Properties

Label 3150.2.bf.f.1601.1
Level $3150$
Weight $2$
Character 3150.1601
Analytic conductor $25.153$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1601.1
Character \(\chi\) \(=\) 3150.1601
Dual form 3150.2.bf.f.1151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.54232 + 0.732536i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.54232 + 0.732536i) q^{7} -1.00000i q^{8} +(-2.07577 + 1.19845i) q^{11} -5.67714i q^{13} +(2.56798 + 0.636766i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.03596 - 1.79434i) q^{17} +(5.12164 + 2.95698i) q^{19} +2.39690 q^{22} +(1.61233 + 0.930877i) q^{23} +(-2.83857 + 4.91654i) q^{26} +(-1.90555 - 1.83545i) q^{28} -4.88913i q^{29} +(-3.92008 + 2.26326i) q^{31} +(0.866025 - 0.500000i) q^{32} +2.07192i q^{34} +(1.48455 - 2.57132i) q^{37} +(-2.95698 - 5.12164i) q^{38} -7.04428 q^{41} +8.55956 q^{43} +(-2.07577 - 1.19845i) q^{44} +(-0.930877 - 1.61233i) q^{46} +(-2.78941 + 4.83140i) q^{47} +(5.92678 - 3.72468i) q^{49} +(4.91654 - 2.83857i) q^{52} +(-3.62931 + 2.09538i) q^{53} +(0.732536 + 2.54232i) q^{56} +(-2.44457 + 4.23411i) q^{58} +(1.00312 + 1.73746i) q^{59} +(10.7862 + 6.22739i) q^{61} +4.52651 q^{62} -1.00000 q^{64} +(-3.81111 - 6.60103i) q^{67} +(1.03596 - 1.79434i) q^{68} +9.14126i q^{71} +(-0.937339 + 0.541173i) q^{73} +(-2.57132 + 1.48455i) q^{74} +5.91397i q^{76} +(4.39937 - 4.56742i) q^{77} +(-8.38392 + 14.5214i) q^{79} +(6.10053 + 3.52214i) q^{82} -13.6122 q^{83} +(-7.41279 - 4.27978i) q^{86} +(1.19845 + 2.07577i) q^{88} +(-6.63129 + 11.4857i) q^{89} +(4.15870 + 14.4331i) q^{91} +1.86175i q^{92} +(4.83140 - 2.78941i) q^{94} +12.8260i q^{97} +(-6.99508 + 0.262276i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{4} - 16 q^{16} - 48 q^{19} + 24 q^{31} - 16 q^{46} + 56 q^{49} + 48 q^{61} - 32 q^{64} - 8 q^{79} - 56 q^{91} + 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 0.500000i −0.612372 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) −2.54232 + 0.732536i −0.960907 + 0.276872i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.07577 + 1.19845i −0.625869 + 0.361346i −0.779150 0.626837i \(-0.784349\pi\)
0.153282 + 0.988183i \(0.451016\pi\)
\(12\) 0 0
\(13\) 5.67714i 1.57455i −0.616599 0.787277i \(-0.711490\pi\)
0.616599 0.787277i \(-0.288510\pi\)
\(14\) 2.56798 + 0.636766i 0.686322 + 0.170183i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.03596 1.79434i −0.251258 0.435191i 0.712615 0.701556i \(-0.247511\pi\)
−0.963872 + 0.266365i \(0.914177\pi\)
\(18\) 0 0
\(19\) 5.12164 + 2.95698i 1.17499 + 0.678378i 0.954849 0.297091i \(-0.0960164\pi\)
0.220136 + 0.975469i \(0.429350\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.39690 0.511020
\(23\) 1.61233 + 0.930877i 0.336193 + 0.194101i 0.658587 0.752504i \(-0.271154\pi\)
−0.322394 + 0.946606i \(0.604488\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.83857 + 4.91654i −0.556689 + 0.964214i
\(27\) 0 0
\(28\) −1.90555 1.83545i −0.360116 0.346867i
\(29\) 4.88913i 0.907889i −0.891030 0.453944i \(-0.850016\pi\)
0.891030 0.453944i \(-0.149984\pi\)
\(30\) 0 0
\(31\) −3.92008 + 2.26326i −0.704067 + 0.406493i −0.808860 0.588001i \(-0.799915\pi\)
0.104794 + 0.994494i \(0.466582\pi\)
\(32\) 0.866025 0.500000i 0.153093 0.0883883i
\(33\) 0 0
\(34\) 2.07192i 0.355332i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.48455 2.57132i 0.244059 0.422723i −0.717807 0.696242i \(-0.754854\pi\)
0.961867 + 0.273519i \(0.0881875\pi\)
\(38\) −2.95698 5.12164i −0.479686 0.830840i
\(39\) 0 0
\(40\) 0 0
\(41\) −7.04428 −1.10013 −0.550066 0.835121i \(-0.685397\pi\)
−0.550066 + 0.835121i \(0.685397\pi\)
\(42\) 0 0
\(43\) 8.55956 1.30532 0.652660 0.757651i \(-0.273653\pi\)
0.652660 + 0.757651i \(0.273653\pi\)
\(44\) −2.07577 1.19845i −0.312934 0.180673i
\(45\) 0 0
\(46\) −0.930877 1.61233i −0.137250 0.237725i
\(47\) −2.78941 + 4.83140i −0.406877 + 0.704732i −0.994538 0.104375i \(-0.966716\pi\)
0.587661 + 0.809107i \(0.300049\pi\)
\(48\) 0 0
\(49\) 5.92678 3.72468i 0.846683 0.532097i
\(50\) 0 0
\(51\) 0 0
\(52\) 4.91654 2.83857i 0.681802 0.393639i
\(53\) −3.62931 + 2.09538i −0.498524 + 0.287823i −0.728104 0.685467i \(-0.759598\pi\)
0.229580 + 0.973290i \(0.426265\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.732536 + 2.54232i 0.0978892 + 0.339732i
\(57\) 0 0
\(58\) −2.44457 + 4.23411i −0.320987 + 0.555966i
\(59\) 1.00312 + 1.73746i 0.130595 + 0.226198i 0.923906 0.382619i \(-0.124978\pi\)
−0.793311 + 0.608817i \(0.791644\pi\)
\(60\) 0 0
\(61\) 10.7862 + 6.22739i 1.38103 + 0.797335i 0.992281 0.124011i \(-0.0395757\pi\)
0.388744 + 0.921346i \(0.372909\pi\)
\(62\) 4.52651 0.574868
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.81111 6.60103i −0.465601 0.806445i 0.533627 0.845720i \(-0.320829\pi\)
−0.999228 + 0.0392750i \(0.987495\pi\)
\(68\) 1.03596 1.79434i 0.125629 0.217596i
\(69\) 0 0
\(70\) 0 0
\(71\) 9.14126i 1.08487i 0.840099 + 0.542434i \(0.182497\pi\)
−0.840099 + 0.542434i \(0.817503\pi\)
\(72\) 0 0
\(73\) −0.937339 + 0.541173i −0.109707 + 0.0633395i −0.553850 0.832617i \(-0.686842\pi\)
0.444142 + 0.895956i \(0.353508\pi\)
\(74\) −2.57132 + 1.48455i −0.298910 + 0.172576i
\(75\) 0 0
\(76\) 5.91397i 0.678378i
\(77\) 4.39937 4.56742i 0.501355 0.520505i
\(78\) 0 0
\(79\) −8.38392 + 14.5214i −0.943265 + 1.63378i −0.184076 + 0.982912i \(0.558929\pi\)
−0.759189 + 0.650870i \(0.774404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.10053 + 3.52214i 0.673690 + 0.388955i
\(83\) −13.6122 −1.49414 −0.747068 0.664747i \(-0.768539\pi\)
−0.747068 + 0.664747i \(0.768539\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.41279 4.27978i −0.799342 0.461500i
\(87\) 0 0
\(88\) 1.19845 + 2.07577i 0.127755 + 0.221278i
\(89\) −6.63129 + 11.4857i −0.702916 + 1.21749i 0.264523 + 0.964379i \(0.414786\pi\)
−0.967438 + 0.253106i \(0.918548\pi\)
\(90\) 0 0
\(91\) 4.15870 + 14.4331i 0.435951 + 1.51300i
\(92\) 1.86175i 0.194101i
\(93\) 0 0
\(94\) 4.83140 2.78941i 0.498321 0.287706i
\(95\) 0 0
\(96\) 0 0
\(97\) 12.8260i 1.30229i 0.758955 + 0.651143i \(0.225710\pi\)
−0.758955 + 0.651143i \(0.774290\pi\)
\(98\) −6.99508 + 0.262276i −0.706610 + 0.0264939i
\(99\) 0 0
\(100\) 0 0
\(101\) −4.45573 7.71756i −0.443362 0.767926i 0.554574 0.832134i \(-0.312881\pi\)
−0.997937 + 0.0642084i \(0.979548\pi\)
\(102\) 0 0
\(103\) 9.37021 + 5.40989i 0.923274 + 0.533053i 0.884678 0.466202i \(-0.154378\pi\)
0.0385960 + 0.999255i \(0.487711\pi\)
\(104\) −5.67714 −0.556689
\(105\) 0 0
\(106\) 4.19077 0.407043
\(107\) −4.82989 2.78854i −0.466923 0.269578i 0.248028 0.968753i \(-0.420218\pi\)
−0.714951 + 0.699175i \(0.753551\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.73205i 0.0957826 + 0.165900i 0.909935 0.414751i \(-0.136131\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.636766 2.56798i 0.0601687 0.242651i
\(113\) 14.5030i 1.36432i −0.731202 0.682161i \(-0.761040\pi\)
0.731202 0.682161i \(-0.238960\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.23411 2.44457i 0.393127 0.226972i
\(117\) 0 0
\(118\) 2.00624i 0.184690i
\(119\) 3.94816 + 3.80290i 0.361928 + 0.348612i
\(120\) 0 0
\(121\) −2.62745 + 4.55087i −0.238859 + 0.413715i
\(122\) −6.22739 10.7862i −0.563801 0.976532i
\(123\) 0 0
\(124\) −3.92008 2.26326i −0.352033 0.203247i
\(125\) 0 0
\(126\) 0 0
\(127\) −19.2462 −1.70783 −0.853913 0.520416i \(-0.825777\pi\)
−0.853913 + 0.520416i \(0.825777\pi\)
\(128\) 0.866025 + 0.500000i 0.0765466 + 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.34970 4.06980i 0.205294 0.355580i −0.744932 0.667140i \(-0.767518\pi\)
0.950226 + 0.311560i \(0.100851\pi\)
\(132\) 0 0
\(133\) −15.1870 3.76581i −1.31688 0.326537i
\(134\) 7.62222i 0.658459i
\(135\) 0 0
\(136\) −1.79434 + 1.03596i −0.153863 + 0.0888330i
\(137\) −19.8185 + 11.4422i −1.69321 + 0.977577i −0.741317 + 0.671155i \(0.765798\pi\)
−0.951896 + 0.306421i \(0.900868\pi\)
\(138\) 0 0
\(139\) 9.13862i 0.775127i −0.921843 0.387564i \(-0.873317\pi\)
0.921843 0.387564i \(-0.126683\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.57063 7.91656i 0.383559 0.664343i
\(143\) 6.80375 + 11.7844i 0.568958 + 0.985465i
\(144\) 0 0
\(145\) 0 0
\(146\) 1.08235 0.0895756
\(147\) 0 0
\(148\) 2.96911 0.244059
\(149\) −9.05052 5.22532i −0.741448 0.428075i 0.0811477 0.996702i \(-0.474141\pi\)
−0.822596 + 0.568627i \(0.807475\pi\)
\(150\) 0 0
\(151\) 8.85937 + 15.3449i 0.720965 + 1.24875i 0.960613 + 0.277888i \(0.0896346\pi\)
−0.239648 + 0.970860i \(0.577032\pi\)
\(152\) 2.95698 5.12164i 0.239843 0.415420i
\(153\) 0 0
\(154\) −6.09368 + 1.75581i −0.491042 + 0.141487i
\(155\) 0 0
\(156\) 0 0
\(157\) −5.26801 + 3.04149i −0.420433 + 0.242737i −0.695263 0.718756i \(-0.744712\pi\)
0.274830 + 0.961493i \(0.411379\pi\)
\(158\) 14.5214 8.38392i 1.15526 0.666989i
\(159\) 0 0
\(160\) 0 0
\(161\) −4.78095 1.18550i −0.376792 0.0934305i
\(162\) 0 0
\(163\) 0.468670 0.811759i 0.0367090 0.0635819i −0.847087 0.531454i \(-0.821646\pi\)
0.883796 + 0.467872i \(0.154979\pi\)
\(164\) −3.52214 6.10053i −0.275033 0.476371i
\(165\) 0 0
\(166\) 11.7885 + 6.80611i 0.914968 + 0.528257i
\(167\) 17.2101 1.33176 0.665879 0.746060i \(-0.268057\pi\)
0.665879 + 0.746060i \(0.268057\pi\)
\(168\) 0 0
\(169\) −19.2299 −1.47922
\(170\) 0 0
\(171\) 0 0
\(172\) 4.27978 + 7.41279i 0.326330 + 0.565220i
\(173\) −0.988114 + 1.71146i −0.0751249 + 0.130120i −0.901141 0.433527i \(-0.857269\pi\)
0.826016 + 0.563647i \(0.190602\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.39690i 0.180673i
\(177\) 0 0
\(178\) 11.4857 6.63129i 0.860893 0.497037i
\(179\) −0.768461 + 0.443671i −0.0574375 + 0.0331615i −0.528444 0.848968i \(-0.677224\pi\)
0.471006 + 0.882130i \(0.343891\pi\)
\(180\) 0 0
\(181\) 4.89973i 0.364194i 0.983281 + 0.182097i \(0.0582885\pi\)
−0.983281 + 0.182097i \(0.941712\pi\)
\(182\) 3.61500 14.5788i 0.267962 1.08065i
\(183\) 0 0
\(184\) 0.930877 1.61233i 0.0686252 0.118862i
\(185\) 0 0
\(186\) 0 0
\(187\) 4.30084 + 2.48309i 0.314509 + 0.181582i
\(188\) −5.57882 −0.406877
\(189\) 0 0
\(190\) 0 0
\(191\) 2.44949 + 1.41421i 0.177239 + 0.102329i 0.585995 0.810315i \(-0.300704\pi\)
−0.408756 + 0.912644i \(0.634037\pi\)
\(192\) 0 0
\(193\) −9.74828 16.8845i −0.701697 1.21537i −0.967871 0.251449i \(-0.919093\pi\)
0.266174 0.963925i \(-0.414240\pi\)
\(194\) 6.41301 11.1077i 0.460428 0.797484i
\(195\) 0 0
\(196\) 6.18906 + 3.27040i 0.442076 + 0.233600i
\(197\) 27.1576i 1.93490i 0.253069 + 0.967448i \(0.418560\pi\)
−0.253069 + 0.967448i \(0.581440\pi\)
\(198\) 0 0
\(199\) 3.00000 1.73205i 0.212664 0.122782i −0.389885 0.920864i \(-0.627485\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.91147i 0.627009i
\(203\) 3.58146 + 12.4297i 0.251369 + 0.872396i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.40989 9.37021i −0.376925 0.652853i
\(207\) 0 0
\(208\) 4.91654 + 2.83857i 0.340901 + 0.196819i
\(209\) −14.1752 −0.980516
\(210\) 0 0
\(211\) −4.06071 −0.279551 −0.139775 0.990183i \(-0.544638\pi\)
−0.139775 + 0.990183i \(0.544638\pi\)
\(212\) −3.62931 2.09538i −0.249262 0.143912i
\(213\) 0 0
\(214\) 2.78854 + 4.82989i 0.190620 + 0.330164i
\(215\) 0 0
\(216\) 0 0
\(217\) 8.30817 8.62552i 0.563996 0.585538i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −10.1867 + 5.88130i −0.685232 + 0.395619i
\(222\) 0 0
\(223\) 16.1486i 1.08139i 0.841218 + 0.540696i \(0.181839\pi\)
−0.841218 + 0.540696i \(0.818161\pi\)
\(224\) −1.83545 + 1.90555i −0.122636 + 0.127320i
\(225\) 0 0
\(226\) −7.25148 + 12.5599i −0.482361 + 0.835474i
\(227\) 0.839901 + 1.45475i 0.0557462 + 0.0965552i 0.892552 0.450945i \(-0.148913\pi\)
−0.836806 + 0.547500i \(0.815580\pi\)
\(228\) 0 0
\(229\) 10.9143 + 6.30136i 0.721236 + 0.416406i 0.815207 0.579169i \(-0.196623\pi\)
−0.0939717 + 0.995575i \(0.529956\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.88913 −0.320987
\(233\) 12.5715 + 7.25818i 0.823589 + 0.475499i 0.851653 0.524107i \(-0.175601\pi\)
−0.0280635 + 0.999606i \(0.508934\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.00312 + 1.73746i −0.0652977 + 0.113099i
\(237\) 0 0
\(238\) −1.51776 5.26749i −0.0983816 0.341441i
\(239\) 0.207089i 0.0133955i 0.999978 + 0.00669774i \(0.00213197\pi\)
−0.999978 + 0.00669774i \(0.997868\pi\)
\(240\) 0 0
\(241\) 9.04172 5.22024i 0.582428 0.336265i −0.179669 0.983727i \(-0.557503\pi\)
0.762098 + 0.647462i \(0.224169\pi\)
\(242\) 4.55087 2.62745i 0.292541 0.168899i
\(243\) 0 0
\(244\) 12.4548i 0.797335i
\(245\) 0 0
\(246\) 0 0
\(247\) 16.7872 29.0763i 1.06814 1.85008i
\(248\) 2.26326 + 3.92008i 0.143717 + 0.248925i
\(249\) 0 0
\(250\) 0 0
\(251\) −28.6464 −1.80815 −0.904074 0.427377i \(-0.859438\pi\)
−0.904074 + 0.427377i \(0.859438\pi\)
\(252\) 0 0
\(253\) −4.46243 −0.280551
\(254\) 16.6677 + 9.62311i 1.04583 + 0.603808i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) −12.4595 + 21.5805i −0.777202 + 1.34615i 0.156346 + 0.987702i \(0.450029\pi\)
−0.933548 + 0.358452i \(0.883305\pi\)
\(258\) 0 0
\(259\) −1.89063 + 7.62462i −0.117478 + 0.473771i
\(260\) 0 0
\(261\) 0 0
\(262\) −4.06980 + 2.34970i −0.251433 + 0.145165i
\(263\) 16.6197 9.59538i 1.02481 0.591677i 0.109319 0.994007i \(-0.465133\pi\)
0.915495 + 0.402330i \(0.131800\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 11.2694 + 10.8548i 0.690970 + 0.665548i
\(267\) 0 0
\(268\) 3.81111 6.60103i 0.232801 0.403222i
\(269\) 2.42744 + 4.20446i 0.148004 + 0.256350i 0.930490 0.366318i \(-0.119382\pi\)
−0.782486 + 0.622668i \(0.786049\pi\)
\(270\) 0 0
\(271\) −11.8344 6.83257i −0.718886 0.415049i 0.0954567 0.995434i \(-0.469569\pi\)
−0.814342 + 0.580385i \(0.802902\pi\)
\(272\) 2.07192 0.125629
\(273\) 0 0
\(274\) 22.8845 1.38250
\(275\) 0 0
\(276\) 0 0
\(277\) −6.06491 10.5047i −0.364405 0.631168i 0.624276 0.781204i \(-0.285394\pi\)
−0.988680 + 0.150036i \(0.952061\pi\)
\(278\) −4.56931 + 7.91427i −0.274049 + 0.474667i
\(279\) 0 0
\(280\) 0 0
\(281\) 32.6206i 1.94598i 0.230839 + 0.972992i \(0.425853\pi\)
−0.230839 + 0.972992i \(0.574147\pi\)
\(282\) 0 0
\(283\) −10.9647 + 6.33045i −0.651781 + 0.376306i −0.789138 0.614216i \(-0.789473\pi\)
0.137357 + 0.990522i \(0.456139\pi\)
\(284\) −7.91656 + 4.57063i −0.469761 + 0.271217i
\(285\) 0 0
\(286\) 13.6075i 0.804628i
\(287\) 17.9088 5.16019i 1.05712 0.304596i
\(288\) 0 0
\(289\) 6.35357 11.0047i 0.373739 0.647335i
\(290\) 0 0
\(291\) 0 0
\(292\) −0.937339 0.541173i −0.0548536 0.0316698i
\(293\) −25.1151 −1.46724 −0.733621 0.679559i \(-0.762171\pi\)
−0.733621 + 0.679559i \(0.762171\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.57132 1.48455i −0.149455 0.0862880i
\(297\) 0 0
\(298\) 5.22532 + 9.05052i 0.302695 + 0.524283i
\(299\) 5.28472 9.15340i 0.305623 0.529355i
\(300\) 0 0
\(301\) −21.7611 + 6.27018i −1.25429 + 0.361407i
\(302\) 17.7187i 1.01960i
\(303\) 0 0
\(304\) −5.12164 + 2.95698i −0.293746 + 0.169595i
\(305\) 0 0
\(306\) 0 0
\(307\) 18.5674i 1.05970i 0.848092 + 0.529849i \(0.177751\pi\)
−0.848092 + 0.529849i \(0.822249\pi\)
\(308\) 6.15518 + 1.52626i 0.350724 + 0.0869668i
\(309\) 0 0
\(310\) 0 0
\(311\) 6.21831 + 10.7704i 0.352608 + 0.610735i 0.986706 0.162518i \(-0.0519616\pi\)
−0.634098 + 0.773253i \(0.718628\pi\)
\(312\) 0 0
\(313\) −10.7504 6.20675i −0.607649 0.350826i 0.164396 0.986394i \(-0.447433\pi\)
−0.772045 + 0.635568i \(0.780766\pi\)
\(314\) 6.08297 0.343282
\(315\) 0 0
\(316\) −16.7678 −0.943265
\(317\) 21.3444 + 12.3232i 1.19882 + 0.692141i 0.960293 0.278995i \(-0.0900012\pi\)
0.238530 + 0.971135i \(0.423335\pi\)
\(318\) 0 0
\(319\) 5.85937 + 10.1487i 0.328062 + 0.568219i
\(320\) 0 0
\(321\) 0 0
\(322\) 3.54767 + 3.41715i 0.197704 + 0.190430i
\(323\) 12.2533i 0.681791i
\(324\) 0 0
\(325\) 0 0
\(326\) −0.811759 + 0.468670i −0.0449592 + 0.0259572i
\(327\) 0 0
\(328\) 7.04428i 0.388955i
\(329\) 3.55240 14.3263i 0.195850 0.789835i
\(330\) 0 0
\(331\) 12.5788 21.7871i 0.691392 1.19753i −0.279989 0.960003i \(-0.590331\pi\)
0.971382 0.237524i \(-0.0763357\pi\)
\(332\) −6.80611 11.7885i −0.373534 0.646980i
\(333\) 0 0
\(334\) −14.9044 8.60505i −0.815531 0.470847i
\(335\) 0 0
\(336\) 0 0
\(337\) −14.4214 −0.785584 −0.392792 0.919627i \(-0.628491\pi\)
−0.392792 + 0.919627i \(0.628491\pi\)
\(338\) 16.6536 + 9.61494i 0.905834 + 0.522984i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.42479 9.39601i 0.293769 0.508823i
\(342\) 0 0
\(343\) −12.3393 + 13.8109i −0.666261 + 0.745719i
\(344\) 8.55956i 0.461500i
\(345\) 0 0
\(346\) 1.71146 0.988114i 0.0920088 0.0531213i
\(347\) 12.5457 7.24329i 0.673491 0.388840i −0.123907 0.992294i \(-0.539543\pi\)
0.797398 + 0.603454i \(0.206209\pi\)
\(348\) 0 0
\(349\) 2.12483i 0.113739i 0.998382 + 0.0568697i \(0.0181119\pi\)
−0.998382 + 0.0568697i \(0.981888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.19845 + 2.07577i −0.0638775 + 0.110639i
\(353\) 7.26335 + 12.5805i 0.386589 + 0.669592i 0.991988 0.126330i \(-0.0403199\pi\)
−0.605399 + 0.795922i \(0.706987\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −13.2626 −0.702916
\(357\) 0 0
\(358\) 0.887342 0.0468975
\(359\) 3.42054 + 1.97485i 0.180529 + 0.104228i 0.587541 0.809194i \(-0.300096\pi\)
−0.407012 + 0.913423i \(0.633429\pi\)
\(360\) 0 0
\(361\) 7.98749 + 13.8347i 0.420394 + 0.728144i
\(362\) 2.44986 4.24329i 0.128762 0.223022i
\(363\) 0 0
\(364\) −10.4201 + 10.8181i −0.546160 + 0.567022i
\(365\) 0 0
\(366\) 0 0
\(367\) −23.9826 + 13.8464i −1.25188 + 0.722775i −0.971483 0.237109i \(-0.923800\pi\)
−0.280400 + 0.959883i \(0.590467\pi\)
\(368\) −1.61233 + 0.930877i −0.0840483 + 0.0485253i
\(369\) 0 0
\(370\) 0 0
\(371\) 7.69193 7.98574i 0.399345 0.414599i
\(372\) 0 0
\(373\) −6.06491 + 10.5047i −0.314029 + 0.543914i −0.979231 0.202750i \(-0.935012\pi\)
0.665202 + 0.746664i \(0.268346\pi\)
\(374\) −2.48309 4.30084i −0.128398 0.222391i
\(375\) 0 0
\(376\) 4.83140 + 2.78941i 0.249160 + 0.143853i
\(377\) −27.7563 −1.42952
\(378\) 0 0
\(379\) 18.6821 0.959636 0.479818 0.877368i \(-0.340703\pi\)
0.479818 + 0.877368i \(0.340703\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.41421 2.44949i −0.0723575 0.125327i
\(383\) −5.51463 + 9.55162i −0.281784 + 0.488065i −0.971824 0.235706i \(-0.924260\pi\)
0.690040 + 0.723771i \(0.257593\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.4966i 0.992349i
\(387\) 0 0
\(388\) −11.1077 + 6.41301i −0.563906 + 0.325571i
\(389\) 29.7662 17.1855i 1.50921 0.871341i 0.509264 0.860611i \(-0.329918\pi\)
0.999942 0.0107299i \(-0.00341551\pi\)
\(390\) 0 0
\(391\) 3.85741i 0.195078i
\(392\) −3.72468 5.92678i −0.188125 0.299348i
\(393\) 0 0
\(394\) 13.5788 23.5191i 0.684089 1.18488i
\(395\) 0 0
\(396\) 0 0
\(397\) 26.8561 + 15.5054i 1.34787 + 0.778191i 0.987947 0.154792i \(-0.0494706\pi\)
0.359920 + 0.932983i \(0.382804\pi\)
\(398\) −3.46410 −0.173640
\(399\) 0 0
\(400\) 0 0
\(401\) −8.20771 4.73872i −0.409873 0.236641i 0.280862 0.959748i \(-0.409380\pi\)
−0.690735 + 0.723108i \(0.742713\pi\)
\(402\) 0 0
\(403\) 12.8488 + 22.2548i 0.640045 + 1.10859i
\(404\) 4.45573 7.71756i 0.221681 0.383963i
\(405\) 0 0
\(406\) 3.11323 12.5552i 0.154507 0.623104i
\(407\) 7.11665i 0.352759i
\(408\) 0 0
\(409\) −8.20805 + 4.73892i −0.405862 + 0.234324i −0.689010 0.724752i \(-0.741954\pi\)
0.283148 + 0.959076i \(0.408621\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.8198i 0.533053i
\(413\) −3.82301 3.68235i −0.188118 0.181197i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.83857 4.91654i −0.139172 0.241053i
\(417\) 0 0
\(418\) 12.2760 + 7.08758i 0.600441 + 0.346665i
\(419\) −2.54445 −0.124305 −0.0621523 0.998067i \(-0.519796\pi\)
−0.0621523 + 0.998067i \(0.519796\pi\)
\(420\) 0 0
\(421\) −5.08573 −0.247863 −0.123932 0.992291i \(-0.539550\pi\)
−0.123932 + 0.992291i \(0.539550\pi\)
\(422\) 3.51668 + 2.03035i 0.171189 + 0.0988361i
\(423\) 0 0
\(424\) 2.09538 + 3.62931i 0.101761 + 0.176255i
\(425\) 0 0
\(426\) 0 0
\(427\) −31.9836 7.93077i −1.54780 0.383797i
\(428\) 5.57707i 0.269578i
\(429\) 0 0
\(430\) 0 0
\(431\) 11.5164 6.64902i 0.554727 0.320272i −0.196299 0.980544i \(-0.562892\pi\)
0.751027 + 0.660272i \(0.229559\pi\)
\(432\) 0 0
\(433\) 12.7895i 0.614626i −0.951608 0.307313i \(-0.900570\pi\)
0.951608 0.307313i \(-0.0994299\pi\)
\(434\) −11.5078 + 3.31583i −0.552394 + 0.159165i
\(435\) 0 0
\(436\) −1.00000 + 1.73205i −0.0478913 + 0.0829502i
\(437\) 5.50517 + 9.53524i 0.263348 + 0.456132i
\(438\) 0 0
\(439\) −0.323211 0.186606i −0.0154260 0.00890623i 0.492267 0.870444i \(-0.336168\pi\)
−0.507693 + 0.861538i \(0.669502\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.7626 0.559490
\(443\) 21.3328 + 12.3165i 1.01355 + 0.585175i 0.912230 0.409679i \(-0.134359\pi\)
0.101323 + 0.994854i \(0.467693\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 8.07432 13.9851i 0.382330 0.662215i
\(447\) 0 0
\(448\) 2.54232 0.732536i 0.120113 0.0346091i
\(449\) 40.6223i 1.91708i 0.284950 + 0.958542i \(0.408023\pi\)
−0.284950 + 0.958542i \(0.591977\pi\)
\(450\) 0 0
\(451\) 14.6223 8.44220i 0.688538 0.397528i
\(452\) 12.5599 7.25148i 0.590769 0.341081i
\(453\) 0 0
\(454\) 1.67980i 0.0788370i
\(455\) 0 0
\(456\) 0 0
\(457\) −0.495276 + 0.857843i −0.0231680 + 0.0401282i −0.877377 0.479802i \(-0.840709\pi\)
0.854209 + 0.519930i \(0.174042\pi\)
\(458\) −6.30136 10.9143i −0.294443 0.509991i
\(459\) 0 0
\(460\) 0 0
\(461\) 20.7397 0.965945 0.482972 0.875636i \(-0.339557\pi\)
0.482972 + 0.875636i \(0.339557\pi\)
\(462\) 0 0
\(463\) −1.46421 −0.0680476 −0.0340238 0.999421i \(-0.510832\pi\)
−0.0340238 + 0.999421i \(0.510832\pi\)
\(464\) 4.23411 + 2.44457i 0.196564 + 0.113486i
\(465\) 0 0
\(466\) −7.25818 12.5715i −0.336229 0.582365i
\(467\) −18.1340 + 31.4090i −0.839142 + 1.45344i 0.0514705 + 0.998675i \(0.483609\pi\)
−0.890613 + 0.454762i \(0.849724\pi\)
\(468\) 0 0
\(469\) 14.5246 + 13.9902i 0.670682 + 0.646006i
\(470\) 0 0
\(471\) 0 0
\(472\) 1.73746 1.00312i 0.0799730 0.0461724i
\(473\) −17.7677 + 10.2582i −0.816959 + 0.471672i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.31933 + 5.32066i −0.0604714 + 0.243872i
\(477\) 0 0
\(478\) 0.103545 0.179344i 0.00473602 0.00820302i
\(479\) −5.16288 8.94237i −0.235898 0.408587i 0.723635 0.690183i \(-0.242470\pi\)
−0.959533 + 0.281595i \(0.909136\pi\)
\(480\) 0 0
\(481\) −14.5978 8.42802i −0.665600 0.384285i
\(482\) −10.4405 −0.475551
\(483\) 0 0
\(484\) −5.25489 −0.238859
\(485\) 0 0
\(486\) 0 0
\(487\) 10.1645 + 17.6055i 0.460599 + 0.797781i 0.998991 0.0449135i \(-0.0143012\pi\)
−0.538392 + 0.842695i \(0.680968\pi\)
\(488\) 6.22739 10.7862i 0.281901 0.488266i
\(489\) 0 0
\(490\) 0 0
\(491\) 34.6034i 1.56163i −0.624764 0.780814i \(-0.714805\pi\)
0.624764 0.780814i \(-0.285195\pi\)
\(492\) 0 0
\(493\) −8.77276 + 5.06495i −0.395105 + 0.228114i
\(494\) −29.0763 + 16.7872i −1.30820 + 0.755292i
\(495\) 0 0
\(496\) 4.52651i 0.203247i
\(497\) −6.69630 23.2400i −0.300370 1.04246i
\(498\) 0 0
\(499\) 1.47545 2.55555i 0.0660501 0.114402i −0.831109 0.556109i \(-0.812294\pi\)
0.897159 + 0.441707i \(0.145627\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 24.8085 + 14.3232i 1.10726 + 0.639277i
\(503\) 31.8907 1.42193 0.710967 0.703225i \(-0.248258\pi\)
0.710967 + 0.703225i \(0.248258\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.86458 + 2.23121i 0.171801 + 0.0991896i
\(507\) 0 0
\(508\) −9.62311 16.6677i −0.426957 0.739510i
\(509\) 0.421199 0.729538i 0.0186693 0.0323362i −0.856540 0.516081i \(-0.827390\pi\)
0.875209 + 0.483745i \(0.160724\pi\)
\(510\) 0 0
\(511\) 1.98659 2.06247i 0.0878815 0.0912383i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 21.5805 12.4595i 0.951875 0.549565i
\(515\) 0 0
\(516\) 0 0
\(517\) 13.3718i 0.588093i
\(518\) 5.44964 5.65780i 0.239443 0.248589i
\(519\) 0 0
\(520\) 0 0
\(521\) 2.81499 + 4.87571i 0.123327 + 0.213609i 0.921078 0.389379i \(-0.127310\pi\)
−0.797751 + 0.602987i \(0.793977\pi\)
\(522\) 0 0
\(523\) 15.4482 + 8.91899i 0.675500 + 0.390000i 0.798158 0.602449i \(-0.205808\pi\)
−0.122657 + 0.992449i \(0.539142\pi\)
\(524\) 4.69940 0.205294
\(525\) 0 0
\(526\) −19.1908 −0.836757
\(527\) 8.12210 + 4.68930i 0.353804 + 0.204269i
\(528\) 0 0
\(529\) −9.76694 16.9168i −0.424649 0.735514i
\(530\) 0 0
\(531\) 0 0
\(532\) −4.33219 15.0352i −0.187824 0.651858i
\(533\) 39.9913i 1.73222i
\(534\) 0 0
\(535\) 0 0
\(536\) −6.60103 + 3.81111i −0.285121 + 0.164615i
\(537\) 0 0
\(538\) 4.85489i 0.209309i
\(539\) −7.83882 + 14.8345i −0.337642 + 0.638968i
\(540\) 0 0
\(541\) 12.7120 22.0179i 0.546533 0.946623i −0.451976 0.892030i \(-0.649281\pi\)
0.998509 0.0545925i \(-0.0173860\pi\)
\(542\) 6.83257 + 11.8344i 0.293484 + 0.508329i
\(543\) 0 0
\(544\) −1.79434 1.03596i −0.0769316 0.0444165i
\(545\) 0 0
\(546\) 0 0
\(547\) 12.1182 0.518138 0.259069 0.965859i \(-0.416584\pi\)
0.259069 + 0.965859i \(0.416584\pi\)
\(548\) −19.8185 11.4422i −0.846606 0.488788i
\(549\) 0 0
\(550\) 0 0
\(551\) 14.4571 25.0404i 0.615892 1.06676i
\(552\) 0 0
\(553\) 10.6772 43.0595i 0.454040 1.83108i
\(554\) 12.1298i 0.515346i
\(555\) 0 0
\(556\) 7.91427 4.56931i 0.335640 0.193782i
\(557\) −1.33331 + 0.769786i −0.0564941 + 0.0326169i −0.527981 0.849256i \(-0.677051\pi\)
0.471487 + 0.881873i \(0.343717\pi\)
\(558\) 0 0
\(559\) 48.5938i 2.05530i
\(560\) 0 0
\(561\) 0 0
\(562\) 16.3103 28.2503i 0.688009 1.19167i
\(563\) −11.2322 19.4548i −0.473381 0.819920i 0.526155 0.850389i \(-0.323633\pi\)
−0.999536 + 0.0304689i \(0.990300\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 12.6609 0.532177
\(567\) 0 0
\(568\) 9.14126 0.383559
\(569\) −9.35810 5.40290i −0.392312 0.226501i 0.290849 0.956769i \(-0.406062\pi\)
−0.683161 + 0.730267i \(0.739395\pi\)
\(570\) 0 0
\(571\) 3.98169 + 6.89649i 0.166629 + 0.288609i 0.937232 0.348705i \(-0.113379\pi\)
−0.770604 + 0.637314i \(0.780045\pi\)
\(572\) −6.80375 + 11.7844i −0.284479 + 0.492732i
\(573\) 0 0
\(574\) −18.0896 4.48555i −0.755045 0.187223i
\(575\) 0 0
\(576\) 0 0
\(577\) −21.2980 + 12.2964i −0.886648 + 0.511906i −0.872845 0.487998i \(-0.837727\pi\)
−0.0138033 + 0.999905i \(0.504394\pi\)
\(578\) −11.0047 + 6.35357i −0.457735 + 0.264274i
\(579\) 0 0
\(580\) 0 0
\(581\) 34.6066 9.97144i 1.43573 0.413685i
\(582\) 0 0
\(583\) 5.02242 8.69908i 0.208007 0.360279i
\(584\) 0.541173 + 0.937339i 0.0223939 + 0.0387874i
\(585\) 0 0
\(586\) 21.7503 + 12.5576i 0.898498 + 0.518748i
\(587\) −12.8469 −0.530248 −0.265124 0.964214i \(-0.585413\pi\)
−0.265124 + 0.964214i \(0.585413\pi\)
\(588\) 0 0
\(589\) −26.7696 −1.10302
\(590\) 0 0
\(591\) 0 0
\(592\) 1.48455 + 2.57132i 0.0610148 + 0.105681i
\(593\) −8.58155 + 14.8637i −0.352402 + 0.610379i −0.986670 0.162735i \(-0.947968\pi\)
0.634268 + 0.773114i \(0.281302\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.4506i 0.428075i
\(597\) 0 0
\(598\) −9.15340 + 5.28472i −0.374310 + 0.216108i
\(599\) 16.9813 9.80416i 0.693837 0.400587i −0.111211 0.993797i \(-0.535473\pi\)
0.805048 + 0.593210i \(0.202140\pi\)
\(600\) 0 0
\(601\) 28.2340i 1.15169i −0.817560 0.575844i \(-0.804674\pi\)
0.817560 0.575844i \(-0.195326\pi\)
\(602\) 21.9808 + 5.45043i 0.895870 + 0.222143i
\(603\) 0 0
\(604\) −8.85937 + 15.3449i −0.360483 + 0.624374i
\(605\) 0 0
\(606\) 0 0
\(607\) −8.35987 4.82657i −0.339316 0.195904i 0.320653 0.947197i \(-0.396098\pi\)
−0.659970 + 0.751292i \(0.729431\pi\)
\(608\) 5.91397 0.239843
\(609\) 0 0
\(610\) 0 0
\(611\) 27.4285 + 15.8359i 1.10964 + 0.640650i
\(612\) 0 0
\(613\) 4.52794 + 7.84262i 0.182882 + 0.316761i 0.942861 0.333187i \(-0.108124\pi\)
−0.759979 + 0.649948i \(0.774791\pi\)
\(614\) 9.28370 16.0798i 0.374660 0.648930i
\(615\) 0 0
\(616\) −4.56742 4.39937i −0.184026 0.177256i
\(617\) 25.3122i 1.01903i −0.860462 0.509515i \(-0.829825\pi\)
0.860462 0.509515i \(-0.170175\pi\)
\(618\) 0 0
\(619\) −33.6634 + 19.4356i −1.35304 + 0.781181i −0.988675 0.150073i \(-0.952049\pi\)
−0.364370 + 0.931254i \(0.618716\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.4366i 0.498663i
\(623\) 8.44516 34.0581i 0.338348 1.36451i
\(624\) 0 0
\(625\) 0 0
\(626\) 6.20675 + 10.7504i 0.248072 + 0.429673i
\(627\) 0 0
\(628\) −5.26801 3.04149i −0.210216 0.121369i
\(629\) −6.15177 −0.245287
\(630\) 0 0
\(631\) −23.8670 −0.950129 −0.475065 0.879951i \(-0.657575\pi\)
−0.475065 + 0.879951i \(0.657575\pi\)
\(632\) 14.5214 + 8.38392i 0.577629 + 0.333494i
\(633\) 0 0
\(634\) −12.3232 21.3444i −0.489417 0.847696i
\(635\) 0 0
\(636\) 0 0
\(637\) −21.1455 33.6472i −0.837816 1.33315i
\(638\) 11.7187i 0.463949i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.2066 17.4398i 1.19309 0.688830i 0.234083 0.972217i \(-0.424791\pi\)
0.959006 + 0.283386i \(0.0914578\pi\)
\(642\) 0 0
\(643\) 6.25944i 0.246848i −0.992354 0.123424i \(-0.960612\pi\)
0.992354 0.123424i \(-0.0393875\pi\)
\(644\) −1.36380 4.73317i −0.0537413 0.186513i
\(645\) 0 0
\(646\) −6.12664 + 10.6117i −0.241050 + 0.417510i
\(647\) −18.7511 32.4778i −0.737181 1.27683i −0.953760 0.300570i \(-0.902823\pi\)
0.216579 0.976265i \(-0.430510\pi\)
\(648\) 0 0
\(649\) −4.16451 2.40438i −0.163471 0.0943801i
\(650\) 0 0
\(651\) 0 0
\(652\) 0.937339 0.0367090
\(653\) −22.9077 13.2258i −0.896449 0.517565i −0.0204023 0.999792i \(-0.506495\pi\)
−0.876046 + 0.482227i \(0.839828\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.52214 6.10053i 0.137516 0.238186i
\(657\) 0 0
\(658\) −10.2396 + 10.6307i −0.399182 + 0.414430i
\(659\) 19.9524i 0.777234i −0.921399 0.388617i \(-0.872953\pi\)
0.921399 0.388617i \(-0.127047\pi\)
\(660\) 0 0
\(661\) 4.71203 2.72049i 0.183277 0.105815i −0.405555 0.914071i \(-0.632922\pi\)
0.588831 + 0.808256i \(0.299588\pi\)
\(662\) −21.7871 + 12.5788i −0.846779 + 0.488888i
\(663\) 0 0
\(664\) 13.6122i 0.528257i
\(665\) 0 0
\(666\) 0 0
\(667\) 4.55118 7.88287i 0.176222 0.305226i
\(668\) 8.60505 + 14.9044i 0.332939 + 0.576668i
\(669\) 0 0
\(670\) 0 0
\(671\) −29.8528 −1.15245
\(672\) 0 0
\(673\) −8.50635 −0.327896 −0.163948 0.986469i \(-0.552423\pi\)
−0.163948 + 0.986469i \(0.552423\pi\)
\(674\) 12.4893 + 7.21070i 0.481070 + 0.277746i
\(675\) 0 0
\(676\) −9.61494 16.6536i −0.369805 0.640522i
\(677\) 2.54320 4.40495i 0.0977430 0.169296i −0.813007 0.582254i \(-0.802171\pi\)
0.910750 + 0.412958i \(0.135504\pi\)
\(678\) 0 0
\(679\) −9.39552 32.6079i −0.360567 1.25138i
\(680\) 0 0
\(681\) 0 0
\(682\) −9.39601 + 5.42479i −0.359792 + 0.207726i
\(683\) 18.1991 10.5073i 0.696370 0.402050i −0.109624 0.993973i \(-0.534965\pi\)
0.805994 + 0.591924i \(0.201631\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.5916 5.79094i 0.671651 0.221099i
\(687\) 0 0
\(688\) −4.27978 + 7.41279i −0.163165 + 0.282610i
\(689\) 11.8958 + 20.6041i 0.453193 + 0.784953i
\(690\) 0 0
\(691\) −21.5723 12.4548i −0.820649 0.473802i 0.0299912 0.999550i \(-0.490452\pi\)
−0.850640 + 0.525748i \(0.823785\pi\)
\(692\) −1.97623 −0.0751249
\(693\) 0 0
\(694\) −14.4866 −0.549903
\(695\) 0 0
\(696\) 0 0
\(697\) 7.29761 + 12.6398i 0.276417 + 0.478767i
\(698\) 1.06241 1.84015i 0.0402129 0.0696508i
\(699\) 0 0
\(700\) 0 0
\(701\) 22.5321i 0.851025i 0.904953 + 0.425512i \(0.139906\pi\)
−0.904953 + 0.425512i \(0.860094\pi\)
\(702\) 0 0
\(703\) 15.2067 8.77961i 0.573532 0.331129i
\(704\) 2.07577 1.19845i 0.0782336 0.0451682i
\(705\) 0 0
\(706\) 14.5267i 0.546719i
\(707\) 16.9813 + 16.3565i 0.638647 + 0.615150i
\(708\) 0 0
\(709\) 12.4504 21.5648i 0.467586 0.809882i −0.531728 0.846915i \(-0.678457\pi\)
0.999314 + 0.0370327i \(0.0117906\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 11.4857 + 6.63129i 0.430446 + 0.248518i
\(713\) −8.42726 −0.315603
\(714\) 0 0
\(715\) 0 0
\(716\) −0.768461 0.443671i −0.0287187 0.0165808i
\(717\) 0 0
\(718\) −1.97485 3.42054i −0.0737007 0.127653i
\(719\) −19.9241 + 34.5096i −0.743045 + 1.28699i 0.208057 + 0.978117i \(0.433286\pi\)
−0.951103 + 0.308875i \(0.900047\pi\)
\(720\) 0 0
\(721\) −27.7850 6.88967i −1.03477 0.256585i
\(722\) 15.9750i 0.594527i
\(723\) 0 0
\(724\) −4.24329 + 2.44986i −0.157701 + 0.0910485i
\(725\) 0 0
\(726\) 0 0
\(727\) 0.124004i 0.00459906i −0.999997 0.00229953i \(-0.999268\pi\)
0.999997 0.00229953i \(-0.000731964\pi\)
\(728\) 14.4331 4.15870i 0.534926 0.154132i
\(729\) 0 0
\(730\) 0 0
\(731\) −8.86738 15.3587i −0.327972 0.568064i
\(732\) 0 0
\(733\) 24.7231 + 14.2739i 0.913168 + 0.527218i 0.881449 0.472279i \(-0.156569\pi\)
0.0317189 + 0.999497i \(0.489902\pi\)
\(734\) 27.6927 1.02216
\(735\) 0 0
\(736\) 1.86175 0.0686252
\(737\) 15.8220 + 9.13483i 0.582811 + 0.336486i
\(738\) 0 0
\(739\) −9.51807 16.4858i −0.350128 0.606439i 0.636144 0.771571i \(-0.280529\pi\)
−0.986272 + 0.165131i \(0.947195\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −10.6543 + 3.06989i −0.391131 + 0.112699i
\(743\) 17.0800i 0.626605i 0.949653 + 0.313303i \(0.101435\pi\)
−0.949653 + 0.313303i \(0.898565\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.5047 6.06491i 0.384605 0.222052i
\(747\) 0 0
\(748\) 4.96618i 0.181582i
\(749\) 14.3218 + 3.55129i 0.523308 + 0.129761i
\(750\) 0 0
\(751\) 8.44463 14.6265i 0.308149 0.533730i −0.669809 0.742534i \(-0.733624\pi\)
0.977957 + 0.208804i \(0.0669572\pi\)
\(752\) −2.78941 4.83140i −0.101719 0.176183i
\(753\) 0 0
\(754\) 24.0376 + 13.8781i 0.875399 + 0.505412i
\(755\) 0 0
\(756\) 0 0
\(757\) −33.4057 −1.21415 −0.607076 0.794644i \(-0.707658\pi\)
−0.607076 + 0.794644i \(0.707658\pi\)
\(758\) −16.1792 9.34106i −0.587655 0.339282i
\(759\) 0 0
\(760\) 0 0
\(761\) 16.3074 28.2453i 0.591143 1.02389i −0.402935 0.915228i \(-0.632010\pi\)
0.994079 0.108662i \(-0.0346566\pi\)
\(762\) 0 0
\(763\) −3.81111 3.67089i −0.137971 0.132895i
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 9.55162 5.51463i 0.345114 0.199252i
\(767\) 9.86379 5.69486i 0.356161 0.205630i
\(768\) 0 0
\(769\) 22.4396i 0.809192i 0.914495 + 0.404596i \(0.132588\pi\)
−0.914495 + 0.404596i \(0.867412\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.74828 16.8845i 0.350848 0.607687i
\(773\) −19.9165 34.4965i −0.716348 1.24075i −0.962437 0.271505i \(-0.912479\pi\)
0.246089 0.969247i \(-0.420855\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.8260 0.460428
\(777\) 0 0
\(778\) −34.3710 −1.23226
\(779\) −36.0783 20.8298i −1.29264 0.746306i
\(780\) 0 0
\(781\) −10.9553 18.9752i −0.392012 0.678985i
\(782\) −1.92871 + 3.34062i −0.0689704 + 0.119460i
\(783\) 0 0
\(784\) 0.262276 + 6.99508i 0.00936701 + 0.249824i
\(785\) 0 0
\(786\) 0 0
\(787\) −14.4912 + 8.36649i −0.516555 + 0.298233i −0.735524 0.677499i \(-0.763064\pi\)
0.218969 + 0.975732i \(0.429731\pi\)
\(788\) −23.5191 + 13.5788i −0.837835 + 0.483724i
\(789\) 0 0
\(790\) 0 0
\(791\) 10.6239 + 36.8711i 0.377743 + 1.31099i
\(792\) 0 0
\(793\) 35.3537 61.2344i 1.25545 2.17450i
\(794\) −15.5054 26.8561i −0.550264 0.953086i
\(795\) 0 0
\(796\) 3.00000 + 1.73205i 0.106332 + 0.0613909i
\(797\) −2.68812 −0.0952182 −0.0476091 0.998866i \(-0.515160\pi\)
−0.0476091 + 0.998866i \(0.515160\pi\)
\(798\) 0 0
\(799\) 11.5589 0.408924
\(800\) 0 0
\(801\) 0 0
\(802\) 4.73872 + 8.20771i 0.167330 + 0.289824i
\(803\) 1.29714 2.24670i 0.0457749 0.0792845i
\(804\) 0 0
\(805\) 0 0
\(806\) 25.6976i 0.905161i
\(807\) 0 0
\(808\) −7.71756 + 4.45573i −0.271503 + 0.156752i
\(809\) 27.2213 15.7162i 0.957051 0.552554i 0.0617867 0.998089i \(-0.480320\pi\)
0.895264 + 0.445536i \(0.146987\pi\)
\(810\) 0 0
\(811\) 49.3830i 1.73407i −0.498246 0.867036i \(-0.666022\pi\)
0.498246 0.867036i \(-0.333978\pi\)
\(812\) −8.97374 + 9.31651i −0.314916 + 0.326945i
\(813\) 0 0
\(814\) 3.55832 6.16320i 0.124719 0.216020i
\(815\) 0 0
\(816\) 0 0
\(817\) 43.8390 + 25.3105i 1.53373 + 0.885501i
\(818\) 9.47784 0.331385
\(819\) 0 0
\(820\) 0 0
\(821\) −38.7342 22.3632i −1.35183 0.780480i −0.363325 0.931662i \(-0.618359\pi\)
−0.988506 + 0.151182i \(0.951692\pi\)
\(822\) 0 0
\(823\) −6.57071 11.3808i −0.229041 0.396710i 0.728483 0.685063i \(-0.240226\pi\)
−0.957524 + 0.288353i \(0.906892\pi\)
\(824\) 5.40989 9.37021i 0.188463 0.326427i
\(825\) 0 0
\(826\) 1.46965 + 5.10052i 0.0511355 + 0.177470i
\(827\) 47.4125i 1.64869i −0.566086 0.824346i \(-0.691543\pi\)
0.566086 0.824346i \(-0.308457\pi\)
\(828\) 0 0
\(829\) −7.11516 + 4.10794i −0.247120 + 0.142675i −0.618445 0.785828i \(-0.712237\pi\)
0.371325 + 0.928503i \(0.378904\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.67714i 0.196819i
\(833\) −12.8233 6.77603i −0.444300 0.234775i
\(834\) 0 0
\(835\) 0 0
\(836\) −7.08758 12.2760i −0.245129 0.424576i
\(837\) 0 0
\(838\) 2.20356 + 1.27223i 0.0761208 + 0.0439483i
\(839\) −17.6943 −0.610875 −0.305437 0.952212i \(-0.598803\pi\)
−0.305437 + 0.952212i \(0.598803\pi\)
\(840\) 0 0
\(841\) 5.09640 0.175738
\(842\) 4.40437 + 2.54286i 0.151785 + 0.0876328i
\(843\) 0 0
\(844\) −2.03035 3.51668i −0.0698877 0.121049i
\(845\) 0 0
\(846\) 0 0
\(847\) 3.34613 13.4945i 0.114975 0.463675i
\(848\) 4.19077i 0.143912i
\(849\) 0 0
\(850\) 0 0
\(851\) 4.78717 2.76388i 0.164102 0.0947444i
\(852\) 0 0
\(853\) 40.1129i 1.37344i 0.726922 + 0.686720i \(0.240950\pi\)
−0.726922 + 0.686720i \(0.759050\pi\)
\(854\) 23.7333 + 22.8601i 0.812135 + 0.782255i
\(855\) 0 0
\(856\) −2.78854 + 4.82989i −0.0953102 + 0.165082i
\(857\) −22.0654 38.2184i −0.753740 1.30552i −0.945998 0.324171i \(-0.894915\pi\)
0.192259 0.981344i \(-0.438419\pi\)
\(858\) 0 0
\(859\) 12.4339 + 7.17873i 0.424240 + 0.244935i 0.696890 0.717178i \(-0.254567\pi\)
−0.272650 + 0.962113i \(0.587900\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −13.2980 −0.452933
\(863\) −11.0876 6.40143i −0.377426 0.217907i 0.299272 0.954168i \(-0.403256\pi\)
−0.676698 + 0.736261i \(0.736590\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −6.39477 + 11.0761i −0.217303 + 0.376380i
\(867\) 0 0
\(868\) 11.6240 + 2.88233i 0.394544 + 0.0978326i
\(869\) 40.1908i 1.36338i
\(870\) 0 0
\(871\) −37.4750 + 21.6362i −1.26979 + 0.733114i
\(872\) 1.73205 1.00000i 0.0586546 0.0338643i
\(873\) 0 0
\(874\) 11.0103i 0.372431i
\(875\) 0 0
\(876\) 0 0
\(877\) 29.1360 50.4650i 0.983853 1.70408i 0.336925 0.941532i \(-0.390613\pi\)
0.646928 0.762551i \(-0.276053\pi\)
\(878\) 0.186606 + 0.323211i 0.00629765 + 0.0109079i
\(879\) 0 0
\(880\) 0 0
\(881\) −1.80667 −0.0608682 −0.0304341 0.999537i \(-0.509689\pi\)
−0.0304341 + 0.999537i \(0.509689\pi\)
\(882\) 0 0
\(883\) −13.8997 −0.467761 −0.233881 0.972265i \(-0.575142\pi\)
−0.233881 + 0.972265i \(0.575142\pi\)
\(884\) −10.1867 5.88130i −0.342616 0.197809i
\(885\) 0 0
\(886\) −12.3165 21.3328i −0.413781 0.716690i
\(887\) −13.2459 + 22.9426i −0.444754 + 0.770337i −0.998035 0.0626581i \(-0.980042\pi\)
0.553281 + 0.832995i \(0.313376\pi\)
\(888\) 0 0
\(889\) 48.9301 14.0985i 1.64106 0.472850i
\(890\) 0 0
\(891\) 0 0
\(892\) −13.9851 + 8.07432i −0.468257 + 0.270348i
\(893\) −28.5727 + 16.4965i −0.956150 + 0.552033i
\(894\) 0 0
\(895\) 0 0
\(896\) −2.56798 0.636766i −0.0857902 0.0212728i
\(897\) 0 0
\(898\) 20.3111 35.1799i 0.677792 1.17397i
\(899\) 11.0654 + 19.1658i 0.369050 + 0.639214i
\(900\) 0 0
\(901\) 7.51966 + 4.34148i 0.250516 + 0.144636i
\(902\) −16.8844 −0.562189
\(903\) 0 0
\(904\) −14.5030 −0.482361
\(905\) 0 0
\(906\) 0 0
\(907\) 18.3650 + 31.8091i 0.609800 + 1.05620i 0.991273 + 0.131824i \(0.0420834\pi\)
−0.381474 + 0.924380i \(0.624583\pi\)
\(908\) −0.839901 + 1.45475i −0.0278731 + 0.0482776i
\(909\) 0 0
\(910\) 0 0
\(911\) 39.8018i 1.31869i 0.751839 + 0.659347i \(0.229167\pi\)
−0.751839 + 0.659347i \(0.770833\pi\)
\(912\) 0 0
\(913\) 28.2559 16.3135i 0.935134 0.539900i
\(914\) 0.857843 0.495276i 0.0283749 0.0163823i
\(915\) 0 0
\(916\) 12.6027i 0.416406i
\(917\) −2.99242 + 12.0680i −0.0988184 + 0.398520i
\(918\) 0 0
\(919\) −30.1692 + 52.2545i −0.995189 + 1.72372i −0.412747 + 0.910846i \(0.635430\pi\)
−0.582442 + 0.812872i \(0.697903\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −17.9611 10.3699i −0.591518 0.341513i
\(923\) 51.8962 1.70818
\(924\) 0 0
\(925\) 0 0
\(926\) 1.26804 + 0.732105i 0.0416705 + 0.0240585i
\(927\) 0 0
\(928\) −2.44457 4.23411i −0.0802468 0.138992i
\(929\) 22.1749 38.4080i 0.727533 1.26012i −0.230389 0.973099i \(-0.574000\pi\)
0.957923 0.287026i \(-0.0926667\pi\)
\(930\) 0 0
\(931\) 41.3687 1.55109i 1.35580 0.0508350i
\(932\) 14.5164i 0.475499i
\(933\) 0 0
\(934\) 31.4090 18.1340i 1.02774 0.593363i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.54073i 0.0830021i 0.999138 + 0.0415010i \(0.0132140\pi\)
−0.999138 + 0.0415010i \(0.986786\pi\)
\(938\) −5.58355 19.3781i −0.182309 0.632718i
\(939\) 0 0
\(940\) 0 0
\(941\) 16.9488 + 29.3563i 0.552516 + 0.956987i 0.998092 + 0.0617423i \(0.0196657\pi\)
−0.445576 + 0.895244i \(0.647001\pi\)
\(942\) 0 0
\(943\) −11.3577 6.55736i −0.369857 0.213537i
\(944\) −2.00624 −0.0652977
\(945\) 0 0
\(946\) 20.5164 0.667045
\(947\) 42.8759 + 24.7544i 1.39328 + 0.804411i 0.993677 0.112277i \(-0.0358145\pi\)
0.399604 + 0.916688i \(0.369148\pi\)
\(948\) 0 0
\(949\) 3.07231 + 5.32140i 0.0997315 + 0.172740i
\(950\) 0 0
\(951\) 0 0
\(952\) 3.80290 3.94816i 0.123253 0.127961i
\(953\) 17.0625i 0.552709i 0.961056 + 0.276355i \(0.0891264\pi\)
−0.961056 + 0.276355i \(0.910874\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −0.179344 + 0.103545i −0.00580041 + 0.00334887i
\(957\) 0 0
\(958\) 10.3258i 0.333610i
\(959\) 42.0032 43.6076i 1.35636 1.40816i
\(960\) 0 0
\(961\) −5.25533 + 9.10250i −0.169527 + 0.293629i
\(962\) 8.42802 + 14.5978i 0.271730 + 0.470651i
\(963\) 0 0
\(964\) 9.04172 + 5.22024i 0.291214 + 0.168133i
\(965\) 0 0
\(966\) 0 0
\(967\) 49.6639 1.59708 0.798541 0.601940i \(-0.205606\pi\)
0.798541 + 0.601940i \(0.205606\pi\)
\(968\) 4.55087 + 2.62745i 0.146270 + 0.0844493i
\(969\) 0 0
\(970\) 0 0
\(971\) −14.2930 + 24.7561i −0.458683 + 0.794462i −0.998892 0.0470689i \(-0.985012\pi\)
0.540209 + 0.841531i \(0.318345\pi\)
\(972\) 0 0
\(973\) 6.69436 + 23.2333i 0.214611 + 0.744825i
\(974\) 20.3291i 0.651386i
\(975\) 0 0
\(976\) −10.7862 + 6.22739i −0.345256 + 0.199334i
\(977\) −24.0067 + 13.8603i −0.768042 + 0.443429i −0.832176 0.554512i \(-0.812905\pi\)
0.0641335 + 0.997941i \(0.479572\pi\)
\(978\) 0 0
\(979\) 31.7890i 1.01598i
\(980\) 0 0
\(981\) 0 0
\(982\) −17.3017 + 29.9674i −0.552119 + 0.956298i
\(983\) −21.4134 37.0891i −0.682982 1.18296i −0.974067 0.226262i \(-0.927350\pi\)
0.291085 0.956697i \(-0.405984\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10.1299 0.322602
\(987\) 0 0
\(988\) 33.5744 1.06814
\(989\) 13.8008 + 7.96789i 0.438840 + 0.253364i
\(990\) 0 0
\(991\) −21.3723 37.0179i −0.678914 1.17591i −0.975308 0.220848i \(-0.929118\pi\)
0.296394 0.955066i \(-0.404216\pi\)
\(992\) −2.26326 + 3.92008i −0.0718585 + 0.124463i
\(993\) 0 0
\(994\) −5.82084 + 23.4746i −0.184626 + 0.744568i
\(995\) 0 0
\(996\) 0 0
\(997\) −28.4679 + 16.4359i −0.901587 + 0.520531i −0.877715 0.479184i \(-0.840933\pi\)
−0.0238722 + 0.999715i \(0.507599\pi\)
\(998\) −2.55555 + 1.47545i −0.0808945 + 0.0467045i
\(999\) 0 0
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 3150.2.bf.f.1601.1 32
3.2 odd 2 inner 3150.2.bf.f.1601.11 32
5.2 odd 4 630.2.bo.b.89.2 yes 16
5.3 odd 4 630.2.bo.a.89.4 16
5.4 even 2 inner 3150.2.bf.f.1601.12 32
7.3 odd 6 inner 3150.2.bf.f.1151.11 32
15.2 even 4 630.2.bo.a.89.7 yes 16
15.8 even 4 630.2.bo.b.89.5 yes 16
15.14 odd 2 inner 3150.2.bf.f.1601.2 32
21.17 even 6 inner 3150.2.bf.f.1151.1 32
35.2 odd 12 4410.2.d.a.4409.16 16
35.3 even 12 630.2.bo.a.269.7 yes 16
35.12 even 12 4410.2.d.a.4409.1 16
35.17 even 12 630.2.bo.b.269.5 yes 16
35.23 odd 12 4410.2.d.b.4409.15 16
35.24 odd 6 inner 3150.2.bf.f.1151.2 32
35.33 even 12 4410.2.d.b.4409.2 16
105.2 even 12 4410.2.d.b.4409.1 16
105.17 odd 12 630.2.bo.a.269.4 yes 16
105.23 even 12 4410.2.d.a.4409.2 16
105.38 odd 12 630.2.bo.b.269.2 yes 16
105.47 odd 12 4410.2.d.b.4409.16 16
105.59 even 6 inner 3150.2.bf.f.1151.12 32
105.68 odd 12 4410.2.d.a.4409.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.4 16 5.3 odd 4
630.2.bo.a.89.7 yes 16 15.2 even 4
630.2.bo.a.269.4 yes 16 105.17 odd 12
630.2.bo.a.269.7 yes 16 35.3 even 12
630.2.bo.b.89.2 yes 16 5.2 odd 4
630.2.bo.b.89.5 yes 16 15.8 even 4
630.2.bo.b.269.2 yes 16 105.38 odd 12
630.2.bo.b.269.5 yes 16 35.17 even 12
3150.2.bf.f.1151.1 32 21.17 even 6 inner
3150.2.bf.f.1151.2 32 35.24 odd 6 inner
3150.2.bf.f.1151.11 32 7.3 odd 6 inner
3150.2.bf.f.1151.12 32 105.59 even 6 inner
3150.2.bf.f.1601.1 32 1.1 even 1 trivial
3150.2.bf.f.1601.2 32 15.14 odd 2 inner
3150.2.bf.f.1601.11 32 3.2 odd 2 inner
3150.2.bf.f.1601.12 32 5.4 even 2 inner
4410.2.d.a.4409.1 16 35.12 even 12
4410.2.d.a.4409.2 16 105.23 even 12
4410.2.d.a.4409.15 16 105.68 odd 12
4410.2.d.a.4409.16 16 35.2 odd 12
4410.2.d.b.4409.1 16 105.2 even 12
4410.2.d.b.4409.2 16 35.33 even 12
4410.2.d.b.4409.15 16 35.23 odd 12
4410.2.d.b.4409.16 16 105.47 odd 12