Properties

Label 4235.2.a.bp.1.12
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $18$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + 6290 x^{10} - 9228 x^{9} - 12411 x^{8} + 14224 x^{7} + 14618 x^{6} - 10744 x^{5} + \cdots - 176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(1.11352\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.11352 q^{2} +2.27506 q^{3} -0.760083 q^{4} +1.00000 q^{5} +2.53332 q^{6} +1.00000 q^{7} -3.07340 q^{8} +2.17591 q^{9} +O(q^{10})\) \(q+1.11352 q^{2} +2.27506 q^{3} -0.760083 q^{4} +1.00000 q^{5} +2.53332 q^{6} +1.00000 q^{7} -3.07340 q^{8} +2.17591 q^{9} +1.11352 q^{10} -1.72924 q^{12} +4.17534 q^{13} +1.11352 q^{14} +2.27506 q^{15} -1.90211 q^{16} +5.01458 q^{17} +2.42291 q^{18} +0.312580 q^{19} -0.760083 q^{20} +2.27506 q^{21} -1.02983 q^{23} -6.99217 q^{24} +1.00000 q^{25} +4.64930 q^{26} -1.87485 q^{27} -0.760083 q^{28} +2.15091 q^{29} +2.53332 q^{30} +10.5165 q^{31} +4.02876 q^{32} +5.58381 q^{34} +1.00000 q^{35} -1.65387 q^{36} -2.11848 q^{37} +0.348062 q^{38} +9.49915 q^{39} -3.07340 q^{40} -11.3552 q^{41} +2.53332 q^{42} -7.14249 q^{43} +2.17591 q^{45} -1.14673 q^{46} +3.94104 q^{47} -4.32742 q^{48} +1.00000 q^{49} +1.11352 q^{50} +11.4085 q^{51} -3.17360 q^{52} -1.36714 q^{53} -2.08768 q^{54} -3.07340 q^{56} +0.711138 q^{57} +2.39507 q^{58} +11.8053 q^{59} -1.72924 q^{60} -11.9389 q^{61} +11.7103 q^{62} +2.17591 q^{63} +8.29031 q^{64} +4.17534 q^{65} +11.7596 q^{67} -3.81150 q^{68} -2.34293 q^{69} +1.11352 q^{70} +8.76864 q^{71} -6.68744 q^{72} +8.87242 q^{73} -2.35896 q^{74} +2.27506 q^{75} -0.237586 q^{76} +10.5775 q^{78} -1.61328 q^{79} -1.90211 q^{80} -10.7931 q^{81} -12.6441 q^{82} -14.3618 q^{83} -1.72924 q^{84} +5.01458 q^{85} -7.95328 q^{86} +4.89346 q^{87} +12.0930 q^{89} +2.42291 q^{90} +4.17534 q^{91} +0.782759 q^{92} +23.9258 q^{93} +4.38841 q^{94} +0.312580 q^{95} +9.16569 q^{96} -6.89342 q^{97} +1.11352 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9} + 2 q^{10} + 15 q^{12} + 8 q^{13} + 2 q^{14} + 5 q^{15} + 44 q^{16} - 5 q^{17} + 2 q^{18} + 15 q^{19} + 24 q^{20} + 5 q^{21} + 4 q^{23} + 8 q^{24} + 18 q^{25} + 14 q^{26} + 20 q^{27} + 24 q^{28} - 6 q^{29} - q^{30} + 22 q^{31} - 6 q^{32} + 44 q^{34} + 18 q^{35} + 83 q^{36} + 26 q^{37} - 11 q^{38} - 38 q^{39} + 6 q^{40} + 7 q^{41} - q^{42} + 10 q^{43} + 37 q^{45} - 40 q^{46} - q^{47} - 15 q^{48} + 18 q^{49} + 2 q^{50} - 11 q^{51} + 18 q^{52} + 23 q^{53} + 13 q^{54} + 6 q^{56} - 16 q^{57} + 2 q^{58} + 30 q^{59} + 15 q^{60} + 17 q^{61} + 57 q^{62} + 37 q^{63} + 64 q^{64} + 8 q^{65} + 29 q^{67} - 66 q^{68} + 54 q^{69} + 2 q^{70} - 2 q^{71} - 77 q^{72} + 3 q^{73} - 48 q^{74} + 5 q^{75} + 47 q^{76} + 10 q^{78} - 18 q^{79} + 44 q^{80} + 110 q^{81} + 56 q^{82} + 9 q^{83} + 15 q^{84} - 5 q^{85} + 25 q^{86} + 23 q^{87} + 59 q^{89} + 2 q^{90} + 8 q^{91} - 74 q^{92} + 33 q^{93} - 19 q^{94} + 15 q^{95} + 14 q^{96} + 30 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11352 0.787374 0.393687 0.919244i \(-0.371199\pi\)
0.393687 + 0.919244i \(0.371199\pi\)
\(3\) 2.27506 1.31351 0.656754 0.754105i \(-0.271929\pi\)
0.656754 + 0.754105i \(0.271929\pi\)
\(4\) −0.760083 −0.380042
\(5\) 1.00000 0.447214
\(6\) 2.53332 1.03422
\(7\) 1.00000 0.377964
\(8\) −3.07340 −1.08661
\(9\) 2.17591 0.725304
\(10\) 1.11352 0.352125
\(11\) 0 0
\(12\) −1.72924 −0.499188
\(13\) 4.17534 1.15803 0.579015 0.815317i \(-0.303437\pi\)
0.579015 + 0.815317i \(0.303437\pi\)
\(14\) 1.11352 0.297600
\(15\) 2.27506 0.587419
\(16\) −1.90211 −0.475527
\(17\) 5.01458 1.21621 0.608107 0.793855i \(-0.291929\pi\)
0.608107 + 0.793855i \(0.291929\pi\)
\(18\) 2.42291 0.571086
\(19\) 0.312580 0.0717107 0.0358553 0.999357i \(-0.488584\pi\)
0.0358553 + 0.999357i \(0.488584\pi\)
\(20\) −0.760083 −0.169960
\(21\) 2.27506 0.496459
\(22\) 0 0
\(23\) −1.02983 −0.214735 −0.107368 0.994219i \(-0.534242\pi\)
−0.107368 + 0.994219i \(0.534242\pi\)
\(24\) −6.99217 −1.42727
\(25\) 1.00000 0.200000
\(26\) 4.64930 0.911803
\(27\) −1.87485 −0.360815
\(28\) −0.760083 −0.143642
\(29\) 2.15091 0.399414 0.199707 0.979856i \(-0.436001\pi\)
0.199707 + 0.979856i \(0.436001\pi\)
\(30\) 2.53332 0.462519
\(31\) 10.5165 1.88883 0.944414 0.328760i \(-0.106631\pi\)
0.944414 + 0.328760i \(0.106631\pi\)
\(32\) 4.02876 0.712192
\(33\) 0 0
\(34\) 5.58381 0.957616
\(35\) 1.00000 0.169031
\(36\) −1.65387 −0.275646
\(37\) −2.11848 −0.348276 −0.174138 0.984721i \(-0.555714\pi\)
−0.174138 + 0.984721i \(0.555714\pi\)
\(38\) 0.348062 0.0564631
\(39\) 9.49915 1.52108
\(40\) −3.07340 −0.485946
\(41\) −11.3552 −1.77338 −0.886689 0.462367i \(-0.847000\pi\)
−0.886689 + 0.462367i \(0.847000\pi\)
\(42\) 2.53332 0.390900
\(43\) −7.14249 −1.08922 −0.544610 0.838689i \(-0.683322\pi\)
−0.544610 + 0.838689i \(0.683322\pi\)
\(44\) 0 0
\(45\) 2.17591 0.324366
\(46\) −1.14673 −0.169077
\(47\) 3.94104 0.574860 0.287430 0.957802i \(-0.407199\pi\)
0.287430 + 0.957802i \(0.407199\pi\)
\(48\) −4.32742 −0.624609
\(49\) 1.00000 0.142857
\(50\) 1.11352 0.157475
\(51\) 11.4085 1.59751
\(52\) −3.17360 −0.440099
\(53\) −1.36714 −0.187791 −0.0938954 0.995582i \(-0.529932\pi\)
−0.0938954 + 0.995582i \(0.529932\pi\)
\(54\) −2.08768 −0.284097
\(55\) 0 0
\(56\) −3.07340 −0.410700
\(57\) 0.711138 0.0941926
\(58\) 2.39507 0.314489
\(59\) 11.8053 1.53692 0.768461 0.639897i \(-0.221023\pi\)
0.768461 + 0.639897i \(0.221023\pi\)
\(60\) −1.72924 −0.223244
\(61\) −11.9389 −1.52861 −0.764307 0.644853i \(-0.776919\pi\)
−0.764307 + 0.644853i \(0.776919\pi\)
\(62\) 11.7103 1.48721
\(63\) 2.17591 0.274139
\(64\) 8.29031 1.03629
\(65\) 4.17534 0.517887
\(66\) 0 0
\(67\) 11.7596 1.43666 0.718331 0.695701i \(-0.244906\pi\)
0.718331 + 0.695701i \(0.244906\pi\)
\(68\) −3.81150 −0.462212
\(69\) −2.34293 −0.282056
\(70\) 1.11352 0.133091
\(71\) 8.76864 1.04065 0.520323 0.853969i \(-0.325812\pi\)
0.520323 + 0.853969i \(0.325812\pi\)
\(72\) −6.68744 −0.788122
\(73\) 8.87242 1.03844 0.519219 0.854641i \(-0.326223\pi\)
0.519219 + 0.854641i \(0.326223\pi\)
\(74\) −2.35896 −0.274224
\(75\) 2.27506 0.262702
\(76\) −0.237586 −0.0272530
\(77\) 0 0
\(78\) 10.5775 1.19766
\(79\) −1.61328 −0.181508 −0.0907538 0.995873i \(-0.528928\pi\)
−0.0907538 + 0.995873i \(0.528928\pi\)
\(80\) −1.90211 −0.212662
\(81\) −10.7931 −1.19924
\(82\) −12.6441 −1.39631
\(83\) −14.3618 −1.57642 −0.788208 0.615410i \(-0.788991\pi\)
−0.788208 + 0.615410i \(0.788991\pi\)
\(84\) −1.72924 −0.188675
\(85\) 5.01458 0.543907
\(86\) −7.95328 −0.857624
\(87\) 4.89346 0.524634
\(88\) 0 0
\(89\) 12.0930 1.28185 0.640926 0.767603i \(-0.278551\pi\)
0.640926 + 0.767603i \(0.278551\pi\)
\(90\) 2.42291 0.255397
\(91\) 4.17534 0.437694
\(92\) 0.782759 0.0816082
\(93\) 23.9258 2.48099
\(94\) 4.38841 0.452630
\(95\) 0.312580 0.0320700
\(96\) 9.16569 0.935470
\(97\) −6.89342 −0.699921 −0.349961 0.936764i \(-0.613805\pi\)
−0.349961 + 0.936764i \(0.613805\pi\)
\(98\) 1.11352 0.112482
\(99\) 0 0
\(100\) −0.760083 −0.0760083
\(101\) −8.55470 −0.851224 −0.425612 0.904906i \(-0.639941\pi\)
−0.425612 + 0.904906i \(0.639941\pi\)
\(102\) 12.7035 1.25784
\(103\) 6.76626 0.666700 0.333350 0.942803i \(-0.391821\pi\)
0.333350 + 0.942803i \(0.391821\pi\)
\(104\) −12.8325 −1.25833
\(105\) 2.27506 0.222023
\(106\) −1.52233 −0.147862
\(107\) 8.94463 0.864710 0.432355 0.901703i \(-0.357683\pi\)
0.432355 + 0.901703i \(0.357683\pi\)
\(108\) 1.42504 0.137125
\(109\) 13.0660 1.25150 0.625749 0.780025i \(-0.284793\pi\)
0.625749 + 0.780025i \(0.284793\pi\)
\(110\) 0 0
\(111\) −4.81968 −0.457463
\(112\) −1.90211 −0.179732
\(113\) 7.00687 0.659151 0.329576 0.944129i \(-0.393094\pi\)
0.329576 + 0.944129i \(0.393094\pi\)
\(114\) 0.791863 0.0741648
\(115\) −1.02983 −0.0960324
\(116\) −1.63487 −0.151794
\(117\) 9.08517 0.839924
\(118\) 13.1454 1.21013
\(119\) 5.01458 0.459686
\(120\) −6.99217 −0.638295
\(121\) 0 0
\(122\) −13.2941 −1.20359
\(123\) −25.8337 −2.32935
\(124\) −7.99345 −0.717833
\(125\) 1.00000 0.0894427
\(126\) 2.42291 0.215850
\(127\) 4.24376 0.376573 0.188286 0.982114i \(-0.439707\pi\)
0.188286 + 0.982114i \(0.439707\pi\)
\(128\) 1.17386 0.103755
\(129\) −16.2496 −1.43070
\(130\) 4.64930 0.407771
\(131\) 11.3525 0.991870 0.495935 0.868360i \(-0.334825\pi\)
0.495935 + 0.868360i \(0.334825\pi\)
\(132\) 0 0
\(133\) 0.312580 0.0271041
\(134\) 13.0945 1.13119
\(135\) −1.87485 −0.161361
\(136\) −15.4118 −1.32155
\(137\) −0.885628 −0.0756643 −0.0378322 0.999284i \(-0.512045\pi\)
−0.0378322 + 0.999284i \(0.512045\pi\)
\(138\) −2.60889 −0.222084
\(139\) 14.2443 1.20818 0.604092 0.796915i \(-0.293536\pi\)
0.604092 + 0.796915i \(0.293536\pi\)
\(140\) −0.760083 −0.0642387
\(141\) 8.96611 0.755083
\(142\) 9.76402 0.819378
\(143\) 0 0
\(144\) −4.13882 −0.344902
\(145\) 2.15091 0.178623
\(146\) 9.87957 0.817639
\(147\) 2.27506 0.187644
\(148\) 1.61022 0.132359
\(149\) −7.94306 −0.650720 −0.325360 0.945590i \(-0.605486\pi\)
−0.325360 + 0.945590i \(0.605486\pi\)
\(150\) 2.53332 0.206845
\(151\) −14.3899 −1.17103 −0.585515 0.810662i \(-0.699108\pi\)
−0.585515 + 0.810662i \(0.699108\pi\)
\(152\) −0.960681 −0.0779215
\(153\) 10.9113 0.882125
\(154\) 0 0
\(155\) 10.5165 0.844709
\(156\) −7.22015 −0.578074
\(157\) −12.1593 −0.970416 −0.485208 0.874399i \(-0.661256\pi\)
−0.485208 + 0.874399i \(0.661256\pi\)
\(158\) −1.79641 −0.142914
\(159\) −3.11032 −0.246665
\(160\) 4.02876 0.318502
\(161\) −1.02983 −0.0811622
\(162\) −12.0183 −0.944249
\(163\) 16.3139 1.27780 0.638902 0.769288i \(-0.279389\pi\)
0.638902 + 0.769288i \(0.279389\pi\)
\(164\) 8.63086 0.673957
\(165\) 0 0
\(166\) −15.9921 −1.24123
\(167\) −10.5623 −0.817337 −0.408669 0.912683i \(-0.634007\pi\)
−0.408669 + 0.912683i \(0.634007\pi\)
\(168\) −6.99217 −0.539458
\(169\) 4.43343 0.341033
\(170\) 5.58381 0.428259
\(171\) 0.680146 0.0520120
\(172\) 5.42889 0.413949
\(173\) −8.13127 −0.618209 −0.309104 0.951028i \(-0.600029\pi\)
−0.309104 + 0.951028i \(0.600029\pi\)
\(174\) 5.44894 0.413083
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 26.8579 2.01876
\(178\) 13.4657 1.00930
\(179\) −21.2805 −1.59058 −0.795291 0.606228i \(-0.792682\pi\)
−0.795291 + 0.606228i \(0.792682\pi\)
\(180\) −1.65387 −0.123273
\(181\) −2.81598 −0.209310 −0.104655 0.994509i \(-0.533374\pi\)
−0.104655 + 0.994509i \(0.533374\pi\)
\(182\) 4.64930 0.344629
\(183\) −27.1616 −2.00785
\(184\) 3.16508 0.233333
\(185\) −2.11848 −0.155754
\(186\) 26.6418 1.95347
\(187\) 0 0
\(188\) −2.99552 −0.218470
\(189\) −1.87485 −0.136375
\(190\) 0.348062 0.0252511
\(191\) −1.48362 −0.107351 −0.0536755 0.998558i \(-0.517094\pi\)
−0.0536755 + 0.998558i \(0.517094\pi\)
\(192\) 18.8610 1.36117
\(193\) 7.36948 0.530467 0.265233 0.964184i \(-0.414551\pi\)
0.265233 + 0.964184i \(0.414551\pi\)
\(194\) −7.67593 −0.551100
\(195\) 9.49915 0.680248
\(196\) −0.760083 −0.0542916
\(197\) 4.74986 0.338413 0.169207 0.985581i \(-0.445879\pi\)
0.169207 + 0.985581i \(0.445879\pi\)
\(198\) 0 0
\(199\) 18.5306 1.31360 0.656801 0.754064i \(-0.271909\pi\)
0.656801 + 0.754064i \(0.271909\pi\)
\(200\) −3.07340 −0.217322
\(201\) 26.7538 1.88707
\(202\) −9.52579 −0.670232
\(203\) 2.15091 0.150964
\(204\) −8.67139 −0.607119
\(205\) −11.3552 −0.793078
\(206\) 7.53434 0.524942
\(207\) −2.24083 −0.155748
\(208\) −7.94194 −0.550674
\(209\) 0 0
\(210\) 2.53332 0.174816
\(211\) −1.09345 −0.0752763 −0.0376382 0.999291i \(-0.511983\pi\)
−0.0376382 + 0.999291i \(0.511983\pi\)
\(212\) 1.03914 0.0713683
\(213\) 19.9492 1.36690
\(214\) 9.95998 0.680851
\(215\) −7.14249 −0.487114
\(216\) 5.76216 0.392065
\(217\) 10.5165 0.713910
\(218\) 14.5492 0.985397
\(219\) 20.1853 1.36400
\(220\) 0 0
\(221\) 20.9375 1.40841
\(222\) −5.36678 −0.360195
\(223\) −19.5847 −1.31149 −0.655743 0.754984i \(-0.727644\pi\)
−0.655743 + 0.754984i \(0.727644\pi\)
\(224\) 4.02876 0.269183
\(225\) 2.17591 0.145061
\(226\) 7.80226 0.518999
\(227\) −9.66593 −0.641551 −0.320775 0.947155i \(-0.603943\pi\)
−0.320775 + 0.947155i \(0.603943\pi\)
\(228\) −0.540524 −0.0357971
\(229\) −15.4487 −1.02088 −0.510440 0.859914i \(-0.670517\pi\)
−0.510440 + 0.859914i \(0.670517\pi\)
\(230\) −1.14673 −0.0756135
\(231\) 0 0
\(232\) −6.61060 −0.434007
\(233\) 9.47282 0.620585 0.310293 0.950641i \(-0.399573\pi\)
0.310293 + 0.950641i \(0.399573\pi\)
\(234\) 10.1165 0.661335
\(235\) 3.94104 0.257085
\(236\) −8.97302 −0.584094
\(237\) −3.67030 −0.238412
\(238\) 5.58381 0.361945
\(239\) −26.7655 −1.73132 −0.865659 0.500634i \(-0.833100\pi\)
−0.865659 + 0.500634i \(0.833100\pi\)
\(240\) −4.32742 −0.279333
\(241\) 0.595851 0.0383821 0.0191911 0.999816i \(-0.493891\pi\)
0.0191911 + 0.999816i \(0.493891\pi\)
\(242\) 0 0
\(243\) −18.9305 −1.21439
\(244\) 9.07452 0.580937
\(245\) 1.00000 0.0638877
\(246\) −28.7662 −1.83407
\(247\) 1.30512 0.0830431
\(248\) −32.3215 −2.05242
\(249\) −32.6741 −2.07063
\(250\) 1.11352 0.0704249
\(251\) −12.8329 −0.810005 −0.405002 0.914316i \(-0.632729\pi\)
−0.405002 + 0.914316i \(0.632729\pi\)
\(252\) −1.65387 −0.104184
\(253\) 0 0
\(254\) 4.72549 0.296504
\(255\) 11.4085 0.714427
\(256\) −15.2735 −0.954594
\(257\) −16.9113 −1.05490 −0.527449 0.849587i \(-0.676852\pi\)
−0.527449 + 0.849587i \(0.676852\pi\)
\(258\) −18.0942 −1.12650
\(259\) −2.11848 −0.131636
\(260\) −3.17360 −0.196818
\(261\) 4.68020 0.289697
\(262\) 12.6412 0.780973
\(263\) −25.4109 −1.56690 −0.783450 0.621455i \(-0.786542\pi\)
−0.783450 + 0.621455i \(0.786542\pi\)
\(264\) 0 0
\(265\) −1.36714 −0.0839826
\(266\) 0.348062 0.0213411
\(267\) 27.5123 1.68372
\(268\) −8.93826 −0.545991
\(269\) −3.23176 −0.197044 −0.0985221 0.995135i \(-0.531412\pi\)
−0.0985221 + 0.995135i \(0.531412\pi\)
\(270\) −2.08768 −0.127052
\(271\) −23.6985 −1.43958 −0.719791 0.694190i \(-0.755763\pi\)
−0.719791 + 0.694190i \(0.755763\pi\)
\(272\) −9.53827 −0.578342
\(273\) 9.49915 0.574915
\(274\) −0.986161 −0.0595762
\(275\) 0 0
\(276\) 1.78083 0.107193
\(277\) 0.376998 0.0226516 0.0113258 0.999936i \(-0.496395\pi\)
0.0113258 + 0.999936i \(0.496395\pi\)
\(278\) 15.8612 0.951293
\(279\) 22.8831 1.36997
\(280\) −3.07340 −0.183671
\(281\) 9.76106 0.582296 0.291148 0.956678i \(-0.405963\pi\)
0.291148 + 0.956678i \(0.405963\pi\)
\(282\) 9.98390 0.594533
\(283\) −15.9320 −0.947056 −0.473528 0.880779i \(-0.657020\pi\)
−0.473528 + 0.880779i \(0.657020\pi\)
\(284\) −6.66490 −0.395489
\(285\) 0.711138 0.0421242
\(286\) 0 0
\(287\) −11.3552 −0.670274
\(288\) 8.76624 0.516556
\(289\) 8.14599 0.479176
\(290\) 2.39507 0.140644
\(291\) −15.6830 −0.919352
\(292\) −6.74377 −0.394649
\(293\) 4.04334 0.236214 0.118107 0.993001i \(-0.462317\pi\)
0.118107 + 0.993001i \(0.462317\pi\)
\(294\) 2.53332 0.147746
\(295\) 11.8053 0.687332
\(296\) 6.51093 0.378440
\(297\) 0 0
\(298\) −8.84472 −0.512361
\(299\) −4.29990 −0.248670
\(300\) −1.72924 −0.0998375
\(301\) −7.14249 −0.411686
\(302\) −16.0233 −0.922039
\(303\) −19.4625 −1.11809
\(304\) −0.594560 −0.0341004
\(305\) −11.9389 −0.683617
\(306\) 12.1499 0.694563
\(307\) −27.9625 −1.59590 −0.797951 0.602722i \(-0.794083\pi\)
−0.797951 + 0.602722i \(0.794083\pi\)
\(308\) 0 0
\(309\) 15.3937 0.875716
\(310\) 11.7103 0.665102
\(311\) −25.7900 −1.46242 −0.731209 0.682154i \(-0.761043\pi\)
−0.731209 + 0.682154i \(0.761043\pi\)
\(312\) −29.1947 −1.65282
\(313\) −5.19429 −0.293599 −0.146799 0.989166i \(-0.546897\pi\)
−0.146799 + 0.989166i \(0.546897\pi\)
\(314\) −13.5396 −0.764081
\(315\) 2.17591 0.122599
\(316\) 1.22622 0.0689805
\(317\) 18.8735 1.06004 0.530021 0.847984i \(-0.322184\pi\)
0.530021 + 0.847984i \(0.322184\pi\)
\(318\) −3.46340 −0.194218
\(319\) 0 0
\(320\) 8.29031 0.463442
\(321\) 20.3496 1.13580
\(322\) −1.14673 −0.0639050
\(323\) 1.56745 0.0872155
\(324\) 8.20368 0.455760
\(325\) 4.17534 0.231606
\(326\) 18.1658 1.00611
\(327\) 29.7260 1.64385
\(328\) 34.8989 1.92697
\(329\) 3.94104 0.217276
\(330\) 0 0
\(331\) −13.2955 −0.730785 −0.365393 0.930854i \(-0.619065\pi\)
−0.365393 + 0.930854i \(0.619065\pi\)
\(332\) 10.9162 0.599103
\(333\) −4.60963 −0.252606
\(334\) −11.7613 −0.643550
\(335\) 11.7596 0.642495
\(336\) −4.32742 −0.236080
\(337\) −6.70480 −0.365234 −0.182617 0.983184i \(-0.558457\pi\)
−0.182617 + 0.983184i \(0.558457\pi\)
\(338\) 4.93669 0.268521
\(339\) 15.9411 0.865801
\(340\) −3.81150 −0.206707
\(341\) 0 0
\(342\) 0.757353 0.0409530
\(343\) 1.00000 0.0539949
\(344\) 21.9517 1.18356
\(345\) −2.34293 −0.126139
\(346\) −9.05429 −0.486762
\(347\) 20.0331 1.07543 0.537717 0.843126i \(-0.319287\pi\)
0.537717 + 0.843126i \(0.319287\pi\)
\(348\) −3.71944 −0.199383
\(349\) −2.33534 −0.125008 −0.0625039 0.998045i \(-0.519909\pi\)
−0.0625039 + 0.998045i \(0.519909\pi\)
\(350\) 1.11352 0.0595199
\(351\) −7.82813 −0.417835
\(352\) 0 0
\(353\) −4.08902 −0.217636 −0.108818 0.994062i \(-0.534707\pi\)
−0.108818 + 0.994062i \(0.534707\pi\)
\(354\) 29.9066 1.58952
\(355\) 8.76864 0.465391
\(356\) −9.19166 −0.487157
\(357\) 11.4085 0.603801
\(358\) −23.6962 −1.25238
\(359\) 22.3068 1.17731 0.588654 0.808385i \(-0.299658\pi\)
0.588654 + 0.808385i \(0.299658\pi\)
\(360\) −6.68744 −0.352459
\(361\) −18.9023 −0.994858
\(362\) −3.13563 −0.164805
\(363\) 0 0
\(364\) −3.17360 −0.166342
\(365\) 8.87242 0.464403
\(366\) −30.2449 −1.58093
\(367\) −7.37748 −0.385101 −0.192551 0.981287i \(-0.561676\pi\)
−0.192551 + 0.981287i \(0.561676\pi\)
\(368\) 1.95885 0.102112
\(369\) −24.7078 −1.28624
\(370\) −2.35896 −0.122637
\(371\) −1.36714 −0.0709783
\(372\) −18.1856 −0.942879
\(373\) 6.98127 0.361476 0.180738 0.983531i \(-0.442151\pi\)
0.180738 + 0.983531i \(0.442151\pi\)
\(374\) 0 0
\(375\) 2.27506 0.117484
\(376\) −12.1124 −0.624648
\(377\) 8.98078 0.462534
\(378\) −2.08768 −0.107378
\(379\) 10.5931 0.544129 0.272065 0.962279i \(-0.412294\pi\)
0.272065 + 0.962279i \(0.412294\pi\)
\(380\) −0.237586 −0.0121879
\(381\) 9.65482 0.494632
\(382\) −1.65204 −0.0845255
\(383\) −18.8440 −0.962883 −0.481442 0.876478i \(-0.659887\pi\)
−0.481442 + 0.876478i \(0.659887\pi\)
\(384\) 2.67060 0.136283
\(385\) 0 0
\(386\) 8.20603 0.417676
\(387\) −15.5414 −0.790016
\(388\) 5.23957 0.265999
\(389\) −17.5458 −0.889606 −0.444803 0.895629i \(-0.646726\pi\)
−0.444803 + 0.895629i \(0.646726\pi\)
\(390\) 10.5775 0.535610
\(391\) −5.16418 −0.261164
\(392\) −3.07340 −0.155230
\(393\) 25.8276 1.30283
\(394\) 5.28904 0.266458
\(395\) −1.61328 −0.0811727
\(396\) 0 0
\(397\) 13.0385 0.654384 0.327192 0.944958i \(-0.393898\pi\)
0.327192 + 0.944958i \(0.393898\pi\)
\(398\) 20.6341 1.03430
\(399\) 0.711138 0.0356014
\(400\) −1.90211 −0.0951054
\(401\) −31.2032 −1.55821 −0.779107 0.626891i \(-0.784327\pi\)
−0.779107 + 0.626891i \(0.784327\pi\)
\(402\) 29.7908 1.48583
\(403\) 43.9101 2.18732
\(404\) 6.50228 0.323501
\(405\) −10.7931 −0.536316
\(406\) 2.39507 0.118865
\(407\) 0 0
\(408\) −35.0628 −1.73587
\(409\) 4.00622 0.198095 0.0990473 0.995083i \(-0.468420\pi\)
0.0990473 + 0.995083i \(0.468420\pi\)
\(410\) −12.6441 −0.624450
\(411\) −2.01486 −0.0993857
\(412\) −5.14292 −0.253374
\(413\) 11.8053 0.580902
\(414\) −2.49520 −0.122632
\(415\) −14.3618 −0.704994
\(416\) 16.8214 0.824739
\(417\) 32.4066 1.58696
\(418\) 0 0
\(419\) 29.4722 1.43981 0.719904 0.694073i \(-0.244186\pi\)
0.719904 + 0.694073i \(0.244186\pi\)
\(420\) −1.72924 −0.0843781
\(421\) 20.4435 0.996357 0.498178 0.867075i \(-0.334003\pi\)
0.498178 + 0.867075i \(0.334003\pi\)
\(422\) −1.21758 −0.0592707
\(423\) 8.57536 0.416948
\(424\) 4.20175 0.204055
\(425\) 5.01458 0.243243
\(426\) 22.2138 1.07626
\(427\) −11.9389 −0.577762
\(428\) −6.79866 −0.328626
\(429\) 0 0
\(430\) −7.95328 −0.383541
\(431\) −14.6351 −0.704948 −0.352474 0.935822i \(-0.614659\pi\)
−0.352474 + 0.935822i \(0.614659\pi\)
\(432\) 3.56617 0.171577
\(433\) 35.0436 1.68409 0.842044 0.539409i \(-0.181352\pi\)
0.842044 + 0.539409i \(0.181352\pi\)
\(434\) 11.7103 0.562114
\(435\) 4.89346 0.234623
\(436\) −9.93126 −0.475621
\(437\) −0.321905 −0.0153988
\(438\) 22.4767 1.07398
\(439\) −30.3987 −1.45085 −0.725425 0.688301i \(-0.758357\pi\)
−0.725425 + 0.688301i \(0.758357\pi\)
\(440\) 0 0
\(441\) 2.17591 0.103615
\(442\) 23.3143 1.10895
\(443\) −21.3935 −1.01644 −0.508219 0.861228i \(-0.669696\pi\)
−0.508219 + 0.861228i \(0.669696\pi\)
\(444\) 3.66335 0.173855
\(445\) 12.0930 0.573262
\(446\) −21.8078 −1.03263
\(447\) −18.0710 −0.854727
\(448\) 8.29031 0.391680
\(449\) 16.2274 0.765816 0.382908 0.923786i \(-0.374923\pi\)
0.382908 + 0.923786i \(0.374923\pi\)
\(450\) 2.42291 0.114217
\(451\) 0 0
\(452\) −5.32581 −0.250505
\(453\) −32.7378 −1.53816
\(454\) −10.7632 −0.505140
\(455\) 4.17534 0.195743
\(456\) −2.18561 −0.102351
\(457\) 23.0581 1.07861 0.539306 0.842110i \(-0.318687\pi\)
0.539306 + 0.842110i \(0.318687\pi\)
\(458\) −17.2024 −0.803814
\(459\) −9.40159 −0.438828
\(460\) 0.782759 0.0364963
\(461\) −8.06731 −0.375732 −0.187866 0.982195i \(-0.560157\pi\)
−0.187866 + 0.982195i \(0.560157\pi\)
\(462\) 0 0
\(463\) −26.7289 −1.24220 −0.621100 0.783732i \(-0.713314\pi\)
−0.621100 + 0.783732i \(0.713314\pi\)
\(464\) −4.09126 −0.189932
\(465\) 23.9258 1.10953
\(466\) 10.5481 0.488633
\(467\) −24.7786 −1.14662 −0.573308 0.819340i \(-0.694340\pi\)
−0.573308 + 0.819340i \(0.694340\pi\)
\(468\) −6.90548 −0.319206
\(469\) 11.7596 0.543007
\(470\) 4.38841 0.202422
\(471\) −27.6631 −1.27465
\(472\) −36.2824 −1.67003
\(473\) 0 0
\(474\) −4.08694 −0.187719
\(475\) 0.312580 0.0143421
\(476\) −3.81150 −0.174700
\(477\) −2.97477 −0.136205
\(478\) −29.8038 −1.36320
\(479\) 3.03628 0.138731 0.0693655 0.997591i \(-0.477903\pi\)
0.0693655 + 0.997591i \(0.477903\pi\)
\(480\) 9.16569 0.418355
\(481\) −8.84537 −0.403314
\(482\) 0.663489 0.0302211
\(483\) −2.34293 −0.106607
\(484\) 0 0
\(485\) −6.89342 −0.313014
\(486\) −21.0794 −0.956183
\(487\) 39.8582 1.80615 0.903073 0.429487i \(-0.141306\pi\)
0.903073 + 0.429487i \(0.141306\pi\)
\(488\) 36.6928 1.66101
\(489\) 37.1152 1.67841
\(490\) 1.11352 0.0503035
\(491\) −25.2397 −1.13905 −0.569525 0.821974i \(-0.692873\pi\)
−0.569525 + 0.821974i \(0.692873\pi\)
\(492\) 19.6358 0.885248
\(493\) 10.7859 0.485773
\(494\) 1.45328 0.0653860
\(495\) 0 0
\(496\) −20.0036 −0.898188
\(497\) 8.76864 0.393327
\(498\) −36.3831 −1.63036
\(499\) 25.3540 1.13500 0.567501 0.823373i \(-0.307910\pi\)
0.567501 + 0.823373i \(0.307910\pi\)
\(500\) −0.760083 −0.0339919
\(501\) −24.0300 −1.07358
\(502\) −14.2896 −0.637777
\(503\) 19.9335 0.888789 0.444394 0.895831i \(-0.353419\pi\)
0.444394 + 0.895831i \(0.353419\pi\)
\(504\) −6.68744 −0.297882
\(505\) −8.55470 −0.380679
\(506\) 0 0
\(507\) 10.0863 0.447950
\(508\) −3.22561 −0.143113
\(509\) 7.03658 0.311891 0.155945 0.987766i \(-0.450158\pi\)
0.155945 + 0.987766i \(0.450158\pi\)
\(510\) 12.7035 0.562521
\(511\) 8.87242 0.392493
\(512\) −19.3550 −0.855378
\(513\) −0.586040 −0.0258743
\(514\) −18.8310 −0.830600
\(515\) 6.76626 0.298157
\(516\) 12.3511 0.543725
\(517\) 0 0
\(518\) −2.35896 −0.103647
\(519\) −18.4991 −0.812023
\(520\) −12.8325 −0.562741
\(521\) −24.3393 −1.06632 −0.533161 0.846014i \(-0.678996\pi\)
−0.533161 + 0.846014i \(0.678996\pi\)
\(522\) 5.21147 0.228100
\(523\) −21.2450 −0.928980 −0.464490 0.885578i \(-0.653762\pi\)
−0.464490 + 0.885578i \(0.653762\pi\)
\(524\) −8.62882 −0.376952
\(525\) 2.27506 0.0992919
\(526\) −28.2954 −1.23374
\(527\) 52.7360 2.29722
\(528\) 0 0
\(529\) −21.9394 −0.953889
\(530\) −1.52233 −0.0661258
\(531\) 25.6873 1.11474
\(532\) −0.237586 −0.0103007
\(533\) −47.4116 −2.05362
\(534\) 30.6353 1.32572
\(535\) 8.94463 0.386710
\(536\) −36.1419 −1.56109
\(537\) −48.4146 −2.08924
\(538\) −3.59862 −0.155148
\(539\) 0 0
\(540\) 1.42504 0.0613241
\(541\) −38.9615 −1.67509 −0.837543 0.546371i \(-0.816009\pi\)
−0.837543 + 0.546371i \(0.816009\pi\)
\(542\) −26.3887 −1.13349
\(543\) −6.40652 −0.274930
\(544\) 20.2026 0.866177
\(545\) 13.0660 0.559687
\(546\) 10.5775 0.452673
\(547\) 1.11963 0.0478718 0.0239359 0.999713i \(-0.492380\pi\)
0.0239359 + 0.999713i \(0.492380\pi\)
\(548\) 0.673151 0.0287556
\(549\) −25.9779 −1.10871
\(550\) 0 0
\(551\) 0.672331 0.0286423
\(552\) 7.20077 0.306485
\(553\) −1.61328 −0.0686034
\(554\) 0.419793 0.0178353
\(555\) −4.81968 −0.204584
\(556\) −10.8268 −0.459160
\(557\) −13.1234 −0.556058 −0.278029 0.960573i \(-0.589681\pi\)
−0.278029 + 0.960573i \(0.589681\pi\)
\(558\) 25.4807 1.07868
\(559\) −29.8223 −1.26135
\(560\) −1.90211 −0.0803787
\(561\) 0 0
\(562\) 10.8691 0.458485
\(563\) 19.6066 0.826322 0.413161 0.910658i \(-0.364425\pi\)
0.413161 + 0.910658i \(0.364425\pi\)
\(564\) −6.81499 −0.286963
\(565\) 7.00687 0.294781
\(566\) −17.7405 −0.745688
\(567\) −10.7931 −0.453269
\(568\) −26.9495 −1.13078
\(569\) −34.5612 −1.44888 −0.724441 0.689337i \(-0.757902\pi\)
−0.724441 + 0.689337i \(0.757902\pi\)
\(570\) 0.791863 0.0331675
\(571\) 0.701299 0.0293484 0.0146742 0.999892i \(-0.495329\pi\)
0.0146742 + 0.999892i \(0.495329\pi\)
\(572\) 0 0
\(573\) −3.37533 −0.141007
\(574\) −12.6441 −0.527756
\(575\) −1.02983 −0.0429470
\(576\) 18.0390 0.751624
\(577\) −0.0841312 −0.00350243 −0.00175121 0.999998i \(-0.500557\pi\)
−0.00175121 + 0.999998i \(0.500557\pi\)
\(578\) 9.07069 0.377291
\(579\) 16.7660 0.696772
\(580\) −1.63487 −0.0678843
\(581\) −14.3618 −0.595829
\(582\) −17.4632 −0.723874
\(583\) 0 0
\(584\) −27.2684 −1.12838
\(585\) 9.08517 0.375625
\(586\) 4.50232 0.185989
\(587\) 38.9212 1.60645 0.803226 0.595675i \(-0.203115\pi\)
0.803226 + 0.595675i \(0.203115\pi\)
\(588\) −1.72924 −0.0713125
\(589\) 3.28726 0.135449
\(590\) 13.1454 0.541188
\(591\) 10.8062 0.444509
\(592\) 4.02958 0.165615
\(593\) −26.3594 −1.08245 −0.541225 0.840878i \(-0.682039\pi\)
−0.541225 + 0.840878i \(0.682039\pi\)
\(594\) 0 0
\(595\) 5.01458 0.205578
\(596\) 6.03738 0.247301
\(597\) 42.1584 1.72543
\(598\) −4.78800 −0.195796
\(599\) −22.1910 −0.906700 −0.453350 0.891333i \(-0.649771\pi\)
−0.453350 + 0.891333i \(0.649771\pi\)
\(600\) −6.99217 −0.285454
\(601\) −14.5234 −0.592421 −0.296210 0.955123i \(-0.595723\pi\)
−0.296210 + 0.955123i \(0.595723\pi\)
\(602\) −7.95328 −0.324151
\(603\) 25.5878 1.04202
\(604\) 10.9375 0.445040
\(605\) 0 0
\(606\) −21.6718 −0.880356
\(607\) 42.3001 1.71691 0.858454 0.512890i \(-0.171425\pi\)
0.858454 + 0.512890i \(0.171425\pi\)
\(608\) 1.25931 0.0510717
\(609\) 4.89346 0.198293
\(610\) −13.2941 −0.538262
\(611\) 16.4552 0.665704
\(612\) −8.29348 −0.335244
\(613\) 25.2536 1.01998 0.509992 0.860179i \(-0.329648\pi\)
0.509992 + 0.860179i \(0.329648\pi\)
\(614\) −31.1366 −1.25657
\(615\) −25.8337 −1.04172
\(616\) 0 0
\(617\) 33.9481 1.36670 0.683350 0.730091i \(-0.260522\pi\)
0.683350 + 0.730091i \(0.260522\pi\)
\(618\) 17.1411 0.689516
\(619\) −31.4892 −1.26566 −0.632830 0.774291i \(-0.718107\pi\)
−0.632830 + 0.774291i \(0.718107\pi\)
\(620\) −7.99345 −0.321025
\(621\) 1.93078 0.0774797
\(622\) −28.7176 −1.15147
\(623\) 12.0930 0.484495
\(624\) −18.0684 −0.723315
\(625\) 1.00000 0.0400000
\(626\) −5.78393 −0.231172
\(627\) 0 0
\(628\) 9.24207 0.368799
\(629\) −10.6233 −0.423578
\(630\) 2.42291 0.0965311
\(631\) 28.5949 1.13834 0.569172 0.822219i \(-0.307264\pi\)
0.569172 + 0.822219i \(0.307264\pi\)
\(632\) 4.95823 0.197228
\(633\) −2.48767 −0.0988761
\(634\) 21.0160 0.834651
\(635\) 4.24376 0.168408
\(636\) 2.36411 0.0937429
\(637\) 4.17534 0.165433
\(638\) 0 0
\(639\) 19.0798 0.754785
\(640\) 1.17386 0.0464008
\(641\) −28.2997 −1.11777 −0.558885 0.829245i \(-0.688771\pi\)
−0.558885 + 0.829245i \(0.688771\pi\)
\(642\) 22.6596 0.894303
\(643\) 15.8817 0.626314 0.313157 0.949701i \(-0.398613\pi\)
0.313157 + 0.949701i \(0.398613\pi\)
\(644\) 0.782759 0.0308450
\(645\) −16.2496 −0.639828
\(646\) 1.74538 0.0686713
\(647\) −16.0475 −0.630894 −0.315447 0.948943i \(-0.602154\pi\)
−0.315447 + 0.948943i \(0.602154\pi\)
\(648\) 33.1716 1.30310
\(649\) 0 0
\(650\) 4.64930 0.182361
\(651\) 23.9258 0.937726
\(652\) −12.3999 −0.485618
\(653\) −21.5898 −0.844874 −0.422437 0.906392i \(-0.638825\pi\)
−0.422437 + 0.906392i \(0.638825\pi\)
\(654\) 33.1004 1.29433
\(655\) 11.3525 0.443578
\(656\) 21.5987 0.843289
\(657\) 19.3056 0.753183
\(658\) 4.38841 0.171078
\(659\) −34.0482 −1.32633 −0.663164 0.748474i \(-0.730787\pi\)
−0.663164 + 0.748474i \(0.730787\pi\)
\(660\) 0 0
\(661\) 38.5458 1.49926 0.749629 0.661858i \(-0.230232\pi\)
0.749629 + 0.661858i \(0.230232\pi\)
\(662\) −14.8047 −0.575401
\(663\) 47.6342 1.84996
\(664\) 44.1396 1.71295
\(665\) 0.312580 0.0121213
\(666\) −5.13289 −0.198896
\(667\) −2.21508 −0.0857682
\(668\) 8.02824 0.310622
\(669\) −44.5563 −1.72265
\(670\) 13.0945 0.505884
\(671\) 0 0
\(672\) 9.16569 0.353574
\(673\) −34.2289 −1.31943 −0.659715 0.751516i \(-0.729323\pi\)
−0.659715 + 0.751516i \(0.729323\pi\)
\(674\) −7.46590 −0.287576
\(675\) −1.87485 −0.0721630
\(676\) −3.36977 −0.129607
\(677\) 38.9730 1.49785 0.748926 0.662653i \(-0.230570\pi\)
0.748926 + 0.662653i \(0.230570\pi\)
\(678\) 17.7506 0.681709
\(679\) −6.89342 −0.264545
\(680\) −15.4118 −0.591015
\(681\) −21.9906 −0.842682
\(682\) 0 0
\(683\) 38.8225 1.48550 0.742750 0.669569i \(-0.233521\pi\)
0.742750 + 0.669569i \(0.233521\pi\)
\(684\) −0.516967 −0.0197667
\(685\) −0.885628 −0.0338381
\(686\) 1.11352 0.0425142
\(687\) −35.1468 −1.34093
\(688\) 13.5858 0.517953
\(689\) −5.70826 −0.217467
\(690\) −2.60889 −0.0993189
\(691\) −4.42354 −0.168279 −0.0841397 0.996454i \(-0.526814\pi\)
−0.0841397 + 0.996454i \(0.526814\pi\)
\(692\) 6.18044 0.234945
\(693\) 0 0
\(694\) 22.3072 0.846769
\(695\) 14.2443 0.540316
\(696\) −15.0395 −0.570072
\(697\) −56.9413 −2.15681
\(698\) −2.60044 −0.0984280
\(699\) 21.5513 0.815144
\(700\) −0.760083 −0.0287284
\(701\) −15.9428 −0.602150 −0.301075 0.953601i \(-0.597345\pi\)
−0.301075 + 0.953601i \(0.597345\pi\)
\(702\) −8.71675 −0.328992
\(703\) −0.662194 −0.0249751
\(704\) 0 0
\(705\) 8.96611 0.337683
\(706\) −4.55318 −0.171361
\(707\) −8.55470 −0.321733
\(708\) −20.4142 −0.767213
\(709\) 37.8285 1.42068 0.710340 0.703859i \(-0.248541\pi\)
0.710340 + 0.703859i \(0.248541\pi\)
\(710\) 9.76402 0.366437
\(711\) −3.51035 −0.131648
\(712\) −37.1665 −1.39287
\(713\) −10.8303 −0.405597
\(714\) 12.7035 0.475417
\(715\) 0 0
\(716\) 16.1750 0.604487
\(717\) −60.8933 −2.27410
\(718\) 24.8390 0.926982
\(719\) 32.2686 1.20342 0.601708 0.798716i \(-0.294487\pi\)
0.601708 + 0.798716i \(0.294487\pi\)
\(720\) −4.13882 −0.154245
\(721\) 6.76626 0.251989
\(722\) −21.0480 −0.783325
\(723\) 1.35560 0.0504152
\(724\) 2.14038 0.0795464
\(725\) 2.15091 0.0798828
\(726\) 0 0
\(727\) −30.5177 −1.13184 −0.565919 0.824461i \(-0.691479\pi\)
−0.565919 + 0.824461i \(0.691479\pi\)
\(728\) −12.8325 −0.475603
\(729\) −10.6887 −0.395879
\(730\) 9.87957 0.365659
\(731\) −35.8166 −1.32472
\(732\) 20.6451 0.763065
\(733\) 29.9784 1.10728 0.553638 0.832757i \(-0.313239\pi\)
0.553638 + 0.832757i \(0.313239\pi\)
\(734\) −8.21494 −0.303219
\(735\) 2.27506 0.0839170
\(736\) −4.14895 −0.152932
\(737\) 0 0
\(738\) −27.5125 −1.01275
\(739\) −27.2743 −1.00330 −0.501651 0.865070i \(-0.667274\pi\)
−0.501651 + 0.865070i \(0.667274\pi\)
\(740\) 1.61022 0.0591929
\(741\) 2.96924 0.109078
\(742\) −1.52233 −0.0558865
\(743\) 8.95224 0.328426 0.164213 0.986425i \(-0.447492\pi\)
0.164213 + 0.986425i \(0.447492\pi\)
\(744\) −73.5335 −2.69587
\(745\) −7.94306 −0.291011
\(746\) 7.77375 0.284617
\(747\) −31.2501 −1.14338
\(748\) 0 0
\(749\) 8.94463 0.326830
\(750\) 2.53332 0.0925037
\(751\) 7.39028 0.269675 0.134838 0.990868i \(-0.456949\pi\)
0.134838 + 0.990868i \(0.456949\pi\)
\(752\) −7.49628 −0.273361
\(753\) −29.1956 −1.06395
\(754\) 10.0002 0.364187
\(755\) −14.3899 −0.523700
\(756\) 1.42504 0.0518283
\(757\) 46.1150 1.67608 0.838039 0.545610i \(-0.183702\pi\)
0.838039 + 0.545610i \(0.183702\pi\)
\(758\) 11.7955 0.428433
\(759\) 0 0
\(760\) −0.960681 −0.0348475
\(761\) −11.4980 −0.416803 −0.208402 0.978043i \(-0.566826\pi\)
−0.208402 + 0.978043i \(0.566826\pi\)
\(762\) 10.7508 0.389460
\(763\) 13.0660 0.473022
\(764\) 1.12768 0.0407979
\(765\) 10.9113 0.394498
\(766\) −20.9831 −0.758150
\(767\) 49.2912 1.77980
\(768\) −34.7482 −1.25387
\(769\) −12.6422 −0.455888 −0.227944 0.973674i \(-0.573200\pi\)
−0.227944 + 0.973674i \(0.573200\pi\)
\(770\) 0 0
\(771\) −38.4743 −1.38562
\(772\) −5.60142 −0.201599
\(773\) 38.8039 1.39568 0.697839 0.716254i \(-0.254145\pi\)
0.697839 + 0.716254i \(0.254145\pi\)
\(774\) −17.3056 −0.622038
\(775\) 10.5165 0.377765
\(776\) 21.1862 0.760541
\(777\) −4.81968 −0.172905
\(778\) −19.5375 −0.700453
\(779\) −3.54939 −0.127170
\(780\) −7.22015 −0.258523
\(781\) 0 0
\(782\) −5.75039 −0.205634
\(783\) −4.03264 −0.144115
\(784\) −1.90211 −0.0679324
\(785\) −12.1593 −0.433983
\(786\) 28.7594 1.02581
\(787\) −30.2463 −1.07816 −0.539082 0.842253i \(-0.681229\pi\)
−0.539082 + 0.842253i \(0.681229\pi\)
\(788\) −3.61029 −0.128611
\(789\) −57.8113 −2.05814
\(790\) −1.79641 −0.0639133
\(791\) 7.00687 0.249136
\(792\) 0 0
\(793\) −49.8487 −1.77018
\(794\) 14.5186 0.515245
\(795\) −3.11032 −0.110312
\(796\) −14.0848 −0.499223
\(797\) 24.9560 0.883987 0.441993 0.897018i \(-0.354271\pi\)
0.441993 + 0.897018i \(0.354271\pi\)
\(798\) 0.791863 0.0280317
\(799\) 19.7626 0.699152
\(800\) 4.02876 0.142438
\(801\) 26.3132 0.929733
\(802\) −34.7453 −1.22690
\(803\) 0 0
\(804\) −20.3351 −0.717164
\(805\) −1.02983 −0.0362968
\(806\) 48.8946 1.72224
\(807\) −7.35247 −0.258819
\(808\) 26.2920 0.924948
\(809\) 52.4442 1.84384 0.921919 0.387382i \(-0.126621\pi\)
0.921919 + 0.387382i \(0.126621\pi\)
\(810\) −12.0183 −0.422281
\(811\) 50.8600 1.78594 0.892968 0.450119i \(-0.148619\pi\)
0.892968 + 0.450119i \(0.148619\pi\)
\(812\) −1.63487 −0.0573727
\(813\) −53.9156 −1.89090
\(814\) 0 0
\(815\) 16.3139 0.571451
\(816\) −21.7002 −0.759658
\(817\) −2.23260 −0.0781087
\(818\) 4.46098 0.155975
\(819\) 9.08517 0.317461
\(820\) 8.63086 0.301403
\(821\) 36.9164 1.28839 0.644196 0.764861i \(-0.277192\pi\)
0.644196 + 0.764861i \(0.277192\pi\)
\(822\) −2.24358 −0.0782538
\(823\) 9.01322 0.314181 0.157090 0.987584i \(-0.449789\pi\)
0.157090 + 0.987584i \(0.449789\pi\)
\(824\) −20.7954 −0.724442
\(825\) 0 0
\(826\) 13.1454 0.457387
\(827\) −2.33667 −0.0812540 −0.0406270 0.999174i \(-0.512936\pi\)
−0.0406270 + 0.999174i \(0.512936\pi\)
\(828\) 1.70321 0.0591908
\(829\) 13.5307 0.469941 0.234970 0.972003i \(-0.424501\pi\)
0.234970 + 0.972003i \(0.424501\pi\)
\(830\) −15.9921 −0.555094
\(831\) 0.857693 0.0297530
\(832\) 34.6148 1.20005
\(833\) 5.01458 0.173745
\(834\) 36.0853 1.24953
\(835\) −10.5623 −0.365524
\(836\) 0 0
\(837\) −19.7170 −0.681518
\(838\) 32.8177 1.13367
\(839\) −8.47250 −0.292503 −0.146252 0.989247i \(-0.546721\pi\)
−0.146252 + 0.989247i \(0.546721\pi\)
\(840\) −6.99217 −0.241253
\(841\) −24.3736 −0.840468
\(842\) 22.7642 0.784506
\(843\) 22.2070 0.764851
\(844\) 0.831114 0.0286081
\(845\) 4.43343 0.152515
\(846\) 9.54879 0.328294
\(847\) 0 0
\(848\) 2.60044 0.0892996
\(849\) −36.2462 −1.24397
\(850\) 5.58381 0.191523
\(851\) 2.18168 0.0747870
\(852\) −15.1631 −0.519478
\(853\) −20.0059 −0.684988 −0.342494 0.939520i \(-0.611272\pi\)
−0.342494 + 0.939520i \(0.611272\pi\)
\(854\) −13.2941 −0.454915
\(855\) 0.680146 0.0232605
\(856\) −27.4904 −0.939602
\(857\) 27.4144 0.936458 0.468229 0.883607i \(-0.344892\pi\)
0.468229 + 0.883607i \(0.344892\pi\)
\(858\) 0 0
\(859\) 3.23732 0.110456 0.0552279 0.998474i \(-0.482411\pi\)
0.0552279 + 0.998474i \(0.482411\pi\)
\(860\) 5.42889 0.185124
\(861\) −25.8337 −0.880410
\(862\) −16.2964 −0.555058
\(863\) −19.9925 −0.680553 −0.340276 0.940325i \(-0.610521\pi\)
−0.340276 + 0.940325i \(0.610521\pi\)
\(864\) −7.55333 −0.256970
\(865\) −8.13127 −0.276471
\(866\) 39.0216 1.32601
\(867\) 18.5326 0.629401
\(868\) −7.99345 −0.271315
\(869\) 0 0
\(870\) 5.44894 0.184736
\(871\) 49.1002 1.66370
\(872\) −40.1570 −1.35989
\(873\) −14.9995 −0.507656
\(874\) −0.358446 −0.0121246
\(875\) 1.00000 0.0338062
\(876\) −15.3425 −0.518375
\(877\) 3.34642 0.113001 0.0565003 0.998403i \(-0.482006\pi\)
0.0565003 + 0.998403i \(0.482006\pi\)
\(878\) −33.8494 −1.14236
\(879\) 9.19885 0.310269
\(880\) 0 0
\(881\) 47.4904 1.59999 0.799996 0.600005i \(-0.204835\pi\)
0.799996 + 0.600005i \(0.204835\pi\)
\(882\) 2.42291 0.0815837
\(883\) −32.5105 −1.09407 −0.547034 0.837111i \(-0.684243\pi\)
−0.547034 + 0.837111i \(0.684243\pi\)
\(884\) −15.9143 −0.535255
\(885\) 26.8579 0.902817
\(886\) −23.8220 −0.800317
\(887\) 56.2965 1.89025 0.945126 0.326705i \(-0.105938\pi\)
0.945126 + 0.326705i \(0.105938\pi\)
\(888\) 14.8128 0.497084
\(889\) 4.24376 0.142331
\(890\) 13.4657 0.451372
\(891\) 0 0
\(892\) 14.8860 0.498419
\(893\) 1.23189 0.0412236
\(894\) −20.1223 −0.672990
\(895\) −21.2805 −0.711330
\(896\) 1.17386 0.0392158
\(897\) −9.78254 −0.326630
\(898\) 18.0694 0.602984
\(899\) 22.6202 0.754424
\(900\) −1.65387 −0.0551291
\(901\) −6.85562 −0.228394
\(902\) 0 0
\(903\) −16.2496 −0.540754
\(904\) −21.5349 −0.716240
\(905\) −2.81598 −0.0936062
\(906\) −36.4541 −1.21111
\(907\) 23.4679 0.779238 0.389619 0.920976i \(-0.372607\pi\)
0.389619 + 0.920976i \(0.372607\pi\)
\(908\) 7.34691 0.243816
\(909\) −18.6143 −0.617397
\(910\) 4.64930 0.154123
\(911\) 7.44129 0.246541 0.123270 0.992373i \(-0.460662\pi\)
0.123270 + 0.992373i \(0.460662\pi\)
\(912\) −1.35266 −0.0447911
\(913\) 0 0
\(914\) 25.6755 0.849271
\(915\) −27.1616 −0.897936
\(916\) 11.7423 0.387976
\(917\) 11.3525 0.374892
\(918\) −10.4688 −0.345522
\(919\) −17.1308 −0.565093 −0.282546 0.959254i \(-0.591179\pi\)
−0.282546 + 0.959254i \(0.591179\pi\)
\(920\) 3.16508 0.104350
\(921\) −63.6164 −2.09623
\(922\) −8.98307 −0.295842
\(923\) 36.6120 1.20510
\(924\) 0 0
\(925\) −2.11848 −0.0696552
\(926\) −29.7631 −0.978076
\(927\) 14.7228 0.483560
\(928\) 8.66552 0.284459
\(929\) −28.3985 −0.931725 −0.465862 0.884857i \(-0.654256\pi\)
−0.465862 + 0.884857i \(0.654256\pi\)
\(930\) 26.6418 0.873618
\(931\) 0.312580 0.0102444
\(932\) −7.20013 −0.235848
\(933\) −58.6739 −1.92090
\(934\) −27.5913 −0.902816
\(935\) 0 0
\(936\) −27.9223 −0.912669
\(937\) −29.1310 −0.951669 −0.475834 0.879535i \(-0.657854\pi\)
−0.475834 + 0.879535i \(0.657854\pi\)
\(938\) 13.0945 0.427550
\(939\) −11.8173 −0.385645
\(940\) −2.99552 −0.0977030
\(941\) 2.40040 0.0782508 0.0391254 0.999234i \(-0.487543\pi\)
0.0391254 + 0.999234i \(0.487543\pi\)
\(942\) −30.8033 −1.00363
\(943\) 11.6939 0.380806
\(944\) −22.4550 −0.730848
\(945\) −1.87485 −0.0609889
\(946\) 0 0
\(947\) −9.37029 −0.304493 −0.152247 0.988343i \(-0.548651\pi\)
−0.152247 + 0.988343i \(0.548651\pi\)
\(948\) 2.78974 0.0906064
\(949\) 37.0453 1.20254
\(950\) 0.348062 0.0112926
\(951\) 42.9385 1.39238
\(952\) −15.4118 −0.499499
\(953\) 18.0336 0.584165 0.292082 0.956393i \(-0.405652\pi\)
0.292082 + 0.956393i \(0.405652\pi\)
\(954\) −3.31246 −0.107245
\(955\) −1.48362 −0.0480089
\(956\) 20.3440 0.657973
\(957\) 0 0
\(958\) 3.38094 0.109233
\(959\) −0.885628 −0.0285984
\(960\) 18.8610 0.608735
\(961\) 79.5977 2.56767
\(962\) −9.84945 −0.317559
\(963\) 19.4627 0.627178
\(964\) −0.452896 −0.0145868
\(965\) 7.36948 0.237232
\(966\) −2.60889 −0.0839398
\(967\) −48.0532 −1.54529 −0.772643 0.634840i \(-0.781066\pi\)
−0.772643 + 0.634840i \(0.781066\pi\)
\(968\) 0 0
\(969\) 3.56606 0.114558
\(970\) −7.67593 −0.246459
\(971\) −43.3729 −1.39190 −0.695951 0.718089i \(-0.745017\pi\)
−0.695951 + 0.718089i \(0.745017\pi\)
\(972\) 14.3888 0.461520
\(973\) 14.2443 0.456650
\(974\) 44.3827 1.42211
\(975\) 9.49915 0.304216
\(976\) 22.7090 0.726897
\(977\) −54.0777 −1.73010 −0.865049 0.501688i \(-0.832713\pi\)
−0.865049 + 0.501688i \(0.832713\pi\)
\(978\) 41.3283 1.32153
\(979\) 0 0
\(980\) −0.760083 −0.0242800
\(981\) 28.4305 0.907717
\(982\) −28.1047 −0.896858
\(983\) −42.6369 −1.35991 −0.679954 0.733255i \(-0.738000\pi\)
−0.679954 + 0.733255i \(0.738000\pi\)
\(984\) 79.3972 2.53109
\(985\) 4.74986 0.151343
\(986\) 12.0103 0.382485
\(987\) 8.96611 0.285394
\(988\) −0.992003 −0.0315598
\(989\) 7.35557 0.233894
\(990\) 0 0
\(991\) −52.3376 −1.66256 −0.831280 0.555855i \(-0.812391\pi\)
−0.831280 + 0.555855i \(0.812391\pi\)
\(992\) 42.3687 1.34521
\(993\) −30.2480 −0.959892
\(994\) 9.76402 0.309696
\(995\) 18.5306 0.587460
\(996\) 24.8350 0.786927
\(997\) −10.1811 −0.322438 −0.161219 0.986919i \(-0.551543\pi\)
−0.161219 + 0.986919i \(0.551543\pi\)
\(998\) 28.2321 0.893672
\(999\) 3.97183 0.125663
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 4235.2.a.bp.1.12 18
11.7 odd 10 385.2.n.f.71.4 36
11.8 odd 10 385.2.n.f.141.4 yes 36
11.10 odd 2 4235.2.a.bo.1.7 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.n.f.71.4 36 11.7 odd 10
385.2.n.f.141.4 yes 36 11.8 odd 10
4235.2.a.bo.1.7 18 11.10 odd 2
4235.2.a.bp.1.12 18 1.1 even 1 trivial