Properties

Label 9075.2.a.cu.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.456850\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.18890 q^{2} -1.00000 q^{3} +2.79129 q^{4} +2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.18890 q^{2} -1.00000 q^{3} +2.79129 q^{4} +2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} -2.79129 q^{12} -4.37780 q^{13} +2.00000 q^{14} -1.79129 q^{16} +1.73205 q^{17} -2.18890 q^{18} +3.46410 q^{19} +0.913701 q^{21} -8.58258 q^{23} +1.73205 q^{24} +9.58258 q^{26} -1.00000 q^{27} -2.55040 q^{28} +6.20520 q^{29} +0.582576 q^{31} +7.38505 q^{32} -3.79129 q^{34} +2.79129 q^{36} +1.58258 q^{37} -7.58258 q^{38} +4.37780 q^{39} -3.46410 q^{41} -2.00000 q^{42} +6.20520 q^{43} +18.7864 q^{46} -12.5826 q^{47} +1.79129 q^{48} -6.16515 q^{49} -1.73205 q^{51} -12.2197 q^{52} +6.16515 q^{53} +2.18890 q^{54} +1.58258 q^{56} -3.46410 q^{57} -13.5826 q^{58} +4.41742 q^{59} -7.02355 q^{61} -1.27520 q^{62} -0.913701 q^{63} -12.5826 q^{64} +13.5826 q^{67} +4.83465 q^{68} +8.58258 q^{69} +8.00000 q^{71} -1.73205 q^{72} +15.6838 q^{73} -3.46410 q^{74} +9.66930 q^{76} -9.58258 q^{78} -6.10985 q^{79} +1.00000 q^{81} +7.58258 q^{82} +15.8745 q^{83} +2.55040 q^{84} -13.5826 q^{86} -6.20520 q^{87} -12.7477 q^{89} +4.00000 q^{91} -23.9564 q^{92} -0.582576 q^{93} +27.5420 q^{94} -7.38505 q^{96} -13.5826 q^{97} +13.4949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9} - 2 q^{12} + 8 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} - 4 q^{27} - 16 q^{31} - 6 q^{34} + 2 q^{36} - 12 q^{37} - 12 q^{38} - 8 q^{42} - 32 q^{47} - 2 q^{48} + 12 q^{49} - 12 q^{53} - 12 q^{56} - 36 q^{58} + 36 q^{59} - 32 q^{64} + 36 q^{67} + 16 q^{69} + 32 q^{71} - 20 q^{78} + 4 q^{81} + 12 q^{82} - 36 q^{86} + 4 q^{89} + 16 q^{91} - 50 q^{92} + 16 q^{93} - 36 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18890 −1.54779 −0.773893 0.633316i \(-0.781693\pi\)
−0.773893 + 0.633316i \(0.781693\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.79129 1.39564
\(5\) 0 0
\(6\) 2.18890 0.893615
\(7\) −0.913701 −0.345346 −0.172673 0.984979i \(-0.555240\pi\)
−0.172673 + 0.984979i \(0.555240\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −2.79129 −0.805775
\(13\) −4.37780 −1.21418 −0.607092 0.794632i \(-0.707664\pi\)
−0.607092 + 0.794632i \(0.707664\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.79129 −0.447822
\(17\) 1.73205 0.420084 0.210042 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) −2.18890 −0.515929
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 0.913701 0.199386
\(22\) 0 0
\(23\) −8.58258 −1.78959 −0.894795 0.446476i \(-0.852679\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 1.73205 0.353553
\(25\) 0 0
\(26\) 9.58258 1.87930
\(27\) −1.00000 −0.192450
\(28\) −2.55040 −0.481981
\(29\) 6.20520 1.15228 0.576139 0.817352i \(-0.304559\pi\)
0.576139 + 0.817352i \(0.304559\pi\)
\(30\) 0 0
\(31\) 0.582576 0.104634 0.0523168 0.998631i \(-0.483339\pi\)
0.0523168 + 0.998631i \(0.483339\pi\)
\(32\) 7.38505 1.30551
\(33\) 0 0
\(34\) −3.79129 −0.650201
\(35\) 0 0
\(36\) 2.79129 0.465215
\(37\) 1.58258 0.260174 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(38\) −7.58258 −1.23006
\(39\) 4.37780 0.701009
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) −2.00000 −0.308607
\(43\) 6.20520 0.946285 0.473142 0.880986i \(-0.343120\pi\)
0.473142 + 0.880986i \(0.343120\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 18.7864 2.76990
\(47\) −12.5826 −1.83536 −0.917679 0.397323i \(-0.869939\pi\)
−0.917679 + 0.397323i \(0.869939\pi\)
\(48\) 1.79129 0.258550
\(49\) −6.16515 −0.880736
\(50\) 0 0
\(51\) −1.73205 −0.242536
\(52\) −12.2197 −1.69457
\(53\) 6.16515 0.846849 0.423424 0.905931i \(-0.360828\pi\)
0.423424 + 0.905931i \(0.360828\pi\)
\(54\) 2.18890 0.297872
\(55\) 0 0
\(56\) 1.58258 0.211481
\(57\) −3.46410 −0.458831
\(58\) −13.5826 −1.78348
\(59\) 4.41742 0.575100 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(60\) 0 0
\(61\) −7.02355 −0.899274 −0.449637 0.893211i \(-0.648447\pi\)
−0.449637 + 0.893211i \(0.648447\pi\)
\(62\) −1.27520 −0.161951
\(63\) −0.913701 −0.115115
\(64\) −12.5826 −1.57282
\(65\) 0 0
\(66\) 0 0
\(67\) 13.5826 1.65938 0.829688 0.558228i \(-0.188518\pi\)
0.829688 + 0.558228i \(0.188518\pi\)
\(68\) 4.83465 0.586288
\(69\) 8.58258 1.03322
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.73205 −0.204124
\(73\) 15.6838 1.83565 0.917825 0.396984i \(-0.129943\pi\)
0.917825 + 0.396984i \(0.129943\pi\)
\(74\) −3.46410 −0.402694
\(75\) 0 0
\(76\) 9.66930 1.10915
\(77\) 0 0
\(78\) −9.58258 −1.08501
\(79\) −6.10985 −0.687412 −0.343706 0.939077i \(-0.611682\pi\)
−0.343706 + 0.939077i \(0.611682\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.58258 0.837355
\(83\) 15.8745 1.74245 0.871227 0.490881i \(-0.163325\pi\)
0.871227 + 0.490881i \(0.163325\pi\)
\(84\) 2.55040 0.278272
\(85\) 0 0
\(86\) −13.5826 −1.46465
\(87\) −6.20520 −0.665268
\(88\) 0 0
\(89\) −12.7477 −1.35126 −0.675628 0.737243i \(-0.736128\pi\)
−0.675628 + 0.737243i \(0.736128\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −23.9564 −2.49763
\(93\) −0.582576 −0.0604103
\(94\) 27.5420 2.84074
\(95\) 0 0
\(96\) −7.38505 −0.753734
\(97\) −13.5826 −1.37910 −0.689551 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(98\) 13.4949 1.36319
\(99\) 0 0
\(100\) 0 0
\(101\) 8.75560 0.871215 0.435608 0.900137i \(-0.356534\pi\)
0.435608 + 0.900137i \(0.356534\pi\)
\(102\) 3.79129 0.375393
\(103\) 11.1652 1.10014 0.550068 0.835120i \(-0.314602\pi\)
0.550068 + 0.835120i \(0.314602\pi\)
\(104\) 7.58258 0.743533
\(105\) 0 0
\(106\) −13.4949 −1.31074
\(107\) −7.74655 −0.748888 −0.374444 0.927250i \(-0.622166\pi\)
−0.374444 + 0.927250i \(0.622166\pi\)
\(108\) −2.79129 −0.268592
\(109\) 6.92820 0.663602 0.331801 0.943349i \(-0.392344\pi\)
0.331801 + 0.943349i \(0.392344\pi\)
\(110\) 0 0
\(111\) −1.58258 −0.150211
\(112\) 1.63670 0.154654
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 7.58258 0.710173
\(115\) 0 0
\(116\) 17.3205 1.60817
\(117\) −4.37780 −0.404728
\(118\) −9.66930 −0.890132
\(119\) −1.58258 −0.145074
\(120\) 0 0
\(121\) 0 0
\(122\) 15.3739 1.39188
\(123\) 3.46410 0.312348
\(124\) 1.62614 0.146031
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 2.55040 0.226312 0.113156 0.993577i \(-0.463904\pi\)
0.113156 + 0.993577i \(0.463904\pi\)
\(128\) 12.7719 1.12889
\(129\) −6.20520 −0.546338
\(130\) 0 0
\(131\) 0.190700 0.0166616 0.00833079 0.999965i \(-0.497348\pi\)
0.00833079 + 0.999965i \(0.497348\pi\)
\(132\) 0 0
\(133\) −3.16515 −0.274453
\(134\) −29.7309 −2.56836
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −17.3303 −1.48063 −0.740314 0.672261i \(-0.765323\pi\)
−0.740314 + 0.672261i \(0.765323\pi\)
\(138\) −18.7864 −1.59921
\(139\) −13.0381 −1.10587 −0.552937 0.833223i \(-0.686493\pi\)
−0.552937 + 0.833223i \(0.686493\pi\)
\(140\) 0 0
\(141\) 12.5826 1.05964
\(142\) −17.5112 −1.46951
\(143\) 0 0
\(144\) −1.79129 −0.149274
\(145\) 0 0
\(146\) −34.3303 −2.84120
\(147\) 6.16515 0.508493
\(148\) 4.41742 0.363110
\(149\) −13.1334 −1.07593 −0.537965 0.842967i \(-0.680807\pi\)
−0.537965 + 0.842967i \(0.680807\pi\)
\(150\) 0 0
\(151\) −13.0381 −1.06102 −0.530511 0.847678i \(-0.678000\pi\)
−0.530511 + 0.847678i \(0.678000\pi\)
\(152\) −6.00000 −0.486664
\(153\) 1.73205 0.140028
\(154\) 0 0
\(155\) 0 0
\(156\) 12.2197 0.978359
\(157\) 16.7477 1.33661 0.668307 0.743886i \(-0.267019\pi\)
0.668307 + 0.743886i \(0.267019\pi\)
\(158\) 13.3739 1.06397
\(159\) −6.16515 −0.488928
\(160\) 0 0
\(161\) 7.84190 0.618029
\(162\) −2.18890 −0.171976
\(163\) −6.74773 −0.528523 −0.264261 0.964451i \(-0.585128\pi\)
−0.264261 + 0.964451i \(0.585128\pi\)
\(164\) −9.66930 −0.755046
\(165\) 0 0
\(166\) −34.7477 −2.69695
\(167\) −11.4014 −0.882263 −0.441132 0.897442i \(-0.645423\pi\)
−0.441132 + 0.897442i \(0.645423\pi\)
\(168\) −1.58258 −0.122098
\(169\) 6.16515 0.474242
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) 17.3205 1.32068
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 13.5826 1.02969
\(175\) 0 0
\(176\) 0 0
\(177\) −4.41742 −0.332034
\(178\) 27.9035 2.09146
\(179\) −1.58258 −0.118287 −0.0591436 0.998249i \(-0.518837\pi\)
−0.0591436 + 0.998249i \(0.518837\pi\)
\(180\) 0 0
\(181\) 13.1652 0.978558 0.489279 0.872127i \(-0.337260\pi\)
0.489279 + 0.872127i \(0.337260\pi\)
\(182\) −8.75560 −0.649009
\(183\) 7.02355 0.519196
\(184\) 14.8655 1.09590
\(185\) 0 0
\(186\) 1.27520 0.0935022
\(187\) 0 0
\(188\) −35.1216 −2.56151
\(189\) 0.913701 0.0664619
\(190\) 0 0
\(191\) 4.74773 0.343533 0.171767 0.985138i \(-0.445052\pi\)
0.171767 + 0.985138i \(0.445052\pi\)
\(192\) 12.5826 0.908069
\(193\) 21.6983 1.56188 0.780939 0.624607i \(-0.214741\pi\)
0.780939 + 0.624607i \(0.214741\pi\)
\(194\) 29.7309 2.13456
\(195\) 0 0
\(196\) −17.2087 −1.22919
\(197\) 19.3386 1.37782 0.688909 0.724847i \(-0.258090\pi\)
0.688909 + 0.724847i \(0.258090\pi\)
\(198\) 0 0
\(199\) 11.7477 0.832774 0.416387 0.909187i \(-0.363296\pi\)
0.416387 + 0.909187i \(0.363296\pi\)
\(200\) 0 0
\(201\) −13.5826 −0.958041
\(202\) −19.1652 −1.34846
\(203\) −5.66970 −0.397935
\(204\) −4.83465 −0.338493
\(205\) 0 0
\(206\) −24.4394 −1.70277
\(207\) −8.58258 −0.596530
\(208\) 7.84190 0.543738
\(209\) 0 0
\(210\) 0 0
\(211\) 18.1389 1.24873 0.624365 0.781133i \(-0.285358\pi\)
0.624365 + 0.781133i \(0.285358\pi\)
\(212\) 17.2087 1.18190
\(213\) −8.00000 −0.548151
\(214\) 16.9564 1.15912
\(215\) 0 0
\(216\) 1.73205 0.117851
\(217\) −0.532300 −0.0361349
\(218\) −15.1652 −1.02711
\(219\) −15.6838 −1.05981
\(220\) 0 0
\(221\) −7.58258 −0.510059
\(222\) 3.46410 0.232495
\(223\) −20.3303 −1.36142 −0.680709 0.732554i \(-0.738328\pi\)
−0.680709 + 0.732554i \(0.738328\pi\)
\(224\) −6.74773 −0.450851
\(225\) 0 0
\(226\) 19.7001 1.31043
\(227\) 6.30055 0.418182 0.209091 0.977896i \(-0.432949\pi\)
0.209091 + 0.977896i \(0.432949\pi\)
\(228\) −9.66930 −0.640365
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.7477 −0.705623
\(233\) 17.6066 1.15344 0.576722 0.816940i \(-0.304332\pi\)
0.576722 + 0.816940i \(0.304332\pi\)
\(234\) 9.58258 0.626433
\(235\) 0 0
\(236\) 12.3303 0.802634
\(237\) 6.10985 0.396878
\(238\) 3.46410 0.224544
\(239\) −26.9898 −1.74583 −0.872913 0.487876i \(-0.837772\pi\)
−0.872913 + 0.487876i \(0.837772\pi\)
\(240\) 0 0
\(241\) 5.00545 0.322430 0.161215 0.986919i \(-0.448459\pi\)
0.161215 + 0.986919i \(0.448459\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −19.6048 −1.25507
\(245\) 0 0
\(246\) −7.58258 −0.483447
\(247\) −15.1652 −0.964935
\(248\) −1.00905 −0.0640748
\(249\) −15.8745 −1.00601
\(250\) 0 0
\(251\) 3.58258 0.226130 0.113065 0.993588i \(-0.463933\pi\)
0.113065 + 0.993588i \(0.463933\pi\)
\(252\) −2.55040 −0.160660
\(253\) 0 0
\(254\) −5.58258 −0.350282
\(255\) 0 0
\(256\) −2.79129 −0.174455
\(257\) −24.1652 −1.50738 −0.753690 0.657230i \(-0.771728\pi\)
−0.753690 + 0.657230i \(0.771728\pi\)
\(258\) 13.5826 0.845614
\(259\) −1.44600 −0.0898501
\(260\) 0 0
\(261\) 6.20520 0.384092
\(262\) −0.417424 −0.0257886
\(263\) 4.47315 0.275826 0.137913 0.990444i \(-0.455961\pi\)
0.137913 + 0.990444i \(0.455961\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.92820 0.424795
\(267\) 12.7477 0.780148
\(268\) 37.9129 2.31590
\(269\) −27.1652 −1.65629 −0.828144 0.560515i \(-0.810603\pi\)
−0.828144 + 0.560515i \(0.810603\pi\)
\(270\) 0 0
\(271\) 9.76465 0.593161 0.296580 0.955008i \(-0.404154\pi\)
0.296580 + 0.955008i \(0.404154\pi\)
\(272\) −3.10260 −0.188123
\(273\) −4.00000 −0.242091
\(274\) 37.9343 2.29170
\(275\) 0 0
\(276\) 23.9564 1.44201
\(277\) 8.75560 0.526073 0.263037 0.964786i \(-0.415276\pi\)
0.263037 + 0.964786i \(0.415276\pi\)
\(278\) 28.5390 1.71166
\(279\) 0.582576 0.0348779
\(280\) 0 0
\(281\) −13.3241 −0.794850 −0.397425 0.917635i \(-0.630096\pi\)
−0.397425 + 0.917635i \(0.630096\pi\)
\(282\) −27.5420 −1.64010
\(283\) 31.5583 1.87595 0.937974 0.346707i \(-0.112700\pi\)
0.937974 + 0.346707i \(0.112700\pi\)
\(284\) 22.3303 1.32506
\(285\) 0 0
\(286\) 0 0
\(287\) 3.16515 0.186833
\(288\) 7.38505 0.435168
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) 13.5826 0.796225
\(292\) 43.7780 2.56191
\(293\) 1.73205 0.101187 0.0505937 0.998719i \(-0.483889\pi\)
0.0505937 + 0.998719i \(0.483889\pi\)
\(294\) −13.4949 −0.787039
\(295\) 0 0
\(296\) −2.74110 −0.159323
\(297\) 0 0
\(298\) 28.7477 1.66531
\(299\) 37.5728 2.17289
\(300\) 0 0
\(301\) −5.66970 −0.326796
\(302\) 28.5390 1.64224
\(303\) −8.75560 −0.502996
\(304\) −6.20520 −0.355893
\(305\) 0 0
\(306\) −3.79129 −0.216734
\(307\) −26.0761 −1.48824 −0.744121 0.668045i \(-0.767131\pi\)
−0.744121 + 0.668045i \(0.767131\pi\)
\(308\) 0 0
\(309\) −11.1652 −0.635163
\(310\) 0 0
\(311\) −2.41742 −0.137080 −0.0685398 0.997648i \(-0.521834\pi\)
−0.0685398 + 0.997648i \(0.521834\pi\)
\(312\) −7.58258 −0.429279
\(313\) 3.58258 0.202499 0.101250 0.994861i \(-0.467716\pi\)
0.101250 + 0.994861i \(0.467716\pi\)
\(314\) −36.6591 −2.06879
\(315\) 0 0
\(316\) −17.0544 −0.959383
\(317\) −11.8348 −0.664711 −0.332356 0.943154i \(-0.607843\pi\)
−0.332356 + 0.943154i \(0.607843\pi\)
\(318\) 13.4949 0.756757
\(319\) 0 0
\(320\) 0 0
\(321\) 7.74655 0.432370
\(322\) −17.1652 −0.956576
\(323\) 6.00000 0.333849
\(324\) 2.79129 0.155072
\(325\) 0 0
\(326\) 14.7701 0.818041
\(327\) −6.92820 −0.383131
\(328\) 6.00000 0.331295
\(329\) 11.4967 0.633834
\(330\) 0 0
\(331\) 0.252273 0.0138662 0.00693309 0.999976i \(-0.497793\pi\)
0.00693309 + 0.999976i \(0.497793\pi\)
\(332\) 44.3103 2.43184
\(333\) 1.58258 0.0867246
\(334\) 24.9564 1.36556
\(335\) 0 0
\(336\) −1.63670 −0.0892893
\(337\) −20.9753 −1.14260 −0.571299 0.820742i \(-0.693560\pi\)
−0.571299 + 0.820742i \(0.693560\pi\)
\(338\) −13.4949 −0.734026
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) −7.58258 −0.410019
\(343\) 12.0290 0.649505
\(344\) −10.7477 −0.579479
\(345\) 0 0
\(346\) 15.1652 0.815284
\(347\) −15.0562 −0.808257 −0.404128 0.914702i \(-0.632425\pi\)
−0.404128 + 0.914702i \(0.632425\pi\)
\(348\) −17.3205 −0.928477
\(349\) 25.9808 1.39072 0.695359 0.718662i \(-0.255245\pi\)
0.695359 + 0.718662i \(0.255245\pi\)
\(350\) 0 0
\(351\) 4.37780 0.233670
\(352\) 0 0
\(353\) −36.1652 −1.92488 −0.962438 0.271500i \(-0.912480\pi\)
−0.962438 + 0.271500i \(0.912480\pi\)
\(354\) 9.66930 0.513918
\(355\) 0 0
\(356\) −35.5826 −1.88587
\(357\) 1.58258 0.0837588
\(358\) 3.46410 0.183083
\(359\) 1.82740 0.0964465 0.0482233 0.998837i \(-0.484644\pi\)
0.0482233 + 0.998837i \(0.484644\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) −28.8172 −1.51460
\(363\) 0 0
\(364\) 11.1652 0.585213
\(365\) 0 0
\(366\) −15.3739 −0.803605
\(367\) 28.7477 1.50062 0.750310 0.661087i \(-0.229904\pi\)
0.750310 + 0.661087i \(0.229904\pi\)
\(368\) 15.3739 0.801418
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) −5.63310 −0.292456
\(372\) −1.62614 −0.0843112
\(373\) −13.8564 −0.717458 −0.358729 0.933442i \(-0.616790\pi\)
−0.358729 + 0.933442i \(0.616790\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 21.7937 1.12392
\(377\) −27.1652 −1.39908
\(378\) −2.00000 −0.102869
\(379\) 11.7477 0.603440 0.301720 0.953397i \(-0.402439\pi\)
0.301720 + 0.953397i \(0.402439\pi\)
\(380\) 0 0
\(381\) −2.55040 −0.130661
\(382\) −10.3923 −0.531717
\(383\) 30.3303 1.54981 0.774903 0.632080i \(-0.217799\pi\)
0.774903 + 0.632080i \(0.217799\pi\)
\(384\) −12.7719 −0.651764
\(385\) 0 0
\(386\) −47.4955 −2.41745
\(387\) 6.20520 0.315428
\(388\) −37.9129 −1.92473
\(389\) −22.7477 −1.15336 −0.576678 0.816972i \(-0.695651\pi\)
−0.576678 + 0.816972i \(0.695651\pi\)
\(390\) 0 0
\(391\) −14.8655 −0.751778
\(392\) 10.6784 0.539338
\(393\) −0.190700 −0.00961956
\(394\) −42.3303 −2.13257
\(395\) 0 0
\(396\) 0 0
\(397\) 4.33030 0.217332 0.108666 0.994078i \(-0.465342\pi\)
0.108666 + 0.994078i \(0.465342\pi\)
\(398\) −25.7146 −1.28896
\(399\) 3.16515 0.158456
\(400\) 0 0
\(401\) −22.3303 −1.11512 −0.557561 0.830136i \(-0.688263\pi\)
−0.557561 + 0.830136i \(0.688263\pi\)
\(402\) 29.7309 1.48284
\(403\) −2.55040 −0.127045
\(404\) 24.4394 1.21591
\(405\) 0 0
\(406\) 12.4104 0.615918
\(407\) 0 0
\(408\) 3.00000 0.148522
\(409\) 22.8981 1.13224 0.566118 0.824324i \(-0.308445\pi\)
0.566118 + 0.824324i \(0.308445\pi\)
\(410\) 0 0
\(411\) 17.3303 0.854841
\(412\) 31.1652 1.53540
\(413\) −4.03620 −0.198609
\(414\) 18.7864 0.923302
\(415\) 0 0
\(416\) −32.3303 −1.58512
\(417\) 13.0381 0.638476
\(418\) 0 0
\(419\) 32.3303 1.57944 0.789719 0.613468i \(-0.210226\pi\)
0.789719 + 0.613468i \(0.210226\pi\)
\(420\) 0 0
\(421\) −2.66970 −0.130113 −0.0650565 0.997882i \(-0.520723\pi\)
−0.0650565 + 0.997882i \(0.520723\pi\)
\(422\) −39.7042 −1.93277
\(423\) −12.5826 −0.611786
\(424\) −10.6784 −0.518587
\(425\) 0 0
\(426\) 17.5112 0.848421
\(427\) 6.41742 0.310561
\(428\) −21.6229 −1.04518
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0616 0.966334 0.483167 0.875528i \(-0.339486\pi\)
0.483167 + 0.875528i \(0.339486\pi\)
\(432\) 1.79129 0.0861834
\(433\) 4.41742 0.212288 0.106144 0.994351i \(-0.466150\pi\)
0.106144 + 0.994351i \(0.466150\pi\)
\(434\) 1.16515 0.0559291
\(435\) 0 0
\(436\) 19.3386 0.926151
\(437\) −29.7309 −1.42222
\(438\) 34.3303 1.64037
\(439\) 9.57395 0.456940 0.228470 0.973551i \(-0.426628\pi\)
0.228470 + 0.973551i \(0.426628\pi\)
\(440\) 0 0
\(441\) −6.16515 −0.293579
\(442\) 16.5975 0.789463
\(443\) −7.16515 −0.340427 −0.170213 0.985407i \(-0.554446\pi\)
−0.170213 + 0.985407i \(0.554446\pi\)
\(444\) −4.41742 −0.209642
\(445\) 0 0
\(446\) 44.5010 2.10718
\(447\) 13.1334 0.621189
\(448\) 11.4967 0.543168
\(449\) 2.83485 0.133785 0.0668924 0.997760i \(-0.478692\pi\)
0.0668924 + 0.997760i \(0.478692\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −25.1216 −1.18162
\(453\) 13.0381 0.612581
\(454\) −13.7913 −0.647257
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 22.4213 1.04882 0.524412 0.851464i \(-0.324285\pi\)
0.524412 + 0.851464i \(0.324285\pi\)
\(458\) −37.2113 −1.73877
\(459\) −1.73205 −0.0808452
\(460\) 0 0
\(461\) 8.03260 0.374116 0.187058 0.982349i \(-0.440105\pi\)
0.187058 + 0.982349i \(0.440105\pi\)
\(462\) 0 0
\(463\) 35.4955 1.64961 0.824807 0.565415i \(-0.191284\pi\)
0.824807 + 0.565415i \(0.191284\pi\)
\(464\) −11.1153 −0.516015
\(465\) 0 0
\(466\) −38.5390 −1.78529
\(467\) 3.41742 0.158140 0.0790698 0.996869i \(-0.474805\pi\)
0.0790698 + 0.996869i \(0.474805\pi\)
\(468\) −12.2197 −0.564856
\(469\) −12.4104 −0.573059
\(470\) 0 0
\(471\) −16.7477 −0.771695
\(472\) −7.65120 −0.352175
\(473\) 0 0
\(474\) −13.3739 −0.614282
\(475\) 0 0
\(476\) −4.41742 −0.202472
\(477\) 6.16515 0.282283
\(478\) 59.0780 2.70217
\(479\) −27.3712 −1.25062 −0.625311 0.780375i \(-0.715028\pi\)
−0.625311 + 0.780375i \(0.715028\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) −10.9564 −0.499052
\(483\) −7.84190 −0.356819
\(484\) 0 0
\(485\) 0 0
\(486\) 2.18890 0.0992906
\(487\) −3.58258 −0.162342 −0.0811710 0.996700i \(-0.525866\pi\)
−0.0811710 + 0.996700i \(0.525866\pi\)
\(488\) 12.1652 0.550691
\(489\) 6.74773 0.305143
\(490\) 0 0
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 9.66930 0.435926
\(493\) 10.7477 0.484053
\(494\) 33.1950 1.49351
\(495\) 0 0
\(496\) −1.04356 −0.0468573
\(497\) −7.30960 −0.327881
\(498\) 34.7477 1.55708
\(499\) 26.3303 1.17871 0.589353 0.807876i \(-0.299383\pi\)
0.589353 + 0.807876i \(0.299383\pi\)
\(500\) 0 0
\(501\) 11.4014 0.509375
\(502\) −7.84190 −0.350001
\(503\) −0.818350 −0.0364884 −0.0182442 0.999834i \(-0.505808\pi\)
−0.0182442 + 0.999834i \(0.505808\pi\)
\(504\) 1.58258 0.0704935
\(505\) 0 0
\(506\) 0 0
\(507\) −6.16515 −0.273804
\(508\) 7.11890 0.315850
\(509\) 0.747727 0.0331424 0.0165712 0.999863i \(-0.494725\pi\)
0.0165712 + 0.999863i \(0.494725\pi\)
\(510\) 0 0
\(511\) −14.3303 −0.633935
\(512\) −19.4340 −0.858868
\(513\) −3.46410 −0.152944
\(514\) 52.8951 2.33310
\(515\) 0 0
\(516\) −17.3205 −0.762493
\(517\) 0 0
\(518\) 3.16515 0.139069
\(519\) 6.92820 0.304114
\(520\) 0 0
\(521\) 9.25227 0.405349 0.202675 0.979246i \(-0.435037\pi\)
0.202675 + 0.979246i \(0.435037\pi\)
\(522\) −13.5826 −0.594493
\(523\) 4.56850 0.199767 0.0998833 0.994999i \(-0.468153\pi\)
0.0998833 + 0.994999i \(0.468153\pi\)
\(524\) 0.532300 0.0232536
\(525\) 0 0
\(526\) −9.79129 −0.426920
\(527\) 1.00905 0.0439549
\(528\) 0 0
\(529\) 50.6606 2.20264
\(530\) 0 0
\(531\) 4.41742 0.191700
\(532\) −8.83485 −0.383039
\(533\) 15.1652 0.656876
\(534\) −27.9035 −1.20750
\(535\) 0 0
\(536\) −23.5257 −1.01616
\(537\) 1.58258 0.0682932
\(538\) 59.4618 2.56358
\(539\) 0 0
\(540\) 0 0
\(541\) −29.9216 −1.28643 −0.643215 0.765685i \(-0.722400\pi\)
−0.643215 + 0.765685i \(0.722400\pi\)
\(542\) −21.3739 −0.918086
\(543\) −13.1652 −0.564971
\(544\) 12.7913 0.548422
\(545\) 0 0
\(546\) 8.75560 0.374705
\(547\) 3.99640 0.170874 0.0854369 0.996344i \(-0.472771\pi\)
0.0854369 + 0.996344i \(0.472771\pi\)
\(548\) −48.3739 −2.06643
\(549\) −7.02355 −0.299758
\(550\) 0 0
\(551\) 21.4955 0.915737
\(552\) −14.8655 −0.632716
\(553\) 5.58258 0.237395
\(554\) −19.1652 −0.814249
\(555\) 0 0
\(556\) −36.3930 −1.54341
\(557\) −29.6356 −1.25570 −0.627850 0.778335i \(-0.716065\pi\)
−0.627850 + 0.778335i \(0.716065\pi\)
\(558\) −1.27520 −0.0539835
\(559\) −27.1652 −1.14896
\(560\) 0 0
\(561\) 0 0
\(562\) 29.1652 1.23026
\(563\) −11.8383 −0.498925 −0.249463 0.968384i \(-0.580254\pi\)
−0.249463 + 0.968384i \(0.580254\pi\)
\(564\) 35.1216 1.47889
\(565\) 0 0
\(566\) −69.0780 −2.90357
\(567\) −0.913701 −0.0383718
\(568\) −13.8564 −0.581402
\(569\) 21.5076 0.901646 0.450823 0.892613i \(-0.351131\pi\)
0.450823 + 0.892613i \(0.351131\pi\)
\(570\) 0 0
\(571\) −27.0852 −1.13348 −0.566739 0.823897i \(-0.691795\pi\)
−0.566739 + 0.823897i \(0.691795\pi\)
\(572\) 0 0
\(573\) −4.74773 −0.198339
\(574\) −6.92820 −0.289178
\(575\) 0 0
\(576\) −12.5826 −0.524274
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 30.6446 1.27465
\(579\) −21.6983 −0.901751
\(580\) 0 0
\(581\) −14.5045 −0.601750
\(582\) −29.7309 −1.23239
\(583\) 0 0
\(584\) −27.1652 −1.12410
\(585\) 0 0
\(586\) −3.79129 −0.156617
\(587\) 19.7477 0.815076 0.407538 0.913188i \(-0.366387\pi\)
0.407538 + 0.913188i \(0.366387\pi\)
\(588\) 17.2087 0.709675
\(589\) 2.01810 0.0831544
\(590\) 0 0
\(591\) −19.3386 −0.795484
\(592\) −2.83485 −0.116512
\(593\) −27.7128 −1.13803 −0.569014 0.822328i \(-0.692675\pi\)
−0.569014 + 0.822328i \(0.692675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −36.6591 −1.50162
\(597\) −11.7477 −0.480802
\(598\) −82.2432 −3.36317
\(599\) −20.7477 −0.847729 −0.423865 0.905726i \(-0.639327\pi\)
−0.423865 + 0.905726i \(0.639327\pi\)
\(600\) 0 0
\(601\) −33.1950 −1.35405 −0.677026 0.735959i \(-0.736732\pi\)
−0.677026 + 0.735959i \(0.736732\pi\)
\(602\) 12.4104 0.505810
\(603\) 13.5826 0.553125
\(604\) −36.3930 −1.48081
\(605\) 0 0
\(606\) 19.1652 0.778531
\(607\) −22.0797 −0.896188 −0.448094 0.893986i \(-0.647897\pi\)
−0.448094 + 0.893986i \(0.647897\pi\)
\(608\) 25.5826 1.03751
\(609\) 5.66970 0.229748
\(610\) 0 0
\(611\) 55.0840 2.22846
\(612\) 4.83465 0.195429
\(613\) −8.75560 −0.353636 −0.176818 0.984244i \(-0.556580\pi\)
−0.176818 + 0.984244i \(0.556580\pi\)
\(614\) 57.0780 2.30348
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 24.4394 0.983097
\(619\) −41.4955 −1.66784 −0.833922 0.551883i \(-0.813910\pi\)
−0.833922 + 0.551883i \(0.813910\pi\)
\(620\) 0 0
\(621\) 8.58258 0.344407
\(622\) 5.29150 0.212170
\(623\) 11.6476 0.466651
\(624\) −7.84190 −0.313927
\(625\) 0 0
\(626\) −7.84190 −0.313426
\(627\) 0 0
\(628\) 46.7477 1.86544
\(629\) 2.74110 0.109295
\(630\) 0 0
\(631\) −30.5826 −1.21747 −0.608737 0.793372i \(-0.708323\pi\)
−0.608737 + 0.793372i \(0.708323\pi\)
\(632\) 10.5826 0.420952
\(633\) −18.1389 −0.720955
\(634\) 25.9053 1.02883
\(635\) 0 0
\(636\) −17.2087 −0.682370
\(637\) 26.9898 1.06938
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 20.7477 0.819486 0.409743 0.912201i \(-0.365618\pi\)
0.409743 + 0.912201i \(0.365618\pi\)
\(642\) −16.9564 −0.669217
\(643\) −15.4955 −0.611081 −0.305541 0.952179i \(-0.598837\pi\)
−0.305541 + 0.952179i \(0.598837\pi\)
\(644\) 21.8890 0.862548
\(645\) 0 0
\(646\) −13.1334 −0.516727
\(647\) −2.25227 −0.0885460 −0.0442730 0.999019i \(-0.514097\pi\)
−0.0442730 + 0.999019i \(0.514097\pi\)
\(648\) −1.73205 −0.0680414
\(649\) 0 0
\(650\) 0 0
\(651\) 0.532300 0.0208625
\(652\) −18.8348 −0.737630
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 15.1652 0.593004
\(655\) 0 0
\(656\) 6.20520 0.242272
\(657\) 15.6838 0.611884
\(658\) −25.1652 −0.981040
\(659\) −12.9427 −0.504176 −0.252088 0.967704i \(-0.581117\pi\)
−0.252088 + 0.967704i \(0.581117\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −0.552200 −0.0214619
\(663\) 7.58258 0.294483
\(664\) −27.4955 −1.06703
\(665\) 0 0
\(666\) −3.46410 −0.134231
\(667\) −53.2566 −2.06210
\(668\) −31.8245 −1.23133
\(669\) 20.3303 0.786015
\(670\) 0 0
\(671\) 0 0
\(672\) 6.74773 0.260299
\(673\) 6.01450 0.231842 0.115921 0.993258i \(-0.463018\pi\)
0.115921 + 0.993258i \(0.463018\pi\)
\(674\) 45.9129 1.76850
\(675\) 0 0
\(676\) 17.2087 0.661874
\(677\) −22.6120 −0.869050 −0.434525 0.900660i \(-0.643084\pi\)
−0.434525 + 0.900660i \(0.643084\pi\)
\(678\) −19.7001 −0.756578
\(679\) 12.4104 0.476268
\(680\) 0 0
\(681\) −6.30055 −0.241438
\(682\) 0 0
\(683\) −9.66970 −0.370001 −0.185000 0.982738i \(-0.559229\pi\)
−0.185000 + 0.982738i \(0.559229\pi\)
\(684\) 9.66930 0.369715
\(685\) 0 0
\(686\) −26.3303 −1.00530
\(687\) −17.0000 −0.648590
\(688\) −11.1153 −0.423767
\(689\) −26.9898 −1.02823
\(690\) 0 0
\(691\) 8.58258 0.326497 0.163248 0.986585i \(-0.447803\pi\)
0.163248 + 0.986585i \(0.447803\pi\)
\(692\) −19.3386 −0.735144
\(693\) 0 0
\(694\) 32.9564 1.25101
\(695\) 0 0
\(696\) 10.7477 0.407392
\(697\) −6.00000 −0.227266
\(698\) −56.8693 −2.15254
\(699\) −17.6066 −0.665941
\(700\) 0 0
\(701\) −4.56850 −0.172550 −0.0862750 0.996271i \(-0.527496\pi\)
−0.0862750 + 0.996271i \(0.527496\pi\)
\(702\) −9.58258 −0.361671
\(703\) 5.48220 0.206765
\(704\) 0 0
\(705\) 0 0
\(706\) 79.1619 2.97930
\(707\) −8.00000 −0.300871
\(708\) −12.3303 −0.463401
\(709\) −45.0000 −1.69001 −0.845005 0.534758i \(-0.820403\pi\)
−0.845005 + 0.534758i \(0.820403\pi\)
\(710\) 0 0
\(711\) −6.10985 −0.229137
\(712\) 22.0797 0.827472
\(713\) −5.00000 −0.187251
\(714\) −3.46410 −0.129641
\(715\) 0 0
\(716\) −4.41742 −0.165087
\(717\) 26.9898 1.00795
\(718\) −4.00000 −0.149279
\(719\) 20.3303 0.758192 0.379096 0.925357i \(-0.376235\pi\)
0.379096 + 0.925357i \(0.376235\pi\)
\(720\) 0 0
\(721\) −10.2016 −0.379928
\(722\) 15.3223 0.570237
\(723\) −5.00545 −0.186155
\(724\) 36.7477 1.36572
\(725\) 0 0
\(726\) 0 0
\(727\) −13.1652 −0.488268 −0.244134 0.969741i \(-0.578504\pi\)
−0.244134 + 0.969741i \(0.578504\pi\)
\(728\) −6.92820 −0.256776
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.7477 0.397519
\(732\) 19.6048 0.724613
\(733\) −34.8317 −1.28654 −0.643269 0.765640i \(-0.722422\pi\)
−0.643269 + 0.765640i \(0.722422\pi\)
\(734\) −62.9259 −2.32264
\(735\) 0 0
\(736\) −63.3828 −2.33632
\(737\) 0 0
\(738\) 7.58258 0.279118
\(739\) −15.0562 −0.553850 −0.276925 0.960892i \(-0.589315\pi\)
−0.276925 + 0.960892i \(0.589315\pi\)
\(740\) 0 0
\(741\) 15.1652 0.557106
\(742\) 12.3303 0.452660
\(743\) −49.6972 −1.82321 −0.911606 0.411065i \(-0.865157\pi\)
−0.911606 + 0.411065i \(0.865157\pi\)
\(744\) 1.00905 0.0369936
\(745\) 0 0
\(746\) 30.3303 1.11047
\(747\) 15.8745 0.580818
\(748\) 0 0
\(749\) 7.07803 0.258626
\(750\) 0 0
\(751\) −44.0780 −1.60843 −0.804215 0.594338i \(-0.797414\pi\)
−0.804215 + 0.594338i \(0.797414\pi\)
\(752\) 22.5390 0.821913
\(753\) −3.58258 −0.130556
\(754\) 59.4618 2.16547
\(755\) 0 0
\(756\) 2.55040 0.0927572
\(757\) −25.1652 −0.914643 −0.457321 0.889301i \(-0.651191\pi\)
−0.457321 + 0.889301i \(0.651191\pi\)
\(758\) −25.7146 −0.933997
\(759\) 0 0
\(760\) 0 0
\(761\) −9.47860 −0.343599 −0.171800 0.985132i \(-0.554958\pi\)
−0.171800 + 0.985132i \(0.554958\pi\)
\(762\) 5.58258 0.202235
\(763\) −6.33030 −0.229172
\(764\) 13.2523 0.479450
\(765\) 0 0
\(766\) −66.3900 −2.39877
\(767\) −19.3386 −0.698277
\(768\) 2.79129 0.100722
\(769\) 7.40495 0.267029 0.133515 0.991047i \(-0.457374\pi\)
0.133515 + 0.991047i \(0.457374\pi\)
\(770\) 0 0
\(771\) 24.1652 0.870287
\(772\) 60.5662 2.17983
\(773\) 6.49545 0.233625 0.116813 0.993154i \(-0.462732\pi\)
0.116813 + 0.993154i \(0.462732\pi\)
\(774\) −13.5826 −0.488216
\(775\) 0 0
\(776\) 23.5257 0.844524
\(777\) 1.44600 0.0518750
\(778\) 49.7925 1.78515
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 32.5390 1.16359
\(783\) −6.20520 −0.221756
\(784\) 11.0436 0.394413
\(785\) 0 0
\(786\) 0.417424 0.0148890
\(787\) −28.0942 −1.00145 −0.500725 0.865606i \(-0.666933\pi\)
−0.500725 + 0.865606i \(0.666933\pi\)
\(788\) 53.9796 1.92294
\(789\) −4.47315 −0.159248
\(790\) 0 0
\(791\) 8.22330 0.292387
\(792\) 0 0
\(793\) 30.7477 1.09188
\(794\) −9.47860 −0.336383
\(795\) 0 0
\(796\) 32.7913 1.16226
\(797\) −32.3303 −1.14520 −0.572599 0.819836i \(-0.694065\pi\)
−0.572599 + 0.819836i \(0.694065\pi\)
\(798\) −6.92820 −0.245256
\(799\) −21.7937 −0.771004
\(800\) 0 0
\(801\) −12.7477 −0.450419
\(802\) 48.8788 1.72597
\(803\) 0 0
\(804\) −37.9129 −1.33708
\(805\) 0 0
\(806\) 5.58258 0.196638
\(807\) 27.1652 0.956259
\(808\) −15.1652 −0.533508
\(809\) −1.82740 −0.0642480 −0.0321240 0.999484i \(-0.510227\pi\)
−0.0321240 + 0.999484i \(0.510227\pi\)
\(810\) 0 0
\(811\) −28.3405 −0.995168 −0.497584 0.867416i \(-0.665779\pi\)
−0.497584 + 0.867416i \(0.665779\pi\)
\(812\) −15.8258 −0.555375
\(813\) −9.76465 −0.342461
\(814\) 0 0
\(815\) 0 0
\(816\) 3.10260 0.108613
\(817\) 21.4955 0.752031
\(818\) −50.1216 −1.75246
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −50.5155 −1.76300 −0.881502 0.472180i \(-0.843467\pi\)
−0.881502 + 0.472180i \(0.843467\pi\)
\(822\) −37.9343 −1.32311
\(823\) 0.834849 0.0291010 0.0145505 0.999894i \(-0.495368\pi\)
0.0145505 + 0.999894i \(0.495368\pi\)
\(824\) −19.3386 −0.673692
\(825\) 0 0
\(826\) 8.83485 0.307404
\(827\) 47.2421 1.64277 0.821385 0.570374i \(-0.193202\pi\)
0.821385 + 0.570374i \(0.193202\pi\)
\(828\) −23.9564 −0.832544
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) −8.75560 −0.303729
\(832\) 55.0840 1.90970
\(833\) −10.6784 −0.369983
\(834\) −28.5390 −0.988225
\(835\) 0 0
\(836\) 0 0
\(837\) −0.582576 −0.0201368
\(838\) −70.7678 −2.44463
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 9.50455 0.327743
\(842\) 5.84370 0.201387
\(843\) 13.3241 0.458907
\(844\) 50.6308 1.74278
\(845\) 0 0
\(846\) 27.5420 0.946914
\(847\) 0 0
\(848\) −11.0436 −0.379237
\(849\) −31.5583 −1.08308
\(850\) 0 0
\(851\) −13.5826 −0.465605
\(852\) −22.3303 −0.765024
\(853\) 42.1413 1.44289 0.721446 0.692471i \(-0.243478\pi\)
0.721446 + 0.692471i \(0.243478\pi\)
\(854\) −14.0471 −0.480682
\(855\) 0 0
\(856\) 13.4174 0.458598
\(857\) −17.4159 −0.594914 −0.297457 0.954735i \(-0.596139\pi\)
−0.297457 + 0.954735i \(0.596139\pi\)
\(858\) 0 0
\(859\) −24.6606 −0.841409 −0.420705 0.907198i \(-0.638217\pi\)
−0.420705 + 0.907198i \(0.638217\pi\)
\(860\) 0 0
\(861\) −3.16515 −0.107868
\(862\) −43.9129 −1.49568
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) −7.38505 −0.251245
\(865\) 0 0
\(866\) −9.66930 −0.328576
\(867\) 14.0000 0.475465
\(868\) −1.48580 −0.0504314
\(869\) 0 0
\(870\) 0 0
\(871\) −59.4618 −2.01479
\(872\) −12.0000 −0.406371
\(873\) −13.5826 −0.459701
\(874\) 65.0780 2.20130
\(875\) 0 0
\(876\) −43.7780 −1.47912
\(877\) −41.7599 −1.41013 −0.705066 0.709142i \(-0.749083\pi\)
−0.705066 + 0.709142i \(0.749083\pi\)
\(878\) −20.9564 −0.707246
\(879\) −1.73205 −0.0584206
\(880\) 0 0
\(881\) 25.4955 0.858964 0.429482 0.903075i \(-0.358696\pi\)
0.429482 + 0.903075i \(0.358696\pi\)
\(882\) 13.4949 0.454397
\(883\) 48.2432 1.62351 0.811756 0.583997i \(-0.198512\pi\)
0.811756 + 0.583997i \(0.198512\pi\)
\(884\) −21.1652 −0.711861
\(885\) 0 0
\(886\) 15.6838 0.526908
\(887\) −2.01810 −0.0677612 −0.0338806 0.999426i \(-0.510787\pi\)
−0.0338806 + 0.999426i \(0.510787\pi\)
\(888\) 2.74110 0.0919853
\(889\) −2.33030 −0.0781558
\(890\) 0 0
\(891\) 0 0
\(892\) −56.7477 −1.90005
\(893\) −43.5873 −1.45859
\(894\) −28.7477 −0.961468
\(895\) 0 0
\(896\) −11.6697 −0.389857
\(897\) −37.5728 −1.25452
\(898\) −6.20520 −0.207070
\(899\) 3.61500 0.120567
\(900\) 0 0
\(901\) 10.6784 0.355748
\(902\) 0 0
\(903\) 5.66970 0.188676
\(904\) 15.5885 0.518464
\(905\) 0 0
\(906\) −28.5390 −0.948145
\(907\) −29.1652 −0.968413 −0.484206 0.874954i \(-0.660892\pi\)
−0.484206 + 0.874954i \(0.660892\pi\)
\(908\) 17.5867 0.583634
\(909\) 8.75560 0.290405
\(910\) 0 0
\(911\) −44.7477 −1.48256 −0.741279 0.671197i \(-0.765781\pi\)
−0.741279 + 0.671197i \(0.765781\pi\)
\(912\) 6.20520 0.205475
\(913\) 0 0
\(914\) −49.0780 −1.62336
\(915\) 0 0
\(916\) 47.4519 1.56785
\(917\) −0.174243 −0.00575401
\(918\) 3.79129 0.125131
\(919\) 27.9035 0.920452 0.460226 0.887802i \(-0.347768\pi\)
0.460226 + 0.887802i \(0.347768\pi\)
\(920\) 0 0
\(921\) 26.0761 0.859237
\(922\) −17.5826 −0.579051
\(923\) −35.0224 −1.15278
\(924\) 0 0
\(925\) 0 0
\(926\) −77.6960 −2.55325
\(927\) 11.1652 0.366712
\(928\) 45.8258 1.50430
\(929\) −16.3303 −0.535780 −0.267890 0.963450i \(-0.586326\pi\)
−0.267890 + 0.963450i \(0.586326\pi\)
\(930\) 0 0
\(931\) −21.3567 −0.699938
\(932\) 49.1450 1.60980
\(933\) 2.41742 0.0791429
\(934\) −7.48040 −0.244766
\(935\) 0 0
\(936\) 7.58258 0.247844
\(937\) −27.1805 −0.887949 −0.443974 0.896040i \(-0.646432\pi\)
−0.443974 + 0.896040i \(0.646432\pi\)
\(938\) 27.1652 0.886974
\(939\) −3.58258 −0.116913
\(940\) 0 0
\(941\) 3.99640 0.130279 0.0651395 0.997876i \(-0.479251\pi\)
0.0651395 + 0.997876i \(0.479251\pi\)
\(942\) 36.6591 1.19442
\(943\) 29.7309 0.968172
\(944\) −7.91288 −0.257542
\(945\) 0 0
\(946\) 0 0
\(947\) −20.9129 −0.679577 −0.339789 0.940502i \(-0.610356\pi\)
−0.339789 + 0.940502i \(0.610356\pi\)
\(948\) 17.0544 0.553900
\(949\) −68.6606 −2.22882
\(950\) 0 0
\(951\) 11.8348 0.383771
\(952\) 2.74110 0.0888396
\(953\) 20.7846 0.673280 0.336640 0.941634i \(-0.390710\pi\)
0.336640 + 0.941634i \(0.390710\pi\)
\(954\) −13.4949 −0.436914
\(955\) 0 0
\(956\) −75.3363 −2.43655
\(957\) 0 0
\(958\) 59.9129 1.93570
\(959\) 15.8347 0.511329
\(960\) 0 0
\(961\) −30.6606 −0.989052
\(962\) 15.1652 0.488944
\(963\) −7.74655 −0.249629
\(964\) 13.9717 0.449997
\(965\) 0 0
\(966\) 17.1652 0.552280
\(967\) −57.6344 −1.85340 −0.926699 0.375804i \(-0.877367\pi\)
−0.926699 + 0.375804i \(0.877367\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 16.4174 0.526860 0.263430 0.964678i \(-0.415146\pi\)
0.263430 + 0.964678i \(0.415146\pi\)
\(972\) −2.79129 −0.0895306
\(973\) 11.9129 0.381909
\(974\) 7.84190 0.251271
\(975\) 0 0
\(976\) 12.5812 0.402715
\(977\) 6.16515 0.197241 0.0986203 0.995125i \(-0.468557\pi\)
0.0986203 + 0.995125i \(0.468557\pi\)
\(978\) −14.7701 −0.472296
\(979\) 0 0
\(980\) 0 0
\(981\) 6.92820 0.221201
\(982\) −60.6606 −1.93576
\(983\) −30.5826 −0.975433 −0.487716 0.873002i \(-0.662170\pi\)
−0.487716 + 0.873002i \(0.662170\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −23.5257 −0.749211
\(987\) −11.4967 −0.365944
\(988\) −42.3303 −1.34671
\(989\) −53.2566 −1.69346
\(990\) 0 0
\(991\) 45.2432 1.43720 0.718599 0.695425i \(-0.244784\pi\)
0.718599 + 0.695425i \(0.244784\pi\)
\(992\) 4.30235 0.136600
\(993\) −0.252273 −0.00800564
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −44.3103 −1.40403
\(997\) 37.9542 1.20202 0.601011 0.799241i \(-0.294765\pi\)
0.601011 + 0.799241i \(0.294765\pi\)
\(998\) −57.6344 −1.82439
\(999\) −1.58258 −0.0500705
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 9075.2.a.cu.1.1 4
5.4 even 2 1815.2.a.t.1.4 yes 4
11.10 odd 2 inner 9075.2.a.cu.1.4 4
15.14 odd 2 5445.2.a.bl.1.1 4
55.54 odd 2 1815.2.a.t.1.1 4
165.164 even 2 5445.2.a.bl.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.t.1.1 4 55.54 odd 2
1815.2.a.t.1.4 yes 4 5.4 even 2
5445.2.a.bl.1.1 4 15.14 odd 2
5445.2.a.bl.1.4 4 165.164 even 2
9075.2.a.cu.1.1 4 1.1 even 1 trivial
9075.2.a.cu.1.4 4 11.10 odd 2 inner