Properties

Label 1300.2.bs.a.293.1
Level $1300$
Weight $2$
Character 1300.293
Analytic conductor $10.381$
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] = [1300,2,Mod(193,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bs (of order \(12\), degree \(4\), 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 293.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1300.293
Dual form 1300.2.bs.a.457.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.86603 + 0.500000i) q^{3} +(-3.23205 + 1.86603i) q^{7} +(0.633975 - 0.366025i) q^{9} +O(q^{10})\) \(q+(-1.86603 + 0.500000i) q^{3} +(-3.23205 + 1.86603i) q^{7} +(0.633975 - 0.366025i) q^{9} +(-0.598076 - 2.23205i) q^{11} +(-3.00000 - 2.00000i) q^{13} +(1.13397 - 4.23205i) q^{17} +(0.866025 + 0.232051i) q^{19} +(5.09808 - 5.09808i) q^{21} +(1.86603 + 6.96410i) q^{23} +(3.09808 - 3.09808i) q^{27} +(-7.96410 - 4.59808i) q^{29} +(5.73205 + 5.73205i) q^{31} +(2.23205 + 3.86603i) q^{33} +(-0.232051 - 0.133975i) q^{37} +(6.59808 + 2.23205i) q^{39} +(-0.133975 + 0.0358984i) q^{41} +(11.3301 + 3.03590i) q^{43} -0.535898i q^{47} +(3.46410 - 6.00000i) q^{49} +8.46410i q^{51} +(1.53590 + 1.53590i) q^{53} -1.73205 q^{57} +(-1.79423 + 6.69615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-1.36603 + 2.36603i) q^{63} +(2.76795 - 4.79423i) q^{67} +(-6.96410 - 12.0622i) q^{69} +(1.13397 - 4.23205i) q^{71} +12.9282 q^{73} +(6.09808 + 6.09808i) q^{77} -4.53590i q^{79} +(-5.33013 + 9.23205i) q^{81} -3.46410i q^{83} +(17.1603 + 4.59808i) q^{87} +(14.7942 - 3.96410i) q^{89} +(13.4282 + 0.866025i) q^{91} +(-13.5622 - 7.83013i) q^{93} +(2.23205 + 3.86603i) q^{97} +(-1.19615 - 1.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9} + 8 q^{11} - 12 q^{13} + 8 q^{17} + 10 q^{21} + 4 q^{23} + 2 q^{27} - 18 q^{29} + 16 q^{31} + 2 q^{33} + 6 q^{37} + 16 q^{39} - 4 q^{41} + 28 q^{43} + 20 q^{53} + 24 q^{59} + 2 q^{61} - 2 q^{63} + 18 q^{67} - 14 q^{69} + 8 q^{71} + 24 q^{73} + 14 q^{77} - 4 q^{81} + 34 q^{87} + 28 q^{89} + 26 q^{91} - 30 q^{93} + 2 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −1.86603 + 0.500000i −1.07735 + 0.288675i −0.753510 0.657437i \(-0.771641\pi\)
−0.323840 + 0.946112i \(0.604974\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.23205 + 1.86603i −1.22160 + 0.705291i −0.965259 0.261295i \(-0.915850\pi\)
−0.256341 + 0.966586i \(0.582517\pi\)
\(8\) 0 0
\(9\) 0.633975 0.366025i 0.211325 0.122008i
\(10\) 0 0
\(11\) −0.598076 2.23205i −0.180327 0.672989i −0.995583 0.0938879i \(-0.970070\pi\)
0.815256 0.579101i \(-0.196596\pi\)
\(12\) 0 0
\(13\) −3.00000 2.00000i −0.832050 0.554700i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.13397 4.23205i 0.275029 1.02642i −0.680793 0.732476i \(-0.738364\pi\)
0.955822 0.293947i \(-0.0949689\pi\)
\(18\) 0 0
\(19\) 0.866025 + 0.232051i 0.198680 + 0.0532361i 0.356787 0.934186i \(-0.383872\pi\)
−0.158107 + 0.987422i \(0.550539\pi\)
\(20\) 0 0
\(21\) 5.09808 5.09808i 1.11249 1.11249i
\(22\) 0 0
\(23\) 1.86603 + 6.96410i 0.389093 + 1.45212i 0.831612 + 0.555357i \(0.187418\pi\)
−0.442519 + 0.896759i \(0.645915\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.09808 3.09808i 0.596225 0.596225i
\(28\) 0 0
\(29\) −7.96410 4.59808i −1.47890 0.853841i −0.479182 0.877716i \(-0.659066\pi\)
−0.999715 + 0.0238745i \(0.992400\pi\)
\(30\) 0 0
\(31\) 5.73205 + 5.73205i 1.02951 + 1.02951i 0.999551 + 0.0299555i \(0.00953655\pi\)
0.0299555 + 0.999551i \(0.490463\pi\)
\(32\) 0 0
\(33\) 2.23205 + 3.86603i 0.388550 + 0.672989i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.232051 0.133975i −0.0381489 0.0220253i 0.480804 0.876828i \(-0.340345\pi\)
−0.518953 + 0.854803i \(0.673678\pi\)
\(38\) 0 0
\(39\) 6.59808 + 2.23205i 1.05654 + 0.357414i
\(40\) 0 0
\(41\) −0.133975 + 0.0358984i −0.0209233 + 0.00560639i −0.269266 0.963066i \(-0.586781\pi\)
0.248342 + 0.968672i \(0.420114\pi\)
\(42\) 0 0
\(43\) 11.3301 + 3.03590i 1.72783 + 0.462970i 0.979681 0.200563i \(-0.0642770\pi\)
0.748147 + 0.663533i \(0.230944\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.535898i 0.0781688i −0.999236 0.0390844i \(-0.987556\pi\)
0.999236 0.0390844i \(-0.0124441\pi\)
\(48\) 0 0
\(49\) 3.46410 6.00000i 0.494872 0.857143i
\(50\) 0 0
\(51\) 8.46410i 1.18521i
\(52\) 0 0
\(53\) 1.53590 + 1.53590i 0.210972 + 0.210972i 0.804680 0.593708i \(-0.202337\pi\)
−0.593708 + 0.804680i \(0.702337\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.73205 −0.229416
\(58\) 0 0
\(59\) −1.79423 + 6.69615i −0.233589 + 0.871765i 0.745191 + 0.666851i \(0.232358\pi\)
−0.978780 + 0.204914i \(0.934309\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) −1.36603 + 2.36603i −0.172103 + 0.298091i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.76795 4.79423i 0.338159 0.585708i −0.645928 0.763399i \(-0.723529\pi\)
0.984086 + 0.177690i \(0.0568626\pi\)
\(68\) 0 0
\(69\) −6.96410 12.0622i −0.838379 1.45212i
\(70\) 0 0
\(71\) 1.13397 4.23205i 0.134578 0.502252i −0.865421 0.501045i \(-0.832949\pi\)
0.999999 0.00120705i \(-0.000384217\pi\)
\(72\) 0 0
\(73\) 12.9282 1.51313 0.756566 0.653917i \(-0.226876\pi\)
0.756566 + 0.653917i \(0.226876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.09808 + 6.09808i 0.694940 + 0.694940i
\(78\) 0 0
\(79\) 4.53590i 0.510328i −0.966898 0.255164i \(-0.917870\pi\)
0.966898 0.255164i \(-0.0821295\pi\)
\(80\) 0 0
\(81\) −5.33013 + 9.23205i −0.592236 + 1.02578i
\(82\) 0 0
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17.1603 + 4.59808i 1.83977 + 0.492966i
\(88\) 0 0
\(89\) 14.7942 3.96410i 1.56819 0.420194i 0.632942 0.774199i \(-0.281847\pi\)
0.935243 + 0.354005i \(0.115181\pi\)
\(90\) 0 0
\(91\) 13.4282 + 0.866025i 1.40766 + 0.0907841i
\(92\) 0 0
\(93\) −13.5622 7.83013i −1.40633 0.811946i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.23205 + 3.86603i 0.226630 + 0.392535i 0.956807 0.290723i \(-0.0938957\pi\)
−0.730177 + 0.683258i \(0.760562\pi\)
\(98\) 0 0
\(99\) −1.19615 1.19615i −0.120218 0.120218i
\(100\) 0 0
\(101\) 1.50000 + 0.866025i 0.149256 + 0.0861727i 0.572768 0.819718i \(-0.305870\pi\)
−0.423512 + 0.905890i \(0.639203\pi\)
\(102\) 0 0
\(103\) −6.66025 + 6.66025i −0.656254 + 0.656254i −0.954492 0.298237i \(-0.903601\pi\)
0.298237 + 0.954492i \(0.403601\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.59808 5.96410i −0.154492 0.576571i −0.999148 0.0412627i \(-0.986862\pi\)
0.844656 0.535309i \(-0.179805\pi\)
\(108\) 0 0
\(109\) 9.39230 9.39230i 0.899620 0.899620i −0.0957826 0.995402i \(-0.530535\pi\)
0.995402 + 0.0957826i \(0.0305354\pi\)
\(110\) 0 0
\(111\) 0.500000 + 0.133975i 0.0474579 + 0.0127163i
\(112\) 0 0
\(113\) 3.52628 13.1603i 0.331724 1.23801i −0.575652 0.817695i \(-0.695252\pi\)
0.907377 0.420318i \(-0.138081\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.63397 0.169873i −0.243511 0.0157048i
\(118\) 0 0
\(119\) 4.23205 + 15.7942i 0.387951 + 1.44785i
\(120\) 0 0
\(121\) 4.90192 2.83013i 0.445629 0.257284i
\(122\) 0 0
\(123\) 0.232051 0.133975i 0.0209233 0.0120801i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.7942 + 4.50000i −1.49025 + 0.399310i −0.909820 0.415002i \(-0.863781\pi\)
−0.580426 + 0.814313i \(0.697114\pi\)
\(128\) 0 0
\(129\) −22.6603 −1.99512
\(130\) 0 0
\(131\) 13.8564 1.21064 0.605320 0.795982i \(-0.293045\pi\)
0.605320 + 0.795982i \(0.293045\pi\)
\(132\) 0 0
\(133\) −3.23205 + 0.866025i −0.280254 + 0.0750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.696152 0.401924i 0.0594763 0.0343387i −0.469967 0.882684i \(-0.655734\pi\)
0.529443 + 0.848345i \(0.322401\pi\)
\(138\) 0 0
\(139\) −8.42820 + 4.86603i −0.714871 + 0.412731i −0.812862 0.582456i \(-0.802092\pi\)
0.0979911 + 0.995187i \(0.468758\pi\)
\(140\) 0 0
\(141\) 0.267949 + 1.00000i 0.0225654 + 0.0842152i
\(142\) 0 0
\(143\) −2.66987 + 7.89230i −0.223266 + 0.659988i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.46410 + 12.9282i −0.285714 + 1.06630i
\(148\) 0 0
\(149\) 18.2583 + 4.89230i 1.49578 + 0.400793i 0.911684 0.410891i \(-0.134782\pi\)
0.584096 + 0.811684i \(0.301449\pi\)
\(150\) 0 0
\(151\) 0.803848 0.803848i 0.0654162 0.0654162i −0.673642 0.739058i \(-0.735271\pi\)
0.739058 + 0.673642i \(0.235271\pi\)
\(152\) 0 0
\(153\) −0.830127 3.09808i −0.0671118 0.250465i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.39230 7.39230i 0.589970 0.589970i −0.347653 0.937623i \(-0.613021\pi\)
0.937623 + 0.347653i \(0.113021\pi\)
\(158\) 0 0
\(159\) −3.63397 2.09808i −0.288193 0.166388i
\(160\) 0 0
\(161\) −19.0263 19.0263i −1.49948 1.49948i
\(162\) 0 0
\(163\) −0.767949 1.33013i −0.0601504 0.104184i 0.834382 0.551186i \(-0.185825\pi\)
−0.894533 + 0.447003i \(0.852491\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.76795 + 3.33013i 0.446337 + 0.257693i 0.706282 0.707931i \(-0.250371\pi\)
−0.259945 + 0.965623i \(0.583704\pi\)
\(168\) 0 0
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) 0 0
\(171\) 0.633975 0.169873i 0.0484812 0.0129905i
\(172\) 0 0
\(173\) −8.59808 2.30385i −0.653700 0.175158i −0.0832986 0.996525i \(-0.526546\pi\)
−0.570401 + 0.821366i \(0.693212\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 13.3923i 1.00663i
\(178\) 0 0
\(179\) 5.96410 10.3301i 0.445778 0.772110i −0.552328 0.833627i \(-0.686260\pi\)
0.998106 + 0.0615168i \(0.0195938\pi\)
\(180\) 0 0
\(181\) 14.9282i 1.10960i 0.831982 + 0.554802i \(0.187206\pi\)
−0.831982 + 0.554802i \(0.812794\pi\)
\(182\) 0 0
\(183\) −1.36603 1.36603i −0.100980 0.100980i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.1244 −0.740366
\(188\) 0 0
\(189\) −4.23205 + 15.7942i −0.307836 + 1.14886i
\(190\) 0 0
\(191\) −11.9641 20.7224i −0.865692 1.49942i −0.866358 0.499423i \(-0.833545\pi\)
0.000666402 1.00000i \(-0.499788\pi\)
\(192\) 0 0
\(193\) −12.1603 + 21.0622i −0.875314 + 1.51609i −0.0188866 + 0.999822i \(0.506012\pi\)
−0.856428 + 0.516267i \(0.827321\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.16025 12.4019i 0.510147 0.883600i −0.489784 0.871844i \(-0.662924\pi\)
0.999931 0.0117566i \(-0.00374234\pi\)
\(198\) 0 0
\(199\) 4.96410 + 8.59808i 0.351896 + 0.609501i 0.986582 0.163269i \(-0.0522038\pi\)
−0.634686 + 0.772770i \(0.718870\pi\)
\(200\) 0 0
\(201\) −2.76795 + 10.3301i −0.195236 + 0.728631i
\(202\) 0 0
\(203\) 34.3205 2.40883
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.73205 + 3.73205i 0.259395 + 0.259395i
\(208\) 0 0
\(209\) 2.07180i 0.143309i
\(210\) 0 0
\(211\) 1.96410 3.40192i 0.135214 0.234198i −0.790465 0.612507i \(-0.790161\pi\)
0.925679 + 0.378309i \(0.123494\pi\)
\(212\) 0 0
\(213\) 8.46410i 0.579951i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −29.2224 7.83013i −1.98375 0.531544i
\(218\) 0 0
\(219\) −24.1244 + 6.46410i −1.63017 + 0.436804i
\(220\) 0 0
\(221\) −11.8660 + 10.4282i −0.798195 + 0.701477i
\(222\) 0 0
\(223\) −14.0885 8.13397i −0.943433 0.544691i −0.0523981 0.998626i \(-0.516686\pi\)
−0.891035 + 0.453935i \(0.850020\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.23205 + 5.59808i 0.214519 + 0.371557i 0.953124 0.302581i \(-0.0978484\pi\)
−0.738605 + 0.674139i \(0.764515\pi\)
\(228\) 0 0
\(229\) −16.8564 16.8564i −1.11390 1.11390i −0.992618 0.121285i \(-0.961299\pi\)
−0.121285 0.992618i \(-0.538701\pi\)
\(230\) 0 0
\(231\) −14.4282 8.33013i −0.949306 0.548082i
\(232\) 0 0
\(233\) 11.0000 11.0000i 0.720634 0.720634i −0.248100 0.968734i \(-0.579806\pi\)
0.968734 + 0.248100i \(0.0798063\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.26795 + 8.46410i 0.147319 + 0.549802i
\(238\) 0 0
\(239\) 1.19615 1.19615i 0.0773727 0.0773727i −0.667361 0.744734i \(-0.732576\pi\)
0.744734 + 0.667361i \(0.232576\pi\)
\(240\) 0 0
\(241\) −17.0622 4.57180i −1.09907 0.294495i −0.336685 0.941617i \(-0.609306\pi\)
−0.762386 + 0.647122i \(0.775972\pi\)
\(242\) 0 0
\(243\) 1.92820 7.19615i 0.123694 0.461633i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.13397 2.42820i −0.135782 0.154503i
\(248\) 0 0
\(249\) 1.73205 + 6.46410i 0.109764 + 0.409646i
\(250\) 0 0
\(251\) −3.10770 + 1.79423i −0.196156 + 0.113251i −0.594861 0.803828i \(-0.702793\pi\)
0.398705 + 0.917079i \(0.369460\pi\)
\(252\) 0 0
\(253\) 14.4282 8.33013i 0.907093 0.523711i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.5263 + 4.16025i −0.968503 + 0.259510i −0.708196 0.706016i \(-0.750490\pi\)
−0.260307 + 0.965526i \(0.583824\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −6.73205 −0.416703
\(262\) 0 0
\(263\) −10.7942 + 2.89230i −0.665601 + 0.178347i −0.575772 0.817610i \(-0.695299\pi\)
−0.0898284 + 0.995957i \(0.528632\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −25.6244 + 14.7942i −1.56819 + 0.905392i
\(268\) 0 0
\(269\) −9.57180 + 5.52628i −0.583603 + 0.336943i −0.762564 0.646913i \(-0.776060\pi\)
0.178961 + 0.983856i \(0.442726\pi\)
\(270\) 0 0
\(271\) −4.20577 15.6962i −0.255482 0.953473i −0.967821 0.251638i \(-0.919031\pi\)
0.712339 0.701836i \(-0.247636\pi\)
\(272\) 0 0
\(273\) −25.4904 + 5.09808i −1.54275 + 0.308550i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.99038 11.1603i 0.179675 0.670555i −0.816033 0.578005i \(-0.803832\pi\)
0.995708 0.0925500i \(-0.0295018\pi\)
\(278\) 0 0
\(279\) 5.73205 + 1.53590i 0.343169 + 0.0919518i
\(280\) 0 0
\(281\) 13.3923 13.3923i 0.798918 0.798918i −0.184007 0.982925i \(-0.558907\pi\)
0.982925 + 0.184007i \(0.0589069\pi\)
\(282\) 0 0
\(283\) −6.52628 24.3564i −0.387947 1.44784i −0.833469 0.552567i \(-0.813648\pi\)
0.445522 0.895271i \(-0.353018\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.366025 0.366025i 0.0216058 0.0216058i
\(288\) 0 0
\(289\) −1.90192 1.09808i −0.111878 0.0645927i
\(290\) 0 0
\(291\) −6.09808 6.09808i −0.357476 0.357476i
\(292\) 0 0
\(293\) −7.76795 13.4545i −0.453808 0.786019i 0.544810 0.838559i \(-0.316602\pi\)
−0.998619 + 0.0525400i \(0.983268\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.76795 5.06218i −0.508768 0.293737i
\(298\) 0 0
\(299\) 8.33013 24.6244i 0.481744 1.42406i
\(300\) 0 0
\(301\) −42.2846 + 11.3301i −2.43724 + 0.653058i
\(302\) 0 0
\(303\) −3.23205 0.866025i −0.185676 0.0497519i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.60770i 0.548340i −0.961681 0.274170i \(-0.911597\pi\)
0.961681 0.274170i \(-0.0884031\pi\)
\(308\) 0 0
\(309\) 9.09808 15.7583i 0.517571 0.896460i
\(310\) 0 0
\(311\) 32.2487i 1.82866i 0.404974 + 0.914328i \(0.367281\pi\)
−0.404974 + 0.914328i \(0.632719\pi\)
\(312\) 0 0
\(313\) 14.3205 + 14.3205i 0.809443 + 0.809443i 0.984549 0.175107i \(-0.0560270\pi\)
−0.175107 + 0.984549i \(0.556027\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.9282 0.950783 0.475391 0.879774i \(-0.342306\pi\)
0.475391 + 0.879774i \(0.342306\pi\)
\(318\) 0 0
\(319\) −5.50000 + 20.5263i −0.307941 + 1.14925i
\(320\) 0 0
\(321\) 5.96410 + 10.3301i 0.332884 + 0.576571i
\(322\) 0 0
\(323\) 1.96410 3.40192i 0.109286 0.189288i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.8301 + 22.2224i −0.709508 + 1.22890i
\(328\) 0 0
\(329\) 1.00000 + 1.73205i 0.0551318 + 0.0954911i
\(330\) 0 0
\(331\) 1.66987 6.23205i 0.0917845 0.342544i −0.904728 0.425990i \(-0.859926\pi\)
0.996512 + 0.0834456i \(0.0265925\pi\)
\(332\) 0 0
\(333\) −0.196152 −0.0107491
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 13.9282 + 13.9282i 0.758718 + 0.758718i 0.976089 0.217371i \(-0.0697483\pi\)
−0.217371 + 0.976089i \(0.569748\pi\)
\(338\) 0 0
\(339\) 26.3205i 1.42953i
\(340\) 0 0
\(341\) 9.36603 16.2224i 0.507199 0.878494i
\(342\) 0 0
\(343\) 0.267949i 0.0144679i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.3301 + 4.10770i 0.822964 + 0.220513i 0.645642 0.763640i \(-0.276590\pi\)
0.177322 + 0.984153i \(0.443256\pi\)
\(348\) 0 0
\(349\) −24.5263 + 6.57180i −1.31286 + 0.351780i −0.846299 0.532708i \(-0.821174\pi\)
−0.466563 + 0.884488i \(0.654508\pi\)
\(350\) 0 0
\(351\) −15.4904 + 3.09808i −0.826815 + 0.165363i
\(352\) 0 0
\(353\) −10.6244 6.13397i −0.565477 0.326479i 0.189864 0.981810i \(-0.439195\pi\)
−0.755341 + 0.655332i \(0.772529\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.7942 27.3564i −0.835919 1.44785i
\(358\) 0 0
\(359\) 23.5885 + 23.5885i 1.24495 + 1.24495i 0.957922 + 0.287029i \(0.0926677\pi\)
0.287029 + 0.957922i \(0.407332\pi\)
\(360\) 0 0
\(361\) −15.7583 9.09808i −0.829386 0.478846i
\(362\) 0 0
\(363\) −7.73205 + 7.73205i −0.405827 + 0.405827i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.79423 + 17.8923i 0.250257 + 0.933971i 0.970668 + 0.240424i \(0.0772864\pi\)
−0.720411 + 0.693547i \(0.756047\pi\)
\(368\) 0 0
\(369\) −0.0717968 + 0.0717968i −0.00373759 + 0.00373759i
\(370\) 0 0
\(371\) −7.83013 2.09808i −0.406520 0.108927i
\(372\) 0 0
\(373\) −1.00962 + 3.76795i −0.0522761 + 0.195097i −0.987126 0.159947i \(-0.948868\pi\)
0.934849 + 0.355044i \(0.115534\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.6962 + 29.7224i 0.756890 + 1.53078i
\(378\) 0 0
\(379\) 4.47372 + 16.6962i 0.229800 + 0.857624i 0.980425 + 0.196895i \(0.0630859\pi\)
−0.750625 + 0.660729i \(0.770247\pi\)
\(380\) 0 0
\(381\) 29.0885 16.7942i 1.49025 0.860394i
\(382\) 0 0
\(383\) 9.69615 5.59808i 0.495450 0.286048i −0.231382 0.972863i \(-0.574325\pi\)
0.726833 + 0.686815i \(0.240992\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.29423 2.22243i 0.421619 0.112973i
\(388\) 0 0
\(389\) 23.8564 1.20957 0.604784 0.796390i \(-0.293259\pi\)
0.604784 + 0.796390i \(0.293259\pi\)
\(390\) 0 0
\(391\) 31.5885 1.59750
\(392\) 0 0
\(393\) −25.8564 + 6.92820i −1.30428 + 0.349482i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 31.6244 18.2583i 1.58718 0.916359i 0.593412 0.804899i \(-0.297780\pi\)
0.993769 0.111460i \(-0.0355529\pi\)
\(398\) 0 0
\(399\) 5.59808 3.23205i 0.280254 0.161805i
\(400\) 0 0
\(401\) 4.13397 + 15.4282i 0.206441 + 0.770448i 0.989006 + 0.147878i \(0.0472444\pi\)
−0.782565 + 0.622569i \(0.786089\pi\)
\(402\) 0 0
\(403\) −5.73205 28.6603i −0.285534 1.42767i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.160254 + 0.598076i −0.00794350 + 0.0296455i
\(408\) 0 0
\(409\) −9.06218 2.42820i −0.448096 0.120067i 0.0277124 0.999616i \(-0.491178\pi\)
−0.475808 + 0.879549i \(0.657844\pi\)
\(410\) 0 0
\(411\) −1.09808 + 1.09808i −0.0541641 + 0.0541641i
\(412\) 0 0
\(413\) −6.69615 24.9904i −0.329496 1.22970i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.2942 13.2942i 0.651021 0.651021i
\(418\) 0 0
\(419\) −0.356406 0.205771i −0.0174116 0.0100526i 0.491269 0.871008i \(-0.336533\pi\)
−0.508681 + 0.860955i \(0.669867\pi\)
\(420\) 0 0
\(421\) 27.9282 + 27.9282i 1.36114 + 1.36114i 0.872470 + 0.488667i \(0.162517\pi\)
0.488667 + 0.872470i \(0.337483\pi\)
\(422\) 0 0
\(423\) −0.196152 0.339746i −0.00953726 0.0165190i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.23205 1.86603i −0.156410 0.0903033i
\(428\) 0 0
\(429\) 1.03590 16.0622i 0.0500136 0.775489i
\(430\) 0 0
\(431\) 27.5263 7.37564i 1.32589 0.355272i 0.474711 0.880142i \(-0.342552\pi\)
0.851183 + 0.524869i \(0.175886\pi\)
\(432\) 0 0
\(433\) 26.7224 + 7.16025i 1.28420 + 0.344100i 0.835454 0.549560i \(-0.185205\pi\)
0.448744 + 0.893660i \(0.351871\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.46410i 0.309220i
\(438\) 0 0
\(439\) −0.964102 + 1.66987i −0.0460141 + 0.0796987i −0.888115 0.459621i \(-0.847985\pi\)
0.842101 + 0.539320i \(0.181319\pi\)
\(440\) 0 0
\(441\) 5.07180i 0.241514i
\(442\) 0 0
\(443\) 11.0526 + 11.0526i 0.525123 + 0.525123i 0.919114 0.393991i \(-0.128906\pi\)
−0.393991 + 0.919114i \(0.628906\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −36.5167 −1.72718
\(448\) 0 0
\(449\) 1.99038 7.42820i 0.0939319 0.350559i −0.902923 0.429801i \(-0.858584\pi\)
0.996855 + 0.0792428i \(0.0252502\pi\)
\(450\) 0 0
\(451\) 0.160254 + 0.277568i 0.00754607 + 0.0130702i
\(452\) 0 0
\(453\) −1.09808 + 1.90192i −0.0515921 + 0.0893602i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.30385 + 7.45448i −0.201325 + 0.348706i −0.948956 0.315409i \(-0.897858\pi\)
0.747630 + 0.664115i \(0.231192\pi\)
\(458\) 0 0
\(459\) −9.59808 16.6244i −0.448000 0.775958i
\(460\) 0 0
\(461\) 1.99038 7.42820i 0.0927013 0.345966i −0.903960 0.427618i \(-0.859353\pi\)
0.996661 + 0.0816519i \(0.0260196\pi\)
\(462\) 0 0
\(463\) 2.14359 0.0996212 0.0498106 0.998759i \(-0.484138\pi\)
0.0498106 + 0.998759i \(0.484138\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.12436 + 4.12436i 0.190852 + 0.190852i 0.796064 0.605212i \(-0.206912\pi\)
−0.605212 + 0.796064i \(0.706912\pi\)
\(468\) 0 0
\(469\) 20.6603i 0.954002i
\(470\) 0 0
\(471\) −10.0981 + 17.4904i −0.465295 + 0.805914i
\(472\) 0 0
\(473\) 27.1051i 1.24629i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.53590 + 0.411543i 0.0703240 + 0.0188432i
\(478\) 0 0
\(479\) 26.9904 7.23205i 1.23322 0.330441i 0.417389 0.908728i \(-0.362945\pi\)
0.815833 + 0.578287i \(0.196279\pi\)
\(480\) 0 0
\(481\) 0.428203 + 0.866025i 0.0195244 + 0.0394874i
\(482\) 0 0
\(483\) 45.0167 + 25.9904i 2.04833 + 1.18260i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0.160254 + 0.277568i 0.00726180 + 0.0125778i 0.869633 0.493698i \(-0.164355\pi\)
−0.862372 + 0.506276i \(0.831022\pi\)
\(488\) 0 0
\(489\) 2.09808 + 2.09808i 0.0948783 + 0.0948783i
\(490\) 0 0
\(491\) −7.03590 4.06218i −0.317526 0.183324i 0.332763 0.943010i \(-0.392019\pi\)
−0.650289 + 0.759687i \(0.725352\pi\)
\(492\) 0 0
\(493\) −28.4904 + 28.4904i −1.28314 + 1.28314i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.23205 + 15.7942i 0.189833 + 0.708468i
\(498\) 0 0
\(499\) −7.19615 + 7.19615i −0.322144 + 0.322144i −0.849589 0.527445i \(-0.823150\pi\)
0.527445 + 0.849589i \(0.323150\pi\)
\(500\) 0 0
\(501\) −12.4282 3.33013i −0.555251 0.148779i
\(502\) 0 0
\(503\) −2.93782 + 10.9641i −0.130991 + 0.488865i −0.999982 0.00596240i \(-0.998102\pi\)
0.868991 + 0.494828i \(0.164769\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.3301 19.8923i −0.680835 0.883448i
\(508\) 0 0
\(509\) −6.79423 25.3564i −0.301149 1.12390i −0.936210 0.351441i \(-0.885692\pi\)
0.635061 0.772462i \(-0.280975\pi\)
\(510\) 0 0
\(511\) −41.7846 + 24.1244i −1.84844 + 1.06720i
\(512\) 0 0
\(513\) 3.40192 1.96410i 0.150199 0.0867172i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.19615 + 0.320508i −0.0526067 + 0.0140959i
\(518\) 0 0
\(519\) 17.1962 0.754827
\(520\) 0 0
\(521\) −23.8564 −1.04517 −0.522584 0.852588i \(-0.675032\pi\)
−0.522584 + 0.852588i \(0.675032\pi\)
\(522\) 0 0
\(523\) 3.06218 0.820508i 0.133900 0.0358783i −0.191247 0.981542i \(-0.561253\pi\)
0.325146 + 0.945664i \(0.394586\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.7583 17.7583i 1.33985 0.773565i
\(528\) 0 0
\(529\) −25.0981 + 14.4904i −1.09122 + 0.630017i
\(530\) 0 0
\(531\) 1.31347 + 4.90192i 0.0569996 + 0.212725i
\(532\) 0 0
\(533\) 0.473721 + 0.160254i 0.0205191 + 0.00694137i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.96410 + 22.2583i −0.257370 + 0.960518i
\(538\) 0 0
\(539\) −15.4641 4.14359i −0.666086 0.178477i
\(540\) 0 0
\(541\) 27.9282 27.9282i 1.20073 1.20073i 0.226782 0.973946i \(-0.427180\pi\)
0.973946 0.226782i \(-0.0728204\pi\)
\(542\) 0 0
\(543\) −7.46410 27.8564i −0.320315 1.19543i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.19615 + 1.19615i −0.0511438 + 0.0511438i −0.732216 0.681072i \(-0.761514\pi\)
0.681072 + 0.732216i \(0.261514\pi\)
\(548\) 0 0
\(549\) 0.633975 + 0.366025i 0.0270574 + 0.0156216i
\(550\) 0 0
\(551\) −5.83013 5.83013i −0.248372 0.248372i
\(552\) 0 0
\(553\) 8.46410 + 14.6603i 0.359930 + 0.623417i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.2321 + 12.2583i 0.899631 + 0.519402i 0.877080 0.480344i \(-0.159488\pi\)
0.0225505 + 0.999746i \(0.492821\pi\)
\(558\) 0 0
\(559\) −27.9186 31.7679i −1.18083 1.34364i
\(560\) 0 0
\(561\) 18.8923 5.06218i 0.797634 0.213725i
\(562\) 0 0
\(563\) −10.5263 2.82051i −0.443630 0.118870i 0.0300874 0.999547i \(-0.490421\pi\)
−0.473717 + 0.880677i \(0.657088\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 39.7846i 1.67080i
\(568\) 0 0
\(569\) 15.4282 26.7224i 0.646784 1.12026i −0.337102 0.941468i \(-0.609447\pi\)
0.983886 0.178795i \(-0.0572198\pi\)
\(570\) 0 0
\(571\) 34.1051i 1.42725i −0.700525 0.713627i \(-0.747051\pi\)
0.700525 0.713627i \(-0.252949\pi\)
\(572\) 0 0
\(573\) 32.6865 + 32.6865i 1.36550 + 1.36550i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.0718 0.627447 0.313724 0.949514i \(-0.398423\pi\)
0.313724 + 0.949514i \(0.398423\pi\)
\(578\) 0 0
\(579\) 12.1603 45.3827i 0.505363 1.88604i
\(580\) 0 0
\(581\) 6.46410 + 11.1962i 0.268176 + 0.464495i
\(582\) 0 0
\(583\) 2.50962 4.34679i 0.103938 0.180026i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.16025 + 7.20577i −0.171712 + 0.297414i −0.939019 0.343867i \(-0.888263\pi\)
0.767306 + 0.641281i \(0.221597\pi\)
\(588\) 0 0
\(589\) 3.63397 + 6.29423i 0.149735 + 0.259349i
\(590\) 0 0
\(591\) −7.16025 + 26.7224i −0.294533 + 1.09921i
\(592\) 0 0
\(593\) 20.9282 0.859418 0.429709 0.902967i \(-0.358616\pi\)
0.429709 + 0.902967i \(0.358616\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −13.5622 13.5622i −0.555063 0.555063i
\(598\) 0 0
\(599\) 26.3923i 1.07836i −0.842190 0.539180i \(-0.818734\pi\)
0.842190 0.539180i \(-0.181266\pi\)
\(600\) 0 0
\(601\) −18.4282 + 31.9186i −0.751702 + 1.30199i 0.195295 + 0.980745i \(0.437434\pi\)
−0.946997 + 0.321242i \(0.895900\pi\)
\(602\) 0 0
\(603\) 4.05256i 0.165033i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −21.0622 5.64359i −0.854887 0.229066i −0.195346 0.980734i \(-0.562583\pi\)
−0.659542 + 0.751668i \(0.729250\pi\)
\(608\) 0 0
\(609\) −64.0429 + 17.1603i −2.59515 + 0.695369i
\(610\) 0 0
\(611\) −1.07180 + 1.60770i −0.0433603 + 0.0650404i
\(612\) 0 0
\(613\) 5.76795 + 3.33013i 0.232965 + 0.134503i 0.611939 0.790905i \(-0.290390\pi\)
−0.378974 + 0.925407i \(0.623723\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.0885 26.1340i −0.607438 1.05211i −0.991661 0.128874i \(-0.958864\pi\)
0.384223 0.923240i \(-0.374470\pi\)
\(618\) 0 0
\(619\) 30.1244 + 30.1244i 1.21080 + 1.21080i 0.970764 + 0.240036i \(0.0771593\pi\)
0.240036 + 0.970764i \(0.422841\pi\)
\(620\) 0 0
\(621\) 27.3564 + 15.7942i 1.09777 + 0.633801i
\(622\) 0 0
\(623\) −40.4186 + 40.4186i −1.61934 + 1.61934i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.03590 + 3.86603i 0.0413698 + 0.154394i
\(628\) 0 0
\(629\) −0.830127 + 0.830127i −0.0330993 + 0.0330993i
\(630\) 0 0
\(631\) 21.7942 + 5.83975i 0.867615 + 0.232477i 0.665056 0.746794i \(-0.268408\pi\)
0.202559 + 0.979270i \(0.435074\pi\)
\(632\) 0 0
\(633\) −1.96410 + 7.33013i −0.0780660 + 0.291346i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −22.3923 + 11.0718i −0.887215 + 0.438681i
\(638\) 0 0
\(639\) −0.830127 3.09808i −0.0328393 0.122558i
\(640\) 0 0
\(641\) −1.28461 + 0.741670i −0.0507390 + 0.0292942i −0.525155 0.851007i \(-0.675993\pi\)
0.474416 + 0.880301i \(0.342659\pi\)
\(642\) 0 0
\(643\) −39.2321 + 22.6506i −1.54716 + 0.893254i −0.548804 + 0.835951i \(0.684917\pi\)
−0.998357 + 0.0573029i \(0.981750\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.7942 + 3.96410i −0.581621 + 0.155845i −0.537621 0.843187i \(-0.680677\pi\)
−0.0440001 + 0.999032i \(0.514010\pi\)
\(648\) 0 0
\(649\) 16.0192 0.628810
\(650\) 0 0
\(651\) 58.4449 2.29063
\(652\) 0 0
\(653\) 36.1865 9.69615i 1.41609 0.379440i 0.531994 0.846748i \(-0.321443\pi\)
0.884094 + 0.467308i \(0.154776\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.19615 4.73205i 0.319762 0.184615i
\(658\) 0 0
\(659\) 26.2128 15.1340i 1.02111 0.589536i 0.106682 0.994293i \(-0.465977\pi\)
0.914424 + 0.404757i \(0.132644\pi\)
\(660\) 0 0
\(661\) 3.59808 + 13.4282i 0.139949 + 0.522297i 0.999928 + 0.0119679i \(0.00380959\pi\)
−0.859979 + 0.510329i \(0.829524\pi\)
\(662\) 0 0
\(663\) 16.9282 25.3923i 0.657437 0.986155i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17.1603 64.0429i 0.664448 2.47975i
\(668\) 0 0
\(669\) 30.3564 + 8.13397i 1.17365 + 0.314478i
\(670\) 0 0
\(671\) 1.63397 1.63397i 0.0630789 0.0630789i
\(672\) 0 0
\(673\) −1.00962 3.76795i −0.0389180 0.145244i 0.943733 0.330708i \(-0.107288\pi\)
−0.982651 + 0.185465i \(0.940621\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.3205 + 24.3205i −0.934713 + 0.934713i −0.997996 0.0632826i \(-0.979843\pi\)
0.0632826 + 0.997996i \(0.479843\pi\)
\(678\) 0 0
\(679\) −14.4282 8.33013i −0.553704 0.319681i
\(680\) 0 0
\(681\) −8.83013 8.83013i −0.338371 0.338371i
\(682\) 0 0
\(683\) 11.7679 + 20.3827i 0.450288 + 0.779922i 0.998404 0.0564808i \(-0.0179880\pi\)
−0.548116 + 0.836403i \(0.684655\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 39.8827 + 23.0263i 1.52162 + 0.878507i
\(688\) 0 0
\(689\) −1.53590 7.67949i −0.0585131 0.292565i
\(690\) 0 0
\(691\) 12.0622 3.23205i 0.458867 0.122953i −0.0219785 0.999758i \(-0.506997\pi\)
0.480845 + 0.876805i \(0.340330\pi\)
\(692\) 0 0
\(693\) 6.09808 + 1.63397i 0.231647 + 0.0620696i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.607695i 0.0230181i
\(698\) 0 0
\(699\) −15.0263 + 26.0263i −0.568346 + 0.984404i
\(700\) 0 0
\(701\) 42.9282i 1.62138i −0.585479 0.810688i \(-0.699093\pi\)
0.585479 0.810688i \(-0.300907\pi\)
\(702\) 0 0
\(703\) −0.169873 0.169873i −0.00640688 0.00640688i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.46410 −0.243108
\(708\) 0 0
\(709\) −0.401924 + 1.50000i −0.0150946 + 0.0563337i −0.973062 0.230542i \(-0.925950\pi\)
0.957968 + 0.286876i \(0.0926167\pi\)
\(710\) 0 0
\(711\) −1.66025 2.87564i −0.0622644 0.107845i
\(712\) 0 0
\(713\) −29.2224 + 50.6147i −1.09439 + 1.89554i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.63397 + 2.83013i −0.0610219 + 0.105693i
\(718\) 0 0
\(719\) 5.89230 + 10.2058i 0.219746 + 0.380611i 0.954730 0.297473i \(-0.0961438\pi\)
−0.734984 + 0.678084i \(0.762811\pi\)
\(720\) 0 0
\(721\) 9.09808 33.9545i 0.338830 1.26453i
\(722\) 0 0
\(723\) 34.1244 1.26910
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.5885 + 17.5885i 0.652320 + 0.652320i 0.953551 0.301231i \(-0.0973976\pi\)
−0.301231 + 0.953551i \(0.597398\pi\)
\(728\) 0 0
\(729\) 17.5885i 0.651424i
\(730\) 0 0
\(731\) 25.6962 44.5070i 0.950407 1.64615i
\(732\) 0 0
\(733\) 30.6410i 1.13175i −0.824490 0.565876i \(-0.808538\pi\)
0.824490 0.565876i \(-0.191462\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.3564 3.31089i −0.455154 0.121958i
\(738\) 0 0
\(739\) −33.2583 + 8.91154i −1.22343 + 0.327816i −0.812017 0.583634i \(-0.801630\pi\)
−0.411410 + 0.911450i \(0.634964\pi\)
\(740\) 0 0
\(741\) 5.19615 + 3.46410i 0.190885 + 0.127257i
\(742\) 0 0
\(743\) 21.4808 + 12.4019i 0.788053 + 0.454982i 0.839277 0.543705i \(-0.182979\pi\)
−0.0512238 + 0.998687i \(0.516312\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.26795 2.19615i −0.0463918 0.0803530i
\(748\) 0 0
\(749\) 16.2942 + 16.2942i 0.595378 + 0.595378i
\(750\) 0 0
\(751\) −16.7487 9.66987i −0.611169 0.352859i 0.162254 0.986749i \(-0.448124\pi\)
−0.773423 + 0.633890i \(0.781457\pi\)
\(752\) 0 0
\(753\) 4.90192 4.90192i 0.178636 0.178636i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.93782 14.6962i −0.143123 0.534141i −0.999832 0.0183410i \(-0.994162\pi\)
0.856709 0.515800i \(-0.172505\pi\)
\(758\) 0 0
\(759\) −22.7583 + 22.7583i −0.826075 + 0.826075i
\(760\) 0 0
\(761\) 11.3301 + 3.03590i 0.410717 + 0.110051i 0.458261 0.888818i \(-0.348473\pi\)
−0.0475437 + 0.998869i \(0.515139\pi\)
\(762\) 0 0
\(763\) −12.8301 + 47.8827i −0.464482 + 1.73347i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.7750 16.5000i 0.677926 0.595780i
\(768\) 0 0
\(769\) −3.33013 12.4282i −0.120087 0.448172i 0.879530 0.475844i \(-0.157857\pi\)
−0.999617 + 0.0276717i \(0.991191\pi\)
\(770\) 0 0
\(771\) 26.8923 15.5263i 0.968503 0.559165i
\(772\) 0 0
\(773\) 15.4808 8.93782i 0.556804 0.321471i −0.195058 0.980792i \(-0.562489\pi\)
0.751862 + 0.659321i \(0.229156\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.86603 + 0.500000i −0.0669433 + 0.0179374i
\(778\) 0 0
\(779\) −0.124356 −0.00445550
\(780\) 0 0
\(781\) −10.1244 −0.362278
\(782\) 0 0
\(783\) −38.9186 + 10.4282i −1.39084 + 0.372674i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6.91154 3.99038i 0.246370 0.142242i −0.371731 0.928340i \(-0.621236\pi\)
0.618101 + 0.786099i \(0.287902\pi\)
\(788\) 0 0
\(789\) 18.6962 10.7942i 0.665601 0.384285i
\(790\) 0 0
\(791\) 13.1603 + 49.1147i 0.467925 + 1.74632i
\(792\) 0 0
\(793\) 0.232051 3.59808i 0.00824037 0.127771i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.72243 + 17.6244i −0.167277 + 0.624287i 0.830462 + 0.557076i \(0.188077\pi\)
−0.997739 + 0.0672109i \(0.978590\pi\)
\(798\) 0 0
\(799\) −2.26795 0.607695i −0.0802343 0.0214987i
\(800\) 0 0
\(801\) 7.92820 7.92820i 0.280129 0.280129i
\(802\) 0 0
\(803\) −7.73205 28.8564i −0.272858 1.01832i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0981 15.0981i 0.531477 0.531477i
\(808\) 0 0
\(809\) 9.10770 + 5.25833i 0.320210 + 0.184873i 0.651486 0.758661i \(-0.274146\pi\)
−0.331276 + 0.943534i \(0.607479\pi\)
\(810\) 0 0
\(811\) 7.87564 + 7.87564i 0.276551 + 0.276551i 0.831731 0.555179i \(-0.187350\pi\)
−0.555179 + 0.831731i \(0.687350\pi\)
\(812\) 0 0
\(813\) 15.6962 + 27.1865i 0.550488 + 0.953473i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.10770 + 5.25833i 0.318638 + 0.183966i
\(818\) 0 0
\(819\) 8.83013 4.36603i 0.308550 0.152561i
\(820\) 0 0
\(821\) −9.59808 + 2.57180i −0.334975 + 0.0897563i −0.422386 0.906416i \(-0.638807\pi\)
0.0874109 + 0.996172i \(0.472141\pi\)
\(822\) 0 0
\(823\) −34.9186 9.35641i −1.21719 0.326144i −0.407608 0.913157i \(-0.633637\pi\)
−0.809577 + 0.587013i \(0.800304\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.4641i 1.09411i −0.837095 0.547057i \(-0.815748\pi\)
0.837095 0.547057i \(-0.184252\pi\)
\(828\) 0 0
\(829\) 7.96410 13.7942i 0.276605 0.479093i −0.693934 0.720039i \(-0.744124\pi\)
0.970539 + 0.240945i \(0.0774574\pi\)
\(830\) 0 0
\(831\) 22.3205i 0.774290i
\(832\) 0 0
\(833\) −21.4641 21.4641i −0.743687 0.743687i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 35.5167 1.22764
\(838\) 0 0
\(839\) −6.47372 + 24.1603i −0.223498 + 0.834105i 0.759503 + 0.650504i \(0.225442\pi\)
−0.983001 + 0.183601i \(0.941225\pi\)
\(840\) 0 0
\(841\) 27.7846 + 48.1244i 0.958090 + 1.65946i
\(842\) 0 0
\(843\) −18.2942 + 31.6865i −0.630087 + 1.09134i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.5622 + 18.2942i −0.362921 + 0.628597i
\(848\) 0 0
\(849\) 24.3564 + 42.1865i 0.835910 + 1.44784i
\(850\) 0 0
\(851\) 0.500000 1.86603i 0.0171398 0.0639665i
\(852\) 0 0
\(853\) −27.5692 −0.943952 −0.471976 0.881611i \(-0.656459\pi\)
−0.471976 + 0.881611i \(0.656459\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.3923 29.3923i −1.00402 1.00402i −0.999992 0.00403013i \(-0.998717\pi\)
−0.00403013 0.999992i \(-0.501283\pi\)
\(858\) 0 0
\(859\) 30.1051i 1.02717i −0.858038 0.513587i \(-0.828316\pi\)
0.858038 0.513587i \(-0.171684\pi\)
\(860\) 0 0
\(861\) −0.500000 + 0.866025i −0.0170400 + 0.0295141i
\(862\) 0 0
\(863\) 28.2487i 0.961597i −0.876831 0.480799i \(-0.840347\pi\)
0.876831 0.480799i \(-0.159653\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.09808 + 1.09808i 0.139178 + 0.0372926i
\(868\) 0 0
\(869\) −10.1244 + 2.71281i −0.343445 + 0.0920259i
\(870\) 0 0
\(871\) −17.8923 + 8.84679i −0.606258 + 0.299762i
\(872\) 0 0
\(873\) 2.83013 + 1.63397i 0.0957853 + 0.0553017i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17.5526 + 30.4019i 0.592708 + 1.02660i 0.993866 + 0.110591i \(0.0352744\pi\)
−0.401158 + 0.916009i \(0.631392\pi\)
\(878\) 0 0
\(879\) 21.2224 + 21.2224i 0.715815 + 0.715815i
\(880\) 0 0
\(881\) 4.96410 + 2.86603i 0.167245 + 0.0965588i 0.581286 0.813699i \(-0.302550\pi\)
−0.414041 + 0.910258i \(0.635883\pi\)
\(882\) 0 0
\(883\) 27.1962 27.1962i 0.915223 0.915223i −0.0814537 0.996677i \(-0.525956\pi\)
0.996677 + 0.0814537i \(0.0259563\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.7942 + 47.7487i 0.429588 + 1.60324i 0.753695 + 0.657224i \(0.228270\pi\)
−0.324107 + 0.946020i \(0.605064\pi\)
\(888\) 0 0
\(889\) 45.8827 45.8827i 1.53886 1.53886i
\(890\) 0 0
\(891\) 23.7942 + 6.37564i 0.797137 + 0.213592i
\(892\) 0 0
\(893\) 0.124356 0.464102i 0.00416140 0.0155306i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −3.23205 + 50.1147i −0.107915 + 1.67328i
\(898\) 0 0
\(899\) −19.2942 72.0070i −0.643499 2.40157i
\(900\) 0 0
\(901\) 8.24167 4.75833i 0.274570 0.158523i
\(902\) 0 0
\(903\) 73.2391 42.2846i 2.43724 1.40714i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.5263 6.03590i 0.747973 0.200419i 0.135354 0.990797i \(-0.456783\pi\)
0.612619 + 0.790379i \(0.290116\pi\)
\(908\) 0 0
\(909\) 1.26795 0.0420552
\(910\) 0 0
\(911\) −42.9282 −1.42227 −0.711137 0.703053i \(-0.751820\pi\)
−0.711137 + 0.703053i \(0.751820\pi\)
\(912\) 0 0
\(913\) −7.73205 + 2.07180i −0.255894 + 0.0685665i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.7846 + 25.8564i −1.47892 + 0.853854i
\(918\) 0 0
\(919\) 10.7487 6.20577i 0.354567 0.204710i −0.312128 0.950040i \(-0.601042\pi\)
0.666695 + 0.745331i \(0.267708\pi\)
\(920\) 0 0
\(921\) 4.80385 + 17.9282i 0.158292 + 0.590754i
\(922\) 0 0
\(923\) −11.8660 + 10.4282i −0.390575 + 0.343248i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.78461 + 6.66025i −0.0586143 + 0.218751i
\(928\) 0 0
\(929\) 14.7942 + 3.96410i 0.485383 + 0.130058i 0.493208 0.869911i \(-0.335824\pi\)
−0.00782508 + 0.999969i \(0.502491\pi\)
\(930\) 0 0
\(931\) 4.39230 4.39230i 0.143952 0.143952i
\(932\) 0 0
\(933\) −16.1244 60.1769i −0.527888 1.97010i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.215390 + 0.215390i −0.00703649 + 0.00703649i −0.710616 0.703580i \(-0.751584\pi\)
0.703580 + 0.710616i \(0.251584\pi\)
\(938\) 0 0
\(939\) −33.8827 19.5622i −1.10572 0.638388i
\(940\) 0 0
\(941\) −19.0000 19.0000i −0.619382 0.619382i 0.325991 0.945373i \(-0.394302\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) 0 0
\(943\) −0.500000 0.866025i −0.0162822 0.0282017i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0167 + 24.2583i 1.36536 + 0.788290i 0.990331 0.138724i \(-0.0443001\pi\)
0.375027 + 0.927014i \(0.377633\pi\)
\(948\) 0 0
\(949\) −38.7846 25.8564i −1.25900 0.839334i
\(950\) 0 0
\(951\) −31.5885 + 8.46410i −1.02433 + 0.274467i
\(952\) 0 0
\(953\) −20.9904 5.62436i −0.679945 0.182191i −0.0977146 0.995214i \(-0.531153\pi\)
−0.582231 + 0.813024i \(0.697820\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 41.0526i 1.32704i
\(958\) 0 0
\(959\) −1.50000 + 2.59808i −0.0484375 + 0.0838963i
\(960\) 0 0
\(961\) 34.7128i 1.11977i
\(962\) 0 0
\(963\) −3.19615 3.19615i −0.102995 0.102995i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −58.6410 −1.88577 −0.942884 0.333121i \(-0.891898\pi\)
−0.942884 + 0.333121i \(0.891898\pi\)
\(968\) 0 0
\(969\) −1.96410 + 7.33013i −0.0630960 + 0.235478i
\(970\) 0 0
\(971\) −16.5000 28.5788i −0.529510 0.917139i −0.999408 0.0344175i \(-0.989042\pi\)
0.469897 0.882721i \(-0.344291\pi\)
\(972\) 0 0
\(973\) 18.1603 31.4545i 0.582191 1.00838i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.6244 42.6506i 0.787803 1.36451i −0.139507 0.990221i \(-0.544552\pi\)
0.927310 0.374294i \(-0.122115\pi\)
\(978\) 0 0
\(979\) −17.6962 30.6506i −0.565571 0.979599i
\(980\) 0 0
\(981\) 2.51666 9.39230i 0.0803508 0.299873i
\(982\) 0 0
\(983\) 29.0718 0.927246 0.463623 0.886032i \(-0.346549\pi\)
0.463623 + 0.886032i \(0.346549\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.73205 2.73205i −0.0869621 0.0869621i
\(988\) 0 0
\(989\) 84.5692i 2.68914i
\(990\) 0 0
\(991\) −24.9641 + 43.2391i −0.793011 + 1.37354i 0.131084 + 0.991371i \(0.458154\pi\)
−0.924095 + 0.382164i \(0.875179\pi\)
\(992\) 0 0
\(993\) 12.4641i 0.395536i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.33013 + 1.69615i 0.200477 + 0.0537177i 0.357660 0.933852i \(-0.383575\pi\)
−0.157183 + 0.987569i \(0.550241\pi\)
\(998\) 0 0
\(999\) −1.13397 + 0.303848i −0.0358774 + 0.00961331i
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 1300.2.bs.a.293.1 4
5.2 odd 4 1300.2.bn.b.657.1 4
5.3 odd 4 260.2.bf.a.137.1 yes 4
5.4 even 2 260.2.bk.b.33.1 yes 4
13.2 odd 12 1300.2.bn.b.93.1 4
65.2 even 12 inner 1300.2.bs.a.457.1 4
65.28 even 12 260.2.bk.b.197.1 yes 4
65.54 odd 12 260.2.bf.a.93.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.bf.a.93.1 4 65.54 odd 12
260.2.bf.a.137.1 yes 4 5.3 odd 4
260.2.bk.b.33.1 yes 4 5.4 even 2
260.2.bk.b.197.1 yes 4 65.28 even 12
1300.2.bn.b.93.1 4 13.2 odd 12
1300.2.bn.b.657.1 4 5.2 odd 4
1300.2.bs.a.293.1 4 1.1 even 1 trivial
1300.2.bs.a.457.1 4 65.2 even 12 inner