Properties

Label 360.2.q.c.241.1
Level $360$
Weight $2$
Character 360.241
Analytic conductor $2.875$
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] = [360,2,Mod(121,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, 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("360.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 241.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 360.241
Dual form 360.2.q.c.121.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.724745 + 1.57313i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +(-1.94949 - 2.28024i) q^{9} +O(q^{10})\) \(q+(-0.724745 + 1.57313i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +(-1.94949 - 2.28024i) q^{9} +(-1.72474 + 0.158919i) q^{15} -2.00000 q^{17} -6.89898 q^{19} +(-3.44949 - 4.87832i) q^{21} +(3.72474 + 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(5.00000 - 1.41421i) q^{27} +(0.949490 - 1.64456i) q^{29} +(-0.550510 - 0.953512i) q^{31} -3.44949 q^{35} -6.00000 q^{37} +(4.94949 + 8.57277i) q^{41} +(-5.89898 + 10.2173i) q^{43} +(1.00000 - 2.82843i) q^{45} +(4.72474 - 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} +(1.44949 - 3.14626i) q^{51} +7.79796 q^{53} +(5.00000 - 10.8530i) q^{57} +(0.550510 + 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(10.1742 - 1.89097i) q^{63} +(6.62372 + 11.4726i) q^{67} +(-12.8485 + 1.18386i) q^{69} +9.79796 q^{71} +13.7980 q^{73} +(-1.00000 - 1.41421i) q^{75} +(3.44949 - 5.97469i) q^{79} +(-1.39898 + 8.89060i) q^{81} +(2.72474 - 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} +(1.89898 + 2.68556i) q^{87} +2.79796 q^{89} +(1.89898 - 0.174973i) q^{93} +(-3.44949 - 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{15} - 8 q^{17} - 8 q^{19} - 4 q^{21} + 10 q^{23} - 2 q^{25} + 20 q^{27} - 6 q^{29} - 12 q^{31} - 4 q^{35} - 24 q^{37} + 10 q^{41} - 4 q^{43} + 4 q^{45} + 14 q^{47} - 4 q^{51} - 8 q^{53} + 20 q^{57} + 12 q^{59} + 6 q^{61} + 26 q^{63} + 2 q^{67} - 22 q^{69} + 16 q^{73} - 4 q^{75} + 4 q^{79} + 14 q^{81} + 6 q^{83} - 4 q^{85} - 12 q^{87} - 28 q^{89} - 12 q^{93} - 4 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.724745 + 1.57313i −0.418432 + 0.908248i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −1.72474 + 2.98735i −0.651892 + 1.12911i 0.330771 + 0.943711i \(0.392691\pi\)
−0.982663 + 0.185399i \(0.940642\pi\)
\(8\) 0 0
\(9\) −1.94949 2.28024i −0.649830 0.760080i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) −1.72474 + 0.158919i −0.445327 + 0.0410326i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.89898 −1.58273 −0.791367 0.611341i \(-0.790630\pi\)
−0.791367 + 0.611341i \(0.790630\pi\)
\(20\) 0 0
\(21\) −3.44949 4.87832i −0.752740 1.06454i
\(22\) 0 0
\(23\) 3.72474 + 6.45145i 0.776663 + 1.34522i 0.933855 + 0.357652i \(0.116422\pi\)
−0.157192 + 0.987568i \(0.550244\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 5.00000 1.41421i 0.962250 0.272166i
\(28\) 0 0
\(29\) 0.949490 1.64456i 0.176316 0.305388i −0.764300 0.644861i \(-0.776915\pi\)
0.940616 + 0.339473i \(0.110249\pi\)
\(30\) 0 0
\(31\) −0.550510 0.953512i −0.0988746 0.171256i 0.812345 0.583178i \(-0.198191\pi\)
−0.911219 + 0.411922i \(0.864858\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.44949 −0.583070
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.94949 + 8.57277i 0.772980 + 1.33884i 0.935923 + 0.352206i \(0.114568\pi\)
−0.162942 + 0.986636i \(0.552098\pi\)
\(42\) 0 0
\(43\) −5.89898 + 10.2173i −0.899586 + 1.55813i −0.0715617 + 0.997436i \(0.522798\pi\)
−0.828024 + 0.560692i \(0.810535\pi\)
\(44\) 0 0
\(45\) 1.00000 2.82843i 0.149071 0.421637i
\(46\) 0 0
\(47\) 4.72474 8.18350i 0.689175 1.19369i −0.282930 0.959140i \(-0.591307\pi\)
0.972105 0.234545i \(-0.0753602\pi\)
\(48\) 0 0
\(49\) −2.44949 4.24264i −0.349927 0.606092i
\(50\) 0 0
\(51\) 1.44949 3.14626i 0.202969 0.440565i
\(52\) 0 0
\(53\) 7.79796 1.07113 0.535566 0.844493i \(-0.320098\pi\)
0.535566 + 0.844493i \(0.320098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000 10.8530i 0.662266 1.43752i
\(58\) 0 0
\(59\) 0.550510 + 0.953512i 0.0716703 + 0.124137i 0.899633 0.436646i \(-0.143834\pi\)
−0.827963 + 0.560783i \(0.810500\pi\)
\(60\) 0 0
\(61\) 1.50000 2.59808i 0.192055 0.332650i −0.753876 0.657017i \(-0.771818\pi\)
0.945931 + 0.324367i \(0.105151\pi\)
\(62\) 0 0
\(63\) 10.1742 1.89097i 1.28183 0.238240i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.62372 + 11.4726i 0.809217 + 1.40160i 0.913407 + 0.407047i \(0.133442\pi\)
−0.104190 + 0.994557i \(0.533225\pi\)
\(68\) 0 0
\(69\) −12.8485 + 1.18386i −1.54677 + 0.142520i
\(70\) 0 0
\(71\) 9.79796 1.16280 0.581402 0.813617i \(-0.302504\pi\)
0.581402 + 0.813617i \(0.302504\pi\)
\(72\) 0 0
\(73\) 13.7980 1.61493 0.807464 0.589916i \(-0.200839\pi\)
0.807464 + 0.589916i \(0.200839\pi\)
\(74\) 0 0
\(75\) −1.00000 1.41421i −0.115470 0.163299i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.44949 5.97469i 0.388098 0.672205i −0.604096 0.796912i \(-0.706466\pi\)
0.992194 + 0.124706i \(0.0397989\pi\)
\(80\) 0 0
\(81\) −1.39898 + 8.89060i −0.155442 + 0.987845i
\(82\) 0 0
\(83\) 2.72474 4.71940i 0.299080 0.518021i −0.676846 0.736125i \(-0.736654\pi\)
0.975926 + 0.218104i \(0.0699871\pi\)
\(84\) 0 0
\(85\) −1.00000 1.73205i −0.108465 0.187867i
\(86\) 0 0
\(87\) 1.89898 + 2.68556i 0.203592 + 0.287923i
\(88\) 0 0
\(89\) 2.79796 0.296583 0.148292 0.988944i \(-0.452623\pi\)
0.148292 + 0.988944i \(0.452623\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.89898 0.174973i 0.196915 0.0181438i
\(94\) 0 0
\(95\) −3.44949 5.97469i −0.353910 0.612990i
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.00000 1.73205i 0.0995037 0.172345i −0.811976 0.583691i \(-0.801608\pi\)
0.911479 + 0.411346i \(0.134941\pi\)
\(102\) 0 0
\(103\) −5.00000 8.66025i −0.492665 0.853320i 0.507300 0.861770i \(-0.330644\pi\)
−0.999964 + 0.00844953i \(0.997310\pi\)
\(104\) 0 0
\(105\) 2.50000 5.42650i 0.243975 0.529573i
\(106\) 0 0
\(107\) −12.3485 −1.19377 −0.596886 0.802326i \(-0.703595\pi\)
−0.596886 + 0.802326i \(0.703595\pi\)
\(108\) 0 0
\(109\) −10.7980 −1.03426 −0.517128 0.855908i \(-0.672999\pi\)
−0.517128 + 0.855908i \(0.672999\pi\)
\(110\) 0 0
\(111\) 4.34847 9.43879i 0.412738 0.895891i
\(112\) 0 0
\(113\) 4.89898 + 8.48528i 0.460857 + 0.798228i 0.999004 0.0446231i \(-0.0142087\pi\)
−0.538147 + 0.842851i \(0.680875\pi\)
\(114\) 0 0
\(115\) −3.72474 + 6.45145i −0.347334 + 0.601601i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.44949 5.97469i 0.316214 0.547699i
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) −17.0732 + 1.57313i −1.53944 + 0.141845i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.348469 0.0309216 0.0154608 0.999880i \(-0.495078\pi\)
0.0154608 + 0.999880i \(0.495078\pi\)
\(128\) 0 0
\(129\) −11.7980 16.6848i −1.03875 1.46902i
\(130\) 0 0
\(131\) −1.44949 2.51059i −0.126643 0.219351i 0.795731 0.605650i \(-0.207087\pi\)
−0.922374 + 0.386299i \(0.873753\pi\)
\(132\) 0 0
\(133\) 11.8990 20.6096i 1.03177 1.78708i
\(134\) 0 0
\(135\) 3.72474 + 3.62302i 0.320575 + 0.311820i
\(136\) 0 0
\(137\) −9.79796 + 16.9706i −0.837096 + 1.44989i 0.0552162 + 0.998474i \(0.482415\pi\)
−0.892312 + 0.451419i \(0.850918\pi\)
\(138\) 0 0
\(139\) −9.79796 16.9706i −0.831052 1.43942i −0.897205 0.441615i \(-0.854406\pi\)
0.0661527 0.997810i \(-0.478928\pi\)
\(140\) 0 0
\(141\) 9.44949 + 13.3636i 0.795791 + 1.12542i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.89898 0.157702
\(146\) 0 0
\(147\) 8.44949 0.778539i 0.696902 0.0642128i
\(148\) 0 0
\(149\) 10.5000 + 18.1865i 0.860194 + 1.48990i 0.871742 + 0.489966i \(0.162991\pi\)
−0.0115483 + 0.999933i \(0.503676\pi\)
\(150\) 0 0
\(151\) −6.00000 + 10.3923i −0.488273 + 0.845714i −0.999909 0.0134886i \(-0.995706\pi\)
0.511636 + 0.859202i \(0.329040\pi\)
\(152\) 0 0
\(153\) 3.89898 + 4.56048i 0.315214 + 0.368693i
\(154\) 0 0
\(155\) 0.550510 0.953512i 0.0442180 0.0765879i
\(156\) 0 0
\(157\) 11.8990 + 20.6096i 0.949642 + 1.64483i 0.746179 + 0.665745i \(0.231886\pi\)
0.203463 + 0.979083i \(0.434780\pi\)
\(158\) 0 0
\(159\) −5.65153 + 12.2672i −0.448196 + 0.972854i
\(160\) 0 0
\(161\) −25.6969 −2.02520
\(162\) 0 0
\(163\) −7.79796 −0.610783 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.17423 7.22999i −0.323012 0.559473i 0.658096 0.752934i \(-0.271362\pi\)
−0.981108 + 0.193461i \(0.938029\pi\)
\(168\) 0 0
\(169\) 6.50000 11.2583i 0.500000 0.866025i
\(170\) 0 0
\(171\) 13.4495 + 15.7313i 1.02851 + 1.20300i
\(172\) 0 0
\(173\) 6.00000 10.3923i 0.456172 0.790112i −0.542583 0.840002i \(-0.682554\pi\)
0.998755 + 0.0498898i \(0.0158870\pi\)
\(174\) 0 0
\(175\) −1.72474 2.98735i −0.130378 0.225822i
\(176\) 0 0
\(177\) −1.89898 + 0.174973i −0.142736 + 0.0131518i
\(178\) 0 0
\(179\) −13.7980 −1.03131 −0.515654 0.856797i \(-0.672451\pi\)
−0.515654 + 0.856797i \(0.672451\pi\)
\(180\) 0 0
\(181\) −9.69694 −0.720768 −0.360384 0.932804i \(-0.617354\pi\)
−0.360384 + 0.932804i \(0.617354\pi\)
\(182\) 0 0
\(183\) 3.00000 + 4.24264i 0.221766 + 0.313625i
\(184\) 0 0
\(185\) −3.00000 5.19615i −0.220564 0.382029i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.39898 + 17.3759i −0.319979 + 1.26391i
\(190\) 0 0
\(191\) 7.44949 12.9029i 0.539026 0.933621i −0.459931 0.887955i \(-0.652126\pi\)
0.998957 0.0456658i \(-0.0145409\pi\)
\(192\) 0 0
\(193\) 1.10102 + 1.90702i 0.0792532 + 0.137271i 0.902928 0.429792i \(-0.141413\pi\)
−0.823675 + 0.567063i \(0.808080\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.5959 −1.11116 −0.555582 0.831462i \(-0.687504\pi\)
−0.555582 + 0.831462i \(0.687504\pi\)
\(198\) 0 0
\(199\) 22.8990 1.62327 0.811633 0.584168i \(-0.198579\pi\)
0.811633 + 0.584168i \(0.198579\pi\)
\(200\) 0 0
\(201\) −22.8485 + 2.10527i −1.61161 + 0.148494i
\(202\) 0 0
\(203\) 3.27526 + 5.67291i 0.229878 + 0.398160i
\(204\) 0 0
\(205\) −4.94949 + 8.57277i −0.345687 + 0.598748i
\(206\) 0 0
\(207\) 7.44949 21.0703i 0.517775 1.46449i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.00000 10.3923i −0.413057 0.715436i 0.582165 0.813070i \(-0.302206\pi\)
−0.995222 + 0.0976347i \(0.968872\pi\)
\(212\) 0 0
\(213\) −7.10102 + 15.4135i −0.486554 + 1.05611i
\(214\) 0 0
\(215\) −11.7980 −0.804614
\(216\) 0 0
\(217\) 3.79796 0.257822
\(218\) 0 0
\(219\) −10.0000 + 21.7060i −0.675737 + 1.46676i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.72474 + 4.71940i −0.182462 + 0.316034i −0.942718 0.333589i \(-0.891740\pi\)
0.760256 + 0.649623i \(0.225073\pi\)
\(224\) 0 0
\(225\) 2.94949 0.548188i 0.196633 0.0365459i
\(226\) 0 0
\(227\) 4.10102 7.10318i 0.272194 0.471454i −0.697229 0.716848i \(-0.745584\pi\)
0.969423 + 0.245394i \(0.0789173\pi\)
\(228\) 0 0
\(229\) 2.05051 + 3.55159i 0.135502 + 0.234696i 0.925789 0.378041i \(-0.123402\pi\)
−0.790287 + 0.612736i \(0.790069\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.7980 −1.29701 −0.648504 0.761211i \(-0.724605\pi\)
−0.648504 + 0.761211i \(0.724605\pi\)
\(234\) 0 0
\(235\) 9.44949 0.616417
\(236\) 0 0
\(237\) 6.89898 + 9.75663i 0.448137 + 0.633761i
\(238\) 0 0
\(239\) −1.10102 1.90702i −0.0712191 0.123355i 0.828217 0.560408i \(-0.189356\pi\)
−0.899436 + 0.437053i \(0.856022\pi\)
\(240\) 0 0
\(241\) 1.84847 3.20164i 0.119070 0.206236i −0.800329 0.599561i \(-0.795342\pi\)
0.919400 + 0.393325i \(0.128675\pi\)
\(242\) 0 0
\(243\) −12.9722 8.64420i −0.832167 0.554526i
\(244\) 0 0
\(245\) 2.44949 4.24264i 0.156492 0.271052i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 5.44949 + 7.70674i 0.345347 + 0.488395i
\(250\) 0 0
\(251\) 2.89898 0.182982 0.0914910 0.995806i \(-0.470837\pi\)
0.0914910 + 0.995806i \(0.470837\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.44949 0.317837i 0.216015 0.0199037i
\(256\) 0 0
\(257\) −13.8990 24.0737i −0.866995 1.50168i −0.865053 0.501680i \(-0.832715\pi\)
−0.00194150 0.999998i \(-0.500618\pi\)
\(258\) 0 0
\(259\) 10.3485 17.9241i 0.643023 1.11375i
\(260\) 0 0
\(261\) −5.60102 + 1.04100i −0.346694 + 0.0644362i
\(262\) 0 0
\(263\) −9.00000 + 15.5885i −0.554964 + 0.961225i 0.442943 + 0.896550i \(0.353935\pi\)
−0.997906 + 0.0646755i \(0.979399\pi\)
\(264\) 0 0
\(265\) 3.89898 + 6.75323i 0.239512 + 0.414848i
\(266\) 0 0
\(267\) −2.02781 + 4.40156i −0.124100 + 0.269371i
\(268\) 0 0
\(269\) 22.5959 1.37770 0.688849 0.724905i \(-0.258116\pi\)
0.688849 + 0.724905i \(0.258116\pi\)
\(270\) 0 0
\(271\) 30.8990 1.87698 0.938490 0.345307i \(-0.112225\pi\)
0.938490 + 0.345307i \(0.112225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000 12.1244i 0.420589 0.728482i −0.575408 0.817867i \(-0.695157\pi\)
0.995997 + 0.0893846i \(0.0284900\pi\)
\(278\) 0 0
\(279\) −1.10102 + 3.11416i −0.0659164 + 0.186440i
\(280\) 0 0
\(281\) 4.05051 7.01569i 0.241633 0.418521i −0.719546 0.694444i \(-0.755650\pi\)
0.961180 + 0.275923i \(0.0889836\pi\)
\(282\) 0 0
\(283\) 1.82577 + 3.16232i 0.108530 + 0.187980i 0.915175 0.403056i \(-0.132052\pi\)
−0.806645 + 0.591037i \(0.798719\pi\)
\(284\) 0 0
\(285\) 11.8990 1.09638i 0.704835 0.0649437i
\(286\) 0 0
\(287\) −34.1464 −2.01560
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.00000 2.82843i −0.117242 0.165805i
\(292\) 0 0
\(293\) −8.79796 15.2385i −0.513982 0.890243i −0.999868 0.0162213i \(-0.994836\pi\)
0.485886 0.874022i \(-0.338497\pi\)
\(294\) 0 0
\(295\) −0.550510 + 0.953512i −0.0320519 + 0.0555156i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −20.3485 35.2446i −1.17287 2.03146i
\(302\) 0 0
\(303\) 2.00000 + 2.82843i 0.114897 + 0.162489i
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) −4.75255 −0.271242 −0.135621 0.990761i \(-0.543303\pi\)
−0.135621 + 0.990761i \(0.543303\pi\)
\(308\) 0 0
\(309\) 17.2474 1.58919i 0.981173 0.0904056i
\(310\) 0 0
\(311\) 3.44949 + 5.97469i 0.195603 + 0.338794i 0.947098 0.320945i \(-0.104000\pi\)
−0.751495 + 0.659738i \(0.770667\pi\)
\(312\) 0 0
\(313\) 7.79796 13.5065i 0.440767 0.763430i −0.556980 0.830526i \(-0.688040\pi\)
0.997747 + 0.0670957i \(0.0213733\pi\)
\(314\) 0 0
\(315\) 6.72474 + 7.86566i 0.378896 + 0.443180i
\(316\) 0 0
\(317\) 12.8990 22.3417i 0.724479 1.25483i −0.234709 0.972066i \(-0.575414\pi\)
0.959188 0.282769i \(-0.0912528\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8.94949 19.4258i 0.499512 1.08424i
\(322\) 0 0
\(323\) 13.7980 0.767739
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.82577 16.9866i 0.432766 0.939362i
\(328\) 0 0
\(329\) 16.2980 + 28.2289i 0.898536 + 1.55631i
\(330\) 0 0
\(331\) 3.44949 5.97469i 0.189601 0.328399i −0.755516 0.655130i \(-0.772614\pi\)
0.945117 + 0.326731i \(0.105947\pi\)
\(332\) 0 0
\(333\) 11.6969 + 13.6814i 0.640988 + 0.749738i
\(334\) 0 0
\(335\) −6.62372 + 11.4726i −0.361893 + 0.626817i
\(336\) 0 0
\(337\) 14.8990 + 25.8058i 0.811599 + 1.40573i 0.911744 + 0.410758i \(0.134736\pi\)
−0.100145 + 0.994973i \(0.531931\pi\)
\(338\) 0 0
\(339\) −16.8990 + 1.55708i −0.917827 + 0.0845689i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.24745 −0.391325
\(344\) 0 0
\(345\) −7.44949 10.5352i −0.401067 0.567194i
\(346\) 0 0
\(347\) 12.7980 + 22.1667i 0.687030 + 1.18997i 0.972794 + 0.231671i \(0.0744194\pi\)
−0.285764 + 0.958300i \(0.592247\pi\)
\(348\) 0 0
\(349\) 1.15153 1.99451i 0.0616400 0.106764i −0.833559 0.552431i \(-0.813700\pi\)
0.895199 + 0.445667i \(0.147034\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 4.89898 + 8.48528i 0.260011 + 0.450352i
\(356\) 0 0
\(357\) 6.89898 + 9.75663i 0.365133 + 0.516376i
\(358\) 0 0
\(359\) 28.6969 1.51457 0.757283 0.653087i \(-0.226526\pi\)
0.757283 + 0.653087i \(0.226526\pi\)
\(360\) 0 0
\(361\) 28.5959 1.50505
\(362\) 0 0
\(363\) −18.9722 + 1.74810i −0.995782 + 0.0917517i
\(364\) 0 0
\(365\) 6.89898 + 11.9494i 0.361109 + 0.625459i
\(366\) 0 0
\(367\) −11.0000 + 19.0526i −0.574195 + 0.994535i 0.421933 + 0.906627i \(0.361352\pi\)
−0.996129 + 0.0879086i \(0.971982\pi\)
\(368\) 0 0
\(369\) 9.89898 27.9985i 0.515320 1.45755i
\(370\) 0 0
\(371\) −13.4495 + 23.2952i −0.698263 + 1.20943i
\(372\) 0 0
\(373\) −10.7980 18.7026i −0.559097 0.968385i −0.997572 0.0696412i \(-0.977815\pi\)
0.438475 0.898743i \(-0.355519\pi\)
\(374\) 0 0
\(375\) 0.724745 1.57313i 0.0374257 0.0812362i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 26.8990 1.38171 0.690854 0.722994i \(-0.257235\pi\)
0.690854 + 0.722994i \(0.257235\pi\)
\(380\) 0 0
\(381\) −0.252551 + 0.548188i −0.0129386 + 0.0280845i
\(382\) 0 0
\(383\) −1.00000 1.73205i −0.0510976 0.0885037i 0.839345 0.543599i \(-0.182939\pi\)
−0.890443 + 0.455095i \(0.849605\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 34.7980 6.46750i 1.76888 0.328762i
\(388\) 0 0
\(389\) 8.29796 14.3725i 0.420723 0.728714i −0.575287 0.817952i \(-0.695110\pi\)
0.996010 + 0.0892375i \(0.0284430\pi\)
\(390\) 0 0
\(391\) −7.44949 12.9029i −0.376737 0.652527i
\(392\) 0 0
\(393\) 5.00000 0.460702i 0.252217 0.0232393i
\(394\) 0 0
\(395\) 6.89898 0.347125
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 23.7980 + 33.6554i 1.19139 + 1.68488i
\(400\) 0 0
\(401\) −12.7980 22.1667i −0.639100 1.10695i −0.985631 0.168914i \(-0.945974\pi\)
0.346531 0.938038i \(-0.387359\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −8.39898 + 3.23375i −0.417349 + 0.160686i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.79796 + 8.31031i 0.237244 + 0.410918i 0.959922 0.280266i \(-0.0904226\pi\)
−0.722679 + 0.691184i \(0.757089\pi\)
\(410\) 0 0
\(411\) −19.5959 27.7128i −0.966595 1.36697i
\(412\) 0 0
\(413\) −3.79796 −0.186885
\(414\) 0 0
\(415\) 5.44949 0.267505
\(416\) 0 0
\(417\) 33.7980 3.11416i 1.65509 0.152501i
\(418\) 0 0
\(419\) −10.0000 17.3205i −0.488532 0.846162i 0.511381 0.859354i \(-0.329134\pi\)
−0.999913 + 0.0131919i \(0.995801\pi\)
\(420\) 0 0
\(421\) −12.7980 + 22.1667i −0.623734 + 1.08034i 0.365050 + 0.930988i \(0.381052\pi\)
−0.988784 + 0.149352i \(0.952281\pi\)
\(422\) 0 0
\(423\) −27.8712 + 5.18010i −1.35514 + 0.251865i
\(424\) 0 0
\(425\) 1.00000 1.73205i 0.0485071 0.0840168i
\(426\) 0 0
\(427\) 5.17423 + 8.96204i 0.250399 + 0.433703i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.8990 0.717659 0.358829 0.933403i \(-0.383176\pi\)
0.358829 + 0.933403i \(0.383176\pi\)
\(432\) 0 0
\(433\) −11.7980 −0.566974 −0.283487 0.958976i \(-0.591491\pi\)
−0.283487 + 0.958976i \(0.591491\pi\)
\(434\) 0 0
\(435\) −1.37628 + 2.98735i −0.0659874 + 0.143232i
\(436\) 0 0
\(437\) −25.6969 44.5084i −1.22925 2.12913i
\(438\) 0 0
\(439\) 8.89898 15.4135i 0.424725 0.735645i −0.571670 0.820484i \(-0.693704\pi\)
0.996395 + 0.0848384i \(0.0270374\pi\)
\(440\) 0 0
\(441\) −4.89898 + 13.8564i −0.233285 + 0.659829i
\(442\) 0 0
\(443\) 4.07321 7.05501i 0.193524 0.335194i −0.752892 0.658145i \(-0.771342\pi\)
0.946416 + 0.322951i \(0.104675\pi\)
\(444\) 0 0
\(445\) 1.39898 + 2.42310i 0.0663180 + 0.114866i
\(446\) 0 0
\(447\) −36.2196 + 3.33729i −1.71313 + 0.157848i
\(448\) 0 0
\(449\) 21.5959 1.01917 0.509587 0.860419i \(-0.329798\pi\)
0.509587 + 0.860419i \(0.329798\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −12.0000 16.9706i −0.563809 0.797347i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.79796 + 15.2385i −0.411551 + 0.712828i −0.995060 0.0992796i \(-0.968346\pi\)
0.583508 + 0.812107i \(0.301680\pi\)
\(458\) 0 0
\(459\) −10.0000 + 2.82843i −0.466760 + 0.132020i
\(460\) 0 0
\(461\) −6.05051 + 10.4798i −0.281800 + 0.488093i −0.971828 0.235690i \(-0.924265\pi\)
0.690028 + 0.723783i \(0.257598\pi\)
\(462\) 0 0
\(463\) −6.10102 10.5673i −0.283538 0.491103i 0.688715 0.725032i \(-0.258175\pi\)
−0.972254 + 0.233929i \(0.924842\pi\)
\(464\) 0 0
\(465\) 1.10102 + 1.55708i 0.0510586 + 0.0722078i
\(466\) 0 0
\(467\) 7.79796 0.360847 0.180423 0.983589i \(-0.442253\pi\)
0.180423 + 0.983589i \(0.442253\pi\)
\(468\) 0 0
\(469\) −45.6969 −2.11009
\(470\) 0 0
\(471\) −41.0454 + 3.78194i −1.89127 + 0.174263i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.44949 5.97469i 0.158273 0.274138i
\(476\) 0 0
\(477\) −15.2020 17.7812i −0.696054 0.814146i
\(478\) 0 0
\(479\) −15.2474 + 26.4094i −0.696674 + 1.20667i 0.272939 + 0.962031i \(0.412004\pi\)
−0.969613 + 0.244643i \(0.921329\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 18.6237 40.4247i 0.847409 1.83939i
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −13.5959 −0.616090 −0.308045 0.951372i \(-0.599675\pi\)
−0.308045 + 0.951372i \(0.599675\pi\)
\(488\) 0 0
\(489\) 5.65153 12.2672i 0.255571 0.554743i
\(490\) 0 0
\(491\) 6.89898 + 11.9494i 0.311347 + 0.539268i 0.978654 0.205514i \(-0.0658867\pi\)
−0.667308 + 0.744782i \(0.732553\pi\)
\(492\) 0 0
\(493\) −1.89898 + 3.28913i −0.0855257 + 0.148135i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.8990 + 29.2699i −0.758023 + 1.31293i
\(498\) 0 0
\(499\) 14.0000 + 24.2487i 0.626726 + 1.08552i 0.988204 + 0.153141i \(0.0489388\pi\)
−0.361478 + 0.932381i \(0.617728\pi\)
\(500\) 0 0
\(501\) 14.3990 1.32673i 0.643299 0.0592738i
\(502\) 0 0
\(503\) 25.0454 1.11672 0.558360 0.829599i \(-0.311431\pi\)
0.558360 + 0.829599i \(0.311431\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 13.0000 + 18.3848i 0.577350 + 0.816497i
\(508\) 0 0
\(509\) 2.05051 + 3.55159i 0.0908873 + 0.157421i 0.907885 0.419220i \(-0.137696\pi\)
−0.816997 + 0.576641i \(0.804363\pi\)
\(510\) 0 0
\(511\) −23.7980 + 41.2193i −1.05276 + 1.82343i
\(512\) 0 0
\(513\) −34.4949 + 9.75663i −1.52299 + 0.430766i
\(514\) 0 0
\(515\) 5.00000 8.66025i 0.220326 0.381616i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 + 16.9706i 0.526742 + 0.744925i
\(520\) 0 0
\(521\) 9.20204 0.403149 0.201574 0.979473i \(-0.435394\pi\)
0.201574 + 0.979473i \(0.435394\pi\)
\(522\) 0 0
\(523\) −34.3485 −1.50195 −0.750977 0.660329i \(-0.770417\pi\)
−0.750977 + 0.660329i \(0.770417\pi\)
\(524\) 0 0
\(525\) 5.94949 0.548188i 0.259657 0.0239249i
\(526\) 0 0
\(527\) 1.10102 + 1.90702i 0.0479612 + 0.0830712i
\(528\) 0 0
\(529\) −16.2474 + 28.1414i −0.706411 + 1.22354i
\(530\) 0 0
\(531\) 1.10102 3.11416i 0.0477802 0.135143i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.17423 10.6941i −0.266935 0.462346i
\(536\) 0 0
\(537\) 10.0000 21.7060i 0.431532 0.936684i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.1010 0.520264 0.260132 0.965573i \(-0.416234\pi\)
0.260132 + 0.965573i \(0.416234\pi\)
\(542\) 0 0
\(543\) 7.02781 15.2546i 0.301592 0.654636i
\(544\) 0 0
\(545\) −5.39898 9.35131i −0.231267 0.400566i
\(546\) 0 0
\(547\) 8.37628 14.5081i 0.358144 0.620323i −0.629507 0.776995i \(-0.716743\pi\)
0.987651 + 0.156672i \(0.0500765\pi\)
\(548\) 0 0
\(549\) −8.84847 + 1.64456i −0.377643 + 0.0701883i
\(550\) 0 0
\(551\) −6.55051 + 11.3458i −0.279061 + 0.483348i
\(552\) 0 0
\(553\) 11.8990 + 20.6096i 0.505996 + 0.876411i
\(554\) 0 0
\(555\) 10.3485 0.953512i 0.439268 0.0404743i
\(556\) 0 0
\(557\) 43.7980 1.85578 0.927890 0.372855i \(-0.121621\pi\)
0.927890 + 0.372855i \(0.121621\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.17423 + 8.96204i 0.218068 + 0.377705i 0.954217 0.299114i \(-0.0966912\pi\)
−0.736149 + 0.676819i \(0.763358\pi\)
\(564\) 0 0
\(565\) −4.89898 + 8.48528i −0.206102 + 0.356978i
\(566\) 0 0
\(567\) −24.1464 19.5133i −1.01405 0.819480i
\(568\) 0 0
\(569\) −5.00000 + 8.66025i −0.209611 + 0.363057i −0.951592 0.307364i \(-0.900553\pi\)
0.741981 + 0.670421i \(0.233886\pi\)
\(570\) 0 0
\(571\) −12.8990 22.3417i −0.539805 0.934971i −0.998914 0.0465904i \(-0.985164\pi\)
0.459109 0.888380i \(-0.348169\pi\)
\(572\) 0 0
\(573\) 14.8990 + 21.0703i 0.622414 + 0.880226i
\(574\) 0 0
\(575\) −7.44949 −0.310665
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 0 0
\(579\) −3.79796 + 0.349945i −0.157838 + 0.0145432i
\(580\) 0 0
\(581\) 9.39898 + 16.2795i 0.389935 + 0.675388i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.82577 + 6.62642i −0.157906 + 0.273502i −0.934113 0.356976i \(-0.883808\pi\)
0.776207 + 0.630478i \(0.217141\pi\)
\(588\) 0 0
\(589\) 3.79796 + 6.57826i 0.156492 + 0.271052i
\(590\) 0 0
\(591\) 11.3031 24.5344i 0.464946 1.00921i
\(592\) 0 0
\(593\) −31.3939 −1.28919 −0.644596 0.764523i \(-0.722974\pi\)
−0.644596 + 0.764523i \(0.722974\pi\)
\(594\) 0 0
\(595\) 6.89898 0.282831
\(596\) 0 0
\(597\) −16.5959 + 36.0231i −0.679226 + 1.47433i
\(598\) 0 0
\(599\) 12.3485 + 21.3882i 0.504545 + 0.873897i 0.999986 + 0.00525583i \(0.00167299\pi\)
−0.495441 + 0.868641i \(0.664994\pi\)
\(600\) 0 0
\(601\) −7.00000 + 12.1244i −0.285536 + 0.494563i −0.972739 0.231903i \(-0.925505\pi\)
0.687203 + 0.726465i \(0.258838\pi\)
\(602\) 0 0
\(603\) 13.2474 37.4694i 0.539478 1.52587i
\(604\) 0 0
\(605\) −5.50000 + 9.52628i −0.223607 + 0.387298i
\(606\) 0 0
\(607\) −1.07321 1.85886i −0.0435604 0.0754489i 0.843423 0.537250i \(-0.180537\pi\)
−0.886984 + 0.461801i \(0.847203\pi\)
\(608\) 0 0
\(609\) −11.2980 + 1.04100i −0.457816 + 0.0421834i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −29.5959 −1.19537 −0.597684 0.801732i \(-0.703912\pi\)
−0.597684 + 0.801732i \(0.703912\pi\)
\(614\) 0 0
\(615\) −9.89898 13.9993i −0.399165 0.564505i
\(616\) 0 0
\(617\) 17.5959 + 30.4770i 0.708385 + 1.22696i 0.965456 + 0.260566i \(0.0839092\pi\)
−0.257071 + 0.966393i \(0.582757\pi\)
\(618\) 0 0
\(619\) 12.3485 21.3882i 0.496327 0.859663i −0.503664 0.863900i \(-0.668015\pi\)
0.999991 + 0.00423617i \(0.00134842\pi\)
\(620\) 0 0
\(621\) 27.7474 + 26.9897i 1.11347 + 1.08306i
\(622\) 0 0
\(623\) −4.82577 + 8.35847i −0.193340 + 0.334875i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −15.5959 −0.620864 −0.310432 0.950596i \(-0.600474\pi\)
−0.310432 + 0.950596i \(0.600474\pi\)
\(632\) 0 0
\(633\) 20.6969 1.90702i 0.822629 0.0757974i
\(634\) 0 0
\(635\) 0.174235 + 0.301783i 0.00691429 + 0.0119759i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −19.1010 22.3417i −0.755625 0.883824i
\(640\) 0 0
\(641\) 6.94949 12.0369i 0.274488 0.475428i −0.695518 0.718509i \(-0.744825\pi\)
0.970006 + 0.243081i \(0.0781582\pi\)
\(642\) 0 0
\(643\) 10.0732 + 17.4473i 0.397249 + 0.688055i 0.993385 0.114828i \(-0.0366317\pi\)
−0.596137 + 0.802883i \(0.703298\pi\)
\(644\) 0 0
\(645\) 8.55051 18.5597i 0.336676 0.730789i
\(646\) 0 0
\(647\) 6.55051 0.257527 0.128764 0.991675i \(-0.458899\pi\)
0.128764 + 0.991675i \(0.458899\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.75255 + 5.97469i −0.107881 + 0.234167i
\(652\) 0 0
\(653\) −13.8990 24.0737i −0.543909 0.942078i −0.998675 0.0514670i \(-0.983610\pi\)
0.454766 0.890611i \(-0.349723\pi\)
\(654\) 0 0
\(655\) 1.44949 2.51059i 0.0566363 0.0980969i
\(656\) 0 0
\(657\) −26.8990 31.4626i −1.04943 1.22747i
\(658\) 0 0
\(659\) −5.10102 + 8.83523i −0.198708 + 0.344172i −0.948110 0.317944i \(-0.897008\pi\)
0.749402 + 0.662115i \(0.230341\pi\)
\(660\) 0 0
\(661\) −5.00000 8.66025i −0.194477 0.336845i 0.752252 0.658876i \(-0.228968\pi\)
−0.946729 + 0.322031i \(0.895634\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 23.7980 0.922845
\(666\) 0 0
\(667\) 14.1464 0.547752
\(668\) 0 0
\(669\) −5.44949 7.70674i −0.210689 0.297960i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000 13.8564i 0.308377 0.534125i −0.669630 0.742695i \(-0.733547\pi\)
0.978008 + 0.208569i \(0.0668807\pi\)
\(674\) 0 0
\(675\) −1.27526 + 5.03723i −0.0490846 + 0.193883i
\(676\) 0 0
\(677\) −7.10102 + 12.2993i −0.272914 + 0.472702i −0.969607 0.244668i \(-0.921321\pi\)
0.696692 + 0.717370i \(0.254654\pi\)
\(678\) 0 0
\(679\) −3.44949 5.97469i −0.132379 0.229288i
\(680\) 0 0
\(681\) 8.20204 + 11.5994i 0.314303 + 0.444491i
\(682\) 0 0
\(683\) 23.3939 0.895142 0.447571 0.894248i \(-0.352289\pi\)
0.447571 + 0.894248i \(0.352289\pi\)
\(684\) 0 0
\(685\) −19.5959 −0.748722
\(686\) 0 0
\(687\) −7.07321 + 0.651729i −0.269860 + 0.0248650i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −19.4495 + 33.6875i −0.739893 + 1.28153i 0.212649 + 0.977129i \(0.431791\pi\)
−0.952543 + 0.304404i \(0.901543\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.79796 16.9706i 0.371658 0.643730i
\(696\) 0 0
\(697\) −9.89898 17.1455i −0.374951 0.649433i
\(698\) 0 0
\(699\) 14.3485 31.1448i 0.542709 1.17800i
\(700\) 0 0
\(701\) 30.3939 1.14796 0.573980 0.818869i \(-0.305399\pi\)
0.573980 + 0.818869i \(0.305399\pi\)
\(702\) 0 0
\(703\) 41.3939 1.56120
\(704\) 0 0
\(705\) −6.84847 + 14.8653i −0.257928 + 0.559859i
\(706\) 0 0
\(707\) 3.44949 + 5.97469i 0.129731 + 0.224701i
\(708\) 0 0
\(709\) −4.84847 + 8.39780i −0.182088 + 0.315386i −0.942591 0.333948i \(-0.891619\pi\)
0.760503 + 0.649334i \(0.224952\pi\)
\(710\) 0 0
\(711\) −20.3485 + 3.78194i −0.763127 + 0.141834i
\(712\) 0 0
\(713\) 4.10102 7.10318i 0.153584 0.266016i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.79796 0.349945i 0.141837 0.0130689i
\(718\) 0 0
\(719\) −28.2929 −1.05515 −0.527573 0.849510i \(-0.676898\pi\)
−0.527573 + 0.849510i \(0.676898\pi\)
\(720\) 0 0
\(721\) 34.4949 1.28466
\(722\) 0 0
\(723\) 3.69694 + 5.22826i 0.137491 + 0.194441i
\(724\) 0 0
\(725\) 0.949490 + 1.64456i 0.0352632 + 0.0610776i
\(726\) 0 0
\(727\) 15.6237 27.0611i 0.579452 1.00364i −0.416090 0.909323i \(-0.636600\pi\)
0.995542 0.0943168i \(-0.0300667\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 11.7980 20.4347i 0.436363 0.755803i
\(732\) 0 0
\(733\) 14.7980 + 25.6308i 0.546575 + 0.946696i 0.998506 + 0.0546429i \(0.0174020\pi\)
−0.451931 + 0.892053i \(0.649265\pi\)
\(734\) 0 0
\(735\) 4.89898 + 6.92820i 0.180702 + 0.255551i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −33.1010 −1.21764 −0.608820 0.793308i \(-0.708357\pi\)
−0.608820 + 0.793308i \(0.708357\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.52270 + 6.10150i 0.129235 + 0.223842i 0.923381 0.383886i \(-0.125414\pi\)
−0.794145 + 0.607728i \(0.792081\pi\)
\(744\) 0 0
\(745\) −10.5000 + 18.1865i −0.384690 + 0.666303i
\(746\) 0 0
\(747\) −16.0732 + 2.98735i −0.588088 + 0.109301i
\(748\) 0 0
\(749\) 21.2980 36.8891i 0.778210 1.34790i
\(750\) 0 0
\(751\) −13.1010 22.6916i −0.478063 0.828029i 0.521621 0.853177i \(-0.325328\pi\)
−0.999684 + 0.0251480i \(0.991994\pi\)
\(752\) 0 0
\(753\) −2.10102 + 4.56048i −0.0765654 + 0.166193i
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 31.5959 1.14837 0.574187 0.818724i \(-0.305318\pi\)
0.574187 + 0.818724i \(0.305318\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.5000 30.3109i −0.634375 1.09877i −0.986647 0.162872i \(-0.947924\pi\)
0.352273 0.935897i \(-0.385409\pi\)
\(762\) 0 0
\(763\) 18.6237 32.2572i 0.674224 1.16779i
\(764\) 0 0
\(765\) −2.00000 + 5.65685i −0.0723102 + 0.204524i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.60102 + 4.50510i 0.0937952 + 0.162458i 0.909105 0.416567i \(-0.136767\pi\)
−0.815310 + 0.579025i \(0.803433\pi\)
\(770\) 0 0
\(771\) 47.9444 4.41761i 1.72667 0.159096i
\(772\) 0 0
\(773\) 6.20204 0.223072 0.111536 0.993760i \(-0.464423\pi\)
0.111536 + 0.993760i \(0.464423\pi\)
\(774\) 0 0
\(775\) 1.10102 0.0395498
\(776\) 0 0
\(777\) 20.6969 + 29.2699i 0.742499 + 1.05005i
\(778\) 0 0
\(779\) −34.1464 59.1433i −1.22342 2.11903i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 2.42168 9.56560i 0.0865439 0.341847i
\(784\) 0 0
\(785\) −11.8990 + 20.6096i −0.424693 + 0.735589i
\(786\) 0 0
\(787\) 24.7980 + 42.9513i 0.883952 + 1.53105i 0.846910 + 0.531736i \(0.178460\pi\)
0.0370414 + 0.999314i \(0.488207\pi\)
\(788\) 0 0
\(789\) −18.0000 25.4558i −0.640817 0.906252i
\(790\) 0 0
\(791\) −33.7980 −1.20172
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −13.4495 + 1.23924i −0.477004 + 0.0439513i
\(796\) 0 0
\(797\) 4.00000 + 6.92820i 0.141687 + 0.245410i 0.928132 0.372251i \(-0.121414\pi\)
−0.786445 + 0.617661i \(0.788081\pi\)
\(798\) 0 0
\(799\) −9.44949 + 16.3670i −0.334299 + 0.579023i
\(800\) 0 0
\(801\) −5.45459 6.38002i −0.192729 0.225427i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −12.8485 22.2542i −0.452849 0.784358i
\(806\) 0 0
\(807\) −16.3763 + 35.5464i −0.576472 + 1.25129i
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 13.1010 0.460039 0.230020 0.973186i \(-0.426121\pi\)
0.230020 + 0.973186i \(0.426121\pi\)
\(812\) 0 0
\(813\) −22.3939 + 48.6082i −0.785388 + 1.70476i
\(814\) 0 0
\(815\) −3.89898 6.75323i −0.136575 0.236555i
\(816\) 0 0
\(817\) 40.6969 70.4892i 1.42381 2.46610i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.39898 + 2.42310i −0.0488247 + 0.0845669i −0.889405 0.457120i \(-0.848881\pi\)
0.840580 + 0.541687i \(0.182214\pi\)
\(822\) 0 0
\(823\) 4.17423 + 7.22999i 0.145505 + 0.252021i 0.929561 0.368668i \(-0.120186\pi\)
−0.784056 + 0.620690i \(0.786853\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 53.5403 1.86178 0.930889 0.365301i \(-0.119034\pi\)
0.930889 + 0.365301i \(0.119034\pi\)
\(828\) 0 0
\(829\) 50.1918 1.74323 0.871617 0.490187i \(-0.163072\pi\)
0.871617 + 0.490187i \(0.163072\pi\)
\(830\) 0 0
\(831\) 14.0000 + 19.7990i 0.485655 + 0.686819i
\(832\) 0 0
\(833\) 4.89898 + 8.48528i 0.169740 + 0.293998i
\(834\) 0 0
\(835\) 4.17423 7.22999i 0.144455 0.250204i
\(836\) 0 0
\(837\) −4.10102 3.98902i −0.141752 0.137881i
\(838\) 0 0
\(839\) −2.34847 + 4.06767i −0.0810782 + 0.140431i −0.903713 0.428138i \(-0.859170\pi\)
0.822635 + 0.568570i \(0.192503\pi\)
\(840\) 0 0
\(841\) 12.6969 + 21.9917i 0.437825 + 0.758336i
\(842\) 0 0
\(843\) 8.10102 + 11.4566i 0.279014 + 0.394585i
\(844\) 0 0
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −37.9444 −1.30378
\(848\) 0 0
\(849\) −6.29796 + 0.580296i −0.216145 + 0.0199157i
\(850\) 0 0
\(851\) −22.3485 38.7087i −0.766096 1.32692i
\(852\) 0 0
\(853\) −9.10102 + 15.7634i −0.311613 + 0.539730i −0.978712 0.205240i \(-0.934203\pi\)
0.667099 + 0.744969i \(0.267536\pi\)
\(854\) 0 0
\(855\) −6.89898 + 19.5133i −0.235940 + 0.667339i
\(856\) 0 0
\(857\) −0.202041 + 0.349945i −0.00690159 + 0.0119539i −0.869456 0.494011i \(-0.835530\pi\)
0.862554 + 0.505965i \(0.168864\pi\)
\(858\) 0 0
\(859\) −19.5959 33.9411i −0.668604 1.15806i −0.978295 0.207219i \(-0.933559\pi\)
0.309691 0.950837i \(-0.399775\pi\)
\(860\) 0 0
\(861\) 24.7474 53.7168i 0.843391 1.83067i
\(862\) 0 0
\(863\) 57.4495 1.95560 0.977802 0.209532i \(-0.0671942\pi\)
0.977802 + 0.209532i \(0.0671942\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 9.42168 20.4507i 0.319977 0.694543i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 5.89898 1.09638i 0.199650 0.0371067i
\(874\) 0 0
\(875\) 1.72474 2.98735i 0.0583070 0.100991i
\(876\) 0 0
\(877\) 0.202041 + 0.349945i 0.00682244 + 0.0118168i 0.869416 0.494080i \(-0.164495\pi\)
−0.862594 + 0.505897i \(0.831162\pi\)
\(878\) 0 0
\(879\) 30.3485 2.79632i 1.02363 0.0943175i
\(880\) 0 0
\(881\) 6.10102 0.205549 0.102774 0.994705i \(-0.467228\pi\)
0.102774 + 0.994705i \(0.467228\pi\)
\(882\) 0 0
\(883\) −0.954592 −0.0321246 −0.0160623 0.999871i \(-0.505113\pi\)
−0.0160623 + 0.999871i \(0.505113\pi\)
\(884\) 0 0
\(885\) −1.10102 1.55708i −0.0370104 0.0523406i
\(886\) 0 0
\(887\) −0.797959 1.38211i −0.0267928 0.0464066i 0.852318 0.523024i \(-0.175196\pi\)
−0.879111 + 0.476617i \(0.841863\pi\)
\(888\) 0 0
\(889\) −0.601021 + 1.04100i −0.0201576 + 0.0349140i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32.5959 + 56.4578i −1.09078 + 1.88929i
\(894\) 0 0
\(895\) −6.89898 11.9494i −0.230607 0.399424i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.09082 −0.0697326
\(900\) 0 0
\(901\) −15.5959 −0.519575
\(902\) 0 0
\(903\) 70.1918 6.46750i 2.33584 0.215225i
\(904\) 0 0
\(905\) −4.84847 8.39780i −0.161169 0.279152i
\(906\) 0 0
\(907\) 10.8258 18.7508i 0.359464 0.622609i −0.628408 0.777884i \(-0.716293\pi\)
0.987871 + 0.155275i \(0.0496263\pi\)
\(908\) 0 0
\(909\) −5.89898 + 1.09638i −0.195657 + 0.0363645i
\(910\) 0 0
\(911\) −5.24745 + 9.08885i −0.173856 + 0.301127i −0.939765 0.341822i \(-0.888956\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −2.17423 + 4.71940i −0.0718780 + 0.156018i
\(916\) 0 0
\(917\) 10.0000 0.330229
\(918\) 0 0
\(919\) 24.6969 0.814677 0.407338 0.913277i \(-0.366457\pi\)
0.407338 + 0.913277i \(0.366457\pi\)
\(920\) 0 0
\(921\) 3.44439 7.47639i 0.113496 0.246355i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.00000 5.19615i 0.0986394 0.170848i
\(926\) 0 0
\(927\) −10.0000 + 28.2843i −0.328443 + 0.928977i
\(928\) 0 0
\(929\) −18.7980 + 32.5590i −0.616741 + 1.06823i 0.373335 + 0.927696i \(0.378214\pi\)
−0.990076 + 0.140530i \(0.955119\pi\)
\(930\) 0 0
\(931\) 16.8990 + 29.2699i 0.553842 + 0.959282i
\(932\) 0 0
\(933\) −11.8990 + 1.09638i −0.389555 + 0.0358937i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40.9898 −1.33908 −0.669539 0.742777i \(-0.733508\pi\)
−0.669539 + 0.742777i \(0.733508\pi\)
\(938\) 0 0
\(939\) 15.5959 + 22.0560i 0.508954 + 0.719769i
\(940\) 0 0
\(941\) 0.848469 + 1.46959i 0.0276593 + 0.0479073i 0.879524 0.475855i \(-0.157861\pi\)
−0.851864 + 0.523762i \(0.824528\pi\)
\(942\) 0 0
\(943\) −36.8712 + 63.8627i −1.20069 + 2.07966i
\(944\) 0 0
\(945\) −17.2474 + 4.87832i −0.561060 + 0.158692i
\(946\) 0 0
\(947\) 5.92679 10.2655i 0.192595 0.333584i −0.753515 0.657431i \(-0.771643\pi\)
0.946109 + 0.323847i \(0.104976\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 25.7980 + 36.4838i 0.836556 + 1.18307i
\(952\) 0 0
\(953\) −14.2020 −0.460049 −0.230025 0.973185i \(-0.573881\pi\)
−0.230025 + 0.973185i \(0.573881\pi\)
\(954\) 0 0
\(955\) 14.8990 0.482120
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −33.7980 58.5398i −1.09139 1.89035i
\(960\) 0 0
\(961\) 14.8939 25.7970i 0.480448 0.832160i
\(962\) 0 0
\(963\) 24.0732 + 28.1575i 0.775748 + 0.907361i
\(964\) 0 0
\(965\) −1.10102 + 1.90702i −0.0354431 + 0.0613893i
\(966\) 0 0
\(967\) −24.4217 42.2996i −0.785348 1.36026i −0.928791 0.370605i \(-0.879150\pi\)
0.143442 0.989659i \(-0.454183\pi\)
\(968\) 0 0
\(969\) −10.0000 + 21.7060i −0.321246 + 0.697298i
\(970\) 0 0
\(971\) 31.3031 1.00456 0.502282 0.864704i \(-0.332494\pi\)
0.502282 + 0.864704i \(0.332494\pi\)
\(972\) 0 0
\(973\) 67.5959 2.16703
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.69694 + 11.5994i 0.214254 + 0.371099i 0.953042 0.302840i \(-0.0979346\pi\)
−0.738788 + 0.673938i \(0.764601\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 21.0505 + 24.6219i 0.672091 + 0.786118i
\(982\) 0 0
\(983\) 5.72474 9.91555i 0.182591 0.316257i −0.760171 0.649723i \(-0.774885\pi\)
0.942762 + 0.333466i \(0.108218\pi\)
\(984\) 0 0
\(985\) −7.79796 13.5065i −0.248464 0.430352i
\(986\) 0 0
\(987\) −56.2196 + 5.18010i −1.78949 + 0.164884i
\(988\) 0 0
\(989\) −87.8888 −2.79470
\(990\) 0 0
\(991\) 28.6969 0.911588 0.455794 0.890085i \(-0.349355\pi\)
0.455794 + 0.890085i \(0.349355\pi\)
\(992\) 0 0
\(993\) 6.89898 + 9.75663i 0.218932 + 0.309617i
\(994\) 0 0
\(995\) 11.4495 + 19.8311i 0.362973 + 0.628688i
\(996\) 0 0
\(997\) 11.6969 20.2597i 0.370446 0.641631i −0.619188 0.785242i \(-0.712538\pi\)
0.989634 + 0.143612i \(0.0458716\pi\)
\(998\) 0 0
\(999\) −30.0000 + 8.48528i −0.949158 + 0.268462i
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 360.2.q.c.241.1 yes 4
3.2 odd 2 1080.2.q.c.721.1 4
4.3 odd 2 720.2.q.g.241.2 4
9.2 odd 6 3240.2.a.o.1.2 2
9.4 even 3 inner 360.2.q.c.121.1 4
9.5 odd 6 1080.2.q.c.361.1 4
9.7 even 3 3240.2.a.j.1.2 2
12.11 even 2 2160.2.q.g.721.2 4
36.7 odd 6 6480.2.a.bc.1.1 2
36.11 even 6 6480.2.a.bl.1.1 2
36.23 even 6 2160.2.q.g.1441.2 4
36.31 odd 6 720.2.q.g.481.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.c.121.1 4 9.4 even 3 inner
360.2.q.c.241.1 yes 4 1.1 even 1 trivial
720.2.q.g.241.2 4 4.3 odd 2
720.2.q.g.481.2 4 36.31 odd 6
1080.2.q.c.361.1 4 9.5 odd 6
1080.2.q.c.721.1 4 3.2 odd 2
2160.2.q.g.721.2 4 12.11 even 2
2160.2.q.g.1441.2 4 36.23 even 6
3240.2.a.j.1.2 2 9.7 even 3
3240.2.a.o.1.2 2 9.2 odd 6
6480.2.a.bc.1.1 2 36.7 odd 6
6480.2.a.bl.1.1 2 36.11 even 6