Properties

Label 416.2.w.c.257.2
Level $416$
Weight $2$
Character 416.257
Analytic conductor $3.322$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(225,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.w (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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.2
Root \(0.500000 + 1.19293i\) of defining polynomial
Character \(\chi\) \(=\) 416.257
Dual form 416.2.w.c.225.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.619657 - 1.07328i) q^{3} -2.00000i q^{5} +(4.00552 + 2.31259i) q^{7} +(0.732051 - 1.26795i) q^{9} +(-1.85897 + 1.07328i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(-2.14655 + 1.23931i) q^{15} +(0.232051 - 0.401924i) q^{17} +(4.00552 + 2.31259i) q^{19} -5.73205i q^{21} +(-2.76621 - 4.79122i) q^{23} +1.00000 q^{25} -5.53242 q^{27} +(-1.50000 - 2.59808i) q^{29} -9.25036i q^{31} +(2.30385 + 1.33013i) q^{33} +(4.62518 - 8.01105i) q^{35} +(6.69615 - 3.86603i) q^{37} +(-3.09828 + 3.21983i) q^{39} +(-6.23205 + 3.59808i) q^{41} +(-1.85897 + 3.21983i) q^{43} +(-2.53590 - 1.46410i) q^{45} +11.7290i q^{47} +(7.19615 + 12.4641i) q^{49} -0.575167 q^{51} +2.53590 q^{53} +(2.14655 + 3.71794i) q^{55} -5.73205i q^{57} +(8.29863 + 4.79122i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(5.86450 - 3.38587i) q^{63} +(-6.92820 + 2.00000i) q^{65} +(-4.00552 + 2.31259i) q^{67} +(-3.42820 + 5.93782i) q^{69} +(-5.57691 - 3.21983i) q^{71} +6.00000i q^{73} +(-0.619657 - 1.07328i) q^{75} -9.92820 q^{77} +4.29311 q^{79} +(1.23205 + 2.13397i) q^{81} +4.29311i q^{83} +(-0.803848 - 0.464102i) q^{85} +(-1.85897 + 3.21983i) q^{87} +(-1.03590 + 0.598076i) q^{89} +(4.00552 - 16.1881i) q^{91} +(-9.92820 + 5.73205i) q^{93} +(4.62518 - 8.01105i) q^{95} +(11.8923 + 6.86603i) q^{97} +3.14277i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.619657 1.07328i −0.357759 0.619657i 0.629827 0.776735i \(-0.283126\pi\)
−0.987586 + 0.157079i \(0.949792\pi\)
\(4\) 0 0
\(5\) 2.00000i 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 4.00552 + 2.31259i 1.51395 + 0.874077i 0.999867 + 0.0163346i \(0.00519970\pi\)
0.514079 + 0.857743i \(0.328134\pi\)
\(8\) 0 0
\(9\) 0.732051 1.26795i 0.244017 0.422650i
\(10\) 0 0
\(11\) −1.85897 + 1.07328i −0.560501 + 0.323605i −0.753346 0.657624i \(-0.771562\pi\)
0.192846 + 0.981229i \(0.438228\pi\)
\(12\) 0 0
\(13\) −1.00000 3.46410i −0.277350 0.960769i
\(14\) 0 0
\(15\) −2.14655 + 1.23931i −0.554238 + 0.319989i
\(16\) 0 0
\(17\) 0.232051 0.401924i 0.0562806 0.0974808i −0.836512 0.547948i \(-0.815409\pi\)
0.892793 + 0.450467i \(0.148743\pi\)
\(18\) 0 0
\(19\) 4.00552 + 2.31259i 0.918930 + 0.530545i 0.883294 0.468820i \(-0.155321\pi\)
0.0356367 + 0.999365i \(0.488654\pi\)
\(20\) 0 0
\(21\) 5.73205i 1.25084i
\(22\) 0 0
\(23\) −2.76621 4.79122i −0.576795 0.999038i −0.995844 0.0910745i \(-0.970970\pi\)
0.419049 0.907964i \(-0.362363\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.53242 −1.06472
\(28\) 0 0
\(29\) −1.50000 2.59808i −0.278543 0.482451i 0.692480 0.721437i \(-0.256518\pi\)
−0.971023 + 0.238987i \(0.923185\pi\)
\(30\) 0 0
\(31\) 9.25036i 1.66141i −0.556710 0.830707i \(-0.687936\pi\)
0.556710 0.830707i \(-0.312064\pi\)
\(32\) 0 0
\(33\) 2.30385 + 1.33013i 0.401048 + 0.231545i
\(34\) 0 0
\(35\) 4.62518 8.01105i 0.781798 1.35411i
\(36\) 0 0
\(37\) 6.69615 3.86603i 1.10084 0.635571i 0.164399 0.986394i \(-0.447432\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(38\) 0 0
\(39\) −3.09828 + 3.21983i −0.496123 + 0.515586i
\(40\) 0 0
\(41\) −6.23205 + 3.59808i −0.973283 + 0.561925i −0.900235 0.435403i \(-0.856606\pi\)
−0.0730473 + 0.997328i \(0.523272\pi\)
\(42\) 0 0
\(43\) −1.85897 + 3.21983i −0.283490 + 0.491020i −0.972242 0.233978i \(-0.924826\pi\)
0.688752 + 0.724997i \(0.258159\pi\)
\(44\) 0 0
\(45\) −2.53590 1.46410i −0.378029 0.218255i
\(46\) 0 0
\(47\) 11.7290i 1.71085i 0.517927 + 0.855425i \(0.326704\pi\)
−0.517927 + 0.855425i \(0.673296\pi\)
\(48\) 0 0
\(49\) 7.19615 + 12.4641i 1.02802 + 1.78059i
\(50\) 0 0
\(51\) −0.575167 −0.0805396
\(52\) 0 0
\(53\) 2.53590 0.348332 0.174166 0.984716i \(-0.444277\pi\)
0.174166 + 0.984716i \(0.444277\pi\)
\(54\) 0 0
\(55\) 2.14655 + 3.71794i 0.289441 + 0.501327i
\(56\) 0 0
\(57\) 5.73205i 0.759229i
\(58\) 0 0
\(59\) 8.29863 + 4.79122i 1.08039 + 0.623763i 0.931002 0.365015i \(-0.118936\pi\)
0.149388 + 0.988779i \(0.452270\pi\)
\(60\) 0 0
\(61\) −1.50000 + 2.59808i −0.192055 + 0.332650i −0.945931 0.324367i \(-0.894849\pi\)
0.753876 + 0.657017i \(0.228182\pi\)
\(62\) 0 0
\(63\) 5.86450 3.38587i 0.738857 0.426579i
\(64\) 0 0
\(65\) −6.92820 + 2.00000i −0.859338 + 0.248069i
\(66\) 0 0
\(67\) −4.00552 + 2.31259i −0.489353 + 0.282528i −0.724306 0.689479i \(-0.757840\pi\)
0.234953 + 0.972007i \(0.424506\pi\)
\(68\) 0 0
\(69\) −3.42820 + 5.93782i −0.412707 + 0.714830i
\(70\) 0 0
\(71\) −5.57691 3.21983i −0.661858 0.382124i 0.131127 0.991366i \(-0.458141\pi\)
−0.792984 + 0.609242i \(0.791474\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) −0.619657 1.07328i −0.0715518 0.123931i
\(76\) 0 0
\(77\) −9.92820 −1.13142
\(78\) 0 0
\(79\) 4.29311 0.483012 0.241506 0.970399i \(-0.422359\pi\)
0.241506 + 0.970399i \(0.422359\pi\)
\(80\) 0 0
\(81\) 1.23205 + 2.13397i 0.136895 + 0.237108i
\(82\) 0 0
\(83\) 4.29311i 0.471230i 0.971846 + 0.235615i \(0.0757104\pi\)
−0.971846 + 0.235615i \(0.924290\pi\)
\(84\) 0 0
\(85\) −0.803848 0.464102i −0.0871895 0.0503389i
\(86\) 0 0
\(87\) −1.85897 + 3.21983i −0.199303 + 0.345202i
\(88\) 0 0
\(89\) −1.03590 + 0.598076i −0.109805 + 0.0633960i −0.553897 0.832585i \(-0.686860\pi\)
0.444092 + 0.895981i \(0.353526\pi\)
\(90\) 0 0
\(91\) 4.00552 16.1881i 0.419893 1.69698i
\(92\) 0 0
\(93\) −9.92820 + 5.73205i −1.02951 + 0.594386i
\(94\) 0 0
\(95\) 4.62518 8.01105i 0.474534 0.821916i
\(96\) 0 0
\(97\) 11.8923 + 6.86603i 1.20748 + 0.697139i 0.962208 0.272315i \(-0.0877893\pi\)
0.245272 + 0.969454i \(0.421123\pi\)
\(98\) 0 0
\(99\) 3.14277i 0.315861i
\(100\) 0 0
\(101\) 9.69615 + 16.7942i 0.964803 + 1.67109i 0.710143 + 0.704058i \(0.248630\pi\)
0.254660 + 0.967031i \(0.418036\pi\)
\(102\) 0 0
\(103\) −8.58622 −0.846025 −0.423013 0.906124i \(-0.639027\pi\)
−0.423013 + 0.906124i \(0.639027\pi\)
\(104\) 0 0
\(105\) −11.4641 −1.11878
\(106\) 0 0
\(107\) 1.85897 + 3.21983i 0.179713 + 0.311273i 0.941782 0.336223i \(-0.109150\pi\)
−0.762069 + 0.647496i \(0.775816\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) −8.29863 4.79122i −0.787671 0.454762i
\(112\) 0 0
\(113\) 6.23205 10.7942i 0.586262 1.01544i −0.408455 0.912779i \(-0.633932\pi\)
0.994717 0.102657i \(-0.0327344\pi\)
\(114\) 0 0
\(115\) −9.58244 + 5.53242i −0.893567 + 0.515901i
\(116\) 0 0
\(117\) −5.12436 1.26795i −0.473747 0.117222i
\(118\) 0 0
\(119\) 1.85897 1.07328i 0.170412 0.0983872i
\(120\) 0 0
\(121\) −3.19615 + 5.53590i −0.290559 + 0.503263i
\(122\) 0 0
\(123\) 7.72347 + 4.45915i 0.696401 + 0.402068i
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 1.85897 + 3.21983i 0.164957 + 0.285714i 0.936640 0.350293i \(-0.113918\pi\)
−0.771683 + 0.636007i \(0.780585\pi\)
\(128\) 0 0
\(129\) 4.60770 0.405685
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 10.6962 + 18.5263i 0.927474 + 1.60643i
\(134\) 0 0
\(135\) 11.0648i 0.952310i
\(136\) 0 0
\(137\) −11.4282 6.59808i −0.976377 0.563712i −0.0752028 0.997168i \(-0.523960\pi\)
−0.901174 + 0.433457i \(0.857294\pi\)
\(138\) 0 0
\(139\) −4.00552 + 6.93777i −0.339744 + 0.588454i −0.984384 0.176032i \(-0.943674\pi\)
0.644640 + 0.764486i \(0.277007\pi\)
\(140\) 0 0
\(141\) 12.5885 7.26795i 1.06014 0.612072i
\(142\) 0 0
\(143\) 5.57691 + 5.36639i 0.466365 + 0.448760i
\(144\) 0 0
\(145\) −5.19615 + 3.00000i −0.431517 + 0.249136i
\(146\) 0 0
\(147\) 8.91829 15.4469i 0.735568 1.27404i
\(148\) 0 0
\(149\) 10.0359 + 5.79423i 0.822173 + 0.474682i 0.851165 0.524898i \(-0.175897\pi\)
−0.0289923 + 0.999580i \(0.509230\pi\)
\(150\) 0 0
\(151\) 9.25036i 0.752784i −0.926461 0.376392i \(-0.877165\pi\)
0.926461 0.376392i \(-0.122835\pi\)
\(152\) 0 0
\(153\) −0.339746 0.588457i −0.0274668 0.0475740i
\(154\) 0 0
\(155\) −18.5007 −1.48601
\(156\) 0 0
\(157\) 4.39230 0.350544 0.175272 0.984520i \(-0.443920\pi\)
0.175272 + 0.984520i \(0.443920\pi\)
\(158\) 0 0
\(159\) −1.57139 2.72172i −0.124619 0.215847i
\(160\) 0 0
\(161\) 25.5885i 2.01665i
\(162\) 0 0
\(163\) −21.5990 12.4702i −1.69177 0.976741i −0.953094 0.302675i \(-0.902120\pi\)
−0.738671 0.674066i \(-0.764546\pi\)
\(164\) 0 0
\(165\) 2.66025 4.60770i 0.207100 0.358709i
\(166\) 0 0
\(167\) 19.4525 11.2309i 1.50528 0.869072i 0.505295 0.862947i \(-0.331384\pi\)
0.999981 0.00612500i \(-0.00194966\pi\)
\(168\) 0 0
\(169\) −11.0000 + 6.92820i −0.846154 + 0.532939i
\(170\) 0 0
\(171\) 5.86450 3.38587i 0.448469 0.258924i
\(172\) 0 0
\(173\) −4.96410 + 8.59808i −0.377414 + 0.653700i −0.990685 0.136173i \(-0.956520\pi\)
0.613271 + 0.789872i \(0.289853\pi\)
\(174\) 0 0
\(175\) 4.00552 + 2.31259i 0.302789 + 0.174815i
\(176\) 0 0
\(177\) 11.8756i 0.892628i
\(178\) 0 0
\(179\) 1.85897 + 3.21983i 0.138946 + 0.240661i 0.927098 0.374819i \(-0.122295\pi\)
−0.788152 + 0.615481i \(0.788962\pi\)
\(180\) 0 0
\(181\) −16.3923 −1.21843 −0.609215 0.793005i \(-0.708515\pi\)
−0.609215 + 0.793005i \(0.708515\pi\)
\(182\) 0 0
\(183\) 3.71794 0.274838
\(184\) 0 0
\(185\) −7.73205 13.3923i −0.568472 0.984622i
\(186\) 0 0
\(187\) 0.996219i 0.0728508i
\(188\) 0 0
\(189\) −22.1603 12.7942i −1.61192 0.930643i
\(190\) 0 0
\(191\) −6.48415 + 11.2309i −0.469177 + 0.812638i −0.999379 0.0352331i \(-0.988783\pi\)
0.530202 + 0.847871i \(0.322116\pi\)
\(192\) 0 0
\(193\) 17.0885 9.86603i 1.23005 0.710172i 0.263013 0.964792i \(-0.415284\pi\)
0.967041 + 0.254620i \(0.0819505\pi\)
\(194\) 0 0
\(195\) 6.43966 + 6.19657i 0.461154 + 0.443745i
\(196\) 0 0
\(197\) 20.4282 11.7942i 1.45545 0.840304i 0.456667 0.889638i \(-0.349043\pi\)
0.998782 + 0.0493338i \(0.0157098\pi\)
\(198\) 0 0
\(199\) −6.15208 + 10.6557i −0.436109 + 0.755363i −0.997385 0.0722651i \(-0.976977\pi\)
0.561276 + 0.827629i \(0.310311\pi\)
\(200\) 0 0
\(201\) 4.96410 + 2.86603i 0.350141 + 0.202154i
\(202\) 0 0
\(203\) 13.8755i 0.973872i
\(204\) 0 0
\(205\) 7.19615 + 12.4641i 0.502601 + 0.870531i
\(206\) 0 0
\(207\) −8.10003 −0.562991
\(208\) 0 0
\(209\) −9.92820 −0.686748
\(210\) 0 0
\(211\) 13.5880 + 23.5350i 0.935434 + 1.62022i 0.773859 + 0.633359i \(0.218324\pi\)
0.161575 + 0.986860i \(0.448343\pi\)
\(212\) 0 0
\(213\) 7.98076i 0.546833i
\(214\) 0 0
\(215\) 6.43966 + 3.71794i 0.439181 + 0.253561i
\(216\) 0 0
\(217\) 21.3923 37.0526i 1.45220 2.51529i
\(218\) 0 0
\(219\) 6.43966 3.71794i 0.435152 0.251235i
\(220\) 0 0
\(221\) −1.62436 0.401924i −0.109266 0.0270363i
\(222\) 0 0
\(223\) −4.00552 + 2.31259i −0.268230 + 0.154863i −0.628083 0.778146i \(-0.716160\pi\)
0.359853 + 0.933009i \(0.382827\pi\)
\(224\) 0 0
\(225\) 0.732051 1.26795i 0.0488034 0.0845299i
\(226\) 0 0
\(227\) −12.0166 6.93777i −0.797568 0.460476i 0.0450520 0.998985i \(-0.485655\pi\)
−0.842620 + 0.538509i \(0.818988\pi\)
\(228\) 0 0
\(229\) 0.928203i 0.0613374i −0.999530 0.0306687i \(-0.990236\pi\)
0.999530 0.0306687i \(-0.00976368\pi\)
\(230\) 0 0
\(231\) 6.15208 + 10.6557i 0.404777 + 0.701094i
\(232\) 0 0
\(233\) −14.5359 −0.952278 −0.476139 0.879370i \(-0.657964\pi\)
−0.476139 + 0.879370i \(0.657964\pi\)
\(234\) 0 0
\(235\) 23.4580 1.53023
\(236\) 0 0
\(237\) −2.66025 4.60770i −0.172802 0.299302i
\(238\) 0 0
\(239\) 4.29311i 0.277698i −0.990314 0.138849i \(-0.955660\pi\)
0.990314 0.138849i \(-0.0443403\pi\)
\(240\) 0 0
\(241\) 18.6962 + 10.7942i 1.20433 + 0.695317i 0.961514 0.274756i \(-0.0885971\pi\)
0.242811 + 0.970074i \(0.421930\pi\)
\(242\) 0 0
\(243\) −6.77174 + 11.7290i −0.434407 + 0.752415i
\(244\) 0 0
\(245\) 24.9282 14.3923i 1.59260 0.919491i
\(246\) 0 0
\(247\) 4.00552 16.1881i 0.254865 1.03003i
\(248\) 0 0
\(249\) 4.60770 2.66025i 0.292001 0.168587i
\(250\) 0 0
\(251\) −12.9238 + 22.3847i −0.815744 + 1.41291i 0.0930485 + 0.995662i \(0.470339\pi\)
−0.908792 + 0.417248i \(0.862995\pi\)
\(252\) 0 0
\(253\) 10.2846 + 5.93782i 0.646588 + 0.373308i
\(254\) 0 0
\(255\) 1.15033i 0.0720368i
\(256\) 0 0
\(257\) 0.232051 + 0.401924i 0.0144749 + 0.0250713i 0.873172 0.487412i \(-0.162059\pi\)
−0.858697 + 0.512483i \(0.828726\pi\)
\(258\) 0 0
\(259\) 35.7621 2.22215
\(260\) 0 0
\(261\) −4.39230 −0.271877
\(262\) 0 0
\(263\) −12.0166 20.8133i −0.740974 1.28340i −0.952052 0.305935i \(-0.901031\pi\)
0.211079 0.977469i \(-0.432302\pi\)
\(264\) 0 0
\(265\) 5.07180i 0.311558i
\(266\) 0 0
\(267\) 1.28380 + 0.741204i 0.0785675 + 0.0453609i
\(268\) 0 0
\(269\) −2.30385 + 3.99038i −0.140468 + 0.243298i −0.927673 0.373394i \(-0.878194\pi\)
0.787205 + 0.616692i \(0.211527\pi\)
\(270\) 0 0
\(271\) −20.0276 + 11.5630i −1.21659 + 0.702399i −0.964187 0.265222i \(-0.914555\pi\)
−0.252404 + 0.967622i \(0.581221\pi\)
\(272\) 0 0
\(273\) −19.8564 + 5.73205i −1.20176 + 0.346919i
\(274\) 0 0
\(275\) −1.85897 + 1.07328i −0.112100 + 0.0647210i
\(276\) 0 0
\(277\) 1.69615 2.93782i 0.101912 0.176517i −0.810560 0.585655i \(-0.800837\pi\)
0.912472 + 0.409138i \(0.134171\pi\)
\(278\) 0 0
\(279\) −11.7290 6.77174i −0.702196 0.405413i
\(280\) 0 0
\(281\) 10.0000i 0.596550i −0.954480 0.298275i \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 0 0
\(283\) −9.87002 17.0954i −0.586712 1.01621i −0.994660 0.103209i \(-0.967089\pi\)
0.407948 0.913005i \(-0.366245\pi\)
\(284\) 0 0
\(285\) −11.4641 −0.679075
\(286\) 0 0
\(287\) −33.2835 −1.96466
\(288\) 0 0
\(289\) 8.39230 + 14.5359i 0.493665 + 0.855053i
\(290\) 0 0
\(291\) 17.0183i 0.997631i
\(292\) 0 0
\(293\) −21.8205 12.5981i −1.27477 0.735987i −0.298886 0.954289i \(-0.596615\pi\)
−0.975881 + 0.218301i \(0.929948\pi\)
\(294\) 0 0
\(295\) 9.58244 16.5973i 0.557911 0.966330i
\(296\) 0 0
\(297\) 10.2846 5.93782i 0.596774 0.344547i
\(298\) 0 0
\(299\) −13.8311 + 14.3737i −0.799871 + 0.831250i
\(300\) 0 0
\(301\) −14.8923 + 8.59808i −0.858378 + 0.495585i
\(302\) 0 0
\(303\) 12.0166 20.8133i 0.690334 1.19569i
\(304\) 0 0
\(305\) 5.19615 + 3.00000i 0.297531 + 0.171780i
\(306\) 0 0
\(307\) 5.62140i 0.320830i −0.987050 0.160415i \(-0.948717\pi\)
0.987050 0.160415i \(-0.0512833\pi\)
\(308\) 0 0
\(309\) 5.32051 + 9.21539i 0.302673 + 0.524245i
\(310\) 0 0
\(311\) 18.5007 1.04908 0.524540 0.851386i \(-0.324237\pi\)
0.524540 + 0.851386i \(0.324237\pi\)
\(312\) 0 0
\(313\) −8.39230 −0.474361 −0.237181 0.971466i \(-0.576223\pi\)
−0.237181 + 0.971466i \(0.576223\pi\)
\(314\) 0 0
\(315\) −6.77174 11.7290i −0.381544 0.660854i
\(316\) 0 0
\(317\) 4.00000i 0.224662i −0.993671 0.112331i \(-0.964168\pi\)
0.993671 0.112331i \(-0.0358318\pi\)
\(318\) 0 0
\(319\) 5.57691 + 3.21983i 0.312247 + 0.180276i
\(320\) 0 0
\(321\) 2.30385 3.99038i 0.128588 0.222721i
\(322\) 0 0
\(323\) 1.85897 1.07328i 0.103436 0.0597187i
\(324\) 0 0
\(325\) −1.00000 3.46410i −0.0554700 0.192154i
\(326\) 0 0
\(327\) 6.43966 3.71794i 0.356114 0.205603i
\(328\) 0 0
\(329\) −27.1244 + 46.9808i −1.49541 + 2.59013i
\(330\) 0 0
\(331\) −2.43414 1.40535i −0.133792 0.0772450i 0.431610 0.902060i \(-0.357946\pi\)
−0.565402 + 0.824815i \(0.691279\pi\)
\(332\) 0 0
\(333\) 11.3205i 0.620360i
\(334\) 0 0
\(335\) 4.62518 + 8.01105i 0.252701 + 0.437690i
\(336\) 0 0
\(337\) −0.392305 −0.0213702 −0.0106851 0.999943i \(-0.503401\pi\)
−0.0106851 + 0.999943i \(0.503401\pi\)
\(338\) 0 0
\(339\) −15.4469 −0.838962
\(340\) 0 0
\(341\) 9.92820 + 17.1962i 0.537642 + 0.931224i
\(342\) 0 0
\(343\) 34.1908i 1.84613i
\(344\) 0 0
\(345\) 11.8756 + 6.85641i 0.639363 + 0.369137i
\(346\) 0 0
\(347\) 7.39139 12.8023i 0.396791 0.687262i −0.596537 0.802585i \(-0.703457\pi\)
0.993328 + 0.115324i \(0.0367905\pi\)
\(348\) 0 0
\(349\) 8.30385 4.79423i 0.444495 0.256629i −0.261008 0.965337i \(-0.584055\pi\)
0.705502 + 0.708708i \(0.250721\pi\)
\(350\) 0 0
\(351\) 5.53242 + 19.1649i 0.295299 + 1.02295i
\(352\) 0 0
\(353\) −7.96410 + 4.59808i −0.423886 + 0.244731i −0.696739 0.717325i \(-0.745366\pi\)
0.272852 + 0.962056i \(0.412033\pi\)
\(354\) 0 0
\(355\) −6.43966 + 11.1538i −0.341782 + 0.591983i
\(356\) 0 0
\(357\) −2.30385 1.33013i −0.121933 0.0703978i
\(358\) 0 0
\(359\) 4.29311i 0.226582i 0.993562 + 0.113291i \(0.0361392\pi\)
−0.993562 + 0.113291i \(0.963861\pi\)
\(360\) 0 0
\(361\) 1.19615 + 2.07180i 0.0629554 + 0.109042i
\(362\) 0 0
\(363\) 7.92207 0.415801
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −0.287584 0.498110i −0.0150117 0.0260011i 0.858422 0.512944i \(-0.171445\pi\)
−0.873434 + 0.486943i \(0.838112\pi\)
\(368\) 0 0
\(369\) 10.5359i 0.548477i
\(370\) 0 0
\(371\) 10.1576 + 5.86450i 0.527357 + 0.304469i
\(372\) 0 0
\(373\) −4.69615 + 8.13397i −0.243158 + 0.421161i −0.961612 0.274413i \(-0.911517\pi\)
0.718454 + 0.695574i \(0.244850\pi\)
\(374\) 0 0
\(375\) −12.8793 + 7.43588i −0.665086 + 0.383987i
\(376\) 0 0
\(377\) −7.50000 + 7.79423i −0.386270 + 0.401423i
\(378\) 0 0
\(379\) 2.43414 1.40535i 0.125033 0.0721880i −0.436179 0.899860i \(-0.643669\pi\)
0.561212 + 0.827672i \(0.310335\pi\)
\(380\) 0 0
\(381\) 2.30385 3.99038i 0.118030 0.204433i
\(382\) 0 0
\(383\) 23.1704 + 13.3774i 1.18395 + 0.683555i 0.956926 0.290334i \(-0.0937663\pi\)
0.227026 + 0.973889i \(0.427100\pi\)
\(384\) 0 0
\(385\) 19.8564i 1.01198i
\(386\) 0 0
\(387\) 2.72172 + 4.71416i 0.138353 + 0.239634i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −2.56761 −0.129849
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.58622i 0.432019i
\(396\) 0 0
\(397\) −21.6962 12.5263i −1.08890 0.628676i −0.155616 0.987818i \(-0.549736\pi\)
−0.933283 + 0.359142i \(0.883069\pi\)
\(398\) 0 0
\(399\) 13.2559 22.9599i 0.663624 1.14943i
\(400\) 0 0
\(401\) −31.1603 + 17.9904i −1.55607 + 0.898397i −0.558442 + 0.829544i \(0.688601\pi\)
−0.997627 + 0.0688532i \(0.978066\pi\)
\(402\) 0 0
\(403\) −32.0442 + 9.25036i −1.59624 + 0.460793i
\(404\) 0 0
\(405\) 4.26795 2.46410i 0.212076 0.122442i
\(406\) 0 0
\(407\) −8.29863 + 14.3737i −0.411348 + 0.712476i
\(408\) 0 0
\(409\) 24.6962 + 14.2583i 1.22115 + 0.705029i 0.965162 0.261652i \(-0.0842673\pi\)
0.255984 + 0.966681i \(0.417601\pi\)
\(410\) 0 0
\(411\) 16.3542i 0.806692i
\(412\) 0 0
\(413\) 22.1603 + 38.3827i 1.09043 + 1.88869i
\(414\) 0 0
\(415\) 8.58622 0.421481
\(416\) 0 0
\(417\) 9.92820 0.486186
\(418\) 0 0
\(419\) −12.0166 20.8133i −0.587048 1.01680i −0.994617 0.103623i \(-0.966957\pi\)
0.407569 0.913175i \(-0.366377\pi\)
\(420\) 0 0
\(421\) 0.928203i 0.0452379i 0.999744 + 0.0226189i \(0.00720044\pi\)
−0.999744 + 0.0226189i \(0.992800\pi\)
\(422\) 0 0
\(423\) 14.8718 + 8.58622i 0.723090 + 0.417476i
\(424\) 0 0
\(425\) 0.232051 0.401924i 0.0112561 0.0194962i
\(426\) 0 0
\(427\) −12.0166 + 6.93777i −0.581523 + 0.335742i
\(428\) 0 0
\(429\) 2.30385 9.31089i 0.111231 0.449534i
\(430\) 0 0
\(431\) 18.4562 10.6557i 0.889006 0.513268i 0.0153885 0.999882i \(-0.495101\pi\)
0.873617 + 0.486614i \(0.161768\pi\)
\(432\) 0 0
\(433\) −6.69615 + 11.5981i −0.321797 + 0.557368i −0.980859 0.194720i \(-0.937620\pi\)
0.659062 + 0.752088i \(0.270953\pi\)
\(434\) 0 0
\(435\) 6.43966 + 3.71794i 0.308758 + 0.178262i
\(436\) 0 0
\(437\) 25.5885i 1.22406i
\(438\) 0 0
\(439\) −9.87002 17.0954i −0.471070 0.815918i 0.528382 0.849007i \(-0.322799\pi\)
−0.999452 + 0.0330889i \(0.989466\pi\)
\(440\) 0 0
\(441\) 21.0718 1.00342
\(442\) 0 0
\(443\) 40.6304 1.93041 0.965205 0.261496i \(-0.0842157\pi\)
0.965205 + 0.261496i \(0.0842157\pi\)
\(444\) 0 0
\(445\) 1.19615 + 2.07180i 0.0567031 + 0.0982126i
\(446\) 0 0
\(447\) 14.3617i 0.679287i
\(448\) 0 0
\(449\) 32.5526 + 18.7942i 1.53625 + 0.886954i 0.999054 + 0.0434975i \(0.0138501\pi\)
0.537197 + 0.843457i \(0.319483\pi\)
\(450\) 0 0
\(451\) 7.72347 13.3774i 0.363684 0.629919i
\(452\) 0 0
\(453\) −9.92820 + 5.73205i −0.466468 + 0.269315i
\(454\) 0 0
\(455\) −32.3763 8.01105i −1.51782 0.375564i
\(456\) 0 0
\(457\) −20.8923 + 12.0622i −0.977301 + 0.564245i −0.901454 0.432875i \(-0.857499\pi\)
−0.0758467 + 0.997119i \(0.524166\pi\)
\(458\) 0 0
\(459\) −1.28380 + 2.22361i −0.0599228 + 0.103789i
\(460\) 0 0
\(461\) −2.76795 1.59808i −0.128916 0.0744298i 0.434155 0.900838i \(-0.357047\pi\)
−0.563071 + 0.826408i \(0.690380\pi\)
\(462\) 0 0
\(463\) 31.3801i 1.45835i −0.684325 0.729177i \(-0.739903\pi\)
0.684325 0.729177i \(-0.260097\pi\)
\(464\) 0 0
\(465\) 11.4641 + 19.8564i 0.531635 + 0.920819i
\(466\) 0 0
\(467\) 9.25036 0.428056 0.214028 0.976828i \(-0.431342\pi\)
0.214028 + 0.976828i \(0.431342\pi\)
\(468\) 0 0
\(469\) −21.3923 −0.987805
\(470\) 0 0
\(471\) −2.72172 4.71416i −0.125410 0.217217i
\(472\) 0 0
\(473\) 7.98076i 0.366956i
\(474\) 0 0
\(475\) 4.00552 + 2.31259i 0.183786 + 0.106109i
\(476\) 0 0
\(477\) 1.85641 3.21539i 0.0849990 0.147223i
\(478\) 0 0
\(479\) −9.29485 + 5.36639i −0.424693 + 0.245196i −0.697083 0.716990i \(-0.745519\pi\)
0.272390 + 0.962187i \(0.412186\pi\)
\(480\) 0 0
\(481\) −20.0885 19.3301i −0.915955 0.881378i
\(482\) 0 0
\(483\) −27.4635 + 15.8561i −1.24963 + 0.721476i
\(484\) 0 0
\(485\) 13.7321 23.7846i 0.623540 1.08000i
\(486\) 0 0
\(487\) −2.43414 1.40535i −0.110301 0.0636825i 0.443834 0.896109i \(-0.353618\pi\)
−0.554136 + 0.832426i \(0.686951\pi\)
\(488\) 0 0
\(489\) 30.9090i 1.39775i
\(490\) 0 0
\(491\) 6.48415 + 11.2309i 0.292626 + 0.506843i 0.974430 0.224692i \(-0.0721377\pi\)
−0.681804 + 0.731535i \(0.738804\pi\)
\(492\) 0 0
\(493\) −1.39230 −0.0627063
\(494\) 0 0
\(495\) 6.28555 0.282514
\(496\) 0 0
\(497\) −14.8923 25.7942i −0.668011 1.15703i
\(498\) 0 0
\(499\) 9.25036i 0.414103i −0.978330 0.207052i \(-0.933613\pi\)
0.978330 0.207052i \(-0.0663868\pi\)
\(500\) 0 0
\(501\) −24.1077 13.9186i −1.07705 0.621836i
\(502\) 0 0
\(503\) −15.7345 + 27.2530i −0.701567 + 1.21515i 0.266349 + 0.963877i \(0.414183\pi\)
−0.967916 + 0.251274i \(0.919151\pi\)
\(504\) 0 0
\(505\) 33.5885 19.3923i 1.49467 0.862946i
\(506\) 0 0
\(507\) 14.2521 + 7.51294i 0.632958 + 0.333661i
\(508\) 0 0
\(509\) −23.5526 + 13.5981i −1.04395 + 0.602724i −0.920949 0.389683i \(-0.872585\pi\)
−0.123000 + 0.992407i \(0.539251\pi\)
\(510\) 0 0
\(511\) −13.8755 + 24.0331i −0.613818 + 1.06316i
\(512\) 0 0
\(513\) −22.1603 12.7942i −0.978399 0.564879i
\(514\) 0 0
\(515\) 17.1724i 0.756708i
\(516\) 0 0
\(517\) −12.5885 21.8038i −0.553640 0.958932i
\(518\) 0 0
\(519\) 12.3042 0.540093
\(520\) 0 0
\(521\) 6.24871 0.273761 0.136881 0.990588i \(-0.456292\pi\)
0.136881 + 0.990588i \(0.456292\pi\)
\(522\) 0 0
\(523\) −3.43036 5.94155i −0.149999 0.259806i 0.781228 0.624246i \(-0.214594\pi\)
−0.931227 + 0.364440i \(0.881260\pi\)
\(524\) 0 0
\(525\) 5.73205i 0.250167i
\(526\) 0 0
\(527\) −3.71794 2.14655i −0.161956 0.0935054i
\(528\) 0 0
\(529\) −3.80385 + 6.58846i −0.165385 + 0.286455i
\(530\) 0 0
\(531\) 12.1500 7.01483i 0.527267 0.304418i
\(532\) 0 0
\(533\) 18.6962 + 17.9904i 0.809820 + 0.779250i
\(534\) 0 0
\(535\) 6.43966 3.71794i 0.278411 0.160741i
\(536\) 0 0
\(537\) 2.30385 3.99038i 0.0994184 0.172198i
\(538\) 0 0
\(539\) −26.7549 15.4469i −1.15241 0.665346i
\(540\) 0 0
\(541\) 6.92820i 0.297867i −0.988847 0.148933i \(-0.952416\pi\)
0.988847 0.148933i \(-0.0475840\pi\)
\(542\) 0 0
\(543\) 10.1576 + 17.5935i 0.435905 + 0.755009i
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −8.58622 −0.367120 −0.183560 0.983008i \(-0.558762\pi\)
−0.183560 + 0.983008i \(0.558762\pi\)
\(548\) 0 0
\(549\) 2.19615 + 3.80385i 0.0937295 + 0.162344i
\(550\) 0 0
\(551\) 13.8755i 0.591118i
\(552\) 0 0
\(553\) 17.1962 + 9.92820i 0.731255 + 0.422190i
\(554\) 0 0
\(555\) −9.58244 + 16.5973i −0.406752 + 0.704515i
\(556\) 0 0
\(557\) 19.7487 11.4019i 0.836780 0.483115i −0.0193886 0.999812i \(-0.506172\pi\)
0.856168 + 0.516697i \(0.172839\pi\)
\(558\) 0 0
\(559\) 13.0128 + 3.21983i 0.550383 + 0.136184i
\(560\) 0 0
\(561\) 1.06922 0.617314i 0.0451425 0.0260630i
\(562\) 0 0
\(563\) 19.4525 33.6926i 0.819823 1.41998i −0.0859889 0.996296i \(-0.527405\pi\)
0.905812 0.423679i \(-0.139262\pi\)
\(564\) 0 0
\(565\) −21.5885 12.4641i −0.908233 0.524369i
\(566\) 0 0
\(567\) 11.3969i 0.478626i
\(568\) 0 0
\(569\) −4.96410 8.59808i −0.208106 0.360450i 0.743012 0.669278i \(-0.233397\pi\)
−0.951118 + 0.308828i \(0.900063\pi\)
\(570\) 0 0
\(571\) −23.4580 −0.981686 −0.490843 0.871248i \(-0.663311\pi\)
−0.490843 + 0.871248i \(0.663311\pi\)
\(572\) 0 0
\(573\) 16.0718 0.671409
\(574\) 0 0
\(575\) −2.76621 4.79122i −0.115359 0.199808i
\(576\) 0 0
\(577\) 43.8564i 1.82577i −0.408221 0.912883i \(-0.633851\pi\)
0.408221 0.912883i \(-0.366149\pi\)
\(578\) 0 0
\(579\) −21.1780 12.2271i −0.880126 0.508141i
\(580\) 0 0
\(581\) −9.92820 + 17.1962i −0.411891 + 0.713417i
\(582\) 0 0
\(583\) −4.71416 + 2.72172i −0.195241 + 0.112722i
\(584\) 0 0
\(585\) −2.53590 + 10.2487i −0.104846 + 0.423732i
\(586\) 0 0
\(587\) 19.4525 11.2309i 0.802889 0.463548i −0.0415915 0.999135i \(-0.513243\pi\)
0.844480 + 0.535587i \(0.179909\pi\)
\(588\) 0 0
\(589\) 21.3923 37.0526i 0.881455 1.52672i
\(590\) 0 0
\(591\) −25.3170 14.6167i −1.04140 0.601253i
\(592\) 0 0
\(593\) 22.7846i 0.935652i −0.883821 0.467826i \(-0.845037\pi\)
0.883821 0.467826i \(-0.154963\pi\)
\(594\) 0 0
\(595\) −2.14655 3.71794i −0.0880001 0.152421i
\(596\) 0 0
\(597\) 15.2487 0.624088
\(598\) 0 0
\(599\) 12.8793 0.526235 0.263117 0.964764i \(-0.415249\pi\)
0.263117 + 0.964764i \(0.415249\pi\)
\(600\) 0 0
\(601\) −2.69615 4.66987i −0.109978 0.190488i 0.805783 0.592211i \(-0.201745\pi\)
−0.915761 + 0.401723i \(0.868411\pi\)
\(602\) 0 0
\(603\) 6.77174i 0.275766i
\(604\) 0 0
\(605\) 11.0718 + 6.39230i 0.450133 + 0.259884i
\(606\) 0 0
\(607\) 2.43414 4.21605i 0.0987986 0.171124i −0.812389 0.583116i \(-0.801833\pi\)
0.911188 + 0.411992i \(0.135167\pi\)
\(608\) 0 0
\(609\) −14.8923 + 8.59808i −0.603467 + 0.348412i
\(610\) 0 0
\(611\) 40.6304 11.7290i 1.64373 0.474504i
\(612\) 0 0
\(613\) 0.696152 0.401924i 0.0281173 0.0162335i −0.485875 0.874028i \(-0.661499\pi\)
0.513993 + 0.857795i \(0.328166\pi\)
\(614\) 0 0
\(615\) 8.91829 15.4469i 0.359620 0.622880i
\(616\) 0 0
\(617\) 10.0359 + 5.79423i 0.404030 + 0.233267i 0.688221 0.725501i \(-0.258392\pi\)
−0.284191 + 0.958768i \(0.591725\pi\)
\(618\) 0 0
\(619\) 9.25036i 0.371803i −0.982568 0.185902i \(-0.940479\pi\)
0.982568 0.185902i \(-0.0595206\pi\)
\(620\) 0 0
\(621\) 15.3038 + 26.5070i 0.614122 + 1.06369i
\(622\) 0 0
\(623\) −5.53242 −0.221652
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 6.15208 + 10.6557i 0.245690 + 0.425548i
\(628\) 0 0
\(629\) 3.58846i 0.143081i
\(630\) 0 0
\(631\) −24.7418 14.2847i −0.984955 0.568664i −0.0811925 0.996698i \(-0.525873\pi\)
−0.903762 + 0.428034i \(0.859206\pi\)
\(632\) 0 0
\(633\) 16.8397 29.1673i 0.669320 1.15930i
\(634\) 0 0
\(635\) 6.43966 3.71794i 0.255550 0.147542i
\(636\) 0 0
\(637\) 35.9808 37.3923i 1.42561 1.48154i
\(638\) 0 0
\(639\) −8.16517 + 4.71416i −0.323009 + 0.186489i
\(640\) 0 0
\(641\) 9.69615 16.7942i 0.382975 0.663332i −0.608511 0.793545i \(-0.708233\pi\)
0.991486 + 0.130213i \(0.0415662\pi\)
\(642\) 0 0
\(643\) 26.4673 + 15.2809i 1.04377 + 0.602620i 0.920898 0.389803i \(-0.127457\pi\)
0.122870 + 0.992423i \(0.460790\pi\)
\(644\) 0 0
\(645\) 9.21539i 0.362856i
\(646\) 0 0
\(647\) −5.57691 9.65949i −0.219251 0.379754i 0.735328 0.677711i \(-0.237028\pi\)
−0.954579 + 0.297957i \(0.903695\pi\)
\(648\) 0 0
\(649\) −20.5692 −0.807412
\(650\) 0 0
\(651\) −53.0236 −2.07816
\(652\) 0 0
\(653\) −7.62436 13.2058i −0.298364 0.516782i 0.677398 0.735617i \(-0.263108\pi\)
−0.975762 + 0.218835i \(0.929774\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.60770 + 4.39230i 0.296804 + 0.171360i
\(658\) 0 0
\(659\) −3.67345 + 6.36260i −0.143097 + 0.247852i −0.928661 0.370928i \(-0.879040\pi\)
0.785564 + 0.618780i \(0.212373\pi\)
\(660\) 0 0
\(661\) −11.3038 + 6.52628i −0.439669 + 0.253843i −0.703457 0.710738i \(-0.748361\pi\)
0.263788 + 0.964581i \(0.415028\pi\)
\(662\) 0 0
\(663\) 0.575167 + 1.99244i 0.0223377 + 0.0773799i
\(664\) 0 0
\(665\) 37.0526 21.3923i 1.43684 0.829558i
\(666\) 0 0
\(667\) −8.29863 + 14.3737i −0.321324 + 0.556550i
\(668\) 0 0
\(669\) 4.96410 + 2.86603i 0.191923 + 0.110807i
\(670\) 0 0
\(671\) 6.43966i 0.248600i
\(672\) 0 0
\(673\) −11.8923 20.5981i −0.458415 0.793997i 0.540463 0.841368i \(-0.318249\pi\)
−0.998877 + 0.0473705i \(0.984916\pi\)
\(674\) 0 0
\(675\) −5.53242 −0.212943
\(676\) 0 0
\(677\) 9.46410 0.363735 0.181867 0.983323i \(-0.441786\pi\)
0.181867 + 0.983323i \(0.441786\pi\)
\(678\) 0 0
\(679\) 31.7566 + 55.0041i 1.21871 + 2.11086i
\(680\) 0 0
\(681\) 17.1962i 0.658958i
\(682\) 0 0
\(683\) 8.29863 + 4.79122i 0.317538 + 0.183331i 0.650295 0.759682i \(-0.274645\pi\)
−0.332756 + 0.943013i \(0.607979\pi\)
\(684\) 0 0
\(685\) −13.1962 + 22.8564i −0.504199 + 0.873298i
\(686\) 0 0
\(687\) −0.996219 + 0.575167i −0.0380081 + 0.0219440i
\(688\) 0 0
\(689\) −2.53590 8.78461i −0.0966100 0.334667i
\(690\) 0 0
\(691\) 34.4783 19.9061i 1.31162 0.757263i 0.329254 0.944241i \(-0.393203\pi\)
0.982364 + 0.186979i \(0.0598695\pi\)
\(692\) 0 0
\(693\) −7.26795 + 12.5885i −0.276087 + 0.478196i
\(694\) 0 0
\(695\) 13.8755 + 8.01105i 0.526329 + 0.303876i
\(696\) 0 0
\(697\) 3.33975i 0.126502i
\(698\) 0 0
\(699\) 9.00727 + 15.6010i 0.340686 + 0.590086i
\(700\) 0 0
\(701\) 32.1051 1.21259 0.606297 0.795238i \(-0.292654\pi\)
0.606297 + 0.795238i \(0.292654\pi\)
\(702\) 0 0
\(703\) 35.7621 1.34879
\(704\) 0 0
\(705\) −14.5359 25.1769i −0.547454 0.948217i
\(706\) 0 0
\(707\) 89.6929i 3.37325i
\(708\) 0 0
\(709\) −23.3038 13.4545i −0.875194 0.505294i −0.00612347 0.999981i \(-0.501949\pi\)
−0.869071 + 0.494688i \(0.835283\pi\)
\(710\) 0 0
\(711\) 3.14277 5.44344i 0.117863 0.204145i
\(712\) 0 0
\(713\) −44.3205 + 25.5885i −1.65982 + 0.958295i
\(714\) 0 0
\(715\) 10.7328 11.1538i 0.401383 0.417129i
\(716\) 0 0
\(717\) −4.60770 + 2.66025i −0.172078 + 0.0993490i
\(718\) 0 0
\(719\) −6.48415 + 11.2309i −0.241818 + 0.418841i −0.961232 0.275740i \(-0.911077\pi\)
0.719414 + 0.694581i \(0.244410\pi\)
\(720\) 0 0
\(721\) −34.3923 19.8564i −1.28084 0.739491i
\(722\) 0 0
\(723\) 26.7549i 0.995024i
\(724\) 0 0
\(725\) −1.50000 2.59808i −0.0557086 0.0964901i
\(726\) 0 0
\(727\) −17.1724 −0.636890 −0.318445 0.947941i \(-0.603161\pi\)
−0.318445 + 0.947941i \(0.603161\pi\)
\(728\) 0 0
\(729\) 24.1769 0.895441
\(730\) 0 0
\(731\) 0.862751 + 1.49433i 0.0319100 + 0.0552698i
\(732\) 0 0
\(733\) 15.7128i 0.580366i 0.956971 + 0.290183i \(0.0937162\pi\)
−0.956971 + 0.290183i \(0.906284\pi\)
\(734\) 0 0
\(735\) −30.8939 17.8366i −1.13954 0.657912i
\(736\) 0 0
\(737\) 4.96410 8.59808i 0.182855 0.316714i
\(738\) 0 0
\(739\) 31.1814 18.0026i 1.14703 0.662237i 0.198867 0.980027i \(-0.436274\pi\)
0.948161 + 0.317790i \(0.102941\pi\)
\(740\) 0 0
\(741\) −19.8564 + 5.73205i −0.729443 + 0.210572i
\(742\) 0 0
\(743\) −37.0459 + 21.3885i −1.35908 + 0.784667i −0.989500 0.144530i \(-0.953833\pi\)
−0.369583 + 0.929198i \(0.620500\pi\)
\(744\) 0 0
\(745\) 11.5885 20.0718i 0.424568 0.735374i
\(746\) 0 0
\(747\) 5.44344 + 3.14277i 0.199165 + 0.114988i
\(748\) 0 0
\(749\) 17.1962i 0.628334i
\(750\) 0 0
\(751\) −0.287584 0.498110i −0.0104941 0.0181763i 0.860731 0.509061i \(-0.170007\pi\)
−0.871225 + 0.490884i \(0.836674\pi\)
\(752\) 0 0
\(753\) 32.0333 1.16736
\(754\) 0 0
\(755\) −18.5007 −0.673310
\(756\) 0 0
\(757\) 11.3038 + 19.5788i 0.410845 + 0.711605i 0.994982 0.100050i \(-0.0319003\pi\)
−0.584137 + 0.811655i \(0.698567\pi\)
\(758\) 0 0
\(759\) 14.7176i 0.534217i
\(760\) 0 0
\(761\) −2.76795 1.59808i −0.100338 0.0579302i 0.448991 0.893536i \(-0.351783\pi\)
−0.549329 + 0.835606i \(0.685117\pi\)
\(762\) 0 0
\(763\) −13.8755 + 24.0331i −0.502328 + 0.870058i
\(764\) 0 0
\(765\) −1.17691 + 0.679492i −0.0425514 + 0.0245671i
\(766\) 0 0
\(767\) 8.29863 33.5385i 0.299646 1.21101i
\(768\) 0 0
\(769\) −36.4808 + 21.0622i −1.31553 + 0.759522i −0.983006 0.183573i \(-0.941234\pi\)
−0.332524 + 0.943095i \(0.607900\pi\)
\(770\) 0 0
\(771\) 0.287584 0.498110i 0.0103571 0.0179390i
\(772\) 0 0
\(773\) 14.5526 + 8.40192i 0.523419 + 0.302196i 0.738332 0.674437i \(-0.235614\pi\)
−0.214913 + 0.976633i \(0.568947\pi\)
\(774\) 0 0
\(775\) 9.25036i 0.332283i
\(776\) 0 0
\(777\) −22.1603 38.3827i −0.794995 1.37697i
\(778\) 0 0
\(779\) −33.2835 −1.19251
\(780\) 0 0
\(781\) 13.8231 0.494629
\(782\) 0 0
\(783\) 8.29863 + 14.3737i 0.296569 + 0.513673i
\(784\) 0 0
\(785\) 8.78461i 0.313536i
\(786\) 0 0
\(787\) 23.3245 + 13.4664i 0.831429 + 0.480026i 0.854342 0.519712i \(-0.173961\pi\)
−0.0229126 + 0.999737i \(0.507294\pi\)
\(788\) 0 0
\(789\) −14.8923 + 25.7942i −0.530180 + 0.918299i
\(790\) 0 0
\(791\) 49.9253 28.8244i 1.77514 1.02488i
\(792\) 0 0
\(793\) 10.5000 + 2.59808i 0.372866 + 0.0922604i
\(794\) 0 0
\(795\) −5.44344 + 3.14277i −0.193059 + 0.111463i
\(796\) 0 0
\(797\) −15.3564 + 26.5981i −0.543952 + 0.942152i 0.454720 + 0.890634i \(0.349739\pi\)
−0.998672 + 0.0515179i \(0.983594\pi\)
\(798\) 0 0
\(799\) 4.71416 + 2.72172i 0.166775 + 0.0962876i
\(800\) 0 0
\(801\) 1.75129i 0.0618787i
\(802\) 0 0
\(803\) −6.43966 11.1538i −0.227251 0.393610i
\(804\) 0 0
\(805\) −51.1769 −1.80375
\(806\) 0 0
\(807\) 5.71038 0.201015
\(808\) 0 0
\(809\) 8.76795 + 15.1865i 0.308265 + 0.533930i 0.977983 0.208685i \(-0.0669184\pi\)
−0.669718 + 0.742615i \(0.733585\pi\)
\(810\) 0 0
\(811\) 16.6862i 0.585933i −0.956123 0.292967i \(-0.905358\pi\)
0.956123 0.292967i \(-0.0946425\pi\)
\(812\) 0 0
\(813\) 24.8205 + 14.3301i 0.870493 + 0.502579i
\(814\) 0 0
\(815\) −24.9404 + 43.1980i −0.873624 + 1.51316i
\(816\) 0 0
\(817\) −14.8923 + 8.59808i −0.521016 + 0.300809i
\(818\) 0 0
\(819\) −17.5935 16.9293i −0.614766 0.591559i
\(820\) 0 0
\(821\) 38.4282 22.1865i 1.34115 0.774315i 0.354177 0.935178i \(-0.384761\pi\)
0.986977 + 0.160863i \(0.0514278\pi\)
\(822\) 0 0
\(823\) 21.5990 37.4106i 0.752894 1.30405i −0.193520 0.981096i \(-0.561991\pi\)
0.946414 0.322955i \(-0.104676\pi\)
\(824\) 0 0
\(825\) 2.30385 + 1.33013i 0.0802097 + 0.0463091i
\(826\) 0 0
\(827\) 27.7511i 0.965000i −0.875896 0.482500i \(-0.839729\pi\)
0.875896 0.482500i \(-0.160271\pi\)
\(828\) 0 0
\(829\) −22.2846 38.5981i −0.773976 1.34057i −0.935368 0.353676i \(-0.884932\pi\)
0.161391 0.986890i \(-0.448402\pi\)
\(830\) 0 0
\(831\) −4.20413 −0.145840
\(832\) 0 0
\(833\) 6.67949 0.231431
\(834\) 0 0
\(835\) −22.4618 38.9049i −0.777321 1.34636i
\(836\) 0 0
\(837\) 51.1769i 1.76893i
\(838\) 0 0
\(839\) −18.4562 10.6557i −0.637180 0.367876i 0.146347 0.989233i \(-0.453248\pi\)
−0.783527 + 0.621357i \(0.786582\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) −10.7328 + 6.19657i −0.369656 + 0.213421i
\(844\) 0 0
\(845\) 13.8564 + 22.0000i 0.476675 + 0.756823i
\(846\) 0 0
\(847\) −25.6045 + 14.7828i −0.879782 + 0.507943i
\(848\) 0 0
\(849\) −12.2321 + 21.1865i −0.419803 + 0.727120i
\(850\) 0 0
\(851\) −37.0459 21.3885i −1.26992 0.733188i
\(852\) 0 0
\(853\) 35.0718i 1.20084i 0.799687 + 0.600418i \(0.204999\pi\)
−0.799687 + 0.600418i \(0.795001\pi\)
\(854\) 0 0
\(855\) −6.77174 11.7290i −0.231588 0.401123i
\(856\) 0 0
\(857\) 39.0333 1.33335 0.666676 0.745347i \(-0.267716\pi\)
0.666676 + 0.745347i \(0.267716\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 20.6244 + 35.7224i 0.702876 + 1.21742i
\(862\) 0 0
\(863\) 12.8793i 0.438417i −0.975678 0.219209i \(-0.929652\pi\)
0.975678 0.219209i \(-0.0703475\pi\)
\(864\) 0 0
\(865\) 17.1962 + 9.92820i 0.584687 + 0.337569i
\(866\) 0 0
\(867\) 10.4007 18.0145i 0.353226 0.611806i
\(868\) 0 0
\(869\) −7.98076 + 4.60770i −0.270729 + 0.156305i
\(870\) 0 0
\(871\) 12.0166 + 11.5630i 0.407166 + 0.391796i
\(872\) 0 0
\(873\) 17.4115 10.0526i 0.589291 0.340228i
\(874\) 0 0
\(875\) 27.7511 48.0663i 0.938158 1.62494i
\(876\) 0 0
\(877\) −17.6769 10.2058i −0.596907 0.344624i 0.170917 0.985285i \(-0.445327\pi\)
−0.767824 + 0.640661i \(0.778660\pi\)
\(878\) 0 0
\(879\) 31.2259i 1.05322i
\(880\) 0 0
\(881\) −4.83975 8.38269i −0.163055 0.282420i 0.772908 0.634518i \(-0.218802\pi\)
−0.935963 + 0.352099i \(0.885468\pi\)
\(882\) 0 0
\(883\) −40.6304 −1.36732 −0.683661 0.729800i \(-0.739613\pi\)
−0.683661 + 0.729800i \(0.739613\pi\)
\(884\) 0 0
\(885\) −23.7513 −0.798391
\(886\) 0 0
\(887\) 20.3597 + 35.2640i 0.683612 + 1.18405i 0.973871 + 0.227102i \(0.0729251\pi\)
−0.290259 + 0.956948i \(0.593742\pi\)
\(888\) 0 0
\(889\) 17.1962i 0.576740i
\(890\) 0 0
\(891\) −4.58069 2.64466i −0.153459 0.0885996i
\(892\) 0 0
\(893\) −27.1244 + 46.9808i −0.907682 + 1.57215i
\(894\) 0 0
\(895\) 6.43966 3.71794i 0.215254 0.124277i
\(896\) 0 0
\(897\) 23.9974 + 5.93782i 0.801251 + 0.198258i
\(898\) 0 0
\(899\) −24.0331 + 13.8755i −0.801550 + 0.462775i
\(900\) 0 0
\(901\) 0.588457 1.01924i 0.0196044 0.0339557i
\(902\) 0 0
\(903\) 18.4562 + 10.6557i 0.614185 + 0.354600i
\(904\) 0 0
\(905\) 32.7846i 1.08980i
\(906\) 0 0
\(907\) 17.8811 + 30.9709i 0.593731 + 1.02837i 0.993725 + 0.111855i \(0.0356791\pi\)
−0.399993 + 0.916518i \(0.630988\pi\)
\(908\) 0 0
\(909\) 28.3923 0.941713
\(910\) 0 0
\(911\) 37.0015 1.22591 0.612956 0.790117i \(-0.289980\pi\)
0.612956 + 0.790117i \(0.289980\pi\)
\(912\) 0 0
\(913\) −4.60770 7.98076i −0.152492 0.264125i
\(914\) 0 0
\(915\) 7.43588i 0.245823i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.287584 0.498110i 0.00948651 0.0164311i −0.861243 0.508193i \(-0.830314\pi\)
0.870730 + 0.491762i \(0.163647\pi\)
\(920\) 0 0
\(921\) −6.03332 + 3.48334i −0.198805 + 0.114780i
\(922\) 0 0
\(923\) −5.57691 + 22.5388i −0.183566 + 0.741874i
\(924\) 0 0
\(925\) 6.69615 3.86603i 0.220168 0.127114i
\(926\) 0 0
\(927\) −6.28555 + 10.8869i −0.206444 + 0.357572i
\(928\) 0 0
\(929\) −5.55256 3.20577i −0.182174 0.105178i 0.406140 0.913811i \(-0.366875\pi\)
−0.588313 + 0.808633i \(0.700208\pi\)
\(930\) 0 0
\(931\) 66.5670i 2.18165i
\(932\) 0 0
\(933\) −11.4641 19.8564i −0.375318 0.650070i
\(934\) 0 0
\(935\) 1.99244 0.0651597
\(936\) 0 0
\(937\) −37.1769 −1.21452 −0.607258 0.794504i \(-0.707731\pi\)
−0.607258 + 0.794504i \(0.707731\pi\)
\(938\) 0 0
\(939\) 5.20035 + 9.00727i 0.169707 + 0.293941i
\(940\) 0 0
\(941\) 36.7846i 1.19914i 0.800321 + 0.599572i \(0.204663\pi\)
−0.800321 + 0.599572i \(0.795337\pi\)
\(942\) 0 0
\(943\) 34.4783 + 19.9061i 1.12277 + 0.648231i
\(944\) 0 0
\(945\) −25.5885 + 44.3205i −0.832393 + 1.44175i
\(946\) 0 0
\(947\) −29.6101 + 17.0954i −0.962198 + 0.555525i −0.896849 0.442337i \(-0.854149\pi\)
−0.0653489 + 0.997862i \(0.520816\pi\)
\(948\) 0 0
\(949\) 20.7846 6.00000i 0.674697 0.194768i
\(950\) 0 0
\(951\) −4.29311 + 2.47863i −0.139214 + 0.0803750i
\(952\) 0 0
\(953\) 0.356406 0.617314i 0.0115451 0.0199968i −0.860195 0.509965i \(-0.829658\pi\)
0.871740 + 0.489968i \(0.162992\pi\)
\(954\) 0 0
\(955\) 22.4618 + 12.9683i 0.726846 + 0.419644i
\(956\) 0 0
\(957\) 7.98076i 0.257981i
\(958\) 0 0
\(959\) −30.5173 52.8575i −0.985455 1.70686i
\(960\) 0 0
\(961\) −54.5692 −1.76030
\(962\) 0 0
\(963\) 5.44344 0.175413
\(964\) 0 0
\(965\) −19.7321 34.1769i −0.635197 1.10019i
\(966\) 0 0
\(967\) 16.5083i 0.530871i −0.964129 0.265435i \(-0.914484\pi\)
0.964129 0.265435i \(-0.0855157\pi\)
\(968\) 0 0
\(969\) −2.30385 1.33013i −0.0740102 0.0427298i
\(970\) 0 0
\(971\) 16.6418 28.8244i 0.534059 0.925018i −0.465149 0.885232i \(-0.653999\pi\)
0.999208 0.0397855i \(-0.0126675\pi\)
\(972\) 0 0
\(973\) −32.0885 + 18.5263i −1.02871 + 0.593925i
\(974\) 0 0
\(975\) −3.09828 + 3.21983i −0.0992245 + 0.103117i
\(976\) 0 0
\(977\) −1.03590 + 0.598076i −0.0331413 + 0.0191342i −0.516479 0.856300i \(-0.672758\pi\)
0.483338 + 0.875434i \(0.339424\pi\)
\(978\) 0 0
\(979\) 1.28380 2.22361i 0.0410305 0.0710670i
\(980\) 0 0
\(981\) 7.60770 + 4.39230i 0.242895 + 0.140236i
\(982\) 0 0
\(983\) 10.5787i 0.337407i 0.985667 + 0.168703i \(0.0539580\pi\)
−0.985667 + 0.168703i \(0.946042\pi\)
\(984\) 0 0
\(985\) −23.5885 40.8564i −0.751591 1.30179i
\(986\) 0 0
\(987\) 67.2312 2.13999
\(988\) 0 0
\(989\) 20.5692 0.654063
\(990\) 0 0
\(991\) −17.3059 29.9747i −0.549740 0.952178i −0.998292 0.0584210i \(-0.981393\pi\)
0.448552 0.893757i \(-0.351940\pi\)
\(992\) 0 0
\(993\) 3.48334i 0.110540i
\(994\) 0 0
\(995\) 21.3114 + 12.3042i 0.675618 + 0.390068i
\(996\) 0 0
\(997\) 14.5000 25.1147i 0.459220 0.795392i −0.539700 0.841857i \(-0.681462\pi\)
0.998920 + 0.0464655i \(0.0147958\pi\)
\(998\) 0 0
\(999\) −37.0459 + 21.3885i −1.17208 + 0.676702i
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 416.2.w.c.257.2 yes 8
4.3 odd 2 inner 416.2.w.c.257.3 yes 8
8.3 odd 2 832.2.w.h.257.2 8
8.5 even 2 832.2.w.h.257.3 8
13.2 odd 12 5408.2.a.bg.1.3 4
13.4 even 6 inner 416.2.w.c.225.2 8
13.11 odd 12 5408.2.a.bk.1.3 4
52.11 even 12 5408.2.a.bk.1.2 4
52.15 even 12 5408.2.a.bg.1.2 4
52.43 odd 6 inner 416.2.w.c.225.3 yes 8
104.43 odd 6 832.2.w.h.641.2 8
104.69 even 6 832.2.w.h.641.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.w.c.225.2 8 13.4 even 6 inner
416.2.w.c.225.3 yes 8 52.43 odd 6 inner
416.2.w.c.257.2 yes 8 1.1 even 1 trivial
416.2.w.c.257.3 yes 8 4.3 odd 2 inner
832.2.w.h.257.2 8 8.3 odd 2
832.2.w.h.257.3 8 8.5 even 2
832.2.w.h.641.2 8 104.43 odd 6
832.2.w.h.641.3 8 104.69 even 6
5408.2.a.bg.1.2 4 52.15 even 12
5408.2.a.bg.1.3 4 13.2 odd 12
5408.2.a.bk.1.2 4 52.11 even 12
5408.2.a.bk.1.3 4 13.11 odd 12