Properties

Label 1300.2.bb.e.1049.2
Level $1300$
Weight $2$
Character 1300.1049
Analytic conductor $10.381$
Analytic rank $0$
Dimension $4$
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] = [1300,2,Mod(549,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1300.549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1049.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1300.1049
Dual form 1300.2.bb.e.549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.59808 + 1.50000i) q^{3} +(-2.59808 + 1.50000i) q^{7} +(3.00000 + 5.19615i) q^{9} +O(q^{10})\) \(q+(2.59808 + 1.50000i) q^{3} +(-2.59808 + 1.50000i) q^{7} +(3.00000 + 5.19615i) q^{9} +(-1.50000 + 2.59808i) q^{11} +(-3.46410 + 1.00000i) q^{13} +(6.06218 - 3.50000i) q^{17} +(0.500000 + 0.866025i) q^{19} -9.00000 q^{21} +(6.06218 + 3.50000i) q^{23} +9.00000i q^{27} +(-2.50000 + 4.33013i) q^{29} -4.00000 q^{31} +(-7.79423 + 4.50000i) q^{33} +(-2.59808 - 1.50000i) q^{37} +(-10.5000 - 2.59808i) q^{39} +(-3.50000 + 6.06218i) q^{41} +(-7.79423 + 4.50000i) q^{43} -8.00000i q^{47} +(1.00000 - 1.73205i) q^{49} +21.0000 q^{51} -6.00000i q^{53} +3.00000i q^{57} +(2.50000 + 4.33013i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-15.5885 - 9.00000i) q^{63} +(11.2583 + 6.50000i) q^{67} +(10.5000 + 18.1865i) q^{69} +(1.50000 + 2.59808i) q^{71} -14.0000i q^{73} -9.00000i q^{77} +8.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000i q^{83} +(-12.9904 + 7.50000i) q^{87} +(3.50000 - 6.06218i) q^{89} +(7.50000 - 7.79423i) q^{91} +(-10.3923 - 6.00000i) q^{93} +(9.52628 - 5.50000i) q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 6 q^{11} + 2 q^{19} - 36 q^{21} - 10 q^{29} - 16 q^{31} - 42 q^{39} - 14 q^{41} + 4 q^{49} + 84 q^{51} + 10 q^{59} + 10 q^{61} + 42 q^{69} + 6 q^{71} + 32 q^{79} - 18 q^{81} + 14 q^{89} + 30 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.59808 + 1.50000i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.59808 + 1.50000i −0.981981 + 0.566947i −0.902867 0.429919i \(-0.858542\pi\)
−0.0791130 + 0.996866i \(0.525209\pi\)
\(8\) 0 0
\(9\) 3.00000 + 5.19615i 1.00000 + 1.73205i
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −3.46410 + 1.00000i −0.960769 + 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.06218 3.50000i 1.47029 0.848875i 0.470850 0.882213i \(-0.343947\pi\)
0.999444 + 0.0333386i \(0.0106140\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) −9.00000 −1.96396
\(22\) 0 0
\(23\) 6.06218 + 3.50000i 1.26405 + 0.729800i 0.973856 0.227167i \(-0.0729463\pi\)
0.290196 + 0.956967i \(0.406280\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) −2.50000 + 4.33013i −0.464238 + 0.804084i −0.999167 0.0408130i \(-0.987005\pi\)
0.534928 + 0.844897i \(0.320339\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −7.79423 + 4.50000i −1.35680 + 0.783349i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.59808 1.50000i −0.427121 0.246598i 0.270998 0.962580i \(-0.412646\pi\)
−0.698119 + 0.715981i \(0.745980\pi\)
\(38\) 0 0
\(39\) −10.5000 2.59808i −1.68135 0.416025i
\(40\) 0 0
\(41\) −3.50000 + 6.06218i −0.546608 + 0.946753i 0.451896 + 0.892071i \(0.350748\pi\)
−0.998504 + 0.0546823i \(0.982585\pi\)
\(42\) 0 0
\(43\) −7.79423 + 4.50000i −1.18861 + 0.686244i −0.957990 0.286801i \(-0.907408\pi\)
−0.230618 + 0.973044i \(0.574075\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 1.00000 1.73205i 0.142857 0.247436i
\(50\) 0 0
\(51\) 21.0000 2.94059
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000i 0.397360i
\(58\) 0 0
\(59\) 2.50000 + 4.33013i 0.325472 + 0.563735i 0.981608 0.190909i \(-0.0611434\pi\)
−0.656136 + 0.754643i \(0.727810\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) −15.5885 9.00000i −1.96396 1.13389i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2583 + 6.50000i 1.37542 + 0.794101i 0.991605 0.129307i \(-0.0412752\pi\)
0.383819 + 0.923408i \(0.374609\pi\)
\(68\) 0 0
\(69\) 10.5000 + 18.1865i 1.26405 + 2.18940i
\(70\) 0 0
\(71\) 1.50000 + 2.59808i 0.178017 + 0.308335i 0.941201 0.337846i \(-0.109698\pi\)
−0.763184 + 0.646181i \(0.776365\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.00000i 1.02565i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.9904 + 7.50000i −1.39272 + 0.804084i
\(88\) 0 0
\(89\) 3.50000 6.06218i 0.370999 0.642590i −0.618720 0.785611i \(-0.712349\pi\)
0.989720 + 0.143022i \(0.0456819\pi\)
\(90\) 0 0
\(91\) 7.50000 7.79423i 0.786214 0.817057i
\(92\) 0 0
\(93\) −10.3923 6.00000i −1.07763 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.52628 5.50000i 0.967247 0.558440i 0.0688512 0.997627i \(-0.478067\pi\)
0.898396 + 0.439187i \(0.144733\pi\)
\(98\) 0 0
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) 4.50000 7.79423i 0.447767 0.775555i −0.550474 0.834853i \(-0.685553\pi\)
0.998240 + 0.0592978i \(0.0188862\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.59808 1.50000i −0.251166 0.145010i 0.369132 0.929377i \(-0.379655\pi\)
−0.620298 + 0.784366i \(0.712988\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −4.50000 7.79423i −0.427121 0.739795i
\(112\) 0 0
\(113\) 11.2583 6.50000i 1.05909 0.611469i 0.133913 0.990993i \(-0.457246\pi\)
0.925182 + 0.379525i \(0.123912\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −15.5885 15.0000i −1.44115 1.38675i
\(118\) 0 0
\(119\) −10.5000 + 18.1865i −0.962533 + 1.66716i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) −18.1865 + 10.5000i −1.63982 + 0.946753i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.866025 + 0.500000i 0.0768473 + 0.0443678i 0.537931 0.842989i \(-0.319206\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) −27.0000 −2.37722
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −2.59808 1.50000i −0.225282 0.130066i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.59808 1.50000i 0.221969 0.128154i −0.384893 0.922961i \(-0.625762\pi\)
0.606861 + 0.794808i \(0.292428\pi\)
\(138\) 0 0
\(139\) 6.50000 + 11.2583i 0.551323 + 0.954919i 0.998179 + 0.0603135i \(0.0192101\pi\)
−0.446857 + 0.894606i \(0.647457\pi\)
\(140\) 0 0
\(141\) 12.0000 20.7846i 1.01058 1.75038i
\(142\) 0 0
\(143\) 2.59808 10.5000i 0.217262 0.878054i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.19615 3.00000i 0.428571 0.247436i
\(148\) 0 0
\(149\) 1.50000 + 2.59808i 0.122885 + 0.212843i 0.920904 0.389789i \(-0.127452\pi\)
−0.798019 + 0.602632i \(0.794119\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 36.3731 + 21.0000i 2.94059 + 1.69775i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000i 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 0 0
\(159\) 9.00000 15.5885i 0.713746 1.23625i
\(160\) 0 0
\(161\) −21.0000 −1.65503
\(162\) 0 0
\(163\) 9.52628 5.50000i 0.746156 0.430793i −0.0781474 0.996942i \(-0.524900\pi\)
0.824303 + 0.566149i \(0.191567\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.866025 + 0.500000i 0.0670151 + 0.0386912i 0.533133 0.846031i \(-0.321014\pi\)
−0.466118 + 0.884723i \(0.654348\pi\)
\(168\) 0 0
\(169\) 11.0000 6.92820i 0.846154 0.532939i
\(170\) 0 0
\(171\) −3.00000 + 5.19615i −0.229416 + 0.397360i
\(172\) 0 0
\(173\) −12.9904 + 7.50000i −0.987640 + 0.570214i −0.904568 0.426329i \(-0.859807\pi\)
−0.0830722 + 0.996544i \(0.526473\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.0000i 1.12747i
\(178\) 0 0
\(179\) 9.50000 16.4545i 0.710063 1.22987i −0.254770 0.967002i \(-0.582000\pi\)
0.964833 0.262864i \(-0.0846670\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 15.0000i 1.10883i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 21.0000i 1.53567i
\(188\) 0 0
\(189\) −13.5000 23.3827i −0.981981 1.70084i
\(190\) 0 0
\(191\) 1.50000 + 2.59808i 0.108536 + 0.187990i 0.915177 0.403051i \(-0.132050\pi\)
−0.806641 + 0.591041i \(0.798717\pi\)
\(192\) 0 0
\(193\) 12.9904 + 7.50000i 0.935068 + 0.539862i 0.888411 0.459049i \(-0.151810\pi\)
0.0466572 + 0.998911i \(0.485143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −19.9186 11.5000i −1.41914 0.819341i −0.422917 0.906168i \(-0.638994\pi\)
−0.996223 + 0.0868274i \(0.972327\pi\)
\(198\) 0 0
\(199\) 4.50000 + 7.79423i 0.318997 + 0.552518i 0.980279 0.197619i \(-0.0633208\pi\)
−0.661282 + 0.750137i \(0.729987\pi\)
\(200\) 0 0
\(201\) 19.5000 + 33.7750i 1.37542 + 2.38230i
\(202\) 0 0
\(203\) 15.0000i 1.05279i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 42.0000i 2.91920i
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 2.50000 4.33013i 0.172107 0.298098i −0.767049 0.641588i \(-0.778276\pi\)
0.939156 + 0.343490i \(0.111609\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.3923 6.00000i 0.705476 0.407307i
\(218\) 0 0
\(219\) 21.0000 36.3731i 1.41905 2.45786i
\(220\) 0 0
\(221\) −17.5000 + 18.1865i −1.17718 + 1.22336i
\(222\) 0 0
\(223\) 19.9186 + 11.5000i 1.33385 + 0.770097i 0.985887 0.167412i \(-0.0535411\pi\)
0.347960 + 0.937509i \(0.386874\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.866025 0.500000i 0.0574801 0.0331862i −0.470985 0.882141i \(-0.656101\pi\)
0.528465 + 0.848955i \(0.322768\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 13.5000 23.3827i 0.888235 1.53847i
\(232\) 0 0
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 20.7846 + 12.0000i 1.35011 + 0.779484i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.0322078 + 0.0557856i 0.881680 0.471848i \(-0.156413\pi\)
−0.849472 + 0.527633i \(0.823079\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.59808 2.50000i −0.165312 0.159071i
\(248\) 0 0
\(249\) −18.0000 + 31.1769i −1.14070 + 1.97576i
\(250\) 0 0
\(251\) −2.50000 4.33013i −0.157799 0.273315i 0.776276 0.630393i \(-0.217106\pi\)
−0.934075 + 0.357078i \(0.883773\pi\)
\(252\) 0 0
\(253\) −18.1865 + 10.5000i −1.14338 + 0.660129i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.4545 9.50000i −1.02640 0.592594i −0.110450 0.993882i \(-0.535229\pi\)
−0.915952 + 0.401288i \(0.868563\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) 6.06218 + 3.50000i 0.373810 + 0.215819i 0.675122 0.737706i \(-0.264091\pi\)
−0.301312 + 0.953526i \(0.597424\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 18.1865 10.5000i 1.11300 0.642590i
\(268\) 0 0
\(269\) 1.50000 + 2.59808i 0.0914566 + 0.158408i 0.908124 0.418701i \(-0.137514\pi\)
−0.816668 + 0.577108i \(0.804181\pi\)
\(270\) 0 0
\(271\) −11.5000 + 19.9186i −0.698575 + 1.20997i 0.270385 + 0.962752i \(0.412849\pi\)
−0.968960 + 0.247216i \(0.920484\pi\)
\(272\) 0 0
\(273\) 31.1769 9.00000i 1.88691 0.544705i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.9186 11.5000i 1.19679 0.690968i 0.236953 0.971521i \(-0.423851\pi\)
0.959839 + 0.280553i \(0.0905179\pi\)
\(278\) 0 0
\(279\) −12.0000 20.7846i −0.718421 1.24434i
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −0.866025 0.500000i −0.0514799 0.0297219i 0.474039 0.880504i \(-0.342796\pi\)
−0.525519 + 0.850782i \(0.676129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.0000i 1.23959i
\(288\) 0 0
\(289\) 16.0000 27.7128i 0.941176 1.63017i
\(290\) 0 0
\(291\) 33.0000 1.93449
\(292\) 0 0
\(293\) 7.79423 4.50000i 0.455344 0.262893i −0.254741 0.967009i \(-0.581990\pi\)
0.710084 + 0.704117i \(0.248657\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −23.3827 13.5000i −1.35680 0.783349i
\(298\) 0 0
\(299\) −24.5000 6.06218i −1.41687 0.350585i
\(300\) 0 0
\(301\) 13.5000 23.3827i 0.778127 1.34776i
\(302\) 0 0
\(303\) 23.3827 13.5000i 1.34330 0.775555i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000i 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) 24.0000 41.5692i 1.36531 2.36479i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) −7.50000 12.9904i −0.419919 0.727322i
\(320\) 0 0
\(321\) −4.50000 7.79423i −0.251166 0.435031i
\(322\) 0 0
\(323\) 6.06218 + 3.50000i 0.337309 + 0.194745i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 36.3731 + 21.0000i 2.01144 + 1.16130i
\(328\) 0 0
\(329\) 12.0000 + 20.7846i 0.661581 + 1.14589i
\(330\) 0 0
\(331\) −6.50000 11.2583i −0.357272 0.618814i 0.630232 0.776407i \(-0.282960\pi\)
−0.987504 + 0.157593i \(0.949627\pi\)
\(332\) 0 0
\(333\) 18.0000i 0.986394i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 39.0000 2.11819
\(340\) 0 0
\(341\) 6.00000 10.3923i 0.324918 0.562775i
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.2583 6.50000i 0.604379 0.348938i −0.166383 0.986061i \(-0.553209\pi\)
0.770762 + 0.637123i \(0.219876\pi\)
\(348\) 0 0
\(349\) −12.5000 + 21.6506i −0.669110 + 1.15893i 0.309044 + 0.951048i \(0.399991\pi\)
−0.978153 + 0.207884i \(0.933342\pi\)
\(350\) 0 0
\(351\) −9.00000 31.1769i −0.480384 1.66410i
\(352\) 0 0
\(353\) −18.1865 10.5000i −0.967972 0.558859i −0.0693543 0.997592i \(-0.522094\pi\)
−0.898617 + 0.438733i \(0.855427\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −54.5596 + 31.5000i −2.88760 + 1.66716i
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 9.00000 15.5885i 0.473684 0.820445i
\(362\) 0 0
\(363\) 6.00000i 0.314918i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.79423 + 4.50000i 0.406855 + 0.234898i 0.689438 0.724345i \(-0.257858\pi\)
−0.282582 + 0.959243i \(0.591191\pi\)
\(368\) 0 0
\(369\) −42.0000 −2.18643
\(370\) 0 0
\(371\) 9.00000 + 15.5885i 0.467257 + 0.809312i
\(372\) 0 0
\(373\) −23.3827 + 13.5000i −1.21071 + 0.699004i −0.962914 0.269809i \(-0.913039\pi\)
−0.247796 + 0.968812i \(0.579706\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.33013 17.5000i 0.223013 0.901296i
\(378\) 0 0
\(379\) −4.50000 + 7.79423i −0.231149 + 0.400363i −0.958147 0.286278i \(-0.907582\pi\)
0.726997 + 0.686640i \(0.240915\pi\)
\(380\) 0 0
\(381\) 1.50000 + 2.59808i 0.0768473 + 0.133103i
\(382\) 0 0
\(383\) −11.2583 + 6.50000i −0.575274 + 0.332134i −0.759253 0.650796i \(-0.774435\pi\)
0.183979 + 0.982930i \(0.441102\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −46.7654 27.0000i −2.37722 1.37249i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 49.0000 2.47804
\(392\) 0 0
\(393\) 10.3923 + 6.00000i 0.524222 + 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −28.5788 + 16.5000i −1.43433 + 0.828111i −0.997447 0.0714068i \(-0.977251\pi\)
−0.436884 + 0.899518i \(0.643918\pi\)
\(398\) 0 0
\(399\) −4.50000 7.79423i −0.225282 0.390199i
\(400\) 0 0
\(401\) −7.50000 + 12.9904i −0.374532 + 0.648709i −0.990257 0.139253i \(-0.955530\pi\)
0.615725 + 0.787961i \(0.288863\pi\)
\(402\) 0 0
\(403\) 13.8564 4.00000i 0.690237 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.79423 4.50000i 0.386346 0.223057i
\(408\) 0 0
\(409\) −0.500000 0.866025i −0.0247234 0.0428222i 0.853399 0.521258i \(-0.174537\pi\)
−0.878122 + 0.478436i \(0.841204\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) −12.9904 7.50000i −0.639215 0.369051i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 39.0000i 1.90984i
\(418\) 0 0
\(419\) −16.5000 + 28.5788i −0.806078 + 1.39617i 0.109483 + 0.993989i \(0.465080\pi\)
−0.915561 + 0.402179i \(0.868253\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 41.5692 24.0000i 2.02116 1.16692i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.9904 7.50000i −0.628649 0.362950i
\(428\) 0 0
\(429\) 22.5000 23.3827i 1.08631 1.12893i
\(430\) 0 0
\(431\) 4.50000 7.79423i 0.216757 0.375435i −0.737057 0.675830i \(-0.763785\pi\)
0.953815 + 0.300395i \(0.0971186\pi\)
\(432\) 0 0
\(433\) 0.866025 0.500000i 0.0416185 0.0240285i −0.479046 0.877790i \(-0.659017\pi\)
0.520665 + 0.853761i \(0.325684\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.00000i 0.334855i
\(438\) 0 0
\(439\) 1.50000 2.59808i 0.0715911 0.123999i −0.828008 0.560717i \(-0.810526\pi\)
0.899599 + 0.436717i \(0.143859\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.00000i 0.425685i
\(448\) 0 0
\(449\) −10.5000 18.1865i −0.495526 0.858276i 0.504461 0.863434i \(-0.331691\pi\)
−0.999987 + 0.00515887i \(0.998358\pi\)
\(450\) 0 0
\(451\) −10.5000 18.1865i −0.494426 0.856370i
\(452\) 0 0
\(453\) −20.7846 12.0000i −0.976546 0.563809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.9186 11.5000i −0.931752 0.537947i −0.0443868 0.999014i \(-0.514133\pi\)
−0.887365 + 0.461067i \(0.847467\pi\)
\(458\) 0 0
\(459\) 31.5000 + 54.5596i 1.47029 + 2.54662i
\(460\) 0 0
\(461\) −5.50000 9.52628i −0.256161 0.443683i 0.709050 0.705159i \(-0.249124\pi\)
−0.965210 + 0.261476i \(0.915791\pi\)
\(462\) 0 0
\(463\) 28.0000i 1.30127i −0.759390 0.650635i \(-0.774503\pi\)
0.759390 0.650635i \(-0.225497\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000i 0.925490i −0.886492 0.462745i \(-0.846865\pi\)
0.886492 0.462745i \(-0.153135\pi\)
\(468\) 0 0
\(469\) −39.0000 −1.80085
\(470\) 0 0
\(471\) 9.00000 15.5885i 0.414698 0.718278i
\(472\) 0 0
\(473\) 27.0000i 1.24146i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 31.1769 18.0000i 1.42749 0.824163i
\(478\) 0 0
\(479\) −0.500000 + 0.866025i −0.0228456 + 0.0395697i −0.877222 0.480085i \(-0.840606\pi\)
0.854377 + 0.519654i \(0.173939\pi\)
\(480\) 0 0
\(481\) 10.5000 + 2.59808i 0.478759 + 0.118462i
\(482\) 0 0
\(483\) −54.5596 31.5000i −2.48255 1.43330i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.7224 8.50000i 0.667137 0.385172i −0.127854 0.991793i \(-0.540809\pi\)
0.794991 + 0.606621i \(0.207476\pi\)
\(488\) 0 0
\(489\) 33.0000 1.49231
\(490\) 0 0
\(491\) −11.5000 + 19.9186i −0.518988 + 0.898913i 0.480769 + 0.876847i \(0.340358\pi\)
−0.999757 + 0.0220657i \(0.992976\pi\)
\(492\) 0 0
\(493\) 35.0000i 1.57632i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.79423 4.50000i −0.349619 0.201853i
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 1.50000 + 2.59808i 0.0670151 + 0.116073i
\(502\) 0 0
\(503\) 9.52628 5.50000i 0.424756 0.245233i −0.272354 0.962197i \(-0.587802\pi\)
0.697110 + 0.716964i \(0.254469\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 38.9711 1.50000i 1.73077 0.0666173i
\(508\) 0 0
\(509\) 7.50000 12.9904i 0.332432 0.575789i −0.650556 0.759458i \(-0.725464\pi\)
0.982988 + 0.183669i \(0.0587976\pi\)
\(510\) 0 0
\(511\) 21.0000 + 36.3731i 0.928985 + 1.60905i
\(512\) 0 0
\(513\) −7.79423 + 4.50000i −0.344124 + 0.198680i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 20.7846 + 12.0000i 0.914106 + 0.527759i
\(518\) 0 0
\(519\) −45.0000 −1.97528
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 19.9186 + 11.5000i 0.870979 + 0.502860i 0.867673 0.497135i \(-0.165615\pi\)
0.00330547 + 0.999995i \(0.498948\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.2487 + 14.0000i −1.05629 + 0.609850i
\(528\) 0 0
\(529\) 13.0000 + 22.5167i 0.565217 + 0.978985i
\(530\) 0 0
\(531\) −15.0000 + 25.9808i −0.650945 + 1.12747i
\(532\) 0 0
\(533\) 6.06218 24.5000i 0.262582 1.06121i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 49.3634 28.5000i 2.13019 1.22987i
\(538\) 0 0
\(539\) 3.00000 + 5.19615i 0.129219 + 0.223814i
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) −36.3731 21.0000i −1.56092 0.901196i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.0000i 0.684111i 0.939680 + 0.342055i \(0.111123\pi\)
−0.939680 + 0.342055i \(0.888877\pi\)
\(548\) 0 0
\(549\) −15.0000 + 25.9808i −0.640184 + 1.10883i
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −20.7846 + 12.0000i −0.883852 + 0.510292i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.4545 9.50000i −0.697199 0.402528i 0.109104 0.994030i \(-0.465202\pi\)
−0.806303 + 0.591502i \(0.798535\pi\)
\(558\) 0 0
\(559\) 22.5000 23.3827i 0.951649 0.988982i
\(560\) 0 0
\(561\) −31.5000 + 54.5596i −1.32993 + 2.30351i
\(562\) 0 0
\(563\) −38.9711 + 22.5000i −1.64244 + 0.948262i −0.662474 + 0.749085i \(0.730494\pi\)
−0.979963 + 0.199177i \(0.936173\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 27.0000i 1.13389i
\(568\) 0 0
\(569\) 9.50000 16.4545i 0.398261 0.689808i −0.595251 0.803540i \(-0.702947\pi\)
0.993511 + 0.113732i \(0.0362806\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 9.00000i 0.375980i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 22.5000 + 38.9711i 0.935068 + 1.61959i
\(580\) 0 0
\(581\) −18.0000 31.1769i −0.746766 1.29344i
\(582\) 0 0
\(583\) 15.5885 + 9.00000i 0.645608 + 0.372742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.0429 + 18.5000i 1.32255 + 0.763577i 0.984135 0.177419i \(-0.0567748\pi\)
0.338418 + 0.940996i \(0.390108\pi\)
\(588\) 0 0
\(589\) −2.00000 3.46410i −0.0824086 0.142736i
\(590\) 0 0
\(591\) −34.5000 59.7558i −1.41914 2.45802i
\(592\) 0 0
\(593\) 26.0000i 1.06769i 0.845582 + 0.533846i \(0.179254\pi\)
−0.845582 + 0.533846i \(0.820746\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 27.0000i 1.10504i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 6.50000 11.2583i 0.265141 0.459237i −0.702460 0.711723i \(-0.747915\pi\)
0.967600 + 0.252486i \(0.0812483\pi\)
\(602\) 0 0
\(603\) 78.0000i 3.17641i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.79423 4.50000i 0.316358 0.182649i −0.333410 0.942782i \(-0.608199\pi\)
0.649768 + 0.760133i \(0.274866\pi\)
\(608\) 0 0
\(609\) 22.5000 38.9711i 0.911746 1.57919i
\(610\) 0 0
\(611\) 8.00000 + 27.7128i 0.323645 + 1.12114i
\(612\) 0 0
\(613\) 26.8468 + 15.5000i 1.08433 + 0.626039i 0.932062 0.362300i \(-0.118008\pi\)
0.152270 + 0.988339i \(0.451342\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.1147 + 14.5000i −1.01108 + 0.583748i −0.911508 0.411282i \(-0.865081\pi\)
−0.0995732 + 0.995030i \(0.531748\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) −31.5000 + 54.5596i −1.26405 + 2.18940i
\(622\) 0 0
\(623\) 21.0000i 0.841347i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −7.79423 4.50000i −0.311272 0.179713i
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 7.50000 + 12.9904i 0.298570 + 0.517139i 0.975809 0.218624i \(-0.0701569\pi\)
−0.677239 + 0.735763i \(0.736824\pi\)
\(632\) 0 0
\(633\) 12.9904 7.50000i 0.516321 0.298098i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.73205 + 7.00000i −0.0686264 + 0.277350i
\(638\) 0 0
\(639\) −9.00000 + 15.5885i −0.356034 + 0.616670i
\(640\) 0 0
\(641\) 10.5000 + 18.1865i 0.414725 + 0.718325i 0.995400 0.0958109i \(-0.0305444\pi\)
−0.580674 + 0.814136i \(0.697211\pi\)
\(642\) 0 0
\(643\) 6.06218 3.50000i 0.239069 0.138027i −0.375680 0.926750i \(-0.622591\pi\)
0.614749 + 0.788723i \(0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.7224 + 8.50000i 0.578799 + 0.334169i 0.760656 0.649155i \(-0.224878\pi\)
−0.181857 + 0.983325i \(0.558211\pi\)
\(648\) 0 0
\(649\) −15.0000 −0.588802
\(650\) 0 0
\(651\) 36.0000 1.41095
\(652\) 0 0
\(653\) 9.52628 + 5.50000i 0.372792 + 0.215232i 0.674678 0.738113i \(-0.264283\pi\)
−0.301885 + 0.953344i \(0.597616\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 72.7461 42.0000i 2.83810 1.63858i
\(658\) 0 0
\(659\) −7.50000 12.9904i −0.292159 0.506033i 0.682161 0.731202i \(-0.261040\pi\)
−0.974320 + 0.225168i \(0.927707\pi\)
\(660\) 0 0
\(661\) 8.50000 14.7224i 0.330612 0.572636i −0.652020 0.758202i \(-0.726078\pi\)
0.982632 + 0.185565i \(0.0594116\pi\)
\(662\) 0 0
\(663\) −72.7461 + 21.0000i −2.82523 + 0.815572i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.3109 + 17.5000i −1.17364 + 0.677603i
\(668\) 0 0
\(669\) 34.5000 + 59.7558i 1.33385 + 2.31029i
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −11.2583 6.50000i −0.433977 0.250557i 0.267063 0.963679i \(-0.413947\pi\)
−0.701039 + 0.713123i \(0.747280\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.0000i 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) −16.5000 + 28.5788i −0.633212 + 1.09676i
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) 26.8468 15.5000i 1.02726 0.593091i 0.111064 0.993813i \(-0.464574\pi\)
0.916200 + 0.400722i \(0.131241\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −67.5500 39.0000i −2.57719 1.48794i
\(688\) 0 0
\(689\) 6.00000 + 20.7846i 0.228582 + 0.791831i
\(690\) 0 0
\(691\) 12.5000 21.6506i 0.475522 0.823629i −0.524084 0.851666i \(-0.675592\pi\)
0.999607 + 0.0280373i \(0.00892572\pi\)
\(692\) 0 0
\(693\) 46.7654 27.0000i 1.77647 1.02565i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 49.0000i 1.85601i
\(698\) 0 0
\(699\) −27.0000 + 46.7654i −1.02123 + 1.76883i
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 3.00000i 0.113147i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.0000i 1.01544i
\(708\) 0 0
\(709\) 17.5000 + 30.3109i 0.657226 + 1.13835i 0.981331 + 0.192328i \(0.0616038\pi\)
−0.324104 + 0.946021i \(0.605063\pi\)
\(710\) 0 0
\(711\) 24.0000 + 41.5692i 0.900070 + 1.55897i
\(712\) 0 0
\(713\) −24.2487 14.0000i −0.908121 0.524304i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 41.5692 + 24.0000i 1.55243 + 0.896296i
\(718\) 0 0
\(719\) 0.500000 + 0.866025i 0.0186469 + 0.0322973i 0.875198 0.483764i \(-0.160731\pi\)
−0.856551 + 0.516062i \(0.827398\pi\)
\(720\) 0 0
\(721\) 24.0000 + 41.5692i 0.893807 + 1.54812i
\(722\) 0 0
\(723\) 3.00000i 0.111571i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −31.5000 + 54.5596i −1.16507 + 2.01796i
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −33.7750 + 19.5000i −1.24412 + 0.718292i
\(738\) 0 0
\(739\) 19.5000 33.7750i 0.717319 1.24243i −0.244739 0.969589i \(-0.578702\pi\)
0.962058 0.272844i \(-0.0879643\pi\)
\(740\) 0 0
\(741\) −3.00000 10.3923i −0.110208 0.381771i
\(742\) 0 0
\(743\) −0.866025 0.500000i −0.0317714 0.0183432i 0.484030 0.875051i \(-0.339172\pi\)
−0.515802 + 0.856708i \(0.672506\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −62.3538 + 36.0000i −2.28141 + 1.31717i
\(748\) 0 0
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) 6.50000 11.2583i 0.237188 0.410822i −0.722718 0.691143i \(-0.757107\pi\)
0.959906 + 0.280321i \(0.0904408\pi\)
\(752\) 0 0
\(753\) 15.0000i 0.546630i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.866025 + 0.500000i 0.0314762 + 0.0181728i 0.515656 0.856796i \(-0.327548\pi\)
−0.484179 + 0.874969i \(0.660882\pi\)
\(758\) 0 0
\(759\) −63.0000 −2.28676
\(760\) 0 0
\(761\) −25.5000 44.1673i −0.924374 1.60106i −0.792564 0.609788i \(-0.791255\pi\)
−0.131810 0.991275i \(-0.542079\pi\)
\(762\) 0 0
\(763\) −36.3731 + 21.0000i −1.31679 + 0.760251i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.9904 12.5000i −0.469055 0.451349i
\(768\) 0 0
\(769\) −2.50000 + 4.33013i −0.0901523 + 0.156148i −0.907575 0.419890i \(-0.862069\pi\)
0.817423 + 0.576038i \(0.195402\pi\)
\(770\) 0 0
\(771\) −28.5000 49.3634i −1.02640 1.77778i
\(772\) 0 0
\(773\) 21.6506 12.5000i 0.778719 0.449594i −0.0572570 0.998359i \(-0.518235\pi\)
0.835976 + 0.548766i \(0.184902\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 23.3827 + 13.5000i 0.838849 + 0.484310i
\(778\) 0 0
\(779\) −7.00000 −0.250801
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) −38.9711 22.5000i −1.39272 0.804084i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.866025 0.500000i 0.0308705 0.0178231i −0.484485 0.874799i \(-0.660993\pi\)
0.515356 + 0.856976i \(0.327660\pi\)
\(788\) 0 0
\(789\) 10.5000 + 18.1865i 0.373810 + 0.647458i
\(790\) 0 0
\(791\) −19.5000 + 33.7750i −0.693340 + 1.20090i
\(792\) 0 0
\(793\) −12.9904 12.5000i −0.461302 0.443888i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.06218 3.50000i 0.214733 0.123976i −0.388776 0.921332i \(-0.627102\pi\)
0.603509 + 0.797356i \(0.293769\pi\)
\(798\) 0 0
\(799\) −28.0000 48.4974i −0.990569 1.71572i
\(800\) 0 0
\(801\) 42.0000 1.48400
\(802\) 0 0
\(803\) 36.3731 + 21.0000i 1.28358 + 0.741074i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000i 0.316815i
\(808\) 0 0
\(809\) −12.5000 + 21.6506i −0.439477 + 0.761196i −0.997649 0.0685291i \(-0.978169\pi\)
0.558173 + 0.829725i \(0.311503\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) −59.7558 + 34.5000i −2.09573 + 1.20997i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.79423 4.50000i −0.272686 0.157435i
\(818\) 0 0
\(819\) 63.0000 + 15.5885i 2.20140 + 0.544705i
\(820\) 0 0
\(821\) −7.50000 + 12.9904i −0.261752 + 0.453367i −0.966708 0.255884i \(-0.917634\pi\)
0.704956 + 0.709251i \(0.250967\pi\)
\(822\) 0 0
\(823\) 40.7032 23.5000i 1.41882 0.819159i 0.422628 0.906303i \(-0.361108\pi\)
0.996196 + 0.0871445i \(0.0277742\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) −6.50000 + 11.2583i −0.225754 + 0.391018i −0.956545 0.291583i \(-0.905818\pi\)
0.730791 + 0.682601i \(0.239151\pi\)
\(830\) 0 0
\(831\) 69.0000 2.39358
\(832\) 0 0
\(833\) 14.0000i 0.485071i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 36.0000i 1.24434i
\(838\) 0 0
\(839\) 6.50000 + 11.2583i 0.224405 + 0.388681i 0.956141 0.292908i \(-0.0946228\pi\)
−0.731736 + 0.681588i \(0.761290\pi\)
\(840\) 0 0
\(841\) 2.00000 + 3.46410i 0.0689655 + 0.119452i
\(842\) 0 0
\(843\) −25.9808 15.0000i −0.894825 0.516627i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.19615 3.00000i −0.178542 0.103081i
\(848\) 0 0
\(849\) −1.50000 2.59808i −0.0514799 0.0891657i
\(850\) 0 0
\(851\) −10.5000 18.1865i −0.359935 0.623426i
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 31.5000 54.5596i 1.07352 1.85939i
\(862\) 0 0
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 83.1384 48.0000i 2.82353 1.63017i
\(868\) 0 0
\(869\) −12.0000 + 20.7846i −0.407072 + 0.705070i
\(870\) 0 0
\(871\) −45.5000 11.2583i −1.54171 0.381474i
\(872\) 0 0
\(873\) 57.1577 + 33.0000i 1.93449 + 1.11688i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.33013 + 2.50000i −0.146218 + 0.0844190i −0.571324 0.820724i \(-0.693570\pi\)
0.425106 + 0.905143i \(0.360237\pi\)
\(878\) 0 0
\(879\) 27.0000 0.910687
\(880\) 0 0
\(881\) −5.50000 + 9.52628i −0.185300 + 0.320949i −0.943677 0.330867i \(-0.892659\pi\)
0.758378 + 0.651815i \(0.225992\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.4545 9.50000i −0.552487 0.318979i 0.197637 0.980275i \(-0.436673\pi\)
−0.750125 + 0.661296i \(0.770007\pi\)
\(888\) 0 0
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) −13.5000 23.3827i −0.452267 0.783349i
\(892\) 0 0
\(893\) 6.92820 4.00000i 0.231843 0.133855i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −54.5596 52.5000i −1.82169 1.75292i
\(898\) 0 0
\(899\) 10.0000 17.3205i 0.333519 0.577671i
\(900\) 0 0
\(901\) −21.0000 36.3731i −0.699611 1.21176i
\(902\) 0 0
\(903\) 70.1481 40.5000i 2.33438 1.34776i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.7224 + 8.50000i 0.488850 + 0.282238i 0.724097 0.689698i \(-0.242257\pi\)
−0.235247 + 0.971936i \(0.575590\pi\)
\(908\) 0 0
\(909\) 54.0000 1.79107
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −31.1769 18.0000i −1.03181 0.595713i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.3923 + 6.00000i −0.343184 + 0.198137i
\(918\) 0 0
\(919\) −21.5000 37.2391i −0.709220 1.22840i −0.965147 0.261708i \(-0.915714\pi\)
0.255927 0.966696i \(-0.417619\pi\)
\(920\) 0 0
\(921\) 42.0000 72.7461i 1.38395 2.39707i
\(922\) 0 0
\(923\) −7.79423 7.50000i −0.256550 0.246866i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 83.1384 48.0000i 2.73062 1.57653i
\(928\) 0 0
\(929\) −26.5000 45.8993i −0.869437 1.50591i −0.862573 0.505932i \(-0.831149\pi\)
−0.00686358 0.999976i \(-0.502185\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −62.3538 36.0000i −2.04137 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 50.0000i 1.63343i −0.577042 0.816714i \(-0.695793\pi\)
0.577042 0.816714i \(-0.304207\pi\)
\(938\) 0 0
\(939\) 9.00000 15.5885i 0.293704 0.508710i
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −42.4352 + 24.5000i −1.38188 + 0.797830i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.6314 27.5000i −1.54781 0.893630i −0.998309 0.0581388i \(-0.981483\pi\)
−0.549504 0.835491i \(-0.685183\pi\)
\(948\) 0 0
\(949\) 14.0000 + 48.4974i 0.454459 + 1.57429i
\(950\) 0 0
\(951\) −3.00000 + 5.19615i −0.0972817 + 0.168497i
\(952\) 0 0
\(953\) −26.8468 + 15.5000i −0.869653 + 0.502094i −0.867233 0.497903i \(-0.834104\pi\)
−0.00241992 + 0.999997i \(0.500770\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 45.0000i 1.45464i
\(958\) 0 0
\(959\) −4.50000 + 7.79423i −0.145313 + 0.251689i
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 18.0000i 0.580042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 0 0
\(969\) 10.5000 + 18.1865i 0.337309 + 0.584236i
\(970\) 0 0
\(971\) 21.5000 + 37.2391i 0.689968 + 1.19506i 0.971848 + 0.235610i \(0.0757087\pi\)
−0.281880 + 0.959450i \(0.590958\pi\)
\(972\) 0 0
\(973\) −33.7750 19.5000i −1.08278 0.625141i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.59808 1.50000i −0.0831198 0.0479893i 0.457864 0.889022i \(-0.348615\pi\)
−0.540984 + 0.841033i \(0.681948\pi\)
\(978\) 0 0
\(979\) 10.5000 + 18.1865i 0.335581 + 0.581244i
\(980\) 0 0
\(981\) 42.0000 + 72.7461i 1.34096 + 2.32261i
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 72.0000i 2.29179i
\(988\) 0 0
\(989\) −63.0000 −2.00328
\(990\) 0 0
\(991\) 6.50000 11.2583i 0.206479 0.357633i −0.744124 0.668042i \(-0.767133\pi\)
0.950603 + 0.310409i \(0.100466\pi\)
\(992\) 0 0
\(993\) 39.0000i 1.23763i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −11.2583 + 6.50000i −0.356555 + 0.205857i −0.667568 0.744548i \(-0.732665\pi\)
0.311014 + 0.950405i \(0.399332\pi\)
\(998\) 0 0
\(999\) 13.5000 23.3827i 0.427121 0.739795i
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 1300.2.bb.e.1049.2 4
5.2 odd 4 260.2.i.d.61.1 2
5.3 odd 4 1300.2.i.a.1101.1 2
5.4 even 2 inner 1300.2.bb.e.1049.1 4
13.3 even 3 inner 1300.2.bb.e.549.1 4
15.2 even 4 2340.2.q.a.1621.1 2
20.7 even 4 1040.2.q.b.321.1 2
65.3 odd 12 1300.2.i.a.601.1 2
65.7 even 12 3380.2.f.a.3041.2 2
65.17 odd 12 3380.2.a.a.1.1 1
65.22 odd 12 3380.2.a.b.1.1 1
65.29 even 6 inner 1300.2.bb.e.549.2 4
65.32 even 12 3380.2.f.a.3041.1 2
65.42 odd 12 260.2.i.d.81.1 yes 2
195.107 even 12 2340.2.q.a.2161.1 2
260.107 even 12 1040.2.q.b.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.d.61.1 2 5.2 odd 4
260.2.i.d.81.1 yes 2 65.42 odd 12
1040.2.q.b.81.1 2 260.107 even 12
1040.2.q.b.321.1 2 20.7 even 4
1300.2.i.a.601.1 2 65.3 odd 12
1300.2.i.a.1101.1 2 5.3 odd 4
1300.2.bb.e.549.1 4 13.3 even 3 inner
1300.2.bb.e.549.2 4 65.29 even 6 inner
1300.2.bb.e.1049.1 4 5.4 even 2 inner
1300.2.bb.e.1049.2 4 1.1 even 1 trivial
2340.2.q.a.1621.1 2 15.2 even 4
2340.2.q.a.2161.1 2 195.107 even 12
3380.2.a.a.1.1 1 65.17 odd 12
3380.2.a.b.1.1 1 65.22 odd 12
3380.2.f.a.3041.1 2 65.32 even 12
3380.2.f.a.3041.2 2 65.7 even 12