Properties

Label 1040.2.da.e.881.2
Level $1040$
Weight $2$
Character 1040.881
Analytic conductor $8.304$
Analytic rank $0$
Dimension $12$
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] = [1040,2,Mod(641,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.641");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.da (of order \(6\), 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 881.2
Root \(0.759479 + 1.19298i\) of defining polynomial
Character \(\chi\) \(=\) 1040.881
Dual form 1040.2.da.e.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.653409 - 1.13174i) q^{3} -1.00000i q^{5} +(-2.51573 - 1.45245i) q^{7} +(0.646115 - 1.11910i) q^{9} +O(q^{10})\) \(q+(-0.653409 - 1.13174i) q^{3} -1.00000i q^{5} +(-2.51573 - 1.45245i) q^{7} +(0.646115 - 1.11910i) q^{9} +(-0.174603 + 0.100807i) q^{11} +(2.15455 - 2.89101i) q^{13} +(-1.13174 + 0.653409i) q^{15} +(-1.53093 + 2.65165i) q^{17} +(0.0122700 + 0.00708410i) q^{19} +3.79619i q^{21} +(-2.36476 - 4.09588i) q^{23} -1.00000 q^{25} -5.60916 q^{27} +(1.05550 + 1.82818i) q^{29} +1.79645i q^{31} +(0.228175 + 0.131737i) q^{33} +(-1.45245 + 2.51573i) q^{35} +(-0.503609 + 0.290759i) q^{37} +(-4.67966 - 0.549370i) q^{39} +(-6.87817 + 3.97111i) q^{41} +(-4.63035 + 8.02001i) q^{43} +(-1.11910 - 0.646115i) q^{45} -9.37618i q^{47} +(0.719250 + 1.24578i) q^{49} +4.00129 q^{51} -4.70777 q^{53} +(0.100807 + 0.174603i) q^{55} -0.0185152i q^{57} +(-5.03504 - 2.90698i) q^{59} +(-1.95119 + 3.37956i) q^{61} +(-3.25089 + 1.87690i) q^{63} +(-2.89101 - 2.15455i) q^{65} +(1.80553 - 1.04242i) q^{67} +(-3.09031 + 5.35257i) q^{69} +(5.65201 + 3.26319i) q^{71} -12.2027i q^{73} +(0.653409 + 1.13174i) q^{75} +0.585672 q^{77} +10.8293 q^{79} +(1.72673 + 2.99078i) q^{81} +15.2892i q^{83} +(2.65165 + 1.53093i) q^{85} +(1.37935 - 2.38910i) q^{87} +(-2.67609 + 1.54504i) q^{89} +(-9.61931 + 4.14360i) q^{91} +(2.03310 - 1.17381i) q^{93} +(0.00708410 - 0.0122700i) q^{95} +(-15.9695 - 9.22000i) q^{97} +0.260532i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{7} + 2 q^{9} - 2 q^{13} - 8 q^{17} + 24 q^{19} + 2 q^{23} - 12 q^{25} + 12 q^{27} + 12 q^{29} + 4 q^{35} + 24 q^{37} - 28 q^{39} + 24 q^{41} + 18 q^{43} - 12 q^{45} + 24 q^{49} - 68 q^{53} - 2 q^{55} + 48 q^{59} + 18 q^{61} + 36 q^{63} + 16 q^{65} - 18 q^{67} - 8 q^{69} + 64 q^{77} + 12 q^{79} + 14 q^{81} - 18 q^{87} + 30 q^{89} - 76 q^{91} - 12 q^{93} + 10 q^{95} - 84 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −0.653409 1.13174i −0.377246 0.653409i 0.613415 0.789761i \(-0.289795\pi\)
−0.990660 + 0.136352i \(0.956462\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −2.51573 1.45245i −0.950855 0.548976i −0.0575085 0.998345i \(-0.518316\pi\)
−0.893346 + 0.449369i \(0.851649\pi\)
\(8\) 0 0
\(9\) 0.646115 1.11910i 0.215372 0.373034i
\(10\) 0 0
\(11\) −0.174603 + 0.100807i −0.0526449 + 0.0303946i −0.526091 0.850428i \(-0.676343\pi\)
0.473447 + 0.880823i \(0.343010\pi\)
\(12\) 0 0
\(13\) 2.15455 2.89101i 0.597564 0.801821i
\(14\) 0 0
\(15\) −1.13174 + 0.653409i −0.292213 + 0.168709i
\(16\) 0 0
\(17\) −1.53093 + 2.65165i −0.371305 + 0.643119i −0.989767 0.142696i \(-0.954423\pi\)
0.618462 + 0.785815i \(0.287756\pi\)
\(18\) 0 0
\(19\) 0.0122700 + 0.00708410i 0.00281494 + 0.00162520i 0.501407 0.865212i \(-0.332816\pi\)
−0.498592 + 0.866837i \(0.666149\pi\)
\(20\) 0 0
\(21\) 3.79619i 0.828396i
\(22\) 0 0
\(23\) −2.36476 4.09588i −0.493086 0.854050i 0.506882 0.862015i \(-0.330798\pi\)
−0.999968 + 0.00796522i \(0.997465\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.60916 −1.07948
\(28\) 0 0
\(29\) 1.05550 + 1.82818i 0.196001 + 0.339484i 0.947228 0.320560i \(-0.103871\pi\)
−0.751227 + 0.660044i \(0.770538\pi\)
\(30\) 0 0
\(31\) 1.79645i 0.322651i 0.986901 + 0.161326i \(0.0515769\pi\)
−0.986901 + 0.161326i \(0.948423\pi\)
\(32\) 0 0
\(33\) 0.228175 + 0.131737i 0.0397201 + 0.0229324i
\(34\) 0 0
\(35\) −1.45245 + 2.51573i −0.245510 + 0.425235i
\(36\) 0 0
\(37\) −0.503609 + 0.290759i −0.0827928 + 0.0478004i −0.540825 0.841135i \(-0.681888\pi\)
0.458032 + 0.888936i \(0.348554\pi\)
\(38\) 0 0
\(39\) −4.67966 0.549370i −0.749345 0.0879695i
\(40\) 0 0
\(41\) −6.87817 + 3.97111i −1.07419 + 0.620184i −0.929323 0.369267i \(-0.879609\pi\)
−0.144867 + 0.989451i \(0.546275\pi\)
\(42\) 0 0
\(43\) −4.63035 + 8.02001i −0.706122 + 1.22304i 0.260163 + 0.965565i \(0.416224\pi\)
−0.966285 + 0.257475i \(0.917110\pi\)
\(44\) 0 0
\(45\) −1.11910 0.646115i −0.166826 0.0963171i
\(46\) 0 0
\(47\) 9.37618i 1.36766i −0.729643 0.683828i \(-0.760314\pi\)
0.729643 0.683828i \(-0.239686\pi\)
\(48\) 0 0
\(49\) 0.719250 + 1.24578i 0.102750 + 0.177968i
\(50\) 0 0
\(51\) 4.00129 0.560293
\(52\) 0 0
\(53\) −4.70777 −0.646662 −0.323331 0.946286i \(-0.604803\pi\)
−0.323331 + 0.946286i \(0.604803\pi\)
\(54\) 0 0
\(55\) 0.100807 + 0.174603i 0.0135929 + 0.0235435i
\(56\) 0 0
\(57\) 0.0185152i 0.00245240i
\(58\) 0 0
\(59\) −5.03504 2.90698i −0.655506 0.378457i 0.135057 0.990838i \(-0.456878\pi\)
−0.790563 + 0.612381i \(0.790212\pi\)
\(60\) 0 0
\(61\) −1.95119 + 3.37956i −0.249824 + 0.432708i −0.963477 0.267792i \(-0.913706\pi\)
0.713653 + 0.700499i \(0.247039\pi\)
\(62\) 0 0
\(63\) −3.25089 + 1.87690i −0.409574 + 0.236468i
\(64\) 0 0
\(65\) −2.89101 2.15455i −0.358585 0.267239i
\(66\) 0 0
\(67\) 1.80553 1.04242i 0.220581 0.127352i −0.385638 0.922650i \(-0.626019\pi\)
0.606219 + 0.795298i \(0.292685\pi\)
\(68\) 0 0
\(69\) −3.09031 + 5.35257i −0.372029 + 0.644373i
\(70\) 0 0
\(71\) 5.65201 + 3.26319i 0.670770 + 0.387269i 0.796368 0.604812i \(-0.206752\pi\)
−0.125598 + 0.992081i \(0.540085\pi\)
\(72\) 0 0
\(73\) 12.2027i 1.42822i −0.700033 0.714111i \(-0.746831\pi\)
0.700033 0.714111i \(-0.253169\pi\)
\(74\) 0 0
\(75\) 0.653409 + 1.13174i 0.0754491 + 0.130682i
\(76\) 0 0
\(77\) 0.585672 0.0667436
\(78\) 0 0
\(79\) 10.8293 1.21839 0.609195 0.793020i \(-0.291493\pi\)
0.609195 + 0.793020i \(0.291493\pi\)
\(80\) 0 0
\(81\) 1.72673 + 2.99078i 0.191859 + 0.332309i
\(82\) 0 0
\(83\) 15.2892i 1.67821i 0.543971 + 0.839104i \(0.316920\pi\)
−0.543971 + 0.839104i \(0.683080\pi\)
\(84\) 0 0
\(85\) 2.65165 + 1.53093i 0.287612 + 0.166053i
\(86\) 0 0
\(87\) 1.37935 2.38910i 0.147881 0.256138i
\(88\) 0 0
\(89\) −2.67609 + 1.54504i −0.283665 + 0.163774i −0.635081 0.772445i \(-0.719033\pi\)
0.351417 + 0.936219i \(0.385700\pi\)
\(90\) 0 0
\(91\) −9.61931 + 4.14360i −1.00838 + 0.434368i
\(92\) 0 0
\(93\) 2.03310 1.17381i 0.210823 0.121719i
\(94\) 0 0
\(95\) 0.00708410 0.0122700i 0.000726813 0.00125888i
\(96\) 0 0
\(97\) −15.9695 9.22000i −1.62146 0.936149i −0.986531 0.163577i \(-0.947697\pi\)
−0.634927 0.772572i \(-0.718970\pi\)
\(98\) 0 0
\(99\) 0.260532i 0.0261845i
\(100\) 0 0
\(101\) 2.79773 + 4.84581i 0.278384 + 0.482176i 0.970983 0.239147i \(-0.0768678\pi\)
−0.692599 + 0.721323i \(0.743534\pi\)
\(102\) 0 0
\(103\) −1.97220 −0.194326 −0.0971632 0.995268i \(-0.530977\pi\)
−0.0971632 + 0.995268i \(0.530977\pi\)
\(104\) 0 0
\(105\) 3.79619 0.370470
\(106\) 0 0
\(107\) −0.667577 1.15628i −0.0645371 0.111781i 0.831951 0.554848i \(-0.187224\pi\)
−0.896489 + 0.443067i \(0.853890\pi\)
\(108\) 0 0
\(109\) 4.66841i 0.447153i −0.974686 0.223576i \(-0.928227\pi\)
0.974686 0.223576i \(-0.0717732\pi\)
\(110\) 0 0
\(111\) 0.658125 + 0.379969i 0.0624664 + 0.0360650i
\(112\) 0 0
\(113\) 5.46901 9.47260i 0.514481 0.891107i −0.485378 0.874304i \(-0.661318\pi\)
0.999859 0.0168025i \(-0.00534866\pi\)
\(114\) 0 0
\(115\) −4.09588 + 2.36476i −0.381943 + 0.220515i
\(116\) 0 0
\(117\) −1.84325 4.27908i −0.170409 0.395601i
\(118\) 0 0
\(119\) 7.70280 4.44721i 0.706114 0.407675i
\(120\) 0 0
\(121\) −5.47968 + 9.49108i −0.498152 + 0.862825i
\(122\) 0 0
\(123\) 8.98851 + 5.18952i 0.810467 + 0.467923i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −6.05980 10.4959i −0.537720 0.931358i −0.999026 0.0441173i \(-0.985952\pi\)
0.461306 0.887241i \(-0.347381\pi\)
\(128\) 0 0
\(129\) 12.1020 1.06553
\(130\) 0 0
\(131\) 14.1852 1.23937 0.619685 0.784850i \(-0.287260\pi\)
0.619685 + 0.784850i \(0.287260\pi\)
\(132\) 0 0
\(133\) −0.0205787 0.0356433i −0.00178440 0.00309067i
\(134\) 0 0
\(135\) 5.60916i 0.482759i
\(136\) 0 0
\(137\) −15.5623 8.98490i −1.32958 0.767632i −0.344343 0.938844i \(-0.611898\pi\)
−0.985234 + 0.171212i \(0.945232\pi\)
\(138\) 0 0
\(139\) 0.0521243 0.0902819i 0.00442113 0.00765761i −0.863806 0.503824i \(-0.831926\pi\)
0.868228 + 0.496166i \(0.165259\pi\)
\(140\) 0 0
\(141\) −10.6114 + 6.12648i −0.893639 + 0.515942i
\(142\) 0 0
\(143\) −0.0847563 + 0.721974i −0.00708768 + 0.0603745i
\(144\) 0 0
\(145\) 1.82818 1.05550i 0.151822 0.0876545i
\(146\) 0 0
\(147\) 0.939928 1.62800i 0.0775239 0.134275i
\(148\) 0 0
\(149\) −15.4101 8.89704i −1.26245 0.728874i −0.288900 0.957359i \(-0.593289\pi\)
−0.973547 + 0.228485i \(0.926623\pi\)
\(150\) 0 0
\(151\) 12.1805i 0.991238i −0.868540 0.495619i \(-0.834941\pi\)
0.868540 0.495619i \(-0.165059\pi\)
\(152\) 0 0
\(153\) 1.97831 + 3.42654i 0.159937 + 0.277019i
\(154\) 0 0
\(155\) 1.79645 0.144294
\(156\) 0 0
\(157\) 9.95336 0.794364 0.397182 0.917740i \(-0.369988\pi\)
0.397182 + 0.917740i \(0.369988\pi\)
\(158\) 0 0
\(159\) 3.07610 + 5.32796i 0.243950 + 0.422534i
\(160\) 0 0
\(161\) 13.7388i 1.08277i
\(162\) 0 0
\(163\) 7.80493 + 4.50618i 0.611329 + 0.352951i 0.773485 0.633814i \(-0.218512\pi\)
−0.162156 + 0.986765i \(0.551845\pi\)
\(164\) 0 0
\(165\) 0.131737 0.228175i 0.0102557 0.0177634i
\(166\) 0 0
\(167\) 13.9204 8.03692i 1.07719 0.621915i 0.147052 0.989129i \(-0.453021\pi\)
0.930137 + 0.367213i \(0.119688\pi\)
\(168\) 0 0
\(169\) −3.71586 12.4576i −0.285835 0.958279i
\(170\) 0 0
\(171\) 0.0158557 0.00915428i 0.00121251 0.000700045i
\(172\) 0 0
\(173\) −7.38307 + 12.7879i −0.561324 + 0.972242i 0.436057 + 0.899919i \(0.356375\pi\)
−0.997381 + 0.0723233i \(0.976959\pi\)
\(174\) 0 0
\(175\) 2.51573 + 1.45245i 0.190171 + 0.109795i
\(176\) 0 0
\(177\) 7.59778i 0.571084i
\(178\) 0 0
\(179\) 1.75375 + 3.03758i 0.131081 + 0.227039i 0.924094 0.382166i \(-0.124822\pi\)
−0.793012 + 0.609205i \(0.791488\pi\)
\(180\) 0 0
\(181\) 15.8577 1.17869 0.589346 0.807881i \(-0.299385\pi\)
0.589346 + 0.807881i \(0.299385\pi\)
\(182\) 0 0
\(183\) 5.09969 0.376980
\(184\) 0 0
\(185\) 0.290759 + 0.503609i 0.0213770 + 0.0370261i
\(186\) 0 0
\(187\) 0.617316i 0.0451426i
\(188\) 0 0
\(189\) 14.1111 + 8.14705i 1.02643 + 0.592611i
\(190\) 0 0
\(191\) 2.16545 3.75067i 0.156686 0.271389i −0.776985 0.629519i \(-0.783252\pi\)
0.933672 + 0.358130i \(0.116585\pi\)
\(192\) 0 0
\(193\) 18.4890 10.6746i 1.33087 0.768377i 0.345435 0.938443i \(-0.387731\pi\)
0.985433 + 0.170066i \(0.0543981\pi\)
\(194\) 0 0
\(195\) −0.549370 + 4.67966i −0.0393412 + 0.335117i
\(196\) 0 0
\(197\) 18.0261 10.4074i 1.28431 0.741494i 0.306673 0.951815i \(-0.400784\pi\)
0.977632 + 0.210321i \(0.0674510\pi\)
\(198\) 0 0
\(199\) 12.4432 21.5522i 0.882072 1.52779i 0.0330394 0.999454i \(-0.489481\pi\)
0.849033 0.528340i \(-0.177185\pi\)
\(200\) 0 0
\(201\) −2.35950 1.36226i −0.166426 0.0960863i
\(202\) 0 0
\(203\) 6.13226i 0.430401i
\(204\) 0 0
\(205\) 3.97111 + 6.87817i 0.277355 + 0.480392i
\(206\) 0 0
\(207\) −6.11162 −0.424787
\(208\) 0 0
\(209\) −0.00285652 −0.000197589
\(210\) 0 0
\(211\) −4.84952 8.39962i −0.333855 0.578254i 0.649409 0.760439i \(-0.275016\pi\)
−0.983264 + 0.182185i \(0.941683\pi\)
\(212\) 0 0
\(213\) 8.52878i 0.584383i
\(214\) 0 0
\(215\) 8.02001 + 4.63035i 0.546960 + 0.315787i
\(216\) 0 0
\(217\) 2.60926 4.51936i 0.177128 0.306794i
\(218\) 0 0
\(219\) −13.8103 + 7.97337i −0.933212 + 0.538790i
\(220\) 0 0
\(221\) 4.36748 + 10.1390i 0.293788 + 0.682025i
\(222\) 0 0
\(223\) −1.14775 + 0.662655i −0.0768591 + 0.0443746i −0.537937 0.842985i \(-0.680796\pi\)
0.461078 + 0.887360i \(0.347463\pi\)
\(224\) 0 0
\(225\) −0.646115 + 1.11910i −0.0430743 + 0.0746069i
\(226\) 0 0
\(227\) 1.25811 + 0.726372i 0.0835039 + 0.0482110i 0.541171 0.840913i \(-0.317981\pi\)
−0.457667 + 0.889124i \(0.651315\pi\)
\(228\) 0 0
\(229\) 27.4918i 1.81671i −0.418199 0.908355i \(-0.637339\pi\)
0.418199 0.908355i \(-0.362661\pi\)
\(230\) 0 0
\(231\) −0.382683 0.662827i −0.0251787 0.0436108i
\(232\) 0 0
\(233\) 23.9607 1.56972 0.784859 0.619674i \(-0.212735\pi\)
0.784859 + 0.619674i \(0.212735\pi\)
\(234\) 0 0
\(235\) −9.37618 −0.611635
\(236\) 0 0
\(237\) −7.07595 12.2559i −0.459632 0.796106i
\(238\) 0 0
\(239\) 4.99651i 0.323197i −0.986857 0.161599i \(-0.948335\pi\)
0.986857 0.161599i \(-0.0516650\pi\)
\(240\) 0 0
\(241\) 11.4047 + 6.58448i 0.734638 + 0.424144i 0.820117 0.572196i \(-0.193908\pi\)
−0.0854782 + 0.996340i \(0.527242\pi\)
\(242\) 0 0
\(243\) −6.15722 + 10.6646i −0.394986 + 0.684136i
\(244\) 0 0
\(245\) 1.24578 0.719250i 0.0795898 0.0459512i
\(246\) 0 0
\(247\) 0.0469165 0.0202097i 0.00298523 0.00128591i
\(248\) 0 0
\(249\) 17.3034 9.99009i 1.09656 0.633097i
\(250\) 0 0
\(251\) −7.66087 + 13.2690i −0.483550 + 0.837533i −0.999822 0.0188920i \(-0.993986\pi\)
0.516272 + 0.856425i \(0.327319\pi\)
\(252\) 0 0
\(253\) 0.825789 + 0.476770i 0.0519169 + 0.0299743i
\(254\) 0 0
\(255\) 4.00129i 0.250570i
\(256\) 0 0
\(257\) 7.30055 + 12.6449i 0.455395 + 0.788768i 0.998711 0.0507606i \(-0.0161645\pi\)
−0.543315 + 0.839529i \(0.682831\pi\)
\(258\) 0 0
\(259\) 1.68926 0.104965
\(260\) 0 0
\(261\) 2.72790 0.168853
\(262\) 0 0
\(263\) −6.25088 10.8269i −0.385446 0.667612i 0.606385 0.795171i \(-0.292619\pi\)
−0.991831 + 0.127559i \(0.959286\pi\)
\(264\) 0 0
\(265\) 4.70777i 0.289196i
\(266\) 0 0
\(267\) 3.49716 + 2.01908i 0.214023 + 0.123566i
\(268\) 0 0
\(269\) −10.1182 + 17.5252i −0.616916 + 1.06853i 0.373129 + 0.927779i \(0.378285\pi\)
−0.990045 + 0.140750i \(0.955049\pi\)
\(270\) 0 0
\(271\) 3.03983 1.75505i 0.184657 0.106611i −0.404822 0.914395i \(-0.632667\pi\)
0.589479 + 0.807784i \(0.299333\pi\)
\(272\) 0 0
\(273\) 10.9748 + 8.17906i 0.664225 + 0.495019i
\(274\) 0 0
\(275\) 0.174603 0.100807i 0.0105290 0.00607891i
\(276\) 0 0
\(277\) 1.70447 2.95224i 0.102412 0.177383i −0.810266 0.586062i \(-0.800677\pi\)
0.912678 + 0.408680i \(0.134011\pi\)
\(278\) 0 0
\(279\) 2.01041 + 1.16071i 0.120360 + 0.0694899i
\(280\) 0 0
\(281\) 25.4204i 1.51646i 0.651989 + 0.758228i \(0.273935\pi\)
−0.651989 + 0.758228i \(0.726065\pi\)
\(282\) 0 0
\(283\) −10.5336 18.2447i −0.626155 1.08453i −0.988316 0.152417i \(-0.951294\pi\)
0.362162 0.932115i \(-0.382039\pi\)
\(284\) 0 0
\(285\) −0.0185152 −0.00109675
\(286\) 0 0
\(287\) 23.0715 1.36186
\(288\) 0 0
\(289\) 3.81251 + 6.60346i 0.224265 + 0.388439i
\(290\) 0 0
\(291\) 24.0977i 1.41263i
\(292\) 0 0
\(293\) −20.5039 11.8379i −1.19785 0.691579i −0.237774 0.971320i \(-0.576418\pi\)
−0.960075 + 0.279742i \(0.909751\pi\)
\(294\) 0 0
\(295\) −2.90698 + 5.03504i −0.169251 + 0.293151i
\(296\) 0 0
\(297\) 0.979378 0.565444i 0.0568293 0.0328104i
\(298\) 0 0
\(299\) −16.9362 1.98823i −0.979446 0.114982i
\(300\) 0 0
\(301\) 23.2974 13.4508i 1.34284 0.775289i
\(302\) 0 0
\(303\) 3.65612 6.33258i 0.210038 0.363797i
\(304\) 0 0
\(305\) 3.37956 + 1.95119i 0.193513 + 0.111725i
\(306\) 0 0
\(307\) 25.7386i 1.46898i −0.678621 0.734488i \(-0.737422\pi\)
0.678621 0.734488i \(-0.262578\pi\)
\(308\) 0 0
\(309\) 1.28865 + 2.23201i 0.0733088 + 0.126975i
\(310\) 0 0
\(311\) −6.94438 −0.393779 −0.196890 0.980426i \(-0.563084\pi\)
−0.196890 + 0.980426i \(0.563084\pi\)
\(312\) 0 0
\(313\) −17.9872 −1.01669 −0.508347 0.861152i \(-0.669743\pi\)
−0.508347 + 0.861152i \(0.669743\pi\)
\(314\) 0 0
\(315\) 1.87690 + 3.25089i 0.105752 + 0.183167i
\(316\) 0 0
\(317\) 1.52532i 0.0856707i −0.999082 0.0428353i \(-0.986361\pi\)
0.999082 0.0428353i \(-0.0136391\pi\)
\(318\) 0 0
\(319\) −0.368588 0.212804i −0.0206370 0.0119148i
\(320\) 0 0
\(321\) −0.872401 + 1.51104i −0.0486927 + 0.0843381i
\(322\) 0 0
\(323\) −0.0375691 + 0.0216905i −0.00209040 + 0.00120689i
\(324\) 0 0
\(325\) −2.15455 + 2.89101i −0.119513 + 0.160364i
\(326\) 0 0
\(327\) −5.28341 + 3.05038i −0.292173 + 0.168686i
\(328\) 0 0
\(329\) −13.6185 + 23.5879i −0.750811 + 1.30044i
\(330\) 0 0
\(331\) 17.1459 + 9.89916i 0.942421 + 0.544107i 0.890719 0.454555i \(-0.150202\pi\)
0.0517029 + 0.998663i \(0.483535\pi\)
\(332\) 0 0
\(333\) 0.751454i 0.0411794i
\(334\) 0 0
\(335\) −1.04242 1.80553i −0.0569538 0.0986468i
\(336\) 0 0
\(337\) 19.4107 1.05737 0.528684 0.848819i \(-0.322686\pi\)
0.528684 + 0.848819i \(0.322686\pi\)
\(338\) 0 0
\(339\) −14.2940 −0.776342
\(340\) 0 0
\(341\) −0.181095 0.313665i −0.00980683 0.0169859i
\(342\) 0 0
\(343\) 16.1557i 0.872323i
\(344\) 0 0
\(345\) 5.35257 + 3.09031i 0.288172 + 0.166376i
\(346\) 0 0
\(347\) −9.95435 + 17.2414i −0.534377 + 0.925569i 0.464816 + 0.885407i \(0.346121\pi\)
−0.999193 + 0.0401615i \(0.987213\pi\)
\(348\) 0 0
\(349\) 12.8255 7.40482i 0.686535 0.396371i −0.115778 0.993275i \(-0.536936\pi\)
0.802313 + 0.596904i \(0.203603\pi\)
\(350\) 0 0
\(351\) −12.0852 + 16.2161i −0.645060 + 0.865553i
\(352\) 0 0
\(353\) −16.6040 + 9.58631i −0.883740 + 0.510228i −0.871890 0.489702i \(-0.837105\pi\)
−0.0118503 + 0.999930i \(0.503772\pi\)
\(354\) 0 0
\(355\) 3.26319 5.65201i 0.173192 0.299978i
\(356\) 0 0
\(357\) −10.0661 5.81169i −0.532757 0.307587i
\(358\) 0 0
\(359\) 17.0500i 0.899865i −0.893062 0.449933i \(-0.851448\pi\)
0.893062 0.449933i \(-0.148552\pi\)
\(360\) 0 0
\(361\) −9.49990 16.4543i −0.499995 0.866016i
\(362\) 0 0
\(363\) 14.3219 0.751703
\(364\) 0 0
\(365\) −12.2027 −0.638720
\(366\) 0 0
\(367\) −12.7091 22.0128i −0.663408 1.14906i −0.979714 0.200400i \(-0.935776\pi\)
0.316306 0.948657i \(-0.397557\pi\)
\(368\) 0 0
\(369\) 10.2632i 0.534280i
\(370\) 0 0
\(371\) 11.8435 + 6.83782i 0.614882 + 0.355002i
\(372\) 0 0
\(373\) 3.09970 5.36884i 0.160497 0.277988i −0.774550 0.632512i \(-0.782024\pi\)
0.935047 + 0.354524i \(0.115357\pi\)
\(374\) 0 0
\(375\) 1.13174 0.653409i 0.0584426 0.0337419i
\(376\) 0 0
\(377\) 7.55941 + 0.887438i 0.389329 + 0.0457054i
\(378\) 0 0
\(379\) −6.51857 + 3.76350i −0.334837 + 0.193318i −0.657986 0.753030i \(-0.728592\pi\)
0.323150 + 0.946348i \(0.395258\pi\)
\(380\) 0 0
\(381\) −7.91904 + 13.7162i −0.405705 + 0.702702i
\(382\) 0 0
\(383\) −25.8268 14.9111i −1.31969 0.761923i −0.336011 0.941858i \(-0.609078\pi\)
−0.983679 + 0.179935i \(0.942411\pi\)
\(384\) 0 0
\(385\) 0.585672i 0.0298486i
\(386\) 0 0
\(387\) 5.98348 + 10.3637i 0.304157 + 0.526816i
\(388\) 0 0
\(389\) −17.0850 −0.866245 −0.433122 0.901335i \(-0.642588\pi\)
−0.433122 + 0.901335i \(0.642588\pi\)
\(390\) 0 0
\(391\) 14.4811 0.732341
\(392\) 0 0
\(393\) −9.26876 16.0540i −0.467547 0.809815i
\(394\) 0 0
\(395\) 10.8293i 0.544881i
\(396\) 0 0
\(397\) 5.38344 + 3.10813i 0.270187 + 0.155993i 0.628973 0.777427i \(-0.283476\pi\)
−0.358786 + 0.933420i \(0.616809\pi\)
\(398\) 0 0
\(399\) −0.0268926 + 0.0465793i −0.00134631 + 0.00233188i
\(400\) 0 0
\(401\) −1.18917 + 0.686569i −0.0593844 + 0.0342856i −0.529398 0.848373i \(-0.677582\pi\)
0.470014 + 0.882659i \(0.344249\pi\)
\(402\) 0 0
\(403\) 5.19354 + 3.87052i 0.258709 + 0.192805i
\(404\) 0 0
\(405\) 2.99078 1.72673i 0.148613 0.0858018i
\(406\) 0 0
\(407\) 0.0586212 0.101535i 0.00290575 0.00503290i
\(408\) 0 0
\(409\) −23.1731 13.3790i −1.14584 0.661549i −0.197968 0.980209i \(-0.563434\pi\)
−0.947869 + 0.318659i \(0.896767\pi\)
\(410\) 0 0
\(411\) 23.4832i 1.15834i
\(412\) 0 0
\(413\) 8.44451 + 14.6263i 0.415527 + 0.719714i
\(414\) 0 0
\(415\) 15.2892 0.750517
\(416\) 0 0
\(417\) −0.136234 −0.00667140
\(418\) 0 0
\(419\) −13.6575 23.6555i −0.667212 1.15565i −0.978680 0.205389i \(-0.934154\pi\)
0.311468 0.950257i \(-0.399179\pi\)
\(420\) 0 0
\(421\) 18.4971i 0.901493i −0.892652 0.450746i \(-0.851158\pi\)
0.892652 0.450746i \(-0.148842\pi\)
\(422\) 0 0
\(423\) −10.4929 6.05809i −0.510183 0.294554i
\(424\) 0 0
\(425\) 1.53093 2.65165i 0.0742610 0.128624i
\(426\) 0 0
\(427\) 9.81730 5.66802i 0.475093 0.274295i
\(428\) 0 0
\(429\) 0.872465 0.375822i 0.0421230 0.0181449i
\(430\) 0 0
\(431\) −7.27873 + 4.20237i −0.350604 + 0.202421i −0.664951 0.746887i \(-0.731548\pi\)
0.314347 + 0.949308i \(0.398214\pi\)
\(432\) 0 0
\(433\) −8.75170 + 15.1584i −0.420580 + 0.728465i −0.995996 0.0893951i \(-0.971507\pi\)
0.575417 + 0.817861i \(0.304840\pi\)
\(434\) 0 0
\(435\) −2.38910 1.37935i −0.114548 0.0661345i
\(436\) 0 0
\(437\) 0.0670087i 0.00320546i
\(438\) 0 0
\(439\) 18.8296 + 32.6138i 0.898687 + 1.55657i 0.829174 + 0.558991i \(0.188811\pi\)
0.0695136 + 0.997581i \(0.477855\pi\)
\(440\) 0 0
\(441\) 1.85887 0.0885177
\(442\) 0 0
\(443\) 31.5697 1.49992 0.749960 0.661483i \(-0.230073\pi\)
0.749960 + 0.661483i \(0.230073\pi\)
\(444\) 0 0
\(445\) 1.54504 + 2.67609i 0.0732419 + 0.126859i
\(446\) 0 0
\(447\) 23.2536i 1.09986i
\(448\) 0 0
\(449\) −13.4173 7.74649i −0.633202 0.365580i 0.148789 0.988869i \(-0.452463\pi\)
−0.781991 + 0.623289i \(0.785796\pi\)
\(450\) 0 0
\(451\) 0.800635 1.38674i 0.0377004 0.0652990i
\(452\) 0 0
\(453\) −13.7852 + 7.95886i −0.647683 + 0.373940i
\(454\) 0 0
\(455\) 4.14360 + 9.61931i 0.194255 + 0.450960i
\(456\) 0 0
\(457\) 15.9160 9.18912i 0.744520 0.429849i −0.0791904 0.996860i \(-0.525233\pi\)
0.823710 + 0.567011i \(0.191900\pi\)
\(458\) 0 0
\(459\) 8.58723 14.8735i 0.400817 0.694236i
\(460\) 0 0
\(461\) −20.7199 11.9626i −0.965020 0.557154i −0.0673055 0.997732i \(-0.521440\pi\)
−0.897714 + 0.440578i \(0.854774\pi\)
\(462\) 0 0
\(463\) 29.8775i 1.38852i −0.719723 0.694262i \(-0.755731\pi\)
0.719723 0.694262i \(-0.244269\pi\)
\(464\) 0 0
\(465\) −1.17381 2.03310i −0.0544343 0.0942829i
\(466\) 0 0
\(467\) 15.2809 0.707116 0.353558 0.935413i \(-0.384972\pi\)
0.353558 + 0.935413i \(0.384972\pi\)
\(468\) 0 0
\(469\) −6.05630 −0.279654
\(470\) 0 0
\(471\) −6.50361 11.2646i −0.299670 0.519044i
\(472\) 0 0
\(473\) 1.86709i 0.0858491i
\(474\) 0 0
\(475\) −0.0122700 0.00708410i −0.000562987 0.000325041i
\(476\) 0 0
\(477\) −3.04176 + 5.26848i −0.139273 + 0.241227i
\(478\) 0 0
\(479\) −20.8839 + 12.0573i −0.954209 + 0.550913i −0.894386 0.447296i \(-0.852387\pi\)
−0.0598232 + 0.998209i \(0.519054\pi\)
\(480\) 0 0
\(481\) −0.244463 + 2.08239i −0.0111465 + 0.0949489i
\(482\) 0 0
\(483\) 15.5487 8.97706i 0.707491 0.408470i
\(484\) 0 0
\(485\) −9.22000 + 15.9695i −0.418659 + 0.725138i
\(486\) 0 0
\(487\) −9.28811 5.36249i −0.420884 0.242998i 0.274571 0.961567i \(-0.411464\pi\)
−0.695456 + 0.718569i \(0.744797\pi\)
\(488\) 0 0
\(489\) 11.7775i 0.532597i
\(490\) 0 0
\(491\) 16.7535 + 29.0178i 0.756073 + 1.30956i 0.944839 + 0.327535i \(0.106218\pi\)
−0.188766 + 0.982022i \(0.560449\pi\)
\(492\) 0 0
\(493\) −6.46358 −0.291105
\(494\) 0 0
\(495\) 0.260532 0.0117101
\(496\) 0 0
\(497\) −9.47927 16.4186i −0.425203 0.736474i
\(498\) 0 0
\(499\) 0.744396i 0.0333237i −0.999861 0.0166619i \(-0.994696\pi\)
0.999861 0.0166619i \(-0.00530388\pi\)
\(500\) 0 0
\(501\) −18.1914 10.5028i −0.812730 0.469230i
\(502\) 0 0
\(503\) −5.38218 + 9.32221i −0.239980 + 0.415657i −0.960708 0.277561i \(-0.910474\pi\)
0.720729 + 0.693217i \(0.243807\pi\)
\(504\) 0 0
\(505\) 4.84581 2.79773i 0.215636 0.124497i
\(506\) 0 0
\(507\) −11.6708 + 12.3453i −0.518317 + 0.548274i
\(508\) 0 0
\(509\) 24.4548 14.1190i 1.08394 0.625814i 0.151985 0.988383i \(-0.451434\pi\)
0.931957 + 0.362569i \(0.118100\pi\)
\(510\) 0 0
\(511\) −17.7239 + 30.6987i −0.784060 + 1.35803i
\(512\) 0 0
\(513\) −0.0688245 0.0397358i −0.00303868 0.00175438i
\(514\) 0 0
\(515\) 1.97220i 0.0869054i
\(516\) 0 0
\(517\) 0.945188 + 1.63711i 0.0415693 + 0.0720002i
\(518\) 0 0
\(519\) 19.2966 0.847029
\(520\) 0 0
\(521\) −3.86998 −0.169547 −0.0847734 0.996400i \(-0.527017\pi\)
−0.0847734 + 0.996400i \(0.527017\pi\)
\(522\) 0 0
\(523\) −19.6421 34.0211i −0.858889 1.48764i −0.872990 0.487738i \(-0.837822\pi\)
0.0141014 0.999901i \(-0.495511\pi\)
\(524\) 0 0
\(525\) 3.79619i 0.165679i
\(526\) 0 0
\(527\) −4.76354 2.75023i −0.207503 0.119802i
\(528\) 0 0
\(529\) 0.315843 0.547055i 0.0137323 0.0237850i
\(530\) 0 0
\(531\) −6.50642 + 3.75648i −0.282355 + 0.163018i
\(532\) 0 0
\(533\) −3.33881 + 28.4408i −0.144620 + 1.23191i
\(534\) 0 0
\(535\) −1.15628 + 0.667577i −0.0499902 + 0.0288619i
\(536\) 0 0
\(537\) 2.29183 3.96956i 0.0988996 0.171299i
\(538\) 0 0
\(539\) −0.251167 0.145011i −0.0108185 0.00624608i
\(540\) 0 0
\(541\) 9.24167i 0.397330i 0.980067 + 0.198665i \(0.0636606\pi\)
−0.980067 + 0.198665i \(0.936339\pi\)
\(542\) 0 0
\(543\) −10.3615 17.9467i −0.444656 0.770167i
\(544\) 0 0
\(545\) −4.66841 −0.199973
\(546\) 0 0
\(547\) −8.91834 −0.381321 −0.190660 0.981656i \(-0.561063\pi\)
−0.190660 + 0.981656i \(0.561063\pi\)
\(548\) 0 0
\(549\) 2.52138 + 4.36716i 0.107610 + 0.186386i
\(550\) 0 0
\(551\) 0.0299091i 0.00127417i
\(552\) 0 0
\(553\) −27.2435 15.7291i −1.15851 0.668867i
\(554\) 0 0
\(555\) 0.379969 0.658125i 0.0161288 0.0279358i
\(556\) 0 0
\(557\) −2.17367 + 1.25497i −0.0921012 + 0.0531746i −0.545343 0.838213i \(-0.683601\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(558\) 0 0
\(559\) 13.2096 + 30.6659i 0.558706 + 1.29703i
\(560\) 0 0
\(561\) −0.698639 + 0.403359i −0.0294965 + 0.0170298i
\(562\) 0 0
\(563\) −6.96884 + 12.0704i −0.293702 + 0.508706i −0.974682 0.223595i \(-0.928221\pi\)
0.680980 + 0.732302i \(0.261554\pi\)
\(564\) 0 0
\(565\) −9.47260 5.46901i −0.398515 0.230083i
\(566\) 0 0
\(567\) 10.0320i 0.421303i
\(568\) 0 0
\(569\) −12.3977 21.4735i −0.519739 0.900215i −0.999737 0.0229448i \(-0.992696\pi\)
0.479998 0.877270i \(-0.340638\pi\)
\(570\) 0 0
\(571\) 2.78548 0.116569 0.0582843 0.998300i \(-0.481437\pi\)
0.0582843 + 0.998300i \(0.481437\pi\)
\(572\) 0 0
\(573\) −5.65969 −0.236437
\(574\) 0 0
\(575\) 2.36476 + 4.09588i 0.0986172 + 0.170810i
\(576\) 0 0
\(577\) 29.0538i 1.20952i 0.796406 + 0.604762i \(0.206732\pi\)
−0.796406 + 0.604762i \(0.793268\pi\)
\(578\) 0 0
\(579\) −24.1617 13.9498i −1.00413 0.579733i
\(580\) 0 0
\(581\) 22.2069 38.4634i 0.921296 1.59573i
\(582\) 0 0
\(583\) 0.821993 0.474578i 0.0340435 0.0196550i
\(584\) 0 0
\(585\) −4.27908 + 1.84325i −0.176918 + 0.0762092i
\(586\) 0 0
\(587\) −3.23339 + 1.86680i −0.133456 + 0.0770510i −0.565242 0.824925i \(-0.691217\pi\)
0.431785 + 0.901976i \(0.357884\pi\)
\(588\) 0 0
\(589\) −0.0127262 + 0.0220424i −0.000524374 + 0.000908242i
\(590\) 0 0
\(591\) −23.5568 13.6005i −0.968997 0.559451i
\(592\) 0 0
\(593\) 12.5374i 0.514848i 0.966299 + 0.257424i \(0.0828737\pi\)
−0.966299 + 0.257424i \(0.917126\pi\)
\(594\) 0 0
\(595\) −4.44721 7.70280i −0.182318 0.315784i
\(596\) 0 0
\(597\) −32.5219 −1.33103
\(598\) 0 0
\(599\) −9.65827 −0.394626 −0.197313 0.980341i \(-0.563222\pi\)
−0.197313 + 0.980341i \(0.563222\pi\)
\(600\) 0 0
\(601\) −22.0400 38.1744i −0.899031 1.55717i −0.828736 0.559640i \(-0.810939\pi\)
−0.0702951 0.997526i \(-0.522394\pi\)
\(602\) 0 0
\(603\) 2.69410i 0.109712i
\(604\) 0 0
\(605\) 9.49108 + 5.47968i 0.385867 + 0.222781i
\(606\) 0 0
\(607\) −3.27578 + 5.67382i −0.132960 + 0.230293i −0.924816 0.380414i \(-0.875781\pi\)
0.791856 + 0.610707i \(0.209115\pi\)
\(608\) 0 0
\(609\) −6.94011 + 4.00687i −0.281227 + 0.162367i
\(610\) 0 0
\(611\) −27.1066 20.2014i −1.09662 0.817262i
\(612\) 0 0
\(613\) −15.1583 + 8.75167i −0.612240 + 0.353477i −0.773842 0.633379i \(-0.781667\pi\)
0.161602 + 0.986856i \(0.448334\pi\)
\(614\) 0 0
\(615\) 5.18952 8.98851i 0.209262 0.362452i
\(616\) 0 0
\(617\) −7.28725 4.20729i −0.293374 0.169379i 0.346089 0.938202i \(-0.387510\pi\)
−0.639462 + 0.768823i \(0.720843\pi\)
\(618\) 0 0
\(619\) 32.8802i 1.32157i −0.750576 0.660784i \(-0.770224\pi\)
0.750576 0.660784i \(-0.229776\pi\)
\(620\) 0 0
\(621\) 13.2643 + 22.9744i 0.532278 + 0.921933i
\(622\) 0 0
\(623\) 8.97640 0.359632
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.00186647 + 0.00323282i 7.45397e−5 + 0.000129107i
\(628\) 0 0
\(629\) 1.78052i 0.0709942i
\(630\) 0 0
\(631\) −2.46259 1.42178i −0.0980342 0.0566001i 0.450181 0.892937i \(-0.351359\pi\)
−0.548216 + 0.836337i \(0.684693\pi\)
\(632\) 0 0
\(633\) −6.33744 + 10.9768i −0.251891 + 0.436287i
\(634\) 0 0
\(635\) −10.4959 + 6.05980i −0.416516 + 0.240476i
\(636\) 0 0
\(637\) 5.15121 + 0.604727i 0.204098 + 0.0239602i
\(638\) 0 0
\(639\) 7.30369 4.21679i 0.288930 0.166814i
\(640\) 0 0
\(641\) 16.3090 28.2480i 0.644166 1.11573i −0.340327 0.940307i \(-0.610538\pi\)
0.984493 0.175422i \(-0.0561289\pi\)
\(642\) 0 0
\(643\) 18.5719 + 10.7225i 0.732406 + 0.422855i 0.819302 0.573363i \(-0.194361\pi\)
−0.0868960 + 0.996217i \(0.527695\pi\)
\(644\) 0 0
\(645\) 12.1020i 0.476518i
\(646\) 0 0
\(647\) 20.7500 + 35.9401i 0.815768 + 1.41295i 0.908775 + 0.417287i \(0.137019\pi\)
−0.0930064 + 0.995666i \(0.529648\pi\)
\(648\) 0 0
\(649\) 1.17218 0.0460121
\(650\) 0 0
\(651\) −6.81964 −0.267283
\(652\) 0 0
\(653\) 18.2686 + 31.6421i 0.714905 + 1.23825i 0.962996 + 0.269515i \(0.0868634\pi\)
−0.248091 + 0.968737i \(0.579803\pi\)
\(654\) 0 0
\(655\) 14.1852i 0.554263i
\(656\) 0 0
\(657\) −13.6561 7.88436i −0.532776 0.307598i
\(658\) 0 0
\(659\) −19.4024 + 33.6059i −0.755810 + 1.30910i 0.189161 + 0.981946i \(0.439423\pi\)
−0.944971 + 0.327155i \(0.893910\pi\)
\(660\) 0 0
\(661\) −29.7925 + 17.2007i −1.15880 + 0.669031i −0.951015 0.309144i \(-0.899957\pi\)
−0.207781 + 0.978175i \(0.566624\pi\)
\(662\) 0 0
\(663\) 8.62096 11.5678i 0.334810 0.449255i
\(664\) 0 0
\(665\) −0.0356433 + 0.0205787i −0.00138219 + 0.000798006i
\(666\) 0 0
\(667\) 4.99200 8.64640i 0.193291 0.334790i
\(668\) 0 0
\(669\) 1.49990 + 0.865968i 0.0579895 + 0.0334803i
\(670\) 0 0
\(671\) 0.786776i 0.0303731i
\(672\) 0 0
\(673\) 17.0929 + 29.6058i 0.658884 + 1.14122i 0.980905 + 0.194488i \(0.0623044\pi\)
−0.322021 + 0.946732i \(0.604362\pi\)
\(674\) 0 0
\(675\) 5.60916 0.215897
\(676\) 0 0
\(677\) −10.1215 −0.389001 −0.194500 0.980902i \(-0.562309\pi\)
−0.194500 + 0.980902i \(0.562309\pi\)
\(678\) 0 0
\(679\) 26.7833 + 46.3900i 1.02785 + 1.78028i
\(680\) 0 0
\(681\) 1.89847i 0.0727495i
\(682\) 0 0
\(683\) 31.4636 + 18.1655i 1.20392 + 0.695083i 0.961424 0.275069i \(-0.0887007\pi\)
0.242495 + 0.970153i \(0.422034\pi\)
\(684\) 0 0
\(685\) −8.98490 + 15.5623i −0.343295 + 0.594605i
\(686\) 0 0
\(687\) −31.1135 + 17.9634i −1.18705 + 0.685346i
\(688\) 0 0
\(689\) −10.1431 + 13.6102i −0.386422 + 0.518508i
\(690\) 0 0
\(691\) 12.4030 7.16085i 0.471830 0.272411i −0.245175 0.969479i \(-0.578845\pi\)
0.717006 + 0.697067i \(0.245512\pi\)
\(692\) 0 0
\(693\) 0.378411 0.655428i 0.0143747 0.0248976i
\(694\) 0 0
\(695\) −0.0902819 0.0521243i −0.00342459 0.00197719i
\(696\) 0 0
\(697\) 24.3180i 0.921109i
\(698\) 0 0
\(699\) −15.6561 27.1172i −0.592169 1.02567i
\(700\) 0 0
\(701\) 45.9203 1.73438 0.867192 0.497973i \(-0.165922\pi\)
0.867192 + 0.497973i \(0.165922\pi\)
\(702\) 0 0
\(703\) −0.00823906 −0.000310742
\(704\) 0 0
\(705\) 6.12648 + 10.6114i 0.230736 + 0.399647i
\(706\) 0 0
\(707\) 16.2543i 0.611305i
\(708\) 0 0
\(709\) 27.9608 + 16.1432i 1.05009 + 0.606270i 0.922675 0.385579i \(-0.125998\pi\)
0.127416 + 0.991849i \(0.459332\pi\)
\(710\) 0 0
\(711\) 6.99696 12.1191i 0.262407 0.454502i
\(712\) 0 0
\(713\) 7.35802 4.24816i 0.275560 0.159095i
\(714\) 0 0
\(715\) 0.721974 + 0.0847563i 0.0270003 + 0.00316971i
\(716\) 0 0
\(717\) −5.65473 + 3.26476i −0.211180 + 0.121925i
\(718\) 0 0
\(719\) 9.49588 16.4473i 0.354136 0.613382i −0.632833 0.774288i \(-0.718108\pi\)
0.986970 + 0.160906i \(0.0514415\pi\)
\(720\) 0 0
\(721\) 4.96151 + 2.86453i 0.184776 + 0.106681i
\(722\) 0 0
\(723\) 17.2094i 0.640025i
\(724\) 0 0
\(725\) −1.05550 1.82818i −0.0392003 0.0678969i
\(726\) 0 0
\(727\) −34.8696 −1.29324 −0.646622 0.762811i \(-0.723819\pi\)
−0.646622 + 0.762811i \(0.723819\pi\)
\(728\) 0 0
\(729\) 26.4531 0.979744
\(730\) 0 0
\(731\) −14.1775 24.5561i −0.524373 0.908241i
\(732\) 0 0
\(733\) 7.05777i 0.260685i −0.991469 0.130342i \(-0.958392\pi\)
0.991469 0.130342i \(-0.0416076\pi\)
\(734\) 0 0
\(735\) −1.62800 0.939928i −0.0600498 0.0346698i
\(736\) 0 0
\(737\) −0.210168 + 0.364022i −0.00774164 + 0.0134089i
\(738\) 0 0
\(739\) 11.3087 6.52909i 0.415998 0.240177i −0.277366 0.960764i \(-0.589461\pi\)
0.693364 + 0.720588i \(0.256128\pi\)
\(740\) 0 0
\(741\) −0.0535277 0.0398919i −0.00196639 0.00146547i
\(742\) 0 0
\(743\) −25.2360 + 14.5700i −0.925819 + 0.534522i −0.885487 0.464664i \(-0.846175\pi\)
−0.0403324 + 0.999186i \(0.512842\pi\)
\(744\) 0 0
\(745\) −8.89704 + 15.4101i −0.325962 + 0.564583i
\(746\) 0 0
\(747\) 17.1102 + 9.87858i 0.626029 + 0.361438i
\(748\) 0 0
\(749\) 3.87850i 0.141717i
\(750\) 0 0
\(751\) 8.12718 + 14.0767i 0.296565 + 0.513666i 0.975348 0.220673i \(-0.0708255\pi\)
−0.678783 + 0.734339i \(0.737492\pi\)
\(752\) 0 0
\(753\) 20.0227 0.729668
\(754\) 0 0
\(755\) −12.1805 −0.443295
\(756\) 0 0
\(757\) 15.9791 + 27.6766i 0.580770 + 1.00592i 0.995388 + 0.0959276i \(0.0305817\pi\)
−0.414618 + 0.909995i \(0.636085\pi\)
\(758\) 0 0
\(759\) 1.24610i 0.0452306i
\(760\) 0 0
\(761\) −15.0342 8.68000i −0.544990 0.314650i 0.202109 0.979363i \(-0.435220\pi\)
−0.747099 + 0.664713i \(0.768554\pi\)
\(762\) 0 0
\(763\) −6.78066 + 11.7444i −0.245476 + 0.425177i
\(764\) 0 0
\(765\) 3.42654 1.97831i 0.123887 0.0715260i
\(766\) 0 0
\(767\) −19.2523 + 8.29311i −0.695161 + 0.299447i
\(768\) 0 0
\(769\) 7.25269 4.18734i 0.261539 0.150999i −0.363498 0.931595i \(-0.618417\pi\)
0.625036 + 0.780596i \(0.285084\pi\)
\(770\) 0 0
\(771\) 9.54048 16.5246i 0.343592 0.595119i
\(772\) 0 0
\(773\) 31.2609 + 18.0485i 1.12438 + 0.649160i 0.942515 0.334164i \(-0.108454\pi\)
0.181863 + 0.983324i \(0.441787\pi\)
\(774\) 0 0
\(775\) 1.79645i 0.0645302i
\(776\) 0 0
\(777\) −1.10377 1.91179i −0.0395977 0.0685852i
\(778\) 0 0
\(779\) −0.112527 −0.00403170
\(780\) 0 0
\(781\) −1.31581 −0.0470835
\(782\) 0 0
\(783\) −5.92047 10.2545i −0.211580 0.366468i
\(784\) 0 0
\(785\) 9.95336i 0.355250i
\(786\) 0 0
\(787\) −41.4824 23.9499i −1.47869 0.853721i −0.478979 0.877827i \(-0.658993\pi\)
−0.999709 + 0.0241057i \(0.992326\pi\)
\(788\) 0 0
\(789\) −8.16876 + 14.1487i −0.290816 + 0.503707i
\(790\) 0 0
\(791\) −27.5170 + 15.8870i −0.978393 + 0.564876i
\(792\) 0 0
\(793\) 5.56640 + 12.9223i 0.197669 + 0.458885i
\(794\) 0 0
\(795\) 5.32796 3.07610i 0.188963 0.109098i
\(796\) 0 0
\(797\) 11.1525 19.3168i 0.395043 0.684235i −0.598063 0.801449i \(-0.704063\pi\)
0.993107 + 0.117214i \(0.0373963\pi\)
\(798\) 0 0
\(799\) 24.8623 + 14.3543i 0.879566 + 0.507818i
\(800\) 0 0
\(801\) 3.99309i 0.141089i
\(802\) 0 0
\(803\) 1.23012 + 2.13064i 0.0434101 + 0.0751886i
\(804\) 0 0
\(805\) 13.7388 0.484230
\(806\) 0 0
\(807\) 26.4452 0.930915
\(808\) 0 0
\(809\) 2.07676 + 3.59705i 0.0730148 + 0.126465i 0.900221 0.435433i \(-0.143405\pi\)
−0.827206 + 0.561898i \(0.810071\pi\)
\(810\) 0 0
\(811\) 35.8894i 1.26025i −0.776494 0.630124i \(-0.783004\pi\)
0.776494 0.630124i \(-0.216996\pi\)
\(812\) 0 0
\(813\) −3.97250 2.29352i −0.139322 0.0804374i
\(814\) 0 0
\(815\) 4.50618 7.80493i 0.157845 0.273395i
\(816\) 0 0
\(817\) −0.113629 + 0.0656038i −0.00397538 + 0.00229519i
\(818\) 0 0
\(819\) −1.57805 + 13.4422i −0.0551417 + 0.469710i
\(820\) 0 0
\(821\) 29.6474 17.1169i 1.03470 0.597385i 0.116373 0.993206i \(-0.462873\pi\)
0.918328 + 0.395821i \(0.129540\pi\)
\(822\) 0 0
\(823\) −27.6762 + 47.9366i −0.964731 + 1.67096i −0.254395 + 0.967100i \(0.581876\pi\)
−0.710336 + 0.703863i \(0.751457\pi\)
\(824\) 0 0
\(825\) −0.228175 0.131737i −0.00794402 0.00458648i
\(826\) 0 0
\(827\) 45.3781i 1.57795i 0.614425 + 0.788975i \(0.289388\pi\)
−0.614425 + 0.788975i \(0.710612\pi\)
\(828\) 0 0
\(829\) 6.81278 + 11.8001i 0.236618 + 0.409834i 0.959742 0.280885i \(-0.0906278\pi\)
−0.723124 + 0.690718i \(0.757295\pi\)
\(830\) 0 0
\(831\) −4.45487 −0.154538
\(832\) 0 0
\(833\) −4.40448 −0.152606
\(834\) 0 0
\(835\) −8.03692 13.9204i −0.278129 0.481734i
\(836\) 0 0
\(837\) 10.0765i 0.348296i
\(838\) 0 0
\(839\) −35.5499 20.5247i −1.22732 0.708592i −0.260850 0.965379i \(-0.584003\pi\)
−0.966468 + 0.256787i \(0.917336\pi\)
\(840\) 0 0
\(841\) 12.2718 21.2554i 0.423167 0.732947i
\(842\) 0 0
\(843\) 28.7692 16.6099i 0.990865 0.572076i
\(844\) 0 0
\(845\) −12.4576 + 3.71586i −0.428555 + 0.127829i
\(846\) 0 0
\(847\) 27.5707 15.9180i 0.947341 0.546948i
\(848\) 0 0
\(849\) −13.7654 + 23.8424i −0.472428 + 0.818270i
\(850\) 0 0
\(851\) 2.38183 + 1.37515i 0.0816480 + 0.0471395i
\(852\) 0 0
\(853\) 55.0548i 1.88504i −0.334148 0.942521i \(-0.608448\pi\)
0.334148 0.942521i \(-0.391552\pi\)
\(854\) 0 0
\(855\) −0.00915428 0.0158557i −0.000313070 0.000542253i
\(856\) 0 0
\(857\) 27.9165 0.953609 0.476805 0.879009i \(-0.341795\pi\)
0.476805 + 0.879009i \(0.341795\pi\)
\(858\) 0 0
\(859\) −22.1918 −0.757174 −0.378587 0.925566i \(-0.623590\pi\)
−0.378587 + 0.925566i \(0.623590\pi\)
\(860\) 0 0
\(861\) −15.0751 26.1108i −0.513758 0.889854i
\(862\) 0 0
\(863\) 11.5089i 0.391768i 0.980627 + 0.195884i \(0.0627576\pi\)
−0.980627 + 0.195884i \(0.937242\pi\)
\(864\) 0 0
\(865\) 12.7879 + 7.38307i 0.434800 + 0.251032i
\(866\) 0 0
\(867\) 4.98225 8.62952i 0.169206 0.293074i
\(868\) 0 0
\(869\) −1.89083 + 1.09167i −0.0641420 + 0.0370324i
\(870\) 0 0
\(871\) 0.876445 7.46576i 0.0296972 0.252968i
\(872\) 0 0
\(873\) −20.6363 + 11.9144i −0.698432 + 0.403240i
\(874\) 0 0
\(875\) 1.45245 2.51573i 0.0491019 0.0850470i
\(876\) 0 0
\(877\) 29.8575 + 17.2383i 1.00822 + 0.582094i 0.910669 0.413136i \(-0.135567\pi\)
0.0975480 + 0.995231i \(0.468900\pi\)
\(878\) 0 0
\(879\) 30.9400i 1.04358i
\(880\) 0 0
\(881\) 13.6960 + 23.7222i 0.461430 + 0.799220i 0.999032 0.0439786i \(-0.0140033\pi\)
−0.537603 + 0.843198i \(0.680670\pi\)
\(882\) 0 0
\(883\) 27.8229 0.936316 0.468158 0.883645i \(-0.344918\pi\)
0.468158 + 0.883645i \(0.344918\pi\)
\(884\) 0 0
\(885\) 7.59778 0.255397
\(886\) 0 0
\(887\) 8.94929 + 15.5006i 0.300488 + 0.520460i 0.976247 0.216663i \(-0.0695172\pi\)
−0.675759 + 0.737123i \(0.736184\pi\)
\(888\) 0 0
\(889\) 35.2063i 1.18078i
\(890\) 0 0
\(891\) −0.602985 0.348134i −0.0202008 0.0116629i
\(892\) 0 0
\(893\) 0.0664218 0.115046i 0.00222272 0.00384987i
\(894\) 0 0
\(895\) 3.03758 1.75375i 0.101535 0.0586213i
\(896\) 0 0
\(897\) 8.81611 + 20.4665i 0.294361 + 0.683355i
\(898\) 0 0
\(899\) −3.28422 + 1.89615i −0.109535 + 0.0632401i
\(900\) 0 0
\(901\) 7.20726 12.4833i 0.240109 0.415881i
\(902\) 0 0
\(903\) −30.4454 17.5777i −1.01316 0.584949i
\(904\) 0 0
\(905\) 15.8577i 0.527127i
\(906\) 0 0
\(907\) 1.74344 + 3.01972i 0.0578898 + 0.100268i 0.893518 0.449027i \(-0.148229\pi\)
−0.835628 + 0.549296i \(0.814896\pi\)
\(908\) 0 0
\(909\) 7.23061 0.239824
\(910\) 0 0
\(911\) −16.9862 −0.562778 −0.281389 0.959594i \(-0.590795\pi\)
−0.281389 + 0.959594i \(0.590795\pi\)
\(912\) 0 0
\(913\) −1.54126 2.66955i −0.0510084 0.0883491i
\(914\) 0 0
\(915\) 5.09969i 0.168591i
\(916\) 0 0
\(917\) −35.6862 20.6034i −1.17846 0.680385i
\(918\) 0 0
\(919\) 10.7765 18.6654i 0.355482 0.615714i −0.631718 0.775198i \(-0.717650\pi\)
0.987200 + 0.159485i \(0.0509833\pi\)
\(920\) 0 0
\(921\) −29.1293 + 16.8178i −0.959842 + 0.554165i
\(922\) 0 0
\(923\) 21.6114 9.30932i 0.711349 0.306420i
\(924\) 0 0
\(925\) 0.503609 0.290759i 0.0165586 0.00956009i
\(926\) 0 0
\(927\) −1.27427 + 2.20709i −0.0418524 + 0.0724905i
\(928\) 0 0
\(929\) 17.6458 + 10.1878i 0.578941 + 0.334252i 0.760712 0.649089i \(-0.224850\pi\)
−0.181771 + 0.983341i \(0.558183\pi\)
\(930\) 0 0
\(931\) 0.0203809i 0.000667959i
\(932\) 0 0
\(933\) 4.53752 + 7.85921i 0.148552 + 0.257299i
\(934\) 0 0
\(935\) −0.617316 −0.0201884
\(936\) 0 0
\(937\) 20.1601 0.658602 0.329301 0.944225i \(-0.393187\pi\)
0.329301 + 0.944225i \(0.393187\pi\)
\(938\) 0 0
\(939\) 11.7530 + 20.3567i 0.383544 + 0.664317i
\(940\) 0 0
\(941\) 14.2121i 0.463300i 0.972799 + 0.231650i \(0.0744124\pi\)
−0.972799 + 0.231650i \(0.925588\pi\)
\(942\) 0 0
\(943\) 32.5304 + 18.7814i 1.05934 + 0.611608i
\(944\) 0 0
\(945\) 8.14705 14.1111i 0.265024 0.459034i
\(946\) 0 0
\(947\) 1.63562 0.944323i 0.0531504 0.0306864i −0.473189 0.880961i \(-0.656897\pi\)
0.526340 + 0.850274i \(0.323564\pi\)
\(948\) 0 0
\(949\) −35.2782 26.2913i −1.14518 0.853453i
\(950\) 0 0
\(951\) −1.72626 + 0.996659i −0.0559780 + 0.0323189i
\(952\) 0 0
\(953\) 16.0550 27.8081i 0.520072 0.900792i −0.479655 0.877457i \(-0.659238\pi\)
0.999728 0.0233346i \(-0.00742831\pi\)
\(954\) 0 0
\(955\) −3.75067 2.16545i −0.121369 0.0700723i
\(956\) 0 0
\(957\) 0.556192i 0.0179791i
\(958\) 0 0
\(959\) 26.1003 + 45.2071i 0.842823 + 1.45981i
\(960\) 0 0
\(961\) 27.7728 0.895896
\(962\) 0 0
\(963\) −1.72532 −0.0555978
\(964\) 0 0
\(965\) −10.6746 18.4890i −0.343629 0.595182i
\(966\) 0 0
\(967\) 54.0621i 1.73852i 0.494357 + 0.869259i \(0.335404\pi\)
−0.494357 + 0.869259i \(0.664596\pi\)
\(968\) 0 0
\(969\) 0.0490959 + 0.0283455i 0.00157719 + 0.000910590i
\(970\) 0 0
\(971\) 24.3278 42.1369i 0.780715 1.35224i −0.150811 0.988563i \(-0.548188\pi\)
0.931526 0.363676i \(-0.118478\pi\)
\(972\) 0 0
\(973\) −0.262261 + 0.151416i −0.00840770 + 0.00485419i
\(974\) 0 0
\(975\) 4.67966 + 0.549370i 0.149869 + 0.0175939i
\(976\) 0 0
\(977\) −14.4838 + 8.36225i −0.463379 + 0.267532i −0.713464 0.700692i \(-0.752875\pi\)
0.250085 + 0.968224i \(0.419541\pi\)
\(978\) 0 0
\(979\) 0.311503 0.539539i 0.00995567 0.0172437i
\(980\) 0 0
\(981\) −5.22444 3.01633i −0.166803 0.0963040i
\(982\) 0 0
\(983\) 35.7213i 1.13933i −0.821876 0.569666i \(-0.807072\pi\)
0.821876 0.569666i \(-0.192928\pi\)
\(984\) 0 0
\(985\) −10.4074 18.0261i −0.331606 0.574359i
\(986\) 0 0
\(987\) 35.5937 1.13296
\(988\) 0 0
\(989\) 43.7987 1.39272
\(990\) 0 0
\(991\) 0.608309 + 1.05362i 0.0193236 + 0.0334694i 0.875525 0.483172i \(-0.160515\pi\)
−0.856202 + 0.516641i \(0.827182\pi\)
\(992\) 0 0
\(993\) 25.8728i 0.821048i
\(994\) 0 0
\(995\) −21.5522 12.4432i −0.683250 0.394475i
\(996\) 0 0
\(997\) −19.1849 + 33.2291i −0.607590 + 1.05238i 0.384046 + 0.923314i \(0.374530\pi\)
−0.991636 + 0.129063i \(0.958803\pi\)
\(998\) 0 0
\(999\) 2.82482 1.63091i 0.0893734 0.0515998i
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 1040.2.da.e.881.2 12
4.3 odd 2 520.2.bu.a.361.5 yes 12
13.4 even 6 inner 1040.2.da.e.641.2 12
52.11 even 12 6760.2.a.bj.1.2 6
52.15 even 12 6760.2.a.bg.1.2 6
52.43 odd 6 520.2.bu.a.121.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.bu.a.121.5 12 52.43 odd 6
520.2.bu.a.361.5 yes 12 4.3 odd 2
1040.2.da.e.641.2 12 13.4 even 6 inner
1040.2.da.e.881.2 12 1.1 even 1 trivial
6760.2.a.bg.1.2 6 52.15 even 12
6760.2.a.bj.1.2 6 52.11 even 12