Properties

Label 390.2.b.d.181.3
Level $390$
Weight $2$
Character 390.181
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(181,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.3
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 390.181
Dual form 390.2.b.d.181.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.12311i q^{11} -1.00000 q^{12} +(3.56155 + 0.561553i) q^{13} +3.12311 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} -3.12311i q^{21} +5.12311 q^{22} -5.12311 q^{23} -1.00000i q^{24} -1.00000 q^{25} +(-0.561553 + 3.56155i) q^{26} +1.00000 q^{27} +3.12311i q^{28} +2.00000 q^{29} +1.00000 q^{30} -3.12311i q^{31} +1.00000i q^{32} -5.12311i q^{33} +2.00000i q^{34} -3.12311 q^{35} -1.00000 q^{36} +5.12311i q^{37} -6.00000 q^{38} +(3.56155 + 0.561553i) q^{39} -1.00000 q^{40} -0.876894i q^{41} +3.12311 q^{42} +6.24621 q^{43} +5.12311i q^{44} -1.00000i q^{45} -5.12311i q^{46} +6.24621i q^{47} +1.00000 q^{48} -2.75379 q^{49} -1.00000i q^{50} +2.00000 q^{51} +(-3.56155 - 0.561553i) q^{52} -13.3693 q^{53} +1.00000i q^{54} -5.12311 q^{55} -3.12311 q^{56} +6.00000i q^{57} +2.00000i q^{58} +1.12311i q^{59} +1.00000i q^{60} +10.0000 q^{61} +3.12311 q^{62} -3.12311i q^{63} -1.00000 q^{64} +(0.561553 - 3.56155i) q^{65} +5.12311 q^{66} -4.87689i q^{67} -2.00000 q^{68} -5.12311 q^{69} -3.12311i q^{70} -10.2462i q^{71} -1.00000i q^{72} +13.1231i q^{73} -5.12311 q^{74} -1.00000 q^{75} -6.00000i q^{76} -16.0000 q^{77} +(-0.561553 + 3.56155i) q^{78} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +0.876894 q^{82} +6.24621i q^{83} +3.12311i q^{84} -2.00000i q^{85} +6.24621i q^{86} +2.00000 q^{87} -5.12311 q^{88} -3.12311i q^{89} +1.00000 q^{90} +(1.75379 - 11.1231i) q^{91} +5.12311 q^{92} -3.12311i q^{93} -6.24621 q^{94} +6.00000 q^{95} +1.00000i q^{96} +13.1231i q^{97} -2.75379i q^{98} -5.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 8 q^{17} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} - 24 q^{38} + 6 q^{39} - 4 q^{40} - 4 q^{42} - 8 q^{43} + 4 q^{48} - 44 q^{49} + 8 q^{51} - 6 q^{52} - 4 q^{53} - 4 q^{55} + 4 q^{56} + 40 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} + 4 q^{66} - 8 q^{68} - 4 q^{69} - 4 q^{74} - 4 q^{75} - 64 q^{77} + 6 q^{78} + 32 q^{79} + 4 q^{81} + 20 q^{82} + 8 q^{87} - 4 q^{88} + 4 q^{90} + 40 q^{91} + 4 q^{92} + 8 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 3.12311i 1.18042i −0.807249 0.590211i \(-0.799044\pi\)
0.807249 0.590211i \(-0.200956\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.12311i 1.54467i −0.635213 0.772337i \(-0.719088\pi\)
0.635213 0.772337i \(-0.280912\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.56155 + 0.561553i 0.987797 + 0.155747i
\(14\) 3.12311 0.834685
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 3.12311i 0.681518i
\(22\) 5.12311 1.09225
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) −0.561553 + 3.56155i −0.110130 + 0.698478i
\(27\) 1.00000 0.192450
\(28\) 3.12311i 0.590211i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.12311i 0.560926i −0.959865 0.280463i \(-0.909512\pi\)
0.959865 0.280463i \(-0.0904881\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.12311i 0.891818i
\(34\) 2.00000i 0.342997i
\(35\) −3.12311 −0.527901
\(36\) −1.00000 −0.166667
\(37\) 5.12311i 0.842233i 0.907006 + 0.421117i \(0.138362\pi\)
−0.907006 + 0.421117i \(0.861638\pi\)
\(38\) −6.00000 −0.973329
\(39\) 3.56155 + 0.561553i 0.570305 + 0.0899204i
\(40\) −1.00000 −0.158114
\(41\) 0.876894i 0.136948i −0.997653 0.0684739i \(-0.978187\pi\)
0.997653 0.0684739i \(-0.0218130\pi\)
\(42\) 3.12311 0.481906
\(43\) 6.24621 0.952538 0.476269 0.879300i \(-0.341989\pi\)
0.476269 + 0.879300i \(0.341989\pi\)
\(44\) 5.12311i 0.772337i
\(45\) 1.00000i 0.149071i
\(46\) 5.12311i 0.755361i
\(47\) 6.24621i 0.911104i 0.890209 + 0.455552i \(0.150558\pi\)
−0.890209 + 0.455552i \(0.849442\pi\)
\(48\) 1.00000 0.144338
\(49\) −2.75379 −0.393398
\(50\) 1.00000i 0.141421i
\(51\) 2.00000 0.280056
\(52\) −3.56155 0.561553i −0.493899 0.0778734i
\(53\) −13.3693 −1.83642 −0.918208 0.396098i \(-0.870364\pi\)
−0.918208 + 0.396098i \(0.870364\pi\)
\(54\) 1.00000i 0.136083i
\(55\) −5.12311 −0.690799
\(56\) −3.12311 −0.417343
\(57\) 6.00000i 0.794719i
\(58\) 2.00000i 0.262613i
\(59\) 1.12311i 0.146216i 0.997324 + 0.0731079i \(0.0232918\pi\)
−0.997324 + 0.0731079i \(0.976708\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 3.12311 0.396635
\(63\) 3.12311i 0.393474i
\(64\) −1.00000 −0.125000
\(65\) 0.561553 3.56155i 0.0696521 0.441756i
\(66\) 5.12311 0.630611
\(67\) 4.87689i 0.595807i −0.954596 0.297904i \(-0.903713\pi\)
0.954596 0.297904i \(-0.0962874\pi\)
\(68\) −2.00000 −0.242536
\(69\) −5.12311 −0.616749
\(70\) 3.12311i 0.373283i
\(71\) 10.2462i 1.21600i −0.793936 0.608001i \(-0.791972\pi\)
0.793936 0.608001i \(-0.208028\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 13.1231i 1.53594i 0.640484 + 0.767972i \(0.278734\pi\)
−0.640484 + 0.767972i \(0.721266\pi\)
\(74\) −5.12311 −0.595549
\(75\) −1.00000 −0.115470
\(76\) 6.00000i 0.688247i
\(77\) −16.0000 −1.82337
\(78\) −0.561553 + 3.56155i −0.0635833 + 0.403266i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 0.876894 0.0968368
\(83\) 6.24621i 0.685611i 0.939406 + 0.342805i \(0.111377\pi\)
−0.939406 + 0.342805i \(0.888623\pi\)
\(84\) 3.12311i 0.340759i
\(85\) 2.00000i 0.216930i
\(86\) 6.24621i 0.673546i
\(87\) 2.00000 0.214423
\(88\) −5.12311 −0.546125
\(89\) 3.12311i 0.331049i −0.986206 0.165524i \(-0.947068\pi\)
0.986206 0.165524i \(-0.0529316\pi\)
\(90\) 1.00000 0.105409
\(91\) 1.75379 11.1231i 0.183847 1.16602i
\(92\) 5.12311 0.534121
\(93\) 3.12311i 0.323851i
\(94\) −6.24621 −0.644247
\(95\) 6.00000 0.615587
\(96\) 1.00000i 0.102062i
\(97\) 13.1231i 1.33245i 0.745751 + 0.666225i \(0.232091\pi\)
−0.745751 + 0.666225i \(0.767909\pi\)
\(98\) 2.75379i 0.278175i
\(99\) 5.12311i 0.514891i
\(100\) 1.00000 0.100000
\(101\) −12.2462 −1.21854 −0.609272 0.792961i \(-0.708538\pi\)
−0.609272 + 0.792961i \(0.708538\pi\)
\(102\) 2.00000i 0.198030i
\(103\) −13.1231 −1.29306 −0.646529 0.762889i \(-0.723780\pi\)
−0.646529 + 0.762889i \(0.723780\pi\)
\(104\) 0.561553 3.56155i 0.0550648 0.349239i
\(105\) −3.12311 −0.304784
\(106\) 13.3693i 1.29854i
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.87689i 0.275557i −0.990463 0.137778i \(-0.956004\pi\)
0.990463 0.137778i \(-0.0439961\pi\)
\(110\) 5.12311i 0.488469i
\(111\) 5.12311i 0.486264i
\(112\) 3.12311i 0.295106i
\(113\) 12.2462 1.15203 0.576013 0.817440i \(-0.304608\pi\)
0.576013 + 0.817440i \(0.304608\pi\)
\(114\) −6.00000 −0.561951
\(115\) 5.12311i 0.477732i
\(116\) −2.00000 −0.185695
\(117\) 3.56155 + 0.561553i 0.329266 + 0.0519156i
\(118\) −1.12311 −0.103390
\(119\) 6.24621i 0.572589i
\(120\) −1.00000 −0.0912871
\(121\) −15.2462 −1.38602
\(122\) 10.0000i 0.905357i
\(123\) 0.876894i 0.0790669i
\(124\) 3.12311i 0.280463i
\(125\) 1.00000i 0.0894427i
\(126\) 3.12311 0.278228
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 6.24621 0.549948
\(130\) 3.56155 + 0.561553i 0.312369 + 0.0492514i
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 5.12311i 0.445909i
\(133\) 18.7386 1.62485
\(134\) 4.87689 0.421300
\(135\) 1.00000i 0.0860663i
\(136\) 2.00000i 0.171499i
\(137\) 10.4924i 0.896428i 0.893926 + 0.448214i \(0.147940\pi\)
−0.893926 + 0.448214i \(0.852060\pi\)
\(138\) 5.12311i 0.436108i
\(139\) −16.4924 −1.39887 −0.699435 0.714697i \(-0.746565\pi\)
−0.699435 + 0.714697i \(0.746565\pi\)
\(140\) 3.12311 0.263951
\(141\) 6.24621i 0.526026i
\(142\) 10.2462 0.859843
\(143\) 2.87689 18.2462i 0.240578 1.52582i
\(144\) 1.00000 0.0833333
\(145\) 2.00000i 0.166091i
\(146\) −13.1231 −1.08608
\(147\) −2.75379 −0.227129
\(148\) 5.12311i 0.421117i
\(149\) 14.0000i 1.14692i 0.819232 + 0.573462i \(0.194400\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 13.3693i 1.08798i −0.839092 0.543990i \(-0.816913\pi\)
0.839092 0.543990i \(-0.183087\pi\)
\(152\) 6.00000 0.486664
\(153\) 2.00000 0.161690
\(154\) 16.0000i 1.28932i
\(155\) −3.12311 −0.250854
\(156\) −3.56155 0.561553i −0.285152 0.0449602i
\(157\) 21.3693 1.70546 0.852729 0.522354i \(-0.174946\pi\)
0.852729 + 0.522354i \(0.174946\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −13.3693 −1.06026
\(160\) 1.00000 0.0790569
\(161\) 16.0000i 1.26098i
\(162\) 1.00000i 0.0785674i
\(163\) 7.12311i 0.557925i 0.960302 + 0.278962i \(0.0899905\pi\)
−0.960302 + 0.278962i \(0.910010\pi\)
\(164\) 0.876894i 0.0684739i
\(165\) −5.12311 −0.398833
\(166\) −6.24621 −0.484800
\(167\) 22.2462i 1.72146i 0.509059 + 0.860732i \(0.329994\pi\)
−0.509059 + 0.860732i \(0.670006\pi\)
\(168\) −3.12311 −0.240953
\(169\) 12.3693 + 4.00000i 0.951486 + 0.307692i
\(170\) 2.00000 0.153393
\(171\) 6.00000i 0.458831i
\(172\) −6.24621 −0.476269
\(173\) 23.1231 1.75802 0.879009 0.476806i \(-0.158206\pi\)
0.879009 + 0.476806i \(0.158206\pi\)
\(174\) 2.00000i 0.151620i
\(175\) 3.12311i 0.236085i
\(176\) 5.12311i 0.386169i
\(177\) 1.12311i 0.0844178i
\(178\) 3.12311 0.234087
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −20.2462 −1.50489 −0.752445 0.658656i \(-0.771125\pi\)
−0.752445 + 0.658656i \(0.771125\pi\)
\(182\) 11.1231 + 1.75379i 0.824499 + 0.129999i
\(183\) 10.0000 0.739221
\(184\) 5.12311i 0.377680i
\(185\) 5.12311 0.376658
\(186\) 3.12311 0.228997
\(187\) 10.2462i 0.749277i
\(188\) 6.24621i 0.455552i
\(189\) 3.12311i 0.227173i
\(190\) 6.00000i 0.435286i
\(191\) 16.4924 1.19335 0.596675 0.802483i \(-0.296488\pi\)
0.596675 + 0.802483i \(0.296488\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.1231i 1.80840i 0.427109 + 0.904200i \(0.359532\pi\)
−0.427109 + 0.904200i \(0.640468\pi\)
\(194\) −13.1231 −0.942184
\(195\) 0.561553 3.56155i 0.0402136 0.255048i
\(196\) 2.75379 0.196699
\(197\) 16.2462i 1.15749i −0.815507 0.578747i \(-0.803542\pi\)
0.815507 0.578747i \(-0.196458\pi\)
\(198\) 5.12311 0.364083
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 4.87689i 0.343990i
\(202\) 12.2462i 0.861640i
\(203\) 6.24621i 0.438398i
\(204\) −2.00000 −0.140028
\(205\) −0.876894 −0.0612450
\(206\) 13.1231i 0.914330i
\(207\) −5.12311 −0.356080
\(208\) 3.56155 + 0.561553i 0.246949 + 0.0389367i
\(209\) 30.7386 2.12624
\(210\) 3.12311i 0.215515i
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 13.3693 0.918208
\(213\) 10.2462i 0.702059i
\(214\) 8.00000i 0.546869i
\(215\) 6.24621i 0.425988i
\(216\) 1.00000i 0.0680414i
\(217\) −9.75379 −0.662130
\(218\) 2.87689 0.194848
\(219\) 13.1231i 0.886777i
\(220\) 5.12311 0.345400
\(221\) 7.12311 + 1.12311i 0.479152 + 0.0755483i
\(222\) −5.12311 −0.343840
\(223\) 9.36932i 0.627416i −0.949520 0.313708i \(-0.898429\pi\)
0.949520 0.313708i \(-0.101571\pi\)
\(224\) 3.12311 0.208671
\(225\) −1.00000 −0.0666667
\(226\) 12.2462i 0.814606i
\(227\) 8.00000i 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 6.00000i 0.397360i
\(229\) 5.12311i 0.338544i 0.985569 + 0.169272i \(0.0541417\pi\)
−0.985569 + 0.169272i \(0.945858\pi\)
\(230\) −5.12311 −0.337808
\(231\) −16.0000 −1.05272
\(232\) 2.00000i 0.131306i
\(233\) −7.75379 −0.507968 −0.253984 0.967208i \(-0.581741\pi\)
−0.253984 + 0.967208i \(0.581741\pi\)
\(234\) −0.561553 + 3.56155i −0.0367099 + 0.232826i
\(235\) 6.24621 0.407458
\(236\) 1.12311i 0.0731079i
\(237\) 8.00000 0.519656
\(238\) 6.24621 0.404882
\(239\) 4.49242i 0.290591i −0.989388 0.145295i \(-0.953587\pi\)
0.989388 0.145295i \(-0.0464132\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 14.2462i 0.917679i 0.888519 + 0.458840i \(0.151735\pi\)
−0.888519 + 0.458840i \(0.848265\pi\)
\(242\) 15.2462i 0.980064i
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 2.75379i 0.175933i
\(246\) 0.876894 0.0559087
\(247\) −3.36932 + 21.3693i −0.214384 + 1.35970i
\(248\) −3.12311 −0.198317
\(249\) 6.24621i 0.395838i
\(250\) −1.00000 −0.0632456
\(251\) 26.2462 1.65665 0.828323 0.560251i \(-0.189295\pi\)
0.828323 + 0.560251i \(0.189295\pi\)
\(252\) 3.12311i 0.196737i
\(253\) 26.2462i 1.65009i
\(254\) 13.1231i 0.823417i
\(255\) 2.00000i 0.125245i
\(256\) 1.00000 0.0625000
\(257\) −12.2462 −0.763898 −0.381949 0.924183i \(-0.624747\pi\)
−0.381949 + 0.924183i \(0.624747\pi\)
\(258\) 6.24621i 0.388872i
\(259\) 16.0000 0.994192
\(260\) −0.561553 + 3.56155i −0.0348260 + 0.220878i
\(261\) 2.00000 0.123797
\(262\) 4.00000i 0.247121i
\(263\) −27.3693 −1.68766 −0.843832 0.536607i \(-0.819706\pi\)
−0.843832 + 0.536607i \(0.819706\pi\)
\(264\) −5.12311 −0.315305
\(265\) 13.3693i 0.821271i
\(266\) 18.7386i 1.14894i
\(267\) 3.12311i 0.191131i
\(268\) 4.87689i 0.297904i
\(269\) −16.2462 −0.990549 −0.495274 0.868737i \(-0.664933\pi\)
−0.495274 + 0.868737i \(0.664933\pi\)
\(270\) 1.00000 0.0608581
\(271\) 23.1231i 1.40463i −0.711867 0.702314i \(-0.752150\pi\)
0.711867 0.702314i \(-0.247850\pi\)
\(272\) 2.00000 0.121268
\(273\) 1.75379 11.1231i 0.106144 0.673201i
\(274\) −10.4924 −0.633870
\(275\) 5.12311i 0.308935i
\(276\) 5.12311 0.308375
\(277\) −29.8617 −1.79422 −0.897109 0.441809i \(-0.854337\pi\)
−0.897109 + 0.441809i \(0.854337\pi\)
\(278\) 16.4924i 0.989150i
\(279\) 3.12311i 0.186975i
\(280\) 3.12311i 0.186641i
\(281\) 3.12311i 0.186309i 0.995652 + 0.0931544i \(0.0296950\pi\)
−0.995652 + 0.0931544i \(0.970305\pi\)
\(282\) −6.24621 −0.371956
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 10.2462i 0.608001i
\(285\) 6.00000 0.355409
\(286\) 18.2462 + 2.87689i 1.07892 + 0.170114i
\(287\) −2.73863 −0.161656
\(288\) 1.00000i 0.0589256i
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 13.1231i 0.769290i
\(292\) 13.1231i 0.767972i
\(293\) 28.7386i 1.67893i −0.543415 0.839464i \(-0.682869\pi\)
0.543415 0.839464i \(-0.317131\pi\)
\(294\) 2.75379i 0.160604i
\(295\) 1.12311 0.0653897
\(296\) 5.12311 0.297774
\(297\) 5.12311i 0.297273i
\(298\) −14.0000 −0.810998
\(299\) −18.2462 2.87689i −1.05521 0.166375i
\(300\) 1.00000 0.0577350
\(301\) 19.5076i 1.12440i
\(302\) 13.3693 0.769318
\(303\) −12.2462 −0.703526
\(304\) 6.00000i 0.344124i
\(305\) 10.0000i 0.572598i
\(306\) 2.00000i 0.114332i
\(307\) 31.1231i 1.77629i −0.459564 0.888145i \(-0.651994\pi\)
0.459564 0.888145i \(-0.348006\pi\)
\(308\) 16.0000 0.911685
\(309\) −13.1231 −0.746547
\(310\) 3.12311i 0.177380i
\(311\) 8.49242 0.481561 0.240781 0.970580i \(-0.422597\pi\)
0.240781 + 0.970580i \(0.422597\pi\)
\(312\) 0.561553 3.56155i 0.0317917 0.201633i
\(313\) 16.2462 0.918290 0.459145 0.888361i \(-0.348156\pi\)
0.459145 + 0.888361i \(0.348156\pi\)
\(314\) 21.3693i 1.20594i
\(315\) −3.12311 −0.175967
\(316\) −8.00000 −0.450035
\(317\) 6.00000i 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 13.3693i 0.749714i
\(319\) 10.2462i 0.573678i
\(320\) 1.00000i 0.0559017i
\(321\) −8.00000 −0.446516
\(322\) −16.0000 −0.891645
\(323\) 12.0000i 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) −3.56155 0.561553i −0.197559 0.0311493i
\(326\) −7.12311 −0.394512
\(327\) 2.87689i 0.159093i
\(328\) −0.876894 −0.0484184
\(329\) 19.5076 1.07549
\(330\) 5.12311i 0.282018i
\(331\) 7.75379i 0.426187i 0.977032 + 0.213093i \(0.0683539\pi\)
−0.977032 + 0.213093i \(0.931646\pi\)
\(332\) 6.24621i 0.342805i
\(333\) 5.12311i 0.280744i
\(334\) −22.2462 −1.21726
\(335\) −4.87689 −0.266453
\(336\) 3.12311i 0.170379i
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −4.00000 + 12.3693i −0.217571 + 0.672802i
\(339\) 12.2462 0.665123
\(340\) 2.00000i 0.108465i
\(341\) −16.0000 −0.866449
\(342\) −6.00000 −0.324443
\(343\) 13.2614i 0.716046i
\(344\) 6.24621i 0.336773i
\(345\) 5.12311i 0.275819i
\(346\) 23.1231i 1.24311i
\(347\) −14.2462 −0.764777 −0.382388 0.924002i \(-0.624898\pi\)
−0.382388 + 0.924002i \(0.624898\pi\)
\(348\) −2.00000 −0.107211
\(349\) 19.3693i 1.03682i −0.855133 0.518408i \(-0.826525\pi\)
0.855133 0.518408i \(-0.173475\pi\)
\(350\) −3.12311 −0.166937
\(351\) 3.56155 + 0.561553i 0.190102 + 0.0299735i
\(352\) 5.12311 0.273062
\(353\) 12.2462i 0.651800i 0.945404 + 0.325900i \(0.105667\pi\)
−0.945404 + 0.325900i \(0.894333\pi\)
\(354\) −1.12311 −0.0596924
\(355\) −10.2462 −0.543812
\(356\) 3.12311i 0.165524i
\(357\) 6.24621i 0.330585i
\(358\) 16.4924i 0.871652i
\(359\) 17.7538i 0.937009i −0.883461 0.468505i \(-0.844793\pi\)
0.883461 0.468505i \(-0.155207\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.0000 −0.894737
\(362\) 20.2462i 1.06412i
\(363\) −15.2462 −0.800219
\(364\) −1.75379 + 11.1231i −0.0919235 + 0.583009i
\(365\) 13.1231 0.686895
\(366\) 10.0000i 0.522708i
\(367\) 8.63068 0.450518 0.225259 0.974299i \(-0.427677\pi\)
0.225259 + 0.974299i \(0.427677\pi\)
\(368\) −5.12311 −0.267060
\(369\) 0.876894i 0.0456493i
\(370\) 5.12311i 0.266338i
\(371\) 41.7538i 2.16775i
\(372\) 3.12311i 0.161925i
\(373\) 7.12311 0.368820 0.184410 0.982849i \(-0.440963\pi\)
0.184410 + 0.982849i \(0.440963\pi\)
\(374\) 10.2462 0.529819
\(375\) 1.00000i 0.0516398i
\(376\) 6.24621 0.322124
\(377\) 7.12311 + 1.12311i 0.366859 + 0.0578429i
\(378\) 3.12311 0.160635
\(379\) 6.00000i 0.308199i 0.988055 + 0.154100i \(0.0492477\pi\)
−0.988055 + 0.154100i \(0.950752\pi\)
\(380\) −6.00000 −0.307794
\(381\) −13.1231 −0.672317
\(382\) 16.4924i 0.843826i
\(383\) 1.75379i 0.0896144i −0.998996 0.0448072i \(-0.985733\pi\)
0.998996 0.0448072i \(-0.0142674\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 16.0000i 0.815436i
\(386\) −25.1231 −1.27873
\(387\) 6.24621 0.317513
\(388\) 13.1231i 0.666225i
\(389\) −0.246211 −0.0124834 −0.00624170 0.999981i \(-0.501987\pi\)
−0.00624170 + 0.999981i \(0.501987\pi\)
\(390\) 3.56155 + 0.561553i 0.180346 + 0.0284353i
\(391\) −10.2462 −0.518173
\(392\) 2.75379i 0.139087i
\(393\) 4.00000 0.201773
\(394\) 16.2462 0.818472
\(395\) 8.00000i 0.402524i
\(396\) 5.12311i 0.257446i
\(397\) 15.3693i 0.771364i 0.922632 + 0.385682i \(0.126034\pi\)
−0.922632 + 0.385682i \(0.873966\pi\)
\(398\) 18.2462i 0.914600i
\(399\) 18.7386 0.938105
\(400\) −1.00000 −0.0500000
\(401\) 1.36932i 0.0683804i 0.999415 + 0.0341902i \(0.0108852\pi\)
−0.999415 + 0.0341902i \(0.989115\pi\)
\(402\) 4.87689 0.243237
\(403\) 1.75379 11.1231i 0.0873624 0.554081i
\(404\) 12.2462 0.609272
\(405\) 1.00000i 0.0496904i
\(406\) 6.24621 0.309994
\(407\) 26.2462 1.30098
\(408\) 2.00000i 0.0990148i
\(409\) 8.49242i 0.419923i −0.977710 0.209962i \(-0.932666\pi\)
0.977710 0.209962i \(-0.0673339\pi\)
\(410\) 0.876894i 0.0433067i
\(411\) 10.4924i 0.517553i
\(412\) 13.1231 0.646529
\(413\) 3.50758 0.172597
\(414\) 5.12311i 0.251787i
\(415\) 6.24621 0.306614
\(416\) −0.561553 + 3.56155i −0.0275324 + 0.174619i
\(417\) −16.4924 −0.807637
\(418\) 30.7386i 1.50348i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 3.12311 0.152392
\(421\) 0.630683i 0.0307376i 0.999882 + 0.0153688i \(0.00489224\pi\)
−0.999882 + 0.0153688i \(0.995108\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 6.24621i 0.303701i
\(424\) 13.3693i 0.649271i
\(425\) −2.00000 −0.0970143
\(426\) 10.2462 0.496431
\(427\) 31.2311i 1.51138i
\(428\) 8.00000 0.386695
\(429\) 2.87689 18.2462i 0.138898 0.880935i
\(430\) 6.24621 0.301219
\(431\) 32.4924i 1.56510i 0.622585 + 0.782552i \(0.286083\pi\)
−0.622585 + 0.782552i \(0.713917\pi\)
\(432\) 1.00000 0.0481125
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 9.75379i 0.468197i
\(435\) 2.00000i 0.0958927i
\(436\) 2.87689i 0.137778i
\(437\) 30.7386i 1.47043i
\(438\) −13.1231 −0.627046
\(439\) −36.4924 −1.74169 −0.870844 0.491559i \(-0.836427\pi\)
−0.870844 + 0.491559i \(0.836427\pi\)
\(440\) 5.12311i 0.244234i
\(441\) −2.75379 −0.131133
\(442\) −1.12311 + 7.12311i −0.0534207 + 0.338812i
\(443\) −3.50758 −0.166650 −0.0833250 0.996522i \(-0.526554\pi\)
−0.0833250 + 0.996522i \(0.526554\pi\)
\(444\) 5.12311i 0.243132i
\(445\) −3.12311 −0.148049
\(446\) 9.36932 0.443650
\(447\) 14.0000i 0.662177i
\(448\) 3.12311i 0.147553i
\(449\) 28.8769i 1.36278i 0.731918 + 0.681392i \(0.238625\pi\)
−0.731918 + 0.681392i \(0.761375\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −4.49242 −0.211540
\(452\) −12.2462 −0.576013
\(453\) 13.3693i 0.628145i
\(454\) 8.00000 0.375459
\(455\) −11.1231 1.75379i −0.521459 0.0822189i
\(456\) 6.00000 0.280976
\(457\) 31.3693i 1.46739i −0.679476 0.733697i \(-0.737793\pi\)
0.679476 0.733697i \(-0.262207\pi\)
\(458\) −5.12311 −0.239387
\(459\) 2.00000 0.0933520
\(460\) 5.12311i 0.238866i
\(461\) 18.4924i 0.861278i −0.902524 0.430639i \(-0.858288\pi\)
0.902524 0.430639i \(-0.141712\pi\)
\(462\) 16.0000i 0.744387i
\(463\) 21.8617i 1.01600i 0.861357 + 0.508001i \(0.169615\pi\)
−0.861357 + 0.508001i \(0.830385\pi\)
\(464\) 2.00000 0.0928477
\(465\) −3.12311 −0.144831
\(466\) 7.75379i 0.359187i
\(467\) −22.2462 −1.02943 −0.514716 0.857361i \(-0.672103\pi\)
−0.514716 + 0.857361i \(0.672103\pi\)
\(468\) −3.56155 0.561553i −0.164633 0.0259578i
\(469\) −15.2311 −0.703305
\(470\) 6.24621i 0.288116i
\(471\) 21.3693 0.984646
\(472\) 1.12311 0.0516951
\(473\) 32.0000i 1.47136i
\(474\) 8.00000i 0.367452i
\(475\) 6.00000i 0.275299i
\(476\) 6.24621i 0.286295i
\(477\) −13.3693 −0.612139
\(478\) 4.49242 0.205479
\(479\) 12.4924i 0.570793i 0.958409 + 0.285397i \(0.0921253\pi\)
−0.958409 + 0.285397i \(0.907875\pi\)
\(480\) 1.00000 0.0456435
\(481\) −2.87689 + 18.2462i −0.131175 + 0.831956i
\(482\) −14.2462 −0.648897
\(483\) 16.0000i 0.728025i
\(484\) 15.2462 0.693010
\(485\) 13.1231 0.595890
\(486\) 1.00000i 0.0453609i
\(487\) 17.3693i 0.787079i −0.919308 0.393539i \(-0.871250\pi\)
0.919308 0.393539i \(-0.128750\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 7.12311i 0.322118i
\(490\) −2.75379 −0.124403
\(491\) 38.7386 1.74825 0.874125 0.485701i \(-0.161436\pi\)
0.874125 + 0.485701i \(0.161436\pi\)
\(492\) 0.876894i 0.0395335i
\(493\) 4.00000 0.180151
\(494\) −21.3693 3.36932i −0.961451 0.151593i
\(495\) −5.12311 −0.230266
\(496\) 3.12311i 0.140232i
\(497\) −32.0000 −1.43540
\(498\) −6.24621 −0.279899
\(499\) 34.4924i 1.54409i −0.635566 0.772046i \(-0.719233\pi\)
0.635566 0.772046i \(-0.280767\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 22.2462i 0.993887i
\(502\) 26.2462i 1.17143i
\(503\) 35.3693 1.57704 0.788520 0.615009i \(-0.210848\pi\)
0.788520 + 0.615009i \(0.210848\pi\)
\(504\) −3.12311 −0.139114
\(505\) 12.2462i 0.544949i
\(506\) −26.2462 −1.16679
\(507\) 12.3693 + 4.00000i 0.549341 + 0.177646i
\(508\) 13.1231 0.582244
\(509\) 32.2462i 1.42929i −0.699488 0.714644i \(-0.746589\pi\)
0.699488 0.714644i \(-0.253411\pi\)
\(510\) 2.00000 0.0885615
\(511\) 40.9848 1.81306
\(512\) 1.00000i 0.0441942i
\(513\) 6.00000i 0.264906i
\(514\) 12.2462i 0.540157i
\(515\) 13.1231i 0.578273i
\(516\) −6.24621 −0.274974
\(517\) 32.0000 1.40736
\(518\) 16.0000i 0.703000i
\(519\) 23.1231 1.01499
\(520\) −3.56155 0.561553i −0.156184 0.0246257i
\(521\) 0.246211 0.0107867 0.00539336 0.999985i \(-0.498283\pi\)
0.00539336 + 0.999985i \(0.498283\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) −26.7386 −1.16920 −0.584599 0.811322i \(-0.698748\pi\)
−0.584599 + 0.811322i \(0.698748\pi\)
\(524\) −4.00000 −0.174741
\(525\) 3.12311i 0.136304i
\(526\) 27.3693i 1.19336i
\(527\) 6.24621i 0.272089i
\(528\) 5.12311i 0.222955i
\(529\) 3.24621 0.141140
\(530\) −13.3693 −0.580726
\(531\) 1.12311i 0.0487386i
\(532\) −18.7386 −0.812423
\(533\) 0.492423 3.12311i 0.0213292 0.135277i
\(534\) 3.12311 0.135150
\(535\) 8.00000i 0.345870i
\(536\) −4.87689 −0.210650
\(537\) 16.4924 0.711701
\(538\) 16.2462i 0.700424i
\(539\) 14.1080i 0.607672i
\(540\) 1.00000i 0.0430331i
\(541\) 10.8769i 0.467634i −0.972281 0.233817i \(-0.924878\pi\)
0.972281 0.233817i \(-0.0751217\pi\)
\(542\) 23.1231 0.993222
\(543\) −20.2462 −0.868848
\(544\) 2.00000i 0.0857493i
\(545\) −2.87689 −0.123233
\(546\) 11.1231 + 1.75379i 0.476025 + 0.0750552i
\(547\) 20.9848 0.897247 0.448624 0.893721i \(-0.351914\pi\)
0.448624 + 0.893721i \(0.351914\pi\)
\(548\) 10.4924i 0.448214i
\(549\) 10.0000 0.426790
\(550\) −5.12311 −0.218450
\(551\) 12.0000i 0.511217i
\(552\) 5.12311i 0.218054i
\(553\) 24.9848i 1.06246i
\(554\) 29.8617i 1.26870i
\(555\) 5.12311 0.217464
\(556\) 16.4924 0.699435
\(557\) 11.7538i 0.498024i −0.968500 0.249012i \(-0.919894\pi\)
0.968500 0.249012i \(-0.0801059\pi\)
\(558\) 3.12311 0.132212
\(559\) 22.2462 + 3.50758i 0.940914 + 0.148355i
\(560\) −3.12311 −0.131975
\(561\) 10.2462i 0.432595i
\(562\) −3.12311 −0.131740
\(563\) −32.9848 −1.39015 −0.695073 0.718939i \(-0.744628\pi\)
−0.695073 + 0.718939i \(0.744628\pi\)
\(564\) 6.24621i 0.263013i
\(565\) 12.2462i 0.515202i
\(566\) 4.00000i 0.168133i
\(567\) 3.12311i 0.131158i
\(568\) −10.2462 −0.429921
\(569\) −12.7386 −0.534031 −0.267016 0.963692i \(-0.586038\pi\)
−0.267016 + 0.963692i \(0.586038\pi\)
\(570\) 6.00000i 0.251312i
\(571\) −8.49242 −0.355397 −0.177698 0.984085i \(-0.556865\pi\)
−0.177698 + 0.984085i \(0.556865\pi\)
\(572\) −2.87689 + 18.2462i −0.120289 + 0.762912i
\(573\) 16.4924 0.688981
\(574\) 2.73863i 0.114308i
\(575\) 5.12311 0.213648
\(576\) −1.00000 −0.0416667
\(577\) 27.3693i 1.13940i −0.821853 0.569700i \(-0.807059\pi\)
0.821853 0.569700i \(-0.192941\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 25.1231i 1.04408i
\(580\) 2.00000i 0.0830455i
\(581\) 19.5076 0.809311
\(582\) −13.1231 −0.543970
\(583\) 68.4924i 2.83667i
\(584\) 13.1231 0.543038
\(585\) 0.561553 3.56155i 0.0232174 0.147252i
\(586\) 28.7386 1.18718
\(587\) 16.4924i 0.680715i 0.940296 + 0.340358i \(0.110548\pi\)
−0.940296 + 0.340358i \(0.889452\pi\)
\(588\) 2.75379 0.113564
\(589\) 18.7386 0.772112
\(590\) 1.12311i 0.0462375i
\(591\) 16.2462i 0.668280i
\(592\) 5.12311i 0.210558i
\(593\) 5.50758i 0.226169i 0.993585 + 0.113085i \(0.0360731\pi\)
−0.993585 + 0.113085i \(0.963927\pi\)
\(594\) 5.12311 0.210204
\(595\) −6.24621 −0.256070
\(596\) 14.0000i 0.573462i
\(597\) 18.2462 0.746768
\(598\) 2.87689 18.2462i 0.117645 0.746143i
\(599\) −36.4924 −1.49104 −0.745520 0.666483i \(-0.767799\pi\)
−0.745520 + 0.666483i \(0.767799\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 19.5076 0.795070
\(603\) 4.87689i 0.198602i
\(604\) 13.3693i 0.543990i
\(605\) 15.2462i 0.619847i
\(606\) 12.2462i 0.497468i
\(607\) 15.3693 0.623821 0.311911 0.950111i \(-0.399031\pi\)
0.311911 + 0.950111i \(0.399031\pi\)
\(608\) −6.00000 −0.243332
\(609\) 6.24621i 0.253109i
\(610\) 10.0000 0.404888
\(611\) −3.50758 + 22.2462i −0.141901 + 0.899985i
\(612\) −2.00000 −0.0808452
\(613\) 39.3693i 1.59011i −0.606536 0.795056i \(-0.707441\pi\)
0.606536 0.795056i \(-0.292559\pi\)
\(614\) 31.1231 1.25603
\(615\) −0.876894 −0.0353598
\(616\) 16.0000i 0.644658i
\(617\) 40.7386i 1.64008i 0.572309 + 0.820038i \(0.306048\pi\)
−0.572309 + 0.820038i \(0.693952\pi\)
\(618\) 13.1231i 0.527889i
\(619\) 38.9848i 1.56693i −0.621434 0.783467i \(-0.713450\pi\)
0.621434 0.783467i \(-0.286550\pi\)
\(620\) 3.12311 0.125427
\(621\) −5.12311 −0.205583
\(622\) 8.49242i 0.340515i
\(623\) −9.75379 −0.390777
\(624\) 3.56155 + 0.561553i 0.142576 + 0.0224801i
\(625\) 1.00000 0.0400000
\(626\) 16.2462i 0.649329i
\(627\) 30.7386 1.22758
\(628\) −21.3693 −0.852729
\(629\) 10.2462i 0.408543i
\(630\) 3.12311i 0.124428i
\(631\) 35.6155i 1.41783i 0.705293 + 0.708916i \(0.250815\pi\)
−0.705293 + 0.708916i \(0.749185\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 4.00000 0.158986
\(634\) 6.00000 0.238290
\(635\) 13.1231i 0.520775i
\(636\) 13.3693 0.530128
\(637\) −9.80776 1.54640i −0.388598 0.0612705i
\(638\) 10.2462 0.405651
\(639\) 10.2462i 0.405334i
\(640\) −1.00000 −0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 8.00000i 0.315735i
\(643\) 17.3693i 0.684979i −0.939522 0.342489i \(-0.888730\pi\)
0.939522 0.342489i \(-0.111270\pi\)
\(644\) 16.0000i 0.630488i
\(645\) 6.24621i 0.245944i
\(646\) −12.0000 −0.472134
\(647\) 29.6155 1.16431 0.582153 0.813079i \(-0.302210\pi\)
0.582153 + 0.813079i \(0.302210\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 5.75379 0.225856
\(650\) 0.561553 3.56155i 0.0220259 0.139696i
\(651\) −9.75379 −0.382281
\(652\) 7.12311i 0.278962i
\(653\) −13.8617 −0.542452 −0.271226 0.962516i \(-0.587429\pi\)
−0.271226 + 0.962516i \(0.587429\pi\)
\(654\) 2.87689 0.112495
\(655\) 4.00000i 0.156293i
\(656\) 0.876894i 0.0342370i
\(657\) 13.1231i 0.511981i
\(658\) 19.5076i 0.760485i
\(659\) −21.7538 −0.847407 −0.423704 0.905801i \(-0.639270\pi\)
−0.423704 + 0.905801i \(0.639270\pi\)
\(660\) 5.12311 0.199417
\(661\) 9.12311i 0.354848i −0.984135 0.177424i \(-0.943224\pi\)
0.984135 0.177424i \(-0.0567763\pi\)
\(662\) −7.75379 −0.301360
\(663\) 7.12311 + 1.12311i 0.276638 + 0.0436178i
\(664\) 6.24621 0.242400
\(665\) 18.7386i 0.726653i
\(666\) −5.12311 −0.198516
\(667\) −10.2462 −0.396735
\(668\) 22.2462i 0.860732i
\(669\) 9.36932i 0.362239i
\(670\) 4.87689i 0.188411i
\(671\) 51.2311i 1.97775i
\(672\) 3.12311 0.120476
\(673\) 26.9848 1.04019 0.520095 0.854109i \(-0.325897\pi\)
0.520095 + 0.854109i \(0.325897\pi\)
\(674\) 6.00000i 0.231111i
\(675\) −1.00000 −0.0384900
\(676\) −12.3693 4.00000i −0.475743 0.153846i
\(677\) 12.8769 0.494899 0.247450 0.968901i \(-0.420408\pi\)
0.247450 + 0.968901i \(0.420408\pi\)
\(678\) 12.2462i 0.470313i
\(679\) 40.9848 1.57285
\(680\) −2.00000 −0.0766965
\(681\) 8.00000i 0.306561i
\(682\) 16.0000i 0.612672i
\(683\) 16.9848i 0.649907i 0.945730 + 0.324954i \(0.105349\pi\)
−0.945730 + 0.324954i \(0.894651\pi\)
\(684\) 6.00000i 0.229416i
\(685\) 10.4924 0.400895
\(686\) 13.2614 0.506321
\(687\) 5.12311i 0.195459i
\(688\) 6.24621 0.238135
\(689\) −47.6155 7.50758i −1.81401 0.286016i
\(690\) −5.12311 −0.195033
\(691\) 28.7386i 1.09327i −0.837371 0.546635i \(-0.815909\pi\)
0.837371 0.546635i \(-0.184091\pi\)
\(692\) −23.1231 −0.879009
\(693\) −16.0000 −0.607790
\(694\) 14.2462i 0.540779i
\(695\) 16.4924i 0.625593i
\(696\) 2.00000i 0.0758098i
\(697\) 1.75379i 0.0664295i
\(698\) 19.3693 0.733139
\(699\) −7.75379 −0.293275
\(700\) 3.12311i 0.118042i
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) −0.561553 + 3.56155i −0.0211944 + 0.134422i
\(703\) −30.7386 −1.15933
\(704\) 5.12311i 0.193084i
\(705\) 6.24621 0.235246
\(706\) −12.2462 −0.460892
\(707\) 38.2462i 1.43840i
\(708\) 1.12311i 0.0422089i
\(709\) 1.61553i 0.0606724i 0.999540 + 0.0303362i \(0.00965780\pi\)
−0.999540 + 0.0303362i \(0.990342\pi\)
\(710\) 10.2462i 0.384533i
\(711\) 8.00000 0.300023
\(712\) −3.12311 −0.117043
\(713\) 16.0000i 0.599205i
\(714\) 6.24621 0.233759
\(715\) −18.2462 2.87689i −0.682370 0.107590i
\(716\) −16.4924 −0.616351
\(717\) 4.49242i 0.167773i
\(718\) 17.7538 0.662566
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 40.9848i 1.52636i
\(722\) 17.0000i 0.632674i
\(723\) 14.2462i 0.529822i
\(724\) 20.2462 0.752445
\(725\) −2.00000 −0.0742781
\(726\) 15.2462i 0.565840i
\(727\) 19.8617 0.736631 0.368316 0.929701i \(-0.379935\pi\)
0.368316 + 0.929701i \(0.379935\pi\)
\(728\) −11.1231 1.75379i −0.412250 0.0649997i
\(729\) 1.00000 0.0370370
\(730\) 13.1231i 0.485708i
\(731\) 12.4924 0.462049
\(732\) −10.0000 −0.369611
\(733\) 35.3693i 1.30640i −0.757187 0.653198i \(-0.773427\pi\)
0.757187 0.653198i \(-0.226573\pi\)
\(734\) 8.63068i 0.318564i
\(735\) 2.75379i 0.101575i
\(736\) 5.12311i 0.188840i
\(737\) −24.9848 −0.920329
\(738\) 0.876894 0.0322789
\(739\) 37.2311i 1.36957i 0.728747 + 0.684783i \(0.240103\pi\)
−0.728747 + 0.684783i \(0.759897\pi\)
\(740\) −5.12311 −0.188329
\(741\) −3.36932 + 21.3693i −0.123775 + 0.785021i
\(742\) −41.7538 −1.53283
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) −3.12311 −0.114499
\(745\) 14.0000 0.512920
\(746\) 7.12311i 0.260795i
\(747\) 6.24621i 0.228537i
\(748\) 10.2462i 0.374639i
\(749\) 24.9848i 0.912926i
\(750\) −1.00000 −0.0365148
\(751\) −26.2462 −0.957738 −0.478869 0.877886i \(-0.658953\pi\)
−0.478869 + 0.877886i \(0.658953\pi\)
\(752\) 6.24621i 0.227776i
\(753\) 26.2462 0.956465
\(754\) −1.12311 + 7.12311i −0.0409011 + 0.259408i
\(755\) −13.3693 −0.486559
\(756\) 3.12311i 0.113586i
\(757\) −3.12311 −0.113511 −0.0567556 0.998388i \(-0.518076\pi\)
−0.0567556 + 0.998388i \(0.518076\pi\)
\(758\) −6.00000 −0.217930
\(759\) 26.2462i 0.952677i
\(760\) 6.00000i 0.217643i
\(761\) 3.12311i 0.113212i 0.998397 + 0.0566062i \(0.0180280\pi\)
−0.998397 + 0.0566062i \(0.981972\pi\)
\(762\) 13.1231i 0.475400i
\(763\) −8.98485 −0.325273
\(764\) −16.4924 −0.596675
\(765\) 2.00000i 0.0723102i
\(766\) 1.75379 0.0633670
\(767\) −0.630683 + 4.00000i −0.0227726 + 0.144432i
\(768\) 1.00000 0.0360844
\(769\) 32.9848i 1.18946i 0.803924 + 0.594732i \(0.202742\pi\)
−0.803924 + 0.594732i \(0.797258\pi\)
\(770\) −16.0000 −0.576600
\(771\) −12.2462 −0.441037
\(772\) 25.1231i 0.904200i
\(773\) 16.2462i 0.584336i 0.956367 + 0.292168i \(0.0943766\pi\)
−0.956367 + 0.292168i \(0.905623\pi\)
\(774\) 6.24621i 0.224515i
\(775\) 3.12311i 0.112185i
\(776\) 13.1231 0.471092
\(777\) 16.0000 0.573997
\(778\) 0.246211i 0.00882710i
\(779\) 5.26137 0.188508
\(780\) −0.561553 + 3.56155i −0.0201068 + 0.127524i
\(781\) −52.4924 −1.87833
\(782\) 10.2462i 0.366404i
\(783\) 2.00000 0.0714742
\(784\) −2.75379 −0.0983496
\(785\) 21.3693i 0.762704i
\(786\) 4.00000i 0.142675i
\(787\) 12.3845i 0.441459i 0.975335 + 0.220729i \(0.0708438\pi\)
−0.975335 + 0.220729i \(0.929156\pi\)
\(788\) 16.2462i 0.578747i
\(789\) −27.3693 −0.974373
\(790\) 8.00000 0.284627
\(791\) 38.2462i 1.35988i
\(792\) −5.12311 −0.182042
\(793\) 35.6155 + 5.61553i 1.26474 + 0.199413i
\(794\) −15.3693 −0.545437
\(795\) 13.3693i 0.474161i
\(796\) −18.2462 −0.646720
\(797\) 25.8617 0.916070 0.458035 0.888934i \(-0.348553\pi\)
0.458035 + 0.888934i \(0.348553\pi\)
\(798\) 18.7386i 0.663340i
\(799\) 12.4924i 0.441950i
\(800\) 1.00000i 0.0353553i
\(801\) 3.12311i 0.110350i
\(802\) −1.36932 −0.0483523
\(803\) 67.2311 2.37253
\(804\) 4.87689i 0.171995i
\(805\) 16.0000 0.563926
\(806\) 11.1231 + 1.75379i 0.391795 + 0.0617746i
\(807\) −16.2462 −0.571894
\(808\) 12.2462i 0.430820i
\(809\) −13.5076 −0.474901 −0.237451 0.971400i \(-0.576312\pi\)
−0.237451 + 0.971400i \(0.576312\pi\)
\(810\) 1.00000 0.0351364
\(811\) 27.7538i 0.974567i 0.873244 + 0.487284i \(0.162012\pi\)
−0.873244 + 0.487284i \(0.837988\pi\)
\(812\) 6.24621i 0.219199i
\(813\) 23.1231i 0.810963i
\(814\) 26.2462i 0.919929i
\(815\) 7.12311 0.249512
\(816\) 2.00000 0.0700140
\(817\) 37.4773i 1.31116i
\(818\) 8.49242 0.296931
\(819\) 1.75379 11.1231i 0.0612823 0.388673i
\(820\) 0.876894 0.0306225
\(821\) 44.2462i 1.54420i −0.635499 0.772102i \(-0.719206\pi\)
0.635499 0.772102i \(-0.280794\pi\)
\(822\) −10.4924 −0.365965
\(823\) −42.8769 −1.49459 −0.747297 0.664490i \(-0.768649\pi\)
−0.747297 + 0.664490i \(0.768649\pi\)
\(824\) 13.1231i 0.457165i
\(825\) 5.12311i 0.178364i
\(826\) 3.50758i 0.122044i
\(827\) 1.26137i 0.0438620i −0.999759 0.0219310i \(-0.993019\pi\)
0.999759 0.0219310i \(-0.00698141\pi\)
\(828\) 5.12311 0.178040
\(829\) −24.2462 −0.842106 −0.421053 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(830\) 6.24621i 0.216809i
\(831\) −29.8617 −1.03589
\(832\) −3.56155 0.561553i −0.123475 0.0194683i
\(833\) −5.50758 −0.190826
\(834\) 16.4924i 0.571086i
\(835\) 22.2462 0.769862
\(836\) −30.7386 −1.06312
\(837\) 3.12311i 0.107950i
\(838\) 28.0000i 0.967244i
\(839\) 46.7386i 1.61360i 0.590827 + 0.806798i \(0.298802\pi\)
−0.590827 + 0.806798i \(0.701198\pi\)
\(840\) 3.12311i 0.107757i
\(841\) −25.0000 −0.862069
\(842\) −0.630683 −0.0217348
\(843\) 3.12311i 0.107565i
\(844\) −4.00000 −0.137686
\(845\) 4.00000 12.3693i 0.137604 0.425517i
\(846\) −6.24621 −0.214749
\(847\) 47.6155i 1.63609i
\(848\) −13.3693 −0.459104
\(849\) −4.00000 −0.137280
\(850\) 2.00000i 0.0685994i
\(851\) 26.2462i 0.899709i
\(852\) 10.2462i 0.351029i
\(853\) 35.8617i 1.22788i −0.789352 0.613941i \(-0.789583\pi\)
0.789352 0.613941i \(-0.210417\pi\)
\(854\) 31.2311 1.06870
\(855\) 6.00000 0.205196
\(856\) 8.00000i 0.273434i
\(857\) −30.4924 −1.04160 −0.520801 0.853678i \(-0.674367\pi\)
−0.520801 + 0.853678i \(0.674367\pi\)
\(858\) 18.2462 + 2.87689i 0.622915 + 0.0982156i
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 6.24621i 0.212994i
\(861\) −2.73863 −0.0933324
\(862\) −32.4924 −1.10670
\(863\) 6.24621i 0.212624i 0.994333 + 0.106312i \(0.0339042\pi\)
−0.994333 + 0.106312i \(0.966096\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 23.1231i 0.786209i
\(866\) 18.0000i 0.611665i
\(867\) −13.0000 −0.441503
\(868\) 9.75379 0.331065
\(869\) 40.9848i 1.39032i
\(870\) 2.00000 0.0678064
\(871\) 2.73863 17.3693i 0.0927951 0.588537i
\(872\) −2.87689 −0.0974239
\(873\) 13.1231i 0.444150i
\(874\) 30.7386 1.03975
\(875\) 3.12311 0.105580
\(876\) 13.1231i 0.443389i
\(877\) 18.8769i 0.637427i 0.947851 + 0.318714i \(0.103251\pi\)
−0.947851 + 0.318714i \(0.896749\pi\)
\(878\) 36.4924i 1.23156i
\(879\) 28.7386i 0.969330i
\(880\) −5.12311 −0.172700
\(881\) 28.2462 0.951639 0.475820 0.879543i \(-0.342152\pi\)
0.475820 + 0.879543i \(0.342152\pi\)
\(882\) 2.75379i 0.0927249i
\(883\) −9.75379 −0.328241 −0.164121 0.986440i \(-0.552479\pi\)
−0.164121 + 0.986440i \(0.552479\pi\)
\(884\) −7.12311 1.12311i −0.239576 0.0377741i
\(885\) 1.12311 0.0377528
\(886\) 3.50758i 0.117839i
\(887\) −18.8769 −0.633824 −0.316912 0.948455i \(-0.602646\pi\)
−0.316912 + 0.948455i \(0.602646\pi\)
\(888\) 5.12311 0.171920
\(889\) 40.9848i 1.37459i
\(890\) 3.12311i 0.104687i
\(891\) 5.12311i 0.171630i
\(892\) 9.36932i 0.313708i
\(893\) −37.4773 −1.25413
\(894\) −14.0000 −0.468230
\(895\) 16.4924i 0.551281i
\(896\) −3.12311 −0.104336
\(897\) −18.2462 2.87689i −0.609223 0.0960567i
\(898\) −28.8769 −0.963634
\(899\) 6.24621i 0.208323i
\(900\) 1.00000 0.0333333
\(901\) −26.7386 −0.890793
\(902\) 4.49242i 0.149581i
\(903\) 19.5076i 0.649172i
\(904\) 12.2462i 0.407303i
\(905\) 20.2462i 0.673007i
\(906\) 13.3693 0.444166
\(907\) 25.7538 0.855141 0.427570 0.903982i \(-0.359370\pi\)
0.427570 + 0.903982i \(0.359370\pi\)
\(908\) 8.00000i 0.265489i
\(909\) −12.2462 −0.406181
\(910\) 1.75379 11.1231i 0.0581375 0.368727i
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 6.00000i 0.198680i
\(913\) 32.0000 1.05905
\(914\) 31.3693 1.03760
\(915\) 10.0000i 0.330590i
\(916\) 5.12311i 0.169272i
\(917\) 12.4924i 0.412536i
\(918\) 2.00000i 0.0660098i
\(919\) −21.7538 −0.717591 −0.358796 0.933416i \(-0.616813\pi\)
−0.358796 + 0.933416i \(0.616813\pi\)
\(920\) 5.12311 0.168904
\(921\) 31.1231i 1.02554i
\(922\) 18.4924 0.609016
\(923\) 5.75379 36.4924i 0.189388 1.20116i
\(924\) 16.0000 0.526361
\(925\) 5.12311i 0.168447i
\(926\) −21.8617 −0.718421
\(927\) −13.1231 −0.431019
\(928\) 2.00000i 0.0656532i
\(929\) 28.1080i 0.922192i −0.887350 0.461096i \(-0.847456\pi\)
0.887350 0.461096i \(-0.152544\pi\)
\(930\) 3.12311i 0.102411i
\(931\) 16.5227i 0.541511i
\(932\) 7.75379 0.253984
\(933\) 8.49242 0.278029
\(934\) 22.2462i 0.727918i
\(935\) −10.2462 −0.335087
\(936\) 0.561553 3.56155i 0.0183549 0.116413i
\(937\) −20.2462 −0.661415 −0.330707 0.943733i \(-0.607287\pi\)
−0.330707 + 0.943733i \(0.607287\pi\)
\(938\) 15.2311i 0.497312i
\(939\) 16.2462 0.530175
\(940\) −6.24621 −0.203729
\(941\) 14.0000i 0.456387i 0.973616 + 0.228193i \(0.0732819\pi\)
−0.973616 + 0.228193i \(0.926718\pi\)
\(942\) 21.3693i 0.696250i
\(943\) 4.49242i 0.146293i
\(944\) 1.12311i 0.0365540i
\(945\) −3.12311 −0.101595
\(946\) 32.0000 1.04041
\(947\) 8.49242i 0.275967i −0.990435 0.137983i \(-0.955938\pi\)
0.990435 0.137983i \(-0.0440620\pi\)
\(948\) −8.00000 −0.259828
\(949\) −7.36932 + 46.7386i −0.239218 + 1.51720i
\(950\) 6.00000 0.194666
\(951\) 6.00000i 0.194563i
\(952\) −6.24621 −0.202441
\(953\) 22.9848 0.744552 0.372276 0.928122i \(-0.378577\pi\)
0.372276 + 0.928122i \(0.378577\pi\)
\(954\) 13.3693i 0.432848i
\(955\) 16.4924i 0.533682i
\(956\) 4.49242i 0.145295i
\(957\) 10.2462i 0.331213i
\(958\) −12.4924 −0.403612
\(959\) 32.7689 1.05816
\(960\) 1.00000i 0.0322749i
\(961\) 21.2462 0.685362
\(962\) −18.2462 2.87689i −0.588281 0.0927548i
\(963\) −8.00000 −0.257796
\(964\) 14.2462i 0.458840i
\(965\) 25.1231 0.808741
\(966\) −16.0000 −0.514792
\(967\) 47.6155i 1.53121i 0.643310 + 0.765606i \(0.277561\pi\)
−0.643310 + 0.765606i \(0.722439\pi\)
\(968\) 15.2462i 0.490032i
\(969\) 12.0000i 0.385496i
\(970\) 13.1231i 0.421358i
\(971\) 37.7538 1.21158 0.605788 0.795626i \(-0.292858\pi\)
0.605788 + 0.795626i \(0.292858\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 51.5076i 1.65126i
\(974\) 17.3693 0.556549
\(975\) −3.56155 0.561553i −0.114061 0.0179841i
\(976\) 10.0000 0.320092
\(977\) 4.73863i 0.151602i 0.997123 + 0.0758012i \(0.0241514\pi\)
−0.997123 + 0.0758012i \(0.975849\pi\)
\(978\) −7.12311 −0.227772
\(979\) −16.0000 −0.511362
\(980\) 2.75379i 0.0879666i
\(981\) 2.87689i 0.0918522i
\(982\) 38.7386i 1.23620i
\(983\) 60.4924i 1.92941i 0.263335 + 0.964704i \(0.415177\pi\)
−0.263335 + 0.964704i \(0.584823\pi\)
\(984\) −0.876894 −0.0279544
\(985\) −16.2462 −0.517647
\(986\) 4.00000i 0.127386i
\(987\) 19.5076 0.620933
\(988\) 3.36932 21.3693i 0.107192 0.679849i
\(989\) −32.0000 −1.01754
\(990\) 5.12311i 0.162823i
\(991\) −30.7386 −0.976445 −0.488222 0.872719i \(-0.662355\pi\)
−0.488222 + 0.872719i \(0.662355\pi\)
\(992\) 3.12311 0.0991587
\(993\) 7.75379i 0.246059i
\(994\) 32.0000i 1.01498i
\(995\) 18.2462i 0.578444i
\(996\) 6.24621i 0.197919i
\(997\) 38.3542 1.21469 0.607344 0.794439i \(-0.292235\pi\)
0.607344 + 0.794439i \(0.292235\pi\)
\(998\) 34.4924 1.09184
\(999\) 5.12311i 0.162088i
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 390.2.b.d.181.3 yes 4
3.2 odd 2 1170.2.b.f.181.1 4
4.3 odd 2 3120.2.g.o.961.2 4
5.2 odd 4 1950.2.f.l.649.2 4
5.3 odd 4 1950.2.f.o.649.3 4
5.4 even 2 1950.2.b.h.1351.2 4
13.5 odd 4 5070.2.a.bh.1.2 2
13.8 odd 4 5070.2.a.bd.1.1 2
13.12 even 2 inner 390.2.b.d.181.2 4
39.38 odd 2 1170.2.b.f.181.4 4
52.51 odd 2 3120.2.g.o.961.3 4
65.12 odd 4 1950.2.f.o.649.1 4
65.38 odd 4 1950.2.f.l.649.4 4
65.64 even 2 1950.2.b.h.1351.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.d.181.2 4 13.12 even 2 inner
390.2.b.d.181.3 yes 4 1.1 even 1 trivial
1170.2.b.f.181.1 4 3.2 odd 2
1170.2.b.f.181.4 4 39.38 odd 2
1950.2.b.h.1351.2 4 5.4 even 2
1950.2.b.h.1351.3 4 65.64 even 2
1950.2.f.l.649.2 4 5.2 odd 4
1950.2.f.l.649.4 4 65.38 odd 4
1950.2.f.o.649.1 4 65.12 odd 4
1950.2.f.o.649.3 4 5.3 odd 4
3120.2.g.o.961.2 4 4.3 odd 2
3120.2.g.o.961.3 4 52.51 odd 2
5070.2.a.bd.1.1 2 13.8 odd 4
5070.2.a.bh.1.2 2 13.5 odd 4