Properties

Label 768.2.f.e.383.4
Level $768$
Weight $2$
Character 768.383
Analytic conductor $6.133$
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] = [768,2,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.f (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: \(6.13251087523\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.4
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 768.383
Dual form 768.2.f.e.383.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.61803 + 0.618034i) q^{3} -3.23607 q^{5} -1.23607i q^{7} +(2.23607 + 2.00000i) q^{9} +O(q^{10})\) \(q+(1.61803 + 0.618034i) q^{3} -3.23607 q^{5} -1.23607i q^{7} +(2.23607 + 2.00000i) q^{9} -5.23607i q^{11} -4.47214i q^{13} +(-5.23607 - 2.00000i) q^{15} -2.47214i q^{17} -0.763932 q^{19} +(0.763932 - 2.00000i) q^{21} +2.47214 q^{23} +5.47214 q^{25} +(2.38197 + 4.61803i) q^{27} +4.76393 q^{29} +5.23607i q^{31} +(3.23607 - 8.47214i) q^{33} +4.00000i q^{35} -8.47214i q^{37} +(2.76393 - 7.23607i) q^{39} -6.47214i q^{41} -7.23607 q^{43} +(-7.23607 - 6.47214i) q^{45} +8.00000 q^{47} +5.47214 q^{49} +(1.52786 - 4.00000i) q^{51} -3.23607 q^{53} +16.9443i q^{55} +(-1.23607 - 0.472136i) q^{57} +1.23607i q^{59} +0.472136i q^{61} +(2.47214 - 2.76393i) q^{63} +14.4721i q^{65} -9.70820 q^{67} +(4.00000 + 1.52786i) q^{69} -15.4164 q^{71} +2.00000 q^{73} +(8.85410 + 3.38197i) q^{75} -6.47214 q^{77} +0.291796i q^{79} +(1.00000 + 8.94427i) q^{81} +2.76393i q^{83} +8.00000i q^{85} +(7.70820 + 2.94427i) q^{87} -4.00000i q^{89} -5.52786 q^{91} +(-3.23607 + 8.47214i) q^{93} +2.47214 q^{95} +0.472136 q^{97} +(10.4721 - 11.7082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 12 q^{15} - 12 q^{19} + 12 q^{21} - 8 q^{23} + 4 q^{25} + 14 q^{27} + 28 q^{29} + 4 q^{33} + 20 q^{39} - 20 q^{43} - 20 q^{45} + 32 q^{47} + 4 q^{49} + 24 q^{51} - 4 q^{53} + 4 q^{57} - 8 q^{63} - 12 q^{67} + 16 q^{69} - 8 q^{71} + 8 q^{73} + 22 q^{75} - 8 q^{77} + 4 q^{81} + 4 q^{87} - 40 q^{91} - 4 q^{93} - 8 q^{95} - 16 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 1.23607i 0.467190i −0.972334 0.233595i \(-0.924951\pi\)
0.972334 0.233595i \(-0.0750489\pi\)
\(8\) 0 0
\(9\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(10\) 0 0
\(11\) 5.23607i 1.57873i −0.613922 0.789367i \(-0.710409\pi\)
0.613922 0.789367i \(-0.289591\pi\)
\(12\) 0 0
\(13\) 4.47214i 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) 0 0
\(15\) −5.23607 2.00000i −1.35195 0.516398i
\(16\) 0 0
\(17\) 2.47214i 0.599581i −0.954005 0.299791i \(-0.903083\pi\)
0.954005 0.299791i \(-0.0969168\pi\)
\(18\) 0 0
\(19\) −0.763932 −0.175258 −0.0876290 0.996153i \(-0.527929\pi\)
−0.0876290 + 0.996153i \(0.527929\pi\)
\(20\) 0 0
\(21\) 0.763932 2.00000i 0.166704 0.436436i
\(22\) 0 0
\(23\) 2.47214 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 2.38197 + 4.61803i 0.458410 + 0.888741i
\(28\) 0 0
\(29\) 4.76393 0.884640 0.442320 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) 5.23607i 0.940426i 0.882553 + 0.470213i \(0.155823\pi\)
−0.882553 + 0.470213i \(0.844177\pi\)
\(32\) 0 0
\(33\) 3.23607 8.47214i 0.563327 1.47481i
\(34\) 0 0
\(35\) 4.00000i 0.676123i
\(36\) 0 0
\(37\) 8.47214i 1.39281i −0.717649 0.696405i \(-0.754782\pi\)
0.717649 0.696405i \(-0.245218\pi\)
\(38\) 0 0
\(39\) 2.76393 7.23607i 0.442583 1.15870i
\(40\) 0 0
\(41\) 6.47214i 1.01078i −0.862892 0.505389i \(-0.831349\pi\)
0.862892 0.505389i \(-0.168651\pi\)
\(42\) 0 0
\(43\) −7.23607 −1.10349 −0.551745 0.834013i \(-0.686038\pi\)
−0.551745 + 0.834013i \(0.686038\pi\)
\(44\) 0 0
\(45\) −7.23607 6.47214i −1.07869 0.964809i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 5.47214 0.781734
\(50\) 0 0
\(51\) 1.52786 4.00000i 0.213944 0.560112i
\(52\) 0 0
\(53\) −3.23607 −0.444508 −0.222254 0.974989i \(-0.571341\pi\)
−0.222254 + 0.974989i \(0.571341\pi\)
\(54\) 0 0
\(55\) 16.9443i 2.28477i
\(56\) 0 0
\(57\) −1.23607 0.472136i −0.163721 0.0625359i
\(58\) 0 0
\(59\) 1.23607i 0.160922i 0.996758 + 0.0804612i \(0.0256393\pi\)
−0.996758 + 0.0804612i \(0.974361\pi\)
\(60\) 0 0
\(61\) 0.472136i 0.0604508i 0.999543 + 0.0302254i \(0.00962251\pi\)
−0.999543 + 0.0302254i \(0.990377\pi\)
\(62\) 0 0
\(63\) 2.47214 2.76393i 0.311460 0.348223i
\(64\) 0 0
\(65\) 14.4721i 1.79505i
\(66\) 0 0
\(67\) −9.70820 −1.18605 −0.593023 0.805186i \(-0.702066\pi\)
−0.593023 + 0.805186i \(0.702066\pi\)
\(68\) 0 0
\(69\) 4.00000 + 1.52786i 0.481543 + 0.183933i
\(70\) 0 0
\(71\) −15.4164 −1.82959 −0.914796 0.403917i \(-0.867648\pi\)
−0.914796 + 0.403917i \(0.867648\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 8.85410 + 3.38197i 1.02238 + 0.390516i
\(76\) 0 0
\(77\) −6.47214 −0.737568
\(78\) 0 0
\(79\) 0.291796i 0.0328296i 0.999865 + 0.0164148i \(0.00522523\pi\)
−0.999865 + 0.0164148i \(0.994775\pi\)
\(80\) 0 0
\(81\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(82\) 0 0
\(83\) 2.76393i 0.303381i 0.988428 + 0.151690i \(0.0484717\pi\)
−0.988428 + 0.151690i \(0.951528\pi\)
\(84\) 0 0
\(85\) 8.00000i 0.867722i
\(86\) 0 0
\(87\) 7.70820 + 2.94427i 0.826406 + 0.315659i
\(88\) 0 0
\(89\) 4.00000i 0.423999i −0.977270 0.212000i \(-0.932002\pi\)
0.977270 0.212000i \(-0.0679975\pi\)
\(90\) 0 0
\(91\) −5.52786 −0.579478
\(92\) 0 0
\(93\) −3.23607 + 8.47214i −0.335565 + 0.878520i
\(94\) 0 0
\(95\) 2.47214 0.253636
\(96\) 0 0
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) 0 0
\(99\) 10.4721 11.7082i 1.05249 1.17672i
\(100\) 0 0
\(101\) 1.70820 0.169973 0.0849863 0.996382i \(-0.472915\pi\)
0.0849863 + 0.996382i \(0.472915\pi\)
\(102\) 0 0
\(103\) 14.1803i 1.39723i −0.715498 0.698615i \(-0.753800\pi\)
0.715498 0.698615i \(-0.246200\pi\)
\(104\) 0 0
\(105\) −2.47214 + 6.47214i −0.241256 + 0.631616i
\(106\) 0 0
\(107\) 14.1803i 1.37087i 0.728136 + 0.685433i \(0.240387\pi\)
−0.728136 + 0.685433i \(0.759613\pi\)
\(108\) 0 0
\(109\) 8.47214i 0.811483i 0.913988 + 0.405742i \(0.132987\pi\)
−0.913988 + 0.405742i \(0.867013\pi\)
\(110\) 0 0
\(111\) 5.23607 13.7082i 0.496986 1.30113i
\(112\) 0 0
\(113\) 8.00000i 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 8.94427 10.0000i 0.826898 0.924500i
\(118\) 0 0
\(119\) −3.05573 −0.280118
\(120\) 0 0
\(121\) −16.4164 −1.49240
\(122\) 0 0
\(123\) 4.00000 10.4721i 0.360668 0.944241i
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 13.2361i 1.17451i 0.809402 + 0.587256i \(0.199792\pi\)
−0.809402 + 0.587256i \(0.800208\pi\)
\(128\) 0 0
\(129\) −11.7082 4.47214i −1.03085 0.393750i
\(130\) 0 0
\(131\) 6.76393i 0.590967i −0.955348 0.295484i \(-0.904519\pi\)
0.955348 0.295484i \(-0.0954808\pi\)
\(132\) 0 0
\(133\) 0.944272i 0.0818788i
\(134\) 0 0
\(135\) −7.70820 14.9443i −0.663417 1.28620i
\(136\) 0 0
\(137\) 1.52786i 0.130534i −0.997868 0.0652671i \(-0.979210\pi\)
0.997868 0.0652671i \(-0.0207899\pi\)
\(138\) 0 0
\(139\) 9.70820 0.823439 0.411720 0.911311i \(-0.364928\pi\)
0.411720 + 0.911311i \(0.364928\pi\)
\(140\) 0 0
\(141\) 12.9443 + 4.94427i 1.09010 + 0.416383i
\(142\) 0 0
\(143\) −23.4164 −1.95818
\(144\) 0 0
\(145\) −15.4164 −1.28026
\(146\) 0 0
\(147\) 8.85410 + 3.38197i 0.730274 + 0.278940i
\(148\) 0 0
\(149\) 9.70820 0.795327 0.397664 0.917531i \(-0.369821\pi\)
0.397664 + 0.917531i \(0.369821\pi\)
\(150\) 0 0
\(151\) 19.7082i 1.60383i 0.597438 + 0.801915i \(0.296186\pi\)
−0.597438 + 0.801915i \(0.703814\pi\)
\(152\) 0 0
\(153\) 4.94427 5.52786i 0.399721 0.446901i
\(154\) 0 0
\(155\) 16.9443i 1.36100i
\(156\) 0 0
\(157\) 4.47214i 0.356915i −0.983948 0.178458i \(-0.942889\pi\)
0.983948 0.178458i \(-0.0571108\pi\)
\(158\) 0 0
\(159\) −5.23607 2.00000i −0.415247 0.158610i
\(160\) 0 0
\(161\) 3.05573i 0.240825i
\(162\) 0 0
\(163\) 4.18034 0.327429 0.163715 0.986508i \(-0.447652\pi\)
0.163715 + 0.986508i \(0.447652\pi\)
\(164\) 0 0
\(165\) −10.4721 + 27.4164i −0.815255 + 2.13436i
\(166\) 0 0
\(167\) 13.5279 1.04682 0.523409 0.852082i \(-0.324660\pi\)
0.523409 + 0.852082i \(0.324660\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) −1.70820 1.52786i −0.130630 0.116839i
\(172\) 0 0
\(173\) 22.6525 1.72224 0.861118 0.508405i \(-0.169765\pi\)
0.861118 + 0.508405i \(0.169765\pi\)
\(174\) 0 0
\(175\) 6.76393i 0.511305i
\(176\) 0 0
\(177\) −0.763932 + 2.00000i −0.0574206 + 0.150329i
\(178\) 0 0
\(179\) 6.18034i 0.461940i 0.972961 + 0.230970i \(0.0741900\pi\)
−0.972961 + 0.230970i \(0.925810\pi\)
\(180\) 0 0
\(181\) 20.4721i 1.52168i 0.648938 + 0.760841i \(0.275213\pi\)
−0.648938 + 0.760841i \(0.724787\pi\)
\(182\) 0 0
\(183\) −0.291796 + 0.763932i −0.0215702 + 0.0564715i
\(184\) 0 0
\(185\) 27.4164i 2.01569i
\(186\) 0 0
\(187\) −12.9443 −0.946579
\(188\) 0 0
\(189\) 5.70820 2.94427i 0.415211 0.214164i
\(190\) 0 0
\(191\) 12.9443 0.936615 0.468307 0.883566i \(-0.344864\pi\)
0.468307 + 0.883566i \(0.344864\pi\)
\(192\) 0 0
\(193\) −11.8885 −0.855756 −0.427878 0.903836i \(-0.640739\pi\)
−0.427878 + 0.903836i \(0.640739\pi\)
\(194\) 0 0
\(195\) −8.94427 + 23.4164i −0.640513 + 1.67688i
\(196\) 0 0
\(197\) 4.76393 0.339416 0.169708 0.985494i \(-0.445718\pi\)
0.169708 + 0.985494i \(0.445718\pi\)
\(198\) 0 0
\(199\) 6.18034i 0.438113i −0.975712 0.219056i \(-0.929702\pi\)
0.975712 0.219056i \(-0.0702979\pi\)
\(200\) 0 0
\(201\) −15.7082 6.00000i −1.10797 0.423207i
\(202\) 0 0
\(203\) 5.88854i 0.413295i
\(204\) 0 0
\(205\) 20.9443i 1.46281i
\(206\) 0 0
\(207\) 5.52786 + 4.94427i 0.384213 + 0.343651i
\(208\) 0 0
\(209\) 4.00000i 0.276686i
\(210\) 0 0
\(211\) 11.2361 0.773523 0.386761 0.922180i \(-0.373594\pi\)
0.386761 + 0.922180i \(0.373594\pi\)
\(212\) 0 0
\(213\) −24.9443 9.52786i −1.70915 0.652838i
\(214\) 0 0
\(215\) 23.4164 1.59699
\(216\) 0 0
\(217\) 6.47214 0.439357
\(218\) 0 0
\(219\) 3.23607 + 1.23607i 0.218673 + 0.0835257i
\(220\) 0 0
\(221\) −11.0557 −0.743689
\(222\) 0 0
\(223\) 10.1803i 0.681726i 0.940113 + 0.340863i \(0.110719\pi\)
−0.940113 + 0.340863i \(0.889281\pi\)
\(224\) 0 0
\(225\) 12.2361 + 10.9443i 0.815738 + 0.729618i
\(226\) 0 0
\(227\) 2.18034i 0.144714i −0.997379 0.0723571i \(-0.976948\pi\)
0.997379 0.0723571i \(-0.0230521\pi\)
\(228\) 0 0
\(229\) 11.5279i 0.761783i −0.924620 0.380891i \(-0.875617\pi\)
0.924620 0.380891i \(-0.124383\pi\)
\(230\) 0 0
\(231\) −10.4721 4.00000i −0.689016 0.263181i
\(232\) 0 0
\(233\) 24.9443i 1.63415i 0.576529 + 0.817077i \(0.304407\pi\)
−0.576529 + 0.817077i \(0.695593\pi\)
\(234\) 0 0
\(235\) −25.8885 −1.68878
\(236\) 0 0
\(237\) −0.180340 + 0.472136i −0.0117143 + 0.0306685i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) −3.90983 + 15.0902i −0.250816 + 0.968035i
\(244\) 0 0
\(245\) −17.7082 −1.13134
\(246\) 0 0
\(247\) 3.41641i 0.217381i
\(248\) 0 0
\(249\) −1.70820 + 4.47214i −0.108253 + 0.283410i
\(250\) 0 0
\(251\) 23.7082i 1.49645i 0.663446 + 0.748224i \(0.269093\pi\)
−0.663446 + 0.748224i \(0.730907\pi\)
\(252\) 0 0
\(253\) 12.9443i 0.813799i
\(254\) 0 0
\(255\) −4.94427 + 12.9443i −0.309622 + 0.810602i
\(256\) 0 0
\(257\) 20.9443i 1.30647i −0.757156 0.653234i \(-0.773412\pi\)
0.757156 0.653234i \(-0.226588\pi\)
\(258\) 0 0
\(259\) −10.4721 −0.650707
\(260\) 0 0
\(261\) 10.6525 + 9.52786i 0.659372 + 0.589760i
\(262\) 0 0
\(263\) 10.4721 0.645740 0.322870 0.946443i \(-0.395352\pi\)
0.322870 + 0.946443i \(0.395352\pi\)
\(264\) 0 0
\(265\) 10.4721 0.643298
\(266\) 0 0
\(267\) 2.47214 6.47214i 0.151292 0.396088i
\(268\) 0 0
\(269\) −16.1803 −0.986533 −0.493266 0.869878i \(-0.664197\pi\)
−0.493266 + 0.869878i \(0.664197\pi\)
\(270\) 0 0
\(271\) 0.291796i 0.0177253i 0.999961 + 0.00886267i \(0.00282111\pi\)
−0.999961 + 0.00886267i \(0.997179\pi\)
\(272\) 0 0
\(273\) −8.94427 3.41641i −0.541332 0.206770i
\(274\) 0 0
\(275\) 28.6525i 1.72781i
\(276\) 0 0
\(277\) 7.52786i 0.452306i 0.974092 + 0.226153i \(0.0726149\pi\)
−0.974092 + 0.226153i \(0.927385\pi\)
\(278\) 0 0
\(279\) −10.4721 + 11.7082i −0.626950 + 0.700952i
\(280\) 0 0
\(281\) 24.9443i 1.48805i −0.668151 0.744025i \(-0.732914\pi\)
0.668151 0.744025i \(-0.267086\pi\)
\(282\) 0 0
\(283\) 20.7639 1.23429 0.617144 0.786850i \(-0.288290\pi\)
0.617144 + 0.786850i \(0.288290\pi\)
\(284\) 0 0
\(285\) 4.00000 + 1.52786i 0.236940 + 0.0905029i
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 0.763932 + 0.291796i 0.0447825 + 0.0171054i
\(292\) 0 0
\(293\) 20.7639 1.21304 0.606521 0.795068i \(-0.292565\pi\)
0.606521 + 0.795068i \(0.292565\pi\)
\(294\) 0 0
\(295\) 4.00000i 0.232889i
\(296\) 0 0
\(297\) 24.1803 12.4721i 1.40309 0.723707i
\(298\) 0 0
\(299\) 11.0557i 0.639369i
\(300\) 0 0
\(301\) 8.94427i 0.515539i
\(302\) 0 0
\(303\) 2.76393 + 1.05573i 0.158784 + 0.0606500i
\(304\) 0 0
\(305\) 1.52786i 0.0874852i
\(306\) 0 0
\(307\) −27.5967 −1.57503 −0.787515 0.616296i \(-0.788633\pi\)
−0.787515 + 0.616296i \(0.788633\pi\)
\(308\) 0 0
\(309\) 8.76393 22.9443i 0.498563 1.30525i
\(310\) 0 0
\(311\) 5.52786 0.313456 0.156728 0.987642i \(-0.449905\pi\)
0.156728 + 0.987642i \(0.449905\pi\)
\(312\) 0 0
\(313\) 0.472136 0.0266867 0.0133434 0.999911i \(-0.495753\pi\)
0.0133434 + 0.999911i \(0.495753\pi\)
\(314\) 0 0
\(315\) −8.00000 + 8.94427i −0.450749 + 0.503953i
\(316\) 0 0
\(317\) −13.1246 −0.737152 −0.368576 0.929598i \(-0.620155\pi\)
−0.368576 + 0.929598i \(0.620155\pi\)
\(318\) 0 0
\(319\) 24.9443i 1.39661i
\(320\) 0 0
\(321\) −8.76393 + 22.9443i −0.489155 + 1.28062i
\(322\) 0 0
\(323\) 1.88854i 0.105081i
\(324\) 0 0
\(325\) 24.4721i 1.35747i
\(326\) 0 0
\(327\) −5.23607 + 13.7082i −0.289555 + 0.758065i
\(328\) 0 0
\(329\) 9.88854i 0.545173i
\(330\) 0 0
\(331\) 32.5410 1.78862 0.894308 0.447452i \(-0.147668\pi\)
0.894308 + 0.447452i \(0.147668\pi\)
\(332\) 0 0
\(333\) 16.9443 18.9443i 0.928540 1.03814i
\(334\) 0 0
\(335\) 31.4164 1.71646
\(336\) 0 0
\(337\) 22.3607 1.21806 0.609032 0.793146i \(-0.291558\pi\)
0.609032 + 0.793146i \(0.291558\pi\)
\(338\) 0 0
\(339\) 4.94427 12.9443i 0.268536 0.703036i
\(340\) 0 0
\(341\) 27.4164 1.48468
\(342\) 0 0
\(343\) 15.4164i 0.832408i
\(344\) 0 0
\(345\) −12.9443 4.94427i −0.696896 0.266191i
\(346\) 0 0
\(347\) 23.7082i 1.27272i 0.771391 + 0.636362i \(0.219561\pi\)
−0.771391 + 0.636362i \(0.780439\pi\)
\(348\) 0 0
\(349\) 28.4721i 1.52408i −0.647531 0.762039i \(-0.724198\pi\)
0.647531 0.762039i \(-0.275802\pi\)
\(350\) 0 0
\(351\) 20.6525 10.6525i 1.10235 0.568587i
\(352\) 0 0
\(353\) 9.88854i 0.526314i −0.964753 0.263157i \(-0.915236\pi\)
0.964753 0.263157i \(-0.0847637\pi\)
\(354\) 0 0
\(355\) 49.8885 2.64781
\(356\) 0 0
\(357\) −4.94427 1.88854i −0.261679 0.0999523i
\(358\) 0 0
\(359\) −31.4164 −1.65809 −0.829047 0.559178i \(-0.811117\pi\)
−0.829047 + 0.559178i \(0.811117\pi\)
\(360\) 0 0
\(361\) −18.4164 −0.969285
\(362\) 0 0
\(363\) −26.5623 10.1459i −1.39416 0.532522i
\(364\) 0 0
\(365\) −6.47214 −0.338767
\(366\) 0 0
\(367\) 5.23607i 0.273321i 0.990618 + 0.136660i \(0.0436369\pi\)
−0.990618 + 0.136660i \(0.956363\pi\)
\(368\) 0 0
\(369\) 12.9443 14.4721i 0.673852 0.753389i
\(370\) 0 0
\(371\) 4.00000i 0.207670i
\(372\) 0 0
\(373\) 2.58359i 0.133773i 0.997761 + 0.0668867i \(0.0213066\pi\)
−0.997761 + 0.0668867i \(0.978693\pi\)
\(374\) 0 0
\(375\) −2.47214 0.944272i −0.127661 0.0487620i
\(376\) 0 0
\(377\) 21.3050i 1.09726i
\(378\) 0 0
\(379\) 10.6525 0.547181 0.273590 0.961846i \(-0.411789\pi\)
0.273590 + 0.961846i \(0.411789\pi\)
\(380\) 0 0
\(381\) −8.18034 + 21.4164i −0.419092 + 1.09720i
\(382\) 0 0
\(383\) −28.9443 −1.47898 −0.739492 0.673166i \(-0.764934\pi\)
−0.739492 + 0.673166i \(0.764934\pi\)
\(384\) 0 0
\(385\) 20.9443 1.06742
\(386\) 0 0
\(387\) −16.1803 14.4721i −0.822493 0.735660i
\(388\) 0 0
\(389\) −32.1803 −1.63161 −0.815804 0.578328i \(-0.803705\pi\)
−0.815804 + 0.578328i \(0.803705\pi\)
\(390\) 0 0
\(391\) 6.11146i 0.309070i
\(392\) 0 0
\(393\) 4.18034 10.9443i 0.210870 0.552065i
\(394\) 0 0
\(395\) 0.944272i 0.0475115i
\(396\) 0 0
\(397\) 17.4164i 0.874104i −0.899436 0.437052i \(-0.856022\pi\)
0.899436 0.437052i \(-0.143978\pi\)
\(398\) 0 0
\(399\) −0.583592 + 1.52786i −0.0292161 + 0.0764889i
\(400\) 0 0
\(401\) 28.3607i 1.41626i 0.706080 + 0.708132i \(0.250462\pi\)
−0.706080 + 0.708132i \(0.749538\pi\)
\(402\) 0 0
\(403\) 23.4164 1.16645
\(404\) 0 0
\(405\) −3.23607 28.9443i −0.160802 1.43825i
\(406\) 0 0
\(407\) −44.3607 −2.19888
\(408\) 0 0
\(409\) 31.8885 1.57679 0.788394 0.615171i \(-0.210913\pi\)
0.788394 + 0.615171i \(0.210913\pi\)
\(410\) 0 0
\(411\) 0.944272 2.47214i 0.0465775 0.121941i
\(412\) 0 0
\(413\) 1.52786 0.0751813
\(414\) 0 0
\(415\) 8.94427i 0.439057i
\(416\) 0 0
\(417\) 15.7082 + 6.00000i 0.769234 + 0.293821i
\(418\) 0 0
\(419\) 10.1803i 0.497342i −0.968588 0.248671i \(-0.920006\pi\)
0.968588 0.248671i \(-0.0799938\pi\)
\(420\) 0 0
\(421\) 21.4164i 1.04377i −0.853015 0.521886i \(-0.825229\pi\)
0.853015 0.521886i \(-0.174771\pi\)
\(422\) 0 0
\(423\) 17.8885 + 16.0000i 0.869771 + 0.777947i
\(424\) 0 0
\(425\) 13.5279i 0.656198i
\(426\) 0 0
\(427\) 0.583592 0.0282420
\(428\) 0 0
\(429\) −37.8885 14.4721i −1.82928 0.698721i
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 16.4721 0.791600 0.395800 0.918337i \(-0.370467\pi\)
0.395800 + 0.918337i \(0.370467\pi\)
\(434\) 0 0
\(435\) −24.9443 9.52786i −1.19599 0.456826i
\(436\) 0 0
\(437\) −1.88854 −0.0903413
\(438\) 0 0
\(439\) 22.1803i 1.05861i −0.848432 0.529305i \(-0.822453\pi\)
0.848432 0.529305i \(-0.177547\pi\)
\(440\) 0 0
\(441\) 12.2361 + 10.9443i 0.582670 + 0.521156i
\(442\) 0 0
\(443\) 16.2918i 0.774047i −0.922070 0.387023i \(-0.873503\pi\)
0.922070 0.387023i \(-0.126497\pi\)
\(444\) 0 0
\(445\) 12.9443i 0.613617i
\(446\) 0 0
\(447\) 15.7082 + 6.00000i 0.742973 + 0.283790i
\(448\) 0 0
\(449\) 0.583592i 0.0275414i 0.999905 + 0.0137707i \(0.00438349\pi\)
−0.999905 + 0.0137707i \(0.995617\pi\)
\(450\) 0 0
\(451\) −33.8885 −1.59575
\(452\) 0 0
\(453\) −12.1803 + 31.8885i −0.572282 + 1.49825i
\(454\) 0 0
\(455\) 17.8885 0.838628
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 11.4164 5.88854i 0.532872 0.274854i
\(460\) 0 0
\(461\) −8.18034 −0.380996 −0.190498 0.981688i \(-0.561010\pi\)
−0.190498 + 0.981688i \(0.561010\pi\)
\(462\) 0 0
\(463\) 34.7639i 1.61562i −0.589445 0.807808i \(-0.700654\pi\)
0.589445 0.807808i \(-0.299346\pi\)
\(464\) 0 0
\(465\) 10.4721 27.4164i 0.485634 1.27141i
\(466\) 0 0
\(467\) 10.7639i 0.498095i 0.968491 + 0.249048i \(0.0801176\pi\)
−0.968491 + 0.249048i \(0.919882\pi\)
\(468\) 0 0
\(469\) 12.0000i 0.554109i
\(470\) 0 0
\(471\) 2.76393 7.23607i 0.127355 0.333420i
\(472\) 0 0
\(473\) 37.8885i 1.74212i
\(474\) 0 0
\(475\) −4.18034 −0.191807
\(476\) 0 0
\(477\) −7.23607 6.47214i −0.331317 0.296339i
\(478\) 0 0
\(479\) 6.11146 0.279240 0.139620 0.990205i \(-0.455412\pi\)
0.139620 + 0.990205i \(0.455412\pi\)
\(480\) 0 0
\(481\) −37.8885 −1.72757
\(482\) 0 0
\(483\) 1.88854 4.94427i 0.0859317 0.224972i
\(484\) 0 0
\(485\) −1.52786 −0.0693767
\(486\) 0 0
\(487\) 32.6525i 1.47962i 0.672813 + 0.739812i \(0.265086\pi\)
−0.672813 + 0.739812i \(0.734914\pi\)
\(488\) 0 0
\(489\) 6.76393 + 2.58359i 0.305876 + 0.116834i
\(490\) 0 0
\(491\) 32.0689i 1.44725i 0.690194 + 0.723624i \(0.257525\pi\)
−0.690194 + 0.723624i \(0.742475\pi\)
\(492\) 0 0
\(493\) 11.7771i 0.530413i
\(494\) 0 0
\(495\) −33.8885 + 37.8885i −1.52318 + 1.70296i
\(496\) 0 0
\(497\) 19.0557i 0.854766i
\(498\) 0 0
\(499\) 27.2361 1.21925 0.609627 0.792688i \(-0.291319\pi\)
0.609627 + 0.792688i \(0.291319\pi\)
\(500\) 0 0
\(501\) 21.8885 + 8.36068i 0.977908 + 0.373528i
\(502\) 0 0
\(503\) 21.5279 0.959880 0.479940 0.877301i \(-0.340658\pi\)
0.479940 + 0.877301i \(0.340658\pi\)
\(504\) 0 0
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) −11.3262 4.32624i −0.503016 0.192135i
\(508\) 0 0
\(509\) 25.7082 1.13950 0.569748 0.821819i \(-0.307041\pi\)
0.569748 + 0.821819i \(0.307041\pi\)
\(510\) 0 0
\(511\) 2.47214i 0.109361i
\(512\) 0 0
\(513\) −1.81966 3.52786i −0.0803400 0.155759i
\(514\) 0 0
\(515\) 45.8885i 2.02209i
\(516\) 0 0
\(517\) 41.8885i 1.84226i
\(518\) 0 0
\(519\) 36.6525 + 14.0000i 1.60887 + 0.614532i
\(520\) 0 0
\(521\) 24.3607i 1.06726i 0.845718 + 0.533630i \(0.179173\pi\)
−0.845718 + 0.533630i \(0.820827\pi\)
\(522\) 0 0
\(523\) 11.8197 0.516838 0.258419 0.966033i \(-0.416799\pi\)
0.258419 + 0.966033i \(0.416799\pi\)
\(524\) 0 0
\(525\) 4.18034 10.9443i 0.182445 0.477647i
\(526\) 0 0
\(527\) 12.9443 0.563861
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) −2.47214 + 2.76393i −0.107282 + 0.119944i
\(532\) 0 0
\(533\) −28.9443 −1.25372
\(534\) 0 0
\(535\) 45.8885i 1.98393i
\(536\) 0 0
\(537\) −3.81966 + 10.0000i −0.164831 + 0.431532i
\(538\) 0 0
\(539\) 28.6525i 1.23415i
\(540\) 0 0
\(541\) 26.3607i 1.13333i 0.823947 + 0.566667i \(0.191767\pi\)
−0.823947 + 0.566667i \(0.808233\pi\)
\(542\) 0 0
\(543\) −12.6525 + 33.1246i −0.542970 + 1.42151i
\(544\) 0 0
\(545\) 27.4164i 1.17439i
\(546\) 0 0
\(547\) 7.23607 0.309392 0.154696 0.987962i \(-0.450560\pi\)
0.154696 + 0.987962i \(0.450560\pi\)
\(548\) 0 0
\(549\) −0.944272 + 1.05573i −0.0403005 + 0.0450574i
\(550\) 0 0
\(551\) −3.63932 −0.155040
\(552\) 0 0
\(553\) 0.360680 0.0153377
\(554\) 0 0
\(555\) −16.9443 + 44.3607i −0.719244 + 1.88301i
\(556\) 0 0
\(557\) 14.6525 0.620845 0.310423 0.950599i \(-0.399529\pi\)
0.310423 + 0.950599i \(0.399529\pi\)
\(558\) 0 0
\(559\) 32.3607i 1.36871i
\(560\) 0 0
\(561\) −20.9443 8.00000i −0.884268 0.337760i
\(562\) 0 0
\(563\) 30.5410i 1.28715i 0.765383 + 0.643575i \(0.222550\pi\)
−0.765383 + 0.643575i \(0.777450\pi\)
\(564\) 0 0
\(565\) 25.8885i 1.08914i
\(566\) 0 0
\(567\) 11.0557 1.23607i 0.464297 0.0519100i
\(568\) 0 0
\(569\) 46.4721i 1.94821i −0.226089 0.974107i \(-0.572594\pi\)
0.226089 0.974107i \(-0.427406\pi\)
\(570\) 0 0
\(571\) −5.12461 −0.214458 −0.107229 0.994234i \(-0.534198\pi\)
−0.107229 + 0.994234i \(0.534198\pi\)
\(572\) 0 0
\(573\) 20.9443 + 8.00000i 0.874960 + 0.334205i
\(574\) 0 0
\(575\) 13.5279 0.564151
\(576\) 0 0
\(577\) 28.4721 1.18531 0.592655 0.805456i \(-0.298080\pi\)
0.592655 + 0.805456i \(0.298080\pi\)
\(578\) 0 0
\(579\) −19.2361 7.34752i −0.799424 0.305353i
\(580\) 0 0
\(581\) 3.41641 0.141736
\(582\) 0 0
\(583\) 16.9443i 0.701760i
\(584\) 0 0
\(585\) −28.9443 + 32.3607i −1.19670 + 1.33795i
\(586\) 0 0
\(587\) 21.5967i 0.891393i −0.895184 0.445697i \(-0.852956\pi\)
0.895184 0.445697i \(-0.147044\pi\)
\(588\) 0 0
\(589\) 4.00000i 0.164817i
\(590\) 0 0
\(591\) 7.70820 + 2.94427i 0.317073 + 0.121111i
\(592\) 0 0
\(593\) 28.9443i 1.18860i −0.804244 0.594299i \(-0.797429\pi\)
0.804244 0.594299i \(-0.202571\pi\)
\(594\) 0 0
\(595\) 9.88854 0.405391
\(596\) 0 0
\(597\) 3.81966 10.0000i 0.156328 0.409273i
\(598\) 0 0
\(599\) −7.41641 −0.303026 −0.151513 0.988455i \(-0.548415\pi\)
−0.151513 + 0.988455i \(0.548415\pi\)
\(600\) 0 0
\(601\) −33.7771 −1.37780 −0.688898 0.724858i \(-0.741905\pi\)
−0.688898 + 0.724858i \(0.741905\pi\)
\(602\) 0 0
\(603\) −21.7082 19.4164i −0.884026 0.790697i
\(604\) 0 0
\(605\) 53.1246 2.15982
\(606\) 0 0
\(607\) 34.7639i 1.41102i −0.708698 0.705512i \(-0.750717\pi\)
0.708698 0.705512i \(-0.249283\pi\)
\(608\) 0 0
\(609\) 3.63932 9.52786i 0.147473 0.386089i
\(610\) 0 0
\(611\) 35.7771i 1.44739i
\(612\) 0 0
\(613\) 19.3050i 0.779720i 0.920874 + 0.389860i \(0.127477\pi\)
−0.920874 + 0.389860i \(0.872523\pi\)
\(614\) 0 0
\(615\) −12.9443 + 33.8885i −0.521963 + 1.36652i
\(616\) 0 0
\(617\) 21.8885i 0.881200i −0.897704 0.440600i \(-0.854766\pi\)
0.897704 0.440600i \(-0.145234\pi\)
\(618\) 0 0
\(619\) −29.1246 −1.17062 −0.585308 0.810811i \(-0.699027\pi\)
−0.585308 + 0.810811i \(0.699027\pi\)
\(620\) 0 0
\(621\) 5.88854 + 11.4164i 0.236299 + 0.458125i
\(622\) 0 0
\(623\) −4.94427 −0.198088
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) −2.47214 + 6.47214i −0.0987276 + 0.258472i
\(628\) 0 0
\(629\) −20.9443 −0.835103
\(630\) 0 0
\(631\) 19.1246i 0.761339i −0.924711 0.380669i \(-0.875694\pi\)
0.924711 0.380669i \(-0.124306\pi\)
\(632\) 0 0
\(633\) 18.1803 + 6.94427i 0.722604 + 0.276010i
\(634\) 0 0
\(635\) 42.8328i 1.69977i
\(636\) 0 0
\(637\) 24.4721i 0.969621i
\(638\) 0 0
\(639\) −34.4721 30.8328i −1.36370 1.21973i
\(640\) 0 0
\(641\) 25.3050i 0.999485i 0.866174 + 0.499743i \(0.166572\pi\)
−0.866174 + 0.499743i \(0.833428\pi\)
\(642\) 0 0
\(643\) 13.3475 0.526375 0.263187 0.964745i \(-0.415226\pi\)
0.263187 + 0.964745i \(0.415226\pi\)
\(644\) 0 0
\(645\) 37.8885 + 14.4721i 1.49186 + 0.569840i
\(646\) 0 0
\(647\) 33.3050 1.30935 0.654676 0.755909i \(-0.272805\pi\)
0.654676 + 0.755909i \(0.272805\pi\)
\(648\) 0 0
\(649\) 6.47214 0.254054
\(650\) 0 0
\(651\) 10.4721 + 4.00000i 0.410435 + 0.156772i
\(652\) 0 0
\(653\) −14.2918 −0.559281 −0.279641 0.960105i \(-0.590215\pi\)
−0.279641 + 0.960105i \(0.590215\pi\)
\(654\) 0 0
\(655\) 21.8885i 0.855256i
\(656\) 0 0
\(657\) 4.47214 + 4.00000i 0.174475 + 0.156055i
\(658\) 0 0
\(659\) 28.2918i 1.10209i 0.834475 + 0.551046i \(0.185771\pi\)
−0.834475 + 0.551046i \(0.814229\pi\)
\(660\) 0 0
\(661\) 51.3050i 1.99553i 0.0668107 + 0.997766i \(0.478718\pi\)
−0.0668107 + 0.997766i \(0.521282\pi\)
\(662\) 0 0
\(663\) −17.8885 6.83282i −0.694733 0.265365i
\(664\) 0 0
\(665\) 3.05573i 0.118496i
\(666\) 0 0
\(667\) 11.7771 0.456011
\(668\) 0 0
\(669\) −6.29180 + 16.4721i −0.243255 + 0.636850i
\(670\) 0 0
\(671\) 2.47214 0.0954358
\(672\) 0 0
\(673\) −25.4164 −0.979731 −0.489865 0.871798i \(-0.662954\pi\)
−0.489865 + 0.871798i \(0.662954\pi\)
\(674\) 0 0
\(675\) 13.0344 + 25.2705i 0.501696 + 0.972662i
\(676\) 0 0
\(677\) −14.2918 −0.549278 −0.274639 0.961547i \(-0.588558\pi\)
−0.274639 + 0.961547i \(0.588558\pi\)
\(678\) 0 0
\(679\) 0.583592i 0.0223962i
\(680\) 0 0
\(681\) 1.34752 3.52786i 0.0516372 0.135188i
\(682\) 0 0
\(683\) 13.2361i 0.506464i −0.967406 0.253232i \(-0.918506\pi\)
0.967406 0.253232i \(-0.0814936\pi\)
\(684\) 0 0
\(685\) 4.94427i 0.188911i
\(686\) 0 0
\(687\) 7.12461 18.6525i 0.271821 0.711636i
\(688\) 0 0
\(689\) 14.4721i 0.551344i
\(690\) 0 0
\(691\) −29.7082 −1.13015 −0.565077 0.825038i \(-0.691153\pi\)
−0.565077 + 0.825038i \(0.691153\pi\)
\(692\) 0 0
\(693\) −14.4721 12.9443i −0.549751 0.491712i
\(694\) 0 0
\(695\) −31.4164 −1.19169
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) −15.4164 + 40.3607i −0.583102 + 1.52658i
\(700\) 0 0
\(701\) −14.2918 −0.539794 −0.269897 0.962889i \(-0.586990\pi\)
−0.269897 + 0.962889i \(0.586990\pi\)
\(702\) 0 0
\(703\) 6.47214i 0.244101i
\(704\) 0 0
\(705\) −41.8885 16.0000i −1.57761 0.602595i
\(706\) 0 0
\(707\) 2.11146i 0.0794095i
\(708\) 0 0
\(709\) 7.52786i 0.282715i 0.989959 + 0.141357i \(0.0451467\pi\)
−0.989959 + 0.141357i \(0.954853\pi\)
\(710\) 0 0
\(711\) −0.583592 + 0.652476i −0.0218864 + 0.0244698i
\(712\) 0 0
\(713\) 12.9443i 0.484767i
\(714\) 0 0
\(715\) 75.7771 2.83390
\(716\) 0 0
\(717\) −12.9443 4.94427i −0.483413 0.184647i
\(718\) 0 0
\(719\) 30.8328 1.14987 0.574935 0.818199i \(-0.305027\pi\)
0.574935 + 0.818199i \(0.305027\pi\)
\(720\) 0 0
\(721\) −17.5279 −0.652772
\(722\) 0 0
\(723\) −3.23607 1.23607i −0.120351 0.0459699i
\(724\) 0 0
\(725\) 26.0689 0.968174
\(726\) 0 0
\(727\) 14.1803i 0.525920i −0.964807 0.262960i \(-0.915301\pi\)
0.964807 0.262960i \(-0.0846987\pi\)
\(728\) 0 0
\(729\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(730\) 0 0
\(731\) 17.8885i 0.661632i
\(732\) 0 0
\(733\) 29.4164i 1.08652i 0.839565 + 0.543260i \(0.182810\pi\)
−0.839565 + 0.543260i \(0.817190\pi\)
\(734\) 0 0
\(735\) −28.6525 10.9443i −1.05686 0.403686i
\(736\) 0 0
\(737\) 50.8328i 1.87245i
\(738\) 0 0
\(739\) 13.1246 0.482797 0.241398 0.970426i \(-0.422394\pi\)
0.241398 + 0.970426i \(0.422394\pi\)
\(740\) 0 0
\(741\) −2.11146 + 5.52786i −0.0775663 + 0.203071i
\(742\) 0 0
\(743\) 7.41641 0.272082 0.136041 0.990703i \(-0.456562\pi\)
0.136041 + 0.990703i \(0.456562\pi\)
\(744\) 0 0
\(745\) −31.4164 −1.15101
\(746\) 0 0
\(747\) −5.52786 + 6.18034i −0.202254 + 0.226127i
\(748\) 0 0
\(749\) 17.5279 0.640454
\(750\) 0 0
\(751\) 17.5967i 0.642114i −0.947060 0.321057i \(-0.895962\pi\)
0.947060 0.321057i \(-0.104038\pi\)
\(752\) 0 0
\(753\) −14.6525 + 38.3607i −0.533966 + 1.39794i
\(754\) 0 0
\(755\) 63.7771i 2.32109i
\(756\) 0 0
\(757\) 5.41641i 0.196863i −0.995144 0.0984313i \(-0.968618\pi\)
0.995144 0.0984313i \(-0.0313825\pi\)
\(758\) 0 0
\(759\) 8.00000 20.9443i 0.290382 0.760229i
\(760\) 0 0
\(761\) 42.2492i 1.53153i 0.643119 + 0.765767i \(0.277640\pi\)
−0.643119 + 0.765767i \(0.722360\pi\)
\(762\) 0 0
\(763\) 10.4721 0.379117
\(764\) 0 0
\(765\) −16.0000 + 17.8885i −0.578481 + 0.646762i
\(766\) 0 0
\(767\) 5.52786 0.199600
\(768\) 0 0
\(769\) 49.7771 1.79501 0.897504 0.441007i \(-0.145378\pi\)
0.897504 + 0.441007i \(0.145378\pi\)
\(770\) 0 0
\(771\) 12.9443 33.8885i 0.466177 1.22047i
\(772\) 0 0
\(773\) 7.81966 0.281254 0.140627 0.990063i \(-0.455088\pi\)
0.140627 + 0.990063i \(0.455088\pi\)
\(774\) 0 0
\(775\) 28.6525i 1.02923i
\(776\) 0 0
\(777\) −16.9443 6.47214i −0.607872 0.232187i
\(778\) 0 0
\(779\) 4.94427i 0.177147i
\(780\) 0 0
\(781\) 80.7214i 2.88844i
\(782\) 0 0
\(783\) 11.3475 + 22.0000i 0.405527 + 0.786216i
\(784\) 0 0
\(785\) 14.4721i 0.516533i
\(786\) 0 0
\(787\) 28.1803 1.00452 0.502260 0.864716i \(-0.332502\pi\)
0.502260 + 0.864716i \(0.332502\pi\)
\(788\) 0 0
\(789\) 16.9443 + 6.47214i 0.603232 + 0.230414i
\(790\) 0 0
\(791\) −9.88854 −0.351596
\(792\) 0 0
\(793\) 2.11146 0.0749800
\(794\) 0 0
\(795\) 16.9443 + 6.47214i 0.600951 + 0.229543i
\(796\) 0 0
\(797\) −50.0689 −1.77353 −0.886765 0.462220i \(-0.847053\pi\)
−0.886765 + 0.462220i \(0.847053\pi\)
\(798\) 0 0
\(799\) 19.7771i 0.699663i
\(800\) 0 0
\(801\) 8.00000 8.94427i 0.282666 0.316030i
\(802\) 0 0
\(803\) 10.4721i 0.369554i
\(804\) 0 0
\(805\) 9.88854i 0.348525i
\(806\) 0 0
\(807\) −26.1803 10.0000i −0.921592 0.352017i
\(808\) 0 0
\(809\) 3.41641i 0.120115i 0.998195 + 0.0600573i \(0.0191283\pi\)
−0.998195 + 0.0600573i \(0.980872\pi\)
\(810\) 0 0
\(811\) −52.1803 −1.83230 −0.916150 0.400836i \(-0.868720\pi\)
−0.916150 + 0.400836i \(0.868720\pi\)
\(812\) 0 0
\(813\) −0.180340 + 0.472136i −0.00632480 + 0.0165585i
\(814\) 0 0
\(815\) −13.5279 −0.473860
\(816\) 0 0
\(817\) 5.52786 0.193395
\(818\) 0 0
\(819\) −12.3607 11.0557i −0.431917 0.386318i
\(820\) 0 0
\(821\) 50.4296 1.76000 0.880002 0.474970i \(-0.157541\pi\)
0.880002 + 0.474970i \(0.157541\pi\)
\(822\) 0 0
\(823\) 51.1246i 1.78209i −0.453913 0.891046i \(-0.649972\pi\)
0.453913 0.891046i \(-0.350028\pi\)
\(824\) 0 0
\(825\) 17.7082 46.3607i 0.616521 1.61407i
\(826\) 0 0
\(827\) 19.1246i 0.665028i 0.943098 + 0.332514i \(0.107897\pi\)
−0.943098 + 0.332514i \(0.892103\pi\)
\(828\) 0 0
\(829\) 4.47214i 0.155324i −0.996980 0.0776619i \(-0.975255\pi\)
0.996980 0.0776619i \(-0.0247455\pi\)
\(830\) 0 0
\(831\) −4.65248 + 12.1803i −0.161393 + 0.422531i
\(832\) 0 0
\(833\) 13.5279i 0.468713i
\(834\) 0 0
\(835\) −43.7771 −1.51497
\(836\) 0 0
\(837\) −24.1803 + 12.4721i −0.835795 + 0.431100i
\(838\) 0 0
\(839\) −31.4164 −1.08461 −0.542307 0.840180i \(-0.682449\pi\)
−0.542307 + 0.840180i \(0.682449\pi\)
\(840\) 0 0
\(841\) −6.30495 −0.217412
\(842\) 0 0
\(843\) 15.4164 40.3607i 0.530969 1.39010i
\(844\) 0 0
\(845\) 22.6525 0.779269
\(846\) 0 0
\(847\) 20.2918i 0.697234i
\(848\) 0 0
\(849\) 33.5967 + 12.8328i 1.15304 + 0.440421i
\(850\) 0 0
\(851\) 20.9443i 0.717960i
\(852\) 0 0
\(853\) 39.3050i 1.34578i −0.739744 0.672888i \(-0.765054\pi\)
0.739744 0.672888i \(-0.234946\pi\)
\(854\) 0 0
\(855\) 5.52786 + 4.94427i 0.189049 + 0.169091i
\(856\) 0 0
\(857\) 7.63932i 0.260954i 0.991451 + 0.130477i \(0.0416509\pi\)
−0.991451 + 0.130477i \(0.958349\pi\)
\(858\) 0 0
\(859\) 16.7639 0.571978 0.285989 0.958233i \(-0.407678\pi\)
0.285989 + 0.958233i \(0.407678\pi\)
\(860\) 0 0
\(861\) −12.9443 4.94427i −0.441140 0.168500i
\(862\) 0 0
\(863\) 22.8328 0.777238 0.388619 0.921399i \(-0.372952\pi\)
0.388619 + 0.921399i \(0.372952\pi\)
\(864\) 0 0
\(865\) −73.3050 −2.49244
\(866\) 0 0
\(867\) 17.6180 + 6.72949i 0.598340 + 0.228545i
\(868\) 0 0
\(869\) 1.52786 0.0518292
\(870\) 0 0
\(871\) 43.4164i 1.47111i
\(872\) 0 0
\(873\) 1.05573 + 0.944272i 0.0357310 + 0.0319588i
\(874\) 0 0
\(875\) 1.88854i 0.0638444i
\(876\) 0 0
\(877\) 18.5836i 0.627523i −0.949502 0.313762i \(-0.898411\pi\)
0.949502 0.313762i \(-0.101589\pi\)
\(878\) 0 0
\(879\) 33.5967 + 12.8328i 1.13319 + 0.432840i
\(880\) 0 0
\(881\) 24.0000i 0.808581i −0.914631 0.404290i \(-0.867519\pi\)
0.914631 0.404290i \(-0.132481\pi\)
\(882\) 0 0
\(883\) −0.763932 −0.0257084 −0.0128542 0.999917i \(-0.504092\pi\)
−0.0128542 + 0.999917i \(0.504092\pi\)
\(884\) 0 0
\(885\) 2.47214 6.47214i 0.0830999 0.217558i
\(886\) 0 0
\(887\) −20.3607 −0.683645 −0.341822 0.939765i \(-0.611044\pi\)
−0.341822 + 0.939765i \(0.611044\pi\)
\(888\) 0 0
\(889\) 16.3607 0.548720
\(890\) 0 0
\(891\) 46.8328 5.23607i 1.56896 0.175415i
\(892\) 0 0
\(893\) −6.11146 −0.204512
\(894\) 0 0
\(895\) 20.0000i 0.668526i
\(896\) 0 0
\(897\) 6.83282 17.8885i 0.228141 0.597281i
\(898\) 0 0
\(899\) 24.9443i 0.831938i
\(900\) 0 0
\(901\) 8.00000i 0.266519i
\(902\) 0 0
\(903\) −5.52786 + 14.4721i −0.183956 + 0.481603i
\(904\) 0 0
\(905\) 66.2492i 2.20220i
\(906\) 0 0
\(907\) −39.2361 −1.30281 −0.651406 0.758729i \(-0.725821\pi\)
−0.651406 + 0.758729i \(0.725821\pi\)
\(908\) 0 0
\(909\) 3.81966 + 3.41641i 0.126690 + 0.113315i
\(910\) 0 0
\(911\) −33.8885 −1.12278 −0.561389 0.827552i \(-0.689733\pi\)
−0.561389 + 0.827552i \(0.689733\pi\)
\(912\) 0 0
\(913\) 14.4721 0.478958
\(914\) 0 0
\(915\) 0.944272 2.47214i 0.0312167 0.0817263i
\(916\) 0 0
\(917\) −8.36068 −0.276094
\(918\) 0 0
\(919\) 58.5410i 1.93109i 0.260236 + 0.965545i \(0.416200\pi\)
−0.260236 + 0.965545i \(0.583800\pi\)
\(920\) 0 0
\(921\) −44.6525 17.0557i −1.47135 0.562005i
\(922\) 0 0
\(923\) 68.9443i 2.26933i
\(924\) 0 0
\(925\) 46.3607i 1.52433i
\(926\) 0 0
\(927\) 28.3607 31.7082i 0.931487 1.04143i
\(928\) 0 0
\(929\) 47.4164i 1.55568i 0.628461 + 0.777841i \(0.283685\pi\)
−0.628461 + 0.777841i \(0.716315\pi\)
\(930\) 0 0
\(931\) −4.18034 −0.137005
\(932\) 0 0
\(933\) 8.94427 + 3.41641i 0.292822 + 0.111848i
\(934\) 0 0
\(935\) 41.8885 1.36990
\(936\) 0 0
\(937\) 14.3607 0.469143 0.234571 0.972099i \(-0.424631\pi\)
0.234571 + 0.972099i \(0.424631\pi\)
\(938\) 0 0
\(939\) 0.763932 + 0.291796i 0.0249300 + 0.00952240i
\(940\) 0 0
\(941\) 1.70820 0.0556859 0.0278429 0.999612i \(-0.491136\pi\)
0.0278429 + 0.999612i \(0.491136\pi\)
\(942\) 0 0
\(943\) 16.0000i 0.521032i
\(944\) 0 0
\(945\) −18.4721 + 9.52786i −0.600899 + 0.309941i
\(946\) 0 0
\(947\) 24.0689i 0.782134i 0.920362 + 0.391067i \(0.127894\pi\)
−0.920362 + 0.391067i \(0.872106\pi\)
\(948\) 0 0
\(949\) 8.94427i 0.290343i
\(950\) 0 0
\(951\) −21.2361 8.11146i −0.688627 0.263032i
\(952\) 0 0
\(953\) 24.3607i 0.789120i −0.918870 0.394560i \(-0.870897\pi\)
0.918870 0.394560i \(-0.129103\pi\)
\(954\) 0 0
\(955\) −41.8885 −1.35548
\(956\) 0 0
\(957\) 15.4164 40.3607i 0.498342 1.30468i
\(958\) 0 0
\(959\) −1.88854 −0.0609843
\(960\) 0 0
\(961\) 3.58359 0.115600
\(962\) 0 0
\(963\) −28.3607 + 31.7082i −0.913910 + 1.02178i
\(964\) 0 0
\(965\) 38.4721 1.23846
\(966\) 0 0
\(967\) 9.23607i 0.297012i −0.988912 0.148506i \(-0.952554\pi\)
0.988912 0.148506i \(-0.0474464\pi\)
\(968\) 0 0
\(969\) −1.16718 + 3.05573i −0.0374954 + 0.0981641i
\(970\) 0 0
\(971\) 17.0132i 0.545978i −0.962017 0.272989i \(-0.911988\pi\)
0.962017 0.272989i \(-0.0880123\pi\)
\(972\) 0 0
\(973\) 12.0000i 0.384702i
\(974\) 0 0
\(975\) 15.1246 39.5967i 0.484375 1.26811i
\(976\) 0 0
\(977\) 60.3607i 1.93111i −0.260200 0.965555i \(-0.583789\pi\)
0.260200 0.965555i \(-0.416211\pi\)
\(978\) 0 0
\(979\) −20.9443 −0.669382
\(980\) 0 0
\(981\) −16.9443 + 18.9443i −0.540989 + 0.604844i
\(982\) 0 0
\(983\) 21.5279 0.686632 0.343316 0.939220i \(-0.388450\pi\)
0.343316 + 0.939220i \(0.388450\pi\)
\(984\) 0 0
\(985\) −15.4164 −0.491208
\(986\) 0 0
\(987\) 6.11146 16.0000i 0.194530 0.509286i
\(988\) 0 0
\(989\) −17.8885 −0.568823
\(990\) 0 0
\(991\) 50.1803i 1.59403i 0.603959 + 0.797016i \(0.293589\pi\)
−0.603959 + 0.797016i \(0.706411\pi\)
\(992\) 0 0
\(993\) 52.6525 + 20.1115i 1.67088 + 0.638218i
\(994\) 0 0
\(995\) 20.0000i 0.634043i
\(996\) 0 0
\(997\) 18.3607i 0.581489i −0.956801 0.290744i \(-0.906097\pi\)
0.956801 0.290744i \(-0.0939029\pi\)
\(998\) 0 0
\(999\) 39.1246 20.1803i 1.23785 0.638478i
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 768.2.f.e.383.4 4
3.2 odd 2 768.2.f.f.383.3 4
4.3 odd 2 768.2.f.b.383.1 4
8.3 odd 2 768.2.f.f.383.4 4
8.5 even 2 768.2.f.c.383.1 4
12.11 even 2 768.2.f.c.383.2 4
16.3 odd 4 384.2.c.b.383.3 yes 4
16.5 even 4 384.2.c.a.383.3 4
16.11 odd 4 384.2.c.d.383.2 yes 4
16.13 even 4 384.2.c.c.383.2 yes 4
24.5 odd 2 768.2.f.b.383.2 4
24.11 even 2 inner 768.2.f.e.383.3 4
48.5 odd 4 384.2.c.d.383.1 yes 4
48.11 even 4 384.2.c.a.383.4 yes 4
48.29 odd 4 384.2.c.b.383.4 yes 4
48.35 even 4 384.2.c.c.383.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.c.a.383.3 4 16.5 even 4
384.2.c.a.383.4 yes 4 48.11 even 4
384.2.c.b.383.3 yes 4 16.3 odd 4
384.2.c.b.383.4 yes 4 48.29 odd 4
384.2.c.c.383.1 yes 4 48.35 even 4
384.2.c.c.383.2 yes 4 16.13 even 4
384.2.c.d.383.1 yes 4 48.5 odd 4
384.2.c.d.383.2 yes 4 16.11 odd 4
768.2.f.b.383.1 4 4.3 odd 2
768.2.f.b.383.2 4 24.5 odd 2
768.2.f.c.383.1 4 8.5 even 2
768.2.f.c.383.2 4 12.11 even 2
768.2.f.e.383.3 4 24.11 even 2 inner
768.2.f.e.383.4 4 1.1 even 1 trivial
768.2.f.f.383.3 4 3.2 odd 2
768.2.f.f.383.4 4 8.3 odd 2