Properties

Label 4235.2.a.bi.1.2
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 67x^{6} - 110x^{5} - 132x^{4} + 168x^{3} + 94x^{2} - 54x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.09068\) 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)
 
\(f(q)\) \(=\) \(q-2.09068 q^{2} -1.17179 q^{3} +2.37096 q^{4} +1.00000 q^{5} +2.44983 q^{6} +1.00000 q^{7} -0.775558 q^{8} -1.62692 q^{9} +O(q^{10})\) \(q-2.09068 q^{2} -1.17179 q^{3} +2.37096 q^{4} +1.00000 q^{5} +2.44983 q^{6} +1.00000 q^{7} -0.775558 q^{8} -1.62692 q^{9} -2.09068 q^{10} -2.77826 q^{12} -1.06949 q^{13} -2.09068 q^{14} -1.17179 q^{15} -3.12047 q^{16} -5.01686 q^{17} +3.40137 q^{18} -1.36590 q^{19} +2.37096 q^{20} -1.17179 q^{21} +4.12817 q^{23} +0.908789 q^{24} +1.00000 q^{25} +2.23596 q^{26} +5.42176 q^{27} +2.37096 q^{28} +6.92308 q^{29} +2.44983 q^{30} +1.54746 q^{31} +8.07504 q^{32} +10.4887 q^{34} +1.00000 q^{35} -3.85735 q^{36} +10.1396 q^{37} +2.85566 q^{38} +1.25321 q^{39} -0.775558 q^{40} -3.70892 q^{41} +2.44983 q^{42} -7.68270 q^{43} -1.62692 q^{45} -8.63070 q^{46} -12.0349 q^{47} +3.65652 q^{48} +1.00000 q^{49} -2.09068 q^{50} +5.87868 q^{51} -2.53572 q^{52} +3.81239 q^{53} -11.3352 q^{54} -0.775558 q^{56} +1.60054 q^{57} -14.4740 q^{58} -8.72837 q^{59} -2.77826 q^{60} -12.4241 q^{61} -3.23525 q^{62} -1.62692 q^{63} -10.6414 q^{64} -1.06949 q^{65} +12.2819 q^{67} -11.8948 q^{68} -4.83733 q^{69} -2.09068 q^{70} +7.29340 q^{71} +1.26177 q^{72} -7.32603 q^{73} -21.1988 q^{74} -1.17179 q^{75} -3.23849 q^{76} -2.62007 q^{78} -5.41792 q^{79} -3.12047 q^{80} -1.47239 q^{81} +7.75417 q^{82} +0.0466470 q^{83} -2.77826 q^{84} -5.01686 q^{85} +16.0621 q^{86} -8.11237 q^{87} -3.14060 q^{89} +3.40137 q^{90} -1.06949 q^{91} +9.78772 q^{92} -1.81329 q^{93} +25.1612 q^{94} -1.36590 q^{95} -9.46222 q^{96} +12.8210 q^{97} -2.09068 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 4 q^{3} + 12 q^{4} + 10 q^{5} - 8 q^{6} + 10 q^{7} - 6 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} - 4 q^{3} + 12 q^{4} + 10 q^{5} - 8 q^{6} + 10 q^{7} - 6 q^{8} + 14 q^{9} - 2 q^{10} - 4 q^{12} - 18 q^{13} - 2 q^{14} - 4 q^{15} + 4 q^{16} - 4 q^{17} - 12 q^{18} - 14 q^{19} + 12 q^{20} - 4 q^{21} - 4 q^{23} - 46 q^{24} + 10 q^{25} + 2 q^{26} - 34 q^{27} + 12 q^{28} - 36 q^{29} - 8 q^{30} - 18 q^{31} - 4 q^{32} - 32 q^{34} + 10 q^{35} + 22 q^{36} + 4 q^{37} + 18 q^{38} - 6 q^{40} - 38 q^{41} - 8 q^{42} - 6 q^{43} + 14 q^{45} - 28 q^{46} - 18 q^{47} + 16 q^{48} + 10 q^{49} - 2 q^{50} + 4 q^{51} - 26 q^{52} - 26 q^{53} - 2 q^{54} - 6 q^{56} - 22 q^{57} + 10 q^{58} - 14 q^{59} - 4 q^{60} - 60 q^{61} - 22 q^{62} + 14 q^{63} - 18 q^{65} - 10 q^{67} - 2 q^{68} - 8 q^{69} - 2 q^{70} + 54 q^{72} - 18 q^{73} + 20 q^{74} - 4 q^{75} - 38 q^{76} + 40 q^{78} - 40 q^{79} + 4 q^{80} + 18 q^{81} - 36 q^{82} - 2 q^{83} - 4 q^{84} - 4 q^{85} + 42 q^{86} + 32 q^{87} + 2 q^{89} - 12 q^{90} - 18 q^{91} - 64 q^{92} + 26 q^{93} + 68 q^{94} - 14 q^{95} - 28 q^{96} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.09068 −1.47834 −0.739168 0.673521i \(-0.764781\pi\)
−0.739168 + 0.673521i \(0.764781\pi\)
\(3\) −1.17179 −0.676531 −0.338266 0.941051i \(-0.609840\pi\)
−0.338266 + 0.941051i \(0.609840\pi\)
\(4\) 2.37096 1.18548
\(5\) 1.00000 0.447214
\(6\) 2.44983 1.00014
\(7\) 1.00000 0.377964
\(8\) −0.775558 −0.274201
\(9\) −1.62692 −0.542306
\(10\) −2.09068 −0.661132
\(11\) 0 0
\(12\) −2.77826 −0.802014
\(13\) −1.06949 −0.296623 −0.148311 0.988941i \(-0.547384\pi\)
−0.148311 + 0.988941i \(0.547384\pi\)
\(14\) −2.09068 −0.558759
\(15\) −1.17179 −0.302554
\(16\) −3.12047 −0.780118
\(17\) −5.01686 −1.21677 −0.608383 0.793643i \(-0.708182\pi\)
−0.608383 + 0.793643i \(0.708182\pi\)
\(18\) 3.40137 0.801710
\(19\) −1.36590 −0.313359 −0.156679 0.987650i \(-0.550079\pi\)
−0.156679 + 0.987650i \(0.550079\pi\)
\(20\) 2.37096 0.530163
\(21\) −1.17179 −0.255705
\(22\) 0 0
\(23\) 4.12817 0.860783 0.430391 0.902642i \(-0.358376\pi\)
0.430391 + 0.902642i \(0.358376\pi\)
\(24\) 0.908789 0.185506
\(25\) 1.00000 0.200000
\(26\) 2.23596 0.438509
\(27\) 5.42176 1.04342
\(28\) 2.37096 0.448069
\(29\) 6.92308 1.28558 0.642792 0.766041i \(-0.277776\pi\)
0.642792 + 0.766041i \(0.277776\pi\)
\(30\) 2.44983 0.447277
\(31\) 1.54746 0.277932 0.138966 0.990297i \(-0.455622\pi\)
0.138966 + 0.990297i \(0.455622\pi\)
\(32\) 8.07504 1.42748
\(33\) 0 0
\(34\) 10.4887 1.79879
\(35\) 1.00000 0.169031
\(36\) −3.85735 −0.642892
\(37\) 10.1396 1.66694 0.833472 0.552561i \(-0.186350\pi\)
0.833472 + 0.552561i \(0.186350\pi\)
\(38\) 2.85566 0.463249
\(39\) 1.25321 0.200675
\(40\) −0.775558 −0.122627
\(41\) −3.70892 −0.579235 −0.289618 0.957142i \(-0.593528\pi\)
−0.289618 + 0.957142i \(0.593528\pi\)
\(42\) 2.44983 0.378018
\(43\) −7.68270 −1.17160 −0.585801 0.810455i \(-0.699220\pi\)
−0.585801 + 0.810455i \(0.699220\pi\)
\(44\) 0 0
\(45\) −1.62692 −0.242526
\(46\) −8.63070 −1.27253
\(47\) −12.0349 −1.75548 −0.877738 0.479141i \(-0.840948\pi\)
−0.877738 + 0.479141i \(0.840948\pi\)
\(48\) 3.65652 0.527774
\(49\) 1.00000 0.142857
\(50\) −2.09068 −0.295667
\(51\) 5.87868 0.823180
\(52\) −2.53572 −0.351640
\(53\) 3.81239 0.523672 0.261836 0.965112i \(-0.415672\pi\)
0.261836 + 0.965112i \(0.415672\pi\)
\(54\) −11.3352 −1.54252
\(55\) 0 0
\(56\) −0.775558 −0.103638
\(57\) 1.60054 0.211997
\(58\) −14.4740 −1.90052
\(59\) −8.72837 −1.13634 −0.568168 0.822912i \(-0.692348\pi\)
−0.568168 + 0.822912i \(0.692348\pi\)
\(60\) −2.77826 −0.358671
\(61\) −12.4241 −1.59074 −0.795372 0.606122i \(-0.792724\pi\)
−0.795372 + 0.606122i \(0.792724\pi\)
\(62\) −3.23525 −0.410877
\(63\) −1.62692 −0.204972
\(64\) −10.6414 −1.33018
\(65\) −1.06949 −0.132654
\(66\) 0 0
\(67\) 12.2819 1.50047 0.750237 0.661169i \(-0.229939\pi\)
0.750237 + 0.661169i \(0.229939\pi\)
\(68\) −11.8948 −1.44245
\(69\) −4.83733 −0.582346
\(70\) −2.09068 −0.249885
\(71\) 7.29340 0.865567 0.432784 0.901498i \(-0.357531\pi\)
0.432784 + 0.901498i \(0.357531\pi\)
\(72\) 1.26177 0.148701
\(73\) −7.32603 −0.857447 −0.428723 0.903436i \(-0.641037\pi\)
−0.428723 + 0.903436i \(0.641037\pi\)
\(74\) −21.1988 −2.46431
\(75\) −1.17179 −0.135306
\(76\) −3.23849 −0.371480
\(77\) 0 0
\(78\) −2.62007 −0.296665
\(79\) −5.41792 −0.609564 −0.304782 0.952422i \(-0.598584\pi\)
−0.304782 + 0.952422i \(0.598584\pi\)
\(80\) −3.12047 −0.348879
\(81\) −1.47239 −0.163599
\(82\) 7.75417 0.856305
\(83\) 0.0466470 0.00512017 0.00256009 0.999997i \(-0.499185\pi\)
0.00256009 + 0.999997i \(0.499185\pi\)
\(84\) −2.77826 −0.303133
\(85\) −5.01686 −0.544154
\(86\) 16.0621 1.73202
\(87\) −8.11237 −0.869737
\(88\) 0 0
\(89\) −3.14060 −0.332903 −0.166452 0.986050i \(-0.553231\pi\)
−0.166452 + 0.986050i \(0.553231\pi\)
\(90\) 3.40137 0.358536
\(91\) −1.06949 −0.112113
\(92\) 9.78772 1.02044
\(93\) −1.81329 −0.188030
\(94\) 25.1612 2.59518
\(95\) −1.36590 −0.140138
\(96\) −9.46222 −0.965733
\(97\) 12.8210 1.30178 0.650889 0.759173i \(-0.274396\pi\)
0.650889 + 0.759173i \(0.274396\pi\)
\(98\) −2.09068 −0.211191
\(99\) 0 0
\(100\) 2.37096 0.237096
\(101\) 6.58237 0.654970 0.327485 0.944856i \(-0.393799\pi\)
0.327485 + 0.944856i \(0.393799\pi\)
\(102\) −12.2905 −1.21694
\(103\) 1.42092 0.140007 0.0700036 0.997547i \(-0.477699\pi\)
0.0700036 + 0.997547i \(0.477699\pi\)
\(104\) 0.829451 0.0813344
\(105\) −1.17179 −0.114355
\(106\) −7.97050 −0.774163
\(107\) 10.5639 1.02125 0.510623 0.859805i \(-0.329415\pi\)
0.510623 + 0.859805i \(0.329415\pi\)
\(108\) 12.8548 1.23695
\(109\) −8.32464 −0.797356 −0.398678 0.917091i \(-0.630531\pi\)
−0.398678 + 0.917091i \(0.630531\pi\)
\(110\) 0 0
\(111\) −11.8815 −1.12774
\(112\) −3.12047 −0.294857
\(113\) 11.9777 1.12677 0.563383 0.826196i \(-0.309500\pi\)
0.563383 + 0.826196i \(0.309500\pi\)
\(114\) −3.34622 −0.313403
\(115\) 4.12817 0.384954
\(116\) 16.4143 1.52403
\(117\) 1.73997 0.160860
\(118\) 18.2483 1.67989
\(119\) −5.01686 −0.459894
\(120\) 0.908789 0.0829607
\(121\) 0 0
\(122\) 25.9749 2.35165
\(123\) 4.34606 0.391871
\(124\) 3.66896 0.329483
\(125\) 1.00000 0.0894427
\(126\) 3.40137 0.303018
\(127\) 1.16924 0.103753 0.0518767 0.998653i \(-0.483480\pi\)
0.0518767 + 0.998653i \(0.483480\pi\)
\(128\) 6.09774 0.538969
\(129\) 9.00249 0.792625
\(130\) 2.23596 0.196107
\(131\) −0.200825 −0.0175462 −0.00877310 0.999962i \(-0.502793\pi\)
−0.00877310 + 0.999962i \(0.502793\pi\)
\(132\) 0 0
\(133\) −1.36590 −0.118438
\(134\) −25.6776 −2.21821
\(135\) 5.42176 0.466631
\(136\) 3.89086 0.333639
\(137\) −5.42631 −0.463601 −0.231801 0.972763i \(-0.574462\pi\)
−0.231801 + 0.972763i \(0.574462\pi\)
\(138\) 10.1133 0.860904
\(139\) 3.01028 0.255329 0.127664 0.991817i \(-0.459252\pi\)
0.127664 + 0.991817i \(0.459252\pi\)
\(140\) 2.37096 0.200383
\(141\) 14.1024 1.18763
\(142\) −15.2482 −1.27960
\(143\) 0 0
\(144\) 5.07675 0.423062
\(145\) 6.92308 0.574930
\(146\) 15.3164 1.26760
\(147\) −1.17179 −0.0966473
\(148\) 24.0406 1.97613
\(149\) −5.65447 −0.463233 −0.231616 0.972807i \(-0.574401\pi\)
−0.231616 + 0.972807i \(0.574401\pi\)
\(150\) 2.44983 0.200028
\(151\) −18.1008 −1.47302 −0.736512 0.676425i \(-0.763528\pi\)
−0.736512 + 0.676425i \(0.763528\pi\)
\(152\) 1.05933 0.0859233
\(153\) 8.16201 0.659859
\(154\) 0 0
\(155\) 1.54746 0.124295
\(156\) 2.97132 0.237896
\(157\) −14.4299 −1.15164 −0.575818 0.817578i \(-0.695316\pi\)
−0.575818 + 0.817578i \(0.695316\pi\)
\(158\) 11.3272 0.901140
\(159\) −4.46730 −0.354280
\(160\) 8.07504 0.638388
\(161\) 4.12817 0.325345
\(162\) 3.07830 0.241854
\(163\) 7.53849 0.590460 0.295230 0.955426i \(-0.404604\pi\)
0.295230 + 0.955426i \(0.404604\pi\)
\(164\) −8.79369 −0.686672
\(165\) 0 0
\(166\) −0.0975241 −0.00756934
\(167\) 11.5314 0.892325 0.446162 0.894952i \(-0.352790\pi\)
0.446162 + 0.894952i \(0.352790\pi\)
\(168\) 0.908789 0.0701146
\(169\) −11.8562 −0.912015
\(170\) 10.4887 0.804443
\(171\) 2.22220 0.169936
\(172\) −18.2154 −1.38891
\(173\) −10.0459 −0.763779 −0.381889 0.924208i \(-0.624727\pi\)
−0.381889 + 0.924208i \(0.624727\pi\)
\(174\) 16.9604 1.28576
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 10.2278 0.768767
\(178\) 6.56601 0.492143
\(179\) −16.3072 −1.21886 −0.609428 0.792842i \(-0.708601\pi\)
−0.609428 + 0.792842i \(0.708601\pi\)
\(180\) −3.85735 −0.287510
\(181\) −11.7382 −0.872492 −0.436246 0.899827i \(-0.643692\pi\)
−0.436246 + 0.899827i \(0.643692\pi\)
\(182\) 2.23596 0.165741
\(183\) 14.5584 1.07619
\(184\) −3.20164 −0.236028
\(185\) 10.1396 0.745480
\(186\) 3.79102 0.277971
\(187\) 0 0
\(188\) −28.5343 −2.08108
\(189\) 5.42176 0.394375
\(190\) 2.85566 0.207171
\(191\) 11.5859 0.838329 0.419164 0.907910i \(-0.362323\pi\)
0.419164 + 0.907910i \(0.362323\pi\)
\(192\) 12.4695 0.899905
\(193\) 11.0124 0.792687 0.396344 0.918102i \(-0.370279\pi\)
0.396344 + 0.918102i \(0.370279\pi\)
\(194\) −26.8047 −1.92447
\(195\) 1.25321 0.0897444
\(196\) 2.37096 0.169354
\(197\) 12.3315 0.878585 0.439293 0.898344i \(-0.355229\pi\)
0.439293 + 0.898344i \(0.355229\pi\)
\(198\) 0 0
\(199\) −9.46213 −0.670752 −0.335376 0.942084i \(-0.608863\pi\)
−0.335376 + 0.942084i \(0.608863\pi\)
\(200\) −0.775558 −0.0548403
\(201\) −14.3918 −1.01512
\(202\) −13.7616 −0.968266
\(203\) 6.92308 0.485905
\(204\) 13.9381 0.975863
\(205\) −3.70892 −0.259042
\(206\) −2.97069 −0.206978
\(207\) −6.71619 −0.466807
\(208\) 3.33731 0.231401
\(209\) 0 0
\(210\) 2.44983 0.169055
\(211\) −10.4327 −0.718218 −0.359109 0.933296i \(-0.616919\pi\)
−0.359109 + 0.933296i \(0.616919\pi\)
\(212\) 9.03902 0.620802
\(213\) −8.54630 −0.585583
\(214\) −22.0857 −1.50975
\(215\) −7.68270 −0.523956
\(216\) −4.20489 −0.286107
\(217\) 1.54746 0.105048
\(218\) 17.4042 1.17876
\(219\) 8.58454 0.580090
\(220\) 0 0
\(221\) 5.36547 0.360921
\(222\) 24.8404 1.66718
\(223\) −3.36259 −0.225176 −0.112588 0.993642i \(-0.535914\pi\)
−0.112588 + 0.993642i \(0.535914\pi\)
\(224\) 8.07504 0.539536
\(225\) −1.62692 −0.108461
\(226\) −25.0415 −1.66574
\(227\) 4.82165 0.320024 0.160012 0.987115i \(-0.448847\pi\)
0.160012 + 0.987115i \(0.448847\pi\)
\(228\) 3.79482 0.251318
\(229\) −13.2867 −0.878011 −0.439005 0.898484i \(-0.644669\pi\)
−0.439005 + 0.898484i \(0.644669\pi\)
\(230\) −8.63070 −0.569091
\(231\) 0 0
\(232\) −5.36925 −0.352509
\(233\) −22.6748 −1.48548 −0.742738 0.669582i \(-0.766473\pi\)
−0.742738 + 0.669582i \(0.766473\pi\)
\(234\) −3.63773 −0.237806
\(235\) −12.0349 −0.785073
\(236\) −20.6946 −1.34710
\(237\) 6.34864 0.412389
\(238\) 10.4887 0.679879
\(239\) −7.21013 −0.466385 −0.233192 0.972431i \(-0.574917\pi\)
−0.233192 + 0.972431i \(0.574917\pi\)
\(240\) 3.65652 0.236028
\(241\) 13.2512 0.853587 0.426793 0.904349i \(-0.359643\pi\)
0.426793 + 0.904349i \(0.359643\pi\)
\(242\) 0 0
\(243\) −14.5399 −0.932738
\(244\) −29.4570 −1.88579
\(245\) 1.00000 0.0638877
\(246\) −9.08623 −0.579317
\(247\) 1.46081 0.0929493
\(248\) −1.20015 −0.0762093
\(249\) −0.0546603 −0.00346395
\(250\) −2.09068 −0.132226
\(251\) 25.5786 1.61451 0.807253 0.590206i \(-0.200953\pi\)
0.807253 + 0.590206i \(0.200953\pi\)
\(252\) −3.85735 −0.242990
\(253\) 0 0
\(254\) −2.44452 −0.153383
\(255\) 5.87868 0.368137
\(256\) 8.53436 0.533397
\(257\) −6.44721 −0.402166 −0.201083 0.979574i \(-0.564446\pi\)
−0.201083 + 0.979574i \(0.564446\pi\)
\(258\) −18.8214 −1.17177
\(259\) 10.1396 0.630046
\(260\) −2.53572 −0.157258
\(261\) −11.2633 −0.697179
\(262\) 0.419862 0.0259392
\(263\) 17.7885 1.09688 0.548442 0.836189i \(-0.315221\pi\)
0.548442 + 0.836189i \(0.315221\pi\)
\(264\) 0 0
\(265\) 3.81239 0.234193
\(266\) 2.85566 0.175092
\(267\) 3.68012 0.225220
\(268\) 29.1199 1.77878
\(269\) 14.8592 0.905984 0.452992 0.891515i \(-0.350357\pi\)
0.452992 + 0.891515i \(0.350357\pi\)
\(270\) −11.3352 −0.689837
\(271\) −13.1746 −0.800303 −0.400151 0.916449i \(-0.631042\pi\)
−0.400151 + 0.916449i \(0.631042\pi\)
\(272\) 15.6550 0.949221
\(273\) 1.25321 0.0758479
\(274\) 11.3447 0.685359
\(275\) 0 0
\(276\) −11.4691 −0.690360
\(277\) 4.92847 0.296123 0.148061 0.988978i \(-0.452697\pi\)
0.148061 + 0.988978i \(0.452697\pi\)
\(278\) −6.29355 −0.377462
\(279\) −2.51759 −0.150724
\(280\) −0.775558 −0.0463485
\(281\) −25.6183 −1.52826 −0.764130 0.645062i \(-0.776832\pi\)
−0.764130 + 0.645062i \(0.776832\pi\)
\(282\) −29.4836 −1.75572
\(283\) −25.2779 −1.50262 −0.751309 0.659951i \(-0.770577\pi\)
−0.751309 + 0.659951i \(0.770577\pi\)
\(284\) 17.2923 1.02611
\(285\) 1.60054 0.0948078
\(286\) 0 0
\(287\) −3.70892 −0.218930
\(288\) −13.1374 −0.774129
\(289\) 8.16884 0.480520
\(290\) −14.4740 −0.849941
\(291\) −15.0235 −0.880694
\(292\) −17.3697 −1.01649
\(293\) 23.8860 1.39543 0.697716 0.716374i \(-0.254200\pi\)
0.697716 + 0.716374i \(0.254200\pi\)
\(294\) 2.44983 0.142877
\(295\) −8.72837 −0.508185
\(296\) −7.86387 −0.457078
\(297\) 0 0
\(298\) 11.8217 0.684814
\(299\) −4.41503 −0.255328
\(300\) −2.77826 −0.160403
\(301\) −7.68270 −0.442824
\(302\) 37.8431 2.17763
\(303\) −7.71313 −0.443107
\(304\) 4.26225 0.244457
\(305\) −12.4241 −0.711402
\(306\) −17.0642 −0.975494
\(307\) 27.5818 1.57418 0.787088 0.616841i \(-0.211588\pi\)
0.787088 + 0.616841i \(0.211588\pi\)
\(308\) 0 0
\(309\) −1.66501 −0.0947193
\(310\) −3.23525 −0.183750
\(311\) 12.7943 0.725501 0.362750 0.931886i \(-0.381838\pi\)
0.362750 + 0.931886i \(0.381838\pi\)
\(312\) −0.971940 −0.0550253
\(313\) −3.66737 −0.207292 −0.103646 0.994614i \(-0.533051\pi\)
−0.103646 + 0.994614i \(0.533051\pi\)
\(314\) 30.1685 1.70250
\(315\) −1.62692 −0.0916664
\(316\) −12.8457 −0.722625
\(317\) 29.3633 1.64921 0.824603 0.565712i \(-0.191399\pi\)
0.824603 + 0.565712i \(0.191399\pi\)
\(318\) 9.33972 0.523746
\(319\) 0 0
\(320\) −10.6414 −0.594873
\(321\) −12.3786 −0.690905
\(322\) −8.63070 −0.480970
\(323\) 6.85251 0.381284
\(324\) −3.49098 −0.193943
\(325\) −1.06949 −0.0593246
\(326\) −15.7606 −0.872899
\(327\) 9.75470 0.539436
\(328\) 2.87648 0.158827
\(329\) −12.0349 −0.663507
\(330\) 0 0
\(331\) −23.9178 −1.31464 −0.657321 0.753611i \(-0.728310\pi\)
−0.657321 + 0.753611i \(0.728310\pi\)
\(332\) 0.110598 0.00606986
\(333\) −16.4963 −0.903993
\(334\) −24.1085 −1.31916
\(335\) 12.2819 0.671033
\(336\) 3.65652 0.199480
\(337\) −25.3419 −1.38046 −0.690232 0.723588i \(-0.742491\pi\)
−0.690232 + 0.723588i \(0.742491\pi\)
\(338\) 24.7876 1.34827
\(339\) −14.0353 −0.762292
\(340\) −11.8948 −0.645084
\(341\) 0 0
\(342\) −4.64592 −0.251223
\(343\) 1.00000 0.0539949
\(344\) 5.95839 0.321255
\(345\) −4.83733 −0.260433
\(346\) 21.0029 1.12912
\(347\) −34.8652 −1.87166 −0.935831 0.352449i \(-0.885349\pi\)
−0.935831 + 0.352449i \(0.885349\pi\)
\(348\) −19.2341 −1.03106
\(349\) −0.759664 −0.0406639 −0.0203319 0.999793i \(-0.506472\pi\)
−0.0203319 + 0.999793i \(0.506472\pi\)
\(350\) −2.09068 −0.111752
\(351\) −5.79851 −0.309502
\(352\) 0 0
\(353\) −36.1431 −1.92371 −0.961853 0.273567i \(-0.911796\pi\)
−0.961853 + 0.273567i \(0.911796\pi\)
\(354\) −21.3831 −1.13650
\(355\) 7.29340 0.387093
\(356\) −7.44625 −0.394650
\(357\) 5.87868 0.311133
\(358\) 34.0931 1.80188
\(359\) 8.32590 0.439424 0.219712 0.975565i \(-0.429488\pi\)
0.219712 + 0.975565i \(0.429488\pi\)
\(360\) 1.26177 0.0665011
\(361\) −17.1343 −0.901806
\(362\) 24.5408 1.28984
\(363\) 0 0
\(364\) −2.53572 −0.132908
\(365\) −7.32603 −0.383462
\(366\) −30.4370 −1.59097
\(367\) −34.3230 −1.79165 −0.895823 0.444411i \(-0.853413\pi\)
−0.895823 + 0.444411i \(0.853413\pi\)
\(368\) −12.8818 −0.671512
\(369\) 6.03410 0.314123
\(370\) −21.1988 −1.10207
\(371\) 3.81239 0.197929
\(372\) −4.29924 −0.222905
\(373\) 11.5541 0.598249 0.299125 0.954214i \(-0.403305\pi\)
0.299125 + 0.954214i \(0.403305\pi\)
\(374\) 0 0
\(375\) −1.17179 −0.0605108
\(376\) 9.33379 0.481354
\(377\) −7.40416 −0.381333
\(378\) −11.3352 −0.583019
\(379\) −26.1411 −1.34278 −0.671390 0.741104i \(-0.734302\pi\)
−0.671390 + 0.741104i \(0.734302\pi\)
\(380\) −3.23849 −0.166131
\(381\) −1.37010 −0.0701924
\(382\) −24.2225 −1.23933
\(383\) −13.0351 −0.666064 −0.333032 0.942916i \(-0.608072\pi\)
−0.333032 + 0.942916i \(0.608072\pi\)
\(384\) −7.14525 −0.364630
\(385\) 0 0
\(386\) −23.0234 −1.17186
\(387\) 12.4991 0.635366
\(388\) 30.3981 1.54323
\(389\) 19.3799 0.982602 0.491301 0.870990i \(-0.336522\pi\)
0.491301 + 0.870990i \(0.336522\pi\)
\(390\) −2.62007 −0.132672
\(391\) −20.7104 −1.04737
\(392\) −0.775558 −0.0391716
\(393\) 0.235324 0.0118705
\(394\) −25.7813 −1.29884
\(395\) −5.41792 −0.272605
\(396\) 0 0
\(397\) 27.8036 1.39542 0.697712 0.716378i \(-0.254202\pi\)
0.697712 + 0.716378i \(0.254202\pi\)
\(398\) 19.7823 0.991598
\(399\) 1.60054 0.0801273
\(400\) −3.12047 −0.156024
\(401\) −22.9886 −1.14800 −0.573998 0.818857i \(-0.694608\pi\)
−0.573998 + 0.818857i \(0.694608\pi\)
\(402\) 30.0887 1.50069
\(403\) −1.65499 −0.0824410
\(404\) 15.6065 0.776453
\(405\) −1.47239 −0.0731637
\(406\) −14.4740 −0.718331
\(407\) 0 0
\(408\) −4.55926 −0.225717
\(409\) −35.9430 −1.77727 −0.888634 0.458618i \(-0.848345\pi\)
−0.888634 + 0.458618i \(0.848345\pi\)
\(410\) 7.75417 0.382951
\(411\) 6.35848 0.313641
\(412\) 3.36894 0.165976
\(413\) −8.72837 −0.429495
\(414\) 14.0414 0.690098
\(415\) 0.0466470 0.00228981
\(416\) −8.63616 −0.423423
\(417\) −3.52741 −0.172738
\(418\) 0 0
\(419\) −19.1444 −0.935263 −0.467632 0.883923i \(-0.654893\pi\)
−0.467632 + 0.883923i \(0.654893\pi\)
\(420\) −2.77826 −0.135565
\(421\) 28.7275 1.40009 0.700046 0.714098i \(-0.253163\pi\)
0.700046 + 0.714098i \(0.253163\pi\)
\(422\) 21.8115 1.06177
\(423\) 19.5798 0.952004
\(424\) −2.95673 −0.143591
\(425\) −5.01686 −0.243353
\(426\) 17.8676 0.865689
\(427\) −12.4241 −0.601245
\(428\) 25.0465 1.21067
\(429\) 0 0
\(430\) 16.0621 0.774584
\(431\) 12.9759 0.625025 0.312513 0.949914i \(-0.398829\pi\)
0.312513 + 0.949914i \(0.398829\pi\)
\(432\) −16.9184 −0.813989
\(433\) −37.5382 −1.80397 −0.901985 0.431767i \(-0.857890\pi\)
−0.901985 + 0.431767i \(0.857890\pi\)
\(434\) −3.23525 −0.155297
\(435\) −8.11237 −0.388958
\(436\) −19.7374 −0.945250
\(437\) −5.63866 −0.269734
\(438\) −17.9476 −0.857568
\(439\) −16.2174 −0.774015 −0.387007 0.922077i \(-0.626491\pi\)
−0.387007 + 0.922077i \(0.626491\pi\)
\(440\) 0 0
\(441\) −1.62692 −0.0774722
\(442\) −11.2175 −0.533562
\(443\) −34.5866 −1.64326 −0.821629 0.570022i \(-0.806935\pi\)
−0.821629 + 0.570022i \(0.806935\pi\)
\(444\) −28.1705 −1.33691
\(445\) −3.14060 −0.148879
\(446\) 7.03012 0.332886
\(447\) 6.62584 0.313391
\(448\) −10.6414 −0.502759
\(449\) −11.6118 −0.547995 −0.273998 0.961730i \(-0.588346\pi\)
−0.273998 + 0.961730i \(0.588346\pi\)
\(450\) 3.40137 0.160342
\(451\) 0 0
\(452\) 28.3986 1.33576
\(453\) 21.2103 0.996546
\(454\) −10.0805 −0.473103
\(455\) −1.06949 −0.0501384
\(456\) −1.24131 −0.0581298
\(457\) −24.1449 −1.12945 −0.564726 0.825279i \(-0.691018\pi\)
−0.564726 + 0.825279i \(0.691018\pi\)
\(458\) 27.7783 1.29800
\(459\) −27.2002 −1.26960
\(460\) 9.78772 0.456355
\(461\) 12.7118 0.592046 0.296023 0.955181i \(-0.404340\pi\)
0.296023 + 0.955181i \(0.404340\pi\)
\(462\) 0 0
\(463\) −20.0814 −0.933263 −0.466632 0.884452i \(-0.654533\pi\)
−0.466632 + 0.884452i \(0.654533\pi\)
\(464\) −21.6033 −1.00291
\(465\) −1.81329 −0.0840894
\(466\) 47.4058 2.19603
\(467\) −19.9131 −0.921467 −0.460734 0.887538i \(-0.652414\pi\)
−0.460734 + 0.887538i \(0.652414\pi\)
\(468\) 4.12540 0.190697
\(469\) 12.2819 0.567126
\(470\) 25.1612 1.16060
\(471\) 16.9088 0.779117
\(472\) 6.76936 0.311585
\(473\) 0 0
\(474\) −13.2730 −0.609649
\(475\) −1.36590 −0.0626717
\(476\) −11.8948 −0.545195
\(477\) −6.20244 −0.283990
\(478\) 15.0741 0.689474
\(479\) −32.1313 −1.46812 −0.734058 0.679087i \(-0.762376\pi\)
−0.734058 + 0.679087i \(0.762376\pi\)
\(480\) −9.46222 −0.431889
\(481\) −10.8442 −0.494454
\(482\) −27.7041 −1.26189
\(483\) −4.83733 −0.220106
\(484\) 0 0
\(485\) 12.8210 0.582173
\(486\) 30.3984 1.37890
\(487\) 6.86275 0.310981 0.155490 0.987837i \(-0.450304\pi\)
0.155490 + 0.987837i \(0.450304\pi\)
\(488\) 9.63562 0.436184
\(489\) −8.83350 −0.399465
\(490\) −2.09068 −0.0944475
\(491\) 16.5239 0.745711 0.372856 0.927889i \(-0.378379\pi\)
0.372856 + 0.927889i \(0.378379\pi\)
\(492\) 10.3043 0.464555
\(493\) −34.7321 −1.56425
\(494\) −3.05410 −0.137410
\(495\) 0 0
\(496\) −4.82880 −0.216820
\(497\) 7.29340 0.327154
\(498\) 0.114277 0.00512089
\(499\) 13.3608 0.598111 0.299056 0.954236i \(-0.403328\pi\)
0.299056 + 0.954236i \(0.403328\pi\)
\(500\) 2.37096 0.106033
\(501\) −13.5123 −0.603685
\(502\) −53.4767 −2.38678
\(503\) 41.5574 1.85295 0.926475 0.376355i \(-0.122823\pi\)
0.926475 + 0.376355i \(0.122823\pi\)
\(504\) 1.26177 0.0562037
\(505\) 6.58237 0.292911
\(506\) 0 0
\(507\) 13.8929 0.617006
\(508\) 2.77222 0.122998
\(509\) 20.1615 0.893644 0.446822 0.894623i \(-0.352556\pi\)
0.446822 + 0.894623i \(0.352556\pi\)
\(510\) −12.2905 −0.544231
\(511\) −7.32603 −0.324085
\(512\) −30.0381 −1.32751
\(513\) −7.40557 −0.326964
\(514\) 13.4791 0.594537
\(515\) 1.42092 0.0626132
\(516\) 21.3445 0.939641
\(517\) 0 0
\(518\) −21.1988 −0.931420
\(519\) 11.7717 0.516720
\(520\) 0.829451 0.0363738
\(521\) 15.8795 0.695693 0.347846 0.937552i \(-0.386913\pi\)
0.347846 + 0.937552i \(0.386913\pi\)
\(522\) 23.5479 1.03067
\(523\) −19.5991 −0.857007 −0.428504 0.903540i \(-0.640959\pi\)
−0.428504 + 0.903540i \(0.640959\pi\)
\(524\) −0.476149 −0.0208007
\(525\) −1.17179 −0.0511409
\(526\) −37.1900 −1.62156
\(527\) −7.76338 −0.338178
\(528\) 0 0
\(529\) −5.95822 −0.259053
\(530\) −7.97050 −0.346216
\(531\) 14.2003 0.616242
\(532\) −3.23849 −0.140406
\(533\) 3.96664 0.171814
\(534\) −7.69396 −0.332950
\(535\) 10.5639 0.456715
\(536\) −9.52534 −0.411432
\(537\) 19.1085 0.824593
\(538\) −31.0660 −1.33935
\(539\) 0 0
\(540\) 12.8548 0.553181
\(541\) −22.2854 −0.958124 −0.479062 0.877781i \(-0.659023\pi\)
−0.479062 + 0.877781i \(0.659023\pi\)
\(542\) 27.5440 1.18312
\(543\) 13.7546 0.590268
\(544\) −40.5113 −1.73691
\(545\) −8.32464 −0.356589
\(546\) −2.62007 −0.112129
\(547\) 38.5829 1.64969 0.824843 0.565361i \(-0.191263\pi\)
0.824843 + 0.565361i \(0.191263\pi\)
\(548\) −12.8656 −0.549590
\(549\) 20.2130 0.862669
\(550\) 0 0
\(551\) −9.45622 −0.402848
\(552\) 3.75163 0.159680
\(553\) −5.41792 −0.230393
\(554\) −10.3039 −0.437769
\(555\) −11.8815 −0.504341
\(556\) 7.13725 0.302687
\(557\) −22.8525 −0.968292 −0.484146 0.874987i \(-0.660870\pi\)
−0.484146 + 0.874987i \(0.660870\pi\)
\(558\) 5.26348 0.222821
\(559\) 8.21657 0.347524
\(560\) −3.12047 −0.131864
\(561\) 0 0
\(562\) 53.5598 2.25928
\(563\) 7.37512 0.310824 0.155412 0.987850i \(-0.450329\pi\)
0.155412 + 0.987850i \(0.450329\pi\)
\(564\) 33.4361 1.40792
\(565\) 11.9777 0.503905
\(566\) 52.8482 2.22137
\(567\) −1.47239 −0.0618346
\(568\) −5.65646 −0.237340
\(569\) 21.4100 0.897554 0.448777 0.893644i \(-0.351860\pi\)
0.448777 + 0.893644i \(0.351860\pi\)
\(570\) −3.34622 −0.140158
\(571\) 11.6530 0.487661 0.243831 0.969818i \(-0.421596\pi\)
0.243831 + 0.969818i \(0.421596\pi\)
\(572\) 0 0
\(573\) −13.5762 −0.567156
\(574\) 7.75417 0.323653
\(575\) 4.12817 0.172157
\(576\) 17.3127 0.721362
\(577\) 32.0583 1.33460 0.667302 0.744788i \(-0.267449\pi\)
0.667302 + 0.744788i \(0.267449\pi\)
\(578\) −17.0785 −0.710371
\(579\) −12.9041 −0.536278
\(580\) 16.4143 0.681568
\(581\) 0.0466470 0.00193524
\(582\) 31.4094 1.30196
\(583\) 0 0
\(584\) 5.68176 0.235113
\(585\) 1.73997 0.0719389
\(586\) −49.9380 −2.06292
\(587\) −8.23754 −0.340000 −0.170000 0.985444i \(-0.554377\pi\)
−0.170000 + 0.985444i \(0.554377\pi\)
\(588\) −2.77826 −0.114573
\(589\) −2.11367 −0.0870923
\(590\) 18.2483 0.751269
\(591\) −14.4499 −0.594390
\(592\) −31.6404 −1.30041
\(593\) 0.976706 0.0401085 0.0200543 0.999799i \(-0.493616\pi\)
0.0200543 + 0.999799i \(0.493616\pi\)
\(594\) 0 0
\(595\) −5.01686 −0.205671
\(596\) −13.4065 −0.549153
\(597\) 11.0876 0.453785
\(598\) 9.23044 0.377461
\(599\) −12.9440 −0.528876 −0.264438 0.964403i \(-0.585186\pi\)
−0.264438 + 0.964403i \(0.585186\pi\)
\(600\) 0.908789 0.0371011
\(601\) 9.36714 0.382093 0.191047 0.981581i \(-0.438812\pi\)
0.191047 + 0.981581i \(0.438812\pi\)
\(602\) 16.0621 0.654643
\(603\) −19.9817 −0.813716
\(604\) −42.9163 −1.74624
\(605\) 0 0
\(606\) 16.1257 0.655062
\(607\) −3.92443 −0.159288 −0.0796439 0.996823i \(-0.525378\pi\)
−0.0796439 + 0.996823i \(0.525378\pi\)
\(608\) −11.0297 −0.447312
\(609\) −8.11237 −0.328730
\(610\) 25.9749 1.05169
\(611\) 12.8712 0.520714
\(612\) 19.3518 0.782250
\(613\) −47.1030 −1.90247 −0.951236 0.308463i \(-0.900185\pi\)
−0.951236 + 0.308463i \(0.900185\pi\)
\(614\) −57.6648 −2.32716
\(615\) 4.34606 0.175250
\(616\) 0 0
\(617\) −29.1976 −1.17545 −0.587725 0.809061i \(-0.699976\pi\)
−0.587725 + 0.809061i \(0.699976\pi\)
\(618\) 3.48102 0.140027
\(619\) −14.4885 −0.582342 −0.291171 0.956671i \(-0.594045\pi\)
−0.291171 + 0.956671i \(0.594045\pi\)
\(620\) 3.66896 0.147349
\(621\) 22.3819 0.898156
\(622\) −26.7489 −1.07253
\(623\) −3.14060 −0.125826
\(624\) −3.91061 −0.156550
\(625\) 1.00000 0.0400000
\(626\) 7.66731 0.306447
\(627\) 0 0
\(628\) −34.2128 −1.36524
\(629\) −50.8690 −2.02828
\(630\) 3.40137 0.135514
\(631\) −4.24628 −0.169042 −0.0845209 0.996422i \(-0.526936\pi\)
−0.0845209 + 0.996422i \(0.526936\pi\)
\(632\) 4.20191 0.167143
\(633\) 12.2249 0.485897
\(634\) −61.3893 −2.43808
\(635\) 1.16924 0.0463999
\(636\) −10.5918 −0.419992
\(637\) −1.06949 −0.0423747
\(638\) 0 0
\(639\) −11.8658 −0.469402
\(640\) 6.09774 0.241034
\(641\) −5.47306 −0.216173 −0.108086 0.994142i \(-0.534472\pi\)
−0.108086 + 0.994142i \(0.534472\pi\)
\(642\) 25.8797 1.02139
\(643\) −4.59595 −0.181247 −0.0906233 0.995885i \(-0.528886\pi\)
−0.0906233 + 0.995885i \(0.528886\pi\)
\(644\) 9.78772 0.385690
\(645\) 9.00249 0.354473
\(646\) −14.3264 −0.563666
\(647\) 46.8490 1.84183 0.920913 0.389769i \(-0.127445\pi\)
0.920913 + 0.389769i \(0.127445\pi\)
\(648\) 1.14192 0.0448590
\(649\) 0 0
\(650\) 2.23596 0.0877017
\(651\) −1.81329 −0.0710685
\(652\) 17.8735 0.699979
\(653\) −19.7735 −0.773798 −0.386899 0.922122i \(-0.626454\pi\)
−0.386899 + 0.922122i \(0.626454\pi\)
\(654\) −20.3940 −0.797469
\(655\) −0.200825 −0.00784689
\(656\) 11.5736 0.451872
\(657\) 11.9188 0.464998
\(658\) 25.1612 0.980887
\(659\) −0.186499 −0.00726498 −0.00363249 0.999993i \(-0.501156\pi\)
−0.00363249 + 0.999993i \(0.501156\pi\)
\(660\) 0 0
\(661\) 5.06632 0.197057 0.0985285 0.995134i \(-0.468586\pi\)
0.0985285 + 0.995134i \(0.468586\pi\)
\(662\) 50.0046 1.94348
\(663\) −6.28719 −0.244174
\(664\) −0.0361775 −0.00140396
\(665\) −1.36590 −0.0529673
\(666\) 34.4886 1.33641
\(667\) 28.5796 1.10661
\(668\) 27.3404 1.05783
\(669\) 3.94024 0.152339
\(670\) −25.6776 −0.992012
\(671\) 0 0
\(672\) −9.46222 −0.365013
\(673\) −10.1700 −0.392023 −0.196012 0.980602i \(-0.562799\pi\)
−0.196012 + 0.980602i \(0.562799\pi\)
\(674\) 52.9820 2.04079
\(675\) 5.42176 0.208684
\(676\) −28.1105 −1.08117
\(677\) 3.28103 0.126100 0.0630500 0.998010i \(-0.479917\pi\)
0.0630500 + 0.998010i \(0.479917\pi\)
\(678\) 29.3433 1.12692
\(679\) 12.8210 0.492026
\(680\) 3.89086 0.149208
\(681\) −5.64994 −0.216506
\(682\) 0 0
\(683\) −1.11730 −0.0427523 −0.0213762 0.999772i \(-0.506805\pi\)
−0.0213762 + 0.999772i \(0.506805\pi\)
\(684\) 5.26875 0.201456
\(685\) −5.42631 −0.207329
\(686\) −2.09068 −0.0798227
\(687\) 15.5692 0.594002
\(688\) 23.9737 0.913987
\(689\) −4.07731 −0.155333
\(690\) 10.1133 0.385008
\(691\) −40.3162 −1.53370 −0.766850 0.641826i \(-0.778177\pi\)
−0.766850 + 0.641826i \(0.778177\pi\)
\(692\) −23.8185 −0.905444
\(693\) 0 0
\(694\) 72.8921 2.76695
\(695\) 3.01028 0.114186
\(696\) 6.29161 0.238483
\(697\) 18.6071 0.704794
\(698\) 1.58822 0.0601149
\(699\) 26.5700 1.00497
\(700\) 2.37096 0.0896138
\(701\) 4.75394 0.179554 0.0897769 0.995962i \(-0.471385\pi\)
0.0897769 + 0.995962i \(0.471385\pi\)
\(702\) 12.1229 0.457548
\(703\) −13.8497 −0.522351
\(704\) 0 0
\(705\) 14.1024 0.531126
\(706\) 75.5639 2.84388
\(707\) 6.58237 0.247555
\(708\) 24.2497 0.911358
\(709\) −47.5235 −1.78478 −0.892392 0.451261i \(-0.850974\pi\)
−0.892392 + 0.451261i \(0.850974\pi\)
\(710\) −15.2482 −0.572254
\(711\) 8.81450 0.330570
\(712\) 2.43572 0.0912826
\(713\) 6.38817 0.239239
\(714\) −12.2905 −0.459959
\(715\) 0 0
\(716\) −38.6636 −1.44493
\(717\) 8.44874 0.315524
\(718\) −17.4068 −0.649617
\(719\) 15.6866 0.585011 0.292506 0.956264i \(-0.405511\pi\)
0.292506 + 0.956264i \(0.405511\pi\)
\(720\) 5.07675 0.189199
\(721\) 1.42092 0.0529178
\(722\) 35.8225 1.33317
\(723\) −15.5276 −0.577478
\(724\) −27.8307 −1.03432
\(725\) 6.92308 0.257117
\(726\) 0 0
\(727\) −35.7434 −1.32565 −0.662824 0.748775i \(-0.730642\pi\)
−0.662824 + 0.748775i \(0.730642\pi\)
\(728\) 0.829451 0.0307415
\(729\) 21.4549 0.794625
\(730\) 15.3164 0.566886
\(731\) 38.5430 1.42557
\(732\) 34.5174 1.27580
\(733\) −28.5664 −1.05512 −0.527562 0.849517i \(-0.676894\pi\)
−0.527562 + 0.849517i \(0.676894\pi\)
\(734\) 71.7585 2.64866
\(735\) −1.17179 −0.0432220
\(736\) 33.3351 1.22875
\(737\) 0 0
\(738\) −12.6154 −0.464379
\(739\) −7.97795 −0.293474 −0.146737 0.989176i \(-0.546877\pi\)
−0.146737 + 0.989176i \(0.546877\pi\)
\(740\) 24.0406 0.883752
\(741\) −1.71176 −0.0628831
\(742\) −7.97050 −0.292606
\(743\) 44.8508 1.64542 0.822708 0.568464i \(-0.192462\pi\)
0.822708 + 0.568464i \(0.192462\pi\)
\(744\) 1.40631 0.0515580
\(745\) −5.65447 −0.207164
\(746\) −24.1560 −0.884414
\(747\) −0.0758908 −0.00277670
\(748\) 0 0
\(749\) 10.5639 0.385995
\(750\) 2.44983 0.0894553
\(751\) 49.7175 1.81422 0.907108 0.420897i \(-0.138285\pi\)
0.907108 + 0.420897i \(0.138285\pi\)
\(752\) 37.5547 1.36948
\(753\) −29.9726 −1.09226
\(754\) 15.4797 0.563739
\(755\) −18.1008 −0.658756
\(756\) 12.8548 0.467523
\(757\) 0.797363 0.0289807 0.0144903 0.999895i \(-0.495387\pi\)
0.0144903 + 0.999895i \(0.495387\pi\)
\(758\) 54.6528 1.98508
\(759\) 0 0
\(760\) 1.05933 0.0384261
\(761\) −38.6346 −1.40050 −0.700251 0.713897i \(-0.746929\pi\)
−0.700251 + 0.713897i \(0.746929\pi\)
\(762\) 2.86445 0.103768
\(763\) −8.32464 −0.301372
\(764\) 27.4698 0.993822
\(765\) 8.16201 0.295098
\(766\) 27.2523 0.984667
\(767\) 9.33490 0.337064
\(768\) −10.0004 −0.360860
\(769\) −37.9357 −1.36800 −0.683999 0.729483i \(-0.739761\pi\)
−0.683999 + 0.729483i \(0.739761\pi\)
\(770\) 0 0
\(771\) 7.55475 0.272078
\(772\) 26.1099 0.939715
\(773\) 1.46474 0.0526830 0.0263415 0.999653i \(-0.491614\pi\)
0.0263415 + 0.999653i \(0.491614\pi\)
\(774\) −26.1317 −0.939285
\(775\) 1.54746 0.0555864
\(776\) −9.94346 −0.356949
\(777\) −11.8815 −0.426246
\(778\) −40.5173 −1.45262
\(779\) 5.06600 0.181508
\(780\) 2.97132 0.106390
\(781\) 0 0
\(782\) 43.2990 1.54837
\(783\) 37.5352 1.34140
\(784\) −3.12047 −0.111445
\(785\) −14.4299 −0.515027
\(786\) −0.491989 −0.0175487
\(787\) −12.2969 −0.438339 −0.219169 0.975687i \(-0.570335\pi\)
−0.219169 + 0.975687i \(0.570335\pi\)
\(788\) 29.2376 1.04154
\(789\) −20.8443 −0.742076
\(790\) 11.3272 0.403002
\(791\) 11.9777 0.425877
\(792\) 0 0
\(793\) 13.2874 0.471851
\(794\) −58.1286 −2.06291
\(795\) −4.46730 −0.158439
\(796\) −22.4343 −0.795163
\(797\) −24.9795 −0.884818 −0.442409 0.896813i \(-0.645876\pi\)
−0.442409 + 0.896813i \(0.645876\pi\)
\(798\) −3.34622 −0.118455
\(799\) 60.3775 2.13600
\(800\) 8.07504 0.285496
\(801\) 5.10950 0.180535
\(802\) 48.0619 1.69712
\(803\) 0 0
\(804\) −34.1223 −1.20340
\(805\) 4.12817 0.145499
\(806\) 3.46006 0.121876
\(807\) −17.4119 −0.612926
\(808\) −5.10501 −0.179594
\(809\) −29.7899 −1.04736 −0.523679 0.851916i \(-0.675441\pi\)
−0.523679 + 0.851916i \(0.675441\pi\)
\(810\) 3.07830 0.108161
\(811\) 17.8361 0.626311 0.313156 0.949702i \(-0.398614\pi\)
0.313156 + 0.949702i \(0.398614\pi\)
\(812\) 16.4143 0.576030
\(813\) 15.4379 0.541430
\(814\) 0 0
\(815\) 7.53849 0.264062
\(816\) −18.3443 −0.642177
\(817\) 10.4938 0.367131
\(818\) 75.1455 2.62740
\(819\) 1.73997 0.0607995
\(820\) −8.79369 −0.307089
\(821\) −37.5866 −1.31178 −0.655890 0.754857i \(-0.727706\pi\)
−0.655890 + 0.754857i \(0.727706\pi\)
\(822\) −13.2936 −0.463667
\(823\) −4.98553 −0.173785 −0.0868923 0.996218i \(-0.527694\pi\)
−0.0868923 + 0.996218i \(0.527694\pi\)
\(824\) −1.10201 −0.0383902
\(825\) 0 0
\(826\) 18.2483 0.634938
\(827\) 8.03345 0.279351 0.139675 0.990197i \(-0.455394\pi\)
0.139675 + 0.990197i \(0.455394\pi\)
\(828\) −15.9238 −0.553391
\(829\) −31.0863 −1.07967 −0.539836 0.841770i \(-0.681514\pi\)
−0.539836 + 0.841770i \(0.681514\pi\)
\(830\) −0.0975241 −0.00338511
\(831\) −5.77511 −0.200336
\(832\) 11.3809 0.394561
\(833\) −5.01686 −0.173824
\(834\) 7.37469 0.255365
\(835\) 11.5314 0.399060
\(836\) 0 0
\(837\) 8.38995 0.289999
\(838\) 40.0248 1.38263
\(839\) −16.6694 −0.575492 −0.287746 0.957707i \(-0.592906\pi\)
−0.287746 + 0.957707i \(0.592906\pi\)
\(840\) 0.908789 0.0313562
\(841\) 18.9290 0.652724
\(842\) −60.0601 −2.06981
\(843\) 30.0192 1.03392
\(844\) −24.7355 −0.851432
\(845\) −11.8562 −0.407865
\(846\) −40.9353 −1.40738
\(847\) 0 0
\(848\) −11.8964 −0.408526
\(849\) 29.6203 1.01657
\(850\) 10.4887 0.359758
\(851\) 41.8581 1.43488
\(852\) −20.2629 −0.694197
\(853\) −13.9370 −0.477195 −0.238598 0.971119i \(-0.576688\pi\)
−0.238598 + 0.971119i \(0.576688\pi\)
\(854\) 25.9749 0.888842
\(855\) 2.22220 0.0759977
\(856\) −8.19289 −0.280027
\(857\) −53.5529 −1.82933 −0.914666 0.404211i \(-0.867546\pi\)
−0.914666 + 0.404211i \(0.867546\pi\)
\(858\) 0 0
\(859\) 13.4377 0.458488 0.229244 0.973369i \(-0.426375\pi\)
0.229244 + 0.973369i \(0.426375\pi\)
\(860\) −18.2154 −0.621139
\(861\) 4.34606 0.148113
\(862\) −27.1284 −0.923997
\(863\) −23.5880 −0.802943 −0.401472 0.915871i \(-0.631501\pi\)
−0.401472 + 0.915871i \(0.631501\pi\)
\(864\) 43.7809 1.48946
\(865\) −10.0459 −0.341572
\(866\) 78.4805 2.66688
\(867\) −9.57214 −0.325087
\(868\) 3.66896 0.124533
\(869\) 0 0
\(870\) 16.9604 0.575011
\(871\) −13.1354 −0.445075
\(872\) 6.45625 0.218636
\(873\) −20.8588 −0.705962
\(874\) 11.7887 0.398757
\(875\) 1.00000 0.0338062
\(876\) 20.3536 0.687684
\(877\) −41.4253 −1.39883 −0.699416 0.714715i \(-0.746556\pi\)
−0.699416 + 0.714715i \(0.746556\pi\)
\(878\) 33.9055 1.14425
\(879\) −27.9892 −0.944053
\(880\) 0 0
\(881\) 21.9662 0.740059 0.370029 0.929020i \(-0.379348\pi\)
0.370029 + 0.929020i \(0.379348\pi\)
\(882\) 3.40137 0.114530
\(883\) −21.2030 −0.713539 −0.356770 0.934192i \(-0.616122\pi\)
−0.356770 + 0.934192i \(0.616122\pi\)
\(884\) 12.7213 0.427864
\(885\) 10.2278 0.343803
\(886\) 72.3096 2.42929
\(887\) 7.91729 0.265837 0.132918 0.991127i \(-0.457565\pi\)
0.132918 + 0.991127i \(0.457565\pi\)
\(888\) 9.21478 0.309228
\(889\) 1.16924 0.0392151
\(890\) 6.56601 0.220093
\(891\) 0 0
\(892\) −7.97257 −0.266941
\(893\) 16.4385 0.550093
\(894\) −13.8525 −0.463298
\(895\) −16.3072 −0.545089
\(896\) 6.09774 0.203711
\(897\) 5.17347 0.172737
\(898\) 24.2766 0.810122
\(899\) 10.7132 0.357305
\(900\) −3.85735 −0.128578
\(901\) −19.1262 −0.637186
\(902\) 0 0
\(903\) 9.00249 0.299584
\(904\) −9.28939 −0.308960
\(905\) −11.7382 −0.390190
\(906\) −44.3440 −1.47323
\(907\) 14.2135 0.471951 0.235975 0.971759i \(-0.424171\pi\)
0.235975 + 0.971759i \(0.424171\pi\)
\(908\) 11.4319 0.379382
\(909\) −10.7090 −0.355194
\(910\) 2.23596 0.0741215
\(911\) 47.1079 1.56076 0.780378 0.625308i \(-0.215027\pi\)
0.780378 + 0.625308i \(0.215027\pi\)
\(912\) −4.99444 −0.165382
\(913\) 0 0
\(914\) 50.4794 1.66971
\(915\) 14.5584 0.481286
\(916\) −31.5023 −1.04086
\(917\) −0.200825 −0.00663184
\(918\) 56.8670 1.87689
\(919\) 27.3086 0.900827 0.450414 0.892820i \(-0.351277\pi\)
0.450414 + 0.892820i \(0.351277\pi\)
\(920\) −3.20164 −0.105555
\(921\) −32.3200 −1.06498
\(922\) −26.5763 −0.875243
\(923\) −7.80021 −0.256747
\(924\) 0 0
\(925\) 10.1396 0.333389
\(926\) 41.9839 1.37968
\(927\) −2.31172 −0.0759267
\(928\) 55.9041 1.83514
\(929\) −44.6107 −1.46363 −0.731815 0.681503i \(-0.761326\pi\)
−0.731815 + 0.681503i \(0.761326\pi\)
\(930\) 3.79102 0.124312
\(931\) −1.36590 −0.0447655
\(932\) −53.7610 −1.76100
\(933\) −14.9922 −0.490824
\(934\) 41.6320 1.36224
\(935\) 0 0
\(936\) −1.34945 −0.0441081
\(937\) 44.6166 1.45756 0.728781 0.684747i \(-0.240087\pi\)
0.728781 + 0.684747i \(0.240087\pi\)
\(938\) −25.6776 −0.838403
\(939\) 4.29737 0.140239
\(940\) −28.5343 −0.930687
\(941\) 47.7722 1.55733 0.778665 0.627440i \(-0.215897\pi\)
0.778665 + 0.627440i \(0.215897\pi\)
\(942\) −35.3510 −1.15180
\(943\) −15.3110 −0.498596
\(944\) 27.2366 0.886476
\(945\) 5.42176 0.176370
\(946\) 0 0
\(947\) −44.1814 −1.43570 −0.717851 0.696197i \(-0.754874\pi\)
−0.717851 + 0.696197i \(0.754874\pi\)
\(948\) 15.0524 0.488878
\(949\) 7.83511 0.254338
\(950\) 2.85566 0.0926499
\(951\) −34.4075 −1.11574
\(952\) 3.89086 0.126104
\(953\) −27.6676 −0.896241 −0.448121 0.893973i \(-0.647906\pi\)
−0.448121 + 0.893973i \(0.647906\pi\)
\(954\) 12.9673 0.419833
\(955\) 11.5859 0.374912
\(956\) −17.0949 −0.552890
\(957\) 0 0
\(958\) 67.1764 2.17037
\(959\) −5.42631 −0.175225
\(960\) 12.4695 0.402450
\(961\) −28.6054 −0.922754
\(962\) 22.6718 0.730969
\(963\) −17.1865 −0.553828
\(964\) 31.4181 1.01191
\(965\) 11.0124 0.354501
\(966\) 10.1133 0.325391
\(967\) 51.9072 1.66922 0.834611 0.550840i \(-0.185693\pi\)
0.834611 + 0.550840i \(0.185693\pi\)
\(968\) 0 0
\(969\) −8.02968 −0.257951
\(970\) −26.8047 −0.860648
\(971\) −28.4215 −0.912089 −0.456045 0.889957i \(-0.650734\pi\)
−0.456045 + 0.889957i \(0.650734\pi\)
\(972\) −34.4736 −1.10574
\(973\) 3.01028 0.0965052
\(974\) −14.3478 −0.459735
\(975\) 1.25321 0.0401349
\(976\) 38.7691 1.24097
\(977\) 7.22406 0.231118 0.115559 0.993301i \(-0.463134\pi\)
0.115559 + 0.993301i \(0.463134\pi\)
\(978\) 18.4681 0.590543
\(979\) 0 0
\(980\) 2.37096 0.0757375
\(981\) 13.5435 0.432411
\(982\) −34.5462 −1.10241
\(983\) −47.4725 −1.51414 −0.757068 0.653336i \(-0.773369\pi\)
−0.757068 + 0.653336i \(0.773369\pi\)
\(984\) −3.37062 −0.107451
\(985\) 12.3315 0.392915
\(986\) 72.6138 2.31249
\(987\) 14.1024 0.448883
\(988\) 3.46353 0.110190
\(989\) −31.7155 −1.00849
\(990\) 0 0
\(991\) 39.7713 1.26338 0.631688 0.775223i \(-0.282362\pi\)
0.631688 + 0.775223i \(0.282362\pi\)
\(992\) 12.4958 0.396742
\(993\) 28.0265 0.889396
\(994\) −15.2482 −0.483643
\(995\) −9.46213 −0.299970
\(996\) −0.129597 −0.00410645
\(997\) 51.0645 1.61723 0.808614 0.588339i \(-0.200218\pi\)
0.808614 + 0.588339i \(0.200218\pi\)
\(998\) −27.9332 −0.884210
\(999\) 54.9746 1.73932
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.bi.1.2 10
11.10 odd 2 4235.2.a.bk.1.9 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.bi.1.2 10 1.1 even 1 trivial
4235.2.a.bk.1.9 yes 10 11.10 odd 2