) −6.32008 −0.213658
\(876\) 0 0
\(877\) 20.3159i 0.686020i 0.939332 + 0.343010i \(0.111446\pi\)
−0.939332 + 0.343010i \(0.888554\pi\)
\(878\) 0 0
\(879\) 14.5004i 0.489086i
\(880\) 0 0
\(881\) −24.2538 −0.817131 −0.408565 0.912729i \(-0.633971\pi\)
−0.408565 + 0.912729i \(0.633971\pi\)
\(882\) 0 0
\(883\) −17.1454 −0.576988 −0.288494 0.957482i \(-0.593154\pi\)
−0.288494 + 0.957482i \(0.593154\pi\)
\(884\) 0 0
\(885\) 0.708229 0.0238069
\(886\) 0 0
\(887\) −39.3223 −1.32031 −0.660156 0.751128i \(-0.729510\pi\)
−0.660156 + 0.751128i \(0.729510\pi\)
\(888\) 0 0
\(889\) 6.22189i 0.208676i
\(890\) 0 0
\(891\) 0.393402i 0.0131795i
\(892\) 0 0
\(893\) 38.2198 1.27898
\(894\) 0 0
\(895\) 5.55845i 0.185799i
\(896\) 0 0
\(897\) −18.7227 7.19759i −0.625134 0.240321i
\(898\) 0 0
\(899\) 25.1845i 0.839951i
\(900\) 0 0
\(901\) −84.7468 −2.82332
\(902\) 0 0
\(903\) 0.998337i 0.0332226i
\(904\) 0 0
\(905\) 1.33670i 0.0444336i
\(906\) 0 0
\(907\) 16.8783 0.560436 0.280218 0.959936i \(-0.409593\pi\)
0.280218 + 0.959936i \(0.409593\pi\)
\(908\) 0 0
\(909\) 16.1174 0.534579
\(910\) 0 0
\(911\) −1.49371 −0.0494888 −0.0247444 0.999694i \(-0.507877\pi\)
−0.0247444 + 0.999694i \(0.507877\pi\)
\(912\) 0 0
\(913\) −1.08248 −0.0358248
\(914\) 0 0
\(915\) 4.44824i 0.147054i
\(916\) 0 0
\(917\) 3.12650i 0.103246i
\(918\) 0 0
\(919\) −21.7861 −0.718658 −0.359329 0.933211i \(-0.616994\pi\)
−0.359329 + 0.933211i \(0.616994\pi\)
\(920\) 0 0
\(921\) 0.124834i 0.00411343i
\(922\) 0 0
\(923\) −10.8592 + 28.2474i −0.357434 + 0.929776i
\(924\) 0 0
\(925\) 49.1044i 1.61454i
\(926\) 0 0
\(927\) −18.3392 −0.602340
\(928\) 0 0
\(929\) 11.6631i 0.382653i −0.981526 0.191326i \(-0.938721\pi\)
0.981526 0.191326i \(-0.0612789\pi\)
\(930\) 0 0
\(931\) 6.32008i 0.207132i
\(932\) 0 0
\(933\) 4.92680 0.161296
\(934\) 0 0
\(935\) 1.89682 0.0620326
\(936\) 0 0
\(937\) 3.25784 0.106429 0.0532144 0.998583i \(-0.483053\pi\)
0.0532144 + 0.998583i \(0.483053\pi\)
\(938\) 0 0
\(939\) −32.8701 −1.07267
\(940\) 0 0
\(941\) 42.1411i 1.37376i −0.726770 0.686881i \(-0.758979\pi\)
0.726770 0.686881i \(-0.241021\pi\)
\(942\) 0 0
\(943\) 37.8950i 1.23403i
\(944\) 0 0
\(945\) −0.660872 −0.0214982
\(946\) 0 0
\(947\) 23.3451i 0.758615i 0.925271 + 0.379308i \(0.123838\pi\)
−0.925271 + 0.379308i \(0.876162\pi\)
\(948\) 0 0
\(949\) 9.56613 24.8839i 0.310530 0.807766i
\(950\) 0 0
\(951\) 25.0302i 0.811661i
\(952\) 0 0
\(953\) 22.0579 0.714527 0.357263 0.934004i \(-0.383710\pi\)
0.357263 + 0.934004i \(0.383710\pi\)
\(954\) 0 0
\(955\) 10.6633i 0.345056i
\(956\) 0 0
\(957\) 1.17065i 0.0378416i
\(958\) 0 0
\(959\) 9.51990 0.307414
\(960\) 0 0
\(961\) −40.6291 −1.31062
\(962\) 0 0
\(963\) −4.58755 −0.147832
\(964\) 0 0
\(965\) −5.19006 −0.167074
\(966\) 0 0
\(967\) 24.5178i 0.788439i −0.919016 0.394219i \(-0.871015\pi\)
0.919016 0.394219i \(-0.128985\pi\)
\(968\) 0 0
\(969\) 46.1099i 1.48126i
\(970\) 0 0
\(971\) −39.7504 −1.27565 −0.637826 0.770181i \(-0.720166\pi\)
−0.637826 + 0.770181i \(0.720166\pi\)
\(972\) 0 0
\(973\) 16.0266i 0.513791i
\(974\) 0 0
\(975\) 15.3573 + 5.90382i 0.491828 + 0.189073i
\(976\) 0 0
\(977\) 50.1289i 1.60376i 0.597482 + 0.801882i \(0.296168\pi\)
−0.597482 + 0.801882i \(0.703832\pi\)
\(978\) 0 0
\(979\) −2.30066 −0.0735293
\(980\) 0 0
\(981\) 8.47088i 0.270454i
\(982\) 0 0
\(983\) 22.6425i 0.722185i 0.932530 + 0.361092i \(0.117596\pi\)
−0.932530 + 0.361092i \(0.882404\pi\)
\(984\) 0 0
\(985\) 11.6983 0.372739
\(986\) 0 0
\(987\) −6.04736 −0.192490
\(988\) 0 0
\(989\) 5.55400 0.176607
\(990\) 0 0
\(991\) 10.6821 0.339329 0.169664 0.985502i \(-0.445732\pi\)
0.169664 + 0.985502i \(0.445732\pi\)
\(992\) 0 0
\(993\) 26.0793i 0.827600i
\(994\) 0 0
\(995\) 15.7956i 0.500754i
\(996\) 0 0
\(997\) −2.26400 −0.0717015 −0.0358508 0.999357i \(-0.511414\pi\)
−0.0358508 + 0.999357i \(0.511414\pi\)
\(998\) 0 0
\(999\) 10.7608i 0.340458i
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 4368.2.h.t.337.4 10
4.3 odd 2 2184.2.h.e.337.4 10
13.12 even 2 inner 4368.2.h.t.337.7 10
52.51 odd 2 2184.2.h.e.337.7 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.h.e.337.4 10 4.3 odd 2
2184.2.h.e.337.7 yes 10 52.51 odd 2
4368.2.h.t.337.4 10 1.1 even 1 trivial
4368.2.h.t.337.7 10 13.12 even 2 inner