Properties

Label 7360.2.a.be.1.2
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7360,2,Mod(1,7360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7360.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.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: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3680)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.64575 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.64575 q^{7} -2.00000 q^{9} +1.64575 q^{11} -2.64575 q^{13} +1.00000 q^{15} -7.29150 q^{17} +5.64575 q^{19} -1.64575 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +3.00000 q^{29} -0.645751 q^{31} -1.64575 q^{33} -1.64575 q^{35} +9.64575 q^{37} +2.64575 q^{39} +4.29150 q^{41} +1.64575 q^{43} +2.00000 q^{45} -3.00000 q^{47} -4.29150 q^{49} +7.29150 q^{51} -5.29150 q^{53} -1.64575 q^{55} -5.64575 q^{57} +5.29150 q^{59} +8.35425 q^{61} -3.29150 q^{63} +2.64575 q^{65} -6.93725 q^{67} +1.00000 q^{69} -1.35425 q^{71} -4.64575 q^{73} -1.00000 q^{75} +2.70850 q^{77} -1.64575 q^{79} +1.00000 q^{81} -14.5830 q^{83} +7.29150 q^{85} -3.00000 q^{87} -1.64575 q^{89} -4.35425 q^{91} +0.645751 q^{93} -5.64575 q^{95} +2.93725 q^{97} -3.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9} - 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} - 2 q^{41} - 2 q^{43} + 4 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 6 q^{57} + 22 q^{61} + 4 q^{63} + 2 q^{67} + 2 q^{69} - 8 q^{71} - 4 q^{73} - 2 q^{75} + 16 q^{77} + 2 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 6 q^{87} + 2 q^{89} - 14 q^{91} - 4 q^{93} - 6 q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.64575 0.622036 0.311018 0.950404i \(-0.399330\pi\)
0.311018 + 0.950404i \(0.399330\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.64575 0.496213 0.248106 0.968733i \(-0.420192\pi\)
0.248106 + 0.968733i \(0.420192\pi\)
\(12\) 0 0
\(13\) −2.64575 −0.733799 −0.366900 0.930261i \(-0.619581\pi\)
−0.366900 + 0.930261i \(0.619581\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −7.29150 −1.76845 −0.884225 0.467062i \(-0.845312\pi\)
−0.884225 + 0.467062i \(0.845312\pi\)
\(18\) 0 0
\(19\) 5.64575 1.29522 0.647612 0.761970i \(-0.275768\pi\)
0.647612 + 0.761970i \(0.275768\pi\)
\(20\) 0 0
\(21\) −1.64575 −0.359132
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −0.645751 −0.115980 −0.0579902 0.998317i \(-0.518469\pi\)
−0.0579902 + 0.998317i \(0.518469\pi\)
\(32\) 0 0
\(33\) −1.64575 −0.286489
\(34\) 0 0
\(35\) −1.64575 −0.278183
\(36\) 0 0
\(37\) 9.64575 1.58575 0.792876 0.609383i \(-0.208583\pi\)
0.792876 + 0.609383i \(0.208583\pi\)
\(38\) 0 0
\(39\) 2.64575 0.423659
\(40\) 0 0
\(41\) 4.29150 0.670220 0.335110 0.942179i \(-0.391226\pi\)
0.335110 + 0.942179i \(0.391226\pi\)
\(42\) 0 0
\(43\) 1.64575 0.250975 0.125487 0.992095i \(-0.459951\pi\)
0.125487 + 0.992095i \(0.459951\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −4.29150 −0.613072
\(50\) 0 0
\(51\) 7.29150 1.02101
\(52\) 0 0
\(53\) −5.29150 −0.726844 −0.363422 0.931625i \(-0.618392\pi\)
−0.363422 + 0.931625i \(0.618392\pi\)
\(54\) 0 0
\(55\) −1.64575 −0.221913
\(56\) 0 0
\(57\) −5.64575 −0.747798
\(58\) 0 0
\(59\) 5.29150 0.688895 0.344447 0.938806i \(-0.388066\pi\)
0.344447 + 0.938806i \(0.388066\pi\)
\(60\) 0 0
\(61\) 8.35425 1.06965 0.534826 0.844962i \(-0.320377\pi\)
0.534826 + 0.844962i \(0.320377\pi\)
\(62\) 0 0
\(63\) −3.29150 −0.414690
\(64\) 0 0
\(65\) 2.64575 0.328165
\(66\) 0 0
\(67\) −6.93725 −0.847520 −0.423760 0.905774i \(-0.639290\pi\)
−0.423760 + 0.905774i \(0.639290\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.35425 −0.160720 −0.0803599 0.996766i \(-0.525607\pi\)
−0.0803599 + 0.996766i \(0.525607\pi\)
\(72\) 0 0
\(73\) −4.64575 −0.543744 −0.271872 0.962333i \(-0.587643\pi\)
−0.271872 + 0.962333i \(0.587643\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 2.70850 0.308662
\(78\) 0 0
\(79\) −1.64575 −0.185161 −0.0925807 0.995705i \(-0.529512\pi\)
−0.0925807 + 0.995705i \(0.529512\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.5830 −1.60069 −0.800346 0.599538i \(-0.795351\pi\)
−0.800346 + 0.599538i \(0.795351\pi\)
\(84\) 0 0
\(85\) 7.29150 0.790875
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −1.64575 −0.174449 −0.0872246 0.996189i \(-0.527800\pi\)
−0.0872246 + 0.996189i \(0.527800\pi\)
\(90\) 0 0
\(91\) −4.35425 −0.456449
\(92\) 0 0
\(93\) 0.645751 0.0669613
\(94\) 0 0
\(95\) −5.64575 −0.579242
\(96\) 0 0
\(97\) 2.93725 0.298233 0.149116 0.988820i \(-0.452357\pi\)
0.149116 + 0.988820i \(0.452357\pi\)
\(98\) 0 0
\(99\) −3.29150 −0.330808
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 7.29150 0.718453 0.359227 0.933250i \(-0.383041\pi\)
0.359227 + 0.933250i \(0.383041\pi\)
\(104\) 0 0
\(105\) 1.64575 0.160609
\(106\) 0 0
\(107\) −18.2288 −1.76224 −0.881120 0.472892i \(-0.843210\pi\)
−0.881120 + 0.472892i \(0.843210\pi\)
\(108\) 0 0
\(109\) 2.35425 0.225496 0.112748 0.993624i \(-0.464035\pi\)
0.112748 + 0.993624i \(0.464035\pi\)
\(110\) 0 0
\(111\) −9.64575 −0.915534
\(112\) 0 0
\(113\) −0.354249 −0.0333249 −0.0166625 0.999861i \(-0.505304\pi\)
−0.0166625 + 0.999861i \(0.505304\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 5.29150 0.489200
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −8.29150 −0.753773
\(122\) 0 0
\(123\) −4.29150 −0.386952
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) −1.64575 −0.144900
\(130\) 0 0
\(131\) −5.35425 −0.467803 −0.233901 0.972260i \(-0.575149\pi\)
−0.233901 + 0.972260i \(0.575149\pi\)
\(132\) 0 0
\(133\) 9.29150 0.805675
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 9.87451 0.843636 0.421818 0.906680i \(-0.361392\pi\)
0.421818 + 0.906680i \(0.361392\pi\)
\(138\) 0 0
\(139\) −5.93725 −0.503591 −0.251796 0.967780i \(-0.581021\pi\)
−0.251796 + 0.967780i \(0.581021\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) −4.35425 −0.364121
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 4.29150 0.353957
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −17.2288 −1.40206 −0.701028 0.713134i \(-0.747275\pi\)
−0.701028 + 0.713134i \(0.747275\pi\)
\(152\) 0 0
\(153\) 14.5830 1.17897
\(154\) 0 0
\(155\) 0.645751 0.0518680
\(156\) 0 0
\(157\) 18.2288 1.45481 0.727407 0.686207i \(-0.240725\pi\)
0.727407 + 0.686207i \(0.240725\pi\)
\(158\) 0 0
\(159\) 5.29150 0.419643
\(160\) 0 0
\(161\) −1.64575 −0.129703
\(162\) 0 0
\(163\) 6.87451 0.538453 0.269227 0.963077i \(-0.413232\pi\)
0.269227 + 0.963077i \(0.413232\pi\)
\(164\) 0 0
\(165\) 1.64575 0.128122
\(166\) 0 0
\(167\) 4.58301 0.354644 0.177322 0.984153i \(-0.443257\pi\)
0.177322 + 0.984153i \(0.443257\pi\)
\(168\) 0 0
\(169\) −6.00000 −0.461538
\(170\) 0 0
\(171\) −11.2915 −0.863483
\(172\) 0 0
\(173\) −12.5830 −0.956668 −0.478334 0.878178i \(-0.658759\pi\)
−0.478334 + 0.878178i \(0.658759\pi\)
\(174\) 0 0
\(175\) 1.64575 0.124407
\(176\) 0 0
\(177\) −5.29150 −0.397734
\(178\) 0 0
\(179\) 7.35425 0.549682 0.274841 0.961490i \(-0.411375\pi\)
0.274841 + 0.961490i \(0.411375\pi\)
\(180\) 0 0
\(181\) 2.93725 0.218324 0.109162 0.994024i \(-0.465183\pi\)
0.109162 + 0.994024i \(0.465183\pi\)
\(182\) 0 0
\(183\) −8.35425 −0.617564
\(184\) 0 0
\(185\) −9.64575 −0.709170
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 8.22876 0.598554
\(190\) 0 0
\(191\) 19.8745 1.43807 0.719034 0.694974i \(-0.244584\pi\)
0.719034 + 0.694974i \(0.244584\pi\)
\(192\) 0 0
\(193\) −21.2288 −1.52808 −0.764040 0.645169i \(-0.776787\pi\)
−0.764040 + 0.645169i \(0.776787\pi\)
\(194\) 0 0
\(195\) −2.64575 −0.189466
\(196\) 0 0
\(197\) 0.0627461 0.00447047 0.00223524 0.999998i \(-0.499289\pi\)
0.00223524 + 0.999998i \(0.499289\pi\)
\(198\) 0 0
\(199\) 5.64575 0.400217 0.200108 0.979774i \(-0.435871\pi\)
0.200108 + 0.979774i \(0.435871\pi\)
\(200\) 0 0
\(201\) 6.93725 0.489316
\(202\) 0 0
\(203\) 4.93725 0.346527
\(204\) 0 0
\(205\) −4.29150 −0.299732
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 9.29150 0.642707
\(210\) 0 0
\(211\) 9.87451 0.679789 0.339895 0.940464i \(-0.389609\pi\)
0.339895 + 0.940464i \(0.389609\pi\)
\(212\) 0 0
\(213\) 1.35425 0.0927916
\(214\) 0 0
\(215\) −1.64575 −0.112239
\(216\) 0 0
\(217\) −1.06275 −0.0721439
\(218\) 0 0
\(219\) 4.64575 0.313931
\(220\) 0 0
\(221\) 19.2915 1.29769
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −16.9373 −1.12417 −0.562083 0.827081i \(-0.690000\pi\)
−0.562083 + 0.827081i \(0.690000\pi\)
\(228\) 0 0
\(229\) −0.937254 −0.0619355 −0.0309677 0.999520i \(-0.509859\pi\)
−0.0309677 + 0.999520i \(0.509859\pi\)
\(230\) 0 0
\(231\) −2.70850 −0.178206
\(232\) 0 0
\(233\) −5.22876 −0.342547 −0.171274 0.985224i \(-0.554788\pi\)
−0.171274 + 0.985224i \(0.554788\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 0 0
\(237\) 1.64575 0.106903
\(238\) 0 0
\(239\) 5.35425 0.346338 0.173169 0.984892i \(-0.444599\pi\)
0.173169 + 0.984892i \(0.444599\pi\)
\(240\) 0 0
\(241\) −15.8745 −1.02257 −0.511283 0.859412i \(-0.670830\pi\)
−0.511283 + 0.859412i \(0.670830\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 4.29150 0.274174
\(246\) 0 0
\(247\) −14.9373 −0.950435
\(248\) 0 0
\(249\) 14.5830 0.924160
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −1.64575 −0.103467
\(254\) 0 0
\(255\) −7.29150 −0.456612
\(256\) 0 0
\(257\) −24.6458 −1.53736 −0.768680 0.639634i \(-0.779086\pi\)
−0.768680 + 0.639634i \(0.779086\pi\)
\(258\) 0 0
\(259\) 15.8745 0.986394
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −26.2288 −1.61733 −0.808667 0.588266i \(-0.799811\pi\)
−0.808667 + 0.588266i \(0.799811\pi\)
\(264\) 0 0
\(265\) 5.29150 0.325054
\(266\) 0 0
\(267\) 1.64575 0.100718
\(268\) 0 0
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) 4.70850 0.286021 0.143010 0.989721i \(-0.454322\pi\)
0.143010 + 0.989721i \(0.454322\pi\)
\(272\) 0 0
\(273\) 4.35425 0.263531
\(274\) 0 0
\(275\) 1.64575 0.0992425
\(276\) 0 0
\(277\) 4.64575 0.279136 0.139568 0.990212i \(-0.455429\pi\)
0.139568 + 0.990212i \(0.455429\pi\)
\(278\) 0 0
\(279\) 1.29150 0.0773202
\(280\) 0 0
\(281\) −8.22876 −0.490886 −0.245443 0.969411i \(-0.578933\pi\)
−0.245443 + 0.969411i \(0.578933\pi\)
\(282\) 0 0
\(283\) −6.70850 −0.398779 −0.199389 0.979920i \(-0.563896\pi\)
−0.199389 + 0.979920i \(0.563896\pi\)
\(284\) 0 0
\(285\) 5.64575 0.334425
\(286\) 0 0
\(287\) 7.06275 0.416901
\(288\) 0 0
\(289\) 36.1660 2.12741
\(290\) 0 0
\(291\) −2.93725 −0.172185
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −5.29150 −0.308083
\(296\) 0 0
\(297\) 8.22876 0.477481
\(298\) 0 0
\(299\) 2.64575 0.153008
\(300\) 0 0
\(301\) 2.70850 0.156115
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) −8.35425 −0.478363
\(306\) 0 0
\(307\) 23.8745 1.36259 0.681295 0.732009i \(-0.261417\pi\)
0.681295 + 0.732009i \(0.261417\pi\)
\(308\) 0 0
\(309\) −7.29150 −0.414799
\(310\) 0 0
\(311\) −10.0627 −0.570606 −0.285303 0.958437i \(-0.592094\pi\)
−0.285303 + 0.958437i \(0.592094\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 3.29150 0.185455
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 4.93725 0.276433
\(320\) 0 0
\(321\) 18.2288 1.01743
\(322\) 0 0
\(323\) −41.1660 −2.29054
\(324\) 0 0
\(325\) −2.64575 −0.146760
\(326\) 0 0
\(327\) −2.35425 −0.130190
\(328\) 0 0
\(329\) −4.93725 −0.272200
\(330\) 0 0
\(331\) −33.9373 −1.86536 −0.932680 0.360705i \(-0.882536\pi\)
−0.932680 + 0.360705i \(0.882536\pi\)
\(332\) 0 0
\(333\) −19.2915 −1.05717
\(334\) 0 0
\(335\) 6.93725 0.379023
\(336\) 0 0
\(337\) −1.77124 −0.0964858 −0.0482429 0.998836i \(-0.515362\pi\)
−0.0482429 + 0.998836i \(0.515362\pi\)
\(338\) 0 0
\(339\) 0.354249 0.0192401
\(340\) 0 0
\(341\) −1.06275 −0.0575509
\(342\) 0 0
\(343\) −18.5830 −1.00339
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 8.58301 0.460760 0.230380 0.973101i \(-0.426003\pi\)
0.230380 + 0.973101i \(0.426003\pi\)
\(348\) 0 0
\(349\) −18.1660 −0.972404 −0.486202 0.873846i \(-0.661618\pi\)
−0.486202 + 0.873846i \(0.661618\pi\)
\(350\) 0 0
\(351\) −13.2288 −0.706099
\(352\) 0 0
\(353\) −28.6458 −1.52466 −0.762330 0.647189i \(-0.775945\pi\)
−0.762330 + 0.647189i \(0.775945\pi\)
\(354\) 0 0
\(355\) 1.35425 0.0718761
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) −12.2288 −0.645409 −0.322705 0.946500i \(-0.604592\pi\)
−0.322705 + 0.946500i \(0.604592\pi\)
\(360\) 0 0
\(361\) 12.8745 0.677606
\(362\) 0 0
\(363\) 8.29150 0.435191
\(364\) 0 0
\(365\) 4.64575 0.243170
\(366\) 0 0
\(367\) −25.8745 −1.35064 −0.675319 0.737526i \(-0.735994\pi\)
−0.675319 + 0.737526i \(0.735994\pi\)
\(368\) 0 0
\(369\) −8.58301 −0.446813
\(370\) 0 0
\(371\) −8.70850 −0.452123
\(372\) 0 0
\(373\) 11.1660 0.578154 0.289077 0.957306i \(-0.406652\pi\)
0.289077 + 0.957306i \(0.406652\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −7.93725 −0.408789
\(378\) 0 0
\(379\) −21.8745 −1.12362 −0.561809 0.827267i \(-0.689894\pi\)
−0.561809 + 0.827267i \(0.689894\pi\)
\(380\) 0 0
\(381\) −1.00000 −0.0512316
\(382\) 0 0
\(383\) 16.2288 0.829251 0.414625 0.909992i \(-0.363913\pi\)
0.414625 + 0.909992i \(0.363913\pi\)
\(384\) 0 0
\(385\) −2.70850 −0.138038
\(386\) 0 0
\(387\) −3.29150 −0.167316
\(388\) 0 0
\(389\) −7.29150 −0.369694 −0.184847 0.982767i \(-0.559179\pi\)
−0.184847 + 0.982767i \(0.559179\pi\)
\(390\) 0 0
\(391\) 7.29150 0.368747
\(392\) 0 0
\(393\) 5.35425 0.270086
\(394\) 0 0
\(395\) 1.64575 0.0828067
\(396\) 0 0
\(397\) 15.9373 0.799868 0.399934 0.916544i \(-0.369033\pi\)
0.399934 + 0.916544i \(0.369033\pi\)
\(398\) 0 0
\(399\) −9.29150 −0.465157
\(400\) 0 0
\(401\) −37.7490 −1.88510 −0.942548 0.334071i \(-0.891577\pi\)
−0.942548 + 0.334071i \(0.891577\pi\)
\(402\) 0 0
\(403\) 1.70850 0.0851063
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 15.8745 0.786870
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) −9.87451 −0.487074
\(412\) 0 0
\(413\) 8.70850 0.428517
\(414\) 0 0
\(415\) 14.5830 0.715852
\(416\) 0 0
\(417\) 5.93725 0.290749
\(418\) 0 0
\(419\) 30.9373 1.51138 0.755692 0.654927i \(-0.227301\pi\)
0.755692 + 0.654927i \(0.227301\pi\)
\(420\) 0 0
\(421\) −4.22876 −0.206097 −0.103048 0.994676i \(-0.532860\pi\)
−0.103048 + 0.994676i \(0.532860\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −7.29150 −0.353690
\(426\) 0 0
\(427\) 13.7490 0.665362
\(428\) 0 0
\(429\) 4.35425 0.210225
\(430\) 0 0
\(431\) −27.6458 −1.33165 −0.665824 0.746108i \(-0.731920\pi\)
−0.665824 + 0.746108i \(0.731920\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) −5.64575 −0.270073
\(438\) 0 0
\(439\) −14.6458 −0.699004 −0.349502 0.936936i \(-0.613649\pi\)
−0.349502 + 0.936936i \(0.613649\pi\)
\(440\) 0 0
\(441\) 8.58301 0.408715
\(442\) 0 0
\(443\) −32.8745 −1.56192 −0.780958 0.624584i \(-0.785268\pi\)
−0.780958 + 0.624584i \(0.785268\pi\)
\(444\) 0 0
\(445\) 1.64575 0.0780161
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 9.41699 0.444415 0.222208 0.974999i \(-0.428674\pi\)
0.222208 + 0.974999i \(0.428674\pi\)
\(450\) 0 0
\(451\) 7.06275 0.332572
\(452\) 0 0
\(453\) 17.2288 0.809478
\(454\) 0 0
\(455\) 4.35425 0.204130
\(456\) 0 0
\(457\) −27.7490 −1.29804 −0.649022 0.760770i \(-0.724822\pi\)
−0.649022 + 0.760770i \(0.724822\pi\)
\(458\) 0 0
\(459\) −36.4575 −1.70169
\(460\) 0 0
\(461\) −19.7085 −0.917916 −0.458958 0.888458i \(-0.651777\pi\)
−0.458958 + 0.888458i \(0.651777\pi\)
\(462\) 0 0
\(463\) 18.4575 0.857793 0.428897 0.903354i \(-0.358902\pi\)
0.428897 + 0.903354i \(0.358902\pi\)
\(464\) 0 0
\(465\) −0.645751 −0.0299460
\(466\) 0 0
\(467\) 4.58301 0.212076 0.106038 0.994362i \(-0.466183\pi\)
0.106038 + 0.994362i \(0.466183\pi\)
\(468\) 0 0
\(469\) −11.4170 −0.527188
\(470\) 0 0
\(471\) −18.2288 −0.839937
\(472\) 0 0
\(473\) 2.70850 0.124537
\(474\) 0 0
\(475\) 5.64575 0.259045
\(476\) 0 0
\(477\) 10.5830 0.484563
\(478\) 0 0
\(479\) 7.29150 0.333157 0.166579 0.986028i \(-0.446728\pi\)
0.166579 + 0.986028i \(0.446728\pi\)
\(480\) 0 0
\(481\) −25.5203 −1.16362
\(482\) 0 0
\(483\) 1.64575 0.0748843
\(484\) 0 0
\(485\) −2.93725 −0.133374
\(486\) 0 0
\(487\) −15.7085 −0.711820 −0.355910 0.934520i \(-0.615829\pi\)
−0.355910 + 0.934520i \(0.615829\pi\)
\(488\) 0 0
\(489\) −6.87451 −0.310876
\(490\) 0 0
\(491\) 17.3542 0.783186 0.391593 0.920138i \(-0.371924\pi\)
0.391593 + 0.920138i \(0.371924\pi\)
\(492\) 0 0
\(493\) −21.8745 −0.985178
\(494\) 0 0
\(495\) 3.29150 0.147942
\(496\) 0 0
\(497\) −2.22876 −0.0999734
\(498\) 0 0
\(499\) 2.06275 0.0923412 0.0461706 0.998934i \(-0.485298\pi\)
0.0461706 + 0.998934i \(0.485298\pi\)
\(500\) 0 0
\(501\) −4.58301 −0.204754
\(502\) 0 0
\(503\) −42.4575 −1.89309 −0.946543 0.322576i \(-0.895451\pi\)
−0.946543 + 0.322576i \(0.895451\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 6.00000 0.266469
\(508\) 0 0
\(509\) 20.8745 0.925246 0.462623 0.886555i \(-0.346908\pi\)
0.462623 + 0.886555i \(0.346908\pi\)
\(510\) 0 0
\(511\) −7.64575 −0.338228
\(512\) 0 0
\(513\) 28.2288 1.24633
\(514\) 0 0
\(515\) −7.29150 −0.321302
\(516\) 0 0
\(517\) −4.93725 −0.217140
\(518\) 0 0
\(519\) 12.5830 0.552333
\(520\) 0 0
\(521\) 40.4575 1.77248 0.886238 0.463230i \(-0.153310\pi\)
0.886238 + 0.463230i \(0.153310\pi\)
\(522\) 0 0
\(523\) −22.8118 −0.997489 −0.498744 0.866749i \(-0.666205\pi\)
−0.498744 + 0.866749i \(0.666205\pi\)
\(524\) 0 0
\(525\) −1.64575 −0.0718265
\(526\) 0 0
\(527\) 4.70850 0.205105
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.5830 −0.459263
\(532\) 0 0
\(533\) −11.3542 −0.491807
\(534\) 0 0
\(535\) 18.2288 0.788098
\(536\) 0 0
\(537\) −7.35425 −0.317359
\(538\) 0 0
\(539\) −7.06275 −0.304214
\(540\) 0 0
\(541\) −40.0405 −1.72148 −0.860738 0.509048i \(-0.829998\pi\)
−0.860738 + 0.509048i \(0.829998\pi\)
\(542\) 0 0
\(543\) −2.93725 −0.126050
\(544\) 0 0
\(545\) −2.35425 −0.100845
\(546\) 0 0
\(547\) −16.1660 −0.691209 −0.345604 0.938380i \(-0.612326\pi\)
−0.345604 + 0.938380i \(0.612326\pi\)
\(548\) 0 0
\(549\) −16.7085 −0.713101
\(550\) 0 0
\(551\) 16.9373 0.721551
\(552\) 0 0
\(553\) −2.70850 −0.115177
\(554\) 0 0
\(555\) 9.64575 0.409439
\(556\) 0 0
\(557\) 25.5203 1.08133 0.540664 0.841239i \(-0.318173\pi\)
0.540664 + 0.841239i \(0.318173\pi\)
\(558\) 0 0
\(559\) −4.35425 −0.184165
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −42.5830 −1.79466 −0.897330 0.441361i \(-0.854496\pi\)
−0.897330 + 0.441361i \(0.854496\pi\)
\(564\) 0 0
\(565\) 0.354249 0.0149034
\(566\) 0 0
\(567\) 1.64575 0.0691151
\(568\) 0 0
\(569\) 11.4170 0.478625 0.239313 0.970943i \(-0.423078\pi\)
0.239313 + 0.970943i \(0.423078\pi\)
\(570\) 0 0
\(571\) −24.4575 −1.02352 −0.511758 0.859130i \(-0.671005\pi\)
−0.511758 + 0.859130i \(0.671005\pi\)
\(572\) 0 0
\(573\) −19.8745 −0.830269
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −21.3542 −0.888989 −0.444495 0.895782i \(-0.646617\pi\)
−0.444495 + 0.895782i \(0.646617\pi\)
\(578\) 0 0
\(579\) 21.2288 0.882237
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) −8.70850 −0.360669
\(584\) 0 0
\(585\) −5.29150 −0.218777
\(586\) 0 0
\(587\) 20.8745 0.861583 0.430792 0.902451i \(-0.358234\pi\)
0.430792 + 0.902451i \(0.358234\pi\)
\(588\) 0 0
\(589\) −3.64575 −0.150221
\(590\) 0 0
\(591\) −0.0627461 −0.00258103
\(592\) 0 0
\(593\) 19.7490 0.810995 0.405497 0.914096i \(-0.367098\pi\)
0.405497 + 0.914096i \(0.367098\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) −5.64575 −0.231065
\(598\) 0 0
\(599\) −39.8745 −1.62923 −0.814614 0.580003i \(-0.803051\pi\)
−0.814614 + 0.580003i \(0.803051\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) 13.8745 0.565014
\(604\) 0 0
\(605\) 8.29150 0.337098
\(606\) 0 0
\(607\) 18.7085 0.759354 0.379677 0.925119i \(-0.376035\pi\)
0.379677 + 0.925119i \(0.376035\pi\)
\(608\) 0 0
\(609\) −4.93725 −0.200068
\(610\) 0 0
\(611\) 7.93725 0.321107
\(612\) 0 0
\(613\) −47.2915 −1.91009 −0.955043 0.296468i \(-0.904191\pi\)
−0.955043 + 0.296468i \(0.904191\pi\)
\(614\) 0 0
\(615\) 4.29150 0.173050
\(616\) 0 0
\(617\) 36.9373 1.48704 0.743519 0.668715i \(-0.233155\pi\)
0.743519 + 0.668715i \(0.233155\pi\)
\(618\) 0 0
\(619\) 15.2915 0.614617 0.307309 0.951610i \(-0.400572\pi\)
0.307309 + 0.951610i \(0.400572\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −2.70850 −0.108514
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.29150 −0.371067
\(628\) 0 0
\(629\) −70.3320 −2.80432
\(630\) 0 0
\(631\) 21.5203 0.856708 0.428354 0.903611i \(-0.359094\pi\)
0.428354 + 0.903611i \(0.359094\pi\)
\(632\) 0 0
\(633\) −9.87451 −0.392476
\(634\) 0 0
\(635\) −1.00000 −0.0396838
\(636\) 0 0
\(637\) 11.3542 0.449872
\(638\) 0 0
\(639\) 2.70850 0.107147
\(640\) 0 0
\(641\) 40.9373 1.61692 0.808462 0.588548i \(-0.200300\pi\)
0.808462 + 0.588548i \(0.200300\pi\)
\(642\) 0 0
\(643\) 17.2915 0.681910 0.340955 0.940080i \(-0.389250\pi\)
0.340955 + 0.940080i \(0.389250\pi\)
\(644\) 0 0
\(645\) 1.64575 0.0648014
\(646\) 0 0
\(647\) −22.2915 −0.876369 −0.438185 0.898885i \(-0.644378\pi\)
−0.438185 + 0.898885i \(0.644378\pi\)
\(648\) 0 0
\(649\) 8.70850 0.341838
\(650\) 0 0
\(651\) 1.06275 0.0416523
\(652\) 0 0
\(653\) 0.520259 0.0203593 0.0101797 0.999948i \(-0.496760\pi\)
0.0101797 + 0.999948i \(0.496760\pi\)
\(654\) 0 0
\(655\) 5.35425 0.209208
\(656\) 0 0
\(657\) 9.29150 0.362496
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 7.16601 0.278726 0.139363 0.990241i \(-0.455495\pi\)
0.139363 + 0.990241i \(0.455495\pi\)
\(662\) 0 0
\(663\) −19.2915 −0.749220
\(664\) 0 0
\(665\) −9.29150 −0.360309
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 13.7490 0.530775
\(672\) 0 0
\(673\) 13.1033 0.505094 0.252547 0.967585i \(-0.418732\pi\)
0.252547 + 0.967585i \(0.418732\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 4.83399 0.185511
\(680\) 0 0
\(681\) 16.9373 0.649037
\(682\) 0 0
\(683\) 30.0405 1.14947 0.574734 0.818340i \(-0.305105\pi\)
0.574734 + 0.818340i \(0.305105\pi\)
\(684\) 0 0
\(685\) −9.87451 −0.377286
\(686\) 0 0
\(687\) 0.937254 0.0357585
\(688\) 0 0
\(689\) 14.0000 0.533358
\(690\) 0 0
\(691\) 13.8745 0.527811 0.263906 0.964549i \(-0.414989\pi\)
0.263906 + 0.964549i \(0.414989\pi\)
\(692\) 0 0
\(693\) −5.41699 −0.205775
\(694\) 0 0
\(695\) 5.93725 0.225213
\(696\) 0 0
\(697\) −31.2915 −1.18525
\(698\) 0 0
\(699\) 5.22876 0.197770
\(700\) 0 0
\(701\) −11.0627 −0.417834 −0.208917 0.977933i \(-0.566994\pi\)
−0.208917 + 0.977933i \(0.566994\pi\)
\(702\) 0 0
\(703\) 54.4575 2.05390
\(704\) 0 0
\(705\) −3.00000 −0.112987
\(706\) 0 0
\(707\) 13.1660 0.495159
\(708\) 0 0
\(709\) −34.3542 −1.29020 −0.645100 0.764098i \(-0.723184\pi\)
−0.645100 + 0.764098i \(0.723184\pi\)
\(710\) 0 0
\(711\) 3.29150 0.123441
\(712\) 0 0
\(713\) 0.645751 0.0241836
\(714\) 0 0
\(715\) 4.35425 0.162840
\(716\) 0 0
\(717\) −5.35425 −0.199958
\(718\) 0 0
\(719\) 14.1255 0.526792 0.263396 0.964688i \(-0.415157\pi\)
0.263396 + 0.964688i \(0.415157\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 15.8745 0.590379
\(724\) 0 0
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) −18.4575 −0.684551 −0.342276 0.939600i \(-0.611198\pi\)
−0.342276 + 0.939600i \(0.611198\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 10.5830 0.390892 0.195446 0.980714i \(-0.437385\pi\)
0.195446 + 0.980714i \(0.437385\pi\)
\(734\) 0 0
\(735\) −4.29150 −0.158294
\(736\) 0 0
\(737\) −11.4170 −0.420550
\(738\) 0 0
\(739\) 8.06275 0.296593 0.148296 0.988943i \(-0.452621\pi\)
0.148296 + 0.988943i \(0.452621\pi\)
\(740\) 0 0
\(741\) 14.9373 0.548734
\(742\) 0 0
\(743\) −43.2915 −1.58821 −0.794106 0.607780i \(-0.792060\pi\)
−0.794106 + 0.607780i \(0.792060\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 29.1660 1.06713
\(748\) 0 0
\(749\) −30.0000 −1.09618
\(750\) 0 0
\(751\) −53.9778 −1.96968 −0.984838 0.173474i \(-0.944501\pi\)
−0.984838 + 0.173474i \(0.944501\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 17.2288 0.627019
\(756\) 0 0
\(757\) 53.6458 1.94979 0.974894 0.222669i \(-0.0714771\pi\)
0.974894 + 0.222669i \(0.0714771\pi\)
\(758\) 0 0
\(759\) 1.64575 0.0597370
\(760\) 0 0
\(761\) 16.1660 0.586017 0.293009 0.956110i \(-0.405344\pi\)
0.293009 + 0.956110i \(0.405344\pi\)
\(762\) 0 0
\(763\) 3.87451 0.140267
\(764\) 0 0
\(765\) −14.5830 −0.527250
\(766\) 0 0
\(767\) −14.0000 −0.505511
\(768\) 0 0
\(769\) 15.1660 0.546900 0.273450 0.961886i \(-0.411835\pi\)
0.273450 + 0.961886i \(0.411835\pi\)
\(770\) 0 0
\(771\) 24.6458 0.887595
\(772\) 0 0
\(773\) −38.2288 −1.37499 −0.687496 0.726188i \(-0.741290\pi\)
−0.687496 + 0.726188i \(0.741290\pi\)
\(774\) 0 0
\(775\) −0.645751 −0.0231961
\(776\) 0 0
\(777\) −15.8745 −0.569495
\(778\) 0 0
\(779\) 24.2288 0.868085
\(780\) 0 0
\(781\) −2.22876 −0.0797512
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) −18.2288 −0.650612
\(786\) 0 0
\(787\) −10.5830 −0.377243 −0.188622 0.982050i \(-0.560402\pi\)
−0.188622 + 0.982050i \(0.560402\pi\)
\(788\) 0 0
\(789\) 26.2288 0.933768
\(790\) 0 0
\(791\) −0.583005 −0.0207293
\(792\) 0 0
\(793\) −22.1033 −0.784910
\(794\) 0 0
\(795\) −5.29150 −0.187670
\(796\) 0 0
\(797\) −32.2288 −1.14160 −0.570801 0.821089i \(-0.693367\pi\)
−0.570801 + 0.821089i \(0.693367\pi\)
\(798\) 0 0
\(799\) 21.8745 0.773864
\(800\) 0 0
\(801\) 3.29150 0.116300
\(802\) 0 0
\(803\) −7.64575 −0.269813
\(804\) 0 0
\(805\) 1.64575 0.0580051
\(806\) 0 0
\(807\) 5.00000 0.176008
\(808\) 0 0
\(809\) −43.7490 −1.53813 −0.769067 0.639168i \(-0.779279\pi\)
−0.769067 + 0.639168i \(0.779279\pi\)
\(810\) 0 0
\(811\) 13.2288 0.464524 0.232262 0.972653i \(-0.425387\pi\)
0.232262 + 0.972653i \(0.425387\pi\)
\(812\) 0 0
\(813\) −4.70850 −0.165134
\(814\) 0 0
\(815\) −6.87451 −0.240804
\(816\) 0 0
\(817\) 9.29150 0.325069
\(818\) 0 0
\(819\) 8.70850 0.304300
\(820\) 0 0
\(821\) 18.5830 0.648551 0.324276 0.945963i \(-0.394879\pi\)
0.324276 + 0.945963i \(0.394879\pi\)
\(822\) 0 0
\(823\) −25.5830 −0.891768 −0.445884 0.895091i \(-0.647111\pi\)
−0.445884 + 0.895091i \(0.647111\pi\)
\(824\) 0 0
\(825\) −1.64575 −0.0572977
\(826\) 0 0
\(827\) −52.4575 −1.82413 −0.912063 0.410050i \(-0.865511\pi\)
−0.912063 + 0.410050i \(0.865511\pi\)
\(828\) 0 0
\(829\) 35.0405 1.21701 0.608504 0.793551i \(-0.291770\pi\)
0.608504 + 0.793551i \(0.291770\pi\)
\(830\) 0 0
\(831\) −4.64575 −0.161159
\(832\) 0 0
\(833\) 31.2915 1.08419
\(834\) 0 0
\(835\) −4.58301 −0.158601
\(836\) 0 0
\(837\) −3.22876 −0.111602
\(838\) 0 0
\(839\) 9.06275 0.312881 0.156440 0.987687i \(-0.449998\pi\)
0.156440 + 0.987687i \(0.449998\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 8.22876 0.283413
\(844\) 0 0
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) −13.6458 −0.468874
\(848\) 0 0
\(849\) 6.70850 0.230235
\(850\) 0 0
\(851\) −9.64575 −0.330652
\(852\) 0 0
\(853\) 17.7490 0.607715 0.303857 0.952718i \(-0.401725\pi\)
0.303857 + 0.952718i \(0.401725\pi\)
\(854\) 0 0
\(855\) 11.2915 0.386161
\(856\) 0 0
\(857\) 22.0627 0.753649 0.376825 0.926285i \(-0.377016\pi\)
0.376825 + 0.926285i \(0.377016\pi\)
\(858\) 0 0
\(859\) 16.7712 0.572227 0.286114 0.958196i \(-0.407636\pi\)
0.286114 + 0.958196i \(0.407636\pi\)
\(860\) 0 0
\(861\) −7.06275 −0.240698
\(862\) 0 0
\(863\) −42.8745 −1.45947 −0.729733 0.683733i \(-0.760355\pi\)
−0.729733 + 0.683733i \(0.760355\pi\)
\(864\) 0 0
\(865\) 12.5830 0.427835
\(866\) 0 0
\(867\) −36.1660 −1.22826
\(868\) 0 0
\(869\) −2.70850 −0.0918795
\(870\) 0 0
\(871\) 18.3542 0.621910
\(872\) 0 0
\(873\) −5.87451 −0.198822
\(874\) 0 0
\(875\) −1.64575 −0.0556365
\(876\) 0 0
\(877\) 23.8745 0.806185 0.403092 0.915159i \(-0.367935\pi\)
0.403092 + 0.915159i \(0.367935\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.8745 1.27602 0.638012 0.770026i \(-0.279757\pi\)
0.638012 + 0.770026i \(0.279757\pi\)
\(882\) 0 0
\(883\) −4.58301 −0.154230 −0.0771152 0.997022i \(-0.524571\pi\)
−0.0771152 + 0.997022i \(0.524571\pi\)
\(884\) 0 0
\(885\) 5.29150 0.177872
\(886\) 0 0
\(887\) 26.0405 0.874355 0.437178 0.899375i \(-0.355978\pi\)
0.437178 + 0.899375i \(0.355978\pi\)
\(888\) 0 0
\(889\) 1.64575 0.0551967
\(890\) 0 0
\(891\) 1.64575 0.0551347
\(892\) 0 0
\(893\) −16.9373 −0.566784
\(894\) 0 0
\(895\) −7.35425 −0.245825
\(896\) 0 0
\(897\) −2.64575 −0.0883391
\(898\) 0 0
\(899\) −1.93725 −0.0646110
\(900\) 0 0
\(901\) 38.5830 1.28539
\(902\) 0 0
\(903\) −2.70850 −0.0901331
\(904\) 0 0
\(905\) −2.93725 −0.0976376
\(906\) 0 0
\(907\) −47.3948 −1.57372 −0.786859 0.617133i \(-0.788294\pi\)
−0.786859 + 0.617133i \(0.788294\pi\)
\(908\) 0 0
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) 23.5203 0.779261 0.389630 0.920971i \(-0.372603\pi\)
0.389630 + 0.920971i \(0.372603\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 0 0
\(915\) 8.35425 0.276183
\(916\) 0 0
\(917\) −8.81176 −0.290990
\(918\) 0 0
\(919\) 29.0627 0.958692 0.479346 0.877626i \(-0.340874\pi\)
0.479346 + 0.877626i \(0.340874\pi\)
\(920\) 0 0
\(921\) −23.8745 −0.786692
\(922\) 0 0
\(923\) 3.58301 0.117936
\(924\) 0 0
\(925\) 9.64575 0.317150
\(926\) 0 0
\(927\) −14.5830 −0.478969
\(928\) 0 0
\(929\) 25.4575 0.835234 0.417617 0.908623i \(-0.362865\pi\)
0.417617 + 0.908623i \(0.362865\pi\)
\(930\) 0 0
\(931\) −24.2288 −0.794065
\(932\) 0 0
\(933\) 10.0627 0.329440
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −12.9373 −0.422642 −0.211321 0.977417i \(-0.567776\pi\)
−0.211321 + 0.977417i \(0.567776\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.7490 0.643800 0.321900 0.946774i \(-0.395679\pi\)
0.321900 + 0.946774i \(0.395679\pi\)
\(942\) 0 0
\(943\) −4.29150 −0.139751
\(944\) 0 0
\(945\) −8.22876 −0.267681
\(946\) 0 0
\(947\) −15.4575 −0.502302 −0.251151 0.967948i \(-0.580809\pi\)
−0.251151 + 0.967948i \(0.580809\pi\)
\(948\) 0 0
\(949\) 12.2915 0.398999
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) 0 0
\(953\) 21.2915 0.689699 0.344850 0.938658i \(-0.387930\pi\)
0.344850 + 0.938658i \(0.387930\pi\)
\(954\) 0 0
\(955\) −19.8745 −0.643124
\(956\) 0 0
\(957\) −4.93725 −0.159599
\(958\) 0 0
\(959\) 16.2510 0.524772
\(960\) 0 0
\(961\) −30.5830 −0.986549
\(962\) 0 0
\(963\) 36.4575 1.17483
\(964\) 0 0
\(965\) 21.2288 0.683378
\(966\) 0 0
\(967\) −52.8745 −1.70033 −0.850165 0.526517i \(-0.823498\pi\)
−0.850165 + 0.526517i \(0.823498\pi\)
\(968\) 0 0
\(969\) 41.1660 1.32244
\(970\) 0 0
\(971\) 21.7490 0.697959 0.348980 0.937130i \(-0.386528\pi\)
0.348980 + 0.937130i \(0.386528\pi\)
\(972\) 0 0
\(973\) −9.77124 −0.313252
\(974\) 0 0
\(975\) 2.64575 0.0847319
\(976\) 0 0
\(977\) −21.6458 −0.692509 −0.346254 0.938141i \(-0.612547\pi\)
−0.346254 + 0.938141i \(0.612547\pi\)
\(978\) 0 0
\(979\) −2.70850 −0.0865640
\(980\) 0 0
\(981\) −4.70850 −0.150331
\(982\) 0 0
\(983\) −1.54249 −0.0491977 −0.0245988 0.999697i \(-0.507831\pi\)
−0.0245988 + 0.999697i \(0.507831\pi\)
\(984\) 0 0
\(985\) −0.0627461 −0.00199926
\(986\) 0 0
\(987\) 4.93725 0.157155
\(988\) 0 0
\(989\) −1.64575 −0.0523318
\(990\) 0 0
\(991\) 47.7490 1.51680 0.758399 0.651791i \(-0.225982\pi\)
0.758399 + 0.651791i \(0.225982\pi\)
\(992\) 0 0
\(993\) 33.9373 1.07697
\(994\) 0 0
\(995\) −5.64575 −0.178982
\(996\) 0 0
\(997\) −51.6235 −1.63493 −0.817467 0.575976i \(-0.804622\pi\)
−0.817467 + 0.575976i \(0.804622\pi\)
\(998\) 0 0
\(999\) 48.2288 1.52589
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 7360.2.a.be.1.2 2
4.3 odd 2 7360.2.a.br.1.1 2
8.3 odd 2 3680.2.a.m.1.1 2
8.5 even 2 3680.2.a.o.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3680.2.a.m.1.1 2 8.3 odd 2
3680.2.a.o.1.2 yes 2 8.5 even 2
7360.2.a.be.1.2 2 1.1 even 1 trivial
7360.2.a.br.1.1 2 4.3 odd 2