Properties

Label 912.2.bn.n.449.4
Level $912$
Weight $2$
Character 912.449
Analytic conductor $7.282$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(65,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.4
Root \(-1.72340 + 0.172852i\) of defining polynomial
Character \(\chi\) \(=\) 912.449
Dual form 912.2.bn.n.65.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.712008 + 1.57894i) q^{3} +(1.02997 - 0.594652i) q^{5} -1.04533 q^{7} +(-1.98609 - 2.24843i) q^{9} +O(q^{10})\) \(q+(-0.712008 + 1.57894i) q^{3} +(1.02997 - 0.594652i) q^{5} -1.04533 q^{7} +(-1.98609 - 2.24843i) q^{9} +3.41364i q^{11} +(0.793296 + 0.458009i) q^{13} +(0.205573 + 2.04965i) q^{15} +(-2.59249 + 1.49677i) q^{17} +(-4.30418 + 0.688533i) q^{19} +(0.744282 - 1.65051i) q^{21} +(-1.62874 - 0.940355i) q^{23} +(-1.79278 + 3.10518i) q^{25} +(4.96425 - 1.53501i) q^{27} +(0.797081 - 1.38058i) q^{29} +8.98511i q^{31} +(-5.38993 - 2.43054i) q^{33} +(-1.07665 + 0.621605i) q^{35} +4.54422i q^{37} +(-1.28800 + 0.926458i) q^{39} +(-0.469860 - 0.813822i) q^{41} +(2.73567 + 4.73832i) q^{43} +(-3.38264 - 1.13478i) q^{45} +(-7.82414 - 4.51727i) q^{47} -5.90729 q^{49} +(-0.517439 - 5.15909i) q^{51} +(0.0418792 - 0.0725369i) q^{53} +(2.02993 + 3.51594i) q^{55} +(1.97746 - 7.28627i) q^{57} +(-5.12477 - 8.87636i) q^{59} +(-0.766777 + 1.32810i) q^{61} +(2.07611 + 2.35035i) q^{63} +1.08942 q^{65} +(-8.67930 - 5.01100i) q^{67} +(2.64444 - 1.90214i) q^{69} +(-1.80899 - 3.13327i) q^{71} +(4.38049 + 7.58723i) q^{73} +(-3.62642 - 5.04160i) q^{75} -3.56837i q^{77} +(2.95719 - 1.70734i) q^{79} +(-1.11091 + 8.93117i) q^{81} -7.04823i q^{83} +(-1.78012 + 3.08325i) q^{85} +(1.61233 + 2.24153i) q^{87} +(-2.48869 + 4.31053i) q^{89} +(-0.829253 - 0.478770i) q^{91} +(-14.1869 - 6.39747i) q^{93} +(-4.02372 + 3.26865i) q^{95} +(-7.48961 + 4.32413i) q^{97} +(7.67534 - 6.77979i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 3 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 3 q^{5} - 5 q^{9} - 3 q^{13} - 12 q^{15} - 3 q^{17} - 11 q^{19} - 12 q^{21} + 3 q^{23} + 11 q^{25} - 4 q^{27} + 5 q^{29} + 14 q^{33} - 24 q^{35} + 9 q^{39} + 6 q^{41} - 13 q^{43} + 33 q^{45} - 27 q^{47} + 8 q^{49} + 18 q^{51} - 7 q^{53} + 12 q^{55} - 36 q^{57} + 10 q^{59} - q^{61} + 26 q^{63} - 30 q^{65} + 24 q^{67} - 41 q^{69} - 27 q^{71} + 2 q^{73} - 21 q^{75} + 21 q^{79} - 13 q^{81} - 5 q^{85} + 23 q^{87} + 25 q^{89} + 78 q^{91} + 22 q^{93} - 13 q^{95} - 60 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.712008 + 1.57894i −0.411078 + 0.911600i
\(4\) 0 0
\(5\) 1.02997 0.594652i 0.460615 0.265936i −0.251688 0.967808i \(-0.580986\pi\)
0.712303 + 0.701872i \(0.247652\pi\)
\(6\) 0 0
\(7\) −1.04533 −0.395096 −0.197548 0.980293i \(-0.563298\pi\)
−0.197548 + 0.980293i \(0.563298\pi\)
\(8\) 0 0
\(9\) −1.98609 2.24843i −0.662029 0.749478i
\(10\) 0 0
\(11\) 3.41364i 1.02925i 0.857415 + 0.514626i \(0.172069\pi\)
−0.857415 + 0.514626i \(0.827931\pi\)
\(12\) 0 0
\(13\) 0.793296 + 0.458009i 0.220021 + 0.127029i 0.605960 0.795495i \(-0.292789\pi\)
−0.385939 + 0.922524i \(0.626123\pi\)
\(14\) 0 0
\(15\) 0.205573 + 2.04965i 0.0530787 + 0.529217i
\(16\) 0 0
\(17\) −2.59249 + 1.49677i −0.628771 + 0.363021i −0.780276 0.625436i \(-0.784921\pi\)
0.151505 + 0.988456i \(0.451588\pi\)
\(18\) 0 0
\(19\) −4.30418 + 0.688533i −0.987445 + 0.157960i
\(20\) 0 0
\(21\) 0.744282 1.65051i 0.162416 0.360170i
\(22\) 0 0
\(23\) −1.62874 0.940355i −0.339616 0.196077i 0.320486 0.947253i \(-0.396154\pi\)
−0.660102 + 0.751176i \(0.729487\pi\)
\(24\) 0 0
\(25\) −1.79278 + 3.10518i −0.358556 + 0.621037i
\(26\) 0 0
\(27\) 4.96425 1.53501i 0.955370 0.295412i
\(28\) 0 0
\(29\) 0.797081 1.38058i 0.148014 0.256368i −0.782479 0.622677i \(-0.786045\pi\)
0.930493 + 0.366309i \(0.119379\pi\)
\(30\) 0 0
\(31\) 8.98511i 1.61377i 0.590706 + 0.806887i \(0.298849\pi\)
−0.590706 + 0.806887i \(0.701151\pi\)
\(32\) 0 0
\(33\) −5.38993 2.43054i −0.938266 0.423103i
\(34\) 0 0
\(35\) −1.07665 + 0.621605i −0.181987 + 0.105070i
\(36\) 0 0
\(37\) 4.54422i 0.747066i 0.927617 + 0.373533i \(0.121854\pi\)
−0.927617 + 0.373533i \(0.878146\pi\)
\(38\) 0 0
\(39\) −1.28800 + 0.926458i −0.206245 + 0.148352i
\(40\) 0 0
\(41\) −0.469860 0.813822i −0.0733798 0.127098i 0.827001 0.562201i \(-0.190045\pi\)
−0.900381 + 0.435103i \(0.856712\pi\)
\(42\) 0 0
\(43\) 2.73567 + 4.73832i 0.417185 + 0.722586i 0.995655 0.0931175i \(-0.0296832\pi\)
−0.578470 + 0.815704i \(0.696350\pi\)
\(44\) 0 0
\(45\) −3.38264 1.13478i −0.504254 0.169163i
\(46\) 0 0
\(47\) −7.82414 4.51727i −1.14127 0.658911i −0.194524 0.980898i \(-0.562316\pi\)
−0.946744 + 0.321986i \(0.895649\pi\)
\(48\) 0 0
\(49\) −5.90729 −0.843899
\(50\) 0 0
\(51\) −0.517439 5.15909i −0.0724560 0.722417i
\(52\) 0 0
\(53\) 0.0418792 0.0725369i 0.00575255 0.00996371i −0.863135 0.504974i \(-0.831502\pi\)
0.868887 + 0.495010i \(0.164836\pi\)
\(54\) 0 0
\(55\) 2.02993 + 3.51594i 0.273715 + 0.474089i
\(56\) 0 0
\(57\) 1.97746 7.28627i 0.261921 0.965089i
\(58\) 0 0
\(59\) −5.12477 8.87636i −0.667188 1.15560i −0.978687 0.205358i \(-0.934164\pi\)
0.311499 0.950247i \(-0.399169\pi\)
\(60\) 0 0
\(61\) −0.766777 + 1.32810i −0.0981757 + 0.170045i −0.910930 0.412562i \(-0.864634\pi\)
0.812754 + 0.582607i \(0.197967\pi\)
\(62\) 0 0
\(63\) 2.07611 + 2.35035i 0.261565 + 0.296116i
\(64\) 0 0
\(65\) 1.08942 0.135126
\(66\) 0 0
\(67\) −8.67930 5.01100i −1.06035 0.612191i −0.134817 0.990870i \(-0.543045\pi\)
−0.925528 + 0.378680i \(0.876378\pi\)
\(68\) 0 0
\(69\) 2.64444 1.90214i 0.318353 0.228991i
\(70\) 0 0
\(71\) −1.80899 3.13327i −0.214688 0.371851i 0.738488 0.674267i \(-0.235540\pi\)
−0.953176 + 0.302416i \(0.902207\pi\)
\(72\) 0 0
\(73\) 4.38049 + 7.58723i 0.512698 + 0.888018i 0.999892 + 0.0147244i \(0.00468709\pi\)
−0.487194 + 0.873294i \(0.661980\pi\)
\(74\) 0 0
\(75\) −3.62642 5.04160i −0.418743 0.582154i
\(76\) 0 0
\(77\) 3.56837i 0.406654i
\(78\) 0 0
\(79\) 2.95719 1.70734i 0.332710 0.192090i −0.324334 0.945943i \(-0.605140\pi\)
0.657044 + 0.753853i \(0.271807\pi\)
\(80\) 0 0
\(81\) −1.11091 + 8.93117i −0.123434 + 0.992353i
\(82\) 0 0
\(83\) 7.04823i 0.773644i −0.922154 0.386822i \(-0.873573\pi\)
0.922154 0.386822i \(-0.126427\pi\)
\(84\) 0 0
\(85\) −1.78012 + 3.08325i −0.193081 + 0.334426i
\(86\) 0 0
\(87\) 1.61233 + 2.24153i 0.172860 + 0.240317i
\(88\) 0 0
\(89\) −2.48869 + 4.31053i −0.263800 + 0.456916i −0.967249 0.253831i \(-0.918309\pi\)
0.703448 + 0.710746i \(0.251643\pi\)
\(90\) 0 0
\(91\) −0.829253 0.478770i −0.0869294 0.0501887i
\(92\) 0 0
\(93\) −14.1869 6.39747i −1.47112 0.663387i
\(94\) 0 0
\(95\) −4.02372 + 3.26865i −0.412825 + 0.335356i
\(96\) 0 0
\(97\) −7.48961 + 4.32413i −0.760455 + 0.439049i −0.829459 0.558568i \(-0.811351\pi\)
0.0690043 + 0.997616i \(0.478018\pi\)
\(98\) 0 0
\(99\) 7.67534 6.77979i 0.771401 0.681395i
\(100\) 0 0
\(101\) 15.4238 + 8.90495i 1.53473 + 0.886076i 0.999134 + 0.0416008i \(0.0132458\pi\)
0.535595 + 0.844475i \(0.320088\pi\)
\(102\) 0 0
\(103\) 13.3812i 1.31849i 0.751929 + 0.659244i \(0.229123\pi\)
−0.751929 + 0.659244i \(0.770877\pi\)
\(104\) 0 0
\(105\) −0.214891 2.14255i −0.0209712 0.209092i
\(106\) 0 0
\(107\) −6.74732 −0.652288 −0.326144 0.945320i \(-0.605749\pi\)
−0.326144 + 0.945320i \(0.605749\pi\)
\(108\) 0 0
\(109\) −1.56805 + 0.905316i −0.150192 + 0.0867135i −0.573213 0.819407i \(-0.694303\pi\)
0.423021 + 0.906120i \(0.360970\pi\)
\(110\) 0 0
\(111\) −7.17504 3.23552i −0.681025 0.307102i
\(112\) 0 0
\(113\) −0.00792875 −0.000745874 −0.000372937 1.00000i \(-0.500119\pi\)
−0.000372937 1.00000i \(0.500119\pi\)
\(114\) 0 0
\(115\) −2.23673 −0.208576
\(116\) 0 0
\(117\) −0.545751 2.69332i −0.0504547 0.248997i
\(118\) 0 0
\(119\) 2.71000 1.56462i 0.248425 0.143428i
\(120\) 0 0
\(121\) −0.652944 −0.0593585
\(122\) 0 0
\(123\) 1.61952 0.162432i 0.146027 0.0146460i
\(124\) 0 0
\(125\) 10.2108i 0.913284i
\(126\) 0 0
\(127\) 13.4312 + 7.75450i 1.19182 + 0.688100i 0.958720 0.284351i \(-0.0917782\pi\)
0.233105 + 0.972452i \(0.425111\pi\)
\(128\) 0 0
\(129\) −9.42932 + 0.945729i −0.830206 + 0.0832668i
\(130\) 0 0
\(131\) 15.0950 8.71513i 1.31886 0.761444i 0.335315 0.942106i \(-0.391157\pi\)
0.983545 + 0.180662i \(0.0578241\pi\)
\(132\) 0 0
\(133\) 4.49927 0.719742i 0.390136 0.0624095i
\(134\) 0 0
\(135\) 4.20022 4.53300i 0.361497 0.390139i
\(136\) 0 0
\(137\) 12.6485 + 7.30262i 1.08064 + 0.623905i 0.931068 0.364846i \(-0.118878\pi\)
0.149568 + 0.988751i \(0.452212\pi\)
\(138\) 0 0
\(139\) 8.42559 14.5936i 0.714649 1.23781i −0.248445 0.968646i \(-0.579920\pi\)
0.963095 0.269163i \(-0.0867471\pi\)
\(140\) 0 0
\(141\) 12.7033 9.13750i 1.06981 0.769516i
\(142\) 0 0
\(143\) −1.56348 + 2.70803i −0.130745 + 0.226457i
\(144\) 0 0
\(145\) 1.89594i 0.157449i
\(146\) 0 0
\(147\) 4.20604 9.32724i 0.346908 0.769298i
\(148\) 0 0
\(149\) 2.37577 1.37165i 0.194631 0.112370i −0.399518 0.916725i \(-0.630822\pi\)
0.594149 + 0.804355i \(0.297489\pi\)
\(150\) 0 0
\(151\) 2.29062i 0.186408i 0.995647 + 0.0932038i \(0.0297108\pi\)
−0.995647 + 0.0932038i \(0.970289\pi\)
\(152\) 0 0
\(153\) 8.51431 + 2.85631i 0.688341 + 0.230919i
\(154\) 0 0
\(155\) 5.34301 + 9.25436i 0.429161 + 0.743328i
\(156\) 0 0
\(157\) 6.60941 + 11.4478i 0.527488 + 0.913637i 0.999487 + 0.0320372i \(0.0101995\pi\)
−0.471998 + 0.881599i \(0.656467\pi\)
\(158\) 0 0
\(159\) 0.0847129 + 0.117772i 0.00671817 + 0.00933989i
\(160\) 0 0
\(161\) 1.70257 + 0.982978i 0.134181 + 0.0774695i
\(162\) 0 0
\(163\) 1.23577 0.0967932 0.0483966 0.998828i \(-0.484589\pi\)
0.0483966 + 0.998828i \(0.484589\pi\)
\(164\) 0 0
\(165\) −6.99677 + 0.701752i −0.544698 + 0.0546313i
\(166\) 0 0
\(167\) −4.40067 + 7.62219i −0.340534 + 0.589823i −0.984532 0.175205i \(-0.943941\pi\)
0.643998 + 0.765027i \(0.277275\pi\)
\(168\) 0 0
\(169\) −6.08045 10.5317i −0.467727 0.810127i
\(170\) 0 0
\(171\) 10.0966 + 8.31016i 0.772106 + 0.635494i
\(172\) 0 0
\(173\) −13.0208 22.5527i −0.989952 1.71465i −0.617433 0.786623i \(-0.711827\pi\)
−0.372519 0.928024i \(-0.621506\pi\)
\(174\) 0 0
\(175\) 1.87404 3.24593i 0.141664 0.245369i
\(176\) 0 0
\(177\) 17.6641 1.77165i 1.32772 0.133165i
\(178\) 0 0
\(179\) 22.7770 1.70243 0.851217 0.524815i \(-0.175865\pi\)
0.851217 + 0.524815i \(0.175865\pi\)
\(180\) 0 0
\(181\) 5.61211 + 3.24015i 0.417145 + 0.240839i 0.693855 0.720115i \(-0.255911\pi\)
−0.276710 + 0.960953i \(0.589244\pi\)
\(182\) 0 0
\(183\) −1.55103 2.15631i −0.114655 0.159399i
\(184\) 0 0
\(185\) 2.70223 + 4.68040i 0.198672 + 0.344110i
\(186\) 0 0
\(187\) −5.10945 8.84982i −0.373640 0.647163i
\(188\) 0 0
\(189\) −5.18926 + 1.60458i −0.377463 + 0.116716i
\(190\) 0 0
\(191\) 16.8390i 1.21843i 0.793007 + 0.609213i \(0.208514\pi\)
−0.793007 + 0.609213i \(0.791486\pi\)
\(192\) 0 0
\(193\) 12.8046 7.39275i 0.921696 0.532142i 0.0375205 0.999296i \(-0.488054\pi\)
0.884176 + 0.467154i \(0.154721\pi\)
\(194\) 0 0
\(195\) −0.775679 + 1.72013i −0.0555475 + 0.123181i
\(196\) 0 0
\(197\) 8.97140i 0.639186i 0.947555 + 0.319593i \(0.103546\pi\)
−0.947555 + 0.319593i \(0.896454\pi\)
\(198\) 0 0
\(199\) 1.73567 3.00627i 0.123038 0.213108i −0.797926 0.602755i \(-0.794070\pi\)
0.920964 + 0.389647i \(0.127403\pi\)
\(200\) 0 0
\(201\) 14.0918 10.1362i 0.993958 0.714953i
\(202\) 0 0
\(203\) −0.833210 + 1.44316i −0.0584799 + 0.101290i
\(204\) 0 0
\(205\) −0.967881 0.558806i −0.0675997 0.0390287i
\(206\) 0 0
\(207\) 1.12050 + 5.52975i 0.0778802 + 0.384344i
\(208\) 0 0
\(209\) −2.35040 14.6929i −0.162581 1.01633i
\(210\) 0 0
\(211\) 3.14092 1.81341i 0.216230 0.124840i −0.387973 0.921671i \(-0.626825\pi\)
0.604203 + 0.796830i \(0.293491\pi\)
\(212\) 0 0
\(213\) 6.23525 0.625375i 0.427233 0.0428500i
\(214\) 0 0
\(215\) 5.63529 + 3.25354i 0.384324 + 0.221889i
\(216\) 0 0
\(217\) 9.39238i 0.637596i
\(218\) 0 0
\(219\) −15.0987 + 1.51435i −1.02028 + 0.102330i
\(220\) 0 0
\(221\) −2.74215 −0.184457
\(222\) 0 0
\(223\) −14.1524 + 8.17089i −0.947714 + 0.547163i −0.892370 0.451304i \(-0.850959\pi\)
−0.0553443 + 0.998467i \(0.517626\pi\)
\(224\) 0 0
\(225\) 10.5424 2.13623i 0.702828 0.142415i
\(226\) 0 0
\(227\) 0.0153826 0.00102098 0.000510491 1.00000i \(-0.499838\pi\)
0.000510491 1.00000i \(0.499838\pi\)
\(228\) 0 0
\(229\) 27.3381 1.80655 0.903275 0.429062i \(-0.141156\pi\)
0.903275 + 0.429062i \(0.141156\pi\)
\(230\) 0 0
\(231\) 5.63424 + 2.54071i 0.370705 + 0.167166i
\(232\) 0 0
\(233\) −18.6729 + 10.7808i −1.22330 + 0.706273i −0.965620 0.259957i \(-0.916292\pi\)
−0.257681 + 0.966230i \(0.582958\pi\)
\(234\) 0 0
\(235\) −10.7448 −0.700914
\(236\) 0 0
\(237\) 0.590231 + 5.88486i 0.0383396 + 0.382263i
\(238\) 0 0
\(239\) 12.0836i 0.781626i 0.920470 + 0.390813i \(0.127806\pi\)
−0.920470 + 0.390813i \(0.872194\pi\)
\(240\) 0 0
\(241\) −17.6746 10.2044i −1.13852 0.657325i −0.192457 0.981305i \(-0.561646\pi\)
−0.946064 + 0.323980i \(0.894979\pi\)
\(242\) 0 0
\(243\) −13.3108 8.11312i −0.853888 0.520457i
\(244\) 0 0
\(245\) −6.08431 + 3.51278i −0.388713 + 0.224423i
\(246\) 0 0
\(247\) −3.72984 1.42514i −0.237324 0.0906797i
\(248\) 0 0
\(249\) 11.1287 + 5.01840i 0.705254 + 0.318028i
\(250\) 0 0
\(251\) 23.9785 + 13.8440i 1.51351 + 0.873824i 0.999875 + 0.0158130i \(0.00503365\pi\)
0.513632 + 0.858011i \(0.328300\pi\)
\(252\) 0 0
\(253\) 3.21003 5.55994i 0.201813 0.349550i
\(254\) 0 0
\(255\) −3.60081 5.00600i −0.225491 0.313488i
\(256\) 0 0
\(257\) 14.7065 25.4724i 0.917367 1.58893i 0.113969 0.993484i \(-0.463644\pi\)
0.803398 0.595442i \(-0.203023\pi\)
\(258\) 0 0
\(259\) 4.75020i 0.295163i
\(260\) 0 0
\(261\) −4.68723 + 0.949780i −0.290132 + 0.0587899i
\(262\) 0 0
\(263\) −8.78482 + 5.07192i −0.541695 + 0.312748i −0.745766 0.666208i \(-0.767916\pi\)
0.204071 + 0.978956i \(0.434583\pi\)
\(264\) 0 0
\(265\) 0.0996141i 0.00611925i
\(266\) 0 0
\(267\) −5.03410 6.99862i −0.308082 0.428309i
\(268\) 0 0
\(269\) −10.2191 17.6999i −0.623067 1.07918i −0.988911 0.148507i \(-0.952553\pi\)
0.365845 0.930676i \(-0.380780\pi\)
\(270\) 0 0
\(271\) 6.39147 + 11.0703i 0.388254 + 0.672476i 0.992215 0.124538i \(-0.0397450\pi\)
−0.603961 + 0.797014i \(0.706412\pi\)
\(272\) 0 0
\(273\) 1.34638 0.968451i 0.0814868 0.0586133i
\(274\) 0 0
\(275\) −10.6000 6.11990i −0.639203 0.369044i
\(276\) 0 0
\(277\) 9.32291 0.560159 0.280080 0.959977i \(-0.409639\pi\)
0.280080 + 0.959977i \(0.409639\pi\)
\(278\) 0 0
\(279\) 20.2024 17.8452i 1.20949 1.06837i
\(280\) 0 0
\(281\) 2.29185 3.96959i 0.136720 0.236806i −0.789533 0.613708i \(-0.789677\pi\)
0.926253 + 0.376902i \(0.123011\pi\)
\(282\) 0 0
\(283\) 9.95495 + 17.2425i 0.591761 + 1.02496i 0.993995 + 0.109423i \(0.0349002\pi\)
−0.402235 + 0.915537i \(0.631766\pi\)
\(284\) 0 0
\(285\) −2.29607 8.68051i −0.136008 0.514189i
\(286\) 0 0
\(287\) 0.491158 + 0.850710i 0.0289921 + 0.0502158i
\(288\) 0 0
\(289\) −4.01934 + 6.96170i −0.236432 + 0.409512i
\(290\) 0 0
\(291\) −1.49486 14.9044i −0.0876305 0.873714i
\(292\) 0 0
\(293\) −26.0913 −1.52427 −0.762136 0.647417i \(-0.775849\pi\)
−0.762136 + 0.647417i \(0.775849\pi\)
\(294\) 0 0
\(295\) −10.5567 6.09491i −0.614634 0.354859i
\(296\) 0 0
\(297\) 5.23996 + 16.9462i 0.304053 + 0.983316i
\(298\) 0 0
\(299\) −0.861383 1.49196i −0.0498150 0.0862822i
\(300\) 0 0
\(301\) −2.85967 4.95309i −0.164829 0.285491i
\(302\) 0 0
\(303\) −25.0423 + 18.0129i −1.43864 + 1.03481i
\(304\) 0 0
\(305\) 1.82386i 0.104434i
\(306\) 0 0
\(307\) −3.94327 + 2.27665i −0.225054 + 0.129935i −0.608288 0.793716i \(-0.708144\pi\)
0.383234 + 0.923651i \(0.374810\pi\)
\(308\) 0 0
\(309\) −21.1281 9.52751i −1.20193 0.542001i
\(310\) 0 0
\(311\) 18.2836i 1.03677i −0.855147 0.518385i \(-0.826533\pi\)
0.855147 0.518385i \(-0.173467\pi\)
\(312\) 0 0
\(313\) 0.796717 1.37995i 0.0450331 0.0779997i −0.842630 0.538493i \(-0.818994\pi\)
0.887663 + 0.460493i \(0.152327\pi\)
\(314\) 0 0
\(315\) 3.53596 + 1.18622i 0.199229 + 0.0668358i
\(316\) 0 0
\(317\) 3.67389 6.36336i 0.206346 0.357402i −0.744215 0.667940i \(-0.767176\pi\)
0.950561 + 0.310539i \(0.100509\pi\)
\(318\) 0 0
\(319\) 4.71282 + 2.72095i 0.263867 + 0.152344i
\(320\) 0 0
\(321\) 4.80415 10.6536i 0.268141 0.594625i
\(322\) 0 0
\(323\) 10.1279 8.22739i 0.563534 0.457784i
\(324\) 0 0
\(325\) −2.84441 + 1.64222i −0.157779 + 0.0910939i
\(326\) 0 0
\(327\) −0.312970 3.12045i −0.0173073 0.172561i
\(328\) 0 0
\(329\) 8.17878 + 4.72202i 0.450911 + 0.260334i
\(330\) 0 0
\(331\) 7.27646i 0.399950i −0.979801 0.199975i \(-0.935914\pi\)
0.979801 0.199975i \(-0.0640862\pi\)
\(332\) 0 0
\(333\) 10.2174 9.02523i 0.559909 0.494579i
\(334\) 0 0
\(335\) −11.9192 −0.651215
\(336\) 0 0
\(337\) −15.5410 + 8.97260i −0.846572 + 0.488768i −0.859493 0.511148i \(-0.829220\pi\)
0.0129208 + 0.999917i \(0.495887\pi\)
\(338\) 0 0
\(339\) 0.00564533 0.0125190i 0.000306612 0.000679939i
\(340\) 0 0
\(341\) −30.6719 −1.66098
\(342\) 0 0
\(343\) 13.4923 0.728518
\(344\) 0 0
\(345\) 1.59257 3.53166i 0.0857412 0.190138i
\(346\) 0 0
\(347\) −21.8410 + 12.6099i −1.17249 + 0.676936i −0.954265 0.298962i \(-0.903359\pi\)
−0.218223 + 0.975899i \(0.570026\pi\)
\(348\) 0 0
\(349\) −34.0609 −1.82324 −0.911620 0.411035i \(-0.865167\pi\)
−0.911620 + 0.411035i \(0.865167\pi\)
\(350\) 0 0
\(351\) 4.64116 + 1.05596i 0.247727 + 0.0563629i
\(352\) 0 0
\(353\) 11.2144i 0.596884i 0.954428 + 0.298442i \(0.0964670\pi\)
−0.954428 + 0.298442i \(0.903533\pi\)
\(354\) 0 0
\(355\) −3.72641 2.15144i −0.197777 0.114187i
\(356\) 0 0
\(357\) 0.540893 + 5.39294i 0.0286271 + 0.285425i
\(358\) 0 0
\(359\) 31.0882 17.9488i 1.64077 0.947300i 0.660212 0.751079i \(-0.270466\pi\)
0.980560 0.196221i \(-0.0628670\pi\)
\(360\) 0 0
\(361\) 18.0518 5.92713i 0.950097 0.311954i
\(362\) 0 0
\(363\) 0.464901 1.03096i 0.0244010 0.0541112i
\(364\) 0 0
\(365\) 9.02352 + 5.20973i 0.472312 + 0.272690i
\(366\) 0 0
\(367\) 13.7083 23.7434i 0.715566 1.23940i −0.247175 0.968971i \(-0.579502\pi\)
0.962741 0.270425i \(-0.0871643\pi\)
\(368\) 0 0
\(369\) −0.896640 + 2.67277i −0.0466772 + 0.139139i
\(370\) 0 0
\(371\) −0.0437775 + 0.0758248i −0.00227281 + 0.00393663i
\(372\) 0 0
\(373\) 37.4573i 1.93946i 0.244173 + 0.969732i \(0.421483\pi\)
−0.244173 + 0.969732i \(0.578517\pi\)
\(374\) 0 0
\(375\) −16.1223 7.27020i −0.832550 0.375431i
\(376\) 0 0
\(377\) 1.26464 0.730141i 0.0651324 0.0376042i
\(378\) 0 0
\(379\) 1.41500i 0.0726837i 0.999339 + 0.0363419i \(0.0115705\pi\)
−0.999339 + 0.0363419i \(0.988429\pi\)
\(380\) 0 0
\(381\) −21.8070 + 15.6857i −1.11721 + 0.803605i
\(382\) 0 0
\(383\) 16.5830 + 28.7225i 0.847349 + 1.46765i 0.883565 + 0.468308i \(0.155136\pi\)
−0.0362158 + 0.999344i \(0.511530\pi\)
\(384\) 0 0
\(385\) −2.12194 3.67530i −0.108144 0.187311i
\(386\) 0 0
\(387\) 5.22051 15.5617i 0.265373 0.791045i
\(388\) 0 0
\(389\) 25.7534 + 14.8687i 1.30575 + 0.753875i 0.981384 0.192057i \(-0.0615158\pi\)
0.324366 + 0.945932i \(0.394849\pi\)
\(390\) 0 0
\(391\) 5.62999 0.284721
\(392\) 0 0
\(393\) 3.01285 + 30.0394i 0.151978 + 1.51529i
\(394\) 0 0
\(395\) 2.03054 3.51700i 0.102168 0.176959i
\(396\) 0 0
\(397\) 6.46894 + 11.2045i 0.324667 + 0.562339i 0.981445 0.191745i \(-0.0614146\pi\)
−0.656778 + 0.754084i \(0.728081\pi\)
\(398\) 0 0
\(399\) −2.06709 + 7.61653i −0.103484 + 0.381303i
\(400\) 0 0
\(401\) −2.87324 4.97659i −0.143483 0.248519i 0.785323 0.619086i \(-0.212497\pi\)
−0.928806 + 0.370567i \(0.879163\pi\)
\(402\) 0 0
\(403\) −4.11526 + 7.12785i −0.204996 + 0.355063i
\(404\) 0 0
\(405\) 4.16674 + 9.85942i 0.207047 + 0.489918i
\(406\) 0 0
\(407\) −15.5123 −0.768918
\(408\) 0 0
\(409\) −12.6583 7.30828i −0.625913 0.361371i 0.153254 0.988187i \(-0.451025\pi\)
−0.779168 + 0.626815i \(0.784358\pi\)
\(410\) 0 0
\(411\) −20.5362 + 14.7717i −1.01298 + 0.728634i
\(412\) 0 0
\(413\) 5.35706 + 9.27870i 0.263604 + 0.456575i
\(414\) 0 0
\(415\) −4.19124 7.25944i −0.205740 0.356352i
\(416\) 0 0
\(417\) 17.0432 + 23.6942i 0.834610 + 1.16031i
\(418\) 0 0
\(419\) 2.03348i 0.0993419i 0.998766 + 0.0496709i \(0.0158173\pi\)
−0.998766 + 0.0496709i \(0.984183\pi\)
\(420\) 0 0
\(421\) 26.2813 15.1735i 1.28087 0.739512i 0.303865 0.952715i \(-0.401723\pi\)
0.977008 + 0.213203i \(0.0683895\pi\)
\(422\) 0 0
\(423\) 5.38265 + 26.5638i 0.261714 + 1.29157i
\(424\) 0 0
\(425\) 10.7335i 0.520653i
\(426\) 0 0
\(427\) 0.801532 1.38829i 0.0387889 0.0671843i
\(428\) 0 0
\(429\) −3.16259 4.39677i −0.152691 0.212278i
\(430\) 0 0
\(431\) 15.1542 26.2478i 0.729950 1.26431i −0.226954 0.973905i \(-0.572877\pi\)
0.956904 0.290405i \(-0.0937899\pi\)
\(432\) 0 0
\(433\) −20.0370 11.5684i −0.962917 0.555941i −0.0658477 0.997830i \(-0.520975\pi\)
−0.897070 + 0.441889i \(0.854308\pi\)
\(434\) 0 0
\(435\) 2.99357 + 1.34993i 0.143531 + 0.0647240i
\(436\) 0 0
\(437\) 7.65786 + 2.92601i 0.366325 + 0.139970i
\(438\) 0 0
\(439\) −34.5058 + 19.9219i −1.64687 + 0.950822i −0.668566 + 0.743653i \(0.733092\pi\)
−0.978305 + 0.207169i \(0.933575\pi\)
\(440\) 0 0
\(441\) 11.7324 + 13.2822i 0.558686 + 0.632483i
\(442\) 0 0
\(443\) −17.7959 10.2744i −0.845507 0.488154i 0.0136253 0.999907i \(-0.495663\pi\)
−0.859132 + 0.511753i \(0.828996\pi\)
\(444\) 0 0
\(445\) 5.91961i 0.280616i
\(446\) 0 0
\(447\) 0.474184 + 4.72782i 0.0224281 + 0.223618i
\(448\) 0 0
\(449\) −7.29168 −0.344116 −0.172058 0.985087i \(-0.555042\pi\)
−0.172058 + 0.985087i \(0.555042\pi\)
\(450\) 0 0
\(451\) 2.77810 1.60393i 0.130815 0.0755263i
\(452\) 0 0
\(453\) −3.61674 1.63094i −0.169929 0.0766281i
\(454\) 0 0
\(455\) −1.13880 −0.0533880
\(456\) 0 0
\(457\) −15.3652 −0.718754 −0.359377 0.933193i \(-0.617011\pi\)
−0.359377 + 0.933193i \(0.617011\pi\)
\(458\) 0 0
\(459\) −10.5722 + 11.4098i −0.493468 + 0.532566i
\(460\) 0 0
\(461\) −2.41846 + 1.39630i −0.112639 + 0.0650322i −0.555261 0.831676i \(-0.687382\pi\)
0.442622 + 0.896708i \(0.354048\pi\)
\(462\) 0 0
\(463\) 6.11119 0.284011 0.142006 0.989866i \(-0.454645\pi\)
0.142006 + 0.989866i \(0.454645\pi\)
\(464\) 0 0
\(465\) −18.4163 + 1.84709i −0.854037 + 0.0856569i
\(466\) 0 0
\(467\) 4.17320i 0.193113i 0.995328 + 0.0965564i \(0.0307828\pi\)
−0.995328 + 0.0965564i \(0.969217\pi\)
\(468\) 0 0
\(469\) 9.07271 + 5.23813i 0.418939 + 0.241874i
\(470\) 0 0
\(471\) −22.7814 + 2.28489i −1.04971 + 0.105282i
\(472\) 0 0
\(473\) −16.1749 + 9.33859i −0.743723 + 0.429389i
\(474\) 0 0
\(475\) 5.57841 14.5996i 0.255955 0.669878i
\(476\) 0 0
\(477\) −0.246270 + 0.0499021i −0.0112759 + 0.00228486i
\(478\) 0 0
\(479\) 3.06865 + 1.77169i 0.140210 + 0.0809504i 0.568464 0.822708i \(-0.307538\pi\)
−0.428254 + 0.903658i \(0.640871\pi\)
\(480\) 0 0
\(481\) −2.08130 + 3.60491i −0.0948990 + 0.164370i
\(482\) 0 0
\(483\) −2.76430 + 1.98836i −0.125780 + 0.0904735i
\(484\) 0 0
\(485\) −5.14270 + 8.90742i −0.233518 + 0.404465i
\(486\) 0 0
\(487\) 18.1819i 0.823900i −0.911207 0.411950i \(-0.864848\pi\)
0.911207 0.411950i \(-0.135152\pi\)
\(488\) 0 0
\(489\) −0.879880 + 1.95121i −0.0397896 + 0.0882367i
\(490\) 0 0
\(491\) −11.4107 + 6.58799i −0.514959 + 0.297312i −0.734870 0.678208i \(-0.762757\pi\)
0.219911 + 0.975520i \(0.429423\pi\)
\(492\) 0 0
\(493\) 4.77220i 0.214929i
\(494\) 0 0
\(495\) 3.87374 11.5471i 0.174111 0.519004i
\(496\) 0 0
\(497\) 1.89099 + 3.27529i 0.0848225 + 0.146917i
\(498\) 0 0
\(499\) 9.34894 + 16.1928i 0.418516 + 0.724891i 0.995790 0.0916594i \(-0.0292171\pi\)
−0.577275 + 0.816550i \(0.695884\pi\)
\(500\) 0 0
\(501\) −8.90165 12.3755i −0.397696 0.552894i
\(502\) 0 0
\(503\) −30.5448 17.6351i −1.36193 0.786309i −0.372046 0.928214i \(-0.621344\pi\)
−0.989880 + 0.141906i \(0.954677\pi\)
\(504\) 0 0
\(505\) 21.1814 0.942559
\(506\) 0 0
\(507\) 20.9582 2.10203i 0.930785 0.0933545i
\(508\) 0 0
\(509\) 19.9499 34.5542i 0.884262 1.53159i 0.0377043 0.999289i \(-0.487996\pi\)
0.846557 0.532297i \(-0.178671\pi\)
\(510\) 0 0
\(511\) −4.57904 7.93114i −0.202565 0.350853i
\(512\) 0 0
\(513\) −20.3101 + 10.0250i −0.896712 + 0.442614i
\(514\) 0 0
\(515\) 7.95714 + 13.7822i 0.350634 + 0.607315i
\(516\) 0 0
\(517\) 15.4203 26.7088i 0.678186 1.17465i
\(518\) 0 0
\(519\) 44.8802 4.50133i 1.97002 0.197586i
\(520\) 0 0
\(521\) −16.2560 −0.712188 −0.356094 0.934450i \(-0.615892\pi\)
−0.356094 + 0.934450i \(0.615892\pi\)
\(522\) 0 0
\(523\) 5.37315 + 3.10219i 0.234952 + 0.135649i 0.612854 0.790196i \(-0.290021\pi\)
−0.377903 + 0.925845i \(0.623355\pi\)
\(524\) 0 0
\(525\) 3.79079 + 5.27012i 0.165444 + 0.230007i
\(526\) 0 0
\(527\) −13.4487 23.2938i −0.585833 1.01469i
\(528\) 0 0
\(529\) −9.73147 16.8554i −0.423107 0.732843i
\(530\) 0 0
\(531\) −9.77967 + 29.1519i −0.424401 + 1.26509i
\(532\) 0 0
\(533\) 0.860802i 0.0372855i
\(534\) 0 0
\(535\) −6.94951 + 4.01230i −0.300454 + 0.173467i
\(536\) 0 0
\(537\) −16.2174 + 35.9635i −0.699833 + 1.55194i
\(538\) 0 0
\(539\) 20.1654i 0.868584i
\(540\) 0 0
\(541\) −17.9064 + 31.0147i −0.769855 + 1.33343i 0.167786 + 0.985823i \(0.446338\pi\)
−0.937641 + 0.347605i \(0.886995\pi\)
\(542\) 0 0
\(543\) −9.11187 + 6.55415i −0.391028 + 0.281266i
\(544\) 0 0
\(545\) −1.07669 + 1.86489i −0.0461205 + 0.0798831i
\(546\) 0 0
\(547\) −23.7443 13.7088i −1.01523 0.586144i −0.102512 0.994732i \(-0.532688\pi\)
−0.912719 + 0.408587i \(0.866021\pi\)
\(548\) 0 0
\(549\) 4.50902 0.913670i 0.192440 0.0389945i
\(550\) 0 0
\(551\) −2.48020 + 6.49110i −0.105660 + 0.276530i
\(552\) 0 0
\(553\) −3.09123 + 1.78472i −0.131453 + 0.0758942i
\(554\) 0 0
\(555\) −9.31407 + 0.934169i −0.395360 + 0.0396533i
\(556\) 0 0
\(557\) 29.5249 + 17.0462i 1.25101 + 0.722271i 0.971310 0.237816i \(-0.0764315\pi\)
0.279700 + 0.960087i \(0.409765\pi\)
\(558\) 0 0
\(559\) 5.01185i 0.211979i
\(560\) 0 0
\(561\) 17.6113 1.76635i 0.743549 0.0745754i
\(562\) 0 0
\(563\) −21.5649 −0.908854 −0.454427 0.890784i \(-0.650156\pi\)
−0.454427 + 0.890784i \(0.650156\pi\)
\(564\) 0 0
\(565\) −0.00816634 + 0.00471484i −0.000343561 + 0.000198355i
\(566\) 0 0
\(567\) 1.16126 9.33600i 0.0487684 0.392075i
\(568\) 0 0
\(569\) −13.8187 −0.579311 −0.289656 0.957131i \(-0.593541\pi\)
−0.289656 + 0.957131i \(0.593541\pi\)
\(570\) 0 0
\(571\) 27.0772 1.13314 0.566572 0.824012i \(-0.308269\pi\)
0.566572 + 0.824012i \(0.308269\pi\)
\(572\) 0 0
\(573\) −26.5877 11.9895i −1.11072 0.500868i
\(574\) 0 0
\(575\) 5.83995 3.37170i 0.243543 0.140609i
\(576\) 0 0
\(577\) −38.4381 −1.60020 −0.800099 0.599868i \(-0.795220\pi\)
−0.800099 + 0.599868i \(0.795220\pi\)
\(578\) 0 0
\(579\) 2.55570 + 25.4814i 0.106211 + 1.05897i
\(580\) 0 0
\(581\) 7.36771i 0.305664i
\(582\) 0 0
\(583\) 0.247615 + 0.142961i 0.0102552 + 0.00592082i
\(584\) 0 0
\(585\) −2.16369 2.44950i −0.0894577 0.101274i
\(586\) 0 0
\(587\) −6.18916 + 3.57331i −0.255454 + 0.147486i −0.622259 0.782811i \(-0.713785\pi\)
0.366805 + 0.930298i \(0.380452\pi\)
\(588\) 0 0
\(589\) −6.18654 38.6735i −0.254912 1.59351i
\(590\) 0 0
\(591\) −14.1653 6.38771i −0.582682 0.262755i
\(592\) 0 0
\(593\) −29.1089 16.8060i −1.19536 0.690141i −0.235843 0.971791i \(-0.575785\pi\)
−0.959517 + 0.281650i \(0.909118\pi\)
\(594\) 0 0
\(595\) 1.86080 3.22301i 0.0762856 0.132130i
\(596\) 0 0
\(597\) 3.51090 + 4.88100i 0.143691 + 0.199766i
\(598\) 0 0
\(599\) 6.75537 11.7006i 0.276017 0.478076i −0.694374 0.719614i \(-0.744319\pi\)
0.970391 + 0.241539i \(0.0776520\pi\)
\(600\) 0 0
\(601\) 12.3030i 0.501850i 0.968007 + 0.250925i \(0.0807347\pi\)
−0.968007 + 0.250925i \(0.919265\pi\)
\(602\) 0 0
\(603\) 5.97096 + 29.4671i 0.243156 + 1.19999i
\(604\) 0 0
\(605\) −0.672510 + 0.388274i −0.0273414 + 0.0157856i
\(606\) 0 0
\(607\) 11.7542i 0.477087i −0.971132 0.238543i \(-0.923330\pi\)
0.971132 0.238543i \(-0.0766699\pi\)
\(608\) 0 0
\(609\) −1.68541 2.34313i −0.0682963 0.0949485i
\(610\) 0 0
\(611\) −4.13790 7.16706i −0.167402 0.289948i
\(612\) 0 0
\(613\) −8.26759 14.3199i −0.333925 0.578375i 0.649353 0.760487i \(-0.275040\pi\)
−0.983278 + 0.182112i \(0.941707\pi\)
\(614\) 0 0
\(615\) 1.57146 1.13035i 0.0633674 0.0455801i
\(616\) 0 0
\(617\) 26.7140 + 15.4233i 1.07546 + 0.620920i 0.929669 0.368395i \(-0.120093\pi\)
0.145795 + 0.989315i \(0.453426\pi\)
\(618\) 0 0
\(619\) −15.7056 −0.631260 −0.315630 0.948882i \(-0.602216\pi\)
−0.315630 + 0.948882i \(0.602216\pi\)
\(620\) 0 0
\(621\) −9.52893 2.16802i −0.382383 0.0869998i
\(622\) 0 0
\(623\) 2.60149 4.50592i 0.104227 0.180526i
\(624\) 0 0
\(625\) −2.89201 5.00911i −0.115680 0.200364i
\(626\) 0 0
\(627\) 24.8727 + 6.75033i 0.993320 + 0.269582i
\(628\) 0 0
\(629\) −6.80167 11.7808i −0.271200 0.469733i
\(630\) 0 0
\(631\) −4.14864 + 7.18566i −0.165155 + 0.286056i −0.936710 0.350106i \(-0.886146\pi\)
0.771555 + 0.636162i \(0.219479\pi\)
\(632\) 0 0
\(633\) 0.626902 + 6.25049i 0.0249171 + 0.248434i
\(634\) 0 0
\(635\) 18.4449 0.731963
\(636\) 0 0
\(637\) −4.68623 2.70560i −0.185675 0.107200i
\(638\) 0 0
\(639\) −3.45213 + 10.2904i −0.136564 + 0.407080i
\(640\) 0 0
\(641\) −10.8249 18.7493i −0.427559 0.740553i 0.569097 0.822270i \(-0.307293\pi\)
−0.996656 + 0.0817172i \(0.973960\pi\)
\(642\) 0 0
\(643\) 15.0677 + 26.0980i 0.594211 + 1.02920i 0.993658 + 0.112447i \(0.0358689\pi\)
−0.399447 + 0.916756i \(0.630798\pi\)
\(644\) 0 0
\(645\) −9.14951 + 6.58123i −0.360262 + 0.259136i
\(646\) 0 0
\(647\) 37.9584i 1.49230i 0.665778 + 0.746150i \(0.268100\pi\)
−0.665778 + 0.746150i \(0.731900\pi\)
\(648\) 0 0
\(649\) 30.3007 17.4941i 1.18941 0.686705i
\(650\) 0 0
\(651\) 14.8300 + 6.68745i 0.581233 + 0.262102i
\(652\) 0 0
\(653\) 3.08483i 0.120719i 0.998177 + 0.0603593i \(0.0192246\pi\)
−0.998177 + 0.0603593i \(0.980775\pi\)
\(654\) 0 0
\(655\) 10.3649 17.9526i 0.404991 0.701465i
\(656\) 0 0
\(657\) 8.35934 24.9181i 0.326129 0.972150i
\(658\) 0 0
\(659\) −19.4544 + 33.6961i −0.757837 + 1.31261i 0.186114 + 0.982528i \(0.440411\pi\)
−0.943951 + 0.330084i \(0.892923\pi\)
\(660\) 0 0
\(661\) 24.5116 + 14.1518i 0.953392 + 0.550441i 0.894133 0.447801i \(-0.147793\pi\)
0.0592592 + 0.998243i \(0.481126\pi\)
\(662\) 0 0
\(663\) 1.95243 4.32968i 0.0758261 0.168151i
\(664\) 0 0
\(665\) 4.20610 3.41681i 0.163106 0.132498i
\(666\) 0 0
\(667\) −2.59648 + 1.49908i −0.100536 + 0.0580445i
\(668\) 0 0
\(669\) −2.82470 28.1635i −0.109209 1.08886i
\(670\) 0 0
\(671\) −4.53364 2.61750i −0.175019 0.101047i
\(672\) 0 0
\(673\) 9.62600i 0.371055i 0.982639 + 0.185528i \(0.0593994\pi\)
−0.982639 + 0.185528i \(0.940601\pi\)
\(674\) 0 0
\(675\) −4.13332 + 18.1668i −0.159092 + 0.699242i
\(676\) 0 0
\(677\) 34.0999 1.31056 0.655282 0.755384i \(-0.272550\pi\)
0.655282 + 0.755384i \(0.272550\pi\)
\(678\) 0 0
\(679\) 7.82909 4.52013i 0.300453 0.173467i
\(680\) 0 0
\(681\) −0.0109526 + 0.0242882i −0.000419703 + 0.000930727i
\(682\) 0 0
\(683\) 3.12003 0.119385 0.0596924 0.998217i \(-0.480988\pi\)
0.0596924 + 0.998217i \(0.480988\pi\)
\(684\) 0 0
\(685\) 17.3701 0.663676
\(686\) 0 0
\(687\) −19.4649 + 43.1651i −0.742633 + 1.64685i
\(688\) 0 0
\(689\) 0.0664452 0.0383621i 0.00253136 0.00146148i
\(690\) 0 0
\(691\) −28.4071 −1.08066 −0.540328 0.841454i \(-0.681700\pi\)
−0.540328 + 0.841454i \(0.681700\pi\)
\(692\) 0 0
\(693\) −8.02325 + 7.08710i −0.304778 + 0.269217i
\(694\) 0 0
\(695\) 20.0412i 0.760205i
\(696\) 0 0
\(697\) 2.43621 + 1.40655i 0.0922782 + 0.0532768i
\(698\) 0 0
\(699\) −3.72695 37.1593i −0.140966 1.40550i
\(700\) 0 0
\(701\) −13.6634 + 7.88854i −0.516058 + 0.297946i −0.735320 0.677720i \(-0.762968\pi\)
0.219262 + 0.975666i \(0.429635\pi\)
\(702\) 0 0
\(703\) −3.12885 19.5591i −0.118007 0.737687i
\(704\) 0 0
\(705\) 7.65039 16.9654i 0.288130 0.638953i
\(706\) 0 0
\(707\) −16.1230 9.30859i −0.606366 0.350086i
\(708\) 0 0
\(709\) 25.4121 44.0150i 0.954371 1.65302i 0.218569 0.975822i \(-0.429861\pi\)
0.735802 0.677197i \(-0.236805\pi\)
\(710\) 0 0
\(711\) −9.71207 3.25813i −0.364231 0.122189i
\(712\) 0 0
\(713\) 8.44919 14.6344i 0.316425 0.548063i
\(714\) 0 0
\(715\) 3.71890i 0.139079i
\(716\) 0 0
\(717\) −19.0793 8.60365i −0.712530 0.321309i
\(718\) 0 0
\(719\) 35.3153 20.3893i 1.31704 0.760393i 0.333788 0.942648i \(-0.391673\pi\)
0.983251 + 0.182255i \(0.0583396\pi\)
\(720\) 0 0
\(721\) 13.9877i 0.520930i
\(722\) 0 0
\(723\) 28.6966 20.6414i 1.06724 0.767664i
\(724\) 0 0
\(725\) 2.85798 + 4.95017i 0.106143 + 0.183845i
\(726\) 0 0
\(727\) −11.9402 20.6811i −0.442839 0.767020i 0.555060 0.831810i \(-0.312695\pi\)
−0.997899 + 0.0647907i \(0.979362\pi\)
\(728\) 0 0
\(729\) 22.2875 15.2403i 0.825463 0.564456i
\(730\) 0 0
\(731\) −14.1844 8.18935i −0.524628 0.302894i
\(732\) 0 0
\(733\) 3.85130 0.142251 0.0711255 0.997467i \(-0.477341\pi\)
0.0711255 + 0.997467i \(0.477341\pi\)
\(734\) 0 0
\(735\) −1.21438 12.1079i −0.0447930 0.446606i
\(736\) 0 0
\(737\) 17.1057 29.6280i 0.630098 1.09136i
\(738\) 0 0
\(739\) 3.89096 + 6.73933i 0.143131 + 0.247910i 0.928674 0.370897i \(-0.120950\pi\)
−0.785543 + 0.618807i \(0.787616\pi\)
\(740\) 0 0
\(741\) 4.90589 4.87447i 0.180222 0.179068i
\(742\) 0 0
\(743\) 16.6890 + 28.9062i 0.612260 + 1.06046i 0.990859 + 0.134904i \(0.0430726\pi\)
−0.378599 + 0.925561i \(0.623594\pi\)
\(744\) 0 0
\(745\) 1.63131 2.82551i 0.0597666 0.103519i
\(746\) 0 0
\(747\) −15.8475 + 13.9984i −0.579829 + 0.512175i
\(748\) 0 0
\(749\) 7.05315 0.257717
\(750\) 0 0
\(751\) −33.3286 19.2423i −1.21618 0.702161i −0.252081 0.967706i \(-0.581115\pi\)
−0.964099 + 0.265545i \(0.914448\pi\)
\(752\) 0 0
\(753\) −38.9316 + 28.0035i −1.41875 + 1.02050i
\(754\) 0 0
\(755\) 1.36212 + 2.35926i 0.0495726 + 0.0858622i
\(756\) 0 0
\(757\) 5.74237 + 9.94608i 0.208710 + 0.361497i 0.951308 0.308240i \(-0.0997401\pi\)
−0.742598 + 0.669737i \(0.766407\pi\)
\(758\) 0 0
\(759\) 6.49323 + 9.02717i 0.235689 + 0.327665i
\(760\) 0 0
\(761\) 1.17421i 0.0425651i −0.999774 0.0212825i \(-0.993225\pi\)
0.999774 0.0212825i \(-0.00677496\pi\)
\(762\) 0 0
\(763\) 1.63913 0.946351i 0.0593404 0.0342602i
\(764\) 0 0
\(765\) 10.4680 2.12114i 0.378470 0.0766899i
\(766\) 0 0
\(767\) 9.38877i 0.339009i
\(768\) 0 0
\(769\) 12.8688 22.2895i 0.464062 0.803779i −0.535097 0.844791i \(-0.679725\pi\)
0.999159 + 0.0410120i \(0.0130582\pi\)
\(770\) 0 0
\(771\) 29.7482 + 41.3572i 1.07136 + 1.48944i
\(772\) 0 0
\(773\) −0.919746 + 1.59305i −0.0330810 + 0.0572979i −0.882092 0.471077i \(-0.843865\pi\)
0.849011 + 0.528375i \(0.177199\pi\)
\(774\) 0 0
\(775\) −27.9004 16.1083i −1.00221 0.578628i
\(776\) 0 0
\(777\) 7.50027 + 3.38218i 0.269071 + 0.121335i
\(778\) 0 0
\(779\) 2.58270 + 3.17932i 0.0925350 + 0.113911i
\(780\) 0 0
\(781\) 10.6959 6.17525i 0.382728 0.220968i
\(782\) 0 0
\(783\) 1.83770 8.07709i 0.0656741 0.288652i
\(784\) 0 0
\(785\) 13.6149 + 7.86059i 0.485938 + 0.280557i
\(786\) 0 0
\(787\) 6.20260i 0.221099i 0.993871 + 0.110549i \(0.0352610\pi\)
−0.993871 + 0.110549i \(0.964739\pi\)
\(788\) 0 0
\(789\) −1.75338 17.4819i −0.0624219 0.622373i
\(790\) 0 0
\(791\) 0.00828813 0.000294692
\(792\) 0 0
\(793\) −1.21656 + 0.702382i −0.0432014 + 0.0249423i
\(794\) 0 0
\(795\) 0.157284 + 0.0709261i 0.00557831 + 0.00251549i
\(796\) 0 0
\(797\) −7.63497 −0.270444 −0.135222 0.990815i \(-0.543175\pi\)
−0.135222 + 0.990815i \(0.543175\pi\)
\(798\) 0 0
\(799\) 27.0453 0.956795
\(800\) 0 0
\(801\) 14.6347 2.96545i 0.517092 0.104779i
\(802\) 0 0
\(803\) −25.9001 + 14.9534i −0.913994 + 0.527695i
\(804\) 0 0
\(805\) 2.33812 0.0824078
\(806\) 0 0
\(807\) 35.2231 3.53276i 1.23991 0.124359i
\(808\) 0 0
\(809\) 34.3075i 1.20619i 0.797670 + 0.603094i \(0.206066\pi\)
−0.797670 + 0.603094i \(0.793934\pi\)
\(810\) 0 0
\(811\) 33.5409 + 19.3649i 1.17778 + 0.679993i 0.955500 0.294990i \(-0.0953163\pi\)
0.222281 + 0.974983i \(0.428650\pi\)
\(812\) 0 0
\(813\) −22.0302 + 2.20955i −0.772632 + 0.0774923i
\(814\) 0 0
\(815\) 1.27280 0.734854i 0.0445844 0.0257408i
\(816\) 0 0
\(817\) −15.0373 18.5109i −0.526088 0.647616i
\(818\) 0 0
\(819\) 0.570489 + 2.81540i 0.0199345 + 0.0983780i
\(820\) 0 0
\(821\) 10.9969 + 6.34907i 0.383795 + 0.221584i 0.679468 0.733705i \(-0.262211\pi\)
−0.295673 + 0.955289i \(0.595544\pi\)
\(822\) 0 0
\(823\) 12.3998 21.4770i 0.432228 0.748641i −0.564837 0.825203i \(-0.691061\pi\)
0.997065 + 0.0765614i \(0.0243941\pi\)
\(824\) 0 0
\(825\) 17.2102 12.3793i 0.599183 0.430992i
\(826\) 0 0
\(827\) 10.2728 17.7929i 0.357219 0.618721i −0.630276 0.776371i \(-0.717058\pi\)
0.987495 + 0.157650i \(0.0503917\pi\)
\(828\) 0 0
\(829\) 9.70009i 0.336898i 0.985710 + 0.168449i \(0.0538758\pi\)
−0.985710 + 0.168449i \(0.946124\pi\)
\(830\) 0 0
\(831\) −6.63799 + 14.7203i −0.230269 + 0.510641i
\(832\) 0 0
\(833\) 15.3146 8.84188i 0.530619 0.306353i
\(834\) 0 0
\(835\) 10.4675i 0.362242i
\(836\) 0 0
\(837\) 13.7922 + 44.6043i 0.476728 + 1.54175i
\(838\) 0 0
\(839\) 18.3395 + 31.7649i 0.633149 + 1.09665i 0.986904 + 0.161309i \(0.0515715\pi\)
−0.353755 + 0.935338i \(0.615095\pi\)
\(840\) 0 0
\(841\) 13.2293 + 22.9139i 0.456184 + 0.790133i
\(842\) 0 0
\(843\) 4.63593 + 6.44506i 0.159670 + 0.221980i
\(844\) 0 0
\(845\) −12.5253 7.23150i −0.430885 0.248771i
\(846\) 0 0
\(847\) 0.682540 0.0234523
\(848\) 0 0
\(849\) −34.3128 + 3.44146i −1.17761 + 0.118111i
\(850\) 0 0
\(851\) 4.27318 7.40137i 0.146483 0.253716i
\(852\) 0 0
\(853\) 11.7924 + 20.4251i 0.403765 + 0.699341i 0.994177 0.107761i \(-0.0343682\pi\)
−0.590412 + 0.807102i \(0.701035\pi\)
\(854\) 0 0
\(855\) 15.3408 + 2.55524i 0.524644 + 0.0873873i
\(856\) 0 0
\(857\) 13.5699 + 23.5038i 0.463539 + 0.802874i 0.999134 0.0416012i \(-0.0132459\pi\)
−0.535595 + 0.844475i \(0.679913\pi\)
\(858\) 0 0
\(859\) 10.4649 18.1257i 0.357058 0.618442i −0.630410 0.776262i \(-0.717113\pi\)
0.987468 + 0.157820i \(0.0504466\pi\)
\(860\) 0 0
\(861\) −1.69293 + 0.169795i −0.0576948 + 0.00578659i
\(862\) 0 0
\(863\) 9.52529 0.324245 0.162122 0.986771i \(-0.448166\pi\)
0.162122 + 0.986771i \(0.448166\pi\)
\(864\) 0 0
\(865\) −26.8220 15.4857i −0.911974 0.526528i
\(866\) 0 0
\(867\) −8.13028 11.3031i −0.276119 0.383872i
\(868\) 0 0
\(869\) 5.82823 + 10.0948i 0.197709 + 0.342442i
\(870\) 0 0
\(871\) −4.59017 7.95040i −0.155532 0.269389i
\(872\) 0 0
\(873\) 24.5975 + 8.25179i 0.832501 + 0.279281i
\(874\) 0 0
\(875\) 10.6737i 0.360835i
\(876\) 0 0
\(877\) −9.96965 + 5.75598i −0.336651 + 0.194366i −0.658790 0.752327i \(-0.728931\pi\)
0.322139 + 0.946692i \(0.395598\pi\)
\(878\) 0 0
\(879\) 18.5772 41.1966i 0.626595 1.38953i
\(880\) 0 0
\(881\) 42.0306i 1.41605i 0.706189 + 0.708024i \(0.250413\pi\)
−0.706189 + 0.708024i \(0.749587\pi\)
\(882\) 0 0
\(883\) 10.6448 18.4374i 0.358227 0.620468i −0.629438 0.777051i \(-0.716715\pi\)
0.987665 + 0.156583i \(0.0500480\pi\)
\(884\) 0 0
\(885\) 17.1399 12.3287i 0.576152 0.414426i
\(886\) 0 0
\(887\) −23.6484 + 40.9603i −0.794036 + 1.37531i 0.129413 + 0.991591i \(0.458691\pi\)
−0.923449 + 0.383720i \(0.874643\pi\)
\(888\) 0 0
\(889\) −14.0400 8.10599i −0.470886 0.271866i
\(890\) 0 0
\(891\) −30.4878 3.79224i −1.02138 0.127045i
\(892\) 0 0
\(893\) 36.7868 + 14.0559i 1.23102 + 0.470364i
\(894\) 0 0
\(895\) 23.4596 13.5444i 0.784166 0.452739i
\(896\) 0 0
\(897\) 2.96902 0.297783i 0.0991327 0.00994267i
\(898\) 0 0
\(899\) 12.4047 + 7.16186i 0.413720 + 0.238861i
\(900\) 0 0
\(901\) 0.250735i 0.00835318i
\(902\) 0 0
\(903\) 9.85673 0.988596i 0.328011 0.0328984i
\(904\) 0 0
\(905\) 7.70705 0.256191
\(906\) 0 0
\(907\) 9.37924 5.41511i 0.311433 0.179806i −0.336135 0.941814i \(-0.609120\pi\)
0.647567 + 0.762008i \(0.275786\pi\)
\(908\) 0 0
\(909\) −10.6109 52.3655i −0.351941 1.73685i
\(910\) 0 0
\(911\) 16.1178 0.534008 0.267004 0.963695i \(-0.413966\pi\)
0.267004 + 0.963695i \(0.413966\pi\)
\(912\) 0 0
\(913\) 24.0601 0.796274
\(914\) 0 0
\(915\) −2.87976 1.29860i −0.0952020 0.0429305i
\(916\) 0 0
\(917\) −15.7793 + 9.11016i −0.521077 + 0.300844i
\(918\) 0 0
\(919\) −26.9311 −0.888376 −0.444188 0.895934i \(-0.646508\pi\)
−0.444188 + 0.895934i \(0.646508\pi\)
\(920\) 0 0
\(921\) −0.787044 7.84717i −0.0259340 0.258573i
\(922\) 0 0
\(923\) 3.31414i 0.109086i
\(924\) 0 0
\(925\) −14.1107 8.14679i −0.463955 0.267865i
\(926\) 0 0
\(927\) 30.0867 26.5762i 0.988177 0.872877i
\(928\) 0 0
\(929\) −29.8822 + 17.2525i −0.980405 + 0.566037i −0.902392 0.430915i \(-0.858191\pi\)
−0.0780125 + 0.996952i \(0.524857\pi\)
\(930\) 0 0
\(931\) 25.4260 4.06736i 0.833304 0.133302i
\(932\) 0 0
\(933\) 28.8687 + 13.0181i 0.945120 + 0.426194i
\(934\) 0 0
\(935\) −10.5251 6.07668i −0.344208 0.198729i
\(936\) 0 0
\(937\) 16.1803 28.0252i 0.528588 0.915542i −0.470856 0.882210i \(-0.656055\pi\)
0.999444 0.0333319i \(-0.0106118\pi\)
\(938\) 0 0
\(939\) 1.61159 + 2.24051i 0.0525924 + 0.0731162i
\(940\) 0 0
\(941\) −17.0330 + 29.5020i −0.555260 + 0.961738i 0.442624 + 0.896708i \(0.354048\pi\)
−0.997883 + 0.0650306i \(0.979286\pi\)
\(942\) 0 0
\(943\) 1.76734i 0.0575525i
\(944\) 0 0
\(945\) −4.39060 + 4.73847i −0.142826 + 0.154142i
\(946\) 0 0
\(947\) −24.8522 + 14.3484i −0.807589 + 0.466262i −0.846118 0.532996i \(-0.821066\pi\)
0.0385292 + 0.999257i \(0.487733\pi\)
\(948\) 0 0
\(949\) 8.02522i 0.260510i
\(950\) 0 0
\(951\) 7.43151 + 10.3316i 0.240983 + 0.335025i
\(952\) 0 0
\(953\) 22.8722 + 39.6159i 0.740905 + 1.28328i 0.952084 + 0.305837i \(0.0989364\pi\)
−0.211179 + 0.977447i \(0.567730\pi\)
\(954\) 0 0
\(955\) 10.0133 + 17.3436i 0.324023 + 0.561225i
\(956\) 0 0
\(957\) −7.65178 + 5.50391i −0.247347 + 0.177916i
\(958\) 0 0
\(959\) −13.2218 7.63363i −0.426955 0.246503i
\(960\) 0 0
\(961\) −49.7322 −1.60426
\(962\) 0 0
\(963\) 13.4008 + 15.1709i 0.431834 + 0.488875i
\(964\) 0 0
\(965\) 8.79222 15.2286i 0.283031 0.490225i
\(966\) 0 0
\(967\) 9.59096 + 16.6120i 0.308424 + 0.534207i 0.978018 0.208521i \(-0.0668650\pi\)
−0.669593 + 0.742728i \(0.733532\pi\)
\(968\) 0 0
\(969\) 5.77935 + 21.8494i 0.185660 + 0.701903i
\(970\) 0 0
\(971\) 3.55749 + 6.16176i 0.114165 + 0.197740i 0.917446 0.397861i \(-0.130247\pi\)
−0.803280 + 0.595601i \(0.796914\pi\)
\(972\) 0 0
\(973\) −8.80750 + 15.2550i −0.282355 + 0.489054i
\(974\) 0 0
\(975\) −0.567720 5.66042i −0.0181816 0.181278i
\(976\) 0 0
\(977\) 16.4357 0.525823 0.262912 0.964820i \(-0.415317\pi\)
0.262912 + 0.964820i \(0.415317\pi\)
\(978\) 0 0
\(979\) −14.7146 8.49549i −0.470281 0.271517i
\(980\) 0 0
\(981\) 5.14983 + 1.72763i 0.164422 + 0.0551588i
\(982\) 0 0
\(983\) −30.2278 52.3560i −0.964116 1.66990i −0.711972 0.702207i \(-0.752198\pi\)
−0.252143 0.967690i \(-0.581135\pi\)
\(984\) 0 0
\(985\) 5.33486 + 9.24025i 0.169983 + 0.294419i
\(986\) 0 0
\(987\) −13.2791 + 9.55167i −0.422680 + 0.304033i
\(988\) 0 0
\(989\) 10.2900i 0.327203i
\(990\) 0 0
\(991\) 46.8229 27.0332i 1.48738 0.858738i 0.487482 0.873133i \(-0.337916\pi\)
0.999896 + 0.0143949i \(0.00458218\pi\)
\(992\) 0 0
\(993\) 11.4891 + 5.18090i 0.364595 + 0.164411i
\(994\) 0 0
\(995\) 4.12847i 0.130881i
\(996\) 0 0
\(997\) −12.2536 + 21.2238i −0.388074 + 0.672164i −0.992190 0.124732i \(-0.960193\pi\)
0.604116 + 0.796896i \(0.293526\pi\)
\(998\) 0 0
\(999\) 6.97541 + 22.5586i 0.220692 + 0.713724i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 912.2.bn.n.449.4 16
3.2 odd 2 912.2.bn.o.449.1 16
4.3 odd 2 456.2.bf.d.449.5 yes 16
12.11 even 2 456.2.bf.c.449.8 yes 16
19.8 odd 6 912.2.bn.o.65.1 16
57.8 even 6 inner 912.2.bn.n.65.4 16
76.27 even 6 456.2.bf.c.65.8 16
228.179 odd 6 456.2.bf.d.65.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.bf.c.65.8 16 76.27 even 6
456.2.bf.c.449.8 yes 16 12.11 even 2
456.2.bf.d.65.5 yes 16 228.179 odd 6
456.2.bf.d.449.5 yes 16 4.3 odd 2
912.2.bn.n.65.4 16 57.8 even 6 inner
912.2.bn.n.449.4 16 1.1 even 1 trivial
912.2.bn.o.65.1 16 19.8 odd 6
912.2.bn.o.449.1 16 3.2 odd 2