Properties

Label 2352.2.q.bg.1537.2
Level $2352$
Weight $2$
Character 2352.1537
Analytic conductor $18.781$
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] = [2352,2,Mod(961,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (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: \(18.7808145554\)
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: no (minimal twist has level 1176)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1537
Dual form 2352.2.q.bg.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(1.70711 + 2.95680i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(1.70711 + 2.95680i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(0.414214 - 0.717439i) q^{11} +4.24264 q^{13} +3.41421 q^{15} +(3.70711 - 6.42090i) q^{17} +(-3.41421 - 5.91359i) q^{19} +(-2.41421 - 4.18154i) q^{23} +(-3.32843 + 5.76500i) q^{25} -1.00000 q^{27} +2.82843 q^{29} +(1.41421 - 2.44949i) q^{31} +(-0.414214 - 0.717439i) q^{33} +(0.828427 + 1.43488i) q^{37} +(2.12132 - 3.67423i) q^{39} -10.2426 q^{41} +11.3137 q^{43} +(1.70711 - 2.95680i) q^{45} +(2.24264 + 3.88437i) q^{47} +(-3.70711 - 6.42090i) q^{51} +(1.00000 - 1.73205i) q^{53} +2.82843 q^{55} -6.82843 q^{57} +(-4.24264 + 7.34847i) q^{59} +(5.53553 + 9.58783i) q^{61} +(7.24264 + 12.5446i) q^{65} +(5.65685 - 9.79796i) q^{67} -4.82843 q^{69} +10.4853 q^{71} +(3.87868 - 6.71807i) q^{73} +(3.32843 + 5.76500i) q^{75} +(6.82843 + 11.8272i) q^{79} +(-0.500000 + 0.866025i) q^{81} -4.00000 q^{83} +25.3137 q^{85} +(1.41421 - 2.44949i) q^{87} +(2.87868 + 4.98602i) q^{89} +(-1.41421 - 2.44949i) q^{93} +(11.6569 - 20.1903i) q^{95} +0.242641 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{9} - 4 q^{11} + 8 q^{15} + 12 q^{17} - 8 q^{19} - 4 q^{23} - 2 q^{25} - 4 q^{27} + 4 q^{33} - 8 q^{37} - 24 q^{41} + 4 q^{45} - 8 q^{47} - 12 q^{51} + 4 q^{53} - 16 q^{57} + 8 q^{61} + 12 q^{65} - 8 q^{69} + 8 q^{71} + 24 q^{73} + 2 q^{75} + 16 q^{79} - 2 q^{81} - 16 q^{83} + 56 q^{85} + 20 q^{89} + 24 q^{95} - 16 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) 1.70711 + 2.95680i 0.763441 + 1.32232i 0.941067 + 0.338221i \(0.109825\pi\)
−0.177625 + 0.984098i \(0.556842\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 0.414214 0.717439i 0.124890 0.216316i −0.796800 0.604243i \(-0.793476\pi\)
0.921690 + 0.387927i \(0.126809\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 0 0
\(17\) 3.70711 6.42090i 0.899105 1.55730i 0.0704656 0.997514i \(-0.477551\pi\)
0.828640 0.559782i \(-0.189115\pi\)
\(18\) 0 0
\(19\) −3.41421 5.91359i −0.783274 1.35667i −0.930025 0.367497i \(-0.880215\pi\)
0.146750 0.989174i \(-0.453119\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.41421 4.18154i −0.503398 0.871911i −0.999992 0.00392850i \(-0.998750\pi\)
0.496594 0.867983i \(-0.334584\pi\)
\(24\) 0 0
\(25\) −3.32843 + 5.76500i −0.665685 + 1.15300i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) 1.41421 2.44949i 0.254000 0.439941i −0.710623 0.703573i \(-0.751587\pi\)
0.964623 + 0.263631i \(0.0849203\pi\)
\(32\) 0 0
\(33\) −0.414214 0.717439i −0.0721053 0.124890i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.828427 + 1.43488i 0.136193 + 0.235892i 0.926052 0.377395i \(-0.123180\pi\)
−0.789860 + 0.613287i \(0.789847\pi\)
\(38\) 0 0
\(39\) 2.12132 3.67423i 0.339683 0.588348i
\(40\) 0 0
\(41\) −10.2426 −1.59963 −0.799816 0.600245i \(-0.795070\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(42\) 0 0
\(43\) 11.3137 1.72532 0.862662 0.505781i \(-0.168795\pi\)
0.862662 + 0.505781i \(0.168795\pi\)
\(44\) 0 0
\(45\) 1.70711 2.95680i 0.254480 0.440773i
\(46\) 0 0
\(47\) 2.24264 + 3.88437i 0.327123 + 0.566593i 0.981940 0.189194i \(-0.0605876\pi\)
−0.654817 + 0.755788i \(0.727254\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.70711 6.42090i −0.519099 0.899105i
\(52\) 0 0
\(53\) 1.00000 1.73205i 0.137361 0.237915i −0.789136 0.614218i \(-0.789471\pi\)
0.926497 + 0.376303i \(0.122805\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) −6.82843 −0.904447
\(58\) 0 0
\(59\) −4.24264 + 7.34847i −0.552345 + 0.956689i 0.445760 + 0.895152i \(0.352933\pi\)
−0.998105 + 0.0615367i \(0.980400\pi\)
\(60\) 0 0
\(61\) 5.53553 + 9.58783i 0.708752 + 1.22760i 0.965320 + 0.261069i \(0.0840750\pi\)
−0.256568 + 0.966526i \(0.582592\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.24264 + 12.5446i 0.898339 + 1.55597i
\(66\) 0 0
\(67\) 5.65685 9.79796i 0.691095 1.19701i −0.280385 0.959888i \(-0.590462\pi\)
0.971480 0.237124i \(-0.0762046\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) 3.87868 6.71807i 0.453965 0.786291i −0.544663 0.838655i \(-0.683342\pi\)
0.998628 + 0.0523644i \(0.0166757\pi\)
\(74\) 0 0
\(75\) 3.32843 + 5.76500i 0.384334 + 0.665685i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.82843 + 11.8272i 0.768258 + 1.33066i 0.938507 + 0.345261i \(0.112210\pi\)
−0.170249 + 0.985401i \(0.554457\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 25.3137 2.74566
\(86\) 0 0
\(87\) 1.41421 2.44949i 0.151620 0.262613i
\(88\) 0 0
\(89\) 2.87868 + 4.98602i 0.305139 + 0.528517i 0.977292 0.211895i \(-0.0679636\pi\)
−0.672153 + 0.740412i \(0.734630\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.41421 2.44949i −0.146647 0.254000i
\(94\) 0 0
\(95\) 11.6569 20.1903i 1.19597 2.07148i
\(96\) 0 0
\(97\) 0.242641 0.0246364 0.0123182 0.999924i \(-0.496079\pi\)
0.0123182 + 0.999924i \(0.496079\pi\)
\(98\) 0 0
\(99\) −0.828427 −0.0832601
\(100\) 0 0
\(101\) −5.36396 + 9.29065i −0.533734 + 0.924455i 0.465489 + 0.885053i \(0.345878\pi\)
−0.999223 + 0.0394011i \(0.987455\pi\)
\(102\) 0 0
\(103\) −7.07107 12.2474i −0.696733 1.20678i −0.969593 0.244723i \(-0.921303\pi\)
0.272860 0.962054i \(-0.412030\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.41421 + 11.1097i 0.620085 + 1.07402i 0.989469 + 0.144743i \(0.0462354\pi\)
−0.369384 + 0.929277i \(0.620431\pi\)
\(108\) 0 0
\(109\) 1.65685 2.86976i 0.158698 0.274873i −0.775702 0.631100i \(-0.782604\pi\)
0.934399 + 0.356227i \(0.115937\pi\)
\(110\) 0 0
\(111\) 1.65685 0.157262
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 8.24264 14.2767i 0.768630 1.33131i
\(116\) 0 0
\(117\) −2.12132 3.67423i −0.196116 0.339683i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.15685 + 8.93193i 0.468805 + 0.811994i
\(122\) 0 0
\(123\) −5.12132 + 8.87039i −0.461774 + 0.799816i
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 5.65685 9.79796i 0.498058 0.862662i
\(130\) 0 0
\(131\) 2.00000 + 3.46410i 0.174741 + 0.302660i 0.940072 0.340977i \(-0.110758\pi\)
−0.765331 + 0.643637i \(0.777425\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.70711 2.95680i −0.146924 0.254480i
\(136\) 0 0
\(137\) 4.58579 7.94282i 0.391790 0.678600i −0.600896 0.799328i \(-0.705189\pi\)
0.992686 + 0.120727i \(0.0385226\pi\)
\(138\) 0 0
\(139\) −20.9706 −1.77870 −0.889350 0.457227i \(-0.848843\pi\)
−0.889350 + 0.457227i \(0.848843\pi\)
\(140\) 0 0
\(141\) 4.48528 0.377729
\(142\) 0 0
\(143\) 1.75736 3.04384i 0.146958 0.254538i
\(144\) 0 0
\(145\) 4.82843 + 8.36308i 0.400979 + 0.694516i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.656854 1.13770i −0.0538116 0.0932044i 0.837865 0.545878i \(-0.183804\pi\)
−0.891676 + 0.452673i \(0.850470\pi\)
\(150\) 0 0
\(151\) −4.82843 + 8.36308i −0.392932 + 0.680578i −0.992835 0.119495i \(-0.961872\pi\)
0.599903 + 0.800073i \(0.295206\pi\)
\(152\) 0 0
\(153\) −7.41421 −0.599404
\(154\) 0 0
\(155\) 9.65685 0.775657
\(156\) 0 0
\(157\) 0.121320 0.210133i 0.00968242 0.0167704i −0.861144 0.508362i \(-0.830251\pi\)
0.870826 + 0.491591i \(0.163585\pi\)
\(158\) 0 0
\(159\) −1.00000 1.73205i −0.0793052 0.137361i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.82843 + 4.89898i 0.221540 + 0.383718i 0.955276 0.295717i \(-0.0955585\pi\)
−0.733736 + 0.679435i \(0.762225\pi\)
\(164\) 0 0
\(165\) 1.41421 2.44949i 0.110096 0.190693i
\(166\) 0 0
\(167\) −14.8284 −1.14746 −0.573729 0.819045i \(-0.694504\pi\)
−0.573729 + 0.819045i \(0.694504\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −3.41421 + 5.91359i −0.261091 + 0.452224i
\(172\) 0 0
\(173\) −8.29289 14.3637i −0.630497 1.09205i −0.987450 0.157931i \(-0.949518\pi\)
0.356953 0.934122i \(-0.383816\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.24264 + 7.34847i 0.318896 + 0.552345i
\(178\) 0 0
\(179\) −3.24264 + 5.61642i −0.242366 + 0.419791i −0.961388 0.275197i \(-0.911257\pi\)
0.719022 + 0.694988i \(0.244590\pi\)
\(180\) 0 0
\(181\) 7.07107 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(182\) 0 0
\(183\) 11.0711 0.818397
\(184\) 0 0
\(185\) −2.82843 + 4.89898i −0.207950 + 0.360180i
\(186\) 0 0
\(187\) −3.07107 5.31925i −0.224579 0.388982i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.07107 + 10.5154i 0.439287 + 0.760867i 0.997635 0.0687396i \(-0.0218978\pi\)
−0.558348 + 0.829607i \(0.688564\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.0719816 + 0.124676i −0.899770 0.436365i \(-0.856266\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 14.4853 1.03731
\(196\) 0 0
\(197\) −2.68629 −0.191390 −0.0956952 0.995411i \(-0.530507\pi\)
−0.0956952 + 0.995411i \(0.530507\pi\)
\(198\) 0 0
\(199\) 2.82843 4.89898i 0.200502 0.347279i −0.748188 0.663486i \(-0.769076\pi\)
0.948690 + 0.316207i \(0.102409\pi\)
\(200\) 0 0
\(201\) −5.65685 9.79796i −0.399004 0.691095i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −17.4853 30.2854i −1.22123 2.11522i
\(206\) 0 0
\(207\) −2.41421 + 4.18154i −0.167799 + 0.290637i
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −1.65685 −0.114063 −0.0570313 0.998372i \(-0.518163\pi\)
−0.0570313 + 0.998372i \(0.518163\pi\)
\(212\) 0 0
\(213\) 5.24264 9.08052i 0.359220 0.622187i
\(214\) 0 0
\(215\) 19.3137 + 33.4523i 1.31718 + 2.28143i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.87868 6.71807i −0.262097 0.453965i
\(220\) 0 0
\(221\) 15.7279 27.2416i 1.05797 1.83247i
\(222\) 0 0
\(223\) 13.6569 0.914531 0.457265 0.889330i \(-0.348829\pi\)
0.457265 + 0.889330i \(0.348829\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 2.58579 4.47871i 0.171625 0.297263i −0.767363 0.641213i \(-0.778432\pi\)
0.938988 + 0.343950i \(0.111765\pi\)
\(228\) 0 0
\(229\) −12.7071 22.0094i −0.839709 1.45442i −0.890138 0.455691i \(-0.849392\pi\)
0.0504286 0.998728i \(-0.483941\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.41421 + 12.8418i 0.485721 + 0.841294i 0.999865 0.0164099i \(-0.00522367\pi\)
−0.514144 + 0.857704i \(0.671890\pi\)
\(234\) 0 0
\(235\) −7.65685 + 13.2621i −0.499478 + 0.865121i
\(236\) 0 0
\(237\) 13.6569 0.887108
\(238\) 0 0
\(239\) 12.8284 0.829802 0.414901 0.909867i \(-0.363816\pi\)
0.414901 + 0.909867i \(0.363816\pi\)
\(240\) 0 0
\(241\) 6.94975 12.0373i 0.447673 0.775392i −0.550561 0.834795i \(-0.685586\pi\)
0.998234 + 0.0594029i \(0.0189197\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −14.4853 25.0892i −0.921676 1.59639i
\(248\) 0 0
\(249\) −2.00000 + 3.46410i −0.126745 + 0.219529i
\(250\) 0 0
\(251\) −6.14214 −0.387688 −0.193844 0.981032i \(-0.562096\pi\)
−0.193844 + 0.981032i \(0.562096\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 12.6569 21.9223i 0.792603 1.37283i
\(256\) 0 0
\(257\) −1.70711 2.95680i −0.106486 0.184440i 0.807858 0.589377i \(-0.200627\pi\)
−0.914345 + 0.404937i \(0.867293\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.41421 2.44949i −0.0875376 0.151620i
\(262\) 0 0
\(263\) −10.0711 + 17.4436i −0.621009 + 1.07562i 0.368290 + 0.929711i \(0.379943\pi\)
−0.989298 + 0.145907i \(0.953390\pi\)
\(264\) 0 0
\(265\) 6.82843 0.419467
\(266\) 0 0
\(267\) 5.75736 0.352345
\(268\) 0 0
\(269\) −0.0502525 + 0.0870399i −0.00306395 + 0.00530692i −0.867553 0.497344i \(-0.834309\pi\)
0.864489 + 0.502651i \(0.167642\pi\)
\(270\) 0 0
\(271\) −3.07107 5.31925i −0.186554 0.323121i 0.757545 0.652783i \(-0.226399\pi\)
−0.944099 + 0.329662i \(0.893065\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.75736 + 4.77589i 0.166275 + 0.287997i
\(276\) 0 0
\(277\) 14.3137 24.7921i 0.860027 1.48961i −0.0118739 0.999930i \(-0.503780\pi\)
0.871901 0.489682i \(-0.162887\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) 1.17157 0.0698902 0.0349451 0.999389i \(-0.488874\pi\)
0.0349451 + 0.999389i \(0.488874\pi\)
\(282\) 0 0
\(283\) 13.0711 22.6398i 0.776994 1.34579i −0.156672 0.987651i \(-0.550077\pi\)
0.933667 0.358143i \(-0.116590\pi\)
\(284\) 0 0
\(285\) −11.6569 20.1903i −0.690492 1.19597i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −18.9853 32.8835i −1.11678 1.93432i
\(290\) 0 0
\(291\) 0.121320 0.210133i 0.00711192 0.0123182i
\(292\) 0 0
\(293\) 1.75736 0.102666 0.0513330 0.998682i \(-0.483653\pi\)
0.0513330 + 0.998682i \(0.483653\pi\)
\(294\) 0 0
\(295\) −28.9706 −1.68673
\(296\) 0 0
\(297\) −0.414214 + 0.717439i −0.0240351 + 0.0416300i
\(298\) 0 0
\(299\) −10.2426 17.7408i −0.592347 1.02598i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.36396 + 9.29065i 0.308152 + 0.533734i
\(304\) 0 0
\(305\) −18.8995 + 32.7349i −1.08218 + 1.87439i
\(306\) 0 0
\(307\) −11.5147 −0.657180 −0.328590 0.944473i \(-0.606573\pi\)
−0.328590 + 0.944473i \(0.606573\pi\)
\(308\) 0 0
\(309\) −14.1421 −0.804518
\(310\) 0 0
\(311\) −11.8995 + 20.6105i −0.674758 + 1.16872i 0.301781 + 0.953377i \(0.402419\pi\)
−0.976539 + 0.215339i \(0.930914\pi\)
\(312\) 0 0
\(313\) 14.3640 + 24.8791i 0.811899 + 1.40625i 0.911533 + 0.411226i \(0.134899\pi\)
−0.0996342 + 0.995024i \(0.531767\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.0000 + 19.0526i 0.617822 + 1.07010i 0.989882 + 0.141890i \(0.0453179\pi\)
−0.372061 + 0.928208i \(0.621349\pi\)
\(318\) 0 0
\(319\) 1.17157 2.02922i 0.0655955 0.113615i
\(320\) 0 0
\(321\) 12.8284 0.716013
\(322\) 0 0
\(323\) −50.6274 −2.81698
\(324\) 0 0
\(325\) −14.1213 + 24.4588i −0.783310 + 1.35673i
\(326\) 0 0
\(327\) −1.65685 2.86976i −0.0916242 0.158698i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.65685 + 6.33386i 0.200999 + 0.348140i 0.948851 0.315726i \(-0.102248\pi\)
−0.747852 + 0.663866i \(0.768915\pi\)
\(332\) 0 0
\(333\) 0.828427 1.43488i 0.0453975 0.0786308i
\(334\) 0 0
\(335\) 38.6274 2.11044
\(336\) 0 0
\(337\) 10.3431 0.563427 0.281714 0.959499i \(-0.409097\pi\)
0.281714 + 0.959499i \(0.409097\pi\)
\(338\) 0 0
\(339\) −5.00000 + 8.66025i −0.271563 + 0.470360i
\(340\) 0 0
\(341\) −1.17157 2.02922i −0.0634442 0.109889i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8.24264 14.2767i −0.443769 0.768630i
\(346\) 0 0
\(347\) −1.24264 + 2.15232i −0.0667084 + 0.115542i −0.897451 0.441115i \(-0.854583\pi\)
0.830742 + 0.556657i \(0.187916\pi\)
\(348\) 0 0
\(349\) −10.5858 −0.566644 −0.283322 0.959025i \(-0.591437\pi\)
−0.283322 + 0.959025i \(0.591437\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) −13.8492 + 23.9876i −0.737121 + 1.27673i 0.216666 + 0.976246i \(0.430482\pi\)
−0.953787 + 0.300485i \(0.902852\pi\)
\(354\) 0 0
\(355\) 17.8995 + 31.0028i 0.950007 + 1.64546i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.4142 + 24.9662i 0.760753 + 1.31766i 0.942463 + 0.334311i \(0.108504\pi\)
−0.181710 + 0.983352i \(0.558163\pi\)
\(360\) 0 0
\(361\) −13.8137 + 23.9260i −0.727037 + 1.25927i
\(362\) 0 0
\(363\) 10.3137 0.541329
\(364\) 0 0
\(365\) 26.4853 1.38630
\(366\) 0 0
\(367\) −2.34315 + 4.05845i −0.122311 + 0.211849i −0.920679 0.390321i \(-0.872364\pi\)
0.798368 + 0.602171i \(0.205697\pi\)
\(368\) 0 0
\(369\) 5.12132 + 8.87039i 0.266605 + 0.461774i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.65685 4.60181i −0.137567 0.238273i 0.789008 0.614383i \(-0.210595\pi\)
−0.926575 + 0.376110i \(0.877261\pi\)
\(374\) 0 0
\(375\) −2.82843 + 4.89898i −0.146059 + 0.252982i
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 23.3137 1.19754 0.598772 0.800919i \(-0.295655\pi\)
0.598772 + 0.800919i \(0.295655\pi\)
\(380\) 0 0
\(381\) −10.0000 + 17.3205i −0.512316 + 0.887357i
\(382\) 0 0
\(383\) 4.48528 + 7.76874i 0.229187 + 0.396964i 0.957567 0.288209i \(-0.0930599\pi\)
−0.728380 + 0.685173i \(0.759727\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.65685 9.79796i −0.287554 0.498058i
\(388\) 0 0
\(389\) 17.8995 31.0028i 0.907540 1.57191i 0.0900701 0.995935i \(-0.471291\pi\)
0.817470 0.575971i \(-0.195376\pi\)
\(390\) 0 0
\(391\) −35.7990 −1.81043
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −23.3137 + 40.3805i −1.17304 + 2.03176i
\(396\) 0 0
\(397\) −8.12132 14.0665i −0.407597 0.705979i 0.587023 0.809571i \(-0.300300\pi\)
−0.994620 + 0.103591i \(0.966967\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.585786 1.01461i −0.0292528 0.0506673i 0.851028 0.525120i \(-0.175979\pi\)
−0.880281 + 0.474452i \(0.842646\pi\)
\(402\) 0 0
\(403\) 6.00000 10.3923i 0.298881 0.517678i
\(404\) 0 0
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 1.37258 0.0680364
\(408\) 0 0
\(409\) −4.12132 + 7.13834i −0.203786 + 0.352968i −0.949745 0.313024i \(-0.898658\pi\)
0.745959 + 0.665992i \(0.231991\pi\)
\(410\) 0 0
\(411\) −4.58579 7.94282i −0.226200 0.391790i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.82843 11.8272i −0.335194 0.580574i
\(416\) 0 0
\(417\) −10.4853 + 18.1610i −0.513466 + 0.889350i
\(418\) 0 0
\(419\) 7.51472 0.367118 0.183559 0.983009i \(-0.441238\pi\)
0.183559 + 0.983009i \(0.441238\pi\)
\(420\) 0 0
\(421\) −25.3137 −1.23371 −0.616857 0.787075i \(-0.711594\pi\)
−0.616857 + 0.787075i \(0.711594\pi\)
\(422\) 0 0
\(423\) 2.24264 3.88437i 0.109041 0.188864i
\(424\) 0 0
\(425\) 24.6777 + 42.7430i 1.19704 + 2.07334i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.75736 3.04384i −0.0848461 0.146958i
\(430\) 0 0
\(431\) −4.07107 + 7.05130i −0.196096 + 0.339649i −0.947259 0.320468i \(-0.896160\pi\)
0.751163 + 0.660117i \(0.229493\pi\)
\(432\) 0 0
\(433\) −0.928932 −0.0446416 −0.0223208 0.999751i \(-0.507106\pi\)
−0.0223208 + 0.999751i \(0.507106\pi\)
\(434\) 0 0
\(435\) 9.65685 0.463011
\(436\) 0 0
\(437\) −16.4853 + 28.5533i −0.788598 + 1.36589i
\(438\) 0 0
\(439\) −6.34315 10.9867i −0.302742 0.524364i 0.674014 0.738718i \(-0.264569\pi\)
−0.976756 + 0.214354i \(0.931235\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.58579 9.67487i −0.265389 0.459667i 0.702277 0.711904i \(-0.252167\pi\)
−0.967665 + 0.252237i \(0.918834\pi\)
\(444\) 0 0
\(445\) −9.82843 + 17.0233i −0.465912 + 0.806983i
\(446\) 0 0
\(447\) −1.31371 −0.0621363
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) −4.24264 + 7.34847i −0.199778 + 0.346026i
\(452\) 0 0
\(453\) 4.82843 + 8.36308i 0.226859 + 0.392932i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.3137 + 21.3280i 0.576011 + 0.997680i 0.995931 + 0.0901192i \(0.0287248\pi\)
−0.419920 + 0.907561i \(0.637942\pi\)
\(458\) 0 0
\(459\) −3.70711 + 6.42090i −0.173033 + 0.299702i
\(460\) 0 0
\(461\) −16.3848 −0.763115 −0.381558 0.924345i \(-0.624612\pi\)
−0.381558 + 0.924345i \(0.624612\pi\)
\(462\) 0 0
\(463\) −1.65685 −0.0770005 −0.0385003 0.999259i \(-0.512258\pi\)
−0.0385003 + 0.999259i \(0.512258\pi\)
\(464\) 0 0
\(465\) 4.82843 8.36308i 0.223913 0.387829i
\(466\) 0 0
\(467\) 9.41421 + 16.3059i 0.435638 + 0.754547i 0.997347 0.0727876i \(-0.0231895\pi\)
−0.561710 + 0.827334i \(0.689856\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.121320 0.210133i −0.00559015 0.00968242i
\(472\) 0 0
\(473\) 4.68629 8.11689i 0.215476 0.373215i
\(474\) 0 0
\(475\) 45.4558 2.08566
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −11.4142 + 19.7700i −0.521529 + 0.903314i 0.478158 + 0.878274i \(0.341305\pi\)
−0.999686 + 0.0250403i \(0.992029\pi\)
\(480\) 0 0
\(481\) 3.51472 + 6.08767i 0.160257 + 0.277574i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.414214 + 0.717439i 0.0188085 + 0.0325772i
\(486\) 0 0
\(487\) 2.48528 4.30463i 0.112619 0.195062i −0.804207 0.594350i \(-0.797409\pi\)
0.916825 + 0.399288i \(0.130743\pi\)
\(488\) 0 0
\(489\) 5.65685 0.255812
\(490\) 0 0
\(491\) −19.1716 −0.865201 −0.432600 0.901586i \(-0.642404\pi\)
−0.432600 + 0.901586i \(0.642404\pi\)
\(492\) 0 0
\(493\) 10.4853 18.1610i 0.472233 0.817932i
\(494\) 0 0
\(495\) −1.41421 2.44949i −0.0635642 0.110096i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12.4853 21.6251i −0.558918 0.968074i −0.997587 0.0694257i \(-0.977883\pi\)
0.438669 0.898649i \(-0.355450\pi\)
\(500\) 0 0
\(501\) −7.41421 + 12.8418i −0.331243 + 0.573729i
\(502\) 0 0
\(503\) −19.3137 −0.861156 −0.430578 0.902553i \(-0.641690\pi\)
−0.430578 + 0.902553i \(0.641690\pi\)
\(504\) 0 0
\(505\) −36.6274 −1.62990
\(506\) 0 0
\(507\) 2.50000 4.33013i 0.111029 0.192308i
\(508\) 0 0
\(509\) −18.1924 31.5101i −0.806363 1.39666i −0.915367 0.402621i \(-0.868099\pi\)
0.109003 0.994041i \(-0.465234\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.41421 + 5.91359i 0.150741 + 0.261091i
\(514\) 0 0
\(515\) 24.1421 41.8154i 1.06383 1.84261i
\(516\) 0 0
\(517\) 3.71573 0.163418
\(518\) 0 0
\(519\) −16.5858 −0.728035
\(520\) 0 0
\(521\) 7.12132 12.3345i 0.311991 0.540384i −0.666803 0.745234i \(-0.732338\pi\)
0.978793 + 0.204851i \(0.0656709\pi\)
\(522\) 0 0
\(523\) 2.48528 + 4.30463i 0.108674 + 0.188228i 0.915233 0.402924i \(-0.132006\pi\)
−0.806559 + 0.591153i \(0.798673\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.4853 18.1610i −0.456746 0.791107i
\(528\) 0 0
\(529\) −0.156854 + 0.271680i −0.00681975 + 0.0118122i
\(530\) 0 0
\(531\) 8.48528 0.368230
\(532\) 0 0
\(533\) −43.4558 −1.88228
\(534\) 0 0
\(535\) −21.8995 + 37.9310i −0.946798 + 1.63990i
\(536\) 0 0
\(537\) 3.24264 + 5.61642i 0.139930 + 0.242366i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.00000 + 15.5885i 0.386940 + 0.670200i 0.992036 0.125952i \(-0.0401986\pi\)
−0.605096 + 0.796152i \(0.706865\pi\)
\(542\) 0 0
\(543\) 3.53553 6.12372i 0.151724 0.262794i
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) 3.02944 0.129529 0.0647647 0.997901i \(-0.479370\pi\)
0.0647647 + 0.997901i \(0.479370\pi\)
\(548\) 0 0
\(549\) 5.53553 9.58783i 0.236251 0.409198i
\(550\) 0 0
\(551\) −9.65685 16.7262i −0.411396 0.712558i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.82843 + 4.89898i 0.120060 + 0.207950i
\(556\) 0 0
\(557\) −16.3137 + 28.2562i −0.691234 + 1.19725i 0.280200 + 0.959942i \(0.409599\pi\)
−0.971434 + 0.237311i \(0.923734\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −6.14214 −0.259321
\(562\) 0 0
\(563\) −0.928932 + 1.60896i −0.0391498 + 0.0678095i −0.884936 0.465712i \(-0.845798\pi\)
0.845787 + 0.533521i \(0.179132\pi\)
\(564\) 0 0
\(565\) −17.0711 29.5680i −0.718185 1.24393i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.92893 + 5.07306i 0.122787 + 0.212674i 0.920866 0.389880i \(-0.127483\pi\)
−0.798079 + 0.602553i \(0.794150\pi\)
\(570\) 0 0
\(571\) −20.8284 + 36.0759i −0.871643 + 1.50973i −0.0113458 + 0.999936i \(0.503612\pi\)
−0.860297 + 0.509794i \(0.829722\pi\)
\(572\) 0 0
\(573\) 12.1421 0.507245
\(574\) 0 0
\(575\) 32.1421 1.34042
\(576\) 0 0
\(577\) 6.60660 11.4430i 0.275036 0.476377i −0.695108 0.718905i \(-0.744643\pi\)
0.970144 + 0.242528i \(0.0779767\pi\)
\(578\) 0 0
\(579\) 1.00000 + 1.73205i 0.0415586 + 0.0719816i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.828427 1.43488i −0.0343099 0.0594266i
\(584\) 0 0
\(585\) 7.24264 12.5446i 0.299446 0.518656i
\(586\) 0 0
\(587\) −35.7990 −1.47758 −0.738791 0.673934i \(-0.764603\pi\)
−0.738791 + 0.673934i \(0.764603\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 0 0
\(591\) −1.34315 + 2.32640i −0.0552496 + 0.0956952i
\(592\) 0 0
\(593\) 23.7071 + 41.0619i 0.973534 + 1.68621i 0.684690 + 0.728835i \(0.259938\pi\)
0.288844 + 0.957376i \(0.406729\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.82843 4.89898i −0.115760 0.200502i
\(598\) 0 0
\(599\) 11.7279 20.3134i 0.479190 0.829981i −0.520525 0.853846i \(-0.674264\pi\)
0.999715 + 0.0238650i \(0.00759718\pi\)
\(600\) 0 0
\(601\) 12.2426 0.499388 0.249694 0.968325i \(-0.419670\pi\)
0.249694 + 0.968325i \(0.419670\pi\)
\(602\) 0 0
\(603\) −11.3137 −0.460730
\(604\) 0 0
\(605\) −17.6066 + 30.4955i −0.715810 + 1.23982i
\(606\) 0 0
\(607\) −12.4853 21.6251i −0.506762 0.877737i −0.999969 0.00782569i \(-0.997509\pi\)
0.493207 0.869912i \(-0.335824\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.51472 + 16.4800i 0.384924 + 0.666708i
\(612\) 0 0
\(613\) 5.17157 8.95743i 0.208878 0.361787i −0.742483 0.669864i \(-0.766352\pi\)
0.951361 + 0.308077i \(0.0996856\pi\)
\(614\) 0 0
\(615\) −34.9706 −1.41015
\(616\) 0 0
\(617\) −4.48528 −0.180571 −0.0902853 0.995916i \(-0.528778\pi\)
−0.0902853 + 0.995916i \(0.528778\pi\)
\(618\) 0 0
\(619\) −16.8284 + 29.1477i −0.676392 + 1.17154i 0.299668 + 0.954043i \(0.403124\pi\)
−0.976060 + 0.217501i \(0.930209\pi\)
\(620\) 0 0
\(621\) 2.41421 + 4.18154i 0.0968791 + 0.167799i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.98528 + 12.0989i 0.279411 + 0.483954i
\(626\) 0 0
\(627\) −2.82843 + 4.89898i −0.112956 + 0.195646i
\(628\) 0 0
\(629\) 12.2843 0.489806
\(630\) 0 0
\(631\) 3.02944 0.120600 0.0603000 0.998180i \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) 0 0
\(633\) −0.828427 + 1.43488i −0.0329270 + 0.0570313i
\(634\) 0 0
\(635\) −34.1421 59.1359i −1.35489 2.34674i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5.24264 9.08052i −0.207396 0.359220i
\(640\) 0 0
\(641\) −17.5563 + 30.4085i −0.693434 + 1.20106i 0.277272 + 0.960792i \(0.410570\pi\)
−0.970706 + 0.240272i \(0.922764\pi\)
\(642\) 0 0
\(643\) 31.7990 1.25403 0.627015 0.779007i \(-0.284277\pi\)
0.627015 + 0.779007i \(0.284277\pi\)
\(644\) 0 0
\(645\) 38.6274 1.52095
\(646\) 0 0
\(647\) 0.100505 0.174080i 0.00395126 0.00684379i −0.864043 0.503418i \(-0.832076\pi\)
0.867994 + 0.496574i \(0.165409\pi\)
\(648\) 0 0
\(649\) 3.51472 + 6.08767i 0.137965 + 0.238962i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.0711 39.9603i −0.902841 1.56377i −0.823790 0.566895i \(-0.808144\pi\)
−0.0790508 0.996871i \(-0.525189\pi\)
\(654\) 0 0
\(655\) −6.82843 + 11.8272i −0.266809 + 0.462126i
\(656\) 0 0
\(657\) −7.75736 −0.302643
\(658\) 0 0
\(659\) −21.5147 −0.838094 −0.419047 0.907964i \(-0.637636\pi\)
−0.419047 + 0.907964i \(0.637636\pi\)
\(660\) 0 0
\(661\) 2.46447 4.26858i 0.0958566 0.166029i −0.814109 0.580712i \(-0.802774\pi\)
0.909966 + 0.414683i \(0.136108\pi\)
\(662\) 0 0
\(663\) −15.7279 27.2416i −0.610822 1.05797i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.82843 11.8272i −0.264398 0.457950i
\(668\) 0 0
\(669\) 6.82843 11.8272i 0.264002 0.457265i
\(670\) 0 0
\(671\) 9.17157 0.354065
\(672\) 0 0
\(673\) −23.3137 −0.898677 −0.449339 0.893361i \(-0.648340\pi\)
−0.449339 + 0.893361i \(0.648340\pi\)
\(674\) 0 0
\(675\) 3.32843 5.76500i 0.128111 0.221895i
\(676\) 0 0
\(677\) 13.8492 + 23.9876i 0.532270 + 0.921918i 0.999290 + 0.0376716i \(0.0119941\pi\)
−0.467021 + 0.884246i \(0.654673\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2.58579 4.47871i −0.0990876 0.171625i
\(682\) 0 0
\(683\) −0.899495 + 1.55797i −0.0344182 + 0.0596141i −0.882721 0.469897i \(-0.844291\pi\)
0.848303 + 0.529511i \(0.177624\pi\)
\(684\) 0 0
\(685\) 31.3137 1.19644
\(686\) 0 0
\(687\) −25.4142 −0.969613
\(688\) 0 0
\(689\) 4.24264 7.34847i 0.161632 0.279954i
\(690\) 0 0
\(691\) −4.34315 7.52255i −0.165221 0.286171i 0.771513 0.636214i \(-0.219500\pi\)
−0.936734 + 0.350043i \(0.886167\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35.7990 62.0057i −1.35793 2.35201i
\(696\) 0 0
\(697\) −37.9706 + 65.7669i −1.43824 + 2.49110i
\(698\) 0 0
\(699\) 14.8284 0.560863
\(700\) 0 0
\(701\) −6.14214 −0.231985 −0.115993 0.993250i \(-0.537005\pi\)
−0.115993 + 0.993250i \(0.537005\pi\)
\(702\) 0 0
\(703\) 5.65685 9.79796i 0.213352 0.369537i
\(704\) 0 0
\(705\) 7.65685 + 13.2621i 0.288374 + 0.499478i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.31371 5.73951i −0.124449 0.215552i 0.797068 0.603889i \(-0.206383\pi\)
−0.921517 + 0.388337i \(0.873050\pi\)
\(710\) 0 0
\(711\) 6.82843 11.8272i 0.256086 0.443554i
\(712\) 0 0
\(713\) −13.6569 −0.511453
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 6.41421 11.1097i 0.239543 0.414901i
\(718\) 0 0
\(719\) 3.31371 + 5.73951i 0.123580 + 0.214048i 0.921177 0.389143i \(-0.127229\pi\)
−0.797597 + 0.603191i \(0.793896\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −6.94975 12.0373i −0.258464 0.447673i
\(724\) 0 0
\(725\) −9.41421 + 16.3059i −0.349635 + 0.605586i
\(726\) 0 0
\(727\) −9.85786 −0.365608 −0.182804 0.983149i \(-0.558517\pi\)
−0.182804 + 0.983149i \(0.558517\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 41.9411 72.6442i 1.55125 2.68684i
\(732\) 0 0
\(733\) −4.02082 6.96426i −0.148512 0.257231i 0.782166 0.623071i \(-0.214115\pi\)
−0.930678 + 0.365840i \(0.880782\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.68629 8.11689i −0.172622 0.298990i
\(738\) 0 0
\(739\) 16.4853 28.5533i 0.606421 1.05035i −0.385404 0.922748i \(-0.625938\pi\)
0.991825 0.127604i \(-0.0407286\pi\)
\(740\) 0 0
\(741\) −28.9706 −1.06426
\(742\) 0 0
\(743\) 4.82843 0.177138 0.0885689 0.996070i \(-0.471771\pi\)
0.0885689 + 0.996070i \(0.471771\pi\)
\(744\) 0 0
\(745\) 2.24264 3.88437i 0.0821640 0.142312i
\(746\) 0 0
\(747\) 2.00000 + 3.46410i 0.0731762 + 0.126745i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 10.1421 + 17.5667i 0.370092 + 0.641018i 0.989579 0.143989i \(-0.0459929\pi\)
−0.619488 + 0.785006i \(0.712660\pi\)
\(752\) 0 0
\(753\) −3.07107 + 5.31925i −0.111916 + 0.193844i
\(754\) 0 0
\(755\) −32.9706 −1.19992
\(756\) 0 0
\(757\) 41.9411 1.52438 0.762188 0.647356i \(-0.224125\pi\)
0.762188 + 0.647356i \(0.224125\pi\)
\(758\) 0 0
\(759\) −2.00000 + 3.46410i −0.0725954 + 0.125739i
\(760\) 0 0
\(761\) −15.9497 27.6258i −0.578178 1.00143i −0.995688 0.0927614i \(-0.970431\pi\)
0.417510 0.908672i \(-0.362903\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −12.6569 21.9223i −0.457610 0.792603i
\(766\) 0 0
\(767\) −18.0000 + 31.1769i −0.649942 + 1.12573i
\(768\) 0 0
\(769\) −34.8701 −1.25745 −0.628723 0.777629i \(-0.716422\pi\)
−0.628723 + 0.777629i \(0.716422\pi\)
\(770\) 0 0
\(771\) −3.41421 −0.122960
\(772\) 0 0
\(773\) 16.6777 28.8866i 0.599854 1.03898i −0.392988 0.919544i \(-0.628559\pi\)
0.992842 0.119434i \(-0.0381080\pi\)
\(774\) 0 0
\(775\) 9.41421 + 16.3059i 0.338169 + 0.585725i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.9706 + 60.5708i 1.25295 + 2.17017i
\(780\) 0 0
\(781\) 4.34315 7.52255i 0.155410 0.269178i
\(782\) 0 0
\(783\) −2.82843 −0.101080
\(784\) 0 0
\(785\) 0.828427 0.0295678
\(786\) 0 0
\(787\) −7.65685 + 13.2621i −0.272937 + 0.472741i −0.969613 0.244645i \(-0.921329\pi\)
0.696675 + 0.717387i \(0.254662\pi\)
\(788\) 0 0
\(789\) 10.0711 + 17.4436i 0.358540 + 0.621009i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 23.4853 + 40.6777i 0.833987 + 1.44451i
\(794\) 0 0
\(795\) 3.41421 5.91359i 0.121090 0.209733i
\(796\) 0 0
\(797\) 39.4142 1.39612 0.698062 0.716038i \(-0.254046\pi\)
0.698062 + 0.716038i \(0.254046\pi\)
\(798\) 0 0
\(799\) 33.2548 1.17647
\(800\) 0 0
\(801\) 2.87868 4.98602i 0.101713 0.176172i
\(802\) 0 0
\(803\) −3.21320 5.56543i −0.113391 0.196400i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.0502525 + 0.0870399i 0.00176897 + 0.00306395i
\(808\) 0 0
\(809\) 2.31371 4.00746i 0.0813457 0.140895i −0.822483 0.568790i \(-0.807412\pi\)
0.903828 + 0.427896i \(0.140745\pi\)
\(810\) 0 0
\(811\) 25.6569 0.900934 0.450467 0.892793i \(-0.351257\pi\)
0.450467 + 0.892793i \(0.351257\pi\)
\(812\) 0 0
\(813\) −6.14214 −0.215414
\(814\) 0 0
\(815\) −9.65685 + 16.7262i −0.338265 + 0.585892i
\(816\) 0 0
\(817\) −38.6274 66.9046i −1.35140 2.34070i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.34315 + 9.25460i 0.186477 + 0.322988i 0.944073 0.329736i \(-0.106960\pi\)
−0.757596 + 0.652724i \(0.773626\pi\)
\(822\) 0 0
\(823\) −12.4853 + 21.6251i −0.435210 + 0.753805i −0.997313 0.0732621i \(-0.976659\pi\)
0.562103 + 0.827067i \(0.309992\pi\)
\(824\) 0 0
\(825\) 5.51472 0.191998
\(826\) 0 0
\(827\) −40.1421 −1.39588 −0.697939 0.716157i \(-0.745900\pi\)
−0.697939 + 0.716157i \(0.745900\pi\)
\(828\) 0 0
\(829\) 17.1924 29.7781i 0.597116 1.03424i −0.396128 0.918195i \(-0.629646\pi\)
0.993244 0.116041i \(-0.0370203\pi\)
\(830\) 0 0
\(831\) −14.3137 24.7921i −0.496537 0.860027i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −25.3137 43.8446i −0.876017 1.51731i
\(836\) 0 0
\(837\) −1.41421 + 2.44949i −0.0488824 + 0.0846668i
\(838\) 0 0
\(839\) 23.7990 0.821632 0.410816 0.911718i \(-0.365244\pi\)
0.410816 + 0.911718i \(0.365244\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0.585786 1.01461i 0.0201756 0.0349451i
\(844\) 0 0
\(845\) 8.53553 + 14.7840i 0.293631 + 0.508584i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −13.0711 22.6398i −0.448598 0.776994i
\(850\) 0 0
\(851\) 4.00000 6.92820i 0.137118 0.237496i
\(852\) 0 0
\(853\) −4.92893 −0.168763 −0.0843817 0.996434i \(-0.526892\pi\)
−0.0843817 + 0.996434i \(0.526892\pi\)
\(854\) 0 0
\(855\) −23.3137 −0.797312
\(856\) 0 0
\(857\) −5.60660 + 9.71092i −0.191518 + 0.331719i −0.945753 0.324885i \(-0.894674\pi\)
0.754236 + 0.656604i \(0.228008\pi\)
\(858\) 0 0
\(859\) −1.27208 2.20330i −0.0434027 0.0751757i 0.843508 0.537117i \(-0.180487\pi\)
−0.886911 + 0.461941i \(0.847153\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.0416 + 46.8375i 0.920508 + 1.59437i 0.798631 + 0.601821i \(0.205558\pi\)
0.121877 + 0.992545i \(0.461109\pi\)
\(864\) 0 0
\(865\) 28.3137 49.0408i 0.962695 1.66744i
\(866\) 0 0
\(867\) −37.9706 −1.28955
\(868\) 0 0
\(869\) 11.3137 0.383791
\(870\) 0 0
\(871\) 24.0000 41.5692i 0.813209 1.40852i
\(872\) 0 0
\(873\) −0.121320 0.210133i −0.00410607 0.00711192i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.1421 + 24.4949i 0.477546 + 0.827134i 0.999669 0.0257364i \(-0.00819306\pi\)
−0.522123 + 0.852870i \(0.674860\pi\)
\(878\) 0 0
\(879\) 0.878680 1.52192i 0.0296371 0.0513330i
\(880\) 0 0
\(881\) −30.0416 −1.01213 −0.506064 0.862496i \(-0.668900\pi\)
−0.506064 + 0.862496i \(0.668900\pi\)
\(882\) 0 0
\(883\) 10.3431 0.348075 0.174037 0.984739i \(-0.444319\pi\)
0.174037 + 0.984739i \(0.444319\pi\)
\(884\) 0 0
\(885\) −14.4853 + 25.0892i −0.486917 + 0.843366i
\(886\) 0 0
\(887\) 19.8995 + 34.4669i 0.668160 + 1.15729i 0.978418 + 0.206634i \(0.0662510\pi\)
−0.310259 + 0.950652i \(0.600416\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.414214 + 0.717439i 0.0138767 + 0.0240351i
\(892\) 0 0
\(893\) 15.3137 26.5241i 0.512454 0.887596i
\(894\) 0 0
\(895\) −22.1421 −0.740130
\(896\) 0 0
\(897\) −20.4853 −0.683984
\(898\) 0 0
\(899\) 4.00000 6.92820i 0.133407 0.231069i
\(900\) 0 0
\(901\) −7.41421 12.8418i −0.247003 0.427822i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.0711 + 20.9077i 0.401256 + 0.694996i
\(906\) 0 0
\(907\) 26.8284 46.4682i 0.890823 1.54295i 0.0519331 0.998651i \(-0.483462\pi\)
0.838890 0.544301i \(-0.183205\pi\)
\(908\) 0 0
\(909\) 10.7279 0.355823
\(910\) 0 0
\(911\) −6.48528 −0.214867 −0.107433 0.994212i \(-0.534263\pi\)
−0.107433 + 0.994212i \(0.534263\pi\)
\(912\) 0 0
\(913\) −1.65685 + 2.86976i −0.0548339 + 0.0949751i
\(914\) 0 0
\(915\) 18.8995 + 32.7349i 0.624798 + 1.08218i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.6569 + 27.1185i 0.516472 + 0.894556i 0.999817 + 0.0191258i \(0.00608830\pi\)
−0.483345 + 0.875430i \(0.660578\pi\)
\(920\) 0 0
\(921\) −5.75736 + 9.97204i −0.189711 + 0.328590i
\(922\) 0 0
\(923\) 44.4853 1.46425
\(924\) 0 0
\(925\) −11.0294 −0.362646
\(926\) 0 0
\(927\) −7.07107 + 12.2474i −0.232244 + 0.402259i
\(928\) 0 0
\(929\) −7.94975 13.7694i −0.260823 0.451758i 0.705638 0.708572i \(-0.250660\pi\)
−0.966461 + 0.256814i \(0.917327\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 11.8995 + 20.6105i 0.389572 + 0.674758i
\(934\) 0 0
\(935\) 10.4853 18.1610i 0.342905 0.593930i
\(936\) 0 0
\(937\) −51.3553 −1.67771 −0.838853 0.544358i \(-0.816773\pi\)
−0.838853 + 0.544358i \(0.816773\pi\)
\(938\) 0 0
\(939\) 28.7279 0.937500
\(940\) 0 0
\(941\) −7.36396 + 12.7548i −0.240058 + 0.415793i −0.960731 0.277483i \(-0.910500\pi\)
0.720672 + 0.693276i \(0.243833\pi\)
\(942\) 0 0
\(943\) 24.7279 + 42.8300i 0.805252 + 1.39474i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.3848 50.8959i −0.954877 1.65390i −0.734650 0.678447i \(-0.762653\pi\)
−0.220227 0.975449i \(-0.570680\pi\)
\(948\) 0 0
\(949\) 16.4558 28.5024i 0.534179 0.925226i
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −20.7279 + 35.9018i −0.670740 + 1.16176i
\(956\) 0 0
\(957\) −1.17157 2.02922i −0.0378716 0.0655955i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 11.5000 + 19.9186i 0.370968 + 0.642535i
\(962\) 0 0
\(963\) 6.41421 11.1097i 0.206695 0.358006i
\(964\) 0 0
\(965\) −6.82843 −0.219815
\(966\) 0 0
\(967\) 31.3137 1.00698 0.503490 0.864001i \(-0.332049\pi\)
0.503490 + 0.864001i \(0.332049\pi\)
\(968\) 0 0
\(969\) −25.3137 + 43.8446i −0.813193 + 1.40849i
\(970\) 0 0
\(971\) 10.0000 + 17.3205i 0.320915 + 0.555842i 0.980677 0.195633i \(-0.0626762\pi\)
−0.659762 + 0.751475i \(0.729343\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 14.1213 + 24.4588i 0.452244 + 0.783310i
\(976\) 0 0
\(977\) 3.89949 6.75412i 0.124756 0.216084i −0.796882 0.604136i \(-0.793519\pi\)
0.921637 + 0.388052i \(0.126852\pi\)
\(978\) 0 0
\(979\) 4.76955 0.152436
\(980\) 0 0
\(981\) −3.31371 −0.105799
\(982\) 0 0
\(983\) −0.686292 + 1.18869i −0.0218893 + 0.0379134i −0.876763 0.480924i \(-0.840301\pi\)
0.854873 + 0.518837i \(0.173635\pi\)
\(984\) 0 0
\(985\) −4.58579 7.94282i −0.146115 0.253079i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −27.3137 47.3087i −0.868525 1.50433i
\(990\) 0 0
\(991\) −9.31371 + 16.1318i −0.295860 + 0.512444i −0.975185 0.221394i \(-0.928939\pi\)
0.679325 + 0.733838i \(0.262273\pi\)
\(992\) 0 0
\(993\) 7.31371 0.232094
\(994\) 0 0
\(995\) 19.3137 0.612286
\(996\) 0 0
\(997\) −27.1924 + 47.0986i −0.861192 + 1.49163i 0.00958804 + 0.999954i \(0.496948\pi\)
−0.870780 + 0.491674i \(0.836385\pi\)
\(998\) 0 0
\(999\) −0.828427 1.43488i −0.0262103 0.0453975i
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 2352.2.q.bg.1537.2 4
4.3 odd 2 1176.2.q.m.361.2 4
7.2 even 3 inner 2352.2.q.bg.961.2 4
7.3 odd 6 2352.2.a.bg.1.2 2
7.4 even 3 2352.2.a.z.1.1 2
7.5 odd 6 2352.2.q.ba.961.1 4
7.6 odd 2 2352.2.q.ba.1537.1 4
12.11 even 2 3528.2.s.bc.361.1 4
21.11 odd 6 7056.2.a.cw.1.2 2
21.17 even 6 7056.2.a.ce.1.1 2
28.3 even 6 1176.2.a.l.1.2 2
28.11 odd 6 1176.2.a.m.1.1 yes 2
28.19 even 6 1176.2.q.n.961.1 4
28.23 odd 6 1176.2.q.m.961.2 4
28.27 even 2 1176.2.q.n.361.1 4
56.3 even 6 9408.2.a.dv.1.1 2
56.11 odd 6 9408.2.a.dr.1.2 2
56.45 odd 6 9408.2.a.dh.1.1 2
56.53 even 6 9408.2.a.ed.1.2 2
84.11 even 6 3528.2.a.bm.1.2 2
84.23 even 6 3528.2.s.bc.3313.1 4
84.47 odd 6 3528.2.s.bl.3313.2 4
84.59 odd 6 3528.2.a.bc.1.1 2
84.83 odd 2 3528.2.s.bl.361.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.l.1.2 2 28.3 even 6
1176.2.a.m.1.1 yes 2 28.11 odd 6
1176.2.q.m.361.2 4 4.3 odd 2
1176.2.q.m.961.2 4 28.23 odd 6
1176.2.q.n.361.1 4 28.27 even 2
1176.2.q.n.961.1 4 28.19 even 6
2352.2.a.z.1.1 2 7.4 even 3
2352.2.a.bg.1.2 2 7.3 odd 6
2352.2.q.ba.961.1 4 7.5 odd 6
2352.2.q.ba.1537.1 4 7.6 odd 2
2352.2.q.bg.961.2 4 7.2 even 3 inner
2352.2.q.bg.1537.2 4 1.1 even 1 trivial
3528.2.a.bc.1.1 2 84.59 odd 6
3528.2.a.bm.1.2 2 84.11 even 6
3528.2.s.bc.361.1 4 12.11 even 2
3528.2.s.bc.3313.1 4 84.23 even 6
3528.2.s.bl.361.2 4 84.83 odd 2
3528.2.s.bl.3313.2 4 84.47 odd 6
7056.2.a.ce.1.1 2 21.17 even 6
7056.2.a.cw.1.2 2 21.11 odd 6
9408.2.a.dh.1.1 2 56.45 odd 6
9408.2.a.dr.1.2 2 56.11 odd 6
9408.2.a.dv.1.1 2 56.3 even 6
9408.2.a.ed.1.2 2 56.53 even 6