Properties

Label 2175.2.c.h.349.2
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (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: \(17.3674624396\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.h.349.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155i q^{2} -1.00000i q^{3} -0.438447 q^{4} -1.56155 q^{6} +5.12311i q^{7} -2.43845i q^{8} -1.00000 q^{9} -1.43845 q^{11} +0.438447i q^{12} +2.00000i q^{13} +8.00000 q^{14} -4.68466 q^{16} -7.12311i q^{17} +1.56155i q^{18} -5.12311 q^{19} +5.12311 q^{21} +2.24621i q^{22} -6.56155i q^{23} -2.43845 q^{24} +3.12311 q^{26} +1.00000i q^{27} -2.24621i q^{28} -1.00000 q^{29} +4.00000 q^{31} +2.43845i q^{32} +1.43845i q^{33} -11.1231 q^{34} +0.438447 q^{36} -1.68466i q^{37} +8.00000i q^{38} +2.00000 q^{39} -1.68466 q^{41} -8.00000i q^{42} -7.68466i q^{43} +0.630683 q^{44} -10.2462 q^{46} -13.1231i q^{47} +4.68466i q^{48} -19.2462 q^{49} -7.12311 q^{51} -0.876894i q^{52} +3.43845i q^{53} +1.56155 q^{54} +12.4924 q^{56} +5.12311i q^{57} +1.56155i q^{58} -12.0000 q^{59} +0.876894 q^{61} -6.24621i q^{62} -5.12311i q^{63} -5.56155 q^{64} +2.24621 q^{66} -11.3693i q^{67} +3.12311i q^{68} -6.56155 q^{69} -2.87689 q^{71} +2.43845i q^{72} -1.68466i q^{73} -2.63068 q^{74} +2.24621 q^{76} -7.36932i q^{77} -3.12311i q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.63068i q^{82} +2.56155i q^{83} -2.24621 q^{84} -12.0000 q^{86} +1.00000i q^{87} +3.50758i q^{88} -12.2462 q^{89} -10.2462 q^{91} +2.87689i q^{92} -4.00000i q^{93} -20.4924 q^{94} +2.43845 q^{96} -5.68466i q^{97} +30.0540i q^{98} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} + 2 q^{6} - 4 q^{9} - 14 q^{11} + 32 q^{14} + 6 q^{16} - 4 q^{19} + 4 q^{21} - 18 q^{24} - 4 q^{26} - 4 q^{29} + 16 q^{31} - 28 q^{34} + 10 q^{36} + 8 q^{39} + 18 q^{41} + 52 q^{44} - 8 q^{46}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.56155i − 1.10418i −0.833783 0.552092i \(-0.813830\pi\)
0.833783 0.552092i \(-0.186170\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −0.438447 −0.219224
\(5\) 0 0
\(6\) −1.56155 −0.637501
\(7\) 5.12311i 1.93635i 0.250270 + 0.968176i \(0.419480\pi\)
−0.250270 + 0.968176i \(0.580520\pi\)
\(8\) − 2.43845i − 0.862121i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.43845 −0.433708 −0.216854 0.976204i \(-0.569580\pi\)
−0.216854 + 0.976204i \(0.569580\pi\)
\(12\) 0.438447i 0.126569i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 8.00000 2.13809
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) − 7.12311i − 1.72761i −0.503829 0.863803i \(-0.668076\pi\)
0.503829 0.863803i \(-0.331924\pi\)
\(18\) 1.56155i 0.368062i
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 0 0
\(21\) 5.12311 1.11795
\(22\) 2.24621i 0.478894i
\(23\) − 6.56155i − 1.36818i −0.729398 0.684089i \(-0.760200\pi\)
0.729398 0.684089i \(-0.239800\pi\)
\(24\) −2.43845 −0.497746
\(25\) 0 0
\(26\) 3.12311 0.612491
\(27\) 1.00000i 0.192450i
\(28\) − 2.24621i − 0.424494i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 2.43845i 0.431061i
\(33\) 1.43845i 0.250402i
\(34\) −11.1231 −1.90760
\(35\) 0 0
\(36\) 0.438447 0.0730745
\(37\) − 1.68466i − 0.276956i −0.990366 0.138478i \(-0.955779\pi\)
0.990366 0.138478i \(-0.0442210\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −1.68466 −0.263099 −0.131550 0.991310i \(-0.541995\pi\)
−0.131550 + 0.991310i \(0.541995\pi\)
\(42\) − 8.00000i − 1.23443i
\(43\) − 7.68466i − 1.17190i −0.810347 0.585950i \(-0.800722\pi\)
0.810347 0.585950i \(-0.199278\pi\)
\(44\) 0.630683 0.0950791
\(45\) 0 0
\(46\) −10.2462 −1.51072
\(47\) − 13.1231i − 1.91420i −0.289755 0.957101i \(-0.593574\pi\)
0.289755 0.957101i \(-0.406426\pi\)
\(48\) 4.68466i 0.676172i
\(49\) −19.2462 −2.74946
\(50\) 0 0
\(51\) −7.12311 −0.997434
\(52\) − 0.876894i − 0.121603i
\(53\) 3.43845i 0.472307i 0.971716 + 0.236154i \(0.0758868\pi\)
−0.971716 + 0.236154i \(0.924113\pi\)
\(54\) 1.56155 0.212500
\(55\) 0 0
\(56\) 12.4924 1.66937
\(57\) 5.12311i 0.678572i
\(58\) 1.56155i 0.205042i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) − 6.24621i − 0.793270i
\(63\) − 5.12311i − 0.645451i
\(64\) −5.56155 −0.695194
\(65\) 0 0
\(66\) 2.24621 0.276489
\(67\) − 11.3693i − 1.38898i −0.719501 0.694492i \(-0.755629\pi\)
0.719501 0.694492i \(-0.244371\pi\)
\(68\) 3.12311i 0.378732i
\(69\) −6.56155 −0.789918
\(70\) 0 0
\(71\) −2.87689 −0.341425 −0.170712 0.985321i \(-0.554607\pi\)
−0.170712 + 0.985321i \(0.554607\pi\)
\(72\) 2.43845i 0.287374i
\(73\) − 1.68466i − 0.197174i −0.995128 0.0985872i \(-0.968568\pi\)
0.995128 0.0985872i \(-0.0314323\pi\)
\(74\) −2.63068 −0.305811
\(75\) 0 0
\(76\) 2.24621 0.257658
\(77\) − 7.36932i − 0.839812i
\(78\) − 3.12311i − 0.353622i
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.63068i 0.290510i
\(83\) 2.56155i 0.281167i 0.990069 + 0.140583i \(0.0448978\pi\)
−0.990069 + 0.140583i \(0.955102\pi\)
\(84\) −2.24621 −0.245082
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 1.00000i 0.107211i
\(88\) 3.50758i 0.373909i
\(89\) −12.2462 −1.29810 −0.649048 0.760748i \(-0.724833\pi\)
−0.649048 + 0.760748i \(0.724833\pi\)
\(90\) 0 0
\(91\) −10.2462 −1.07409
\(92\) 2.87689i 0.299937i
\(93\) − 4.00000i − 0.414781i
\(94\) −20.4924 −2.11363
\(95\) 0 0
\(96\) 2.43845 0.248873
\(97\) − 5.68466i − 0.577190i −0.957451 0.288595i \(-0.906812\pi\)
0.957451 0.288595i \(-0.0931880\pi\)
\(98\) 30.0540i 3.03591i
\(99\) 1.43845 0.144569
\(100\) 0 0
\(101\) −8.56155 −0.851906 −0.425953 0.904745i \(-0.640061\pi\)
−0.425953 + 0.904745i \(0.640061\pi\)
\(102\) 11.1231i 1.10135i
\(103\) 2.87689i 0.283469i 0.989905 + 0.141734i \(0.0452679\pi\)
−0.989905 + 0.141734i \(0.954732\pi\)
\(104\) 4.87689 0.478219
\(105\) 0 0
\(106\) 5.36932 0.521514
\(107\) 16.4924i 1.59438i 0.603727 + 0.797191i \(0.293682\pi\)
−0.603727 + 0.797191i \(0.706318\pi\)
\(108\) − 0.438447i − 0.0421896i
\(109\) 5.68466 0.544492 0.272246 0.962228i \(-0.412234\pi\)
0.272246 + 0.962228i \(0.412234\pi\)
\(110\) 0 0
\(111\) −1.68466 −0.159901
\(112\) − 24.0000i − 2.26779i
\(113\) 4.87689i 0.458780i 0.973335 + 0.229390i \(0.0736731\pi\)
−0.973335 + 0.229390i \(0.926327\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 0.438447 0.0407088
\(117\) − 2.00000i − 0.184900i
\(118\) 18.7386i 1.72503i
\(119\) 36.4924 3.34525
\(120\) 0 0
\(121\) −8.93087 −0.811897
\(122\) − 1.36932i − 0.123972i
\(123\) 1.68466i 0.151901i
\(124\) −1.75379 −0.157495
\(125\) 0 0
\(126\) −8.00000 −0.712697
\(127\) 4.31534i 0.382925i 0.981500 + 0.191462i \(0.0613230\pi\)
−0.981500 + 0.191462i \(0.938677\pi\)
\(128\) 13.5616i 1.19868i
\(129\) −7.68466 −0.676596
\(130\) 0 0
\(131\) 10.2462 0.895216 0.447608 0.894230i \(-0.352276\pi\)
0.447608 + 0.894230i \(0.352276\pi\)
\(132\) − 0.630683i − 0.0548939i
\(133\) − 26.2462i − 2.27584i
\(134\) −17.7538 −1.53369
\(135\) 0 0
\(136\) −17.3693 −1.48941
\(137\) − 15.1231i − 1.29205i −0.763315 0.646027i \(-0.776429\pi\)
0.763315 0.646027i \(-0.223571\pi\)
\(138\) 10.2462i 0.872215i
\(139\) 17.9309 1.52088 0.760438 0.649410i \(-0.224984\pi\)
0.760438 + 0.649410i \(0.224984\pi\)
\(140\) 0 0
\(141\) −13.1231 −1.10516
\(142\) 4.49242i 0.376996i
\(143\) − 2.87689i − 0.240578i
\(144\) 4.68466 0.390388
\(145\) 0 0
\(146\) −2.63068 −0.217717
\(147\) 19.2462i 1.58740i
\(148\) 0.738634i 0.0607153i
\(149\) −0.246211 −0.0201704 −0.0100852 0.999949i \(-0.503210\pi\)
−0.0100852 + 0.999949i \(0.503210\pi\)
\(150\) 0 0
\(151\) 4.31534 0.351178 0.175589 0.984464i \(-0.443817\pi\)
0.175589 + 0.984464i \(0.443817\pi\)
\(152\) 12.4924i 1.01327i
\(153\) 7.12311i 0.575869i
\(154\) −11.5076 −0.927307
\(155\) 0 0
\(156\) −0.876894 −0.0702077
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) − 18.7386i − 1.49077i
\(159\) 3.43845 0.272687
\(160\) 0 0
\(161\) 33.6155 2.64927
\(162\) − 1.56155i − 0.122687i
\(163\) 17.9309i 1.40445i 0.711953 + 0.702227i \(0.247811\pi\)
−0.711953 + 0.702227i \(0.752189\pi\)
\(164\) 0.738634 0.0576776
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 12.4924i − 0.963811i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 5.12311 0.391774
\(172\) 3.36932i 0.256908i
\(173\) − 12.5616i − 0.955037i −0.878622 0.477519i \(-0.841536\pi\)
0.878622 0.477519i \(-0.158464\pi\)
\(174\) 1.56155 0.118381
\(175\) 0 0
\(176\) 6.73863 0.507944
\(177\) 12.0000i 0.901975i
\(178\) 19.1231i 1.43334i
\(179\) −1.12311 −0.0839449 −0.0419724 0.999119i \(-0.513364\pi\)
−0.0419724 + 0.999119i \(0.513364\pi\)
\(180\) 0 0
\(181\) 25.6847 1.90913 0.954563 0.298010i \(-0.0963228\pi\)
0.954563 + 0.298010i \(0.0963228\pi\)
\(182\) 16.0000i 1.18600i
\(183\) − 0.876894i − 0.0648219i
\(184\) −16.0000 −1.17954
\(185\) 0 0
\(186\) −6.24621 −0.457994
\(187\) 10.2462i 0.749277i
\(188\) 5.75379i 0.419638i
\(189\) −5.12311 −0.372651
\(190\) 0 0
\(191\) −9.93087 −0.718573 −0.359286 0.933227i \(-0.616980\pi\)
−0.359286 + 0.933227i \(0.616980\pi\)
\(192\) 5.56155i 0.401371i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) −8.87689 −0.637324
\(195\) 0 0
\(196\) 8.43845 0.602746
\(197\) 4.56155i 0.324997i 0.986709 + 0.162499i \(0.0519553\pi\)
−0.986709 + 0.162499i \(0.948045\pi\)
\(198\) − 2.24621i − 0.159631i
\(199\) 11.0540 0.783596 0.391798 0.920051i \(-0.371853\pi\)
0.391798 + 0.920051i \(0.371853\pi\)
\(200\) 0 0
\(201\) −11.3693 −0.801930
\(202\) 13.3693i 0.940662i
\(203\) − 5.12311i − 0.359572i
\(204\) 3.12311 0.218661
\(205\) 0 0
\(206\) 4.49242 0.313002
\(207\) 6.56155i 0.456059i
\(208\) − 9.36932i − 0.649645i
\(209\) 7.36932 0.509746
\(210\) 0 0
\(211\) −2.87689 −0.198054 −0.0990268 0.995085i \(-0.531573\pi\)
−0.0990268 + 0.995085i \(0.531573\pi\)
\(212\) − 1.50758i − 0.103541i
\(213\) 2.87689i 0.197122i
\(214\) 25.7538 1.76049
\(215\) 0 0
\(216\) 2.43845 0.165915
\(217\) 20.4924i 1.39112i
\(218\) − 8.87689i − 0.601219i
\(219\) −1.68466 −0.113839
\(220\) 0 0
\(221\) 14.2462 0.958304
\(222\) 2.63068i 0.176560i
\(223\) 18.2462i 1.22186i 0.791686 + 0.610928i \(0.209204\pi\)
−0.791686 + 0.610928i \(0.790796\pi\)
\(224\) −12.4924 −0.834685
\(225\) 0 0
\(226\) 7.61553 0.506577
\(227\) − 3.19224i − 0.211876i −0.994373 0.105938i \(-0.966215\pi\)
0.994373 0.105938i \(-0.0337845\pi\)
\(228\) − 2.24621i − 0.148759i
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −7.36932 −0.484865
\(232\) 2.43845i 0.160092i
\(233\) 10.3153i 0.675780i 0.941186 + 0.337890i \(0.109713\pi\)
−0.941186 + 0.337890i \(0.890287\pi\)
\(234\) −3.12311 −0.204164
\(235\) 0 0
\(236\) 5.26137 0.342486
\(237\) − 12.0000i − 0.779484i
\(238\) − 56.9848i − 3.69378i
\(239\) 13.1231 0.848863 0.424432 0.905460i \(-0.360474\pi\)
0.424432 + 0.905460i \(0.360474\pi\)
\(240\) 0 0
\(241\) −25.0540 −1.61387 −0.806934 0.590641i \(-0.798875\pi\)
−0.806934 + 0.590641i \(0.798875\pi\)
\(242\) 13.9460i 0.896484i
\(243\) − 1.00000i − 0.0641500i
\(244\) −0.384472 −0.0246133
\(245\) 0 0
\(246\) 2.63068 0.167726
\(247\) − 10.2462i − 0.651951i
\(248\) − 9.75379i − 0.619366i
\(249\) 2.56155 0.162332
\(250\) 0 0
\(251\) 2.24621 0.141780 0.0708898 0.997484i \(-0.477416\pi\)
0.0708898 + 0.997484i \(0.477416\pi\)
\(252\) 2.24621i 0.141498i
\(253\) 9.43845i 0.593390i
\(254\) 6.73863 0.422819
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) 11.4384i 0.713511i 0.934198 + 0.356755i \(0.116117\pi\)
−0.934198 + 0.356755i \(0.883883\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 8.63068 0.536285
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) − 16.0000i − 0.988483i
\(263\) 5.75379i 0.354794i 0.984139 + 0.177397i \(0.0567676\pi\)
−0.984139 + 0.177397i \(0.943232\pi\)
\(264\) 3.50758 0.215876
\(265\) 0 0
\(266\) −40.9848 −2.51294
\(267\) 12.2462i 0.749456i
\(268\) 4.98485i 0.304498i
\(269\) −11.7538 −0.716641 −0.358321 0.933599i \(-0.616651\pi\)
−0.358321 + 0.933599i \(0.616651\pi\)
\(270\) 0 0
\(271\) 17.1231 1.04015 0.520077 0.854119i \(-0.325903\pi\)
0.520077 + 0.854119i \(0.325903\pi\)
\(272\) 33.3693i 2.02331i
\(273\) 10.2462i 0.620129i
\(274\) −23.6155 −1.42667
\(275\) 0 0
\(276\) 2.87689 0.173169
\(277\) 0.876894i 0.0526875i 0.999653 + 0.0263437i \(0.00838644\pi\)
−0.999653 + 0.0263437i \(0.991614\pi\)
\(278\) − 28.0000i − 1.67933i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −23.6155 −1.40878 −0.704392 0.709811i \(-0.748780\pi\)
−0.704392 + 0.709811i \(0.748780\pi\)
\(282\) 20.4924i 1.22031i
\(283\) − 6.87689i − 0.408789i −0.978889 0.204394i \(-0.934477\pi\)
0.978889 0.204394i \(-0.0655225\pi\)
\(284\) 1.26137 0.0748483
\(285\) 0 0
\(286\) −4.49242 −0.265643
\(287\) − 8.63068i − 0.509453i
\(288\) − 2.43845i − 0.143687i
\(289\) −33.7386 −1.98463
\(290\) 0 0
\(291\) −5.68466 −0.333241
\(292\) 0.738634i 0.0432253i
\(293\) 21.3693i 1.24841i 0.781261 + 0.624204i \(0.214577\pi\)
−0.781261 + 0.624204i \(0.785423\pi\)
\(294\) 30.0540 1.75278
\(295\) 0 0
\(296\) −4.10795 −0.238770
\(297\) − 1.43845i − 0.0834672i
\(298\) 0.384472i 0.0222719i
\(299\) 13.1231 0.758929
\(300\) 0 0
\(301\) 39.3693 2.26921
\(302\) − 6.73863i − 0.387765i
\(303\) 8.56155i 0.491848i
\(304\) 24.0000 1.37649
\(305\) 0 0
\(306\) 11.1231 0.635866
\(307\) − 31.6847i − 1.80834i −0.427174 0.904169i \(-0.640491\pi\)
0.427174 0.904169i \(-0.359509\pi\)
\(308\) 3.23106i 0.184107i
\(309\) 2.87689 0.163661
\(310\) 0 0
\(311\) 16.3153 0.925158 0.462579 0.886578i \(-0.346924\pi\)
0.462579 + 0.886578i \(0.346924\pi\)
\(312\) − 4.87689i − 0.276100i
\(313\) 21.3693i 1.20787i 0.797035 + 0.603933i \(0.206400\pi\)
−0.797035 + 0.603933i \(0.793600\pi\)
\(314\) −21.8617 −1.23373
\(315\) 0 0
\(316\) −5.26137 −0.295975
\(317\) 4.87689i 0.273914i 0.990577 + 0.136957i \(0.0437322\pi\)
−0.990577 + 0.136957i \(0.956268\pi\)
\(318\) − 5.36932i − 0.301096i
\(319\) 1.43845 0.0805376
\(320\) 0 0
\(321\) 16.4924 0.920517
\(322\) − 52.4924i − 2.92529i
\(323\) 36.4924i 2.03049i
\(324\) −0.438447 −0.0243582
\(325\) 0 0
\(326\) 28.0000 1.55078
\(327\) − 5.68466i − 0.314362i
\(328\) 4.10795i 0.226824i
\(329\) 67.2311 3.70657
\(330\) 0 0
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) − 1.12311i − 0.0616384i
\(333\) 1.68466i 0.0923187i
\(334\) 0 0
\(335\) 0 0
\(336\) −24.0000 −1.30931
\(337\) − 7.75379i − 0.422376i −0.977445 0.211188i \(-0.932267\pi\)
0.977445 0.211188i \(-0.0677332\pi\)
\(338\) − 14.0540i − 0.764435i
\(339\) 4.87689 0.264877
\(340\) 0 0
\(341\) −5.75379 −0.311585
\(342\) − 8.00000i − 0.432590i
\(343\) − 62.7386i − 3.38757i
\(344\) −18.7386 −1.01032
\(345\) 0 0
\(346\) −19.6155 −1.05454
\(347\) − 12.1771i − 0.653700i −0.945076 0.326850i \(-0.894013\pi\)
0.945076 0.326850i \(-0.105987\pi\)
\(348\) − 0.438447i − 0.0235032i
\(349\) −0.0691303 −0.00370046 −0.00185023 0.999998i \(-0.500589\pi\)
−0.00185023 + 0.999998i \(0.500589\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 3.50758i − 0.186955i
\(353\) − 30.4924i − 1.62295i −0.584389 0.811474i \(-0.698666\pi\)
0.584389 0.811474i \(-0.301334\pi\)
\(354\) 18.7386 0.995947
\(355\) 0 0
\(356\) 5.36932 0.284573
\(357\) − 36.4924i − 1.93138i
\(358\) 1.75379i 0.0926906i
\(359\) 3.19224 0.168480 0.0842399 0.996446i \(-0.473154\pi\)
0.0842399 + 0.996446i \(0.473154\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) − 40.1080i − 2.10803i
\(363\) 8.93087i 0.468749i
\(364\) 4.49242 0.235467
\(365\) 0 0
\(366\) −1.36932 −0.0715753
\(367\) − 19.6847i − 1.02753i −0.857931 0.513765i \(-0.828250\pi\)
0.857931 0.513765i \(-0.171750\pi\)
\(368\) 30.7386i 1.60236i
\(369\) 1.68466 0.0876998
\(370\) 0 0
\(371\) −17.6155 −0.914553
\(372\) 1.75379i 0.0909297i
\(373\) − 13.3693i − 0.692237i −0.938191 0.346118i \(-0.887500\pi\)
0.938191 0.346118i \(-0.112500\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −32.0000 −1.65027
\(377\) − 2.00000i − 0.103005i
\(378\) 8.00000i 0.411476i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 4.31534 0.221082
\(382\) 15.5076i 0.793437i
\(383\) − 11.0540i − 0.564832i −0.959292 0.282416i \(-0.908864\pi\)
0.959292 0.282416i \(-0.0911358\pi\)
\(384\) 13.5616 0.692060
\(385\) 0 0
\(386\) 15.6155 0.794809
\(387\) 7.68466i 0.390633i
\(388\) 2.49242i 0.126534i
\(389\) −14.8078 −0.750783 −0.375392 0.926866i \(-0.622492\pi\)
−0.375392 + 0.926866i \(0.622492\pi\)
\(390\) 0 0
\(391\) −46.7386 −2.36367
\(392\) 46.9309i 2.37037i
\(393\) − 10.2462i − 0.516853i
\(394\) 7.12311 0.358857
\(395\) 0 0
\(396\) −0.630683 −0.0316930
\(397\) 24.2462i 1.21688i 0.793599 + 0.608441i \(0.208205\pi\)
−0.793599 + 0.608441i \(0.791795\pi\)
\(398\) − 17.2614i − 0.865234i
\(399\) −26.2462 −1.31395
\(400\) 0 0
\(401\) 35.6155 1.77855 0.889277 0.457368i \(-0.151208\pi\)
0.889277 + 0.457368i \(0.151208\pi\)
\(402\) 17.7538i 0.885479i
\(403\) 8.00000i 0.398508i
\(404\) 3.75379 0.186758
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 2.42329i 0.120118i
\(408\) 17.3693i 0.859909i
\(409\) −17.3693 −0.858857 −0.429429 0.903101i \(-0.641285\pi\)
−0.429429 + 0.903101i \(0.641285\pi\)
\(410\) 0 0
\(411\) −15.1231 −0.745968
\(412\) − 1.26137i − 0.0621431i
\(413\) − 61.4773i − 3.02510i
\(414\) 10.2462 0.503574
\(415\) 0 0
\(416\) −4.87689 −0.239109
\(417\) − 17.9309i − 0.878078i
\(418\) − 11.5076i − 0.562854i
\(419\) 27.3693 1.33708 0.668539 0.743677i \(-0.266920\pi\)
0.668539 + 0.743677i \(0.266920\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.49242i 0.218688i
\(423\) 13.1231i 0.638067i
\(424\) 8.38447 0.407186
\(425\) 0 0
\(426\) 4.49242 0.217659
\(427\) 4.49242i 0.217404i
\(428\) − 7.23106i − 0.349526i
\(429\) −2.87689 −0.138898
\(430\) 0 0
\(431\) −21.1231 −1.01746 −0.508732 0.860925i \(-0.669886\pi\)
−0.508732 + 0.860925i \(0.669886\pi\)
\(432\) − 4.68466i − 0.225391i
\(433\) 4.06913i 0.195550i 0.995209 + 0.0977750i \(0.0311725\pi\)
−0.995209 + 0.0977750i \(0.968827\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −2.49242 −0.119365
\(437\) 33.6155i 1.60805i
\(438\) 2.63068i 0.125699i
\(439\) −12.4924 −0.596231 −0.298115 0.954530i \(-0.596358\pi\)
−0.298115 + 0.954530i \(0.596358\pi\)
\(440\) 0 0
\(441\) 19.2462 0.916486
\(442\) − 22.2462i − 1.05814i
\(443\) − 6.24621i − 0.296766i −0.988930 0.148383i \(-0.952593\pi\)
0.988930 0.148383i \(-0.0474069\pi\)
\(444\) 0.738634 0.0350540
\(445\) 0 0
\(446\) 28.4924 1.34916
\(447\) 0.246211i 0.0116454i
\(448\) − 28.4924i − 1.34614i
\(449\) −11.4384 −0.539814 −0.269907 0.962886i \(-0.586993\pi\)
−0.269907 + 0.962886i \(0.586993\pi\)
\(450\) 0 0
\(451\) 2.42329 0.114108
\(452\) − 2.13826i − 0.100575i
\(453\) − 4.31534i − 0.202752i
\(454\) −4.98485 −0.233950
\(455\) 0 0
\(456\) 12.4924 0.585011
\(457\) − 8.87689i − 0.415244i −0.978209 0.207622i \(-0.933428\pi\)
0.978209 0.207622i \(-0.0665723\pi\)
\(458\) 34.3542i 1.60526i
\(459\) 7.12311 0.332478
\(460\) 0 0
\(461\) −18.1771 −0.846591 −0.423296 0.905992i \(-0.639127\pi\)
−0.423296 + 0.905992i \(0.639127\pi\)
\(462\) 11.5076i 0.535381i
\(463\) − 25.6155i − 1.19045i −0.803557 0.595227i \(-0.797062\pi\)
0.803557 0.595227i \(-0.202938\pi\)
\(464\) 4.68466 0.217480
\(465\) 0 0
\(466\) 16.1080 0.746186
\(467\) − 3.36932i − 0.155913i −0.996957 0.0779567i \(-0.975160\pi\)
0.996957 0.0779567i \(-0.0248396\pi\)
\(468\) 0.876894i 0.0405345i
\(469\) 58.2462 2.68956
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 29.2614i 1.34686i
\(473\) 11.0540i 0.508262i
\(474\) −18.7386 −0.860694
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) − 3.43845i − 0.157436i
\(478\) − 20.4924i − 0.937302i
\(479\) 22.2462 1.01646 0.508228 0.861223i \(-0.330301\pi\)
0.508228 + 0.861223i \(0.330301\pi\)
\(480\) 0 0
\(481\) 3.36932 0.153628
\(482\) 39.1231i 1.78201i
\(483\) − 33.6155i − 1.52956i
\(484\) 3.91571 0.177987
\(485\) 0 0
\(486\) −1.56155 −0.0708335
\(487\) 34.2462i 1.55184i 0.630829 + 0.775922i \(0.282715\pi\)
−0.630829 + 0.775922i \(0.717285\pi\)
\(488\) − 2.13826i − 0.0967945i
\(489\) 17.9309 0.810862
\(490\) 0 0
\(491\) 6.73863 0.304110 0.152055 0.988372i \(-0.451411\pi\)
0.152055 + 0.988372i \(0.451411\pi\)
\(492\) − 0.738634i − 0.0333002i
\(493\) 7.12311i 0.320809i
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −18.7386 −0.841389
\(497\) − 14.7386i − 0.661118i
\(498\) − 4.00000i − 0.179244i
\(499\) −42.7386 −1.91324 −0.956622 0.291332i \(-0.905902\pi\)
−0.956622 + 0.291332i \(0.905902\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.50758i − 0.156551i
\(503\) 39.3693i 1.75539i 0.479219 + 0.877696i \(0.340920\pi\)
−0.479219 + 0.877696i \(0.659080\pi\)
\(504\) −12.4924 −0.556457
\(505\) 0 0
\(506\) 14.7386 0.655212
\(507\) − 9.00000i − 0.399704i
\(508\) − 1.89205i − 0.0839461i
\(509\) −18.4924 −0.819662 −0.409831 0.912161i \(-0.634412\pi\)
−0.409831 + 0.912161i \(0.634412\pi\)
\(510\) 0 0
\(511\) 8.63068 0.381799
\(512\) 11.4233i 0.504843i
\(513\) − 5.12311i − 0.226191i
\(514\) 17.8617 0.787848
\(515\) 0 0
\(516\) 3.36932 0.148326
\(517\) 18.8769i 0.830205i
\(518\) − 13.4773i − 0.592157i
\(519\) −12.5616 −0.551391
\(520\) 0 0
\(521\) −0.246211 −0.0107867 −0.00539336 0.999985i \(-0.501717\pi\)
−0.00539336 + 0.999985i \(0.501717\pi\)
\(522\) − 1.56155i − 0.0683473i
\(523\) − 18.7386i − 0.819383i −0.912224 0.409692i \(-0.865636\pi\)
0.912224 0.409692i \(-0.134364\pi\)
\(524\) −4.49242 −0.196252
\(525\) 0 0
\(526\) 8.98485 0.391758
\(527\) − 28.4924i − 1.24115i
\(528\) − 6.73863i − 0.293261i
\(529\) −20.0540 −0.871912
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 11.5076i 0.498917i
\(533\) − 3.36932i − 0.145941i
\(534\) 19.1231 0.827538
\(535\) 0 0
\(536\) −27.7235 −1.19747
\(537\) 1.12311i 0.0484656i
\(538\) 18.3542i 0.791304i
\(539\) 27.6847 1.19246
\(540\) 0 0
\(541\) −40.7386 −1.75149 −0.875745 0.482773i \(-0.839629\pi\)
−0.875745 + 0.482773i \(0.839629\pi\)
\(542\) − 26.7386i − 1.14852i
\(543\) − 25.6847i − 1.10223i
\(544\) 17.3693 0.744703
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) − 42.7386i − 1.82737i −0.406421 0.913686i \(-0.633223\pi\)
0.406421 0.913686i \(-0.366777\pi\)
\(548\) 6.63068i 0.283249i
\(549\) −0.876894 −0.0374249
\(550\) 0 0
\(551\) 5.12311 0.218252
\(552\) 16.0000i 0.681005i
\(553\) 61.4773i 2.61428i
\(554\) 1.36932 0.0581767
\(555\) 0 0
\(556\) −7.86174 −0.333412
\(557\) − 20.4233i − 0.865363i −0.901547 0.432681i \(-0.857568\pi\)
0.901547 0.432681i \(-0.142432\pi\)
\(558\) 6.24621i 0.264423i
\(559\) 15.3693 0.650053
\(560\) 0 0
\(561\) 10.2462 0.432595
\(562\) 36.8769i 1.55556i
\(563\) − 9.12311i − 0.384493i −0.981347 0.192247i \(-0.938423\pi\)
0.981347 0.192247i \(-0.0615773\pi\)
\(564\) 5.75379 0.242278
\(565\) 0 0
\(566\) −10.7386 −0.451378
\(567\) 5.12311i 0.215150i
\(568\) 7.01515i 0.294349i
\(569\) 16.2462 0.681077 0.340538 0.940231i \(-0.389391\pi\)
0.340538 + 0.940231i \(0.389391\pi\)
\(570\) 0 0
\(571\) 39.0540 1.63436 0.817179 0.576384i \(-0.195537\pi\)
0.817179 + 0.576384i \(0.195537\pi\)
\(572\) 1.26137i 0.0527404i
\(573\) 9.93087i 0.414868i
\(574\) −13.4773 −0.562530
\(575\) 0 0
\(576\) 5.56155 0.231731
\(577\) − 30.4924i − 1.26942i −0.772752 0.634708i \(-0.781120\pi\)
0.772752 0.634708i \(-0.218880\pi\)
\(578\) 52.6847i 2.19139i
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −13.1231 −0.544438
\(582\) 8.87689i 0.367959i
\(583\) − 4.94602i − 0.204843i
\(584\) −4.10795 −0.169988
\(585\) 0 0
\(586\) 33.3693 1.37847
\(587\) 24.4924i 1.01091i 0.862853 + 0.505455i \(0.168675\pi\)
−0.862853 + 0.505455i \(0.831325\pi\)
\(588\) − 8.43845i − 0.347996i
\(589\) −20.4924 −0.844376
\(590\) 0 0
\(591\) 4.56155 0.187637
\(592\) 7.89205i 0.324361i
\(593\) − 36.2462i − 1.48845i −0.667927 0.744227i \(-0.732818\pi\)
0.667927 0.744227i \(-0.267182\pi\)
\(594\) −2.24621 −0.0921632
\(595\) 0 0
\(596\) 0.107951 0.00442183
\(597\) − 11.0540i − 0.452409i
\(598\) − 20.4924i − 0.837997i
\(599\) 20.9848 0.857418 0.428709 0.903443i \(-0.358969\pi\)
0.428709 + 0.903443i \(0.358969\pi\)
\(600\) 0 0
\(601\) 41.3693 1.68749 0.843745 0.536745i \(-0.180346\pi\)
0.843745 + 0.536745i \(0.180346\pi\)
\(602\) − 61.4773i − 2.50563i
\(603\) 11.3693i 0.462994i
\(604\) −1.89205 −0.0769864
\(605\) 0 0
\(606\) 13.3693 0.543091
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) − 12.4924i − 0.506635i
\(609\) −5.12311 −0.207599
\(610\) 0 0
\(611\) 26.2462 1.06181
\(612\) − 3.12311i − 0.126244i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −49.4773 −1.99674
\(615\) 0 0
\(616\) −17.9697 −0.724019
\(617\) 8.24621i 0.331980i 0.986127 + 0.165990i \(0.0530819\pi\)
−0.986127 + 0.165990i \(0.946918\pi\)
\(618\) − 4.49242i − 0.180712i
\(619\) −29.1231 −1.17056 −0.585278 0.810833i \(-0.699015\pi\)
−0.585278 + 0.810833i \(0.699015\pi\)
\(620\) 0 0
\(621\) 6.56155 0.263306
\(622\) − 25.4773i − 1.02155i
\(623\) − 62.7386i − 2.51357i
\(624\) −9.36932 −0.375073
\(625\) 0 0
\(626\) 33.3693 1.33371
\(627\) − 7.36932i − 0.294302i
\(628\) 6.13826i 0.244943i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) − 29.2614i − 1.16395i
\(633\) 2.87689i 0.114346i
\(634\) 7.61553 0.302451
\(635\) 0 0
\(636\) −1.50758 −0.0597793
\(637\) − 38.4924i − 1.52513i
\(638\) − 2.24621i − 0.0889284i
\(639\) 2.87689 0.113808
\(640\) 0 0
\(641\) 19.4384 0.767773 0.383886 0.923380i \(-0.374585\pi\)
0.383886 + 0.923380i \(0.374585\pi\)
\(642\) − 25.7538i − 1.01642i
\(643\) 18.7386i 0.738980i 0.929235 + 0.369490i \(0.120468\pi\)
−0.929235 + 0.369490i \(0.879532\pi\)
\(644\) −14.7386 −0.580784
\(645\) 0 0
\(646\) 56.9848 2.24204
\(647\) 5.93087i 0.233167i 0.993181 + 0.116583i \(0.0371942\pi\)
−0.993181 + 0.116583i \(0.962806\pi\)
\(648\) − 2.43845i − 0.0957913i
\(649\) 17.2614 0.677568
\(650\) 0 0
\(651\) 20.4924 0.803161
\(652\) − 7.86174i − 0.307889i
\(653\) − 12.2462i − 0.479231i −0.970868 0.239616i \(-0.922979\pi\)
0.970868 0.239616i \(-0.0770214\pi\)
\(654\) −8.87689 −0.347114
\(655\) 0 0
\(656\) 7.89205 0.308133
\(657\) 1.68466i 0.0657248i
\(658\) − 104.985i − 4.09274i
\(659\) −6.56155 −0.255602 −0.127801 0.991800i \(-0.540792\pi\)
−0.127801 + 0.991800i \(0.540792\pi\)
\(660\) 0 0
\(661\) 1.05398 0.0409949 0.0204974 0.999790i \(-0.493475\pi\)
0.0204974 + 0.999790i \(0.493475\pi\)
\(662\) − 16.0000i − 0.621858i
\(663\) − 14.2462i − 0.553277i
\(664\) 6.24621 0.242400
\(665\) 0 0
\(666\) 2.63068 0.101937
\(667\) 6.56155i 0.254064i
\(668\) 0 0
\(669\) 18.2462 0.705439
\(670\) 0 0
\(671\) −1.26137 −0.0486945
\(672\) 12.4924i 0.481906i
\(673\) 6.00000i 0.231283i 0.993291 + 0.115642i \(0.0368924\pi\)
−0.993291 + 0.115642i \(0.963108\pi\)
\(674\) −12.1080 −0.466381
\(675\) 0 0
\(676\) −3.94602 −0.151770
\(677\) 13.8617i 0.532750i 0.963869 + 0.266375i \(0.0858259\pi\)
−0.963869 + 0.266375i \(0.914174\pi\)
\(678\) − 7.61553i − 0.292473i
\(679\) 29.1231 1.11764
\(680\) 0 0
\(681\) −3.19224 −0.122327
\(682\) 8.98485i 0.344047i
\(683\) 21.4384i 0.820319i 0.912014 + 0.410160i \(0.134527\pi\)
−0.912014 + 0.410160i \(0.865473\pi\)
\(684\) −2.24621 −0.0858860
\(685\) 0 0
\(686\) −97.9697 −3.74050
\(687\) 22.0000i 0.839352i
\(688\) 36.0000i 1.37249i
\(689\) −6.87689 −0.261989
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 5.50758i 0.209367i
\(693\) 7.36932i 0.279937i
\(694\) −19.0152 −0.721805
\(695\) 0 0
\(696\) 2.43845 0.0924291
\(697\) 12.0000i 0.454532i
\(698\) 0.107951i 0.00408599i
\(699\) 10.3153 0.390162
\(700\) 0 0
\(701\) 3.75379 0.141779 0.0708893 0.997484i \(-0.477416\pi\)
0.0708893 + 0.997484i \(0.477416\pi\)
\(702\) 3.12311i 0.117874i
\(703\) 8.63068i 0.325512i
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −47.6155 −1.79203
\(707\) − 43.8617i − 1.64959i
\(708\) − 5.26137i − 0.197734i
\(709\) 12.4233 0.466567 0.233283 0.972409i \(-0.425053\pi\)
0.233283 + 0.972409i \(0.425053\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 29.8617i 1.11912i
\(713\) − 26.2462i − 0.982928i
\(714\) −56.9848 −2.13260
\(715\) 0 0
\(716\) 0.492423 0.0184027
\(717\) − 13.1231i − 0.490091i
\(718\) − 4.98485i − 0.186033i
\(719\) 4.49242 0.167539 0.0837695 0.996485i \(-0.473304\pi\)
0.0837695 + 0.996485i \(0.473304\pi\)
\(720\) 0 0
\(721\) −14.7386 −0.548895
\(722\) − 11.3153i − 0.421113i
\(723\) 25.0540i 0.931767i
\(724\) −11.2614 −0.418525
\(725\) 0 0
\(726\) 13.9460 0.517586
\(727\) − 38.7386i − 1.43674i −0.695663 0.718368i \(-0.744889\pi\)
0.695663 0.718368i \(-0.255111\pi\)
\(728\) 24.9848i 0.926000i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −54.7386 −2.02458
\(732\) 0.384472i 0.0142105i
\(733\) 22.9848i 0.848965i 0.905436 + 0.424482i \(0.139544\pi\)
−0.905436 + 0.424482i \(0.860456\pi\)
\(734\) −30.7386 −1.13458
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) 16.3542i 0.602413i
\(738\) − 2.63068i − 0.0968368i
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 0 0
\(741\) −10.2462 −0.376404
\(742\) 27.5076i 1.00983i
\(743\) − 34.8769i − 1.27951i −0.768579 0.639755i \(-0.779036\pi\)
0.768579 0.639755i \(-0.220964\pi\)
\(744\) −9.75379 −0.357591
\(745\) 0 0
\(746\) −20.8769 −0.764357
\(747\) − 2.56155i − 0.0937223i
\(748\) − 4.49242i − 0.164259i
\(749\) −84.4924 −3.08729
\(750\) 0 0
\(751\) 48.4924 1.76951 0.884757 0.466053i \(-0.154324\pi\)
0.884757 + 0.466053i \(0.154324\pi\)
\(752\) 61.4773i 2.24185i
\(753\) − 2.24621i − 0.0818565i
\(754\) −3.12311 −0.113737
\(755\) 0 0
\(756\) 2.24621 0.0816939
\(757\) 5.68466i 0.206612i 0.994650 + 0.103306i \(0.0329422\pi\)
−0.994650 + 0.103306i \(0.967058\pi\)
\(758\) 12.4924i 0.453745i
\(759\) 9.43845 0.342594
\(760\) 0 0
\(761\) −24.8769 −0.901787 −0.450893 0.892578i \(-0.648895\pi\)
−0.450893 + 0.892578i \(0.648895\pi\)
\(762\) − 6.73863i − 0.244115i
\(763\) 29.1231i 1.05433i
\(764\) 4.35416 0.157528
\(765\) 0 0
\(766\) −17.2614 −0.623679
\(767\) − 24.0000i − 0.866590i
\(768\) − 10.0540i − 0.362792i
\(769\) −43.6155 −1.57282 −0.786408 0.617707i \(-0.788062\pi\)
−0.786408 + 0.617707i \(0.788062\pi\)
\(770\) 0 0
\(771\) 11.4384 0.411946
\(772\) − 4.38447i − 0.157801i
\(773\) − 24.7386i − 0.889787i −0.895584 0.444893i \(-0.853242\pi\)
0.895584 0.444893i \(-0.146758\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −13.8617 −0.497607
\(777\) − 8.63068i − 0.309624i
\(778\) 23.1231i 0.829004i
\(779\) 8.63068 0.309226
\(780\) 0 0
\(781\) 4.13826 0.148079
\(782\) 72.9848i 2.60993i
\(783\) − 1.00000i − 0.0357371i
\(784\) 90.1619 3.22007
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) − 31.2311i − 1.11327i −0.830758 0.556633i \(-0.812093\pi\)
0.830758 0.556633i \(-0.187907\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) 5.75379 0.204840
\(790\) 0 0
\(791\) −24.9848 −0.888359
\(792\) − 3.50758i − 0.124636i
\(793\) 1.75379i 0.0622789i
\(794\) 37.8617 1.34366
\(795\) 0 0
\(796\) −4.84658 −0.171783
\(797\) 38.4924i 1.36347i 0.731598 + 0.681736i \(0.238775\pi\)
−0.731598 + 0.681736i \(0.761225\pi\)
\(798\) 40.9848i 1.45085i
\(799\) −93.4773 −3.30699
\(800\) 0 0
\(801\) 12.2462 0.432699
\(802\) − 55.6155i − 1.96385i
\(803\) 2.42329i 0.0855161i
\(804\) 4.98485 0.175802
\(805\) 0 0
\(806\) 12.4924 0.440027
\(807\) 11.7538i 0.413753i
\(808\) 20.8769i 0.734447i
\(809\) 22.1771 0.779705 0.389852 0.920877i \(-0.372526\pi\)
0.389852 + 0.920877i \(0.372526\pi\)
\(810\) 0 0
\(811\) 44.1771 1.55127 0.775634 0.631183i \(-0.217431\pi\)
0.775634 + 0.631183i \(0.217431\pi\)
\(812\) 2.24621i 0.0788266i
\(813\) − 17.1231i − 0.600534i
\(814\) 3.78410 0.132633
\(815\) 0 0
\(816\) 33.3693 1.16816
\(817\) 39.3693i 1.37736i
\(818\) 27.1231i 0.948337i
\(819\) 10.2462 0.358032
\(820\) 0 0
\(821\) 55.6155 1.94100 0.970498 0.241111i \(-0.0775116\pi\)
0.970498 + 0.241111i \(0.0775116\pi\)
\(822\) 23.6155i 0.823686i
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 7.01515 0.244385
\(825\) 0 0
\(826\) −96.0000 −3.34027
\(827\) 53.6155i 1.86439i 0.361951 + 0.932197i \(0.382111\pi\)
−0.361951 + 0.932197i \(0.617889\pi\)
\(828\) − 2.87689i − 0.0999790i
\(829\) 4.24621 0.147477 0.0737385 0.997278i \(-0.476507\pi\)
0.0737385 + 0.997278i \(0.476507\pi\)
\(830\) 0 0
\(831\) 0.876894 0.0304191
\(832\) − 11.1231i − 0.385624i
\(833\) 137.093i 4.74998i
\(834\) −28.0000 −0.969561
\(835\) 0 0
\(836\) −3.23106 −0.111748
\(837\) 4.00000i 0.138260i
\(838\) − 42.7386i − 1.47638i
\(839\) −42.7386 −1.47550 −0.737751 0.675073i \(-0.764112\pi\)
−0.737751 + 0.675073i \(0.764112\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 15.6155i 0.538147i
\(843\) 23.6155i 0.813362i
\(844\) 1.26137 0.0434180
\(845\) 0 0
\(846\) 20.4924 0.704544
\(847\) − 45.7538i − 1.57212i
\(848\) − 16.1080i − 0.553149i
\(849\) −6.87689 −0.236014
\(850\) 0 0
\(851\) −11.0540 −0.378925
\(852\) − 1.26137i − 0.0432137i
\(853\) − 19.4384i − 0.665560i −0.943005 0.332780i \(-0.892013\pi\)
0.943005 0.332780i \(-0.107987\pi\)
\(854\) 7.01515 0.240054
\(855\) 0 0
\(856\) 40.2159 1.37455
\(857\) − 15.4384i − 0.527367i −0.964609 0.263684i \(-0.915062\pi\)
0.964609 0.263684i \(-0.0849375\pi\)
\(858\) 4.49242i 0.153369i
\(859\) 23.3693 0.797351 0.398675 0.917092i \(-0.369470\pi\)
0.398675 + 0.917092i \(0.369470\pi\)
\(860\) 0 0
\(861\) −8.63068 −0.294133
\(862\) 32.9848i 1.12347i
\(863\) − 40.9848i − 1.39514i −0.716516 0.697570i \(-0.754265\pi\)
0.716516 0.697570i \(-0.245735\pi\)
\(864\) −2.43845 −0.0829577
\(865\) 0 0
\(866\) 6.35416 0.215923
\(867\) 33.7386i 1.14582i
\(868\) − 8.98485i − 0.304966i
\(869\) −17.2614 −0.585552
\(870\) 0 0
\(871\) 22.7386 0.770469
\(872\) − 13.8617i − 0.469418i
\(873\) 5.68466i 0.192397i
\(874\) 52.4924 1.77558
\(875\) 0 0
\(876\) 0.738634 0.0249561
\(877\) 8.87689i 0.299751i 0.988705 + 0.149876i \(0.0478874\pi\)
−0.988705 + 0.149876i \(0.952113\pi\)
\(878\) 19.5076i 0.658349i
\(879\) 21.3693 0.720769
\(880\) 0 0
\(881\) −8.06913 −0.271856 −0.135928 0.990719i \(-0.543402\pi\)
−0.135928 + 0.990719i \(0.543402\pi\)
\(882\) − 30.0540i − 1.01197i
\(883\) 42.7386i 1.43827i 0.694871 + 0.719135i \(0.255462\pi\)
−0.694871 + 0.719135i \(0.744538\pi\)
\(884\) −6.24621 −0.210083
\(885\) 0 0
\(886\) −9.75379 −0.327685
\(887\) − 43.8617i − 1.47273i −0.676583 0.736367i \(-0.736540\pi\)
0.676583 0.736367i \(-0.263460\pi\)
\(888\) 4.10795i 0.137854i
\(889\) −22.1080 −0.741477
\(890\) 0 0
\(891\) −1.43845 −0.0481898
\(892\) − 8.00000i − 0.267860i
\(893\) 67.2311i 2.24980i
\(894\) 0.384472 0.0128587
\(895\) 0 0
\(896\) −69.4773 −2.32107
\(897\) − 13.1231i − 0.438168i
\(898\) 17.8617i 0.596054i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 24.4924 0.815961
\(902\) − 3.78410i − 0.125997i
\(903\) − 39.3693i − 1.31013i
\(904\) 11.8920 0.395524
\(905\) 0 0
\(906\) −6.73863 −0.223876
\(907\) 31.0540i 1.03113i 0.856850 + 0.515565i \(0.172418\pi\)
−0.856850 + 0.515565i \(0.827582\pi\)
\(908\) 1.39963i 0.0464482i
\(909\) 8.56155 0.283969
\(910\) 0 0
\(911\) −18.5616 −0.614972 −0.307486 0.951553i \(-0.599488\pi\)
−0.307486 + 0.951553i \(0.599488\pi\)
\(912\) − 24.0000i − 0.794719i
\(913\) − 3.68466i − 0.121944i
\(914\) −13.8617 −0.458506
\(915\) 0 0
\(916\) 9.64584 0.318707
\(917\) 52.4924i 1.73345i
\(918\) − 11.1231i − 0.367117i
\(919\) −30.7386 −1.01397 −0.506987 0.861954i \(-0.669241\pi\)
−0.506987 + 0.861954i \(0.669241\pi\)
\(920\) 0 0
\(921\) −31.6847 −1.04404
\(922\) 28.3845i 0.934793i
\(923\) − 5.75379i − 0.189388i
\(924\) 3.23106 0.106294
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) − 2.87689i − 0.0944896i
\(928\) − 2.43845i − 0.0800460i
\(929\) −4.87689 −0.160006 −0.0800029 0.996795i \(-0.525493\pi\)
−0.0800029 + 0.996795i \(0.525493\pi\)
\(930\) 0 0
\(931\) 98.6004 3.23150
\(932\) − 4.52273i − 0.148147i
\(933\) − 16.3153i − 0.534140i
\(934\) −5.26137 −0.172157
\(935\) 0 0
\(936\) −4.87689 −0.159406
\(937\) 39.1231i 1.27810i 0.769167 + 0.639048i \(0.220672\pi\)
−0.769167 + 0.639048i \(0.779328\pi\)
\(938\) − 90.9545i − 2.96977i
\(939\) 21.3693 0.697361
\(940\) 0 0
\(941\) −0.738634 −0.0240788 −0.0120394 0.999928i \(-0.503832\pi\)
−0.0120394 + 0.999928i \(0.503832\pi\)
\(942\) 21.8617i 0.712294i
\(943\) 11.0540i 0.359967i
\(944\) 56.2159 1.82967
\(945\) 0 0
\(946\) 17.2614 0.561215
\(947\) − 3.36932i − 0.109488i −0.998500 0.0547440i \(-0.982566\pi\)
0.998500 0.0547440i \(-0.0174343\pi\)
\(948\) 5.26137i 0.170881i
\(949\) 3.36932 0.109373
\(950\) 0 0
\(951\) 4.87689 0.158144
\(952\) − 88.9848i − 2.88402i
\(953\) 34.4924i 1.11732i 0.829397 + 0.558660i \(0.188684\pi\)
−0.829397 + 0.558660i \(0.811316\pi\)
\(954\) −5.36932 −0.173838
\(955\) 0 0
\(956\) −5.75379 −0.186091
\(957\) − 1.43845i − 0.0464984i
\(958\) − 34.7386i − 1.12235i
\(959\) 77.4773 2.50187
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 5.26137i − 0.169633i
\(963\) − 16.4924i − 0.531461i
\(964\) 10.9848 0.353798
\(965\) 0 0
\(966\) −52.4924 −1.68892
\(967\) 27.0540i 0.869997i 0.900431 + 0.434999i \(0.143251\pi\)
−0.900431 + 0.434999i \(0.856749\pi\)
\(968\) 21.7775i 0.699954i
\(969\) 36.4924 1.17231
\(970\) 0 0
\(971\) −44.6695 −1.43351 −0.716756 0.697324i \(-0.754374\pi\)
−0.716756 + 0.697324i \(0.754374\pi\)
\(972\) 0.438447i 0.0140632i
\(973\) 91.8617i 2.94495i
\(974\) 53.4773 1.71352
\(975\) 0 0
\(976\) −4.10795 −0.131492
\(977\) − 7.43845i − 0.237977i −0.992896 0.118989i \(-0.962035\pi\)
0.992896 0.118989i \(-0.0379652\pi\)
\(978\) − 28.0000i − 0.895341i
\(979\) 17.6155 0.562995
\(980\) 0 0
\(981\) −5.68466 −0.181497
\(982\) − 10.5227i − 0.335794i
\(983\) − 27.8617i − 0.888651i −0.895865 0.444326i \(-0.853443\pi\)
0.895865 0.444326i \(-0.146557\pi\)
\(984\) 4.10795 0.130957
\(985\) 0 0
\(986\) 11.1231 0.354232
\(987\) − 67.2311i − 2.13999i
\(988\) 4.49242i 0.142923i
\(989\) −50.4233 −1.60337
\(990\) 0 0
\(991\) 4.94602 0.157116 0.0785578 0.996910i \(-0.474968\pi\)
0.0785578 + 0.996910i \(0.474968\pi\)
\(992\) 9.75379i 0.309683i
\(993\) − 10.2462i − 0.325154i
\(994\) −23.0152 −0.729996
\(995\) 0 0
\(996\) −1.12311 −0.0355870
\(997\) 54.0388i 1.71143i 0.517450 + 0.855713i \(0.326881\pi\)
−0.517450 + 0.855713i \(0.673119\pi\)
\(998\) 66.7386i 2.11257i
\(999\) 1.68466 0.0533002
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 2175.2.c.h.349.2 4
5.2 odd 4 2175.2.a.m.1.2 2
5.3 odd 4 435.2.a.h.1.1 2
5.4 even 2 inner 2175.2.c.h.349.3 4
15.2 even 4 6525.2.a.bc.1.1 2
15.8 even 4 1305.2.a.i.1.2 2
20.3 even 4 6960.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.h.1.1 2 5.3 odd 4
1305.2.a.i.1.2 2 15.8 even 4
2175.2.a.m.1.2 2 5.2 odd 4
2175.2.c.h.349.2 4 1.1 even 1 trivial
2175.2.c.h.349.3 4 5.4 even 2 inner
6525.2.a.bc.1.1 2 15.2 even 4
6960.2.a.bx.1.1 2 20.3 even 4