Properties

Label 85.2.d.a.16.1
Level $85$
Weight $2$
Character 85.16
Analytic conductor $0.679$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,2,Mod(16,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.d (of order \(2\), degree \(1\), 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: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.1
Root \(-0.854638 + 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 85.16
Dual form 85.2.d.a.16.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.17009 q^{2} -0.539189i q^{3} +2.70928 q^{4} +1.00000i q^{5} +1.17009i q^{6} -4.87936i q^{7} -1.53919 q^{8} +2.70928 q^{9} -2.17009i q^{10} -3.17009i q^{11} -1.46081i q^{12} +2.63090 q^{13} +10.5886i q^{14} +0.539189 q^{15} -2.07838 q^{16} +(-3.24846 + 2.53919i) q^{17} -5.87936 q^{18} +1.07838 q^{19} +2.70928i q^{20} -2.63090 q^{21} +6.87936i q^{22} +5.21953i q^{23} +0.829914i q^{24} -1.00000 q^{25} -5.70928 q^{26} -3.07838i q^{27} -13.2195i q^{28} -2.92162i q^{29} -1.17009 q^{30} +4.09171i q^{31} +7.58864 q^{32} -1.70928 q^{33} +(7.04945 - 5.51026i) q^{34} +4.87936 q^{35} +7.34017 q^{36} +5.26180i q^{37} -2.34017 q^{38} -1.41855i q^{39} -1.53919i q^{40} -5.60197i q^{41} +5.70928 q^{42} +3.36910 q^{43} -8.58864i q^{44} +2.70928i q^{45} -11.3268i q^{46} -6.78765 q^{47} +1.12064i q^{48} -16.8082 q^{49} +2.17009 q^{50} +(1.36910 + 1.75154i) q^{51} +7.12783 q^{52} +3.75872 q^{53} +6.68035i q^{54} +3.17009 q^{55} +7.51026i q^{56} -0.581449i q^{57} +6.34017i q^{58} -2.34017 q^{59} +1.46081 q^{60} +12.2557i q^{61} -8.87936i q^{62} -13.2195i q^{63} -12.3112 q^{64} +2.63090i q^{65} +3.70928 q^{66} +10.2062 q^{67} +(-8.80098 + 6.87936i) q^{68} +2.81432 q^{69} -10.5886 q^{70} +4.06505i q^{71} -4.17009 q^{72} +11.0784i q^{73} -11.4186i q^{74} +0.539189i q^{75} +2.92162 q^{76} -15.4680 q^{77} +3.07838i q^{78} +6.92881i q^{79} -2.07838i q^{80} +6.46800 q^{81} +12.1568i q^{82} +8.23287 q^{83} -7.12783 q^{84} +(-2.53919 - 3.24846i) q^{85} -7.31124 q^{86} -1.57531 q^{87} +4.87936i q^{88} +7.15449 q^{89} -5.87936i q^{90} -12.8371i q^{91} +14.1412i q^{92} +2.20620 q^{93} +14.7298 q^{94} +1.07838i q^{95} -4.09171i q^{96} -8.18342i q^{97} +36.4752 q^{98} -8.58864i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} + 2 q^{9} + 8 q^{13} - 6 q^{16} - 2 q^{17} - 10 q^{18} - 8 q^{21} - 6 q^{25} - 20 q^{26} + 4 q^{30} + 6 q^{32} + 4 q^{33} + 6 q^{34} + 4 q^{35} + 22 q^{36} + 8 q^{38} + 20 q^{42}+ \cdots + 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17009 −1.53448 −0.767241 0.641358i \(-0.778371\pi\)
−0.767241 + 0.641358i \(0.778371\pi\)
\(3\) 0.539189i 0.311301i −0.987812 0.155650i \(-0.950253\pi\)
0.987812 0.155650i \(-0.0497473\pi\)
\(4\) 2.70928 1.35464
\(5\) 1.00000i 0.447214i
\(6\) 1.17009i 0.477686i
\(7\) 4.87936i 1.84423i −0.386921 0.922113i \(-0.626462\pi\)
0.386921 0.922113i \(-0.373538\pi\)
\(8\) −1.53919 −0.544185
\(9\) 2.70928 0.903092
\(10\) 2.17009i 0.686242i
\(11\) 3.17009i 0.955817i −0.878410 0.477909i \(-0.841395\pi\)
0.878410 0.477909i \(-0.158605\pi\)
\(12\) 1.46081i 0.421700i
\(13\) 2.63090 0.729680 0.364840 0.931070i \(-0.381124\pi\)
0.364840 + 0.931070i \(0.381124\pi\)
\(14\) 10.5886i 2.82993i
\(15\) 0.539189 0.139218
\(16\) −2.07838 −0.519594
\(17\) −3.24846 + 2.53919i −0.787868 + 0.615844i
\(18\) −5.87936 −1.38578
\(19\) 1.07838 0.247397 0.123698 0.992320i \(-0.460524\pi\)
0.123698 + 0.992320i \(0.460524\pi\)
\(20\) 2.70928i 0.605812i
\(21\) −2.63090 −0.574109
\(22\) 6.87936i 1.46668i
\(23\) 5.21953i 1.08835i 0.838972 + 0.544174i \(0.183157\pi\)
−0.838972 + 0.544174i \(0.816843\pi\)
\(24\) 0.829914i 0.169405i
\(25\) −1.00000 −0.200000
\(26\) −5.70928 −1.11968
\(27\) 3.07838i 0.592434i
\(28\) 13.2195i 2.49826i
\(29\) 2.92162i 0.542532i −0.962504 0.271266i \(-0.912558\pi\)
0.962504 0.271266i \(-0.0874422\pi\)
\(30\) −1.17009 −0.213628
\(31\) 4.09171i 0.734893i 0.930045 + 0.367446i \(0.119768\pi\)
−0.930045 + 0.367446i \(0.880232\pi\)
\(32\) 7.58864 1.34149
\(33\) −1.70928 −0.297547
\(34\) 7.04945 5.51026i 1.20897 0.945002i
\(35\) 4.87936 0.824763
\(36\) 7.34017 1.22336
\(37\) 5.26180i 0.865034i 0.901626 + 0.432517i \(0.142374\pi\)
−0.901626 + 0.432517i \(0.857626\pi\)
\(38\) −2.34017 −0.379626
\(39\) 1.41855i 0.227150i
\(40\) 1.53919i 0.243367i
\(41\) 5.60197i 0.874880i −0.899247 0.437440i \(-0.855885\pi\)
0.899247 0.437440i \(-0.144115\pi\)
\(42\) 5.70928 0.880960
\(43\) 3.36910 0.513783 0.256892 0.966440i \(-0.417302\pi\)
0.256892 + 0.966440i \(0.417302\pi\)
\(44\) 8.58864i 1.29479i
\(45\) 2.70928i 0.403875i
\(46\) 11.3268i 1.67005i
\(47\) −6.78765 −0.990081 −0.495040 0.868870i \(-0.664847\pi\)
−0.495040 + 0.868870i \(0.664847\pi\)
\(48\) 1.12064i 0.161750i
\(49\) −16.8082 −2.40117
\(50\) 2.17009 0.306897
\(51\) 1.36910 + 1.75154i 0.191713 + 0.245264i
\(52\) 7.12783 0.988452
\(53\) 3.75872 0.516300 0.258150 0.966105i \(-0.416887\pi\)
0.258150 + 0.966105i \(0.416887\pi\)
\(54\) 6.68035i 0.909080i
\(55\) 3.17009 0.427454
\(56\) 7.51026i 1.00360i
\(57\) 0.581449i 0.0770148i
\(58\) 6.34017i 0.832505i
\(59\) −2.34017 −0.304665 −0.152332 0.988329i \(-0.548678\pi\)
−0.152332 + 0.988329i \(0.548678\pi\)
\(60\) 1.46081 0.188590
\(61\) 12.2557i 1.56918i 0.620018 + 0.784588i \(0.287125\pi\)
−0.620018 + 0.784588i \(0.712875\pi\)
\(62\) 8.87936i 1.12768i
\(63\) 13.2195i 1.66550i
\(64\) −12.3112 −1.53891
\(65\) 2.63090i 0.326323i
\(66\) 3.70928 0.456580
\(67\) 10.2062 1.24689 0.623443 0.781869i \(-0.285733\pi\)
0.623443 + 0.781869i \(0.285733\pi\)
\(68\) −8.80098 + 6.87936i −1.06728 + 0.834245i
\(69\) 2.81432 0.338804
\(70\) −10.5886 −1.26558
\(71\) 4.06505i 0.482432i 0.970471 + 0.241216i \(0.0775463\pi\)
−0.970471 + 0.241216i \(0.922454\pi\)
\(72\) −4.17009 −0.491449
\(73\) 11.0784i 1.29663i 0.761374 + 0.648313i \(0.224525\pi\)
−0.761374 + 0.648313i \(0.775475\pi\)
\(74\) 11.4186i 1.32738i
\(75\) 0.539189i 0.0622602i
\(76\) 2.92162 0.335133
\(77\) −15.4680 −1.76274
\(78\) 3.07838i 0.348558i
\(79\) 6.92881i 0.779552i 0.920910 + 0.389776i \(0.127448\pi\)
−0.920910 + 0.389776i \(0.872552\pi\)
\(80\) 2.07838i 0.232370i
\(81\) 6.46800 0.718667
\(82\) 12.1568i 1.34249i
\(83\) 8.23287 0.903674 0.451837 0.892100i \(-0.350769\pi\)
0.451837 + 0.892100i \(0.350769\pi\)
\(84\) −7.12783 −0.777710
\(85\) −2.53919 3.24846i −0.275414 0.352345i
\(86\) −7.31124 −0.788392
\(87\) −1.57531 −0.168891
\(88\) 4.87936i 0.520142i
\(89\) 7.15449 0.758374 0.379187 0.925320i \(-0.376204\pi\)
0.379187 + 0.925320i \(0.376204\pi\)
\(90\) 5.87936i 0.619739i
\(91\) 12.8371i 1.34569i
\(92\) 14.1412i 1.47432i
\(93\) 2.20620 0.228773
\(94\) 14.7298 1.51926
\(95\) 1.07838i 0.110639i
\(96\) 4.09171i 0.417608i
\(97\) 8.18342i 0.830900i −0.909616 0.415450i \(-0.863624\pi\)
0.909616 0.415450i \(-0.136376\pi\)
\(98\) 36.4752 3.68455
\(99\) 8.58864i 0.863191i
\(100\) −2.70928 −0.270928
\(101\) −2.47414 −0.246186 −0.123093 0.992395i \(-0.539281\pi\)
−0.123093 + 0.992395i \(0.539281\pi\)
\(102\) −2.97107 3.80098i −0.294180 0.376354i
\(103\) −19.6514 −1.93631 −0.968156 0.250348i \(-0.919455\pi\)
−0.968156 + 0.250348i \(0.919455\pi\)
\(104\) −4.04945 −0.397081
\(105\) 2.63090i 0.256749i
\(106\) −8.15676 −0.792254
\(107\) 12.6381i 1.22177i −0.791719 0.610885i \(-0.790814\pi\)
0.791719 0.610885i \(-0.209186\pi\)
\(108\) 8.34017i 0.802534i
\(109\) 8.15676i 0.781275i −0.920544 0.390638i \(-0.872255\pi\)
0.920544 0.390638i \(-0.127745\pi\)
\(110\) −6.87936 −0.655921
\(111\) 2.83710 0.269286
\(112\) 10.1412i 0.958249i
\(113\) 17.0205i 1.60116i 0.599229 + 0.800578i \(0.295474\pi\)
−0.599229 + 0.800578i \(0.704526\pi\)
\(114\) 1.26180i 0.118178i
\(115\) −5.21953 −0.486724
\(116\) 7.91548i 0.734934i
\(117\) 7.12783 0.658968
\(118\) 5.07838 0.467503
\(119\) 12.3896 + 15.8504i 1.13575 + 1.45301i
\(120\) −0.829914 −0.0757604
\(121\) 0.950552 0.0864138
\(122\) 26.5958i 2.40787i
\(123\) −3.02052 −0.272351
\(124\) 11.0856i 0.995513i
\(125\) 1.00000i 0.0894427i
\(126\) 28.6875i 2.55569i
\(127\) −8.04945 −0.714273 −0.357137 0.934052i \(-0.616247\pi\)
−0.357137 + 0.934052i \(0.616247\pi\)
\(128\) 11.5392 1.01993
\(129\) 1.81658i 0.159941i
\(130\) 5.70928i 0.500737i
\(131\) 11.6937i 1.02168i 0.859675 + 0.510841i \(0.170666\pi\)
−0.859675 + 0.510841i \(0.829334\pi\)
\(132\) −4.63090 −0.403068
\(133\) 5.26180i 0.456256i
\(134\) −22.1483 −1.91333
\(135\) 3.07838 0.264945
\(136\) 5.00000 3.90829i 0.428746 0.335133i
\(137\) 1.95055 0.166647 0.0833234 0.996523i \(-0.473447\pi\)
0.0833234 + 0.996523i \(0.473447\pi\)
\(138\) −6.10731 −0.519889
\(139\) 2.00719i 0.170247i −0.996370 0.0851237i \(-0.972871\pi\)
0.996370 0.0851237i \(-0.0271286\pi\)
\(140\) 13.2195 1.11725
\(141\) 3.65983i 0.308213i
\(142\) 8.82150i 0.740284i
\(143\) 8.34017i 0.697440i
\(144\) −5.63090 −0.469241
\(145\) 2.92162 0.242628
\(146\) 24.0410i 1.98965i
\(147\) 9.06278i 0.747485i
\(148\) 14.2557i 1.17181i
\(149\) −14.2823 −1.17005 −0.585026 0.811014i \(-0.698916\pi\)
−0.585026 + 0.811014i \(0.698916\pi\)
\(150\) 1.17009i 0.0955372i
\(151\) 12.8638 1.04684 0.523419 0.852075i \(-0.324656\pi\)
0.523419 + 0.852075i \(0.324656\pi\)
\(152\) −1.65983 −0.134630
\(153\) −8.80098 + 6.87936i −0.711517 + 0.556163i
\(154\) 33.5669 2.70490
\(155\) −4.09171 −0.328654
\(156\) 3.84324i 0.307706i
\(157\) 3.75872 0.299979 0.149989 0.988688i \(-0.452076\pi\)
0.149989 + 0.988688i \(0.452076\pi\)
\(158\) 15.0361i 1.19621i
\(159\) 2.02666i 0.160725i
\(160\) 7.58864i 0.599934i
\(161\) 25.4680 2.00716
\(162\) −14.0361 −1.10278
\(163\) 8.69594i 0.681119i −0.940223 0.340559i \(-0.889384\pi\)
0.940223 0.340559i \(-0.110616\pi\)
\(164\) 15.1773i 1.18515i
\(165\) 1.70928i 0.133067i
\(166\) −17.8660 −1.38667
\(167\) 1.37629i 0.106501i 0.998581 + 0.0532503i \(0.0169581\pi\)
−0.998581 + 0.0532503i \(0.983042\pi\)
\(168\) 4.04945 0.312422
\(169\) −6.07838 −0.467568
\(170\) 5.51026 + 7.04945i 0.422618 + 0.540668i
\(171\) 2.92162 0.223422
\(172\) 9.12783 0.695990
\(173\) 17.3607i 1.31991i −0.751306 0.659954i \(-0.770576\pi\)
0.751306 0.659954i \(-0.229424\pi\)
\(174\) 3.41855 0.259160
\(175\) 4.87936i 0.368845i
\(176\) 6.58864i 0.496637i
\(177\) 1.26180i 0.0948423i
\(178\) −15.5259 −1.16371
\(179\) 6.83710 0.511029 0.255514 0.966805i \(-0.417755\pi\)
0.255514 + 0.966805i \(0.417755\pi\)
\(180\) 7.34017i 0.547104i
\(181\) 15.0205i 1.11647i 0.829684 + 0.558233i \(0.188521\pi\)
−0.829684 + 0.558233i \(0.811479\pi\)
\(182\) 27.8576i 2.06494i
\(183\) 6.60811 0.488486
\(184\) 8.03385i 0.592263i
\(185\) −5.26180 −0.386855
\(186\) −4.78765 −0.351048
\(187\) 8.04945 + 10.2979i 0.588634 + 0.753058i
\(188\) −18.3896 −1.34120
\(189\) −15.0205 −1.09258
\(190\) 2.34017i 0.169774i
\(191\) −24.2823 −1.75701 −0.878503 0.477736i \(-0.841457\pi\)
−0.878503 + 0.477736i \(0.841457\pi\)
\(192\) 6.63809i 0.479063i
\(193\) 11.8576i 0.853530i −0.904363 0.426765i \(-0.859653\pi\)
0.904363 0.426765i \(-0.140347\pi\)
\(194\) 17.7587i 1.27500i
\(195\) 1.41855 0.101585
\(196\) −45.5380 −3.25271
\(197\) 18.2557i 1.30066i −0.759651 0.650331i \(-0.774630\pi\)
0.759651 0.650331i \(-0.225370\pi\)
\(198\) 18.6381i 1.32455i
\(199\) 3.72487i 0.264049i 0.991246 + 0.132025i \(0.0421478\pi\)
−0.991246 + 0.132025i \(0.957852\pi\)
\(200\) 1.53919 0.108837
\(201\) 5.50307i 0.388157i
\(202\) 5.36910 0.377769
\(203\) −14.2557 −1.00055
\(204\) 3.70928 + 4.74539i 0.259701 + 0.332244i
\(205\) 5.60197 0.391258
\(206\) 42.6453 2.97124
\(207\) 14.1412i 0.982878i
\(208\) −5.46800 −0.379138
\(209\) 3.41855i 0.236466i
\(210\) 5.70928i 0.393977i
\(211\) 22.2485i 1.53165i −0.643051 0.765824i \(-0.722332\pi\)
0.643051 0.765824i \(-0.277668\pi\)
\(212\) 10.1834 0.699400
\(213\) 2.19183 0.150182
\(214\) 27.4257i 1.87478i
\(215\) 3.36910i 0.229771i
\(216\) 4.73820i 0.322394i
\(217\) 19.9649 1.35531
\(218\) 17.7009i 1.19885i
\(219\) 5.97334 0.403641
\(220\) 8.58864 0.579046
\(221\) −8.54638 + 6.68035i −0.574892 + 0.449369i
\(222\) −6.15676 −0.413214
\(223\) −2.19183 −0.146776 −0.0733878 0.997303i \(-0.523381\pi\)
−0.0733878 + 0.997303i \(0.523381\pi\)
\(224\) 37.0277i 2.47402i
\(225\) −2.70928 −0.180618
\(226\) 36.9360i 2.45695i
\(227\) 9.55971i 0.634500i 0.948342 + 0.317250i \(0.102759\pi\)
−0.948342 + 0.317250i \(0.897241\pi\)
\(228\) 1.57531i 0.104327i
\(229\) 7.36910 0.486964 0.243482 0.969905i \(-0.421710\pi\)
0.243482 + 0.969905i \(0.421710\pi\)
\(230\) 11.3268 0.746870
\(231\) 8.34017i 0.548743i
\(232\) 4.49693i 0.295238i
\(233\) 9.44521i 0.618776i 0.950936 + 0.309388i \(0.100124\pi\)
−0.950936 + 0.309388i \(0.899876\pi\)
\(234\) −15.4680 −1.01117
\(235\) 6.78765i 0.442778i
\(236\) −6.34017 −0.412710
\(237\) 3.73594 0.242675
\(238\) −26.8865 34.3968i −1.74280 2.22961i
\(239\) 6.25565 0.404644 0.202322 0.979319i \(-0.435151\pi\)
0.202322 + 0.979319i \(0.435151\pi\)
\(240\) −1.12064 −0.0723369
\(241\) 2.49693i 0.160841i −0.996761 0.0804207i \(-0.974374\pi\)
0.996761 0.0804207i \(-0.0256264\pi\)
\(242\) −2.06278 −0.132600
\(243\) 12.7226i 0.816156i
\(244\) 33.2039i 2.12566i
\(245\) 16.8082i 1.07383i
\(246\) 6.55479 0.417918
\(247\) 2.83710 0.180520
\(248\) 6.29791i 0.399918i
\(249\) 4.43907i 0.281315i
\(250\) 2.17009i 0.137248i
\(251\) −11.8166 −0.745856 −0.372928 0.927860i \(-0.621646\pi\)
−0.372928 + 0.927860i \(0.621646\pi\)
\(252\) 35.8154i 2.25616i
\(253\) 16.5464 1.04026
\(254\) 17.4680 1.09604
\(255\) −1.75154 + 1.36910i −0.109685 + 0.0857365i
\(256\) −0.418551 −0.0261594
\(257\) 14.9444 0.932207 0.466103 0.884730i \(-0.345658\pi\)
0.466103 + 0.884730i \(0.345658\pi\)
\(258\) 3.94214i 0.245427i
\(259\) 25.6742 1.59532
\(260\) 7.12783i 0.442049i
\(261\) 7.91548i 0.489956i
\(262\) 25.3763i 1.56775i
\(263\) −12.9444 −0.798186 −0.399093 0.916910i \(-0.630675\pi\)
−0.399093 + 0.916910i \(0.630675\pi\)
\(264\) 2.63090 0.161921
\(265\) 3.75872i 0.230897i
\(266\) 11.4186i 0.700116i
\(267\) 3.85762i 0.236083i
\(268\) 27.6514 1.68908
\(269\) 7.47641i 0.455845i −0.973679 0.227922i \(-0.926807\pi\)
0.973679 0.227922i \(-0.0731932\pi\)
\(270\) −6.68035 −0.406553
\(271\) −2.15676 −0.131014 −0.0655068 0.997852i \(-0.520866\pi\)
−0.0655068 + 0.997852i \(0.520866\pi\)
\(272\) 6.75154 5.27739i 0.409372 0.319989i
\(273\) −6.92162 −0.418916
\(274\) −4.23287 −0.255717
\(275\) 3.17009i 0.191163i
\(276\) 7.62475 0.458956
\(277\) 12.1568i 0.730429i 0.930923 + 0.365214i \(0.119004\pi\)
−0.930923 + 0.365214i \(0.880996\pi\)
\(278\) 4.35577i 0.261242i
\(279\) 11.0856i 0.663675i
\(280\) −7.51026 −0.448824
\(281\) 13.1194 0.782639 0.391319 0.920255i \(-0.372019\pi\)
0.391319 + 0.920255i \(0.372019\pi\)
\(282\) 7.94214i 0.472948i
\(283\) 13.9577i 0.829701i −0.909889 0.414851i \(-0.863834\pi\)
0.909889 0.414851i \(-0.136166\pi\)
\(284\) 11.0133i 0.653521i
\(285\) 0.581449 0.0344421
\(286\) 18.0989i 1.07021i
\(287\) −27.3340 −1.61348
\(288\) 20.5597 1.21149
\(289\) 4.10504 16.4969i 0.241473 0.970408i
\(290\) −6.34017 −0.372308
\(291\) −4.41241 −0.258660
\(292\) 30.0144i 1.75646i
\(293\) 4.73820 0.276809 0.138404 0.990376i \(-0.455803\pi\)
0.138404 + 0.990376i \(0.455803\pi\)
\(294\) 19.6670i 1.14700i
\(295\) 2.34017i 0.136250i
\(296\) 8.09890i 0.470739i
\(297\) −9.75872 −0.566259
\(298\) 30.9939 1.79543
\(299\) 13.7321i 0.794146i
\(300\) 1.46081i 0.0843400i
\(301\) 16.4391i 0.947532i
\(302\) −27.9155 −1.60636
\(303\) 1.33403i 0.0766380i
\(304\) −2.24128 −0.128546
\(305\) −12.2557 −0.701757
\(306\) 19.0989 14.9288i 1.09181 0.853423i
\(307\) −21.5936 −1.23241 −0.616205 0.787586i \(-0.711331\pi\)
−0.616205 + 0.787586i \(0.711331\pi\)
\(308\) −41.9071 −2.38788
\(309\) 10.5958i 0.602775i
\(310\) 8.87936 0.504314
\(311\) 24.2628i 1.37582i 0.725796 + 0.687910i \(0.241472\pi\)
−0.725796 + 0.687910i \(0.758528\pi\)
\(312\) 2.18342i 0.123612i
\(313\) 4.07223i 0.230176i 0.993355 + 0.115088i \(0.0367151\pi\)
−0.993355 + 0.115088i \(0.963285\pi\)
\(314\) −8.15676 −0.460312
\(315\) 13.2195 0.744836
\(316\) 18.7721i 1.05601i
\(317\) 8.05786i 0.452574i 0.974061 + 0.226287i \(0.0726588\pi\)
−0.974061 + 0.226287i \(0.927341\pi\)
\(318\) 4.39803i 0.246629i
\(319\) −9.26180 −0.518561
\(320\) 12.3112i 0.688219i
\(321\) −6.81432 −0.380338
\(322\) −55.2678 −3.07995
\(323\) −3.50307 + 2.73820i −0.194916 + 0.152358i
\(324\) 17.5236 0.973533
\(325\) −2.63090 −0.145936
\(326\) 18.8710i 1.04517i
\(327\) −4.39803 −0.243212
\(328\) 8.62249i 0.476097i
\(329\) 33.1194i 1.82593i
\(330\) 3.70928i 0.204189i
\(331\) −10.0722 −0.553620 −0.276810 0.960925i \(-0.589277\pi\)
−0.276810 + 0.960925i \(0.589277\pi\)
\(332\) 22.3051 1.22415
\(333\) 14.2557i 0.781205i
\(334\) 2.98667i 0.163423i
\(335\) 10.2062i 0.557624i
\(336\) 5.46800 0.298304
\(337\) 11.2351i 0.612017i −0.952029 0.306008i \(-0.901006\pi\)
0.952029 0.306008i \(-0.0989936\pi\)
\(338\) 13.1906 0.717474
\(339\) 9.17727 0.498441
\(340\) −6.87936 8.80098i −0.373086 0.477300i
\(341\) 12.9711 0.702423
\(342\) −6.34017 −0.342837
\(343\) 47.8576i 2.58407i
\(344\) −5.18568 −0.279593
\(345\) 2.81432i 0.151518i
\(346\) 37.6742i 2.02538i
\(347\) 8.74927i 0.469685i −0.972033 0.234843i \(-0.924543\pi\)
0.972033 0.234843i \(-0.0754575\pi\)
\(348\) −4.26794 −0.228786
\(349\) −26.9093 −1.44042 −0.720212 0.693754i \(-0.755955\pi\)
−0.720212 + 0.693754i \(0.755955\pi\)
\(350\) 10.5886i 0.565986i
\(351\) 8.09890i 0.432287i
\(352\) 24.0566i 1.28222i
\(353\) 18.3135 0.974730 0.487365 0.873198i \(-0.337958\pi\)
0.487365 + 0.873198i \(0.337958\pi\)
\(354\) 2.73820i 0.145534i
\(355\) −4.06505 −0.215750
\(356\) 19.3835 1.02732
\(357\) 8.54638 6.68035i 0.452322 0.353561i
\(358\) −14.8371 −0.784165
\(359\) −9.57531 −0.505365 −0.252683 0.967549i \(-0.581313\pi\)
−0.252683 + 0.967549i \(0.581313\pi\)
\(360\) 4.17009i 0.219783i
\(361\) −17.8371 −0.938795
\(362\) 32.5958i 1.71320i
\(363\) 0.512527i 0.0269007i
\(364\) 34.7792i 1.82293i
\(365\) −11.0784 −0.579869
\(366\) −14.3402 −0.749573
\(367\) 12.1145i 0.632371i 0.948697 + 0.316186i \(0.102402\pi\)
−0.948697 + 0.316186i \(0.897598\pi\)
\(368\) 10.8482i 0.565500i
\(369\) 15.1773i 0.790097i
\(370\) 11.4186 0.593622
\(371\) 18.3402i 0.952174i
\(372\) 5.97721 0.309904
\(373\) 30.4619 1.57726 0.788628 0.614871i \(-0.210792\pi\)
0.788628 + 0.614871i \(0.210792\pi\)
\(374\) −17.4680 22.3474i −0.903249 1.15555i
\(375\) −0.539189 −0.0278436
\(376\) 10.4475 0.538788
\(377\) 7.68649i 0.395874i
\(378\) 32.5958 1.67655
\(379\) 0.986669i 0.0506818i −0.999679 0.0253409i \(-0.991933\pi\)
0.999679 0.0253409i \(-0.00806712\pi\)
\(380\) 2.92162i 0.149876i
\(381\) 4.34017i 0.222354i
\(382\) 52.6947 2.69610
\(383\) −24.9588 −1.27533 −0.637667 0.770312i \(-0.720100\pi\)
−0.637667 + 0.770312i \(0.720100\pi\)
\(384\) 6.22180i 0.317505i
\(385\) 15.4680i 0.788322i
\(386\) 25.7321i 1.30973i
\(387\) 9.12783 0.463993
\(388\) 22.1711i 1.12557i
\(389\) 33.8082 1.71414 0.857071 0.515198i \(-0.172282\pi\)
0.857071 + 0.515198i \(0.172282\pi\)
\(390\) −3.07838 −0.155880
\(391\) −13.2534 16.9555i −0.670252 0.857475i
\(392\) 25.8710 1.30668
\(393\) 6.30510 0.318050
\(394\) 39.6163i 1.99584i
\(395\) −6.92881 −0.348626
\(396\) 23.2690i 1.16931i
\(397\) 28.5236i 1.43156i 0.698327 + 0.715779i \(0.253928\pi\)
−0.698327 + 0.715779i \(0.746072\pi\)
\(398\) 8.08330i 0.405179i
\(399\) −2.83710 −0.142033
\(400\) 2.07838 0.103919
\(401\) 33.0928i 1.65257i −0.563250 0.826287i \(-0.690449\pi\)
0.563250 0.826287i \(-0.309551\pi\)
\(402\) 11.9421i 0.595620i
\(403\) 10.7649i 0.536236i
\(404\) −6.70313 −0.333493
\(405\) 6.46800i 0.321397i
\(406\) 30.9360 1.53533
\(407\) 16.6803 0.826814
\(408\) −2.10731 2.69594i −0.104327 0.133469i
\(409\) −30.1978 −1.49318 −0.746592 0.665282i \(-0.768311\pi\)
−0.746592 + 0.665282i \(0.768311\pi\)
\(410\) −12.1568 −0.600379
\(411\) 1.05172i 0.0518773i
\(412\) −53.2411 −2.62300
\(413\) 11.4186i 0.561870i
\(414\) 30.6875i 1.50821i
\(415\) 8.23287i 0.404135i
\(416\) 19.9649 0.978861
\(417\) −1.08225 −0.0529982
\(418\) 7.41855i 0.362853i
\(419\) 18.1639i 0.887367i 0.896184 + 0.443683i \(0.146329\pi\)
−0.896184 + 0.443683i \(0.853671\pi\)
\(420\) 7.12783i 0.347802i
\(421\) 0.760991 0.0370884 0.0185442 0.999828i \(-0.494097\pi\)
0.0185442 + 0.999828i \(0.494097\pi\)
\(422\) 48.2811i 2.35029i
\(423\) −18.3896 −0.894134
\(424\) −5.78539 −0.280963
\(425\) 3.24846 2.53919i 0.157574 0.123169i
\(426\) −4.75646 −0.230451
\(427\) 59.7998 2.89391
\(428\) 34.2401i 1.65506i
\(429\) −4.49693 −0.217114
\(430\) 7.31124i 0.352579i
\(431\) 6.34736i 0.305742i 0.988246 + 0.152871i \(0.0488518\pi\)
−0.988246 + 0.152871i \(0.951148\pi\)
\(432\) 6.39803i 0.307825i
\(433\) 3.62475 0.174195 0.0870973 0.996200i \(-0.472241\pi\)
0.0870973 + 0.996200i \(0.472241\pi\)
\(434\) −43.3256 −2.07970
\(435\) 1.57531i 0.0755302i
\(436\) 22.0989i 1.05835i
\(437\) 5.62863i 0.269254i
\(438\) −12.9627 −0.619380
\(439\) 6.40522i 0.305704i −0.988249 0.152852i \(-0.951154\pi\)
0.988249 0.152852i \(-0.0488459\pi\)
\(440\) −4.87936 −0.232614
\(441\) −45.5380 −2.16847
\(442\) 18.5464 14.4969i 0.882161 0.689549i
\(443\) 27.4824 1.30573 0.652864 0.757476i \(-0.273568\pi\)
0.652864 + 0.757476i \(0.273568\pi\)
\(444\) 7.68649 0.364785
\(445\) 7.15449i 0.339155i
\(446\) 4.75646 0.225225
\(447\) 7.70086i 0.364238i
\(448\) 60.0710i 2.83809i
\(449\) 28.7526i 1.35692i −0.734638 0.678459i \(-0.762648\pi\)
0.734638 0.678459i \(-0.237352\pi\)
\(450\) 5.87936 0.277156
\(451\) −17.7587 −0.836226
\(452\) 46.1133i 2.16899i
\(453\) 6.93600i 0.325882i
\(454\) 20.7454i 0.973630i
\(455\) 12.8371 0.601813
\(456\) 0.894960i 0.0419104i
\(457\) −31.4101 −1.46930 −0.734652 0.678444i \(-0.762655\pi\)
−0.734652 + 0.678444i \(0.762655\pi\)
\(458\) −15.9916 −0.747238
\(459\) 7.81658 + 10.0000i 0.364847 + 0.466760i
\(460\) −14.1412 −0.659335
\(461\) −7.75872 −0.361360 −0.180680 0.983542i \(-0.557830\pi\)
−0.180680 + 0.983542i \(0.557830\pi\)
\(462\) 18.0989i 0.842037i
\(463\) 24.2329 1.12620 0.563098 0.826390i \(-0.309609\pi\)
0.563098 + 0.826390i \(0.309609\pi\)
\(464\) 6.07223i 0.281896i
\(465\) 2.20620i 0.102310i
\(466\) 20.4969i 0.949502i
\(467\) −12.3174 −0.569981 −0.284990 0.958530i \(-0.591990\pi\)
−0.284990 + 0.958530i \(0.591990\pi\)
\(468\) 19.3112 0.892663
\(469\) 49.7998i 2.29954i
\(470\) 14.7298i 0.679435i
\(471\) 2.02666i 0.0933837i
\(472\) 3.60197 0.165794
\(473\) 10.6803i 0.491083i
\(474\) −8.10731 −0.372381
\(475\) −1.07838 −0.0494794
\(476\) 33.5669 + 42.9432i 1.53854 + 1.96830i
\(477\) 10.1834 0.466267
\(478\) −13.5753 −0.620920
\(479\) 2.14957i 0.0982162i 0.998793 + 0.0491081i \(0.0156379\pi\)
−0.998793 + 0.0491081i \(0.984362\pi\)
\(480\) 4.09171 0.186760
\(481\) 13.8432i 0.631198i
\(482\) 5.41855i 0.246808i
\(483\) 13.7321i 0.624830i
\(484\) 2.57531 0.117059
\(485\) 8.18342 0.371590
\(486\) 27.6092i 1.25238i
\(487\) 40.0833i 1.81635i −0.418593 0.908174i \(-0.637477\pi\)
0.418593 0.908174i \(-0.362523\pi\)
\(488\) 18.8638i 0.853922i
\(489\) −4.68876 −0.212033
\(490\) 36.4752i 1.64778i
\(491\) −2.25565 −0.101796 −0.0508981 0.998704i \(-0.516208\pi\)
−0.0508981 + 0.998704i \(0.516208\pi\)
\(492\) −8.18342 −0.368937
\(493\) 7.41855 + 9.49079i 0.334115 + 0.427443i
\(494\) −6.15676 −0.277006
\(495\) 8.58864 0.386031
\(496\) 8.50412i 0.381846i
\(497\) 19.8348 0.889714
\(498\) 9.63317i 0.431672i
\(499\) 42.5452i 1.90458i −0.305190 0.952291i \(-0.598720\pi\)
0.305190 0.952291i \(-0.401280\pi\)
\(500\) 2.70928i 0.121162i
\(501\) 0.742080 0.0331537
\(502\) 25.6430 1.14450
\(503\) 9.55971i 0.426246i −0.977025 0.213123i \(-0.931636\pi\)
0.977025 0.213123i \(-0.0683636\pi\)
\(504\) 20.3474i 0.906343i
\(505\) 2.47414i 0.110098i
\(506\) −35.9071 −1.59626
\(507\) 3.27739i 0.145554i
\(508\) −21.8082 −0.967581
\(509\) 28.3545 1.25679 0.628397 0.777893i \(-0.283712\pi\)
0.628397 + 0.777893i \(0.283712\pi\)
\(510\) 3.80098 2.97107i 0.168310 0.131561i
\(511\) 54.0554 2.39127
\(512\) −22.1701 −0.979789
\(513\) 3.31965i 0.146566i
\(514\) −32.4307 −1.43046
\(515\) 19.6514i 0.865945i
\(516\) 4.92162i 0.216662i
\(517\) 21.5174i 0.946336i
\(518\) −55.7152 −2.44799
\(519\) −9.36069 −0.410889
\(520\) 4.04945i 0.177580i
\(521\) 15.1050i 0.661764i 0.943672 + 0.330882i \(0.107346\pi\)
−0.943672 + 0.330882i \(0.892654\pi\)
\(522\) 17.1773i 0.751829i
\(523\) −10.8865 −0.476036 −0.238018 0.971261i \(-0.576498\pi\)
−0.238018 + 0.971261i \(0.576498\pi\)
\(524\) 31.6814i 1.38401i
\(525\) 2.63090 0.114822
\(526\) 28.0905 1.22480
\(527\) −10.3896 13.2918i −0.452579 0.578999i
\(528\) 3.55252 0.154604
\(529\) −4.24354 −0.184502
\(530\) 8.15676i 0.354307i
\(531\) −6.34017 −0.275140
\(532\) 14.2557i 0.618061i
\(533\) 14.7382i 0.638383i
\(534\) 8.37137i 0.362265i
\(535\) 12.6381 0.546392
\(536\) −15.7093 −0.678537
\(537\) 3.68649i 0.159084i
\(538\) 16.2245i 0.699486i
\(539\) 53.2834i 2.29508i
\(540\) 8.34017 0.358904
\(541\) 6.86830i 0.295291i 0.989040 + 0.147646i \(0.0471695\pi\)
−0.989040 + 0.147646i \(0.952830\pi\)
\(542\) 4.68035 0.201038
\(543\) 8.09890 0.347557
\(544\) −24.6514 + 19.2690i −1.05692 + 0.826151i
\(545\) 8.15676 0.349397
\(546\) 15.0205 0.642819
\(547\) 5.89988i 0.252261i −0.992014 0.126130i \(-0.959744\pi\)
0.992014 0.126130i \(-0.0402558\pi\)
\(548\) 5.28458 0.225746
\(549\) 33.2039i 1.41711i
\(550\) 6.87936i 0.293337i
\(551\) 3.15061i 0.134221i
\(552\) −4.33176 −0.184372
\(553\) 33.8082 1.43767
\(554\) 26.3812i 1.12083i
\(555\) 2.83710i 0.120428i
\(556\) 5.43802i 0.230624i
\(557\) −17.7359 −0.751496 −0.375748 0.926722i \(-0.622614\pi\)
−0.375748 + 0.926722i \(0.622614\pi\)
\(558\) 24.0566i 1.01840i
\(559\) 8.86376 0.374897
\(560\) −10.1412 −0.428542
\(561\) 5.55252 4.34017i 0.234428 0.183242i
\(562\) −28.4703 −1.20095
\(563\) −38.8020 −1.63531 −0.817655 0.575708i \(-0.804726\pi\)
−0.817655 + 0.575708i \(0.804726\pi\)
\(564\) 9.91548i 0.417517i
\(565\) −17.0205 −0.716059
\(566\) 30.2895i 1.27316i
\(567\) 31.5597i 1.32538i
\(568\) 6.25687i 0.262533i
\(569\) 12.1568 0.509638 0.254819 0.966989i \(-0.417984\pi\)
0.254819 + 0.966989i \(0.417984\pi\)
\(570\) −1.26180 −0.0528508
\(571\) 15.4569i 0.646853i −0.946253 0.323426i \(-0.895165\pi\)
0.946253 0.323426i \(-0.104835\pi\)
\(572\) 22.5958i 0.944779i
\(573\) 13.0928i 0.546958i
\(574\) 59.3172 2.47585
\(575\) 5.21953i 0.217670i
\(576\) −33.3545 −1.38977
\(577\) −2.36296 −0.0983713 −0.0491856 0.998790i \(-0.515663\pi\)
−0.0491856 + 0.998790i \(0.515663\pi\)
\(578\) −8.90829 + 35.7998i −0.370536 + 1.48907i
\(579\) −6.39350 −0.265705
\(580\) 7.91548 0.328672
\(581\) 40.1711i 1.66658i
\(582\) 9.57531 0.396909
\(583\) 11.9155i 0.493489i
\(584\) 17.0517i 0.705605i
\(585\) 7.12783i 0.294699i
\(586\) −10.2823 −0.424758
\(587\) −3.65142 −0.150710 −0.0753550 0.997157i \(-0.524009\pi\)
−0.0753550 + 0.997157i \(0.524009\pi\)
\(588\) 24.5536i 1.01257i
\(589\) 4.41241i 0.181810i
\(590\) 5.07838i 0.209074i
\(591\) −9.84324 −0.404897
\(592\) 10.9360i 0.449467i
\(593\) −1.38735 −0.0569718 −0.0284859 0.999594i \(-0.509069\pi\)
−0.0284859 + 0.999594i \(0.509069\pi\)
\(594\) 21.1773 0.868914
\(595\) −15.8504 + 12.3896i −0.649804 + 0.507925i
\(596\) −38.6947 −1.58500
\(597\) 2.00841 0.0821988
\(598\) 29.7998i 1.21860i
\(599\) −0.451356 −0.0184419 −0.00922095 0.999957i \(-0.502935\pi\)
−0.00922095 + 0.999957i \(0.502935\pi\)
\(600\) 0.829914i 0.0338811i
\(601\) 22.1301i 0.902705i 0.892346 + 0.451353i \(0.149058\pi\)
−0.892346 + 0.451353i \(0.850942\pi\)
\(602\) 35.6742i 1.45397i
\(603\) 27.6514 1.12605
\(604\) 34.8515 1.41809
\(605\) 0.950552i 0.0386454i
\(606\) 2.89496i 0.117600i
\(607\) 10.2667i 0.416713i 0.978053 + 0.208357i \(0.0668114\pi\)
−0.978053 + 0.208357i \(0.933189\pi\)
\(608\) 8.18342 0.331881
\(609\) 7.68649i 0.311472i
\(610\) 26.5958 1.07683
\(611\) −17.8576 −0.722442
\(612\) −23.8443 + 18.6381i −0.963848 + 0.753400i
\(613\) −9.05172 −0.365595 −0.182798 0.983151i \(-0.558515\pi\)
−0.182798 + 0.983151i \(0.558515\pi\)
\(614\) 46.8599 1.89111
\(615\) 3.02052i 0.121799i
\(616\) 23.8082 0.959259
\(617\) 17.8166i 0.717269i 0.933478 + 0.358634i \(0.116757\pi\)
−0.933478 + 0.358634i \(0.883243\pi\)
\(618\) 22.9939i 0.924949i
\(619\) 38.3884i 1.54296i 0.636254 + 0.771480i \(0.280483\pi\)
−0.636254 + 0.771480i \(0.719517\pi\)
\(620\) −11.0856 −0.445207
\(621\) 16.0677 0.644775
\(622\) 52.6525i 2.11117i
\(623\) 34.9093i 1.39861i
\(624\) 2.94828i 0.118026i
\(625\) 1.00000 0.0400000
\(626\) 8.83710i 0.353202i
\(627\) −1.84324 −0.0736121
\(628\) 10.1834 0.406363
\(629\) −13.3607 17.0928i −0.532726 0.681533i
\(630\) −28.6875 −1.14294
\(631\) −27.4863 −1.09421 −0.547105 0.837064i \(-0.684270\pi\)
−0.547105 + 0.837064i \(0.684270\pi\)
\(632\) 10.6647i 0.424221i
\(633\) −11.9961 −0.476803
\(634\) 17.4863i 0.694468i
\(635\) 8.04945i 0.319433i
\(636\) 5.49079i 0.217724i
\(637\) −44.2206 −1.75208
\(638\) 20.0989 0.795723
\(639\) 11.0133i 0.435681i
\(640\) 11.5392i 0.456126i
\(641\) 9.79976i 0.387067i 0.981094 + 0.193534i \(0.0619949\pi\)
−0.981094 + 0.193534i \(0.938005\pi\)
\(642\) 14.7877 0.583622
\(643\) 16.9372i 0.667939i 0.942584 + 0.333969i \(0.108388\pi\)
−0.942584 + 0.333969i \(0.891612\pi\)
\(644\) 68.9998 2.71897
\(645\) 1.81658 0.0715279
\(646\) 7.60197 5.94214i 0.299095 0.233790i
\(647\) −2.98545 −0.117370 −0.0586850 0.998277i \(-0.518691\pi\)
−0.0586850 + 0.998277i \(0.518691\pi\)
\(648\) −9.95547 −0.391088
\(649\) 7.41855i 0.291204i
\(650\) 5.70928 0.223936
\(651\) 10.7649i 0.421908i
\(652\) 23.5597i 0.922669i
\(653\) 40.1978i 1.57306i 0.617551 + 0.786531i \(0.288125\pi\)
−0.617551 + 0.786531i \(0.711875\pi\)
\(654\) 9.54411 0.373204
\(655\) −11.6937 −0.456910
\(656\) 11.6430i 0.454583i
\(657\) 30.0144i 1.17097i
\(658\) 71.8720i 2.80186i
\(659\) 43.9832 1.71334 0.856671 0.515864i \(-0.172529\pi\)
0.856671 + 0.515864i \(0.172529\pi\)
\(660\) 4.63090i 0.180257i
\(661\) 26.1133 1.01569 0.507844 0.861449i \(-0.330443\pi\)
0.507844 + 0.861449i \(0.330443\pi\)
\(662\) 21.8576 0.849521
\(663\) 3.60197 + 4.60811i 0.139889 + 0.178964i
\(664\) −12.6719 −0.491766
\(665\) 5.26180 0.204044
\(666\) 30.9360i 1.19875i
\(667\) 15.2495 0.590463
\(668\) 3.72875i 0.144270i
\(669\) 1.18181i 0.0456914i
\(670\) 22.1483i 0.855665i
\(671\) 38.8515 1.49984
\(672\) −19.9649 −0.770164
\(673\) 13.3340i 0.513989i −0.966413 0.256995i \(-0.917268\pi\)
0.966413 0.256995i \(-0.0827322\pi\)
\(674\) 24.3812i 0.939129i
\(675\) 3.07838i 0.118487i
\(676\) −16.4680 −0.633385
\(677\) 26.5113i 1.01891i −0.860497 0.509456i \(-0.829847\pi\)
0.860497 0.509456i \(-0.170153\pi\)
\(678\) −19.9155 −0.764849
\(679\) −39.9299 −1.53237
\(680\) 3.90829 + 5.00000i 0.149876 + 0.191741i
\(681\) 5.15449 0.197520
\(682\) −28.1483 −1.07786
\(683\) 5.71646i 0.218734i 0.994001 + 0.109367i \(0.0348824\pi\)
−0.994001 + 0.109367i \(0.965118\pi\)
\(684\) 7.91548 0.302656
\(685\) 1.95055i 0.0745267i
\(686\) 103.855i 3.96521i
\(687\) 3.97334i 0.151592i
\(688\) −7.00227 −0.266959
\(689\) 9.88882 0.376734
\(690\) 6.10731i 0.232501i
\(691\) 43.8504i 1.66815i −0.551652 0.834075i \(-0.686002\pi\)
0.551652 0.834075i \(-0.313998\pi\)
\(692\) 47.0349i 1.78800i
\(693\) −41.9071 −1.59192
\(694\) 18.9867i 0.720724i
\(695\) 2.00719 0.0761370
\(696\) 2.42469 0.0919078
\(697\) 14.2245 + 18.1978i 0.538790 + 0.689291i
\(698\) 58.3956 2.21031
\(699\) 5.09275 0.192626
\(700\) 13.2195i 0.499651i
\(701\) 0.0806452 0.00304593 0.00152296 0.999999i \(-0.499515\pi\)
0.00152296 + 0.999999i \(0.499515\pi\)
\(702\) 17.5753i 0.663337i
\(703\) 5.67420i 0.214007i
\(704\) 39.0277i 1.47091i
\(705\) −3.65983 −0.137837
\(706\) −39.7419 −1.49571
\(707\) 12.0722i 0.454023i
\(708\) 3.41855i 0.128477i
\(709\) 10.8227i 0.406456i −0.979131 0.203228i \(-0.934857\pi\)
0.979131 0.203228i \(-0.0651433\pi\)
\(710\) 8.82150 0.331065
\(711\) 18.7721i 0.704007i
\(712\) −11.0121 −0.412696
\(713\) −21.3568 −0.799819
\(714\) −18.5464 + 14.4969i −0.694081 + 0.542534i
\(715\) 8.34017 0.311905
\(716\) 18.5236 0.692259
\(717\) 3.37298i 0.125966i
\(718\) 20.7792 0.775474
\(719\) 43.7659i 1.63219i −0.577916 0.816097i \(-0.696134\pi\)
0.577916 0.816097i \(-0.303866\pi\)
\(720\) 5.63090i 0.209851i
\(721\) 95.8864i 3.57100i
\(722\) 38.7081 1.44056
\(723\) −1.34632 −0.0500700
\(724\) 40.6947i 1.51241i
\(725\) 2.92162i 0.108506i
\(726\) 1.11223i 0.0412786i
\(727\) −3.59809 −0.133446 −0.0667229 0.997772i \(-0.521254\pi\)
−0.0667229 + 0.997772i \(0.521254\pi\)
\(728\) 19.7587i 0.732307i
\(729\) 12.5441 0.464597
\(730\) 24.0410 0.889799
\(731\) −10.9444 + 8.55479i −0.404794 + 0.316410i
\(732\) 17.9032 0.661721
\(733\) −39.8264 −1.47102 −0.735511 0.677513i \(-0.763058\pi\)
−0.735511 + 0.677513i \(0.763058\pi\)
\(734\) 26.2895i 0.970363i
\(735\) −9.06278 −0.334286
\(736\) 39.6092i 1.46001i
\(737\) 32.3545i 1.19180i
\(738\) 32.9360i 1.21239i
\(739\) 13.7587 0.506123 0.253061 0.967450i \(-0.418563\pi\)
0.253061 + 0.967450i \(0.418563\pi\)
\(740\) −14.2557 −0.524048
\(741\) 1.52973i 0.0561962i
\(742\) 39.7998i 1.46110i
\(743\) 9.34963i 0.343005i −0.985184 0.171502i \(-0.945138\pi\)
0.985184 0.171502i \(-0.0548621\pi\)
\(744\) −3.39576 −0.124495
\(745\) 14.2823i 0.523264i
\(746\) −66.1049 −2.42027
\(747\) 22.3051 0.816101
\(748\) 21.8082 + 27.8999i 0.797386 + 1.02012i
\(749\) −61.6658 −2.25322
\(750\) 1.17009 0.0427255
\(751\) 53.7392i 1.96097i 0.196586 + 0.980487i \(0.437014\pi\)
−0.196586 + 0.980487i \(0.562986\pi\)
\(752\) 14.1073 0.514441
\(753\) 6.37137i 0.232186i
\(754\) 16.6803i 0.607462i
\(755\) 12.8638i 0.468160i
\(756\) −40.6947 −1.48005
\(757\) −41.5136 −1.50884 −0.754418 0.656394i \(-0.772081\pi\)
−0.754418 + 0.656394i \(0.772081\pi\)
\(758\) 2.14116i 0.0777703i
\(759\) 8.92162i 0.323834i
\(760\) 1.65983i 0.0602083i
\(761\) −18.0372 −0.653847 −0.326923 0.945051i \(-0.606012\pi\)
−0.326923 + 0.945051i \(0.606012\pi\)
\(762\) 9.41855i 0.341198i
\(763\) −39.7998 −1.44085
\(764\) −65.7875 −2.38011
\(765\) −6.87936 8.80098i −0.248724 0.318200i
\(766\) 54.1627 1.95698
\(767\) −6.15676 −0.222308
\(768\) 0.225678i 0.00814345i
\(769\) −23.0843 −0.832443 −0.416221 0.909263i \(-0.636646\pi\)
−0.416221 + 0.909263i \(0.636646\pi\)
\(770\) 33.5669i 1.20967i
\(771\) 8.05786i 0.290197i
\(772\) 32.1256i 1.15622i
\(773\) −27.4101 −0.985874 −0.492937 0.870065i \(-0.664077\pi\)
−0.492937 + 0.870065i \(0.664077\pi\)
\(774\) −19.8082 −0.711990
\(775\) 4.09171i 0.146979i
\(776\) 12.5958i 0.452164i
\(777\) 13.8432i 0.496624i
\(778\) −73.3667 −2.63032
\(779\) 6.04104i 0.216443i
\(780\) 3.84324 0.137610
\(781\) 12.8865 0.461117
\(782\) 28.7610 + 36.7948i 1.02849 + 1.31578i
\(783\) −8.99386 −0.321414
\(784\) 34.9337 1.24763
\(785\) 3.75872i 0.134155i
\(786\) −13.6826 −0.488043
\(787\) 30.6069i 1.09102i 0.838105 + 0.545509i \(0.183664\pi\)
−0.838105 + 0.545509i \(0.816336\pi\)
\(788\) 49.4596i 1.76192i
\(789\) 6.97948i 0.248476i
\(790\) 15.0361 0.534961
\(791\) 83.0493 2.95289
\(792\) 13.2195i 0.469736i
\(793\) 32.2434i 1.14500i
\(794\) 61.8987i 2.19670i
\(795\) 2.02666 0.0718783
\(796\) 10.0917i 0.357691i
\(797\) 22.9770 0.813888 0.406944 0.913453i \(-0.366594\pi\)
0.406944 + 0.913453i \(0.366594\pi\)
\(798\) 6.15676 0.217947
\(799\) 22.0494 17.2351i 0.780053 0.609735i
\(800\) −7.58864 −0.268299
\(801\) 19.3835 0.684882
\(802\) 71.8141i 2.53585i
\(803\) 35.1194 1.23934
\(804\) 14.9093i 0.525812i
\(805\) 25.4680i 0.897629i
\(806\) 23.3607i 0.822845i
\(807\) −4.03120 −0.141905
\(808\) 3.80817 0.133971
\(809\) 42.7670i 1.50361i 0.659388 + 0.751803i \(0.270816\pi\)
−0.659388 + 0.751803i \(0.729184\pi\)
\(810\) 14.0361i 0.493179i
\(811\) 9.41136i 0.330478i 0.986254 + 0.165239i \(0.0528395\pi\)
−0.986254 + 0.165239i \(0.947161\pi\)
\(812\) −38.6225 −1.35538
\(813\) 1.16290i 0.0407846i
\(814\) −36.1978 −1.26873
\(815\) 8.69594 0.304606
\(816\) −2.84551 3.64035i −0.0996128 0.127438i
\(817\) 3.63317 0.127108
\(818\) 65.5318 2.29127
\(819\) 34.7792i 1.21529i
\(820\) 15.1773 0.530013
\(821\) 32.6681i 1.14012i −0.821602 0.570062i \(-0.806919\pi\)
0.821602 0.570062i \(-0.193081\pi\)
\(822\) 2.28231i 0.0796048i
\(823\) 15.9265i 0.555164i −0.960702 0.277582i \(-0.910467\pi\)
0.960702 0.277582i \(-0.0895331\pi\)
\(824\) 30.2472 1.05371
\(825\) 1.70928 0.0595093
\(826\) 24.7792i 0.862180i
\(827\) 6.01560i 0.209183i −0.994515 0.104591i \(-0.966647\pi\)
0.994515 0.104591i \(-0.0333535\pi\)
\(828\) 38.3123i 1.33144i
\(829\) 11.0472 0.383684 0.191842 0.981426i \(-0.438554\pi\)
0.191842 + 0.981426i \(0.438554\pi\)
\(830\) 17.8660i 0.620139i
\(831\) 6.55479 0.227383
\(832\) −32.3896 −1.12291
\(833\) 54.6007 42.6791i 1.89180 1.47874i
\(834\) 2.34858 0.0813248
\(835\) −1.37629 −0.0476285
\(836\) 9.26180i 0.320326i
\(837\) 12.5958 0.435375
\(838\) 39.4173i 1.36165i
\(839\) 35.9805i 1.24219i −0.783737 0.621093i \(-0.786689\pi\)
0.783737 0.621093i \(-0.213311\pi\)
\(840\) 4.04945i 0.139719i
\(841\) 20.4641 0.705659
\(842\) −1.65142 −0.0569116
\(843\) 7.07384i 0.243636i
\(844\) 60.2772i 2.07483i
\(845\) 6.07838i 0.209103i
\(846\) 39.9071 1.37203
\(847\) 4.63809i 0.159367i
\(848\) −7.81205 −0.268267
\(849\) −7.52586 −0.258287
\(850\) −7.04945 + 5.51026i −0.241794 + 0.189000i
\(851\) −27.4641 −0.941458
\(852\) 5.93827 0.203442
\(853\) 28.7792i 0.985382i −0.870204 0.492691i \(-0.836013\pi\)
0.870204 0.492691i \(-0.163987\pi\)
\(854\) −129.771 −4.44066
\(855\) 2.92162i 0.0999174i
\(856\) 19.4524i 0.664869i
\(857\) 17.0661i 0.582967i −0.956576 0.291483i \(-0.905851\pi\)
0.956576 0.291483i \(-0.0941488\pi\)
\(858\) 9.75872 0.333157
\(859\) −18.9360 −0.646088 −0.323044 0.946384i \(-0.604706\pi\)
−0.323044 + 0.946384i \(0.604706\pi\)
\(860\) 9.12783i 0.311256i
\(861\) 14.7382i 0.502277i
\(862\) 13.7743i 0.469155i
\(863\) 30.8332 1.04958 0.524788 0.851233i \(-0.324145\pi\)
0.524788 + 0.851233i \(0.324145\pi\)
\(864\) 23.3607i 0.794747i
\(865\) 17.3607 0.590281
\(866\) −7.86603 −0.267299
\(867\) −8.89496 2.21339i −0.302089 0.0751707i
\(868\) 54.0905 1.83595
\(869\) 21.9649 0.745109
\(870\) 3.41855i 0.115900i
\(871\) 26.8515 0.909828
\(872\) 12.5548i 0.425159i
\(873\) 22.1711i 0.750379i
\(874\) 12.2146i 0.413165i
\(875\) −4.87936 −0.164953
\(876\) 16.1834 0.546787
\(877\) 4.30898i 0.145504i 0.997350 + 0.0727519i \(0.0231781\pi\)
−0.997350 + 0.0727519i \(0.976822\pi\)
\(878\) 13.8999i 0.469098i
\(879\) 2.55479i 0.0861708i
\(880\) −6.58864 −0.222103
\(881\) 12.2245i 0.411852i −0.978568 0.205926i \(-0.933979\pi\)
0.978568 0.205926i \(-0.0660207\pi\)
\(882\) 98.8213 3.32749
\(883\) −28.2329 −0.950112 −0.475056 0.879956i \(-0.657572\pi\)
−0.475056 + 0.879956i \(0.657572\pi\)
\(884\) −23.1545 + 18.0989i −0.778770 + 0.608732i
\(885\) −1.26180 −0.0424148
\(886\) −59.6391 −2.00362
\(887\) 5.17396i 0.173725i 0.996220 + 0.0868623i \(0.0276840\pi\)
−0.996220 + 0.0868623i \(0.972316\pi\)
\(888\) −4.36683 −0.146541
\(889\) 39.2762i 1.31728i
\(890\) 15.5259i 0.520428i
\(891\) 20.5041i 0.686914i
\(892\) −5.93827 −0.198828
\(893\) −7.31965 −0.244943
\(894\) 16.7115i 0.558918i
\(895\) 6.83710i 0.228539i
\(896\) 56.3039i 1.88098i
\(897\) 7.40417 0.247218
\(898\) 62.3956i 2.08217i
\(899\) 11.9544 0.398702
\(900\) −7.34017 −0.244672
\(901\) −12.2101 + 9.54411i −0.406777 + 0.317960i
\(902\) 38.5380 1.28317
\(903\) −8.86376 −0.294968
\(904\) 26.1978i 0.871326i
\(905\) −15.0205 −0.499299
\(906\) 15.0517i 0.500060i
\(907\) 6.32457i 0.210004i −0.994472 0.105002i \(-0.966515\pi\)
0.994472 0.105002i \(-0.0334849\pi\)
\(908\) 25.8999i 0.859518i
\(909\) −6.70313 −0.222329
\(910\) −27.8576 −0.923471
\(911\) 27.6526i 0.916173i 0.888908 + 0.458086i \(0.151465\pi\)
−0.888908 + 0.458086i \(0.848535\pi\)
\(912\) 1.20847i 0.0400165i
\(913\) 26.0989i 0.863747i
\(914\) 68.1627 2.25462
\(915\) 6.60811i 0.218457i
\(916\) 19.9649 0.659660
\(917\) 57.0577 1.88421
\(918\) −16.9627 21.7009i −0.559851 0.716235i
\(919\) −47.5318 −1.56793 −0.783965 0.620805i \(-0.786806\pi\)
−0.783965 + 0.620805i \(0.786806\pi\)
\(920\) 8.03385 0.264868
\(921\) 11.6430i 0.383650i
\(922\) 16.8371 0.554500
\(923\) 10.6947i 0.352021i
\(924\) 22.5958i 0.743348i
\(925\) 5.26180i 0.173007i
\(926\) −52.5874 −1.72813
\(927\) −53.2411 −1.74867
\(928\) 22.1711i 0.727803i
\(929\) 43.7875i 1.43662i −0.695723 0.718310i \(-0.744916\pi\)
0.695723 0.718310i \(-0.255084\pi\)
\(930\) 4.78765i 0.156993i
\(931\) −18.1256 −0.594041
\(932\) 25.5897i 0.838218i
\(933\) 13.0823 0.428294
\(934\) 26.7298 0.874626
\(935\) −10.2979 + 8.04945i −0.336778 + 0.263245i
\(936\) −10.9711 −0.358601
\(937\) 14.2367 0.465094 0.232547 0.972585i \(-0.425294\pi\)
0.232547 + 0.972585i \(0.425294\pi\)
\(938\) 108.070i 3.52860i
\(939\) 2.19570 0.0716541
\(940\) 18.3896i 0.599803i
\(941\) 35.2183i 1.14808i 0.818826 + 0.574042i \(0.194625\pi\)
−0.818826 + 0.574042i \(0.805375\pi\)
\(942\) 4.39803i 0.143296i
\(943\) 29.2397 0.952175
\(944\) 4.86376 0.158302
\(945\) 15.0205i 0.488618i
\(946\) 23.1773i 0.753558i
\(947\) 49.1227i 1.59627i 0.602476 + 0.798137i \(0.294181\pi\)
−0.602476 + 0.798137i \(0.705819\pi\)
\(948\) 10.1217 0.328737
\(949\) 29.1461i 0.946122i
\(950\) 2.34017 0.0759252
\(951\) 4.34471 0.140887
\(952\) −19.0700 24.3968i −0.618061 0.790705i
\(953\) −17.0556 −0.552485 −0.276242 0.961088i \(-0.589089\pi\)
−0.276242 + 0.961088i \(0.589089\pi\)
\(954\) −22.0989 −0.715478
\(955\) 24.2823i 0.785757i
\(956\) 16.9483 0.548147
\(957\) 4.99386i 0.161428i
\(958\) 4.66475i 0.150711i
\(959\) 9.51745i 0.307334i
\(960\) −6.63809 −0.214243
\(961\) 14.2579 0.459933
\(962\) 30.0410i 0.968562i
\(963\) 34.2401i 1.10337i
\(964\) 6.76487i 0.217882i
\(965\) 11.8576 0.381710
\(966\) 29.7998i 0.958792i
\(967\) −25.1955 −0.810233 −0.405117 0.914265i \(-0.632769\pi\)
−0.405117 + 0.914265i \(0.632769\pi\)
\(968\) −1.46308 −0.0470251
\(969\) 1.47641 + 1.88882i 0.0474291 + 0.0606776i
\(970\) −17.7587 −0.570198
\(971\) 1.67420 0.0537277 0.0268639 0.999639i \(-0.491448\pi\)
0.0268639 + 0.999639i \(0.491448\pi\)
\(972\) 34.4690i 1.10560i
\(973\) −9.79380 −0.313975
\(974\) 86.9842i 2.78715i
\(975\) 1.41855i 0.0454300i
\(976\) 25.4719i 0.815335i
\(977\) −39.9109 −1.27686 −0.638432 0.769678i \(-0.720417\pi\)
−0.638432 + 0.769678i \(0.720417\pi\)
\(978\) 10.1750 0.325361
\(979\) 22.6803i 0.724867i
\(980\) 45.5380i 1.45466i
\(981\) 22.0989i 0.705563i
\(982\) 4.89496 0.156204
\(983\) 5.46081i 0.174173i −0.996201 0.0870864i \(-0.972244\pi\)
0.996201 0.0870864i \(-0.0277556\pi\)
\(984\) 4.64915 0.148209
\(985\) 18.2557 0.581673
\(986\) −16.0989 20.5958i −0.512693 0.655905i
\(987\) 17.8576 0.568414
\(988\) 7.68649 0.244540
\(989\) 17.5851i 0.559175i
\(990\) −18.6381 −0.592357
\(991\) 42.0749i 1.33655i −0.743913 0.668276i \(-0.767032\pi\)
0.743913 0.668276i \(-0.232968\pi\)
\(992\) 31.0505i 0.985854i
\(993\) 5.43084i 0.172342i
\(994\) −43.0433 −1.36525
\(995\) −3.72487 −0.118086
\(996\) 12.0267i 0.381079i
\(997\) 54.6681i 1.73135i −0.500602 0.865677i \(-0.666888\pi\)
0.500602 0.865677i \(-0.333112\pi\)
\(998\) 92.3267i 2.92255i
\(999\) 16.1978 0.512476
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 85.2.d.a.16.1 6
3.2 odd 2 765.2.g.b.271.5 6
4.3 odd 2 1360.2.c.f.1121.4 6
5.2 odd 4 425.2.c.b.424.1 6
5.3 odd 4 425.2.c.a.424.6 6
5.4 even 2 425.2.d.c.101.6 6
17.4 even 4 1445.2.a.k.1.3 3
17.13 even 4 1445.2.a.j.1.3 3
17.16 even 2 inner 85.2.d.a.16.2 yes 6
51.50 odd 2 765.2.g.b.271.6 6
68.67 odd 2 1360.2.c.f.1121.3 6
85.4 even 4 7225.2.a.r.1.1 3
85.33 odd 4 425.2.c.b.424.6 6
85.64 even 4 7225.2.a.q.1.1 3
85.67 odd 4 425.2.c.a.424.1 6
85.84 even 2 425.2.d.c.101.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.d.a.16.1 6 1.1 even 1 trivial
85.2.d.a.16.2 yes 6 17.16 even 2 inner
425.2.c.a.424.1 6 85.67 odd 4
425.2.c.a.424.6 6 5.3 odd 4
425.2.c.b.424.1 6 5.2 odd 4
425.2.c.b.424.6 6 85.33 odd 4
425.2.d.c.101.5 6 85.84 even 2
425.2.d.c.101.6 6 5.4 even 2
765.2.g.b.271.5 6 3.2 odd 2
765.2.g.b.271.6 6 51.50 odd 2
1360.2.c.f.1121.3 6 68.67 odd 2
1360.2.c.f.1121.4 6 4.3 odd 2
1445.2.a.j.1.3 3 17.13 even 4
1445.2.a.k.1.3 3 17.4 even 4
7225.2.a.q.1.1 3 85.64 even 4
7225.2.a.r.1.1 3 85.4 even 4