Properties

Label 3450.2.d.v.2899.4
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $4$
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] = [3450,2,Mod(2899,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3450.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.4
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.v.2899.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.12311i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.12311i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.12311 q^{11} -1.00000i q^{12} -2.00000i q^{13} -3.12311 q^{14} +1.00000 q^{16} -1.12311i q^{17} -1.00000i q^{18} -4.00000 q^{19} -3.12311 q^{21} -3.12311i q^{22} -1.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -3.12311i q^{28} -2.00000 q^{29} +1.00000i q^{32} -3.12311i q^{33} +1.12311 q^{34} +1.00000 q^{36} +1.12311i q^{37} -4.00000i q^{38} +2.00000 q^{39} +2.00000 q^{41} -3.12311i q^{42} +3.12311 q^{44} +1.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -2.75379 q^{49} +1.12311 q^{51} +2.00000i q^{52} -12.2462i q^{53} +1.00000 q^{54} +3.12311 q^{56} -4.00000i q^{57} -2.00000i q^{58} -2.24621 q^{59} +9.12311 q^{61} -3.12311i q^{63} -1.00000 q^{64} +3.12311 q^{66} -8.00000i q^{67} +1.12311i q^{68} +1.00000 q^{69} -10.2462 q^{71} +1.00000i q^{72} +4.24621i q^{73} -1.12311 q^{74} +4.00000 q^{76} -9.75379i q^{77} +2.00000i q^{78} -3.12311 q^{79} +1.00000 q^{81} +2.00000i q^{82} +13.3693i q^{83} +3.12311 q^{84} -2.00000i q^{87} +3.12311i q^{88} +5.12311 q^{89} +6.24621 q^{91} +1.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} -16.2462i q^{97} -2.75379i q^{98} +3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{11} + 4 q^{14} + 4 q^{16} - 16 q^{19} + 4 q^{21} + 4 q^{24} + 8 q^{26} - 8 q^{29} - 12 q^{34} + 4 q^{36} + 8 q^{39} + 8 q^{41} - 4 q^{44} + 4 q^{46} - 44 q^{49} - 12 q^{51} + 4 q^{54} - 4 q^{56} + 24 q^{59} + 20 q^{61} - 4 q^{64} - 4 q^{66} + 4 q^{69} - 8 q^{71} + 12 q^{74} + 16 q^{76} + 4 q^{79} + 4 q^{81} - 4 q^{84} + 4 q^{89} - 8 q^{91} + 32 q^{94} - 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 3.12311i 1.18042i 0.807249 + 0.590211i \(0.200956\pi\)
−0.807249 + 0.590211i \(0.799044\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −3.12311 −0.834685
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.12311i − 0.272393i −0.990682 0.136197i \(-0.956512\pi\)
0.990682 0.136197i \(-0.0434879\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −3.12311 −0.681518
\(22\) − 3.12311i − 0.665848i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 3.12311i − 0.590211i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.12311i − 0.543663i
\(34\) 1.12311 0.192611
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.12311i 0.184637i 0.995730 + 0.0923187i \(0.0294279\pi\)
−0.995730 + 0.0923187i \(0.970572\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 3.12311i − 0.481906i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 3.12311 0.470826
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −2.75379 −0.393398
\(50\) 0 0
\(51\) 1.12311 0.157266
\(52\) 2.00000i 0.277350i
\(53\) − 12.2462i − 1.68215i −0.540921 0.841073i \(-0.681924\pi\)
0.540921 0.841073i \(-0.318076\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.12311 0.417343
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) −2.24621 −0.292432 −0.146216 0.989253i \(-0.546709\pi\)
−0.146216 + 0.989253i \(0.546709\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 0 0
\(63\) − 3.12311i − 0.393474i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.12311 0.384428
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 1.12311i 0.136197i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.2462 −1.21600 −0.608001 0.793936i \(-0.708028\pi\)
−0.608001 + 0.793936i \(0.708028\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 4.24621i 0.496981i 0.968634 + 0.248491i \(0.0799345\pi\)
−0.968634 + 0.248491i \(0.920065\pi\)
\(74\) −1.12311 −0.130558
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 9.75379i − 1.11155i
\(78\) 2.00000i 0.226455i
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 13.3693i 1.46747i 0.679434 + 0.733737i \(0.262225\pi\)
−0.679434 + 0.733737i \(0.737775\pi\)
\(84\) 3.12311 0.340759
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.00000i − 0.214423i
\(88\) 3.12311i 0.332924i
\(89\) 5.12311 0.543048 0.271524 0.962432i \(-0.412472\pi\)
0.271524 + 0.962432i \(0.412472\pi\)
\(90\) 0 0
\(91\) 6.24621 0.654781
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 16.2462i − 1.64955i −0.565459 0.824776i \(-0.691301\pi\)
0.565459 0.824776i \(-0.308699\pi\)
\(98\) − 2.75379i − 0.278175i
\(99\) 3.12311 0.313884
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 1.12311i 0.111204i
\(103\) 9.36932i 0.923186i 0.887092 + 0.461593i \(0.152722\pi\)
−0.887092 + 0.461593i \(0.847278\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 12.2462 1.18946
\(107\) − 13.3693i − 1.29246i −0.763142 0.646230i \(-0.776345\pi\)
0.763142 0.646230i \(-0.223655\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −2.87689 −0.275557 −0.137778 0.990463i \(-0.543996\pi\)
−0.137778 + 0.990463i \(0.543996\pi\)
\(110\) 0 0
\(111\) −1.12311 −0.106600
\(112\) 3.12311i 0.295106i
\(113\) 9.12311i 0.858230i 0.903250 + 0.429115i \(0.141174\pi\)
−0.903250 + 0.429115i \(0.858826\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) − 2.24621i − 0.206781i
\(119\) 3.50758 0.321539
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 9.12311i 0.825967i
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 3.12311 0.278228
\(127\) − 5.75379i − 0.510566i −0.966866 0.255283i \(-0.917831\pi\)
0.966866 0.255283i \(-0.0821687\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 3.12311i 0.271831i
\(133\) − 12.4924i − 1.08323i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.12311 −0.0963055
\(137\) − 10.8769i − 0.929276i −0.885501 0.464638i \(-0.846184\pi\)
0.885501 0.464638i \(-0.153816\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) − 10.2462i − 0.859843i
\(143\) 6.24621i 0.522334i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.24621 −0.351419
\(147\) − 2.75379i − 0.227129i
\(148\) − 1.12311i − 0.0923187i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 6.24621 0.508309 0.254155 0.967164i \(-0.418203\pi\)
0.254155 + 0.967164i \(0.418203\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 1.12311i 0.0907977i
\(154\) 9.75379 0.785983
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 9.12311i 0.728103i 0.931379 + 0.364052i \(0.118607\pi\)
−0.931379 + 0.364052i \(0.881393\pi\)
\(158\) − 3.12311i − 0.248461i
\(159\) 12.2462 0.971188
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 1.00000i 0.0785674i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −13.3693 −1.03766
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 3.12311i 0.240953i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) − 20.2462i − 1.53929i −0.638471 0.769645i \(-0.720433\pi\)
0.638471 0.769645i \(-0.279567\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −3.12311 −0.235413
\(177\) − 2.24621i − 0.168836i
\(178\) 5.12311i 0.383993i
\(179\) 8.49242 0.634753 0.317377 0.948300i \(-0.397198\pi\)
0.317377 + 0.948300i \(0.397198\pi\)
\(180\) 0 0
\(181\) 10.8769 0.808473 0.404237 0.914654i \(-0.367537\pi\)
0.404237 + 0.914654i \(0.367537\pi\)
\(182\) 6.24621i 0.463000i
\(183\) 9.12311i 0.674399i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.50758i 0.256499i
\(188\) 8.00000i 0.583460i
\(189\) 3.12311 0.227173
\(190\) 0 0
\(191\) 12.4924 0.903920 0.451960 0.892038i \(-0.350725\pi\)
0.451960 + 0.892038i \(0.350725\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 8.24621i − 0.593575i −0.954944 0.296788i \(-0.904085\pi\)
0.954944 0.296788i \(-0.0959153\pi\)
\(194\) 16.2462 1.16641
\(195\) 0 0
\(196\) 2.75379 0.196699
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 3.12311i 0.221949i
\(199\) −19.1231 −1.35560 −0.677801 0.735246i \(-0.737067\pi\)
−0.677801 + 0.735246i \(0.737067\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0.246211i 0.0173234i
\(203\) − 6.24621i − 0.438398i
\(204\) −1.12311 −0.0786331
\(205\) 0 0
\(206\) −9.36932 −0.652791
\(207\) 1.00000i 0.0695048i
\(208\) − 2.00000i − 0.138675i
\(209\) 12.4924 0.864119
\(210\) 0 0
\(211\) −16.4924 −1.13539 −0.567693 0.823241i \(-0.692164\pi\)
−0.567693 + 0.823241i \(0.692164\pi\)
\(212\) 12.2462i 0.841073i
\(213\) − 10.2462i − 0.702059i
\(214\) 13.3693 0.913908
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) − 2.87689i − 0.194848i
\(219\) −4.24621 −0.286932
\(220\) 0 0
\(221\) −2.24621 −0.151097
\(222\) − 1.12311i − 0.0753779i
\(223\) 10.2462i 0.686137i 0.939310 + 0.343069i \(0.111466\pi\)
−0.939310 + 0.343069i \(0.888534\pi\)
\(224\) −3.12311 −0.208671
\(225\) 0 0
\(226\) −9.12311 −0.606860
\(227\) − 5.36932i − 0.356374i −0.983997 0.178187i \(-0.942977\pi\)
0.983997 0.178187i \(-0.0570232\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −23.3693 −1.54429 −0.772144 0.635448i \(-0.780816\pi\)
−0.772144 + 0.635448i \(0.780816\pi\)
\(230\) 0 0
\(231\) 9.75379 0.641752
\(232\) 2.00000i 0.131306i
\(233\) − 28.7386i − 1.88273i −0.337389 0.941365i \(-0.609544\pi\)
0.337389 0.941365i \(-0.390456\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 2.24621 0.146216
\(237\) − 3.12311i − 0.202868i
\(238\) 3.50758i 0.227362i
\(239\) 18.2462 1.18025 0.590125 0.807312i \(-0.299079\pi\)
0.590125 + 0.807312i \(0.299079\pi\)
\(240\) 0 0
\(241\) 30.4924 1.96419 0.982095 0.188387i \(-0.0603260\pi\)
0.982095 + 0.188387i \(0.0603260\pi\)
\(242\) − 1.24621i − 0.0801095i
\(243\) 1.00000i 0.0641500i
\(244\) −9.12311 −0.584047
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) −13.3693 −0.847246
\(250\) 0 0
\(251\) 17.3693 1.09634 0.548171 0.836366i \(-0.315324\pi\)
0.548171 + 0.836366i \(0.315324\pi\)
\(252\) 3.12311i 0.196737i
\(253\) 3.12311i 0.196348i
\(254\) 5.75379 0.361025
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.24621i 0.514385i 0.966360 + 0.257192i \(0.0827974\pi\)
−0.966360 + 0.257192i \(0.917203\pi\)
\(258\) 0 0
\(259\) −3.50758 −0.217950
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 4.00000i − 0.247121i
\(263\) 6.24621i 0.385158i 0.981281 + 0.192579i \(0.0616851\pi\)
−0.981281 + 0.192579i \(0.938315\pi\)
\(264\) −3.12311 −0.192214
\(265\) 0 0
\(266\) 12.4924 0.765960
\(267\) 5.12311i 0.313529i
\(268\) 8.00000i 0.488678i
\(269\) −0.246211 −0.0150118 −0.00750588 0.999972i \(-0.502389\pi\)
−0.00750588 + 0.999972i \(0.502389\pi\)
\(270\) 0 0
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) − 1.12311i − 0.0680983i
\(273\) 6.24621i 0.378038i
\(274\) 10.8769 0.657097
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 3.75379i 0.225543i 0.993621 + 0.112772i \(0.0359729\pi\)
−0.993621 + 0.112772i \(0.964027\pi\)
\(278\) − 16.4924i − 0.989150i
\(279\) 0 0
\(280\) 0 0
\(281\) −21.1231 −1.26010 −0.630049 0.776555i \(-0.716965\pi\)
−0.630049 + 0.776555i \(0.716965\pi\)
\(282\) 8.00000i 0.476393i
\(283\) − 14.2462i − 0.846849i −0.905931 0.423425i \(-0.860828\pi\)
0.905931 0.423425i \(-0.139172\pi\)
\(284\) 10.2462 0.608001
\(285\) 0 0
\(286\) −6.24621 −0.369346
\(287\) 6.24621i 0.368702i
\(288\) − 1.00000i − 0.0589256i
\(289\) 15.7386 0.925802
\(290\) 0 0
\(291\) 16.2462 0.952370
\(292\) − 4.24621i − 0.248491i
\(293\) − 28.2462i − 1.65016i −0.565015 0.825081i \(-0.691130\pi\)
0.565015 0.825081i \(-0.308870\pi\)
\(294\) 2.75379 0.160604
\(295\) 0 0
\(296\) 1.12311 0.0652792
\(297\) 3.12311i 0.181221i
\(298\) 10.0000i 0.579284i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 6.24621i 0.359429i
\(303\) 0.246211i 0.0141445i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −1.12311 −0.0642037
\(307\) − 32.4924i − 1.85444i −0.374516 0.927220i \(-0.622191\pi\)
0.374516 0.927220i \(-0.377809\pi\)
\(308\) 9.75379i 0.555774i
\(309\) −9.36932 −0.533002
\(310\) 0 0
\(311\) −22.7386 −1.28939 −0.644695 0.764440i \(-0.723016\pi\)
−0.644695 + 0.764440i \(0.723016\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 22.4924i 1.27135i 0.771958 + 0.635673i \(0.219278\pi\)
−0.771958 + 0.635673i \(0.780722\pi\)
\(314\) −9.12311 −0.514847
\(315\) 0 0
\(316\) 3.12311 0.175688
\(317\) − 28.7386i − 1.61412i −0.590468 0.807061i \(-0.701057\pi\)
0.590468 0.807061i \(-0.298943\pi\)
\(318\) 12.2462i 0.686733i
\(319\) 6.24621 0.349721
\(320\) 0 0
\(321\) 13.3693 0.746203
\(322\) 3.12311i 0.174044i
\(323\) 4.49242i 0.249965i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 2.87689i − 0.159093i
\(328\) − 2.00000i − 0.110432i
\(329\) 24.9848 1.37746
\(330\) 0 0
\(331\) 0.492423 0.0270660 0.0135330 0.999908i \(-0.495692\pi\)
0.0135330 + 0.999908i \(0.495692\pi\)
\(332\) − 13.3693i − 0.733737i
\(333\) − 1.12311i − 0.0615458i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −3.12311 −0.170379
\(337\) 26.4924i 1.44313i 0.692345 + 0.721567i \(0.256578\pi\)
−0.692345 + 0.721567i \(0.743422\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −9.12311 −0.495499
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000i 0.216295i
\(343\) 13.2614i 0.716046i
\(344\) 0 0
\(345\) 0 0
\(346\) 20.2462 1.08844
\(347\) 14.7386i 0.791211i 0.918420 + 0.395606i \(0.129465\pi\)
−0.918420 + 0.395606i \(0.870535\pi\)
\(348\) 2.00000i 0.107211i
\(349\) −24.7386 −1.32423 −0.662114 0.749403i \(-0.730341\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 3.12311i − 0.166462i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 2.24621 0.119385
\(355\) 0 0
\(356\) −5.12311 −0.271524
\(357\) 3.50758i 0.185641i
\(358\) 8.49242i 0.448838i
\(359\) −20.4924 −1.08155 −0.540774 0.841168i \(-0.681869\pi\)
−0.540774 + 0.841168i \(0.681869\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 10.8769i 0.571677i
\(363\) − 1.24621i − 0.0654091i
\(364\) −6.24621 −0.327390
\(365\) 0 0
\(366\) −9.12311 −0.476872
\(367\) 4.87689i 0.254572i 0.991866 + 0.127286i \(0.0406266\pi\)
−0.991866 + 0.127286i \(0.959373\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 38.2462 1.98564
\(372\) 0 0
\(373\) 11.3693i 0.588681i 0.955701 + 0.294340i \(0.0951000\pi\)
−0.955701 + 0.294340i \(0.904900\pi\)
\(374\) −3.50758 −0.181373
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 4.00000i 0.206010i
\(378\) 3.12311i 0.160635i
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 5.75379 0.294776
\(382\) 12.4924i 0.639168i
\(383\) − 30.2462i − 1.54551i −0.634705 0.772755i \(-0.718878\pi\)
0.634705 0.772755i \(-0.281122\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 8.24621 0.419721
\(387\) 0 0
\(388\) 16.2462i 0.824776i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −1.12311 −0.0567979
\(392\) 2.75379i 0.139087i
\(393\) − 4.00000i − 0.201773i
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −3.12311 −0.156942
\(397\) − 1.50758i − 0.0756631i −0.999284 0.0378316i \(-0.987955\pi\)
0.999284 0.0378316i \(-0.0120450\pi\)
\(398\) − 19.1231i − 0.958555i
\(399\) 12.4924 0.625403
\(400\) 0 0
\(401\) 7.36932 0.368006 0.184003 0.982926i \(-0.441094\pi\)
0.184003 + 0.982926i \(0.441094\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 0 0
\(404\) −0.246211 −0.0122495
\(405\) 0 0
\(406\) 6.24621 0.309994
\(407\) − 3.50758i − 0.173864i
\(408\) − 1.12311i − 0.0556020i
\(409\) −0.246211 −0.0121744 −0.00608718 0.999981i \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) 0 0
\(411\) 10.8769 0.536518
\(412\) − 9.36932i − 0.461593i
\(413\) − 7.01515i − 0.345193i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 16.4924i − 0.807637i
\(418\) 12.4924i 0.611024i
\(419\) −39.6155 −1.93535 −0.967673 0.252210i \(-0.918843\pi\)
−0.967673 + 0.252210i \(0.918843\pi\)
\(420\) 0 0
\(421\) −11.3693 −0.554107 −0.277053 0.960855i \(-0.589358\pi\)
−0.277053 + 0.960855i \(0.589358\pi\)
\(422\) − 16.4924i − 0.802839i
\(423\) 8.00000i 0.388973i
\(424\) −12.2462 −0.594729
\(425\) 0 0
\(426\) 10.2462 0.496431
\(427\) 28.4924i 1.37884i
\(428\) 13.3693i 0.646230i
\(429\) −6.24621 −0.301570
\(430\) 0 0
\(431\) 9.75379 0.469823 0.234912 0.972017i \(-0.424520\pi\)
0.234912 + 0.972017i \(0.424520\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 22.4924i 1.08092i 0.841371 + 0.540458i \(0.181749\pi\)
−0.841371 + 0.540458i \(0.818251\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.87689 0.137778
\(437\) 4.00000i 0.191346i
\(438\) − 4.24621i − 0.202892i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 2.75379 0.131133
\(442\) − 2.24621i − 0.106841i
\(443\) − 8.49242i − 0.403487i −0.979438 0.201744i \(-0.935339\pi\)
0.979438 0.201744i \(-0.0646608\pi\)
\(444\) 1.12311 0.0533002
\(445\) 0 0
\(446\) −10.2462 −0.485172
\(447\) 10.0000i 0.472984i
\(448\) − 3.12311i − 0.147553i
\(449\) −22.4924 −1.06148 −0.530742 0.847534i \(-0.678087\pi\)
−0.530742 + 0.847534i \(0.678087\pi\)
\(450\) 0 0
\(451\) −6.24621 −0.294123
\(452\) − 9.12311i − 0.429115i
\(453\) 6.24621i 0.293473i
\(454\) 5.36932 0.251995
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 16.7386i 0.783000i 0.920178 + 0.391500i \(0.128044\pi\)
−0.920178 + 0.391500i \(0.871956\pi\)
\(458\) − 23.3693i − 1.09198i
\(459\) −1.12311 −0.0524221
\(460\) 0 0
\(461\) −16.7386 −0.779596 −0.389798 0.920900i \(-0.627455\pi\)
−0.389798 + 0.920900i \(0.627455\pi\)
\(462\) 9.75379i 0.453787i
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 28.7386 1.33129
\(467\) 21.3693i 0.988854i 0.869219 + 0.494427i \(0.164622\pi\)
−0.869219 + 0.494427i \(0.835378\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 24.9848 1.15369
\(470\) 0 0
\(471\) −9.12311 −0.420371
\(472\) 2.24621i 0.103390i
\(473\) 0 0
\(474\) 3.12311 0.143449
\(475\) 0 0
\(476\) −3.50758 −0.160770
\(477\) 12.2462i 0.560715i
\(478\) 18.2462i 0.834562i
\(479\) −12.4924 −0.570793 −0.285397 0.958409i \(-0.592125\pi\)
−0.285397 + 0.958409i \(0.592125\pi\)
\(480\) 0 0
\(481\) 2.24621 0.102418
\(482\) 30.4924i 1.38889i
\(483\) 3.12311i 0.142106i
\(484\) 1.24621 0.0566460
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 7.50758i 0.340201i 0.985427 + 0.170100i \(0.0544092\pi\)
−0.985427 + 0.170100i \(0.945591\pi\)
\(488\) − 9.12311i − 0.412984i
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) 2.24621i 0.101164i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) − 32.0000i − 1.43540i
\(498\) − 13.3693i − 0.599093i
\(499\) −36.9848 −1.65567 −0.827835 0.560972i \(-0.810427\pi\)
−0.827835 + 0.560972i \(0.810427\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 17.3693i 0.775231i
\(503\) − 42.7386i − 1.90562i −0.303565 0.952811i \(-0.598177\pi\)
0.303565 0.952811i \(-0.401823\pi\)
\(504\) −3.12311 −0.139114
\(505\) 0 0
\(506\) −3.12311 −0.138839
\(507\) 9.00000i 0.399704i
\(508\) 5.75379i 0.255283i
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −13.2614 −0.586648
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −8.24621 −0.363725
\(515\) 0 0
\(516\) 0 0
\(517\) 24.9848i 1.09883i
\(518\) − 3.50758i − 0.154114i
\(519\) 20.2462 0.888710
\(520\) 0 0
\(521\) 17.1231 0.750177 0.375088 0.926989i \(-0.377612\pi\)
0.375088 + 0.926989i \(0.377612\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 4.49242i − 0.196440i −0.995165 0.0982200i \(-0.968685\pi\)
0.995165 0.0982200i \(-0.0313149\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −6.24621 −0.272348
\(527\) 0 0
\(528\) − 3.12311i − 0.135916i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 2.24621 0.0974773
\(532\) 12.4924i 0.541615i
\(533\) − 4.00000i − 0.173259i
\(534\) −5.12311 −0.221698
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 8.49242i 0.366475i
\(538\) − 0.246211i − 0.0106149i
\(539\) 8.60037 0.370444
\(540\) 0 0
\(541\) −32.2462 −1.38637 −0.693186 0.720758i \(-0.743794\pi\)
−0.693186 + 0.720758i \(0.743794\pi\)
\(542\) − 6.24621i − 0.268298i
\(543\) 10.8769i 0.466772i
\(544\) 1.12311 0.0481528
\(545\) 0 0
\(546\) −6.24621 −0.267313
\(547\) − 32.4924i − 1.38928i −0.719360 0.694638i \(-0.755565\pi\)
0.719360 0.694638i \(-0.244435\pi\)
\(548\) 10.8769i 0.464638i
\(549\) −9.12311 −0.389365
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 1.00000i − 0.0425628i
\(553\) − 9.75379i − 0.414773i
\(554\) −3.75379 −0.159483
\(555\) 0 0
\(556\) 16.4924 0.699435
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.50758 −0.148090
\(562\) − 21.1231i − 0.891024i
\(563\) 13.3693i 0.563450i 0.959495 + 0.281725i \(0.0909065\pi\)
−0.959495 + 0.281725i \(0.909093\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 14.2462 0.598813
\(567\) 3.12311i 0.131158i
\(568\) 10.2462i 0.429921i
\(569\) −18.8769 −0.791361 −0.395680 0.918388i \(-0.629491\pi\)
−0.395680 + 0.918388i \(0.629491\pi\)
\(570\) 0 0
\(571\) −14.7386 −0.616793 −0.308396 0.951258i \(-0.599792\pi\)
−0.308396 + 0.951258i \(0.599792\pi\)
\(572\) − 6.24621i − 0.261167i
\(573\) 12.4924i 0.521878i
\(574\) −6.24621 −0.260712
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 18.4924i − 0.769850i −0.922948 0.384925i \(-0.874227\pi\)
0.922948 0.384925i \(-0.125773\pi\)
\(578\) 15.7386i 0.654641i
\(579\) 8.24621 0.342701
\(580\) 0 0
\(581\) −41.7538 −1.73224
\(582\) 16.2462i 0.673427i
\(583\) 38.2462i 1.58400i
\(584\) 4.24621 0.175709
\(585\) 0 0
\(586\) 28.2462 1.16684
\(587\) − 26.2462i − 1.08330i −0.840605 0.541649i \(-0.817800\pi\)
0.840605 0.541649i \(-0.182200\pi\)
\(588\) 2.75379i 0.113564i
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 1.12311i 0.0461594i
\(593\) − 24.2462i − 0.995673i −0.867271 0.497836i \(-0.834128\pi\)
0.867271 0.497836i \(-0.165872\pi\)
\(594\) −3.12311 −0.128143
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) − 19.1231i − 0.782657i
\(598\) − 2.00000i − 0.0817861i
\(599\) −10.2462 −0.418649 −0.209324 0.977846i \(-0.567126\pi\)
−0.209324 + 0.977846i \(0.567126\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) −6.24621 −0.254155
\(605\) 0 0
\(606\) −0.246211 −0.0100016
\(607\) − 22.7386i − 0.922933i −0.887158 0.461466i \(-0.847324\pi\)
0.887158 0.461466i \(-0.152676\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 6.24621 0.253109
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) − 1.12311i − 0.0453989i
\(613\) − 34.8769i − 1.40866i −0.709870 0.704332i \(-0.751247\pi\)
0.709870 0.704332i \(-0.248753\pi\)
\(614\) 32.4924 1.31129
\(615\) 0 0
\(616\) −9.75379 −0.392991
\(617\) − 34.1080i − 1.37313i −0.727066 0.686567i \(-0.759117\pi\)
0.727066 0.686567i \(-0.240883\pi\)
\(618\) − 9.36932i − 0.376889i
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 22.7386i − 0.911736i
\(623\) 16.0000i 0.641026i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −22.4924 −0.898978
\(627\) 12.4924i 0.498899i
\(628\) − 9.12311i − 0.364052i
\(629\) 1.26137 0.0502940
\(630\) 0 0
\(631\) −14.6307 −0.582438 −0.291219 0.956656i \(-0.594061\pi\)
−0.291219 + 0.956656i \(0.594061\pi\)
\(632\) 3.12311i 0.124230i
\(633\) − 16.4924i − 0.655515i
\(634\) 28.7386 1.14136
\(635\) 0 0
\(636\) −12.2462 −0.485594
\(637\) 5.50758i 0.218218i
\(638\) 6.24621i 0.247290i
\(639\) 10.2462 0.405334
\(640\) 0 0
\(641\) 15.3693 0.607052 0.303526 0.952823i \(-0.401836\pi\)
0.303526 + 0.952823i \(0.401836\pi\)
\(642\) 13.3693i 0.527645i
\(643\) − 3.50758i − 0.138325i −0.997605 0.0691627i \(-0.977967\pi\)
0.997605 0.0691627i \(-0.0220328\pi\)
\(644\) −3.12311 −0.123068
\(645\) 0 0
\(646\) −4.49242 −0.176752
\(647\) 44.4924i 1.74918i 0.484865 + 0.874589i \(0.338869\pi\)
−0.484865 + 0.874589i \(0.661131\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 7.01515 0.275369
\(650\) 0 0
\(651\) 0 0
\(652\) − 12.0000i − 0.469956i
\(653\) 22.4924i 0.880197i 0.897950 + 0.440098i \(0.145056\pi\)
−0.897950 + 0.440098i \(0.854944\pi\)
\(654\) 2.87689 0.112495
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) − 4.24621i − 0.165660i
\(658\) 24.9848i 0.974011i
\(659\) −17.3693 −0.676613 −0.338306 0.941036i \(-0.609854\pi\)
−0.338306 + 0.941036i \(0.609854\pi\)
\(660\) 0 0
\(661\) −23.8617 −0.928114 −0.464057 0.885805i \(-0.653607\pi\)
−0.464057 + 0.885805i \(0.653607\pi\)
\(662\) 0.492423i 0.0191385i
\(663\) − 2.24621i − 0.0872356i
\(664\) 13.3693 0.518830
\(665\) 0 0
\(666\) 1.12311 0.0435195
\(667\) 2.00000i 0.0774403i
\(668\) 8.00000i 0.309529i
\(669\) −10.2462 −0.396141
\(670\) 0 0
\(671\) −28.4924 −1.09994
\(672\) − 3.12311i − 0.120476i
\(673\) − 18.9848i − 0.731812i −0.930652 0.365906i \(-0.880759\pi\)
0.930652 0.365906i \(-0.119241\pi\)
\(674\) −26.4924 −1.02045
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 13.5076i − 0.519138i −0.965725 0.259569i \(-0.916419\pi\)
0.965725 0.259569i \(-0.0835806\pi\)
\(678\) − 9.12311i − 0.350371i
\(679\) 50.7386 1.94717
\(680\) 0 0
\(681\) 5.36932 0.205753
\(682\) 0 0
\(683\) − 34.2462i − 1.31039i −0.755458 0.655197i \(-0.772585\pi\)
0.755458 0.655197i \(-0.227415\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −13.2614 −0.506321
\(687\) − 23.3693i − 0.891595i
\(688\) 0 0
\(689\) −24.4924 −0.933087
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 20.2462i 0.769645i
\(693\) 9.75379i 0.370516i
\(694\) −14.7386 −0.559471
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) − 2.24621i − 0.0850813i
\(698\) − 24.7386i − 0.936371i
\(699\) 28.7386 1.08699
\(700\) 0 0
\(701\) −19.7538 −0.746090 −0.373045 0.927813i \(-0.621686\pi\)
−0.373045 + 0.927813i \(0.621686\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) − 4.49242i − 0.169435i
\(704\) 3.12311 0.117706
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0.768944i 0.0289191i
\(708\) 2.24621i 0.0844178i
\(709\) 11.3693 0.426984 0.213492 0.976945i \(-0.431516\pi\)
0.213492 + 0.976945i \(0.431516\pi\)
\(710\) 0 0
\(711\) 3.12311 0.117126
\(712\) − 5.12311i − 0.191997i
\(713\) 0 0
\(714\) −3.50758 −0.131268
\(715\) 0 0
\(716\) −8.49242 −0.317377
\(717\) 18.2462i 0.681417i
\(718\) − 20.4924i − 0.764770i
\(719\) 26.2462 0.978819 0.489409 0.872054i \(-0.337212\pi\)
0.489409 + 0.872054i \(0.337212\pi\)
\(720\) 0 0
\(721\) −29.2614 −1.08975
\(722\) − 3.00000i − 0.111648i
\(723\) 30.4924i 1.13403i
\(724\) −10.8769 −0.404237
\(725\) 0 0
\(726\) 1.24621 0.0462512
\(727\) − 29.8617i − 1.10751i −0.832679 0.553755i \(-0.813194\pi\)
0.832679 0.553755i \(-0.186806\pi\)
\(728\) − 6.24621i − 0.231500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 9.12311i − 0.337200i
\(733\) − 43.8617i − 1.62007i −0.586381 0.810035i \(-0.699448\pi\)
0.586381 0.810035i \(-0.300552\pi\)
\(734\) −4.87689 −0.180009
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 24.9848i 0.920329i
\(738\) − 2.00000i − 0.0736210i
\(739\) −7.50758 −0.276171 −0.138085 0.990420i \(-0.544095\pi\)
−0.138085 + 0.990420i \(0.544095\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 38.2462i 1.40406i
\(743\) − 34.7386i − 1.27444i −0.770683 0.637218i \(-0.780085\pi\)
0.770683 0.637218i \(-0.219915\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −11.3693 −0.416260
\(747\) − 13.3693i − 0.489158i
\(748\) − 3.50758i − 0.128250i
\(749\) 41.7538 1.52565
\(750\) 0 0
\(751\) −13.8617 −0.505822 −0.252911 0.967490i \(-0.581388\pi\)
−0.252911 + 0.967490i \(0.581388\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 17.3693i 0.632973i
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −3.12311 −0.113586
\(757\) − 47.8617i − 1.73956i −0.493436 0.869782i \(-0.664259\pi\)
0.493436 0.869782i \(-0.335741\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) −3.12311 −0.113362
\(760\) 0 0
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 5.75379i 0.208438i
\(763\) − 8.98485i − 0.325273i
\(764\) −12.4924 −0.451960
\(765\) 0 0
\(766\) 30.2462 1.09284
\(767\) 4.49242i 0.162212i
\(768\) 1.00000i 0.0360844i
\(769\) −28.7386 −1.03634 −0.518171 0.855277i \(-0.673387\pi\)
−0.518171 + 0.855277i \(0.673387\pi\)
\(770\) 0 0
\(771\) −8.24621 −0.296980
\(772\) 8.24621i 0.296788i
\(773\) 19.7538i 0.710494i 0.934772 + 0.355247i \(0.115603\pi\)
−0.934772 + 0.355247i \(0.884397\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.2462 −0.583205
\(777\) − 3.50758i − 0.125834i
\(778\) 18.0000i 0.645331i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) − 1.12311i − 0.0401622i
\(783\) 2.00000i 0.0714742i
\(784\) −2.75379 −0.0983496
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) 48.0000i 1.71102i 0.517790 + 0.855508i \(0.326755\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(788\) 10.0000i 0.356235i
\(789\) −6.24621 −0.222371
\(790\) 0 0
\(791\) −28.4924 −1.01307
\(792\) − 3.12311i − 0.110975i
\(793\) − 18.2462i − 0.647942i
\(794\) 1.50758 0.0535019
\(795\) 0 0
\(796\) 19.1231 0.677801
\(797\) 10.4924i 0.371661i 0.982582 + 0.185830i \(0.0594975\pi\)
−0.982582 + 0.185830i \(0.940503\pi\)
\(798\) 12.4924i 0.442227i
\(799\) −8.98485 −0.317861
\(800\) 0 0
\(801\) −5.12311 −0.181016
\(802\) 7.36932i 0.260220i
\(803\) − 13.2614i − 0.467983i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) − 0.246211i − 0.00866705i
\(808\) − 0.246211i − 0.00866168i
\(809\) 12.2462 0.430554 0.215277 0.976553i \(-0.430935\pi\)
0.215277 + 0.976553i \(0.430935\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 6.24621i 0.219199i
\(813\) − 6.24621i − 0.219064i
\(814\) 3.50758 0.122941
\(815\) 0 0
\(816\) 1.12311 0.0393166
\(817\) 0 0
\(818\) − 0.246211i − 0.00860857i
\(819\) −6.24621 −0.218260
\(820\) 0 0
\(821\) −36.2462 −1.26500 −0.632501 0.774560i \(-0.717971\pi\)
−0.632501 + 0.774560i \(0.717971\pi\)
\(822\) 10.8769i 0.379375i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 9.36932 0.326396
\(825\) 0 0
\(826\) 7.01515 0.244088
\(827\) 8.87689i 0.308680i 0.988018 + 0.154340i \(0.0493251\pi\)
−0.988018 + 0.154340i \(0.950675\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) −15.7538 −0.547152 −0.273576 0.961850i \(-0.588206\pi\)
−0.273576 + 0.961850i \(0.588206\pi\)
\(830\) 0 0
\(831\) −3.75379 −0.130217
\(832\) 2.00000i 0.0693375i
\(833\) 3.09280i 0.107159i
\(834\) 16.4924 0.571086
\(835\) 0 0
\(836\) −12.4924 −0.432059
\(837\) 0 0
\(838\) − 39.6155i − 1.36850i
\(839\) 9.75379 0.336738 0.168369 0.985724i \(-0.446150\pi\)
0.168369 + 0.985724i \(0.446150\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 11.3693i − 0.391813i
\(843\) − 21.1231i − 0.727518i
\(844\) 16.4924 0.567693
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) − 3.89205i − 0.133732i
\(848\) − 12.2462i − 0.420537i
\(849\) 14.2462 0.488929
\(850\) 0 0
\(851\) 1.12311 0.0384996
\(852\) 10.2462i 0.351029i
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) −28.4924 −0.974991
\(855\) 0 0
\(856\) −13.3693 −0.456954
\(857\) − 24.7386i − 0.845056i −0.906350 0.422528i \(-0.861143\pi\)
0.906350 0.422528i \(-0.138857\pi\)
\(858\) − 6.24621i − 0.213242i
\(859\) −15.5076 −0.529112 −0.264556 0.964370i \(-0.585225\pi\)
−0.264556 + 0.964370i \(0.585225\pi\)
\(860\) 0 0
\(861\) −6.24621 −0.212870
\(862\) 9.75379i 0.332215i
\(863\) 16.9848i 0.578171i 0.957303 + 0.289085i \(0.0933512\pi\)
−0.957303 + 0.289085i \(0.906649\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −22.4924 −0.764324
\(867\) 15.7386i 0.534512i
\(868\) 0 0
\(869\) 9.75379 0.330875
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 2.87689i 0.0974239i
\(873\) 16.2462i 0.549851i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 4.24621 0.143466
\(877\) 52.7386i 1.78086i 0.455123 + 0.890429i \(0.349595\pi\)
−0.455123 + 0.890429i \(0.650405\pi\)
\(878\) 0 0
\(879\) 28.2462 0.952721
\(880\) 0 0
\(881\) −9.61553 −0.323955 −0.161978 0.986794i \(-0.551787\pi\)
−0.161978 + 0.986794i \(0.551787\pi\)
\(882\) 2.75379i 0.0927249i
\(883\) − 44.9848i − 1.51386i −0.653496 0.756930i \(-0.726698\pi\)
0.653496 0.756930i \(-0.273302\pi\)
\(884\) 2.24621 0.0755483
\(885\) 0 0
\(886\) 8.49242 0.285309
\(887\) 19.5076i 0.655000i 0.944851 + 0.327500i \(0.106206\pi\)
−0.944851 + 0.327500i \(0.893794\pi\)
\(888\) 1.12311i 0.0376890i
\(889\) 17.9697 0.602684
\(890\) 0 0
\(891\) −3.12311 −0.104628
\(892\) − 10.2462i − 0.343069i
\(893\) 32.0000i 1.07084i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 3.12311 0.104336
\(897\) − 2.00000i − 0.0667781i
\(898\) − 22.4924i − 0.750582i
\(899\) 0 0
\(900\) 0 0
\(901\) −13.7538 −0.458205
\(902\) − 6.24621i − 0.207976i
\(903\) 0 0
\(904\) 9.12311 0.303430
\(905\) 0 0
\(906\) −6.24621 −0.207516
\(907\) 54.2462i 1.80122i 0.434632 + 0.900608i \(0.356878\pi\)
−0.434632 + 0.900608i \(0.643122\pi\)
\(908\) 5.36932i 0.178187i
\(909\) −0.246211 −0.00816631
\(910\) 0 0
\(911\) −30.2462 −1.00210 −0.501051 0.865418i \(-0.667053\pi\)
−0.501051 + 0.865418i \(0.667053\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 41.7538i − 1.38185i
\(914\) −16.7386 −0.553665
\(915\) 0 0
\(916\) 23.3693 0.772144
\(917\) − 12.4924i − 0.412536i
\(918\) − 1.12311i − 0.0370680i
\(919\) −19.1231 −0.630813 −0.315407 0.948957i \(-0.602141\pi\)
−0.315407 + 0.948957i \(0.602141\pi\)
\(920\) 0 0
\(921\) 32.4924 1.07066
\(922\) − 16.7386i − 0.551258i
\(923\) 20.4924i 0.674516i
\(924\) −9.75379 −0.320876
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) − 9.36932i − 0.307729i
\(928\) − 2.00000i − 0.0656532i
\(929\) −34.9848 −1.14782 −0.573908 0.818920i \(-0.694573\pi\)
−0.573908 + 0.818920i \(0.694573\pi\)
\(930\) 0 0
\(931\) 11.0152 0.361007
\(932\) 28.7386i 0.941365i
\(933\) − 22.7386i − 0.744429i
\(934\) −21.3693 −0.699225
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 32.7386i 1.06952i 0.845003 + 0.534762i \(0.179599\pi\)
−0.845003 + 0.534762i \(0.820401\pi\)
\(938\) 24.9848i 0.815784i
\(939\) −22.4924 −0.734012
\(940\) 0 0
\(941\) 26.4924 0.863628 0.431814 0.901963i \(-0.357874\pi\)
0.431814 + 0.901963i \(0.357874\pi\)
\(942\) − 9.12311i − 0.297247i
\(943\) − 2.00000i − 0.0651290i
\(944\) −2.24621 −0.0731079
\(945\) 0 0
\(946\) 0 0
\(947\) 5.75379i 0.186973i 0.995621 + 0.0934865i \(0.0298012\pi\)
−0.995621 + 0.0934865i \(0.970199\pi\)
\(948\) 3.12311i 0.101434i
\(949\) 8.49242 0.275676
\(950\) 0 0
\(951\) 28.7386 0.931914
\(952\) − 3.50758i − 0.113681i
\(953\) − 23.8617i − 0.772958i −0.922298 0.386479i \(-0.873691\pi\)
0.922298 0.386479i \(-0.126309\pi\)
\(954\) −12.2462 −0.396486
\(955\) 0 0
\(956\) −18.2462 −0.590125
\(957\) 6.24621i 0.201911i
\(958\) − 12.4924i − 0.403612i
\(959\) 33.9697 1.09694
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 2.24621i 0.0724208i
\(963\) 13.3693i 0.430820i
\(964\) −30.4924 −0.982095
\(965\) 0 0
\(966\) −3.12311 −0.100484
\(967\) 60.9848i 1.96114i 0.196168 + 0.980570i \(0.437150\pi\)
−0.196168 + 0.980570i \(0.562850\pi\)
\(968\) 1.24621i 0.0400547i
\(969\) −4.49242 −0.144317
\(970\) 0 0
\(971\) 43.1231 1.38389 0.691943 0.721952i \(-0.256755\pi\)
0.691943 + 0.721952i \(0.256755\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 51.5076i − 1.65126i
\(974\) −7.50758 −0.240558
\(975\) 0 0
\(976\) 9.12311 0.292023
\(977\) − 35.8617i − 1.14732i −0.819094 0.573659i \(-0.805523\pi\)
0.819094 0.573659i \(-0.194477\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 2.87689 0.0918522
\(982\) − 12.0000i − 0.382935i
\(983\) 9.75379i 0.311098i 0.987828 + 0.155549i \(0.0497146\pi\)
−0.987828 + 0.155549i \(0.950285\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −2.24621 −0.0715339
\(987\) 24.9848i 0.795276i
\(988\) − 8.00000i − 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) 61.4773 1.95289 0.976445 0.215767i \(-0.0692252\pi\)
0.976445 + 0.215767i \(0.0692252\pi\)
\(992\) 0 0
\(993\) 0.492423i 0.0156266i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 13.3693 0.423623
\(997\) 3.75379i 0.118884i 0.998232 + 0.0594418i \(0.0189321\pi\)
−0.998232 + 0.0594418i \(0.981068\pi\)
\(998\) − 36.9848i − 1.17073i
\(999\) 1.12311 0.0355335
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 3450.2.d.v.2899.4 4
5.2 odd 4 3450.2.a.bi.1.1 2
5.3 odd 4 690.2.a.l.1.2 2
5.4 even 2 inner 3450.2.d.v.2899.1 4
15.8 even 4 2070.2.a.t.1.2 2
20.3 even 4 5520.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.l.1.2 2 5.3 odd 4
2070.2.a.t.1.2 2 15.8 even 4
3450.2.a.bi.1.1 2 5.2 odd 4
3450.2.d.v.2899.1 4 5.4 even 2 inner
3450.2.d.v.2899.4 4 1.1 even 1 trivial
5520.2.a.bs.1.1 2 20.3 even 4