Properties

Label 768.2.n.b.97.1
Level $768$
Weight $2$
Character 768.97
Analytic conductor $6.133$
Analytic rank $0$
Dimension $32$
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] = [768,2,Mod(97,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.n (of order \(8\), degree \(4\), 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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 97.1
Character \(\chi\) \(=\) 768.97
Dual form 768.2.n.b.673.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.923880 + 0.382683i) q^{3} +(-1.36206 + 3.28830i) q^{5} +(2.73097 + 2.73097i) q^{7} +(0.707107 - 0.707107i) q^{9} +O(q^{10})\) \(q+(-0.923880 + 0.382683i) q^{3} +(-1.36206 + 3.28830i) q^{5} +(2.73097 + 2.73097i) q^{7} +(0.707107 - 0.707107i) q^{9} +(3.01609 + 1.24931i) q^{11} +(-0.932498 - 2.25125i) q^{13} -3.55923i q^{15} -0.517450i q^{17} +(1.52739 + 3.68744i) q^{19} +(-3.56818 - 1.47799i) q^{21} +(-2.39792 + 2.39792i) q^{23} +(-5.42220 - 5.42220i) q^{25} +(-0.382683 + 0.923880i) q^{27} +(-7.09056 + 2.93701i) q^{29} +1.50132 q^{31} -3.26460 q^{33} +(-12.7000 + 5.26051i) q^{35} +(3.40814 - 8.22797i) q^{37} +(1.72303 + 1.72303i) q^{39} +(-3.21656 + 3.21656i) q^{41} +(-1.31346 - 0.544054i) q^{43} +(1.36206 + 3.28830i) q^{45} +4.67448i q^{47} +7.91635i q^{49} +(0.198019 + 0.478061i) q^{51} +(4.19534 + 1.73777i) q^{53} +(-8.21621 + 8.21621i) q^{55} +(-2.82224 - 2.82224i) q^{57} +(-0.680868 + 1.64376i) q^{59} +(-6.71487 + 2.78139i) q^{61} +3.86217 q^{63} +8.67291 q^{65} +(11.1312 - 4.61070i) q^{67} +(1.29774 - 3.13303i) q^{69} +(-1.86620 - 1.86620i) q^{71} +(-9.06859 + 9.06859i) q^{73} +(7.08445 + 2.93447i) q^{75} +(4.82504 + 11.6487i) q^{77} -10.4412i q^{79} -1.00000i q^{81} +(-1.89340 - 4.57106i) q^{83} +(1.70153 + 0.704798i) q^{85} +(5.42688 - 5.42688i) q^{87} +(-2.70762 - 2.70762i) q^{89} +(3.60147 - 8.69471i) q^{91} +(-1.38704 + 0.574529i) q^{93} -14.2058 q^{95} +3.73293 q^{97} +(3.01609 - 1.24931i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{23} + 48 q^{31} - 48 q^{35} - 16 q^{43} + 16 q^{51} + 32 q^{53} - 32 q^{55} + 64 q^{59} + 32 q^{61} - 16 q^{63} + 16 q^{67} + 32 q^{69} - 64 q^{71} + 32 q^{75} + 32 q^{77} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{8}\right)\)

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
\(3\) −0.923880 + 0.382683i −0.533402 + 0.220942i
\(4\) 0 0
\(5\) −1.36206 + 3.28830i −0.609132 + 1.47057i 0.254814 + 0.966990i \(0.417986\pi\)
−0.863946 + 0.503584i \(0.832014\pi\)
\(6\) 0 0
\(7\) 2.73097 + 2.73097i 1.03221 + 1.03221i 0.999464 + 0.0327442i \(0.0104247\pi\)
0.0327442 + 0.999464i \(0.489575\pi\)
\(8\) 0 0
\(9\) 0.707107 0.707107i 0.235702 0.235702i
\(10\) 0 0
\(11\) 3.01609 + 1.24931i 0.909387 + 0.376680i 0.787822 0.615903i \(-0.211209\pi\)
0.121565 + 0.992583i \(0.461209\pi\)
\(12\) 0 0
\(13\) −0.932498 2.25125i −0.258628 0.624384i 0.740220 0.672365i \(-0.234721\pi\)
−0.998848 + 0.0479806i \(0.984721\pi\)
\(14\) 0 0
\(15\) 3.55923i 0.918990i
\(16\) 0 0
\(17\) 0.517450i 0.125500i −0.998029 0.0627500i \(-0.980013\pi\)
0.998029 0.0627500i \(-0.0199871\pi\)
\(18\) 0 0
\(19\) 1.52739 + 3.68744i 0.350407 + 0.845956i 0.996570 + 0.0827570i \(0.0263725\pi\)
−0.646163 + 0.763199i \(0.723627\pi\)
\(20\) 0 0
\(21\) −3.56818 1.47799i −0.778640 0.322523i
\(22\) 0 0
\(23\) −2.39792 + 2.39792i −0.500000 + 0.500000i −0.911438 0.411438i \(-0.865027\pi\)
0.411438 + 0.911438i \(0.365027\pi\)
\(24\) 0 0
\(25\) −5.42220 5.42220i −1.08444 1.08444i
\(26\) 0 0
\(27\) −0.382683 + 0.923880i −0.0736475 + 0.177801i
\(28\) 0 0
\(29\) −7.09056 + 2.93701i −1.31668 + 0.545389i −0.926828 0.375487i \(-0.877476\pi\)
−0.389857 + 0.920875i \(0.627476\pi\)
\(30\) 0 0
\(31\) 1.50132 0.269644 0.134822 0.990870i \(-0.456954\pi\)
0.134822 + 0.990870i \(0.456954\pi\)
\(32\) 0 0
\(33\) −3.26460 −0.568293
\(34\) 0 0
\(35\) −12.7000 + 5.26051i −2.14669 + 0.889188i
\(36\) 0 0
\(37\) 3.40814 8.22797i 0.560294 1.35267i −0.349237 0.937034i \(-0.613559\pi\)
0.909531 0.415635i \(-0.136441\pi\)
\(38\) 0 0
\(39\) 1.72303 + 1.72303i 0.275906 + 0.275906i
\(40\) 0 0
\(41\) −3.21656 + 3.21656i −0.502342 + 0.502342i −0.912165 0.409823i \(-0.865590\pi\)
0.409823 + 0.912165i \(0.365590\pi\)
\(42\) 0 0
\(43\) −1.31346 0.544054i −0.200301 0.0829675i 0.280278 0.959919i \(-0.409573\pi\)
−0.480579 + 0.876951i \(0.659573\pi\)
\(44\) 0 0
\(45\) 1.36206 + 3.28830i 0.203044 + 0.490191i
\(46\) 0 0
\(47\) 4.67448i 0.681843i 0.940092 + 0.340921i \(0.110739\pi\)
−0.940092 + 0.340921i \(0.889261\pi\)
\(48\) 0 0
\(49\) 7.91635i 1.13091i
\(50\) 0 0
\(51\) 0.198019 + 0.478061i 0.0277283 + 0.0669419i
\(52\) 0 0
\(53\) 4.19534 + 1.73777i 0.576275 + 0.238701i 0.651733 0.758448i \(-0.274042\pi\)
−0.0754586 + 0.997149i \(0.524042\pi\)
\(54\) 0 0
\(55\) −8.21621 + 8.21621i −1.10787 + 1.10787i
\(56\) 0 0
\(57\) −2.82224 2.82224i −0.373815 0.373815i
\(58\) 0 0
\(59\) −0.680868 + 1.64376i −0.0886415 + 0.213999i −0.961983 0.273109i \(-0.911948\pi\)
0.873342 + 0.487108i \(0.161948\pi\)
\(60\) 0 0
\(61\) −6.71487 + 2.78139i −0.859751 + 0.356121i −0.768610 0.639717i \(-0.779051\pi\)
−0.0911409 + 0.995838i \(0.529051\pi\)
\(62\) 0 0
\(63\) 3.86217 0.486587
\(64\) 0 0
\(65\) 8.67291 1.07574
\(66\) 0 0
\(67\) 11.1312 4.61070i 1.35989 0.563286i 0.420864 0.907124i \(-0.361727\pi\)
0.939029 + 0.343837i \(0.111727\pi\)
\(68\) 0 0
\(69\) 1.29774 3.13303i 0.156230 0.377172i
\(70\) 0 0
\(71\) −1.86620 1.86620i −0.221477 0.221477i 0.587643 0.809120i \(-0.300056\pi\)
−0.809120 + 0.587643i \(0.800056\pi\)
\(72\) 0 0
\(73\) −9.06859 + 9.06859i −1.06140 + 1.06140i −0.0634101 + 0.997988i \(0.520198\pi\)
−0.997988 + 0.0634101i \(0.979802\pi\)
\(74\) 0 0
\(75\) 7.08445 + 2.93447i 0.818042 + 0.338844i
\(76\) 0 0
\(77\) 4.82504 + 11.6487i 0.549864 + 1.32749i
\(78\) 0 0
\(79\) 10.4412i 1.17473i −0.809323 0.587364i \(-0.800166\pi\)
0.809323 0.587364i \(-0.199834\pi\)
\(80\) 0 0
\(81\) 1.00000i 0.111111i
\(82\) 0 0
\(83\) −1.89340 4.57106i −0.207827 0.501739i 0.785253 0.619175i \(-0.212533\pi\)
−0.993081 + 0.117435i \(0.962533\pi\)
\(84\) 0 0
\(85\) 1.70153 + 0.704798i 0.184557 + 0.0764460i
\(86\) 0 0
\(87\) 5.42688 5.42688i 0.581823 0.581823i
\(88\) 0 0
\(89\) −2.70762 2.70762i −0.287007 0.287007i 0.548888 0.835896i \(-0.315051\pi\)
−0.835896 + 0.548888i \(0.815051\pi\)
\(90\) 0 0
\(91\) 3.60147 8.69471i 0.377536 0.911453i
\(92\) 0 0
\(93\) −1.38704 + 0.574529i −0.143829 + 0.0595759i
\(94\) 0 0
\(95\) −14.2058 −1.45749
\(96\) 0 0
\(97\) 3.73293 0.379022 0.189511 0.981879i \(-0.439310\pi\)
0.189511 + 0.981879i \(0.439310\pi\)
\(98\) 0 0
\(99\) 3.01609 1.24931i 0.303129 0.125560i
\(100\) 0 0
\(101\) 6.36949 15.3773i 0.633788 1.53010i −0.201036 0.979584i \(-0.564431\pi\)
0.834824 0.550517i \(-0.185569\pi\)
\(102\) 0 0
\(103\) 7.42244 + 7.42244i 0.731355 + 0.731355i 0.970888 0.239533i \(-0.0769944\pi\)
−0.239533 + 0.970888i \(0.576994\pi\)
\(104\) 0 0
\(105\) 9.72015 9.72015i 0.948589 0.948589i
\(106\) 0 0
\(107\) 7.85343 + 3.25300i 0.759220 + 0.314479i 0.728497 0.685049i \(-0.240219\pi\)
0.0307226 + 0.999528i \(0.490219\pi\)
\(108\) 0 0
\(109\) −3.33028 8.04001i −0.318983 0.770094i −0.999308 0.0371832i \(-0.988161\pi\)
0.680325 0.732910i \(-0.261839\pi\)
\(110\) 0 0
\(111\) 8.90589i 0.845310i
\(112\) 0 0
\(113\) 17.4463i 1.64121i 0.571494 + 0.820606i \(0.306364\pi\)
−0.571494 + 0.820606i \(0.693636\pi\)
\(114\) 0 0
\(115\) −4.61897 11.1512i −0.430721 1.03985i
\(116\) 0 0
\(117\) −2.25125 0.932498i −0.208128 0.0862095i
\(118\) 0 0
\(119\) 1.41314 1.41314i 0.129542 0.129542i
\(120\) 0 0
\(121\) −0.242116 0.242116i −0.0220105 0.0220105i
\(122\) 0 0
\(123\) 1.74079 4.20263i 0.156962 0.378939i
\(124\) 0 0
\(125\) 8.77369 3.63418i 0.784743 0.325051i
\(126\) 0 0
\(127\) −1.84791 −0.163975 −0.0819876 0.996633i \(-0.526127\pi\)
−0.0819876 + 0.996633i \(0.526127\pi\)
\(128\) 0 0
\(129\) 1.42168 0.125172
\(130\) 0 0
\(131\) −7.59353 + 3.14534i −0.663450 + 0.274810i −0.688889 0.724866i \(-0.741901\pi\)
0.0254395 + 0.999676i \(0.491901\pi\)
\(132\) 0 0
\(133\) −5.89902 + 14.2415i −0.511510 + 1.23490i
\(134\) 0 0
\(135\) −2.51676 2.51676i −0.216608 0.216608i
\(136\) 0 0
\(137\) 7.98582 7.98582i 0.682274 0.682274i −0.278238 0.960512i \(-0.589750\pi\)
0.960512 + 0.278238i \(0.0897503\pi\)
\(138\) 0 0
\(139\) 1.52261 + 0.630686i 0.129146 + 0.0534941i 0.446321 0.894873i \(-0.352734\pi\)
−0.317174 + 0.948367i \(0.602734\pi\)
\(140\) 0 0
\(141\) −1.78885 4.31865i −0.150648 0.363696i
\(142\) 0 0
\(143\) 7.95496i 0.665227i
\(144\) 0 0
\(145\) 27.3163i 2.26850i
\(146\) 0 0
\(147\) −3.02945 7.31375i −0.249865 0.603228i
\(148\) 0 0
\(149\) 3.85004 + 1.59474i 0.315408 + 0.130646i 0.534771 0.844997i \(-0.320398\pi\)
−0.219363 + 0.975643i \(0.570398\pi\)
\(150\) 0 0
\(151\) 0.409447 0.409447i 0.0333203 0.0333203i −0.690250 0.723571i \(-0.742500\pi\)
0.723571 + 0.690250i \(0.242500\pi\)
\(152\) 0 0
\(153\) −0.365892 0.365892i −0.0295806 0.0295806i
\(154\) 0 0
\(155\) −2.04488 + 4.93678i −0.164249 + 0.396532i
\(156\) 0 0
\(157\) 12.1456 5.03087i 0.969324 0.401507i 0.158864 0.987301i \(-0.449217\pi\)
0.810461 + 0.585793i \(0.199217\pi\)
\(158\) 0 0
\(159\) −4.54101 −0.360125
\(160\) 0 0
\(161\) −13.0973 −1.03221
\(162\) 0 0
\(163\) −15.6056 + 6.46405i −1.22233 + 0.506304i −0.898148 0.439693i \(-0.855087\pi\)
−0.324177 + 0.945996i \(0.605087\pi\)
\(164\) 0 0
\(165\) 4.44658 10.7350i 0.346166 0.835718i
\(166\) 0 0
\(167\) 3.65825 + 3.65825i 0.283084 + 0.283084i 0.834338 0.551254i \(-0.185850\pi\)
−0.551254 + 0.834338i \(0.685850\pi\)
\(168\) 0 0
\(169\) 4.99382 4.99382i 0.384140 0.384140i
\(170\) 0 0
\(171\) 3.68744 + 1.52739i 0.281985 + 0.116802i
\(172\) 0 0
\(173\) 4.69766 + 11.3412i 0.357156 + 0.862252i 0.995697 + 0.0926640i \(0.0295382\pi\)
−0.638541 + 0.769588i \(0.720462\pi\)
\(174\) 0 0
\(175\) 29.6157i 2.23874i
\(176\) 0 0
\(177\) 1.77919i 0.133732i
\(178\) 0 0
\(179\) 2.26551 + 5.46943i 0.169332 + 0.408804i 0.985651 0.168798i \(-0.0539885\pi\)
−0.816318 + 0.577602i \(0.803989\pi\)
\(180\) 0 0
\(181\) 14.5765 + 6.03777i 1.08346 + 0.448784i 0.851722 0.523994i \(-0.175559\pi\)
0.231738 + 0.972778i \(0.425559\pi\)
\(182\) 0 0
\(183\) 5.13934 5.13934i 0.379911 0.379911i
\(184\) 0 0
\(185\) 22.4140 + 22.4140i 1.64791 + 1.64791i
\(186\) 0 0
\(187\) 0.646454 1.56068i 0.0472734 0.114128i
\(188\) 0 0
\(189\) −3.56818 + 1.47799i −0.259547 + 0.107508i
\(190\) 0 0
\(191\) 12.3765 0.895535 0.447768 0.894150i \(-0.352219\pi\)
0.447768 + 0.894150i \(0.352219\pi\)
\(192\) 0 0
\(193\) 21.9343 1.57887 0.789434 0.613835i \(-0.210374\pi\)
0.789434 + 0.613835i \(0.210374\pi\)
\(194\) 0 0
\(195\) −8.01273 + 3.31898i −0.573803 + 0.237677i
\(196\) 0 0
\(197\) −6.52756 + 15.7589i −0.465070 + 1.12278i 0.501220 + 0.865320i \(0.332885\pi\)
−0.966290 + 0.257458i \(0.917115\pi\)
\(198\) 0 0
\(199\) −5.80270 5.80270i −0.411343 0.411343i 0.470863 0.882206i \(-0.343942\pi\)
−0.882206 + 0.470863i \(0.843942\pi\)
\(200\) 0 0
\(201\) −8.51946 + 8.51946i −0.600916 + 0.600916i
\(202\) 0 0
\(203\) −27.3850 11.3432i −1.92205 0.796138i
\(204\) 0 0
\(205\) −6.19587 14.9582i −0.432738 1.04472i
\(206\) 0 0
\(207\) 3.39117i 0.235702i
\(208\) 0 0
\(209\) 13.0298i 0.901293i
\(210\) 0 0
\(211\) 7.65444 + 18.4794i 0.526953 + 1.27218i 0.933510 + 0.358552i \(0.116729\pi\)
−0.406557 + 0.913626i \(0.633271\pi\)
\(212\) 0 0
\(213\) 2.43831 + 1.00998i 0.167070 + 0.0692026i
\(214\) 0 0
\(215\) 3.57803 3.57803i 0.244020 0.244020i
\(216\) 0 0
\(217\) 4.10004 + 4.10004i 0.278329 + 0.278329i
\(218\) 0 0
\(219\) 4.90788 11.8487i 0.331644 0.800659i
\(220\) 0 0
\(221\) −1.16491 + 0.482521i −0.0783602 + 0.0324579i
\(222\) 0 0
\(223\) 29.0773 1.94716 0.973580 0.228346i \(-0.0733317\pi\)
0.973580 + 0.228346i \(0.0733317\pi\)
\(224\) 0 0
\(225\) −7.66815 −0.511210
\(226\) 0 0
\(227\) 23.2583 9.63391i 1.54371 0.639425i 0.561544 0.827447i \(-0.310208\pi\)
0.982165 + 0.188022i \(0.0602077\pi\)
\(228\) 0 0
\(229\) −2.92749 + 7.06760i −0.193454 + 0.467040i −0.990607 0.136737i \(-0.956338\pi\)
0.797153 + 0.603777i \(0.206338\pi\)
\(230\) 0 0
\(231\) −8.91550 8.91550i −0.586597 0.586597i
\(232\) 0 0
\(233\) 13.0873 13.0873i 0.857375 0.857375i −0.133653 0.991028i \(-0.542671\pi\)
0.991028 + 0.133653i \(0.0426708\pi\)
\(234\) 0 0
\(235\) −15.3711 6.36692i −1.00270 0.415332i
\(236\) 0 0
\(237\) 3.99568 + 9.64642i 0.259547 + 0.626602i
\(238\) 0 0
\(239\) 4.10909i 0.265795i 0.991130 + 0.132897i \(0.0424281\pi\)
−0.991130 + 0.132897i \(0.957572\pi\)
\(240\) 0 0
\(241\) 14.2144i 0.915632i −0.889047 0.457816i \(-0.848632\pi\)
0.889047 0.457816i \(-0.151368\pi\)
\(242\) 0 0
\(243\) 0.382683 + 0.923880i 0.0245492 + 0.0592669i
\(244\) 0 0
\(245\) −26.0314 10.7825i −1.66308 0.688871i
\(246\) 0 0
\(247\) 6.87706 6.87706i 0.437577 0.437577i
\(248\) 0 0
\(249\) 3.49854 + 3.49854i 0.221711 + 0.221711i
\(250\) 0 0
\(251\) 2.36893 5.71910i 0.149525 0.360986i −0.831314 0.555803i \(-0.812411\pi\)
0.980840 + 0.194816i \(0.0624110\pi\)
\(252\) 0 0
\(253\) −10.2281 + 4.23661i −0.643034 + 0.266353i
\(254\) 0 0
\(255\) −1.84172 −0.115333
\(256\) 0 0
\(257\) −8.28191 −0.516612 −0.258306 0.966063i \(-0.583164\pi\)
−0.258306 + 0.966063i \(0.583164\pi\)
\(258\) 0 0
\(259\) 31.7778 13.1628i 1.97458 0.817896i
\(260\) 0 0
\(261\) −2.93701 + 7.09056i −0.181796 + 0.438895i
\(262\) 0 0
\(263\) −15.4068 15.4068i −0.950023 0.950023i 0.0487861 0.998809i \(-0.484465\pi\)
−0.998809 + 0.0487861i \(0.984465\pi\)
\(264\) 0 0
\(265\) −11.4286 + 11.4286i −0.702055 + 0.702055i
\(266\) 0 0
\(267\) 3.53768 + 1.46535i 0.216502 + 0.0896782i
\(268\) 0 0
\(269\) 3.09893 + 7.48148i 0.188945 + 0.456154i 0.989757 0.142762i \(-0.0455983\pi\)
−0.800812 + 0.598916i \(0.795598\pi\)
\(270\) 0 0
\(271\) 28.7832i 1.74846i 0.485515 + 0.874228i \(0.338632\pi\)
−0.485515 + 0.874228i \(0.661368\pi\)
\(272\) 0 0
\(273\) 9.41108i 0.569585i
\(274\) 0 0
\(275\) −9.57988 23.1279i −0.577688 1.39466i
\(276\) 0 0
\(277\) 12.1515 + 5.03333i 0.730115 + 0.302424i 0.716599 0.697485i \(-0.245698\pi\)
0.0135158 + 0.999909i \(0.495698\pi\)
\(278\) 0 0
\(279\) 1.06159 1.06159i 0.0635558 0.0635558i
\(280\) 0 0
\(281\) −21.6306 21.6306i −1.29037 1.29037i −0.934555 0.355820i \(-0.884202\pi\)
−0.355820 0.934555i \(-0.615798\pi\)
\(282\) 0 0
\(283\) −11.4777 + 27.7097i −0.682281 + 1.64717i 0.0775002 + 0.996992i \(0.475306\pi\)
−0.759781 + 0.650179i \(0.774694\pi\)
\(284\) 0 0
\(285\) 13.1245 5.43633i 0.777426 0.322020i
\(286\) 0 0
\(287\) −17.5686 −1.03704
\(288\) 0 0
\(289\) 16.7322 0.984250
\(290\) 0 0
\(291\) −3.44878 + 1.42853i −0.202171 + 0.0837420i
\(292\) 0 0
\(293\) 1.84257 4.44836i 0.107644 0.259876i −0.860874 0.508818i \(-0.830083\pi\)
0.968519 + 0.248942i \(0.0800827\pi\)
\(294\) 0 0
\(295\) −4.47780 4.47780i −0.260708 0.260708i
\(296\) 0 0
\(297\) −2.30842 + 2.30842i −0.133948 + 0.133948i
\(298\) 0 0
\(299\) 7.63436 + 3.16226i 0.441507 + 0.182878i
\(300\) 0 0
\(301\) −2.10123 5.07282i −0.121113 0.292392i
\(302\) 0 0
\(303\) 16.6443i 0.956190i
\(304\) 0 0
\(305\) 25.8690i 1.48125i
\(306\) 0 0
\(307\) 11.7403 + 28.3435i 0.670052 + 1.61765i 0.781519 + 0.623881i \(0.214445\pi\)
−0.111467 + 0.993768i \(0.535555\pi\)
\(308\) 0 0
\(309\) −9.69789 4.01700i −0.551694 0.228519i
\(310\) 0 0
\(311\) −11.6121 + 11.6121i −0.658459 + 0.658459i −0.955015 0.296556i \(-0.904162\pi\)
0.296556 + 0.955015i \(0.404162\pi\)
\(312\) 0 0
\(313\) −6.84409 6.84409i −0.386851 0.386851i 0.486712 0.873563i \(-0.338196\pi\)
−0.873563 + 0.486712i \(0.838196\pi\)
\(314\) 0 0
\(315\) −5.26051 + 12.7000i −0.296396 + 0.715563i
\(316\) 0 0
\(317\) 23.6795 9.80835i 1.32997 0.550892i 0.399324 0.916810i \(-0.369245\pi\)
0.930647 + 0.365918i \(0.119245\pi\)
\(318\) 0 0
\(319\) −25.0550 −1.40281
\(320\) 0 0
\(321\) −8.50049 −0.474451
\(322\) 0 0
\(323\) 1.90806 0.790346i 0.106167 0.0439760i
\(324\) 0 0
\(325\) −7.15054 + 17.2629i −0.396640 + 0.957575i
\(326\) 0 0
\(327\) 6.15356 + 6.15356i 0.340293 + 0.340293i
\(328\) 0 0
\(329\) −12.7658 + 12.7658i −0.703803 + 0.703803i
\(330\) 0 0
\(331\) −21.7107 8.99285i −1.19333 0.494292i −0.304489 0.952516i \(-0.598486\pi\)
−0.888837 + 0.458224i \(0.848486\pi\)
\(332\) 0 0
\(333\) −3.40814 8.22797i −0.186765 0.450890i
\(334\) 0 0
\(335\) 42.8828i 2.34294i
\(336\) 0 0
\(337\) 24.2394i 1.32040i 0.751088 + 0.660202i \(0.229529\pi\)
−0.751088 + 0.660202i \(0.770471\pi\)
\(338\) 0 0
\(339\) −6.67642 16.1183i −0.362613 0.875426i
\(340\) 0 0
\(341\) 4.52811 + 1.87561i 0.245211 + 0.101570i
\(342\) 0 0
\(343\) −2.50251 + 2.50251i −0.135123 + 0.135123i
\(344\) 0 0
\(345\) 8.53475 + 8.53475i 0.459495 + 0.459495i
\(346\) 0 0
\(347\) −1.42236 + 3.43389i −0.0763565 + 0.184341i −0.957449 0.288604i \(-0.906809\pi\)
0.881092 + 0.472945i \(0.156809\pi\)
\(348\) 0 0
\(349\) −3.17452 + 1.31493i −0.169928 + 0.0703866i −0.466026 0.884771i \(-0.654315\pi\)
0.296098 + 0.955158i \(0.404315\pi\)
\(350\) 0 0
\(351\) 2.43674 0.130063
\(352\) 0 0
\(353\) −7.88845 −0.419860 −0.209930 0.977716i \(-0.567324\pi\)
−0.209930 + 0.977716i \(0.567324\pi\)
\(354\) 0 0
\(355\) 8.67850 3.59475i 0.460607 0.190790i
\(356\) 0 0
\(357\) −0.764784 + 1.84635i −0.0404767 + 0.0977193i
\(358\) 0 0
\(359\) 2.60336 + 2.60336i 0.137400 + 0.137400i 0.772462 0.635062i \(-0.219025\pi\)
−0.635062 + 0.772462i \(0.719025\pi\)
\(360\) 0 0
\(361\) 2.17074 2.17074i 0.114250 0.114250i
\(362\) 0 0
\(363\) 0.316339 + 0.131032i 0.0166035 + 0.00687740i
\(364\) 0 0
\(365\) −17.4683 42.1722i −0.914333 2.20740i
\(366\) 0 0
\(367\) 1.41682i 0.0739575i 0.999316 + 0.0369787i \(0.0117734\pi\)
−0.999316 + 0.0369787i \(0.988227\pi\)
\(368\) 0 0
\(369\) 4.54890i 0.236806i
\(370\) 0 0
\(371\) 6.71155 + 16.2031i 0.348447 + 0.841224i
\(372\) 0 0
\(373\) −30.8483 12.7778i −1.59726 0.661609i −0.606239 0.795282i \(-0.707323\pi\)
−0.991025 + 0.133674i \(0.957323\pi\)
\(374\) 0 0
\(375\) −6.71510 + 6.71510i −0.346766 + 0.346766i
\(376\) 0 0
\(377\) 13.2239 + 13.2239i 0.681064 + 0.681064i
\(378\) 0 0
\(379\) −4.79982 + 11.5878i −0.246550 + 0.595224i −0.997907 0.0646720i \(-0.979400\pi\)
0.751357 + 0.659896i \(0.229400\pi\)
\(380\) 0 0
\(381\) 1.70724 0.707163i 0.0874647 0.0362291i
\(382\) 0 0
\(383\) 13.2502 0.677053 0.338527 0.940957i \(-0.390071\pi\)
0.338527 + 0.940957i \(0.390071\pi\)
\(384\) 0 0
\(385\) −44.8763 −2.28711
\(386\) 0 0
\(387\) −1.31346 + 0.544054i −0.0667671 + 0.0276558i
\(388\) 0 0
\(389\) 1.30717 3.15579i 0.0662762 0.160005i −0.887271 0.461248i \(-0.847402\pi\)
0.953547 + 0.301243i \(0.0974017\pi\)
\(390\) 0 0
\(391\) 1.24080 + 1.24080i 0.0627500 + 0.0627500i
\(392\) 0 0
\(393\) 5.81184 5.81184i 0.293168 0.293168i
\(394\) 0 0
\(395\) 34.3339 + 14.2215i 1.72752 + 0.715564i
\(396\) 0 0
\(397\) 2.75561 + 6.65262i 0.138300 + 0.333886i 0.977821 0.209441i \(-0.0671645\pi\)
−0.839521 + 0.543327i \(0.817164\pi\)
\(398\) 0 0
\(399\) 15.4149i 0.771710i
\(400\) 0 0
\(401\) 22.5969i 1.12844i −0.825626 0.564218i \(-0.809178\pi\)
0.825626 0.564218i \(-0.190822\pi\)
\(402\) 0 0
\(403\) −1.39997 3.37984i −0.0697377 0.168362i
\(404\) 0 0
\(405\) 3.28830 + 1.36206i 0.163397 + 0.0676813i
\(406\) 0 0
\(407\) 20.5585 20.5585i 1.01905 1.01905i
\(408\) 0 0
\(409\) −3.48618 3.48618i −0.172380 0.172380i 0.615644 0.788024i \(-0.288896\pi\)
−0.788024 + 0.615644i \(0.788896\pi\)
\(410\) 0 0
\(411\) −4.32189 + 10.4340i −0.213183 + 0.514670i
\(412\) 0 0
\(413\) −6.34848 + 2.62963i −0.312388 + 0.129395i
\(414\) 0 0
\(415\) 17.6100 0.864439
\(416\) 0 0
\(417\) −1.64806 −0.0807059
\(418\) 0 0
\(419\) 9.10369 3.77087i 0.444744 0.184219i −0.149061 0.988828i \(-0.547625\pi\)
0.593805 + 0.804609i \(0.297625\pi\)
\(420\) 0 0
\(421\) 0.766817 1.85126i 0.0373724 0.0902249i −0.904091 0.427339i \(-0.859451\pi\)
0.941464 + 0.337114i \(0.109451\pi\)
\(422\) 0 0
\(423\) 3.30535 + 3.30535i 0.160712 + 0.160712i
\(424\) 0 0
\(425\) −2.80572 + 2.80572i −0.136097 + 0.136097i
\(426\) 0 0
\(427\) −25.9340 10.7422i −1.25503 0.519851i
\(428\) 0 0
\(429\) 3.04423 + 7.34942i 0.146977 + 0.354834i
\(430\) 0 0
\(431\) 11.5510i 0.556394i −0.960524 0.278197i \(-0.910263\pi\)
0.960524 0.278197i \(-0.0897368\pi\)
\(432\) 0 0
\(433\) 3.12797i 0.150321i −0.997171 0.0751603i \(-0.976053\pi\)
0.997171 0.0751603i \(-0.0239468\pi\)
\(434\) 0 0
\(435\) 10.4535 + 25.2370i 0.501207 + 1.21002i
\(436\) 0 0
\(437\) −12.5047 5.17962i −0.598182 0.247775i
\(438\) 0 0
\(439\) −8.10867 + 8.10867i −0.387006 + 0.387006i −0.873618 0.486612i \(-0.838232\pi\)
0.486612 + 0.873618i \(0.338232\pi\)
\(440\) 0 0
\(441\) 5.59770 + 5.59770i 0.266557 + 0.266557i
\(442\) 0 0
\(443\) 9.34973 22.5722i 0.444219 1.07244i −0.530235 0.847851i \(-0.677896\pi\)
0.974454 0.224588i \(-0.0721037\pi\)
\(444\) 0 0
\(445\) 12.5914 5.21554i 0.596891 0.247240i
\(446\) 0 0
\(447\) −4.16726 −0.197105
\(448\) 0 0
\(449\) −28.6259 −1.35094 −0.675469 0.737388i \(-0.736059\pi\)
−0.675469 + 0.737388i \(0.736059\pi\)
\(450\) 0 0
\(451\) −13.7199 + 5.68297i −0.646045 + 0.267601i
\(452\) 0 0
\(453\) −0.221591 + 0.534968i −0.0104113 + 0.0251350i
\(454\) 0 0
\(455\) 23.6854 + 23.6854i 1.11039 + 1.11039i
\(456\) 0 0
\(457\) −18.7182 + 18.7182i −0.875602 + 0.875602i −0.993076 0.117474i \(-0.962520\pi\)
0.117474 + 0.993076i \(0.462520\pi\)
\(458\) 0 0
\(459\) 0.478061 + 0.198019i 0.0223140 + 0.00924275i
\(460\) 0 0
\(461\) −2.60820 6.29674i −0.121476 0.293268i 0.851431 0.524467i \(-0.175735\pi\)
−0.972907 + 0.231198i \(0.925735\pi\)
\(462\) 0 0
\(463\) 5.96406i 0.277174i 0.990350 + 0.138587i \(0.0442560\pi\)
−0.990350 + 0.138587i \(0.955744\pi\)
\(464\) 0 0
\(465\) 5.34354i 0.247801i
\(466\) 0 0
\(467\) 8.88493 + 21.4501i 0.411146 + 0.992593i 0.984831 + 0.173518i \(0.0555133\pi\)
−0.573685 + 0.819076i \(0.694487\pi\)
\(468\) 0 0
\(469\) 42.9906 + 17.8073i 1.98512 + 0.822264i
\(470\) 0 0
\(471\) −9.29584 + 9.29584i −0.428330 + 0.428330i
\(472\) 0 0
\(473\) −3.28184 3.28184i −0.150899 0.150899i
\(474\) 0 0
\(475\) 11.7122 28.2758i 0.537394 1.29738i
\(476\) 0 0
\(477\) 4.19534 1.73777i 0.192092 0.0795669i
\(478\) 0 0
\(479\) 22.6413 1.03451 0.517254 0.855832i \(-0.326954\pi\)
0.517254 + 0.855832i \(0.326954\pi\)
\(480\) 0 0
\(481\) −21.7013 −0.989494
\(482\) 0 0
\(483\) 12.1003 5.01210i 0.550582 0.228059i
\(484\) 0 0
\(485\) −5.08448 + 12.2750i −0.230874 + 0.557380i
\(486\) 0 0
\(487\) −2.79015 2.79015i −0.126434 0.126434i 0.641058 0.767492i \(-0.278496\pi\)
−0.767492 + 0.641058i \(0.778496\pi\)
\(488\) 0 0
\(489\) 11.9440 11.9440i 0.540127 0.540127i
\(490\) 0 0
\(491\) 19.9243 + 8.25292i 0.899172 + 0.372449i 0.783902 0.620885i \(-0.213227\pi\)
0.115270 + 0.993334i \(0.463227\pi\)
\(492\) 0 0
\(493\) 1.51975 + 3.66901i 0.0684463 + 0.165244i
\(494\) 0 0
\(495\) 11.6195i 0.522256i
\(496\) 0 0
\(497\) 10.1930i 0.457221i
\(498\) 0 0
\(499\) −14.7704 35.6588i −0.661212 1.59631i −0.795907 0.605419i \(-0.793005\pi\)
0.134695 0.990887i \(-0.456995\pi\)
\(500\) 0 0
\(501\) −4.77973 1.97983i −0.213543 0.0884523i
\(502\) 0 0
\(503\) 26.5194 26.5194i 1.18244 1.18244i 0.203333 0.979110i \(-0.434823\pi\)
0.979110 0.203333i \(-0.0651773\pi\)
\(504\) 0 0
\(505\) 41.8897 + 41.8897i 1.86407 + 1.86407i
\(506\) 0 0
\(507\) −2.70263 + 6.52473i −0.120028 + 0.289774i
\(508\) 0 0
\(509\) −33.0361 + 13.6840i −1.46430 + 0.606534i −0.965551 0.260212i \(-0.916207\pi\)
−0.498750 + 0.866746i \(0.666207\pi\)
\(510\) 0 0
\(511\) −49.5320 −2.19117
\(512\) 0 0
\(513\) −3.99125 −0.176218
\(514\) 0 0
\(515\) −34.5171 + 14.2974i −1.52100 + 0.630020i
\(516\) 0 0
\(517\) −5.83986 + 14.0987i −0.256837 + 0.620059i
\(518\) 0 0
\(519\) −8.68014 8.68014i −0.381016 0.381016i
\(520\) 0 0
\(521\) −2.51581 + 2.51581i −0.110220 + 0.110220i −0.760066 0.649846i \(-0.774833\pi\)
0.649846 + 0.760066i \(0.274833\pi\)
\(522\) 0 0
\(523\) 21.6449 + 8.96561i 0.946465 + 0.392039i 0.801901 0.597457i \(-0.203822\pi\)
0.144564 + 0.989495i \(0.453822\pi\)
\(524\) 0 0
\(525\) 11.3334 + 27.3613i 0.494632 + 1.19415i
\(526\) 0 0
\(527\) 0.776856i 0.0338404i
\(528\) 0 0
\(529\) 11.5000i 0.500000i
\(530\) 0 0
\(531\) 0.680868 + 1.64376i 0.0295472 + 0.0713331i
\(532\) 0 0
\(533\) 10.2407 + 4.24184i 0.443574 + 0.183734i
\(534\) 0 0
\(535\) −21.3937 + 21.3937i −0.924930 + 0.924930i
\(536\) 0 0
\(537\) −4.18612 4.18612i −0.180644 0.180644i
\(538\) 0 0
\(539\) −9.88995 + 23.8764i −0.425990 + 1.02843i
\(540\) 0 0
\(541\) −9.88526 + 4.09461i −0.425001 + 0.176041i −0.584924 0.811088i \(-0.698875\pi\)
0.159923 + 0.987130i \(0.448875\pi\)
\(542\) 0 0
\(543\) −15.7775 −0.677075
\(544\) 0 0
\(545\) 30.9741 1.32678
\(546\) 0 0
\(547\) −4.92501 + 2.04001i −0.210578 + 0.0872244i −0.485479 0.874248i \(-0.661355\pi\)
0.274901 + 0.961473i \(0.411355\pi\)
\(548\) 0 0
\(549\) −2.78139 + 6.71487i −0.118707 + 0.286584i
\(550\) 0 0
\(551\) −21.6601 21.6601i −0.922750 0.922750i
\(552\) 0 0
\(553\) 28.5146 28.5146i 1.21256 1.21256i
\(554\) 0 0
\(555\) −29.2853 12.1304i −1.24309 0.514905i
\(556\) 0 0
\(557\) 1.88066 + 4.54032i 0.0796862 + 0.192380i 0.958702 0.284414i \(-0.0917990\pi\)
−0.879015 + 0.476793i \(0.841799\pi\)
\(558\) 0 0
\(559\) 3.46426i 0.146523i
\(560\) 0 0
\(561\) 1.68926i 0.0713208i
\(562\) 0 0
\(563\) −4.16303 10.0504i −0.175451 0.423575i 0.811552 0.584281i \(-0.198623\pi\)
−0.987002 + 0.160705i \(0.948623\pi\)
\(564\) 0 0
\(565\) −57.3688 23.7630i −2.41353 0.999715i
\(566\) 0 0
\(567\) 2.73097 2.73097i 0.114690 0.114690i
\(568\) 0 0
\(569\) −1.48801 1.48801i −0.0623805 0.0623805i 0.675228 0.737609i \(-0.264045\pi\)
−0.737609 + 0.675228i \(0.764045\pi\)
\(570\) 0 0
\(571\) −10.9476 + 26.4299i −0.458145 + 1.10606i 0.511003 + 0.859579i \(0.329274\pi\)
−0.969148 + 0.246480i \(0.920726\pi\)
\(572\) 0 0
\(573\) −11.4344 + 4.73630i −0.477680 + 0.197862i
\(574\) 0 0
\(575\) 26.0040 1.08444
\(576\) 0 0
\(577\) 28.1723 1.17283 0.586413 0.810012i \(-0.300539\pi\)
0.586413 + 0.810012i \(0.300539\pi\)
\(578\) 0 0
\(579\) −20.2647 + 8.39391i −0.842172 + 0.348839i
\(580\) 0 0
\(581\) 7.31261 17.6542i 0.303378 0.732420i
\(582\) 0 0
\(583\) 10.4825 + 10.4825i 0.434143 + 0.434143i
\(584\) 0 0
\(585\) 6.13268 6.13268i 0.253555 0.253555i
\(586\) 0 0
\(587\) −26.4589 10.9597i −1.09208 0.452353i −0.237347 0.971425i \(-0.576278\pi\)
−0.854731 + 0.519072i \(0.826278\pi\)
\(588\) 0 0
\(589\) 2.29309 + 5.53601i 0.0944851 + 0.228107i
\(590\) 0 0
\(591\) 17.0573i 0.701645i
\(592\) 0 0
\(593\) 1.34616i 0.0552802i 0.999618 + 0.0276401i \(0.00879924\pi\)
−0.999618 + 0.0276401i \(0.991201\pi\)
\(594\) 0 0
\(595\) 2.72205 + 6.57160i 0.111593 + 0.269409i
\(596\) 0 0
\(597\) 7.58160 + 3.14040i 0.310294 + 0.128528i
\(598\) 0 0
\(599\) 8.34761 8.34761i 0.341074 0.341074i −0.515697 0.856771i \(-0.672467\pi\)
0.856771 + 0.515697i \(0.172467\pi\)
\(600\) 0 0
\(601\) −25.9145 25.9145i −1.05707 1.05707i −0.998270 0.0588044i \(-0.981271\pi\)
−0.0588044 0.998270i \(-0.518729\pi\)
\(602\) 0 0
\(603\) 4.61070 11.1312i 0.187762 0.453298i
\(604\) 0 0
\(605\) 1.12593 0.466374i 0.0457754 0.0189608i
\(606\) 0 0
\(607\) 2.09569 0.0850615 0.0425307 0.999095i \(-0.486458\pi\)
0.0425307 + 0.999095i \(0.486458\pi\)
\(608\) 0 0
\(609\) 29.6413 1.20112
\(610\) 0 0
\(611\) 10.5234 4.35894i 0.425732 0.176344i
\(612\) 0 0
\(613\) 8.75402 21.1341i 0.353572 0.853597i −0.642602 0.766200i \(-0.722145\pi\)
0.996174 0.0873972i \(-0.0278549\pi\)
\(614\) 0 0
\(615\) 11.4485 + 11.4485i 0.461647 + 0.461647i
\(616\) 0 0
\(617\) 7.54724 7.54724i 0.303841 0.303841i −0.538674 0.842514i \(-0.681074\pi\)
0.842514 + 0.538674i \(0.181074\pi\)
\(618\) 0 0
\(619\) 30.8044 + 12.7596i 1.23813 + 0.512851i 0.903130 0.429366i \(-0.141263\pi\)
0.335002 + 0.942217i \(0.391263\pi\)
\(620\) 0 0
\(621\) −1.29774 3.13303i −0.0520766 0.125724i
\(622\) 0 0
\(623\) 14.7888i 0.592502i
\(624\) 0 0
\(625\) 4.54021i 0.181608i
\(626\) 0 0
\(627\) −4.98630 12.0380i −0.199134 0.480751i
\(628\) 0 0
\(629\) −4.25756 1.76354i −0.169760 0.0703169i
\(630\) 0 0
\(631\) −4.03006 + 4.03006i −0.160434 + 0.160434i −0.782759 0.622325i \(-0.786188\pi\)
0.622325 + 0.782759i \(0.286188\pi\)
\(632\) 0 0
\(633\) −14.1436 14.1436i −0.562156 0.562156i
\(634\) 0 0
\(635\) 2.51696 6.07648i 0.0998825 0.241138i
\(636\) 0 0
\(637\) 17.8217 7.38198i 0.706120 0.292485i
\(638\) 0 0
\(639\) −2.63920 −0.104405
\(640\) 0 0
\(641\) 10.3489 0.408757 0.204378 0.978892i \(-0.434483\pi\)
0.204378 + 0.978892i \(0.434483\pi\)
\(642\) 0 0
\(643\) −8.10100 + 3.35554i −0.319472 + 0.132330i −0.536656 0.843801i \(-0.680313\pi\)
0.217184 + 0.976131i \(0.430313\pi\)
\(644\) 0 0
\(645\) −1.93642 + 4.67493i −0.0762464 + 0.184075i
\(646\) 0 0
\(647\) −20.0358 20.0358i −0.787689 0.787689i 0.193426 0.981115i \(-0.438040\pi\)
−0.981115 + 0.193426i \(0.938040\pi\)
\(648\) 0 0
\(649\) −4.10712 + 4.10712i −0.161219 + 0.161219i
\(650\) 0 0
\(651\) −5.35696 2.21893i −0.209956 0.0869666i
\(652\) 0 0
\(653\) −11.1811 26.9936i −0.437552 1.05634i −0.976792 0.214191i \(-0.931289\pi\)
0.539240 0.842152i \(-0.318711\pi\)
\(654\) 0 0
\(655\) 29.2540i 1.14305i
\(656\) 0 0
\(657\) 12.8249i 0.500348i
\(658\) 0 0
\(659\) −10.0306 24.2161i −0.390738 0.943325i −0.989779 0.142607i \(-0.954451\pi\)
0.599041 0.800718i \(-0.295549\pi\)
\(660\) 0 0
\(661\) 13.8010 + 5.71657i 0.536798 + 0.222349i 0.634578 0.772859i \(-0.281174\pi\)
−0.0977799 + 0.995208i \(0.531174\pi\)
\(662\) 0 0
\(663\) 0.891582 0.891582i 0.0346262 0.0346262i
\(664\) 0 0
\(665\) −38.7956 38.7956i −1.50443 1.50443i
\(666\) 0 0
\(667\) 9.95988 24.0453i 0.385648 0.931037i
\(668\) 0 0
\(669\) −26.8639 + 11.1274i −1.03862 + 0.430210i
\(670\) 0 0
\(671\) −23.7275 −0.915990
\(672\) 0 0
\(673\) 31.4646 1.21287 0.606436 0.795132i \(-0.292599\pi\)
0.606436 + 0.795132i \(0.292599\pi\)
\(674\) 0 0
\(675\) 7.08445 2.93447i 0.272681 0.112948i
\(676\) 0 0
\(677\) 6.25070 15.0905i 0.240234 0.579976i −0.757072 0.653332i \(-0.773371\pi\)
0.997306 + 0.0733553i \(0.0233707\pi\)
\(678\) 0 0
\(679\) 10.1945 + 10.1945i 0.391229 + 0.391229i
\(680\) 0 0
\(681\) −17.8011 + 17.8011i −0.682141 + 0.682141i
\(682\) 0 0
\(683\) −13.3397 5.52548i −0.510429 0.211426i 0.112578 0.993643i \(-0.464089\pi\)
−0.623007 + 0.782216i \(0.714089\pi\)
\(684\) 0 0
\(685\) 15.3826 + 37.1370i 0.587740 + 1.41893i
\(686\) 0 0
\(687\) 7.64991i 0.291862i
\(688\) 0 0
\(689\) 11.0652i 0.421552i
\(690\) 0 0
\(691\) −9.47463 22.8738i −0.360432 0.870160i −0.995237 0.0974874i \(-0.968919\pi\)
0.634805 0.772673i \(-0.281081\pi\)
\(692\) 0 0
\(693\) 11.6487 + 4.82504i 0.442496 + 0.183288i
\(694\) 0 0
\(695\) −4.14777 + 4.14777i −0.157334 + 0.157334i
\(696\) 0 0
\(697\) 1.66441 + 1.66441i 0.0630439 + 0.0630439i
\(698\) 0 0
\(699\) −7.08278 + 17.0993i −0.267895 + 0.646756i
\(700\) 0 0
\(701\) 5.36748 2.22328i 0.202727 0.0839723i −0.279010 0.960288i \(-0.590006\pi\)
0.481737 + 0.876316i \(0.340006\pi\)
\(702\) 0 0
\(703\) 35.5457 1.34063
\(704\) 0 0
\(705\) 16.6376 0.626607
\(706\) 0 0
\(707\) 59.3898 24.6001i 2.23358 0.925180i
\(708\) 0 0
\(709\) 14.4269 34.8295i 0.541812 1.30805i −0.381631 0.924315i \(-0.624638\pi\)
0.923443 0.383735i \(-0.125362\pi\)
\(710\) 0 0
\(711\) −7.38305 7.38305i −0.276886 0.276886i
\(712\) 0 0
\(713\) −3.60003 + 3.60003i −0.134822 + 0.134822i
\(714\) 0 0
\(715\) 26.1583 + 10.8351i 0.978266 + 0.405211i
\(716\) 0 0
\(717\) −1.57248 3.79630i −0.0587254 0.141776i
\(718\) 0 0
\(719\) 38.6849i 1.44270i −0.692569 0.721351i \(-0.743521\pi\)
0.692569 0.721351i \(-0.256479\pi\)
\(720\) 0 0
\(721\) 40.5409i 1.50982i
\(722\) 0 0
\(723\) 5.43963 + 13.1324i 0.202302 + 0.488400i
\(724\) 0 0
\(725\) 54.3715 + 22.5214i 2.01931 + 0.836425i
\(726\) 0 0
\(727\) 31.8871 31.8871i 1.18263 1.18263i 0.203566 0.979061i \(-0.434747\pi\)
0.979061 0.203566i \(-0.0652532\pi\)
\(728\) 0 0
\(729\) −0.707107 0.707107i −0.0261891 0.0261891i
\(730\) 0 0
\(731\) −0.281521 + 0.679651i −0.0104124 + 0.0251378i
\(732\) 0 0
\(733\) 18.4333 7.63532i 0.680849 0.282017i −0.0153326 0.999882i \(-0.504881\pi\)
0.696182 + 0.717865i \(0.254881\pi\)
\(734\) 0 0
\(735\) 28.1761 1.03929
\(736\) 0 0
\(737\) 39.3330 1.44885
\(738\) 0 0
\(739\) −19.0069 + 7.87292i −0.699180 + 0.289610i −0.703819 0.710379i \(-0.748523\pi\)
0.00463865 + 0.999989i \(0.498523\pi\)
\(740\) 0 0
\(741\) −3.72184 + 8.98531i −0.136725 + 0.330084i
\(742\) 0 0
\(743\) 17.5266 + 17.5266i 0.642987 + 0.642987i 0.951289 0.308302i \(-0.0997606\pi\)
−0.308302 + 0.951289i \(0.599761\pi\)
\(744\) 0 0
\(745\) −10.4880 + 10.4880i −0.384250 + 0.384250i
\(746\) 0 0
\(747\) −4.57106 1.89340i −0.167246 0.0692757i
\(748\) 0 0
\(749\) 12.5636 + 30.3313i 0.459065 + 1.10828i
\(750\) 0 0
\(751\) 40.4908i 1.47753i 0.673963 + 0.738765i \(0.264591\pi\)
−0.673963 + 0.738765i \(0.735409\pi\)
\(752\) 0 0
\(753\) 6.19031i 0.225587i
\(754\) 0 0
\(755\) 0.788695 + 1.90408i 0.0287035 + 0.0692965i
\(756\) 0 0
\(757\) 29.7102 + 12.3064i 1.07984 + 0.447283i 0.850452 0.526053i \(-0.176329\pi\)
0.229385 + 0.973336i \(0.426329\pi\)
\(758\) 0 0
\(759\) 7.82823 7.82823i 0.284147 0.284147i
\(760\) 0 0
\(761\) −15.8052 15.8052i −0.572939 0.572939i 0.360010 0.932949i \(-0.382773\pi\)
−0.932949 + 0.360010i \(0.882773\pi\)
\(762\) 0 0
\(763\) 12.8621 31.0519i 0.465640 1.12415i
\(764\) 0 0
\(765\) 1.70153 0.704798i 0.0615190 0.0254820i
\(766\) 0 0
\(767\) 4.33542 0.156543
\(768\) 0 0
\(769\) 2.99003 0.107823 0.0539116 0.998546i \(-0.482831\pi\)
0.0539116 + 0.998546i \(0.482831\pi\)
\(770\) 0 0
\(771\) 7.65149 3.16935i 0.275562 0.114141i
\(772\) 0 0
\(773\) −1.59236 + 3.84429i −0.0572731 + 0.138269i −0.949926 0.312476i \(-0.898842\pi\)
0.892653 + 0.450745i \(0.148842\pi\)
\(774\) 0 0
\(775\) −8.14044 8.14044i −0.292413 0.292413i
\(776\) 0 0
\(777\) −24.3217 + 24.3217i −0.872535 + 0.872535i
\(778\) 0 0
\(779\) −16.7738 6.94792i −0.600983 0.248935i
\(780\) 0 0
\(781\) −3.29718 7.96009i −0.117982 0.284834i
\(782\) 0 0
\(783\) 7.67477i 0.274274i
\(784\) 0 0
\(785\) 46.7908i 1.67003i
\(786\) 0 0
\(787\) 3.14740 + 7.59851i 0.112193 + 0.270857i 0.969996 0.243121i \(-0.0781712\pi\)
−0.857803 + 0.513978i \(0.828171\pi\)
\(788\) 0 0
\(789\) 20.1299 + 8.33809i 0.716645 + 0.296844i
\(790\) 0 0
\(791\) −47.6453 + 47.6453i −1.69407 + 1.69407i
\(792\) 0 0
\(793\) 12.5232 + 12.5232i 0.444712 + 0.444712i
\(794\) 0 0
\(795\) 6.18513 14.9322i 0.219364 0.529591i
\(796\) 0 0
\(797\) 13.9175 5.76480i 0.492982 0.204200i −0.122321 0.992491i \(-0.539034\pi\)
0.615303 + 0.788291i \(0.289034\pi\)
\(798\) 0 0
\(799\) 2.41881 0.0855712
\(800\) 0 0
\(801\) −3.82915 −0.135297
\(802\) 0 0
\(803\) −38.6812 + 16.0223i −1.36503 + 0.565413i
\(804\) 0 0
\(805\) 17.8393 43.0678i 0.628751 1.51794i
\(806\) 0 0
\(807\) −5.72608 5.72608i −0.201568 0.201568i
\(808\) 0 0
\(809\) 30.4595 30.4595i 1.07090 1.07090i 0.0736122 0.997287i \(-0.476547\pi\)
0.997287 0.0736122i \(-0.0234527\pi\)
\(810\) 0 0
\(811\) 19.1190 + 7.91934i 0.671358 + 0.278086i 0.692209 0.721697i \(-0.256638\pi\)
−0.0208509 + 0.999783i \(0.506638\pi\)
\(812\) 0 0
\(813\) −11.0149 26.5922i −0.386308 0.932631i
\(814\) 0 0
\(815\) 60.1204i 2.10593i
\(816\) 0 0
\(817\) 5.67430i 0.198519i
\(818\) 0 0
\(819\) −3.60147 8.69471i −0.125845 0.303818i
\(820\) 0 0
\(821\) −32.8139 13.5920i −1.14521 0.474363i −0.272287 0.962216i \(-0.587780\pi\)
−0.872926 + 0.487853i \(0.837780\pi\)
\(822\) 0 0
\(823\) −5.67489 + 5.67489i −0.197814 + 0.197814i −0.799062 0.601248i \(-0.794670\pi\)
0.601248 + 0.799062i \(0.294670\pi\)
\(824\) 0 0
\(825\) 17.7013 + 17.7013i 0.616280 + 0.616280i
\(826\) 0 0
\(827\) 21.2654 51.3392i 0.739471 1.78524i 0.131443 0.991324i \(-0.458039\pi\)
0.608027 0.793916i \(-0.291961\pi\)
\(828\) 0 0
\(829\) 14.6369 6.06280i 0.508361 0.210570i −0.113735 0.993511i \(-0.536281\pi\)
0.622096 + 0.782941i \(0.286281\pi\)
\(830\) 0 0
\(831\) −13.1527 −0.456263
\(832\) 0 0
\(833\) 4.09631 0.141929
\(834\) 0 0
\(835\) −17.0122 + 7.04668i −0.588731 + 0.243860i
\(836\) 0 0
\(837\) −0.574529 + 1.38704i −0.0198586 + 0.0479430i
\(838\) 0 0
\(839\) 39.7201 + 39.7201i 1.37129 + 1.37129i 0.858535 + 0.512756i \(0.171375\pi\)
0.512756 + 0.858535i \(0.328625\pi\)
\(840\) 0 0
\(841\) 21.1440 21.1440i 0.729103 0.729103i
\(842\) 0 0
\(843\) 28.2618 + 11.7064i 0.973387 + 0.403190i
\(844\) 0 0
\(845\) 9.61931 + 23.2231i 0.330914 + 0.798898i
\(846\) 0 0
\(847\) 1.32242i 0.0454389i
\(848\) 0 0
\(849\) 29.9928i 1.02935i
\(850\) 0 0
\(851\) 11.5576 + 27.9024i 0.396188 + 0.956482i
\(852\) 0 0
\(853\) −17.7015 7.33222i −0.606089 0.251050i 0.0584662 0.998289i \(-0.481379\pi\)
−0.664555 + 0.747239i \(0.731379\pi\)
\(854\) 0 0
\(855\) −10.0450 + 10.0450i −0.343533 + 0.343533i
\(856\) 0 0
\(857\) 34.8578 + 34.8578i 1.19072 + 1.19072i 0.976866 + 0.213853i \(0.0686015\pi\)
0.213853 + 0.976866i \(0.431399\pi\)
\(858\) 0 0
\(859\) 13.9729 33.7335i 0.476748 1.15097i −0.484378 0.874859i \(-0.660954\pi\)
0.961125 0.276112i \(-0.0890461\pi\)
\(860\) 0 0
\(861\) 16.2313 6.72321i 0.553160 0.229127i
\(862\) 0 0
\(863\) −35.2947 −1.20144 −0.600722 0.799458i \(-0.705120\pi\)
−0.600722 + 0.799458i \(0.705120\pi\)
\(864\) 0 0
\(865\) −43.6916 −1.48556
\(866\) 0 0
\(867\) −15.4586 + 6.40315i −0.525001 + 0.217462i
\(868\) 0 0
\(869\) 13.0443 31.4917i 0.442497 1.06828i
\(870\) 0 0
\(871\) −20.7597 20.7597i −0.703414 0.703414i
\(872\) 0 0
\(873\) 2.63958 2.63958i 0.0893363 0.0893363i
\(874\) 0 0
\(875\) 33.8855 + 14.0358i 1.14554 + 0.474498i
\(876\) 0 0
\(877\) 3.81781 + 9.21700i 0.128918 + 0.311236i 0.975138 0.221598i \(-0.0711271\pi\)
−0.846220 + 0.532834i \(0.821127\pi\)
\(878\) 0 0
\(879\) 4.81487i 0.162402i
\(880\) 0 0
\(881\) 20.1037i 0.677312i 0.940910 + 0.338656i \(0.109972\pi\)
−0.940910 + 0.338656i \(0.890028\pi\)
\(882\) 0 0
\(883\) −9.85279 23.7867i −0.331573 0.800487i −0.998468 0.0553357i \(-0.982377\pi\)
0.666895 0.745152i \(-0.267623\pi\)
\(884\) 0 0
\(885\) 5.85053 + 2.42337i 0.196663 + 0.0814607i
\(886\) 0 0
\(887\) −14.1672 + 14.1672i −0.475688 + 0.475688i −0.903749 0.428062i \(-0.859197\pi\)
0.428062 + 0.903749i \(0.359197\pi\)
\(888\) 0 0
\(889\) −5.04657 5.04657i −0.169256 0.169256i
\(890\) 0 0
\(891\) 1.24931 3.01609i 0.0418534 0.101043i
\(892\) 0 0
\(893\) −17.2368 + 7.13973i −0.576809 + 0.238922i
\(894\) 0 0
\(895\) −21.0709 −0.704322
\(896\) 0 0
\(897\) −8.26337 −0.275906
\(898\) 0 0
\(899\) −10.6452 + 4.40938i −0.355037 + 0.147061i
\(900\) 0 0
\(901\) 0.899208 2.17088i 0.0299569 0.0723225i
\(902\) 0 0
\(903\) 3.88257 + 3.88257i 0.129204 + 0.129204i
\(904\) 0 0
\(905\) −39.7080 + 39.7080i −1.31994 + 1.31994i
\(906\) 0 0
\(907\) −1.27402 0.527716i −0.0423031 0.0175225i 0.361432 0.932399i \(-0.382288\pi\)
−0.403735 + 0.914876i \(0.632288\pi\)
\(908\) 0 0
\(909\) −6.36949 15.3773i −0.211263 0.510033i
\(910\) 0 0
\(911\) 3.64101i 0.120632i 0.998179 + 0.0603161i \(0.0192109\pi\)
−0.998179 + 0.0603161i \(0.980789\pi\)
\(912\) 0 0
\(913\) 16.1522i 0.534560i
\(914\) 0 0
\(915\) 9.89962 + 23.8998i 0.327271 + 0.790103i
\(916\) 0 0
\(917\) −29.3275 12.1478i −0.968479 0.401157i
\(918\) 0 0
\(919\) −16.8542 + 16.8542i −0.555970 + 0.555970i −0.928158 0.372187i \(-0.878608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(920\) 0 0
\(921\) −21.6932 21.6932i −0.714815 0.714815i
\(922\) 0 0
\(923\) −2.46105 + 5.94151i −0.0810065 + 0.195567i
\(924\) 0 0
\(925\) −63.0933 + 26.1341i −2.07450 + 0.859284i
\(926\) 0 0
\(927\) 10.4969 0.344764
\(928\) 0 0
\(929\) −29.1718 −0.957096 −0.478548 0.878061i \(-0.658837\pi\)
−0.478548 + 0.878061i \(0.658837\pi\)
\(930\) 0 0
\(931\) −29.1910 + 12.0913i −0.956697 + 0.396277i
\(932\) 0 0
\(933\) 6.28440 15.1719i 0.205742 0.496705i
\(934\) 0 0
\(935\) 4.25147 + 4.25147i 0.139038 + 0.139038i
\(936\) 0 0
\(937\) 16.6591 16.6591i 0.544230 0.544230i −0.380536 0.924766i \(-0.624261\pi\)
0.924766 + 0.380536i \(0.124261\pi\)
\(938\) 0 0
\(939\) 8.94223 + 3.70399i 0.291819 + 0.120875i
\(940\) 0 0
\(941\) −2.44955 5.91373i −0.0798529 0.192782i 0.878911 0.476986i \(-0.158271\pi\)
−0.958764 + 0.284204i \(0.908271\pi\)
\(942\) 0 0
\(943\) 15.4261i 0.502342i
\(944\) 0 0
\(945\) 13.7464i 0.447169i
\(946\) 0 0
\(947\) −4.96711 11.9917i −0.161409 0.389676i 0.822396 0.568915i \(-0.192637\pi\)
−0.983806 + 0.179238i \(0.942637\pi\)
\(948\) 0 0
\(949\) 28.8721 + 11.9592i 0.937228 + 0.388212i
\(950\) 0 0
\(951\) −18.1235 + 18.1235i −0.587694 + 0.587694i
\(952\) 0 0
\(953\) −6.25983 6.25983i −0.202776 0.202776i 0.598412 0.801188i \(-0.295798\pi\)
−0.801188 + 0.598412i \(0.795798\pi\)
\(954\) 0 0
\(955\) −16.8576 + 40.6978i −0.545499 + 1.31695i
\(956\) 0 0
\(957\) 23.1478 9.58815i 0.748263 0.309941i
\(958\) 0 0
\(959\) 43.6180 1.40850
\(960\) 0 0
\(961\) −28.7460 −0.927292
\(962\) 0 0
\(963\) 7.85343 3.25300i 0.253073 0.104826i
\(964\) 0 0
\(965\) −29.8759 + 72.1268i −0.961739 + 2.32184i
\(966\) 0 0
\(967\) −4.21628 4.21628i −0.135586 0.135586i 0.636056 0.771643i \(-0.280565\pi\)
−0.771643 + 0.636056i \(0.780565\pi\)
\(968\) 0 0
\(969\) −1.46037 + 1.46037i −0.0469138 + 0.0469138i
\(970\) 0 0
\(971\) −35.3365 14.6369i −1.13400 0.469720i −0.264864 0.964286i \(-0.585327\pi\)
−0.869140 + 0.494566i \(0.835327\pi\)
\(972\) 0 0
\(973\) 2.43582 + 5.88058i 0.0780887 + 0.188523i
\(974\) 0 0
\(975\) 18.6853i 0.598407i
\(976\) 0 0
\(977\) 60.3138i 1.92961i −0.262970 0.964804i \(-0.584702\pi\)
0.262970 0.964804i \(-0.415298\pi\)
\(978\) 0 0
\(979\) −4.78379 11.5491i −0.152891 0.369111i
\(980\) 0 0
\(981\) −8.04001 3.33028i −0.256698 0.106328i
\(982\) 0 0
\(983\) −38.0148 + 38.0148i −1.21248 + 1.21248i −0.242277 + 0.970207i \(0.577894\pi\)
−0.970207 + 0.242277i \(0.922106\pi\)
\(984\) 0 0
\(985\) −42.9292 42.9292i −1.36784 1.36784i
\(986\) 0 0
\(987\) 6.90882 16.6794i 0.219910 0.530910i
\(988\) 0 0
\(989\) 4.45417 1.84498i 0.141634 0.0586669i
\(990\) 0 0
\(991\) −51.7734 −1.64464 −0.822319 0.569027i \(-0.807320\pi\)
−0.822319 + 0.569027i \(0.807320\pi\)
\(992\) 0 0
\(993\) 23.4995 0.745733
\(994\) 0 0
\(995\) 26.9847 11.1774i 0.855472 0.354348i
\(996\) 0 0
\(997\) −13.2018 + 31.8719i −0.418105 + 1.00939i 0.564791 + 0.825234i \(0.308957\pi\)
−0.982896 + 0.184161i \(0.941043\pi\)
\(998\) 0 0
\(999\) 6.29742 + 6.29742i 0.199241 + 0.199241i
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 768.2.n.b.97.1 32
4.3 odd 2 768.2.n.a.97.5 32
8.3 odd 2 96.2.n.a.85.5 yes 32
8.5 even 2 384.2.n.a.49.8 32
24.5 odd 2 1152.2.v.c.433.1 32
24.11 even 2 288.2.v.d.181.4 32
32.3 odd 8 768.2.n.a.673.5 32
32.13 even 8 384.2.n.a.337.8 32
32.19 odd 8 96.2.n.a.61.5 32
32.29 even 8 inner 768.2.n.b.673.1 32
96.77 odd 8 1152.2.v.c.721.1 32
96.83 even 8 288.2.v.d.253.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.n.a.61.5 32 32.19 odd 8
96.2.n.a.85.5 yes 32 8.3 odd 2
288.2.v.d.181.4 32 24.11 even 2
288.2.v.d.253.4 32 96.83 even 8
384.2.n.a.49.8 32 8.5 even 2
384.2.n.a.337.8 32 32.13 even 8
768.2.n.a.97.5 32 4.3 odd 2
768.2.n.a.673.5 32 32.3 odd 8
768.2.n.b.97.1 32 1.1 even 1 trivial
768.2.n.b.673.1 32 32.29 even 8 inner
1152.2.v.c.433.1 32 24.5 odd 2
1152.2.v.c.721.1 32 96.77 odd 8