Properties

Label 700.2.i.f.401.3
Level $700$
Weight $2$
Character 700.401
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(401,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("700.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.i (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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 401.3
Root \(-1.52274 + 1.63746i\) of defining polynomial
Character \(\chi\) \(=\) 700.401
Dual form 700.2.i.f.501.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 1.50000i) q^{3} +(-2.38876 + 1.13746i) q^{7} +O(q^{10})\) \(q+(0.866025 + 1.50000i) q^{3} +(-2.38876 + 1.13746i) q^{7} +(2.63746 + 4.56821i) q^{11} -2.62685 q^{13} +(0.209313 + 0.362541i) q^{17} +(-1.63746 + 2.83616i) q^{19} +(-3.77492 - 2.59808i) q^{21} +(-3.91150 + 6.77492i) q^{23} +5.19615 q^{27} +4.27492 q^{29} +(-1.63746 - 2.83616i) q^{31} +(-4.56821 + 7.91238i) q^{33} +(-4.98684 + 8.63746i) q^{37} +(-2.27492 - 3.94027i) q^{39} -3.72508 q^{41} +2.15068 q^{43} +(3.25479 - 5.63746i) q^{47} +(4.41238 - 5.43424i) q^{49} +(-0.362541 + 0.627940i) q^{51} +(-2.83616 - 4.91238i) q^{53} -5.67232 q^{57} +(1.63746 + 2.83616i) q^{59} +(6.77492 - 11.7345i) q^{61} +(1.76082 + 3.04983i) q^{67} -13.5498 q^{69} -4.54983 q^{71} +(3.25479 + 5.63746i) q^{73} +(-11.4964 - 7.91238i) q^{77} +(-3.63746 + 6.30026i) q^{79} +(4.50000 + 7.79423i) q^{81} +7.40437 q^{83} +(3.70219 + 6.41238i) q^{87} +(3.50000 - 6.06218i) q^{89} +(6.27492 - 2.98793i) q^{91} +(2.83616 - 4.91238i) q^{93} +6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0.866025 + 1.50000i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.38876 + 1.13746i −0.902867 + 0.429919i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.63746 + 4.56821i 0.795224 + 1.37737i 0.922697 + 0.385526i \(0.125980\pi\)
−0.127473 + 0.991842i \(0.540687\pi\)
\(12\) 0 0
\(13\) −2.62685 −0.728557 −0.364278 0.931290i \(-0.618684\pi\)
−0.364278 + 0.931290i \(0.618684\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.209313 + 0.362541i 0.0507659 + 0.0879292i 0.890292 0.455391i \(-0.150500\pi\)
−0.839526 + 0.543320i \(0.817167\pi\)
\(18\) 0 0
\(19\) −1.63746 + 2.83616i −0.375659 + 0.650660i −0.990425 0.138049i \(-0.955917\pi\)
0.614767 + 0.788709i \(0.289250\pi\)
\(20\) 0 0
\(21\) −3.77492 2.59808i −0.823754 0.566947i
\(22\) 0 0
\(23\) −3.91150 + 6.77492i −0.815604 + 1.41267i 0.0932891 + 0.995639i \(0.470262\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 4.27492 0.793832 0.396916 0.917855i \(-0.370080\pi\)
0.396916 + 0.917855i \(0.370080\pi\)
\(30\) 0 0
\(31\) −1.63746 2.83616i −0.294096 0.509390i 0.680678 0.732583i \(-0.261685\pi\)
−0.974774 + 0.223193i \(0.928352\pi\)
\(32\) 0 0
\(33\) −4.56821 + 7.91238i −0.795224 + 1.37737i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.98684 + 8.63746i −0.819831 + 1.41999i 0.0859750 + 0.996297i \(0.472599\pi\)
−0.905806 + 0.423692i \(0.860734\pi\)
\(38\) 0 0
\(39\) −2.27492 3.94027i −0.364278 0.630949i
\(40\) 0 0
\(41\) −3.72508 −0.581760 −0.290880 0.956760i \(-0.593948\pi\)
−0.290880 + 0.956760i \(0.593948\pi\)
\(42\) 0 0
\(43\) 2.15068 0.327975 0.163988 0.986462i \(-0.447564\pi\)
0.163988 + 0.986462i \(0.447564\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.25479 5.63746i 0.474760 0.822308i −0.524823 0.851212i \(-0.675868\pi\)
0.999582 + 0.0289038i \(0.00920165\pi\)
\(48\) 0 0
\(49\) 4.41238 5.43424i 0.630339 0.776320i
\(50\) 0 0
\(51\) −0.362541 + 0.627940i −0.0507659 + 0.0879292i
\(52\) 0 0
\(53\) −2.83616 4.91238i −0.389577 0.674767i 0.602816 0.797880i \(-0.294045\pi\)
−0.992393 + 0.123114i \(0.960712\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.67232 −0.751318
\(58\) 0 0
\(59\) 1.63746 + 2.83616i 0.213179 + 0.369237i 0.952708 0.303888i \(-0.0982849\pi\)
−0.739529 + 0.673125i \(0.764952\pi\)
\(60\) 0 0
\(61\) 6.77492 11.7345i 0.867439 1.50245i 0.00283468 0.999996i \(-0.499098\pi\)
0.864605 0.502453i \(-0.167569\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.76082 + 3.04983i 0.215119 + 0.372597i 0.953309 0.301996i \(-0.0976528\pi\)
−0.738191 + 0.674592i \(0.764319\pi\)
\(68\) 0 0
\(69\) −13.5498 −1.63121
\(70\) 0 0
\(71\) −4.54983 −0.539966 −0.269983 0.962865i \(-0.587018\pi\)
−0.269983 + 0.962865i \(0.587018\pi\)
\(72\) 0 0
\(73\) 3.25479 + 5.63746i 0.380944 + 0.659815i 0.991197 0.132392i \(-0.0422657\pi\)
−0.610253 + 0.792206i \(0.708932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −11.4964 7.91238i −1.31014 0.901699i
\(78\) 0 0
\(79\) −3.63746 + 6.30026i −0.409246 + 0.708835i −0.994805 0.101795i \(-0.967542\pi\)
0.585559 + 0.810630i \(0.300875\pi\)
\(80\) 0 0
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 7.40437 0.812736 0.406368 0.913710i \(-0.366795\pi\)
0.406368 + 0.913710i \(0.366795\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.70219 + 6.41238i 0.396916 + 0.687479i
\(88\) 0 0
\(89\) 3.50000 6.06218i 0.370999 0.642590i −0.618720 0.785611i \(-0.712349\pi\)
0.989720 + 0.143022i \(0.0456819\pi\)
\(90\) 0 0
\(91\) 6.27492 2.98793i 0.657790 0.313220i
\(92\) 0 0
\(93\) 2.83616 4.91238i 0.294096 0.509390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820 0.703452 0.351726 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.77492 + 11.7345i 0.674129 + 1.16763i 0.976723 + 0.214507i \(0.0688144\pi\)
−0.302593 + 0.953120i \(0.597852\pi\)
\(102\) 0 0
\(103\) 5.64355 9.77492i 0.556076 0.963151i −0.441743 0.897141i \(-0.645640\pi\)
0.997819 0.0660098i \(-0.0210268\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.76082 3.04983i 0.170225 0.294839i −0.768273 0.640122i \(-0.778884\pi\)
0.938499 + 0.345283i \(0.112217\pi\)
\(108\) 0 0
\(109\) 5.77492 + 10.0025i 0.553137 + 0.958061i 0.998046 + 0.0624852i \(0.0199026\pi\)
−0.444909 + 0.895576i \(0.646764\pi\)
\(110\) 0 0
\(111\) −17.2749 −1.63966
\(112\) 0 0
\(113\) −4.30136 −0.404637 −0.202319 0.979320i \(-0.564848\pi\)
−0.202319 + 0.979320i \(0.564848\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.912376 0.627940i −0.0836374 0.0575632i
\(120\) 0 0
\(121\) −8.41238 + 14.5707i −0.764761 + 1.32461i
\(122\) 0 0
\(123\) −3.22602 5.58762i −0.290880 0.503819i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.6460 1.38836 0.694179 0.719802i \(-0.255768\pi\)
0.694179 + 0.719802i \(0.255768\pi\)
\(128\) 0 0
\(129\) 1.86254 + 3.22602i 0.163988 + 0.284035i
\(130\) 0 0
\(131\) 5.36254 9.28819i 0.468527 0.811513i −0.530826 0.847481i \(-0.678118\pi\)
0.999353 + 0.0359678i \(0.0114514\pi\)
\(132\) 0 0
\(133\) 0.685484 8.63746i 0.0594390 0.748963i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.6592 18.4622i −0.910674 1.57733i −0.813115 0.582103i \(-0.802230\pi\)
−0.0975588 0.995230i \(-0.531103\pi\)
\(138\) 0 0
\(139\) 13.0997 1.11110 0.555550 0.831483i \(-0.312508\pi\)
0.555550 + 0.831483i \(0.312508\pi\)
\(140\) 0 0
\(141\) 11.2749 0.949519
\(142\) 0 0
\(143\) −6.92820 12.0000i −0.579365 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.9726 + 1.91238i 0.987482 + 0.157730i
\(148\) 0 0
\(149\) 3.77492 6.53835i 0.309253 0.535642i −0.668946 0.743311i \(-0.733254\pi\)
0.978199 + 0.207669i \(0.0665876\pi\)
\(150\) 0 0
\(151\) 6.36254 + 11.0202i 0.517776 + 0.896815i 0.999787 + 0.0206494i \(0.00657337\pi\)
−0.482011 + 0.876165i \(0.660093\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.10411 1.91238i −0.0881176 0.152624i 0.818598 0.574367i \(-0.194752\pi\)
−0.906715 + 0.421743i \(0.861418\pi\)
\(158\) 0 0
\(159\) 4.91238 8.50848i 0.389577 0.674767i
\(160\) 0 0
\(161\) 1.63746 20.6328i 0.129050 1.62610i
\(162\) 0 0
\(163\) 2.83616 4.91238i 0.222145 0.384767i −0.733314 0.679890i \(-0.762027\pi\)
0.955459 + 0.295123i \(0.0953607\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.476171 0.0368472 0.0184236 0.999830i \(-0.494135\pi\)
0.0184236 + 0.999830i \(0.494135\pi\)
\(168\) 0 0
\(169\) −6.09967 −0.469205
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.2405 17.7371i 0.778573 1.34853i −0.154190 0.988041i \(-0.549277\pi\)
0.932764 0.360488i \(-0.117390\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.83616 + 4.91238i −0.213179 + 0.369237i
\(178\) 0 0
\(179\) 3.63746 + 6.30026i 0.271876 + 0.470904i 0.969342 0.245714i \(-0.0790225\pi\)
−0.697466 + 0.716618i \(0.745689\pi\)
\(180\) 0 0
\(181\) −24.2749 −1.80434 −0.902170 0.431380i \(-0.858027\pi\)
−0.902170 + 0.431380i \(0.858027\pi\)
\(182\) 0 0
\(183\) 23.4690 1.73488
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.10411 + 1.91238i −0.0807406 + 0.139847i
\(188\) 0 0
\(189\) −12.4124 + 5.91041i −0.902867 + 0.429919i
\(190\) 0 0
\(191\) −0.0876242 + 0.151770i −0.00634026 + 0.0109817i −0.869178 0.494499i \(-0.835352\pi\)
0.862838 + 0.505481i \(0.168685\pi\)
\(192\) 0 0
\(193\) 10.6592 + 18.4622i 0.767263 + 1.32894i 0.939042 + 0.343803i \(0.111715\pi\)
−0.171778 + 0.985136i \(0.554951\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.60271 −0.612918 −0.306459 0.951884i \(-0.599144\pi\)
−0.306459 + 0.951884i \(0.599144\pi\)
\(198\) 0 0
\(199\) −8.63746 14.9605i −0.612293 1.06052i −0.990853 0.134946i \(-0.956914\pi\)
0.378560 0.925577i \(-0.376419\pi\)
\(200\) 0 0
\(201\) −3.04983 + 5.28247i −0.215119 + 0.372597i
\(202\) 0 0
\(203\) −10.2118 + 4.86254i −0.716725 + 0.341284i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −17.2749 −1.19493
\(210\) 0 0
\(211\) 25.6495 1.76578 0.882892 0.469576i \(-0.155593\pi\)
0.882892 + 0.469576i \(0.155593\pi\)
\(212\) 0 0
\(213\) −3.94027 6.82475i −0.269983 0.467624i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.13752 + 4.91238i 0.484526 + 0.333474i
\(218\) 0 0
\(219\) −5.63746 + 9.76436i −0.380944 + 0.659815i
\(220\) 0 0
\(221\) −0.549834 0.952341i −0.0369859 0.0640614i
\(222\) 0 0
\(223\) 8.71780 0.583787 0.291893 0.956451i \(-0.405715\pi\)
0.291893 + 0.956451i \(0.405715\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.76436 16.9124i −0.648084 1.12251i −0.983580 0.180472i \(-0.942237\pi\)
0.335496 0.942041i \(-0.391096\pi\)
\(228\) 0 0
\(229\) −1.63746 + 2.83616i −0.108206 + 0.187419i −0.915044 0.403355i \(-0.867844\pi\)
0.806837 + 0.590774i \(0.201177\pi\)
\(230\) 0 0
\(231\) 1.91238 24.0969i 0.125825 1.58546i
\(232\) 0 0
\(233\) 7.13752 12.3625i 0.467594 0.809897i −0.531720 0.846920i \(-0.678454\pi\)
0.999314 + 0.0370231i \(0.0117875\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −12.6005 −0.818492
\(238\) 0 0
\(239\) 0.549834 0.0355658 0.0177829 0.999842i \(-0.494339\pi\)
0.0177829 + 0.999842i \(0.494339\pi\)
\(240\) 0 0
\(241\) −4.91238 8.50848i −0.316434 0.548080i 0.663307 0.748347i \(-0.269152\pi\)
−0.979741 + 0.200267i \(0.935819\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.30136 7.45017i 0.273689 0.474043i
\(248\) 0 0
\(249\) 6.41238 + 11.1066i 0.406368 + 0.703850i
\(250\) 0 0
\(251\) −20.5498 −1.29709 −0.648547 0.761175i \(-0.724623\pi\)
−0.648547 + 0.761175i \(0.724623\pi\)
\(252\) 0 0
\(253\) −41.2657 −2.59435
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.82409 + 10.0876i −0.363297 + 0.629249i −0.988501 0.151212i \(-0.951682\pi\)
0.625204 + 0.780461i \(0.285016\pi\)
\(258\) 0 0
\(259\) 2.08762 26.3052i 0.129719 1.63452i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.389855 + 0.675248i 0.0240395 + 0.0416376i 0.877795 0.479037i \(-0.159014\pi\)
−0.853755 + 0.520674i \(0.825681\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.1244 0.741999
\(268\) 0 0
\(269\) 7.22508 + 12.5142i 0.440521 + 0.763005i 0.997728 0.0673687i \(-0.0214604\pi\)
−0.557207 + 0.830374i \(0.688127\pi\)
\(270\) 0 0
\(271\) −4.91238 + 8.50848i −0.298406 + 0.516854i −0.975771 0.218793i \(-0.929788\pi\)
0.677366 + 0.735646i \(0.263121\pi\)
\(272\) 0 0
\(273\) 9.91613 + 6.82475i 0.600152 + 0.413053i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.13752 + 12.3625i 0.428852 + 0.742793i 0.996771 0.0802909i \(-0.0255849\pi\)
−0.567920 + 0.823084i \(0.692252\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 5.46301 + 9.46221i 0.324742 + 0.562470i 0.981460 0.191666i \(-0.0613891\pi\)
−0.656718 + 0.754136i \(0.728056\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.89834 4.23713i 0.525252 0.250110i
\(288\) 0 0
\(289\) 8.41238 14.5707i 0.494846 0.857098i
\(290\) 0 0
\(291\) 6.00000 + 10.3923i 0.351726 + 0.609208i
\(292\) 0 0
\(293\) −6.92820 −0.404750 −0.202375 0.979308i \(-0.564866\pi\)
−0.202375 + 0.979308i \(0.564866\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.7046 + 23.7371i 0.795224 + 1.37737i
\(298\) 0 0
\(299\) 10.2749 17.7967i 0.594214 1.02921i
\(300\) 0 0
\(301\) −5.13746 + 2.44631i −0.296118 + 0.141003i
\(302\) 0 0
\(303\) −11.7345 + 20.3248i −0.674129 + 1.16763i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −26.5145 −1.51326 −0.756631 0.653843i \(-0.773156\pi\)
−0.756631 + 0.653843i \(0.773156\pi\)
\(308\) 0 0
\(309\) 19.5498 1.11215
\(310\) 0 0
\(311\) 4.91238 + 8.50848i 0.278555 + 0.482472i 0.971026 0.238974i \(-0.0768111\pi\)
−0.692471 + 0.721446i \(0.743478\pi\)
\(312\) 0 0
\(313\) −16.7501 + 29.0120i −0.946772 + 1.63986i −0.194609 + 0.980881i \(0.562344\pi\)
−0.752163 + 0.658977i \(0.770990\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.8098 22.1873i 0.719472 1.24616i −0.241737 0.970342i \(-0.577717\pi\)
0.961209 0.275821i \(-0.0889496\pi\)
\(318\) 0 0
\(319\) 11.2749 + 19.5287i 0.631274 + 1.09340i
\(320\) 0 0
\(321\) 6.09967 0.340450
\(322\) 0 0
\(323\) −1.37097 −0.0762827
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0025 + 17.3248i −0.553137 + 0.958061i
\(328\) 0 0
\(329\) −1.36254 + 17.1687i −0.0751193 + 0.946543i
\(330\) 0 0
\(331\) −8.91238 + 15.4367i −0.489868 + 0.848477i −0.999932 0.0116596i \(-0.996289\pi\)
0.510064 + 0.860137i \(0.329622\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.30136 0.234310 0.117155 0.993114i \(-0.462623\pi\)
0.117155 + 0.993114i \(0.462623\pi\)
\(338\) 0 0
\(339\) −3.72508 6.45203i −0.202319 0.350426i
\(340\) 0 0
\(341\) 8.63746 14.9605i 0.467745 0.810157i
\(342\) 0 0
\(343\) −4.35890 + 18.0000i −0.235358 + 0.971909i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.06218 + 10.5000i 0.325435 + 0.563670i 0.981600 0.190947i \(-0.0611560\pi\)
−0.656165 + 0.754617i \(0.727823\pi\)
\(348\) 0 0
\(349\) −3.72508 −0.199399 −0.0996996 0.995018i \(-0.531788\pi\)
−0.0996996 + 0.995018i \(0.531788\pi\)
\(350\) 0 0
\(351\) −13.6495 −0.728557
\(352\) 0 0
\(353\) 4.09204 + 7.08762i 0.217797 + 0.377236i 0.954134 0.299379i \(-0.0967795\pi\)
−0.736337 + 0.676615i \(0.763446\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.151770 1.91238i 0.00803249 0.101214i
\(358\) 0 0
\(359\) −18.1873 + 31.5013i −0.959889 + 1.66258i −0.237127 + 0.971479i \(0.576206\pi\)
−0.722762 + 0.691097i \(0.757128\pi\)
\(360\) 0 0
\(361\) 4.13746 + 7.16629i 0.217761 + 0.377173i
\(362\) 0 0
\(363\) −29.1413 −1.52952
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.01670 5.22508i −0.157471 0.272747i 0.776485 0.630135i \(-0.217001\pi\)
−0.933956 + 0.357388i \(0.883667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.3625 + 8.50848i 0.641831 + 0.441739i
\(372\) 0 0
\(373\) 4.98684 8.63746i 0.258209 0.447231i −0.707553 0.706660i \(-0.750201\pi\)
0.965762 + 0.259429i \(0.0835344\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11.2296 −0.578352
\(378\) 0 0
\(379\) −21.6495 −1.11206 −0.556030 0.831162i \(-0.687676\pi\)
−0.556030 + 0.831162i \(0.687676\pi\)
\(380\) 0 0
\(381\) 13.5498 + 23.4690i 0.694179 + 1.20235i
\(382\) 0 0
\(383\) −3.07425 + 5.32475i −0.157087 + 0.272082i −0.933817 0.357751i \(-0.883544\pi\)
0.776730 + 0.629833i \(0.216877\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.1873 28.0372i −0.820728 1.42154i −0.905141 0.425112i \(-0.860235\pi\)
0.0844123 0.996431i \(-0.473099\pi\)
\(390\) 0 0
\(391\) −3.27492 −0.165620
\(392\) 0 0
\(393\) 18.5764 0.937055
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.40547 9.36254i 0.271293 0.469892i −0.697901 0.716195i \(-0.745882\pi\)
0.969193 + 0.246302i \(0.0792156\pi\)
\(398\) 0 0
\(399\) 13.5498 6.45203i 0.678340 0.323006i
\(400\) 0 0
\(401\) −1.50000 + 2.59808i −0.0749064 + 0.129742i −0.901046 0.433724i \(-0.857199\pi\)
0.826139 + 0.563466i \(0.190532\pi\)
\(402\) 0 0
\(403\) 4.30136 + 7.45017i 0.214266 + 0.371119i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −52.6103 −2.60780
\(408\) 0 0
\(409\) 10.0498 + 17.4068i 0.496932 + 0.860712i 0.999994 0.00353862i \(-0.00112638\pi\)
−0.503061 + 0.864251i \(0.667793\pi\)
\(410\) 0 0
\(411\) 18.4622 31.9775i 0.910674 1.57733i
\(412\) 0 0
\(413\) −7.13752 4.91238i −0.351214 0.241722i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.3446 + 19.6495i 0.555550 + 0.962240i
\(418\) 0 0
\(419\) −13.0997 −0.639961 −0.319980 0.947424i \(-0.603676\pi\)
−0.319980 + 0.947424i \(0.603676\pi\)
\(420\) 0 0
\(421\) −4.27492 −0.208347 −0.104173 0.994559i \(-0.533220\pi\)
−0.104173 + 0.994559i \(0.533220\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.83616 + 35.7371i −0.137251 + 1.72944i
\(428\) 0 0
\(429\) 12.0000 20.7846i 0.579365 1.00349i
\(430\) 0 0
\(431\) 9.18729 + 15.9129i 0.442536 + 0.766495i 0.997877 0.0651276i \(-0.0207454\pi\)
−0.555341 + 0.831623i \(0.687412\pi\)
\(432\) 0 0
\(433\) 18.1578 0.872606 0.436303 0.899800i \(-0.356288\pi\)
0.436303 + 0.899800i \(0.356288\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.8098 22.1873i −0.612778 1.06136i
\(438\) 0 0
\(439\) −11.9124 + 20.6328i −0.568547 + 0.984752i 0.428163 + 0.903701i \(0.359161\pi\)
−0.996710 + 0.0810504i \(0.974173\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.06218 + 10.5000i −0.288023 + 0.498870i −0.973338 0.229377i \(-0.926331\pi\)
0.685315 + 0.728247i \(0.259665\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.0767 0.618507
\(448\) 0 0
\(449\) −3.17525 −0.149849 −0.0749246 0.997189i \(-0.523872\pi\)
−0.0749246 + 0.997189i \(0.523872\pi\)
\(450\) 0 0
\(451\) −9.82475 17.0170i −0.462629 0.801298i
\(452\) 0 0
\(453\) −11.0202 + 19.0876i −0.517776 + 0.896815i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.685484 + 1.18729i −0.0320656 + 0.0555392i −0.881613 0.471973i \(-0.843542\pi\)
0.849547 + 0.527512i \(0.176875\pi\)
\(458\) 0 0
\(459\) 1.08762 + 1.88382i 0.0507659 + 0.0879292i
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −2.15068 −0.0999505 −0.0499752 0.998750i \(-0.515914\pi\)
−0.0499752 + 0.998750i \(0.515914\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.85177 13.5997i 0.363337 0.629318i −0.625171 0.780488i \(-0.714971\pi\)
0.988508 + 0.151170i \(0.0483041\pi\)
\(468\) 0 0
\(469\) −7.67525 5.28247i −0.354410 0.243922i
\(470\) 0 0
\(471\) 1.91238 3.31233i 0.0881176 0.152624i
\(472\) 0 0
\(473\) 5.67232 + 9.82475i 0.260814 + 0.451743i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.91238 8.50848i −0.224452 0.388763i 0.731703 0.681624i \(-0.238726\pi\)
−0.956155 + 0.292861i \(0.905393\pi\)
\(480\) 0 0
\(481\) 13.0997 22.6893i 0.597293 1.03454i
\(482\) 0 0
\(483\) 32.3673 15.4124i 1.47277 0.701287i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.46519 + 2.53779i 0.0663943 + 0.114998i 0.897312 0.441398i \(-0.145517\pi\)
−0.830917 + 0.556396i \(0.812184\pi\)
\(488\) 0 0
\(489\) 9.82475 0.444291
\(490\) 0 0
\(491\) −28.5498 −1.28844 −0.644218 0.764842i \(-0.722817\pi\)
−0.644218 + 0.764842i \(0.722817\pi\)
\(492\) 0 0
\(493\) 0.894797 + 1.54983i 0.0402996 + 0.0698010i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.8685 5.17525i 0.487518 0.232142i
\(498\) 0 0
\(499\) −0.812707 + 1.40765i −0.0363818 + 0.0630151i −0.883643 0.468161i \(-0.844917\pi\)
0.847261 + 0.531177i \(0.178250\pi\)
\(500\) 0 0
\(501\) 0.412376 + 0.714256i 0.0184236 + 0.0319106i
\(502\) 0 0
\(503\) −31.7682 −1.41647 −0.708236 0.705975i \(-0.750509\pi\)
−0.708236 + 0.705975i \(0.750509\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.28247 9.14950i −0.234603 0.406344i
\(508\) 0 0
\(509\) 7.22508 12.5142i 0.320246 0.554683i −0.660293 0.751008i \(-0.729568\pi\)
0.980539 + 0.196326i \(0.0629010\pi\)
\(510\) 0 0
\(511\) −14.1873 9.76436i −0.627609 0.431950i
\(512\) 0 0
\(513\) −8.50848 + 14.7371i −0.375659 + 0.650660i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 34.3375 1.51016
\(518\) 0 0
\(519\) 35.4743 1.55715
\(520\) 0 0
\(521\) −4.91238 8.50848i −0.215215 0.372763i 0.738124 0.674665i \(-0.235712\pi\)
−0.953339 + 0.301902i \(0.902379\pi\)
\(522\) 0 0
\(523\) 3.67341 6.36254i 0.160627 0.278215i −0.774467 0.632615i \(-0.781982\pi\)
0.935094 + 0.354400i \(0.115315\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.685484 1.18729i 0.0298602 0.0517193i
\(528\) 0 0
\(529\) −19.0997 33.0816i −0.830420 1.43833i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.78523 0.423845
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −6.30026 + 10.9124i −0.271876 + 0.470904i
\(538\) 0 0
\(539\) 36.4622 + 5.82409i 1.57054 + 0.250861i
\(540\) 0 0
\(541\) 8.77492 15.1986i 0.377263 0.653439i −0.613400 0.789773i \(-0.710199\pi\)
0.990663 + 0.136334i \(0.0435319\pi\)
\(542\) 0 0
\(543\) −21.0227 36.4124i −0.902170 1.56260i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.5386 0.878168 0.439084 0.898446i \(-0.355303\pi\)
0.439084 + 0.898446i \(0.355303\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.00000 + 12.1244i −0.298210 + 0.516515i
\(552\) 0 0
\(553\) 1.52274 19.1873i 0.0647534 0.815927i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.98684 + 8.63746i 0.211299 + 0.365981i 0.952121 0.305720i \(-0.0988972\pi\)
−0.740822 + 0.671701i \(0.765564\pi\)
\(558\) 0 0
\(559\) −5.64950 −0.238949
\(560\) 0 0
\(561\) −3.82475 −0.161481
\(562\) 0 0
\(563\) −11.3159 19.5997i −0.476907 0.826028i 0.522743 0.852491i \(-0.324909\pi\)
−0.999650 + 0.0264630i \(0.991576\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −19.6150 13.5000i −0.823754 0.566947i
\(568\) 0 0
\(569\) −4.18729 + 7.25260i −0.175540 + 0.304045i −0.940348 0.340214i \(-0.889501\pi\)
0.764808 + 0.644259i \(0.222834\pi\)
\(570\) 0 0
\(571\) −3.63746 6.30026i −0.152223 0.263658i 0.779821 0.626002i \(-0.215310\pi\)
−0.932044 + 0.362344i \(0.881976\pi\)
\(572\) 0 0
\(573\) −0.303539 −0.0126805
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.94136 3.36254i −0.0808200 0.139984i 0.822782 0.568357i \(-0.192421\pi\)
−0.903602 + 0.428372i \(0.859087\pi\)
\(578\) 0 0
\(579\) −18.4622 + 31.9775i −0.767263 + 1.32894i
\(580\) 0 0
\(581\) −17.6873 + 8.42217i −0.733793 + 0.349410i
\(582\) 0 0
\(583\) 14.9605 25.9124i 0.619601 1.07318i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.8997 −0.862623 −0.431311 0.902203i \(-0.641949\pi\)
−0.431311 + 0.902203i \(0.641949\pi\)
\(588\) 0 0
\(589\) 10.7251 0.441919
\(590\) 0 0
\(591\) −7.45017 12.9041i −0.306459 0.530802i
\(592\) 0 0
\(593\) 16.6926 28.9124i 0.685482 1.18729i −0.287804 0.957689i \(-0.592925\pi\)
0.973285 0.229600i \(-0.0737416\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.9605 25.9124i 0.612293 1.06052i
\(598\) 0 0
\(599\) −2.63746 4.56821i −0.107764 0.186652i 0.807100 0.590414i \(-0.201036\pi\)
−0.914864 + 0.403762i \(0.867702\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.70109 9.87459i 0.231400 0.400797i −0.726820 0.686828i \(-0.759003\pi\)
0.958220 + 0.286031i \(0.0923360\pi\)
\(608\) 0 0
\(609\) −16.1375 11.1066i −0.653923 0.450061i
\(610\) 0 0
\(611\) −8.54983 + 14.8087i −0.345889 + 0.599098i
\(612\) 0 0
\(613\) −14.1808 24.5619i −0.572757 0.992045i −0.996281 0.0861600i \(-0.972540\pi\)
0.423524 0.905885i \(-0.360793\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.2920 1.25977 0.629884 0.776689i \(-0.283102\pi\)
0.629884 + 0.776689i \(0.283102\pi\)
\(618\) 0 0
\(619\) 4.46221 + 7.72877i 0.179351 + 0.310646i 0.941659 0.336570i \(-0.109267\pi\)
−0.762307 + 0.647215i \(0.775933\pi\)
\(620\) 0 0
\(621\) −20.3248 + 35.2035i −0.815604 + 1.41267i
\(622\) 0 0
\(623\) −1.46519 + 18.4622i −0.0587017 + 0.739673i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −14.9605 25.9124i −0.597466 1.03484i
\(628\) 0 0
\(629\) −4.17525 −0.166478
\(630\) 0 0
\(631\) 33.0997 1.31768 0.658839 0.752284i \(-0.271048\pi\)
0.658839 + 0.752284i \(0.271048\pi\)
\(632\) 0 0
\(633\) 22.2131 + 38.4743i 0.882892 + 1.52921i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −11.5906 + 14.2749i −0.459238 + 0.565593i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.04983 + 1.81837i 0.0414660 + 0.0718212i 0.886014 0.463659i \(-0.153464\pi\)
−0.844548 + 0.535481i \(0.820131\pi\)
\(642\) 0 0
\(643\) 31.4071 1.23857 0.619287 0.785164i \(-0.287422\pi\)
0.619287 + 0.785164i \(0.287422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.4666 + 23.3248i 0.529425 + 0.916991i 0.999411 + 0.0343169i \(0.0109255\pi\)
−0.469986 + 0.882674i \(0.655741\pi\)
\(648\) 0 0
\(649\) −8.63746 + 14.9605i −0.339050 + 0.587252i
\(650\) 0 0
\(651\) −1.18729 + 14.9605i −0.0465337 + 0.586349i
\(652\) 0 0
\(653\) 14.1808 24.5619i 0.554938 0.961181i −0.442970 0.896536i \(-0.646075\pi\)
0.997908 0.0646444i \(-0.0205913\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.5498 −1.57960 −0.789799 0.613366i \(-0.789815\pi\)
−0.789799 + 0.613366i \(0.789815\pi\)
\(660\) 0 0
\(661\) 0.225083 + 0.389855i 0.00875471 + 0.0151636i 0.870370 0.492399i \(-0.163880\pi\)
−0.861615 + 0.507563i \(0.830547\pi\)
\(662\) 0 0
\(663\) 0.952341 1.64950i 0.0369859 0.0640614i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.7213 + 28.9622i −0.647453 + 1.12142i
\(668\) 0 0
\(669\) 7.54983 + 13.0767i 0.291893 + 0.505574i
\(670\) 0 0
\(671\) 71.4743 2.75923
\(672\) 0 0
\(673\) 31.2920 1.20622 0.603109 0.797659i \(-0.293928\pi\)
0.603109 + 0.797659i \(0.293928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23.2021 40.1873i 0.891731 1.54452i 0.0539317 0.998545i \(-0.482825\pi\)
0.837799 0.545979i \(-0.183842\pi\)
\(678\) 0 0
\(679\) −16.5498 + 7.88054i −0.635124 + 0.302428i
\(680\) 0 0
\(681\) 16.9124 29.2931i 0.648084 1.12251i
\(682\) 0 0
\(683\) 9.58382 + 16.5997i 0.366715 + 0.635169i 0.989050 0.147582i \(-0.0471491\pi\)
−0.622335 + 0.782751i \(0.713816\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.67232 −0.216413
\(688\) 0 0
\(689\) 7.45017 + 12.9041i 0.283829 + 0.491606i
\(690\) 0 0
\(691\) −15.1873 + 26.3052i −0.577752 + 1.00070i 0.417985 + 0.908454i \(0.362737\pi\)
−0.995737 + 0.0922416i \(0.970597\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.779710 1.35050i −0.0295336 0.0511537i
\(698\) 0 0
\(699\) 24.7251 0.935189
\(700\) 0 0
\(701\) 8.82475 0.333306 0.166653 0.986016i \(-0.446704\pi\)
0.166653 + 0.986016i \(0.446704\pi\)
\(702\) 0 0
\(703\) −16.3315 28.2870i −0.615954 1.06686i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.5312 20.3248i −1.11063 0.764391i
\(708\) 0 0
\(709\) 5.22508 9.05011i 0.196232 0.339884i −0.751072 0.660221i \(-0.770463\pi\)
0.947304 + 0.320337i \(0.103796\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.6197 0.959465
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.476171 + 0.824752i 0.0177829 + 0.0308009i
\(718\) 0 0
\(719\) 15.1873 26.3052i 0.566390 0.981017i −0.430528 0.902577i \(-0.641673\pi\)
0.996919 0.0784400i \(-0.0249939\pi\)
\(720\) 0 0
\(721\) −2.36254 + 29.7693i −0.0879856 + 1.10867i
\(722\) 0 0
\(723\) 8.50848 14.7371i 0.316434 0.548080i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3.10302 −0.115085 −0.0575423 0.998343i \(-0.518326\pi\)
−0.0575423 + 0.998343i \(0.518326\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0.450166 + 0.779710i 0.0166500 + 0.0288386i
\(732\) 0 0
\(733\) 18.8432 32.6375i 0.695991 1.20549i −0.273854 0.961771i \(-0.588299\pi\)
0.969845 0.243721i \(-0.0783681\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.28819 + 16.0876i −0.342135 + 0.592595i
\(738\) 0 0
\(739\) 10.4622 + 18.1211i 0.384859 + 0.666595i 0.991750 0.128190i \(-0.0409169\pi\)
−0.606891 + 0.794785i \(0.707584\pi\)
\(740\) 0 0
\(741\) 14.9003 0.547377
\(742\) 0 0
\(743\) 6.45203 0.236702 0.118351 0.992972i \(-0.462239\pi\)
0.118351 + 0.992972i \(0.462239\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.737127 + 9.28819i −0.0269341 + 0.339383i
\(750\) 0 0
\(751\) 7.36254 12.7523i 0.268663 0.465338i −0.699854 0.714286i \(-0.746752\pi\)
0.968517 + 0.248948i \(0.0800849\pi\)
\(752\) 0 0
\(753\) −17.7967 30.8248i −0.648547 1.12332i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −35.5934 −1.29366 −0.646831 0.762633i \(-0.723906\pi\)
−0.646831 + 0.762633i \(0.723906\pi\)
\(758\) 0 0
\(759\) −35.7371 61.8985i −1.29718 2.24677i
\(760\) 0 0
\(761\) −11.4622 + 19.8531i −0.415505 + 0.719675i −0.995481 0.0949578i \(-0.969728\pi\)
0.579977 + 0.814633i \(0.303062\pi\)
\(762\) 0 0
\(763\) −25.1723 17.3248i −0.911298 0.627198i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.30136 7.45017i −0.155313 0.269010i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −20.1752 −0.726594
\(772\) 0 0
\(773\) −20.1567 34.9124i −0.724985 1.25571i −0.958980 0.283473i \(-0.908513\pi\)
0.233995 0.972238i \(-0.424820\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 41.2657 19.6495i 1.48040 0.704922i
\(778\) 0 0
\(779\) 6.09967 10.5649i 0.218543 0.378528i
\(780\) 0 0
\(781\) −12.0000 20.7846i −0.429394 0.743732i
\(782\) 0 0
\(783\) 22.2131 0.793832
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −0.866025 1.50000i −0.0308705 0.0534692i 0.850177 0.526496i \(-0.176495\pi\)
−0.881048 + 0.473027i \(0.843161\pi\)
\(788\) 0 0
\(789\) −0.675248 + 1.16956i −0.0240395 + 0.0416376i
\(790\) 0 0
\(791\) 10.2749 4.89261i 0.365334 0.173961i
\(792\) 0 0
\(793\) −17.7967 + 30.8248i −0.631979 + 1.09462i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.8229 1.65855 0.829276 0.558839i \(-0.188753\pi\)
0.829276 + 0.558839i \(0.188753\pi\)
\(798\) 0 0
\(799\) 2.72508 0.0964065
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.1687 + 29.7371i −0.605872 + 1.04940i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.5142 + 21.6752i −0.440521 + 0.763005i
\(808\) 0 0
\(809\) 8.59967 + 14.8951i 0.302348 + 0.523683i 0.976667 0.214757i \(-0.0688961\pi\)
−0.674319 + 0.738440i \(0.735563\pi\)
\(810\) 0 0
\(811\) 7.45017 0.261611 0.130805 0.991408i \(-0.458244\pi\)
0.130805 + 0.991408i \(0.458244\pi\)
\(812\) 0 0
\(813\) −17.0170 −0.596811
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.52165 + 6.09967i −0.123207 + 0.213400i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.1873 17.6449i 0.355539 0.615812i −0.631671 0.775237i \(-0.717631\pi\)
0.987210 + 0.159425i \(0.0509640\pi\)
\(822\) 0 0
\(823\) 23.0791 + 39.9743i 0.804488 + 1.39341i 0.916636 + 0.399723i \(0.130894\pi\)
−0.112147 + 0.993692i \(0.535773\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.0547 0.523505 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(828\) 0 0
\(829\) 25.4622 + 44.1018i 0.884339 + 1.53172i 0.846469 + 0.532438i \(0.178724\pi\)
0.0378699 + 0.999283i \(0.487943\pi\)
\(830\) 0 0
\(831\) −12.3625 + 21.4125i −0.428852 + 0.742793i
\(832\) 0 0
\(833\) 2.89371 + 0.462210i 0.100261 + 0.0160146i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.50848 14.7371i −0.294096 0.509390i
\(838\) 0 0
\(839\) 41.0997 1.41892 0.709459 0.704747i \(-0.248939\pi\)
0.709459 + 0.704747i \(0.248939\pi\)
\(840\) 0 0
\(841\) −10.7251 −0.369830
\(842\) 0 0
\(843\) 5.19615 + 9.00000i 0.178965 + 0.309976i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.52165 44.3746i 0.121005 1.52473i
\(848\) 0 0
\(849\) −9.46221 + 16.3890i −0.324742 + 0.562470i
\(850\) 0 0
\(851\) −39.0120 67.5708i −1.33732 2.31630i
\(852\) 0 0
\(853\) −13.1342 −0.449708 −0.224854 0.974392i \(-0.572190\pi\)
−0.224854 + 0.974392i \(0.572190\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.8432 + 32.6375i 0.643673 + 1.11487i 0.984606 + 0.174787i \(0.0559236\pi\)
−0.340933 + 0.940087i \(0.610743\pi\)
\(858\) 0 0
\(859\) 1.18729 2.05645i 0.0405099 0.0701652i −0.845060 0.534672i \(-0.820435\pi\)
0.885569 + 0.464507i \(0.153768\pi\)
\(860\) 0 0
\(861\) 14.0619 + 9.67805i 0.479228 + 0.329827i
\(862\) 0 0
\(863\) −8.21286 + 14.2251i −0.279569 + 0.484227i −0.971278 0.237949i \(-0.923525\pi\)
0.691709 + 0.722177i \(0.256858\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 29.1413 0.989691
\(868\) 0 0
\(869\) −38.3746 −1.30177
\(870\) 0 0
\(871\) −4.62541 8.01145i −0.156726 0.271458i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −11.4389 + 19.8127i −0.386263 + 0.669028i −0.991944 0.126681i \(-0.959568\pi\)
0.605680 + 0.795708i \(0.292901\pi\)
\(878\) 0 0
\(879\) −6.00000 10.3923i −0.202375 0.350524i
\(880\) 0 0
\(881\) −43.0241 −1.44952 −0.724759 0.689002i \(-0.758049\pi\)
−0.724759 + 0.689002i \(0.758049\pi\)
\(882\) 0 0
\(883\) −55.5407 −1.86909 −0.934547 0.355840i \(-0.884195\pi\)
−0.934547 + 0.355840i \(0.884195\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −19.6150 + 33.9743i −0.658609 + 1.14074i 0.322367 + 0.946615i \(0.395521\pi\)
−0.980976 + 0.194129i \(0.937812\pi\)
\(888\) 0 0
\(889\) −37.3746 + 17.7967i −1.25350 + 0.596881i
\(890\) 0 0
\(891\) −23.7371 + 41.1139i −0.795224 + 1.37737i
\(892\) 0 0
\(893\) 10.6592 + 18.4622i 0.356695 + 0.617814i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 35.5934 1.18843
\(898\) 0 0
\(899\) −7.00000 12.1244i −0.233463 0.404370i
\(900\) 0 0
\(901\) 1.18729 2.05645i 0.0395545 0.0685103i
\(902\) 0 0
\(903\) −8.11863 5.58762i −0.270171 0.185944i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.9285 36.2492i −0.694918 1.20363i −0.970208 0.242273i \(-0.922107\pi\)
0.275290 0.961361i \(-0.411226\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −25.0997 −0.831589 −0.415795 0.909459i \(-0.636496\pi\)
−0.415795 + 0.909459i \(0.636496\pi\)
\(912\) 0 0
\(913\) 19.5287 + 33.8248i 0.646307 + 1.11944i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.24490 + 28.2870i −0.0741332 + 0.934118i
\(918\) 0 0
\(919\) 23.4622 40.6377i 0.773947 1.34052i −0.161438 0.986883i \(-0.551613\pi\)
0.935384 0.353632i \(-0.115054\pi\)
\(920\) 0 0
\(921\) −22.9622 39.7717i −0.756631 1.31052i
\(922\) 0 0
\(923\) 11.9517 0.393396
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.0498 41.6555i 0.789049 1.36667i −0.137500 0.990502i \(-0.543907\pi\)
0.926550 0.376172i \(-0.122760\pi\)
\(930\) 0 0
\(931\) 8.18729 + 21.4125i 0.268328 + 0.701768i
\(932\) 0 0
\(933\) −8.50848 + 14.7371i −0.278555 + 0.482472i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.3638 −0.795931 −0.397965 0.917400i \(-0.630284\pi\)
−0.397965 + 0.917400i \(0.630284\pi\)
\(938\) 0 0
\(939\) −58.0241 −1.89354
\(940\) 0 0
\(941\) 1.63746 + 2.83616i 0.0533796 + 0.0924562i 0.891481 0.453059i \(-0.149667\pi\)
−0.838101 + 0.545515i \(0.816334\pi\)
\(942\) 0 0
\(943\) 14.5707 25.2371i 0.474486 0.821834i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.28247 + 9.14950i −0.171657 + 0.297319i −0.938999 0.343919i \(-0.888245\pi\)
0.767342 + 0.641238i \(0.221579\pi\)
\(948\) 0 0
\(949\) −8.54983 14.8087i −0.277539 0.480712i
\(950\) 0 0
\(951\) 44.3746 1.43894
\(952\) 0 0
\(953\) −22.6893 −0.734978 −0.367489 0.930028i \(-0.619782\pi\)
−0.367489 + 0.930028i \(0.619782\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −19.5287 + 33.8248i −0.631274 + 1.09340i
\(958\) 0 0
\(959\) 46.4622 + 31.9775i 1.50034 + 1.03261i
\(960\) 0 0
\(961\) 10.1375 17.5586i 0.327015 0.566406i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.15068 0.0691611 0.0345806 0.999402i \(-0.488990\pi\)
0.0345806 + 0.999402i \(0.488990\pi\)
\(968\) 0 0
\(969\) −1.18729 2.05645i −0.0381413 0.0660628i
\(970\) 0 0
\(971\) 18.4622 31.9775i 0.592481 1.02621i −0.401417 0.915896i \(-0.631482\pi\)
0.993897 0.110311i \(-0.0351846\pi\)
\(972\) 0 0
\(973\) −31.2920 + 14.9003i −1.00318 + 0.477683i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.9605 25.9124i −0.478629 0.829010i 0.521070 0.853514i \(-0.325533\pi\)
−0.999700 + 0.0245034i \(0.992200\pi\)
\(978\) 0 0
\(979\) 36.9244 1.18011
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.9641 + 39.7749i 0.732440 + 1.26862i 0.955838 + 0.293895i \(0.0949518\pi\)
−0.223398 + 0.974727i \(0.571715\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −26.9331 + 12.8248i −0.857290 + 0.408216i
\(988\) 0 0
\(989\) −8.41238 + 14.5707i −0.267498 + 0.463320i
\(990\) 0 0
\(991\) −16.7371 28.9896i −0.531672 0.920884i −0.999316 0.0369667i \(-0.988230\pi\)
0.467644 0.883917i \(-0.345103\pi\)
\(992\) 0 0
\(993\) −30.8734 −0.979737
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.209313 + 0.362541i 0.00662902 + 0.0114818i 0.869321 0.494248i \(-0.164557\pi\)
−0.862692 + 0.505730i \(0.831223\pi\)
\(998\) 0 0
\(999\) −25.9124 + 44.8816i −0.819831 + 1.41999i
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 700.2.i.f.401.3 8
5.2 odd 4 140.2.q.b.9.1 yes 4
5.3 odd 4 140.2.q.a.9.2 4
5.4 even 2 inner 700.2.i.f.401.2 8
7.2 even 3 4900.2.a.be.1.1 4
7.4 even 3 inner 700.2.i.f.501.3 8
7.5 odd 6 4900.2.a.bf.1.3 4
15.2 even 4 1260.2.bm.b.289.2 4
15.8 even 4 1260.2.bm.a.289.1 4
20.3 even 4 560.2.bw.e.289.2 4
20.7 even 4 560.2.bw.a.289.1 4
35.2 odd 12 980.2.e.f.589.3 4
35.3 even 12 980.2.q.b.949.1 4
35.4 even 6 inner 700.2.i.f.501.2 8
35.9 even 6 4900.2.a.be.1.3 4
35.12 even 12 980.2.e.c.589.2 4
35.13 even 4 980.2.q.g.569.1 4
35.17 even 12 980.2.q.g.949.1 4
35.18 odd 12 140.2.q.b.109.2 yes 4
35.19 odd 6 4900.2.a.bf.1.1 4
35.23 odd 12 980.2.e.f.589.1 4
35.27 even 4 980.2.q.b.569.2 4
35.32 odd 12 140.2.q.a.109.2 yes 4
35.33 even 12 980.2.e.c.589.4 4
105.32 even 12 1260.2.bm.a.109.1 4
105.53 even 12 1260.2.bm.b.109.1 4
140.67 even 12 560.2.bw.e.529.2 4
140.123 even 12 560.2.bw.a.529.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.q.a.9.2 4 5.3 odd 4
140.2.q.a.109.2 yes 4 35.32 odd 12
140.2.q.b.9.1 yes 4 5.2 odd 4
140.2.q.b.109.2 yes 4 35.18 odd 12
560.2.bw.a.289.1 4 20.7 even 4
560.2.bw.a.529.2 4 140.123 even 12
560.2.bw.e.289.2 4 20.3 even 4
560.2.bw.e.529.2 4 140.67 even 12
700.2.i.f.401.2 8 5.4 even 2 inner
700.2.i.f.401.3 8 1.1 even 1 trivial
700.2.i.f.501.2 8 35.4 even 6 inner
700.2.i.f.501.3 8 7.4 even 3 inner
980.2.e.c.589.2 4 35.12 even 12
980.2.e.c.589.4 4 35.33 even 12
980.2.e.f.589.1 4 35.23 odd 12
980.2.e.f.589.3 4 35.2 odd 12
980.2.q.b.569.2 4 35.27 even 4
980.2.q.b.949.1 4 35.3 even 12
980.2.q.g.569.1 4 35.13 even 4
980.2.q.g.949.1 4 35.17 even 12
1260.2.bm.a.109.1 4 105.32 even 12
1260.2.bm.a.289.1 4 15.8 even 4
1260.2.bm.b.109.1 4 105.53 even 12
1260.2.bm.b.289.2 4 15.2 even 4
4900.2.a.be.1.1 4 7.2 even 3
4900.2.a.be.1.3 4 35.9 even 6
4900.2.a.bf.1.1 4 35.19 odd 6
4900.2.a.bf.1.3 4 7.5 odd 6