Properties

Label 1305.2.a.i.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +5.12311 q^{7} -2.43845 q^{8} +O(q^{10})\) \(q+1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +5.12311 q^{7} -2.43845 q^{8} -1.56155 q^{10} +1.43845 q^{11} -2.00000 q^{13} +8.00000 q^{14} -4.68466 q^{16} +7.12311 q^{17} +5.12311 q^{19} -0.438447 q^{20} +2.24621 q^{22} -6.56155 q^{23} +1.00000 q^{25} -3.12311 q^{26} +2.24621 q^{28} -1.00000 q^{29} +4.00000 q^{31} -2.43845 q^{32} +11.1231 q^{34} -5.12311 q^{35} -1.68466 q^{37} +8.00000 q^{38} +2.43845 q^{40} +1.68466 q^{41} +7.68466 q^{43} +0.630683 q^{44} -10.2462 q^{46} +13.1231 q^{47} +19.2462 q^{49} +1.56155 q^{50} -0.876894 q^{52} +3.43845 q^{53} -1.43845 q^{55} -12.4924 q^{56} -1.56155 q^{58} -12.0000 q^{59} +0.876894 q^{61} +6.24621 q^{62} +5.56155 q^{64} +2.00000 q^{65} -11.3693 q^{67} +3.12311 q^{68} -8.00000 q^{70} +2.87689 q^{71} +1.68466 q^{73} -2.63068 q^{74} +2.24621 q^{76} +7.36932 q^{77} -12.0000 q^{79} +4.68466 q^{80} +2.63068 q^{82} +2.56155 q^{83} -7.12311 q^{85} +12.0000 q^{86} -3.50758 q^{88} -12.2462 q^{89} -10.2462 q^{91} -2.87689 q^{92} +20.4924 q^{94} -5.12311 q^{95} -5.68466 q^{97} +30.0540 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8} + q^{10} + 7 q^{11} - 4 q^{13} + 16 q^{14} + 3 q^{16} + 6 q^{17} + 2 q^{19} - 5 q^{20} - 12 q^{22} - 9 q^{23} + 2 q^{25} + 2 q^{26} - 12 q^{28} - 2 q^{29} + 8 q^{31} - 9 q^{32} + 14 q^{34} - 2 q^{35} + 9 q^{37} + 16 q^{38} + 9 q^{40} - 9 q^{41} + 3 q^{43} + 26 q^{44} - 4 q^{46} + 18 q^{47} + 22 q^{49} - q^{50} - 10 q^{52} + 11 q^{53} - 7 q^{55} + 8 q^{56} + q^{58} - 24 q^{59} + 10 q^{61} - 4 q^{62} + 7 q^{64} + 4 q^{65} + 2 q^{67} - 2 q^{68} - 16 q^{70} + 14 q^{71} - 9 q^{73} - 30 q^{74} - 12 q^{76} - 10 q^{77} - 24 q^{79} - 3 q^{80} + 30 q^{82} + q^{83} - 6 q^{85} + 24 q^{86} - 40 q^{88} - 8 q^{89} - 4 q^{91} - 14 q^{92} + 8 q^{94} - 2 q^{95} + q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 1.10418 0.552092 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 5.12311 1.93635 0.968176 0.250270i \(-0.0805195\pi\)
0.968176 + 0.250270i \(0.0805195\pi\)
\(8\) −2.43845 −0.862121
\(9\) 0 0
\(10\) −1.56155 −0.493806
\(11\) 1.43845 0.433708 0.216854 0.976204i \(-0.430420\pi\)
0.216854 + 0.976204i \(0.430420\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 8.00000 2.13809
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 7.12311 1.72761 0.863803 0.503829i \(-0.168076\pi\)
0.863803 + 0.503829i \(0.168076\pi\)
\(18\) 0 0
\(19\) 5.12311 1.17532 0.587661 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) −0.438447 −0.0980398
\(21\) 0 0
\(22\) 2.24621 0.478894
\(23\) −6.56155 −1.36818 −0.684089 0.729398i \(-0.739800\pi\)
−0.684089 + 0.729398i \(0.739800\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.12311 −0.612491
\(27\) 0 0
\(28\) 2.24621 0.424494
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −2.43845 −0.431061
\(33\) 0 0
\(34\) 11.1231 1.90760
\(35\) −5.12311 −0.865963
\(36\) 0 0
\(37\) −1.68466 −0.276956 −0.138478 0.990366i \(-0.544221\pi\)
−0.138478 + 0.990366i \(0.544221\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 2.43845 0.385552
\(41\) 1.68466 0.263099 0.131550 0.991310i \(-0.458005\pi\)
0.131550 + 0.991310i \(0.458005\pi\)
\(42\) 0 0
\(43\) 7.68466 1.17190 0.585950 0.810347i \(-0.300722\pi\)
0.585950 + 0.810347i \(0.300722\pi\)
\(44\) 0.630683 0.0950791
\(45\) 0 0
\(46\) −10.2462 −1.51072
\(47\) 13.1231 1.91420 0.957101 0.289755i \(-0.0935738\pi\)
0.957101 + 0.289755i \(0.0935738\pi\)
\(48\) 0 0
\(49\) 19.2462 2.74946
\(50\) 1.56155 0.220837
\(51\) 0 0
\(52\) −0.876894 −0.121603
\(53\) 3.43845 0.472307 0.236154 0.971716i \(-0.424113\pi\)
0.236154 + 0.971716i \(0.424113\pi\)
\(54\) 0 0
\(55\) −1.43845 −0.193960
\(56\) −12.4924 −1.66937
\(57\) 0 0
\(58\) −1.56155 −0.205042
\(59\) −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 \(0.482122\pi\)
\(62\) 6.24621 0.793270
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −11.3693 −1.38898 −0.694492 0.719501i \(-0.744371\pi\)
−0.694492 + 0.719501i \(0.744371\pi\)
\(68\) 3.12311 0.378732
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 2.87689 0.341425 0.170712 0.985321i \(-0.445393\pi\)
0.170712 + 0.985321i \(0.445393\pi\)
\(72\) 0 0
\(73\) 1.68466 0.197174 0.0985872 0.995128i \(-0.468568\pi\)
0.0985872 + 0.995128i \(0.468568\pi\)
\(74\) −2.63068 −0.305811
\(75\) 0 0
\(76\) 2.24621 0.257658
\(77\) 7.36932 0.839812
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 4.68466 0.523761
\(81\) 0 0
\(82\) 2.63068 0.290510
\(83\) 2.56155 0.281167 0.140583 0.990069i \(-0.455102\pi\)
0.140583 + 0.990069i \(0.455102\pi\)
\(84\) 0 0
\(85\) −7.12311 −0.772609
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −3.50758 −0.373909
\(89\) −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
\(92\) −2.87689 −0.299937
\(93\) 0 0
\(94\) 20.4924 2.11363
\(95\) −5.12311 −0.525620
\(96\) 0 0
\(97\) −5.68466 −0.577190 −0.288595 0.957451i \(-0.593188\pi\)
−0.288595 + 0.957451i \(0.593188\pi\)
\(98\) 30.0540 3.03591
\(99\) 0 0
\(100\) 0.438447 0.0438447
\(101\) 8.56155 0.851906 0.425953 0.904745i \(-0.359939\pi\)
0.425953 + 0.904745i \(0.359939\pi\)
\(102\) 0 0
\(103\) −2.87689 −0.283469 −0.141734 0.989905i \(-0.545268\pi\)
−0.141734 + 0.989905i \(0.545268\pi\)
\(104\) 4.87689 0.478219
\(105\) 0 0
\(106\) 5.36932 0.521514
\(107\) −16.4924 −1.59438 −0.797191 0.603727i \(-0.793682\pi\)
−0.797191 + 0.603727i \(0.793682\pi\)
\(108\) 0 0
\(109\) −5.68466 −0.544492 −0.272246 0.962228i \(-0.587766\pi\)
−0.272246 + 0.962228i \(0.587766\pi\)
\(110\) −2.24621 −0.214168
\(111\) 0 0
\(112\) −24.0000 −2.26779
\(113\) 4.87689 0.458780 0.229390 0.973335i \(-0.426327\pi\)
0.229390 + 0.973335i \(0.426327\pi\)
\(114\) 0 0
\(115\) 6.56155 0.611868
\(116\) −0.438447 −0.0407088
\(117\) 0 0
\(118\) −18.7386 −1.72503
\(119\) 36.4924 3.34525
\(120\) 0 0
\(121\) −8.93087 −0.811897
\(122\) 1.36932 0.123972
\(123\) 0 0
\(124\) 1.75379 0.157495
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.31534 0.382925 0.191462 0.981500i \(-0.438677\pi\)
0.191462 + 0.981500i \(0.438677\pi\)
\(128\) 13.5616 1.19868
\(129\) 0 0
\(130\) 3.12311 0.273914
\(131\) −10.2462 −0.895216 −0.447608 0.894230i \(-0.647724\pi\)
−0.447608 + 0.894230i \(0.647724\pi\)
\(132\) 0 0
\(133\) 26.2462 2.27584
\(134\) −17.7538 −1.53369
\(135\) 0 0
\(136\) −17.3693 −1.48941
\(137\) 15.1231 1.29205 0.646027 0.763315i \(-0.276429\pi\)
0.646027 + 0.763315i \(0.276429\pi\)
\(138\) 0 0
\(139\) −17.9309 −1.52088 −0.760438 0.649410i \(-0.775016\pi\)
−0.760438 + 0.649410i \(0.775016\pi\)
\(140\) −2.24621 −0.189839
\(141\) 0 0
\(142\) 4.49242 0.376996
\(143\) −2.87689 −0.240578
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 2.63068 0.217717
\(147\) 0 0
\(148\) −0.738634 −0.0607153
\(149\) −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 \(0.443817\pi\)
\(152\) −12.4924 −1.01327
\(153\) 0 0
\(154\) 11.5076 0.927307
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −18.7386 −1.49077
\(159\) 0 0
\(160\) 2.43845 0.192776
\(161\) −33.6155 −2.64927
\(162\) 0 0
\(163\) −17.9309 −1.40445 −0.702227 0.711953i \(-0.747811\pi\)
−0.702227 + 0.711953i \(0.747811\pi\)
\(164\) 0.738634 0.0576776
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −11.1231 −0.853103
\(171\) 0 0
\(172\) 3.36932 0.256908
\(173\) −12.5616 −0.955037 −0.477519 0.878622i \(-0.658464\pi\)
−0.477519 + 0.878622i \(0.658464\pi\)
\(174\) 0 0
\(175\) 5.12311 0.387270
\(176\) −6.73863 −0.507944
\(177\) 0 0
\(178\) −19.1231 −1.43334
\(179\) −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 \(0.0963228\pi\)
\(182\) −16.0000 −1.18600
\(183\) 0 0
\(184\) 16.0000 1.17954
\(185\) 1.68466 0.123859
\(186\) 0 0
\(187\) 10.2462 0.749277
\(188\) 5.75379 0.419638
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 9.93087 0.718573 0.359286 0.933227i \(-0.383020\pi\)
0.359286 + 0.933227i \(0.383020\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −8.87689 −0.637324
\(195\) 0 0
\(196\) 8.43845 0.602746
\(197\) −4.56155 −0.324997 −0.162499 0.986709i \(-0.551955\pi\)
−0.162499 + 0.986709i \(0.551955\pi\)
\(198\) 0 0
\(199\) −11.0540 −0.783596 −0.391798 0.920051i \(-0.628147\pi\)
−0.391798 + 0.920051i \(0.628147\pi\)
\(200\) −2.43845 −0.172424
\(201\) 0 0
\(202\) 13.3693 0.940662
\(203\) −5.12311 −0.359572
\(204\) 0 0
\(205\) −1.68466 −0.117662
\(206\) −4.49242 −0.313002
\(207\) 0 0
\(208\) 9.36932 0.649645
\(209\) 7.36932 0.509746
\(210\) 0 0
\(211\) −2.87689 −0.198054 −0.0990268 0.995085i \(-0.531573\pi\)
−0.0990268 + 0.995085i \(0.531573\pi\)
\(212\) 1.50758 0.103541
\(213\) 0 0
\(214\) −25.7538 −1.76049
\(215\) −7.68466 −0.524089
\(216\) 0 0
\(217\) 20.4924 1.39112
\(218\) −8.87689 −0.601219
\(219\) 0 0
\(220\) −0.630683 −0.0425206
\(221\) −14.2462 −0.958304
\(222\) 0 0
\(223\) −18.2462 −1.22186 −0.610928 0.791686i \(-0.709204\pi\)
−0.610928 + 0.791686i \(0.709204\pi\)
\(224\) −12.4924 −0.834685
\(225\) 0 0
\(226\) 7.61553 0.506577
\(227\) 3.19224 0.211876 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 10.2462 0.675615
\(231\) 0 0
\(232\) 2.43845 0.160092
\(233\) 10.3153 0.675780 0.337890 0.941186i \(-0.390287\pi\)
0.337890 + 0.941186i \(0.390287\pi\)
\(234\) 0 0
\(235\) −13.1231 −0.856057
\(236\) −5.26137 −0.342486
\(237\) 0 0
\(238\) 56.9848 3.69378
\(239\) 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 \(0.798875\pi\)
\(242\) −13.9460 −0.896484
\(243\) 0 0
\(244\) 0.384472 0.0246133
\(245\) −19.2462 −1.22960
\(246\) 0 0
\(247\) −10.2462 −0.651951
\(248\) −9.75379 −0.619366
\(249\) 0 0
\(250\) −1.56155 −0.0987613
\(251\) −2.24621 −0.141780 −0.0708898 0.997484i \(-0.522584\pi\)
−0.0708898 + 0.997484i \(0.522584\pi\)
\(252\) 0 0
\(253\) −9.43845 −0.593390
\(254\) 6.73863 0.422819
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −11.4384 −0.713511 −0.356755 0.934198i \(-0.616117\pi\)
−0.356755 + 0.934198i \(0.616117\pi\)
\(258\) 0 0
\(259\) −8.63068 −0.536285
\(260\) 0.876894 0.0543827
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) 5.75379 0.354794 0.177397 0.984139i \(-0.443232\pi\)
0.177397 + 0.984139i \(0.443232\pi\)
\(264\) 0 0
\(265\) −3.43845 −0.211222
\(266\) 40.9848 2.51294
\(267\) 0 0
\(268\) −4.98485 −0.304498
\(269\) −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 \(0.325903\pi\)
\(272\) −33.3693 −2.02331
\(273\) 0 0
\(274\) 23.6155 1.42667
\(275\) 1.43845 0.0867416
\(276\) 0 0
\(277\) 0.876894 0.0526875 0.0263437 0.999653i \(-0.491614\pi\)
0.0263437 + 0.999653i \(0.491614\pi\)
\(278\) −28.0000 −1.67933
\(279\) 0 0
\(280\) 12.4924 0.746565
\(281\) 23.6155 1.40878 0.704392 0.709811i \(-0.251220\pi\)
0.704392 + 0.709811i \(0.251220\pi\)
\(282\) 0 0
\(283\) 6.87689 0.408789 0.204394 0.978889i \(-0.434477\pi\)
0.204394 + 0.978889i \(0.434477\pi\)
\(284\) 1.26137 0.0748483
\(285\) 0 0
\(286\) −4.49242 −0.265643
\(287\) 8.63068 0.509453
\(288\) 0 0
\(289\) 33.7386 1.98463
\(290\) 1.56155 0.0916975
\(291\) 0 0
\(292\) 0.738634 0.0432253
\(293\) 21.3693 1.24841 0.624204 0.781261i \(-0.285423\pi\)
0.624204 + 0.781261i \(0.285423\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 4.10795 0.238770
\(297\) 0 0
\(298\) −0.384472 −0.0222719
\(299\) 13.1231 0.758929
\(300\) 0 0
\(301\) 39.3693 2.26921
\(302\) 6.73863 0.387765
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) −0.876894 −0.0502108
\(306\) 0 0
\(307\) −31.6847 −1.80834 −0.904169 0.427174i \(-0.859509\pi\)
−0.904169 + 0.427174i \(0.859509\pi\)
\(308\) 3.23106 0.184107
\(309\) 0 0
\(310\) −6.24621 −0.354761
\(311\) −16.3153 −0.925158 −0.462579 0.886578i \(-0.653076\pi\)
−0.462579 + 0.886578i \(0.653076\pi\)
\(312\) 0 0
\(313\) −21.3693 −1.20787 −0.603933 0.797035i \(-0.706400\pi\)
−0.603933 + 0.797035i \(0.706400\pi\)
\(314\) −21.8617 −1.23373
\(315\) 0 0
\(316\) −5.26137 −0.295975
\(317\) −4.87689 −0.273914 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(318\) 0 0
\(319\) −1.43845 −0.0805376
\(320\) −5.56155 −0.310900
\(321\) 0 0
\(322\) −52.4924 −2.92529
\(323\) 36.4924 2.03049
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −28.0000 −1.55078
\(327\) 0 0
\(328\) −4.10795 −0.226824
\(329\) 67.2311 3.70657
\(330\) 0 0
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) 1.12311 0.0616384
\(333\) 0 0
\(334\) 0 0
\(335\) 11.3693 0.621172
\(336\) 0 0
\(337\) −7.75379 −0.422376 −0.211188 0.977445i \(-0.567733\pi\)
−0.211188 + 0.977445i \(0.567733\pi\)
\(338\) −14.0540 −0.764435
\(339\) 0 0
\(340\) −3.12311 −0.169374
\(341\) 5.75379 0.311585
\(342\) 0 0
\(343\) 62.7386 3.38757
\(344\) −18.7386 −1.01032
\(345\) 0 0
\(346\) −19.6155 −1.05454
\(347\) 12.1771 0.653700 0.326850 0.945076i \(-0.394013\pi\)
0.326850 + 0.945076i \(0.394013\pi\)
\(348\) 0 0
\(349\) 0.0691303 0.00370046 0.00185023 0.999998i \(-0.499411\pi\)
0.00185023 + 0.999998i \(0.499411\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −3.50758 −0.186955
\(353\) −30.4924 −1.62295 −0.811474 0.584389i \(-0.801334\pi\)
−0.811474 + 0.584389i \(0.801334\pi\)
\(354\) 0 0
\(355\) −2.87689 −0.152690
\(356\) −5.36932 −0.284573
\(357\) 0 0
\(358\) −1.75379 −0.0926906
\(359\) 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
\(362\) 40.1080 2.10803
\(363\) 0 0
\(364\) −4.49242 −0.235467
\(365\) −1.68466 −0.0881791
\(366\) 0 0
\(367\) −19.6847 −1.02753 −0.513765 0.857931i \(-0.671750\pi\)
−0.513765 + 0.857931i \(0.671750\pi\)
\(368\) 30.7386 1.60236
\(369\) 0 0
\(370\) 2.63068 0.136763
\(371\) 17.6155 0.914553
\(372\) 0 0
\(373\) 13.3693 0.692237 0.346118 0.938191i \(-0.387500\pi\)
0.346118 + 0.938191i \(0.387500\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −32.0000 −1.65027
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −2.24621 −0.115228
\(381\) 0 0
\(382\) 15.5076 0.793437
\(383\) −11.0540 −0.564832 −0.282416 0.959292i \(-0.591136\pi\)
−0.282416 + 0.959292i \(0.591136\pi\)
\(384\) 0 0
\(385\) −7.36932 −0.375575
\(386\) −15.6155 −0.794809
\(387\) 0 0
\(388\) −2.49242 −0.126534
\(389\) −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
\(392\) −46.9309 −2.37037
\(393\) 0 0
\(394\) −7.12311 −0.358857
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 24.2462 1.21688 0.608441 0.793599i \(-0.291795\pi\)
0.608441 + 0.793599i \(0.291795\pi\)
\(398\) −17.2614 −0.865234
\(399\) 0 0
\(400\) −4.68466 −0.234233
\(401\) −35.6155 −1.77855 −0.889277 0.457368i \(-0.848792\pi\)
−0.889277 + 0.457368i \(0.848792\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 3.75379 0.186758
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −2.42329 −0.120118
\(408\) 0 0
\(409\) 17.3693 0.858857 0.429429 0.903101i \(-0.358715\pi\)
0.429429 + 0.903101i \(0.358715\pi\)
\(410\) −2.63068 −0.129920
\(411\) 0 0
\(412\) −1.26137 −0.0621431
\(413\) −61.4773 −3.02510
\(414\) 0 0
\(415\) −2.56155 −0.125742
\(416\) 4.87689 0.239109
\(417\) 0 0
\(418\) 11.5076 0.562854
\(419\) 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 \(0.578356\pi\)
\(422\) −4.49242 −0.218688
\(423\) 0 0
\(424\) −8.38447 −0.407186
\(425\) 7.12311 0.345521
\(426\) 0 0
\(427\) 4.49242 0.217404
\(428\) −7.23106 −0.349526
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) 21.1231 1.01746 0.508732 0.860925i \(-0.330114\pi\)
0.508732 + 0.860925i \(0.330114\pi\)
\(432\) 0 0
\(433\) −4.06913 −0.195550 −0.0977750 0.995209i \(-0.531173\pi\)
−0.0977750 + 0.995209i \(0.531173\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −2.49242 −0.119365
\(437\) −33.6155 −1.60805
\(438\) 0 0
\(439\) 12.4924 0.596231 0.298115 0.954530i \(-0.403642\pi\)
0.298115 + 0.954530i \(0.403642\pi\)
\(440\) 3.50758 0.167217
\(441\) 0 0
\(442\) −22.2462 −1.05814
\(443\) −6.24621 −0.296766 −0.148383 0.988930i \(-0.547407\pi\)
−0.148383 + 0.988930i \(0.547407\pi\)
\(444\) 0 0
\(445\) 12.2462 0.580526
\(446\) −28.4924 −1.34916
\(447\) 0 0
\(448\) 28.4924 1.34614
\(449\) −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
\(452\) 2.13826 0.100575
\(453\) 0 0
\(454\) 4.98485 0.233950
\(455\) 10.2462 0.480350
\(456\) 0 0
\(457\) −8.87689 −0.415244 −0.207622 0.978209i \(-0.566572\pi\)
−0.207622 + 0.978209i \(0.566572\pi\)
\(458\) 34.3542 1.60526
\(459\) 0 0
\(460\) 2.87689 0.134136
\(461\) 18.1771 0.846591 0.423296 0.905992i \(-0.360873\pi\)
0.423296 + 0.905992i \(0.360873\pi\)
\(462\) 0 0
\(463\) 25.6155 1.19045 0.595227 0.803557i \(-0.297062\pi\)
0.595227 + 0.803557i \(0.297062\pi\)
\(464\) 4.68466 0.217480
\(465\) 0 0
\(466\) 16.1080 0.746186
\(467\) 3.36932 0.155913 0.0779567 0.996957i \(-0.475160\pi\)
0.0779567 + 0.996957i \(0.475160\pi\)
\(468\) 0 0
\(469\) −58.2462 −2.68956
\(470\) −20.4924 −0.945245
\(471\) 0 0
\(472\) 29.2614 1.34686
\(473\) 11.0540 0.508262
\(474\) 0 0
\(475\) 5.12311 0.235064
\(476\) 16.0000 0.733359
\(477\) 0 0
\(478\) 20.4924 0.937302
\(479\) 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
\(482\) −39.1231 −1.78201
\(483\) 0 0
\(484\) −3.91571 −0.177987
\(485\) 5.68466 0.258127
\(486\) 0 0
\(487\) 34.2462 1.55184 0.775922 0.630829i \(-0.217285\pi\)
0.775922 + 0.630829i \(0.217285\pi\)
\(488\) −2.13826 −0.0967945
\(489\) 0 0
\(490\) −30.0540 −1.35770
\(491\) −6.73863 −0.304110 −0.152055 0.988372i \(-0.548589\pi\)
−0.152055 + 0.988372i \(0.548589\pi\)
\(492\) 0 0
\(493\) −7.12311 −0.320809
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −18.7386 −0.841389
\(497\) 14.7386 0.661118
\(498\) 0 0
\(499\) 42.7386 1.91324 0.956622 0.291332i \(-0.0940984\pi\)
0.956622 + 0.291332i \(0.0940984\pi\)
\(500\) −0.438447 −0.0196080
\(501\) 0 0
\(502\) −3.50758 −0.156551
\(503\) 39.3693 1.75539 0.877696 0.479219i \(-0.159080\pi\)
0.877696 + 0.479219i \(0.159080\pi\)
\(504\) 0 0
\(505\) −8.56155 −0.380984
\(506\) −14.7386 −0.655212
\(507\) 0 0
\(508\) 1.89205 0.0839461
\(509\) −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
\(512\) −11.4233 −0.504843
\(513\) 0 0
\(514\) −17.8617 −0.787848
\(515\) 2.87689 0.126771
\(516\) 0 0
\(517\) 18.8769 0.830205
\(518\) −13.4773 −0.592157
\(519\) 0 0
\(520\) −4.87689 −0.213866
\(521\) 0.246211 0.0107867 0.00539336 0.999985i \(-0.498283\pi\)
0.00539336 + 0.999985i \(0.498283\pi\)
\(522\) 0 0
\(523\) 18.7386 0.819383 0.409692 0.912224i \(-0.365636\pi\)
0.409692 + 0.912224i \(0.365636\pi\)
\(524\) −4.49242 −0.196252
\(525\) 0 0
\(526\) 8.98485 0.391758
\(527\) 28.4924 1.24115
\(528\) 0 0
\(529\) 20.0540 0.871912
\(530\) −5.36932 −0.233228
\(531\) 0 0
\(532\) 11.5076 0.498917
\(533\) −3.36932 −0.145941
\(534\) 0 0
\(535\) 16.4924 0.713030
\(536\) 27.7235 1.19747
\(537\) 0 0
\(538\) −18.3542 −0.791304
\(539\) 27.6847 1.19246
\(540\) 0 0
\(541\) −40.7386 −1.75149 −0.875745 0.482773i \(-0.839629\pi\)
−0.875745 + 0.482773i \(0.839629\pi\)
\(542\) 26.7386 1.14852
\(543\) 0 0
\(544\) −17.3693 −0.744703
\(545\) 5.68466 0.243504
\(546\) 0 0
\(547\) −42.7386 −1.82737 −0.913686 0.406421i \(-0.866777\pi\)
−0.913686 + 0.406421i \(0.866777\pi\)
\(548\) 6.63068 0.283249
\(549\) 0 0
\(550\) 2.24621 0.0957788
\(551\) −5.12311 −0.218252
\(552\) 0 0
\(553\) −61.4773 −2.61428
\(554\) 1.36932 0.0581767
\(555\) 0 0
\(556\) −7.86174 −0.333412
\(557\) 20.4233 0.865363 0.432681 0.901547i \(-0.357568\pi\)
0.432681 + 0.901547i \(0.357568\pi\)
\(558\) 0 0
\(559\) −15.3693 −0.650053
\(560\) 24.0000 1.01419
\(561\) 0 0
\(562\) 36.8769 1.55556
\(563\) −9.12311 −0.384493 −0.192247 0.981347i \(-0.561577\pi\)
−0.192247 + 0.981347i \(0.561577\pi\)
\(564\) 0 0
\(565\) −4.87689 −0.205172
\(566\) 10.7386 0.451378
\(567\) 0 0
\(568\) −7.01515 −0.294349
\(569\) 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 \(0.195537\pi\)
\(572\) −1.26137 −0.0527404
\(573\) 0 0
\(574\) 13.4773 0.562530
\(575\) −6.56155 −0.273636
\(576\) 0 0
\(577\) −30.4924 −1.26942 −0.634708 0.772752i \(-0.718880\pi\)
−0.634708 + 0.772752i \(0.718880\pi\)
\(578\) 52.6847 2.19139
\(579\) 0 0
\(580\) 0.438447 0.0182055
\(581\) 13.1231 0.544438
\(582\) 0 0
\(583\) 4.94602 0.204843
\(584\) −4.10795 −0.169988
\(585\) 0 0
\(586\) 33.3693 1.37847
\(587\) −24.4924 −1.01091 −0.505455 0.862853i \(-0.668675\pi\)
−0.505455 + 0.862853i \(0.668675\pi\)
\(588\) 0 0
\(589\) 20.4924 0.844376
\(590\) 18.7386 0.771457
\(591\) 0 0
\(592\) 7.89205 0.324361
\(593\) −36.2462 −1.48845 −0.744227 0.667927i \(-0.767182\pi\)
−0.744227 + 0.667927i \(0.767182\pi\)
\(594\) 0 0
\(595\) −36.4924 −1.49604
\(596\) −0.107951 −0.00442183
\(597\) 0 0
\(598\) 20.4924 0.837997
\(599\) 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 \(0.180346\pi\)
\(602\) 61.4773 2.50563
\(603\) 0 0
\(604\) 1.89205 0.0769864
\(605\) 8.93087 0.363091
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −12.4924 −0.506635
\(609\) 0 0
\(610\) −1.36932 −0.0554420
\(611\) −26.2462 −1.06181
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −49.4773 −1.99674
\(615\) 0 0
\(616\) −17.9697 −0.724019
\(617\) −8.24621 −0.331980 −0.165990 0.986127i \(-0.553082\pi\)
−0.165990 + 0.986127i \(0.553082\pi\)
\(618\) 0 0
\(619\) 29.1231 1.17056 0.585278 0.810833i \(-0.300985\pi\)
0.585278 + 0.810833i \(0.300985\pi\)
\(620\) −1.75379 −0.0704339
\(621\) 0 0
\(622\) −25.4773 −1.02155
\(623\) −62.7386 −2.51357
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −33.3693 −1.33371
\(627\) 0 0
\(628\) −6.13826 −0.244943
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 29.2614 1.16395
\(633\) 0 0
\(634\) −7.61553 −0.302451
\(635\) −4.31534 −0.171249
\(636\) 0 0
\(637\) −38.4924 −1.52513
\(638\) −2.24621 −0.0889284
\(639\) 0 0
\(640\) −13.5616 −0.536067
\(641\) −19.4384 −0.767773 −0.383886 0.923380i \(-0.625415\pi\)
−0.383886 + 0.923380i \(0.625415\pi\)
\(642\) 0 0
\(643\) −18.7386 −0.738980 −0.369490 0.929235i \(-0.620468\pi\)
−0.369490 + 0.929235i \(0.620468\pi\)
\(644\) −14.7386 −0.580784
\(645\) 0 0
\(646\) 56.9848 2.24204
\(647\) −5.93087 −0.233167 −0.116583 0.993181i \(-0.537194\pi\)
−0.116583 + 0.993181i \(0.537194\pi\)
\(648\) 0 0
\(649\) −17.2614 −0.677568
\(650\) −3.12311 −0.122498
\(651\) 0 0
\(652\) −7.86174 −0.307889
\(653\) −12.2462 −0.479231 −0.239616 0.970868i \(-0.577021\pi\)
−0.239616 + 0.970868i \(0.577021\pi\)
\(654\) 0 0
\(655\) 10.2462 0.400353
\(656\) −7.89205 −0.308133
\(657\) 0 0
\(658\) 104.985 4.09274
\(659\) −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 \(0.493475\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) −6.24621 −0.242400
\(665\) −26.2462 −1.01778
\(666\) 0 0
\(667\) 6.56155 0.254064
\(668\) 0 0
\(669\) 0 0
\(670\) 17.7538 0.685889
\(671\) 1.26137 0.0486945
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −12.1080 −0.466381
\(675\) 0 0
\(676\) −3.94602 −0.151770
\(677\) −13.8617 −0.532750 −0.266375 0.963869i \(-0.585826\pi\)
−0.266375 + 0.963869i \(0.585826\pi\)
\(678\) 0 0
\(679\) −29.1231 −1.11764
\(680\) 17.3693 0.666083
\(681\) 0 0
\(682\) 8.98485 0.344047
\(683\) 21.4384 0.820319 0.410160 0.912014i \(-0.365473\pi\)
0.410160 + 0.912014i \(0.365473\pi\)
\(684\) 0 0
\(685\) −15.1231 −0.577824
\(686\) 97.9697 3.74050
\(687\) 0 0
\(688\) −36.0000 −1.37249
\(689\) −6.87689 −0.261989
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −5.50758 −0.209367
\(693\) 0 0
\(694\) 19.0152 0.721805
\(695\) 17.9309 0.680157
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0.107951 0.00408599
\(699\) 0 0
\(700\) 2.24621 0.0848988
\(701\) −3.75379 −0.141779 −0.0708893 0.997484i \(-0.522584\pi\)
−0.0708893 + 0.997484i \(0.522584\pi\)
\(702\) 0 0
\(703\) −8.63068 −0.325512
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −47.6155 −1.79203
\(707\) 43.8617 1.64959
\(708\) 0 0
\(709\) −12.4233 −0.466567 −0.233283 0.972409i \(-0.574947\pi\)
−0.233283 + 0.972409i \(0.574947\pi\)
\(710\) −4.49242 −0.168598
\(711\) 0 0
\(712\) 29.8617 1.11912
\(713\) −26.2462 −0.982928
\(714\) 0 0
\(715\) 2.87689 0.107590
\(716\) −0.492423 −0.0184027
\(717\) 0 0
\(718\) 4.98485 0.186033
\(719\) 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
\(722\) 11.3153 0.421113
\(723\) 0 0
\(724\) 11.2614 0.418525
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −38.7386 −1.43674 −0.718368 0.695663i \(-0.755111\pi\)
−0.718368 + 0.695663i \(0.755111\pi\)
\(728\) 24.9848 0.926000
\(729\) 0 0
\(730\) −2.63068 −0.0973660
\(731\) 54.7386 2.02458
\(732\) 0 0
\(733\) −22.9848 −0.848965 −0.424482 0.905436i \(-0.639544\pi\)
−0.424482 + 0.905436i \(0.639544\pi\)
\(734\) −30.7386 −1.13458
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) −16.3542 −0.602413
\(738\) 0 0
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 0.738634 0.0271527
\(741\) 0 0
\(742\) 27.5076 1.00983
\(743\) −34.8769 −1.27951 −0.639755 0.768579i \(-0.720964\pi\)
−0.639755 + 0.768579i \(0.720964\pi\)
\(744\) 0 0
\(745\) 0.246211 0.00902048
\(746\) 20.8769 0.764357
\(747\) 0 0
\(748\) 4.49242 0.164259
\(749\) −84.4924 −3.08729
\(750\) 0 0
\(751\) 48.4924 1.76951 0.884757 0.466053i \(-0.154324\pi\)
0.884757 + 0.466053i \(0.154324\pi\)
\(752\) −61.4773 −2.24185
\(753\) 0 0
\(754\) 3.12311 0.113737
\(755\) −4.31534 −0.157051
\(756\) 0 0
\(757\) 5.68466 0.206612 0.103306 0.994650i \(-0.467058\pi\)
0.103306 + 0.994650i \(0.467058\pi\)
\(758\) 12.4924 0.453745
\(759\) 0 0
\(760\) 12.4924 0.453148
\(761\) 24.8769 0.901787 0.450893 0.892578i \(-0.351105\pi\)
0.450893 + 0.892578i \(0.351105\pi\)
\(762\) 0 0
\(763\) −29.1231 −1.05433
\(764\) 4.35416 0.157528
\(765\) 0 0
\(766\) −17.2614 −0.623679
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 43.6155 1.57282 0.786408 0.617707i \(-0.211938\pi\)
0.786408 + 0.617707i \(0.211938\pi\)
\(770\) −11.5076 −0.414704
\(771\) 0 0
\(772\) −4.38447 −0.157801
\(773\) −24.7386 −0.889787 −0.444893 0.895584i \(-0.646758\pi\)
−0.444893 + 0.895584i \(0.646758\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 13.8617 0.497607
\(777\) 0 0
\(778\) −23.1231 −0.829004
\(779\) 8.63068 0.309226
\(780\) 0 0
\(781\) 4.13826 0.148079
\(782\) −72.9848 −2.60993
\(783\) 0 0
\(784\) −90.1619 −3.22007
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −31.2311 −1.11327 −0.556633 0.830758i \(-0.687907\pi\)
−0.556633 + 0.830758i \(0.687907\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 18.7386 0.666691
\(791\) 24.9848 0.888359
\(792\) 0 0
\(793\) −1.75379 −0.0622789
\(794\) 37.8617 1.34366
\(795\) 0 0
\(796\) −4.84658 −0.171783
\(797\) −38.4924 −1.36347 −0.681736 0.731598i \(-0.738775\pi\)
−0.681736 + 0.731598i \(0.738775\pi\)
\(798\) 0 0
\(799\) 93.4773 3.30699
\(800\) −2.43845 −0.0862121
\(801\) 0 0
\(802\) −55.6155 −1.96385
\(803\) 2.42329 0.0855161
\(804\) 0 0
\(805\) 33.6155 1.18479
\(806\) −12.4924 −0.440027
\(807\) 0 0
\(808\) −20.8769 −0.734447
\(809\) 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 \(0.217431\pi\)
\(812\) −2.24621 −0.0788266
\(813\) 0 0
\(814\) −3.78410 −0.132633
\(815\) 17.9309 0.628091
\(816\) 0 0
\(817\) 39.3693 1.37736
\(818\) 27.1231 0.948337
\(819\) 0 0
\(820\) −0.738634 −0.0257942
\(821\) −55.6155 −1.94100 −0.970498 0.241111i \(-0.922488\pi\)
−0.970498 + 0.241111i \(0.922488\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 7.01515 0.244385
\(825\) 0 0
\(826\) −96.0000 −3.34027
\(827\) −53.6155 −1.86439 −0.932197 0.361951i \(-0.882111\pi\)
−0.932197 + 0.361951i \(0.882111\pi\)
\(828\) 0 0
\(829\) −4.24621 −0.147477 −0.0737385 0.997278i \(-0.523493\pi\)
−0.0737385 + 0.997278i \(0.523493\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −11.1231 −0.385624
\(833\) 137.093 4.74998
\(834\) 0 0
\(835\) 0 0
\(836\) 3.23106 0.111748
\(837\) 0 0
\(838\) 42.7386 1.47638
\(839\) −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
\(842\) −15.6155 −0.538147
\(843\) 0 0
\(844\) −1.26137 −0.0434180
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −45.7538 −1.57212
\(848\) −16.1080 −0.553149
\(849\) 0 0
\(850\) 11.1231 0.381519
\(851\) 11.0540 0.378925
\(852\) 0 0
\(853\) 19.4384 0.665560 0.332780 0.943005i \(-0.392013\pi\)
0.332780 + 0.943005i \(0.392013\pi\)
\(854\) 7.01515 0.240054
\(855\) 0 0
\(856\) 40.2159 1.37455
\(857\) 15.4384 0.527367 0.263684 0.964609i \(-0.415062\pi\)
0.263684 + 0.964609i \(0.415062\pi\)
\(858\) 0 0
\(859\) −23.3693 −0.797351 −0.398675 0.917092i \(-0.630530\pi\)
−0.398675 + 0.917092i \(0.630530\pi\)
\(860\) −3.36932 −0.114893
\(861\) 0 0
\(862\) 32.9848 1.12347
\(863\) −40.9848 −1.39514 −0.697570 0.716516i \(-0.745735\pi\)
−0.697570 + 0.716516i \(0.745735\pi\)
\(864\) 0 0
\(865\) 12.5616 0.427106
\(866\) −6.35416 −0.215923
\(867\) 0 0
\(868\) 8.98485 0.304966
\(869\) −17.2614 −0.585552
\(870\) 0 0
\(871\) 22.7386 0.770469
\(872\) 13.8617 0.469418
\(873\) 0 0
\(874\) −52.4924 −1.77558
\(875\) −5.12311 −0.173193
\(876\) 0 0
\(877\) 8.87689 0.299751 0.149876 0.988705i \(-0.452113\pi\)
0.149876 + 0.988705i \(0.452113\pi\)
\(878\) 19.5076 0.658349
\(879\) 0 0
\(880\) 6.73863 0.227159
\(881\) 8.06913 0.271856 0.135928 0.990719i \(-0.456598\pi\)
0.135928 + 0.990719i \(0.456598\pi\)
\(882\) 0 0
\(883\) −42.7386 −1.43827 −0.719135 0.694871i \(-0.755462\pi\)
−0.719135 + 0.694871i \(0.755462\pi\)
\(884\) −6.24621 −0.210083
\(885\) 0 0
\(886\) −9.75379 −0.327685
\(887\) 43.8617 1.47273 0.736367 0.676583i \(-0.236540\pi\)
0.736367 + 0.676583i \(0.236540\pi\)
\(888\) 0 0
\(889\) 22.1080 0.741477
\(890\) 19.1231 0.641008
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 67.2311 2.24980
\(894\) 0 0
\(895\) 1.12311 0.0375413
\(896\) 69.4773 2.32107
\(897\) 0 0
\(898\) −17.8617 −0.596054
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 24.4924 0.815961
\(902\) 3.78410 0.125997
\(903\) 0 0
\(904\) −11.8920 −0.395524
\(905\) −25.6847 −0.853787
\(906\) 0 0
\(907\) 31.0540 1.03113 0.515565 0.856850i \(-0.327582\pi\)
0.515565 + 0.856850i \(0.327582\pi\)
\(908\) 1.39963 0.0464482
\(909\) 0 0
\(910\) 16.0000 0.530395
\(911\) 18.5616 0.614972 0.307486 0.951553i \(-0.400512\pi\)
0.307486 + 0.951553i \(0.400512\pi\)
\(912\) 0 0
\(913\) 3.68466 0.121944
\(914\) −13.8617 −0.458506
\(915\) 0 0
\(916\) 9.64584 0.318707
\(917\) −52.4924 −1.73345
\(918\) 0 0
\(919\) 30.7386 1.01397 0.506987 0.861954i \(-0.330759\pi\)
0.506987 + 0.861954i \(0.330759\pi\)
\(920\) −16.0000 −0.527504
\(921\) 0 0
\(922\) 28.3845 0.934793
\(923\) −5.75379 −0.189388
\(924\) 0 0
\(925\) −1.68466 −0.0553912
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) 2.43845 0.0800460
\(929\) −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
\(932\) 4.52273 0.148147
\(933\) 0 0
\(934\) 5.26137 0.172157
\(935\) −10.2462 −0.335087
\(936\) 0 0
\(937\) 39.1231 1.27810 0.639048 0.769167i \(-0.279328\pi\)
0.639048 + 0.769167i \(0.279328\pi\)
\(938\) −90.9545 −2.96977
\(939\) 0 0
\(940\) −5.75379 −0.187668
\(941\) 0.738634 0.0240788 0.0120394 0.999928i \(-0.496168\pi\)
0.0120394 + 0.999928i \(0.496168\pi\)
\(942\) 0 0
\(943\) −11.0540 −0.359967
\(944\) 56.2159 1.82967
\(945\) 0 0
\(946\) 17.2614 0.561215
\(947\) 3.36932 0.109488 0.0547440 0.998500i \(-0.482566\pi\)
0.0547440 + 0.998500i \(0.482566\pi\)
\(948\) 0 0
\(949\) −3.36932 −0.109373
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) −88.9848 −2.88402
\(953\) 34.4924 1.11732 0.558660 0.829397i \(-0.311316\pi\)
0.558660 + 0.829397i \(0.311316\pi\)
\(954\) 0 0
\(955\) −9.93087 −0.321355
\(956\) 5.75379 0.186091
\(957\) 0 0
\(958\) 34.7386 1.12235
\(959\) 77.4773 2.50187
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 5.26137 0.169633
\(963\) 0 0
\(964\) −10.9848 −0.353798
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 27.0540 0.869997 0.434999 0.900431i \(-0.356749\pi\)
0.434999 + 0.900431i \(0.356749\pi\)
\(968\) 21.7775 0.699954
\(969\) 0 0
\(970\) 8.87689 0.285020
\(971\) 44.6695 1.43351 0.716756 0.697324i \(-0.245626\pi\)
0.716756 + 0.697324i \(0.245626\pi\)
\(972\) 0 0
\(973\) −91.8617 −2.94495
\(974\) 53.4773 1.71352
\(975\) 0 0
\(976\) −4.10795 −0.131492
\(977\) 7.43845 0.237977 0.118989 0.992896i \(-0.462035\pi\)
0.118989 + 0.992896i \(0.462035\pi\)
\(978\) 0 0
\(979\) −17.6155 −0.562995
\(980\) −8.43845 −0.269556
\(981\) 0 0
\(982\) −10.5227 −0.335794
\(983\) −27.8617 −0.888651 −0.444326 0.895865i \(-0.646557\pi\)
−0.444326 + 0.895865i \(0.646557\pi\)
\(984\) 0 0
\(985\) 4.56155 0.145343
\(986\) −11.1231 −0.354232
\(987\) 0 0
\(988\) −4.49242 −0.142923
\(989\) −50.4233 −1.60337
\(990\) 0 0
\(991\) 4.94602 0.157116 0.0785578 0.996910i \(-0.474968\pi\)
0.0785578 + 0.996910i \(0.474968\pi\)
\(992\) −9.75379 −0.309683
\(993\) 0 0
\(994\) 23.0152 0.729996
\(995\) 11.0540 0.350435
\(996\) 0 0
\(997\) 54.0388 1.71143 0.855713 0.517450i \(-0.173119\pi\)
0.855713 + 0.517450i \(0.173119\pi\)
\(998\) 66.7386 2.11257
\(999\) 0 0
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 1305.2.a.i.1.2 2
3.2 odd 2 435.2.a.h.1.1 2
5.4 even 2 6525.2.a.bc.1.1 2
12.11 even 2 6960.2.a.bx.1.1 2
15.2 even 4 2175.2.c.h.349.2 4
15.8 even 4 2175.2.c.h.349.3 4
15.14 odd 2 2175.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.h.1.1 2 3.2 odd 2
1305.2.a.i.1.2 2 1.1 even 1 trivial
2175.2.a.m.1.2 2 15.14 odd 2
2175.2.c.h.349.2 4 15.2 even 4
2175.2.c.h.349.3 4 15.8 even 4
6525.2.a.bc.1.1 2 5.4 even 2
6960.2.a.bx.1.1 2 12.11 even 2