Properties

Label 672.2.j.c.239.4
Level $672$
Weight $2$
Character 672.239
Analytic conductor $5.366$
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] = [672,2,Mod(239,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.j (of order \(2\), degree \(1\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.4
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 672.239
Dual form 672.2.j.c.239.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.61803 + 0.618034i) q^{3} +3.23607 q^{5} -1.00000i q^{7} +(2.23607 + 2.00000i) q^{9} +O(q^{10})\) \(q+(1.61803 + 0.618034i) q^{3} +3.23607 q^{5} -1.00000i q^{7} +(2.23607 + 2.00000i) q^{9} +4.47214i q^{11} -1.23607i q^{13} +(5.23607 + 2.00000i) q^{15} -2.00000i q^{17} -7.23607 q^{19} +(0.618034 - 1.61803i) q^{21} +0.472136 q^{23} +5.47214 q^{25} +(2.38197 + 4.61803i) q^{27} -8.94427i q^{31} +(-2.76393 + 7.23607i) q^{33} -3.23607i q^{35} -6.47214i q^{37} +(0.763932 - 2.00000i) q^{39} +4.47214i q^{41} +0.472136 q^{43} +(7.23607 + 6.47214i) q^{45} -2.47214 q^{47} -1.00000 q^{49} +(1.23607 - 3.23607i) q^{51} -10.4721 q^{53} +14.4721i q^{55} +(-11.7082 - 4.47214i) q^{57} -1.23607i q^{59} +11.7082i q^{61} +(2.00000 - 2.23607i) q^{63} -4.00000i q^{65} +10.9443 q^{67} +(0.763932 + 0.291796i) q^{69} +6.94427 q^{71} -0.472136 q^{73} +(8.85410 + 3.38197i) q^{75} +4.47214 q^{77} +4.94427i q^{79} +(1.00000 + 8.94427i) q^{81} -13.2361i q^{83} -6.47214i q^{85} -6.00000i q^{89} -1.23607 q^{91} +(5.52786 - 14.4721i) q^{93} -23.4164 q^{95} +3.52786 q^{97} +(-8.94427 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} + 12 q^{15} - 20 q^{19} - 2 q^{21} - 16 q^{23} + 4 q^{25} + 14 q^{27} - 20 q^{33} + 12 q^{39} - 16 q^{43} + 20 q^{45} + 8 q^{47} - 4 q^{49} - 4 q^{51} - 24 q^{53} - 20 q^{57} + 8 q^{63} + 8 q^{67} + 12 q^{69} - 8 q^{71} + 16 q^{73} + 22 q^{75} + 4 q^{81} + 4 q^{91} + 40 q^{93} - 40 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.61803 + 0.618034i 0.934172 + 0.356822i
\(4\) 0 0
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(10\) 0 0
\(11\) 4.47214i 1.34840i 0.738549 + 0.674200i \(0.235511\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 1.23607i 0.342824i −0.985199 0.171412i \(-0.945167\pi\)
0.985199 0.171412i \(-0.0548329\pi\)
\(14\) 0 0
\(15\) 5.23607 + 2.00000i 1.35195 + 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 0 0
\(21\) 0.618034 1.61803i 0.134866 0.353084i
\(22\) 0 0
\(23\) 0.472136 0.0984472 0.0492236 0.998788i \(-0.484325\pi\)
0.0492236 + 0.998788i \(0.484325\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 2.38197 + 4.61803i 0.458410 + 0.888741i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.94427i 1.60644i −0.595683 0.803219i \(-0.703119\pi\)
0.595683 0.803219i \(-0.296881\pi\)
\(32\) 0 0
\(33\) −2.76393 + 7.23607i −0.481139 + 1.25964i
\(34\) 0 0
\(35\) 3.23607i 0.546995i
\(36\) 0 0
\(37\) 6.47214i 1.06401i −0.846740 0.532006i \(-0.821438\pi\)
0.846740 0.532006i \(-0.178562\pi\)
\(38\) 0 0
\(39\) 0.763932 2.00000i 0.122327 0.320256i
\(40\) 0 0
\(41\) 4.47214i 0.698430i 0.937043 + 0.349215i \(0.113552\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 0.472136 0.0720001 0.0360000 0.999352i \(-0.488538\pi\)
0.0360000 + 0.999352i \(0.488538\pi\)
\(44\) 0 0
\(45\) 7.23607 + 6.47214i 1.07869 + 0.964809i
\(46\) 0 0
\(47\) −2.47214 −0.360598 −0.180299 0.983612i \(-0.557707\pi\)
−0.180299 + 0.983612i \(0.557707\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.23607 3.23607i 0.173084 0.453140i
\(52\) 0 0
\(53\) −10.4721 −1.43846 −0.719229 0.694773i \(-0.755505\pi\)
−0.719229 + 0.694773i \(0.755505\pi\)
\(54\) 0 0
\(55\) 14.4721i 1.95142i
\(56\) 0 0
\(57\) −11.7082 4.47214i −1.55079 0.592349i
\(58\) 0 0
\(59\) 1.23607i 0.160922i −0.996758 0.0804612i \(-0.974361\pi\)
0.996758 0.0804612i \(-0.0256393\pi\)
\(60\) 0 0
\(61\) 11.7082i 1.49908i 0.661958 + 0.749541i \(0.269726\pi\)
−0.661958 + 0.749541i \(0.730274\pi\)
\(62\) 0 0
\(63\) 2.00000 2.23607i 0.251976 0.281718i
\(64\) 0 0
\(65\) 4.00000i 0.496139i
\(66\) 0 0
\(67\) 10.9443 1.33706 0.668528 0.743687i \(-0.266925\pi\)
0.668528 + 0.743687i \(0.266925\pi\)
\(68\) 0 0
\(69\) 0.763932 + 0.291796i 0.0919666 + 0.0351281i
\(70\) 0 0
\(71\) 6.94427 0.824133 0.412067 0.911154i \(-0.364807\pi\)
0.412067 + 0.911154i \(0.364807\pi\)
\(72\) 0 0
\(73\) −0.472136 −0.0552593 −0.0276297 0.999618i \(-0.508796\pi\)
−0.0276297 + 0.999618i \(0.508796\pi\)
\(74\) 0 0
\(75\) 8.85410 + 3.38197i 1.02238 + 0.390516i
\(76\) 0 0
\(77\) 4.47214 0.509647
\(78\) 0 0
\(79\) 4.94427i 0.556274i 0.960541 + 0.278137i \(0.0897169\pi\)
−0.960541 + 0.278137i \(0.910283\pi\)
\(80\) 0 0
\(81\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(82\) 0 0
\(83\) 13.2361i 1.45285i −0.687247 0.726424i \(-0.741181\pi\)
0.687247 0.726424i \(-0.258819\pi\)
\(84\) 0 0
\(85\) 6.47214i 0.702002i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 0 0
\(93\) 5.52786 14.4721i 0.573213 1.50069i
\(94\) 0 0
\(95\) −23.4164 −2.40247
\(96\) 0 0
\(97\) 3.52786 0.358200 0.179100 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 0 0
\(99\) −8.94427 + 10.0000i −0.898933 + 1.00504i
\(100\) 0 0
\(101\) −6.29180 −0.626057 −0.313029 0.949744i \(-0.601344\pi\)
−0.313029 + 0.949744i \(0.601344\pi\)
\(102\) 0 0
\(103\) 1.52786i 0.150545i −0.997163 0.0752725i \(-0.976017\pi\)
0.997163 0.0752725i \(-0.0239827\pi\)
\(104\) 0 0
\(105\) 2.00000 5.23607i 0.195180 0.510988i
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 7.41641i 0.710363i 0.934797 + 0.355182i \(0.115581\pi\)
−0.934797 + 0.355182i \(0.884419\pi\)
\(110\) 0 0
\(111\) 4.00000 10.4721i 0.379663 0.993971i
\(112\) 0 0
\(113\) 12.9443i 1.21769i −0.793287 0.608847i \(-0.791632\pi\)
0.793287 0.608847i \(-0.208368\pi\)
\(114\) 0 0
\(115\) 1.52786 0.142474
\(116\) 0 0
\(117\) 2.47214 2.76393i 0.228549 0.255526i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −2.76393 + 7.23607i −0.249215 + 0.652454i
\(124\) 0 0
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) 16.9443i 1.50356i −0.659413 0.751780i \(-0.729195\pi\)
0.659413 0.751780i \(-0.270805\pi\)
\(128\) 0 0
\(129\) 0.763932 + 0.291796i 0.0672605 + 0.0256912i
\(130\) 0 0
\(131\) 2.76393i 0.241486i −0.992684 0.120743i \(-0.961472\pi\)
0.992684 0.120743i \(-0.0385277\pi\)
\(132\) 0 0
\(133\) 7.23607i 0.627447i
\(134\) 0 0
\(135\) 7.70820 + 14.9443i 0.663417 + 1.28620i
\(136\) 0 0
\(137\) 16.9443i 1.44765i 0.689985 + 0.723823i \(0.257617\pi\)
−0.689985 + 0.723823i \(0.742383\pi\)
\(138\) 0 0
\(139\) −10.6525 −0.903531 −0.451766 0.892137i \(-0.649206\pi\)
−0.451766 + 0.892137i \(0.649206\pi\)
\(140\) 0 0
\(141\) −4.00000 1.52786i −0.336861 0.128669i
\(142\) 0 0
\(143\) 5.52786 0.462263
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.61803 0.618034i −0.133453 0.0509746i
\(148\) 0 0
\(149\) −23.4164 −1.91835 −0.959173 0.282819i \(-0.908731\pi\)
−0.959173 + 0.282819i \(0.908731\pi\)
\(150\) 0 0
\(151\) 8.94427i 0.727875i 0.931423 + 0.363937i \(0.118568\pi\)
−0.931423 + 0.363937i \(0.881432\pi\)
\(152\) 0 0
\(153\) 4.00000 4.47214i 0.323381 0.361551i
\(154\) 0 0
\(155\) 28.9443i 2.32486i
\(156\) 0 0
\(157\) 3.70820i 0.295947i −0.988991 0.147973i \(-0.952725\pi\)
0.988991 0.147973i \(-0.0472750\pi\)
\(158\) 0 0
\(159\) −16.9443 6.47214i −1.34377 0.513274i
\(160\) 0 0
\(161\) 0.472136i 0.0372095i
\(162\) 0 0
\(163\) −8.47214 −0.663589 −0.331794 0.943352i \(-0.607654\pi\)
−0.331794 + 0.943352i \(0.607654\pi\)
\(164\) 0 0
\(165\) −8.94427 + 23.4164i −0.696311 + 1.82296i
\(166\) 0 0
\(167\) 0.944272 0.0730700 0.0365350 0.999332i \(-0.488368\pi\)
0.0365350 + 0.999332i \(0.488368\pi\)
\(168\) 0 0
\(169\) 11.4721 0.882472
\(170\) 0 0
\(171\) −16.1803 14.4721i −1.23734 1.10671i
\(172\) 0 0
\(173\) −17.7082 −1.34633 −0.673165 0.739492i \(-0.735066\pi\)
−0.673165 + 0.739492i \(0.735066\pi\)
\(174\) 0 0
\(175\) 5.47214i 0.413655i
\(176\) 0 0
\(177\) 0.763932 2.00000i 0.0574206 0.150329i
\(178\) 0 0
\(179\) 14.9443i 1.11699i 0.829509 + 0.558494i \(0.188620\pi\)
−0.829509 + 0.558494i \(0.811380\pi\)
\(180\) 0 0
\(181\) 17.2361i 1.28115i −0.767897 0.640573i \(-0.778697\pi\)
0.767897 0.640573i \(-0.221303\pi\)
\(182\) 0 0
\(183\) −7.23607 + 18.9443i −0.534906 + 1.40040i
\(184\) 0 0
\(185\) 20.9443i 1.53985i
\(186\) 0 0
\(187\) 8.94427 0.654070
\(188\) 0 0
\(189\) 4.61803 2.38197i 0.335913 0.173263i
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 0 0
\(193\) −11.5279 −0.829794 −0.414897 0.909868i \(-0.636182\pi\)
−0.414897 + 0.909868i \(0.636182\pi\)
\(194\) 0 0
\(195\) 2.47214 6.47214i 0.177033 0.463479i
\(196\) 0 0
\(197\) 22.4721 1.60107 0.800537 0.599284i \(-0.204548\pi\)
0.800537 + 0.599284i \(0.204548\pi\)
\(198\) 0 0
\(199\) 10.4721i 0.742350i 0.928563 + 0.371175i \(0.121045\pi\)
−0.928563 + 0.371175i \(0.878955\pi\)
\(200\) 0 0
\(201\) 17.7082 + 6.76393i 1.24904 + 0.477091i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.4721i 1.01078i
\(206\) 0 0
\(207\) 1.05573 + 0.944272i 0.0733782 + 0.0656314i
\(208\) 0 0
\(209\) 32.3607i 2.23844i
\(210\) 0 0
\(211\) 9.05573 0.623422 0.311711 0.950177i \(-0.399098\pi\)
0.311711 + 0.950177i \(0.399098\pi\)
\(212\) 0 0
\(213\) 11.2361 + 4.29180i 0.769883 + 0.294069i
\(214\) 0 0
\(215\) 1.52786 0.104199
\(216\) 0 0
\(217\) −8.94427 −0.607177
\(218\) 0 0
\(219\) −0.763932 0.291796i −0.0516217 0.0197178i
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 0 0
\(223\) 0.583592i 0.0390802i 0.999809 + 0.0195401i \(0.00622021\pi\)
−0.999809 + 0.0195401i \(0.993780\pi\)
\(224\) 0 0
\(225\) 12.2361 + 10.9443i 0.815738 + 0.729618i
\(226\) 0 0
\(227\) 0.291796i 0.0193672i 0.999953 + 0.00968359i \(0.00308243\pi\)
−0.999953 + 0.00968359i \(0.996918\pi\)
\(228\) 0 0
\(229\) 4.29180i 0.283610i −0.989895 0.141805i \(-0.954709\pi\)
0.989895 0.141805i \(-0.0452906\pi\)
\(230\) 0 0
\(231\) 7.23607 + 2.76393i 0.476098 + 0.181853i
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) −3.05573 + 8.00000i −0.198491 + 0.519656i
\(238\) 0 0
\(239\) −13.4164 −0.867835 −0.433918 0.900953i \(-0.642869\pi\)
−0.433918 + 0.900953i \(0.642869\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) 0 0
\(243\) −3.90983 + 15.0902i −0.250816 + 0.968035i
\(244\) 0 0
\(245\) −3.23607 −0.206745
\(246\) 0 0
\(247\) 8.94427i 0.569110i
\(248\) 0 0
\(249\) 8.18034 21.4164i 0.518408 1.35721i
\(250\) 0 0
\(251\) 8.29180i 0.523374i 0.965153 + 0.261687i \(0.0842787\pi\)
−0.965153 + 0.261687i \(0.915721\pi\)
\(252\) 0 0
\(253\) 2.11146i 0.132746i
\(254\) 0 0
\(255\) 4.00000 10.4721i 0.250490 0.655791i
\(256\) 0 0
\(257\) 1.41641i 0.0883531i 0.999024 + 0.0441765i \(0.0140664\pi\)
−0.999024 + 0.0441765i \(0.985934\pi\)
\(258\) 0 0
\(259\) −6.47214 −0.402159
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) −33.8885 −2.08176
\(266\) 0 0
\(267\) 3.70820 9.70820i 0.226938 0.594132i
\(268\) 0 0
\(269\) 16.1803 0.986533 0.493266 0.869878i \(-0.335803\pi\)
0.493266 + 0.869878i \(0.335803\pi\)
\(270\) 0 0
\(271\) 14.4721i 0.879120i −0.898213 0.439560i \(-0.855134\pi\)
0.898213 0.439560i \(-0.144866\pi\)
\(272\) 0 0
\(273\) −2.00000 0.763932i −0.121046 0.0462353i
\(274\) 0 0
\(275\) 24.4721i 1.47573i
\(276\) 0 0
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) 17.8885 20.0000i 1.07096 1.19737i
\(280\) 0 0
\(281\) 17.8885i 1.06714i 0.845756 + 0.533571i \(0.179150\pi\)
−0.845756 + 0.533571i \(0.820850\pi\)
\(282\) 0 0
\(283\) 3.23607 0.192364 0.0961821 0.995364i \(-0.469337\pi\)
0.0961821 + 0.995364i \(0.469337\pi\)
\(284\) 0 0
\(285\) −37.8885 14.4721i −2.24432 0.857255i
\(286\) 0 0
\(287\) 4.47214 0.263982
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 5.70820 + 2.18034i 0.334621 + 0.127814i
\(292\) 0 0
\(293\) 29.1246 1.70148 0.850739 0.525588i \(-0.176155\pi\)
0.850739 + 0.525588i \(0.176155\pi\)
\(294\) 0 0
\(295\) 4.00000i 0.232889i
\(296\) 0 0
\(297\) −20.6525 + 10.6525i −1.19838 + 0.618119i
\(298\) 0 0
\(299\) 0.583592i 0.0337500i
\(300\) 0 0
\(301\) 0.472136i 0.0272135i
\(302\) 0 0
\(303\) −10.1803 3.88854i −0.584845 0.223391i
\(304\) 0 0
\(305\) 37.8885i 2.16949i
\(306\) 0 0
\(307\) 17.1246 0.977353 0.488677 0.872465i \(-0.337480\pi\)
0.488677 + 0.872465i \(0.337480\pi\)
\(308\) 0 0
\(309\) 0.944272 2.47214i 0.0537178 0.140635i
\(310\) 0 0
\(311\) 20.3607 1.15455 0.577274 0.816550i \(-0.304116\pi\)
0.577274 + 0.816550i \(0.304116\pi\)
\(312\) 0 0
\(313\) 2.94427 0.166420 0.0832100 0.996532i \(-0.473483\pi\)
0.0832100 + 0.996532i \(0.473483\pi\)
\(314\) 0 0
\(315\) 6.47214 7.23607i 0.364664 0.407706i
\(316\) 0 0
\(317\) 2.47214 0.138849 0.0694245 0.997587i \(-0.477884\pi\)
0.0694245 + 0.997587i \(0.477884\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.23607 + 3.23607i −0.0689906 + 0.180620i
\(322\) 0 0
\(323\) 14.4721i 0.805251i
\(324\) 0 0
\(325\) 6.76393i 0.375195i
\(326\) 0 0
\(327\) −4.58359 + 12.0000i −0.253473 + 0.663602i
\(328\) 0 0
\(329\) 2.47214i 0.136293i
\(330\) 0 0
\(331\) 30.3607 1.66877 0.834387 0.551179i \(-0.185822\pi\)
0.834387 + 0.551179i \(0.185822\pi\)
\(332\) 0 0
\(333\) 12.9443 14.4721i 0.709342 0.793068i
\(334\) 0 0
\(335\) 35.4164 1.93501
\(336\) 0 0
\(337\) −16.4721 −0.897294 −0.448647 0.893709i \(-0.648094\pi\)
−0.448647 + 0.893709i \(0.648094\pi\)
\(338\) 0 0
\(339\) 8.00000 20.9443i 0.434500 1.13754i
\(340\) 0 0
\(341\) 40.0000 2.16612
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.47214 + 0.944272i 0.133095 + 0.0508379i
\(346\) 0 0
\(347\) 7.52786i 0.404117i 0.979373 + 0.202058i \(0.0647631\pi\)
−0.979373 + 0.202058i \(0.935237\pi\)
\(348\) 0 0
\(349\) 22.1803i 1.18729i −0.804728 0.593643i \(-0.797689\pi\)
0.804728 0.593643i \(-0.202311\pi\)
\(350\) 0 0
\(351\) 5.70820 2.94427i 0.304681 0.157154i
\(352\) 0 0
\(353\) 20.4721i 1.08962i 0.838559 + 0.544811i \(0.183399\pi\)
−0.838559 + 0.544811i \(0.816601\pi\)
\(354\) 0 0
\(355\) 22.4721 1.19270
\(356\) 0 0
\(357\) −3.23607 1.23607i −0.171271 0.0654197i
\(358\) 0 0
\(359\) −13.4164 −0.708091 −0.354045 0.935228i \(-0.615194\pi\)
−0.354045 + 0.935228i \(0.615194\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) 0 0
\(363\) −14.5623 5.56231i −0.764323 0.291945i
\(364\) 0 0
\(365\) −1.52786 −0.0799721
\(366\) 0 0
\(367\) 2.47214i 0.129044i −0.997916 0.0645222i \(-0.979448\pi\)
0.997916 0.0645222i \(-0.0205523\pi\)
\(368\) 0 0
\(369\) −8.94427 + 10.0000i −0.465620 + 0.520579i
\(370\) 0 0
\(371\) 10.4721i 0.543686i
\(372\) 0 0
\(373\) 33.8885i 1.75468i 0.479867 + 0.877341i \(0.340685\pi\)
−0.479867 + 0.877341i \(0.659315\pi\)
\(374\) 0 0
\(375\) 2.47214 + 0.944272i 0.127661 + 0.0487620i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.47214 −0.229718 −0.114859 0.993382i \(-0.536642\pi\)
−0.114859 + 0.993382i \(0.536642\pi\)
\(380\) 0 0
\(381\) 10.4721 27.4164i 0.536504 1.40459i
\(382\) 0 0
\(383\) 30.4721 1.55705 0.778527 0.627611i \(-0.215967\pi\)
0.778527 + 0.627611i \(0.215967\pi\)
\(384\) 0 0
\(385\) 14.4721 0.737568
\(386\) 0 0
\(387\) 1.05573 + 0.944272i 0.0536657 + 0.0480000i
\(388\) 0 0
\(389\) −28.9443 −1.46753 −0.733766 0.679402i \(-0.762239\pi\)
−0.733766 + 0.679402i \(0.762239\pi\)
\(390\) 0 0
\(391\) 0.944272i 0.0477539i
\(392\) 0 0
\(393\) 1.70820 4.47214i 0.0861675 0.225589i
\(394\) 0 0
\(395\) 16.0000i 0.805047i
\(396\) 0 0
\(397\) 30.7639i 1.54400i 0.635624 + 0.771999i \(0.280743\pi\)
−0.635624 + 0.771999i \(0.719257\pi\)
\(398\) 0 0
\(399\) −4.47214 + 11.7082i −0.223887 + 0.586143i
\(400\) 0 0
\(401\) 17.8885i 0.893311i 0.894706 + 0.446656i \(0.147385\pi\)
−0.894706 + 0.446656i \(0.852615\pi\)
\(402\) 0 0
\(403\) −11.0557 −0.550725
\(404\) 0 0
\(405\) 3.23607 + 28.9443i 0.160802 + 1.43825i
\(406\) 0 0
\(407\) 28.9443 1.43471
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −10.4721 + 27.4164i −0.516552 + 1.35235i
\(412\) 0 0
\(413\) −1.23607 −0.0608229
\(414\) 0 0
\(415\) 42.8328i 2.10258i
\(416\) 0 0
\(417\) −17.2361 6.58359i −0.844054 0.322400i
\(418\) 0 0
\(419\) 35.7082i 1.74446i −0.489096 0.872230i \(-0.662673\pi\)
0.489096 0.872230i \(-0.337327\pi\)
\(420\) 0 0
\(421\) 8.94427i 0.435917i 0.975958 + 0.217959i \(0.0699398\pi\)
−0.975958 + 0.217959i \(0.930060\pi\)
\(422\) 0 0
\(423\) −5.52786 4.94427i −0.268774 0.240399i
\(424\) 0 0
\(425\) 10.9443i 0.530875i
\(426\) 0 0
\(427\) 11.7082 0.566600
\(428\) 0 0
\(429\) 8.94427 + 3.41641i 0.431834 + 0.164946i
\(430\) 0 0
\(431\) 12.4721 0.600762 0.300381 0.953819i \(-0.402886\pi\)
0.300381 + 0.953819i \(0.402886\pi\)
\(432\) 0 0
\(433\) 26.3607 1.26681 0.633407 0.773819i \(-0.281656\pi\)
0.633407 + 0.773819i \(0.281656\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.41641 −0.163429
\(438\) 0 0
\(439\) 33.8885i 1.61741i 0.588213 + 0.808706i \(0.299832\pi\)
−0.588213 + 0.808706i \(0.700168\pi\)
\(440\) 0 0
\(441\) −2.23607 2.00000i −0.106479 0.0952381i
\(442\) 0 0
\(443\) 2.94427i 0.139887i 0.997551 + 0.0699433i \(0.0222818\pi\)
−0.997551 + 0.0699433i \(0.977718\pi\)
\(444\) 0 0
\(445\) 19.4164i 0.920426i
\(446\) 0 0
\(447\) −37.8885 14.4721i −1.79207 0.684509i
\(448\) 0 0
\(449\) 13.8885i 0.655441i −0.944775 0.327720i \(-0.893720\pi\)
0.944775 0.327720i \(-0.106280\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) −5.52786 + 14.4721i −0.259722 + 0.679960i
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 21.4164 1.00182 0.500909 0.865500i \(-0.332999\pi\)
0.500909 + 0.865500i \(0.332999\pi\)
\(458\) 0 0
\(459\) 9.23607 4.76393i 0.431103 0.222361i
\(460\) 0 0
\(461\) 8.18034 0.380996 0.190498 0.981688i \(-0.438990\pi\)
0.190498 + 0.981688i \(0.438990\pi\)
\(462\) 0 0
\(463\) 12.9443i 0.601571i 0.953692 + 0.300786i \(0.0972489\pi\)
−0.953692 + 0.300786i \(0.902751\pi\)
\(464\) 0 0
\(465\) 17.8885 46.8328i 0.829561 2.17182i
\(466\) 0 0
\(467\) 0.291796i 0.0135027i 0.999977 + 0.00675136i \(0.00214904\pi\)
−0.999977 + 0.00675136i \(0.997851\pi\)
\(468\) 0 0
\(469\) 10.9443i 0.505360i
\(470\) 0 0
\(471\) 2.29180 6.00000i 0.105600 0.276465i
\(472\) 0 0
\(473\) 2.11146i 0.0970849i
\(474\) 0 0
\(475\) −39.5967 −1.81682
\(476\) 0 0
\(477\) −23.4164 20.9443i −1.07216 0.958972i
\(478\) 0 0
\(479\) −34.4721 −1.57507 −0.787536 0.616269i \(-0.788644\pi\)
−0.787536 + 0.616269i \(0.788644\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0.291796 0.763932i 0.0132772 0.0347601i
\(484\) 0 0
\(485\) 11.4164 0.518392
\(486\) 0 0
\(487\) 29.8885i 1.35438i 0.735809 + 0.677190i \(0.236802\pi\)
−0.735809 + 0.677190i \(0.763198\pi\)
\(488\) 0 0
\(489\) −13.7082 5.23607i −0.619906 0.236783i
\(490\) 0 0
\(491\) 27.8885i 1.25859i 0.777166 + 0.629296i \(0.216657\pi\)
−0.777166 + 0.629296i \(0.783343\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −28.9443 + 32.3607i −1.30095 + 1.45450i
\(496\) 0 0
\(497\) 6.94427i 0.311493i
\(498\) 0 0
\(499\) −27.8885 −1.24846 −0.624231 0.781240i \(-0.714588\pi\)
−0.624231 + 0.781240i \(0.714588\pi\)
\(500\) 0 0
\(501\) 1.52786 + 0.583592i 0.0682599 + 0.0260730i
\(502\) 0 0
\(503\) 12.5836 0.561075 0.280537 0.959843i \(-0.409487\pi\)
0.280537 + 0.959843i \(0.409487\pi\)
\(504\) 0 0
\(505\) −20.3607 −0.906038
\(506\) 0 0
\(507\) 18.5623 + 7.09017i 0.824381 + 0.314886i
\(508\) 0 0
\(509\) −5.12461 −0.227144 −0.113572 0.993530i \(-0.536229\pi\)
−0.113572 + 0.993530i \(0.536229\pi\)
\(510\) 0 0
\(511\) 0.472136i 0.0208861i
\(512\) 0 0
\(513\) −17.2361 33.4164i −0.760991 1.47537i
\(514\) 0 0
\(515\) 4.94427i 0.217871i
\(516\) 0 0
\(517\) 11.0557i 0.486230i
\(518\) 0 0
\(519\) −28.6525 10.9443i −1.25770 0.480400i
\(520\) 0 0
\(521\) 6.58359i 0.288432i 0.989546 + 0.144216i \(0.0460661\pi\)
−0.989546 + 0.144216i \(0.953934\pi\)
\(522\) 0 0
\(523\) −25.7082 −1.12414 −0.562071 0.827089i \(-0.689995\pi\)
−0.562071 + 0.827089i \(0.689995\pi\)
\(524\) 0 0
\(525\) 3.38197 8.85410i 0.147601 0.386425i
\(526\) 0 0
\(527\) −17.8885 −0.779237
\(528\) 0 0
\(529\) −22.7771 −0.990308
\(530\) 0 0
\(531\) 2.47214 2.76393i 0.107282 0.119944i
\(532\) 0 0
\(533\) 5.52786 0.239438
\(534\) 0 0
\(535\) 6.47214i 0.279815i
\(536\) 0 0
\(537\) −9.23607 + 24.1803i −0.398566 + 1.04346i
\(538\) 0 0
\(539\) 4.47214i 0.192629i
\(540\) 0 0
\(541\) 8.94427i 0.384544i −0.981342 0.192272i \(-0.938414\pi\)
0.981342 0.192272i \(-0.0615856\pi\)
\(542\) 0 0
\(543\) 10.6525 27.8885i 0.457141 1.19681i
\(544\) 0 0
\(545\) 24.0000i 1.02805i
\(546\) 0 0
\(547\) −32.4721 −1.38841 −0.694204 0.719778i \(-0.744244\pi\)
−0.694204 + 0.719778i \(0.744244\pi\)
\(548\) 0 0
\(549\) −23.4164 + 26.1803i −0.999388 + 1.11735i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.94427 0.210252
\(554\) 0 0
\(555\) 12.9443 33.8885i 0.549454 1.43849i
\(556\) 0 0
\(557\) 2.47214 0.104748 0.0523739 0.998628i \(-0.483321\pi\)
0.0523739 + 0.998628i \(0.483321\pi\)
\(558\) 0 0
\(559\) 0.583592i 0.0246833i
\(560\) 0 0
\(561\) 14.4721 + 5.52786i 0.611014 + 0.233387i
\(562\) 0 0
\(563\) 9.81966i 0.413849i −0.978357 0.206925i \(-0.933654\pi\)
0.978357 0.206925i \(-0.0663455\pi\)
\(564\) 0 0
\(565\) 41.8885i 1.76226i
\(566\) 0 0
\(567\) 8.94427 1.00000i 0.375624 0.0419961i
\(568\) 0 0
\(569\) 13.8885i 0.582238i −0.956687 0.291119i \(-0.905972\pi\)
0.956687 0.291119i \(-0.0940276\pi\)
\(570\) 0 0
\(571\) 23.5279 0.984610 0.492305 0.870423i \(-0.336154\pi\)
0.492305 + 0.870423i \(0.336154\pi\)
\(572\) 0 0
\(573\) −3.23607 1.23607i −0.135189 0.0516375i
\(574\) 0 0
\(575\) 2.58359 0.107743
\(576\) 0 0
\(577\) 9.05573 0.376995 0.188497 0.982074i \(-0.439638\pi\)
0.188497 + 0.982074i \(0.439638\pi\)
\(578\) 0 0
\(579\) −18.6525 7.12461i −0.775170 0.296089i
\(580\) 0 0
\(581\) −13.2361 −0.549125
\(582\) 0 0
\(583\) 46.8328i 1.93962i
\(584\) 0 0
\(585\) 8.00000 8.94427i 0.330759 0.369800i
\(586\) 0 0
\(587\) 23.7082i 0.978542i 0.872132 + 0.489271i \(0.162737\pi\)
−0.872132 + 0.489271i \(0.837263\pi\)
\(588\) 0 0
\(589\) 64.7214i 2.66680i
\(590\) 0 0
\(591\) 36.3607 + 13.8885i 1.49568 + 0.571298i
\(592\) 0 0
\(593\) 9.41641i 0.386686i 0.981131 + 0.193343i \(0.0619329\pi\)
−0.981131 + 0.193343i \(0.938067\pi\)
\(594\) 0 0
\(595\) −6.47214 −0.265332
\(596\) 0 0
\(597\) −6.47214 + 16.9443i −0.264887 + 0.693483i
\(598\) 0 0
\(599\) −1.05573 −0.0431359 −0.0215679 0.999767i \(-0.506866\pi\)
−0.0215679 + 0.999767i \(0.506866\pi\)
\(600\) 0 0
\(601\) 7.52786 0.307068 0.153534 0.988143i \(-0.450935\pi\)
0.153534 + 0.988143i \(0.450935\pi\)
\(602\) 0 0
\(603\) 24.4721 + 21.8885i 0.996582 + 0.891370i
\(604\) 0 0
\(605\) −29.1246 −1.18408
\(606\) 0 0
\(607\) 2.47214i 0.100341i −0.998741 0.0501705i \(-0.984024\pi\)
0.998741 0.0501705i \(-0.0159765\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.05573i 0.123622i
\(612\) 0 0
\(613\) 0.583592i 0.0235711i −0.999931 0.0117855i \(-0.996248\pi\)
0.999931 0.0117855i \(-0.00375154\pi\)
\(614\) 0 0
\(615\) −8.94427 + 23.4164i −0.360668 + 0.944241i
\(616\) 0 0
\(617\) 32.0000i 1.28827i −0.764911 0.644136i \(-0.777217\pi\)
0.764911 0.644136i \(-0.222783\pi\)
\(618\) 0 0
\(619\) −5.12461 −0.205976 −0.102988 0.994683i \(-0.532840\pi\)
−0.102988 + 0.994683i \(0.532840\pi\)
\(620\) 0 0
\(621\) 1.12461 + 2.18034i 0.0451291 + 0.0874940i
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 20.0000 52.3607i 0.798723 2.09108i
\(628\) 0 0
\(629\) −12.9443 −0.516122
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 14.6525 + 5.59675i 0.582384 + 0.222451i
\(634\) 0 0
\(635\) 54.8328i 2.17597i
\(636\) 0 0
\(637\) 1.23607i 0.0489748i
\(638\) 0 0
\(639\) 15.5279 + 13.8885i 0.614273 + 0.549422i
\(640\) 0 0
\(641\) 40.0000i 1.57991i −0.613168 0.789953i \(-0.710105\pi\)
0.613168 0.789953i \(-0.289895\pi\)
\(642\) 0 0
\(643\) 32.1803 1.26907 0.634534 0.772895i \(-0.281192\pi\)
0.634534 + 0.772895i \(0.281192\pi\)
\(644\) 0 0
\(645\) 2.47214 + 0.944272i 0.0973403 + 0.0371807i
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 5.52786 0.216988
\(650\) 0 0
\(651\) −14.4721 5.52786i −0.567208 0.216654i
\(652\) 0 0
\(653\) 10.8328 0.423921 0.211960 0.977278i \(-0.432015\pi\)
0.211960 + 0.977278i \(0.432015\pi\)
\(654\) 0 0
\(655\) 8.94427i 0.349482i
\(656\) 0 0
\(657\) −1.05573 0.944272i −0.0411879 0.0368396i
\(658\) 0 0
\(659\) 46.3607i 1.80596i −0.429687 0.902978i \(-0.641376\pi\)
0.429687 0.902978i \(-0.358624\pi\)
\(660\) 0 0
\(661\) 35.1246i 1.36619i −0.730330 0.683095i \(-0.760634\pi\)
0.730330 0.683095i \(-0.239366\pi\)
\(662\) 0 0
\(663\) −4.00000 1.52786i −0.155347 0.0593373i
\(664\) 0 0
\(665\) 23.4164i 0.908049i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.360680 + 0.944272i −0.0139447 + 0.0365077i
\(670\) 0 0
\(671\) −52.3607 −2.02136
\(672\) 0 0
\(673\) 22.9443 0.884437 0.442218 0.896907i \(-0.354192\pi\)
0.442218 + 0.896907i \(0.354192\pi\)
\(674\) 0 0
\(675\) 13.0344 + 25.2705i 0.501696 + 0.972662i
\(676\) 0 0
\(677\) 24.1803 0.929326 0.464663 0.885488i \(-0.346175\pi\)
0.464663 + 0.885488i \(0.346175\pi\)
\(678\) 0 0
\(679\) 3.52786i 0.135387i
\(680\) 0 0
\(681\) −0.180340 + 0.472136i −0.00691064 + 0.0180923i
\(682\) 0 0
\(683\) 42.9443i 1.64322i 0.570052 + 0.821608i \(0.306923\pi\)
−0.570052 + 0.821608i \(0.693077\pi\)
\(684\) 0 0
\(685\) 54.8328i 2.09505i
\(686\) 0 0
\(687\) 2.65248 6.94427i 0.101198 0.264940i
\(688\) 0 0
\(689\) 12.9443i 0.493137i
\(690\) 0 0
\(691\) −2.65248 −0.100905 −0.0504525 0.998726i \(-0.516066\pi\)
−0.0504525 + 0.998726i \(0.516066\pi\)
\(692\) 0 0
\(693\) 10.0000 + 8.94427i 0.379869 + 0.339765i
\(694\) 0 0
\(695\) −34.4721 −1.30760
\(696\) 0 0
\(697\) 8.94427 0.338788
\(698\) 0 0
\(699\) −9.88854 + 25.8885i −0.374019 + 0.979195i
\(700\) 0 0
\(701\) −5.88854 −0.222407 −0.111204 0.993798i \(-0.535471\pi\)
−0.111204 + 0.993798i \(0.535471\pi\)
\(702\) 0 0
\(703\) 46.8328i 1.76633i
\(704\) 0 0
\(705\) −12.9443 4.94427i −0.487509 0.186212i
\(706\) 0 0
\(707\) 6.29180i 0.236627i
\(708\) 0 0
\(709\) 10.4721i 0.393289i −0.980475 0.196645i \(-0.936995\pi\)
0.980475 0.196645i \(-0.0630045\pi\)
\(710\) 0 0
\(711\) −9.88854 + 11.0557i −0.370849 + 0.414622i
\(712\) 0 0
\(713\) 4.22291i 0.158149i
\(714\) 0 0
\(715\) 17.8885 0.668994
\(716\) 0 0
\(717\) −21.7082 8.29180i −0.810708 0.309663i
\(718\) 0 0
\(719\) 32.3607 1.20685 0.603425 0.797420i \(-0.293802\pi\)
0.603425 + 0.797420i \(0.293802\pi\)
\(720\) 0 0
\(721\) −1.52786 −0.0569006
\(722\) 0 0
\(723\) −5.70820 2.18034i −0.212290 0.0810877i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11.4164i 0.423411i −0.977333 0.211706i \(-0.932098\pi\)
0.977333 0.211706i \(-0.0679018\pi\)
\(728\) 0 0
\(729\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(730\) 0 0
\(731\) 0.944272i 0.0349252i
\(732\) 0 0
\(733\) 42.5410i 1.57129i −0.618679 0.785644i \(-0.712332\pi\)
0.618679 0.785644i \(-0.287668\pi\)
\(734\) 0 0
\(735\) −5.23607 2.00000i −0.193135 0.0737711i
\(736\) 0 0
\(737\) 48.9443i 1.80289i
\(738\) 0 0
\(739\) −4.47214 −0.164510 −0.0822551 0.996611i \(-0.526212\pi\)
−0.0822551 + 0.996611i \(0.526212\pi\)
\(740\) 0 0
\(741\) −5.52786 + 14.4721i −0.203071 + 0.531647i
\(742\) 0 0
\(743\) 47.3050 1.73545 0.867725 0.497044i \(-0.165581\pi\)
0.867725 + 0.497044i \(0.165581\pi\)
\(744\) 0 0
\(745\) −75.7771 −2.77626
\(746\) 0 0
\(747\) 26.4721 29.5967i 0.968565 1.08289i
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 26.8328i 0.979143i 0.871963 + 0.489572i \(0.162847\pi\)
−0.871963 + 0.489572i \(0.837153\pi\)
\(752\) 0 0
\(753\) −5.12461 + 13.4164i −0.186751 + 0.488921i
\(754\) 0 0
\(755\) 28.9443i 1.05339i
\(756\) 0 0
\(757\) 8.58359i 0.311976i −0.987759 0.155988i \(-0.950144\pi\)
0.987759 0.155988i \(-0.0498561\pi\)
\(758\) 0 0
\(759\) −1.30495 + 3.41641i −0.0473667 + 0.124008i
\(760\) 0 0
\(761\) 51.3050i 1.85980i −0.367809 0.929902i \(-0.619892\pi\)
0.367809 0.929902i \(-0.380108\pi\)
\(762\) 0 0
\(763\) 7.41641 0.268492
\(764\) 0 0
\(765\) 12.9443 14.4721i 0.468001 0.523241i
\(766\) 0 0
\(767\) −1.52786 −0.0551680
\(768\) 0 0
\(769\) 22.3607 0.806347 0.403173 0.915124i \(-0.367907\pi\)
0.403173 + 0.915124i \(0.367907\pi\)
\(770\) 0 0
\(771\) −0.875388 + 2.29180i −0.0315263 + 0.0825370i
\(772\) 0 0
\(773\) −6.65248 −0.239273 −0.119636 0.992818i \(-0.538173\pi\)
−0.119636 + 0.992818i \(0.538173\pi\)
\(774\) 0 0
\(775\) 48.9443i 1.75813i
\(776\) 0 0
\(777\) −10.4721 4.00000i −0.375686 0.143499i
\(778\) 0 0
\(779\) 32.3607i 1.15944i
\(780\) 0 0
\(781\) 31.0557i 1.11126i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0000i 0.428298i
\(786\) 0 0
\(787\) 17.1246 0.610426 0.305213 0.952284i \(-0.401272\pi\)
0.305213 + 0.952284i \(0.401272\pi\)
\(788\) 0 0
\(789\) −22.6525 8.65248i −0.806449 0.308036i
\(790\) 0 0
\(791\) −12.9443 −0.460245
\(792\) 0 0
\(793\) 14.4721 0.513921
\(794\) 0 0
\(795\) −54.8328 20.9443i −1.94472 0.742817i
\(796\) 0 0
\(797\) −26.0689 −0.923407 −0.461704 0.887034i \(-0.652762\pi\)
−0.461704 + 0.887034i \(0.652762\pi\)
\(798\) 0 0
\(799\) 4.94427i 0.174916i
\(800\) 0 0
\(801\) 12.0000 13.4164i 0.423999 0.474045i
\(802\) 0 0
\(803\) 2.11146i 0.0745117i
\(804\) 0 0
\(805\) 1.52786i 0.0538501i
\(806\) 0 0
\(807\) 26.1803 + 10.0000i 0.921592 + 0.352017i
\(808\) 0 0
\(809\) 40.7214i 1.43169i −0.698261 0.715843i \(-0.746042\pi\)
0.698261 0.715843i \(-0.253958\pi\)
\(810\) 0 0
\(811\) 42.0689 1.47724 0.738619 0.674123i \(-0.235478\pi\)
0.738619 + 0.674123i \(0.235478\pi\)
\(812\) 0 0
\(813\) 8.94427 23.4164i 0.313689 0.821249i
\(814\) 0 0
\(815\) −27.4164 −0.960355
\(816\) 0 0
\(817\) −3.41641 −0.119525
\(818\) 0 0
\(819\) −2.76393 2.47214i −0.0965796 0.0863834i
\(820\) 0 0
\(821\) −13.5279 −0.472126 −0.236063 0.971738i \(-0.575857\pi\)
−0.236063 + 0.971738i \(0.575857\pi\)
\(822\) 0 0
\(823\) 22.8328i 0.795902i −0.917407 0.397951i \(-0.869721\pi\)
0.917407 0.397951i \(-0.130279\pi\)
\(824\) 0 0
\(825\) −15.1246 + 39.5967i −0.526571 + 1.37858i
\(826\) 0 0
\(827\) 2.00000i 0.0695468i 0.999395 + 0.0347734i \(0.0110710\pi\)
−0.999395 + 0.0347734i \(0.988929\pi\)
\(828\) 0 0
\(829\) 24.6525i 0.856216i 0.903728 + 0.428108i \(0.140820\pi\)
−0.903728 + 0.428108i \(0.859180\pi\)
\(830\) 0 0
\(831\) 7.41641 19.4164i 0.257272 0.673548i
\(832\) 0 0
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 3.05573 0.105748
\(836\) 0 0
\(837\) 41.3050 21.3050i 1.42771 0.736407i
\(838\) 0 0
\(839\) −55.7771 −1.92564 −0.962819 0.270146i \(-0.912928\pi\)
−0.962819 + 0.270146i \(0.912928\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −11.0557 + 28.9443i −0.380780 + 0.996894i
\(844\) 0 0
\(845\) 37.1246 1.27713
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) 5.23607 + 2.00000i 0.179701 + 0.0686398i
\(850\) 0 0
\(851\) 3.05573i 0.104749i
\(852\) 0 0
\(853\) 19.1246i 0.654814i −0.944883 0.327407i \(-0.893825\pi\)
0.944883 0.327407i \(-0.106175\pi\)
\(854\) 0 0
\(855\) −52.3607 46.8328i −1.79070 1.60165i
\(856\) 0 0
\(857\) 39.3050i 1.34263i 0.741171 + 0.671316i \(0.234271\pi\)
−0.741171 + 0.671316i \(0.765729\pi\)
\(858\) 0 0
\(859\) 14.0689 0.480024 0.240012 0.970770i \(-0.422849\pi\)
0.240012 + 0.970770i \(0.422849\pi\)
\(860\) 0 0
\(861\) 7.23607 + 2.76393i 0.246605 + 0.0941946i
\(862\) 0 0
\(863\) 50.7214 1.72658 0.863288 0.504712i \(-0.168401\pi\)
0.863288 + 0.504712i \(0.168401\pi\)
\(864\) 0 0
\(865\) −57.3050 −1.94843
\(866\) 0 0
\(867\) 21.0344 + 8.03444i 0.714367 + 0.272864i
\(868\) 0 0
\(869\) −22.1115 −0.750080
\(870\) 0 0
\(871\) 13.5279i 0.458374i
\(872\) 0 0
\(873\) 7.88854 + 7.05573i 0.266987 + 0.238800i
\(874\) 0 0
\(875\) 1.52786i 0.0516512i
\(876\) 0 0
\(877\) 31.4164i 1.06086i 0.847730 + 0.530428i \(0.177969\pi\)
−0.847730 + 0.530428i \(0.822031\pi\)
\(878\) 0 0
\(879\) 47.1246 + 18.0000i 1.58947 + 0.607125i
\(880\) 0 0
\(881\) 13.4164i 0.452010i 0.974126 + 0.226005i \(0.0725666\pi\)
−0.974126 + 0.226005i \(0.927433\pi\)
\(882\) 0 0
\(883\) −2.94427 −0.0990826 −0.0495413 0.998772i \(-0.515776\pi\)
−0.0495413 + 0.998772i \(0.515776\pi\)
\(884\) 0 0
\(885\) 2.47214 6.47214i 0.0830999 0.217558i
\(886\) 0 0
\(887\) −5.88854 −0.197718 −0.0988590 0.995101i \(-0.531519\pi\)
−0.0988590 + 0.995101i \(0.531519\pi\)
\(888\) 0 0
\(889\) −16.9443 −0.568293
\(890\) 0 0
\(891\) −40.0000 + 4.47214i −1.34005 + 0.149822i
\(892\) 0 0
\(893\) 17.8885 0.598617
\(894\) 0 0
\(895\) 48.3607i 1.61652i
\(896\) 0 0
\(897\) 0.360680 0.944272i 0.0120427 0.0315283i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 20.9443i 0.697755i
\(902\) 0 0
\(903\) 0.291796 0.763932i 0.00971037 0.0254221i
\(904\) 0 0
\(905\) 55.7771i 1.85409i
\(906\) 0 0
\(907\) −42.7214 −1.41854 −0.709270 0.704937i \(-0.750975\pi\)
−0.709270 + 0.704937i \(0.750975\pi\)
\(908\) 0 0
\(909\) −14.0689 12.5836i −0.466635 0.417371i
\(910\) 0 0
\(911\) −54.3607 −1.80105 −0.900525 0.434805i \(-0.856817\pi\)
−0.900525 + 0.434805i \(0.856817\pi\)
\(912\) 0 0
\(913\) 59.1935 1.95902
\(914\) 0 0
\(915\) −23.4164 + 61.3050i −0.774123 + 2.02668i
\(916\) 0 0
\(917\) −2.76393 −0.0912731
\(918\) 0 0
\(919\) 27.0557i 0.892486i 0.894912 + 0.446243i \(0.147238\pi\)
−0.894912 + 0.446243i \(0.852762\pi\)
\(920\) 0 0
\(921\) 27.7082 + 10.5836i 0.913016 + 0.348741i
\(922\) 0 0
\(923\) 8.58359i 0.282532i
\(924\) 0 0
\(925\) 35.4164i 1.16448i
\(926\) 0 0
\(927\) 3.05573 3.41641i 0.100363 0.112210i
\(928\) 0 0
\(929\) 54.9443i 1.80266i −0.433130 0.901332i \(-0.642591\pi\)
0.433130 0.901332i \(-0.357409\pi\)
\(930\) 0 0
\(931\) 7.23607 0.237153
\(932\) 0 0
\(933\) 32.9443 + 12.5836i 1.07855 + 0.411968i
\(934\) 0 0
\(935\) 28.9443 0.946579
\(936\) 0 0
\(937\) 21.4164 0.699644 0.349822 0.936816i \(-0.386242\pi\)
0.349822 + 0.936816i \(0.386242\pi\)
\(938\) 0 0
\(939\) 4.76393 + 1.81966i 0.155465 + 0.0593824i
\(940\) 0 0
\(941\) 26.0689 0.849821 0.424911 0.905235i \(-0.360306\pi\)
0.424911 + 0.905235i \(0.360306\pi\)
\(942\) 0 0
\(943\) 2.11146i 0.0687585i
\(944\) 0 0
\(945\) 14.9443 7.70820i 0.486137 0.250748i
\(946\) 0 0
\(947\) 16.4721i 0.535272i 0.963520 + 0.267636i \(0.0862425\pi\)
−0.963520 + 0.267636i \(0.913757\pi\)
\(948\) 0 0
\(949\) 0.583592i 0.0189442i
\(950\) 0 0
\(951\) 4.00000 + 1.52786i 0.129709 + 0.0495444i
\(952\) 0 0
\(953\) 33.8885i 1.09776i 0.835902 + 0.548879i \(0.184945\pi\)
−0.835902 + 0.548879i \(0.815055\pi\)
\(954\) 0 0
\(955\) −6.47214 −0.209433
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.9443 0.547159
\(960\) 0 0
\(961\) −49.0000 −1.58065
\(962\) 0 0
\(963\) −4.00000 + 4.47214i −0.128898 + 0.144113i
\(964\) 0 0
\(965\) −37.3050 −1.20089
\(966\) 0 0
\(967\) 0.944272i 0.0303657i 0.999885 + 0.0151829i \(0.00483304\pi\)
−0.999885 + 0.0151829i \(0.995167\pi\)
\(968\) 0 0
\(969\) −8.94427 + 23.4164i −0.287331 + 0.752243i
\(970\) 0 0
\(971\) 28.2918i 0.907927i −0.891020 0.453963i \(-0.850010\pi\)
0.891020 0.453963i \(-0.149990\pi\)
\(972\) 0 0
\(973\) 10.6525i 0.341503i
\(974\) 0 0
\(975\) 4.18034 10.9443i 0.133878 0.350497i
\(976\) 0 0
\(977\) 25.8885i 0.828248i 0.910220 + 0.414124i \(0.135912\pi\)
−0.910220 + 0.414124i \(0.864088\pi\)
\(978\) 0 0
\(979\) 26.8328 0.857581
\(980\) 0 0
\(981\) −14.8328 + 16.5836i −0.473575 + 0.529473i
\(982\) 0 0
\(983\) −52.9443 −1.68866 −0.844330 0.535824i \(-0.820001\pi\)
−0.844330 + 0.535824i \(0.820001\pi\)
\(984\) 0 0
\(985\) 72.7214 2.31710
\(986\) 0 0
\(987\) −1.52786 + 4.00000i −0.0486324 + 0.127321i
\(988\) 0 0
\(989\) 0.222912 0.00708820
\(990\) 0 0
\(991\) 40.0000i 1.27064i 0.772248 + 0.635321i \(0.219132\pi\)
−0.772248 + 0.635321i \(0.780868\pi\)
\(992\) 0 0
\(993\) 49.1246 + 18.7639i 1.55892 + 0.595455i
\(994\) 0 0
\(995\) 33.8885i 1.07434i
\(996\) 0 0
\(997\) 43.1246i 1.36577i 0.730526 + 0.682885i \(0.239275\pi\)
−0.730526 + 0.682885i \(0.760725\pi\)
\(998\) 0 0
\(999\) 29.8885 15.4164i 0.945632 0.487754i
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 672.2.j.c.239.4 4
3.2 odd 2 672.2.j.b.239.3 4
4.3 odd 2 168.2.j.a.155.1 4
8.3 odd 2 672.2.j.b.239.4 4
8.5 even 2 168.2.j.c.155.1 yes 4
12.11 even 2 168.2.j.c.155.3 yes 4
24.5 odd 2 168.2.j.a.155.3 yes 4
24.11 even 2 inner 672.2.j.c.239.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.a.155.1 4 4.3 odd 2
168.2.j.a.155.3 yes 4 24.5 odd 2
168.2.j.c.155.1 yes 4 8.5 even 2
168.2.j.c.155.3 yes 4 12.11 even 2
672.2.j.b.239.3 4 3.2 odd 2
672.2.j.b.239.4 4 8.3 odd 2
672.2.j.c.239.3 4 24.11 even 2 inner
672.2.j.c.239.4 4 1.1 even 1 trivial