Properties

Label 380.2.d.a.379.4
Level $380$
Weight $2$
Character 380.379
Analytic conductor $3.034$
Analytic rank $0$
Dimension $16$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(379,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 13x^{8} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 379.4
Root \(-1.13320 + 0.846083i\) of defining polynomial
Character \(\chi\) \(=\) 380.379
Dual form 380.2.d.a.379.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 - 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 + 1.72677i) q^{6} +(0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +O(q^{10})\) \(q+(-1.13320 + 0.846083i) q^{2} -1.52380i q^{3} +(0.568286 - 1.91756i) q^{4} +2.23607 q^{5} +(1.28926 + 1.72677i) q^{6} +(0.978437 + 2.65380i) q^{8} +0.678024 q^{9} +(-2.53391 + 1.89190i) q^{10} +2.92978i q^{11} +(-2.92199 - 0.865956i) q^{12} +2.42350 q^{13} -3.40733i q^{15} +(-3.35410 - 2.17945i) q^{16} +(-0.768338 + 0.573665i) q^{18} -4.35890i q^{19} +(1.27073 - 4.28780i) q^{20} +(-2.47884 - 3.32002i) q^{22} +(4.04387 - 1.49094i) q^{24} +5.00000 q^{25} +(-2.74631 + 2.05048i) q^{26} -5.60458i q^{27} +(2.88288 + 3.86118i) q^{30} +(5.64487 - 0.368097i) q^{32} +4.46440 q^{33} +(0.385312 - 1.30016i) q^{36} +0.802795 q^{37} +(3.68799 + 4.93951i) q^{38} -3.69294i q^{39} +(2.18785 + 5.93408i) q^{40} +(5.61803 + 1.66495i) q^{44} +1.51611 q^{45} +(-3.32105 + 5.11099i) q^{48} -7.00000 q^{49} +(-5.66600 + 4.23042i) q^{50} +(1.37724 - 4.64722i) q^{52} +13.2466 q^{53} +(4.74195 + 6.35112i) q^{54} +6.55118i q^{55} -6.64210 q^{57} +(-6.53377 - 1.93634i) q^{60} -15.5804 q^{61} +(-6.08532 + 5.19315i) q^{64} +5.41911 q^{65} +(-5.05906 + 3.77726i) q^{66} +3.36134i q^{67} +(0.663404 + 1.79934i) q^{72} +(-0.909728 + 0.679231i) q^{74} -7.61901i q^{75} +(-8.35847 - 2.47710i) q^{76} +(3.12453 + 4.18484i) q^{78} +(-7.50000 - 4.87340i) q^{80} -6.50621 q^{81} +(-7.77505 + 2.86660i) q^{88} +(-1.71806 + 1.28275i) q^{90} -9.74679i q^{95} +(-0.560907 - 8.60166i) q^{96} -19.4685 q^{97} +(7.93240 - 5.92258i) q^{98} +1.98646i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 48 q^{9} + 8 q^{24} + 80 q^{25} + 24 q^{26} - 40 q^{30} - 56 q^{36} + 72 q^{44} - 112 q^{49} + 88 q^{54} - 104 q^{66} - 120 q^{80} + 144 q^{81} + 136 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13320 + 0.846083i −0.801294 + 0.598271i
\(3\) 1.52380i 0.879768i −0.898055 0.439884i \(-0.855020\pi\)
0.898055 0.439884i \(-0.144980\pi\)
\(4\) 0.568286 1.91756i 0.284143 0.958782i
\(5\) 2.23607 1.00000
\(6\) 1.28926 + 1.72677i 0.526340 + 0.704953i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0.978437 + 2.65380i 0.345930 + 0.938260i
\(9\) 0.678024 0.226008
\(10\) −2.53391 + 1.89190i −0.801294 + 0.598271i
\(11\) 2.92978i 0.883361i 0.897172 + 0.441680i \(0.145618\pi\)
−0.897172 + 0.441680i \(0.854382\pi\)
\(12\) −2.92199 0.865956i −0.843506 0.249980i
\(13\) 2.42350 0.672158 0.336079 0.941834i \(-0.390899\pi\)
0.336079 + 0.941834i \(0.390899\pi\)
\(14\) 0 0
\(15\) 3.40733i 0.879768i
\(16\) −3.35410 2.17945i −0.838525 0.544862i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −0.768338 + 0.573665i −0.181099 + 0.135214i
\(19\) 4.35890i 1.00000i
\(20\) 1.27073 4.28780i 0.284143 0.958782i
\(21\) 0 0
\(22\) −2.47884 3.32002i −0.528489 0.707832i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 4.04387 1.49094i 0.825452 0.304338i
\(25\) 5.00000 1.00000
\(26\) −2.74631 + 2.05048i −0.538596 + 0.402133i
\(27\) 5.60458i 1.07860i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 2.88288 + 3.86118i 0.526340 + 0.704953i
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.64487 0.368097i 0.997881 0.0650709i
\(33\) 4.46440 0.777153
\(34\) 0 0
\(35\) 0 0
\(36\) 0.385312 1.30016i 0.0642186 0.216693i
\(37\) 0.802795 0.131979 0.0659893 0.997820i \(-0.478980\pi\)
0.0659893 + 0.997820i \(0.478980\pi\)
\(38\) 3.68799 + 4.93951i 0.598271 + 0.801294i
\(39\) 3.69294i 0.591343i
\(40\) 2.18785 + 5.93408i 0.345930 + 0.938260i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 5.61803 + 1.66495i 0.846950 + 0.251001i
\(45\) 1.51611 0.226008
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −3.32105 + 5.11099i −0.479353 + 0.737708i
\(49\) −7.00000 −1.00000
\(50\) −5.66600 + 4.23042i −0.801294 + 0.598271i
\(51\) 0 0
\(52\) 1.37724 4.64722i 0.190989 0.644453i
\(53\) 13.2466 1.81956 0.909781 0.415090i \(-0.136250\pi\)
0.909781 + 0.415090i \(0.136250\pi\)
\(54\) 4.74195 + 6.35112i 0.645297 + 0.864278i
\(55\) 6.55118i 0.883361i
\(56\) 0 0
\(57\) −6.64210 −0.879768
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −6.53377 1.93634i −0.843506 0.249980i
\(61\) −15.5804 −1.99486 −0.997430 0.0716414i \(-0.977176\pi\)
−0.997430 + 0.0716414i \(0.977176\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −6.08532 + 5.19315i −0.760665 + 0.649144i
\(65\) 5.41911 0.672158
\(66\) −5.05906 + 3.77726i −0.622728 + 0.464948i
\(67\) 3.36134i 0.410653i 0.978694 + 0.205326i \(0.0658256\pi\)
−0.978694 + 0.205326i \(0.934174\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.663404 + 1.79934i 0.0781829 + 0.212055i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −0.909728 + 0.679231i −0.105754 + 0.0789591i
\(75\) 7.61901i 0.879768i
\(76\) −8.35847 2.47710i −0.958782 0.284143i
\(77\) 0 0
\(78\) 3.12453 + 4.18484i 0.353784 + 0.473839i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −7.50000 4.87340i −0.838525 0.544862i
\(81\) −6.50621 −0.722912
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −7.77505 + 2.86660i −0.828823 + 0.305581i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.71806 + 1.28275i −0.181099 + 0.135214i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.74679i 1.00000i
\(96\) −0.560907 8.60166i −0.0572473 0.877904i
\(97\) −19.4685 −1.97673 −0.988364 0.152109i \(-0.951394\pi\)
−0.988364 + 0.152109i \(0.951394\pi\)
\(98\) 7.93240 5.92258i 0.801294 0.598271i
\(99\) 1.98646i 0.199647i
\(100\) 2.84143 9.58782i 0.284143 0.958782i
\(101\) −0.868264 −0.0863955 −0.0431977 0.999067i \(-0.513755\pi\)
−0.0431977 + 0.999067i \(0.513755\pi\)
\(102\) 0 0
\(103\) 16.9447i 1.66961i −0.550548 0.834803i \(-0.685581\pi\)
0.550548 0.834803i \(-0.314419\pi\)
\(104\) 2.37124 + 6.43149i 0.232519 + 0.630659i
\(105\) 0 0
\(106\) −15.0111 + 11.2077i −1.45800 + 1.08859i
\(107\) 16.9906i 1.64255i 0.570534 + 0.821274i \(0.306736\pi\)
−0.570534 + 0.821274i \(0.693264\pi\)
\(108\) −10.7471 3.18501i −1.03414 0.306477i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −5.54284 7.42380i −0.528489 0.707832i
\(111\) 1.22330i 0.116111i
\(112\) 0 0
\(113\) −16.3361 −1.53677 −0.768386 0.639987i \(-0.778940\pi\)
−0.768386 + 0.639987i \(0.778940\pi\)
\(114\) 7.52683 5.61977i 0.704953 0.526340i
\(115\) 0 0
\(116\) 0 0
\(117\) 1.64319 0.151913
\(118\) 0 0
\(119\) 0 0
\(120\) 9.04237 3.33385i 0.825452 0.304338i
\(121\) 2.41641 0.219673
\(122\) 17.6557 13.1823i 1.59847 1.19347i
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 19.7066i 1.74868i 0.485315 + 0.874339i \(0.338705\pi\)
−0.485315 + 0.874339i \(0.661295\pi\)
\(128\) 2.50205 11.0336i 0.221152 0.975239i
\(129\) 0 0
\(130\) −6.14094 + 4.58502i −0.538596 + 0.402133i
\(131\) 19.4936i 1.70316i 0.524222 + 0.851581i \(0.324356\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 2.53706 8.56078i 0.220823 0.745120i
\(133\) 0 0
\(134\) −2.84397 3.80907i −0.245682 0.329054i
\(135\) 12.5322i 1.07860i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 21.8917i 1.85683i 0.371546 + 0.928414i \(0.378828\pi\)
−0.371546 + 0.928414i \(0.621172\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.10031i 0.593758i
\(144\) −2.27416 1.47772i −0.189514 0.123143i
\(145\) 0 0
\(146\) 0 0
\(147\) 10.6666i 0.879768i
\(148\) 0.456217 1.53941i 0.0375008 0.126539i
\(149\) 22.9364 1.87902 0.939512 0.342516i \(-0.111279\pi\)
0.939512 + 0.342516i \(0.111279\pi\)
\(150\) 6.44632 + 8.63387i 0.526340 + 0.704953i
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 11.5677 4.26491i 0.938260 0.345930i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −7.08144 2.09864i −0.566969 0.168026i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 20.1852i 1.60079i
\(160\) 12.6223 0.823090i 0.997881 0.0650709i
\(161\) 0 0
\(162\) 7.37284 5.50480i 0.579265 0.432498i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 9.98271 0.777153
\(166\) 0 0
\(167\) 25.8018i 1.99660i 0.0582442 + 0.998302i \(0.481450\pi\)
−0.0582442 + 0.998302i \(0.518550\pi\)
\(168\) 0 0
\(169\) −7.12665 −0.548204
\(170\) 0 0
\(171\) 2.95544i 0.226008i
\(172\) 0 0
\(173\) −25.4017 −1.93126 −0.965628 0.259928i \(-0.916301\pi\)
−0.965628 + 0.259928i \(0.916301\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.38530 9.82677i 0.481310 0.740721i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0.861583 2.90724i 0.0642186 0.216693i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 23.7414i 1.75501i
\(184\) 0 0
\(185\) 1.79510 0.131979
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 8.24660 + 11.0451i 0.598271 + 0.801294i
\(191\) 26.1534i 1.89239i −0.323592 0.946197i \(-0.604891\pi\)
0.323592 0.946197i \(-0.395109\pi\)
\(192\) 7.91334 + 9.27283i 0.571096 + 0.669209i
\(193\) 17.8629 1.28580 0.642901 0.765950i \(-0.277731\pi\)
0.642901 + 0.765950i \(0.277731\pi\)
\(194\) 22.0617 16.4720i 1.58394 1.18262i
\(195\) 8.25766i 0.591343i
\(196\) −3.97800 + 13.4229i −0.284143 + 0.958782i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −1.68071 2.25106i −0.119443 0.159976i
\(199\) 8.71780i 0.617988i 0.951064 + 0.308994i \(0.0999924\pi\)
−0.951064 + 0.308994i \(0.900008\pi\)
\(200\) 4.89218 + 13.2690i 0.345930 + 0.938260i
\(201\) 5.12202 0.361279
\(202\) 0.983917 0.734624i 0.0692281 0.0516879i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 14.3366 + 19.2017i 0.998878 + 1.33785i
\(207\) 0 0
\(208\) −8.12866 5.28189i −0.563621 0.366234i
\(209\) 12.7706 0.883361
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 7.52786 25.4012i 0.517016 1.74456i
\(213\) 0 0
\(214\) −14.3755 19.2538i −0.982689 1.31616i
\(215\) 0 0
\(216\) 14.8735 5.48373i 1.01201 0.373121i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 12.5623 + 3.72294i 0.846950 + 0.251001i
\(221\) 0 0
\(222\) 1.03501 + 1.38625i 0.0694657 + 0.0930387i
\(223\) 28.8494i 1.93190i −0.258728 0.965950i \(-0.583303\pi\)
0.258728 0.965950i \(-0.416697\pi\)
\(224\) 0 0
\(225\) 3.39012 0.226008
\(226\) 18.5121 13.8217i 1.23140 0.919406i
\(227\) 3.45331i 0.229205i 0.993411 + 0.114602i \(0.0365594\pi\)
−0.993411 + 0.114602i \(0.963441\pi\)
\(228\) −3.77461 + 12.7367i −0.249980 + 0.843506i
\(229\) −6.48779 −0.428725 −0.214362 0.976754i \(-0.568767\pi\)
−0.214362 + 0.976754i \(0.568767\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −1.86207 + 1.39028i −0.121727 + 0.0908853i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.4936i 1.26094i −0.776215 0.630468i \(-0.782863\pi\)
0.776215 0.630468i \(-0.217137\pi\)
\(240\) −7.42610 + 11.4285i −0.479353 + 0.737708i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −2.73827 + 2.04448i −0.176023 + 0.131424i
\(243\) 6.89957i 0.442608i
\(244\) −8.85410 + 29.8763i −0.566826 + 1.91264i
\(245\) −15.6525 −1.00000
\(246\) 0 0
\(247\) 10.5638i 0.672158i
\(248\) 0 0
\(249\) 0 0
\(250\) −12.6696 + 9.45950i −0.801294 + 0.598271i
\(251\) 26.1534i 1.65079i 0.564557 + 0.825394i \(0.309047\pi\)
−0.564557 + 0.825394i \(0.690953\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −16.6734 22.3315i −1.04618 1.40121i
\(255\) 0 0
\(256\) 6.50000 + 14.6202i 0.406250 + 0.913762i
\(257\) −31.9123 −1.99064 −0.995318 0.0966558i \(-0.969185\pi\)
−0.995318 + 0.0966558i \(0.969185\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3.07960 10.3915i 0.190989 0.644453i
\(261\) 0 0
\(262\) −16.4932 22.0901i −1.01895 1.36473i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 4.36814 + 11.8476i 0.268840 + 0.729172i
\(265\) 29.6203 1.81956
\(266\) 0 0
\(267\) 0 0
\(268\) 6.44558 + 1.91020i 0.393727 + 0.116684i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 10.6033 + 14.2015i 0.645297 + 0.864278i
\(271\) 24.1298i 1.46578i −0.680345 0.732892i \(-0.738170\pi\)
0.680345 0.732892i \(-0.261830\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.6489i 0.883361i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −18.5222 24.8077i −1.11089 1.48787i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) −14.8522 −0.879768
\(286\) −6.00746 8.04608i −0.355228 0.475774i
\(287\) 0 0
\(288\) 3.82736 0.249579i 0.225529 0.0147066i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 29.6662i 1.73906i
\(292\) 0 0
\(293\) −24.0848 −1.40705 −0.703525 0.710670i \(-0.748392\pi\)
−0.703525 + 0.710670i \(0.748392\pi\)
\(294\) −9.02485 12.0874i −0.526340 0.704953i
\(295\) 0 0
\(296\) 0.785484 + 2.13046i 0.0456553 + 0.123830i
\(297\) 16.4202 0.952796
\(298\) −25.9915 + 19.4061i −1.50565 + 1.12417i
\(299\) 0 0
\(300\) −14.6099 4.32978i −0.843506 0.249980i
\(301\) 0 0
\(302\) 0 0
\(303\) 1.32306i 0.0760080i
\(304\) −9.50000 + 14.6202i −0.544862 + 0.838525i
\(305\) −34.8387 −1.99486
\(306\) 0 0
\(307\) 23.8053i 1.35864i −0.733842 0.679320i \(-0.762275\pi\)
0.733842 0.679320i \(-0.237725\pi\)
\(308\) 0 0
\(309\) −25.8203 −1.46887
\(310\) 0 0
\(311\) 31.3726i 1.77898i −0.456955 0.889490i \(-0.651060\pi\)
0.456955 0.889490i \(-0.348940\pi\)
\(312\) 9.80032 3.61330i 0.554834 0.204563i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.8607 0.609998 0.304999 0.952353i \(-0.401344\pi\)
0.304999 + 0.952353i \(0.401344\pi\)
\(318\) 17.0784 + 22.8739i 0.957708 + 1.28270i
\(319\) 0 0
\(320\) −13.6072 + 11.6122i −0.760665 + 0.649144i
\(321\) 25.8904 1.44506
\(322\) 0 0
\(323\) 0 0
\(324\) −3.69739 + 12.4761i −0.205410 + 0.693115i
\(325\) 12.1175 0.672158
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) −11.3124 + 8.44620i −0.622728 + 0.464948i
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0.544315 0.0298283
\(334\) −21.8305 29.2386i −1.19451 1.59987i
\(335\) 7.51618i 0.410653i
\(336\) 0 0
\(337\) −24.7733 −1.34949 −0.674744 0.738052i \(-0.735746\pi\)
−0.674744 + 0.738052i \(0.735746\pi\)
\(338\) 8.07592 6.02974i 0.439272 0.327975i
\(339\) 24.8930i 1.35200i
\(340\) 0 0
\(341\) 0 0
\(342\) 2.50055 + 3.34911i 0.135214 + 0.181099i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 28.7852 21.4920i 1.54750 1.15541i
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −13.4164 −0.718164 −0.359082 0.933306i \(-0.616910\pi\)
−0.359082 + 0.933306i \(0.616910\pi\)
\(350\) 0 0
\(351\) 13.5827i 0.724991i
\(352\) 1.07844 + 16.5382i 0.0574811 + 0.881489i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.5131i 1.34653i 0.739401 + 0.673265i \(0.235109\pi\)
−0.739401 + 0.673265i \(0.764891\pi\)
\(360\) 1.48342 + 4.02345i 0.0781829 + 0.212055i
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 3.68213i 0.193262i
\(364\) 0 0
\(365\) 0 0
\(366\) −20.0872 26.9038i −1.04997 1.40628i
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −2.03421 + 1.51881i −0.105754 + 0.0789591i
\(371\) 0 0
\(372\) 0 0
\(373\) 38.1342 1.97452 0.987258 0.159130i \(-0.0508689\pi\)
0.987258 + 0.159130i \(0.0508689\pi\)
\(374\) 0 0
\(375\) 17.0366i 0.879768i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) −18.6901 5.53897i −0.958782 0.284143i
\(381\) 30.0290 1.53843
\(382\) 22.1280 + 29.6370i 1.13216 + 1.51636i
\(383\) 3.31535i 0.169407i −0.996406 0.0847033i \(-0.973006\pi\)
0.996406 0.0847033i \(-0.0269942\pi\)
\(384\) −16.8130 3.81263i −0.857984 0.194562i
\(385\) 0 0
\(386\) −20.2423 + 15.1135i −1.03030 + 0.769258i
\(387\) 0 0
\(388\) −11.0637 + 37.3321i −0.561673 + 1.89525i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 6.98666 + 9.35758i 0.353784 + 0.473839i
\(391\) 0 0
\(392\) −6.84906 18.5766i −0.345930 0.938260i
\(393\) 29.7044 1.49839
\(394\) 0 0
\(395\) 0 0
\(396\) 3.80916 + 1.12888i 0.191418 + 0.0567282i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −7.37598 9.87901i −0.369725 0.495190i
\(399\) 0 0
\(400\) −16.7705 10.8972i −0.838525 0.544862i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −5.80427 + 4.33365i −0.289491 + 0.216143i
\(403\) 0 0
\(404\) −0.493422 + 1.66495i −0.0245487 + 0.0828344i
\(405\) −14.5483 −0.722912
\(406\) 0 0
\(407\) 2.35201i 0.116585i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −32.4925 9.62941i −1.60079 0.474407i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 13.6803 0.892083i 0.670733 0.0437379i
\(417\) 33.3586 1.63358
\(418\) −14.4716 + 10.8050i −0.707832 + 0.528489i
\(419\) 19.4936i 0.952324i 0.879358 + 0.476162i \(0.157972\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 12.9610 + 35.1539i 0.629440 + 1.70722i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 32.5807 + 9.65555i 1.57485 + 0.466719i
\(429\) 10.8195 0.522369
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −12.2149 + 18.7983i −0.587690 + 0.904436i
\(433\) 39.3143 1.88932 0.944662 0.328044i \(-0.106389\pi\)
0.944662 + 0.328044i \(0.106389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −17.3855 + 6.40992i −0.828823 + 0.305581i
\(441\) −4.74617 −0.226008
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −2.34576 0.695185i −0.111325 0.0329920i
\(445\) 0 0
\(446\) 24.4090 + 32.6922i 1.15580 + 1.54802i
\(447\) 34.9506i 1.65311i
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −3.84169 + 2.86833i −0.181099 + 0.135214i
\(451\) 0 0
\(452\) −9.28358 + 31.3255i −0.436663 + 1.47343i
\(453\) 0 0
\(454\) −2.92179 3.91330i −0.137126 0.183660i
\(455\) 0 0
\(456\) −6.49888 17.6268i −0.304338 0.825452i
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 7.35196 5.48921i 0.343535 0.256494i
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0.933803 3.15093i 0.0431651 0.145652i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21.7945i 1.00000i
\(476\) 0 0
\(477\) 8.98152 0.411236
\(478\) 16.4932 + 22.0901i 0.754381 + 1.01038i
\(479\) 43.0918i 1.96891i −0.175630 0.984456i \(-0.556196\pi\)
0.175630 0.984456i \(-0.443804\pi\)
\(480\) −1.25423 19.2339i −0.0572473 0.877904i
\(481\) 1.94557 0.0887105
\(482\) 0 0
\(483\) 0 0
\(484\) 1.37321 4.63362i 0.0624187 0.210619i
\(485\) −43.5329 −1.97673
\(486\) 5.83761 + 7.81860i 0.264800 + 0.354659i
\(487\) 44.1113i 1.99887i −0.0335531 0.999437i \(-0.510682\pi\)
0.0335531 0.999437i \(-0.489318\pi\)
\(488\) −15.2444 41.3472i −0.690082 1.87170i
\(489\) 0 0
\(490\) 17.7374 13.2433i 0.801294 0.598271i
\(491\) 26.1534i 1.18029i 0.807299 + 0.590143i \(0.200929\pi\)
−0.807299 + 0.590143i \(0.799071\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 8.93785 + 11.9709i 0.402133 + 0.538596i
\(495\) 4.44186i 0.199647i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.31303i 0.193078i −0.995329 0.0965389i \(-0.969223\pi\)
0.995329 0.0965389i \(-0.0307772\pi\)
\(500\) 6.35363 21.4390i 0.284143 0.958782i
\(501\) 39.3169 1.75655
\(502\) −22.1280 29.6370i −0.987619 1.32277i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −1.94150 −0.0863955
\(506\) 0 0
\(507\) 10.8596i 0.482292i
\(508\) 37.7887 + 11.1990i 1.67660 + 0.496875i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −19.7357 11.0681i −0.872203 0.489144i
\(513\) −24.4298 −1.07860
\(514\) 36.1631 27.0005i 1.59508 1.19094i
\(515\) 37.8894i 1.66961i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 38.7072i 1.69906i
\(520\) 5.30226 + 14.3812i 0.232519 + 0.630659i
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 43.9846i 1.92331i −0.274256 0.961657i \(-0.588432\pi\)
0.274256 0.961657i \(-0.411568\pi\)
\(524\) 37.3802 + 11.0779i 1.63296 + 0.483942i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −14.9741 9.72994i −0.651662 0.423441i
\(529\) −23.0000 −1.00000
\(530\) −33.5657 + 25.0612i −1.45800 + 1.08859i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 37.9922i 1.64255i
\(536\) −8.92033 + 3.28886i −0.385299 + 0.142057i
\(537\) 0 0
\(538\) 0 0
\(539\) 20.5084i 0.883361i
\(540\) −24.0314 7.12189i −1.03414 0.306477i
\(541\) 37.6485 1.61864 0.809318 0.587371i \(-0.199837\pi\)
0.809318 + 0.587371i \(0.199837\pi\)
\(542\) 20.4159 + 27.3439i 0.876936 + 1.17452i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 34.8418i 1.48973i 0.667216 + 0.744864i \(0.267486\pi\)
−0.667216 + 0.744864i \(0.732514\pi\)
\(548\) 0 0
\(549\) −10.5639 −0.450855
\(550\) −12.3942 16.6001i −0.528489 0.707832i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.73539i 0.116111i
\(556\) 41.9787 + 12.4407i 1.78029 + 0.527605i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 47.0322i 1.98217i 0.133223 + 0.991086i \(0.457467\pi\)
−0.133223 + 0.991086i \(0.542533\pi\)
\(564\) 0 0
\(565\) −36.5286 −1.53677
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 16.8305 12.5662i 0.704953 0.526340i
\(571\) 39.4704i 1.65178i −0.563829 0.825891i \(-0.690672\pi\)
0.563829 0.825891i \(-0.309328\pi\)
\(572\) 13.6153 + 4.03501i 0.569284 + 0.168712i
\(573\) −39.8526 −1.66487
\(574\) 0 0
\(575\) 0 0
\(576\) −4.12600 + 3.52109i −0.171917 + 0.146712i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −19.2644 + 14.3834i −0.801294 + 0.598271i
\(579\) 27.2196i 1.13121i
\(580\) 0 0
\(581\) 0 0
\(582\) −25.1001 33.6177i −1.04043 1.39350i
\(583\) 38.8096i 1.60733i
\(584\) 0 0
\(585\) 3.67429 0.151913
\(586\) 27.2929 20.3778i 1.12746 0.841798i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 20.4539 + 6.06169i 0.843506 + 0.249980i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.69266 1.74965i −0.110667 0.0719102i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) −18.6074 + 13.8928i −0.763469 + 0.570030i
\(595\) 0 0
\(596\) 13.0344 43.9820i 0.533912 1.80157i
\(597\) 13.2842 0.543686
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 20.2194 7.45472i 0.825452 0.304338i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 2.27907i 0.0928109i
\(604\) 0 0
\(605\) 5.40325 0.219673
\(606\) −1.11942 1.49930i −0.0454734 0.0609047i
\(607\) 23.6673i 0.960628i 0.877097 + 0.480314i \(0.159477\pi\)
−0.877097 + 0.480314i \(0.840523\pi\)
\(608\) −1.60450 24.6054i −0.0650709 0.997881i
\(609\) 0 0
\(610\) 39.4793 29.4765i 1.59847 1.19347i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 20.1413 + 26.9762i 0.812835 + 1.08867i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 29.2596 21.8461i 1.17699 0.878781i
\(619\) 34.9941i 1.40653i −0.710928 0.703265i \(-0.751725\pi\)
0.710928 0.703265i \(-0.248275\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 26.5439 + 35.5515i 1.06431 + 1.42549i
\(623\) 0 0
\(624\) −8.04857 + 12.3865i −0.322201 + 0.495856i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 19.4599i 0.777153i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.0275i 0.438997i 0.975613 + 0.219499i \(0.0704421\pi\)
−0.975613 + 0.219499i \(0.929558\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −12.3074 + 9.18906i −0.488788 + 0.364944i
\(635\) 44.0653i 1.74868i
\(636\) −38.7064 11.4710i −1.53481 0.454854i
\(637\) −16.9645 −0.672158
\(638\) 0 0
\(639\) 0 0
\(640\) 5.59475 24.6718i 0.221152 0.975239i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) −29.3390 + 21.9054i −1.15792 + 0.864539i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −6.36591 17.2662i −0.250077 0.678280i
\(649\) 0 0
\(650\) −13.7316 + 10.2524i −0.538596 + 0.402133i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 43.5890i 1.70316i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 5.67303 19.1425i 0.220823 0.745120i
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.616818 + 0.460536i −0.0239012 + 0.0178454i
\(667\) 0 0
\(668\) 49.4766 + 14.6628i 1.91431 + 0.567321i
\(669\) −43.9608 −1.69962
\(670\) −6.35932 8.51734i −0.245682 0.329054i
\(671\) 45.6470i 1.76218i
\(672\) 0 0
\(673\) −44.1613 −1.70229 −0.851147 0.524928i \(-0.824092\pi\)
−0.851147 + 0.524928i \(0.824092\pi\)
\(674\) 28.0731 20.9603i 1.08134 0.807360i
\(675\) 28.0229i 1.07860i
\(676\) −4.04998 + 13.6658i −0.155768 + 0.525608i
\(677\) 50.5780 1.94387 0.971936 0.235246i \(-0.0755896\pi\)
0.971936 + 0.235246i \(0.0755896\pi\)
\(678\) −21.0616 28.2088i −0.808864 1.08335i
\(679\) 0 0
\(680\) 0 0
\(681\) 5.26217 0.201647
\(682\) 0 0
\(683\) 28.7466i 1.09996i 0.835179 + 0.549979i \(0.185364\pi\)
−0.835179 + 0.549979i \(0.814636\pi\)
\(684\) −5.66725 1.67954i −0.216693 0.0642186i
\(685\) 0 0
\(686\) 0 0
\(687\) 9.88611i 0.377178i
\(688\) 0 0
\(689\) 32.1031 1.22303
\(690\) 0 0
\(691\) 52.5727i 1.99996i 0.00630823 + 0.999980i \(0.497992\pi\)
−0.00630823 + 0.999980i \(0.502008\pi\)
\(692\) −14.4354 + 48.7094i −0.548753 + 1.85165i
\(693\) 0 0
\(694\) 0 0
\(695\) 48.9513i 1.85683i
\(696\) 0 0
\(697\) 0 0
\(698\) 15.2035 11.3514i 0.575460 0.429657i
\(699\) 0 0
\(700\) 0 0
\(701\) −21.1999 −0.800709 −0.400354 0.916360i \(-0.631113\pi\)
−0.400354 + 0.916360i \(0.631113\pi\)
\(702\) 11.4921 + 15.3919i 0.433741 + 0.580931i
\(703\) 3.49930i 0.131979i
\(704\) −15.2148 17.8286i −0.573429 0.671942i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 40.2492 1.51159 0.755796 0.654808i \(-0.227250\pi\)
0.755796 + 0.654808i \(0.227250\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 15.8768i 0.593758i
\(716\) 0 0
\(717\) −29.7044 −1.10933
\(718\) −21.5862 28.9114i −0.805590 1.07897i
\(719\) 13.7940i 0.514429i 0.966354 + 0.257214i \(0.0828047\pi\)
−0.966354 + 0.257214i \(0.917195\pi\)
\(720\) −5.08518 3.30428i −0.189514 0.123143i
\(721\) 0 0
\(722\) 21.5308 16.0756i 0.801294 0.598271i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 3.11539 + 4.17259i 0.115623 + 0.154859i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −30.0322 −1.11230
\(730\) 0 0
\(731\) 0 0
\(732\) 45.5257 + 13.4919i 1.68268 + 0.498675i
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 23.8513i 0.879768i
\(736\) 0 0
\(737\) −9.84797 −0.362755
\(738\) 0 0
\(739\) 8.71780i 0.320689i −0.987061 0.160345i \(-0.948739\pi\)
0.987061 0.160345i \(-0.0512606\pi\)
\(740\) 1.02013 3.44223i 0.0375008 0.126539i
\(741\) −16.0971 −0.591343
\(742\) 0 0
\(743\) 1.62663i 0.0596754i 0.999555 + 0.0298377i \(0.00949905\pi\)
−0.999555 + 0.0298377i \(0.990501\pi\)
\(744\) 0 0
\(745\) 51.2874 1.87902
\(746\) −43.2137 + 32.2647i −1.58217 + 1.18130i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 14.4144 + 19.3059i 0.526340 + 0.704953i
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 39.8526 1.45231
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 25.8661 9.53662i 0.938260 0.345930i
\(761\) 9.29755 0.337036 0.168518 0.985699i \(-0.446102\pi\)
0.168518 + 0.985699i \(0.446102\pi\)
\(762\) −34.0289 + 25.4070i −1.23274 + 0.920399i
\(763\) 0 0
\(764\) −50.1508 14.8626i −1.81439 0.537710i
\(765\) 0 0
\(766\) 2.80506 + 3.75696i 0.101351 + 0.135744i
\(767\) 0 0
\(768\) 22.2783 9.90472i 0.803899 0.357406i
\(769\) −29.2192 −1.05367 −0.526836 0.849967i \(-0.676622\pi\)
−0.526836 + 0.849967i \(0.676622\pi\)
\(770\) 0 0
\(771\) 48.6281i 1.75130i
\(772\) 10.1512 34.2533i 0.365351 1.23280i
\(773\) 28.9919 1.04277 0.521383 0.853323i \(-0.325416\pi\)
0.521383 + 0.853323i \(0.325416\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −19.0487 51.6656i −0.683809 1.85469i
\(777\) 0 0
\(778\) −6.79920 + 5.07650i −0.243763 + 0.182001i
\(779\) 0 0
\(780\) −15.8346 4.69271i −0.566969 0.168026i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 23.4787 + 15.2561i 0.838525 + 0.544862i
\(785\) 0 0
\(786\) −33.6610 + 25.1324i −1.20065 + 0.896443i
\(787\) 25.6990i 0.916070i −0.888934 0.458035i \(-0.848553\pi\)
0.888934 0.458035i \(-0.151447\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −5.27167 + 1.94363i −0.187321 + 0.0690637i
\(793\) −37.7590 −1.34086
\(794\) 0 0
\(795\) 45.1355i 1.60079i
\(796\) 16.7169 + 4.95420i 0.592516 + 0.175597i
\(797\) 28.9016 1.02375 0.511873 0.859061i \(-0.328952\pi\)
0.511873 + 0.859061i \(0.328952\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.2243 1.84048i 0.997881 0.0650709i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 2.91077 9.82180i 0.102655 0.346388i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.849541 2.30420i −0.0298868 0.0810615i
\(809\) −22.3607 −0.786160 −0.393080 0.919504i \(-0.628590\pi\)
−0.393080 + 0.919504i \(0.628590\pi\)
\(810\) 16.4862 12.3091i 0.579265 0.432498i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) −36.7691 −1.28955
\(814\) −1.99000 2.66530i −0.0697493 0.0934187i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 44.9678 16.5793i 1.56653 0.577567i
\(825\) 22.3220 0.777153
\(826\) 0 0
\(827\) 56.1750i 1.95340i −0.214614 0.976699i \(-0.568849\pi\)
0.214614 0.976699i \(-0.431151\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −14.7478 + 12.5856i −0.511287 + 0.436327i
\(833\) 0 0
\(834\) −37.8020 + 28.2242i −1.30898 + 0.977323i
\(835\) 57.6946i 1.99660i
\(836\) 7.25735 24.4884i 0.251001 0.846950i
\(837\) 0 0
\(838\) −16.4932 22.0901i −0.569748 0.763091i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −15.9357 −0.548204
\(846\) 0 0
\(847\) 0 0
\(848\) −44.4305 28.8703i −1.52575 0.991410i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 6.60857i 0.226008i
\(856\) −45.0898 + 16.6243i −1.54114 + 0.568206i
\(857\) −10.2359 −0.349651 −0.174825 0.984599i \(-0.555936\pi\)
−0.174825 + 0.984599i \(0.555936\pi\)
\(858\) −12.2606 + 9.15418i −0.418571 + 0.312519i
\(859\) 58.4808i 1.99534i −0.0682391 0.997669i \(-0.521738\pi\)
0.0682391 0.997669i \(-0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 44.0875i 1.50075i 0.661010 + 0.750377i \(0.270128\pi\)
−0.661010 + 0.750377i \(0.729872\pi\)
\(864\) −2.06303 31.6371i −0.0701857 1.07632i
\(865\) −56.7999 −1.93126
\(866\) −44.5510 + 33.2632i −1.51390 + 1.13033i
\(867\) 25.9047i 0.879768i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.14620i 0.276024i
\(872\) 0 0
\(873\) −13.2001 −0.446757
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.3454 1.39614 0.698068 0.716032i \(-0.254044\pi\)
0.698068 + 0.716032i \(0.254044\pi\)
\(878\) 0 0
\(879\) 36.7005i 1.23788i
\(880\) 14.2780 21.9733i 0.481310 0.740721i
\(881\) −58.6434 −1.97575 −0.987874 0.155261i \(-0.950378\pi\)
−0.987874 + 0.155261i \(0.950378\pi\)
\(882\) 5.37836 4.01566i 0.181099 0.135214i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.8527i 0.565858i −0.959141 0.282929i \(-0.908694\pi\)
0.959141 0.282929i \(-0.0913060\pi\)
\(888\) 3.24640 1.19692i 0.108942 0.0401661i
\(889\) 0 0
\(890\) 0 0
\(891\) 19.0617i 0.638592i
\(892\) −55.3206 16.3947i −1.85227 0.548936i
\(893\) 0 0
\(894\) 29.5711 + 39.6060i 0.989005 + 1.32462i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.92656 6.50078i 0.0642186 0.216693i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −15.9838 43.3528i −0.531615 1.44189i
\(905\) 0 0
\(906\) 0 0
\(907\) 17.0826i 0.567219i 0.958940 + 0.283610i \(0.0915320\pi\)
−0.958940 + 0.283610i \(0.908468\pi\)
\(908\) 6.62195 + 1.96247i 0.219757 + 0.0651269i
\(909\) −0.588704 −0.0195261
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 22.2783 + 14.4761i 0.737708 + 0.479353i
\(913\) 0 0
\(914\) 0 0
\(915\) 53.0874i 1.75501i
\(916\) −3.68692 + 12.4407i −0.121819 + 0.411054i
\(917\) 0 0
\(918\) 0 0
\(919\) 58.4808i 1.92910i 0.263896 + 0.964551i \(0.414993\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −36.2746 −1.19529
\(922\) 20.3976 15.2295i 0.671759 0.501557i
\(923\) 0 0
\(924\) 0 0
\(925\) 4.01397 0.131979
\(926\) 0 0
\(927\) 11.4889i 0.377345i
\(928\) 0 0
\(929\) 31.3050 1.02708 0.513541 0.858065i \(-0.328333\pi\)
0.513541 + 0.858065i \(0.328333\pi\)
\(930\) 0 0
\(931\) 30.5123i 1.00000i
\(932\) 0 0
\(933\) −47.8057 −1.56509
\(934\) 0 0
\(935\) 0 0
\(936\) 1.60776 + 4.36071i 0.0525513 + 0.142534i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 18.4400 + 24.6975i 0.598271 + 0.801294i
\(951\) 16.5496i 0.536657i
\(952\) 0 0
\(953\) −47.5673 −1.54086 −0.770428 0.637527i \(-0.779957\pi\)
−0.770428 + 0.637527i \(0.779957\pi\)
\(954\) −10.1779 + 7.59912i −0.329521 + 0.246030i
\(955\) 58.4808i 1.89239i
\(956\) −37.3802 11.0779i −1.20896 0.358286i
\(957\) 0 0
\(958\) 36.4592 + 48.8316i 1.17794 + 1.57768i
\(959\) 0 0
\(960\) 17.6948 + 20.7347i 0.571096 + 0.669209i
\(961\) −31.0000 −1.00000
\(962\) −2.20472 + 1.64612i −0.0710832 + 0.0530729i
\(963\) 11.5201i 0.371229i
\(964\) 0 0
\(965\) 39.9427 1.28580
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 2.36430 + 6.41267i 0.0759916 + 0.206111i
\(969\) 0 0
\(970\) 49.3315 36.8325i 1.58394 1.18262i
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −13.2304 3.92093i −0.424364 0.125764i
\(973\) 0 0
\(974\) 37.3218 + 49.9869i 1.19587 + 1.60168i
\(975\) 18.4647i 0.591343i
\(976\) 52.2581 + 33.9566i 1.67274 + 1.08692i
\(977\) 30.8771 0.987846 0.493923 0.869506i \(-0.335562\pi\)
0.493923 + 0.869506i \(0.335562\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.89508 + 30.0146i −0.284143 + 0.958782i
\(981\) 0 0
\(982\) −22.1280 29.6370i −0.706131 0.945756i
\(983\) 44.2033i 1.40987i −0.709274 0.704933i \(-0.750977\pi\)
0.709274 0.704933i \(-0.249023\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −20.2567 6.00325i −0.644453 0.190989i
\(989\) 0 0
\(990\) −3.75818 5.03352i −0.119443 0.159976i
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.4936i 0.617988i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 3.64918 + 4.88753i 0.115513 + 0.154712i
\(999\) 4.49933i 0.142353i
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 380.2.d.a.379.4 yes 16
4.3 odd 2 inner 380.2.d.a.379.3 16
5.4 even 2 inner 380.2.d.a.379.13 yes 16
19.18 odd 2 inner 380.2.d.a.379.13 yes 16
20.19 odd 2 inner 380.2.d.a.379.14 yes 16
76.75 even 2 inner 380.2.d.a.379.14 yes 16
95.94 odd 2 CM 380.2.d.a.379.4 yes 16
380.379 even 2 inner 380.2.d.a.379.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.d.a.379.3 16 4.3 odd 2 inner
380.2.d.a.379.3 16 380.379 even 2 inner
380.2.d.a.379.4 yes 16 1.1 even 1 trivial
380.2.d.a.379.4 yes 16 95.94 odd 2 CM
380.2.d.a.379.13 yes 16 5.4 even 2 inner
380.2.d.a.379.13 yes 16 19.18 odd 2 inner
380.2.d.a.379.14 yes 16 20.19 odd 2 inner
380.2.d.a.379.14 yes 16 76.75 even 2 inner