Properties

Label 28.5.h.a.17.2
Level $28$
Weight $5$
Character 28.17
Analytic conductor $2.894$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,5,Mod(5,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 28.h (of order \(6\), 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: \(2.89435896635\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.11337408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.2
Root \(-0.391571i\) of defining polynomial
Character \(\chi\) \(=\) 28.17
Dual form 28.5.h.a.5.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.35901 - 0.784623i) q^{3} +(38.3327 - 22.1314i) q^{5} +(42.3967 + 24.5667i) q^{7} +(-39.2687 - 68.0154i) q^{9} +O(q^{10})\) \(q+(-1.35901 - 0.784623i) q^{3} +(38.3327 - 22.1314i) q^{5} +(42.3967 + 24.5667i) q^{7} +(-39.2687 - 68.0154i) q^{9} +(-11.7557 + 20.3615i) q^{11} +136.269i q^{13} -69.4592 q^{15} +(-227.333 - 131.251i) q^{17} +(-387.162 + 223.528i) q^{19} +(-38.3418 - 66.6517i) q^{21} +(374.585 + 648.800i) q^{23} +(667.098 - 1155.45i) q^{25} +250.353i q^{27} +406.524 q^{29} +(-584.199 - 337.287i) q^{31} +(31.9521 - 18.4476i) q^{33} +(2168.87 + 3.40909i) q^{35} +(-372.666 - 645.476i) q^{37} +(106.920 - 185.191i) q^{39} +2476.20i q^{41} -2636.68 q^{43} +(-3010.55 - 1738.14i) q^{45} +(-579.099 + 334.343i) q^{47} +(1193.96 + 2083.09i) q^{49} +(205.964 + 356.741i) q^{51} +(1014.61 - 1757.36i) q^{53} +1040.68i q^{55} +701.541 q^{57} +(-1014.22 - 585.561i) q^{59} +(-2022.05 + 1167.43i) q^{61} +(6.04891 - 3848.33i) q^{63} +(3015.83 + 5223.58i) q^{65} +(3779.79 - 6546.78i) q^{67} -1175.63i q^{69} +7575.36 q^{71} +(3284.18 + 1896.12i) q^{73} +(-1813.18 + 1046.84i) q^{75} +(-998.615 + 574.460i) q^{77} +(-3897.33 - 6750.37i) q^{79} +(-2984.33 + 5169.02i) q^{81} +2181.41i q^{83} -11619.0 q^{85} +(-552.469 - 318.968i) q^{87} +(8050.38 - 4647.89i) q^{89} +(-3347.69 + 5777.37i) q^{91} +(529.287 + 916.751i) q^{93} +(-9893.98 + 17136.9i) q^{95} -9981.90i q^{97} +1846.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} - 27 q^{5} + 66 q^{7} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} - 27 q^{5} + 66 q^{7} + 90 q^{9} + 135 q^{11} - 486 q^{15} - 1107 q^{17} - 747 q^{19} + 2169 q^{21} + 243 q^{23} + 1878 q^{25} - 540 q^{29} - 5355 q^{31} - 1863 q^{33} + 6021 q^{35} + 2355 q^{37} + 6588 q^{39} - 948 q^{43} - 14418 q^{45} - 9747 q^{47} + 8430 q^{49} - 891 q^{51} + 6291 q^{53} - 6894 q^{57} - 2943 q^{59} + 4041 q^{61} + 15138 q^{63} + 7668 q^{65} + 1659 q^{67} + 2268 q^{71} - 8703 q^{73} - 3438 q^{75} - 10665 q^{77} - 7773 q^{79} - 7479 q^{81} - 702 q^{85} + 40014 q^{87} + 14985 q^{89} - 20952 q^{91} + 1449 q^{93} - 20655 q^{95} + 8100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.35901 0.784623i −0.151001 0.0871803i 0.422596 0.906318i \(-0.361119\pi\)
−0.573597 + 0.819138i \(0.694452\pi\)
\(4\) 0 0
\(5\) 38.3327 22.1314i 1.53331 0.885256i 0.534102 0.845420i \(-0.320650\pi\)
0.999206 0.0398360i \(-0.0126836\pi\)
\(6\) 0 0
\(7\) 42.3967 + 24.5667i 0.865238 + 0.501361i
\(8\) 0 0
\(9\) −39.2687 68.0154i −0.484799 0.839697i
\(10\) 0 0
\(11\) −11.7557 + 20.3615i −0.0971545 + 0.168276i −0.910506 0.413496i \(-0.864307\pi\)
0.813351 + 0.581773i \(0.197641\pi\)
\(12\) 0 0
\(13\) 136.269i 0.806328i 0.915128 + 0.403164i \(0.132090\pi\)
−0.915128 + 0.403164i \(0.867910\pi\)
\(14\) 0 0
\(15\) −69.4592 −0.308708
\(16\) 0 0
\(17\) −227.333 131.251i −0.786618 0.454154i 0.0521523 0.998639i \(-0.483392\pi\)
−0.838771 + 0.544485i \(0.816725\pi\)
\(18\) 0 0
\(19\) −387.162 + 223.528i −1.07247 + 0.619191i −0.928855 0.370442i \(-0.879206\pi\)
−0.143615 + 0.989634i \(0.545873\pi\)
\(20\) 0 0
\(21\) −38.3418 66.6517i −0.0869429 0.151138i
\(22\) 0 0
\(23\) 374.585 + 648.800i 0.708100 + 1.22646i 0.965561 + 0.260176i \(0.0837805\pi\)
−0.257462 + 0.966288i \(0.582886\pi\)
\(24\) 0 0
\(25\) 667.098 1155.45i 1.06736 1.84872i
\(26\) 0 0
\(27\) 250.353i 0.343420i
\(28\) 0 0
\(29\) 406.524 0.483382 0.241691 0.970353i \(-0.422298\pi\)
0.241691 + 0.970353i \(0.422298\pi\)
\(30\) 0 0
\(31\) −584.199 337.287i −0.607907 0.350975i 0.164239 0.986421i \(-0.447483\pi\)
−0.772146 + 0.635445i \(0.780817\pi\)
\(32\) 0 0
\(33\) 31.9521 18.4476i 0.0293408 0.0169399i
\(34\) 0 0
\(35\) 2168.87 + 3.40909i 1.77051 + 0.00278293i
\(36\) 0 0
\(37\) −372.666 645.476i −0.272218 0.471495i 0.697212 0.716865i \(-0.254424\pi\)
−0.969429 + 0.245370i \(0.921090\pi\)
\(38\) 0 0
\(39\) 106.920 185.191i 0.0702959 0.121756i
\(40\) 0 0
\(41\) 2476.20i 1.47305i 0.676410 + 0.736526i \(0.263535\pi\)
−0.676410 + 0.736526i \(0.736465\pi\)
\(42\) 0 0
\(43\) −2636.68 −1.42601 −0.713003 0.701161i \(-0.752665\pi\)
−0.713003 + 0.701161i \(0.752665\pi\)
\(44\) 0 0
\(45\) −3010.55 1738.14i −1.48669 0.858343i
\(46\) 0 0
\(47\) −579.099 + 334.343i −0.262154 + 0.151355i −0.625317 0.780371i \(-0.715030\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(48\) 0 0
\(49\) 1193.96 + 2083.09i 0.497275 + 0.867593i
\(50\) 0 0
\(51\) 205.964 + 356.741i 0.0791866 + 0.137155i
\(52\) 0 0
\(53\) 1014.61 1757.36i 0.361200 0.625617i −0.626959 0.779053i \(-0.715700\pi\)
0.988159 + 0.153436i \(0.0490338\pi\)
\(54\) 0 0
\(55\) 1040.68i 0.344026i
\(56\) 0 0
\(57\) 701.541 0.215925
\(58\) 0 0
\(59\) −1014.22 585.561i −0.291359 0.168216i 0.347195 0.937793i \(-0.387134\pi\)
−0.638555 + 0.769576i \(0.720467\pi\)
\(60\) 0 0
\(61\) −2022.05 + 1167.43i −0.543416 + 0.313741i −0.746462 0.665428i \(-0.768249\pi\)
0.203047 + 0.979169i \(0.434916\pi\)
\(62\) 0 0
\(63\) 6.04891 3848.33i 0.00152404 0.969597i
\(64\) 0 0
\(65\) 3015.83 + 5223.58i 0.713807 + 1.23635i
\(66\) 0 0
\(67\) 3779.79 6546.78i 0.842011 1.45841i −0.0461815 0.998933i \(-0.514705\pi\)
0.888192 0.459472i \(-0.151961\pi\)
\(68\) 0 0
\(69\) 1175.63i 0.246929i
\(70\) 0 0
\(71\) 7575.36 1.50275 0.751375 0.659876i \(-0.229391\pi\)
0.751375 + 0.659876i \(0.229391\pi\)
\(72\) 0 0
\(73\) 3284.18 + 1896.12i 0.616284 + 0.355812i 0.775421 0.631445i \(-0.217538\pi\)
−0.159137 + 0.987257i \(0.550871\pi\)
\(74\) 0 0
\(75\) −1813.18 + 1046.84i −0.322343 + 0.186105i
\(76\) 0 0
\(77\) −998.615 + 574.460i −0.168429 + 0.0968898i
\(78\) 0 0
\(79\) −3897.33 6750.37i −0.624472 1.08162i −0.988643 0.150285i \(-0.951981\pi\)
0.364171 0.931332i \(-0.381353\pi\)
\(80\) 0 0
\(81\) −2984.33 + 5169.02i −0.454860 + 0.787840i
\(82\) 0 0
\(83\) 2181.41i 0.316651i 0.987387 + 0.158326i \(0.0506096\pi\)
−0.987387 + 0.158326i \(0.949390\pi\)
\(84\) 0 0
\(85\) −11619.0 −1.60817
\(86\) 0 0
\(87\) −552.469 318.968i −0.0729910 0.0421414i
\(88\) 0 0
\(89\) 8050.38 4647.89i 1.01633 0.586781i 0.103294 0.994651i \(-0.467062\pi\)
0.913040 + 0.407870i \(0.133728\pi\)
\(90\) 0 0
\(91\) −3347.69 + 5777.37i −0.404261 + 0.697666i
\(92\) 0 0
\(93\) 529.287 + 916.751i 0.0611963 + 0.105995i
\(94\) 0 0
\(95\) −9893.98 + 17136.9i −1.09629 + 1.89882i
\(96\) 0 0
\(97\) 9981.90i 1.06089i −0.847720 0.530444i \(-0.822025\pi\)
0.847720 0.530444i \(-0.177975\pi\)
\(98\) 0 0
\(99\) 1846.52 0.188402
\(100\) 0 0
\(101\) −675.552 390.030i −0.0662241 0.0382345i 0.466522 0.884509i \(-0.345507\pi\)
−0.532746 + 0.846275i \(0.678840\pi\)
\(102\) 0 0
\(103\) 2383.90 1376.35i 0.224706 0.129734i −0.383422 0.923573i \(-0.625254\pi\)
0.608127 + 0.793840i \(0.291921\pi\)
\(104\) 0 0
\(105\) −2944.84 1706.38i −0.267106 0.154774i
\(106\) 0 0
\(107\) −7429.46 12868.2i −0.648917 1.12396i −0.983382 0.181550i \(-0.941889\pi\)
0.334464 0.942408i \(-0.391445\pi\)
\(108\) 0 0
\(109\) −1378.33 + 2387.33i −0.116011 + 0.200937i −0.918183 0.396155i \(-0.870344\pi\)
0.802172 + 0.597092i \(0.203677\pi\)
\(110\) 0 0
\(111\) 1169.61i 0.0949281i
\(112\) 0 0
\(113\) 6264.79 0.490625 0.245312 0.969444i \(-0.421109\pi\)
0.245312 + 0.969444i \(0.421109\pi\)
\(114\) 0 0
\(115\) 28717.7 + 16580.2i 2.17147 + 1.25370i
\(116\) 0 0
\(117\) 9268.43 5351.13i 0.677071 0.390907i
\(118\) 0 0
\(119\) −6413.76 11149.4i −0.452917 0.787331i
\(120\) 0 0
\(121\) 7044.11 + 12200.8i 0.481122 + 0.833328i
\(122\) 0 0
\(123\) 1942.88 3365.17i 0.128421 0.222432i
\(124\) 0 0
\(125\) 31391.0i 2.00902i
\(126\) 0 0
\(127\) 12855.0 0.797010 0.398505 0.917166i \(-0.369529\pi\)
0.398505 + 0.917166i \(0.369529\pi\)
\(128\) 0 0
\(129\) 3583.27 + 2068.80i 0.215328 + 0.124320i
\(130\) 0 0
\(131\) −21412.5 + 12362.5i −1.24774 + 0.720383i −0.970658 0.240464i \(-0.922701\pi\)
−0.277082 + 0.960846i \(0.589367\pi\)
\(132\) 0 0
\(133\) −21905.7 34.4320i −1.23838 0.00194652i
\(134\) 0 0
\(135\) 5540.67 + 9596.73i 0.304015 + 0.526569i
\(136\) 0 0
\(137\) −661.078 + 1145.02i −0.0352218 + 0.0610059i −0.883099 0.469187i \(-0.844547\pi\)
0.847877 + 0.530193i \(0.177880\pi\)
\(138\) 0 0
\(139\) 5119.78i 0.264985i 0.991184 + 0.132493i \(0.0422981\pi\)
−0.991184 + 0.132493i \(0.957702\pi\)
\(140\) 0 0
\(141\) 1049.33 0.0527807
\(142\) 0 0
\(143\) −2774.64 1601.94i −0.135686 0.0783384i
\(144\) 0 0
\(145\) 15583.2 8996.95i 0.741173 0.427917i
\(146\) 0 0
\(147\) 11.8445 3767.74i 0.000548128 0.174360i
\(148\) 0 0
\(149\) −5104.93 8842.00i −0.229942 0.398270i 0.727849 0.685737i \(-0.240520\pi\)
−0.957791 + 0.287467i \(0.907187\pi\)
\(150\) 0 0
\(151\) −14672.1 + 25412.8i −0.643483 + 1.11455i 0.341166 + 0.940003i \(0.389178\pi\)
−0.984650 + 0.174543i \(0.944155\pi\)
\(152\) 0 0
\(153\) 20616.2i 0.880695i
\(154\) 0 0
\(155\) −29858.6 −1.24281
\(156\) 0 0
\(157\) −26033.1 15030.2i −1.05615 0.609770i −0.131787 0.991278i \(-0.542072\pi\)
−0.924366 + 0.381508i \(0.875405\pi\)
\(158\) 0 0
\(159\) −2757.73 + 1592.17i −0.109083 + 0.0629791i
\(160\) 0 0
\(161\) −57.7005 + 36709.3i −0.00222602 + 1.41620i
\(162\) 0 0
\(163\) −15439.7 26742.3i −0.581115 1.00652i −0.995348 0.0963500i \(-0.969283\pi\)
0.414232 0.910171i \(-0.364050\pi\)
\(164\) 0 0
\(165\) 816.541 1414.29i 0.0299923 0.0519482i
\(166\) 0 0
\(167\) 12858.8i 0.461070i −0.973064 0.230535i \(-0.925952\pi\)
0.973064 0.230535i \(-0.0740477\pi\)
\(168\) 0 0
\(169\) 9991.63 0.349835
\(170\) 0 0
\(171\) 30406.7 + 17555.3i 1.03987 + 0.600367i
\(172\) 0 0
\(173\) 13588.8 7845.51i 0.454035 0.262137i −0.255498 0.966810i \(-0.582239\pi\)
0.709533 + 0.704672i \(0.248906\pi\)
\(174\) 0 0
\(175\) 56668.2 32598.8i 1.85039 1.06445i
\(176\) 0 0
\(177\) 918.890 + 1591.56i 0.0293303 + 0.0508016i
\(178\) 0 0
\(179\) 15993.2 27701.0i 0.499148 0.864550i −0.500851 0.865533i \(-0.666980\pi\)
1.00000 0.000983488i \(0.000313054\pi\)
\(180\) 0 0
\(181\) 47967.6i 1.46417i 0.681215 + 0.732083i \(0.261452\pi\)
−0.681215 + 0.732083i \(0.738548\pi\)
\(182\) 0 0
\(183\) 3663.97 0.109408
\(184\) 0 0
\(185\) −28570.6 16495.2i −0.834787 0.481965i
\(186\) 0 0
\(187\) 5344.91 3085.88i 0.152847 0.0882462i
\(188\) 0 0
\(189\) −6150.35 + 10614.2i −0.172177 + 0.297141i
\(190\) 0 0
\(191\) −2227.93 3858.88i −0.0610709 0.105778i 0.833873 0.551956i \(-0.186118\pi\)
−0.894944 + 0.446178i \(0.852785\pi\)
\(192\) 0 0
\(193\) 14699.9 25460.9i 0.394638 0.683533i −0.598417 0.801185i \(-0.704203\pi\)
0.993055 + 0.117652i \(0.0375366\pi\)
\(194\) 0 0
\(195\) 9465.17i 0.248920i
\(196\) 0 0
\(197\) 23816.0 0.613671 0.306836 0.951763i \(-0.400730\pi\)
0.306836 + 0.951763i \(0.400730\pi\)
\(198\) 0 0
\(199\) −42280.9 24410.9i −1.06767 0.616421i −0.140128 0.990133i \(-0.544751\pi\)
−0.927545 + 0.373713i \(0.878085\pi\)
\(200\) 0 0
\(201\) −10273.5 + 5931.41i −0.254288 + 0.146813i
\(202\) 0 0
\(203\) 17235.3 + 9986.94i 0.418241 + 0.242349i
\(204\) 0 0
\(205\) 54801.8 + 94919.4i 1.30403 + 2.25864i
\(206\) 0 0
\(207\) 29418.9 50955.1i 0.686572 1.18918i
\(208\) 0 0
\(209\) 10510.9i 0.240629i
\(210\) 0 0
\(211\) −9582.07 −0.215226 −0.107613 0.994193i \(-0.534321\pi\)
−0.107613 + 0.994193i \(0.534321\pi\)
\(212\) 0 0
\(213\) −10295.0 5943.80i −0.226916 0.131010i
\(214\) 0 0
\(215\) −101071. + 58353.5i −2.18651 + 1.26238i
\(216\) 0 0
\(217\) −16482.1 28651.7i −0.350019 0.608458i
\(218\) 0 0
\(219\) −2975.48 5153.68i −0.0620396 0.107456i
\(220\) 0 0
\(221\) 17885.5 30978.5i 0.366197 0.634273i
\(222\) 0 0
\(223\) 54978.9i 1.10557i 0.833324 + 0.552785i \(0.186435\pi\)
−0.833324 + 0.552785i \(0.813565\pi\)
\(224\) 0 0
\(225\) −104784. −2.06981
\(226\) 0 0
\(227\) −7518.24 4340.66i −0.145903 0.0842372i 0.425271 0.905066i \(-0.360179\pi\)
−0.571174 + 0.820829i \(0.693512\pi\)
\(228\) 0 0
\(229\) −6262.86 + 3615.86i −0.119427 + 0.0689511i −0.558523 0.829489i \(-0.688632\pi\)
0.439097 + 0.898440i \(0.355299\pi\)
\(230\) 0 0
\(231\) 1807.86 + 2.84164i 0.0338798 + 5.32531e-5i
\(232\) 0 0
\(233\) −4422.96 7660.79i −0.0814707 0.141111i 0.822411 0.568894i \(-0.192628\pi\)
−0.903882 + 0.427782i \(0.859295\pi\)
\(234\) 0 0
\(235\) −14799.0 + 25632.5i −0.267976 + 0.464147i
\(236\) 0 0
\(237\) 12231.7i 0.217767i
\(238\) 0 0
\(239\) 88491.7 1.54920 0.774599 0.632453i \(-0.217952\pi\)
0.774599 + 0.632453i \(0.217952\pi\)
\(240\) 0 0
\(241\) −53365.8 30810.7i −0.918816 0.530479i −0.0355589 0.999368i \(-0.511321\pi\)
−0.883257 + 0.468889i \(0.844654\pi\)
\(242\) 0 0
\(243\) 25673.3 14822.5i 0.434779 0.251020i
\(244\) 0 0
\(245\) 91869.3 + 53426.6i 1.53052 + 0.890072i
\(246\) 0 0
\(247\) −30460.0 52758.4i −0.499271 0.864763i
\(248\) 0 0
\(249\) 1711.59 2964.55i 0.0276058 0.0478146i
\(250\) 0 0
\(251\) 57796.3i 0.917386i 0.888595 + 0.458693i \(0.151682\pi\)
−0.888595 + 0.458693i \(0.848318\pi\)
\(252\) 0 0
\(253\) −17614.0 −0.275180
\(254\) 0 0
\(255\) 15790.3 + 9116.56i 0.242835 + 0.140201i
\(256\) 0 0
\(257\) 60734.3 35065.0i 0.919534 0.530893i 0.0360479 0.999350i \(-0.488523\pi\)
0.883486 + 0.468457i \(0.155190\pi\)
\(258\) 0 0
\(259\) 57.4050 36521.2i 0.000855756 0.544435i
\(260\) 0 0
\(261\) −15963.7 27649.9i −0.234343 0.405894i
\(262\) 0 0
\(263\) −11120.9 + 19262.0i −0.160779 + 0.278477i −0.935148 0.354257i \(-0.884734\pi\)
0.774369 + 0.632734i \(0.218067\pi\)
\(264\) 0 0
\(265\) 89819.0i 1.27902i
\(266\) 0 0
\(267\) −14587.4 −0.204623
\(268\) 0 0
\(269\) 40332.3 + 23285.9i 0.557376 + 0.321801i 0.752092 0.659058i \(-0.229045\pi\)
−0.194715 + 0.980860i \(0.562378\pi\)
\(270\) 0 0
\(271\) 71958.3 41545.1i 0.979811 0.565694i 0.0775980 0.996985i \(-0.475275\pi\)
0.902213 + 0.431291i \(0.141942\pi\)
\(272\) 0 0
\(273\) 9082.59 5224.82i 0.121866 0.0701045i
\(274\) 0 0
\(275\) 15684.4 + 27166.2i 0.207397 + 0.359222i
\(276\) 0 0
\(277\) 4514.06 7818.59i 0.0588313 0.101899i −0.835110 0.550083i \(-0.814596\pi\)
0.893941 + 0.448185i \(0.147929\pi\)
\(278\) 0 0
\(279\) 52979.4i 0.680610i
\(280\) 0 0
\(281\) 64188.4 0.812912 0.406456 0.913670i \(-0.366764\pi\)
0.406456 + 0.913670i \(0.366764\pi\)
\(282\) 0 0
\(283\) 19563.7 + 11295.1i 0.244274 + 0.141032i 0.617140 0.786854i \(-0.288291\pi\)
−0.372865 + 0.927885i \(0.621625\pi\)
\(284\) 0 0
\(285\) 26892.0 15526.1i 0.331080 0.191149i
\(286\) 0 0
\(287\) −60832.0 + 104983.i −0.738530 + 1.27454i
\(288\) 0 0
\(289\) −7307.06 12656.2i −0.0874877 0.151533i
\(290\) 0 0
\(291\) −7832.02 + 13565.5i −0.0924885 + 0.160195i
\(292\) 0 0
\(293\) 130473.i 1.51980i −0.650040 0.759900i \(-0.725248\pi\)
0.650040 0.759900i \(-0.274752\pi\)
\(294\) 0 0
\(295\) −51837.2 −0.595658
\(296\) 0 0
\(297\) −5097.56 2943.08i −0.0577896 0.0333648i
\(298\) 0 0
\(299\) −88411.6 + 51044.5i −0.988933 + 0.570961i
\(300\) 0 0
\(301\) −111787. 64774.5i −1.23383 0.714943i
\(302\) 0 0
\(303\) 612.053 + 1060.11i 0.00666659 + 0.0115469i
\(304\) 0 0
\(305\) −51673.7 + 89501.6i −0.555482 + 0.962124i
\(306\) 0 0
\(307\) 56173.9i 0.596015i 0.954563 + 0.298008i \(0.0963221\pi\)
−0.954563 + 0.298008i \(0.903678\pi\)
\(308\) 0 0
\(309\) −4319.65 −0.0452410
\(310\) 0 0
\(311\) 40087.2 + 23144.4i 0.414462 + 0.239290i 0.692705 0.721221i \(-0.256419\pi\)
−0.278243 + 0.960511i \(0.589752\pi\)
\(312\) 0 0
\(313\) −119814. + 69174.4i −1.22298 + 0.706085i −0.965551 0.260213i \(-0.916207\pi\)
−0.257424 + 0.966299i \(0.582874\pi\)
\(314\) 0 0
\(315\) −84937.1 147651.i −0.856005 1.48804i
\(316\) 0 0
\(317\) 29152.7 + 50493.9i 0.290108 + 0.502482i 0.973835 0.227256i \(-0.0729754\pi\)
−0.683727 + 0.729738i \(0.739642\pi\)
\(318\) 0 0
\(319\) −4778.97 + 8277.42i −0.0469627 + 0.0813418i
\(320\) 0 0
\(321\) 23317.3i 0.226291i
\(322\) 0 0
\(323\) 117353. 1.12483
\(324\) 0 0
\(325\) 157452. + 90905.1i 1.49067 + 0.860640i
\(326\) 0 0
\(327\) 3746.31 2162.93i 0.0350355 0.0202278i
\(328\) 0 0
\(329\) −32765.6 51.5018i −0.302709 0.000475806i
\(330\) 0 0
\(331\) −494.602 856.675i −0.00451440 0.00781916i 0.863759 0.503904i \(-0.168104\pi\)
−0.868274 + 0.496085i \(0.834770\pi\)
\(332\) 0 0
\(333\) −29268.2 + 50694.1i −0.263942 + 0.457161i
\(334\) 0 0
\(335\) 334608.i 2.98158i
\(336\) 0 0
\(337\) −55958.7 −0.492728 −0.246364 0.969177i \(-0.579236\pi\)
−0.246364 + 0.969177i \(0.579236\pi\)
\(338\) 0 0
\(339\) −8513.89 4915.49i −0.0740847 0.0427728i
\(340\) 0 0
\(341\) 13735.3 7930.09i 0.118122 0.0681976i
\(342\) 0 0
\(343\) −554.769 + 117648.i −0.00471546 + 0.999989i
\(344\) 0 0
\(345\) −26018.3 45065.1i −0.218596 0.378619i
\(346\) 0 0
\(347\) −65322.2 + 113141.i −0.542502 + 0.939641i 0.456257 + 0.889848i \(0.349190\pi\)
−0.998760 + 0.0497936i \(0.984144\pi\)
\(348\) 0 0
\(349\) 110948.i 0.910899i 0.890262 + 0.455450i \(0.150522\pi\)
−0.890262 + 0.455450i \(0.849478\pi\)
\(350\) 0 0
\(351\) −34115.5 −0.276910
\(352\) 0 0
\(353\) 14727.9 + 8503.18i 0.118193 + 0.0682389i 0.557931 0.829887i \(-0.311595\pi\)
−0.439738 + 0.898126i \(0.644929\pi\)
\(354\) 0 0
\(355\) 290384. 167653.i 2.30418 1.33032i
\(356\) 0 0
\(357\) −31.7265 + 20184.5i −0.000248935 + 0.158373i
\(358\) 0 0
\(359\) −35732.4 61890.3i −0.277251 0.480212i 0.693450 0.720505i \(-0.256090\pi\)
−0.970701 + 0.240293i \(0.922757\pi\)
\(360\) 0 0
\(361\) 34769.0 60221.8i 0.266795 0.462103i
\(362\) 0 0
\(363\) 22107.9i 0.167777i
\(364\) 0 0
\(365\) 167855. 1.25994
\(366\) 0 0
\(367\) 147543. + 85183.8i 1.09543 + 0.632448i 0.935017 0.354602i \(-0.115383\pi\)
0.160415 + 0.987050i \(0.448717\pi\)
\(368\) 0 0
\(369\) 168420. 97237.2i 1.23692 0.714134i
\(370\) 0 0
\(371\) 86188.6 49580.5i 0.626184 0.360216i
\(372\) 0 0
\(373\) −75346.0 130503.i −0.541555 0.938001i −0.998815 0.0486680i \(-0.984502\pi\)
0.457260 0.889333i \(-0.348831\pi\)
\(374\) 0 0
\(375\) −24630.1 + 42660.5i −0.175147 + 0.303364i
\(376\) 0 0
\(377\) 55396.8i 0.389764i
\(378\) 0 0
\(379\) −211477. −1.47226 −0.736129 0.676841i \(-0.763348\pi\)
−0.736129 + 0.676841i \(0.763348\pi\)
\(380\) 0 0
\(381\) −17470.0 10086.3i −0.120349 0.0694836i
\(382\) 0 0
\(383\) −102769. + 59333.6i −0.700590 + 0.404486i −0.807567 0.589776i \(-0.799216\pi\)
0.106977 + 0.994261i \(0.465883\pi\)
\(384\) 0 0
\(385\) −25566.0 + 44121.4i −0.172481 + 0.297665i
\(386\) 0 0
\(387\) 103539. + 179335.i 0.691326 + 1.19741i
\(388\) 0 0
\(389\) −107858. + 186816.i −0.712778 + 1.23457i 0.251033 + 0.967979i \(0.419230\pi\)
−0.963810 + 0.266588i \(0.914104\pi\)
\(390\) 0 0
\(391\) 196658.i 1.28635i
\(392\) 0 0
\(393\) 38799.6 0.251213
\(394\) 0 0
\(395\) −298790. 172507.i −1.91502 1.10564i
\(396\) 0 0
\(397\) −134011. + 77371.1i −0.850273 + 0.490905i −0.860743 0.509040i \(-0.830001\pi\)
0.0104699 + 0.999945i \(0.496667\pi\)
\(398\) 0 0
\(399\) 29743.0 + 17234.5i 0.186827 + 0.108256i
\(400\) 0 0
\(401\) −100029. 173255.i −0.622067 1.07745i −0.989100 0.147244i \(-0.952960\pi\)
0.367033 0.930208i \(-0.380374\pi\)
\(402\) 0 0
\(403\) 45962.0 79608.5i 0.283001 0.490173i
\(404\) 0 0
\(405\) 264190.i 1.61067i
\(406\) 0 0
\(407\) 17523.8 0.105789
\(408\) 0 0
\(409\) 190382. + 109917.i 1.13810 + 0.657080i 0.945958 0.324288i \(-0.105125\pi\)
0.192137 + 0.981368i \(0.438458\pi\)
\(410\) 0 0
\(411\) 1796.82 1037.39i 0.0106370 0.00614129i
\(412\) 0 0
\(413\) −28614.4 49741.9i −0.167758 0.291623i
\(414\) 0 0
\(415\) 48277.7 + 83619.4i 0.280318 + 0.485524i
\(416\) 0 0
\(417\) 4017.10 6957.81i 0.0231015 0.0400130i
\(418\) 0 0
\(419\) 31359.8i 0.178626i 0.996004 + 0.0893132i \(0.0284672\pi\)
−0.996004 + 0.0893132i \(0.971533\pi\)
\(420\) 0 0
\(421\) −73690.7 −0.415766 −0.207883 0.978154i \(-0.566657\pi\)
−0.207883 + 0.978154i \(0.566657\pi\)
\(422\) 0 0
\(423\) 45481.0 + 26258.5i 0.254184 + 0.146753i
\(424\) 0 0
\(425\) −303306. + 175114.i −1.67920 + 0.969489i
\(426\) 0 0
\(427\) −114408. 179.830i −0.627481 0.000986291i
\(428\) 0 0
\(429\) 2513.84 + 4354.10i 0.0136591 + 0.0236583i
\(430\) 0 0
\(431\) 132586. 229646.i 0.713746 1.23624i −0.249695 0.968325i \(-0.580330\pi\)
0.963441 0.267920i \(-0.0863364\pi\)
\(432\) 0 0
\(433\) 124873.i 0.666027i 0.942922 + 0.333014i \(0.108066\pi\)
−0.942922 + 0.333014i \(0.891934\pi\)
\(434\) 0 0
\(435\) −28236.8 −0.149224
\(436\) 0 0
\(437\) −290050. 167460.i −1.51883 0.876898i
\(438\) 0 0
\(439\) −120989. + 69852.8i −0.627792 + 0.362456i −0.779896 0.625909i \(-0.784728\pi\)
0.152105 + 0.988364i \(0.451395\pi\)
\(440\) 0 0
\(441\) 94797.1 163008.i 0.487437 0.838169i
\(442\) 0 0
\(443\) 110595. + 191556.i 0.563544 + 0.976087i 0.997183 + 0.0750005i \(0.0238959\pi\)
−0.433639 + 0.901087i \(0.642771\pi\)
\(444\) 0 0
\(445\) 205729. 356332.i 1.03890 1.79943i
\(446\) 0 0
\(447\) 16021.8i 0.0801855i
\(448\) 0 0
\(449\) 255878. 1.26923 0.634615 0.772828i \(-0.281159\pi\)
0.634615 + 0.772828i \(0.281159\pi\)
\(450\) 0 0
\(451\) −50419.0 29109.4i −0.247880 0.143114i
\(452\) 0 0
\(453\) 39878.9 23024.1i 0.194333 0.112198i
\(454\) 0 0
\(455\) −464.555 + 295551.i −0.00224396 + 1.42761i
\(456\) 0 0
\(457\) 159964. + 277065.i 0.765929 + 1.32663i 0.939754 + 0.341852i \(0.111054\pi\)
−0.173825 + 0.984777i \(0.555613\pi\)
\(458\) 0 0
\(459\) 32859.0 56913.5i 0.155966 0.270141i
\(460\) 0 0
\(461\) 369557.i 1.73892i −0.494002 0.869461i \(-0.664466\pi\)
0.494002 0.869461i \(-0.335534\pi\)
\(462\) 0 0
\(463\) 32715.0 0.152611 0.0763053 0.997084i \(-0.475688\pi\)
0.0763053 + 0.997084i \(0.475688\pi\)
\(464\) 0 0
\(465\) 40578.0 + 23427.7i 0.187666 + 0.108349i
\(466\) 0 0
\(467\) 265325. 153186.i 1.21659 0.702400i 0.252405 0.967622i \(-0.418779\pi\)
0.964187 + 0.265222i \(0.0854452\pi\)
\(468\) 0 0
\(469\) 321083. 184705.i 1.45973 0.839717i
\(470\) 0 0
\(471\) 23586.1 + 40852.3i 0.106320 + 0.184151i
\(472\) 0 0
\(473\) 30996.0 53686.7i 0.138543 0.239963i
\(474\) 0 0
\(475\) 596460.i 2.64359i
\(476\) 0 0
\(477\) −159370. −0.700438
\(478\) 0 0
\(479\) −226801. 130944.i −0.988493 0.570707i −0.0836697 0.996494i \(-0.526664\pi\)
−0.904824 + 0.425787i \(0.859997\pi\)
\(480\) 0 0
\(481\) 87958.7 50783.0i 0.380180 0.219497i
\(482\) 0 0
\(483\) 28881.3 49842.8i 0.123801 0.213653i
\(484\) 0 0
\(485\) −220913. 382633.i −0.939158 1.62667i
\(486\) 0 0
\(487\) −41127.7 + 71235.3i −0.173411 + 0.300357i −0.939610 0.342246i \(-0.888812\pi\)
0.766199 + 0.642603i \(0.222146\pi\)
\(488\) 0 0
\(489\) 48457.2i 0.202647i
\(490\) 0 0
\(491\) −102923. −0.426924 −0.213462 0.976951i \(-0.568474\pi\)
−0.213462 + 0.976951i \(0.568474\pi\)
\(492\) 0 0
\(493\) −92416.2 53356.5i −0.380237 0.219530i
\(494\) 0 0
\(495\) 70782.3 40866.2i 0.288878 0.166784i
\(496\) 0 0
\(497\) 321170. + 186101.i 1.30024 + 0.753420i
\(498\) 0 0
\(499\) −75028.2 129953.i −0.301317 0.521896i 0.675118 0.737710i \(-0.264093\pi\)
−0.976434 + 0.215814i \(0.930759\pi\)
\(500\) 0 0
\(501\) −10089.3 + 17475.2i −0.0401962 + 0.0696219i
\(502\) 0 0
\(503\) 91806.6i 0.362859i 0.983404 + 0.181430i \(0.0580725\pi\)
−0.983404 + 0.181430i \(0.941928\pi\)
\(504\) 0 0
\(505\) −34527.6 −0.135389
\(506\) 0 0
\(507\) −13578.7 7839.66i −0.0528253 0.0304987i
\(508\) 0 0
\(509\) −412600. + 238215.i −1.59255 + 0.919460i −0.599684 + 0.800237i \(0.704707\pi\)
−0.992867 + 0.119224i \(0.961959\pi\)
\(510\) 0 0
\(511\) 92656.9 + 161071.i 0.354843 + 0.616843i
\(512\) 0 0
\(513\) −55961.0 96927.3i −0.212643 0.368308i
\(514\) 0 0
\(515\) 60921.0 105518.i 0.229696 0.397844i
\(516\) 0 0
\(517\) 15721.7i 0.0588192i
\(518\) 0 0
\(519\) −24623.1 −0.0914128
\(520\) 0 0
\(521\) 147768. + 85314.0i 0.544384 + 0.314300i 0.746854 0.664988i \(-0.231563\pi\)
−0.202470 + 0.979289i \(0.564897\pi\)
\(522\) 0 0
\(523\) 85040.5 49098.2i 0.310901 0.179499i −0.336428 0.941709i \(-0.609219\pi\)
0.647330 + 0.762210i \(0.275886\pi\)
\(524\) 0 0
\(525\) −102590. 161.254i −0.372209 0.000585048i
\(526\) 0 0
\(527\) 88538.3 + 153353.i 0.318794 + 0.552167i
\(528\) 0 0
\(529\) −140707. + 243711.i −0.502810 + 0.870892i
\(530\) 0 0
\(531\) 91977.0i 0.326205i
\(532\) 0 0
\(533\) −337430. −1.18776
\(534\) 0 0
\(535\) −569582. 328849.i −1.98998 1.14892i
\(536\) 0 0
\(537\) −43469.7 + 25097.3i −0.150743 + 0.0870318i
\(538\) 0 0
\(539\) −56450.5 177.461i −0.194308 0.000610838i
\(540\) 0 0
\(541\) 52833.7 + 91510.6i 0.180516 + 0.312663i 0.942056 0.335454i \(-0.108890\pi\)
−0.761540 + 0.648118i \(0.775557\pi\)
\(542\) 0 0
\(543\) 37636.4 65188.2i 0.127646 0.221090i
\(544\) 0 0
\(545\) 122017.i 0.410798i
\(546\) 0 0
\(547\) −127121. −0.424855 −0.212428 0.977177i \(-0.568137\pi\)
−0.212428 + 0.977177i \(0.568137\pi\)
\(548\) 0 0
\(549\) 158807. + 91687.0i 0.526895 + 0.304203i
\(550\) 0 0
\(551\) −157391. + 90869.5i −0.518413 + 0.299306i
\(552\) 0 0
\(553\) 600.340 381938.i 0.00196312 1.24894i
\(554\) 0 0
\(555\) 25885.1 + 44834.3i 0.0840356 + 0.145554i
\(556\) 0 0
\(557\) −15246.5 + 26407.6i −0.0491427 + 0.0851176i −0.889550 0.456837i \(-0.848982\pi\)
0.840408 + 0.541955i \(0.182316\pi\)
\(558\) 0 0
\(559\) 359300.i 1.14983i
\(560\) 0 0
\(561\) −9685.01 −0.0307733
\(562\) 0 0
\(563\) 459404. + 265237.i 1.44936 + 0.836791i 0.998444 0.0557725i \(-0.0177622\pi\)
0.450921 + 0.892564i \(0.351095\pi\)
\(564\) 0 0
\(565\) 240146. 138649.i 0.752279 0.434329i
\(566\) 0 0
\(567\) −253511. + 145834.i −0.788554 + 0.453621i
\(568\) 0 0
\(569\) 169366. + 293351.i 0.523121 + 0.906072i 0.999638 + 0.0269067i \(0.00856571\pi\)
−0.476517 + 0.879165i \(0.658101\pi\)
\(570\) 0 0
\(571\) 53455.6 92587.8i 0.163954 0.283976i −0.772330 0.635222i \(-0.780909\pi\)
0.936283 + 0.351246i \(0.114242\pi\)
\(572\) 0 0
\(573\) 6992.33i 0.0212967i
\(574\) 0 0
\(575\) 999538. 3.02318
\(576\) 0 0
\(577\) −104337. 60238.8i −0.313390 0.180936i 0.335052 0.942200i \(-0.391246\pi\)
−0.648443 + 0.761264i \(0.724579\pi\)
\(578\) 0 0
\(579\) −39954.4 + 23067.7i −0.119181 + 0.0688093i
\(580\) 0 0
\(581\) −53590.0 + 92484.6i −0.158757 + 0.273979i
\(582\) 0 0
\(583\) 23854.9 + 41317.9i 0.0701844 + 0.121563i
\(584\) 0 0
\(585\) 236856. 410247.i 0.692106 1.19876i
\(586\) 0 0
\(587\) 525818.i 1.52602i −0.646389 0.763008i \(-0.723722\pi\)
0.646389 0.763008i \(-0.276278\pi\)
\(588\) 0 0
\(589\) 301573. 0.869283
\(590\) 0 0
\(591\) −32366.0 18686.5i −0.0926648 0.0535000i
\(592\) 0 0
\(593\) −64283.0 + 37113.8i −0.182804 + 0.105542i −0.588610 0.808417i \(-0.700324\pi\)
0.405805 + 0.913960i \(0.366991\pi\)
\(594\) 0 0
\(595\) −492609. 285441.i −1.39145 0.806274i
\(596\) 0 0
\(597\) 38306.7 + 66349.1i 0.107480 + 0.186160i
\(598\) 0 0
\(599\) −10990.7 + 19036.5i −0.0306318 + 0.0530558i −0.880935 0.473237i \(-0.843085\pi\)
0.850303 + 0.526293i \(0.176419\pi\)
\(600\) 0 0
\(601\) 77418.2i 0.214336i 0.994241 + 0.107168i \(0.0341782\pi\)
−0.994241 + 0.107168i \(0.965822\pi\)
\(602\) 0 0
\(603\) −593710. −1.63282
\(604\) 0 0
\(605\) 540039. + 311792.i 1.47542 + 0.851832i
\(606\) 0 0
\(607\) 155765. 89931.2i 0.422760 0.244081i −0.273498 0.961873i \(-0.588181\pi\)
0.696257 + 0.717792i \(0.254847\pi\)
\(608\) 0 0
\(609\) −15586.9 27095.5i −0.0420266 0.0730572i
\(610\) 0 0
\(611\) −45560.7 78913.5i −0.122042 0.211382i
\(612\) 0 0
\(613\) −314637. + 544968.i −0.837316 + 1.45027i 0.0548147 + 0.998497i \(0.482543\pi\)
−0.892131 + 0.451777i \(0.850790\pi\)
\(614\) 0 0
\(615\) 171995.i 0.454742i
\(616\) 0 0
\(617\) −52877.1 −0.138899 −0.0694493 0.997585i \(-0.522124\pi\)
−0.0694493 + 0.997585i \(0.522124\pi\)
\(618\) 0 0
\(619\) 447040. + 258099.i 1.16672 + 0.673603i 0.952904 0.303271i \(-0.0980789\pi\)
0.213811 + 0.976875i \(0.431412\pi\)
\(620\) 0 0
\(621\) −162429. + 93778.6i −0.421193 + 0.243176i
\(622\) 0 0
\(623\) 455493. + 715.955i 1.17356 + 0.00184463i
\(624\) 0 0
\(625\) −277790. 481147.i −0.711143 1.23174i
\(626\) 0 0
\(627\) −8247.09 + 14284.4i −0.0209781 + 0.0363351i
\(628\) 0 0
\(629\) 195651.i 0.494515i
\(630\) 0 0
\(631\) 193439. 0.485831 0.242915 0.970047i \(-0.421896\pi\)
0.242915 + 0.970047i \(0.421896\pi\)
\(632\) 0 0
\(633\) 13022.1 + 7518.31i 0.0324993 + 0.0187635i
\(634\) 0 0
\(635\) 492766. 284499.i 1.22206 0.705558i
\(636\) 0 0
\(637\) −283862. + 162700.i −0.699565 + 0.400967i
\(638\) 0 0
\(639\) −297475. 515242.i −0.728532 1.26185i
\(640\) 0 0
\(641\) −376563. + 652226.i −0.916477 + 1.58738i −0.111751 + 0.993736i \(0.535646\pi\)
−0.804725 + 0.593648i \(0.797687\pi\)
\(642\) 0 0
\(643\) 7110.23i 0.0171974i −0.999963 0.00859868i \(-0.997263\pi\)
0.999963 0.00859868i \(-0.00273708\pi\)
\(644\) 0 0
\(645\) 183142. 0.440219
\(646\) 0 0
\(647\) −341808. 197343.i −0.816534 0.471426i 0.0326859 0.999466i \(-0.489594\pi\)
−0.849220 + 0.528040i \(0.822927\pi\)
\(648\) 0 0
\(649\) 23845.8 13767.4i 0.0566137 0.0326860i
\(650\) 0 0
\(651\) −81.5306 + 51870.0i −0.000192380 + 0.122392i
\(652\) 0 0
\(653\) −115578. 200186.i −0.271049 0.469470i 0.698082 0.716018i \(-0.254037\pi\)
−0.969131 + 0.246548i \(0.920704\pi\)
\(654\) 0 0
\(655\) −547198. + 947775.i −1.27545 + 2.20914i
\(656\) 0 0
\(657\) 297833.i 0.689989i
\(658\) 0 0
\(659\) −772498. −1.77880 −0.889399 0.457131i \(-0.848877\pi\)
−0.889399 + 0.457131i \(0.848877\pi\)
\(660\) 0 0
\(661\) −195287. 112749.i −0.446962 0.258053i 0.259584 0.965720i \(-0.416414\pi\)
−0.706546 + 0.707667i \(0.749748\pi\)
\(662\) 0 0
\(663\) −48612.9 + 28066.7i −0.110592 + 0.0638504i
\(664\) 0 0
\(665\) −840468. + 483484.i −1.90054 + 1.09330i
\(666\) 0 0
\(667\) 152278. + 263753.i 0.342282 + 0.592851i
\(668\) 0 0
\(669\) 43137.7 74716.6i 0.0963839 0.166942i
\(670\) 0 0
\(671\) 54895.8i 0.121925i
\(672\) 0 0
\(673\) 59564.6 0.131510 0.0657549 0.997836i \(-0.479054\pi\)
0.0657549 + 0.997836i \(0.479054\pi\)
\(674\) 0 0
\(675\) 289270. + 167010.i 0.634887 + 0.366552i
\(676\) 0 0
\(677\) 687131. 396715.i 1.49921 0.865569i 0.499210 0.866481i \(-0.333624\pi\)
1.00000 0.000912440i \(0.000290439\pi\)
\(678\) 0 0
\(679\) 245222. 423199.i 0.531888 0.917921i
\(680\) 0 0
\(681\) 6811.56 + 11798.0i 0.0146876 + 0.0254397i
\(682\) 0 0
\(683\) 46304.3 80201.3i 0.0992612 0.171925i −0.812118 0.583493i \(-0.801685\pi\)
0.911379 + 0.411568i \(0.135019\pi\)
\(684\) 0 0
\(685\) 58522.3i 0.124721i
\(686\) 0 0
\(687\) 11348.4 0.0240447
\(688\) 0 0
\(689\) 239474. + 138261.i 0.504453 + 0.291246i
\(690\) 0 0
\(691\) −162273. + 93688.2i −0.339852 + 0.196213i −0.660206 0.751084i \(-0.729531\pi\)
0.320355 + 0.947298i \(0.396198\pi\)
\(692\) 0 0
\(693\) 78286.5 + 45362.9i 0.163012 + 0.0944571i
\(694\) 0 0
\(695\) 113308. + 196255.i 0.234580 + 0.406304i
\(696\) 0 0
\(697\) 325003. 562921.i 0.668993 1.15873i
\(698\) 0 0
\(699\) 13881.4i 0.0284105i
\(700\) 0 0
\(701\) 326056. 0.663522 0.331761 0.943363i \(-0.392357\pi\)
0.331761 + 0.943363i \(0.392357\pi\)
\(702\) 0 0
\(703\) 288564. + 166603.i 0.583891 + 0.337110i
\(704\) 0 0
\(705\) 40223.8 23223.2i 0.0809290 0.0467244i
\(706\) 0 0
\(707\) −19059.4 33132.0i −0.0381303 0.0662841i
\(708\) 0 0
\(709\) −32372.6 56070.9i −0.0643998 0.111544i 0.832028 0.554734i \(-0.187180\pi\)
−0.896428 + 0.443190i \(0.853847\pi\)
\(710\) 0 0
\(711\) −306086. + 530157.i −0.605487 + 1.04873i
\(712\) 0 0
\(713\) 505371.i 0.994102i
\(714\) 0 0
\(715\) −141813. −0.277398
\(716\) 0 0
\(717\) −120261. 69432.6i −0.233930 0.135060i
\(718\) 0 0
\(719\) −616355. + 355853.i −1.19227 + 0.688355i −0.958820 0.284014i \(-0.908334\pi\)
−0.233446 + 0.972370i \(0.575000\pi\)
\(720\) 0 0
\(721\) 134882. + 212.011i 0.259468 + 0.000407838i
\(722\) 0 0
\(723\) 48349.6 + 83744.0i 0.0924946 + 0.160205i
\(724\) 0 0
\(725\) 271191. 469717.i 0.515941 0.893635i
\(726\) 0 0
\(727\) 763134.i 1.44388i −0.691954 0.721941i \(-0.743250\pi\)
0.691954 0.721941i \(-0.256750\pi\)
\(728\) 0 0
\(729\) 436942. 0.822183
\(730\) 0 0
\(731\) 599404. + 346066.i 1.12172 + 0.647626i
\(732\) 0 0
\(733\) −17848.0 + 10304.6i −0.0332187 + 0.0191788i −0.516517 0.856277i \(-0.672772\pi\)
0.483299 + 0.875456i \(0.339439\pi\)
\(734\) 0 0
\(735\) −82931.3 144690.i −0.153513 0.267832i
\(736\) 0 0
\(737\) 88868.0 + 153924.i 0.163610 + 0.283381i
\(738\) 0 0
\(739\) 393194. 681032.i 0.719976 1.24703i −0.241033 0.970517i \(-0.577486\pi\)
0.961009 0.276518i \(-0.0891805\pi\)
\(740\) 0 0
\(741\) 95598.6i 0.174107i
\(742\) 0 0
\(743\) 740479. 1.34133 0.670664 0.741761i \(-0.266009\pi\)
0.670664 + 0.741761i \(0.266009\pi\)
\(744\) 0 0
\(745\) −391372. 225959.i −0.705143 0.407114i
\(746\) 0 0
\(747\) 148370. 85661.3i 0.265891 0.153512i
\(748\) 0 0
\(749\) 1144.42 728086.i 0.00203997 1.29783i
\(750\) 0 0
\(751\) 247292. + 428322.i 0.438460 + 0.759436i 0.997571 0.0696574i \(-0.0221906\pi\)
−0.559111 + 0.829093i \(0.688857\pi\)
\(752\) 0 0
\(753\) 45348.3 78545.5i 0.0799780 0.138526i
\(754\) 0 0
\(755\) 1.29885e6i 2.27859i
\(756\) 0 0
\(757\) −139399. −0.243258 −0.121629 0.992576i \(-0.538812\pi\)
−0.121629 + 0.992576i \(0.538812\pi\)
\(758\) 0 0
\(759\) 23937.5 + 13820.3i 0.0415524 + 0.0239903i
\(760\) 0 0
\(761\) −207340. + 119708.i −0.358025 + 0.206706i −0.668214 0.743969i \(-0.732941\pi\)
0.310189 + 0.950675i \(0.399608\pi\)
\(762\) 0 0
\(763\) −117085. + 67354.1i −0.201119 + 0.115695i
\(764\) 0 0
\(765\) 456265. + 790274.i 0.779640 + 1.35038i
\(766\) 0 0
\(767\) 79794.1 138208.i 0.135638 0.234931i
\(768\) 0 0
\(769\) 98701.5i 0.166906i −0.996512 0.0834528i \(-0.973405\pi\)
0.996512 0.0834528i \(-0.0265948\pi\)
\(770\) 0 0
\(771\) −110051. −0.185134
\(772\) 0 0
\(773\) −507921. 293248.i −0.850036 0.490769i 0.0106268 0.999944i \(-0.496617\pi\)
−0.860663 + 0.509175i \(0.829951\pi\)
\(774\) 0 0
\(775\) −779435. + 450007.i −1.29771 + 0.749232i
\(776\) 0 0
\(777\) −28733.4 + 49587.5i −0.0475932 + 0.0821354i
\(778\) 0 0
\(779\) −553500. 958690.i −0.912100 1.57980i
\(780\) 0 0
\(781\) −89053.6 + 154245.i −0.145999 + 0.252877i
\(782\) 0 0
\(783\) 101775.i 0.166003i
\(784\) 0 0
\(785\) −1.33056e6 −2.15921
\(786\) 0 0
\(787\) −129536. 74787.4i −0.209141 0.120748i 0.391771 0.920063i \(-0.371863\pi\)
−0.600912 + 0.799315i \(0.705196\pi\)
\(788\) 0 0
\(789\) 30226.8 17451.4i 0.0485554 0.0280335i
\(790\) 0 0
\(791\) 265606. + 153905.i 0.424507 + 0.245980i
\(792\) 0 0
\(793\) −159085. 275544.i −0.252978 0.438171i
\(794\) 0 0
\(795\) −70474.1 + 122065.i −0.111505 + 0.193133i
\(796\) 0 0
\(797\) 241092.i 0.379547i −0.981828 0.189774i \(-0.939225\pi\)
0.981828 0.189774i \(-0.0607754\pi\)
\(798\) 0 0
\(799\) 175531. 0.274954
\(800\) 0 0
\(801\) −632256. 365033.i −0.985436 0.568941i
\(802\) 0 0
\(803\) −77215.6 + 44580.4i −0.119750 + 0.0691374i
\(804\) 0 0
\(805\) 810215. + 1.40844e6i 1.25028 + 2.17344i
\(806\) 0 0
\(807\) −36541.3 63291.3i −0.0561095 0.0971845i
\(808\) 0 0
\(809\) −100933. + 174820.i −0.154218 + 0.267113i −0.932774 0.360462i \(-0.882619\pi\)
0.778556 + 0.627575i \(0.215952\pi\)
\(810\) 0 0
\(811\) 1.16747e6i 1.77503i 0.460779 + 0.887515i \(0.347570\pi\)
−0.460779 + 0.887515i \(0.652430\pi\)
\(812\) 0 0
\(813\) −130389. −0.197270
\(814\) 0 0
\(815\) −1.18369e6 683402.i −1.78206 1.02887i
\(816\) 0 0
\(817\) 1.02082e6 589373.i 1.52935 0.882970i
\(818\) 0 0
\(819\) 524410. + 824.281i 0.781814 + 0.00122887i
\(820\) 0 0
\(821\) 324585. + 562198.i 0.481551 + 0.834071i 0.999776 0.0211734i \(-0.00674022\pi\)
−0.518225 + 0.855245i \(0.673407\pi\)
\(822\) 0 0
\(823\) 245486. 425193.i 0.362432 0.627750i −0.625929 0.779880i \(-0.715280\pi\)
0.988360 + 0.152130i \(0.0486133\pi\)
\(824\) 0 0
\(825\) 49225.3i 0.0723237i
\(826\) 0 0
\(827\) −709017. −1.03668 −0.518341 0.855174i \(-0.673450\pi\)
−0.518341 + 0.855174i \(0.673450\pi\)
\(828\) 0 0
\(829\) 534645. + 308677.i 0.777958 + 0.449154i 0.835706 0.549177i \(-0.185059\pi\)
−0.0577480 + 0.998331i \(0.518392\pi\)
\(830\) 0 0
\(831\) −12269.3 + 7083.68i −0.0177671 + 0.0102579i
\(832\) 0 0
\(833\) 1981.33 630262.i 0.00285540 0.908304i
\(834\) 0 0
\(835\) −284583. 492912.i −0.408165 0.706963i
\(836\) 0 0
\(837\) 84441.0 146256.i 0.120532 0.208768i
\(838\) 0 0
\(839\) 493961.i 0.701728i 0.936426 + 0.350864i \(0.114112\pi\)
−0.936426 + 0.350864i \(0.885888\pi\)
\(840\) 0 0
\(841\) −542019. −0.766342
\(842\) 0 0
\(843\) −87232.4 50363.7i −0.122750 0.0708699i
\(844\) 0 0
\(845\) 383006. 221129.i 0.536404 0.309693i
\(846\) 0 0
\(847\) −1085.07 + 690322.i −0.00151248 + 0.962243i
\(848\) 0 0
\(849\) −17724.8 30700.2i −0.0245904 0.0425918i
\(850\) 0 0
\(851\) 279190. 483571.i 0.385514 0.667730i
\(852\) 0 0
\(853\) 505142.i 0.694249i 0.937819 + 0.347124i \(0.112842\pi\)
−0.937819 + 0.347124i \(0.887158\pi\)
\(854\) 0 0
\(855\) 1.55410e6 2.12591
\(856\) 0 0
\(857\) −969980. 560018.i −1.32069 0.762501i −0.336852 0.941557i \(-0.609362\pi\)
−0.983839 + 0.179056i \(0.942696\pi\)
\(858\) 0 0
\(859\) −605141. + 349379.i −0.820107 + 0.473489i −0.850453 0.526050i \(-0.823672\pi\)
0.0303463 + 0.999539i \(0.490339\pi\)
\(860\) 0 0
\(861\) 165043. 94941.9i 0.222633 0.128071i
\(862\) 0 0
\(863\) −479533. 830576.i −0.643868 1.11521i −0.984562 0.175038i \(-0.943995\pi\)
0.340693 0.940175i \(-0.389338\pi\)
\(864\) 0 0
\(865\) 347264. 601479.i 0.464117 0.803875i
\(866\) 0 0
\(867\) 22933.1i 0.0305088i
\(868\) 0 0
\(869\) 183263. 0.242681
\(870\) 0 0
\(871\) 892127. + 515069.i 1.17595 + 0.678937i
\(872\) 0 0
\(873\) −678923. + 391976.i −0.890824 + 0.514318i
\(874\) 0 0
\(875\) 771172. 1.33087e6i 1.00724 1.73828i
\(876\) 0 0
\(877\) 498442. + 863326.i 0.648060 + 1.12247i 0.983586 + 0.180442i \(0.0577527\pi\)
−0.335526 + 0.942031i \(0.608914\pi\)
\(878\) 0 0
\(879\) −102372. + 177314.i −0.132497 + 0.229491i
\(880\) 0 0
\(881\) 494568.i 0.637198i 0.947890 + 0.318599i \(0.103212\pi\)
−0.947890 + 0.318599i \(0.896788\pi\)
\(882\) 0 0
\(883\) 887278. 1.13799 0.568995 0.822341i \(-0.307333\pi\)
0.568995 + 0.822341i \(0.307333\pi\)
\(884\) 0 0
\(885\) 70447.1 + 40672.6i 0.0899448 + 0.0519297i
\(886\) 0 0
\(887\) 542062. 312960.i 0.688973 0.397779i −0.114254 0.993452i \(-0.536448\pi\)
0.803227 + 0.595673i \(0.203115\pi\)
\(888\) 0 0
\(889\) 545008. + 315804.i 0.689604 + 0.399589i
\(890\) 0 0
\(891\) −70165.8 121531.i −0.0883833 0.153084i
\(892\) 0 0
\(893\) 149470. 258890.i 0.187435 0.324647i
\(894\) 0 0
\(895\) 1.41581e6i 1.76750i
\(896\) 0 0
\(897\) 160203. 0.199106
\(898\) 0 0
\(899\) −237491. 137115.i −0.293851 0.169655i
\(900\) 0 0
\(901\) −461309. + 266337.i −0.568253 + 0.328081i
\(902\) 0 0
\(903\) 101095. + 175739.i 0.123981 + 0.215523i
\(904\) 0 0
\(905\) 1.06159e6 + 1.83873e6i 1.29616 + 2.24502i
\(906\) 0 0
\(907\) 634131. 1.09835e6i 0.770840 1.33513i −0.166263 0.986081i \(-0.553170\pi\)
0.937103 0.349052i \(-0.113496\pi\)
\(908\) 0 0
\(909\) 61263.9i 0.0741442i
\(910\) 0 0
\(911\) −1.09985e6 −1.32525 −0.662625 0.748951i \(-0.730558\pi\)
−0.662625 + 0.748951i \(0.730558\pi\)
\(912\) 0 0
\(913\) −44416.7 25644.0i −0.0532850 0.0307641i
\(914\) 0 0
\(915\) 140450. 81088.8i 0.167756 0.0968542i
\(916\) 0 0
\(917\) −1.21152e6 1904.30i −1.44076 0.00226463i
\(918\) 0 0
\(919\) −430660. 745925.i −0.509922 0.883211i −0.999934 0.0114950i \(-0.996341\pi\)
0.490012 0.871716i \(-0.336992\pi\)
\(920\) 0 0
\(921\) 44075.3 76340.6i 0.0519608 0.0899987i
\(922\) 0 0
\(923\) 1.03229e6i 1.21171i
\(924\) 0 0
\(925\) −994418. −1.16221
\(926\) 0 0
\(927\) −187226. 108095.i −0.217874 0.125790i
\(928\) 0 0
\(929\) −326247. + 188359.i −0.378021 + 0.218250i −0.676957 0.736023i \(-0.736702\pi\)
0.298936 + 0.954273i \(0.403368\pi\)
\(930\) 0 0
\(931\) −927884. 539610.i −1.07052 0.622560i
\(932\) 0 0
\(933\) −36319.2 62906.7i −0.0417227 0.0722659i
\(934\) 0 0
\(935\) 136590. 236580.i 0.156241 0.270617i
\(936\) 0 0
\(937\) 417813.i 0.475885i −0.971279 0.237943i \(-0.923527\pi\)
0.971279 0.237943i \(-0.0764731\pi\)
\(938\) 0 0
\(939\) 217103. 0.246227
\(940\) 0 0
\(941\) 213031. + 122994.i 0.240582 + 0.138900i 0.615444 0.788180i \(-0.288977\pi\)
−0.374862 + 0.927081i \(0.622310\pi\)
\(942\) 0 0
\(943\) −1.60656e6 + 927546.i −1.80665 + 1.04307i
\(944\) 0 0
\(945\) −853.478 + 542985.i −0.000955716 + 0.608029i
\(946\) 0 0
\(947\) 722964. + 1.25221e6i 0.806152 + 1.39630i 0.915511 + 0.402294i \(0.131787\pi\)
−0.109359 + 0.994002i \(0.534880\pi\)
\(948\) 0 0
\(949\) −258384. + 447533.i −0.286901 + 0.496928i
\(950\) 0 0
\(951\) 91495.4i 0.101167i
\(952\) 0 0
\(953\) 907377. 0.999084 0.499542 0.866290i \(-0.333502\pi\)
0.499542 + 0.866290i \(0.333502\pi\)
\(954\) 0 0
\(955\) −170805. 98614.2i −0.187281 0.108127i
\(956\) 0 0
\(957\) 12989.3 7499.38i 0.0141828 0.00818845i
\(958\) 0 0
\(959\) −56156.9 + 32304.6i −0.0610612 + 0.0351259i
\(960\) 0 0
\(961\) −234235. 405707.i −0.253633 0.439305i
\(962\) 0 0
\(963\) −583491. + 1.01064e6i −0.629189 + 1.08979i
\(964\) 0 0
\(965\) 1.30132e6i 1.39742i
\(966\) 0 0
\(967\) −50286.2 −0.0537770 −0.0268885 0.999638i \(-0.508560\pi\)
−0.0268885 + 0.999638i \(0.508560\pi\)
\(968\) 0 0
\(969\) −159483. 92077.6i −0.169851 0.0980633i
\(970\) 0 0
\(971\) 677500. 391155.i 0.718573 0.414868i −0.0956544 0.995415i \(-0.530494\pi\)
0.814227 + 0.580546i \(0.197161\pi\)
\(972\) 0 0
\(973\) −125776. + 217062.i −0.132853 + 0.229275i
\(974\) 0 0
\(975\) −142652. 247081.i −0.150062 0.259914i
\(976\) 0 0
\(977\) −638714. + 1.10629e6i −0.669141 + 1.15899i 0.309004 + 0.951061i \(0.400004\pi\)
−0.978145 + 0.207925i \(0.933329\pi\)
\(978\) 0 0
\(979\) 218557.i 0.228033i
\(980\) 0 0
\(981\) 216501. 0.224968
\(982\) 0 0
\(983\) 1.36250e6 + 786638.i 1.41003 + 0.814081i 0.995391 0.0959049i \(-0.0305745\pi\)
0.414639 + 0.909986i \(0.363908\pi\)
\(984\) 0 0
\(985\) 912930. 527080.i 0.940947 0.543256i
\(986\) 0 0
\(987\) 44488.2 + 25778.6i 0.0456679 + 0.0264621i
\(988\) 0 0
\(989\) −987661. 1.71068e6i −1.00975 1.74894i
\(990\) 0 0
\(991\) −162173. + 280892.i −0.165132 + 0.286017i −0.936702 0.350127i \(-0.886138\pi\)
0.771570 + 0.636144i \(0.219472\pi\)
\(992\) 0 0
\(993\) 1552.30i 0.00157427i
\(994\) 0 0
\(995\) −2.16099e6 −2.18276
\(996\) 0 0
\(997\) −1.57129e6 907185.i −1.58076 0.912652i −0.994748 0.102350i \(-0.967364\pi\)
−0.586012 0.810303i \(-0.699303\pi\)
\(998\) 0 0
\(999\) 161597. 93298.2i 0.161921 0.0934851i
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 28.5.h.a.17.2 yes 6
3.2 odd 2 252.5.z.f.73.1 6
4.3 odd 2 112.5.s.c.17.2 6
5.2 odd 4 700.5.o.a.549.3 12
5.3 odd 4 700.5.o.a.549.4 12
5.4 even 2 700.5.s.a.101.2 6
7.2 even 3 196.5.h.c.117.2 6
7.3 odd 6 196.5.b.a.97.3 6
7.4 even 3 196.5.b.a.97.4 6
7.5 odd 6 inner 28.5.h.a.5.2 6
7.6 odd 2 196.5.h.c.129.2 6
21.5 even 6 252.5.z.f.145.1 6
28.3 even 6 784.5.c.e.97.4 6
28.11 odd 6 784.5.c.e.97.3 6
28.19 even 6 112.5.s.c.33.2 6
35.12 even 12 700.5.o.a.649.4 12
35.19 odd 6 700.5.s.a.201.2 6
35.33 even 12 700.5.o.a.649.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.5.h.a.5.2 6 7.5 odd 6 inner
28.5.h.a.17.2 yes 6 1.1 even 1 trivial
112.5.s.c.17.2 6 4.3 odd 2
112.5.s.c.33.2 6 28.19 even 6
196.5.b.a.97.3 6 7.3 odd 6
196.5.b.a.97.4 6 7.4 even 3
196.5.h.c.117.2 6 7.2 even 3
196.5.h.c.129.2 6 7.6 odd 2
252.5.z.f.73.1 6 3.2 odd 2
252.5.z.f.145.1 6 21.5 even 6
700.5.o.a.549.3 12 5.2 odd 4
700.5.o.a.549.4 12 5.3 odd 4
700.5.o.a.649.3 12 35.33 even 12
700.5.o.a.649.4 12 35.12 even 12
700.5.s.a.101.2 6 5.4 even 2
700.5.s.a.201.2 6 35.19 odd 6
784.5.c.e.97.3 6 28.11 odd 6
784.5.c.e.97.4 6 28.3 even 6