Properties

Label 304.6.i.d.49.6
Level $304$
Weight $6$
Character 304.49
Analytic conductor $48.757$
Analytic rank $0$
Dimension $18$
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] = [304,6,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), 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: \(48.7566812231\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} - 2546775754 x^{12} + 7879525640 x^{11} + 976140188391 x^{10} + \cdots + 66\!\cdots\!83 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.6
Root \(-6.49256 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 304.49
Dual form 304.6.i.d.273.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.99628 - 6.92176i) q^{3} +(31.6056 - 54.7425i) q^{5} +80.2775 q^{7} +(89.5595 + 155.122i) q^{9} +O(q^{10})\) \(q+(3.99628 - 6.92176i) q^{3} +(31.6056 - 54.7425i) q^{5} +80.2775 q^{7} +(89.5595 + 155.122i) q^{9} +475.169 q^{11} +(-337.473 - 584.520i) q^{13} +(-252.609 - 437.532i) q^{15} +(866.499 - 1500.82i) q^{17} +(574.366 + 1464.99i) q^{19} +(320.811 - 555.661i) q^{21} +(2424.72 + 4199.73i) q^{23} +(-435.326 - 754.007i) q^{25} +3373.81 q^{27} +(2394.08 + 4146.67i) q^{29} +127.218 q^{31} +(1898.91 - 3289.01i) q^{33} +(2537.22 - 4394.59i) q^{35} -13949.4 q^{37} -5394.54 q^{39} +(7883.24 - 13654.2i) q^{41} +(-964.112 + 1669.89i) q^{43} +11322.3 q^{45} +(-8099.51 - 14028.8i) q^{47} -10362.5 q^{49} +(-6925.54 - 11995.4i) q^{51} +(8925.98 + 15460.3i) q^{53} +(15018.0 - 26011.9i) q^{55} +(12435.6 + 1878.90i) q^{57} +(-8424.75 + 14592.1i) q^{59} +(11641.7 + 20164.0i) q^{61} +(7189.61 + 12452.8i) q^{63} -42664.1 q^{65} +(-13618.4 - 23587.8i) q^{67} +38759.3 q^{69} +(37449.3 - 64864.1i) q^{71} +(34900.9 - 60450.1i) q^{73} -6958.73 q^{75} +38145.4 q^{77} +(-11408.5 + 19760.1i) q^{79} +(-8280.29 + 14341.9i) q^{81} +58008.8 q^{83} +(-54772.4 - 94868.6i) q^{85} +38269.6 q^{87} +(8203.84 + 14209.5i) q^{89} +(-27091.5 - 46923.8i) q^{91} +(508.400 - 880.575i) q^{93} +(98350.5 + 14859.7i) q^{95} +(65710.8 - 113814. i) q^{97} +(42555.9 + 73709.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9} + 320 q^{11} + 227 q^{13} + 101 q^{15} + 179 q^{17} + 868 q^{19} - 5700 q^{21} + 3425 q^{23} - 7054 q^{25} - 14722 q^{27} - 7349 q^{29} + 9960 q^{31} - 2998 q^{33} - 15888 q^{35} + 26444 q^{37} + 30246 q^{39} - 7311 q^{41} + 8283 q^{43} - 62164 q^{45} - 37603 q^{47} + 124738 q^{49} - 47227 q^{51} - 20337 q^{53} - 716 q^{55} - 57555 q^{57} + 74455 q^{59} - 7569 q^{61} + 52544 q^{63} + 188998 q^{65} + 26177 q^{67} + 116282 q^{69} + 53463 q^{71} - 14103 q^{73} - 120912 q^{75} - 31960 q^{77} - 31825 q^{79} - 21137 q^{81} - 82600 q^{83} - 50787 q^{85} + 339766 q^{87} - 155197 q^{89} + 2800 q^{91} - 46460 q^{93} - 49315 q^{95} + 111241 q^{97} + 193544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 3.99628 6.92176i 0.256361 0.444031i −0.708903 0.705306i \(-0.750810\pi\)
0.965264 + 0.261275i \(0.0841430\pi\)
\(4\) 0 0
\(5\) 31.6056 54.7425i 0.565378 0.979263i −0.431637 0.902048i \(-0.642064\pi\)
0.997014 0.0772156i \(-0.0246030\pi\)
\(6\) 0 0
\(7\) 80.2775 0.619225 0.309613 0.950863i \(-0.399801\pi\)
0.309613 + 0.950863i \(0.399801\pi\)
\(8\) 0 0
\(9\) 89.5595 + 155.122i 0.368558 + 0.638361i
\(10\) 0 0
\(11\) 475.169 1.18404 0.592020 0.805923i \(-0.298331\pi\)
0.592020 + 0.805923i \(0.298331\pi\)
\(12\) 0 0
\(13\) −337.473 584.520i −0.553835 0.959270i −0.997993 0.0633218i \(-0.979831\pi\)
0.444158 0.895948i \(-0.353503\pi\)
\(14\) 0 0
\(15\) −252.609 437.532i −0.289882 0.502090i
\(16\) 0 0
\(17\) 866.499 1500.82i 0.727186 1.25952i −0.230881 0.972982i \(-0.574161\pi\)
0.958068 0.286542i \(-0.0925058\pi\)
\(18\) 0 0
\(19\) 574.366 + 1464.99i 0.365010 + 0.931004i
\(20\) 0 0
\(21\) 320.811 555.661i 0.158745 0.274955i
\(22\) 0 0
\(23\) 2424.72 + 4199.73i 0.955743 + 1.65540i 0.732658 + 0.680597i \(0.238279\pi\)
0.223085 + 0.974799i \(0.428387\pi\)
\(24\) 0 0
\(25\) −435.326 754.007i −0.139304 0.241282i
\(26\) 0 0
\(27\) 3373.81 0.890658
\(28\) 0 0
\(29\) 2394.08 + 4146.67i 0.528620 + 0.915597i 0.999443 + 0.0333693i \(0.0106237\pi\)
−0.470823 + 0.882228i \(0.656043\pi\)
\(30\) 0 0
\(31\) 127.218 0.0237764 0.0118882 0.999929i \(-0.496216\pi\)
0.0118882 + 0.999929i \(0.496216\pi\)
\(32\) 0 0
\(33\) 1898.91 3289.01i 0.303542 0.525751i
\(34\) 0 0
\(35\) 2537.22 4394.59i 0.350096 0.606384i
\(36\) 0 0
\(37\) −13949.4 −1.67514 −0.837570 0.546331i \(-0.816024\pi\)
−0.837570 + 0.546331i \(0.816024\pi\)
\(38\) 0 0
\(39\) −5394.54 −0.567927
\(40\) 0 0
\(41\) 7883.24 13654.2i 0.732394 1.26854i −0.223463 0.974712i \(-0.571736\pi\)
0.955857 0.293831i \(-0.0949304\pi\)
\(42\) 0 0
\(43\) −964.112 + 1669.89i −0.0795163 + 0.137726i −0.903041 0.429554i \(-0.858671\pi\)
0.823525 + 0.567280i \(0.192004\pi\)
\(44\) 0 0
\(45\) 11322.3 0.833498
\(46\) 0 0
\(47\) −8099.51 14028.8i −0.534828 0.926349i −0.999172 0.0406941i \(-0.987043\pi\)
0.464344 0.885655i \(-0.346290\pi\)
\(48\) 0 0
\(49\) −10362.5 −0.616560
\(50\) 0 0
\(51\) −6925.54 11995.4i −0.372845 0.645786i
\(52\) 0 0
\(53\) 8925.98 + 15460.3i 0.436482 + 0.756009i 0.997415 0.0718520i \(-0.0228909\pi\)
−0.560933 + 0.827861i \(0.689558\pi\)
\(54\) 0 0
\(55\) 15018.0 26011.9i 0.669430 1.15949i
\(56\) 0 0
\(57\) 12435.6 + 1878.90i 0.506969 + 0.0765978i
\(58\) 0 0
\(59\) −8424.75 + 14592.1i −0.315084 + 0.545742i −0.979455 0.201661i \(-0.935366\pi\)
0.664371 + 0.747403i \(0.268699\pi\)
\(60\) 0 0
\(61\) 11641.7 + 20164.0i 0.400583 + 0.693830i 0.993796 0.111215i \(-0.0354743\pi\)
−0.593214 + 0.805045i \(0.702141\pi\)
\(62\) 0 0
\(63\) 7189.61 + 12452.8i 0.228220 + 0.395289i
\(64\) 0 0
\(65\) −42664.1 −1.25250
\(66\) 0 0
\(67\) −13618.4 23587.8i −0.370630 0.641950i 0.619033 0.785365i \(-0.287525\pi\)
−0.989663 + 0.143415i \(0.954192\pi\)
\(68\) 0 0
\(69\) 38759.3 0.980062
\(70\) 0 0
\(71\) 37449.3 64864.1i 0.881654 1.52707i 0.0321524 0.999483i \(-0.489764\pi\)
0.849501 0.527586i \(-0.176903\pi\)
\(72\) 0 0
\(73\) 34900.9 60450.1i 0.766530 1.32767i −0.172903 0.984939i \(-0.555315\pi\)
0.939434 0.342731i \(-0.111352\pi\)
\(74\) 0 0
\(75\) −6958.73 −0.142849
\(76\) 0 0
\(77\) 38145.4 0.733188
\(78\) 0 0
\(79\) −11408.5 + 19760.1i −0.205665 + 0.356222i −0.950344 0.311200i \(-0.899269\pi\)
0.744680 + 0.667422i \(0.232602\pi\)
\(80\) 0 0
\(81\) −8280.29 + 14341.9i −0.140227 + 0.242881i
\(82\) 0 0
\(83\) 58008.8 0.924270 0.462135 0.886810i \(-0.347084\pi\)
0.462135 + 0.886810i \(0.347084\pi\)
\(84\) 0 0
\(85\) −54772.4 94868.6i −0.822270 1.42421i
\(86\) 0 0
\(87\) 38269.6 0.542071
\(88\) 0 0
\(89\) 8203.84 + 14209.5i 0.109785 + 0.190153i 0.915683 0.401901i \(-0.131651\pi\)
−0.805898 + 0.592054i \(0.798317\pi\)
\(90\) 0 0
\(91\) −27091.5 46923.8i −0.342948 0.594004i
\(92\) 0 0
\(93\) 508.400 880.575i 0.00609535 0.0105575i
\(94\) 0 0
\(95\) 98350.5 + 14859.7i 1.11807 + 0.168928i
\(96\) 0 0
\(97\) 65710.8 113814.i 0.709100 1.22820i −0.256092 0.966653i \(-0.582435\pi\)
0.965191 0.261544i \(-0.0842318\pi\)
\(98\) 0 0
\(99\) 42555.9 + 73709.0i 0.436387 + 0.755845i
\(100\) 0 0
\(101\) −46068.0 79792.2i −0.449362 0.778318i 0.548983 0.835834i \(-0.315015\pi\)
−0.998345 + 0.0575159i \(0.981682\pi\)
\(102\) 0 0
\(103\) −149801. −1.39130 −0.695651 0.718380i \(-0.744884\pi\)
−0.695651 + 0.718380i \(0.744884\pi\)
\(104\) 0 0
\(105\) −20278.8 35124.0i −0.179502 0.310907i
\(106\) 0 0
\(107\) −62370.7 −0.526649 −0.263324 0.964707i \(-0.584819\pi\)
−0.263324 + 0.964707i \(0.584819\pi\)
\(108\) 0 0
\(109\) −27370.0 + 47406.1i −0.220652 + 0.382180i −0.955006 0.296586i \(-0.904152\pi\)
0.734354 + 0.678767i \(0.237485\pi\)
\(110\) 0 0
\(111\) −55745.6 + 96554.3i −0.429441 + 0.743813i
\(112\) 0 0
\(113\) −84395.3 −0.621759 −0.310879 0.950449i \(-0.600624\pi\)
−0.310879 + 0.950449i \(0.600624\pi\)
\(114\) 0 0
\(115\) 306538. 2.16142
\(116\) 0 0
\(117\) 60447.8 104699.i 0.408240 0.707093i
\(118\) 0 0
\(119\) 69560.4 120482.i 0.450292 0.779929i
\(120\) 0 0
\(121\) 64734.8 0.401952
\(122\) 0 0
\(123\) −63007.2 109132.i −0.375515 0.650411i
\(124\) 0 0
\(125\) 142500. 0.815717
\(126\) 0 0
\(127\) −59200.7 102539.i −0.325700 0.564128i 0.655954 0.754801i \(-0.272266\pi\)
−0.981654 + 0.190672i \(0.938933\pi\)
\(128\) 0 0
\(129\) 7705.72 + 13346.7i 0.0407698 + 0.0706154i
\(130\) 0 0
\(131\) 37228.3 64481.4i 0.189538 0.328289i −0.755559 0.655081i \(-0.772634\pi\)
0.945096 + 0.326792i \(0.105968\pi\)
\(132\) 0 0
\(133\) 46108.6 + 117606.i 0.226023 + 0.576501i
\(134\) 0 0
\(135\) 106631. 184691.i 0.503559 0.872189i
\(136\) 0 0
\(137\) −70227.1 121637.i −0.319671 0.553686i 0.660748 0.750607i \(-0.270239\pi\)
−0.980419 + 0.196921i \(0.936906\pi\)
\(138\) 0 0
\(139\) 112322. + 194548.i 0.493093 + 0.854061i 0.999968 0.00795774i \(-0.00253305\pi\)
−0.506876 + 0.862019i \(0.669200\pi\)
\(140\) 0 0
\(141\) −129472. −0.548437
\(142\) 0 0
\(143\) −160357. 277746.i −0.655763 1.13581i
\(144\) 0 0
\(145\) 302665. 1.19548
\(146\) 0 0
\(147\) −41411.5 + 71726.9i −0.158062 + 0.273772i
\(148\) 0 0
\(149\) −146994. + 254601.i −0.542417 + 0.939494i 0.456347 + 0.889802i \(0.349157\pi\)
−0.998765 + 0.0496923i \(0.984176\pi\)
\(150\) 0 0
\(151\) −366703. −1.30880 −0.654398 0.756150i \(-0.727078\pi\)
−0.654398 + 0.756150i \(0.727078\pi\)
\(152\) 0 0
\(153\) 310413. 1.07204
\(154\) 0 0
\(155\) 4020.81 6964.26i 0.0134426 0.0232833i
\(156\) 0 0
\(157\) 14338.2 24834.5i 0.0464243 0.0804093i −0.841880 0.539666i \(-0.818551\pi\)
0.888304 + 0.459256i \(0.151884\pi\)
\(158\) 0 0
\(159\) 142683. 0.447588
\(160\) 0 0
\(161\) 194650. + 337144.i 0.591820 + 1.02506i
\(162\) 0 0
\(163\) −12767.3 −0.0376383 −0.0188192 0.999823i \(-0.505991\pi\)
−0.0188192 + 0.999823i \(0.505991\pi\)
\(164\) 0 0
\(165\) −120032. 207902.i −0.343232 0.594495i
\(166\) 0 0
\(167\) 121547. + 210526.i 0.337252 + 0.584138i 0.983915 0.178638i \(-0.0571692\pi\)
−0.646663 + 0.762776i \(0.723836\pi\)
\(168\) 0 0
\(169\) −42129.2 + 72969.8i −0.113466 + 0.196529i
\(170\) 0 0
\(171\) −175812. + 220301.i −0.459789 + 0.576137i
\(172\) 0 0
\(173\) −213672. + 370091.i −0.542792 + 0.940143i 0.455951 + 0.890005i \(0.349299\pi\)
−0.998742 + 0.0501376i \(0.984034\pi\)
\(174\) 0 0
\(175\) −34946.9 60529.7i −0.0862607 0.149408i
\(176\) 0 0
\(177\) 67335.2 + 116628.i 0.161551 + 0.279814i
\(178\) 0 0
\(179\) −41174.1 −0.0960488 −0.0480244 0.998846i \(-0.515293\pi\)
−0.0480244 + 0.998846i \(0.515293\pi\)
\(180\) 0 0
\(181\) −241411. 418136.i −0.547723 0.948684i −0.998430 0.0560122i \(-0.982161\pi\)
0.450707 0.892672i \(-0.351172\pi\)
\(182\) 0 0
\(183\) 186094. 0.410776
\(184\) 0 0
\(185\) −440879. + 763624.i −0.947087 + 1.64040i
\(186\) 0 0
\(187\) 411734. 713144.i 0.861018 1.49133i
\(188\) 0 0
\(189\) 270841. 0.551518
\(190\) 0 0
\(191\) −218201. −0.432786 −0.216393 0.976306i \(-0.569429\pi\)
−0.216393 + 0.976306i \(0.569429\pi\)
\(192\) 0 0
\(193\) 357570. 619329.i 0.690983 1.19682i −0.280533 0.959844i \(-0.590511\pi\)
0.971516 0.236973i \(-0.0761554\pi\)
\(194\) 0 0
\(195\) −170498. + 295310.i −0.321094 + 0.556150i
\(196\) 0 0
\(197\) 17740.5 0.0325686 0.0162843 0.999867i \(-0.494816\pi\)
0.0162843 + 0.999867i \(0.494816\pi\)
\(198\) 0 0
\(199\) 290101. + 502469.i 0.519297 + 0.899449i 0.999748 + 0.0224275i \(0.00713948\pi\)
−0.480451 + 0.877021i \(0.659527\pi\)
\(200\) 0 0
\(201\) −217692. −0.380061
\(202\) 0 0
\(203\) 192191. + 332884.i 0.327335 + 0.566961i
\(204\) 0 0
\(205\) −498309. 863096.i −0.828159 1.43441i
\(206\) 0 0
\(207\) −434313. + 752252.i −0.704493 + 1.22022i
\(208\) 0 0
\(209\) 272921. + 696119.i 0.432186 + 1.10235i
\(210\) 0 0
\(211\) −418159. + 724272.i −0.646599 + 1.11994i 0.337330 + 0.941386i \(0.390476\pi\)
−0.983930 + 0.178556i \(0.942857\pi\)
\(212\) 0 0
\(213\) −299316. 518430.i −0.452044 0.782963i
\(214\) 0 0
\(215\) 60942.6 + 105556.i 0.0899135 + 0.155735i
\(216\) 0 0
\(217\) 10212.8 0.0147229
\(218\) 0 0
\(219\) −278947. 483151.i −0.393017 0.680726i
\(220\) 0 0
\(221\) −1.16968e6 −1.61096
\(222\) 0 0
\(223\) −349535. + 605413.i −0.470684 + 0.815248i −0.999438 0.0335271i \(-0.989326\pi\)
0.528754 + 0.848775i \(0.322659\pi\)
\(224\) 0 0
\(225\) 77975.2 135057.i 0.102683 0.177853i
\(226\) 0 0
\(227\) −813403. −1.04771 −0.523855 0.851808i \(-0.675507\pi\)
−0.523855 + 0.851808i \(0.675507\pi\)
\(228\) 0 0
\(229\) −1.18902e6 −1.49830 −0.749151 0.662399i \(-0.769538\pi\)
−0.749151 + 0.662399i \(0.769538\pi\)
\(230\) 0 0
\(231\) 152440. 264033.i 0.187961 0.325558i
\(232\) 0 0
\(233\) −106736. + 184872.i −0.128801 + 0.223091i −0.923212 0.384290i \(-0.874446\pi\)
0.794411 + 0.607381i \(0.207780\pi\)
\(234\) 0 0
\(235\) −1.02396e6 −1.20952
\(236\) 0 0
\(237\) 91182.9 + 157933.i 0.105449 + 0.182643i
\(238\) 0 0
\(239\) 965775. 1.09366 0.546829 0.837245i \(-0.315835\pi\)
0.546829 + 0.837245i \(0.315835\pi\)
\(240\) 0 0
\(241\) 391235. + 677639.i 0.433906 + 0.751547i 0.997206 0.0747059i \(-0.0238018\pi\)
−0.563300 + 0.826252i \(0.690468\pi\)
\(242\) 0 0
\(243\) 476099. + 824627.i 0.517227 + 0.895863i
\(244\) 0 0
\(245\) −327514. + 567270.i −0.348590 + 0.603775i
\(246\) 0 0
\(247\) 662485. 830123.i 0.690929 0.865765i
\(248\) 0 0
\(249\) 231819. 401523.i 0.236947 0.410404i
\(250\) 0 0
\(251\) 448780. + 777310.i 0.449624 + 0.778771i 0.998361 0.0572236i \(-0.0182248\pi\)
−0.548738 + 0.835995i \(0.684891\pi\)
\(252\) 0 0
\(253\) 1.15215e6 + 1.99558e6i 1.13164 + 1.96006i
\(254\) 0 0
\(255\) −875543. −0.843193
\(256\) 0 0
\(257\) −552328. 956660.i −0.521632 0.903493i −0.999683 0.0251612i \(-0.991990\pi\)
0.478051 0.878332i \(-0.341343\pi\)
\(258\) 0 0
\(259\) −1.11982e6 −1.03729
\(260\) 0 0
\(261\) −428825. + 742747.i −0.389654 + 0.674901i
\(262\) 0 0
\(263\) 306009. 530023.i 0.272800 0.472504i −0.696778 0.717287i \(-0.745384\pi\)
0.969578 + 0.244783i \(0.0787169\pi\)
\(264\) 0 0
\(265\) 1.12844e6 0.987109
\(266\) 0 0
\(267\) 131139. 0.112578
\(268\) 0 0
\(269\) −248820. + 430969.i −0.209655 + 0.363133i −0.951606 0.307321i \(-0.900567\pi\)
0.741951 + 0.670454i \(0.233901\pi\)
\(270\) 0 0
\(271\) −724980. + 1.25570e6i −0.599657 + 1.03864i 0.393214 + 0.919447i \(0.371363\pi\)
−0.992871 + 0.119190i \(0.961970\pi\)
\(272\) 0 0
\(273\) −433060. −0.351675
\(274\) 0 0
\(275\) −206853. 358281.i −0.164942 0.285688i
\(276\) 0 0
\(277\) −87028.2 −0.0681492 −0.0340746 0.999419i \(-0.510848\pi\)
−0.0340746 + 0.999419i \(0.510848\pi\)
\(278\) 0 0
\(279\) 11393.6 + 19734.3i 0.00876297 + 0.0151779i
\(280\) 0 0
\(281\) 949563. + 1.64469e6i 0.717394 + 1.24256i 0.962029 + 0.272948i \(0.0879987\pi\)
−0.244635 + 0.969615i \(0.578668\pi\)
\(282\) 0 0
\(283\) 727473. 1.26002e6i 0.539946 0.935214i −0.458960 0.888457i \(-0.651778\pi\)
0.998906 0.0467574i \(-0.0148888\pi\)
\(284\) 0 0
\(285\) 495891. 621374.i 0.361638 0.453149i
\(286\) 0 0
\(287\) 632846. 1.09612e6i 0.453517 0.785514i
\(288\) 0 0
\(289\) −791713. 1.37129e6i −0.557600 0.965792i
\(290\) 0 0
\(291\) −525197. 909668.i −0.363572 0.629724i
\(292\) 0 0
\(293\) 795288. 0.541197 0.270598 0.962692i \(-0.412778\pi\)
0.270598 + 0.962692i \(0.412778\pi\)
\(294\) 0 0
\(295\) 532538. + 922383.i 0.356283 + 0.617101i
\(296\) 0 0
\(297\) 1.60313e6 1.05458
\(298\) 0 0
\(299\) 1.63655e6 2.83459e6i 1.05865 1.83363i
\(300\) 0 0
\(301\) −77396.5 + 134055.i −0.0492385 + 0.0852836i
\(302\) 0 0
\(303\) −736403. −0.460796
\(304\) 0 0
\(305\) 1.47177e6 0.905923
\(306\) 0 0
\(307\) −964887. + 1.67123e6i −0.584293 + 1.01202i 0.410670 + 0.911784i \(0.365295\pi\)
−0.994963 + 0.100241i \(0.968039\pi\)
\(308\) 0 0
\(309\) −598646. + 1.03689e6i −0.356676 + 0.617781i
\(310\) 0 0
\(311\) −1.42463e6 −0.835220 −0.417610 0.908626i \(-0.637132\pi\)
−0.417610 + 0.908626i \(0.637132\pi\)
\(312\) 0 0
\(313\) 1.12004e6 + 1.93997e6i 0.646209 + 1.11927i 0.984021 + 0.178053i \(0.0569800\pi\)
−0.337812 + 0.941214i \(0.609687\pi\)
\(314\) 0 0
\(315\) 908928. 0.516123
\(316\) 0 0
\(317\) −278493. 482363.i −0.155656 0.269604i 0.777642 0.628708i \(-0.216416\pi\)
−0.933298 + 0.359104i \(0.883082\pi\)
\(318\) 0 0
\(319\) 1.13759e6 + 1.97037e6i 0.625908 + 1.08410i
\(320\) 0 0
\(321\) −249251. + 431715.i −0.135012 + 0.233848i
\(322\) 0 0
\(323\) 2.69638e6 + 407395.i 1.43805 + 0.217275i
\(324\) 0 0
\(325\) −293821. + 508913.i −0.154303 + 0.267261i
\(326\) 0 0
\(327\) 218756. + 378896.i 0.113133 + 0.195953i
\(328\) 0 0
\(329\) −650208. 1.12619e6i −0.331179 0.573619i
\(330\) 0 0
\(331\) 1.85079e6 0.928511 0.464256 0.885701i \(-0.346322\pi\)
0.464256 + 0.885701i \(0.346322\pi\)
\(332\) 0 0
\(333\) −1.24930e6 2.16385e6i −0.617386 1.06934i
\(334\) 0 0
\(335\) −1.72168e6 −0.838184
\(336\) 0 0
\(337\) 1.16626e6 2.02003e6i 0.559399 0.968907i −0.438148 0.898903i \(-0.644365\pi\)
0.997547 0.0700041i \(-0.0223013\pi\)
\(338\) 0 0
\(339\) −337267. + 584163.i −0.159395 + 0.276080i
\(340\) 0 0
\(341\) 60450.3 0.0281522
\(342\) 0 0
\(343\) −2.18110e6 −1.00101
\(344\) 0 0
\(345\) 1.22501e6 2.12178e6i 0.554106 0.959739i
\(346\) 0 0
\(347\) −1.33391e6 + 2.31041e6i −0.594708 + 1.03006i 0.398880 + 0.917003i \(0.369399\pi\)
−0.993588 + 0.113062i \(0.963934\pi\)
\(348\) 0 0
\(349\) −2.33528e6 −1.02630 −0.513152 0.858298i \(-0.671522\pi\)
−0.513152 + 0.858298i \(0.671522\pi\)
\(350\) 0 0
\(351\) −1.13857e6 1.97206e6i −0.493278 0.854382i
\(352\) 0 0
\(353\) −2.57168e6 −1.09845 −0.549225 0.835675i \(-0.685077\pi\)
−0.549225 + 0.835675i \(0.685077\pi\)
\(354\) 0 0
\(355\) −2.36722e6 4.10014e6i −0.996935 1.72674i
\(356\) 0 0
\(357\) −555965. 962960.i −0.230875 0.399887i
\(358\) 0 0
\(359\) 2.06014e6 3.56826e6i 0.843645 1.46124i −0.0431473 0.999069i \(-0.513738\pi\)
0.886793 0.462168i \(-0.152928\pi\)
\(360\) 0 0
\(361\) −1.81631e6 + 1.68288e6i −0.733536 + 0.679651i
\(362\) 0 0
\(363\) 258698. 448078.i 0.103045 0.178479i
\(364\) 0 0
\(365\) −2.20613e6 3.82112e6i −0.866759 1.50127i
\(366\) 0 0
\(367\) 1.76851e6 + 3.06315e6i 0.685398 + 1.18714i 0.973312 + 0.229487i \(0.0737050\pi\)
−0.287914 + 0.957656i \(0.592962\pi\)
\(368\) 0 0
\(369\) 2.82408e6 1.07972
\(370\) 0 0
\(371\) 716555. + 1.24111e6i 0.270281 + 0.468140i
\(372\) 0 0
\(373\) −3.14142e6 −1.16910 −0.584552 0.811356i \(-0.698730\pi\)
−0.584552 + 0.811356i \(0.698730\pi\)
\(374\) 0 0
\(375\) 569469. 986350.i 0.209118 0.362204i
\(376\) 0 0
\(377\) 1.61587e6 2.79878e6i 0.585537 1.01418i
\(378\) 0 0
\(379\) −1.24166e6 −0.444021 −0.222011 0.975044i \(-0.571262\pi\)
−0.222011 + 0.975044i \(0.571262\pi\)
\(380\) 0 0
\(381\) −946329. −0.333987
\(382\) 0 0
\(383\) −1.41220e6 + 2.44600e6i −0.491924 + 0.852038i −0.999957 0.00929990i \(-0.997040\pi\)
0.508032 + 0.861338i \(0.330373\pi\)
\(384\) 0 0
\(385\) 1.20561e6 2.08817e6i 0.414528 0.717984i
\(386\) 0 0
\(387\) −345382. −0.117225
\(388\) 0 0
\(389\) 2.26664e6 + 3.92593e6i 0.759465 + 1.31543i 0.943124 + 0.332442i \(0.107873\pi\)
−0.183658 + 0.982990i \(0.558794\pi\)
\(390\) 0 0
\(391\) 8.40405e6 2.78001
\(392\) 0 0
\(393\) −297550. 515371.i −0.0971802 0.168321i
\(394\) 0 0
\(395\) 721143. + 1.24906e6i 0.232557 + 0.402800i
\(396\) 0 0
\(397\) 968813. 1.67803e6i 0.308506 0.534348i −0.669530 0.742785i \(-0.733504\pi\)
0.978036 + 0.208437i \(0.0668377\pi\)
\(398\) 0 0
\(399\) 998302. + 150833.i 0.313928 + 0.0474313i
\(400\) 0 0
\(401\) 2.21857e6 3.84268e6i 0.688990 1.19336i −0.283176 0.959068i \(-0.591388\pi\)
0.972165 0.234297i \(-0.0752788\pi\)
\(402\) 0 0
\(403\) −42932.8 74361.7i −0.0131682 0.0228080i
\(404\) 0 0
\(405\) 523407. + 906567.i 0.158563 + 0.274639i
\(406\) 0 0
\(407\) −6.62832e6 −1.98343
\(408\) 0 0
\(409\) 3.20721e6 + 5.55506e6i 0.948025 + 1.64203i 0.749579 + 0.661915i \(0.230256\pi\)
0.198446 + 0.980112i \(0.436411\pi\)
\(410\) 0 0
\(411\) −1.12259e6 −0.327805
\(412\) 0 0
\(413\) −676317. + 1.17142e6i −0.195108 + 0.337937i
\(414\) 0 0
\(415\) 1.83340e6 3.17555e6i 0.522562 0.905103i
\(416\) 0 0
\(417\) 1.79548e6 0.505639
\(418\) 0 0
\(419\) −2.16492e6 −0.602429 −0.301215 0.953556i \(-0.597392\pi\)
−0.301215 + 0.953556i \(0.597392\pi\)
\(420\) 0 0
\(421\) 3.39931e6 5.88777e6i 0.934727 1.61900i 0.159608 0.987180i \(-0.448977\pi\)
0.775119 0.631815i \(-0.217690\pi\)
\(422\) 0 0
\(423\) 1.45078e6 2.51282e6i 0.394230 0.682826i
\(424\) 0 0
\(425\) −1.50884e6 −0.405201
\(426\) 0 0
\(427\) 934567. + 1.61872e6i 0.248051 + 0.429637i
\(428\) 0 0
\(429\) −2.56332e6 −0.672449
\(430\) 0 0
\(431\) −101699. 176148.i −0.0263708 0.0456756i 0.852539 0.522664i \(-0.175062\pi\)
−0.878910 + 0.476988i \(0.841728\pi\)
\(432\) 0 0
\(433\) 487850. + 844981.i 0.125045 + 0.216584i 0.921751 0.387783i \(-0.126759\pi\)
−0.796705 + 0.604368i \(0.793426\pi\)
\(434\) 0 0
\(435\) 1.20953e6 2.09497e6i 0.306475 0.530830i
\(436\) 0 0
\(437\) −4.75990e6 + 5.96437e6i −1.19232 + 1.49404i
\(438\) 0 0
\(439\) 1.96789e6 3.40848e6i 0.487348 0.844111i −0.512546 0.858660i \(-0.671298\pi\)
0.999894 + 0.0145483i \(0.00463102\pi\)
\(440\) 0 0
\(441\) −928063. 1.60745e6i −0.227238 0.393588i
\(442\) 0 0
\(443\) −1.39091e6 2.40913e6i −0.336737 0.583246i 0.647080 0.762422i \(-0.275990\pi\)
−0.983817 + 0.179176i \(0.942657\pi\)
\(444\) 0 0
\(445\) 1.03715e6 0.248279
\(446\) 0 0
\(447\) 1.17486e6 + 2.03491e6i 0.278110 + 0.481700i
\(448\) 0 0
\(449\) 6.91764e6 1.61936 0.809678 0.586875i \(-0.199642\pi\)
0.809678 + 0.586875i \(0.199642\pi\)
\(450\) 0 0
\(451\) 3.74587e6 6.48804e6i 0.867184 1.50201i
\(452\) 0 0
\(453\) −1.46545e6 + 2.53823e6i −0.335525 + 0.581146i
\(454\) 0 0
\(455\) −3.42497e6 −0.775582
\(456\) 0 0
\(457\) 6.44928e6 1.44451 0.722256 0.691626i \(-0.243105\pi\)
0.722256 + 0.691626i \(0.243105\pi\)
\(458\) 0 0
\(459\) 2.92340e6 5.06348e6i 0.647675 1.12181i
\(460\) 0 0
\(461\) 610341. 1.05714e6i 0.133758 0.231676i −0.791364 0.611345i \(-0.790629\pi\)
0.925122 + 0.379669i \(0.123962\pi\)
\(462\) 0 0
\(463\) −847245. −0.183678 −0.0918388 0.995774i \(-0.529274\pi\)
−0.0918388 + 0.995774i \(0.529274\pi\)
\(464\) 0 0
\(465\) −32136.6 55662.2i −0.00689235 0.0119379i
\(466\) 0 0
\(467\) 76212.7 0.0161709 0.00808547 0.999967i \(-0.497426\pi\)
0.00808547 + 0.999967i \(0.497426\pi\)
\(468\) 0 0
\(469\) −1.09325e6 1.89357e6i −0.229503 0.397512i
\(470\) 0 0
\(471\) −114599. 198491.i −0.0238028 0.0412276i
\(472\) 0 0
\(473\) −458116. + 793481.i −0.0941505 + 0.163074i
\(474\) 0 0
\(475\) 854578. 1.07082e6i 0.173787 0.217763i
\(476\) 0 0
\(477\) −1.59881e6 + 2.76923e6i −0.321738 + 0.557266i
\(478\) 0 0
\(479\) 1.33045e6 + 2.30441e6i 0.264947 + 0.458902i 0.967550 0.252680i \(-0.0813121\pi\)
−0.702602 + 0.711583i \(0.747979\pi\)
\(480\) 0 0
\(481\) 4.70754e6 + 8.15370e6i 0.927750 + 1.60691i
\(482\) 0 0
\(483\) 3.11150e6 0.606879
\(484\) 0 0
\(485\) −4.15366e6 7.19434e6i −0.801819 1.38879i
\(486\) 0 0
\(487\) −5.81426e6 −1.11089 −0.555446 0.831552i \(-0.687453\pi\)
−0.555446 + 0.831552i \(0.687453\pi\)
\(488\) 0 0
\(489\) −51021.7 + 88372.2i −0.00964901 + 0.0167126i
\(490\) 0 0
\(491\) −2.27968e6 + 3.94853e6i −0.426747 + 0.739148i −0.996582 0.0826113i \(-0.973674\pi\)
0.569834 + 0.821760i \(0.307007\pi\)
\(492\) 0 0
\(493\) 8.29787e6 1.53762
\(494\) 0 0
\(495\) 5.38002e6 0.986895
\(496\) 0 0
\(497\) 3.00634e6 5.20713e6i 0.545942 0.945600i
\(498\) 0 0
\(499\) −3.38613e6 + 5.86495e6i −0.608769 + 1.05442i 0.382675 + 0.923883i \(0.375003\pi\)
−0.991444 + 0.130536i \(0.958330\pi\)
\(500\) 0 0
\(501\) 1.94295e6 0.345833
\(502\) 0 0
\(503\) −2.62169e6 4.54089e6i −0.462020 0.800242i 0.537042 0.843556i \(-0.319542\pi\)
−0.999062 + 0.0433140i \(0.986208\pi\)
\(504\) 0 0
\(505\) −5.82403e6 −1.01624
\(506\) 0 0
\(507\) 336720. + 583215.i 0.0581766 + 0.100765i
\(508\) 0 0
\(509\) −4.68578e6 8.11601e6i −0.801655 1.38851i −0.918526 0.395360i \(-0.870620\pi\)
0.116872 0.993147i \(-0.462713\pi\)
\(510\) 0 0
\(511\) 2.80175e6 4.85278e6i 0.474655 0.822126i
\(512\) 0 0
\(513\) 1.93780e6 + 4.94261e6i 0.325099 + 0.829206i
\(514\) 0 0
\(515\) −4.73455e6 + 8.20047e6i −0.786611 + 1.36245i
\(516\) 0 0
\(517\) −3.84864e6 6.66604e6i −0.633258 1.09683i
\(518\) 0 0
\(519\) 1.70779e6 + 2.95798e6i 0.278302 + 0.482032i
\(520\) 0 0
\(521\) 244496. 0.0394619 0.0197309 0.999805i \(-0.493719\pi\)
0.0197309 + 0.999805i \(0.493719\pi\)
\(522\) 0 0
\(523\) 2.04939e6 + 3.54964e6i 0.327619 + 0.567453i 0.982039 0.188678i \(-0.0604204\pi\)
−0.654420 + 0.756131i \(0.727087\pi\)
\(524\) 0 0
\(525\) −558630. −0.0884557
\(526\) 0 0
\(527\) 110235. 190932.i 0.0172899 0.0299469i
\(528\) 0 0
\(529\) −8.54032e6 + 1.47923e7i −1.32689 + 2.29824i
\(530\) 0 0
\(531\) −3.01807e6 −0.464507
\(532\) 0 0
\(533\) −1.06415e7 −1.62250
\(534\) 0 0
\(535\) −1.97126e6 + 3.41433e6i −0.297756 + 0.515728i
\(536\) 0 0
\(537\) −164543. + 284997.i −0.0246232 + 0.0426486i
\(538\) 0 0
\(539\) −4.92395e6 −0.730032
\(540\) 0 0
\(541\) 227501. + 394043.i 0.0334187 + 0.0578829i 0.882251 0.470779i \(-0.156027\pi\)
−0.848832 + 0.528662i \(0.822694\pi\)
\(542\) 0 0
\(543\) −3.85898e6 −0.561660
\(544\) 0 0
\(545\) 1.73009e6 + 2.99660e6i 0.249503 + 0.432153i
\(546\) 0 0
\(547\) 4.19066e6 + 7.25843e6i 0.598844 + 1.03723i 0.992992 + 0.118182i \(0.0377065\pi\)
−0.394148 + 0.919047i \(0.628960\pi\)
\(548\) 0 0
\(549\) −2.08525e6 + 3.61176e6i −0.295276 + 0.511433i
\(550\) 0 0
\(551\) −4.69976e6 + 5.88901e6i −0.659473 + 0.826349i
\(552\) 0 0
\(553\) −915844. + 1.58629e6i −0.127353 + 0.220582i
\(554\) 0 0
\(555\) 3.52375e6 + 6.10331e6i 0.485593 + 0.841071i
\(556\) 0 0
\(557\) −2.14702e6 3.71874e6i −0.293222 0.507876i 0.681347 0.731960i \(-0.261394\pi\)
−0.974570 + 0.224084i \(0.928061\pi\)
\(558\) 0 0
\(559\) 1.30145e6 0.176156
\(560\) 0 0
\(561\) −3.29080e6 5.69984e6i −0.441464 0.764637i
\(562\) 0 0
\(563\) 5.43360e6 0.722465 0.361232 0.932476i \(-0.382356\pi\)
0.361232 + 0.932476i \(0.382356\pi\)
\(564\) 0 0
\(565\) −2.66736e6 + 4.62001e6i −0.351529 + 0.608866i
\(566\) 0 0
\(567\) −664720. + 1.15133e6i −0.0868323 + 0.150398i
\(568\) 0 0
\(569\) 4.53732e6 0.587514 0.293757 0.955880i \(-0.405094\pi\)
0.293757 + 0.955880i \(0.405094\pi\)
\(570\) 0 0
\(571\) 1.75132e6 0.224789 0.112395 0.993664i \(-0.464148\pi\)
0.112395 + 0.993664i \(0.464148\pi\)
\(572\) 0 0
\(573\) −871992. + 1.51033e6i −0.110950 + 0.192170i
\(574\) 0 0
\(575\) 2.11108e6 3.65650e6i 0.266278 0.461208i
\(576\) 0 0
\(577\) 2.69353e6 0.336808 0.168404 0.985718i \(-0.446139\pi\)
0.168404 + 0.985718i \(0.446139\pi\)
\(578\) 0 0
\(579\) −2.85790e6 4.95002e6i −0.354283 0.613636i
\(580\) 0 0
\(581\) 4.65680e6 0.572331
\(582\) 0 0
\(583\) 4.24135e6 + 7.34624e6i 0.516812 + 0.895145i
\(584\) 0 0
\(585\) −3.82098e6 6.61812e6i −0.461620 0.799549i
\(586\) 0 0
\(587\) −1.95802e6 + 3.39138e6i −0.234542 + 0.406239i −0.959140 0.282934i \(-0.908692\pi\)
0.724597 + 0.689172i \(0.242026\pi\)
\(588\) 0 0
\(589\) 73069.9 + 186374.i 0.00867862 + 0.0221359i
\(590\) 0 0
\(591\) 70895.9 122795.i 0.00834934 0.0144615i
\(592\) 0 0
\(593\) −5.66866e6 9.81841e6i −0.661978 1.14658i −0.980095 0.198528i \(-0.936384\pi\)
0.318117 0.948051i \(-0.396950\pi\)
\(594\) 0 0
\(595\) −4.39699e6 7.61581e6i −0.509170 0.881909i
\(596\) 0 0
\(597\) 4.63729e6 0.532511
\(598\) 0 0
\(599\) −3.86543e6 6.69512e6i −0.440180 0.762414i 0.557522 0.830162i \(-0.311752\pi\)
−0.997702 + 0.0677475i \(0.978419\pi\)
\(600\) 0 0
\(601\) −6.87953e6 −0.776913 −0.388457 0.921467i \(-0.626992\pi\)
−0.388457 + 0.921467i \(0.626992\pi\)
\(602\) 0 0
\(603\) 2.43932e6 4.22503e6i 0.273197 0.473191i
\(604\) 0 0
\(605\) 2.04598e6 3.54374e6i 0.227255 0.393617i
\(606\) 0 0
\(607\) −1.19598e7 −1.31750 −0.658750 0.752362i \(-0.728915\pi\)
−0.658750 + 0.752362i \(0.728915\pi\)
\(608\) 0 0
\(609\) 3.07219e6 0.335664
\(610\) 0 0
\(611\) −5.46673e6 + 9.46865e6i −0.592413 + 1.02609i
\(612\) 0 0
\(613\) 1.49181e6 2.58389e6i 0.160347 0.277730i −0.774646 0.632395i \(-0.782072\pi\)
0.934993 + 0.354666i \(0.115405\pi\)
\(614\) 0 0
\(615\) −7.96552e6 −0.849232
\(616\) 0 0
\(617\) −2.62649e6 4.54922e6i −0.277756 0.481087i 0.693071 0.720869i \(-0.256257\pi\)
−0.970827 + 0.239782i \(0.922924\pi\)
\(618\) 0 0
\(619\) 1.09695e7 1.15069 0.575346 0.817910i \(-0.304867\pi\)
0.575346 + 0.817910i \(0.304867\pi\)
\(620\) 0 0
\(621\) 8.18053e6 + 1.41691e7i 0.851241 + 1.47439i
\(622\) 0 0
\(623\) 658583. + 1.14070e6i 0.0679815 + 0.117747i
\(624\) 0 0
\(625\) 5.86419e6 1.01571e7i 0.600493 1.04008i
\(626\) 0 0
\(627\) 5.90904e6 + 892794.i 0.600272 + 0.0906949i
\(628\) 0 0
\(629\) −1.20871e7 + 2.09355e7i −1.21814 + 2.10988i
\(630\) 0 0
\(631\) 3.73436e6 + 6.46811e6i 0.373373 + 0.646701i 0.990082 0.140490i \(-0.0448678\pi\)
−0.616709 + 0.787191i \(0.711534\pi\)
\(632\) 0 0
\(633\) 3.34216e6 + 5.78879e6i 0.331526 + 0.574220i
\(634\) 0 0
\(635\) −7.48429e6 −0.736573
\(636\) 0 0
\(637\) 3.49707e6 + 6.05710e6i 0.341473 + 0.591448i
\(638\) 0 0
\(639\) 1.34158e7 1.29976
\(640\) 0 0
\(641\) −8.26419e6 + 1.43140e7i −0.794430 + 1.37599i 0.128771 + 0.991674i \(0.458897\pi\)
−0.923201 + 0.384318i \(0.874437\pi\)
\(642\) 0 0
\(643\) 6.29383e6 1.09012e7i 0.600326 1.03980i −0.392445 0.919775i \(-0.628371\pi\)
0.992771 0.120020i \(-0.0382959\pi\)
\(644\) 0 0
\(645\) 974175. 0.0922014
\(646\) 0 0
\(647\) −1.21912e7 −1.14494 −0.572472 0.819924i \(-0.694015\pi\)
−0.572472 + 0.819924i \(0.694015\pi\)
\(648\) 0 0
\(649\) −4.00318e6 + 6.93371e6i −0.373073 + 0.646181i
\(650\) 0 0
\(651\) 40813.1 70690.4i 0.00377439 0.00653744i
\(652\) 0 0
\(653\) −7.52249e6 −0.690365 −0.345182 0.938536i \(-0.612183\pi\)
−0.345182 + 0.938536i \(0.612183\pi\)
\(654\) 0 0
\(655\) −2.35325e6 4.07594e6i −0.214321 0.371214i
\(656\) 0 0
\(657\) 1.25028e7 1.13004
\(658\) 0 0
\(659\) −8.02028e6 1.38915e7i −0.719409 1.24605i −0.961234 0.275733i \(-0.911079\pi\)
0.241825 0.970320i \(-0.422254\pi\)
\(660\) 0 0
\(661\) 6.26368e6 + 1.08490e7i 0.557604 + 0.965798i 0.997696 + 0.0678454i \(0.0216125\pi\)
−0.440092 + 0.897953i \(0.645054\pi\)
\(662\) 0 0
\(663\) −4.67436e6 + 8.09623e6i −0.412989 + 0.715318i
\(664\) 0 0
\(665\) 7.89533e6 + 1.19290e6i 0.692335 + 0.104605i
\(666\) 0 0
\(667\) −1.16099e7 + 2.01090e7i −1.01045 + 1.75015i
\(668\) 0 0
\(669\) 2.79368e6 + 4.83880e6i 0.241330 + 0.417996i
\(670\) 0 0
\(671\) 5.53178e6 + 9.58133e6i 0.474306 + 0.821523i
\(672\) 0 0
\(673\) −7.50054e6 −0.638344 −0.319172 0.947697i \(-0.603405\pi\)
−0.319172 + 0.947697i \(0.603405\pi\)
\(674\) 0 0
\(675\) −1.46871e6 2.54388e6i −0.124073 0.214900i
\(676\) 0 0
\(677\) 1.19677e7 1.00355 0.501777 0.864997i \(-0.332680\pi\)
0.501777 + 0.864997i \(0.332680\pi\)
\(678\) 0 0
\(679\) 5.27510e6 9.13673e6i 0.439092 0.760530i
\(680\) 0 0
\(681\) −3.25058e6 + 5.63017e6i −0.268592 + 0.465215i
\(682\) 0 0
\(683\) −8.11577e6 −0.665699 −0.332850 0.942980i \(-0.608010\pi\)
−0.332850 + 0.942980i \(0.608010\pi\)
\(684\) 0 0
\(685\) −8.87827e6 −0.722939
\(686\) 0 0
\(687\) −4.75164e6 + 8.23009e6i −0.384107 + 0.665293i
\(688\) 0 0
\(689\) 6.02455e6 1.04348e7i 0.483478 0.837408i
\(690\) 0 0
\(691\) 1.23034e7 0.980235 0.490117 0.871657i \(-0.336954\pi\)
0.490117 + 0.871657i \(0.336954\pi\)
\(692\) 0 0
\(693\) 3.41628e6 + 5.91717e6i 0.270222 + 0.468038i
\(694\) 0 0
\(695\) 1.42000e7 1.11513
\(696\) 0 0
\(697\) −1.36616e7 2.36626e7i −1.06517 1.84494i
\(698\) 0 0
\(699\) 853093. + 1.47760e6i 0.0660394 + 0.114384i
\(700\) 0 0
\(701\) −7.04732e6 + 1.22063e7i −0.541663 + 0.938188i 0.457146 + 0.889392i \(0.348872\pi\)
−0.998809 + 0.0487959i \(0.984462\pi\)
\(702\) 0 0
\(703\) −8.01205e6 2.04358e7i −0.611442 1.55956i
\(704\) 0 0
\(705\) −4.09202e6 + 7.08759e6i −0.310074 + 0.537064i
\(706\) 0 0
\(707\) −3.69823e6 6.40552e6i −0.278256 0.481954i
\(708\) 0 0
\(709\) −2.13108e6 3.69113e6i −0.159215 0.275768i 0.775371 0.631506i \(-0.217563\pi\)
−0.934586 + 0.355738i \(0.884230\pi\)
\(710\) 0 0
\(711\) −4.08695e6 −0.303197
\(712\) 0 0
\(713\) 308469. + 534283.i 0.0227241 + 0.0393593i
\(714\) 0 0
\(715\) −2.02727e7 −1.48302
\(716\) 0 0
\(717\) 3.85950e6 6.68486e6i 0.280371 0.485617i
\(718\) 0 0
\(719\) −6.85763e6 + 1.18778e7i −0.494711 + 0.856865i −0.999981 0.00609644i \(-0.998059\pi\)
0.505270 + 0.862961i \(0.331393\pi\)
\(720\) 0 0
\(721\) −1.20256e7 −0.861529
\(722\) 0 0
\(723\) 6.25394e6 0.444946
\(724\) 0 0
\(725\) 2.08441e6 3.61031e6i 0.147278 0.255093i
\(726\) 0 0
\(727\) 1.05919e7 1.83457e7i 0.743256 1.28736i −0.207749 0.978182i \(-0.566614\pi\)
0.951005 0.309175i \(-0.100053\pi\)
\(728\) 0 0
\(729\) 3.58627e6 0.249933
\(730\) 0 0
\(731\) 1.67080e6 + 2.89392e6i 0.115646 + 0.200305i
\(732\) 0 0
\(733\) 6.01308e6 0.413368 0.206684 0.978408i \(-0.433733\pi\)
0.206684 + 0.978408i \(0.433733\pi\)
\(734\) 0 0
\(735\) 2.61767e6 + 4.53394e6i 0.178730 + 0.309569i
\(736\) 0 0
\(737\) −6.47107e6 1.12082e7i −0.438841 0.760095i
\(738\) 0 0
\(739\) 8.41424e6 1.45739e7i 0.566766 0.981667i −0.430117 0.902773i \(-0.641528\pi\)
0.996883 0.0788942i \(-0.0251389\pi\)
\(740\) 0 0
\(741\) −3.09844e6 7.90296e6i −0.207299 0.528742i
\(742\) 0 0
\(743\) 2.01935e6 3.49762e6i 0.134196 0.232434i −0.791094 0.611695i \(-0.790488\pi\)
0.925290 + 0.379260i \(0.123821\pi\)
\(744\) 0 0
\(745\) 9.29165e6 + 1.60936e7i 0.613341 + 1.06234i
\(746\) 0 0
\(747\) 5.19524e6 + 8.99842e6i 0.340647 + 0.590018i
\(748\) 0 0
\(749\) −5.00696e6 −0.326114
\(750\) 0 0
\(751\) 351347. + 608550.i 0.0227319 + 0.0393728i 0.877168 0.480184i \(-0.159430\pi\)
−0.854436 + 0.519557i \(0.826097\pi\)
\(752\) 0 0
\(753\) 7.17380e6 0.461064
\(754\) 0 0
\(755\) −1.15899e7 + 2.00742e7i −0.739964 + 1.28166i
\(756\) 0 0
\(757\) 1.19882e7 2.07642e7i 0.760354 1.31697i −0.182314 0.983240i \(-0.558359\pi\)
0.942668 0.333731i \(-0.108308\pi\)
\(758\) 0 0
\(759\) 1.84172e7 1.16043
\(760\) 0 0
\(761\) −1.29063e7 −0.807869 −0.403934 0.914788i \(-0.632358\pi\)
−0.403934 + 0.914788i \(0.632358\pi\)
\(762\) 0 0
\(763\) −2.19719e6 + 3.80565e6i −0.136633 + 0.236656i
\(764\) 0 0
\(765\) 9.81078e6 1.69928e7i 0.606108 1.04981i
\(766\) 0 0
\(767\) 1.13725e7 0.698019
\(768\) 0 0
\(769\) −1.05367e7 1.82501e7i −0.642524 1.11288i −0.984867 0.173310i \(-0.944554\pi\)
0.342343 0.939575i \(-0.388779\pi\)
\(770\) 0 0
\(771\) −8.82902e6 −0.534905
\(772\) 0 0
\(773\) 4.35234e6 + 7.53848e6i 0.261984 + 0.453769i 0.966769 0.255652i \(-0.0822900\pi\)
−0.704785 + 0.709421i \(0.748957\pi\)
\(774\) 0 0
\(775\) −55381.5 95923.6i −0.00331215 0.00573682i
\(776\) 0 0
\(777\) −4.47512e6 + 7.75113e6i −0.265921 + 0.460588i
\(778\) 0 0
\(779\) 2.45311e7 + 3.70640e6i 1.44835 + 0.218831i
\(780\) 0 0
\(781\) 1.77948e7 3.08214e7i 1.04391 1.80811i
\(782\) 0 0
\(783\) 8.07717e6 + 1.39901e7i 0.470820 + 0.815484i
\(784\) 0 0
\(785\) −906334. 1.56982e6i −0.0524946 0.0909232i
\(786\) 0 0
\(787\) 2.69329e6 0.155005 0.0775026 0.996992i \(-0.475305\pi\)
0.0775026 + 0.996992i \(0.475305\pi\)
\(788\) 0 0
\(789\) −2.44579e6 4.23624e6i −0.139871 0.242263i
\(790\) 0 0
\(791\) −6.77504e6 −0.385009
\(792\) 0 0
\(793\) 7.85752e6 1.36096e7i 0.443713 0.768534i
\(794\) 0 0
\(795\) 4.50957e6 7.81081e6i 0.253057 0.438307i
\(796\) 0 0
\(797\) −4.67101e6 −0.260474 −0.130237 0.991483i \(-0.541574\pi\)
−0.130237 + 0.991483i \(0.541574\pi\)
\(798\) 0 0
\(799\) −2.80729e7 −1.55568
\(800\) 0 0
\(801\) −1.46946e6 + 2.54519e6i −0.0809240 + 0.140165i
\(802\) 0 0
\(803\) 1.65838e7 2.87240e7i 0.907603 1.57201i
\(804\) 0 0
\(805\) 2.46081e7 1.33841
\(806\) 0 0
\(807\) 1.98871e6 + 3.44455e6i 0.107495 + 0.186186i
\(808\) 0 0
\(809\) −1.82792e7 −0.981941 −0.490970 0.871176i \(-0.663358\pi\)
−0.490970 + 0.871176i \(0.663358\pi\)
\(810\) 0 0
\(811\) 9.38560e6 + 1.62563e7i 0.501083 + 0.867902i 0.999999 + 0.00125119i \(0.000398266\pi\)
−0.498916 + 0.866650i \(0.666268\pi\)
\(812\) 0 0
\(813\) 5.79445e6 + 1.00363e7i 0.307458 + 0.532533i
\(814\) 0 0
\(815\) −403518. + 698914.i −0.0212799 + 0.0368578i
\(816\) 0 0
\(817\) −3.00013e6 453289.i −0.157248 0.0237585i
\(818\) 0 0
\(819\) 4.85260e6 8.40494e6i 0.252793 0.437850i
\(820\) 0 0
\(821\) 5.00143e6 + 8.66273e6i 0.258962 + 0.448536i 0.965964 0.258676i \(-0.0832861\pi\)
−0.707002 + 0.707212i \(0.749953\pi\)
\(822\) 0 0
\(823\) 5.31371e6 + 9.20361e6i 0.273463 + 0.473651i 0.969746 0.244116i \(-0.0784977\pi\)
−0.696283 + 0.717767i \(0.745164\pi\)
\(824\) 0 0
\(825\) −3.30658e6 −0.169139
\(826\) 0 0
\(827\) 2.74700e6 + 4.75795e6i 0.139668 + 0.241911i 0.927371 0.374144i \(-0.122063\pi\)
−0.787703 + 0.616055i \(0.788730\pi\)
\(828\) 0 0
\(829\) 6.66282e6 0.336722 0.168361 0.985725i \(-0.446153\pi\)
0.168361 + 0.985725i \(0.446153\pi\)
\(830\) 0 0
\(831\) −347789. + 602388.i −0.0174708 + 0.0302603i
\(832\) 0 0
\(833\) −8.97912e6 + 1.55523e7i −0.448354 + 0.776572i
\(834\) 0 0
\(835\) 1.53663e7 0.762699
\(836\) 0 0
\(837\) 429211. 0.0211766
\(838\) 0 0
\(839\) 3.62638e6 6.28107e6i 0.177856 0.308055i −0.763290 0.646056i \(-0.776417\pi\)
0.941146 + 0.338001i \(0.109751\pi\)
\(840\) 0 0
\(841\) −1.20767e6 + 2.09174e6i −0.0588786 + 0.101981i
\(842\) 0 0
\(843\) 1.51789e7 0.735649
\(844\) 0 0
\(845\) 2.66303e6 + 4.61251e6i 0.128302 + 0.222226i
\(846\) 0 0
\(847\) 5.19674e6 0.248899
\(848\) 0 0
\(849\) −5.81436e6 1.00708e7i −0.276843 0.479506i
\(850\) 0 0
\(851\) −3.38233e7 5.85837e7i −1.60100 2.77302i
\(852\) 0 0
\(853\) 1.39644e7 2.41871e7i 0.657130 1.13818i −0.324225 0.945980i \(-0.605104\pi\)
0.981355 0.192202i \(-0.0615630\pi\)
\(854\) 0 0
\(855\) 6.50315e6 + 1.65871e7i 0.304235 + 0.775989i
\(856\) 0 0
\(857\) −1.35275e6 + 2.34304e6i −0.0629169 + 0.108975i −0.895768 0.444522i \(-0.853374\pi\)
0.832851 + 0.553497i \(0.186707\pi\)
\(858\) 0 0
\(859\) −1.45030e7 2.51199e7i −0.670616 1.16154i −0.977730 0.209868i \(-0.932697\pi\)
0.307114 0.951673i \(-0.400637\pi\)
\(860\) 0 0
\(861\) −5.05806e6 8.76081e6i −0.232528 0.402751i
\(862\) 0 0
\(863\) 8.44738e6 0.386096 0.193048 0.981189i \(-0.438163\pi\)
0.193048 + 0.981189i \(0.438163\pi\)
\(864\) 0 0
\(865\) 1.35065e7 + 2.33939e7i 0.613765 + 1.06307i
\(866\) 0 0
\(867\) −1.26556e7 −0.571789
\(868\) 0 0
\(869\) −5.42096e6 + 9.38937e6i −0.243515 + 0.421781i
\(870\) 0 0
\(871\) −9.19171e6 + 1.59205e7i −0.410536 + 0.711068i
\(872\) 0 0
\(873\) 2.35401e7 1.04538
\(874\) 0 0
\(875\) 1.14395e7 0.505113
\(876\) 0 0
\(877\) −1.25597e7 + 2.17540e7i −0.551416 + 0.955080i 0.446757 + 0.894655i \(0.352579\pi\)
−0.998173 + 0.0604245i \(0.980755\pi\)
\(878\) 0 0
\(879\) 3.17819e6 5.50479e6i 0.138742 0.240308i
\(880\) 0 0
\(881\) −1.95758e7 −0.849728 −0.424864 0.905257i \(-0.639678\pi\)
−0.424864 + 0.905257i \(0.639678\pi\)
\(882\) 0 0
\(883\) 1.40024e6 + 2.42528e6i 0.0604366 + 0.104679i 0.894661 0.446746i \(-0.147417\pi\)
−0.834224 + 0.551426i \(0.814084\pi\)
\(884\) 0 0
\(885\) 8.51268e6 0.365349
\(886\) 0 0
\(887\) −2.11985e7 3.67169e7i −0.904682 1.56696i −0.821343 0.570435i \(-0.806775\pi\)
−0.0833393 0.996521i \(-0.526559\pi\)
\(888\) 0 0
\(889\) −4.75248e6 8.23154e6i −0.201681 0.349322i
\(890\) 0 0
\(891\) −3.93454e6 + 6.81482e6i −0.166035 + 0.287581i
\(892\) 0 0
\(893\) 1.59000e7 1.99234e7i 0.667217 0.836053i
\(894\) 0 0
\(895\) −1.30133e6 + 2.25397e6i −0.0543039 + 0.0940571i
\(896\) 0 0
\(897\) −1.30802e7 2.26556e7i −0.542793 0.940145i
\(898\) 0 0
\(899\) 304571. + 527533.i 0.0125687 + 0.0217696i
\(900\) 0 0
\(901\) 3.09374e7 1.26962
\(902\) 0 0
\(903\) 618595. + 1.07144e6i 0.0252457 + 0.0437268i
\(904\) 0 0
\(905\) −3.05198e7 −1.23868
\(906\) 0 0
\(907\) −1.00865e7 + 1.74703e7i −0.407118 + 0.705149i −0.994565 0.104113i \(-0.966800\pi\)
0.587447 + 0.809262i \(0.300133\pi\)
\(908\) 0 0
\(909\) 8.25167e6 1.42923e7i 0.331232 0.573710i
\(910\) 0 0
\(911\) 3.89915e7 1.55659 0.778294 0.627900i \(-0.216085\pi\)
0.778294 + 0.627900i \(0.216085\pi\)
\(912\) 0 0
\(913\) 2.75640e7 1.09437
\(914\) 0 0
\(915\) 5.88161e6 1.01873e7i 0.232244 0.402258i
\(916\) 0 0
\(917\) 2.98860e6 5.17640e6i 0.117366 0.203285i
\(918\) 0 0
\(919\) −9.96447e6 −0.389193 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(920\) 0 0
\(921\) 7.71191e6 + 1.33574e7i 0.299580 + 0.518888i
\(922\) 0 0
\(923\) −5.05525e7 −1.95316
\(924\) 0 0
\(925\) 6.07253e6 + 1.05179e7i 0.233354 + 0.404181i
\(926\) 0 0
\(927\) −1.34161e7 2.32374e7i −0.512775 0.888153i
\(928\) 0 0
\(929\) 6.00913e6 1.04081e7i 0.228440 0.395670i −0.728906 0.684614i \(-0.759971\pi\)
0.957346 + 0.288944i \(0.0933042\pi\)
\(930\) 0 0
\(931\) −5.95188e6 1.51810e7i −0.225050 0.574020i
\(932\) 0 0
\(933\) −5.69322e6 + 9.86094e6i −0.214118 + 0.370864i
\(934\) 0 0
\(935\) −2.60262e7 4.50786e7i −0.973601 1.68633i
\(936\) 0 0
\(937\) 1.90689e7 + 3.30284e7i 0.709541 + 1.22896i 0.965027 + 0.262149i \(0.0844311\pi\)
−0.255486 + 0.966813i \(0.582236\pi\)
\(938\) 0 0
\(939\) 1.79040e7 0.662652
\(940\) 0 0
\(941\) 8.78331e6 + 1.52131e7i 0.323358 + 0.560073i 0.981179 0.193102i \(-0.0618548\pi\)
−0.657820 + 0.753175i \(0.728521\pi\)
\(942\) 0 0
\(943\) 7.64584e7 2.79992
\(944\) 0 0
\(945\) 8.56009e6 1.48265e7i 0.311816 0.540081i
\(946\) 0 0
\(947\) −2.41786e7 + 4.18786e7i −0.876106 + 1.51746i −0.0205267 + 0.999789i \(0.506534\pi\)
−0.855580 + 0.517671i \(0.826799\pi\)
\(948\) 0 0
\(949\) −4.71124e7 −1.69812
\(950\) 0 0
\(951\) −4.45174e6 −0.159617
\(952\) 0 0
\(953\) −1.89643e7 + 3.28471e7i −0.676402 + 1.17156i 0.299656 + 0.954047i \(0.403128\pi\)
−0.976057 + 0.217514i \(0.930205\pi\)
\(954\) 0 0
\(955\) −6.89637e6 + 1.19449e7i −0.244688 + 0.423812i
\(956\) 0 0
\(957\) 1.81846e7 0.641834
\(958\) 0 0
\(959\) −5.63765e6 9.76470e6i −0.197948 0.342856i
\(960\) 0 0
\(961\) −2.86130e7 −0.999435
\(962\) 0 0
\(963\) −5.58589e6 9.67504e6i −0.194100 0.336192i
\(964\) 0 0
\(965\) −2.26024e7 3.91485e7i −0.781333 1.35331i
\(966\) 0 0
\(967\) 8.31734e6 1.44061e7i 0.286034 0.495426i −0.686825 0.726823i \(-0.740996\pi\)
0.972859 + 0.231397i \(0.0743295\pi\)
\(968\) 0 0
\(969\) 1.35954e7 1.70356e7i 0.465138 0.582838i
\(970\) 0 0
\(971\) 1.41242e7 2.44638e7i 0.480746 0.832677i −0.519010 0.854768i \(-0.673699\pi\)
0.999756 + 0.0220914i \(0.00703248\pi\)
\(972\) 0 0
\(973\) 9.01694e6 + 1.56178e7i 0.305335 + 0.528856i
\(974\) 0 0
\(975\) 2.34838e6 + 4.06752e6i 0.0791147 + 0.137031i
\(976\) 0 0
\(977\) −3.01881e7 −1.01181 −0.505906 0.862589i \(-0.668842\pi\)
−0.505906 + 0.862589i \(0.668842\pi\)
\(978\) 0 0
\(979\) 3.89821e6 + 6.75190e6i 0.129990 + 0.225149i
\(980\) 0 0
\(981\) −9.80496e6 −0.325292
\(982\) 0 0
\(983\) 2321.02 4020.12i 7.66117e−5 0.000132695i −0.865987 0.500066i \(-0.833309\pi\)
0.866064 + 0.499934i \(0.166642\pi\)
\(984\) 0 0
\(985\) 560698. 971158.i 0.0184136 0.0318933i
\(986\) 0 0
\(987\) −1.03936e7 −0.339606
\(988\) 0 0
\(989\) −9.35079e6 −0.303989
\(990\) 0 0
\(991\) −1.56708e7 + 2.71426e7i −0.506882 + 0.877945i 0.493087 + 0.869980i \(0.335869\pi\)
−0.999968 + 0.00796469i \(0.997465\pi\)
\(992\) 0 0
\(993\) 7.39627e6 1.28107e7i 0.238034 0.412288i
\(994\) 0 0
\(995\) 3.66752e7 1.17440
\(996\) 0 0
\(997\) −1.11856e7 1.93740e7i −0.356387 0.617280i 0.630968 0.775809i \(-0.282658\pi\)
−0.987354 + 0.158529i \(0.949325\pi\)
\(998\) 0 0
\(999\) −4.70626e7 −1.49198
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 304.6.i.d.49.6 18
4.3 odd 2 76.6.e.a.49.4 yes 18
12.11 even 2 684.6.k.f.505.3 18
19.7 even 3 inner 304.6.i.d.273.6 18
76.7 odd 6 76.6.e.a.45.4 18
228.83 even 6 684.6.k.f.577.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.4 18 76.7 odd 6
76.6.e.a.49.4 yes 18 4.3 odd 2
304.6.i.d.49.6 18 1.1 even 1 trivial
304.6.i.d.273.6 18 19.7 even 3 inner
684.6.k.f.505.3 18 12.11 even 2
684.6.k.f.577.3 18 228.83 even 6