Properties

Label 912.3.be.c.145.1
Level $912$
Weight $3$
Character 912.145
Analytic conductor $24.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,3,Mod(145,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.145
Dual form 912.3.be.c.673.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +1.00000 q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +1.00000 q^{7} +(1.50000 - 2.59808i) q^{9} -16.0000 q^{11} +(13.5000 + 7.79423i) q^{13} +(-3.00000 - 1.73205i) q^{15} +(-11.0000 - 19.0526i) q^{17} -19.0000 q^{19} +(-1.50000 + 0.866025i) q^{21} +(20.0000 - 34.6410i) q^{23} +(10.5000 - 18.1865i) q^{25} +5.19615i q^{27} +(15.0000 + 8.66025i) q^{29} +50.2295i q^{31} +(24.0000 - 13.8564i) q^{33} +(1.00000 + 1.73205i) q^{35} +15.5885i q^{37} -27.0000 q^{39} +(-12.0000 + 6.92820i) q^{41} +(-24.5000 - 42.4352i) q^{43} +6.00000 q^{45} +(23.0000 - 39.8372i) q^{47} -48.0000 q^{49} +(33.0000 + 19.0526i) q^{51} +(-42.0000 - 24.2487i) q^{53} +(-16.0000 - 27.7128i) q^{55} +(28.5000 - 16.4545i) q^{57} +(57.0000 - 32.9090i) q^{59} +(48.5000 - 84.0045i) q^{61} +(1.50000 - 2.59808i) q^{63} +31.1769i q^{65} +(22.5000 + 12.9904i) q^{67} +69.2820i q^{69} +(42.0000 - 24.2487i) q^{71} +(-17.5000 - 30.3109i) q^{73} +36.3731i q^{75} -16.0000 q^{77} +(76.5000 - 44.1673i) q^{79} +(-4.50000 - 7.79423i) q^{81} +146.000 q^{83} +(22.0000 - 38.1051i) q^{85} -30.0000 q^{87} +(-33.0000 - 19.0526i) q^{89} +(13.5000 + 7.79423i) q^{91} +(-43.5000 - 75.3442i) q^{93} +(-19.0000 - 32.9090i) q^{95} +(54.0000 - 31.1769i) q^{97} +(-24.0000 + 41.5692i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 32 q^{11} + 27 q^{13} - 6 q^{15} - 22 q^{17} - 38 q^{19} - 3 q^{21} + 40 q^{23} + 21 q^{25} + 30 q^{29} + 48 q^{33} + 2 q^{35} - 54 q^{39} - 24 q^{41} - 49 q^{43} + 12 q^{45} + 46 q^{47} - 96 q^{49} + 66 q^{51} - 84 q^{53} - 32 q^{55} + 57 q^{57} + 114 q^{59} + 97 q^{61} + 3 q^{63} + 45 q^{67} + 84 q^{71} - 35 q^{73} - 32 q^{77} + 153 q^{79} - 9 q^{81} + 292 q^{83} + 44 q^{85} - 60 q^{87} - 66 q^{89} + 27 q^{91} - 87 q^{93} - 38 q^{95} + 108 q^{97} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.50000 + 0.866025i −0.500000 + 0.288675i
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.200000 + 0.346410i 0.948528 0.316693i \(-0.102572\pi\)
−0.748528 + 0.663103i \(0.769239\pi\)
\(6\) 0 0
\(7\) 1.00000 0.142857 0.0714286 0.997446i \(-0.477244\pi\)
0.0714286 + 0.997446i \(0.477244\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.166667 0.288675i
\(10\) 0 0
\(11\) −16.0000 −1.45455 −0.727273 0.686349i \(-0.759212\pi\)
−0.727273 + 0.686349i \(0.759212\pi\)
\(12\) 0 0
\(13\) 13.5000 + 7.79423i 1.03846 + 0.599556i 0.919397 0.393330i \(-0.128677\pi\)
0.119064 + 0.992887i \(0.462011\pi\)
\(14\) 0 0
\(15\) −3.00000 1.73205i −0.200000 0.115470i
\(16\) 0 0
\(17\) −11.0000 19.0526i −0.647059 1.12074i −0.983822 0.179149i \(-0.942665\pi\)
0.336763 0.941589i \(-0.390668\pi\)
\(18\) 0 0
\(19\) −19.0000 −1.00000
\(20\) 0 0
\(21\) −1.50000 + 0.866025i −0.0714286 + 0.0412393i
\(22\) 0 0
\(23\) 20.0000 34.6410i 0.869565 1.50613i 0.00712357 0.999975i \(-0.497732\pi\)
0.862442 0.506157i \(-0.168934\pi\)
\(24\) 0 0
\(25\) 10.5000 18.1865i 0.420000 0.727461i
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 15.0000 + 8.66025i 0.517241 + 0.298629i 0.735805 0.677193i \(-0.236804\pi\)
−0.218564 + 0.975823i \(0.570137\pi\)
\(30\) 0 0
\(31\) 50.2295i 1.62031i 0.586219 + 0.810153i \(0.300616\pi\)
−0.586219 + 0.810153i \(0.699384\pi\)
\(32\) 0 0
\(33\) 24.0000 13.8564i 0.727273 0.419891i
\(34\) 0 0
\(35\) 1.00000 + 1.73205i 0.0285714 + 0.0494872i
\(36\) 0 0
\(37\) 15.5885i 0.421310i 0.977561 + 0.210655i \(0.0675596\pi\)
−0.977561 + 0.210655i \(0.932440\pi\)
\(38\) 0 0
\(39\) −27.0000 −0.692308
\(40\) 0 0
\(41\) −12.0000 + 6.92820i −0.292683 + 0.168981i −0.639151 0.769081i \(-0.720714\pi\)
0.346468 + 0.938062i \(0.387381\pi\)
\(42\) 0 0
\(43\) −24.5000 42.4352i −0.569767 0.986866i −0.996589 0.0825301i \(-0.973700\pi\)
0.426821 0.904336i \(-0.359633\pi\)
\(44\) 0 0
\(45\) 6.00000 0.133333
\(46\) 0 0
\(47\) 23.0000 39.8372i 0.489362 0.847599i −0.510563 0.859840i \(-0.670563\pi\)
0.999925 + 0.0122408i \(0.00389646\pi\)
\(48\) 0 0
\(49\) −48.0000 −0.979592
\(50\) 0 0
\(51\) 33.0000 + 19.0526i 0.647059 + 0.373580i
\(52\) 0 0
\(53\) −42.0000 24.2487i −0.792453 0.457523i 0.0483725 0.998829i \(-0.484597\pi\)
−0.840825 + 0.541307i \(0.817930\pi\)
\(54\) 0 0
\(55\) −16.0000 27.7128i −0.290909 0.503869i
\(56\) 0 0
\(57\) 28.5000 16.4545i 0.500000 0.288675i
\(58\) 0 0
\(59\) 57.0000 32.9090i 0.966102 0.557779i 0.0680561 0.997681i \(-0.478320\pi\)
0.898046 + 0.439902i \(0.144987\pi\)
\(60\) 0 0
\(61\) 48.5000 84.0045i 0.795082 1.37712i −0.127705 0.991812i \(-0.540761\pi\)
0.922787 0.385310i \(-0.125906\pi\)
\(62\) 0 0
\(63\) 1.50000 2.59808i 0.0238095 0.0412393i
\(64\) 0 0
\(65\) 31.1769i 0.479645i
\(66\) 0 0
\(67\) 22.5000 + 12.9904i 0.335821 + 0.193886i 0.658422 0.752649i \(-0.271224\pi\)
−0.322602 + 0.946535i \(0.604557\pi\)
\(68\) 0 0
\(69\) 69.2820i 1.00409i
\(70\) 0 0
\(71\) 42.0000 24.2487i 0.591549 0.341531i −0.174161 0.984717i \(-0.555721\pi\)
0.765710 + 0.643186i \(0.222388\pi\)
\(72\) 0 0
\(73\) −17.5000 30.3109i −0.239726 0.415218i 0.720910 0.693029i \(-0.243724\pi\)
−0.960636 + 0.277811i \(0.910391\pi\)
\(74\) 0 0
\(75\) 36.3731i 0.484974i
\(76\) 0 0
\(77\) −16.0000 −0.207792
\(78\) 0 0
\(79\) 76.5000 44.1673i 0.968354 0.559080i 0.0696203 0.997574i \(-0.477821\pi\)
0.898734 + 0.438494i \(0.144488\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 146.000 1.75904 0.879518 0.475865i \(-0.157865\pi\)
0.879518 + 0.475865i \(0.157865\pi\)
\(84\) 0 0
\(85\) 22.0000 38.1051i 0.258824 0.448296i
\(86\) 0 0
\(87\) −30.0000 −0.344828
\(88\) 0 0
\(89\) −33.0000 19.0526i −0.370787 0.214074i 0.303015 0.952986i \(-0.402007\pi\)
−0.673802 + 0.738912i \(0.735340\pi\)
\(90\) 0 0
\(91\) 13.5000 + 7.79423i 0.148352 + 0.0856509i
\(92\) 0 0
\(93\) −43.5000 75.3442i −0.467742 0.810153i
\(94\) 0 0
\(95\) −19.0000 32.9090i −0.200000 0.346410i
\(96\) 0 0
\(97\) 54.0000 31.1769i 0.556701 0.321411i −0.195119 0.980780i \(-0.562509\pi\)
0.751820 + 0.659368i \(0.229176\pi\)
\(98\) 0 0
\(99\) −24.0000 + 41.5692i −0.242424 + 0.419891i
\(100\) 0 0
\(101\) −14.0000 + 24.2487i −0.138614 + 0.240086i −0.926972 0.375130i \(-0.877598\pi\)
0.788358 + 0.615216i \(0.210931\pi\)
\(102\) 0 0
\(103\) 126.440i 1.22757i −0.789473 0.613785i \(-0.789646\pi\)
0.789473 0.613785i \(-0.210354\pi\)
\(104\) 0 0
\(105\) −3.00000 1.73205i −0.0285714 0.0164957i
\(106\) 0 0
\(107\) 83.1384i 0.776995i −0.921450 0.388497i \(-0.872994\pi\)
0.921450 0.388497i \(-0.127006\pi\)
\(108\) 0 0
\(109\) −66.0000 + 38.1051i −0.605505 + 0.349588i −0.771204 0.636588i \(-0.780345\pi\)
0.165699 + 0.986176i \(0.447012\pi\)
\(110\) 0 0
\(111\) −13.5000 23.3827i −0.121622 0.210655i
\(112\) 0 0
\(113\) 183.597i 1.62476i −0.583131 0.812378i \(-0.698173\pi\)
0.583131 0.812378i \(-0.301827\pi\)
\(114\) 0 0
\(115\) 80.0000 0.695652
\(116\) 0 0
\(117\) 40.5000 23.3827i 0.346154 0.199852i
\(118\) 0 0
\(119\) −11.0000 19.0526i −0.0924370 0.160106i
\(120\) 0 0
\(121\) 135.000 1.11570
\(122\) 0 0
\(123\) 12.0000 20.7846i 0.0975610 0.168981i
\(124\) 0 0
\(125\) 92.0000 0.736000
\(126\) 0 0
\(127\) −24.0000 13.8564i −0.188976 0.109106i 0.402527 0.915408i \(-0.368132\pi\)
−0.591503 + 0.806303i \(0.701465\pi\)
\(128\) 0 0
\(129\) 73.5000 + 42.4352i 0.569767 + 0.328955i
\(130\) 0 0
\(131\) −64.0000 110.851i −0.488550 0.846193i 0.511364 0.859364i \(-0.329141\pi\)
−0.999913 + 0.0131717i \(0.995807\pi\)
\(132\) 0 0
\(133\) −19.0000 −0.142857
\(134\) 0 0
\(135\) −9.00000 + 5.19615i −0.0666667 + 0.0384900i
\(136\) 0 0
\(137\) 58.0000 100.459i 0.423358 0.733277i −0.572908 0.819620i \(-0.694185\pi\)
0.996265 + 0.0863428i \(0.0275180\pi\)
\(138\) 0 0
\(139\) −87.5000 + 151.554i −0.629496 + 1.09032i 0.358156 + 0.933662i \(0.383405\pi\)
−0.987653 + 0.156658i \(0.949928\pi\)
\(140\) 0 0
\(141\) 79.6743i 0.565066i
\(142\) 0 0
\(143\) −216.000 124.708i −1.51049 0.872082i
\(144\) 0 0
\(145\) 34.6410i 0.238904i
\(146\) 0 0
\(147\) 72.0000 41.5692i 0.489796 0.282784i
\(148\) 0 0
\(149\) −74.0000 128.172i −0.496644 0.860213i 0.503348 0.864084i \(-0.332101\pi\)
−0.999993 + 0.00387051i \(0.998768\pi\)
\(150\) 0 0
\(151\) 193.990i 1.28470i 0.766411 + 0.642350i \(0.222040\pi\)
−0.766411 + 0.642350i \(0.777960\pi\)
\(152\) 0 0
\(153\) −66.0000 −0.431373
\(154\) 0 0
\(155\) −87.0000 + 50.2295i −0.561290 + 0.324061i
\(156\) 0 0
\(157\) 96.5000 + 167.143i 0.614650 + 1.06460i 0.990446 + 0.137902i \(0.0440359\pi\)
−0.375796 + 0.926702i \(0.622631\pi\)
\(158\) 0 0
\(159\) 84.0000 0.528302
\(160\) 0 0
\(161\) 20.0000 34.6410i 0.124224 0.215162i
\(162\) 0 0
\(163\) −233.000 −1.42945 −0.714724 0.699407i \(-0.753448\pi\)
−0.714724 + 0.699407i \(0.753448\pi\)
\(164\) 0 0
\(165\) 48.0000 + 27.7128i 0.290909 + 0.167956i
\(166\) 0 0
\(167\) −63.0000 36.3731i −0.377246 0.217803i 0.299374 0.954136i \(-0.403222\pi\)
−0.676619 + 0.736333i \(0.736556\pi\)
\(168\) 0 0
\(169\) 37.0000 + 64.0859i 0.218935 + 0.379206i
\(170\) 0 0
\(171\) −28.5000 + 49.3634i −0.166667 + 0.288675i
\(172\) 0 0
\(173\) 99.0000 57.1577i 0.572254 0.330391i −0.185795 0.982589i \(-0.559486\pi\)
0.758049 + 0.652197i \(0.226153\pi\)
\(174\) 0 0
\(175\) 10.5000 18.1865i 0.0600000 0.103923i
\(176\) 0 0
\(177\) −57.0000 + 98.7269i −0.322034 + 0.557779i
\(178\) 0 0
\(179\) 83.1384i 0.464461i 0.972661 + 0.232230i \(0.0746023\pi\)
−0.972661 + 0.232230i \(0.925398\pi\)
\(180\) 0 0
\(181\) 42.0000 + 24.2487i 0.232044 + 0.133971i 0.611515 0.791233i \(-0.290561\pi\)
−0.379471 + 0.925204i \(0.623894\pi\)
\(182\) 0 0
\(183\) 168.009i 0.918082i
\(184\) 0 0
\(185\) −27.0000 + 15.5885i −0.145946 + 0.0842619i
\(186\) 0 0
\(187\) 176.000 + 304.841i 0.941176 + 1.63017i
\(188\) 0 0
\(189\) 5.19615i 0.0274929i
\(190\) 0 0
\(191\) −70.0000 −0.366492 −0.183246 0.983067i \(-0.558661\pi\)
−0.183246 + 0.983067i \(0.558661\pi\)
\(192\) 0 0
\(193\) −262.500 + 151.554i −1.36010 + 0.785256i −0.989637 0.143590i \(-0.954135\pi\)
−0.370466 + 0.928846i \(0.620802\pi\)
\(194\) 0 0
\(195\) −27.0000 46.7654i −0.138462 0.239822i
\(196\) 0 0
\(197\) −2.00000 −0.0101523 −0.00507614 0.999987i \(-0.501616\pi\)
−0.00507614 + 0.999987i \(0.501616\pi\)
\(198\) 0 0
\(199\) 98.5000 170.607i 0.494975 0.857322i −0.505008 0.863114i \(-0.668511\pi\)
0.999983 + 0.00579283i \(0.00184392\pi\)
\(200\) 0 0
\(201\) −45.0000 −0.223881
\(202\) 0 0
\(203\) 15.0000 + 8.66025i 0.0738916 + 0.0426613i
\(204\) 0 0
\(205\) −24.0000 13.8564i −0.117073 0.0675922i
\(206\) 0 0
\(207\) −60.0000 103.923i −0.289855 0.502044i
\(208\) 0 0
\(209\) 304.000 1.45455
\(210\) 0 0
\(211\) 316.500 182.731i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(212\) 0 0
\(213\) −42.0000 + 72.7461i −0.197183 + 0.341531i
\(214\) 0 0
\(215\) 49.0000 84.8705i 0.227907 0.394746i
\(216\) 0 0
\(217\) 50.2295i 0.231472i
\(218\) 0 0
\(219\) 52.5000 + 30.3109i 0.239726 + 0.138406i
\(220\) 0 0
\(221\) 342.946i 1.55179i
\(222\) 0 0
\(223\) −211.500 + 122.110i −0.948430 + 0.547577i −0.892593 0.450863i \(-0.851116\pi\)
−0.0558375 + 0.998440i \(0.517783\pi\)
\(224\) 0 0
\(225\) −31.5000 54.5596i −0.140000 0.242487i
\(226\) 0 0
\(227\) 148.956i 0.656195i 0.944644 + 0.328098i \(0.106408\pi\)
−0.944644 + 0.328098i \(0.893592\pi\)
\(228\) 0 0
\(229\) 83.0000 0.362445 0.181223 0.983442i \(-0.441995\pi\)
0.181223 + 0.983442i \(0.441995\pi\)
\(230\) 0 0
\(231\) 24.0000 13.8564i 0.103896 0.0599844i
\(232\) 0 0
\(233\) 40.0000 + 69.2820i 0.171674 + 0.297348i 0.939005 0.343903i \(-0.111749\pi\)
−0.767331 + 0.641251i \(0.778416\pi\)
\(234\) 0 0
\(235\) 92.0000 0.391489
\(236\) 0 0
\(237\) −76.5000 + 132.502i −0.322785 + 0.559080i
\(238\) 0 0
\(239\) −58.0000 −0.242678 −0.121339 0.992611i \(-0.538719\pi\)
−0.121339 + 0.992611i \(0.538719\pi\)
\(240\) 0 0
\(241\) −244.500 141.162i −1.01452 0.585735i −0.102010 0.994783i \(-0.532527\pi\)
−0.912513 + 0.409048i \(0.865861\pi\)
\(242\) 0 0
\(243\) 13.5000 + 7.79423i 0.0555556 + 0.0320750i
\(244\) 0 0
\(245\) −48.0000 83.1384i −0.195918 0.339341i
\(246\) 0 0
\(247\) −256.500 148.090i −1.03846 0.599556i
\(248\) 0 0
\(249\) −219.000 + 126.440i −0.879518 + 0.507790i
\(250\) 0 0
\(251\) −133.000 + 230.363i −0.529880 + 0.917780i 0.469512 + 0.882926i \(0.344430\pi\)
−0.999392 + 0.0348538i \(0.988903\pi\)
\(252\) 0 0
\(253\) −320.000 + 554.256i −1.26482 + 2.19074i
\(254\) 0 0
\(255\) 76.2102i 0.298864i
\(256\) 0 0
\(257\) −162.000 93.5307i −0.630350 0.363933i 0.150537 0.988604i \(-0.451900\pi\)
−0.780888 + 0.624671i \(0.785233\pi\)
\(258\) 0 0
\(259\) 15.5885i 0.0601871i
\(260\) 0 0
\(261\) 45.0000 25.9808i 0.172414 0.0995431i
\(262\) 0 0
\(263\) −82.0000 142.028i −0.311787 0.540031i 0.666962 0.745092i \(-0.267594\pi\)
−0.978749 + 0.205060i \(0.934261\pi\)
\(264\) 0 0
\(265\) 96.9948i 0.366018i
\(266\) 0 0
\(267\) 66.0000 0.247191
\(268\) 0 0
\(269\) −447.000 + 258.076i −1.66171 + 0.959389i −0.689811 + 0.723989i \(0.742307\pi\)
−0.971899 + 0.235400i \(0.924360\pi\)
\(270\) 0 0
\(271\) 7.00000 + 12.1244i 0.0258303 + 0.0447393i 0.878652 0.477464i \(-0.158444\pi\)
−0.852821 + 0.522203i \(0.825110\pi\)
\(272\) 0 0
\(273\) −27.0000 −0.0989011
\(274\) 0 0
\(275\) −168.000 + 290.985i −0.610909 + 1.05813i
\(276\) 0 0
\(277\) 206.000 0.743682 0.371841 0.928296i \(-0.378727\pi\)
0.371841 + 0.928296i \(0.378727\pi\)
\(278\) 0 0
\(279\) 130.500 + 75.3442i 0.467742 + 0.270051i
\(280\) 0 0
\(281\) 87.0000 + 50.2295i 0.309609 + 0.178753i 0.646751 0.762701i \(-0.276127\pi\)
−0.337143 + 0.941454i \(0.609461\pi\)
\(282\) 0 0
\(283\) 271.000 + 469.386i 0.957597 + 1.65861i 0.728310 + 0.685248i \(0.240306\pi\)
0.229288 + 0.973359i \(0.426360\pi\)
\(284\) 0 0
\(285\) 57.0000 + 32.9090i 0.200000 + 0.115470i
\(286\) 0 0
\(287\) −12.0000 + 6.92820i −0.0418118 + 0.0241401i
\(288\) 0 0
\(289\) −97.5000 + 168.875i −0.337370 + 0.584342i
\(290\) 0 0
\(291\) −54.0000 + 93.5307i −0.185567 + 0.321411i
\(292\) 0 0
\(293\) 481.510i 1.64338i 0.569935 + 0.821690i \(0.306968\pi\)
−0.569935 + 0.821690i \(0.693032\pi\)
\(294\) 0 0
\(295\) 114.000 + 65.8179i 0.386441 + 0.223112i
\(296\) 0 0
\(297\) 83.1384i 0.279927i
\(298\) 0 0
\(299\) 540.000 311.769i 1.80602 1.04271i
\(300\) 0 0
\(301\) −24.5000 42.4352i −0.0813953 0.140981i
\(302\) 0 0
\(303\) 48.4974i 0.160058i
\(304\) 0 0
\(305\) 194.000 0.636066
\(306\) 0 0
\(307\) 138.000 79.6743i 0.449511 0.259526i −0.258112 0.966115i \(-0.583100\pi\)
0.707624 + 0.706589i \(0.249767\pi\)
\(308\) 0 0
\(309\) 109.500 + 189.660i 0.354369 + 0.613785i
\(310\) 0 0
\(311\) −322.000 −1.03537 −0.517685 0.855571i \(-0.673206\pi\)
−0.517685 + 0.855571i \(0.673206\pi\)
\(312\) 0 0
\(313\) −13.0000 + 22.5167i −0.0415335 + 0.0719382i −0.886045 0.463599i \(-0.846558\pi\)
0.844511 + 0.535538i \(0.179891\pi\)
\(314\) 0 0
\(315\) 6.00000 0.0190476
\(316\) 0 0
\(317\) −291.000 168.009i −0.917981 0.529997i −0.0349907 0.999388i \(-0.511140\pi\)
−0.882990 + 0.469391i \(0.844473\pi\)
\(318\) 0 0
\(319\) −240.000 138.564i −0.752351 0.434370i
\(320\) 0 0
\(321\) 72.0000 + 124.708i 0.224299 + 0.388497i
\(322\) 0 0
\(323\) 209.000 + 361.999i 0.647059 + 1.12074i
\(324\) 0 0
\(325\) 283.500 163.679i 0.872308 0.503627i
\(326\) 0 0
\(327\) 66.0000 114.315i 0.201835 0.349588i
\(328\) 0 0
\(329\) 23.0000 39.8372i 0.0699088 0.121086i
\(330\) 0 0
\(331\) 219.970i 0.664563i 0.943180 + 0.332282i \(0.107818\pi\)
−0.943180 + 0.332282i \(0.892182\pi\)
\(332\) 0 0
\(333\) 40.5000 + 23.3827i 0.121622 + 0.0702183i
\(334\) 0 0
\(335\) 51.9615i 0.155109i
\(336\) 0 0
\(337\) −505.500 + 291.851i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 159.000 + 275.396i 0.469027 + 0.812378i
\(340\) 0 0
\(341\) 803.672i 2.35681i
\(342\) 0 0
\(343\) −97.0000 −0.282799
\(344\) 0 0
\(345\) −120.000 + 69.2820i −0.347826 + 0.200817i
\(346\) 0 0
\(347\) −235.000 407.032i −0.677233 1.17300i −0.975811 0.218617i \(-0.929845\pi\)
0.298577 0.954385i \(-0.403488\pi\)
\(348\) 0 0
\(349\) −295.000 −0.845272 −0.422636 0.906299i \(-0.638895\pi\)
−0.422636 + 0.906299i \(0.638895\pi\)
\(350\) 0 0
\(351\) −40.5000 + 70.1481i −0.115385 + 0.199852i
\(352\) 0 0
\(353\) 262.000 0.742210 0.371105 0.928591i \(-0.378979\pi\)
0.371105 + 0.928591i \(0.378979\pi\)
\(354\) 0 0
\(355\) 84.0000 + 48.4974i 0.236620 + 0.136612i
\(356\) 0 0
\(357\) 33.0000 + 19.0526i 0.0924370 + 0.0533685i
\(358\) 0 0
\(359\) −232.000 401.836i −0.646240 1.11932i −0.984014 0.178093i \(-0.943007\pi\)
0.337774 0.941227i \(-0.390326\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) −202.500 + 116.913i −0.557851 + 0.322076i
\(364\) 0 0
\(365\) 35.0000 60.6218i 0.0958904 0.166087i
\(366\) 0 0
\(367\) 116.500 201.784i 0.317439 0.549820i −0.662514 0.749049i \(-0.730511\pi\)
0.979953 + 0.199229i \(0.0638439\pi\)
\(368\) 0 0
\(369\) 41.5692i 0.112654i
\(370\) 0 0
\(371\) −42.0000 24.2487i −0.113208 0.0653604i
\(372\) 0 0
\(373\) 533.472i 1.43022i −0.699013 0.715109i \(-0.746377\pi\)
0.699013 0.715109i \(-0.253623\pi\)
\(374\) 0 0
\(375\) −138.000 + 79.6743i −0.368000 + 0.212465i
\(376\) 0 0
\(377\) 135.000 + 233.827i 0.358090 + 0.620230i
\(378\) 0 0
\(379\) 161.081i 0.425015i −0.977159 0.212508i \(-0.931837\pi\)
0.977159 0.212508i \(-0.0681630\pi\)
\(380\) 0 0
\(381\) 48.0000 0.125984
\(382\) 0 0
\(383\) 498.000 287.520i 1.30026 0.750706i 0.319812 0.947481i \(-0.396380\pi\)
0.980449 + 0.196775i \(0.0630468\pi\)
\(384\) 0 0
\(385\) −16.0000 27.7128i −0.0415584 0.0719813i
\(386\) 0 0
\(387\) −147.000 −0.379845
\(388\) 0 0
\(389\) 76.0000 131.636i 0.195373 0.338396i −0.751650 0.659562i \(-0.770742\pi\)
0.947023 + 0.321167i \(0.104075\pi\)
\(390\) 0 0
\(391\) −880.000 −2.25064
\(392\) 0 0
\(393\) 192.000 + 110.851i 0.488550 + 0.282064i
\(394\) 0 0
\(395\) 153.000 + 88.3346i 0.387342 + 0.223632i
\(396\) 0 0
\(397\) 249.500 + 432.147i 0.628463 + 1.08853i 0.987860 + 0.155346i \(0.0496492\pi\)
−0.359397 + 0.933185i \(0.617017\pi\)
\(398\) 0 0
\(399\) 28.5000 16.4545i 0.0714286 0.0412393i
\(400\) 0 0
\(401\) 87.0000 50.2295i 0.216958 0.125261i −0.387583 0.921835i \(-0.626690\pi\)
0.604541 + 0.796574i \(0.293357\pi\)
\(402\) 0 0
\(403\) −391.500 + 678.098i −0.971464 + 1.68263i
\(404\) 0 0
\(405\) 9.00000 15.5885i 0.0222222 0.0384900i
\(406\) 0 0
\(407\) 249.415i 0.612814i
\(408\) 0 0
\(409\) 342.000 + 197.454i 0.836186 + 0.482772i 0.855966 0.517032i \(-0.172963\pi\)
−0.0197801 + 0.999804i \(0.506297\pi\)
\(410\) 0 0
\(411\) 200.918i 0.488851i
\(412\) 0 0
\(413\) 57.0000 32.9090i 0.138015 0.0796827i
\(414\) 0 0
\(415\) 146.000 + 252.879i 0.351807 + 0.609348i
\(416\) 0 0
\(417\) 303.109i 0.726880i
\(418\) 0 0
\(419\) −628.000 −1.49881 −0.749403 0.662114i \(-0.769660\pi\)
−0.749403 + 0.662114i \(0.769660\pi\)
\(420\) 0 0
\(421\) 558.000 322.161i 1.32542 0.765229i 0.340829 0.940125i \(-0.389292\pi\)
0.984587 + 0.174896i \(0.0559590\pi\)
\(422\) 0 0
\(423\) −69.0000 119.512i −0.163121 0.282533i
\(424\) 0 0
\(425\) −462.000 −1.08706
\(426\) 0 0
\(427\) 48.5000 84.0045i 0.113583 0.196732i
\(428\) 0 0
\(429\) 432.000 1.00699
\(430\) 0 0
\(431\) 393.000 + 226.899i 0.911833 + 0.526447i 0.881020 0.473078i \(-0.156857\pi\)
0.0308125 + 0.999525i \(0.490191\pi\)
\(432\) 0 0
\(433\) 433.500 + 250.281i 1.00115 + 0.578017i 0.908590 0.417690i \(-0.137160\pi\)
0.0925650 + 0.995707i \(0.470493\pi\)
\(434\) 0 0
\(435\) −30.0000 51.9615i −0.0689655 0.119452i
\(436\) 0 0
\(437\) −380.000 + 658.179i −0.869565 + 1.50613i
\(438\) 0 0
\(439\) −292.500 + 168.875i −0.666287 + 0.384681i −0.794668 0.607044i \(-0.792355\pi\)
0.128381 + 0.991725i \(0.459022\pi\)
\(440\) 0 0
\(441\) −72.0000 + 124.708i −0.163265 + 0.282784i
\(442\) 0 0
\(443\) 365.000 632.199i 0.823928 1.42708i −0.0788082 0.996890i \(-0.525111\pi\)
0.902736 0.430195i \(-0.141555\pi\)
\(444\) 0 0
\(445\) 76.2102i 0.171259i
\(446\) 0 0
\(447\) 222.000 + 128.172i 0.496644 + 0.286738i
\(448\) 0 0
\(449\) 446.869i 0.995254i 0.867391 + 0.497627i \(0.165795\pi\)
−0.867391 + 0.497627i \(0.834205\pi\)
\(450\) 0 0
\(451\) 192.000 110.851i 0.425721 0.245790i
\(452\) 0 0
\(453\) −168.000 290.985i −0.370861 0.642350i
\(454\) 0 0
\(455\) 31.1769i 0.0685207i
\(456\) 0 0
\(457\) −397.000 −0.868709 −0.434354 0.900742i \(-0.643023\pi\)
−0.434354 + 0.900742i \(0.643023\pi\)
\(458\) 0 0
\(459\) 99.0000 57.1577i 0.215686 0.124527i
\(460\) 0 0
\(461\) −287.000 497.099i −0.622560 1.07830i −0.989007 0.147866i \(-0.952759\pi\)
0.366448 0.930439i \(-0.380574\pi\)
\(462\) 0 0
\(463\) 367.000 0.792657 0.396328 0.918109i \(-0.370284\pi\)
0.396328 + 0.918109i \(0.370284\pi\)
\(464\) 0 0
\(465\) 87.0000 150.688i 0.187097 0.324061i
\(466\) 0 0
\(467\) −424.000 −0.907923 −0.453961 0.891021i \(-0.649990\pi\)
−0.453961 + 0.891021i \(0.649990\pi\)
\(468\) 0 0
\(469\) 22.5000 + 12.9904i 0.0479744 + 0.0276980i
\(470\) 0 0
\(471\) −289.500 167.143i −0.614650 0.354868i
\(472\) 0 0
\(473\) 392.000 + 678.964i 0.828753 + 1.43544i
\(474\) 0 0
\(475\) −199.500 + 345.544i −0.420000 + 0.727461i
\(476\) 0 0
\(477\) −126.000 + 72.7461i −0.264151 + 0.152508i
\(478\) 0 0
\(479\) −364.000 + 630.466i −0.759916 + 1.31621i 0.182976 + 0.983117i \(0.441427\pi\)
−0.942893 + 0.333097i \(0.891906\pi\)
\(480\) 0 0
\(481\) −121.500 + 210.444i −0.252599 + 0.437514i
\(482\) 0 0
\(483\) 69.2820i 0.143441i
\(484\) 0 0
\(485\) 108.000 + 62.3538i 0.222680 + 0.128565i
\(486\) 0 0
\(487\) 270.200i 0.554825i −0.960751 0.277413i \(-0.910523\pi\)
0.960751 0.277413i \(-0.0894769\pi\)
\(488\) 0 0
\(489\) 349.500 201.784i 0.714724 0.412646i
\(490\) 0 0
\(491\) 380.000 + 658.179i 0.773931 + 1.34049i 0.935393 + 0.353609i \(0.115046\pi\)
−0.161463 + 0.986879i \(0.551621\pi\)
\(492\) 0 0
\(493\) 381.051i 0.772923i
\(494\) 0 0
\(495\) −96.0000 −0.193939
\(496\) 0 0
\(497\) 42.0000 24.2487i 0.0845070 0.0487902i
\(498\) 0 0
\(499\) −231.500 400.970i −0.463928 0.803547i 0.535225 0.844710i \(-0.320227\pi\)
−0.999152 + 0.0411632i \(0.986894\pi\)
\(500\) 0 0
\(501\) 126.000 0.251497
\(502\) 0 0
\(503\) 65.0000 112.583i 0.129225 0.223824i −0.794152 0.607720i \(-0.792084\pi\)
0.923376 + 0.383896i \(0.125418\pi\)
\(504\) 0 0
\(505\) −56.0000 −0.110891
\(506\) 0 0
\(507\) −111.000 64.0859i −0.218935 0.126402i
\(508\) 0 0
\(509\) −459.000 265.004i −0.901768 0.520636i −0.0239947 0.999712i \(-0.507638\pi\)
−0.877773 + 0.479076i \(0.840972\pi\)
\(510\) 0 0
\(511\) −17.5000 30.3109i −0.0342466 0.0593168i
\(512\) 0 0
\(513\) 98.7269i 0.192450i
\(514\) 0 0
\(515\) 219.000 126.440i 0.425243 0.245514i
\(516\) 0 0
\(517\) −368.000 + 637.395i −0.711799 + 1.23287i
\(518\) 0 0
\(519\) −99.0000 + 171.473i −0.190751 + 0.330391i
\(520\) 0 0
\(521\) 602.754i 1.15692i −0.815712 0.578458i \(-0.803655\pi\)
0.815712 0.578458i \(-0.196345\pi\)
\(522\) 0 0
\(523\) 277.500 + 160.215i 0.530593 + 0.306338i 0.741258 0.671220i \(-0.234230\pi\)
−0.210665 + 0.977558i \(0.567563\pi\)
\(524\) 0 0
\(525\) 36.3731i 0.0692820i
\(526\) 0 0
\(527\) 957.000 552.524i 1.81594 1.04843i
\(528\) 0 0
\(529\) −535.500 927.513i −1.01229 1.75333i
\(530\) 0 0
\(531\) 197.454i 0.371853i
\(532\) 0 0
\(533\) −216.000 −0.405253
\(534\) 0 0
\(535\) 144.000 83.1384i 0.269159 0.155399i
\(536\) 0 0
\(537\) −72.0000 124.708i −0.134078 0.232230i
\(538\) 0 0
\(539\) 768.000 1.42486
\(540\) 0 0
\(541\) −182.500 + 316.099i −0.337338 + 0.584287i −0.983931 0.178548i \(-0.942860\pi\)
0.646593 + 0.762835i \(0.276193\pi\)
\(542\) 0 0
\(543\) −84.0000 −0.154696
\(544\) 0 0
\(545\) −132.000 76.2102i −0.242202 0.139835i
\(546\) 0 0
\(547\) 199.500 + 115.181i 0.364717 + 0.210569i 0.671148 0.741324i \(-0.265802\pi\)
−0.306431 + 0.951893i \(0.599135\pi\)
\(548\) 0 0
\(549\) −145.500 252.013i −0.265027 0.459041i
\(550\) 0 0
\(551\) −285.000 164.545i −0.517241 0.298629i
\(552\) 0 0
\(553\) 76.5000 44.1673i 0.138336 0.0798685i
\(554\) 0 0
\(555\) 27.0000 46.7654i 0.0486486 0.0842619i
\(556\) 0 0
\(557\) −320.000 + 554.256i −0.574506 + 0.995074i 0.421589 + 0.906787i \(0.361473\pi\)
−0.996095 + 0.0882870i \(0.971861\pi\)
\(558\) 0 0
\(559\) 763.834i 1.36643i
\(560\) 0 0
\(561\) −528.000 304.841i −0.941176 0.543388i
\(562\) 0 0
\(563\) 96.9948i 0.172282i −0.996283 0.0861411i \(-0.972546\pi\)
0.996283 0.0861411i \(-0.0274536\pi\)
\(564\) 0 0
\(565\) 318.000 183.597i 0.562832 0.324951i
\(566\) 0 0
\(567\) −4.50000 7.79423i −0.00793651 0.0137464i
\(568\) 0 0
\(569\) 152.420i 0.267874i 0.990990 + 0.133937i \(0.0427620\pi\)
−0.990990 + 0.133937i \(0.957238\pi\)
\(570\) 0 0
\(571\) −335.000 −0.586690 −0.293345 0.956007i \(-0.594768\pi\)
−0.293345 + 0.956007i \(0.594768\pi\)
\(572\) 0 0
\(573\) 105.000 60.6218i 0.183246 0.105797i
\(574\) 0 0
\(575\) −420.000 727.461i −0.730435 1.26515i
\(576\) 0 0
\(577\) 746.000 1.29289 0.646447 0.762959i \(-0.276254\pi\)
0.646447 + 0.762959i \(0.276254\pi\)
\(578\) 0 0
\(579\) 262.500 454.663i 0.453368 0.785256i
\(580\) 0 0
\(581\) 146.000 0.251291
\(582\) 0 0
\(583\) 672.000 + 387.979i 1.15266 + 0.665488i
\(584\) 0 0
\(585\) 81.0000 + 46.7654i 0.138462 + 0.0799408i
\(586\) 0 0
\(587\) 89.0000 + 154.153i 0.151618 + 0.262611i 0.931823 0.362914i \(-0.118218\pi\)
−0.780204 + 0.625525i \(0.784885\pi\)
\(588\) 0 0
\(589\) 954.360i 1.62031i
\(590\) 0 0
\(591\) 3.00000 1.73205i 0.00507614 0.00293071i
\(592\) 0 0
\(593\) −137.000 + 237.291i −0.231029 + 0.400153i −0.958111 0.286397i \(-0.907542\pi\)
0.727082 + 0.686550i \(0.240876\pi\)
\(594\) 0 0
\(595\) 22.0000 38.1051i 0.0369748 0.0640422i
\(596\) 0 0
\(597\) 341.214i 0.571548i
\(598\) 0 0
\(599\) 252.000 + 145.492i 0.420701 + 0.242892i 0.695377 0.718645i \(-0.255237\pi\)
−0.274676 + 0.961537i \(0.588571\pi\)
\(600\) 0 0
\(601\) 1020.18i 1.69747i −0.528820 0.848734i \(-0.677366\pi\)
0.528820 0.848734i \(-0.322634\pi\)
\(602\) 0 0
\(603\) 67.5000 38.9711i 0.111940 0.0646288i
\(604\) 0 0
\(605\) 135.000 + 233.827i 0.223140 + 0.386491i
\(606\) 0 0
\(607\) 157.617i 0.259665i −0.991536 0.129832i \(-0.958556\pi\)
0.991536 0.129832i \(-0.0414440\pi\)
\(608\) 0 0
\(609\) −30.0000 −0.0492611
\(610\) 0 0
\(611\) 621.000 358.535i 1.01637 0.586800i
\(612\) 0 0
\(613\) 401.000 + 694.552i 0.654160 + 1.13304i 0.982104 + 0.188341i \(0.0603110\pi\)
−0.327944 + 0.944697i \(0.606356\pi\)
\(614\) 0 0
\(615\) 48.0000 0.0780488
\(616\) 0 0
\(617\) 382.000 661.643i 0.619125 1.07236i −0.370521 0.928824i \(-0.620821\pi\)
0.989646 0.143531i \(-0.0458458\pi\)
\(618\) 0 0
\(619\) 127.000 0.205170 0.102585 0.994724i \(-0.467289\pi\)
0.102585 + 0.994724i \(0.467289\pi\)
\(620\) 0 0
\(621\) 180.000 + 103.923i 0.289855 + 0.167348i
\(622\) 0 0
\(623\) −33.0000 19.0526i −0.0529695 0.0305820i
\(624\) 0 0
\(625\) −170.500 295.315i −0.272800 0.472503i
\(626\) 0 0
\(627\) −456.000 + 263.272i −0.727273 + 0.419891i
\(628\) 0 0
\(629\) 297.000 171.473i 0.472178 0.272612i
\(630\) 0 0
\(631\) 161.500 279.726i 0.255943 0.443306i −0.709208 0.704999i \(-0.750947\pi\)
0.965151 + 0.261693i \(0.0842807\pi\)
\(632\) 0 0
\(633\) −316.500 + 548.194i −0.500000 + 0.866025i
\(634\) 0 0
\(635\) 55.4256i 0.0872845i
\(636\) 0 0
\(637\) −648.000 374.123i −1.01727 0.587320i
\(638\) 0 0
\(639\) 145.492i 0.227687i
\(640\) 0 0
\(641\) −873.000 + 504.027i −1.36193 + 0.786313i −0.989881 0.141898i \(-0.954679\pi\)
−0.372053 + 0.928211i \(0.621346\pi\)
\(642\) 0 0
\(643\) −222.500 385.381i −0.346034 0.599349i 0.639507 0.768785i \(-0.279139\pi\)
−0.985541 + 0.169437i \(0.945805\pi\)
\(644\) 0 0
\(645\) 169.741i 0.263164i
\(646\) 0 0
\(647\) −106.000 −0.163833 −0.0819165 0.996639i \(-0.526104\pi\)
−0.0819165 + 0.996639i \(0.526104\pi\)
\(648\) 0 0
\(649\) −912.000 + 526.543i −1.40524 + 0.811315i
\(650\) 0 0
\(651\) −43.5000 75.3442i −0.0668203 0.115736i
\(652\) 0 0
\(653\) 328.000 0.502297 0.251149 0.967949i \(-0.419192\pi\)
0.251149 + 0.967949i \(0.419192\pi\)
\(654\) 0 0
\(655\) 128.000 221.703i 0.195420 0.338477i
\(656\) 0 0
\(657\) −105.000 −0.159817
\(658\) 0 0
\(659\) −348.000 200.918i −0.528073 0.304883i 0.212158 0.977235i \(-0.431951\pi\)
−0.740231 + 0.672352i \(0.765284\pi\)
\(660\) 0 0
\(661\) 828.000 + 478.046i 1.25265 + 0.723216i 0.971635 0.236488i \(-0.0759963\pi\)
0.281013 + 0.959704i \(0.409330\pi\)
\(662\) 0 0
\(663\) 297.000 + 514.419i 0.447964 + 0.775896i
\(664\) 0 0
\(665\) −19.0000 32.9090i −0.0285714 0.0494872i
\(666\) 0 0
\(667\) 600.000 346.410i 0.899550 0.519356i
\(668\) 0 0
\(669\) 211.500 366.329i 0.316143 0.547577i
\(670\) 0 0
\(671\) −776.000 + 1344.07i −1.15648 + 2.00309i
\(672\) 0 0
\(673\) 521.347i 0.774662i −0.921941 0.387331i \(-0.873397\pi\)
0.921941 0.387331i \(-0.126603\pi\)
\(674\) 0 0
\(675\) 94.5000 + 54.5596i 0.140000 + 0.0808290i
\(676\) 0 0
\(677\) 439.941i 0.649839i −0.945742 0.324919i \(-0.894663\pi\)
0.945742 0.324919i \(-0.105337\pi\)
\(678\) 0 0
\(679\) 54.0000 31.1769i 0.0795287 0.0459159i
\(680\) 0 0
\(681\) −129.000 223.435i −0.189427 0.328098i
\(682\) 0 0
\(683\) 717.069i 1.04988i 0.851139 + 0.524941i \(0.175913\pi\)
−0.851139 + 0.524941i \(0.824087\pi\)
\(684\) 0 0
\(685\) 232.000 0.338686
\(686\) 0 0
\(687\) −124.500 + 71.8801i −0.181223 + 0.104629i
\(688\) 0 0
\(689\) −378.000 654.715i −0.548621 0.950240i
\(690\) 0 0
\(691\) −422.000 −0.610709 −0.305355 0.952239i \(-0.598775\pi\)
−0.305355 + 0.952239i \(0.598775\pi\)
\(692\) 0 0
\(693\) −24.0000 + 41.5692i −0.0346320 + 0.0599844i
\(694\) 0 0
\(695\) −350.000 −0.503597
\(696\) 0 0
\(697\) 264.000 + 152.420i 0.378766 + 0.218681i
\(698\) 0 0
\(699\) −120.000 69.2820i −0.171674 0.0991159i
\(700\) 0 0
\(701\) −185.000 320.429i −0.263909 0.457103i 0.703368 0.710825i \(-0.251678\pi\)
−0.967277 + 0.253722i \(0.918345\pi\)
\(702\) 0 0
\(703\) 296.181i 0.421310i
\(704\) 0 0
\(705\) −138.000 + 79.6743i −0.195745 + 0.113013i
\(706\) 0 0
\(707\) −14.0000 + 24.2487i −0.0198020 + 0.0342980i
\(708\) 0 0
\(709\) 45.5000 78.8083i 0.0641749 0.111154i −0.832153 0.554546i \(-0.812892\pi\)
0.896328 + 0.443392i \(0.146225\pi\)
\(710\) 0 0
\(711\) 265.004i 0.372720i
\(712\) 0 0
\(713\) 1740.00 + 1004.59i 2.44039 + 1.40896i
\(714\) 0 0
\(715\) 498.831i 0.697665i
\(716\) 0 0
\(717\) 87.0000 50.2295i 0.121339 0.0700551i
\(718\) 0 0
\(719\) −502.000 869.490i −0.698192 1.20930i −0.969093 0.246697i \(-0.920655\pi\)
0.270901 0.962607i \(-0.412678\pi\)
\(720\) 0 0
\(721\) 126.440i 0.175367i
\(722\) 0 0
\(723\) 489.000 0.676349
\(724\) 0 0
\(725\) 315.000 181.865i 0.434483 0.250849i
\(726\) 0 0
\(727\) −147.500 255.477i −0.202889 0.351413i 0.746569 0.665308i \(-0.231700\pi\)
−0.949458 + 0.313894i \(0.898366\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) −539.000 + 933.575i −0.737346 + 1.27712i
\(732\) 0 0
\(733\) −250.000 −0.341064 −0.170532 0.985352i \(-0.554549\pi\)
−0.170532 + 0.985352i \(0.554549\pi\)
\(734\) 0 0
\(735\) 144.000 + 83.1384i 0.195918 + 0.113114i
\(736\) 0 0
\(737\) −360.000 207.846i −0.488467 0.282016i
\(738\) 0 0
\(739\) −285.500 494.501i −0.386333 0.669148i 0.605620 0.795754i \(-0.292925\pi\)
−0.991953 + 0.126606i \(0.959592\pi\)
\(740\) 0 0
\(741\) 513.000 0.692308
\(742\) 0 0
\(743\) −6.00000 + 3.46410i −0.00807537 + 0.00466232i −0.504032 0.863685i \(-0.668151\pi\)
0.495957 + 0.868347i \(0.334817\pi\)
\(744\) 0 0
\(745\) 148.000 256.344i 0.198658 0.344085i
\(746\) 0 0
\(747\) 219.000 379.319i 0.293173 0.507790i
\(748\) 0 0
\(749\) 83.1384i 0.110999i
\(750\) 0 0
\(751\) 841.500 + 485.840i 1.12051 + 0.646924i 0.941530 0.336929i \(-0.109388\pi\)
0.178976 + 0.983853i \(0.442722\pi\)
\(752\) 0 0
\(753\) 460.726i 0.611853i
\(754\) 0 0
\(755\) −336.000 + 193.990i −0.445033 + 0.256940i
\(756\) 0 0
\(757\) −119.500 206.980i −0.157860 0.273421i 0.776237 0.630442i \(-0.217126\pi\)
−0.934097 + 0.357020i \(0.883793\pi\)
\(758\) 0 0
\(759\) 1108.51i 1.46049i
\(760\) 0 0
\(761\) 1402.00 1.84231 0.921156 0.389193i \(-0.127246\pi\)
0.921156 + 0.389193i \(0.127246\pi\)
\(762\) 0 0
\(763\) −66.0000 + 38.1051i −0.0865007 + 0.0499412i
\(764\) 0 0
\(765\) −66.0000 114.315i −0.0862745 0.149432i
\(766\) 0 0
\(767\) 1026.00 1.33768
\(768\) 0 0
\(769\) 51.5000 89.2006i 0.0669701 0.115996i −0.830596 0.556875i \(-0.812000\pi\)
0.897566 + 0.440880i \(0.145333\pi\)
\(770\) 0 0
\(771\) 324.000 0.420233
\(772\) 0 0
\(773\) −69.0000 39.8372i −0.0892626 0.0515358i 0.454704 0.890642i \(-0.349745\pi\)
−0.543967 + 0.839107i \(0.683078\pi\)
\(774\) 0 0
\(775\) 913.500 + 527.409i 1.17871 + 0.680528i
\(776\) 0 0
\(777\) −13.5000 23.3827i −0.0173745 0.0300935i
\(778\) 0 0
\(779\) 228.000 131.636i 0.292683 0.168981i
\(780\) 0 0
\(781\) −672.000 + 387.979i −0.860435 + 0.496773i
\(782\) 0 0
\(783\) −45.0000 + 77.9423i −0.0574713 + 0.0995431i
\(784\) 0 0
\(785\) −193.000 + 334.286i −0.245860 + 0.425842i
\(786\) 0 0
\(787\) 188.794i 0.239890i −0.992781 0.119945i \(-0.961728\pi\)
0.992781 0.119945i \(-0.0382718\pi\)
\(788\) 0 0
\(789\) 246.000 + 142.028i 0.311787 + 0.180010i
\(790\) 0 0
\(791\) 183.597i 0.232108i
\(792\) 0 0
\(793\) 1309.50 756.040i 1.65132 0.953392i
\(794\) 0 0
\(795\) 84.0000 + 145.492i 0.105660 + 0.183009i
\(796\) 0 0
\(797\) 630.466i 0.791050i −0.918455 0.395525i \(-0.870563\pi\)
0.918455 0.395525i \(-0.129437\pi\)
\(798\) 0 0
\(799\) −1012.00 −1.26658
\(800\) 0 0
\(801\) −99.0000 + 57.1577i −0.123596 + 0.0713579i
\(802\) 0 0
\(803\) 280.000 + 484.974i 0.348692 + 0.603953i
\(804\) 0 0
\(805\) 80.0000 0.0993789
\(806\) 0 0
\(807\) 447.000 774.227i 0.553903 0.959389i
\(808\) 0 0
\(809\) 1054.00 1.30284 0.651422 0.758716i \(-0.274173\pi\)
0.651422 + 0.758716i \(0.274173\pi\)
\(810\) 0 0
\(811\) −960.000 554.256i −1.18372 0.683423i −0.226851 0.973930i \(-0.572843\pi\)
−0.956873 + 0.290506i \(0.906176\pi\)
\(812\) 0 0
\(813\) −21.0000 12.1244i −0.0258303 0.0149131i
\(814\) 0 0
\(815\) −233.000 403.568i −0.285890 0.495175i
\(816\) 0 0
\(817\) 465.500 + 806.270i 0.569767 + 0.986866i
\(818\) 0 0
\(819\) 40.5000 23.3827i 0.0494505 0.0285503i
\(820\) 0 0
\(821\) 487.000 843.509i 0.593179 1.02742i −0.400622 0.916243i \(-0.631206\pi\)
0.993801 0.111173i \(-0.0354607\pi\)
\(822\) 0 0
\(823\) −317.000 + 549.060i −0.385176 + 0.667145i −0.991794 0.127849i \(-0.959193\pi\)
0.606617 + 0.794994i \(0.292526\pi\)
\(824\) 0 0
\(825\) 581.969i 0.705417i
\(826\) 0 0
\(827\) −1155.00 666.840i −1.39661 0.806336i −0.402578 0.915386i \(-0.631886\pi\)
−0.994036 + 0.109050i \(0.965219\pi\)
\(828\) 0 0
\(829\) 296.181i 0.357275i 0.983915 + 0.178637i \(0.0571689\pi\)
−0.983915 + 0.178637i \(0.942831\pi\)
\(830\) 0 0
\(831\) −309.000 + 178.401i −0.371841 + 0.214683i
\(832\) 0 0
\(833\) 528.000 + 914.523i 0.633854 + 1.09787i
\(834\) 0 0
\(835\) 145.492i 0.174242i
\(836\) 0 0
\(837\) −261.000 −0.311828
\(838\) 0 0
\(839\) −1395.00 + 805.404i −1.66269 + 0.959957i −0.691273 + 0.722594i \(0.742950\pi\)
−0.971421 + 0.237363i \(0.923717\pi\)
\(840\) 0 0
\(841\) −270.500 468.520i −0.321641 0.557098i
\(842\) 0 0
\(843\) −174.000 −0.206406
\(844\) 0 0
\(845\) −74.0000 + 128.172i −0.0875740 + 0.151683i
\(846\) 0 0
\(847\) 135.000 0.159386
\(848\) 0 0
\(849\) −813.000 469.386i −0.957597 0.552869i
\(850\) 0 0
\(851\) 540.000 + 311.769i 0.634548 + 0.366356i
\(852\) 0 0
\(853\) 219.500 + 380.185i 0.257327 + 0.445704i 0.965525 0.260310i \(-0.0838249\pi\)
−0.708198 + 0.706014i \(0.750492\pi\)
\(854\) 0 0
\(855\) −114.000 −0.133333
\(856\) 0 0
\(857\) 531.000 306.573i 0.619603 0.357728i −0.157111 0.987581i \(-0.550218\pi\)
0.776715 + 0.629853i \(0.216885\pi\)
\(858\) 0 0
\(859\) 278.500 482.376i 0.324214 0.561555i −0.657139 0.753770i \(-0.728233\pi\)
0.981353 + 0.192214i \(0.0615668\pi\)
\(860\) 0 0
\(861\) 12.0000 20.7846i 0.0139373 0.0241401i
\(862\) 0 0
\(863\) 1340.61i 1.55343i 0.629854 + 0.776713i \(0.283115\pi\)
−0.629854 + 0.776713i \(0.716885\pi\)
\(864\) 0 0
\(865\) 198.000 + 114.315i 0.228902 + 0.132156i
\(866\) 0 0
\(867\) 337.750i 0.389562i
\(868\) 0 0
\(869\) −1224.00 + 706.677i −1.40852 + 0.813207i
\(870\) 0 0
\(871\) 202.500 + 350.740i 0.232491 + 0.402687i
\(872\) 0 0
\(873\) 187.061i 0.214274i
\(874\) 0 0
\(875\) 92.0000 0.105143
\(876\) 0 0
\(877\) −397.500 + 229.497i −0.453250 + 0.261684i −0.709202 0.705006i \(-0.750944\pi\)
0.255952 + 0.966690i \(0.417611\pi\)
\(878\) 0 0
\(879\) −417.000 722.265i −0.474403 0.821690i
\(880\) 0 0
\(881\) 910.000 1.03292 0.516459 0.856312i \(-0.327250\pi\)
0.516459 + 0.856312i \(0.327250\pi\)
\(882\) 0 0
\(883\) 2.50000 4.33013i 0.00283126 0.00490388i −0.864606 0.502450i \(-0.832432\pi\)
0.867438 + 0.497546i \(0.165765\pi\)
\(884\) 0 0
\(885\) −228.000 −0.257627
\(886\) 0 0
\(887\) 927.000 + 535.204i 1.04510 + 0.603386i 0.921272 0.388918i \(-0.127151\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(888\) 0 0
\(889\) −24.0000 13.8564i −0.0269966 0.0155865i
\(890\) 0 0
\(891\) 72.0000 + 124.708i 0.0808081 + 0.139964i
\(892\) 0 0
\(893\) −437.000 + 756.906i −0.489362 + 0.847599i
\(894\) 0 0
\(895\) −144.000 + 83.1384i −0.160894 + 0.0928921i
\(896\) 0 0
\(897\) −540.000 + 935.307i −0.602007 + 1.04271i
\(898\) 0 0
\(899\) −435.000 + 753.442i −0.483871 + 0.838089i
\(900\) 0 0
\(901\) 1066.94i 1.18418i
\(902\) 0 0
\(903\) 73.5000 + 42.4352i 0.0813953 + 0.0469936i
\(904\) 0 0
\(905\) 96.9948i 0.107177i
\(906\) 0 0
\(907\) −180.000 + 103.923i −0.198456 + 0.114579i −0.595935 0.803032i \(-0.703219\pi\)
0.397479 + 0.917611i \(0.369885\pi\)
\(908\) 0 0
\(909\) 42.0000 + 72.7461i 0.0462046 + 0.0800288i
\(910\) 0 0
\(911\) 782.887i 0.859371i 0.902979 + 0.429685i \(0.141376\pi\)
−0.902979 + 0.429685i \(0.858624\pi\)
\(912\) 0 0
\(913\) −2336.00 −2.55860
\(914\) 0 0
\(915\) −291.000 + 168.009i −0.318033 + 0.183616i
\(916\) 0 0
\(917\) −64.0000 110.851i −0.0697928 0.120885i
\(918\) 0 0
\(919\) 1225.00 1.33297 0.666485 0.745518i \(-0.267798\pi\)
0.666485 + 0.745518i \(0.267798\pi\)
\(920\) 0 0
\(921\) −138.000 + 239.023i −0.149837 + 0.259526i
\(922\) 0 0
\(923\) 756.000 0.819068
\(924\) 0 0
\(925\) 283.500 + 163.679i 0.306486 + 0.176950i
\(926\) 0 0
\(927\) −328.500 189.660i −0.354369 0.204595i
\(928\) 0 0
\(929\) 145.000 + 251.147i 0.156082 + 0.270342i 0.933452 0.358701i \(-0.116780\pi\)
−0.777371 + 0.629043i \(0.783447\pi\)
\(930\) 0 0
\(931\) 912.000 0.979592
\(932\) 0 0
\(933\) 483.000 278.860i 0.517685 0.298886i
\(934\) 0 0
\(935\) −352.000 + 609.682i −0.376471 + 0.652066i
\(936\) 0 0
\(937\) 276.500 478.912i 0.295091 0.511112i −0.679915 0.733291i \(-0.737983\pi\)
0.975006 + 0.222179i \(0.0713168\pi\)
\(938\) 0 0
\(939\) 45.0333i 0.0479588i
\(940\) 0 0
\(941\) 972.000 + 561.184i 1.03294 + 0.596370i 0.917827 0.396982i \(-0.129942\pi\)
0.115117 + 0.993352i \(0.463276\pi\)
\(942\) 0 0
\(943\) 554.256i 0.587758i
\(944\) 0 0
\(945\) −9.00000 + 5.19615i −0.00952381 + 0.00549857i
\(946\) 0 0
\(947\) −370.000 640.859i −0.390707 0.676725i 0.601836 0.798620i \(-0.294436\pi\)
−0.992543 + 0.121895i \(0.961103\pi\)
\(948\) 0 0
\(949\) 545.596i 0.574917i
\(950\) 0 0
\(951\) 582.000 0.611987
\(952\) 0 0
\(953\) −1086.00 + 627.002i −1.13956 + 0.657925i −0.946321 0.323227i \(-0.895232\pi\)
−0.193238 + 0.981152i \(0.561899\pi\)
\(954\) 0 0
\(955\) −70.0000 121.244i −0.0732984 0.126957i
\(956\) 0 0
\(957\) 480.000 0.501567
\(958\) 0 0
\(959\) 58.0000 100.459i 0.0604797 0.104754i
\(960\) 0 0
\(961\) −1562.00 −1.62539
\(962\) 0 0
\(963\) −216.000 124.708i −0.224299 0.129499i
\(964\) 0 0
\(965\) −525.000 303.109i −0.544041 0.314102i
\(966\) 0 0
\(967\) 167.500 + 290.119i 0.173216 + 0.300019i 0.939542 0.342432i \(-0.111251\pi\)
−0.766326 + 0.642451i \(0.777917\pi\)
\(968\) 0 0
\(969\) −627.000 361.999i −0.647059 0.373580i
\(970\) 0 0
\(971\) −1083.00 + 625.270i −1.11535 + 0.643945i −0.940209 0.340599i \(-0.889370\pi\)
−0.175136 + 0.984544i \(0.556037\pi\)
\(972\) 0 0
\(973\) −87.5000 + 151.554i −0.0899281 + 0.155760i
\(974\) 0 0
\(975\) −283.500 + 491.036i −0.290769 + 0.503627i
\(976\) 0 0
\(977\) 1292.11i 1.32253i 0.750153 + 0.661264i \(0.229980\pi\)
−0.750153 + 0.661264i \(0.770020\pi\)
\(978\) 0 0
\(979\) 528.000 + 304.841i 0.539326 + 0.311380i
\(980\) 0 0
\(981\) 228.631i 0.233059i
\(982\) 0 0
\(983\) 414.000 239.023i 0.421160 0.243157i −0.274414 0.961612i \(-0.588484\pi\)
0.695573 + 0.718455i \(0.255150\pi\)
\(984\) 0 0
\(985\) −2.00000 3.46410i −0.00203046 0.00351685i
\(986\) 0 0
\(987\) 79.6743i 0.0807237i
\(988\) 0 0
\(989\) −1960.00 −1.98180
\(990\) 0 0
\(991\) −955.500 + 551.658i −0.964178 + 0.556668i −0.897456 0.441103i \(-0.854587\pi\)
−0.0667213 + 0.997772i \(0.521254\pi\)
\(992\) 0 0
\(993\) −190.500 329.956i −0.191843 0.332282i
\(994\) 0 0
\(995\) 394.000 0.395980
\(996\) 0 0
\(997\) 270.500 468.520i 0.271314 0.469930i −0.697885 0.716210i \(-0.745875\pi\)
0.969199 + 0.246281i \(0.0792085\pi\)
\(998\) 0 0
\(999\) −81.0000 −0.0810811
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 912.3.be.c.145.1 2
4.3 odd 2 228.3.l.b.145.1 2
12.11 even 2 684.3.y.a.145.1 2
19.8 odd 6 inner 912.3.be.c.673.1 2
76.27 even 6 228.3.l.b.217.1 yes 2
228.179 odd 6 684.3.y.a.217.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.3.l.b.145.1 2 4.3 odd 2
228.3.l.b.217.1 yes 2 76.27 even 6
684.3.y.a.145.1 2 12.11 even 2
684.3.y.a.217.1 2 228.179 odd 6
912.3.be.c.145.1 2 1.1 even 1 trivial
912.3.be.c.673.1 2 19.8 odd 6 inner