Properties

Label 160.6.n.a.127.5
Level $160$
Weight $6$
Character 160.127
Analytic conductor $25.661$
Analytic rank $0$
Dimension $14$
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] = [160,6,Mod(63,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.n (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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} + \cdots + 69451154208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.5
Root \(2.04998 - 2.04998i\) of defining polynomial
Character \(\chi\) \(=\) 160.127
Dual form 160.6.n.a.63.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.65790 + 5.65790i) q^{3} +(-54.5514 - 12.2124i) q^{5} +(-23.8754 + 23.8754i) q^{7} -178.976i q^{9} +O(q^{10})\) \(q+(5.65790 + 5.65790i) q^{3} +(-54.5514 - 12.2124i) q^{5} +(-23.8754 + 23.8754i) q^{7} -178.976i q^{9} +218.294i q^{11} +(152.629 - 152.629i) q^{13} +(-239.550 - 377.743i) q^{15} +(318.212 + 318.212i) q^{17} +2458.73 q^{19} -270.169 q^{21} +(512.053 + 512.053i) q^{23} +(2826.71 + 1332.41i) q^{25} +(2387.50 - 2387.50i) q^{27} +5946.31i q^{29} -3065.42i q^{31} +(-1235.09 + 1235.09i) q^{33} +(1594.01 - 1010.86i) q^{35} +(1314.27 + 1314.27i) q^{37} +1727.12 q^{39} +6094.04 q^{41} +(1906.24 + 1906.24i) q^{43} +(-2185.74 + 9763.41i) q^{45} +(8043.61 - 8043.61i) q^{47} +15666.9i q^{49} +3600.83i q^{51} +(-8094.04 + 8094.04i) q^{53} +(2665.90 - 11908.2i) q^{55} +(13911.3 + 13911.3i) q^{57} +41644.1 q^{59} +43094.2 q^{61} +(4273.13 + 4273.13i) q^{63} +(-10190.1 + 6462.16i) q^{65} +(41625.7 - 41625.7i) q^{67} +5794.29i q^{69} -23788.6i q^{71} +(9336.77 - 9336.77i) q^{73} +(8454.62 + 23531.9i) q^{75} +(-5211.86 - 5211.86i) q^{77} -86090.7 q^{79} -16474.8 q^{81} +(75841.2 + 75841.2i) q^{83} +(-13472.8 - 21245.1i) q^{85} +(-33643.6 + 33643.6i) q^{87} +18677.4i q^{89} +7288.16i q^{91} +(17343.8 - 17343.8i) q^{93} +(-134127. - 30027.1i) q^{95} +(-97740.2 - 97740.2i) q^{97} +39069.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 10 q^{3} + 42 q^{5} - 66 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 10 q^{3} + 42 q^{5} - 66 q^{7} - 414 q^{13} - 278 q^{15} + 1222 q^{17} - 5672 q^{19} + 5924 q^{21} - 2902 q^{23} - 4466 q^{25} + 2168 q^{27} - 2444 q^{33} + 2618 q^{35} - 1790 q^{37} + 11076 q^{39} + 11644 q^{41} + 3982 q^{43} + 14704 q^{45} + 1278 q^{47} + 5882 q^{53} - 65608 q^{55} - 14552 q^{57} + 8504 q^{59} + 20564 q^{61} - 19422 q^{63} + 40798 q^{65} - 107926 q^{67} - 16418 q^{73} - 66586 q^{75} - 13348 q^{77} + 146544 q^{79} + 173806 q^{81} + 36398 q^{83} - 66262 q^{85} - 124384 q^{87} - 306620 q^{93} - 173768 q^{95} - 60314 q^{97} + 388628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 5.65790 + 5.65790i 0.362954 + 0.362954i 0.864900 0.501945i \(-0.167382\pi\)
−0.501945 + 0.864900i \(0.667382\pi\)
\(4\) 0 0
\(5\) −54.5514 12.2124i −0.975845 0.218463i
\(6\) 0 0
\(7\) −23.8754 + 23.8754i −0.184164 + 0.184164i −0.793168 0.609003i \(-0.791570\pi\)
0.609003 + 0.793168i \(0.291570\pi\)
\(8\) 0 0
\(9\) 178.976i 0.736528i
\(10\) 0 0
\(11\) 218.294i 0.543951i 0.962304 + 0.271976i \(0.0876770\pi\)
−0.962304 + 0.271976i \(0.912323\pi\)
\(12\) 0 0
\(13\) 152.629 152.629i 0.250483 0.250483i −0.570685 0.821169i \(-0.693322\pi\)
0.821169 + 0.570685i \(0.193322\pi\)
\(14\) 0 0
\(15\) −239.550 377.743i −0.274895 0.433479i
\(16\) 0 0
\(17\) 318.212 + 318.212i 0.267051 + 0.267051i 0.827911 0.560860i \(-0.189529\pi\)
−0.560860 + 0.827911i \(0.689529\pi\)
\(18\) 0 0
\(19\) 2458.73 1.56253 0.781264 0.624201i \(-0.214575\pi\)
0.781264 + 0.624201i \(0.214575\pi\)
\(20\) 0 0
\(21\) −270.169 −0.133687
\(22\) 0 0
\(23\) 512.053 + 512.053i 0.201834 + 0.201834i 0.800786 0.598951i \(-0.204416\pi\)
−0.598951 + 0.800786i \(0.704416\pi\)
\(24\) 0 0
\(25\) 2826.71 + 1332.41i 0.904548 + 0.426371i
\(26\) 0 0
\(27\) 2387.50 2387.50i 0.630281 0.630281i
\(28\) 0 0
\(29\) 5946.31i 1.31296i 0.754342 + 0.656482i \(0.227956\pi\)
−0.754342 + 0.656482i \(0.772044\pi\)
\(30\) 0 0
\(31\) 3065.42i 0.572909i −0.958094 0.286454i \(-0.907523\pi\)
0.958094 0.286454i \(-0.0924767\pi\)
\(32\) 0 0
\(33\) −1235.09 + 1235.09i −0.197430 + 0.197430i
\(34\) 0 0
\(35\) 1594.01 1010.86i 0.219949 0.139483i
\(36\) 0 0
\(37\) 1314.27 + 1314.27i 0.157827 + 0.157827i 0.781603 0.623776i \(-0.214402\pi\)
−0.623776 + 0.781603i \(0.714402\pi\)
\(38\) 0 0
\(39\) 1727.12 0.181828
\(40\) 0 0
\(41\) 6094.04 0.566168 0.283084 0.959095i \(-0.408642\pi\)
0.283084 + 0.959095i \(0.408642\pi\)
\(42\) 0 0
\(43\) 1906.24 + 1906.24i 0.157220 + 0.157220i 0.781334 0.624114i \(-0.214540\pi\)
−0.624114 + 0.781334i \(0.714540\pi\)
\(44\) 0 0
\(45\) −2185.74 + 9763.41i −0.160904 + 0.718737i
\(46\) 0 0
\(47\) 8043.61 8043.61i 0.531137 0.531137i −0.389774 0.920911i \(-0.627447\pi\)
0.920911 + 0.389774i \(0.127447\pi\)
\(48\) 0 0
\(49\) 15666.9i 0.932167i
\(50\) 0 0
\(51\) 3600.83i 0.193855i
\(52\) 0 0
\(53\) −8094.04 + 8094.04i −0.395800 + 0.395800i −0.876749 0.480949i \(-0.840292\pi\)
0.480949 + 0.876749i \(0.340292\pi\)
\(54\) 0 0
\(55\) 2665.90 11908.2i 0.118833 0.530812i
\(56\) 0 0
\(57\) 13911.3 + 13911.3i 0.567126 + 0.567126i
\(58\) 0 0
\(59\) 41644.1 1.55748 0.778742 0.627344i \(-0.215858\pi\)
0.778742 + 0.627344i \(0.215858\pi\)
\(60\) 0 0
\(61\) 43094.2 1.48284 0.741420 0.671041i \(-0.234153\pi\)
0.741420 + 0.671041i \(0.234153\pi\)
\(62\) 0 0
\(63\) 4273.13 + 4273.13i 0.135642 + 0.135642i
\(64\) 0 0
\(65\) −10190.1 + 6462.16i −0.299154 + 0.189712i
\(66\) 0 0
\(67\) 41625.7 41625.7i 1.13285 1.13285i 0.143154 0.989700i \(-0.454276\pi\)
0.989700 0.143154i \(-0.0457244\pi\)
\(68\) 0 0
\(69\) 5794.29i 0.146513i
\(70\) 0 0
\(71\) 23788.6i 0.560045i −0.959993 0.280023i \(-0.909658\pi\)
0.959993 0.280023i \(-0.0903419\pi\)
\(72\) 0 0
\(73\) 9336.77 9336.77i 0.205064 0.205064i −0.597102 0.802166i \(-0.703681\pi\)
0.802166 + 0.597102i \(0.203681\pi\)
\(74\) 0 0
\(75\) 8454.62 + 23531.9i 0.173556 + 0.483063i
\(76\) 0 0
\(77\) −5211.86 5211.86i −0.100177 0.100177i
\(78\) 0 0
\(79\) −86090.7 −1.55199 −0.775994 0.630740i \(-0.782751\pi\)
−0.775994 + 0.630740i \(0.782751\pi\)
\(80\) 0 0
\(81\) −16474.8 −0.279002
\(82\) 0 0
\(83\) 75841.2 + 75841.2i 1.20840 + 1.20840i 0.971546 + 0.236852i \(0.0761156\pi\)
0.236852 + 0.971546i \(0.423884\pi\)
\(84\) 0 0
\(85\) −13472.8 21245.1i −0.202260 0.318942i
\(86\) 0 0
\(87\) −33643.6 + 33643.6i −0.476546 + 0.476546i
\(88\) 0 0
\(89\) 18677.4i 0.249943i 0.992160 + 0.124971i \(0.0398839\pi\)
−0.992160 + 0.124971i \(0.960116\pi\)
\(90\) 0 0
\(91\) 7288.16i 0.0922602i
\(92\) 0 0
\(93\) 17343.8 17343.8i 0.207940 0.207940i
\(94\) 0 0
\(95\) −134127. 30027.1i −1.52479 0.341354i
\(96\) 0 0
\(97\) −97740.2 97740.2i −1.05474 1.05474i −0.998413 0.0563240i \(-0.982062\pi\)
−0.0563240 0.998413i \(-0.517938\pi\)
\(98\) 0 0
\(99\) 39069.5 0.400635
\(100\) 0 0
\(101\) −14938.3 −0.145713 −0.0728565 0.997342i \(-0.523211\pi\)
−0.0728565 + 0.997342i \(0.523211\pi\)
\(102\) 0 0
\(103\) −85212.1 85212.1i −0.791422 0.791422i 0.190303 0.981725i \(-0.439053\pi\)
−0.981725 + 0.190303i \(0.939053\pi\)
\(104\) 0 0
\(105\) 14738.1 + 3299.42i 0.130457 + 0.0292055i
\(106\) 0 0
\(107\) −78160.1 + 78160.1i −0.659972 + 0.659972i −0.955373 0.295401i \(-0.904547\pi\)
0.295401 + 0.955373i \(0.404547\pi\)
\(108\) 0 0
\(109\) 36099.2i 0.291026i −0.989356 0.145513i \(-0.953517\pi\)
0.989356 0.145513i \(-0.0464832\pi\)
\(110\) 0 0
\(111\) 14872.0i 0.114568i
\(112\) 0 0
\(113\) −10180.6 + 10180.6i −0.0750028 + 0.0750028i −0.743613 0.668610i \(-0.766889\pi\)
0.668610 + 0.743613i \(0.266889\pi\)
\(114\) 0 0
\(115\) −21679.8 34186.6i −0.152866 0.241052i
\(116\) 0 0
\(117\) −27317.0 27317.0i −0.184488 0.184488i
\(118\) 0 0
\(119\) −15194.9 −0.0983627
\(120\) 0 0
\(121\) 113399. 0.704117
\(122\) 0 0
\(123\) 34479.5 + 34479.5i 0.205493 + 0.205493i
\(124\) 0 0
\(125\) −137929. 107206.i −0.789553 0.613682i
\(126\) 0 0
\(127\) −88663.6 + 88663.6i −0.487794 + 0.487794i −0.907609 0.419816i \(-0.862095\pi\)
0.419816 + 0.907609i \(0.362095\pi\)
\(128\) 0 0
\(129\) 21570.7i 0.114127i
\(130\) 0 0
\(131\) 4797.45i 0.0244249i 0.999925 + 0.0122124i \(0.00388744\pi\)
−0.999925 + 0.0122124i \(0.996113\pi\)
\(132\) 0 0
\(133\) −58703.3 + 58703.3i −0.287762 + 0.287762i
\(134\) 0 0
\(135\) −159399. + 101084.i −0.752749 + 0.477364i
\(136\) 0 0
\(137\) −149951. 149951.i −0.682569 0.682569i 0.278009 0.960578i \(-0.410325\pi\)
−0.960578 + 0.278009i \(0.910325\pi\)
\(138\) 0 0
\(139\) 22452.1 0.0985642 0.0492821 0.998785i \(-0.484307\pi\)
0.0492821 + 0.998785i \(0.484307\pi\)
\(140\) 0 0
\(141\) 91019.9 0.385557
\(142\) 0 0
\(143\) 33318.0 + 33318.0i 0.136251 + 0.136251i
\(144\) 0 0
\(145\) 72618.9 324380.i 0.286833 1.28125i
\(146\) 0 0
\(147\) −88641.9 + 88641.9i −0.338334 + 0.338334i
\(148\) 0 0
\(149\) 107764.i 0.397656i −0.980034 0.198828i \(-0.936286\pi\)
0.980034 0.198828i \(-0.0637135\pi\)
\(150\) 0 0
\(151\) 140313.i 0.500789i −0.968144 0.250394i \(-0.919440\pi\)
0.968144 0.250394i \(-0.0805603\pi\)
\(152\) 0 0
\(153\) 56952.5 56952.5i 0.196691 0.196691i
\(154\) 0 0
\(155\) −37436.2 + 167223.i −0.125159 + 0.559070i
\(156\) 0 0
\(157\) −151141. 151141.i −0.489366 0.489366i 0.418740 0.908106i \(-0.362472\pi\)
−0.908106 + 0.418740i \(0.862472\pi\)
\(158\) 0 0
\(159\) −91590.5 −0.287315
\(160\) 0 0
\(161\) −24451.0 −0.0743415
\(162\) 0 0
\(163\) 126439. + 126439.i 0.372746 + 0.372746i 0.868476 0.495730i \(-0.165100\pi\)
−0.495730 + 0.868476i \(0.665100\pi\)
\(164\) 0 0
\(165\) 82459.1 52292.3i 0.235792 0.149530i
\(166\) 0 0
\(167\) 125948. 125948.i 0.349463 0.349463i −0.510447 0.859909i \(-0.670520\pi\)
0.859909 + 0.510447i \(0.170520\pi\)
\(168\) 0 0
\(169\) 324702.i 0.874516i
\(170\) 0 0
\(171\) 440055.i 1.15085i
\(172\) 0 0
\(173\) −399318. + 399318.i −1.01439 + 1.01439i −0.0144914 + 0.999895i \(0.504613\pi\)
−0.999895 + 0.0144914i \(0.995387\pi\)
\(174\) 0 0
\(175\) −99300.8 + 35677.1i −0.245108 + 0.0880632i
\(176\) 0 0
\(177\) 235618. + 235618.i 0.565296 + 0.565296i
\(178\) 0 0
\(179\) 457843. 1.06803 0.534016 0.845474i \(-0.320682\pi\)
0.534016 + 0.845474i \(0.320682\pi\)
\(180\) 0 0
\(181\) 625293. 1.41869 0.709344 0.704863i \(-0.248991\pi\)
0.709344 + 0.704863i \(0.248991\pi\)
\(182\) 0 0
\(183\) 243823. + 243823.i 0.538204 + 0.538204i
\(184\) 0 0
\(185\) −55644.8 87745.7i −0.119535 0.188494i
\(186\) 0 0
\(187\) −69463.9 + 69463.9i −0.145263 + 0.145263i
\(188\) 0 0
\(189\) 114005.i 0.232151i
\(190\) 0 0
\(191\) 426964.i 0.846853i −0.905930 0.423426i \(-0.860827\pi\)
0.905930 0.423426i \(-0.139173\pi\)
\(192\) 0 0
\(193\) −519101. + 519101.i −1.00313 + 1.00313i −0.00313800 + 0.999995i \(0.500999\pi\)
−0.999995 + 0.00313800i \(0.999001\pi\)
\(194\) 0 0
\(195\) −94216.8 21092.3i −0.177436 0.0397226i
\(196\) 0 0
\(197\) 645637. + 645637.i 1.18529 + 1.18529i 0.978356 + 0.206930i \(0.0663472\pi\)
0.206930 + 0.978356i \(0.433653\pi\)
\(198\) 0 0
\(199\) 824647. 1.47617 0.738083 0.674710i \(-0.235731\pi\)
0.738083 + 0.674710i \(0.235731\pi\)
\(200\) 0 0
\(201\) 471028. 0.822349
\(202\) 0 0
\(203\) −141971. 141971.i −0.241801 0.241801i
\(204\) 0 0
\(205\) −332438. 74423.0i −0.552492 0.123687i
\(206\) 0 0
\(207\) 91645.4 91645.4i 0.148657 0.148657i
\(208\) 0 0
\(209\) 536727.i 0.849939i
\(210\) 0 0
\(211\) 1.27068e6i 1.96485i −0.186670 0.982423i \(-0.559769\pi\)
0.186670 0.982423i \(-0.440231\pi\)
\(212\) 0 0
\(213\) 134594. 134594.i 0.203271 0.203271i
\(214\) 0 0
\(215\) −80708.4 127268.i −0.119076 0.187769i
\(216\) 0 0
\(217\) 73188.1 + 73188.1i 0.105509 + 0.105509i
\(218\) 0 0
\(219\) 105653. 0.148858
\(220\) 0 0
\(221\) 97136.9 0.133784
\(222\) 0 0
\(223\) 946335. + 946335.i 1.27433 + 1.27433i 0.943792 + 0.330540i \(0.107231\pi\)
0.330540 + 0.943792i \(0.392769\pi\)
\(224\) 0 0
\(225\) 238470. 505915.i 0.314034 0.666225i
\(226\) 0 0
\(227\) −751480. + 751480.i −0.967950 + 0.967950i −0.999502 0.0315523i \(-0.989955\pi\)
0.0315523 + 0.999502i \(0.489955\pi\)
\(228\) 0 0
\(229\) 792900.i 0.999147i 0.866271 + 0.499574i \(0.166510\pi\)
−0.866271 + 0.499574i \(0.833490\pi\)
\(230\) 0 0
\(231\) 58976.4i 0.0727190i
\(232\) 0 0
\(233\) −310134. + 310134.i −0.374248 + 0.374248i −0.869022 0.494774i \(-0.835251\pi\)
0.494774 + 0.869022i \(0.335251\pi\)
\(234\) 0 0
\(235\) −537022. + 340558.i −0.634341 + 0.402274i
\(236\) 0 0
\(237\) −487092. 487092.i −0.563301 0.563301i
\(238\) 0 0
\(239\) 1.36178e6 1.54210 0.771048 0.636777i \(-0.219733\pi\)
0.771048 + 0.636777i \(0.219733\pi\)
\(240\) 0 0
\(241\) 77351.9 0.0857883 0.0428942 0.999080i \(-0.486342\pi\)
0.0428942 + 0.999080i \(0.486342\pi\)
\(242\) 0 0
\(243\) −673375. 673375.i −0.731546 0.731546i
\(244\) 0 0
\(245\) 191331. 854653.i 0.203644 0.909651i
\(246\) 0 0
\(247\) 375274. 375274.i 0.391387 0.391387i
\(248\) 0 0
\(249\) 858204.i 0.877187i
\(250\) 0 0
\(251\) 312063.i 0.312649i −0.987706 0.156325i \(-0.950035\pi\)
0.987706 0.156325i \(-0.0499646\pi\)
\(252\) 0 0
\(253\) −111778. + 111778.i −0.109788 + 0.109788i
\(254\) 0 0
\(255\) 43974.9 196430.i 0.0423501 0.189173i
\(256\) 0 0
\(257\) −981036. 981036.i −0.926514 0.926514i 0.0709649 0.997479i \(-0.477392\pi\)
−0.997479 + 0.0709649i \(0.977392\pi\)
\(258\) 0 0
\(259\) −62757.5 −0.0581321
\(260\) 0 0
\(261\) 1.06425e6 0.967035
\(262\) 0 0
\(263\) 1.26719e6 + 1.26719e6i 1.12968 + 1.12968i 0.990230 + 0.139446i \(0.0445321\pi\)
0.139446 + 0.990230i \(0.455468\pi\)
\(264\) 0 0
\(265\) 540389. 342693.i 0.472707 0.299772i
\(266\) 0 0
\(267\) −105675. + 105675.i −0.0907178 + 0.0907178i
\(268\) 0 0
\(269\) 396279.i 0.333903i 0.985965 + 0.166951i \(0.0533923\pi\)
−0.985965 + 0.166951i \(0.946608\pi\)
\(270\) 0 0
\(271\) 490416.i 0.405640i −0.979216 0.202820i \(-0.934989\pi\)
0.979216 0.202820i \(-0.0650107\pi\)
\(272\) 0 0
\(273\) −41235.7 + 41235.7i −0.0334863 + 0.0334863i
\(274\) 0 0
\(275\) −290857. + 617055.i −0.231925 + 0.492030i
\(276\) 0 0
\(277\) 130834. + 130834.i 0.102452 + 0.102452i 0.756475 0.654023i \(-0.226920\pi\)
−0.654023 + 0.756475i \(0.726920\pi\)
\(278\) 0 0
\(279\) −548637. −0.421963
\(280\) 0 0
\(281\) 1.58558e6 1.19790 0.598952 0.800785i \(-0.295584\pi\)
0.598952 + 0.800785i \(0.295584\pi\)
\(282\) 0 0
\(283\) −269779. 269779.i −0.200236 0.200236i 0.599865 0.800101i \(-0.295221\pi\)
−0.800101 + 0.599865i \(0.795221\pi\)
\(284\) 0 0
\(285\) −588989. 928770.i −0.429532 0.677323i
\(286\) 0 0
\(287\) −145498. + 145498.i −0.104268 + 0.104268i
\(288\) 0 0
\(289\) 1.21734e6i 0.857367i
\(290\) 0 0
\(291\) 1.10601e6i 0.765643i
\(292\) 0 0
\(293\) −1.67347e6 + 1.67347e6i −1.13880 + 1.13880i −0.150135 + 0.988665i \(0.547971\pi\)
−0.988665 + 0.150135i \(0.952029\pi\)
\(294\) 0 0
\(295\) −2.27175e6 508576.i −1.51986 0.340252i
\(296\) 0 0
\(297\) 521177. + 521177.i 0.342842 + 0.342842i
\(298\) 0 0
\(299\) 156308. 0.101112
\(300\) 0 0
\(301\) −91024.7 −0.0579086
\(302\) 0 0
\(303\) −84519.5 84519.5i −0.0528872 0.0528872i
\(304\) 0 0
\(305\) −2.35085e6 526285.i −1.44702 0.323945i
\(306\) 0 0
\(307\) −1.87184e6 + 1.87184e6i −1.13350 + 1.13350i −0.143914 + 0.989590i \(0.545969\pi\)
−0.989590 + 0.143914i \(0.954031\pi\)
\(308\) 0 0
\(309\) 964243.i 0.574500i
\(310\) 0 0
\(311\) 2.01273e6i 1.18001i 0.807400 + 0.590005i \(0.200874\pi\)
−0.807400 + 0.590005i \(0.799126\pi\)
\(312\) 0 0
\(313\) 1.09008e6 1.09008e6i 0.628921 0.628921i −0.318876 0.947797i \(-0.603305\pi\)
0.947797 + 0.318876i \(0.103305\pi\)
\(314\) 0 0
\(315\) −180920. 285291.i −0.102733 0.161999i
\(316\) 0 0
\(317\) −2.18740e6 2.18740e6i −1.22259 1.22259i −0.966709 0.255877i \(-0.917636\pi\)
−0.255877 0.966709i \(-0.582364\pi\)
\(318\) 0 0
\(319\) −1.29804e6 −0.714188
\(320\) 0 0
\(321\) −884444. −0.479080
\(322\) 0 0
\(323\) 782400. + 782400.i 0.417275 + 0.417275i
\(324\) 0 0
\(325\) 634803. 228074.i 0.333373 0.119775i
\(326\) 0 0
\(327\) 204246. 204246.i 0.105629 0.105629i
\(328\) 0 0
\(329\) 384089.i 0.195633i
\(330\) 0 0
\(331\) 1.45657e6i 0.730738i −0.930863 0.365369i \(-0.880943\pi\)
0.930863 0.365369i \(-0.119057\pi\)
\(332\) 0 0
\(333\) 235223. 235223.i 0.116244 0.116244i
\(334\) 0 0
\(335\) −2.77909e6 + 1.76239e6i −1.35298 + 0.858004i
\(336\) 0 0
\(337\) −1.63889e6 1.63889e6i −0.786096 0.786096i 0.194756 0.980852i \(-0.437609\pi\)
−0.980852 + 0.194756i \(0.937609\pi\)
\(338\) 0 0
\(339\) −115202. −0.0544452
\(340\) 0 0
\(341\) 669162. 0.311634
\(342\) 0 0
\(343\) −775328. 775328.i −0.355836 0.355836i
\(344\) 0 0
\(345\) 70762.3 316087.i 0.0320077 0.142974i
\(346\) 0 0
\(347\) −1.26972e6 + 1.26972e6i −0.566087 + 0.566087i −0.931030 0.364943i \(-0.881088\pi\)
0.364943 + 0.931030i \(0.381088\pi\)
\(348\) 0 0
\(349\) 26961.5i 0.0118490i 0.999982 + 0.00592449i \(0.00188584\pi\)
−0.999982 + 0.00592449i \(0.998114\pi\)
\(350\) 0 0
\(351\) 728804.i 0.315750i
\(352\) 0 0
\(353\) −1.76514e6 + 1.76514e6i −0.753949 + 0.753949i −0.975214 0.221265i \(-0.928981\pi\)
0.221265 + 0.975214i \(0.428981\pi\)
\(354\) 0 0
\(355\) −290516. + 1.29770e6i −0.122349 + 0.546517i
\(356\) 0 0
\(357\) −85971.3 85971.3i −0.0357012 0.0357012i
\(358\) 0 0
\(359\) −1.46738e6 −0.600905 −0.300453 0.953797i \(-0.597138\pi\)
−0.300453 + 0.953797i \(0.597138\pi\)
\(360\) 0 0
\(361\) 3.56928e6 1.44149
\(362\) 0 0
\(363\) 641599. + 641599.i 0.255562 + 0.255562i
\(364\) 0 0
\(365\) −623359. + 395310.i −0.244910 + 0.155312i
\(366\) 0 0
\(367\) 3.11806e6 3.11806e6i 1.20842 1.20842i 0.236886 0.971537i \(-0.423873\pi\)
0.971537 0.236886i \(-0.0761268\pi\)
\(368\) 0 0
\(369\) 1.09069e6i 0.416999i
\(370\) 0 0
\(371\) 386497.i 0.145785i
\(372\) 0 0
\(373\) 2.88408e6 2.88408e6i 1.07333 1.07333i 0.0762441 0.997089i \(-0.475707\pi\)
0.997089 0.0762441i \(-0.0242928\pi\)
\(374\) 0 0
\(375\) −173830. 1.38695e6i −0.0638331 0.509311i
\(376\) 0 0
\(377\) 907580. + 907580.i 0.328875 + 0.328875i
\(378\) 0 0
\(379\) 2.02320e6 0.723505 0.361753 0.932274i \(-0.382179\pi\)
0.361753 + 0.932274i \(0.382179\pi\)
\(380\) 0 0
\(381\) −1.00330e6 −0.354094
\(382\) 0 0
\(383\) 2.69527e6 + 2.69527e6i 0.938871 + 0.938871i 0.998236 0.0593656i \(-0.0189078\pi\)
−0.0593656 + 0.998236i \(0.518908\pi\)
\(384\) 0 0
\(385\) 220665. + 347964.i 0.0758720 + 0.119642i
\(386\) 0 0
\(387\) 341172. 341172.i 0.115797 0.115797i
\(388\) 0 0
\(389\) 4.00903e6i 1.34328i 0.740880 + 0.671638i \(0.234409\pi\)
−0.740880 + 0.671638i \(0.765591\pi\)
\(390\) 0 0
\(391\) 325883.i 0.107800i
\(392\) 0 0
\(393\) −27143.5 + 27143.5i −0.00886512 + 0.00886512i
\(394\) 0 0
\(395\) 4.69637e6 + 1.05138e6i 1.51450 + 0.339051i
\(396\) 0 0
\(397\) 107569. + 107569.i 0.0342541 + 0.0342541i 0.724026 0.689772i \(-0.242289\pi\)
−0.689772 + 0.724026i \(0.742289\pi\)
\(398\) 0 0
\(399\) −664275. −0.208889
\(400\) 0 0
\(401\) −3.65913e6 −1.13636 −0.568182 0.822903i \(-0.692353\pi\)
−0.568182 + 0.822903i \(0.692353\pi\)
\(402\) 0 0
\(403\) −467872. 467872.i −0.143504 0.143504i
\(404\) 0 0
\(405\) 898722. + 201197.i 0.272262 + 0.0609514i
\(406\) 0 0
\(407\) −286897. + 286897.i −0.0858500 + 0.0858500i
\(408\) 0 0
\(409\) 297521.i 0.0879445i 0.999033 + 0.0439723i \(0.0140013\pi\)
−0.999033 + 0.0439723i \(0.985999\pi\)
\(410\) 0 0
\(411\) 1.69681e6i 0.495483i
\(412\) 0 0
\(413\) −994270. + 994270.i −0.286833 + 0.286833i
\(414\) 0 0
\(415\) −3.21104e6 5.06345e6i −0.915220 1.44320i
\(416\) 0 0
\(417\) 127032. + 127032.i 0.0357743 + 0.0357743i
\(418\) 0 0
\(419\) 6.52109e6 1.81462 0.907309 0.420464i \(-0.138133\pi\)
0.907309 + 0.420464i \(0.138133\pi\)
\(420\) 0 0
\(421\) 4.57043e6 1.25676 0.628379 0.777907i \(-0.283719\pi\)
0.628379 + 0.777907i \(0.283719\pi\)
\(422\) 0 0
\(423\) −1.43962e6 1.43962e6i −0.391197 0.391197i
\(424\) 0 0
\(425\) 475506. + 1.32348e6i 0.127698 + 0.355424i
\(426\) 0 0
\(427\) −1.02889e6 + 1.02889e6i −0.273087 + 0.273087i
\(428\) 0 0
\(429\) 377020.i 0.0989056i
\(430\) 0 0
\(431\) 4.64706e6i 1.20499i 0.798121 + 0.602497i \(0.205827\pi\)
−0.798121 + 0.602497i \(0.794173\pi\)
\(432\) 0 0
\(433\) −1.28765e6 + 1.28765e6i −0.330049 + 0.330049i −0.852605 0.522556i \(-0.824979\pi\)
0.522556 + 0.852605i \(0.324979\pi\)
\(434\) 0 0
\(435\) 2.24618e6 1.42444e6i 0.569143 0.360928i
\(436\) 0 0
\(437\) 1.25900e6 + 1.25900e6i 0.315372 + 0.315372i
\(438\) 0 0
\(439\) −2.49221e6 −0.617196 −0.308598 0.951193i \(-0.599860\pi\)
−0.308598 + 0.951193i \(0.599860\pi\)
\(440\) 0 0
\(441\) 2.80401e6 0.686567
\(442\) 0 0
\(443\) 108531. + 108531.i 0.0262751 + 0.0262751i 0.720122 0.693847i \(-0.244086\pi\)
−0.693847 + 0.720122i \(0.744086\pi\)
\(444\) 0 0
\(445\) 228096. 1.01888e6i 0.0546031 0.243905i
\(446\) 0 0
\(447\) 609717. 609717.i 0.144331 0.144331i
\(448\) 0 0
\(449\) 1.27740e6i 0.299029i 0.988760 + 0.149514i \(0.0477710\pi\)
−0.988760 + 0.149514i \(0.952229\pi\)
\(450\) 0 0
\(451\) 1.33029e6i 0.307968i
\(452\) 0 0
\(453\) 793875. 793875.i 0.181764 0.181764i
\(454\) 0 0
\(455\) 89006.2 397580.i 0.0201554 0.0900317i
\(456\) 0 0
\(457\) 4.12352e6 + 4.12352e6i 0.923586 + 0.923586i 0.997281 0.0736949i \(-0.0234791\pi\)
−0.0736949 + 0.997281i \(0.523479\pi\)
\(458\) 0 0
\(459\) 1.51946e6 0.336635
\(460\) 0 0
\(461\) −8.86506e6 −1.94280 −0.971402 0.237439i \(-0.923692\pi\)
−0.971402 + 0.237439i \(0.923692\pi\)
\(462\) 0 0
\(463\) −3.00796e6 3.00796e6i −0.652108 0.652108i 0.301392 0.953500i \(-0.402549\pi\)
−0.953500 + 0.301392i \(0.902549\pi\)
\(464\) 0 0
\(465\) −1.15794e6 + 734320.i −0.248344 + 0.157490i
\(466\) 0 0
\(467\) 2.19836e6 2.19836e6i 0.466452 0.466452i −0.434311 0.900763i \(-0.643008\pi\)
0.900763 + 0.434311i \(0.143008\pi\)
\(468\) 0 0
\(469\) 1.98766e6i 0.417263i
\(470\) 0 0
\(471\) 1.71028e6i 0.355235i
\(472\) 0 0
\(473\) −416122. + 416122.i −0.0855199 + 0.0855199i
\(474\) 0 0
\(475\) 6.95014e6 + 3.27604e6i 1.41338 + 0.666217i
\(476\) 0 0
\(477\) 1.44864e6 + 1.44864e6i 0.291518 + 0.291518i
\(478\) 0 0
\(479\) −3.12528e6 −0.622372 −0.311186 0.950349i \(-0.600726\pi\)
−0.311186 + 0.950349i \(0.600726\pi\)
\(480\) 0 0
\(481\) 401191. 0.0790659
\(482\) 0 0
\(483\) −138341. 138341.i −0.0269826 0.0269826i
\(484\) 0 0
\(485\) 4.13822e6 + 6.52551e6i 0.798839 + 1.25968i
\(486\) 0 0
\(487\) 5.74900e6 5.74900e6i 1.09842 1.09842i 0.103828 0.994595i \(-0.466891\pi\)
0.994595 0.103828i \(-0.0331091\pi\)
\(488\) 0 0
\(489\) 1.43076e6i 0.270580i
\(490\) 0 0
\(491\) 1.06215e6i 0.198830i −0.995046 0.0994149i \(-0.968303\pi\)
0.995046 0.0994149i \(-0.0316971\pi\)
\(492\) 0 0
\(493\) −1.89219e6 + 1.89219e6i −0.350629 + 0.350629i
\(494\) 0 0
\(495\) −2.13129e6 477133.i −0.390958 0.0875238i
\(496\) 0 0
\(497\) 567963. + 567963.i 0.103140 + 0.103140i
\(498\) 0 0
\(499\) 1.96563e6 0.353386 0.176693 0.984266i \(-0.443460\pi\)
0.176693 + 0.984266i \(0.443460\pi\)
\(500\) 0 0
\(501\) 1.42520e6 0.253678
\(502\) 0 0
\(503\) 4.00724e6 + 4.00724e6i 0.706196 + 0.706196i 0.965733 0.259537i \(-0.0835700\pi\)
−0.259537 + 0.965733i \(0.583570\pi\)
\(504\) 0 0
\(505\) 814906. + 182433.i 0.142193 + 0.0318328i
\(506\) 0 0
\(507\) −1.83713e6 + 1.83713e6i −0.317410 + 0.317410i
\(508\) 0 0
\(509\) 4.63858e6i 0.793580i −0.917909 0.396790i \(-0.870124\pi\)
0.917909 0.396790i \(-0.129876\pi\)
\(510\) 0 0
\(511\) 445839.i 0.0755310i
\(512\) 0 0
\(513\) 5.87023e6 5.87023e6i 0.984831 0.984831i
\(514\) 0 0
\(515\) 3.60779e6 + 5.68909e6i 0.599409 + 0.945201i
\(516\) 0 0
\(517\) 1.75587e6 + 1.75587e6i 0.288913 + 0.288913i
\(518\) 0 0
\(519\) −4.51860e6 −0.736352
\(520\) 0 0
\(521\) −6.43145e6 −1.03804 −0.519021 0.854762i \(-0.673703\pi\)
−0.519021 + 0.854762i \(0.673703\pi\)
\(522\) 0 0
\(523\) −7.62526e6 7.62526e6i −1.21899 1.21899i −0.967986 0.251006i \(-0.919239\pi\)
−0.251006 0.967986i \(-0.580761\pi\)
\(524\) 0 0
\(525\) −763691. 359976.i −0.120926 0.0570001i
\(526\) 0 0
\(527\) 975454. 975454.i 0.152996 0.152996i
\(528\) 0 0
\(529\) 5.91195e6i 0.918526i
\(530\) 0 0
\(531\) 7.45331e6i 1.14713i
\(532\) 0 0
\(533\) 930127. 930127.i 0.141816 0.141816i
\(534\) 0 0
\(535\) 5.21827e6 3.30922e6i 0.788210 0.499852i
\(536\) 0 0
\(537\) 2.59043e6 + 2.59043e6i 0.387647 + 0.387647i
\(538\) 0 0
\(539\) −3.42000e6 −0.507053
\(540\) 0 0
\(541\) −7.69592e6 −1.13049 −0.565246 0.824923i \(-0.691219\pi\)
−0.565246 + 0.824923i \(0.691219\pi\)
\(542\) 0 0
\(543\) 3.53784e6 + 3.53784e6i 0.514919 + 0.514919i
\(544\) 0 0
\(545\) −440859. + 1.96926e6i −0.0635782 + 0.283996i
\(546\) 0 0
\(547\) −8.31090e6 + 8.31090e6i −1.18763 + 1.18763i −0.209904 + 0.977722i \(0.567315\pi\)
−0.977722 + 0.209904i \(0.932685\pi\)
\(548\) 0 0
\(549\) 7.71285e6i 1.09215i
\(550\) 0 0
\(551\) 1.46204e7i 2.05154i
\(552\) 0 0
\(553\) 2.05545e6 2.05545e6i 0.285821 0.285821i
\(554\) 0 0
\(555\) 181623. 811289.i 0.0250288 0.111800i
\(556\) 0 0
\(557\) −2.90207e6 2.90207e6i −0.396341 0.396341i 0.480599 0.876940i \(-0.340419\pi\)
−0.876940 + 0.480599i \(0.840419\pi\)
\(558\) 0 0
\(559\) 581896. 0.0787619
\(560\) 0 0
\(561\) −786039. −0.105448
\(562\) 0 0
\(563\) −7.08912e6 7.08912e6i −0.942586 0.942586i 0.0558525 0.998439i \(-0.482212\pi\)
−0.998439 + 0.0558525i \(0.982212\pi\)
\(564\) 0 0
\(565\) 679697. 431037.i 0.0895765 0.0568059i
\(566\) 0 0
\(567\) 393342. 393342.i 0.0513822 0.0513822i
\(568\) 0 0
\(569\) 1.02793e7i 1.33102i −0.746389 0.665509i \(-0.768214\pi\)
0.746389 0.665509i \(-0.231786\pi\)
\(570\) 0 0
\(571\) 9.92423e6i 1.27382i 0.770940 + 0.636908i \(0.219787\pi\)
−0.770940 + 0.636908i \(0.780213\pi\)
\(572\) 0 0
\(573\) 2.41572e6 2.41572e6i 0.307369 0.307369i
\(574\) 0 0
\(575\) 765162. + 2.12969e6i 0.0965126 + 0.268625i
\(576\) 0 0
\(577\) −7.82091e6 7.82091e6i −0.977953 0.977953i 0.0218096 0.999762i \(-0.493057\pi\)
−0.999762 + 0.0218096i \(0.993057\pi\)
\(578\) 0 0
\(579\) −5.87404e6 −0.728183
\(580\) 0 0
\(581\) −3.62148e6 −0.445088
\(582\) 0 0
\(583\) −1.76688e6 1.76688e6i −0.215296 0.215296i
\(584\) 0 0
\(585\) 1.15657e6 + 1.82379e6i 0.139728 + 0.220335i
\(586\) 0 0
\(587\) −4.94337e6 + 4.94337e6i −0.592144 + 0.592144i −0.938210 0.346066i \(-0.887517\pi\)
0.346066 + 0.938210i \(0.387517\pi\)
\(588\) 0 0
\(589\) 7.53705e6i 0.895185i
\(590\) 0 0
\(591\) 7.30590e6i 0.860409i
\(592\) 0 0
\(593\) −5.40438e6 + 5.40438e6i −0.631116 + 0.631116i −0.948348 0.317232i \(-0.897247\pi\)
0.317232 + 0.948348i \(0.397247\pi\)
\(594\) 0 0
\(595\) 828904. + 185567.i 0.0959868 + 0.0214886i
\(596\) 0 0
\(597\) 4.66577e6 + 4.66577e6i 0.535781 + 0.535781i
\(598\) 0 0
\(599\) −3.02994e6 −0.345039 −0.172519 0.985006i \(-0.555191\pi\)
−0.172519 + 0.985006i \(0.555191\pi\)
\(600\) 0 0
\(601\) −8.33982e6 −0.941825 −0.470913 0.882180i \(-0.656075\pi\)
−0.470913 + 0.882180i \(0.656075\pi\)
\(602\) 0 0
\(603\) −7.45001e6 7.45001e6i −0.834379 0.834379i
\(604\) 0 0
\(605\) −6.18606e6 1.38487e6i −0.687109 0.153823i
\(606\) 0 0
\(607\) 4.99673e6 4.99673e6i 0.550445 0.550445i −0.376124 0.926569i \(-0.622743\pi\)
0.926569 + 0.376124i \(0.122743\pi\)
\(608\) 0 0
\(609\) 1.60651e6i 0.175526i
\(610\) 0 0
\(611\) 2.45538e6i 0.266082i
\(612\) 0 0
\(613\) 3.24993e6 3.24993e6i 0.349320 0.349320i −0.510536 0.859856i \(-0.670553\pi\)
0.859856 + 0.510536i \(0.170553\pi\)
\(614\) 0 0
\(615\) −1.45982e6 2.30198e6i −0.155637 0.245422i
\(616\) 0 0
\(617\) 1.74236e6 + 1.74236e6i 0.184258 + 0.184258i 0.793208 0.608951i \(-0.208409\pi\)
−0.608951 + 0.793208i \(0.708409\pi\)
\(618\) 0 0
\(619\) −1.08664e7 −1.13988 −0.569941 0.821686i \(-0.693034\pi\)
−0.569941 + 0.821686i \(0.693034\pi\)
\(620\) 0 0
\(621\) 2.44505e6 0.254425
\(622\) 0 0
\(623\) −445929. 445929.i −0.0460305 0.0460305i
\(624\) 0 0
\(625\) 6.21499e6 + 7.53268e6i 0.636415 + 0.771347i
\(626\) 0 0
\(627\) −3.03675e6 + 3.03675e6i −0.308489 + 0.308489i
\(628\) 0 0
\(629\) 836434.i 0.0842956i
\(630\) 0 0
\(631\) 6.31541e6i 0.631434i 0.948853 + 0.315717i \(0.102245\pi\)
−0.948853 + 0.315717i \(0.897755\pi\)
\(632\) 0 0
\(633\) 7.18935e6 7.18935e6i 0.713149 0.713149i
\(634\) 0 0
\(635\) 5.91952e6 3.75393e6i 0.582576 0.369447i
\(636\) 0 0
\(637\) 2.39123e6 + 2.39123e6i 0.233492 + 0.233492i
\(638\) 0 0
\(639\) −4.25760e6 −0.412489
\(640\) 0 0
\(641\) −1.01885e7 −0.979408 −0.489704 0.871889i \(-0.662895\pi\)
−0.489704 + 0.871889i \(0.662895\pi\)
\(642\) 0 0
\(643\) −1.11631e6 1.11631e6i −0.106478 0.106478i 0.651861 0.758339i \(-0.273989\pi\)
−0.758339 + 0.651861i \(0.773989\pi\)
\(644\) 0 0
\(645\) 263430. 1.17671e6i 0.0249325 0.111371i
\(646\) 0 0
\(647\) 1.08225e7 1.08225e7i 1.01640 1.01640i 0.0165382 0.999863i \(-0.494735\pi\)
0.999863 0.0165382i \(-0.00526451\pi\)
\(648\) 0 0
\(649\) 9.09066e6i 0.847196i
\(650\) 0 0
\(651\) 828182.i 0.0765902i
\(652\) 0 0
\(653\) 5.19640e6 5.19640e6i 0.476892 0.476892i −0.427244 0.904136i \(-0.640516\pi\)
0.904136 + 0.427244i \(0.140516\pi\)
\(654\) 0 0
\(655\) 58588.5 261708.i 0.00533592 0.0238349i
\(656\) 0 0
\(657\) −1.67106e6 1.67106e6i −0.151035 0.151035i
\(658\) 0 0
\(659\) −5.17149e6 −0.463876 −0.231938 0.972731i \(-0.574507\pi\)
−0.231938 + 0.972731i \(0.574507\pi\)
\(660\) 0 0
\(661\) −8.28147e6 −0.737231 −0.368616 0.929582i \(-0.620168\pi\)
−0.368616 + 0.929582i \(0.620168\pi\)
\(662\) 0 0
\(663\) 549591. + 549591.i 0.0485574 + 0.0485574i
\(664\) 0 0
\(665\) 3.91926e6 2.48544e6i 0.343676 0.217946i
\(666\) 0 0
\(667\) −3.04483e6 + 3.04483e6i −0.265001 + 0.265001i
\(668\) 0 0
\(669\) 1.07085e7i 0.925049i
\(670\) 0 0
\(671\) 9.40721e6i 0.806593i
\(672\) 0 0
\(673\) 3.42754e6 3.42754e6i 0.291706 0.291706i −0.546048 0.837754i \(-0.683868\pi\)
0.837754 + 0.546048i \(0.183868\pi\)
\(674\) 0 0
\(675\) 9.92991e6 3.56765e6i 0.838853 0.301386i
\(676\) 0 0
\(677\) −8.26736e6 8.26736e6i −0.693258 0.693258i 0.269689 0.962947i \(-0.413079\pi\)
−0.962947 + 0.269689i \(0.913079\pi\)
\(678\) 0 0
\(679\) 4.66718e6 0.388490
\(680\) 0 0
\(681\) −8.50360e6 −0.702643
\(682\) 0 0
\(683\) −3.18611e6 3.18611e6i −0.261342 0.261342i 0.564257 0.825599i \(-0.309163\pi\)
−0.825599 + 0.564257i \(0.809163\pi\)
\(684\) 0 0
\(685\) 6.34876e6 + 1.00113e7i 0.516966 + 0.815198i
\(686\) 0 0
\(687\) −4.48615e6 + 4.48615e6i −0.362645 + 0.362645i
\(688\) 0 0
\(689\) 2.47077e6i 0.198283i
\(690\) 0 0
\(691\) 1.80913e6i 0.144137i −0.997400 0.0720685i \(-0.977040\pi\)
0.997400 0.0720685i \(-0.0229600\pi\)
\(692\) 0 0
\(693\) −932799. + 932799.i −0.0737828 + 0.0737828i
\(694\) 0 0
\(695\) −1.22479e6 274194.i −0.0961834 0.0215326i
\(696\) 0 0
\(697\) 1.93920e6 + 1.93920e6i 0.151196 + 0.151196i
\(698\) 0 0
\(699\) −3.50941e6 −0.271670
\(700\) 0 0
\(701\) 8.47023e6 0.651029 0.325514 0.945537i \(-0.394463\pi\)
0.325514 + 0.945537i \(0.394463\pi\)
\(702\) 0 0
\(703\) 3.23144e6 + 3.23144e6i 0.246608 + 0.246608i
\(704\) 0 0
\(705\) −4.96526e6 1.11157e6i −0.376244 0.0842297i
\(706\) 0 0
\(707\) 356658. 356658.i 0.0268351 0.0268351i
\(708\) 0 0
\(709\) 1.88589e7i 1.40896i −0.709722 0.704482i \(-0.751179\pi\)
0.709722 0.704482i \(-0.248821\pi\)
\(710\) 0 0
\(711\) 1.54082e7i 1.14308i
\(712\) 0 0
\(713\) 1.56966e6 1.56966e6i 0.115633 0.115633i
\(714\) 0 0
\(715\) −1.41065e6 2.22444e6i −0.103194 0.162725i
\(716\) 0 0
\(717\) 7.70480e6 + 7.70480e6i 0.559711 + 0.559711i
\(718\) 0 0
\(719\) 1.02992e7 0.742989 0.371495 0.928435i \(-0.378845\pi\)
0.371495 + 0.928435i \(0.378845\pi\)
\(720\) 0 0
\(721\) 4.06895e6 0.291504
\(722\) 0 0
\(723\) 437649. + 437649.i 0.0311373 + 0.0311373i
\(724\) 0 0
\(725\) −7.92293e6 + 1.68085e7i −0.559810 + 1.18764i
\(726\) 0 0
\(727\) 1.33721e7 1.33721e7i 0.938346 0.938346i −0.0598611 0.998207i \(-0.519066\pi\)
0.998207 + 0.0598611i \(0.0190658\pi\)
\(728\) 0 0
\(729\) 3.61641e6i 0.252034i
\(730\) 0 0
\(731\) 1.21318e6i 0.0839715i
\(732\) 0 0
\(733\) 1.34325e6 1.34325e6i 0.0923418 0.0923418i −0.659427 0.751769i \(-0.729201\pi\)
0.751769 + 0.659427i \(0.229201\pi\)
\(734\) 0 0
\(735\) 5.91808e6 3.75301e6i 0.404075 0.256248i
\(736\) 0 0
\(737\) 9.08663e6 + 9.08663e6i 0.616218 + 0.616218i
\(738\) 0 0
\(739\) 3.77236e6 0.254099 0.127049 0.991896i \(-0.459449\pi\)
0.127049 + 0.991896i \(0.459449\pi\)
\(740\) 0 0
\(741\) 4.24653e6 0.284111
\(742\) 0 0
\(743\) 4.63047e6 + 4.63047e6i 0.307718 + 0.307718i 0.844024 0.536306i \(-0.180181\pi\)
−0.536306 + 0.844024i \(0.680181\pi\)
\(744\) 0 0
\(745\) −1.31606e6 + 5.87867e6i −0.0868729 + 0.388051i
\(746\) 0 0
\(747\) 1.35738e7 1.35738e7i 0.890019 0.890019i
\(748\) 0 0
\(749\) 3.73221e6i 0.243087i
\(750\) 0 0
\(751\) 2.08661e7i 1.35002i −0.737807 0.675012i \(-0.764138\pi\)
0.737807 0.675012i \(-0.235862\pi\)
\(752\) 0 0
\(753\) 1.76562e6 1.76562e6i 0.113477 0.113477i
\(754\) 0 0
\(755\) −1.71356e6 + 7.65426e6i −0.109404 + 0.488692i
\(756\) 0 0
\(757\) −964426. 964426.i −0.0611687 0.0611687i 0.675861 0.737029i \(-0.263772\pi\)
−0.737029 + 0.675861i \(0.763772\pi\)
\(758\) 0 0
\(759\) −1.26486e6 −0.0796962
\(760\) 0 0
\(761\) 1.49933e7 0.938505 0.469252 0.883064i \(-0.344523\pi\)
0.469252 + 0.883064i \(0.344523\pi\)
\(762\) 0 0
\(763\) 861883. + 861883.i 0.0535966 + 0.0535966i
\(764\) 0 0
\(765\) −3.80237e6 + 2.41131e6i −0.234909 + 0.148970i
\(766\) 0 0
\(767\) 6.35610e6 6.35610e6i 0.390124 0.390124i
\(768\) 0 0
\(769\) 4.51581e6i 0.275372i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(770\) 0 0
\(771\) 1.11012e7i 0.672565i
\(772\) 0 0
\(773\) −1.29303e7 + 1.29303e7i −0.778320 + 0.778320i −0.979545 0.201225i \(-0.935508\pi\)
0.201225 + 0.979545i \(0.435508\pi\)
\(774\) 0 0
\(775\) 4.08439e6 8.66505e6i 0.244272 0.518223i
\(776\) 0 0
\(777\) −355075. 355075.i −0.0210993 0.0210993i
\(778\) 0 0
\(779\) 1.49836e7 0.884653
\(780\) 0 0
\(781\) 5.19291e6 0.304637
\(782\) 0 0
\(783\) 1.41968e7 + 1.41968e7i 0.827536 + 0.827536i
\(784\) 0 0
\(785\) 6.39917e6 + 1.00908e7i 0.370637 + 0.584454i
\(786\) 0 0
\(787\) −3.11176e6 + 3.11176e6i −0.179089 + 0.179089i −0.790959 0.611869i \(-0.790418\pi\)
0.611869 + 0.790959i \(0.290418\pi\)
\(788\) 0 0
\(789\) 1.43393e7i 0.820042i
\(790\) 0 0
\(791\) 486132.i 0.0276257i
\(792\) 0 0
\(793\) 6.57743e6 6.57743e6i 0.371427 0.371427i
\(794\) 0 0
\(795\) 4.99639e6 + 1.11854e6i 0.280375 + 0.0627675i
\(796\) 0 0
\(797\) 1.47157e7 + 1.47157e7i 0.820607 + 0.820607i 0.986195 0.165588i \(-0.0529521\pi\)
−0.165588 + 0.986195i \(0.552952\pi\)
\(798\) 0 0
\(799\) 5.11915e6 0.283682
\(800\) 0 0
\(801\) 3.34280e6 0.184090
\(802\) 0 0
\(803\) 2.03816e6 + 2.03816e6i 0.111545 + 0.111545i
\(804\) 0 0
\(805\) 1.33383e6 + 298605.i 0.0725458 + 0.0162408i
\(806\) 0 0
\(807\) −2.24211e6 + 2.24211e6i −0.121192 + 0.121192i
\(808\) 0 0
\(809\) 1.92301e7i 1.03303i 0.856280 + 0.516513i \(0.172770\pi\)
−0.856280 + 0.516513i \(0.827230\pi\)
\(810\) 0 0
\(811\) 1.25439e7i 0.669701i 0.942271 + 0.334850i \(0.108686\pi\)
−0.942271 + 0.334850i \(0.891314\pi\)
\(812\) 0 0
\(813\) 2.77472e6 2.77472e6i 0.147229 0.147229i
\(814\) 0 0
\(815\) −5.35331e6 8.44158e6i −0.282312 0.445174i
\(816\) 0 0
\(817\) 4.68695e6 + 4.68695e6i 0.245660 + 0.245660i
\(818\) 0 0
\(819\) 1.30441e6 0.0679523
\(820\) 0 0
\(821\) −5.56071e6 −0.287920 −0.143960 0.989583i \(-0.545984\pi\)
−0.143960 + 0.989583i \(0.545984\pi\)
\(822\) 0 0
\(823\) −1.87493e7 1.87493e7i −0.964909 0.964909i 0.0344961 0.999405i \(-0.489017\pi\)
−0.999405 + 0.0344961i \(0.989017\pi\)
\(824\) 0 0
\(825\) −5.13687e6 + 1.84559e6i −0.262763 + 0.0944063i
\(826\) 0 0
\(827\) 1.46263e7 1.46263e7i 0.743652 0.743652i −0.229627 0.973279i \(-0.573750\pi\)
0.973279 + 0.229627i \(0.0737505\pi\)
\(828\) 0 0
\(829\) 1.77285e7i 0.895954i 0.894045 + 0.447977i \(0.147855\pi\)
−0.894045 + 0.447977i \(0.852145\pi\)
\(830\) 0 0
\(831\) 1.48049e6i 0.0743707i
\(832\) 0 0
\(833\) −4.98541e6 + 4.98541e6i −0.248936 + 0.248936i
\(834\) 0 0
\(835\) −8.40878e6 + 5.33252e6i −0.417366 + 0.264677i
\(836\) 0 0
\(837\) −7.31868e6 7.31868e6i −0.361093 0.361093i
\(838\) 0 0
\(839\) −3.43040e7 −1.68244 −0.841221 0.540692i \(-0.818163\pi\)
−0.841221 + 0.540692i \(0.818163\pi\)
\(840\) 0 0
\(841\) −1.48475e7 −0.723874
\(842\) 0 0
\(843\) 8.97105e6 + 8.97105e6i 0.434785 + 0.434785i
\(844\) 0 0
\(845\) 3.96540e6 1.77129e7i 0.191049 0.853393i
\(846\) 0 0
\(847\) −2.70744e6 + 2.70744e6i −0.129673 + 0.129673i
\(848\) 0 0
\(849\) 3.05277e6i 0.145353i
\(850\) 0 0
\(851\) 1.34595e6i 0.0637097i
\(852\) 0 0
\(853\) −5.28231e6 + 5.28231e6i −0.248572 + 0.248572i −0.820384 0.571813i \(-0.806240\pi\)
0.571813 + 0.820384i \(0.306240\pi\)
\(854\) 0 0
\(855\) −5.37414e6 + 2.40056e7i −0.251417 + 1.12305i
\(856\) 0 0
\(857\) −2.66196e7 2.66196e7i −1.23808 1.23808i −0.960785 0.277296i \(-0.910562\pi\)
−0.277296 0.960785i \(-0.589438\pi\)
\(858\) 0 0
\(859\) 1.67064e7 0.772503 0.386252 0.922393i \(-0.373770\pi\)
0.386252 + 0.922393i \(0.373770\pi\)
\(860\) 0 0
\(861\) −1.64642e6 −0.0756891
\(862\) 0 0
\(863\) −1.80862e7 1.80862e7i −0.826646 0.826646i 0.160406 0.987051i \(-0.448720\pi\)
−0.987051 + 0.160406i \(0.948720\pi\)
\(864\) 0 0
\(865\) 2.66600e7 1.69067e7i 1.21149 0.768279i
\(866\) 0 0
\(867\) 6.88758e6 6.88758e6i 0.311185 0.311185i
\(868\) 0 0
\(869\) 1.87931e7i 0.844206i
\(870\) 0 0
\(871\) 1.27066e7i 0.567522i
\(872\) 0 0
\(873\) −1.74932e7 + 1.74932e7i −0.776843 + 0.776843i
\(874\) 0 0
\(875\) 5.85270e6 733533.i 0.258426 0.0323891i
\(876\) 0 0
\(877\) −4.69228e6 4.69228e6i −0.206008 0.206008i 0.596560 0.802568i \(-0.296534\pi\)
−0.802568 + 0.596560i \(0.796534\pi\)
\(878\) 0 0
\(879\) −1.89366e7 −0.826666
\(880\) 0 0
\(881\) 3.72179e7 1.61552 0.807760 0.589511i \(-0.200680\pi\)
0.807760 + 0.589511i \(0.200680\pi\)
\(882\) 0 0
\(883\) 1.77903e7 + 1.77903e7i 0.767859 + 0.767859i 0.977729 0.209871i \(-0.0673043\pi\)
−0.209871 + 0.977729i \(0.567304\pi\)
\(884\) 0 0
\(885\) −9.97584e6 1.57308e7i −0.428145 0.675137i
\(886\) 0 0
\(887\) −3.08604e7 + 3.08604e7i −1.31702 + 1.31702i −0.400899 + 0.916122i \(0.631302\pi\)
−0.916122 + 0.400899i \(0.868698\pi\)
\(888\) 0 0
\(889\) 4.23376e6i 0.179669i
\(890\) 0 0
\(891\) 3.59634e6i 0.151763i
\(892\) 0 0
\(893\) 1.97771e7 1.97771e7i 0.829916 0.829916i
\(894\) 0 0
\(895\) −2.49760e7 5.59138e6i −1.04223 0.233325i
\(896\) 0 0
\(897\) 884377. + 884377.i 0.0366992 + 0.0366992i
\(898\) 0 0
\(899\) 1.82279e7 0.752208
\(900\) 0 0
\(901\) −5.15125e6 −0.211398
\(902\) 0 0
\(903\) −515009. 515009.i −0.0210182 0.0210182i
\(904\) 0 0
\(905\) −3.41106e7 7.63634e6i −1.38442 0.309930i
\(906\) 0 0
\(907\) −2.23972e7 + 2.23972e7i −0.904016 + 0.904016i −0.995781 0.0917650i \(-0.970749\pi\)
0.0917650 + 0.995781i \(0.470749\pi\)
\(908\) 0 0
\(909\) 2.67360e6i 0.107322i
\(910\) 0 0
\(911\) 1.43417e7i 0.572538i −0.958149 0.286269i \(-0.907585\pi\)
0.958149 0.286269i \(-0.0924151\pi\)
\(912\) 0 0
\(913\) −1.65557e7 + 1.65557e7i −0.657310 + 0.657310i
\(914\) 0 0
\(915\) −1.03232e7 1.62785e7i −0.407626 0.642781i
\(916\) 0 0
\(917\) −114541. 114541.i −0.00449819 0.00449819i
\(918\) 0 0
\(919\) 1.70726e7 0.666823 0.333411 0.942781i \(-0.391800\pi\)
0.333411 + 0.942781i \(0.391800\pi\)
\(920\) 0 0
\(921\) −2.11814e7 −0.822821
\(922\) 0 0
\(923\) −3.63083e6 3.63083e6i −0.140282 0.140282i
\(924\) 0 0
\(925\) 1.96392e6 + 5.46621e6i 0.0754690 + 0.210054i
\(926\) 0 0
\(927\) −1.52509e7 + 1.52509e7i −0.582905 + 0.582905i
\(928\) 0 0
\(929\) 2.41131e7i 0.916672i −0.888779 0.458336i \(-0.848446\pi\)
0.888779 0.458336i \(-0.151554\pi\)
\(930\) 0 0
\(931\) 3.85208e7i 1.45654i
\(932\) 0 0
\(933\) −1.13879e7 + 1.13879e7i −0.428290 + 0.428290i
\(934\) 0 0
\(935\) 4.63767e6 2.94103e6i 0.173489 0.110020i
\(936\) 0 0
\(937\) −1.17497e7 1.17497e7i −0.437198 0.437198i 0.453870 0.891068i \(-0.350043\pi\)
−0.891068 + 0.453870i \(0.850043\pi\)
\(938\) 0 0
\(939\) 1.23351e7 0.456539
\(940\) 0 0
\(941\) 2.10237e7 0.773988 0.386994 0.922082i \(-0.373513\pi\)
0.386994 + 0.922082i \(0.373513\pi\)
\(942\) 0 0
\(943\) 3.12047e6 + 3.12047e6i 0.114272 + 0.114272i
\(944\) 0 0
\(945\) 1.39228e6 6.21914e6i 0.0507162 0.226543i
\(946\) 0 0
\(947\) 6.44103e6 6.44103e6i 0.233389 0.233389i −0.580717 0.814106i \(-0.697228\pi\)
0.814106 + 0.580717i \(0.197228\pi\)
\(948\) 0 0
\(949\) 2.85013e6i 0.102730i
\(950\) 0 0
\(951\) 2.47522e7i 0.887486i
\(952\) 0 0
\(953\) 1.92602e7 1.92602e7i 0.686957 0.686957i −0.274601 0.961558i \(-0.588546\pi\)
0.961558 + 0.274601i \(0.0885458\pi\)
\(954\) 0 0
\(955\) −5.21427e6 + 2.32915e7i −0.185006 + 0.826397i
\(956\) 0 0
\(957\) −7.34421e6 7.34421e6i −0.259218 0.259218i
\(958\) 0 0
\(959\) 7.16027e6 0.251410
\(960\) 0 0
\(961\) 1.92324e7 0.671776
\(962\) 0 0
\(963\) 1.39888e7 + 1.39888e7i 0.486088 + 0.486088i
\(964\) 0 0
\(965\) 3.46572e7 2.19782e7i 1.19805 0.759756i
\(966\) 0 0
\(967\) −2.31321e7 + 2.31321e7i −0.795515 + 0.795515i −0.982385 0.186870i \(-0.940166\pi\)
0.186870 + 0.982385i \(0.440166\pi\)
\(968\) 0 0
\(969\) 8.85348e6i 0.302904i
\(970\) 0 0
\(971\) 3.13091e7i 1.06567i −0.846219 0.532835i \(-0.821127\pi\)
0.846219 0.532835i \(-0.178873\pi\)
\(972\) 0 0
\(973\) −536052. + 536052.i −0.0181520 + 0.0181520i
\(974\) 0 0
\(975\) 4.88207e6 + 2.30123e6i 0.164472 + 0.0775263i
\(976\) 0 0
\(977\) 1.59117e7 + 1.59117e7i 0.533310 + 0.533310i 0.921556 0.388246i \(-0.126919\pi\)
−0.388246 + 0.921556i \(0.626919\pi\)
\(978\) 0 0
\(979\) −4.07715e6 −0.135957
\(980\) 0 0
\(981\) −6.46090e6 −0.214349
\(982\) 0 0
\(983\) −5.17343e6 5.17343e6i −0.170763 0.170763i 0.616551 0.787315i \(-0.288529\pi\)
−0.787315 + 0.616551i \(0.788529\pi\)
\(984\) 0 0
\(985\) −2.73356e7 4.31052e7i −0.897715 1.41560i
\(986\) 0 0
\(987\) −2.17314e6 + 2.17314e6i −0.0710059 + 0.0710059i
\(988\) 0 0
\(989\) 1.95220e6i 0.0634647i
\(990\) 0 0
\(991\) 1.98249e7i 0.641250i −0.947206 0.320625i \(-0.896107\pi\)
0.947206 0.320625i \(-0.103893\pi\)
\(992\) 0 0
\(993\) 8.24113e6 8.24113e6i 0.265225 0.265225i
\(994\) 0 0
\(995\) −4.49856e7 1.00709e7i −1.44051 0.322487i
\(996\) 0 0
\(997\) −3.59461e7 3.59461e7i −1.14529 1.14529i −0.987468 0.157818i \(-0.949554\pi\)
−0.157818 0.987468i \(-0.550446\pi\)
\(998\) 0 0
\(999\) 6.27564e6 0.198950
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 160.6.n.a.127.5 yes 14
4.3 odd 2 160.6.n.b.127.3 yes 14
5.3 odd 4 160.6.n.b.63.3 yes 14
20.3 even 4 inner 160.6.n.a.63.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.n.a.63.5 14 20.3 even 4 inner
160.6.n.a.127.5 yes 14 1.1 even 1 trivial
160.6.n.b.63.3 yes 14 5.3 odd 4
160.6.n.b.127.3 yes 14 4.3 odd 2