Properties

Label 1008.2.r.i.673.3
Level $1008$
Weight $2$
Character 1008.673
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 673.3
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1008.673
Dual form 1008.2.r.i.337.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.70574 + 0.300767i) q^{3} +(-0.266044 - 0.460802i) q^{5} +(0.500000 - 0.866025i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\) \(q+(1.70574 + 0.300767i) q^{3} +(-0.266044 - 0.460802i) q^{5} +(0.500000 - 0.866025i) q^{7} +(2.81908 + 1.02606i) q^{9} +(-1.11334 + 1.92836i) q^{11} +(-1.03209 - 1.78763i) q^{13} +(-0.315207 - 0.866025i) q^{15} +0.815207 q^{17} +7.94356 q^{19} +(1.11334 - 1.32683i) q^{21} +(3.40033 + 5.88954i) q^{23} +(2.35844 - 4.08494i) q^{25} +(4.50000 + 2.59808i) q^{27} +(3.73783 - 6.47410i) q^{29} +(1.14543 + 1.98394i) q^{31} +(-2.47906 + 2.95442i) q^{33} -0.532089 q^{35} -10.2909 q^{37} +(-1.22281 - 3.35965i) q^{39} +(1.29813 + 2.24843i) q^{41} +(0.145430 - 0.251892i) q^{43} +(-0.277189 - 1.57202i) q^{45} +(0.213011 - 0.368946i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(1.39053 + 0.245188i) q^{51} +1.41147 q^{53} +1.18479 q^{55} +(13.5496 + 2.38917i) q^{57} +(1.71554 + 2.97140i) q^{59} +(5.23783 - 9.07218i) q^{61} +(2.29813 - 1.92836i) q^{63} +(-0.549163 + 0.951178i) q^{65} +(-2.10220 - 3.64111i) q^{67} +(4.02869 + 11.0687i) q^{69} -12.0865 q^{71} -9.09152 q^{73} +(5.25150 - 6.25849i) q^{75} +(1.11334 + 1.92836i) q^{77} +(3.73055 - 6.46151i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-8.76264 + 15.1773i) q^{83} +(-0.216881 - 0.375650i) q^{85} +(8.32295 - 9.91890i) q^{87} -2.09833 q^{89} -2.06418 q^{91} +(1.35710 + 3.72859i) q^{93} +(-2.11334 - 3.66041i) q^{95} +(-5.94222 + 10.2922i) q^{97} +(-5.11721 + 4.29385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} + 3 q^{7} + 3 q^{13} - 9 q^{15} + 12 q^{17} + 18 q^{19} + 6 q^{23} + 6 q^{25} + 27 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} + 6 q^{35} - 30 q^{37} - 18 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 9 q^{47} - 3 q^{49} - 9 q^{51} - 12 q^{53} + 27 q^{57} + 3 q^{59} + 12 q^{61} - 15 q^{65} - 12 q^{67} - 27 q^{69} - 42 q^{71} + 6 q^{73} - 9 q^{75} - 15 q^{79} - 6 q^{83} + 15 q^{85} + 9 q^{87} - 36 q^{89} + 6 q^{91} + 9 q^{93} - 6 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 1.70574 + 0.300767i 0.984808 + 0.173648i
\(4\) 0 0
\(5\) −0.266044 0.460802i −0.118979 0.206077i 0.800385 0.599487i \(-0.204629\pi\)
−0.919363 + 0.393410i \(0.871295\pi\)
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.188982 0.327327i
\(8\) 0 0
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 0 0
\(11\) −1.11334 + 1.92836i −0.335685 + 0.581423i −0.983616 0.180276i \(-0.942301\pi\)
0.647931 + 0.761699i \(0.275634\pi\)
\(12\) 0 0
\(13\) −1.03209 1.78763i −0.286250 0.495799i 0.686662 0.726977i \(-0.259075\pi\)
−0.972912 + 0.231178i \(0.925742\pi\)
\(14\) 0 0
\(15\) −0.315207 0.866025i −0.0813862 0.223607i
\(16\) 0 0
\(17\) 0.815207 0.197717 0.0988584 0.995102i \(-0.468481\pi\)
0.0988584 + 0.995102i \(0.468481\pi\)
\(18\) 0 0
\(19\) 7.94356 1.82238 0.911189 0.411988i \(-0.135166\pi\)
0.911189 + 0.411988i \(0.135166\pi\)
\(20\) 0 0
\(21\) 1.11334 1.32683i 0.242951 0.289538i
\(22\) 0 0
\(23\) 3.40033 + 5.88954i 0.709018 + 1.22805i 0.965222 + 0.261432i \(0.0841947\pi\)
−0.256204 + 0.966623i \(0.582472\pi\)
\(24\) 0 0
\(25\) 2.35844 4.08494i 0.471688 0.816988i
\(26\) 0 0
\(27\) 4.50000 + 2.59808i 0.866025 + 0.500000i
\(28\) 0 0
\(29\) 3.73783 6.47410i 0.694097 1.20221i −0.276387 0.961046i \(-0.589137\pi\)
0.970484 0.241165i \(-0.0775294\pi\)
\(30\) 0 0
\(31\) 1.14543 + 1.98394i 0.205725 + 0.356327i 0.950364 0.311142i \(-0.100711\pi\)
−0.744638 + 0.667468i \(0.767378\pi\)
\(32\) 0 0
\(33\) −2.47906 + 2.95442i −0.431548 + 0.514299i
\(34\) 0 0
\(35\) −0.532089 −0.0899394
\(36\) 0 0
\(37\) −10.2909 −1.69181 −0.845903 0.533336i \(-0.820938\pi\)
−0.845903 + 0.533336i \(0.820938\pi\)
\(38\) 0 0
\(39\) −1.22281 3.35965i −0.195807 0.537974i
\(40\) 0 0
\(41\) 1.29813 + 2.24843i 0.202734 + 0.351146i 0.949409 0.314044i \(-0.101684\pi\)
−0.746674 + 0.665190i \(0.768351\pi\)
\(42\) 0 0
\(43\) 0.145430 0.251892i 0.0221778 0.0384131i −0.854723 0.519084i \(-0.826273\pi\)
0.876901 + 0.480670i \(0.159607\pi\)
\(44\) 0 0
\(45\) −0.277189 1.57202i −0.0413209 0.234342i
\(46\) 0 0
\(47\) 0.213011 0.368946i 0.0310709 0.0538163i −0.850072 0.526667i \(-0.823442\pi\)
0.881143 + 0.472850i \(0.156775\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 1.39053 + 0.245188i 0.194713 + 0.0343332i
\(52\) 0 0
\(53\) 1.41147 0.193881 0.0969404 0.995290i \(-0.469094\pi\)
0.0969404 + 0.995290i \(0.469094\pi\)
\(54\) 0 0
\(55\) 1.18479 0.159757
\(56\) 0 0
\(57\) 13.5496 + 2.38917i 1.79469 + 0.316453i
\(58\) 0 0
\(59\) 1.71554 + 2.97140i 0.223344 + 0.386843i 0.955821 0.293948i \(-0.0949694\pi\)
−0.732477 + 0.680791i \(0.761636\pi\)
\(60\) 0 0
\(61\) 5.23783 9.07218i 0.670635 1.16157i −0.307089 0.951681i \(-0.599355\pi\)
0.977724 0.209893i \(-0.0673116\pi\)
\(62\) 0 0
\(63\) 2.29813 1.92836i 0.289538 0.242951i
\(64\) 0 0
\(65\) −0.549163 + 0.951178i −0.0681153 + 0.117979i
\(66\) 0 0
\(67\) −2.10220 3.64111i −0.256824 0.444833i 0.708565 0.705645i \(-0.249343\pi\)
−0.965389 + 0.260813i \(0.916009\pi\)
\(68\) 0 0
\(69\) 4.02869 + 11.0687i 0.484997 + 1.33252i
\(70\) 0 0
\(71\) −12.0865 −1.43440 −0.717200 0.696868i \(-0.754577\pi\)
−0.717200 + 0.696868i \(0.754577\pi\)
\(72\) 0 0
\(73\) −9.09152 −1.06408 −0.532041 0.846719i \(-0.678575\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(74\) 0 0
\(75\) 5.25150 6.25849i 0.606391 0.722668i
\(76\) 0 0
\(77\) 1.11334 + 1.92836i 0.126877 + 0.219757i
\(78\) 0 0
\(79\) 3.73055 6.46151i 0.419720 0.726976i −0.576191 0.817315i \(-0.695462\pi\)
0.995911 + 0.0903388i \(0.0287950\pi\)
\(80\) 0 0
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) −8.76264 + 15.1773i −0.961825 + 1.66593i −0.243911 + 0.969798i \(0.578430\pi\)
−0.717914 + 0.696132i \(0.754903\pi\)
\(84\) 0 0
\(85\) −0.216881 0.375650i −0.0235241 0.0407449i
\(86\) 0 0
\(87\) 8.32295 9.91890i 0.892314 1.06342i
\(88\) 0 0
\(89\) −2.09833 −0.222422 −0.111211 0.993797i \(-0.535473\pi\)
−0.111211 + 0.993797i \(0.535473\pi\)
\(90\) 0 0
\(91\) −2.06418 −0.216385
\(92\) 0 0
\(93\) 1.35710 + 3.72859i 0.140724 + 0.386637i
\(94\) 0 0
\(95\) −2.11334 3.66041i −0.216824 0.375551i
\(96\) 0 0
\(97\) −5.94222 + 10.2922i −0.603341 + 1.04502i 0.388970 + 0.921250i \(0.372831\pi\)
−0.992311 + 0.123767i \(0.960503\pi\)
\(98\) 0 0
\(99\) −5.11721 + 4.29385i −0.514299 + 0.431548i
\(100\) 0 0
\(101\) −3.68866 + 6.38895i −0.367036 + 0.635724i −0.989101 0.147241i \(-0.952961\pi\)
0.622065 + 0.782966i \(0.286294\pi\)
\(102\) 0 0
\(103\) −3.19594 5.53553i −0.314905 0.545431i 0.664512 0.747277i \(-0.268639\pi\)
−0.979417 + 0.201846i \(0.935306\pi\)
\(104\) 0 0
\(105\) −0.907604 0.160035i −0.0885731 0.0156178i
\(106\) 0 0
\(107\) 0.0196004 0.00189484 0.000947419 1.00000i \(-0.499698\pi\)
0.000947419 1.00000i \(0.499698\pi\)
\(108\) 0 0
\(109\) 1.24897 0.119630 0.0598148 0.998209i \(-0.480949\pi\)
0.0598148 + 0.998209i \(0.480949\pi\)
\(110\) 0 0
\(111\) −17.5535 3.09516i −1.66610 0.293779i
\(112\) 0 0
\(113\) 6.10607 + 10.5760i 0.574410 + 0.994908i 0.996105 + 0.0881703i \(0.0281020\pi\)
−0.421695 + 0.906738i \(0.638565\pi\)
\(114\) 0 0
\(115\) 1.80928 3.13376i 0.168716 0.292225i
\(116\) 0 0
\(117\) −1.07532 6.09845i −0.0994136 0.563802i
\(118\) 0 0
\(119\) 0.407604 0.705990i 0.0373650 0.0647180i
\(120\) 0 0
\(121\) 3.02094 + 5.23243i 0.274631 + 0.475675i
\(122\) 0 0
\(123\) 1.53802 + 4.22567i 0.138678 + 0.381016i
\(124\) 0 0
\(125\) −5.17024 −0.462441
\(126\) 0 0
\(127\) −1.41921 −0.125935 −0.0629675 0.998016i \(-0.520056\pi\)
−0.0629675 + 0.998016i \(0.520056\pi\)
\(128\) 0 0
\(129\) 0.323826 0.385920i 0.0285113 0.0339784i
\(130\) 0 0
\(131\) −4.61721 7.99724i −0.403408 0.698722i 0.590727 0.806871i \(-0.298841\pi\)
−0.994135 + 0.108149i \(0.965508\pi\)
\(132\) 0 0
\(133\) 3.97178 6.87933i 0.344397 0.596513i
\(134\) 0 0
\(135\) 2.76481i 0.237957i
\(136\) 0 0
\(137\) 0.975185 1.68907i 0.0833157 0.144307i −0.821357 0.570415i \(-0.806782\pi\)
0.904672 + 0.426108i \(0.140116\pi\)
\(138\) 0 0
\(139\) −5.05556 8.75649i −0.428807 0.742715i 0.567961 0.823056i \(-0.307733\pi\)
−0.996767 + 0.0803403i \(0.974399\pi\)
\(140\) 0 0
\(141\) 0.474308 0.565258i 0.0399439 0.0476033i
\(142\) 0 0
\(143\) 4.59627 0.384359
\(144\) 0 0
\(145\) −3.97771 −0.330331
\(146\) 0 0
\(147\) −0.592396 1.62760i −0.0488600 0.134242i
\(148\) 0 0
\(149\) −6.81567 11.8051i −0.558362 0.967111i −0.997633 0.0687567i \(-0.978097\pi\)
0.439272 0.898354i \(-0.355237\pi\)
\(150\) 0 0
\(151\) −4.92262 + 8.52623i −0.400597 + 0.693854i −0.993798 0.111200i \(-0.964531\pi\)
0.593201 + 0.805054i \(0.297864\pi\)
\(152\) 0 0
\(153\) 2.29813 + 0.836452i 0.185793 + 0.0676231i
\(154\) 0 0
\(155\) 0.609470 1.05563i 0.0489538 0.0847905i
\(156\) 0 0
\(157\) −7.58765 13.1422i −0.605560 1.04886i −0.991963 0.126531i \(-0.959616\pi\)
0.386402 0.922330i \(-0.373718\pi\)
\(158\) 0 0
\(159\) 2.40760 + 0.424525i 0.190935 + 0.0336671i
\(160\) 0 0
\(161\) 6.80066 0.535967
\(162\) 0 0
\(163\) 21.5699 1.68948 0.844741 0.535176i \(-0.179755\pi\)
0.844741 + 0.535176i \(0.179755\pi\)
\(164\) 0 0
\(165\) 2.02094 + 0.356347i 0.157330 + 0.0277416i
\(166\) 0 0
\(167\) 9.97565 + 17.2783i 0.771939 + 1.33704i 0.936499 + 0.350671i \(0.114046\pi\)
−0.164560 + 0.986367i \(0.552620\pi\)
\(168\) 0 0
\(169\) 4.36959 7.56834i 0.336122 0.582180i
\(170\) 0 0
\(171\) 22.3935 + 8.15058i 1.71248 + 0.623290i
\(172\) 0 0
\(173\) −0.515015 + 0.892032i −0.0391558 + 0.0678199i −0.884939 0.465707i \(-0.845800\pi\)
0.845783 + 0.533527i \(0.179134\pi\)
\(174\) 0 0
\(175\) −2.35844 4.08494i −0.178281 0.308792i
\(176\) 0 0
\(177\) 2.03256 + 5.58440i 0.152776 + 0.419749i
\(178\) 0 0
\(179\) −26.4962 −1.98042 −0.990209 0.139593i \(-0.955421\pi\)
−0.990209 + 0.139593i \(0.955421\pi\)
\(180\) 0 0
\(181\) −1.35504 −0.100719 −0.0503596 0.998731i \(-0.516037\pi\)
−0.0503596 + 0.998731i \(0.516037\pi\)
\(182\) 0 0
\(183\) 11.6630 13.8994i 0.862152 1.02747i
\(184\) 0 0
\(185\) 2.73783 + 4.74205i 0.201289 + 0.348643i
\(186\) 0 0
\(187\) −0.907604 + 1.57202i −0.0663706 + 0.114957i
\(188\) 0 0
\(189\) 4.50000 2.59808i 0.327327 0.188982i
\(190\) 0 0
\(191\) −3.10354 + 5.37549i −0.224564 + 0.388957i −0.956189 0.292751i \(-0.905429\pi\)
0.731624 + 0.681708i \(0.238763\pi\)
\(192\) 0 0
\(193\) 3.34864 + 5.80002i 0.241040 + 0.417494i 0.961011 0.276510i \(-0.0891780\pi\)
−0.719971 + 0.694005i \(0.755845\pi\)
\(194\) 0 0
\(195\) −1.22281 + 1.45729i −0.0875673 + 0.104359i
\(196\) 0 0
\(197\) −19.0223 −1.35528 −0.677641 0.735393i \(-0.736998\pi\)
−0.677641 + 0.735393i \(0.736998\pi\)
\(198\) 0 0
\(199\) −7.99319 −0.566622 −0.283311 0.959028i \(-0.591433\pi\)
−0.283311 + 0.959028i \(0.591433\pi\)
\(200\) 0 0
\(201\) −2.49067 6.84305i −0.175678 0.482672i
\(202\) 0 0
\(203\) −3.73783 6.47410i −0.262344 0.454393i
\(204\) 0 0
\(205\) 0.690722 1.19637i 0.0482421 0.0835578i
\(206\) 0 0
\(207\) 3.54277 + 20.0920i 0.246239 + 1.39649i
\(208\) 0 0
\(209\) −8.84389 + 15.3181i −0.611745 + 1.05957i
\(210\) 0 0
\(211\) −11.3648 19.6845i −0.782388 1.35514i −0.930547 0.366172i \(-0.880668\pi\)
0.148160 0.988963i \(-0.452665\pi\)
\(212\) 0 0
\(213\) −20.6163 3.63522i −1.41261 0.249081i
\(214\) 0 0
\(215\) −0.154763 −0.0105548
\(216\) 0 0
\(217\) 2.29086 0.155514
\(218\) 0 0
\(219\) −15.5077 2.73443i −1.04792 0.184776i
\(220\) 0 0
\(221\) −0.841367 1.45729i −0.0565964 0.0980279i
\(222\) 0 0
\(223\) −10.2442 + 17.7435i −0.686004 + 1.18819i 0.287116 + 0.957896i \(0.407303\pi\)
−0.973120 + 0.230298i \(0.926030\pi\)
\(224\) 0 0
\(225\) 10.8400 9.09586i 0.722668 0.606391i
\(226\) 0 0
\(227\) 5.81655 10.0746i 0.386058 0.668672i −0.605857 0.795573i \(-0.707170\pi\)
0.991915 + 0.126901i \(0.0405031\pi\)
\(228\) 0 0
\(229\) −1.20233 2.08250i −0.0794524 0.137616i 0.823561 0.567227i \(-0.191984\pi\)
−0.903014 + 0.429612i \(0.858651\pi\)
\(230\) 0 0
\(231\) 1.31908 + 3.62414i 0.0867890 + 0.238451i
\(232\) 0 0
\(233\) −16.6313 −1.08956 −0.544778 0.838580i \(-0.683386\pi\)
−0.544778 + 0.838580i \(0.683386\pi\)
\(234\) 0 0
\(235\) −0.226682 −0.0147871
\(236\) 0 0
\(237\) 8.30675 9.89960i 0.539581 0.643048i
\(238\) 0 0
\(239\) 13.3229 + 23.0760i 0.861790 + 1.49266i 0.870200 + 0.492699i \(0.163990\pi\)
−0.00840979 + 0.999965i \(0.502677\pi\)
\(240\) 0 0
\(241\) 6.27972 10.8768i 0.404512 0.700635i −0.589753 0.807584i \(-0.700775\pi\)
0.994265 + 0.106949i \(0.0341081\pi\)
\(242\) 0 0
\(243\) 10.0201 + 11.9415i 0.642788 + 0.766044i
\(244\) 0 0
\(245\) −0.266044 + 0.460802i −0.0169970 + 0.0294396i
\(246\) 0 0
\(247\) −8.19846 14.2002i −0.521656 0.903534i
\(248\) 0 0
\(249\) −19.5116 + 23.2530i −1.23650 + 1.47360i
\(250\) 0 0
\(251\) −18.7023 −1.18048 −0.590240 0.807228i \(-0.700967\pi\)
−0.590240 + 0.807228i \(0.700967\pi\)
\(252\) 0 0
\(253\) −15.1429 −0.952026
\(254\) 0 0
\(255\) −0.256959 0.705990i −0.0160914 0.0442108i
\(256\) 0 0
\(257\) 6.57398 + 11.3865i 0.410073 + 0.710268i 0.994897 0.100892i \(-0.0321697\pi\)
−0.584824 + 0.811160i \(0.698836\pi\)
\(258\) 0 0
\(259\) −5.14543 + 8.91215i −0.319721 + 0.553774i
\(260\) 0 0
\(261\) 17.1800 14.4158i 1.06342 0.892314i
\(262\) 0 0
\(263\) 11.4474 19.8275i 0.705879 1.22262i −0.260494 0.965475i \(-0.583886\pi\)
0.966373 0.257143i \(-0.0827811\pi\)
\(264\) 0 0
\(265\) −0.375515 0.650411i −0.0230677 0.0399544i
\(266\) 0 0
\(267\) −3.57919 0.631108i −0.219043 0.0386232i
\(268\) 0 0
\(269\) −26.3432 −1.60617 −0.803086 0.595863i \(-0.796810\pi\)
−0.803086 + 0.595863i \(0.796810\pi\)
\(270\) 0 0
\(271\) −10.7246 −0.651474 −0.325737 0.945460i \(-0.605612\pi\)
−0.325737 + 0.945460i \(0.605612\pi\)
\(272\) 0 0
\(273\) −3.52094 0.620838i −0.213097 0.0375748i
\(274\) 0 0
\(275\) 5.25150 + 9.09586i 0.316677 + 0.548501i
\(276\) 0 0
\(277\) 2.49138 4.31520i 0.149693 0.259275i −0.781421 0.624004i \(-0.785505\pi\)
0.931114 + 0.364729i \(0.118838\pi\)
\(278\) 0 0
\(279\) 1.19341 + 6.76817i 0.0714476 + 0.405200i
\(280\) 0 0
\(281\) 4.30928 7.46389i 0.257070 0.445258i −0.708386 0.705826i \(-0.750576\pi\)
0.965456 + 0.260567i \(0.0839096\pi\)
\(282\) 0 0
\(283\) −1.18227 2.04775i −0.0702784 0.121726i 0.828745 0.559627i \(-0.189055\pi\)
−0.899023 + 0.437901i \(0.855722\pi\)
\(284\) 0 0
\(285\) −2.50387 6.87933i −0.148316 0.407496i
\(286\) 0 0
\(287\) 2.59627 0.153253
\(288\) 0 0
\(289\) −16.3354 −0.960908
\(290\) 0 0
\(291\) −13.2314 + 15.7686i −0.775640 + 0.924372i
\(292\) 0 0
\(293\) 8.73783 + 15.1344i 0.510469 + 0.884159i 0.999926 + 0.0121313i \(0.00386162\pi\)
−0.489457 + 0.872027i \(0.662805\pi\)
\(294\) 0 0
\(295\) 0.912818 1.58105i 0.0531463 0.0920522i
\(296\) 0 0
\(297\) −10.0201 + 5.78509i −0.581423 + 0.335685i
\(298\) 0 0
\(299\) 7.01889 12.1571i 0.405913 0.703061i
\(300\) 0 0
\(301\) −0.145430 0.251892i −0.00838243 0.0145188i
\(302\) 0 0
\(303\) −8.21348 + 9.78844i −0.471852 + 0.562331i
\(304\) 0 0
\(305\) −5.57398 −0.319165
\(306\) 0 0
\(307\) −25.6928 −1.46637 −0.733184 0.680030i \(-0.761967\pi\)
−0.733184 + 0.680030i \(0.761967\pi\)
\(308\) 0 0
\(309\) −3.78652 10.4034i −0.215408 0.591828i
\(310\) 0 0
\(311\) −3.91400 6.77925i −0.221943 0.384416i 0.733455 0.679738i \(-0.237906\pi\)
−0.955398 + 0.295322i \(0.904573\pi\)
\(312\) 0 0
\(313\) 9.51620 16.4825i 0.537887 0.931648i −0.461130 0.887332i \(-0.652556\pi\)
0.999018 0.0443156i \(-0.0141107\pi\)
\(314\) 0 0
\(315\) −1.50000 0.545955i −0.0845154 0.0307611i
\(316\) 0 0
\(317\) 15.5890 27.0009i 0.875565 1.51652i 0.0194055 0.999812i \(-0.493823\pi\)
0.856160 0.516711i \(-0.172844\pi\)
\(318\) 0 0
\(319\) 8.32295 + 14.4158i 0.465996 + 0.807128i
\(320\) 0 0
\(321\) 0.0334331 + 0.00589515i 0.00186605 + 0.000329035i
\(322\) 0 0
\(323\) 6.47565 0.360315
\(324\) 0 0
\(325\) −9.73648 −0.540083
\(326\) 0 0
\(327\) 2.13041 + 0.375650i 0.117812 + 0.0207735i
\(328\) 0 0
\(329\) −0.213011 0.368946i −0.0117437 0.0203406i
\(330\) 0 0
\(331\) 13.4427 23.2834i 0.738877 1.27977i −0.214125 0.976806i \(-0.568690\pi\)
0.953001 0.302966i \(-0.0979768\pi\)
\(332\) 0 0
\(333\) −29.0107 10.5590i −1.58978 0.578632i
\(334\) 0 0
\(335\) −1.11856 + 1.93739i −0.0611132 + 0.105851i
\(336\) 0 0
\(337\) 3.08512 + 5.34359i 0.168057 + 0.291084i 0.937737 0.347347i \(-0.112917\pi\)
−0.769679 + 0.638431i \(0.779584\pi\)
\(338\) 0 0
\(339\) 7.23442 + 19.8764i 0.392920 + 1.07954i
\(340\) 0 0
\(341\) −5.10101 −0.276235
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 4.02869 4.80120i 0.216897 0.258488i
\(346\) 0 0
\(347\) −7.06330 12.2340i −0.379178 0.656755i 0.611765 0.791040i \(-0.290460\pi\)
−0.990943 + 0.134284i \(0.957126\pi\)
\(348\) 0 0
\(349\) −12.3628 + 21.4130i −0.661764 + 1.14621i 0.318387 + 0.947961i \(0.396859\pi\)
−0.980152 + 0.198249i \(0.936475\pi\)
\(350\) 0 0
\(351\) 10.7258i 0.572500i
\(352\) 0 0
\(353\) −9.55097 + 16.5428i −0.508347 + 0.880483i 0.491606 + 0.870818i \(0.336410\pi\)
−0.999953 + 0.00966532i \(0.996923\pi\)
\(354\) 0 0
\(355\) 3.21554 + 5.56947i 0.170663 + 0.295597i
\(356\) 0 0
\(357\) 0.907604 1.08164i 0.0480355 0.0572465i
\(358\) 0 0
\(359\) 19.3233 1.01984 0.509921 0.860221i \(-0.329675\pi\)
0.509921 + 0.860221i \(0.329675\pi\)
\(360\) 0 0
\(361\) 44.1002 2.32106
\(362\) 0 0
\(363\) 3.57919 + 9.83375i 0.187859 + 0.516138i
\(364\) 0 0
\(365\) 2.41875 + 4.18939i 0.126603 + 0.219283i
\(366\) 0 0
\(367\) −17.0180 + 29.4761i −0.888333 + 1.53864i −0.0464873 + 0.998919i \(0.514803\pi\)
−0.841845 + 0.539719i \(0.818531\pi\)
\(368\) 0 0
\(369\) 1.35251 + 7.67047i 0.0704089 + 0.399309i
\(370\) 0 0
\(371\) 0.705737 1.22237i 0.0366400 0.0634624i
\(372\) 0 0
\(373\) 10.5765 + 18.3190i 0.547631 + 0.948524i 0.998436 + 0.0559019i \(0.0178034\pi\)
−0.450806 + 0.892622i \(0.648863\pi\)
\(374\) 0 0
\(375\) −8.81908 1.55504i −0.455415 0.0803020i
\(376\) 0 0
\(377\) −15.4311 −0.794741
\(378\) 0 0
\(379\) 11.4757 0.589465 0.294732 0.955580i \(-0.404770\pi\)
0.294732 + 0.955580i \(0.404770\pi\)
\(380\) 0 0
\(381\) −2.42081 0.426854i −0.124022 0.0218684i
\(382\) 0 0
\(383\) −2.41400 4.18117i −0.123350 0.213648i 0.797737 0.603006i \(-0.206030\pi\)
−0.921087 + 0.389358i \(0.872697\pi\)
\(384\) 0 0
\(385\) 0.592396 1.02606i 0.0301913 0.0522929i
\(386\) 0 0
\(387\) 0.668434 0.560882i 0.0339784 0.0285113i
\(388\) 0 0
\(389\) −8.01367 + 13.8801i −0.406309 + 0.703748i −0.994473 0.104994i \(-0.966518\pi\)
0.588164 + 0.808742i \(0.299851\pi\)
\(390\) 0 0
\(391\) 2.77197 + 4.80120i 0.140185 + 0.242807i
\(392\) 0 0
\(393\) −5.47044 15.0299i −0.275947 0.758158i
\(394\) 0 0
\(395\) −3.96997 −0.199751
\(396\) 0 0
\(397\) 35.5827 1.78584 0.892921 0.450213i \(-0.148652\pi\)
0.892921 + 0.450213i \(0.148652\pi\)
\(398\) 0 0
\(399\) 8.84389 10.5397i 0.442748 0.527647i
\(400\) 0 0
\(401\) −14.0792 24.3859i −0.703081 1.21777i −0.967380 0.253332i \(-0.918474\pi\)
0.264298 0.964441i \(-0.414860\pi\)
\(402\) 0 0
\(403\) 2.36437 4.09521i 0.117778 0.203997i
\(404\) 0 0
\(405\) 0.831566 4.71605i 0.0413209 0.234342i
\(406\) 0 0
\(407\) 11.4572 19.8445i 0.567914 0.983656i
\(408\) 0 0
\(409\) 13.3969 + 23.2042i 0.662435 + 1.14737i 0.979974 + 0.199126i \(0.0638104\pi\)
−0.317538 + 0.948245i \(0.602856\pi\)
\(410\) 0 0
\(411\) 2.17143 2.58781i 0.107109 0.127647i
\(412\) 0 0
\(413\) 3.43107 0.168832
\(414\) 0 0
\(415\) 9.32501 0.457747
\(416\) 0 0
\(417\) −5.98979 16.4568i −0.293321 0.805893i
\(418\) 0 0
\(419\) 10.0608 + 17.4258i 0.491501 + 0.851305i 0.999952 0.00978617i \(-0.00311509\pi\)
−0.508451 + 0.861091i \(0.669782\pi\)
\(420\) 0 0
\(421\) 4.52616 7.83954i 0.220591 0.382076i −0.734396 0.678721i \(-0.762535\pi\)
0.954988 + 0.296645i \(0.0958679\pi\)
\(422\) 0 0
\(423\) 0.979055 0.821525i 0.0476033 0.0399439i
\(424\) 0 0
\(425\) 1.92262 3.33007i 0.0932607 0.161532i
\(426\) 0 0
\(427\) −5.23783 9.07218i −0.253476 0.439034i
\(428\) 0 0
\(429\) 7.84002 + 1.38241i 0.378520 + 0.0667433i
\(430\) 0 0
\(431\) 19.1952 0.924601 0.462301 0.886723i \(-0.347024\pi\)
0.462301 + 0.886723i \(0.347024\pi\)
\(432\) 0 0
\(433\) −25.5080 −1.22584 −0.612919 0.790146i \(-0.710005\pi\)
−0.612919 + 0.790146i \(0.710005\pi\)
\(434\) 0 0
\(435\) −6.78493 1.19637i −0.325312 0.0573614i
\(436\) 0 0
\(437\) 27.0107 + 46.7840i 1.29210 + 2.23798i
\(438\) 0 0
\(439\) −9.45471 + 16.3760i −0.451249 + 0.781585i −0.998464 0.0554064i \(-0.982355\pi\)
0.547215 + 0.836992i \(0.315688\pi\)
\(440\) 0 0
\(441\) −0.520945 2.95442i −0.0248069 0.140687i
\(442\) 0 0
\(443\) 10.3131 17.8629i 0.489992 0.848692i −0.509941 0.860209i \(-0.670333\pi\)
0.999934 + 0.0115174i \(0.00366619\pi\)
\(444\) 0 0
\(445\) 0.558248 + 0.966914i 0.0264635 + 0.0458361i
\(446\) 0 0
\(447\) −8.07516 22.1863i −0.381942 1.04938i
\(448\) 0 0
\(449\) 34.3327 1.62026 0.810131 0.586248i \(-0.199396\pi\)
0.810131 + 0.586248i \(0.199396\pi\)
\(450\) 0 0
\(451\) −5.78106 −0.272219
\(452\) 0 0
\(453\) −10.9611 + 13.0629i −0.514998 + 0.613750i
\(454\) 0 0
\(455\) 0.549163 + 0.951178i 0.0257452 + 0.0445919i
\(456\) 0 0
\(457\) 0.922152 1.59721i 0.0431364 0.0747145i −0.843651 0.536892i \(-0.819598\pi\)
0.886788 + 0.462177i \(0.152932\pi\)
\(458\) 0 0
\(459\) 3.66843 + 2.11797i 0.171228 + 0.0988584i
\(460\) 0 0
\(461\) 16.5582 28.6797i 0.771194 1.33575i −0.165714 0.986174i \(-0.552993\pi\)
0.936909 0.349574i \(-0.113674\pi\)
\(462\) 0 0
\(463\) −2.14631 3.71751i −0.0997473 0.172767i 0.811833 0.583890i \(-0.198470\pi\)
−0.911580 + 0.411123i \(0.865137\pi\)
\(464\) 0 0
\(465\) 1.35710 1.61732i 0.0629338 0.0750016i
\(466\) 0 0
\(467\) −31.7347 −1.46851 −0.734254 0.678875i \(-0.762468\pi\)
−0.734254 + 0.678875i \(0.762468\pi\)
\(468\) 0 0
\(469\) −4.20439 −0.194141
\(470\) 0 0
\(471\) −8.98979 24.6992i −0.414228 1.13808i
\(472\) 0 0
\(473\) 0.323826 + 0.560882i 0.0148895 + 0.0257894i
\(474\) 0 0
\(475\) 18.7344 32.4490i 0.859594 1.48886i
\(476\) 0 0
\(477\) 3.97906 + 1.44826i 0.182188 + 0.0663112i
\(478\) 0 0
\(479\) −9.54845 + 16.5384i −0.436280 + 0.755659i −0.997399 0.0720762i \(-0.977038\pi\)
0.561119 + 0.827735i \(0.310371\pi\)
\(480\) 0 0
\(481\) 10.6211 + 18.3963i 0.484280 + 0.838797i
\(482\) 0 0
\(483\) 11.6001 + 2.04542i 0.527825 + 0.0930697i
\(484\) 0 0
\(485\) 6.32358 0.287139
\(486\) 0 0
\(487\) 1.40198 0.0635297 0.0317649 0.999495i \(-0.489887\pi\)
0.0317649 + 0.999495i \(0.489887\pi\)
\(488\) 0 0
\(489\) 36.7925 + 6.48751i 1.66381 + 0.293375i
\(490\) 0 0
\(491\) −10.4586 18.1148i −0.471989 0.817509i 0.527497 0.849557i \(-0.323130\pi\)
−0.999486 + 0.0320477i \(0.989797\pi\)
\(492\) 0 0
\(493\) 3.04710 5.27774i 0.137235 0.237697i
\(494\) 0 0
\(495\) 3.34002 + 1.21567i 0.150123 + 0.0546402i
\(496\) 0 0
\(497\) −6.04323 + 10.4672i −0.271076 + 0.469518i
\(498\) 0 0
\(499\) −11.3105 19.5903i −0.506326 0.876982i −0.999973 0.00731977i \(-0.997670\pi\)
0.493647 0.869662i \(-0.335663\pi\)
\(500\) 0 0
\(501\) 11.8191 + 32.4726i 0.528037 + 1.45077i
\(502\) 0 0
\(503\) 21.2327 0.946718 0.473359 0.880870i \(-0.343041\pi\)
0.473359 + 0.880870i \(0.343041\pi\)
\(504\) 0 0
\(505\) 3.92539 0.174678
\(506\) 0 0
\(507\) 9.72967 11.5954i 0.432110 0.514969i
\(508\) 0 0
\(509\) 2.94697 + 5.10430i 0.130622 + 0.226244i 0.923917 0.382594i \(-0.124969\pi\)
−0.793295 + 0.608838i \(0.791636\pi\)
\(510\) 0 0
\(511\) −4.54576 + 7.87349i −0.201093 + 0.348303i
\(512\) 0 0
\(513\) 35.7460 + 20.6380i 1.57823 + 0.911189i
\(514\) 0 0
\(515\) −1.70052 + 2.94539i −0.0749340 + 0.129789i
\(516\) 0 0
\(517\) 0.474308 + 0.821525i 0.0208600 + 0.0361306i
\(518\) 0 0
\(519\) −1.14677 + 1.36667i −0.0503378 + 0.0599902i
\(520\) 0 0
\(521\) 42.5039 1.86213 0.931065 0.364852i \(-0.118881\pi\)
0.931065 + 0.364852i \(0.118881\pi\)
\(522\) 0 0
\(523\) 23.7716 1.03946 0.519729 0.854331i \(-0.326033\pi\)
0.519729 + 0.854331i \(0.326033\pi\)
\(524\) 0 0
\(525\) −2.79426 7.67717i −0.121952 0.335059i
\(526\) 0 0
\(527\) 0.933763 + 1.61732i 0.0406753 + 0.0704518i
\(528\) 0 0
\(529\) −11.6245 + 20.1342i −0.505412 + 0.875400i
\(530\) 0 0
\(531\) 1.78740 + 10.1368i 0.0775665 + 0.439902i
\(532\) 0 0
\(533\) 2.67958 4.64117i 0.116065 0.201031i
\(534\) 0 0
\(535\) −0.00521457 0.00903189i −0.000225445 0.000390483i
\(536\) 0 0
\(537\) −45.1955 7.96919i −1.95033 0.343896i
\(538\) 0 0
\(539\) 2.22668 0.0959100
\(540\) 0 0
\(541\) 22.6560 0.974058 0.487029 0.873386i \(-0.338081\pi\)
0.487029 + 0.873386i \(0.338081\pi\)
\(542\) 0 0
\(543\) −2.31134 0.407551i −0.0991890 0.0174897i
\(544\) 0 0
\(545\) −0.332282 0.575529i −0.0142334 0.0246529i
\(546\) 0 0
\(547\) −2.13176 + 3.69232i −0.0911474 + 0.157872i −0.907994 0.418983i \(-0.862387\pi\)
0.816847 + 0.576855i \(0.195720\pi\)
\(548\) 0 0
\(549\) 24.0744 20.2009i 1.02747 0.862152i
\(550\) 0 0
\(551\) 29.6917 51.4275i 1.26491 2.19088i
\(552\) 0 0
\(553\) −3.73055 6.46151i −0.158639 0.274771i
\(554\) 0 0
\(555\) 3.24376 + 8.91215i 0.137690 + 0.378300i
\(556\) 0 0
\(557\) −2.90404 −0.123048 −0.0615240 0.998106i \(-0.519596\pi\)
−0.0615240 + 0.998106i \(0.519596\pi\)
\(558\) 0 0
\(559\) −0.600385 −0.0253936
\(560\) 0 0
\(561\) −2.02094 + 2.40847i −0.0853243 + 0.101686i
\(562\) 0 0
\(563\) 5.16978 + 8.95432i 0.217880 + 0.377380i 0.954160 0.299298i \(-0.0967525\pi\)
−0.736280 + 0.676678i \(0.763419\pi\)
\(564\) 0 0
\(565\) 3.24897 5.62738i 0.136685 0.236746i
\(566\) 0 0
\(567\) 8.45723 3.07818i 0.355170 0.129271i
\(568\) 0 0
\(569\) 17.3530 30.0562i 0.727475 1.26002i −0.230473 0.973079i \(-0.574027\pi\)
0.957947 0.286944i \(-0.0926395\pi\)
\(570\) 0 0
\(571\) 21.2135 + 36.7428i 0.887756 + 1.53764i 0.842521 + 0.538663i \(0.181070\pi\)
0.0452350 + 0.998976i \(0.485596\pi\)
\(572\) 0 0
\(573\) −6.91060 + 8.23573i −0.288694 + 0.344052i
\(574\) 0 0
\(575\) 32.0779 1.33774
\(576\) 0 0
\(577\) 23.7811 0.990018 0.495009 0.868888i \(-0.335165\pi\)
0.495009 + 0.868888i \(0.335165\pi\)
\(578\) 0 0
\(579\) 3.96744 + 10.9005i 0.164881 + 0.453008i
\(580\) 0 0
\(581\) 8.76264 + 15.1773i 0.363536 + 0.629662i
\(582\) 0 0
\(583\) −1.57145 + 2.72183i −0.0650829 + 0.112727i
\(584\) 0 0
\(585\) −2.52410 + 2.11797i −0.104359 + 0.0875673i
\(586\) 0 0
\(587\) 6.10560 10.5752i 0.252005 0.436486i −0.712073 0.702106i \(-0.752243\pi\)
0.964078 + 0.265620i \(0.0855767\pi\)
\(588\) 0 0
\(589\) 9.09879 + 15.7596i 0.374909 + 0.649362i
\(590\) 0 0
\(591\) −32.4470 5.72129i −1.33469 0.235342i
\(592\) 0 0
\(593\) −14.2139 −0.583694 −0.291847 0.956465i \(-0.594270\pi\)
−0.291847 + 0.956465i \(0.594270\pi\)
\(594\) 0 0
\(595\) −0.433763 −0.0177825
\(596\) 0 0
\(597\) −13.6343 2.40409i −0.558014 0.0983930i
\(598\) 0 0
\(599\) −9.34730 16.1900i −0.381920 0.661505i 0.609417 0.792850i \(-0.291404\pi\)
−0.991337 + 0.131345i \(0.958070\pi\)
\(600\) 0 0
\(601\) −19.7096 + 34.1380i −0.803972 + 1.39252i 0.113011 + 0.993594i \(0.463950\pi\)
−0.916983 + 0.398926i \(0.869383\pi\)
\(602\) 0 0
\(603\) −2.19026 12.4216i −0.0891941 0.505845i
\(604\) 0 0
\(605\) 1.60741 2.78412i 0.0653506 0.113190i
\(606\) 0 0
\(607\) 15.1682 + 26.2721i 0.615658 + 1.06635i 0.990269 + 0.139168i \(0.0444429\pi\)
−0.374611 + 0.927182i \(0.622224\pi\)
\(608\) 0 0
\(609\) −4.42855 12.1673i −0.179454 0.493045i
\(610\) 0 0
\(611\) −0.879385 −0.0355761
\(612\) 0 0
\(613\) 4.91622 0.198564 0.0992822 0.995059i \(-0.468345\pi\)
0.0992822 + 0.995059i \(0.468345\pi\)
\(614\) 0 0
\(615\) 1.53802 1.83294i 0.0620189 0.0739112i
\(616\) 0 0
\(617\) −15.9440 27.6159i −0.641882 1.11177i −0.985012 0.172485i \(-0.944820\pi\)
0.343130 0.939288i \(-0.388513\pi\)
\(618\) 0 0
\(619\) −0.109470 + 0.189608i −0.00439999 + 0.00762100i −0.868217 0.496185i \(-0.834734\pi\)
0.863817 + 0.503806i \(0.168067\pi\)
\(620\) 0 0
\(621\) 35.3373i 1.41804i
\(622\) 0 0
\(623\) −1.04916 + 1.81720i −0.0420338 + 0.0728047i
\(624\) 0 0
\(625\) −10.4167 18.0422i −0.416668 0.721689i
\(626\) 0 0
\(627\) −19.6925 + 23.4686i −0.786444 + 0.937247i
\(628\) 0 0
\(629\) −8.38919 −0.334499
\(630\) 0 0
\(631\) 22.1685 0.882514 0.441257 0.897381i \(-0.354533\pi\)
0.441257 + 0.897381i \(0.354533\pi\)
\(632\) 0 0
\(633\) −13.4650 36.9947i −0.535185 1.47041i
\(634\) 0 0
\(635\) 0.377574 + 0.653978i 0.0149836 + 0.0259523i
\(636\) 0 0
\(637\) −1.03209 + 1.78763i −0.0408929 + 0.0708285i
\(638\) 0 0
\(639\) −34.0727 12.4014i −1.34789 0.490594i
\(640\) 0 0
\(641\) 15.4996 26.8461i 0.612197 1.06036i −0.378672 0.925531i \(-0.623619\pi\)
0.990869 0.134825i \(-0.0430474\pi\)
\(642\) 0 0
\(643\) 21.6591 + 37.5147i 0.854152 + 1.47943i 0.877430 + 0.479705i \(0.159256\pi\)
−0.0232784 + 0.999729i \(0.507410\pi\)
\(644\) 0 0
\(645\) −0.263985 0.0465477i −0.0103944 0.00183281i
\(646\) 0 0
\(647\) −24.8658 −0.977574 −0.488787 0.872403i \(-0.662561\pi\)
−0.488787 + 0.872403i \(0.662561\pi\)
\(648\) 0 0
\(649\) −7.63991 −0.299893
\(650\) 0 0
\(651\) 3.90760 + 0.689016i 0.153151 + 0.0270047i
\(652\) 0 0
\(653\) −10.8883 18.8591i −0.426092 0.738014i 0.570429 0.821347i \(-0.306777\pi\)
−0.996522 + 0.0833330i \(0.973443\pi\)
\(654\) 0 0
\(655\) −2.45677 + 4.25524i −0.0959938 + 0.166266i
\(656\) 0 0
\(657\) −25.6297 9.32845i −0.999910 0.363937i
\(658\) 0 0
\(659\) −20.6177 + 35.7109i −0.803151 + 1.39110i 0.114382 + 0.993437i \(0.463511\pi\)
−0.917532 + 0.397661i \(0.869822\pi\)
\(660\) 0 0
\(661\) −11.6959 20.2580i −0.454919 0.787943i 0.543764 0.839238i \(-0.316999\pi\)
−0.998684 + 0.0512948i \(0.983665\pi\)
\(662\) 0 0
\(663\) −0.996845 2.73881i −0.0387142 0.106367i
\(664\) 0 0
\(665\) −4.22668 −0.163904
\(666\) 0 0
\(667\) 50.8394 1.96851
\(668\) 0 0
\(669\) −22.8106 + 27.1846i −0.881910 + 1.05102i
\(670\) 0 0
\(671\) 11.6630 + 20.2009i 0.450244 + 0.779845i
\(672\) 0 0
\(673\) 2.22621 3.85592i 0.0858143 0.148635i −0.819924 0.572473i \(-0.805984\pi\)
0.905738 + 0.423838i \(0.139317\pi\)
\(674\) 0 0
\(675\) 21.2260 12.2548i 0.816988 0.471688i
\(676\) 0 0
\(677\) 21.6172 37.4421i 0.830817 1.43902i −0.0665744 0.997781i \(-0.521207\pi\)
0.897391 0.441236i \(-0.145460\pi\)
\(678\) 0 0
\(679\) 5.94222 + 10.2922i 0.228041 + 0.394979i
\(680\) 0 0
\(681\) 12.9516 15.4351i 0.496307 0.591475i
\(682\) 0 0
\(683\) −1.79797 −0.0687975 −0.0343987 0.999408i \(-0.510952\pi\)
−0.0343987 + 0.999408i \(0.510952\pi\)
\(684\) 0 0
\(685\) −1.03777 −0.0396512
\(686\) 0 0
\(687\) −1.42452 3.91382i −0.0543487 0.149322i
\(688\) 0 0
\(689\) −1.45677 2.52319i −0.0554984 0.0961260i
\(690\) 0 0
\(691\) −2.78833 + 4.82953i −0.106073 + 0.183724i −0.914176 0.405317i \(-0.867161\pi\)
0.808103 + 0.589041i \(0.200494\pi\)
\(692\) 0 0
\(693\) 1.15998 + 6.57856i 0.0440639 + 0.249899i
\(694\) 0 0
\(695\) −2.69001 + 4.65923i −0.102038 + 0.176735i
\(696\) 0 0
\(697\) 1.05825 + 1.83294i 0.0400840 + 0.0694275i
\(698\) 0 0
\(699\) −28.3687 5.00217i −1.07300 0.189199i
\(700\) 0 0
\(701\) −11.3737 −0.429579 −0.214789 0.976660i \(-0.568907\pi\)
−0.214789 + 0.976660i \(0.568907\pi\)
\(702\) 0 0
\(703\) −81.7461 −3.08311
\(704\) 0 0
\(705\) −0.386659 0.0681784i −0.0145624 0.00256775i
\(706\) 0 0
\(707\) 3.68866 + 6.38895i 0.138726 + 0.240281i
\(708\) 0 0
\(709\) −23.2053 + 40.1928i −0.871494 + 1.50947i −0.0110435 + 0.999939i \(0.503515\pi\)
−0.860451 + 0.509533i \(0.829818\pi\)
\(710\) 0 0
\(711\) 17.1466 14.3877i 0.643048 0.539581i
\(712\) 0 0
\(713\) −7.78968 + 13.4921i −0.291726 + 0.505284i
\(714\) 0 0
\(715\) −1.22281 2.11797i −0.0457305 0.0792076i
\(716\) 0 0
\(717\) 15.7849 + 43.3687i 0.589499 + 1.61964i
\(718\) 0 0
\(719\) −29.6878 −1.10717 −0.553584 0.832793i \(-0.686740\pi\)
−0.553584 + 0.832793i \(0.686740\pi\)
\(720\) 0 0
\(721\) −6.39187 −0.238046
\(722\) 0 0
\(723\) 13.9829 16.6642i 0.520031 0.619748i
\(724\) 0 0
\(725\) −17.6309 30.5376i −0.654795 1.13414i
\(726\) 0 0
\(727\) 14.7904 25.6177i 0.548545 0.950108i −0.449829 0.893115i \(-0.648515\pi\)
0.998375 0.0569937i \(-0.0181515\pi\)
\(728\) 0 0
\(729\) 13.5000 + 23.3827i 0.500000 + 0.866025i
\(730\) 0 0
\(731\) 0.118555 0.205344i 0.00438493 0.00759492i
\(732\) 0 0
\(733\) 2.31790 + 4.01471i 0.0856134 + 0.148287i 0.905652 0.424021i \(-0.139382\pi\)
−0.820039 + 0.572308i \(0.806048\pi\)
\(734\) 0 0
\(735\) −0.592396 + 0.705990i −0.0218509 + 0.0260408i
\(736\) 0 0
\(737\) 9.36184 0.344848
\(738\) 0 0
\(739\) 31.3492 1.15320 0.576599 0.817027i \(-0.304380\pi\)
0.576599 + 0.817027i \(0.304380\pi\)
\(740\) 0 0
\(741\) −9.71348 26.6876i −0.356834 0.980392i
\(742\) 0 0
\(743\) 19.9183 + 34.4996i 0.730733 + 1.26567i 0.956570 + 0.291502i \(0.0941549\pi\)
−0.225837 + 0.974165i \(0.572512\pi\)
\(744\) 0 0
\(745\) −3.62654 + 6.28136i −0.132866 + 0.230131i
\(746\) 0 0
\(747\) −40.2754 + 33.7951i −1.47360 + 1.23650i
\(748\) 0 0
\(749\) 0.00980018 0.0169744i 0.000358091 0.000620231i
\(750\) 0 0
\(751\) 4.20115 + 7.27661i 0.153302 + 0.265527i 0.932440 0.361326i \(-0.117676\pi\)
−0.779137 + 0.626853i \(0.784343\pi\)
\(752\) 0 0
\(753\) −31.9013 5.62505i −1.16255 0.204988i
\(754\) 0 0
\(755\) 5.23854 0.190650
\(756\) 0 0
\(757\) −34.9299 −1.26955 −0.634775 0.772697i \(-0.718907\pi\)
−0.634775 + 0.772697i \(0.718907\pi\)
\(758\) 0 0
\(759\) −25.8298 4.55449i −0.937563 0.165318i
\(760\) 0 0
\(761\) −6.01279 10.4145i −0.217964 0.377524i 0.736222 0.676740i \(-0.236608\pi\)
−0.954185 + 0.299216i \(0.903275\pi\)
\(762\) 0 0
\(763\) 0.624485 1.08164i 0.0226079 0.0391580i
\(764\) 0 0
\(765\) −0.225966 1.28152i −0.00816983 0.0463334i
\(766\) 0 0
\(767\) 3.54117 6.13349i 0.127864 0.221468i
\(768\) 0 0
\(769\) −26.1695 45.3270i −0.943697 1.63453i −0.758339 0.651861i \(-0.773989\pi\)
−0.185359 0.982671i \(-0.559345\pi\)
\(770\) 0 0
\(771\) 7.78880 + 21.3996i 0.280507 + 0.770686i
\(772\) 0 0
\(773\) 38.2181 1.37461 0.687305 0.726369i \(-0.258794\pi\)
0.687305 + 0.726369i \(0.258794\pi\)
\(774\) 0 0
\(775\) 10.8057 0.388153
\(776\) 0 0
\(777\) −11.4572 + 13.6542i −0.411026 + 0.489842i
\(778\) 0 0
\(779\) 10.3118 + 17.8606i 0.369459 + 0.639921i
\(780\) 0 0
\(781\) 13.4564 23.3071i 0.481506 0.833993i
\(782\) 0 0
\(783\) 33.6404 19.4223i 1.20221 0.694097i
\(784\) 0 0
\(785\) −4.03730 + 6.99281i −0.144098 + 0.249584i
\(786\) 0 0
\(787\) 19.4809 + 33.7419i 0.694418 + 1.20277i 0.970377 + 0.241598i \(0.0776714\pi\)
−0.275959 + 0.961170i \(0.588995\pi\)
\(788\) 0 0
\(789\) 25.4898 30.3775i 0.907461 1.08147i
\(790\) 0 0
\(791\) 12.2121 0.434213
\(792\) 0 0
\(793\) −21.6236 −0.767877
\(794\) 0 0
\(795\) −0.444907 1.22237i −0.0157792 0.0433531i
\(796\) 0 0
\(797\) 12.4863 + 21.6270i 0.442288 + 0.766066i 0.997859 0.0654034i \(-0.0208334\pi\)
−0.555570 + 0.831469i \(0.687500\pi\)
\(798\) 0 0
\(799\) 0.173648 0.300767i 0.00614323 0.0106404i
\(800\) 0 0
\(801\) −5.91534 2.15301i −0.209008 0.0760728i
\(802\) 0 0
\(803\) 10.1220 17.5317i 0.357196 0.618682i
\(804\) 0 0
\(805\) −1.80928 3.13376i −0.0637687 0.110451i
\(806\) 0 0
\(807\) −44.9345 7.92317i −1.58177 0.278909i
\(808\) 0 0
\(809\) −6.21450 −0.218490 −0.109245 0.994015i \(-0.534843\pi\)
−0.109245 + 0.994015i \(0.534843\pi\)
\(810\) 0 0
\(811\) −29.5689 −1.03831 −0.519153 0.854682i \(-0.673752\pi\)
−0.519153 + 0.854682i \(0.673752\pi\)
\(812\) 0 0
\(813\) −18.2934 3.22562i −0.641577 0.113127i
\(814\) 0 0
\(815\) −5.73854 9.93944i −0.201012 0.348164i
\(816\) 0 0
\(817\) 1.15523 2.00092i 0.0404164 0.0700032i
\(818\) 0 0
\(819\) −5.81908 2.11797i −0.203335 0.0740079i
\(820\) 0 0
\(821\) 9.10788 15.7753i 0.317867 0.550562i −0.662176 0.749349i \(-0.730367\pi\)
0.980043 + 0.198787i \(0.0637001\pi\)
\(822\) 0 0
\(823\) 12.7939 + 22.1596i 0.445966 + 0.772435i 0.998119 0.0613074i \(-0.0195270\pi\)
−0.552153 + 0.833743i \(0.686194\pi\)
\(824\) 0 0
\(825\) 6.22193 + 17.0946i 0.216620 + 0.595158i
\(826\) 0 0
\(827\) −32.3432 −1.12468 −0.562341 0.826905i \(-0.690099\pi\)
−0.562341 + 0.826905i \(0.690099\pi\)
\(828\) 0 0
\(829\) 13.7314 0.476912 0.238456 0.971153i \(-0.423359\pi\)
0.238456 + 0.971153i \(0.423359\pi\)
\(830\) 0 0
\(831\) 5.54751 6.61127i 0.192441 0.229342i
\(832\) 0 0
\(833\) −0.407604 0.705990i −0.0141226 0.0244611i
\(834\) 0 0
\(835\) 5.30793 9.19361i 0.183689 0.318158i
\(836\) 0 0
\(837\) 11.9037i 0.411450i
\(838\) 0 0
\(839\) −18.5077 + 32.0563i −0.638958 + 1.10671i 0.346704 + 0.937975i \(0.387301\pi\)
−0.985662 + 0.168733i \(0.946032\pi\)
\(840\) 0 0
\(841\) −13.4427 23.2834i −0.463541 0.802876i
\(842\) 0 0
\(843\) 9.59539 11.4353i 0.330483 0.393854i
\(844\) 0 0
\(845\) −4.65002 −0.159965
\(846\) 0 0
\(847\) 6.04189 0.207602
\(848\) 0 0
\(849\) −1.40074 3.84850i −0.0480733 0.132080i
\(850\) 0 0
\(851\) −34.9923 60.6085i −1.19952 2.07763i
\(852\) 0 0
\(853\) −11.4893 + 19.9001i −0.393387 + 0.681366i −0.992894 0.119003i \(-0.962030\pi\)
0.599507 + 0.800370i \(0.295363\pi\)
\(854\) 0 0
\(855\) −2.20187 12.4874i −0.0753023 0.427060i
\(856\) 0 0
\(857\) 2.22328 3.85083i 0.0759457 0.131542i −0.825551 0.564327i \(-0.809136\pi\)
0.901497 + 0.432785i \(0.142469\pi\)
\(858\) 0 0
\(859\) −11.5196 19.9525i −0.393044 0.680772i 0.599806 0.800146i \(-0.295245\pi\)
−0.992849 + 0.119374i \(0.961911\pi\)
\(860\) 0 0
\(861\) 4.42855 + 0.780873i 0.150925 + 0.0266121i
\(862\) 0 0
\(863\) −21.2395 −0.723000 −0.361500 0.932372i \(-0.617735\pi\)
−0.361500 + 0.932372i \(0.617735\pi\)
\(864\) 0 0
\(865\) 0.548067 0.0186348
\(866\) 0 0
\(867\) −27.8640 4.91317i −0.946310 0.166860i
\(868\) 0 0
\(869\) 8.30675 + 14.3877i 0.281787 + 0.488070i
\(870\) 0 0
\(871\) −4.33931 + 7.51590i −0.147032 + 0.254667i
\(872\) 0 0
\(873\) −27.3120 + 22.9175i −0.924372 + 0.775640i
\(874\) 0 0
\(875\) −2.58512 + 4.47756i −0.0873931 + 0.151369i
\(876\) 0 0
\(877\) 23.2237 + 40.2247i 0.784210 + 1.35829i 0.929470 + 0.368898i \(0.120265\pi\)
−0.145260 + 0.989394i \(0.546402\pi\)
\(878\) 0 0
\(879\) 10.3525 + 28.4433i 0.349182 + 0.959368i
\(880\) 0 0
\(881\) 18.3301 0.617555 0.308778 0.951134i \(-0.400080\pi\)
0.308778 + 0.951134i \(0.400080\pi\)
\(882\) 0 0
\(883\) −14.4894 −0.487606 −0.243803 0.969825i \(-0.578395\pi\)
−0.243803 + 0.969825i \(0.578395\pi\)
\(884\) 0 0
\(885\) 2.03256 2.42231i 0.0683236 0.0814249i
\(886\) 0 0
\(887\) 7.49360 + 12.9793i 0.251611 + 0.435802i 0.963969 0.266013i \(-0.0857065\pi\)
−0.712359 + 0.701815i \(0.752373\pi\)
\(888\) 0 0
\(889\) −0.709607 + 1.22908i −0.0237995 + 0.0412219i
\(890\) 0 0
\(891\) −18.8316 + 6.85413i −0.630881 + 0.229622i
\(892\) 0 0
\(893\) 1.69207 2.93075i 0.0566228 0.0980736i
\(894\) 0 0
\(895\) 7.04916 + 12.2095i 0.235628 + 0.408119i
\(896\) 0 0
\(897\) 15.6288 18.6257i 0.521831 0.621894i
\(898\) 0 0
\(899\) 17.1257 0.571173
\(900\) 0 0
\(901\) 1.15064 0.0383335
\(902\) 0 0
\(903\) −0.172304 0.473401i −0.00573392 0.0157538i
\(904\) 0 0
\(905\) 0.360500 + 0.624404i 0.0119834 + 0.0207559i
\(906\) 0 0
\(907\) 13.0790 22.6535i 0.434282 0.752199i −0.562955 0.826488i \(-0.690335\pi\)
0.997237 + 0.0742892i \(0.0236688\pi\)
\(908\) 0 0
\(909\) −16.9541 + 14.2262i −0.562331 + 0.471852i
\(910\) 0 0
\(911\) 13.0432 22.5915i 0.432142 0.748491i −0.564916 0.825148i \(-0.691091\pi\)
0.997058 + 0.0766573i \(0.0244247\pi\)
\(912\) 0 0
\(913\) −19.5116 33.7951i −0.645740 1.11845i
\(914\) 0 0
\(915\) −9.50774 1.67647i −0.314316 0.0554224i
\(916\) 0 0
\(917\) −9.23442 −0.304947
\(918\) 0 0
\(919\) 0.709141 0.0233924 0.0116962 0.999932i \(-0.496277\pi\)
0.0116962 + 0.999932i \(0.496277\pi\)
\(920\) 0 0
\(921\) −43.8252 7.72757i −1.44409 0.254632i
\(922\) 0 0
\(923\) 12.4743 + 21.6061i 0.410597 + 0.711175i
\(924\) 0 0
\(925\) −24.2704 + 42.0375i −0.798005 + 1.38219i
\(926\) 0 0
\(927\) −3.32981 18.8843i −0.109365 0.620242i
\(928\) 0 0
\(929\) 20.0929 34.8019i 0.659225 1.14181i −0.321591 0.946879i \(-0.604218\pi\)
0.980816 0.194933i \(-0.0624490\pi\)
\(930\) 0 0
\(931\) −3.97178 6.87933i −0.130170 0.225461i
\(932\) 0 0
\(933\) −4.63728 12.7408i −0.151818 0.417116i
\(934\) 0 0
\(935\) 0.965852 0.0315867
\(936\) 0 0
\(937\) 29.5389 0.964994 0.482497 0.875898i \(-0.339730\pi\)
0.482497 + 0.875898i \(0.339730\pi\)
\(938\) 0 0
\(939\) 21.1895 25.2527i 0.691495 0.824091i
\(940\) 0 0
\(941\) 0.512326 + 0.887375i 0.0167014 + 0.0289276i 0.874255 0.485466i \(-0.161350\pi\)
−0.857554 + 0.514394i \(0.828017\pi\)
\(942\) 0 0
\(943\) −8.82816 + 15.2908i −0.287485 + 0.497938i
\(944\) 0 0
\(945\) −2.39440 1.38241i −0.0778898 0.0449697i
\(946\) 0 0
\(947\) −29.1891 + 50.5571i −0.948519 + 1.64288i −0.199973 + 0.979801i \(0.564085\pi\)
−0.748547 + 0.663082i \(0.769248\pi\)
\(948\) 0 0
\(949\) 9.38326 + 16.2523i 0.304593 + 0.527571i
\(950\) 0 0
\(951\) 34.7117 41.3678i 1.12560 1.34144i
\(952\) 0 0
\(953\) −33.7743 −1.09406 −0.547028 0.837115i \(-0.684241\pi\)
−0.547028 + 0.837115i \(0.684241\pi\)
\(954\) 0 0
\(955\) 3.30272 0.106873
\(956\) 0 0
\(957\) 9.86097 + 27.0928i 0.318760 + 0.875785i
\(958\) 0 0
\(959\) −0.975185 1.68907i −0.0314904 0.0545429i
\(960\) 0 0
\(961\) 12.8760 22.3019i 0.415354 0.719415i
\(962\) 0 0
\(963\) 0.0552549 + 0.0201112i 0.00178057 + 0.000648073i
\(964\) 0 0
\(965\) 1.78177 3.08612i 0.0573573 0.0993458i
\(966\) 0 0
\(967\) 6.28699 + 10.8894i 0.202176 + 0.350179i 0.949229 0.314585i \(-0.101865\pi\)
−0.747053 + 0.664764i \(0.768532\pi\)
\(968\) 0 0
\(969\) 11.0458 + 1.94767i 0.354841 + 0.0625680i
\(970\) 0 0
\(971\) 2.92808 0.0939666 0.0469833 0.998896i \(-0.485039\pi\)
0.0469833 + 0.998896i \(0.485039\pi\)
\(972\) 0 0
\(973\) −10.1111 −0.324148
\(974\) 0 0
\(975\) −16.6079 2.92842i −0.531878 0.0937844i
\(976\) 0 0
\(977\) −20.3425 35.2343i −0.650816 1.12725i −0.982925 0.184005i \(-0.941094\pi\)
0.332109 0.943241i \(-0.392240\pi\)
\(978\) 0 0
\(979\) 2.33615 4.04633i 0.0746637 0.129321i
\(980\) 0 0
\(981\) 3.52094 + 1.28152i 0.112415 + 0.0409158i
\(982\) 0 0
\(983\) −2.78446 + 4.82283i −0.0888106 + 0.153824i −0.907009 0.421112i \(-0.861640\pi\)
0.818198 + 0.574936i \(0.194973\pi\)
\(984\) 0 0
\(985\) 5.06077 + 8.76552i 0.161250 + 0.279293i
\(986\) 0 0
\(987\) −0.252374 0.693392i −0.00803315 0.0220709i
\(988\) 0 0
\(989\) 1.97804 0.0628979
\(990\) 0 0
\(991\) 31.1307 0.988900 0.494450 0.869206i \(-0.335370\pi\)
0.494450 + 0.869206i \(0.335370\pi\)
\(992\) 0 0
\(993\) 29.9326 35.6723i 0.949882 1.13202i
\(994\) 0 0
\(995\) 2.12654 + 3.68328i 0.0674160 + 0.116768i
\(996\) 0 0
\(997\) −14.2317 + 24.6501i −0.450724 + 0.780676i −0.998431 0.0559938i \(-0.982167\pi\)
0.547708 + 0.836670i \(0.315501\pi\)
\(998\) 0 0
\(999\) −46.3089 26.7364i −1.46515 0.845903i
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 1008.2.r.i.673.3 6
3.2 odd 2 3024.2.r.h.2017.3 6
4.3 odd 2 504.2.r.c.169.1 6
9.2 odd 6 9072.2.a.cc.1.1 3
9.4 even 3 inner 1008.2.r.i.337.3 6
9.5 odd 6 3024.2.r.h.1009.3 6
9.7 even 3 9072.2.a.br.1.3 3
12.11 even 2 1512.2.r.c.505.3 6
36.7 odd 6 4536.2.a.s.1.3 3
36.11 even 6 4536.2.a.v.1.1 3
36.23 even 6 1512.2.r.c.1009.3 6
36.31 odd 6 504.2.r.c.337.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.1 6 4.3 odd 2
504.2.r.c.337.1 yes 6 36.31 odd 6
1008.2.r.i.337.3 6 9.4 even 3 inner
1008.2.r.i.673.3 6 1.1 even 1 trivial
1512.2.r.c.505.3 6 12.11 even 2
1512.2.r.c.1009.3 6 36.23 even 6
3024.2.r.h.1009.3 6 9.5 odd 6
3024.2.r.h.2017.3 6 3.2 odd 2
4536.2.a.s.1.3 3 36.7 odd 6
4536.2.a.v.1.1 3 36.11 even 6
9072.2.a.br.1.3 3 9.7 even 3
9072.2.a.cc.1.1 3 9.2 odd 6