Properties

Label 1560.2.bg.f.601.2
Level $1560$
Weight $2$
Character 1560.601
Analytic conductor $12.457$
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] = [1560,2,Mod(601,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bg (of order \(3\), 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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 601.2
Root \(-0.780776 - 1.35234i\) of defining polynomial
Character \(\chi\) \(=\) 1560.601
Dual form 1560.2.bg.f.841.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +1.00000 q^{5} +(-0.219224 + 0.379706i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +1.00000 q^{5} +(-0.219224 + 0.379706i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-2.28078 - 3.95042i) q^{11} +(3.34233 - 1.35234i) q^{13} +(0.500000 + 0.866025i) q^{15} +(1.00000 - 1.73205i) q^{17} +(2.56155 - 4.43674i) q^{19} -0.438447 q^{21} +(-2.28078 - 3.95042i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(1.84233 + 3.19101i) q^{29} +4.12311 q^{31} +(2.28078 - 3.95042i) q^{33} +(-0.219224 + 0.379706i) q^{35} +(-2.28078 - 3.95042i) q^{37} +(2.84233 + 2.21837i) q^{39} +(1.00000 + 1.73205i) q^{41} +(3.06155 - 5.30277i) q^{43} +(-0.500000 + 0.866025i) q^{45} +6.56155 q^{47} +(3.40388 + 5.89570i) q^{49} +2.00000 q^{51} +1.12311 q^{53} +(-2.28078 - 3.95042i) q^{55} +5.12311 q^{57} +(1.84233 - 3.19101i) q^{59} +(-5.34233 + 9.25319i) q^{61} +(-0.219224 - 0.379706i) q^{63} +(3.34233 - 1.35234i) q^{65} +(1.21922 + 2.11176i) q^{67} +(2.28078 - 3.95042i) q^{69} +4.43845 q^{73} +(0.500000 + 0.866025i) q^{75} +2.00000 q^{77} -0.753789 q^{79} +(-0.500000 - 0.866025i) q^{81} -11.3693 q^{83} +(1.00000 - 1.73205i) q^{85} +(-1.84233 + 3.19101i) q^{87} +(-1.00000 - 1.73205i) q^{89} +(-0.219224 + 1.56557i) q^{91} +(2.06155 + 3.57071i) q^{93} +(2.56155 - 4.43674i) q^{95} +(-4.21922 + 7.30791i) q^{97} +4.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9} - 5 q^{11} + q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{19} - 10 q^{21} - 5 q^{23} + 4 q^{25} - 4 q^{27} - 5 q^{29} + 5 q^{33} - 5 q^{35} - 5 q^{37} - q^{39} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 18 q^{47} - 7 q^{49} + 8 q^{51} - 12 q^{53} - 5 q^{55} + 4 q^{57} - 5 q^{59} - 9 q^{61} - 5 q^{63} + q^{65} + 9 q^{67} + 5 q^{69} + 26 q^{73} + 2 q^{75} + 8 q^{77} - 36 q^{79} - 2 q^{81} + 4 q^{83} + 4 q^{85} + 5 q^{87} - 4 q^{89} - 5 q^{91} + 2 q^{95} - 21 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.219224 + 0.379706i −0.0828587 + 0.143516i −0.904477 0.426523i \(-0.859738\pi\)
0.821618 + 0.570038i \(0.193072\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.28078 3.95042i −0.687680 1.19110i −0.972587 0.232541i \(-0.925296\pi\)
0.284907 0.958555i \(-0.408037\pi\)
\(12\) 0 0
\(13\) 3.34233 1.35234i 0.926995 0.375073i
\(14\) 0 0
\(15\) 0.500000 + 0.866025i 0.129099 + 0.223607i
\(16\) 0 0
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 2.56155 4.43674i 0.587661 1.01786i −0.406877 0.913483i \(-0.633382\pi\)
0.994538 0.104375i \(-0.0332843\pi\)
\(20\) 0 0
\(21\) −0.438447 −0.0956770
\(22\) 0 0
\(23\) −2.28078 3.95042i −0.475575 0.823720i 0.524034 0.851697i \(-0.324427\pi\)
−0.999609 + 0.0279778i \(0.991093\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.84233 + 3.19101i 0.342112 + 0.592555i 0.984825 0.173552i \(-0.0555245\pi\)
−0.642713 + 0.766107i \(0.722191\pi\)
\(30\) 0 0
\(31\) 4.12311 0.740532 0.370266 0.928926i \(-0.379267\pi\)
0.370266 + 0.928926i \(0.379267\pi\)
\(32\) 0 0
\(33\) 2.28078 3.95042i 0.397032 0.687680i
\(34\) 0 0
\(35\) −0.219224 + 0.379706i −0.0370556 + 0.0641821i
\(36\) 0 0
\(37\) −2.28078 3.95042i −0.374957 0.649445i 0.615363 0.788244i \(-0.289009\pi\)
−0.990321 + 0.138798i \(0.955676\pi\)
\(38\) 0 0
\(39\) 2.84233 + 2.21837i 0.455137 + 0.355223i
\(40\) 0 0
\(41\) 1.00000 + 1.73205i 0.156174 + 0.270501i 0.933486 0.358614i \(-0.116751\pi\)
−0.777312 + 0.629115i \(0.783417\pi\)
\(42\) 0 0
\(43\) 3.06155 5.30277i 0.466882 0.808664i −0.532402 0.846492i \(-0.678710\pi\)
0.999284 + 0.0378277i \(0.0120438\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 0 0
\(47\) 6.56155 0.957101 0.478550 0.878060i \(-0.341162\pi\)
0.478550 + 0.878060i \(0.341162\pi\)
\(48\) 0 0
\(49\) 3.40388 + 5.89570i 0.486269 + 0.842242i
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 1.12311 0.154270 0.0771352 0.997021i \(-0.475423\pi\)
0.0771352 + 0.997021i \(0.475423\pi\)
\(54\) 0 0
\(55\) −2.28078 3.95042i −0.307540 0.532675i
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) 0 0
\(59\) 1.84233 3.19101i 0.239851 0.415434i −0.720820 0.693122i \(-0.756235\pi\)
0.960671 + 0.277688i \(0.0895681\pi\)
\(60\) 0 0
\(61\) −5.34233 + 9.25319i −0.684015 + 1.18475i 0.289730 + 0.957108i \(0.406434\pi\)
−0.973745 + 0.227641i \(0.926899\pi\)
\(62\) 0 0
\(63\) −0.219224 0.379706i −0.0276196 0.0478385i
\(64\) 0 0
\(65\) 3.34233 1.35234i 0.414565 0.167738i
\(66\) 0 0
\(67\) 1.21922 + 2.11176i 0.148952 + 0.257992i 0.930840 0.365426i \(-0.119077\pi\)
−0.781888 + 0.623418i \(0.785743\pi\)
\(68\) 0 0
\(69\) 2.28078 3.95042i 0.274573 0.475575i
\(70\) 0 0
\(71\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0 0
\(73\) 4.43845 0.519481 0.259740 0.965678i \(-0.416363\pi\)
0.259740 + 0.965678i \(0.416363\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −0.753789 −0.0848079 −0.0424039 0.999101i \(-0.513502\pi\)
−0.0424039 + 0.999101i \(0.513502\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −11.3693 −1.24794 −0.623972 0.781446i \(-0.714482\pi\)
−0.623972 + 0.781446i \(0.714482\pi\)
\(84\) 0 0
\(85\) 1.00000 1.73205i 0.108465 0.187867i
\(86\) 0 0
\(87\) −1.84233 + 3.19101i −0.197518 + 0.342112i
\(88\) 0 0
\(89\) −1.00000 1.73205i −0.106000 0.183597i 0.808146 0.588982i \(-0.200471\pi\)
−0.914146 + 0.405385i \(0.867138\pi\)
\(90\) 0 0
\(91\) −0.219224 + 1.56557i −0.0229809 + 0.164116i
\(92\) 0 0
\(93\) 2.06155 + 3.57071i 0.213773 + 0.370266i
\(94\) 0 0
\(95\) 2.56155 4.43674i 0.262810 0.455200i
\(96\) 0 0
\(97\) −4.21922 + 7.30791i −0.428397 + 0.742006i −0.996731 0.0807924i \(-0.974255\pi\)
0.568334 + 0.822798i \(0.307588\pi\)
\(98\) 0 0
\(99\) 4.56155 0.458453
\(100\) 0 0
\(101\) −4.12311 7.14143i −0.410264 0.710599i 0.584654 0.811283i \(-0.301230\pi\)
−0.994918 + 0.100684i \(0.967897\pi\)
\(102\) 0 0
\(103\) −3.56155 −0.350930 −0.175465 0.984486i \(-0.556143\pi\)
−0.175465 + 0.984486i \(0.556143\pi\)
\(104\) 0 0
\(105\) −0.438447 −0.0427881
\(106\) 0 0
\(107\) 1.43845 + 2.49146i 0.139060 + 0.240859i 0.927141 0.374713i \(-0.122259\pi\)
−0.788081 + 0.615571i \(0.788925\pi\)
\(108\) 0 0
\(109\) 17.8078 1.70567 0.852837 0.522177i \(-0.174880\pi\)
0.852837 + 0.522177i \(0.174880\pi\)
\(110\) 0 0
\(111\) 2.28078 3.95042i 0.216482 0.374957i
\(112\) 0 0
\(113\) −3.15767 + 5.46925i −0.297049 + 0.514503i −0.975459 0.220179i \(-0.929336\pi\)
0.678411 + 0.734683i \(0.262669\pi\)
\(114\) 0 0
\(115\) −2.28078 3.95042i −0.212683 0.368379i
\(116\) 0 0
\(117\) −0.500000 + 3.57071i −0.0462250 + 0.330113i
\(118\) 0 0
\(119\) 0.438447 + 0.759413i 0.0401924 + 0.0696153i
\(120\) 0 0
\(121\) −4.90388 + 8.49377i −0.445807 + 0.772161i
\(122\) 0 0
\(123\) −1.00000 + 1.73205i −0.0901670 + 0.156174i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.34233 2.32498i −0.119112 0.206309i 0.800304 0.599595i \(-0.204672\pi\)
−0.919416 + 0.393286i \(0.871338\pi\)
\(128\) 0 0
\(129\) 6.12311 0.539109
\(130\) 0 0
\(131\) −2.56155 −0.223804 −0.111902 0.993719i \(-0.535694\pi\)
−0.111902 + 0.993719i \(0.535694\pi\)
\(132\) 0 0
\(133\) 1.12311 + 1.94528i 0.0973856 + 0.168677i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.28078 3.95042i 0.194860 0.337507i −0.751995 0.659169i \(-0.770908\pi\)
0.946855 + 0.321662i \(0.104241\pi\)
\(138\) 0 0
\(139\) 8.90388 15.4220i 0.755217 1.30807i −0.190049 0.981775i \(-0.560865\pi\)
0.945266 0.326300i \(-0.105802\pi\)
\(140\) 0 0
\(141\) 3.28078 + 5.68247i 0.276291 + 0.478550i
\(142\) 0 0
\(143\) −12.9654 10.1192i −1.08422 0.846211i
\(144\) 0 0
\(145\) 1.84233 + 3.19101i 0.152997 + 0.264999i
\(146\) 0 0
\(147\) −3.40388 + 5.89570i −0.280747 + 0.486269i
\(148\) 0 0
\(149\) 6.96543 12.0645i 0.570631 0.988361i −0.425871 0.904784i \(-0.640032\pi\)
0.996501 0.0835772i \(-0.0266345\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 1.00000 + 1.73205i 0.0808452 + 0.140028i
\(154\) 0 0
\(155\) 4.12311 0.331176
\(156\) 0 0
\(157\) 11.2462 0.897545 0.448773 0.893646i \(-0.351861\pi\)
0.448773 + 0.893646i \(0.351861\pi\)
\(158\) 0 0
\(159\) 0.561553 + 0.972638i 0.0445340 + 0.0771352i
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −10.0616 + 17.4271i −0.788082 + 1.36500i 0.139059 + 0.990284i \(0.455592\pi\)
−0.927141 + 0.374713i \(0.877741\pi\)
\(164\) 0 0
\(165\) 2.28078 3.95042i 0.177558 0.307540i
\(166\) 0 0
\(167\) −10.7192 18.5662i −0.829478 1.43670i −0.898448 0.439080i \(-0.855304\pi\)
0.0689695 0.997619i \(-0.478029\pi\)
\(168\) 0 0
\(169\) 9.34233 9.03996i 0.718641 0.695382i
\(170\) 0 0
\(171\) 2.56155 + 4.43674i 0.195887 + 0.339286i
\(172\) 0 0
\(173\) 2.00000 3.46410i 0.152057 0.263371i −0.779926 0.625871i \(-0.784744\pi\)
0.931984 + 0.362500i \(0.118077\pi\)
\(174\) 0 0
\(175\) −0.219224 + 0.379706i −0.0165717 + 0.0287031i
\(176\) 0 0
\(177\) 3.68466 0.276956
\(178\) 0 0
\(179\) −5.40388 9.35980i −0.403905 0.699584i 0.590288 0.807193i \(-0.299014\pi\)
−0.994193 + 0.107608i \(0.965681\pi\)
\(180\) 0 0
\(181\) −3.12311 −0.232139 −0.116069 0.993241i \(-0.537029\pi\)
−0.116069 + 0.993241i \(0.537029\pi\)
\(182\) 0 0
\(183\) −10.6847 −0.789833
\(184\) 0 0
\(185\) −2.28078 3.95042i −0.167686 0.290441i
\(186\) 0 0
\(187\) −9.12311 −0.667148
\(188\) 0 0
\(189\) 0.219224 0.379706i 0.0159462 0.0276196i
\(190\) 0 0
\(191\) −10.0000 + 17.3205i −0.723575 + 1.25327i 0.235983 + 0.971757i \(0.424169\pi\)
−0.959558 + 0.281511i \(0.909164\pi\)
\(192\) 0 0
\(193\) 9.02699 + 15.6352i 0.649777 + 1.12545i 0.983176 + 0.182661i \(0.0584710\pi\)
−0.333399 + 0.942786i \(0.608196\pi\)
\(194\) 0 0
\(195\) 2.84233 + 2.21837i 0.203543 + 0.158861i
\(196\) 0 0
\(197\) 4.00000 + 6.92820i 0.284988 + 0.493614i 0.972606 0.232458i \(-0.0746770\pi\)
−0.687618 + 0.726073i \(0.741344\pi\)
\(198\) 0 0
\(199\) −6.78078 + 11.7446i −0.480676 + 0.832556i −0.999754 0.0221709i \(-0.992942\pi\)
0.519078 + 0.854727i \(0.326276\pi\)
\(200\) 0 0
\(201\) −1.21922 + 2.11176i −0.0859974 + 0.148952i
\(202\) 0 0
\(203\) −1.61553 −0.113388
\(204\) 0 0
\(205\) 1.00000 + 1.73205i 0.0698430 + 0.120972i
\(206\) 0 0
\(207\) 4.56155 0.317050
\(208\) 0 0
\(209\) −23.3693 −1.61649
\(210\) 0 0
\(211\) 7.02699 + 12.1711i 0.483758 + 0.837893i 0.999826 0.0186544i \(-0.00593824\pi\)
−0.516068 + 0.856547i \(0.672605\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.06155 5.30277i 0.208796 0.361646i
\(216\) 0 0
\(217\) −0.903882 + 1.56557i −0.0613595 + 0.106278i
\(218\) 0 0
\(219\) 2.21922 + 3.84381i 0.149961 + 0.259740i
\(220\) 0 0
\(221\) 1.00000 7.14143i 0.0672673 0.480384i
\(222\) 0 0
\(223\) −8.80776 15.2555i −0.589812 1.02158i −0.994257 0.107021i \(-0.965869\pi\)
0.404445 0.914562i \(-0.367465\pi\)
\(224\) 0 0
\(225\) −0.500000 + 0.866025i −0.0333333 + 0.0577350i
\(226\) 0 0
\(227\) −12.6847 + 21.9705i −0.841910 + 1.45823i 0.0463672 + 0.998924i \(0.485236\pi\)
−0.888278 + 0.459307i \(0.848098\pi\)
\(228\) 0 0
\(229\) 20.7386 1.37045 0.685224 0.728333i \(-0.259704\pi\)
0.685224 + 0.728333i \(0.259704\pi\)
\(230\) 0 0
\(231\) 1.00000 + 1.73205i 0.0657952 + 0.113961i
\(232\) 0 0
\(233\) −21.0540 −1.37929 −0.689646 0.724147i \(-0.742234\pi\)
−0.689646 + 0.724147i \(0.742234\pi\)
\(234\) 0 0
\(235\) 6.56155 0.428029
\(236\) 0 0
\(237\) −0.376894 0.652800i −0.0244819 0.0424039i
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 0.0345652 0.0598686i 0.00222654 0.00385648i −0.864910 0.501927i \(-0.832625\pi\)
0.867137 + 0.498071i \(0.165958\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 3.40388 + 5.89570i 0.217466 + 0.376662i
\(246\) 0 0
\(247\) 2.56155 18.2931i 0.162988 1.16397i
\(248\) 0 0
\(249\) −5.68466 9.84612i −0.360251 0.623972i
\(250\) 0 0
\(251\) −5.28078 + 9.14657i −0.333320 + 0.577327i −0.983161 0.182744i \(-0.941502\pi\)
0.649841 + 0.760070i \(0.274835\pi\)
\(252\) 0 0
\(253\) −10.4039 + 18.0201i −0.654086 + 1.13291i
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) −8.28078 14.3427i −0.516541 0.894675i −0.999816 0.0192060i \(-0.993886\pi\)
0.483275 0.875469i \(-0.339447\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −3.68466 −0.228075
\(262\) 0 0
\(263\) −1.96543 3.40423i −0.121194 0.209914i 0.799045 0.601271i \(-0.205339\pi\)
−0.920239 + 0.391357i \(0.872006\pi\)
\(264\) 0 0
\(265\) 1.12311 0.0689918
\(266\) 0 0
\(267\) 1.00000 1.73205i 0.0611990 0.106000i
\(268\) 0 0
\(269\) 4.12311 7.14143i 0.251390 0.435421i −0.712519 0.701653i \(-0.752446\pi\)
0.963909 + 0.266233i \(0.0857789\pi\)
\(270\) 0 0
\(271\) −12.8693 22.2903i −0.781755 1.35404i −0.930918 0.365227i \(-0.880991\pi\)
0.149163 0.988813i \(-0.452342\pi\)
\(272\) 0 0
\(273\) −1.46543 + 0.592932i −0.0886922 + 0.0358859i
\(274\) 0 0
\(275\) −2.28078 3.95042i −0.137536 0.238219i
\(276\) 0 0
\(277\) −5.71922 + 9.90599i −0.343635 + 0.595193i −0.985105 0.171955i \(-0.944992\pi\)
0.641470 + 0.767148i \(0.278325\pi\)
\(278\) 0 0
\(279\) −2.06155 + 3.57071i −0.123422 + 0.213773i
\(280\) 0 0
\(281\) −31.8617 −1.90071 −0.950356 0.311165i \(-0.899281\pi\)
−0.950356 + 0.311165i \(0.899281\pi\)
\(282\) 0 0
\(283\) −1.50000 2.59808i −0.0891657 0.154440i 0.817993 0.575228i \(-0.195087\pi\)
−0.907159 + 0.420789i \(0.861753\pi\)
\(284\) 0 0
\(285\) 5.12311 0.303467
\(286\) 0 0
\(287\) −0.876894 −0.0517614
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) −8.43845 −0.494671
\(292\) 0 0
\(293\) 6.56155 11.3649i 0.383330 0.663947i −0.608206 0.793779i \(-0.708111\pi\)
0.991536 + 0.129832i \(0.0414439\pi\)
\(294\) 0 0
\(295\) 1.84233 3.19101i 0.107265 0.185788i
\(296\) 0 0
\(297\) 2.28078 + 3.95042i 0.132344 + 0.229227i
\(298\) 0 0
\(299\) −12.9654 10.1192i −0.749810 0.585209i
\(300\) 0 0
\(301\) 1.34233 + 2.32498i 0.0773706 + 0.134010i
\(302\) 0 0
\(303\) 4.12311 7.14143i 0.236866 0.410264i
\(304\) 0 0
\(305\) −5.34233 + 9.25319i −0.305901 + 0.529836i
\(306\) 0 0
\(307\) −14.9309 −0.852150 −0.426075 0.904688i \(-0.640104\pi\)
−0.426075 + 0.904688i \(0.640104\pi\)
\(308\) 0 0
\(309\) −1.78078 3.08440i −0.101305 0.175465i
\(310\) 0 0
\(311\) −5.36932 −0.304466 −0.152233 0.988345i \(-0.548646\pi\)
−0.152233 + 0.988345i \(0.548646\pi\)
\(312\) 0 0
\(313\) −14.0540 −0.794378 −0.397189 0.917737i \(-0.630014\pi\)
−0.397189 + 0.917737i \(0.630014\pi\)
\(314\) 0 0
\(315\) −0.219224 0.379706i −0.0123519 0.0213940i
\(316\) 0 0
\(317\) 1.75379 0.0985026 0.0492513 0.998786i \(-0.484316\pi\)
0.0492513 + 0.998786i \(0.484316\pi\)
\(318\) 0 0
\(319\) 8.40388 14.5560i 0.470527 0.814977i
\(320\) 0 0
\(321\) −1.43845 + 2.49146i −0.0802863 + 0.139060i
\(322\) 0 0
\(323\) −5.12311 8.87348i −0.285057 0.493734i
\(324\) 0 0
\(325\) 3.34233 1.35234i 0.185399 0.0750146i
\(326\) 0 0
\(327\) 8.90388 + 15.4220i 0.492386 + 0.852837i
\(328\) 0 0
\(329\) −1.43845 + 2.49146i −0.0793042 + 0.137359i
\(330\) 0 0
\(331\) −4.78078 + 8.28055i −0.262775 + 0.455140i −0.966978 0.254859i \(-0.917971\pi\)
0.704203 + 0.709999i \(0.251304\pi\)
\(332\) 0 0
\(333\) 4.56155 0.249972
\(334\) 0 0
\(335\) 1.21922 + 2.11176i 0.0666133 + 0.115378i
\(336\) 0 0
\(337\) −21.8078 −1.18794 −0.593972 0.804485i \(-0.702441\pi\)
−0.593972 + 0.804485i \(0.702441\pi\)
\(338\) 0 0
\(339\) −6.31534 −0.343002
\(340\) 0 0
\(341\) −9.40388 16.2880i −0.509249 0.882045i
\(342\) 0 0
\(343\) −6.05398 −0.326884
\(344\) 0 0
\(345\) 2.28078 3.95042i 0.122793 0.212683i
\(346\) 0 0
\(347\) −5.68466 + 9.84612i −0.305168 + 0.528567i −0.977299 0.211866i \(-0.932046\pi\)
0.672130 + 0.740433i \(0.265379\pi\)
\(348\) 0 0
\(349\) 10.3423 + 17.9134i 0.553612 + 0.958884i 0.998010 + 0.0630549i \(0.0200843\pi\)
−0.444398 + 0.895830i \(0.646582\pi\)
\(350\) 0 0
\(351\) −3.34233 + 1.35234i −0.178400 + 0.0721828i
\(352\) 0 0
\(353\) 5.87689 + 10.1791i 0.312796 + 0.541778i 0.978966 0.204021i \(-0.0654012\pi\)
−0.666171 + 0.745799i \(0.732068\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.438447 + 0.759413i −0.0232051 + 0.0401924i
\(358\) 0 0
\(359\) 16.8769 0.890728 0.445364 0.895349i \(-0.353074\pi\)
0.445364 + 0.895349i \(0.353074\pi\)
\(360\) 0 0
\(361\) −3.62311 6.27540i −0.190690 0.330284i
\(362\) 0 0
\(363\) −9.80776 −0.514774
\(364\) 0 0
\(365\) 4.43845 0.232319
\(366\) 0 0
\(367\) −10.5885 18.3399i −0.552717 0.957334i −0.998077 0.0619828i \(-0.980258\pi\)
0.445360 0.895352i \(-0.353076\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −0.246211 + 0.426450i −0.0127827 + 0.0221402i
\(372\) 0 0
\(373\) −3.62311 + 6.27540i −0.187597 + 0.324928i −0.944449 0.328659i \(-0.893403\pi\)
0.756851 + 0.653587i \(0.226737\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 10.4730 + 8.17394i 0.539388 + 0.420979i
\(378\) 0 0
\(379\) 11.5885 + 20.0719i 0.595263 + 1.03103i 0.993510 + 0.113748i \(0.0362856\pi\)
−0.398246 + 0.917279i \(0.630381\pi\)
\(380\) 0 0
\(381\) 1.34233 2.32498i 0.0687696 0.119112i
\(382\) 0 0
\(383\) −8.52699 + 14.7692i −0.435709 + 0.754670i −0.997353 0.0727090i \(-0.976836\pi\)
0.561644 + 0.827379i \(0.310169\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 3.06155 + 5.30277i 0.155627 + 0.269555i
\(388\) 0 0
\(389\) 29.3002 1.48558 0.742789 0.669525i \(-0.233502\pi\)
0.742789 + 0.669525i \(0.233502\pi\)
\(390\) 0 0
\(391\) −9.12311 −0.461375
\(392\) 0 0
\(393\) −1.28078 2.21837i −0.0646066 0.111902i
\(394\) 0 0
\(395\) −0.753789 −0.0379272
\(396\) 0 0
\(397\) 6.06155 10.4989i 0.304221 0.526926i −0.672867 0.739764i \(-0.734937\pi\)
0.977087 + 0.212838i \(0.0682707\pi\)
\(398\) 0 0
\(399\) −1.12311 + 1.94528i −0.0562256 + 0.0973856i
\(400\) 0 0
\(401\) 2.87689 + 4.98293i 0.143665 + 0.248836i 0.928874 0.370395i \(-0.120778\pi\)
−0.785209 + 0.619231i \(0.787444\pi\)
\(402\) 0 0
\(403\) 13.7808 5.57586i 0.686469 0.277753i
\(404\) 0 0
\(405\) −0.500000 0.866025i −0.0248452 0.0430331i
\(406\) 0 0
\(407\) −10.4039 + 18.0201i −0.515701 + 0.893221i
\(408\) 0 0
\(409\) −16.9039 + 29.2784i −0.835843 + 1.44772i 0.0574989 + 0.998346i \(0.481687\pi\)
−0.893342 + 0.449377i \(0.851646\pi\)
\(410\) 0 0
\(411\) 4.56155 0.225005
\(412\) 0 0
\(413\) 0.807764 + 1.39909i 0.0397475 + 0.0688446i
\(414\) 0 0
\(415\) −11.3693 −0.558098
\(416\) 0 0
\(417\) 17.8078 0.872050
\(418\) 0 0
\(419\) −7.56155 13.0970i −0.369406 0.639830i 0.620067 0.784549i \(-0.287106\pi\)
−0.989473 + 0.144719i \(0.953772\pi\)
\(420\) 0 0
\(421\) −35.8078 −1.74516 −0.872582 0.488468i \(-0.837556\pi\)
−0.872582 + 0.488468i \(0.837556\pi\)
\(422\) 0 0
\(423\) −3.28078 + 5.68247i −0.159517 + 0.276291i
\(424\) 0 0
\(425\) 1.00000 1.73205i 0.0485071 0.0840168i
\(426\) 0 0
\(427\) −2.34233 4.05703i −0.113353 0.196334i
\(428\) 0 0
\(429\) 2.28078 16.2880i 0.110117 0.786392i
\(430\) 0 0
\(431\) 6.24621 + 10.8188i 0.300869 + 0.521121i 0.976333 0.216272i \(-0.0693898\pi\)
−0.675464 + 0.737393i \(0.736057\pi\)
\(432\) 0 0
\(433\) −15.7116 + 27.2134i −0.755054 + 1.30779i 0.190294 + 0.981727i \(0.439056\pi\)
−0.945348 + 0.326064i \(0.894278\pi\)
\(434\) 0 0
\(435\) −1.84233 + 3.19101i −0.0883329 + 0.152997i
\(436\) 0 0
\(437\) −23.3693 −1.11791
\(438\) 0 0
\(439\) 14.1501 + 24.5087i 0.675347 + 1.16974i 0.976367 + 0.216118i \(0.0693395\pi\)
−0.301020 + 0.953618i \(0.597327\pi\)
\(440\) 0 0
\(441\) −6.80776 −0.324179
\(442\) 0 0
\(443\) −29.3693 −1.39538 −0.697689 0.716401i \(-0.745788\pi\)
−0.697689 + 0.716401i \(0.745788\pi\)
\(444\) 0 0
\(445\) −1.00000 1.73205i −0.0474045 0.0821071i
\(446\) 0 0
\(447\) 13.9309 0.658908
\(448\) 0 0
\(449\) 10.6847 18.5064i 0.504240 0.873370i −0.495748 0.868467i \(-0.665106\pi\)
0.999988 0.00490311i \(-0.00156071\pi\)
\(450\) 0 0
\(451\) 4.56155 7.90084i 0.214795 0.372036i
\(452\) 0 0
\(453\) 8.00000 + 13.8564i 0.375873 + 0.651031i
\(454\) 0 0
\(455\) −0.219224 + 1.56557i −0.0102774 + 0.0733950i
\(456\) 0 0
\(457\) −2.46543 4.27026i −0.115328 0.199754i 0.802583 0.596541i \(-0.203459\pi\)
−0.917911 + 0.396787i \(0.870125\pi\)
\(458\) 0 0
\(459\) −1.00000 + 1.73205i −0.0466760 + 0.0808452i
\(460\) 0 0
\(461\) 13.4039 23.2162i 0.624281 1.08129i −0.364398 0.931243i \(-0.618725\pi\)
0.988679 0.150043i \(-0.0479413\pi\)
\(462\) 0 0
\(463\) 1.80776 0.0840139 0.0420070 0.999117i \(-0.486625\pi\)
0.0420070 + 0.999117i \(0.486625\pi\)
\(464\) 0 0
\(465\) 2.06155 + 3.57071i 0.0956022 + 0.165588i
\(466\) 0 0
\(467\) 21.6155 1.00025 0.500124 0.865954i \(-0.333288\pi\)
0.500124 + 0.865954i \(0.333288\pi\)
\(468\) 0 0
\(469\) −1.06913 −0.0493679
\(470\) 0 0
\(471\) 5.62311 + 9.73950i 0.259099 + 0.448773i
\(472\) 0 0
\(473\) −27.9309 −1.28426
\(474\) 0 0
\(475\) 2.56155 4.43674i 0.117532 0.203572i
\(476\) 0 0
\(477\) −0.561553 + 0.972638i −0.0257117 + 0.0445340i
\(478\) 0 0
\(479\) 10.9309 + 18.9328i 0.499444 + 0.865063i 1.00000 0.000641682i \(-0.000204254\pi\)
−0.500556 + 0.865704i \(0.666871\pi\)
\(480\) 0 0
\(481\) −12.9654 10.1192i −0.591173 0.461396i
\(482\) 0 0
\(483\) 1.00000 + 1.73205i 0.0455016 + 0.0788110i
\(484\) 0 0
\(485\) −4.21922 + 7.30791i −0.191585 + 0.331835i
\(486\) 0 0
\(487\) −13.6847 + 23.7025i −0.620111 + 1.07406i 0.369354 + 0.929289i \(0.379579\pi\)
−0.989465 + 0.144775i \(0.953754\pi\)
\(488\) 0 0
\(489\) −20.1231 −0.909998
\(490\) 0 0
\(491\) 2.24621 + 3.89055i 0.101370 + 0.175578i 0.912249 0.409635i \(-0.134344\pi\)
−0.810879 + 0.585213i \(0.801011\pi\)
\(492\) 0 0
\(493\) 7.36932 0.331897
\(494\) 0 0
\(495\) 4.56155 0.205027
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16.4924 0.738302 0.369151 0.929369i \(-0.379648\pi\)
0.369151 + 0.929369i \(0.379648\pi\)
\(500\) 0 0
\(501\) 10.7192 18.5662i 0.478900 0.829478i
\(502\) 0 0
\(503\) −16.6847 + 28.8987i −0.743932 + 1.28853i 0.206760 + 0.978392i \(0.433708\pi\)
−0.950692 + 0.310137i \(0.899625\pi\)
\(504\) 0 0
\(505\) −4.12311 7.14143i −0.183476 0.317789i
\(506\) 0 0
\(507\) 12.5000 + 3.57071i 0.555144 + 0.158581i
\(508\) 0 0
\(509\) 9.15767 + 15.8616i 0.405907 + 0.703051i 0.994427 0.105432i \(-0.0336225\pi\)
−0.588520 + 0.808483i \(0.700289\pi\)
\(510\) 0 0
\(511\) −0.973012 + 1.68531i −0.0430435 + 0.0745536i
\(512\) 0 0
\(513\) −2.56155 + 4.43674i −0.113095 + 0.195887i
\(514\) 0 0
\(515\) −3.56155 −0.156941
\(516\) 0 0
\(517\) −14.9654 25.9209i −0.658179 1.14000i
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −24.7386 −1.08382 −0.541910 0.840437i \(-0.682298\pi\)
−0.541910 + 0.840437i \(0.682298\pi\)
\(522\) 0 0
\(523\) −11.2808 19.5389i −0.493274 0.854375i 0.506696 0.862125i \(-0.330867\pi\)
−0.999970 + 0.00774928i \(0.997533\pi\)
\(524\) 0 0
\(525\) −0.438447 −0.0191354
\(526\) 0 0
\(527\) 4.12311 7.14143i 0.179605 0.311086i
\(528\) 0 0
\(529\) 1.09612 1.89853i 0.0476573 0.0825449i
\(530\) 0 0
\(531\) 1.84233 + 3.19101i 0.0799503 + 0.138478i
\(532\) 0 0
\(533\) 5.68466 + 4.43674i 0.246230 + 0.192177i
\(534\) 0 0
\(535\) 1.43845 + 2.49146i 0.0621895 + 0.107715i
\(536\) 0 0
\(537\) 5.40388 9.35980i 0.233195 0.403905i
\(538\) 0 0
\(539\) 15.5270 26.8935i 0.668795 1.15839i
\(540\) 0 0
\(541\) 5.94602 0.255640 0.127820 0.991797i \(-0.459202\pi\)
0.127820 + 0.991797i \(0.459202\pi\)
\(542\) 0 0
\(543\) −1.56155 2.70469i −0.0670126 0.116069i
\(544\) 0 0
\(545\) 17.8078 0.762801
\(546\) 0 0
\(547\) 28.6847 1.22647 0.613234 0.789902i \(-0.289868\pi\)
0.613234 + 0.789902i \(0.289868\pi\)
\(548\) 0 0
\(549\) −5.34233 9.25319i −0.228005 0.394916i
\(550\) 0 0
\(551\) 18.8769 0.804183
\(552\) 0 0
\(553\) 0.165248 0.286218i 0.00702707 0.0121712i
\(554\) 0 0
\(555\) 2.28078 3.95042i 0.0968136 0.167686i
\(556\) 0 0
\(557\) −17.6847 30.6307i −0.749323 1.29787i −0.948148 0.317830i \(-0.897046\pi\)
0.198825 0.980035i \(-0.436288\pi\)
\(558\) 0 0
\(559\) 3.06155 21.8639i 0.129490 0.924743i
\(560\) 0 0
\(561\) −4.56155 7.90084i −0.192589 0.333574i
\(562\) 0 0
\(563\) −9.80776 + 16.9875i −0.413348 + 0.715940i −0.995253 0.0973167i \(-0.968974\pi\)
0.581905 + 0.813256i \(0.302307\pi\)
\(564\) 0 0
\(565\) −3.15767 + 5.46925i −0.132844 + 0.230093i
\(566\) 0 0
\(567\) 0.438447 0.0184131
\(568\) 0 0
\(569\) −18.5616 32.1496i −0.778141 1.34778i −0.933012 0.359845i \(-0.882830\pi\)
0.154871 0.987935i \(-0.450504\pi\)
\(570\) 0 0
\(571\) 39.2311 1.64177 0.820884 0.571095i \(-0.193481\pi\)
0.820884 + 0.571095i \(0.193481\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −2.28078 3.95042i −0.0951150 0.164744i
\(576\) 0 0
\(577\) −38.9848 −1.62296 −0.811480 0.584380i \(-0.801338\pi\)
−0.811480 + 0.584380i \(0.801338\pi\)
\(578\) 0 0
\(579\) −9.02699 + 15.6352i −0.375149 + 0.649777i
\(580\) 0 0
\(581\) 2.49242 4.31700i 0.103403 0.179099i
\(582\) 0 0
\(583\) −2.56155 4.43674i −0.106089 0.183751i
\(584\) 0 0
\(585\) −0.500000 + 3.57071i −0.0206725 + 0.147631i
\(586\) 0 0
\(587\) 8.31534 + 14.4026i 0.343211 + 0.594459i 0.985027 0.172400i \(-0.0551521\pi\)
−0.641816 + 0.766859i \(0.721819\pi\)
\(588\) 0 0
\(589\) 10.5616 18.2931i 0.435181 0.753756i
\(590\) 0 0
\(591\) −4.00000 + 6.92820i −0.164538 + 0.284988i
\(592\) 0 0
\(593\) 43.3002 1.77813 0.889063 0.457785i \(-0.151357\pi\)
0.889063 + 0.457785i \(0.151357\pi\)
\(594\) 0 0
\(595\) 0.438447 + 0.759413i 0.0179746 + 0.0311329i
\(596\) 0 0
\(597\) −13.5616 −0.555037
\(598\) 0 0
\(599\) 3.36932 0.137667 0.0688333 0.997628i \(-0.478072\pi\)
0.0688333 + 0.997628i \(0.478072\pi\)
\(600\) 0 0
\(601\) 10.0346 + 17.3804i 0.409318 + 0.708960i 0.994814 0.101716i \(-0.0324332\pi\)
−0.585495 + 0.810676i \(0.699100\pi\)
\(602\) 0 0
\(603\) −2.43845 −0.0993012
\(604\) 0 0
\(605\) −4.90388 + 8.49377i −0.199371 + 0.345321i
\(606\) 0 0
\(607\) 15.6847 27.1666i 0.636621 1.10266i −0.349549 0.936918i \(-0.613665\pi\)
0.986169 0.165741i \(-0.0530016\pi\)
\(608\) 0 0
\(609\) −0.807764 1.39909i −0.0327323 0.0566939i
\(610\) 0 0
\(611\) 21.9309 8.87348i 0.887228 0.358983i
\(612\) 0 0
\(613\) 20.3078 + 35.1741i 0.820223 + 1.42067i 0.905516 + 0.424312i \(0.139484\pi\)
−0.0852933 + 0.996356i \(0.527183\pi\)
\(614\) 0 0
\(615\) −1.00000 + 1.73205i −0.0403239 + 0.0698430i
\(616\) 0 0
\(617\) −7.71922 + 13.3701i −0.310764 + 0.538259i −0.978528 0.206114i \(-0.933918\pi\)
0.667764 + 0.744373i \(0.267252\pi\)
\(618\) 0 0
\(619\) 33.4233 1.34340 0.671698 0.740825i \(-0.265565\pi\)
0.671698 + 0.740825i \(0.265565\pi\)
\(620\) 0 0
\(621\) 2.28078 + 3.95042i 0.0915244 + 0.158525i
\(622\) 0 0
\(623\) 0.876894 0.0351320
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −11.6847 20.2384i −0.466640 0.808245i
\(628\) 0 0
\(629\) −9.12311 −0.363762
\(630\) 0 0
\(631\) −1.21922 + 2.11176i −0.0485365 + 0.0840677i −0.889273 0.457377i \(-0.848789\pi\)
0.840736 + 0.541445i \(0.182122\pi\)
\(632\) 0 0
\(633\) −7.02699 + 12.1711i −0.279298 + 0.483758i
\(634\) 0 0
\(635\) −1.34233 2.32498i −0.0532687 0.0922641i
\(636\) 0 0
\(637\) 19.3499 + 15.1021i 0.766671 + 0.598369i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.0000 + 29.4449i −0.671460 + 1.16300i 0.306031 + 0.952022i \(0.400999\pi\)
−0.977490 + 0.210981i \(0.932334\pi\)
\(642\) 0 0
\(643\) −1.41146 + 2.44472i −0.0556625 + 0.0964103i −0.892514 0.451020i \(-0.851060\pi\)
0.836851 + 0.547430i \(0.184394\pi\)
\(644\) 0 0
\(645\) 6.12311 0.241097
\(646\) 0 0
\(647\) 18.0540 + 31.2704i 0.709775 + 1.22937i 0.964940 + 0.262469i \(0.0845367\pi\)
−0.255165 + 0.966897i \(0.582130\pi\)
\(648\) 0 0
\(649\) −16.8078 −0.659762
\(650\) 0 0
\(651\) −1.80776 −0.0708519
\(652\) 0 0
\(653\) 3.87689 + 6.71498i 0.151715 + 0.262777i 0.931858 0.362823i \(-0.118187\pi\)
−0.780143 + 0.625601i \(0.784854\pi\)
\(654\) 0 0
\(655\) −2.56155 −0.100088
\(656\) 0 0
\(657\) −2.21922 + 3.84381i −0.0865802 + 0.149961i
\(658\) 0 0
\(659\) 16.0885 27.8662i 0.626721 1.08551i −0.361485 0.932378i \(-0.617730\pi\)
0.988205 0.153134i \(-0.0489366\pi\)
\(660\) 0 0
\(661\) 5.21922 + 9.03996i 0.203004 + 0.351614i 0.949495 0.313782i \(-0.101596\pi\)
−0.746491 + 0.665396i \(0.768263\pi\)
\(662\) 0 0
\(663\) 6.68466 2.70469i 0.259611 0.105041i
\(664\) 0 0
\(665\) 1.12311 + 1.94528i 0.0435522 + 0.0754346i
\(666\) 0 0
\(667\) 8.40388 14.5560i 0.325400 0.563609i
\(668\) 0 0
\(669\) 8.80776 15.2555i 0.340528 0.589812i
\(670\) 0 0
\(671\) 48.7386 1.88153
\(672\) 0 0
\(673\) −16.3423 28.3057i −0.629950 1.09111i −0.987561 0.157235i \(-0.949742\pi\)
0.357611 0.933871i \(-0.383592\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 10.2462 0.393794 0.196897 0.980424i \(-0.436914\pi\)
0.196897 + 0.980424i \(0.436914\pi\)
\(678\) 0 0
\(679\) −1.84991 3.20413i −0.0709929 0.122963i
\(680\) 0 0
\(681\) −25.3693 −0.972154
\(682\) 0 0
\(683\) −13.1231 + 22.7299i −0.502142 + 0.869735i 0.497855 + 0.867260i \(0.334121\pi\)
−0.999997 + 0.00247477i \(0.999212\pi\)
\(684\) 0 0
\(685\) 2.28078 3.95042i 0.0871440 0.150938i
\(686\) 0 0
\(687\) 10.3693 + 17.9602i 0.395614 + 0.685224i
\(688\) 0 0
\(689\) 3.75379 1.51883i 0.143008 0.0578626i
\(690\) 0 0
\(691\) 11.1501 + 19.3125i 0.424170 + 0.734683i 0.996342 0.0854498i \(-0.0272327\pi\)
−0.572173 + 0.820133i \(0.693899\pi\)
\(692\) 0 0
\(693\) −1.00000 + 1.73205i −0.0379869 + 0.0657952i
\(694\) 0 0
\(695\) 8.90388 15.4220i 0.337743 0.584989i
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −10.5270 18.2333i −0.398167 0.689646i
\(700\) 0 0
\(701\) −35.6847 −1.34779 −0.673895 0.738827i \(-0.735380\pi\)
−0.673895 + 0.738827i \(0.735380\pi\)
\(702\) 0 0
\(703\) −23.3693 −0.881390
\(704\) 0 0
\(705\) 3.28078 + 5.68247i 0.123561 + 0.214014i
\(706\) 0 0
\(707\) 3.61553 0.135976
\(708\) 0 0
\(709\) 21.9039 37.9386i 0.822618 1.42482i −0.0811090 0.996705i \(-0.525846\pi\)
0.903727 0.428110i \(-0.140820\pi\)
\(710\) 0 0
\(711\) 0.376894 0.652800i 0.0141346 0.0244819i
\(712\) 0 0
\(713\) −9.40388 16.2880i −0.352178 0.609990i
\(714\) 0 0
\(715\) −12.9654 10.1192i −0.484880 0.378437i
\(716\) 0 0
\(717\) 2.00000 + 3.46410i 0.0746914 + 0.129369i
\(718\) 0 0
\(719\) −9.75379 + 16.8941i −0.363755 + 0.630042i −0.988576 0.150726i \(-0.951839\pi\)
0.624821 + 0.780768i \(0.285172\pi\)
\(720\) 0 0
\(721\) 0.780776 1.35234i 0.0290776 0.0503639i
\(722\) 0 0
\(723\) 0.0691303 0.00257098
\(724\) 0 0
\(725\) 1.84233 + 3.19101i 0.0684224 + 0.118511i
\(726\) 0 0
\(727\) −24.6847 −0.915503 −0.457752 0.889080i \(-0.651345\pi\)
−0.457752 + 0.889080i \(0.651345\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.12311 10.6055i −0.226471 0.392260i
\(732\) 0 0
\(733\) 17.4233 0.643544 0.321772 0.946817i \(-0.395722\pi\)
0.321772 + 0.946817i \(0.395722\pi\)
\(734\) 0 0
\(735\) −3.40388 + 5.89570i −0.125554 + 0.217466i
\(736\) 0 0
\(737\) 5.56155 9.63289i 0.204862 0.354832i
\(738\) 0 0
\(739\) 24.2462 + 41.9957i 0.891911 + 1.54484i 0.837581 + 0.546313i \(0.183969\pi\)
0.0543300 + 0.998523i \(0.482698\pi\)
\(740\) 0 0
\(741\) 17.1231 6.92820i 0.629033 0.254514i
\(742\) 0 0
\(743\) 12.4039 + 21.4842i 0.455054 + 0.788177i 0.998691 0.0511433i \(-0.0162865\pi\)
−0.543637 + 0.839320i \(0.682953\pi\)
\(744\) 0 0
\(745\) 6.96543 12.0645i 0.255194 0.442009i
\(746\) 0 0
\(747\) 5.68466 9.84612i 0.207991 0.360251i
\(748\) 0 0
\(749\) −1.26137 −0.0460893
\(750\) 0 0
\(751\) −11.5270 19.9653i −0.420626 0.728545i 0.575375 0.817890i \(-0.304856\pi\)
−0.996001 + 0.0893445i \(0.971523\pi\)
\(752\) 0 0
\(753\) −10.5616 −0.384884
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −17.4924 30.2978i −0.635773 1.10119i −0.986351 0.164657i \(-0.947348\pi\)
0.350578 0.936534i \(-0.385985\pi\)
\(758\) 0 0
\(759\) −20.8078 −0.755274
\(760\) 0 0
\(761\) 10.8769 18.8393i 0.394287 0.682925i −0.598723 0.800956i \(-0.704325\pi\)
0.993010 + 0.118031i \(0.0376582\pi\)
\(762\) 0 0
\(763\) −3.90388 + 6.76172i −0.141330 + 0.244791i
\(764\) 0 0
\(765\) 1.00000 + 1.73205i 0.0361551 + 0.0626224i
\(766\) 0 0
\(767\) 1.84233 13.1569i 0.0665227 0.475067i
\(768\) 0 0
\(769\) 0.596118 + 1.03251i 0.0214966 + 0.0372331i 0.876574 0.481268i \(-0.159824\pi\)
−0.855077 + 0.518501i \(0.826490\pi\)
\(770\) 0 0
\(771\) 8.28078 14.3427i 0.298225 0.516541i
\(772\) 0 0
\(773\) 18.4384 31.9363i 0.663185 1.14867i −0.316589 0.948563i \(-0.602538\pi\)
0.979774 0.200107i \(-0.0641290\pi\)
\(774\) 0 0
\(775\) 4.12311 0.148106
\(776\) 0 0
\(777\) 1.00000 + 1.73205i 0.0358748 + 0.0621370i
\(778\) 0 0
\(779\) 10.2462 0.367109
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.84233 3.19101i −0.0658395 0.114037i
\(784\) 0 0
\(785\) 11.2462 0.401394
\(786\) 0 0
\(787\) −11.8153 + 20.4648i −0.421171 + 0.729490i −0.996054 0.0887454i \(-0.971714\pi\)
0.574883 + 0.818236i \(0.305048\pi\)
\(788\) 0 0
\(789\) 1.96543 3.40423i 0.0699713 0.121194i
\(790\) 0 0
\(791\) −1.38447 2.39798i −0.0492262 0.0852622i
\(792\) 0 0
\(793\) −5.34233 + 38.1519i −0.189712 + 1.35481i
\(794\) 0 0
\(795\) 0.561553 + 0.972638i 0.0199162 + 0.0344959i
\(796\) 0 0
\(797\) −13.3153 + 23.0628i −0.471654 + 0.816928i −0.999474 0.0324280i \(-0.989676\pi\)
0.527820 + 0.849356i \(0.323009\pi\)
\(798\) 0 0
\(799\) 6.56155 11.3649i 0.232131 0.402063i
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) −10.1231 17.5337i −0.357237 0.618752i
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 8.24621 0.290280
\(808\) 0 0
\(809\) 21.4384 + 37.1325i 0.753736 + 1.30551i 0.946000 + 0.324165i \(0.105083\pi\)
−0.192265 + 0.981343i \(0.561583\pi\)
\(810\) 0 0
\(811\) −16.1922 −0.568586 −0.284293 0.958737i \(-0.591759\pi\)
−0.284293 + 0.958737i \(0.591759\pi\)
\(812\) 0 0
\(813\) 12.8693 22.2903i 0.451347 0.781755i
\(814\) 0 0
\(815\) −10.0616 + 17.4271i −0.352441 + 0.610445i
\(816\) 0 0
\(817\) −15.6847 27.1666i −0.548737 0.950440i
\(818\) 0 0
\(819\) −1.24621 0.972638i −0.0435461 0.0339867i
\(820\) 0 0
\(821\) −2.84233 4.92306i −0.0991980 0.171816i 0.812155 0.583442i \(-0.198294\pi\)
−0.911353 + 0.411626i \(0.864961\pi\)
\(822\) 0 0
\(823\) 11.9309 20.6649i 0.415884 0.720332i −0.579637 0.814875i \(-0.696806\pi\)
0.995521 + 0.0945428i \(0.0301389\pi\)
\(824\) 0 0
\(825\) 2.28078 3.95042i 0.0794064 0.137536i
\(826\) 0 0
\(827\) 56.7386 1.97300 0.986498 0.163774i \(-0.0523669\pi\)
0.986498 + 0.163774i \(0.0523669\pi\)
\(828\) 0 0
\(829\) −18.6577 32.3160i −0.648008 1.12238i −0.983598 0.180374i \(-0.942269\pi\)
0.335590 0.942008i \(-0.391064\pi\)
\(830\) 0 0
\(831\) −11.4384 −0.396795
\(832\) 0 0
\(833\) 13.6155 0.471750
\(834\) 0 0
\(835\) −10.7192 18.5662i −0.370954 0.642511i
\(836\) 0 0
\(837\) −4.12311 −0.142515
\(838\) 0 0
\(839\) 14.1231 24.4619i 0.487584 0.844520i −0.512314 0.858798i \(-0.671212\pi\)
0.999898 + 0.0142782i \(0.00454504\pi\)
\(840\) 0 0
\(841\) 7.71165 13.3570i 0.265919 0.460585i
\(842\) 0 0
\(843\) −15.9309 27.5931i −0.548688 0.950356i
\(844\) 0 0
\(845\) 9.34233 9.03996i 0.321386 0.310984i
\(846\) 0 0
\(847\) −2.15009 3.72407i −0.0738781 0.127961i
\(848\) 0 0
\(849\) 1.50000 2.59808i 0.0514799 0.0891657i
\(850\) 0 0
\(851\) −10.4039 + 18.0201i −0.356640 + 0.617719i
\(852\) 0 0
\(853\) 50.6155 1.73304 0.866521 0.499140i \(-0.166351\pi\)
0.866521 + 0.499140i \(0.166351\pi\)
\(854\) 0 0
\(855\) 2.56155 + 4.43674i 0.0876033 + 0.151733i
\(856\) 0 0
\(857\) 54.4233 1.85906 0.929532 0.368741i \(-0.120211\pi\)
0.929532 + 0.368741i \(0.120211\pi\)
\(858\) 0 0
\(859\) 1.80776 0.0616801 0.0308401 0.999524i \(-0.490182\pi\)
0.0308401 + 0.999524i \(0.490182\pi\)
\(860\) 0 0
\(861\) −0.438447 0.759413i −0.0149422 0.0258807i
\(862\) 0 0
\(863\) 55.3002 1.88244 0.941220 0.337794i \(-0.109681\pi\)
0.941220 + 0.337794i \(0.109681\pi\)
\(864\) 0 0
\(865\) 2.00000 3.46410i 0.0680020 0.117783i
\(866\) 0 0
\(867\) −6.50000 + 11.2583i −0.220752 + 0.382353i
\(868\) 0 0
\(869\) 1.71922 + 2.97778i 0.0583207 + 0.101014i
\(870\) 0 0
\(871\) 6.93087 + 5.40938i 0.234844 + 0.183290i
\(872\) 0 0
\(873\) −4.21922 7.30791i −0.142799 0.247335i
\(874\) 0 0
\(875\) −0.219224 + 0.379706i −0.00741111 + 0.0128364i
\(876\) 0 0
\(877\) 7.15767 12.3974i 0.241697 0.418632i −0.719501 0.694492i \(-0.755629\pi\)
0.961198 + 0.275860i \(0.0889625\pi\)
\(878\) 0 0
\(879\) 13.1231 0.442631
\(880\) 0 0
\(881\) −24.5616 42.5419i −0.827500 1.43327i −0.899994 0.435903i \(-0.856429\pi\)
0.0724940 0.997369i \(-0.476904\pi\)
\(882\) 0 0
\(883\) 11.8769 0.399689 0.199845 0.979828i \(-0.435956\pi\)
0.199845 + 0.979828i \(0.435956\pi\)
\(884\) 0 0
\(885\) 3.68466 0.123858
\(886\) 0 0
\(887\) 20.8963 + 36.1935i 0.701629 + 1.21526i 0.967894 + 0.251358i \(0.0808770\pi\)
−0.266265 + 0.963900i \(0.585790\pi\)
\(888\) 0 0
\(889\) 1.17708 0.0394780
\(890\) 0 0
\(891\) −2.28078 + 3.95042i −0.0764089 + 0.132344i
\(892\) 0 0
\(893\) 16.8078 29.1119i 0.562450 0.974193i
\(894\) 0 0
\(895\) −5.40388 9.35980i −0.180632 0.312864i
\(896\) 0 0
\(897\) 2.28078 16.2880i 0.0761529 0.543841i
\(898\) 0 0
\(899\) 7.59612 + 13.1569i 0.253345 + 0.438806i
\(900\) 0 0
\(901\) 1.12311 1.94528i 0.0374161 0.0648065i
\(902\) 0 0
\(903\) −1.34233 + 2.32498i −0.0446699 + 0.0773706i
\(904\) 0 0
\(905\) −3.12311 −0.103816
\(906\) 0 0
\(907\) −18.7192 32.4226i −0.621562 1.07658i −0.989195 0.146606i \(-0.953165\pi\)
0.367633 0.929971i \(-0.380168\pi\)
\(908\) 0 0
\(909\) 8.24621 0.273510
\(910\) 0 0
\(911\) 37.6155 1.24626 0.623129 0.782119i \(-0.285861\pi\)
0.623129 + 0.782119i \(0.285861\pi\)
\(912\) 0 0
\(913\) 25.9309 + 44.9136i 0.858187 + 1.48642i
\(914\) 0 0
\(915\) −10.6847 −0.353224
\(916\) 0 0
\(917\) 0.561553 0.972638i 0.0185441 0.0321193i
\(918\) 0 0
\(919\) −16.4924 + 28.5657i −0.544035 + 0.942296i 0.454632 + 0.890679i \(0.349771\pi\)
−0.998667 + 0.0516167i \(0.983563\pi\)
\(920\) 0 0
\(921\) −7.46543 12.9305i −0.245994 0.426075i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.28078 3.95042i −0.0749915 0.129889i
\(926\) 0 0
\(927\) 1.78078 3.08440i 0.0584884 0.101305i
\(928\) 0 0
\(929\) −15.6155 + 27.0469i −0.512329 + 0.887380i 0.487569 + 0.873084i \(0.337884\pi\)
−0.999898 + 0.0142951i \(0.995450\pi\)
\(930\) 0 0
\(931\) 34.8769 1.14304
\(932\) 0 0
\(933\) −2.68466 4.64996i −0.0878918 0.152233i
\(934\) 0 0
\(935\) −9.12311 −0.298357
\(936\) 0 0
\(937\) −24.2462 −0.792089 −0.396045 0.918231i \(-0.629617\pi\)
−0.396045 + 0.918231i \(0.629617\pi\)
\(938\) 0 0
\(939\) −7.02699 12.1711i −0.229317 0.397189i
\(940\) 0 0
\(941\) 56.3542 1.83709 0.918547 0.395313i \(-0.129364\pi\)
0.918547 + 0.395313i \(0.129364\pi\)
\(942\) 0 0
\(943\) 4.56155 7.90084i 0.148545 0.257287i
\(944\) 0 0
\(945\) 0.219224 0.379706i 0.00713134 0.0123519i
\(946\) 0 0
\(947\) −15.3153 26.5269i −0.497682 0.862010i 0.502315 0.864685i \(-0.332482\pi\)
−0.999996 + 0.00267484i \(0.999149\pi\)
\(948\) 0 0
\(949\) 14.8348 6.00231i 0.481556 0.194843i
\(950\) 0 0
\(951\) 0.876894 + 1.51883i 0.0284352 + 0.0492513i
\(952\) 0 0
\(953\) −10.5961 + 18.3530i −0.343242 + 0.594512i −0.985033 0.172367i \(-0.944858\pi\)
0.641791 + 0.766880i \(0.278192\pi\)
\(954\) 0 0
\(955\) −10.0000 + 17.3205i −0.323592 + 0.560478i
\(956\) 0 0
\(957\) 16.8078 0.543318
\(958\) 0 0
\(959\) 1.00000 + 1.73205i 0.0322917 + 0.0559308i
\(960\) 0 0
\(961\) −14.0000 −0.451613
\(962\) 0 0
\(963\) −2.87689 −0.0927066
\(964\) 0 0
\(965\) 9.02699 + 15.6352i 0.290589 + 0.503315i
\(966\) 0 0
\(967\) 10.2462 0.329496 0.164748 0.986336i \(-0.447319\pi\)
0.164748 + 0.986336i \(0.447319\pi\)
\(968\) 0 0
\(969\) 5.12311 8.87348i 0.164578 0.285057i
\(970\) 0 0
\(971\) −18.2462 + 31.6034i −0.585549 + 1.01420i 0.409258 + 0.912419i \(0.365788\pi\)
−0.994807 + 0.101782i \(0.967546\pi\)
\(972\) 0 0
\(973\) 3.90388 + 6.76172i 0.125153 + 0.216771i
\(974\) 0 0
\(975\) 2.84233 + 2.21837i 0.0910274 + 0.0710447i
\(976\) 0 0
\(977\) 0.650093 + 1.12599i 0.0207983 + 0.0360238i 0.876237 0.481880i \(-0.160046\pi\)
−0.855439 + 0.517904i \(0.826713\pi\)
\(978\) 0 0
\(979\) −4.56155 + 7.90084i −0.145788 + 0.252512i
\(980\) 0 0
\(981\) −8.90388 + 15.4220i −0.284279 + 0.492386i
\(982\) 0 0
\(983\) 1.19224 0.0380264 0.0190132 0.999819i \(-0.493948\pi\)
0.0190132 + 0.999819i \(0.493948\pi\)
\(984\) 0 0
\(985\) 4.00000 + 6.92820i 0.127451 + 0.220751i
\(986\) 0 0
\(987\) −2.87689 −0.0915726
\(988\) 0 0
\(989\) −27.9309 −0.888150
\(990\) 0 0
\(991\) −17.8423 30.9038i −0.566780 0.981692i −0.996882 0.0789116i \(-0.974856\pi\)
0.430101 0.902781i \(-0.358478\pi\)
\(992\) 0 0
\(993\) −9.56155 −0.303427
\(994\) 0 0
\(995\) −6.78078 + 11.7446i −0.214965 + 0.372330i
\(996\) 0 0
\(997\) −18.2732 + 31.6501i −0.578718 + 1.00237i 0.416909 + 0.908948i \(0.363113\pi\)
−0.995627 + 0.0934206i \(0.970220\pi\)
\(998\) 0 0
\(999\) 2.28078 + 3.95042i 0.0721606 + 0.124986i
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 1560.2.bg.f.601.2 4
13.9 even 3 inner 1560.2.bg.f.841.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.bg.f.601.2 4 1.1 even 1 trivial
1560.2.bg.f.841.2 yes 4 13.9 even 3 inner