Properties

Label 912.2.ch.d.671.1
Level $912$
Weight $2$
Character 912.671
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(47,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 671.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 912.671
Dual form 912.2.ch.d.719.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.592396 + 1.62760i) q^{3} +(4.00387 + 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\) \(q+(-0.592396 + 1.62760i) q^{3} +(4.00387 + 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(6.73783 - 2.45237i) q^{13} +(3.50000 - 2.59808i) q^{19} +(-6.13429 + 5.14728i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(4.50000 - 2.59808i) q^{27} +(9.12108 + 5.26606i) q^{31} -12.1138 q^{37} +12.4192i q^{39} +(-2.77584 + 0.489456i) q^{43} +(7.18732 + 12.4488i) q^{49} +(2.15523 + 7.23567i) q^{57} +(1.60694 - 9.11343i) q^{61} +(-4.74376 - 13.0334i) q^{63} +(-5.60994 + 6.68566i) q^{67} +(11.1912 + 4.07326i) q^{73} -8.66025i q^{75} +(-6.03462 + 16.5800i) q^{79} +(1.56283 + 8.86327i) q^{81} +(32.6464 + 5.75643i) q^{91} +(-13.9743 + 11.7258i) q^{93} +(-3.83022 + 3.21394i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 21 q^{13} + 21 q^{19} - 27 q^{21} + 27 q^{27} - 15 q^{43} + 21 q^{49} + 42 q^{61} - 36 q^{63} + 15 q^{67} + 21 q^{73} - 12 q^{79} + 48 q^{91} - 54 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.592396 + 1.62760i −0.342020 + 0.939693i
\(4\) 0 0
\(5\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(6\) 0 0
\(7\) 4.00387 + 2.31164i 1.51332 + 0.873716i 0.999878 + 0.0155920i \(0.00496330\pi\)
0.513442 + 0.858124i \(0.328370\pi\)
\(8\) 0 0
\(9\) −2.29813 1.92836i −0.766044 0.642788i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 6.73783 2.45237i 1.86874 0.680165i 0.898011 0.439972i \(-0.145012\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(18\) 0 0
\(19\) 3.50000 2.59808i 0.802955 0.596040i
\(20\) 0 0
\(21\) −6.13429 + 5.14728i −1.33861 + 1.12323i
\(22\) 0 0
\(23\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(24\) 0 0
\(25\) −4.69846 + 1.71010i −0.939693 + 0.342020i
\(26\) 0 0
\(27\) 4.50000 2.59808i 0.866025 0.500000i
\(28\) 0 0
\(29\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(30\) 0 0
\(31\) 9.12108 + 5.26606i 1.63819 + 0.945812i 0.981455 + 0.191695i \(0.0613985\pi\)
0.656740 + 0.754117i \(0.271935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.1138 −1.99150 −0.995749 0.0921098i \(-0.970639\pi\)
−0.995749 + 0.0921098i \(0.970639\pi\)
\(38\) 0 0
\(39\) 12.4192i 1.98867i
\(40\) 0 0
\(41\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(42\) 0 0
\(43\) −2.77584 + 0.489456i −0.423312 + 0.0746414i −0.381246 0.924473i \(-0.624505\pi\)
−0.0420659 + 0.999115i \(0.513394\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(48\) 0 0
\(49\) 7.18732 + 12.4488i 1.02676 + 1.77840i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.15523 + 7.23567i 0.285467 + 0.958388i
\(58\) 0 0
\(59\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(60\) 0 0
\(61\) 1.60694 9.11343i 0.205748 1.16686i −0.690510 0.723323i \(-0.742614\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) −4.74376 13.0334i −0.597657 1.64205i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.60994 + 6.68566i −0.685363 + 0.816784i −0.990787 0.135433i \(-0.956757\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(72\) 0 0
\(73\) 11.1912 + 4.07326i 1.30983 + 0.476739i 0.900186 0.435506i \(-0.143431\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.03462 + 16.5800i −0.678947 + 1.86539i −0.225018 + 0.974355i \(0.572244\pi\)
−0.453930 + 0.891038i \(0.649978\pi\)
\(80\) 0 0
\(81\) 1.56283 + 8.86327i 0.173648 + 0.984808i
\(82\) 0 0
\(83\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(90\) 0 0
\(91\) 32.6464 + 5.75643i 3.42227 + 0.603438i
\(92\) 0 0
\(93\) −13.9743 + 11.7258i −1.44907 + 1.21591i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.83022 + 3.21394i −0.388900 + 0.326326i −0.816185 0.577791i \(-0.803915\pi\)
0.427284 + 0.904117i \(0.359470\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(102\) 0 0
\(103\) −16.5364 + 9.54731i −1.62938 + 0.940724i −0.645105 + 0.764094i \(0.723187\pi\)
−0.984277 + 0.176631i \(0.943480\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) −2.95202 16.7417i −0.282752 1.60357i −0.713206 0.700954i \(-0.752758\pi\)
0.430454 0.902613i \(-0.358354\pi\)
\(110\) 0 0
\(111\) 7.17617 19.7164i 0.681132 1.87140i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −20.2135 7.35710i −1.86874 0.680165i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.70115 21.1587i −0.683367 1.87753i −0.383375 0.923593i \(-0.625238\pi\)
−0.299991 0.953942i \(-0.596984\pi\)
\(128\) 0 0
\(129\) 0.847763 4.80790i 0.0746414 0.423312i
\(130\) 0 0
\(131\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(132\) 0 0
\(133\) 20.0194 2.31164i 1.73590 0.200444i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(138\) 0 0
\(139\) −4.36050 11.9804i −0.369853 1.01616i −0.975417 0.220366i \(-0.929275\pi\)
0.605564 0.795796i \(-0.292947\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −24.5194 + 4.32342i −2.02232 + 0.356590i
\(148\) 0 0
\(149\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(150\) 0 0
\(151\) 8.66025i 0.704761i 0.935857 + 0.352381i \(0.114628\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.51889 19.9566i −0.280838 1.59271i −0.719785 0.694197i \(-0.755760\pi\)
0.438948 0.898513i \(-0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.4696 6.04460i 0.820039 0.473450i −0.0303908 0.999538i \(-0.509675\pi\)
0.850430 + 0.526088i \(0.176342\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(168\) 0 0
\(169\) 29.4256 24.6910i 2.26351 1.89931i
\(170\) 0 0
\(171\) −13.0535 0.778544i −0.998226 0.0595368i
\(172\) 0 0
\(173\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(174\) 0 0
\(175\) −22.7652 4.01411i −1.72088 0.303438i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 5.36231 + 4.49951i 0.398577 + 0.334446i 0.819943 0.572444i \(-0.194005\pi\)
−0.421366 + 0.906891i \(0.638449\pi\)
\(182\) 0 0
\(183\) 13.8810 + 8.01422i 1.02612 + 0.592428i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 24.0232 1.74743
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −18.1789 6.61656i −1.30854 0.476271i −0.408773 0.912636i \(-0.634043\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 9.86319 11.7545i 0.699183 0.833254i −0.293251 0.956036i \(-0.594737\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) 0 0
\(201\) −7.55825 13.0913i −0.533118 0.923387i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.32934 + 6.35127i 0.366887 + 0.437239i 0.917630 0.397436i \(-0.130100\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.3464 + 42.1692i 1.65274 + 2.86263i
\(218\) 0 0
\(219\) −13.2592 + 15.8017i −0.895976 + 1.06778i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.186137 + 0.0328209i −0.0124646 + 0.00219785i −0.179877 0.983689i \(-0.557570\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) 14.0954 + 5.13030i 0.939693 + 0.342020i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 3.49289 0.230817 0.115408 0.993318i \(-0.463182\pi\)
0.115408 + 0.993318i \(0.463182\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −23.4106 19.6438i −1.52068 1.27600i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −3.45558 + 1.25773i −0.222594 + 0.0810175i −0.450910 0.892570i \(-0.648900\pi\)
0.228316 + 0.973587i \(0.426678\pi\)
\(242\) 0 0
\(243\) −15.3516 2.70691i −0.984808 0.173648i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.2110 26.0887i 1.09511 1.65998i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(258\) 0 0
\(259\) −48.5021 28.0027i −3.01377 1.74000i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(270\) 0 0
\(271\) −15.3516 + 2.70691i −0.932545 + 0.164433i −0.619224 0.785214i \(-0.712553\pi\)
−0.313321 + 0.949647i \(0.601442\pi\)
\(272\) 0 0
\(273\) −28.7087 + 49.7250i −1.73753 + 3.00949i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −15.5000 26.8468i −0.931305 1.61307i −0.781094 0.624413i \(-0.785338\pi\)
−0.150210 0.988654i \(-0.547995\pi\)
\(278\) 0 0
\(279\) −10.8066 29.6909i −0.646974 1.77755i
\(280\) 0 0
\(281\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(282\) 0 0
\(283\) 14.4734 + 17.2488i 0.860356 + 1.02533i 0.999386 + 0.0350443i \(0.0111572\pi\)
−0.139030 + 0.990288i \(0.544398\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.95202 16.7417i 0.173648 0.984808i
\(290\) 0 0
\(291\) −2.96198 8.13798i −0.173634 0.477057i
\(292\) 0 0
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.2456 4.45702i −0.705823 0.256898i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.592396 1.62760i 0.0338098 0.0928918i −0.921639 0.388048i \(-0.873149\pi\)
0.955449 + 0.295156i \(0.0953717\pi\)
\(308\) 0 0
\(309\) −5.74304 32.5704i −0.326710 1.85287i
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 26.8116 + 22.4976i 1.51548 + 1.27164i 0.852134 + 0.523324i \(0.175308\pi\)
0.663345 + 0.748314i \(0.269136\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −27.4636 + 23.0447i −1.52341 + 1.27829i
\(326\) 0 0
\(327\) 28.9975 + 5.11305i 1.60357 + 0.282752i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −29.3041 + 16.9187i −1.61070 + 0.929938i −0.621492 + 0.783420i \(0.713473\pi\)
−0.989208 + 0.146518i \(0.953193\pi\)
\(332\) 0 0
\(333\) 27.8391 + 23.3598i 1.52558 + 1.28011i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.34436 7.62424i −0.0732319 0.415319i −0.999281 0.0379157i \(-0.987928\pi\)
0.926049 0.377403i \(-0.123183\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 34.0949i 1.84095i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(348\) 0 0
\(349\) −16.4957 + 28.5714i −0.882996 + 1.52939i −0.0350017 + 0.999387i \(0.511144\pi\)
−0.847994 + 0.530006i \(0.822190\pi\)
\(350\) 0 0
\(351\) 23.9488 28.5410i 1.27829 1.52341i
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(360\) 0 0
\(361\) 5.50000 18.1865i 0.289474 0.957186i
\(362\) 0 0
\(363\) 12.2467 + 14.5951i 0.642788 + 0.766044i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.10426 + 25.0137i 0.475238 + 1.30571i 0.913493 + 0.406855i \(0.133375\pi\)
−0.438254 + 0.898851i \(0.644403\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −19.0000 + 32.9090i −0.983783 + 1.70396i −0.336557 + 0.941663i \(0.609263\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.4953i 0.744576i 0.928117 + 0.372288i \(0.121427\pi\)
−0.928117 + 0.372288i \(0.878573\pi\)
\(380\) 0 0
\(381\) 39.0000 1.99803
\(382\) 0 0
\(383\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.32311 + 4.22800i 0.372255 + 0.214921i
\(388\) 0 0
\(389\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.1573 9.36208i 0.559968 0.469869i −0.318332 0.947979i \(-0.603123\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 0 0
\(399\) −8.09698 + 33.9528i −0.405356 + 1.69977i
\(400\) 0 0
\(401\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(402\) 0 0
\(403\) 74.3706 + 13.1135i 3.70466 + 0.653232i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −29.1097 24.4259i −1.43938 1.20778i −0.939895 0.341463i \(-0.889078\pi\)
−0.499486 0.866322i \(-0.666478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.0823 1.08138
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 38.5274 + 14.0228i 1.87771 + 0.683431i 0.953291 + 0.302053i \(0.0976721\pi\)
0.924419 + 0.381377i \(0.124550\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 27.5009 32.7743i 1.33086 1.58606i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(432\) 0 0
\(433\) 2.14249 12.1507i 0.102962 0.583924i −0.889053 0.457804i \(-0.848636\pi\)
0.992015 0.126121i \(-0.0402527\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −14.2447 16.9762i −0.679862 0.810228i 0.310228 0.950662i \(-0.399595\pi\)
−0.990090 + 0.140434i \(0.955150\pi\)
\(440\) 0 0
\(441\) 7.48839 42.4688i 0.356590 2.02232i
\(442\) 0 0
\(443\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −14.0954 5.13030i −0.662259 0.241043i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.82058 0.225497 0.112749 0.993624i \(-0.464034\pi\)
0.112749 + 0.993624i \(0.464034\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(462\) 0 0
\(463\) −11.3270 6.53964i −0.526409 0.303923i 0.213144 0.977021i \(-0.431630\pi\)
−0.739553 + 0.673098i \(0.764963\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) −37.9163 + 13.8004i −1.75081 + 0.637243i
\(470\) 0 0
\(471\) 34.5658 + 6.09489i 1.59271 + 0.280838i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0016 + 18.1923i −0.550673 + 0.834721i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(480\) 0 0
\(481\) −81.6207 + 29.7075i −3.72158 + 1.35455i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −31.5000 18.1865i −1.42740 0.824110i −0.430486 0.902597i \(-0.641658\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 3.63604 + 20.6210i 0.164427 + 0.932514i
\(490\) 0 0
\(491\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.3666 6.41242i 1.62799 0.287059i 0.716258 0.697835i \(-0.245853\pi\)
0.911736 + 0.410776i \(0.134742\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.7554 + 62.5199i 1.01060 + 2.77660i
\(508\) 0 0
\(509\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(510\) 0 0
\(511\) 35.3922 + 42.1788i 1.56566 + 1.86588i
\(512\) 0 0
\(513\) 9.00000 20.7846i 0.397360 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 18.8692 22.4874i 0.825091 0.983306i −0.174908 0.984585i \(-0.555963\pi\)
0.999999 + 0.00127919i \(0.000407178\pi\)
\(524\) 0 0
\(525\) 20.0194 34.6745i 0.873716 1.51332i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.6129 + 7.86646i 0.939693 + 0.342020i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −31.2977 26.2619i −1.34559 1.12909i −0.980151 0.198254i \(-0.936473\pi\)
−0.365444 0.930834i \(-0.619083\pi\)
\(542\) 0 0
\(543\) −10.5000 + 6.06218i −0.450598 + 0.260153i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.96720 1.58116i −0.383410 0.0676055i −0.0213785 0.999771i \(-0.506805\pi\)
−0.362031 + 0.932166i \(0.617917\pi\)
\(548\) 0 0
\(549\) −21.2670 + 17.8451i −0.907653 + 0.761611i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −62.4887 + 52.4342i −2.65729 + 2.22973i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(558\) 0 0
\(559\) −17.5028 + 10.1053i −0.740291 + 0.427407i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −14.2313 + 39.1001i −0.597657 + 1.64205i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 44.7821i 1.87407i −0.349231 0.937037i \(-0.613557\pi\)
0.349231 0.937037i \(-0.386443\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.50000 + 9.52628i −0.228968 + 0.396584i −0.957503 0.288425i \(-0.906868\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 0 0
\(579\) 21.5382 25.6682i 0.895096 1.06673i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(588\) 0 0
\(589\) 45.6054 5.26606i 1.87914 0.216984i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.2886 + 23.0166i 0.543868 + 0.942007i
\(598\) 0 0
\(599\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(600\) 0 0
\(601\) 18.2114 31.5431i 0.742859 1.28667i −0.208329 0.978059i \(-0.566802\pi\)
0.951188 0.308611i \(-0.0998642\pi\)
\(602\) 0 0
\(603\) 25.7848 4.54655i 1.05004 0.185150i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 35.4771i 1.43997i −0.693990 0.719985i \(-0.744149\pi\)
0.693990 0.719985i \(-0.255851\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.16146 + 46.2860i 0.329638 + 1.86947i 0.474843 + 0.880071i \(0.342505\pi\)
−0.145204 + 0.989402i \(0.546384\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(618\) 0 0
\(619\) −21.5315 + 12.4312i −0.865425 + 0.499653i −0.865825 0.500347i \(-0.833206\pi\)
0.000400419 1.00000i \(0.499873\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.1511 16.0697i 0.766044 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 9.55959 + 1.68561i 0.380561 + 0.0671032i 0.360657 0.932699i \(-0.382553\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) 0 0
\(633\) −13.4944 + 4.91155i −0.536353 + 0.195217i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 78.9559 + 66.2519i 3.12835 + 2.62500i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(642\) 0 0
\(643\) 5.34096 14.6742i 0.210627 0.578692i −0.788723 0.614749i \(-0.789257\pi\)
0.999350 + 0.0360565i \(0.0114796\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −83.0572 + 14.6452i −3.25527 + 0.573991i
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −17.8641 30.9416i −0.696946 1.20715i
\(658\) 0 0
\(659\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(660\) 0 0
\(661\) 6.59863 37.4227i 0.256657 1.45557i −0.535126 0.844772i \(-0.679736\pi\)
0.791783 0.610802i \(-0.209153\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.0568475 0.322398i 0.00219785 0.0124646i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.8619 + 44.7941i 0.996903 + 1.72669i 0.566557 + 0.824023i \(0.308275\pi\)
0.430346 + 0.902664i \(0.358391\pi\)
\(674\) 0 0
\(675\) −16.7001 + 19.9024i −0.642788 + 0.766044i
\(676\) 0 0
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −22.7652 + 4.01411i −0.873647 + 0.154048i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.06917 + 5.68501i −0.0789439 + 0.216897i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 16.5000 + 9.52628i 0.627690 + 0.362397i 0.779857 0.625958i \(-0.215292\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(702\) 0 0
\(703\) −42.3983 + 31.4726i −1.59908 + 1.18701i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −48.4702 + 17.6417i −1.82034 + 0.662548i −0.825108 + 0.564975i \(0.808886\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) 45.8405 26.4661i 1.71915 0.992554i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(720\) 0 0
\(721\) −88.2796 −3.28770
\(722\) 0 0
\(723\) 6.36937i 0.236879i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 37.3417 6.58434i 1.38493 0.244200i 0.568991 0.822344i \(-0.307334\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 13.5000 23.3827i 0.500000 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.50000 6.06218i −0.129275 0.223912i 0.794121 0.607760i \(-0.207932\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −27.8684 33.2123i −1.02516 1.22173i −0.974818 0.223001i \(-0.928415\pi\)
−0.0503375 0.998732i \(-0.516030\pi\)
\(740\) 0 0
\(741\) 32.2661 + 43.4673i 1.18532 + 1.59681i
\(742\) 0 0
\(743\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −34.8505 + 41.5332i −1.27171 + 1.51557i −0.523655 + 0.851930i \(0.675432\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.5775 3.84991i −0.384447 0.139927i 0.142564 0.989786i \(-0.454465\pi\)
−0.527011 + 0.849858i \(0.676688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 26.8813 73.8557i 0.973168 2.67376i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −41.5474 34.8624i −1.49824 1.25717i −0.883493 0.468445i \(-0.844814\pi\)
−0.614745 0.788726i \(-0.710741\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(774\) 0 0
\(775\) −51.8606 9.14442i −1.86289 0.328477i
\(776\) 0 0
\(777\) 74.3096 62.3531i 2.66584 2.23691i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −48.2816 + 27.8754i −1.72105 + 0.993650i −0.804269 + 0.594265i \(0.797443\pi\)
−0.916783 + 0.399385i \(0.869224\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −11.5222 65.3455i −0.409165 2.32049i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) −14.8099 40.6899i −0.520046 1.42882i −0.870469 0.492223i \(-0.836184\pi\)
0.350423 0.936592i \(-0.386038\pi\)
\(812\) 0 0
\(813\) 4.68850 26.5898i 0.164433 0.932545i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.44381 + 8.92495i −0.295412 + 0.312245i
\(818\) 0 0
\(819\) −63.9252 76.1831i −2.23373 2.66205i
\(820\) 0 0
\(821\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(822\) 0 0
\(823\) −19.5491 53.7106i −0.681438 1.87224i −0.422608 0.906313i \(-0.638885\pi\)
−0.258830 0.965923i \(-0.583337\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(828\) 0 0
\(829\) 15.6890 27.1741i 0.544900 0.943795i −0.453713 0.891148i \(-0.649901\pi\)
0.998613 0.0526472i \(-0.0167659\pi\)
\(830\) 0 0
\(831\) 52.8778 9.32379i 1.83431 0.323439i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 54.7265 1.89162
\(838\) 0 0
\(839\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(840\) 0 0
\(841\) 5.03580 + 28.5594i 0.173648 + 0.984808i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0426 25.4280i 1.51332 0.873716i
\(848\) 0 0
\(849\) −36.6480 + 13.3388i −1.25776 + 0.457786i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −30.6243 + 25.6969i −1.04856 + 0.879844i −0.992941 0.118609i \(-0.962157\pi\)
−0.0556158 + 0.998452i \(0.517712\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(858\) 0 0
\(859\) −2.83346 0.499616i −0.0966765 0.0170467i 0.125101 0.992144i \(-0.460075\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 25.5000 + 14.7224i 0.866025 + 0.500000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −21.4031 + 58.8045i −0.725216 + 1.99251i
\(872\) 0 0
\(873\) 15.0000 0.507673
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.4360 5.25427i −0.487469 0.177424i 0.0865807 0.996245i \(-0.472406\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) −33.8120 + 40.2955i −1.13786 + 1.35605i −0.212415 + 0.977180i \(0.568133\pi\)
−0.925449 + 0.378873i \(0.876312\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(888\) 0 0
\(889\) 18.0769 102.519i 0.606279 3.43838i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 14.5085 17.2905i 0.482811 0.575392i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.9402 2.10537i 0.396466 0.0699077i 0.0281394 0.999604i \(-0.491042\pi\)
0.368327 + 0.929696i \(0.379931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −49.6301 28.6539i −1.63714 0.945206i −0.981810 0.189867i \(-0.939194\pi\)
−0.655335 0.755338i \(-0.727472\pi\)
\(920\) 0 0
\(921\) 2.29813 + 1.92836i 0.0757261 + 0.0635417i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 56.9163 20.7158i 1.87140 0.681132i
\(926\) 0 0
\(927\) 56.4136 + 9.94724i 1.85287 + 0.326710i
\(928\) 0 0
\(929\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(930\) 0 0
\(931\) 57.4986 + 24.8976i 1.88444 + 0.815986i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.14512 1.87267i 0.168084 0.0611775i −0.256608 0.966516i \(-0.582605\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −52.5000 + 30.3109i −1.71327 + 0.989158i
\(940\) 0 0
\(941\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(948\) 0 0
\(949\) 85.3934 2.77199
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 39.9628 + 69.2175i 1.28912 + 2.23282i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 20.7340 + 24.7098i 0.666761 + 0.794615i 0.988339 0.152268i \(-0.0486578\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(972\) 0 0
\(973\) 10.2354 58.0478i 0.328131 1.86093i
\(974\) 0 0
\(975\) −21.2381 58.3513i −0.680165 1.86874i
\(976\) 0 0
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −25.5000 + 44.1673i −0.814152 + 1.41015i
\(982\) 0 0
\(983\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 17.4948 48.0667i 0.555742 1.52689i −0.270011 0.962857i \(-0.587027\pi\)
0.825753 0.564031i \(-0.190750\pi\)
\(992\) 0 0
\(993\) −10.1772 57.7178i −0.322964 1.83162i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −36.5715 30.6871i −1.15823 0.971871i −0.158352 0.987383i \(-0.550618\pi\)
−0.999880 + 0.0155113i \(0.995062\pi\)
\(998\) 0 0
\(999\) −54.5121 + 31.4726i −1.72469 + 0.995749i
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 912.2.ch.d.671.1 yes 6
3.2 odd 2 CM 912.2.ch.d.671.1 yes 6
4.3 odd 2 912.2.ch.c.671.1 6
12.11 even 2 912.2.ch.c.671.1 6
19.16 even 9 912.2.ch.c.719.1 yes 6
57.35 odd 18 912.2.ch.c.719.1 yes 6
76.35 odd 18 inner 912.2.ch.d.719.1 yes 6
228.35 even 18 inner 912.2.ch.d.719.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.ch.c.671.1 6 4.3 odd 2
912.2.ch.c.671.1 6 12.11 even 2
912.2.ch.c.719.1 yes 6 19.16 even 9
912.2.ch.c.719.1 yes 6 57.35 odd 18
912.2.ch.d.671.1 yes 6 1.1 even 1 trivial
912.2.ch.d.671.1 yes 6 3.2 odd 2 CM
912.2.ch.d.719.1 yes 6 76.35 odd 18 inner
912.2.ch.d.719.1 yes 6 228.35 even 18 inner