Properties

Label 4020.2.f.b.401.6
Level $4020$
Weight $2$
Character 4020.401
Analytic conductor $32.100$
Analytic rank $0$
Dimension $46$
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] = [4020,2,Mod(401,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("4020.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.f (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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.6
Character \(\chi\) \(=\) 4020.401
Dual form 4020.2.f.b.401.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.65640 + 0.506295i) q^{3} +1.00000 q^{5} +4.66467i q^{7} +(2.48733 - 1.67726i) q^{9} +O(q^{10})\) \(q+(-1.65640 + 0.506295i) q^{3} +1.00000 q^{5} +4.66467i q^{7} +(2.48733 - 1.67726i) q^{9} +1.85459 q^{11} -5.81559i q^{13} +(-1.65640 + 0.506295i) q^{15} -2.48476i q^{17} -4.91357 q^{19} +(-2.36170 - 7.72657i) q^{21} -4.23397i q^{23} +1.00000 q^{25} +(-3.27083 + 4.03753i) q^{27} +2.15242i q^{29} +6.26861i q^{31} +(-3.07194 + 0.938969i) q^{33} +4.66467i q^{35} -6.13976 q^{37} +(2.94440 + 9.63295i) q^{39} -6.27504 q^{41} +4.35463i q^{43} +(2.48733 - 1.67726i) q^{45} +2.64731i q^{47} -14.7592 q^{49} +(1.25802 + 4.11575i) q^{51} +2.24742 q^{53} +1.85459 q^{55} +(8.13884 - 2.48772i) q^{57} +8.02689i q^{59} +0.752813i q^{61} +(7.82385 + 11.6026i) q^{63} -5.81559i q^{65} +(-8.18218 + 0.228007i) q^{67} +(2.14364 + 7.01316i) q^{69} -7.41023i q^{71} -11.4070 q^{73} +(-1.65640 + 0.506295i) q^{75} +8.65104i q^{77} +7.19579i q^{79} +(3.37362 - 8.34378i) q^{81} -12.4884i q^{83} -2.48476i q^{85} +(-1.08976 - 3.56527i) q^{87} +8.65585i q^{89} +27.1278 q^{91} +(-3.17377 - 10.3833i) q^{93} -4.91357 q^{95} +0.0915269i q^{97} +(4.61297 - 3.11062i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q + 46 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q + 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} + 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} + 4 q^{41} - 4 q^{45} - 62 q^{49} + 8 q^{51} + 8 q^{53} + 12 q^{57} + 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.65640 + 0.506295i −0.956324 + 0.292310i
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.66467i 1.76308i 0.472108 + 0.881540i \(0.343493\pi\)
−0.472108 + 0.881540i \(0.656507\pi\)
\(8\) 0 0
\(9\) 2.48733 1.67726i 0.829110 0.559085i
\(10\) 0 0
\(11\) 1.85459 0.559179 0.279590 0.960120i \(-0.409802\pi\)
0.279590 + 0.960120i \(0.409802\pi\)
\(12\) 0 0
\(13\) 5.81559i 1.61295i −0.591266 0.806477i \(-0.701372\pi\)
0.591266 0.806477i \(-0.298628\pi\)
\(14\) 0 0
\(15\) −1.65640 + 0.506295i −0.427681 + 0.130725i
\(16\) 0 0
\(17\) 2.48476i 0.602642i −0.953523 0.301321i \(-0.902572\pi\)
0.953523 0.301321i \(-0.0974276\pi\)
\(18\) 0 0
\(19\) −4.91357 −1.12725 −0.563625 0.826031i \(-0.690594\pi\)
−0.563625 + 0.826031i \(0.690594\pi\)
\(20\) 0 0
\(21\) −2.36170 7.72657i −0.515366 1.68608i
\(22\) 0 0
\(23\) 4.23397i 0.882845i −0.897299 0.441422i \(-0.854474\pi\)
0.897299 0.441422i \(-0.145526\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.27083 + 4.03753i −0.629472 + 0.777024i
\(28\) 0 0
\(29\) 2.15242i 0.399695i 0.979827 + 0.199847i \(0.0640446\pi\)
−0.979827 + 0.199847i \(0.935955\pi\)
\(30\) 0 0
\(31\) 6.26861i 1.12588i 0.826499 + 0.562938i \(0.190329\pi\)
−0.826499 + 0.562938i \(0.809671\pi\)
\(32\) 0 0
\(33\) −3.07194 + 0.938969i −0.534756 + 0.163453i
\(34\) 0 0
\(35\) 4.66467i 0.788474i
\(36\) 0 0
\(37\) −6.13976 −1.00937 −0.504685 0.863303i \(-0.668392\pi\)
−0.504685 + 0.863303i \(0.668392\pi\)
\(38\) 0 0
\(39\) 2.94440 + 9.63295i 0.471482 + 1.54251i
\(40\) 0 0
\(41\) −6.27504 −0.979996 −0.489998 0.871724i \(-0.663002\pi\)
−0.489998 + 0.871724i \(0.663002\pi\)
\(42\) 0 0
\(43\) 4.35463i 0.664075i 0.943266 + 0.332037i \(0.107736\pi\)
−0.943266 + 0.332037i \(0.892264\pi\)
\(44\) 0 0
\(45\) 2.48733 1.67726i 0.370789 0.250031i
\(46\) 0 0
\(47\) 2.64731i 0.386151i 0.981184 + 0.193075i \(0.0618462\pi\)
−0.981184 + 0.193075i \(0.938154\pi\)
\(48\) 0 0
\(49\) −14.7592 −2.10845
\(50\) 0 0
\(51\) 1.25802 + 4.11575i 0.176158 + 0.576321i
\(52\) 0 0
\(53\) 2.24742 0.308707 0.154353 0.988016i \(-0.450671\pi\)
0.154353 + 0.988016i \(0.450671\pi\)
\(54\) 0 0
\(55\) 1.85459 0.250072
\(56\) 0 0
\(57\) 8.13884 2.48772i 1.07802 0.329506i
\(58\) 0 0
\(59\) 8.02689i 1.04501i 0.852635 + 0.522506i \(0.175003\pi\)
−0.852635 + 0.522506i \(0.824997\pi\)
\(60\) 0 0
\(61\) 0.752813i 0.0963879i 0.998838 + 0.0481939i \(0.0153466\pi\)
−0.998838 + 0.0481939i \(0.984653\pi\)
\(62\) 0 0
\(63\) 7.82385 + 11.6026i 0.985713 + 1.46179i
\(64\) 0 0
\(65\) 5.81559i 0.721335i
\(66\) 0 0
\(67\) −8.18218 + 0.228007i −0.999612 + 0.0278555i
\(68\) 0 0
\(69\) 2.14364 + 7.01316i 0.258064 + 0.844285i
\(70\) 0 0
\(71\) 7.41023i 0.879433i −0.898137 0.439716i \(-0.855079\pi\)
0.898137 0.439716i \(-0.144921\pi\)
\(72\) 0 0
\(73\) −11.4070 −1.33508 −0.667541 0.744573i \(-0.732653\pi\)
−0.667541 + 0.744573i \(0.732653\pi\)
\(74\) 0 0
\(75\) −1.65640 + 0.506295i −0.191265 + 0.0584619i
\(76\) 0 0
\(77\) 8.65104i 0.985878i
\(78\) 0 0
\(79\) 7.19579i 0.809590i 0.914407 + 0.404795i \(0.132657\pi\)
−0.914407 + 0.404795i \(0.867343\pi\)
\(80\) 0 0
\(81\) 3.37362 8.34378i 0.374847 0.927087i
\(82\) 0 0
\(83\) 12.4884i 1.37078i −0.728174 0.685392i \(-0.759631\pi\)
0.728174 0.685392i \(-0.240369\pi\)
\(84\) 0 0
\(85\) 2.48476i 0.269510i
\(86\) 0 0
\(87\) −1.08976 3.56527i −0.116835 0.382237i
\(88\) 0 0
\(89\) 8.65585i 0.917518i 0.888561 + 0.458759i \(0.151706\pi\)
−0.888561 + 0.458759i \(0.848294\pi\)
\(90\) 0 0
\(91\) 27.1278 2.84377
\(92\) 0 0
\(93\) −3.17377 10.3833i −0.329104 1.07670i
\(94\) 0 0
\(95\) −4.91357 −0.504121
\(96\) 0 0
\(97\) 0.0915269i 0.00929315i 0.999989 + 0.00464657i \(0.00147906\pi\)
−0.999989 + 0.00464657i \(0.998521\pi\)
\(98\) 0 0
\(99\) 4.61297 3.11062i 0.463621 0.312629i
\(100\) 0 0
\(101\) 14.9663 1.48920 0.744601 0.667510i \(-0.232640\pi\)
0.744601 + 0.667510i \(0.232640\pi\)
\(102\) 0 0
\(103\) −3.38243 −0.333281 −0.166640 0.986018i \(-0.553292\pi\)
−0.166640 + 0.986018i \(0.553292\pi\)
\(104\) 0 0
\(105\) −2.36170 7.72657i −0.230479 0.754036i
\(106\) 0 0
\(107\) 9.57713i 0.925856i −0.886396 0.462928i \(-0.846799\pi\)
0.886396 0.462928i \(-0.153201\pi\)
\(108\) 0 0
\(109\) 1.60653i 0.153878i −0.997036 0.0769389i \(-0.975485\pi\)
0.997036 0.0769389i \(-0.0245146\pi\)
\(110\) 0 0
\(111\) 10.1699 3.10853i 0.965285 0.295049i
\(112\) 0 0
\(113\) −11.9965 −1.12853 −0.564267 0.825592i \(-0.690841\pi\)
−0.564267 + 0.825592i \(0.690841\pi\)
\(114\) 0 0
\(115\) 4.23397i 0.394820i
\(116\) 0 0
\(117\) −9.75423 14.4653i −0.901779 1.33732i
\(118\) 0 0
\(119\) 11.5906 1.06251
\(120\) 0 0
\(121\) −7.56051 −0.687319
\(122\) 0 0
\(123\) 10.3940 3.17702i 0.937193 0.286462i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −18.0413 −1.60090 −0.800452 0.599397i \(-0.795407\pi\)
−0.800452 + 0.599397i \(0.795407\pi\)
\(128\) 0 0
\(129\) −2.20473 7.21301i −0.194115 0.635070i
\(130\) 0 0
\(131\) 15.2675i 1.33393i 0.745091 + 0.666963i \(0.232406\pi\)
−0.745091 + 0.666963i \(0.767594\pi\)
\(132\) 0 0
\(133\) 22.9202i 1.98743i
\(134\) 0 0
\(135\) −3.27083 + 4.03753i −0.281508 + 0.347495i
\(136\) 0 0
\(137\) −2.32666 −0.198780 −0.0993901 0.995049i \(-0.531689\pi\)
−0.0993901 + 0.995049i \(0.531689\pi\)
\(138\) 0 0
\(139\) 3.57809i 0.303490i −0.988420 0.151745i \(-0.951511\pi\)
0.988420 0.151745i \(-0.0484892\pi\)
\(140\) 0 0
\(141\) −1.34032 4.38502i −0.112876 0.369285i
\(142\) 0 0
\(143\) 10.7855i 0.901930i
\(144\) 0 0
\(145\) 2.15242i 0.178749i
\(146\) 0 0
\(147\) 24.4471 7.47250i 2.01636 0.616322i
\(148\) 0 0
\(149\) 10.9654i 0.898318i 0.893452 + 0.449159i \(0.148276\pi\)
−0.893452 + 0.449159i \(0.851724\pi\)
\(150\) 0 0
\(151\) −1.63035 −0.132676 −0.0663379 0.997797i \(-0.521132\pi\)
−0.0663379 + 0.997797i \(0.521132\pi\)
\(152\) 0 0
\(153\) −4.16757 6.18041i −0.336928 0.499656i
\(154\) 0 0
\(155\) 6.26861i 0.503507i
\(156\) 0 0
\(157\) 11.6770 0.931925 0.465963 0.884804i \(-0.345708\pi\)
0.465963 + 0.884804i \(0.345708\pi\)
\(158\) 0 0
\(159\) −3.72263 + 1.13786i −0.295224 + 0.0902380i
\(160\) 0 0
\(161\) 19.7501 1.55653
\(162\) 0 0
\(163\) −1.29655 −0.101554 −0.0507768 0.998710i \(-0.516170\pi\)
−0.0507768 + 0.998710i \(0.516170\pi\)
\(164\) 0 0
\(165\) −3.07194 + 0.938969i −0.239150 + 0.0730986i
\(166\) 0 0
\(167\) 8.36731i 0.647482i 0.946146 + 0.323741i \(0.104941\pi\)
−0.946146 + 0.323741i \(0.895059\pi\)
\(168\) 0 0
\(169\) −20.8211 −1.60162
\(170\) 0 0
\(171\) −12.2217 + 8.24131i −0.934614 + 0.630229i
\(172\) 0 0
\(173\) 14.4609i 1.09944i 0.835347 + 0.549722i \(0.185267\pi\)
−0.835347 + 0.549722i \(0.814733\pi\)
\(174\) 0 0
\(175\) 4.66467i 0.352616i
\(176\) 0 0
\(177\) −4.06398 13.2958i −0.305467 0.999370i
\(178\) 0 0
\(179\) −17.3463 −1.29652 −0.648262 0.761417i \(-0.724504\pi\)
−0.648262 + 0.761417i \(0.724504\pi\)
\(180\) 0 0
\(181\) −0.0961398 −0.00714602 −0.00357301 0.999994i \(-0.501137\pi\)
−0.00357301 + 0.999994i \(0.501137\pi\)
\(182\) 0 0
\(183\) −0.381146 1.24696i −0.0281751 0.0921780i
\(184\) 0 0
\(185\) −6.13976 −0.451404
\(186\) 0 0
\(187\) 4.60820i 0.336985i
\(188\) 0 0
\(189\) −18.8338 15.2574i −1.36996 1.10981i
\(190\) 0 0
\(191\) −27.4064 −1.98306 −0.991531 0.129874i \(-0.958543\pi\)
−0.991531 + 0.129874i \(0.958543\pi\)
\(192\) 0 0
\(193\) 9.74008 0.701106 0.350553 0.936543i \(-0.385994\pi\)
0.350553 + 0.936543i \(0.385994\pi\)
\(194\) 0 0
\(195\) 2.94440 + 9.63295i 0.210853 + 0.689830i
\(196\) 0 0
\(197\) −4.56079 −0.324943 −0.162471 0.986713i \(-0.551947\pi\)
−0.162471 + 0.986713i \(0.551947\pi\)
\(198\) 0 0
\(199\) −7.49985 −0.531651 −0.265825 0.964021i \(-0.585644\pi\)
−0.265825 + 0.964021i \(0.585644\pi\)
\(200\) 0 0
\(201\) 13.4375 4.52027i 0.947810 0.318835i
\(202\) 0 0
\(203\) −10.0403 −0.704694
\(204\) 0 0
\(205\) −6.27504 −0.438267
\(206\) 0 0
\(207\) −7.10146 10.5313i −0.493586 0.731975i
\(208\) 0 0
\(209\) −9.11264 −0.630334
\(210\) 0 0
\(211\) −17.7738 −1.22360 −0.611799 0.791013i \(-0.709554\pi\)
−0.611799 + 0.791013i \(0.709554\pi\)
\(212\) 0 0
\(213\) 3.75177 + 12.2743i 0.257067 + 0.841022i
\(214\) 0 0
\(215\) 4.35463i 0.296983i
\(216\) 0 0
\(217\) −29.2410 −1.98501
\(218\) 0 0
\(219\) 18.8945 5.77529i 1.27677 0.390258i
\(220\) 0 0
\(221\) −14.4503 −0.972034
\(222\) 0 0
\(223\) −20.8007 −1.39292 −0.696460 0.717596i \(-0.745243\pi\)
−0.696460 + 0.717596i \(0.745243\pi\)
\(224\) 0 0
\(225\) 2.48733 1.67726i 0.165822 0.111817i
\(226\) 0 0
\(227\) 28.9431i 1.92102i −0.278247 0.960510i \(-0.589753\pi\)
0.278247 0.960510i \(-0.410247\pi\)
\(228\) 0 0
\(229\) 1.77352i 0.117198i 0.998282 + 0.0585988i \(0.0186633\pi\)
−0.998282 + 0.0585988i \(0.981337\pi\)
\(230\) 0 0
\(231\) −4.37998 14.3296i −0.288182 0.942818i
\(232\) 0 0
\(233\) −14.1983 −0.930160 −0.465080 0.885269i \(-0.653974\pi\)
−0.465080 + 0.885269i \(0.653974\pi\)
\(234\) 0 0
\(235\) 2.64731i 0.172692i
\(236\) 0 0
\(237\) −3.64320 11.9191i −0.236651 0.774230i
\(238\) 0 0
\(239\) 11.7153 0.757799 0.378900 0.925438i \(-0.376303\pi\)
0.378900 + 0.925438i \(0.376303\pi\)
\(240\) 0 0
\(241\) 27.4018 1.76510 0.882551 0.470216i \(-0.155824\pi\)
0.882551 + 0.470216i \(0.155824\pi\)
\(242\) 0 0
\(243\) −1.36366 + 15.5287i −0.0874787 + 0.996166i
\(244\) 0 0
\(245\) −14.7592 −0.942929
\(246\) 0 0
\(247\) 28.5753i 1.81820i
\(248\) 0 0
\(249\) 6.32283 + 20.6859i 0.400693 + 1.31091i
\(250\) 0 0
\(251\) 22.0033 1.38883 0.694417 0.719573i \(-0.255662\pi\)
0.694417 + 0.719573i \(0.255662\pi\)
\(252\) 0 0
\(253\) 7.85227i 0.493668i
\(254\) 0 0
\(255\) 1.25802 + 4.11575i 0.0787803 + 0.257738i
\(256\) 0 0
\(257\) 2.73411i 0.170549i −0.996357 0.0852746i \(-0.972823\pi\)
0.996357 0.0852746i \(-0.0271768\pi\)
\(258\) 0 0
\(259\) 28.6400i 1.77960i
\(260\) 0 0
\(261\) 3.61016 + 5.35378i 0.223463 + 0.331391i
\(262\) 0 0
\(263\) 30.0129i 1.85068i −0.379144 0.925338i \(-0.623781\pi\)
0.379144 0.925338i \(-0.376219\pi\)
\(264\) 0 0
\(265\) 2.24742 0.138058
\(266\) 0 0
\(267\) −4.38242 14.3376i −0.268199 0.877444i
\(268\) 0 0
\(269\) 2.75306i 0.167857i −0.996472 0.0839285i \(-0.973253\pi\)
0.996472 0.0839285i \(-0.0267467\pi\)
\(270\) 0 0
\(271\) 25.2052i 1.53111i 0.643372 + 0.765554i \(0.277535\pi\)
−0.643372 + 0.765554i \(0.722465\pi\)
\(272\) 0 0
\(273\) −44.9346 + 13.7347i −2.71956 + 0.831261i
\(274\) 0 0
\(275\) 1.85459 0.111836
\(276\) 0 0
\(277\) −17.8472 −1.07234 −0.536169 0.844111i \(-0.680129\pi\)
−0.536169 + 0.844111i \(0.680129\pi\)
\(278\) 0 0
\(279\) 10.5141 + 15.5921i 0.629461 + 0.933475i
\(280\) 0 0
\(281\) −7.62012 −0.454578 −0.227289 0.973827i \(-0.572986\pi\)
−0.227289 + 0.973827i \(0.572986\pi\)
\(282\) 0 0
\(283\) 22.9886 1.36653 0.683265 0.730171i \(-0.260560\pi\)
0.683265 + 0.730171i \(0.260560\pi\)
\(284\) 0 0
\(285\) 8.13884 2.48772i 0.482103 0.147360i
\(286\) 0 0
\(287\) 29.2710i 1.72781i
\(288\) 0 0
\(289\) 10.8260 0.636823
\(290\) 0 0
\(291\) −0.0463396 0.151605i −0.00271648 0.00888726i
\(292\) 0 0
\(293\) 3.48606i 0.203658i −0.994802 0.101829i \(-0.967531\pi\)
0.994802 0.101829i \(-0.0324694\pi\)
\(294\) 0 0
\(295\) 8.02689i 0.467344i
\(296\) 0 0
\(297\) −6.06604 + 7.48796i −0.351987 + 0.434495i
\(298\) 0 0
\(299\) −24.6230 −1.42399
\(300\) 0 0
\(301\) −20.3129 −1.17082
\(302\) 0 0
\(303\) −24.7902 + 7.57736i −1.42416 + 0.435308i
\(304\) 0 0
\(305\) 0.752813i 0.0431060i
\(306\) 0 0
\(307\) 21.0950 1.20396 0.601979 0.798512i \(-0.294379\pi\)
0.601979 + 0.798512i \(0.294379\pi\)
\(308\) 0 0
\(309\) 5.60266 1.71251i 0.318724 0.0974212i
\(310\) 0 0
\(311\) 17.2041 0.975553 0.487776 0.872969i \(-0.337808\pi\)
0.487776 + 0.872969i \(0.337808\pi\)
\(312\) 0 0
\(313\) 2.74318i 0.155054i −0.996990 0.0775269i \(-0.975298\pi\)
0.996990 0.0775269i \(-0.0247024\pi\)
\(314\) 0 0
\(315\) 7.82385 + 11.6026i 0.440824 + 0.653732i
\(316\) 0 0
\(317\) 12.6964i 0.713101i −0.934276 0.356551i \(-0.883953\pi\)
0.934276 0.356551i \(-0.116047\pi\)
\(318\) 0 0
\(319\) 3.99185i 0.223501i
\(320\) 0 0
\(321\) 4.84885 + 15.8636i 0.270637 + 0.885418i
\(322\) 0 0
\(323\) 12.2090i 0.679328i
\(324\) 0 0
\(325\) 5.81559i 0.322591i
\(326\) 0 0
\(327\) 0.813379 + 2.66106i 0.0449799 + 0.147157i
\(328\) 0 0
\(329\) −12.3489 −0.680815
\(330\) 0 0
\(331\) 6.01948i 0.330861i −0.986221 0.165430i \(-0.947099\pi\)
0.986221 0.165430i \(-0.0529013\pi\)
\(332\) 0 0
\(333\) −15.2716 + 10.2980i −0.836880 + 0.564325i
\(334\) 0 0
\(335\) −8.18218 + 0.228007i −0.447040 + 0.0124573i
\(336\) 0 0
\(337\) 32.3648i 1.76302i 0.472165 + 0.881510i \(0.343473\pi\)
−0.472165 + 0.881510i \(0.656527\pi\)
\(338\) 0 0
\(339\) 19.8710 6.07376i 1.07924 0.329881i
\(340\) 0 0
\(341\) 11.6257i 0.629566i
\(342\) 0 0
\(343\) 36.1940i 1.95429i
\(344\) 0 0
\(345\) 2.14364 + 7.01316i 0.115410 + 0.377576i
\(346\) 0 0
\(347\) −25.6844 −1.37881 −0.689406 0.724375i \(-0.742128\pi\)
−0.689406 + 0.724375i \(0.742128\pi\)
\(348\) 0 0
\(349\) −27.1555 −1.45360 −0.726800 0.686849i \(-0.758993\pi\)
−0.726800 + 0.686849i \(0.758993\pi\)
\(350\) 0 0
\(351\) 23.4806 + 19.0218i 1.25330 + 1.01531i
\(352\) 0 0
\(353\) −26.6584 −1.41888 −0.709441 0.704765i \(-0.751052\pi\)
−0.709441 + 0.704765i \(0.751052\pi\)
\(354\) 0 0
\(355\) 7.41023i 0.393294i
\(356\) 0 0
\(357\) −19.1986 + 5.86825i −1.01610 + 0.310581i
\(358\) 0 0
\(359\) 20.8930i 1.10269i −0.834277 0.551346i \(-0.814115\pi\)
0.834277 0.551346i \(-0.185885\pi\)
\(360\) 0 0
\(361\) 5.14315 0.270692
\(362\) 0 0
\(363\) 12.5232 3.82785i 0.657299 0.200910i
\(364\) 0 0
\(365\) −11.4070 −0.597067
\(366\) 0 0
\(367\) 17.1786i 0.896714i 0.893855 + 0.448357i \(0.147991\pi\)
−0.893855 + 0.448357i \(0.852009\pi\)
\(368\) 0 0
\(369\) −15.6081 + 10.5248i −0.812524 + 0.547901i
\(370\) 0 0
\(371\) 10.4835i 0.544275i
\(372\) 0 0
\(373\) 5.24645i 0.271651i −0.990733 0.135825i \(-0.956631\pi\)
0.990733 0.135825i \(-0.0433686\pi\)
\(374\) 0 0
\(375\) −1.65640 + 0.506295i −0.0855362 + 0.0261450i
\(376\) 0 0
\(377\) 12.5176 0.644689
\(378\) 0 0
\(379\) 0.138169i 0.00709728i −0.999994 0.00354864i \(-0.998870\pi\)
0.999994 0.00354864i \(-0.00112957\pi\)
\(380\) 0 0
\(381\) 29.8836 9.13421i 1.53098 0.467960i
\(382\) 0 0
\(383\) 8.63739 0.441350 0.220675 0.975347i \(-0.429174\pi\)
0.220675 + 0.975347i \(0.429174\pi\)
\(384\) 0 0
\(385\) 8.65104i 0.440898i
\(386\) 0 0
\(387\) 7.30383 + 10.8314i 0.371274 + 0.550591i
\(388\) 0 0
\(389\) 23.0040i 1.16635i −0.812347 0.583174i \(-0.801810\pi\)
0.812347 0.583174i \(-0.198190\pi\)
\(390\) 0 0
\(391\) −10.5204 −0.532039
\(392\) 0 0
\(393\) −7.72985 25.2891i −0.389919 1.27566i
\(394\) 0 0
\(395\) 7.19579i 0.362060i
\(396\) 0 0
\(397\) 16.9997 0.853189 0.426594 0.904443i \(-0.359713\pi\)
0.426594 + 0.904443i \(0.359713\pi\)
\(398\) 0 0
\(399\) 11.6044 + 37.9650i 0.580946 + 1.90063i
\(400\) 0 0
\(401\) 10.6389 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(402\) 0 0
\(403\) 36.4557 1.81599
\(404\) 0 0
\(405\) 3.37362 8.34378i 0.167637 0.414606i
\(406\) 0 0
\(407\) −11.3867 −0.564419
\(408\) 0 0
\(409\) 16.8187i 0.831629i −0.909449 0.415815i \(-0.863497\pi\)
0.909449 0.415815i \(-0.136503\pi\)
\(410\) 0 0
\(411\) 3.85389 1.17798i 0.190098 0.0581054i
\(412\) 0 0
\(413\) −37.4428 −1.84244
\(414\) 0 0
\(415\) 12.4884i 0.613033i
\(416\) 0 0
\(417\) 1.81157 + 5.92675i 0.0887130 + 0.290234i
\(418\) 0 0
\(419\) 16.3818i 0.800304i 0.916449 + 0.400152i \(0.131043\pi\)
−0.916449 + 0.400152i \(0.868957\pi\)
\(420\) 0 0
\(421\) 26.4074 1.28702 0.643509 0.765439i \(-0.277478\pi\)
0.643509 + 0.765439i \(0.277478\pi\)
\(422\) 0 0
\(423\) 4.44022 + 6.58475i 0.215891 + 0.320161i
\(424\) 0 0
\(425\) 2.48476i 0.120528i
\(426\) 0 0
\(427\) −3.51163 −0.169940
\(428\) 0 0
\(429\) 5.46065 + 17.8651i 0.263643 + 0.862537i
\(430\) 0 0
\(431\) 6.53866i 0.314956i 0.987522 + 0.157478i \(0.0503363\pi\)
−0.987522 + 0.157478i \(0.949664\pi\)
\(432\) 0 0
\(433\) 11.2765i 0.541916i 0.962591 + 0.270958i \(0.0873404\pi\)
−0.962591 + 0.270958i \(0.912660\pi\)
\(434\) 0 0
\(435\) −1.08976 3.56527i −0.0522500 0.170942i
\(436\) 0 0
\(437\) 20.8039i 0.995186i
\(438\) 0 0
\(439\) 7.07504 0.337673 0.168837 0.985644i \(-0.445999\pi\)
0.168837 + 0.985644i \(0.445999\pi\)
\(440\) 0 0
\(441\) −36.7110 + 24.7549i −1.74814 + 1.17881i
\(442\) 0 0
\(443\) 7.56348 0.359352 0.179676 0.983726i \(-0.442495\pi\)
0.179676 + 0.983726i \(0.442495\pi\)
\(444\) 0 0
\(445\) 8.65585i 0.410327i
\(446\) 0 0
\(447\) −5.55171 18.1631i −0.262587 0.859083i
\(448\) 0 0
\(449\) 21.6674i 1.02255i −0.859418 0.511273i \(-0.829174\pi\)
0.859418 0.511273i \(-0.170826\pi\)
\(450\) 0 0
\(451\) −11.6376 −0.547993
\(452\) 0 0
\(453\) 2.70051 0.825438i 0.126881 0.0387824i
\(454\) 0 0
\(455\) 27.1278 1.27177
\(456\) 0 0
\(457\) −33.2853 −1.55702 −0.778511 0.627631i \(-0.784025\pi\)
−0.778511 + 0.627631i \(0.784025\pi\)
\(458\) 0 0
\(459\) 10.0323 + 8.12722i 0.468267 + 0.379346i
\(460\) 0 0
\(461\) 37.0985i 1.72785i −0.503619 0.863926i \(-0.667999\pi\)
0.503619 0.863926i \(-0.332001\pi\)
\(462\) 0 0
\(463\) 26.6282i 1.23752i −0.785581 0.618759i \(-0.787636\pi\)
0.785581 0.618759i \(-0.212364\pi\)
\(464\) 0 0
\(465\) −3.17377 10.3833i −0.147180 0.481516i
\(466\) 0 0
\(467\) 21.3309i 0.987077i −0.869724 0.493538i \(-0.835703\pi\)
0.869724 0.493538i \(-0.164297\pi\)
\(468\) 0 0
\(469\) −1.06358 38.1672i −0.0491115 1.76240i
\(470\) 0 0
\(471\) −19.3418 + 5.91200i −0.891222 + 0.272411i
\(472\) 0 0
\(473\) 8.07604i 0.371337i
\(474\) 0 0
\(475\) −4.91357 −0.225450
\(476\) 0 0
\(477\) 5.59007 3.76950i 0.255952 0.172593i
\(478\) 0 0
\(479\) 36.4557i 1.66570i −0.553496 0.832852i \(-0.686706\pi\)
0.553496 0.832852i \(-0.313294\pi\)
\(480\) 0 0
\(481\) 35.7063i 1.62807i
\(482\) 0 0
\(483\) −32.7141 + 9.99939i −1.48854 + 0.454988i
\(484\) 0 0
\(485\) 0.0915269i 0.00415602i
\(486\) 0 0
\(487\) 6.07562i 0.275313i −0.990480 0.137656i \(-0.956043\pi\)
0.990480 0.137656i \(-0.0439569\pi\)
\(488\) 0 0
\(489\) 2.14761 0.656437i 0.0971182 0.0296851i
\(490\) 0 0
\(491\) 44.1905i 1.99429i 0.0755275 + 0.997144i \(0.475936\pi\)
−0.0755275 + 0.997144i \(0.524064\pi\)
\(492\) 0 0
\(493\) 5.34824 0.240873
\(494\) 0 0
\(495\) 4.61297 3.11062i 0.207338 0.139812i
\(496\) 0 0
\(497\) 34.5663 1.55051
\(498\) 0 0
\(499\) 10.8944i 0.487703i −0.969813 0.243851i \(-0.921589\pi\)
0.969813 0.243851i \(-0.0784109\pi\)
\(500\) 0 0
\(501\) −4.23633 13.8596i −0.189265 0.619202i
\(502\) 0 0
\(503\) 7.22373 0.322090 0.161045 0.986947i \(-0.448514\pi\)
0.161045 + 0.986947i \(0.448514\pi\)
\(504\) 0 0
\(505\) 14.9663 0.665991
\(506\) 0 0
\(507\) 34.4880 10.5416i 1.53167 0.468169i
\(508\) 0 0
\(509\) 16.1181i 0.714423i −0.934024 0.357212i \(-0.883728\pi\)
0.934024 0.357212i \(-0.116272\pi\)
\(510\) 0 0
\(511\) 53.2097i 2.35386i
\(512\) 0 0
\(513\) 16.0714 19.8387i 0.709572 0.875900i
\(514\) 0 0
\(515\) −3.38243 −0.149048
\(516\) 0 0
\(517\) 4.90967i 0.215927i
\(518\) 0 0
\(519\) −7.32150 23.9531i −0.321378 1.05143i
\(520\) 0 0
\(521\) −40.0366 −1.75403 −0.877017 0.480460i \(-0.840470\pi\)
−0.877017 + 0.480460i \(0.840470\pi\)
\(522\) 0 0
\(523\) 5.70290 0.249370 0.124685 0.992196i \(-0.460208\pi\)
0.124685 + 0.992196i \(0.460208\pi\)
\(524\) 0 0
\(525\) −2.36170 7.72657i −0.103073 0.337215i
\(526\) 0 0
\(527\) 15.5760 0.678500
\(528\) 0 0
\(529\) 5.07346 0.220585
\(530\) 0 0
\(531\) 13.4632 + 19.9655i 0.584251 + 0.866431i
\(532\) 0 0
\(533\) 36.4930i 1.58069i
\(534\) 0 0
\(535\) 9.57713i 0.414055i
\(536\) 0 0
\(537\) 28.7325 8.78236i 1.23990 0.378987i
\(538\) 0 0
\(539\) −27.3722 −1.17900
\(540\) 0 0
\(541\) 21.7799i 0.936389i 0.883625 + 0.468195i \(0.155095\pi\)
−0.883625 + 0.468195i \(0.844905\pi\)
\(542\) 0 0
\(543\) 0.159246 0.0486751i 0.00683391 0.00208885i
\(544\) 0 0
\(545\) 1.60653i 0.0688162i
\(546\) 0 0
\(547\) 4.43896i 0.189796i 0.995487 + 0.0948981i \(0.0302525\pi\)
−0.995487 + 0.0948981i \(0.969747\pi\)
\(548\) 0 0
\(549\) 1.26266 + 1.87250i 0.0538891 + 0.0799162i
\(550\) 0 0
\(551\) 10.5761i 0.450556i
\(552\) 0 0
\(553\) −33.5660 −1.42737
\(554\) 0 0
\(555\) 10.1699 3.10853i 0.431689 0.131950i
\(556\) 0 0
\(557\) 9.31693i 0.394771i 0.980326 + 0.197386i \(0.0632451\pi\)
−0.980326 + 0.197386i \(0.936755\pi\)
\(558\) 0 0
\(559\) 25.3247 1.07112
\(560\) 0 0
\(561\) 2.33311 + 7.63302i 0.0985039 + 0.322266i
\(562\) 0 0
\(563\) −43.4048 −1.82929 −0.914647 0.404253i \(-0.867532\pi\)
−0.914647 + 0.404253i \(0.867532\pi\)
\(564\) 0 0
\(565\) −11.9965 −0.504696
\(566\) 0 0
\(567\) 38.9210 + 15.7369i 1.63453 + 0.660886i
\(568\) 0 0
\(569\) 28.8447i 1.20923i −0.796517 0.604617i \(-0.793326\pi\)
0.796517 0.604617i \(-0.206674\pi\)
\(570\) 0 0
\(571\) 26.4854 1.10838 0.554189 0.832391i \(-0.313028\pi\)
0.554189 + 0.832391i \(0.313028\pi\)
\(572\) 0 0
\(573\) 45.3961 13.8758i 1.89645 0.579668i
\(574\) 0 0
\(575\) 4.23397i 0.176569i
\(576\) 0 0
\(577\) 7.95947i 0.331357i −0.986180 0.165678i \(-0.947019\pi\)
0.986180 0.165678i \(-0.0529814\pi\)
\(578\) 0 0
\(579\) −16.1335 + 4.93135i −0.670484 + 0.204940i
\(580\) 0 0
\(581\) 58.2545 2.41680
\(582\) 0 0
\(583\) 4.16803 0.172622
\(584\) 0 0
\(585\) −9.75423 14.4653i −0.403288 0.598066i
\(586\) 0 0
\(587\) 34.0836 1.40678 0.703390 0.710804i \(-0.251669\pi\)
0.703390 + 0.710804i \(0.251669\pi\)
\(588\) 0 0
\(589\) 30.8012i 1.26914i
\(590\) 0 0
\(591\) 7.55450 2.30911i 0.310750 0.0949839i
\(592\) 0 0
\(593\) −22.7130 −0.932712 −0.466356 0.884597i \(-0.654433\pi\)
−0.466356 + 0.884597i \(0.654433\pi\)
\(594\) 0 0
\(595\) 11.5906 0.475167
\(596\) 0 0
\(597\) 12.4228 3.79714i 0.508430 0.155407i
\(598\) 0 0
\(599\) −34.0904 −1.39289 −0.696447 0.717608i \(-0.745237\pi\)
−0.696447 + 0.717608i \(0.745237\pi\)
\(600\) 0 0
\(601\) 30.6322 1.24951 0.624756 0.780820i \(-0.285198\pi\)
0.624756 + 0.780820i \(0.285198\pi\)
\(602\) 0 0
\(603\) −19.9693 + 14.2907i −0.813215 + 0.581964i
\(604\) 0 0
\(605\) −7.56051 −0.307378
\(606\) 0 0
\(607\) 26.3533 1.06965 0.534825 0.844963i \(-0.320378\pi\)
0.534825 + 0.844963i \(0.320378\pi\)
\(608\) 0 0
\(609\) 16.6308 5.08338i 0.673916 0.205989i
\(610\) 0 0
\(611\) 15.3957 0.622843
\(612\) 0 0
\(613\) 1.43510 0.0579633 0.0289817 0.999580i \(-0.490774\pi\)
0.0289817 + 0.999580i \(0.490774\pi\)
\(614\) 0 0
\(615\) 10.3940 3.17702i 0.419126 0.128110i
\(616\) 0 0
\(617\) 7.43295i 0.299239i −0.988744 0.149620i \(-0.952195\pi\)
0.988744 0.149620i \(-0.0478049\pi\)
\(618\) 0 0
\(619\) 0.974193 0.0391561 0.0195781 0.999808i \(-0.493768\pi\)
0.0195781 + 0.999808i \(0.493768\pi\)
\(620\) 0 0
\(621\) 17.0948 + 13.8486i 0.685991 + 0.555726i
\(622\) 0 0
\(623\) −40.3767 −1.61766
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.0942 4.61369i 0.602804 0.184253i
\(628\) 0 0
\(629\) 15.2558i 0.608289i
\(630\) 0 0
\(631\) 47.9821i 1.91014i 0.296382 + 0.955070i \(0.404220\pi\)
−0.296382 + 0.955070i \(0.595780\pi\)
\(632\) 0 0
\(633\) 29.4405 8.99879i 1.17016 0.357670i
\(634\) 0 0
\(635\) −18.0413 −0.715946
\(636\) 0 0
\(637\) 85.8333i 3.40084i
\(638\) 0 0
\(639\) −12.4289 18.4317i −0.491678 0.729147i
\(640\) 0 0
\(641\) −13.2638 −0.523888 −0.261944 0.965083i \(-0.584364\pi\)
−0.261944 + 0.965083i \(0.584364\pi\)
\(642\) 0 0
\(643\) −5.43363 −0.214282 −0.107141 0.994244i \(-0.534170\pi\)
−0.107141 + 0.994244i \(0.534170\pi\)
\(644\) 0 0
\(645\) −2.20473 7.21301i −0.0868111 0.284012i
\(646\) 0 0
\(647\) −9.91923 −0.389965 −0.194983 0.980807i \(-0.562465\pi\)
−0.194983 + 0.980807i \(0.562465\pi\)
\(648\) 0 0
\(649\) 14.8866i 0.584349i
\(650\) 0 0
\(651\) 48.4349 14.8046i 1.89831 0.580238i
\(652\) 0 0
\(653\) 1.30639 0.0511230 0.0255615 0.999673i \(-0.491863\pi\)
0.0255615 + 0.999673i \(0.491863\pi\)
\(654\) 0 0
\(655\) 15.2675i 0.596549i
\(656\) 0 0
\(657\) −28.3729 + 19.1324i −1.10693 + 0.746425i
\(658\) 0 0
\(659\) 7.09051i 0.276207i −0.990418 0.138103i \(-0.955899\pi\)
0.990418 0.138103i \(-0.0441006\pi\)
\(660\) 0 0
\(661\) 33.7859i 1.31412i −0.753839 0.657060i \(-0.771800\pi\)
0.753839 0.657060i \(-0.228200\pi\)
\(662\) 0 0
\(663\) 23.9355 7.31613i 0.929579 0.284135i
\(664\) 0 0
\(665\) 22.9202i 0.888807i
\(666\) 0 0
\(667\) 9.11330 0.352868
\(668\) 0 0
\(669\) 34.4544 10.5313i 1.33208 0.407164i
\(670\) 0 0
\(671\) 1.39616i 0.0538981i
\(672\) 0 0
\(673\) 13.9854i 0.539097i 0.962987 + 0.269548i \(0.0868744\pi\)
−0.962987 + 0.269548i \(0.913126\pi\)
\(674\) 0 0
\(675\) −3.27083 + 4.03753i −0.125894 + 0.155405i
\(676\) 0 0
\(677\) 8.35018 0.320924 0.160462 0.987042i \(-0.448702\pi\)
0.160462 + 0.987042i \(0.448702\pi\)
\(678\) 0 0
\(679\) −0.426943 −0.0163846
\(680\) 0 0
\(681\) 14.6537 + 47.9413i 0.561533 + 1.83712i
\(682\) 0 0
\(683\) −30.6554 −1.17300 −0.586498 0.809951i \(-0.699494\pi\)
−0.586498 + 0.809951i \(0.699494\pi\)
\(684\) 0 0
\(685\) −2.32666 −0.0888972
\(686\) 0 0
\(687\) −0.897926 2.93767i −0.0342580 0.112079i
\(688\) 0 0
\(689\) 13.0701i 0.497930i
\(690\) 0 0
\(691\) −20.0932 −0.764382 −0.382191 0.924083i \(-0.624830\pi\)
−0.382191 + 0.924083i \(0.624830\pi\)
\(692\) 0 0
\(693\) 14.5100 + 21.5180i 0.551190 + 0.817401i
\(694\) 0 0
\(695\) 3.57809i 0.135725i
\(696\) 0 0
\(697\) 15.5919i 0.590587i
\(698\) 0 0
\(699\) 23.5180 7.18852i 0.889534 0.271895i
\(700\) 0 0
\(701\) 0.865169 0.0326770 0.0163385 0.999867i \(-0.494799\pi\)
0.0163385 + 0.999867i \(0.494799\pi\)
\(702\) 0 0
\(703\) 30.1681 1.13781
\(704\) 0 0
\(705\) −1.34032 4.38502i −0.0504795 0.165149i
\(706\) 0 0
\(707\) 69.8128i 2.62558i
\(708\) 0 0
\(709\) −43.6220 −1.63826 −0.819130 0.573608i \(-0.805543\pi\)
−0.819130 + 0.573608i \(0.805543\pi\)
\(710\) 0 0
\(711\) 12.0692 + 17.8983i 0.452630 + 0.671239i
\(712\) 0 0
\(713\) 26.5411 0.993973
\(714\) 0 0
\(715\) 10.7855i 0.403355i
\(716\) 0 0
\(717\) −19.4052 + 5.93140i −0.724701 + 0.221512i
\(718\) 0 0
\(719\) 29.3270i 1.09371i 0.837226 + 0.546857i \(0.184176\pi\)
−0.837226 + 0.546857i \(0.815824\pi\)
\(720\) 0 0
\(721\) 15.7779i 0.587601i
\(722\) 0 0
\(723\) −45.3883 + 13.8734i −1.68801 + 0.515957i
\(724\) 0 0
\(725\) 2.15242i 0.0799389i
\(726\) 0 0
\(727\) 32.6139i 1.20958i 0.796383 + 0.604792i \(0.206744\pi\)
−0.796383 + 0.604792i \(0.793256\pi\)
\(728\) 0 0
\(729\) −5.60334 26.4122i −0.207531 0.978228i
\(730\) 0 0
\(731\) 10.8202 0.400199
\(732\) 0 0
\(733\) 12.2144i 0.451150i 0.974226 + 0.225575i \(0.0724260\pi\)
−0.974226 + 0.225575i \(0.927574\pi\)
\(734\) 0 0
\(735\) 24.4471 7.47250i 0.901746 0.275627i
\(736\) 0 0
\(737\) −15.1746 + 0.422859i −0.558962 + 0.0155762i
\(738\) 0 0
\(739\) 53.9114i 1.98316i 0.129489 + 0.991581i \(0.458666\pi\)
−0.129489 + 0.991581i \(0.541334\pi\)
\(740\) 0 0
\(741\) −14.4675 47.3321i −0.531478 1.73879i
\(742\) 0 0
\(743\) 48.5913i 1.78264i 0.453373 + 0.891321i \(0.350221\pi\)
−0.453373 + 0.891321i \(0.649779\pi\)
\(744\) 0 0
\(745\) 10.9654i 0.401740i
\(746\) 0 0
\(747\) −20.9463 31.0629i −0.766385 1.13653i
\(748\) 0 0
\(749\) 44.6742 1.63236
\(750\) 0 0
\(751\) −6.20564 −0.226447 −0.113223 0.993570i \(-0.536118\pi\)
−0.113223 + 0.993570i \(0.536118\pi\)
\(752\) 0 0
\(753\) −36.4462 + 11.1401i −1.32817 + 0.405970i
\(754\) 0 0
\(755\) −1.63035 −0.0593345
\(756\) 0 0
\(757\) 15.9766i 0.580679i −0.956924 0.290339i \(-0.906232\pi\)
0.956924 0.290339i \(-0.0937682\pi\)
\(758\) 0 0
\(759\) 3.97557 + 13.0065i 0.144304 + 0.472107i
\(760\) 0 0
\(761\) 37.6989i 1.36658i 0.730145 + 0.683292i \(0.239452\pi\)
−0.730145 + 0.683292i \(0.760548\pi\)
\(762\) 0 0
\(763\) 7.49394 0.271299
\(764\) 0 0
\(765\) −4.16757 6.18041i −0.150679 0.223453i
\(766\) 0 0
\(767\) 46.6811 1.68556
\(768\) 0 0
\(769\) 10.1126i 0.364669i 0.983237 + 0.182334i \(0.0583653\pi\)
−0.983237 + 0.182334i \(0.941635\pi\)
\(770\) 0 0
\(771\) 1.38427 + 4.52879i 0.0498532 + 0.163100i
\(772\) 0 0
\(773\) 23.3296i 0.839106i 0.907731 + 0.419553i \(0.137813\pi\)
−0.907731 + 0.419553i \(0.862187\pi\)
\(774\) 0 0
\(775\) 6.26861i 0.225175i
\(776\) 0 0
\(777\) 14.5003 + 47.4393i 0.520195 + 1.70188i
\(778\) 0 0
\(779\) 30.8328 1.10470
\(780\) 0 0
\(781\) 13.7429i 0.491760i
\(782\) 0 0
\(783\) −8.69047 7.04020i −0.310572 0.251596i
\(784\) 0 0
\(785\) 11.6770 0.416770
\(786\) 0 0
\(787\) 37.3847i 1.33262i −0.745675 0.666310i \(-0.767873\pi\)
0.745675 0.666310i \(-0.232127\pi\)
\(788\) 0 0
\(789\) 15.1954 + 49.7134i 0.540970 + 1.76985i
\(790\) 0 0
\(791\) 55.9597i 1.98970i
\(792\) 0 0
\(793\) 4.37805 0.155469
\(794\) 0 0
\(795\) −3.72263 + 1.13786i −0.132028 + 0.0403557i
\(796\) 0 0
\(797\) 28.7033i 1.01672i 0.861144 + 0.508361i \(0.169748\pi\)
−0.861144 + 0.508361i \(0.830252\pi\)
\(798\) 0 0
\(799\) 6.57793 0.232710
\(800\) 0 0
\(801\) 14.5181 + 21.5300i 0.512971 + 0.760724i
\(802\) 0 0
\(803\) −21.1552 −0.746550
\(804\) 0 0
\(805\) 19.7501 0.696100
\(806\) 0 0
\(807\) 1.39386 + 4.56017i 0.0490662 + 0.160526i
\(808\) 0 0
\(809\) 47.1622 1.65814 0.829068 0.559148i \(-0.188872\pi\)
0.829068 + 0.559148i \(0.188872\pi\)
\(810\) 0 0
\(811\) 5.04581i 0.177182i 0.996068 + 0.0885911i \(0.0282364\pi\)
−0.996068 + 0.0885911i \(0.971764\pi\)
\(812\) 0 0
\(813\) −12.7613 41.7499i −0.447558 1.46423i
\(814\) 0 0
\(815\) −1.29655 −0.0454162
\(816\) 0 0
\(817\) 21.3968i 0.748578i
\(818\) 0 0
\(819\) 67.4758 45.5003i 2.35780 1.58991i
\(820\) 0 0
\(821\) 26.9863i 0.941827i −0.882179 0.470914i \(-0.843924\pi\)
0.882179 0.470914i \(-0.156076\pi\)
\(822\) 0 0
\(823\) −3.79168 −0.132170 −0.0660848 0.997814i \(-0.521051\pi\)
−0.0660848 + 0.997814i \(0.521051\pi\)
\(824\) 0 0
\(825\) −3.07194 + 0.938969i −0.106951 + 0.0326907i
\(826\) 0 0
\(827\) 11.7113i 0.407243i 0.979050 + 0.203621i \(0.0652711\pi\)
−0.979050 + 0.203621i \(0.934729\pi\)
\(828\) 0 0
\(829\) 10.7941 0.374894 0.187447 0.982275i \(-0.439979\pi\)
0.187447 + 0.982275i \(0.439979\pi\)
\(830\) 0 0
\(831\) 29.5622 9.03598i 1.02550 0.313455i
\(832\) 0 0
\(833\) 36.6730i 1.27064i
\(834\) 0 0
\(835\) 8.36731i 0.289563i
\(836\) 0 0
\(837\) −25.3097 20.5036i −0.874832 0.708707i
\(838\) 0 0
\(839\) 37.7995i 1.30498i 0.757796 + 0.652492i \(0.226276\pi\)
−0.757796 + 0.652492i \(0.773724\pi\)
\(840\) 0 0
\(841\) 24.3671 0.840244
\(842\) 0 0
\(843\) 12.6220 3.85803i 0.434724 0.132878i
\(844\) 0 0
\(845\) −20.8211 −0.716266
\(846\) 0 0
\(847\) 35.2673i 1.21180i
\(848\) 0 0
\(849\) −38.0783 + 11.6390i −1.30684 + 0.399450i
\(850\) 0 0
\(851\) 25.9956i 0.891118i
\(852\) 0 0
\(853\) 18.9904 0.650219 0.325109 0.945676i \(-0.394599\pi\)
0.325109 + 0.945676i \(0.394599\pi\)
\(854\) 0 0
\(855\) −12.2217 + 8.24131i −0.417972 + 0.281847i
\(856\) 0 0
\(857\) −36.7014 −1.25370 −0.626848 0.779142i \(-0.715655\pi\)
−0.626848 + 0.779142i \(0.715655\pi\)
\(858\) 0 0
\(859\) 37.4450 1.27761 0.638804 0.769370i \(-0.279430\pi\)
0.638804 + 0.769370i \(0.279430\pi\)
\(860\) 0 0
\(861\) 14.8198 + 48.4845i 0.505056 + 1.65235i
\(862\) 0 0
\(863\) 27.3882i 0.932307i −0.884704 0.466153i \(-0.845639\pi\)
0.884704 0.466153i \(-0.154361\pi\)
\(864\) 0 0
\(865\) 14.4609i 0.491687i
\(866\) 0 0
\(867\) −17.9322 + 5.48115i −0.609009 + 0.186149i
\(868\) 0 0
\(869\) 13.3452i 0.452706i
\(870\) 0 0
\(871\) 1.32599 + 47.5842i 0.0449296 + 1.61233i
\(872\) 0 0
\(873\) 0.153514 + 0.227658i 0.00519566 + 0.00770504i
\(874\) 0 0
\(875\) 4.66467i 0.157695i
\(876\) 0 0
\(877\) 53.6724 1.81239 0.906194 0.422861i \(-0.138974\pi\)
0.906194 + 0.422861i \(0.138974\pi\)
\(878\) 0 0
\(879\) 1.76498 + 5.77431i 0.0595311 + 0.194763i
\(880\) 0 0
\(881\) 15.1960i 0.511965i −0.966681 0.255983i \(-0.917601\pi\)
0.966681 0.255983i \(-0.0823990\pi\)
\(882\) 0 0
\(883\) 53.2856i 1.79320i 0.442839 + 0.896601i \(0.353971\pi\)
−0.442839 + 0.896601i \(0.646029\pi\)
\(884\) 0 0
\(885\) −4.06398 13.2958i −0.136609 0.446932i
\(886\) 0 0
\(887\) 25.4042i 0.852988i 0.904490 + 0.426494i \(0.140252\pi\)
−0.904490 + 0.426494i \(0.859748\pi\)
\(888\) 0 0
\(889\) 84.1566i 2.82252i
\(890\) 0 0
\(891\) 6.25668 15.4743i 0.209607 0.518407i
\(892\) 0 0
\(893\) 13.0078i 0.435288i
\(894\) 0 0
\(895\) −17.3463 −0.579823
\(896\) 0 0
\(897\) 40.7856 12.4665i 1.36179 0.416245i
\(898\) 0 0
\(899\) −13.4927 −0.450006
\(900\) 0 0
\(901\) 5.58429i 0.186040i
\(902\) 0 0
\(903\) 33.6463 10.2843i 1.11968 0.342241i
\(904\) 0 0
\(905\) −0.0961398 −0.00319580
\(906\) 0 0
\(907\) 48.2592 1.60242 0.801210 0.598383i \(-0.204190\pi\)
0.801210 + 0.598383i \(0.204190\pi\)
\(908\) 0 0
\(909\) 37.2261 25.1023i 1.23471 0.832591i
\(910\) 0 0
\(911\) 14.6148i 0.484210i 0.970250 + 0.242105i \(0.0778378\pi\)
−0.970250 + 0.242105i \(0.922162\pi\)
\(912\) 0 0
\(913\) 23.1609i 0.766513i
\(914\) 0 0
\(915\) −0.381146 1.24696i −0.0126003 0.0412233i
\(916\) 0 0
\(917\) −71.2178 −2.35182
\(918\) 0 0
\(919\) 3.67252i 0.121145i −0.998164 0.0605727i \(-0.980707\pi\)
0.998164 0.0605727i \(-0.0192927\pi\)
\(920\) 0 0
\(921\) −34.9419 + 10.6803i −1.15137 + 0.351929i
\(922\) 0 0
\(923\) −43.0949 −1.41848
\(924\) 0 0
\(925\) −6.13976 −0.201874
\(926\) 0 0
\(927\) −8.41322 + 5.67320i −0.276326 + 0.186332i
\(928\) 0 0
\(929\) 39.4217 1.29338 0.646692 0.762751i \(-0.276152\pi\)
0.646692 + 0.762751i \(0.276152\pi\)
\(930\) 0 0
\(931\) 72.5202 2.37675
\(932\) 0 0
\(933\) −28.4968 + 8.71034i −0.932944 + 0.285164i
\(934\) 0 0
\(935\) 4.60820i 0.150704i
\(936\) 0 0
\(937\) 47.8618i 1.56358i −0.623545 0.781788i \(-0.714308\pi\)
0.623545 0.781788i \(-0.285692\pi\)
\(938\) 0 0
\(939\) 1.38886 + 4.54381i 0.0453238 + 0.148282i
\(940\) 0 0
\(941\) 39.3746 1.28358 0.641788 0.766882i \(-0.278193\pi\)
0.641788 + 0.766882i \(0.278193\pi\)
\(942\) 0 0
\(943\) 26.5683i 0.865184i
\(944\) 0 0
\(945\) −18.8338 15.2574i −0.612663 0.496322i
\(946\) 0 0
\(947\) 23.2261i 0.754746i −0.926061 0.377373i \(-0.876827\pi\)
0.926061 0.377373i \(-0.123173\pi\)
\(948\) 0 0
\(949\) 66.3381i 2.15343i
\(950\) 0 0
\(951\) 6.42813 + 21.0303i 0.208446 + 0.681956i
\(952\) 0 0
\(953\) 42.1506i 1.36539i 0.730702 + 0.682696i \(0.239193\pi\)
−0.730702 + 0.682696i \(0.760807\pi\)
\(954\) 0 0
\(955\) −27.4064 −0.886852
\(956\) 0 0
\(957\) −2.02106 6.61211i −0.0653315 0.213739i
\(958\) 0 0
\(959\) 10.8531i 0.350466i
\(960\) 0 0
\(961\) −8.29547 −0.267596
\(962\) 0 0
\(963\) −16.0633 23.8215i −0.517633 0.767637i
\(964\) 0 0
\(965\) 9.74008 0.313544
\(966\) 0 0
\(967\) −24.4938 −0.787669 −0.393834 0.919181i \(-0.628852\pi\)
−0.393834 + 0.919181i \(0.628852\pi\)
\(968\) 0 0
\(969\) −6.18137 20.2230i −0.198574 0.649657i
\(970\) 0 0
\(971\) 27.2521i 0.874561i 0.899325 + 0.437280i \(0.144058\pi\)
−0.899325 + 0.437280i \(0.855942\pi\)
\(972\) 0 0
\(973\) 16.6906 0.535077
\(974\) 0 0
\(975\) 2.94440 + 9.63295i 0.0942964 + 0.308501i
\(976\) 0 0
\(977\) 25.8133i 0.825841i 0.910767 + 0.412920i \(0.135491\pi\)
−0.910767 + 0.412920i \(0.864509\pi\)
\(978\) 0 0
\(979\) 16.0530i 0.513057i
\(980\) 0 0
\(981\) −2.69456 3.99597i −0.0860308 0.127582i
\(982\) 0 0
\(983\) −46.2340 −1.47464 −0.737318 0.675545i \(-0.763908\pi\)
−0.737318 + 0.675545i \(0.763908\pi\)
\(984\) 0 0
\(985\) −4.56079 −0.145319
\(986\) 0 0
\(987\) 20.4547 6.25217i 0.651079 0.199009i
\(988\) 0 0
\(989\) 18.4374 0.586275
\(990\) 0 0
\(991\) 48.2546i 1.53286i −0.642328 0.766429i \(-0.722031\pi\)
0.642328 0.766429i \(-0.277969\pi\)
\(992\) 0 0
\(993\) 3.04764 + 9.97068i 0.0967138 + 0.316410i
\(994\) 0 0
\(995\) −7.49985 −0.237761
\(996\) 0 0
\(997\) 6.02279 0.190744 0.0953718 0.995442i \(-0.469596\pi\)
0.0953718 + 0.995442i \(0.469596\pi\)
\(998\) 0 0
\(999\) 20.0821 24.7895i 0.635370 0.784305i
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 4020.2.f.b.401.6 yes 46
3.2 odd 2 4020.2.f.a.401.42 yes 46
67.66 odd 2 4020.2.f.a.401.41 46
201.200 even 2 inner 4020.2.f.b.401.5 yes 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.f.a.401.41 46 67.66 odd 2
4020.2.f.a.401.42 yes 46 3.2 odd 2
4020.2.f.b.401.5 yes 46 201.200 even 2 inner
4020.2.f.b.401.6 yes 46 1.1 even 1 trivial