Properties

Label 320.6.c.g.129.2
Level $320$
Weight $6$
Character 320.129
Analytic conductor $51.323$
Analytic rank $0$
Dimension $2$
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] = [320,6,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.2
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.6.c.g.129.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+19.8997i q^{3} +(45.0000 + 33.1662i) q^{5} -59.6992i q^{7} -153.000 q^{9} +O(q^{10})\) \(q+19.8997i q^{3} +(45.0000 + 33.1662i) q^{5} -59.6992i q^{7} -153.000 q^{9} +252.000 q^{11} -119.398i q^{13} +(-660.000 + 895.489i) q^{15} +689.858i q^{17} -220.000 q^{19} +1188.00 q^{21} +2434.40i q^{23} +(925.000 + 2984.96i) q^{25} +1790.98i q^{27} +6930.00 q^{29} -6752.00 q^{31} +5014.74i q^{33} +(1980.00 - 2686.47i) q^{35} +13969.6i q^{37} +2376.00 q^{39} -198.000 q^{41} +417.895i q^{43} +(-6885.00 - 5074.44i) q^{45} -10540.2i q^{47} +13243.0 q^{49} -13728.0 q^{51} -5823.99i q^{53} +(11340.0 + 8357.89i) q^{55} -4377.94i q^{57} -24660.0 q^{59} +5698.00 q^{61} +9133.98i q^{63} +(3960.00 - 5372.93i) q^{65} +43640.1i q^{67} -48444.0 q^{69} -53352.0 q^{71} -70922.7i q^{73} +(-59400.0 + 18407.3i) q^{75} -15044.2i q^{77} -51920.0 q^{79} -72819.0 q^{81} +61841.8i q^{83} +(-22880.0 + 31043.6i) q^{85} +137905. i q^{87} -9990.00 q^{89} -7128.00 q^{91} -134363. i q^{93} +(-9900.00 - 7296.57i) q^{95} +101250. i q^{97} -38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 90 q^{5} - 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 90 q^{5} - 306 q^{9} + 504 q^{11} - 1320 q^{15} - 440 q^{19} + 2376 q^{21} + 1850 q^{25} + 13860 q^{29} - 13504 q^{31} + 3960 q^{35} + 4752 q^{39} - 396 q^{41} - 13770 q^{45} + 26486 q^{49} - 27456 q^{51} + 22680 q^{55} - 49320 q^{59} + 11396 q^{61} + 7920 q^{65} - 96888 q^{69} - 106704 q^{71} - 118800 q^{75} - 103840 q^{79} - 145638 q^{81} - 45760 q^{85} - 19980 q^{89} - 14256 q^{91} - 19800 q^{95} - 77112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 19.8997i 1.27657i 0.769800 + 0.638285i \(0.220356\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) 0 0
\(5\) 45.0000 + 33.1662i 0.804984 + 0.593296i
\(6\) 0 0
\(7\) 59.6992i 0.460494i −0.973132 0.230247i \(-0.926047\pi\)
0.973132 0.230247i \(-0.0739534\pi\)
\(8\) 0 0
\(9\) −153.000 −0.629630
\(10\) 0 0
\(11\) 252.000 0.627941 0.313970 0.949433i \(-0.398341\pi\)
0.313970 + 0.949433i \(0.398341\pi\)
\(12\) 0 0
\(13\) 119.398i 0.195948i −0.995189 0.0979739i \(-0.968764\pi\)
0.995189 0.0979739i \(-0.0312362\pi\)
\(14\) 0 0
\(15\) −660.000 + 895.489i −0.757383 + 1.02762i
\(16\) 0 0
\(17\) 689.858i 0.578945i 0.957186 + 0.289473i \(0.0934799\pi\)
−0.957186 + 0.289473i \(0.906520\pi\)
\(18\) 0 0
\(19\) −220.000 −0.139810 −0.0699051 0.997554i \(-0.522270\pi\)
−0.0699051 + 0.997554i \(0.522270\pi\)
\(20\) 0 0
\(21\) 1188.00 0.587852
\(22\) 0 0
\(23\) 2434.40i 0.959561i 0.877388 + 0.479781i \(0.159284\pi\)
−0.877388 + 0.479781i \(0.840716\pi\)
\(24\) 0 0
\(25\) 925.000 + 2984.96i 0.296000 + 0.955188i
\(26\) 0 0
\(27\) 1790.98i 0.472804i
\(28\) 0 0
\(29\) 6930.00 1.53016 0.765082 0.643932i \(-0.222698\pi\)
0.765082 + 0.643932i \(0.222698\pi\)
\(30\) 0 0
\(31\) −6752.00 −1.26191 −0.630955 0.775820i \(-0.717337\pi\)
−0.630955 + 0.775820i \(0.717337\pi\)
\(32\) 0 0
\(33\) 5014.74i 0.801610i
\(34\) 0 0
\(35\) 1980.00 2686.47i 0.273209 0.370690i
\(36\) 0 0
\(37\) 13969.6i 1.67757i 0.544464 + 0.838785i \(0.316733\pi\)
−0.544464 + 0.838785i \(0.683267\pi\)
\(38\) 0 0
\(39\) 2376.00 0.250141
\(40\) 0 0
\(41\) −198.000 −0.0183952 −0.00919762 0.999958i \(-0.502928\pi\)
−0.00919762 + 0.999958i \(0.502928\pi\)
\(42\) 0 0
\(43\) 417.895i 0.0344664i 0.999851 + 0.0172332i \(0.00548577\pi\)
−0.999851 + 0.0172332i \(0.994514\pi\)
\(44\) 0 0
\(45\) −6885.00 5074.44i −0.506842 0.373557i
\(46\) 0 0
\(47\) 10540.2i 0.695994i −0.937496 0.347997i \(-0.886862\pi\)
0.937496 0.347997i \(-0.113138\pi\)
\(48\) 0 0
\(49\) 13243.0 0.787945
\(50\) 0 0
\(51\) −13728.0 −0.739064
\(52\) 0 0
\(53\) 5823.99i 0.284794i −0.989810 0.142397i \(-0.954519\pi\)
0.989810 0.142397i \(-0.0454810\pi\)
\(54\) 0 0
\(55\) 11340.0 + 8357.89i 0.505483 + 0.372555i
\(56\) 0 0
\(57\) 4377.94i 0.178477i
\(58\) 0 0
\(59\) −24660.0 −0.922281 −0.461140 0.887327i \(-0.652560\pi\)
−0.461140 + 0.887327i \(0.652560\pi\)
\(60\) 0 0
\(61\) 5698.00 0.196064 0.0980320 0.995183i \(-0.468745\pi\)
0.0980320 + 0.995183i \(0.468745\pi\)
\(62\) 0 0
\(63\) 9133.98i 0.289941i
\(64\) 0 0
\(65\) 3960.00 5372.93i 0.116255 0.157735i
\(66\) 0 0
\(67\) 43640.1i 1.18768i 0.804583 + 0.593840i \(0.202389\pi\)
−0.804583 + 0.593840i \(0.797611\pi\)
\(68\) 0 0
\(69\) −48444.0 −1.22495
\(70\) 0 0
\(71\) −53352.0 −1.25604 −0.628022 0.778196i \(-0.716135\pi\)
−0.628022 + 0.778196i \(0.716135\pi\)
\(72\) 0 0
\(73\) 70922.7i 1.55768i −0.627223 0.778840i \(-0.715808\pi\)
0.627223 0.778840i \(-0.284192\pi\)
\(74\) 0 0
\(75\) −59400.0 + 18407.3i −1.21936 + 0.377865i
\(76\) 0 0
\(77\) 15044.2i 0.289163i
\(78\) 0 0
\(79\) −51920.0 −0.935981 −0.467990 0.883734i \(-0.655022\pi\)
−0.467990 + 0.883734i \(0.655022\pi\)
\(80\) 0 0
\(81\) −72819.0 −1.23320
\(82\) 0 0
\(83\) 61841.8i 0.985342i 0.870216 + 0.492671i \(0.163979\pi\)
−0.870216 + 0.492671i \(0.836021\pi\)
\(84\) 0 0
\(85\) −22880.0 + 31043.6i −0.343486 + 0.466042i
\(86\) 0 0
\(87\) 137905.i 1.95336i
\(88\) 0 0
\(89\) −9990.00 −0.133687 −0.0668437 0.997763i \(-0.521293\pi\)
−0.0668437 + 0.997763i \(0.521293\pi\)
\(90\) 0 0
\(91\) −7128.00 −0.0902328
\(92\) 0 0
\(93\) 134363.i 1.61092i
\(94\) 0 0
\(95\) −9900.00 7296.57i −0.112545 0.0829488i
\(96\) 0 0
\(97\) 101250.i 1.09261i 0.837586 + 0.546305i \(0.183966\pi\)
−0.837586 + 0.546305i \(0.816034\pi\)
\(98\) 0 0
\(99\) −38556.0 −0.395370
\(100\) 0 0
\(101\) 109098. 1.06418 0.532088 0.846689i \(-0.321408\pi\)
0.532088 + 0.846689i \(0.321408\pi\)
\(102\) 0 0
\(103\) 70624.2i 0.655935i 0.944689 + 0.327967i \(0.106364\pi\)
−0.944689 + 0.327967i \(0.893636\pi\)
\(104\) 0 0
\(105\) 53460.0 + 39401.5i 0.473212 + 0.348770i
\(106\) 0 0
\(107\) 97117.4i 0.820045i −0.912075 0.410022i \(-0.865521\pi\)
0.912075 0.410022i \(-0.134479\pi\)
\(108\) 0 0
\(109\) 21010.0 0.169379 0.0846895 0.996407i \(-0.473010\pi\)
0.0846895 + 0.996407i \(0.473010\pi\)
\(110\) 0 0
\(111\) −277992. −2.14153
\(112\) 0 0
\(113\) 105018.i 0.773688i −0.922145 0.386844i \(-0.873565\pi\)
0.922145 0.386844i \(-0.126435\pi\)
\(114\) 0 0
\(115\) −80740.0 + 109548.i −0.569304 + 0.772432i
\(116\) 0 0
\(117\) 18268.0i 0.123375i
\(118\) 0 0
\(119\) 41184.0 0.266601
\(120\) 0 0
\(121\) −97547.0 −0.605690
\(122\) 0 0
\(123\) 3940.15i 0.0234828i
\(124\) 0 0
\(125\) −57375.0 + 165002.i −0.328434 + 0.944527i
\(126\) 0 0
\(127\) 87220.6i 0.479855i −0.970791 0.239927i \(-0.922876\pi\)
0.970791 0.239927i \(-0.0771236\pi\)
\(128\) 0 0
\(129\) −8316.00 −0.0439987
\(130\) 0 0
\(131\) 192852. 0.981852 0.490926 0.871201i \(-0.336659\pi\)
0.490926 + 0.871201i \(0.336659\pi\)
\(132\) 0 0
\(133\) 13133.8i 0.0643817i
\(134\) 0 0
\(135\) −59400.0 + 80594.0i −0.280512 + 0.380599i
\(136\) 0 0
\(137\) 143570.i 0.653525i 0.945106 + 0.326763i \(0.105958\pi\)
−0.945106 + 0.326763i \(0.894042\pi\)
\(138\) 0 0
\(139\) −318340. −1.39751 −0.698754 0.715362i \(-0.746262\pi\)
−0.698754 + 0.715362i \(0.746262\pi\)
\(140\) 0 0
\(141\) 209748. 0.888485
\(142\) 0 0
\(143\) 30088.4i 0.123044i
\(144\) 0 0
\(145\) 311850. + 229842.i 1.23176 + 0.907841i
\(146\) 0 0
\(147\) 263532.i 1.00587i
\(148\) 0 0
\(149\) −84150.0 −0.310519 −0.155260 0.987874i \(-0.549621\pi\)
−0.155260 + 0.987874i \(0.549621\pi\)
\(150\) 0 0
\(151\) 155848. 0.556236 0.278118 0.960547i \(-0.410289\pi\)
0.278118 + 0.960547i \(0.410289\pi\)
\(152\) 0 0
\(153\) 105548.i 0.364521i
\(154\) 0 0
\(155\) −303840. 223939.i −1.01582 0.748686i
\(156\) 0 0
\(157\) 356643.i 1.15474i 0.816482 + 0.577371i \(0.195921\pi\)
−0.816482 + 0.577371i \(0.804079\pi\)
\(158\) 0 0
\(159\) 115896. 0.363560
\(160\) 0 0
\(161\) 145332. 0.441872
\(162\) 0 0
\(163\) 144890.i 0.427139i 0.976928 + 0.213570i \(0.0685090\pi\)
−0.976928 + 0.213570i \(0.931491\pi\)
\(164\) 0 0
\(165\) −166320. + 225663.i −0.475592 + 0.645284i
\(166\) 0 0
\(167\) 18102.1i 0.0502272i −0.999685 0.0251136i \(-0.992005\pi\)
0.999685 0.0251136i \(-0.00799474\pi\)
\(168\) 0 0
\(169\) 357037. 0.961604
\(170\) 0 0
\(171\) 33660.0 0.0880286
\(172\) 0 0
\(173\) 492572.i 1.25128i −0.780112 0.625640i \(-0.784838\pi\)
0.780112 0.625640i \(-0.215162\pi\)
\(174\) 0 0
\(175\) 178200. 55221.8i 0.439858 0.136306i
\(176\) 0 0
\(177\) 490728.i 1.17736i
\(178\) 0 0
\(179\) 444420. 1.03672 0.518359 0.855163i \(-0.326543\pi\)
0.518359 + 0.855163i \(0.326543\pi\)
\(180\) 0 0
\(181\) −156902. −0.355985 −0.177993 0.984032i \(-0.556960\pi\)
−0.177993 + 0.984032i \(0.556960\pi\)
\(182\) 0 0
\(183\) 113389.i 0.250289i
\(184\) 0 0
\(185\) −463320. + 628633.i −0.995295 + 1.35042i
\(186\) 0 0
\(187\) 173844.i 0.363543i
\(188\) 0 0
\(189\) 106920. 0.217723
\(190\) 0 0
\(191\) −332352. −0.659196 −0.329598 0.944121i \(-0.606913\pi\)
−0.329598 + 0.944121i \(0.606913\pi\)
\(192\) 0 0
\(193\) 786120.i 1.51913i −0.650430 0.759566i \(-0.725411\pi\)
0.650430 0.759566i \(-0.274589\pi\)
\(194\) 0 0
\(195\) 106920. + 78803.0i 0.201360 + 0.148408i
\(196\) 0 0
\(197\) 59606.4i 0.109428i 0.998502 + 0.0547138i \(0.0174247\pi\)
−0.998502 + 0.0547138i \(0.982575\pi\)
\(198\) 0 0
\(199\) 395800. 0.708505 0.354253 0.935150i \(-0.384735\pi\)
0.354253 + 0.935150i \(0.384735\pi\)
\(200\) 0 0
\(201\) −868428. −1.51616
\(202\) 0 0
\(203\) 413716.i 0.704631i
\(204\) 0 0
\(205\) −8910.00 6566.92i −0.0148079 0.0109138i
\(206\) 0 0
\(207\) 372464.i 0.604168i
\(208\) 0 0
\(209\) −55440.0 −0.0877925
\(210\) 0 0
\(211\) −251548. −0.388969 −0.194484 0.980906i \(-0.562303\pi\)
−0.194484 + 0.980906i \(0.562303\pi\)
\(212\) 0 0
\(213\) 1.06169e6i 1.60343i
\(214\) 0 0
\(215\) −13860.0 + 18805.3i −0.0204488 + 0.0277449i
\(216\) 0 0
\(217\) 403089.i 0.581101i
\(218\) 0 0
\(219\) 1.41134e6 1.98849
\(220\) 0 0
\(221\) 82368.0 0.113443
\(222\) 0 0
\(223\) 288765.i 0.388851i −0.980917 0.194425i \(-0.937716\pi\)
0.980917 0.194425i \(-0.0622842\pi\)
\(224\) 0 0
\(225\) −141525. 456699.i −0.186370 0.601415i
\(226\) 0 0
\(227\) 1.16414e6i 1.49948i −0.661731 0.749741i \(-0.730178\pi\)
0.661731 0.749741i \(-0.269822\pi\)
\(228\) 0 0
\(229\) −547670. −0.690129 −0.345064 0.938579i \(-0.612143\pi\)
−0.345064 + 0.938579i \(0.612143\pi\)
\(230\) 0 0
\(231\) 299376. 0.369137
\(232\) 0 0
\(233\) 48104.3i 0.0580489i −0.999579 0.0290245i \(-0.990760\pi\)
0.999579 0.0290245i \(-0.00924007\pi\)
\(234\) 0 0
\(235\) 349580. 474311.i 0.412930 0.560264i
\(236\) 0 0
\(237\) 1.03319e6i 1.19484i
\(238\) 0 0
\(239\) 1.00584e6 1.13903 0.569514 0.821982i \(-0.307132\pi\)
0.569514 + 0.821982i \(0.307132\pi\)
\(240\) 0 0
\(241\) 895202. 0.992838 0.496419 0.868083i \(-0.334648\pi\)
0.496419 + 0.868083i \(0.334648\pi\)
\(242\) 0 0
\(243\) 1.01387e6i 1.10146i
\(244\) 0 0
\(245\) 595935. + 439221.i 0.634284 + 0.467485i
\(246\) 0 0
\(247\) 26267.7i 0.0273955i
\(248\) 0 0
\(249\) −1.23064e6 −1.25786
\(250\) 0 0
\(251\) 558252. 0.559301 0.279651 0.960102i \(-0.409781\pi\)
0.279651 + 0.960102i \(0.409781\pi\)
\(252\) 0 0
\(253\) 613469.i 0.602548i
\(254\) 0 0
\(255\) −617760. 455306.i −0.594935 0.438483i
\(256\) 0 0
\(257\) 787924.i 0.744135i 0.928206 + 0.372067i \(0.121351\pi\)
−0.928206 + 0.372067i \(0.878649\pi\)
\(258\) 0 0
\(259\) 833976. 0.772510
\(260\) 0 0
\(261\) −1.06029e6 −0.963437
\(262\) 0 0
\(263\) 1.63173e6i 1.45465i −0.686291 0.727327i \(-0.740762\pi\)
0.686291 0.727327i \(-0.259238\pi\)
\(264\) 0 0
\(265\) 193160. 262080.i 0.168967 0.229255i
\(266\) 0 0
\(267\) 198798.i 0.170661i
\(268\) 0 0
\(269\) 1.73637e6 1.46306 0.731529 0.681810i \(-0.238807\pi\)
0.731529 + 0.681810i \(0.238807\pi\)
\(270\) 0 0
\(271\) 1.72005e6 1.42271 0.711357 0.702831i \(-0.248081\pi\)
0.711357 + 0.702831i \(0.248081\pi\)
\(272\) 0 0
\(273\) 141845.i 0.115188i
\(274\) 0 0
\(275\) 233100. + 752211.i 0.185871 + 0.599802i
\(276\) 0 0
\(277\) 1.27243e6i 0.996402i 0.867062 + 0.498201i \(0.166006\pi\)
−0.867062 + 0.498201i \(0.833994\pi\)
\(278\) 0 0
\(279\) 1.03306e6 0.794536
\(280\) 0 0
\(281\) 1.46500e6 1.10681 0.553404 0.832913i \(-0.313329\pi\)
0.553404 + 0.832913i \(0.313329\pi\)
\(282\) 0 0
\(283\) 1.65051e6i 1.22504i 0.790455 + 0.612521i \(0.209844\pi\)
−0.790455 + 0.612521i \(0.790156\pi\)
\(284\) 0 0
\(285\) 145200. 197008.i 0.105890 0.143672i
\(286\) 0 0
\(287\) 11820.5i 0.00847089i
\(288\) 0 0
\(289\) 943953. 0.664823
\(290\) 0 0
\(291\) −2.01485e6 −1.39479
\(292\) 0 0
\(293\) 2.38772e6i 1.62485i 0.583064 + 0.812426i \(0.301854\pi\)
−0.583064 + 0.812426i \(0.698146\pi\)
\(294\) 0 0
\(295\) −1.10970e6 817880.i −0.742422 0.547185i
\(296\) 0 0
\(297\) 451326.i 0.296893i
\(298\) 0 0
\(299\) 290664. 0.188024
\(300\) 0 0
\(301\) 24948.0 0.0158716
\(302\) 0 0
\(303\) 2.17102e6i 1.35849i
\(304\) 0 0
\(305\) 256410. + 188981.i 0.157828 + 0.116324i
\(306\) 0 0
\(307\) 928264.i 0.562115i −0.959691 0.281058i \(-0.909315\pi\)
0.959691 0.281058i \(-0.0906852\pi\)
\(308\) 0 0
\(309\) −1.40540e6 −0.837346
\(310\) 0 0
\(311\) −568152. −0.333092 −0.166546 0.986034i \(-0.553261\pi\)
−0.166546 + 0.986034i \(0.553261\pi\)
\(312\) 0 0
\(313\) 1.72244e6i 0.993766i 0.867818 + 0.496883i \(0.165522\pi\)
−0.867818 + 0.496883i \(0.834478\pi\)
\(314\) 0 0
\(315\) −302940. + 411029.i −0.172021 + 0.233398i
\(316\) 0 0
\(317\) 131643.i 0.0735785i 0.999323 + 0.0367893i \(0.0117130\pi\)
−0.999323 + 0.0367893i \(0.988287\pi\)
\(318\) 0 0
\(319\) 1.74636e6 0.960853
\(320\) 0 0
\(321\) 1.93261e6 1.04684
\(322\) 0 0
\(323\) 151769.i 0.0809424i
\(324\) 0 0
\(325\) 356400. 110444.i 0.187167 0.0580006i
\(326\) 0 0
\(327\) 418094.i 0.216224i
\(328\) 0 0
\(329\) −629244. −0.320501
\(330\) 0 0
\(331\) −1.58055e6 −0.792935 −0.396468 0.918049i \(-0.629764\pi\)
−0.396468 + 0.918049i \(0.629764\pi\)
\(332\) 0 0
\(333\) 2.13735e6i 1.05625i
\(334\) 0 0
\(335\) −1.44738e6 + 1.96381e6i −0.704645 + 0.956063i
\(336\) 0 0
\(337\) 1.22885e6i 0.589419i −0.955587 0.294709i \(-0.904777\pi\)
0.955587 0.294709i \(-0.0952228\pi\)
\(338\) 0 0
\(339\) 2.08982e6 0.987667
\(340\) 0 0
\(341\) −1.70150e6 −0.792405
\(342\) 0 0
\(343\) 1.79396e6i 0.823338i
\(344\) 0 0
\(345\) −2.17998e6 1.60671e6i −0.986063 0.726756i
\(346\) 0 0
\(347\) 3.84224e6i 1.71301i 0.516137 + 0.856506i \(0.327370\pi\)
−0.516137 + 0.856506i \(0.672630\pi\)
\(348\) 0 0
\(349\) 1.59445e6 0.700725 0.350362 0.936614i \(-0.386058\pi\)
0.350362 + 0.936614i \(0.386058\pi\)
\(350\) 0 0
\(351\) 213840. 0.0926448
\(352\) 0 0
\(353\) 295365.i 0.126160i 0.998008 + 0.0630802i \(0.0200924\pi\)
−0.998008 + 0.0630802i \(0.979908\pi\)
\(354\) 0 0
\(355\) −2.40084e6 1.76949e6i −1.01110 0.745206i
\(356\) 0 0
\(357\) 819551.i 0.340334i
\(358\) 0 0
\(359\) −1.10484e6 −0.452442 −0.226221 0.974076i \(-0.572637\pi\)
−0.226221 + 0.974076i \(0.572637\pi\)
\(360\) 0 0
\(361\) −2.42770e6 −0.980453
\(362\) 0 0
\(363\) 1.94116e6i 0.773206i
\(364\) 0 0
\(365\) 2.35224e6 3.19152e6i 0.924165 1.25391i
\(366\) 0 0
\(367\) 1.83760e6i 0.712174i −0.934453 0.356087i \(-0.884111\pi\)
0.934453 0.356087i \(-0.115889\pi\)
\(368\) 0 0
\(369\) 30294.0 0.0115822
\(370\) 0 0
\(371\) −347688. −0.131146
\(372\) 0 0
\(373\) 2.93350e6i 1.09173i 0.837874 + 0.545864i \(0.183798\pi\)
−0.837874 + 0.545864i \(0.816202\pi\)
\(374\) 0 0
\(375\) −3.28350e6 1.14175e6i −1.20575 0.419268i
\(376\) 0 0
\(377\) 827432.i 0.299832i
\(378\) 0 0
\(379\) 5.09342e6 1.82143 0.910713 0.413040i \(-0.135533\pi\)
0.910713 + 0.413040i \(0.135533\pi\)
\(380\) 0 0
\(381\) 1.73567e6 0.612568
\(382\) 0 0
\(383\) 3.17485e6i 1.10593i 0.833205 + 0.552964i \(0.186503\pi\)
−0.833205 + 0.552964i \(0.813497\pi\)
\(384\) 0 0
\(385\) 498960. 676989.i 0.171559 0.232772i
\(386\) 0 0
\(387\) 63937.9i 0.0217011i
\(388\) 0 0
\(389\) −1.79991e6 −0.603083 −0.301541 0.953453i \(-0.597501\pi\)
−0.301541 + 0.953453i \(0.597501\pi\)
\(390\) 0 0
\(391\) −1.67939e6 −0.555533
\(392\) 0 0
\(393\) 3.83771e6i 1.25340i
\(394\) 0 0
\(395\) −2.33640e6 1.72199e6i −0.753450 0.555314i
\(396\) 0 0
\(397\) 4.90405e6i 1.56163i −0.624760 0.780817i \(-0.714803\pi\)
0.624760 0.780817i \(-0.285197\pi\)
\(398\) 0 0
\(399\) −261360. −0.0821877
\(400\) 0 0
\(401\) −642798. −0.199624 −0.0998122 0.995006i \(-0.531824\pi\)
−0.0998122 + 0.995006i \(0.531824\pi\)
\(402\) 0 0
\(403\) 806179.i 0.247268i
\(404\) 0 0
\(405\) −3.27686e6 2.41513e6i −0.992704 0.731650i
\(406\) 0 0
\(407\) 3.52035e6i 1.05341i
\(408\) 0 0
\(409\) −2.05711e6 −0.608064 −0.304032 0.952662i \(-0.598333\pi\)
−0.304032 + 0.952662i \(0.598333\pi\)
\(410\) 0 0
\(411\) −2.85701e6 −0.834271
\(412\) 0 0
\(413\) 1.47218e6i 0.424704i
\(414\) 0 0
\(415\) −2.05106e6 + 2.78288e6i −0.584599 + 0.793185i
\(416\) 0 0
\(417\) 6.33489e6i 1.78402i
\(418\) 0 0
\(419\) −2.93742e6 −0.817393 −0.408697 0.912670i \(-0.634017\pi\)
−0.408697 + 0.912670i \(0.634017\pi\)
\(420\) 0 0
\(421\) −2.71770e6 −0.747303 −0.373651 0.927569i \(-0.621894\pi\)
−0.373651 + 0.927569i \(0.621894\pi\)
\(422\) 0 0
\(423\) 1.61266e6i 0.438219i
\(424\) 0 0
\(425\) −2.05920e6 + 638119.i −0.553001 + 0.171368i
\(426\) 0 0
\(427\) 340166.i 0.0902862i
\(428\) 0 0
\(429\) 598752. 0.157074
\(430\) 0 0
\(431\) −4.99435e6 −1.29505 −0.647524 0.762045i \(-0.724196\pi\)
−0.647524 + 0.762045i \(0.724196\pi\)
\(432\) 0 0
\(433\) 2.08183e6i 0.533612i 0.963750 + 0.266806i \(0.0859684\pi\)
−0.963750 + 0.266806i \(0.914032\pi\)
\(434\) 0 0
\(435\) −4.57380e6 + 6.20574e6i −1.15892 + 1.57243i
\(436\) 0 0
\(437\) 535569.i 0.134156i
\(438\) 0 0
\(439\) 4.70404e6 1.16496 0.582478 0.812846i \(-0.302083\pi\)
0.582478 + 0.812846i \(0.302083\pi\)
\(440\) 0 0
\(441\) −2.02618e6 −0.496114
\(442\) 0 0
\(443\) 5.70103e6i 1.38021i 0.723711 + 0.690103i \(0.242435\pi\)
−0.723711 + 0.690103i \(0.757565\pi\)
\(444\) 0 0
\(445\) −449550. 331331.i −0.107616 0.0793162i
\(446\) 0 0
\(447\) 1.67456e6i 0.396399i
\(448\) 0 0
\(449\) 6.20325e6 1.45212 0.726062 0.687630i \(-0.241349\pi\)
0.726062 + 0.687630i \(0.241349\pi\)
\(450\) 0 0
\(451\) −49896.0 −0.0115511
\(452\) 0 0
\(453\) 3.10134e6i 0.710074i
\(454\) 0 0
\(455\) −320760. 236409.i −0.0726360 0.0535347i
\(456\) 0 0
\(457\) 2.15371e6i 0.482388i 0.970477 + 0.241194i \(0.0775391\pi\)
−0.970477 + 0.241194i \(0.922461\pi\)
\(458\) 0 0
\(459\) −1.23552e6 −0.273727
\(460\) 0 0
\(461\) 3.85130e6 0.844024 0.422012 0.906590i \(-0.361324\pi\)
0.422012 + 0.906590i \(0.361324\pi\)
\(462\) 0 0
\(463\) 2.08213e6i 0.451394i −0.974198 0.225697i \(-0.927534\pi\)
0.974198 0.225697i \(-0.0724659\pi\)
\(464\) 0 0
\(465\) 4.45632e6 6.04634e6i 0.955749 1.29676i
\(466\) 0 0
\(467\) 1.30822e6i 0.277579i −0.990322 0.138790i \(-0.955679\pi\)
0.990322 0.138790i \(-0.0443212\pi\)
\(468\) 0 0
\(469\) 2.60528e6 0.546919
\(470\) 0 0
\(471\) −7.09711e6 −1.47411
\(472\) 0 0
\(473\) 105309.i 0.0216429i
\(474\) 0 0
\(475\) −203500. 656692.i −0.0413838 0.133545i
\(476\) 0 0
\(477\) 891071.i 0.179315i
\(478\) 0 0
\(479\) 6.76368e6 1.34693 0.673464 0.739220i \(-0.264806\pi\)
0.673464 + 0.739220i \(0.264806\pi\)
\(480\) 0 0
\(481\) 1.66795e6 0.328716
\(482\) 0 0
\(483\) 2.89207e6i 0.564080i
\(484\) 0 0
\(485\) −3.35808e6 + 4.55625e6i −0.648241 + 0.879534i
\(486\) 0 0
\(487\) 6.67193e6i 1.27476i 0.770549 + 0.637381i \(0.219982\pi\)
−0.770549 + 0.637381i \(0.780018\pi\)
\(488\) 0 0
\(489\) −2.88328e6 −0.545273
\(490\) 0 0
\(491\) −6.87575e6 −1.28711 −0.643556 0.765399i \(-0.722542\pi\)
−0.643556 + 0.765399i \(0.722542\pi\)
\(492\) 0 0
\(493\) 4.78072e6i 0.885881i
\(494\) 0 0
\(495\) −1.73502e6 1.27876e6i −0.318267 0.234572i
\(496\) 0 0
\(497\) 3.18507e6i 0.578400i
\(498\) 0 0
\(499\) 6.94010e6 1.24771 0.623856 0.781539i \(-0.285565\pi\)
0.623856 + 0.781539i \(0.285565\pi\)
\(500\) 0 0
\(501\) 360228. 0.0641185
\(502\) 0 0
\(503\) 921007.i 0.162309i 0.996702 + 0.0811546i \(0.0258607\pi\)
−0.996702 + 0.0811546i \(0.974139\pi\)
\(504\) 0 0
\(505\) 4.90941e6 + 3.61837e6i 0.856645 + 0.631371i
\(506\) 0 0
\(507\) 7.10495e6i 1.22755i
\(508\) 0 0
\(509\) −4.97979e6 −0.851955 −0.425977 0.904734i \(-0.640070\pi\)
−0.425977 + 0.904734i \(0.640070\pi\)
\(510\) 0 0
\(511\) −4.23403e6 −0.717302
\(512\) 0 0
\(513\) 394015.i 0.0661027i
\(514\) 0 0
\(515\) −2.34234e6 + 3.17809e6i −0.389163 + 0.528017i
\(516\) 0 0
\(517\) 2.65614e6i 0.437043i
\(518\) 0 0
\(519\) 9.80206e6 1.59735
\(520\) 0 0
\(521\) −147798. −0.0238547 −0.0119274 0.999929i \(-0.503797\pi\)
−0.0119274 + 0.999929i \(0.503797\pi\)
\(522\) 0 0
\(523\) 1.23884e7i 1.98043i −0.139543 0.990216i \(-0.544563\pi\)
0.139543 0.990216i \(-0.455437\pi\)
\(524\) 0 0
\(525\) 1.09890e6 + 3.54614e6i 0.174004 + 0.561509i
\(526\) 0 0
\(527\) 4.65792e6i 0.730576i
\(528\) 0 0
\(529\) 510027. 0.0792417
\(530\) 0 0
\(531\) 3.77298e6 0.580695
\(532\) 0 0
\(533\) 23640.9i 0.00360451i
\(534\) 0 0
\(535\) 3.22102e6 4.37028e6i 0.486529 0.660123i
\(536\) 0 0
\(537\) 8.84385e6i 1.32344i
\(538\) 0 0
\(539\) 3.33724e6 0.494783
\(540\) 0 0
\(541\) 9.99810e6 1.46867 0.734335 0.678787i \(-0.237494\pi\)
0.734335 + 0.678787i \(0.237494\pi\)
\(542\) 0 0
\(543\) 3.12231e6i 0.454440i
\(544\) 0 0
\(545\) 945450. + 696823.i 0.136348 + 0.100492i
\(546\) 0 0
\(547\) 1.18580e7i 1.69451i −0.531189 0.847253i \(-0.678255\pi\)
0.531189 0.847253i \(-0.321745\pi\)
\(548\) 0 0
\(549\) −871794. −0.123448
\(550\) 0 0
\(551\) −1.52460e6 −0.213933
\(552\) 0 0
\(553\) 3.09958e6i 0.431013i
\(554\) 0 0
\(555\) −1.25096e7 9.21995e6i −1.72390 1.27056i
\(556\) 0 0
\(557\) 904550.i 0.123536i 0.998091 + 0.0617681i \(0.0196739\pi\)
−0.998091 + 0.0617681i \(0.980326\pi\)
\(558\) 0 0
\(559\) 49896.0 0.00675361
\(560\) 0 0
\(561\) −3.45946e6 −0.464088
\(562\) 0 0
\(563\) 8.68719e6i 1.15507i 0.816366 + 0.577535i \(0.195985\pi\)
−0.816366 + 0.577535i \(0.804015\pi\)
\(564\) 0 0
\(565\) 3.48304e6 4.72579e6i 0.459026 0.622807i
\(566\) 0 0
\(567\) 4.34724e6i 0.567879i
\(568\) 0 0
\(569\) −2.27007e6 −0.293940 −0.146970 0.989141i \(-0.546952\pi\)
−0.146970 + 0.989141i \(0.546952\pi\)
\(570\) 0 0
\(571\) 1.43807e7 1.84582 0.922908 0.385021i \(-0.125806\pi\)
0.922908 + 0.385021i \(0.125806\pi\)
\(572\) 0 0
\(573\) 6.61372e6i 0.841510i
\(574\) 0 0
\(575\) −7.26660e6 + 2.25182e6i −0.916562 + 0.284030i
\(576\) 0 0
\(577\) 5.63943e6i 0.705173i −0.935779 0.352586i \(-0.885302\pi\)
0.935779 0.352586i \(-0.114698\pi\)
\(578\) 0 0
\(579\) 1.56436e7 1.93928
\(580\) 0 0
\(581\) 3.69191e6 0.453744
\(582\) 0 0
\(583\) 1.46765e6i 0.178834i
\(584\) 0 0
\(585\) −605880. + 822059.i −0.0731976 + 0.0993146i
\(586\) 0 0
\(587\) 1.28473e6i 0.153893i −0.997035 0.0769464i \(-0.975483\pi\)
0.997035 0.0769464i \(-0.0245170\pi\)
\(588\) 0 0
\(589\) 1.48544e6 0.176428
\(590\) 0 0
\(591\) −1.18615e6 −0.139692
\(592\) 0 0
\(593\) 7.00943e6i 0.818552i −0.912411 0.409276i \(-0.865781\pi\)
0.912411 0.409276i \(-0.134219\pi\)
\(594\) 0 0
\(595\) 1.85328e6 + 1.36592e6i 0.214609 + 0.158173i
\(596\) 0 0
\(597\) 7.87632e6i 0.904456i
\(598\) 0 0
\(599\) 8.80020e6 1.00213 0.501067 0.865409i \(-0.332941\pi\)
0.501067 + 0.865409i \(0.332941\pi\)
\(600\) 0 0
\(601\) −1.07670e7 −1.21593 −0.607965 0.793964i \(-0.708014\pi\)
−0.607965 + 0.793964i \(0.708014\pi\)
\(602\) 0 0
\(603\) 6.67694e6i 0.747798i
\(604\) 0 0
\(605\) −4.38962e6 3.23527e6i −0.487571 0.359353i
\(606\) 0 0
\(607\) 1.51219e7i 1.66584i −0.553391 0.832921i \(-0.686667\pi\)
0.553391 0.832921i \(-0.313333\pi\)
\(608\) 0 0
\(609\) 8.23284e6 0.899511
\(610\) 0 0
\(611\) −1.25849e6 −0.136379
\(612\) 0 0
\(613\) 8.31622e6i 0.893871i −0.894566 0.446936i \(-0.852515\pi\)
0.894566 0.446936i \(-0.147485\pi\)
\(614\) 0 0
\(615\) 130680. 177307.i 0.0139323 0.0189033i
\(616\) 0 0
\(617\) 1.21083e7i 1.28047i −0.768178 0.640237i \(-0.778836\pi\)
0.768178 0.640237i \(-0.221164\pi\)
\(618\) 0 0
\(619\) 9.73238e6 1.02092 0.510461 0.859901i \(-0.329475\pi\)
0.510461 + 0.859901i \(0.329475\pi\)
\(620\) 0 0
\(621\) −4.35996e6 −0.453684
\(622\) 0 0
\(623\) 596395.i 0.0615622i
\(624\) 0 0
\(625\) −8.05437e6 + 5.52218e6i −0.824768 + 0.565471i
\(626\) 0 0
\(627\) 1.10324e6i 0.112073i
\(628\) 0 0
\(629\) −9.63706e6 −0.971220
\(630\) 0 0
\(631\) 8.60145e6 0.859999 0.430000 0.902829i \(-0.358514\pi\)
0.430000 + 0.902829i \(0.358514\pi\)
\(632\) 0 0
\(633\) 5.00574e6i 0.496546i
\(634\) 0 0
\(635\) 2.89278e6 3.92493e6i 0.284696 0.386276i
\(636\) 0 0
\(637\) 1.58119e6i 0.154396i
\(638\) 0 0
\(639\) 8.16286e6 0.790842
\(640\) 0 0
\(641\) −6.42440e6 −0.617572 −0.308786 0.951132i \(-0.599923\pi\)
−0.308786 + 0.951132i \(0.599923\pi\)
\(642\) 0 0
\(643\) 3.64721e6i 0.347883i −0.984756 0.173941i \(-0.944350\pi\)
0.984756 0.173941i \(-0.0556503\pi\)
\(644\) 0 0
\(645\) −374220. 275811.i −0.0354183 0.0261043i
\(646\) 0 0
\(647\) 3.78036e6i 0.355036i 0.984118 + 0.177518i \(0.0568068\pi\)
−0.984118 + 0.177518i \(0.943193\pi\)
\(648\) 0 0
\(649\) −6.21432e6 −0.579138
\(650\) 0 0
\(651\) −8.02138e6 −0.741816
\(652\) 0 0
\(653\) 1.66957e7i 1.53223i −0.642706 0.766113i \(-0.722188\pi\)
0.642706 0.766113i \(-0.277812\pi\)
\(654\) 0 0
\(655\) 8.67834e6 + 6.39618e6i 0.790375 + 0.582529i
\(656\) 0 0
\(657\) 1.08512e7i 0.980761i
\(658\) 0 0
\(659\) −1.22166e6 −0.109581 −0.0547907 0.998498i \(-0.517449\pi\)
−0.0547907 + 0.998498i \(0.517449\pi\)
\(660\) 0 0
\(661\) −1.62789e7 −1.44918 −0.724589 0.689182i \(-0.757970\pi\)
−0.724589 + 0.689182i \(0.757970\pi\)
\(662\) 0 0
\(663\) 1.63910e6i 0.144818i
\(664\) 0 0
\(665\) −435600. + 591023.i −0.0381974 + 0.0518263i
\(666\) 0 0
\(667\) 1.68704e7i 1.46829i
\(668\) 0 0
\(669\) 5.74636e6 0.496395
\(670\) 0 0
\(671\) 1.43590e6 0.123117
\(672\) 0 0
\(673\) 1.43928e7i 1.22492i −0.790503 0.612459i \(-0.790181\pi\)
0.790503 0.612459i \(-0.209819\pi\)
\(674\) 0 0
\(675\) −5.34600e6 + 1.65665e6i −0.451616 + 0.139950i
\(676\) 0 0
\(677\) 2.62429e6i 0.220059i 0.993928 + 0.110030i \(0.0350946\pi\)
−0.993928 + 0.110030i \(0.964905\pi\)
\(678\) 0 0
\(679\) 6.04454e6 0.503140
\(680\) 0 0
\(681\) 2.31661e7 1.91419
\(682\) 0 0
\(683\) 1.03039e7i 0.845184i −0.906320 0.422592i \(-0.861120\pi\)
0.906320 0.422592i \(-0.138880\pi\)
\(684\) 0 0
\(685\) −4.76168e6 + 6.46065e6i −0.387734 + 0.526078i
\(686\) 0 0
\(687\) 1.08985e7i 0.880998i
\(688\) 0 0
\(689\) −695376. −0.0558048
\(690\) 0 0
\(691\) 4.50285e6 0.358751 0.179375 0.983781i \(-0.442592\pi\)
0.179375 + 0.983781i \(0.442592\pi\)
\(692\) 0 0
\(693\) 2.30176e6i 0.182066i
\(694\) 0 0
\(695\) −1.43253e7 1.05581e7i −1.12497 0.829136i
\(696\) 0 0
\(697\) 136592.i 0.0106498i
\(698\) 0 0
\(699\) 957264. 0.0741035
\(700\) 0 0
\(701\) 4.88090e6 0.375150 0.187575 0.982250i \(-0.439937\pi\)
0.187575 + 0.982250i \(0.439937\pi\)
\(702\) 0 0
\(703\) 3.07332e6i 0.234541i
\(704\) 0 0
\(705\) 9.43866e6 + 6.95655e6i 0.715217 + 0.527134i
\(706\) 0 0
\(707\) 6.51307e6i 0.490046i
\(708\) 0 0
\(709\) 9.96961e6 0.744839 0.372420 0.928064i \(-0.378528\pi\)
0.372420 + 0.928064i \(0.378528\pi\)
\(710\) 0 0
\(711\) 7.94376e6 0.589321
\(712\) 0 0
\(713\) 1.64371e7i 1.21088i
\(714\) 0 0
\(715\) 997920. 1.35398e6i 0.0730013 0.0990482i
\(716\) 0 0
\(717\) 2.00160e7i 1.45405i
\(718\) 0 0
\(719\) 1.19167e7 0.859675 0.429838 0.902906i \(-0.358571\pi\)
0.429838 + 0.902906i \(0.358571\pi\)
\(720\) 0 0
\(721\) 4.21621e6 0.302054
\(722\) 0 0
\(723\) 1.78143e7i 1.26743i
\(724\) 0 0
\(725\) 6.41025e6 + 2.06858e7i 0.452929 + 1.46160i
\(726\) 0 0
\(727\) 1.38269e6i 0.0970264i −0.998823 0.0485132i \(-0.984552\pi\)
0.998823 0.0485132i \(-0.0154483\pi\)
\(728\) 0 0
\(729\) 2.48079e6 0.172890
\(730\) 0 0
\(731\) −288288. −0.0199541
\(732\) 0 0
\(733\) 6.09661e6i 0.419110i −0.977797 0.209555i \(-0.932798\pi\)
0.977797 0.209555i \(-0.0672016\pi\)
\(734\) 0 0
\(735\) −8.74038e6 + 1.18590e7i −0.596777 + 0.809707i
\(736\) 0 0
\(737\) 1.09973e7i 0.745793i
\(738\) 0 0
\(739\) 6.16946e6 0.415562 0.207781 0.978175i \(-0.433376\pi\)
0.207781 + 0.978175i \(0.433376\pi\)
\(740\) 0 0
\(741\) −522720. −0.0349723
\(742\) 0 0
\(743\) 1.57574e7i 1.04716i −0.851978 0.523578i \(-0.824597\pi\)
0.851978 0.523578i \(-0.175403\pi\)
\(744\) 0 0
\(745\) −3.78675e6 2.79094e6i −0.249963 0.184230i
\(746\) 0 0
\(747\) 9.46179e6i 0.620400i
\(748\) 0 0
\(749\) −5.79784e6 −0.377626
\(750\) 0 0
\(751\) 1.51816e7 0.982243 0.491122 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(752\) 0 0
\(753\) 1.11091e7i 0.713987i
\(754\) 0 0
\(755\) 7.01316e6 + 5.16889e6i 0.447761 + 0.330012i
\(756\) 0 0
\(757\) 652274.i 0.0413705i −0.999786 0.0206852i \(-0.993415\pi\)
0.999786 0.0206852i \(-0.00658478\pi\)
\(758\) 0 0
\(759\) −1.22079e7 −0.769194
\(760\) 0 0
\(761\) 4.51420e6 0.282566 0.141283 0.989969i \(-0.454877\pi\)
0.141283 + 0.989969i \(0.454877\pi\)
\(762\) 0 0
\(763\) 1.25428e6i 0.0779980i
\(764\) 0 0
\(765\) 3.50064e6 4.74967e6i 0.216269 0.293434i
\(766\) 0 0
\(767\) 2.94437e6i 0.180719i
\(768\) 0 0
\(769\) −1.20799e7 −0.736625 −0.368312 0.929702i \(-0.620064\pi\)
−0.368312 + 0.929702i \(0.620064\pi\)
\(770\) 0 0
\(771\) −1.56795e7 −0.949939
\(772\) 0 0
\(773\) 1.04245e7i 0.627492i −0.949507 0.313746i \(-0.898416\pi\)
0.949507 0.313746i \(-0.101584\pi\)
\(774\) 0 0
\(775\) −6.24560e6 2.01545e7i −0.373525 1.20536i
\(776\) 0 0
\(777\) 1.65959e7i 0.986163i
\(778\) 0 0
\(779\) 43560.0 0.00257184
\(780\) 0 0
\(781\) −1.34447e7 −0.788721
\(782\) 0 0
\(783\) 1.24115e7i 0.723467i
\(784\) 0 0
\(785\) −1.18285e7 + 1.60489e7i −0.685104 + 0.929549i
\(786\) 0 0
\(787\) 3.45366e7i 1.98766i −0.110913 0.993830i \(-0.535378\pi\)
0.110913 0.993830i \(-0.464622\pi\)
\(788\) 0 0
\(789\) 3.24711e7 1.85697
\(790\) 0 0
\(791\) −6.26947e6 −0.356279
\(792\) 0 0
\(793\) 680333.i 0.0384183i
\(794\) 0 0
\(795\) 5.21532e6 + 3.84384e6i 0.292660 + 0.215698i
\(796\) 0 0
\(797\) 2.09287e7i 1.16707i −0.812089 0.583533i \(-0.801670\pi\)
0.812089 0.583533i \(-0.198330\pi\)
\(798\) 0 0
\(799\) 7.27126e6 0.402942
\(800\) 0 0
\(801\) 1.52847e6 0.0841735
\(802\) 0 0
\(803\) 1.78725e7i 0.978131i
\(804\) 0 0
\(805\) 6.53994e6 + 4.82012e6i 0.355700 + 0.262161i
\(806\) 0 0
\(807\) 3.45533e7i 1.86770i
\(808\) 0 0
\(809\) 2.48797e7 1.33651 0.668257 0.743930i \(-0.267041\pi\)
0.668257 + 0.743930i \(0.267041\pi\)
\(810\) 0 0
\(811\) −3.95415e6 −0.211106 −0.105553 0.994414i \(-0.533661\pi\)
−0.105553 + 0.994414i \(0.533661\pi\)
\(812\) 0 0
\(813\) 3.42285e7i 1.81619i
\(814\) 0 0
\(815\) −4.80546e6 + 6.52005e6i −0.253420 + 0.343841i
\(816\) 0 0
\(817\) 91936.8i 0.00481875i
\(818\) 0 0
\(819\) 1.09058e6 0.0568132
\(820\) 0 0
\(821\) 3.43550e6 0.177882 0.0889410 0.996037i \(-0.471652\pi\)
0.0889410 + 0.996037i \(0.471652\pi\)
\(822\) 0 0
\(823\) 3.94833e6i 0.203195i −0.994826 0.101598i \(-0.967605\pi\)
0.994826 0.101598i \(-0.0323954\pi\)
\(824\) 0 0
\(825\) −1.49688e7 + 4.63863e6i −0.765688 + 0.237277i
\(826\) 0 0
\(827\) 3.38176e7i 1.71941i 0.510791 + 0.859705i \(0.329353\pi\)
−0.510791 + 0.859705i \(0.670647\pi\)
\(828\) 0 0
\(829\) −1.52015e7 −0.768244 −0.384122 0.923282i \(-0.625496\pi\)
−0.384122 + 0.923282i \(0.625496\pi\)
\(830\) 0 0
\(831\) −2.53210e7 −1.27198
\(832\) 0 0
\(833\) 9.13579e6i 0.456177i
\(834\) 0 0
\(835\) 600380. 814596.i 0.0297996 0.0404321i
\(836\) 0 0
\(837\) 1.20927e7i 0.596635i
\(838\) 0 0
\(839\) −2.89012e7 −1.41746 −0.708729 0.705481i \(-0.750731\pi\)
−0.708729 + 0.705481i \(0.750731\pi\)
\(840\) 0 0
\(841\) 2.75138e7 1.34140
\(842\) 0 0
\(843\) 2.91532e7i 1.41292i
\(844\) 0 0
\(845\) 1.60667e7 + 1.18416e7i 0.774077 + 0.570516i
\(846\) 0 0
\(847\) 5.82348e6i 0.278917i
\(848\) 0 0
\(849\) −3.28446e7 −1.56385
\(850\) 0 0
\(851\) −3.40077e7 −1.60973
\(852\) 0 0
\(853\) 2.02107e7i 0.951062i −0.879699 0.475531i \(-0.842256\pi\)
0.879699 0.475531i \(-0.157744\pi\)
\(854\) 0 0
\(855\) 1.51470e6 + 1.11638e6i 0.0708617 + 0.0522270i
\(856\) 0 0
\(857\) 1.70522e7i 0.793101i −0.918013 0.396550i \(-0.870207\pi\)
0.918013 0.396550i \(-0.129793\pi\)
\(858\) 0 0
\(859\) 1.95505e7 0.904015 0.452008 0.892014i \(-0.350708\pi\)
0.452008 + 0.892014i \(0.350708\pi\)
\(860\) 0 0
\(861\) −235224. −0.0108137
\(862\) 0 0
\(863\) 2.70896e7i 1.23816i 0.785330 + 0.619078i \(0.212493\pi\)
−0.785330 + 0.619078i \(0.787507\pi\)
\(864\) 0 0
\(865\) 1.63368e7 2.21657e7i 0.742379 1.00726i
\(866\) 0 0
\(867\) 1.87844e7i 0.848692i
\(868\) 0 0
\(869\) −1.30838e7 −0.587741
\(870\) 0 0
\(871\) 5.21057e6 0.232723
\(872\) 0 0
\(873\) 1.54912e7i 0.687940i
\(874\) 0 0
\(875\) 9.85050e6 + 3.42524e6i 0.434949 + 0.151242i
\(876\) 0 0
\(877\) 1.98285e6i 0.0870545i 0.999052 + 0.0435272i \(0.0138595\pi\)
−0.999052 + 0.0435272i \(0.986140\pi\)
\(878\) 0 0
\(879\) −4.75150e7 −2.07424
\(880\) 0 0
\(881\) −4.22840e7 −1.83542 −0.917712 0.397247i \(-0.869966\pi\)
−0.917712 + 0.397247i \(0.869966\pi\)
\(882\) 0 0
\(883\) 134502.i 0.00580535i −0.999996 0.00290267i \(-0.999076\pi\)
0.999996 0.00290267i \(-0.000923951\pi\)
\(884\) 0 0
\(885\) 1.62756e7 2.20828e7i 0.698520 0.947753i
\(886\) 0 0
\(887\) 3.87668e6i 0.165444i 0.996573 + 0.0827219i \(0.0263613\pi\)
−0.996573 + 0.0827219i \(0.973639\pi\)
\(888\) 0 0
\(889\) −5.20700e6 −0.220970
\(890\) 0 0
\(891\) −1.83504e7 −0.774374
\(892\) 0 0
\(893\) 2.31885e6i 0.0973070i
\(894\) 0 0
\(895\) 1.99989e7 + 1.47397e7i 0.834543 + 0.615081i
\(896\) 0 0
\(897\) 5.78414e6i 0.240026i
\(898\) 0 0
\(899\) −4.67914e7 −1.93093
\(900\) 0 0
\(901\) 4.01773e6 0.164880
\(902\) 0 0
\(903\) 496459.i 0.0202611i
\(904\) 0 0
\(905\) −7.06059e6 5.20385e6i −0.286563 0.211205i
\(906\) 0 0
\(907\) 2.87363e7i 1.15988i 0.814660 + 0.579939i \(0.196924\pi\)
−0.814660 + 0.579939i \(0.803076\pi\)
\(908\) 0 0
\(909\) −1.66920e7 −0.670037
\(910\) 0 0
\(911\) −1.87675e6 −0.0749223 −0.0374611 0.999298i \(-0.511927\pi\)
−0.0374611 + 0.999298i \(0.511927\pi\)
\(912\) 0 0
\(913\) 1.55841e7i 0.618736i
\(914\) 0 0
\(915\) −3.76068e6 + 5.10249e6i −0.148496 + 0.201479i
\(916\) 0 0
\(917\) 1.15131e7i 0.452137i
\(918\) 0 0
\(919\) 6.76852e6 0.264366 0.132183 0.991225i \(-0.457801\pi\)
0.132183 + 0.991225i \(0.457801\pi\)
\(920\) 0 0
\(921\) 1.84722e7 0.717579
\(922\) 0 0
\(923\) 6.37015e6i 0.246119i
\(924\) 0 0
\(925\) −4.16988e7 + 1.29219e7i −1.60239 + 0.496560i
\(926\) 0 0
\(927\) 1.08055e7i 0.412996i
\(928\) 0 0
\(929\) 1.15356e7 0.438530 0.219265 0.975665i \(-0.429634\pi\)
0.219265 + 0.975665i \(0.429634\pi\)
\(930\) 0 0
\(931\) −2.91346e6 −0.110163
\(932\) 0 0
\(933\) 1.13061e7i 0.425214i
\(934\) 0 0
\(935\) −5.76576e6 + 7.82299e6i −0.215689 + 0.292647i
\(936\) 0 0
\(937\) 3.92632e7i 1.46096i −0.682936 0.730478i \(-0.739297\pi\)
0.682936 0.730478i \(-0.260703\pi\)
\(938\) 0 0
\(939\) −3.42762e7 −1.26861
\(940\) 0 0
\(941\) −2.94919e7 −1.08575 −0.542874 0.839814i \(-0.682664\pi\)
−0.542874 + 0.839814i \(0.682664\pi\)
\(942\) 0 0
\(943\) 482012.i 0.0176514i
\(944\) 0 0
\(945\) 4.81140e6 + 3.54614e6i 0.175264 + 0.129174i
\(946\) 0 0
\(947\) 2.09628e7i 0.759581i −0.925072 0.379791i \(-0.875996\pi\)
0.925072 0.379791i \(-0.124004\pi\)
\(948\) 0 0
\(949\) −8.46806e6 −0.305224
\(950\) 0 0
\(951\) −2.61967e6 −0.0939281
\(952\) 0 0
\(953\) 1.64122e7i 0.585375i 0.956208 + 0.292687i \(0.0945495\pi\)
−0.956208 + 0.292687i \(0.905451\pi\)
\(954\) 0 0
\(955\) −1.49558e7 1.10229e7i −0.530643 0.391099i
\(956\) 0 0
\(957\) 3.47521e7i 1.22660i
\(958\) 0 0
\(959\) 8.57102e6 0.300944
\(960\) 0 0
\(961\) 1.69604e7 0.592416
\(962\) 0 0
\(963\) 1.48590e7i 0.516325i
\(964\) 0 0
\(965\) 2.60726e7 3.53754e7i 0.901295 1.22288i
\(966\) 0 0
\(967\) 4.71911e7i 1.62291i −0.584416 0.811454i \(-0.698676\pi\)
0.584416 0.811454i \(-0.301324\pi\)
\(968\) 0 0
\(969\) 3.02016e6 0.103329
\(970\) 0 0
\(971\) 3.84771e7 1.30965 0.654823 0.755783i \(-0.272743\pi\)
0.654823 + 0.755783i \(0.272743\pi\)
\(972\) 0 0
\(973\) 1.90047e7i 0.643544i
\(974\) 0 0
\(975\) 2.19780e6 + 7.09227e6i 0.0740417 + 0.238932i
\(976\) 0 0
\(977\) 2.70184e7i 0.905572i −0.891619 0.452786i \(-0.850430\pi\)
0.891619 0.452786i \(-0.149570\pi\)
\(978\) 0 0
\(979\) −2.51748e6 −0.0839478
\(980\) 0 0
\(981\) −3.21453e6 −0.106646
\(982\) 0 0
\(983\) 2.88475e7i 0.952192i 0.879393 + 0.476096i \(0.157949\pi\)
−0.879393 + 0.476096i \(0.842051\pi\)
\(984\) 0 0
\(985\) −1.97692e6 + 2.68229e6i −0.0649230 + 0.0880876i
\(986\) 0 0
\(987\) 1.25218e7i 0.409142i
\(988\) 0 0
\(989\) −1.01732e6 −0.0330726
\(990\) 0 0
\(991\) 5.21596e7 1.68714 0.843569 0.537021i \(-0.180450\pi\)
0.843569 + 0.537021i \(0.180450\pi\)
\(992\) 0 0
\(993\) 3.14525e7i 1.01224i
\(994\) 0 0
\(995\) 1.78110e7 + 1.31272e7i 0.570336 + 0.420353i
\(996\) 0 0
\(997\) 9.78148e6i 0.311650i −0.987785 0.155825i \(-0.950196\pi\)
0.987785 0.155825i \(-0.0498036\pi\)
\(998\) 0 0
\(999\) −2.50193e7 −0.793161
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 320.6.c.g.129.2 2
4.3 odd 2 320.6.c.f.129.1 2
5.4 even 2 inner 320.6.c.g.129.1 2
8.3 odd 2 5.6.b.a.4.1 2
8.5 even 2 80.6.c.a.49.1 2
20.19 odd 2 320.6.c.f.129.2 2
24.5 odd 2 720.6.f.f.289.2 2
24.11 even 2 45.6.b.b.19.2 2
40.3 even 4 25.6.a.c.1.1 2
40.13 odd 4 400.6.a.t.1.2 2
40.19 odd 2 5.6.b.a.4.2 yes 2
40.27 even 4 25.6.a.c.1.2 2
40.29 even 2 80.6.c.a.49.2 2
40.37 odd 4 400.6.a.t.1.1 2
56.27 even 2 245.6.b.a.99.1 2
120.29 odd 2 720.6.f.f.289.1 2
120.59 even 2 45.6.b.b.19.1 2
120.83 odd 4 225.6.a.n.1.2 2
120.107 odd 4 225.6.a.n.1.1 2
280.139 even 2 245.6.b.a.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.b.a.4.1 2 8.3 odd 2
5.6.b.a.4.2 yes 2 40.19 odd 2
25.6.a.c.1.1 2 40.3 even 4
25.6.a.c.1.2 2 40.27 even 4
45.6.b.b.19.1 2 120.59 even 2
45.6.b.b.19.2 2 24.11 even 2
80.6.c.a.49.1 2 8.5 even 2
80.6.c.a.49.2 2 40.29 even 2
225.6.a.n.1.1 2 120.107 odd 4
225.6.a.n.1.2 2 120.83 odd 4
245.6.b.a.99.1 2 56.27 even 2
245.6.b.a.99.2 2 280.139 even 2
320.6.c.f.129.1 2 4.3 odd 2
320.6.c.f.129.2 2 20.19 odd 2
320.6.c.g.129.1 2 5.4 even 2 inner
320.6.c.g.129.2 2 1.1 even 1 trivial
400.6.a.t.1.1 2 40.37 odd 4
400.6.a.t.1.2 2 40.13 odd 4
720.6.f.f.289.1 2 120.29 odd 2
720.6.f.f.289.2 2 24.5 odd 2