Properties

Label 1456.2.r.p.417.2
Level $1456$
Weight $2$
Character 1456.417
Analytic conductor $11.626$
Analytic rank $0$
Dimension $10$
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] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.2
Root \(-0.862625 - 1.49411i\) of defining polynomial
Character \(\chi\) \(=\) 1456.417
Dual form 1456.2.r.p.625.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.673208 + 1.16603i) q^{3} +(-1.09358 - 1.89414i) q^{5} +(2.19729 + 1.47375i) q^{7} +(0.593582 + 1.02811i) q^{9} +O(q^{10})\) \(q+(-0.673208 + 1.16603i) q^{3} +(-1.09358 - 1.89414i) q^{5} +(2.19729 + 1.47375i) q^{7} +(0.593582 + 1.02811i) q^{9} +(-0.524077 + 0.907729i) q^{11} +1.00000 q^{13} +2.94483 q^{15} +(2.64562 - 4.58236i) q^{17} +(0.378453 + 0.655500i) q^{19} +(-3.19767 + 1.56996i) q^{21} +(0.326792 + 0.566020i) q^{23} +(0.108157 - 0.187333i) q^{25} -5.63766 q^{27} -3.10408 q^{29} +(0.513956 - 0.890198i) q^{31} +(-0.705626 - 1.22218i) q^{33} +(0.388575 - 5.77363i) q^{35} +(5.44661 + 9.43381i) q^{37} +(-0.673208 + 1.16603i) q^{39} +7.32040 q^{41} -0.887771 q^{43} +(1.29826 - 2.24865i) q^{45} +(1.16875 + 2.02434i) q^{47} +(2.65613 + 6.47650i) q^{49} +(3.56211 + 6.16976i) q^{51} +(-2.44407 + 4.23325i) q^{53} +2.29249 q^{55} -1.01911 q^{57} +(-0.524077 + 0.907729i) q^{59} +(6.24989 + 10.8251i) q^{61} +(-0.210913 + 3.13385i) q^{63} +(-1.09358 - 1.89414i) q^{65} +(2.23944 - 3.87883i) q^{67} -0.879996 q^{69} +6.60274 q^{71} +(4.14174 - 7.17370i) q^{73} +(0.145624 + 0.252229i) q^{75} +(-2.48931 + 1.22218i) q^{77} +(1.07007 + 1.85342i) q^{79} +(2.01457 - 3.48935i) q^{81} +6.66558 q^{83} -11.5728 q^{85} +(2.08969 - 3.61946i) q^{87} +(2.88388 + 4.99503i) q^{89} +(2.19729 + 1.47375i) q^{91} +(0.691998 + 1.19858i) q^{93} +(0.827739 - 1.43369i) q^{95} -2.88777 q^{97} -1.24433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - q^{7} - 3 q^{9} + 11 q^{11} + 10 q^{13} + 5 q^{17} + 9 q^{19} + 2 q^{21} + 10 q^{23} - 9 q^{25} - 6 q^{29} - 6 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 28 q^{41} - 4 q^{43} + 32 q^{45} + q^{47} - 11 q^{49} - 8 q^{51} - 17 q^{53} - 32 q^{57} + 11 q^{59} + 11 q^{61} - 5 q^{63} - 2 q^{65} + 13 q^{67} + 36 q^{69} - 30 q^{71} - 20 q^{75} - 46 q^{77} + 2 q^{79} + 19 q^{81} - 12 q^{83} - 44 q^{85} - 8 q^{87} + 4 q^{89} - q^{91} - 18 q^{93} - 12 q^{95} - 24 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.673208 + 1.16603i −0.388677 + 0.673208i −0.992272 0.124083i \(-0.960401\pi\)
0.603595 + 0.797291i \(0.293734\pi\)
\(4\) 0 0
\(5\) −1.09358 1.89414i −0.489065 0.847085i 0.510856 0.859666i \(-0.329328\pi\)
−0.999921 + 0.0125813i \(0.995995\pi\)
\(6\) 0 0
\(7\) 2.19729 + 1.47375i 0.830496 + 0.557025i
\(8\) 0 0
\(9\) 0.593582 + 1.02811i 0.197861 + 0.342705i
\(10\) 0 0
\(11\) −0.524077 + 0.907729i −0.158015 + 0.273691i −0.934153 0.356873i \(-0.883843\pi\)
0.776138 + 0.630564i \(0.217176\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.94483 0.760352
\(16\) 0 0
\(17\) 2.64562 4.58236i 0.641658 1.11138i −0.343404 0.939188i \(-0.611580\pi\)
0.985063 0.172197i \(-0.0550865\pi\)
\(18\) 0 0
\(19\) 0.378453 + 0.655500i 0.0868231 + 0.150382i 0.906167 0.422921i \(-0.138995\pi\)
−0.819344 + 0.573303i \(0.805662\pi\)
\(20\) 0 0
\(21\) −3.19767 + 1.56996i −0.697788 + 0.342594i
\(22\) 0 0
\(23\) 0.326792 + 0.566020i 0.0681408 + 0.118023i 0.898083 0.439826i \(-0.144960\pi\)
−0.829942 + 0.557850i \(0.811627\pi\)
\(24\) 0 0
\(25\) 0.108157 0.187333i 0.0216314 0.0374667i
\(26\) 0 0
\(27\) −5.63766 −1.08497
\(28\) 0 0
\(29\) −3.10408 −0.576414 −0.288207 0.957568i \(-0.593059\pi\)
−0.288207 + 0.957568i \(0.593059\pi\)
\(30\) 0 0
\(31\) 0.513956 0.890198i 0.0923092 0.159884i −0.816173 0.577807i \(-0.803909\pi\)
0.908482 + 0.417923i \(0.137242\pi\)
\(32\) 0 0
\(33\) −0.705626 1.22218i −0.122834 0.212754i
\(34\) 0 0
\(35\) 0.388575 5.77363i 0.0656811 0.975922i
\(36\) 0 0
\(37\) 5.44661 + 9.43381i 0.895418 + 1.55091i 0.833287 + 0.552841i \(0.186456\pi\)
0.0621309 + 0.998068i \(0.480210\pi\)
\(38\) 0 0
\(39\) −0.673208 + 1.16603i −0.107800 + 0.186714i
\(40\) 0 0
\(41\) 7.32040 1.14325 0.571627 0.820514i \(-0.306312\pi\)
0.571627 + 0.820514i \(0.306312\pi\)
\(42\) 0 0
\(43\) −0.887771 −0.135384 −0.0676919 0.997706i \(-0.521563\pi\)
−0.0676919 + 0.997706i \(0.521563\pi\)
\(44\) 0 0
\(45\) 1.29826 2.24865i 0.193533 0.335210i
\(46\) 0 0
\(47\) 1.16875 + 2.02434i 0.170480 + 0.295281i 0.938588 0.345040i \(-0.112135\pi\)
−0.768108 + 0.640321i \(0.778801\pi\)
\(48\) 0 0
\(49\) 2.65613 + 6.47650i 0.379447 + 0.925214i
\(50\) 0 0
\(51\) 3.56211 + 6.16976i 0.498795 + 0.863939i
\(52\) 0 0
\(53\) −2.44407 + 4.23325i −0.335719 + 0.581482i −0.983623 0.180240i \(-0.942313\pi\)
0.647904 + 0.761722i \(0.275646\pi\)
\(54\) 0 0
\(55\) 2.29249 0.309119
\(56\) 0 0
\(57\) −1.01911 −0.134985
\(58\) 0 0
\(59\) −0.524077 + 0.907729i −0.0682291 + 0.118176i −0.898122 0.439747i \(-0.855068\pi\)
0.829893 + 0.557923i \(0.188402\pi\)
\(60\) 0 0
\(61\) 6.24989 + 10.8251i 0.800217 + 1.38602i 0.919473 + 0.393153i \(0.128616\pi\)
−0.119256 + 0.992864i \(0.538051\pi\)
\(62\) 0 0
\(63\) −0.210913 + 3.13385i −0.0265726 + 0.394828i
\(64\) 0 0
\(65\) −1.09358 1.89414i −0.135642 0.234939i
\(66\) 0 0
\(67\) 2.23944 3.87883i 0.273592 0.473875i −0.696187 0.717860i \(-0.745122\pi\)
0.969779 + 0.243986i \(0.0784550\pi\)
\(68\) 0 0
\(69\) −0.879996 −0.105939
\(70\) 0 0
\(71\) 6.60274 0.783601 0.391801 0.920050i \(-0.371852\pi\)
0.391801 + 0.920050i \(0.371852\pi\)
\(72\) 0 0
\(73\) 4.14174 7.17370i 0.484754 0.839618i −0.515093 0.857134i \(-0.672243\pi\)
0.999847 + 0.0175164i \(0.00557593\pi\)
\(74\) 0 0
\(75\) 0.145624 + 0.252229i 0.0168152 + 0.0291249i
\(76\) 0 0
\(77\) −2.48931 + 1.22218i −0.283683 + 0.139280i
\(78\) 0 0
\(79\) 1.07007 + 1.85342i 0.120392 + 0.208526i 0.919922 0.392100i \(-0.128251\pi\)
−0.799530 + 0.600626i \(0.794918\pi\)
\(80\) 0 0
\(81\) 2.01457 3.48935i 0.223842 0.387705i
\(82\) 0 0
\(83\) 6.66558 0.731642 0.365821 0.930685i \(-0.380788\pi\)
0.365821 + 0.930685i \(0.380788\pi\)
\(84\) 0 0
\(85\) −11.5728 −1.25525
\(86\) 0 0
\(87\) 2.08969 3.61946i 0.224039 0.388047i
\(88\) 0 0
\(89\) 2.88388 + 4.99503i 0.305691 + 0.529472i 0.977415 0.211329i \(-0.0677792\pi\)
−0.671724 + 0.740802i \(0.734446\pi\)
\(90\) 0 0
\(91\) 2.19729 + 1.47375i 0.230338 + 0.154491i
\(92\) 0 0
\(93\) 0.691998 + 1.19858i 0.0717569 + 0.124287i
\(94\) 0 0
\(95\) 0.827739 1.43369i 0.0849242 0.147093i
\(96\) 0 0
\(97\) −2.88777 −0.293209 −0.146604 0.989195i \(-0.546834\pi\)
−0.146604 + 0.989195i \(0.546834\pi\)
\(98\) 0 0
\(99\) −1.24433 −0.125060
\(100\) 0 0
\(101\) 5.62716 9.74653i 0.559924 0.969816i −0.437579 0.899180i \(-0.644164\pi\)
0.997502 0.0706359i \(-0.0225028\pi\)
\(102\) 0 0
\(103\) 10.1167 + 17.5226i 0.996828 + 1.72656i 0.567341 + 0.823483i \(0.307972\pi\)
0.429487 + 0.903073i \(0.358694\pi\)
\(104\) 0 0
\(105\) 6.47064 + 4.33994i 0.631470 + 0.423535i
\(106\) 0 0
\(107\) 4.52758 + 7.84201i 0.437698 + 0.758115i 0.997512 0.0705034i \(-0.0224606\pi\)
−0.559813 + 0.828619i \(0.689127\pi\)
\(108\) 0 0
\(109\) −7.55070 + 13.0782i −0.723226 + 1.25266i 0.236474 + 0.971638i \(0.424008\pi\)
−0.959700 + 0.281026i \(0.909325\pi\)
\(110\) 0 0
\(111\) −14.6668 −1.39211
\(112\) 0 0
\(113\) 3.10408 0.292008 0.146004 0.989284i \(-0.453359\pi\)
0.146004 + 0.989284i \(0.453359\pi\)
\(114\) 0 0
\(115\) 0.714748 1.23798i 0.0666506 0.115442i
\(116\) 0 0
\(117\) 0.593582 + 1.02811i 0.0548767 + 0.0950492i
\(118\) 0 0
\(119\) 12.5664 6.16976i 1.15196 0.565581i
\(120\) 0 0
\(121\) 4.95069 + 8.57484i 0.450062 + 0.779531i
\(122\) 0 0
\(123\) −4.92815 + 8.53581i −0.444356 + 0.769648i
\(124\) 0 0
\(125\) −11.4089 −1.02045
\(126\) 0 0
\(127\) −8.78914 −0.779910 −0.389955 0.920834i \(-0.627509\pi\)
−0.389955 + 0.920834i \(0.627509\pi\)
\(128\) 0 0
\(129\) 0.597654 1.03517i 0.0526205 0.0911414i
\(130\) 0 0
\(131\) −5.25723 9.10580i −0.459327 0.795577i 0.539599 0.841922i \(-0.318576\pi\)
−0.998925 + 0.0463451i \(0.985243\pi\)
\(132\) 0 0
\(133\) −0.134473 + 1.99807i −0.0116603 + 0.173254i
\(134\) 0 0
\(135\) 6.16525 + 10.6785i 0.530620 + 0.919061i
\(136\) 0 0
\(137\) −4.36583 + 7.56183i −0.372998 + 0.646051i −0.990025 0.140891i \(-0.955003\pi\)
0.617028 + 0.786942i \(0.288337\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −3.14726 −0.265047
\(142\) 0 0
\(143\) −0.524077 + 0.907729i −0.0438256 + 0.0759081i
\(144\) 0 0
\(145\) 3.39457 + 5.87957i 0.281904 + 0.488272i
\(146\) 0 0
\(147\) −9.33992 1.26290i −0.770343 0.104163i
\(148\) 0 0
\(149\) −7.69632 13.3304i −0.630507 1.09207i −0.987448 0.157944i \(-0.949514\pi\)
0.356941 0.934127i \(-0.383820\pi\)
\(150\) 0 0
\(151\) −6.83786 + 11.8435i −0.556457 + 0.963812i 0.441331 + 0.897344i \(0.354506\pi\)
−0.997789 + 0.0664680i \(0.978827\pi\)
\(152\) 0 0
\(153\) 6.28158 0.507836
\(154\) 0 0
\(155\) −2.24821 −0.180581
\(156\) 0 0
\(157\) −1.69378 + 2.93371i −0.135178 + 0.234136i −0.925666 0.378343i \(-0.876494\pi\)
0.790487 + 0.612478i \(0.209827\pi\)
\(158\) 0 0
\(159\) −3.29074 5.69972i −0.260972 0.452017i
\(160\) 0 0
\(161\) −0.116117 + 1.72532i −0.00915128 + 0.135974i
\(162\) 0 0
\(163\) −6.90502 11.9598i −0.540843 0.936767i −0.998856 0.0478219i \(-0.984772\pi\)
0.458013 0.888946i \(-0.348561\pi\)
\(164\) 0 0
\(165\) −1.54332 + 2.67311i −0.120147 + 0.208101i
\(166\) 0 0
\(167\) −16.3783 −1.26739 −0.633695 0.773583i \(-0.718462\pi\)
−0.633695 + 0.773583i \(0.718462\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.449286 + 0.778186i −0.0343578 + 0.0595094i
\(172\) 0 0
\(173\) −2.06273 3.57275i −0.156826 0.271631i 0.776896 0.629629i \(-0.216793\pi\)
−0.933723 + 0.357997i \(0.883460\pi\)
\(174\) 0 0
\(175\) 0.513734 0.252229i 0.0388346 0.0190667i
\(176\) 0 0
\(177\) −0.705626 1.22218i −0.0530381 0.0918647i
\(178\) 0 0
\(179\) 7.20679 12.4825i 0.538661 0.932988i −0.460316 0.887755i \(-0.652264\pi\)
0.998976 0.0452324i \(-0.0144028\pi\)
\(180\) 0 0
\(181\) 18.1014 1.34547 0.672733 0.739885i \(-0.265120\pi\)
0.672733 + 0.739885i \(0.265120\pi\)
\(182\) 0 0
\(183\) −16.8299 −1.24410
\(184\) 0 0
\(185\) 11.9126 20.6333i 0.875834 1.51699i
\(186\) 0 0
\(187\) 2.77302 + 4.80302i 0.202784 + 0.351232i
\(188\) 0 0
\(189\) −12.3876 8.30850i −0.901062 0.604355i
\(190\) 0 0
\(191\) 2.77068 + 4.79895i 0.200479 + 0.347240i 0.948683 0.316229i \(-0.102417\pi\)
−0.748204 + 0.663469i \(0.769083\pi\)
\(192\) 0 0
\(193\) 4.37044 7.56983i 0.314591 0.544888i −0.664759 0.747058i \(-0.731466\pi\)
0.979351 + 0.202170i \(0.0647992\pi\)
\(194\) 0 0
\(195\) 2.94483 0.210884
\(196\) 0 0
\(197\) −5.46874 −0.389632 −0.194816 0.980840i \(-0.562411\pi\)
−0.194816 + 0.980840i \(0.562411\pi\)
\(198\) 0 0
\(199\) 9.76839 16.9193i 0.692463 1.19938i −0.278566 0.960417i \(-0.589859\pi\)
0.971029 0.238963i \(-0.0768075\pi\)
\(200\) 0 0
\(201\) 3.01522 + 5.22252i 0.212677 + 0.368368i
\(202\) 0 0
\(203\) −6.82056 4.57464i −0.478709 0.321077i
\(204\) 0 0
\(205\) −8.00546 13.8659i −0.559125 0.968433i
\(206\) 0 0
\(207\) −0.387956 + 0.671959i −0.0269648 + 0.0467044i
\(208\) 0 0
\(209\) −0.793355 −0.0548775
\(210\) 0 0
\(211\) −16.6905 −1.14902 −0.574511 0.818497i \(-0.694808\pi\)
−0.574511 + 0.818497i \(0.694808\pi\)
\(212\) 0 0
\(213\) −4.44502 + 7.69900i −0.304568 + 0.527527i
\(214\) 0 0
\(215\) 0.970850 + 1.68156i 0.0662114 + 0.114682i
\(216\) 0 0
\(217\) 2.44124 1.19858i 0.165722 0.0813647i
\(218\) 0 0
\(219\) 5.57650 + 9.65878i 0.376825 + 0.652680i
\(220\) 0 0
\(221\) 2.64562 4.58236i 0.177964 0.308243i
\(222\) 0 0
\(223\) 5.34217 0.357738 0.178869 0.983873i \(-0.442756\pi\)
0.178869 + 0.983873i \(0.442756\pi\)
\(224\) 0 0
\(225\) 0.256800 0.0171200
\(226\) 0 0
\(227\) 10.0608 17.4258i 0.667757 1.15659i −0.310774 0.950484i \(-0.600588\pi\)
0.978530 0.206104i \(-0.0660786\pi\)
\(228\) 0 0
\(229\) −12.6249 21.8669i −0.834275 1.44501i −0.894619 0.446829i \(-0.852553\pi\)
0.0603445 0.998178i \(-0.480780\pi\)
\(230\) 0 0
\(231\) 0.250725 3.72540i 0.0164965 0.245113i
\(232\) 0 0
\(233\) 0.396678 + 0.687066i 0.0259872 + 0.0450112i 0.878727 0.477326i \(-0.158394\pi\)
−0.852739 + 0.522337i \(0.825060\pi\)
\(234\) 0 0
\(235\) 2.55626 4.42757i 0.166752 0.288823i
\(236\) 0 0
\(237\) −2.88152 −0.187175
\(238\) 0 0
\(239\) −20.0488 −1.29685 −0.648425 0.761279i \(-0.724572\pi\)
−0.648425 + 0.761279i \(0.724572\pi\)
\(240\) 0 0
\(241\) −6.90602 + 11.9616i −0.444856 + 0.770513i −0.998042 0.0625446i \(-0.980078\pi\)
0.553186 + 0.833058i \(0.313412\pi\)
\(242\) 0 0
\(243\) −5.74404 9.94897i −0.368480 0.638227i
\(244\) 0 0
\(245\) 9.36269 12.1137i 0.598161 0.773913i
\(246\) 0 0
\(247\) 0.378453 + 0.655500i 0.0240804 + 0.0417085i
\(248\) 0 0
\(249\) −4.48732 + 7.77227i −0.284372 + 0.492547i
\(250\) 0 0
\(251\) 26.1095 1.64802 0.824010 0.566576i \(-0.191732\pi\)
0.824010 + 0.566576i \(0.191732\pi\)
\(252\) 0 0
\(253\) −0.685057 −0.0430692
\(254\) 0 0
\(255\) 7.79092 13.4943i 0.487886 0.845044i
\(256\) 0 0
\(257\) −5.30990 9.19701i −0.331222 0.573694i 0.651530 0.758623i \(-0.274128\pi\)
−0.982752 + 0.184930i \(0.940794\pi\)
\(258\) 0 0
\(259\) −1.93531 + 28.7557i −0.120254 + 1.78679i
\(260\) 0 0
\(261\) −1.84253 3.19135i −0.114050 0.197540i
\(262\) 0 0
\(263\) 5.17888 8.97008i 0.319343 0.553119i −0.661008 0.750379i \(-0.729871\pi\)
0.980351 + 0.197260i \(0.0632044\pi\)
\(264\) 0 0
\(265\) 10.6912 0.656753
\(266\) 0 0
\(267\) −7.76581 −0.475260
\(268\) 0 0
\(269\) −5.98503 + 10.3664i −0.364914 + 0.632049i −0.988762 0.149496i \(-0.952235\pi\)
0.623849 + 0.781545i \(0.285568\pi\)
\(270\) 0 0
\(271\) −1.37845 2.38755i −0.0837351 0.145033i 0.821116 0.570761i \(-0.193352\pi\)
−0.904852 + 0.425727i \(0.860018\pi\)
\(272\) 0 0
\(273\) −3.19767 + 1.56996i −0.193532 + 0.0950184i
\(274\) 0 0
\(275\) 0.113365 + 0.196354i 0.00683618 + 0.0118406i
\(276\) 0 0
\(277\) 11.9637 20.7218i 0.718831 1.24505i −0.242632 0.970118i \(-0.578011\pi\)
0.961463 0.274933i \(-0.0886558\pi\)
\(278\) 0 0
\(279\) 1.22030 0.0730574
\(280\) 0 0
\(281\) −3.87870 −0.231384 −0.115692 0.993285i \(-0.536909\pi\)
−0.115692 + 0.993285i \(0.536909\pi\)
\(282\) 0 0
\(283\) −3.10499 + 5.37801i −0.184573 + 0.319689i −0.943432 0.331565i \(-0.892423\pi\)
0.758860 + 0.651254i \(0.225757\pi\)
\(284\) 0 0
\(285\) 1.11448 + 1.93034i 0.0660162 + 0.114343i
\(286\) 0 0
\(287\) 16.0850 + 10.7884i 0.949468 + 0.636821i
\(288\) 0 0
\(289\) −5.49866 9.52395i −0.323450 0.560232i
\(290\) 0 0
\(291\) 1.94407 3.36723i 0.113963 0.197390i
\(292\) 0 0
\(293\) 16.5754 0.968347 0.484174 0.874972i \(-0.339120\pi\)
0.484174 + 0.874972i \(0.339120\pi\)
\(294\) 0 0
\(295\) 2.29249 0.133474
\(296\) 0 0
\(297\) 2.95457 5.11747i 0.171442 0.296946i
\(298\) 0 0
\(299\) 0.326792 + 0.566020i 0.0188989 + 0.0327338i
\(300\) 0 0
\(301\) −1.95069 1.30835i −0.112436 0.0754121i
\(302\) 0 0
\(303\) 7.57650 + 13.1229i 0.435259 + 0.753890i
\(304\) 0 0
\(305\) 13.6695 23.6763i 0.782716 1.35570i
\(306\) 0 0
\(307\) 7.05788 0.402815 0.201407 0.979508i \(-0.435449\pi\)
0.201407 + 0.979508i \(0.435449\pi\)
\(308\) 0 0
\(309\) −27.2426 −1.54978
\(310\) 0 0
\(311\) 10.5551 18.2820i 0.598525 1.03668i −0.394514 0.918890i \(-0.629087\pi\)
0.993039 0.117785i \(-0.0375795\pi\)
\(312\) 0 0
\(313\) −0.990260 1.71518i −0.0559728 0.0969477i 0.836681 0.547690i \(-0.184493\pi\)
−0.892654 + 0.450742i \(0.851159\pi\)
\(314\) 0 0
\(315\) 6.16660 3.02762i 0.347449 0.170587i
\(316\) 0 0
\(317\) 9.02297 + 15.6282i 0.506781 + 0.877770i 0.999969 + 0.00784727i \(0.00249789\pi\)
−0.493189 + 0.869922i \(0.664169\pi\)
\(318\) 0 0
\(319\) 1.62678 2.81767i 0.0910822 0.157759i
\(320\) 0 0
\(321\) −12.1920 −0.680492
\(322\) 0 0
\(323\) 4.00498 0.222843
\(324\) 0 0
\(325\) 0.108157 0.187333i 0.00599947 0.0103914i
\(326\) 0 0
\(327\) −10.1664 17.6087i −0.562202 0.973763i
\(328\) 0 0
\(329\) −0.415285 + 6.17051i −0.0228954 + 0.340191i
\(330\) 0 0
\(331\) −7.33689 12.7079i −0.403272 0.698488i 0.590847 0.806784i \(-0.298794\pi\)
−0.994119 + 0.108296i \(0.965460\pi\)
\(332\) 0 0
\(333\) −6.46602 + 11.1995i −0.354336 + 0.613728i
\(334\) 0 0
\(335\) −9.79606 −0.535216
\(336\) 0 0
\(337\) 12.8080 0.697698 0.348849 0.937179i \(-0.386573\pi\)
0.348849 + 0.937179i \(0.386573\pi\)
\(338\) 0 0
\(339\) −2.08969 + 3.61946i −0.113497 + 0.196582i
\(340\) 0 0
\(341\) 0.538705 + 0.933065i 0.0291725 + 0.0505283i
\(342\) 0 0
\(343\) −3.70846 + 18.1452i −0.200238 + 0.979747i
\(344\) 0 0
\(345\) 0.962348 + 1.66684i 0.0518111 + 0.0897394i
\(346\) 0 0
\(347\) 10.1027 17.4984i 0.542342 0.939363i −0.456428 0.889761i \(-0.650871\pi\)
0.998769 0.0496025i \(-0.0157954\pi\)
\(348\) 0 0
\(349\) −18.4434 −0.987252 −0.493626 0.869674i \(-0.664329\pi\)
−0.493626 + 0.869674i \(0.664329\pi\)
\(350\) 0 0
\(351\) −5.63766 −0.300916
\(352\) 0 0
\(353\) 4.07218 7.05322i 0.216740 0.375405i −0.737069 0.675817i \(-0.763791\pi\)
0.953810 + 0.300412i \(0.0971242\pi\)
\(354\) 0 0
\(355\) −7.22064 12.5065i −0.383232 0.663777i
\(356\) 0 0
\(357\) −1.26570 + 18.8064i −0.0669879 + 0.995339i
\(358\) 0 0
\(359\) −16.3050 28.2411i −0.860545 1.49051i −0.871404 0.490566i \(-0.836790\pi\)
0.0108595 0.999941i \(-0.496543\pi\)
\(360\) 0 0
\(361\) 9.21355 15.9583i 0.484923 0.839912i
\(362\) 0 0
\(363\) −13.3314 −0.699715
\(364\) 0 0
\(365\) −18.1173 −0.948304
\(366\) 0 0
\(367\) −1.58006 + 2.73675i −0.0824786 + 0.142857i −0.904314 0.426868i \(-0.859617\pi\)
0.821835 + 0.569725i \(0.192950\pi\)
\(368\) 0 0
\(369\) 4.34526 + 7.52621i 0.226205 + 0.391799i
\(370\) 0 0
\(371\) −11.6091 + 5.69972i −0.602713 + 0.295915i
\(372\) 0 0
\(373\) 0.738849 + 1.27972i 0.0382561 + 0.0662616i 0.884520 0.466503i \(-0.154486\pi\)
−0.846263 + 0.532765i \(0.821153\pi\)
\(374\) 0 0
\(375\) 7.68059 13.3032i 0.396624 0.686972i
\(376\) 0 0
\(377\) −3.10408 −0.159868
\(378\) 0 0
\(379\) −10.7254 −0.550927 −0.275463 0.961312i \(-0.588831\pi\)
−0.275463 + 0.961312i \(0.588831\pi\)
\(380\) 0 0
\(381\) 5.91692 10.2484i 0.303133 0.525042i
\(382\) 0 0
\(383\) −10.7054 18.5424i −0.547023 0.947471i −0.998477 0.0551766i \(-0.982428\pi\)
0.451454 0.892294i \(-0.350906\pi\)
\(384\) 0 0
\(385\) 5.03725 + 3.37855i 0.256722 + 0.172187i
\(386\) 0 0
\(387\) −0.526965 0.912730i −0.0267871 0.0463967i
\(388\) 0 0
\(389\) −17.3909 + 30.1220i −0.881755 + 1.52725i −0.0323675 + 0.999476i \(0.510305\pi\)
−0.849388 + 0.527769i \(0.823029\pi\)
\(390\) 0 0
\(391\) 3.45828 0.174893
\(392\) 0 0
\(393\) 14.1568 0.714119
\(394\) 0 0
\(395\) 2.34042 4.05373i 0.117759 0.203965i
\(396\) 0 0
\(397\) −2.22605 3.85564i −0.111722 0.193509i 0.804742 0.593624i \(-0.202303\pi\)
−0.916465 + 0.400115i \(0.868970\pi\)
\(398\) 0 0
\(399\) −2.23928 1.50191i −0.112104 0.0751897i
\(400\) 0 0
\(401\) 6.87687 + 11.9111i 0.343415 + 0.594811i 0.985064 0.172186i \(-0.0550831\pi\)
−0.641650 + 0.766998i \(0.721750\pi\)
\(402\) 0 0
\(403\) 0.513956 0.890198i 0.0256020 0.0443439i
\(404\) 0 0
\(405\) −8.81241 −0.437892
\(406\) 0 0
\(407\) −11.4178 −0.565959
\(408\) 0 0
\(409\) 1.74603 3.02422i 0.0863358 0.149538i −0.819624 0.572902i \(-0.805818\pi\)
0.905960 + 0.423364i \(0.139151\pi\)
\(410\) 0 0
\(411\) −5.87822 10.1814i −0.289951 0.502210i
\(412\) 0 0
\(413\) −2.48931 + 1.22218i −0.122491 + 0.0601396i
\(414\) 0 0
\(415\) −7.28935 12.6255i −0.357820 0.619763i
\(416\) 0 0
\(417\) −2.69283 + 4.66412i −0.131869 + 0.228403i
\(418\) 0 0
\(419\) 3.56737 0.174278 0.0871388 0.996196i \(-0.472228\pi\)
0.0871388 + 0.996196i \(0.472228\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −1.38750 + 2.40323i −0.0674627 + 0.116849i
\(424\) 0 0
\(425\) −0.572285 0.991227i −0.0277599 0.0480816i
\(426\) 0 0
\(427\) −2.22073 + 32.9967i −0.107469 + 1.59682i
\(428\) 0 0
\(429\) −0.705626 1.22218i −0.0340680 0.0590074i
\(430\) 0 0
\(431\) −5.68211 + 9.84171i −0.273698 + 0.474059i −0.969806 0.243879i \(-0.921580\pi\)
0.696108 + 0.717937i \(0.254913\pi\)
\(432\) 0 0
\(433\) −21.2136 −1.01946 −0.509731 0.860334i \(-0.670255\pi\)
−0.509731 + 0.860334i \(0.670255\pi\)
\(434\) 0 0
\(435\) −9.14101 −0.438278
\(436\) 0 0
\(437\) −0.247351 + 0.428424i −0.0118324 + 0.0204943i
\(438\) 0 0
\(439\) −12.2503 21.2182i −0.584676 1.01269i −0.994916 0.100711i \(-0.967888\pi\)
0.410239 0.911978i \(-0.365445\pi\)
\(440\) 0 0
\(441\) −5.08195 + 6.57513i −0.241997 + 0.313102i
\(442\) 0 0
\(443\) 20.2344 + 35.0470i 0.961366 + 1.66513i 0.719077 + 0.694930i \(0.244565\pi\)
0.242288 + 0.970204i \(0.422102\pi\)
\(444\) 0 0
\(445\) 6.30753 10.9250i 0.299005 0.517893i
\(446\) 0 0
\(447\) 20.7249 0.980254
\(448\) 0 0
\(449\) −27.7638 −1.31025 −0.655127 0.755519i \(-0.727385\pi\)
−0.655127 + 0.755519i \(0.727385\pi\)
\(450\) 0 0
\(451\) −3.83646 + 6.64494i −0.180652 + 0.312898i
\(452\) 0 0
\(453\) −9.20661 15.9463i −0.432564 0.749223i
\(454\) 0 0
\(455\) 0.388575 5.77363i 0.0182167 0.270672i
\(456\) 0 0
\(457\) 5.59696 + 9.69422i 0.261815 + 0.453476i 0.966724 0.255821i \(-0.0823457\pi\)
−0.704910 + 0.709297i \(0.749012\pi\)
\(458\) 0 0
\(459\) −14.9151 + 25.8338i −0.696179 + 1.20582i
\(460\) 0 0
\(461\) −9.29773 −0.433038 −0.216519 0.976278i \(-0.569470\pi\)
−0.216519 + 0.976278i \(0.569470\pi\)
\(462\) 0 0
\(463\) −28.2439 −1.31260 −0.656302 0.754499i \(-0.727880\pi\)
−0.656302 + 0.754499i \(0.727880\pi\)
\(464\) 0 0
\(465\) 1.51351 2.62148i 0.0701875 0.121568i
\(466\) 0 0
\(467\) 11.1303 + 19.2783i 0.515050 + 0.892093i 0.999847 + 0.0174663i \(0.00555997\pi\)
−0.484797 + 0.874626i \(0.661107\pi\)
\(468\) 0 0
\(469\) 10.6371 5.22252i 0.491177 0.241154i
\(470\) 0 0
\(471\) −2.28053 3.95000i −0.105081 0.182006i
\(472\) 0 0
\(473\) 0.465261 0.805855i 0.0213927 0.0370533i
\(474\) 0 0
\(475\) 0.163729 0.00751242
\(476\) 0 0
\(477\) −5.80302 −0.265702
\(478\) 0 0
\(479\) −16.4382 + 28.4718i −0.751081 + 1.30091i 0.196219 + 0.980560i \(0.437134\pi\)
−0.947299 + 0.320350i \(0.896200\pi\)
\(480\) 0 0
\(481\) 5.44661 + 9.43381i 0.248344 + 0.430145i
\(482\) 0 0
\(483\) −1.93360 1.29689i −0.0879820 0.0590107i
\(484\) 0 0
\(485\) 3.15801 + 5.46984i 0.143398 + 0.248373i
\(486\) 0 0
\(487\) −13.9462 + 24.1555i −0.631962 + 1.09459i 0.355188 + 0.934795i \(0.384417\pi\)
−0.987150 + 0.159796i \(0.948916\pi\)
\(488\) 0 0
\(489\) 18.5941 0.840852
\(490\) 0 0
\(491\) −10.6571 −0.480948 −0.240474 0.970656i \(-0.577303\pi\)
−0.240474 + 0.970656i \(0.577303\pi\)
\(492\) 0 0
\(493\) −8.21224 + 14.2240i −0.369861 + 0.640618i
\(494\) 0 0
\(495\) 1.36078 + 2.35694i 0.0611625 + 0.105936i
\(496\) 0 0
\(497\) 14.5081 + 9.73078i 0.650777 + 0.436485i
\(498\) 0 0
\(499\) −12.2557 21.2275i −0.548641 0.950274i −0.998368 0.0571077i \(-0.981812\pi\)
0.449727 0.893166i \(-0.351521\pi\)
\(500\) 0 0
\(501\) 11.0260 19.0976i 0.492605 0.853217i
\(502\) 0 0
\(503\) 38.0054 1.69458 0.847288 0.531134i \(-0.178234\pi\)
0.847288 + 0.531134i \(0.178234\pi\)
\(504\) 0 0
\(505\) −24.6151 −1.09536
\(506\) 0 0
\(507\) −0.673208 + 1.16603i −0.0298982 + 0.0517852i
\(508\) 0 0
\(509\) −19.9250 34.5112i −0.883161 1.52968i −0.847807 0.530305i \(-0.822077\pi\)
−0.0353545 0.999375i \(-0.511256\pi\)
\(510\) 0 0
\(511\) 19.6728 9.65878i 0.870274 0.427279i
\(512\) 0 0
\(513\) −2.13359 3.69549i −0.0942004 0.163160i
\(514\) 0 0
\(515\) 22.1269 38.3249i 0.975027 1.68880i
\(516\) 0 0
\(517\) −2.45007 −0.107754
\(518\) 0 0
\(519\) 5.55459 0.243819
\(520\) 0 0
\(521\) 9.81670 17.0030i 0.430077 0.744916i −0.566802 0.823854i \(-0.691820\pi\)
0.996880 + 0.0789382i \(0.0251530\pi\)
\(522\) 0 0
\(523\) 11.4162 + 19.7734i 0.499195 + 0.864632i 1.00000 0.000928862i \(-0.000295666\pi\)
−0.500804 + 0.865561i \(0.666962\pi\)
\(524\) 0 0
\(525\) −0.0517436 + 0.768832i −0.00225828 + 0.0335546i
\(526\) 0 0
\(527\) −2.71947 4.71026i −0.118462 0.205182i
\(528\) 0 0
\(529\) 11.2864 19.5486i 0.490714 0.849941i
\(530\) 0 0
\(531\) −1.24433 −0.0539994
\(532\) 0 0
\(533\) 7.32040 0.317082
\(534\) 0 0
\(535\) 9.90257 17.1518i 0.428125 0.741535i
\(536\) 0 0
\(537\) 9.70333 + 16.8067i 0.418730 + 0.725261i
\(538\) 0 0
\(539\) −7.27092 0.983142i −0.313181 0.0423469i
\(540\) 0 0
\(541\) 4.82334 + 8.35427i 0.207372 + 0.359178i 0.950886 0.309542i \(-0.100176\pi\)
−0.743514 + 0.668720i \(0.766842\pi\)
\(542\) 0 0
\(543\) −12.1860 + 21.1068i −0.522952 + 0.905779i
\(544\) 0 0
\(545\) 33.0292 1.41482
\(546\) 0 0
\(547\) 43.8570 1.87519 0.937596 0.347728i \(-0.113047\pi\)
0.937596 + 0.347728i \(0.113047\pi\)
\(548\) 0 0
\(549\) −7.41965 + 12.8512i −0.316663 + 0.548476i
\(550\) 0 0
\(551\) −1.17475 2.03473i −0.0500461 0.0866823i
\(552\) 0 0
\(553\) −0.380221 + 5.64950i −0.0161686 + 0.240241i
\(554\) 0 0
\(555\) 16.0394 + 27.7810i 0.680833 + 1.17924i
\(556\) 0 0
\(557\) −7.45977 + 12.9207i −0.316080 + 0.547467i −0.979667 0.200633i \(-0.935700\pi\)
0.663586 + 0.748100i \(0.269034\pi\)
\(558\) 0 0
\(559\) −0.887771 −0.0375487
\(560\) 0 0
\(561\) −7.46729 −0.315269
\(562\) 0 0
\(563\) −8.63486 + 14.9560i −0.363916 + 0.630321i −0.988602 0.150555i \(-0.951894\pi\)
0.624686 + 0.780876i \(0.285227\pi\)
\(564\) 0 0
\(565\) −3.39457 5.87957i −0.142811 0.247355i
\(566\) 0 0
\(567\) 9.56902 4.69811i 0.401861 0.197302i
\(568\) 0 0
\(569\) −13.2662 22.9777i −0.556148 0.963277i −0.997813 0.0660972i \(-0.978945\pi\)
0.441665 0.897180i \(-0.354388\pi\)
\(570\) 0 0
\(571\) −0.992844 + 1.71966i −0.0415492 + 0.0719654i −0.886052 0.463586i \(-0.846563\pi\)
0.844503 + 0.535551i \(0.179896\pi\)
\(572\) 0 0
\(573\) −7.46097 −0.311686
\(574\) 0 0
\(575\) 0.141379 0.00589593
\(576\) 0 0
\(577\) −5.94915 + 10.3042i −0.247666 + 0.428971i −0.962878 0.269937i \(-0.912997\pi\)
0.715212 + 0.698908i \(0.246330\pi\)
\(578\) 0 0
\(579\) 5.88443 + 10.1921i 0.244549 + 0.423571i
\(580\) 0 0
\(581\) 14.6462 + 9.82339i 0.607626 + 0.407543i
\(582\) 0 0
\(583\) −2.56176 4.43711i −0.106097 0.183766i
\(584\) 0 0
\(585\) 1.29826 2.24865i 0.0536765 0.0929704i
\(586\) 0 0
\(587\) −33.5122 −1.38320 −0.691598 0.722283i \(-0.743093\pi\)
−0.691598 + 0.722283i \(0.743093\pi\)
\(588\) 0 0
\(589\) 0.778033 0.0320583
\(590\) 0 0
\(591\) 3.68160 6.37672i 0.151441 0.262303i
\(592\) 0 0
\(593\) −17.6408 30.5547i −0.724419 1.25473i −0.959213 0.282686i \(-0.908775\pi\)
0.234793 0.972045i \(-0.424559\pi\)
\(594\) 0 0
\(595\) −25.4288 17.0554i −1.04248 0.699205i
\(596\) 0 0
\(597\) 13.1523 + 22.7805i 0.538288 + 0.932343i
\(598\) 0 0
\(599\) −12.5034 + 21.6565i −0.510876 + 0.884863i 0.489045 + 0.872259i \(0.337345\pi\)
−0.999921 + 0.0126040i \(0.995988\pi\)
\(600\) 0 0
\(601\) −28.4688 −1.16127 −0.580634 0.814165i \(-0.697195\pi\)
−0.580634 + 0.814165i \(0.697195\pi\)
\(602\) 0 0
\(603\) 5.31717 0.216532
\(604\) 0 0
\(605\) 10.8280 18.7546i 0.440219 0.762482i
\(606\) 0 0
\(607\) 18.0234 + 31.2175i 0.731549 + 1.26708i 0.956221 + 0.292646i \(0.0945356\pi\)
−0.224672 + 0.974434i \(0.572131\pi\)
\(608\) 0 0
\(609\) 9.92583 4.87330i 0.402215 0.197476i
\(610\) 0 0
\(611\) 1.16875 + 2.02434i 0.0472827 + 0.0818961i
\(612\) 0 0
\(613\) −9.16264 + 15.8702i −0.370075 + 0.640989i −0.989577 0.144006i \(-0.954001\pi\)
0.619501 + 0.784996i \(0.287335\pi\)
\(614\) 0 0
\(615\) 21.5573 0.869276
\(616\) 0 0
\(617\) 44.3782 1.78660 0.893299 0.449463i \(-0.148385\pi\)
0.893299 + 0.449463i \(0.148385\pi\)
\(618\) 0 0
\(619\) −12.5043 + 21.6580i −0.502588 + 0.870509i 0.497407 + 0.867517i \(0.334286\pi\)
−0.999996 + 0.00299144i \(0.999048\pi\)
\(620\) 0 0
\(621\) −1.84234 3.19103i −0.0739307 0.128052i
\(622\) 0 0
\(623\) −1.02471 + 15.2256i −0.0410541 + 0.610002i
\(624\) 0 0
\(625\) 11.9358 + 20.6734i 0.477433 + 0.826938i
\(626\) 0 0
\(627\) 0.534093 0.925076i 0.0213296 0.0369440i
\(628\) 0 0
\(629\) 57.6388 2.29821
\(630\) 0 0
\(631\) 18.4638 0.735032 0.367516 0.930017i \(-0.380208\pi\)
0.367516 + 0.930017i \(0.380208\pi\)
\(632\) 0 0
\(633\) 11.2362 19.4616i 0.446598 0.773531i
\(634\) 0 0
\(635\) 9.61165 + 16.6479i 0.381427 + 0.660650i
\(636\) 0 0
\(637\) 2.65613 + 6.47650i 0.105240 + 0.256608i
\(638\) 0 0
\(639\) 3.91927 + 6.78837i 0.155044 + 0.268544i
\(640\) 0 0
\(641\) 10.6284 18.4088i 0.419795 0.727106i −0.576124 0.817362i \(-0.695435\pi\)
0.995919 + 0.0902567i \(0.0287687\pi\)
\(642\) 0 0
\(643\) 36.0554 1.42188 0.710942 0.703251i \(-0.248269\pi\)
0.710942 + 0.703251i \(0.248269\pi\)
\(644\) 0 0
\(645\) −2.61434 −0.102939
\(646\) 0 0
\(647\) 19.9117 34.4881i 0.782809 1.35587i −0.147490 0.989064i \(-0.547119\pi\)
0.930299 0.366802i \(-0.119547\pi\)
\(648\) 0 0
\(649\) −0.549314 0.951440i −0.0215625 0.0373473i
\(650\) 0 0
\(651\) −0.245883 + 3.65345i −0.00963691 + 0.143190i
\(652\) 0 0
\(653\) 16.2335 + 28.1172i 0.635265 + 1.10031i 0.986459 + 0.164008i \(0.0524423\pi\)
−0.351195 + 0.936303i \(0.614224\pi\)
\(654\) 0 0
\(655\) −11.4984 + 19.9159i −0.449281 + 0.778177i
\(656\) 0 0
\(657\) 9.83384 0.383655
\(658\) 0 0
\(659\) −23.5230 −0.916327 −0.458164 0.888868i \(-0.651493\pi\)
−0.458164 + 0.888868i \(0.651493\pi\)
\(660\) 0 0
\(661\) −7.01944 + 12.1580i −0.273025 + 0.472893i −0.969635 0.244557i \(-0.921357\pi\)
0.696610 + 0.717450i \(0.254691\pi\)
\(662\) 0 0
\(663\) 3.56211 + 6.16976i 0.138341 + 0.239614i
\(664\) 0 0
\(665\) 3.93167 1.93034i 0.152464 0.0748553i
\(666\) 0 0
\(667\) −1.01439 1.75698i −0.0392773 0.0680303i
\(668\) 0 0
\(669\) −3.59639 + 6.22913i −0.139044 + 0.240832i
\(670\) 0 0
\(671\) −13.1017 −0.505786
\(672\) 0 0
\(673\) −47.1937 −1.81918 −0.909592 0.415502i \(-0.863606\pi\)
−0.909592 + 0.415502i \(0.863606\pi\)
\(674\) 0 0
\(675\) −0.609753 + 1.05612i −0.0234694 + 0.0406502i
\(676\) 0 0
\(677\) 4.79438 + 8.30411i 0.184263 + 0.319153i 0.943328 0.331862i \(-0.107677\pi\)
−0.759065 + 0.651015i \(0.774344\pi\)
\(678\) 0 0
\(679\) −6.34526 4.25585i −0.243509 0.163325i
\(680\) 0 0
\(681\) 13.5460 + 23.4623i 0.519083 + 0.899078i
\(682\) 0 0
\(683\) 23.6581 40.9769i 0.905250 1.56794i 0.0846691 0.996409i \(-0.473017\pi\)
0.820581 0.571530i \(-0.193650\pi\)
\(684\) 0 0
\(685\) 19.0976 0.729680
\(686\) 0 0
\(687\) 33.9967 1.29705
\(688\) 0 0
\(689\) −2.44407 + 4.23325i −0.0931117 + 0.161274i
\(690\) 0 0
\(691\) −13.5559 23.4796i −0.515692 0.893205i −0.999834 0.0182158i \(-0.994201\pi\)
0.484142 0.874990i \(-0.339132\pi\)
\(692\) 0 0
\(693\) −2.73415 1.83383i −0.103862 0.0696615i
\(694\) 0 0
\(695\) −4.37433 7.57656i −0.165928 0.287395i
\(696\) 0 0
\(697\) 19.3670 33.5447i 0.733578 1.27059i
\(698\) 0 0
\(699\) −1.06819 −0.0404025
\(700\) 0 0
\(701\) −1.79821 −0.0679176 −0.0339588 0.999423i \(-0.510811\pi\)
−0.0339588 + 0.999423i \(0.510811\pi\)
\(702\) 0 0
\(703\) −4.12258 + 7.14051i −0.155486 + 0.269310i
\(704\) 0 0
\(705\) 3.44179 + 5.96135i 0.129625 + 0.224517i
\(706\) 0 0
\(707\) 26.7284 13.1229i 1.00523 0.493537i
\(708\) 0 0
\(709\) 14.1615 + 24.5284i 0.531846 + 0.921185i 0.999309 + 0.0371721i \(0.0118350\pi\)
−0.467462 + 0.884013i \(0.654832\pi\)
\(710\) 0 0
\(711\) −1.27035 + 2.20031i −0.0476418 + 0.0825180i
\(712\) 0 0
\(713\) 0.671827 0.0251601
\(714\) 0 0
\(715\) 2.29249 0.0857341
\(716\) 0 0
\(717\) 13.4970 23.3775i 0.504055 0.873050i
\(718\) 0 0
\(719\) −20.9485 36.2839i −0.781249 1.35316i −0.931215 0.364471i \(-0.881250\pi\)
0.149966 0.988691i \(-0.452084\pi\)
\(720\) 0 0
\(721\) −3.59469 + 53.4117i −0.133873 + 1.98916i
\(722\) 0 0
\(723\) −9.29838 16.1053i −0.345810 0.598961i
\(724\) 0 0
\(725\) −0.335728 + 0.581499i −0.0124686 + 0.0215963i
\(726\) 0 0
\(727\) −19.5123 −0.723670 −0.361835 0.932242i \(-0.617850\pi\)
−0.361835 + 0.932242i \(0.617850\pi\)
\(728\) 0 0
\(729\) 27.5552 1.02056
\(730\) 0 0
\(731\) −2.34871 + 4.06808i −0.0868701 + 0.150463i
\(732\) 0 0
\(733\) −8.87698 15.3754i −0.327879 0.567902i 0.654212 0.756311i \(-0.273000\pi\)
−0.982091 + 0.188409i \(0.939667\pi\)
\(734\) 0 0
\(735\) 7.82185 + 19.0722i 0.288513 + 0.703488i
\(736\) 0 0
\(737\) 2.34728 + 4.06562i 0.0864633 + 0.149759i
\(738\) 0 0
\(739\) 22.1571 38.3772i 0.815061 1.41173i −0.0942227 0.995551i \(-0.530037\pi\)
0.909284 0.416176i \(-0.136630\pi\)
\(740\) 0 0
\(741\) −1.01911 −0.0374380
\(742\) 0 0
\(743\) 7.16727 0.262941 0.131471 0.991320i \(-0.458030\pi\)
0.131471 + 0.991320i \(0.458030\pi\)
\(744\) 0 0
\(745\) −16.8331 + 29.1558i −0.616718 + 1.06819i
\(746\) 0 0
\(747\) 3.95657 + 6.85297i 0.144763 + 0.250737i
\(748\) 0 0
\(749\) −1.60875 + 23.9036i −0.0587826 + 0.873420i
\(750\) 0 0
\(751\) −16.9532 29.3639i −0.618632 1.07150i −0.989736 0.142911i \(-0.954354\pi\)
0.371103 0.928592i \(-0.378980\pi\)
\(752\) 0 0
\(753\) −17.5772 + 30.4445i −0.640547 + 1.10946i
\(754\) 0 0
\(755\) 29.9110 1.08857
\(756\) 0 0
\(757\) −0.906670 −0.0329535 −0.0164767 0.999864i \(-0.505245\pi\)
−0.0164767 + 0.999864i \(0.505245\pi\)
\(758\) 0 0
\(759\) 0.461186 0.798798i 0.0167400 0.0289945i
\(760\) 0 0
\(761\) −10.1247 17.5365i −0.367020 0.635697i 0.622079 0.782955i \(-0.286288\pi\)
−0.989098 + 0.147258i \(0.952955\pi\)
\(762\) 0 0
\(763\) −35.8650 + 17.6087i −1.29840 + 0.637477i
\(764\) 0 0
\(765\) −6.86942 11.8982i −0.248364 0.430180i
\(766\) 0 0
\(767\) −0.524077 + 0.907729i −0.0189233 + 0.0327762i
\(768\) 0 0
\(769\) −36.9094 −1.33099 −0.665494 0.746403i \(-0.731779\pi\)
−0.665494 + 0.746403i \(0.731779\pi\)
\(770\) 0 0
\(771\) 14.2987 0.514954
\(772\) 0 0
\(773\) 4.94018 8.55665i 0.177686 0.307761i −0.763402 0.645924i \(-0.776472\pi\)
0.941088 + 0.338163i \(0.109806\pi\)
\(774\) 0 0
\(775\) −0.111176 0.192562i −0.00399355 0.00691704i
\(776\) 0 0
\(777\) −32.2272 21.6152i −1.15614 0.775441i
\(778\) 0 0
\(779\) 2.77043 + 4.79852i 0.0992609 + 0.171925i
\(780\) 0 0
\(781\) −3.46035 + 5.99350i −0.123821 + 0.214464i
\(782\) 0 0
\(783\) 17.4998 0.625391
\(784\) 0 0
\(785\) 7.40915 0.264444
\(786\) 0 0
\(787\) 18.8411 32.6337i 0.671611 1.16326i −0.305836 0.952084i \(-0.598936\pi\)
0.977447 0.211180i \(-0.0677307\pi\)
\(788\) 0 0
\(789\) 6.97292 + 12.0775i 0.248243 + 0.429969i
\(790\) 0 0
\(791\) 6.82056 + 4.57464i 0.242511 + 0.162656i
\(792\) 0 0
\(793\) 6.24989 + 10.8251i 0.221940 + 0.384412i
\(794\) 0 0
\(795\) −7.19738 + 12.4662i −0.255265 + 0.442131i
\(796\) 0 0
\(797\) 28.3837 1.00540 0.502701 0.864460i \(-0.332340\pi\)
0.502701 + 0.864460i \(0.332340\pi\)
\(798\) 0 0
\(799\) 12.3683 0.437560
\(800\) 0 0
\(801\) −3.42364 + 5.92992i −0.120968 + 0.209524i
\(802\) 0 0
\(803\) 4.34118 + 7.51915i 0.153197 + 0.265345i
\(804\) 0 0
\(805\) 3.39498 1.66684i 0.119657 0.0587482i
\(806\) 0 0
\(807\) −8.05834 13.9574i −0.283667 0.491325i
\(808\) 0 0
\(809\) 5.87327 10.1728i 0.206493 0.357657i −0.744114 0.668052i \(-0.767128\pi\)
0.950607 + 0.310396i \(0.100461\pi\)
\(810\) 0 0
\(811\) −2.01940 −0.0709108 −0.0354554 0.999371i \(-0.511288\pi\)
−0.0354554 + 0.999371i \(0.511288\pi\)
\(812\) 0 0
\(813\) 3.71194 0.130184
\(814\) 0 0
\(815\) −15.1024 + 26.1581i −0.529014 + 0.916280i
\(816\) 0 0
\(817\) −0.335980 0.581934i −0.0117544 0.0203593i
\(818\) 0 0
\(819\) −0.210913 + 3.13385i −0.00736991 + 0.109506i
\(820\) 0 0
\(821\) 7.54208 + 13.0633i 0.263220 + 0.455911i 0.967096 0.254412i \(-0.0818820\pi\)
−0.703875 + 0.710323i \(0.748549\pi\)
\(822\) 0 0
\(823\) −7.38828 + 12.7969i −0.257539 + 0.446071i −0.965582 0.260098i \(-0.916245\pi\)
0.708043 + 0.706170i \(0.249578\pi\)
\(824\) 0 0
\(825\) −0.305274 −0.0106283
\(826\) 0 0
\(827\) −13.0407 −0.453471 −0.226736 0.973956i \(-0.572805\pi\)
−0.226736 + 0.973956i \(0.572805\pi\)
\(828\) 0 0
\(829\) 12.7291 22.0474i 0.442099 0.765738i −0.555746 0.831352i \(-0.687567\pi\)
0.997845 + 0.0656144i \(0.0209007\pi\)
\(830\) 0 0
\(831\) 16.1082 + 27.9002i 0.558786 + 0.967845i
\(832\) 0 0
\(833\) 36.7047 + 4.96305i 1.27174 + 0.171960i
\(834\) 0 0
\(835\) 17.9110 + 31.0228i 0.619835 + 1.07359i
\(836\) 0 0
\(837\) −2.89751 + 5.01864i −0.100153 + 0.173469i
\(838\) 0 0
\(839\) −32.1703 −1.11064 −0.555321 0.831636i \(-0.687404\pi\)
−0.555321 + 0.831636i \(0.687404\pi\)
\(840\) 0 0
\(841\) −19.3647 −0.667747
\(842\) 0 0
\(843\) 2.61117 4.52268i 0.0899335 0.155769i
\(844\) 0 0
\(845\) −1.09358 1.89414i −0.0376204 0.0651604i
\(846\) 0 0
\(847\) −1.75909 + 26.1374i −0.0604431 + 0.898093i
\(848\) 0 0
\(849\) −4.18061 7.24104i −0.143478 0.248512i
\(850\) 0 0
\(851\) −3.55982 + 6.16579i −0.122029 + 0.211360i
\(852\) 0 0
\(853\) −19.3910 −0.663934 −0.331967 0.943291i \(-0.607712\pi\)
−0.331967 + 0.943291i \(0.607712\pi\)
\(854\) 0 0
\(855\) 1.96532 0.0672127
\(856\) 0 0
\(857\) −8.71210 + 15.0898i −0.297600 + 0.515458i −0.975586 0.219616i \(-0.929519\pi\)
0.677987 + 0.735074i \(0.262853\pi\)
\(858\) 0 0
\(859\) −17.7459 30.7367i −0.605481 1.04872i −0.991975 0.126432i \(-0.959648\pi\)
0.386495 0.922292i \(-0.373686\pi\)
\(860\) 0 0
\(861\) −23.4082 + 11.4927i −0.797749 + 0.391672i
\(862\) 0 0
\(863\) 28.0010 + 48.4991i 0.953164 + 1.65093i 0.738516 + 0.674236i \(0.235527\pi\)
0.214648 + 0.976691i \(0.431140\pi\)
\(864\) 0 0
\(865\) −4.51153 + 7.81420i −0.153397 + 0.265691i
\(866\) 0 0
\(867\) 14.8070 0.502871
\(868\) 0 0
\(869\) −2.24320 −0.0760953
\(870\) 0 0
\(871\) 2.23944 3.87883i 0.0758807 0.131429i
\(872\) 0 0
\(873\) −1.71413 2.96896i −0.0580145 0.100484i
\(874\) 0 0
\(875\) −25.0687 16.8139i −0.847476 0.568414i
\(876\) 0 0
\(877\) −12.6031 21.8292i −0.425577 0.737120i 0.570898 0.821021i \(-0.306595\pi\)
−0.996474 + 0.0839011i \(0.973262\pi\)
\(878\) 0 0
\(879\) −11.1587 + 19.3275i −0.376374 + 0.651899i
\(880\) 0 0
\(881\) −18.6082 −0.626925 −0.313463 0.949601i \(-0.601489\pi\)
−0.313463 + 0.949601i \(0.601489\pi\)
\(882\) 0 0
\(883\) 11.2552 0.378768 0.189384 0.981903i \(-0.439351\pi\)
0.189384 + 0.981903i \(0.439351\pi\)
\(884\) 0 0
\(885\) −1.54332 + 2.67311i −0.0518781 + 0.0898556i
\(886\) 0 0
\(887\) −19.6056 33.9579i −0.658292 1.14020i −0.981057 0.193717i \(-0.937946\pi\)
0.322765 0.946479i \(-0.395388\pi\)
\(888\) 0 0
\(889\) −19.3123 12.9530i −0.647712 0.434429i
\(890\) 0 0
\(891\) 2.11159 + 3.65738i 0.0707408 + 0.122527i
\(892\) 0 0
\(893\) −0.884638 + 1.53224i −0.0296033 + 0.0512744i
\(894\) 0 0
\(895\) −31.5249 −1.05376
\(896\) 0 0
\(897\) −0.879996 −0.0293822
\(898\) 0 0
\(899\) −1.59536 + 2.76325i −0.0532083 + 0.0921595i
\(900\) 0 0
\(901\) 12.9322 + 22.3992i 0.430834 + 0.746226i
\(902\) 0 0
\(903\) 2.83879 1.39377i 0.0944692 0.0463816i
\(904\) 0 0
\(905\) −19.7954 34.2866i −0.658020 1.13972i
\(906\) 0 0
\(907\) 10.7985 18.7035i 0.358558 0.621040i −0.629162 0.777274i \(-0.716602\pi\)
0.987720 + 0.156234i \(0.0499353\pi\)
\(908\) 0 0
\(909\) 13.3607 0.443147
\(910\) 0 0
\(911\) 32.4434 1.07490 0.537449 0.843297i \(-0.319388\pi\)
0.537449 + 0.843297i \(0.319388\pi\)
\(912\) 0 0
\(913\) −3.49328 + 6.05054i −0.115611 + 0.200244i
\(914\) 0 0
\(915\) 18.4049 + 31.8782i 0.608447 + 1.05386i
\(916\) 0 0
\(917\) 1.86802 27.7559i 0.0616873 0.916580i
\(918\) 0 0
\(919\) 17.8686 + 30.9493i 0.589430 + 1.02092i 0.994307 + 0.106552i \(0.0339811\pi\)
−0.404877 + 0.914371i \(0.632686\pi\)
\(920\) 0 0
\(921\) −4.75142 + 8.22971i −0.156565 + 0.271178i
\(922\) 0 0
\(923\) 6.60274 0.217332
\(924\) 0 0
\(925\) 2.35636 0.0774765
\(926\) 0 0
\(927\) −12.0102 + 20.8022i −0.394466 + 0.683235i
\(928\) 0 0
\(929\) −5.88847 10.1991i −0.193194 0.334622i 0.753113 0.657891i \(-0.228551\pi\)
−0.946307 + 0.323269i \(0.895218\pi\)
\(930\) 0 0
\(931\) −3.24012 + 4.19214i −0.106191 + 0.137392i
\(932\) 0 0
\(933\) 14.2116 + 24.6151i 0.465265 + 0.805863i
\(934\) 0 0
\(935\) 6.06506 10.5050i 0.198349 0.343550i
\(936\) 0 0
\(937\) 18.9937 0.620497 0.310248 0.950655i \(-0.399588\pi\)
0.310248 + 0.950655i \(0.399588\pi\)
\(938\) 0 0
\(939\) 2.66660 0.0870213
\(940\) 0 0
\(941\) −3.40932 + 5.90511i −0.111141 + 0.192501i −0.916230 0.400652i \(-0.868784\pi\)
0.805090 + 0.593153i \(0.202117\pi\)
\(942\) 0 0
\(943\) 2.39225 + 4.14349i 0.0779023 + 0.134931i
\(944\) 0 0
\(945\) −2.19065 + 32.5498i −0.0712620 + 1.05884i
\(946\) 0 0
\(947\) 0.529958 + 0.917914i 0.0172213 + 0.0298282i 0.874508 0.485012i \(-0.161185\pi\)
−0.857286 + 0.514840i \(0.827851\pi\)
\(948\) 0 0
\(949\) 4.14174 7.17370i 0.134446 0.232868i
\(950\) 0 0
\(951\) −24.2973 −0.787895
\(952\) 0 0
\(953\) −40.4127 −1.30910 −0.654548 0.756020i \(-0.727141\pi\)
−0.654548 + 0.756020i \(0.727141\pi\)
\(954\) 0 0
\(955\) 6.05993 10.4961i 0.196095 0.339646i
\(956\) 0 0
\(957\) 2.19032 + 3.79375i 0.0708031 + 0.122635i
\(958\) 0 0
\(959\) −20.7372 + 10.1814i −0.669639 + 0.328774i
\(960\) 0 0
\(961\) 14.9717 + 25.9317i 0.482958 + 0.836508i
\(962\) 0 0
\(963\) −5.37498 + 9.30975i −0.173206 + 0.300002i
\(964\) 0 0
\(965\) −19.1177 −0.615422
\(966\) 0 0
\(967\) 36.2949 1.16717 0.583583 0.812053i \(-0.301650\pi\)
0.583583 + 0.812053i \(0.301650\pi\)
\(968\) 0 0
\(969\) −2.69618 + 4.66993i −0.0866139 + 0.150020i
\(970\) 0 0
\(971\) −10.7218 18.5708i −0.344080 0.595964i 0.641106 0.767452i \(-0.278476\pi\)
−0.985186 + 0.171488i \(0.945142\pi\)
\(972\) 0 0
\(973\) 8.78914 + 5.89500i 0.281767 + 0.188985i
\(974\) 0 0
\(975\) 0.145624 + 0.252229i 0.00466371 + 0.00807778i
\(976\) 0 0
\(977\) 19.9138 34.4918i 0.637100 1.10349i −0.348966 0.937135i \(-0.613467\pi\)
0.986066 0.166354i \(-0.0531994\pi\)
\(978\) 0 0
\(979\) −6.04551 −0.193215
\(980\) 0 0
\(981\) −17.9278 −0.572392
\(982\) 0 0
\(983\) 7.94071 13.7537i 0.253269 0.438675i −0.711155 0.703036i \(-0.751827\pi\)
0.964424 + 0.264360i \(0.0851608\pi\)
\(984\) 0 0
\(985\) 5.98052 + 10.3586i 0.190555 + 0.330051i
\(986\) 0 0
\(987\) −6.91543 4.63827i −0.220120 0.147638i
\(988\) 0 0
\(989\) −0.290116 0.502496i −0.00922516 0.0159785i
\(990\) 0 0
\(991\) −8.83435 + 15.3016i −0.280633 + 0.486070i −0.971541 0.236873i \(-0.923878\pi\)
0.690908 + 0.722943i \(0.257211\pi\)
\(992\) 0 0
\(993\) 19.7570 0.626970
\(994\) 0 0
\(995\) −42.7301 −1.35464
\(996\) 0 0
\(997\) 12.4304 21.5301i 0.393675 0.681865i −0.599256 0.800558i \(-0.704537\pi\)
0.992931 + 0.118692i \(0.0378701\pi\)
\(998\) 0 0
\(999\) −30.7062 53.1847i −0.971500 1.68269i
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 1456.2.r.p.417.2 10
4.3 odd 2 91.2.e.c.53.1 10
7.2 even 3 inner 1456.2.r.p.625.2 10
12.11 even 2 819.2.j.h.235.5 10
28.3 even 6 637.2.a.k.1.5 5
28.11 odd 6 637.2.a.l.1.5 5
28.19 even 6 637.2.e.m.79.1 10
28.23 odd 6 91.2.e.c.79.1 yes 10
28.27 even 2 637.2.e.m.508.1 10
52.51 odd 2 1183.2.e.f.508.5 10
84.11 even 6 5733.2.a.bl.1.1 5
84.23 even 6 819.2.j.h.352.5 10
84.59 odd 6 5733.2.a.bm.1.1 5
364.51 odd 6 1183.2.e.f.170.5 10
364.207 odd 6 8281.2.a.bw.1.1 5
364.311 even 6 8281.2.a.bx.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.1 10 4.3 odd 2
91.2.e.c.79.1 yes 10 28.23 odd 6
637.2.a.k.1.5 5 28.3 even 6
637.2.a.l.1.5 5 28.11 odd 6
637.2.e.m.79.1 10 28.19 even 6
637.2.e.m.508.1 10 28.27 even 2
819.2.j.h.235.5 10 12.11 even 2
819.2.j.h.352.5 10 84.23 even 6
1183.2.e.f.170.5 10 364.51 odd 6
1183.2.e.f.508.5 10 52.51 odd 2
1456.2.r.p.417.2 10 1.1 even 1 trivial
1456.2.r.p.625.2 10 7.2 even 3 inner
5733.2.a.bl.1.1 5 84.11 even 6
5733.2.a.bm.1.1 5 84.59 odd 6
8281.2.a.bw.1.1 5 364.207 odd 6
8281.2.a.bx.1.1 5 364.311 even 6