Properties

Label 378.2.d.c.377.4
Level $378$
Weight $2$
Character 378.377
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(377,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.d (of order \(2\), degree \(1\), 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 377.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 378.377
Dual form 378.2.d.c.377.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +3.46410 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +3.46410i q^{10} -6.00000i q^{11} -1.73205i q^{13} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -1.73205 q^{17} +6.92820i q^{19} -3.46410 q^{20} +6.00000 q^{22} +3.00000i q^{23} +7.00000 q^{25} +1.73205 q^{26} +(-2.00000 - 1.73205i) q^{28} -3.00000i q^{29} +5.19615i q^{31} +1.00000i q^{32} -1.73205i q^{34} +(6.92820 + 6.00000i) q^{35} -2.00000 q^{37} -6.92820 q^{38} -3.46410i q^{40} +6.92820 q^{41} -11.0000 q^{43} +6.00000i q^{44} -3.00000 q^{46} -6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +7.00000i q^{50} +1.73205i q^{52} -3.00000i q^{53} -20.7846i q^{55} +(1.73205 - 2.00000i) q^{56} +3.00000 q^{58} -8.66025 q^{59} -13.8564i q^{61} -5.19615 q^{62} -1.00000 q^{64} -6.00000i q^{65} -7.00000 q^{67} +1.73205 q^{68} +(-6.00000 + 6.92820i) q^{70} -3.00000i q^{71} +6.92820i q^{73} -2.00000i q^{74} -6.92820i q^{76} +(10.3923 - 12.0000i) q^{77} +8.00000 q^{79} +3.46410 q^{80} +6.92820i q^{82} -3.46410 q^{83} -6.00000 q^{85} -11.0000i q^{86} -6.00000 q^{88} +5.19615 q^{89} +(3.00000 - 3.46410i) q^{91} -3.00000i q^{92} -6.92820i q^{94} +24.0000i q^{95} -6.92820i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} + 24 q^{22} + 28 q^{25} - 8 q^{28} - 8 q^{37} - 44 q^{43} - 12 q^{46} + 4 q^{49} + 12 q^{58} - 4 q^{64} - 28 q^{67} - 24 q^{70} + 32 q^{79} - 24 q^{85} - 24 q^{88} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.46410i 1.09545i
\(11\) 6.00000i 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) −1.73205 + 2.00000i −0.462910 + 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) −3.46410 −0.774597
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 1.73205 0.339683
\(27\) 0 0
\(28\) −2.00000 1.73205i −0.377964 0.327327i
\(29\) 3.00000i 0.557086i −0.960424 0.278543i \(-0.910149\pi\)
0.960424 0.278543i \(-0.0898515\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i 0.884454 + 0.466628i \(0.154531\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.73205i 0.297044i
\(35\) 6.92820 + 6.00000i 1.17108 + 1.01419i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.92820 −1.12390
\(39\) 0 0
\(40\) 3.46410i 0.547723i
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 6.00000i 0.904534i
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 7.00000i 0.989949i
\(51\) 0 0
\(52\) 1.73205i 0.240192i
\(53\) 3.00000i 0.412082i −0.978543 0.206041i \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) 0 0
\(55\) 20.7846i 2.80260i
\(56\) 1.73205 2.00000i 0.231455 0.267261i
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −8.66025 −1.12747 −0.563735 0.825956i \(-0.690636\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i −0.461644 0.887066i \(-0.652740\pi\)
0.461644 0.887066i \(-0.347260\pi\)
\(62\) −5.19615 −0.659912
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 1.73205 0.210042
\(69\) 0 0
\(70\) −6.00000 + 6.92820i −0.717137 + 0.828079i
\(71\) 3.00000i 0.356034i −0.984027 0.178017i \(-0.943032\pi\)
0.984027 0.178017i \(-0.0569683\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 2.00000i 0.232495i
\(75\) 0 0
\(76\) 6.92820i 0.794719i
\(77\) 10.3923 12.0000i 1.18431 1.36753i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 3.46410 0.387298
\(81\) 0 0
\(82\) 6.92820i 0.765092i
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 11.0000i 1.18616i
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) 3.00000 3.46410i 0.314485 0.363137i
\(92\) 3.00000i 0.312772i
\(93\) 0 0
\(94\) 6.92820i 0.714590i
\(95\) 24.0000i 2.46235i
\(96\) 0 0
\(97\) 6.92820i 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) −6.92820 + 1.00000i −0.699854 + 0.101015i
\(99\) 0 0
\(100\) −7.00000 −0.700000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.66025i 0.853320i −0.904412 0.426660i \(-0.859690\pi\)
0.904412 0.426660i \(-0.140310\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 20.7846 1.98173
\(111\) 0 0
\(112\) 2.00000 + 1.73205i 0.188982 + 0.163663i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 10.3923i 0.969087i
\(116\) 3.00000i 0.278543i
\(117\) 0 0
\(118\) 8.66025i 0.797241i
\(119\) −3.46410 3.00000i −0.317554 0.275010i
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 13.8564 1.25450
\(123\) 0 0
\(124\) 5.19615i 0.466628i
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −5.19615 −0.453990 −0.226995 0.973896i \(-0.572890\pi\)
−0.226995 + 0.973896i \(0.572890\pi\)
\(132\) 0 0
\(133\) −12.0000 + 13.8564i −1.04053 + 1.20150i
\(134\) 7.00000i 0.604708i
\(135\) 0 0
\(136\) 1.73205i 0.148522i
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i −0.989150 0.146911i \(-0.953067\pi\)
0.989150 0.146911i \(-0.0469330\pi\)
\(140\) −6.92820 6.00000i −0.585540 0.507093i
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) −6.92820 −0.573382
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 15.0000i 1.22885i −0.788976 0.614424i \(-0.789388\pi\)
0.788976 0.614424i \(-0.210612\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.92820 0.561951
\(153\) 0 0
\(154\) 12.0000 + 10.3923i 0.966988 + 0.837436i
\(155\) 18.0000i 1.44579i
\(156\) 0 0
\(157\) 15.5885i 1.24409i 0.782980 + 0.622047i \(0.213699\pi\)
−0.782980 + 0.622047i \(0.786301\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 3.46410i 0.273861i
\(161\) −5.19615 + 6.00000i −0.409514 + 0.472866i
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −6.92820 −0.541002
\(165\) 0 0
\(166\) 3.46410i 0.268866i
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 6.00000i 0.460179i
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) −20.7846 −1.58022 −0.790112 0.612962i \(-0.789978\pi\)
−0.790112 + 0.612962i \(0.789978\pi\)
\(174\) 0 0
\(175\) 14.0000 + 12.1244i 1.05830 + 0.916515i
\(176\) 6.00000i 0.452267i
\(177\) 0 0
\(178\) 5.19615i 0.389468i
\(179\) 6.00000i 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 1.73205i 0.128742i −0.997926 0.0643712i \(-0.979496\pi\)
0.997926 0.0643712i \(-0.0205042\pi\)
\(182\) 3.46410 + 3.00000i 0.256776 + 0.222375i
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 10.3923i 0.759961i
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) 24.0000i 1.73658i 0.496058 + 0.868290i \(0.334780\pi\)
−0.496058 + 0.868290i \(0.665220\pi\)
\(192\) 0 0
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 6.92820 0.497416
\(195\) 0 0
\(196\) −1.00000 6.92820i −0.0714286 0.494872i
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) 7.00000i 0.494975i
\(201\) 0 0
\(202\) 0 0
\(203\) 5.19615 6.00000i 0.364698 0.421117i
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 8.66025 0.603388
\(207\) 0 0
\(208\) 1.73205i 0.120096i
\(209\) 41.5692 2.87540
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 3.00000i 0.206041i
\(213\) 0 0
\(214\) 0 0
\(215\) −38.1051 −2.59875
\(216\) 0 0
\(217\) −9.00000 + 10.3923i −0.610960 + 0.705476i
\(218\) 4.00000i 0.270914i
\(219\) 0 0
\(220\) 20.7846i 1.40130i
\(221\) 3.00000i 0.201802i
\(222\) 0 0
\(223\) 10.3923i 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) −1.73205 + 2.00000i −0.115728 + 0.133631i
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 25.9808 1.72440 0.862202 0.506565i \(-0.169085\pi\)
0.862202 + 0.506565i \(0.169085\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i 0.726900 + 0.686743i \(0.240960\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −10.3923 −0.685248
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 8.66025 0.563735
\(237\) 0 0
\(238\) 3.00000 3.46410i 0.194461 0.224544i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i −0.894891 0.446285i \(-0.852747\pi\)
0.894891 0.446285i \(-0.147253\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 0 0
\(244\) 13.8564i 0.887066i
\(245\) 3.46410 + 24.0000i 0.221313 + 1.53330i
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 5.19615 0.329956
\(249\) 0 0
\(250\) 6.92820i 0.438178i
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 14.0000i 0.878438i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.7846 −1.29651 −0.648254 0.761424i \(-0.724501\pi\)
−0.648254 + 0.761424i \(0.724501\pi\)
\(258\) 0 0
\(259\) −4.00000 3.46410i −0.248548 0.215249i
\(260\) 6.00000i 0.372104i
\(261\) 0 0
\(262\) 5.19615i 0.321019i
\(263\) 9.00000i 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 0 0
\(265\) 10.3923i 0.638394i
\(266\) −13.8564 12.0000i −0.849591 0.735767i
\(267\) 0 0
\(268\) 7.00000 0.427593
\(269\) −3.46410 −0.211210 −0.105605 0.994408i \(-0.533678\pi\)
−0.105605 + 0.994408i \(0.533678\pi\)
\(270\) 0 0
\(271\) 12.1244i 0.736502i −0.929726 0.368251i \(-0.879957\pi\)
0.929726 0.368251i \(-0.120043\pi\)
\(272\) −1.73205 −0.105021
\(273\) 0 0
\(274\) 0 0
\(275\) 42.0000i 2.53270i
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 3.46410 0.207763
\(279\) 0 0
\(280\) 6.00000 6.92820i 0.358569 0.414039i
\(281\) 30.0000i 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) 20.7846i 1.23552i −0.786368 0.617758i \(-0.788041\pi\)
0.786368 0.617758i \(-0.211959\pi\)
\(284\) 3.00000i 0.178017i
\(285\) 0 0
\(286\) 10.3923i 0.614510i
\(287\) 13.8564 + 12.0000i 0.817918 + 0.708338i
\(288\) 0 0
\(289\) −14.0000 −0.823529
\(290\) 10.3923 0.610257
\(291\) 0 0
\(292\) 6.92820i 0.405442i
\(293\) 17.3205 1.01187 0.505937 0.862570i \(-0.331147\pi\)
0.505937 + 0.862570i \(0.331147\pi\)
\(294\) 0 0
\(295\) −30.0000 −1.74667
\(296\) 2.00000i 0.116248i
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 5.19615 0.300501
\(300\) 0 0
\(301\) −22.0000 19.0526i −1.26806 1.09817i
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) 6.92820i 0.397360i
\(305\) 48.0000i 2.74847i
\(306\) 0 0
\(307\) 31.1769i 1.77936i 0.456584 + 0.889680i \(0.349073\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) −10.3923 + 12.0000i −0.592157 + 0.683763i
\(309\) 0 0
\(310\) −18.0000 −1.02233
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 0 0
\(313\) 13.8564i 0.783210i 0.920133 + 0.391605i \(0.128080\pi\)
−0.920133 + 0.391605i \(0.871920\pi\)
\(314\) −15.5885 −0.879708
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) −3.46410 −0.193649
\(321\) 0 0
\(322\) −6.00000 5.19615i −0.334367 0.289570i
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 12.1244i 0.672538i
\(326\) 11.0000i 0.609234i
\(327\) 0 0
\(328\) 6.92820i 0.382546i
\(329\) −13.8564 12.0000i −0.763928 0.661581i
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 3.46410 0.190117
\(333\) 0 0
\(334\) 17.3205i 0.947736i
\(335\) −24.2487 −1.32485
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 10.0000i 0.543928i
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 31.1769 1.68832
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 11.0000i 0.593080i
\(345\) 0 0
\(346\) 20.7846i 1.11739i
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 19.0526i 1.01986i −0.860216 0.509930i \(-0.829671\pi\)
0.860216 0.509930i \(-0.170329\pi\)
\(350\) −12.1244 + 14.0000i −0.648074 + 0.748331i
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) 8.66025 0.460939 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(354\) 0 0
\(355\) 10.3923i 0.551566i
\(356\) −5.19615 −0.275396
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) 27.0000i 1.42501i 0.701669 + 0.712503i \(0.252438\pi\)
−0.701669 + 0.712503i \(0.747562\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 1.73205 0.0910346
\(363\) 0 0
\(364\) −3.00000 + 3.46410i −0.157243 + 0.181568i
\(365\) 24.0000i 1.25622i
\(366\) 0 0
\(367\) 19.0526i 0.994535i 0.867597 + 0.497268i \(0.165663\pi\)
−0.867597 + 0.497268i \(0.834337\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 6.92820i 0.360180i
\(371\) 5.19615 6.00000i 0.269771 0.311504i
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −10.3923 −0.537373
\(375\) 0 0
\(376\) 6.92820i 0.357295i
\(377\) −5.19615 −0.267615
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 24.0000i 1.23117i
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −3.46410 −0.177007 −0.0885037 0.996076i \(-0.528208\pi\)
−0.0885037 + 0.996076i \(0.528208\pi\)
\(384\) 0 0
\(385\) 36.0000 41.5692i 1.83473 2.11856i
\(386\) 13.0000i 0.661683i
\(387\) 0 0
\(388\) 6.92820i 0.351726i
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) 5.19615i 0.262781i
\(392\) 6.92820 1.00000i 0.349927 0.0505076i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 27.7128 1.39438
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −8.66025 −0.434099
\(399\) 0 0
\(400\) 7.00000 0.350000
\(401\) 18.0000i 0.898877i −0.893311 0.449439i \(-0.851624\pi\)
0.893311 0.449439i \(-0.148376\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) 0 0
\(405\) 0 0
\(406\) 6.00000 + 5.19615i 0.297775 + 0.257881i
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 3.46410i 0.171289i −0.996326 0.0856444i \(-0.972705\pi\)
0.996326 0.0856444i \(-0.0272949\pi\)
\(410\) 24.0000i 1.18528i
\(411\) 0 0
\(412\) 8.66025i 0.426660i
\(413\) −17.3205 15.0000i −0.852286 0.738102i
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 1.73205 0.0849208
\(417\) 0 0
\(418\) 41.5692i 2.03322i
\(419\) 29.4449 1.43848 0.719238 0.694764i \(-0.244491\pi\)
0.719238 + 0.694764i \(0.244491\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 7.00000i 0.340755i
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) −12.1244 −0.588118
\(426\) 0 0
\(427\) 24.0000 27.7128i 1.16144 1.34112i
\(428\) 0 0
\(429\) 0 0
\(430\) 38.1051i 1.83759i
\(431\) 12.0000i 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 0 0
\(433\) 20.7846i 0.998845i 0.866359 + 0.499422i \(0.166454\pi\)
−0.866359 + 0.499422i \(0.833546\pi\)
\(434\) −10.3923 9.00000i −0.498847 0.432014i
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −20.7846 −0.994263
\(438\) 0 0
\(439\) 5.19615i 0.247999i 0.992282 + 0.123999i \(0.0395721\pi\)
−0.992282 + 0.123999i \(0.960428\pi\)
\(440\) −20.7846 −0.990867
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) 18.0000i 0.855206i −0.903967 0.427603i \(-0.859358\pi\)
0.903967 0.427603i \(-0.140642\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 10.3923 0.492090
\(447\) 0 0
\(448\) −2.00000 1.73205i −0.0944911 0.0818317i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 0 0
\(451\) 41.5692i 1.95742i
\(452\) 6.00000i 0.282216i
\(453\) 0 0
\(454\) 25.9808i 1.21934i
\(455\) 10.3923 12.0000i 0.487199 0.562569i
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) −20.7846 −0.971201
\(459\) 0 0
\(460\) 10.3923i 0.484544i
\(461\) 6.92820 0.322679 0.161339 0.986899i \(-0.448419\pi\)
0.161339 + 0.986899i \(0.448419\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 3.00000i 0.139272i
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −31.1769 −1.44270 −0.721348 0.692573i \(-0.756477\pi\)
−0.721348 + 0.692573i \(0.756477\pi\)
\(468\) 0 0
\(469\) −14.0000 12.1244i −0.646460 0.559851i
\(470\) 24.0000i 1.10704i
\(471\) 0 0
\(472\) 8.66025i 0.398621i
\(473\) 66.0000i 3.03468i
\(474\) 0 0
\(475\) 48.4974i 2.22521i
\(476\) 3.46410 + 3.00000i 0.158777 + 0.137505i
\(477\) 0 0
\(478\) 0 0
\(479\) −24.2487 −1.10795 −0.553976 0.832533i \(-0.686890\pi\)
−0.553976 + 0.832533i \(0.686890\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 13.8564 0.631142
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 24.0000i 1.08978i
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −13.8564 −0.627250
\(489\) 0 0
\(490\) −24.0000 + 3.46410i −1.08421 + 0.156492i
\(491\) 18.0000i 0.812329i −0.913800 0.406164i \(-0.866866\pi\)
0.913800 0.406164i \(-0.133134\pi\)
\(492\) 0 0
\(493\) 5.19615i 0.234023i
\(494\) 12.0000i 0.539906i
\(495\) 0 0
\(496\) 5.19615i 0.233314i
\(497\) 5.19615 6.00000i 0.233079 0.269137i
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −6.92820 −0.309839
\(501\) 0 0
\(502\) 3.46410i 0.154610i
\(503\) 17.3205 0.772283 0.386142 0.922440i \(-0.373808\pi\)
0.386142 + 0.922440i \(0.373808\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 18.0000i 0.800198i
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) −3.46410 −0.153544 −0.0767718 0.997049i \(-0.524461\pi\)
−0.0767718 + 0.997049i \(0.524461\pi\)
\(510\) 0 0
\(511\) −12.0000 + 13.8564i −0.530849 + 0.612971i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 20.7846i 0.916770i
\(515\) 30.0000i 1.32196i
\(516\) 0 0
\(517\) 41.5692i 1.82821i
\(518\) 3.46410 4.00000i 0.152204 0.175750i
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 1.73205 0.0758825 0.0379413 0.999280i \(-0.487920\pi\)
0.0379413 + 0.999280i \(0.487920\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i −0.925525 0.378686i \(-0.876376\pi\)
0.925525 0.378686i \(-0.123624\pi\)
\(524\) 5.19615 0.226995
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) 9.00000i 0.392046i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 10.3923 0.451413
\(531\) 0 0
\(532\) 12.0000 13.8564i 0.520266 0.600751i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.00000i 0.302354i
\(537\) 0 0
\(538\) 3.46410i 0.149348i
\(539\) 41.5692 6.00000i 1.79051 0.258438i
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 12.1244 0.520786
\(543\) 0 0
\(544\) 1.73205i 0.0742611i
\(545\) 13.8564 0.593543
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 42.0000 1.79089
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) 16.0000 + 13.8564i 0.680389 + 0.589234i
\(554\) 20.0000i 0.849719i
\(555\) 0 0
\(556\) 3.46410i 0.146911i
\(557\) 39.0000i 1.65248i 0.563316 + 0.826242i \(0.309525\pi\)
−0.563316 + 0.826242i \(0.690475\pi\)
\(558\) 0 0
\(559\) 19.0526i 0.805837i
\(560\) 6.92820 + 6.00000i 0.292770 + 0.253546i
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 36.3731 1.53294 0.766471 0.642279i \(-0.222011\pi\)
0.766471 + 0.642279i \(0.222011\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 20.7846 0.873642
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) 18.0000i 0.754599i −0.926091 0.377300i \(-0.876853\pi\)
0.926091 0.377300i \(-0.123147\pi\)
\(570\) 0 0
\(571\) 41.0000 1.71580 0.857898 0.513820i \(-0.171770\pi\)
0.857898 + 0.513820i \(0.171770\pi\)
\(572\) 10.3923 0.434524
\(573\) 0 0
\(574\) −12.0000 + 13.8564i −0.500870 + 0.578355i
\(575\) 21.0000i 0.875761i
\(576\) 0 0
\(577\) 38.1051i 1.58634i 0.609002 + 0.793168i \(0.291570\pi\)
−0.609002 + 0.793168i \(0.708430\pi\)
\(578\) 14.0000i 0.582323i
\(579\) 0 0
\(580\) 10.3923i 0.431517i
\(581\) −6.92820 6.00000i −0.287430 0.248922i
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) 17.3205i 0.715504i
\(587\) −25.9808 −1.07234 −0.536170 0.844110i \(-0.680130\pi\)
−0.536170 + 0.844110i \(0.680130\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 30.0000i 1.23508i
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 6.92820 0.284507 0.142254 0.989830i \(-0.454565\pi\)
0.142254 + 0.989830i \(0.454565\pi\)
\(594\) 0 0
\(595\) −12.0000 10.3923i −0.491952 0.426043i
\(596\) 15.0000i 0.614424i
\(597\) 0 0
\(598\) 5.19615i 0.212486i
\(599\) 9.00000i 0.367730i 0.982952 + 0.183865i \(0.0588609\pi\)
−0.982952 + 0.183865i \(0.941139\pi\)
\(600\) 0 0
\(601\) 31.1769i 1.27173i 0.771799 + 0.635866i \(0.219357\pi\)
−0.771799 + 0.635866i \(0.780643\pi\)
\(602\) 19.0526 22.0000i 0.776524 0.896653i
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −86.6025 −3.52089
\(606\) 0 0
\(607\) 1.73205i 0.0703018i −0.999382 0.0351509i \(-0.988809\pi\)
0.999382 0.0351509i \(-0.0111912\pi\)
\(608\) −6.92820 −0.280976
\(609\) 0 0
\(610\) 48.0000 1.94346
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −31.1769 −1.25820
\(615\) 0 0
\(616\) −12.0000 10.3923i −0.483494 0.418718i
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 38.1051i 1.53157i −0.643094 0.765787i \(-0.722350\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 18.0000i 0.722897i
\(621\) 0 0
\(622\) 13.8564i 0.555591i
\(623\) 10.3923 + 9.00000i 0.416359 + 0.360577i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −13.8564 −0.553813
\(627\) 0 0
\(628\) 15.5885i 0.622047i
\(629\) 3.46410 0.138123
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 48.4974 1.92456
\(636\) 0 0
\(637\) 12.0000 1.73205i 0.475457 0.0686264i
\(638\) 18.0000i 0.712627i
\(639\) 0 0
\(640\) 3.46410i 0.136931i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 10.3923i 0.409832i 0.978780 + 0.204916i \(0.0656922\pi\)
−0.978780 + 0.204916i \(0.934308\pi\)
\(644\) 5.19615 6.00000i 0.204757 0.236433i
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −10.3923 −0.408564 −0.204282 0.978912i \(-0.565486\pi\)
−0.204282 + 0.978912i \(0.565486\pi\)
\(648\) 0 0
\(649\) 51.9615i 2.03967i
\(650\) 12.1244 0.475556
\(651\) 0 0
\(652\) −11.0000 −0.430793
\(653\) 27.0000i 1.05659i 0.849060 + 0.528296i \(0.177169\pi\)
−0.849060 + 0.528296i \(0.822831\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 6.92820 0.270501
\(657\) 0 0
\(658\) 12.0000 13.8564i 0.467809 0.540179i
\(659\) 48.0000i 1.86981i −0.354892 0.934907i \(-0.615482\pi\)
0.354892 0.934907i \(-0.384518\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 17.0000i 0.660724i
\(663\) 0 0
\(664\) 3.46410i 0.134433i
\(665\) −41.5692 + 48.0000i −1.61199 + 1.86136i
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 17.3205 0.670151
\(669\) 0 0
\(670\) 24.2487i 0.936809i
\(671\) −83.1384 −3.20952
\(672\) 0 0
\(673\) 31.0000 1.19496 0.597481 0.801883i \(-0.296168\pi\)
0.597481 + 0.801883i \(0.296168\pi\)
\(674\) 13.0000i 0.500741i
\(675\) 0 0
\(676\) −10.0000 −0.384615
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 0 0
\(679\) 12.0000 13.8564i 0.460518 0.531760i
\(680\) 6.00000i 0.230089i
\(681\) 0 0
\(682\) 31.1769i 1.19383i
\(683\) 42.0000i 1.60709i −0.595247 0.803543i \(-0.702946\pi\)
0.595247 0.803543i \(-0.297054\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15.5885 10.0000i −0.595170 0.381802i
\(687\) 0 0
\(688\) −11.0000 −0.419371
\(689\) −5.19615 −0.197958
\(690\) 0 0
\(691\) 10.3923i 0.395342i −0.980268 0.197671i \(-0.936662\pi\)
0.980268 0.197671i \(-0.0633378\pi\)
\(692\) 20.7846 0.790112
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 12.0000i 0.455186i
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 19.0526 0.721150
\(699\) 0 0
\(700\) −14.0000 12.1244i −0.529150 0.458258i
\(701\) 30.0000i 1.13308i 0.824033 + 0.566542i \(0.191719\pi\)
−0.824033 + 0.566542i \(0.808281\pi\)
\(702\) 0 0
\(703\) 13.8564i 0.522604i
\(704\) 6.00000i 0.226134i
\(705\) 0 0
\(706\) 8.66025i 0.325933i
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 10.3923 0.390016
\(711\) 0 0
\(712\) 5.19615i 0.194734i
\(713\) −15.5885 −0.583792
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) 6.00000i 0.224231i
\(717\) 0 0
\(718\) −27.0000 −1.00763
\(719\) 6.92820 0.258378 0.129189 0.991620i \(-0.458763\pi\)
0.129189 + 0.991620i \(0.458763\pi\)
\(720\) 0 0
\(721\) 15.0000 17.3205i 0.558629 0.645049i
\(722\) 29.0000i 1.07927i
\(723\) 0 0
\(724\) 1.73205i 0.0643712i
\(725\) 21.0000i 0.779920i
\(726\) 0 0
\(727\) 25.9808i 0.963573i 0.876289 + 0.481787i \(0.160012\pi\)
−0.876289 + 0.481787i \(0.839988\pi\)
\(728\) −3.46410 3.00000i −0.128388 0.111187i
\(729\) 0 0
\(730\) −24.0000 −0.888280
\(731\) 19.0526 0.704684
\(732\) 0 0
\(733\) 36.3731i 1.34347i −0.740792 0.671735i \(-0.765549\pi\)
0.740792 0.671735i \(-0.234451\pi\)
\(734\) −19.0526 −0.703243
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 42.0000i 1.54709i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 6.92820 0.254686
\(741\) 0 0
\(742\) 6.00000 + 5.19615i 0.220267 + 0.190757i
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 0 0
\(745\) 51.9615i 1.90372i
\(746\) 4.00000i 0.146450i
\(747\) 0 0
\(748\) 10.3923i 0.379980i
\(749\) 0 0
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 5.19615i 0.189233i
\(755\) −27.7128 −1.00857
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −19.0526 −0.690655 −0.345327 0.938482i \(-0.612232\pi\)
−0.345327 + 0.938482i \(0.612232\pi\)
\(762\) 0 0
\(763\) 8.00000 + 6.92820i 0.289619 + 0.250818i
\(764\) 24.0000i 0.868290i
\(765\) 0 0
\(766\) 3.46410i 0.125163i
\(767\) 15.0000i 0.541619i
\(768\) 0 0
\(769\) 3.46410i 0.124919i −0.998048 0.0624593i \(-0.980106\pi\)
0.998048 0.0624593i \(-0.0198944\pi\)
\(770\) 41.5692 + 36.0000i 1.49805 + 1.29735i
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) −38.1051 −1.37055 −0.685273 0.728286i \(-0.740317\pi\)
−0.685273 + 0.728286i \(0.740317\pi\)
\(774\) 0 0
\(775\) 36.3731i 1.30656i
\(776\) −6.92820 −0.248708
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 48.0000i 1.71978i
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 5.19615 0.185814
\(783\) 0 0
\(784\) 1.00000 + 6.92820i 0.0357143 + 0.247436i
\(785\) 54.0000i 1.92734i
\(786\) 0 0
\(787\) 3.46410i 0.123482i 0.998092 + 0.0617409i \(0.0196653\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 27.7128i 0.985978i
\(791\) −10.3923 + 12.0000i −0.369508 + 0.426671i
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 8.66025i 0.306955i
\(797\) 31.1769 1.10434 0.552171 0.833731i \(-0.313799\pi\)
0.552171 + 0.833731i \(0.313799\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 7.00000i 0.247487i
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 41.5692 1.46695
\(804\) 0 0
\(805\) −18.0000 + 20.7846i −0.634417 + 0.732561i
\(806\) 9.00000i 0.317011i
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000i 0.843795i −0.906644 0.421898i \(-0.861364\pi\)
0.906644 0.421898i \(-0.138636\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i −0.836881 0.547385i \(-0.815623\pi\)
0.836881 0.547385i \(-0.184377\pi\)
\(812\) −5.19615 + 6.00000i −0.182349 + 0.210559i
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 38.1051 1.33476
\(816\) 0 0
\(817\) 76.2102i 2.66626i
\(818\) 3.46410 0.121119
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) 33.0000i 1.15171i 0.817553 + 0.575854i \(0.195330\pi\)
−0.817553 + 0.575854i \(0.804670\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −8.66025 −0.301694
\(825\) 0 0
\(826\) 15.0000 17.3205i 0.521917 0.602658i
\(827\) 48.0000i 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) 27.7128i 0.962506i −0.876582 0.481253i \(-0.840182\pi\)
0.876582 0.481253i \(-0.159818\pi\)
\(830\) 12.0000i 0.416526i
\(831\) 0 0
\(832\) 1.73205i 0.0600481i
\(833\) −1.73205 12.0000i −0.0600120 0.415775i
\(834\) 0 0
\(835\) −60.0000 −2.07639
\(836\) −41.5692 −1.43770
\(837\) 0 0
\(838\) 29.4449i 1.01716i
\(839\) −17.3205 −0.597970 −0.298985 0.954258i \(-0.596648\pi\)
−0.298985 + 0.954258i \(0.596648\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 40.0000i 1.37849i
\(843\) 0 0
\(844\) −7.00000 −0.240950
\(845\) 34.6410 1.19169
\(846\) 0 0
\(847\) −50.0000 43.3013i −1.71802 1.48785i
\(848\) 3.00000i 0.103020i
\(849\) 0 0
\(850\) 12.1244i 0.415862i
\(851\) 6.00000i 0.205677i
\(852\) 0 0
\(853\) 12.1244i 0.415130i −0.978221 0.207565i \(-0.933446\pi\)
0.978221 0.207565i \(-0.0665539\pi\)
\(854\) 27.7128 + 24.0000i 0.948313 + 0.821263i
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5167 −0.769154 −0.384577 0.923093i \(-0.625653\pi\)
−0.384577 + 0.923093i \(0.625653\pi\)
\(858\) 0 0
\(859\) 45.0333i 1.53652i 0.640140 + 0.768259i \(0.278876\pi\)
−0.640140 + 0.768259i \(0.721124\pi\)
\(860\) 38.1051 1.29937
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 45.0000i 1.53182i −0.642949 0.765909i \(-0.722289\pi\)
0.642949 0.765909i \(-0.277711\pi\)
\(864\) 0 0
\(865\) −72.0000 −2.44807
\(866\) −20.7846 −0.706290
\(867\) 0 0
\(868\) 9.00000 10.3923i 0.305480 0.352738i
\(869\) 48.0000i 1.62829i
\(870\) 0 0
\(871\) 12.1244i 0.410818i
\(872\) 4.00000i 0.135457i
\(873\) 0 0
\(874\) 20.7846i 0.703050i
\(875\) 13.8564 + 12.0000i 0.468432 + 0.405674i
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −5.19615 −0.175362
\(879\) 0 0
\(880\) 20.7846i 0.700649i
\(881\) −5.19615 −0.175063 −0.0875314 0.996162i \(-0.527898\pi\)
−0.0875314 + 0.996162i \(0.527898\pi\)
\(882\) 0 0
\(883\) 25.0000 0.841317 0.420658 0.907219i \(-0.361799\pi\)
0.420658 + 0.907219i \(0.361799\pi\)
\(884\) 3.00000i 0.100901i
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) −48.4974 −1.62838 −0.814192 0.580596i \(-0.802820\pi\)
−0.814192 + 0.580596i \(0.802820\pi\)
\(888\) 0 0
\(889\) 28.0000 + 24.2487i 0.939090 + 0.813276i
\(890\) 18.0000i 0.603361i
\(891\) 0 0
\(892\) 10.3923i 0.347960i
\(893\) 48.0000i 1.60626i
\(894\) 0 0
\(895\) 20.7846i 0.694753i
\(896\) 1.73205 2.00000i 0.0578638 0.0668153i
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 15.5885 0.519904
\(900\) 0 0
\(901\) 5.19615i 0.173109i
\(902\) 41.5692 1.38410
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 6.00000i 0.199447i
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −25.9808 −0.862202
\(909\) 0 0
\(910\) 12.0000 + 10.3923i 0.397796 + 0.344502i
\(911\) 24.0000i 0.795155i 0.917568 + 0.397578i \(0.130149\pi\)
−0.917568 + 0.397578i \(0.869851\pi\)
\(912\) 0 0
\(913\) 20.7846i 0.687870i
\(914\) 17.0000i 0.562310i
\(915\) 0 0
\(916\) 20.7846i 0.686743i
\(917\) −10.3923 9.00000i −0.343184 0.297206i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 10.3923 0.342624
\(921\) 0 0
\(922\) 6.92820i 0.228168i
\(923\) −5.19615 −0.171033
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) 4.00000i 0.131448i
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −6.92820 −0.227307 −0.113653 0.993520i \(-0.536255\pi\)
−0.113653 + 0.993520i \(0.536255\pi\)
\(930\) 0 0
\(931\) −48.0000 + 6.92820i −1.57314 + 0.227063i
\(932\) 24.0000i 0.786146i
\(933\) 0 0
\(934\) 31.1769i 1.02014i
\(935\) 36.0000i 1.17733i
\(936\) 0 0
\(937\) 24.2487i 0.792171i −0.918214 0.396085i \(-0.870368\pi\)
0.918214 0.396085i \(-0.129632\pi\)
\(938\) 12.1244 14.0000i 0.395874 0.457116i
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −24.2487 −0.790485 −0.395243 0.918577i \(-0.629340\pi\)
−0.395243 + 0.918577i \(0.629340\pi\)
\(942\) 0 0
\(943\) 20.7846i 0.676840i
\(944\) −8.66025 −0.281867
\(945\) 0 0
\(946\) −66.0000 −2.14585
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) −48.4974 −1.57346
\(951\) 0 0
\(952\) −3.00000 + 3.46410i −0.0972306 + 0.112272i
\(953\) 18.0000i 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 83.1384i 2.69030i
\(956\) 0 0
\(957\) 0 0
\(958\) 24.2487i 0.783440i
\(959\) 0 0
\(960\) 0 0
\(961\) 4.00000 0.129032
\(962\) −3.46410 −0.111687
\(963\) 0 0
\(964\) 13.8564i 0.446285i
\(965\) 45.0333 1.44967
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) 22.5167 0.722594 0.361297 0.932451i \(-0.382334\pi\)
0.361297 + 0.932451i \(0.382334\pi\)
\(972\) 0 0
\(973\) 6.00000 6.92820i 0.192351 0.222108i
\(974\) 28.0000i 0.897178i
\(975\) 0 0
\(976\) 13.8564i 0.443533i
\(977\) 24.0000i 0.767828i −0.923369 0.383914i \(-0.874576\pi\)
0.923369 0.383914i \(-0.125424\pi\)
\(978\) 0 0
\(979\) 31.1769i 0.996419i
\(980\) −3.46410 24.0000i −0.110657 0.766652i
\(981\) 0 0
\(982\) 18.0000 0.574403
\(983\) 38.1051 1.21536 0.607682 0.794180i \(-0.292099\pi\)
0.607682 + 0.794180i \(0.292099\pi\)
\(984\) 0 0
\(985\) 20.7846i 0.662253i
\(986\) −5.19615 −0.165479
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 33.0000i 1.04934i
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) −5.19615 −0.164978
\(993\) 0 0
\(994\) 6.00000 + 5.19615i 0.190308 + 0.164812i
\(995\) 30.0000i 0.951064i
\(996\) 0 0
\(997\) 46.7654i 1.48107i 0.672015 + 0.740537i \(0.265429\pi\)
−0.672015 + 0.740537i \(0.734571\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 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 378.2.d.c.377.4 yes 4
3.2 odd 2 inner 378.2.d.c.377.1 4
4.3 odd 2 3024.2.k.f.1889.3 4
7.6 odd 2 inner 378.2.d.c.377.3 yes 4
9.2 odd 6 1134.2.m.d.377.1 4
9.4 even 3 1134.2.m.a.755.1 4
9.5 odd 6 1134.2.m.a.755.2 4
9.7 even 3 1134.2.m.d.377.2 4
12.11 even 2 3024.2.k.f.1889.1 4
21.20 even 2 inner 378.2.d.c.377.2 yes 4
28.27 even 2 3024.2.k.f.1889.2 4
63.13 odd 6 1134.2.m.d.755.1 4
63.20 even 6 1134.2.m.a.377.1 4
63.34 odd 6 1134.2.m.a.377.2 4
63.41 even 6 1134.2.m.d.755.2 4
84.83 odd 2 3024.2.k.f.1889.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.d.c.377.1 4 3.2 odd 2 inner
378.2.d.c.377.2 yes 4 21.20 even 2 inner
378.2.d.c.377.3 yes 4 7.6 odd 2 inner
378.2.d.c.377.4 yes 4 1.1 even 1 trivial
1134.2.m.a.377.1 4 63.20 even 6
1134.2.m.a.377.2 4 63.34 odd 6
1134.2.m.a.755.1 4 9.4 even 3
1134.2.m.a.755.2 4 9.5 odd 6
1134.2.m.d.377.1 4 9.2 odd 6
1134.2.m.d.377.2 4 9.7 even 3
1134.2.m.d.755.1 4 63.13 odd 6
1134.2.m.d.755.2 4 63.41 even 6
3024.2.k.f.1889.1 4 12.11 even 2
3024.2.k.f.1889.2 4 28.27 even 2
3024.2.k.f.1889.3 4 4.3 odd 2
3024.2.k.f.1889.4 4 84.83 odd 2