Properties

Label 900.3.c.k.451.3
Level $900$
Weight $3$
Character 900.451
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
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] = [900,3,Mod(451,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 451.3
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 900.451
Dual form 900.3.c.k.451.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.618034 - 1.90211i) q^{2} +(-3.23607 - 2.35114i) q^{4} -5.25731i q^{7} +(-6.47214 + 4.70228i) q^{8} +O(q^{10})\) \(q+(0.618034 - 1.90211i) q^{2} +(-3.23607 - 2.35114i) q^{4} -5.25731i q^{7} +(-6.47214 + 4.70228i) q^{8} +19.9192i q^{11} +8.47214 q^{13} +(-10.0000 - 3.24920i) q^{14} +(4.94427 + 15.2169i) q^{16} +11.8885 q^{17} +15.2169i q^{19} +(37.8885 + 12.3107i) q^{22} +0.555029i q^{23} +(5.23607 - 16.1150i) q^{26} +(-12.3607 + 17.0130i) q^{28} +10.9443 q^{29} +8.29451i q^{31} +32.0000 q^{32} +(7.34752 - 22.6134i) q^{34} +18.3607 q^{37} +(28.9443 + 9.40456i) q^{38} +14.5836 q^{41} +22.2703i q^{43} +(46.8328 - 64.4598i) q^{44} +(1.05573 + 0.343027i) q^{46} -53.3902i q^{47} +21.3607 q^{49} +(-27.4164 - 19.9192i) q^{52} -66.3607 q^{53} +(24.7214 + 34.0260i) q^{56} +(6.76393 - 20.8172i) q^{58} +17.4370i q^{59} +90.1378 q^{61} +(15.7771 + 5.12629i) q^{62} +(19.7771 - 60.8676i) q^{64} +50.2220i q^{67} +(-38.4721 - 27.9516i) q^{68} -80.7868i q^{71} +5.55418 q^{73} +(11.3475 - 34.9241i) q^{74} +(35.7771 - 49.2429i) q^{76} +104.721 q^{77} +13.8448i q^{79} +(9.01316 - 27.7396i) q^{82} +76.2155i q^{83} +(42.3607 + 13.7638i) q^{86} +(-93.6656 - 128.920i) q^{88} +111.443 q^{89} -44.5407i q^{91} +(1.30495 - 1.79611i) q^{92} +(-101.554 - 32.9970i) q^{94} +92.8328 q^{97} +(13.2016 - 40.6304i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{4} - 8 q^{8} + 16 q^{13} - 40 q^{14} - 16 q^{16} - 24 q^{17} + 80 q^{22} + 12 q^{26} + 40 q^{28} + 8 q^{29} + 128 q^{32} + 92 q^{34} - 16 q^{37} + 80 q^{38} + 112 q^{41} + 80 q^{44} + 40 q^{46} - 4 q^{49} - 56 q^{52} - 176 q^{53} - 80 q^{56} + 36 q^{58} + 128 q^{61} - 80 q^{62} - 64 q^{64} - 136 q^{68} - 264 q^{73} + 108 q^{74} + 240 q^{77} - 116 q^{82} + 80 q^{86} - 160 q^{88} + 88 q^{89} - 120 q^{92} - 120 q^{94} + 264 q^{97} + 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 1.90211i 0.309017 0.951057i
\(3\) 0 0
\(4\) −3.23607 2.35114i −0.809017 0.587785i
\(5\) 0 0
\(6\) 0 0
\(7\) 5.25731i 0.751044i −0.926813 0.375522i \(-0.877463\pi\)
0.926813 0.375522i \(-0.122537\pi\)
\(8\) −6.47214 + 4.70228i −0.809017 + 0.587785i
\(9\) 0 0
\(10\) 0 0
\(11\) 19.9192i 1.81084i 0.424522 + 0.905418i \(0.360442\pi\)
−0.424522 + 0.905418i \(0.639558\pi\)
\(12\) 0 0
\(13\) 8.47214 0.651703 0.325851 0.945421i \(-0.394349\pi\)
0.325851 + 0.945421i \(0.394349\pi\)
\(14\) −10.0000 3.24920i −0.714286 0.232085i
\(15\) 0 0
\(16\) 4.94427 + 15.2169i 0.309017 + 0.951057i
\(17\) 11.8885 0.699326 0.349663 0.936876i \(-0.386296\pi\)
0.349663 + 0.936876i \(0.386296\pi\)
\(18\) 0 0
\(19\) 15.2169i 0.800890i 0.916321 + 0.400445i \(0.131144\pi\)
−0.916321 + 0.400445i \(0.868856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 37.8885 + 12.3107i 1.72221 + 0.559579i
\(23\) 0.555029i 0.0241317i 0.999927 + 0.0120659i \(0.00384077\pi\)
−0.999927 + 0.0120659i \(0.996159\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.23607 16.1150i 0.201387 0.619806i
\(27\) 0 0
\(28\) −12.3607 + 17.0130i −0.441453 + 0.607608i
\(29\) 10.9443 0.377389 0.188694 0.982036i \(-0.439574\pi\)
0.188694 + 0.982036i \(0.439574\pi\)
\(30\) 0 0
\(31\) 8.29451i 0.267565i 0.991011 + 0.133782i \(0.0427123\pi\)
−0.991011 + 0.133782i \(0.957288\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 7.34752 22.6134i 0.216104 0.665099i
\(35\) 0 0
\(36\) 0 0
\(37\) 18.3607 0.496235 0.248117 0.968730i \(-0.420188\pi\)
0.248117 + 0.968730i \(0.420188\pi\)
\(38\) 28.9443 + 9.40456i 0.761691 + 0.247489i
\(39\) 0 0
\(40\) 0 0
\(41\) 14.5836 0.355697 0.177849 0.984058i \(-0.443086\pi\)
0.177849 + 0.984058i \(0.443086\pi\)
\(42\) 0 0
\(43\) 22.2703i 0.517915i 0.965889 + 0.258957i \(0.0833789\pi\)
−0.965889 + 0.258957i \(0.916621\pi\)
\(44\) 46.8328 64.4598i 1.06438 1.46500i
\(45\) 0 0
\(46\) 1.05573 + 0.343027i 0.0229506 + 0.00745711i
\(47\) 53.3902i 1.13596i −0.823042 0.567981i \(-0.807725\pi\)
0.823042 0.567981i \(-0.192275\pi\)
\(48\) 0 0
\(49\) 21.3607 0.435932
\(50\) 0 0
\(51\) 0 0
\(52\) −27.4164 19.9192i −0.527239 0.383061i
\(53\) −66.3607 −1.25209 −0.626044 0.779788i \(-0.715327\pi\)
−0.626044 + 0.779788i \(0.715327\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 24.7214 + 34.0260i 0.441453 + 0.607608i
\(57\) 0 0
\(58\) 6.76393 20.8172i 0.116620 0.358918i
\(59\) 17.4370i 0.295543i 0.989022 + 0.147771i \(0.0472100\pi\)
−0.989022 + 0.147771i \(0.952790\pi\)
\(60\) 0 0
\(61\) 90.1378 1.47767 0.738834 0.673887i \(-0.235377\pi\)
0.738834 + 0.673887i \(0.235377\pi\)
\(62\) 15.7771 + 5.12629i 0.254469 + 0.0826820i
\(63\) 0 0
\(64\) 19.7771 60.8676i 0.309017 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) 50.2220i 0.749582i 0.927109 + 0.374791i \(0.122285\pi\)
−0.927109 + 0.374791i \(0.877715\pi\)
\(68\) −38.4721 27.9516i −0.565767 0.411054i
\(69\) 0 0
\(70\) 0 0
\(71\) 80.7868i 1.13784i −0.822392 0.568921i \(-0.807361\pi\)
0.822392 0.568921i \(-0.192639\pi\)
\(72\) 0 0
\(73\) 5.55418 0.0760846 0.0380423 0.999276i \(-0.487888\pi\)
0.0380423 + 0.999276i \(0.487888\pi\)
\(74\) 11.3475 34.9241i 0.153345 0.471947i
\(75\) 0 0
\(76\) 35.7771 49.2429i 0.470751 0.647933i
\(77\) 104.721 1.36002
\(78\) 0 0
\(79\) 13.8448i 0.175251i 0.996154 + 0.0876253i \(0.0279278\pi\)
−0.996154 + 0.0876253i \(0.972072\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.01316 27.7396i 0.109917 0.338288i
\(83\) 76.2155i 0.918260i 0.888369 + 0.459130i \(0.151839\pi\)
−0.888369 + 0.459130i \(0.848161\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 42.3607 + 13.7638i 0.492566 + 0.160044i
\(87\) 0 0
\(88\) −93.6656 128.920i −1.06438 1.46500i
\(89\) 111.443 1.25217 0.626083 0.779757i \(-0.284657\pi\)
0.626083 + 0.779757i \(0.284657\pi\)
\(90\) 0 0
\(91\) 44.5407i 0.489458i
\(92\) 1.30495 1.79611i 0.0141843 0.0195230i
\(93\) 0 0
\(94\) −101.554 32.9970i −1.08036 0.351031i
\(95\) 0 0
\(96\) 0 0
\(97\) 92.8328 0.957039 0.478520 0.878077i \(-0.341174\pi\)
0.478520 + 0.878077i \(0.341174\pi\)
\(98\) 13.2016 40.6304i 0.134710 0.414596i
\(99\) 0 0
\(100\) 0 0
\(101\) −64.1115 −0.634767 −0.317383 0.948297i \(-0.602804\pi\)
−0.317383 + 0.948297i \(0.602804\pi\)
\(102\) 0 0
\(103\) 137.769i 1.33757i 0.743458 + 0.668783i \(0.233184\pi\)
−0.743458 + 0.668783i \(0.766816\pi\)
\(104\) −54.8328 + 39.8384i −0.527239 + 0.383061i
\(105\) 0 0
\(106\) −41.0132 + 126.226i −0.386917 + 1.19081i
\(107\) 51.3320i 0.479739i 0.970805 + 0.239869i \(0.0771046\pi\)
−0.970805 + 0.239869i \(0.922895\pi\)
\(108\) 0 0
\(109\) 133.469 1.22449 0.612243 0.790669i \(-0.290267\pi\)
0.612243 + 0.790669i \(0.290267\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 80.0000 25.9936i 0.714286 0.232085i
\(113\) 170.721 1.51081 0.755404 0.655259i \(-0.227441\pi\)
0.755404 + 0.655259i \(0.227441\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −35.4164 25.7315i −0.305314 0.221824i
\(117\) 0 0
\(118\) 33.1672 + 10.7767i 0.281078 + 0.0913277i
\(119\) 62.5018i 0.525225i
\(120\) 0 0
\(121\) −275.774 −2.27912
\(122\) 55.7082 171.452i 0.456625 1.40535i
\(123\) 0 0
\(124\) 19.5016 26.8416i 0.157271 0.216464i
\(125\) 0 0
\(126\) 0 0
\(127\) 198.637i 1.56407i −0.623235 0.782035i \(-0.714182\pi\)
0.623235 0.782035i \(-0.285818\pi\)
\(128\) −103.554 75.2365i −0.809017 0.587785i
\(129\) 0 0
\(130\) 0 0
\(131\) 7.77041i 0.0593161i 0.999560 + 0.0296580i \(0.00944183\pi\)
−0.999560 + 0.0296580i \(0.990558\pi\)
\(132\) 0 0
\(133\) 80.0000 0.601504
\(134\) 95.5279 + 31.0389i 0.712895 + 0.231633i
\(135\) 0 0
\(136\) −76.9443 + 55.9033i −0.565767 + 0.411054i
\(137\) 0.832816 0.00607895 0.00303947 0.999995i \(-0.499033\pi\)
0.00303947 + 0.999995i \(0.499033\pi\)
\(138\) 0 0
\(139\) 237.658i 1.70977i 0.518817 + 0.854885i \(0.326373\pi\)
−0.518817 + 0.854885i \(0.673627\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −153.666 49.9290i −1.08215 0.351613i
\(143\) 168.758i 1.18013i
\(144\) 0 0
\(145\) 0 0
\(146\) 3.43267 10.5647i 0.0235114 0.0723607i
\(147\) 0 0
\(148\) −59.4164 43.1685i −0.401462 0.291679i
\(149\) 36.9706 0.248125 0.124062 0.992274i \(-0.460408\pi\)
0.124062 + 0.992274i \(0.460408\pi\)
\(150\) 0 0
\(151\) 282.723i 1.87234i −0.351552 0.936168i \(-0.614346\pi\)
0.351552 0.936168i \(-0.385654\pi\)
\(152\) −71.5542 98.4859i −0.470751 0.647933i
\(153\) 0 0
\(154\) 64.7214 199.192i 0.420269 1.29345i
\(155\) 0 0
\(156\) 0 0
\(157\) −204.748 −1.30413 −0.652063 0.758165i \(-0.726096\pi\)
−0.652063 + 0.758165i \(0.726096\pi\)
\(158\) 26.3344 + 8.55656i 0.166673 + 0.0541554i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.91796 0.0181240
\(162\) 0 0
\(163\) 107.235i 0.657885i 0.944350 + 0.328943i \(0.106692\pi\)
−0.944350 + 0.328943i \(0.893308\pi\)
\(164\) −47.1935 34.2881i −0.287765 0.209074i
\(165\) 0 0
\(166\) 144.971 + 47.1038i 0.873317 + 0.283758i
\(167\) 33.2090i 0.198856i −0.995045 0.0994280i \(-0.968299\pi\)
0.995045 0.0994280i \(-0.0317013\pi\)
\(168\) 0 0
\(169\) −97.2229 −0.575284
\(170\) 0 0
\(171\) 0 0
\(172\) 52.3607 72.0683i 0.304423 0.419002i
\(173\) −226.361 −1.30844 −0.654222 0.756303i \(-0.727004\pi\)
−0.654222 + 0.756303i \(0.727004\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −303.108 + 98.4859i −1.72221 + 0.559579i
\(177\) 0 0
\(178\) 68.8754 211.977i 0.386940 1.19088i
\(179\) 224.337i 1.25328i 0.779308 + 0.626641i \(0.215571\pi\)
−0.779308 + 0.626641i \(0.784429\pi\)
\(180\) 0 0
\(181\) 86.2229 0.476370 0.238185 0.971220i \(-0.423448\pi\)
0.238185 + 0.971220i \(0.423448\pi\)
\(182\) −84.7214 27.5276i −0.465502 0.151251i
\(183\) 0 0
\(184\) −2.60990 3.59222i −0.0141843 0.0195230i
\(185\) 0 0
\(186\) 0 0
\(187\) 236.810i 1.26636i
\(188\) −125.528 + 172.774i −0.667701 + 0.919012i
\(189\) 0 0
\(190\) 0 0
\(191\) 31.0198i 0.162407i −0.996698 0.0812036i \(-0.974124\pi\)
0.996698 0.0812036i \(-0.0258764\pi\)
\(192\) 0 0
\(193\) −110.223 −0.571103 −0.285552 0.958363i \(-0.592177\pi\)
−0.285552 + 0.958363i \(0.592177\pi\)
\(194\) 57.3738 176.579i 0.295741 0.910198i
\(195\) 0 0
\(196\) −69.1246 50.2220i −0.352677 0.256235i
\(197\) −172.525 −0.875760 −0.437880 0.899033i \(-0.644271\pi\)
−0.437880 + 0.899033i \(0.644271\pi\)
\(198\) 0 0
\(199\) 272.208i 1.36788i 0.729538 + 0.683940i \(0.239735\pi\)
−0.729538 + 0.683940i \(0.760265\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −39.6231 + 121.947i −0.196154 + 0.603699i
\(203\) 57.5374i 0.283436i
\(204\) 0 0
\(205\) 0 0
\(206\) 262.053 + 85.1461i 1.27210 + 0.413330i
\(207\) 0 0
\(208\) 41.8885 + 128.920i 0.201387 + 0.619806i
\(209\) −303.108 −1.45028
\(210\) 0 0
\(211\) 205.266i 0.972826i 0.873729 + 0.486413i \(0.161695\pi\)
−0.873729 + 0.486413i \(0.838305\pi\)
\(212\) 214.748 + 156.023i 1.01296 + 0.735959i
\(213\) 0 0
\(214\) 97.6393 + 31.7249i 0.456259 + 0.148247i
\(215\) 0 0
\(216\) 0 0
\(217\) 43.6068 0.200953
\(218\) 82.4884 253.873i 0.378387 1.16456i
\(219\) 0 0
\(220\) 0 0
\(221\) 100.721 0.455753
\(222\) 0 0
\(223\) 235.731i 1.05709i 0.848905 + 0.528545i \(0.177262\pi\)
−0.848905 + 0.528545i \(0.822738\pi\)
\(224\) 168.234i 0.751044i
\(225\) 0 0
\(226\) 105.512 324.731i 0.466865 1.43686i
\(227\) 58.5165i 0.257782i 0.991659 + 0.128891i \(0.0411417\pi\)
−0.991659 + 0.128891i \(0.958858\pi\)
\(228\) 0 0
\(229\) 162.721 0.710574 0.355287 0.934757i \(-0.384383\pi\)
0.355287 + 0.934757i \(0.384383\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −70.8328 + 51.4631i −0.305314 + 0.221824i
\(233\) 319.050 1.36931 0.684656 0.728867i \(-0.259953\pi\)
0.684656 + 0.728867i \(0.259953\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 40.9969 56.4274i 0.173716 0.239099i
\(237\) 0 0
\(238\) −118.885 38.6282i −0.499519 0.162303i
\(239\) 236.810i 0.990837i 0.868654 + 0.495419i \(0.164985\pi\)
−0.868654 + 0.495419i \(0.835015\pi\)
\(240\) 0 0
\(241\) −0.917961 −0.00380897 −0.00190448 0.999998i \(-0.500606\pi\)
−0.00190448 + 0.999998i \(0.500606\pi\)
\(242\) −170.438 + 524.553i −0.704288 + 2.16758i
\(243\) 0 0
\(244\) −291.692 211.927i −1.19546 0.868552i
\(245\) 0 0
\(246\) 0 0
\(247\) 128.920i 0.521942i
\(248\) −39.0031 53.6832i −0.157271 0.216464i
\(249\) 0 0
\(250\) 0 0
\(251\) 136.690i 0.544582i −0.962215 0.272291i \(-0.912219\pi\)
0.962215 0.272291i \(-0.0877813\pi\)
\(252\) 0 0
\(253\) −11.0557 −0.0436985
\(254\) −377.830 122.764i −1.48752 0.483324i
\(255\) 0 0
\(256\) −207.108 + 150.473i −0.809017 + 0.587785i
\(257\) −274.944 −1.06982 −0.534911 0.844908i \(-0.679655\pi\)
−0.534911 + 0.844908i \(0.679655\pi\)
\(258\) 0 0
\(259\) 96.5278i 0.372694i
\(260\) 0 0
\(261\) 0 0
\(262\) 14.7802 + 4.80238i 0.0564130 + 0.0183297i
\(263\) 406.385i 1.54519i −0.634899 0.772596i \(-0.718958\pi\)
0.634899 0.772596i \(-0.281042\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 49.4427 152.169i 0.185875 0.572064i
\(267\) 0 0
\(268\) 118.079 162.522i 0.440593 0.606424i
\(269\) 348.525 1.29563 0.647816 0.761797i \(-0.275683\pi\)
0.647816 + 0.761797i \(0.275683\pi\)
\(270\) 0 0
\(271\) 247.849i 0.914571i 0.889320 + 0.457286i \(0.151178\pi\)
−0.889320 + 0.457286i \(0.848822\pi\)
\(272\) 58.7802 + 180.907i 0.216104 + 0.665099i
\(273\) 0 0
\(274\) 0.514708 1.58411i 0.00187850 0.00578142i
\(275\) 0 0
\(276\) 0 0
\(277\) 54.7539 0.197667 0.0988337 0.995104i \(-0.468489\pi\)
0.0988337 + 0.995104i \(0.468489\pi\)
\(278\) 452.053 + 146.881i 1.62609 + 0.528348i
\(279\) 0 0
\(280\) 0 0
\(281\) 50.3607 0.179220 0.0896098 0.995977i \(-0.471438\pi\)
0.0896098 + 0.995977i \(0.471438\pi\)
\(282\) 0 0
\(283\) 147.336i 0.520621i −0.965525 0.260310i \(-0.916175\pi\)
0.965525 0.260310i \(-0.0838249\pi\)
\(284\) −189.941 + 261.432i −0.668807 + 0.920534i
\(285\) 0 0
\(286\) 320.997 + 104.298i 1.12237 + 0.364679i
\(287\) 76.6705i 0.267145i
\(288\) 0 0
\(289\) −147.663 −0.510943
\(290\) 0 0
\(291\) 0 0
\(292\) −17.9737 13.0586i −0.0615537 0.0447214i
\(293\) 178.859 0.610441 0.305220 0.952282i \(-0.401270\pi\)
0.305220 + 0.952282i \(0.401270\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −118.833 + 86.3371i −0.401462 + 0.291679i
\(297\) 0 0
\(298\) 22.8491 70.3222i 0.0766748 0.235981i
\(299\) 4.70228i 0.0157267i
\(300\) 0 0
\(301\) 117.082 0.388977
\(302\) −537.771 174.732i −1.78070 0.578584i
\(303\) 0 0
\(304\) −231.554 + 75.2365i −0.761691 + 0.247489i
\(305\) 0 0
\(306\) 0 0
\(307\) 284.550i 0.926873i −0.886130 0.463436i \(-0.846616\pi\)
0.886130 0.463436i \(-0.153384\pi\)
\(308\) −338.885 246.215i −1.10028 0.799398i
\(309\) 0 0
\(310\) 0 0
\(311\) 282.199i 0.907392i 0.891157 + 0.453696i \(0.149895\pi\)
−0.891157 + 0.453696i \(0.850105\pi\)
\(312\) 0 0
\(313\) 567.548 1.81325 0.906626 0.421935i \(-0.138649\pi\)
0.906626 + 0.421935i \(0.138649\pi\)
\(314\) −126.541 + 389.453i −0.402997 + 1.24030i
\(315\) 0 0
\(316\) 32.5511 44.8027i 0.103010 0.141781i
\(317\) 161.141 0.508331 0.254165 0.967161i \(-0.418199\pi\)
0.254165 + 0.967161i \(0.418199\pi\)
\(318\) 0 0
\(319\) 218.001i 0.683389i
\(320\) 0 0
\(321\) 0 0
\(322\) 1.80340 5.55029i 0.00560062 0.0172369i
\(323\) 180.907i 0.560083i
\(324\) 0 0
\(325\) 0 0
\(326\) 203.974 + 66.2751i 0.625686 + 0.203298i
\(327\) 0 0
\(328\) −94.3870 + 68.5762i −0.287765 + 0.209074i
\(329\) −280.689 −0.853158
\(330\) 0 0
\(331\) 331.966i 1.00292i −0.865181 0.501459i \(-0.832797\pi\)
0.865181 0.501459i \(-0.167203\pi\)
\(332\) 179.193 246.639i 0.539739 0.742888i
\(333\) 0 0
\(334\) −63.1672 20.5243i −0.189123 0.0614499i
\(335\) 0 0
\(336\) 0 0
\(337\) 269.108 0.798541 0.399271 0.916833i \(-0.369263\pi\)
0.399271 + 0.916833i \(0.369263\pi\)
\(338\) −60.0871 + 184.929i −0.177772 + 0.547127i
\(339\) 0 0
\(340\) 0 0
\(341\) −165.220 −0.484516
\(342\) 0 0
\(343\) 369.908i 1.07845i
\(344\) −104.721 144.137i −0.304423 0.419002i
\(345\) 0 0
\(346\) −139.899 + 430.564i −0.404331 + 1.24440i
\(347\) 503.075i 1.44978i 0.688863 + 0.724892i \(0.258110\pi\)
−0.688863 + 0.724892i \(0.741890\pi\)
\(348\) 0 0
\(349\) −0.504658 −0.00144601 −0.000723006 1.00000i \(-0.500230\pi\)
−0.000723006 1.00000i \(0.500230\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 637.414i 1.81084i
\(353\) −335.994 −0.951824 −0.475912 0.879493i \(-0.657882\pi\)
−0.475912 + 0.879493i \(0.657882\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −360.636 262.018i −1.01302 0.736004i
\(357\) 0 0
\(358\) 426.715 + 138.648i 1.19194 + 0.387285i
\(359\) 98.4859i 0.274334i 0.990548 + 0.137167i \(0.0437997\pi\)
−0.990548 + 0.137167i \(0.956200\pi\)
\(360\) 0 0
\(361\) 129.446 0.358576
\(362\) 53.2887 164.006i 0.147206 0.453054i
\(363\) 0 0
\(364\) −104.721 + 144.137i −0.287696 + 0.395980i
\(365\) 0 0
\(366\) 0 0
\(367\) 498.473i 1.35824i 0.734029 + 0.679118i \(0.237638\pi\)
−0.734029 + 0.679118i \(0.762362\pi\)
\(368\) −8.44582 + 2.74421i −0.0229506 + 0.00745711i
\(369\) 0 0
\(370\) 0 0
\(371\) 348.879i 0.940374i
\(372\) 0 0
\(373\) −600.354 −1.60953 −0.804765 0.593594i \(-0.797709\pi\)
−0.804765 + 0.593594i \(0.797709\pi\)
\(374\) 450.440 + 146.357i 1.20438 + 0.391328i
\(375\) 0 0
\(376\) 251.056 + 345.549i 0.667701 + 0.919012i
\(377\) 92.7214 0.245945
\(378\) 0 0
\(379\) 303.490i 0.800765i −0.916348 0.400383i \(-0.868877\pi\)
0.916348 0.400383i \(-0.131123\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −59.0031 19.1713i −0.154458 0.0501866i
\(383\) 332.583i 0.868362i 0.900826 + 0.434181i \(0.142962\pi\)
−0.900826 + 0.434181i \(0.857038\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −68.1215 + 209.656i −0.176481 + 0.543151i
\(387\) 0 0
\(388\) −300.413 218.263i −0.774261 0.562534i
\(389\) −392.354 −1.00862 −0.504312 0.863522i \(-0.668254\pi\)
−0.504312 + 0.863522i \(0.668254\pi\)
\(390\) 0 0
\(391\) 6.59849i 0.0168759i
\(392\) −138.249 + 100.444i −0.352677 + 0.256235i
\(393\) 0 0
\(394\) −106.626 + 328.162i −0.270625 + 0.832897i
\(395\) 0 0
\(396\) 0 0
\(397\) −334.190 −0.841789 −0.420895 0.907110i \(-0.638284\pi\)
−0.420895 + 0.907110i \(0.638284\pi\)
\(398\) 517.771 + 168.234i 1.30093 + 0.422698i
\(399\) 0 0
\(400\) 0 0
\(401\) −121.003 −0.301753 −0.150877 0.988553i \(-0.548210\pi\)
−0.150877 + 0.988553i \(0.548210\pi\)
\(402\) 0 0
\(403\) 70.2722i 0.174373i
\(404\) 207.469 + 150.735i 0.513537 + 0.373107i
\(405\) 0 0
\(406\) −109.443 35.5601i −0.269563 0.0875864i
\(407\) 365.730i 0.898599i
\(408\) 0 0
\(409\) −607.410 −1.48511 −0.742555 0.669785i \(-0.766386\pi\)
−0.742555 + 0.669785i \(0.766386\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 323.915 445.831i 0.786201 1.08211i
\(413\) 91.6718 0.221966
\(414\) 0 0
\(415\) 0 0
\(416\) 271.108 0.651703
\(417\) 0 0
\(418\) −187.331 + 576.546i −0.448161 + 1.37930i
\(419\) 466.760i 1.11398i −0.830518 0.556992i \(-0.811955\pi\)
0.830518 0.556992i \(-0.188045\pi\)
\(420\) 0 0
\(421\) −73.0883 −0.173606 −0.0868031 0.996225i \(-0.527665\pi\)
−0.0868031 + 0.996225i \(0.527665\pi\)
\(422\) 390.440 + 126.862i 0.925212 + 0.300620i
\(423\) 0 0
\(424\) 429.495 312.047i 1.01296 0.735959i
\(425\) 0 0
\(426\) 0 0
\(427\) 473.882i 1.10979i
\(428\) 120.689 166.114i 0.281983 0.388117i
\(429\) 0 0
\(430\) 0 0
\(431\) 463.630i 1.07571i −0.843038 0.537853i \(-0.819235\pi\)
0.843038 0.537853i \(-0.180765\pi\)
\(432\) 0 0
\(433\) 99.8359 0.230568 0.115284 0.993333i \(-0.463222\pi\)
0.115284 + 0.993333i \(0.463222\pi\)
\(434\) 26.9505 82.9451i 0.0620979 0.191118i
\(435\) 0 0
\(436\) −431.915 313.805i −0.990630 0.719735i
\(437\) −8.44582 −0.0193268
\(438\) 0 0
\(439\) 374.086i 0.852133i −0.904692 0.426066i \(-0.859899\pi\)
0.904692 0.426066i \(-0.140101\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 62.2492 191.583i 0.140835 0.433447i
\(443\) 290.100i 0.654854i 0.944877 + 0.327427i \(0.106182\pi\)
−0.944877 + 0.327427i \(0.893818\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 448.387 + 145.690i 1.00535 + 0.326659i
\(447\) 0 0
\(448\) −320.000 103.974i −0.714286 0.232085i
\(449\) −299.921 −0.667976 −0.333988 0.942577i \(-0.608394\pi\)
−0.333988 + 0.942577i \(0.608394\pi\)
\(450\) 0 0
\(451\) 290.493i 0.644109i
\(452\) −552.466 401.390i −1.22227 0.888031i
\(453\) 0 0
\(454\) 111.305 + 36.1652i 0.245165 + 0.0796590i
\(455\) 0 0
\(456\) 0 0
\(457\) −822.328 −1.79941 −0.899703 0.436503i \(-0.856217\pi\)
−0.899703 + 0.436503i \(0.856217\pi\)
\(458\) 100.567 309.514i 0.219579 0.675796i
\(459\) 0 0
\(460\) 0 0
\(461\) 456.885 0.991075 0.495537 0.868587i \(-0.334971\pi\)
0.495537 + 0.868587i \(0.334971\pi\)
\(462\) 0 0
\(463\) 400.249i 0.864469i 0.901761 + 0.432234i \(0.142275\pi\)
−0.901761 + 0.432234i \(0.857725\pi\)
\(464\) 54.1115 + 166.538i 0.116620 + 0.358918i
\(465\) 0 0
\(466\) 197.183 606.868i 0.423140 1.30229i
\(467\) 913.145i 1.95534i −0.210139 0.977672i \(-0.567392\pi\)
0.210139 0.977672i \(-0.432608\pi\)
\(468\) 0 0
\(469\) 264.033 0.562969
\(470\) 0 0
\(471\) 0 0
\(472\) −81.9938 112.855i −0.173716 0.239099i
\(473\) −443.607 −0.937858
\(474\) 0 0
\(475\) 0 0
\(476\) −146.950 + 202.260i −0.308720 + 0.424916i
\(477\) 0 0
\(478\) 450.440 + 146.357i 0.942342 + 0.306186i
\(479\) 526.131i 1.09840i −0.835692 0.549198i \(-0.814933\pi\)
0.835692 0.549198i \(-0.185067\pi\)
\(480\) 0 0
\(481\) 155.554 0.323397
\(482\) −0.567331 + 1.74606i −0.00117704 + 0.00362254i
\(483\) 0 0
\(484\) 892.423 + 648.384i 1.84385 + 1.33964i
\(485\) 0 0
\(486\) 0 0
\(487\) 443.541i 0.910762i 0.890297 + 0.455381i \(0.150497\pi\)
−0.890297 + 0.455381i \(0.849503\pi\)
\(488\) −583.384 + 423.853i −1.19546 + 0.868552i
\(489\) 0 0
\(490\) 0 0
\(491\) 287.163i 0.584854i −0.956288 0.292427i \(-0.905537\pi\)
0.956288 0.292427i \(-0.0944628\pi\)
\(492\) 0 0
\(493\) 130.111 0.263918
\(494\) 245.220 + 79.6767i 0.496396 + 0.161289i
\(495\) 0 0
\(496\) −126.217 + 41.0103i −0.254469 + 0.0826820i
\(497\) −424.721 −0.854570
\(498\) 0 0
\(499\) 810.936i 1.62512i −0.582876 0.812561i \(-0.698073\pi\)
0.582876 0.812561i \(-0.301927\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −260.000 84.4791i −0.517928 0.168285i
\(503\) 642.471i 1.27728i −0.769506 0.638639i \(-0.779498\pi\)
0.769506 0.638639i \(-0.220502\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.83282 + 21.0292i −0.0135036 + 0.0415598i
\(507\) 0 0
\(508\) −467.023 + 642.802i −0.919337 + 1.26536i
\(509\) 915.050 1.79774 0.898870 0.438216i \(-0.144389\pi\)
0.898870 + 0.438216i \(0.144389\pi\)
\(510\) 0 0
\(511\) 29.2000i 0.0571429i
\(512\) 158.217 + 486.941i 0.309017 + 0.951057i
\(513\) 0 0
\(514\) −169.925 + 522.975i −0.330593 + 1.01746i
\(515\) 0 0
\(516\) 0 0
\(517\) 1063.49 2.05704
\(518\) −183.607 59.6575i −0.354453 0.115169i
\(519\) 0 0
\(520\) 0 0
\(521\) −1006.98 −1.93279 −0.966396 0.257058i \(-0.917247\pi\)
−0.966396 + 0.257058i \(0.917247\pi\)
\(522\) 0 0
\(523\) 774.173i 1.48025i −0.672467 0.740127i \(-0.734765\pi\)
0.672467 0.740127i \(-0.265235\pi\)
\(524\) 18.2693 25.1456i 0.0348651 0.0479877i
\(525\) 0 0
\(526\) −772.991 251.160i −1.46956 0.477490i
\(527\) 98.6096i 0.187115i
\(528\) 0 0
\(529\) 528.692 0.999418
\(530\) 0 0
\(531\) 0 0
\(532\) −258.885 188.091i −0.486627 0.353555i
\(533\) 123.554 0.231809
\(534\) 0 0
\(535\) 0 0
\(536\) −236.158 325.043i −0.440593 0.606424i
\(537\) 0 0
\(538\) 215.400 662.933i 0.400372 1.23222i
\(539\) 425.487i 0.789401i
\(540\) 0 0
\(541\) −259.115 −0.478955 −0.239477 0.970902i \(-0.576976\pi\)
−0.239477 + 0.970902i \(0.576976\pi\)
\(542\) 471.437 + 153.179i 0.869809 + 0.282618i
\(543\) 0 0
\(544\) 380.433 0.699326
\(545\) 0 0
\(546\) 0 0
\(547\) 149.818i 0.273890i −0.990579 0.136945i \(-0.956272\pi\)
0.990579 0.136945i \(-0.0437284\pi\)
\(548\) −2.69505 1.95807i −0.00491797 0.00357312i
\(549\) 0 0
\(550\) 0 0
\(551\) 166.538i 0.302247i
\(552\) 0 0
\(553\) 72.7864 0.131621
\(554\) 33.8398 104.148i 0.0610826 0.187993i
\(555\) 0 0
\(556\) 558.768 769.078i 1.00498 1.38323i
\(557\) 511.698 0.918668 0.459334 0.888264i \(-0.348088\pi\)
0.459334 + 0.888264i \(0.348088\pi\)
\(558\) 0 0
\(559\) 188.677i 0.337526i
\(560\) 0 0
\(561\) 0 0
\(562\) 31.1246 95.7917i 0.0553819 0.170448i
\(563\) 490.726i 0.871627i −0.900037 0.435814i \(-0.856461\pi\)
0.900037 0.435814i \(-0.143539\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −280.249 91.0585i −0.495140 0.160881i
\(567\) 0 0
\(568\) 379.882 + 522.863i 0.668807 + 0.920534i
\(569\) 232.748 0.409047 0.204523 0.978862i \(-0.434436\pi\)
0.204523 + 0.978862i \(0.434436\pi\)
\(570\) 0 0
\(571\) 210.755i 0.369098i 0.982823 + 0.184549i \(0.0590824\pi\)
−0.982823 + 0.184549i \(0.940918\pi\)
\(572\) 396.774 546.113i 0.693661 0.954742i
\(573\) 0 0
\(574\) −145.836 47.3850i −0.254070 0.0825522i
\(575\) 0 0
\(576\) 0 0
\(577\) −341.712 −0.592222 −0.296111 0.955154i \(-0.595690\pi\)
−0.296111 + 0.955154i \(0.595690\pi\)
\(578\) −91.2605 + 280.871i −0.157890 + 0.485936i
\(579\) 0 0
\(580\) 0 0
\(581\) 400.689 0.689654
\(582\) 0 0
\(583\) 1321.85i 2.26733i
\(584\) −35.9474 + 26.1173i −0.0615537 + 0.0447214i
\(585\) 0 0
\(586\) 110.541 340.210i 0.188637 0.580564i
\(587\) 618.412i 1.05351i −0.850016 0.526756i \(-0.823408\pi\)
0.850016 0.526756i \(-0.176592\pi\)
\(588\) 0 0
\(589\) −126.217 −0.214290
\(590\) 0 0
\(591\) 0 0
\(592\) 90.7802 + 279.393i 0.153345 + 0.471947i
\(593\) 120.663 0.203478 0.101739 0.994811i \(-0.467559\pi\)
0.101739 + 0.994811i \(0.467559\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −119.639 86.9231i −0.200737 0.145844i
\(597\) 0 0
\(598\) 8.94427 + 2.90617i 0.0149570 + 0.00485982i
\(599\) 849.927i 1.41891i −0.704751 0.709455i \(-0.748941\pi\)
0.704751 0.709455i \(-0.251059\pi\)
\(600\) 0 0
\(601\) −11.3576 −0.0188978 −0.00944890 0.999955i \(-0.503008\pi\)
−0.00944890 + 0.999955i \(0.503008\pi\)
\(602\) 72.3607 222.703i 0.120200 0.369939i
\(603\) 0 0
\(604\) −664.721 + 914.910i −1.10053 + 1.51475i
\(605\) 0 0
\(606\) 0 0
\(607\) 1115.12i 1.83710i −0.395305 0.918550i \(-0.629361\pi\)
0.395305 0.918550i \(-0.370639\pi\)
\(608\) 486.941i 0.800890i
\(609\) 0 0
\(610\) 0 0
\(611\) 452.329i 0.740309i
\(612\) 0 0
\(613\) −499.475 −0.814805 −0.407402 0.913249i \(-0.633565\pi\)
−0.407402 + 0.913249i \(0.633565\pi\)
\(614\) −541.246 175.862i −0.881508 0.286419i
\(615\) 0 0
\(616\) −677.771 + 492.429i −1.10028 + 0.799398i
\(617\) 545.935 0.884822 0.442411 0.896813i \(-0.354123\pi\)
0.442411 + 0.896813i \(0.354123\pi\)
\(618\) 0 0
\(619\) 455.011i 0.735075i −0.930009 0.367537i \(-0.880201\pi\)
0.930009 0.367537i \(-0.119799\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 536.774 + 174.408i 0.862981 + 0.280399i
\(623\) 585.889i 0.940432i
\(624\) 0 0
\(625\) 0 0
\(626\) 350.764 1079.54i 0.560326 1.72451i
\(627\) 0 0
\(628\) 662.577 + 481.391i 1.05506 + 0.766546i
\(629\) 218.282 0.347030
\(630\) 0 0
\(631\) 267.706i 0.424257i −0.977242 0.212128i \(-0.931960\pi\)
0.977242 0.212128i \(-0.0680395\pi\)
\(632\) −65.1021 89.6054i −0.103010 0.141781i
\(633\) 0 0
\(634\) 99.5905 306.508i 0.157083 0.483451i
\(635\) 0 0
\(636\) 0 0
\(637\) 180.971 0.284098
\(638\) 414.663 + 134.732i 0.649941 + 0.211179i
\(639\) 0 0
\(640\) 0 0
\(641\) 418.571 0.652997 0.326499 0.945198i \(-0.394131\pi\)
0.326499 + 0.945198i \(0.394131\pi\)
\(642\) 0 0
\(643\) 439.339i 0.683265i 0.939834 + 0.341633i \(0.110980\pi\)
−0.939834 + 0.341633i \(0.889020\pi\)
\(644\) −9.44272 6.86054i −0.0146626 0.0106530i
\(645\) 0 0
\(646\) 344.105 + 111.807i 0.532671 + 0.173075i
\(647\) 419.644i 0.648600i −0.945954 0.324300i \(-0.894871\pi\)
0.945954 0.324300i \(-0.105129\pi\)
\(648\) 0 0
\(649\) −347.331 −0.535179
\(650\) 0 0
\(651\) 0 0
\(652\) 252.125 347.021i 0.386695 0.532240i
\(653\) −370.085 −0.566746 −0.283373 0.959010i \(-0.591453\pi\)
−0.283373 + 0.959010i \(0.591453\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 72.1052 + 221.917i 0.109917 + 0.338288i
\(657\) 0 0
\(658\) −173.475 + 533.902i −0.263640 + 0.811401i
\(659\) 322.823i 0.489868i −0.969540 0.244934i \(-0.921234\pi\)
0.969540 0.244934i \(-0.0787664\pi\)
\(660\) 0 0
\(661\) −812.735 −1.22955 −0.614777 0.788701i \(-0.710754\pi\)
−0.614777 + 0.788701i \(0.710754\pi\)
\(662\) −631.437 205.166i −0.953832 0.309919i
\(663\) 0 0
\(664\) −358.387 493.277i −0.539739 0.742888i
\(665\) 0 0
\(666\) 0 0
\(667\) 6.07439i 0.00910703i
\(668\) −78.0789 + 107.466i −0.116885 + 0.160878i
\(669\) 0 0
\(670\) 0 0
\(671\) 1795.47i 2.67581i
\(672\) 0 0
\(673\) −467.378 −0.694469 −0.347235 0.937778i \(-0.612879\pi\)
−0.347235 + 0.937778i \(0.612879\pi\)
\(674\) 166.318 511.875i 0.246763 0.759458i
\(675\) 0 0
\(676\) 314.620 + 228.585i 0.465414 + 0.338143i
\(677\) 548.237 0.809803 0.404902 0.914360i \(-0.367306\pi\)
0.404902 + 0.914360i \(0.367306\pi\)
\(678\) 0 0
\(679\) 488.051i 0.718779i
\(680\) 0 0
\(681\) 0 0
\(682\) −102.111 + 314.267i −0.149724 + 0.460802i
\(683\) 23.9663i 0.0350898i −0.999846 0.0175449i \(-0.994415\pi\)
0.999846 0.0175449i \(-0.00558500\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −703.607 228.616i −1.02567 0.333259i
\(687\) 0 0
\(688\) −338.885 + 110.111i −0.492566 + 0.160044i
\(689\) −562.217 −0.815989
\(690\) 0 0
\(691\) 186.981i 0.270595i 0.990805 + 0.135298i \(0.0431990\pi\)
−0.990805 + 0.135298i \(0.956801\pi\)
\(692\) 732.519 + 532.206i 1.05855 + 0.769084i
\(693\) 0 0
\(694\) 956.906 + 310.917i 1.37883 + 0.448008i
\(695\) 0 0
\(696\) 0 0
\(697\) 173.378 0.248748
\(698\) −0.311896 + 0.959917i −0.000446842 + 0.00137524i
\(699\) 0 0
\(700\) 0 0
\(701\) 706.636 1.00804 0.504020 0.863692i \(-0.331854\pi\)
0.504020 + 0.863692i \(0.331854\pi\)
\(702\) 0 0
\(703\) 279.393i 0.397429i
\(704\) 1212.43 + 393.943i 1.72221 + 0.559579i
\(705\) 0 0
\(706\) −207.656 + 639.098i −0.294130 + 0.905238i
\(707\) 337.054i 0.476738i
\(708\) 0 0
\(709\) −188.597 −0.266005 −0.133002 0.991116i \(-0.542462\pi\)
−0.133002 + 0.991116i \(0.542462\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −721.272 + 524.035i −1.01302 + 0.736004i
\(713\) −4.60369 −0.00645679
\(714\) 0 0
\(715\) 0 0
\(716\) 527.449 725.971i 0.736661 1.01393i
\(717\) 0 0
\(718\) 187.331 + 60.8676i 0.260907 + 0.0847738i
\(719\) 156.085i 0.217086i 0.994092 + 0.108543i \(0.0346186\pi\)
−0.994092 + 0.108543i \(0.965381\pi\)
\(720\) 0 0
\(721\) 724.296 1.00457
\(722\) 80.0019 246.221i 0.110806 0.341026i
\(723\) 0 0
\(724\) −279.023 202.722i −0.385391 0.280003i
\(725\) 0 0
\(726\) 0 0
\(727\) 715.164i 0.983719i −0.870675 0.491859i \(-0.836317\pi\)
0.870675 0.491859i \(-0.163683\pi\)
\(728\) 209.443 + 288.273i 0.287696 + 0.395980i
\(729\) 0 0
\(730\) 0 0
\(731\) 264.762i 0.362191i
\(732\) 0 0
\(733\) −1233.29 −1.68252 −0.841259 0.540632i \(-0.818185\pi\)
−0.841259 + 0.540632i \(0.818185\pi\)
\(734\) 948.152 + 308.073i 1.29176 + 0.419718i
\(735\) 0 0
\(736\) 17.7609i 0.0241317i
\(737\) −1000.38 −1.35737
\(738\) 0 0
\(739\) 8.55656i 0.0115786i −0.999983 0.00578928i \(-0.998157\pi\)
0.999983 0.00578928i \(-0.00184280\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 663.607 + 215.619i 0.894349 + 0.290592i
\(743\) 1010.56i 1.36011i 0.733163 + 0.680053i \(0.238043\pi\)
−0.733163 + 0.680053i \(0.761957\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −371.039 + 1141.94i −0.497372 + 1.53075i
\(747\) 0 0
\(748\) 556.774 766.334i 0.744350 1.02451i
\(749\) 269.868 0.360305
\(750\) 0 0
\(751\) 1104.31i 1.47046i 0.677820 + 0.735228i \(0.262925\pi\)
−0.677820 + 0.735228i \(0.737075\pi\)
\(752\) 812.433 263.976i 1.08036 0.351031i
\(753\) 0 0
\(754\) 57.3050 176.367i 0.0760013 0.233908i
\(755\) 0 0
\(756\) 0 0
\(757\) 875.633 1.15671 0.578357 0.815783i \(-0.303694\pi\)
0.578357 + 0.815783i \(0.303694\pi\)
\(758\) −577.272 187.567i −0.761573 0.247450i
\(759\) 0 0
\(760\) 0 0
\(761\) 647.207 0.850470 0.425235 0.905083i \(-0.360192\pi\)
0.425235 + 0.905083i \(0.360192\pi\)
\(762\) 0 0
\(763\) 701.688i 0.919644i
\(764\) −72.9318 + 100.382i −0.0954605 + 0.131390i
\(765\) 0 0
\(766\) 632.610 + 205.547i 0.825861 + 0.268339i
\(767\) 147.729i 0.192606i
\(768\) 0 0
\(769\) 631.430 0.821106 0.410553 0.911837i \(-0.365336\pi\)
0.410553 + 0.911837i \(0.365336\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 356.689 + 259.150i 0.462032 + 0.335686i
\(773\) −421.522 −0.545306 −0.272653 0.962112i \(-0.587901\pi\)
−0.272653 + 0.962112i \(0.587901\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −600.827 + 436.526i −0.774261 + 0.562534i
\(777\) 0 0
\(778\) −242.488 + 746.303i −0.311682 + 0.959258i
\(779\) 221.917i 0.284874i
\(780\) 0 0
\(781\) 1609.21 2.06044
\(782\) 12.5511 + 4.07809i 0.0160500 + 0.00521495i
\(783\) 0 0
\(784\) 105.613 + 325.043i 0.134710 + 0.414596i
\(785\) 0 0
\(786\) 0 0
\(787\) 838.633i 1.06561i 0.846239 + 0.532804i \(0.178862\pi\)
−0.846239 + 0.532804i \(0.821138\pi\)
\(788\) 558.302 + 405.630i 0.708505 + 0.514759i
\(789\) 0 0
\(790\) 0 0
\(791\) 897.535i 1.13468i
\(792\) 0 0
\(793\) 763.659 0.963001
\(794\) −206.541 + 635.668i −0.260127 + 0.800589i
\(795\) 0 0
\(796\) 640.000 880.884i 0.804020 1.10664i
\(797\) 1213.57 1.52268 0.761339 0.648354i \(-0.224542\pi\)
0.761339 + 0.648354i \(0.224542\pi\)
\(798\) 0 0
\(799\) 634.732i 0.794408i
\(800\) 0 0
\(801\) 0 0
\(802\) −74.7840 + 230.162i −0.0932469 + 0.286985i
\(803\) 110.635i 0.137777i
\(804\) 0 0
\(805\) 0 0
\(806\) 133.666 + 43.4306i 0.165838 + 0.0538841i
\(807\) 0 0
\(808\) 414.938 301.470i 0.513537 0.373107i
\(809\) 229.214 0.283330 0.141665 0.989915i \(-0.454754\pi\)
0.141665 + 0.989915i \(0.454754\pi\)
\(810\) 0 0
\(811\) 454.225i 0.560080i −0.959988 0.280040i \(-0.909652\pi\)
0.959988 0.280040i \(-0.0903478\pi\)
\(812\) −135.279 + 186.195i −0.166599 + 0.229304i
\(813\) 0 0
\(814\) 695.659 + 226.033i 0.854618 + 0.277682i
\(815\) 0 0
\(816\) 0 0
\(817\) −338.885 −0.414792
\(818\) −375.400 + 1155.36i −0.458924 + 1.41242i
\(819\) 0 0
\(820\) 0 0
\(821\) −1130.90 −1.37747 −0.688733 0.725015i \(-0.741833\pi\)
−0.688733 + 0.725015i \(0.741833\pi\)
\(822\) 0 0
\(823\) 780.148i 0.947931i 0.880543 + 0.473966i \(0.157178\pi\)
−0.880543 + 0.473966i \(0.842822\pi\)
\(824\) −647.830 891.661i −0.786201 1.08211i
\(825\) 0 0
\(826\) 56.6563 174.370i 0.0685912 0.211102i
\(827\) 209.175i 0.252932i −0.991971 0.126466i \(-0.959636\pi\)
0.991971 0.126466i \(-0.0403635\pi\)
\(828\) 0 0
\(829\) −508.525 −0.613419 −0.306710 0.951803i \(-0.599228\pi\)
−0.306710 + 0.951803i \(0.599228\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 167.554 515.679i 0.201387 0.619806i
\(833\) 253.947 0.304859
\(834\) 0 0
\(835\) 0 0
\(836\) 980.879 + 712.650i 1.17330 + 0.852453i
\(837\) 0 0
\(838\) −887.830 288.473i −1.05946 0.344240i
\(839\) 274.028i 0.326613i −0.986575 0.163306i \(-0.947784\pi\)
0.986575 0.163306i \(-0.0522159\pi\)
\(840\) 0 0
\(841\) −721.223 −0.857578
\(842\) −45.1710 + 139.022i −0.0536473 + 0.165109i
\(843\) 0 0
\(844\) 482.610 664.256i 0.571813 0.787033i
\(845\) 0 0
\(846\) 0 0
\(847\) 1449.83i 1.71172i
\(848\) −328.105 1009.80i −0.386917 1.19081i
\(849\) 0 0
\(850\) 0 0
\(851\) 10.1907i 0.0119750i
\(852\) 0 0
\(853\) 1583.28 1.85613 0.928066 0.372416i \(-0.121471\pi\)
0.928066 + 0.372416i \(0.121471\pi\)
\(854\) −901.378 292.875i −1.05548 0.342945i
\(855\) 0 0
\(856\) −241.378 332.228i −0.281983 0.388117i
\(857\) −1007.38 −1.17547 −0.587735 0.809054i \(-0.699980\pi\)
−0.587735 + 0.809054i \(0.699980\pi\)
\(858\) 0 0
\(859\) 76.6086i 0.0891835i 0.999005 + 0.0445917i \(0.0141987\pi\)
−0.999005 + 0.0445917i \(0.985801\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −881.876 286.539i −1.02306 0.332412i
\(863\) 255.450i 0.296002i 0.988987 + 0.148001i \(0.0472839\pi\)
−0.988987 + 0.148001i \(0.952716\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 61.7020 189.899i 0.0712494 0.219283i
\(867\) 0 0
\(868\) −141.115 102.526i −0.162574 0.118117i
\(869\) −275.777 −0.317350
\(870\) 0 0
\(871\) 425.487i 0.488504i
\(872\) −863.830 + 627.609i −0.990630 + 0.719735i
\(873\) 0 0
\(874\) −5.21981 + 16.0649i −0.00597232 + 0.0183809i
\(875\) 0 0
\(876\) 0 0
\(877\) −601.522 −0.685886 −0.342943 0.939356i \(-0.611424\pi\)
−0.342943 + 0.939356i \(0.611424\pi\)
\(878\) −711.554 231.198i −0.810426 0.263323i
\(879\) 0 0
\(880\) 0 0
\(881\) −237.850 −0.269977 −0.134989 0.990847i \(-0.543100\pi\)
−0.134989 + 0.990847i \(0.543100\pi\)
\(882\) 0 0
\(883\) 1.30294i 0.00147559i −1.00000 0.000737794i \(-0.999765\pi\)
1.00000 0.000737794i \(-0.000234847\pi\)
\(884\) −325.941 236.810i −0.368712 0.267885i
\(885\) 0 0
\(886\) 551.803 + 179.292i 0.622803 + 0.202361i
\(887\) 536.353i 0.604682i 0.953200 + 0.302341i \(0.0977682\pi\)
−0.953200 + 0.302341i \(0.902232\pi\)
\(888\) 0 0
\(889\) −1044.30 −1.17469
\(890\) 0 0
\(891\) 0 0
\(892\) 554.237 762.842i 0.621342 0.855203i
\(893\) 812.433 0.909780
\(894\) 0 0
\(895\) 0 0
\(896\) −395.542 + 544.417i −0.441453 + 0.607608i
\(897\) 0 0
\(898\) −185.361 + 570.484i −0.206416 + 0.635283i
\(899\) 90.7773i 0.100976i
\(900\) 0 0
\(901\) −788.932 −0.875618
\(902\) 552.551 + 179.535i 0.612584 + 0.199041i
\(903\) 0 0
\(904\) −1104.93 + 802.780i −1.22227 + 0.888031i
\(905\) 0 0
\(906\) 0 0
\(907\) 332.159i 0.366217i 0.983093 + 0.183108i \(0.0586159\pi\)
−0.983093 + 0.183108i \(0.941384\pi\)
\(908\) 137.580 189.363i 0.151520 0.208550i
\(909\) 0 0
\(910\) 0 0
\(911\) 1450.06i 1.59172i −0.605478 0.795862i \(-0.707018\pi\)
0.605478 0.795862i \(-0.292982\pi\)
\(912\) 0 0
\(913\) −1518.15 −1.66282
\(914\) −508.227 + 1564.16i −0.556047 + 1.71134i
\(915\) 0 0
\(916\) −526.577 382.581i −0.574866 0.417665i
\(917\) 40.8514 0.0445490
\(918\) 0 0
\(919\) 814.405i 0.886186i −0.896476 0.443093i \(-0.853881\pi\)
0.896476 0.443093i \(-0.146119\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 282.371 869.048i 0.306259 0.942568i
\(923\) 684.437i 0.741535i
\(924\) 0 0
\(925\) 0 0
\(926\) 761.319 + 247.367i 0.822159 + 0.267136i
\(927\) 0 0
\(928\) 350.217 0.377389
\(929\) −400.039 −0.430612 −0.215306 0.976547i \(-0.569075\pi\)
−0.215306 + 0.976547i \(0.569075\pi\)
\(930\) 0 0
\(931\) 325.043i 0.349134i
\(932\) −1032.47 750.130i −1.10780 0.804861i
\(933\) 0 0
\(934\) −1736.91 564.355i −1.85964 0.604234i
\(935\) 0 0
\(936\) 0 0
\(937\) −249.279 −0.266039 −0.133020 0.991113i \(-0.542467\pi\)
−0.133020 + 0.991113i \(0.542467\pi\)
\(938\) 163.181 502.220i 0.173967 0.535415i
\(939\) 0 0
\(940\) 0 0
\(941\) −724.229 −0.769638 −0.384819 0.922992i \(-0.625736\pi\)
−0.384819 + 0.922992i \(0.625736\pi\)
\(942\) 0 0
\(943\) 8.09432i 0.00858358i
\(944\) −265.337 + 86.2134i −0.281078 + 0.0913277i
\(945\) 0 0
\(946\) −274.164 + 843.790i −0.289814 + 0.891956i
\(947\) 1141.54i 1.20542i 0.797959 + 0.602712i \(0.205913\pi\)
−0.797959 + 0.602712i \(0.794087\pi\)
\(948\) 0 0
\(949\) 47.0557 0.0495845
\(950\) 0 0
\(951\) 0 0
\(952\) 293.901 + 404.520i 0.308720 + 0.424916i
\(953\) 1295.33 1.35921 0.679604 0.733579i \(-0.262152\pi\)
0.679604 + 0.733579i \(0.262152\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 556.774 766.334i 0.582400 0.801604i
\(957\) 0 0
\(958\) −1000.76 325.167i −1.04464 0.339423i
\(959\) 4.37837i 0.00456556i
\(960\) 0 0
\(961\) 892.201 0.928409
\(962\) 96.1378 295.882i 0.0999353 0.307569i
\(963\) 0 0
\(964\) 2.97058 + 2.15825i 0.00308152 + 0.00223885i
\(965\) 0 0
\(966\) 0 0
\(967\) 398.477i 0.412075i −0.978544 0.206037i \(-0.933943\pi\)
0.978544 0.206037i \(-0.0660569\pi\)
\(968\) 1784.85 1296.77i 1.84385 1.33964i
\(969\) 0 0
\(970\) 0 0
\(971\) 928.093i 0.955811i −0.878411 0.477906i \(-0.841396\pi\)
0.878411 0.477906i \(-0.158604\pi\)
\(972\) 0 0
\(973\) 1249.44 1.28411
\(974\) 843.666 + 274.124i 0.866186 + 0.281441i
\(975\) 0 0
\(976\) 445.666 + 1371.62i 0.456625 + 1.40535i
\(977\) −1378.05 −1.41049 −0.705247 0.708962i \(-0.749164\pi\)
−0.705247 + 0.708962i \(0.749164\pi\)
\(978\) 0 0
\(979\) 2219.85i 2.26747i
\(980\) 0 0
\(981\) 0 0
\(982\) −546.217 177.477i −0.556229 0.180730i
\(983\) 311.291i 0.316675i −0.987385 0.158337i \(-0.949387\pi\)
0.987385 0.158337i \(-0.0506134\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 80.4133 247.487i 0.0815551 0.251001i
\(987\) 0 0
\(988\) 303.108 417.193i 0.306790 0.422260i
\(989\) −12.3607 −0.0124982
\(990\) 0 0
\(991\) 961.147i 0.969876i −0.874549 0.484938i \(-0.838842\pi\)
0.874549 0.484938i \(-0.161158\pi\)
\(992\) 265.424i 0.267565i
\(993\) 0 0
\(994\) −262.492 + 807.868i −0.264077 + 0.812745i
\(995\) 0 0
\(996\) 0 0
\(997\) 1089.68 1.09296 0.546479 0.837473i \(-0.315968\pi\)
0.546479 + 0.837473i \(0.315968\pi\)
\(998\) −1542.49 501.186i −1.54558 0.502190i
\(999\) 0 0
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 900.3.c.k.451.3 4
3.2 odd 2 100.3.b.f.51.2 4
4.3 odd 2 inner 900.3.c.k.451.4 4
5.2 odd 4 900.3.f.e.199.8 8
5.3 odd 4 900.3.f.e.199.1 8
5.4 even 2 180.3.c.a.91.2 4
12.11 even 2 100.3.b.f.51.1 4
15.2 even 4 100.3.d.b.99.1 8
15.8 even 4 100.3.d.b.99.8 8
15.14 odd 2 20.3.b.a.11.3 4
20.3 even 4 900.3.f.e.199.7 8
20.7 even 4 900.3.f.e.199.2 8
20.19 odd 2 180.3.c.a.91.1 4
24.5 odd 2 1600.3.b.s.1151.3 4
24.11 even 2 1600.3.b.s.1151.2 4
40.19 odd 2 2880.3.e.e.2431.1 4
40.29 even 2 2880.3.e.e.2431.2 4
60.23 odd 4 100.3.d.b.99.2 8
60.47 odd 4 100.3.d.b.99.7 8
60.59 even 2 20.3.b.a.11.4 yes 4
120.29 odd 2 320.3.b.c.191.2 4
120.53 even 4 1600.3.h.n.1599.4 8
120.59 even 2 320.3.b.c.191.3 4
120.77 even 4 1600.3.h.n.1599.6 8
120.83 odd 4 1600.3.h.n.1599.5 8
120.107 odd 4 1600.3.h.n.1599.3 8
240.29 odd 4 1280.3.g.e.1151.4 8
240.59 even 4 1280.3.g.e.1151.3 8
240.149 odd 4 1280.3.g.e.1151.5 8
240.179 even 4 1280.3.g.e.1151.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.3 4 15.14 odd 2
20.3.b.a.11.4 yes 4 60.59 even 2
100.3.b.f.51.1 4 12.11 even 2
100.3.b.f.51.2 4 3.2 odd 2
100.3.d.b.99.1 8 15.2 even 4
100.3.d.b.99.2 8 60.23 odd 4
100.3.d.b.99.7 8 60.47 odd 4
100.3.d.b.99.8 8 15.8 even 4
180.3.c.a.91.1 4 20.19 odd 2
180.3.c.a.91.2 4 5.4 even 2
320.3.b.c.191.2 4 120.29 odd 2
320.3.b.c.191.3 4 120.59 even 2
900.3.c.k.451.3 4 1.1 even 1 trivial
900.3.c.k.451.4 4 4.3 odd 2 inner
900.3.f.e.199.1 8 5.3 odd 4
900.3.f.e.199.2 8 20.7 even 4
900.3.f.e.199.7 8 20.3 even 4
900.3.f.e.199.8 8 5.2 odd 4
1280.3.g.e.1151.3 8 240.59 even 4
1280.3.g.e.1151.4 8 240.29 odd 4
1280.3.g.e.1151.5 8 240.149 odd 4
1280.3.g.e.1151.6 8 240.179 even 4
1600.3.b.s.1151.2 4 24.11 even 2
1600.3.b.s.1151.3 4 24.5 odd 2
1600.3.h.n.1599.3 8 120.107 odd 4
1600.3.h.n.1599.4 8 120.53 even 4
1600.3.h.n.1599.5 8 120.83 odd 4
1600.3.h.n.1599.6 8 120.77 even 4
2880.3.e.e.2431.1 4 40.19 odd 2
2880.3.e.e.2431.2 4 40.29 even 2