Properties

Label 912.3.o.c.721.4
Level $912$
Weight $3$
Character 912.721
Analytic conductor $24.850$
Analytic rank $0$
Dimension $6$
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] = [912,3,Mod(721,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.219615408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.4
Root \(0.500000 - 2.79345i\) of defining polynomial
Character \(\chi\) \(=\) 912.721
Dual form 912.3.o.c.721.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} -7.39180 q^{5} -3.28502 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} -7.39180 q^{5} -3.28502 q^{7} -3.00000 q^{9} +0.714984 q^{11} +25.4210i q^{13} -12.8030i q^{15} -26.0686 q^{17} +(15.7836 - 10.5773i) q^{19} -5.68981i q^{21} -8.46325 q^{23} +29.6386 q^{25} -5.19615i q^{27} +23.1292i q^{29} -18.8627i q^{31} +1.23839i q^{33} +24.2822 q^{35} -48.5501i q^{37} -44.0304 q^{39} -35.3808i q^{41} +24.5014 q^{43} +22.1754 q^{45} +46.0990 q^{47} -38.2087 q^{49} -45.1522i q^{51} -51.2119i q^{53} -5.28502 q^{55} +(18.3203 + 27.3380i) q^{57} -101.314i q^{59} +38.5721 q^{61} +9.85505 q^{63} -187.907i q^{65} -44.2837i q^{67} -14.6588i q^{69} +74.3410i q^{71} +112.200 q^{73} +51.3356i q^{75} -2.34873 q^{77} +125.500i q^{79} +9.00000 q^{81} +95.5672 q^{83} +192.694 q^{85} -40.0609 q^{87} -28.0827i q^{89} -83.5083i q^{91} +32.6711 q^{93} +(-116.669 + 78.1849i) q^{95} +55.7955i q^{97} -2.14495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) −7.39180 −1.47836 −0.739180 0.673508i \(-0.764787\pi\)
−0.739180 + 0.673508i \(0.764787\pi\)
\(6\) 0 0
\(7\) −3.28502 −0.469288 −0.234644 0.972081i \(-0.575392\pi\)
−0.234644 + 0.972081i \(0.575392\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 0.714984 0.0649986 0.0324993 0.999472i \(-0.489653\pi\)
0.0324993 + 0.999472i \(0.489653\pi\)
\(12\) 0 0
\(13\) 25.4210i 1.95546i 0.209866 + 0.977730i \(0.432697\pi\)
−0.209866 + 0.977730i \(0.567303\pi\)
\(14\) 0 0
\(15\) 12.8030i 0.853531i
\(16\) 0 0
\(17\) −26.0686 −1.53345 −0.766724 0.641977i \(-0.778114\pi\)
−0.766724 + 0.641977i \(0.778114\pi\)
\(18\) 0 0
\(19\) 15.7836 10.5773i 0.830715 0.556697i
\(20\) 0 0
\(21\) 5.68981i 0.270944i
\(22\) 0 0
\(23\) −8.46325 −0.367967 −0.183984 0.982929i \(-0.558899\pi\)
−0.183984 + 0.982929i \(0.558899\pi\)
\(24\) 0 0
\(25\) 29.6386 1.18555
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 23.1292i 0.797557i 0.917047 + 0.398778i \(0.130566\pi\)
−0.917047 + 0.398778i \(0.869434\pi\)
\(30\) 0 0
\(31\) 18.8627i 0.608473i −0.952596 0.304237i \(-0.901599\pi\)
0.952596 0.304237i \(-0.0984014\pi\)
\(32\) 0 0
\(33\) 1.23839i 0.0375269i
\(34\) 0 0
\(35\) 24.2822 0.693776
\(36\) 0 0
\(37\) 48.5501i 1.31217i −0.754689 0.656083i \(-0.772212\pi\)
0.754689 0.656083i \(-0.227788\pi\)
\(38\) 0 0
\(39\) −44.0304 −1.12899
\(40\) 0 0
\(41\) 35.3808i 0.862946i −0.902126 0.431473i \(-0.857994\pi\)
0.902126 0.431473i \(-0.142006\pi\)
\(42\) 0 0
\(43\) 24.5014 0.569801 0.284900 0.958557i \(-0.408040\pi\)
0.284900 + 0.958557i \(0.408040\pi\)
\(44\) 0 0
\(45\) 22.1754 0.492786
\(46\) 0 0
\(47\) 46.0990 0.980831 0.490415 0.871489i \(-0.336845\pi\)
0.490415 + 0.871489i \(0.336845\pi\)
\(48\) 0 0
\(49\) −38.2087 −0.779769
\(50\) 0 0
\(51\) 45.1522i 0.885336i
\(52\) 0 0
\(53\) 51.2119i 0.966262i −0.875548 0.483131i \(-0.839500\pi\)
0.875548 0.483131i \(-0.160500\pi\)
\(54\) 0 0
\(55\) −5.28502 −0.0960912
\(56\) 0 0
\(57\) 18.3203 + 27.3380i 0.321409 + 0.479614i
\(58\) 0 0
\(59\) 101.314i 1.71719i −0.512657 0.858594i \(-0.671339\pi\)
0.512657 0.858594i \(-0.328661\pi\)
\(60\) 0 0
\(61\) 38.5721 0.632329 0.316165 0.948704i \(-0.397605\pi\)
0.316165 + 0.948704i \(0.397605\pi\)
\(62\) 0 0
\(63\) 9.85505 0.156429
\(64\) 0 0
\(65\) 187.907i 2.89087i
\(66\) 0 0
\(67\) 44.2837i 0.660950i −0.943815 0.330475i \(-0.892791\pi\)
0.943815 0.330475i \(-0.107209\pi\)
\(68\) 0 0
\(69\) 14.6588i 0.212446i
\(70\) 0 0
\(71\) 74.3410i 1.04706i 0.852008 + 0.523528i \(0.175385\pi\)
−0.852008 + 0.523528i \(0.824615\pi\)
\(72\) 0 0
\(73\) 112.200 1.53699 0.768494 0.639857i \(-0.221006\pi\)
0.768494 + 0.639857i \(0.221006\pi\)
\(74\) 0 0
\(75\) 51.3356i 0.684475i
\(76\) 0 0
\(77\) −2.34873 −0.0305030
\(78\) 0 0
\(79\) 125.500i 1.58861i 0.607519 + 0.794305i \(0.292165\pi\)
−0.607519 + 0.794305i \(0.707835\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 95.5672 1.15141 0.575706 0.817657i \(-0.304727\pi\)
0.575706 + 0.817657i \(0.304727\pi\)
\(84\) 0 0
\(85\) 192.694 2.26699
\(86\) 0 0
\(87\) −40.0609 −0.460470
\(88\) 0 0
\(89\) 28.0827i 0.315536i −0.987476 0.157768i \(-0.949570\pi\)
0.987476 0.157768i \(-0.0504298\pi\)
\(90\) 0 0
\(91\) 83.5083i 0.917674i
\(92\) 0 0
\(93\) 32.6711 0.351302
\(94\) 0 0
\(95\) −116.669 + 78.1849i −1.22810 + 0.822999i
\(96\) 0 0
\(97\) 55.7955i 0.575212i 0.957749 + 0.287606i \(0.0928593\pi\)
−0.957749 + 0.287606i \(0.907141\pi\)
\(98\) 0 0
\(99\) −2.14495 −0.0216662
\(100\) 0 0
\(101\) −131.018 −1.29721 −0.648603 0.761127i \(-0.724646\pi\)
−0.648603 + 0.761127i \(0.724646\pi\)
\(102\) 0 0
\(103\) 74.6582i 0.724837i 0.932015 + 0.362418i \(0.118049\pi\)
−0.932015 + 0.362418i \(0.881951\pi\)
\(104\) 0 0
\(105\) 42.0579i 0.400552i
\(106\) 0 0
\(107\) 171.811i 1.60571i −0.596173 0.802856i \(-0.703313\pi\)
0.596173 0.802856i \(-0.296687\pi\)
\(108\) 0 0
\(109\) 34.9582i 0.320717i 0.987059 + 0.160359i \(0.0512651\pi\)
−0.987059 + 0.160359i \(0.948735\pi\)
\(110\) 0 0
\(111\) 84.0913 0.757579
\(112\) 0 0
\(113\) 42.8041i 0.378797i 0.981900 + 0.189399i \(0.0606538\pi\)
−0.981900 + 0.189399i \(0.939346\pi\)
\(114\) 0 0
\(115\) 62.5586 0.543988
\(116\) 0 0
\(117\) 76.2629i 0.651820i
\(118\) 0 0
\(119\) 85.6358 0.719628
\(120\) 0 0
\(121\) −120.489 −0.995775
\(122\) 0 0
\(123\) 61.2814 0.498222
\(124\) 0 0
\(125\) −34.2879 −0.274303
\(126\) 0 0
\(127\) 176.342i 1.38852i −0.719724 0.694260i \(-0.755732\pi\)
0.719724 0.694260i \(-0.244268\pi\)
\(128\) 0 0
\(129\) 42.4377i 0.328975i
\(130\) 0 0
\(131\) −202.114 −1.54285 −0.771427 0.636317i \(-0.780457\pi\)
−0.771427 + 0.636317i \(0.780457\pi\)
\(132\) 0 0
\(133\) −51.8493 + 34.7464i −0.389845 + 0.261251i
\(134\) 0 0
\(135\) 38.4089i 0.284510i
\(136\) 0 0
\(137\) −11.2891 −0.0824021 −0.0412011 0.999151i \(-0.513118\pi\)
−0.0412011 + 0.999151i \(0.513118\pi\)
\(138\) 0 0
\(139\) 56.6427 0.407502 0.203751 0.979023i \(-0.434687\pi\)
0.203751 + 0.979023i \(0.434687\pi\)
\(140\) 0 0
\(141\) 79.8459i 0.566283i
\(142\) 0 0
\(143\) 18.1756i 0.127102i
\(144\) 0 0
\(145\) 170.966i 1.17908i
\(146\) 0 0
\(147\) 66.1794i 0.450200i
\(148\) 0 0
\(149\) −5.96590 −0.0400396 −0.0200198 0.999800i \(-0.506373\pi\)
−0.0200198 + 0.999800i \(0.506373\pi\)
\(150\) 0 0
\(151\) 28.0300i 0.185629i 0.995683 + 0.0928146i \(0.0295864\pi\)
−0.995683 + 0.0928146i \(0.970414\pi\)
\(152\) 0 0
\(153\) 78.2058 0.511149
\(154\) 0 0
\(155\) 139.429i 0.899542i
\(156\) 0 0
\(157\) 141.317 0.900108 0.450054 0.893001i \(-0.351405\pi\)
0.450054 + 0.893001i \(0.351405\pi\)
\(158\) 0 0
\(159\) 88.7015 0.557871
\(160\) 0 0
\(161\) 27.8019 0.172683
\(162\) 0 0
\(163\) 181.550 1.11380 0.556902 0.830578i \(-0.311990\pi\)
0.556902 + 0.830578i \(0.311990\pi\)
\(164\) 0 0
\(165\) 9.15392i 0.0554783i
\(166\) 0 0
\(167\) 301.737i 1.80681i −0.428791 0.903404i \(-0.641060\pi\)
0.428791 0.903404i \(-0.358940\pi\)
\(168\) 0 0
\(169\) −477.226 −2.82382
\(170\) 0 0
\(171\) −47.3508 + 31.7318i −0.276905 + 0.185566i
\(172\) 0 0
\(173\) 195.469i 1.12988i 0.825132 + 0.564940i \(0.191101\pi\)
−0.825132 + 0.564940i \(0.808899\pi\)
\(174\) 0 0
\(175\) −97.3634 −0.556362
\(176\) 0 0
\(177\) 175.481 0.991418
\(178\) 0 0
\(179\) 91.2818i 0.509954i −0.966947 0.254977i \(-0.917932\pi\)
0.966947 0.254977i \(-0.0820679\pi\)
\(180\) 0 0
\(181\) 15.7783i 0.0871732i −0.999050 0.0435866i \(-0.986122\pi\)
0.999050 0.0435866i \(-0.0138784\pi\)
\(182\) 0 0
\(183\) 66.8088i 0.365075i
\(184\) 0 0
\(185\) 358.873i 1.93985i
\(186\) 0 0
\(187\) −18.6386 −0.0996719
\(188\) 0 0
\(189\) 17.0694i 0.0903145i
\(190\) 0 0
\(191\) −193.939 −1.01539 −0.507694 0.861538i \(-0.669502\pi\)
−0.507694 + 0.861538i \(0.669502\pi\)
\(192\) 0 0
\(193\) 150.941i 0.782077i −0.920374 0.391039i \(-0.872116\pi\)
0.920374 0.391039i \(-0.127884\pi\)
\(194\) 0 0
\(195\) 325.464 1.66905
\(196\) 0 0
\(197\) 68.0052 0.345204 0.172602 0.984992i \(-0.444783\pi\)
0.172602 + 0.984992i \(0.444783\pi\)
\(198\) 0 0
\(199\) −29.4815 −0.148148 −0.0740741 0.997253i \(-0.523600\pi\)
−0.0740741 + 0.997253i \(0.523600\pi\)
\(200\) 0 0
\(201\) 76.7015 0.381600
\(202\) 0 0
\(203\) 75.9796i 0.374284i
\(204\) 0 0
\(205\) 261.528i 1.27574i
\(206\) 0 0
\(207\) 25.3898 0.122656
\(208\) 0 0
\(209\) 11.2850 7.56257i 0.0539953 0.0361845i
\(210\) 0 0
\(211\) 9.92686i 0.0470467i −0.999723 0.0235234i \(-0.992512\pi\)
0.999723 0.0235234i \(-0.00748841\pi\)
\(212\) 0 0
\(213\) −128.762 −0.604518
\(214\) 0 0
\(215\) −181.110 −0.842370
\(216\) 0 0
\(217\) 61.9642i 0.285549i
\(218\) 0 0
\(219\) 194.336i 0.887380i
\(220\) 0 0
\(221\) 662.690i 2.99860i
\(222\) 0 0
\(223\) 177.702i 0.796871i −0.917196 0.398435i \(-0.869553\pi\)
0.917196 0.398435i \(-0.130447\pi\)
\(224\) 0 0
\(225\) −88.9159 −0.395182
\(226\) 0 0
\(227\) 79.1694i 0.348764i −0.984678 0.174382i \(-0.944207\pi\)
0.984678 0.174382i \(-0.0557927\pi\)
\(228\) 0 0
\(229\) 161.424 0.704908 0.352454 0.935829i \(-0.385347\pi\)
0.352454 + 0.935829i \(0.385347\pi\)
\(230\) 0 0
\(231\) 4.06813i 0.0176109i
\(232\) 0 0
\(233\) −196.958 −0.845315 −0.422658 0.906289i \(-0.638903\pi\)
−0.422658 + 0.906289i \(0.638903\pi\)
\(234\) 0 0
\(235\) −340.755 −1.45002
\(236\) 0 0
\(237\) −217.373 −0.917184
\(238\) 0 0
\(239\) −179.160 −0.749621 −0.374811 0.927101i \(-0.622292\pi\)
−0.374811 + 0.927101i \(0.622292\pi\)
\(240\) 0 0
\(241\) 464.116i 1.92579i −0.269870 0.962897i \(-0.586981\pi\)
0.269870 0.962897i \(-0.413019\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 282.431 1.15278
\(246\) 0 0
\(247\) 268.884 + 401.234i 1.08860 + 1.62443i
\(248\) 0 0
\(249\) 165.527i 0.664768i
\(250\) 0 0
\(251\) −140.760 −0.560798 −0.280399 0.959883i \(-0.590467\pi\)
−0.280399 + 0.959883i \(0.590467\pi\)
\(252\) 0 0
\(253\) −6.05109 −0.0239174
\(254\) 0 0
\(255\) 333.755i 1.30884i
\(256\) 0 0
\(257\) 184.717i 0.718743i −0.933195 0.359371i \(-0.882991\pi\)
0.933195 0.359371i \(-0.117009\pi\)
\(258\) 0 0
\(259\) 159.488i 0.615784i
\(260\) 0 0
\(261\) 69.3875i 0.265852i
\(262\) 0 0
\(263\) −254.011 −0.965822 −0.482911 0.875669i \(-0.660421\pi\)
−0.482911 + 0.875669i \(0.660421\pi\)
\(264\) 0 0
\(265\) 378.548i 1.42848i
\(266\) 0 0
\(267\) 48.6407 0.182175
\(268\) 0 0
\(269\) 54.2105i 0.201526i −0.994910 0.100763i \(-0.967872\pi\)
0.994910 0.100763i \(-0.0321284\pi\)
\(270\) 0 0
\(271\) −397.966 −1.46851 −0.734254 0.678875i \(-0.762468\pi\)
−0.734254 + 0.678875i \(0.762468\pi\)
\(272\) 0 0
\(273\) 144.641 0.529819
\(274\) 0 0
\(275\) 21.1912 0.0770587
\(276\) 0 0
\(277\) −310.820 −1.12209 −0.561046 0.827784i \(-0.689601\pi\)
−0.561046 + 0.827784i \(0.689601\pi\)
\(278\) 0 0
\(279\) 56.5880i 0.202824i
\(280\) 0 0
\(281\) 189.796i 0.675429i 0.941249 + 0.337715i \(0.109654\pi\)
−0.941249 + 0.337715i \(0.890346\pi\)
\(282\) 0 0
\(283\) 352.032 1.24393 0.621965 0.783045i \(-0.286335\pi\)
0.621965 + 0.783045i \(0.286335\pi\)
\(284\) 0 0
\(285\) −135.420 202.077i −0.475159 0.709041i
\(286\) 0 0
\(287\) 116.227i 0.404970i
\(288\) 0 0
\(289\) 390.572 1.35146
\(290\) 0 0
\(291\) −96.6407 −0.332099
\(292\) 0 0
\(293\) 238.729i 0.814774i 0.913256 + 0.407387i \(0.133560\pi\)
−0.913256 + 0.407387i \(0.866440\pi\)
\(294\) 0 0
\(295\) 748.893i 2.53862i
\(296\) 0 0
\(297\) 3.71517i 0.0125090i
\(298\) 0 0
\(299\) 215.144i 0.719546i
\(300\) 0 0
\(301\) −80.4876 −0.267401
\(302\) 0 0
\(303\) 226.929i 0.748942i
\(304\) 0 0
\(305\) −285.117 −0.934809
\(306\) 0 0
\(307\) 78.3760i 0.255296i −0.991820 0.127648i \(-0.959257\pi\)
0.991820 0.127648i \(-0.0407428\pi\)
\(308\) 0 0
\(309\) −129.312 −0.418485
\(310\) 0 0
\(311\) 257.012 0.826404 0.413202 0.910639i \(-0.364410\pi\)
0.413202 + 0.910639i \(0.364410\pi\)
\(312\) 0 0
\(313\) 152.676 0.487784 0.243892 0.969802i \(-0.421576\pi\)
0.243892 + 0.969802i \(0.421576\pi\)
\(314\) 0 0
\(315\) −72.8465 −0.231259
\(316\) 0 0
\(317\) 293.190i 0.924889i 0.886648 + 0.462444i \(0.153028\pi\)
−0.886648 + 0.462444i \(0.846972\pi\)
\(318\) 0 0
\(319\) 16.5370i 0.0518400i
\(320\) 0 0
\(321\) 297.586 0.927058
\(322\) 0 0
\(323\) −411.456 + 275.734i −1.27386 + 0.853666i
\(324\) 0 0
\(325\) 753.443i 2.31829i
\(326\) 0 0
\(327\) −60.5494 −0.185166
\(328\) 0 0
\(329\) −151.436 −0.460292
\(330\) 0 0
\(331\) 515.401i 1.55710i 0.627580 + 0.778552i \(0.284046\pi\)
−0.627580 + 0.778552i \(0.715954\pi\)
\(332\) 0 0
\(333\) 145.650i 0.437389i
\(334\) 0 0
\(335\) 327.336i 0.977122i
\(336\) 0 0
\(337\) 452.163i 1.34173i 0.741580 + 0.670865i \(0.234077\pi\)
−0.741580 + 0.670865i \(0.765923\pi\)
\(338\) 0 0
\(339\) −74.1388 −0.218699
\(340\) 0 0
\(341\) 13.4865i 0.0395499i
\(342\) 0 0
\(343\) 286.482 0.835224
\(344\) 0 0
\(345\) 108.355i 0.314072i
\(346\) 0 0
\(347\) −378.380 −1.09043 −0.545217 0.838295i \(-0.683553\pi\)
−0.545217 + 0.838295i \(0.683553\pi\)
\(348\) 0 0
\(349\) −95.1859 −0.272739 −0.136369 0.990658i \(-0.543543\pi\)
−0.136369 + 0.990658i \(0.543543\pi\)
\(350\) 0 0
\(351\) 132.091 0.376328
\(352\) 0 0
\(353\) 457.473 1.29596 0.647979 0.761659i \(-0.275615\pi\)
0.647979 + 0.761659i \(0.275615\pi\)
\(354\) 0 0
\(355\) 549.514i 1.54793i
\(356\) 0 0
\(357\) 148.326i 0.415478i
\(358\) 0 0
\(359\) −308.486 −0.859294 −0.429647 0.902997i \(-0.641362\pi\)
−0.429647 + 0.902997i \(0.641362\pi\)
\(360\) 0 0
\(361\) 137.243 333.894i 0.380176 0.924914i
\(362\) 0 0
\(363\) 208.693i 0.574911i
\(364\) 0 0
\(365\) −829.360 −2.27222
\(366\) 0 0
\(367\) 432.928 1.17964 0.589820 0.807535i \(-0.299199\pi\)
0.589820 + 0.807535i \(0.299199\pi\)
\(368\) 0 0
\(369\) 106.142i 0.287649i
\(370\) 0 0
\(371\) 168.232i 0.453455i
\(372\) 0 0
\(373\) 649.540i 1.74139i −0.491820 0.870697i \(-0.663668\pi\)
0.491820 0.870697i \(-0.336332\pi\)
\(374\) 0 0
\(375\) 59.3883i 0.158369i
\(376\) 0 0
\(377\) −587.966 −1.55959
\(378\) 0 0
\(379\) 103.303i 0.272567i −0.990670 0.136283i \(-0.956484\pi\)
0.990670 0.136283i \(-0.0435158\pi\)
\(380\) 0 0
\(381\) 305.434 0.801663
\(382\) 0 0
\(383\) 526.913i 1.37575i 0.725828 + 0.687876i \(0.241457\pi\)
−0.725828 + 0.687876i \(0.758543\pi\)
\(384\) 0 0
\(385\) 17.3614 0.0450944
\(386\) 0 0
\(387\) −73.5043 −0.189934
\(388\) 0 0
\(389\) 93.9846 0.241606 0.120803 0.992677i \(-0.461453\pi\)
0.120803 + 0.992677i \(0.461453\pi\)
\(390\) 0 0
\(391\) 220.625 0.564259
\(392\) 0 0
\(393\) 350.072i 0.890768i
\(394\) 0 0
\(395\) 927.672i 2.34854i
\(396\) 0 0
\(397\) 218.605 0.550643 0.275321 0.961352i \(-0.411216\pi\)
0.275321 + 0.961352i \(0.411216\pi\)
\(398\) 0 0
\(399\) −60.1826 89.8057i −0.150834 0.225077i
\(400\) 0 0
\(401\) 0.984510i 0.00245514i −0.999999 0.00122757i \(-0.999609\pi\)
0.999999 0.00122757i \(-0.000390747\pi\)
\(402\) 0 0
\(403\) 479.508 1.18985
\(404\) 0 0
\(405\) −66.5262 −0.164262
\(406\) 0 0
\(407\) 34.7126i 0.0852889i
\(408\) 0 0
\(409\) 711.378i 1.73931i 0.493659 + 0.869655i \(0.335659\pi\)
−0.493659 + 0.869655i \(0.664341\pi\)
\(410\) 0 0
\(411\) 19.5533i 0.0475749i
\(412\) 0 0
\(413\) 332.818i 0.805855i
\(414\) 0 0
\(415\) −706.413 −1.70220
\(416\) 0 0
\(417\) 98.1081i 0.235271i
\(418\) 0 0
\(419\) 692.753 1.65335 0.826675 0.562680i \(-0.190230\pi\)
0.826675 + 0.562680i \(0.190230\pi\)
\(420\) 0 0
\(421\) 205.304i 0.487658i −0.969818 0.243829i \(-0.921596\pi\)
0.969818 0.243829i \(-0.0784035\pi\)
\(422\) 0 0
\(423\) −138.297 −0.326944
\(424\) 0 0
\(425\) −772.638 −1.81797
\(426\) 0 0
\(427\) −126.710 −0.296744
\(428\) 0 0
\(429\) −31.4811 −0.0733824
\(430\) 0 0
\(431\) 534.388i 1.23988i −0.784649 0.619940i \(-0.787157\pi\)
0.784649 0.619940i \(-0.212843\pi\)
\(432\) 0 0
\(433\) 561.059i 1.29575i 0.761747 + 0.647874i \(0.224342\pi\)
−0.761747 + 0.647874i \(0.775658\pi\)
\(434\) 0 0
\(435\) 296.122 0.680740
\(436\) 0 0
\(437\) −133.581 + 89.5180i −0.305676 + 0.204847i
\(438\) 0 0
\(439\) 422.164i 0.961649i −0.876817 0.480825i \(-0.840337\pi\)
0.876817 0.480825i \(-0.159663\pi\)
\(440\) 0 0
\(441\) 114.626 0.259923
\(442\) 0 0
\(443\) 780.825 1.76259 0.881293 0.472571i \(-0.156674\pi\)
0.881293 + 0.472571i \(0.156674\pi\)
\(444\) 0 0
\(445\) 207.582i 0.466476i
\(446\) 0 0
\(447\) 10.3332i 0.0231169i
\(448\) 0 0
\(449\) 671.171i 1.49481i −0.664367 0.747406i \(-0.731299\pi\)
0.664367 0.747406i \(-0.268701\pi\)
\(450\) 0 0
\(451\) 25.2967i 0.0560903i
\(452\) 0 0
\(453\) −48.5494 −0.107173
\(454\) 0 0
\(455\) 617.277i 1.35665i
\(456\) 0 0
\(457\) −371.441 −0.812781 −0.406390 0.913700i \(-0.633213\pi\)
−0.406390 + 0.913700i \(0.633213\pi\)
\(458\) 0 0
\(459\) 135.456i 0.295112i
\(460\) 0 0
\(461\) 301.718 0.654487 0.327243 0.944940i \(-0.393880\pi\)
0.327243 + 0.944940i \(0.393880\pi\)
\(462\) 0 0
\(463\) −665.736 −1.43788 −0.718938 0.695074i \(-0.755371\pi\)
−0.718938 + 0.695074i \(0.755371\pi\)
\(464\) 0 0
\(465\) −241.498 −0.519351
\(466\) 0 0
\(467\) 95.4717 0.204436 0.102218 0.994762i \(-0.467406\pi\)
0.102218 + 0.994762i \(0.467406\pi\)
\(468\) 0 0
\(469\) 145.473i 0.310176i
\(470\) 0 0
\(471\) 244.768i 0.519678i
\(472\) 0 0
\(473\) 17.5181 0.0370362
\(474\) 0 0
\(475\) 467.804 313.495i 0.984851 0.659990i
\(476\) 0 0
\(477\) 153.636i 0.322087i
\(478\) 0 0
\(479\) 39.3541 0.0821590 0.0410795 0.999156i \(-0.486920\pi\)
0.0410795 + 0.999156i \(0.486920\pi\)
\(480\) 0 0
\(481\) 1234.19 2.56589
\(482\) 0 0
\(483\) 48.1543i 0.0996984i
\(484\) 0 0
\(485\) 412.429i 0.850369i
\(486\) 0 0
\(487\) 344.218i 0.706813i −0.935470 0.353407i \(-0.885023\pi\)
0.935470 0.353407i \(-0.114977\pi\)
\(488\) 0 0
\(489\) 314.454i 0.643055i
\(490\) 0 0
\(491\) 582.955 1.18728 0.593641 0.804730i \(-0.297690\pi\)
0.593641 + 0.804730i \(0.297690\pi\)
\(492\) 0 0
\(493\) 602.945i 1.22301i
\(494\) 0 0
\(495\) 15.8550 0.0320304
\(496\) 0 0
\(497\) 244.211i 0.491371i
\(498\) 0 0
\(499\) 166.892 0.334453 0.167226 0.985919i \(-0.446519\pi\)
0.167226 + 0.985919i \(0.446519\pi\)
\(500\) 0 0
\(501\) 522.624 1.04316
\(502\) 0 0
\(503\) −559.547 −1.11242 −0.556210 0.831042i \(-0.687745\pi\)
−0.556210 + 0.831042i \(0.687745\pi\)
\(504\) 0 0
\(505\) 968.457 1.91774
\(506\) 0 0
\(507\) 826.580i 1.63034i
\(508\) 0 0
\(509\) 371.176i 0.729226i −0.931159 0.364613i \(-0.881201\pi\)
0.931159 0.364613i \(-0.118799\pi\)
\(510\) 0 0
\(511\) −368.579 −0.721290
\(512\) 0 0
\(513\) −54.9610 82.0139i −0.107136 0.159871i
\(514\) 0 0
\(515\) 551.858i 1.07157i
\(516\) 0 0
\(517\) 32.9601 0.0637526
\(518\) 0 0
\(519\) −338.563 −0.652337
\(520\) 0 0
\(521\) 967.227i 1.85648i −0.371981 0.928240i \(-0.621321\pi\)
0.371981 0.928240i \(-0.378679\pi\)
\(522\) 0 0
\(523\) 316.578i 0.605311i −0.953100 0.302656i \(-0.902127\pi\)
0.953100 0.302656i \(-0.0978732\pi\)
\(524\) 0 0
\(525\) 168.638i 0.321216i
\(526\) 0 0
\(527\) 491.724i 0.933062i
\(528\) 0 0
\(529\) −457.373 −0.864600
\(530\) 0 0
\(531\) 303.942i 0.572396i
\(532\) 0 0
\(533\) 899.415 1.68746
\(534\) 0 0
\(535\) 1269.99i 2.37382i
\(536\) 0 0
\(537\) 158.105 0.294422
\(538\) 0 0
\(539\) −27.3186 −0.0506838
\(540\) 0 0
\(541\) −700.118 −1.29412 −0.647059 0.762440i \(-0.724001\pi\)
−0.647059 + 0.762440i \(0.724001\pi\)
\(542\) 0 0
\(543\) 27.3289 0.0503294
\(544\) 0 0
\(545\) 258.404i 0.474135i
\(546\) 0 0
\(547\) 782.925i 1.43131i −0.698456 0.715653i \(-0.746129\pi\)
0.698456 0.715653i \(-0.253871\pi\)
\(548\) 0 0
\(549\) −115.716 −0.210776
\(550\) 0 0
\(551\) 244.643 + 365.061i 0.443998 + 0.662543i
\(552\) 0 0
\(553\) 412.270i 0.745515i
\(554\) 0 0
\(555\) −621.586 −1.11997
\(556\) 0 0
\(557\) −358.027 −0.642777 −0.321388 0.946947i \(-0.604150\pi\)
−0.321388 + 0.946947i \(0.604150\pi\)
\(558\) 0 0
\(559\) 622.850i 1.11422i
\(560\) 0 0
\(561\) 32.2831i 0.0575456i
\(562\) 0 0
\(563\) 982.338i 1.74483i −0.488769 0.872413i \(-0.662554\pi\)
0.488769 0.872413i \(-0.337446\pi\)
\(564\) 0 0
\(565\) 316.399i 0.559998i
\(566\) 0 0
\(567\) −29.5651 −0.0521431
\(568\) 0 0
\(569\) 26.6426i 0.0468236i 0.999726 + 0.0234118i \(0.00745289\pi\)
−0.999726 + 0.0234118i \(0.992547\pi\)
\(570\) 0 0
\(571\) −309.230 −0.541559 −0.270779 0.962641i \(-0.587281\pi\)
−0.270779 + 0.962641i \(0.587281\pi\)
\(572\) 0 0
\(573\) 335.912i 0.586234i
\(574\) 0 0
\(575\) −250.839 −0.436242
\(576\) 0 0
\(577\) 163.804 0.283889 0.141945 0.989875i \(-0.454665\pi\)
0.141945 + 0.989875i \(0.454665\pi\)
\(578\) 0 0
\(579\) 261.437 0.451532
\(580\) 0 0
\(581\) −313.940 −0.540344
\(582\) 0 0
\(583\) 36.6157i 0.0628056i
\(584\) 0 0
\(585\) 563.720i 0.963624i
\(586\) 0 0
\(587\) 216.974 0.369632 0.184816 0.982773i \(-0.440831\pi\)
0.184816 + 0.982773i \(0.440831\pi\)
\(588\) 0 0
\(589\) −199.515 297.721i −0.338736 0.505468i
\(590\) 0 0
\(591\) 117.788i 0.199304i
\(592\) 0 0
\(593\) −893.250 −1.50632 −0.753162 0.657835i \(-0.771472\pi\)
−0.753162 + 0.657835i \(0.771472\pi\)
\(594\) 0 0
\(595\) −633.002 −1.06387
\(596\) 0 0
\(597\) 51.0634i 0.0855333i
\(598\) 0 0
\(599\) 1086.97i 1.81464i −0.420446 0.907318i \(-0.638126\pi\)
0.420446 0.907318i \(-0.361874\pi\)
\(600\) 0 0
\(601\) 60.8027i 0.101169i −0.998720 0.0505846i \(-0.983892\pi\)
0.998720 0.0505846i \(-0.0161085\pi\)
\(602\) 0 0
\(603\) 132.851i 0.220317i
\(604\) 0 0
\(605\) 890.629 1.47211
\(606\) 0 0
\(607\) 394.292i 0.649575i −0.945787 0.324788i \(-0.894707\pi\)
0.945787 0.324788i \(-0.105293\pi\)
\(608\) 0 0
\(609\) 131.601 0.216093
\(610\) 0 0
\(611\) 1171.88i 1.91798i
\(612\) 0 0
\(613\) −899.074 −1.46668 −0.733340 0.679862i \(-0.762039\pi\)
−0.733340 + 0.679862i \(0.762039\pi\)
\(614\) 0 0
\(615\) −452.979 −0.736552
\(616\) 0 0
\(617\) 593.943 0.962630 0.481315 0.876548i \(-0.340159\pi\)
0.481315 + 0.876548i \(0.340159\pi\)
\(618\) 0 0
\(619\) 459.655 0.742576 0.371288 0.928518i \(-0.378916\pi\)
0.371288 + 0.928518i \(0.378916\pi\)
\(620\) 0 0
\(621\) 43.9763i 0.0708154i
\(622\) 0 0
\(623\) 92.2521i 0.148077i
\(624\) 0 0
\(625\) −487.517 −0.780027
\(626\) 0 0
\(627\) 13.0988 + 19.5462i 0.0208911 + 0.0311742i
\(628\) 0 0
\(629\) 1265.63i 2.01214i
\(630\) 0 0
\(631\) −856.677 −1.35765 −0.678825 0.734300i \(-0.737510\pi\)
−0.678825 + 0.734300i \(0.737510\pi\)
\(632\) 0 0
\(633\) 17.1938 0.0271624
\(634\) 0 0
\(635\) 1303.48i 2.05273i
\(636\) 0 0
\(637\) 971.302i 1.52481i
\(638\) 0 0
\(639\) 223.023i 0.349019i
\(640\) 0 0
\(641\) 264.843i 0.413171i 0.978429 + 0.206585i \(0.0662352\pi\)
−0.978429 + 0.206585i \(0.933765\pi\)
\(642\) 0 0
\(643\) −355.596 −0.553026 −0.276513 0.961010i \(-0.589179\pi\)
−0.276513 + 0.961010i \(0.589179\pi\)
\(644\) 0 0
\(645\) 313.691i 0.486342i
\(646\) 0 0
\(647\) 589.871 0.911701 0.455851 0.890056i \(-0.349335\pi\)
0.455851 + 0.890056i \(0.349335\pi\)
\(648\) 0 0
\(649\) 72.4379i 0.111615i
\(650\) 0 0
\(651\) −107.325 −0.164862
\(652\) 0 0
\(653\) 891.917 1.36588 0.682938 0.730477i \(-0.260702\pi\)
0.682938 + 0.730477i \(0.260702\pi\)
\(654\) 0 0
\(655\) 1493.99 2.28089
\(656\) 0 0
\(657\) −336.600 −0.512329
\(658\) 0 0
\(659\) 828.284i 1.25688i 0.777858 + 0.628440i \(0.216306\pi\)
−0.777858 + 0.628440i \(0.783694\pi\)
\(660\) 0 0
\(661\) 769.980i 1.16487i −0.812877 0.582436i \(-0.802100\pi\)
0.812877 0.582436i \(-0.197900\pi\)
\(662\) 0 0
\(663\) 1147.81 1.73124
\(664\) 0 0
\(665\) 383.260 256.839i 0.576330 0.386223i
\(666\) 0 0
\(667\) 195.748i 0.293475i
\(668\) 0 0
\(669\) 307.789 0.460073
\(670\) 0 0
\(671\) 27.5784 0.0411005
\(672\) 0 0
\(673\) 372.195i 0.553038i −0.961008 0.276519i \(-0.910819\pi\)
0.961008 0.276519i \(-0.0891809\pi\)
\(674\) 0 0
\(675\) 154.007i 0.228158i
\(676\) 0 0
\(677\) 484.617i 0.715830i 0.933754 + 0.357915i \(0.116512\pi\)
−0.933754 + 0.357915i \(0.883488\pi\)
\(678\) 0 0
\(679\) 183.289i 0.269940i
\(680\) 0 0
\(681\) 137.125 0.201359
\(682\) 0 0
\(683\) 779.041i 1.14062i −0.821431 0.570308i \(-0.806824\pi\)
0.821431 0.570308i \(-0.193176\pi\)
\(684\) 0 0
\(685\) 83.4467 0.121820
\(686\) 0 0
\(687\) 279.594i 0.406979i
\(688\) 0 0
\(689\) 1301.86 1.88949
\(690\) 0 0
\(691\) 76.9998 0.111432 0.0557162 0.998447i \(-0.482256\pi\)
0.0557162 + 0.998447i \(0.482256\pi\)
\(692\) 0 0
\(693\) 7.04620 0.0101677
\(694\) 0 0
\(695\) −418.691 −0.602434
\(696\) 0 0
\(697\) 922.328i 1.32328i
\(698\) 0 0
\(699\) 341.142i 0.488043i
\(700\) 0 0
\(701\) 754.168 1.07585 0.537923 0.842994i \(-0.319209\pi\)
0.537923 + 0.842994i \(0.319209\pi\)
\(702\) 0 0
\(703\) −513.527 766.295i −0.730479 1.09004i
\(704\) 0 0
\(705\) 590.204i 0.837169i
\(706\) 0 0
\(707\) 430.396 0.608763
\(708\) 0 0
\(709\) 48.7186 0.0687146 0.0343573 0.999410i \(-0.489062\pi\)
0.0343573 + 0.999410i \(0.489062\pi\)
\(710\) 0 0
\(711\) 376.500i 0.529537i
\(712\) 0 0
\(713\) 159.640i 0.223898i
\(714\) 0 0
\(715\) 134.350i 0.187903i
\(716\) 0 0
\(717\) 310.313i 0.432794i
\(718\) 0 0
\(719\) 583.791 0.811949 0.405974 0.913884i \(-0.366932\pi\)
0.405974 + 0.913884i \(0.366932\pi\)
\(720\) 0 0
\(721\) 245.253i 0.340157i
\(722\) 0 0
\(723\) 803.873 1.11186
\(724\) 0 0
\(725\) 685.517i 0.945540i
\(726\) 0 0
\(727\) 1005.23 1.38271 0.691357 0.722513i \(-0.257013\pi\)
0.691357 + 0.722513i \(0.257013\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) −638.718 −0.873759
\(732\) 0 0
\(733\) 850.053 1.15969 0.579845 0.814727i \(-0.303113\pi\)
0.579845 + 0.814727i \(0.303113\pi\)
\(734\) 0 0
\(735\) 489.184i 0.665557i
\(736\) 0 0
\(737\) 31.6621i 0.0429608i
\(738\) 0 0
\(739\) −1215.10 −1.64425 −0.822127 0.569305i \(-0.807212\pi\)
−0.822127 + 0.569305i \(0.807212\pi\)
\(740\) 0 0
\(741\) −694.958 + 465.721i −0.937866 + 0.628503i
\(742\) 0 0
\(743\) 661.859i 0.890792i −0.895334 0.445396i \(-0.853063\pi\)
0.895334 0.445396i \(-0.146937\pi\)
\(744\) 0 0
\(745\) 44.0987 0.0591929
\(746\) 0 0
\(747\) −286.702 −0.383804
\(748\) 0 0
\(749\) 564.402i 0.753541i
\(750\) 0 0
\(751\) 1354.03i 1.80297i −0.432812 0.901484i \(-0.642479\pi\)
0.432812 0.901484i \(-0.357521\pi\)
\(752\) 0 0
\(753\) 243.804i 0.323777i
\(754\) 0 0
\(755\) 207.192i 0.274426i
\(756\) 0 0
\(757\) −972.577 −1.28478 −0.642389 0.766379i \(-0.722057\pi\)
−0.642389 + 0.766379i \(0.722057\pi\)
\(758\) 0 0
\(759\) 10.4808i 0.0138087i
\(760\) 0 0
\(761\) −644.913 −0.847454 −0.423727 0.905790i \(-0.639278\pi\)
−0.423727 + 0.905790i \(0.639278\pi\)
\(762\) 0 0
\(763\) 114.838i 0.150509i
\(764\) 0 0
\(765\) −578.081 −0.755662
\(766\) 0 0
\(767\) 2575.50 3.35789
\(768\) 0 0
\(769\) −665.867 −0.865887 −0.432943 0.901421i \(-0.642525\pi\)
−0.432943 + 0.901421i \(0.642525\pi\)
\(770\) 0 0
\(771\) 319.939 0.414966
\(772\) 0 0
\(773\) 127.536i 0.164988i 0.996592 + 0.0824942i \(0.0262886\pi\)
−0.996592 + 0.0824942i \(0.973711\pi\)
\(774\) 0 0
\(775\) 559.064i 0.721373i
\(776\) 0 0
\(777\) −276.241 −0.355523
\(778\) 0 0
\(779\) −374.232 558.436i −0.480400 0.716863i
\(780\) 0 0
\(781\) 53.1526i 0.0680572i
\(782\) 0 0
\(783\) 120.183 0.153490
\(784\) 0 0
\(785\) −1044.59 −1.33068
\(786\) 0 0
\(787\) 627.730i 0.797625i −0.917033 0.398812i \(-0.869423\pi\)
0.917033 0.398812i \(-0.130577\pi\)
\(788\) 0 0
\(789\) 439.960i 0.557618i
\(790\) 0 0
\(791\) 140.612i 0.177765i
\(792\) 0 0
\(793\) 980.540i 1.23649i
\(794\) 0 0
\(795\) −655.664 −0.824734
\(796\) 0 0
\(797\) 1418.77i 1.78014i 0.455827 + 0.890069i \(0.349344\pi\)
−0.455827 + 0.890069i \(0.650656\pi\)
\(798\) 0 0
\(799\) −1201.74 −1.50405
\(800\) 0 0
\(801\) 84.2481i 0.105179i
\(802\) 0 0
\(803\) 80.2213 0.0999020
\(804\) 0 0
\(805\) −205.506 −0.255287
\(806\) 0 0
\(807\) 93.8954 0.116351
\(808\) 0 0
\(809\) −753.612 −0.931535 −0.465768 0.884907i \(-0.654222\pi\)
−0.465768 + 0.884907i \(0.654222\pi\)
\(810\) 0 0
\(811\) 225.670i 0.278261i 0.990274 + 0.139131i \(0.0444308\pi\)
−0.990274 + 0.139131i \(0.955569\pi\)
\(812\) 0 0
\(813\) 689.297i 0.847844i
\(814\) 0 0
\(815\) −1341.98 −1.64660
\(816\) 0 0
\(817\) 386.720 259.158i 0.473342 0.317207i
\(818\) 0 0
\(819\) 250.525i 0.305891i
\(820\) 0 0
\(821\) 515.673 0.628104 0.314052 0.949406i \(-0.398313\pi\)
0.314052 + 0.949406i \(0.398313\pi\)
\(822\) 0 0
\(823\) 1436.24 1.74513 0.872564 0.488500i \(-0.162456\pi\)
0.872564 + 0.488500i \(0.162456\pi\)
\(824\) 0 0
\(825\) 36.7042i 0.0444899i
\(826\) 0 0
\(827\) 134.792i 0.162989i 0.996674 + 0.0814944i \(0.0259693\pi\)
−0.996674 + 0.0814944i \(0.974031\pi\)
\(828\) 0 0
\(829\) 376.958i 0.454714i −0.973811 0.227357i \(-0.926992\pi\)
0.973811 0.227357i \(-0.0730084\pi\)
\(830\) 0 0
\(831\) 538.355i 0.647840i
\(832\) 0 0
\(833\) 996.047 1.19573
\(834\) 0 0
\(835\) 2230.38i 2.67111i
\(836\) 0 0
\(837\) −98.0133 −0.117101
\(838\) 0 0
\(839\) 572.973i 0.682924i −0.939896 0.341462i \(-0.889078\pi\)
0.939896 0.341462i \(-0.110922\pi\)
\(840\) 0 0
\(841\) 306.042 0.363903
\(842\) 0 0
\(843\) −328.736 −0.389959
\(844\) 0 0
\(845\) 3527.56 4.17463
\(846\) 0 0
\(847\) 395.808 0.467305
\(848\) 0 0
\(849\) 609.737i 0.718183i
\(850\) 0 0
\(851\) 410.892i 0.482834i
\(852\) 0 0
\(853\) −1076.65 −1.26219 −0.631096 0.775705i \(-0.717395\pi\)
−0.631096 + 0.775705i \(0.717395\pi\)
\(854\) 0 0
\(855\) 350.007 234.555i 0.409365 0.274333i
\(856\) 0 0
\(857\) 1506.83i 1.75826i −0.476578 0.879132i \(-0.658123\pi\)
0.476578 0.879132i \(-0.341877\pi\)
\(858\) 0 0
\(859\) 929.147 1.08166 0.540831 0.841131i \(-0.318110\pi\)
0.540831 + 0.841131i \(0.318110\pi\)
\(860\) 0 0
\(861\) −201.310 −0.233810
\(862\) 0 0
\(863\) 211.241i 0.244775i −0.992482 0.122388i \(-0.960945\pi\)
0.992482 0.122388i \(-0.0390551\pi\)
\(864\) 0 0
\(865\) 1444.87i 1.67037i
\(866\) 0 0
\(867\) 676.491i 0.780266i
\(868\) 0 0
\(869\) 89.7306i 0.103257i
\(870\) 0 0
\(871\) 1125.73 1.29246
\(872\) 0 0
\(873\) 167.387i 0.191737i
\(874\) 0 0
\(875\) 112.636 0.128727
\(876\) 0 0
\(877\) 229.052i 0.261176i −0.991437 0.130588i \(-0.958313\pi\)
0.991437 0.130588i \(-0.0416865\pi\)
\(878\) 0 0
\(879\) −413.491 −0.470410
\(880\) 0 0
\(881\) −1354.01 −1.53690 −0.768449 0.639911i \(-0.778971\pi\)
−0.768449 + 0.639911i \(0.778971\pi\)
\(882\) 0 0
\(883\) 391.523 0.443401 0.221701 0.975115i \(-0.428839\pi\)
0.221701 + 0.975115i \(0.428839\pi\)
\(884\) 0 0
\(885\) −1297.12 −1.46567
\(886\) 0 0
\(887\) 697.322i 0.786158i −0.919505 0.393079i \(-0.871410\pi\)
0.919505 0.393079i \(-0.128590\pi\)
\(888\) 0 0
\(889\) 579.287i 0.651616i
\(890\) 0 0
\(891\) 6.43486 0.00722206
\(892\) 0 0
\(893\) 727.608 487.601i 0.814791 0.546026i
\(894\) 0 0
\(895\) 674.736i 0.753895i
\(896\) 0 0
\(897\) 372.641 0.415430
\(898\) 0 0
\(899\) 436.278 0.485292
\(900\) 0 0
\(901\) 1335.02i 1.48171i
\(902\) 0 0
\(903\) 139.409i 0.154384i
\(904\) 0 0
\(905\) 116.630i 0.128873i
\(906\) 0 0
\(907\) 626.344i 0.690567i 0.938498 + 0.345283i \(0.112217\pi\)
−0.938498 + 0.345283i \(0.887783\pi\)
\(908\) 0 0
\(909\) 393.053 0.432402
\(910\) 0 0
\(911\) 517.549i 0.568111i 0.958808 + 0.284056i \(0.0916800\pi\)
−0.958808 + 0.284056i \(0.908320\pi\)
\(912\) 0 0
\(913\) 68.3290 0.0748401
\(914\) 0 0
\(915\) 493.837i 0.539712i
\(916\) 0 0
\(917\) 663.948 0.724043
\(918\) 0 0
\(919\) −900.211 −0.979555 −0.489777 0.871848i \(-0.662922\pi\)
−0.489777 + 0.871848i \(0.662922\pi\)
\(920\) 0 0
\(921\) 135.751 0.147395
\(922\) 0 0
\(923\) −1889.82 −2.04748
\(924\) 0 0
\(925\) 1438.96i 1.55563i
\(926\) 0 0
\(927\) 223.975i 0.241612i
\(928\) 0 0
\(929\) −734.220 −0.790333 −0.395167 0.918609i \(-0.629313\pi\)
−0.395167 + 0.918609i \(0.629313\pi\)
\(930\) 0 0
\(931\) −603.070 + 404.143i −0.647766 + 0.434095i
\(932\) 0 0
\(933\) 445.157i 0.477125i
\(934\) 0 0
\(935\) 137.773 0.147351
\(936\) 0 0
\(937\) 832.710 0.888698 0.444349 0.895854i \(-0.353435\pi\)
0.444349 + 0.895854i \(0.353435\pi\)
\(938\) 0 0
\(939\) 264.443i 0.281622i
\(940\) 0 0
\(941\) 1383.55i 1.47030i −0.677905 0.735149i \(-0.737112\pi\)
0.677905 0.735149i \(-0.262888\pi\)
\(942\) 0 0
\(943\) 299.437i 0.317536i
\(944\) 0 0
\(945\) 126.174i 0.133517i
\(946\) 0 0
\(947\) −322.935 −0.341008 −0.170504 0.985357i \(-0.554540\pi\)
−0.170504 + 0.985357i \(0.554540\pi\)
\(948\) 0 0
\(949\) 2852.24i 3.00552i
\(950\) 0 0
\(951\) −507.820 −0.533985
\(952\) 0 0
\(953\) 1815.82i 1.90537i 0.303962 + 0.952684i \(0.401691\pi\)
−0.303962 + 0.952684i \(0.598309\pi\)
\(954\) 0 0
\(955\) 1433.56 1.50111
\(956\) 0 0
\(957\) −28.6429 −0.0299299
\(958\) 0 0
\(959\) 37.0848 0.0386703
\(960\) 0 0
\(961\) 605.200 0.629760
\(962\) 0 0
\(963\) 515.434i 0.535237i
\(964\) 0 0
\(965\) 1115.72i 1.15619i
\(966\) 0 0
\(967\) −738.252 −0.763446 −0.381723 0.924277i \(-0.624669\pi\)
−0.381723 + 0.924277i \(0.624669\pi\)
\(968\) 0 0
\(969\) −477.586 712.663i −0.492864 0.735462i
\(970\) 0 0
\(971\) 1140.17i 1.17422i 0.809507 + 0.587111i \(0.199735\pi\)
−0.809507 + 0.587111i \(0.800265\pi\)
\(972\) 0 0
\(973\) −186.072 −0.191236
\(974\) 0 0
\(975\) −1305.00 −1.33846
\(976\) 0 0
\(977\) 1559.45i 1.59616i 0.602550 + 0.798081i \(0.294151\pi\)
−0.602550 + 0.798081i \(0.705849\pi\)
\(978\) 0 0
\(979\) 20.0787i 0.0205094i
\(980\) 0 0
\(981\) 104.875i 0.106906i
\(982\) 0 0
\(983\) 635.108i 0.646091i 0.946383 + 0.323046i \(0.104707\pi\)
−0.946383 + 0.323046i \(0.895293\pi\)
\(984\) 0 0
\(985\) −502.680 −0.510335
\(986\) 0 0
\(987\) 262.295i 0.265750i
\(988\) 0 0
\(989\) −207.362 −0.209668
\(990\) 0 0
\(991\) 700.746i 0.707110i −0.935414 0.353555i \(-0.884973\pi\)
0.935414 0.353555i \(-0.115027\pi\)
\(992\) 0 0
\(993\) −892.702 −0.898995
\(994\) 0 0
\(995\) 217.921 0.219016
\(996\) 0 0
\(997\) −1462.90 −1.46730 −0.733649 0.679529i \(-0.762184\pi\)
−0.733649 + 0.679529i \(0.762184\pi\)
\(998\) 0 0
\(999\) −252.274 −0.252526
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 912.3.o.c.721.4 6
3.2 odd 2 2736.3.o.m.721.6 6
4.3 odd 2 228.3.h.a.37.1 6
12.11 even 2 684.3.h.e.37.6 6
19.18 odd 2 inner 912.3.o.c.721.1 6
57.56 even 2 2736.3.o.m.721.5 6
76.75 even 2 228.3.h.a.37.4 yes 6
228.227 odd 2 684.3.h.e.37.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.3.h.a.37.1 6 4.3 odd 2
228.3.h.a.37.4 yes 6 76.75 even 2
684.3.h.e.37.5 6 228.227 odd 2
684.3.h.e.37.6 6 12.11 even 2
912.3.o.c.721.1 6 19.18 odd 2 inner
912.3.o.c.721.4 6 1.1 even 1 trivial
2736.3.o.m.721.5 6 57.56 even 2
2736.3.o.m.721.6 6 3.2 odd 2