Properties

Label 8008.2.a.z.1.7
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.a (trivial)

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: yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + 11690 x^{7} - 7119 x^{6} - 22246 x^{5} + 11137 x^{4} + 20034 x^{3} - 8392 x^{2} + \cdots + 2560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.619952\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.619952 q^{3} -1.29231 q^{5} -1.00000 q^{7} -2.61566 q^{9} +O(q^{10})\) \(q-0.619952 q^{3} -1.29231 q^{5} -1.00000 q^{7} -2.61566 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.801172 q^{15} -6.20824 q^{17} -7.57687 q^{19} +0.619952 q^{21} -2.07849 q^{23} -3.32993 q^{25} +3.48144 q^{27} +4.36889 q^{29} -9.32165 q^{31} +0.619952 q^{33} +1.29231 q^{35} -6.40784 q^{37} -0.619952 q^{39} -12.4066 q^{41} +9.61081 q^{43} +3.38025 q^{45} -6.59201 q^{47} +1.00000 q^{49} +3.84881 q^{51} +8.49683 q^{53} +1.29231 q^{55} +4.69729 q^{57} +10.3403 q^{59} -1.17828 q^{61} +2.61566 q^{63} -1.29231 q^{65} -1.17214 q^{67} +1.28856 q^{69} -13.9076 q^{71} +12.0315 q^{73} +2.06439 q^{75} +1.00000 q^{77} +11.4620 q^{79} +5.68866 q^{81} -3.04207 q^{83} +8.02299 q^{85} -2.70850 q^{87} -1.94288 q^{89} -1.00000 q^{91} +5.77897 q^{93} +9.79169 q^{95} -16.4101 q^{97} +2.61566 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.619952 −0.357929 −0.178965 0.983856i \(-0.557275\pi\)
−0.178965 + 0.983856i \(0.557275\pi\)
\(4\) 0 0
\(5\) −1.29231 −0.577940 −0.288970 0.957338i \(-0.593313\pi\)
−0.288970 + 0.957338i \(0.593313\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.61566 −0.871887
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0.801172 0.206862
\(16\) 0 0
\(17\) −6.20824 −1.50572 −0.752859 0.658181i \(-0.771326\pi\)
−0.752859 + 0.658181i \(0.771326\pi\)
\(18\) 0 0
\(19\) −7.57687 −1.73825 −0.869127 0.494589i \(-0.835318\pi\)
−0.869127 + 0.494589i \(0.835318\pi\)
\(20\) 0 0
\(21\) 0.619952 0.135285
\(22\) 0 0
\(23\) −2.07849 −0.433395 −0.216697 0.976239i \(-0.569528\pi\)
−0.216697 + 0.976239i \(0.569528\pi\)
\(24\) 0 0
\(25\) −3.32993 −0.665985
\(26\) 0 0
\(27\) 3.48144 0.670003
\(28\) 0 0
\(29\) 4.36889 0.811283 0.405641 0.914032i \(-0.367048\pi\)
0.405641 + 0.914032i \(0.367048\pi\)
\(30\) 0 0
\(31\) −9.32165 −1.67422 −0.837109 0.547036i \(-0.815756\pi\)
−0.837109 + 0.547036i \(0.815756\pi\)
\(32\) 0 0
\(33\) 0.619952 0.107920
\(34\) 0 0
\(35\) 1.29231 0.218441
\(36\) 0 0
\(37\) −6.40784 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(38\) 0 0
\(39\) −0.619952 −0.0992717
\(40\) 0 0
\(41\) −12.4066 −1.93758 −0.968790 0.247884i \(-0.920265\pi\)
−0.968790 + 0.247884i \(0.920265\pi\)
\(42\) 0 0
\(43\) 9.61081 1.46564 0.732818 0.680425i \(-0.238205\pi\)
0.732818 + 0.680425i \(0.238205\pi\)
\(44\) 0 0
\(45\) 3.38025 0.503898
\(46\) 0 0
\(47\) −6.59201 −0.961544 −0.480772 0.876846i \(-0.659644\pi\)
−0.480772 + 0.876846i \(0.659644\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.84881 0.538941
\(52\) 0 0
\(53\) 8.49683 1.16713 0.583565 0.812066i \(-0.301657\pi\)
0.583565 + 0.812066i \(0.301657\pi\)
\(54\) 0 0
\(55\) 1.29231 0.174255
\(56\) 0 0
\(57\) 4.69729 0.622172
\(58\) 0 0
\(59\) 10.3403 1.34620 0.673098 0.739553i \(-0.264963\pi\)
0.673098 + 0.739553i \(0.264963\pi\)
\(60\) 0 0
\(61\) −1.17828 −0.150864 −0.0754319 0.997151i \(-0.524034\pi\)
−0.0754319 + 0.997151i \(0.524034\pi\)
\(62\) 0 0
\(63\) 2.61566 0.329542
\(64\) 0 0
\(65\) −1.29231 −0.160292
\(66\) 0 0
\(67\) −1.17214 −0.143200 −0.0715999 0.997433i \(-0.522810\pi\)
−0.0715999 + 0.997433i \(0.522810\pi\)
\(68\) 0 0
\(69\) 1.28856 0.155125
\(70\) 0 0
\(71\) −13.9076 −1.65053 −0.825263 0.564749i \(-0.808973\pi\)
−0.825263 + 0.564749i \(0.808973\pi\)
\(72\) 0 0
\(73\) 12.0315 1.40818 0.704090 0.710111i \(-0.251355\pi\)
0.704090 + 0.710111i \(0.251355\pi\)
\(74\) 0 0
\(75\) 2.06439 0.238376
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 11.4620 1.28958 0.644789 0.764361i \(-0.276945\pi\)
0.644789 + 0.764361i \(0.276945\pi\)
\(80\) 0 0
\(81\) 5.68866 0.632073
\(82\) 0 0
\(83\) −3.04207 −0.333911 −0.166955 0.985964i \(-0.553394\pi\)
−0.166955 + 0.985964i \(0.553394\pi\)
\(84\) 0 0
\(85\) 8.02299 0.870215
\(86\) 0 0
\(87\) −2.70850 −0.290382
\(88\) 0 0
\(89\) −1.94288 −0.205945 −0.102973 0.994684i \(-0.532835\pi\)
−0.102973 + 0.994684i \(0.532835\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 5.77897 0.599252
\(94\) 0 0
\(95\) 9.79169 1.00461
\(96\) 0 0
\(97\) −16.4101 −1.66619 −0.833095 0.553130i \(-0.813433\pi\)
−0.833095 + 0.553130i \(0.813433\pi\)
\(98\) 0 0
\(99\) 2.61566 0.262884
\(100\) 0 0
\(101\) 1.57255 0.156475 0.0782373 0.996935i \(-0.475071\pi\)
0.0782373 + 0.996935i \(0.475071\pi\)
\(102\) 0 0
\(103\) −18.8754 −1.85985 −0.929925 0.367749i \(-0.880128\pi\)
−0.929925 + 0.367749i \(0.880128\pi\)
\(104\) 0 0
\(105\) −0.801172 −0.0781863
\(106\) 0 0
\(107\) −9.75305 −0.942863 −0.471432 0.881903i \(-0.656263\pi\)
−0.471432 + 0.881903i \(0.656263\pi\)
\(108\) 0 0
\(109\) −7.17646 −0.687380 −0.343690 0.939083i \(-0.611677\pi\)
−0.343690 + 0.939083i \(0.611677\pi\)
\(110\) 0 0
\(111\) 3.97255 0.377058
\(112\) 0 0
\(113\) 17.7361 1.66847 0.834234 0.551411i \(-0.185910\pi\)
0.834234 + 0.551411i \(0.185910\pi\)
\(114\) 0 0
\(115\) 2.68606 0.250476
\(116\) 0 0
\(117\) −2.61566 −0.241818
\(118\) 0 0
\(119\) 6.20824 0.569108
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 7.69147 0.693516
\(124\) 0 0
\(125\) 10.7649 0.962840
\(126\) 0 0
\(127\) −16.9800 −1.50673 −0.753363 0.657604i \(-0.771570\pi\)
−0.753363 + 0.657604i \(0.771570\pi\)
\(128\) 0 0
\(129\) −5.95824 −0.524594
\(130\) 0 0
\(131\) −1.80885 −0.158040 −0.0790200 0.996873i \(-0.525179\pi\)
−0.0790200 + 0.996873i \(0.525179\pi\)
\(132\) 0 0
\(133\) 7.57687 0.656998
\(134\) 0 0
\(135\) −4.49911 −0.387222
\(136\) 0 0
\(137\) −2.48116 −0.211979 −0.105990 0.994367i \(-0.533801\pi\)
−0.105990 + 0.994367i \(0.533801\pi\)
\(138\) 0 0
\(139\) 14.3192 1.21454 0.607270 0.794495i \(-0.292264\pi\)
0.607270 + 0.794495i \(0.292264\pi\)
\(140\) 0 0
\(141\) 4.08673 0.344165
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −5.64598 −0.468873
\(146\) 0 0
\(147\) −0.619952 −0.0511328
\(148\) 0 0
\(149\) −24.0105 −1.96702 −0.983509 0.180862i \(-0.942111\pi\)
−0.983509 + 0.180862i \(0.942111\pi\)
\(150\) 0 0
\(151\) 18.4303 1.49984 0.749918 0.661531i \(-0.230093\pi\)
0.749918 + 0.661531i \(0.230093\pi\)
\(152\) 0 0
\(153\) 16.2386 1.31282
\(154\) 0 0
\(155\) 12.0465 0.967597
\(156\) 0 0
\(157\) −8.83286 −0.704939 −0.352469 0.935823i \(-0.614658\pi\)
−0.352469 + 0.935823i \(0.614658\pi\)
\(158\) 0 0
\(159\) −5.26763 −0.417750
\(160\) 0 0
\(161\) 2.07849 0.163808
\(162\) 0 0
\(163\) −13.3803 −1.04802 −0.524012 0.851711i \(-0.675565\pi\)
−0.524012 + 0.851711i \(0.675565\pi\)
\(164\) 0 0
\(165\) −0.801172 −0.0623711
\(166\) 0 0
\(167\) −6.34806 −0.491227 −0.245614 0.969368i \(-0.578989\pi\)
−0.245614 + 0.969368i \(0.578989\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 19.8185 1.51556
\(172\) 0 0
\(173\) −5.41887 −0.411989 −0.205994 0.978553i \(-0.566043\pi\)
−0.205994 + 0.978553i \(0.566043\pi\)
\(174\) 0 0
\(175\) 3.32993 0.251719
\(176\) 0 0
\(177\) −6.41050 −0.481843
\(178\) 0 0
\(179\) −1.09721 −0.0820093 −0.0410046 0.999159i \(-0.513056\pi\)
−0.0410046 + 0.999159i \(0.513056\pi\)
\(180\) 0 0
\(181\) 18.0611 1.34247 0.671236 0.741243i \(-0.265764\pi\)
0.671236 + 0.741243i \(0.265764\pi\)
\(182\) 0 0
\(183\) 0.730479 0.0539986
\(184\) 0 0
\(185\) 8.28094 0.608827
\(186\) 0 0
\(187\) 6.20824 0.453991
\(188\) 0 0
\(189\) −3.48144 −0.253237
\(190\) 0 0
\(191\) 17.6107 1.27426 0.637131 0.770755i \(-0.280121\pi\)
0.637131 + 0.770755i \(0.280121\pi\)
\(192\) 0 0
\(193\) 4.44303 0.319816 0.159908 0.987132i \(-0.448880\pi\)
0.159908 + 0.987132i \(0.448880\pi\)
\(194\) 0 0
\(195\) 0.801172 0.0573731
\(196\) 0 0
\(197\) −4.87740 −0.347500 −0.173750 0.984790i \(-0.555588\pi\)
−0.173750 + 0.984790i \(0.555588\pi\)
\(198\) 0 0
\(199\) 11.9307 0.845744 0.422872 0.906189i \(-0.361022\pi\)
0.422872 + 0.906189i \(0.361022\pi\)
\(200\) 0 0
\(201\) 0.726671 0.0512554
\(202\) 0 0
\(203\) −4.36889 −0.306636
\(204\) 0 0
\(205\) 16.0332 1.11980
\(206\) 0 0
\(207\) 5.43662 0.377871
\(208\) 0 0
\(209\) 7.57687 0.524103
\(210\) 0 0
\(211\) −23.9152 −1.64639 −0.823195 0.567758i \(-0.807811\pi\)
−0.823195 + 0.567758i \(0.807811\pi\)
\(212\) 0 0
\(213\) 8.62202 0.590771
\(214\) 0 0
\(215\) −12.4202 −0.847049
\(216\) 0 0
\(217\) 9.32165 0.632795
\(218\) 0 0
\(219\) −7.45894 −0.504028
\(220\) 0 0
\(221\) −6.20824 −0.417611
\(222\) 0 0
\(223\) −5.23055 −0.350263 −0.175132 0.984545i \(-0.556035\pi\)
−0.175132 + 0.984545i \(0.556035\pi\)
\(224\) 0 0
\(225\) 8.70996 0.580664
\(226\) 0 0
\(227\) 13.6686 0.907214 0.453607 0.891202i \(-0.350137\pi\)
0.453607 + 0.891202i \(0.350137\pi\)
\(228\) 0 0
\(229\) −18.3511 −1.21267 −0.606336 0.795209i \(-0.707361\pi\)
−0.606336 + 0.795209i \(0.707361\pi\)
\(230\) 0 0
\(231\) −0.619952 −0.0407898
\(232\) 0 0
\(233\) 16.2260 1.06300 0.531500 0.847058i \(-0.321628\pi\)
0.531500 + 0.847058i \(0.321628\pi\)
\(234\) 0 0
\(235\) 8.51895 0.555715
\(236\) 0 0
\(237\) −7.10589 −0.461577
\(238\) 0 0
\(239\) 5.47650 0.354245 0.177123 0.984189i \(-0.443321\pi\)
0.177123 + 0.984189i \(0.443321\pi\)
\(240\) 0 0
\(241\) 21.1742 1.36395 0.681976 0.731375i \(-0.261121\pi\)
0.681976 + 0.731375i \(0.261121\pi\)
\(242\) 0 0
\(243\) −13.9710 −0.896240
\(244\) 0 0
\(245\) −1.29231 −0.0825629
\(246\) 0 0
\(247\) −7.57687 −0.482105
\(248\) 0 0
\(249\) 1.88594 0.119517
\(250\) 0 0
\(251\) 10.5021 0.662888 0.331444 0.943475i \(-0.392464\pi\)
0.331444 + 0.943475i \(0.392464\pi\)
\(252\) 0 0
\(253\) 2.07849 0.130673
\(254\) 0 0
\(255\) −4.97386 −0.311475
\(256\) 0 0
\(257\) −17.2850 −1.07821 −0.539106 0.842238i \(-0.681238\pi\)
−0.539106 + 0.842238i \(0.681238\pi\)
\(258\) 0 0
\(259\) 6.40784 0.398164
\(260\) 0 0
\(261\) −11.4275 −0.707347
\(262\) 0 0
\(263\) −10.2976 −0.634979 −0.317489 0.948262i \(-0.602840\pi\)
−0.317489 + 0.948262i \(0.602840\pi\)
\(264\) 0 0
\(265\) −10.9806 −0.674531
\(266\) 0 0
\(267\) 1.20449 0.0737138
\(268\) 0 0
\(269\) 10.0830 0.614770 0.307385 0.951585i \(-0.400546\pi\)
0.307385 + 0.951585i \(0.400546\pi\)
\(270\) 0 0
\(271\) 1.32823 0.0806841 0.0403421 0.999186i \(-0.487155\pi\)
0.0403421 + 0.999186i \(0.487155\pi\)
\(272\) 0 0
\(273\) 0.619952 0.0375212
\(274\) 0 0
\(275\) 3.32993 0.200802
\(276\) 0 0
\(277\) 24.3437 1.46267 0.731335 0.682018i \(-0.238897\pi\)
0.731335 + 0.682018i \(0.238897\pi\)
\(278\) 0 0
\(279\) 24.3823 1.45973
\(280\) 0 0
\(281\) 7.36292 0.439235 0.219617 0.975586i \(-0.429519\pi\)
0.219617 + 0.975586i \(0.429519\pi\)
\(282\) 0 0
\(283\) −21.8568 −1.29925 −0.649626 0.760254i \(-0.725074\pi\)
−0.649626 + 0.760254i \(0.725074\pi\)
\(284\) 0 0
\(285\) −6.07037 −0.359578
\(286\) 0 0
\(287\) 12.4066 0.732336
\(288\) 0 0
\(289\) 21.5422 1.26719
\(290\) 0 0
\(291\) 10.1735 0.596378
\(292\) 0 0
\(293\) 1.47001 0.0858792 0.0429396 0.999078i \(-0.486328\pi\)
0.0429396 + 0.999078i \(0.486328\pi\)
\(294\) 0 0
\(295\) −13.3629 −0.778020
\(296\) 0 0
\(297\) −3.48144 −0.202014
\(298\) 0 0
\(299\) −2.07849 −0.120202
\(300\) 0 0
\(301\) −9.61081 −0.553958
\(302\) 0 0
\(303\) −0.974905 −0.0560068
\(304\) 0 0
\(305\) 1.52271 0.0871902
\(306\) 0 0
\(307\) −14.3314 −0.817938 −0.408969 0.912548i \(-0.634112\pi\)
−0.408969 + 0.912548i \(0.634112\pi\)
\(308\) 0 0
\(309\) 11.7018 0.665695
\(310\) 0 0
\(311\) −19.8291 −1.12440 −0.562201 0.827000i \(-0.690045\pi\)
−0.562201 + 0.827000i \(0.690045\pi\)
\(312\) 0 0
\(313\) 29.2120 1.65116 0.825580 0.564285i \(-0.190848\pi\)
0.825580 + 0.564285i \(0.190848\pi\)
\(314\) 0 0
\(315\) −3.38025 −0.190456
\(316\) 0 0
\(317\) 0.747824 0.0420020 0.0210010 0.999779i \(-0.493315\pi\)
0.0210010 + 0.999779i \(0.493315\pi\)
\(318\) 0 0
\(319\) −4.36889 −0.244611
\(320\) 0 0
\(321\) 6.04642 0.337478
\(322\) 0 0
\(323\) 47.0390 2.61732
\(324\) 0 0
\(325\) −3.32993 −0.184711
\(326\) 0 0
\(327\) 4.44906 0.246033
\(328\) 0 0
\(329\) 6.59201 0.363430
\(330\) 0 0
\(331\) −9.67424 −0.531744 −0.265872 0.964008i \(-0.585660\pi\)
−0.265872 + 0.964008i \(0.585660\pi\)
\(332\) 0 0
\(333\) 16.7607 0.918483
\(334\) 0 0
\(335\) 1.51477 0.0827609
\(336\) 0 0
\(337\) −2.06916 −0.112714 −0.0563572 0.998411i \(-0.517949\pi\)
−0.0563572 + 0.998411i \(0.517949\pi\)
\(338\) 0 0
\(339\) −10.9955 −0.597193
\(340\) 0 0
\(341\) 9.32165 0.504796
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.66523 −0.0896527
\(346\) 0 0
\(347\) −13.4310 −0.721013 −0.360506 0.932757i \(-0.617396\pi\)
−0.360506 + 0.932757i \(0.617396\pi\)
\(348\) 0 0
\(349\) −0.515138 −0.0275747 −0.0137874 0.999905i \(-0.504389\pi\)
−0.0137874 + 0.999905i \(0.504389\pi\)
\(350\) 0 0
\(351\) 3.48144 0.185825
\(352\) 0 0
\(353\) −29.2092 −1.55465 −0.777324 0.629101i \(-0.783423\pi\)
−0.777324 + 0.629101i \(0.783423\pi\)
\(354\) 0 0
\(355\) 17.9729 0.953905
\(356\) 0 0
\(357\) −3.84881 −0.203700
\(358\) 0 0
\(359\) −10.3605 −0.546807 −0.273403 0.961899i \(-0.588149\pi\)
−0.273403 + 0.961899i \(0.588149\pi\)
\(360\) 0 0
\(361\) 38.4090 2.02153
\(362\) 0 0
\(363\) −0.619952 −0.0325390
\(364\) 0 0
\(365\) −15.5484 −0.813843
\(366\) 0 0
\(367\) −23.4510 −1.22413 −0.612067 0.790806i \(-0.709662\pi\)
−0.612067 + 0.790806i \(0.709662\pi\)
\(368\) 0 0
\(369\) 32.4513 1.68935
\(370\) 0 0
\(371\) −8.49683 −0.441134
\(372\) 0 0
\(373\) 9.83239 0.509102 0.254551 0.967059i \(-0.418072\pi\)
0.254551 + 0.967059i \(0.418072\pi\)
\(374\) 0 0
\(375\) −6.67370 −0.344628
\(376\) 0 0
\(377\) 4.36889 0.225009
\(378\) 0 0
\(379\) −12.3709 −0.635452 −0.317726 0.948183i \(-0.602919\pi\)
−0.317726 + 0.948183i \(0.602919\pi\)
\(380\) 0 0
\(381\) 10.5267 0.539302
\(382\) 0 0
\(383\) 30.8124 1.57444 0.787221 0.616671i \(-0.211519\pi\)
0.787221 + 0.616671i \(0.211519\pi\)
\(384\) 0 0
\(385\) −1.29231 −0.0658624
\(386\) 0 0
\(387\) −25.1386 −1.27787
\(388\) 0 0
\(389\) −12.3490 −0.626118 −0.313059 0.949734i \(-0.601354\pi\)
−0.313059 + 0.949734i \(0.601354\pi\)
\(390\) 0 0
\(391\) 12.9037 0.652570
\(392\) 0 0
\(393\) 1.12140 0.0565671
\(394\) 0 0
\(395\) −14.8125 −0.745298
\(396\) 0 0
\(397\) 28.6331 1.43705 0.718526 0.695500i \(-0.244817\pi\)
0.718526 + 0.695500i \(0.244817\pi\)
\(398\) 0 0
\(399\) −4.69729 −0.235159
\(400\) 0 0
\(401\) 35.5777 1.77666 0.888332 0.459201i \(-0.151864\pi\)
0.888332 + 0.459201i \(0.151864\pi\)
\(402\) 0 0
\(403\) −9.32165 −0.464344
\(404\) 0 0
\(405\) −7.35153 −0.365300
\(406\) 0 0
\(407\) 6.40784 0.317625
\(408\) 0 0
\(409\) 37.0587 1.83244 0.916219 0.400679i \(-0.131226\pi\)
0.916219 + 0.400679i \(0.131226\pi\)
\(410\) 0 0
\(411\) 1.53820 0.0758737
\(412\) 0 0
\(413\) −10.3403 −0.508814
\(414\) 0 0
\(415\) 3.93131 0.192981
\(416\) 0 0
\(417\) −8.87723 −0.434720
\(418\) 0 0
\(419\) −14.1674 −0.692121 −0.346061 0.938212i \(-0.612481\pi\)
−0.346061 + 0.938212i \(0.612481\pi\)
\(420\) 0 0
\(421\) −3.61952 −0.176405 −0.0882023 0.996103i \(-0.528112\pi\)
−0.0882023 + 0.996103i \(0.528112\pi\)
\(422\) 0 0
\(423\) 17.2425 0.838358
\(424\) 0 0
\(425\) 20.6730 1.00279
\(426\) 0 0
\(427\) 1.17828 0.0570211
\(428\) 0 0
\(429\) 0.619952 0.0299315
\(430\) 0 0
\(431\) −7.79753 −0.375594 −0.187797 0.982208i \(-0.560135\pi\)
−0.187797 + 0.982208i \(0.560135\pi\)
\(432\) 0 0
\(433\) −19.6345 −0.943572 −0.471786 0.881713i \(-0.656391\pi\)
−0.471786 + 0.881713i \(0.656391\pi\)
\(434\) 0 0
\(435\) 3.50023 0.167823
\(436\) 0 0
\(437\) 15.7484 0.753350
\(438\) 0 0
\(439\) 20.1122 0.959901 0.479951 0.877295i \(-0.340655\pi\)
0.479951 + 0.877295i \(0.340655\pi\)
\(440\) 0 0
\(441\) −2.61566 −0.124555
\(442\) 0 0
\(443\) 28.7559 1.36623 0.683117 0.730309i \(-0.260624\pi\)
0.683117 + 0.730309i \(0.260624\pi\)
\(444\) 0 0
\(445\) 2.51081 0.119024
\(446\) 0 0
\(447\) 14.8854 0.704053
\(448\) 0 0
\(449\) 34.9243 1.64818 0.824090 0.566459i \(-0.191687\pi\)
0.824090 + 0.566459i \(0.191687\pi\)
\(450\) 0 0
\(451\) 12.4066 0.584202
\(452\) 0 0
\(453\) −11.4259 −0.536835
\(454\) 0 0
\(455\) 1.29231 0.0605846
\(456\) 0 0
\(457\) 11.7566 0.549951 0.274976 0.961451i \(-0.411330\pi\)
0.274976 + 0.961451i \(0.411330\pi\)
\(458\) 0 0
\(459\) −21.6136 −1.00884
\(460\) 0 0
\(461\) 25.1726 1.17241 0.586203 0.810164i \(-0.300622\pi\)
0.586203 + 0.810164i \(0.300622\pi\)
\(462\) 0 0
\(463\) 13.2344 0.615056 0.307528 0.951539i \(-0.400498\pi\)
0.307528 + 0.951539i \(0.400498\pi\)
\(464\) 0 0
\(465\) −7.46824 −0.346331
\(466\) 0 0
\(467\) 16.0162 0.741139 0.370570 0.928805i \(-0.379162\pi\)
0.370570 + 0.928805i \(0.379162\pi\)
\(468\) 0 0
\(469\) 1.17214 0.0541245
\(470\) 0 0
\(471\) 5.47595 0.252318
\(472\) 0 0
\(473\) −9.61081 −0.441906
\(474\) 0 0
\(475\) 25.2304 1.15765
\(476\) 0 0
\(477\) −22.2248 −1.01760
\(478\) 0 0
\(479\) 3.90965 0.178636 0.0893181 0.996003i \(-0.471531\pi\)
0.0893181 + 0.996003i \(0.471531\pi\)
\(480\) 0 0
\(481\) −6.40784 −0.292172
\(482\) 0 0
\(483\) −1.28856 −0.0586316
\(484\) 0 0
\(485\) 21.2070 0.962958
\(486\) 0 0
\(487\) 18.7629 0.850228 0.425114 0.905140i \(-0.360234\pi\)
0.425114 + 0.905140i \(0.360234\pi\)
\(488\) 0 0
\(489\) 8.29513 0.375119
\(490\) 0 0
\(491\) 14.9958 0.676750 0.338375 0.941011i \(-0.390123\pi\)
0.338375 + 0.941011i \(0.390123\pi\)
\(492\) 0 0
\(493\) −27.1231 −1.22156
\(494\) 0 0
\(495\) −3.38025 −0.151931
\(496\) 0 0
\(497\) 13.9076 0.623840
\(498\) 0 0
\(499\) −40.7120 −1.82252 −0.911260 0.411831i \(-0.864889\pi\)
−0.911260 + 0.411831i \(0.864889\pi\)
\(500\) 0 0
\(501\) 3.93549 0.175825
\(502\) 0 0
\(503\) −5.90482 −0.263283 −0.131641 0.991297i \(-0.542025\pi\)
−0.131641 + 0.991297i \(0.542025\pi\)
\(504\) 0 0
\(505\) −2.03223 −0.0904329
\(506\) 0 0
\(507\) −0.619952 −0.0275330
\(508\) 0 0
\(509\) 12.1013 0.536380 0.268190 0.963366i \(-0.413574\pi\)
0.268190 + 0.963366i \(0.413574\pi\)
\(510\) 0 0
\(511\) −12.0315 −0.532242
\(512\) 0 0
\(513\) −26.3784 −1.16464
\(514\) 0 0
\(515\) 24.3929 1.07488
\(516\) 0 0
\(517\) 6.59201 0.289916
\(518\) 0 0
\(519\) 3.35944 0.147463
\(520\) 0 0
\(521\) 31.7226 1.38979 0.694896 0.719110i \(-0.255450\pi\)
0.694896 + 0.719110i \(0.255450\pi\)
\(522\) 0 0
\(523\) −34.7025 −1.51744 −0.758718 0.651419i \(-0.774174\pi\)
−0.758718 + 0.651419i \(0.774174\pi\)
\(524\) 0 0
\(525\) −2.06439 −0.0900975
\(526\) 0 0
\(527\) 57.8710 2.52090
\(528\) 0 0
\(529\) −18.6799 −0.812169
\(530\) 0 0
\(531\) −27.0468 −1.17373
\(532\) 0 0
\(533\) −12.4066 −0.537388
\(534\) 0 0
\(535\) 12.6040 0.544918
\(536\) 0 0
\(537\) 0.680217 0.0293535
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −17.7064 −0.761258 −0.380629 0.924728i \(-0.624293\pi\)
−0.380629 + 0.924728i \(0.624293\pi\)
\(542\) 0 0
\(543\) −11.1970 −0.480510
\(544\) 0 0
\(545\) 9.27423 0.397264
\(546\) 0 0
\(547\) 2.21418 0.0946713 0.0473356 0.998879i \(-0.484927\pi\)
0.0473356 + 0.998879i \(0.484927\pi\)
\(548\) 0 0
\(549\) 3.08199 0.131536
\(550\) 0 0
\(551\) −33.1025 −1.41022
\(552\) 0 0
\(553\) −11.4620 −0.487414
\(554\) 0 0
\(555\) −5.13378 −0.217917
\(556\) 0 0
\(557\) 0.770744 0.0326575 0.0163287 0.999867i \(-0.494802\pi\)
0.0163287 + 0.999867i \(0.494802\pi\)
\(558\) 0 0
\(559\) 9.61081 0.406494
\(560\) 0 0
\(561\) −3.84881 −0.162497
\(562\) 0 0
\(563\) −15.6229 −0.658426 −0.329213 0.944256i \(-0.606783\pi\)
−0.329213 + 0.944256i \(0.606783\pi\)
\(564\) 0 0
\(565\) −22.9205 −0.964274
\(566\) 0 0
\(567\) −5.68866 −0.238901
\(568\) 0 0
\(569\) −31.2760 −1.31116 −0.655580 0.755126i \(-0.727576\pi\)
−0.655580 + 0.755126i \(0.727576\pi\)
\(570\) 0 0
\(571\) −45.5926 −1.90799 −0.953996 0.299820i \(-0.903073\pi\)
−0.953996 + 0.299820i \(0.903073\pi\)
\(572\) 0 0
\(573\) −10.9178 −0.456096
\(574\) 0 0
\(575\) 6.92121 0.288634
\(576\) 0 0
\(577\) 14.1204 0.587838 0.293919 0.955830i \(-0.405040\pi\)
0.293919 + 0.955830i \(0.405040\pi\)
\(578\) 0 0
\(579\) −2.75446 −0.114472
\(580\) 0 0
\(581\) 3.04207 0.126206
\(582\) 0 0
\(583\) −8.49683 −0.351903
\(584\) 0 0
\(585\) 3.38025 0.139756
\(586\) 0 0
\(587\) 34.8879 1.43998 0.719989 0.693986i \(-0.244147\pi\)
0.719989 + 0.693986i \(0.244147\pi\)
\(588\) 0 0
\(589\) 70.6289 2.91021
\(590\) 0 0
\(591\) 3.02375 0.124380
\(592\) 0 0
\(593\) −19.5512 −0.802871 −0.401436 0.915887i \(-0.631489\pi\)
−0.401436 + 0.915887i \(0.631489\pi\)
\(594\) 0 0
\(595\) −8.02299 −0.328910
\(596\) 0 0
\(597\) −7.39645 −0.302717
\(598\) 0 0
\(599\) −18.8081 −0.768477 −0.384238 0.923234i \(-0.625536\pi\)
−0.384238 + 0.923234i \(0.625536\pi\)
\(600\) 0 0
\(601\) −48.7858 −1.99001 −0.995006 0.0998129i \(-0.968176\pi\)
−0.995006 + 0.0998129i \(0.968176\pi\)
\(602\) 0 0
\(603\) 3.06592 0.124854
\(604\) 0 0
\(605\) −1.29231 −0.0525400
\(606\) 0 0
\(607\) −34.8493 −1.41449 −0.707245 0.706969i \(-0.750062\pi\)
−0.707245 + 0.706969i \(0.750062\pi\)
\(608\) 0 0
\(609\) 2.70850 0.109754
\(610\) 0 0
\(611\) −6.59201 −0.266684
\(612\) 0 0
\(613\) 3.56338 0.143924 0.0719618 0.997407i \(-0.477074\pi\)
0.0719618 + 0.997407i \(0.477074\pi\)
\(614\) 0 0
\(615\) −9.93978 −0.400811
\(616\) 0 0
\(617\) −7.35703 −0.296183 −0.148091 0.988974i \(-0.547313\pi\)
−0.148091 + 0.988974i \(0.547313\pi\)
\(618\) 0 0
\(619\) −38.9999 −1.56754 −0.783770 0.621051i \(-0.786706\pi\)
−0.783770 + 0.621051i \(0.786706\pi\)
\(620\) 0 0
\(621\) −7.23612 −0.290376
\(622\) 0 0
\(623\) 1.94288 0.0778400
\(624\) 0 0
\(625\) 2.73805 0.109522
\(626\) 0 0
\(627\) −4.69729 −0.187592
\(628\) 0 0
\(629\) 39.7814 1.58619
\(630\) 0 0
\(631\) −33.6259 −1.33863 −0.669313 0.742981i \(-0.733411\pi\)
−0.669313 + 0.742981i \(0.733411\pi\)
\(632\) 0 0
\(633\) 14.8263 0.589291
\(634\) 0 0
\(635\) 21.9434 0.870798
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 36.3775 1.43907
\(640\) 0 0
\(641\) −26.2385 −1.03636 −0.518179 0.855272i \(-0.673390\pi\)
−0.518179 + 0.855272i \(0.673390\pi\)
\(642\) 0 0
\(643\) −36.2912 −1.43119 −0.715593 0.698517i \(-0.753843\pi\)
−0.715593 + 0.698517i \(0.753843\pi\)
\(644\) 0 0
\(645\) 7.69991 0.303184
\(646\) 0 0
\(647\) 26.2318 1.03128 0.515638 0.856806i \(-0.327555\pi\)
0.515638 + 0.856806i \(0.327555\pi\)
\(648\) 0 0
\(649\) −10.3403 −0.405893
\(650\) 0 0
\(651\) −5.77897 −0.226496
\(652\) 0 0
\(653\) −24.4011 −0.954890 −0.477445 0.878662i \(-0.658437\pi\)
−0.477445 + 0.878662i \(0.658437\pi\)
\(654\) 0 0
\(655\) 2.33760 0.0913376
\(656\) 0 0
\(657\) −31.4703 −1.22777
\(658\) 0 0
\(659\) 42.4192 1.65242 0.826209 0.563364i \(-0.190493\pi\)
0.826209 + 0.563364i \(0.190493\pi\)
\(660\) 0 0
\(661\) −30.6437 −1.19190 −0.595950 0.803021i \(-0.703224\pi\)
−0.595950 + 0.803021i \(0.703224\pi\)
\(662\) 0 0
\(663\) 3.84881 0.149475
\(664\) 0 0
\(665\) −9.79169 −0.379705
\(666\) 0 0
\(667\) −9.08069 −0.351606
\(668\) 0 0
\(669\) 3.24269 0.125370
\(670\) 0 0
\(671\) 1.17828 0.0454871
\(672\) 0 0
\(673\) −40.7296 −1.57001 −0.785006 0.619488i \(-0.787340\pi\)
−0.785006 + 0.619488i \(0.787340\pi\)
\(674\) 0 0
\(675\) −11.5929 −0.446212
\(676\) 0 0
\(677\) 44.3760 1.70551 0.852754 0.522313i \(-0.174931\pi\)
0.852754 + 0.522313i \(0.174931\pi\)
\(678\) 0 0
\(679\) 16.4101 0.629761
\(680\) 0 0
\(681\) −8.47385 −0.324718
\(682\) 0 0
\(683\) 14.0658 0.538213 0.269107 0.963110i \(-0.413272\pi\)
0.269107 + 0.963110i \(0.413272\pi\)
\(684\) 0 0
\(685\) 3.20643 0.122511
\(686\) 0 0
\(687\) 11.3768 0.434051
\(688\) 0 0
\(689\) 8.49683 0.323704
\(690\) 0 0
\(691\) −17.9219 −0.681782 −0.340891 0.940103i \(-0.610729\pi\)
−0.340891 + 0.940103i \(0.610729\pi\)
\(692\) 0 0
\(693\) −2.61566 −0.0993607
\(694\) 0 0
\(695\) −18.5049 −0.701932
\(696\) 0 0
\(697\) 77.0229 2.91745
\(698\) 0 0
\(699\) −10.0593 −0.380479
\(700\) 0 0
\(701\) 39.6269 1.49669 0.748343 0.663312i \(-0.230850\pi\)
0.748343 + 0.663312i \(0.230850\pi\)
\(702\) 0 0
\(703\) 48.5514 1.83115
\(704\) 0 0
\(705\) −5.28134 −0.198907
\(706\) 0 0
\(707\) −1.57255 −0.0591418
\(708\) 0 0
\(709\) 11.2174 0.421278 0.210639 0.977564i \(-0.432446\pi\)
0.210639 + 0.977564i \(0.432446\pi\)
\(710\) 0 0
\(711\) −29.9807 −1.12436
\(712\) 0 0
\(713\) 19.3749 0.725597
\(714\) 0 0
\(715\) 1.29231 0.0483298
\(716\) 0 0
\(717\) −3.39516 −0.126795
\(718\) 0 0
\(719\) −1.80817 −0.0674333 −0.0337166 0.999431i \(-0.510734\pi\)
−0.0337166 + 0.999431i \(0.510734\pi\)
\(720\) 0 0
\(721\) 18.8754 0.702957
\(722\) 0 0
\(723\) −13.1270 −0.488198
\(724\) 0 0
\(725\) −14.5481 −0.540303
\(726\) 0 0
\(727\) −28.6656 −1.06315 −0.531573 0.847012i \(-0.678399\pi\)
−0.531573 + 0.847012i \(0.678399\pi\)
\(728\) 0 0
\(729\) −8.40462 −0.311282
\(730\) 0 0
\(731\) −59.6662 −2.20683
\(732\) 0 0
\(733\) 32.3537 1.19501 0.597505 0.801865i \(-0.296159\pi\)
0.597505 + 0.801865i \(0.296159\pi\)
\(734\) 0 0
\(735\) 0.801172 0.0295517
\(736\) 0 0
\(737\) 1.17214 0.0431764
\(738\) 0 0
\(739\) −4.88480 −0.179690 −0.0898452 0.995956i \(-0.528637\pi\)
−0.0898452 + 0.995956i \(0.528637\pi\)
\(740\) 0 0
\(741\) 4.69729 0.172559
\(742\) 0 0
\(743\) 36.2509 1.32992 0.664958 0.746881i \(-0.268449\pi\)
0.664958 + 0.746881i \(0.268449\pi\)
\(744\) 0 0
\(745\) 31.0291 1.13682
\(746\) 0 0
\(747\) 7.95703 0.291133
\(748\) 0 0
\(749\) 9.75305 0.356369
\(750\) 0 0
\(751\) −12.2303 −0.446289 −0.223144 0.974785i \(-0.571632\pi\)
−0.223144 + 0.974785i \(0.571632\pi\)
\(752\) 0 0
\(753\) −6.51081 −0.237267
\(754\) 0 0
\(755\) −23.8177 −0.866815
\(756\) 0 0
\(757\) 35.6868 1.29706 0.648529 0.761190i \(-0.275384\pi\)
0.648529 + 0.761190i \(0.275384\pi\)
\(758\) 0 0
\(759\) −1.28856 −0.0467718
\(760\) 0 0
\(761\) −50.8042 −1.84165 −0.920825 0.389977i \(-0.872483\pi\)
−0.920825 + 0.389977i \(0.872483\pi\)
\(762\) 0 0
\(763\) 7.17646 0.259805
\(764\) 0 0
\(765\) −20.9854 −0.758729
\(766\) 0 0
\(767\) 10.3403 0.373367
\(768\) 0 0
\(769\) −7.95698 −0.286936 −0.143468 0.989655i \(-0.545825\pi\)
−0.143468 + 0.989655i \(0.545825\pi\)
\(770\) 0 0
\(771\) 10.7159 0.385924
\(772\) 0 0
\(773\) 39.3667 1.41592 0.707960 0.706252i \(-0.249616\pi\)
0.707960 + 0.706252i \(0.249616\pi\)
\(774\) 0 0
\(775\) 31.0404 1.11500
\(776\) 0 0
\(777\) −3.97255 −0.142515
\(778\) 0 0
\(779\) 94.0029 3.36800
\(780\) 0 0
\(781\) 13.9076 0.497652
\(782\) 0 0
\(783\) 15.2100 0.543562
\(784\) 0 0
\(785\) 11.4148 0.407412
\(786\) 0 0
\(787\) −17.7885 −0.634090 −0.317045 0.948410i \(-0.602691\pi\)
−0.317045 + 0.948410i \(0.602691\pi\)
\(788\) 0 0
\(789\) 6.38403 0.227277
\(790\) 0 0
\(791\) −17.7361 −0.630621
\(792\) 0 0
\(793\) −1.17828 −0.0418421
\(794\) 0 0
\(795\) 6.80742 0.241434
\(796\) 0 0
\(797\) −6.10907 −0.216394 −0.108197 0.994129i \(-0.534508\pi\)
−0.108197 + 0.994129i \(0.534508\pi\)
\(798\) 0 0
\(799\) 40.9248 1.44782
\(800\) 0 0
\(801\) 5.08192 0.179561
\(802\) 0 0
\(803\) −12.0315 −0.424582
\(804\) 0 0
\(805\) −2.68606 −0.0946710
\(806\) 0 0
\(807\) −6.25096 −0.220044
\(808\) 0 0
\(809\) 6.33308 0.222659 0.111330 0.993784i \(-0.464489\pi\)
0.111330 + 0.993784i \(0.464489\pi\)
\(810\) 0 0
\(811\) −31.4720 −1.10513 −0.552565 0.833470i \(-0.686351\pi\)
−0.552565 + 0.833470i \(0.686351\pi\)
\(812\) 0 0
\(813\) −0.823437 −0.0288792
\(814\) 0 0
\(815\) 17.2915 0.605695
\(816\) 0 0
\(817\) −72.8199 −2.54765
\(818\) 0 0
\(819\) 2.61566 0.0913986
\(820\) 0 0
\(821\) 11.3983 0.397804 0.198902 0.980019i \(-0.436262\pi\)
0.198902 + 0.980019i \(0.436262\pi\)
\(822\) 0 0
\(823\) −8.64642 −0.301395 −0.150698 0.988580i \(-0.548152\pi\)
−0.150698 + 0.988580i \(0.548152\pi\)
\(824\) 0 0
\(825\) −2.06439 −0.0718730
\(826\) 0 0
\(827\) −28.7132 −0.998456 −0.499228 0.866471i \(-0.666383\pi\)
−0.499228 + 0.866471i \(0.666383\pi\)
\(828\) 0 0
\(829\) 19.5172 0.677860 0.338930 0.940812i \(-0.389935\pi\)
0.338930 + 0.940812i \(0.389935\pi\)
\(830\) 0 0
\(831\) −15.0919 −0.523533
\(832\) 0 0
\(833\) −6.20824 −0.215103
\(834\) 0 0
\(835\) 8.20367 0.283900
\(836\) 0 0
\(837\) −32.4527 −1.12173
\(838\) 0 0
\(839\) −38.0423 −1.31337 −0.656683 0.754167i \(-0.728041\pi\)
−0.656683 + 0.754167i \(0.728041\pi\)
\(840\) 0 0
\(841\) −9.91278 −0.341820
\(842\) 0 0
\(843\) −4.56465 −0.157215
\(844\) 0 0
\(845\) −1.29231 −0.0444569
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 13.5502 0.465040
\(850\) 0 0
\(851\) 13.3186 0.456556
\(852\) 0 0
\(853\) −11.9466 −0.409045 −0.204523 0.978862i \(-0.565564\pi\)
−0.204523 + 0.978862i \(0.565564\pi\)
\(854\) 0 0
\(855\) −25.6117 −0.875903
\(856\) 0 0
\(857\) 13.2108 0.451272 0.225636 0.974212i \(-0.427554\pi\)
0.225636 + 0.974212i \(0.427554\pi\)
\(858\) 0 0
\(859\) −11.9509 −0.407759 −0.203879 0.978996i \(-0.565355\pi\)
−0.203879 + 0.978996i \(0.565355\pi\)
\(860\) 0 0
\(861\) −7.69147 −0.262125
\(862\) 0 0
\(863\) −13.2473 −0.450943 −0.225472 0.974250i \(-0.572392\pi\)
−0.225472 + 0.974250i \(0.572392\pi\)
\(864\) 0 0
\(865\) 7.00287 0.238105
\(866\) 0 0
\(867\) −13.3551 −0.453564
\(868\) 0 0
\(869\) −11.4620 −0.388822
\(870\) 0 0
\(871\) −1.17214 −0.0397165
\(872\) 0 0
\(873\) 42.9232 1.45273
\(874\) 0 0
\(875\) −10.7649 −0.363919
\(876\) 0 0
\(877\) 18.2730 0.617034 0.308517 0.951219i \(-0.400167\pi\)
0.308517 + 0.951219i \(0.400167\pi\)
\(878\) 0 0
\(879\) −0.911338 −0.0307387
\(880\) 0 0
\(881\) −14.5003 −0.488529 −0.244265 0.969709i \(-0.578547\pi\)
−0.244265 + 0.969709i \(0.578547\pi\)
\(882\) 0 0
\(883\) 17.8291 0.599998 0.299999 0.953940i \(-0.403014\pi\)
0.299999 + 0.953940i \(0.403014\pi\)
\(884\) 0 0
\(885\) 8.28437 0.278476
\(886\) 0 0
\(887\) 22.8407 0.766917 0.383458 0.923558i \(-0.374733\pi\)
0.383458 + 0.923558i \(0.374733\pi\)
\(888\) 0 0
\(889\) 16.9800 0.569489
\(890\) 0 0
\(891\) −5.68866 −0.190577
\(892\) 0 0
\(893\) 49.9469 1.67141
\(894\) 0 0
\(895\) 1.41794 0.0473964
\(896\) 0 0
\(897\) 1.28856 0.0430238
\(898\) 0 0
\(899\) −40.7253 −1.35826
\(900\) 0 0
\(901\) −52.7504 −1.75737
\(902\) 0 0
\(903\) 5.95824 0.198278
\(904\) 0 0
\(905\) −23.3406 −0.775869
\(906\) 0 0
\(907\) 32.8704 1.09144 0.545722 0.837966i \(-0.316255\pi\)
0.545722 + 0.837966i \(0.316255\pi\)
\(908\) 0 0
\(909\) −4.11326 −0.136428
\(910\) 0 0
\(911\) 5.22055 0.172964 0.0864822 0.996253i \(-0.472437\pi\)
0.0864822 + 0.996253i \(0.472437\pi\)
\(912\) 0 0
\(913\) 3.04207 0.100678
\(914\) 0 0
\(915\) −0.944008 −0.0312079
\(916\) 0 0
\(917\) 1.80885 0.0597335
\(918\) 0 0
\(919\) 26.5120 0.874549 0.437274 0.899328i \(-0.355944\pi\)
0.437274 + 0.899328i \(0.355944\pi\)
\(920\) 0 0
\(921\) 8.88479 0.292764
\(922\) 0 0
\(923\) −13.9076 −0.457773
\(924\) 0 0
\(925\) 21.3376 0.701578
\(926\) 0 0
\(927\) 49.3717 1.62158
\(928\) 0 0
\(929\) 1.34919 0.0442653 0.0221327 0.999755i \(-0.492954\pi\)
0.0221327 + 0.999755i \(0.492954\pi\)
\(930\) 0 0
\(931\) −7.57687 −0.248322
\(932\) 0 0
\(933\) 12.2931 0.402457
\(934\) 0 0
\(935\) −8.02299 −0.262380
\(936\) 0 0
\(937\) 8.61167 0.281331 0.140666 0.990057i \(-0.455076\pi\)
0.140666 + 0.990057i \(0.455076\pi\)
\(938\) 0 0
\(939\) −18.1100 −0.590999
\(940\) 0 0
\(941\) 40.6016 1.32358 0.661788 0.749691i \(-0.269798\pi\)
0.661788 + 0.749691i \(0.269798\pi\)
\(942\) 0 0
\(943\) 25.7869 0.839736
\(944\) 0 0
\(945\) 4.49911 0.146356
\(946\) 0 0
\(947\) 2.77733 0.0902511 0.0451255 0.998981i \(-0.485631\pi\)
0.0451255 + 0.998981i \(0.485631\pi\)
\(948\) 0 0
\(949\) 12.0315 0.390559
\(950\) 0 0
\(951\) −0.463615 −0.0150337
\(952\) 0 0
\(953\) 29.7363 0.963252 0.481626 0.876377i \(-0.340046\pi\)
0.481626 + 0.876377i \(0.340046\pi\)
\(954\) 0 0
\(955\) −22.7585 −0.736447
\(956\) 0 0
\(957\) 2.70850 0.0875534
\(958\) 0 0
\(959\) 2.48116 0.0801207
\(960\) 0 0
\(961\) 55.8931 1.80300
\(962\) 0 0
\(963\) 25.5107 0.822070
\(964\) 0 0
\(965\) −5.74178 −0.184835
\(966\) 0 0
\(967\) 27.1425 0.872843 0.436421 0.899742i \(-0.356246\pi\)
0.436421 + 0.899742i \(0.356246\pi\)
\(968\) 0 0
\(969\) −29.1619 −0.936816
\(970\) 0 0
\(971\) 23.5397 0.755423 0.377712 0.925923i \(-0.376711\pi\)
0.377712 + 0.925923i \(0.376711\pi\)
\(972\) 0 0
\(973\) −14.3192 −0.459053
\(974\) 0 0
\(975\) 2.06439 0.0661135
\(976\) 0 0
\(977\) 22.1425 0.708402 0.354201 0.935169i \(-0.384753\pi\)
0.354201 + 0.935169i \(0.384753\pi\)
\(978\) 0 0
\(979\) 1.94288 0.0620948
\(980\) 0 0
\(981\) 18.7712 0.599317
\(982\) 0 0
\(983\) −18.7248 −0.597229 −0.298615 0.954374i \(-0.596525\pi\)
−0.298615 + 0.954374i \(0.596525\pi\)
\(984\) 0 0
\(985\) 6.30312 0.200834
\(986\) 0 0
\(987\) −4.08673 −0.130082
\(988\) 0 0
\(989\) −19.9760 −0.635198
\(990\) 0 0
\(991\) −24.4591 −0.776968 −0.388484 0.921456i \(-0.627001\pi\)
−0.388484 + 0.921456i \(0.627001\pi\)
\(992\) 0 0
\(993\) 5.99756 0.190327
\(994\) 0 0
\(995\) −15.4182 −0.488790
\(996\) 0 0
\(997\) 53.9538 1.70873 0.854366 0.519671i \(-0.173945\pi\)
0.854366 + 0.519671i \(0.173945\pi\)
\(998\) 0 0
\(999\) −22.3085 −0.705810
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 8008.2.a.z.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.z.1.7 15 1.1 even 1 trivial