Properties

Label 3360.2.d.a.2911.4
Level $3360$
Weight $2$
Character 3360.2911
Analytic conductor $26.830$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3360,2,Mod(2911,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.2911"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2911.4
Root \(-0.400519i\) of defining polynomial
Character \(\chi\) \(=\) 3360.2911
Dual form 3360.2.d.a.2911.12

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000i q^{5} +(-0.400519 + 2.61526i) q^{7} +1.00000 q^{9} +3.87335i q^{11} +2.15545i q^{13} +1.00000i q^{15} -4.79356i q^{17} +5.31954 q^{19} +(0.400519 - 2.61526i) q^{21} +5.67917i q^{23} -1.00000 q^{25} -1.00000 q^{27} +1.79858 q^{29} -0.221571 q^{31} -3.87335i q^{33} +(2.61526 + 0.400519i) q^{35} -7.50006 q^{37} -2.15545i q^{39} +11.1072i q^{41} -11.2593i q^{43} -1.00000i q^{45} +7.11469 q^{47} +(-6.67917 - 2.09492i) q^{49} +4.79356i q^{51} -1.97985 q^{53} +3.87335 q^{55} -5.31954 q^{57} -5.27574 q^{59} -7.99669i q^{61} +(-0.400519 + 2.61526i) q^{63} +2.15545 q^{65} +3.00246i q^{67} -5.67917i q^{69} +6.50426i q^{71} -3.16961i q^{73} +1.00000 q^{75} +(-10.1298 - 1.55135i) q^{77} +7.99118i q^{79} +1.00000 q^{81} -10.4453 q^{83} -4.79356 q^{85} -1.79858 q^{87} +11.4560i q^{89} +(-5.63706 - 0.863298i) q^{91} +0.221571 q^{93} -5.31954i q^{95} +0.716023i q^{97} +3.87335i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77}+ \cdots + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\) \(1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −0.400519 + 2.61526i −0.151382 + 0.988475i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.87335i 1.16786i 0.811804 + 0.583930i \(0.198486\pi\)
−0.811804 + 0.583930i \(0.801514\pi\)
\(12\) 0 0
\(13\) 2.15545i 0.597814i 0.954282 + 0.298907i \(0.0966220\pi\)
−0.954282 + 0.298907i \(0.903378\pi\)
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 0 0
\(17\) 4.79356i 1.16261i −0.813686 0.581304i \(-0.802543\pi\)
0.813686 0.581304i \(-0.197457\pi\)
\(18\) 0 0
\(19\) 5.31954 1.22039 0.610193 0.792253i \(-0.291092\pi\)
0.610193 + 0.792253i \(0.291092\pi\)
\(20\) 0 0
\(21\) 0.400519 2.61526i 0.0874004 0.570697i
\(22\) 0 0
\(23\) 5.67917i 1.18419i 0.805869 + 0.592094i \(0.201699\pi\)
−0.805869 + 0.592094i \(0.798301\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.79858 0.333987 0.166994 0.985958i \(-0.446594\pi\)
0.166994 + 0.985958i \(0.446594\pi\)
\(30\) 0 0
\(31\) −0.221571 −0.0397954 −0.0198977 0.999802i \(-0.506334\pi\)
−0.0198977 + 0.999802i \(0.506334\pi\)
\(32\) 0 0
\(33\) 3.87335i 0.674264i
\(34\) 0 0
\(35\) 2.61526 + 0.400519i 0.442060 + 0.0677001i
\(36\) 0 0
\(37\) −7.50006 −1.23300 −0.616501 0.787354i \(-0.711450\pi\)
−0.616501 + 0.787354i \(0.711450\pi\)
\(38\) 0 0
\(39\) 2.15545i 0.345148i
\(40\) 0 0
\(41\) 11.1072i 1.73465i 0.497738 + 0.867327i \(0.334164\pi\)
−0.497738 + 0.867327i \(0.665836\pi\)
\(42\) 0 0
\(43\) 11.2593i 1.71703i −0.512788 0.858515i \(-0.671387\pi\)
0.512788 0.858515i \(-0.328613\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 7.11469 1.03778 0.518892 0.854840i \(-0.326345\pi\)
0.518892 + 0.854840i \(0.326345\pi\)
\(48\) 0 0
\(49\) −6.67917 2.09492i −0.954167 0.299275i
\(50\) 0 0
\(51\) 4.79356i 0.671232i
\(52\) 0 0
\(53\) −1.97985 −0.271954 −0.135977 0.990712i \(-0.543417\pi\)
−0.135977 + 0.990712i \(0.543417\pi\)
\(54\) 0 0
\(55\) 3.87335 0.522283
\(56\) 0 0
\(57\) −5.31954 −0.704590
\(58\) 0 0
\(59\) −5.27574 −0.686843 −0.343421 0.939181i \(-0.611586\pi\)
−0.343421 + 0.939181i \(0.611586\pi\)
\(60\) 0 0
\(61\) 7.99669i 1.02387i −0.859024 0.511935i \(-0.828929\pi\)
0.859024 0.511935i \(-0.171071\pi\)
\(62\) 0 0
\(63\) −0.400519 + 2.61526i −0.0504607 + 0.329492i
\(64\) 0 0
\(65\) 2.15545 0.267350
\(66\) 0 0
\(67\) 3.00246i 0.366809i 0.983038 + 0.183405i \(0.0587118\pi\)
−0.983038 + 0.183405i \(0.941288\pi\)
\(68\) 0 0
\(69\) 5.67917i 0.683692i
\(70\) 0 0
\(71\) 6.50426i 0.771913i 0.922517 + 0.385957i \(0.126129\pi\)
−0.922517 + 0.385957i \(0.873871\pi\)
\(72\) 0 0
\(73\) 3.16961i 0.370974i −0.982647 0.185487i \(-0.940614\pi\)
0.982647 0.185487i \(-0.0593863\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −10.1298 1.55135i −1.15440 0.176793i
\(78\) 0 0
\(79\) 7.99118i 0.899078i 0.893261 + 0.449539i \(0.148412\pi\)
−0.893261 + 0.449539i \(0.851588\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.4453 −1.14652 −0.573262 0.819372i \(-0.694322\pi\)
−0.573262 + 0.819372i \(0.694322\pi\)
\(84\) 0 0
\(85\) −4.79356 −0.519934
\(86\) 0 0
\(87\) −1.79858 −0.192828
\(88\) 0 0
\(89\) 11.4560i 1.21434i 0.794574 + 0.607168i \(0.207694\pi\)
−0.794574 + 0.607168i \(0.792306\pi\)
\(90\) 0 0
\(91\) −5.63706 0.863298i −0.590924 0.0904982i
\(92\) 0 0
\(93\) 0.221571 0.0229759
\(94\) 0 0
\(95\) 5.31954i 0.545773i
\(96\) 0 0
\(97\) 0.716023i 0.0727012i 0.999339 + 0.0363506i \(0.0115733\pi\)
−0.999339 + 0.0363506i \(0.988427\pi\)
\(98\) 0 0
\(99\) 3.87335i 0.389286i
\(100\) 0 0
\(101\) 5.18641i 0.516067i 0.966136 + 0.258033i \(0.0830744\pi\)
−0.966136 + 0.258033i \(0.916926\pi\)
\(102\) 0 0
\(103\) −3.19896 −0.315203 −0.157602 0.987503i \(-0.550376\pi\)
−0.157602 + 0.987503i \(0.550376\pi\)
\(104\) 0 0
\(105\) −2.61526 0.400519i −0.255223 0.0390867i
\(106\) 0 0
\(107\) 8.49311i 0.821060i −0.911847 0.410530i \(-0.865344\pi\)
0.911847 0.410530i \(-0.134656\pi\)
\(108\) 0 0
\(109\) −8.39403 −0.804002 −0.402001 0.915639i \(-0.631685\pi\)
−0.402001 + 0.915639i \(0.631685\pi\)
\(110\) 0 0
\(111\) 7.50006 0.711874
\(112\) 0 0
\(113\) −10.0556 −0.945952 −0.472976 0.881075i \(-0.656820\pi\)
−0.472976 + 0.881075i \(0.656820\pi\)
\(114\) 0 0
\(115\) 5.67917 0.529585
\(116\) 0 0
\(117\) 2.15545i 0.199271i
\(118\) 0 0
\(119\) 12.5364 + 1.91991i 1.14921 + 0.175998i
\(120\) 0 0
\(121\) −4.00285 −0.363895
\(122\) 0 0
\(123\) 11.1072i 1.00150i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 19.4226i 1.72347i 0.507356 + 0.861737i \(0.330623\pi\)
−0.507356 + 0.861737i \(0.669377\pi\)
\(128\) 0 0
\(129\) 11.2593i 0.991328i
\(130\) 0 0
\(131\) −4.52170 −0.395063 −0.197531 0.980297i \(-0.563292\pi\)
−0.197531 + 0.980297i \(0.563292\pi\)
\(132\) 0 0
\(133\) −2.13058 + 13.9120i −0.184744 + 1.20632i
\(134\) 0 0
\(135\) 1.00000i 0.0860663i
\(136\) 0 0
\(137\) 12.7163 1.08643 0.543215 0.839594i \(-0.317207\pi\)
0.543215 + 0.839594i \(0.317207\pi\)
\(138\) 0 0
\(139\) 2.52657 0.214301 0.107150 0.994243i \(-0.465827\pi\)
0.107150 + 0.994243i \(0.465827\pi\)
\(140\) 0 0
\(141\) −7.11469 −0.599165
\(142\) 0 0
\(143\) −8.34881 −0.698162
\(144\) 0 0
\(145\) 1.79858i 0.149364i
\(146\) 0 0
\(147\) 6.67917 + 2.09492i 0.550889 + 0.172786i
\(148\) 0 0
\(149\) −3.27960 −0.268675 −0.134338 0.990936i \(-0.542891\pi\)
−0.134338 + 0.990936i \(0.542891\pi\)
\(150\) 0 0
\(151\) 4.74673i 0.386284i 0.981171 + 0.193142i \(0.0618677\pi\)
−0.981171 + 0.193142i \(0.938132\pi\)
\(152\) 0 0
\(153\) 4.79356i 0.387536i
\(154\) 0 0
\(155\) 0.221571i 0.0177970i
\(156\) 0 0
\(157\) 15.5590i 1.24174i −0.783912 0.620872i \(-0.786779\pi\)
0.783912 0.620872i \(-0.213221\pi\)
\(158\) 0 0
\(159\) 1.97985 0.157012
\(160\) 0 0
\(161\) −14.8525 2.27462i −1.17054 0.179265i
\(162\) 0 0
\(163\) 21.4106i 1.67701i 0.544894 + 0.838505i \(0.316570\pi\)
−0.544894 + 0.838505i \(0.683430\pi\)
\(164\) 0 0
\(165\) −3.87335 −0.301540
\(166\) 0 0
\(167\) 6.86529 0.531252 0.265626 0.964076i \(-0.414421\pi\)
0.265626 + 0.964076i \(0.414421\pi\)
\(168\) 0 0
\(169\) 8.35404 0.642619
\(170\) 0 0
\(171\) 5.31954 0.406795
\(172\) 0 0
\(173\) 13.9225i 1.05851i 0.848463 + 0.529255i \(0.177529\pi\)
−0.848463 + 0.529255i \(0.822471\pi\)
\(174\) 0 0
\(175\) 0.400519 2.61526i 0.0302764 0.197695i
\(176\) 0 0
\(177\) 5.27574 0.396549
\(178\) 0 0
\(179\) 6.99942i 0.523162i 0.965182 + 0.261581i \(0.0842438\pi\)
−0.965182 + 0.261581i \(0.915756\pi\)
\(180\) 0 0
\(181\) 1.97879i 0.147082i −0.997292 0.0735412i \(-0.976570\pi\)
0.997292 0.0735412i \(-0.0234300\pi\)
\(182\) 0 0
\(183\) 7.99669i 0.591132i
\(184\) 0 0
\(185\) 7.50006i 0.551415i
\(186\) 0 0
\(187\) 18.5671 1.35776
\(188\) 0 0
\(189\) 0.400519 2.61526i 0.0291335 0.190232i
\(190\) 0 0
\(191\) 20.5288i 1.48541i 0.669616 + 0.742707i \(0.266459\pi\)
−0.669616 + 0.742707i \(0.733541\pi\)
\(192\) 0 0
\(193\) −13.1050 −0.943321 −0.471661 0.881780i \(-0.656345\pi\)
−0.471661 + 0.881780i \(0.656345\pi\)
\(194\) 0 0
\(195\) −2.15545 −0.154355
\(196\) 0 0
\(197\) 6.78520 0.483426 0.241713 0.970348i \(-0.422291\pi\)
0.241713 + 0.970348i \(0.422291\pi\)
\(198\) 0 0
\(199\) 6.62783 0.469835 0.234917 0.972015i \(-0.424518\pi\)
0.234917 + 0.972015i \(0.424518\pi\)
\(200\) 0 0
\(201\) 3.00246i 0.211777i
\(202\) 0 0
\(203\) −0.720364 + 4.70375i −0.0505597 + 0.330138i
\(204\) 0 0
\(205\) 11.1072 0.775761
\(206\) 0 0
\(207\) 5.67917i 0.394730i
\(208\) 0 0
\(209\) 20.6044i 1.42524i
\(210\) 0 0
\(211\) 10.7000i 0.736618i −0.929703 0.368309i \(-0.879937\pi\)
0.929703 0.368309i \(-0.120063\pi\)
\(212\) 0 0
\(213\) 6.50426i 0.445664i
\(214\) 0 0
\(215\) −11.2593 −0.767879
\(216\) 0 0
\(217\) 0.0887435 0.579467i 0.00602430 0.0393368i
\(218\) 0 0
\(219\) 3.16961i 0.214182i
\(220\) 0 0
\(221\) 10.3323 0.695023
\(222\) 0 0
\(223\) −26.0191 −1.74236 −0.871182 0.490960i \(-0.836646\pi\)
−0.871182 + 0.490960i \(0.836646\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −13.0050 −0.863173 −0.431586 0.902072i \(-0.642046\pi\)
−0.431586 + 0.902072i \(0.642046\pi\)
\(228\) 0 0
\(229\) 21.0268i 1.38949i 0.719255 + 0.694746i \(0.244483\pi\)
−0.719255 + 0.694746i \(0.755517\pi\)
\(230\) 0 0
\(231\) 10.1298 + 1.55135i 0.666493 + 0.102071i
\(232\) 0 0
\(233\) −22.2270 −1.45614 −0.728070 0.685503i \(-0.759583\pi\)
−0.728070 + 0.685503i \(0.759583\pi\)
\(234\) 0 0
\(235\) 7.11469i 0.464111i
\(236\) 0 0
\(237\) 7.99118i 0.519083i
\(238\) 0 0
\(239\) 13.6516i 0.883048i 0.897249 + 0.441524i \(0.145562\pi\)
−0.897249 + 0.441524i \(0.854438\pi\)
\(240\) 0 0
\(241\) 18.7440i 1.20741i −0.797209 0.603704i \(-0.793691\pi\)
0.797209 0.603704i \(-0.206309\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.09492 + 6.67917i −0.133840 + 0.426716i
\(246\) 0 0
\(247\) 11.4660i 0.729564i
\(248\) 0 0
\(249\) 10.4453 0.661946
\(250\) 0 0
\(251\) −13.1144 −0.827773 −0.413887 0.910328i \(-0.635829\pi\)
−0.413887 + 0.910328i \(0.635829\pi\)
\(252\) 0 0
\(253\) −21.9974 −1.38297
\(254\) 0 0
\(255\) 4.79356 0.300184
\(256\) 0 0
\(257\) 10.4950i 0.654663i 0.944910 + 0.327331i \(0.106149\pi\)
−0.944910 + 0.327331i \(0.893851\pi\)
\(258\) 0 0
\(259\) 3.00392 19.6146i 0.186654 1.21879i
\(260\) 0 0
\(261\) 1.79858 0.111329
\(262\) 0 0
\(263\) 27.9160i 1.72138i 0.509134 + 0.860688i \(0.329966\pi\)
−0.509134 + 0.860688i \(0.670034\pi\)
\(264\) 0 0
\(265\) 1.97985i 0.121621i
\(266\) 0 0
\(267\) 11.4560i 0.701097i
\(268\) 0 0
\(269\) 1.37287i 0.0837051i −0.999124 0.0418525i \(-0.986674\pi\)
0.999124 0.0418525i \(-0.0133260\pi\)
\(270\) 0 0
\(271\) −2.00052 −0.121523 −0.0607616 0.998152i \(-0.519353\pi\)
−0.0607616 + 0.998152i \(0.519353\pi\)
\(272\) 0 0
\(273\) 5.63706 + 0.863298i 0.341170 + 0.0522492i
\(274\) 0 0
\(275\) 3.87335i 0.233572i
\(276\) 0 0
\(277\) −3.87788 −0.232999 −0.116500 0.993191i \(-0.537167\pi\)
−0.116500 + 0.993191i \(0.537167\pi\)
\(278\) 0 0
\(279\) −0.221571 −0.0132651
\(280\) 0 0
\(281\) −15.4583 −0.922167 −0.461083 0.887357i \(-0.652539\pi\)
−0.461083 + 0.887357i \(0.652539\pi\)
\(282\) 0 0
\(283\) 22.1130 1.31448 0.657240 0.753681i \(-0.271724\pi\)
0.657240 + 0.753681i \(0.271724\pi\)
\(284\) 0 0
\(285\) 5.31954i 0.315102i
\(286\) 0 0
\(287\) −29.0482 4.44865i −1.71466 0.262595i
\(288\) 0 0
\(289\) −5.97818 −0.351658
\(290\) 0 0
\(291\) 0.716023i 0.0419740i
\(292\) 0 0
\(293\) 24.2993i 1.41958i −0.704412 0.709792i \(-0.748789\pi\)
0.704412 0.709792i \(-0.251211\pi\)
\(294\) 0 0
\(295\) 5.27574i 0.307165i
\(296\) 0 0
\(297\) 3.87335i 0.224755i
\(298\) 0 0
\(299\) −12.2412 −0.707924
\(300\) 0 0
\(301\) 29.4460 + 4.50957i 1.69724 + 0.259927i
\(302\) 0 0
\(303\) 5.18641i 0.297951i
\(304\) 0 0
\(305\) −7.99669 −0.457889
\(306\) 0 0
\(307\) −29.4220 −1.67920 −0.839602 0.543202i \(-0.817212\pi\)
−0.839602 + 0.543202i \(0.817212\pi\)
\(308\) 0 0
\(309\) 3.19896 0.181983
\(310\) 0 0
\(311\) −26.2914 −1.49085 −0.745424 0.666590i \(-0.767753\pi\)
−0.745424 + 0.666590i \(0.767753\pi\)
\(312\) 0 0
\(313\) 31.6179i 1.78715i −0.448914 0.893575i \(-0.648189\pi\)
0.448914 0.893575i \(-0.351811\pi\)
\(314\) 0 0
\(315\) 2.61526 + 0.400519i 0.147353 + 0.0225667i
\(316\) 0 0
\(317\) 26.4489 1.48552 0.742760 0.669558i \(-0.233516\pi\)
0.742760 + 0.669558i \(0.233516\pi\)
\(318\) 0 0
\(319\) 6.96652i 0.390050i
\(320\) 0 0
\(321\) 8.49311i 0.474039i
\(322\) 0 0
\(323\) 25.4995i 1.41883i
\(324\) 0 0
\(325\) 2.15545i 0.119563i
\(326\) 0 0
\(327\) 8.39403 0.464191
\(328\) 0 0
\(329\) −2.84957 + 18.6068i −0.157102 + 1.02582i
\(330\) 0 0
\(331\) 8.39519i 0.461441i 0.973020 + 0.230721i \(0.0741084\pi\)
−0.973020 + 0.230721i \(0.925892\pi\)
\(332\) 0 0
\(333\) −7.50006 −0.411001
\(334\) 0 0
\(335\) 3.00246 0.164042
\(336\) 0 0
\(337\) −23.9891 −1.30677 −0.653384 0.757027i \(-0.726651\pi\)
−0.653384 + 0.757027i \(0.726651\pi\)
\(338\) 0 0
\(339\) 10.0556 0.546146
\(340\) 0 0
\(341\) 0.858224i 0.0464754i
\(342\) 0 0
\(343\) 8.15390 16.6287i 0.440269 0.897866i
\(344\) 0 0
\(345\) −5.67917 −0.305756
\(346\) 0 0
\(347\) 11.0986i 0.595803i 0.954597 + 0.297901i \(0.0962867\pi\)
−0.954597 + 0.297901i \(0.903713\pi\)
\(348\) 0 0
\(349\) 10.0912i 0.540168i 0.962837 + 0.270084i \(0.0870515\pi\)
−0.962837 + 0.270084i \(0.912949\pi\)
\(350\) 0 0
\(351\) 2.15545i 0.115049i
\(352\) 0 0
\(353\) 11.3661i 0.604959i −0.953156 0.302479i \(-0.902186\pi\)
0.953156 0.302479i \(-0.0978143\pi\)
\(354\) 0 0
\(355\) 6.50426 0.345210
\(356\) 0 0
\(357\) −12.5364 1.91991i −0.663496 0.101612i
\(358\) 0 0
\(359\) 32.1591i 1.69729i 0.528960 + 0.848647i \(0.322582\pi\)
−0.528960 + 0.848647i \(0.677418\pi\)
\(360\) 0 0
\(361\) 9.29750 0.489342
\(362\) 0 0
\(363\) 4.00285 0.210095
\(364\) 0 0
\(365\) −3.16961 −0.165905
\(366\) 0 0
\(367\) 16.4005 0.856101 0.428050 0.903755i \(-0.359201\pi\)
0.428050 + 0.903755i \(0.359201\pi\)
\(368\) 0 0
\(369\) 11.1072i 0.578218i
\(370\) 0 0
\(371\) 0.792968 5.17783i 0.0411689 0.268819i
\(372\) 0 0
\(373\) 9.52954 0.493421 0.246710 0.969089i \(-0.420650\pi\)
0.246710 + 0.969089i \(0.420650\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 3.87674i 0.199662i
\(378\) 0 0
\(379\) 15.1323i 0.777296i −0.921386 0.388648i \(-0.872942\pi\)
0.921386 0.388648i \(-0.127058\pi\)
\(380\) 0 0
\(381\) 19.4226i 0.995048i
\(382\) 0 0
\(383\) −17.1555 −0.876607 −0.438304 0.898827i \(-0.644420\pi\)
−0.438304 + 0.898827i \(0.644420\pi\)
\(384\) 0 0
\(385\) −1.55135 + 10.1298i −0.0790642 + 0.516263i
\(386\) 0 0
\(387\) 11.2593i 0.572343i
\(388\) 0 0
\(389\) −23.7357 −1.20345 −0.601724 0.798704i \(-0.705519\pi\)
−0.601724 + 0.798704i \(0.705519\pi\)
\(390\) 0 0
\(391\) 27.2234 1.37675
\(392\) 0 0
\(393\) 4.52170 0.228090
\(394\) 0 0
\(395\) 7.99118 0.402080
\(396\) 0 0
\(397\) 15.2336i 0.764554i 0.924048 + 0.382277i \(0.124860\pi\)
−0.924048 + 0.382277i \(0.875140\pi\)
\(398\) 0 0
\(399\) 2.13058 13.9120i 0.106662 0.696470i
\(400\) 0 0
\(401\) 30.1291 1.50458 0.752288 0.658834i \(-0.228950\pi\)
0.752288 + 0.658834i \(0.228950\pi\)
\(402\) 0 0
\(403\) 0.477586i 0.0237902i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 29.0504i 1.43997i
\(408\) 0 0
\(409\) 9.56863i 0.473138i 0.971615 + 0.236569i \(0.0760230\pi\)
−0.971615 + 0.236569i \(0.923977\pi\)
\(410\) 0 0
\(411\) −12.7163 −0.627250
\(412\) 0 0
\(413\) 2.11303 13.7974i 0.103976 0.678927i
\(414\) 0 0
\(415\) 10.4453i 0.512741i
\(416\) 0 0
\(417\) −2.52657 −0.123727
\(418\) 0 0
\(419\) 22.4745 1.09795 0.548975 0.835839i \(-0.315018\pi\)
0.548975 + 0.835839i \(0.315018\pi\)
\(420\) 0 0
\(421\) −14.5415 −0.708710 −0.354355 0.935111i \(-0.615299\pi\)
−0.354355 + 0.935111i \(0.615299\pi\)
\(422\) 0 0
\(423\) 7.11469 0.345928
\(424\) 0 0
\(425\) 4.79356i 0.232522i
\(426\) 0 0
\(427\) 20.9134 + 3.20283i 1.01207 + 0.154996i
\(428\) 0 0
\(429\) 8.34881 0.403084
\(430\) 0 0
\(431\) 0.493919i 0.0237912i −0.999929 0.0118956i \(-0.996213\pi\)
0.999929 0.0118956i \(-0.00378658\pi\)
\(432\) 0 0
\(433\) 38.4919i 1.84980i −0.380204 0.924902i \(-0.624146\pi\)
0.380204 0.924902i \(-0.375854\pi\)
\(434\) 0 0
\(435\) 1.79858i 0.0862352i
\(436\) 0 0
\(437\) 30.2106i 1.44517i
\(438\) 0 0
\(439\) 38.0565 1.81634 0.908170 0.418602i \(-0.137480\pi\)
0.908170 + 0.418602i \(0.137480\pi\)
\(440\) 0 0
\(441\) −6.67917 2.09492i −0.318056 0.0997582i
\(442\) 0 0
\(443\) 29.2504i 1.38973i −0.719141 0.694864i \(-0.755465\pi\)
0.719141 0.694864i \(-0.244535\pi\)
\(444\) 0 0
\(445\) 11.4560 0.543067
\(446\) 0 0
\(447\) 3.27960 0.155120
\(448\) 0 0
\(449\) 23.6966 1.11831 0.559155 0.829063i \(-0.311126\pi\)
0.559155 + 0.829063i \(0.311126\pi\)
\(450\) 0 0
\(451\) −43.0221 −2.02583
\(452\) 0 0
\(453\) 4.74673i 0.223021i
\(454\) 0 0
\(455\) −0.863298 + 5.63706i −0.0404720 + 0.264269i
\(456\) 0 0
\(457\) 38.9971 1.82421 0.912103 0.409961i \(-0.134458\pi\)
0.912103 + 0.409961i \(0.134458\pi\)
\(458\) 0 0
\(459\) 4.79356i 0.223744i
\(460\) 0 0
\(461\) 15.4289i 0.718595i 0.933223 + 0.359298i \(0.116984\pi\)
−0.933223 + 0.359298i \(0.883016\pi\)
\(462\) 0 0
\(463\) 15.5650i 0.723365i −0.932301 0.361682i \(-0.882202\pi\)
0.932301 0.361682i \(-0.117798\pi\)
\(464\) 0 0
\(465\) 0.221571i 0.0102751i
\(466\) 0 0
\(467\) −34.9489 −1.61724 −0.808621 0.588330i \(-0.799786\pi\)
−0.808621 + 0.588330i \(0.799786\pi\)
\(468\) 0 0
\(469\) −7.85222 1.20254i −0.362582 0.0555283i
\(470\) 0 0
\(471\) 15.5590i 0.716921i
\(472\) 0 0
\(473\) 43.6113 2.00525
\(474\) 0 0
\(475\) −5.31954 −0.244077
\(476\) 0 0
\(477\) −1.97985 −0.0906512
\(478\) 0 0
\(479\) −12.0326 −0.549782 −0.274891 0.961475i \(-0.588642\pi\)
−0.274891 + 0.961475i \(0.588642\pi\)
\(480\) 0 0
\(481\) 16.1660i 0.737106i
\(482\) 0 0
\(483\) 14.8525 + 2.27462i 0.675812 + 0.103499i
\(484\) 0 0
\(485\) 0.716023 0.0325130
\(486\) 0 0
\(487\) 33.1682i 1.50299i 0.659736 + 0.751497i \(0.270668\pi\)
−0.659736 + 0.751497i \(0.729332\pi\)
\(488\) 0 0
\(489\) 21.4106i 0.968222i
\(490\) 0 0
\(491\) 3.74110i 0.168834i 0.996431 + 0.0844169i \(0.0269027\pi\)
−0.996431 + 0.0844169i \(0.973097\pi\)
\(492\) 0 0
\(493\) 8.62158i 0.388296i
\(494\) 0 0
\(495\) 3.87335 0.174094
\(496\) 0 0
\(497\) −17.0103 2.60508i −0.763017 0.116854i
\(498\) 0 0
\(499\) 28.7989i 1.28921i −0.764514 0.644607i \(-0.777021\pi\)
0.764514 0.644607i \(-0.222979\pi\)
\(500\) 0 0
\(501\) −6.86529 −0.306719
\(502\) 0 0
\(503\) −30.5630 −1.36274 −0.681369 0.731940i \(-0.738615\pi\)
−0.681369 + 0.731940i \(0.738615\pi\)
\(504\) 0 0
\(505\) 5.18641 0.230792
\(506\) 0 0
\(507\) −8.35404 −0.371016
\(508\) 0 0
\(509\) 24.5626i 1.08872i 0.838852 + 0.544359i \(0.183227\pi\)
−0.838852 + 0.544359i \(0.816773\pi\)
\(510\) 0 0
\(511\) 8.28934 + 1.26949i 0.366699 + 0.0561588i
\(512\) 0 0
\(513\) −5.31954 −0.234863
\(514\) 0 0
\(515\) 3.19896i 0.140963i
\(516\) 0 0
\(517\) 27.5577i 1.21199i
\(518\) 0 0
\(519\) 13.9225i 0.611131i
\(520\) 0 0
\(521\) 32.4634i 1.42225i −0.703066 0.711125i \(-0.748186\pi\)
0.703066 0.711125i \(-0.251814\pi\)
\(522\) 0 0
\(523\) −33.8522 −1.48026 −0.740128 0.672466i \(-0.765235\pi\)
−0.740128 + 0.672466i \(0.765235\pi\)
\(524\) 0 0
\(525\) −0.400519 + 2.61526i −0.0174801 + 0.114139i
\(526\) 0 0
\(527\) 1.06211i 0.0462664i
\(528\) 0 0
\(529\) −9.25296 −0.402303
\(530\) 0 0
\(531\) −5.27574 −0.228948
\(532\) 0 0
\(533\) −23.9410 −1.03700
\(534\) 0 0
\(535\) −8.49311 −0.367189
\(536\) 0 0
\(537\) 6.99942i 0.302047i
\(538\) 0 0
\(539\) 8.11437 25.8708i 0.349511 1.11433i
\(540\) 0 0
\(541\) 32.2557 1.38678 0.693390 0.720563i \(-0.256116\pi\)
0.693390 + 0.720563i \(0.256116\pi\)
\(542\) 0 0
\(543\) 1.97879i 0.0849180i
\(544\) 0 0
\(545\) 8.39403i 0.359561i
\(546\) 0 0
\(547\) 10.1007i 0.431877i 0.976407 + 0.215938i \(0.0692810\pi\)
−0.976407 + 0.215938i \(0.930719\pi\)
\(548\) 0 0
\(549\) 7.99669i 0.341290i
\(550\) 0 0
\(551\) 9.56760 0.407593
\(552\) 0 0
\(553\) −20.8990 3.20062i −0.888716 0.136104i
\(554\) 0 0
\(555\) 7.50006i 0.318360i
\(556\) 0 0
\(557\) −30.5437 −1.29418 −0.647089 0.762414i \(-0.724014\pi\)
−0.647089 + 0.762414i \(0.724014\pi\)
\(558\) 0 0
\(559\) 24.2689 1.02646
\(560\) 0 0
\(561\) −18.5671 −0.783905
\(562\) 0 0
\(563\) −1.16461 −0.0490825 −0.0245413 0.999699i \(-0.507813\pi\)
−0.0245413 + 0.999699i \(0.507813\pi\)
\(564\) 0 0
\(565\) 10.0556i 0.423043i
\(566\) 0 0
\(567\) −0.400519 + 2.61526i −0.0168202 + 0.109831i
\(568\) 0 0
\(569\) 33.8001 1.41697 0.708487 0.705724i \(-0.249378\pi\)
0.708487 + 0.705724i \(0.249378\pi\)
\(570\) 0 0
\(571\) 20.7283i 0.867452i −0.901045 0.433726i \(-0.857199\pi\)
0.901045 0.433726i \(-0.142801\pi\)
\(572\) 0 0
\(573\) 20.5288i 0.857604i
\(574\) 0 0
\(575\) 5.67917i 0.236838i
\(576\) 0 0
\(577\) 14.3167i 0.596011i −0.954564 0.298005i \(-0.903679\pi\)
0.954564 0.298005i \(-0.0963213\pi\)
\(578\) 0 0
\(579\) 13.1050 0.544627
\(580\) 0 0
\(581\) 4.18356 27.3173i 0.173563 1.13331i
\(582\) 0 0
\(583\) 7.66866i 0.317604i
\(584\) 0 0
\(585\) 2.15545 0.0891168
\(586\) 0 0
\(587\) 10.4316 0.430558 0.215279 0.976553i \(-0.430934\pi\)
0.215279 + 0.976553i \(0.430934\pi\)
\(588\) 0 0
\(589\) −1.17866 −0.0485657
\(590\) 0 0
\(591\) −6.78520 −0.279106
\(592\) 0 0
\(593\) 42.2732i 1.73595i 0.496606 + 0.867976i \(0.334579\pi\)
−0.496606 + 0.867976i \(0.665421\pi\)
\(594\) 0 0
\(595\) 1.91991 12.5364i 0.0787087 0.513942i
\(596\) 0 0
\(597\) −6.62783 −0.271259
\(598\) 0 0
\(599\) 39.9371i 1.63179i −0.578202 0.815894i \(-0.696245\pi\)
0.578202 0.815894i \(-0.303755\pi\)
\(600\) 0 0
\(601\) 27.0538i 1.10355i −0.833994 0.551773i \(-0.813951\pi\)
0.833994 0.551773i \(-0.186049\pi\)
\(602\) 0 0
\(603\) 3.00246i 0.122270i
\(604\) 0 0
\(605\) 4.00285i 0.162739i
\(606\) 0 0
\(607\) 18.6004 0.754968 0.377484 0.926016i \(-0.376789\pi\)
0.377484 + 0.926016i \(0.376789\pi\)
\(608\) 0 0
\(609\) 0.720364 4.70375i 0.0291906 0.190605i
\(610\) 0 0
\(611\) 15.3354i 0.620402i
\(612\) 0 0
\(613\) −2.72099 −0.109900 −0.0549499 0.998489i \(-0.517500\pi\)
−0.0549499 + 0.998489i \(0.517500\pi\)
\(614\) 0 0
\(615\) −11.1072 −0.447886
\(616\) 0 0
\(617\) 9.75305 0.392643 0.196321 0.980540i \(-0.437100\pi\)
0.196321 + 0.980540i \(0.437100\pi\)
\(618\) 0 0
\(619\) 46.0457 1.85073 0.925367 0.379073i \(-0.123757\pi\)
0.925367 + 0.379073i \(0.123757\pi\)
\(620\) 0 0
\(621\) 5.67917i 0.227897i
\(622\) 0 0
\(623\) −29.9605 4.58835i −1.20034 0.183828i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 20.6044i 0.822862i
\(628\) 0 0
\(629\) 35.9520i 1.43350i
\(630\) 0 0
\(631\) 13.4133i 0.533973i −0.963700 0.266987i \(-0.913972\pi\)
0.963700 0.266987i \(-0.0860280\pi\)
\(632\) 0 0
\(633\) 10.7000i 0.425287i
\(634\) 0 0
\(635\) 19.4226 0.770761
\(636\) 0 0
\(637\) 4.51550 14.3966i 0.178911 0.570414i
\(638\) 0 0
\(639\) 6.50426i 0.257304i
\(640\) 0 0
\(641\) −20.6816 −0.816876 −0.408438 0.912786i \(-0.633926\pi\)
−0.408438 + 0.912786i \(0.633926\pi\)
\(642\) 0 0
\(643\) 9.99396 0.394123 0.197062 0.980391i \(-0.436860\pi\)
0.197062 + 0.980391i \(0.436860\pi\)
\(644\) 0 0
\(645\) 11.2593 0.443335
\(646\) 0 0
\(647\) −14.4620 −0.568558 −0.284279 0.958742i \(-0.591754\pi\)
−0.284279 + 0.958742i \(0.591754\pi\)
\(648\) 0 0
\(649\) 20.4348i 0.802136i
\(650\) 0 0
\(651\) −0.0887435 + 0.579467i −0.00347813 + 0.0227111i
\(652\) 0 0
\(653\) 43.4639 1.70087 0.850437 0.526076i \(-0.176337\pi\)
0.850437 + 0.526076i \(0.176337\pi\)
\(654\) 0 0
\(655\) 4.52170i 0.176677i
\(656\) 0 0
\(657\) 3.16961i 0.123658i
\(658\) 0 0
\(659\) 43.7090i 1.70266i −0.524631 0.851330i \(-0.675797\pi\)
0.524631 0.851330i \(-0.324203\pi\)
\(660\) 0 0
\(661\) 32.8390i 1.27729i 0.769503 + 0.638644i \(0.220504\pi\)
−0.769503 + 0.638644i \(0.779496\pi\)
\(662\) 0 0
\(663\) −10.3323 −0.401272
\(664\) 0 0
\(665\) 13.9120 + 2.13058i 0.539483 + 0.0826202i
\(666\) 0 0
\(667\) 10.2144i 0.395504i
\(668\) 0 0
\(669\) 26.0191 1.00595
\(670\) 0 0
\(671\) 30.9740 1.19574
\(672\) 0 0
\(673\) 19.6011 0.755568 0.377784 0.925894i \(-0.376686\pi\)
0.377784 + 0.925894i \(0.376686\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 30.8816i 1.18688i −0.804880 0.593438i \(-0.797770\pi\)
0.804880 0.593438i \(-0.202230\pi\)
\(678\) 0 0
\(679\) −1.87259 0.286781i −0.0718633 0.0110056i
\(680\) 0 0
\(681\) 13.0050 0.498353
\(682\) 0 0
\(683\) 13.7409i 0.525781i 0.964826 + 0.262891i \(0.0846758\pi\)
−0.964826 + 0.262891i \(0.915324\pi\)
\(684\) 0 0
\(685\) 12.7163i 0.485866i
\(686\) 0 0
\(687\) 21.0268i 0.802224i
\(688\) 0 0
\(689\) 4.26747i 0.162578i
\(690\) 0 0
\(691\) 47.6561 1.81292 0.906462 0.422288i \(-0.138773\pi\)
0.906462 + 0.422288i \(0.138773\pi\)
\(692\) 0 0
\(693\) −10.1298 1.55135i −0.384800 0.0589309i
\(694\) 0 0
\(695\) 2.52657i 0.0958383i
\(696\) 0 0
\(697\) 53.2430 2.01672
\(698\) 0 0
\(699\) 22.2270 0.840703
\(700\) 0 0
\(701\) 10.7523 0.406109 0.203055 0.979167i \(-0.434913\pi\)
0.203055 + 0.979167i \(0.434913\pi\)
\(702\) 0 0
\(703\) −39.8969 −1.50474
\(704\) 0 0
\(705\) 7.11469i 0.267955i
\(706\) 0 0
\(707\) −13.5638 2.07725i −0.510119 0.0781232i
\(708\) 0 0
\(709\) 18.9409 0.711340 0.355670 0.934612i \(-0.384253\pi\)
0.355670 + 0.934612i \(0.384253\pi\)
\(710\) 0 0
\(711\) 7.99118i 0.299693i
\(712\) 0 0
\(713\) 1.25834i 0.0471252i
\(714\) 0 0
\(715\) 8.34881i 0.312228i
\(716\) 0 0
\(717\) 13.6516i 0.509828i
\(718\) 0 0
\(719\) 38.4552 1.43414 0.717068 0.697003i \(-0.245484\pi\)
0.717068 + 0.697003i \(0.245484\pi\)
\(720\) 0 0
\(721\) 1.28125 8.36612i 0.0477161 0.311570i
\(722\) 0 0
\(723\) 18.7440i 0.697097i
\(724\) 0 0
\(725\) −1.79858 −0.0667975
\(726\) 0 0
\(727\) 5.37304 0.199275 0.0996375 0.995024i \(-0.468232\pi\)
0.0996375 + 0.995024i \(0.468232\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −53.9722 −1.99623
\(732\) 0 0
\(733\) 19.9952i 0.738540i −0.929322 0.369270i \(-0.879608\pi\)
0.929322 0.369270i \(-0.120392\pi\)
\(734\) 0 0
\(735\) 2.09492 6.67917i 0.0772724 0.246365i
\(736\) 0 0
\(737\) −11.6296 −0.428381
\(738\) 0 0
\(739\) 21.4299i 0.788312i 0.919044 + 0.394156i \(0.128963\pi\)
−0.919044 + 0.394156i \(0.871037\pi\)
\(740\) 0 0
\(741\) 11.4660i 0.421214i
\(742\) 0 0
\(743\) 1.35726i 0.0497930i −0.999690 0.0248965i \(-0.992074\pi\)
0.999690 0.0248965i \(-0.00792563\pi\)
\(744\) 0 0
\(745\) 3.27960i 0.120155i
\(746\) 0 0
\(747\) −10.4453 −0.382175
\(748\) 0 0
\(749\) 22.2117 + 3.40165i 0.811598 + 0.124294i
\(750\) 0 0
\(751\) 23.5955i 0.861012i −0.902588 0.430506i \(-0.858335\pi\)
0.902588 0.430506i \(-0.141665\pi\)
\(752\) 0 0
\(753\) 13.1144 0.477915
\(754\) 0 0
\(755\) 4.74673 0.172751
\(756\) 0 0
\(757\) 22.2907 0.810170 0.405085 0.914279i \(-0.367242\pi\)
0.405085 + 0.914279i \(0.367242\pi\)
\(758\) 0 0
\(759\) 21.9974 0.798456
\(760\) 0 0
\(761\) 27.8234i 1.00860i 0.863530 + 0.504298i \(0.168249\pi\)
−0.863530 + 0.504298i \(0.831751\pi\)
\(762\) 0 0
\(763\) 3.36197 21.9526i 0.121711 0.794736i
\(764\) 0 0
\(765\) −4.79356 −0.173311
\(766\) 0 0
\(767\) 11.3716i 0.410604i
\(768\) 0 0
\(769\) 17.5470i 0.632760i 0.948633 + 0.316380i \(0.102467\pi\)
−0.948633 + 0.316380i \(0.897533\pi\)
\(770\) 0 0
\(771\) 10.4950i 0.377970i
\(772\) 0 0
\(773\) 27.3646i 0.984235i −0.870529 0.492118i \(-0.836223\pi\)
0.870529 0.492118i \(-0.163777\pi\)
\(774\) 0 0
\(775\) 0.221571 0.00795908
\(776\) 0 0
\(777\) −3.00392 + 19.6146i −0.107765 + 0.703670i
\(778\) 0 0
\(779\) 59.0852i 2.11695i
\(780\) 0 0
\(781\) −25.1933 −0.901486
\(782\) 0 0
\(783\) −1.79858 −0.0642759
\(784\) 0 0
\(785\) −15.5590 −0.555325
\(786\) 0 0
\(787\) −37.1840 −1.32547 −0.662733 0.748855i \(-0.730604\pi\)
−0.662733 + 0.748855i \(0.730604\pi\)
\(788\) 0 0
\(789\) 27.9160i 0.993836i
\(790\) 0 0
\(791\) 4.02746 26.2980i 0.143200 0.935050i
\(792\) 0 0
\(793\) 17.2364 0.612084
\(794\) 0 0
\(795\) 1.97985i 0.0702181i
\(796\) 0 0
\(797\) 39.4892i 1.39878i 0.714741 + 0.699389i \(0.246544\pi\)
−0.714741 + 0.699389i \(0.753456\pi\)
\(798\) 0 0
\(799\) 34.1047i 1.20654i
\(800\) 0 0
\(801\) 11.4560i 0.404779i
\(802\) 0 0
\(803\) 12.2770 0.433246
\(804\) 0 0
\(805\) −2.27462 + 14.8525i −0.0801697 + 0.523482i
\(806\) 0 0
\(807\) 1.37287i 0.0483272i
\(808\) 0 0
\(809\) 44.5833 1.56746 0.783732 0.621099i \(-0.213314\pi\)
0.783732 + 0.621099i \(0.213314\pi\)
\(810\) 0 0
\(811\) −42.6704 −1.49836 −0.749181 0.662365i \(-0.769553\pi\)
−0.749181 + 0.662365i \(0.769553\pi\)
\(812\) 0 0
\(813\) 2.00052 0.0701614
\(814\) 0 0
\(815\) 21.4106 0.749982
\(816\) 0 0
\(817\) 59.8944i 2.09544i
\(818\) 0 0
\(819\) −5.63706 0.863298i −0.196975 0.0301661i
\(820\) 0 0
\(821\) 2.16342 0.0755039 0.0377519 0.999287i \(-0.487980\pi\)
0.0377519 + 0.999287i \(0.487980\pi\)
\(822\) 0 0
\(823\) 24.8339i 0.865655i 0.901477 + 0.432828i \(0.142484\pi\)
−0.901477 + 0.432828i \(0.857516\pi\)
\(824\) 0 0
\(825\) 3.87335i 0.134853i
\(826\) 0 0
\(827\) 15.1462i 0.526685i −0.964702 0.263343i \(-0.915175\pi\)
0.964702 0.263343i \(-0.0848250\pi\)
\(828\) 0 0
\(829\) 38.5463i 1.33877i 0.742917 + 0.669383i \(0.233442\pi\)
−0.742917 + 0.669383i \(0.766558\pi\)
\(830\) 0 0
\(831\) 3.87788 0.134522
\(832\) 0 0
\(833\) −10.0421 + 32.0170i −0.347939 + 1.10932i
\(834\) 0 0
\(835\) 6.86529i 0.237583i
\(836\) 0 0
\(837\) 0.221571 0.00765863
\(838\) 0 0
\(839\) 26.9316 0.929783 0.464892 0.885368i \(-0.346093\pi\)
0.464892 + 0.885368i \(0.346093\pi\)
\(840\) 0 0
\(841\) −25.7651 −0.888452
\(842\) 0 0
\(843\) 15.4583 0.532413
\(844\) 0 0
\(845\) 8.35404i 0.287388i
\(846\) 0 0
\(847\) 1.60322 10.4685i 0.0550872 0.359702i
\(848\) 0 0
\(849\) −22.1130 −0.758915
\(850\) 0 0
\(851\) 42.5941i 1.46011i
\(852\) 0 0
\(853\) 28.2964i 0.968852i −0.874832 0.484426i \(-0.839029\pi\)
0.874832 0.484426i \(-0.160971\pi\)
\(854\) 0 0
\(855\) 5.31954i 0.181924i
\(856\) 0 0
\(857\) 44.2477i 1.51147i 0.654875 + 0.755737i \(0.272721\pi\)
−0.654875 + 0.755737i \(0.727279\pi\)
\(858\) 0 0
\(859\) −5.70764 −0.194742 −0.0973710 0.995248i \(-0.531043\pi\)
−0.0973710 + 0.995248i \(0.531043\pi\)
\(860\) 0 0
\(861\) 29.0482 + 4.44865i 0.989962 + 0.151610i
\(862\) 0 0
\(863\) 22.5353i 0.767110i 0.923518 + 0.383555i \(0.125300\pi\)
−0.923518 + 0.383555i \(0.874700\pi\)
\(864\) 0 0
\(865\) 13.9225 0.473380
\(866\) 0 0
\(867\) 5.97818 0.203030
\(868\) 0 0
\(869\) −30.9526 −1.05000
\(870\) 0 0
\(871\) −6.47165 −0.219283
\(872\) 0 0
\(873\) 0.716023i 0.0242337i
\(874\) 0 0
\(875\) −2.61526 0.400519i −0.0884119 0.0135400i
\(876\) 0 0
\(877\) 42.9633 1.45077 0.725384 0.688344i \(-0.241662\pi\)
0.725384 + 0.688344i \(0.241662\pi\)
\(878\) 0 0
\(879\) 24.2993i 0.819597i
\(880\) 0 0
\(881\) 37.3812i 1.25940i 0.776837 + 0.629702i \(0.216823\pi\)
−0.776837 + 0.629702i \(0.783177\pi\)
\(882\) 0 0
\(883\) 22.2753i 0.749624i −0.927101 0.374812i \(-0.877707\pi\)
0.927101 0.374812i \(-0.122293\pi\)
\(884\) 0 0
\(885\) 5.27574i 0.177342i
\(886\) 0 0
\(887\) 10.0896 0.338774 0.169387 0.985550i \(-0.445821\pi\)
0.169387 + 0.985550i \(0.445821\pi\)
\(888\) 0 0
\(889\) −50.7950 7.77910i −1.70361 0.260903i
\(890\) 0 0
\(891\) 3.87335i 0.129762i
\(892\) 0 0
\(893\) 37.8469 1.26650
\(894\) 0 0
\(895\) 6.99942 0.233965
\(896\) 0 0
\(897\) 12.2412 0.408720
\(898\) 0 0
\(899\) −0.398513 −0.0132912
\(900\) 0 0
\(901\) 9.49053i 0.316175i
\(902\) 0 0
\(903\) −29.4460 4.50957i −0.979903 0.150069i
\(904\) 0 0
\(905\) −1.97879 −0.0657772
\(906\) 0 0
\(907\) 9.27869i 0.308094i 0.988064 + 0.154047i \(0.0492307\pi\)
−0.988064 + 0.154047i \(0.950769\pi\)
\(908\) 0 0
\(909\) 5.18641i 0.172022i
\(910\) 0 0
\(911\) 12.0695i 0.399880i 0.979808 + 0.199940i \(0.0640748\pi\)
−0.979808 + 0.199940i \(0.935925\pi\)
\(912\) 0 0
\(913\) 40.4585i 1.33898i
\(914\) 0 0
\(915\) 7.99669 0.264362
\(916\) 0 0
\(917\) 1.81103 11.8254i 0.0598054 0.390510i
\(918\) 0 0
\(919\) 41.5233i 1.36973i −0.728672 0.684863i \(-0.759862\pi\)
0.728672 0.684863i \(-0.240138\pi\)
\(920\) 0 0
\(921\) 29.4220 0.969489
\(922\) 0 0
\(923\) −14.0196 −0.461460
\(924\) 0 0
\(925\) 7.50006 0.246600
\(926\) 0 0
\(927\) −3.19896 −0.105068
\(928\) 0 0
\(929\) 1.38568i 0.0454625i −0.999742 0.0227313i \(-0.992764\pi\)
0.999742 0.0227313i \(-0.00723621\pi\)
\(930\) 0 0
\(931\) −35.5301 11.1440i −1.16445 0.365231i
\(932\) 0 0
\(933\) 26.2914 0.860742
\(934\) 0 0
\(935\) 18.5671i 0.607210i
\(936\) 0 0
\(937\) 20.3283i 0.664096i −0.943263 0.332048i \(-0.892260\pi\)
0.943263 0.332048i \(-0.107740\pi\)
\(938\) 0 0
\(939\) 31.6179i 1.03181i
\(940\) 0 0
\(941\) 17.6899i 0.576675i −0.957529 0.288338i \(-0.906897\pi\)
0.957529 0.288338i \(-0.0931026\pi\)
\(942\) 0 0
\(943\) −63.0797 −2.05416
\(944\) 0 0
\(945\) −2.61526 0.400519i −0.0850744 0.0130289i
\(946\) 0 0
\(947\) 39.4890i 1.28322i 0.767031 + 0.641610i \(0.221733\pi\)
−0.767031 + 0.641610i \(0.778267\pi\)
\(948\) 0 0
\(949\) 6.83192 0.221774
\(950\) 0 0
\(951\) −26.4489 −0.857666
\(952\) 0 0
\(953\) 8.25093 0.267274 0.133637 0.991030i \(-0.457334\pi\)
0.133637 + 0.991030i \(0.457334\pi\)
\(954\) 0 0
\(955\) 20.5288 0.664297
\(956\) 0 0
\(957\) 6.96652i 0.225196i
\(958\) 0 0
\(959\) −5.09313 + 33.2565i −0.164466 + 1.07391i
\(960\) 0 0
\(961\) −30.9509 −0.998416
\(962\) 0 0
\(963\) 8.49311i 0.273687i
\(964\) 0 0
\(965\) 13.1050i 0.421866i
\(966\) 0 0
\(967\) 49.0779i 1.57824i −0.614241 0.789119i \(-0.710538\pi\)
0.614241 0.789119i \(-0.289462\pi\)
\(968\) 0 0
\(969\) 25.4995i 0.819162i
\(970\) 0 0
\(971\) −9.07094 −0.291100 −0.145550 0.989351i \(-0.546495\pi\)
−0.145550 + 0.989351i \(0.546495\pi\)
\(972\) 0 0
\(973\) −1.01194 + 6.60764i −0.0324413 + 0.211831i
\(974\) 0 0
\(975\) 2.15545i 0.0690296i
\(976\) 0 0
\(977\) 19.3704 0.619713 0.309857 0.950783i \(-0.399719\pi\)
0.309857 + 0.950783i \(0.399719\pi\)
\(978\) 0 0
\(979\) −44.3732 −1.41817
\(980\) 0 0
\(981\) −8.39403 −0.268001
\(982\) 0 0
\(983\) −31.0078 −0.988995 −0.494497 0.869179i \(-0.664648\pi\)
−0.494497 + 0.869179i \(0.664648\pi\)
\(984\) 0 0
\(985\) 6.78520i 0.216194i
\(986\) 0 0
\(987\) 2.84957 18.6068i 0.0907028 0.592260i
\(988\) 0 0
\(989\) 63.9436 2.03329
\(990\) 0 0
\(991\) 27.8278i 0.883981i −0.897020 0.441990i \(-0.854273\pi\)
0.897020 0.441990i \(-0.145727\pi\)
\(992\) 0 0
\(993\) 8.39519i 0.266413i
\(994\) 0 0
\(995\) 6.62783i 0.210116i
\(996\) 0 0
\(997\) 6.73019i 0.213147i 0.994305 + 0.106574i \(0.0339880\pi\)
−0.994305 + 0.106574i \(0.966012\pi\)
\(998\) 0 0
\(999\) 7.50006 0.237291
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 3360.2.d.a.2911.4 16
4.3 odd 2 3360.2.d.d.2911.5 yes 16
7.6 odd 2 3360.2.d.d.2911.13 yes 16
28.27 even 2 inner 3360.2.d.a.2911.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.d.a.2911.4 16 1.1 even 1 trivial
3360.2.d.a.2911.12 yes 16 28.27 even 2 inner
3360.2.d.d.2911.5 yes 16 4.3 odd 2
3360.2.d.d.2911.13 yes 16 7.6 odd 2