Properties

Label 912.2.bn.h.449.1
Level $912$
Weight $2$
Character 912.449
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
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] = [912,2,Mod(65,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 912.449
Dual form 912.2.bn.h.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.72474 + 0.158919i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.449490 q^{7} +(2.94949 - 0.548188i) q^{9} +O(q^{10})\) \(q+(-1.72474 + 0.158919i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.449490 q^{7} +(2.94949 - 0.548188i) q^{9} +3.14626i q^{11} +(-3.00000 - 1.73205i) q^{13} +(2.00000 - 1.41421i) q^{15} +(-0.550510 + 0.317837i) q^{17} +(-3.17423 - 2.98735i) q^{19} +(-0.775255 + 0.0714323i) q^{21} +(6.12372 + 3.53553i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(-5.00000 + 1.41421i) q^{27} +(1.22474 - 2.12132i) q^{29} -4.24264i q^{31} +(-0.500000 - 5.42650i) q^{33} +(-0.550510 + 0.317837i) q^{35} -7.70674i q^{37} +(5.44949 + 2.51059i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(-4.44949 - 7.70674i) q^{43} +(-3.22474 + 2.75699i) q^{45} +(-11.5732 - 6.68180i) q^{47} -6.79796 q^{49} +(0.898979 - 0.635674i) q^{51} +(-5.44949 + 9.43879i) q^{53} +(-2.22474 - 3.85337i) q^{55} +(5.94949 + 4.64796i) q^{57} +(-5.72474 - 9.91555i) q^{59} +(0.775255 - 1.34278i) q^{61} +(1.32577 - 0.246405i) q^{63} +4.89898 q^{65} +(-2.17423 - 1.25529i) q^{67} +(-11.1237 - 5.12472i) q^{69} +(-3.00000 - 5.19615i) q^{71} +(-4.39898 - 7.61926i) q^{73} +(2.17423 - 4.71940i) q^{75} +1.41421i q^{77} +(7.34847 - 4.24264i) q^{79} +(8.39898 - 3.23375i) q^{81} +17.0027i q^{83} +(0.449490 - 0.778539i) q^{85} +(-1.77526 + 3.85337i) q^{87} +(3.55051 - 6.14966i) q^{89} +(-1.34847 - 0.778539i) q^{91} +(0.674235 + 7.31747i) q^{93} +(6.00000 + 1.41421i) q^{95} +(2.84847 - 1.64456i) q^{97} +(1.72474 + 9.27987i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 8 q^{7} + 2 q^{9} - 12 q^{13} + 8 q^{15} - 12 q^{17} + 2 q^{19} - 8 q^{21} - 6 q^{25} - 20 q^{27} - 2 q^{33} - 12 q^{35} + 12 q^{39} - 6 q^{41} - 8 q^{43} - 8 q^{45} - 12 q^{47} + 12 q^{49} - 16 q^{51} - 12 q^{53} - 4 q^{55} + 14 q^{57} - 18 q^{59} + 8 q^{61} + 20 q^{63} + 6 q^{67} - 20 q^{69} - 12 q^{71} + 2 q^{73} - 6 q^{75} + 14 q^{81} - 8 q^{85} - 12 q^{87} + 24 q^{89} + 24 q^{91} - 12 q^{93} + 24 q^{95} - 18 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.72474 + 0.158919i −0.995782 + 0.0917517i
\(4\) 0 0
\(5\) −1.22474 + 0.707107i −0.547723 + 0.316228i −0.748203 0.663470i \(-0.769083\pi\)
0.200480 + 0.979698i \(0.435750\pi\)
\(6\) 0 0
\(7\) 0.449490 0.169891 0.0849456 0.996386i \(-0.472928\pi\)
0.0849456 + 0.996386i \(0.472928\pi\)
\(8\) 0 0
\(9\) 2.94949 0.548188i 0.983163 0.182729i
\(10\) 0 0
\(11\) 3.14626i 0.948634i 0.880354 + 0.474317i \(0.157305\pi\)
−0.880354 + 0.474317i \(0.842695\pi\)
\(12\) 0 0
\(13\) −3.00000 1.73205i −0.832050 0.480384i 0.0225039 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) 0 0
\(15\) 2.00000 1.41421i 0.516398 0.365148i
\(16\) 0 0
\(17\) −0.550510 + 0.317837i −0.133518 + 0.0770869i −0.565271 0.824905i \(-0.691229\pi\)
0.431753 + 0.901992i \(0.357895\pi\)
\(18\) 0 0
\(19\) −3.17423 2.98735i −0.728219 0.685344i
\(20\) 0 0
\(21\) −0.775255 + 0.0714323i −0.169175 + 0.0155878i
\(22\) 0 0
\(23\) 6.12372 + 3.53553i 1.27688 + 0.737210i 0.976274 0.216537i \(-0.0694763\pi\)
0.300610 + 0.953747i \(0.402810\pi\)
\(24\) 0 0
\(25\) −1.50000 + 2.59808i −0.300000 + 0.519615i
\(26\) 0 0
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) 0 0
\(29\) 1.22474 2.12132i 0.227429 0.393919i −0.729616 0.683857i \(-0.760301\pi\)
0.957046 + 0.289938i \(0.0936346\pi\)
\(30\) 0 0
\(31\) 4.24264i 0.762001i −0.924575 0.381000i \(-0.875580\pi\)
0.924575 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) −0.500000 5.42650i −0.0870388 0.944633i
\(34\) 0 0
\(35\) −0.550510 + 0.317837i −0.0930532 + 0.0537243i
\(36\) 0 0
\(37\) 7.70674i 1.26698i −0.773751 0.633490i \(-0.781622\pi\)
0.773751 0.633490i \(-0.218378\pi\)
\(38\) 0 0
\(39\) 5.44949 + 2.51059i 0.872617 + 0.402016i
\(40\) 0 0
\(41\) −1.50000 2.59808i −0.234261 0.405751i 0.724797 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −4.44949 7.70674i −0.678541 1.17527i −0.975420 0.220352i \(-0.929279\pi\)
0.296880 0.954915i \(-0.404054\pi\)
\(44\) 0 0
\(45\) −3.22474 + 2.75699i −0.480717 + 0.410989i
\(46\) 0 0
\(47\) −11.5732 6.68180i −1.68813 0.974640i −0.955952 0.293524i \(-0.905172\pi\)
−0.732175 0.681117i \(-0.761495\pi\)
\(48\) 0 0
\(49\) −6.79796 −0.971137
\(50\) 0 0
\(51\) 0.898979 0.635674i 0.125882 0.0890122i
\(52\) 0 0
\(53\) −5.44949 + 9.43879i −0.748545 + 1.29652i 0.199975 + 0.979801i \(0.435914\pi\)
−0.948520 + 0.316717i \(0.897419\pi\)
\(54\) 0 0
\(55\) −2.22474 3.85337i −0.299985 0.519588i
\(56\) 0 0
\(57\) 5.94949 + 4.64796i 0.788029 + 0.615638i
\(58\) 0 0
\(59\) −5.72474 9.91555i −0.745298 1.29089i −0.950055 0.312082i \(-0.898974\pi\)
0.204757 0.978813i \(-0.434360\pi\)
\(60\) 0 0
\(61\) 0.775255 1.34278i 0.0992612 0.171926i −0.812118 0.583493i \(-0.801685\pi\)
0.911379 + 0.411568i \(0.135019\pi\)
\(62\) 0 0
\(63\) 1.32577 0.246405i 0.167031 0.0310441i
\(64\) 0 0
\(65\) 4.89898 0.607644
\(66\) 0 0
\(67\) −2.17423 1.25529i −0.265625 0.153359i 0.361273 0.932460i \(-0.382342\pi\)
−0.626898 + 0.779101i \(0.715676\pi\)
\(68\) 0 0
\(69\) −11.1237 5.12472i −1.33914 0.616944i
\(70\) 0 0
\(71\) −3.00000 5.19615i −0.356034 0.616670i 0.631260 0.775571i \(-0.282538\pi\)
−0.987294 + 0.158901i \(0.949205\pi\)
\(72\) 0 0
\(73\) −4.39898 7.61926i −0.514862 0.891766i −0.999851 0.0172466i \(-0.994510\pi\)
0.484990 0.874520i \(-0.338823\pi\)
\(74\) 0 0
\(75\) 2.17423 4.71940i 0.251059 0.544949i
\(76\) 0 0
\(77\) 1.41421i 0.161165i
\(78\) 0 0
\(79\) 7.34847 4.24264i 0.826767 0.477334i −0.0259772 0.999663i \(-0.508270\pi\)
0.852745 + 0.522328i \(0.174936\pi\)
\(80\) 0 0
\(81\) 8.39898 3.23375i 0.933220 0.359306i
\(82\) 0 0
\(83\) 17.0027i 1.86629i 0.359506 + 0.933143i \(0.382945\pi\)
−0.359506 + 0.933143i \(0.617055\pi\)
\(84\) 0 0
\(85\) 0.449490 0.778539i 0.0487540 0.0844444i
\(86\) 0 0
\(87\) −1.77526 + 3.85337i −0.190327 + 0.413125i
\(88\) 0 0
\(89\) 3.55051 6.14966i 0.376353 0.651863i −0.614175 0.789170i \(-0.710511\pi\)
0.990529 + 0.137307i \(0.0438445\pi\)
\(90\) 0 0
\(91\) −1.34847 0.778539i −0.141358 0.0816131i
\(92\) 0 0
\(93\) 0.674235 + 7.31747i 0.0699149 + 0.758787i
\(94\) 0 0
\(95\) 6.00000 + 1.41421i 0.615587 + 0.145095i
\(96\) 0 0
\(97\) 2.84847 1.64456i 0.289218 0.166980i −0.348371 0.937357i \(-0.613265\pi\)
0.637589 + 0.770377i \(0.279932\pi\)
\(98\) 0 0
\(99\) 1.72474 + 9.27987i 0.173343 + 0.932662i
\(100\) 0 0
\(101\) 8.57321 + 4.94975i 0.853067 + 0.492518i 0.861684 0.507445i \(-0.169410\pi\)
−0.00861771 + 0.999963i \(0.502743\pi\)
\(102\) 0 0
\(103\) 5.02118i 0.494752i 0.968920 + 0.247376i \(0.0795682\pi\)
−0.968920 + 0.247376i \(0.920432\pi\)
\(104\) 0 0
\(105\) 0.898979 0.635674i 0.0877314 0.0620355i
\(106\) 0 0
\(107\) 4.89898 0.473602 0.236801 0.971558i \(-0.423901\pi\)
0.236801 + 0.971558i \(0.423901\pi\)
\(108\) 0 0
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 1.22474 + 13.2922i 0.116248 + 1.26164i
\(112\) 0 0
\(113\) −18.7980 −1.76836 −0.884182 0.467143i \(-0.845283\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) −10.0000 −0.932505
\(116\) 0 0
\(117\) −9.79796 3.46410i −0.905822 0.320256i
\(118\) 0 0
\(119\) −0.247449 + 0.142865i −0.0226836 + 0.0130964i
\(120\) 0 0
\(121\) 1.10102 0.100093
\(122\) 0 0
\(123\) 3.00000 + 4.24264i 0.270501 + 0.382546i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 9.00000 + 5.19615i 0.798621 + 0.461084i 0.842989 0.537931i \(-0.180794\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) 8.89898 + 12.5851i 0.783511 + 1.10805i
\(130\) 0 0
\(131\) −1.92679 + 1.11243i −0.168344 + 0.0971935i −0.581805 0.813328i \(-0.697653\pi\)
0.413461 + 0.910522i \(0.364320\pi\)
\(132\) 0 0
\(133\) −1.42679 1.34278i −0.123718 0.116434i
\(134\) 0 0
\(135\) 5.12372 5.26758i 0.440980 0.453362i
\(136\) 0 0
\(137\) 3.39898 + 1.96240i 0.290394 + 0.167659i 0.638120 0.769937i \(-0.279712\pi\)
−0.347725 + 0.937596i \(0.613046\pi\)
\(138\) 0 0
\(139\) −7.17423 + 12.4261i −0.608511 + 1.05397i 0.382975 + 0.923759i \(0.374899\pi\)
−0.991486 + 0.130213i \(0.958434\pi\)
\(140\) 0 0
\(141\) 21.0227 + 9.68520i 1.77043 + 0.815641i
\(142\) 0 0
\(143\) 5.44949 9.43879i 0.455709 0.789312i
\(144\) 0 0
\(145\) 3.46410i 0.287678i
\(146\) 0 0
\(147\) 11.7247 1.08032i 0.967041 0.0891035i
\(148\) 0 0
\(149\) 1.77526 1.02494i 0.145435 0.0839667i −0.425517 0.904950i \(-0.639908\pi\)
0.570952 + 0.820984i \(0.306574\pi\)
\(150\) 0 0
\(151\) 9.61377i 0.782357i 0.920315 + 0.391179i \(0.127933\pi\)
−0.920315 + 0.391179i \(0.872067\pi\)
\(152\) 0 0
\(153\) −1.44949 + 1.23924i −0.117184 + 0.100187i
\(154\) 0 0
\(155\) 3.00000 + 5.19615i 0.240966 + 0.417365i
\(156\) 0 0
\(157\) −5.34847 9.26382i −0.426854 0.739333i 0.569737 0.821827i \(-0.307045\pi\)
−0.996592 + 0.0824935i \(0.973712\pi\)
\(158\) 0 0
\(159\) 7.89898 17.1455i 0.626430 1.35973i
\(160\) 0 0
\(161\) 2.75255 + 1.58919i 0.216931 + 0.125245i
\(162\) 0 0
\(163\) −18.3485 −1.43716 −0.718582 0.695443i \(-0.755208\pi\)
−0.718582 + 0.695443i \(0.755208\pi\)
\(164\) 0 0
\(165\) 4.44949 + 6.29253i 0.346392 + 0.489873i
\(166\) 0 0
\(167\) 2.44949 4.24264i 0.189547 0.328305i −0.755552 0.655089i \(-0.772631\pi\)
0.945099 + 0.326783i \(0.105965\pi\)
\(168\) 0 0
\(169\) −0.500000 0.866025i −0.0384615 0.0666173i
\(170\) 0 0
\(171\) −11.0000 7.07107i −0.841191 0.540738i
\(172\) 0 0
\(173\) 0.550510 + 0.953512i 0.0418545 + 0.0724942i 0.886194 0.463315i \(-0.153340\pi\)
−0.844339 + 0.535809i \(0.820007\pi\)
\(174\) 0 0
\(175\) −0.674235 + 1.16781i −0.0509673 + 0.0882780i
\(176\) 0 0
\(177\) 11.4495 + 16.1920i 0.860596 + 1.21707i
\(178\) 0 0
\(179\) 7.65153 0.571902 0.285951 0.958244i \(-0.407690\pi\)
0.285951 + 0.958244i \(0.407690\pi\)
\(180\) 0 0
\(181\) 18.6742 + 10.7816i 1.38804 + 0.801388i 0.993095 0.117314i \(-0.0374285\pi\)
0.394950 + 0.918703i \(0.370762\pi\)
\(182\) 0 0
\(183\) −1.12372 + 2.43916i −0.0830681 + 0.180308i
\(184\) 0 0
\(185\) 5.44949 + 9.43879i 0.400654 + 0.693954i
\(186\) 0 0
\(187\) −1.00000 1.73205i −0.0731272 0.126660i
\(188\) 0 0
\(189\) −2.24745 + 0.635674i −0.163478 + 0.0462385i
\(190\) 0 0
\(191\) 8.19955i 0.593299i −0.954986 0.296649i \(-0.904131\pi\)
0.954986 0.296649i \(-0.0958693\pi\)
\(192\) 0 0
\(193\) −22.3485 + 12.9029i −1.60868 + 0.928771i −0.619012 + 0.785382i \(0.712467\pi\)
−0.989666 + 0.143389i \(0.954200\pi\)
\(194\) 0 0
\(195\) −8.44949 + 0.778539i −0.605081 + 0.0557523i
\(196\) 0 0
\(197\) 20.2918i 1.44573i −0.690989 0.722865i \(-0.742825\pi\)
0.690989 0.722865i \(-0.257175\pi\)
\(198\) 0 0
\(199\) −1.44949 + 2.51059i −0.102752 + 0.177971i −0.912817 0.408368i \(-0.866098\pi\)
0.810066 + 0.586339i \(0.199431\pi\)
\(200\) 0 0
\(201\) 3.94949 + 1.81954i 0.278576 + 0.128340i
\(202\) 0 0
\(203\) 0.550510 0.953512i 0.0386382 0.0669234i
\(204\) 0 0
\(205\) 3.67423 + 2.12132i 0.256620 + 0.148159i
\(206\) 0 0
\(207\) 20.0000 + 7.07107i 1.39010 + 0.491473i
\(208\) 0 0
\(209\) 9.39898 9.98698i 0.650141 0.690814i
\(210\) 0 0
\(211\) 1.34847 0.778539i 0.0928325 0.0535968i −0.452865 0.891579i \(-0.649598\pi\)
0.545698 + 0.837982i \(0.316265\pi\)
\(212\) 0 0
\(213\) 6.00000 + 8.48528i 0.411113 + 0.581402i
\(214\) 0 0
\(215\) 10.8990 + 6.29253i 0.743304 + 0.429147i
\(216\) 0 0
\(217\) 1.90702i 0.129457i
\(218\) 0 0
\(219\) 8.79796 + 12.4422i 0.594511 + 0.840765i
\(220\) 0 0
\(221\) 2.20204 0.148125
\(222\) 0 0
\(223\) 15.6742 9.04952i 1.04962 0.606001i 0.127080 0.991892i \(-0.459439\pi\)
0.922544 + 0.385892i \(0.126106\pi\)
\(224\) 0 0
\(225\) −3.00000 + 8.48528i −0.200000 + 0.565685i
\(226\) 0 0
\(227\) −0.550510 −0.0365386 −0.0182693 0.999833i \(-0.505816\pi\)
−0.0182693 + 0.999833i \(0.505816\pi\)
\(228\) 0 0
\(229\) 0.898979 0.0594062 0.0297031 0.999559i \(-0.490544\pi\)
0.0297031 + 0.999559i \(0.490544\pi\)
\(230\) 0 0
\(231\) −0.224745 2.43916i −0.0147871 0.160485i
\(232\) 0 0
\(233\) −15.3990 + 8.89060i −1.00882 + 0.582443i −0.910846 0.412746i \(-0.864570\pi\)
−0.0979745 + 0.995189i \(0.531236\pi\)
\(234\) 0 0
\(235\) 18.8990 1.23283
\(236\) 0 0
\(237\) −12.0000 + 8.48528i −0.779484 + 0.551178i
\(238\) 0 0
\(239\) 21.0703i 1.36293i 0.731852 + 0.681463i \(0.238656\pi\)
−0.731852 + 0.681463i \(0.761344\pi\)
\(240\) 0 0
\(241\) −5.84847 3.37662i −0.376733 0.217507i 0.299663 0.954045i \(-0.403126\pi\)
−0.676396 + 0.736538i \(0.736459\pi\)
\(242\) 0 0
\(243\) −13.9722 + 6.91215i −0.896317 + 0.443415i
\(244\) 0 0
\(245\) 8.32577 4.80688i 0.531914 0.307100i
\(246\) 0 0
\(247\) 4.34847 + 14.4600i 0.276686 + 0.920066i
\(248\) 0 0
\(249\) −2.70204 29.3253i −0.171235 1.85841i
\(250\) 0 0
\(251\) 3.27526 + 1.89097i 0.206732 + 0.119357i 0.599792 0.800156i \(-0.295250\pi\)
−0.393060 + 0.919513i \(0.628583\pi\)
\(252\) 0 0
\(253\) −11.1237 + 19.2669i −0.699343 + 1.21130i
\(254\) 0 0
\(255\) −0.651531 + 1.41421i −0.0408004 + 0.0885615i
\(256\) 0 0
\(257\) −7.50000 + 12.9904i −0.467837 + 0.810318i −0.999325 0.0367485i \(-0.988300\pi\)
0.531487 + 0.847066i \(0.321633\pi\)
\(258\) 0 0
\(259\) 3.46410i 0.215249i
\(260\) 0 0
\(261\) 2.44949 6.92820i 0.151620 0.428845i
\(262\) 0 0
\(263\) −7.77526 + 4.48905i −0.479443 + 0.276806i −0.720184 0.693783i \(-0.755943\pi\)
0.240741 + 0.970589i \(0.422609\pi\)
\(264\) 0 0
\(265\) 15.4135i 0.946843i
\(266\) 0 0
\(267\) −5.14643 + 11.1708i −0.314956 + 0.683645i
\(268\) 0 0
\(269\) 12.2474 + 21.2132i 0.746740 + 1.29339i 0.949377 + 0.314138i \(0.101715\pi\)
−0.202637 + 0.979254i \(0.564951\pi\)
\(270\) 0 0
\(271\) −10.0227 17.3598i −0.608836 1.05453i −0.991433 0.130619i \(-0.958303\pi\)
0.382597 0.923915i \(-0.375030\pi\)
\(272\) 0 0
\(273\) 2.44949 + 1.12848i 0.148250 + 0.0682990i
\(274\) 0 0
\(275\) −8.17423 4.71940i −0.492925 0.284590i
\(276\) 0 0
\(277\) 20.2474 1.21655 0.608276 0.793726i \(-0.291862\pi\)
0.608276 + 0.793726i \(0.291862\pi\)
\(278\) 0 0
\(279\) −2.32577 12.5136i −0.139240 0.749171i
\(280\) 0 0
\(281\) 11.2980 19.5686i 0.673980 1.16737i −0.302786 0.953058i \(-0.597917\pi\)
0.976766 0.214309i \(-0.0687498\pi\)
\(282\) 0 0
\(283\) −4.72474 8.18350i −0.280857 0.486458i 0.690739 0.723104i \(-0.257285\pi\)
−0.971596 + 0.236646i \(0.923952\pi\)
\(284\) 0 0
\(285\) −10.5732 1.48565i −0.626303 0.0880021i
\(286\) 0 0
\(287\) −0.674235 1.16781i −0.0397988 0.0689336i
\(288\) 0 0
\(289\) −8.29796 + 14.3725i −0.488115 + 0.845440i
\(290\) 0 0
\(291\) −4.65153 + 3.28913i −0.272678 + 0.192812i
\(292\) 0 0
\(293\) −19.3485 −1.13035 −0.565175 0.824971i \(-0.691191\pi\)
−0.565175 + 0.824971i \(0.691191\pi\)
\(294\) 0 0
\(295\) 14.0227 + 8.09601i 0.816433 + 0.471368i
\(296\) 0 0
\(297\) −4.44949 15.7313i −0.258186 0.912824i
\(298\) 0 0
\(299\) −12.2474 21.2132i −0.708288 1.22679i
\(300\) 0 0
\(301\) −2.00000 3.46410i −0.115278 0.199667i
\(302\) 0 0
\(303\) −15.5732 7.17461i −0.894658 0.412170i
\(304\) 0 0
\(305\) 2.19275i 0.125557i
\(306\) 0 0
\(307\) −18.5227 + 10.6941i −1.05715 + 0.610344i −0.924642 0.380838i \(-0.875635\pi\)
−0.132505 + 0.991182i \(0.542302\pi\)
\(308\) 0 0
\(309\) −0.797959 8.66025i −0.0453943 0.492665i
\(310\) 0 0
\(311\) 15.5563i 0.882120i −0.897478 0.441060i \(-0.854603\pi\)
0.897478 0.441060i \(-0.145397\pi\)
\(312\) 0 0
\(313\) 9.50000 16.4545i 0.536972 0.930062i −0.462093 0.886831i \(-0.652902\pi\)
0.999065 0.0432311i \(-0.0137652\pi\)
\(314\) 0 0
\(315\) −1.44949 + 1.23924i −0.0816695 + 0.0698233i
\(316\) 0 0
\(317\) −3.00000 + 5.19615i −0.168497 + 0.291845i −0.937892 0.346929i \(-0.887225\pi\)
0.769395 + 0.638774i \(0.220558\pi\)
\(318\) 0 0
\(319\) 6.67423 + 3.85337i 0.373685 + 0.215747i
\(320\) 0 0
\(321\) −8.44949 + 0.778539i −0.471605 + 0.0434538i
\(322\) 0 0
\(323\) 2.69694 + 0.635674i 0.150062 + 0.0353699i
\(324\) 0 0
\(325\) 9.00000 5.19615i 0.499230 0.288231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.20204 3.00340i −0.286798 0.165583i
\(330\) 0 0
\(331\) 23.2952i 1.28042i −0.768200 0.640210i \(-0.778847\pi\)
0.768200 0.640210i \(-0.221153\pi\)
\(332\) 0 0
\(333\) −4.22474 22.7310i −0.231515 1.24565i
\(334\) 0 0
\(335\) 3.55051 0.193985
\(336\) 0 0
\(337\) −17.8485 + 10.3048i −0.972268 + 0.561339i −0.899927 0.436041i \(-0.856380\pi\)
−0.0723411 + 0.997380i \(0.523047\pi\)
\(338\) 0 0
\(339\) 32.4217 2.98735i 1.76090 0.162250i
\(340\) 0 0
\(341\) 13.3485 0.722860
\(342\) 0 0
\(343\) −6.20204 −0.334879
\(344\) 0 0
\(345\) 17.2474 1.58919i 0.928571 0.0855589i
\(346\) 0 0
\(347\) −3.27526 + 1.89097i −0.175825 + 0.101513i −0.585330 0.810795i \(-0.699035\pi\)
0.409505 + 0.912308i \(0.365702\pi\)
\(348\) 0 0
\(349\) 32.4949 1.73941 0.869706 0.493570i \(-0.164308\pi\)
0.869706 + 0.493570i \(0.164308\pi\)
\(350\) 0 0
\(351\) 17.4495 + 4.41761i 0.931385 + 0.235795i
\(352\) 0 0
\(353\) 24.9951i 1.33036i 0.746684 + 0.665179i \(0.231645\pi\)
−0.746684 + 0.665179i \(0.768355\pi\)
\(354\) 0 0
\(355\) 7.34847 + 4.24264i 0.390016 + 0.225176i
\(356\) 0 0
\(357\) 0.404082 0.285729i 0.0213863 0.0151224i
\(358\) 0 0
\(359\) −20.8207 + 12.0208i −1.09887 + 0.634434i −0.935925 0.352200i \(-0.885434\pi\)
−0.162948 + 0.986635i \(0.552100\pi\)
\(360\) 0 0
\(361\) 1.15153 + 18.9651i 0.0606069 + 0.998162i
\(362\) 0 0
\(363\) −1.89898 + 0.174973i −0.0996706 + 0.00918368i
\(364\) 0 0
\(365\) 10.7753 + 6.22110i 0.564003 + 0.325627i
\(366\) 0 0
\(367\) −4.32577 + 7.49245i −0.225803 + 0.391102i −0.956560 0.291535i \(-0.905834\pi\)
0.730757 + 0.682638i \(0.239167\pi\)
\(368\) 0 0
\(369\) −5.84847 6.84072i −0.304459 0.356113i
\(370\) 0 0
\(371\) −2.44949 + 4.24264i −0.127171 + 0.220267i
\(372\) 0 0
\(373\) 25.4558i 1.31805i −0.752119 0.659027i \(-0.770968\pi\)
0.752119 0.659027i \(-0.229032\pi\)
\(374\) 0 0
\(375\) 1.79796 + 19.5133i 0.0928462 + 1.00766i
\(376\) 0 0
\(377\) −7.34847 + 4.24264i −0.378465 + 0.218507i
\(378\) 0 0
\(379\) 1.55708i 0.0799817i −0.999200 0.0399909i \(-0.987267\pi\)
0.999200 0.0399909i \(-0.0127329\pi\)
\(380\) 0 0
\(381\) −16.3485 7.53177i −0.837557 0.385864i
\(382\) 0 0
\(383\) −10.2247 17.7098i −0.522460 0.904927i −0.999659 0.0261318i \(-0.991681\pi\)
0.477198 0.878796i \(-0.341652\pi\)
\(384\) 0 0
\(385\) −1.00000 1.73205i −0.0509647 0.0882735i
\(386\) 0 0
\(387\) −17.3485 20.2918i −0.881872 1.03149i
\(388\) 0 0
\(389\) −13.1010 7.56388i −0.664248 0.383504i 0.129646 0.991560i \(-0.458616\pi\)
−0.793894 + 0.608057i \(0.791949\pi\)
\(390\) 0 0
\(391\) −4.49490 −0.227317
\(392\) 0 0
\(393\) 3.14643 2.22486i 0.158716 0.112229i
\(394\) 0 0
\(395\) −6.00000 + 10.3923i −0.301893 + 0.522894i
\(396\) 0 0
\(397\) 2.67423 + 4.63191i 0.134216 + 0.232469i 0.925298 0.379242i \(-0.123815\pi\)
−0.791082 + 0.611711i \(0.790482\pi\)
\(398\) 0 0
\(399\) 2.67423 + 2.08921i 0.133879 + 0.104591i
\(400\) 0 0
\(401\) −3.39898 5.88721i −0.169737 0.293993i 0.768590 0.639741i \(-0.220958\pi\)
−0.938327 + 0.345748i \(0.887625\pi\)
\(402\) 0 0
\(403\) −7.34847 + 12.7279i −0.366053 + 0.634023i
\(404\) 0 0
\(405\) −8.00000 + 9.89949i −0.397523 + 0.491910i
\(406\) 0 0
\(407\) 24.2474 1.20190
\(408\) 0 0
\(409\) 3.15153 + 1.81954i 0.155833 + 0.0899703i 0.575889 0.817528i \(-0.304656\pi\)
−0.420056 + 0.907498i \(0.637989\pi\)
\(410\) 0 0
\(411\) −6.17423 2.84448i −0.304553 0.140308i
\(412\) 0 0
\(413\) −2.57321 4.45694i −0.126620 0.219312i
\(414\) 0 0
\(415\) −12.0227 20.8239i −0.590171 1.02221i
\(416\) 0 0
\(417\) 10.3990 22.5720i 0.509240 1.10536i
\(418\) 0 0
\(419\) 24.5344i 1.19859i 0.800530 + 0.599293i \(0.204552\pi\)
−0.800530 + 0.599293i \(0.795448\pi\)
\(420\) 0 0
\(421\) −3.97730 + 2.29629i −0.193842 + 0.111914i −0.593780 0.804628i \(-0.702365\pi\)
0.399938 + 0.916542i \(0.369032\pi\)
\(422\) 0 0
\(423\) −37.7980 13.3636i −1.83780 0.649760i
\(424\) 0 0
\(425\) 1.90702i 0.0925042i
\(426\) 0 0
\(427\) 0.348469 0.603566i 0.0168636 0.0292086i
\(428\) 0 0
\(429\) −7.89898 + 17.1455i −0.381366 + 0.827794i
\(430\) 0 0
\(431\) −1.65153 + 2.86054i −0.0795514 + 0.137787i −0.903056 0.429522i \(-0.858682\pi\)
0.823505 + 0.567309i \(0.192015\pi\)
\(432\) 0 0
\(433\) −17.6969 10.2173i −0.850461 0.491014i 0.0103456 0.999946i \(-0.496707\pi\)
−0.860806 + 0.508933i \(0.830040\pi\)
\(434\) 0 0
\(435\) −0.550510 5.97469i −0.0263949 0.286465i
\(436\) 0 0
\(437\) −8.87628 29.5163i −0.424610 1.41196i
\(438\) 0 0
\(439\) 9.67423 5.58542i 0.461726 0.266578i −0.251044 0.967976i \(-0.580774\pi\)
0.712770 + 0.701398i \(0.247440\pi\)
\(440\) 0 0
\(441\) −20.0505 + 3.72656i −0.954786 + 0.177455i
\(442\) 0 0
\(443\) 16.3207 + 9.42274i 0.775418 + 0.447688i 0.834804 0.550547i \(-0.185581\pi\)
−0.0593859 + 0.998235i \(0.518914\pi\)
\(444\) 0 0
\(445\) 10.0424i 0.476053i
\(446\) 0 0
\(447\) −2.89898 + 2.04989i −0.137117 + 0.0969564i
\(448\) 0 0
\(449\) 30.7980 1.45345 0.726723 0.686931i \(-0.241042\pi\)
0.726723 + 0.686931i \(0.241042\pi\)
\(450\) 0 0
\(451\) 8.17423 4.71940i 0.384910 0.222228i
\(452\) 0 0
\(453\) −1.52781 16.5813i −0.0717826 0.779057i
\(454\) 0 0
\(455\) 2.20204 0.103233
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) 2.30306 2.36773i 0.107498 0.110516i
\(460\) 0 0
\(461\) −33.2474 + 19.1954i −1.54849 + 0.894020i −0.550231 + 0.835013i \(0.685460\pi\)
−0.998258 + 0.0590072i \(0.981207\pi\)
\(462\) 0 0
\(463\) 19.7980 0.920089 0.460045 0.887896i \(-0.347833\pi\)
0.460045 + 0.887896i \(0.347833\pi\)
\(464\) 0 0
\(465\) −6.00000 8.48528i −0.278243 0.393496i
\(466\) 0 0
\(467\) 12.6172i 0.583853i −0.956441 0.291926i \(-0.905704\pi\)
0.956441 0.291926i \(-0.0942962\pi\)
\(468\) 0 0
\(469\) −0.977296 0.564242i −0.0451273 0.0260543i
\(470\) 0 0
\(471\) 10.6969 + 15.1278i 0.492889 + 0.697050i
\(472\) 0 0
\(473\) 24.2474 13.9993i 1.11490 0.643687i
\(474\) 0 0
\(475\) 12.5227 3.76588i 0.574581 0.172791i
\(476\) 0 0
\(477\) −10.8990 + 30.8270i −0.499030 + 1.41147i
\(478\) 0 0
\(479\) −0.853572 0.492810i −0.0390007 0.0225171i 0.480373 0.877064i \(-0.340501\pi\)
−0.519374 + 0.854547i \(0.673835\pi\)
\(480\) 0 0
\(481\) −13.3485 + 23.1202i −0.608638 + 1.05419i
\(482\) 0 0
\(483\) −5.00000 2.30351i −0.227508 0.104813i
\(484\) 0 0
\(485\) −2.32577 + 4.02834i −0.105608 + 0.182918i
\(486\) 0 0
\(487\) 25.0273i 1.13409i −0.823686 0.567046i \(-0.808086\pi\)
0.823686 0.567046i \(-0.191914\pi\)
\(488\) 0 0
\(489\) 31.6464 2.91591i 1.43110 0.131862i
\(490\) 0 0
\(491\) 13.8990 8.02458i 0.627252 0.362144i −0.152435 0.988314i \(-0.548711\pi\)
0.779687 + 0.626169i \(0.215378\pi\)
\(492\) 0 0
\(493\) 1.55708i 0.0701273i
\(494\) 0 0
\(495\) −8.67423 10.1459i −0.389878 0.456024i
\(496\) 0 0
\(497\) −1.34847 2.33562i −0.0604871 0.104767i
\(498\) 0 0
\(499\) 3.72474 + 6.45145i 0.166742 + 0.288806i 0.937273 0.348597i \(-0.113342\pi\)
−0.770530 + 0.637403i \(0.780008\pi\)
\(500\) 0 0
\(501\) −3.55051 + 7.70674i −0.158625 + 0.344312i
\(502\) 0 0
\(503\) 13.4722 + 7.77817i 0.600695 + 0.346812i 0.769315 0.638870i \(-0.220598\pi\)
−0.168620 + 0.985681i \(0.553931\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 1.00000 + 1.41421i 0.0444116 + 0.0628074i
\(508\) 0 0
\(509\) 8.69694 15.0635i 0.385485 0.667680i −0.606351 0.795197i \(-0.707367\pi\)
0.991836 + 0.127517i \(0.0407008\pi\)
\(510\) 0 0
\(511\) −1.97730 3.42478i −0.0874704 0.151503i
\(512\) 0 0
\(513\) 20.0959 + 10.4477i 0.887256 + 0.461276i
\(514\) 0 0
\(515\) −3.55051 6.14966i −0.156454 0.270987i
\(516\) 0 0
\(517\) 21.0227 36.4124i 0.924577 1.60142i
\(518\) 0 0
\(519\) −1.10102 1.55708i −0.0483294 0.0683481i
\(520\) 0 0
\(521\) −25.8990 −1.13465 −0.567327 0.823492i \(-0.692023\pi\)
−0.567327 + 0.823492i \(0.692023\pi\)
\(522\) 0 0
\(523\) 5.69694 + 3.28913i 0.249110 + 0.143824i 0.619357 0.785110i \(-0.287394\pi\)
−0.370247 + 0.928933i \(0.620727\pi\)
\(524\) 0 0
\(525\) 0.977296 2.12132i 0.0426527 0.0925820i
\(526\) 0 0
\(527\) 1.34847 + 2.33562i 0.0587402 + 0.101741i
\(528\) 0 0
\(529\) 13.5000 + 23.3827i 0.586957 + 1.01664i
\(530\) 0 0
\(531\) −22.3207 26.1076i −0.968634 1.13297i
\(532\) 0 0
\(533\) 10.3923i 0.450141i
\(534\) 0 0
\(535\) −6.00000 + 3.46410i −0.259403 + 0.149766i
\(536\) 0 0
\(537\) −13.1969 + 1.21597i −0.569490 + 0.0524730i
\(538\) 0 0
\(539\) 21.3882i 0.921254i
\(540\) 0 0
\(541\) −5.34847 + 9.26382i −0.229949 + 0.398283i −0.957793 0.287460i \(-0.907189\pi\)
0.727844 + 0.685743i \(0.240522\pi\)
\(542\) 0 0
\(543\) −33.9217 15.6278i −1.45572 0.670652i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.3939 + 18.7026i 1.38506 + 0.799666i 0.992754 0.120168i \(-0.0383432\pi\)
0.392309 + 0.919834i \(0.371677\pi\)
\(548\) 0 0
\(549\) 1.55051 4.38551i 0.0661742 0.187169i
\(550\) 0 0
\(551\) −10.2247 + 3.07483i −0.435589 + 0.130992i
\(552\) 0 0
\(553\) 3.30306 1.90702i 0.140460 0.0810949i
\(554\) 0 0
\(555\) −10.8990 15.4135i −0.462636 0.654266i
\(556\) 0 0
\(557\) 0.247449 + 0.142865i 0.0104847 + 0.00605337i 0.505233 0.862983i \(-0.331406\pi\)
−0.494748 + 0.869036i \(0.664740\pi\)
\(558\) 0 0
\(559\) 30.8270i 1.30384i
\(560\) 0 0
\(561\) 2.00000 + 2.82843i 0.0844401 + 0.119416i
\(562\) 0 0
\(563\) −40.8434 −1.72134 −0.860671 0.509161i \(-0.829956\pi\)
−0.860671 + 0.509161i \(0.829956\pi\)
\(564\) 0 0
\(565\) 23.0227 13.2922i 0.968572 0.559206i
\(566\) 0 0
\(567\) 3.77526 1.45354i 0.158546 0.0610428i
\(568\) 0 0
\(569\) −34.2929 −1.43763 −0.718816 0.695201i \(-0.755315\pi\)
−0.718816 + 0.695201i \(0.755315\pi\)
\(570\) 0 0
\(571\) −33.0454 −1.38291 −0.691454 0.722421i \(-0.743029\pi\)
−0.691454 + 0.722421i \(0.743029\pi\)
\(572\) 0 0
\(573\) 1.30306 + 14.1421i 0.0544362 + 0.590796i
\(574\) 0 0
\(575\) −18.3712 + 10.6066i −0.766131 + 0.442326i
\(576\) 0 0
\(577\) 18.5959 0.774158 0.387079 0.922047i \(-0.373484\pi\)
0.387079 + 0.922047i \(0.373484\pi\)
\(578\) 0 0
\(579\) 36.4949 25.8058i 1.51668 1.07245i
\(580\) 0 0
\(581\) 7.64253i 0.317065i
\(582\) 0 0
\(583\) −29.6969 17.1455i −1.22992 0.710096i
\(584\) 0 0
\(585\) 14.4495 2.68556i 0.597413 0.111034i
\(586\) 0 0
\(587\) 13.8990 8.02458i 0.573672 0.331210i −0.184942 0.982749i \(-0.559210\pi\)
0.758615 + 0.651540i \(0.225877\pi\)
\(588\) 0 0
\(589\) −12.6742 + 13.4671i −0.522233 + 0.554904i
\(590\) 0 0
\(591\) 3.22474 + 34.9982i 0.132648 + 1.43963i
\(592\) 0 0
\(593\) 15.0959 + 8.71563i 0.619915 + 0.357908i 0.776836 0.629703i \(-0.216823\pi\)
−0.156921 + 0.987611i \(0.550157\pi\)
\(594\) 0 0
\(595\) 0.202041 0.349945i 0.00828287 0.0143464i
\(596\) 0 0
\(597\) 2.10102 4.56048i 0.0859890 0.186648i
\(598\) 0 0
\(599\) 2.57321 4.45694i 0.105139 0.182106i −0.808656 0.588282i \(-0.799805\pi\)
0.913795 + 0.406176i \(0.133138\pi\)
\(600\) 0 0
\(601\) 14.0314i 0.572352i −0.958177 0.286176i \(-0.907616\pi\)
0.958177 0.286176i \(-0.0923842\pi\)
\(602\) 0 0
\(603\) −7.10102 2.51059i −0.289176 0.102239i
\(604\) 0 0
\(605\) −1.34847 + 0.778539i −0.0548231 + 0.0316521i
\(606\) 0 0
\(607\) 11.5208i 0.467614i −0.972283 0.233807i \(-0.924882\pi\)
0.972283 0.233807i \(-0.0751185\pi\)
\(608\) 0 0
\(609\) −0.797959 + 1.73205i −0.0323349 + 0.0701862i
\(610\) 0 0
\(611\) 23.1464 + 40.0908i 0.936404 + 1.62190i
\(612\) 0 0
\(613\) −23.8990 41.3942i −0.965271 1.67190i −0.708886 0.705323i \(-0.750802\pi\)
−0.256385 0.966575i \(-0.582531\pi\)
\(614\) 0 0
\(615\) −6.67423 3.07483i −0.269131 0.123989i
\(616\) 0 0
\(617\) 8.05051 + 4.64796i 0.324101 + 0.187120i 0.653219 0.757169i \(-0.273418\pi\)
−0.329118 + 0.944289i \(0.606751\pi\)
\(618\) 0 0
\(619\) 30.6969 1.23381 0.616907 0.787036i \(-0.288385\pi\)
0.616907 + 0.787036i \(0.288385\pi\)
\(620\) 0 0
\(621\) −35.6186 9.01742i −1.42933 0.361856i
\(622\) 0 0
\(623\) 1.59592 2.76421i 0.0639391 0.110746i
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −14.6237 + 18.7187i −0.584015 + 0.747552i
\(628\) 0 0
\(629\) 2.44949 + 4.24264i 0.0976676 + 0.169165i
\(630\) 0 0
\(631\) −17.1237 + 29.6592i −0.681685 + 1.18071i 0.292782 + 0.956179i \(0.405419\pi\)
−0.974466 + 0.224533i \(0.927914\pi\)
\(632\) 0 0
\(633\) −2.20204 + 1.55708i −0.0875233 + 0.0618883i
\(634\) 0 0
\(635\) −14.6969 −0.583230
\(636\) 0 0
\(637\) 20.3939 + 11.7744i 0.808035 + 0.466519i
\(638\) 0 0
\(639\) −11.6969 13.6814i −0.462724 0.541229i
\(640\) 0 0
\(641\) −13.1969 22.8578i −0.521248 0.902828i −0.999695 0.0247111i \(-0.992133\pi\)
0.478447 0.878116i \(-0.341200\pi\)
\(642\) 0 0
\(643\) 5.07321 + 8.78706i 0.200068 + 0.346528i 0.948550 0.316627i \(-0.102550\pi\)
−0.748482 + 0.663155i \(0.769217\pi\)
\(644\) 0 0
\(645\) −19.7980 9.12096i −0.779544 0.359137i
\(646\) 0 0
\(647\) 14.7778i 0.580976i −0.956879 0.290488i \(-0.906182\pi\)
0.956879 0.290488i \(-0.0938176\pi\)
\(648\) 0 0
\(649\) 31.1969 18.0116i 1.22459 0.707016i
\(650\) 0 0
\(651\) 0.303062 + 3.28913i 0.0118779 + 0.128911i
\(652\) 0 0
\(653\) 8.19955i 0.320873i 0.987046 + 0.160437i \(0.0512902\pi\)
−0.987046 + 0.160437i \(0.948710\pi\)
\(654\) 0 0
\(655\) 1.57321 2.72489i 0.0614706 0.106470i
\(656\) 0 0
\(657\) −17.1515 20.0614i −0.669145 0.782672i
\(658\) 0 0
\(659\) 12.0000 20.7846i 0.467454 0.809653i −0.531855 0.846836i \(-0.678505\pi\)
0.999309 + 0.0371821i \(0.0118382\pi\)
\(660\) 0 0
\(661\) 25.7196 + 14.8492i 1.00038 + 0.577569i 0.908360 0.418189i \(-0.137335\pi\)
0.0920180 + 0.995757i \(0.470668\pi\)
\(662\) 0 0
\(663\) −3.79796 + 0.349945i −0.147501 + 0.0135908i
\(664\) 0 0
\(665\) 2.69694 + 0.635674i 0.104583 + 0.0246504i
\(666\) 0 0
\(667\) 15.0000 8.66025i 0.580802 0.335326i
\(668\) 0 0
\(669\) −25.5959 + 18.0990i −0.989595 + 0.699750i
\(670\) 0 0
\(671\) 4.22474 + 2.43916i 0.163094 + 0.0941626i
\(672\) 0 0
\(673\) 3.46410i 0.133531i 0.997769 + 0.0667657i \(0.0212680\pi\)
−0.997769 + 0.0667657i \(0.978732\pi\)
\(674\) 0 0
\(675\) 3.82577 15.1117i 0.147254 0.581650i
\(676\) 0 0
\(677\) −32.6969 −1.25665 −0.628323 0.777953i \(-0.716258\pi\)
−0.628323 + 0.777953i \(0.716258\pi\)
\(678\) 0 0
\(679\) 1.28036 0.739215i 0.0491356 0.0283685i
\(680\) 0 0
\(681\) 0.949490 0.0874863i 0.0363845 0.00335248i
\(682\) 0 0
\(683\) −17.3939 −0.665558 −0.332779 0.943005i \(-0.607986\pi\)
−0.332779 + 0.943005i \(0.607986\pi\)
\(684\) 0 0
\(685\) −5.55051 −0.212074
\(686\) 0 0
\(687\) −1.55051 + 0.142865i −0.0591557 + 0.00545062i
\(688\) 0 0
\(689\) 32.6969 18.8776i 1.24565 0.719179i
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 0 0
\(693\) 0.775255 + 4.17121i 0.0294495 + 0.158451i
\(694\) 0 0
\(695\) 20.2918i 0.769712i
\(696\) 0 0
\(697\) 1.65153 + 0.953512i 0.0625562 + 0.0361168i
\(698\) 0 0
\(699\) 25.1464 17.7812i 0.951125 0.672547i
\(700\) 0 0
\(701\) −8.57321 + 4.94975i −0.323806 + 0.186949i −0.653088 0.757282i \(-0.726527\pi\)
0.329282 + 0.944232i \(0.393193\pi\)
\(702\) 0 0
\(703\) −23.0227 + 24.4630i −0.868318 + 0.922640i
\(704\) 0 0
\(705\) −32.5959 + 3.00340i −1.22763 + 0.113115i
\(706\) 0 0
\(707\) 3.85357 + 2.22486i 0.144928 + 0.0836745i
\(708\) 0 0
\(709\) 8.67423 15.0242i 0.325768 0.564246i −0.655900 0.754848i \(-0.727711\pi\)
0.981667 + 0.190602i \(0.0610439\pi\)
\(710\) 0 0
\(711\) 19.3485 16.5420i 0.725624 0.620372i
\(712\) 0 0
\(713\) 15.0000 25.9808i 0.561754 0.972987i
\(714\) 0 0
\(715\) 15.4135i 0.576432i
\(716\) 0 0
\(717\) −3.34847 36.3410i −0.125051 1.35718i
\(718\) 0 0
\(719\) −26.1464 + 15.0956i −0.975097 + 0.562973i −0.900786 0.434262i \(-0.857009\pi\)
−0.0743109 + 0.997235i \(0.523676\pi\)
\(720\) 0 0
\(721\) 2.25697i 0.0840539i
\(722\) 0 0
\(723\) 10.6237 + 4.89437i 0.395101 + 0.182024i
\(724\) 0 0
\(725\) 3.67423 + 6.36396i 0.136458 + 0.236352i
\(726\) 0 0
\(727\) 4.00000 + 6.92820i 0.148352 + 0.256953i 0.930618 0.365991i \(-0.119270\pi\)
−0.782267 + 0.622944i \(0.785937\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 4.89898 + 2.82843i 0.181195 + 0.104613i
\(732\) 0 0
\(733\) −20.0454 −0.740394 −0.370197 0.928953i \(-0.620710\pi\)
−0.370197 + 0.928953i \(0.620710\pi\)
\(734\) 0 0
\(735\) −13.5959 + 9.61377i −0.501493 + 0.354609i
\(736\) 0 0
\(737\) 3.94949 6.84072i 0.145481 0.251981i
\(738\) 0 0
\(739\) 12.1742 + 21.0864i 0.447836 + 0.775676i 0.998245 0.0592200i \(-0.0188613\pi\)
−0.550408 + 0.834895i \(0.685528\pi\)
\(740\) 0 0
\(741\) −9.79796 24.2487i −0.359937 0.890799i
\(742\) 0 0
\(743\) 2.32577 + 4.02834i 0.0853241 + 0.147786i 0.905529 0.424284i \(-0.139474\pi\)
−0.820205 + 0.572069i \(0.806141\pi\)
\(744\) 0 0
\(745\) −1.44949 + 2.51059i −0.0531052 + 0.0919809i
\(746\) 0 0
\(747\) 9.32066 + 50.1492i 0.341025 + 1.83486i
\(748\) 0 0
\(749\) 2.20204 0.0804608
\(750\) 0 0
\(751\) −17.6969 10.2173i −0.645770 0.372836i 0.141063 0.990001i \(-0.454948\pi\)
−0.786834 + 0.617165i \(0.788281\pi\)
\(752\) 0 0
\(753\) −5.94949 2.74094i −0.216811 0.0998854i
\(754\) 0 0
\(755\) −6.79796 11.7744i −0.247403 0.428515i
\(756\) 0 0
\(757\) 19.6969 + 34.1161i 0.715897 + 1.23997i 0.962612 + 0.270883i \(0.0873155\pi\)
−0.246715 + 0.969088i \(0.579351\pi\)
\(758\) 0 0
\(759\) 16.1237 34.9982i 0.585254 1.27035i
\(760\) 0 0
\(761\) 31.5734i 1.14453i 0.820067 + 0.572267i \(0.193936\pi\)
−0.820067 + 0.572267i \(0.806064\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.898979 2.54270i 0.0325027 0.0919314i
\(766\) 0 0
\(767\) 39.6622i 1.43212i
\(768\) 0 0
\(769\) −1.79796 + 3.11416i −0.0648361 + 0.112299i −0.896621 0.442798i \(-0.853986\pi\)
0.831785 + 0.555098i \(0.187319\pi\)
\(770\) 0 0
\(771\) 10.8712 23.5970i 0.391516 0.849825i
\(772\) 0 0
\(773\) −1.22474 + 2.12132i −0.0440510 + 0.0762986i −0.887210 0.461365i \(-0.847360\pi\)
0.843159 + 0.537664i \(0.180693\pi\)
\(774\) 0 0
\(775\) 11.0227 + 6.36396i 0.395947 + 0.228600i
\(776\) 0 0
\(777\) 0.550510 + 5.97469i 0.0197494 + 0.214341i
\(778\) 0 0
\(779\) −3.00000 + 12.7279i −0.107486 + 0.456025i
\(780\) 0 0
\(781\) 16.3485 9.43879i 0.584994 0.337747i
\(782\) 0 0
\(783\) −3.12372 + 12.3387i −0.111633 + 0.440947i
\(784\) 0 0
\(785\) 13.1010 + 7.56388i 0.467595 + 0.269966i
\(786\) 0 0
\(787\) 7.53177i 0.268479i −0.990949 0.134239i \(-0.957141\pi\)
0.990949 0.134239i \(-0.0428591\pi\)
\(788\) 0 0
\(789\) 12.6969 8.97809i 0.452023 0.319629i
\(790\) 0 0
\(791\) −8.44949 −0.300429
\(792\) 0 0
\(793\) −4.65153 + 2.68556i −0.165181 + 0.0953671i
\(794\) 0 0
\(795\) 2.44949 + 26.5843i 0.0868744 + 0.942849i
\(796\) 0 0
\(797\) 37.3485 1.32295 0.661475 0.749967i \(-0.269931\pi\)
0.661475 + 0.749967i \(0.269931\pi\)
\(798\) 0 0
\(799\) 8.49490 0.300528
\(800\) 0 0
\(801\) 7.10102 20.0847i 0.250902 0.709659i
\(802\) 0 0
\(803\) 23.9722 13.8404i 0.845960 0.488415i
\(804\) 0 0
\(805\) −4.49490 −0.158424
\(806\) 0 0
\(807\) −24.4949 34.6410i −0.862261 1.21942i
\(808\) 0 0
\(809\) 8.02458i 0.282129i 0.990000 + 0.141065i \(0.0450525\pi\)
−0.990000 + 0.141065i \(0.954947\pi\)
\(810\) 0 0
\(811\) −3.00000 1.73205i −0.105344 0.0608205i 0.446402 0.894832i \(-0.352705\pi\)
−0.551746 + 0.834012i \(0.686038\pi\)
\(812\) 0 0
\(813\) 20.0454 + 28.3485i 0.703023 + 0.994225i
\(814\) 0 0
\(815\) 22.4722 12.9743i 0.787167 0.454471i
\(816\) 0 0
\(817\) −8.89898 + 37.7552i −0.311336 + 1.32089i
\(818\) 0 0
\(819\) −4.40408 1.55708i −0.153891 0.0544087i
\(820\) 0 0
\(821\) −5.44949 3.14626i −0.190189 0.109805i 0.401882 0.915691i \(-0.368356\pi\)
−0.592071 + 0.805886i \(0.701689\pi\)
\(822\) 0 0
\(823\) −24.3485 + 42.1728i −0.848734 + 1.47005i 0.0336040 + 0.999435i \(0.489301\pi\)
−0.882338 + 0.470616i \(0.844032\pi\)
\(824\) 0 0
\(825\) 14.8485 + 6.84072i 0.516957 + 0.238163i
\(826\) 0 0
\(827\) 10.6237 18.4008i 0.369423 0.639860i −0.620052 0.784560i \(-0.712889\pi\)
0.989475 + 0.144701i \(0.0462220\pi\)
\(828\) 0 0
\(829\) 19.2275i 0.667800i 0.942609 + 0.333900i \(0.108365\pi\)
−0.942609 + 0.333900i \(0.891635\pi\)
\(830\) 0 0
\(831\) −34.9217 + 3.21770i −1.21142 + 0.111621i
\(832\) 0 0
\(833\) 3.74235 2.16064i 0.129665 0.0748619i
\(834\) 0 0
\(835\) 6.92820i 0.239760i
\(836\) 0 0
\(837\) 6.00000 + 21.2132i 0.207390 + 0.733236i
\(838\) 0 0
\(839\) −6.67423 11.5601i −0.230420 0.399099i 0.727512 0.686095i \(-0.240677\pi\)
−0.957932 + 0.286996i \(0.907343\pi\)
\(840\) 0 0
\(841\) 11.5000 + 19.9186i 0.396552 + 0.686848i
\(842\) 0 0
\(843\) −16.3763 + 35.5464i −0.564029 + 1.22428i
\(844\) 0 0
\(845\) 1.22474 + 0.707107i 0.0421325 + 0.0243252i
\(846\) 0 0
\(847\) 0.494897 0.0170049
\(848\) 0 0
\(849\) 9.44949 + 13.3636i 0.324306 + 0.458637i
\(850\) 0 0
\(851\) 27.2474 47.1940i 0.934031 1.61779i
\(852\) 0 0
\(853\) 5.00000 + 8.66025i 0.171197 + 0.296521i 0.938839 0.344358i \(-0.111903\pi\)
−0.767642 + 0.640879i \(0.778570\pi\)
\(854\) 0 0
\(855\) 18.4722 + 0.882079i 0.631736 + 0.0301665i
\(856\) 0 0
\(857\) 8.29796 + 14.3725i 0.283453 + 0.490955i 0.972233 0.234016i \(-0.0751867\pi\)
−0.688780 + 0.724970i \(0.741853\pi\)
\(858\) 0 0
\(859\) 9.42168 16.3188i 0.321464 0.556791i −0.659327 0.751857i \(-0.729159\pi\)
0.980790 + 0.195065i \(0.0624919\pi\)
\(860\) 0 0
\(861\) 1.34847 + 1.90702i 0.0459557 + 0.0649912i
\(862\) 0 0
\(863\) −16.6515 −0.566825 −0.283412 0.958998i \(-0.591467\pi\)
−0.283412 + 0.958998i \(0.591467\pi\)
\(864\) 0 0
\(865\) −1.34847 0.778539i −0.0458493 0.0264711i
\(866\) 0 0
\(867\) 12.0278 26.1076i 0.408486 0.886660i
\(868\) 0 0
\(869\) 13.3485 + 23.1202i 0.452816 + 0.784300i
\(870\) 0 0
\(871\) 4.34847 + 7.53177i 0.147342 + 0.255204i
\(872\) 0 0
\(873\) 7.50000 6.41212i 0.253837 0.217017i
\(874\) 0 0
\(875\) 5.08540i 0.171918i
\(876\) 0 0
\(877\) −36.3712 + 20.9989i −1.22817 + 0.709083i −0.966646 0.256115i \(-0.917557\pi\)
−0.261521 + 0.965198i \(0.584224\pi\)
\(878\) 0 0
\(879\) 33.3712 3.07483i 1.12558 0.103712i
\(880\) 0 0
\(881\) 10.2815i 0.346394i 0.984887 + 0.173197i \(0.0554098\pi\)
−0.984887 + 0.173197i \(0.944590\pi\)
\(882\) 0 0
\(883\) −10.4217 + 18.0509i −0.350718 + 0.607461i −0.986375 0.164510i \(-0.947396\pi\)
0.635658 + 0.771971i \(0.280729\pi\)
\(884\) 0 0
\(885\) −25.4722 11.7351i −0.856238 0.394471i
\(886\) 0 0
\(887\) −28.0454 + 48.5761i −0.941673 + 1.63102i −0.179393 + 0.983778i \(0.557413\pi\)
−0.762280 + 0.647247i \(0.775920\pi\)
\(888\) 0 0
\(889\) 4.04541 + 2.33562i 0.135679 + 0.0783341i
\(890\) 0 0
\(891\) 10.1742 + 26.4254i 0.340850 + 0.885285i
\(892\) 0 0
\(893\) 16.7753 + 55.7828i 0.561363 + 1.86670i
\(894\) 0 0
\(895\) −9.37117 + 5.41045i −0.313244 + 0.180851i
\(896\) 0 0
\(897\) 24.4949 + 34.6410i 0.817861 + 1.15663i
\(898\) 0 0
\(899\) −9.00000 5.19615i −0.300167 0.173301i
\(900\) 0 0
\(901\) 6.92820i 0.230812i
\(902\) 0 0
\(903\) 4.00000 + 5.65685i 0.133112 + 0.188248i
\(904\) 0 0
\(905\) −30.4949 −1.01368
\(906\) 0 0
\(907\) 32.9166 19.0044i 1.09298 0.631031i 0.158610 0.987341i \(-0.449299\pi\)
0.934368 + 0.356311i \(0.115966\pi\)
\(908\) 0 0
\(909\) 28.0000 + 9.89949i 0.928701 + 0.328346i
\(910\) 0 0
\(911\) 28.6515 0.949268 0.474634 0.880183i \(-0.342581\pi\)
0.474634 + 0.880183i \(0.342581\pi\)
\(912\) 0 0
\(913\) −53.4949 −1.77042
\(914\) 0 0
\(915\) −0.348469 3.78194i −0.0115200 0.125027i
\(916\) 0 0
\(917\) −0.866070 + 0.500026i −0.0286002 + 0.0165123i
\(918\) 0 0
\(919\) 36.6969 1.21052 0.605260 0.796028i \(-0.293069\pi\)
0.605260 + 0.796028i \(0.293069\pi\)
\(920\) 0 0
\(921\) 30.2474 21.3882i 0.996687 0.704764i
\(922\) 0 0
\(923\) 20.7846i 0.684134i
\(924\) 0 0
\(925\) 20.0227 + 11.5601i 0.658342 + 0.380094i
\(926\) 0 0
\(927\) 2.75255 + 14.8099i 0.0904056 + 0.486422i
\(928\) 0 0
\(929\) −24.0959 + 13.9118i −0.790561 + 0.456431i −0.840160 0.542338i \(-0.817539\pi\)
0.0495987 + 0.998769i \(0.484206\pi\)
\(930\) 0 0
\(931\) 21.5783 + 20.3079i 0.707201 + 0.665563i
\(932\) 0 0
\(933\) 2.47219 + 26.8307i 0.0809360 + 0.878399i
\(934\) 0 0
\(935\) 2.44949 + 1.41421i 0.0801069 + 0.0462497i
\(936\) 0 0
\(937\) −11.5000 + 19.9186i −0.375689 + 0.650712i −0.990430 0.138017i \(-0.955927\pi\)
0.614741 + 0.788729i \(0.289260\pi\)
\(938\) 0 0
\(939\) −13.7702 + 29.8895i −0.449372 + 0.975407i
\(940\) 0 0
\(941\) 21.4949 37.2303i 0.700714 1.21367i −0.267503 0.963557i \(-0.586198\pi\)
0.968216 0.250114i \(-0.0804683\pi\)
\(942\) 0 0
\(943\) 21.2132i 0.690797i
\(944\) 0 0
\(945\) 2.30306 2.36773i 0.0749186 0.0770221i
\(946\) 0 0
\(947\) 6.55051 3.78194i 0.212863 0.122896i −0.389778 0.920909i \(-0.627448\pi\)
0.602641 + 0.798012i \(0.294115\pi\)
\(948\) 0 0
\(949\) 30.4770i 0.989326i
\(950\) 0 0
\(951\) 4.34847 9.43879i 0.141009 0.306074i
\(952\) 0 0
\(953\) −12.3990 21.4757i −0.401642 0.695665i 0.592282 0.805731i \(-0.298227\pi\)
−0.993924 + 0.110066i \(0.964894\pi\)
\(954\) 0 0
\(955\) 5.79796 + 10.0424i 0.187618 + 0.324963i
\(956\) 0 0
\(957\) −12.1237 5.58542i −0.391904 0.180551i
\(958\) 0 0
\(959\) 1.52781 + 0.882079i 0.0493354 + 0.0284838i
\(960\) 0 0
\(961\) 13.0000 0.419355
\(962\) 0 0
\(963\) 14.4495 2.68556i 0.465628 0.0865410i
\(964\) 0 0
\(965\) 18.2474 31.6055i 0.587406 1.01742i
\(966\) 0 0
\(967\) −25.6969 44.5084i −0.826358 1.43129i −0.900877 0.434074i \(-0.857076\pi\)
0.0745193 0.997220i \(-0.476258\pi\)
\(968\) 0 0
\(969\) −4.75255 0.667783i −0.152674 0.0214523i
\(970\) 0 0
\(971\) 4.07321 + 7.05501i 0.130716 + 0.226406i 0.923953 0.382507i \(-0.124939\pi\)
−0.793237 + 0.608913i \(0.791606\pi\)
\(972\) 0 0
\(973\) −3.22474 + 5.58542i −0.103381 + 0.179060i
\(974\) 0 0
\(975\) −14.6969 + 10.3923i −0.470679 + 0.332820i
\(976\) 0 0
\(977\) 40.1010 1.28295 0.641473 0.767146i \(-0.278324\pi\)
0.641473 + 0.767146i \(0.278324\pi\)
\(978\) 0 0
\(979\) 19.3485 + 11.1708i 0.618380 + 0.357022i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 28.8990 + 50.0545i 0.921734 + 1.59649i 0.796731 + 0.604334i \(0.206561\pi\)
0.125003 + 0.992156i \(0.460106\pi\)
\(984\) 0 0
\(985\) 14.3485 + 24.8523i 0.457180 + 0.791859i
\(986\) 0 0
\(987\) 9.44949 + 4.35340i 0.300781 + 0.138570i
\(988\) 0 0
\(989\) 62.9253i 2.00091i
\(990\) 0 0
\(991\) −15.3031 + 8.83523i −0.486118 + 0.280660i −0.722962 0.690887i \(-0.757220\pi\)
0.236845 + 0.971548i \(0.423887\pi\)
\(992\) 0 0
\(993\) 3.70204 + 40.1783i 0.117481 + 1.27502i
\(994\) 0 0
\(995\) 4.09978i 0.129972i
\(996\) 0 0
\(997\) 21.7196 37.6195i 0.687868 1.19142i −0.284658 0.958629i \(-0.591880\pi\)
0.972526 0.232793i \(-0.0747865\pi\)
\(998\) 0 0
\(999\) 10.8990 + 38.5337i 0.344828 + 1.21915i
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 912.2.bn.h.449.1 4
3.2 odd 2 912.2.bn.g.449.2 4
4.3 odd 2 114.2.h.f.107.2 yes 4
12.11 even 2 114.2.h.e.107.1 yes 4
19.8 odd 6 912.2.bn.g.65.1 4
57.8 even 6 inner 912.2.bn.h.65.1 4
76.27 even 6 114.2.h.e.65.2 4
228.179 odd 6 114.2.h.f.65.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.h.e.65.2 4 76.27 even 6
114.2.h.e.107.1 yes 4 12.11 even 2
114.2.h.f.65.2 yes 4 228.179 odd 6
114.2.h.f.107.2 yes 4 4.3 odd 2
912.2.bn.g.65.1 4 19.8 odd 6
912.2.bn.g.449.2 4 3.2 odd 2
912.2.bn.h.65.1 4 57.8 even 6 inner
912.2.bn.h.449.1 4 1.1 even 1 trivial