Properties

Label 400.2.n.b.207.1
Level $400$
Weight $2$
Character 400.207
Analytic conductor $3.194$
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] = [400,2,Mod(143,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \( 2^{3} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 207.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 400.207
Dual form 400.2.n.b.143.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.73205 - 1.73205i) q^{3} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(-1.73205 - 1.73205i) q^{3} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} -3.46410i q^{11} +(-1.00000 + 1.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} -6.92820 q^{19} -6.00000 q^{21} +(-1.73205 - 1.73205i) q^{23} -4.00000i q^{29} +3.46410i q^{31} +(-6.00000 + 6.00000i) q^{33} +(-5.00000 - 5.00000i) q^{37} +3.46410 q^{39} +2.00000 q^{41} +(-1.73205 - 1.73205i) q^{43} +(1.73205 - 1.73205i) q^{47} +1.00000i q^{49} +3.46410i q^{51} +(7.00000 - 7.00000i) q^{53} +(12.0000 + 12.0000i) q^{57} +6.92820 q^{59} +6.00000 q^{61} +(5.19615 + 5.19615i) q^{63} +(-5.19615 + 5.19615i) q^{67} +6.00000i q^{69} -10.3923i q^{71} +(7.00000 - 7.00000i) q^{73} +(-6.00000 - 6.00000i) q^{77} +9.00000 q^{81} +(12.1244 + 12.1244i) q^{83} +(-6.92820 + 6.92820i) q^{87} -8.00000i q^{89} +3.46410i q^{91} +(6.00000 - 6.00000i) q^{93} +(7.00000 + 7.00000i) q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{13} - 4 q^{17} - 24 q^{21} - 24 q^{33} - 20 q^{37} + 8 q^{41} + 28 q^{53} + 48 q^{57} + 24 q^{61} + 28 q^{73} - 24 q^{77} + 36 q^{81} + 24 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73205 1.73205i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.73205 1.73205i 0.654654 0.654654i −0.299456 0.954110i \(-0.596805\pi\)
0.954110 + 0.299456i \(0.0968053\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.00000i −0.277350 + 0.277350i −0.832050 0.554700i \(-0.812833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.00000i −0.242536 0.242536i 0.575363 0.817898i \(-0.304861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) −1.73205 1.73205i −0.361158 0.361158i 0.503081 0.864239i \(-0.332200\pi\)
−0.864239 + 0.503081i \(0.832200\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i −0.928477 0.371391i \(-0.878881\pi\)
0.928477 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) −6.00000 + 6.00000i −1.04447 + 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 5.00000i −0.821995 0.821995i 0.164399 0.986394i \(-0.447432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.73205 1.73205i −0.264135 0.264135i 0.562596 0.826732i \(-0.309803\pi\)
−0.826732 + 0.562596i \(0.809803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.73205 1.73205i 0.252646 0.252646i −0.569409 0.822054i \(-0.692828\pi\)
0.822054 + 0.569409i \(0.192828\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 3.46410i 0.485071i
\(52\) 0 0
\(53\) 7.00000 7.00000i 0.961524 0.961524i −0.0377628 0.999287i \(-0.512023\pi\)
0.999287 + 0.0377628i \(0.0120231\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.0000 + 12.0000i 1.58944 + 1.58944i
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 5.19615 + 5.19615i 0.654654 + 0.654654i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.19615 + 5.19615i −0.634811 + 0.634811i −0.949271 0.314460i \(-0.898177\pi\)
0.314460 + 0.949271i \(0.398177\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 10.3923i 1.23334i −0.787222 0.616670i \(-0.788481\pi\)
0.787222 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) 7.00000 7.00000i 0.819288 0.819288i −0.166717 0.986005i \(-0.553317\pi\)
0.986005 + 0.166717i \(0.0533166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 6.00000i −0.683763 0.683763i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.1244 + 12.1244i 1.33082 + 1.33082i 0.904636 + 0.426185i \(0.140143\pi\)
0.426185 + 0.904636i \(0.359857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.92820 + 6.92820i −0.742781 + 0.742781i
\(88\) 0 0
\(89\) 8.00000i 0.847998i −0.905663 0.423999i \(-0.860626\pi\)
0.905663 0.423999i \(-0.139374\pi\)
\(90\) 0 0
\(91\) 3.46410i 0.363137i
\(92\) 0 0
\(93\) 6.00000 6.00000i 0.622171 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 + 7.00000i 0.710742 + 0.710742i 0.966691 0.255948i \(-0.0823876\pi\)
−0.255948 + 0.966691i \(0.582388\pi\)
\(98\) 0 0
\(99\) 10.3923 1.04447
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −1.73205 1.73205i −0.170664 0.170664i 0.616607 0.787271i \(-0.288507\pi\)
−0.787271 + 0.616607i \(0.788507\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.66025 8.66025i 0.837218 0.837218i −0.151274 0.988492i \(-0.548337\pi\)
0.988492 + 0.151274i \(0.0483374\pi\)
\(108\) 0 0
\(109\) 12.0000i 1.14939i −0.818367 0.574696i \(-0.805120\pi\)
0.818367 0.574696i \(-0.194880\pi\)
\(110\) 0 0
\(111\) 17.3205i 1.64399i
\(112\) 0 0
\(113\) −9.00000 + 9.00000i −0.846649 + 0.846649i −0.989713 0.143065i \(-0.954304\pi\)
0.143065 + 0.989713i \(0.454304\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.00000 3.00000i −0.277350 0.277350i
\(118\) 0 0
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) −3.46410 3.46410i −0.312348 0.312348i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.1244 + 12.1244i −1.07586 + 1.07586i −0.0789869 + 0.996876i \(0.525169\pi\)
−0.996876 + 0.0789869i \(0.974831\pi\)
\(128\) 0 0
\(129\) 6.00000i 0.528271i
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) −12.0000 + 12.0000i −1.04053 + 1.04053i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 9.00000i −0.768922 0.768922i 0.208995 0.977917i \(-0.432981\pi\)
−0.977917 + 0.208995i \(0.932981\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 3.46410 + 3.46410i 0.289683 + 0.289683i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.73205 1.73205i 0.142857 0.142857i
\(148\) 0 0
\(149\) 4.00000i 0.327693i −0.986486 0.163846i \(-0.947610\pi\)
0.986486 0.163846i \(-0.0523901\pi\)
\(150\) 0 0
\(151\) 10.3923i 0.845714i −0.906196 0.422857i \(-0.861027\pi\)
0.906196 0.422857i \(-0.138973\pi\)
\(152\) 0 0
\(153\) 3.00000 3.00000i 0.242536 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.0000 + 11.0000i 0.877896 + 0.877896i 0.993317 0.115421i \(-0.0368217\pi\)
−0.115421 + 0.993317i \(0.536822\pi\)
\(158\) 0 0
\(159\) −24.2487 −1.92305
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −1.73205 1.73205i −0.135665 0.135665i 0.636013 0.771678i \(-0.280582\pi\)
−0.771678 + 0.636013i \(0.780582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.5885 15.5885i 1.20627 1.20627i 0.234045 0.972226i \(-0.424804\pi\)
0.972226 0.234045i \(-0.0751964\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 20.7846i 1.58944i
\(172\) 0 0
\(173\) 15.0000 15.0000i 1.14043 1.14043i 0.152057 0.988372i \(-0.451410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 12.0000i −0.901975 0.901975i
\(178\) 0 0
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −10.3923 10.3923i −0.768221 0.768221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.46410 + 3.46410i −0.253320 + 0.253320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.2487i 1.75458i −0.479965 0.877288i \(-0.659351\pi\)
0.479965 0.877288i \(-0.340649\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.00000i −0.0719816 + 0.0719816i −0.742181 0.670199i \(-0.766209\pi\)
0.670199 + 0.742181i \(0.266209\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 + 3.00000i 0.213741 + 0.213741i 0.805855 0.592113i \(-0.201706\pi\)
−0.592113 + 0.805855i \(0.701706\pi\)
\(198\) 0 0
\(199\) 13.8564 0.982255 0.491127 0.871088i \(-0.336585\pi\)
0.491127 + 0.871088i \(0.336585\pi\)
\(200\) 0 0
\(201\) 18.0000 1.26962
\(202\) 0 0
\(203\) −6.92820 6.92820i −0.486265 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.19615 5.19615i 0.361158 0.361158i
\(208\) 0 0
\(209\) 24.0000i 1.66011i
\(210\) 0 0
\(211\) 10.3923i 0.715436i 0.933830 + 0.357718i \(0.116445\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(212\) 0 0
\(213\) −18.0000 + 18.0000i −1.23334 + 1.23334i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 + 6.00000i 0.407307 + 0.407307i
\(218\) 0 0
\(219\) −24.2487 −1.63858
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −15.5885 15.5885i −1.04388 1.04388i −0.998992 0.0448883i \(-0.985707\pi\)
−0.0448883 0.998992i \(-0.514293\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.66025 8.66025i 0.574801 0.574801i −0.358665 0.933466i \(-0.616768\pi\)
0.933466 + 0.358665i \(0.116768\pi\)
\(228\) 0 0
\(229\) 20.0000i 1.32164i 0.750546 + 0.660819i \(0.229791\pi\)
−0.750546 + 0.660819i \(0.770209\pi\)
\(230\) 0 0
\(231\) 20.7846i 1.36753i
\(232\) 0 0
\(233\) −1.00000 + 1.00000i −0.0655122 + 0.0655122i −0.739104 0.673592i \(-0.764751\pi\)
0.673592 + 0.739104i \(0.264751\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 27.7128 1.79259 0.896296 0.443455i \(-0.146248\pi\)
0.896296 + 0.443455i \(0.146248\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) −15.5885 15.5885i −1.00000 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.92820 6.92820i 0.440831 0.440831i
\(248\) 0 0
\(249\) 42.0000i 2.66164i
\(250\) 0 0
\(251\) 3.46410i 0.218652i −0.994006 0.109326i \(-0.965131\pi\)
0.994006 0.109326i \(-0.0348693\pi\)
\(252\) 0 0
\(253\) −6.00000 + 6.00000i −0.377217 + 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.00000 9.00000i −0.561405 0.561405i 0.368302 0.929706i \(-0.379939\pi\)
−0.929706 + 0.368302i \(0.879939\pi\)
\(258\) 0 0
\(259\) −17.3205 −1.07624
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 12.1244 + 12.1244i 0.747620 + 0.747620i 0.974032 0.226412i \(-0.0726995\pi\)
−0.226412 + 0.974032i \(0.572700\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −13.8564 + 13.8564i −0.847998 + 0.847998i
\(268\) 0 0
\(269\) 4.00000i 0.243884i 0.992537 + 0.121942i \(0.0389122\pi\)
−0.992537 + 0.121942i \(0.961088\pi\)
\(270\) 0 0
\(271\) 3.46410i 0.210429i 0.994450 + 0.105215i \(0.0335529\pi\)
−0.994450 + 0.105215i \(0.966447\pi\)
\(272\) 0 0
\(273\) 6.00000 6.00000i 0.363137 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.0000 13.0000i −0.781094 0.781094i 0.198921 0.980015i \(-0.436256\pi\)
−0.980015 + 0.198921i \(0.936256\pi\)
\(278\) 0 0
\(279\) −10.3923 −0.622171
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 12.1244 + 12.1244i 0.720718 + 0.720718i 0.968751 0.248033i \(-0.0797843\pi\)
−0.248033 + 0.968751i \(0.579784\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.46410 3.46410i 0.204479 0.204479i
\(288\) 0 0
\(289\) 15.0000i 0.882353i
\(290\) 0 0
\(291\) 24.2487i 1.42148i
\(292\) 0 0
\(293\) −9.00000 + 9.00000i −0.525786 + 0.525786i −0.919313 0.393527i \(-0.871255\pi\)
0.393527 + 0.919313i \(0.371255\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.46410 0.200334
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 17.3205 + 17.3205i 0.995037 + 0.995037i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.19615 + 5.19615i −0.296560 + 0.296560i −0.839665 0.543105i \(-0.817249\pi\)
0.543105 + 0.839665i \(0.317249\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 17.3205i 0.982156i 0.871116 + 0.491078i \(0.163397\pi\)
−0.871116 + 0.491078i \(0.836603\pi\)
\(312\) 0 0
\(313\) 7.00000 7.00000i 0.395663 0.395663i −0.481037 0.876700i \(-0.659740\pi\)
0.876700 + 0.481037i \(0.159740\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.0000 + 19.0000i 1.06715 + 1.06715i 0.997577 + 0.0695692i \(0.0221625\pi\)
0.0695692 + 0.997577i \(0.477838\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) −30.0000 −1.67444
\(322\) 0 0
\(323\) 6.92820 + 6.92820i 0.385496 + 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.7846 + 20.7846i −1.14939 + 1.14939i
\(328\) 0 0
\(329\) 6.00000i 0.330791i
\(330\) 0 0
\(331\) 24.2487i 1.33283i 0.745581 + 0.666415i \(0.232172\pi\)
−0.745581 + 0.666415i \(0.767828\pi\)
\(332\) 0 0
\(333\) 15.0000 15.0000i 0.821995 0.821995i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 1.00000i −0.0544735 0.0544735i 0.679345 0.733819i \(-0.262264\pi\)
−0.733819 + 0.679345i \(0.762264\pi\)
\(338\) 0 0
\(339\) 31.1769 1.69330
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 13.8564 + 13.8564i 0.748176 + 0.748176i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.19615 + 5.19615i −0.278944 + 0.278944i −0.832687 0.553743i \(-0.813199\pi\)
0.553743 + 0.832687i \(0.313199\pi\)
\(348\) 0 0
\(349\) 12.0000i 0.642345i 0.947021 + 0.321173i \(0.104077\pi\)
−0.947021 + 0.321173i \(0.895923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 + 1.00000i −0.0532246 + 0.0532246i −0.733218 0.679994i \(-0.761983\pi\)
0.679994 + 0.733218i \(0.261983\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6.00000 + 6.00000i 0.317554 + 0.317554i
\(358\) 0 0
\(359\) 13.8564 0.731313 0.365657 0.930750i \(-0.380844\pi\)
0.365657 + 0.930750i \(0.380844\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 1.73205 + 1.73205i 0.0909091 + 0.0909091i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.5885 15.5885i 0.813711 0.813711i −0.171477 0.985188i \(-0.554854\pi\)
0.985188 + 0.171477i \(0.0548540\pi\)
\(368\) 0 0
\(369\) 6.00000i 0.312348i
\(370\) 0 0
\(371\) 24.2487i 1.25893i
\(372\) 0 0
\(373\) −1.00000 + 1.00000i −0.0517780 + 0.0517780i −0.732522 0.680744i \(-0.761657\pi\)
0.680744 + 0.732522i \(0.261657\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000 + 4.00000i 0.206010 + 0.206010i
\(378\) 0 0
\(379\) 6.92820 0.355878 0.177939 0.984042i \(-0.443057\pi\)
0.177939 + 0.984042i \(0.443057\pi\)
\(380\) 0 0
\(381\) 42.0000 2.15173
\(382\) 0 0
\(383\) −15.5885 15.5885i −0.796533 0.796533i 0.186014 0.982547i \(-0.440443\pi\)
−0.982547 + 0.186014i \(0.940443\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.19615 5.19615i 0.264135 0.264135i
\(388\) 0 0
\(389\) 20.0000i 1.01404i −0.861934 0.507020i \(-0.830747\pi\)
0.861934 0.507020i \(-0.169253\pi\)
\(390\) 0 0
\(391\) 3.46410i 0.175187i
\(392\) 0 0
\(393\) 18.0000 18.0000i 0.907980 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.0000 + 11.0000i 0.552074 + 0.552074i 0.927039 0.374965i \(-0.122345\pi\)
−0.374965 + 0.927039i \(0.622345\pi\)
\(398\) 0 0
\(399\) 41.5692 2.08106
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) −3.46410 3.46410i −0.172559 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.3205 + 17.3205i −0.858546 + 0.858546i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 31.1769i 1.53784i
\(412\) 0 0
\(413\) 12.0000 12.0000i 0.590481 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 12.0000i −0.587643 0.587643i
\(418\) 0 0
\(419\) −6.92820 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 5.19615 + 5.19615i 0.252646 + 0.252646i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.3923 10.3923i 0.502919 0.502919i
\(428\) 0 0
\(429\) 12.0000i 0.579365i
\(430\) 0 0
\(431\) 3.46410i 0.166860i 0.996514 + 0.0834300i \(0.0265875\pi\)
−0.996514 + 0.0834300i \(0.973413\pi\)
\(432\) 0 0
\(433\) 7.00000 7.00000i 0.336399 0.336399i −0.518611 0.855010i \(-0.673551\pi\)
0.855010 + 0.518611i \(0.173551\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 + 12.0000i 0.574038 + 0.574038i
\(438\) 0 0
\(439\) −13.8564 −0.661330 −0.330665 0.943748i \(-0.607273\pi\)
−0.330665 + 0.943748i \(0.607273\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.1244 + 12.1244i 0.576046 + 0.576046i 0.933811 0.357766i \(-0.116461\pi\)
−0.357766 + 0.933811i \(0.616461\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.92820 + 6.92820i −0.327693 + 0.327693i
\(448\) 0 0
\(449\) 8.00000i 0.377543i 0.982021 + 0.188772i \(0.0604506\pi\)
−0.982021 + 0.188772i \(0.939549\pi\)
\(450\) 0 0
\(451\) 6.92820i 0.326236i
\(452\) 0 0
\(453\) −18.0000 + 18.0000i −0.845714 + 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 17.0000i −0.795226 0.795226i 0.187112 0.982339i \(-0.440087\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −1.73205 1.73205i −0.0804952 0.0804952i 0.665713 0.746208i \(-0.268128\pi\)
−0.746208 + 0.665713i \(0.768128\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.0526 + 19.0526i −0.881647 + 0.881647i −0.993702 0.112055i \(-0.964257\pi\)
0.112055 + 0.993702i \(0.464257\pi\)
\(468\) 0 0
\(469\) 18.0000i 0.831163i
\(470\) 0 0
\(471\) 38.1051i 1.75579i
\(472\) 0 0
\(473\) −6.00000 + 6.00000i −0.275880 + 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 21.0000 + 21.0000i 0.961524 + 0.961524i
\(478\) 0 0
\(479\) −27.7128 −1.26623 −0.633115 0.774057i \(-0.718224\pi\)
−0.633115 + 0.774057i \(0.718224\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 10.3923 + 10.3923i 0.472866 + 0.472866i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.73205 1.73205i 0.0784867 0.0784867i −0.666774 0.745260i \(-0.732325\pi\)
0.745260 + 0.666774i \(0.232325\pi\)
\(488\) 0 0
\(489\) 6.00000i 0.271329i
\(490\) 0 0
\(491\) 24.2487i 1.09433i 0.837025 + 0.547165i \(0.184293\pi\)
−0.837025 + 0.547165i \(0.815707\pi\)
\(492\) 0 0
\(493\) −4.00000 + 4.00000i −0.180151 + 0.180151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.0000 18.0000i −0.807410 0.807410i
\(498\) 0 0
\(499\) 20.7846 0.930447 0.465223 0.885193i \(-0.345974\pi\)
0.465223 + 0.885193i \(0.345974\pi\)
\(500\) 0 0
\(501\) −54.0000 −2.41254
\(502\) 0 0
\(503\) −1.73205 1.73205i −0.0772283 0.0772283i 0.667437 0.744666i \(-0.267391\pi\)
−0.744666 + 0.667437i \(0.767391\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.0526 19.0526i 0.846154 0.846154i
\(508\) 0 0
\(509\) 20.0000i 0.886484i −0.896402 0.443242i \(-0.853828\pi\)
0.896402 0.443242i \(-0.146172\pi\)
\(510\) 0 0
\(511\) 24.2487i 1.07270i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.00000 6.00000i −0.263880 0.263880i
\(518\) 0 0
\(519\) −51.9615 −2.28086
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 12.1244 + 12.1244i 0.530161 + 0.530161i 0.920620 0.390459i \(-0.127684\pi\)
−0.390459 + 0.920620i \(0.627684\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46410 3.46410i 0.150899 0.150899i
\(528\) 0 0
\(529\) 17.0000i 0.739130i
\(530\) 0 0
\(531\) 20.7846i 0.901975i
\(532\) 0 0
\(533\) −2.00000 + 2.00000i −0.0866296 + 0.0866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 + 12.0000i 0.517838 + 0.517838i
\(538\) 0 0
\(539\) 3.46410 0.149209
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) −10.3923 10.3923i −0.445976 0.445976i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.66025 8.66025i 0.370286 0.370286i −0.497296 0.867581i \(-0.665674\pi\)
0.867581 + 0.497296i \(0.165674\pi\)
\(548\) 0 0
\(549\) 18.0000i 0.768221i
\(550\) 0 0
\(551\) 27.7128i 1.18061i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.0000 29.0000i −1.22877 1.22877i −0.964430 0.264340i \(-0.914846\pi\)
−0.264340 0.964430i \(-0.585154\pi\)
\(558\) 0 0
\(559\) 3.46410 0.146516
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −29.4449 29.4449i −1.24095 1.24095i −0.959606 0.281347i \(-0.909219\pi\)
−0.281347 0.959606i \(-0.590781\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 15.5885 15.5885i 0.654654 0.654654i
\(568\) 0 0
\(569\) 16.0000i 0.670755i 0.942084 + 0.335377i \(0.108864\pi\)
−0.942084 + 0.335377i \(0.891136\pi\)
\(570\) 0 0
\(571\) 31.1769i 1.30471i −0.757912 0.652357i \(-0.773780\pi\)
0.757912 0.652357i \(-0.226220\pi\)
\(572\) 0 0
\(573\) −42.0000 + 42.0000i −1.75458 + 1.75458i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000 + 7.00000i 0.291414 + 0.291414i 0.837639 0.546225i \(-0.183936\pi\)
−0.546225 + 0.837639i \(0.683936\pi\)
\(578\) 0 0
\(579\) 3.46410 0.143963
\(580\) 0 0
\(581\) 42.0000 1.74245
\(582\) 0 0
\(583\) −24.2487 24.2487i −1.00428 1.00428i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.19615 + 5.19615i −0.214468 + 0.214468i −0.806162 0.591694i \(-0.798459\pi\)
0.591694 + 0.806162i \(0.298459\pi\)
\(588\) 0 0
\(589\) 24.0000i 0.988903i
\(590\) 0 0
\(591\) 10.3923i 0.427482i
\(592\) 0 0
\(593\) −33.0000 + 33.0000i −1.35515 + 1.35515i −0.475352 + 0.879796i \(0.657679\pi\)
−0.879796 + 0.475352i \(0.842321\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.0000 24.0000i −0.982255 0.982255i
\(598\) 0 0
\(599\) −13.8564 −0.566157 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 0 0
\(601\) 18.0000 0.734235 0.367118 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) 0 0
\(603\) −15.5885 15.5885i −0.634811 0.634811i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −25.9808 + 25.9808i −1.05453 + 1.05453i −0.0561015 + 0.998425i \(0.517867\pi\)
−0.998425 + 0.0561015i \(0.982133\pi\)
\(608\) 0 0
\(609\) 24.0000i 0.972529i
\(610\) 0 0
\(611\) 3.46410i 0.140143i
\(612\) 0 0
\(613\) 31.0000 31.0000i 1.25208 1.25208i 0.297291 0.954787i \(-0.403917\pi\)
0.954787 0.297291i \(-0.0960833\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.00000 + 7.00000i 0.281809 + 0.281809i 0.833830 0.552021i \(-0.186143\pi\)
−0.552021 + 0.833830i \(0.686143\pi\)
\(618\) 0 0
\(619\) −20.7846 −0.835404 −0.417702 0.908584i \(-0.637164\pi\)
−0.417702 + 0.908584i \(0.637164\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.8564 13.8564i −0.555145 0.555145i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 41.5692 41.5692i 1.66011 1.66011i
\(628\) 0 0
\(629\) 10.0000i 0.398726i
\(630\) 0 0
\(631\) 17.3205i 0.689519i 0.938691 + 0.344759i \(0.112039\pi\)
−0.938691 + 0.344759i \(0.887961\pi\)
\(632\) 0 0
\(633\) 18.0000 18.0000i 0.715436 0.715436i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 1.00000i −0.0396214 0.0396214i
\(638\) 0 0
\(639\) 31.1769 1.23334
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −15.5885 15.5885i −0.614749 0.614749i 0.329431 0.944180i \(-0.393143\pi\)
−0.944180 + 0.329431i \(0.893143\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.73205 1.73205i 0.0680939 0.0680939i −0.672240 0.740334i \(-0.734668\pi\)
0.740334 + 0.672240i \(0.234668\pi\)
\(648\) 0 0
\(649\) 24.0000i 0.942082i
\(650\) 0 0
\(651\) 20.7846i 0.814613i
\(652\) 0 0
\(653\) 23.0000 23.0000i 0.900060 0.900060i −0.0953813 0.995441i \(-0.530407\pi\)
0.995441 + 0.0953813i \(0.0304070\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 21.0000 + 21.0000i 0.819288 + 0.819288i
\(658\) 0 0
\(659\) −34.6410 −1.34942 −0.674711 0.738082i \(-0.735732\pi\)
−0.674711 + 0.738082i \(0.735732\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) −3.46410 3.46410i −0.134535 0.134535i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.92820 + 6.92820i −0.268261 + 0.268261i
\(668\) 0 0
\(669\) 54.0000i 2.08776i
\(670\) 0 0
\(671\) 20.7846i 0.802381i
\(672\) 0 0
\(673\) −25.0000 + 25.0000i −0.963679 + 0.963679i −0.999363 0.0356839i \(-0.988639\pi\)
0.0356839 + 0.999363i \(0.488639\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.0000 + 35.0000i 1.34516 + 1.34516i 0.890838 + 0.454321i \(0.150118\pi\)
0.454321 + 0.890838i \(0.349882\pi\)
\(678\) 0 0
\(679\) 24.2487 0.930580
\(680\) 0 0
\(681\) −30.0000 −1.14960
\(682\) 0 0
\(683\) −1.73205 1.73205i −0.0662751 0.0662751i 0.673192 0.739467i \(-0.264923\pi\)
−0.739467 + 0.673192i \(0.764923\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 34.6410 34.6410i 1.32164 1.32164i
\(688\) 0 0
\(689\) 14.0000i 0.533358i
\(690\) 0 0
\(691\) 45.0333i 1.71315i −0.516024 0.856574i \(-0.672588\pi\)
0.516024 0.856574i \(-0.327412\pi\)
\(692\) 0 0
\(693\) 18.0000 18.0000i 0.683763 0.683763i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.00000 2.00000i −0.0757554 0.0757554i
\(698\) 0 0
\(699\) 3.46410 0.131024
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) 34.6410 + 34.6410i 1.30651 + 1.30651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.3205 + 17.3205i −0.651405 + 0.651405i
\(708\) 0 0
\(709\) 4.00000i 0.150223i 0.997175 + 0.0751116i \(0.0239313\pi\)
−0.997175 + 0.0751116i \(0.976069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.00000 6.00000i 0.224702 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −48.0000 48.0000i −1.79259 1.79259i
\(718\) 0 0
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) 38.1051 + 38.1051i 1.41714 + 1.41714i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.73205 1.73205i 0.0642382 0.0642382i −0.674258 0.738496i \(-0.735536\pi\)
0.738496 + 0.674258i \(0.235536\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 3.46410i 0.128124i
\(732\) 0 0
\(733\) −1.00000 + 1.00000i −0.0369358 + 0.0369358i −0.725333 0.688398i \(-0.758314\pi\)
0.688398 + 0.725333i \(0.258314\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.0000 + 18.0000i 0.663039 + 0.663039i
\(738\) 0 0
\(739\) 20.7846 0.764574 0.382287 0.924044i \(-0.375137\pi\)
0.382287 + 0.924044i \(0.375137\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 12.1244 + 12.1244i 0.444799 + 0.444799i 0.893621 0.448822i \(-0.148156\pi\)
−0.448822 + 0.893621i \(0.648156\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −36.3731 + 36.3731i −1.33082 + 1.33082i
\(748\) 0 0
\(749\) 30.0000i 1.09618i
\(750\) 0 0
\(751\) 3.46410i 0.126407i 0.998001 + 0.0632034i \(0.0201317\pi\)
−0.998001 + 0.0632034i \(0.979868\pi\)
\(752\) 0 0
\(753\) −6.00000 + 6.00000i −0.218652 + 0.218652i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −29.0000 29.0000i −1.05402 1.05402i −0.998455 0.0555680i \(-0.982303\pi\)
−0.0555680 0.998455i \(-0.517697\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) −20.7846 20.7846i −0.752453 0.752453i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.92820 + 6.92820i −0.250163 + 0.250163i
\(768\) 0 0
\(769\) 48.0000i 1.73092i −0.500974 0.865462i \(-0.667025\pi\)
0.500974 0.865462i \(-0.332975\pi\)
\(770\) 0 0
\(771\) 31.1769i 1.12281i
\(772\) 0 0
\(773\) −1.00000 + 1.00000i −0.0359675 + 0.0359675i −0.724862 0.688894i \(-0.758096\pi\)
0.688894 + 0.724862i \(0.258096\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 30.0000 + 30.0000i 1.07624 + 1.07624i
\(778\) 0 0
\(779\) −13.8564 −0.496457
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −5.19615 + 5.19615i −0.185223 + 0.185223i −0.793627 0.608404i \(-0.791810\pi\)
0.608404 + 0.793627i \(0.291810\pi\)
\(788\) 0 0
\(789\) 42.0000i 1.49524i
\(790\) 0 0
\(791\) 31.1769i 1.10852i
\(792\) 0 0
\(793\) −6.00000 + 6.00000i −0.213066 + 0.213066i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.00000 + 3.00000i 0.106265 + 0.106265i 0.758240 0.651975i \(-0.226059\pi\)
−0.651975 + 0.758240i \(0.726059\pi\)
\(798\) 0 0
\(799\) −3.46410 −0.122551
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) −24.2487 24.2487i −0.855718 0.855718i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.92820 6.92820i 0.243884 0.243884i
\(808\) 0 0
\(809\) 8.00000i 0.281265i 0.990062 + 0.140633i \(0.0449136\pi\)
−0.990062 + 0.140633i \(0.955086\pi\)
\(810\) 0 0
\(811\) 24.2487i 0.851487i 0.904844 + 0.425744i \(0.139987\pi\)
−0.904844 + 0.425744i \(0.860013\pi\)
\(812\) 0 0
\(813\) 6.00000 6.00000i 0.210429 0.210429i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.0000 + 12.0000i 0.419827 + 0.419827i
\(818\) 0 0
\(819\) −10.3923 −0.363137
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 25.9808 + 25.9808i 0.905632 + 0.905632i 0.995916 0.0902837i \(-0.0287774\pi\)
−0.0902837 + 0.995916i \(0.528777\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.19615 + 5.19615i −0.180688 + 0.180688i −0.791656 0.610968i \(-0.790781\pi\)
0.610968 + 0.791656i \(0.290781\pi\)
\(828\) 0 0
\(829\) 12.0000i 0.416777i −0.978046 0.208389i \(-0.933178\pi\)
0.978046 0.208389i \(-0.0668219\pi\)
\(830\) 0 0
\(831\) 45.0333i 1.56219i
\(832\) 0 0
\(833\) 1.00000 1.00000i 0.0346479 0.0346479i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.5692 1.43513 0.717564 0.696492i \(-0.245257\pi\)
0.717564 + 0.696492i \(0.245257\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) 24.2487 + 24.2487i 0.835170 + 0.835170i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.73205 + 1.73205i −0.0595140 + 0.0595140i
\(848\) 0 0
\(849\) 42.0000i 1.44144i
\(850\) 0 0
\(851\) 17.3205i 0.593739i
\(852\) 0 0
\(853\) −1.00000 + 1.00000i −0.0342393 + 0.0342393i −0.724019 0.689780i \(-0.757707\pi\)
0.689780 + 0.724019i \(0.257707\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.0000 25.0000i −0.853984 0.853984i 0.136637 0.990621i \(-0.456370\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) 0 0
\(859\) 6.92820 0.236387 0.118194 0.992991i \(-0.462290\pi\)
0.118194 + 0.992991i \(0.462290\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −1.73205 1.73205i −0.0589597 0.0589597i 0.677012 0.735972i \(-0.263274\pi\)
−0.735972 + 0.677012i \(0.763274\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −25.9808 + 25.9808i −0.882353 + 0.882353i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 10.3923i 0.352130i
\(872\) 0 0
\(873\) −21.0000 + 21.0000i −0.710742 + 0.710742i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.00000 5.00000i −0.168838 0.168838i 0.617630 0.786468i \(-0.288093\pi\)
−0.786468 + 0.617630i \(0.788093\pi\)
\(878\) 0 0
\(879\) 31.1769 1.05157
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) 25.9808 + 25.9808i 0.874322 + 0.874322i 0.992940 0.118618i \(-0.0378463\pi\)
−0.118618 + 0.992940i \(0.537846\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25.9808 + 25.9808i −0.872349 + 0.872349i −0.992728 0.120379i \(-0.961589\pi\)
0.120379 + 0.992728i \(0.461589\pi\)
\(888\) 0 0
\(889\) 42.0000i 1.40863i
\(890\) 0 0
\(891\) 31.1769i 1.04447i
\(892\) 0 0
\(893\) −12.0000 + 12.0000i −0.401565 + 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 6.00000i −0.200334 0.200334i
\(898\) 0 0
\(899\) 13.8564 0.462137
\(900\) 0 0
\(901\) −14.0000 −0.466408
\(902\) 0 0
\(903\) 10.3923 + 10.3923i 0.345834 + 0.345834i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 36.3731 36.3731i 1.20775 1.20775i 0.235993 0.971755i \(-0.424166\pi\)
0.971755 0.235993i \(-0.0758343\pi\)
\(908\) 0 0
\(909\) 30.0000i 0.995037i
\(910\) 0 0
\(911\) 31.1769i 1.03294i 0.856306 + 0.516469i \(0.172754\pi\)
−0.856306 + 0.516469i \(0.827246\pi\)
\(912\) 0 0
\(913\) 42.0000 42.0000i 1.39000 1.39000i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.0000 + 18.0000i 0.594412 + 0.594412i
\(918\) 0 0
\(919\) 41.5692 1.37124 0.685621 0.727959i \(-0.259531\pi\)
0.685621 + 0.727959i \(0.259531\pi\)
\(920\) 0 0
\(921\) 18.0000 0.593120
\(922\) 0 0
\(923\) 10.3923 + 10.3923i 0.342067 + 0.342067i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 5.19615 5.19615i 0.170664 0.170664i
\(928\) 0 0
\(929\) 56.0000i 1.83730i 0.395072 + 0.918650i \(0.370720\pi\)
−0.395072 + 0.918650i \(0.629280\pi\)
\(930\) 0 0
\(931\) 6.92820i 0.227063i
\(932\) 0 0
\(933\) 30.0000 30.0000i 0.982156 0.982156i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0000 + 23.0000i 0.751377 + 0.751377i 0.974736 0.223359i \(-0.0717022\pi\)
−0.223359 + 0.974736i \(0.571702\pi\)
\(938\) 0 0
\(939\) −24.2487 −0.791327
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) −3.46410 3.46410i −0.112807 0.112807i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.19615 + 5.19615i −0.168852 + 0.168852i −0.786475 0.617622i \(-0.788096\pi\)
0.617622 + 0.786475i \(0.288096\pi\)
\(948\) 0 0
\(949\) 14.0000i 0.454459i
\(950\) 0 0
\(951\) 65.8179i 2.13429i
\(952\) 0 0
\(953\) −9.00000 + 9.00000i −0.291539 + 0.291539i −0.837688 0.546149i \(-0.816093\pi\)
0.546149 + 0.837688i \(0.316093\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.0000 + 24.0000i 0.775810 + 0.775810i
\(958\) 0 0
\(959\) −31.1769 −1.00676
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 25.9808 + 25.9808i 0.837218 + 0.837218i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 15.5885 15.5885i 0.501291 0.501291i −0.410548 0.911839i \(-0.634663\pi\)
0.911839 + 0.410548i \(0.134663\pi\)
\(968\) 0 0
\(969\) 24.0000i 0.770991i
\(970\) 0 0
\(971\) 3.46410i 0.111168i −0.998454 0.0555842i \(-0.982298\pi\)
0.998454 0.0555842i \(-0.0177021\pi\)
\(972\) 0 0
\(973\) 12.0000 12.0000i 0.384702 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.0000 + 23.0000i 0.735835 + 0.735835i 0.971769 0.235934i \(-0.0758149\pi\)
−0.235934 + 0.971769i \(0.575815\pi\)
\(978\) 0 0
\(979\) −27.7128 −0.885705
\(980\) 0 0
\(981\) 36.0000 1.14939
\(982\) 0 0
\(983\) −15.5885 15.5885i −0.497195 0.497195i 0.413369 0.910564i \(-0.364352\pi\)
−0.910564 + 0.413369i \(0.864352\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −10.3923 + 10.3923i −0.330791 + 0.330791i
\(988\) 0 0
\(989\) 6.00000i 0.190789i
\(990\) 0 0
\(991\) 3.46410i 0.110041i 0.998485 + 0.0550204i \(0.0175224\pi\)
−0.998485 + 0.0550204i \(0.982478\pi\)
\(992\) 0 0
\(993\) 42.0000 42.0000i 1.33283 1.33283i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.00000 5.00000i −0.158352 0.158352i 0.623484 0.781836i \(-0.285717\pi\)
−0.781836 + 0.623484i \(0.785717\pi\)
\(998\) 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 400.2.n.b.207.1 4
3.2 odd 2 3600.2.x.e.3007.2 4
4.3 odd 2 inner 400.2.n.b.207.2 4
5.2 odd 4 80.2.n.b.63.1 yes 4
5.3 odd 4 inner 400.2.n.b.143.2 4
5.4 even 2 80.2.n.b.47.2 yes 4
8.3 odd 2 1600.2.n.r.1407.1 4
8.5 even 2 1600.2.n.r.1407.2 4
12.11 even 2 3600.2.x.e.3007.1 4
15.2 even 4 720.2.x.d.703.2 4
15.8 even 4 3600.2.x.e.2143.1 4
15.14 odd 2 720.2.x.d.127.1 4
20.3 even 4 inner 400.2.n.b.143.1 4
20.7 even 4 80.2.n.b.63.2 yes 4
20.19 odd 2 80.2.n.b.47.1 4
40.3 even 4 1600.2.n.r.1343.2 4
40.13 odd 4 1600.2.n.r.1343.1 4
40.19 odd 2 320.2.n.i.127.2 4
40.27 even 4 320.2.n.i.63.1 4
40.29 even 2 320.2.n.i.127.1 4
40.37 odd 4 320.2.n.i.63.2 4
60.23 odd 4 3600.2.x.e.2143.2 4
60.47 odd 4 720.2.x.d.703.1 4
60.59 even 2 720.2.x.d.127.2 4
80.19 odd 4 1280.2.o.q.127.2 4
80.27 even 4 1280.2.o.q.383.1 4
80.29 even 4 1280.2.o.q.127.1 4
80.37 odd 4 1280.2.o.q.383.2 4
80.59 odd 4 1280.2.o.r.127.1 4
80.67 even 4 1280.2.o.r.383.2 4
80.69 even 4 1280.2.o.r.127.2 4
80.77 odd 4 1280.2.o.r.383.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.n.b.47.1 4 20.19 odd 2
80.2.n.b.47.2 yes 4 5.4 even 2
80.2.n.b.63.1 yes 4 5.2 odd 4
80.2.n.b.63.2 yes 4 20.7 even 4
320.2.n.i.63.1 4 40.27 even 4
320.2.n.i.63.2 4 40.37 odd 4
320.2.n.i.127.1 4 40.29 even 2
320.2.n.i.127.2 4 40.19 odd 2
400.2.n.b.143.1 4 20.3 even 4 inner
400.2.n.b.143.2 4 5.3 odd 4 inner
400.2.n.b.207.1 4 1.1 even 1 trivial
400.2.n.b.207.2 4 4.3 odd 2 inner
720.2.x.d.127.1 4 15.14 odd 2
720.2.x.d.127.2 4 60.59 even 2
720.2.x.d.703.1 4 60.47 odd 4
720.2.x.d.703.2 4 15.2 even 4
1280.2.o.q.127.1 4 80.29 even 4
1280.2.o.q.127.2 4 80.19 odd 4
1280.2.o.q.383.1 4 80.27 even 4
1280.2.o.q.383.2 4 80.37 odd 4
1280.2.o.r.127.1 4 80.59 odd 4
1280.2.o.r.127.2 4 80.69 even 4
1280.2.o.r.383.1 4 80.77 odd 4
1280.2.o.r.383.2 4 80.67 even 4
1600.2.n.r.1343.1 4 40.13 odd 4
1600.2.n.r.1343.2 4 40.3 even 4
1600.2.n.r.1407.1 4 8.3 odd 2
1600.2.n.r.1407.2 4 8.5 even 2
3600.2.x.e.2143.1 4 15.8 even 4
3600.2.x.e.2143.2 4 60.23 odd 4
3600.2.x.e.3007.1 4 12.11 even 2
3600.2.x.e.3007.2 4 3.2 odd 2