Properties

Label 1520.2.a.r.1.2
Level $1520$
Weight $2$
Character 1520.1
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $3$
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] = [1520,2,Mod(1,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 1520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.363328 q^{3} -1.00000 q^{5} -1.14134 q^{7} -2.86799 q^{9} +O(q^{10})\) \(q-0.363328 q^{3} -1.00000 q^{5} -1.14134 q^{7} -2.86799 q^{9} +2.72666 q^{11} -4.64600 q^{13} +0.363328 q^{15} -0.858664 q^{17} +1.00000 q^{19} +0.414680 q^{21} +4.41468 q^{23} +1.00000 q^{25} +2.13201 q^{27} +9.42401 q^{29} +10.2827 q^{31} -0.990671 q^{33} +1.14134 q^{35} +6.77801 q^{37} +1.68802 q^{39} -7.55602 q^{41} -9.29200 q^{43} +2.86799 q^{45} +7.00933 q^{47} -5.69735 q^{49} +0.311977 q^{51} -8.64600 q^{53} -2.72666 q^{55} -0.363328 q^{57} +5.14134 q^{59} +9.45331 q^{61} +3.27334 q^{63} +4.64600 q^{65} +13.6553 q^{67} -1.60398 q^{69} +5.45331 q^{71} +6.87732 q^{73} -0.363328 q^{75} -3.11203 q^{77} -17.2920 q^{79} +7.82936 q^{81} +14.2827 q^{83} +0.858664 q^{85} -3.42401 q^{87} +13.0093 q^{89} +5.30265 q^{91} -3.73599 q^{93} -1.00000 q^{95} -9.68463 q^{97} -7.82003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} + 5 q^{7} + 4 q^{9} + 4 q^{11} + 5 q^{13} - q^{15} - 11 q^{17} + 3 q^{19} - 3 q^{21} + 9 q^{23} + 3 q^{25} + 19 q^{27} + 3 q^{29} + 14 q^{31} - 24 q^{33} - 5 q^{35} + 14 q^{37} + 5 q^{39} - 10 q^{41} + 10 q^{43} - 4 q^{45} + 4 q^{49} + q^{51} - 7 q^{53} - 4 q^{55} + q^{57} + 7 q^{59} + 20 q^{61} + 14 q^{63} - 5 q^{65} + q^{67} + 33 q^{69} + 8 q^{71} - 13 q^{73} + q^{75} + 16 q^{77} - 14 q^{79} + 15 q^{81} + 26 q^{83} + 11 q^{85} + 15 q^{87} + 18 q^{89} + 37 q^{91} + 14 q^{93} - 3 q^{95} - 6 q^{97} - 36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.363328 −0.209768 −0.104884 0.994484i \(-0.533447\pi\)
−0.104884 + 0.994484i \(0.533447\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.14134 −0.431385 −0.215692 0.976461i \(-0.569201\pi\)
−0.215692 + 0.976461i \(0.569201\pi\)
\(8\) 0 0
\(9\) −2.86799 −0.955998
\(10\) 0 0
\(11\) 2.72666 0.822118 0.411059 0.911609i \(-0.365159\pi\)
0.411059 + 0.911609i \(0.365159\pi\)
\(12\) 0 0
\(13\) −4.64600 −1.28857 −0.644284 0.764786i \(-0.722845\pi\)
−0.644284 + 0.764786i \(0.722845\pi\)
\(14\) 0 0
\(15\) 0.363328 0.0938109
\(16\) 0 0
\(17\) −0.858664 −0.208257 −0.104128 0.994564i \(-0.533205\pi\)
−0.104128 + 0.994564i \(0.533205\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.414680 0.0904905
\(22\) 0 0
\(23\) 4.41468 0.920524 0.460262 0.887783i \(-0.347755\pi\)
0.460262 + 0.887783i \(0.347755\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.13201 0.410305
\(28\) 0 0
\(29\) 9.42401 1.74999 0.874997 0.484128i \(-0.160863\pi\)
0.874997 + 0.484128i \(0.160863\pi\)
\(30\) 0 0
\(31\) 10.2827 1.84682 0.923411 0.383812i \(-0.125389\pi\)
0.923411 + 0.383812i \(0.125389\pi\)
\(32\) 0 0
\(33\) −0.990671 −0.172454
\(34\) 0 0
\(35\) 1.14134 0.192921
\(36\) 0 0
\(37\) 6.77801 1.11430 0.557149 0.830413i \(-0.311895\pi\)
0.557149 + 0.830413i \(0.311895\pi\)
\(38\) 0 0
\(39\) 1.68802 0.270300
\(40\) 0 0
\(41\) −7.55602 −1.18005 −0.590026 0.807384i \(-0.700882\pi\)
−0.590026 + 0.807384i \(0.700882\pi\)
\(42\) 0 0
\(43\) −9.29200 −1.41702 −0.708508 0.705702i \(-0.750632\pi\)
−0.708508 + 0.705702i \(0.750632\pi\)
\(44\) 0 0
\(45\) 2.86799 0.427535
\(46\) 0 0
\(47\) 7.00933 1.02242 0.511208 0.859457i \(-0.329198\pi\)
0.511208 + 0.859457i \(0.329198\pi\)
\(48\) 0 0
\(49\) −5.69735 −0.813907
\(50\) 0 0
\(51\) 0.311977 0.0436855
\(52\) 0 0
\(53\) −8.64600 −1.18762 −0.593810 0.804605i \(-0.702377\pi\)
−0.593810 + 0.804605i \(0.702377\pi\)
\(54\) 0 0
\(55\) −2.72666 −0.367662
\(56\) 0 0
\(57\) −0.363328 −0.0481240
\(58\) 0 0
\(59\) 5.14134 0.669345 0.334672 0.942335i \(-0.391374\pi\)
0.334672 + 0.942335i \(0.391374\pi\)
\(60\) 0 0
\(61\) 9.45331 1.21037 0.605186 0.796084i \(-0.293099\pi\)
0.605186 + 0.796084i \(0.293099\pi\)
\(62\) 0 0
\(63\) 3.27334 0.412403
\(64\) 0 0
\(65\) 4.64600 0.576265
\(66\) 0 0
\(67\) 13.6553 1.66826 0.834132 0.551565i \(-0.185969\pi\)
0.834132 + 0.551565i \(0.185969\pi\)
\(68\) 0 0
\(69\) −1.60398 −0.193096
\(70\) 0 0
\(71\) 5.45331 0.647189 0.323595 0.946196i \(-0.395109\pi\)
0.323595 + 0.946196i \(0.395109\pi\)
\(72\) 0 0
\(73\) 6.87732 0.804930 0.402465 0.915435i \(-0.368154\pi\)
0.402465 + 0.915435i \(0.368154\pi\)
\(74\) 0 0
\(75\) −0.363328 −0.0419535
\(76\) 0 0
\(77\) −3.11203 −0.354649
\(78\) 0 0
\(79\) −17.2920 −1.94550 −0.972751 0.231852i \(-0.925521\pi\)
−0.972751 + 0.231852i \(0.925521\pi\)
\(80\) 0 0
\(81\) 7.82936 0.869929
\(82\) 0 0
\(83\) 14.2827 1.56773 0.783863 0.620933i \(-0.213246\pi\)
0.783863 + 0.620933i \(0.213246\pi\)
\(84\) 0 0
\(85\) 0.858664 0.0931352
\(86\) 0 0
\(87\) −3.42401 −0.367092
\(88\) 0 0
\(89\) 13.0093 1.37899 0.689493 0.724292i \(-0.257833\pi\)
0.689493 + 0.724292i \(0.257833\pi\)
\(90\) 0 0
\(91\) 5.30265 0.555869
\(92\) 0 0
\(93\) −3.73599 −0.387404
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −9.68463 −0.983326 −0.491663 0.870786i \(-0.663611\pi\)
−0.491663 + 0.870786i \(0.663611\pi\)
\(98\) 0 0
\(99\) −7.82003 −0.785943
\(100\) 0 0
\(101\) 1.45331 0.144610 0.0723050 0.997383i \(-0.476964\pi\)
0.0723050 + 0.997383i \(0.476964\pi\)
\(102\) 0 0
\(103\) −2.23132 −0.219859 −0.109929 0.993939i \(-0.535062\pi\)
−0.109929 + 0.993939i \(0.535062\pi\)
\(104\) 0 0
\(105\) −0.414680 −0.0404686
\(106\) 0 0
\(107\) −13.6553 −1.32011 −0.660055 0.751217i \(-0.729467\pi\)
−0.660055 + 0.751217i \(0.729467\pi\)
\(108\) 0 0
\(109\) 20.1693 1.93187 0.965935 0.258784i \(-0.0833216\pi\)
0.965935 + 0.258784i \(0.0833216\pi\)
\(110\) 0 0
\(111\) −2.46264 −0.233744
\(112\) 0 0
\(113\) −10.0514 −0.945552 −0.472776 0.881183i \(-0.656748\pi\)
−0.472776 + 0.881183i \(0.656748\pi\)
\(114\) 0 0
\(115\) −4.41468 −0.411671
\(116\) 0 0
\(117\) 13.3247 1.23187
\(118\) 0 0
\(119\) 0.980024 0.0898387
\(120\) 0 0
\(121\) −3.56534 −0.324122
\(122\) 0 0
\(123\) 2.74531 0.247537
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.0700 1.24851 0.624256 0.781220i \(-0.285402\pi\)
0.624256 + 0.781220i \(0.285402\pi\)
\(128\) 0 0
\(129\) 3.37605 0.297244
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −1.14134 −0.0989664
\(134\) 0 0
\(135\) −2.13201 −0.183494
\(136\) 0 0
\(137\) −4.85866 −0.415104 −0.207552 0.978224i \(-0.566550\pi\)
−0.207552 + 0.978224i \(0.566550\pi\)
\(138\) 0 0
\(139\) 4.17997 0.354540 0.177270 0.984162i \(-0.443273\pi\)
0.177270 + 0.984162i \(0.443273\pi\)
\(140\) 0 0
\(141\) −2.54669 −0.214470
\(142\) 0 0
\(143\) −12.6680 −1.05936
\(144\) 0 0
\(145\) −9.42401 −0.782621
\(146\) 0 0
\(147\) 2.07001 0.170731
\(148\) 0 0
\(149\) 1.45331 0.119060 0.0595300 0.998227i \(-0.481040\pi\)
0.0595300 + 0.998227i \(0.481040\pi\)
\(150\) 0 0
\(151\) −7.27334 −0.591896 −0.295948 0.955204i \(-0.595636\pi\)
−0.295948 + 0.955204i \(0.595636\pi\)
\(152\) 0 0
\(153\) 2.46264 0.199093
\(154\) 0 0
\(155\) −10.2827 −0.825924
\(156\) 0 0
\(157\) 17.0093 1.35749 0.678746 0.734373i \(-0.262524\pi\)
0.678746 + 0.734373i \(0.262524\pi\)
\(158\) 0 0
\(159\) 3.14134 0.249124
\(160\) 0 0
\(161\) −5.03863 −0.397100
\(162\) 0 0
\(163\) 10.1800 0.797357 0.398678 0.917091i \(-0.369469\pi\)
0.398678 + 0.917091i \(0.369469\pi\)
\(164\) 0 0
\(165\) 0.990671 0.0771237
\(166\) 0 0
\(167\) −6.07001 −0.469711 −0.234856 0.972030i \(-0.575462\pi\)
−0.234856 + 0.972030i \(0.575462\pi\)
\(168\) 0 0
\(169\) 8.58532 0.660409
\(170\) 0 0
\(171\) −2.86799 −0.219321
\(172\) 0 0
\(173\) −5.22199 −0.397021 −0.198510 0.980099i \(-0.563610\pi\)
−0.198510 + 0.980099i \(0.563610\pi\)
\(174\) 0 0
\(175\) −1.14134 −0.0862769
\(176\) 0 0
\(177\) −1.86799 −0.140407
\(178\) 0 0
\(179\) 1.55602 0.116302 0.0581510 0.998308i \(-0.481480\pi\)
0.0581510 + 0.998308i \(0.481480\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −3.43466 −0.253897
\(184\) 0 0
\(185\) −6.77801 −0.498329
\(186\) 0 0
\(187\) −2.34128 −0.171211
\(188\) 0 0
\(189\) −2.43334 −0.176999
\(190\) 0 0
\(191\) 5.03863 0.364583 0.182291 0.983245i \(-0.441649\pi\)
0.182291 + 0.983245i \(0.441649\pi\)
\(192\) 0 0
\(193\) −9.68463 −0.697115 −0.348558 0.937287i \(-0.613328\pi\)
−0.348558 + 0.937287i \(0.613328\pi\)
\(194\) 0 0
\(195\) −1.68802 −0.120882
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −19.8867 −1.40973 −0.704864 0.709343i \(-0.748992\pi\)
−0.704864 + 0.709343i \(0.748992\pi\)
\(200\) 0 0
\(201\) −4.96137 −0.349948
\(202\) 0 0
\(203\) −10.7560 −0.754920
\(204\) 0 0
\(205\) 7.55602 0.527735
\(206\) 0 0
\(207\) −12.6613 −0.880019
\(208\) 0 0
\(209\) 2.72666 0.188607
\(210\) 0 0
\(211\) −4.31198 −0.296849 −0.148424 0.988924i \(-0.547420\pi\)
−0.148424 + 0.988924i \(0.547420\pi\)
\(212\) 0 0
\(213\) −1.98134 −0.135759
\(214\) 0 0
\(215\) 9.29200 0.633709
\(216\) 0 0
\(217\) −11.7360 −0.796691
\(218\) 0 0
\(219\) −2.49873 −0.168848
\(220\) 0 0
\(221\) 3.98935 0.268353
\(222\) 0 0
\(223\) 18.5327 1.24104 0.620519 0.784191i \(-0.286922\pi\)
0.620519 + 0.784191i \(0.286922\pi\)
\(224\) 0 0
\(225\) −2.86799 −0.191200
\(226\) 0 0
\(227\) −4.82597 −0.320311 −0.160155 0.987092i \(-0.551200\pi\)
−0.160155 + 0.987092i \(0.551200\pi\)
\(228\) 0 0
\(229\) 23.7360 1.56852 0.784259 0.620434i \(-0.213043\pi\)
0.784259 + 0.620434i \(0.213043\pi\)
\(230\) 0 0
\(231\) 1.13069 0.0743939
\(232\) 0 0
\(233\) −14.5653 −0.954207 −0.477104 0.878847i \(-0.658313\pi\)
−0.477104 + 0.878847i \(0.658313\pi\)
\(234\) 0 0
\(235\) −7.00933 −0.457238
\(236\) 0 0
\(237\) 6.28267 0.408103
\(238\) 0 0
\(239\) −2.43334 −0.157399 −0.0786997 0.996898i \(-0.525077\pi\)
−0.0786997 + 0.996898i \(0.525077\pi\)
\(240\) 0 0
\(241\) 18.1986 1.17228 0.586138 0.810211i \(-0.300648\pi\)
0.586138 + 0.810211i \(0.300648\pi\)
\(242\) 0 0
\(243\) −9.24065 −0.592788
\(244\) 0 0
\(245\) 5.69735 0.363990
\(246\) 0 0
\(247\) −4.64600 −0.295618
\(248\) 0 0
\(249\) −5.18930 −0.328858
\(250\) 0 0
\(251\) 15.1120 0.953863 0.476931 0.878940i \(-0.341749\pi\)
0.476931 + 0.878940i \(0.341749\pi\)
\(252\) 0 0
\(253\) 12.0373 0.756779
\(254\) 0 0
\(255\) −0.311977 −0.0195367
\(256\) 0 0
\(257\) 12.2313 0.762969 0.381484 0.924375i \(-0.375413\pi\)
0.381484 + 0.924375i \(0.375413\pi\)
\(258\) 0 0
\(259\) −7.73599 −0.480691
\(260\) 0 0
\(261\) −27.0280 −1.67299
\(262\) 0 0
\(263\) −22.5840 −1.39259 −0.696295 0.717756i \(-0.745169\pi\)
−0.696295 + 0.717756i \(0.745169\pi\)
\(264\) 0 0
\(265\) 8.64600 0.531120
\(266\) 0 0
\(267\) −4.72666 −0.289267
\(268\) 0 0
\(269\) −13.1120 −0.799455 −0.399727 0.916634i \(-0.630895\pi\)
−0.399727 + 0.916634i \(0.630895\pi\)
\(270\) 0 0
\(271\) 20.8960 1.26934 0.634670 0.772783i \(-0.281136\pi\)
0.634670 + 0.772783i \(0.281136\pi\)
\(272\) 0 0
\(273\) −1.92660 −0.116603
\(274\) 0 0
\(275\) 2.72666 0.164424
\(276\) 0 0
\(277\) −20.1800 −1.21250 −0.606248 0.795275i \(-0.707326\pi\)
−0.606248 + 0.795275i \(0.707326\pi\)
\(278\) 0 0
\(279\) −29.4906 −1.76556
\(280\) 0 0
\(281\) −22.4626 −1.34001 −0.670004 0.742357i \(-0.733708\pi\)
−0.670004 + 0.742357i \(0.733708\pi\)
\(282\) 0 0
\(283\) 0.829359 0.0493003 0.0246501 0.999696i \(-0.492153\pi\)
0.0246501 + 0.999696i \(0.492153\pi\)
\(284\) 0 0
\(285\) 0.363328 0.0215217
\(286\) 0 0
\(287\) 8.62395 0.509056
\(288\) 0 0
\(289\) −16.2627 −0.956629
\(290\) 0 0
\(291\) 3.51870 0.206270
\(292\) 0 0
\(293\) −31.2886 −1.82790 −0.913950 0.405827i \(-0.866984\pi\)
−0.913950 + 0.405827i \(0.866984\pi\)
\(294\) 0 0
\(295\) −5.14134 −0.299340
\(296\) 0 0
\(297\) 5.81325 0.337319
\(298\) 0 0
\(299\) −20.5106 −1.18616
\(300\) 0 0
\(301\) 10.6053 0.611279
\(302\) 0 0
\(303\) −0.528030 −0.0303345
\(304\) 0 0
\(305\) −9.45331 −0.541295
\(306\) 0 0
\(307\) 3.11929 0.178027 0.0890136 0.996030i \(-0.471629\pi\)
0.0890136 + 0.996030i \(0.471629\pi\)
\(308\) 0 0
\(309\) 0.810702 0.0461192
\(310\) 0 0
\(311\) 21.4240 1.21484 0.607422 0.794379i \(-0.292204\pi\)
0.607422 + 0.794379i \(0.292204\pi\)
\(312\) 0 0
\(313\) −12.1320 −0.685742 −0.342871 0.939383i \(-0.611399\pi\)
−0.342871 + 0.939383i \(0.611399\pi\)
\(314\) 0 0
\(315\) −3.27334 −0.184432
\(316\) 0 0
\(317\) 16.7487 0.940701 0.470350 0.882480i \(-0.344128\pi\)
0.470350 + 0.882480i \(0.344128\pi\)
\(318\) 0 0
\(319\) 25.6960 1.43870
\(320\) 0 0
\(321\) 4.96137 0.276916
\(322\) 0 0
\(323\) −0.858664 −0.0477773
\(324\) 0 0
\(325\) −4.64600 −0.257714
\(326\) 0 0
\(327\) −7.32808 −0.405244
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 5.97070 0.328179 0.164090 0.986445i \(-0.447531\pi\)
0.164090 + 0.986445i \(0.447531\pi\)
\(332\) 0 0
\(333\) −19.4393 −1.06527
\(334\) 0 0
\(335\) −13.6553 −0.746070
\(336\) 0 0
\(337\) −20.3340 −1.10766 −0.553832 0.832628i \(-0.686835\pi\)
−0.553832 + 0.832628i \(0.686835\pi\)
\(338\) 0 0
\(339\) 3.65194 0.198346
\(340\) 0 0
\(341\) 28.0373 1.51831
\(342\) 0 0
\(343\) 14.4919 0.782492
\(344\) 0 0
\(345\) 1.60398 0.0863553
\(346\) 0 0
\(347\) −20.8294 −1.11818 −0.559089 0.829107i \(-0.688849\pi\)
−0.559089 + 0.829107i \(0.688849\pi\)
\(348\) 0 0
\(349\) −20.0187 −1.07157 −0.535787 0.844353i \(-0.679985\pi\)
−0.535787 + 0.844353i \(0.679985\pi\)
\(350\) 0 0
\(351\) −9.90531 −0.528706
\(352\) 0 0
\(353\) −27.1413 −1.44459 −0.722294 0.691586i \(-0.756912\pi\)
−0.722294 + 0.691586i \(0.756912\pi\)
\(354\) 0 0
\(355\) −5.45331 −0.289432
\(356\) 0 0
\(357\) −0.356070 −0.0188452
\(358\) 0 0
\(359\) 4.59465 0.242496 0.121248 0.992622i \(-0.461310\pi\)
0.121248 + 0.992622i \(0.461310\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.29539 0.0679904
\(364\) 0 0
\(365\) −6.87732 −0.359975
\(366\) 0 0
\(367\) 34.1214 1.78112 0.890560 0.454865i \(-0.150313\pi\)
0.890560 + 0.454865i \(0.150313\pi\)
\(368\) 0 0
\(369\) 21.6706 1.12813
\(370\) 0 0
\(371\) 9.86799 0.512321
\(372\) 0 0
\(373\) −7.81664 −0.404730 −0.202365 0.979310i \(-0.564863\pi\)
−0.202365 + 0.979310i \(0.564863\pi\)
\(374\) 0 0
\(375\) 0.363328 0.0187622
\(376\) 0 0
\(377\) −43.7839 −2.25499
\(378\) 0 0
\(379\) 23.0573 1.18437 0.592187 0.805801i \(-0.298265\pi\)
0.592187 + 0.805801i \(0.298265\pi\)
\(380\) 0 0
\(381\) −5.11203 −0.261897
\(382\) 0 0
\(383\) −16.8807 −0.862564 −0.431282 0.902217i \(-0.641939\pi\)
−0.431282 + 0.902217i \(0.641939\pi\)
\(384\) 0 0
\(385\) 3.11203 0.158604
\(386\) 0 0
\(387\) 26.6494 1.35466
\(388\) 0 0
\(389\) −18.3013 −0.927914 −0.463957 0.885858i \(-0.653571\pi\)
−0.463957 + 0.885858i \(0.653571\pi\)
\(390\) 0 0
\(391\) −3.79073 −0.191705
\(392\) 0 0
\(393\) −1.45331 −0.0733099
\(394\) 0 0
\(395\) 17.2920 0.870055
\(396\) 0 0
\(397\) 12.1800 0.611295 0.305648 0.952145i \(-0.401127\pi\)
0.305648 + 0.952145i \(0.401127\pi\)
\(398\) 0 0
\(399\) 0.414680 0.0207599
\(400\) 0 0
\(401\) 16.5840 0.828166 0.414083 0.910239i \(-0.364102\pi\)
0.414083 + 0.910239i \(0.364102\pi\)
\(402\) 0 0
\(403\) −47.7733 −2.37976
\(404\) 0 0
\(405\) −7.82936 −0.389044
\(406\) 0 0
\(407\) 18.4813 0.916084
\(408\) 0 0
\(409\) 18.5653 0.917997 0.458999 0.888437i \(-0.348208\pi\)
0.458999 + 0.888437i \(0.348208\pi\)
\(410\) 0 0
\(411\) 1.76529 0.0870753
\(412\) 0 0
\(413\) −5.86799 −0.288745
\(414\) 0 0
\(415\) −14.2827 −0.701109
\(416\) 0 0
\(417\) −1.51870 −0.0743711
\(418\) 0 0
\(419\) −2.84802 −0.139135 −0.0695674 0.997577i \(-0.522162\pi\)
−0.0695674 + 0.997577i \(0.522162\pi\)
\(420\) 0 0
\(421\) 24.5360 1.19581 0.597907 0.801566i \(-0.295999\pi\)
0.597907 + 0.801566i \(0.295999\pi\)
\(422\) 0 0
\(423\) −20.1027 −0.977427
\(424\) 0 0
\(425\) −0.858664 −0.0416513
\(426\) 0 0
\(427\) −10.7894 −0.522136
\(428\) 0 0
\(429\) 4.60266 0.222218
\(430\) 0 0
\(431\) 7.53736 0.363062 0.181531 0.983385i \(-0.441895\pi\)
0.181531 + 0.983385i \(0.441895\pi\)
\(432\) 0 0
\(433\) −0.392633 −0.0188687 −0.00943437 0.999955i \(-0.503003\pi\)
−0.00943437 + 0.999955i \(0.503003\pi\)
\(434\) 0 0
\(435\) 3.42401 0.164169
\(436\) 0 0
\(437\) 4.41468 0.211183
\(438\) 0 0
\(439\) −5.29200 −0.252573 −0.126287 0.991994i \(-0.540306\pi\)
−0.126287 + 0.991994i \(0.540306\pi\)
\(440\) 0 0
\(441\) 16.3400 0.778093
\(442\) 0 0
\(443\) 22.0187 1.04614 0.523069 0.852290i \(-0.324787\pi\)
0.523069 + 0.852290i \(0.324787\pi\)
\(444\) 0 0
\(445\) −13.0093 −0.616701
\(446\) 0 0
\(447\) −0.528030 −0.0249749
\(448\) 0 0
\(449\) −4.90663 −0.231558 −0.115779 0.993275i \(-0.536936\pi\)
−0.115779 + 0.993275i \(0.536936\pi\)
\(450\) 0 0
\(451\) −20.6027 −0.970141
\(452\) 0 0
\(453\) 2.64261 0.124161
\(454\) 0 0
\(455\) −5.30265 −0.248592
\(456\) 0 0
\(457\) 26.9800 1.26207 0.631036 0.775753i \(-0.282630\pi\)
0.631036 + 0.775753i \(0.282630\pi\)
\(458\) 0 0
\(459\) −1.83068 −0.0854487
\(460\) 0 0
\(461\) −27.1307 −1.26360 −0.631801 0.775131i \(-0.717684\pi\)
−0.631801 + 0.775131i \(0.717684\pi\)
\(462\) 0 0
\(463\) 16.9253 0.786585 0.393292 0.919413i \(-0.371336\pi\)
0.393292 + 0.919413i \(0.371336\pi\)
\(464\) 0 0
\(465\) 3.73599 0.173252
\(466\) 0 0
\(467\) 42.6213 1.97228 0.986140 0.165917i \(-0.0530585\pi\)
0.986140 + 0.165917i \(0.0530585\pi\)
\(468\) 0 0
\(469\) −15.5853 −0.719663
\(470\) 0 0
\(471\) −6.17997 −0.284758
\(472\) 0 0
\(473\) −25.3361 −1.16495
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 24.7967 1.13536
\(478\) 0 0
\(479\) 25.3107 1.15647 0.578237 0.815869i \(-0.303741\pi\)
0.578237 + 0.815869i \(0.303741\pi\)
\(480\) 0 0
\(481\) −31.4906 −1.43585
\(482\) 0 0
\(483\) 1.83068 0.0832987
\(484\) 0 0
\(485\) 9.68463 0.439757
\(486\) 0 0
\(487\) −25.4020 −1.15107 −0.575536 0.817776i \(-0.695207\pi\)
−0.575536 + 0.817776i \(0.695207\pi\)
\(488\) 0 0
\(489\) −3.69867 −0.167260
\(490\) 0 0
\(491\) −20.6240 −0.930746 −0.465373 0.885115i \(-0.654080\pi\)
−0.465373 + 0.885115i \(0.654080\pi\)
\(492\) 0 0
\(493\) −8.09206 −0.364448
\(494\) 0 0
\(495\) 7.82003 0.351484
\(496\) 0 0
\(497\) −6.22406 −0.279187
\(498\) 0 0
\(499\) −17.3693 −0.777555 −0.388778 0.921332i \(-0.627103\pi\)
−0.388778 + 0.921332i \(0.627103\pi\)
\(500\) 0 0
\(501\) 2.20541 0.0985303
\(502\) 0 0
\(503\) 14.9987 0.668758 0.334379 0.942439i \(-0.391473\pi\)
0.334379 + 0.942439i \(0.391473\pi\)
\(504\) 0 0
\(505\) −1.45331 −0.0646716
\(506\) 0 0
\(507\) −3.11929 −0.138533
\(508\) 0 0
\(509\) −19.9160 −0.882759 −0.441380 0.897320i \(-0.645511\pi\)
−0.441380 + 0.897320i \(0.645511\pi\)
\(510\) 0 0
\(511\) −7.84934 −0.347234
\(512\) 0 0
\(513\) 2.13201 0.0941304
\(514\) 0 0
\(515\) 2.23132 0.0983237
\(516\) 0 0
\(517\) 19.1120 0.840546
\(518\) 0 0
\(519\) 1.89730 0.0832821
\(520\) 0 0
\(521\) −14.8294 −0.649686 −0.324843 0.945768i \(-0.605311\pi\)
−0.324843 + 0.945768i \(0.605311\pi\)
\(522\) 0 0
\(523\) −11.9966 −0.524575 −0.262288 0.964990i \(-0.584477\pi\)
−0.262288 + 0.964990i \(0.584477\pi\)
\(524\) 0 0
\(525\) 0.414680 0.0180981
\(526\) 0 0
\(527\) −8.82936 −0.384613
\(528\) 0 0
\(529\) −3.51060 −0.152635
\(530\) 0 0
\(531\) −14.7453 −0.639892
\(532\) 0 0
\(533\) 35.1053 1.52058
\(534\) 0 0
\(535\) 13.6553 0.590371
\(536\) 0 0
\(537\) −0.565344 −0.0243964
\(538\) 0 0
\(539\) −15.5347 −0.669128
\(540\) 0 0
\(541\) −2.01866 −0.0867889 −0.0433944 0.999058i \(-0.513817\pi\)
−0.0433944 + 0.999058i \(0.513817\pi\)
\(542\) 0 0
\(543\) −7.99322 −0.343022
\(544\) 0 0
\(545\) −20.1693 −0.863959
\(546\) 0 0
\(547\) 18.6940 0.799296 0.399648 0.916669i \(-0.369132\pi\)
0.399648 + 0.916669i \(0.369132\pi\)
\(548\) 0 0
\(549\) −27.1120 −1.15711
\(550\) 0 0
\(551\) 9.42401 0.401476
\(552\) 0 0
\(553\) 19.7360 0.839259
\(554\) 0 0
\(555\) 2.46264 0.104533
\(556\) 0 0
\(557\) 11.0866 0.469754 0.234877 0.972025i \(-0.424531\pi\)
0.234877 + 0.972025i \(0.424531\pi\)
\(558\) 0 0
\(559\) 43.1706 1.82592
\(560\) 0 0
\(561\) 0.850654 0.0359146
\(562\) 0 0
\(563\) −2.69396 −0.113537 −0.0567685 0.998387i \(-0.518080\pi\)
−0.0567685 + 0.998387i \(0.518080\pi\)
\(564\) 0 0
\(565\) 10.0514 0.422864
\(566\) 0 0
\(567\) −8.93593 −0.375274
\(568\) 0 0
\(569\) 29.4134 1.23307 0.616536 0.787327i \(-0.288535\pi\)
0.616536 + 0.787327i \(0.288535\pi\)
\(570\) 0 0
\(571\) 33.0466 1.38296 0.691479 0.722396i \(-0.256959\pi\)
0.691479 + 0.722396i \(0.256959\pi\)
\(572\) 0 0
\(573\) −1.83068 −0.0764777
\(574\) 0 0
\(575\) 4.41468 0.184105
\(576\) 0 0
\(577\) −0.396022 −0.0164866 −0.00824331 0.999966i \(-0.502624\pi\)
−0.00824331 + 0.999966i \(0.502624\pi\)
\(578\) 0 0
\(579\) 3.51870 0.146232
\(580\) 0 0
\(581\) −16.3013 −0.676293
\(582\) 0 0
\(583\) −23.5747 −0.976363
\(584\) 0 0
\(585\) −13.3247 −0.550908
\(586\) 0 0
\(587\) 14.0773 0.581031 0.290515 0.956870i \(-0.406173\pi\)
0.290515 + 0.956870i \(0.406173\pi\)
\(588\) 0 0
\(589\) 10.2827 0.423690
\(590\) 0 0
\(591\) 0.726656 0.0298907
\(592\) 0 0
\(593\) 5.43466 0.223175 0.111587 0.993755i \(-0.464407\pi\)
0.111587 + 0.993755i \(0.464407\pi\)
\(594\) 0 0
\(595\) −0.980024 −0.0401771
\(596\) 0 0
\(597\) 7.22538 0.295715
\(598\) 0 0
\(599\) −11.6333 −0.475323 −0.237662 0.971348i \(-0.576381\pi\)
−0.237662 + 0.971348i \(0.576381\pi\)
\(600\) 0 0
\(601\) 8.12136 0.331277 0.165639 0.986187i \(-0.447031\pi\)
0.165639 + 0.986187i \(0.447031\pi\)
\(602\) 0 0
\(603\) −39.1634 −1.59486
\(604\) 0 0
\(605\) 3.56534 0.144952
\(606\) 0 0
\(607\) −10.5327 −0.427507 −0.213754 0.976888i \(-0.568569\pi\)
−0.213754 + 0.976888i \(0.568569\pi\)
\(608\) 0 0
\(609\) 3.90794 0.158358
\(610\) 0 0
\(611\) −32.5653 −1.31745
\(612\) 0 0
\(613\) 47.3293 1.91161 0.955807 0.293996i \(-0.0949854\pi\)
0.955807 + 0.293996i \(0.0949854\pi\)
\(614\) 0 0
\(615\) −2.74531 −0.110702
\(616\) 0 0
\(617\) −33.3693 −1.34340 −0.671698 0.740825i \(-0.734435\pi\)
−0.671698 + 0.740825i \(0.734435\pi\)
\(618\) 0 0
\(619\) 33.8760 1.36159 0.680796 0.732473i \(-0.261634\pi\)
0.680796 + 0.732473i \(0.261634\pi\)
\(620\) 0 0
\(621\) 9.41213 0.377696
\(622\) 0 0
\(623\) −14.8480 −0.594873
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.990671 −0.0395636
\(628\) 0 0
\(629\) −5.82003 −0.232060
\(630\) 0 0
\(631\) −27.8573 −1.10898 −0.554492 0.832189i \(-0.687087\pi\)
−0.554492 + 0.832189i \(0.687087\pi\)
\(632\) 0 0
\(633\) 1.56666 0.0622693
\(634\) 0 0
\(635\) −14.0700 −0.558351
\(636\) 0 0
\(637\) 26.4699 1.04878
\(638\) 0 0
\(639\) −15.6401 −0.618711
\(640\) 0 0
\(641\) −4.74531 −0.187429 −0.0937143 0.995599i \(-0.529874\pi\)
−0.0937143 + 0.995599i \(0.529874\pi\)
\(642\) 0 0
\(643\) 6.28267 0.247764 0.123882 0.992297i \(-0.460466\pi\)
0.123882 + 0.992297i \(0.460466\pi\)
\(644\) 0 0
\(645\) −3.37605 −0.132932
\(646\) 0 0
\(647\) −16.1507 −0.634948 −0.317474 0.948267i \(-0.602835\pi\)
−0.317474 + 0.948267i \(0.602835\pi\)
\(648\) 0 0
\(649\) 14.0187 0.550280
\(650\) 0 0
\(651\) 4.26401 0.167120
\(652\) 0 0
\(653\) 29.8760 1.16914 0.584569 0.811344i \(-0.301264\pi\)
0.584569 + 0.811344i \(0.301264\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) −19.7241 −0.769511
\(658\) 0 0
\(659\) 17.4427 0.679470 0.339735 0.940521i \(-0.389663\pi\)
0.339735 + 0.940521i \(0.389663\pi\)
\(660\) 0 0
\(661\) −2.87732 −0.111915 −0.0559574 0.998433i \(-0.517821\pi\)
−0.0559574 + 0.998433i \(0.517821\pi\)
\(662\) 0 0
\(663\) −1.44944 −0.0562918
\(664\) 0 0
\(665\) 1.14134 0.0442591
\(666\) 0 0
\(667\) 41.6040 1.61091
\(668\) 0 0
\(669\) −6.73344 −0.260330
\(670\) 0 0
\(671\) 25.7759 0.995069
\(672\) 0 0
\(673\) 10.9393 0.421680 0.210840 0.977521i \(-0.432380\pi\)
0.210840 + 0.977521i \(0.432380\pi\)
\(674\) 0 0
\(675\) 2.13201 0.0820610
\(676\) 0 0
\(677\) 19.0900 0.733688 0.366844 0.930283i \(-0.380438\pi\)
0.366844 + 0.930283i \(0.380438\pi\)
\(678\) 0 0
\(679\) 11.0534 0.424191
\(680\) 0 0
\(681\) 1.75341 0.0671909
\(682\) 0 0
\(683\) −25.7687 −0.986011 −0.493006 0.870026i \(-0.664102\pi\)
−0.493006 + 0.870026i \(0.664102\pi\)
\(684\) 0 0
\(685\) 4.85866 0.185640
\(686\) 0 0
\(687\) −8.62395 −0.329024
\(688\) 0 0
\(689\) 40.1693 1.53033
\(690\) 0 0
\(691\) 27.2334 1.03601 0.518004 0.855378i \(-0.326675\pi\)
0.518004 + 0.855378i \(0.326675\pi\)
\(692\) 0 0
\(693\) 8.92528 0.339043
\(694\) 0 0
\(695\) −4.17997 −0.158555
\(696\) 0 0
\(697\) 6.48808 0.245753
\(698\) 0 0
\(699\) 5.29200 0.200162
\(700\) 0 0
\(701\) 49.6960 1.87699 0.938497 0.345288i \(-0.112219\pi\)
0.938497 + 0.345288i \(0.112219\pi\)
\(702\) 0 0
\(703\) 6.77801 0.255637
\(704\) 0 0
\(705\) 2.54669 0.0959138
\(706\) 0 0
\(707\) −1.65872 −0.0623825
\(708\) 0 0
\(709\) 5.64006 0.211817 0.105908 0.994376i \(-0.466225\pi\)
0.105908 + 0.994376i \(0.466225\pi\)
\(710\) 0 0
\(711\) 49.5933 1.85990
\(712\) 0 0
\(713\) 45.3947 1.70005
\(714\) 0 0
\(715\) 12.6680 0.473758
\(716\) 0 0
\(717\) 0.884100 0.0330173
\(718\) 0 0
\(719\) 33.4240 1.24651 0.623253 0.782021i \(-0.285811\pi\)
0.623253 + 0.782021i \(0.285811\pi\)
\(720\) 0 0
\(721\) 2.54669 0.0948436
\(722\) 0 0
\(723\) −6.61208 −0.245906
\(724\) 0 0
\(725\) 9.42401 0.349999
\(726\) 0 0
\(727\) 14.3306 0.531494 0.265747 0.964043i \(-0.414381\pi\)
0.265747 + 0.964043i \(0.414381\pi\)
\(728\) 0 0
\(729\) −20.1307 −0.745581
\(730\) 0 0
\(731\) 7.97871 0.295103
\(732\) 0 0
\(733\) −40.8853 −1.51013 −0.755067 0.655648i \(-0.772396\pi\)
−0.755067 + 0.655648i \(0.772396\pi\)
\(734\) 0 0
\(735\) −2.07001 −0.0763534
\(736\) 0 0
\(737\) 37.2334 1.37151
\(738\) 0 0
\(739\) 28.8294 1.06051 0.530253 0.847840i \(-0.322097\pi\)
0.530253 + 0.847840i \(0.322097\pi\)
\(740\) 0 0
\(741\) 1.68802 0.0620111
\(742\) 0 0
\(743\) 8.05135 0.295375 0.147688 0.989034i \(-0.452817\pi\)
0.147688 + 0.989034i \(0.452817\pi\)
\(744\) 0 0
\(745\) −1.45331 −0.0532453
\(746\) 0 0
\(747\) −40.9626 −1.49874
\(748\) 0 0
\(749\) 15.5853 0.569475
\(750\) 0 0
\(751\) −47.3107 −1.72639 −0.863195 0.504870i \(-0.831540\pi\)
−0.863195 + 0.504870i \(0.831540\pi\)
\(752\) 0 0
\(753\) −5.49063 −0.200090
\(754\) 0 0
\(755\) 7.27334 0.264704
\(756\) 0 0
\(757\) 6.20541 0.225539 0.112770 0.993621i \(-0.464028\pi\)
0.112770 + 0.993621i \(0.464028\pi\)
\(758\) 0 0
\(759\) −4.37350 −0.158748
\(760\) 0 0
\(761\) 43.8280 1.58877 0.794383 0.607418i \(-0.207795\pi\)
0.794383 + 0.607418i \(0.207795\pi\)
\(762\) 0 0
\(763\) −23.0200 −0.833379
\(764\) 0 0
\(765\) −2.46264 −0.0890370
\(766\) 0 0
\(767\) −23.8867 −0.862497
\(768\) 0 0
\(769\) −3.22538 −0.116310 −0.0581551 0.998308i \(-0.518522\pi\)
−0.0581551 + 0.998308i \(0.518522\pi\)
\(770\) 0 0
\(771\) −4.44398 −0.160046
\(772\) 0 0
\(773\) 6.16470 0.221729 0.110864 0.993836i \(-0.464638\pi\)
0.110864 + 0.993836i \(0.464638\pi\)
\(774\) 0 0
\(775\) 10.2827 0.369364
\(776\) 0 0
\(777\) 2.81070 0.100833
\(778\) 0 0
\(779\) −7.55602 −0.270722
\(780\) 0 0
\(781\) 14.8693 0.532066
\(782\) 0 0
\(783\) 20.0921 0.718031
\(784\) 0 0
\(785\) −17.0093 −0.607089
\(786\) 0 0
\(787\) −35.1527 −1.25306 −0.626530 0.779397i \(-0.715525\pi\)
−0.626530 + 0.779397i \(0.715525\pi\)
\(788\) 0 0
\(789\) 8.20541 0.292120
\(790\) 0 0
\(791\) 11.4720 0.407896
\(792\) 0 0
\(793\) −43.9201 −1.55965
\(794\) 0 0
\(795\) −3.14134 −0.111412
\(796\) 0 0
\(797\) −30.3420 −1.07477 −0.537385 0.843337i \(-0.680588\pi\)
−0.537385 + 0.843337i \(0.680588\pi\)
\(798\) 0 0
\(799\) −6.01866 −0.212925
\(800\) 0 0
\(801\) −37.3107 −1.31831
\(802\) 0 0
\(803\) 18.7521 0.661747
\(804\) 0 0
\(805\) 5.03863 0.177588
\(806\) 0 0
\(807\) 4.76397 0.167700
\(808\) 0 0
\(809\) −4.96137 −0.174432 −0.0872162 0.996189i \(-0.527797\pi\)
−0.0872162 + 0.996189i \(0.527797\pi\)
\(810\) 0 0
\(811\) −29.8094 −1.04675 −0.523375 0.852103i \(-0.675327\pi\)
−0.523375 + 0.852103i \(0.675327\pi\)
\(812\) 0 0
\(813\) −7.59210 −0.266267
\(814\) 0 0
\(815\) −10.1800 −0.356589
\(816\) 0 0
\(817\) −9.29200 −0.325086
\(818\) 0 0
\(819\) −15.2080 −0.531409
\(820\) 0 0
\(821\) −11.1520 −0.389207 −0.194603 0.980882i \(-0.562342\pi\)
−0.194603 + 0.980882i \(0.562342\pi\)
\(822\) 0 0
\(823\) 17.1413 0.597509 0.298755 0.954330i \(-0.403429\pi\)
0.298755 + 0.954330i \(0.403429\pi\)
\(824\) 0 0
\(825\) −0.990671 −0.0344907
\(826\) 0 0
\(827\) −24.8633 −0.864581 −0.432291 0.901734i \(-0.642295\pi\)
−0.432291 + 0.901734i \(0.642295\pi\)
\(828\) 0 0
\(829\) −18.7746 −0.652069 −0.326035 0.945358i \(-0.605713\pi\)
−0.326035 + 0.945358i \(0.605713\pi\)
\(830\) 0 0
\(831\) 7.33195 0.254343
\(832\) 0 0
\(833\) 4.89211 0.169502
\(834\) 0 0
\(835\) 6.07001 0.210061
\(836\) 0 0
\(837\) 21.9227 0.757761
\(838\) 0 0
\(839\) −29.0280 −1.00216 −0.501079 0.865402i \(-0.667063\pi\)
−0.501079 + 0.865402i \(0.667063\pi\)
\(840\) 0 0
\(841\) 59.8119 2.06248
\(842\) 0 0
\(843\) 8.16131 0.281091
\(844\) 0 0
\(845\) −8.58532 −0.295344
\(846\) 0 0
\(847\) 4.06926 0.139821
\(848\) 0 0
\(849\) −0.301330 −0.0103416
\(850\) 0 0
\(851\) 29.9227 1.02574
\(852\) 0 0
\(853\) 40.7894 1.39660 0.698301 0.715804i \(-0.253940\pi\)
0.698301 + 0.715804i \(0.253940\pi\)
\(854\) 0 0
\(855\) 2.86799 0.0980833
\(856\) 0 0
\(857\) −32.6354 −1.11480 −0.557401 0.830243i \(-0.688201\pi\)
−0.557401 + 0.830243i \(0.688201\pi\)
\(858\) 0 0
\(859\) −33.4906 −1.14269 −0.571343 0.820712i \(-0.693577\pi\)
−0.571343 + 0.820712i \(0.693577\pi\)
\(860\) 0 0
\(861\) −3.13333 −0.106783
\(862\) 0 0
\(863\) −5.13795 −0.174898 −0.0874489 0.996169i \(-0.527871\pi\)
−0.0874489 + 0.996169i \(0.527871\pi\)
\(864\) 0 0
\(865\) 5.22199 0.177553
\(866\) 0 0
\(867\) 5.90870 0.200670
\(868\) 0 0
\(869\) −47.1493 −1.59943
\(870\) 0 0
\(871\) −63.4427 −2.14967
\(872\) 0 0
\(873\) 27.7755 0.940057
\(874\) 0 0
\(875\) 1.14134 0.0385842
\(876\) 0 0
\(877\) −16.0220 −0.541026 −0.270513 0.962716i \(-0.587193\pi\)
−0.270513 + 0.962716i \(0.587193\pi\)
\(878\) 0 0
\(879\) 11.3680 0.383434
\(880\) 0 0
\(881\) 11.5306 0.388475 0.194238 0.980955i \(-0.437777\pi\)
0.194238 + 0.980955i \(0.437777\pi\)
\(882\) 0 0
\(883\) 39.5161 1.32982 0.664911 0.746923i \(-0.268470\pi\)
0.664911 + 0.746923i \(0.268470\pi\)
\(884\) 0 0
\(885\) 1.86799 0.0627919
\(886\) 0 0
\(887\) 20.8153 0.698910 0.349455 0.936953i \(-0.386367\pi\)
0.349455 + 0.936953i \(0.386367\pi\)
\(888\) 0 0
\(889\) −16.0586 −0.538588
\(890\) 0 0
\(891\) 21.3480 0.715184
\(892\) 0 0
\(893\) 7.00933 0.234558
\(894\) 0 0
\(895\) −1.55602 −0.0520119
\(896\) 0 0
\(897\) 7.45208 0.248818
\(898\) 0 0
\(899\) 96.9040 3.23193
\(900\) 0 0
\(901\) 7.42401 0.247330
\(902\) 0 0
\(903\) −3.85320 −0.128227
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) −32.0339 −1.06367 −0.531835 0.846848i \(-0.678497\pi\)
−0.531835 + 0.846848i \(0.678497\pi\)
\(908\) 0 0
\(909\) −4.16809 −0.138247
\(910\) 0 0
\(911\) 2.12136 0.0702838 0.0351419 0.999382i \(-0.488812\pi\)
0.0351419 + 0.999382i \(0.488812\pi\)
\(912\) 0 0
\(913\) 38.9439 1.28886
\(914\) 0 0
\(915\) 3.43466 0.113546
\(916\) 0 0
\(917\) −4.56534 −0.150761
\(918\) 0 0
\(919\) 26.4333 0.871955 0.435978 0.899957i \(-0.356403\pi\)
0.435978 + 0.899957i \(0.356403\pi\)
\(920\) 0 0
\(921\) −1.13333 −0.0373444
\(922\) 0 0
\(923\) −25.3361 −0.833948
\(924\) 0 0
\(925\) 6.77801 0.222860
\(926\) 0 0
\(927\) 6.39941 0.210184
\(928\) 0 0
\(929\) 17.7907 0.583695 0.291847 0.956465i \(-0.405730\pi\)
0.291847 + 0.956465i \(0.405730\pi\)
\(930\) 0 0
\(931\) −5.69735 −0.186723
\(932\) 0 0
\(933\) −7.78395 −0.254835
\(934\) 0 0
\(935\) 2.34128 0.0765681
\(936\) 0 0
\(937\) −7.46396 −0.243837 −0.121918 0.992540i \(-0.538905\pi\)
−0.121918 + 0.992540i \(0.538905\pi\)
\(938\) 0 0
\(939\) 4.40790 0.143846
\(940\) 0 0
\(941\) −2.67869 −0.0873229 −0.0436615 0.999046i \(-0.513902\pi\)
−0.0436615 + 0.999046i \(0.513902\pi\)
\(942\) 0 0
\(943\) −33.3574 −1.08627
\(944\) 0 0
\(945\) 2.43334 0.0791565
\(946\) 0 0
\(947\) −42.5840 −1.38379 −0.691897 0.721996i \(-0.743225\pi\)
−0.691897 + 0.721996i \(0.743225\pi\)
\(948\) 0 0
\(949\) −31.9520 −1.03721
\(950\) 0 0
\(951\) −6.08528 −0.197329
\(952\) 0 0
\(953\) 60.0046 1.94374 0.971870 0.235517i \(-0.0756784\pi\)
0.971870 + 0.235517i \(0.0756784\pi\)
\(954\) 0 0
\(955\) −5.03863 −0.163046
\(956\) 0 0
\(957\) −9.33609 −0.301793
\(958\) 0 0
\(959\) 5.54537 0.179069
\(960\) 0 0
\(961\) 74.7333 2.41075
\(962\) 0 0
\(963\) 39.1634 1.26202
\(964\) 0 0
\(965\) 9.68463 0.311759
\(966\) 0 0
\(967\) 61.1307 1.96583 0.982915 0.184059i \(-0.0589237\pi\)
0.982915 + 0.184059i \(0.0589237\pi\)
\(968\) 0 0
\(969\) 0.311977 0.0100221
\(970\) 0 0
\(971\) −54.6213 −1.75288 −0.876441 0.481510i \(-0.840089\pi\)
−0.876441 + 0.481510i \(0.840089\pi\)
\(972\) 0 0
\(973\) −4.77075 −0.152943
\(974\) 0 0
\(975\) 1.68802 0.0540600
\(976\) 0 0
\(977\) −19.3060 −0.617655 −0.308827 0.951118i \(-0.599937\pi\)
−0.308827 + 0.951118i \(0.599937\pi\)
\(978\) 0 0
\(979\) 35.4720 1.13369
\(980\) 0 0
\(981\) −57.8455 −1.84686
\(982\) 0 0
\(983\) −25.1820 −0.803182 −0.401591 0.915819i \(-0.631543\pi\)
−0.401591 + 0.915819i \(0.631543\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 2.90663 0.0925189
\(988\) 0 0
\(989\) −41.0212 −1.30440
\(990\) 0 0
\(991\) 5.82003 0.184879 0.0924397 0.995718i \(-0.470533\pi\)
0.0924397 + 0.995718i \(0.470533\pi\)
\(992\) 0 0
\(993\) −2.16932 −0.0688414
\(994\) 0 0
\(995\) 19.8867 0.630449
\(996\) 0 0
\(997\) 8.97201 0.284147 0.142073 0.989856i \(-0.454623\pi\)
0.142073 + 0.989856i \(0.454623\pi\)
\(998\) 0 0
\(999\) 14.4508 0.457202
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 1520.2.a.r.1.2 3
4.3 odd 2 760.2.a.h.1.2 3
5.4 even 2 7600.2.a.bo.1.2 3
8.3 odd 2 6080.2.a.bw.1.2 3
8.5 even 2 6080.2.a.bs.1.2 3
12.11 even 2 6840.2.a.bj.1.3 3
20.3 even 4 3800.2.d.k.3649.4 6
20.7 even 4 3800.2.d.k.3649.3 6
20.19 odd 2 3800.2.a.y.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.2 3 4.3 odd 2
1520.2.a.r.1.2 3 1.1 even 1 trivial
3800.2.a.y.1.2 3 20.19 odd 2
3800.2.d.k.3649.3 6 20.7 even 4
3800.2.d.k.3649.4 6 20.3 even 4
6080.2.a.bs.1.2 3 8.5 even 2
6080.2.a.bw.1.2 3 8.3 odd 2
6840.2.a.bj.1.3 3 12.11 even 2
7600.2.a.bo.1.2 3 5.4 even 2