Properties

Label 60.6.i.a.17.7
Level $60$
Weight $6$
Character 60.17
Analytic conductor $9.623$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,6,Mod(17,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.i (of order \(4\), degree \(2\), 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: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} - 161859774 x^{13} + 12872525097 x^{12} - 60314430972 x^{11} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{14}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.7
Root \(-4.98882 + 11.3442i\) of defining polynomial
Character \(\chi\) \(=\) 60.17
Dual form 60.6.i.a.53.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.61326 - 14.8902i) q^{3} +(-31.2988 + 46.3183i) q^{5} +(-91.3999 - 91.3999i) q^{7} +(-200.436 - 137.385i) q^{9} +O(q^{10})\) \(q+(4.61326 - 14.8902i) q^{3} +(-31.2988 + 46.3183i) q^{5} +(-91.3999 - 91.3999i) q^{7} +(-200.436 - 137.385i) q^{9} +99.2471i q^{11} +(-452.655 + 452.655i) q^{13} +(545.298 + 679.724i) q^{15} +(-1264.35 + 1264.35i) q^{17} -912.778i q^{19} +(-1782.61 + 939.311i) q^{21} +(-1746.66 - 1746.66i) q^{23} +(-1165.77 - 2899.42i) q^{25} +(-2970.35 + 2350.73i) q^{27} -299.372 q^{29} +2593.02 q^{31} +(1477.81 + 457.853i) q^{33} +(7094.20 - 1372.78i) q^{35} +(5912.19 + 5912.19i) q^{37} +(4651.90 + 8828.33i) q^{39} -19485.6i q^{41} +(-8602.82 + 8602.82i) q^{43} +(12636.8 - 4983.85i) q^{45} +(17376.5 - 17376.5i) q^{47} -99.1062i q^{49} +(12993.6 + 24659.2i) q^{51} +(-19439.8 - 19439.8i) q^{53} +(-4596.96 - 3106.32i) q^{55} +(-13591.4 - 4210.88i) q^{57} -220.999 q^{59} +43926.7 q^{61} +(5762.85 + 30876.8i) q^{63} +(-6798.62 - 35133.7i) q^{65} +(-38693.6 - 38693.6i) q^{67} +(-34066.0 + 17950.3i) q^{69} +61863.4i q^{71} +(-14032.3 + 14032.3i) q^{73} +(-48550.8 + 3982.71i) q^{75} +(9071.18 - 9071.18i) q^{77} +64815.3i q^{79} +(21299.9 + 55073.6i) q^{81} +(-5901.84 - 5901.84i) q^{83} +(-18989.8 - 98135.1i) q^{85} +(-1381.08 + 4457.71i) q^{87} -125772. q^{89} +82745.2 q^{91} +(11962.3 - 38610.5i) q^{93} +(42278.3 + 28568.9i) q^{95} +(-70605.0 - 70605.0i) q^{97} +(13635.0 - 19892.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 4.61326 14.8902i 0.295941 0.955206i
\(4\) 0 0
\(5\) −31.2988 + 46.3183i −0.559891 + 0.828567i
\(6\) 0 0
\(7\) −91.3999 91.3999i −0.705019 0.705019i 0.260465 0.965483i \(-0.416124\pi\)
−0.965483 + 0.260465i \(0.916124\pi\)
\(8\) 0 0
\(9\) −200.436 137.385i −0.824838 0.565369i
\(10\) 0 0
\(11\) 99.2471i 0.247307i 0.992325 + 0.123653i \(0.0394611\pi\)
−0.992325 + 0.123653i \(0.960539\pi\)
\(12\) 0 0
\(13\) −452.655 + 452.655i −0.742863 + 0.742863i −0.973128 0.230265i \(-0.926041\pi\)
0.230265 + 0.973128i \(0.426041\pi\)
\(14\) 0 0
\(15\) 545.298 + 679.724i 0.625757 + 0.780018i
\(16\) 0 0
\(17\) −1264.35 + 1264.35i −1.06107 + 1.06107i −0.0630620 + 0.998010i \(0.520087\pi\)
−0.998010 + 0.0630620i \(0.979913\pi\)
\(18\) 0 0
\(19\) 912.778i 0.580071i −0.957016 0.290035i \(-0.906333\pi\)
0.957016 0.290035i \(-0.0936671\pi\)
\(20\) 0 0
\(21\) −1782.61 + 939.311i −0.882082 + 0.464794i
\(22\) 0 0
\(23\) −1746.66 1746.66i −0.688478 0.688478i 0.273418 0.961895i \(-0.411846\pi\)
−0.961895 + 0.273418i \(0.911846\pi\)
\(24\) 0 0
\(25\) −1165.77 2899.42i −0.373045 0.927813i
\(26\) 0 0
\(27\) −2970.35 + 2350.73i −0.784148 + 0.620574i
\(28\) 0 0
\(29\) −299.372 −0.0661023 −0.0330512 0.999454i \(-0.510522\pi\)
−0.0330512 + 0.999454i \(0.510522\pi\)
\(30\) 0 0
\(31\) 2593.02 0.484620 0.242310 0.970199i \(-0.422095\pi\)
0.242310 + 0.970199i \(0.422095\pi\)
\(32\) 0 0
\(33\) 1477.81 + 457.853i 0.236229 + 0.0731882i
\(34\) 0 0
\(35\) 7094.20 1372.78i 0.978889 0.189422i
\(36\) 0 0
\(37\) 5912.19 + 5912.19i 0.709977 + 0.709977i 0.966530 0.256553i \(-0.0825868\pi\)
−0.256553 + 0.966530i \(0.582587\pi\)
\(38\) 0 0
\(39\) 4651.90 + 8828.33i 0.489744 + 0.929430i
\(40\) 0 0
\(41\) 19485.6i 1.81032i −0.425074 0.905159i \(-0.639752\pi\)
0.425074 0.905159i \(-0.360248\pi\)
\(42\) 0 0
\(43\) −8602.82 + 8602.82i −0.709528 + 0.709528i −0.966436 0.256908i \(-0.917296\pi\)
0.256908 + 0.966436i \(0.417296\pi\)
\(44\) 0 0
\(45\) 12636.8 4983.85i 0.930265 0.366888i
\(46\) 0 0
\(47\) 17376.5 17376.5i 1.14741 1.14741i 0.160348 0.987061i \(-0.448738\pi\)
0.987061 0.160348i \(-0.0512617\pi\)
\(48\) 0 0
\(49\) 99.1062i 0.00589672i
\(50\) 0 0
\(51\) 12993.6 + 24659.2i 0.699528 + 1.32756i
\(52\) 0 0
\(53\) −19439.8 19439.8i −0.950609 0.950609i 0.0482276 0.998836i \(-0.484643\pi\)
−0.998836 + 0.0482276i \(0.984643\pi\)
\(54\) 0 0
\(55\) −4596.96 3106.32i −0.204910 0.138465i
\(56\) 0 0
\(57\) −13591.4 4210.88i −0.554087 0.171667i
\(58\) 0 0
\(59\) −220.999 −0.00826535 −0.00413267 0.999991i \(-0.501315\pi\)
−0.00413267 + 0.999991i \(0.501315\pi\)
\(60\) 0 0
\(61\) 43926.7 1.51149 0.755743 0.654868i \(-0.227276\pi\)
0.755743 + 0.654868i \(0.227276\pi\)
\(62\) 0 0
\(63\) 5762.85 + 30876.8i 0.182930 + 0.980122i
\(64\) 0 0
\(65\) −6798.62 35133.7i −0.199589 1.03143i
\(66\) 0 0
\(67\) −38693.6 38693.6i −1.05306 1.05306i −0.998511 0.0545473i \(-0.982628\pi\)
−0.0545473 0.998511i \(-0.517372\pi\)
\(68\) 0 0
\(69\) −34066.0 + 17950.3i −0.861387 + 0.453890i
\(70\) 0 0
\(71\) 61863.4i 1.45642i 0.685352 + 0.728212i \(0.259648\pi\)
−0.685352 + 0.728212i \(0.740352\pi\)
\(72\) 0 0
\(73\) −14032.3 + 14032.3i −0.308193 + 0.308193i −0.844208 0.536015i \(-0.819929\pi\)
0.536015 + 0.844208i \(0.319929\pi\)
\(74\) 0 0
\(75\) −48550.8 + 3982.71i −0.996652 + 0.0817570i
\(76\) 0 0
\(77\) 9071.18 9071.18i 0.174356 0.174356i
\(78\) 0 0
\(79\) 64815.3i 1.16845i 0.811592 + 0.584225i \(0.198601\pi\)
−0.811592 + 0.584225i \(0.801399\pi\)
\(80\) 0 0
\(81\) 21299.9 + 55073.6i 0.360715 + 0.932676i
\(82\) 0 0
\(83\) −5901.84 5901.84i −0.0940355 0.0940355i 0.658524 0.752560i \(-0.271181\pi\)
−0.752560 + 0.658524i \(0.771181\pi\)
\(84\) 0 0
\(85\) −18989.8 98135.1i −0.285084 1.47325i
\(86\) 0 0
\(87\) −1381.08 + 4457.71i −0.0195624 + 0.0631413i
\(88\) 0 0
\(89\) −125772. −1.68310 −0.841551 0.540177i \(-0.818357\pi\)
−0.841551 + 0.540177i \(0.818357\pi\)
\(90\) 0 0
\(91\) 82745.2 1.04746
\(92\) 0 0
\(93\) 11962.3 38610.5i 0.143419 0.462912i
\(94\) 0 0
\(95\) 42278.3 + 28568.9i 0.480627 + 0.324776i
\(96\) 0 0
\(97\) −70605.0 70605.0i −0.761915 0.761915i 0.214754 0.976668i \(-0.431105\pi\)
−0.976668 + 0.214754i \(0.931105\pi\)
\(98\) 0 0
\(99\) 13635.0 19892.7i 0.139820 0.203988i
\(100\) 0 0
\(101\) 15327.6i 0.149510i 0.997202 + 0.0747552i \(0.0238175\pi\)
−0.997202 + 0.0747552i \(0.976182\pi\)
\(102\) 0 0
\(103\) −110125. + 110125.i −1.02280 + 1.02280i −0.0230675 + 0.999734i \(0.507343\pi\)
−0.999734 + 0.0230675i \(0.992657\pi\)
\(104\) 0 0
\(105\) 12286.5 111967.i 0.108756 0.991098i
\(106\) 0 0
\(107\) 44058.7 44058.7i 0.372025 0.372025i −0.496189 0.868214i \(-0.665268\pi\)
0.868214 + 0.496189i \(0.165268\pi\)
\(108\) 0 0
\(109\) 121373.i 0.978487i 0.872147 + 0.489243i \(0.162727\pi\)
−0.872147 + 0.489243i \(0.837273\pi\)
\(110\) 0 0
\(111\) 115308. 60759.2i 0.888286 0.468063i
\(112\) 0 0
\(113\) 5465.09 + 5465.09i 0.0402626 + 0.0402626i 0.726951 0.686689i \(-0.240937\pi\)
−0.686689 + 0.726951i \(0.740937\pi\)
\(114\) 0 0
\(115\) 135571. 26233.9i 0.955922 0.184977i
\(116\) 0 0
\(117\) 152916. 28540.3i 1.03273 0.192750i
\(118\) 0 0
\(119\) 231123. 1.49615
\(120\) 0 0
\(121\) 151201. 0.938839
\(122\) 0 0
\(123\) −290145. 89892.3i −1.72923 0.535747i
\(124\) 0 0
\(125\) 170783. + 36752.1i 0.977619 + 0.210381i
\(126\) 0 0
\(127\) 58474.8 + 58474.8i 0.321706 + 0.321706i 0.849421 0.527715i \(-0.176951\pi\)
−0.527715 + 0.849421i \(0.676951\pi\)
\(128\) 0 0
\(129\) 88410.5 + 167785.i 0.467767 + 0.887724i
\(130\) 0 0
\(131\) 114053.i 0.580668i 0.956925 + 0.290334i \(0.0937664\pi\)
−0.956925 + 0.290334i \(0.906234\pi\)
\(132\) 0 0
\(133\) −83427.8 + 83427.8i −0.408961 + 0.408961i
\(134\) 0 0
\(135\) −15913.5 211157.i −0.0751504 0.997172i
\(136\) 0 0
\(137\) −219342. + 219342.i −0.998437 + 0.998437i −0.999999 0.00156166i \(-0.999503\pi\)
0.00156166 + 0.999999i \(0.499503\pi\)
\(138\) 0 0
\(139\) 334777.i 1.46966i −0.678249 0.734832i \(-0.737261\pi\)
0.678249 0.734832i \(-0.262739\pi\)
\(140\) 0 0
\(141\) −178577. 338902.i −0.756447 1.43558i
\(142\) 0 0
\(143\) −44924.7 44924.7i −0.183715 0.183715i
\(144\) 0 0
\(145\) 9370.00 13866.4i 0.0370101 0.0547702i
\(146\) 0 0
\(147\) −1475.71 457.203i −0.00563259 0.00174508i
\(148\) 0 0
\(149\) −225318. −0.831438 −0.415719 0.909493i \(-0.636470\pi\)
−0.415719 + 0.909493i \(0.636470\pi\)
\(150\) 0 0
\(151\) −84670.2 −0.302196 −0.151098 0.988519i \(-0.548281\pi\)
−0.151098 + 0.988519i \(0.548281\pi\)
\(152\) 0 0
\(153\) 427123. 79718.3i 1.47511 0.275315i
\(154\) 0 0
\(155\) −81158.4 + 120104.i −0.271334 + 0.401540i
\(156\) 0 0
\(157\) −72732.0 72732.0i −0.235492 0.235492i 0.579488 0.814981i \(-0.303252\pi\)
−0.814981 + 0.579488i \(0.803252\pi\)
\(158\) 0 0
\(159\) −379143. + 199781.i −1.18935 + 0.626703i
\(160\) 0 0
\(161\) 319290.i 0.970780i
\(162\) 0 0
\(163\) −75660.4 + 75660.4i −0.223049 + 0.223049i −0.809781 0.586732i \(-0.800414\pi\)
0.586732 + 0.809781i \(0.300414\pi\)
\(164\) 0 0
\(165\) −67460.6 + 54119.3i −0.192904 + 0.154754i
\(166\) 0 0
\(167\) −116674. + 116674.i −0.323730 + 0.323730i −0.850196 0.526466i \(-0.823517\pi\)
0.526466 + 0.850196i \(0.323517\pi\)
\(168\) 0 0
\(169\) 38499.3i 0.103690i
\(170\) 0 0
\(171\) −125402. + 182953.i −0.327954 + 0.478464i
\(172\) 0 0
\(173\) 283207. + 283207.i 0.719431 + 0.719431i 0.968489 0.249057i \(-0.0801208\pi\)
−0.249057 + 0.968489i \(0.580121\pi\)
\(174\) 0 0
\(175\) −158456. + 371557.i −0.391122 + 0.917130i
\(176\) 0 0
\(177\) −1019.53 + 3290.72i −0.00244605 + 0.00789511i
\(178\) 0 0
\(179\) 49873.7 0.116343 0.0581713 0.998307i \(-0.481473\pi\)
0.0581713 + 0.998307i \(0.481473\pi\)
\(180\) 0 0
\(181\) −124517. −0.282509 −0.141255 0.989973i \(-0.545114\pi\)
−0.141255 + 0.989973i \(0.545114\pi\)
\(182\) 0 0
\(183\) 202646. 654077.i 0.447311 1.44378i
\(184\) 0 0
\(185\) −458887. + 88797.9i −0.985773 + 0.190754i
\(186\) 0 0
\(187\) −125483. 125483.i −0.262410 0.262410i
\(188\) 0 0
\(189\) 486346. + 56632.7i 0.990356 + 0.115322i
\(190\) 0 0
\(191\) 392746.i 0.778984i −0.921030 0.389492i \(-0.872651\pi\)
0.921030 0.389492i \(-0.127349\pi\)
\(192\) 0 0
\(193\) −230496. + 230496.i −0.445420 + 0.445420i −0.893829 0.448409i \(-0.851991\pi\)
0.448409 + 0.893829i \(0.351991\pi\)
\(194\) 0 0
\(195\) −554512. 60848.4i −1.04430 0.114594i
\(196\) 0 0
\(197\) 340304. 340304.i 0.624743 0.624743i −0.321998 0.946740i \(-0.604354\pi\)
0.946740 + 0.321998i \(0.104354\pi\)
\(198\) 0 0
\(199\) 1.08977e6i 1.95076i −0.220535 0.975379i \(-0.570780\pi\)
0.220535 0.975379i \(-0.429220\pi\)
\(200\) 0 0
\(201\) −754659. + 397652.i −1.31753 + 0.694245i
\(202\) 0 0
\(203\) 27362.6 + 27362.6i 0.0466034 + 0.0466034i
\(204\) 0 0
\(205\) 902541. + 609877.i 1.49997 + 1.01358i
\(206\) 0 0
\(207\) 110129. + 590059.i 0.178638 + 0.957127i
\(208\) 0 0
\(209\) 90590.6 0.143455
\(210\) 0 0
\(211\) 664266. 1.02715 0.513577 0.858043i \(-0.328320\pi\)
0.513577 + 0.858043i \(0.328320\pi\)
\(212\) 0 0
\(213\) 921157. + 285392.i 1.39118 + 0.431015i
\(214\) 0 0
\(215\) −129209. 667726.i −0.190633 0.985149i
\(216\) 0 0
\(217\) −237002. 237002.i −0.341666 0.341666i
\(218\) 0 0
\(219\) 144209. + 273679.i 0.203181 + 0.385595i
\(220\) 0 0
\(221\) 1.14463e6i 1.57646i
\(222\) 0 0
\(223\) 252258. 252258.i 0.339690 0.339690i −0.516560 0.856251i \(-0.672788\pi\)
0.856251 + 0.516560i \(0.172788\pi\)
\(224\) 0 0
\(225\) −164675. + 741305.i −0.216855 + 0.976204i
\(226\) 0 0
\(227\) 887226. 887226.i 1.14280 1.14280i 0.154863 0.987936i \(-0.450507\pi\)
0.987936 0.154863i \(-0.0494935\pi\)
\(228\) 0 0
\(229\) 333704.i 0.420507i 0.977647 + 0.210253i \(0.0674289\pi\)
−0.977647 + 0.210253i \(0.932571\pi\)
\(230\) 0 0
\(231\) −93223.9 176919.i −0.114947 0.218145i
\(232\) 0 0
\(233\) −424519. 424519.i −0.512280 0.512280i 0.402944 0.915225i \(-0.367987\pi\)
−0.915225 + 0.402944i \(0.867987\pi\)
\(234\) 0 0
\(235\) 260986. + 1.34872e6i 0.308281 + 1.59313i
\(236\) 0 0
\(237\) 965113. + 299010.i 1.11611 + 0.345792i
\(238\) 0 0
\(239\) 800931. 0.906986 0.453493 0.891260i \(-0.350178\pi\)
0.453493 + 0.891260i \(0.350178\pi\)
\(240\) 0 0
\(241\) −1.63579e6 −1.81419 −0.907097 0.420922i \(-0.861707\pi\)
−0.907097 + 0.420922i \(0.861707\pi\)
\(242\) 0 0
\(243\) 918318. 63090.3i 0.997648 0.0685404i
\(244\) 0 0
\(245\) 4590.43 + 3101.91i 0.00488583 + 0.00330152i
\(246\) 0 0
\(247\) 413173. + 413173.i 0.430913 + 0.430913i
\(248\) 0 0
\(249\) −115106. + 60652.8i −0.117652 + 0.0619944i
\(250\) 0 0
\(251\) 742347.i 0.743742i −0.928284 0.371871i \(-0.878716\pi\)
0.928284 0.371871i \(-0.121284\pi\)
\(252\) 0 0
\(253\) 173351. 173351.i 0.170265 0.170265i
\(254\) 0 0
\(255\) −1.54886e6 169961.i −1.49163 0.163681i
\(256\) 0 0
\(257\) −1.02107e6 + 1.02107e6i −0.964326 + 0.964326i −0.999385 0.0350592i \(-0.988838\pi\)
0.0350592 + 0.999385i \(0.488838\pi\)
\(258\) 0 0
\(259\) 1.08075e6i 1.00109i
\(260\) 0 0
\(261\) 60004.9 + 41129.2i 0.0545237 + 0.0373722i
\(262\) 0 0
\(263\) 540568. + 540568.i 0.481905 + 0.481905i 0.905740 0.423835i \(-0.139316\pi\)
−0.423835 + 0.905740i \(0.639316\pi\)
\(264\) 0 0
\(265\) 1.50886e6 291975.i 1.31988 0.255406i
\(266\) 0 0
\(267\) −580221. + 1.87278e6i −0.498099 + 1.60771i
\(268\) 0 0
\(269\) −749271. −0.631333 −0.315666 0.948870i \(-0.602228\pi\)
−0.315666 + 0.948870i \(0.602228\pi\)
\(270\) 0 0
\(271\) −1.24923e6 −1.03329 −0.516643 0.856201i \(-0.672819\pi\)
−0.516643 + 0.856201i \(0.672819\pi\)
\(272\) 0 0
\(273\) 381725. 1.23209e6i 0.309988 1.00054i
\(274\) 0 0
\(275\) 287759. 115699.i 0.229455 0.0922566i
\(276\) 0 0
\(277\) −785236. 785236.i −0.614895 0.614895i 0.329322 0.944218i \(-0.393180\pi\)
−0.944218 + 0.329322i \(0.893180\pi\)
\(278\) 0 0
\(279\) −519733. 356241.i −0.399733 0.273989i
\(280\) 0 0
\(281\) 280724.i 0.212087i −0.994362 0.106043i \(-0.966182\pi\)
0.994362 0.106043i \(-0.0338182\pi\)
\(282\) 0 0
\(283\) −1.51150e6 + 1.51150e6i −1.12187 + 1.12187i −0.130412 + 0.991460i \(0.541630\pi\)
−0.991460 + 0.130412i \(0.958370\pi\)
\(284\) 0 0
\(285\) 620437. 497736.i 0.452466 0.362984i
\(286\) 0 0
\(287\) −1.78098e6 + 1.78098e6i −1.27631 + 1.27631i
\(288\) 0 0
\(289\) 1.77730e6i 1.25175i
\(290\) 0 0
\(291\) −1.37704e6 + 725603.i −0.953267 + 0.502304i
\(292\) 0 0
\(293\) −1.38666e6 1.38666e6i −0.943628 0.943628i 0.0548654 0.998494i \(-0.482527\pi\)
−0.998494 + 0.0548654i \(0.982527\pi\)
\(294\) 0 0
\(295\) 6917.02 10236.3i 0.00462769 0.00684839i
\(296\) 0 0
\(297\) −233303. 294798.i −0.153472 0.193925i
\(298\) 0 0
\(299\) 1.58127e6 1.02289
\(300\) 0 0
\(301\) 1.57259e6 1.00046
\(302\) 0 0
\(303\) 228231. + 70710.3i 0.142813 + 0.0442462i
\(304\) 0 0
\(305\) −1.37486e6 + 2.03461e6i −0.846267 + 1.25237i
\(306\) 0 0
\(307\) −171663. 171663.i −0.103952 0.103952i 0.653218 0.757170i \(-0.273418\pi\)
−0.757170 + 0.653218i \(0.773418\pi\)
\(308\) 0 0
\(309\) 1.13174e6 + 2.14781e6i 0.674297 + 1.27968i
\(310\) 0 0
\(311\) 1.87250e6i 1.09779i 0.835890 + 0.548897i \(0.184952\pi\)
−0.835890 + 0.548897i \(0.815048\pi\)
\(312\) 0 0
\(313\) −774463. + 774463.i −0.446828 + 0.446828i −0.894299 0.447471i \(-0.852325\pi\)
0.447471 + 0.894299i \(0.352325\pi\)
\(314\) 0 0
\(315\) −1.61053e6 699481.i −0.914518 0.397191i
\(316\) 0 0
\(317\) 1.99126e6 1.99126e6i 1.11296 1.11296i 0.120210 0.992749i \(-0.461643\pi\)
0.992749 0.120210i \(-0.0383567\pi\)
\(318\) 0 0
\(319\) 29711.8i 0.0163476i
\(320\) 0 0
\(321\) −452788. 859297.i −0.245263 0.465458i
\(322\) 0 0
\(323\) 1.15407e6 + 1.15407e6i 0.615497 + 0.615497i
\(324\) 0 0
\(325\) 1.84012e6 + 784745.i 0.966359 + 0.412117i
\(326\) 0 0
\(327\) 1.80726e6 + 559924.i 0.934657 + 0.289574i
\(328\) 0 0
\(329\) −3.17643e6 −1.61789
\(330\) 0 0
\(331\) 3.68012e6 1.84626 0.923129 0.384491i \(-0.125623\pi\)
0.923129 + 0.384491i \(0.125623\pi\)
\(332\) 0 0
\(333\) −372769. 1.99726e6i −0.184217 0.987015i
\(334\) 0 0
\(335\) 3.00329e6 581157.i 1.46213 0.282931i
\(336\) 0 0
\(337\) −1.72073e6 1.72073e6i −0.825349 0.825349i 0.161521 0.986869i \(-0.448360\pi\)
−0.986869 + 0.161521i \(0.948360\pi\)
\(338\) 0 0
\(339\) 106588. 56164.4i 0.0503744 0.0265437i
\(340\) 0 0
\(341\) 257350.i 0.119850i
\(342\) 0 0
\(343\) −1.54522e6 + 1.54522e6i −0.709176 + 0.709176i
\(344\) 0 0
\(345\) 234797. 2.13970e6i 0.106205 0.967845i
\(346\) 0 0
\(347\) 411386. 411386.i 0.183411 0.183411i −0.609429 0.792841i \(-0.708601\pi\)
0.792841 + 0.609429i \(0.208601\pi\)
\(348\) 0 0
\(349\) 1.33706e6i 0.587606i 0.955866 + 0.293803i \(0.0949210\pi\)
−0.955866 + 0.293803i \(0.905079\pi\)
\(350\) 0 0
\(351\) 280471. 2.40861e6i 0.121512 1.04352i
\(352\) 0 0
\(353\) 1.49730e6 + 1.49730e6i 0.639546 + 0.639546i 0.950444 0.310897i \(-0.100629\pi\)
−0.310897 + 0.950444i \(0.600629\pi\)
\(354\) 0 0
\(355\) −2.86540e6 1.93625e6i −1.20674 0.815438i
\(356\) 0 0
\(357\) 1.06623e6 3.44146e6i 0.442772 1.42913i
\(358\) 0 0
\(359\) 815104. 0.333793 0.166896 0.985974i \(-0.446625\pi\)
0.166896 + 0.985974i \(0.446625\pi\)
\(360\) 0 0
\(361\) 1.64294e6 0.663518
\(362\) 0 0
\(363\) 697530. 2.25141e6i 0.277841 0.896785i
\(364\) 0 0
\(365\) −210758. 1.08915e6i −0.0828041 0.427913i
\(366\) 0 0
\(367\) 740709. + 740709.i 0.287066 + 0.287066i 0.835919 0.548853i \(-0.184935\pi\)
−0.548853 + 0.835919i \(0.684935\pi\)
\(368\) 0 0
\(369\) −2.67703e6 + 3.90561e6i −1.02350 + 1.49322i
\(370\) 0 0
\(371\) 3.55359e6i 1.34039i
\(372\) 0 0
\(373\) −1.31784e6 + 1.31784e6i −0.490446 + 0.490446i −0.908447 0.418000i \(-0.862731\pi\)
0.418000 + 0.908447i \(0.362731\pi\)
\(374\) 0 0
\(375\) 1.33511e6 2.37345e6i 0.490275 0.871568i
\(376\) 0 0
\(377\) 135512. 135512.i 0.0491049 0.0491049i
\(378\) 0 0
\(379\) 360145.i 0.128789i 0.997925 + 0.0643947i \(0.0205117\pi\)
−0.997925 + 0.0643947i \(0.979488\pi\)
\(380\) 0 0
\(381\) 1.14046e6 600941.i 0.402502 0.212090i
\(382\) 0 0
\(383\) −877902. 877902.i −0.305808 0.305808i 0.537473 0.843281i \(-0.319379\pi\)
−0.843281 + 0.537473i \(0.819379\pi\)
\(384\) 0 0
\(385\) 136244. + 704079.i 0.0468453 + 0.242086i
\(386\) 0 0
\(387\) 2.90621e6 542415.i 0.986391 0.184100i
\(388\) 0 0
\(389\) 3.46632e6 1.16143 0.580717 0.814105i \(-0.302772\pi\)
0.580717 + 0.814105i \(0.302772\pi\)
\(390\) 0 0
\(391\) 4.41679e6 1.46105
\(392\) 0 0
\(393\) 1.69827e6 + 526156.i 0.554658 + 0.171843i
\(394\) 0 0
\(395\) −3.00213e6 2.02864e6i −0.968138 0.654204i
\(396\) 0 0
\(397\) −808667. 808667.i −0.257510 0.257510i 0.566531 0.824040i \(-0.308285\pi\)
−0.824040 + 0.566531i \(0.808285\pi\)
\(398\) 0 0
\(399\) 857382. + 1.62713e6i 0.269614 + 0.511670i
\(400\) 0 0
\(401\) 3.47909e6i 1.08045i 0.841520 + 0.540226i \(0.181661\pi\)
−0.841520 + 0.540226i \(0.818339\pi\)
\(402\) 0 0
\(403\) −1.17374e6 + 1.17374e6i −0.360006 + 0.360006i
\(404\) 0 0
\(405\) −3.21757e6 737166.i −0.974745 0.223320i
\(406\) 0 0
\(407\) −586768. + 586768.i −0.175582 + 0.175582i
\(408\) 0 0
\(409\) 1.08607e6i 0.321033i 0.987033 + 0.160517i \(0.0513160\pi\)
−0.987033 + 0.160517i \(0.948684\pi\)
\(410\) 0 0
\(411\) 2.25416e6 + 4.27793e6i 0.658235 + 1.24919i
\(412\) 0 0
\(413\) 20199.3 + 20199.3i 0.00582723 + 0.00582723i
\(414\) 0 0
\(415\) 458084. 88642.3i 0.130564 0.0252651i
\(416\) 0 0
\(417\) −4.98489e6 1.54441e6i −1.40383 0.434934i
\(418\) 0 0
\(419\) −6.43635e6 −1.79104 −0.895519 0.445023i \(-0.853196\pi\)
−0.895519 + 0.445023i \(0.853196\pi\)
\(420\) 0 0
\(421\) −2.42261e6 −0.666158 −0.333079 0.942899i \(-0.608088\pi\)
−0.333079 + 0.942899i \(0.608088\pi\)
\(422\) 0 0
\(423\) −5.87014e6 + 1.09561e6i −1.59514 + 0.297717i
\(424\) 0 0
\(425\) 5.13981e6 + 2.19194e6i 1.38030 + 0.588649i
\(426\) 0 0
\(427\) −4.01490e6 4.01490e6i −1.06563 1.06563i
\(428\) 0 0
\(429\) −876186. + 461687.i −0.229855 + 0.121117i
\(430\) 0 0
\(431\) 687785.i 0.178344i −0.996016 0.0891722i \(-0.971578\pi\)
0.996016 0.0891722i \(-0.0284221\pi\)
\(432\) 0 0
\(433\) 72171.9 72171.9i 0.0184990 0.0184990i −0.697797 0.716296i \(-0.745836\pi\)
0.716296 + 0.697797i \(0.245836\pi\)
\(434\) 0 0
\(435\) −163247. 203491.i −0.0413640 0.0515610i
\(436\) 0 0
\(437\) −1.59432e6 + 1.59432e6i −0.399366 + 0.399366i
\(438\) 0 0
\(439\) 5.43596e6i 1.34622i −0.739544 0.673108i \(-0.764959\pi\)
0.739544 0.673108i \(-0.235041\pi\)
\(440\) 0 0
\(441\) −13615.7 + 19864.4i −0.00333383 + 0.00486384i
\(442\) 0 0
\(443\) −1.76348e6 1.76348e6i −0.426934 0.426934i 0.460649 0.887582i \(-0.347617\pi\)
−0.887582 + 0.460649i \(0.847617\pi\)
\(444\) 0 0
\(445\) 3.93653e6 5.82556e6i 0.942353 1.39456i
\(446\) 0 0
\(447\) −1.03945e6 + 3.35502e6i −0.246056 + 0.794194i
\(448\) 0 0
\(449\) −3.66972e6 −0.859047 −0.429524 0.903056i \(-0.641319\pi\)
−0.429524 + 0.903056i \(0.641319\pi\)
\(450\) 0 0
\(451\) 1.93389e6 0.447704
\(452\) 0 0
\(453\) −390606. + 1.26076e6i −0.0894321 + 0.288659i
\(454\) 0 0
\(455\) −2.58983e6 + 3.83261e6i −0.586465 + 0.867894i
\(456\) 0 0
\(457\) 1.80763e6 + 1.80763e6i 0.404874 + 0.404874i 0.879947 0.475073i \(-0.157578\pi\)
−0.475073 + 0.879947i \(0.657578\pi\)
\(458\) 0 0
\(459\) 783409. 6.72770e6i 0.173563 1.49051i
\(460\) 0 0
\(461\) 2.07114e6i 0.453898i −0.973907 0.226949i \(-0.927125\pi\)
0.973907 0.226949i \(-0.0728750\pi\)
\(462\) 0 0
\(463\) −1.80388e6 + 1.80388e6i −0.391070 + 0.391070i −0.875069 0.483999i \(-0.839184\pi\)
0.483999 + 0.875069i \(0.339184\pi\)
\(464\) 0 0
\(465\) 1.41397e6 + 1.76254e6i 0.303255 + 0.378012i
\(466\) 0 0
\(467\) 329212. 329212.i 0.0698526 0.0698526i −0.671317 0.741170i \(-0.734271\pi\)
0.741170 + 0.671317i \(0.234271\pi\)
\(468\) 0 0
\(469\) 7.07319e6i 1.48485i
\(470\) 0 0
\(471\) −1.41853e6 + 747462.i −0.294636 + 0.155252i
\(472\) 0 0
\(473\) −853805. 853805.i −0.175471 0.175471i
\(474\) 0 0
\(475\) −2.64652e6 + 1.06408e6i −0.538197 + 0.216393i
\(476\) 0 0
\(477\) 1.22570e6 + 6.56716e6i 0.246653 + 1.32154i
\(478\) 0 0
\(479\) 7.04336e6 1.40262 0.701311 0.712855i \(-0.252598\pi\)
0.701311 + 0.712855i \(0.252598\pi\)
\(480\) 0 0
\(481\) −5.35236e6 −1.05483
\(482\) 0 0
\(483\) 4.75429e6 + 1.47297e6i 0.927295 + 0.287293i
\(484\) 0 0
\(485\) 5.48016e6 1.06045e6i 1.05789 0.204708i
\(486\) 0 0
\(487\) 1.49566e6 + 1.49566e6i 0.285765 + 0.285765i 0.835403 0.549638i \(-0.185234\pi\)
−0.549638 + 0.835403i \(0.685234\pi\)
\(488\) 0 0
\(489\) 777557. + 1.47564e6i 0.147048 + 0.279067i
\(490\) 0 0
\(491\) 8.24705e6i 1.54381i −0.635735 0.771907i \(-0.719303\pi\)
0.635735 0.771907i \(-0.280697\pi\)
\(492\) 0 0
\(493\) 378511. 378511.i 0.0701393 0.0701393i
\(494\) 0 0
\(495\) 494633. + 1.25417e6i 0.0907340 + 0.230061i
\(496\) 0 0
\(497\) 5.65431e6 5.65431e6i 1.02681 1.02681i
\(498\) 0 0
\(499\) 4.98951e6i 0.897029i 0.893776 + 0.448514i \(0.148047\pi\)
−0.893776 + 0.448514i \(0.851953\pi\)
\(500\) 0 0
\(501\) 1.19905e6 + 2.27554e6i 0.213424 + 0.405033i
\(502\) 0 0
\(503\) −7.34572e6 7.34572e6i −1.29454 1.29454i −0.931950 0.362588i \(-0.881893\pi\)
−0.362588 0.931950i \(-0.618107\pi\)
\(504\) 0 0
\(505\) −709949. 479737.i −0.123879 0.0837094i
\(506\) 0 0
\(507\) −573261. 177607.i −0.0990450 0.0306860i
\(508\) 0 0
\(509\) 2.08281e6 0.356332 0.178166 0.984000i \(-0.442984\pi\)
0.178166 + 0.984000i \(0.442984\pi\)
\(510\) 0 0
\(511\) 2.56511e6 0.434564
\(512\) 0 0
\(513\) 2.14570e6 + 2.71127e6i 0.359977 + 0.454861i
\(514\) 0 0
\(515\) −1.65401e6 8.54755e6i −0.274802 1.42012i
\(516\) 0 0
\(517\) 1.72457e6 + 1.72457e6i 0.283762 + 0.283762i
\(518\) 0 0
\(519\) 5.52352e6 2.91050e6i 0.900114 0.474296i
\(520\) 0 0
\(521\) 6.85982e6i 1.10718i −0.832789 0.553590i \(-0.813257\pi\)
0.832789 0.553590i \(-0.186743\pi\)
\(522\) 0 0
\(523\) 5.98946e6 5.98946e6i 0.957488 0.957488i −0.0416441 0.999133i \(-0.513260\pi\)
0.999133 + 0.0416441i \(0.0132596\pi\)
\(524\) 0 0
\(525\) 4.80156e6 + 4.07353e6i 0.760299 + 0.645018i
\(526\) 0 0
\(527\) −3.27848e6 + 3.27848e6i −0.514216 + 0.514216i
\(528\) 0 0
\(529\) 334669.i 0.0519967i
\(530\) 0 0
\(531\) 44296.2 + 30361.9i 0.00681757 + 0.00467297i
\(532\) 0 0
\(533\) 8.82026e6 + 8.82026e6i 1.34482 + 1.34482i
\(534\) 0 0
\(535\) 661737. + 3.41971e6i 0.0999542 + 0.516541i
\(536\) 0 0
\(537\) 230080. 742629.i 0.0344306 0.111131i
\(538\) 0 0
\(539\) 9836.01 0.00145830
\(540\) 0 0
\(541\) −8.12752e6 −1.19389 −0.596946 0.802281i \(-0.703619\pi\)
−0.596946 + 0.802281i \(0.703619\pi\)
\(542\) 0 0
\(543\) −574431. + 1.85409e6i −0.0836061 + 0.269855i
\(544\) 0 0
\(545\) −5.62178e6 3.79883e6i −0.810742 0.547846i
\(546\) 0 0
\(547\) 4.39180e6 + 4.39180e6i 0.627587 + 0.627587i 0.947460 0.319873i \(-0.103640\pi\)
−0.319873 + 0.947460i \(0.603640\pi\)
\(548\) 0 0
\(549\) −8.80448e6 6.03486e6i −1.24673 0.854548i
\(550\) 0 0
\(551\) 273260.i 0.0383440i
\(552\) 0 0
\(553\) 5.92412e6 5.92412e6i 0.823779 0.823779i
\(554\) 0 0
\(555\) −794751. + 7.24257e6i −0.109521 + 0.998068i
\(556\) 0 0
\(557\) −39334.5 + 39334.5i −0.00537199 + 0.00537199i −0.709788 0.704416i \(-0.751209\pi\)
0.704416 + 0.709788i \(0.251209\pi\)
\(558\) 0 0
\(559\) 7.78821e6i 1.05416i
\(560\) 0 0
\(561\) −2.44735e6 + 1.28958e6i −0.328314 + 0.172998i
\(562\) 0 0
\(563\) −3.42062e6 3.42062e6i −0.454814 0.454814i 0.442135 0.896949i \(-0.354221\pi\)
−0.896949 + 0.442135i \(0.854221\pi\)
\(564\) 0 0
\(565\) −424185. + 82082.7i −0.0559029 + 0.0108176i
\(566\) 0 0
\(567\) 3.08691e6 6.98053e6i 0.403243 0.911865i
\(568\) 0 0
\(569\) 1.14170e7 1.47833 0.739163 0.673526i \(-0.235221\pi\)
0.739163 + 0.673526i \(0.235221\pi\)
\(570\) 0 0
\(571\) −2.32685e6 −0.298661 −0.149331 0.988787i \(-0.547712\pi\)
−0.149331 + 0.988787i \(0.547712\pi\)
\(572\) 0 0
\(573\) −5.84806e6 1.81184e6i −0.744090 0.230533i
\(574\) 0 0
\(575\) −3.02811e6 + 7.10051e6i −0.381946 + 0.895612i
\(576\) 0 0
\(577\) 1.42852e6 + 1.42852e6i 0.178627 + 0.178627i 0.790757 0.612130i \(-0.209687\pi\)
−0.612130 + 0.790757i \(0.709687\pi\)
\(578\) 0 0
\(579\) 2.36879e6 + 4.49546e6i 0.293650 + 0.557285i
\(580\) 0 0
\(581\) 1.07885e6i 0.132594i
\(582\) 0 0
\(583\) 1.92934e6 1.92934e6i 0.235092 0.235092i
\(584\) 0 0
\(585\) −3.46415e6 + 7.97608e6i −0.418512 + 0.963607i
\(586\) 0 0
\(587\) 5.13039e6 5.13039e6i 0.614546 0.614546i −0.329581 0.944127i \(-0.606907\pi\)
0.944127 + 0.329581i \(0.106907\pi\)
\(588\) 0 0
\(589\) 2.36685e6i 0.281114i
\(590\) 0 0
\(591\) −3.49728e6 6.63710e6i −0.411871 0.781645i
\(592\) 0 0
\(593\) 3.57161e6 + 3.57161e6i 0.417087 + 0.417087i 0.884199 0.467111i \(-0.154705\pi\)
−0.467111 + 0.884199i \(0.654705\pi\)
\(594\) 0 0
\(595\) −7.23388e6 + 1.07052e7i −0.837681 + 1.23966i
\(596\) 0 0
\(597\) −1.62269e7 5.02741e6i −1.86338 0.577309i
\(598\) 0 0
\(599\) −1.49341e7 −1.70064 −0.850320 0.526266i \(-0.823592\pi\)
−0.850320 + 0.526266i \(0.823592\pi\)
\(600\) 0 0
\(601\) −6.58569e6 −0.743730 −0.371865 0.928287i \(-0.621282\pi\)
−0.371865 + 0.928287i \(0.621282\pi\)
\(602\) 0 0
\(603\) 2.43967e6 + 1.30715e7i 0.273236 + 1.46397i
\(604\) 0 0
\(605\) −4.73242e6 + 7.00337e6i −0.525647 + 0.777891i
\(606\) 0 0
\(607\) 2.10867e6 + 2.10867e6i 0.232294 + 0.232294i 0.813650 0.581356i \(-0.197477\pi\)
−0.581356 + 0.813650i \(0.697477\pi\)
\(608\) 0 0
\(609\) 533665. 281204.i 0.0583077 0.0307240i
\(610\) 0 0
\(611\) 1.57311e7i 1.70473i
\(612\) 0 0
\(613\) 6.87357e6 6.87357e6i 0.738808 0.738808i −0.233540 0.972347i \(-0.575031\pi\)
0.972347 + 0.233540i \(0.0750309\pi\)
\(614\) 0 0
\(615\) 1.32448e7 1.06255e7i 1.41208 1.13282i
\(616\) 0 0
\(617\) −4.24340e6 + 4.24340e6i −0.448746 + 0.448746i −0.894938 0.446191i \(-0.852780\pi\)
0.446191 + 0.894938i \(0.352780\pi\)
\(618\) 0 0
\(619\) 3.15907e6i 0.331385i −0.986177 0.165692i \(-0.947014\pi\)
0.986177 0.165692i \(-0.0529859\pi\)
\(620\) 0 0
\(621\) 9.29414e6 + 1.08226e6i 0.967120 + 0.112616i
\(622\) 0 0
\(623\) 1.14956e7 + 1.14956e7i 1.18662 + 1.18662i
\(624\) 0 0
\(625\) −7.04761e6 + 6.76008e6i −0.721675 + 0.692232i
\(626\) 0 0
\(627\) 417918. 1.34891e6i 0.0424544 0.137030i
\(628\) 0 0
\(629\) −1.49502e7 −1.50667
\(630\) 0 0
\(631\) −1.35027e7 −1.35004 −0.675019 0.737801i \(-0.735864\pi\)
−0.675019 + 0.737801i \(0.735864\pi\)
\(632\) 0 0
\(633\) 3.06443e6 9.89104e6i 0.303977 0.981144i
\(634\) 0 0
\(635\) −4.53864e6 + 878258.i −0.446675 + 0.0864347i
\(636\) 0 0
\(637\) 44860.9 + 44860.9i 0.00438046 + 0.00438046i
\(638\) 0 0
\(639\) 8.49908e6 1.23996e7i 0.823417 1.20131i
\(640\) 0 0
\(641\) 9.62559e6i 0.925300i −0.886541 0.462650i \(-0.846899\pi\)
0.886541 0.462650i \(-0.153101\pi\)
\(642\) 0 0
\(643\) 7.56761e6 7.56761e6i 0.721824 0.721824i −0.247153 0.968976i \(-0.579495\pi\)
0.968976 + 0.247153i \(0.0794950\pi\)
\(644\) 0 0
\(645\) −1.05386e7 1.15644e6i −0.997437 0.109452i
\(646\) 0 0
\(647\) −1.14833e7 + 1.14833e7i −1.07846 + 1.07846i −0.0818142 + 0.996648i \(0.526071\pi\)
−0.996648 + 0.0818142i \(0.973929\pi\)
\(648\) 0 0
\(649\) 21933.6i 0.00204408i
\(650\) 0 0
\(651\) −4.62235e6 + 2.43565e6i −0.427475 + 0.225249i
\(652\) 0 0
\(653\) −1.08314e6 1.08314e6i −0.0994035 0.0994035i 0.655656 0.755060i \(-0.272392\pi\)
−0.755060 + 0.655656i \(0.772392\pi\)
\(654\) 0 0
\(655\) −5.28273e6 3.56972e6i −0.481122 0.325111i
\(656\) 0 0
\(657\) 4.74041e6 884751.i 0.428452 0.0799664i
\(658\) 0 0
\(659\) 1.50688e7 1.35166 0.675828 0.737060i \(-0.263786\pi\)
0.675828 + 0.737060i \(0.263786\pi\)
\(660\) 0 0
\(661\) 1.82461e6 0.162430 0.0812152 0.996697i \(-0.474120\pi\)
0.0812152 + 0.996697i \(0.474120\pi\)
\(662\) 0 0
\(663\) −1.70437e7 5.28046e6i −1.50585 0.466539i
\(664\) 0 0
\(665\) −1.25304e6 6.47543e6i −0.109878 0.567825i
\(666\) 0 0
\(667\) 522903. + 522903.i 0.0455100 + 0.0455100i
\(668\) 0 0
\(669\) −2.59244e6 4.91991e6i −0.223946 0.425003i
\(670\) 0 0
\(671\) 4.35960e6i 0.373801i
\(672\) 0 0
\(673\) 3.29426e6 3.29426e6i 0.280363 0.280363i −0.552891 0.833254i \(-0.686475\pi\)
0.833254 + 0.552891i \(0.186475\pi\)
\(674\) 0 0
\(675\) 1.02785e7 + 5.87187e6i 0.868300 + 0.496040i
\(676\) 0 0
\(677\) −441807. + 441807.i −0.0370477 + 0.0370477i −0.725388 0.688340i \(-0.758340\pi\)
0.688340 + 0.725388i \(0.258340\pi\)
\(678\) 0 0
\(679\) 1.29066e7i 1.07433i
\(680\) 0 0
\(681\) −9.11796e6 1.73040e7i −0.753407 1.42981i
\(682\) 0 0
\(683\) 1.25040e6 + 1.25040e6i 0.102564 + 0.102564i 0.756527 0.653963i \(-0.226895\pi\)
−0.653963 + 0.756527i \(0.726895\pi\)
\(684\) 0 0
\(685\) −3.29440e6 1.70247e7i −0.268256 1.38629i
\(686\) 0 0
\(687\) 4.96892e6 + 1.53946e6i 0.401671 + 0.124445i
\(688\) 0 0
\(689\) 1.75990e7 1.41234
\(690\) 0 0
\(691\) −8.70803e6 −0.693785 −0.346892 0.937905i \(-0.612763\pi\)
−0.346892 + 0.937905i \(0.612763\pi\)
\(692\) 0 0
\(693\) −3.06443e6 + 571946.i −0.242391 + 0.0452399i
\(694\) 0 0
\(695\) 1.55063e7 + 1.04781e7i 1.21771 + 0.822851i
\(696\) 0 0
\(697\) 2.46366e7 + 2.46366e7i 1.92088 + 1.92088i
\(698\) 0 0
\(699\) −8.27960e6 + 4.36276e6i −0.640938 + 0.337729i
\(700\) 0 0
\(701\) 7.69157e6i 0.591180i 0.955315 + 0.295590i \(0.0955163\pi\)
−0.955315 + 0.295590i \(0.904484\pi\)
\(702\) 0 0
\(703\) 5.39652e6 5.39652e6i 0.411837 0.411837i
\(704\) 0 0
\(705\) 2.12866e7 + 2.33585e6i 1.61300 + 0.177000i
\(706\) 0 0
\(707\) 1.40094e6 1.40094e6i 0.105408 0.105408i
\(708\) 0 0
\(709\) 7.20245e6i 0.538102i 0.963126 + 0.269051i \(0.0867100\pi\)
−0.963126 + 0.269051i \(0.913290\pi\)
\(710\) 0 0
\(711\) 8.90464e6 1.29913e7i 0.660605 0.963781i
\(712\) 0 0
\(713\) −4.52913e6 4.52913e6i −0.333650 0.333650i
\(714\) 0 0
\(715\) 3.48692e6 674743.i 0.255080 0.0493598i
\(716\) 0 0
\(717\) 3.69491e6 1.19260e7i 0.268414 0.866358i
\(718\) 0 0
\(719\) 6.82537e6 0.492384 0.246192 0.969221i \(-0.420821\pi\)
0.246192 + 0.969221i \(0.420821\pi\)
\(720\) 0 0
\(721\) 2.01308e7 1.44219
\(722\) 0 0
\(723\) −7.54631e6 + 2.43572e7i −0.536894 + 1.73293i
\(724\) 0 0
\(725\) 348998. + 868005.i 0.0246591 + 0.0613306i
\(726\) 0 0
\(727\) −6.24584e6 6.24584e6i −0.438283 0.438283i 0.453151 0.891434i \(-0.350300\pi\)
−0.891434 + 0.453151i \(0.850300\pi\)
\(728\) 0 0
\(729\) 3.29702e6 1.39650e7i 0.229775 0.973244i
\(730\) 0 0
\(731\) 2.17539e7i 1.50572i
\(732\) 0 0
\(733\) −1.01323e7 + 1.01323e7i −0.696545 + 0.696545i −0.963664 0.267119i \(-0.913928\pi\)
0.267119 + 0.963664i \(0.413928\pi\)
\(734\) 0 0
\(735\) 67364.9 54042.5i 0.00459955 0.00368992i
\(736\) 0 0
\(737\) 3.84023e6 3.84023e6i 0.260429 0.260429i
\(738\) 0 0
\(739\) 2.01149e7i 1.35490i 0.735570 + 0.677449i \(0.236914\pi\)
−0.735570 + 0.677449i \(0.763086\pi\)
\(740\) 0 0
\(741\) 8.05830e6 4.24615e6i 0.539135 0.284086i
\(742\) 0 0
\(743\) 4.47987e6 + 4.47987e6i 0.297710 + 0.297710i 0.840116 0.542406i \(-0.182487\pi\)
−0.542406 + 0.840116i \(0.682487\pi\)
\(744\) 0 0
\(745\) 7.05218e6 1.04363e7i 0.465514 0.688901i
\(746\) 0 0
\(747\) 372116. + 1.99376e6i 0.0243993 + 0.130729i
\(748\) 0 0
\(749\) −8.05392e6 −0.524569
\(750\) 0 0
\(751\) −8.55756e6 −0.553669 −0.276834 0.960918i \(-0.589285\pi\)
−0.276834 + 0.960918i \(0.589285\pi\)
\(752\) 0 0
\(753\) −1.10537e7 3.42464e6i −0.710427 0.220104i
\(754\) 0 0
\(755\) 2.65008e6 3.92178e6i 0.169197 0.250389i
\(756\) 0 0
\(757\) −7.84285e6 7.84285e6i −0.497433 0.497433i 0.413205 0.910638i \(-0.364409\pi\)
−0.910638 + 0.413205i \(0.864409\pi\)
\(758\) 0 0
\(759\) −1.78152e6 3.38095e6i −0.112250 0.213027i
\(760\) 0 0
\(761\) 1.13075e7i 0.707793i 0.935285 + 0.353897i \(0.115144\pi\)
−0.935285 + 0.353897i \(0.884856\pi\)
\(762\) 0 0
\(763\) 1.10935e7 1.10935e7i 0.689852 0.689852i
\(764\) 0 0
\(765\) −9.67603e6 + 2.22787e7i −0.597783 + 1.37637i
\(766\) 0 0
\(767\) 100036. 100036.i 0.00614002 0.00614002i
\(768\) 0 0
\(769\) 2.51981e7i 1.53657i −0.640109 0.768284i \(-0.721111\pi\)
0.640109 0.768284i \(-0.278889\pi\)
\(770\) 0 0
\(771\) 1.04935e7 + 1.99144e7i 0.635747 + 1.20651i
\(772\) 0 0
\(773\) −2.15965e7 2.15965e7i −1.29997 1.29997i −0.928406 0.371567i \(-0.878821\pi\)
−0.371567 0.928406i \(-0.621179\pi\)
\(774\) 0 0
\(775\) −3.02285e6 7.51824e6i −0.180785 0.449637i
\(776\) 0 0
\(777\) −1.60925e7 4.98578e6i −0.956252 0.296265i
\(778\) 0 0
\(779\) −1.77860e7 −1.05011
\(780\) 0 0
\(781\) −6.13976e6 −0.360183
\(782\) 0 0
\(783\) 889240. 703744.i 0.0518340 0.0410214i
\(784\) 0 0
\(785\) 5.64525e6 1.09239e6i 0.326971 0.0632711i
\(786\) 0 0
\(787\) −1.06749e7 1.06749e7i −0.614368 0.614368i 0.329713 0.944081i \(-0.393048\pi\)
−0.944081 + 0.329713i \(0.893048\pi\)
\(788\) 0 0
\(789\) 1.05430e7 5.55538e6i 0.602934 0.317703i
\(790\) 0 0
\(791\) 999018.i 0.0567717i
\(792\) 0 0
\(793\) −1.98836e7 + 1.98836e7i −1.12283 + 1.12283i
\(794\) 0 0
\(795\) 2.61321e6 2.38142e7i 0.146641 1.33634i
\(796\) 0 0
\(797\) −7.49492e6 + 7.49492e6i −0.417947 + 0.417947i −0.884496 0.466548i \(-0.845497\pi\)
0.466548 + 0.884496i \(0.345497\pi\)
\(798\) 0 0
\(799\) 4.39400e7i 2.43497i
\(800\) 0 0
\(801\) 2.52093e7 + 1.72792e7i 1.38829 + 0.951574i
\(802\) 0 0
\(803\) −1.39267e6 1.39267e6i −0.0762182 0.0762182i
\(804\) 0 0
\(805\) −1.47890e7 9.99341e6i −0.804356 0.543530i
\(806\) 0 0
\(807\) −3.45658e6 + 1.11568e7i −0.186837 + 0.603053i
\(808\) 0 0
\(809\) 3.28460e6 0.176446 0.0882230 0.996101i \(-0.471881\pi\)
0.0882230 + 0.996101i \(0.471881\pi\)
\(810\) 0 0
\(811\) 1.90925e7 1.01932 0.509660 0.860376i \(-0.329771\pi\)
0.509660 + 0.860376i \(0.329771\pi\)
\(812\) 0 0
\(813\) −5.76304e6 + 1.86013e7i −0.305792 + 0.987001i
\(814\) 0 0
\(815\) −1.13638e6 5.87254e6i −0.0599278 0.309694i
\(816\) 0 0
\(817\) 7.85246e6 + 7.85246e6i 0.411576 + 0.411576i
\(818\) 0 0
\(819\) −1.65851e7 1.13679e7i −0.863988 0.592204i
\(820\) 0 0
\(821\) 3.82252e7i 1.97921i 0.143814 + 0.989605i \(0.454063\pi\)
−0.143814 + 0.989605i \(0.545937\pi\)
\(822\) 0 0
\(823\) 1.78614e7 1.78614e7i 0.919212 0.919212i −0.0777600 0.996972i \(-0.524777\pi\)
0.996972 + 0.0777600i \(0.0247768\pi\)
\(824\) 0 0
\(825\) −395272. 4.81853e6i −0.0202191 0.246479i
\(826\) 0 0
\(827\) 1.31871e7 1.31871e7i 0.670479 0.670479i −0.287347 0.957826i \(-0.592773\pi\)
0.957826 + 0.287347i \(0.0927734\pi\)
\(828\) 0 0
\(829\) 5.40847e6i 0.273330i 0.990617 + 0.136665i \(0.0436385\pi\)
−0.990617 + 0.136665i \(0.956362\pi\)
\(830\) 0 0
\(831\) −1.53148e7 + 8.06982e6i −0.769324 + 0.405379i
\(832\) 0 0
\(833\) 125305. + 125305.i 0.00625685 + 0.00625685i
\(834\) 0 0
\(835\) −1.75238e6 9.05589e6i −0.0869784 0.449485i
\(836\) 0 0
\(837\) −7.70216e6 + 6.09549e6i −0.380014 + 0.300743i
\(838\) 0 0
\(839\) −2.58334e7 −1.26700 −0.633501 0.773742i \(-0.718383\pi\)
−0.633501 + 0.773742i \(0.718383\pi\)
\(840\) 0 0
\(841\) −2.04215e7 −0.995630
\(842\) 0 0
\(843\) −4.18003e6 1.29505e6i −0.202586 0.0627651i
\(844\) 0 0
\(845\) 1.78322e6 + 1.20498e6i 0.0859138 + 0.0580549i
\(846\) 0 0
\(847\) −1.38198e7 1.38198e7i −0.661899 0.661899i
\(848\) 0 0
\(849\) 1.55336e7 + 2.94796e7i 0.739611 + 1.40363i
\(850\) 0 0
\(851\) 2.06532e7i 0.977607i
\(852\) 0 0
\(853\) −6.49234e6 + 6.49234e6i −0.305512 + 0.305512i −0.843166 0.537654i \(-0.819311\pi\)
0.537654 + 0.843166i \(0.319311\pi\)
\(854\) 0 0
\(855\) −4.54915e6 1.15346e7i −0.212821 0.539620i
\(856\) 0 0
\(857\) 4.74874e6 4.74874e6i 0.220865 0.220865i −0.587998 0.808862i \(-0.700084\pi\)
0.808862 + 0.587998i \(0.200084\pi\)
\(858\) 0 0
\(859\) 3.44086e6i 0.159105i −0.996831 0.0795525i \(-0.974651\pi\)
0.996831 0.0795525i \(-0.0253491\pi\)
\(860\) 0 0
\(861\) 1.83031e7 + 3.47354e7i 0.841426 + 1.59685i
\(862\) 0 0
\(863\) 1.94180e7 + 1.94180e7i 0.887520 + 0.887520i 0.994284 0.106765i \(-0.0340491\pi\)
−0.106765 + 0.994284i \(0.534049\pi\)
\(864\) 0 0
\(865\) −2.19817e7 + 4.25362e6i −0.998900 + 0.193294i
\(866\) 0 0
\(867\) −2.64643e7 8.19915e6i −1.19568 0.370443i
\(868\) 0 0
\(869\) −6.43273e6 −0.288966
\(870\) 0 0
\(871\) 3.50297e7 1.56456
\(872\) 0 0
\(873\) 4.45171e6 + 2.38518e7i 0.197693 + 1.05922i
\(874\) 0 0
\(875\) −1.22504e7 1.89687e7i −0.540917 0.837563i
\(876\) 0 0
\(877\) 1.15752e6 + 1.15752e6i 0.0508192 + 0.0508192i 0.732060 0.681240i \(-0.238559\pi\)
−0.681240 + 0.732060i \(0.738559\pi\)
\(878\) 0 0
\(879\) −2.70447e7 + 1.42506e7i −1.18062 + 0.622101i
\(880\) 0 0
\(881\) 3.25507e7i 1.41293i −0.707748 0.706465i \(-0.750289\pi\)
0.707748 0.706465i \(-0.249711\pi\)
\(882\) 0 0
\(883\) 6.37239e6 6.37239e6i 0.275043 0.275043i −0.556083 0.831127i \(-0.687696\pi\)
0.831127 + 0.556083i \(0.187696\pi\)
\(884\) 0 0
\(885\) −120511. 150219.i −0.00517210 0.00644712i
\(886\) 0 0
\(887\) 2.01092e7 2.01092e7i 0.858196 0.858196i −0.132930 0.991125i \(-0.542438\pi\)
0.991125 + 0.132930i \(0.0424385\pi\)
\(888\) 0 0
\(889\) 1.06892e7i 0.453618i
\(890\) 0 0
\(891\) −5.46589e6 + 2.11395e6i −0.230657 + 0.0892073i
\(892\) 0 0
\(893\) −1.58609e7 1.58609e7i −0.665578 0.665578i
\(894\) 0 0
\(895\) −1.56099e6 + 2.31006e6i −0.0651392 + 0.0963976i
\(896\) 0 0
\(897\) 7.29482e6 2.35454e7i 0.302715 0.977070i
\(898\) 0 0
\(899\) −776278. −0.0320345
\(900\) 0 0
\(901\) 4.91574e7 2.01733
\(902\) 0 0
\(903\) 7.25479e6 2.34162e7i 0.296077 0.955647i
\(904\) 0 0
\(905\) 3.89724e6 5.76742e6i 0.158174 0.234078i
\(906\) 0 0
\(907\) −1.06262e7 1.06262e7i −0.428904 0.428904i 0.459351 0.888255i \(-0.348082\pi\)
−0.888255 + 0.459351i \(0.848082\pi\)
\(908\) 0 0
\(909\) 2.10578e6 3.07220e6i 0.0845286 0.123322i
\(910\) 0 0
\(911\) 2.34160e7i 0.934795i 0.884047 + 0.467397i \(0.154808\pi\)
−0.884047 + 0.467397i \(0.845192\pi\)
\(912\) 0 0
\(913\) 585740. 585740.i 0.0232556 0.0232556i
\(914\) 0 0
\(915\) 2.39532e7 + 2.98581e7i 0.945824 + 1.17899i
\(916\) 0 0
\(917\) 1.04244e7 1.04244e7i 0.409382 0.409382i
\(918\) 0 0
\(919\) 1.95298e7i 0.762796i 0.924411 + 0.381398i \(0.124557\pi\)
−0.924411 + 0.381398i \(0.875443\pi\)
\(920\) 0 0
\(921\) −3.34803e6 + 1.76417e6i −0.130059 + 0.0685317i
\(922\) 0 0
\(923\) −2.80027e7 2.80027e7i −1.08192 1.08192i
\(924\) 0 0
\(925\) 1.02497e7 2.40341e7i 0.393873 0.923580i
\(926\) 0 0
\(927\) 3.72023e7 6.94345e6i 1.42191 0.265385i
\(928\) 0 0
\(929\) 6.55210e6 0.249081 0.124541 0.992215i \(-0.460254\pi\)
0.124541 + 0.992215i \(0.460254\pi\)
\(930\) 0 0
\(931\) −90462.0 −0.00342052
\(932\) 0 0
\(933\) 2.78819e7 + 8.63834e6i 1.04862 + 0.324882i
\(934\) 0 0
\(935\) 9.73963e6 1.88468e6i 0.364345 0.0705033i
\(936\) 0 0
\(937\) 2.00402e7 + 2.00402e7i 0.745681 + 0.745681i 0.973665 0.227984i \(-0.0732133\pi\)
−0.227984 + 0.973665i \(0.573213\pi\)
\(938\) 0 0
\(939\) 7.95911e6 + 1.51047e7i 0.294578 + 0.559047i
\(940\) 0 0
\(941\) 454846.i 0.0167452i 0.999965 + 0.00837259i \(0.00266511\pi\)
−0.999965 + 0.00837259i \(0.997335\pi\)
\(942\) 0 0
\(943\) −3.40349e7 + 3.40349e7i −1.24636 + 1.24636i
\(944\) 0 0
\(945\) −1.78452e7 + 2.07542e7i −0.650043 + 0.756008i
\(946\) 0 0
\(947\) 5.99645e6 5.99645e6i 0.217280 0.217280i −0.590071 0.807351i \(-0.700900\pi\)
0.807351 + 0.590071i \(0.200900\pi\)
\(948\) 0 0
\(949\) 1.27036e7i 0.457890i
\(950\) 0 0
\(951\) −2.04640e7 3.88364e7i −0.733735 1.39247i
\(952\) 0 0
\(953\) 1.66076e7 + 1.66076e7i 0.592343 + 0.592343i 0.938264 0.345921i \(-0.112433\pi\)
−0.345921 + 0.938264i \(0.612433\pi\)
\(954\) 0 0
\(955\) 1.81913e7 + 1.22925e7i 0.645440 + 0.436146i
\(956\) 0 0
\(957\) −442415. 137069.i −0.0156153 0.00483791i
\(958\) 0 0
\(959\) 4.00957e7 1.40783
\(960\) 0 0
\(961\) −2.19054e7 −0.765144
\(962\) 0 0
\(963\) −1.48839e7 + 2.77794e6i −0.517192 + 0.0965288i
\(964\) 0 0
\(965\) −3.46191e6 1.78904e7i −0.119674 0.618446i
\(966\) 0 0
\(967\) −8.88785e6 8.88785e6i −0.305654 0.305654i 0.537567 0.843221i \(-0.319344\pi\)
−0.843221 + 0.537567i \(0.819344\pi\)
\(968\) 0 0
\(969\) 2.25083e7 1.18603e7i 0.770077 0.405776i
\(970\) 0 0
\(971\) 1.61689e7i 0.550341i −0.961395 0.275171i \(-0.911266\pi\)
0.961395 0.275171i \(-0.0887344\pi\)
\(972\) 0 0
\(973\) −3.05986e7 + 3.05986e7i −1.03614 + 1.03614i
\(974\) 0 0
\(975\) 2.01740e7 2.37796e7i 0.679641 0.801110i
\(976\) 0 0
\(977\) −1.34892e7 + 1.34892e7i −0.452116 + 0.452116i −0.896056 0.443941i \(-0.853580\pi\)
0.443941 + 0.896056i \(0.353580\pi\)
\(978\) 0 0
\(979\) 1.24826e7i 0.416243i
\(980\) 0 0
\(981\) 1.66748e7 2.43274e7i 0.553206 0.807093i
\(982\) 0 0
\(983\) −1.33156e7 1.33156e7i −0.439517 0.439517i 0.452332 0.891849i \(-0.350592\pi\)
−0.891849 + 0.452332i \(0.850592\pi\)
\(984\) 0 0
\(985\) 5.11117e6 + 2.64134e7i 0.167853 + 0.867428i
\(986\) 0 0
\(987\) −1.46537e7 + 4.72976e7i −0.478800 + 1.54542i
\(988\) 0 0
\(989\) 3.00525e7 0.976988
\(990\) 0 0
\(991\) 7.07932e6 0.228985 0.114493 0.993424i \(-0.463476\pi\)
0.114493 + 0.993424i \(0.463476\pi\)
\(992\) 0 0
\(993\) 1.69774e7 5.47977e7i 0.546383 1.76356i
\(994\) 0 0
\(995\) 5.04764e7 + 3.41086e7i 1.61633 + 1.09221i
\(996\) 0 0
\(997\) 2.77833e7 + 2.77833e7i 0.885210 + 0.885210i 0.994058 0.108848i \(-0.0347163\pi\)
−0.108848 + 0.994058i \(0.534716\pi\)
\(998\) 0 0
\(999\) −3.14593e7 3.66328e6i −0.997320 0.116133i
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 60.6.i.a.17.7 20
3.2 odd 2 inner 60.6.i.a.17.9 yes 20
5.2 odd 4 300.6.i.d.293.2 20
5.3 odd 4 inner 60.6.i.a.53.9 yes 20
5.4 even 2 300.6.i.d.257.4 20
15.2 even 4 300.6.i.d.293.4 20
15.8 even 4 inner 60.6.i.a.53.7 yes 20
15.14 odd 2 300.6.i.d.257.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.6.i.a.17.7 20 1.1 even 1 trivial
60.6.i.a.17.9 yes 20 3.2 odd 2 inner
60.6.i.a.53.7 yes 20 15.8 even 4 inner
60.6.i.a.53.9 yes 20 5.3 odd 4 inner
300.6.i.d.257.2 20 15.14 odd 2
300.6.i.d.257.4 20 5.4 even 2
300.6.i.d.293.2 20 5.2 odd 4
300.6.i.d.293.4 20 15.2 even 4