Properties

Label 2205.2.d.l.1324.3
Level $2205$
Weight $2$
Character 2205.1324
Analytic conductor $17.607$
Analytic rank $0$
Dimension $6$
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] = [2205,2,Mod(1324,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.3
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 2205.1324
Dual form 2205.2.d.l.1324.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.193937i q^{2} +1.96239 q^{4} +(-1.48119 + 1.67513i) q^{5} -0.768452i q^{8} +O(q^{10})\) \(q-0.193937i q^{2} +1.96239 q^{4} +(-1.48119 + 1.67513i) q^{5} -0.768452i q^{8} +(0.324869 + 0.287258i) q^{10} -2.00000 q^{11} +1.35026i q^{13} +3.77575 q^{16} -3.35026i q^{17} +5.35026 q^{19} +(-2.90668 + 3.28726i) q^{20} +0.387873i q^{22} +4.96239i q^{23} +(-0.612127 - 4.96239i) q^{25} +0.261865 q^{26} +7.92478 q^{29} -4.57452 q^{31} -2.26916i q^{32} -0.649738 q^{34} -0.775746i q^{37} -1.03761i q^{38} +(1.28726 + 1.13823i) q^{40} +3.73813 q^{41} +12.6253i q^{43} -3.92478 q^{44} +0.962389 q^{46} +9.92478i q^{47} +(-0.962389 + 0.118714i) q^{50} +2.64974i q^{52} +8.57452i q^{53} +(2.96239 - 3.35026i) q^{55} -1.53690i q^{58} +8.62530 q^{59} +8.70052 q^{61} +0.887166i q^{62} +7.11142 q^{64} +(-2.26187 - 2.00000i) q^{65} +9.92478i q^{67} -6.57452i q^{68} -2.00000 q^{71} -9.35026i q^{73} -0.150446 q^{74} +10.4993 q^{76} -10.7005 q^{79} +(-5.59261 + 6.32487i) q^{80} -0.724961i q^{82} -3.22425i q^{83} +(5.61213 + 4.96239i) q^{85} +2.44851 q^{86} +1.53690i q^{88} -1.03761 q^{89} +9.73813i q^{92} +1.92478 q^{94} +(-7.92478 + 8.96239i) q^{95} +18.4993i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 2 q^{5} + 12 q^{10} - 12 q^{11} + 26 q^{16} + 12 q^{19} - 30 q^{20} - 2 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{31} - 24 q^{34} - 4 q^{40} + 4 q^{41} + 20 q^{44} - 16 q^{46} + 16 q^{50} - 4 q^{55} - 32 q^{59} + 12 q^{61} - 26 q^{64} - 32 q^{65} - 12 q^{71} - 88 q^{74} - 4 q^{76} - 24 q^{79} + 46 q^{80} + 32 q^{85} + 8 q^{86} - 28 q^{89} - 32 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.193937i 0.137134i −0.997647 0.0685669i \(-0.978157\pi\)
0.997647 0.0685669i \(-0.0218427\pi\)
\(3\) 0 0
\(4\) 1.96239 0.981194
\(5\) −1.48119 + 1.67513i −0.662410 + 0.749141i
\(6\) 0 0
\(7\) 0 0
\(8\) 0.768452i 0.271689i
\(9\) 0 0
\(10\) 0.324869 + 0.287258i 0.102733 + 0.0908389i
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.35026i 0.374495i 0.982313 + 0.187248i \(0.0599567\pi\)
−0.982313 + 0.187248i \(0.940043\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.77575 0.943937
\(17\) 3.35026i 0.812558i −0.913749 0.406279i \(-0.866826\pi\)
0.913749 0.406279i \(-0.133174\pi\)
\(18\) 0 0
\(19\) 5.35026 1.22743 0.613717 0.789526i \(-0.289674\pi\)
0.613717 + 0.789526i \(0.289674\pi\)
\(20\) −2.90668 + 3.28726i −0.649953 + 0.735053i
\(21\) 0 0
\(22\) 0.387873i 0.0826948i
\(23\) 4.96239i 1.03473i 0.855765 + 0.517365i \(0.173087\pi\)
−0.855765 + 0.517365i \(0.826913\pi\)
\(24\) 0 0
\(25\) −0.612127 4.96239i −0.122425 0.992478i
\(26\) 0.261865 0.0513560
\(27\) 0 0
\(28\) 0 0
\(29\) 7.92478 1.47159 0.735797 0.677202i \(-0.236808\pi\)
0.735797 + 0.677202i \(0.236808\pi\)
\(30\) 0 0
\(31\) −4.57452 −0.821607 −0.410804 0.911724i \(-0.634752\pi\)
−0.410804 + 0.911724i \(0.634752\pi\)
\(32\) 2.26916i 0.401134i
\(33\) 0 0
\(34\) −0.649738 −0.111429
\(35\) 0 0
\(36\) 0 0
\(37\) 0.775746i 0.127532i −0.997965 0.0637660i \(-0.979689\pi\)
0.997965 0.0637660i \(-0.0203111\pi\)
\(38\) 1.03761i 0.168323i
\(39\) 0 0
\(40\) 1.28726 + 1.13823i 0.203533 + 0.179969i
\(41\) 3.73813 0.583799 0.291899 0.956449i \(-0.405713\pi\)
0.291899 + 0.956449i \(0.405713\pi\)
\(42\) 0 0
\(43\) 12.6253i 1.92534i 0.270677 + 0.962670i \(0.412752\pi\)
−0.270677 + 0.962670i \(0.587248\pi\)
\(44\) −3.92478 −0.591682
\(45\) 0 0
\(46\) 0.962389 0.141896
\(47\) 9.92478i 1.44768i 0.689969 + 0.723839i \(0.257624\pi\)
−0.689969 + 0.723839i \(0.742376\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.962389 + 0.118714i −0.136102 + 0.0167887i
\(51\) 0 0
\(52\) 2.64974i 0.367453i
\(53\) 8.57452i 1.17780i 0.808206 + 0.588900i \(0.200439\pi\)
−0.808206 + 0.588900i \(0.799561\pi\)
\(54\) 0 0
\(55\) 2.96239 3.35026i 0.399448 0.451749i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.53690i 0.201805i
\(59\) 8.62530 1.12292 0.561459 0.827504i \(-0.310240\pi\)
0.561459 + 0.827504i \(0.310240\pi\)
\(60\) 0 0
\(61\) 8.70052 1.11399 0.556994 0.830517i \(-0.311955\pi\)
0.556994 + 0.830517i \(0.311955\pi\)
\(62\) 0.887166i 0.112670i
\(63\) 0 0
\(64\) 7.11142 0.888927
\(65\) −2.26187 2.00000i −0.280550 0.248069i
\(66\) 0 0
\(67\) 9.92478i 1.21250i 0.795272 + 0.606252i \(0.207328\pi\)
−0.795272 + 0.606252i \(0.792672\pi\)
\(68\) 6.57452i 0.797277i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 9.35026i 1.09437i −0.837013 0.547183i \(-0.815700\pi\)
0.837013 0.547183i \(-0.184300\pi\)
\(74\) −0.150446 −0.0174889
\(75\) 0 0
\(76\) 10.4993 1.20435
\(77\) 0 0
\(78\) 0 0
\(79\) −10.7005 −1.20390 −0.601951 0.798533i \(-0.705610\pi\)
−0.601951 + 0.798533i \(0.705610\pi\)
\(80\) −5.59261 + 6.32487i −0.625273 + 0.707142i
\(81\) 0 0
\(82\) 0.724961i 0.0800586i
\(83\) 3.22425i 0.353908i −0.984219 0.176954i \(-0.943376\pi\)
0.984219 0.176954i \(-0.0566244\pi\)
\(84\) 0 0
\(85\) 5.61213 + 4.96239i 0.608721 + 0.538247i
\(86\) 2.44851 0.264029
\(87\) 0 0
\(88\) 1.53690i 0.163835i
\(89\) −1.03761 −0.109987 −0.0549933 0.998487i \(-0.517514\pi\)
−0.0549933 + 0.998487i \(0.517514\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.73813i 1.01527i
\(93\) 0 0
\(94\) 1.92478 0.198526
\(95\) −7.92478 + 8.96239i −0.813065 + 0.919522i
\(96\) 0 0
\(97\) 18.4993i 1.87832i 0.343482 + 0.939159i \(0.388394\pi\)
−0.343482 + 0.939159i \(0.611606\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.20123 9.73813i −0.120123 0.973813i
\(101\) −17.6629 −1.75753 −0.878763 0.477259i \(-0.841630\pi\)
−0.878763 + 0.477259i \(0.841630\pi\)
\(102\) 0 0
\(103\) 6.70052i 0.660222i 0.943942 + 0.330111i \(0.107086\pi\)
−0.943942 + 0.330111i \(0.892914\pi\)
\(104\) 1.03761 0.101746
\(105\) 0 0
\(106\) 1.66291 0.161516
\(107\) 13.7381i 1.32812i −0.747681 0.664058i \(-0.768833\pi\)
0.747681 0.664058i \(-0.231167\pi\)
\(108\) 0 0
\(109\) 2.77575 0.265868 0.132934 0.991125i \(-0.457560\pi\)
0.132934 + 0.991125i \(0.457560\pi\)
\(110\) −0.649738 0.574515i −0.0619501 0.0547779i
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0508i 1.13364i 0.823841 + 0.566821i \(0.191827\pi\)
−0.823841 + 0.566821i \(0.808173\pi\)
\(114\) 0 0
\(115\) −8.31265 7.35026i −0.775159 0.685415i
\(116\) 15.5515 1.44392
\(117\) 0 0
\(118\) 1.67276i 0.153990i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 1.68735i 0.152765i
\(123\) 0 0
\(124\) −8.97698 −0.806156
\(125\) 9.21933 + 6.32487i 0.824602 + 0.565713i
\(126\) 0 0
\(127\) 2.70052i 0.239633i 0.992796 + 0.119816i \(0.0382306\pi\)
−0.992796 + 0.119816i \(0.961769\pi\)
\(128\) 5.91748i 0.523037i
\(129\) 0 0
\(130\) −0.387873 + 0.438658i −0.0340187 + 0.0384729i
\(131\) 20.6253 1.80204 0.901020 0.433777i \(-0.142819\pi\)
0.901020 + 0.433777i \(0.142819\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.92478 0.166275
\(135\) 0 0
\(136\) −2.57452 −0.220763
\(137\) 22.4993i 1.92224i −0.276124 0.961122i \(-0.589050\pi\)
0.276124 0.961122i \(-0.410950\pi\)
\(138\) 0 0
\(139\) −3.27504 −0.277785 −0.138893 0.990307i \(-0.544354\pi\)
−0.138893 + 0.990307i \(0.544354\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.387873i 0.0325496i
\(143\) 2.70052i 0.225829i
\(144\) 0 0
\(145\) −11.7381 + 13.2750i −0.974799 + 1.10243i
\(146\) −1.81336 −0.150075
\(147\) 0 0
\(148\) 1.52232i 0.125134i
\(149\) −4.44851 −0.364436 −0.182218 0.983258i \(-0.558328\pi\)
−0.182218 + 0.983258i \(0.558328\pi\)
\(150\) 0 0
\(151\) 1.29948 0.105750 0.0528749 0.998601i \(-0.483162\pi\)
0.0528749 + 0.998601i \(0.483162\pi\)
\(152\) 4.11142i 0.333480i
\(153\) 0 0
\(154\) 0 0
\(155\) 6.77575 7.66291i 0.544241 0.615500i
\(156\) 0 0
\(157\) 2.64974i 0.211472i 0.994394 + 0.105736i \(0.0337199\pi\)
−0.994394 + 0.105736i \(0.966280\pi\)
\(158\) 2.07522i 0.165096i
\(159\) 0 0
\(160\) 3.80114 + 3.36107i 0.300506 + 0.265716i
\(161\) 0 0
\(162\) 0 0
\(163\) 5.29948i 0.415087i −0.978226 0.207544i \(-0.933453\pi\)
0.978226 0.207544i \(-0.0665469\pi\)
\(164\) 7.33567 0.572820
\(165\) 0 0
\(166\) −0.625301 −0.0485327
\(167\) 14.5501i 1.12592i −0.826485 0.562959i \(-0.809663\pi\)
0.826485 0.562959i \(-0.190337\pi\)
\(168\) 0 0
\(169\) 11.1768 0.859753
\(170\) 0.962389 1.08840i 0.0738118 0.0834762i
\(171\) 0 0
\(172\) 24.7757i 1.88913i
\(173\) 4.49929i 0.342075i 0.985265 + 0.171037i \(0.0547119\pi\)
−0.985265 + 0.171037i \(0.945288\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −7.55149 −0.569215
\(177\) 0 0
\(178\) 0.201231i 0.0150829i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −10.6253 −0.789772 −0.394886 0.918730i \(-0.629216\pi\)
−0.394886 + 0.918730i \(0.629216\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.81336 0.281124
\(185\) 1.29948 + 1.14903i 0.0955394 + 0.0844784i
\(186\) 0 0
\(187\) 6.70052i 0.489991i
\(188\) 19.4763i 1.42045i
\(189\) 0 0
\(190\) 1.73813 + 1.53690i 0.126098 + 0.111499i
\(191\) 13.8496 1.00212 0.501059 0.865413i \(-0.332944\pi\)
0.501059 + 0.865413i \(0.332944\pi\)
\(192\) 0 0
\(193\) 15.3258i 1.10318i −0.834116 0.551588i \(-0.814022\pi\)
0.834116 0.551588i \(-0.185978\pi\)
\(194\) 3.58769 0.257581
\(195\) 0 0
\(196\) 0 0
\(197\) 0.574515i 0.0409325i 0.999791 + 0.0204663i \(0.00651507\pi\)
−0.999791 + 0.0204663i \(0.993485\pi\)
\(198\) 0 0
\(199\) −0.201231 −0.0142649 −0.00713244 0.999975i \(-0.502270\pi\)
−0.00713244 + 0.999975i \(0.502270\pi\)
\(200\) −3.81336 + 0.470390i −0.269645 + 0.0332616i
\(201\) 0 0
\(202\) 3.42548i 0.241016i
\(203\) 0 0
\(204\) 0 0
\(205\) −5.53690 + 6.26187i −0.386714 + 0.437348i
\(206\) 1.29948 0.0905388
\(207\) 0 0
\(208\) 5.09825i 0.353500i
\(209\) −10.7005 −0.740171
\(210\) 0 0
\(211\) 6.44851 0.443934 0.221967 0.975054i \(-0.428752\pi\)
0.221967 + 0.975054i \(0.428752\pi\)
\(212\) 16.8265i 1.15565i
\(213\) 0 0
\(214\) −2.66433 −0.182130
\(215\) −21.1490 18.7005i −1.44235 1.27537i
\(216\) 0 0
\(217\) 0 0
\(218\) 0.538319i 0.0364595i
\(219\) 0 0
\(220\) 5.81336 6.57452i 0.391936 0.443254i
\(221\) 4.52373 0.304299
\(222\) 0 0
\(223\) 1.55149i 0.103896i 0.998650 + 0.0519478i \(0.0165429\pi\)
−0.998650 + 0.0519478i \(0.983457\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.33709 0.155461
\(227\) 13.1490i 0.872732i −0.899769 0.436366i \(-0.856265\pi\)
0.899769 0.436366i \(-0.143735\pi\)
\(228\) 0 0
\(229\) 2.77575 0.183426 0.0917132 0.995785i \(-0.470766\pi\)
0.0917132 + 0.995785i \(0.470766\pi\)
\(230\) −1.42548 + 1.61213i −0.0939937 + 0.106300i
\(231\) 0 0
\(232\) 6.08981i 0.399816i
\(233\) 0.0507852i 0.00332705i −0.999999 0.00166353i \(-0.999470\pi\)
0.999999 0.00166353i \(-0.000529517\pi\)
\(234\) 0 0
\(235\) −16.6253 14.7005i −1.08452 0.958956i
\(236\) 16.9262 1.10180
\(237\) 0 0
\(238\) 0 0
\(239\) 5.84955 0.378376 0.189188 0.981941i \(-0.439414\pi\)
0.189188 + 0.981941i \(0.439414\pi\)
\(240\) 0 0
\(241\) 0.0752228 0.00484553 0.00242276 0.999997i \(-0.499229\pi\)
0.00242276 + 0.999997i \(0.499229\pi\)
\(242\) 1.35756i 0.0872670i
\(243\) 0 0
\(244\) 17.0738 1.09304
\(245\) 0 0
\(246\) 0 0
\(247\) 7.22425i 0.459668i
\(248\) 3.51530i 0.223222i
\(249\) 0 0
\(250\) 1.22662 1.78797i 0.0775785 0.113081i
\(251\) 19.2243 1.21342 0.606712 0.794922i \(-0.292488\pi\)
0.606712 + 0.794922i \(0.292488\pi\)
\(252\) 0 0
\(253\) 9.92478i 0.623965i
\(254\) 0.523730 0.0328618
\(255\) 0 0
\(256\) 13.0752 0.817201
\(257\) 7.35026i 0.458497i 0.973368 + 0.229248i \(0.0736268\pi\)
−0.973368 + 0.229248i \(0.926373\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.43866 3.92478i −0.275274 0.243404i
\(261\) 0 0
\(262\) 4.00000i 0.247121i
\(263\) 12.9624i 0.799295i −0.916669 0.399648i \(-0.869133\pi\)
0.916669 0.399648i \(-0.130867\pi\)
\(264\) 0 0
\(265\) −14.3634 12.7005i −0.882339 0.780187i
\(266\) 0 0
\(267\) 0 0
\(268\) 19.4763i 1.18970i
\(269\) −4.11142 −0.250678 −0.125339 0.992114i \(-0.540002\pi\)
−0.125339 + 0.992114i \(0.540002\pi\)
\(270\) 0 0
\(271\) 16.4241 0.997691 0.498846 0.866691i \(-0.333757\pi\)
0.498846 + 0.866691i \(0.333757\pi\)
\(272\) 12.6497i 0.767003i
\(273\) 0 0
\(274\) −4.36344 −0.263605
\(275\) 1.22425 + 9.92478i 0.0738253 + 0.598487i
\(276\) 0 0
\(277\) 11.0738i 0.665361i −0.943040 0.332680i \(-0.892047\pi\)
0.943040 0.332680i \(-0.107953\pi\)
\(278\) 0.635150i 0.0380938i
\(279\) 0 0
\(280\) 0 0
\(281\) −14.3733 −0.857438 −0.428719 0.903438i \(-0.641035\pi\)
−0.428719 + 0.903438i \(0.641035\pi\)
\(282\) 0 0
\(283\) 1.14903i 0.0683028i 0.999417 + 0.0341514i \(0.0108728\pi\)
−0.999417 + 0.0341514i \(0.989127\pi\)
\(284\) −3.92478 −0.232893
\(285\) 0 0
\(286\) −0.523730 −0.0309688
\(287\) 0 0
\(288\) 0 0
\(289\) 5.77575 0.339750
\(290\) 2.57452 + 2.27645i 0.151181 + 0.133678i
\(291\) 0 0
\(292\) 18.3488i 1.07379i
\(293\) 0.649738i 0.0379581i −0.999820 0.0189791i \(-0.993958\pi\)
0.999820 0.0189791i \(-0.00604158\pi\)
\(294\) 0 0
\(295\) −12.7757 + 14.4485i −0.743833 + 0.841225i
\(296\) −0.596124 −0.0346490
\(297\) 0 0
\(298\) 0.862728i 0.0499765i
\(299\) −6.70052 −0.387501
\(300\) 0 0
\(301\) 0 0
\(302\) 0.252016i 0.0145019i
\(303\) 0 0
\(304\) 20.2012 1.15862
\(305\) −12.8872 + 14.5745i −0.737917 + 0.834534i
\(306\) 0 0
\(307\) 24.1016i 1.37555i −0.725924 0.687775i \(-0.758588\pi\)
0.725924 0.687775i \(-0.241412\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.48612 1.31406i −0.0844059 0.0746339i
\(311\) 8.25202 0.467929 0.233964 0.972245i \(-0.424830\pi\)
0.233964 + 0.972245i \(0.424830\pi\)
\(312\) 0 0
\(313\) 14.9018i 0.842297i −0.906992 0.421148i \(-0.861627\pi\)
0.906992 0.421148i \(-0.138373\pi\)
\(314\) 0.513881 0.0290000
\(315\) 0 0
\(316\) −20.9986 −1.18126
\(317\) 10.1260i 0.568733i −0.958716 0.284367i \(-0.908217\pi\)
0.958716 0.284367i \(-0.0917833\pi\)
\(318\) 0 0
\(319\) −15.8496 −0.887405
\(320\) −10.5334 + 11.9126i −0.588835 + 0.665932i
\(321\) 0 0
\(322\) 0 0
\(323\) 17.9248i 0.997361i
\(324\) 0 0
\(325\) 6.70052 0.826531i 0.371678 0.0458477i
\(326\) −1.02776 −0.0569225
\(327\) 0 0
\(328\) 2.87258i 0.158612i
\(329\) 0 0
\(330\) 0 0
\(331\) 27.8496 1.53075 0.765375 0.643585i \(-0.222554\pi\)
0.765375 + 0.643585i \(0.222554\pi\)
\(332\) 6.32724i 0.347252i
\(333\) 0 0
\(334\) −2.82179 −0.154402
\(335\) −16.6253 14.7005i −0.908337 0.803175i
\(336\) 0 0
\(337\) 3.84955i 0.209699i −0.994488 0.104849i \(-0.966564\pi\)
0.994488 0.104849i \(-0.0334360\pi\)
\(338\) 2.16759i 0.117901i
\(339\) 0 0
\(340\) 11.0132 + 9.73813i 0.597273 + 0.528125i
\(341\) 9.14903 0.495448
\(342\) 0 0
\(343\) 0 0
\(344\) 9.70194 0.523093
\(345\) 0 0
\(346\) 0.872577 0.0469101
\(347\) 9.58769i 0.514694i −0.966319 0.257347i \(-0.917152\pi\)
0.966319 0.257347i \(-0.0828484\pi\)
\(348\) 0 0
\(349\) −15.1490 −0.810909 −0.405455 0.914115i \(-0.632887\pi\)
−0.405455 + 0.914115i \(0.632887\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.53832i 0.241893i
\(353\) 20.3488i 1.08306i 0.840681 + 0.541530i \(0.182155\pi\)
−0.840681 + 0.541530i \(0.817845\pi\)
\(354\) 0 0
\(355\) 2.96239 3.35026i 0.157227 0.177813i
\(356\) −2.03620 −0.107918
\(357\) 0 0
\(358\) 1.93937i 0.102499i
\(359\) −31.4010 −1.65728 −0.828642 0.559779i \(-0.810886\pi\)
−0.828642 + 0.559779i \(0.810886\pi\)
\(360\) 0 0
\(361\) 9.62530 0.506595
\(362\) 2.06063i 0.108305i
\(363\) 0 0
\(364\) 0 0
\(365\) 15.6629 + 13.8496i 0.819834 + 0.724919i
\(366\) 0 0
\(367\) 29.4010i 1.53472i 0.641215 + 0.767361i \(0.278431\pi\)
−0.641215 + 0.767361i \(0.721569\pi\)
\(368\) 18.7367i 0.976719i
\(369\) 0 0
\(370\) 0.222839 0.252016i 0.0115849 0.0131017i
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000i 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) 1.29948 0.0671943
\(375\) 0 0
\(376\) 7.62672 0.393318
\(377\) 10.7005i 0.551105i
\(378\) 0 0
\(379\) −10.7005 −0.549649 −0.274824 0.961494i \(-0.588620\pi\)
−0.274824 + 0.961494i \(0.588620\pi\)
\(380\) −15.5515 + 17.5877i −0.797775 + 0.902229i
\(381\) 0 0
\(382\) 2.68594i 0.137424i
\(383\) 16.7757i 0.857201i 0.903494 + 0.428600i \(0.140993\pi\)
−0.903494 + 0.428600i \(0.859007\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.97224 −0.151283
\(387\) 0 0
\(388\) 36.3028i 1.84300i
\(389\) −29.3258 −1.48688 −0.743439 0.668804i \(-0.766807\pi\)
−0.743439 + 0.668804i \(0.766807\pi\)
\(390\) 0 0
\(391\) 16.6253 0.840778
\(392\) 0 0
\(393\) 0 0
\(394\) 0.111420 0.00561324
\(395\) 15.8496 17.9248i 0.797478 0.901893i
\(396\) 0 0
\(397\) 18.3488i 0.920902i −0.887685 0.460451i \(-0.847688\pi\)
0.887685 0.460451i \(-0.152312\pi\)
\(398\) 0.0390260i 0.00195620i
\(399\) 0 0
\(400\) −2.31124 18.7367i −0.115562 0.936836i
\(401\) 37.3258 1.86396 0.931981 0.362506i \(-0.118079\pi\)
0.931981 + 0.362506i \(0.118079\pi\)
\(402\) 0 0
\(403\) 6.17679i 0.307688i
\(404\) −34.6615 −1.72447
\(405\) 0 0
\(406\) 0 0
\(407\) 1.55149i 0.0769046i
\(408\) 0 0
\(409\) −22.3733 −1.10629 −0.553144 0.833086i \(-0.686572\pi\)
−0.553144 + 0.833086i \(0.686572\pi\)
\(410\) 1.21440 + 1.07381i 0.0599752 + 0.0530316i
\(411\) 0 0
\(412\) 13.1490i 0.647806i
\(413\) 0 0
\(414\) 0 0
\(415\) 5.40105 + 4.77575i 0.265127 + 0.234432i
\(416\) 3.06396 0.150223
\(417\) 0 0
\(418\) 2.07522i 0.101502i
\(419\) −23.4763 −1.14689 −0.573445 0.819244i \(-0.694394\pi\)
−0.573445 + 0.819244i \(0.694394\pi\)
\(420\) 0 0
\(421\) −25.2243 −1.22935 −0.614677 0.788779i \(-0.710714\pi\)
−0.614677 + 0.788779i \(0.710714\pi\)
\(422\) 1.25060i 0.0608783i
\(423\) 0 0
\(424\) 6.58910 0.319995
\(425\) −16.6253 + 2.05079i −0.806446 + 0.0994777i
\(426\) 0 0
\(427\) 0 0
\(428\) 26.9596i 1.30314i
\(429\) 0 0
\(430\) −3.62672 + 4.10157i −0.174896 + 0.197795i
\(431\) 19.4010 0.934516 0.467258 0.884121i \(-0.345242\pi\)
0.467258 + 0.884121i \(0.345242\pi\)
\(432\) 0 0
\(433\) 6.49929i 0.312336i 0.987731 + 0.156168i \(0.0499141\pi\)
−0.987731 + 0.156168i \(0.950086\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.44709 0.260868
\(437\) 26.5501i 1.27006i
\(438\) 0 0
\(439\) −14.6497 −0.699194 −0.349597 0.936900i \(-0.613681\pi\)
−0.349597 + 0.936900i \(0.613681\pi\)
\(440\) −2.57452 2.27645i −0.122735 0.108526i
\(441\) 0 0
\(442\) 0.877317i 0.0417297i
\(443\) 19.1392i 0.909330i 0.890663 + 0.454665i \(0.150241\pi\)
−0.890663 + 0.454665i \(0.849759\pi\)
\(444\) 0 0
\(445\) 1.53690 1.73813i 0.0728562 0.0823955i
\(446\) 0.300891 0.0142476
\(447\) 0 0
\(448\) 0 0
\(449\) −32.8021 −1.54803 −0.774013 0.633169i \(-0.781754\pi\)
−0.774013 + 0.633169i \(0.781754\pi\)
\(450\) 0 0
\(451\) −7.47627 −0.352044
\(452\) 23.6483i 1.11232i
\(453\) 0 0
\(454\) −2.55008 −0.119681
\(455\) 0 0
\(456\) 0 0
\(457\) 18.7005i 0.874774i −0.899273 0.437387i \(-0.855904\pi\)
0.899273 0.437387i \(-0.144096\pi\)
\(458\) 0.538319i 0.0251540i
\(459\) 0 0
\(460\) −16.3127 14.4241i −0.760581 0.672526i
\(461\) −6.96239 −0.324271 −0.162135 0.986769i \(-0.551838\pi\)
−0.162135 + 0.986769i \(0.551838\pi\)
\(462\) 0 0
\(463\) 5.29948i 0.246288i −0.992389 0.123144i \(-0.960702\pi\)
0.992389 0.123144i \(-0.0392976\pi\)
\(464\) 29.9219 1.38909
\(465\) 0 0
\(466\) −0.00984911 −0.000456251
\(467\) 13.1490i 0.608465i −0.952598 0.304232i \(-0.901600\pi\)
0.952598 0.304232i \(-0.0983999\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2.85097 + 3.22425i −0.131505 + 0.148724i
\(471\) 0 0
\(472\) 6.62813i 0.305084i
\(473\) 25.2506i 1.16102i
\(474\) 0 0
\(475\) −3.27504 26.5501i −0.150269 1.21820i
\(476\) 0 0
\(477\) 0 0
\(478\) 1.13444i 0.0518882i
\(479\) −5.14903 −0.235265 −0.117633 0.993057i \(-0.537531\pi\)
−0.117633 + 0.993057i \(0.537531\pi\)
\(480\) 0 0
\(481\) 1.04746 0.0477601
\(482\) 0.0145884i 0.000664486i
\(483\) 0 0
\(484\) −13.7367 −0.624396
\(485\) −30.9887 27.4010i −1.40713 1.24422i
\(486\) 0 0
\(487\) 22.1768i 1.00493i −0.864599 0.502463i \(-0.832427\pi\)
0.864599 0.502463i \(-0.167573\pi\)
\(488\) 6.68594i 0.302658i
\(489\) 0 0
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 26.5501i 1.19576i
\(494\) 1.40105 0.0630361
\(495\) 0 0
\(496\) −17.2722 −0.775545
\(497\) 0 0
\(498\) 0 0
\(499\) 6.55008 0.293222 0.146611 0.989194i \(-0.453163\pi\)
0.146611 + 0.989194i \(0.453163\pi\)
\(500\) 18.0919 + 12.4119i 0.809095 + 0.555075i
\(501\) 0 0
\(502\) 3.72829i 0.166402i
\(503\) 8.77575i 0.391291i −0.980675 0.195646i \(-0.937320\pi\)
0.980675 0.195646i \(-0.0626802\pi\)
\(504\) 0 0
\(505\) 26.1622 29.5877i 1.16420 1.31663i
\(506\) −1.92478 −0.0855668
\(507\) 0 0
\(508\) 5.29948i 0.235126i
\(509\) −13.1392 −0.582384 −0.291192 0.956665i \(-0.594052\pi\)
−0.291192 + 0.956665i \(0.594052\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 14.3707i 0.635103i
\(513\) 0 0
\(514\) 1.42548 0.0628754
\(515\) −11.2243 9.92478i −0.494600 0.437338i
\(516\) 0 0
\(517\) 19.8496i 0.872982i
\(518\) 0 0
\(519\) 0 0
\(520\) −1.53690 + 1.73813i −0.0673977 + 0.0762223i
\(521\) −37.6629 −1.65004 −0.825021 0.565102i \(-0.808837\pi\)
−0.825021 + 0.565102i \(0.808837\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 40.4749 1.76815
\(525\) 0 0
\(526\) −2.51388 −0.109610
\(527\) 15.3258i 0.667603i
\(528\) 0 0
\(529\) −1.62530 −0.0706652
\(530\) −2.46310 + 2.78560i −0.106990 + 0.120999i
\(531\) 0 0
\(532\) 0 0
\(533\) 5.04746i 0.218630i
\(534\) 0 0
\(535\) 23.0132 + 20.3488i 0.994946 + 0.879757i
\(536\) 7.62672 0.329424
\(537\) 0 0
\(538\) 0.797355i 0.0343764i
\(539\) 0 0
\(540\) 0 0
\(541\) −22.4749 −0.966269 −0.483135 0.875546i \(-0.660502\pi\)
−0.483135 + 0.875546i \(0.660502\pi\)
\(542\) 3.18523i 0.136817i
\(543\) 0 0
\(544\) −7.60228 −0.325945
\(545\) −4.11142 + 4.64974i −0.176114 + 0.199173i
\(546\) 0 0
\(547\) 25.9248i 1.10846i −0.832362 0.554232i \(-0.813012\pi\)
0.832362 0.554232i \(-0.186988\pi\)
\(548\) 44.1524i 1.88610i
\(549\) 0 0
\(550\) 1.92478 0.237428i 0.0820728 0.0101239i
\(551\) 42.3996 1.80629
\(552\) 0 0
\(553\) 0 0
\(554\) −2.14762 −0.0912435
\(555\) 0 0
\(556\) −6.42690 −0.272561
\(557\) 28.5256i 1.20867i 0.796730 + 0.604335i \(0.206561\pi\)
−0.796730 + 0.604335i \(0.793439\pi\)
\(558\) 0 0
\(559\) −17.0475 −0.721031
\(560\) 0 0
\(561\) 0 0
\(562\) 2.78751i 0.117584i
\(563\) 11.6267i 0.490008i 0.969522 + 0.245004i \(0.0787892\pi\)
−0.969522 + 0.245004i \(0.921211\pi\)
\(564\) 0 0
\(565\) −20.1866 17.8496i −0.849258 0.750936i
\(566\) 0.222839 0.00936663
\(567\) 0 0
\(568\) 1.53690i 0.0644871i
\(569\) 9.32582 0.390959 0.195479 0.980708i \(-0.437374\pi\)
0.195479 + 0.980708i \(0.437374\pi\)
\(570\) 0 0
\(571\) −19.6991 −0.824382 −0.412191 0.911097i \(-0.635236\pi\)
−0.412191 + 0.911097i \(0.635236\pi\)
\(572\) 5.29948i 0.221582i
\(573\) 0 0
\(574\) 0 0
\(575\) 24.6253 3.03761i 1.02695 0.126677i
\(576\) 0 0
\(577\) 32.7974i 1.36537i −0.730712 0.682686i \(-0.760812\pi\)
0.730712 0.682686i \(-0.239188\pi\)
\(578\) 1.12013i 0.0465912i
\(579\) 0 0
\(580\) −23.0348 + 26.0508i −0.956467 + 1.08170i
\(581\) 0 0
\(582\) 0 0
\(583\) 17.1490i 0.710240i
\(584\) −7.18523 −0.297327
\(585\) 0 0
\(586\) −0.126008 −0.00520534
\(587\) 18.8218i 0.776859i 0.921479 + 0.388429i \(0.126982\pi\)
−0.921479 + 0.388429i \(0.873018\pi\)
\(588\) 0 0
\(589\) −24.4749 −1.00847
\(590\) 2.80209 + 2.47768i 0.115360 + 0.102005i
\(591\) 0 0
\(592\) 2.92902i 0.120382i
\(593\) 33.7499i 1.38594i −0.720965 0.692971i \(-0.756301\pi\)
0.720965 0.692971i \(-0.243699\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8.72970 −0.357582
\(597\) 0 0
\(598\) 1.29948i 0.0531395i
\(599\) −20.2981 −0.829356 −0.414678 0.909968i \(-0.636106\pi\)
−0.414678 + 0.909968i \(0.636106\pi\)
\(600\) 0 0
\(601\) 13.8496 0.564935 0.282468 0.959277i \(-0.408847\pi\)
0.282468 + 0.959277i \(0.408847\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.55008 0.103761
\(605\) 10.3684 11.7259i 0.421534 0.476726i
\(606\) 0 0
\(607\) 25.2506i 1.02489i 0.858720 + 0.512445i \(0.171260\pi\)
−0.858720 + 0.512445i \(0.828740\pi\)
\(608\) 12.1406i 0.492366i
\(609\) 0 0
\(610\) 2.82653 + 2.49929i 0.114443 + 0.101193i
\(611\) −13.4010 −0.542148
\(612\) 0 0
\(613\) 9.14903i 0.369526i 0.982783 + 0.184763i \(0.0591517\pi\)
−0.982783 + 0.184763i \(0.940848\pi\)
\(614\) −4.67418 −0.188634
\(615\) 0 0
\(616\) 0 0
\(617\) 15.9492i 0.642091i −0.947064 0.321046i \(-0.895966\pi\)
0.947064 0.321046i \(-0.104034\pi\)
\(618\) 0 0
\(619\) 11.1735 0.449100 0.224550 0.974463i \(-0.427909\pi\)
0.224550 + 0.974463i \(0.427909\pi\)
\(620\) 13.2966 15.0376i 0.534006 0.603925i
\(621\) 0 0
\(622\) 1.60037i 0.0641689i
\(623\) 0 0
\(624\) 0 0
\(625\) −24.2506 + 6.07522i −0.970024 + 0.243009i
\(626\) −2.89000 −0.115507
\(627\) 0 0
\(628\) 5.19982i 0.207495i
\(629\) −2.59895 −0.103627
\(630\) 0 0
\(631\) −14.5501 −0.579229 −0.289615 0.957143i \(-0.593527\pi\)
−0.289615 + 0.957143i \(0.593527\pi\)
\(632\) 8.22284i 0.327087i
\(633\) 0 0
\(634\) −1.96380 −0.0779926
\(635\) −4.52373 4.00000i −0.179519 0.158735i
\(636\) 0 0
\(637\) 0 0
\(638\) 3.07381i 0.121693i
\(639\) 0 0
\(640\) 9.91256 + 8.76494i 0.391828 + 0.346465i
\(641\) 38.7269 1.52962 0.764810 0.644256i \(-0.222833\pi\)
0.764810 + 0.644256i \(0.222833\pi\)
\(642\) 0 0
\(643\) 11.9511i 0.471306i −0.971837 0.235653i \(-0.924277\pi\)
0.971837 0.235653i \(-0.0757229\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.47627 −0.136772
\(647\) 14.5501i 0.572023i −0.958226 0.286011i \(-0.907671\pi\)
0.958226 0.286011i \(-0.0923295\pi\)
\(648\) 0 0
\(649\) −17.2506 −0.677145
\(650\) −0.160295 1.29948i −0.00628727 0.0509697i
\(651\) 0 0
\(652\) 10.3996i 0.407281i
\(653\) 49.9756i 1.95569i 0.209319 + 0.977847i \(0.432875\pi\)
−0.209319 + 0.977847i \(0.567125\pi\)
\(654\) 0 0
\(655\) −30.5501 + 34.5501i −1.19369 + 1.34998i
\(656\) 14.1142 0.551069
\(657\) 0 0
\(658\) 0 0
\(659\) −16.9525 −0.660377 −0.330189 0.943915i \(-0.607112\pi\)
−0.330189 + 0.943915i \(0.607112\pi\)
\(660\) 0 0
\(661\) 15.6531 0.608834 0.304417 0.952539i \(-0.401538\pi\)
0.304417 + 0.952539i \(0.401538\pi\)
\(662\) 5.40105i 0.209918i
\(663\) 0 0
\(664\) −2.47768 −0.0961528
\(665\) 0 0
\(666\) 0 0
\(667\) 39.3258i 1.52270i
\(668\) 28.5529i 1.10475i
\(669\) 0 0
\(670\) −2.85097 + 3.22425i −0.110143 + 0.124564i
\(671\) −17.4010 −0.671760
\(672\) 0 0
\(673\) 26.0263i 1.00324i −0.865088 0.501621i \(-0.832737\pi\)
0.865088 0.501621i \(-0.167263\pi\)
\(674\) −0.746569 −0.0287568
\(675\) 0 0
\(676\) 21.9332 0.843585
\(677\) 35.4518i 1.36252i −0.732039 0.681262i \(-0.761431\pi\)
0.732039 0.681262i \(-0.238569\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.81336 4.31265i 0.146236 0.165383i
\(681\) 0 0
\(682\) 1.77433i 0.0679427i
\(683\) 23.6629i 0.905436i −0.891654 0.452718i \(-0.850454\pi\)
0.891654 0.452718i \(-0.149546\pi\)
\(684\) 0 0
\(685\) 37.6893 + 33.3258i 1.44003 + 1.27331i
\(686\) 0 0
\(687\) 0 0
\(688\) 47.6699i 1.81740i
\(689\) −11.5778 −0.441081
\(690\) 0 0
\(691\) 0.574515 0.0218556 0.0109278 0.999940i \(-0.496522\pi\)
0.0109278 + 0.999940i \(0.496522\pi\)
\(692\) 8.82936i 0.335642i
\(693\) 0 0
\(694\) −1.85940 −0.0705820
\(695\) 4.85097 5.48612i 0.184008 0.208100i
\(696\) 0 0
\(697\) 12.5237i 0.474370i
\(698\) 2.93795i 0.111203i
\(699\) 0 0
\(700\) 0 0
\(701\) −42.7269 −1.61377 −0.806886 0.590707i \(-0.798849\pi\)
−0.806886 + 0.590707i \(0.798849\pi\)
\(702\) 0 0
\(703\) 4.15045i 0.156537i
\(704\) −14.2228 −0.536043
\(705\) 0 0
\(706\) 3.94639 0.148524
\(707\) 0 0
\(708\) 0 0
\(709\) −27.2506 −1.02342 −0.511709 0.859159i \(-0.670987\pi\)
−0.511709 + 0.859159i \(0.670987\pi\)
\(710\) −0.649738 0.574515i −0.0243842 0.0215612i
\(711\) 0 0
\(712\) 0.797355i 0.0298821i
\(713\) 22.7005i 0.850141i
\(714\) 0 0
\(715\) 4.52373 + 4.00000i 0.169178 + 0.149592i
\(716\) 19.6239 0.733379
\(717\) 0 0
\(718\) 6.08981i 0.227270i
\(719\) 10.7005 0.399062 0.199531 0.979891i \(-0.436058\pi\)
0.199531 + 0.979891i \(0.436058\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.86670i 0.0694713i
\(723\) 0 0
\(724\) −20.8510 −0.774920
\(725\) −4.85097 39.3258i −0.180160 1.46052i
\(726\) 0 0
\(727\) 39.9511i 1.48171i −0.671668 0.740853i \(-0.734422\pi\)
0.671668 0.740853i \(-0.265578\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.68594 3.03761i 0.0994109 0.112427i
\(731\) 42.2981 1.56445
\(732\) 0 0
\(733\) 30.3488i 1.12096i −0.828168 0.560480i \(-0.810617\pi\)
0.828168 0.560480i \(-0.189383\pi\)
\(734\) 5.70194 0.210462
\(735\) 0 0
\(736\) 11.2605 0.415066
\(737\) 19.8496i 0.731168i
\(738\) 0 0
\(739\) −37.2506 −1.37029 −0.685143 0.728409i \(-0.740260\pi\)
−0.685143 + 0.728409i \(0.740260\pi\)
\(740\) 2.55008 + 2.25485i 0.0937427 + 0.0828898i
\(741\) 0 0
\(742\) 0 0
\(743\) 26.3634i 0.967181i 0.875294 + 0.483590i \(0.160668\pi\)
−0.875294 + 0.483590i \(0.839332\pi\)
\(744\) 0 0
\(745\) 6.58910 7.45183i 0.241406 0.273014i
\(746\) −3.10299 −0.113608
\(747\) 0 0
\(748\) 13.1490i 0.480776i
\(749\) 0 0
\(750\) 0 0
\(751\) 50.6516 1.84830 0.924152 0.382024i \(-0.124773\pi\)
0.924152 + 0.382024i \(0.124773\pi\)
\(752\) 37.4734i 1.36652i
\(753\) 0 0
\(754\) 2.07522 0.0755752
\(755\) −1.92478 + 2.17679i −0.0700498 + 0.0792216i
\(756\) 0 0
\(757\) 38.9525i 1.41575i 0.706336 + 0.707877i \(0.250347\pi\)
−0.706336 + 0.707877i \(0.749653\pi\)
\(758\) 2.07522i 0.0753755i
\(759\) 0 0
\(760\) 6.88717 + 6.08981i 0.249824 + 0.220901i
\(761\) 48.2130 1.74772 0.873860 0.486178i \(-0.161609\pi\)
0.873860 + 0.486178i \(0.161609\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 27.1782 0.983273
\(765\) 0 0
\(766\) 3.25343 0.117551
\(767\) 11.6464i 0.420528i
\(768\) 0 0
\(769\) 4.44851 0.160417 0.0802086 0.996778i \(-0.474441\pi\)
0.0802086 + 0.996778i \(0.474441\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 30.0752i 1.08243i
\(773\) 39.3014i 1.41357i −0.707427 0.706786i \(-0.750144\pi\)
0.707427 0.706786i \(-0.249856\pi\)
\(774\) 0 0
\(775\) 2.80018 + 22.7005i 0.100586 + 0.815427i
\(776\) 14.2158 0.510318
\(777\) 0 0
\(778\) 5.68735i 0.203901i
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 3.22425i 0.115299i
\(783\) 0 0
\(784\) 0 0
\(785\) −4.43866 3.92478i −0.158423 0.140081i
\(786\) 0 0
\(787\) 0.897015i 0.0319751i −0.999872 0.0159876i \(-0.994911\pi\)
0.999872 0.0159876i \(-0.00508922\pi\)
\(788\) 1.12742i 0.0401628i
\(789\) 0 0
\(790\) −3.47627 3.07381i −0.123680 0.109361i
\(791\) 0 0
\(792\) 0 0
\(793\) 11.7480i 0.417183i
\(794\) −3.55851 −0.126287
\(795\) 0 0
\(796\) −0.394893 −0.0139966
\(797\) 3.19982i 0.113343i 0.998393 + 0.0566717i \(0.0180488\pi\)
−0.998393 + 0.0566717i \(0.981951\pi\)
\(798\) 0 0
\(799\) 33.2506 1.17632
\(800\) −11.2605 + 1.38901i −0.398117 + 0.0491090i
\(801\) 0 0
\(802\) 7.23884i 0.255612i
\(803\) 18.7005i 0.659927i
\(804\) 0 0
\(805\) 0 0
\(806\) −1.19791 −0.0421944
\(807\) 0 0
\(808\) 13.5731i 0.477500i
\(809\) 4.44851 0.156401 0.0782006 0.996938i \(-0.475083\pi\)
0.0782006 + 0.996938i \(0.475083\pi\)
\(810\) 0 0
\(811\) −37.6747 −1.32294 −0.661468 0.749973i \(-0.730066\pi\)
−0.661468 + 0.749973i \(0.730066\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.300891 0.0105462
\(815\) 8.87732 + 7.84955i 0.310959 + 0.274958i
\(816\) 0 0
\(817\) 67.5487i 2.36323i
\(818\) 4.33900i 0.151710i
\(819\) 0 0
\(820\) −10.8656 + 12.2882i −0.379442 + 0.429123i
\(821\) 0.749399 0.0261542 0.0130771 0.999914i \(-0.495837\pi\)
0.0130771 + 0.999914i \(0.495837\pi\)
\(822\) 0 0
\(823\) 26.3996i 0.920233i 0.887858 + 0.460117i \(0.152192\pi\)
−0.887858 + 0.460117i \(0.847808\pi\)
\(824\) 5.14903 0.179375
\(825\) 0 0
\(826\) 0 0
\(827\) 5.43724i 0.189071i −0.995521 0.0945357i \(-0.969863\pi\)
0.995521 0.0945357i \(-0.0301367\pi\)
\(828\) 0 0
\(829\) 22.7757 0.791034 0.395517 0.918459i \(-0.370565\pi\)
0.395517 + 0.918459i \(0.370565\pi\)
\(830\) 0.926192 1.04746i 0.0321486 0.0363579i
\(831\) 0 0
\(832\) 9.60228i 0.332899i
\(833\) 0 0
\(834\) 0 0
\(835\) 24.3733 + 21.5515i 0.843472 + 0.745820i
\(836\) −20.9986 −0.726251
\(837\) 0 0
\(838\) 4.55291i 0.157278i
\(839\) −15.8496 −0.547187 −0.273594 0.961845i \(-0.588212\pi\)
−0.273594 + 0.961845i \(0.588212\pi\)
\(840\) 0 0
\(841\) 33.8021 1.16559
\(842\) 4.89191i 0.168586i
\(843\) 0 0
\(844\) 12.6545 0.435585
\(845\) −16.5550 + 18.7226i −0.569509 + 0.644077i
\(846\) 0 0
\(847\) 0 0
\(848\) 32.3752i 1.11177i
\(849\) 0 0
\(850\) 0.397722 + 3.22425i 0.0136418 + 0.110591i
\(851\) 3.84955 0.131961
\(852\) 0 0
\(853\) 21.0494i 0.720717i −0.932814 0.360358i \(-0.882654\pi\)
0.932814 0.360358i \(-0.117346\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10.5571 −0.360834
\(857\) 50.1524i 1.71317i 0.516004 + 0.856586i \(0.327419\pi\)
−0.516004 + 0.856586i \(0.672581\pi\)
\(858\) 0 0
\(859\) −5.35026 −0.182549 −0.0912743 0.995826i \(-0.529094\pi\)
−0.0912743 + 0.995826i \(0.529094\pi\)
\(860\) −41.5026 36.6977i −1.41523 1.25138i
\(861\) 0 0
\(862\) 3.76257i 0.128154i
\(863\) 33.6893i 1.14680i −0.819277 0.573398i \(-0.805625\pi\)
0.819277 0.573398i \(-0.194375\pi\)
\(864\) 0 0
\(865\) −7.53690 6.66433i −0.256262 0.226594i
\(866\) 1.26045 0.0428319
\(867\) 0 0
\(868\) 0 0
\(869\) 21.4010 0.725981
\(870\) 0 0
\(871\) −13.4010 −0.454077
\(872\) 2.13303i 0.0722334i
\(873\) 0 0
\(874\) 5.14903 0.174169
\(875\) 0 0
\(876\) 0 0
\(877\) 21.5026i 0.726092i −0.931771 0.363046i \(-0.881737\pi\)
0.931771 0.363046i \(-0.118263\pi\)
\(878\) 2.84112i 0.0958832i
\(879\) 0 0
\(880\) 11.1852 12.6497i 0.377054 0.426423i
\(881\) 32.3634 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(882\) 0 0
\(883\) 2.59895i 0.0874617i 0.999043 + 0.0437309i \(0.0139244\pi\)
−0.999043 + 0.0437309i \(0.986076\pi\)
\(884\) 8.87732 0.298576
\(885\) 0 0
\(886\) 3.71179 0.124700
\(887\) 38.2784i 1.28526i 0.766176 + 0.642631i \(0.222157\pi\)
−0.766176 + 0.642631i \(0.777843\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.337088 0.298062i −0.0112992 0.00999106i
\(891\) 0 0
\(892\) 3.04463i 0.101942i
\(893\) 53.1002i 1.77693i
\(894\) 0 0
\(895\) −14.8119 + 16.7513i −0.495109 + 0.559934i
\(896\) 0 0
\(897\) 0 0
\(898\) 6.36153i 0.212287i
\(899\) −36.2520 −1.20907
\(900\) 0 0
\(901\) 28.7269 0.957031
\(902\) 1.44992i 0.0482771i
\(903\) 0 0
\(904\) 9.26045 0.307998
\(905\) 15.7381 17.7988i 0.523153 0.591651i
\(906\) 0 0
\(907\) 49.9972i 1.66013i −0.557668 0.830064i \(-0.688304\pi\)
0.557668 0.830064i \(-0.311696\pi\)
\(908\) 25.8035i 0.856320i
\(909\) 0 0
\(910\) 0 0
\(911\) 24.9525 0.826715 0.413357 0.910569i \(-0.364356\pi\)
0.413357 + 0.910569i \(0.364356\pi\)
\(912\) 0 0
\(913\) 6.44851i 0.213414i
\(914\) −3.62672 −0.119961
\(915\) 0 0
\(916\) 5.44709 0.179977
\(917\) 0 0
\(918\) 0 0
\(919\) 11.6991 0.385918 0.192959 0.981207i \(-0.438192\pi\)
0.192959 + 0.981207i \(0.438192\pi\)
\(920\) −5.64832 + 6.38787i −0.186220 + 0.210602i
\(921\) 0 0
\(922\) 1.35026i 0.0444685i
\(923\) 2.70052i 0.0888888i
\(924\) 0 0
\(925\) −3.84955 + 0.474855i −0.126573 + 0.0156131i
\(926\) −1.02776 −0.0337744
\(927\) 0 0
\(928\) 17.9826i 0.590307i
\(929\) −23.7090 −0.777866 −0.388933 0.921266i \(-0.627156\pi\)
−0.388933 + 0.921266i \(0.627156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.0996603i 0.00326448i
\(933\) 0 0
\(934\) −2.55008 −0.0834411
\(935\) −11.2243 9.92478i −0.367072 0.324575i
\(936\) 0 0
\(937\) 19.9003i 0.650116i 0.945694 + 0.325058i \(0.105384\pi\)
−0.945694 + 0.325058i \(0.894616\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −32.6253 28.8481i −1.06412 0.940923i
\(941\) −6.28821 −0.204990 −0.102495 0.994734i \(-0.532683\pi\)
−0.102495 + 0.994734i \(0.532683\pi\)
\(942\) 0 0
\(943\) 18.5501i 0.604074i
\(944\) 32.5669 1.05996
\(945\) 0 0
\(946\) −4.89701 −0.159216
\(947\) 40.0362i 1.30100i 0.759506 + 0.650501i \(0.225441\pi\)
−0.759506 + 0.650501i \(0.774559\pi\)
\(948\) 0 0
\(949\) 12.6253 0.409835
\(950\) −5.14903 + 0.635150i −0.167057 + 0.0206070i
\(951\) 0 0
\(952\) 0 0
\(953\) 40.9478i 1.32643i 0.748429 + 0.663215i \(0.230808\pi\)
−0.748429 + 0.663215i \(0.769192\pi\)
\(954\) 0 0
\(955\) −20.5139 + 23.1998i −0.663814 + 0.750728i
\(956\) 11.4791 0.371261
\(957\) 0 0
\(958\) 0.998585i 0.0322628i
\(959\) 0 0
\(960\) 0 0
\(961\) −10.0738 −0.324962
\(962\) 0.203141i 0.00654953i
\(963\) 0 0
\(964\) 0.147616 0.00475440
\(965\) 25.6728 + 22.7005i 0.826435 + 0.730756i
\(966\) 0 0
\(967\) 38.2784i 1.23095i 0.788157 + 0.615475i \(0.211036\pi\)
−0.788157 + 0.615475i \(0.788964\pi\)
\(968\) 5.37916i 0.172893i
\(969\) 0 0
\(970\) −5.31406 + 6.00985i −0.170624 + 0.192965i
\(971\) −28.7269 −0.921889 −0.460945 0.887429i \(-0.652489\pi\)
−0.460945 + 0.887429i \(0.652489\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −4.30089 −0.137809
\(975\) 0 0
\(976\) 32.8510 1.05153
\(977\) 41.3014i 1.32135i 0.750673 + 0.660674i \(0.229730\pi\)
−0.750673 + 0.660674i \(0.770270\pi\)
\(978\) 0 0
\(979\) 2.07522 0.0663244
\(980\) 0 0
\(981\) 0 0
\(982\) 0.387873i 0.0123775i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) −0.962389 0.850969i −0.0306643 0.0271141i
\(986\) −5.14903 −0.163979
\(987\) 0 0
\(988\) 14.1768i 0.451024i
\(989\) −62.6516 −1.99221
\(990\) 0 0
\(991\) 25.1002 0.797333 0.398666 0.917096i \(-0.369473\pi\)
0.398666 + 0.917096i \(0.369473\pi\)
\(992\) 10.3803i 0.329575i
\(993\) 0 0
\(994\) 0 0
\(995\) 0.298062 0.337088i 0.00944920 0.0106864i
\(996\) 0 0
\(997\) 32.7974i 1.03870i −0.854561 0.519351i \(-0.826174\pi\)
0.854561 0.519351i \(-0.173826\pi\)
\(998\) 1.27030i 0.0402106i
\(999\) 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 2205.2.d.l.1324.3 6
3.2 odd 2 735.2.d.b.589.4 6
5.4 even 2 inner 2205.2.d.l.1324.4 6
7.6 odd 2 315.2.d.e.64.3 6
15.2 even 4 3675.2.a.bi.1.2 3
15.8 even 4 3675.2.a.bj.1.2 3
15.14 odd 2 735.2.d.b.589.3 6
21.2 odd 6 735.2.q.f.214.4 12
21.5 even 6 735.2.q.e.214.4 12
21.11 odd 6 735.2.q.f.79.3 12
21.17 even 6 735.2.q.e.79.3 12
21.20 even 2 105.2.d.b.64.4 yes 6
28.27 even 2 5040.2.t.v.1009.5 6
35.13 even 4 1575.2.a.w.1.2 3
35.27 even 4 1575.2.a.x.1.2 3
35.34 odd 2 315.2.d.e.64.4 6
84.83 odd 2 1680.2.t.k.1009.1 6
105.44 odd 6 735.2.q.f.214.3 12
105.59 even 6 735.2.q.e.79.4 12
105.62 odd 4 525.2.a.j.1.2 3
105.74 odd 6 735.2.q.f.79.4 12
105.83 odd 4 525.2.a.k.1.2 3
105.89 even 6 735.2.q.e.214.3 12
105.104 even 2 105.2.d.b.64.3 6
140.139 even 2 5040.2.t.v.1009.6 6
420.83 even 4 8400.2.a.dj.1.3 3
420.167 even 4 8400.2.a.dg.1.1 3
420.419 odd 2 1680.2.t.k.1009.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.3 6 105.104 even 2
105.2.d.b.64.4 yes 6 21.20 even 2
315.2.d.e.64.3 6 7.6 odd 2
315.2.d.e.64.4 6 35.34 odd 2
525.2.a.j.1.2 3 105.62 odd 4
525.2.a.k.1.2 3 105.83 odd 4
735.2.d.b.589.3 6 15.14 odd 2
735.2.d.b.589.4 6 3.2 odd 2
735.2.q.e.79.3 12 21.17 even 6
735.2.q.e.79.4 12 105.59 even 6
735.2.q.e.214.3 12 105.89 even 6
735.2.q.e.214.4 12 21.5 even 6
735.2.q.f.79.3 12 21.11 odd 6
735.2.q.f.79.4 12 105.74 odd 6
735.2.q.f.214.3 12 105.44 odd 6
735.2.q.f.214.4 12 21.2 odd 6
1575.2.a.w.1.2 3 35.13 even 4
1575.2.a.x.1.2 3 35.27 even 4
1680.2.t.k.1009.1 6 84.83 odd 2
1680.2.t.k.1009.4 6 420.419 odd 2
2205.2.d.l.1324.3 6 1.1 even 1 trivial
2205.2.d.l.1324.4 6 5.4 even 2 inner
3675.2.a.bi.1.2 3 15.2 even 4
3675.2.a.bj.1.2 3 15.8 even 4
5040.2.t.v.1009.5 6 28.27 even 2
5040.2.t.v.1009.6 6 140.139 even 2
8400.2.a.dg.1.1 3 420.167 even 4
8400.2.a.dj.1.3 3 420.83 even 4