Properties

Label 1183.2.c.g.337.3
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $8$
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] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), 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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11667456256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 44x^{4} + 21x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(-0.710287i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.g.337.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.710287i q^{2} +2.40788 q^{3} +1.49549 q^{4} -1.28971i q^{5} -1.71029i q^{6} +1.00000i q^{7} -2.48280i q^{8} +2.79790 q^{9} +O(q^{10})\) \(q-0.710287i q^{2} +2.40788 q^{3} +1.49549 q^{4} -1.28971i q^{5} -1.71029i q^{6} +1.00000i q^{7} -2.48280i q^{8} +2.79790 q^{9} -0.916066 q^{10} +2.40788i q^{11} +3.60097 q^{12} +0.710287 q^{14} -3.10548i q^{15} +1.22748 q^{16} -3.90338 q^{17} -1.98731i q^{18} -5.89068i q^{19} -1.92876i q^{20} +2.40788i q^{21} +1.71029 q^{22} +6.32395 q^{23} -5.97829i q^{24} +3.33664 q^{25} -0.486640 q^{27} +1.49549i q^{28} +5.61366 q^{29} -2.20578 q^{30} +2.20578i q^{31} -5.83747i q^{32} +5.79790i q^{33} +2.77252i q^{34} +1.28971 q^{35} +4.18424 q^{36} +5.11817i q^{37} -4.18407 q^{38} -3.20210 q^{40} -7.78521i q^{41} +1.71029 q^{42} -0.289713 q^{43} +3.60097i q^{44} -3.60849i q^{45} -4.49182i q^{46} -1.27702i q^{47} +2.95564 q^{48} -1.00000 q^{49} -2.36997i q^{50} -9.39887 q^{51} -13.6225 q^{53} +0.345654i q^{54} +3.10548 q^{55} +2.48280 q^{56} -14.1841i q^{57} -3.98731i q^{58} +4.03056i q^{59} -4.64422i q^{60} -4.60097 q^{61} +1.56674 q^{62} +2.79790i q^{63} -1.69131 q^{64} +4.11817 q^{66} +7.57559i q^{67} -5.83747 q^{68} +15.2273 q^{69} -0.916066i q^{70} +7.22732i q^{71} -6.94662i q^{72} +15.0125i q^{73} +3.63537 q^{74} +8.03424 q^{75} -8.80948i q^{76} -2.40788 q^{77} -9.30758 q^{79} -1.58310i q^{80} -9.56546 q^{81} -5.52973 q^{82} -1.36463i q^{83} +3.60097i q^{84} +5.03424i q^{85} +0.205780i q^{86} +13.5170 q^{87} +5.97829 q^{88} +0.899698i q^{89} -2.56306 q^{90} +9.45742 q^{92} +5.31126i q^{93} -0.907052 q^{94} -7.59729 q^{95} -14.0559i q^{96} +15.6658i q^{97} +0.710287i q^{98} +6.73701i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 10 q^{4} + 14 q^{9} + 22 q^{10} - 24 q^{12} + 2 q^{14} + 38 q^{16} + 8 q^{17} + 10 q^{22} + 4 q^{23} - 10 q^{25} - 52 q^{27} + 2 q^{29} + 8 q^{30} + 14 q^{35} + 68 q^{36} + 46 q^{38} - 34 q^{40} + 10 q^{42} - 6 q^{43} + 22 q^{48} - 8 q^{49} - 14 q^{51} + 4 q^{53} - 6 q^{55} - 12 q^{56} + 16 q^{61} + 10 q^{62} - 28 q^{64} + 12 q^{66} - 66 q^{68} + 36 q^{69} + 40 q^{74} + 14 q^{75} - 2 q^{77} - 52 q^{79} + 48 q^{81} + 28 q^{82} + 26 q^{87} - 6 q^{88} - 52 q^{90} + 24 q^{92} + 66 q^{94} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.710287i − 0.502249i −0.967955 0.251124i \(-0.919200\pi\)
0.967955 0.251124i \(-0.0808003\pi\)
\(3\) 2.40788 1.39019 0.695096 0.718917i \(-0.255362\pi\)
0.695096 + 0.718917i \(0.255362\pi\)
\(4\) 1.49549 0.747746
\(5\) − 1.28971i − 0.576777i −0.957513 0.288389i \(-0.906880\pi\)
0.957513 0.288389i \(-0.0931195\pi\)
\(6\) − 1.71029i − 0.698222i
\(7\) 1.00000i 0.377964i
\(8\) − 2.48280i − 0.877803i
\(9\) 2.79790 0.932632
\(10\) −0.916066 −0.289686
\(11\) 2.40788i 0.726004i 0.931788 + 0.363002i \(0.118248\pi\)
−0.931788 + 0.363002i \(0.881752\pi\)
\(12\) 3.60097 1.03951
\(13\) 0 0
\(14\) 0.710287 0.189832
\(15\) − 3.10548i − 0.801831i
\(16\) 1.22748 0.306871
\(17\) −3.90338 −0.946708 −0.473354 0.880872i \(-0.656957\pi\)
−0.473354 + 0.880872i \(0.656957\pi\)
\(18\) − 1.98731i − 0.468413i
\(19\) − 5.89068i − 1.35142i −0.737170 0.675708i \(-0.763838\pi\)
0.737170 0.675708i \(-0.236162\pi\)
\(20\) − 1.92876i − 0.431283i
\(21\) 2.40788i 0.525443i
\(22\) 1.71029 0.364634
\(23\) 6.32395 1.31863 0.659317 0.751865i \(-0.270845\pi\)
0.659317 + 0.751865i \(0.270845\pi\)
\(24\) − 5.97829i − 1.22031i
\(25\) 3.33664 0.667328
\(26\) 0 0
\(27\) −0.486640 −0.0936539
\(28\) 1.49549i 0.282622i
\(29\) 5.61366 1.04243 0.521215 0.853425i \(-0.325479\pi\)
0.521215 + 0.853425i \(0.325479\pi\)
\(30\) −2.20578 −0.402718
\(31\) 2.20578i 0.396170i 0.980185 + 0.198085i \(0.0634722\pi\)
−0.980185 + 0.198085i \(0.936528\pi\)
\(32\) − 5.83747i − 1.03193i
\(33\) 5.79790i 1.00928i
\(34\) 2.77252i 0.475482i
\(35\) 1.28971 0.218001
\(36\) 4.18424 0.697373
\(37\) 5.11817i 0.841422i 0.907195 + 0.420711i \(0.138219\pi\)
−0.907195 + 0.420711i \(0.861781\pi\)
\(38\) −4.18407 −0.678746
\(39\) 0 0
\(40\) −3.20210 −0.506297
\(41\) − 7.78521i − 1.21584i −0.793996 0.607922i \(-0.792003\pi\)
0.793996 0.607922i \(-0.207997\pi\)
\(42\) 1.71029 0.263903
\(43\) −0.289713 −0.0441809 −0.0220904 0.999756i \(-0.507032\pi\)
−0.0220904 + 0.999756i \(0.507032\pi\)
\(44\) 3.60097i 0.542867i
\(45\) − 3.60849i − 0.537921i
\(46\) − 4.49182i − 0.662282i
\(47\) − 1.27702i − 0.186273i −0.995653 0.0931364i \(-0.970311\pi\)
0.995653 0.0931364i \(-0.0296893\pi\)
\(48\) 2.95564 0.426610
\(49\) −1.00000 −0.142857
\(50\) − 2.36997i − 0.335164i
\(51\) −9.39887 −1.31610
\(52\) 0 0
\(53\) −13.6225 −1.87120 −0.935598 0.353067i \(-0.885139\pi\)
−0.935598 + 0.353067i \(0.885139\pi\)
\(54\) 0.345654i 0.0470375i
\(55\) 3.10548 0.418743
\(56\) 2.48280 0.331778
\(57\) − 14.1841i − 1.87873i
\(58\) − 3.98731i − 0.523559i
\(59\) 4.03056i 0.524734i 0.964968 + 0.262367i \(0.0845031\pi\)
−0.964968 + 0.262367i \(0.915497\pi\)
\(60\) − 4.64422i − 0.599566i
\(61\) −4.60097 −0.589094 −0.294547 0.955637i \(-0.595169\pi\)
−0.294547 + 0.955637i \(0.595169\pi\)
\(62\) 1.56674 0.198976
\(63\) 2.79790i 0.352502i
\(64\) −1.69131 −0.211413
\(65\) 0 0
\(66\) 4.11817 0.506912
\(67\) 7.57559i 0.925505i 0.886487 + 0.462753i \(0.153138\pi\)
−0.886487 + 0.462753i \(0.846862\pi\)
\(68\) −5.83747 −0.707897
\(69\) 15.2273 1.83315
\(70\) − 0.916066i − 0.109491i
\(71\) 7.22732i 0.857726i 0.903370 + 0.428863i \(0.141086\pi\)
−0.903370 + 0.428863i \(0.858914\pi\)
\(72\) − 6.94662i − 0.818668i
\(73\) 15.0125i 1.75708i 0.477665 + 0.878542i \(0.341483\pi\)
−0.477665 + 0.878542i \(0.658517\pi\)
\(74\) 3.63537 0.422603
\(75\) 8.03424 0.927714
\(76\) − 8.80948i − 1.01052i
\(77\) −2.40788 −0.274404
\(78\) 0 0
\(79\) −9.30758 −1.04718 −0.523592 0.851969i \(-0.675408\pi\)
−0.523592 + 0.851969i \(0.675408\pi\)
\(80\) − 1.58310i − 0.176996i
\(81\) −9.56546 −1.06283
\(82\) −5.52973 −0.610656
\(83\) − 1.36463i − 0.149788i −0.997192 0.0748940i \(-0.976138\pi\)
0.997192 0.0748940i \(-0.0238618\pi\)
\(84\) 3.60097i 0.392898i
\(85\) 5.03424i 0.546039i
\(86\) 0.205780i 0.0221898i
\(87\) 13.5170 1.44918
\(88\) 5.97829 0.637288
\(89\) 0.899698i 0.0953678i 0.998862 + 0.0476839i \(0.0151840\pi\)
−0.998862 + 0.0476839i \(0.984816\pi\)
\(90\) −2.56306 −0.270170
\(91\) 0 0
\(92\) 9.45742 0.986004
\(93\) 5.31126i 0.550752i
\(94\) −0.907052 −0.0935553
\(95\) −7.59729 −0.779466
\(96\) − 14.0559i − 1.43458i
\(97\) 15.6658i 1.59062i 0.606205 + 0.795309i \(0.292691\pi\)
−0.606205 + 0.795309i \(0.707309\pi\)
\(98\) 0.710287i 0.0717498i
\(99\) 6.73701i 0.677095i
\(100\) 4.98992 0.498992
\(101\) −0.684905 −0.0681506 −0.0340753 0.999419i \(-0.510849\pi\)
−0.0340753 + 0.999419i \(0.510849\pi\)
\(102\) 6.67589i 0.661012i
\(103\) −17.9249 −1.76619 −0.883097 0.469190i \(-0.844546\pi\)
−0.883097 + 0.469190i \(0.844546\pi\)
\(104\) 0 0
\(105\) 3.10548 0.303064
\(106\) 9.67589i 0.939806i
\(107\) 8.49549 0.821290 0.410645 0.911795i \(-0.365304\pi\)
0.410645 + 0.911795i \(0.365304\pi\)
\(108\) −0.727766 −0.0700293
\(109\) 12.0928i 1.15828i 0.815228 + 0.579139i \(0.196611\pi\)
−0.815228 + 0.579139i \(0.803389\pi\)
\(110\) − 2.20578i − 0.210313i
\(111\) 12.3239i 1.16974i
\(112\) 1.22748i 0.115986i
\(113\) −14.2527 −1.34078 −0.670391 0.742008i \(-0.733874\pi\)
−0.670391 + 0.742008i \(0.733874\pi\)
\(114\) −10.0748 −0.943588
\(115\) − 8.15608i − 0.760558i
\(116\) 8.39519 0.779474
\(117\) 0 0
\(118\) 2.86285 0.263547
\(119\) − 3.90338i − 0.357822i
\(120\) −7.71029 −0.703850
\(121\) 5.20210 0.472918
\(122\) 3.26801i 0.295872i
\(123\) − 18.7459i − 1.69026i
\(124\) 3.29873i 0.296234i
\(125\) − 10.7519i − 0.961677i
\(126\) 1.98731 0.177044
\(127\) 8.82462 0.783058 0.391529 0.920166i \(-0.371946\pi\)
0.391529 + 0.920166i \(0.371946\pi\)
\(128\) − 10.4736i − 0.925747i
\(129\) −0.697596 −0.0614199
\(130\) 0 0
\(131\) −19.4294 −1.69756 −0.848778 0.528749i \(-0.822661\pi\)
−0.848778 + 0.528749i \(0.822661\pi\)
\(132\) 8.67071i 0.754689i
\(133\) 5.89068 0.510787
\(134\) 5.38084 0.464834
\(135\) 0.627626i 0.0540174i
\(136\) 9.69131i 0.831023i
\(137\) − 10.2362i − 0.874536i −0.899331 0.437268i \(-0.855946\pi\)
0.899331 0.437268i \(-0.144054\pi\)
\(138\) − 10.8158i − 0.920699i
\(139\) 8.14723 0.691039 0.345519 0.938412i \(-0.387703\pi\)
0.345519 + 0.938412i \(0.387703\pi\)
\(140\) 1.92876 0.163010
\(141\) − 3.07492i − 0.258955i
\(142\) 5.13347 0.430791
\(143\) 0 0
\(144\) 3.43438 0.286198
\(145\) − 7.24001i − 0.601250i
\(146\) 10.6632 0.882493
\(147\) −2.40788 −0.198599
\(148\) 7.65419i 0.629170i
\(149\) − 9.78888i − 0.801937i −0.916092 0.400968i \(-0.868674\pi\)
0.916092 0.400968i \(-0.131326\pi\)
\(150\) − 5.70661i − 0.465943i
\(151\) − 14.7407i − 1.19958i −0.800158 0.599790i \(-0.795251\pi\)
0.800158 0.599790i \(-0.204749\pi\)
\(152\) −14.6254 −1.18628
\(153\) −10.9212 −0.882930
\(154\) 1.71029i 0.137819i
\(155\) 2.84482 0.228502
\(156\) 0 0
\(157\) 20.5844 1.64282 0.821409 0.570340i \(-0.193189\pi\)
0.821409 + 0.570340i \(0.193189\pi\)
\(158\) 6.61105i 0.525947i
\(159\) −32.8014 −2.60132
\(160\) −7.52866 −0.595193
\(161\) 6.32395i 0.498397i
\(162\) 6.79422i 0.533804i
\(163\) 3.98086i 0.311805i 0.987772 + 0.155902i \(0.0498286\pi\)
−0.987772 + 0.155902i \(0.950171\pi\)
\(164\) − 11.6427i − 0.909144i
\(165\) 7.47763 0.582132
\(166\) −0.969281 −0.0752308
\(167\) − 8.67846i − 0.671559i −0.941941 0.335780i \(-0.891000\pi\)
0.941941 0.335780i \(-0.109000\pi\)
\(168\) 5.97829 0.461235
\(169\) 0 0
\(170\) 3.57575 0.274248
\(171\) − 16.4815i − 1.26037i
\(172\) −0.433264 −0.0330361
\(173\) −0.933934 −0.0710057 −0.0355028 0.999370i \(-0.511303\pi\)
−0.0355028 + 0.999370i \(0.511303\pi\)
\(174\) − 9.60097i − 0.727848i
\(175\) 3.33664i 0.252226i
\(176\) 2.95564i 0.222790i
\(177\) 9.70511i 0.729481i
\(178\) 0.639044 0.0478984
\(179\) −13.5461 −1.01248 −0.506241 0.862392i \(-0.668966\pi\)
−0.506241 + 0.862392i \(0.668966\pi\)
\(180\) − 5.39646i − 0.402229i
\(181\) 8.86269 0.658759 0.329379 0.944198i \(-0.393161\pi\)
0.329379 + 0.944198i \(0.393161\pi\)
\(182\) 0 0
\(183\) −11.0786 −0.818953
\(184\) − 15.7011i − 1.15750i
\(185\) 6.60097 0.485313
\(186\) 3.77252 0.276614
\(187\) − 9.39887i − 0.687313i
\(188\) − 1.90978i − 0.139285i
\(189\) − 0.486640i − 0.0353978i
\(190\) 5.39626i 0.391486i
\(191\) −15.3735 −1.11239 −0.556193 0.831053i \(-0.687739\pi\)
−0.556193 + 0.831053i \(0.687739\pi\)
\(192\) −4.07247 −0.293905
\(193\) 24.4953i 1.76321i 0.471986 + 0.881606i \(0.343537\pi\)
−0.471986 + 0.881606i \(0.656463\pi\)
\(194\) 11.1272 0.798885
\(195\) 0 0
\(196\) −1.49549 −0.106821
\(197\) − 6.71412i − 0.478362i −0.970975 0.239181i \(-0.923121\pi\)
0.970975 0.239181i \(-0.0768789\pi\)
\(198\) 4.78521 0.340070
\(199\) 4.87282 0.345425 0.172712 0.984972i \(-0.444747\pi\)
0.172712 + 0.984972i \(0.444747\pi\)
\(200\) − 8.28421i − 0.585782i
\(201\) 18.2411i 1.28663i
\(202\) 0.486479i 0.0342285i
\(203\) 5.61366i 0.394002i
\(204\) −14.0559 −0.984113
\(205\) −10.0407 −0.701272
\(206\) 12.7318i 0.887069i
\(207\) 17.6938 1.22980
\(208\) 0 0
\(209\) 14.1841 0.981133
\(210\) − 2.20578i − 0.152213i
\(211\) 3.68747 0.253856 0.126928 0.991912i \(-0.459488\pi\)
0.126928 + 0.991912i \(0.459488\pi\)
\(212\) −20.3724 −1.39918
\(213\) 17.4025i 1.19240i
\(214\) − 6.03424i − 0.412492i
\(215\) 0.373647i 0.0254825i
\(216\) 1.20823i 0.0822096i
\(217\) −2.20578 −0.149738
\(218\) 8.58935 0.581744
\(219\) 36.1484i 2.44268i
\(220\) 4.64422 0.313113
\(221\) 0 0
\(222\) 8.75354 0.587499
\(223\) 8.81426i 0.590247i 0.955459 + 0.295123i \(0.0953608\pi\)
−0.955459 + 0.295123i \(0.904639\pi\)
\(224\) 5.83747 0.390032
\(225\) 9.33557 0.622372
\(226\) 10.1235i 0.673406i
\(227\) − 13.4511i − 0.892783i −0.894838 0.446391i \(-0.852709\pi\)
0.894838 0.446391i \(-0.147291\pi\)
\(228\) − 21.2122i − 1.40481i
\(229\) − 6.81576i − 0.450398i −0.974313 0.225199i \(-0.927697\pi\)
0.974313 0.225199i \(-0.0723033\pi\)
\(230\) −5.79316 −0.381989
\(231\) −5.79790 −0.381474
\(232\) − 13.9376i − 0.915049i
\(233\) 25.0672 1.64221 0.821105 0.570777i \(-0.193358\pi\)
0.821105 + 0.570777i \(0.193358\pi\)
\(234\) 0 0
\(235\) −1.64699 −0.107438
\(236\) 6.02767i 0.392368i
\(237\) −22.4116 −1.45579
\(238\) −2.77252 −0.179715
\(239\) − 2.78521i − 0.180160i −0.995935 0.0900800i \(-0.971288\pi\)
0.995935 0.0900800i \(-0.0287123\pi\)
\(240\) − 3.81193i − 0.246059i
\(241\) − 6.97829i − 0.449511i −0.974415 0.224756i \(-0.927842\pi\)
0.974415 0.224756i \(-0.0721584\pi\)
\(242\) − 3.69498i − 0.237523i
\(243\) −21.5726 −1.38388
\(244\) −6.88072 −0.440493
\(245\) 1.28971i 0.0823968i
\(246\) −13.3149 −0.848929
\(247\) 0 0
\(248\) 5.47651 0.347759
\(249\) − 3.28588i − 0.208234i
\(250\) −7.63691 −0.483001
\(251\) 0.783029 0.0494244 0.0247122 0.999695i \(-0.492133\pi\)
0.0247122 + 0.999695i \(0.492133\pi\)
\(252\) 4.18424i 0.263582i
\(253\) 15.2273i 0.957334i
\(254\) − 6.26801i − 0.393290i
\(255\) 12.1218i 0.759099i
\(256\) −10.8219 −0.676368
\(257\) −3.25804 −0.203231 −0.101616 0.994824i \(-0.532401\pi\)
−0.101616 + 0.994824i \(0.532401\pi\)
\(258\) 0.495493i 0.0308480i
\(259\) −5.11817 −0.318028
\(260\) 0 0
\(261\) 15.7064 0.972205
\(262\) 13.8005i 0.852595i
\(263\) 29.5829 1.82416 0.912081 0.410010i \(-0.134475\pi\)
0.912081 + 0.410010i \(0.134475\pi\)
\(264\) 14.3950 0.885953
\(265\) 17.5691i 1.07926i
\(266\) − 4.18407i − 0.256542i
\(267\) 2.16637i 0.132580i
\(268\) 11.3292i 0.692043i
\(269\) 1.23650 0.0753907 0.0376953 0.999289i \(-0.487998\pi\)
0.0376953 + 0.999289i \(0.487998\pi\)
\(270\) 0.445794 0.0271302
\(271\) − 25.2071i − 1.53122i −0.643303 0.765612i \(-0.722436\pi\)
0.643303 0.765612i \(-0.277564\pi\)
\(272\) −4.79133 −0.290517
\(273\) 0 0
\(274\) −7.27062 −0.439234
\(275\) 8.03424i 0.484483i
\(276\) 22.7724 1.37073
\(277\) −9.53602 −0.572964 −0.286482 0.958086i \(-0.592486\pi\)
−0.286482 + 0.958086i \(0.592486\pi\)
\(278\) − 5.78687i − 0.347073i
\(279\) 6.17154i 0.369481i
\(280\) − 3.20210i − 0.191362i
\(281\) − 9.56546i − 0.570628i −0.958434 0.285314i \(-0.907902\pi\)
0.958434 0.285314i \(-0.0920978\pi\)
\(282\) −2.18407 −0.130060
\(283\) −11.4320 −0.679564 −0.339782 0.940504i \(-0.610353\pi\)
−0.339782 + 0.940504i \(0.610353\pi\)
\(284\) 10.8084i 0.641361i
\(285\) −18.2934 −1.08361
\(286\) 0 0
\(287\) 7.78521 0.459546
\(288\) − 16.3326i − 0.962410i
\(289\) −1.76366 −0.103745
\(290\) −5.14249 −0.301977
\(291\) 37.7213i 2.21126i
\(292\) 22.4511i 1.31385i
\(293\) 9.35562i 0.546561i 0.961934 + 0.273281i \(0.0881087\pi\)
−0.961934 + 0.273281i \(0.911891\pi\)
\(294\) 1.71029i 0.0997459i
\(295\) 5.19826 0.302655
\(296\) 12.7074 0.738603
\(297\) − 1.17177i − 0.0679931i
\(298\) −6.95291 −0.402771
\(299\) 0 0
\(300\) 12.0151 0.693695
\(301\) − 0.289713i − 0.0166988i
\(302\) −10.4701 −0.602487
\(303\) −1.64917 −0.0947423
\(304\) − 7.23072i − 0.414711i
\(305\) 5.93393i 0.339776i
\(306\) 7.75721i 0.443450i
\(307\) − 8.11449i − 0.463119i −0.972821 0.231559i \(-0.925617\pi\)
0.972821 0.231559i \(-0.0743827\pi\)
\(308\) −3.60097 −0.205184
\(309\) −43.1611 −2.45535
\(310\) − 2.02064i − 0.114765i
\(311\) 8.64016 0.489938 0.244969 0.969531i \(-0.421222\pi\)
0.244969 + 0.969531i \(0.421222\pi\)
\(312\) 0 0
\(313\) −5.22732 −0.295466 −0.147733 0.989027i \(-0.547198\pi\)
−0.147733 + 0.989027i \(0.547198\pi\)
\(314\) − 14.6209i − 0.825103i
\(315\) 3.60849 0.203315
\(316\) −13.9194 −0.783029
\(317\) 11.1929i 0.628657i 0.949314 + 0.314329i \(0.101779\pi\)
−0.949314 + 0.314329i \(0.898221\pi\)
\(318\) 23.2984i 1.30651i
\(319\) 13.5170i 0.756809i
\(320\) 2.18130i 0.121938i
\(321\) 20.4561 1.14175
\(322\) 4.49182 0.250319
\(323\) 22.9936i 1.27940i
\(324\) −14.3051 −0.794727
\(325\) 0 0
\(326\) 2.82755 0.156604
\(327\) 29.1180i 1.61023i
\(328\) −19.3291 −1.06727
\(329\) 1.27702 0.0704045
\(330\) − 5.31126i − 0.292375i
\(331\) 20.0468i 1.10187i 0.834548 + 0.550935i \(0.185729\pi\)
−0.834548 + 0.550935i \(0.814271\pi\)
\(332\) − 2.04080i − 0.112003i
\(333\) 14.3201i 0.784737i
\(334\) −6.16419 −0.337290
\(335\) 9.77034 0.533811
\(336\) 2.95564i 0.161243i
\(337\) −18.5866 −1.01248 −0.506239 0.862393i \(-0.668965\pi\)
−0.506239 + 0.862393i \(0.668965\pi\)
\(338\) 0 0
\(339\) −34.3188 −1.86394
\(340\) 7.52866i 0.408299i
\(341\) −5.31126 −0.287621
\(342\) −11.7066 −0.633021
\(343\) − 1.00000i − 0.0539949i
\(344\) 0.719301i 0.0387821i
\(345\) − 19.6389i − 1.05732i
\(346\) 0.663361i 0.0356625i
\(347\) 22.3417 1.19936 0.599681 0.800239i \(-0.295294\pi\)
0.599681 + 0.800239i \(0.295294\pi\)
\(348\) 20.2146 1.08362
\(349\) − 8.78632i − 0.470321i −0.971957 0.235160i \(-0.924438\pi\)
0.971957 0.235160i \(-0.0755615\pi\)
\(350\) 2.36997 0.126680
\(351\) 0 0
\(352\) 14.0559 0.749184
\(353\) 7.71898i 0.410840i 0.978674 + 0.205420i \(0.0658560\pi\)
−0.978674 + 0.205420i \(0.934144\pi\)
\(354\) 6.89341 0.366381
\(355\) 9.32118 0.494717
\(356\) 1.34549i 0.0713110i
\(357\) − 9.39887i − 0.497441i
\(358\) 9.62161i 0.508518i
\(359\) 10.5956i 0.559216i 0.960114 + 0.279608i \(0.0902045\pi\)
−0.960114 + 0.279608i \(0.909795\pi\)
\(360\) −8.95915 −0.472189
\(361\) −15.7002 −0.826324
\(362\) − 6.29505i − 0.330861i
\(363\) 12.5261 0.657447
\(364\) 0 0
\(365\) 19.3619 1.01345
\(366\) 7.86898i 0.411318i
\(367\) 5.82067 0.303836 0.151918 0.988393i \(-0.451455\pi\)
0.151918 + 0.988393i \(0.451455\pi\)
\(368\) 7.76255 0.404651
\(369\) − 21.7822i − 1.13394i
\(370\) − 4.68858i − 0.243748i
\(371\) − 13.6225i − 0.707246i
\(372\) 7.94295i 0.411823i
\(373\) 26.0569 1.34917 0.674587 0.738195i \(-0.264322\pi\)
0.674587 + 0.738195i \(0.264322\pi\)
\(374\) −6.67589 −0.345202
\(375\) − 25.8893i − 1.33692i
\(376\) −3.17059 −0.163511
\(377\) 0 0
\(378\) −0.345654 −0.0177785
\(379\) 15.6532i 0.804053i 0.915628 + 0.402026i \(0.131694\pi\)
−0.915628 + 0.402026i \(0.868306\pi\)
\(380\) −11.3617 −0.582843
\(381\) 21.2486 1.08860
\(382\) 10.9196i 0.558694i
\(383\) − 12.5588i − 0.641724i −0.947126 0.320862i \(-0.896027\pi\)
0.947126 0.320862i \(-0.103973\pi\)
\(384\) − 25.2193i − 1.28696i
\(385\) 3.10548i 0.158270i
\(386\) 17.3987 0.885571
\(387\) −0.810588 −0.0412045
\(388\) 23.4280i 1.18938i
\(389\) 0.503007 0.0255035 0.0127517 0.999919i \(-0.495941\pi\)
0.0127517 + 0.999919i \(0.495941\pi\)
\(390\) 0 0
\(391\) −24.6847 −1.24836
\(392\) 2.48280i 0.125400i
\(393\) −46.7838 −2.35993
\(394\) −4.76895 −0.240256
\(395\) 12.0041i 0.603992i
\(396\) 10.0751i 0.506295i
\(397\) 2.35044i 0.117965i 0.998259 + 0.0589827i \(0.0187857\pi\)
−0.998259 + 0.0589827i \(0.981214\pi\)
\(398\) − 3.46110i − 0.173489i
\(399\) 14.1841 0.710092
\(400\) 4.09567 0.204784
\(401\) 34.6046i 1.72807i 0.503429 + 0.864037i \(0.332072\pi\)
−0.503429 + 0.864037i \(0.667928\pi\)
\(402\) 12.9564 0.646208
\(403\) 0 0
\(404\) −1.02427 −0.0509593
\(405\) 12.3367i 0.613016i
\(406\) 3.98731 0.197887
\(407\) −12.3239 −0.610875
\(408\) 23.3355i 1.15528i
\(409\) − 8.02411i − 0.396767i −0.980125 0.198383i \(-0.936431\pi\)
0.980125 0.198383i \(-0.0635691\pi\)
\(410\) 7.13176i 0.352213i
\(411\) − 24.6475i − 1.21577i
\(412\) −26.8066 −1.32067
\(413\) −4.03056 −0.198331
\(414\) − 12.5676i − 0.617666i
\(415\) −1.75999 −0.0863943
\(416\) 0 0
\(417\) 19.6176 0.960676
\(418\) − 10.0748i − 0.492773i
\(419\) −11.6694 −0.570090 −0.285045 0.958514i \(-0.592008\pi\)
−0.285045 + 0.958514i \(0.592008\pi\)
\(420\) 4.64422 0.226615
\(421\) − 18.3381i − 0.893746i −0.894597 0.446873i \(-0.852538\pi\)
0.894597 0.446873i \(-0.147462\pi\)
\(422\) − 2.61916i − 0.127499i
\(423\) − 3.57298i − 0.173724i
\(424\) 33.8220i 1.64254i
\(425\) −13.0242 −0.631764
\(426\) 12.3608 0.598882
\(427\) − 4.60097i − 0.222657i
\(428\) 12.7049 0.614117
\(429\) 0 0
\(430\) 0.265397 0.0127986
\(431\) − 31.2435i − 1.50495i −0.658622 0.752474i \(-0.728860\pi\)
0.658622 0.752474i \(-0.271140\pi\)
\(432\) −0.597343 −0.0287397
\(433\) 30.5513 1.46820 0.734100 0.679041i \(-0.237604\pi\)
0.734100 + 0.679041i \(0.237604\pi\)
\(434\) 1.56674i 0.0752057i
\(435\) − 17.4331i − 0.835853i
\(436\) 18.0847i 0.866099i
\(437\) − 37.2524i − 1.78202i
\(438\) 25.6757 1.22683
\(439\) 12.1472 0.579756 0.289878 0.957064i \(-0.406385\pi\)
0.289878 + 0.957064i \(0.406385\pi\)
\(440\) − 7.71029i − 0.367573i
\(441\) −2.79790 −0.133233
\(442\) 0 0
\(443\) −8.46532 −0.402200 −0.201100 0.979571i \(-0.564452\pi\)
−0.201100 + 0.979571i \(0.564452\pi\)
\(444\) 18.4304i 0.874667i
\(445\) 1.16035 0.0550060
\(446\) 6.26065 0.296451
\(447\) − 23.5705i − 1.11485i
\(448\) − 1.69131i − 0.0799068i
\(449\) 23.3264i 1.10084i 0.834888 + 0.550420i \(0.185533\pi\)
−0.834888 + 0.550420i \(0.814467\pi\)
\(450\) − 6.63093i − 0.312585i
\(451\) 18.7459 0.882708
\(452\) −21.3148 −1.00256
\(453\) − 35.4938i − 1.66765i
\(454\) −9.55416 −0.448399
\(455\) 0 0
\(456\) −35.2162 −1.64915
\(457\) 15.4866i 0.724434i 0.932094 + 0.362217i \(0.117980\pi\)
−0.932094 + 0.362217i \(0.882020\pi\)
\(458\) −4.84115 −0.226212
\(459\) 1.89954 0.0886628
\(460\) − 12.1974i − 0.568705i
\(461\) − 9.28204i − 0.432308i −0.976359 0.216154i \(-0.930649\pi\)
0.976359 0.216154i \(-0.0693513\pi\)
\(462\) 4.11817i 0.191595i
\(463\) − 28.8283i − 1.33976i −0.742467 0.669882i \(-0.766345\pi\)
0.742467 0.669882i \(-0.233655\pi\)
\(464\) 6.89068 0.319892
\(465\) 6.85000 0.317661
\(466\) − 17.8049i − 0.824797i
\(467\) 2.87393 0.132990 0.0664948 0.997787i \(-0.478818\pi\)
0.0664948 + 0.997787i \(0.478818\pi\)
\(468\) 0 0
\(469\) −7.57559 −0.349808
\(470\) 1.16984i 0.0539606i
\(471\) 49.5649 2.28383
\(472\) 10.0071 0.460613
\(473\) − 0.697596i − 0.0320755i
\(474\) 15.9186i 0.731167i
\(475\) − 19.6551i − 0.901837i
\(476\) − 5.83747i − 0.267560i
\(477\) −38.1144 −1.74514
\(478\) −1.97829 −0.0904851
\(479\) 22.6082i 1.03299i 0.856289 + 0.516497i \(0.172764\pi\)
−0.856289 + 0.516497i \(0.827236\pi\)
\(480\) −18.1281 −0.827432
\(481\) 0 0
\(482\) −4.95659 −0.225766
\(483\) 15.2273i 0.692867i
\(484\) 7.77971 0.353623
\(485\) 20.2043 0.917432
\(486\) 15.3227i 0.695053i
\(487\) 2.86803i 0.129963i 0.997886 + 0.0649814i \(0.0206988\pi\)
−0.997886 + 0.0649814i \(0.979301\pi\)
\(488\) 11.4233i 0.517108i
\(489\) 9.58544i 0.433469i
\(490\) 0.916066 0.0413837
\(491\) 22.7201 1.02534 0.512672 0.858585i \(-0.328656\pi\)
0.512672 + 0.858585i \(0.328656\pi\)
\(492\) − 28.0343i − 1.26388i
\(493\) −21.9122 −0.986877
\(494\) 0 0
\(495\) 8.68881 0.390533
\(496\) 2.70756i 0.121573i
\(497\) −7.22732 −0.324190
\(498\) −2.33391 −0.104585
\(499\) − 18.0151i − 0.806466i −0.915097 0.403233i \(-0.867886\pi\)
0.915097 0.403233i \(-0.132114\pi\)
\(500\) − 16.0794i − 0.719091i
\(501\) − 20.8967i − 0.933596i
\(502\) − 0.556175i − 0.0248233i
\(503\) −31.1496 −1.38889 −0.694447 0.719544i \(-0.744351\pi\)
−0.694447 + 0.719544i \(0.744351\pi\)
\(504\) 6.94662 0.309427
\(505\) 0.883331i 0.0393077i
\(506\) 10.8158 0.480819
\(507\) 0 0
\(508\) 13.1972 0.585529
\(509\) − 39.5018i − 1.75089i −0.483319 0.875444i \(-0.660569\pi\)
0.483319 0.875444i \(-0.339431\pi\)
\(510\) 8.60999 0.381257
\(511\) −15.0125 −0.664115
\(512\) − 13.2606i − 0.586042i
\(513\) 2.86664i 0.126565i
\(514\) 2.31414i 0.102073i
\(515\) 23.1180i 1.01870i
\(516\) −1.04325 −0.0459265
\(517\) 3.07492 0.135235
\(518\) 3.63537i 0.159729i
\(519\) −2.24880 −0.0987115
\(520\) 0 0
\(521\) −17.7672 −0.778394 −0.389197 0.921155i \(-0.627247\pi\)
−0.389197 + 0.921155i \(0.627247\pi\)
\(522\) − 11.1561i − 0.488288i
\(523\) −1.78904 −0.0782294 −0.0391147 0.999235i \(-0.512454\pi\)
−0.0391147 + 0.999235i \(0.512454\pi\)
\(524\) −29.0566 −1.26934
\(525\) 8.03424i 0.350643i
\(526\) − 21.0124i − 0.916183i
\(527\) − 8.60999i − 0.375057i
\(528\) 7.11683i 0.309720i
\(529\) 16.9923 0.738797
\(530\) 12.4791 0.542059
\(531\) 11.2771i 0.489384i
\(532\) 8.80948 0.381939
\(533\) 0 0
\(534\) 1.53874 0.0665879
\(535\) − 10.9568i − 0.473702i
\(536\) 18.8087 0.812412
\(537\) −32.6174 −1.40754
\(538\) − 0.878269i − 0.0378649i
\(539\) − 2.40788i − 0.103715i
\(540\) 0.938610i 0.0403913i
\(541\) − 15.4027i − 0.662214i −0.943593 0.331107i \(-0.892578\pi\)
0.943593 0.331107i \(-0.107422\pi\)
\(542\) −17.9043 −0.769055
\(543\) 21.3403 0.915801
\(544\) 22.7858i 0.976935i
\(545\) 15.5962 0.668069
\(546\) 0 0
\(547\) −3.96944 −0.169721 −0.0848605 0.996393i \(-0.527044\pi\)
−0.0848605 + 0.996393i \(0.527044\pi\)
\(548\) − 15.3081i − 0.653931i
\(549\) −12.8730 −0.549408
\(550\) 5.70661 0.243331
\(551\) − 33.0683i − 1.40876i
\(552\) − 37.8064i − 1.60915i
\(553\) − 9.30758i − 0.395799i
\(554\) 6.77331i 0.287770i
\(555\) 15.8944 0.674678
\(556\) 12.1841 0.516722
\(557\) − 18.6901i − 0.791924i −0.918267 0.395962i \(-0.870411\pi\)
0.918267 0.395962i \(-0.129589\pi\)
\(558\) 4.38357 0.185571
\(559\) 0 0
\(560\) 1.58310 0.0668983
\(561\) − 22.6314i − 0.955497i
\(562\) −6.79422 −0.286597
\(563\) −36.3059 −1.53011 −0.765056 0.643964i \(-0.777289\pi\)
−0.765056 + 0.643964i \(0.777289\pi\)
\(564\) − 4.59852i − 0.193633i
\(565\) 18.3819i 0.773333i
\(566\) 8.12002i 0.341310i
\(567\) − 9.56546i − 0.401712i
\(568\) 17.9440 0.752914
\(569\) −10.9760 −0.460136 −0.230068 0.973175i \(-0.573895\pi\)
−0.230068 + 0.973175i \(0.573895\pi\)
\(570\) 12.9936i 0.544240i
\(571\) −31.3363 −1.31138 −0.655692 0.755028i \(-0.727623\pi\)
−0.655692 + 0.755028i \(0.727623\pi\)
\(572\) 0 0
\(573\) −37.0175 −1.54643
\(574\) − 5.52973i − 0.230806i
\(575\) 21.1007 0.879962
\(576\) −4.73210 −0.197171
\(577\) − 42.7876i − 1.78127i −0.454717 0.890636i \(-0.650260\pi\)
0.454717 0.890636i \(-0.349740\pi\)
\(578\) 1.25271i 0.0521057i
\(579\) 58.9819i 2.45120i
\(580\) − 10.8274i − 0.449583i
\(581\) 1.36463 0.0566145
\(582\) 26.7929 1.11060
\(583\) − 32.8014i − 1.35850i
\(584\) 37.2731 1.54237
\(585\) 0 0
\(586\) 6.64517 0.274509
\(587\) − 38.7886i − 1.60098i −0.599349 0.800488i \(-0.704574\pi\)
0.599349 0.800488i \(-0.295426\pi\)
\(588\) −3.60097 −0.148502
\(589\) 12.9936 0.535390
\(590\) − 3.69226i − 0.152008i
\(591\) − 16.1668i − 0.665014i
\(592\) 6.28247i 0.258208i
\(593\) 29.9564i 1.23016i 0.788464 + 0.615081i \(0.210877\pi\)
−0.788464 + 0.615081i \(0.789123\pi\)
\(594\) −0.832293 −0.0341494
\(595\) −5.03424 −0.206384
\(596\) − 14.6392i − 0.599645i
\(597\) 11.7332 0.480207
\(598\) 0 0
\(599\) 41.2539 1.68559 0.842794 0.538236i \(-0.180909\pi\)
0.842794 + 0.538236i \(0.180909\pi\)
\(600\) − 19.9474i − 0.814350i
\(601\) 6.61089 0.269664 0.134832 0.990868i \(-0.456951\pi\)
0.134832 + 0.990868i \(0.456951\pi\)
\(602\) −0.205780 −0.00838695
\(603\) 21.1957i 0.863156i
\(604\) − 22.0446i − 0.896982i
\(605\) − 6.70922i − 0.272769i
\(606\) 1.17138i 0.0475842i
\(607\) −20.8832 −0.847622 −0.423811 0.905751i \(-0.639308\pi\)
−0.423811 + 0.905751i \(0.639308\pi\)
\(608\) −34.3867 −1.39456
\(609\) 13.5170i 0.547738i
\(610\) 4.21479 0.170652
\(611\) 0 0
\(612\) −16.3326 −0.660208
\(613\) − 4.29655i − 0.173536i −0.996229 0.0867680i \(-0.972346\pi\)
0.996229 0.0867680i \(-0.0276539\pi\)
\(614\) −5.76362 −0.232601
\(615\) −24.1768 −0.974902
\(616\) 5.97829i 0.240872i
\(617\) − 30.7380i − 1.23746i −0.785602 0.618732i \(-0.787647\pi\)
0.785602 0.618732i \(-0.212353\pi\)
\(618\) 30.6568i 1.23320i
\(619\) 20.6417i 0.829658i 0.909899 + 0.414829i \(0.136159\pi\)
−0.909899 + 0.414829i \(0.863841\pi\)
\(620\) 4.25441 0.170861
\(621\) −3.07748 −0.123495
\(622\) − 6.13699i − 0.246071i
\(623\) −0.899698 −0.0360457
\(624\) 0 0
\(625\) 2.81636 0.112654
\(626\) 3.71290i 0.148397i
\(627\) 34.1536 1.36396
\(628\) 30.7839 1.22841
\(629\) − 19.9781i − 0.796580i
\(630\) − 2.56306i − 0.102115i
\(631\) − 29.3366i − 1.16787i −0.811799 0.583937i \(-0.801512\pi\)
0.811799 0.583937i \(-0.198488\pi\)
\(632\) 23.1089i 0.919222i
\(633\) 8.87899 0.352908
\(634\) 7.95019 0.315742
\(635\) − 11.3812i − 0.451650i
\(636\) −49.0543 −1.94513
\(637\) 0 0
\(638\) 9.60097 0.380106
\(639\) 20.2213i 0.799943i
\(640\) −13.5080 −0.533950
\(641\) 4.01680 0.158654 0.0793271 0.996849i \(-0.474723\pi\)
0.0793271 + 0.996849i \(0.474723\pi\)
\(642\) − 14.5297i − 0.573443i
\(643\) − 6.87282i − 0.271037i −0.990775 0.135519i \(-0.956730\pi\)
0.990775 0.135519i \(-0.0432701\pi\)
\(644\) 9.45742i 0.372675i
\(645\) 0.899698i 0.0354256i
\(646\) 16.3320 0.642574
\(647\) 33.7431 1.32658 0.663289 0.748363i \(-0.269160\pi\)
0.663289 + 0.748363i \(0.269160\pi\)
\(648\) 23.7491i 0.932955i
\(649\) −9.70511 −0.380959
\(650\) 0 0
\(651\) −5.31126 −0.208165
\(652\) 5.95335i 0.233151i
\(653\) 32.4002 1.26792 0.633958 0.773367i \(-0.281429\pi\)
0.633958 + 0.773367i \(0.281429\pi\)
\(654\) 20.6821 0.808735
\(655\) 25.0584i 0.979112i
\(656\) − 9.55622i − 0.373108i
\(657\) 42.0035i 1.63871i
\(658\) − 0.907052i − 0.0353606i
\(659\) 4.01035 0.156221 0.0781106 0.996945i \(-0.475111\pi\)
0.0781106 + 0.996945i \(0.475111\pi\)
\(660\) 11.1827 0.435287
\(661\) 1.81794i 0.0707097i 0.999375 + 0.0353549i \(0.0112561\pi\)
−0.999375 + 0.0353549i \(0.988744\pi\)
\(662\) 14.2389 0.553412
\(663\) 0 0
\(664\) −3.38811 −0.131484
\(665\) − 7.59729i − 0.294610i
\(666\) 10.1714 0.394133
\(667\) 35.5005 1.37459
\(668\) − 12.9786i − 0.502156i
\(669\) 21.2237i 0.820556i
\(670\) − 6.93974i − 0.268106i
\(671\) − 11.0786i − 0.427684i
\(672\) 14.0559 0.542220
\(673\) 22.9743 0.885594 0.442797 0.896622i \(-0.353986\pi\)
0.442797 + 0.896622i \(0.353986\pi\)
\(674\) 13.2018i 0.508515i
\(675\) −1.62374 −0.0624978
\(676\) 0 0
\(677\) 38.0276 1.46152 0.730760 0.682634i \(-0.239166\pi\)
0.730760 + 0.682634i \(0.239166\pi\)
\(678\) 24.3762i 0.936163i
\(679\) −15.6658 −0.601197
\(680\) 12.4990 0.479315
\(681\) − 32.3887i − 1.24114i
\(682\) 3.77252i 0.144457i
\(683\) 40.7786i 1.56035i 0.625560 + 0.780176i \(0.284870\pi\)
−0.625560 + 0.780176i \(0.715130\pi\)
\(684\) − 24.6480i − 0.942440i
\(685\) −13.2017 −0.504412
\(686\) −0.710287 −0.0271189
\(687\) − 16.4116i − 0.626140i
\(688\) −0.355619 −0.0135578
\(689\) 0 0
\(690\) −13.9492 −0.531038
\(691\) − 3.12868i − 0.119021i −0.998228 0.0595104i \(-0.981046\pi\)
0.998228 0.0595104i \(-0.0189539\pi\)
\(692\) −1.39669 −0.0530942
\(693\) −6.73701 −0.255918
\(694\) − 15.8690i − 0.602378i
\(695\) − 10.5076i − 0.398576i
\(696\) − 33.5601i − 1.27209i
\(697\) 30.3886i 1.15105i
\(698\) −6.24080 −0.236218
\(699\) 60.3590 2.28299
\(700\) 4.98992i 0.188601i
\(701\) −9.61382 −0.363109 −0.181555 0.983381i \(-0.558113\pi\)
−0.181555 + 0.983381i \(0.558113\pi\)
\(702\) 0 0
\(703\) 30.1495 1.13711
\(704\) − 4.07247i − 0.153487i
\(705\) −3.96576 −0.149359
\(706\) 5.48269 0.206344
\(707\) − 0.684905i − 0.0257585i
\(708\) 14.5139i 0.545467i
\(709\) − 28.1294i − 1.05642i −0.849113 0.528211i \(-0.822863\pi\)
0.849113 0.528211i \(-0.177137\pi\)
\(710\) − 6.62071i − 0.248471i
\(711\) −26.0417 −0.976638
\(712\) 2.23377 0.0837142
\(713\) 13.9492i 0.522403i
\(714\) −6.67589 −0.249839
\(715\) 0 0
\(716\) −20.2581 −0.757080
\(717\) − 6.70645i − 0.250457i
\(718\) 7.52594 0.280865
\(719\) −24.9044 −0.928779 −0.464389 0.885631i \(-0.653726\pi\)
−0.464389 + 0.885631i \(0.653726\pi\)
\(720\) − 4.42936i − 0.165073i
\(721\) − 17.9249i − 0.667559i
\(722\) 11.1516i 0.415020i
\(723\) − 16.8029i − 0.624907i
\(724\) 13.2541 0.492584
\(725\) 18.7308 0.695643
\(726\) − 8.89709i − 0.330202i
\(727\) 24.2120 0.897974 0.448987 0.893538i \(-0.351785\pi\)
0.448987 + 0.893538i \(0.351785\pi\)
\(728\) 0 0
\(729\) −23.2479 −0.861032
\(730\) − 13.7525i − 0.509002i
\(731\) 1.13086 0.0418264
\(732\) −16.5680 −0.612369
\(733\) − 12.3989i − 0.457963i −0.973431 0.228981i \(-0.926461\pi\)
0.973431 0.228981i \(-0.0735395\pi\)
\(734\) − 4.13434i − 0.152601i
\(735\) 3.10548i 0.114547i
\(736\) − 36.9159i − 1.36074i
\(737\) −18.2411 −0.671921
\(738\) −15.4716 −0.569518
\(739\) 10.9604i 0.403184i 0.979470 + 0.201592i \(0.0646115\pi\)
−0.979470 + 0.201592i \(0.935388\pi\)
\(740\) 9.87170 0.362891
\(741\) 0 0
\(742\) −9.67589 −0.355213
\(743\) 40.6925i 1.49286i 0.665463 + 0.746431i \(0.268234\pi\)
−0.665463 + 0.746431i \(0.731766\pi\)
\(744\) 13.1868 0.483452
\(745\) −12.6249 −0.462539
\(746\) − 18.5079i − 0.677621i
\(747\) − 3.81810i − 0.139697i
\(748\) − 14.0559i − 0.513936i
\(749\) 8.49549i 0.310419i
\(750\) −18.3888 −0.671464
\(751\) −9.40472 −0.343183 −0.171592 0.985168i \(-0.554891\pi\)
−0.171592 + 0.985168i \(0.554891\pi\)
\(752\) − 1.56753i − 0.0571618i
\(753\) 1.88544 0.0687093
\(754\) 0 0
\(755\) −19.0113 −0.691890
\(756\) − 0.727766i − 0.0264686i
\(757\) −29.5808 −1.07513 −0.537566 0.843222i \(-0.680656\pi\)
−0.537566 + 0.843222i \(0.680656\pi\)
\(758\) 11.1183 0.403834
\(759\) 36.6656i 1.33088i
\(760\) 18.8626i 0.684218i
\(761\) − 16.9417i − 0.614137i −0.951687 0.307068i \(-0.900652\pi\)
0.951687 0.307068i \(-0.0993481\pi\)
\(762\) − 15.0926i − 0.546748i
\(763\) −12.0928 −0.437788
\(764\) −22.9909 −0.831783
\(765\) 14.0853i 0.509254i
\(766\) −8.92034 −0.322305
\(767\) 0 0
\(768\) −26.0578 −0.940281
\(769\) 7.36574i 0.265616i 0.991142 + 0.132808i \(0.0423993\pi\)
−0.991142 + 0.132808i \(0.957601\pi\)
\(770\) 2.20578 0.0794908
\(771\) −7.84498 −0.282530
\(772\) 36.6326i 1.31844i
\(773\) − 44.7734i − 1.61039i −0.593012 0.805194i \(-0.702061\pi\)
0.593012 0.805194i \(-0.297939\pi\)
\(774\) 0.575750i 0.0206949i
\(775\) 7.35989i 0.264375i
\(776\) 38.8950 1.39625
\(777\) −12.3239 −0.442119
\(778\) − 0.357279i − 0.0128091i
\(779\) −45.8602 −1.64311
\(780\) 0 0
\(781\) −17.4025 −0.622712
\(782\) 17.5332i 0.626988i
\(783\) −2.73183 −0.0976277
\(784\) −1.22748 −0.0438387
\(785\) − 26.5480i − 0.947540i
\(786\) 33.2299i 1.18527i
\(787\) 18.7682i 0.669016i 0.942393 + 0.334508i \(0.108570\pi\)
−0.942393 + 0.334508i \(0.891430\pi\)
\(788\) − 10.0409i − 0.357693i
\(789\) 71.2322 2.53594
\(790\) 8.52636 0.303354
\(791\) − 14.2527i − 0.506768i
\(792\) 16.7267 0.594356
\(793\) 0 0
\(794\) 1.66949 0.0592479
\(795\) 42.3044i 1.50038i
\(796\) 7.28726 0.258290
\(797\) −7.21697 −0.255638 −0.127819 0.991797i \(-0.540798\pi\)
−0.127819 + 0.991797i \(0.540798\pi\)
\(798\) − 10.0748i − 0.356643i
\(799\) 4.98470i 0.176346i
\(800\) − 19.4775i − 0.688635i
\(801\) 2.51726i 0.0889431i
\(802\) 24.5792 0.867922
\(803\) −36.1484 −1.27565
\(804\) 27.2795i 0.962073i
\(805\) 8.15608 0.287464
\(806\) 0 0
\(807\) 2.97734 0.104807
\(808\) 1.70048i 0.0598228i
\(809\) 14.8318 0.521459 0.260729 0.965412i \(-0.416037\pi\)
0.260729 + 0.965412i \(0.416037\pi\)
\(810\) 8.76260 0.307886
\(811\) 18.5831i 0.652541i 0.945276 + 0.326271i \(0.105792\pi\)
−0.945276 + 0.326271i \(0.894208\pi\)
\(812\) 8.39519i 0.294613i
\(813\) − 60.6958i − 2.12869i
\(814\) 8.75354i 0.306811i
\(815\) 5.13417 0.179842
\(816\) −11.5370 −0.403875
\(817\) 1.70661i 0.0597067i
\(818\) −5.69942 −0.199275
\(819\) 0 0
\(820\) −15.0158 −0.524374
\(821\) − 26.6236i − 0.929169i −0.885529 0.464585i \(-0.846204\pi\)
0.885529 0.464585i \(-0.153796\pi\)
\(822\) −17.5068 −0.610620
\(823\) −6.75136 −0.235338 −0.117669 0.993053i \(-0.537542\pi\)
−0.117669 + 0.993053i \(0.537542\pi\)
\(824\) 44.5040i 1.55037i
\(825\) 19.3455i 0.673524i
\(826\) 2.86285i 0.0996114i
\(827\) 21.7430i 0.756079i 0.925789 + 0.378039i \(0.123402\pi\)
−0.925789 + 0.378039i \(0.876598\pi\)
\(828\) 26.4609 0.919579
\(829\) 14.6216 0.507828 0.253914 0.967227i \(-0.418282\pi\)
0.253914 + 0.967227i \(0.418282\pi\)
\(830\) 1.25009i 0.0433914i
\(831\) −22.9616 −0.796529
\(832\) 0 0
\(833\) 3.90338 0.135244
\(834\) − 13.9341i − 0.482498i
\(835\) −11.1927 −0.387340
\(836\) 21.2122 0.733639
\(837\) − 1.07342i − 0.0371028i
\(838\) 8.28865i 0.286327i
\(839\) 10.7469i 0.371023i 0.982642 + 0.185511i \(0.0593942\pi\)
−0.982642 + 0.185511i \(0.940606\pi\)
\(840\) − 7.71029i − 0.266030i
\(841\) 2.51320 0.0866620
\(842\) −13.0253 −0.448883
\(843\) − 23.0325i − 0.793282i
\(844\) 5.51458 0.189820
\(845\) 0 0
\(846\) −2.53784 −0.0872527
\(847\) 5.20210i 0.178746i
\(848\) −16.7214 −0.574216
\(849\) −27.5270 −0.944724
\(850\) 9.25088i 0.317303i
\(851\) 32.3670i 1.10953i
\(852\) 26.0254i 0.891615i
\(853\) 31.7709i 1.08782i 0.839145 + 0.543908i \(0.183056\pi\)
−0.839145 + 0.543908i \(0.816944\pi\)
\(854\) −3.26801 −0.111829
\(855\) −21.2564 −0.726955
\(856\) − 21.0926i − 0.720931i
\(857\) 0.648105 0.0221388 0.0110694 0.999939i \(-0.496476\pi\)
0.0110694 + 0.999939i \(0.496476\pi\)
\(858\) 0 0
\(859\) −43.1902 −1.47363 −0.736815 0.676095i \(-0.763671\pi\)
−0.736815 + 0.676095i \(0.763671\pi\)
\(860\) 0.558787i 0.0190545i
\(861\) 18.7459 0.638857
\(862\) −22.1919 −0.755858
\(863\) 21.6094i 0.735592i 0.929906 + 0.367796i \(0.119888\pi\)
−0.929906 + 0.367796i \(0.880112\pi\)
\(864\) 2.84074i 0.0966441i
\(865\) 1.20451i 0.0409545i
\(866\) − 21.7002i − 0.737401i
\(867\) −4.24669 −0.144225
\(868\) −3.29873 −0.111966
\(869\) − 22.4116i − 0.760260i
\(870\) −12.3825 −0.419806
\(871\) 0 0
\(872\) 30.0240 1.01674
\(873\) 43.8312i 1.48346i
\(874\) −26.4599 −0.895018
\(875\) 10.7519 0.363480
\(876\) 54.0597i 1.82651i
\(877\) − 0.880766i − 0.0297413i −0.999889 0.0148707i \(-0.995266\pi\)
0.999889 0.0148707i \(-0.00473366\pi\)
\(878\) − 8.62801i − 0.291181i
\(879\) 22.5272i 0.759825i
\(880\) 3.81193 0.128500
\(881\) 0.140035 0.00471791 0.00235895 0.999997i \(-0.499249\pi\)
0.00235895 + 0.999997i \(0.499249\pi\)
\(882\) 1.98731i 0.0669162i
\(883\) 18.8253 0.633522 0.316761 0.948505i \(-0.397405\pi\)
0.316761 + 0.948505i \(0.397405\pi\)
\(884\) 0 0
\(885\) 12.5168 0.420748
\(886\) 6.01281i 0.202004i
\(887\) −33.3010 −1.11814 −0.559069 0.829121i \(-0.688841\pi\)
−0.559069 + 0.829121i \(0.688841\pi\)
\(888\) 30.5979 1.02680
\(889\) 8.82462i 0.295968i
\(890\) − 0.824183i − 0.0276267i
\(891\) − 23.0325i − 0.771618i
\(892\) 13.1817i 0.441355i
\(893\) −7.52254 −0.251732
\(894\) −16.7418 −0.559929
\(895\) 17.4706i 0.583977i
\(896\) 10.4736 0.349899
\(897\) 0 0
\(898\) 16.5684 0.552896
\(899\) 12.3825i 0.412980i
\(900\) 13.9613 0.465376
\(901\) 53.1738 1.77148
\(902\) − 13.3149i − 0.443339i
\(903\) − 0.697596i − 0.0232145i
\(904\) 35.3866i 1.17694i
\(905\) − 11.4303i − 0.379957i
\(906\) −25.2108 −0.837573
\(907\) −11.5870 −0.384740 −0.192370 0.981322i \(-0.561617\pi\)
−0.192370 + 0.981322i \(0.561617\pi\)
\(908\) − 20.1161i − 0.667575i
\(909\) −1.91629 −0.0635594
\(910\) 0 0
\(911\) 52.5489 1.74102 0.870512 0.492147i \(-0.163788\pi\)
0.870512 + 0.492147i \(0.163788\pi\)
\(912\) − 17.4107i − 0.576527i
\(913\) 3.28588 0.108747
\(914\) 11.0000 0.363846
\(915\) 14.2882i 0.472354i
\(916\) − 10.1929i − 0.336784i
\(917\) − 19.4294i − 0.641616i
\(918\) − 1.34922i − 0.0445308i
\(919\) −20.0534 −0.661502 −0.330751 0.943718i \(-0.607302\pi\)
−0.330751 + 0.943718i \(0.607302\pi\)
\(920\) −20.2499 −0.667621
\(921\) − 19.5387i − 0.643823i
\(922\) −6.59291 −0.217126
\(923\) 0 0
\(924\) −8.67071 −0.285246
\(925\) 17.0775i 0.561504i
\(926\) −20.4764 −0.672895
\(927\) −50.1521 −1.64721
\(928\) − 32.7696i − 1.07571i
\(929\) − 15.9085i − 0.521941i −0.965347 0.260971i \(-0.915957\pi\)
0.965347 0.260971i \(-0.0840426\pi\)
\(930\) − 4.86546i − 0.159545i
\(931\) 5.89068i 0.193059i
\(932\) 37.4879 1.22796
\(933\) 20.8045 0.681108
\(934\) − 2.04131i − 0.0667938i
\(935\) −12.1218 −0.396427
\(936\) 0 0
\(937\) 26.1978 0.855846 0.427923 0.903815i \(-0.359245\pi\)
0.427923 + 0.903815i \(0.359245\pi\)
\(938\) 5.38084i 0.175691i
\(939\) −12.5868 −0.410754
\(940\) −2.46307 −0.0803364
\(941\) − 36.3059i − 1.18354i −0.806108 0.591769i \(-0.798430\pi\)
0.806108 0.591769i \(-0.201570\pi\)
\(942\) − 35.2053i − 1.14705i
\(943\) − 49.2332i − 1.60325i
\(944\) 4.94745i 0.161026i
\(945\) −0.627626 −0.0204167
\(946\) −0.495493 −0.0161099
\(947\) 0.918839i 0.0298582i 0.999889 + 0.0149291i \(0.00475226\pi\)
−0.999889 + 0.0149291i \(0.995248\pi\)
\(948\) −33.5163 −1.08856
\(949\) 0 0
\(950\) −13.9607 −0.452946
\(951\) 26.9513i 0.873954i
\(952\) −9.69131 −0.314097
\(953\) 18.3589 0.594702 0.297351 0.954768i \(-0.403897\pi\)
0.297351 + 0.954768i \(0.403897\pi\)
\(954\) 27.0721i 0.876493i
\(955\) 19.8274i 0.641599i
\(956\) − 4.16526i − 0.134714i
\(957\) 32.5474i 1.05211i
\(958\) 16.0583 0.518819
\(959\) 10.2362 0.330543
\(960\) 5.25232i 0.169518i
\(961\) 26.1345 0.843050
\(962\) 0 0
\(963\) 23.7695 0.765962
\(964\) − 10.4360i − 0.336121i
\(965\) 31.5920 1.01698
\(966\) 10.8158 0.347992
\(967\) 12.5923i 0.404940i 0.979288 + 0.202470i \(0.0648969\pi\)
−0.979288 + 0.202470i \(0.935103\pi\)
\(968\) − 12.9158i − 0.415129i
\(969\) 55.3658i 1.77860i
\(970\) − 14.3509i − 0.460779i
\(971\) −21.3308 −0.684537 −0.342269 0.939602i \(-0.611195\pi\)
−0.342269 + 0.939602i \(0.611195\pi\)
\(972\) −32.2617 −1.03479
\(973\) 8.14723i 0.261188i
\(974\) 2.03712 0.0652736
\(975\) 0 0
\(976\) −5.64762 −0.180776
\(977\) − 42.4279i − 1.35739i −0.734420 0.678695i \(-0.762546\pi\)
0.734420 0.678695i \(-0.237454\pi\)
\(978\) 6.80841 0.217709
\(979\) −2.16637 −0.0692374
\(980\) 1.92876i 0.0616119i
\(981\) 33.8344i 1.08025i
\(982\) − 16.1378i − 0.514977i
\(983\) − 2.09758i − 0.0669023i −0.999440 0.0334511i \(-0.989350\pi\)
0.999440 0.0334511i \(-0.0106498\pi\)
\(984\) −46.5423 −1.48371
\(985\) −8.65930 −0.275908
\(986\) 15.5640i 0.495658i
\(987\) 3.07492 0.0978758
\(988\) 0 0
\(989\) −1.83213 −0.0582584
\(990\) − 6.17154i − 0.196145i
\(991\) −27.6349 −0.877851 −0.438926 0.898523i \(-0.644641\pi\)
−0.438926 + 0.898523i \(0.644641\pi\)
\(992\) 12.8762 0.408819
\(993\) 48.2703i 1.53181i
\(994\) 5.13347i 0.162824i
\(995\) − 6.28454i − 0.199233i
\(996\) − 4.91400i − 0.155706i
\(997\) −2.02783 −0.0642221 −0.0321110 0.999484i \(-0.510223\pi\)
−0.0321110 + 0.999484i \(0.510223\pi\)
\(998\) −12.7959 −0.405047
\(999\) − 2.49070i − 0.0788024i
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 1183.2.c.g.337.3 8
13.2 odd 12 91.2.f.c.22.3 8
13.5 odd 4 1183.2.a.k.1.2 4
13.6 odd 12 91.2.f.c.29.3 yes 8
13.8 odd 4 1183.2.a.l.1.3 4
13.12 even 2 inner 1183.2.c.g.337.6 8
39.2 even 12 819.2.o.h.568.2 8
39.32 even 12 819.2.o.h.757.2 8
52.15 even 12 1456.2.s.q.113.4 8
52.19 even 12 1456.2.s.q.1121.4 8
91.2 odd 12 637.2.h.h.165.2 8
91.6 even 12 637.2.f.i.393.3 8
91.19 even 12 637.2.g.j.263.3 8
91.32 odd 12 637.2.h.h.471.2 8
91.34 even 4 8281.2.a.bt.1.3 4
91.41 even 12 637.2.f.i.295.3 8
91.45 even 12 637.2.h.i.471.2 8
91.54 even 12 637.2.h.i.165.2 8
91.58 odd 12 637.2.g.k.263.3 8
91.67 odd 12 637.2.g.k.373.3 8
91.80 even 12 637.2.g.j.373.3 8
91.83 even 4 8281.2.a.bp.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.3 8 13.2 odd 12
91.2.f.c.29.3 yes 8 13.6 odd 12
637.2.f.i.295.3 8 91.41 even 12
637.2.f.i.393.3 8 91.6 even 12
637.2.g.j.263.3 8 91.19 even 12
637.2.g.j.373.3 8 91.80 even 12
637.2.g.k.263.3 8 91.58 odd 12
637.2.g.k.373.3 8 91.67 odd 12
637.2.h.h.165.2 8 91.2 odd 12
637.2.h.h.471.2 8 91.32 odd 12
637.2.h.i.165.2 8 91.54 even 12
637.2.h.i.471.2 8 91.45 even 12
819.2.o.h.568.2 8 39.2 even 12
819.2.o.h.757.2 8 39.32 even 12
1183.2.a.k.1.2 4 13.5 odd 4
1183.2.a.l.1.3 4 13.8 odd 4
1183.2.c.g.337.3 8 1.1 even 1 trivial
1183.2.c.g.337.6 8 13.12 even 2 inner
1456.2.s.q.113.4 8 52.15 even 12
1456.2.s.q.1121.4 8 52.19 even 12
8281.2.a.bp.1.2 4 91.83 even 4
8281.2.a.bt.1.3 4 91.34 even 4