Properties

Label 8046.2.a.r.1.5
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $14$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + 24455 x^{6} - 12990 x^{5} - 55580 x^{4} + 9808 x^{3} + 53551 x^{2} + \cdots - 7083 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.53522\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.53522 q^{5} +4.84041 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.53522 q^{5} +4.84041 q^{7} +1.00000 q^{8} -1.53522 q^{10} -3.93574 q^{11} +3.57898 q^{13} +4.84041 q^{14} +1.00000 q^{16} +1.48855 q^{17} -7.62957 q^{19} -1.53522 q^{20} -3.93574 q^{22} -5.95631 q^{23} -2.64310 q^{25} +3.57898 q^{26} +4.84041 q^{28} +9.31666 q^{29} +4.91650 q^{31} +1.00000 q^{32} +1.48855 q^{34} -7.43109 q^{35} -1.32731 q^{37} -7.62957 q^{38} -1.53522 q^{40} +3.69856 q^{41} +8.73841 q^{43} -3.93574 q^{44} -5.95631 q^{46} +6.61250 q^{47} +16.4296 q^{49} -2.64310 q^{50} +3.57898 q^{52} +4.60201 q^{53} +6.04222 q^{55} +4.84041 q^{56} +9.31666 q^{58} +1.97402 q^{59} -5.88075 q^{61} +4.91650 q^{62} +1.00000 q^{64} -5.49452 q^{65} +16.0194 q^{67} +1.48855 q^{68} -7.43109 q^{70} -5.26569 q^{71} -3.32646 q^{73} -1.32731 q^{74} -7.62957 q^{76} -19.0506 q^{77} +3.87300 q^{79} -1.53522 q^{80} +3.69856 q^{82} -11.0519 q^{83} -2.28526 q^{85} +8.73841 q^{86} -3.93574 q^{88} -12.0548 q^{89} +17.3237 q^{91} -5.95631 q^{92} +6.61250 q^{94} +11.7131 q^{95} +17.1315 q^{97} +16.4296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.53522 −0.686571 −0.343285 0.939231i \(-0.611540\pi\)
−0.343285 + 0.939231i \(0.611540\pi\)
\(6\) 0 0
\(7\) 4.84041 1.82950 0.914752 0.404017i \(-0.132386\pi\)
0.914752 + 0.404017i \(0.132386\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.53522 −0.485479
\(11\) −3.93574 −1.18667 −0.593335 0.804956i \(-0.702189\pi\)
−0.593335 + 0.804956i \(0.702189\pi\)
\(12\) 0 0
\(13\) 3.57898 0.992630 0.496315 0.868142i \(-0.334686\pi\)
0.496315 + 0.868142i \(0.334686\pi\)
\(14\) 4.84041 1.29365
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.48855 0.361027 0.180514 0.983572i \(-0.442224\pi\)
0.180514 + 0.983572i \(0.442224\pi\)
\(18\) 0 0
\(19\) −7.62957 −1.75034 −0.875172 0.483812i \(-0.839252\pi\)
−0.875172 + 0.483812i \(0.839252\pi\)
\(20\) −1.53522 −0.343285
\(21\) 0 0
\(22\) −3.93574 −0.839102
\(23\) −5.95631 −1.24198 −0.620989 0.783820i \(-0.713269\pi\)
−0.620989 + 0.783820i \(0.713269\pi\)
\(24\) 0 0
\(25\) −2.64310 −0.528621
\(26\) 3.57898 0.701896
\(27\) 0 0
\(28\) 4.84041 0.914752
\(29\) 9.31666 1.73006 0.865030 0.501720i \(-0.167299\pi\)
0.865030 + 0.501720i \(0.167299\pi\)
\(30\) 0 0
\(31\) 4.91650 0.883029 0.441514 0.897254i \(-0.354441\pi\)
0.441514 + 0.897254i \(0.354441\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.48855 0.255285
\(35\) −7.43109 −1.25608
\(36\) 0 0
\(37\) −1.32731 −0.218209 −0.109105 0.994030i \(-0.534798\pi\)
−0.109105 + 0.994030i \(0.534798\pi\)
\(38\) −7.62957 −1.23768
\(39\) 0 0
\(40\) −1.53522 −0.242739
\(41\) 3.69856 0.577619 0.288809 0.957387i \(-0.406741\pi\)
0.288809 + 0.957387i \(0.406741\pi\)
\(42\) 0 0
\(43\) 8.73841 1.33260 0.666298 0.745686i \(-0.267878\pi\)
0.666298 + 0.745686i \(0.267878\pi\)
\(44\) −3.93574 −0.593335
\(45\) 0 0
\(46\) −5.95631 −0.878210
\(47\) 6.61250 0.964532 0.482266 0.876025i \(-0.339814\pi\)
0.482266 + 0.876025i \(0.339814\pi\)
\(48\) 0 0
\(49\) 16.4296 2.34708
\(50\) −2.64310 −0.373791
\(51\) 0 0
\(52\) 3.57898 0.496315
\(53\) 4.60201 0.632134 0.316067 0.948737i \(-0.397638\pi\)
0.316067 + 0.948737i \(0.397638\pi\)
\(54\) 0 0
\(55\) 6.04222 0.814733
\(56\) 4.84041 0.646827
\(57\) 0 0
\(58\) 9.31666 1.22334
\(59\) 1.97402 0.256996 0.128498 0.991710i \(-0.458984\pi\)
0.128498 + 0.991710i \(0.458984\pi\)
\(60\) 0 0
\(61\) −5.88075 −0.752953 −0.376476 0.926426i \(-0.622864\pi\)
−0.376476 + 0.926426i \(0.622864\pi\)
\(62\) 4.91650 0.624396
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.49452 −0.681511
\(66\) 0 0
\(67\) 16.0194 1.95708 0.978539 0.206060i \(-0.0660643\pi\)
0.978539 + 0.206060i \(0.0660643\pi\)
\(68\) 1.48855 0.180514
\(69\) 0 0
\(70\) −7.43109 −0.888185
\(71\) −5.26569 −0.624922 −0.312461 0.949931i \(-0.601153\pi\)
−0.312461 + 0.949931i \(0.601153\pi\)
\(72\) 0 0
\(73\) −3.32646 −0.389333 −0.194667 0.980869i \(-0.562362\pi\)
−0.194667 + 0.980869i \(0.562362\pi\)
\(74\) −1.32731 −0.154297
\(75\) 0 0
\(76\) −7.62957 −0.875172
\(77\) −19.0506 −2.17102
\(78\) 0 0
\(79\) 3.87300 0.435747 0.217873 0.975977i \(-0.430088\pi\)
0.217873 + 0.975977i \(0.430088\pi\)
\(80\) −1.53522 −0.171643
\(81\) 0 0
\(82\) 3.69856 0.408438
\(83\) −11.0519 −1.21311 −0.606554 0.795042i \(-0.707449\pi\)
−0.606554 + 0.795042i \(0.707449\pi\)
\(84\) 0 0
\(85\) −2.28526 −0.247871
\(86\) 8.73841 0.942287
\(87\) 0 0
\(88\) −3.93574 −0.419551
\(89\) −12.0548 −1.27780 −0.638901 0.769289i \(-0.720611\pi\)
−0.638901 + 0.769289i \(0.720611\pi\)
\(90\) 0 0
\(91\) 17.3237 1.81602
\(92\) −5.95631 −0.620989
\(93\) 0 0
\(94\) 6.61250 0.682027
\(95\) 11.7131 1.20174
\(96\) 0 0
\(97\) 17.1315 1.73944 0.869721 0.493543i \(-0.164299\pi\)
0.869721 + 0.493543i \(0.164299\pi\)
\(98\) 16.4296 1.65964
\(99\) 0 0
\(100\) −2.64310 −0.264310
\(101\) 17.0976 1.70128 0.850640 0.525749i \(-0.176215\pi\)
0.850640 + 0.525749i \(0.176215\pi\)
\(102\) 0 0
\(103\) 11.0020 1.08406 0.542029 0.840360i \(-0.317656\pi\)
0.542029 + 0.840360i \(0.317656\pi\)
\(104\) 3.57898 0.350948
\(105\) 0 0
\(106\) 4.60201 0.446986
\(107\) 16.6560 1.61020 0.805098 0.593142i \(-0.202113\pi\)
0.805098 + 0.593142i \(0.202113\pi\)
\(108\) 0 0
\(109\) 11.7121 1.12182 0.560908 0.827878i \(-0.310452\pi\)
0.560908 + 0.827878i \(0.310452\pi\)
\(110\) 6.04222 0.576103
\(111\) 0 0
\(112\) 4.84041 0.457376
\(113\) −18.4572 −1.73631 −0.868154 0.496296i \(-0.834693\pi\)
−0.868154 + 0.496296i \(0.834693\pi\)
\(114\) 0 0
\(115\) 9.14424 0.852705
\(116\) 9.31666 0.865030
\(117\) 0 0
\(118\) 1.97402 0.181724
\(119\) 7.20521 0.660501
\(120\) 0 0
\(121\) 4.49004 0.408185
\(122\) −5.88075 −0.532418
\(123\) 0 0
\(124\) 4.91650 0.441514
\(125\) 11.7338 1.04951
\(126\) 0 0
\(127\) 9.99360 0.886788 0.443394 0.896327i \(-0.353774\pi\)
0.443394 + 0.896327i \(0.353774\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −5.49452 −0.481901
\(131\) −5.53861 −0.483911 −0.241955 0.970287i \(-0.577789\pi\)
−0.241955 + 0.970287i \(0.577789\pi\)
\(132\) 0 0
\(133\) −36.9303 −3.20226
\(134\) 16.0194 1.38386
\(135\) 0 0
\(136\) 1.48855 0.127642
\(137\) −15.2920 −1.30648 −0.653240 0.757151i \(-0.726591\pi\)
−0.653240 + 0.757151i \(0.726591\pi\)
\(138\) 0 0
\(139\) 11.9007 1.00940 0.504701 0.863294i \(-0.331603\pi\)
0.504701 + 0.863294i \(0.331603\pi\)
\(140\) −7.43109 −0.628042
\(141\) 0 0
\(142\) −5.26569 −0.441887
\(143\) −14.0859 −1.17792
\(144\) 0 0
\(145\) −14.3031 −1.18781
\(146\) −3.32646 −0.275300
\(147\) 0 0
\(148\) −1.32731 −0.109105
\(149\) 1.00000 0.0819232
\(150\) 0 0
\(151\) 11.5282 0.938153 0.469077 0.883157i \(-0.344587\pi\)
0.469077 + 0.883157i \(0.344587\pi\)
\(152\) −7.62957 −0.618840
\(153\) 0 0
\(154\) −19.0506 −1.53514
\(155\) −7.54790 −0.606262
\(156\) 0 0
\(157\) −10.4495 −0.833959 −0.416979 0.908916i \(-0.636911\pi\)
−0.416979 + 0.908916i \(0.636911\pi\)
\(158\) 3.87300 0.308120
\(159\) 0 0
\(160\) −1.53522 −0.121370
\(161\) −28.8310 −2.27220
\(162\) 0 0
\(163\) −4.36679 −0.342033 −0.171017 0.985268i \(-0.554705\pi\)
−0.171017 + 0.985268i \(0.554705\pi\)
\(164\) 3.69856 0.288809
\(165\) 0 0
\(166\) −11.0519 −0.857797
\(167\) −10.7819 −0.834332 −0.417166 0.908830i \(-0.636977\pi\)
−0.417166 + 0.908830i \(0.636977\pi\)
\(168\) 0 0
\(169\) −0.190909 −0.0146853
\(170\) −2.28526 −0.175271
\(171\) 0 0
\(172\) 8.73841 0.666298
\(173\) −7.68331 −0.584151 −0.292075 0.956395i \(-0.594346\pi\)
−0.292075 + 0.956395i \(0.594346\pi\)
\(174\) 0 0
\(175\) −12.7937 −0.967113
\(176\) −3.93574 −0.296667
\(177\) 0 0
\(178\) −12.0548 −0.903542
\(179\) 4.84781 0.362342 0.181171 0.983452i \(-0.442011\pi\)
0.181171 + 0.983452i \(0.442011\pi\)
\(180\) 0 0
\(181\) −7.77213 −0.577698 −0.288849 0.957375i \(-0.593273\pi\)
−0.288849 + 0.957375i \(0.593273\pi\)
\(182\) 17.3237 1.28412
\(183\) 0 0
\(184\) −5.95631 −0.439105
\(185\) 2.03772 0.149816
\(186\) 0 0
\(187\) −5.85856 −0.428420
\(188\) 6.61250 0.482266
\(189\) 0 0
\(190\) 11.7131 0.849755
\(191\) 10.0668 0.728411 0.364206 0.931319i \(-0.381340\pi\)
0.364206 + 0.931319i \(0.381340\pi\)
\(192\) 0 0
\(193\) 1.13138 0.0814387 0.0407194 0.999171i \(-0.487035\pi\)
0.0407194 + 0.999171i \(0.487035\pi\)
\(194\) 17.1315 1.22997
\(195\) 0 0
\(196\) 16.4296 1.17354
\(197\) 15.2482 1.08639 0.543195 0.839607i \(-0.317215\pi\)
0.543195 + 0.839607i \(0.317215\pi\)
\(198\) 0 0
\(199\) −18.4763 −1.30975 −0.654875 0.755737i \(-0.727279\pi\)
−0.654875 + 0.755737i \(0.727279\pi\)
\(200\) −2.64310 −0.186896
\(201\) 0 0
\(202\) 17.0976 1.20299
\(203\) 45.0965 3.16515
\(204\) 0 0
\(205\) −5.67810 −0.396576
\(206\) 11.0020 0.766545
\(207\) 0 0
\(208\) 3.57898 0.248158
\(209\) 30.0280 2.07708
\(210\) 0 0
\(211\) 18.2464 1.25614 0.628068 0.778158i \(-0.283846\pi\)
0.628068 + 0.778158i \(0.283846\pi\)
\(212\) 4.60201 0.316067
\(213\) 0 0
\(214\) 16.6560 1.13858
\(215\) −13.4154 −0.914921
\(216\) 0 0
\(217\) 23.7979 1.61550
\(218\) 11.7121 0.793244
\(219\) 0 0
\(220\) 6.04222 0.407366
\(221\) 5.32750 0.358367
\(222\) 0 0
\(223\) 2.90348 0.194432 0.0972158 0.995263i \(-0.469006\pi\)
0.0972158 + 0.995263i \(0.469006\pi\)
\(224\) 4.84041 0.323414
\(225\) 0 0
\(226\) −18.4572 −1.22775
\(227\) −7.73421 −0.513338 −0.256669 0.966499i \(-0.582625\pi\)
−0.256669 + 0.966499i \(0.582625\pi\)
\(228\) 0 0
\(229\) −15.4182 −1.01886 −0.509430 0.860512i \(-0.670144\pi\)
−0.509430 + 0.860512i \(0.670144\pi\)
\(230\) 9.14424 0.602954
\(231\) 0 0
\(232\) 9.31666 0.611669
\(233\) −13.8473 −0.907167 −0.453584 0.891214i \(-0.649855\pi\)
−0.453584 + 0.891214i \(0.649855\pi\)
\(234\) 0 0
\(235\) −10.1516 −0.662220
\(236\) 1.97402 0.128498
\(237\) 0 0
\(238\) 7.20521 0.467044
\(239\) −16.1000 −1.04142 −0.520710 0.853733i \(-0.674333\pi\)
−0.520710 + 0.853733i \(0.674333\pi\)
\(240\) 0 0
\(241\) −14.1158 −0.909282 −0.454641 0.890675i \(-0.650232\pi\)
−0.454641 + 0.890675i \(0.650232\pi\)
\(242\) 4.49004 0.288630
\(243\) 0 0
\(244\) −5.88075 −0.376476
\(245\) −25.2230 −1.61144
\(246\) 0 0
\(247\) −27.3061 −1.73744
\(248\) 4.91650 0.312198
\(249\) 0 0
\(250\) 11.7338 0.742113
\(251\) 12.6088 0.795860 0.397930 0.917416i \(-0.369729\pi\)
0.397930 + 0.917416i \(0.369729\pi\)
\(252\) 0 0
\(253\) 23.4425 1.47382
\(254\) 9.99360 0.627054
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 19.8020 1.23522 0.617609 0.786486i \(-0.288102\pi\)
0.617609 + 0.786486i \(0.288102\pi\)
\(258\) 0 0
\(259\) −6.42475 −0.399214
\(260\) −5.49452 −0.340755
\(261\) 0 0
\(262\) −5.53861 −0.342177
\(263\) −29.6860 −1.83052 −0.915259 0.402866i \(-0.868014\pi\)
−0.915259 + 0.402866i \(0.868014\pi\)
\(264\) 0 0
\(265\) −7.06509 −0.434005
\(266\) −36.9303 −2.26434
\(267\) 0 0
\(268\) 16.0194 0.978539
\(269\) 11.7390 0.715739 0.357870 0.933772i \(-0.383503\pi\)
0.357870 + 0.933772i \(0.383503\pi\)
\(270\) 0 0
\(271\) 6.34546 0.385459 0.192730 0.981252i \(-0.438266\pi\)
0.192730 + 0.981252i \(0.438266\pi\)
\(272\) 1.48855 0.0902568
\(273\) 0 0
\(274\) −15.2920 −0.923820
\(275\) 10.4026 0.627298
\(276\) 0 0
\(277\) 1.59674 0.0959391 0.0479695 0.998849i \(-0.484725\pi\)
0.0479695 + 0.998849i \(0.484725\pi\)
\(278\) 11.9007 0.713755
\(279\) 0 0
\(280\) −7.43109 −0.444093
\(281\) 9.05403 0.540118 0.270059 0.962844i \(-0.412957\pi\)
0.270059 + 0.962844i \(0.412957\pi\)
\(282\) 0 0
\(283\) 27.8974 1.65833 0.829163 0.559006i \(-0.188817\pi\)
0.829163 + 0.559006i \(0.188817\pi\)
\(284\) −5.26569 −0.312461
\(285\) 0 0
\(286\) −14.0859 −0.832918
\(287\) 17.9026 1.05676
\(288\) 0 0
\(289\) −14.7842 −0.869659
\(290\) −14.3031 −0.839908
\(291\) 0 0
\(292\) −3.32646 −0.194667
\(293\) −18.1337 −1.05938 −0.529690 0.848191i \(-0.677692\pi\)
−0.529690 + 0.848191i \(0.677692\pi\)
\(294\) 0 0
\(295\) −3.03056 −0.176446
\(296\) −1.32731 −0.0771486
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) −21.3175 −1.23282
\(300\) 0 0
\(301\) 42.2975 2.43799
\(302\) 11.5282 0.663375
\(303\) 0 0
\(304\) −7.62957 −0.437586
\(305\) 9.02824 0.516955
\(306\) 0 0
\(307\) 25.6884 1.46611 0.733057 0.680167i \(-0.238093\pi\)
0.733057 + 0.680167i \(0.238093\pi\)
\(308\) −19.0506 −1.08551
\(309\) 0 0
\(310\) −7.54790 −0.428692
\(311\) 27.3395 1.55028 0.775141 0.631789i \(-0.217679\pi\)
0.775141 + 0.631789i \(0.217679\pi\)
\(312\) 0 0
\(313\) 22.7702 1.28705 0.643524 0.765426i \(-0.277472\pi\)
0.643524 + 0.765426i \(0.277472\pi\)
\(314\) −10.4495 −0.589698
\(315\) 0 0
\(316\) 3.87300 0.217873
\(317\) −3.03195 −0.170291 −0.0851457 0.996369i \(-0.527136\pi\)
−0.0851457 + 0.996369i \(0.527136\pi\)
\(318\) 0 0
\(319\) −36.6679 −2.05301
\(320\) −1.53522 −0.0858213
\(321\) 0 0
\(322\) −28.8310 −1.60669
\(323\) −11.3570 −0.631922
\(324\) 0 0
\(325\) −9.45961 −0.524725
\(326\) −4.36679 −0.241854
\(327\) 0 0
\(328\) 3.69856 0.204219
\(329\) 32.0072 1.76462
\(330\) 0 0
\(331\) −16.5828 −0.911472 −0.455736 0.890115i \(-0.650624\pi\)
−0.455736 + 0.890115i \(0.650624\pi\)
\(332\) −11.0519 −0.606554
\(333\) 0 0
\(334\) −10.7819 −0.589962
\(335\) −24.5933 −1.34367
\(336\) 0 0
\(337\) −26.3641 −1.43614 −0.718072 0.695969i \(-0.754975\pi\)
−0.718072 + 0.695969i \(0.754975\pi\)
\(338\) −0.190909 −0.0103841
\(339\) 0 0
\(340\) −2.28526 −0.123935
\(341\) −19.3500 −1.04786
\(342\) 0 0
\(343\) 45.6430 2.46449
\(344\) 8.73841 0.471144
\(345\) 0 0
\(346\) −7.68331 −0.413057
\(347\) 31.0644 1.66763 0.833813 0.552047i \(-0.186153\pi\)
0.833813 + 0.552047i \(0.186153\pi\)
\(348\) 0 0
\(349\) −0.508899 −0.0272407 −0.0136204 0.999907i \(-0.504336\pi\)
−0.0136204 + 0.999907i \(0.504336\pi\)
\(350\) −12.7937 −0.683852
\(351\) 0 0
\(352\) −3.93574 −0.209776
\(353\) 2.14478 0.114155 0.0570775 0.998370i \(-0.481822\pi\)
0.0570775 + 0.998370i \(0.481822\pi\)
\(354\) 0 0
\(355\) 8.08398 0.429053
\(356\) −12.0548 −0.638901
\(357\) 0 0
\(358\) 4.84781 0.256215
\(359\) −16.8255 −0.888013 −0.444007 0.896023i \(-0.646443\pi\)
−0.444007 + 0.896023i \(0.646443\pi\)
\(360\) 0 0
\(361\) 39.2104 2.06370
\(362\) −7.77213 −0.408494
\(363\) 0 0
\(364\) 17.3237 0.908010
\(365\) 5.10685 0.267305
\(366\) 0 0
\(367\) −20.1270 −1.05062 −0.525311 0.850910i \(-0.676051\pi\)
−0.525311 + 0.850910i \(0.676051\pi\)
\(368\) −5.95631 −0.310494
\(369\) 0 0
\(370\) 2.03772 0.105936
\(371\) 22.2756 1.15649
\(372\) 0 0
\(373\) 30.5674 1.58272 0.791359 0.611352i \(-0.209374\pi\)
0.791359 + 0.611352i \(0.209374\pi\)
\(374\) −5.85856 −0.302939
\(375\) 0 0
\(376\) 6.61250 0.341014
\(377\) 33.3441 1.71731
\(378\) 0 0
\(379\) −10.5965 −0.544308 −0.272154 0.962254i \(-0.587736\pi\)
−0.272154 + 0.962254i \(0.587736\pi\)
\(380\) 11.7131 0.600868
\(381\) 0 0
\(382\) 10.0668 0.515065
\(383\) 32.0554 1.63795 0.818976 0.573827i \(-0.194542\pi\)
0.818976 + 0.573827i \(0.194542\pi\)
\(384\) 0 0
\(385\) 29.2468 1.49056
\(386\) 1.13138 0.0575859
\(387\) 0 0
\(388\) 17.1315 0.869721
\(389\) 12.0666 0.611801 0.305900 0.952064i \(-0.401043\pi\)
0.305900 + 0.952064i \(0.401043\pi\)
\(390\) 0 0
\(391\) −8.86629 −0.448388
\(392\) 16.4296 0.829819
\(393\) 0 0
\(394\) 15.2482 0.768193
\(395\) −5.94591 −0.299171
\(396\) 0 0
\(397\) 31.4939 1.58063 0.790316 0.612700i \(-0.209917\pi\)
0.790316 + 0.612700i \(0.209917\pi\)
\(398\) −18.4763 −0.926133
\(399\) 0 0
\(400\) −2.64310 −0.132155
\(401\) −23.3772 −1.16740 −0.583701 0.811969i \(-0.698396\pi\)
−0.583701 + 0.811969i \(0.698396\pi\)
\(402\) 0 0
\(403\) 17.5960 0.876521
\(404\) 17.0976 0.850640
\(405\) 0 0
\(406\) 45.0965 2.23810
\(407\) 5.22396 0.258942
\(408\) 0 0
\(409\) −29.6283 −1.46502 −0.732512 0.680754i \(-0.761652\pi\)
−0.732512 + 0.680754i \(0.761652\pi\)
\(410\) −5.67810 −0.280422
\(411\) 0 0
\(412\) 11.0020 0.542029
\(413\) 9.55509 0.470175
\(414\) 0 0
\(415\) 16.9672 0.832885
\(416\) 3.57898 0.175474
\(417\) 0 0
\(418\) 30.0280 1.46872
\(419\) 34.2472 1.67309 0.836543 0.547901i \(-0.184573\pi\)
0.836543 + 0.547901i \(0.184573\pi\)
\(420\) 0 0
\(421\) −27.5967 −1.34498 −0.672490 0.740106i \(-0.734775\pi\)
−0.672490 + 0.740106i \(0.734775\pi\)
\(422\) 18.2464 0.888223
\(423\) 0 0
\(424\) 4.60201 0.223493
\(425\) −3.93440 −0.190846
\(426\) 0 0
\(427\) −28.4652 −1.37753
\(428\) 16.6560 0.805098
\(429\) 0 0
\(430\) −13.4154 −0.646947
\(431\) 7.48689 0.360631 0.180315 0.983609i \(-0.442288\pi\)
0.180315 + 0.983609i \(0.442288\pi\)
\(432\) 0 0
\(433\) −15.5640 −0.747959 −0.373979 0.927437i \(-0.622007\pi\)
−0.373979 + 0.927437i \(0.622007\pi\)
\(434\) 23.7979 1.14233
\(435\) 0 0
\(436\) 11.7121 0.560908
\(437\) 45.4441 2.17389
\(438\) 0 0
\(439\) −22.9930 −1.09739 −0.548697 0.836021i \(-0.684876\pi\)
−0.548697 + 0.836021i \(0.684876\pi\)
\(440\) 6.04222 0.288052
\(441\) 0 0
\(442\) 5.32750 0.253403
\(443\) −12.5727 −0.597346 −0.298673 0.954356i \(-0.596544\pi\)
−0.298673 + 0.954356i \(0.596544\pi\)
\(444\) 0 0
\(445\) 18.5067 0.877301
\(446\) 2.90348 0.137484
\(447\) 0 0
\(448\) 4.84041 0.228688
\(449\) 0.0886056 0.00418156 0.00209078 0.999998i \(-0.499334\pi\)
0.00209078 + 0.999998i \(0.499334\pi\)
\(450\) 0 0
\(451\) −14.5566 −0.685442
\(452\) −18.4572 −0.868154
\(453\) 0 0
\(454\) −7.73421 −0.362985
\(455\) −26.5957 −1.24683
\(456\) 0 0
\(457\) 3.92142 0.183436 0.0917181 0.995785i \(-0.470764\pi\)
0.0917181 + 0.995785i \(0.470764\pi\)
\(458\) −15.4182 −0.720443
\(459\) 0 0
\(460\) 9.14424 0.426353
\(461\) −34.4851 −1.60613 −0.803065 0.595892i \(-0.796799\pi\)
−0.803065 + 0.595892i \(0.796799\pi\)
\(462\) 0 0
\(463\) −16.1591 −0.750978 −0.375489 0.926827i \(-0.622525\pi\)
−0.375489 + 0.926827i \(0.622525\pi\)
\(464\) 9.31666 0.432515
\(465\) 0 0
\(466\) −13.8473 −0.641464
\(467\) 4.05816 0.187789 0.0938946 0.995582i \(-0.470068\pi\)
0.0938946 + 0.995582i \(0.470068\pi\)
\(468\) 0 0
\(469\) 77.5404 3.58048
\(470\) −10.1516 −0.468260
\(471\) 0 0
\(472\) 1.97402 0.0908618
\(473\) −34.3921 −1.58135
\(474\) 0 0
\(475\) 20.1657 0.925268
\(476\) 7.20521 0.330250
\(477\) 0 0
\(478\) −16.1000 −0.736396
\(479\) −7.88158 −0.360119 −0.180059 0.983656i \(-0.557629\pi\)
−0.180059 + 0.983656i \(0.557629\pi\)
\(480\) 0 0
\(481\) −4.75043 −0.216601
\(482\) −14.1158 −0.642959
\(483\) 0 0
\(484\) 4.49004 0.204093
\(485\) −26.3006 −1.19425
\(486\) 0 0
\(487\) 18.0202 0.816572 0.408286 0.912854i \(-0.366127\pi\)
0.408286 + 0.912854i \(0.366127\pi\)
\(488\) −5.88075 −0.266209
\(489\) 0 0
\(490\) −25.2230 −1.13946
\(491\) 25.3709 1.14497 0.572487 0.819914i \(-0.305979\pi\)
0.572487 + 0.819914i \(0.305979\pi\)
\(492\) 0 0
\(493\) 13.8683 0.624599
\(494\) −27.3061 −1.22856
\(495\) 0 0
\(496\) 4.91650 0.220757
\(497\) −25.4881 −1.14330
\(498\) 0 0
\(499\) −25.8660 −1.15792 −0.578961 0.815355i \(-0.696542\pi\)
−0.578961 + 0.815355i \(0.696542\pi\)
\(500\) 11.7338 0.524753
\(501\) 0 0
\(502\) 12.6088 0.562758
\(503\) −10.0085 −0.446256 −0.223128 0.974789i \(-0.571627\pi\)
−0.223128 + 0.974789i \(0.571627\pi\)
\(504\) 0 0
\(505\) −26.2486 −1.16805
\(506\) 23.4425 1.04215
\(507\) 0 0
\(508\) 9.99360 0.443394
\(509\) −15.4778 −0.686041 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(510\) 0 0
\(511\) −16.1014 −0.712286
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 19.8020 0.873431
\(515\) −16.8905 −0.744282
\(516\) 0 0
\(517\) −26.0251 −1.14458
\(518\) −6.42475 −0.282287
\(519\) 0 0
\(520\) −5.49452 −0.240950
\(521\) 40.3112 1.76607 0.883033 0.469310i \(-0.155497\pi\)
0.883033 + 0.469310i \(0.155497\pi\)
\(522\) 0 0
\(523\) −7.02216 −0.307058 −0.153529 0.988144i \(-0.549064\pi\)
−0.153529 + 0.988144i \(0.549064\pi\)
\(524\) −5.53861 −0.241955
\(525\) 0 0
\(526\) −29.6860 −1.29437
\(527\) 7.31847 0.318798
\(528\) 0 0
\(529\) 12.4777 0.542507
\(530\) −7.06509 −0.306888
\(531\) 0 0
\(532\) −36.9303 −1.60113
\(533\) 13.2371 0.573362
\(534\) 0 0
\(535\) −25.5706 −1.10551
\(536\) 16.0194 0.691932
\(537\) 0 0
\(538\) 11.7390 0.506104
\(539\) −64.6625 −2.78521
\(540\) 0 0
\(541\) 26.2296 1.12770 0.563849 0.825878i \(-0.309320\pi\)
0.563849 + 0.825878i \(0.309320\pi\)
\(542\) 6.34546 0.272561
\(543\) 0 0
\(544\) 1.48855 0.0638212
\(545\) −17.9806 −0.770206
\(546\) 0 0
\(547\) 24.4898 1.04711 0.523554 0.851993i \(-0.324606\pi\)
0.523554 + 0.851993i \(0.324606\pi\)
\(548\) −15.2920 −0.653240
\(549\) 0 0
\(550\) 10.4026 0.443567
\(551\) −71.0821 −3.02820
\(552\) 0 0
\(553\) 18.7469 0.797200
\(554\) 1.59674 0.0678392
\(555\) 0 0
\(556\) 11.9007 0.504701
\(557\) 23.6287 1.00118 0.500590 0.865684i \(-0.333116\pi\)
0.500590 + 0.865684i \(0.333116\pi\)
\(558\) 0 0
\(559\) 31.2746 1.32277
\(560\) −7.43109 −0.314021
\(561\) 0 0
\(562\) 9.05403 0.381921
\(563\) 35.6583 1.50282 0.751409 0.659837i \(-0.229375\pi\)
0.751409 + 0.659837i \(0.229375\pi\)
\(564\) 0 0
\(565\) 28.3358 1.19210
\(566\) 27.8974 1.17261
\(567\) 0 0
\(568\) −5.26569 −0.220943
\(569\) 4.39468 0.184234 0.0921172 0.995748i \(-0.470637\pi\)
0.0921172 + 0.995748i \(0.470637\pi\)
\(570\) 0 0
\(571\) −31.2049 −1.30588 −0.652942 0.757408i \(-0.726465\pi\)
−0.652942 + 0.757408i \(0.726465\pi\)
\(572\) −14.0859 −0.588962
\(573\) 0 0
\(574\) 17.9026 0.747239
\(575\) 15.7431 0.656535
\(576\) 0 0
\(577\) −21.0890 −0.877948 −0.438974 0.898500i \(-0.644658\pi\)
−0.438974 + 0.898500i \(0.644658\pi\)
\(578\) −14.7842 −0.614942
\(579\) 0 0
\(580\) −14.3031 −0.593904
\(581\) −53.4960 −2.21939
\(582\) 0 0
\(583\) −18.1123 −0.750135
\(584\) −3.32646 −0.137650
\(585\) 0 0
\(586\) −18.1337 −0.749095
\(587\) −5.57862 −0.230254 −0.115127 0.993351i \(-0.536728\pi\)
−0.115127 + 0.993351i \(0.536728\pi\)
\(588\) 0 0
\(589\) −37.5108 −1.54560
\(590\) −3.03056 −0.124766
\(591\) 0 0
\(592\) −1.32731 −0.0545523
\(593\) 30.1775 1.23924 0.619621 0.784901i \(-0.287286\pi\)
0.619621 + 0.784901i \(0.287286\pi\)
\(594\) 0 0
\(595\) −11.0616 −0.453480
\(596\) 1.00000 0.0409616
\(597\) 0 0
\(598\) −21.3175 −0.871738
\(599\) −20.3036 −0.829582 −0.414791 0.909917i \(-0.636145\pi\)
−0.414791 + 0.909917i \(0.636145\pi\)
\(600\) 0 0
\(601\) −19.3118 −0.787747 −0.393873 0.919165i \(-0.628865\pi\)
−0.393873 + 0.919165i \(0.628865\pi\)
\(602\) 42.2975 1.72392
\(603\) 0 0
\(604\) 11.5282 0.469077
\(605\) −6.89319 −0.280248
\(606\) 0 0
\(607\) 11.3632 0.461220 0.230610 0.973046i \(-0.425928\pi\)
0.230610 + 0.973046i \(0.425928\pi\)
\(608\) −7.62957 −0.309420
\(609\) 0 0
\(610\) 9.02824 0.365543
\(611\) 23.6660 0.957424
\(612\) 0 0
\(613\) −36.8732 −1.48930 −0.744648 0.667458i \(-0.767383\pi\)
−0.744648 + 0.667458i \(0.767383\pi\)
\(614\) 25.6884 1.03670
\(615\) 0 0
\(616\) −19.0506 −0.767570
\(617\) 29.6642 1.19424 0.597119 0.802153i \(-0.296312\pi\)
0.597119 + 0.802153i \(0.296312\pi\)
\(618\) 0 0
\(619\) 12.6706 0.509273 0.254636 0.967037i \(-0.418044\pi\)
0.254636 + 0.967037i \(0.418044\pi\)
\(620\) −7.54790 −0.303131
\(621\) 0 0
\(622\) 27.3395 1.09621
\(623\) −58.3500 −2.33774
\(624\) 0 0
\(625\) −4.79849 −0.191940
\(626\) 22.7702 0.910080
\(627\) 0 0
\(628\) −10.4495 −0.416979
\(629\) −1.97578 −0.0787795
\(630\) 0 0
\(631\) 7.79223 0.310204 0.155102 0.987898i \(-0.450429\pi\)
0.155102 + 0.987898i \(0.450429\pi\)
\(632\) 3.87300 0.154060
\(633\) 0 0
\(634\) −3.03195 −0.120414
\(635\) −15.3424 −0.608843
\(636\) 0 0
\(637\) 58.8011 2.32978
\(638\) −36.6679 −1.45170
\(639\) 0 0
\(640\) −1.53522 −0.0606849
\(641\) −11.8471 −0.467933 −0.233966 0.972245i \(-0.575171\pi\)
−0.233966 + 0.972245i \(0.575171\pi\)
\(642\) 0 0
\(643\) −17.4248 −0.687168 −0.343584 0.939122i \(-0.611641\pi\)
−0.343584 + 0.939122i \(0.611641\pi\)
\(644\) −28.8310 −1.13610
\(645\) 0 0
\(646\) −11.3570 −0.446836
\(647\) −21.0722 −0.828433 −0.414216 0.910178i \(-0.635944\pi\)
−0.414216 + 0.910178i \(0.635944\pi\)
\(648\) 0 0
\(649\) −7.76924 −0.304969
\(650\) −9.45961 −0.371036
\(651\) 0 0
\(652\) −4.36679 −0.171017
\(653\) −5.61278 −0.219645 −0.109823 0.993951i \(-0.535028\pi\)
−0.109823 + 0.993951i \(0.535028\pi\)
\(654\) 0 0
\(655\) 8.50298 0.332239
\(656\) 3.69856 0.144405
\(657\) 0 0
\(658\) 32.0072 1.24777
\(659\) −34.1742 −1.33124 −0.665618 0.746293i \(-0.731832\pi\)
−0.665618 + 0.746293i \(0.731832\pi\)
\(660\) 0 0
\(661\) 19.1521 0.744930 0.372465 0.928046i \(-0.378513\pi\)
0.372465 + 0.928046i \(0.378513\pi\)
\(662\) −16.5828 −0.644508
\(663\) 0 0
\(664\) −11.0519 −0.428899
\(665\) 56.6960 2.19858
\(666\) 0 0
\(667\) −55.4929 −2.14870
\(668\) −10.7819 −0.417166
\(669\) 0 0
\(670\) −24.5933 −0.950120
\(671\) 23.1451 0.893507
\(672\) 0 0
\(673\) 40.2834 1.55281 0.776405 0.630235i \(-0.217041\pi\)
0.776405 + 0.630235i \(0.217041\pi\)
\(674\) −26.3641 −1.01551
\(675\) 0 0
\(676\) −0.190909 −0.00734266
\(677\) −18.5612 −0.713365 −0.356682 0.934226i \(-0.616092\pi\)
−0.356682 + 0.934226i \(0.616092\pi\)
\(678\) 0 0
\(679\) 82.9236 3.18232
\(680\) −2.28526 −0.0876356
\(681\) 0 0
\(682\) −19.3500 −0.740951
\(683\) 3.26406 0.124896 0.0624478 0.998048i \(-0.480109\pi\)
0.0624478 + 0.998048i \(0.480109\pi\)
\(684\) 0 0
\(685\) 23.4765 0.896991
\(686\) 45.6430 1.74266
\(687\) 0 0
\(688\) 8.73841 0.333149
\(689\) 16.4705 0.627476
\(690\) 0 0
\(691\) −17.6155 −0.670124 −0.335062 0.942196i \(-0.608757\pi\)
−0.335062 + 0.942196i \(0.608757\pi\)
\(692\) −7.68331 −0.292075
\(693\) 0 0
\(694\) 31.0644 1.17919
\(695\) −18.2701 −0.693025
\(696\) 0 0
\(697\) 5.50551 0.208536
\(698\) −0.508899 −0.0192621
\(699\) 0 0
\(700\) −12.7937 −0.483557
\(701\) −8.52564 −0.322009 −0.161005 0.986954i \(-0.551473\pi\)
−0.161005 + 0.986954i \(0.551473\pi\)
\(702\) 0 0
\(703\) 10.1268 0.381941
\(704\) −3.93574 −0.148334
\(705\) 0 0
\(706\) 2.14478 0.0807198
\(707\) 82.7596 3.11250
\(708\) 0 0
\(709\) −15.4327 −0.579589 −0.289794 0.957089i \(-0.593587\pi\)
−0.289794 + 0.957089i \(0.593587\pi\)
\(710\) 8.08398 0.303386
\(711\) 0 0
\(712\) −12.0548 −0.451771
\(713\) −29.2842 −1.09670
\(714\) 0 0
\(715\) 21.6250 0.808728
\(716\) 4.84781 0.181171
\(717\) 0 0
\(718\) −16.8255 −0.627920
\(719\) −22.2668 −0.830412 −0.415206 0.909727i \(-0.636290\pi\)
−0.415206 + 0.909727i \(0.636290\pi\)
\(720\) 0 0
\(721\) 53.2541 1.98329
\(722\) 39.2104 1.45926
\(723\) 0 0
\(724\) −7.77213 −0.288849
\(725\) −24.6249 −0.914546
\(726\) 0 0
\(727\) −15.5429 −0.576453 −0.288226 0.957562i \(-0.593066\pi\)
−0.288226 + 0.957562i \(0.593066\pi\)
\(728\) 17.3237 0.642060
\(729\) 0 0
\(730\) 5.10685 0.189013
\(731\) 13.0076 0.481103
\(732\) 0 0
\(733\) 5.75452 0.212548 0.106274 0.994337i \(-0.466108\pi\)
0.106274 + 0.994337i \(0.466108\pi\)
\(734\) −20.1270 −0.742902
\(735\) 0 0
\(736\) −5.95631 −0.219553
\(737\) −63.0481 −2.32241
\(738\) 0 0
\(739\) 30.1581 1.10938 0.554691 0.832056i \(-0.312836\pi\)
0.554691 + 0.832056i \(0.312836\pi\)
\(740\) 2.03772 0.0749080
\(741\) 0 0
\(742\) 22.2756 0.817763
\(743\) −48.8293 −1.79137 −0.895687 0.444686i \(-0.853315\pi\)
−0.895687 + 0.444686i \(0.853315\pi\)
\(744\) 0 0
\(745\) −1.53522 −0.0562461
\(746\) 30.5674 1.11915
\(747\) 0 0
\(748\) −5.85856 −0.214210
\(749\) 80.6219 2.94586
\(750\) 0 0
\(751\) −50.0249 −1.82543 −0.912717 0.408593i \(-0.866020\pi\)
−0.912717 + 0.408593i \(0.866020\pi\)
\(752\) 6.61250 0.241133
\(753\) 0 0
\(754\) 33.3441 1.21432
\(755\) −17.6983 −0.644109
\(756\) 0 0
\(757\) −17.2400 −0.626600 −0.313300 0.949654i \(-0.601434\pi\)
−0.313300 + 0.949654i \(0.601434\pi\)
\(758\) −10.5965 −0.384884
\(759\) 0 0
\(760\) 11.7131 0.424878
\(761\) −29.2370 −1.05984 −0.529920 0.848048i \(-0.677778\pi\)
−0.529920 + 0.848048i \(0.677778\pi\)
\(762\) 0 0
\(763\) 56.6914 2.05237
\(764\) 10.0668 0.364206
\(765\) 0 0
\(766\) 32.0554 1.15821
\(767\) 7.06499 0.255102
\(768\) 0 0
\(769\) 13.5756 0.489549 0.244775 0.969580i \(-0.421286\pi\)
0.244775 + 0.969580i \(0.421286\pi\)
\(770\) 29.2468 1.05398
\(771\) 0 0
\(772\) 1.13138 0.0407194
\(773\) −38.1780 −1.37317 −0.686583 0.727051i \(-0.740890\pi\)
−0.686583 + 0.727051i \(0.740890\pi\)
\(774\) 0 0
\(775\) −12.9948 −0.466787
\(776\) 17.1315 0.614986
\(777\) 0 0
\(778\) 12.0666 0.432608
\(779\) −28.2185 −1.01103
\(780\) 0 0
\(781\) 20.7244 0.741576
\(782\) −8.86629 −0.317058
\(783\) 0 0
\(784\) 16.4296 0.586771
\(785\) 16.0422 0.572572
\(786\) 0 0
\(787\) −15.6175 −0.556705 −0.278352 0.960479i \(-0.589788\pi\)
−0.278352 + 0.960479i \(0.589788\pi\)
\(788\) 15.2482 0.543195
\(789\) 0 0
\(790\) −5.94591 −0.211546
\(791\) −89.3404 −3.17658
\(792\) 0 0
\(793\) −21.0471 −0.747404
\(794\) 31.4939 1.11768
\(795\) 0 0
\(796\) −18.4763 −0.654875
\(797\) 40.5208 1.43532 0.717660 0.696393i \(-0.245213\pi\)
0.717660 + 0.696393i \(0.245213\pi\)
\(798\) 0 0
\(799\) 9.84306 0.348223
\(800\) −2.64310 −0.0934478
\(801\) 0 0
\(802\) −23.3772 −0.825478
\(803\) 13.0921 0.462010
\(804\) 0 0
\(805\) 44.2619 1.56003
\(806\) 17.5960 0.619794
\(807\) 0 0
\(808\) 17.0976 0.601493
\(809\) 2.58457 0.0908687 0.0454343 0.998967i \(-0.485533\pi\)
0.0454343 + 0.998967i \(0.485533\pi\)
\(810\) 0 0
\(811\) −9.29140 −0.326265 −0.163133 0.986604i \(-0.552160\pi\)
−0.163133 + 0.986604i \(0.552160\pi\)
\(812\) 45.0965 1.58258
\(813\) 0 0
\(814\) 5.22396 0.183100
\(815\) 6.70398 0.234830
\(816\) 0 0
\(817\) −66.6704 −2.33250
\(818\) −29.6283 −1.03593
\(819\) 0 0
\(820\) −5.67810 −0.198288
\(821\) 38.5886 1.34675 0.673376 0.739300i \(-0.264843\pi\)
0.673376 + 0.739300i \(0.264843\pi\)
\(822\) 0 0
\(823\) −8.03255 −0.279997 −0.139999 0.990152i \(-0.544710\pi\)
−0.139999 + 0.990152i \(0.544710\pi\)
\(824\) 11.0020 0.383272
\(825\) 0 0
\(826\) 9.55509 0.332464
\(827\) 6.72217 0.233753 0.116876 0.993146i \(-0.462712\pi\)
0.116876 + 0.993146i \(0.462712\pi\)
\(828\) 0 0
\(829\) −54.1155 −1.87951 −0.939754 0.341852i \(-0.888946\pi\)
−0.939754 + 0.341852i \(0.888946\pi\)
\(830\) 16.9672 0.588939
\(831\) 0 0
\(832\) 3.57898 0.124079
\(833\) 24.4563 0.847361
\(834\) 0 0
\(835\) 16.5526 0.572828
\(836\) 30.0280 1.03854
\(837\) 0 0
\(838\) 34.2472 1.18305
\(839\) 32.5525 1.12384 0.561918 0.827193i \(-0.310064\pi\)
0.561918 + 0.827193i \(0.310064\pi\)
\(840\) 0 0
\(841\) 57.8002 1.99311
\(842\) −27.5967 −0.951044
\(843\) 0 0
\(844\) 18.2464 0.628068
\(845\) 0.293087 0.0100825
\(846\) 0 0
\(847\) 21.7336 0.746776
\(848\) 4.60201 0.158034
\(849\) 0 0
\(850\) −3.93440 −0.134949
\(851\) 7.90590 0.271011
\(852\) 0 0
\(853\) −53.9351 −1.84670 −0.923351 0.383957i \(-0.874561\pi\)
−0.923351 + 0.383957i \(0.874561\pi\)
\(854\) −28.4652 −0.974061
\(855\) 0 0
\(856\) 16.6560 0.569290
\(857\) 11.2590 0.384601 0.192301 0.981336i \(-0.438405\pi\)
0.192301 + 0.981336i \(0.438405\pi\)
\(858\) 0 0
\(859\) 35.1537 1.19943 0.599714 0.800214i \(-0.295281\pi\)
0.599714 + 0.800214i \(0.295281\pi\)
\(860\) −13.4154 −0.457461
\(861\) 0 0
\(862\) 7.48689 0.255004
\(863\) 19.4504 0.662100 0.331050 0.943613i \(-0.392597\pi\)
0.331050 + 0.943613i \(0.392597\pi\)
\(864\) 0 0
\(865\) 11.7956 0.401061
\(866\) −15.5640 −0.528887
\(867\) 0 0
\(868\) 23.7979 0.807752
\(869\) −15.2431 −0.517088
\(870\) 0 0
\(871\) 57.3330 1.94266
\(872\) 11.7121 0.396622
\(873\) 0 0
\(874\) 45.4441 1.53717
\(875\) 56.7966 1.92008
\(876\) 0 0
\(877\) −41.0014 −1.38452 −0.692260 0.721648i \(-0.743385\pi\)
−0.692260 + 0.721648i \(0.743385\pi\)
\(878\) −22.9930 −0.775975
\(879\) 0 0
\(880\) 6.04222 0.203683
\(881\) −18.5960 −0.626516 −0.313258 0.949668i \(-0.601420\pi\)
−0.313258 + 0.949668i \(0.601420\pi\)
\(882\) 0 0
\(883\) 28.4135 0.956191 0.478096 0.878308i \(-0.341327\pi\)
0.478096 + 0.878308i \(0.341327\pi\)
\(884\) 5.32750 0.179183
\(885\) 0 0
\(886\) −12.5727 −0.422387
\(887\) 7.21603 0.242290 0.121145 0.992635i \(-0.461343\pi\)
0.121145 + 0.992635i \(0.461343\pi\)
\(888\) 0 0
\(889\) 48.3731 1.62238
\(890\) 18.5067 0.620346
\(891\) 0 0
\(892\) 2.90348 0.0972158
\(893\) −50.4506 −1.68826
\(894\) 0 0
\(895\) −7.44245 −0.248774
\(896\) 4.84041 0.161707
\(897\) 0 0
\(898\) 0.0886056 0.00295681
\(899\) 45.8053 1.52769
\(900\) 0 0
\(901\) 6.85034 0.228218
\(902\) −14.5566 −0.484681
\(903\) 0 0
\(904\) −18.4572 −0.613877
\(905\) 11.9319 0.396631
\(906\) 0 0
\(907\) 13.2991 0.441590 0.220795 0.975320i \(-0.429135\pi\)
0.220795 + 0.975320i \(0.429135\pi\)
\(908\) −7.73421 −0.256669
\(909\) 0 0
\(910\) −26.5957 −0.881639
\(911\) −21.0959 −0.698939 −0.349470 0.936948i \(-0.613638\pi\)
−0.349470 + 0.936948i \(0.613638\pi\)
\(912\) 0 0
\(913\) 43.4976 1.43956
\(914\) 3.92142 0.129709
\(915\) 0 0
\(916\) −15.4182 −0.509430
\(917\) −26.8092 −0.885316
\(918\) 0 0
\(919\) −40.9407 −1.35051 −0.675255 0.737585i \(-0.735966\pi\)
−0.675255 + 0.737585i \(0.735966\pi\)
\(920\) 9.14424 0.301477
\(921\) 0 0
\(922\) −34.4851 −1.13571
\(923\) −18.8458 −0.620316
\(924\) 0 0
\(925\) 3.50823 0.115350
\(926\) −16.1591 −0.531021
\(927\) 0 0
\(928\) 9.31666 0.305834
\(929\) −22.9216 −0.752032 −0.376016 0.926613i \(-0.622706\pi\)
−0.376016 + 0.926613i \(0.622706\pi\)
\(930\) 0 0
\(931\) −125.351 −4.10820
\(932\) −13.8473 −0.453584
\(933\) 0 0
\(934\) 4.05816 0.132787
\(935\) 8.99417 0.294141
\(936\) 0 0
\(937\) 24.5872 0.803228 0.401614 0.915809i \(-0.368449\pi\)
0.401614 + 0.915809i \(0.368449\pi\)
\(938\) 77.5404 2.53178
\(939\) 0 0
\(940\) −10.1516 −0.331110
\(941\) −38.8014 −1.26489 −0.632445 0.774605i \(-0.717949\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(942\) 0 0
\(943\) −22.0298 −0.717389
\(944\) 1.97402 0.0642490
\(945\) 0 0
\(946\) −34.3921 −1.11818
\(947\) −19.0565 −0.619254 −0.309627 0.950858i \(-0.600204\pi\)
−0.309627 + 0.950858i \(0.600204\pi\)
\(948\) 0 0
\(949\) −11.9053 −0.386464
\(950\) 20.1657 0.654263
\(951\) 0 0
\(952\) 7.20521 0.233522
\(953\) −5.14649 −0.166711 −0.0833557 0.996520i \(-0.526564\pi\)
−0.0833557 + 0.996520i \(0.526564\pi\)
\(954\) 0 0
\(955\) −15.4548 −0.500106
\(956\) −16.1000 −0.520710
\(957\) 0 0
\(958\) −7.88158 −0.254642
\(959\) −74.0193 −2.39021
\(960\) 0 0
\(961\) −6.82806 −0.220260
\(962\) −4.75043 −0.153160
\(963\) 0 0
\(964\) −14.1158 −0.454641
\(965\) −1.73692 −0.0559134
\(966\) 0 0
\(967\) 32.2541 1.03722 0.518610 0.855011i \(-0.326450\pi\)
0.518610 + 0.855011i \(0.326450\pi\)
\(968\) 4.49004 0.144315
\(969\) 0 0
\(970\) −26.3006 −0.844463
\(971\) −10.7708 −0.345650 −0.172825 0.984953i \(-0.555290\pi\)
−0.172825 + 0.984953i \(0.555290\pi\)
\(972\) 0 0
\(973\) 57.6041 1.84670
\(974\) 18.0202 0.577404
\(975\) 0 0
\(976\) −5.88075 −0.188238
\(977\) −28.3306 −0.906378 −0.453189 0.891415i \(-0.649714\pi\)
−0.453189 + 0.891415i \(0.649714\pi\)
\(978\) 0 0
\(979\) 47.4444 1.51633
\(980\) −25.2230 −0.805719
\(981\) 0 0
\(982\) 25.3709 0.809618
\(983\) 52.4704 1.67354 0.836772 0.547551i \(-0.184440\pi\)
0.836772 + 0.547551i \(0.184440\pi\)
\(984\) 0 0
\(985\) −23.4093 −0.745883
\(986\) 13.8683 0.441658
\(987\) 0 0
\(988\) −27.3061 −0.868722
\(989\) −52.0487 −1.65505
\(990\) 0 0
\(991\) 4.99379 0.158633 0.0793165 0.996849i \(-0.474726\pi\)
0.0793165 + 0.996849i \(0.474726\pi\)
\(992\) 4.91650 0.156099
\(993\) 0 0
\(994\) −25.4881 −0.808433
\(995\) 28.3652 0.899236
\(996\) 0 0
\(997\) −9.97773 −0.315998 −0.157999 0.987439i \(-0.550504\pi\)
−0.157999 + 0.987439i \(0.550504\pi\)
\(998\) −25.8660 −0.818775
\(999\) 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 8046.2.a.r.1.5 yes 14
3.2 odd 2 8046.2.a.q.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.q.1.10 14 3.2 odd 2
8046.2.a.r.1.5 yes 14 1.1 even 1 trivial