Properties

Label 804.2.o.b.365.1
Level $804$
Weight $2$
Character 804.365
Analytic conductor $6.420$
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] = [804,2,Mod(365,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.365");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.o (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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 365.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 804.365
Dual form 804.2.o.b.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.41421i) q^{3} +2.00000 q^{5} +(-0.949490 - 0.548188i) q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.41421i) q^{3} +2.00000 q^{5} +(-0.949490 - 0.548188i) q^{7} +(-1.00000 + 2.82843i) q^{9} +(-0.724745 + 1.25529i) q^{11} +(4.50000 - 2.59808i) q^{13} +(-2.00000 - 2.82843i) q^{15} +(5.17423 - 2.98735i) q^{17} +(-1.50000 - 2.59808i) q^{19} +(0.174235 + 1.89097i) q^{21} +(0.825765 - 0.476756i) q^{23} -1.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +(4.62372 + 2.66951i) q^{29} +(-1.50000 - 0.866025i) q^{31} +(2.50000 - 0.230351i) q^{33} +(-1.89898 - 1.09638i) q^{35} +(-4.94949 - 8.57277i) q^{37} +(-8.17423 - 3.76588i) q^{39} +(1.72474 - 2.98735i) q^{41} +1.27135i q^{43} +(-2.00000 + 5.65685i) q^{45} +(0.275255 + 0.158919i) q^{47} +(-2.89898 - 5.02118i) q^{49} +(-9.39898 - 4.33013i) q^{51} -6.00000 q^{53} +(-1.44949 + 2.51059i) q^{55} +(-2.17423 + 4.71940i) q^{57} +0.635674i q^{59} +(9.39898 - 5.42650i) q^{61} +(2.50000 - 2.13737i) q^{63} +(9.00000 - 5.19615i) q^{65} +(8.00000 + 1.73205i) q^{67} +(-1.50000 - 0.691053i) q^{69} +(-11.1742 - 6.45145i) q^{71} +(2.94949 + 5.10867i) q^{73} +(1.00000 + 1.41421i) q^{75} +(1.37628 - 0.794593i) q^{77} +(0.398979 + 0.230351i) q^{79} +(-7.00000 - 5.65685i) q^{81} +(10.0732 - 5.81577i) q^{83} +(10.3485 - 5.97469i) q^{85} +(-0.848469 - 9.20844i) q^{87} +4.73545i q^{89} -5.69694 q^{91} +(0.275255 + 2.98735i) q^{93} +(-3.00000 - 5.19615i) q^{95} +(11.8485 - 6.84072i) q^{97} +(-2.82577 - 3.30518i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 8 q^{5} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{5} + 6 q^{7} - 4 q^{9} + 2 q^{11} + 18 q^{13} - 8 q^{15} + 6 q^{17} - 6 q^{19} - 14 q^{21} + 18 q^{23} - 4 q^{25} + 20 q^{27} - 6 q^{29} - 6 q^{31} + 10 q^{33} + 12 q^{35} - 10 q^{37} - 18 q^{39} + 2 q^{41} - 8 q^{45} + 6 q^{47} + 8 q^{49} - 18 q^{51} - 24 q^{53} + 4 q^{55} + 6 q^{57} + 18 q^{61} + 10 q^{63} + 36 q^{65} + 32 q^{67} - 6 q^{69} - 30 q^{71} + 2 q^{73} + 4 q^{75} + 30 q^{77} - 18 q^{79} - 28 q^{81} + 6 q^{83} + 12 q^{85} + 26 q^{87} + 36 q^{91} + 6 q^{93} - 12 q^{95} + 18 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 1.41421i −0.577350 0.816497i
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −0.949490 0.548188i −0.358873 0.207196i 0.309713 0.950830i \(-0.399767\pi\)
−0.668586 + 0.743634i \(0.733100\pi\)
\(8\) 0 0
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 0 0
\(11\) −0.724745 + 1.25529i −0.218519 + 0.378486i −0.954355 0.298674i \(-0.903456\pi\)
0.735837 + 0.677159i \(0.236789\pi\)
\(12\) 0 0
\(13\) 4.50000 2.59808i 1.24808 0.720577i 0.277350 0.960769i \(-0.410544\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) −2.00000 2.82843i −0.516398 0.730297i
\(16\) 0 0
\(17\) 5.17423 2.98735i 1.25494 0.724538i 0.282851 0.959164i \(-0.408720\pi\)
0.972086 + 0.234626i \(0.0753866\pi\)
\(18\) 0 0
\(19\) −1.50000 2.59808i −0.344124 0.596040i 0.641071 0.767482i \(-0.278491\pi\)
−0.985194 + 0.171442i \(0.945157\pi\)
\(20\) 0 0
\(21\) 0.174235 + 1.89097i 0.0380211 + 0.412643i
\(22\) 0 0
\(23\) 0.825765 0.476756i 0.172184 0.0994105i −0.411431 0.911441i \(-0.634971\pi\)
0.583615 + 0.812030i \(0.301638\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 1.41421i 0.962250 0.272166i
\(28\) 0 0
\(29\) 4.62372 + 2.66951i 0.858604 + 0.495715i 0.863545 0.504273i \(-0.168239\pi\)
−0.00494052 + 0.999988i \(0.501573\pi\)
\(30\) 0 0
\(31\) −1.50000 0.866025i −0.269408 0.155543i 0.359211 0.933257i \(-0.383046\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 2.50000 0.230351i 0.435194 0.0400989i
\(34\) 0 0
\(35\) −1.89898 1.09638i −0.320986 0.185321i
\(36\) 0 0
\(37\) −4.94949 8.57277i −0.813691 1.40935i −0.910264 0.414028i \(-0.864121\pi\)
0.0965729 0.995326i \(-0.469212\pi\)
\(38\) 0 0
\(39\) −8.17423 3.76588i −1.30893 0.603024i
\(40\) 0 0
\(41\) 1.72474 2.98735i 0.269360 0.466545i −0.699337 0.714792i \(-0.746521\pi\)
0.968697 + 0.248247i \(0.0798546\pi\)
\(42\) 0 0
\(43\) 1.27135i 0.193879i 0.995290 + 0.0969395i \(0.0309053\pi\)
−0.995290 + 0.0969395i \(0.969095\pi\)
\(44\) 0 0
\(45\) −2.00000 + 5.65685i −0.298142 + 0.843274i
\(46\) 0 0
\(47\) 0.275255 + 0.158919i 0.0401501 + 0.0231807i 0.519941 0.854202i \(-0.325954\pi\)
−0.479791 + 0.877383i \(0.659287\pi\)
\(48\) 0 0
\(49\) −2.89898 5.02118i −0.414140 0.717311i
\(50\) 0 0
\(51\) −9.39898 4.33013i −1.31612 0.606339i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −1.44949 + 2.51059i −0.195449 + 0.338528i
\(56\) 0 0
\(57\) −2.17423 + 4.71940i −0.287984 + 0.625099i
\(58\) 0 0
\(59\) 0.635674i 0.0827578i 0.999144 + 0.0413789i \(0.0131751\pi\)
−0.999144 + 0.0413789i \(0.986825\pi\)
\(60\) 0 0
\(61\) 9.39898 5.42650i 1.20342 0.694793i 0.242103 0.970251i \(-0.422163\pi\)
0.961313 + 0.275458i \(0.0888295\pi\)
\(62\) 0 0
\(63\) 2.50000 2.13737i 0.314970 0.269284i
\(64\) 0 0
\(65\) 9.00000 5.19615i 1.11631 0.644503i
\(66\) 0 0
\(67\) 8.00000 + 1.73205i 0.977356 + 0.211604i
\(68\) 0 0
\(69\) −1.50000 0.691053i −0.180579 0.0831930i
\(70\) 0 0
\(71\) −11.1742 6.45145i −1.32614 0.765646i −0.341438 0.939904i \(-0.610914\pi\)
−0.984700 + 0.174258i \(0.944247\pi\)
\(72\) 0 0
\(73\) 2.94949 + 5.10867i 0.345212 + 0.597924i 0.985392 0.170300i \(-0.0544738\pi\)
−0.640181 + 0.768224i \(0.721140\pi\)
\(74\) 0 0
\(75\) 1.00000 + 1.41421i 0.115470 + 0.163299i
\(76\) 0 0
\(77\) 1.37628 0.794593i 0.156841 0.0905523i
\(78\) 0 0
\(79\) 0.398979 + 0.230351i 0.0448887 + 0.0259165i 0.522276 0.852776i \(-0.325083\pi\)
−0.477388 + 0.878693i \(0.658416\pi\)
\(80\) 0 0
\(81\) −7.00000 5.65685i −0.777778 0.628539i
\(82\) 0 0
\(83\) 10.0732 5.81577i 1.10568 0.638364i 0.167972 0.985792i \(-0.446278\pi\)
0.937707 + 0.347428i \(0.112945\pi\)
\(84\) 0 0
\(85\) 10.3485 5.97469i 1.12245 0.648046i
\(86\) 0 0
\(87\) −0.848469 9.20844i −0.0909654 0.987249i
\(88\) 0 0
\(89\) 4.73545i 0.501957i 0.967993 + 0.250978i \(0.0807523\pi\)
−0.967993 + 0.250978i \(0.919248\pi\)
\(90\) 0 0
\(91\) −5.69694 −0.597201
\(92\) 0 0
\(93\) 0.275255 + 2.98735i 0.0285426 + 0.309773i
\(94\) 0 0
\(95\) −3.00000 5.19615i −0.307794 0.533114i
\(96\) 0 0
\(97\) 11.8485 6.84072i 1.20303 0.694570i 0.241802 0.970326i \(-0.422262\pi\)
0.961228 + 0.275756i \(0.0889282\pi\)
\(98\) 0 0
\(99\) −2.82577 3.30518i −0.284000 0.332183i
\(100\) 0 0
\(101\) −6.72474 + 11.6476i −0.669137 + 1.15898i 0.309009 + 0.951059i \(0.400003\pi\)
−0.978146 + 0.207920i \(0.933331\pi\)
\(102\) 0 0
\(103\) 1.94949 3.37662i 0.192089 0.332708i −0.753853 0.657043i \(-0.771807\pi\)
0.945942 + 0.324335i \(0.105140\pi\)
\(104\) 0 0
\(105\) 0.348469 + 3.78194i 0.0340071 + 0.369079i
\(106\) 0 0
\(107\) 9.12096i 0.881756i 0.897567 + 0.440878i \(0.145333\pi\)
−0.897567 + 0.440878i \(0.854667\pi\)
\(108\) 0 0
\(109\) 5.02118i 0.480942i 0.970656 + 0.240471i \(0.0773019\pi\)
−0.970656 + 0.240471i \(0.922698\pi\)
\(110\) 0 0
\(111\) −7.17423 + 15.5724i −0.680948 + 1.47807i
\(112\) 0 0
\(113\) −3.27526 + 5.67291i −0.308110 + 0.533662i −0.977949 0.208844i \(-0.933030\pi\)
0.669839 + 0.742507i \(0.266363\pi\)
\(114\) 0 0
\(115\) 1.65153 0.953512i 0.154006 0.0889154i
\(116\) 0 0
\(117\) 2.84847 + 15.3260i 0.263341 + 1.41689i
\(118\) 0 0
\(119\) −6.55051 −0.600484
\(120\) 0 0
\(121\) 4.44949 + 7.70674i 0.404499 + 0.700613i
\(122\) 0 0
\(123\) −5.94949 + 0.548188i −0.536447 + 0.0494285i
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 4.39898 7.61926i 0.390346 0.676100i −0.602149 0.798384i \(-0.705689\pi\)
0.992495 + 0.122284i \(0.0390219\pi\)
\(128\) 0 0
\(129\) 1.79796 1.27135i 0.158301 0.111936i
\(130\) 0 0
\(131\) 6.29253i 0.549781i −0.961476 0.274890i \(-0.911358\pi\)
0.961476 0.274890i \(-0.0886416\pi\)
\(132\) 0 0
\(133\) 3.28913i 0.285204i
\(134\) 0 0
\(135\) 10.0000 2.82843i 0.860663 0.243432i
\(136\) 0 0
\(137\) −9.10102 −0.777553 −0.388776 0.921332i \(-0.627102\pi\)
−0.388776 + 0.921332i \(0.627102\pi\)
\(138\) 0 0
\(139\) 7.21393i 0.611878i 0.952051 + 0.305939i \(0.0989703\pi\)
−0.952051 + 0.305939i \(0.901030\pi\)
\(140\) 0 0
\(141\) −0.0505103 0.548188i −0.00425373 0.0461658i
\(142\) 0 0
\(143\) 7.53177i 0.629838i
\(144\) 0 0
\(145\) 9.24745 + 5.33902i 0.767959 + 0.443381i
\(146\) 0 0
\(147\) −4.20204 + 9.12096i −0.346579 + 0.752284i
\(148\) 0 0
\(149\) 14.7778i 1.21065i 0.795980 + 0.605323i \(0.206956\pi\)
−0.795980 + 0.605323i \(0.793044\pi\)
\(150\) 0 0
\(151\) −4.39898 7.61926i −0.357984 0.620046i 0.629640 0.776887i \(-0.283202\pi\)
−0.987624 + 0.156841i \(0.949869\pi\)
\(152\) 0 0
\(153\) 3.27526 + 17.6223i 0.264789 + 1.42468i
\(154\) 0 0
\(155\) −3.00000 1.73205i −0.240966 0.139122i
\(156\) 0 0
\(157\) 6.94949 + 12.0369i 0.554630 + 0.960647i 0.997932 + 0.0642747i \(0.0204734\pi\)
−0.443303 + 0.896372i \(0.646193\pi\)
\(158\) 0 0
\(159\) 6.00000 + 8.48528i 0.475831 + 0.672927i
\(160\) 0 0
\(161\) −1.04541 −0.0823897
\(162\) 0 0
\(163\) −11.2980 + 19.5686i −0.884924 + 1.53273i −0.0391242 + 0.999234i \(0.512457\pi\)
−0.845800 + 0.533500i \(0.820877\pi\)
\(164\) 0 0
\(165\) 5.00000 0.460702i 0.389249 0.0358656i
\(166\) 0 0
\(167\) 22.0732 + 12.7440i 1.70808 + 0.986158i 0.936941 + 0.349487i \(0.113644\pi\)
0.771136 + 0.636671i \(0.219689\pi\)
\(168\) 0 0
\(169\) 7.00000 12.1244i 0.538462 0.932643i
\(170\) 0 0
\(171\) 8.84847 1.64456i 0.676659 0.125763i
\(172\) 0 0
\(173\) −10.6237 + 6.13361i −0.807707 + 0.466330i −0.846159 0.532931i \(-0.821091\pi\)
0.0384521 + 0.999260i \(0.487757\pi\)
\(174\) 0 0
\(175\) 0.949490 + 0.548188i 0.0717747 + 0.0414391i
\(176\) 0 0
\(177\) 0.898979 0.635674i 0.0675714 0.0477802i
\(178\) 0 0
\(179\) 13.5959 1.01621 0.508103 0.861296i \(-0.330347\pi\)
0.508103 + 0.861296i \(0.330347\pi\)
\(180\) 0 0
\(181\) 1.05051 1.81954i 0.0780838 0.135245i −0.824339 0.566096i \(-0.808453\pi\)
0.902423 + 0.430851i \(0.141786\pi\)
\(182\) 0 0
\(183\) −17.0732 7.86566i −1.26209 0.581446i
\(184\) 0 0
\(185\) −9.89898 17.1455i −0.727787 1.26056i
\(186\) 0 0
\(187\) 8.66025i 0.633300i
\(188\) 0 0
\(189\) −5.52270 1.39816i −0.401718 0.101701i
\(190\) 0 0
\(191\) 8.72474 + 15.1117i 0.631300 + 1.09344i 0.987286 + 0.158953i \(0.0508118\pi\)
−0.355986 + 0.934491i \(0.615855\pi\)
\(192\) 0 0
\(193\) −12.8990 −0.928489 −0.464244 0.885707i \(-0.653674\pi\)
−0.464244 + 0.885707i \(0.653674\pi\)
\(194\) 0 0
\(195\) −16.3485 7.53177i −1.17074 0.539361i
\(196\) 0 0
\(197\) −1.62372 + 2.81237i −0.115686 + 0.200373i −0.918054 0.396456i \(-0.870240\pi\)
0.802368 + 0.596830i \(0.203573\pi\)
\(198\) 0 0
\(199\) 5.50000 + 9.52628i 0.389885 + 0.675300i 0.992434 0.122782i \(-0.0391815\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) −5.55051 13.0458i −0.391503 0.920177i
\(202\) 0 0
\(203\) −2.92679 5.06934i −0.205420 0.355798i
\(204\) 0 0
\(205\) 3.44949 5.97469i 0.240923 0.417291i
\(206\) 0 0
\(207\) 0.522704 + 2.81237i 0.0363304 + 0.195473i
\(208\) 0 0
\(209\) 4.34847 0.300790
\(210\) 0 0
\(211\) 2.94949 + 5.10867i 0.203051 + 0.351695i 0.949510 0.313737i \(-0.101581\pi\)
−0.746459 + 0.665432i \(0.768248\pi\)
\(212\) 0 0
\(213\) 2.05051 + 22.2542i 0.140499 + 1.52483i
\(214\) 0 0
\(215\) 2.54270i 0.173411i
\(216\) 0 0
\(217\) 0.949490 + 1.64456i 0.0644556 + 0.111640i
\(218\) 0 0
\(219\) 4.27526 9.27987i 0.288895 0.627076i
\(220\) 0 0
\(221\) 15.5227 26.8861i 1.04417 1.80856i
\(222\) 0 0
\(223\) 20.6969 1.38597 0.692985 0.720952i \(-0.256295\pi\)
0.692985 + 0.720952i \(0.256295\pi\)
\(224\) 0 0
\(225\) 1.00000 2.82843i 0.0666667 0.188562i
\(226\) 0 0
\(227\) −23.4217 13.5225i −1.55455 0.897521i −0.997762 0.0668675i \(-0.978700\pi\)
−0.556790 0.830653i \(-0.687967\pi\)
\(228\) 0 0
\(229\) −8.84847 + 5.10867i −0.584723 + 0.337590i −0.763008 0.646389i \(-0.776278\pi\)
0.178285 + 0.983979i \(0.442945\pi\)
\(230\) 0 0
\(231\) −2.50000 1.15175i −0.164488 0.0757799i
\(232\) 0 0
\(233\) −9.27526 + 16.0652i −0.607642 + 1.05247i 0.383986 + 0.923339i \(0.374551\pi\)
−0.991628 + 0.129128i \(0.958782\pi\)
\(234\) 0 0
\(235\) 0.550510 + 0.317837i 0.0359113 + 0.0207334i
\(236\) 0 0
\(237\) −0.0732141 0.794593i −0.00475577 0.0516144i
\(238\) 0 0
\(239\) 5.62372 9.74058i 0.363768 0.630066i −0.624809 0.780777i \(-0.714823\pi\)
0.988578 + 0.150712i \(0.0481566\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) −1.00000 + 15.5563i −0.0641500 + 0.997940i
\(244\) 0 0
\(245\) −5.79796 10.0424i −0.370418 0.641583i
\(246\) 0 0
\(247\) −13.5000 7.79423i −0.858984 0.495935i
\(248\) 0 0
\(249\) −18.2980 8.42990i −1.15959 0.534223i
\(250\) 0 0
\(251\) −10.0732 17.4473i −0.635816 1.10126i −0.986342 0.164712i \(-0.947331\pi\)
0.350526 0.936553i \(-0.386003\pi\)
\(252\) 0 0
\(253\) 1.38211i 0.0868922i
\(254\) 0 0
\(255\) −18.7980 8.66025i −1.17717 0.542326i
\(256\) 0 0
\(257\) 13.0732 + 7.54782i 0.815485 + 0.470820i 0.848857 0.528623i \(-0.177291\pi\)
−0.0333722 + 0.999443i \(0.510625\pi\)
\(258\) 0 0
\(259\) 10.8530i 0.674373i
\(260\) 0 0
\(261\) −12.1742 + 10.4084i −0.753566 + 0.644261i
\(262\) 0 0
\(263\) 7.84961i 0.484027i −0.970273 0.242014i \(-0.922192\pi\)
0.970273 0.242014i \(-0.0778079\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 6.69694 4.73545i 0.409846 0.289805i
\(268\) 0 0
\(269\) 6.00680i 0.366241i −0.983090 0.183121i \(-0.941380\pi\)
0.983090 0.183121i \(-0.0586199\pi\)
\(270\) 0 0
\(271\) 9.75663i 0.592673i −0.955084 0.296337i \(-0.904235\pi\)
0.955084 0.296337i \(-0.0957650\pi\)
\(272\) 0 0
\(273\) 5.69694 + 8.05669i 0.344794 + 0.487613i
\(274\) 0 0
\(275\) 0.724745 1.25529i 0.0437038 0.0756971i
\(276\) 0 0
\(277\) 25.7980 1.55005 0.775025 0.631931i \(-0.217737\pi\)
0.775025 + 0.631931i \(0.217737\pi\)
\(278\) 0 0
\(279\) 3.94949 3.37662i 0.236450 0.202153i
\(280\) 0 0
\(281\) −3.72474 6.45145i −0.222200 0.384861i 0.733276 0.679931i \(-0.237990\pi\)
−0.955476 + 0.295070i \(0.904657\pi\)
\(282\) 0 0
\(283\) −7.79796 −0.463541 −0.231770 0.972771i \(-0.574452\pi\)
−0.231770 + 0.972771i \(0.574452\pi\)
\(284\) 0 0
\(285\) −4.34847 + 9.43879i −0.257581 + 0.559106i
\(286\) 0 0
\(287\) −3.27526 + 1.89097i −0.193332 + 0.111620i
\(288\) 0 0
\(289\) 9.34847 16.1920i 0.549910 0.952472i
\(290\) 0 0
\(291\) −21.5227 9.91555i −1.26168 0.581260i
\(292\) 0 0
\(293\) 29.2699i 1.70997i 0.518657 + 0.854983i \(0.326432\pi\)
−0.518657 + 0.854983i \(0.673568\pi\)
\(294\) 0 0
\(295\) 1.27135i 0.0740208i
\(296\) 0 0
\(297\) −1.84847 + 7.30142i −0.107259 + 0.423671i
\(298\) 0 0
\(299\) 2.47730 4.29080i 0.143266 0.248144i
\(300\) 0 0
\(301\) 0.696938 1.20713i 0.0401709 0.0695780i
\(302\) 0 0
\(303\) 23.1969 2.13737i 1.33263 0.122789i
\(304\) 0 0
\(305\) 18.7980 10.8530i 1.07637 0.621441i
\(306\) 0 0
\(307\) −4.29796 7.44428i −0.245297 0.424868i 0.716918 0.697158i \(-0.245552\pi\)
−0.962215 + 0.272290i \(0.912219\pi\)
\(308\) 0 0
\(309\) −6.72474 + 0.619620i −0.382557 + 0.0352490i
\(310\) 0 0
\(311\) 14.2020 0.805324 0.402662 0.915349i \(-0.368085\pi\)
0.402662 + 0.915349i \(0.368085\pi\)
\(312\) 0 0
\(313\) 11.6637i 0.659269i −0.944109 0.329634i \(-0.893075\pi\)
0.944109 0.329634i \(-0.106925\pi\)
\(314\) 0 0
\(315\) 5.00000 4.27475i 0.281718 0.240855i
\(316\) 0 0
\(317\) −16.3207 + 9.42274i −0.916660 + 0.529234i −0.882568 0.470185i \(-0.844187\pi\)
−0.0340918 + 0.999419i \(0.510854\pi\)
\(318\) 0 0
\(319\) −6.70204 + 3.86943i −0.375242 + 0.216646i
\(320\) 0 0
\(321\) 12.8990 9.12096i 0.719951 0.509082i
\(322\) 0 0
\(323\) −15.5227 8.96204i −0.863706 0.498661i
\(324\) 0 0
\(325\) −4.50000 + 2.59808i −0.249615 + 0.144115i
\(326\) 0 0
\(327\) 7.10102 5.02118i 0.392687 0.277672i
\(328\) 0 0
\(329\) −0.174235 0.301783i −0.00960587 0.0166378i
\(330\) 0 0
\(331\) −12.0959 6.98358i −0.664852 0.383852i 0.129271 0.991609i \(-0.458736\pi\)
−0.794123 + 0.607757i \(0.792070\pi\)
\(332\) 0 0
\(333\) 29.1969 5.42650i 1.59998 0.297371i
\(334\) 0 0
\(335\) 16.0000 + 3.46410i 0.874173 + 0.189264i
\(336\) 0 0
\(337\) −15.9495 + 9.20844i −0.868824 + 0.501616i −0.866957 0.498382i \(-0.833928\pi\)
−0.00186678 + 0.999998i \(0.500594\pi\)
\(338\) 0 0
\(339\) 11.2980 1.04100i 0.613621 0.0565393i
\(340\) 0 0
\(341\) 2.17423 1.25529i 0.117741 0.0679780i
\(342\) 0 0
\(343\) 14.0314i 0.757623i
\(344\) 0 0
\(345\) −3.00000 1.38211i −0.161515 0.0744101i
\(346\) 0 0
\(347\) −6.07321 + 10.5191i −0.326027 + 0.564696i −0.981720 0.190332i \(-0.939043\pi\)
0.655693 + 0.755028i \(0.272377\pi\)
\(348\) 0 0
\(349\) 2.20204 0.117873 0.0589363 0.998262i \(-0.481229\pi\)
0.0589363 + 0.998262i \(0.481229\pi\)
\(350\) 0 0
\(351\) 18.8258 19.3543i 1.00485 1.03306i
\(352\) 0 0
\(353\) 3.17423 + 5.49794i 0.168947 + 0.292626i 0.938050 0.346500i \(-0.112630\pi\)
−0.769103 + 0.639125i \(0.779297\pi\)
\(354\) 0 0
\(355\) −22.3485 12.9029i −1.18613 0.684815i
\(356\) 0 0
\(357\) 6.55051 + 9.26382i 0.346690 + 0.490293i
\(358\) 0 0
\(359\) 13.5065i 0.712844i 0.934325 + 0.356422i \(0.116003\pi\)
−0.934325 + 0.356422i \(0.883997\pi\)
\(360\) 0 0
\(361\) 5.00000 8.66025i 0.263158 0.455803i
\(362\) 0 0
\(363\) 6.44949 13.9993i 0.338510 0.734771i
\(364\) 0 0
\(365\) 5.89898 + 10.2173i 0.308767 + 0.534800i
\(366\) 0 0
\(367\) 13.5000 + 7.79423i 0.704694 + 0.406855i 0.809093 0.587680i \(-0.199959\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) 6.72474 + 7.86566i 0.350076 + 0.409470i
\(370\) 0 0
\(371\) 5.69694 + 3.28913i 0.295770 + 0.170763i
\(372\) 0 0
\(373\) 0.398979 + 0.230351i 0.0206584 + 0.0119271i 0.510294 0.860000i \(-0.329537\pi\)
−0.489635 + 0.871927i \(0.662870\pi\)
\(374\) 0 0
\(375\) 12.0000 + 16.9706i 0.619677 + 0.876356i
\(376\) 0 0
\(377\) 27.7423 1.42880
\(378\) 0 0
\(379\) −20.6010 + 11.8940i −1.05820 + 0.610954i −0.924935 0.380126i \(-0.875881\pi\)
−0.133269 + 0.991080i \(0.542547\pi\)
\(380\) 0 0
\(381\) −15.1742 + 1.39816i −0.777400 + 0.0716299i
\(382\) 0 0
\(383\) −12.0732 20.9114i −0.616912 1.06852i −0.990046 0.140746i \(-0.955050\pi\)
0.373133 0.927778i \(-0.378283\pi\)
\(384\) 0 0
\(385\) 2.75255 1.58919i 0.140283 0.0809924i
\(386\) 0 0
\(387\) −3.59592 1.27135i −0.182791 0.0646263i
\(388\) 0 0
\(389\) −8.97219 + 5.18010i −0.454908 + 0.262641i −0.709901 0.704302i \(-0.751260\pi\)
0.254993 + 0.966943i \(0.417927\pi\)
\(390\) 0 0
\(391\) 2.84847 4.93369i 0.144053 0.249508i
\(392\) 0 0
\(393\) −8.89898 + 6.29253i −0.448894 + 0.317416i
\(394\) 0 0
\(395\) 0.797959 + 0.460702i 0.0401497 + 0.0231804i
\(396\) 0 0
\(397\) −26.6969 −1.33988 −0.669940 0.742415i \(-0.733680\pi\)
−0.669940 + 0.742415i \(0.733680\pi\)
\(398\) 0 0
\(399\) 4.65153 3.28913i 0.232868 0.164662i
\(400\) 0 0
\(401\) 5.10102 0.254733 0.127366 0.991856i \(-0.459348\pi\)
0.127366 + 0.991856i \(0.459348\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) −14.0000 11.3137i −0.695666 0.562183i
\(406\) 0 0
\(407\) 14.3485 0.711227
\(408\) 0 0
\(409\) −16.5000 9.52628i −0.815872 0.471044i 0.0331186 0.999451i \(-0.489456\pi\)
−0.848991 + 0.528407i \(0.822789\pi\)
\(410\) 0 0
\(411\) 9.10102 + 12.8708i 0.448920 + 0.634869i
\(412\) 0 0
\(413\) 0.348469 0.603566i 0.0171470 0.0296996i
\(414\) 0 0
\(415\) 20.1464 11.6315i 0.988949 0.570970i
\(416\) 0 0
\(417\) 10.2020 7.21393i 0.499596 0.353268i
\(418\) 0 0
\(419\) 22.3207 12.8868i 1.09044 0.629563i 0.156743 0.987639i \(-0.449900\pi\)
0.933692 + 0.358076i \(0.116567\pi\)
\(420\) 0 0
\(421\) −6.50000 11.2583i −0.316791 0.548697i 0.663026 0.748596i \(-0.269272\pi\)
−0.979817 + 0.199899i \(0.935939\pi\)
\(422\) 0 0
\(423\) −0.724745 + 0.619620i −0.0352383 + 0.0301270i
\(424\) 0 0
\(425\) −5.17423 + 2.98735i −0.250987 + 0.144908i
\(426\) 0 0
\(427\) −11.8990 −0.575832
\(428\) 0 0
\(429\) 10.6515 7.53177i 0.514261 0.363637i
\(430\) 0 0
\(431\) 27.7702 + 16.0331i 1.33764 + 0.772287i 0.986457 0.164019i \(-0.0524458\pi\)
0.351184 + 0.936306i \(0.385779\pi\)
\(432\) 0 0
\(433\) −13.5000 7.79423i −0.648769 0.374567i 0.139216 0.990262i \(-0.455542\pi\)
−0.787984 + 0.615695i \(0.788875\pi\)
\(434\) 0 0
\(435\) −1.69694 18.4169i −0.0813620 0.883022i
\(436\) 0 0
\(437\) −2.47730 1.43027i −0.118505 0.0684190i
\(438\) 0 0
\(439\) −4.60102 7.96920i −0.219595 0.380349i 0.735089 0.677970i \(-0.237140\pi\)
−0.954684 + 0.297621i \(0.903807\pi\)
\(440\) 0 0
\(441\) 17.1010 3.17837i 0.814334 0.151351i
\(442\) 0 0
\(443\) −16.6237 + 28.7931i −0.789817 + 1.36800i 0.136262 + 0.990673i \(0.456491\pi\)
−0.926079 + 0.377330i \(0.876842\pi\)
\(444\) 0 0
\(445\) 9.47090i 0.448964i
\(446\) 0 0
\(447\) 20.8990 14.7778i 0.988488 0.698966i
\(448\) 0 0
\(449\) 14.1742 + 8.18350i 0.668923 + 0.386203i 0.795669 0.605732i \(-0.207120\pi\)
−0.126745 + 0.991935i \(0.540453\pi\)
\(450\) 0 0
\(451\) 2.50000 + 4.33013i 0.117720 + 0.203898i
\(452\) 0 0
\(453\) −6.37628 + 13.8404i −0.299584 + 0.650276i
\(454\) 0 0
\(455\) −11.3939 −0.534153
\(456\) 0 0
\(457\) 18.1969 31.5180i 0.851217 1.47435i −0.0288939 0.999582i \(-0.509198\pi\)
0.880111 0.474768i \(-0.157468\pi\)
\(458\) 0 0
\(459\) 21.6464 22.2542i 1.01037 1.03874i
\(460\) 0 0
\(461\) 42.1407i 1.96269i −0.192263 0.981344i \(-0.561583\pi\)
0.192263 0.981344i \(-0.438417\pi\)
\(462\) 0 0
\(463\) −7.19694 + 4.15515i −0.334470 + 0.193106i −0.657824 0.753172i \(-0.728523\pi\)
0.323354 + 0.946278i \(0.395190\pi\)
\(464\) 0 0
\(465\) 0.550510 + 5.97469i 0.0255293 + 0.277070i
\(466\) 0 0
\(467\) −35.1742 + 20.3079i −1.62767 + 0.939735i −0.642883 + 0.765964i \(0.722262\pi\)
−0.984786 + 0.173771i \(0.944405\pi\)
\(468\) 0 0
\(469\) −6.64643 6.03007i −0.306904 0.278443i
\(470\) 0 0
\(471\) 10.0732 21.8649i 0.464149 1.00748i
\(472\) 0 0
\(473\) −1.59592 0.921404i −0.0733804 0.0423662i
\(474\) 0 0
\(475\) 1.50000 + 2.59808i 0.0688247 + 0.119208i
\(476\) 0 0
\(477\) 6.00000 16.9706i 0.274721 0.777029i
\(478\) 0 0
\(479\) −17.7247 + 10.2334i −0.809864 + 0.467575i −0.846909 0.531738i \(-0.821539\pi\)
0.0370445 + 0.999314i \(0.488206\pi\)
\(480\) 0 0
\(481\) −44.5454 25.7183i −2.03110 1.17265i
\(482\) 0 0
\(483\) 1.04541 + 1.47843i 0.0475677 + 0.0672709i
\(484\) 0 0
\(485\) 23.6969 13.6814i 1.07602 0.621242i
\(486\) 0 0
\(487\) −8.60102 + 4.96580i −0.389749 + 0.225022i −0.682052 0.731304i \(-0.738912\pi\)
0.292302 + 0.956326i \(0.405579\pi\)
\(488\) 0 0
\(489\) 38.9722 3.59091i 1.76238 0.162387i
\(490\) 0 0
\(491\) 11.6637i 0.526373i −0.964745 0.263187i \(-0.915226\pi\)
0.964745 0.263187i \(-0.0847735\pi\)
\(492\) 0 0
\(493\) 31.8990 1.43666
\(494\) 0 0
\(495\) −5.65153 6.61037i −0.254017 0.297114i
\(496\) 0 0
\(497\) 7.07321 + 12.2512i 0.317277 + 0.549540i
\(498\) 0 0
\(499\) −3.39898 + 1.96240i −0.152159 + 0.0878492i −0.574147 0.818753i \(-0.694666\pi\)
0.421987 + 0.906602i \(0.361333\pi\)
\(500\) 0 0
\(501\) −4.05051 43.9602i −0.180963 1.96400i
\(502\) 0 0
\(503\) 22.1742 38.4069i 0.988700 1.71248i 0.364528 0.931193i \(-0.381230\pi\)
0.624173 0.781287i \(-0.285436\pi\)
\(504\) 0 0
\(505\) −13.4495 + 23.2952i −0.598494 + 1.03662i
\(506\) 0 0
\(507\) −24.1464 + 2.22486i −1.07238 + 0.0988095i
\(508\) 0 0
\(509\) 2.82843i 0.125368i 0.998033 + 0.0626839i \(0.0199660\pi\)
−0.998033 + 0.0626839i \(0.980034\pi\)
\(510\) 0 0
\(511\) 6.46750i 0.286105i
\(512\) 0 0
\(513\) −11.1742 10.8691i −0.493355 0.479881i
\(514\) 0 0
\(515\) 3.89898 6.75323i 0.171810 0.297583i
\(516\) 0 0
\(517\) −0.398979 + 0.230351i −0.0175471 + 0.0101308i
\(518\) 0 0
\(519\) 19.2980 + 8.89060i 0.847086 + 0.390254i
\(520\) 0 0
\(521\) −40.2929 −1.76526 −0.882631 0.470066i \(-0.844230\pi\)
−0.882631 + 0.470066i \(0.844230\pi\)
\(522\) 0 0
\(523\) 22.6464 + 39.2248i 0.990259 + 1.71518i 0.615712 + 0.787971i \(0.288869\pi\)
0.374547 + 0.927208i \(0.377798\pi\)
\(524\) 0 0
\(525\) −0.174235 1.89097i −0.00760422 0.0825287i
\(526\) 0 0
\(527\) −10.3485 −0.450786
\(528\) 0 0
\(529\) −11.0454 + 19.1312i −0.480235 + 0.831792i
\(530\) 0 0
\(531\) −1.79796 0.635674i −0.0780248 0.0275859i
\(532\) 0 0
\(533\) 17.9241i 0.776378i
\(534\) 0 0
\(535\) 18.2419i 0.788667i
\(536\) 0 0
\(537\) −13.5959 19.2275i −0.586707 0.829729i
\(538\) 0 0
\(539\) 8.40408 0.361989
\(540\) 0 0
\(541\) 38.6766i 1.66284i −0.555648 0.831418i \(-0.687530\pi\)
0.555648 0.831418i \(-0.312470\pi\)
\(542\) 0 0
\(543\) −3.62372 + 0.333891i −0.155509 + 0.0143286i
\(544\) 0 0
\(545\) 10.0424i 0.430167i
\(546\) 0 0
\(547\) 17.8485 + 10.3048i 0.763145 + 0.440602i 0.830424 0.557132i \(-0.188098\pi\)
−0.0672785 + 0.997734i \(0.521432\pi\)
\(548\) 0 0
\(549\) 5.94949 + 32.0108i 0.253918 + 1.36619i
\(550\) 0 0
\(551\) 16.0171i 0.682349i
\(552\) 0 0
\(553\) −0.252551 0.437432i −0.0107396 0.0186015i
\(554\) 0 0
\(555\) −14.3485 + 31.1448i −0.609059 + 1.32202i
\(556\) 0 0
\(557\) −7.07321 4.08372i −0.299702 0.173033i 0.342607 0.939479i \(-0.388690\pi\)
−0.642309 + 0.766446i \(0.722023\pi\)
\(558\) 0 0
\(559\) 3.30306 + 5.72107i 0.139705 + 0.241975i
\(560\) 0 0
\(561\) 12.2474 8.66025i 0.517088 0.365636i
\(562\) 0 0
\(563\) −42.2929 −1.78243 −0.891216 0.453580i \(-0.850147\pi\)
−0.891216 + 0.453580i \(0.850147\pi\)
\(564\) 0 0
\(565\) −6.55051 + 11.3458i −0.275582 + 0.477322i
\(566\) 0 0
\(567\) 3.54541 + 9.20844i 0.148893 + 0.386718i
\(568\) 0 0
\(569\) 23.4217 + 13.5225i 0.981888 + 0.566893i 0.902840 0.429977i \(-0.141478\pi\)
0.0790484 + 0.996871i \(0.474812\pi\)
\(570\) 0 0
\(571\) −18.8485 + 32.6465i −0.788784 + 1.36621i 0.137928 + 0.990442i \(0.455956\pi\)
−0.926712 + 0.375772i \(0.877378\pi\)
\(572\) 0 0
\(573\) 12.6464 27.4504i 0.528312 1.14675i
\(574\) 0 0
\(575\) −0.825765 + 0.476756i −0.0344368 + 0.0198821i
\(576\) 0 0
\(577\) −26.8485 15.5010i −1.11772 0.645314i −0.176899 0.984229i \(-0.556607\pi\)
−0.940817 + 0.338915i \(0.889940\pi\)
\(578\) 0 0
\(579\) 12.8990 + 18.2419i 0.536063 + 0.758108i
\(580\) 0 0
\(581\) −12.7526 −0.529065
\(582\) 0 0
\(583\) 4.34847 7.53177i 0.180095 0.311934i
\(584\) 0 0
\(585\) 5.69694 + 30.6520i 0.235539 + 1.26730i
\(586\) 0 0
\(587\) 8.27526 + 14.3332i 0.341556 + 0.591593i 0.984722 0.174134i \(-0.0557127\pi\)
−0.643166 + 0.765727i \(0.722379\pi\)
\(588\) 0 0
\(589\) 5.19615i 0.214104i
\(590\) 0 0
\(591\) 5.60102 0.516080i 0.230395 0.0212287i
\(592\) 0 0
\(593\) 11.9722 + 20.7364i 0.491639 + 0.851544i 0.999954 0.00962762i \(-0.00306461\pi\)
−0.508315 + 0.861171i \(0.669731\pi\)
\(594\) 0 0
\(595\) −13.1010 −0.537089
\(596\) 0 0
\(597\) 7.97219 17.3045i 0.326280 0.708224i
\(598\) 0 0
\(599\) −8.17423 + 14.1582i −0.333990 + 0.578488i −0.983290 0.182044i \(-0.941729\pi\)
0.649300 + 0.760532i \(0.275062\pi\)
\(600\) 0 0
\(601\) −1.39898 2.42310i −0.0570656 0.0988405i 0.836081 0.548605i \(-0.184841\pi\)
−0.893147 + 0.449765i \(0.851508\pi\)
\(602\) 0 0
\(603\) −12.8990 + 20.8954i −0.525287 + 0.850925i
\(604\) 0 0
\(605\) 8.89898 + 15.4135i 0.361795 + 0.626647i
\(606\) 0 0
\(607\) −9.84847 + 17.0580i −0.399737 + 0.692365i −0.993693 0.112133i \(-0.964232\pi\)
0.593956 + 0.804497i \(0.297565\pi\)
\(608\) 0 0
\(609\) −4.24235 + 9.20844i −0.171909 + 0.373145i
\(610\) 0 0
\(611\) 1.65153 0.0668138
\(612\) 0 0
\(613\) 13.1969 + 22.8578i 0.533019 + 0.923217i 0.999256 + 0.0385567i \(0.0122760\pi\)
−0.466237 + 0.884660i \(0.654391\pi\)
\(614\) 0 0
\(615\) −11.8990 + 1.09638i −0.479813 + 0.0442102i
\(616\) 0 0
\(617\) 26.7272i 1.07600i −0.842946 0.537998i \(-0.819181\pi\)
0.842946 0.537998i \(-0.180819\pi\)
\(618\) 0 0
\(619\) −2.94949 5.10867i −0.118550 0.205335i 0.800643 0.599141i \(-0.204491\pi\)
−0.919193 + 0.393807i \(0.871158\pi\)
\(620\) 0 0
\(621\) 3.45459 3.55159i 0.138628 0.142520i
\(622\) 0 0
\(623\) 2.59592 4.49626i 0.104003 0.180139i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −4.34847 6.14966i −0.173661 0.245594i
\(628\) 0 0
\(629\) −51.2196 29.5717i −2.04226 1.17910i
\(630\) 0 0
\(631\) 23.5454 13.5939i 0.937328 0.541167i 0.0482062 0.998837i \(-0.484650\pi\)
0.889122 + 0.457671i \(0.151316\pi\)
\(632\) 0 0
\(633\) 4.27526 9.27987i 0.169926 0.368842i
\(634\) 0 0
\(635\) 8.79796 15.2385i 0.349136 0.604722i
\(636\) 0 0
\(637\) −26.0908 15.0635i −1.03376 0.596839i
\(638\) 0 0
\(639\) 29.4217 25.1541i 1.16390 0.995079i
\(640\) 0 0
\(641\) 21.5227 37.2784i 0.850096 1.47241i −0.0310256 0.999519i \(-0.509877\pi\)
0.881121 0.472890i \(-0.156789\pi\)
\(642\) 0 0
\(643\) 29.1918 1.15121 0.575607 0.817727i \(-0.304766\pi\)
0.575607 + 0.817727i \(0.304766\pi\)
\(644\) 0 0
\(645\) 3.59592 2.54270i 0.141589 0.100119i
\(646\) 0 0
\(647\) 0.174235 + 0.301783i 0.00684987 + 0.0118643i 0.869430 0.494056i \(-0.164486\pi\)
−0.862580 + 0.505920i \(0.831153\pi\)
\(648\) 0 0
\(649\) −0.797959 0.460702i −0.0313226 0.0180841i
\(650\) 0 0
\(651\) 1.37628 2.98735i 0.0539405 0.117083i
\(652\) 0 0
\(653\) 9.62372 + 16.6688i 0.376605 + 0.652300i 0.990566 0.137037i \(-0.0437579\pi\)
−0.613960 + 0.789337i \(0.710425\pi\)
\(654\) 0 0
\(655\) 12.5851i 0.491739i
\(656\) 0 0
\(657\) −17.3990 + 3.23375i −0.678799 + 0.126161i
\(658\) 0 0
\(659\) 5.72474 + 3.30518i 0.223004 + 0.128752i 0.607341 0.794442i \(-0.292236\pi\)
−0.384336 + 0.923193i \(0.625570\pi\)
\(660\) 0 0
\(661\) 37.1195i 1.44378i 0.692007 + 0.721891i \(0.256727\pi\)
−0.692007 + 0.721891i \(0.743273\pi\)
\(662\) 0 0
\(663\) −53.5454 + 4.93369i −2.07953 + 0.191609i
\(664\) 0 0
\(665\) 6.57826i 0.255094i
\(666\) 0 0
\(667\) 5.09082 0.197117
\(668\) 0 0
\(669\) −20.6969 29.2699i −0.800190 1.13164i
\(670\) 0 0
\(671\) 15.7313i 0.607301i
\(672\) 0 0
\(673\) 4.38551i 0.169049i 0.996421 + 0.0845244i \(0.0269371\pi\)
−0.996421 + 0.0845244i \(0.973063\pi\)
\(674\) 0 0
\(675\) −5.00000 + 1.41421i −0.192450 + 0.0544331i
\(676\) 0 0
\(677\) 17.2753 29.9216i 0.663942 1.14998i −0.315629 0.948883i \(-0.602216\pi\)
0.979571 0.201099i \(-0.0644512\pi\)
\(678\) 0 0
\(679\) −15.0000 −0.575647
\(680\) 0 0
\(681\) 4.29796 + 46.6458i 0.164698 + 1.78747i
\(682\) 0 0
\(683\) 16.1742 + 28.0146i 0.618890 + 1.07195i 0.989689 + 0.143235i \(0.0457506\pi\)
−0.370799 + 0.928713i \(0.620916\pi\)
\(684\) 0 0
\(685\) −18.2020 −0.695464
\(686\) 0 0
\(687\) 16.0732 + 7.40496i 0.613231 + 0.282517i
\(688\) 0 0
\(689\) −27.0000 + 15.5885i −1.02862 + 0.593873i
\(690\) 0 0
\(691\) −13.6464 + 23.6363i −0.519135 + 0.899167i 0.480618 + 0.876930i \(0.340412\pi\)
−0.999753 + 0.0222375i \(0.992921\pi\)
\(692\) 0 0
\(693\) 0.871173 + 4.68729i 0.0330931 + 0.178055i
\(694\) 0 0
\(695\) 14.4279i 0.547280i
\(696\) 0 0
\(697\) 20.6096i 0.780646i
\(698\) 0 0
\(699\) 31.9949 2.94802i 1.21016 0.111504i
\(700\) 0 0
\(701\) −6.82577 + 11.8226i −0.257806 + 0.446532i −0.965654 0.259832i \(-0.916333\pi\)
0.707848 + 0.706365i \(0.249666\pi\)
\(702\) 0 0
\(703\) −14.8485 + 25.7183i −0.560021 + 0.969984i
\(704\) 0 0
\(705\) −0.101021 1.09638i −0.00380465 0.0412919i
\(706\) 0 0
\(707\) 12.7702 7.37285i 0.480271 0.277285i
\(708\) 0 0
\(709\) 12.7474 + 22.0792i 0.478740 + 0.829203i 0.999703 0.0243768i \(-0.00776015\pi\)
−0.520962 + 0.853580i \(0.674427\pi\)
\(710\) 0 0
\(711\) −1.05051 + 0.898133i −0.0393972 + 0.0336826i
\(712\) 0 0
\(713\) −1.65153 −0.0618503
\(714\) 0 0
\(715\) 15.0635i 0.563344i
\(716\) 0 0
\(717\) −19.3990 + 1.78743i −0.724468 + 0.0667528i
\(718\) 0 0
\(719\) −14.6691 + 8.46923i −0.547066 + 0.315849i −0.747938 0.663769i \(-0.768956\pi\)
0.200872 + 0.979618i \(0.435623\pi\)
\(720\) 0 0
\(721\) −3.70204 + 2.13737i −0.137871 + 0.0796000i
\(722\) 0 0
\(723\) −4.00000 5.65685i −0.148762 0.210381i
\(724\) 0 0
\(725\) −4.62372 2.66951i −0.171721 0.0991431i
\(726\) 0 0
\(727\) 45.8939 26.4968i 1.70211 0.982713i 0.758488 0.651687i \(-0.225938\pi\)
0.943621 0.331027i \(-0.107395\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 3.79796 + 6.57826i 0.140473 + 0.243306i
\(732\) 0 0
\(733\) 14.0505 + 8.11207i 0.518967 + 0.299626i 0.736512 0.676424i \(-0.236471\pi\)
−0.217545 + 0.976050i \(0.569805\pi\)
\(734\) 0 0
\(735\) −8.40408 + 18.2419i −0.309989 + 0.672863i
\(736\) 0 0
\(737\) −7.97219 + 8.78706i −0.293660 + 0.323676i
\(738\) 0 0
\(739\) −36.6464 + 21.1578i −1.34806 + 0.778303i −0.987975 0.154616i \(-0.950586\pi\)
−0.360086 + 0.932919i \(0.617253\pi\)
\(740\) 0 0
\(741\) 2.47730 + 26.8861i 0.0910057 + 0.987686i
\(742\) 0 0
\(743\) −6.27526 + 3.62302i −0.230217 + 0.132916i −0.610672 0.791884i \(-0.709101\pi\)
0.380455 + 0.924799i \(0.375767\pi\)
\(744\) 0 0
\(745\) 29.5556i 1.08283i
\(746\) 0 0
\(747\) 6.37628 + 34.3071i 0.233296 + 1.25523i
\(748\) 0 0
\(749\) 5.00000 8.66025i 0.182696 0.316439i
\(750\) 0 0
\(751\) 15.3939 0.561731 0.280865 0.959747i \(-0.409379\pi\)
0.280865 + 0.959747i \(0.409379\pi\)
\(752\) 0 0
\(753\) −14.6010 + 31.6930i −0.532091 + 1.15496i
\(754\) 0 0
\(755\) −8.79796 15.2385i −0.320191 0.554586i
\(756\) 0 0
\(757\) −17.0505 9.84412i −0.619711 0.357791i 0.157045 0.987591i \(-0.449803\pi\)
−0.776757 + 0.629801i \(0.783136\pi\)
\(758\) 0 0
\(759\) 1.95459 1.38211i 0.0709472 0.0501673i
\(760\) 0 0
\(761\) 17.9562i 0.650911i 0.945557 + 0.325456i \(0.105518\pi\)
−0.945557 + 0.325456i \(0.894482\pi\)
\(762\) 0 0
\(763\) 2.75255 4.76756i 0.0996490 0.172597i
\(764\) 0 0
\(765\) 6.55051 + 35.2446i 0.236834 + 1.27427i
\(766\) 0 0
\(767\) 1.65153 + 2.86054i 0.0596333 + 0.103288i
\(768\) 0 0
\(769\) −10.7474 6.20504i −0.387563 0.223760i 0.293541 0.955947i \(-0.405166\pi\)
−0.681104 + 0.732187i \(0.738500\pi\)
\(770\) 0 0
\(771\) −2.39898 26.0361i −0.0863971 0.937669i
\(772\) 0 0
\(773\) 2.97219 + 1.71600i 0.106902 + 0.0617201i 0.552498 0.833514i \(-0.313675\pi\)
−0.445596 + 0.895234i \(0.647008\pi\)
\(774\) 0 0
\(775\) 1.50000 + 0.866025i 0.0538816 + 0.0311086i
\(776\) 0 0
\(777\) 15.3485 10.8530i 0.550623 0.389349i
\(778\) 0 0
\(779\) −10.3485 −0.370772
\(780\) 0 0
\(781\) 16.1969 9.35131i 0.579572 0.334616i
\(782\) 0 0
\(783\) 26.8939 + 6.80861i 0.961109 + 0.243320i
\(784\) 0 0
\(785\) 13.8990 + 24.0737i 0.496076 + 0.859229i
\(786\) 0 0
\(787\) 18.6464 10.7655i 0.664673 0.383749i −0.129382 0.991595i \(-0.541299\pi\)
0.794055 + 0.607845i \(0.207966\pi\)
\(788\) 0 0
\(789\) −11.1010 + 7.84961i −0.395207 + 0.279453i
\(790\) 0 0
\(791\) 6.21964 3.59091i 0.221145 0.127678i
\(792\) 0 0
\(793\) 28.1969 48.8385i 1.00130 1.73431i
\(794\) 0 0
\(795\) 12.0000 + 16.9706i 0.425596 + 0.601884i
\(796\) 0 0
\(797\) 11.1742 + 6.45145i 0.395812 + 0.228522i 0.684675 0.728848i \(-0.259944\pi\)
−0.288864 + 0.957370i \(0.593277\pi\)
\(798\) 0 0
\(799\) 1.89898 0.0671811
\(800\) 0 0
\(801\) −13.3939 4.73545i −0.473249 0.167319i
\(802\) 0 0
\(803\) −8.55051 −0.301741
\(804\) 0 0
\(805\) −2.09082 −0.0736916
\(806\) 0 0
\(807\) −8.49490 + 6.00680i −0.299035 + 0.211449i
\(808\) 0 0
\(809\) 20.2020 0.710266 0.355133 0.934816i \(-0.384436\pi\)
0.355133 + 0.934816i \(0.384436\pi\)
\(810\) 0 0
\(811\) −20.8485 12.0369i −0.732089 0.422672i 0.0870971 0.996200i \(-0.472241\pi\)
−0.819186 + 0.573528i \(0.805574\pi\)
\(812\) 0 0
\(813\) −13.7980 + 9.75663i −0.483916 + 0.342180i
\(814\) 0 0
\(815\) −22.5959 + 39.1373i −0.791500 + 1.37092i
\(816\) 0 0
\(817\) 3.30306 1.90702i 0.115559 0.0667183i
\(818\) 0 0
\(819\) 5.69694 16.1134i 0.199067 0.563047i
\(820\) 0 0
\(821\) 14.7247 8.50134i 0.513897 0.296699i −0.220537 0.975379i \(-0.570781\pi\)
0.734434 + 0.678680i \(0.237448\pi\)
\(822\) 0 0
\(823\) 20.3990 + 35.3321i 0.711064 + 1.23160i 0.964458 + 0.264235i \(0.0851196\pi\)
−0.253394 + 0.967363i \(0.581547\pi\)
\(824\) 0 0
\(825\) −2.50000 + 0.230351i −0.0870388 + 0.00801979i
\(826\) 0 0
\(827\) 37.3207 21.5471i 1.29777 0.749266i 0.317749 0.948175i \(-0.397073\pi\)
0.980018 + 0.198909i \(0.0637398\pi\)
\(828\) 0 0
\(829\) 19.3939 0.673577 0.336789 0.941580i \(-0.390659\pi\)
0.336789 + 0.941580i \(0.390659\pi\)
\(830\) 0 0
\(831\) −25.7980 36.4838i −0.894921 1.26561i
\(832\) 0 0
\(833\) −30.0000 17.3205i −1.03944 0.600120i
\(834\) 0 0
\(835\) 44.1464 + 25.4880i 1.52775 + 0.882047i
\(836\) 0 0
\(837\) −8.72474 2.20881i −0.301571 0.0763475i
\(838\) 0 0
\(839\) −9.52270 5.49794i −0.328760 0.189810i 0.326530 0.945187i \(-0.394120\pi\)
−0.655291 + 0.755377i \(0.727454\pi\)
\(840\) 0 0
\(841\) −0.247449 0.428594i −0.00853271 0.0147791i
\(842\) 0 0
\(843\) −5.39898 + 11.7190i −0.185951 + 0.403625i
\(844\) 0 0
\(845\) 14.0000 24.2487i 0.481615 0.834181i
\(846\) 0 0
\(847\) 9.75663i 0.335242i
\(848\) 0 0
\(849\) 7.79796 + 11.0280i 0.267625 + 0.378479i
\(850\) 0 0
\(851\) −8.17423 4.71940i −0.280209 0.161779i
\(852\) 0 0
\(853\) 21.6464 + 37.4927i 0.741160 + 1.28373i 0.951968 + 0.306199i \(0.0990572\pi\)
−0.210808 + 0.977527i \(0.567610\pi\)
\(854\) 0 0
\(855\) 17.6969 3.28913i 0.605223 0.112486i
\(856\) 0 0
\(857\) −13.5959 −0.464428 −0.232214 0.972665i \(-0.574597\pi\)
−0.232214 + 0.972665i \(0.574597\pi\)
\(858\) 0 0
\(859\) −1.84847 + 3.20164i −0.0630690 + 0.109239i −0.895836 0.444385i \(-0.853422\pi\)
0.832767 + 0.553624i \(0.186755\pi\)
\(860\) 0 0
\(861\) 5.94949 + 2.74094i 0.202758 + 0.0934110i
\(862\) 0 0
\(863\) 2.47848i 0.0843685i 0.999110 + 0.0421843i \(0.0134316\pi\)
−0.999110 + 0.0421843i \(0.986568\pi\)
\(864\) 0 0
\(865\) −21.2474 + 12.2672i −0.722435 + 0.417098i
\(866\) 0 0
\(867\) −32.2474 + 2.97129i −1.09518 + 0.100910i
\(868\) 0 0
\(869\) −0.578317 + 0.333891i −0.0196180 + 0.0113265i
\(870\) 0 0
\(871\) 40.5000 12.9904i 1.37229 0.440162i
\(872\) 0 0
\(873\) 7.50000 + 40.3532i 0.253837 + 1.36575i
\(874\) 0 0
\(875\) 11.3939 + 6.57826i 0.385183 + 0.222386i
\(876\) 0 0
\(877\) −24.6464 42.6889i −0.832251 1.44150i −0.896249 0.443551i \(-0.853719\pi\)
0.0639986 0.997950i \(-0.479615\pi\)
\(878\) 0 0
\(879\) 41.3939 29.2699i 1.39618 0.987249i
\(880\) 0 0
\(881\) −28.3207 + 16.3509i −0.954147 + 0.550877i −0.894367 0.447334i \(-0.852373\pi\)
−0.0597805 + 0.998212i \(0.519040\pi\)
\(882\) 0 0
\(883\) −5.54541 3.20164i −0.186618 0.107744i 0.403780 0.914856i \(-0.367696\pi\)
−0.590398 + 0.807112i \(0.701029\pi\)
\(884\) 0 0
\(885\) 1.79796 1.27135i 0.0604377 0.0427359i
\(886\) 0 0
\(887\) 18.7702 10.8370i 0.630240 0.363869i −0.150605 0.988594i \(-0.548122\pi\)
0.780845 + 0.624725i \(0.214789\pi\)
\(888\) 0 0
\(889\) −8.35357 + 4.82294i −0.280170 + 0.161756i
\(890\) 0 0
\(891\) 12.1742 4.68729i 0.407852 0.157030i
\(892\) 0 0
\(893\) 0.953512i 0.0319081i
\(894\) 0 0
\(895\) 27.1918 0.908923
\(896\) 0 0
\(897\) −8.54541 + 0.787377i −0.285323 + 0.0262898i
\(898\) 0 0
\(899\) −4.62372 8.00853i −0.154210 0.267099i
\(900\) 0 0
\(901\) −31.0454 + 17.9241i −1.03427 + 0.597137i
\(902\) 0 0
\(903\) −2.40408 + 0.221513i −0.0800028 + 0.00737149i
\(904\) 0 0
\(905\) 2.10102 3.63907i 0.0698403 0.120967i
\(906\) 0 0
\(907\) 17.6010 30.4859i 0.584432 1.01227i −0.410514 0.911854i \(-0.634651\pi\)
0.994946 0.100412i \(-0.0320161\pi\)
\(908\) 0 0
\(909\) −26.2196 30.6681i −0.869651 1.01720i
\(910\) 0 0
\(911\) 37.1195i 1.22982i 0.788596 + 0.614912i \(0.210808\pi\)
−0.788596 + 0.614912i \(0.789192\pi\)
\(912\) 0 0
\(913\) 16.8598i 0.557978i
\(914\) 0 0
\(915\) −34.1464 15.7313i −1.12885 0.520061i
\(916\) 0 0
\(917\) −3.44949 + 5.97469i −0.113912 + 0.197302i
\(918\) 0 0
\(919\) 2.84847 1.64456i 0.0939623 0.0542492i −0.452283 0.891875i \(-0.649390\pi\)
0.546245 + 0.837626i \(0.316057\pi\)
\(920\) 0 0
\(921\) −6.22985 + 13.5225i −0.205280 + 0.445582i
\(922\) 0 0
\(923\) −67.0454 −2.20683
\(924\) 0 0
\(925\) 4.94949 + 8.57277i 0.162738 + 0.281871i
\(926\) 0 0
\(927\) 7.60102 + 8.89060i 0.249650 + 0.292006i
\(928\) 0 0
\(929\) −48.4949 −1.59107 −0.795533 0.605910i \(-0.792809\pi\)
−0.795533 + 0.605910i \(0.792809\pi\)
\(930\) 0 0
\(931\) −8.69694 + 15.0635i −0.285031 + 0.493688i
\(932\) 0 0
\(933\) −14.2020 20.0847i −0.464954 0.657544i
\(934\) 0 0
\(935\) 17.3205i 0.566441i
\(936\) 0 0
\(937\) 38.3908i 1.25417i 0.778949 + 0.627087i \(0.215753\pi\)
−0.778949 + 0.627087i \(0.784247\pi\)
\(938\) 0 0
\(939\) −16.4949 + 11.6637i −0.538291 + 0.380629i
\(940\) 0 0
\(941\) 40.2929 1.31351 0.656755 0.754104i \(-0.271929\pi\)
0.656755 + 0.754104i \(0.271929\pi\)
\(942\) 0 0
\(943\) 3.28913i 0.107109i
\(944\) 0 0
\(945\) −11.0454 2.79632i −0.359307 0.0909643i
\(946\) 0 0
\(947\) 36.8338i 1.19694i 0.801146 + 0.598468i \(0.204224\pi\)
−0.801146 + 0.598468i \(0.795776\pi\)
\(948\) 0 0
\(949\) 26.5454 + 15.3260i 0.861700 + 0.497503i
\(950\) 0 0
\(951\) 29.6464 + 13.6582i 0.961351 + 0.442896i
\(952\) 0 0
\(953\) 12.2351i 0.396334i −0.980168 0.198167i \(-0.936501\pi\)
0.980168 0.198167i \(-0.0634989\pi\)
\(954\) 0 0
\(955\) 17.4495 + 30.2234i 0.564652 + 0.978006i
\(956\) 0 0
\(957\) 12.1742 + 5.60869i 0.393537 + 0.181303i
\(958\) 0 0
\(959\) 8.64133 + 4.98907i 0.279043 + 0.161106i
\(960\) 0 0
\(961\) −14.0000 24.2487i −0.451613 0.782216i
\(962\) 0 0
\(963\) −25.7980 9.12096i −0.831328 0.293919i
\(964\) 0 0
\(965\) −25.7980 −0.830466
\(966\) 0 0
\(967\) 6.84847 11.8619i 0.220232 0.381453i −0.734646 0.678450i \(-0.762652\pi\)
0.954878 + 0.296997i \(0.0959852\pi\)
\(968\) 0 0
\(969\) 2.84847 + 30.9145i 0.0915060 + 0.993115i
\(970\) 0 0
\(971\) −30.2196 17.4473i −0.969794 0.559911i −0.0706209 0.997503i \(-0.522498\pi\)
−0.899174 + 0.437592i \(0.855831\pi\)
\(972\) 0 0
\(973\) 3.95459 6.84955i 0.126778 0.219587i
\(974\) 0 0
\(975\) 8.17423 + 3.76588i 0.261785 + 0.120605i
\(976\) 0 0
\(977\) 22.3207 12.8868i 0.714101 0.412287i −0.0984765 0.995139i \(-0.531397\pi\)
0.812578 + 0.582853i \(0.198064\pi\)
\(978\) 0 0
\(979\) −5.94439 3.43199i −0.189983 0.109687i
\(980\) 0 0
\(981\) −14.2020 5.02118i −0.453436 0.160314i
\(982\) 0 0
\(983\) 31.5959 1.00775 0.503877 0.863776i \(-0.331907\pi\)
0.503877 + 0.863776i \(0.331907\pi\)
\(984\) 0 0
\(985\) −3.24745 + 5.62475i −0.103472 + 0.179219i
\(986\) 0 0
\(987\) −0.252551 + 0.548188i −0.00803880 + 0.0174490i
\(988\) 0 0
\(989\) 0.606123 + 1.04984i 0.0192736 + 0.0333828i
\(990\) 0 0
\(991\) 18.5276i 0.588550i −0.955721 0.294275i \(-0.904922\pi\)
0.955721 0.294275i \(-0.0950781\pi\)
\(992\) 0 0
\(993\) 2.21964 + 24.0898i 0.0704382 + 0.764466i
\(994\) 0 0
\(995\) 11.0000 + 19.0526i 0.348723 + 0.604007i
\(996\) 0 0
\(997\) −34.2929 −1.08607 −0.543033 0.839711i \(-0.682724\pi\)
−0.543033 + 0.839711i \(0.682724\pi\)
\(998\) 0 0
\(999\) −36.8712 35.8642i −1.16655 1.13469i
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 804.2.o.b.365.1 4
3.2 odd 2 804.2.o.c.365.1 yes 4
67.38 odd 6 804.2.o.c.641.2 yes 4
201.38 even 6 inner 804.2.o.b.641.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
804.2.o.b.365.1 4 1.1 even 1 trivial
804.2.o.b.641.2 yes 4 201.38 even 6 inner
804.2.o.c.365.1 yes 4 3.2 odd 2
804.2.o.c.641.2 yes 4 67.38 odd 6