Properties

Label 3850.2.c.q.1849.1
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3850,2,Mod(1849,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3850.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not 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: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.q.1849.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.23607i q^{3} -1.00000 q^{4} -3.23607 q^{6} -1.00000i q^{7} +1.00000i q^{8} -7.47214 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -3.23607i q^{3} -1.00000 q^{4} -3.23607 q^{6} -1.00000i q^{7} +1.00000i q^{8} -7.47214 q^{9} +1.00000 q^{11} +3.23607i q^{12} +1.23607i q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.47214i q^{17} +7.47214i q^{18} +2.76393 q^{19} -3.23607 q^{21} -1.00000i q^{22} +4.00000i q^{23} +3.23607 q^{24} +1.23607 q^{26} +14.4721i q^{27} +1.00000i q^{28} +4.47214 q^{29} +2.00000 q^{31} -1.00000i q^{32} -3.23607i q^{33} +6.47214 q^{34} +7.47214 q^{36} +10.9443i q^{37} -2.76393i q^{38} +4.00000 q^{39} +6.47214 q^{41} +3.23607i q^{42} -1.52786i q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000i q^{47} -3.23607i q^{48} -1.00000 q^{49} +20.9443 q^{51} -1.23607i q^{52} -0.472136i q^{53} +14.4721 q^{54} +1.00000 q^{56} -8.94427i q^{57} -4.47214i q^{58} -7.23607 q^{59} -5.23607 q^{61} -2.00000i q^{62} +7.47214i q^{63} -1.00000 q^{64} -3.23607 q^{66} +15.4164i q^{67} -6.47214i q^{68} +12.9443 q^{69} -2.47214 q^{71} -7.47214i q^{72} -4.94427i q^{73} +10.9443 q^{74} -2.76393 q^{76} -1.00000i q^{77} -4.00000i q^{78} +24.4164 q^{81} -6.47214i q^{82} +10.1803i q^{83} +3.23607 q^{84} -1.52786 q^{86} -14.4721i q^{87} +1.00000i q^{88} -10.0000 q^{89} +1.23607 q^{91} -4.00000i q^{92} -6.47214i q^{93} +2.00000 q^{94} -3.23607 q^{96} -3.52786i q^{97} +1.00000i q^{98} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 3.23607i − 1.86834i −0.356822 0.934172i \(-0.616140\pi\)
0.356822 0.934172i \(-0.383860\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.23607 −1.32112
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 3.23607i 0.934172i
\(13\) 1.23607i 0.342824i 0.985199 + 0.171412i \(0.0548329\pi\)
−0.985199 + 0.171412i \(0.945167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.47214i 1.56972i 0.619671 + 0.784862i \(0.287266\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(18\) 7.47214i 1.76120i
\(19\) 2.76393 0.634089 0.317045 0.948411i \(-0.397309\pi\)
0.317045 + 0.948411i \(0.397309\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) − 1.00000i − 0.213201i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 3.23607 0.660560
\(25\) 0 0
\(26\) 1.23607 0.242413
\(27\) 14.4721i 2.78516i
\(28\) 1.00000i 0.188982i
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 3.23607i − 0.563327i
\(34\) 6.47214 1.10996
\(35\) 0 0
\(36\) 7.47214 1.24536
\(37\) 10.9443i 1.79923i 0.436687 + 0.899614i \(0.356152\pi\)
−0.436687 + 0.899614i \(0.643848\pi\)
\(38\) − 2.76393i − 0.448369i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.47214 1.01078 0.505389 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 3.23607i 0.499336i
\(43\) − 1.52786i − 0.232997i −0.993191 0.116499i \(-0.962833\pi\)
0.993191 0.116499i \(-0.0371670\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) − 3.23607i − 0.467086i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 20.9443 2.93278
\(52\) − 1.23607i − 0.171412i
\(53\) − 0.472136i − 0.0648529i −0.999474 0.0324264i \(-0.989677\pi\)
0.999474 0.0324264i \(-0.0103235\pi\)
\(54\) 14.4721 1.96941
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 8.94427i − 1.18470i
\(58\) − 4.47214i − 0.587220i
\(59\) −7.23607 −0.942056 −0.471028 0.882118i \(-0.656117\pi\)
−0.471028 + 0.882118i \(0.656117\pi\)
\(60\) 0 0
\(61\) −5.23607 −0.670410 −0.335205 0.942145i \(-0.608806\pi\)
−0.335205 + 0.942145i \(0.608806\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) 7.47214i 0.941401i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.23607 −0.398332
\(67\) 15.4164i 1.88341i 0.336434 + 0.941707i \(0.390779\pi\)
−0.336434 + 0.941707i \(0.609221\pi\)
\(68\) − 6.47214i − 0.784862i
\(69\) 12.9443 1.55831
\(70\) 0 0
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) − 7.47214i − 0.880600i
\(73\) − 4.94427i − 0.578683i −0.957226 0.289342i \(-0.906564\pi\)
0.957226 0.289342i \(-0.0934364\pi\)
\(74\) 10.9443 1.27225
\(75\) 0 0
\(76\) −2.76393 −0.317045
\(77\) − 1.00000i − 0.113961i
\(78\) − 4.00000i − 0.452911i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) − 6.47214i − 0.714728i
\(83\) 10.1803i 1.11744i 0.829357 + 0.558719i \(0.188707\pi\)
−0.829357 + 0.558719i \(0.811293\pi\)
\(84\) 3.23607 0.353084
\(85\) 0 0
\(86\) −1.52786 −0.164754
\(87\) − 14.4721i − 1.55158i
\(88\) 1.00000i 0.106600i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) − 4.00000i − 0.417029i
\(93\) − 6.47214i − 0.671129i
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) −3.23607 −0.330280
\(97\) − 3.52786i − 0.358200i −0.983831 0.179100i \(-0.942681\pi\)
0.983831 0.179100i \(-0.0573186\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −7.47214 −0.750978
\(100\) 0 0
\(101\) −14.1803 −1.41100 −0.705498 0.708712i \(-0.749277\pi\)
−0.705498 + 0.708712i \(0.749277\pi\)
\(102\) − 20.9443i − 2.07379i
\(103\) 2.94427i 0.290108i 0.989424 + 0.145054i \(0.0463355\pi\)
−0.989424 + 0.145054i \(0.953664\pi\)
\(104\) −1.23607 −0.121206
\(105\) 0 0
\(106\) −0.472136 −0.0458579
\(107\) 6.47214i 0.625685i 0.949805 + 0.312842i \(0.101281\pi\)
−0.949805 + 0.312842i \(0.898719\pi\)
\(108\) − 14.4721i − 1.39258i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 35.4164 3.36158
\(112\) − 1.00000i − 0.0944911i
\(113\) 8.47214i 0.796992i 0.917170 + 0.398496i \(0.130468\pi\)
−0.917170 + 0.398496i \(0.869532\pi\)
\(114\) −8.94427 −0.837708
\(115\) 0 0
\(116\) −4.47214 −0.415227
\(117\) − 9.23607i − 0.853875i
\(118\) 7.23607i 0.666134i
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.23607i 0.474051i
\(123\) − 20.9443i − 1.88848i
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 7.47214 0.665671
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.94427 −0.435319
\(130\) 0 0
\(131\) 9.23607 0.806959 0.403480 0.914989i \(-0.367801\pi\)
0.403480 + 0.914989i \(0.367801\pi\)
\(132\) 3.23607i 0.281664i
\(133\) − 2.76393i − 0.239663i
\(134\) 15.4164 1.33177
\(135\) 0 0
\(136\) −6.47214 −0.554981
\(137\) − 15.8885i − 1.35745i −0.734393 0.678725i \(-0.762533\pi\)
0.734393 0.678725i \(-0.237467\pi\)
\(138\) − 12.9443i − 1.10189i
\(139\) 8.29180 0.703301 0.351650 0.936131i \(-0.385621\pi\)
0.351650 + 0.936131i \(0.385621\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 2.47214i 0.207457i
\(143\) 1.23607i 0.103365i
\(144\) −7.47214 −0.622678
\(145\) 0 0
\(146\) −4.94427 −0.409191
\(147\) 3.23607i 0.266906i
\(148\) − 10.9443i − 0.899614i
\(149\) −22.3607 −1.83186 −0.915929 0.401340i \(-0.868545\pi\)
−0.915929 + 0.401340i \(0.868545\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 2.76393i 0.224184i
\(153\) − 48.3607i − 3.90973i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 18.6525i − 1.48863i −0.667829 0.744315i \(-0.732776\pi\)
0.667829 0.744315i \(-0.267224\pi\)
\(158\) 0 0
\(159\) −1.52786 −0.121168
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) − 24.4164i − 1.91833i
\(163\) 7.41641i 0.580898i 0.956890 + 0.290449i \(0.0938046\pi\)
−0.956890 + 0.290449i \(0.906195\pi\)
\(164\) −6.47214 −0.505389
\(165\) 0 0
\(166\) 10.1803 0.790148
\(167\) 15.4164i 1.19296i 0.802629 + 0.596479i \(0.203434\pi\)
−0.802629 + 0.596479i \(0.796566\pi\)
\(168\) − 3.23607i − 0.249668i
\(169\) 11.4721 0.882472
\(170\) 0 0
\(171\) −20.6525 −1.57933
\(172\) 1.52786i 0.116499i
\(173\) 1.23607i 0.0939765i 0.998895 + 0.0469883i \(0.0149623\pi\)
−0.998895 + 0.0469883i \(0.985038\pi\)
\(174\) −14.4721 −1.09713
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 23.4164i 1.76008i
\(178\) 10.0000i 0.749532i
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) 4.76393 0.354100 0.177050 0.984202i \(-0.443345\pi\)
0.177050 + 0.984202i \(0.443345\pi\)
\(182\) − 1.23607i − 0.0916235i
\(183\) 16.9443i 1.25256i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −6.47214 −0.474560
\(187\) 6.47214i 0.473289i
\(188\) − 2.00000i − 0.145865i
\(189\) 14.4721 1.05269
\(190\) 0 0
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) 3.23607i 0.233543i
\(193\) 2.94427i 0.211933i 0.994370 + 0.105967i \(0.0337937\pi\)
−0.994370 + 0.105967i \(0.966206\pi\)
\(194\) −3.52786 −0.253286
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 7.47214i 0.531022i
\(199\) −1.05573 −0.0748386 −0.0374193 0.999300i \(-0.511914\pi\)
−0.0374193 + 0.999300i \(0.511914\pi\)
\(200\) 0 0
\(201\) 49.8885 3.51887
\(202\) 14.1803i 0.997725i
\(203\) − 4.47214i − 0.313882i
\(204\) −20.9443 −1.46639
\(205\) 0 0
\(206\) 2.94427 0.205137
\(207\) − 29.8885i − 2.07740i
\(208\) 1.23607i 0.0857059i
\(209\) 2.76393 0.191185
\(210\) 0 0
\(211\) −22.4721 −1.54705 −0.773523 0.633768i \(-0.781507\pi\)
−0.773523 + 0.633768i \(0.781507\pi\)
\(212\) 0.472136i 0.0324264i
\(213\) 8.00000i 0.548151i
\(214\) 6.47214 0.442426
\(215\) 0 0
\(216\) −14.4721 −0.984704
\(217\) − 2.00000i − 0.135769i
\(218\) − 10.0000i − 0.677285i
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 35.4164i − 2.37699i
\(223\) 8.47214i 0.567336i 0.958923 + 0.283668i \(0.0915514\pi\)
−0.958923 + 0.283668i \(0.908449\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 8.47214 0.563558
\(227\) 14.7639i 0.979917i 0.871746 + 0.489958i \(0.162988\pi\)
−0.871746 + 0.489958i \(0.837012\pi\)
\(228\) 8.94427i 0.592349i
\(229\) −12.7639 −0.843464 −0.421732 0.906720i \(-0.638578\pi\)
−0.421732 + 0.906720i \(0.638578\pi\)
\(230\) 0 0
\(231\) −3.23607 −0.212918
\(232\) 4.47214i 0.293610i
\(233\) 2.94427i 0.192886i 0.995339 + 0.0964428i \(0.0307465\pi\)
−0.995339 + 0.0964428i \(0.969254\pi\)
\(234\) −9.23607 −0.603781
\(235\) 0 0
\(236\) 7.23607 0.471028
\(237\) 0 0
\(238\) − 6.47214i − 0.419526i
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −11.4164 −0.735395 −0.367698 0.929945i \(-0.619854\pi\)
−0.367698 + 0.929945i \(0.619854\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 35.5967i − 2.28353i
\(244\) 5.23607 0.335205
\(245\) 0 0
\(246\) −20.9443 −1.33536
\(247\) 3.41641i 0.217381i
\(248\) 2.00000i 0.127000i
\(249\) 32.9443 2.08776
\(250\) 0 0
\(251\) 24.7639 1.56309 0.781543 0.623852i \(-0.214433\pi\)
0.781543 + 0.623852i \(0.214433\pi\)
\(252\) − 7.47214i − 0.470700i
\(253\) 4.00000i 0.251478i
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.9443i 0.682685i 0.939939 + 0.341342i \(0.110882\pi\)
−0.939939 + 0.341342i \(0.889118\pi\)
\(258\) 4.94427i 0.307817i
\(259\) 10.9443 0.680044
\(260\) 0 0
\(261\) −33.4164 −2.06842
\(262\) − 9.23607i − 0.570606i
\(263\) 12.9443i 0.798178i 0.916912 + 0.399089i \(0.130674\pi\)
−0.916912 + 0.399089i \(0.869326\pi\)
\(264\) 3.23607 0.199166
\(265\) 0 0
\(266\) −2.76393 −0.169468
\(267\) 32.3607i 1.98044i
\(268\) − 15.4164i − 0.941707i
\(269\) 27.2361 1.66061 0.830306 0.557307i \(-0.188166\pi\)
0.830306 + 0.557307i \(0.188166\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 6.47214i 0.392431i
\(273\) − 4.00000i − 0.242091i
\(274\) −15.8885 −0.959862
\(275\) 0 0
\(276\) −12.9443 −0.779154
\(277\) − 12.4721i − 0.749378i −0.927151 0.374689i \(-0.877749\pi\)
0.927151 0.374689i \(-0.122251\pi\)
\(278\) − 8.29180i − 0.497309i
\(279\) −14.9443 −0.894690
\(280\) 0 0
\(281\) −24.8328 −1.48140 −0.740701 0.671835i \(-0.765506\pi\)
−0.740701 + 0.671835i \(0.765506\pi\)
\(282\) − 6.47214i − 0.385410i
\(283\) − 16.6525i − 0.989887i −0.868925 0.494943i \(-0.835189\pi\)
0.868925 0.494943i \(-0.164811\pi\)
\(284\) 2.47214 0.146694
\(285\) 0 0
\(286\) 1.23607 0.0730902
\(287\) − 6.47214i − 0.382038i
\(288\) 7.47214i 0.440300i
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) −11.4164 −0.669242
\(292\) 4.94427i 0.289342i
\(293\) 4.65248i 0.271801i 0.990723 + 0.135900i \(0.0433927\pi\)
−0.990723 + 0.135900i \(0.956607\pi\)
\(294\) 3.23607 0.188731
\(295\) 0 0
\(296\) −10.9443 −0.636123
\(297\) 14.4721i 0.839759i
\(298\) 22.3607i 1.29532i
\(299\) −4.94427 −0.285935
\(300\) 0 0
\(301\) −1.52786 −0.0880646
\(302\) − 12.0000i − 0.690522i
\(303\) 45.8885i 2.63623i
\(304\) 2.76393 0.158522
\(305\) 0 0
\(306\) −48.3607 −2.76460
\(307\) − 32.0689i − 1.83027i −0.403150 0.915134i \(-0.632085\pi\)
0.403150 0.915134i \(-0.367915\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) 9.52786 0.542021
\(310\) 0 0
\(311\) 5.41641 0.307136 0.153568 0.988138i \(-0.450924\pi\)
0.153568 + 0.988138i \(0.450924\pi\)
\(312\) 4.00000i 0.226455i
\(313\) 28.4721i 1.60934i 0.593722 + 0.804670i \(0.297658\pi\)
−0.593722 + 0.804670i \(0.702342\pi\)
\(314\) −18.6525 −1.05262
\(315\) 0 0
\(316\) 0 0
\(317\) 13.0557i 0.733283i 0.930362 + 0.366641i \(0.119492\pi\)
−0.930362 + 0.366641i \(0.880508\pi\)
\(318\) 1.52786i 0.0856784i
\(319\) 4.47214 0.250392
\(320\) 0 0
\(321\) 20.9443 1.16900
\(322\) − 4.00000i − 0.222911i
\(323\) 17.8885i 0.995345i
\(324\) −24.4164 −1.35647
\(325\) 0 0
\(326\) 7.41641 0.410757
\(327\) − 32.3607i − 1.78955i
\(328\) 6.47214i 0.357364i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 0.944272 0.0519019 0.0259509 0.999663i \(-0.491739\pi\)
0.0259509 + 0.999663i \(0.491739\pi\)
\(332\) − 10.1803i − 0.558719i
\(333\) − 81.7771i − 4.48136i
\(334\) 15.4164 0.843548
\(335\) 0 0
\(336\) −3.23607 −0.176542
\(337\) − 18.0000i − 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) − 11.4721i − 0.624002i
\(339\) 27.4164 1.48905
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 20.6525i 1.11676i
\(343\) 1.00000i 0.0539949i
\(344\) 1.52786 0.0823769
\(345\) 0 0
\(346\) 1.23607 0.0664514
\(347\) 6.47214i 0.347442i 0.984795 + 0.173721i \(0.0555792\pi\)
−0.984795 + 0.173721i \(0.944421\pi\)
\(348\) 14.4721i 0.775788i
\(349\) −8.29180 −0.443850 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) − 1.00000i − 0.0533002i
\(353\) − 34.9443i − 1.85990i −0.367691 0.929948i \(-0.619852\pi\)
0.367691 0.929948i \(-0.380148\pi\)
\(354\) 23.4164 1.24457
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) − 20.9443i − 1.10849i
\(358\) 8.94427i 0.472719i
\(359\) 26.8328 1.41618 0.708091 0.706121i \(-0.249557\pi\)
0.708091 + 0.706121i \(0.249557\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) − 4.76393i − 0.250387i
\(363\) − 3.23607i − 0.169850i
\(364\) −1.23607 −0.0647876
\(365\) 0 0
\(366\) 16.9443 0.885691
\(367\) − 21.4164i − 1.11793i −0.829192 0.558964i \(-0.811199\pi\)
0.829192 0.558964i \(-0.188801\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −48.3607 −2.51756
\(370\) 0 0
\(371\) −0.472136 −0.0245121
\(372\) 6.47214i 0.335565i
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 6.47214 0.334666
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 5.52786i 0.284699i
\(378\) − 14.4721i − 0.744366i
\(379\) −5.52786 −0.283947 −0.141974 0.989870i \(-0.545345\pi\)
−0.141974 + 0.989870i \(0.545345\pi\)
\(380\) 0 0
\(381\) 38.8328 1.98947
\(382\) − 6.47214i − 0.331143i
\(383\) 11.8885i 0.607476i 0.952756 + 0.303738i \(0.0982348\pi\)
−0.952756 + 0.303738i \(0.901765\pi\)
\(384\) 3.23607 0.165140
\(385\) 0 0
\(386\) 2.94427 0.149859
\(387\) 11.4164i 0.580329i
\(388\) 3.52786i 0.179100i
\(389\) −6.58359 −0.333801 −0.166901 0.985974i \(-0.553376\pi\)
−0.166901 + 0.985974i \(0.553376\pi\)
\(390\) 0 0
\(391\) −25.8885 −1.30924
\(392\) − 1.00000i − 0.0505076i
\(393\) − 29.8885i − 1.50768i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 7.47214 0.375489
\(397\) 10.2918i 0.516530i 0.966074 + 0.258265i \(0.0831508\pi\)
−0.966074 + 0.258265i \(0.916849\pi\)
\(398\) 1.05573i 0.0529189i
\(399\) −8.94427 −0.447774
\(400\) 0 0
\(401\) −30.3607 −1.51614 −0.758070 0.652173i \(-0.773857\pi\)
−0.758070 + 0.652173i \(0.773857\pi\)
\(402\) − 49.8885i − 2.48821i
\(403\) 2.47214i 0.123146i
\(404\) 14.1803 0.705498
\(405\) 0 0
\(406\) −4.47214 −0.221948
\(407\) 10.9443i 0.542487i
\(408\) 20.9443i 1.03690i
\(409\) 23.4164 1.15787 0.578933 0.815375i \(-0.303469\pi\)
0.578933 + 0.815375i \(0.303469\pi\)
\(410\) 0 0
\(411\) −51.4164 −2.53618
\(412\) − 2.94427i − 0.145054i
\(413\) 7.23607i 0.356064i
\(414\) −29.8885 −1.46894
\(415\) 0 0
\(416\) 1.23607 0.0606032
\(417\) − 26.8328i − 1.31401i
\(418\) − 2.76393i − 0.135188i
\(419\) −12.7639 −0.623559 −0.311779 0.950155i \(-0.600925\pi\)
−0.311779 + 0.950155i \(0.600925\pi\)
\(420\) 0 0
\(421\) 7.52786 0.366886 0.183443 0.983030i \(-0.441276\pi\)
0.183443 + 0.983030i \(0.441276\pi\)
\(422\) 22.4721i 1.09393i
\(423\) − 14.9443i − 0.726615i
\(424\) 0.472136 0.0229289
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 5.23607i 0.253391i
\(428\) − 6.47214i − 0.312842i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 40.9443 1.97222 0.986108 0.166105i \(-0.0531190\pi\)
0.986108 + 0.166105i \(0.0531190\pi\)
\(432\) 14.4721i 0.696291i
\(433\) 19.5279i 0.938449i 0.883079 + 0.469225i \(0.155467\pi\)
−0.883079 + 0.469225i \(0.844533\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 11.0557i 0.528867i
\(438\) 16.0000i 0.764510i
\(439\) 8.94427 0.426887 0.213443 0.976955i \(-0.431532\pi\)
0.213443 + 0.976955i \(0.431532\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) 8.00000i 0.380521i
\(443\) − 7.05573i − 0.335228i −0.985853 0.167614i \(-0.946394\pi\)
0.985853 0.167614i \(-0.0536062\pi\)
\(444\) −35.4164 −1.68079
\(445\) 0 0
\(446\) 8.47214 0.401167
\(447\) 72.3607i 3.42254i
\(448\) 1.00000i 0.0472456i
\(449\) −1.05573 −0.0498229 −0.0249114 0.999690i \(-0.507930\pi\)
−0.0249114 + 0.999690i \(0.507930\pi\)
\(450\) 0 0
\(451\) 6.47214 0.304761
\(452\) − 8.47214i − 0.398496i
\(453\) − 38.8328i − 1.82452i
\(454\) 14.7639 0.692906
\(455\) 0 0
\(456\) 8.94427 0.418854
\(457\) − 9.05573i − 0.423609i −0.977312 0.211805i \(-0.932066\pi\)
0.977312 0.211805i \(-0.0679340\pi\)
\(458\) 12.7639i 0.596419i
\(459\) −93.6656 −4.37194
\(460\) 0 0
\(461\) 29.2361 1.36166 0.680830 0.732442i \(-0.261619\pi\)
0.680830 + 0.732442i \(0.261619\pi\)
\(462\) 3.23607i 0.150556i
\(463\) − 21.5279i − 1.00048i −0.865885 0.500242i \(-0.833244\pi\)
0.865885 0.500242i \(-0.166756\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) 2.94427 0.136391
\(467\) − 13.1246i − 0.607335i −0.952778 0.303667i \(-0.901789\pi\)
0.952778 0.303667i \(-0.0982111\pi\)
\(468\) 9.23607i 0.426937i
\(469\) 15.4164 0.711864
\(470\) 0 0
\(471\) −60.3607 −2.78127
\(472\) − 7.23607i − 0.333067i
\(473\) − 1.52786i − 0.0702513i
\(474\) 0 0
\(475\) 0 0
\(476\) −6.47214 −0.296650
\(477\) 3.52786i 0.161530i
\(478\) 20.0000i 0.914779i
\(479\) 32.3607 1.47860 0.739299 0.673378i \(-0.235157\pi\)
0.739299 + 0.673378i \(0.235157\pi\)
\(480\) 0 0
\(481\) −13.5279 −0.616818
\(482\) 11.4164i 0.520003i
\(483\) − 12.9443i − 0.588985i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −35.5967 −1.61470
\(487\) 0.944272i 0.0427890i 0.999771 + 0.0213945i \(0.00681061\pi\)
−0.999771 + 0.0213945i \(0.993189\pi\)
\(488\) − 5.23607i − 0.237026i
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 0.944272 0.0426144 0.0213072 0.999773i \(-0.493217\pi\)
0.0213072 + 0.999773i \(0.493217\pi\)
\(492\) 20.9443i 0.944241i
\(493\) 28.9443i 1.30358i
\(494\) 3.41641 0.153711
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 2.47214i 0.110890i
\(498\) − 32.9443i − 1.47627i
\(499\) −12.3607 −0.553340 −0.276670 0.960965i \(-0.589231\pi\)
−0.276670 + 0.960965i \(0.589231\pi\)
\(500\) 0 0
\(501\) 49.8885 2.22886
\(502\) − 24.7639i − 1.10527i
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) −7.47214 −0.332835
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) − 37.1246i − 1.64876i
\(508\) − 12.0000i − 0.532414i
\(509\) 34.0689 1.51008 0.755038 0.655681i \(-0.227618\pi\)
0.755038 + 0.655681i \(0.227618\pi\)
\(510\) 0 0
\(511\) −4.94427 −0.218722
\(512\) − 1.00000i − 0.0441942i
\(513\) 40.0000i 1.76604i
\(514\) 10.9443 0.482731
\(515\) 0 0
\(516\) 4.94427 0.217659
\(517\) 2.00000i 0.0879599i
\(518\) − 10.9443i − 0.480864i
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 34.3607 1.50537 0.752684 0.658382i \(-0.228759\pi\)
0.752684 + 0.658382i \(0.228759\pi\)
\(522\) 33.4164i 1.46260i
\(523\) − 27.7082i − 1.21160i −0.795619 0.605798i \(-0.792854\pi\)
0.795619 0.605798i \(-0.207146\pi\)
\(524\) −9.23607 −0.403480
\(525\) 0 0
\(526\) 12.9443 0.564397
\(527\) 12.9443i 0.563861i
\(528\) − 3.23607i − 0.140832i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 54.0689 2.34639
\(532\) 2.76393i 0.119832i
\(533\) 8.00000i 0.346518i
\(534\) 32.3607 1.40038
\(535\) 0 0
\(536\) −15.4164 −0.665887
\(537\) 28.9443i 1.24904i
\(538\) − 27.2361i − 1.17423i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −9.05573 −0.389336 −0.194668 0.980869i \(-0.562363\pi\)
−0.194668 + 0.980869i \(0.562363\pi\)
\(542\) 16.9443i 0.727819i
\(543\) − 15.4164i − 0.661581i
\(544\) 6.47214 0.277491
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) − 16.9443i − 0.724485i −0.932084 0.362242i \(-0.882011\pi\)
0.932084 0.362242i \(-0.117989\pi\)
\(548\) 15.8885i 0.678725i
\(549\) 39.1246 1.66980
\(550\) 0 0
\(551\) 12.3607 0.526583
\(552\) 12.9443i 0.550945i
\(553\) 0 0
\(554\) −12.4721 −0.529890
\(555\) 0 0
\(556\) −8.29180 −0.351650
\(557\) 28.8328i 1.22169i 0.791752 + 0.610843i \(0.209169\pi\)
−0.791752 + 0.610843i \(0.790831\pi\)
\(558\) 14.9443i 0.632641i
\(559\) 1.88854 0.0798769
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 24.8328i 1.04751i
\(563\) 26.7639i 1.12797i 0.825787 + 0.563983i \(0.190732\pi\)
−0.825787 + 0.563983i \(0.809268\pi\)
\(564\) −6.47214 −0.272526
\(565\) 0 0
\(566\) −16.6525 −0.699956
\(567\) − 24.4164i − 1.02539i
\(568\) − 2.47214i − 0.103729i
\(569\) −16.8328 −0.705668 −0.352834 0.935686i \(-0.614782\pi\)
−0.352834 + 0.935686i \(0.614782\pi\)
\(570\) 0 0
\(571\) −45.8885 −1.92038 −0.960188 0.279355i \(-0.909879\pi\)
−0.960188 + 0.279355i \(0.909879\pi\)
\(572\) − 1.23607i − 0.0516826i
\(573\) − 20.9443i − 0.874960i
\(574\) −6.47214 −0.270142
\(575\) 0 0
\(576\) 7.47214 0.311339
\(577\) − 9.05573i − 0.376995i −0.982074 0.188497i \(-0.939638\pi\)
0.982074 0.188497i \(-0.0603617\pi\)
\(578\) 24.8885i 1.03523i
\(579\) 9.52786 0.395965
\(580\) 0 0
\(581\) 10.1803 0.422352
\(582\) 11.4164i 0.473225i
\(583\) − 0.472136i − 0.0195539i
\(584\) 4.94427 0.204595
\(585\) 0 0
\(586\) 4.65248 0.192192
\(587\) 28.1803i 1.16313i 0.813501 + 0.581564i \(0.197559\pi\)
−0.813501 + 0.581564i \(0.802441\pi\)
\(588\) − 3.23607i − 0.133453i
\(589\) 5.52786 0.227772
\(590\) 0 0
\(591\) −58.2492 −2.39605
\(592\) 10.9443i 0.449807i
\(593\) 24.0000i 0.985562i 0.870153 + 0.492781i \(0.164020\pi\)
−0.870153 + 0.492781i \(0.835980\pi\)
\(594\) 14.4721 0.593799
\(595\) 0 0
\(596\) 22.3607 0.915929
\(597\) 3.41641i 0.139824i
\(598\) 4.94427i 0.202186i
\(599\) −12.3607 −0.505044 −0.252522 0.967591i \(-0.581260\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(600\) 0 0
\(601\) 18.8328 0.768207 0.384103 0.923290i \(-0.374511\pi\)
0.384103 + 0.923290i \(0.374511\pi\)
\(602\) 1.52786i 0.0622711i
\(603\) − 115.193i − 4.69104i
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 45.8885 1.86409
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) − 2.76393i − 0.112092i
\(609\) −14.4721 −0.586441
\(610\) 0 0
\(611\) −2.47214 −0.100012
\(612\) 48.3607i 1.95486i
\(613\) 19.5279i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(614\) −32.0689 −1.29419
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 5.41641i 0.218056i 0.994039 + 0.109028i \(0.0347739\pi\)
−0.994039 + 0.109028i \(0.965226\pi\)
\(618\) − 9.52786i − 0.383267i
\(619\) 48.5410 1.95103 0.975514 0.219937i \(-0.0705851\pi\)
0.975514 + 0.219937i \(0.0705851\pi\)
\(620\) 0 0
\(621\) −57.8885 −2.32299
\(622\) − 5.41641i − 0.217178i
\(623\) 10.0000i 0.400642i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 28.4721 1.13798
\(627\) − 8.94427i − 0.357200i
\(628\) 18.6525i 0.744315i
\(629\) −70.8328 −2.82429
\(630\) 0 0
\(631\) −4.58359 −0.182470 −0.0912350 0.995829i \(-0.529081\pi\)
−0.0912350 + 0.995829i \(0.529081\pi\)
\(632\) 0 0
\(633\) 72.7214i 2.89041i
\(634\) 13.0557 0.518509
\(635\) 0 0
\(636\) 1.52786 0.0605838
\(637\) − 1.23607i − 0.0489748i
\(638\) − 4.47214i − 0.177054i
\(639\) 18.4721 0.730746
\(640\) 0 0
\(641\) 36.4721 1.44056 0.720281 0.693682i \(-0.244013\pi\)
0.720281 + 0.693682i \(0.244013\pi\)
\(642\) − 20.9443i − 0.826604i
\(643\) − 23.2361i − 0.916341i −0.888864 0.458171i \(-0.848505\pi\)
0.888864 0.458171i \(-0.151495\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 17.8885 0.703815
\(647\) − 24.8328i − 0.976279i −0.872766 0.488139i \(-0.837676\pi\)
0.872766 0.488139i \(-0.162324\pi\)
\(648\) 24.4164i 0.959167i
\(649\) −7.23607 −0.284041
\(650\) 0 0
\(651\) −6.47214 −0.253663
\(652\) − 7.41641i − 0.290449i
\(653\) 1.63932i 0.0641516i 0.999485 + 0.0320758i \(0.0102118\pi\)
−0.999485 + 0.0320758i \(0.989788\pi\)
\(654\) −32.3607 −1.26540
\(655\) 0 0
\(656\) 6.47214 0.252694
\(657\) 36.9443i 1.44133i
\(658\) − 2.00000i − 0.0779681i
\(659\) −43.4164 −1.69126 −0.845632 0.533767i \(-0.820776\pi\)
−0.845632 + 0.533767i \(0.820776\pi\)
\(660\) 0 0
\(661\) 37.1246 1.44398 0.721990 0.691903i \(-0.243228\pi\)
0.721990 + 0.691903i \(0.243228\pi\)
\(662\) − 0.944272i − 0.0367002i
\(663\) 25.8885i 1.00543i
\(664\) −10.1803 −0.395074
\(665\) 0 0
\(666\) −81.7771 −3.16880
\(667\) 17.8885i 0.692647i
\(668\) − 15.4164i − 0.596479i
\(669\) 27.4164 1.05998
\(670\) 0 0
\(671\) −5.23607 −0.202136
\(672\) 3.23607i 0.124834i
\(673\) 31.8885i 1.22921i 0.788834 + 0.614607i \(0.210685\pi\)
−0.788834 + 0.614607i \(0.789315\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −11.4721 −0.441236
\(677\) − 32.0689i − 1.23251i −0.787548 0.616254i \(-0.788650\pi\)
0.787548 0.616254i \(-0.211350\pi\)
\(678\) − 27.4164i − 1.05292i
\(679\) −3.52786 −0.135387
\(680\) 0 0
\(681\) 47.7771 1.83082
\(682\) − 2.00000i − 0.0765840i
\(683\) 15.0557i 0.576091i 0.957617 + 0.288046i \(0.0930055\pi\)
−0.957617 + 0.288046i \(0.906994\pi\)
\(684\) 20.6525 0.789667
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 41.3050i 1.57588i
\(688\) − 1.52786i − 0.0582493i
\(689\) 0.583592 0.0222331
\(690\) 0 0
\(691\) −18.6525 −0.709574 −0.354787 0.934947i \(-0.615447\pi\)
−0.354787 + 0.934947i \(0.615447\pi\)
\(692\) − 1.23607i − 0.0469883i
\(693\) 7.47214i 0.283843i
\(694\) 6.47214 0.245679
\(695\) 0 0
\(696\) 14.4721 0.548565
\(697\) 41.8885i 1.58664i
\(698\) 8.29180i 0.313849i
\(699\) 9.52786 0.360377
\(700\) 0 0
\(701\) 46.7214 1.76464 0.882321 0.470649i \(-0.155980\pi\)
0.882321 + 0.470649i \(0.155980\pi\)
\(702\) 17.8885i 0.675160i
\(703\) 30.2492i 1.14087i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −34.9443 −1.31515
\(707\) 14.1803i 0.533307i
\(708\) − 23.4164i − 0.880042i
\(709\) 4.47214 0.167955 0.0839773 0.996468i \(-0.473238\pi\)
0.0839773 + 0.996468i \(0.473238\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 10.0000i − 0.374766i
\(713\) 8.00000i 0.299602i
\(714\) −20.9443 −0.783820
\(715\) 0 0
\(716\) 8.94427 0.334263
\(717\) 64.7214i 2.41706i
\(718\) − 26.8328i − 1.00139i
\(719\) 36.8328 1.37363 0.686816 0.726831i \(-0.259008\pi\)
0.686816 + 0.726831i \(0.259008\pi\)
\(720\) 0 0
\(721\) 2.94427 0.109650
\(722\) 11.3607i 0.422801i
\(723\) 36.9443i 1.37397i
\(724\) −4.76393 −0.177050
\(725\) 0 0
\(726\) −3.23607 −0.120102
\(727\) − 18.0000i − 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 1.23607i 0.0458117i
\(729\) −41.9443 −1.55349
\(730\) 0 0
\(731\) 9.88854 0.365741
\(732\) − 16.9443i − 0.626278i
\(733\) 8.87539i 0.327820i 0.986475 + 0.163910i \(0.0524107\pi\)
−0.986475 + 0.163910i \(0.947589\pi\)
\(734\) −21.4164 −0.790494
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 15.4164i 0.567871i
\(738\) 48.3607i 1.78018i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 11.0557 0.406142
\(742\) 0.472136i 0.0173327i
\(743\) − 13.8885i − 0.509521i −0.967004 0.254761i \(-0.918003\pi\)
0.967004 0.254761i \(-0.0819967\pi\)
\(744\) 6.47214 0.237280
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) − 76.0689i − 2.78321i
\(748\) − 6.47214i − 0.236645i
\(749\) 6.47214 0.236487
\(750\) 0 0
\(751\) 0.944272 0.0344570 0.0172285 0.999852i \(-0.494516\pi\)
0.0172285 + 0.999852i \(0.494516\pi\)
\(752\) 2.00000i 0.0729325i
\(753\) − 80.1378i − 2.92038i
\(754\) 5.52786 0.201313
\(755\) 0 0
\(756\) −14.4721 −0.526346
\(757\) − 39.3050i − 1.42856i −0.699859 0.714281i \(-0.746754\pi\)
0.699859 0.714281i \(-0.253246\pi\)
\(758\) 5.52786i 0.200781i
\(759\) 12.9443 0.469847
\(760\) 0 0
\(761\) 15.4164 0.558844 0.279422 0.960168i \(-0.409857\pi\)
0.279422 + 0.960168i \(0.409857\pi\)
\(762\) − 38.8328i − 1.40676i
\(763\) − 10.0000i − 0.362024i
\(764\) −6.47214 −0.234154
\(765\) 0 0
\(766\) 11.8885 0.429551
\(767\) − 8.94427i − 0.322959i
\(768\) − 3.23607i − 0.116772i
\(769\) 16.5836 0.598020 0.299010 0.954250i \(-0.403344\pi\)
0.299010 + 0.954250i \(0.403344\pi\)
\(770\) 0 0
\(771\) 35.4164 1.27549
\(772\) − 2.94427i − 0.105967i
\(773\) 2.29180i 0.0824302i 0.999150 + 0.0412151i \(0.0131229\pi\)
−0.999150 + 0.0412151i \(0.986877\pi\)
\(774\) 11.4164 0.410354
\(775\) 0 0
\(776\) 3.52786 0.126643
\(777\) − 35.4164i − 1.27056i
\(778\) 6.58359i 0.236033i
\(779\) 17.8885 0.640924
\(780\) 0 0
\(781\) −2.47214 −0.0884600
\(782\) 25.8885i 0.925772i
\(783\) 64.7214i 2.31295i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −29.8885 −1.06609
\(787\) 5.81966i 0.207448i 0.994606 + 0.103724i \(0.0330759\pi\)
−0.994606 + 0.103724i \(0.966924\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 41.8885 1.49127
\(790\) 0 0
\(791\) 8.47214 0.301234
\(792\) − 7.47214i − 0.265511i
\(793\) − 6.47214i − 0.229832i
\(794\) 10.2918 0.365242
\(795\) 0 0
\(796\) 1.05573 0.0374193
\(797\) − 7.59675i − 0.269091i −0.990907 0.134545i \(-0.957043\pi\)
0.990907 0.134545i \(-0.0429574\pi\)
\(798\) 8.94427i 0.316624i
\(799\) −12.9443 −0.457935
\(800\) 0 0
\(801\) 74.7214 2.64015
\(802\) 30.3607i 1.07207i
\(803\) − 4.94427i − 0.174480i
\(804\) −49.8885 −1.75943
\(805\) 0 0
\(806\) 2.47214 0.0870773
\(807\) − 88.1378i − 3.10260i
\(808\) − 14.1803i − 0.498863i
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 0 0
\(811\) 9.23607 0.324322 0.162161 0.986764i \(-0.448154\pi\)
0.162161 + 0.986764i \(0.448154\pi\)
\(812\) 4.47214i 0.156941i
\(813\) 54.8328i 1.92307i
\(814\) 10.9443 0.383597
\(815\) 0 0
\(816\) 20.9443 0.733196
\(817\) − 4.22291i − 0.147741i
\(818\) − 23.4164i − 0.818736i
\(819\) −9.23607 −0.322734
\(820\) 0 0
\(821\) 25.4164 0.887039 0.443519 0.896265i \(-0.353730\pi\)
0.443519 + 0.896265i \(0.353730\pi\)
\(822\) 51.4164i 1.79335i
\(823\) 34.2492i 1.19385i 0.802296 + 0.596926i \(0.203612\pi\)
−0.802296 + 0.596926i \(0.796388\pi\)
\(824\) −2.94427 −0.102569
\(825\) 0 0
\(826\) 7.23607 0.251775
\(827\) 0.944272i 0.0328356i 0.999865 + 0.0164178i \(0.00522618\pi\)
−0.999865 + 0.0164178i \(0.994774\pi\)
\(828\) 29.8885i 1.03870i
\(829\) 1.70820 0.0593284 0.0296642 0.999560i \(-0.490556\pi\)
0.0296642 + 0.999560i \(0.490556\pi\)
\(830\) 0 0
\(831\) −40.3607 −1.40010
\(832\) − 1.23607i − 0.0428529i
\(833\) − 6.47214i − 0.224246i
\(834\) −26.8328 −0.929144
\(835\) 0 0
\(836\) −2.76393 −0.0955926
\(837\) 28.9443i 1.00046i
\(838\) 12.7639i 0.440923i
\(839\) 36.8328 1.27161 0.635805 0.771850i \(-0.280668\pi\)
0.635805 + 0.771850i \(0.280668\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) − 7.52786i − 0.259427i
\(843\) 80.3607i 2.76777i
\(844\) 22.4721 0.773523
\(845\) 0 0
\(846\) −14.9443 −0.513795
\(847\) − 1.00000i − 0.0343604i
\(848\) − 0.472136i − 0.0162132i
\(849\) −53.8885 −1.84945
\(850\) 0 0
\(851\) −43.7771 −1.50066
\(852\) − 8.00000i − 0.274075i
\(853\) − 34.5410i − 1.18266i −0.806429 0.591331i \(-0.798603\pi\)
0.806429 0.591331i \(-0.201397\pi\)
\(854\) 5.23607 0.179175
\(855\) 0 0
\(856\) −6.47214 −0.221213
\(857\) 37.5279i 1.28193i 0.767571 + 0.640964i \(0.221465\pi\)
−0.767571 + 0.640964i \(0.778535\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 25.1246 0.857241 0.428620 0.903485i \(-0.359000\pi\)
0.428620 + 0.903485i \(0.359000\pi\)
\(860\) 0 0
\(861\) −20.9443 −0.713779
\(862\) − 40.9443i − 1.39457i
\(863\) 27.4164i 0.933265i 0.884451 + 0.466633i \(0.154533\pi\)
−0.884451 + 0.466633i \(0.845467\pi\)
\(864\) 14.4721 0.492352
\(865\) 0 0
\(866\) 19.5279 0.663584
\(867\) 80.5410i 2.73532i
\(868\) 2.00000i 0.0678844i
\(869\) 0 0
\(870\) 0 0
\(871\) −19.0557 −0.645679
\(872\) 10.0000i 0.338643i
\(873\) 26.3607i 0.892174i
\(874\) 11.0557 0.373966
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) − 26.9443i − 0.909843i −0.890531 0.454922i \(-0.849667\pi\)
0.890531 0.454922i \(-0.150333\pi\)
\(878\) − 8.94427i − 0.301855i
\(879\) 15.0557 0.507817
\(880\) 0 0
\(881\) −24.8328 −0.836639 −0.418319 0.908300i \(-0.637381\pi\)
−0.418319 + 0.908300i \(0.637381\pi\)
\(882\) − 7.47214i − 0.251600i
\(883\) 50.8328i 1.71066i 0.518083 + 0.855330i \(0.326646\pi\)
−0.518083 + 0.855330i \(0.673354\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −7.05573 −0.237042
\(887\) − 0.360680i − 0.0121104i −0.999982 0.00605522i \(-0.998073\pi\)
0.999982 0.00605522i \(-0.00192745\pi\)
\(888\) 35.4164i 1.18850i
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 24.4164 0.817980
\(892\) − 8.47214i − 0.283668i
\(893\) 5.52786i 0.184983i
\(894\) 72.3607 2.42010
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 16.0000i 0.534224i
\(898\) 1.05573i 0.0352301i
\(899\) 8.94427 0.298308
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) − 6.47214i − 0.215499i
\(903\) 4.94427i 0.164535i
\(904\) −8.47214 −0.281779
\(905\) 0 0
\(906\) −38.8328 −1.29013
\(907\) − 20.3607i − 0.676065i −0.941134 0.338033i \(-0.890239\pi\)
0.941134 0.338033i \(-0.109761\pi\)
\(908\) − 14.7639i − 0.489958i
\(909\) 105.957 3.51439
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) − 8.94427i − 0.296174i
\(913\) 10.1803i 0.336920i
\(914\) −9.05573 −0.299537
\(915\) 0 0
\(916\) 12.7639 0.421732
\(917\) − 9.23607i − 0.305002i
\(918\) 93.6656i 3.09143i
\(919\) 57.8885 1.90957 0.954783 0.297302i \(-0.0960869\pi\)
0.954783 + 0.297302i \(0.0960869\pi\)
\(920\) 0 0
\(921\) −103.777 −3.41957
\(922\) − 29.2361i − 0.962839i
\(923\) − 3.05573i − 0.100581i
\(924\) 3.23607 0.106459
\(925\) 0 0
\(926\) −21.5279 −0.707450
\(927\) − 22.0000i − 0.722575i
\(928\) − 4.47214i − 0.146805i
\(929\) 40.2492 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(930\) 0 0
\(931\) −2.76393 −0.0905842
\(932\) − 2.94427i − 0.0964428i
\(933\) − 17.5279i − 0.573837i
\(934\) −13.1246 −0.429450
\(935\) 0 0
\(936\) 9.23607 0.301890
\(937\) 20.9443i 0.684220i 0.939660 + 0.342110i \(0.111141\pi\)
−0.939660 + 0.342110i \(0.888859\pi\)
\(938\) − 15.4164i − 0.503364i
\(939\) 92.1378 3.00680
\(940\) 0 0
\(941\) −34.1803 −1.11425 −0.557124 0.830430i \(-0.688095\pi\)
−0.557124 + 0.830430i \(0.688095\pi\)
\(942\) 60.3607i 1.96666i
\(943\) 25.8885i 0.843047i
\(944\) −7.23607 −0.235514
\(945\) 0 0
\(946\) −1.52786 −0.0496751
\(947\) 0.944272i 0.0306847i 0.999882 + 0.0153424i \(0.00488382\pi\)
−0.999882 + 0.0153424i \(0.995116\pi\)
\(948\) 0 0
\(949\) 6.11146 0.198386
\(950\) 0 0
\(951\) 42.2492 1.37002
\(952\) 6.47214i 0.209763i
\(953\) 5.05573i 0.163771i 0.996642 + 0.0818855i \(0.0260942\pi\)
−0.996642 + 0.0818855i \(0.973906\pi\)
\(954\) 3.52786 0.114219
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) − 14.4721i − 0.467818i
\(958\) − 32.3607i − 1.04553i
\(959\) −15.8885 −0.513068
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 13.5279i 0.436156i
\(963\) − 48.3607i − 1.55840i
\(964\) 11.4164 0.367698
\(965\) 0 0
\(966\) −12.9443 −0.416475
\(967\) − 10.1115i − 0.325163i −0.986695 0.162581i \(-0.948018\pi\)
0.986695 0.162581i \(-0.0519820\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 57.8885 1.85965
\(970\) 0 0
\(971\) −16.5410 −0.530827 −0.265413 0.964135i \(-0.585508\pi\)
−0.265413 + 0.964135i \(0.585508\pi\)
\(972\) 35.5967i 1.14177i
\(973\) − 8.29180i − 0.265823i
\(974\) 0.944272 0.0302564
\(975\) 0 0
\(976\) −5.23607 −0.167602
\(977\) − 24.8328i − 0.794472i −0.917716 0.397236i \(-0.869969\pi\)
0.917716 0.397236i \(-0.130031\pi\)
\(978\) − 24.0000i − 0.767435i
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −74.7214 −2.38567
\(982\) − 0.944272i − 0.0301329i
\(983\) 14.0000i 0.446531i 0.974758 + 0.223265i \(0.0716716\pi\)
−0.974758 + 0.223265i \(0.928328\pi\)
\(984\) 20.9443 0.667679
\(985\) 0 0
\(986\) 28.9443 0.921773
\(987\) − 6.47214i − 0.206010i
\(988\) − 3.41641i − 0.108690i
\(989\) 6.11146 0.194333
\(990\) 0 0
\(991\) 44.3607 1.40916 0.704582 0.709623i \(-0.251135\pi\)
0.704582 + 0.709623i \(0.251135\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) − 3.05573i − 0.0969706i
\(994\) 2.47214 0.0784114
\(995\) 0 0
\(996\) −32.9443 −1.04388
\(997\) − 1.81966i − 0.0576292i −0.999585 0.0288146i \(-0.990827\pi\)
0.999585 0.0288146i \(-0.00917324\pi\)
\(998\) 12.3607i 0.391270i
\(999\) −158.387 −5.01114
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 3850.2.c.q.1849.1 4
5.2 odd 4 154.2.a.d.1.1 2
5.3 odd 4 3850.2.a.bj.1.2 2
5.4 even 2 inner 3850.2.c.q.1849.4 4
15.2 even 4 1386.2.a.m.1.1 2
20.7 even 4 1232.2.a.p.1.2 2
35.2 odd 12 1078.2.e.q.67.2 4
35.12 even 12 1078.2.e.n.67.1 4
35.17 even 12 1078.2.e.n.177.1 4
35.27 even 4 1078.2.a.w.1.2 2
35.32 odd 12 1078.2.e.q.177.2 4
40.27 even 4 4928.2.a.bk.1.1 2
40.37 odd 4 4928.2.a.bt.1.2 2
55.32 even 4 1694.2.a.l.1.1 2
105.62 odd 4 9702.2.a.cu.1.2 2
140.27 odd 4 8624.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 2 5.2 odd 4
1078.2.a.w.1.2 2 35.27 even 4
1078.2.e.n.67.1 4 35.12 even 12
1078.2.e.n.177.1 4 35.17 even 12
1078.2.e.q.67.2 4 35.2 odd 12
1078.2.e.q.177.2 4 35.32 odd 12
1232.2.a.p.1.2 2 20.7 even 4
1386.2.a.m.1.1 2 15.2 even 4
1694.2.a.l.1.1 2 55.32 even 4
3850.2.a.bj.1.2 2 5.3 odd 4
3850.2.c.q.1849.1 4 1.1 even 1 trivial
3850.2.c.q.1849.4 4 5.4 even 2 inner
4928.2.a.bk.1.1 2 40.27 even 4
4928.2.a.bt.1.2 2 40.37 odd 4
8624.2.a.bf.1.1 2 140.27 odd 4
9702.2.a.cu.1.2 2 105.62 odd 4