Properties

Label 2100.2.bi.k.101.4
Level $2100$
Weight $2$
Character 2100.101
Analytic conductor $16.769$
Analytic rank $0$
Dimension $10$
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] = [2100,2,Mod(101,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2100.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.29471584693248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 101.4
Root \(1.72689 + 0.133595i\) of defining polynomial
Character \(\chi\) \(=\) 2100.101
Dual form 2100.2.bi.k.1601.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.747749 - 1.56233i) q^{3} +(0.456468 + 2.60608i) q^{7} +(-1.88174 - 2.33646i) q^{9} +O(q^{10})\) \(q+(0.747749 - 1.56233i) q^{3} +(0.456468 + 2.60608i) q^{7} +(-1.88174 - 2.33646i) q^{9} +(-0.698384 - 0.403212i) q^{11} +3.86649i q^{13} +(-1.05989 + 1.83578i) q^{17} +(2.70603 - 1.56233i) q^{19} +(4.41287 + 1.23554i) q^{21} +(4.21989 - 2.43635i) q^{23} +(-5.05739 + 1.19282i) q^{27} -6.67701i q^{29} +(5.65010 + 3.26209i) q^{31} +(-1.15217 + 0.789604i) q^{33} +(3.89991 + 6.75485i) q^{37} +(6.04074 + 2.89117i) q^{39} +8.44841 q^{41} +0.819076 q^{43} +(1.38776 + 2.40368i) q^{47} +(-6.58327 + 2.37918i) q^{49} +(2.07556 + 3.02860i) q^{51} +(11.5565 + 6.67216i) q^{53} +(-0.417438 - 5.39594i) q^{57} +(-2.86351 + 4.95974i) q^{59} +(1.79397 - 1.03575i) q^{61} +(5.23004 - 5.97049i) q^{63} +(5.45110 - 9.44157i) q^{67} +(-0.650969 - 8.41464i) q^{69} -10.3850i q^{71} +(-2.67203 - 1.54270i) q^{73} +(0.732012 - 2.00410i) q^{77} +(6.76342 + 11.7146i) q^{79} +(-1.91809 + 8.79323i) q^{81} +12.8948 q^{83} +(-10.4317 - 4.99273i) q^{87} +(1.60530 + 2.78046i) q^{89} +(-10.0764 + 1.76493i) q^{91} +(9.32131 - 6.38809i) q^{93} -1.01388i q^{97} +(0.372090 + 2.39049i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 5 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 5 q^{7} - 3 q^{9} - 6 q^{11} - 6 q^{17} + 3 q^{19} + 10 q^{21} - 24 q^{23} - 8 q^{27} + 15 q^{31} + 20 q^{33} + q^{37} + 15 q^{39} - 8 q^{41} + 26 q^{43} - 14 q^{47} - 13 q^{49} - 44 q^{51} + 24 q^{53} - 18 q^{57} + 42 q^{61} + q^{63} - 7 q^{67} - 14 q^{69} + 3 q^{73} + 26 q^{77} + q^{79} + 41 q^{81} + 8 q^{83} + 26 q^{87} + 28 q^{89} - 11 q^{91} + 47 q^{93} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0.747749 1.56233i 0.431713 0.902011i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.456468 + 2.60608i 0.172529 + 0.985004i
\(8\) 0 0
\(9\) −1.88174 2.33646i −0.627248 0.778820i
\(10\) 0 0
\(11\) −0.698384 0.403212i −0.210571 0.121573i 0.391006 0.920388i \(-0.372127\pi\)
−0.601577 + 0.798815i \(0.705461\pi\)
\(12\) 0 0
\(13\) 3.86649i 1.07237i 0.844100 + 0.536186i \(0.180136\pi\)
−0.844100 + 0.536186i \(0.819864\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.05989 + 1.83578i −0.257061 + 0.445242i −0.965453 0.260577i \(-0.916087\pi\)
0.708393 + 0.705819i \(0.249421\pi\)
\(18\) 0 0
\(19\) 2.70603 1.56233i 0.620807 0.358423i −0.156376 0.987698i \(-0.549981\pi\)
0.777183 + 0.629275i \(0.216648\pi\)
\(20\) 0 0
\(21\) 4.41287 + 1.23554i 0.962968 + 0.269617i
\(22\) 0 0
\(23\) 4.21989 2.43635i 0.879908 0.508015i 0.00927994 0.999957i \(-0.497046\pi\)
0.870628 + 0.491942i \(0.163713\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.05739 + 1.19282i −0.973295 + 0.229558i
\(28\) 0 0
\(29\) 6.67701i 1.23989i −0.784645 0.619945i \(-0.787155\pi\)
0.784645 0.619945i \(-0.212845\pi\)
\(30\) 0 0
\(31\) 5.65010 + 3.26209i 1.01479 + 0.585888i 0.912590 0.408876i \(-0.134079\pi\)
0.102198 + 0.994764i \(0.467413\pi\)
\(32\) 0 0
\(33\) −1.15217 + 0.789604i −0.200566 + 0.137452i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.89991 + 6.75485i 0.641142 + 1.11049i 0.985178 + 0.171534i \(0.0548724\pi\)
−0.344036 + 0.938956i \(0.611794\pi\)
\(38\) 0 0
\(39\) 6.04074 + 2.89117i 0.967292 + 0.462957i
\(40\) 0 0
\(41\) 8.44841 1.31942 0.659710 0.751520i \(-0.270679\pi\)
0.659710 + 0.751520i \(0.270679\pi\)
\(42\) 0 0
\(43\) 0.819076 0.124908 0.0624540 0.998048i \(-0.480107\pi\)
0.0624540 + 0.998048i \(0.480107\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.38776 + 2.40368i 0.202426 + 0.350612i 0.949310 0.314343i \(-0.101784\pi\)
−0.746883 + 0.664955i \(0.768451\pi\)
\(48\) 0 0
\(49\) −6.58327 + 2.37918i −0.940468 + 0.339883i
\(50\) 0 0
\(51\) 2.07556 + 3.02860i 0.290637 + 0.424088i
\(52\) 0 0
\(53\) 11.5565 + 6.67216i 1.58741 + 0.916492i 0.993732 + 0.111789i \(0.0356581\pi\)
0.593678 + 0.804703i \(0.297675\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.417438 5.39594i −0.0552910 0.714710i
\(58\) 0 0
\(59\) −2.86351 + 4.95974i −0.372797 + 0.645703i −0.989995 0.141105i \(-0.954935\pi\)
0.617198 + 0.786808i \(0.288268\pi\)
\(60\) 0 0
\(61\) 1.79397 1.03575i 0.229694 0.132614i −0.380737 0.924683i \(-0.624330\pi\)
0.610431 + 0.792070i \(0.290996\pi\)
\(62\) 0 0
\(63\) 5.23004 5.97049i 0.658923 0.752211i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.45110 9.44157i 0.665957 1.15347i −0.313067 0.949731i \(-0.601357\pi\)
0.979025 0.203741i \(-0.0653101\pi\)
\(68\) 0 0
\(69\) −0.650969 8.41464i −0.0783674 1.01300i
\(70\) 0 0
\(71\) 10.3850i 1.23248i −0.787559 0.616239i \(-0.788656\pi\)
0.787559 0.616239i \(-0.211344\pi\)
\(72\) 0 0
\(73\) −2.67203 1.54270i −0.312737 0.180559i 0.335413 0.942071i \(-0.391124\pi\)
−0.648151 + 0.761512i \(0.724457\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.732012 2.00410i 0.0834205 0.228388i
\(78\) 0 0
\(79\) 6.76342 + 11.7146i 0.760944 + 1.31799i 0.942364 + 0.334589i \(0.108597\pi\)
−0.181420 + 0.983406i \(0.558069\pi\)
\(80\) 0 0
\(81\) −1.91809 + 8.79323i −0.213121 + 0.977026i
\(82\) 0 0
\(83\) 12.8948 1.41539 0.707694 0.706519i \(-0.249735\pi\)
0.707694 + 0.706519i \(0.249735\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −10.4317 4.99273i −1.11839 0.535277i
\(88\) 0 0
\(89\) 1.60530 + 2.78046i 0.170161 + 0.294728i 0.938476 0.345344i \(-0.112238\pi\)
−0.768315 + 0.640072i \(0.778904\pi\)
\(90\) 0 0
\(91\) −10.0764 + 1.76493i −1.05629 + 0.185015i
\(92\) 0 0
\(93\) 9.32131 6.38809i 0.966574 0.662414i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.01388i 0.102944i −0.998674 0.0514721i \(-0.983609\pi\)
0.998674 0.0514721i \(-0.0163913\pi\)
\(98\) 0 0
\(99\) 0.372090 + 2.39049i 0.0373965 + 0.240253i
\(100\) 0 0
\(101\) 4.27796 7.40965i 0.425673 0.737288i −0.570810 0.821082i \(-0.693371\pi\)
0.996483 + 0.0837947i \(0.0267040\pi\)
\(102\) 0 0
\(103\) −8.46835 + 4.88921i −0.834411 + 0.481748i −0.855361 0.518033i \(-0.826664\pi\)
0.0209492 + 0.999781i \(0.493331\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.8336 + 9.14150i −1.53069 + 0.883743i −0.531357 + 0.847148i \(0.678318\pi\)
−0.999330 + 0.0365946i \(0.988349\pi\)
\(108\) 0 0
\(109\) −8.74840 + 15.1527i −0.837945 + 1.45136i 0.0536658 + 0.998559i \(0.482909\pi\)
−0.891610 + 0.452804i \(0.850424\pi\)
\(110\) 0 0
\(111\) 13.4695 1.04202i 1.27846 0.0989039i
\(112\) 0 0
\(113\) 6.96021i 0.654761i −0.944893 0.327381i \(-0.893834\pi\)
0.944893 0.327381i \(-0.106166\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.03391 7.27575i 0.835185 0.672643i
\(118\) 0 0
\(119\) −5.26799 1.92417i −0.482916 0.176389i
\(120\) 0 0
\(121\) −5.17484 8.96309i −0.470440 0.814826i
\(122\) 0 0
\(123\) 6.31729 13.1992i 0.569611 1.19013i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.77139 −0.689600 −0.344800 0.938676i \(-0.612053\pi\)
−0.344800 + 0.938676i \(0.612053\pi\)
\(128\) 0 0
\(129\) 0.612463 1.27967i 0.0539244 0.112668i
\(130\) 0 0
\(131\) 5.93761 + 10.2842i 0.518772 + 0.898539i 0.999762 + 0.0218131i \(0.00694387\pi\)
−0.480990 + 0.876726i \(0.659723\pi\)
\(132\) 0 0
\(133\) 5.30677 + 6.33898i 0.460155 + 0.549659i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.12894 1.22915i −0.181888 0.105013i 0.406291 0.913744i \(-0.366822\pi\)
−0.588179 + 0.808730i \(0.700155\pi\)
\(138\) 0 0
\(139\) 2.81335i 0.238625i −0.992857 0.119313i \(-0.961931\pi\)
0.992857 0.119313i \(-0.0380691\pi\)
\(140\) 0 0
\(141\) 4.79303 0.370796i 0.403646 0.0312267i
\(142\) 0 0
\(143\) 1.55902 2.70030i 0.130372 0.225810i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.20557 + 12.0643i −0.0994339 + 0.995044i
\(148\) 0 0
\(149\) 12.8234 7.40357i 1.05053 0.606524i 0.127733 0.991809i \(-0.459230\pi\)
0.922798 + 0.385285i \(0.125897\pi\)
\(150\) 0 0
\(151\) 0.427898 0.741141i 0.0348218 0.0603132i −0.848089 0.529853i \(-0.822247\pi\)
0.882911 + 0.469540i \(0.155580\pi\)
\(152\) 0 0
\(153\) 6.28366 0.978080i 0.508004 0.0790731i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −21.1042 12.1845i −1.68430 0.972428i −0.958748 0.284256i \(-0.908254\pi\)
−0.725547 0.688173i \(-0.758413\pi\)
\(158\) 0 0
\(159\) 19.0655 13.0660i 1.51199 1.03620i
\(160\) 0 0
\(161\) 8.27557 + 9.88524i 0.652207 + 0.779066i
\(162\) 0 0
\(163\) 0.789402 + 1.36728i 0.0618307 + 0.107094i 0.895284 0.445496i \(-0.146973\pi\)
−0.833453 + 0.552590i \(0.813639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.9264 1.30980 0.654902 0.755714i \(-0.272710\pi\)
0.654902 + 0.755714i \(0.272710\pi\)
\(168\) 0 0
\(169\) −1.94978 −0.149983
\(170\) 0 0
\(171\) −8.74238 3.38264i −0.668546 0.258677i
\(172\) 0 0
\(173\) −2.33037 4.03633i −0.177175 0.306876i 0.763737 0.645528i \(-0.223363\pi\)
−0.940912 + 0.338652i \(0.890029\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.60756 + 8.18238i 0.421490 + 0.615025i
\(178\) 0 0
\(179\) −3.11674 1.79945i −0.232956 0.134497i 0.378979 0.925405i \(-0.376275\pi\)
−0.611935 + 0.790908i \(0.709609\pi\)
\(180\) 0 0
\(181\) 22.5821i 1.67851i −0.543737 0.839256i \(-0.682991\pi\)
0.543737 0.839256i \(-0.317009\pi\)
\(182\) 0 0
\(183\) −0.276741 3.57725i −0.0204573 0.264438i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.48042 0.854719i 0.108259 0.0625033i
\(188\) 0 0
\(189\) −5.41711 12.6355i −0.394037 0.919095i
\(190\) 0 0
\(191\) 2.92355 1.68791i 0.211541 0.122133i −0.390486 0.920609i \(-0.627693\pi\)
0.602027 + 0.798475i \(0.294360\pi\)
\(192\) 0 0
\(193\) −5.54524 + 9.60463i −0.399155 + 0.691357i −0.993622 0.112763i \(-0.964030\pi\)
0.594467 + 0.804120i \(0.297363\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.2585i 1.15837i −0.815197 0.579184i \(-0.803371\pi\)
0.815197 0.579184i \(-0.196629\pi\)
\(198\) 0 0
\(199\) −1.07964 0.623332i −0.0765338 0.0441868i 0.461245 0.887273i \(-0.347403\pi\)
−0.537779 + 0.843086i \(0.680736\pi\)
\(200\) 0 0
\(201\) −10.6748 15.5763i −0.752942 1.09867i
\(202\) 0 0
\(203\) 17.4008 3.04784i 1.22130 0.213917i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −13.6332 5.27501i −0.947573 0.366639i
\(208\) 0 0
\(209\) −2.51980 −0.174298
\(210\) 0 0
\(211\) −7.65466 −0.526968 −0.263484 0.964664i \(-0.584872\pi\)
−0.263484 + 0.964664i \(0.584872\pi\)
\(212\) 0 0
\(213\) −16.2249 7.76541i −1.11171 0.532077i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.92216 + 16.2136i −0.402022 + 1.10065i
\(218\) 0 0
\(219\) −4.40821 + 3.02104i −0.297879 + 0.204143i
\(220\) 0 0
\(221\) −7.09803 4.09805i −0.477465 0.275665i
\(222\) 0 0
\(223\) 20.9369i 1.40204i 0.713143 + 0.701019i \(0.247271\pi\)
−0.713143 + 0.701019i \(0.752729\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.29103 + 3.96818i −0.152061 + 0.263377i −0.931985 0.362497i \(-0.881924\pi\)
0.779924 + 0.625874i \(0.215258\pi\)
\(228\) 0 0
\(229\) 0.845272 0.488018i 0.0558572 0.0322492i −0.471811 0.881700i \(-0.656400\pi\)
0.527669 + 0.849450i \(0.323066\pi\)
\(230\) 0 0
\(231\) −2.58370 2.64220i −0.169995 0.173844i
\(232\) 0 0
\(233\) −17.2382 + 9.95247i −1.12931 + 0.652008i −0.943762 0.330627i \(-0.892740\pi\)
−0.185550 + 0.982635i \(0.559407\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 23.3594 1.80712i 1.51736 0.117385i
\(238\) 0 0
\(239\) 28.2703i 1.82865i 0.404977 + 0.914327i \(0.367279\pi\)
−0.404977 + 0.914327i \(0.632721\pi\)
\(240\) 0 0
\(241\) 11.0949 + 6.40567i 0.714688 + 0.412625i 0.812794 0.582551i \(-0.197945\pi\)
−0.0981065 + 0.995176i \(0.531279\pi\)
\(242\) 0 0
\(243\) 12.3037 + 9.57181i 0.789281 + 0.614032i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.04074 + 10.4629i 0.384363 + 0.665736i
\(248\) 0 0
\(249\) 9.64207 20.1459i 0.611042 1.27670i
\(250\) 0 0
\(251\) 10.3349 0.652335 0.326167 0.945312i \(-0.394243\pi\)
0.326167 + 0.945312i \(0.394243\pi\)
\(252\) 0 0
\(253\) −3.92947 −0.247044
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.2527 26.4185i −0.951439 1.64794i −0.742315 0.670052i \(-0.766272\pi\)
−0.209124 0.977889i \(-0.567061\pi\)
\(258\) 0 0
\(259\) −15.8235 + 13.2469i −0.983223 + 0.823119i
\(260\) 0 0
\(261\) −15.6006 + 12.5644i −0.965651 + 0.777718i
\(262\) 0 0
\(263\) −17.9054 10.3377i −1.10409 0.637447i −0.166799 0.985991i \(-0.553343\pi\)
−0.937293 + 0.348543i \(0.886676\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.54435 0.428919i 0.339309 0.0262494i
\(268\) 0 0
\(269\) 12.4095 21.4938i 0.756618 1.31050i −0.187948 0.982179i \(-0.560184\pi\)
0.944566 0.328322i \(-0.106483\pi\)
\(270\) 0 0
\(271\) 18.8571 10.8871i 1.14548 0.661346i 0.197702 0.980262i \(-0.436652\pi\)
0.947783 + 0.318916i \(0.103319\pi\)
\(272\) 0 0
\(273\) −4.77720 + 17.0623i −0.289129 + 1.03266i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.61126 13.1831i 0.457316 0.792095i −0.541502 0.840699i \(-0.682144\pi\)
0.998818 + 0.0486048i \(0.0154775\pi\)
\(278\) 0 0
\(279\) −3.01030 19.3396i −0.180222 1.15783i
\(280\) 0 0
\(281\) 17.9488i 1.07073i −0.844619 0.535367i \(-0.820173\pi\)
0.844619 0.535367i \(-0.179827\pi\)
\(282\) 0 0
\(283\) 23.3730 + 13.4944i 1.38938 + 0.802159i 0.993245 0.116033i \(-0.0370178\pi\)
0.396135 + 0.918192i \(0.370351\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.85643 + 22.0172i 0.227638 + 1.29963i
\(288\) 0 0
\(289\) 6.25328 + 10.8310i 0.367840 + 0.637117i
\(290\) 0 0
\(291\) −1.58402 0.758129i −0.0928567 0.0444423i
\(292\) 0 0
\(293\) −14.4450 −0.843887 −0.421944 0.906622i \(-0.638652\pi\)
−0.421944 + 0.906622i \(0.638652\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.01296 + 1.20616i 0.232855 + 0.0699884i
\(298\) 0 0
\(299\) 9.42015 + 16.3162i 0.544782 + 0.943589i
\(300\) 0 0
\(301\) 0.373882 + 2.13457i 0.0215502 + 0.123035i
\(302\) 0 0
\(303\) −8.37747 12.2241i −0.481273 0.702259i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.34543i 0.533372i 0.963783 + 0.266686i \(0.0859287\pi\)
−0.963783 + 0.266686i \(0.914071\pi\)
\(308\) 0 0
\(309\) 1.30635 + 16.8862i 0.0743154 + 0.960625i
\(310\) 0 0
\(311\) −14.6664 + 25.4029i −0.831654 + 1.44047i 0.0650722 + 0.997881i \(0.479272\pi\)
−0.896726 + 0.442586i \(0.854061\pi\)
\(312\) 0 0
\(313\) −9.55728 + 5.51790i −0.540209 + 0.311890i −0.745164 0.666881i \(-0.767629\pi\)
0.204954 + 0.978772i \(0.434295\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.6987 + 8.48631i −0.825563 + 0.476639i −0.852331 0.523003i \(-0.824812\pi\)
0.0267682 + 0.999642i \(0.491478\pi\)
\(318\) 0 0
\(319\) −2.69225 + 4.66312i −0.150737 + 0.261085i
\(320\) 0 0
\(321\) 2.44252 + 31.5728i 0.136328 + 1.76222i
\(322\) 0 0
\(323\) 6.62357i 0.368546i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17.1318 + 24.9983i 0.947393 + 1.38241i
\(328\) 0 0
\(329\) −5.63070 + 4.71382i −0.310430 + 0.259881i
\(330\) 0 0
\(331\) 5.82739 + 10.0933i 0.320302 + 0.554780i 0.980550 0.196268i \(-0.0628822\pi\)
−0.660248 + 0.751048i \(0.729549\pi\)
\(332\) 0 0
\(333\) 8.44380 21.8229i 0.462717 1.19589i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −10.1753 −0.554286 −0.277143 0.960829i \(-0.589388\pi\)
−0.277143 + 0.960829i \(0.589388\pi\)
\(338\) 0 0
\(339\) −10.8741 5.20449i −0.590602 0.282669i
\(340\) 0 0
\(341\) −2.63063 4.55638i −0.142456 0.246742i
\(342\) 0 0
\(343\) −9.20538 16.0705i −0.497044 0.867725i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.38025 + 0.796887i 0.0740956 + 0.0427791i 0.536590 0.843843i \(-0.319712\pi\)
−0.462494 + 0.886622i \(0.653045\pi\)
\(348\) 0 0
\(349\) 15.4812i 0.828690i 0.910120 + 0.414345i \(0.135989\pi\)
−0.910120 + 0.414345i \(0.864011\pi\)
\(350\) 0 0
\(351\) −4.61202 19.5544i −0.246171 1.04374i
\(352\) 0 0
\(353\) −6.40056 + 11.0861i −0.340668 + 0.590054i −0.984557 0.175065i \(-0.943986\pi\)
0.643889 + 0.765119i \(0.277320\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.94533 + 6.79153i −0.367586 + 0.359446i
\(358\) 0 0
\(359\) −5.66697 + 3.27183i −0.299091 + 0.172680i −0.642035 0.766676i \(-0.721909\pi\)
0.342943 + 0.939356i \(0.388576\pi\)
\(360\) 0 0
\(361\) −4.61826 + 7.99906i −0.243066 + 0.421003i
\(362\) 0 0
\(363\) −17.8728 + 1.38266i −0.938077 + 0.0725710i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.0004 + 8.66046i 0.783012 + 0.452072i 0.837497 0.546442i \(-0.184018\pi\)
−0.0544847 + 0.998515i \(0.517352\pi\)
\(368\) 0 0
\(369\) −15.8977 19.7394i −0.827603 1.02759i
\(370\) 0 0
\(371\) −12.1130 + 33.1628i −0.628875 + 1.72173i
\(372\) 0 0
\(373\) −0.497933 0.862445i −0.0257820 0.0446557i 0.852847 0.522162i \(-0.174874\pi\)
−0.878629 + 0.477506i \(0.841541\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.8166 1.32962
\(378\) 0 0
\(379\) −20.2985 −1.04266 −0.521331 0.853355i \(-0.674564\pi\)
−0.521331 + 0.853355i \(0.674564\pi\)
\(380\) 0 0
\(381\) −5.81105 + 12.1415i −0.297709 + 0.622026i
\(382\) 0 0
\(383\) −9.93736 17.2120i −0.507776 0.879493i −0.999959 0.00900178i \(-0.997135\pi\)
0.492184 0.870491i \(-0.336199\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.54129 1.91374i −0.0783482 0.0972808i
\(388\) 0 0
\(389\) 1.56923 + 0.905994i 0.0795630 + 0.0459357i 0.539254 0.842143i \(-0.318706\pi\)
−0.459691 + 0.888079i \(0.652040\pi\)
\(390\) 0 0
\(391\) 10.3291i 0.522363i
\(392\) 0 0
\(393\) 20.5072 1.58647i 1.03445 0.0800268i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 30.1320 17.3967i 1.51228 0.873116i 0.512384 0.858757i \(-0.328763\pi\)
0.999897 0.0143590i \(-0.00457078\pi\)
\(398\) 0 0
\(399\) 13.8717 3.55095i 0.694453 0.177770i
\(400\) 0 0
\(401\) −32.5121 + 18.7709i −1.62358 + 0.937374i −0.637627 + 0.770345i \(0.720084\pi\)
−0.985952 + 0.167029i \(0.946583\pi\)
\(402\) 0 0
\(403\) −12.6128 + 21.8461i −0.628290 + 1.08823i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.28997i 0.311782i
\(408\) 0 0
\(409\) 3.24769 + 1.87505i 0.160588 + 0.0927154i 0.578140 0.815937i \(-0.303779\pi\)
−0.417552 + 0.908653i \(0.637112\pi\)
\(410\) 0 0
\(411\) −3.51225 + 2.40702i −0.173246 + 0.118729i
\(412\) 0 0
\(413\) −14.2326 5.19856i −0.700339 0.255804i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.39538 2.10368i −0.215242 0.103018i
\(418\) 0 0
\(419\) −30.0547 −1.46827 −0.734134 0.679005i \(-0.762411\pi\)
−0.734134 + 0.679005i \(0.762411\pi\)
\(420\) 0 0
\(421\) 20.5447 1.00129 0.500643 0.865654i \(-0.333097\pi\)
0.500643 + 0.865654i \(0.333097\pi\)
\(422\) 0 0
\(423\) 3.00468 7.76556i 0.146093 0.377574i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.51813 + 4.20243i 0.170254 + 0.203370i
\(428\) 0 0
\(429\) −3.05300 4.45484i −0.147400 0.215082i
\(430\) 0 0
\(431\) −31.7400 18.3251i −1.52886 0.882688i −0.999410 0.0343478i \(-0.989065\pi\)
−0.529451 0.848340i \(-0.677602\pi\)
\(432\) 0 0
\(433\) 36.4397i 1.75118i −0.483057 0.875589i \(-0.660474\pi\)
0.483057 0.875589i \(-0.339526\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.61278 13.1857i 0.364168 0.630758i
\(438\) 0 0
\(439\) −7.96135 + 4.59649i −0.379975 + 0.219378i −0.677807 0.735240i \(-0.737070\pi\)
0.297833 + 0.954618i \(0.403736\pi\)
\(440\) 0 0
\(441\) 17.9469 + 10.9045i 0.854614 + 0.519264i
\(442\) 0 0
\(443\) −20.7332 + 11.9703i −0.985063 + 0.568727i −0.903795 0.427966i \(-0.859230\pi\)
−0.0812683 + 0.996692i \(0.525897\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.97816 25.5703i −0.0935636 1.20943i
\(448\) 0 0
\(449\) 24.0126i 1.13322i −0.823985 0.566611i \(-0.808254\pi\)
0.823985 0.566611i \(-0.191746\pi\)
\(450\) 0 0
\(451\) −5.90023 3.40650i −0.277831 0.160406i
\(452\) 0 0
\(453\) −0.837945 1.22270i −0.0393701 0.0574476i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.33585 + 5.77786i 0.156044 + 0.270277i 0.933439 0.358737i \(-0.116792\pi\)
−0.777394 + 0.629013i \(0.783459\pi\)
\(458\) 0 0
\(459\) 3.17052 10.5485i 0.147987 0.492362i
\(460\) 0 0
\(461\) 26.6895 1.24305 0.621527 0.783393i \(-0.286513\pi\)
0.621527 + 0.783393i \(0.286513\pi\)
\(462\) 0 0
\(463\) −10.9550 −0.509123 −0.254562 0.967057i \(-0.581931\pi\)
−0.254562 + 0.967057i \(0.581931\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.549726 0.952153i −0.0254383 0.0440604i 0.853026 0.521868i \(-0.174765\pi\)
−0.878464 + 0.477808i \(0.841431\pi\)
\(468\) 0 0
\(469\) 27.0937 + 9.89620i 1.25107 + 0.456964i
\(470\) 0 0
\(471\) −34.8168 + 23.8607i −1.60427 + 1.09944i
\(472\) 0 0
\(473\) −0.572029 0.330261i −0.0263019 0.0151854i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.15717 39.5566i −0.281917 1.81117i
\(478\) 0 0
\(479\) 6.91657 11.9798i 0.316026 0.547373i −0.663629 0.748062i \(-0.730985\pi\)
0.979655 + 0.200689i \(0.0643180\pi\)
\(480\) 0 0
\(481\) −26.1176 + 15.0790i −1.19086 + 0.687543i
\(482\) 0 0
\(483\) 21.6320 5.53749i 0.984292 0.251964i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −15.8416 + 27.4385i −0.717854 + 1.24336i 0.243995 + 0.969776i \(0.421542\pi\)
−0.961849 + 0.273582i \(0.911791\pi\)
\(488\) 0 0
\(489\) 2.72642 0.210920i 0.123293 0.00953813i
\(490\) 0 0
\(491\) 37.3388i 1.68508i −0.538636 0.842538i \(-0.681060\pi\)
0.538636 0.842538i \(-0.318940\pi\)
\(492\) 0 0
\(493\) 12.2575 + 7.07689i 0.552051 + 0.318727i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.0642 4.74044i 1.21400 0.212638i
\(498\) 0 0
\(499\) 5.50065 + 9.52740i 0.246243 + 0.426505i 0.962480 0.271352i \(-0.0874706\pi\)
−0.716237 + 0.697857i \(0.754137\pi\)
\(500\) 0 0
\(501\) 12.6567 26.4446i 0.565460 1.18146i
\(502\) 0 0
\(503\) −20.2246 −0.901772 −0.450886 0.892582i \(-0.648892\pi\)
−0.450886 + 0.892582i \(0.648892\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.45795 + 3.04620i −0.0647497 + 0.135287i
\(508\) 0 0
\(509\) 5.24557 + 9.08560i 0.232506 + 0.402712i 0.958545 0.284941i \(-0.0919742\pi\)
−0.726039 + 0.687654i \(0.758641\pi\)
\(510\) 0 0
\(511\) 2.80069 7.66771i 0.123895 0.339199i
\(512\) 0 0
\(513\) −11.8219 + 11.1291i −0.521949 + 0.491362i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.23825i 0.0984383i
\(518\) 0 0
\(519\) −8.04860 + 0.622652i −0.353295 + 0.0273314i
\(520\) 0 0
\(521\) −6.00130 + 10.3946i −0.262922 + 0.455394i −0.967017 0.254712i \(-0.918019\pi\)
0.704095 + 0.710105i \(0.251353\pi\)
\(522\) 0 0
\(523\) 7.74038 4.46891i 0.338463 0.195412i −0.321129 0.947035i \(-0.604062\pi\)
0.659592 + 0.751624i \(0.270729\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.9769 + 6.91489i −0.521724 + 0.301217i
\(528\) 0 0
\(529\) 0.371651 0.643719i 0.0161587 0.0279878i
\(530\) 0 0
\(531\) 16.9766 2.64249i 0.736723 0.114674i
\(532\) 0 0
\(533\) 32.6657i 1.41491i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.14187 + 3.52383i −0.221888 + 0.152065i
\(538\) 0 0
\(539\) 5.55697 + 0.992874i 0.239356 + 0.0427661i
\(540\) 0 0
\(541\) −15.9695 27.6599i −0.686581 1.18919i −0.972937 0.231069i \(-0.925778\pi\)
0.286357 0.958123i \(-0.407556\pi\)
\(542\) 0 0
\(543\) −35.2806 16.8857i −1.51404 0.724635i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.6313 1.43797 0.718984 0.695026i \(-0.244607\pi\)
0.718984 + 0.695026i \(0.244607\pi\)
\(548\) 0 0
\(549\) −5.79577 2.24252i −0.247357 0.0957085i
\(550\) 0 0
\(551\) −10.4317 18.0682i −0.444405 0.769732i
\(552\) 0 0
\(553\) −27.4418 + 22.9733i −1.16695 + 0.976926i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.17988 + 5.30000i 0.388964 + 0.224568i 0.681711 0.731621i \(-0.261236\pi\)
−0.292747 + 0.956190i \(0.594569\pi\)
\(558\) 0 0
\(559\) 3.16695i 0.133948i
\(560\) 0 0
\(561\) −0.228372 2.95201i −0.00964188 0.124634i
\(562\) 0 0
\(563\) −1.14796 + 1.98833i −0.0483808 + 0.0837980i −0.889202 0.457515i \(-0.848739\pi\)
0.840821 + 0.541313i \(0.182073\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −23.7914 0.984852i −0.999144 0.0413599i
\(568\) 0 0
\(569\) 24.1648 13.9515i 1.01304 0.584878i 0.100959 0.994891i \(-0.467809\pi\)
0.912080 + 0.410012i \(0.134476\pi\)
\(570\) 0 0
\(571\) 3.11649 5.39791i 0.130421 0.225896i −0.793418 0.608677i \(-0.791700\pi\)
0.923839 + 0.382782i \(0.125034\pi\)
\(572\) 0 0
\(573\) −0.450993 5.82969i −0.0188405 0.243539i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −11.4776 6.62662i −0.477820 0.275870i 0.241687 0.970354i \(-0.422299\pi\)
−0.719508 + 0.694484i \(0.755632\pi\)
\(578\) 0 0
\(579\) 10.8592 + 15.8453i 0.451291 + 0.658510i
\(580\) 0 0
\(581\) 5.88606 + 33.6048i 0.244195 + 1.39416i
\(582\) 0 0
\(583\) −5.38059 9.31946i −0.222841 0.385973i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.6209 0.768568 0.384284 0.923215i \(-0.374448\pi\)
0.384284 + 0.923215i \(0.374448\pi\)
\(588\) 0 0
\(589\) 20.3858 0.839982
\(590\) 0 0
\(591\) −25.4011 12.1572i −1.04486 0.500082i
\(592\) 0 0
\(593\) −1.37822 2.38715i −0.0565967 0.0980284i 0.836339 0.548213i \(-0.184692\pi\)
−0.892936 + 0.450184i \(0.851358\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.78115 + 1.22066i −0.0728976 + 0.0499583i
\(598\) 0 0
\(599\) −16.2530 9.38368i −0.664080 0.383407i 0.129750 0.991547i \(-0.458583\pi\)
−0.793830 + 0.608140i \(0.791916\pi\)
\(600\) 0 0
\(601\) 16.5669i 0.675779i −0.941186 0.337890i \(-0.890287\pi\)
0.941186 0.337890i \(-0.109713\pi\)
\(602\) 0 0
\(603\) −32.3174 + 5.03035i −1.31607 + 0.204852i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28.9773 + 16.7301i −1.17615 + 0.679053i −0.955122 0.296213i \(-0.904276\pi\)
−0.221032 + 0.975266i \(0.570943\pi\)
\(608\) 0 0
\(609\) 8.24971 29.4648i 0.334295 1.19397i
\(610\) 0 0
\(611\) −9.29381 + 5.36578i −0.375987 + 0.217076i
\(612\) 0 0
\(613\) −18.7772 + 32.5231i −0.758404 + 1.31359i 0.185260 + 0.982690i \(0.440687\pi\)
−0.943664 + 0.330905i \(0.892646\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.5088i 0.906172i −0.891467 0.453086i \(-0.850323\pi\)
0.891467 0.453086i \(-0.149677\pi\)
\(618\) 0 0
\(619\) −23.3812 13.4991i −0.939770 0.542576i −0.0498816 0.998755i \(-0.515884\pi\)
−0.889888 + 0.456179i \(0.849218\pi\)
\(620\) 0 0
\(621\) −18.4355 + 17.3551i −0.739791 + 0.696438i
\(622\) 0 0
\(623\) −6.51332 + 5.45272i −0.260951 + 0.218459i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.88418 + 3.93676i −0.0752468 + 0.157219i
\(628\) 0 0
\(629\) −16.5339 −0.659249
\(630\) 0 0
\(631\) 9.17965 0.365436 0.182718 0.983165i \(-0.441510\pi\)
0.182718 + 0.983165i \(0.441510\pi\)
\(632\) 0 0
\(633\) −5.72376 + 11.9591i −0.227499 + 0.475331i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.19909 25.4542i −0.364481 1.00853i
\(638\) 0 0
\(639\) −24.2642 + 19.5420i −0.959878 + 0.773069i
\(640\) 0 0
\(641\) −26.6664 15.3959i −1.05326 0.608100i −0.129700 0.991553i \(-0.541401\pi\)
−0.923560 + 0.383453i \(0.874735\pi\)
\(642\) 0 0
\(643\) 3.42929i 0.135238i −0.997711 0.0676189i \(-0.978460\pi\)
0.997711 0.0676189i \(-0.0215402\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.2967 + 43.8152i −0.994516 + 1.72255i −0.406689 + 0.913567i \(0.633317\pi\)
−0.587828 + 0.808986i \(0.700017\pi\)
\(648\) 0 0
\(649\) 3.99966 2.30920i 0.157000 0.0906441i
\(650\) 0 0
\(651\) 20.9027 + 21.3761i 0.819243 + 0.837795i
\(652\) 0 0
\(653\) 4.57643 2.64220i 0.179090 0.103397i −0.407775 0.913082i \(-0.633695\pi\)
0.586865 + 0.809685i \(0.300362\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.42362 + 9.14605i 0.0555408 + 0.356821i
\(658\) 0 0
\(659\) 26.0502i 1.01477i 0.861719 + 0.507385i \(0.169388\pi\)
−0.861719 + 0.507385i \(0.830612\pi\)
\(660\) 0 0
\(661\) −28.3751 16.3823i −1.10366 0.637199i −0.166481 0.986045i \(-0.553241\pi\)
−0.937180 + 0.348845i \(0.886574\pi\)
\(662\) 0 0
\(663\) −11.7101 + 8.02515i −0.454781 + 0.311671i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.2676 28.1763i −0.629883 1.09099i
\(668\) 0 0
\(669\) 32.7103 + 15.6555i 1.26465 + 0.605278i
\(670\) 0 0
\(671\) −1.67050 −0.0644891
\(672\) 0 0
\(673\) 38.5194 1.48481 0.742406 0.669950i \(-0.233685\pi\)
0.742406 + 0.669950i \(0.233685\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.58044 + 16.5938i 0.368206 + 0.637752i 0.989285 0.145996i \(-0.0466387\pi\)
−0.621079 + 0.783748i \(0.713305\pi\)
\(678\) 0 0
\(679\) 2.64225 0.462805i 0.101400 0.0177608i
\(680\) 0 0
\(681\) 4.48649 + 6.54654i 0.171923 + 0.250864i
\(682\) 0 0
\(683\) 25.0821 + 14.4812i 0.959741 + 0.554107i 0.896093 0.443866i \(-0.146393\pi\)
0.0636479 + 0.997972i \(0.479727\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.130393 1.68551i −0.00497482 0.0643062i
\(688\) 0 0
\(689\) −25.7979 + 44.6832i −0.982821 + 1.70230i
\(690\) 0 0
\(691\) 39.4572 22.7806i 1.50102 0.866616i 0.501024 0.865433i \(-0.332957\pi\)
0.999999 0.00118303i \(-0.000376571\pi\)
\(692\) 0 0
\(693\) −6.05995 + 2.06088i −0.230198 + 0.0782862i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.95437 + 15.5094i −0.339171 + 0.587461i
\(698\) 0 0
\(699\) 2.65920 + 34.3737i 0.100580 + 1.30013i
\(700\) 0 0
\(701\) 39.2878i 1.48388i −0.670466 0.741940i \(-0.733906\pi\)
0.670466 0.741940i \(-0.266094\pi\)
\(702\) 0 0
\(703\) 21.1066 + 12.1859i 0.796050 + 0.459600i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 21.2629 + 7.76643i 0.799672 + 0.292087i
\(708\) 0 0
\(709\) −9.88537 17.1220i −0.371253 0.643029i 0.618506 0.785780i \(-0.287738\pi\)
−0.989759 + 0.142752i \(0.954405\pi\)
\(710\) 0 0
\(711\) 14.6436 37.8463i 0.549180 1.41935i
\(712\) 0 0
\(713\) 31.7904 1.19056
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 44.1675 + 21.1391i 1.64947 + 0.789454i
\(718\) 0 0
\(719\) 0.740641 + 1.28283i 0.0276212 + 0.0478414i 0.879506 0.475889i \(-0.157873\pi\)
−0.851884 + 0.523730i \(0.824540\pi\)
\(720\) 0 0
\(721\) −16.6072 19.8374i −0.618484 0.738784i
\(722\) 0 0
\(723\) 18.3040 12.5441i 0.680733 0.466521i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.35193i 0.235580i −0.993039 0.117790i \(-0.962419\pi\)
0.993039 0.117790i \(-0.0375810\pi\)
\(728\) 0 0
\(729\) 24.1544 12.0651i 0.894607 0.446855i
\(730\) 0 0
\(731\) −0.868129 + 1.50364i −0.0321089 + 0.0556142i
\(732\) 0 0
\(733\) 34.0100 19.6357i 1.25619 0.725261i 0.283857 0.958866i \(-0.408386\pi\)
0.972331 + 0.233605i \(0.0750524\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.61392 + 4.39590i −0.280462 + 0.161925i
\(738\) 0 0
\(739\) 11.2904 19.5555i 0.415324 0.719362i −0.580139 0.814518i \(-0.697002\pi\)
0.995462 + 0.0951558i \(0.0303349\pi\)
\(740\) 0 0
\(741\) 20.8634 1.61402i 0.766436 0.0592926i
\(742\) 0 0
\(743\) 26.6330i 0.977070i −0.872544 0.488535i \(-0.837532\pi\)
0.872544 0.488535i \(-0.162468\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −24.2647 30.1282i −0.887799 1.10233i
\(748\) 0 0
\(749\) −31.0510 37.0906i −1.13458 1.35526i
\(750\) 0 0
\(751\) 8.62709 + 14.9426i 0.314807 + 0.545262i 0.979396 0.201947i \(-0.0647268\pi\)
−0.664589 + 0.747209i \(0.731393\pi\)
\(752\) 0 0
\(753\) 7.72793 16.1466i 0.281621 0.588413i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −25.1712 −0.914863 −0.457431 0.889245i \(-0.651230\pi\)
−0.457431 + 0.889245i \(0.651230\pi\)
\(758\) 0 0
\(759\) −2.93826 + 6.13913i −0.106652 + 0.222836i
\(760\) 0 0
\(761\) 0.00796126 + 0.0137893i 0.000288596 + 0.000499862i 0.866170 0.499750i \(-0.166575\pi\)
−0.865881 + 0.500250i \(0.833241\pi\)
\(762\) 0 0
\(763\) −43.4824 15.8823i −1.57417 0.574977i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −19.1768 11.0717i −0.692435 0.399777i
\(768\) 0 0
\(769\) 13.0356i 0.470075i 0.971986 + 0.235037i \(0.0755213\pi\)
−0.971986 + 0.235037i \(0.924479\pi\)
\(770\) 0 0
\(771\) −52.6796 + 4.07537i −1.89721 + 0.146771i
\(772\) 0 0
\(773\) −14.7085 + 25.4758i −0.529027 + 0.916301i 0.470400 + 0.882453i \(0.344110\pi\)
−0.999427 + 0.0338481i \(0.989224\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.86395 + 34.6268i 0.317993 + 1.24223i
\(778\) 0 0
\(779\) 22.8617 13.1992i 0.819105 0.472910i
\(780\) 0 0
\(781\) −4.18738 + 7.25275i −0.149836 + 0.259524i
\(782\) 0 0
\(783\) 7.96445 + 33.7683i 0.284626 + 1.20678i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.95977 1.13148i −0.0698584 0.0403327i 0.464664 0.885487i \(-0.346175\pi\)
−0.534522 + 0.845154i \(0.679508\pi\)
\(788\) 0 0
\(789\) −29.5395 + 20.2441i −1.05164 + 0.720708i
\(790\) 0 0
\(791\) 18.1388 3.17711i 0.644943 0.112965i
\(792\) 0 0
\(793\) 4.00471 + 6.93636i 0.142211 + 0.246318i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.6277 0.695250 0.347625 0.937634i \(-0.386988\pi\)
0.347625 + 0.937634i \(0.386988\pi\)
\(798\) 0 0
\(799\) −5.88350 −0.208143
\(800\) 0 0
\(801\) 3.47567 8.98283i 0.122807 0.317393i
\(802\) 0 0
\(803\) 1.24407 + 2.15479i 0.0439022 + 0.0760409i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.3012 35.4596i −0.855444 1.24824i
\(808\) 0 0
\(809\) 21.2506 + 12.2690i 0.747130 + 0.431356i 0.824656 0.565635i \(-0.191369\pi\)
−0.0775260 + 0.996990i \(0.524702\pi\)
\(810\) 0 0
\(811\) 38.7927i 1.36220i 0.732192 + 0.681098i \(0.238497\pi\)
−0.732192 + 0.681098i \(0.761503\pi\)
\(812\) 0 0
\(813\) −2.90893 37.6018i −0.102021 1.31875i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.21645 1.27967i 0.0775436 0.0447698i
\(818\) 0 0
\(819\) 23.0849 + 20.2219i 0.806650 + 0.706611i
\(820\) 0 0
\(821\) −11.8734 + 6.85513i −0.414386 + 0.239246i −0.692672 0.721252i \(-0.743567\pi\)
0.278287 + 0.960498i \(0.410233\pi\)
\(822\) 0 0
\(823\) 4.04168 7.00040i 0.140884 0.244019i −0.786946 0.617022i \(-0.788339\pi\)
0.927830 + 0.373004i \(0.121672\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.6581i 1.30950i −0.755845 0.654751i \(-0.772774\pi\)
0.755845 0.654751i \(-0.227226\pi\)
\(828\) 0 0
\(829\) 15.4475 + 8.91864i 0.536515 + 0.309757i 0.743665 0.668552i \(-0.233086\pi\)
−0.207150 + 0.978309i \(0.566419\pi\)
\(830\) 0 0
\(831\) −14.9050 21.7489i −0.517049 0.754462i
\(832\) 0 0
\(833\) 2.60988 14.6071i 0.0904269 0.506106i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −32.4658 9.75811i −1.12218 0.337290i
\(838\) 0 0
\(839\) −4.49497 −0.155183 −0.0775917 0.996985i \(-0.524723\pi\)
−0.0775917 + 0.996985i \(0.524723\pi\)
\(840\) 0 0
\(841\) −15.5825 −0.537329
\(842\) 0 0
\(843\) −28.0419 13.4212i −0.965814 0.462250i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.9963 17.5774i 0.721443 0.603966i
\(848\) 0 0
\(849\) 38.5598 26.4259i 1.32337 0.906934i
\(850\) 0 0
\(851\) 32.9144 + 19.0032i 1.12829 + 0.651420i
\(852\) 0 0
\(853\) 33.0503i 1.13162i −0.824535 0.565811i \(-0.808563\pi\)
0.824535 0.565811i \(-0.191437\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.210762 + 0.365051i −0.00719949 + 0.0124699i −0.869603 0.493752i \(-0.835625\pi\)
0.862403 + 0.506222i \(0.168958\pi\)
\(858\) 0 0
\(859\) −17.0294 + 9.83192i −0.581035 + 0.335461i −0.761545 0.648112i \(-0.775559\pi\)
0.180509 + 0.983573i \(0.442225\pi\)
\(860\) 0 0
\(861\) 37.2818 + 10.4383i 1.27056 + 0.355737i
\(862\) 0 0
\(863\) −0.782575 + 0.451820i −0.0266392 + 0.0153801i −0.513260 0.858233i \(-0.671563\pi\)
0.486621 + 0.873613i \(0.338229\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.5974 1.67081i 0.733488 0.0567437i
\(868\) 0 0
\(869\) 10.9084i 0.370041i
\(870\) 0 0
\(871\) 36.5058 + 21.0766i 1.23695 + 0.714154i
\(872\) 0 0
\(873\) −2.36889 + 1.90787i −0.0801749 + 0.0645715i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.65916 13.2661i −0.258632 0.447963i 0.707244 0.706970i \(-0.249938\pi\)
−0.965876 + 0.259006i \(0.916605\pi\)
\(878\) 0 0
\(879\) −10.8013 + 22.5679i −0.364317 + 0.761196i
\(880\) 0 0
\(881\) 43.8614 1.47773 0.738863 0.673855i \(-0.235363\pi\)
0.738863 + 0.673855i \(0.235363\pi\)
\(882\) 0 0
\(883\) 2.91300 0.0980304 0.0490152 0.998798i \(-0.484392\pi\)
0.0490152 + 0.998798i \(0.484392\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.44790 12.9001i −0.250076 0.433145i 0.713470 0.700685i \(-0.247122\pi\)
−0.963547 + 0.267541i \(0.913789\pi\)
\(888\) 0 0
\(889\) −3.54739 20.2528i −0.118976 0.679259i
\(890\) 0 0
\(891\) 4.88510 5.36766i 0.163657 0.179823i
\(892\) 0 0
\(893\) 7.51067 + 4.33629i 0.251335 + 0.145108i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32.5352 2.51697i 1.08632 0.0840391i
\(898\) 0 0
\(899\) 21.7810 37.7258i 0.726437 1.25823i
\(900\) 0 0
\(901\) −24.4972 + 14.1435i −0.816121 + 0.471188i
\(902\) 0 0
\(903\) 3.61448 + 1.01200i 0.120282 + 0.0336772i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.54899 16.5393i 0.317069 0.549180i −0.662806 0.748791i \(-0.730635\pi\)
0.979875 + 0.199611i \(0.0639679\pi\)
\(908\) 0 0
\(909\) −25.3624 + 3.94777i −0.841217 + 0.130939i
\(910\) 0 0
\(911\) 6.16915i 0.204393i −0.994764 0.102197i \(-0.967413\pi\)
0.994764 0.102197i \(-0.0325871\pi\)
\(912\) 0 0
\(913\) −9.00552 5.19934i −0.298039 0.172073i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.0912 + 20.1683i −0.795562 + 0.666016i
\(918\) 0 0
\(919\) −14.8318 25.6894i −0.489255 0.847415i 0.510668 0.859778i \(-0.329398\pi\)
−0.999924 + 0.0123630i \(0.996065\pi\)
\(920\) 0 0
\(921\) 14.6006 + 6.98804i 0.481107 + 0.230264i
\(922\) 0 0
\(923\) 40.1537 1.32168
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 27.3587 + 10.5857i 0.898577 + 0.347681i
\(928\) 0 0
\(929\) −16.8256 29.1428i −0.552030 0.956144i −0.998128 0.0611601i \(-0.980520\pi\)
0.446098 0.894984i \(-0.352813\pi\)
\(930\) 0 0
\(931\) −14.0975 + 16.7234i −0.462027 + 0.548087i
\(932\) 0 0
\(933\) 28.7209 + 41.9087i 0.940281 + 1.37203i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.99276i 0.261112i 0.991441 + 0.130556i \(0.0416762\pi\)
−0.991441 + 0.130556i \(0.958324\pi\)
\(938\) 0 0
\(939\) 1.47433 + 19.0576i 0.0481128 + 0.621922i
\(940\) 0 0
\(941\) 8.69431 15.0590i 0.283426 0.490909i −0.688800 0.724951i \(-0.741862\pi\)
0.972226 + 0.234043i \(0.0751955\pi\)
\(942\) 0 0
\(943\) 35.6514 20.5833i 1.16097 0.670285i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.6302 + 21.1485i −1.19032 + 0.687233i −0.958380 0.285496i \(-0.907842\pi\)
−0.231943 + 0.972729i \(0.574508\pi\)
\(948\) 0 0
\(949\) 5.96483 10.3314i 0.193627 0.335371i
\(950\) 0 0
\(951\) 2.26746 + 29.3099i 0.0735273 + 0.950438i
\(952\) 0 0
\(953\) 40.5612i 1.31391i −0.753932 0.656953i \(-0.771845\pi\)
0.753932 0.656953i \(-0.228155\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.27220 + 7.69303i 0.170426 + 0.248680i
\(958\) 0 0
\(959\) 2.23145 6.10926i 0.0720574 0.197278i
\(960\) 0 0
\(961\) 5.78241 + 10.0154i 0.186529 + 0.323078i
\(962\) 0 0
\(963\) 51.1534 + 19.7925i 1.64840 + 0.637804i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.6713 1.21143 0.605714 0.795682i \(-0.292888\pi\)
0.605714 + 0.795682i \(0.292888\pi\)
\(968\) 0 0
\(969\) 10.3482 + 4.95277i 0.332432 + 0.159106i
\(970\) 0 0
\(971\) 21.5452 + 37.3173i 0.691418 + 1.19757i 0.971373 + 0.237558i \(0.0763469\pi\)
−0.279956 + 0.960013i \(0.590320\pi\)
\(972\) 0 0
\(973\) 7.33180 1.28420i 0.235047 0.0411697i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.7402 11.3970i −0.631545 0.364622i 0.149805 0.988716i \(-0.452135\pi\)
−0.781350 + 0.624093i \(0.785469\pi\)
\(978\) 0 0
\(979\) 2.58910i 0.0827481i
\(980\) 0 0
\(981\) 51.8658 8.07315i 1.65595 0.257756i
\(982\) 0 0
\(983\) 22.5929 39.1321i 0.720602 1.24812i −0.240157 0.970734i \(-0.577199\pi\)
0.960759 0.277385i \(-0.0894677\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.15419 + 12.3218i 0.100399 + 0.392206i
\(988\) 0 0
\(989\) 3.45641 1.99556i 0.109907 0.0634551i
\(990\) 0 0
\(991\) −25.2760 + 43.7793i −0.802919 + 1.39070i 0.114768 + 0.993392i \(0.463388\pi\)
−0.917687 + 0.397304i \(0.869946\pi\)
\(992\) 0 0
\(993\) 20.1265 1.55702i 0.638696 0.0494105i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.5635 + 13.0271i 0.714595 + 0.412571i 0.812760 0.582599i \(-0.197964\pi\)
−0.0981654 + 0.995170i \(0.531297\pi\)
\(998\) 0 0
\(999\) −27.7807 29.5100i −0.878942 0.933656i
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 2100.2.bi.k.101.4 10
3.2 odd 2 2100.2.bi.j.101.2 10
5.2 odd 4 2100.2.bo.h.1949.3 20
5.3 odd 4 2100.2.bo.h.1949.8 20
5.4 even 2 420.2.bh.a.101.2 10
7.5 odd 6 2100.2.bi.j.1601.2 10
15.2 even 4 2100.2.bo.g.1949.9 20
15.8 even 4 2100.2.bo.g.1949.2 20
15.14 odd 2 420.2.bh.b.101.4 yes 10
21.5 even 6 inner 2100.2.bi.k.1601.4 10
35.4 even 6 2940.2.d.b.881.9 10
35.12 even 12 2100.2.bo.g.1349.2 20
35.19 odd 6 420.2.bh.b.341.4 yes 10
35.24 odd 6 2940.2.d.a.881.2 10
35.33 even 12 2100.2.bo.g.1349.9 20
105.47 odd 12 2100.2.bo.h.1349.8 20
105.59 even 6 2940.2.d.b.881.10 10
105.68 odd 12 2100.2.bo.h.1349.3 20
105.74 odd 6 2940.2.d.a.881.1 10
105.89 even 6 420.2.bh.a.341.2 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bh.a.101.2 10 5.4 even 2
420.2.bh.a.341.2 yes 10 105.89 even 6
420.2.bh.b.101.4 yes 10 15.14 odd 2
420.2.bh.b.341.4 yes 10 35.19 odd 6
2100.2.bi.j.101.2 10 3.2 odd 2
2100.2.bi.j.1601.2 10 7.5 odd 6
2100.2.bi.k.101.4 10 1.1 even 1 trivial
2100.2.bi.k.1601.4 10 21.5 even 6 inner
2100.2.bo.g.1349.2 20 35.12 even 12
2100.2.bo.g.1349.9 20 35.33 even 12
2100.2.bo.g.1949.2 20 15.8 even 4
2100.2.bo.g.1949.9 20 15.2 even 4
2100.2.bo.h.1349.3 20 105.68 odd 12
2100.2.bo.h.1349.8 20 105.47 odd 12
2100.2.bo.h.1949.3 20 5.2 odd 4
2100.2.bo.h.1949.8 20 5.3 odd 4
2940.2.d.a.881.1 10 105.74 odd 6
2940.2.d.a.881.2 10 35.24 odd 6
2940.2.d.b.881.9 10 35.4 even 6
2940.2.d.b.881.10 10 105.59 even 6