Properties

Label 2352.2.bl.o.31.3
Level $2352$
Weight $2$
Character 2352.31
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.3
Root \(0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 2352.31
Dual form 2352.2.bl.o.607.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.521114 - 0.300865i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(0.521114 - 0.300865i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(0.141713 + 0.0818181i) q^{11} -3.37849i q^{13} +0.601731i q^{15} +(-1.84677 - 1.06623i) q^{17} +(-0.476834 - 0.825901i) q^{19} +(-6.05870 + 3.49799i) q^{23} +(-2.31896 + 4.01656i) q^{25} +1.00000 q^{27} -1.28897 q^{29} +(-1.67725 + 2.90508i) q^{31} +(-0.141713 + 0.0818181i) q^{33} +(-2.58187 - 4.47192i) q^{37} +(2.92586 + 1.68925i) q^{39} -6.67896i q^{41} +6.09172i q^{43} +(-0.521114 - 0.300865i) q^{45} +(1.84882 + 3.20225i) q^{47} +(1.84677 - 1.06623i) q^{51} +(-6.94028 + 12.0209i) q^{53} +0.0984649 q^{55} +0.953669 q^{57} +(-4.02594 + 6.97313i) q^{59} +(-0.157153 + 0.0907323i) q^{61} +(-1.01647 - 1.76058i) q^{65} +(7.87476 + 4.54649i) q^{67} -6.99598i q^{69} -3.36066i q^{71} +(-10.5172 - 6.07210i) q^{73} +(-2.31896 - 4.01656i) q^{75} +(-6.05213 + 3.49420i) q^{79} +(-0.500000 + 0.866025i) q^{81} -12.9788 q^{83} -1.28317 q^{85} +(0.644486 - 1.11628i) q^{87} +(12.8911 - 7.44266i) q^{89} +(-1.67725 - 2.90508i) q^{93} +(-0.496970 - 0.286926i) q^{95} -9.08274i q^{97} -0.163636i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{9} - 24 q^{23} + 12 q^{25} + 8 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{47} - 8 q^{53} - 64 q^{55} - 24 q^{59} - 48 q^{61} + 8 q^{65} + 48 q^{67} - 48 q^{73} + 12 q^{75} + 24 q^{79} - 4 q^{81} - 64 q^{85} - 8 q^{87} - 48 q^{89} + 16 q^{93} - 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0.521114 0.300865i 0.233049 0.134551i −0.378929 0.925426i \(-0.623707\pi\)
0.611978 + 0.790875i \(0.290374\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 0.141713 + 0.0818181i 0.0427281 + 0.0246691i 0.521212 0.853427i \(-0.325480\pi\)
−0.478484 + 0.878096i \(0.658813\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i −0.883457 0.468513i \(-0.844790\pi\)
0.883457 0.468513i \(-0.155210\pi\)
\(14\) 0 0
\(15\) 0.601731i 0.155366i
\(16\) 0 0
\(17\) −1.84677 1.06623i −0.447907 0.258599i 0.259039 0.965867i \(-0.416594\pi\)
−0.706946 + 0.707268i \(0.749928\pi\)
\(18\) 0 0
\(19\) −0.476834 0.825901i −0.109393 0.189475i 0.806131 0.591737i \(-0.201557\pi\)
−0.915525 + 0.402262i \(0.868224\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.05870 + 3.49799i −1.26333 + 0.729382i −0.973717 0.227763i \(-0.926859\pi\)
−0.289610 + 0.957145i \(0.593525\pi\)
\(24\) 0 0
\(25\) −2.31896 + 4.01656i −0.463792 + 0.803311i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.28897 −0.239356 −0.119678 0.992813i \(-0.538186\pi\)
−0.119678 + 0.992813i \(0.538186\pi\)
\(30\) 0 0
\(31\) −1.67725 + 2.90508i −0.301242 + 0.521767i −0.976418 0.215890i \(-0.930735\pi\)
0.675175 + 0.737657i \(0.264068\pi\)
\(32\) 0 0
\(33\) −0.141713 + 0.0818181i −0.0246691 + 0.0142427i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.58187 4.47192i −0.424456 0.735179i 0.571913 0.820314i \(-0.306201\pi\)
−0.996369 + 0.0851346i \(0.972868\pi\)
\(38\) 0 0
\(39\) 2.92586 + 1.68925i 0.468513 + 0.270496i
\(40\) 0 0
\(41\) 6.67896i 1.04308i −0.853227 0.521539i \(-0.825358\pi\)
0.853227 0.521539i \(-0.174642\pi\)
\(42\) 0 0
\(43\) 6.09172i 0.928978i 0.885579 + 0.464489i \(0.153762\pi\)
−0.885579 + 0.464489i \(0.846238\pi\)
\(44\) 0 0
\(45\) −0.521114 0.300865i −0.0776831 0.0448504i
\(46\) 0 0
\(47\) 1.84882 + 3.20225i 0.269678 + 0.467096i 0.968779 0.247927i \(-0.0797494\pi\)
−0.699101 + 0.715023i \(0.746416\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.84677 1.06623i 0.258599 0.149302i
\(52\) 0 0
\(53\) −6.94028 + 12.0209i −0.953321 + 1.65120i −0.215156 + 0.976580i \(0.569026\pi\)
−0.738165 + 0.674620i \(0.764307\pi\)
\(54\) 0 0
\(55\) 0.0984649 0.0132770
\(56\) 0 0
\(57\) 0.953669 0.126317
\(58\) 0 0
\(59\) −4.02594 + 6.97313i −0.524133 + 0.907824i 0.475473 + 0.879730i \(0.342277\pi\)
−0.999605 + 0.0280937i \(0.991056\pi\)
\(60\) 0 0
\(61\) −0.157153 + 0.0907323i −0.0201214 + 0.0116171i −0.510027 0.860158i \(-0.670365\pi\)
0.489906 + 0.871776i \(0.337031\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.01647 1.76058i −0.126078 0.218373i
\(66\) 0 0
\(67\) 7.87476 + 4.54649i 0.962055 + 0.555443i 0.896805 0.442426i \(-0.145882\pi\)
0.0652500 + 0.997869i \(0.479216\pi\)
\(68\) 0 0
\(69\) 6.99598i 0.842217i
\(70\) 0 0
\(71\) 3.36066i 0.398837i −0.979914 0.199419i \(-0.936095\pi\)
0.979914 0.199419i \(-0.0639054\pi\)
\(72\) 0 0
\(73\) −10.5172 6.07210i −1.23094 0.710686i −0.263717 0.964600i \(-0.584949\pi\)
−0.967227 + 0.253914i \(0.918282\pi\)
\(74\) 0 0
\(75\) −2.31896 4.01656i −0.267770 0.463792i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.05213 + 3.49420i −0.680918 + 0.393128i −0.800201 0.599732i \(-0.795274\pi\)
0.119283 + 0.992860i \(0.461941\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −12.9788 −1.42460 −0.712302 0.701873i \(-0.752348\pi\)
−0.712302 + 0.701873i \(0.752348\pi\)
\(84\) 0 0
\(85\) −1.28317 −0.139179
\(86\) 0 0
\(87\) 0.644486 1.11628i 0.0690961 0.119678i
\(88\) 0 0
\(89\) 12.8911 7.44266i 1.36645 0.788920i 0.375977 0.926629i \(-0.377307\pi\)
0.990473 + 0.137709i \(0.0439739\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.67725 2.90508i −0.173922 0.301242i
\(94\) 0 0
\(95\) −0.496970 0.286926i −0.0509881 0.0294380i
\(96\) 0 0
\(97\) 9.08274i 0.922213i −0.887345 0.461106i \(-0.847453\pi\)
0.887345 0.461106i \(-0.152547\pi\)
\(98\) 0 0
\(99\) 0.163636i 0.0164461i
\(100\) 0 0
\(101\) −7.37548 4.25824i −0.733888 0.423710i 0.0859551 0.996299i \(-0.472606\pi\)
−0.819843 + 0.572589i \(0.805939\pi\)
\(102\) 0 0
\(103\) −8.38043 14.5153i −0.825749 1.43024i −0.901346 0.433100i \(-0.857420\pi\)
0.0755973 0.997138i \(-0.475914\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.80822 + 2.19867i −0.368154 + 0.212554i −0.672652 0.739959i \(-0.734845\pi\)
0.304498 + 0.952513i \(0.401511\pi\)
\(108\) 0 0
\(109\) −5.77263 + 9.99849i −0.552918 + 0.957681i 0.445145 + 0.895459i \(0.353152\pi\)
−0.998062 + 0.0622227i \(0.980181\pi\)
\(110\) 0 0
\(111\) 5.16373 0.490120
\(112\) 0 0
\(113\) −1.09821 −0.103311 −0.0516554 0.998665i \(-0.516450\pi\)
−0.0516554 + 0.998665i \(0.516450\pi\)
\(114\) 0 0
\(115\) −2.10485 + 3.64571i −0.196278 + 0.339964i
\(116\) 0 0
\(117\) −2.92586 + 1.68925i −0.270496 + 0.156171i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.48661 9.50309i −0.498783 0.863917i
\(122\) 0 0
\(123\) 5.78415 + 3.33948i 0.521539 + 0.301111i
\(124\) 0 0
\(125\) 5.79943i 0.518717i
\(126\) 0 0
\(127\) 5.22625i 0.463755i −0.972745 0.231877i \(-0.925513\pi\)
0.972745 0.231877i \(-0.0744868\pi\)
\(128\) 0 0
\(129\) −5.27558 3.04586i −0.464489 0.268173i
\(130\) 0 0
\(131\) 11.2659 + 19.5132i 0.984309 + 1.70487i 0.644966 + 0.764211i \(0.276871\pi\)
0.339343 + 0.940663i \(0.389795\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.521114 0.300865i 0.0448504 0.0258944i
\(136\) 0 0
\(137\) −7.29579 + 12.6367i −0.623322 + 1.07963i 0.365541 + 0.930795i \(0.380884\pi\)
−0.988863 + 0.148830i \(0.952449\pi\)
\(138\) 0 0
\(139\) −8.30816 −0.704689 −0.352345 0.935870i \(-0.614615\pi\)
−0.352345 + 0.935870i \(0.614615\pi\)
\(140\) 0 0
\(141\) −3.69764 −0.311397
\(142\) 0 0
\(143\) 0.276422 0.478777i 0.0231156 0.0400373i
\(144\) 0 0
\(145\) −0.671701 + 0.387807i −0.0557818 + 0.0322056i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.40083 9.35450i −0.442453 0.766351i 0.555418 0.831571i \(-0.312558\pi\)
−0.997871 + 0.0652205i \(0.979225\pi\)
\(150\) 0 0
\(151\) 12.2618 + 7.07937i 0.997854 + 0.576111i 0.907613 0.419809i \(-0.137903\pi\)
0.0902412 + 0.995920i \(0.471236\pi\)
\(152\) 0 0
\(153\) 2.13246i 0.172400i
\(154\) 0 0
\(155\) 2.01850i 0.162130i
\(156\) 0 0
\(157\) 0.946253 + 0.546320i 0.0755192 + 0.0436011i 0.537284 0.843401i \(-0.319450\pi\)
−0.461765 + 0.887002i \(0.652784\pi\)
\(158\) 0 0
\(159\) −6.94028 12.0209i −0.550400 0.953321i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.6627 9.04284i 1.22679 0.708290i 0.260436 0.965491i \(-0.416134\pi\)
0.966358 + 0.257201i \(0.0828003\pi\)
\(164\) 0 0
\(165\) −0.0492325 + 0.0852731i −0.00383274 + 0.00663850i
\(166\) 0 0
\(167\) −19.0196 −1.47178 −0.735889 0.677103i \(-0.763235\pi\)
−0.735889 + 0.677103i \(0.763235\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) −0.476834 + 0.825901i −0.0364644 + 0.0631583i
\(172\) 0 0
\(173\) 7.14939 4.12770i 0.543558 0.313823i −0.202962 0.979187i \(-0.565057\pi\)
0.746520 + 0.665363i \(0.231723\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.02594 6.97313i −0.302608 0.524133i
\(178\) 0 0
\(179\) −4.76008 2.74823i −0.355785 0.205413i 0.311445 0.950264i \(-0.399187\pi\)
−0.667230 + 0.744852i \(0.732520\pi\)
\(180\) 0 0
\(181\) 11.8519i 0.880946i −0.897766 0.440473i \(-0.854811\pi\)
0.897766 0.440473i \(-0.145189\pi\)
\(182\) 0 0
\(183\) 0.181465i 0.0134143i
\(184\) 0 0
\(185\) −2.69089 1.55359i −0.197838 0.114222i
\(186\) 0 0
\(187\) −0.174474 0.302198i −0.0127588 0.0220989i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0470 + 11.5741i −1.45055 + 0.837476i −0.998513 0.0545227i \(-0.982636\pi\)
−0.452038 + 0.891999i \(0.649303\pi\)
\(192\) 0 0
\(193\) 0.256029 0.443455i 0.0184294 0.0319206i −0.856664 0.515875i \(-0.827467\pi\)
0.875093 + 0.483955i \(0.160800\pi\)
\(194\) 0 0
\(195\) 2.03294 0.145582
\(196\) 0 0
\(197\) −12.7934 −0.911495 −0.455748 0.890109i \(-0.650628\pi\)
−0.455748 + 0.890109i \(0.650628\pi\)
\(198\) 0 0
\(199\) 7.90360 13.6894i 0.560271 0.970418i −0.437201 0.899364i \(-0.644030\pi\)
0.997472 0.0710545i \(-0.0226364\pi\)
\(200\) 0 0
\(201\) −7.87476 + 4.54649i −0.555443 + 0.320685i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00947 3.48050i −0.140347 0.243089i
\(206\) 0 0
\(207\) 6.05870 + 3.49799i 0.421109 + 0.243127i
\(208\) 0 0
\(209\) 0.156055i 0.0107945i
\(210\) 0 0
\(211\) 0.191712i 0.0131980i −0.999978 0.00659899i \(-0.997899\pi\)
0.999978 0.00659899i \(-0.00210054\pi\)
\(212\) 0 0
\(213\) 2.91042 + 1.68033i 0.199419 + 0.115134i
\(214\) 0 0
\(215\) 1.83279 + 3.17448i 0.124995 + 0.216498i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.5172 6.07210i 0.710686 0.410315i
\(220\) 0 0
\(221\) −3.60226 + 6.23929i −0.242314 + 0.419700i
\(222\) 0 0
\(223\) 10.0111 0.670392 0.335196 0.942148i \(-0.391197\pi\)
0.335196 + 0.942148i \(0.391197\pi\)
\(224\) 0 0
\(225\) 4.63792 0.309195
\(226\) 0 0
\(227\) −3.06817 + 5.31422i −0.203641 + 0.352717i −0.949699 0.313164i \(-0.898611\pi\)
0.746058 + 0.665881i \(0.231944\pi\)
\(228\) 0 0
\(229\) −8.17961 + 4.72250i −0.540524 + 0.312072i −0.745291 0.666739i \(-0.767690\pi\)
0.204767 + 0.978811i \(0.434356\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.1108 + 19.2445i 0.727895 + 1.26075i 0.957771 + 0.287531i \(0.0928345\pi\)
−0.229876 + 0.973220i \(0.573832\pi\)
\(234\) 0 0
\(235\) 1.92689 + 1.11249i 0.125697 + 0.0725710i
\(236\) 0 0
\(237\) 6.98840i 0.453945i
\(238\) 0 0
\(239\) 20.7962i 1.34520i −0.740008 0.672598i \(-0.765178\pi\)
0.740008 0.672598i \(-0.234822\pi\)
\(240\) 0 0
\(241\) −11.3063 6.52769i −0.728302 0.420486i 0.0894984 0.995987i \(-0.471474\pi\)
−0.817801 + 0.575501i \(0.804807\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.79030 + 1.61098i −0.177543 + 0.102504i
\(248\) 0 0
\(249\) 6.48938 11.2399i 0.411248 0.712302i
\(250\) 0 0
\(251\) −17.8975 −1.12968 −0.564839 0.825201i \(-0.691062\pi\)
−0.564839 + 0.825201i \(0.691062\pi\)
\(252\) 0 0
\(253\) −1.14480 −0.0719727
\(254\) 0 0
\(255\) 0.641585 1.11126i 0.0401776 0.0695896i
\(256\) 0 0
\(257\) 5.78415 3.33948i 0.360805 0.208311i −0.308629 0.951183i \(-0.599870\pi\)
0.669434 + 0.742872i \(0.266537\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.644486 + 1.11628i 0.0398927 + 0.0690961i
\(262\) 0 0
\(263\) −21.1452 12.2082i −1.30387 0.752790i −0.322804 0.946466i \(-0.604626\pi\)
−0.981066 + 0.193676i \(0.937959\pi\)
\(264\) 0 0
\(265\) 8.35236i 0.513081i
\(266\) 0 0
\(267\) 14.8853i 0.910966i
\(268\) 0 0
\(269\) −0.992403 0.572964i −0.0605079 0.0349342i 0.469441 0.882964i \(-0.344456\pi\)
−0.529949 + 0.848030i \(0.677789\pi\)
\(270\) 0 0
\(271\) −2.82578 4.89440i −0.171654 0.297314i 0.767344 0.641235i \(-0.221578\pi\)
−0.938998 + 0.343922i \(0.888244\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.657254 + 0.379466i −0.0396339 + 0.0228826i
\(276\) 0 0
\(277\) 9.63792 16.6934i 0.579087 1.00301i −0.416498 0.909137i \(-0.636743\pi\)
0.995584 0.0938706i \(-0.0299240\pi\)
\(278\) 0 0
\(279\) 3.35449 0.200828
\(280\) 0 0
\(281\) 8.80369 0.525184 0.262592 0.964907i \(-0.415423\pi\)
0.262592 + 0.964907i \(0.415423\pi\)
\(282\) 0 0
\(283\) −13.0725 + 22.6423i −0.777081 + 1.34594i 0.156536 + 0.987672i \(0.449967\pi\)
−0.933617 + 0.358272i \(0.883366\pi\)
\(284\) 0 0
\(285\) 0.496970 0.286926i 0.0294380 0.0169960i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.22630 10.7843i −0.366253 0.634368i
\(290\) 0 0
\(291\) 7.86588 + 4.54137i 0.461106 + 0.266220i
\(292\) 0 0
\(293\) 18.2640i 1.06699i 0.845802 + 0.533497i \(0.179123\pi\)
−0.845802 + 0.533497i \(0.820877\pi\)
\(294\) 0 0
\(295\) 4.84506i 0.282090i
\(296\) 0 0
\(297\) 0.141713 + 0.0818181i 0.00822303 + 0.00474757i
\(298\) 0 0
\(299\) 11.8179 + 20.4693i 0.683449 + 1.18377i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 7.37548 4.25824i 0.423710 0.244629i
\(304\) 0 0
\(305\) −0.0545964 + 0.0945638i −0.00312618 + 0.00541471i
\(306\) 0 0
\(307\) 21.4472 1.22405 0.612027 0.790837i \(-0.290354\pi\)
0.612027 + 0.790837i \(0.290354\pi\)
\(308\) 0 0
\(309\) 16.7609 0.953492
\(310\) 0 0
\(311\) 7.24000 12.5400i 0.410543 0.711081i −0.584407 0.811461i \(-0.698673\pi\)
0.994949 + 0.100380i \(0.0320060\pi\)
\(312\) 0 0
\(313\) −1.28590 + 0.742415i −0.0726834 + 0.0419638i −0.535901 0.844281i \(-0.680028\pi\)
0.463218 + 0.886244i \(0.346695\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.74541 + 9.95135i 0.322695 + 0.558923i 0.981043 0.193790i \(-0.0620780\pi\)
−0.658348 + 0.752713i \(0.728745\pi\)
\(318\) 0 0
\(319\) −0.182664 0.105461i −0.0102272 0.00590469i
\(320\) 0 0
\(321\) 4.39735i 0.245436i
\(322\) 0 0
\(323\) 2.03366i 0.113156i
\(324\) 0 0
\(325\) 13.5699 + 7.83459i 0.752723 + 0.434585i
\(326\) 0 0
\(327\) −5.77263 9.99849i −0.319227 0.552918i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.2703 10.5484i 1.00423 0.579790i 0.0947302 0.995503i \(-0.469801\pi\)
0.909496 + 0.415713i \(0.136468\pi\)
\(332\) 0 0
\(333\) −2.58187 + 4.47192i −0.141485 + 0.245060i
\(334\) 0 0
\(335\) 5.47153 0.298942
\(336\) 0 0
\(337\) −5.62683 −0.306513 −0.153256 0.988186i \(-0.548976\pi\)
−0.153256 + 0.988186i \(0.548976\pi\)
\(338\) 0 0
\(339\) 0.549104 0.951076i 0.0298232 0.0516554i
\(340\) 0 0
\(341\) −0.475376 + 0.274458i −0.0257430 + 0.0148628i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.10485 3.64571i −0.113321 0.196278i
\(346\) 0 0
\(347\) −21.1108 12.1883i −1.13329 0.654305i −0.188529 0.982068i \(-0.560372\pi\)
−0.944760 + 0.327763i \(0.893705\pi\)
\(348\) 0 0
\(349\) 2.84502i 0.152290i 0.997097 + 0.0761451i \(0.0242612\pi\)
−0.997097 + 0.0761451i \(0.975739\pi\)
\(350\) 0 0
\(351\) 3.37849i 0.180331i
\(352\) 0 0
\(353\) 21.3545 + 12.3290i 1.13659 + 0.656208i 0.945583 0.325381i \(-0.105493\pi\)
0.191003 + 0.981589i \(0.438826\pi\)
\(354\) 0 0
\(355\) −1.01111 1.75129i −0.0536640 0.0929488i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.8472 + 17.8096i −1.62805 + 0.939957i −0.643381 + 0.765546i \(0.722469\pi\)
−0.984673 + 0.174411i \(0.944198\pi\)
\(360\) 0 0
\(361\) 9.04526 15.6668i 0.476066 0.824571i
\(362\) 0 0
\(363\) 10.9732 0.575945
\(364\) 0 0
\(365\) −7.30754 −0.382494
\(366\) 0 0
\(367\) 11.6338 20.1504i 0.607280 1.05184i −0.384406 0.923164i \(-0.625594\pi\)
0.991687 0.128676i \(-0.0410728\pi\)
\(368\) 0 0
\(369\) −5.78415 + 3.33948i −0.301111 + 0.173846i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.9866 + 19.0294i 0.568865 + 0.985303i 0.996679 + 0.0814363i \(0.0259507\pi\)
−0.427813 + 0.903867i \(0.640716\pi\)
\(374\) 0 0
\(375\) −5.02246 2.89972i −0.259358 0.149741i
\(376\) 0 0
\(377\) 4.35478i 0.224283i
\(378\) 0 0
\(379\) 19.7876i 1.01642i 0.861233 + 0.508211i \(0.169693\pi\)
−0.861233 + 0.508211i \(0.830307\pi\)
\(380\) 0 0
\(381\) 4.52607 + 2.61313i 0.231877 + 0.133874i
\(382\) 0 0
\(383\) 6.82843 + 11.8272i 0.348916 + 0.604341i 0.986057 0.166407i \(-0.0532164\pi\)
−0.637141 + 0.770747i \(0.719883\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.27558 3.04586i 0.268173 0.154830i
\(388\) 0 0
\(389\) 6.52505 11.3017i 0.330833 0.573020i −0.651842 0.758354i \(-0.726004\pi\)
0.982675 + 0.185335i \(0.0593370\pi\)
\(390\) 0 0
\(391\) 14.9187 0.754470
\(392\) 0 0
\(393\) −22.5319 −1.13658
\(394\) 0 0
\(395\) −2.10257 + 3.64175i −0.105792 + 0.183237i
\(396\) 0 0
\(397\) −5.65172 + 3.26302i −0.283652 + 0.163766i −0.635075 0.772450i \(-0.719031\pi\)
0.351424 + 0.936217i \(0.385698\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.7156 + 27.2201i 0.784797 + 1.35931i 0.929120 + 0.369778i \(0.120566\pi\)
−0.144323 + 0.989531i \(0.546100\pi\)
\(402\) 0 0
\(403\) 9.81478 + 5.66657i 0.488909 + 0.282272i
\(404\) 0 0
\(405\) 0.601731i 0.0299002i
\(406\) 0 0
\(407\) 0.844973i 0.0418838i
\(408\) 0 0
\(409\) 30.1568 + 17.4110i 1.49116 + 0.860921i 0.999949 0.0101207i \(-0.00322157\pi\)
0.491210 + 0.871041i \(0.336555\pi\)
\(410\) 0 0
\(411\) −7.29579 12.6367i −0.359875 0.623322i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.76342 + 3.90486i −0.332003 + 0.191682i
\(416\) 0 0
\(417\) 4.15408 7.19508i 0.203426 0.352345i
\(418\) 0 0
\(419\) 11.9597 0.584271 0.292135 0.956377i \(-0.405634\pi\)
0.292135 + 0.956377i \(0.405634\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) 1.84882 3.20225i 0.0898927 0.155699i
\(424\) 0 0
\(425\) 8.56516 4.94510i 0.415471 0.239873i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.276422 + 0.478777i 0.0133458 + 0.0231156i
\(430\) 0 0
\(431\) −16.5809 9.57300i −0.798675 0.461115i 0.0443327 0.999017i \(-0.485884\pi\)
−0.843008 + 0.537902i \(0.819217\pi\)
\(432\) 0 0
\(433\) 1.66205i 0.0798730i 0.999202 + 0.0399365i \(0.0127156\pi\)
−0.999202 + 0.0399365i \(0.987284\pi\)
\(434\) 0 0
\(435\) 0.775614i 0.0371878i
\(436\) 0 0
\(437\) 5.77799 + 3.33593i 0.276399 + 0.159579i
\(438\) 0 0
\(439\) −8.27932 14.3402i −0.395151 0.684421i 0.597970 0.801519i \(-0.295974\pi\)
−0.993120 + 0.117098i \(0.962641\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.8781 9.16722i 0.754391 0.435548i −0.0728872 0.997340i \(-0.523221\pi\)
0.827278 + 0.561792i \(0.189888\pi\)
\(444\) 0 0
\(445\) 4.47847 7.75695i 0.212300 0.367715i
\(446\) 0 0
\(447\) 10.8017 0.510901
\(448\) 0 0
\(449\) 18.9899 0.896187 0.448093 0.893987i \(-0.352103\pi\)
0.448093 + 0.893987i \(0.352103\pi\)
\(450\) 0 0
\(451\) 0.546460 0.946496i 0.0257318 0.0445687i
\(452\) 0 0
\(453\) −12.2618 + 7.07937i −0.576111 + 0.332618i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.41976 + 14.5835i 0.393860 + 0.682185i 0.992955 0.118493i \(-0.0378062\pi\)
−0.599095 + 0.800678i \(0.704473\pi\)
\(458\) 0 0
\(459\) −1.84677 1.06623i −0.0861998 0.0497675i
\(460\) 0 0
\(461\) 36.8939i 1.71832i 0.511708 + 0.859160i \(0.329013\pi\)
−0.511708 + 0.859160i \(0.670987\pi\)
\(462\) 0 0
\(463\) 35.2394i 1.63771i −0.573997 0.818857i \(-0.694608\pi\)
0.573997 0.818857i \(-0.305392\pi\)
\(464\) 0 0
\(465\) −1.74807 1.00925i −0.0810650 0.0468029i
\(466\) 0 0
\(467\) 17.9123 + 31.0250i 0.828882 + 1.43567i 0.898916 + 0.438122i \(0.144356\pi\)
−0.0700331 + 0.997545i \(0.522310\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.946253 + 0.546320i −0.0436011 + 0.0251731i
\(472\) 0 0
\(473\) −0.498413 + 0.863276i −0.0229170 + 0.0396935i
\(474\) 0 0
\(475\) 4.42304 0.202943
\(476\) 0 0
\(477\) 13.8806 0.635547
\(478\) 0 0
\(479\) 3.05297 5.28790i 0.139494 0.241610i −0.787811 0.615917i \(-0.788786\pi\)
0.927305 + 0.374306i \(0.122119\pi\)
\(480\) 0 0
\(481\) −15.1084 + 8.72281i −0.688882 + 0.397726i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.73268 4.73314i −0.124085 0.214921i
\(486\) 0 0
\(487\) −14.7285 8.50353i −0.667414 0.385332i 0.127682 0.991815i \(-0.459246\pi\)
−0.795096 + 0.606484i \(0.792580\pi\)
\(488\) 0 0
\(489\) 18.0857i 0.817863i
\(490\) 0 0
\(491\) 31.2897i 1.41209i 0.708169 + 0.706043i \(0.249521\pi\)
−0.708169 + 0.706043i \(0.750479\pi\)
\(492\) 0 0
\(493\) 2.38043 + 1.37434i 0.107209 + 0.0618973i
\(494\) 0 0
\(495\) −0.0492325 0.0852731i −0.00221283 0.00383274i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18.4043 10.6257i 0.823891 0.475674i −0.0278654 0.999612i \(-0.508871\pi\)
0.851756 + 0.523938i \(0.175538\pi\)
\(500\) 0 0
\(501\) 9.50978 16.4714i 0.424866 0.735889i
\(502\) 0 0
\(503\) −0.728285 −0.0324726 −0.0162363 0.999868i \(-0.505168\pi\)
−0.0162363 + 0.999868i \(0.505168\pi\)
\(504\) 0 0
\(505\) −5.12462 −0.228043
\(506\) 0 0
\(507\) −0.792893 + 1.37333i −0.0352136 + 0.0609918i
\(508\) 0 0
\(509\) 27.4031 15.8212i 1.21462 0.701263i 0.250860 0.968023i \(-0.419287\pi\)
0.963763 + 0.266761i \(0.0859534\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −0.476834 0.825901i −0.0210528 0.0364644i
\(514\) 0 0
\(515\) −8.73432 5.04276i −0.384880 0.222211i
\(516\) 0 0
\(517\) 0.605068i 0.0266108i
\(518\) 0 0
\(519\) 8.25540i 0.362372i
\(520\) 0 0
\(521\) 16.4836 + 9.51681i 0.722159 + 0.416939i 0.815547 0.578691i \(-0.196436\pi\)
−0.0933876 + 0.995630i \(0.529770\pi\)
\(522\) 0 0
\(523\) 2.45644 + 4.25468i 0.107413 + 0.186044i 0.914721 0.404085i \(-0.132410\pi\)
−0.807309 + 0.590129i \(0.799077\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.19497 3.57667i 0.269857 0.155802i
\(528\) 0 0
\(529\) 12.9719 22.4680i 0.563995 0.976869i
\(530\) 0 0
\(531\) 8.05188 0.349422
\(532\) 0 0
\(533\) −22.5648 −0.977391
\(534\) 0 0
\(535\) −1.32301 + 2.29152i −0.0571987 + 0.0990711i
\(536\) 0 0
\(537\) 4.76008 2.74823i 0.205413 0.118595i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0443 + 24.3254i 0.603811 + 1.04583i 0.992238 + 0.124352i \(0.0396853\pi\)
−0.388427 + 0.921480i \(0.626981\pi\)
\(542\) 0 0
\(543\) 10.2641 + 5.92596i 0.440473 + 0.254307i
\(544\) 0 0
\(545\) 6.94714i 0.297583i
\(546\) 0 0
\(547\) 18.5886i 0.794792i 0.917647 + 0.397396i \(0.130086\pi\)
−0.917647 + 0.397396i \(0.869914\pi\)
\(548\) 0 0
\(549\) 0.157153 + 0.0907323i 0.00670713 + 0.00387236i
\(550\) 0 0
\(551\) 0.614626 + 1.06456i 0.0261840 + 0.0453519i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.69089 1.55359i 0.114222 0.0659461i
\(556\) 0 0
\(557\) 17.9939 31.1663i 0.762424 1.32056i −0.179174 0.983817i \(-0.557342\pi\)
0.941598 0.336740i \(-0.109324\pi\)
\(558\) 0 0
\(559\) 20.5808 0.870476
\(560\) 0 0
\(561\) 0.348948 0.0147326
\(562\) 0 0
\(563\) −12.6731 + 21.9505i −0.534109 + 0.925105i 0.465097 + 0.885260i \(0.346020\pi\)
−0.999206 + 0.0398445i \(0.987314\pi\)
\(564\) 0 0
\(565\) −0.572292 + 0.330413i −0.0240765 + 0.0139006i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.82160 + 15.2795i 0.369821 + 0.640549i 0.989537 0.144277i \(-0.0460857\pi\)
−0.619716 + 0.784826i \(0.712752\pi\)
\(570\) 0 0
\(571\) −25.9138 14.9613i −1.08446 0.626112i −0.152362 0.988325i \(-0.548688\pi\)
−0.932095 + 0.362213i \(0.882021\pi\)
\(572\) 0 0
\(573\) 23.1483i 0.967034i
\(574\) 0 0
\(575\) 32.4468i 1.35313i
\(576\) 0 0
\(577\) −7.36021 4.24942i −0.306410 0.176906i 0.338909 0.940819i \(-0.389942\pi\)
−0.645319 + 0.763913i \(0.723275\pi\)
\(578\) 0 0
\(579\) 0.256029 + 0.443455i 0.0106402 + 0.0184294i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.96706 + 1.13568i −0.0814672 + 0.0470351i
\(584\) 0 0
\(585\) −1.01647 + 1.76058i −0.0420259 + 0.0727910i
\(586\) 0 0
\(587\) −21.1156 −0.871535 −0.435767 0.900059i \(-0.643523\pi\)
−0.435767 + 0.900059i \(0.643523\pi\)
\(588\) 0 0
\(589\) 3.19908 0.131816
\(590\) 0 0
\(591\) 6.39672 11.0794i 0.263126 0.455748i
\(592\) 0 0
\(593\) −31.0085 + 17.9027i −1.27336 + 0.735177i −0.975620 0.219468i \(-0.929568\pi\)
−0.297745 + 0.954645i \(0.596235\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.90360 + 13.6894i 0.323473 + 0.560271i
\(598\) 0 0
\(599\) −8.73895 5.04544i −0.357064 0.206151i 0.310728 0.950499i \(-0.399427\pi\)
−0.667792 + 0.744348i \(0.732760\pi\)
\(600\) 0 0
\(601\) 31.0913i 1.26824i 0.773234 + 0.634120i \(0.218638\pi\)
−0.773234 + 0.634120i \(0.781362\pi\)
\(602\) 0 0
\(603\) 9.09299i 0.370295i
\(604\) 0 0
\(605\) −5.71830 3.30146i −0.232482 0.134224i
\(606\) 0 0
\(607\) −10.8573 18.8053i −0.440683 0.763285i 0.557057 0.830474i \(-0.311930\pi\)
−0.997740 + 0.0671889i \(0.978597\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.8188 6.24622i 0.437681 0.252695i
\(612\) 0 0
\(613\) 10.1405 17.5639i 0.409571 0.709398i −0.585270 0.810838i \(-0.699012\pi\)
0.994842 + 0.101440i \(0.0323449\pi\)
\(614\) 0 0
\(615\) 4.01893 0.162059
\(616\) 0 0
\(617\) 14.1442 0.569423 0.284712 0.958613i \(-0.408102\pi\)
0.284712 + 0.958613i \(0.408102\pi\)
\(618\) 0 0
\(619\) 21.2581 36.8201i 0.854435 1.47992i −0.0227335 0.999742i \(-0.507237\pi\)
0.877168 0.480183i \(-0.159430\pi\)
\(620\) 0 0
\(621\) −6.05870 + 3.49799i −0.243127 + 0.140370i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.84995 17.0606i −0.393998 0.682425i
\(626\) 0 0
\(627\) 0.135147 + 0.0780274i 0.00539727 + 0.00311611i
\(628\) 0 0
\(629\) 11.0115i 0.439056i
\(630\) 0 0
\(631\) 11.2426i 0.447561i 0.974640 + 0.223781i \(0.0718399\pi\)
−0.974640 + 0.223781i \(0.928160\pi\)
\(632\) 0 0
\(633\) 0.166027 + 0.0958559i 0.00659899 + 0.00380993i
\(634\) 0 0
\(635\) −1.57240 2.72347i −0.0623987 0.108078i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.91042 + 1.68033i −0.115134 + 0.0664729i
\(640\) 0 0
\(641\) −0.235818 + 0.408449i −0.00931425 + 0.0161328i −0.870645 0.491912i \(-0.836298\pi\)
0.861331 + 0.508045i \(0.169632\pi\)
\(642\) 0 0
\(643\) 1.10040 0.0433955 0.0216977 0.999765i \(-0.493093\pi\)
0.0216977 + 0.999765i \(0.493093\pi\)
\(644\) 0 0
\(645\) −3.66557 −0.144332
\(646\) 0 0
\(647\) 15.8657 27.4802i 0.623746 1.08036i −0.365036 0.930993i \(-0.618943\pi\)
0.988782 0.149366i \(-0.0477232\pi\)
\(648\) 0 0
\(649\) −1.14106 + 0.658789i −0.0447904 + 0.0258597i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.49210 + 7.78055i 0.175790 + 0.304477i 0.940434 0.339976i \(-0.110419\pi\)
−0.764645 + 0.644452i \(0.777085\pi\)
\(654\) 0 0
\(655\) 11.7417 + 6.77906i 0.458785 + 0.264880i
\(656\) 0 0
\(657\) 12.1442i 0.473791i
\(658\) 0 0
\(659\) 26.0679i 1.01546i −0.861516 0.507731i \(-0.830484\pi\)
0.861516 0.507731i \(-0.169516\pi\)
\(660\) 0 0
\(661\) 1.18105 + 0.681880i 0.0459376 + 0.0265221i 0.522793 0.852460i \(-0.324890\pi\)
−0.476855 + 0.878982i \(0.658223\pi\)
\(662\) 0 0
\(663\) −3.60226 6.23929i −0.139900 0.242314i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.80949 4.50881i 0.302385 0.174582i
\(668\) 0 0
\(669\) −5.00555 + 8.66986i −0.193526 + 0.335196i
\(670\) 0 0
\(671\) −0.0296942 −0.00114633
\(672\) 0 0
\(673\) −17.6874 −0.681799 −0.340899 0.940100i \(-0.610732\pi\)
−0.340899 + 0.940100i \(0.610732\pi\)
\(674\) 0 0
\(675\) −2.31896 + 4.01656i −0.0892568 + 0.154597i
\(676\) 0 0
\(677\) −6.88562 + 3.97541i −0.264636 + 0.152788i −0.626447 0.779464i \(-0.715492\pi\)
0.361812 + 0.932251i \(0.382158\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.06817 5.31422i −0.117572 0.203641i
\(682\) 0 0
\(683\) −15.6132 9.01431i −0.597424 0.344923i 0.170603 0.985340i \(-0.445428\pi\)
−0.768028 + 0.640417i \(0.778762\pi\)
\(684\) 0 0
\(685\) 8.78021i 0.335474i
\(686\) 0 0
\(687\) 9.44500i 0.360349i
\(688\) 0 0
\(689\) 40.6126 + 23.4477i 1.54722 + 0.893286i
\(690\) 0 0
\(691\) 22.2488 + 38.5361i 0.846384 + 1.46598i 0.884413 + 0.466704i \(0.154559\pi\)
−0.0380289 + 0.999277i \(0.512108\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.32950 + 2.49964i −0.164227 + 0.0948167i
\(696\) 0 0
\(697\) −7.12132 + 12.3345i −0.269739 + 0.467202i
\(698\) 0 0
\(699\) −22.2217 −0.840501
\(700\) 0 0
\(701\) −11.8785 −0.448646 −0.224323 0.974515i \(-0.572017\pi\)
−0.224323 + 0.974515i \(0.572017\pi\)
\(702\) 0 0
\(703\) −2.46224 + 4.26473i −0.0928653 + 0.160847i
\(704\) 0 0
\(705\) −1.92689 + 1.11249i −0.0725710 + 0.0418989i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12.4321 21.5331i −0.466899 0.808693i 0.532386 0.846502i \(-0.321295\pi\)
−0.999285 + 0.0378090i \(0.987962\pi\)
\(710\) 0 0
\(711\) 6.05213 + 3.49420i 0.226973 + 0.131043i
\(712\) 0 0
\(713\) 23.4680i 0.878883i
\(714\) 0 0
\(715\) 0.332663i 0.0124409i
\(716\) 0 0
\(717\) 18.0101 + 10.3981i 0.672598 + 0.388325i
\(718\) 0 0
\(719\) −0.302360 0.523703i −0.0112761 0.0195308i 0.860332 0.509734i \(-0.170256\pi\)
−0.871608 + 0.490203i \(0.836923\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.3063 6.52769i 0.420486 0.242767i
\(724\) 0 0
\(725\) 2.98907 5.17723i 0.111011 0.192277i
\(726\) 0 0
\(727\) −17.7628 −0.658786 −0.329393 0.944193i \(-0.606844\pi\)
−0.329393 + 0.944193i \(0.606844\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.49519 11.2500i 0.240233 0.416096i
\(732\) 0 0
\(733\) −13.4467 + 7.76347i −0.496666 + 0.286750i −0.727336 0.686282i \(-0.759242\pi\)
0.230670 + 0.973032i \(0.425908\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.743971 + 1.28860i 0.0274045 + 0.0474660i
\(738\) 0 0
\(739\) 14.8992 + 8.60208i 0.548077 + 0.316433i 0.748346 0.663308i \(-0.230848\pi\)
−0.200269 + 0.979741i \(0.564182\pi\)
\(740\) 0 0
\(741\) 3.22196i 0.118362i
\(742\) 0 0
\(743\) 13.1996i 0.484247i 0.970245 + 0.242123i \(0.0778439\pi\)
−0.970245 + 0.242123i \(0.922156\pi\)
\(744\) 0 0
\(745\) −5.62889 3.24984i −0.206227 0.119065i
\(746\) 0 0
\(747\) 6.48938 + 11.2399i 0.237434 + 0.411248i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 17.5756 10.1473i 0.641344 0.370280i −0.143788 0.989609i \(-0.545928\pi\)
0.785132 + 0.619328i \(0.212595\pi\)
\(752\) 0 0
\(753\) 8.94873 15.4997i 0.326110 0.564839i
\(754\) 0 0
\(755\) 8.51975 0.310065
\(756\) 0 0
\(757\) 24.7011 0.897778 0.448889 0.893587i \(-0.351820\pi\)
0.448889 + 0.893587i \(0.351820\pi\)
\(758\) 0 0
\(759\) 0.572398 0.991422i 0.0207767 0.0359864i
\(760\) 0 0
\(761\) 4.13548 2.38762i 0.149911 0.0865512i −0.423168 0.906051i \(-0.639082\pi\)
0.573079 + 0.819500i \(0.305749\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.641585 + 1.11126i 0.0231965 + 0.0401776i
\(766\) 0 0
\(767\) 23.5587 + 13.6016i 0.850654 + 0.491125i
\(768\) 0 0
\(769\) 14.7721i 0.532696i 0.963877 + 0.266348i \(0.0858170\pi\)
−0.963877 + 0.266348i \(0.914183\pi\)
\(770\) 0 0
\(771\) 6.67896i 0.240537i
\(772\) 0 0
\(773\) −31.9715 18.4588i −1.14994 0.663916i −0.201064 0.979578i \(-0.564440\pi\)
−0.948872 + 0.315662i \(0.897773\pi\)
\(774\) 0 0
\(775\) −7.77894 13.4735i −0.279428 0.483983i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.51616 + 3.18476i −0.197637 + 0.114106i
\(780\) 0 0
\(781\) 0.274963 0.476250i 0.00983895 0.0170416i
\(782\) 0 0
\(783\) −1.28897 −0.0460641
\(784\) 0 0
\(785\) 0.657475 0.0234663
\(786\) 0 0
\(787\) −18.5199 + 32.0774i −0.660164 + 1.14344i 0.320408 + 0.947280i \(0.396180\pi\)
−0.980572 + 0.196158i \(0.937153\pi\)
\(788\) 0 0
\(789\) 21.1452 12.2082i 0.752790 0.434623i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.306539 + 0.530940i 0.0108855 + 0.0188542i
\(794\) 0 0
\(795\) −7.23336 4.17618i −0.256541 0.148114i
\(796\) 0 0
\(797\) 36.9752i 1.30973i 0.755746 + 0.654865i \(0.227275\pi\)
−0.755746 + 0.654865i \(0.772725\pi\)
\(798\) 0 0
\(799\) 7.88509i 0.278954i
\(800\) 0 0
\(801\) −12.8911 7.44266i −0.455483 0.262973i
\(802\) 0 0
\(803\) −0.993616 1.72099i −0.0350639 0.0607325i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.992403 0.572964i 0.0349342 0.0201693i
\(808\) 0 0
\(809\) −2.67050 + 4.62544i −0.0938898 + 0.162622i −0.909145 0.416480i \(-0.863263\pi\)
0.815255 + 0.579102i \(0.196597\pi\)
\(810\) 0 0
\(811\) −37.4180 −1.31392 −0.656961 0.753924i \(-0.728159\pi\)
−0.656961 + 0.753924i \(0.728159\pi\)
\(812\) 0 0
\(813\) 5.65157 0.198209
\(814\) 0 0
\(815\) 5.44135 9.42470i 0.190602 0.330133i
\(816\) 0 0
\(817\) 5.03116 2.90474i 0.176018 0.101624i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.1730 17.6202i −0.355041 0.614949i 0.632084 0.774900i \(-0.282200\pi\)
−0.987125 + 0.159951i \(0.948866\pi\)
\(822\) 0 0
\(823\) −40.1287 23.1683i −1.39880 0.807598i −0.404533 0.914523i \(-0.632566\pi\)
−0.994267 + 0.106926i \(0.965899\pi\)
\(824\) 0 0
\(825\) 0.758932i 0.0264226i
\(826\) 0 0
\(827\) 2.20358i 0.0766260i −0.999266 0.0383130i \(-0.987802\pi\)
0.999266 0.0383130i \(-0.0121984\pi\)
\(828\) 0 0
\(829\) 14.9510 + 8.63194i 0.519268 + 0.299800i 0.736635 0.676290i \(-0.236413\pi\)
−0.217367 + 0.976090i \(0.569747\pi\)
\(830\) 0 0
\(831\) 9.63792 + 16.6934i 0.334336 + 0.579087i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.91136 + 5.72232i −0.342997 + 0.198029i
\(836\) 0 0
\(837\) −1.67725 + 2.90508i −0.0579741 + 0.100414i
\(838\) 0 0
\(839\) 17.4781 0.603410 0.301705 0.953401i \(-0.402444\pi\)
0.301705 + 0.953401i \(0.402444\pi\)
\(840\) 0 0
\(841\) −27.3386 −0.942709
\(842\) 0 0
\(843\) −4.40185 + 7.62422i −0.151608 + 0.262592i
\(844\) 0 0
\(845\) 0.826376 0.477108i 0.0284282 0.0164130i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −13.0725 22.6423i −0.448648 0.777081i
\(850\) 0 0
\(851\) 31.2855 + 18.0627i 1.07245 + 0.619181i
\(852\) 0 0
\(853\) 30.7781i 1.05382i −0.849920 0.526912i \(-0.823350\pi\)
0.849920 0.526912i \(-0.176650\pi\)
\(854\) 0 0
\(855\) 0.573852i 0.0196253i
\(856\) 0 0
\(857\) −20.3399 11.7432i −0.694797 0.401142i 0.110609 0.993864i \(-0.464720\pi\)
−0.805407 + 0.592722i \(0.798053\pi\)
\(858\) 0 0
\(859\) −21.1125 36.5678i −0.720348 1.24768i −0.960860 0.277033i \(-0.910649\pi\)
0.240513 0.970646i \(-0.422684\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.52190 + 3.18807i −0.187968 + 0.108523i −0.591031 0.806649i \(-0.701279\pi\)
0.403063 + 0.915172i \(0.367946\pi\)
\(864\) 0 0
\(865\) 2.48376 4.30200i 0.0844505 0.146273i
\(866\) 0 0
\(867\) 12.4526 0.422912
\(868\) 0 0
\(869\) −1.14356 −0.0387925
\(870\) 0 0
\(871\) 15.3603 26.6048i 0.520464 0.901470i
\(872\) 0 0
\(873\) −7.86588 + 4.54137i −0.266220 + 0.153702i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.10997 + 10.5828i 0.206319 + 0.357355i 0.950552 0.310565i \(-0.100518\pi\)
−0.744233 + 0.667920i \(0.767185\pi\)
\(878\) 0 0
\(879\) −15.8171 9.13200i −0.533497 0.308015i
\(880\) 0 0
\(881\) 45.0545i 1.51793i −0.651134 0.758963i \(-0.725706\pi\)
0.651134 0.758963i \(-0.274294\pi\)
\(882\) 0 0
\(883\) 21.8481i 0.735246i −0.929975 0.367623i \(-0.880172\pi\)
0.929975 0.367623i \(-0.119828\pi\)
\(884\) 0 0
\(885\) −4.19595 2.42253i −0.141045 0.0814325i
\(886\) 0 0
\(887\) −16.1302 27.9383i −0.541599 0.938078i −0.998812 0.0487204i \(-0.984486\pi\)
0.457213 0.889357i \(-0.348848\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.141713 + 0.0818181i −0.00474757 + 0.00274101i
\(892\) 0 0
\(893\) 1.76316 3.05389i 0.0590020 0.102194i
\(894\) 0 0
\(895\) −3.30739 −0.110554
\(896\) 0 0
\(897\) −23.6359 −0.789179
\(898\) 0 0
\(899\) 2.16192 3.74456i 0.0721042 0.124888i
\(900\) 0 0
\(901\) 25.6342 14.7999i 0.853998 0.493056i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.56583 6.17620i −0.118532 0.205304i
\(906\) 0 0
\(907\) 15.2097 + 8.78132i 0.505030 + 0.291579i 0.730788 0.682604i \(-0.239153\pi\)
−0.225759 + 0.974183i \(0.572486\pi\)
\(908\) 0 0
\(909\) 8.51647i 0.282473i
\(910\) 0 0
\(911\) 0.475746i 0.0157622i −0.999969 0.00788108i \(-0.997491\pi\)
0.999969 0.00788108i \(-0.00250865\pi\)
\(912\) 0 0
\(913\) −1.83926 1.06190i −0.0608707 0.0351437i
\(914\) 0 0
\(915\) −0.0545964 0.0945638i −0.00180490 0.00312618i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −28.4809 + 16.4435i −0.939500 + 0.542420i −0.889803 0.456344i \(-0.849159\pi\)
−0.0496962 + 0.998764i \(0.515825\pi\)
\(920\) 0 0
\(921\) −10.7236 + 18.5738i −0.353354 + 0.612027i
\(922\) 0 0
\(923\) −11.3540 −0.373721
\(924\) 0 0
\(925\) 23.9490 0.787437
\(926\) 0 0
\(927\) −8.38043 + 14.5153i −0.275250 + 0.476746i
\(928\) 0 0
\(929\) 11.0677 6.38992i 0.363118 0.209646i −0.307329 0.951603i \(-0.599435\pi\)
0.670448 + 0.741957i \(0.266102\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 7.24000 + 12.5400i 0.237027 + 0.410543i
\(934\) 0 0
\(935\) −0.181842 0.104986i −0.00594687 0.00343342i
\(936\) 0 0
\(937\) 4.03304i 0.131754i 0.997828 + 0.0658768i \(0.0209844\pi\)
−0.997828 + 0.0658768i \(0.979016\pi\)
\(938\) 0 0
\(939\) 1.48483i 0.0484556i
\(940\) 0 0
\(941\) −50.8865 29.3793i −1.65885 0.957739i −0.973246 0.229766i \(-0.926204\pi\)
−0.685606 0.727973i \(-0.740463\pi\)
\(942\) 0 0
\(943\) 23.3629 + 40.4658i 0.760802 + 1.31775i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.6092 6.70260i 0.377250 0.217805i −0.299371 0.954137i \(-0.596777\pi\)
0.676621 + 0.736331i \(0.263444\pi\)
\(948\) 0 0
\(949\) −20.5146 + 35.5323i −0.665931 + 1.15343i
\(950\) 0 0
\(951\) −11.4908 −0.372616
\(952\) 0 0
\(953\) −1.89697 −0.0614490 −0.0307245 0.999528i \(-0.509781\pi\)
−0.0307245 + 0.999528i \(0.509781\pi\)
\(954\) 0 0
\(955\) −6.96452 + 12.0629i −0.225367 + 0.390346i
\(956\) 0 0
\(957\) 0.182664 0.105461i 0.00590469 0.00340908i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 9.87368 + 17.1017i 0.318506 + 0.551669i
\(962\) 0 0
\(963\) 3.80822 + 2.19867i 0.122718 + 0.0708513i
\(964\) 0 0
\(965\) 0.308121i 0.00991877i
\(966\) 0 0
\(967\) 4.84855i 0.155919i 0.996957 + 0.0779595i \(0.0248405\pi\)
−0.996957 + 0.0779595i \(0.975160\pi\)
\(968\) 0 0
\(969\) −1.76121 1.01683i −0.0565781 0.0326654i
\(970\) 0 0
\(971\) −18.8709 32.6854i −0.605596 1.04892i −0.991957 0.126576i \(-0.959601\pi\)
0.386361 0.922348i \(-0.373732\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −13.5699 + 7.83459i −0.434585 + 0.250908i
\(976\) 0 0
\(977\) −12.3343 + 21.3636i −0.394609 + 0.683482i −0.993051 0.117684i \(-0.962453\pi\)
0.598442 + 0.801166i \(0.295787\pi\)
\(978\) 0 0
\(979\) 2.43578 0.0778477
\(980\) 0 0
\(981\) 11.5453 0.368612
\(982\) 0 0
\(983\) −18.8750 + 32.6925i −0.602019 + 1.04273i 0.390495 + 0.920605i \(0.372304\pi\)
−0.992515 + 0.122123i \(0.961030\pi\)
\(984\) 0 0
\(985\) −6.66684 + 3.84910i −0.212423 + 0.122643i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21.3088 36.9079i −0.677580 1.17360i
\(990\) 0 0
\(991\) 48.1078 + 27.7750i 1.52819 + 0.882303i 0.999438 + 0.0335292i \(0.0106747\pi\)
0.528756 + 0.848774i \(0.322659\pi\)
\(992\) 0 0
\(993\) 21.0967i 0.669484i
\(994\) 0 0
\(995\) 9.51167i 0.301540i
\(996\) 0 0
\(997\) −20.9075 12.0710i −0.662148 0.382291i 0.130947 0.991389i \(-0.458198\pi\)
−0.793095 + 0.609098i \(0.791532\pi\)
\(998\) 0 0
\(999\) −2.58187 4.47192i −0.0816866 0.141485i
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 2352.2.bl.o.31.3 8
4.3 odd 2 2352.2.bl.t.31.3 8
7.2 even 3 2352.2.bl.q.607.2 8
7.3 odd 6 2352.2.b.k.1567.4 8
7.4 even 3 2352.2.b.l.1567.5 yes 8
7.5 odd 6 2352.2.bl.t.607.3 8
7.6 odd 2 2352.2.bl.r.31.2 8
21.11 odd 6 7056.2.b.w.1567.4 8
21.17 even 6 7056.2.b.x.1567.5 8
28.3 even 6 2352.2.b.l.1567.4 yes 8
28.11 odd 6 2352.2.b.k.1567.5 yes 8
28.19 even 6 inner 2352.2.bl.o.607.3 8
28.23 odd 6 2352.2.bl.r.607.2 8
28.27 even 2 2352.2.bl.q.31.2 8
84.11 even 6 7056.2.b.x.1567.4 8
84.59 odd 6 7056.2.b.w.1567.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.4 8 7.3 odd 6
2352.2.b.k.1567.5 yes 8 28.11 odd 6
2352.2.b.l.1567.4 yes 8 28.3 even 6
2352.2.b.l.1567.5 yes 8 7.4 even 3
2352.2.bl.o.31.3 8 1.1 even 1 trivial
2352.2.bl.o.607.3 8 28.19 even 6 inner
2352.2.bl.q.31.2 8 28.27 even 2
2352.2.bl.q.607.2 8 7.2 even 3
2352.2.bl.r.31.2 8 7.6 odd 2
2352.2.bl.r.607.2 8 28.23 odd 6
2352.2.bl.t.31.3 8 4.3 odd 2
2352.2.bl.t.607.3 8 7.5 odd 6
7056.2.b.w.1567.4 8 21.11 odd 6
7056.2.b.w.1567.5 8 84.59 odd 6
7056.2.b.x.1567.4 8 84.11 even 6
7056.2.b.x.1567.5 8 21.17 even 6