Properties

Label 3150.2.bf.d.1151.1
Level $3150$
Weight $2$
Character 3150.1151
Analytic conductor $25.153$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
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, 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("3150.1151");
 
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.bf (of order \(6\), degree \(2\), 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 1151.1
Character \(\chi\) \(=\) 3150.1151
Dual form 3150.2.bf.d.1601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.52781 + 2.16005i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.52781 + 2.16005i) q^{7} +1.00000i q^{8} +(-4.29783 - 2.48135i) q^{11} -5.49388i q^{13} +(0.243099 - 2.63456i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.53712 + 2.66237i) q^{17} +(-2.68622 + 1.55089i) q^{19} +4.96270 q^{22} +(-5.34875 + 3.08810i) q^{23} +(2.74694 + 4.75784i) q^{26} +(1.10675 + 2.40314i) q^{28} -6.67885i q^{29} +(-1.01653 - 0.586893i) q^{31} +(0.866025 + 0.500000i) q^{32} -3.07424i q^{34} +(5.35400 + 9.27339i) q^{37} +(1.55089 - 2.68622i) q^{38} +8.39427 q^{41} +8.81025 q^{43} +(-4.29783 + 2.48135i) q^{44} +(3.08810 - 5.34875i) q^{46} +(2.07312 + 3.59075i) q^{47} +(-2.33160 - 6.60028i) q^{49} +(-4.75784 - 2.74694i) q^{52} +(3.85443 + 2.22536i) q^{53} +(-2.16005 - 1.52781i) q^{56} +(3.33943 + 5.78405i) q^{58} +(-3.00381 + 5.20275i) q^{59} +(9.05018 - 5.22512i) q^{61} +1.17379 q^{62} -1.00000 q^{64} +(5.97727 - 10.3529i) q^{67} +(1.53712 + 2.66237i) q^{68} -0.973522i q^{71} +(14.4612 + 8.34916i) q^{73} +(-9.27339 - 5.35400i) q^{74} +3.10178i q^{76} +(11.9261 - 5.49247i) q^{77} +(2.12328 + 3.67763i) q^{79} +(-7.26965 + 4.19713i) q^{82} -14.2841 q^{83} +(-7.62990 + 4.40513i) q^{86} +(2.48135 - 4.29783i) q^{88} +(-7.38517 - 12.7915i) q^{89} +(11.8670 + 8.39360i) q^{91} +6.17620i q^{92} +(-3.59075 - 2.07312i) q^{94} +4.41643i q^{97} +(5.31936 + 4.55021i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 12 q^{4} - 4 q^{7} - 12 q^{16} + 12 q^{19} + 4 q^{28} + 28 q^{37} + 96 q^{43} - 8 q^{46} - 52 q^{49} - 12 q^{52} + 8 q^{58} - 12 q^{61} - 24 q^{64} - 4 q^{67} - 12 q^{73} + 4 q^{79} + 68 q^{91} - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 0.866025i 0.250000 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) −1.52781 + 2.16005i −0.577458 + 0.816421i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.29783 2.48135i −1.29584 0.748156i −0.316160 0.948706i \(-0.602394\pi\)
−0.979683 + 0.200550i \(0.935727\pi\)
\(12\) 0 0
\(13\) 5.49388i 1.52373i −0.647737 0.761864i \(-0.724285\pi\)
0.647737 0.761864i \(-0.275715\pi\)
\(14\) 0.243099 2.63456i 0.0649709 0.704116i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) −1.53712 + 2.66237i −0.372806 + 0.645719i −0.989996 0.141095i \(-0.954938\pi\)
0.617190 + 0.786814i \(0.288271\pi\)
\(18\) 0 0
\(19\) −2.68622 + 1.55089i −0.616261 + 0.355798i −0.775412 0.631456i \(-0.782458\pi\)
0.159151 + 0.987254i \(0.449124\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.96270 1.05805
\(23\) −5.34875 + 3.08810i −1.11529 + 0.643914i −0.940195 0.340638i \(-0.889357\pi\)
−0.175097 + 0.984551i \(0.556024\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.74694 + 4.75784i 0.538719 + 0.933089i
\(27\) 0 0
\(28\) 1.10675 + 2.40314i 0.209156 + 0.454152i
\(29\) 6.67885i 1.24023i −0.784510 0.620116i \(-0.787086\pi\)
0.784510 0.620116i \(-0.212914\pi\)
\(30\) 0 0
\(31\) −1.01653 0.586893i −0.182574 0.105409i 0.405928 0.913905i \(-0.366949\pi\)
−0.588501 + 0.808496i \(0.700282\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 3.07424i 0.527228i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.35400 + 9.27339i 0.880192 + 1.52454i 0.851128 + 0.524959i \(0.175919\pi\)
0.0290640 + 0.999578i \(0.490747\pi\)
\(38\) 1.55089 2.68622i 0.251587 0.435762i
\(39\) 0 0
\(40\) 0 0
\(41\) 8.39427 1.31096 0.655482 0.755211i \(-0.272466\pi\)
0.655482 + 0.755211i \(0.272466\pi\)
\(42\) 0 0
\(43\) 8.81025 1.34355 0.671776 0.740755i \(-0.265532\pi\)
0.671776 + 0.740755i \(0.265532\pi\)
\(44\) −4.29783 + 2.48135i −0.647922 + 0.374078i
\(45\) 0 0
\(46\) 3.08810 5.34875i 0.455316 0.788630i
\(47\) 2.07312 + 3.59075i 0.302396 + 0.523765i 0.976678 0.214709i \(-0.0688803\pi\)
−0.674282 + 0.738474i \(0.735547\pi\)
\(48\) 0 0
\(49\) −2.33160 6.60028i −0.333085 0.942897i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.75784 2.74694i −0.659793 0.380932i
\(53\) 3.85443 + 2.22536i 0.529446 + 0.305676i 0.740791 0.671736i \(-0.234451\pi\)
−0.211345 + 0.977412i \(0.567784\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.16005 1.52781i −0.288648 0.204162i
\(57\) 0 0
\(58\) 3.33943 + 5.78405i 0.438488 + 0.759484i
\(59\) −3.00381 + 5.20275i −0.391062 + 0.677340i −0.992590 0.121512i \(-0.961226\pi\)
0.601528 + 0.798852i \(0.294559\pi\)
\(60\) 0 0
\(61\) 9.05018 5.22512i 1.15876 0.669008i 0.207751 0.978182i \(-0.433386\pi\)
0.951006 + 0.309173i \(0.100052\pi\)
\(62\) 1.17379 0.149071
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.97727 10.3529i 0.730240 1.26481i −0.226540 0.974002i \(-0.572741\pi\)
0.956780 0.290811i \(-0.0939252\pi\)
\(68\) 1.53712 + 2.66237i 0.186403 + 0.322860i
\(69\) 0 0
\(70\) 0 0
\(71\) 0.973522i 0.115536i −0.998330 0.0577679i \(-0.981602\pi\)
0.998330 0.0577679i \(-0.0183983\pi\)
\(72\) 0 0
\(73\) 14.4612 + 8.34916i 1.69255 + 0.977196i 0.952446 + 0.304706i \(0.0985582\pi\)
0.740106 + 0.672490i \(0.234775\pi\)
\(74\) −9.27339 5.35400i −1.07801 0.622389i
\(75\) 0 0
\(76\) 3.10178i 0.355798i
\(77\) 11.9261 5.49247i 1.35910 0.625925i
\(78\) 0 0
\(79\) 2.12328 + 3.67763i 0.238887 + 0.413765i 0.960395 0.278641i \(-0.0898840\pi\)
−0.721508 + 0.692406i \(0.756551\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.26965 + 4.19713i −0.802798 + 0.463496i
\(83\) −14.2841 −1.56789 −0.783944 0.620831i \(-0.786795\pi\)
−0.783944 + 0.620831i \(0.786795\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.62990 + 4.40513i −0.822754 + 0.475017i
\(87\) 0 0
\(88\) 2.48135 4.29783i 0.264513 0.458150i
\(89\) −7.38517 12.7915i −0.782826 1.35590i −0.930289 0.366827i \(-0.880444\pi\)
0.147463 0.989068i \(-0.452889\pi\)
\(90\) 0 0
\(91\) 11.8670 + 8.39360i 1.24400 + 0.879888i
\(92\) 6.17620i 0.643914i
\(93\) 0 0
\(94\) −3.59075 2.07312i −0.370358 0.213826i
\(95\) 0 0
\(96\) 0 0
\(97\) 4.41643i 0.448420i 0.974541 + 0.224210i \(0.0719802\pi\)
−0.974541 + 0.224210i \(0.928020\pi\)
\(98\) 5.31936 + 4.55021i 0.537336 + 0.459641i
\(99\) 0 0
\(100\) 0 0
\(101\) −5.19825 + 9.00364i −0.517245 + 0.895895i 0.482554 + 0.875866i \(0.339709\pi\)
−0.999799 + 0.0200290i \(0.993624\pi\)
\(102\) 0 0
\(103\) 8.86709 5.11942i 0.873701 0.504431i 0.00512447 0.999987i \(-0.498369\pi\)
0.868576 + 0.495556i \(0.165035\pi\)
\(104\) 5.49388 0.538719
\(105\) 0 0
\(106\) −4.45071 −0.432291
\(107\) −5.69797 + 3.28972i −0.550844 + 0.318030i −0.749462 0.662047i \(-0.769688\pi\)
0.198619 + 0.980077i \(0.436355\pi\)
\(108\) 0 0
\(109\) −1.34219 + 2.32474i −0.128558 + 0.222669i −0.923118 0.384516i \(-0.874368\pi\)
0.794560 + 0.607186i \(0.207702\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.63456 + 0.243099i 0.248942 + 0.0229707i
\(113\) 3.55031i 0.333985i 0.985958 + 0.166992i \(0.0534056\pi\)
−0.985958 + 0.166992i \(0.946594\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.78405 3.33943i −0.537036 0.310058i
\(117\) 0 0
\(118\) 6.00761i 0.553046i
\(119\) −3.40241 7.38784i −0.311899 0.677242i
\(120\) 0 0
\(121\) 6.81421 + 11.8026i 0.619474 + 1.07296i
\(122\) −5.22512 + 9.05018i −0.473060 + 0.819365i
\(123\) 0 0
\(124\) −1.01653 + 0.586893i −0.0912869 + 0.0527045i
\(125\) 0 0
\(126\) 0 0
\(127\) −5.51567 −0.489437 −0.244719 0.969594i \(-0.578696\pi\)
−0.244719 + 0.969594i \(0.578696\pi\)
\(128\) 0.866025 0.500000i 0.0765466 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) 10.3068 + 17.8519i 0.900510 + 1.55973i 0.826833 + 0.562447i \(0.190140\pi\)
0.0736773 + 0.997282i \(0.476527\pi\)
\(132\) 0 0
\(133\) 0.754039 8.17182i 0.0653835 0.708586i
\(134\) 11.9545i 1.03272i
\(135\) 0 0
\(136\) −2.66237 1.53712i −0.228296 0.131807i
\(137\) 9.92131 + 5.72807i 0.847635 + 0.489382i 0.859852 0.510543i \(-0.170556\pi\)
−0.0122175 + 0.999925i \(0.503889\pi\)
\(138\) 0 0
\(139\) 1.16700i 0.0989840i −0.998775 0.0494920i \(-0.984240\pi\)
0.998775 0.0494920i \(-0.0157602\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.486761 + 0.843095i 0.0408481 + 0.0707509i
\(143\) −13.6322 + 23.6117i −1.13999 + 1.97451i
\(144\) 0 0
\(145\) 0 0
\(146\) −16.6983 −1.38196
\(147\) 0 0
\(148\) 10.7080 0.880192
\(149\) 13.3404 7.70205i 1.09288 0.630977i 0.158541 0.987352i \(-0.449321\pi\)
0.934343 + 0.356375i \(0.115988\pi\)
\(150\) 0 0
\(151\) −0.511281 + 0.885565i −0.0416075 + 0.0720663i −0.886079 0.463534i \(-0.846581\pi\)
0.844472 + 0.535600i \(0.179915\pi\)
\(152\) −1.55089 2.68622i −0.125794 0.217881i
\(153\) 0 0
\(154\) −7.58207 + 10.7197i −0.610980 + 0.863815i
\(155\) 0 0
\(156\) 0 0
\(157\) 4.64699 + 2.68294i 0.370871 + 0.214122i 0.673839 0.738878i \(-0.264644\pi\)
−0.302968 + 0.953001i \(0.597978\pi\)
\(158\) −3.67763 2.12328i −0.292576 0.168919i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.50143 16.2716i 0.118329 1.28238i
\(162\) 0 0
\(163\) −5.02108 8.69677i −0.393282 0.681184i 0.599599 0.800301i \(-0.295327\pi\)
−0.992880 + 0.119117i \(0.961994\pi\)
\(164\) 4.19713 7.26965i 0.327741 0.567664i
\(165\) 0 0
\(166\) 12.3704 7.14207i 0.960131 0.554332i
\(167\) 2.46005 0.190364 0.0951822 0.995460i \(-0.469657\pi\)
0.0951822 + 0.995460i \(0.469657\pi\)
\(168\) 0 0
\(169\) −17.1827 −1.32175
\(170\) 0 0
\(171\) 0 0
\(172\) 4.40513 7.62990i 0.335888 0.581775i
\(173\) 1.50042 + 2.59880i 0.114075 + 0.197583i 0.917410 0.397944i \(-0.130276\pi\)
−0.803335 + 0.595528i \(0.796943\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.96270i 0.374078i
\(177\) 0 0
\(178\) 12.7915 + 7.38517i 0.958763 + 0.553542i
\(179\) −3.18036 1.83618i −0.237711 0.137243i 0.376413 0.926452i \(-0.377157\pi\)
−0.614124 + 0.789209i \(0.710491\pi\)
\(180\) 0 0
\(181\) 6.13560i 0.456056i −0.973655 0.228028i \(-0.926772\pi\)
0.973655 0.228028i \(-0.0732278\pi\)
\(182\) −14.4739 1.33556i −1.07288 0.0989980i
\(183\) 0 0
\(184\) −3.08810 5.34875i −0.227658 0.394315i
\(185\) 0 0
\(186\) 0 0
\(187\) 13.2125 7.62827i 0.966197 0.557834i
\(188\) 4.14624 0.302396
\(189\) 0 0
\(190\) 0 0
\(191\) 4.95227 2.85920i 0.358334 0.206884i −0.310016 0.950731i \(-0.600334\pi\)
0.668350 + 0.743847i \(0.267001\pi\)
\(192\) 0 0
\(193\) −3.11665 + 5.39819i −0.224341 + 0.388570i −0.956122 0.292970i \(-0.905356\pi\)
0.731780 + 0.681541i \(0.238690\pi\)
\(194\) −2.20821 3.82474i −0.158540 0.274600i
\(195\) 0 0
\(196\) −6.88181 1.28092i −0.491558 0.0914941i
\(197\) 1.32234i 0.0942128i −0.998890 0.0471064i \(-0.985000\pi\)
0.998890 0.0471064i \(-0.0150000\pi\)
\(198\) 0 0
\(199\) −8.27163 4.77563i −0.586360 0.338535i 0.177297 0.984157i \(-0.443265\pi\)
−0.763657 + 0.645622i \(0.776598\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.3965i 0.731495i
\(203\) 14.4266 + 10.2040i 1.01255 + 0.716181i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.11942 + 8.86709i −0.356687 + 0.617800i
\(207\) 0 0
\(208\) −4.75784 + 2.74694i −0.329897 + 0.190466i
\(209\) 15.3932 1.06477
\(210\) 0 0
\(211\) 26.0219 1.79142 0.895711 0.444636i \(-0.146667\pi\)
0.895711 + 0.444636i \(0.146667\pi\)
\(212\) 3.85443 2.22536i 0.264723 0.152838i
\(213\) 0 0
\(214\) 3.28972 5.69797i 0.224881 0.389505i
\(215\) 0 0
\(216\) 0 0
\(217\) 2.82078 1.29909i 0.191487 0.0881878i
\(218\) 2.68437i 0.181809i
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6267 + 8.44475i 0.983900 + 0.568055i
\(222\) 0 0
\(223\) 7.25222i 0.485644i 0.970071 + 0.242822i \(0.0780731\pi\)
−0.970071 + 0.242822i \(0.921927\pi\)
\(224\) −2.40314 + 1.10675i −0.160567 + 0.0739478i
\(225\) 0 0
\(226\) −1.77515 3.07466i −0.118081 0.204523i
\(227\) 13.3604 23.1409i 0.886762 1.53592i 0.0430820 0.999072i \(-0.486282\pi\)
0.843680 0.536846i \(-0.180384\pi\)
\(228\) 0 0
\(229\) 21.0473 12.1517i 1.39085 0.803006i 0.397439 0.917629i \(-0.369899\pi\)
0.993409 + 0.114622i \(0.0365658\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.67885 0.438488
\(233\) 8.01573 4.62788i 0.525128 0.303183i −0.213902 0.976855i \(-0.568617\pi\)
0.739030 + 0.673672i \(0.235284\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00381 + 5.20275i 0.195531 + 0.338670i
\(237\) 0 0
\(238\) 6.64050 + 4.69685i 0.430439 + 0.304452i
\(239\) 0.253367i 0.0163889i 0.999966 + 0.00819446i \(0.00260841\pi\)
−0.999966 + 0.00819446i \(0.997392\pi\)
\(240\) 0 0
\(241\) 2.57538 + 1.48689i 0.165895 + 0.0957792i 0.580649 0.814154i \(-0.302799\pi\)
−0.414754 + 0.909934i \(0.636132\pi\)
\(242\) −11.8026 6.81421i −0.758698 0.438034i
\(243\) 0 0
\(244\) 10.4502i 0.669008i
\(245\) 0 0
\(246\) 0 0
\(247\) 8.52039 + 14.7578i 0.542140 + 0.939013i
\(248\) 0.586893 1.01653i 0.0372677 0.0645496i
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0800 0.825599 0.412800 0.910822i \(-0.364551\pi\)
0.412800 + 0.910822i \(0.364551\pi\)
\(252\) 0 0
\(253\) 30.6507 1.92699
\(254\) 4.77671 2.75784i 0.299718 0.173042i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) 6.29797 + 10.9084i 0.392856 + 0.680447i 0.992825 0.119576i \(-0.0381536\pi\)
−0.599969 + 0.800023i \(0.704820\pi\)
\(258\) 0 0
\(259\) −28.2108 2.60310i −1.75294 0.161749i
\(260\) 0 0
\(261\) 0 0
\(262\) −17.8519 10.3068i −1.10290 0.636757i
\(263\) 14.4383 + 8.33594i 0.890302 + 0.514016i 0.874041 0.485852i \(-0.161490\pi\)
0.0162609 + 0.999868i \(0.494824\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.43289 + 7.45402i 0.210484 + 0.457035i
\(267\) 0 0
\(268\) −5.97727 10.3529i −0.365120 0.632407i
\(269\) 10.0035 17.3265i 0.609923 1.05642i −0.381330 0.924439i \(-0.624534\pi\)
0.991253 0.131978i \(-0.0421328\pi\)
\(270\) 0 0
\(271\) −15.4684 + 8.93068i −0.939638 + 0.542500i −0.889847 0.456259i \(-0.849189\pi\)
−0.0497914 + 0.998760i \(0.515856\pi\)
\(272\) 3.07424 0.186403
\(273\) 0 0
\(274\) −11.4561 −0.692091
\(275\) 0 0
\(276\) 0 0
\(277\) −9.07406 + 15.7167i −0.545207 + 0.944327i 0.453386 + 0.891314i \(0.350216\pi\)
−0.998594 + 0.0530128i \(0.983118\pi\)
\(278\) 0.583502 + 1.01066i 0.0349961 + 0.0606151i
\(279\) 0 0
\(280\) 0 0
\(281\) 15.5129i 0.925425i −0.886508 0.462713i \(-0.846876\pi\)
0.886508 0.462713i \(-0.153124\pi\)
\(282\) 0 0
\(283\) −15.2813 8.82268i −0.908381 0.524454i −0.0284708 0.999595i \(-0.509064\pi\)
−0.879910 + 0.475141i \(0.842397\pi\)
\(284\) −0.843095 0.486761i −0.0500285 0.0288840i
\(285\) 0 0
\(286\) 27.2645i 1.61218i
\(287\) −12.8248 + 18.1320i −0.757026 + 1.07030i
\(288\) 0 0
\(289\) 3.77453 + 6.53767i 0.222031 + 0.384569i
\(290\) 0 0
\(291\) 0 0
\(292\) 14.4612 8.34916i 0.846276 0.488598i
\(293\) −10.4489 −0.610432 −0.305216 0.952283i \(-0.598729\pi\)
−0.305216 + 0.952283i \(0.598729\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.27339 + 5.35400i −0.539005 + 0.311195i
\(297\) 0 0
\(298\) −7.70205 + 13.3404i −0.446168 + 0.772786i
\(299\) 16.9657 + 29.3854i 0.981149 + 1.69940i
\(300\) 0 0
\(301\) −13.4604 + 19.0306i −0.775844 + 1.09690i
\(302\) 1.02256i 0.0588419i
\(303\) 0 0
\(304\) 2.68622 + 1.55089i 0.154065 + 0.0889496i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.724648i 0.0413579i −0.999786 0.0206789i \(-0.993417\pi\)
0.999786 0.0206789i \(-0.00658278\pi\)
\(308\) 1.20643 13.0745i 0.0687426 0.744991i
\(309\) 0 0
\(310\) 0 0
\(311\) 14.1225 24.4609i 0.800813 1.38705i −0.118268 0.992982i \(-0.537734\pi\)
0.919081 0.394068i \(-0.128932\pi\)
\(312\) 0 0
\(313\) −17.7797 + 10.2651i −1.00497 + 0.580218i −0.909714 0.415235i \(-0.863699\pi\)
−0.0952528 + 0.995453i \(0.530366\pi\)
\(314\) −5.36589 −0.302815
\(315\) 0 0
\(316\) 4.24656 0.238887
\(317\) −1.96890 + 1.13674i −0.110584 + 0.0638459i −0.554272 0.832336i \(-0.687003\pi\)
0.443688 + 0.896181i \(0.353670\pi\)
\(318\) 0 0
\(319\) −16.5726 + 28.7045i −0.927886 + 1.60715i
\(320\) 0 0
\(321\) 0 0
\(322\) 6.83551 + 14.8423i 0.380928 + 0.827130i
\(323\) 9.53560i 0.530575i
\(324\) 0 0
\(325\) 0 0
\(326\) 8.69677 + 5.02108i 0.481670 + 0.278092i
\(327\) 0 0
\(328\) 8.39427i 0.463496i
\(329\) −10.9235 1.00795i −0.602233 0.0555699i
\(330\) 0 0
\(331\) 18.0646 + 31.2889i 0.992922 + 1.71979i 0.599317 + 0.800512i \(0.295439\pi\)
0.393605 + 0.919280i \(0.371228\pi\)
\(332\) −7.14207 + 12.3704i −0.391972 + 0.678915i
\(333\) 0 0
\(334\) −2.13047 + 1.23003i −0.116574 + 0.0673040i
\(335\) 0 0
\(336\) 0 0
\(337\) −3.76361 −0.205017 −0.102508 0.994732i \(-0.532687\pi\)
−0.102508 + 0.994732i \(0.532687\pi\)
\(338\) 14.8806 8.59134i 0.809400 0.467308i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.91258 + 5.04473i 0.157725 + 0.273187i
\(342\) 0 0
\(343\) 17.8191 + 5.04761i 0.962143 + 0.272546i
\(344\) 8.81025i 0.475017i
\(345\) 0 0
\(346\) −2.59880 1.50042i −0.139713 0.0806631i
\(347\) 3.50549 + 2.02389i 0.188184 + 0.108648i 0.591132 0.806575i \(-0.298681\pi\)
−0.402948 + 0.915223i \(0.632014\pi\)
\(348\) 0 0
\(349\) 23.9364i 1.28129i 0.767838 + 0.640644i \(0.221333\pi\)
−0.767838 + 0.640644i \(0.778667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.48135 4.29783i −0.132256 0.229075i
\(353\) −12.7409 + 22.0679i −0.678129 + 1.17455i 0.297415 + 0.954748i \(0.403876\pi\)
−0.975544 + 0.219805i \(0.929458\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.7703 −0.782826
\(357\) 0 0
\(358\) 3.67236 0.194091
\(359\) −15.4893 + 8.94277i −0.817496 + 0.471981i −0.849552 0.527505i \(-0.823128\pi\)
0.0320565 + 0.999486i \(0.489794\pi\)
\(360\) 0 0
\(361\) −4.68949 + 8.12243i −0.246815 + 0.427496i
\(362\) 3.06780 + 5.31359i 0.161240 + 0.279276i
\(363\) 0 0
\(364\) 13.2026 6.08035i 0.692003 0.318697i
\(365\) 0 0
\(366\) 0 0
\(367\) −4.57004 2.63851i −0.238554 0.137729i 0.375958 0.926637i \(-0.377314\pi\)
−0.614512 + 0.788907i \(0.710647\pi\)
\(368\) 5.34875 + 3.08810i 0.278823 + 0.160978i
\(369\) 0 0
\(370\) 0 0
\(371\) −10.6957 + 4.92582i −0.555293 + 0.255736i
\(372\) 0 0
\(373\) −7.27390 12.5988i −0.376628 0.652339i 0.613941 0.789352i \(-0.289583\pi\)
−0.990569 + 0.137013i \(0.956250\pi\)
\(374\) −7.62827 + 13.2125i −0.394448 + 0.683205i
\(375\) 0 0
\(376\) −3.59075 + 2.07312i −0.185179 + 0.106913i
\(377\) −36.6928 −1.88977
\(378\) 0 0
\(379\) 3.66669 0.188345 0.0941726 0.995556i \(-0.469979\pi\)
0.0941726 + 0.995556i \(0.469979\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.85920 + 4.95227i −0.146289 + 0.253380i
\(383\) 9.76247 + 16.9091i 0.498839 + 0.864015i 0.999999 0.00134002i \(-0.000426542\pi\)
−0.501160 + 0.865355i \(0.667093\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6.23330i 0.317266i
\(387\) 0 0
\(388\) 3.82474 + 2.20821i 0.194172 + 0.112105i
\(389\) 14.9711 + 8.64356i 0.759064 + 0.438246i 0.828960 0.559309i \(-0.188933\pi\)
−0.0698956 + 0.997554i \(0.522267\pi\)
\(390\) 0 0
\(391\) 18.9871i 0.960220i
\(392\) 6.60028 2.33160i 0.333364 0.117763i
\(393\) 0 0
\(394\) 0.661170 + 1.14518i 0.0333092 + 0.0576933i
\(395\) 0 0
\(396\) 0 0
\(397\) 23.9755 13.8423i 1.20330 0.694724i 0.242011 0.970274i \(-0.422193\pi\)
0.961287 + 0.275549i \(0.0888598\pi\)
\(398\) 9.55125 0.478761
\(399\) 0 0
\(400\) 0 0
\(401\) −32.7521 + 18.9095i −1.63556 + 0.944293i −0.653229 + 0.757161i \(0.726586\pi\)
−0.982335 + 0.187132i \(0.940081\pi\)
\(402\) 0 0
\(403\) −3.22432 + 5.58468i −0.160615 + 0.278193i
\(404\) 5.19825 + 9.00364i 0.258623 + 0.447948i
\(405\) 0 0
\(406\) −17.5958 1.62362i −0.873266 0.0805790i
\(407\) 53.1406i 2.63408i
\(408\) 0 0
\(409\) 31.6028 + 18.2459i 1.56266 + 0.902202i 0.996987 + 0.0775719i \(0.0247167\pi\)
0.565673 + 0.824630i \(0.308617\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.2388i 0.504431i
\(413\) −6.64893 14.4372i −0.327172 0.710407i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.74694 4.75784i 0.134680 0.233272i
\(417\) 0 0
\(418\) −13.3309 + 7.69660i −0.652036 + 0.376453i
\(419\) 21.6669 1.05850 0.529249 0.848466i \(-0.322474\pi\)
0.529249 + 0.848466i \(0.322474\pi\)
\(420\) 0 0
\(421\) −8.84193 −0.430929 −0.215465 0.976512i \(-0.569127\pi\)
−0.215465 + 0.976512i \(0.569127\pi\)
\(422\) −22.5356 + 13.0110i −1.09702 + 0.633363i
\(423\) 0 0
\(424\) −2.22536 + 3.85443i −0.108073 + 0.187188i
\(425\) 0 0
\(426\) 0 0
\(427\) −2.54044 + 27.5318i −0.122941 + 1.33236i
\(428\) 6.57945i 0.318030i
\(429\) 0 0
\(430\) 0 0
\(431\) 25.7481 + 14.8656i 1.24024 + 0.716053i 0.969143 0.246499i \(-0.0792802\pi\)
0.271097 + 0.962552i \(0.412614\pi\)
\(432\) 0 0
\(433\) 26.5666i 1.27671i −0.769741 0.638356i \(-0.779615\pi\)
0.769741 0.638356i \(-0.220385\pi\)
\(434\) −1.79332 + 2.53543i −0.0860822 + 0.121705i
\(435\) 0 0
\(436\) 1.34219 + 2.32474i 0.0642791 + 0.111335i
\(437\) 9.57860 16.5906i 0.458207 0.793637i
\(438\) 0 0
\(439\) 14.4067 8.31774i 0.687597 0.396984i −0.115114 0.993352i \(-0.536723\pi\)
0.802711 + 0.596368i \(0.203390\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.8895 −0.803351
\(443\) −2.83428 + 1.63637i −0.134661 + 0.0777464i −0.565817 0.824531i \(-0.691439\pi\)
0.431156 + 0.902277i \(0.358106\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.62611 6.28060i −0.171701 0.297395i
\(447\) 0 0
\(448\) 1.52781 2.16005i 0.0721822 0.102053i
\(449\) 10.4322i 0.492324i 0.969229 + 0.246162i \(0.0791695\pi\)
−0.969229 + 0.246162i \(0.920831\pi\)
\(450\) 0 0
\(451\) −36.0771 20.8291i −1.69880 0.980805i
\(452\) 3.07466 + 1.77515i 0.144620 + 0.0834962i
\(453\) 0 0
\(454\) 26.7208i 1.25407i
\(455\) 0 0
\(456\) 0 0
\(457\) −16.5394 28.6471i −0.773680 1.34005i −0.935533 0.353238i \(-0.885080\pi\)
0.161853 0.986815i \(-0.448253\pi\)
\(458\) −12.1517 + 21.0473i −0.567811 + 0.983478i
\(459\) 0 0
\(460\) 0 0
\(461\) 11.5639 0.538585 0.269293 0.963058i \(-0.413210\pi\)
0.269293 + 0.963058i \(0.413210\pi\)
\(462\) 0 0
\(463\) −38.6061 −1.79418 −0.897090 0.441848i \(-0.854323\pi\)
−0.897090 + 0.441848i \(0.854323\pi\)
\(464\) −5.78405 + 3.33943i −0.268518 + 0.155029i
\(465\) 0 0
\(466\) −4.62788 + 8.01573i −0.214383 + 0.371322i
\(467\) 2.35692 + 4.08230i 0.109065 + 0.188906i 0.915392 0.402564i \(-0.131881\pi\)
−0.806327 + 0.591471i \(0.798548\pi\)
\(468\) 0 0
\(469\) 13.2307 + 28.7285i 0.610937 + 1.32656i
\(470\) 0 0
\(471\) 0 0
\(472\) −5.20275 3.00381i −0.239476 0.138261i
\(473\) −37.8649 21.8613i −1.74103 1.00519i
\(474\) 0 0
\(475\) 0 0
\(476\) −8.09926 0.747344i −0.371229 0.0342545i
\(477\) 0 0
\(478\) −0.126683 0.219422i −0.00579436 0.0100361i
\(479\) 10.0096 17.3371i 0.457349 0.792152i −0.541471 0.840720i \(-0.682132\pi\)
0.998820 + 0.0485678i \(0.0154657\pi\)
\(480\) 0 0
\(481\) 50.9469 29.4142i 2.32298 1.34117i
\(482\) −2.97379 −0.135452
\(483\) 0 0
\(484\) 13.6284 0.619474
\(485\) 0 0
\(486\) 0 0
\(487\) −2.36544 + 4.09706i −0.107188 + 0.185656i −0.914630 0.404292i \(-0.867518\pi\)
0.807442 + 0.589947i \(0.200851\pi\)
\(488\) 5.22512 + 9.05018i 0.236530 + 0.409682i
\(489\) 0 0
\(490\) 0 0
\(491\) 16.0027i 0.722190i −0.932529 0.361095i \(-0.882403\pi\)
0.932529 0.361095i \(-0.117597\pi\)
\(492\) 0 0
\(493\) 17.7816 + 10.2662i 0.800841 + 0.462366i
\(494\) −14.7578 8.52039i −0.663983 0.383351i
\(495\) 0 0
\(496\) 1.17379i 0.0527045i
\(497\) 2.10285 + 1.48736i 0.0943258 + 0.0667170i
\(498\) 0 0
\(499\) −3.18097 5.50961i −0.142400 0.246644i 0.786000 0.618227i \(-0.212149\pi\)
−0.928400 + 0.371583i \(0.878815\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −11.3276 + 6.53998i −0.505574 + 0.291893i
\(503\) −36.3826 −1.62222 −0.811109 0.584895i \(-0.801136\pi\)
−0.811109 + 0.584895i \(0.801136\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −26.5443 + 15.3253i −1.18004 + 0.681294i
\(507\) 0 0
\(508\) −2.75784 + 4.77671i −0.122359 + 0.211932i
\(509\) −8.55353 14.8151i −0.379128 0.656670i 0.611807 0.791007i \(-0.290443\pi\)
−0.990936 + 0.134337i \(0.957109\pi\)
\(510\) 0 0
\(511\) −40.1285 + 18.4809i −1.77518 + 0.817546i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −10.9084 6.29797i −0.481149 0.277791i
\(515\) 0 0
\(516\) 0 0
\(517\) 20.5766i 0.904956i
\(518\) 25.7329 11.8511i 1.13064 0.520706i
\(519\) 0 0
\(520\) 0 0
\(521\) 20.2375 35.0524i 0.886622 1.53567i 0.0427789 0.999085i \(-0.486379\pi\)
0.843843 0.536590i \(-0.180288\pi\)
\(522\) 0 0
\(523\) −37.8314 + 21.8420i −1.65425 + 0.955083i −0.678957 + 0.734178i \(0.737568\pi\)
−0.975295 + 0.220905i \(0.929099\pi\)
\(524\) 20.6136 0.900510
\(525\) 0 0
\(526\) −16.6719 −0.726929
\(527\) 3.12505 1.80425i 0.136129 0.0785943i
\(528\) 0 0
\(529\) 7.57274 13.1164i 0.329250 0.570277i
\(530\) 0 0
\(531\) 0 0
\(532\) −6.69998 4.73893i −0.290481 0.205458i
\(533\) 46.1171i 1.99755i
\(534\) 0 0
\(535\) 0 0
\(536\) 10.3529 + 5.97727i 0.447179 + 0.258179i
\(537\) 0 0
\(538\) 20.0069i 0.862561i
\(539\) −6.35681 + 34.1524i −0.273807 + 1.47105i
\(540\) 0 0
\(541\) 5.85601 + 10.1429i 0.251770 + 0.436078i 0.964013 0.265855i \(-0.0856541\pi\)
−0.712243 + 0.701933i \(0.752321\pi\)
\(542\) 8.93068 15.4684i 0.383606 0.664425i
\(543\) 0 0
\(544\) −2.66237 + 1.53712i −0.114148 + 0.0659035i
\(545\) 0 0
\(546\) 0 0
\(547\) 34.6501 1.48153 0.740765 0.671764i \(-0.234463\pi\)
0.740765 + 0.671764i \(0.234463\pi\)
\(548\) 9.92131 5.72807i 0.423817 0.244691i
\(549\) 0 0
\(550\) 0 0
\(551\) 10.3582 + 17.9409i 0.441272 + 0.764306i
\(552\) 0 0
\(553\) −11.1878 1.03233i −0.475754 0.0438993i
\(554\) 18.1481i 0.771040i
\(555\) 0 0
\(556\) −1.01066 0.583502i −0.0428613 0.0247460i
\(557\) 30.5822 + 17.6567i 1.29581 + 0.748137i 0.979678 0.200578i \(-0.0642819\pi\)
0.316134 + 0.948715i \(0.397615\pi\)
\(558\) 0 0
\(559\) 48.4025i 2.04721i
\(560\) 0 0
\(561\) 0 0
\(562\) 7.75647 + 13.4346i 0.327187 + 0.566705i
\(563\) 19.3495 33.5143i 0.815483 1.41246i −0.0934975 0.995620i \(-0.529805\pi\)
0.908981 0.416839i \(-0.136862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 17.6454 0.741690
\(567\) 0 0
\(568\) 0.973522 0.0408481
\(569\) −6.20799 + 3.58419i −0.260253 + 0.150257i −0.624450 0.781065i \(-0.714677\pi\)
0.364197 + 0.931322i \(0.381343\pi\)
\(570\) 0 0
\(571\) 10.7717 18.6571i 0.450781 0.780776i −0.547653 0.836705i \(-0.684479\pi\)
0.998435 + 0.0559290i \(0.0178121\pi\)
\(572\) 13.6322 + 23.6117i 0.569993 + 0.987256i
\(573\) 0 0
\(574\) 2.04064 22.1152i 0.0851746 0.923070i
\(575\) 0 0
\(576\) 0 0
\(577\) −13.6179 7.86230i −0.566921 0.327312i 0.188998 0.981978i \(-0.439476\pi\)
−0.755919 + 0.654666i \(0.772809\pi\)
\(578\) −6.53767 3.77453i −0.271931 0.157000i
\(579\) 0 0
\(580\) 0 0
\(581\) 21.8234 30.8544i 0.905389 1.28006i
\(582\) 0 0
\(583\) −11.0438 19.1284i −0.457387 0.792217i
\(584\) −8.34916 + 14.4612i −0.345491 + 0.598408i
\(585\) 0 0
\(586\) 9.04902 5.22446i 0.373812 0.215820i
\(587\) −4.59252 −0.189554 −0.0947769 0.995499i \(-0.530214\pi\)
−0.0947769 + 0.995499i \(0.530214\pi\)
\(588\) 0 0
\(589\) 3.64082 0.150017
\(590\) 0 0
\(591\) 0 0
\(592\) 5.35400 9.27339i 0.220048 0.381134i
\(593\) 1.91286 + 3.31317i 0.0785516 + 0.136055i 0.902625 0.430427i \(-0.141637\pi\)
−0.824074 + 0.566483i \(0.808304\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.4041i 0.630977i
\(597\) 0 0
\(598\) −29.3854 16.9657i −1.20166 0.693777i
\(599\) 13.6589 + 7.88600i 0.558089 + 0.322213i 0.752378 0.658731i \(-0.228907\pi\)
−0.194289 + 0.980944i \(0.562240\pi\)
\(600\) 0 0
\(601\) 1.39673i 0.0569740i −0.999594 0.0284870i \(-0.990931\pi\)
0.999594 0.0284870i \(-0.00906892\pi\)
\(602\) 2.14176 23.2111i 0.0872918 0.946015i
\(603\) 0 0
\(604\) 0.511281 + 0.885565i 0.0208037 + 0.0360331i
\(605\) 0 0
\(606\) 0 0
\(607\) −8.94920 + 5.16682i −0.363237 + 0.209715i −0.670500 0.741910i \(-0.733920\pi\)
0.307263 + 0.951625i \(0.400587\pi\)
\(608\) −3.10178 −0.125794
\(609\) 0 0
\(610\) 0 0
\(611\) 19.7271 11.3895i 0.798075 0.460769i
\(612\) 0 0
\(613\) −6.14772 + 10.6482i −0.248304 + 0.430075i −0.963055 0.269303i \(-0.913207\pi\)
0.714751 + 0.699379i \(0.246540\pi\)
\(614\) 0.362324 + 0.627564i 0.0146222 + 0.0253264i
\(615\) 0 0
\(616\) 5.49247 + 11.9261i 0.221298 + 0.480516i
\(617\) 8.10935i 0.326470i 0.986587 + 0.163235i \(0.0521929\pi\)
−0.986587 + 0.163235i \(0.947807\pi\)
\(618\) 0 0
\(619\) 7.03506 + 4.06170i 0.282763 + 0.163253i 0.634674 0.772780i \(-0.281135\pi\)
−0.351911 + 0.936034i \(0.614468\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 28.2450i 1.13252i
\(623\) 38.9133 + 3.59065i 1.55903 + 0.143857i
\(624\) 0 0
\(625\) 0 0
\(626\) 10.2651 17.7797i 0.410276 0.710619i
\(627\) 0 0
\(628\) 4.64699 2.68294i 0.185435 0.107061i
\(629\) −32.9189 −1.31256
\(630\) 0 0
\(631\) −40.6011 −1.61630 −0.808151 0.588975i \(-0.799532\pi\)
−0.808151 + 0.588975i \(0.799532\pi\)
\(632\) −3.67763 + 2.12328i −0.146288 + 0.0844595i
\(633\) 0 0
\(634\) 1.13674 1.96890i 0.0451459 0.0781950i
\(635\) 0 0
\(636\) 0 0
\(637\) −36.2611 + 12.8095i −1.43672 + 0.507531i
\(638\) 33.1452i 1.31223i
\(639\) 0 0
\(640\) 0 0
\(641\) −32.0260 18.4902i −1.26495 0.730319i −0.290922 0.956747i \(-0.593962\pi\)
−0.974028 + 0.226427i \(0.927295\pi\)
\(642\) 0 0
\(643\) 4.86696i 0.191934i −0.995385 0.0959671i \(-0.969406\pi\)
0.995385 0.0959671i \(-0.0305944\pi\)
\(644\) −13.3409 9.43606i −0.525704 0.371833i
\(645\) 0 0
\(646\) 4.76780 + 8.25808i 0.187587 + 0.324910i
\(647\) 12.4833 21.6217i 0.490769 0.850037i −0.509175 0.860663i \(-0.670049\pi\)
0.999944 + 0.0106266i \(0.00338261\pi\)
\(648\) 0 0
\(649\) 25.8197 14.9070i 1.01351 0.585151i
\(650\) 0 0
\(651\) 0 0
\(652\) −10.0422 −0.393282
\(653\) 12.6352 7.29496i 0.494455 0.285474i −0.231966 0.972724i \(-0.574516\pi\)
0.726421 + 0.687250i \(0.241182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.19713 7.26965i −0.163870 0.283832i
\(657\) 0 0
\(658\) 9.96402 4.58885i 0.388438 0.178892i
\(659\) 33.8468i 1.31848i 0.751931 + 0.659242i \(0.229123\pi\)
−0.751931 + 0.659242i \(0.770877\pi\)
\(660\) 0 0
\(661\) −14.9053 8.60557i −0.579749 0.334718i 0.181285 0.983431i \(-0.441974\pi\)
−0.761034 + 0.648713i \(0.775308\pi\)
\(662\) −31.2889 18.0646i −1.21608 0.702102i
\(663\) 0 0
\(664\) 14.2841i 0.554332i
\(665\) 0 0
\(666\) 0 0
\(667\) 20.6250 + 35.7235i 0.798602 + 1.38322i
\(668\) 1.23003 2.13047i 0.0475911 0.0824302i
\(669\) 0 0
\(670\) 0 0
\(671\) −51.8615 −2.00209
\(672\) 0 0
\(673\) −1.47971 −0.0570387 −0.0285193 0.999593i \(-0.509079\pi\)
−0.0285193 + 0.999593i \(0.509079\pi\)
\(674\) 3.25938 1.88181i 0.125547 0.0724844i
\(675\) 0 0
\(676\) −8.59134 + 14.8806i −0.330436 + 0.572333i
\(677\) 5.92549 + 10.2632i 0.227735 + 0.394448i 0.957136 0.289637i \(-0.0935347\pi\)
−0.729402 + 0.684086i \(0.760201\pi\)
\(678\) 0 0
\(679\) −9.53968 6.74746i −0.366099 0.258944i
\(680\) 0 0
\(681\) 0 0
\(682\) −5.04473 2.91258i −0.193173 0.111528i
\(683\) 9.11732 + 5.26389i 0.348865 + 0.201417i 0.664185 0.747568i \(-0.268779\pi\)
−0.315320 + 0.948985i \(0.602112\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −17.9556 + 4.53821i −0.685549 + 0.173270i
\(687\) 0 0
\(688\) −4.40513 7.62990i −0.167944 0.290887i
\(689\) 12.2258 21.1758i 0.465767 0.806732i
\(690\) 0 0
\(691\) 6.61628 3.81991i 0.251695 0.145316i −0.368845 0.929491i \(-0.620247\pi\)
0.620540 + 0.784175i \(0.286913\pi\)
\(692\) 3.00084 0.114075
\(693\) 0 0
\(694\) −4.04779 −0.153652
\(695\) 0 0
\(696\) 0 0
\(697\) −12.9030 + 22.3486i −0.488736 + 0.846515i
\(698\) −11.9682 20.7296i −0.453004 0.784626i
\(699\) 0 0
\(700\) 0 0
\(701\) 35.2007i 1.32951i −0.747060 0.664757i \(-0.768535\pi\)
0.747060 0.664757i \(-0.231465\pi\)
\(702\) 0 0
\(703\) −28.7640 16.6069i −1.08485 0.626341i
\(704\) 4.29783 + 2.48135i 0.161980 + 0.0935195i
\(705\) 0 0
\(706\) 25.4818i 0.959019i
\(707\) −11.5063 24.9843i −0.432740 0.939631i
\(708\) 0 0
\(709\) 18.1846 + 31.4966i 0.682936 + 1.18288i 0.974081 + 0.226201i \(0.0726306\pi\)
−0.291145 + 0.956679i \(0.594036\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.7915 7.38517i 0.479381 0.276771i
\(713\) 7.24954 0.271497
\(714\) 0 0
\(715\) 0 0
\(716\) −3.18036 + 1.83618i −0.118856 + 0.0686214i
\(717\) 0 0
\(718\) 8.94277 15.4893i 0.333741 0.578057i
\(719\) −0.772550 1.33810i −0.0288113 0.0499026i 0.851260 0.524744i \(-0.175839\pi\)
−0.880072 + 0.474841i \(0.842506\pi\)
\(720\) 0 0
\(721\) −2.48905 + 26.9748i −0.0926971 + 1.00460i
\(722\) 9.37898i 0.349049i
\(723\) 0 0
\(724\) −5.31359 3.06780i −0.197478 0.114014i
\(725\) 0 0
\(726\) 0 0
\(727\) 34.1857i 1.26788i −0.773383 0.633939i \(-0.781437\pi\)
0.773383 0.633939i \(-0.218563\pi\)
\(728\) −8.39360 + 11.8670i −0.311087 + 0.439821i
\(729\) 0 0
\(730\) 0 0
\(731\) −13.5424 + 23.4561i −0.500884 + 0.867557i
\(732\) 0 0
\(733\) −37.9485 + 21.9095i −1.40166 + 0.809248i −0.994563 0.104138i \(-0.966792\pi\)
−0.407095 + 0.913386i \(0.633458\pi\)
\(734\) 5.27703 0.194779
\(735\) 0 0
\(736\) −6.17620 −0.227658
\(737\) −51.3786 + 29.6634i −1.89255 + 1.09267i
\(738\) 0 0
\(739\) 6.86403 11.8888i 0.252497 0.437338i −0.711715 0.702468i \(-0.752081\pi\)
0.964213 + 0.265130i \(0.0854148\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.79984 9.61374i 0.249630 0.352931i
\(743\) 20.8393i 0.764520i 0.924055 + 0.382260i \(0.124854\pi\)
−0.924055 + 0.382260i \(0.875146\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.5988 + 7.27390i 0.461274 + 0.266316i
\(747\) 0 0
\(748\) 15.2565i 0.557834i
\(749\) 1.59946 17.3339i 0.0584429 0.633369i
\(750\) 0 0
\(751\) 21.8346 + 37.8186i 0.796755 + 1.38002i 0.921719 + 0.387859i \(0.126785\pi\)
−0.124964 + 0.992161i \(0.539881\pi\)
\(752\) 2.07312 3.59075i 0.0755989 0.130941i
\(753\) 0 0
\(754\) 31.7769 18.3464i 1.15725 0.668136i
\(755\) 0 0
\(756\) 0 0
\(757\) −40.4115 −1.46878 −0.734391 0.678727i \(-0.762532\pi\)
−0.734391 + 0.678727i \(0.762532\pi\)
\(758\) −3.17545 + 1.83335i −0.115337 + 0.0665901i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.3292 + 31.7471i 0.664432 + 1.15083i 0.979439 + 0.201741i \(0.0646598\pi\)
−0.315007 + 0.949089i \(0.602007\pi\)
\(762\) 0 0
\(763\) −2.97093 6.45094i −0.107555 0.233540i
\(764\) 5.71839i 0.206884i
\(765\) 0 0
\(766\) −16.9091 9.76247i −0.610951 0.352732i
\(767\) 28.5833 + 16.5025i 1.03208 + 0.595872i
\(768\) 0 0
\(769\) 20.4304i 0.736738i −0.929680 0.368369i \(-0.879916\pi\)
0.929680 0.368369i \(-0.120084\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.11665 + 5.39819i 0.112171 + 0.194285i
\(773\) 6.84657 11.8586i 0.246254 0.426525i −0.716229 0.697865i \(-0.754134\pi\)
0.962483 + 0.271340i \(0.0874669\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.41643 −0.158540
\(777\) 0 0
\(778\) −17.2871 −0.619773
\(779\) −22.5488 + 13.0186i −0.807896 + 0.466439i
\(780\) 0 0
\(781\) −2.41565 + 4.18403i −0.0864388 + 0.149716i
\(782\) 9.49356 + 16.4433i 0.339489 + 0.588012i
\(783\) 0 0
\(784\) −4.55021 + 5.31936i −0.162507 + 0.189977i
\(785\) 0 0
\(786\) 0 0
\(787\) −11.6169 6.70701i −0.414097 0.239079i 0.278451 0.960450i \(-0.410179\pi\)
−0.692549 + 0.721371i \(0.743512\pi\)
\(788\) −1.14518 0.661170i −0.0407953 0.0235532i
\(789\) 0 0
\(790\) 0 0
\(791\) −7.66883 5.42419i −0.272672 0.192862i
\(792\) 0 0
\(793\) −28.7062 49.7206i −1.01939 1.76563i
\(794\) −13.8423 + 23.9755i −0.491244 + 0.850860i
\(795\) 0 0
\(796\) −8.27163 + 4.77563i −0.293180 + 0.169268i
\(797\) 53.8858 1.90873 0.954367 0.298636i \(-0.0965316\pi\)
0.954367 + 0.298636i \(0.0965316\pi\)
\(798\) 0 0
\(799\) −12.7465 −0.450940
\(800\) 0 0
\(801\) 0 0
\(802\) 18.9095 32.7521i 0.667716 1.15652i
\(803\) −41.4344 71.7665i −1.46219 2.53259i
\(804\) 0 0
\(805\) 0 0
\(806\) 6.44864i 0.227144i
\(807\) 0 0
\(808\) −9.00364 5.19825i −0.316747 0.182874i
\(809\) −10.1762 5.87522i −0.357775 0.206562i 0.310329 0.950629i \(-0.399561\pi\)
−0.668104 + 0.744068i \(0.732894\pi\)
\(810\) 0 0
\(811\) 22.8579i 0.802649i −0.915936 0.401325i \(-0.868550\pi\)
0.915936 0.401325i \(-0.131450\pi\)
\(812\) 16.0502 7.39182i 0.563253 0.259402i
\(813\) 0 0
\(814\) 26.5703 + 46.0211i 0.931288 + 1.61304i
\(815\) 0 0
\(816\) 0 0
\(817\) −23.6663 + 13.6637i −0.827978 + 0.478033i
\(818\) −36.4918 −1.27591
\(819\) 0 0
\(820\) 0 0
\(821\) −9.58876 + 5.53607i −0.334650 + 0.193210i −0.657904 0.753102i \(-0.728557\pi\)
0.323254 + 0.946312i \(0.395223\pi\)
\(822\) 0 0
\(823\) 0.353401 0.612108i 0.0123188 0.0213368i −0.859800 0.510630i \(-0.829412\pi\)
0.872119 + 0.489294i \(0.162745\pi\)
\(824\) 5.11942 + 8.86709i 0.178343 + 0.308900i
\(825\) 0 0
\(826\) 12.9767 + 9.17849i 0.451518 + 0.319361i
\(827\) 18.8230i 0.654540i 0.944931 + 0.327270i \(0.106129\pi\)
−0.944931 + 0.327270i \(0.893871\pi\)
\(828\) 0 0
\(829\) −26.7423 15.4397i −0.928800 0.536243i −0.0423683 0.999102i \(-0.513490\pi\)
−0.886432 + 0.462859i \(0.846824\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.49388i 0.190466i
\(833\) 21.1563 + 3.93785i 0.733023 + 0.136438i
\(834\) 0 0
\(835\) 0 0
\(836\) 7.69660 13.3309i 0.266193 0.461059i
\(837\) 0 0
\(838\) −18.7641 + 10.8335i −0.648195 + 0.374236i
\(839\) 20.9932 0.724766 0.362383 0.932029i \(-0.381963\pi\)
0.362383 + 0.932029i \(0.381963\pi\)
\(840\) 0 0
\(841\) −15.6071 −0.538174
\(842\) 7.65733 4.42096i 0.263889 0.152356i
\(843\) 0 0
\(844\) 13.0110 22.5356i 0.447856 0.775709i
\(845\) 0 0
\(846\) 0 0
\(847\) −35.9049 3.31306i −1.23371 0.113838i
\(848\) 4.45071i 0.152838i
\(849\) 0 0
\(850\) 0 0
\(851\) −57.2744 33.0674i −1.96334 1.13353i
\(852\) 0 0
\(853\) 18.2167i 0.623729i −0.950127 0.311864i \(-0.899047\pi\)
0.950127 0.311864i \(-0.100953\pi\)
\(854\) −11.5658 25.1135i −0.395774 0.859365i
\(855\) 0 0
\(856\) −3.28972 5.69797i −0.112440 0.194753i
\(857\) 6.19973 10.7382i 0.211779 0.366812i −0.740493 0.672065i \(-0.765408\pi\)
0.952271 + 0.305253i \(0.0987410\pi\)
\(858\) 0 0
\(859\) 6.67438 3.85345i 0.227727 0.131478i −0.381796 0.924247i \(-0.624694\pi\)
0.609523 + 0.792768i \(0.291361\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −29.7313 −1.01265
\(863\) −6.54406 + 3.77821i −0.222762 + 0.128612i −0.607229 0.794527i \(-0.707719\pi\)
0.384466 + 0.923139i \(0.374385\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13.2833 + 23.0074i 0.451386 + 0.781823i
\(867\) 0 0
\(868\) 0.285346 3.09241i 0.00968528 0.104963i
\(869\) 21.0744i 0.714900i
\(870\) 0 0
\(871\) −56.8778 32.8384i −1.92723 1.11269i
\(872\) −2.32474 1.34219i −0.0787255 0.0454522i
\(873\) 0 0
\(874\) 19.1572i 0.648002i
\(875\) 0 0
\(876\) 0 0
\(877\) 15.0598 + 26.0843i 0.508532 + 0.880803i 0.999951 + 0.00987971i \(0.00314486\pi\)
−0.491420 + 0.870923i \(0.663522\pi\)
\(878\) −8.31774 + 14.4067i −0.280710 + 0.486204i
\(879\) 0 0
\(880\) 0 0
\(881\) 29.6642 0.999411 0.499706 0.866195i \(-0.333442\pi\)
0.499706 + 0.866195i \(0.333442\pi\)
\(882\) 0 0
\(883\) 38.9643 1.31125 0.655626 0.755086i \(-0.272405\pi\)
0.655626 + 0.755086i \(0.272405\pi\)
\(884\) 14.6267 8.44475i 0.491950 0.284028i
\(885\) 0 0
\(886\) 1.63637 2.83428i 0.0549750 0.0952195i
\(887\) −10.3423 17.9134i −0.347260 0.601472i 0.638502 0.769620i \(-0.279554\pi\)
−0.985762 + 0.168149i \(0.946221\pi\)
\(888\) 0 0
\(889\) 8.42690 11.9141i 0.282629 0.399586i
\(890\) 0 0
\(891\) 0 0
\(892\) 6.28060 + 3.62611i 0.210290 + 0.121411i
\(893\) −11.1377 6.43036i −0.372709 0.215184i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.243099 + 2.63456i −0.00812137 + 0.0880144i
\(897\) 0 0
\(898\) −5.21608 9.03451i −0.174063 0.301485i
\(899\) −3.91977 + 6.78924i −0.130732 + 0.226434i
\(900\) 0 0
\(901\) −11.8494 + 6.84127i −0.394762 + 0.227916i
\(902\) 41.6583 1.38707
\(903\) 0 0
\(904\) −3.55031 −0.118081
\(905\) 0 0
\(906\) 0 0
\(907\) 6.06658 10.5076i 0.201437 0.348900i −0.747554 0.664201i \(-0.768772\pi\)
0.948992 + 0.315301i \(0.102105\pi\)
\(908\) −13.3604 23.1409i −0.443381 0.767959i
\(909\) 0 0
\(910\) 0 0
\(911\) 24.5869i 0.814600i 0.913294 + 0.407300i \(0.133530\pi\)
−0.913294 + 0.407300i \(0.866470\pi\)
\(912\) 0 0
\(913\) 61.3908 + 35.4440i 2.03174 + 1.17302i
\(914\) 28.6471 + 16.5394i 0.947561 + 0.547074i
\(915\) 0 0
\(916\) 24.3034i 0.803006i
\(917\) −54.3078 5.01115i −1.79340 0.165483i
\(918\) 0 0
\(919\) 11.5230 + 19.9585i 0.380110 + 0.658369i 0.991078 0.133286i \(-0.0425529\pi\)
−0.610968 + 0.791655i \(0.709220\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.0146 + 5.78196i −0.329815 + 0.190419i
\(923\) −5.34841 −0.176045
\(924\) 0 0
\(925\) 0 0
\(926\) 33.4339 19.3031i 1.09871 0.634338i
\(927\) 0 0
\(928\) 3.33943 5.78405i 0.109622 0.189871i
\(929\) −20.1685 34.9328i −0.661706 1.14611i −0.980167 0.198173i \(-0.936499\pi\)
0.318461 0.947936i \(-0.396834\pi\)
\(930\) 0 0
\(931\) 16.4995 + 14.1137i 0.540748 + 0.462559i
\(932\) 9.25577i 0.303183i
\(933\) 0 0
\(934\) −4.08230 2.35692i −0.133577 0.0771207i
\(935\) 0 0
\(936\) 0 0
\(937\) 41.4861i 1.35529i −0.735389 0.677646i \(-0.763000\pi\)
0.735389 0.677646i \(-0.237000\pi\)
\(938\) −25.8224 18.2643i −0.843130 0.596350i
\(939\) 0 0
\(940\) 0 0
\(941\) −9.37786 + 16.2429i −0.305710 + 0.529505i −0.977419 0.211310i \(-0.932227\pi\)
0.671709 + 0.740815i \(0.265560\pi\)
\(942\) 0 0
\(943\) −44.8988 + 25.9223i −1.46211 + 0.844148i
\(944\) 6.00761 0.195531
\(945\) 0 0
\(946\) 43.7227 1.42155
\(947\) 37.5699 21.6910i 1.22086 0.704862i 0.255756 0.966741i \(-0.417676\pi\)
0.965101 + 0.261879i \(0.0843422\pi\)
\(948\) 0 0
\(949\) 45.8693 79.4479i 1.48898 2.57899i
\(950\) 0 0
\(951\) 0 0
\(952\) 7.38784 3.40241i 0.239441 0.110273i
\(953\) 18.1672i 0.588492i −0.955730 0.294246i \(-0.904931\pi\)
0.955730 0.294246i \(-0.0950685\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.219422 + 0.126683i 0.00709661 + 0.00409723i
\(957\) 0 0
\(958\) 20.0191i 0.646789i
\(959\) −27.5308 + 12.6791i −0.889015 + 0.409429i
\(960\) 0 0
\(961\) −14.8111 25.6536i −0.477778 0.827536i
\(962\) −29.4142 + 50.9469i −0.948352 + 1.64259i
\(963\) 0 0
\(964\) 2.57538 1.48689i 0.0829473 0.0478896i
\(965\) 0 0
\(966\) 0 0
\(967\) 29.7712 0.957378 0.478689 0.877985i \(-0.341112\pi\)
0.478689 + 0.877985i \(0.341112\pi\)
\(968\) −11.8026 + 6.81421i −0.379349 + 0.219017i
\(969\) 0 0
\(970\) 0 0
\(971\) 18.5472 + 32.1247i 0.595209 + 1.03093i 0.993517 + 0.113680i \(0.0362640\pi\)
−0.398309 + 0.917251i \(0.630403\pi\)
\(972\) 0 0
\(973\) 2.52078 + 1.78296i 0.0808126 + 0.0571591i
\(974\) 4.73088i 0.151587i
\(975\) 0 0
\(976\) −9.05018 5.22512i −0.289689 0.167252i
\(977\) −0.874971 0.505165i −0.0279928 0.0161617i 0.485938 0.873993i \(-0.338478\pi\)
−0.513931 + 0.857831i \(0.671811\pi\)
\(978\) 0 0
\(979\) 73.3008i 2.34270i
\(980\) 0 0
\(981\) 0 0
\(982\) 8.00133 + 13.8587i 0.255333 + 0.442249i
\(983\) −12.6799 + 21.9623i −0.404427 + 0.700487i −0.994255 0.107042i \(-0.965862\pi\)
0.589828 + 0.807529i \(0.299196\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −20.5324 −0.653884
\(987\) 0 0
\(988\) 17.0408 0.542140
\(989\) −47.1238 + 27.2070i −1.49845 + 0.865131i
\(990\) 0 0
\(991\) 21.3875 37.0442i 0.679396 1.17675i −0.295767 0.955260i \(-0.595575\pi\)
0.975163 0.221488i \(-0.0710914\pi\)
\(992\) −0.586893 1.01653i −0.0186339 0.0322748i
\(993\) 0 0
\(994\) −2.56480 0.236662i −0.0813506 0.00750647i
\(995\) 0 0
\(996\) 0 0
\(997\) −21.9808 12.6906i −0.696139 0.401916i 0.109769 0.993957i \(-0.464989\pi\)
−0.805908 + 0.592041i \(0.798322\pi\)
\(998\) 5.50961 + 3.18097i 0.174404 + 0.100692i
\(999\) 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.bf.d.1151.1 24
3.2 odd 2 inner 3150.2.bf.d.1151.12 yes 24
5.2 odd 4 3150.2.bp.g.899.4 24
5.3 odd 4 3150.2.bp.h.899.9 24
5.4 even 2 3150.2.bf.e.1151.12 yes 24
7.5 odd 6 inner 3150.2.bf.d.1601.12 yes 24
15.2 even 4 3150.2.bp.h.899.4 24
15.8 even 4 3150.2.bp.g.899.9 24
15.14 odd 2 3150.2.bf.e.1151.1 yes 24
21.5 even 6 inner 3150.2.bf.d.1601.1 yes 24
35.12 even 12 3150.2.bp.g.1349.9 24
35.19 odd 6 3150.2.bf.e.1601.1 yes 24
35.33 even 12 3150.2.bp.h.1349.4 24
105.47 odd 12 3150.2.bp.h.1349.9 24
105.68 odd 12 3150.2.bp.g.1349.4 24
105.89 even 6 3150.2.bf.e.1601.12 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.bf.d.1151.1 24 1.1 even 1 trivial
3150.2.bf.d.1151.12 yes 24 3.2 odd 2 inner
3150.2.bf.d.1601.1 yes 24 21.5 even 6 inner
3150.2.bf.d.1601.12 yes 24 7.5 odd 6 inner
3150.2.bf.e.1151.1 yes 24 15.14 odd 2
3150.2.bf.e.1151.12 yes 24 5.4 even 2
3150.2.bf.e.1601.1 yes 24 35.19 odd 6
3150.2.bf.e.1601.12 yes 24 105.89 even 6
3150.2.bp.g.899.4 24 5.2 odd 4
3150.2.bp.g.899.9 24 15.8 even 4
3150.2.bp.g.1349.4 24 105.68 odd 12
3150.2.bp.g.1349.9 24 35.12 even 12
3150.2.bp.h.899.4 24 15.2 even 4
3150.2.bp.h.899.9 24 5.3 odd 4
3150.2.bp.h.1349.4 24 35.33 even 12
3150.2.bp.h.1349.9 24 105.47 odd 12