Properties

Label 912.2.d.b.191.7
Level $912$
Weight $2$
Character 912.191
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(191,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.d (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.7
Character \(\chi\) \(=\) 912.191
Dual form 912.2.d.b.191.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.909651 - 1.47395i) q^{3} +2.62895i q^{5} -3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} +O(q^{10})\) \(q+(-0.909651 - 1.47395i) q^{3} +2.62895i q^{5} -3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} +1.56630 q^{11} +1.23349 q^{13} +(3.87495 - 2.39143i) q^{15} -3.38560i q^{17} +1.00000i q^{19} +(-5.28501 + 3.26165i) q^{21} -1.04413 q^{23} -1.91139 q^{25} +(5.17604 - 0.456722i) q^{27} -7.20096i q^{29} -1.42596i q^{31} +(-1.42479 - 2.30865i) q^{33} +9.42639 q^{35} +4.55367 q^{37} +(-1.12205 - 1.81811i) q^{39} -10.8210i q^{41} -9.96678i q^{43} +(-7.04971 - 3.53612i) q^{45} -0.136958 q^{47} -5.85657 q^{49} +(-4.99022 + 3.07972i) q^{51} +9.69295i q^{53} +4.11773i q^{55} +(1.47395 - 0.909651i) q^{57} -6.50974 q^{59} +6.11212 q^{61} +(9.61503 + 4.82289i) q^{63} +3.24279i q^{65} -14.1977i q^{67} +(0.949792 + 1.53899i) q^{69} -10.0857 q^{71} -9.78698 q^{73} +(1.73870 + 2.81730i) q^{75} -5.61613i q^{77} -10.5281i q^{79} +(-5.38158 - 7.21378i) q^{81} +13.4190 q^{83} +8.90059 q^{85} +(-10.6139 + 6.55036i) q^{87} -4.31626i q^{89} -4.42281i q^{91} +(-2.10180 + 1.29713i) q^{93} -2.62895 q^{95} +6.67044 q^{97} +(-2.10678 + 4.20013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) −0.909651 1.47395i −0.525187 0.850987i
\(4\) 0 0
\(5\) 2.62895i 1.17570i 0.808969 + 0.587852i \(0.200026\pi\)
−0.808969 + 0.587852i \(0.799974\pi\)
\(6\) 0 0
\(7\) 3.58561i 1.35523i −0.735416 0.677616i \(-0.763013\pi\)
0.735416 0.677616i \(-0.236987\pi\)
\(8\) 0 0
\(9\) −1.34507 + 2.68156i −0.448356 + 0.893855i
\(10\) 0 0
\(11\) 1.56630 0.472257 0.236129 0.971722i \(-0.424121\pi\)
0.236129 + 0.971722i \(0.424121\pi\)
\(12\) 0 0
\(13\) 1.23349 0.342109 0.171054 0.985262i \(-0.445283\pi\)
0.171054 + 0.985262i \(0.445283\pi\)
\(14\) 0 0
\(15\) 3.87495 2.39143i 1.00051 0.617465i
\(16\) 0 0
\(17\) 3.38560i 0.821129i −0.911832 0.410565i \(-0.865332\pi\)
0.911832 0.410565i \(-0.134668\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) −5.28501 + 3.26165i −1.15328 + 0.711751i
\(22\) 0 0
\(23\) −1.04413 −0.217716 −0.108858 0.994057i \(-0.534719\pi\)
−0.108858 + 0.994057i \(0.534719\pi\)
\(24\) 0 0
\(25\) −1.91139 −0.382279
\(26\) 0 0
\(27\) 5.17604 0.456722i 0.996130 0.0878963i
\(28\) 0 0
\(29\) 7.20096i 1.33718i −0.743629 0.668592i \(-0.766897\pi\)
0.743629 0.668592i \(-0.233103\pi\)
\(30\) 0 0
\(31\) 1.42596i 0.256111i −0.991767 0.128055i \(-0.959126\pi\)
0.991767 0.128055i \(-0.0408735\pi\)
\(32\) 0 0
\(33\) −1.42479 2.30865i −0.248024 0.401884i
\(34\) 0 0
\(35\) 9.42639 1.59335
\(36\) 0 0
\(37\) 4.55367 0.748619 0.374309 0.927304i \(-0.377880\pi\)
0.374309 + 0.927304i \(0.377880\pi\)
\(38\) 0 0
\(39\) −1.12205 1.81811i −0.179671 0.291130i
\(40\) 0 0
\(41\) 10.8210i 1.68996i −0.534796 0.844981i \(-0.679612\pi\)
0.534796 0.844981i \(-0.320388\pi\)
\(42\) 0 0
\(43\) 9.96678i 1.51992i −0.649970 0.759960i \(-0.725219\pi\)
0.649970 0.759960i \(-0.274781\pi\)
\(44\) 0 0
\(45\) −7.04971 3.53612i −1.05091 0.527134i
\(46\) 0 0
\(47\) −0.136958 −0.0199774 −0.00998870 0.999950i \(-0.503180\pi\)
−0.00998870 + 0.999950i \(0.503180\pi\)
\(48\) 0 0
\(49\) −5.85657 −0.836653
\(50\) 0 0
\(51\) −4.99022 + 3.07972i −0.698770 + 0.431247i
\(52\) 0 0
\(53\) 9.69295i 1.33143i 0.746206 + 0.665715i \(0.231873\pi\)
−0.746206 + 0.665715i \(0.768127\pi\)
\(54\) 0 0
\(55\) 4.11773i 0.555234i
\(56\) 0 0
\(57\) 1.47395 0.909651i 0.195230 0.120486i
\(58\) 0 0
\(59\) −6.50974 −0.847496 −0.423748 0.905780i \(-0.639286\pi\)
−0.423748 + 0.905780i \(0.639286\pi\)
\(60\) 0 0
\(61\) 6.11212 0.782577 0.391288 0.920268i \(-0.372029\pi\)
0.391288 + 0.920268i \(0.372029\pi\)
\(62\) 0 0
\(63\) 9.61503 + 4.82289i 1.21138 + 0.607627i
\(64\) 0 0
\(65\) 3.24279i 0.402218i
\(66\) 0 0
\(67\) 14.1977i 1.73453i −0.497846 0.867265i \(-0.665876\pi\)
0.497846 0.867265i \(-0.334124\pi\)
\(68\) 0 0
\(69\) 0.949792 + 1.53899i 0.114342 + 0.185273i
\(70\) 0 0
\(71\) −10.0857 −1.19695 −0.598474 0.801143i \(-0.704226\pi\)
−0.598474 + 0.801143i \(0.704226\pi\)
\(72\) 0 0
\(73\) −9.78698 −1.14548 −0.572740 0.819737i \(-0.694119\pi\)
−0.572740 + 0.819737i \(0.694119\pi\)
\(74\) 0 0
\(75\) 1.73870 + 2.81730i 0.200768 + 0.325314i
\(76\) 0 0
\(77\) 5.61613i 0.640018i
\(78\) 0 0
\(79\) 10.5281i 1.18450i −0.805753 0.592252i \(-0.798239\pi\)
0.805753 0.592252i \(-0.201761\pi\)
\(80\) 0 0
\(81\) −5.38158 7.21378i −0.597953 0.801531i
\(82\) 0 0
\(83\) 13.4190 1.47293 0.736465 0.676476i \(-0.236494\pi\)
0.736465 + 0.676476i \(0.236494\pi\)
\(84\) 0 0
\(85\) 8.90059 0.965404
\(86\) 0 0
\(87\) −10.6139 + 6.55036i −1.13793 + 0.702272i
\(88\) 0 0
\(89\) 4.31626i 0.457523i −0.973483 0.228761i \(-0.926532\pi\)
0.973483 0.228761i \(-0.0734676\pi\)
\(90\) 0 0
\(91\) 4.42281i 0.463636i
\(92\) 0 0
\(93\) −2.10180 + 1.29713i −0.217947 + 0.134506i
\(94\) 0 0
\(95\) −2.62895 −0.269725
\(96\) 0 0
\(97\) 6.67044 0.677280 0.338640 0.940916i \(-0.390033\pi\)
0.338640 + 0.940916i \(0.390033\pi\)
\(98\) 0 0
\(99\) −2.10678 + 4.20013i −0.211739 + 0.422129i
\(100\) 0 0
\(101\) 16.5666i 1.64844i 0.566268 + 0.824221i \(0.308387\pi\)
−0.566268 + 0.824221i \(0.691613\pi\)
\(102\) 0 0
\(103\) 0.313018i 0.0308426i 0.999881 + 0.0154213i \(0.00490894\pi\)
−0.999881 + 0.0154213i \(0.995091\pi\)
\(104\) 0 0
\(105\) −8.57473 13.8940i −0.836808 1.35592i
\(106\) 0 0
\(107\) 14.7286 1.42387 0.711934 0.702247i \(-0.247820\pi\)
0.711934 + 0.702247i \(0.247820\pi\)
\(108\) 0 0
\(109\) 6.01635 0.576262 0.288131 0.957591i \(-0.406966\pi\)
0.288131 + 0.957591i \(0.406966\pi\)
\(110\) 0 0
\(111\) −4.14225 6.71189i −0.393165 0.637064i
\(112\) 0 0
\(113\) 5.73478i 0.539483i −0.962933 0.269741i \(-0.913062\pi\)
0.962933 0.269741i \(-0.0869382\pi\)
\(114\) 0 0
\(115\) 2.74496i 0.255969i
\(116\) 0 0
\(117\) −1.65913 + 3.30768i −0.153387 + 0.305795i
\(118\) 0 0
\(119\) −12.1394 −1.11282
\(120\) 0 0
\(121\) −8.54671 −0.776973
\(122\) 0 0
\(123\) −15.9497 + 9.84337i −1.43814 + 0.887547i
\(124\) 0 0
\(125\) 8.11980i 0.726257i
\(126\) 0 0
\(127\) 8.79148i 0.780118i 0.920790 + 0.390059i \(0.127545\pi\)
−0.920790 + 0.390059i \(0.872455\pi\)
\(128\) 0 0
\(129\) −14.6906 + 9.06629i −1.29343 + 0.798243i
\(130\) 0 0
\(131\) −8.37456 −0.731688 −0.365844 0.930676i \(-0.619220\pi\)
−0.365844 + 0.930676i \(0.619220\pi\)
\(132\) 0 0
\(133\) 3.58561 0.310911
\(134\) 0 0
\(135\) 1.20070 + 13.6076i 0.103340 + 1.17115i
\(136\) 0 0
\(137\) 17.5826i 1.50218i 0.660201 + 0.751089i \(0.270471\pi\)
−0.660201 + 0.751089i \(0.729529\pi\)
\(138\) 0 0
\(139\) 4.53795i 0.384904i −0.981306 0.192452i \(-0.938356\pi\)
0.981306 0.192452i \(-0.0616440\pi\)
\(140\) 0 0
\(141\) 0.124584 + 0.201870i 0.0104919 + 0.0170005i
\(142\) 0 0
\(143\) 1.93202 0.161563
\(144\) 0 0
\(145\) 18.9310 1.57213
\(146\) 0 0
\(147\) 5.32744 + 8.63230i 0.439399 + 0.711980i
\(148\) 0 0
\(149\) 14.3996i 1.17966i −0.807528 0.589829i \(-0.799195\pi\)
0.807528 0.589829i \(-0.200805\pi\)
\(150\) 0 0
\(151\) 14.0819i 1.14597i 0.819566 + 0.572985i \(0.194215\pi\)
−0.819566 + 0.572985i \(0.805785\pi\)
\(152\) 0 0
\(153\) 9.07871 + 4.55387i 0.733970 + 0.368158i
\(154\) 0 0
\(155\) 3.74879 0.301110
\(156\) 0 0
\(157\) 20.4473 1.63187 0.815936 0.578142i \(-0.196222\pi\)
0.815936 + 0.578142i \(0.196222\pi\)
\(158\) 0 0
\(159\) 14.2869 8.81721i 1.13303 0.699250i
\(160\) 0 0
\(161\) 3.74383i 0.295055i
\(162\) 0 0
\(163\) 12.9161i 1.01166i 0.862632 + 0.505832i \(0.168815\pi\)
−0.862632 + 0.505832i \(0.831185\pi\)
\(164\) 0 0
\(165\) 6.06933 3.74570i 0.472497 0.291602i
\(166\) 0 0
\(167\) 14.8709 1.15074 0.575371 0.817892i \(-0.304858\pi\)
0.575371 + 0.817892i \(0.304858\pi\)
\(168\) 0 0
\(169\) −11.4785 −0.882962
\(170\) 0 0
\(171\) −2.68156 1.34507i −0.205064 0.102860i
\(172\) 0 0
\(173\) 13.3205i 1.01274i 0.862316 + 0.506370i \(0.169013\pi\)
−0.862316 + 0.506370i \(0.830987\pi\)
\(174\) 0 0
\(175\) 6.85350i 0.518076i
\(176\) 0 0
\(177\) 5.92159 + 9.59504i 0.445094 + 0.721208i
\(178\) 0 0
\(179\) 0.567468 0.0424145 0.0212073 0.999775i \(-0.493249\pi\)
0.0212073 + 0.999775i \(0.493249\pi\)
\(180\) 0 0
\(181\) 21.2587 1.58015 0.790074 0.613011i \(-0.210042\pi\)
0.790074 + 0.613011i \(0.210042\pi\)
\(182\) 0 0
\(183\) −5.55990 9.00897i −0.411000 0.665962i
\(184\) 0 0
\(185\) 11.9714i 0.880154i
\(186\) 0 0
\(187\) 5.30287i 0.387784i
\(188\) 0 0
\(189\) −1.63763 18.5592i −0.119120 1.34999i
\(190\) 0 0
\(191\) −8.13476 −0.588611 −0.294305 0.955711i \(-0.595088\pi\)
−0.294305 + 0.955711i \(0.595088\pi\)
\(192\) 0 0
\(193\) 15.0670 1.08455 0.542274 0.840202i \(-0.317564\pi\)
0.542274 + 0.840202i \(0.317564\pi\)
\(194\) 0 0
\(195\) 4.77971 2.94981i 0.342282 0.211240i
\(196\) 0 0
\(197\) 2.78221i 0.198224i −0.995076 0.0991122i \(-0.968400\pi\)
0.995076 0.0991122i \(-0.0316003\pi\)
\(198\) 0 0
\(199\) 12.4471i 0.882351i 0.897421 + 0.441176i \(0.145438\pi\)
−0.897421 + 0.441176i \(0.854562\pi\)
\(200\) 0 0
\(201\) −20.9268 + 12.9150i −1.47606 + 0.910954i
\(202\) 0 0
\(203\) −25.8198 −1.81219
\(204\) 0 0
\(205\) 28.4480 1.98689
\(206\) 0 0
\(207\) 1.40442 2.79990i 0.0976142 0.194606i
\(208\) 0 0
\(209\) 1.56630i 0.108343i
\(210\) 0 0
\(211\) 15.9686i 1.09932i 0.835388 + 0.549661i \(0.185243\pi\)
−0.835388 + 0.549661i \(0.814757\pi\)
\(212\) 0 0
\(213\) 9.17443 + 14.8658i 0.628622 + 1.01859i
\(214\) 0 0
\(215\) 26.2022 1.78697
\(216\) 0 0
\(217\) −5.11295 −0.347089
\(218\) 0 0
\(219\) 8.90274 + 14.4255i 0.601591 + 0.974787i
\(220\) 0 0
\(221\) 4.17611i 0.280915i
\(222\) 0 0
\(223\) 4.22684i 0.283050i −0.989935 0.141525i \(-0.954799\pi\)
0.989935 0.141525i \(-0.0452005\pi\)
\(224\) 0 0
\(225\) 2.57095 5.12552i 0.171397 0.341702i
\(226\) 0 0
\(227\) −15.1403 −1.00489 −0.502447 0.864608i \(-0.667567\pi\)
−0.502447 + 0.864608i \(0.667567\pi\)
\(228\) 0 0
\(229\) −11.3412 −0.749446 −0.374723 0.927137i \(-0.622262\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(230\) 0 0
\(231\) −8.27791 + 5.10872i −0.544646 + 0.336129i
\(232\) 0 0
\(233\) 4.56159i 0.298840i 0.988774 + 0.149420i \(0.0477406\pi\)
−0.988774 + 0.149420i \(0.952259\pi\)
\(234\) 0 0
\(235\) 0.360056i 0.0234875i
\(236\) 0 0
\(237\) −15.5179 + 9.57691i −1.00800 + 0.622087i
\(238\) 0 0
\(239\) 15.6650 1.01328 0.506641 0.862157i \(-0.330887\pi\)
0.506641 + 0.862157i \(0.330887\pi\)
\(240\) 0 0
\(241\) 3.70475 0.238644 0.119322 0.992856i \(-0.461928\pi\)
0.119322 + 0.992856i \(0.461928\pi\)
\(242\) 0 0
\(243\) −5.73740 + 14.4942i −0.368054 + 0.929804i
\(244\) 0 0
\(245\) 15.3966i 0.983655i
\(246\) 0 0
\(247\) 1.23349i 0.0784851i
\(248\) 0 0
\(249\) −12.2066 19.7790i −0.773565 1.25344i
\(250\) 0 0
\(251\) −28.7827 −1.81675 −0.908375 0.418157i \(-0.862676\pi\)
−0.908375 + 0.418157i \(0.862676\pi\)
\(252\) 0 0
\(253\) −1.63542 −0.102818
\(254\) 0 0
\(255\) −8.09643 13.1190i −0.507018 0.821546i
\(256\) 0 0
\(257\) 22.9658i 1.43257i −0.697809 0.716284i \(-0.745841\pi\)
0.697809 0.716284i \(-0.254159\pi\)
\(258\) 0 0
\(259\) 16.3277i 1.01455i
\(260\) 0 0
\(261\) 19.3098 + 9.68578i 1.19525 + 0.599535i
\(262\) 0 0
\(263\) −13.7535 −0.848075 −0.424038 0.905645i \(-0.639388\pi\)
−0.424038 + 0.905645i \(0.639388\pi\)
\(264\) 0 0
\(265\) −25.4823 −1.56537
\(266\) 0 0
\(267\) −6.36196 + 3.92629i −0.389346 + 0.240285i
\(268\) 0 0
\(269\) 2.93383i 0.178879i −0.995992 0.0894393i \(-0.971492\pi\)
0.995992 0.0894393i \(-0.0285075\pi\)
\(270\) 0 0
\(271\) 21.3061i 1.29425i 0.762382 + 0.647127i \(0.224030\pi\)
−0.762382 + 0.647127i \(0.775970\pi\)
\(272\) 0 0
\(273\) −6.51901 + 4.02321i −0.394548 + 0.243496i
\(274\) 0 0
\(275\) −2.99381 −0.180534
\(276\) 0 0
\(277\) −5.89065 −0.353935 −0.176968 0.984217i \(-0.556629\pi\)
−0.176968 + 0.984217i \(0.556629\pi\)
\(278\) 0 0
\(279\) 3.82382 + 1.91802i 0.228926 + 0.114829i
\(280\) 0 0
\(281\) 30.8471i 1.84018i −0.391702 0.920092i \(-0.628113\pi\)
0.391702 0.920092i \(-0.371887\pi\)
\(282\) 0 0
\(283\) 25.9416i 1.54207i 0.636793 + 0.771035i \(0.280261\pi\)
−0.636793 + 0.771035i \(0.719739\pi\)
\(284\) 0 0
\(285\) 2.39143 + 3.87495i 0.141656 + 0.229532i
\(286\) 0 0
\(287\) −38.8000 −2.29029
\(288\) 0 0
\(289\) 5.53770 0.325747
\(290\) 0 0
\(291\) −6.06777 9.83190i −0.355699 0.576356i
\(292\) 0 0
\(293\) 33.1106i 1.93434i 0.254130 + 0.967170i \(0.418211\pi\)
−0.254130 + 0.967170i \(0.581789\pi\)
\(294\) 0 0
\(295\) 17.1138i 0.996404i
\(296\) 0 0
\(297\) 8.10723 0.715364i 0.470429 0.0415096i
\(298\) 0 0
\(299\) −1.28792 −0.0744824
\(300\) 0 0
\(301\) −35.7369 −2.05984
\(302\) 0 0
\(303\) 24.4184 15.0699i 1.40280 0.865741i
\(304\) 0 0
\(305\) 16.0685i 0.920078i
\(306\) 0 0
\(307\) 0.857386i 0.0489336i −0.999701 0.0244668i \(-0.992211\pi\)
0.999701 0.0244668i \(-0.00778880\pi\)
\(308\) 0 0
\(309\) 0.461373 0.284737i 0.0262466 0.0161981i
\(310\) 0 0
\(311\) −18.8346 −1.06801 −0.534006 0.845481i \(-0.679314\pi\)
−0.534006 + 0.845481i \(0.679314\pi\)
\(312\) 0 0
\(313\) 28.4391 1.60747 0.803737 0.594985i \(-0.202842\pi\)
0.803737 + 0.594985i \(0.202842\pi\)
\(314\) 0 0
\(315\) −12.6791 + 25.2775i −0.714389 + 1.42422i
\(316\) 0 0
\(317\) 12.9564i 0.727705i −0.931457 0.363852i \(-0.881461\pi\)
0.931457 0.363852i \(-0.118539\pi\)
\(318\) 0 0
\(319\) 11.2789i 0.631495i
\(320\) 0 0
\(321\) −13.3979 21.7092i −0.747797 1.21169i
\(322\) 0 0
\(323\) 3.38560 0.188380
\(324\) 0 0
\(325\) −2.35768 −0.130781
\(326\) 0 0
\(327\) −5.47278 8.86781i −0.302646 0.490391i
\(328\) 0 0
\(329\) 0.491078i 0.0270740i
\(330\) 0 0
\(331\) 6.45891i 0.355014i 0.984120 + 0.177507i \(0.0568032\pi\)
−0.984120 + 0.177507i \(0.943197\pi\)
\(332\) 0 0
\(333\) −6.12500 + 12.2110i −0.335648 + 0.669157i
\(334\) 0 0
\(335\) 37.3252 2.03929
\(336\) 0 0
\(337\) −16.5456 −0.901297 −0.450648 0.892702i \(-0.648807\pi\)
−0.450648 + 0.892702i \(0.648807\pi\)
\(338\) 0 0
\(339\) −8.45279 + 5.21665i −0.459093 + 0.283330i
\(340\) 0 0
\(341\) 2.23349i 0.120950i
\(342\) 0 0
\(343\) 4.09990i 0.221374i
\(344\) 0 0
\(345\) −4.04594 + 2.49696i −0.217826 + 0.134432i
\(346\) 0 0
\(347\) −16.5061 −0.886095 −0.443047 0.896498i \(-0.646103\pi\)
−0.443047 + 0.896498i \(0.646103\pi\)
\(348\) 0 0
\(349\) −8.12035 −0.434673 −0.217336 0.976097i \(-0.569737\pi\)
−0.217336 + 0.976097i \(0.569737\pi\)
\(350\) 0 0
\(351\) 6.38460 0.563363i 0.340785 0.0300701i
\(352\) 0 0
\(353\) 23.2255i 1.23617i −0.786111 0.618085i \(-0.787909\pi\)
0.786111 0.618085i \(-0.212091\pi\)
\(354\) 0 0
\(355\) 26.5147i 1.40725i
\(356\) 0 0
\(357\) 11.0427 + 17.8929i 0.584439 + 0.946995i
\(358\) 0 0
\(359\) 10.5501 0.556813 0.278406 0.960463i \(-0.410194\pi\)
0.278406 + 0.960463i \(0.410194\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 7.77452 + 12.5974i 0.408057 + 0.661194i
\(364\) 0 0
\(365\) 25.7295i 1.34674i
\(366\) 0 0
\(367\) 16.7774i 0.875772i −0.899030 0.437886i \(-0.855727\pi\)
0.899030 0.437886i \(-0.144273\pi\)
\(368\) 0 0
\(369\) 29.0173 + 14.5550i 1.51058 + 0.757705i
\(370\) 0 0
\(371\) 34.7551 1.80440
\(372\) 0 0
\(373\) 2.88732 0.149500 0.0747498 0.997202i \(-0.476184\pi\)
0.0747498 + 0.997202i \(0.476184\pi\)
\(374\) 0 0
\(375\) 11.9682 7.38619i 0.618035 0.381421i
\(376\) 0 0
\(377\) 8.88231i 0.457462i
\(378\) 0 0
\(379\) 20.0903i 1.03197i −0.856598 0.515985i \(-0.827426\pi\)
0.856598 0.515985i \(-0.172574\pi\)
\(380\) 0 0
\(381\) 12.9582 7.99718i 0.663870 0.409708i
\(382\) 0 0
\(383\) 2.45494 0.125442 0.0627209 0.998031i \(-0.480022\pi\)
0.0627209 + 0.998031i \(0.480022\pi\)
\(384\) 0 0
\(385\) 14.7645 0.752471
\(386\) 0 0
\(387\) 26.7266 + 13.4060i 1.35859 + 0.681465i
\(388\) 0 0
\(389\) 9.41557i 0.477388i 0.971095 + 0.238694i \(0.0767193\pi\)
−0.971095 + 0.238694i \(0.923281\pi\)
\(390\) 0 0
\(391\) 3.53500i 0.178773i
\(392\) 0 0
\(393\) 7.61793 + 12.3437i 0.384274 + 0.622657i
\(394\) 0 0
\(395\) 27.6779 1.39263
\(396\) 0 0
\(397\) −39.3046 −1.97264 −0.986320 0.164841i \(-0.947289\pi\)
−0.986320 + 0.164841i \(0.947289\pi\)
\(398\) 0 0
\(399\) −3.26165 5.28501i −0.163287 0.264581i
\(400\) 0 0
\(401\) 10.2498i 0.511851i −0.966696 0.255926i \(-0.917620\pi\)
0.966696 0.255926i \(-0.0823802\pi\)
\(402\) 0 0
\(403\) 1.75891i 0.0876177i
\(404\) 0 0
\(405\) 18.9647 14.1479i 0.942363 0.703016i
\(406\) 0 0
\(407\) 7.13241 0.353541
\(408\) 0 0
\(409\) −22.7232 −1.12359 −0.561795 0.827276i \(-0.689889\pi\)
−0.561795 + 0.827276i \(0.689889\pi\)
\(410\) 0 0
\(411\) 25.9158 15.9940i 1.27833 0.788925i
\(412\) 0 0
\(413\) 23.3414i 1.14855i
\(414\) 0 0
\(415\) 35.2780i 1.73173i
\(416\) 0 0
\(417\) −6.68872 + 4.12795i −0.327548 + 0.202147i
\(418\) 0 0
\(419\) 36.8117 1.79837 0.899185 0.437569i \(-0.144160\pi\)
0.899185 + 0.437569i \(0.144160\pi\)
\(420\) 0 0
\(421\) 11.3910 0.555164 0.277582 0.960702i \(-0.410467\pi\)
0.277582 + 0.960702i \(0.410467\pi\)
\(422\) 0 0
\(423\) 0.184218 0.367262i 0.00895699 0.0178569i
\(424\) 0 0
\(425\) 6.47122i 0.313900i
\(426\) 0 0
\(427\) 21.9157i 1.06057i
\(428\) 0 0
\(429\) −1.75746 2.84770i −0.0848510 0.137488i
\(430\) 0 0
\(431\) 26.4348 1.27332 0.636659 0.771146i \(-0.280316\pi\)
0.636659 + 0.771146i \(0.280316\pi\)
\(432\) 0 0
\(433\) 11.2338 0.539860 0.269930 0.962880i \(-0.412999\pi\)
0.269930 + 0.962880i \(0.412999\pi\)
\(434\) 0 0
\(435\) −17.2206 27.9034i −0.825664 1.33786i
\(436\) 0 0
\(437\) 1.04413i 0.0499474i
\(438\) 0 0
\(439\) 21.3692i 1.01990i −0.860205 0.509949i \(-0.829664\pi\)
0.860205 0.509949i \(-0.170336\pi\)
\(440\) 0 0
\(441\) 7.87749 15.7048i 0.375118 0.747846i
\(442\) 0 0
\(443\) −15.7849 −0.749963 −0.374981 0.927032i \(-0.622351\pi\)
−0.374981 + 0.927032i \(0.622351\pi\)
\(444\) 0 0
\(445\) 11.3472 0.537911
\(446\) 0 0
\(447\) −21.2243 + 13.0986i −1.00387 + 0.619541i
\(448\) 0 0
\(449\) 10.5162i 0.496292i 0.968723 + 0.248146i \(0.0798213\pi\)
−0.968723 + 0.248146i \(0.920179\pi\)
\(450\) 0 0
\(451\) 16.9490i 0.798097i
\(452\) 0 0
\(453\) 20.7561 12.8096i 0.975205 0.601849i
\(454\) 0 0
\(455\) 11.6274 0.545099
\(456\) 0 0
\(457\) 21.4547 1.00361 0.501804 0.864981i \(-0.332670\pi\)
0.501804 + 0.864981i \(0.332670\pi\)
\(458\) 0 0
\(459\) −1.54628 17.5240i −0.0721742 0.817951i
\(460\) 0 0
\(461\) 27.2511i 1.26921i 0.772836 + 0.634606i \(0.218837\pi\)
−0.772836 + 0.634606i \(0.781163\pi\)
\(462\) 0 0
\(463\) 27.4287i 1.27472i 0.770567 + 0.637359i \(0.219973\pi\)
−0.770567 + 0.637359i \(0.780027\pi\)
\(464\) 0 0
\(465\) −3.41010 5.52554i −0.158139 0.256241i
\(466\) 0 0
\(467\) −21.1682 −0.979546 −0.489773 0.871850i \(-0.662920\pi\)
−0.489773 + 0.871850i \(0.662920\pi\)
\(468\) 0 0
\(469\) −50.9075 −2.35069
\(470\) 0 0
\(471\) −18.5999 30.1383i −0.857039 1.38870i
\(472\) 0 0
\(473\) 15.6110i 0.717793i
\(474\) 0 0
\(475\) 1.91139i 0.0877007i
\(476\) 0 0
\(477\) −25.9923 13.0377i −1.19010 0.596955i
\(478\) 0 0
\(479\) −22.9110 −1.04683 −0.523415 0.852078i \(-0.675342\pi\)
−0.523415 + 0.852078i \(0.675342\pi\)
\(480\) 0 0
\(481\) 5.61691 0.256109
\(482\) 0 0
\(483\) 5.51823 3.40558i 0.251088 0.154959i
\(484\) 0 0
\(485\) 17.5363i 0.796281i
\(486\) 0 0
\(487\) 26.0702i 1.18135i −0.806908 0.590677i \(-0.798861\pi\)
0.806908 0.590677i \(-0.201139\pi\)
\(488\) 0 0
\(489\) 19.0377 11.7491i 0.860913 0.531314i
\(490\) 0 0
\(491\) −2.35468 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(492\) 0 0
\(493\) −24.3796 −1.09800
\(494\) 0 0
\(495\) −11.0420 5.53863i −0.496299 0.248943i
\(496\) 0 0
\(497\) 36.1632i 1.62214i
\(498\) 0 0
\(499\) 8.01852i 0.358958i −0.983762 0.179479i \(-0.942559\pi\)
0.983762 0.179479i \(-0.0574412\pi\)
\(500\) 0 0
\(501\) −13.5273 21.9189i −0.604356 0.979266i
\(502\) 0 0
\(503\) 28.8205 1.28504 0.642522 0.766267i \(-0.277888\pi\)
0.642522 + 0.766267i \(0.277888\pi\)
\(504\) 0 0
\(505\) −43.5529 −1.93808
\(506\) 0 0
\(507\) 10.4414 + 16.9188i 0.463720 + 0.751389i
\(508\) 0 0
\(509\) 35.4676i 1.57207i 0.618179 + 0.786037i \(0.287871\pi\)
−0.618179 + 0.786037i \(0.712129\pi\)
\(510\) 0 0
\(511\) 35.0922i 1.55239i
\(512\) 0 0
\(513\) 0.456722 + 5.17604i 0.0201648 + 0.228528i
\(514\) 0 0
\(515\) −0.822909 −0.0362617
\(516\) 0 0
\(517\) −0.214517 −0.00943447
\(518\) 0 0
\(519\) 19.6338 12.1170i 0.861829 0.531879i
\(520\) 0 0
\(521\) 11.1183i 0.487103i 0.969888 + 0.243551i \(0.0783124\pi\)
−0.969888 + 0.243551i \(0.921688\pi\)
\(522\) 0 0
\(523\) 19.5917i 0.856684i −0.903617 0.428342i \(-0.859098\pi\)
0.903617 0.428342i \(-0.140902\pi\)
\(524\) 0 0
\(525\) 10.1017 6.23430i 0.440876 0.272087i
\(526\) 0 0
\(527\) −4.82775 −0.210300
\(528\) 0 0
\(529\) −21.9098 −0.952600
\(530\) 0 0
\(531\) 8.75605 17.4563i 0.379980 0.757539i
\(532\) 0 0
\(533\) 13.3476i 0.578151i
\(534\) 0 0
\(535\) 38.7208i 1.67405i
\(536\) 0 0
\(537\) −0.516198 0.836420i −0.0222756 0.0360942i
\(538\) 0 0
\(539\) −9.17314 −0.395115
\(540\) 0 0
\(541\) −29.9449 −1.28743 −0.643716 0.765264i \(-0.722608\pi\)
−0.643716 + 0.765264i \(0.722608\pi\)
\(542\) 0 0
\(543\) −19.3380 31.3343i −0.829874 1.34469i
\(544\) 0 0
\(545\) 15.8167i 0.677513i
\(546\) 0 0
\(547\) 17.1892i 0.734955i −0.930032 0.367478i \(-0.880221\pi\)
0.930032 0.367478i \(-0.119779\pi\)
\(548\) 0 0
\(549\) −8.22122 + 16.3900i −0.350873 + 0.699510i
\(550\) 0 0
\(551\) 7.20096 0.306771
\(552\) 0 0
\(553\) −37.7496 −1.60528
\(554\) 0 0
\(555\) 17.6452 10.8898i 0.748999 0.462246i
\(556\) 0 0
\(557\) 39.7539i 1.68443i −0.539144 0.842214i \(-0.681252\pi\)
0.539144 0.842214i \(-0.318748\pi\)
\(558\) 0 0
\(559\) 12.2939i 0.519978i
\(560\) 0 0
\(561\) −7.81617 + 4.82376i −0.329999 + 0.203659i
\(562\) 0 0
\(563\) −8.43451 −0.355472 −0.177736 0.984078i \(-0.556877\pi\)
−0.177736 + 0.984078i \(0.556877\pi\)
\(564\) 0 0
\(565\) 15.0765 0.634272
\(566\) 0 0
\(567\) −25.8658 + 19.2962i −1.08626 + 0.810365i
\(568\) 0 0
\(569\) 28.4016i 1.19066i −0.803482 0.595329i \(-0.797022\pi\)
0.803482 0.595329i \(-0.202978\pi\)
\(570\) 0 0
\(571\) 16.3746i 0.685254i 0.939472 + 0.342627i \(0.111317\pi\)
−0.939472 + 0.342627i \(0.888683\pi\)
\(572\) 0 0
\(573\) 7.39980 + 11.9902i 0.309131 + 0.500900i
\(574\) 0 0
\(575\) 1.99574 0.0832280
\(576\) 0 0
\(577\) 36.4598 1.51784 0.758920 0.651184i \(-0.225727\pi\)
0.758920 + 0.651184i \(0.225727\pi\)
\(578\) 0 0
\(579\) −13.7057 22.2081i −0.569591 0.922936i
\(580\) 0 0
\(581\) 48.1154i 1.99616i
\(582\) 0 0
\(583\) 15.1821i 0.628777i
\(584\) 0 0
\(585\) −8.69574 4.36177i −0.359525 0.180337i
\(586\) 0 0
\(587\) 18.3627 0.757911 0.378955 0.925415i \(-0.376283\pi\)
0.378955 + 0.925415i \(0.376283\pi\)
\(588\) 0 0
\(589\) 1.42596 0.0587558
\(590\) 0 0
\(591\) −4.10085 + 2.53084i −0.168686 + 0.104105i
\(592\) 0 0
\(593\) 37.9486i 1.55836i 0.626798 + 0.779182i \(0.284365\pi\)
−0.626798 + 0.779182i \(0.715635\pi\)
\(594\) 0 0
\(595\) 31.9140i 1.30835i
\(596\) 0 0
\(597\) 18.3464 11.3225i 0.750869 0.463400i
\(598\) 0 0
\(599\) −6.14228 −0.250967 −0.125483 0.992096i \(-0.540048\pi\)
−0.125483 + 0.992096i \(0.540048\pi\)
\(600\) 0 0
\(601\) 27.2065 1.10978 0.554889 0.831924i \(-0.312761\pi\)
0.554889 + 0.831924i \(0.312761\pi\)
\(602\) 0 0
\(603\) 38.0722 + 19.0969i 1.55042 + 0.777688i
\(604\) 0 0
\(605\) 22.4689i 0.913490i
\(606\) 0 0
\(607\) 28.9061i 1.17326i 0.809855 + 0.586630i \(0.199546\pi\)
−0.809855 + 0.586630i \(0.800454\pi\)
\(608\) 0 0
\(609\) 23.4870 + 38.0571i 0.951742 + 1.54215i
\(610\) 0 0
\(611\) −0.168937 −0.00683444
\(612\) 0 0
\(613\) 24.3751 0.984502 0.492251 0.870453i \(-0.336174\pi\)
0.492251 + 0.870453i \(0.336174\pi\)
\(614\) 0 0
\(615\) −25.8778 41.9310i −1.04349 1.69082i
\(616\) 0 0
\(617\) 10.6419i 0.428429i −0.976787 0.214214i \(-0.931281\pi\)
0.976787 0.214214i \(-0.0687191\pi\)
\(618\) 0 0
\(619\) 2.83221i 0.113836i −0.998379 0.0569181i \(-0.981873\pi\)
0.998379 0.0569181i \(-0.0181274\pi\)
\(620\) 0 0
\(621\) −5.40445 + 0.476877i −0.216873 + 0.0191364i
\(622\) 0 0
\(623\) −15.4764 −0.620049
\(624\) 0 0
\(625\) −30.9035 −1.23614
\(626\) 0 0
\(627\) 2.30865 1.42479i 0.0921986 0.0569005i
\(628\) 0 0
\(629\) 15.4169i 0.614713i
\(630\) 0 0
\(631\) 25.1883i 1.00273i 0.865236 + 0.501365i \(0.167169\pi\)
−0.865236 + 0.501365i \(0.832831\pi\)
\(632\) 0 0
\(633\) 23.5369 14.5258i 0.935508 0.577350i
\(634\) 0 0
\(635\) −23.1124 −0.917187
\(636\) 0 0
\(637\) −7.22402 −0.286226
\(638\) 0 0
\(639\) 13.5659 27.0453i 0.536659 1.06990i
\(640\) 0 0
\(641\) 16.6362i 0.657091i 0.944488 + 0.328545i \(0.106558\pi\)
−0.944488 + 0.328545i \(0.893442\pi\)
\(642\) 0 0
\(643\) 40.2224i 1.58622i 0.609081 + 0.793108i \(0.291538\pi\)
−0.609081 + 0.793108i \(0.708462\pi\)
\(644\) 0 0
\(645\) −23.8349 38.6208i −0.938497 1.52069i
\(646\) 0 0
\(647\) −21.8899 −0.860580 −0.430290 0.902691i \(-0.641589\pi\)
−0.430290 + 0.902691i \(0.641589\pi\)
\(648\) 0 0
\(649\) −10.1962 −0.400236
\(650\) 0 0
\(651\) 4.65100 + 7.53624i 0.182287 + 0.295368i
\(652\) 0 0
\(653\) 2.25896i 0.0884001i −0.999023 0.0442000i \(-0.985926\pi\)
0.999023 0.0442000i \(-0.0140739\pi\)
\(654\) 0 0
\(655\) 22.0163i 0.860248i
\(656\) 0 0
\(657\) 13.1642 26.2444i 0.513583 1.02389i
\(658\) 0 0
\(659\) 17.0566 0.664430 0.332215 0.943204i \(-0.392204\pi\)
0.332215 + 0.943204i \(0.392204\pi\)
\(660\) 0 0
\(661\) 28.3209 1.10156 0.550778 0.834652i \(-0.314331\pi\)
0.550778 + 0.834652i \(0.314331\pi\)
\(662\) 0 0
\(663\) −6.15538 + 3.79880i −0.239055 + 0.147533i
\(664\) 0 0
\(665\) 9.42639i 0.365540i
\(666\) 0 0
\(667\) 7.51872i 0.291126i
\(668\) 0 0
\(669\) −6.23015 + 3.84495i −0.240872 + 0.148654i
\(670\) 0 0
\(671\) 9.57341 0.369577
\(672\) 0 0
\(673\) 33.0681 1.27468 0.637341 0.770582i \(-0.280034\pi\)
0.637341 + 0.770582i \(0.280034\pi\)
\(674\) 0 0
\(675\) −9.89345 + 0.872976i −0.380799 + 0.0336009i
\(676\) 0 0
\(677\) 23.9824i 0.921719i 0.887473 + 0.460859i \(0.152459\pi\)
−0.887473 + 0.460859i \(0.847541\pi\)
\(678\) 0 0
\(679\) 23.9176i 0.917872i
\(680\) 0 0
\(681\) 13.7724 + 22.3160i 0.527758 + 0.855151i
\(682\) 0 0
\(683\) −0.0577275 −0.00220888 −0.00110444 0.999999i \(-0.500352\pi\)
−0.00110444 + 0.999999i \(0.500352\pi\)
\(684\) 0 0
\(685\) −46.2237 −1.76612
\(686\) 0 0
\(687\) 10.3165 + 16.7164i 0.393600 + 0.637769i
\(688\) 0 0
\(689\) 11.9562i 0.455493i
\(690\) 0 0
\(691\) 27.4729i 1.04512i 0.852603 + 0.522558i \(0.175022\pi\)
−0.852603 + 0.522558i \(0.824978\pi\)
\(692\) 0 0
\(693\) 15.0600 + 7.55408i 0.572083 + 0.286956i
\(694\) 0 0
\(695\) 11.9301 0.452533
\(696\) 0 0
\(697\) −36.6357 −1.38768
\(698\) 0 0
\(699\) 6.72356 4.14945i 0.254308 0.156947i
\(700\) 0 0
\(701\) 50.3391i 1.90128i −0.310293 0.950641i \(-0.600427\pi\)
0.310293 0.950641i \(-0.399573\pi\)
\(702\) 0 0
\(703\) 4.55367i 0.171745i
\(704\) 0 0
\(705\) −0.530706 + 0.327526i −0.0199875 + 0.0123353i
\(706\) 0 0
\(707\) 59.4014 2.23402
\(708\) 0 0
\(709\) 21.9250 0.823410 0.411705 0.911317i \(-0.364933\pi\)
0.411705 + 0.911317i \(0.364933\pi\)
\(710\) 0 0
\(711\) 28.2318 + 14.1610i 1.05878 + 0.531080i
\(712\) 0 0
\(713\) 1.48889i 0.0557593i
\(714\) 0 0
\(715\) 5.07918i 0.189950i
\(716\) 0 0
\(717\) −14.2496 23.0894i −0.532163 0.862289i
\(718\) 0 0
\(719\) −5.74026 −0.214075 −0.107038 0.994255i \(-0.534137\pi\)
−0.107038 + 0.994255i \(0.534137\pi\)
\(720\) 0 0
\(721\) 1.12236 0.0417988
\(722\) 0 0
\(723\) −3.37003 5.46063i −0.125333 0.203083i
\(724\) 0 0
\(725\) 13.7639i 0.511177i
\(726\) 0 0
\(727\) 27.3443i 1.01414i 0.861904 + 0.507071i \(0.169272\pi\)
−0.861904 + 0.507071i \(0.830728\pi\)
\(728\) 0 0
\(729\) 26.5828 4.72803i 0.984548 0.175112i
\(730\) 0 0
\(731\) −33.7435 −1.24805
\(732\) 0 0
\(733\) −14.2323 −0.525684 −0.262842 0.964839i \(-0.584660\pi\)
−0.262842 + 0.964839i \(0.584660\pi\)
\(734\) 0 0
\(735\) −22.6939 + 14.0056i −0.837077 + 0.516603i
\(736\) 0 0
\(737\) 22.2379i 0.819144i
\(738\) 0 0
\(739\) 33.5861i 1.23548i 0.786381 + 0.617742i \(0.211952\pi\)
−0.786381 + 0.617742i \(0.788048\pi\)
\(740\) 0 0
\(741\) 1.81811 1.12205i 0.0667898 0.0412194i
\(742\) 0 0
\(743\) −3.96649 −0.145517 −0.0727583 0.997350i \(-0.523180\pi\)
−0.0727583 + 0.997350i \(0.523180\pi\)
\(744\) 0 0
\(745\) 37.8558 1.38693
\(746\) 0 0
\(747\) −18.0495 + 35.9840i −0.660397 + 1.31659i
\(748\) 0 0
\(749\) 52.8109i 1.92967i
\(750\) 0 0
\(751\) 29.8793i 1.09031i 0.838335 + 0.545156i \(0.183530\pi\)
−0.838335 + 0.545156i \(0.816470\pi\)
\(752\) 0 0
\(753\) 26.1823 + 42.4244i 0.954134 + 1.54603i
\(754\) 0 0
\(755\) −37.0207 −1.34732
\(756\) 0 0
\(757\) −51.4613 −1.87039 −0.935196 0.354130i \(-0.884777\pi\)
−0.935196 + 0.354130i \(0.884777\pi\)
\(758\) 0 0
\(759\) 1.48766 + 2.41053i 0.0539986 + 0.0874965i
\(760\) 0 0
\(761\) 4.79445i 0.173799i −0.996217 0.0868994i \(-0.972304\pi\)
0.996217 0.0868994i \(-0.0276959\pi\)
\(762\) 0 0
\(763\) 21.5723i 0.780968i
\(764\) 0 0
\(765\) −11.9719 + 23.8675i −0.432845 + 0.862931i
\(766\) 0 0
\(767\) −8.02970 −0.289936
\(768\) 0 0
\(769\) −9.98056 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(770\) 0 0
\(771\) −33.8505 + 20.8909i −1.21910 + 0.752367i
\(772\) 0 0
\(773\) 4.11463i 0.147993i −0.997259 0.0739964i \(-0.976425\pi\)
0.997259 0.0739964i \(-0.0235753\pi\)
\(774\) 0 0
\(775\) 2.72558i 0.0979057i
\(776\) 0 0
\(777\) −24.0662 + 14.8525i −0.863370 + 0.532830i
\(778\) 0 0
\(779\) 10.8210 0.387704
\(780\) 0 0
\(781\) −15.7972 −0.565267
\(782\) 0 0
\(783\) −3.28884 37.2725i −0.117534 1.33201i
\(784\) 0 0
\(785\) 53.7550i 1.91860i
\(786\) 0 0
\(787\) 27.5187i 0.980936i 0.871459 + 0.490468i \(0.163174\pi\)
−0.871459 + 0.490468i \(0.836826\pi\)
\(788\) 0 0
\(789\) 12.5109 + 20.2720i 0.445399 + 0.721701i
\(790\) 0 0
\(791\) −20.5627 −0.731124
\(792\) 0 0
\(793\) 7.53924 0.267726
\(794\) 0 0
\(795\) 23.1800 + 37.5597i 0.822111 + 1.33211i
\(796\) 0 0
\(797\) 40.2348i 1.42519i 0.701575 + 0.712596i \(0.252481\pi\)
−0.701575 + 0.712596i \(0.747519\pi\)
\(798\) 0 0
\(799\) 0.463686i 0.0164040i
\(800\) 0 0
\(801\) 11.5743 + 5.80567i 0.408959 + 0.205133i
\(802\) 0 0
\(803\) −15.3293 −0.540961
\(804\) 0 0
\(805\) −9.84235 −0.346897
\(806\) 0 0
\(807\) −4.32432 + 2.66876i −0.152223 + 0.0939448i
\(808\) 0 0
\(809\) 30.6395i 1.07723i −0.842553 0.538614i \(-0.818948\pi\)
0.842553 0.538614i \(-0.181052\pi\)
\(810\) 0 0
\(811\) 48.3574i 1.69806i −0.528345 0.849030i \(-0.677187\pi\)
0.528345 0.849030i \(-0.322813\pi\)
\(812\) 0 0
\(813\) 31.4042 19.3811i 1.10139 0.679726i
\(814\) 0 0
\(815\) −33.9557 −1.18942
\(816\) 0 0
\(817\) 9.96678 0.348693
\(818\) 0 0
\(819\) 11.8601 + 5.94898i 0.414424 + 0.207874i
\(820\) 0 0
\(821\) 17.2475i 0.601944i 0.953633 + 0.300972i \(0.0973110\pi\)
−0.953633 + 0.300972i \(0.902689\pi\)
\(822\) 0 0
\(823\) 28.4658i 0.992255i −0.868250 0.496128i \(-0.834755\pi\)
0.868250 0.496128i \(-0.165245\pi\)
\(824\) 0 0
\(825\) 2.72333 + 4.41274i 0.0948141 + 0.153632i
\(826\) 0 0
\(827\) 7.03686 0.244695 0.122348 0.992487i \(-0.460958\pi\)
0.122348 + 0.992487i \(0.460958\pi\)
\(828\) 0 0
\(829\) 26.1522 0.908303 0.454151 0.890924i \(-0.349942\pi\)
0.454151 + 0.890924i \(0.349942\pi\)
\(830\) 0 0
\(831\) 5.35844 + 8.68254i 0.185882 + 0.301194i
\(832\) 0 0
\(833\) 19.8280i 0.687000i
\(834\) 0 0
\(835\) 39.0948i 1.35293i
\(836\) 0 0
\(837\) −0.651270 7.38085i −0.0225112 0.255120i
\(838\) 0 0
\(839\) 0.634411 0.0219023 0.0109511 0.999940i \(-0.496514\pi\)
0.0109511 + 0.999940i \(0.496514\pi\)
\(840\) 0 0
\(841\) −22.8538 −0.788062
\(842\) 0 0
\(843\) −45.4672 + 28.0601i −1.56597 + 0.966442i
\(844\) 0 0
\(845\) 30.1764i 1.03810i
\(846\) 0 0
\(847\) 30.6451i 1.05298i
\(848\) 0 0
\(849\) 38.2367 23.5978i 1.31228 0.809876i
\(850\) 0 0
\(851\) −4.75461 −0.162986
\(852\) 0 0
\(853\) 16.9254 0.579516 0.289758 0.957100i \(-0.406425\pi\)
0.289758 + 0.957100i \(0.406425\pi\)
\(854\) 0 0
\(855\) 3.53612 7.04971i 0.120933 0.241095i
\(856\) 0 0
\(857\) 33.9804i 1.16075i −0.814350 0.580374i \(-0.802907\pi\)
0.814350 0.580374i \(-0.197093\pi\)
\(858\) 0 0
\(859\) 16.3639i 0.558330i −0.960243 0.279165i \(-0.909942\pi\)
0.960243 0.279165i \(-0.0900576\pi\)
\(860\) 0 0
\(861\) 35.2945 + 57.1893i 1.20283 + 1.94901i
\(862\) 0 0
\(863\) −12.2331 −0.416419 −0.208210 0.978084i \(-0.566764\pi\)
−0.208210 + 0.978084i \(0.566764\pi\)
\(864\) 0 0
\(865\) −35.0190 −1.19068
\(866\) 0 0
\(867\) −5.03737 8.16230i −0.171078 0.277206i
\(868\) 0 0
\(869\) 16.4902i 0.559391i
\(870\) 0 0
\(871\) 17.5128i 0.593398i
\(872\) 0 0
\(873\) −8.97220 + 17.8872i −0.303663 + 0.605390i
\(874\) 0 0
\(875\) 29.1144 0.984247
\(876\) 0 0
\(877\) 8.59003 0.290065 0.145032 0.989427i \(-0.453671\pi\)
0.145032 + 0.989427i \(0.453671\pi\)
\(878\) 0 0
\(879\) 48.8034 30.1191i 1.64610 1.01589i
\(880\) 0 0
\(881\) 20.1146i 0.677677i −0.940845 0.338838i \(-0.889966\pi\)
0.940845 0.338838i \(-0.110034\pi\)
\(882\) 0 0
\(883\) 32.1457i 1.08179i 0.841091 + 0.540894i \(0.181914\pi\)
−0.841091 + 0.540894i \(0.818086\pi\)
\(884\) 0 0
\(885\) −25.2249 + 15.5676i −0.847926 + 0.523299i
\(886\) 0 0
\(887\) −10.2485 −0.344112 −0.172056 0.985087i \(-0.555041\pi\)
−0.172056 + 0.985087i \(0.555041\pi\)
\(888\) 0 0
\(889\) 31.5228 1.05724
\(890\) 0 0
\(891\) −8.42917 11.2989i −0.282388 0.378529i
\(892\) 0 0
\(893\) 0.136958i 0.00458313i
\(894\) 0 0
\(895\) 1.49185i 0.0498669i
\(896\) 0 0
\(897\) 1.17156 + 1.89833i 0.0391172 + 0.0633835i
\(898\) 0 0
\(899\) −10.2683 −0.342467
\(900\) 0 0
\(901\) 32.8165 1.09328
\(902\) 0 0
\(903\) 32.5082 + 52.6745i 1.08180 + 1.75290i
\(904\) 0 0
\(905\) 55.8882i 1.85779i
\(906\) 0 0
\(907\) 32.9063i 1.09264i −0.837578 0.546318i \(-0.816029\pi\)
0.837578 0.546318i \(-0.183971\pi\)
\(908\) 0 0
\(909\) −44.4245 22.2833i −1.47347 0.739089i
\(910\) 0 0
\(911\) −49.4178 −1.63728 −0.818642 0.574304i \(-0.805273\pi\)
−0.818642 + 0.574304i \(0.805273\pi\)
\(912\) 0 0
\(913\) 21.0182 0.695602
\(914\) 0 0
\(915\) 23.6842 14.6167i 0.782974 0.483214i
\(916\) 0 0
\(917\) 30.0279i 0.991607i
\(918\) 0 0
\(919\) 6.63018i 0.218709i 0.994003 + 0.109355i \(0.0348784\pi\)
−0.994003 + 0.109355i \(0.965122\pi\)
\(920\) 0 0
\(921\) −1.26375 + 0.779923i −0.0416418 + 0.0256993i
\(922\) 0 0
\(923\) −12.4406 −0.409486
\(924\) 0 0
\(925\) −8.70385 −0.286181
\(926\) 0 0
\(927\) −0.839377 0.421030i −0.0275688 0.0138285i
\(928\) 0 0
\(929\) 39.5553i 1.29777i 0.760888 + 0.648884i \(0.224764\pi\)
−0.760888 + 0.648884i \(0.775236\pi\)
\(930\) 0 0
\(931\) 5.85657i 0.191941i
\(932\) 0 0
\(933\) 17.1329 + 27.7613i 0.560907 + 0.908864i
\(934\) 0 0
\(935\) 13.9410 0.455919
\(936\) 0 0
\(937\) 24.2158 0.791096 0.395548 0.918445i \(-0.370555\pi\)
0.395548 + 0.918445i \(0.370555\pi\)
\(938\) 0 0
\(939\) −25.8697 41.9179i −0.844225 1.36794i
\(940\) 0 0
\(941\) 56.6132i 1.84554i 0.385354 + 0.922769i \(0.374079\pi\)
−0.385354 + 0.922769i \(0.625921\pi\)
\(942\) 0 0
\(943\) 11.2985i 0.367931i
\(944\) 0 0
\(945\) 48.7914 4.30524i 1.58718 0.140050i
\(946\) 0 0
\(947\) −5.31711 −0.172783 −0.0863914 0.996261i \(-0.527534\pi\)
−0.0863914 + 0.996261i \(0.527534\pi\)
\(948\) 0 0
\(949\) −12.0721 −0.391878
\(950\) 0 0
\(951\) −19.0971 + 11.7858i −0.619267 + 0.382181i
\(952\) 0 0
\(953\) 20.0357i 0.649021i −0.945882 0.324511i \(-0.894800\pi\)
0.945882 0.324511i \(-0.105200\pi\)
\(954\) 0 0
\(955\) 21.3859i 0.692032i
\(956\) 0 0
\(957\) −16.6245 + 10.2598i −0.537394 + 0.331653i
\(958\) 0 0
\(959\) 63.0441 2.03580
\(960\) 0 0
\(961\) 28.9666 0.934407
\(962\) 0 0
\(963\) −19.8110 + 39.4957i −0.638400 + 1.27273i
\(964\) 0 0
\(965\) 39.6105i 1.27511i
\(966\) 0 0
\(967\) 14.4593i 0.464979i 0.972599 + 0.232490i \(0.0746872\pi\)
−0.972599 + 0.232490i \(0.925313\pi\)
\(968\) 0 0
\(969\) −3.07972 4.99022i −0.0989348 0.160309i
\(970\) 0 0
\(971\) −2.01730 −0.0647382 −0.0323691 0.999476i \(-0.510305\pi\)
−0.0323691 + 0.999476i \(0.510305\pi\)
\(972\) 0 0
\(973\) −16.2713 −0.521634
\(974\) 0 0
\(975\) 2.14467 + 3.47511i 0.0686844 + 0.111293i
\(976\) 0 0
\(977\) 3.17018i 0.101423i −0.998713 0.0507116i \(-0.983851\pi\)
0.998713 0.0507116i \(-0.0161489\pi\)
\(978\) 0 0
\(979\) 6.76056i 0.216068i
\(980\) 0 0
\(981\) −8.09241 + 16.1332i −0.258371 + 0.515095i
\(982\) 0 0
\(983\) −13.9228 −0.444067 −0.222034 0.975039i \(-0.571269\pi\)
−0.222034 + 0.975039i \(0.571269\pi\)
\(984\) 0 0
\(985\) 7.31431 0.233053
\(986\) 0 0
\(987\) 0.723825 0.446710i 0.0230396 0.0142189i
\(988\) 0 0
\(989\) 10.4066i 0.330910i
\(990\) 0 0
\(991\) 9.88678i 0.314064i 0.987594 + 0.157032i \(0.0501926\pi\)
−0.987594 + 0.157032i \(0.949807\pi\)
\(992\) 0 0
\(993\) 9.52013 5.87536i 0.302112 0.186449i
\(994\) 0 0
\(995\) −32.7228 −1.03738
\(996\) 0 0
\(997\) 55.2982 1.75131 0.875657 0.482934i \(-0.160429\pi\)
0.875657 + 0.482934i \(0.160429\pi\)
\(998\) 0 0
\(999\) 23.5700 2.07976i 0.745721 0.0658008i
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.d.b.191.7 24
3.2 odd 2 inner 912.2.d.b.191.17 yes 24
4.3 odd 2 inner 912.2.d.b.191.18 yes 24
12.11 even 2 inner 912.2.d.b.191.8 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.d.b.191.7 24 1.1 even 1 trivial
912.2.d.b.191.8 yes 24 12.11 even 2 inner
912.2.d.b.191.17 yes 24 3.2 odd 2 inner
912.2.d.b.191.18 yes 24 4.3 odd 2 inner