Properties

Label 1728.2.z.a.719.20
Level $1728$
Weight $2$
Character 1728.719
Analytic conductor $13.798$
Analytic rank $0$
Dimension $88$
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] = [1728,2,Mod(143,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.z (of order \(12\), degree \(4\), 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 719.20
Character \(\chi\) \(=\) 1728.719
Dual form 1728.2.z.a.1007.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.759273 + 2.83365i) q^{5} +(-1.41719 - 2.45465i) q^{7} +O(q^{10})\) \(q+(0.759273 + 2.83365i) q^{5} +(-1.41719 - 2.45465i) q^{7} +(0.212907 - 0.794578i) q^{11} +(-0.864050 - 3.22468i) q^{13} +7.28589i q^{17} +(0.951758 + 0.951758i) q^{19} +(5.13217 + 2.96306i) q^{23} +(-3.12293 + 1.80302i) q^{25} +(0.473770 - 1.76813i) q^{29} +(2.05610 + 1.18709i) q^{31} +(5.87957 - 5.87957i) q^{35} +(6.03704 + 6.03704i) q^{37} +(-4.60866 + 7.98243i) q^{41} +(1.50209 + 0.402483i) q^{43} +(2.85832 + 4.95075i) q^{47} +(-0.516861 + 0.895230i) q^{49} +(4.50856 - 4.50856i) q^{53} +2.41321 q^{55} +(-10.6553 + 2.85508i) q^{59} +(8.08106 + 2.16531i) q^{61} +(8.48155 - 4.89682i) q^{65} +(-11.1746 + 2.99423i) q^{67} +2.98978i q^{71} +12.7280i q^{73} +(-2.25214 + 0.603459i) q^{77} +(-8.94529 + 5.16457i) q^{79} +(-3.74713 - 1.00404i) q^{83} +(-20.6456 + 5.53198i) q^{85} +9.25017 q^{89} +(-6.69092 + 6.69092i) q^{91} +(-1.97430 + 3.41959i) q^{95} +(0.148868 + 0.257847i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 88 q + 6 q^{5} + 4 q^{7} - 6 q^{11} - 2 q^{13} + 8 q^{19} - 12 q^{23} + 6 q^{29} - 8 q^{37} + 2 q^{43} - 24 q^{49} + 16 q^{55} - 42 q^{59} - 2 q^{61} + 12 q^{65} + 2 q^{67} + 6 q^{77} + 54 q^{83} + 8 q^{85} - 20 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.759273 + 2.83365i 0.339557 + 1.26725i 0.898843 + 0.438270i \(0.144409\pi\)
−0.559286 + 0.828975i \(0.688925\pi\)
\(6\) 0 0
\(7\) −1.41719 2.45465i −0.535648 0.927769i −0.999132 0.0416640i \(-0.986734\pi\)
0.463484 0.886105i \(-0.346599\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.212907 0.794578i 0.0641937 0.239574i −0.926373 0.376608i \(-0.877090\pi\)
0.990566 + 0.137034i \(0.0437569\pi\)
\(12\) 0 0
\(13\) −0.864050 3.22468i −0.239644 0.894365i −0.976000 0.217770i \(-0.930122\pi\)
0.736356 0.676595i \(-0.236545\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.28589i 1.76709i 0.468348 + 0.883544i \(0.344849\pi\)
−0.468348 + 0.883544i \(0.655151\pi\)
\(18\) 0 0
\(19\) 0.951758 + 0.951758i 0.218348 + 0.218348i 0.807802 0.589454i \(-0.200657\pi\)
−0.589454 + 0.807802i \(0.700657\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.13217 + 2.96306i 1.07013 + 0.617841i 0.928218 0.372038i \(-0.121341\pi\)
0.141915 + 0.989879i \(0.454674\pi\)
\(24\) 0 0
\(25\) −3.12293 + 1.80302i −0.624585 + 0.360605i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.473770 1.76813i 0.0879768 0.328334i −0.907884 0.419221i \(-0.862303\pi\)
0.995861 + 0.0908866i \(0.0289701\pi\)
\(30\) 0 0
\(31\) 2.05610 + 1.18709i 0.369286 + 0.213207i 0.673147 0.739509i \(-0.264942\pi\)
−0.303860 + 0.952717i \(0.598276\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.87957 5.87957i 0.993828 0.993828i
\(36\) 0 0
\(37\) 6.03704 + 6.03704i 0.992483 + 0.992483i 0.999972 0.00748933i \(-0.00238395\pi\)
−0.00748933 + 0.999972i \(0.502384\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.60866 + 7.98243i −0.719751 + 1.24665i 0.241347 + 0.970439i \(0.422411\pi\)
−0.961098 + 0.276207i \(0.910923\pi\)
\(42\) 0 0
\(43\) 1.50209 + 0.402483i 0.229066 + 0.0613780i 0.371526 0.928423i \(-0.378835\pi\)
−0.142460 + 0.989801i \(0.545501\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.85832 + 4.95075i 0.416928 + 0.722141i 0.995629 0.0933994i \(-0.0297733\pi\)
−0.578701 + 0.815540i \(0.696440\pi\)
\(48\) 0 0
\(49\) −0.516861 + 0.895230i −0.0738373 + 0.127890i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.50856 4.50856i 0.619298 0.619298i −0.326053 0.945351i \(-0.605719\pi\)
0.945351 + 0.326053i \(0.105719\pi\)
\(54\) 0 0
\(55\) 2.41321 0.325397
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.6553 + 2.85508i −1.38720 + 0.371700i −0.873732 0.486407i \(-0.838307\pi\)
−0.513471 + 0.858107i \(0.671640\pi\)
\(60\) 0 0
\(61\) 8.08106 + 2.16531i 1.03467 + 0.277240i 0.735904 0.677086i \(-0.236757\pi\)
0.298769 + 0.954325i \(0.403424\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.48155 4.89682i 1.05201 0.607376i
\(66\) 0 0
\(67\) −11.1746 + 2.99423i −1.36520 + 0.365804i −0.865723 0.500524i \(-0.833141\pi\)
−0.499476 + 0.866328i \(0.666474\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.98978i 0.354821i 0.984137 + 0.177411i \(0.0567721\pi\)
−0.984137 + 0.177411i \(0.943228\pi\)
\(72\) 0 0
\(73\) 12.7280i 1.48970i 0.667229 + 0.744852i \(0.267480\pi\)
−0.667229 + 0.744852i \(0.732520\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.25214 + 0.603459i −0.256655 + 0.0687705i
\(78\) 0 0
\(79\) −8.94529 + 5.16457i −1.00642 + 0.581059i −0.910143 0.414295i \(-0.864028\pi\)
−0.0962815 + 0.995354i \(0.530695\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.74713 1.00404i −0.411301 0.110208i 0.0472343 0.998884i \(-0.484959\pi\)
−0.458536 + 0.888676i \(0.651626\pi\)
\(84\) 0 0
\(85\) −20.6456 + 5.53198i −2.23933 + 0.600028i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.25017 0.980516 0.490258 0.871577i \(-0.336903\pi\)
0.490258 + 0.871577i \(0.336903\pi\)
\(90\) 0 0
\(91\) −6.69092 + 6.69092i −0.701399 + 0.701399i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.97430 + 3.41959i −0.202559 + 0.350843i
\(96\) 0 0
\(97\) 0.148868 + 0.257847i 0.0151153 + 0.0261804i 0.873484 0.486853i \(-0.161855\pi\)
−0.858369 + 0.513033i \(0.828522\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.90610 2.11843i −0.786687 0.210792i −0.156956 0.987606i \(-0.550168\pi\)
−0.629731 + 0.776814i \(0.716835\pi\)
\(102\) 0 0
\(103\) 5.72414 9.91450i 0.564016 0.976905i −0.433124 0.901334i \(-0.642589\pi\)
0.997140 0.0755707i \(-0.0240779\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.174948 + 0.174948i 0.0169128 + 0.0169128i 0.715513 0.698600i \(-0.246193\pi\)
−0.698600 + 0.715513i \(0.746193\pi\)
\(108\) 0 0
\(109\) 1.50900 1.50900i 0.144536 0.144536i −0.631136 0.775672i \(-0.717411\pi\)
0.775672 + 0.631136i \(0.217411\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.513633 + 0.296546i 0.0483185 + 0.0278967i 0.523965 0.851740i \(-0.324452\pi\)
−0.475646 + 0.879637i \(0.657786\pi\)
\(114\) 0 0
\(115\) −4.49955 + 16.7925i −0.419585 + 1.56591i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 17.8843 10.3255i 1.63945 0.946537i
\(120\) 0 0
\(121\) 8.94025 + 5.16166i 0.812750 + 0.469242i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.89158 + 2.89158i 0.258631 + 0.258631i
\(126\) 0 0
\(127\) 14.1816i 1.25841i −0.777240 0.629205i \(-0.783381\pi\)
0.777240 0.629205i \(-0.216619\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.03481 + 3.86197i 0.0904119 + 0.337422i 0.996284 0.0861298i \(-0.0274500\pi\)
−0.905872 + 0.423551i \(0.860783\pi\)
\(132\) 0 0
\(133\) 0.987407 3.68505i 0.0856190 0.319535i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.98207 + 6.89714i 0.340211 + 0.589262i 0.984472 0.175543i \(-0.0561682\pi\)
−0.644261 + 0.764806i \(0.722835\pi\)
\(138\) 0 0
\(139\) 2.76404 + 10.3155i 0.234443 + 0.874952i 0.978399 + 0.206724i \(0.0662803\pi\)
−0.743957 + 0.668228i \(0.767053\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.74622 −0.229651
\(144\) 0 0
\(145\) 5.36998 0.445953
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.13642 4.24117i −0.0930990 0.347450i 0.903625 0.428324i \(-0.140896\pi\)
−0.996724 + 0.0808734i \(0.974229\pi\)
\(150\) 0 0
\(151\) 6.93783 + 12.0167i 0.564592 + 0.977903i 0.997087 + 0.0762664i \(0.0242999\pi\)
−0.432495 + 0.901636i \(0.642367\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.80265 + 6.72758i −0.144792 + 0.540372i
\(156\) 0 0
\(157\) −1.61187 6.01560i −0.128642 0.480097i 0.871302 0.490748i \(-0.163276\pi\)
−0.999943 + 0.0106509i \(0.996610\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.7969i 1.32378i
\(162\) 0 0
\(163\) −3.39010 3.39010i −0.265533 0.265533i 0.561764 0.827297i \(-0.310123\pi\)
−0.827297 + 0.561764i \(0.810123\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.58610 + 4.95718i 0.664412 + 0.383598i 0.793956 0.607975i \(-0.208018\pi\)
−0.129544 + 0.991574i \(0.541351\pi\)
\(168\) 0 0
\(169\) 1.60636 0.927433i 0.123566 0.0713410i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.94330 7.25249i 0.147746 0.551396i −0.851872 0.523751i \(-0.824532\pi\)
0.999618 0.0276457i \(-0.00880101\pi\)
\(174\) 0 0
\(175\) 8.85157 + 5.11046i 0.669116 + 0.386314i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.604756 0.604756i 0.0452016 0.0452016i −0.684145 0.729346i \(-0.739824\pi\)
0.729346 + 0.684145i \(0.239824\pi\)
\(180\) 0 0
\(181\) −10.1716 10.1716i −0.756048 0.756048i 0.219553 0.975601i \(-0.429540\pi\)
−0.975601 + 0.219553i \(0.929540\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.5231 + 21.6906i −0.920714 + 1.59472i
\(186\) 0 0
\(187\) 5.78921 + 1.55121i 0.423349 + 0.113436i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.80282 10.0508i −0.419877 0.727249i 0.576049 0.817415i \(-0.304594\pi\)
−0.995927 + 0.0901660i \(0.971260\pi\)
\(192\) 0 0
\(193\) 5.09572 8.82605i 0.366798 0.635313i −0.622265 0.782807i \(-0.713787\pi\)
0.989063 + 0.147494i \(0.0471206\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.32430 4.32430i 0.308094 0.308094i −0.536076 0.844170i \(-0.680094\pi\)
0.844170 + 0.536076i \(0.180094\pi\)
\(198\) 0 0
\(199\) 27.7260 1.96544 0.982720 0.185097i \(-0.0592599\pi\)
0.982720 + 0.185097i \(0.0592599\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.01156 + 1.34284i −0.351743 + 0.0942492i
\(204\) 0 0
\(205\) −26.1186 6.99846i −1.82420 0.488793i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.958882 0.553611i 0.0663272 0.0382940i
\(210\) 0 0
\(211\) 19.0391 5.10152i 1.31071 0.351203i 0.465219 0.885196i \(-0.345976\pi\)
0.845489 + 0.533993i \(0.179309\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.56197i 0.311124i
\(216\) 0 0
\(217\) 6.72933i 0.456817i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.4947 6.29538i 1.58042 0.423473i
\(222\) 0 0
\(223\) −10.9074 + 6.29741i −0.730416 + 0.421706i −0.818574 0.574401i \(-0.805235\pi\)
0.0881582 + 0.996106i \(0.471902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.6445 3.12014i −0.772874 0.207091i −0.149233 0.988802i \(-0.547680\pi\)
−0.623641 + 0.781711i \(0.714347\pi\)
\(228\) 0 0
\(229\) −5.47149 + 1.46608i −0.361566 + 0.0968814i −0.435029 0.900417i \(-0.643262\pi\)
0.0734624 + 0.997298i \(0.476595\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.1291 −0.663580 −0.331790 0.943353i \(-0.607653\pi\)
−0.331790 + 0.943353i \(0.607653\pi\)
\(234\) 0 0
\(235\) −11.8584 + 11.8584i −0.773558 + 0.773558i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.263562 0.456503i 0.0170484 0.0295287i −0.857375 0.514692i \(-0.827906\pi\)
0.874424 + 0.485163i \(0.161240\pi\)
\(240\) 0 0
\(241\) 9.74338 + 16.8760i 0.627626 + 1.08708i 0.988027 + 0.154283i \(0.0493068\pi\)
−0.360400 + 0.932798i \(0.617360\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.92920 0.784878i −0.187140 0.0501440i
\(246\) 0 0
\(247\) 2.24675 3.89148i 0.142957 0.247609i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.9199 + 17.9199i 1.13110 + 1.13110i 0.989995 + 0.141101i \(0.0450643\pi\)
0.141101 + 0.989995i \(0.454936\pi\)
\(252\) 0 0
\(253\) 3.44706 3.44706i 0.216715 0.216715i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.6152 8.43809i −0.911671 0.526353i −0.0307025 0.999529i \(-0.509774\pi\)
−0.880968 + 0.473175i \(0.843108\pi\)
\(258\) 0 0
\(259\) 6.26316 23.3744i 0.389174 1.45242i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −18.8065 + 10.8579i −1.15966 + 0.669528i −0.951222 0.308508i \(-0.900170\pi\)
−0.208435 + 0.978036i \(0.566837\pi\)
\(264\) 0 0
\(265\) 16.1989 + 9.35243i 0.995090 + 0.574515i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.50729 + 3.50729i 0.213843 + 0.213843i 0.805898 0.592055i \(-0.201683\pi\)
−0.592055 + 0.805898i \(0.701683\pi\)
\(270\) 0 0
\(271\) 2.25749i 0.137133i 0.997647 + 0.0685665i \(0.0218425\pi\)
−0.997647 + 0.0685665i \(0.978157\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.767751 + 2.86529i 0.0462971 + 0.172783i
\(276\) 0 0
\(277\) 4.75826 17.7581i 0.285896 1.06698i −0.662286 0.749251i \(-0.730414\pi\)
0.948182 0.317727i \(-0.102920\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.38270 7.59106i −0.261450 0.452845i 0.705177 0.709031i \(-0.250867\pi\)
−0.966627 + 0.256186i \(0.917534\pi\)
\(282\) 0 0
\(283\) −4.18474 15.6177i −0.248757 0.928375i −0.971458 0.237213i \(-0.923766\pi\)
0.722700 0.691161i \(-0.242901\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 26.1254 1.54213
\(288\) 0 0
\(289\) −36.0842 −2.12260
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.24540 12.1120i −0.189598 0.707589i −0.993599 0.112962i \(-0.963966\pi\)
0.804001 0.594628i \(-0.202700\pi\)
\(294\) 0 0
\(295\) −16.1806 28.0256i −0.942070 1.63171i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.12047 19.1098i 0.296124 1.10515i
\(300\) 0 0
\(301\) −1.14079 4.25749i −0.0657540 0.245397i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 24.5429i 1.40532i
\(306\) 0 0
\(307\) −18.9168 18.9168i −1.07964 1.07964i −0.996542 0.0830950i \(-0.973519\pi\)
−0.0830950 0.996542i \(-0.526481\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.08666 2.93678i −0.288438 0.166530i 0.348799 0.937197i \(-0.386589\pi\)
−0.637237 + 0.770668i \(0.719923\pi\)
\(312\) 0 0
\(313\) 14.3490 8.28438i 0.811051 0.468261i −0.0362694 0.999342i \(-0.511547\pi\)
0.847321 + 0.531081i \(0.178214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.80531 6.73750i 0.101396 0.378416i −0.896515 0.443013i \(-0.853910\pi\)
0.997911 + 0.0645973i \(0.0205763\pi\)
\(318\) 0 0
\(319\) −1.30405 0.752894i −0.0730128 0.0421540i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.93441 + 6.93441i −0.385841 + 0.385841i
\(324\) 0 0
\(325\) 8.51253 + 8.51253i 0.472190 + 0.472190i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.10156 14.0323i 0.446653 0.773626i
\(330\) 0 0
\(331\) 21.7853 + 5.83736i 1.19743 + 0.320850i 0.801819 0.597567i \(-0.203866\pi\)
0.395612 + 0.918418i \(0.370533\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.9692 29.3915i −0.927126 1.60583i
\(336\) 0 0
\(337\) 5.04210 8.73317i 0.274661 0.475726i −0.695389 0.718634i \(-0.744768\pi\)
0.970049 + 0.242908i \(0.0781012\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.38099 1.38099i 0.0747849 0.0747849i
\(342\) 0 0
\(343\) −16.9107 −0.913093
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.7569 + 7.70540i −1.54375 + 0.413647i −0.927476 0.373882i \(-0.878026\pi\)
−0.616277 + 0.787530i \(0.711360\pi\)
\(348\) 0 0
\(349\) 8.65259 + 2.31845i 0.463163 + 0.124104i 0.482852 0.875702i \(-0.339601\pi\)
−0.0196892 + 0.999806i \(0.506268\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.1794 + 16.2694i −1.49984 + 0.865933i −1.00000 0.000184008i \(-0.999941\pi\)
−0.499841 + 0.866117i \(0.666608\pi\)
\(354\) 0 0
\(355\) −8.47197 + 2.27006i −0.449645 + 0.120482i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.581843i 0.0307085i 0.999882 + 0.0153542i \(0.00488760\pi\)
−0.999882 + 0.0153542i \(0.995112\pi\)
\(360\) 0 0
\(361\) 17.1883i 0.904648i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −36.0668 + 9.66406i −1.88782 + 0.505840i
\(366\) 0 0
\(367\) 3.68240 2.12603i 0.192220 0.110978i −0.400802 0.916165i \(-0.631268\pi\)
0.593021 + 0.805187i \(0.297935\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −17.4564 4.67743i −0.906291 0.242840i
\(372\) 0 0
\(373\) 6.02298 1.61385i 0.311858 0.0835621i −0.0994955 0.995038i \(-0.531723\pi\)
0.411353 + 0.911476i \(0.365056\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.11102 −0.314733
\(378\) 0 0
\(379\) 8.99791 8.99791i 0.462192 0.462192i −0.437182 0.899373i \(-0.644023\pi\)
0.899373 + 0.437182i \(0.144023\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.35960 + 7.55105i −0.222765 + 0.385840i −0.955647 0.294516i \(-0.904842\pi\)
0.732882 + 0.680356i \(0.238175\pi\)
\(384\) 0 0
\(385\) −3.41998 5.92357i −0.174298 0.301893i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.9041 7.20893i −1.36409 0.365507i −0.498774 0.866732i \(-0.666216\pi\)
−0.865317 + 0.501225i \(0.832883\pi\)
\(390\) 0 0
\(391\) −21.5886 + 37.3925i −1.09178 + 1.89102i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −21.4265 21.4265i −1.07808 1.07808i
\(396\) 0 0
\(397\) 3.82319 3.82319i 0.191880 0.191880i −0.604628 0.796508i \(-0.706678\pi\)
0.796508 + 0.604628i \(0.206678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.9442 + 14.4015i 1.24565 + 0.719178i 0.970239 0.242148i \(-0.0778518\pi\)
0.275414 + 0.961326i \(0.411185\pi\)
\(402\) 0 0
\(403\) 2.05141 7.65596i 0.102188 0.381371i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.08222 3.51157i 0.301485 0.174062i
\(408\) 0 0
\(409\) −9.11179 5.26069i −0.450549 0.260125i 0.257513 0.966275i \(-0.417097\pi\)
−0.708062 + 0.706150i \(0.750430\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 22.1088 + 22.1088i 1.08790 + 1.08790i
\(414\) 0 0
\(415\) 11.3804i 0.558642i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.32320 12.4024i −0.162349 0.605895i −0.998363 0.0571875i \(-0.981787\pi\)
0.836014 0.548708i \(-0.184880\pi\)
\(420\) 0 0
\(421\) 2.46909 9.21476i 0.120336 0.449100i −0.879295 0.476278i \(-0.841985\pi\)
0.999631 + 0.0271782i \(0.00865216\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −13.1366 22.7533i −0.637220 1.10370i
\(426\) 0 0
\(427\) −6.13732 22.9048i −0.297006 1.10844i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.4451 0.792131 0.396065 0.918222i \(-0.370375\pi\)
0.396065 + 0.918222i \(0.370375\pi\)
\(432\) 0 0
\(433\) −3.51715 −0.169023 −0.0845116 0.996422i \(-0.526933\pi\)
−0.0845116 + 0.996422i \(0.526933\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.06447 + 7.70471i 0.0987570 + 0.368566i
\(438\) 0 0
\(439\) −10.4483 18.0969i −0.498668 0.863719i 0.501331 0.865256i \(-0.332844\pi\)
−0.999999 + 0.00153725i \(0.999511\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.86151 10.6793i 0.135954 0.507389i −0.864038 0.503427i \(-0.832072\pi\)
0.999992 0.00396162i \(-0.00126103\pi\)
\(444\) 0 0
\(445\) 7.02341 + 26.2117i 0.332941 + 1.24255i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17.5860i 0.829933i −0.909837 0.414966i \(-0.863793\pi\)
0.909837 0.414966i \(-0.136207\pi\)
\(450\) 0 0
\(451\) 5.36145 + 5.36145i 0.252461 + 0.252461i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −24.0399 13.8795i −1.12701 0.650680i
\(456\) 0 0
\(457\) −18.6606 + 10.7737i −0.872905 + 0.503972i −0.868313 0.496017i \(-0.834795\pi\)
−0.00459262 + 0.999989i \(0.501462\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.55312 16.9925i 0.212060 0.791419i −0.775121 0.631813i \(-0.782311\pi\)
0.987181 0.159606i \(-0.0510223\pi\)
\(462\) 0 0
\(463\) 18.4389 + 10.6457i 0.856930 + 0.494749i 0.862983 0.505233i \(-0.168593\pi\)
−0.00605328 + 0.999982i \(0.501927\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.62864 2.62864i 0.121639 0.121639i −0.643667 0.765306i \(-0.722588\pi\)
0.765306 + 0.643667i \(0.222588\pi\)
\(468\) 0 0
\(469\) 23.1864 + 23.1864i 1.07065 + 1.07065i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.639608 1.10783i 0.0294092 0.0509382i
\(474\) 0 0
\(475\) −4.68831 1.25623i −0.215115 0.0576398i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.7530 + 18.6247i 0.491317 + 0.850986i 0.999950 0.00999740i \(-0.00318233\pi\)
−0.508633 + 0.860983i \(0.669849\pi\)
\(480\) 0 0
\(481\) 14.2512 24.6838i 0.649799 1.12548i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.617616 + 0.617616i −0.0280445 + 0.0280445i
\(486\) 0 0
\(487\) 14.6445 0.663607 0.331804 0.943348i \(-0.392343\pi\)
0.331804 + 0.943348i \(0.392343\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 42.2317 11.3160i 1.90589 0.510682i 0.910658 0.413161i \(-0.135575\pi\)
0.995233 0.0975209i \(-0.0310913\pi\)
\(492\) 0 0
\(493\) 12.8824 + 3.45183i 0.580195 + 0.155463i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.33885 4.23708i 0.329192 0.190059i
\(498\) 0 0
\(499\) −7.91419 + 2.12060i −0.354288 + 0.0949311i −0.431573 0.902078i \(-0.642041\pi\)
0.0772856 + 0.997009i \(0.475375\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.90408i 0.0848988i −0.999099 0.0424494i \(-0.986484\pi\)
0.999099 0.0424494i \(-0.0135161\pi\)
\(504\) 0 0
\(505\) 24.0116i 1.06850i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.27785 1.95009i 0.322585 0.0864364i −0.0938926 0.995582i \(-0.529931\pi\)
0.416478 + 0.909146i \(0.363264\pi\)
\(510\) 0 0
\(511\) 31.2428 18.0381i 1.38210 0.797957i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 32.4404 + 8.69237i 1.42949 + 0.383032i
\(516\) 0 0
\(517\) 4.54231 1.21711i 0.199771 0.0535284i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.5974 1.29669 0.648343 0.761349i \(-0.275462\pi\)
0.648343 + 0.761349i \(0.275462\pi\)
\(522\) 0 0
\(523\) −29.1710 + 29.1710i −1.27556 + 1.27556i −0.332431 + 0.943128i \(0.607869\pi\)
−0.943128 + 0.332431i \(0.892131\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.64900 + 14.9805i −0.376756 + 0.652561i
\(528\) 0 0
\(529\) 6.05948 + 10.4953i 0.263456 + 0.456318i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29.7229 + 7.96422i 1.28744 + 0.344969i
\(534\) 0 0
\(535\) −0.362907 + 0.628573i −0.0156898 + 0.0271756i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.601287 + 0.601287i 0.0258993 + 0.0258993i
\(540\) 0 0
\(541\) 7.87135 7.87135i 0.338416 0.338416i −0.517355 0.855771i \(-0.673083\pi\)
0.855771 + 0.517355i \(0.173083\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.42173 + 3.13024i 0.232241 + 0.134085i
\(546\) 0 0
\(547\) 0.817385 3.05052i 0.0349488 0.130431i −0.946247 0.323444i \(-0.895159\pi\)
0.981196 + 0.193013i \(0.0618260\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.13375 1.23192i 0.0909007 0.0524816i
\(552\) 0 0
\(553\) 25.3544 + 14.6384i 1.07818 + 0.622486i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −22.4110 22.4110i −0.949583 0.949583i 0.0492055 0.998789i \(-0.484331\pi\)
−0.998789 + 0.0492055i \(0.984331\pi\)
\(558\) 0 0
\(559\) 5.19151i 0.219577i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.11646 30.2910i −0.342068 1.27661i −0.896000 0.444053i \(-0.853540\pi\)
0.553933 0.832561i \(-0.313127\pi\)
\(564\) 0 0
\(565\) −0.450319 + 1.68061i −0.0189450 + 0.0707039i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.78787 4.82873i −0.116874 0.202431i 0.801654 0.597789i \(-0.203954\pi\)
−0.918527 + 0.395358i \(0.870621\pi\)
\(570\) 0 0
\(571\) −5.39076 20.1186i −0.225596 0.841937i −0.982165 0.188023i \(-0.939792\pi\)
0.756568 0.653915i \(-0.226874\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −21.3699 −0.891185
\(576\) 0 0
\(577\) 20.3551 0.847394 0.423697 0.905804i \(-0.360732\pi\)
0.423697 + 0.905804i \(0.360732\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.84584 + 10.6208i 0.118065 + 0.440626i
\(582\) 0 0
\(583\) −2.62250 4.54230i −0.108613 0.188123i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.19230 + 30.5741i −0.338132 + 1.26193i 0.562301 + 0.826933i \(0.309916\pi\)
−0.900433 + 0.434994i \(0.856750\pi\)
\(588\) 0 0
\(589\) 0.827087 + 3.08673i 0.0340795 + 0.127186i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.8148i 0.772633i −0.922366 0.386317i \(-0.873747\pi\)
0.922366 0.386317i \(-0.126253\pi\)
\(594\) 0 0
\(595\) 42.8379 + 42.8379i 1.75618 + 1.75618i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.618276 + 0.356962i 0.0252621 + 0.0145851i 0.512578 0.858641i \(-0.328691\pi\)
−0.487316 + 0.873226i \(0.662024\pi\)
\(600\) 0 0
\(601\) −39.9427 + 23.0610i −1.62930 + 0.940676i −0.644996 + 0.764186i \(0.723141\pi\)
−0.984302 + 0.176490i \(0.943526\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.83822 + 29.2526i −0.318669 + 1.18929i
\(606\) 0 0
\(607\) −25.9512 14.9830i −1.05333 0.608139i −0.129749 0.991547i \(-0.541417\pi\)
−0.923579 + 0.383408i \(0.874750\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.4948 13.4948i 0.545943 0.545943i
\(612\) 0 0
\(613\) −4.31839 4.31839i −0.174418 0.174418i 0.614499 0.788917i \(-0.289358\pi\)
−0.788917 + 0.614499i \(0.789358\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.3245 24.8108i 0.576683 0.998844i −0.419174 0.907906i \(-0.637680\pi\)
0.995857 0.0909377i \(-0.0289864\pi\)
\(618\) 0 0
\(619\) 31.5053 + 8.44181i 1.26630 + 0.339305i 0.828614 0.559821i \(-0.189130\pi\)
0.437690 + 0.899126i \(0.355797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.1093 22.7059i −0.525211 0.909693i
\(624\) 0 0
\(625\) −15.0133 + 26.0039i −0.600533 + 1.04015i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −43.9852 + 43.9852i −1.75380 + 1.75380i
\(630\) 0 0
\(631\) −19.5062 −0.776529 −0.388265 0.921548i \(-0.626925\pi\)
−0.388265 + 0.921548i \(0.626925\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 40.1855 10.7677i 1.59471 0.427302i
\(636\) 0 0
\(637\) 3.33342 + 0.893188i 0.132075 + 0.0353894i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.569361 + 0.328721i −0.0224884 + 0.0129837i −0.511202 0.859461i \(-0.670800\pi\)
0.488714 + 0.872444i \(0.337466\pi\)
\(642\) 0 0
\(643\) −23.4845 + 6.29265i −0.926138 + 0.248158i −0.690207 0.723612i \(-0.742481\pi\)
−0.235931 + 0.971770i \(0.575814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.5984i 0.849122i −0.905399 0.424561i \(-0.860428\pi\)
0.905399 0.424561i \(-0.139572\pi\)
\(648\) 0 0
\(649\) 9.07434i 0.356199i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −38.1014 + 10.2092i −1.49102 + 0.399519i −0.910082 0.414427i \(-0.863982\pi\)
−0.580941 + 0.813946i \(0.697315\pi\)
\(654\) 0 0
\(655\) −10.1577 + 5.86458i −0.396896 + 0.229148i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.5220 + 3.62321i 0.526742 + 0.141140i 0.512383 0.858757i \(-0.328763\pi\)
0.0143586 + 0.999897i \(0.495429\pi\)
\(660\) 0 0
\(661\) −40.1488 + 10.7578i −1.56161 + 0.418432i −0.933172 0.359430i \(-0.882971\pi\)
−0.628436 + 0.777861i \(0.716305\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.1918 0.434001
\(666\) 0 0
\(667\) 7.67055 7.67055i 0.297005 0.297005i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.44102 5.96002i 0.132839 0.230084i
\(672\) 0 0
\(673\) 2.88063 + 4.98940i 0.111040 + 0.192327i 0.916190 0.400744i \(-0.131248\pi\)
−0.805150 + 0.593072i \(0.797915\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.1438 + 7.27316i 1.04322 + 0.279530i 0.739447 0.673214i \(-0.235087\pi\)
0.303773 + 0.952744i \(0.401754\pi\)
\(678\) 0 0
\(679\) 0.421949 0.730837i 0.0161929 0.0280469i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.50875 + 4.50875i 0.172522 + 0.172522i 0.788087 0.615564i \(-0.211072\pi\)
−0.615564 + 0.788087i \(0.711072\pi\)
\(684\) 0 0
\(685\) −16.5206 + 16.5206i −0.631219 + 0.631219i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −18.4343 10.6430i −0.702290 0.405467i
\(690\) 0 0
\(691\) 2.65522 9.90941i 0.101009 0.376972i −0.896853 0.442329i \(-0.854152\pi\)
0.997862 + 0.0653578i \(0.0208189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −27.1319 + 15.6646i −1.02917 + 0.594193i
\(696\) 0 0
\(697\) −58.1591 33.5782i −2.20293 1.27186i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −16.5518 16.5518i −0.625154 0.625154i 0.321691 0.946845i \(-0.395749\pi\)
−0.946845 + 0.321691i \(0.895749\pi\)
\(702\) 0 0
\(703\) 11.4916i 0.433414i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00445 + 22.4089i 0.225821 + 0.842774i
\(708\) 0 0
\(709\) 10.7915 40.2743i 0.405282 1.51253i −0.398253 0.917276i \(-0.630383\pi\)
0.803535 0.595257i \(-0.202950\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.03484 + 12.1847i 0.263457 + 0.456320i
\(714\) 0 0
\(715\) −2.08513 7.78182i −0.0779795 0.291023i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −26.8494 −1.00131 −0.500656 0.865646i \(-0.666908\pi\)
−0.500656 + 0.865646i \(0.666908\pi\)
\(720\) 0 0
\(721\) −32.4488 −1.20846
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.70843 + 6.37597i 0.0634497 + 0.236797i
\(726\) 0 0
\(727\) 12.1681 + 21.0758i 0.451290 + 0.781657i 0.998466 0.0553600i \(-0.0176306\pi\)
−0.547176 + 0.837017i \(0.684297\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.93245 + 10.9440i −0.108460 + 0.404780i
\(732\) 0 0
\(733\) 5.85501 + 21.8512i 0.216260 + 0.807093i 0.985719 + 0.168397i \(0.0538591\pi\)
−0.769459 + 0.638696i \(0.779474\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.51661i 0.350549i
\(738\) 0 0
\(739\) 12.6742 + 12.6742i 0.466230 + 0.466230i 0.900691 0.434461i \(-0.143061\pi\)
−0.434461 + 0.900691i \(0.643061\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.2864 + 14.5991i 0.927667 + 0.535589i 0.886073 0.463546i \(-0.153423\pi\)
0.0415943 + 0.999135i \(0.486756\pi\)
\(744\) 0 0
\(745\) 11.1551 6.44042i 0.408692 0.235959i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.181500 0.677369i 0.00663188 0.0247505i
\(750\) 0 0
\(751\) 23.9864 + 13.8486i 0.875277 + 0.505341i 0.869098 0.494639i \(-0.164700\pi\)
0.00617885 + 0.999981i \(0.498033\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −28.7833 + 28.7833i −1.04753 + 1.04753i
\(756\) 0 0
\(757\) −6.84906 6.84906i −0.248933 0.248933i 0.571599 0.820533i \(-0.306323\pi\)
−0.820533 + 0.571599i \(0.806323\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.92685 11.9977i 0.251098 0.434915i −0.712730 0.701438i \(-0.752542\pi\)
0.963828 + 0.266523i \(0.0858750\pi\)
\(762\) 0 0
\(763\) −5.84262 1.56553i −0.211517 0.0566758i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.4134 + 31.8930i 0.664871 + 1.15159i
\(768\) 0 0
\(769\) 5.15368 8.92644i 0.185846 0.321896i −0.758015 0.652237i \(-0.773831\pi\)
0.943861 + 0.330342i \(0.107164\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29.0684 29.0684i 1.04552 1.04552i 0.0466043 0.998913i \(-0.485160\pi\)
0.998913 0.0466043i \(-0.0148400\pi\)
\(774\) 0 0
\(775\) −8.56139 −0.307534
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.9837 + 3.21101i −0.429359 + 0.115046i
\(780\) 0 0
\(781\) 2.37561 + 0.636543i 0.0850060 + 0.0227773i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.8222 9.13496i 0.564719 0.326041i
\(786\) 0 0
\(787\) 35.9593 9.63526i 1.28181 0.343460i 0.447266 0.894401i \(-0.352398\pi\)
0.834544 + 0.550941i \(0.185731\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.68105i 0.0597712i
\(792\) 0 0
\(793\) 27.9297i 0.991814i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.38318 + 2.51421i −0.332369 + 0.0890580i −0.421144 0.906994i \(-0.638371\pi\)
0.0887752 + 0.996052i \(0.471705\pi\)
\(798\) 0 0
\(799\) −36.0706 + 20.8254i −1.27609 + 0.736749i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.1134 + 2.70988i 0.356895 + 0.0956297i
\(804\) 0 0
\(805\) 47.5965 12.7534i 1.67756 0.449500i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.1867 −0.463620 −0.231810 0.972761i \(-0.574465\pi\)
−0.231810 + 0.972761i \(0.574465\pi\)
\(810\) 0 0
\(811\) 11.0093 11.0093i 0.386587 0.386587i −0.486881 0.873468i \(-0.661865\pi\)
0.873468 + 0.486881i \(0.161865\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.03233 12.1804i 0.246332 0.426659i
\(816\) 0 0
\(817\) 1.04656 + 1.81269i 0.0366144 + 0.0634179i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.17943 + 1.11987i 0.145863 + 0.0390839i 0.331012 0.943627i \(-0.392610\pi\)
−0.185149 + 0.982711i \(0.559277\pi\)
\(822\) 0 0
\(823\) −23.8576 + 41.3225i −0.831622 + 1.44041i 0.0651287 + 0.997877i \(0.479254\pi\)
−0.896751 + 0.442535i \(0.854079\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.38625 1.38625i −0.0482045 0.0482045i 0.682594 0.730798i \(-0.260852\pi\)
−0.730798 + 0.682594i \(0.760852\pi\)
\(828\) 0 0
\(829\) −21.8345 + 21.8345i −0.758345 + 0.758345i −0.976021 0.217676i \(-0.930152\pi\)
0.217676 + 0.976021i \(0.430152\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.52255 3.76580i −0.225993 0.130477i
\(834\) 0 0
\(835\) −7.52771 + 28.0938i −0.260507 + 0.972227i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.8914 + 16.1031i −0.962916 + 0.555940i −0.897069 0.441890i \(-0.854308\pi\)
−0.0658470 + 0.997830i \(0.520975\pi\)
\(840\) 0 0
\(841\) 22.2129 + 12.8246i 0.765962 + 0.442228i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.84768 + 3.84768i 0.132364 + 0.132364i
\(846\) 0 0
\(847\) 29.2602i 1.00539i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.0950 + 48.8712i 0.448891 + 1.67528i
\(852\) 0 0
\(853\) −12.0354 + 44.9166i −0.412083 + 1.53791i 0.378525 + 0.925591i \(0.376431\pi\)
−0.790608 + 0.612323i \(0.790235\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.0139 + 17.3446i 0.342068 + 0.592480i 0.984817 0.173598i \(-0.0555392\pi\)
−0.642748 + 0.766077i \(0.722206\pi\)
\(858\) 0 0
\(859\) 4.47358 + 16.6956i 0.152637 + 0.569648i 0.999296 + 0.0375129i \(0.0119435\pi\)
−0.846659 + 0.532135i \(0.821390\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.4242 −0.831410 −0.415705 0.909499i \(-0.636465\pi\)
−0.415705 + 0.909499i \(0.636465\pi\)
\(864\) 0 0
\(865\) 22.0265 0.748923
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.19914 + 8.20731i 0.0746007 + 0.278414i
\(870\) 0 0
\(871\) 19.3109 + 33.4474i 0.654324 + 1.13332i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.99989 11.1957i 0.101415 0.378485i
\(876\) 0 0
\(877\) −9.26415 34.5743i −0.312828 1.16749i −0.925995 0.377537i \(-0.876771\pi\)
0.613167 0.789954i \(-0.289895\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 45.8264i 1.54393i 0.635664 + 0.771966i \(0.280726\pi\)
−0.635664 + 0.771966i \(0.719274\pi\)
\(882\) 0 0
\(883\) −7.22790 7.22790i −0.243238 0.243238i 0.574950 0.818188i \(-0.305021\pi\)
−0.818188 + 0.574950i \(0.805021\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.9502 21.3332i −1.24067 0.716300i −0.271438 0.962456i \(-0.587499\pi\)
−0.969230 + 0.246156i \(0.920832\pi\)
\(888\) 0 0
\(889\) −34.8107 + 20.0980i −1.16751 + 0.674065i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.99149 + 7.43234i −0.0666426 + 0.248714i
\(894\) 0 0
\(895\) 2.17284 + 1.25449i 0.0726300 + 0.0419330i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.07305 3.07305i 0.102492 0.102492i
\(900\) 0 0
\(901\) 32.8489 + 32.8489i 1.09435 + 1.09435i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.0997 36.5457i 0.701376 1.21482i
\(906\) 0 0
\(907\) −38.9844 10.4458i −1.29445 0.346848i −0.455104 0.890438i \(-0.650398\pi\)
−0.839351 + 0.543590i \(0.817065\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 22.6553 + 39.2402i 0.750605 + 1.30009i 0.947530 + 0.319667i \(0.103571\pi\)
−0.196925 + 0.980419i \(0.563095\pi\)
\(912\) 0 0
\(913\) −1.59558 + 2.76362i −0.0528060 + 0.0914626i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.01324 8.01324i 0.264621 0.264621i
\(918\) 0 0
\(919\) 3.91688 0.129206 0.0646030 0.997911i \(-0.479422\pi\)
0.0646030 + 0.997911i \(0.479422\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.64107 2.58332i 0.317340 0.0850309i
\(924\) 0 0
\(925\) −29.7381 7.96831i −0.977784 0.261996i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.7214 7.92207i 0.450185 0.259915i −0.257723 0.966219i \(-0.582972\pi\)
0.707908 + 0.706304i \(0.249639\pi\)
\(930\) 0 0
\(931\) −1.34397 + 0.360116i −0.0440468 + 0.0118023i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 17.5824i 0.575005i
\(936\) 0 0
\(937\) 38.0380i 1.24265i −0.783554 0.621324i \(-0.786595\pi\)
0.783554 0.621324i \(-0.213405\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 49.4756 13.2570i 1.61286 0.432164i 0.663966 0.747762i \(-0.268872\pi\)
0.948893 + 0.315598i \(0.102205\pi\)
\(942\) 0 0
\(943\) −47.3048 + 27.3115i −1.54046 + 0.889384i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.8269 + 4.24080i 0.514305 + 0.137808i 0.506631 0.862163i \(-0.330891\pi\)
0.00767376 + 0.999971i \(0.497557\pi\)
\(948\) 0 0
\(949\) 41.0438 10.9977i 1.33234 0.356999i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.2107 −0.622295 −0.311147 0.950362i \(-0.600713\pi\)
−0.311147 + 0.950362i \(0.600713\pi\)
\(954\) 0 0
\(955\) 24.0744 24.0744i 0.779030 0.779030i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.2867 19.5491i 0.364466 0.631274i
\(960\) 0 0
\(961\) −12.6816 21.9652i −0.409085 0.708556i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 28.8789 + 7.73809i 0.929646 + 0.249098i
\(966\) 0 0
\(967\) 17.0779 29.5798i 0.549188 0.951222i −0.449142 0.893460i \(-0.648270\pi\)
0.998330 0.0577615i \(-0.0183963\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.2022 + 20.2022i 0.648318 + 0.648318i 0.952586 0.304268i \(-0.0984119\pi\)
−0.304268 + 0.952586i \(0.598412\pi\)
\(972\) 0 0
\(973\) 21.4038 21.4038i 0.686175 0.686175i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.37615 + 4.25862i 0.235984 + 0.136245i 0.613329 0.789827i \(-0.289830\pi\)
−0.377346 + 0.926072i \(0.623163\pi\)
\(978\) 0 0
\(979\) 1.96942 7.34998i 0.0629430 0.234906i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 47.6844 27.5306i 1.52090 0.878090i 0.521200 0.853435i \(-0.325485\pi\)
0.999696 0.0246553i \(-0.00784881\pi\)
\(984\) 0 0
\(985\) 15.5369 + 8.97021i 0.495045 + 0.285815i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.51638 + 6.51638i 0.207209 + 0.207209i
\(990\) 0 0
\(991\) 61.9180i 1.96689i 0.181210 + 0.983444i \(0.441999\pi\)
−0.181210 + 0.983444i \(0.558001\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21.0516 + 78.5655i 0.667380 + 2.49069i
\(996\) 0 0
\(997\) −4.65375 + 17.3680i −0.147386 + 0.550051i 0.852252 + 0.523132i \(0.175236\pi\)
−0.999638 + 0.0269194i \(0.991430\pi\)
\(998\) 0 0
\(999\) 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 1728.2.z.a.719.20 88
3.2 odd 2 576.2.y.a.527.15 88
4.3 odd 2 432.2.v.a.395.12 88
9.2 odd 6 inner 1728.2.z.a.143.20 88
9.7 even 3 576.2.y.a.335.19 88
12.11 even 2 144.2.u.a.59.11 yes 88
16.3 odd 4 inner 1728.2.z.a.1583.20 88
16.13 even 4 432.2.v.a.179.5 88
36.7 odd 6 144.2.u.a.11.18 88
36.11 even 6 432.2.v.a.251.5 88
48.29 odd 4 144.2.u.a.131.18 yes 88
48.35 even 4 576.2.y.a.239.19 88
144.29 odd 12 432.2.v.a.35.12 88
144.61 even 12 144.2.u.a.83.11 yes 88
144.83 even 12 inner 1728.2.z.a.1007.20 88
144.115 odd 12 576.2.y.a.47.15 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.u.a.11.18 88 36.7 odd 6
144.2.u.a.59.11 yes 88 12.11 even 2
144.2.u.a.83.11 yes 88 144.61 even 12
144.2.u.a.131.18 yes 88 48.29 odd 4
432.2.v.a.35.12 88 144.29 odd 12
432.2.v.a.179.5 88 16.13 even 4
432.2.v.a.251.5 88 36.11 even 6
432.2.v.a.395.12 88 4.3 odd 2
576.2.y.a.47.15 88 144.115 odd 12
576.2.y.a.239.19 88 48.35 even 4
576.2.y.a.335.19 88 9.7 even 3
576.2.y.a.527.15 88 3.2 odd 2
1728.2.z.a.143.20 88 9.2 odd 6 inner
1728.2.z.a.719.20 88 1.1 even 1 trivial
1728.2.z.a.1007.20 88 144.83 even 12 inner
1728.2.z.a.1583.20 88 16.3 odd 4 inner