Properties

Label 3150.2.bp.h.1349.11
Level $3150$
Weight $2$
Character 3150.1349
Analytic conductor $25.153$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(899,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bp (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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1349.11
Character \(\chi\) \(=\) 3150.1349
Dual form 3150.2.bp.h.899.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.61577 + 0.397202i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(2.61577 + 0.397202i) q^{7} -1.00000 q^{8} +(-0.429853 + 0.248176i) q^{11} -2.74440 q^{13} +(1.65187 - 2.06672i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.16952 + 1.82992i) q^{17} +(-3.12125 - 1.80205i) q^{19} +0.496352i q^{22} +(3.21210 - 5.56351i) q^{23} +(-1.37220 + 2.37672i) q^{26} +(-0.963896 - 2.46392i) q^{28} -8.87959i q^{29} +(-6.90736 + 3.98797i) q^{31} +(0.500000 + 0.866025i) q^{32} +3.65984i q^{34} +(-1.98397 - 1.14545i) q^{37} +(-3.12125 + 1.80205i) q^{38} +2.22816 q^{41} -2.22575i q^{43} +(0.429853 + 0.248176i) q^{44} +(-3.21210 - 5.56351i) q^{46} +(-5.66749 - 3.27213i) q^{47} +(6.68446 + 2.07798i) q^{49} +(1.37220 + 2.37672i) q^{52} +(-3.88322 - 6.72594i) q^{53} +(-2.61577 - 0.397202i) q^{56} +(-7.68995 - 4.43979i) q^{58} +(-3.05194 - 5.28611i) q^{59} +(3.24271 + 1.87218i) q^{61} +7.97593i q^{62} +1.00000 q^{64} +(7.08216 - 4.08889i) q^{67} +(3.16952 + 1.82992i) q^{68} -10.3761i q^{71} +(-6.53361 - 11.3165i) q^{73} +(-1.98397 + 1.14545i) q^{74} +3.60411i q^{76} +(-1.22297 + 0.478431i) q^{77} +(-4.44344 + 7.69627i) q^{79} +(1.11408 - 1.92964i) q^{82} -4.79091i q^{83} +(-1.92756 - 1.11288i) q^{86} +(0.429853 - 0.248176i) q^{88} +(0.743586 - 1.28793i) q^{89} +(-7.17871 - 1.09008i) q^{91} -6.42419 q^{92} +(-5.66749 + 3.27213i) q^{94} +9.05174 q^{97} +(5.14181 - 4.74992i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8} - 12 q^{16} - 24 q^{17} - 12 q^{19} + 8 q^{23} + 12 q^{32} - 12 q^{38} - 8 q^{46} + 24 q^{47} + 52 q^{49} + 32 q^{53} - 12 q^{61} + 24 q^{64} + 24 q^{68} + 16 q^{77} - 4 q^{79} + 68 q^{91} - 16 q^{92} + 24 q^{94} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 0.866025i 0.353553 0.612372i
\(3\) 0 0
\(4\) −0.500000 0.866025i −0.250000 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.61577 + 0.397202i 0.988667 + 0.150128i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −0.429853 + 0.248176i −0.129606 + 0.0748278i −0.563401 0.826184i \(-0.690507\pi\)
0.433795 + 0.901011i \(0.357174\pi\)
\(12\) 0 0
\(13\) −2.74440 −0.761160 −0.380580 0.924748i \(-0.624276\pi\)
−0.380580 + 0.924748i \(0.624276\pi\)
\(14\) 1.65187 2.06672i 0.441481 0.552354i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.16952 + 1.82992i −0.768721 + 0.443821i −0.832418 0.554148i \(-0.813044\pi\)
0.0636974 + 0.997969i \(0.479711\pi\)
\(18\) 0 0
\(19\) −3.12125 1.80205i −0.716064 0.413420i 0.0972384 0.995261i \(-0.468999\pi\)
−0.813302 + 0.581841i \(0.802332\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.496352i 0.105823i
\(23\) 3.21210 5.56351i 0.669768 1.16007i −0.308200 0.951321i \(-0.599727\pi\)
0.977969 0.208751i \(-0.0669400\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.37220 + 2.37672i −0.269111 + 0.466114i
\(27\) 0 0
\(28\) −0.963896 2.46392i −0.182159 0.465637i
\(29\) 8.87959i 1.64890i −0.565937 0.824449i \(-0.691485\pi\)
0.565937 0.824449i \(-0.308515\pi\)
\(30\) 0 0
\(31\) −6.90736 + 3.98797i −1.24060 + 0.716260i −0.969216 0.246211i \(-0.920814\pi\)
−0.271383 + 0.962472i \(0.587481\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.65984i 0.627658i
\(35\) 0 0
\(36\) 0 0
\(37\) −1.98397 1.14545i −0.326163 0.188311i 0.327973 0.944687i \(-0.393634\pi\)
−0.654136 + 0.756377i \(0.726968\pi\)
\(38\) −3.12125 + 1.80205i −0.506334 + 0.292332i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.22816 0.347980 0.173990 0.984747i \(-0.444334\pi\)
0.173990 + 0.984747i \(0.444334\pi\)
\(42\) 0 0
\(43\) 2.22575i 0.339424i −0.985494 0.169712i \(-0.945716\pi\)
0.985494 0.169712i \(-0.0542838\pi\)
\(44\) 0.429853 + 0.248176i 0.0648028 + 0.0374139i
\(45\) 0 0
\(46\) −3.21210 5.56351i −0.473598 0.820295i
\(47\) −5.66749 3.27213i −0.826688 0.477289i 0.0260292 0.999661i \(-0.491714\pi\)
−0.852717 + 0.522373i \(0.825047\pi\)
\(48\) 0 0
\(49\) 6.68446 + 2.07798i 0.954923 + 0.296854i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.37220 + 2.37672i 0.190290 + 0.329592i
\(53\) −3.88322 6.72594i −0.533402 0.923879i −0.999239 0.0390085i \(-0.987580\pi\)
0.465837 0.884870i \(-0.345753\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.61577 0.397202i −0.349546 0.0530784i
\(57\) 0 0
\(58\) −7.68995 4.43979i −1.00974 0.582973i
\(59\) −3.05194 5.28611i −0.397328 0.688193i 0.596067 0.802935i \(-0.296729\pi\)
−0.993395 + 0.114742i \(0.963396\pi\)
\(60\) 0 0
\(61\) 3.24271 + 1.87218i 0.415187 + 0.239708i 0.693016 0.720922i \(-0.256282\pi\)
−0.277829 + 0.960630i \(0.589615\pi\)
\(62\) 7.97593i 1.01294i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.08216 4.08889i 0.865224 0.499537i −0.000534152 1.00000i \(-0.500170\pi\)
0.865758 + 0.500463i \(0.166837\pi\)
\(68\) 3.16952 + 1.82992i 0.384360 + 0.221911i
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3761i 1.23141i −0.787975 0.615707i \(-0.788871\pi\)
0.787975 0.615707i \(-0.211129\pi\)
\(72\) 0 0
\(73\) −6.53361 11.3165i −0.764701 1.32450i −0.940405 0.340058i \(-0.889553\pi\)
0.175704 0.984443i \(-0.443780\pi\)
\(74\) −1.98397 + 1.14545i −0.230632 + 0.133156i
\(75\) 0 0
\(76\) 3.60411i 0.413420i
\(77\) −1.22297 + 0.478431i −0.139370 + 0.0545223i
\(78\) 0 0
\(79\) −4.44344 + 7.69627i −0.499926 + 0.865898i −1.00000 8.52501e-5i \(-0.999973\pi\)
0.500074 + 0.865983i \(0.333306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.11408 1.92964i 0.123030 0.213093i
\(83\) 4.79091i 0.525871i −0.964813 0.262935i \(-0.915309\pi\)
0.964813 0.262935i \(-0.0846906\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.92756 1.11288i −0.207854 0.120005i
\(87\) 0 0
\(88\) 0.429853 0.248176i 0.0458225 0.0264556i
\(89\) 0.743586 1.28793i 0.0788199 0.136520i −0.823921 0.566704i \(-0.808218\pi\)
0.902741 + 0.430184i \(0.141551\pi\)
\(90\) 0 0
\(91\) −7.17871 1.09008i −0.752534 0.114272i
\(92\) −6.42419 −0.669768
\(93\) 0 0
\(94\) −5.66749 + 3.27213i −0.584557 + 0.337494i
\(95\) 0 0
\(96\) 0 0
\(97\) 9.05174 0.919064 0.459532 0.888161i \(-0.348017\pi\)
0.459532 + 0.888161i \(0.348017\pi\)
\(98\) 5.14181 4.74992i 0.519401 0.479815i
\(99\) 0 0
\(100\) 0 0
\(101\) −1.50180 2.60119i −0.149434 0.258828i 0.781584 0.623800i \(-0.214412\pi\)
−0.931019 + 0.364972i \(0.881079\pi\)
\(102\) 0 0
\(103\) 7.18752 12.4491i 0.708207 1.22665i −0.257314 0.966328i \(-0.582838\pi\)
0.965522 0.260323i \(-0.0838291\pi\)
\(104\) 2.74440 0.269111
\(105\) 0 0
\(106\) −7.76645 −0.754344
\(107\) −1.06314 + 1.84141i −0.102778 + 0.178016i −0.912828 0.408344i \(-0.866106\pi\)
0.810050 + 0.586360i \(0.199440\pi\)
\(108\) 0 0
\(109\) 3.95181 + 6.84474i 0.378515 + 0.655607i 0.990846 0.134994i \(-0.0431015\pi\)
−0.612331 + 0.790601i \(0.709768\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.65187 + 2.06672i −0.156087 + 0.195287i
\(113\) 12.2968 1.15678 0.578391 0.815760i \(-0.303681\pi\)
0.578391 + 0.815760i \(0.303681\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.68995 + 4.43979i −0.713994 + 0.412224i
\(117\) 0 0
\(118\) −6.10388 −0.561907
\(119\) −9.01756 + 3.52771i −0.826638 + 0.323384i
\(120\) 0 0
\(121\) −5.37682 + 9.31292i −0.488802 + 0.846629i
\(122\) 3.24271 1.87218i 0.293581 0.169499i
\(123\) 0 0
\(124\) 6.90736 + 3.98797i 0.620299 + 0.358130i
\(125\) 0 0
\(126\) 0 0
\(127\) 0.753445i 0.0668574i 0.999441 + 0.0334287i \(0.0106427\pi\)
−0.999441 + 0.0334287i \(0.989357\pi\)
\(128\) 0.500000 0.866025i 0.0441942 0.0765466i
\(129\) 0 0
\(130\) 0 0
\(131\) −6.35624 + 11.0093i −0.555347 + 0.961890i 0.442529 + 0.896754i \(0.354081\pi\)
−0.997876 + 0.0651355i \(0.979252\pi\)
\(132\) 0 0
\(133\) −7.44868 5.95352i −0.645882 0.516236i
\(134\) 8.17778i 0.706452i
\(135\) 0 0
\(136\) 3.16952 1.82992i 0.271784 0.156914i
\(137\) 2.15740 + 3.73673i 0.184319 + 0.319250i 0.943347 0.331808i \(-0.107659\pi\)
−0.759028 + 0.651058i \(0.774325\pi\)
\(138\) 0 0
\(139\) 0.0681276i 0.00577851i 0.999996 + 0.00288926i \(0.000919680\pi\)
−0.999996 + 0.00288926i \(0.999080\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.98595 5.18804i −0.754084 0.435371i
\(143\) 1.17969 0.681094i 0.0986507 0.0569560i
\(144\) 0 0
\(145\) 0 0
\(146\) −13.0672 −1.08145
\(147\) 0 0
\(148\) 2.29090i 0.188311i
\(149\) −14.4611 8.34911i −1.18470 0.683986i −0.227602 0.973754i \(-0.573088\pi\)
−0.957097 + 0.289768i \(0.906422\pi\)
\(150\) 0 0
\(151\) 6.51016 + 11.2759i 0.529789 + 0.917622i 0.999396 + 0.0347463i \(0.0110623\pi\)
−0.469607 + 0.882876i \(0.655604\pi\)
\(152\) 3.12125 + 1.80205i 0.253167 + 0.146166i
\(153\) 0 0
\(154\) −0.197152 + 1.29834i −0.0158870 + 0.104623i
\(155\) 0 0
\(156\) 0 0
\(157\) 7.40408 + 12.8242i 0.590910 + 1.02349i 0.994110 + 0.108374i \(0.0345645\pi\)
−0.403200 + 0.915112i \(0.632102\pi\)
\(158\) 4.44344 + 7.69627i 0.353501 + 0.612282i
\(159\) 0 0
\(160\) 0 0
\(161\) 10.6119 13.2770i 0.836337 1.04637i
\(162\) 0 0
\(163\) −13.3189 7.68966i −1.04322 0.602301i −0.122473 0.992472i \(-0.539082\pi\)
−0.920742 + 0.390171i \(0.872416\pi\)
\(164\) −1.11408 1.92964i −0.0869950 0.150680i
\(165\) 0 0
\(166\) −4.14905 2.39546i −0.322029 0.185923i
\(167\) 24.5161i 1.89712i −0.316603 0.948558i \(-0.602542\pi\)
0.316603 0.948558i \(-0.397458\pi\)
\(168\) 0 0
\(169\) −5.46825 −0.420635
\(170\) 0 0
\(171\) 0 0
\(172\) −1.92756 + 1.11288i −0.146975 + 0.0848561i
\(173\) −10.8463 6.26213i −0.824631 0.476101i 0.0273795 0.999625i \(-0.491284\pi\)
−0.852011 + 0.523524i \(0.824617\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.496352i 0.0374139i
\(177\) 0 0
\(178\) −0.743586 1.28793i −0.0557341 0.0965343i
\(179\) 1.58961 0.917762i 0.118813 0.0685968i −0.439416 0.898284i \(-0.644814\pi\)
0.558229 + 0.829687i \(0.311481\pi\)
\(180\) 0 0
\(181\) 23.6564i 1.75837i 0.476481 + 0.879185i \(0.341912\pi\)
−0.476481 + 0.879185i \(0.658088\pi\)
\(182\) −4.53340 + 5.67191i −0.336038 + 0.420430i
\(183\) 0 0
\(184\) −3.21210 + 5.56351i −0.236799 + 0.410148i
\(185\) 0 0
\(186\) 0 0
\(187\) 0.908284 1.57319i 0.0664203 0.115043i
\(188\) 6.54425i 0.477289i
\(189\) 0 0
\(190\) 0 0
\(191\) 7.02253 + 4.05446i 0.508133 + 0.293371i 0.732066 0.681234i \(-0.238556\pi\)
−0.223933 + 0.974605i \(0.571890\pi\)
\(192\) 0 0
\(193\) 10.4356 6.02502i 0.751174 0.433691i −0.0749438 0.997188i \(-0.523878\pi\)
0.826118 + 0.563497i \(0.190544\pi\)
\(194\) 4.52587 7.83903i 0.324938 0.562810i
\(195\) 0 0
\(196\) −1.54265 6.82790i −0.110189 0.487707i
\(197\) −12.7463 −0.908137 −0.454068 0.890967i \(-0.650028\pi\)
−0.454068 + 0.890967i \(0.650028\pi\)
\(198\) 0 0
\(199\) −16.4954 + 9.52361i −1.16933 + 0.675111i −0.953521 0.301325i \(-0.902571\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −3.00359 −0.211332
\(203\) 3.52699 23.2269i 0.247546 1.63021i
\(204\) 0 0
\(205\) 0 0
\(206\) −7.18752 12.4491i −0.500778 0.867373i
\(207\) 0 0
\(208\) 1.37220 2.37672i 0.0951450 0.164796i
\(209\) 1.78891 0.123741
\(210\) 0 0
\(211\) −8.92057 −0.614117 −0.307059 0.951691i \(-0.599345\pi\)
−0.307059 + 0.951691i \(0.599345\pi\)
\(212\) −3.88322 + 6.72594i −0.266701 + 0.461939i
\(213\) 0 0
\(214\) 1.06314 + 1.84141i 0.0726748 + 0.125876i
\(215\) 0 0
\(216\) 0 0
\(217\) −19.6521 + 7.68797i −1.33407 + 0.521893i
\(218\) 7.90363 0.535301
\(219\) 0 0
\(220\) 0 0
\(221\) 8.69843 5.02204i 0.585120 0.337819i
\(222\) 0 0
\(223\) −23.5443 −1.57664 −0.788320 0.615265i \(-0.789049\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(224\) 0.963896 + 2.46392i 0.0644030 + 0.164628i
\(225\) 0 0
\(226\) 6.14838 10.6493i 0.408984 0.708382i
\(227\) −8.07522 + 4.66223i −0.535971 + 0.309443i −0.743445 0.668797i \(-0.766809\pi\)
0.207473 + 0.978241i \(0.433476\pi\)
\(228\) 0 0
\(229\) −20.1545 11.6362i −1.33185 0.768944i −0.346266 0.938136i \(-0.612551\pi\)
−0.985583 + 0.169193i \(0.945884\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.87959i 0.582973i
\(233\) 6.31017 10.9295i 0.413393 0.716018i −0.581865 0.813285i \(-0.697677\pi\)
0.995258 + 0.0972676i \(0.0310103\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.05194 + 5.28611i −0.198664 + 0.344097i
\(237\) 0 0
\(238\) −1.45370 + 9.57329i −0.0942292 + 0.620544i
\(239\) 16.1198i 1.04270i 0.853342 + 0.521351i \(0.174572\pi\)
−0.853342 + 0.521351i \(0.825428\pi\)
\(240\) 0 0
\(241\) −18.3222 + 10.5783i −1.18024 + 0.681411i −0.956070 0.293140i \(-0.905300\pi\)
−0.224168 + 0.974550i \(0.571966\pi\)
\(242\) 5.37682 + 9.31292i 0.345635 + 0.598657i
\(243\) 0 0
\(244\) 3.74436i 0.239708i
\(245\) 0 0
\(246\) 0 0
\(247\) 8.56597 + 4.94556i 0.545039 + 0.314679i
\(248\) 6.90736 3.98797i 0.438618 0.253236i
\(249\) 0 0
\(250\) 0 0
\(251\) 6.06317 0.382704 0.191352 0.981522i \(-0.438713\pi\)
0.191352 + 0.981522i \(0.438713\pi\)
\(252\) 0 0
\(253\) 3.18866i 0.200469i
\(254\) 0.652503 + 0.376723i 0.0409416 + 0.0236377i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) 9.51357 + 5.49266i 0.593440 + 0.342623i 0.766457 0.642296i \(-0.222018\pi\)
−0.173016 + 0.984919i \(0.555351\pi\)
\(258\) 0 0
\(259\) −4.73464 3.78426i −0.294196 0.235143i
\(260\) 0 0
\(261\) 0 0
\(262\) 6.35624 + 11.0093i 0.392690 + 0.680159i
\(263\) 0.669365 + 1.15937i 0.0412748 + 0.0714901i 0.885925 0.463829i \(-0.153525\pi\)
−0.844650 + 0.535319i \(0.820191\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.88024 + 3.47399i −0.544482 + 0.213004i
\(267\) 0 0
\(268\) −7.08216 4.08889i −0.432612 0.249769i
\(269\) −0.311161 0.538946i −0.0189718 0.0328601i 0.856384 0.516340i \(-0.172706\pi\)
−0.875355 + 0.483480i \(0.839373\pi\)
\(270\) 0 0
\(271\) 18.4634 + 10.6598i 1.12157 + 0.647539i 0.941801 0.336170i \(-0.109132\pi\)
0.179769 + 0.983709i \(0.442465\pi\)
\(272\) 3.65984i 0.221911i
\(273\) 0 0
\(274\) 4.31480 0.260667
\(275\) 0 0
\(276\) 0 0
\(277\) 1.02805 0.593544i 0.0617694 0.0356626i −0.468797 0.883306i \(-0.655312\pi\)
0.530567 + 0.847643i \(0.321979\pi\)
\(278\) 0.0590003 + 0.0340638i 0.00353860 + 0.00204301i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.97593i 0.117874i −0.998262 0.0589370i \(-0.981229\pi\)
0.998262 0.0589370i \(-0.0187711\pi\)
\(282\) 0 0
\(283\) 13.5654 + 23.4960i 0.806382 + 1.39669i 0.915354 + 0.402650i \(0.131911\pi\)
−0.108972 + 0.994045i \(0.534756\pi\)
\(284\) −8.98595 + 5.18804i −0.533218 + 0.307853i
\(285\) 0 0
\(286\) 1.36219i 0.0805479i
\(287\) 5.82835 + 0.885030i 0.344036 + 0.0522417i
\(288\) 0 0
\(289\) −1.80278 + 3.12250i −0.106046 + 0.183677i
\(290\) 0 0
\(291\) 0 0
\(292\) −6.53361 + 11.3165i −0.382350 + 0.662250i
\(293\) 10.3808i 0.606456i 0.952918 + 0.303228i \(0.0980643\pi\)
−0.952918 + 0.303228i \(0.901936\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.98397 + 1.14545i 0.115316 + 0.0665778i
\(297\) 0 0
\(298\) −14.4611 + 8.34911i −0.837708 + 0.483651i
\(299\) −8.81529 + 15.2685i −0.509801 + 0.883002i
\(300\) 0 0
\(301\) 0.884074 5.82205i 0.0509572 0.335577i
\(302\) 13.0203 0.749235
\(303\) 0 0
\(304\) 3.12125 1.80205i 0.179016 0.103355i
\(305\) 0 0
\(306\) 0 0
\(307\) 14.1364 0.806808 0.403404 0.915022i \(-0.367827\pi\)
0.403404 + 0.915022i \(0.367827\pi\)
\(308\) 1.02582 + 0.819908i 0.0584515 + 0.0467186i
\(309\) 0 0
\(310\) 0 0
\(311\) −3.32643 5.76155i −0.188625 0.326708i 0.756167 0.654378i \(-0.227070\pi\)
−0.944792 + 0.327671i \(0.893736\pi\)
\(312\) 0 0
\(313\) 6.07282 10.5184i 0.343256 0.594537i −0.641779 0.766890i \(-0.721803\pi\)
0.985035 + 0.172352i \(0.0551367\pi\)
\(314\) 14.8082 0.835673
\(315\) 0 0
\(316\) 8.88688 0.499926
\(317\) 15.3605 26.6051i 0.862730 1.49429i −0.00655283 0.999979i \(-0.502086\pi\)
0.869283 0.494314i \(-0.164581\pi\)
\(318\) 0 0
\(319\) 2.20370 + 3.81692i 0.123383 + 0.213706i
\(320\) 0 0
\(321\) 0 0
\(322\) −6.19225 15.8287i −0.345081 0.882099i
\(323\) 13.1905 0.733937
\(324\) 0 0
\(325\) 0 0
\(326\) −13.3189 + 7.68966i −0.737665 + 0.425891i
\(327\) 0 0
\(328\) −2.22816 −0.123030
\(329\) −13.5251 10.8103i −0.745664 0.595989i
\(330\) 0 0
\(331\) 15.5140 26.8710i 0.852724 1.47696i −0.0260166 0.999662i \(-0.508282\pi\)
0.878741 0.477300i \(-0.158384\pi\)
\(332\) −4.14905 + 2.39546i −0.227709 + 0.131468i
\(333\) 0 0
\(334\) −21.2316 12.2581i −1.16174 0.670732i
\(335\) 0 0
\(336\) 0 0
\(337\) 6.91470i 0.376668i −0.982105 0.188334i \(-0.939691\pi\)
0.982105 0.188334i \(-0.0603087\pi\)
\(338\) −2.73413 + 4.73565i −0.148717 + 0.257585i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.97943 3.42848i 0.107192 0.185663i
\(342\) 0 0
\(343\) 16.6596 + 8.09058i 0.899534 + 0.436850i
\(344\) 2.22575i 0.120005i
\(345\) 0 0
\(346\) −10.8463 + 6.26213i −0.583103 + 0.336654i
\(347\) −3.74704 6.49006i −0.201152 0.348405i 0.747748 0.663982i \(-0.231135\pi\)
−0.948900 + 0.315578i \(0.897802\pi\)
\(348\) 0 0
\(349\) 12.4552i 0.666714i −0.942801 0.333357i \(-0.891819\pi\)
0.942801 0.333357i \(-0.108181\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.429853 0.248176i −0.0229113 0.0132278i
\(353\) −16.4027 + 9.47011i −0.873028 + 0.504043i −0.868353 0.495946i \(-0.834821\pi\)
−0.00467471 + 0.999989i \(0.501488\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.48717 −0.0788199
\(357\) 0 0
\(358\) 1.83552i 0.0970105i
\(359\) 23.0590 + 13.3131i 1.21701 + 0.702639i 0.964277 0.264898i \(-0.0853381\pi\)
0.252730 + 0.967537i \(0.418671\pi\)
\(360\) 0 0
\(361\) −3.00520 5.20516i −0.158168 0.273956i
\(362\) 20.4871 + 11.8282i 1.07678 + 0.621677i
\(363\) 0 0
\(364\) 2.64532 + 6.76199i 0.138652 + 0.354425i
\(365\) 0 0
\(366\) 0 0
\(367\) 5.45606 + 9.45017i 0.284804 + 0.493295i 0.972562 0.232646i \(-0.0747382\pi\)
−0.687758 + 0.725940i \(0.741405\pi\)
\(368\) 3.21210 + 5.56351i 0.167442 + 0.290018i
\(369\) 0 0
\(370\) 0 0
\(371\) −7.48604 19.1359i −0.388656 0.993487i
\(372\) 0 0
\(373\) −19.4924 11.2539i −1.00928 0.582707i −0.0982976 0.995157i \(-0.531340\pi\)
−0.910980 + 0.412450i \(0.864673\pi\)
\(374\) −0.908284 1.57319i −0.0469663 0.0813480i
\(375\) 0 0
\(376\) 5.66749 + 3.27213i 0.292278 + 0.168747i
\(377\) 24.3692i 1.25508i
\(378\) 0 0
\(379\) 3.25909 0.167408 0.0837040 0.996491i \(-0.473325\pi\)
0.0837040 + 0.996491i \(0.473325\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.02253 4.05446i 0.359304 0.207444i
\(383\) 15.0013 + 8.66098i 0.766528 + 0.442555i 0.831635 0.555323i \(-0.187405\pi\)
−0.0651064 + 0.997878i \(0.520739\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.0500i 0.613331i
\(387\) 0 0
\(388\) −4.52587 7.83903i −0.229766 0.397967i
\(389\) −27.4515 + 15.8491i −1.39185 + 0.803582i −0.993520 0.113662i \(-0.963742\pi\)
−0.398326 + 0.917244i \(0.630409\pi\)
\(390\) 0 0
\(391\) 23.5115i 1.18903i
\(392\) −6.68446 2.07798i −0.337616 0.104954i
\(393\) 0 0
\(394\) −6.37315 + 11.0386i −0.321075 + 0.556118i
\(395\) 0 0
\(396\) 0 0
\(397\) 16.7561 29.0224i 0.840964 1.45659i −0.0481170 0.998842i \(-0.515322\pi\)
0.889081 0.457750i \(-0.151345\pi\)
\(398\) 19.0472i 0.954751i
\(399\) 0 0
\(400\) 0 0
\(401\) 31.0404 + 17.9212i 1.55008 + 0.894940i 0.998134 + 0.0610611i \(0.0194485\pi\)
0.551947 + 0.833879i \(0.313885\pi\)
\(402\) 0 0
\(403\) 18.9566 10.9446i 0.944295 0.545189i
\(404\) −1.50180 + 2.60119i −0.0747171 + 0.129414i
\(405\) 0 0
\(406\) −18.3516 14.6679i −0.910775 0.727957i
\(407\) 1.13709 0.0563635
\(408\) 0 0
\(409\) 21.3474 12.3249i 1.05556 0.609429i 0.131361 0.991335i \(-0.458065\pi\)
0.924201 + 0.381905i \(0.124732\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14.3750 −0.708207
\(413\) −5.88350 15.0395i −0.289508 0.740044i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.37220 2.37672i −0.0672777 0.116528i
\(417\) 0 0
\(418\) 0.894453 1.54924i 0.0437491 0.0757757i
\(419\) 10.6574 0.520646 0.260323 0.965522i \(-0.416171\pi\)
0.260323 + 0.965522i \(0.416171\pi\)
\(420\) 0 0
\(421\) 22.6815 1.10543 0.552714 0.833371i \(-0.313592\pi\)
0.552714 + 0.833371i \(0.313592\pi\)
\(422\) −4.46028 + 7.72544i −0.217123 + 0.376068i
\(423\) 0 0
\(424\) 3.88322 + 6.72594i 0.188586 + 0.326641i
\(425\) 0 0
\(426\) 0 0
\(427\) 7.73854 + 6.18520i 0.374494 + 0.299323i
\(428\) 2.12628 0.102778
\(429\) 0 0
\(430\) 0 0
\(431\) 26.0439 15.0364i 1.25449 0.724279i 0.282491 0.959270i \(-0.408839\pi\)
0.971998 + 0.234991i \(0.0755061\pi\)
\(432\) 0 0
\(433\) 14.9203 0.717025 0.358512 0.933525i \(-0.383284\pi\)
0.358512 + 0.933525i \(0.383284\pi\)
\(434\) −3.16806 + 20.8632i −0.152072 + 1.00146i
\(435\) 0 0
\(436\) 3.95181 6.84474i 0.189258 0.327804i
\(437\) −20.0515 + 11.5767i −0.959194 + 0.553791i
\(438\) 0 0
\(439\) −11.1126 6.41586i −0.530375 0.306212i 0.210794 0.977530i \(-0.432395\pi\)
−0.741169 + 0.671318i \(0.765728\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0441i 0.477748i
\(443\) 10.2071 17.6792i 0.484953 0.839963i −0.514898 0.857252i \(-0.672170\pi\)
0.999851 + 0.0172887i \(0.00550344\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.7721 + 20.3899i −0.557426 + 0.965491i
\(447\) 0 0
\(448\) 2.61577 + 0.397202i 0.123583 + 0.0187660i
\(449\) 20.8404i 0.983519i −0.870731 0.491760i \(-0.836354\pi\)
0.870731 0.491760i \(-0.163646\pi\)
\(450\) 0 0
\(451\) −0.957782 + 0.552976i −0.0451002 + 0.0260386i
\(452\) −6.14838 10.6493i −0.289196 0.500901i
\(453\) 0 0
\(454\) 9.32446i 0.437619i
\(455\) 0 0
\(456\) 0 0
\(457\) −13.8811 8.01424i −0.649329 0.374890i 0.138870 0.990311i \(-0.455653\pi\)
−0.788199 + 0.615420i \(0.788986\pi\)
\(458\) −20.1545 + 11.6362i −0.941760 + 0.543725i
\(459\) 0 0
\(460\) 0 0
\(461\) 1.98400 0.0924039 0.0462020 0.998932i \(-0.485288\pi\)
0.0462020 + 0.998932i \(0.485288\pi\)
\(462\) 0 0
\(463\) 36.3987i 1.69159i 0.533506 + 0.845796i \(0.320874\pi\)
−0.533506 + 0.845796i \(0.679126\pi\)
\(464\) 7.68995 + 4.43979i 0.356997 + 0.206112i
\(465\) 0 0
\(466\) −6.31017 10.9295i −0.292313 0.506301i
\(467\) 26.9113 + 15.5372i 1.24530 + 0.718977i 0.970169 0.242428i \(-0.0779439\pi\)
0.275136 + 0.961405i \(0.411277\pi\)
\(468\) 0 0
\(469\) 20.1494 7.88253i 0.930413 0.363981i
\(470\) 0 0
\(471\) 0 0
\(472\) 3.05194 + 5.28611i 0.140477 + 0.243313i
\(473\) 0.552378 + 0.956747i 0.0253984 + 0.0439913i
\(474\) 0 0
\(475\) 0 0
\(476\) 7.56386 + 6.04558i 0.346689 + 0.277099i
\(477\) 0 0
\(478\) 13.9601 + 8.05990i 0.638522 + 0.368651i
\(479\) 18.2404 + 31.5933i 0.833426 + 1.44354i 0.895305 + 0.445453i \(0.146957\pi\)
−0.0618788 + 0.998084i \(0.519709\pi\)
\(480\) 0 0
\(481\) 5.44483 + 3.14357i 0.248263 + 0.143335i
\(482\) 21.1567i 0.963660i
\(483\) 0 0
\(484\) 10.7536 0.488802
\(485\) 0 0
\(486\) 0 0
\(487\) −12.2399 + 7.06672i −0.554644 + 0.320224i −0.750993 0.660310i \(-0.770425\pi\)
0.196349 + 0.980534i \(0.437091\pi\)
\(488\) −3.24271 1.87218i −0.146791 0.0847496i
\(489\) 0 0
\(490\) 0 0
\(491\) 32.5466i 1.46881i −0.678714 0.734403i \(-0.737462\pi\)
0.678714 0.734403i \(-0.262538\pi\)
\(492\) 0 0
\(493\) 16.2489 + 28.1440i 0.731815 + 1.26754i
\(494\) 8.56597 4.94556i 0.385401 0.222511i
\(495\) 0 0
\(496\) 7.97593i 0.358130i
\(497\) 4.12140 27.1414i 0.184870 1.21746i
\(498\) 0 0
\(499\) 5.87396 10.1740i 0.262955 0.455451i −0.704071 0.710130i \(-0.748636\pi\)
0.967026 + 0.254679i \(0.0819697\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.03158 5.25086i 0.135306 0.234357i
\(503\) 24.5250i 1.09352i 0.837291 + 0.546758i \(0.184138\pi\)
−0.837291 + 0.546758i \(0.815862\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.76146 + 1.59433i 0.122762 + 0.0708766i
\(507\) 0 0
\(508\) 0.652503 0.376723i 0.0289501 0.0167144i
\(509\) −5.84634 + 10.1262i −0.259135 + 0.448834i −0.966010 0.258503i \(-0.916771\pi\)
0.706876 + 0.707338i \(0.250104\pi\)
\(510\) 0 0
\(511\) −12.5954 32.1966i −0.557189 1.42429i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 9.51357 5.49266i 0.419626 0.242271i
\(515\) 0 0
\(516\) 0 0
\(517\) 3.24825 0.142858
\(518\) −5.64459 + 2.20819i −0.248009 + 0.0970221i
\(519\) 0 0
\(520\) 0 0
\(521\) 14.8674 + 25.7511i 0.651354 + 1.12818i 0.982795 + 0.184702i \(0.0591319\pi\)
−0.331441 + 0.943476i \(0.607535\pi\)
\(522\) 0 0
\(523\) 2.82915 4.90024i 0.123710 0.214272i −0.797518 0.603295i \(-0.793854\pi\)
0.921228 + 0.389023i \(0.127187\pi\)
\(524\) 12.7125 0.555347
\(525\) 0 0
\(526\) 1.33873 0.0583714
\(527\) 14.5953 25.2799i 0.635783 1.10121i
\(528\) 0 0
\(529\) −9.13513 15.8225i −0.397179 0.687935i
\(530\) 0 0
\(531\) 0 0
\(532\) −1.43156 + 9.42751i −0.0620660 + 0.408734i
\(533\) −6.11497 −0.264869
\(534\) 0 0
\(535\) 0 0
\(536\) −7.08216 + 4.08889i −0.305903 + 0.176613i
\(537\) 0 0
\(538\) −0.622322 −0.0268302
\(539\) −3.38904 + 0.765697i −0.145976 + 0.0329809i
\(540\) 0 0
\(541\) 17.6742 30.6126i 0.759874 1.31614i −0.183041 0.983105i \(-0.558594\pi\)
0.942915 0.333035i \(-0.108073\pi\)
\(542\) 18.4634 10.6598i 0.793070 0.457879i
\(543\) 0 0
\(544\) −3.16952 1.82992i −0.135892 0.0784572i
\(545\) 0 0
\(546\) 0 0
\(547\) 21.4806i 0.918445i 0.888321 + 0.459223i \(0.151872\pi\)
−0.888321 + 0.459223i \(0.848128\pi\)
\(548\) 2.15740 3.73673i 0.0921595 0.159625i
\(549\) 0 0
\(550\) 0 0
\(551\) −16.0015 + 27.7154i −0.681687 + 1.18072i
\(552\) 0 0
\(553\) −14.6800 + 18.3667i −0.624256 + 0.781031i
\(554\) 1.18709i 0.0504345i
\(555\) 0 0
\(556\) 0.0590003 0.0340638i 0.00250217 0.00144463i
\(557\) 19.9150 + 34.4939i 0.843828 + 1.46155i 0.886635 + 0.462469i \(0.153036\pi\)
−0.0428076 + 0.999083i \(0.513630\pi\)
\(558\) 0 0
\(559\) 6.10836i 0.258356i
\(560\) 0 0
\(561\) 0 0
\(562\) −1.71120 0.987964i −0.0721828 0.0416747i
\(563\) −8.95567 + 5.17056i −0.377437 + 0.217913i −0.676702 0.736257i \(-0.736592\pi\)
0.299266 + 0.954170i \(0.403258\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 27.1309 1.14040
\(567\) 0 0
\(568\) 10.3761i 0.435371i
\(569\) −24.3464 14.0564i −1.02065 0.589275i −0.106361 0.994328i \(-0.533920\pi\)
−0.914293 + 0.405053i \(0.867253\pi\)
\(570\) 0 0
\(571\) 13.7146 + 23.7544i 0.573938 + 0.994090i 0.996156 + 0.0875958i \(0.0279184\pi\)
−0.422218 + 0.906494i \(0.638748\pi\)
\(572\) −1.17969 0.681094i −0.0493253 0.0284780i
\(573\) 0 0
\(574\) 3.68063 4.60498i 0.153627 0.192208i
\(575\) 0 0
\(576\) 0 0
\(577\) −6.51910 11.2914i −0.271394 0.470068i 0.697825 0.716268i \(-0.254151\pi\)
−0.969219 + 0.246200i \(0.920818\pi\)
\(578\) 1.80278 + 3.12250i 0.0749857 + 0.129879i
\(579\) 0 0
\(580\) 0 0
\(581\) 1.90296 12.5319i 0.0789481 0.519911i
\(582\) 0 0
\(583\) 3.33843 + 1.92744i 0.138264 + 0.0798266i
\(584\) 6.53361 + 11.3165i 0.270363 + 0.468282i
\(585\) 0 0
\(586\) 8.99008 + 5.19042i 0.371377 + 0.214414i
\(587\) 35.0223i 1.44553i −0.691096 0.722763i \(-0.742872\pi\)
0.691096 0.722763i \(-0.257128\pi\)
\(588\) 0 0
\(589\) 28.7461 1.18446
\(590\) 0 0
\(591\) 0 0
\(592\) 1.98397 1.14545i 0.0815409 0.0470776i
\(593\) −14.1919 8.19370i −0.582792 0.336475i 0.179450 0.983767i \(-0.442568\pi\)
−0.762242 + 0.647292i \(0.775901\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.6982i 0.683986i
\(597\) 0 0
\(598\) 8.81529 + 15.2685i 0.360484 + 0.624376i
\(599\) 26.0718 15.0526i 1.06527 0.615032i 0.138382 0.990379i \(-0.455810\pi\)
0.926884 + 0.375347i \(0.122476\pi\)
\(600\) 0 0
\(601\) 1.75569i 0.0716162i 0.999359 + 0.0358081i \(0.0114005\pi\)
−0.999359 + 0.0358081i \(0.988599\pi\)
\(602\) −4.60001 3.67666i −0.187482 0.149849i
\(603\) 0 0
\(604\) 6.51016 11.2759i 0.264895 0.458811i
\(605\) 0 0
\(606\) 0 0
\(607\) 9.03616 15.6511i 0.366766 0.635258i −0.622292 0.782785i \(-0.713798\pi\)
0.989058 + 0.147528i \(0.0471315\pi\)
\(608\) 3.60411i 0.146166i
\(609\) 0 0
\(610\) 0 0
\(611\) 15.5539 + 8.98003i 0.629242 + 0.363293i
\(612\) 0 0
\(613\) 11.1285 6.42507i 0.449478 0.259506i −0.258132 0.966110i \(-0.583107\pi\)
0.707610 + 0.706604i \(0.249774\pi\)
\(614\) 7.06821 12.2425i 0.285250 0.494067i
\(615\) 0 0
\(616\) 1.22297 0.478431i 0.0492749 0.0192765i
\(617\) −37.3633 −1.50419 −0.752094 0.659055i \(-0.770956\pi\)
−0.752094 + 0.659055i \(0.770956\pi\)
\(618\) 0 0
\(619\) −23.7213 + 13.6955i −0.953439 + 0.550468i −0.894147 0.447773i \(-0.852217\pi\)
−0.0592911 + 0.998241i \(0.518884\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.65287 −0.266756
\(623\) 2.45661 3.07356i 0.0984222 0.123140i
\(624\) 0 0
\(625\) 0 0
\(626\) −6.07282 10.5184i −0.242719 0.420401i
\(627\) 0 0
\(628\) 7.40408 12.8242i 0.295455 0.511743i
\(629\) 8.38432 0.334305
\(630\) 0 0
\(631\) −4.09420 −0.162987 −0.0814937 0.996674i \(-0.525969\pi\)
−0.0814937 + 0.996674i \(0.525969\pi\)
\(632\) 4.44344 7.69627i 0.176751 0.306141i
\(633\) 0 0
\(634\) −15.3605 26.6051i −0.610043 1.05662i
\(635\) 0 0
\(636\) 0 0
\(637\) −18.3449 5.70280i −0.726850 0.225953i
\(638\) 4.40740 0.174491
\(639\) 0 0
\(640\) 0 0
\(641\) −28.4700 + 16.4371i −1.12450 + 0.649228i −0.942545 0.334080i \(-0.891574\pi\)
−0.181951 + 0.983308i \(0.558241\pi\)
\(642\) 0 0
\(643\) 48.1790 1.89999 0.949996 0.312261i \(-0.101086\pi\)
0.949996 + 0.312261i \(0.101086\pi\)
\(644\) −16.8042 2.55170i −0.662178 0.100551i
\(645\) 0 0
\(646\) 6.59524 11.4233i 0.259486 0.449443i
\(647\) 4.58478 2.64703i 0.180246 0.104065i −0.407162 0.913356i \(-0.633482\pi\)
0.587408 + 0.809291i \(0.300148\pi\)
\(648\) 0 0
\(649\) 2.62377 + 1.51483i 0.102992 + 0.0594625i
\(650\) 0 0
\(651\) 0 0
\(652\) 15.3793i 0.602301i
\(653\) −25.0603 + 43.4057i −0.980686 + 1.69860i −0.320956 + 0.947094i \(0.604004\pi\)
−0.659729 + 0.751503i \(0.729329\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.11408 + 1.92964i −0.0434975 + 0.0753399i
\(657\) 0 0
\(658\) −16.1245 + 6.30798i −0.628599 + 0.245911i
\(659\) 2.20149i 0.0857579i −0.999080 0.0428790i \(-0.986347\pi\)
0.999080 0.0428790i \(-0.0136530\pi\)
\(660\) 0 0
\(661\) −33.7612 + 19.4921i −1.31316 + 0.758153i −0.982618 0.185638i \(-0.940565\pi\)
−0.330542 + 0.943791i \(0.607231\pi\)
\(662\) −15.5140 26.8710i −0.602967 1.04437i
\(663\) 0 0
\(664\) 4.79091i 0.185923i
\(665\) 0 0
\(666\) 0 0
\(667\) −49.4017 28.5221i −1.91284 1.10438i
\(668\) −21.2316 + 12.2581i −0.821475 + 0.474279i
\(669\) 0 0
\(670\) 0 0
\(671\) −1.85852 −0.0717473
\(672\) 0 0
\(673\) 43.4830i 1.67615i 0.545556 + 0.838074i \(0.316318\pi\)
−0.545556 + 0.838074i \(0.683682\pi\)
\(674\) −5.98830 3.45735i −0.230661 0.133172i
\(675\) 0 0
\(676\) 2.73413 + 4.73565i 0.105159 + 0.182140i
\(677\) −36.0802 20.8309i −1.38668 0.800597i −0.393736 0.919223i \(-0.628818\pi\)
−0.992939 + 0.118626i \(0.962151\pi\)
\(678\) 0 0
\(679\) 23.6772 + 3.59537i 0.908648 + 0.137978i
\(680\) 0 0
\(681\) 0 0
\(682\) −1.97943 3.42848i −0.0757965 0.131283i
\(683\) −23.8637 41.3332i −0.913120 1.58157i −0.809630 0.586940i \(-0.800332\pi\)
−0.103490 0.994631i \(-0.533001\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.3365 10.3824i 0.585548 0.396400i
\(687\) 0 0
\(688\) 1.92756 + 1.11288i 0.0734875 + 0.0424280i
\(689\) 10.6571 + 18.4587i 0.406004 + 0.703220i
\(690\) 0 0
\(691\) 33.6953 + 19.4540i 1.28183 + 0.740066i 0.977183 0.212399i \(-0.0681277\pi\)
0.304648 + 0.952465i \(0.401461\pi\)
\(692\) 12.5243i 0.476101i
\(693\) 0 0
\(694\) −7.49408 −0.284471
\(695\) 0 0
\(696\) 0 0
\(697\) −7.06219 + 4.07736i −0.267500 + 0.154441i
\(698\) −10.7866 6.22762i −0.408277 0.235719i
\(699\) 0 0
\(700\) 0 0
\(701\) 8.73610i 0.329958i 0.986297 + 0.164979i \(0.0527556\pi\)
−0.986297 + 0.164979i \(0.947244\pi\)
\(702\) 0 0
\(703\) 4.12832 + 7.15046i 0.155703 + 0.269685i
\(704\) −0.429853 + 0.248176i −0.0162007 + 0.00935348i
\(705\) 0 0
\(706\) 18.9402i 0.712824i
\(707\) −2.89515 7.40061i −0.108883 0.278329i
\(708\) 0 0
\(709\) 8.25544 14.2988i 0.310039 0.537004i −0.668331 0.743864i \(-0.732991\pi\)
0.978371 + 0.206860i \(0.0663244\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.743586 + 1.28793i −0.0278671 + 0.0482671i
\(713\) 51.2389i 1.91891i
\(714\) 0 0
\(715\) 0 0
\(716\) −1.58961 0.917762i −0.0594066 0.0342984i
\(717\) 0 0
\(718\) 23.0590 13.3131i 0.860554 0.496841i
\(719\) 22.5570 39.0699i 0.841234 1.45706i −0.0476171 0.998866i \(-0.515163\pi\)
0.888852 0.458195i \(-0.151504\pi\)
\(720\) 0 0
\(721\) 23.7457 29.7092i 0.884336 1.10643i
\(722\) −6.01040 −0.223684
\(723\) 0 0
\(724\) 20.4871 11.8282i 0.761396 0.439592i
\(725\) 0 0
\(726\) 0 0
\(727\) 31.9760 1.18593 0.592963 0.805230i \(-0.297958\pi\)
0.592963 + 0.805230i \(0.297958\pi\)
\(728\) 7.17871 + 1.09008i 0.266061 + 0.0404012i
\(729\) 0 0
\(730\) 0 0
\(731\) 4.07295 + 7.05456i 0.150644 + 0.260922i
\(732\) 0 0
\(733\) 23.3606 40.4618i 0.862844 1.49449i −0.00632839 0.999980i \(-0.502014\pi\)
0.869172 0.494509i \(-0.164652\pi\)
\(734\) 10.9121 0.402773
\(735\) 0 0
\(736\) 6.42419 0.236799
\(737\) −2.02953 + 3.51524i −0.0747586 + 0.129486i
\(738\) 0 0
\(739\) −4.59353 7.95623i −0.168976 0.292675i 0.769084 0.639147i \(-0.220713\pi\)
−0.938060 + 0.346473i \(0.887379\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −20.3152 3.08485i −0.745795 0.113248i
\(743\) −28.5353 −1.04686 −0.523430 0.852069i \(-0.675348\pi\)
−0.523430 + 0.852069i \(0.675348\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −19.4924 + 11.2539i −0.713667 + 0.412036i
\(747\) 0 0
\(748\) −1.81657 −0.0664203
\(749\) −3.51234 + 4.39442i −0.128338 + 0.160569i
\(750\) 0 0
\(751\) 21.4749 37.1956i 0.783629 1.35729i −0.146185 0.989257i \(-0.546700\pi\)
0.929814 0.368029i \(-0.119967\pi\)
\(752\) 5.66749 3.27213i 0.206672 0.119322i
\(753\) 0 0
\(754\) 21.1043 + 12.1846i 0.768574 + 0.443736i
\(755\) 0 0
\(756\) 0 0
\(757\) 19.4415i 0.706612i −0.935508 0.353306i \(-0.885057\pi\)
0.935508 0.353306i \(-0.114943\pi\)
\(758\) 1.62954 2.82245i 0.0591877 0.102516i
\(759\) 0 0
\(760\) 0 0
\(761\) −24.9154 + 43.1547i −0.903182 + 1.56436i −0.0798434 + 0.996807i \(0.525442\pi\)
−0.823339 + 0.567550i \(0.807891\pi\)
\(762\) 0 0
\(763\) 7.61827 + 19.4739i 0.275800 + 0.705003i
\(764\) 8.10892i 0.293371i
\(765\) 0 0
\(766\) 15.0013 8.66098i 0.542017 0.312934i
\(767\) 8.37575 + 14.5072i 0.302431 + 0.523825i
\(768\) 0 0
\(769\) 29.5025i 1.06389i −0.846779 0.531944i \(-0.821462\pi\)
0.846779 0.531944i \(-0.178538\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.4356 6.02502i −0.375587 0.216845i
\(773\) −21.9349 + 12.6641i −0.788945 + 0.455498i −0.839591 0.543219i \(-0.817205\pi\)
0.0506458 + 0.998717i \(0.483872\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −9.05174 −0.324938
\(777\) 0 0
\(778\) 31.6982i 1.13644i
\(779\) −6.95465 4.01527i −0.249176 0.143862i
\(780\) 0 0
\(781\) 2.57509 + 4.46019i 0.0921440 + 0.159598i
\(782\) 20.3616 + 11.7558i 0.728129 + 0.420385i
\(783\) 0 0
\(784\) −5.14181 + 4.74992i −0.183636 + 0.169640i
\(785\) 0 0
\(786\) 0 0
\(787\) 3.40220 + 5.89278i 0.121275 + 0.210055i 0.920271 0.391282i \(-0.127968\pi\)
−0.798996 + 0.601337i \(0.794635\pi\)
\(788\) 6.37315 + 11.0386i 0.227034 + 0.393235i
\(789\) 0 0
\(790\) 0 0
\(791\) 32.1655 + 4.88430i 1.14367 + 0.173666i
\(792\) 0 0
\(793\) −8.89931 5.13802i −0.316024 0.182456i
\(794\) −16.7561 29.0224i −0.594651 1.02997i
\(795\) 0 0
\(796\) 16.4954 + 9.52361i 0.584663 + 0.337556i
\(797\) 31.6373i 1.12065i −0.828272 0.560326i \(-0.810676\pi\)
0.828272 0.560326i \(-0.189324\pi\)
\(798\) 0 0
\(799\) 23.9509 0.847323
\(800\) 0 0
\(801\) 0 0
\(802\) 31.0404 17.9212i 1.09607 0.632818i
\(803\) 5.61698 + 3.24297i 0.198219 + 0.114442i
\(804\) 0 0
\(805\) 0 0
\(806\) 21.8892i 0.771013i
\(807\) 0 0
\(808\) 1.50180 + 2.60119i 0.0528330 + 0.0915094i
\(809\) 22.9302 13.2388i 0.806183 0.465450i −0.0394457 0.999222i \(-0.512559\pi\)
0.845629 + 0.533772i \(0.179226\pi\)
\(810\) 0 0
\(811\) 25.4799i 0.894720i −0.894354 0.447360i \(-0.852364\pi\)
0.894354 0.447360i \(-0.147636\pi\)
\(812\) −21.8786 + 8.55899i −0.767788 + 0.300362i
\(813\) 0 0
\(814\) 0.568545 0.984749i 0.0199275 0.0345154i
\(815\) 0 0
\(816\) 0 0
\(817\) −4.01093 + 6.94713i −0.140325 + 0.243049i
\(818\) 24.6499i 0.861863i
\(819\) 0 0
\(820\) 0 0
\(821\) 32.9335 + 19.0141i 1.14939 + 0.663598i 0.948738 0.316065i \(-0.102362\pi\)
0.200649 + 0.979663i \(0.435695\pi\)
\(822\) 0 0
\(823\) 3.09676 1.78791i 0.107946 0.0623228i −0.445055 0.895503i \(-0.646816\pi\)
0.553001 + 0.833181i \(0.313483\pi\)
\(824\) −7.18752 + 12.4491i −0.250389 + 0.433687i
\(825\) 0 0
\(826\) −15.9663 2.42447i −0.555539 0.0843582i
\(827\) 15.0648 0.523855 0.261927 0.965088i \(-0.415642\pi\)
0.261927 + 0.965088i \(0.415642\pi\)
\(828\) 0 0
\(829\) 27.5046 15.8798i 0.955273 0.551527i 0.0605582 0.998165i \(-0.480712\pi\)
0.894715 + 0.446637i \(0.147379\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.74440 −0.0951450
\(833\) −24.9890 + 5.64586i −0.865819 + 0.195617i
\(834\) 0 0
\(835\) 0 0
\(836\) −0.894453 1.54924i −0.0309353 0.0535815i
\(837\) 0 0
\(838\) 5.32868 9.22954i 0.184076 0.318829i
\(839\) 15.1305 0.522362 0.261181 0.965290i \(-0.415888\pi\)
0.261181 + 0.965290i \(0.415888\pi\)
\(840\) 0 0
\(841\) −49.8470 −1.71886
\(842\) 11.3407 19.6427i 0.390828 0.676933i
\(843\) 0 0
\(844\) 4.46028 + 7.72544i 0.153529 + 0.265920i
\(845\) 0 0
\(846\) 0 0
\(847\) −17.7636 + 22.2247i −0.610365 + 0.763651i
\(848\) 7.76645 0.266701
\(849\) 0 0
\(850\) 0 0
\(851\) −12.7454 + 7.35858i −0.436908 + 0.252249i
\(852\) 0 0
\(853\) −12.1586 −0.416302 −0.208151 0.978097i \(-0.566745\pi\)
−0.208151 + 0.978097i \(0.566745\pi\)
\(854\) 9.22581 3.60917i 0.315701 0.123503i
\(855\) 0 0
\(856\) 1.06314 1.84141i 0.0363374 0.0629382i
\(857\) −27.1356 + 15.6668i −0.926936 + 0.535167i −0.885841 0.463989i \(-0.846418\pi\)
−0.0410947 + 0.999155i \(0.513085\pi\)
\(858\) 0 0
\(859\) 12.7872 + 7.38268i 0.436293 + 0.251894i 0.702024 0.712153i \(-0.252280\pi\)
−0.265731 + 0.964047i \(0.585613\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0728i 1.02429i
\(863\) 7.21398 12.4950i 0.245567 0.425334i −0.716724 0.697357i \(-0.754359\pi\)
0.962291 + 0.272023i \(0.0876926\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 7.46016 12.9214i 0.253507 0.439086i
\(867\) 0 0
\(868\) 16.4840 + 13.1752i 0.559504 + 0.447196i
\(869\) 4.41102i 0.149634i
\(870\) 0 0
\(871\) −19.4363 + 11.2216i −0.658574 + 0.380228i
\(872\) −3.95181 6.84474i −0.133825 0.231792i
\(873\) 0 0
\(874\) 23.1535i 0.783179i
\(875\) 0 0
\(876\) 0 0
\(877\) −9.14788 5.28153i −0.308902 0.178345i 0.337533 0.941314i \(-0.390408\pi\)
−0.646435 + 0.762969i \(0.723741\pi\)
\(878\) −11.1126 + 6.41586i −0.375032 + 0.216525i
\(879\) 0 0
\(880\) 0 0
\(881\) −13.4300 −0.452469 −0.226234 0.974073i \(-0.572642\pi\)
−0.226234 + 0.974073i \(0.572642\pi\)
\(882\) 0 0
\(883\) 4.06019i 0.136636i −0.997664 0.0683181i \(-0.978237\pi\)
0.997664 0.0683181i \(-0.0217633\pi\)
\(884\) −8.69843 5.02204i −0.292560 0.168909i
\(885\) 0 0
\(886\) −10.2071 17.6792i −0.342913 0.593943i
\(887\) 28.6095 + 16.5177i 0.960614 + 0.554611i 0.896362 0.443323i \(-0.146201\pi\)
0.0642519 + 0.997934i \(0.479534\pi\)
\(888\) 0 0
\(889\) −0.299270 + 1.97084i −0.0100372 + 0.0660997i
\(890\) 0 0
\(891\) 0 0
\(892\) 11.7721 + 20.3899i 0.394160 + 0.682705i
\(893\) 11.7931 + 20.4262i 0.394641 + 0.683538i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.65187 2.06672i 0.0551851 0.0690442i
\(897\) 0 0
\(898\) −18.0483 10.4202i −0.602280 0.347727i
\(899\) 35.4115 + 61.3345i 1.18104 + 2.04562i
\(900\) 0 0
\(901\) 24.6159 + 14.2120i 0.820074 + 0.473470i
\(902\) 1.10595i 0.0368241i
\(903\) 0 0
\(904\) −12.2968 −0.408984
\(905\) 0 0
\(906\) 0 0
\(907\) −23.6700 + 13.6659i −0.785949 + 0.453768i −0.838535 0.544848i \(-0.816587\pi\)
0.0525853 + 0.998616i \(0.483254\pi\)
\(908\) 8.07522 + 4.66223i 0.267986 + 0.154722i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.06838i 0.0353972i 0.999843 + 0.0176986i \(0.00563393\pi\)
−0.999843 + 0.0176986i \(0.994366\pi\)
\(912\) 0 0
\(913\) 1.18899 + 2.05939i 0.0393498 + 0.0681558i
\(914\) −13.8811 + 8.01424i −0.459145 + 0.265087i
\(915\) 0 0
\(916\) 23.2725i 0.768944i
\(917\) −20.9994 + 26.2731i −0.693460 + 0.867615i
\(918\) 0 0
\(919\) −26.8653 + 46.5320i −0.886203 + 1.53495i −0.0418743 + 0.999123i \(0.513333\pi\)
−0.844329 + 0.535826i \(0.820000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.991998 1.71819i 0.0326697 0.0565856i
\(923\) 28.4761i 0.937303i
\(924\) 0 0
\(925\) 0 0
\(926\) 31.5222 + 18.1994i 1.03588 + 0.598068i
\(927\) 0 0
\(928\) 7.68995 4.43979i 0.252435 0.145743i
\(929\) 4.80716 8.32625i 0.157718 0.273175i −0.776327 0.630330i \(-0.782920\pi\)
0.934045 + 0.357154i \(0.116253\pi\)
\(930\) 0 0
\(931\) −17.1192 18.5316i −0.561061 0.607350i
\(932\) −12.6203 −0.413393
\(933\) 0 0
\(934\) 26.9113 15.5372i 0.880563 0.508394i
\(935\) 0 0
\(936\) 0 0
\(937\) 12.8030 0.418256 0.209128 0.977888i \(-0.432937\pi\)
0.209128 + 0.977888i \(0.432937\pi\)
\(938\) 3.24823 21.3912i 0.106059 0.698446i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.2221 41.9538i −0.789616 1.36766i −0.926202 0.377028i \(-0.876946\pi\)
0.136586 0.990628i \(-0.456387\pi\)
\(942\) 0 0
\(943\) 7.15707 12.3964i 0.233066 0.403682i
\(944\) 6.10388 0.198664
\(945\) 0 0
\(946\) 1.10476 0.0359187
\(947\) 18.9980 32.9055i 0.617353 1.06929i −0.372614 0.927986i \(-0.621539\pi\)
0.989967 0.141300i \(-0.0451281\pi\)
\(948\) 0 0
\(949\) 17.9308 + 31.0571i 0.582060 + 1.00816i
\(950\) 0 0
\(951\) 0 0
\(952\) 9.01756 3.52771i 0.292261 0.114334i
\(953\) 15.2394 0.493652 0.246826 0.969060i \(-0.420612\pi\)
0.246826 + 0.969060i \(0.420612\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 13.9601 8.05990i 0.451503 0.260676i
\(957\) 0 0
\(958\) 36.4809 1.17864
\(959\) 4.15902 + 10.6313i 0.134302 + 0.343303i
\(960\) 0 0
\(961\) 16.3078 28.2459i 0.526057 0.911157i
\(962\) 5.44483 3.14357i 0.175548 0.101353i
\(963\) 0 0
\(964\) 18.3222 + 10.5783i 0.590119 + 0.340705i
\(965\) 0 0
\(966\) 0 0
\(967\) 27.7458i 0.892244i −0.894972 0.446122i \(-0.852805\pi\)
0.894972 0.446122i \(-0.147195\pi\)
\(968\) 5.37682 9.31292i 0.172817 0.299329i
\(969\) 0 0
\(970\) 0 0
\(971\) −10.2730 + 17.7934i −0.329677 + 0.571018i −0.982448 0.186538i \(-0.940273\pi\)
0.652771 + 0.757556i \(0.273607\pi\)
\(972\) 0 0
\(973\) −0.0270604 + 0.178206i −0.000867518 + 0.00571302i
\(974\) 14.1334i 0.452865i
\(975\) 0 0
\(976\) −3.24271 + 1.87218i −0.103797 + 0.0599270i
\(977\) 3.81400 + 6.60604i 0.122021 + 0.211346i 0.920564 0.390591i \(-0.127729\pi\)
−0.798544 + 0.601937i \(0.794396\pi\)
\(978\) 0 0
\(979\) 0.738160i 0.0235917i
\(980\) 0 0
\(981\) 0 0
\(982\) −28.1861 16.2733i −0.899456 0.519301i
\(983\) −20.4984 + 11.8347i −0.653797 + 0.377470i −0.789909 0.613224i \(-0.789872\pi\)
0.136113 + 0.990693i \(0.456539\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 32.4979 1.03494
\(987\) 0 0
\(988\) 9.89113i 0.314679i
\(989\) −12.3830 7.14933i −0.393757 0.227336i
\(990\) 0 0
\(991\) −13.3947 23.2003i −0.425497 0.736982i 0.570970 0.820971i \(-0.306567\pi\)
−0.996467 + 0.0839887i \(0.973234\pi\)
\(992\) −6.90736 3.98797i −0.219309 0.126618i
\(993\) 0 0
\(994\) −21.4444 17.1399i −0.680176 0.543646i
\(995\) 0 0
\(996\) 0 0
\(997\) −21.1111 36.5656i −0.668596 1.15804i −0.978297 0.207209i \(-0.933562\pi\)
0.309700 0.950834i \(-0.399771\pi\)
\(998\) −5.87396 10.1740i −0.185937 0.322053i
\(999\) 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 3150.2.bp.h.1349.11 24
3.2 odd 2 3150.2.bp.g.1349.11 24
5.2 odd 4 3150.2.bf.d.1601.9 yes 24
5.3 odd 4 3150.2.bf.e.1601.4 yes 24
5.4 even 2 3150.2.bp.g.1349.2 24
7.3 odd 6 inner 3150.2.bp.h.899.2 24
15.2 even 4 3150.2.bf.d.1601.4 yes 24
15.8 even 4 3150.2.bf.e.1601.9 yes 24
15.14 odd 2 inner 3150.2.bp.h.1349.2 24
21.17 even 6 3150.2.bp.g.899.2 24
35.3 even 12 3150.2.bf.e.1151.9 yes 24
35.17 even 12 3150.2.bf.d.1151.4 24
35.24 odd 6 3150.2.bp.g.899.11 24
105.17 odd 12 3150.2.bf.d.1151.9 yes 24
105.38 odd 12 3150.2.bf.e.1151.4 yes 24
105.59 even 6 inner 3150.2.bp.h.899.11 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.bf.d.1151.4 24 35.17 even 12
3150.2.bf.d.1151.9 yes 24 105.17 odd 12
3150.2.bf.d.1601.4 yes 24 15.2 even 4
3150.2.bf.d.1601.9 yes 24 5.2 odd 4
3150.2.bf.e.1151.4 yes 24 105.38 odd 12
3150.2.bf.e.1151.9 yes 24 35.3 even 12
3150.2.bf.e.1601.4 yes 24 5.3 odd 4
3150.2.bf.e.1601.9 yes 24 15.8 even 4
3150.2.bp.g.899.2 24 21.17 even 6
3150.2.bp.g.899.11 24 35.24 odd 6
3150.2.bp.g.1349.2 24 5.4 even 2
3150.2.bp.g.1349.11 24 3.2 odd 2
3150.2.bp.h.899.2 24 7.3 odd 6 inner
3150.2.bp.h.899.11 24 105.59 even 6 inner
3150.2.bp.h.1349.2 24 15.14 odd 2 inner
3150.2.bp.h.1349.11 24 1.1 even 1 trivial