Properties

Label 4020.2.q.m.3781.1
Level $4020$
Weight $2$
Character 4020.3781
Analytic conductor $32.100$
Analytic rank $0$
Dimension $24$
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] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3781.1
Character \(\chi\) \(=\) 4020.3781
Dual form 4020.2.q.m.841.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +(-2.18663 - 3.78736i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +(-2.18663 - 3.78736i) q^{7} +1.00000 q^{9} +(1.49547 + 2.59023i) q^{11} +(-1.04087 + 1.80283i) q^{13} +1.00000 q^{15} +(-0.454199 + 0.786696i) q^{17} +(3.53072 - 6.11538i) q^{19} +(-2.18663 - 3.78736i) q^{21} +(3.49083 - 6.04630i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(3.93896 + 6.82248i) q^{29} +(-3.83972 - 6.65058i) q^{31} +(1.49547 + 2.59023i) q^{33} +(-2.18663 - 3.78736i) q^{35} +(-0.417228 + 0.722660i) q^{37} +(-1.04087 + 1.80283i) q^{39} +(-1.33572 - 2.31353i) q^{41} -12.0882 q^{43} +1.00000 q^{45} +(0.436600 + 0.756213i) q^{47} +(-6.06274 + 10.5010i) q^{49} +(-0.454199 + 0.786696i) q^{51} +7.39196 q^{53} +(1.49547 + 2.59023i) q^{55} +(3.53072 - 6.11538i) q^{57} -1.81848 q^{59} +(1.99000 - 3.44678i) q^{61} +(-2.18663 - 3.78736i) q^{63} +(-1.04087 + 1.80283i) q^{65} +(2.50950 - 7.79118i) q^{67} +(3.49083 - 6.04630i) q^{69} +(0.0217755 + 0.0377163i) q^{71} +(4.49932 - 7.79305i) q^{73} +1.00000 q^{75} +(6.54008 - 11.3278i) q^{77} +(-0.453494 - 0.785474i) q^{79} +1.00000 q^{81} +(1.41204 - 2.44573i) q^{83} +(-0.454199 + 0.786696i) q^{85} +(3.93896 + 6.82248i) q^{87} +2.76333 q^{89} +9.10397 q^{91} +(-3.83972 - 6.65058i) q^{93} +(3.53072 - 6.11538i) q^{95} +(0.274039 - 0.474649i) q^{97} +(1.49547 + 2.59023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9} - 2 q^{13} + 24 q^{15} - 6 q^{19} - 3 q^{21} + 2 q^{23} + 24 q^{25} + 24 q^{27} + 7 q^{29} - 8 q^{31} - 3 q^{35} - 10 q^{37} - 2 q^{39} + 2 q^{41} + 28 q^{43} + 24 q^{45} - 3 q^{47} - 17 q^{49} + 36 q^{53} - 6 q^{57} - 10 q^{59} + 9 q^{61} - 3 q^{63} - 2 q^{65} - 46 q^{67} + 2 q^{69} - 12 q^{71} + 6 q^{73} + 24 q^{75} - 5 q^{77} + 2 q^{79} + 24 q^{81} + 11 q^{83} + 7 q^{87} + 52 q^{89} - 22 q^{91} - 8 q^{93} - 6 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.18663 3.78736i −0.826470 1.43149i −0.900791 0.434254i \(-0.857012\pi\)
0.0743207 0.997234i \(-0.476321\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.49547 + 2.59023i 0.450900 + 0.780982i 0.998442 0.0557959i \(-0.0177696\pi\)
−0.547542 + 0.836778i \(0.684436\pi\)
\(12\) 0 0
\(13\) −1.04087 + 1.80283i −0.288684 + 0.500016i −0.973496 0.228704i \(-0.926551\pi\)
0.684812 + 0.728720i \(0.259884\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −0.454199 + 0.786696i −0.110159 + 0.190802i −0.915834 0.401556i \(-0.868469\pi\)
0.805675 + 0.592358i \(0.201803\pi\)
\(18\) 0 0
\(19\) 3.53072 6.11538i 0.810002 1.40297i −0.102859 0.994696i \(-0.532799\pi\)
0.912862 0.408269i \(-0.133868\pi\)
\(20\) 0 0
\(21\) −2.18663 3.78736i −0.477163 0.826470i
\(22\) 0 0
\(23\) 3.49083 6.04630i 0.727889 1.26074i −0.229884 0.973218i \(-0.573835\pi\)
0.957774 0.287523i \(-0.0928319\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.93896 + 6.82248i 0.731447 + 1.26690i 0.956265 + 0.292502i \(0.0944879\pi\)
−0.224818 + 0.974401i \(0.572179\pi\)
\(30\) 0 0
\(31\) −3.83972 6.65058i −0.689633 1.19448i −0.971957 0.235161i \(-0.924438\pi\)
0.282323 0.959319i \(-0.408895\pi\)
\(32\) 0 0
\(33\) 1.49547 + 2.59023i 0.260327 + 0.450900i
\(34\) 0 0
\(35\) −2.18663 3.78736i −0.369609 0.640181i
\(36\) 0 0
\(37\) −0.417228 + 0.722660i −0.0685919 + 0.118805i −0.898282 0.439420i \(-0.855184\pi\)
0.829690 + 0.558225i \(0.188517\pi\)
\(38\) 0 0
\(39\) −1.04087 + 1.80283i −0.166672 + 0.288684i
\(40\) 0 0
\(41\) −1.33572 2.31353i −0.208604 0.361313i 0.742671 0.669657i \(-0.233559\pi\)
−0.951275 + 0.308343i \(0.900225\pi\)
\(42\) 0 0
\(43\) −12.0882 −1.84343 −0.921713 0.387872i \(-0.873210\pi\)
−0.921713 + 0.387872i \(0.873210\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0.436600 + 0.756213i 0.0636846 + 0.110305i 0.896110 0.443833i \(-0.146382\pi\)
−0.832425 + 0.554138i \(0.813048\pi\)
\(48\) 0 0
\(49\) −6.06274 + 10.5010i −0.866105 + 1.50014i
\(50\) 0 0
\(51\) −0.454199 + 0.786696i −0.0636006 + 0.110159i
\(52\) 0 0
\(53\) 7.39196 1.01536 0.507682 0.861545i \(-0.330502\pi\)
0.507682 + 0.861545i \(0.330502\pi\)
\(54\) 0 0
\(55\) 1.49547 + 2.59023i 0.201649 + 0.349266i
\(56\) 0 0
\(57\) 3.53072 6.11538i 0.467655 0.810002i
\(58\) 0 0
\(59\) −1.81848 −0.236746 −0.118373 0.992969i \(-0.537768\pi\)
−0.118373 + 0.992969i \(0.537768\pi\)
\(60\) 0 0
\(61\) 1.99000 3.44678i 0.254793 0.441315i −0.710046 0.704155i \(-0.751326\pi\)
0.964839 + 0.262840i \(0.0846592\pi\)
\(62\) 0 0
\(63\) −2.18663 3.78736i −0.275490 0.477163i
\(64\) 0 0
\(65\) −1.04087 + 1.80283i −0.129104 + 0.223614i
\(66\) 0 0
\(67\) 2.50950 7.79118i 0.306584 0.951844i
\(68\) 0 0
\(69\) 3.49083 6.04630i 0.420247 0.727889i
\(70\) 0 0
\(71\) 0.0217755 + 0.0377163i 0.00258428 + 0.00447610i 0.867315 0.497760i \(-0.165844\pi\)
−0.864730 + 0.502236i \(0.832511\pi\)
\(72\) 0 0
\(73\) 4.49932 7.79305i 0.526606 0.912108i −0.472914 0.881109i \(-0.656798\pi\)
0.999519 0.0309989i \(-0.00986884\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 6.54008 11.3278i 0.745311 1.29092i
\(78\) 0 0
\(79\) −0.453494 0.785474i −0.0510221 0.0883728i 0.839386 0.543535i \(-0.182915\pi\)
−0.890408 + 0.455162i \(0.849581\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.41204 2.44573i 0.154992 0.268454i −0.778064 0.628185i \(-0.783798\pi\)
0.933056 + 0.359731i \(0.117132\pi\)
\(84\) 0 0
\(85\) −0.454199 + 0.786696i −0.0492648 + 0.0853291i
\(86\) 0 0
\(87\) 3.93896 + 6.82248i 0.422301 + 0.731447i
\(88\) 0 0
\(89\) 2.76333 0.292912 0.146456 0.989217i \(-0.453213\pi\)
0.146456 + 0.989217i \(0.453213\pi\)
\(90\) 0 0
\(91\) 9.10397 0.954356
\(92\) 0 0
\(93\) −3.83972 6.65058i −0.398160 0.689633i
\(94\) 0 0
\(95\) 3.53072 6.11538i 0.362244 0.627425i
\(96\) 0 0
\(97\) 0.274039 0.474649i 0.0278244 0.0481933i −0.851778 0.523903i \(-0.824475\pi\)
0.879602 + 0.475710i \(0.157809\pi\)
\(98\) 0 0
\(99\) 1.49547 + 2.59023i 0.150300 + 0.260327i
\(100\) 0 0
\(101\) 0.346371 + 0.599931i 0.0344652 + 0.0596954i 0.882743 0.469855i \(-0.155694\pi\)
−0.848278 + 0.529551i \(0.822361\pi\)
\(102\) 0 0
\(103\) −7.65401 13.2571i −0.754172 1.30626i −0.945785 0.324794i \(-0.894705\pi\)
0.191613 0.981471i \(-0.438628\pi\)
\(104\) 0 0
\(105\) −2.18663 3.78736i −0.213394 0.369609i
\(106\) 0 0
\(107\) 6.42328 0.620962 0.310481 0.950580i \(-0.399510\pi\)
0.310481 + 0.950580i \(0.399510\pi\)
\(108\) 0 0
\(109\) 1.66310 0.159296 0.0796481 0.996823i \(-0.474620\pi\)
0.0796481 + 0.996823i \(0.474620\pi\)
\(110\) 0 0
\(111\) −0.417228 + 0.722660i −0.0396015 + 0.0685919i
\(112\) 0 0
\(113\) −7.20189 12.4740i −0.677496 1.17346i −0.975732 0.218966i \(-0.929732\pi\)
0.298236 0.954492i \(-0.403602\pi\)
\(114\) 0 0
\(115\) 3.49083 6.04630i 0.325522 0.563821i
\(116\) 0 0
\(117\) −1.04087 + 1.80283i −0.0962281 + 0.166672i
\(118\) 0 0
\(119\) 3.97267 0.364174
\(120\) 0 0
\(121\) 1.02715 1.77908i 0.0933776 0.161735i
\(122\) 0 0
\(123\) −1.33572 2.31353i −0.120438 0.208604i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.44194 + 7.69366i 0.394158 + 0.682702i 0.992993 0.118171i \(-0.0377029\pi\)
−0.598835 + 0.800872i \(0.704370\pi\)
\(128\) 0 0
\(129\) −12.0882 −1.06430
\(130\) 0 0
\(131\) −3.70567 −0.323766 −0.161883 0.986810i \(-0.551757\pi\)
−0.161883 + 0.986810i \(0.551757\pi\)
\(132\) 0 0
\(133\) −30.8816 −2.67777
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −17.0338 −1.45530 −0.727648 0.685951i \(-0.759386\pi\)
−0.727648 + 0.685951i \(0.759386\pi\)
\(138\) 0 0
\(139\) 4.20074 0.356302 0.178151 0.984003i \(-0.442988\pi\)
0.178151 + 0.984003i \(0.442988\pi\)
\(140\) 0 0
\(141\) 0.436600 + 0.756213i 0.0367683 + 0.0636846i
\(142\) 0 0
\(143\) −6.22633 −0.520671
\(144\) 0 0
\(145\) 3.93896 + 6.82248i 0.327113 + 0.566576i
\(146\) 0 0
\(147\) −6.06274 + 10.5010i −0.500046 + 0.866105i
\(148\) 0 0
\(149\) 10.5719 0.866084 0.433042 0.901374i \(-0.357440\pi\)
0.433042 + 0.901374i \(0.357440\pi\)
\(150\) 0 0
\(151\) 7.55070 13.0782i 0.614468 1.06429i −0.376010 0.926616i \(-0.622704\pi\)
0.990478 0.137673i \(-0.0439625\pi\)
\(152\) 0 0
\(153\) −0.454199 + 0.786696i −0.0367198 + 0.0636006i
\(154\) 0 0
\(155\) −3.83972 6.65058i −0.308413 0.534188i
\(156\) 0 0
\(157\) 5.40208 9.35667i 0.431133 0.746744i −0.565838 0.824516i \(-0.691447\pi\)
0.996971 + 0.0777723i \(0.0247807\pi\)
\(158\) 0 0
\(159\) 7.39196 0.586221
\(160\) 0 0
\(161\) −30.5327 −2.40631
\(162\) 0 0
\(163\) −6.42050 11.1206i −0.502893 0.871036i −0.999994 0.00334341i \(-0.998936\pi\)
0.497102 0.867692i \(-0.334398\pi\)
\(164\) 0 0
\(165\) 1.49547 + 2.59023i 0.116422 + 0.201649i
\(166\) 0 0
\(167\) 10.1268 + 17.5401i 0.783635 + 1.35730i 0.929811 + 0.368037i \(0.119970\pi\)
−0.146176 + 0.989259i \(0.546697\pi\)
\(168\) 0 0
\(169\) 4.33320 + 7.50532i 0.333323 + 0.577332i
\(170\) 0 0
\(171\) 3.53072 6.11538i 0.270001 0.467655i
\(172\) 0 0
\(173\) 1.68539 2.91918i 0.128138 0.221941i −0.794817 0.606849i \(-0.792433\pi\)
0.922955 + 0.384908i \(0.125767\pi\)
\(174\) 0 0
\(175\) −2.18663 3.78736i −0.165294 0.286298i
\(176\) 0 0
\(177\) −1.81848 −0.136685
\(178\) 0 0
\(179\) −8.54970 −0.639035 −0.319517 0.947580i \(-0.603521\pi\)
−0.319517 + 0.947580i \(0.603521\pi\)
\(180\) 0 0
\(181\) 2.30488 + 3.99218i 0.171321 + 0.296736i 0.938882 0.344240i \(-0.111863\pi\)
−0.767561 + 0.640976i \(0.778530\pi\)
\(182\) 0 0
\(183\) 1.99000 3.44678i 0.147105 0.254793i
\(184\) 0 0
\(185\) −0.417228 + 0.722660i −0.0306752 + 0.0531310i
\(186\) 0 0
\(187\) −2.71696 −0.198684
\(188\) 0 0
\(189\) −2.18663 3.78736i −0.159054 0.275490i
\(190\) 0 0
\(191\) −6.49218 + 11.2448i −0.469758 + 0.813645i −0.999402 0.0345753i \(-0.988992\pi\)
0.529644 + 0.848220i \(0.322325\pi\)
\(192\) 0 0
\(193\) −10.1932 −0.733722 −0.366861 0.930276i \(-0.619568\pi\)
−0.366861 + 0.930276i \(0.619568\pi\)
\(194\) 0 0
\(195\) −1.04087 + 1.80283i −0.0745380 + 0.129104i
\(196\) 0 0
\(197\) −9.06345 15.6984i −0.645744 1.11846i −0.984129 0.177454i \(-0.943214\pi\)
0.338385 0.941008i \(-0.390119\pi\)
\(198\) 0 0
\(199\) 9.84561 17.0531i 0.697937 1.20886i −0.271244 0.962511i \(-0.587435\pi\)
0.969181 0.246351i \(-0.0792317\pi\)
\(200\) 0 0
\(201\) 2.50950 7.79118i 0.177006 0.549547i
\(202\) 0 0
\(203\) 17.2261 29.8365i 1.20904 2.09411i
\(204\) 0 0
\(205\) −1.33572 2.31353i −0.0932907 0.161584i
\(206\) 0 0
\(207\) 3.49083 6.04630i 0.242630 0.420247i
\(208\) 0 0
\(209\) 21.1203 1.46092
\(210\) 0 0
\(211\) −10.5249 + 18.2296i −0.724563 + 1.25498i 0.234590 + 0.972094i \(0.424625\pi\)
−0.959153 + 0.282886i \(0.908708\pi\)
\(212\) 0 0
\(213\) 0.0217755 + 0.0377163i 0.00149203 + 0.00258428i
\(214\) 0 0
\(215\) −12.0882 −0.824405
\(216\) 0 0
\(217\) −16.7921 + 29.0848i −1.13992 + 1.97440i
\(218\) 0 0
\(219\) 4.49932 7.79305i 0.304036 0.526606i
\(220\) 0 0
\(221\) −0.945520 1.63769i −0.0636026 0.110163i
\(222\) 0 0
\(223\) −12.4616 −0.834488 −0.417244 0.908794i \(-0.637004\pi\)
−0.417244 + 0.908794i \(0.637004\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −5.36483 9.29216i −0.356076 0.616742i 0.631225 0.775600i \(-0.282552\pi\)
−0.987302 + 0.158857i \(0.949219\pi\)
\(228\) 0 0
\(229\) −7.19516 + 12.4624i −0.475469 + 0.823537i −0.999605 0.0280976i \(-0.991055\pi\)
0.524136 + 0.851635i \(0.324388\pi\)
\(230\) 0 0
\(231\) 6.54008 11.3278i 0.430306 0.745311i
\(232\) 0 0
\(233\) −1.88321 3.26182i −0.123373 0.213689i 0.797723 0.603025i \(-0.206038\pi\)
−0.921096 + 0.389336i \(0.872705\pi\)
\(234\) 0 0
\(235\) 0.436600 + 0.756213i 0.0284806 + 0.0493299i
\(236\) 0 0
\(237\) −0.453494 0.785474i −0.0294576 0.0510221i
\(238\) 0 0
\(239\) 0.447416 + 0.774947i 0.0289409 + 0.0501272i 0.880133 0.474727i \(-0.157453\pi\)
−0.851192 + 0.524854i \(0.824120\pi\)
\(240\) 0 0
\(241\) −20.3244 −1.30921 −0.654603 0.755972i \(-0.727164\pi\)
−0.654603 + 0.755972i \(0.727164\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.06274 + 10.5010i −0.387334 + 0.670882i
\(246\) 0 0
\(247\) 7.35001 + 12.7306i 0.467670 + 0.810028i
\(248\) 0 0
\(249\) 1.41204 2.44573i 0.0894845 0.154992i
\(250\) 0 0
\(251\) 8.24998 14.2894i 0.520734 0.901938i −0.478975 0.877828i \(-0.658992\pi\)
0.999709 0.0241096i \(-0.00767508\pi\)
\(252\) 0 0
\(253\) 20.8817 1.31282
\(254\) 0 0
\(255\) −0.454199 + 0.786696i −0.0284430 + 0.0492648i
\(256\) 0 0
\(257\) −9.99116 17.3052i −0.623231 1.07947i −0.988880 0.148715i \(-0.952486\pi\)
0.365649 0.930753i \(-0.380847\pi\)
\(258\) 0 0
\(259\) 3.64930 0.226757
\(260\) 0 0
\(261\) 3.93896 + 6.82248i 0.243816 + 0.422301i
\(262\) 0 0
\(263\) 22.0971 1.36257 0.681284 0.732019i \(-0.261422\pi\)
0.681284 + 0.732019i \(0.261422\pi\)
\(264\) 0 0
\(265\) 7.39196 0.454085
\(266\) 0 0
\(267\) 2.76333 0.169113
\(268\) 0 0
\(269\) 19.1061 1.16492 0.582459 0.812860i \(-0.302091\pi\)
0.582459 + 0.812860i \(0.302091\pi\)
\(270\) 0 0
\(271\) 15.1078 0.917736 0.458868 0.888504i \(-0.348255\pi\)
0.458868 + 0.888504i \(0.348255\pi\)
\(272\) 0 0
\(273\) 9.10397 0.550997
\(274\) 0 0
\(275\) 1.49547 + 2.59023i 0.0901801 + 0.156196i
\(276\) 0 0
\(277\) 13.5221 0.812465 0.406233 0.913770i \(-0.366842\pi\)
0.406233 + 0.913770i \(0.366842\pi\)
\(278\) 0 0
\(279\) −3.83972 6.65058i −0.229878 0.398160i
\(280\) 0 0
\(281\) −5.59295 + 9.68728i −0.333648 + 0.577895i −0.983224 0.182402i \(-0.941613\pi\)
0.649577 + 0.760296i \(0.274946\pi\)
\(282\) 0 0
\(283\) 31.3977 1.86640 0.933201 0.359356i \(-0.117004\pi\)
0.933201 + 0.359356i \(0.117004\pi\)
\(284\) 0 0
\(285\) 3.53072 6.11538i 0.209142 0.362244i
\(286\) 0 0
\(287\) −5.84146 + 10.1177i −0.344810 + 0.597229i
\(288\) 0 0
\(289\) 8.08741 + 14.0078i 0.475730 + 0.823988i
\(290\) 0 0
\(291\) 0.274039 0.474649i 0.0160644 0.0278244i
\(292\) 0 0
\(293\) 0.857561 0.0500992 0.0250496 0.999686i \(-0.492026\pi\)
0.0250496 + 0.999686i \(0.492026\pi\)
\(294\) 0 0
\(295\) −1.81848 −0.105876
\(296\) 0 0
\(297\) 1.49547 + 2.59023i 0.0867758 + 0.150300i
\(298\) 0 0
\(299\) 7.26698 + 12.5868i 0.420260 + 0.727912i
\(300\) 0 0
\(301\) 26.4324 + 45.7822i 1.52354 + 2.63884i
\(302\) 0 0
\(303\) 0.346371 + 0.599931i 0.0198985 + 0.0344652i
\(304\) 0 0
\(305\) 1.99000 3.44678i 0.113947 0.197362i
\(306\) 0 0
\(307\) −3.19008 + 5.52538i −0.182067 + 0.315350i −0.942584 0.333968i \(-0.891612\pi\)
0.760517 + 0.649318i \(0.224946\pi\)
\(308\) 0 0
\(309\) −7.65401 13.2571i −0.435421 0.754172i
\(310\) 0 0
\(311\) 21.0122 1.19150 0.595748 0.803172i \(-0.296856\pi\)
0.595748 + 0.803172i \(0.296856\pi\)
\(312\) 0 0
\(313\) −13.7198 −0.775489 −0.387745 0.921767i \(-0.626746\pi\)
−0.387745 + 0.921767i \(0.626746\pi\)
\(314\) 0 0
\(315\) −2.18663 3.78736i −0.123203 0.213394i
\(316\) 0 0
\(317\) 4.53972 7.86302i 0.254976 0.441631i −0.709913 0.704289i \(-0.751266\pi\)
0.964889 + 0.262658i \(0.0845991\pi\)
\(318\) 0 0
\(319\) −11.7812 + 20.4056i −0.659619 + 1.14249i
\(320\) 0 0
\(321\) 6.42328 0.358512
\(322\) 0 0
\(323\) 3.20730 + 5.55520i 0.178459 + 0.309100i
\(324\) 0 0
\(325\) −1.04087 + 1.80283i −0.0577369 + 0.100003i
\(326\) 0 0
\(327\) 1.66310 0.0919697
\(328\) 0 0
\(329\) 1.90937 3.30712i 0.105267 0.182327i
\(330\) 0 0
\(331\) −4.36593 7.56201i −0.239973 0.415646i 0.720733 0.693213i \(-0.243805\pi\)
−0.960706 + 0.277567i \(0.910472\pi\)
\(332\) 0 0
\(333\) −0.417228 + 0.722660i −0.0228640 + 0.0396015i
\(334\) 0 0
\(335\) 2.50950 7.79118i 0.137109 0.425677i
\(336\) 0 0
\(337\) −15.1782 + 26.2894i −0.826807 + 1.43207i 0.0737229 + 0.997279i \(0.476512\pi\)
−0.900530 + 0.434793i \(0.856821\pi\)
\(338\) 0 0
\(339\) −7.20189 12.4740i −0.391153 0.677496i
\(340\) 0 0
\(341\) 11.4843 19.8915i 0.621912 1.07718i
\(342\) 0 0
\(343\) 22.4151 1.21030
\(344\) 0 0
\(345\) 3.49083 6.04630i 0.187940 0.325522i
\(346\) 0 0
\(347\) −2.79561 4.84214i −0.150076 0.259940i 0.781179 0.624307i \(-0.214619\pi\)
−0.931255 + 0.364367i \(0.881285\pi\)
\(348\) 0 0
\(349\) 36.7337 1.96631 0.983154 0.182781i \(-0.0585098\pi\)
0.983154 + 0.182781i \(0.0585098\pi\)
\(350\) 0 0
\(351\) −1.04087 + 1.80283i −0.0555573 + 0.0962281i
\(352\) 0 0
\(353\) −2.21593 + 3.83811i −0.117942 + 0.204282i −0.918952 0.394369i \(-0.870963\pi\)
0.801010 + 0.598651i \(0.204296\pi\)
\(354\) 0 0
\(355\) 0.0217755 + 0.0377163i 0.00115572 + 0.00200177i
\(356\) 0 0
\(357\) 3.97267 0.210256
\(358\) 0 0
\(359\) −8.95476 −0.472614 −0.236307 0.971678i \(-0.575937\pi\)
−0.236307 + 0.971678i \(0.575937\pi\)
\(360\) 0 0
\(361\) −15.4319 26.7289i −0.812208 1.40678i
\(362\) 0 0
\(363\) 1.02715 1.77908i 0.0539116 0.0933776i
\(364\) 0 0
\(365\) 4.49932 7.79305i 0.235505 0.407907i
\(366\) 0 0
\(367\) 8.80302 + 15.2473i 0.459514 + 0.795901i 0.998935 0.0461347i \(-0.0146903\pi\)
−0.539421 + 0.842036i \(0.681357\pi\)
\(368\) 0 0
\(369\) −1.33572 2.31353i −0.0695348 0.120438i
\(370\) 0 0
\(371\) −16.1635 27.9960i −0.839168 1.45348i
\(372\) 0 0
\(373\) 4.17950 + 7.23911i 0.216406 + 0.374827i 0.953707 0.300738i \(-0.0972331\pi\)
−0.737300 + 0.675565i \(0.763900\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −16.3997 −0.844629
\(378\) 0 0
\(379\) −6.77956 + 11.7425i −0.348243 + 0.603174i −0.985937 0.167115i \(-0.946555\pi\)
0.637695 + 0.770289i \(0.279888\pi\)
\(380\) 0 0
\(381\) 4.44194 + 7.69366i 0.227567 + 0.394158i
\(382\) 0 0
\(383\) −5.07543 + 8.79091i −0.259343 + 0.449195i −0.966066 0.258295i \(-0.916839\pi\)
0.706723 + 0.707490i \(0.250173\pi\)
\(384\) 0 0
\(385\) 6.54008 11.3278i 0.333313 0.577316i
\(386\) 0 0
\(387\) −12.0882 −0.614476
\(388\) 0 0
\(389\) −3.92875 + 6.80480i −0.199196 + 0.345017i −0.948268 0.317471i \(-0.897166\pi\)
0.749072 + 0.662488i \(0.230500\pi\)
\(390\) 0 0
\(391\) 3.17107 + 5.49245i 0.160368 + 0.277765i
\(392\) 0 0
\(393\) −3.70567 −0.186926
\(394\) 0 0
\(395\) −0.453494 0.785474i −0.0228178 0.0395215i
\(396\) 0 0
\(397\) 10.6770 0.535861 0.267931 0.963438i \(-0.413660\pi\)
0.267931 + 0.963438i \(0.413660\pi\)
\(398\) 0 0
\(399\) −30.8816 −1.54601
\(400\) 0 0
\(401\) −23.0544 −1.15128 −0.575641 0.817703i \(-0.695247\pi\)
−0.575641 + 0.817703i \(0.695247\pi\)
\(402\) 0 0
\(403\) 15.9865 0.796345
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −2.49580 −0.123712
\(408\) 0 0
\(409\) −9.74620 16.8809i −0.481918 0.834707i 0.517866 0.855462i \(-0.326726\pi\)
−0.999785 + 0.0207545i \(0.993393\pi\)
\(410\) 0 0
\(411\) −17.0338 −0.840215
\(412\) 0 0
\(413\) 3.97635 + 6.88724i 0.195663 + 0.338899i
\(414\) 0 0
\(415\) 1.41204 2.44573i 0.0693144 0.120056i
\(416\) 0 0
\(417\) 4.20074 0.205711
\(418\) 0 0
\(419\) −13.4661 + 23.3240i −0.657863 + 1.13945i 0.323304 + 0.946295i \(0.395206\pi\)
−0.981168 + 0.193158i \(0.938127\pi\)
\(420\) 0 0
\(421\) −11.5808 + 20.0585i −0.564411 + 0.977589i 0.432693 + 0.901541i \(0.357563\pi\)
−0.997104 + 0.0760477i \(0.975770\pi\)
\(422\) 0 0
\(423\) 0.436600 + 0.756213i 0.0212282 + 0.0367683i
\(424\) 0 0
\(425\) −0.454199 + 0.786696i −0.0220319 + 0.0381603i
\(426\) 0 0
\(427\) −17.4056 −0.842316
\(428\) 0 0
\(429\) −6.22633 −0.300610
\(430\) 0 0
\(431\) −5.01576 8.68755i −0.241601 0.418465i 0.719570 0.694420i \(-0.244339\pi\)
−0.961170 + 0.275956i \(0.911006\pi\)
\(432\) 0 0
\(433\) 6.20238 + 10.7428i 0.298067 + 0.516267i 0.975694 0.219139i \(-0.0703247\pi\)
−0.677627 + 0.735406i \(0.736991\pi\)
\(434\) 0 0
\(435\) 3.93896 + 6.82248i 0.188859 + 0.327113i
\(436\) 0 0
\(437\) −24.6503 42.6956i −1.17918 2.04241i
\(438\) 0 0
\(439\) −4.61849 + 7.99945i −0.220428 + 0.381793i −0.954938 0.296805i \(-0.904079\pi\)
0.734510 + 0.678598i \(0.237412\pi\)
\(440\) 0 0
\(441\) −6.06274 + 10.5010i −0.288702 + 0.500046i
\(442\) 0 0
\(443\) −0.837559 1.45070i −0.0397937 0.0689246i 0.845443 0.534066i \(-0.179337\pi\)
−0.885236 + 0.465142i \(0.846003\pi\)
\(444\) 0 0
\(445\) 2.76333 0.130994
\(446\) 0 0
\(447\) 10.5719 0.500034
\(448\) 0 0
\(449\) −10.4119 18.0339i −0.491366 0.851071i 0.508584 0.861012i \(-0.330169\pi\)
−0.999951 + 0.00994101i \(0.996836\pi\)
\(450\) 0 0
\(451\) 3.99505 6.91963i 0.188120 0.325833i
\(452\) 0 0
\(453\) 7.55070 13.0782i 0.354763 0.614468i
\(454\) 0 0
\(455\) 9.10397 0.426801
\(456\) 0 0
\(457\) 17.9737 + 31.1314i 0.840774 + 1.45626i 0.889241 + 0.457439i \(0.151233\pi\)
−0.0484666 + 0.998825i \(0.515433\pi\)
\(458\) 0 0
\(459\) −0.454199 + 0.786696i −0.0212002 + 0.0367198i
\(460\) 0 0
\(461\) 17.9362 0.835370 0.417685 0.908592i \(-0.362842\pi\)
0.417685 + 0.908592i \(0.362842\pi\)
\(462\) 0 0
\(463\) 1.88693 3.26826i 0.0876932 0.151889i −0.818842 0.574018i \(-0.805384\pi\)
0.906536 + 0.422129i \(0.138717\pi\)
\(464\) 0 0
\(465\) −3.83972 6.65058i −0.178063 0.308413i
\(466\) 0 0
\(467\) −14.6753 + 25.4183i −0.679091 + 1.17622i 0.296165 + 0.955137i \(0.404292\pi\)
−0.975255 + 0.221082i \(0.929041\pi\)
\(468\) 0 0
\(469\) −34.9954 + 7.53207i −1.61594 + 0.347799i
\(470\) 0 0
\(471\) 5.40208 9.35667i 0.248915 0.431133i
\(472\) 0 0
\(473\) −18.0774 31.3111i −0.831202 1.43968i
\(474\) 0 0
\(475\) 3.53072 6.11538i 0.162000 0.280593i
\(476\) 0 0
\(477\) 7.39196 0.338455
\(478\) 0 0
\(479\) −7.28475 + 12.6176i −0.332849 + 0.576511i −0.983069 0.183234i \(-0.941343\pi\)
0.650220 + 0.759746i \(0.274677\pi\)
\(480\) 0 0
\(481\) −0.868557 1.50439i −0.0396028 0.0685941i
\(482\) 0 0
\(483\) −30.5327 −1.38929
\(484\) 0 0
\(485\) 0.274039 0.474649i 0.0124435 0.0215527i
\(486\) 0 0
\(487\) −2.94405 + 5.09925i −0.133408 + 0.231069i −0.924988 0.379996i \(-0.875925\pi\)
0.791580 + 0.611065i \(0.209259\pi\)
\(488\) 0 0
\(489\) −6.42050 11.1206i −0.290345 0.502893i
\(490\) 0 0
\(491\) 19.8188 0.894410 0.447205 0.894431i \(-0.352419\pi\)
0.447205 + 0.894431i \(0.352419\pi\)
\(492\) 0 0
\(493\) −7.15629 −0.322303
\(494\) 0 0
\(495\) 1.49547 + 2.59023i 0.0672163 + 0.116422i
\(496\) 0 0
\(497\) 0.0952302 0.164944i 0.00427166 0.00739873i
\(498\) 0 0
\(499\) 5.09991 8.83331i 0.228304 0.395433i −0.729002 0.684512i \(-0.760015\pi\)
0.957305 + 0.289078i \(0.0933488\pi\)
\(500\) 0 0
\(501\) 10.1268 + 17.5401i 0.452432 + 0.783635i
\(502\) 0 0
\(503\) −11.2004 19.3996i −0.499401 0.864987i 0.500599 0.865679i \(-0.333113\pi\)
−1.00000 0.000691938i \(0.999780\pi\)
\(504\) 0 0
\(505\) 0.346371 + 0.599931i 0.0154133 + 0.0266966i
\(506\) 0 0
\(507\) 4.33320 + 7.50532i 0.192444 + 0.333323i
\(508\) 0 0
\(509\) 26.4309 1.17153 0.585765 0.810481i \(-0.300794\pi\)
0.585765 + 0.810481i \(0.300794\pi\)
\(510\) 0 0
\(511\) −39.3535 −1.74089
\(512\) 0 0
\(513\) 3.53072 6.11538i 0.155885 0.270001i
\(514\) 0 0
\(515\) −7.65401 13.2571i −0.337276 0.584179i
\(516\) 0 0
\(517\) −1.30584 + 2.26178i −0.0574308 + 0.0994731i
\(518\) 0 0
\(519\) 1.68539 2.91918i 0.0739803 0.128138i
\(520\) 0 0
\(521\) 20.0477 0.878304 0.439152 0.898413i \(-0.355279\pi\)
0.439152 + 0.898413i \(0.355279\pi\)
\(522\) 0 0
\(523\) −10.3772 + 17.9739i −0.453764 + 0.785943i −0.998616 0.0525894i \(-0.983253\pi\)
0.544852 + 0.838532i \(0.316586\pi\)
\(524\) 0 0
\(525\) −2.18663 3.78736i −0.0954325 0.165294i
\(526\) 0 0
\(527\) 6.97598 0.303878
\(528\) 0 0
\(529\) −12.8719 22.2947i −0.559646 0.969335i
\(530\) 0 0
\(531\) −1.81848 −0.0789153
\(532\) 0 0
\(533\) 5.56122 0.240883
\(534\) 0 0
\(535\) 6.42328 0.277702
\(536\) 0 0
\(537\) −8.54970 −0.368947
\(538\) 0 0
\(539\) −36.2665 −1.56211
\(540\) 0 0
\(541\) 0.500503 0.0215183 0.0107592 0.999942i \(-0.496575\pi\)
0.0107592 + 0.999942i \(0.496575\pi\)
\(542\) 0 0
\(543\) 2.30488 + 3.99218i 0.0989120 + 0.171321i
\(544\) 0 0
\(545\) 1.66310 0.0712395
\(546\) 0 0
\(547\) 17.1918 + 29.7771i 0.735070 + 1.27318i 0.954693 + 0.297594i \(0.0961842\pi\)
−0.219622 + 0.975585i \(0.570483\pi\)
\(548\) 0 0
\(549\) 1.99000 3.44678i 0.0849311 0.147105i
\(550\) 0 0
\(551\) 55.6295 2.36989
\(552\) 0 0
\(553\) −1.98325 + 3.43509i −0.0843364 + 0.146075i
\(554\) 0 0
\(555\) −0.417228 + 0.722660i −0.0177103 + 0.0306752i
\(556\) 0 0
\(557\) −11.7577 20.3650i −0.498192 0.862894i 0.501806 0.864980i \(-0.332669\pi\)
−0.999998 + 0.00208665i \(0.999336\pi\)
\(558\) 0 0
\(559\) 12.5822 21.7929i 0.532168 0.921743i
\(560\) 0 0
\(561\) −2.71696 −0.114710
\(562\) 0 0
\(563\) 43.3432 1.82670 0.913350 0.407176i \(-0.133487\pi\)
0.913350 + 0.407176i \(0.133487\pi\)
\(564\) 0 0
\(565\) −7.20189 12.4740i −0.302986 0.524786i
\(566\) 0 0
\(567\) −2.18663 3.78736i −0.0918300 0.159054i
\(568\) 0 0
\(569\) 13.3419 + 23.1088i 0.559320 + 0.968770i 0.997553 + 0.0699088i \(0.0222708\pi\)
−0.438234 + 0.898861i \(0.644396\pi\)
\(570\) 0 0
\(571\) −0.527129 0.913015i −0.0220597 0.0382085i 0.854785 0.518983i \(-0.173689\pi\)
−0.876844 + 0.480774i \(0.840356\pi\)
\(572\) 0 0
\(573\) −6.49218 + 11.2448i −0.271215 + 0.469758i
\(574\) 0 0
\(575\) 3.49083 6.04630i 0.145578 0.252148i
\(576\) 0 0
\(577\) 9.28518 + 16.0824i 0.386547 + 0.669519i 0.991983 0.126375i \(-0.0403343\pi\)
−0.605435 + 0.795894i \(0.707001\pi\)
\(578\) 0 0
\(579\) −10.1932 −0.423615
\(580\) 0 0
\(581\) −12.3505 −0.512384
\(582\) 0 0
\(583\) 11.0544 + 19.1469i 0.457828 + 0.792982i
\(584\) 0 0
\(585\) −1.04087 + 1.80283i −0.0430345 + 0.0745380i
\(586\) 0 0
\(587\) −10.8014 + 18.7086i −0.445821 + 0.772185i −0.998109 0.0614687i \(-0.980422\pi\)
0.552288 + 0.833653i \(0.313755\pi\)
\(588\) 0 0
\(589\) −54.2278 −2.23442
\(590\) 0 0
\(591\) −9.06345 15.6984i −0.372821 0.645744i
\(592\) 0 0
\(593\) 24.1278 41.7906i 0.990811 1.71614i 0.378275 0.925693i \(-0.376517\pi\)
0.612537 0.790442i \(-0.290149\pi\)
\(594\) 0 0
\(595\) 3.97267 0.162863
\(596\) 0 0
\(597\) 9.84561 17.0531i 0.402954 0.697937i
\(598\) 0 0
\(599\) −13.0066 22.5281i −0.531436 0.920474i −0.999327 0.0366878i \(-0.988319\pi\)
0.467891 0.883786i \(-0.345014\pi\)
\(600\) 0 0
\(601\) −10.4264 + 18.0590i −0.425302 + 0.736644i −0.996449 0.0842039i \(-0.973165\pi\)
0.571147 + 0.820848i \(0.306499\pi\)
\(602\) 0 0
\(603\) 2.50950 7.79118i 0.102195 0.317281i
\(604\) 0 0
\(605\) 1.02715 1.77908i 0.0417597 0.0723300i
\(606\) 0 0
\(607\) −3.02951 5.24727i −0.122964 0.212980i 0.797971 0.602696i \(-0.205907\pi\)
−0.920935 + 0.389715i \(0.872573\pi\)
\(608\) 0 0
\(609\) 17.2261 29.8365i 0.698038 1.20904i
\(610\) 0 0
\(611\) −1.81777 −0.0735390
\(612\) 0 0
\(613\) 13.3261 23.0815i 0.538236 0.932251i −0.460764 0.887523i \(-0.652424\pi\)
0.998999 0.0447284i \(-0.0142423\pi\)
\(614\) 0 0
\(615\) −1.33572 2.31353i −0.0538614 0.0932907i
\(616\) 0 0
\(617\) −23.4714 −0.944924 −0.472462 0.881351i \(-0.656635\pi\)
−0.472462 + 0.881351i \(0.656635\pi\)
\(618\) 0 0
\(619\) −3.97219 + 6.88003i −0.159656 + 0.276532i −0.934745 0.355321i \(-0.884372\pi\)
0.775089 + 0.631852i \(0.217705\pi\)
\(620\) 0 0
\(621\) 3.49083 6.04630i 0.140082 0.242630i
\(622\) 0 0
\(623\) −6.04239 10.4657i −0.242083 0.419300i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 21.1203 0.843463
\(628\) 0 0
\(629\) −0.379009 0.656463i −0.0151121 0.0261749i
\(630\) 0 0
\(631\) −21.0072 + 36.3856i −0.836285 + 1.44849i 0.0566945 + 0.998392i \(0.481944\pi\)
−0.892980 + 0.450097i \(0.851389\pi\)
\(632\) 0 0
\(633\) −10.5249 + 18.2296i −0.418327 + 0.724563i
\(634\) 0 0
\(635\) 4.44194 + 7.69366i 0.176273 + 0.305313i
\(636\) 0 0
\(637\) −12.6210 21.8602i −0.500062 0.866133i
\(638\) 0 0
\(639\) 0.0217755 + 0.0377163i 0.000861426 + 0.00149203i
\(640\) 0 0
\(641\) 20.1762 + 34.9462i 0.796912 + 1.38029i 0.921618 + 0.388098i \(0.126868\pi\)
−0.124707 + 0.992194i \(0.539799\pi\)
\(642\) 0 0
\(643\) 11.4834 0.452862 0.226431 0.974027i \(-0.427294\pi\)
0.226431 + 0.974027i \(0.427294\pi\)
\(644\) 0 0
\(645\) −12.0882 −0.475971
\(646\) 0 0
\(647\) −17.0745 + 29.5740i −0.671270 + 1.16267i 0.306275 + 0.951943i \(0.400917\pi\)
−0.977544 + 0.210730i \(0.932416\pi\)
\(648\) 0 0
\(649\) −2.71948 4.71027i −0.106749 0.184894i
\(650\) 0 0
\(651\) −16.7921 + 29.0848i −0.658135 + 1.13992i
\(652\) 0 0
\(653\) −7.30873 + 12.6591i −0.286013 + 0.495388i −0.972854 0.231419i \(-0.925663\pi\)
0.686842 + 0.726807i \(0.258997\pi\)
\(654\) 0 0
\(655\) −3.70567 −0.144793
\(656\) 0 0
\(657\) 4.49932 7.79305i 0.175535 0.304036i
\(658\) 0 0
\(659\) 9.49306 + 16.4425i 0.369797 + 0.640507i 0.989534 0.144303i \(-0.0460939\pi\)
−0.619736 + 0.784810i \(0.712761\pi\)
\(660\) 0 0
\(661\) −8.42342 −0.327633 −0.163817 0.986491i \(-0.552381\pi\)
−0.163817 + 0.986491i \(0.552381\pi\)
\(662\) 0 0
\(663\) −0.945520 1.63769i −0.0367210 0.0636026i
\(664\) 0 0
\(665\) −30.8816 −1.19754
\(666\) 0 0
\(667\) 55.0011 2.12965
\(668\) 0 0
\(669\) −12.4616 −0.481792
\(670\) 0 0
\(671\) 11.9039 0.459546
\(672\) 0 0
\(673\) 22.6687 0.873814 0.436907 0.899507i \(-0.356074\pi\)
0.436907 + 0.899507i \(0.356074\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 18.3304 + 31.7491i 0.704493 + 1.22022i 0.966874 + 0.255253i \(0.0821589\pi\)
−0.262381 + 0.964964i \(0.584508\pi\)
\(678\) 0 0
\(679\) −2.39689 −0.0919842
\(680\) 0 0
\(681\) −5.36483 9.29216i −0.205581 0.356076i
\(682\) 0 0
\(683\) 16.2991 28.2309i 0.623668 1.08023i −0.365128 0.930957i \(-0.618975\pi\)
0.988797 0.149268i \(-0.0476918\pi\)
\(684\) 0 0
\(685\) −17.0338 −0.650828
\(686\) 0 0
\(687\) −7.19516 + 12.4624i −0.274512 + 0.475469i
\(688\) 0 0
\(689\) −7.69404 + 13.3265i −0.293120 + 0.507698i
\(690\) 0 0
\(691\) 13.9651 + 24.1882i 0.531257 + 0.920164i 0.999335 + 0.0364767i \(0.0116135\pi\)
−0.468077 + 0.883687i \(0.655053\pi\)
\(692\) 0 0
\(693\) 6.54008 11.3278i 0.248437 0.430306i
\(694\) 0 0
\(695\) 4.20074 0.159343
\(696\) 0 0
\(697\) 2.42673 0.0919189
\(698\) 0 0
\(699\) −1.88321 3.26182i −0.0712297 0.123373i
\(700\) 0 0
\(701\) 18.4378 + 31.9352i 0.696386 + 1.20618i 0.969711 + 0.244254i \(0.0785431\pi\)
−0.273325 + 0.961922i \(0.588124\pi\)
\(702\) 0 0
\(703\) 2.94623 + 5.10302i 0.111119 + 0.192464i
\(704\) 0 0
\(705\) 0.436600 + 0.756213i 0.0164433 + 0.0284806i
\(706\) 0 0
\(707\) 1.51477 2.62366i 0.0569688 0.0986729i
\(708\) 0 0
\(709\) −8.62600 + 14.9407i −0.323956 + 0.561109i −0.981301 0.192482i \(-0.938346\pi\)
0.657344 + 0.753590i \(0.271680\pi\)
\(710\) 0 0
\(711\) −0.453494 0.785474i −0.0170074 0.0294576i
\(712\) 0 0
\(713\) −53.6152 −2.00791
\(714\) 0 0
\(715\) −6.22633 −0.232851
\(716\) 0 0
\(717\) 0.447416 + 0.774947i 0.0167091 + 0.0289409i
\(718\) 0 0
\(719\) −16.7156 + 28.9523i −0.623387 + 1.07974i 0.365464 + 0.930825i \(0.380910\pi\)
−0.988850 + 0.148912i \(0.952423\pi\)
\(720\) 0 0
\(721\) −33.4730 + 57.9770i −1.24660 + 2.15918i
\(722\) 0 0
\(723\) −20.3244 −0.755871
\(724\) 0 0
\(725\) 3.93896 + 6.82248i 0.146289 + 0.253381i
\(726\) 0 0
\(727\) 23.3419 40.4294i 0.865703 1.49944i −0.000643607 1.00000i \(-0.500205\pi\)
0.866347 0.499443i \(-0.166462\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.49043 9.50970i 0.203071 0.351729i
\(732\) 0 0
\(733\) −7.73972 13.4056i −0.285873 0.495147i 0.686947 0.726707i \(-0.258950\pi\)
−0.972821 + 0.231560i \(0.925617\pi\)
\(734\) 0 0
\(735\) −6.06274 + 10.5010i −0.223627 + 0.387334i
\(736\) 0 0
\(737\) 23.9338 5.15128i 0.881612 0.189750i
\(738\) 0 0
\(739\) 17.0656 29.5585i 0.627769 1.08733i −0.360230 0.932864i \(-0.617302\pi\)
0.987998 0.154464i \(-0.0493649\pi\)
\(740\) 0 0
\(741\) 7.35001 + 12.7306i 0.270009 + 0.467670i
\(742\) 0 0
\(743\) 17.3705 30.0866i 0.637262 1.10377i −0.348769 0.937209i \(-0.613400\pi\)
0.986031 0.166561i \(-0.0532664\pi\)
\(744\) 0 0
\(745\) 10.5719 0.387325
\(746\) 0 0
\(747\) 1.41204 2.44573i 0.0516639 0.0894845i
\(748\) 0 0
\(749\) −14.0454 24.3273i −0.513206 0.888899i
\(750\) 0 0
\(751\) 37.8526 1.38126 0.690630 0.723208i \(-0.257333\pi\)
0.690630 + 0.723208i \(0.257333\pi\)
\(752\) 0 0
\(753\) 8.24998 14.2894i 0.300646 0.520734i
\(754\) 0 0
\(755\) 7.55070 13.0782i 0.274798 0.475965i
\(756\) 0 0
\(757\) 3.01713 + 5.22582i 0.109659 + 0.189936i 0.915632 0.402017i \(-0.131691\pi\)
−0.805973 + 0.591952i \(0.798357\pi\)
\(758\) 0 0
\(759\) 20.8817 0.757958
\(760\) 0 0
\(761\) −17.2286 −0.624535 −0.312267 0.949994i \(-0.601088\pi\)
−0.312267 + 0.949994i \(0.601088\pi\)
\(762\) 0 0
\(763\) −3.63660 6.29877i −0.131654 0.228031i
\(764\) 0 0
\(765\) −0.454199 + 0.786696i −0.0164216 + 0.0284430i
\(766\) 0 0
\(767\) 1.89279 3.27841i 0.0683448 0.118377i
\(768\) 0 0
\(769\) 5.61143 + 9.71929i 0.202353 + 0.350486i 0.949286 0.314413i \(-0.101808\pi\)
−0.746933 + 0.664900i \(0.768474\pi\)
\(770\) 0 0
\(771\) −9.99116 17.3052i −0.359823 0.623231i
\(772\) 0 0
\(773\) 5.31801 + 9.21106i 0.191275 + 0.331299i 0.945673 0.325119i \(-0.105404\pi\)
−0.754398 + 0.656418i \(0.772071\pi\)
\(774\) 0 0
\(775\) −3.83972 6.65058i −0.137927 0.238896i
\(776\) 0 0
\(777\) 3.64930 0.130918
\(778\) 0 0
\(779\) −18.8642 −0.675880
\(780\) 0 0
\(781\) −0.0651292 + 0.112807i −0.00233050 + 0.00403655i
\(782\) 0 0
\(783\) 3.93896 + 6.82248i 0.140767 + 0.243816i
\(784\) 0 0
\(785\) 5.40208 9.35667i 0.192808 0.333954i
\(786\) 0 0
\(787\) 2.04693 3.54538i 0.0729650 0.126379i −0.827235 0.561857i \(-0.810087\pi\)
0.900200 + 0.435478i \(0.143421\pi\)
\(788\) 0 0
\(789\) 22.0971 0.786679
\(790\) 0 0
\(791\) −31.4958 + 54.5523i −1.11986 + 1.93966i
\(792\) 0 0
\(793\) 4.14265 + 7.17528i 0.147110 + 0.254801i
\(794\) 0 0
\(795\) 7.39196 0.262166
\(796\) 0 0
\(797\) 23.3322 + 40.4126i 0.826470 + 1.43149i 0.900790 + 0.434254i \(0.142988\pi\)
−0.0743202 + 0.997234i \(0.523679\pi\)
\(798\) 0 0
\(799\) −0.793212 −0.0280618
\(800\) 0 0
\(801\) 2.76333 0.0976374
\(802\) 0 0
\(803\) 26.9143 0.949787
\(804\) 0 0
\(805\) −30.5327 −1.07614
\(806\) 0 0
\(807\) 19.1061 0.672565
\(808\) 0 0
\(809\) −23.5590 −0.828291 −0.414145 0.910211i \(-0.635919\pi\)
−0.414145 + 0.910211i \(0.635919\pi\)
\(810\) 0 0
\(811\) −11.3750 19.7020i −0.399429 0.691831i 0.594227 0.804297i \(-0.297458\pi\)
−0.993655 + 0.112467i \(0.964125\pi\)
\(812\) 0 0
\(813\) 15.1078 0.529855
\(814\) 0 0
\(815\) −6.42050 11.1206i −0.224900 0.389539i
\(816\) 0 0
\(817\) −42.6799 + 73.9237i −1.49318 + 2.58626i
\(818\) 0 0
\(819\) 9.10397 0.318119
\(820\) 0 0
\(821\) −6.73838 + 11.6712i −0.235171 + 0.407328i −0.959322 0.282313i \(-0.908898\pi\)
0.724151 + 0.689641i \(0.242232\pi\)
\(822\) 0 0
\(823\) −4.88421 + 8.45970i −0.170253 + 0.294886i −0.938508 0.345257i \(-0.887792\pi\)
0.768255 + 0.640143i \(0.221125\pi\)
\(824\) 0 0
\(825\) 1.49547 + 2.59023i 0.0520655 + 0.0901801i
\(826\) 0 0
\(827\) −0.467810 + 0.810270i −0.0162673 + 0.0281759i −0.874044 0.485846i \(-0.838512\pi\)
0.857777 + 0.514022i \(0.171845\pi\)
\(828\) 0 0
\(829\) −29.4682 −1.02347 −0.511736 0.859143i \(-0.670997\pi\)
−0.511736 + 0.859143i \(0.670997\pi\)
\(830\) 0 0
\(831\) 13.5221 0.469077
\(832\) 0 0
\(833\) −5.50738 9.53906i −0.190819 0.330509i
\(834\) 0 0
\(835\) 10.1268 + 17.5401i 0.350452 + 0.607001i
\(836\) 0 0
\(837\) −3.83972 6.65058i −0.132720 0.229878i
\(838\) 0 0
\(839\) 23.9506 + 41.4837i 0.826867 + 1.43218i 0.900484 + 0.434890i \(0.143213\pi\)
−0.0736164 + 0.997287i \(0.523454\pi\)
\(840\) 0 0
\(841\) −16.5308 + 28.6322i −0.570029 + 0.987319i
\(842\) 0 0
\(843\) −5.59295 + 9.68728i −0.192632 + 0.333648i
\(844\) 0 0
\(845\) 4.33320 + 7.50532i 0.149066 + 0.258191i
\(846\) 0 0
\(847\) −8.98404 −0.308695
\(848\) 0 0
\(849\) 31.3977 1.07757
\(850\) 0 0
\(851\) 2.91295 + 5.04538i 0.0998546 + 0.172953i
\(852\) 0 0
\(853\) −4.06418 + 7.03937i −0.139155 + 0.241024i −0.927177 0.374624i \(-0.877772\pi\)
0.788022 + 0.615647i \(0.211105\pi\)
\(854\) 0 0
\(855\) 3.53072 6.11538i 0.120748 0.209142i
\(856\) 0 0
\(857\) 32.6205 1.11430 0.557148 0.830413i \(-0.311896\pi\)
0.557148 + 0.830413i \(0.311896\pi\)
\(858\) 0 0
\(859\) 15.8967 + 27.5339i 0.542388 + 0.939444i 0.998766 + 0.0496579i \(0.0158131\pi\)
−0.456378 + 0.889786i \(0.650854\pi\)
\(860\) 0 0
\(861\) −5.84146 + 10.1177i −0.199076 + 0.344810i
\(862\) 0 0
\(863\) −54.7907 −1.86510 −0.932549 0.361044i \(-0.882421\pi\)
−0.932549 + 0.361044i \(0.882421\pi\)
\(864\) 0 0
\(865\) 1.68539 2.91918i 0.0573049 0.0992550i
\(866\) 0 0
\(867\) 8.08741 + 14.0078i 0.274663 + 0.475730i
\(868\) 0 0
\(869\) 1.35637 2.34930i 0.0460117 0.0796946i
\(870\) 0 0
\(871\) 11.4341 + 12.6338i 0.387431 + 0.428079i
\(872\) 0 0
\(873\) 0.274039 0.474649i 0.00927481 0.0160644i
\(874\) 0 0
\(875\) −2.18663 3.78736i −0.0739217 0.128036i
\(876\) 0 0
\(877\) 5.01417 8.68480i 0.169317 0.293265i −0.768863 0.639413i \(-0.779177\pi\)
0.938180 + 0.346148i \(0.112511\pi\)
\(878\) 0 0
\(879\) 0.857561 0.0289248
\(880\) 0 0
\(881\) −20.9472 + 36.2816i −0.705729 + 1.22236i 0.260699 + 0.965420i \(0.416047\pi\)
−0.966428 + 0.256938i \(0.917286\pi\)
\(882\) 0 0
\(883\) −2.01930 3.49752i −0.0679547 0.117701i 0.830046 0.557695i \(-0.188314\pi\)
−0.898001 + 0.439994i \(0.854981\pi\)
\(884\) 0 0
\(885\) −1.81848 −0.0611275
\(886\) 0 0
\(887\) 18.3630 31.8056i 0.616569 1.06793i −0.373538 0.927615i \(-0.621855\pi\)
0.990107 0.140314i \(-0.0448112\pi\)
\(888\) 0 0
\(889\) 19.4258 33.6464i 0.651520 1.12846i
\(890\) 0 0
\(891\) 1.49547 + 2.59023i 0.0501000 + 0.0867758i
\(892\) 0 0
\(893\) 6.16604 0.206339
\(894\) 0 0
\(895\) −8.54970 −0.285785
\(896\) 0 0
\(897\) 7.26698 + 12.5868i 0.242637 + 0.420260i
\(898\) 0 0
\(899\) 30.2490 52.3928i 1.00886 1.74740i
\(900\) 0 0
\(901\) −3.35742 + 5.81522i −0.111852 + 0.193733i
\(902\) 0 0
\(903\) 26.4324 + 45.7822i 0.879614 + 1.52354i
\(904\) 0 0
\(905\) 2.30488 + 3.99218i 0.0766169 + 0.132704i
\(906\) 0 0
\(907\) −2.42704 4.20376i −0.0805886 0.139583i 0.822914 0.568165i \(-0.192347\pi\)
−0.903503 + 0.428582i \(0.859013\pi\)
\(908\) 0 0
\(909\) 0.346371 + 0.599931i 0.0114884 + 0.0198985i
\(910\) 0 0
\(911\) 18.1911 0.602697 0.301348 0.953514i \(-0.402563\pi\)
0.301348 + 0.953514i \(0.402563\pi\)
\(912\) 0 0
\(913\) 8.44665 0.279543
\(914\) 0 0
\(915\) 1.99000 3.44678i 0.0657874 0.113947i
\(916\) 0 0
\(917\) 8.10295 + 14.0347i 0.267583 + 0.463467i
\(918\) 0 0
\(919\) −13.8787 + 24.0385i −0.457815 + 0.792958i −0.998845 0.0480449i \(-0.984701\pi\)
0.541031 + 0.841003i \(0.318034\pi\)
\(920\) 0 0
\(921\) −3.19008 + 5.52538i −0.105117 + 0.182067i
\(922\) 0 0
\(923\) −0.0906616 −0.00298416
\(924\) 0 0
\(925\) −0.417228 + 0.722660i −0.0137184 + 0.0237609i
\(926\) 0 0
\(927\) −7.65401 13.2571i −0.251391 0.435421i
\(928\) 0 0
\(929\) 12.8287 0.420897 0.210449 0.977605i \(-0.432508\pi\)
0.210449 + 0.977605i \(0.432508\pi\)
\(930\) 0 0
\(931\) 42.8116 + 74.1519i 1.40309 + 2.43023i
\(932\) 0 0
\(933\) 21.0122 0.687910
\(934\) 0 0
\(935\) −2.71696 −0.0888540
\(936\) 0 0
\(937\) −52.5470 −1.71664 −0.858318 0.513117i \(-0.828491\pi\)
−0.858318 + 0.513117i \(0.828491\pi\)
\(938\) 0 0
\(939\) −13.7198 −0.447729
\(940\) 0 0
\(941\) 56.7928 1.85139 0.925697 0.378266i \(-0.123480\pi\)
0.925697 + 0.378266i \(0.123480\pi\)
\(942\) 0 0
\(943\) −18.6511 −0.607363
\(944\) 0 0
\(945\) −2.18663 3.78736i −0.0711312 0.123203i
\(946\) 0 0
\(947\) −53.5181 −1.73910 −0.869551 0.493842i \(-0.835592\pi\)
−0.869551 + 0.493842i \(0.835592\pi\)
\(948\) 0 0
\(949\) 9.36638 + 16.2230i 0.304046 + 0.526622i
\(950\) 0 0
\(951\) 4.53972 7.86302i 0.147210 0.254976i
\(952\) 0 0
\(953\) 52.1650 1.68979 0.844894 0.534933i \(-0.179663\pi\)
0.844894 + 0.534933i \(0.179663\pi\)
\(954\) 0 0
\(955\) −6.49218 + 11.2448i −0.210082 + 0.363873i
\(956\) 0 0
\(957\) −11.7812 + 20.4056i −0.380831 + 0.659619i
\(958\) 0 0
\(959\) 37.2467 + 64.5132i 1.20276 + 2.08324i
\(960\) 0 0
\(961\) −13.9868 + 24.2259i −0.451188 + 0.781481i
\(962\) 0 0
\(963\) 6.42328 0.206987
\(964\) 0 0
\(965\) −10.1932 −0.328131
\(966\) 0 0
\(967\) 30.3818 + 52.6229i 0.977014 + 1.69224i 0.673122 + 0.739532i \(0.264953\pi\)
0.303892 + 0.952706i \(0.401714\pi\)
\(968\) 0 0
\(969\) 3.20730 + 5.55520i 0.103033 + 0.178459i
\(970\) 0 0
\(971\) −26.0672 45.1497i −0.836537 1.44892i −0.892773 0.450507i \(-0.851243\pi\)
0.0562365 0.998417i \(-0.482090\pi\)
\(972\) 0 0
\(973\) −9.18547 15.9097i −0.294473 0.510042i
\(974\) 0 0
\(975\) −1.04087 + 1.80283i −0.0333344 + 0.0577369i
\(976\) 0 0
\(977\) −0.462261 + 0.800660i −0.0147890 + 0.0256154i −0.873325 0.487138i \(-0.838041\pi\)
0.858536 + 0.512753i \(0.171374\pi\)
\(978\) 0 0
\(979\) 4.13247 + 7.15765i 0.132074 + 0.228759i
\(980\) 0 0
\(981\) 1.66310 0.0530988
\(982\) 0 0
\(983\) 32.6111 1.04013 0.520067 0.854126i \(-0.325907\pi\)
0.520067 + 0.854126i \(0.325907\pi\)
\(984\) 0 0
\(985\) −9.06345 15.6984i −0.288786 0.500191i
\(986\) 0 0
\(987\) 1.90937 3.30712i 0.0607758 0.105267i
\(988\) 0 0
\(989\) −42.1978 + 73.0887i −1.34181 + 2.32408i
\(990\) 0 0
\(991\) −29.1668 −0.926514 −0.463257 0.886224i \(-0.653319\pi\)
−0.463257 + 0.886224i \(0.653319\pi\)
\(992\) 0 0
\(993\) −4.36593 7.56201i −0.138549 0.239973i
\(994\) 0 0
\(995\) 9.84561 17.0531i 0.312127 0.540619i
\(996\) 0 0
\(997\) 17.4009 0.551092 0.275546 0.961288i \(-0.411141\pi\)
0.275546 + 0.961288i \(0.411141\pi\)
\(998\) 0 0
\(999\) −0.417228 + 0.722660i −0.0132005 + 0.0228640i
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 4020.2.q.m.3781.1 yes 24
67.37 even 3 inner 4020.2.q.m.841.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.m.841.1 24 67.37 even 3 inner
4020.2.q.m.3781.1 yes 24 1.1 even 1 trivial