Properties

Label 3549.2.a.q.1.3
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

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: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.630898 q^{5} +2.17009 q^{6} -1.00000 q^{7} +1.53919 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} -0.630898 q^{5} +2.17009 q^{6} -1.00000 q^{7} +1.53919 q^{8} +1.00000 q^{9} -1.36910 q^{10} -5.70928 q^{11} +2.70928 q^{12} -2.17009 q^{14} -0.630898 q^{15} -2.07838 q^{16} -1.07838 q^{17} +2.17009 q^{18} -7.41855 q^{19} -1.70928 q^{20} -1.00000 q^{21} -12.3896 q^{22} -5.41855 q^{23} +1.53919 q^{24} -4.60197 q^{25} +1.00000 q^{27} -2.70928 q^{28} +6.68035 q^{29} -1.36910 q^{30} -6.15676 q^{31} -7.58864 q^{32} -5.70928 q^{33} -2.34017 q^{34} +0.630898 q^{35} +2.70928 q^{36} +3.41855 q^{37} -16.0989 q^{38} -0.971071 q^{40} -1.21235 q^{41} -2.17009 q^{42} +12.6803 q^{43} -15.4680 q^{44} -0.630898 q^{45} -11.7587 q^{46} +6.04945 q^{47} -2.07838 q^{48} +1.00000 q^{49} -9.98667 q^{50} -1.07838 q^{51} -6.00000 q^{53} +2.17009 q^{54} +3.60197 q^{55} -1.53919 q^{56} -7.41855 q^{57} +14.4969 q^{58} -1.36910 q^{59} -1.70928 q^{60} +12.6537 q^{61} -13.3607 q^{62} -1.00000 q^{63} -12.3112 q^{64} -12.3896 q^{66} -0.581449 q^{67} -2.92162 q^{68} -5.41855 q^{69} +1.36910 q^{70} +4.81432 q^{71} +1.53919 q^{72} -3.81658 q^{73} +7.41855 q^{74} -4.60197 q^{75} -20.0989 q^{76} +5.70928 q^{77} -1.65983 q^{79} +1.31124 q^{80} +1.00000 q^{81} -2.63090 q^{82} +12.8865 q^{83} -2.70928 q^{84} +0.680346 q^{85} +27.5174 q^{86} +6.68035 q^{87} -8.78765 q^{88} +6.20620 q^{89} -1.36910 q^{90} -14.6803 q^{92} -6.15676 q^{93} +13.1278 q^{94} +4.68035 q^{95} -7.58864 q^{96} +1.07838 q^{97} +2.17009 q^{98} -5.70928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + q^{4} + 2 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + q^{4} + 2 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 8 q^{10} - 10 q^{11} + q^{12} - q^{14} + 2 q^{15} - 3 q^{16} + q^{18} - 8 q^{19} + 2 q^{20} - 3 q^{21} - 8 q^{22} - 2 q^{23} + 3 q^{24} + 5 q^{25} + 3 q^{27} - q^{28} - 2 q^{29} - 8 q^{30} - 12 q^{31} - 3 q^{32} - 10 q^{33} + 4 q^{34} - 2 q^{35} + q^{36} - 4 q^{37} - 12 q^{38} + 12 q^{40} - 14 q^{41} - q^{42} + 16 q^{43} - 14 q^{44} + 2 q^{45} - 10 q^{46} - 3 q^{48} + 3 q^{49} - 29 q^{50} - 18 q^{53} + q^{54} - 8 q^{55} - 3 q^{56} - 8 q^{57} + 26 q^{58} - 8 q^{59} + 2 q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} - 11 q^{64} - 8 q^{66} - 16 q^{67} - 12 q^{68} - 2 q^{69} + 8 q^{70} + 6 q^{71} + 3 q^{72} - 16 q^{73} + 8 q^{74} + 5 q^{75} - 24 q^{76} + 10 q^{77} - 16 q^{79} - 22 q^{80} + 3 q^{81} - 4 q^{82} - 8 q^{83} - q^{84} - 20 q^{85} + 32 q^{86} - 2 q^{87} - 16 q^{88} - 6 q^{89} - 8 q^{90} - 22 q^{92} - 12 q^{93} + 18 q^{94} - 8 q^{95} - 3 q^{96} + q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.17009 1.53448 0.767241 0.641358i \(-0.221629\pi\)
0.767241 + 0.641358i \(0.221629\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.70928 1.35464
\(5\) −0.630898 −0.282146 −0.141073 0.989999i \(-0.545055\pi\)
−0.141073 + 0.989999i \(0.545055\pi\)
\(6\) 2.17009 0.885934
\(7\) −1.00000 −0.377964
\(8\) 1.53919 0.544185
\(9\) 1.00000 0.333333
\(10\) −1.36910 −0.432948
\(11\) −5.70928 −1.72141 −0.860706 0.509103i \(-0.829977\pi\)
−0.860706 + 0.509103i \(0.829977\pi\)
\(12\) 2.70928 0.782100
\(13\) 0 0
\(14\) −2.17009 −0.579980
\(15\) −0.630898 −0.162897
\(16\) −2.07838 −0.519594
\(17\) −1.07838 −0.261545 −0.130773 0.991412i \(-0.541746\pi\)
−0.130773 + 0.991412i \(0.541746\pi\)
\(18\) 2.17009 0.511494
\(19\) −7.41855 −1.70193 −0.850966 0.525221i \(-0.823983\pi\)
−0.850966 + 0.525221i \(0.823983\pi\)
\(20\) −1.70928 −0.382206
\(21\) −1.00000 −0.218218
\(22\) −12.3896 −2.64148
\(23\) −5.41855 −1.12985 −0.564923 0.825144i \(-0.691094\pi\)
−0.564923 + 0.825144i \(0.691094\pi\)
\(24\) 1.53919 0.314186
\(25\) −4.60197 −0.920394
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.70928 −0.512005
\(29\) 6.68035 1.24051 0.620255 0.784401i \(-0.287029\pi\)
0.620255 + 0.784401i \(0.287029\pi\)
\(30\) −1.36910 −0.249963
\(31\) −6.15676 −1.10579 −0.552893 0.833252i \(-0.686476\pi\)
−0.552893 + 0.833252i \(0.686476\pi\)
\(32\) −7.58864 −1.34149
\(33\) −5.70928 −0.993857
\(34\) −2.34017 −0.401336
\(35\) 0.630898 0.106641
\(36\) 2.70928 0.451546
\(37\) 3.41855 0.562006 0.281003 0.959707i \(-0.409333\pi\)
0.281003 + 0.959707i \(0.409333\pi\)
\(38\) −16.0989 −2.61159
\(39\) 0 0
\(40\) −0.971071 −0.153540
\(41\) −1.21235 −0.189337 −0.0946684 0.995509i \(-0.530179\pi\)
−0.0946684 + 0.995509i \(0.530179\pi\)
\(42\) −2.17009 −0.334852
\(43\) 12.6803 1.93373 0.966867 0.255279i \(-0.0821675\pi\)
0.966867 + 0.255279i \(0.0821675\pi\)
\(44\) −15.4680 −2.33189
\(45\) −0.630898 −0.0940487
\(46\) −11.7587 −1.73373
\(47\) 6.04945 0.882403 0.441201 0.897408i \(-0.354552\pi\)
0.441201 + 0.897408i \(0.354552\pi\)
\(48\) −2.07838 −0.299988
\(49\) 1.00000 0.142857
\(50\) −9.98667 −1.41233
\(51\) −1.07838 −0.151003
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 2.17009 0.295311
\(55\) 3.60197 0.485689
\(56\) −1.53919 −0.205683
\(57\) −7.41855 −0.982611
\(58\) 14.4969 1.90354
\(59\) −1.36910 −0.178242 −0.0891210 0.996021i \(-0.528406\pi\)
−0.0891210 + 0.996021i \(0.528406\pi\)
\(60\) −1.70928 −0.220667
\(61\) 12.6537 1.62014 0.810069 0.586334i \(-0.199430\pi\)
0.810069 + 0.586334i \(0.199430\pi\)
\(62\) −13.3607 −1.69681
\(63\) −1.00000 −0.125988
\(64\) −12.3112 −1.53891
\(65\) 0 0
\(66\) −12.3896 −1.52506
\(67\) −0.581449 −0.0710353 −0.0355177 0.999369i \(-0.511308\pi\)
−0.0355177 + 0.999369i \(0.511308\pi\)
\(68\) −2.92162 −0.354299
\(69\) −5.41855 −0.652317
\(70\) 1.36910 0.163639
\(71\) 4.81432 0.571354 0.285677 0.958326i \(-0.407782\pi\)
0.285677 + 0.958326i \(0.407782\pi\)
\(72\) 1.53919 0.181395
\(73\) −3.81658 −0.446697 −0.223349 0.974739i \(-0.571699\pi\)
−0.223349 + 0.974739i \(0.571699\pi\)
\(74\) 7.41855 0.862389
\(75\) −4.60197 −0.531390
\(76\) −20.0989 −2.30550
\(77\) 5.70928 0.650632
\(78\) 0 0
\(79\) −1.65983 −0.186745 −0.0933726 0.995631i \(-0.529765\pi\)
−0.0933726 + 0.995631i \(0.529765\pi\)
\(80\) 1.31124 0.146601
\(81\) 1.00000 0.111111
\(82\) −2.63090 −0.290534
\(83\) 12.8865 1.41448 0.707241 0.706972i \(-0.249939\pi\)
0.707241 + 0.706972i \(0.249939\pi\)
\(84\) −2.70928 −0.295606
\(85\) 0.680346 0.0737939
\(86\) 27.5174 2.96728
\(87\) 6.68035 0.716208
\(88\) −8.78765 −0.936767
\(89\) 6.20620 0.657856 0.328928 0.944355i \(-0.393313\pi\)
0.328928 + 0.944355i \(0.393313\pi\)
\(90\) −1.36910 −0.144316
\(91\) 0 0
\(92\) −14.6803 −1.53053
\(93\) −6.15676 −0.638426
\(94\) 13.1278 1.35403
\(95\) 4.68035 0.480193
\(96\) −7.58864 −0.774512
\(97\) 1.07838 0.109493 0.0547463 0.998500i \(-0.482565\pi\)
0.0547463 + 0.998500i \(0.482565\pi\)
\(98\) 2.17009 0.219212
\(99\) −5.70928 −0.573804
\(100\) −12.4680 −1.24680
\(101\) 7.23513 0.719923 0.359961 0.932967i \(-0.382790\pi\)
0.359961 + 0.932967i \(0.382790\pi\)
\(102\) −2.34017 −0.231712
\(103\) −13.7587 −1.35569 −0.677844 0.735206i \(-0.737085\pi\)
−0.677844 + 0.735206i \(0.737085\pi\)
\(104\) 0 0
\(105\) 0.630898 0.0615693
\(106\) −13.0205 −1.26466
\(107\) −10.0989 −0.976297 −0.488149 0.872761i \(-0.662328\pi\)
−0.488149 + 0.872761i \(0.662328\pi\)
\(108\) 2.70928 0.260700
\(109\) −17.9421 −1.71855 −0.859273 0.511518i \(-0.829083\pi\)
−0.859273 + 0.511518i \(0.829083\pi\)
\(110\) 7.81658 0.745282
\(111\) 3.41855 0.324474
\(112\) 2.07838 0.196388
\(113\) 10.6803 1.00472 0.502361 0.864658i \(-0.332465\pi\)
0.502361 + 0.864658i \(0.332465\pi\)
\(114\) −16.0989 −1.50780
\(115\) 3.41855 0.318781
\(116\) 18.0989 1.68044
\(117\) 0 0
\(118\) −2.97107 −0.273509
\(119\) 1.07838 0.0988547
\(120\) −0.971071 −0.0886462
\(121\) 21.5958 1.96326
\(122\) 27.4596 2.48607
\(123\) −1.21235 −0.109314
\(124\) −16.6803 −1.49794
\(125\) 6.05786 0.541831
\(126\) −2.17009 −0.193327
\(127\) −6.34017 −0.562599 −0.281300 0.959620i \(-0.590765\pi\)
−0.281300 + 0.959620i \(0.590765\pi\)
\(128\) −11.5392 −1.01993
\(129\) 12.6803 1.11644
\(130\) 0 0
\(131\) −7.51745 −0.656802 −0.328401 0.944538i \(-0.606510\pi\)
−0.328401 + 0.944538i \(0.606510\pi\)
\(132\) −15.4680 −1.34632
\(133\) 7.41855 0.643270
\(134\) −1.26180 −0.109003
\(135\) −0.630898 −0.0542990
\(136\) −1.65983 −0.142329
\(137\) −4.47414 −0.382252 −0.191126 0.981566i \(-0.561214\pi\)
−0.191126 + 0.981566i \(0.561214\pi\)
\(138\) −11.7587 −1.00097
\(139\) −9.75872 −0.827724 −0.413862 0.910340i \(-0.635820\pi\)
−0.413862 + 0.910340i \(0.635820\pi\)
\(140\) 1.70928 0.144460
\(141\) 6.04945 0.509455
\(142\) 10.4475 0.876733
\(143\) 0 0
\(144\) −2.07838 −0.173198
\(145\) −4.21461 −0.350005
\(146\) −8.28231 −0.685449
\(147\) 1.00000 0.0824786
\(148\) 9.26180 0.761315
\(149\) −1.46800 −0.120263 −0.0601316 0.998190i \(-0.519152\pi\)
−0.0601316 + 0.998190i \(0.519152\pi\)
\(150\) −9.98667 −0.815408
\(151\) −13.3607 −1.08728 −0.543639 0.839319i \(-0.682954\pi\)
−0.543639 + 0.839319i \(0.682954\pi\)
\(152\) −11.4186 −0.926167
\(153\) −1.07838 −0.0871817
\(154\) 12.3896 0.998384
\(155\) 3.88428 0.311993
\(156\) 0 0
\(157\) −8.15676 −0.650980 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(158\) −3.60197 −0.286557
\(159\) −6.00000 −0.475831
\(160\) 4.78765 0.378497
\(161\) 5.41855 0.427042
\(162\) 2.17009 0.170498
\(163\) −4.58145 −0.358847 −0.179423 0.983772i \(-0.557423\pi\)
−0.179423 + 0.983772i \(0.557423\pi\)
\(164\) −3.28458 −0.256483
\(165\) 3.60197 0.280413
\(166\) 27.9649 2.17050
\(167\) −7.31124 −0.565761 −0.282881 0.959155i \(-0.591290\pi\)
−0.282881 + 0.959155i \(0.591290\pi\)
\(168\) −1.53919 −0.118751
\(169\) 0 0
\(170\) 1.47641 0.113235
\(171\) −7.41855 −0.567311
\(172\) 34.3545 2.61951
\(173\) −7.60197 −0.577967 −0.288983 0.957334i \(-0.593317\pi\)
−0.288983 + 0.957334i \(0.593317\pi\)
\(174\) 14.4969 1.09901
\(175\) 4.60197 0.347876
\(176\) 11.8660 0.894436
\(177\) −1.36910 −0.102908
\(178\) 13.4680 1.00947
\(179\) −9.10504 −0.680543 −0.340271 0.940327i \(-0.610519\pi\)
−0.340271 + 0.940327i \(0.610519\pi\)
\(180\) −1.70928 −0.127402
\(181\) −15.3607 −1.14175 −0.570876 0.821037i \(-0.693396\pi\)
−0.570876 + 0.821037i \(0.693396\pi\)
\(182\) 0 0
\(183\) 12.6537 0.935387
\(184\) −8.34017 −0.614846
\(185\) −2.15676 −0.158568
\(186\) −13.3607 −0.979653
\(187\) 6.15676 0.450227
\(188\) 16.3896 1.19534
\(189\) −1.00000 −0.0727393
\(190\) 10.1568 0.736848
\(191\) −24.5692 −1.77776 −0.888881 0.458138i \(-0.848517\pi\)
−0.888881 + 0.458138i \(0.848517\pi\)
\(192\) −12.3112 −0.888487
\(193\) −13.5753 −0.977172 −0.488586 0.872516i \(-0.662487\pi\)
−0.488586 + 0.872516i \(0.662487\pi\)
\(194\) 2.34017 0.168015
\(195\) 0 0
\(196\) 2.70928 0.193520
\(197\) 2.94441 0.209780 0.104890 0.994484i \(-0.466551\pi\)
0.104890 + 0.994484i \(0.466551\pi\)
\(198\) −12.3896 −0.880492
\(199\) 10.5236 0.745998 0.372999 0.927832i \(-0.378330\pi\)
0.372999 + 0.927832i \(0.378330\pi\)
\(200\) −7.08330 −0.500865
\(201\) −0.581449 −0.0410123
\(202\) 15.7009 1.10471
\(203\) −6.68035 −0.468868
\(204\) −2.92162 −0.204554
\(205\) 0.764867 0.0534206
\(206\) −29.8576 −2.08028
\(207\) −5.41855 −0.376615
\(208\) 0 0
\(209\) 42.3545 2.92973
\(210\) 1.36910 0.0944770
\(211\) −3.02052 −0.207941 −0.103971 0.994580i \(-0.533155\pi\)
−0.103971 + 0.994580i \(0.533155\pi\)
\(212\) −16.2557 −1.11644
\(213\) 4.81432 0.329871
\(214\) −21.9155 −1.49811
\(215\) −8.00000 −0.545595
\(216\) 1.53919 0.104729
\(217\) 6.15676 0.417948
\(218\) −38.9360 −2.63708
\(219\) −3.81658 −0.257901
\(220\) 9.75872 0.657933
\(221\) 0 0
\(222\) 7.41855 0.497901
\(223\) −22.6225 −1.51491 −0.757457 0.652885i \(-0.773558\pi\)
−0.757457 + 0.652885i \(0.773558\pi\)
\(224\) 7.58864 0.507037
\(225\) −4.60197 −0.306798
\(226\) 23.1773 1.54173
\(227\) 22.9444 1.52287 0.761437 0.648239i \(-0.224494\pi\)
0.761437 + 0.648239i \(0.224494\pi\)
\(228\) −20.0989 −1.33108
\(229\) 21.5441 1.42367 0.711837 0.702344i \(-0.247863\pi\)
0.711837 + 0.702344i \(0.247863\pi\)
\(230\) 7.41855 0.489165
\(231\) 5.70928 0.375643
\(232\) 10.2823 0.675067
\(233\) −7.36069 −0.482215 −0.241107 0.970498i \(-0.577511\pi\)
−0.241107 + 0.970498i \(0.577511\pi\)
\(234\) 0 0
\(235\) −3.81658 −0.248966
\(236\) −3.70928 −0.241453
\(237\) −1.65983 −0.107817
\(238\) 2.34017 0.151691
\(239\) −15.5525 −1.00601 −0.503004 0.864284i \(-0.667772\pi\)
−0.503004 + 0.864284i \(0.667772\pi\)
\(240\) 1.31124 0.0846404
\(241\) 7.33403 0.472426 0.236213 0.971701i \(-0.424094\pi\)
0.236213 + 0.971701i \(0.424094\pi\)
\(242\) 46.8648 3.01258
\(243\) 1.00000 0.0641500
\(244\) 34.2823 2.19470
\(245\) −0.630898 −0.0403066
\(246\) −2.63090 −0.167740
\(247\) 0 0
\(248\) −9.47641 −0.601753
\(249\) 12.8865 0.816652
\(250\) 13.1461 0.831431
\(251\) 13.6742 0.863108 0.431554 0.902087i \(-0.357965\pi\)
0.431554 + 0.902087i \(0.357965\pi\)
\(252\) −2.70928 −0.170668
\(253\) 30.9360 1.94493
\(254\) −13.7587 −0.863299
\(255\) 0.680346 0.0426049
\(256\) −0.418551 −0.0261594
\(257\) 26.1256 1.62967 0.814834 0.579695i \(-0.196828\pi\)
0.814834 + 0.579695i \(0.196828\pi\)
\(258\) 27.5174 1.71316
\(259\) −3.41855 −0.212418
\(260\) 0 0
\(261\) 6.68035 0.413503
\(262\) −16.3135 −1.00785
\(263\) −30.7792 −1.89793 −0.948965 0.315382i \(-0.897867\pi\)
−0.948965 + 0.315382i \(0.897867\pi\)
\(264\) −8.78765 −0.540843
\(265\) 3.78539 0.232534
\(266\) 16.0989 0.987087
\(267\) 6.20620 0.379814
\(268\) −1.57531 −0.0962271
\(269\) 5.39189 0.328749 0.164375 0.986398i \(-0.447439\pi\)
0.164375 + 0.986398i \(0.447439\pi\)
\(270\) −1.36910 −0.0833209
\(271\) 11.4186 0.693628 0.346814 0.937934i \(-0.387264\pi\)
0.346814 + 0.937934i \(0.387264\pi\)
\(272\) 2.24128 0.135897
\(273\) 0 0
\(274\) −9.70928 −0.586559
\(275\) 26.2739 1.58438
\(276\) −14.6803 −0.883653
\(277\) −13.1506 −0.790144 −0.395072 0.918650i \(-0.629280\pi\)
−0.395072 + 0.918650i \(0.629280\pi\)
\(278\) −21.1773 −1.27013
\(279\) −6.15676 −0.368595
\(280\) 0.971071 0.0580326
\(281\) −2.14834 −0.128160 −0.0640798 0.997945i \(-0.520411\pi\)
−0.0640798 + 0.997945i \(0.520411\pi\)
\(282\) 13.1278 0.781751
\(283\) 12.1978 0.725084 0.362542 0.931968i \(-0.381909\pi\)
0.362542 + 0.931968i \(0.381909\pi\)
\(284\) 13.0433 0.773978
\(285\) 4.68035 0.277240
\(286\) 0 0
\(287\) 1.21235 0.0715626
\(288\) −7.58864 −0.447165
\(289\) −15.8371 −0.931594
\(290\) −9.14608 −0.537076
\(291\) 1.07838 0.0632156
\(292\) −10.3402 −0.605113
\(293\) 7.15449 0.417970 0.208985 0.977919i \(-0.432984\pi\)
0.208985 + 0.977919i \(0.432984\pi\)
\(294\) 2.17009 0.126562
\(295\) 0.863763 0.0502903
\(296\) 5.26180 0.305836
\(297\) −5.70928 −0.331286
\(298\) −3.18568 −0.184542
\(299\) 0 0
\(300\) −12.4680 −0.719840
\(301\) −12.6803 −0.730883
\(302\) −28.9939 −1.66841
\(303\) 7.23513 0.415648
\(304\) 15.4186 0.884315
\(305\) −7.98318 −0.457116
\(306\) −2.34017 −0.133779
\(307\) −21.8432 −1.24666 −0.623330 0.781959i \(-0.714221\pi\)
−0.623330 + 0.781959i \(0.714221\pi\)
\(308\) 15.4680 0.881371
\(309\) −13.7587 −0.782706
\(310\) 8.42923 0.478748
\(311\) 7.51745 0.426275 0.213138 0.977022i \(-0.431632\pi\)
0.213138 + 0.977022i \(0.431632\pi\)
\(312\) 0 0
\(313\) 25.7009 1.45270 0.726349 0.687326i \(-0.241215\pi\)
0.726349 + 0.687326i \(0.241215\pi\)
\(314\) −17.7009 −0.998918
\(315\) 0.630898 0.0355471
\(316\) −4.49693 −0.252972
\(317\) −9.36910 −0.526221 −0.263111 0.964766i \(-0.584748\pi\)
−0.263111 + 0.964766i \(0.584748\pi\)
\(318\) −13.0205 −0.730154
\(319\) −38.1399 −2.13543
\(320\) 7.76713 0.434196
\(321\) −10.0989 −0.563665
\(322\) 11.7587 0.655288
\(323\) 8.00000 0.445132
\(324\) 2.70928 0.150515
\(325\) 0 0
\(326\) −9.94214 −0.550644
\(327\) −17.9421 −0.992203
\(328\) −1.86603 −0.103034
\(329\) −6.04945 −0.333517
\(330\) 7.81658 0.430289
\(331\) −21.0472 −1.15686 −0.578429 0.815733i \(-0.696334\pi\)
−0.578429 + 0.815733i \(0.696334\pi\)
\(332\) 34.9132 1.91611
\(333\) 3.41855 0.187335
\(334\) −15.8660 −0.868151
\(335\) 0.366835 0.0200423
\(336\) 2.07838 0.113385
\(337\) −15.4452 −0.841354 −0.420677 0.907210i \(-0.638207\pi\)
−0.420677 + 0.907210i \(0.638207\pi\)
\(338\) 0 0
\(339\) 10.6803 0.580077
\(340\) 1.84324 0.0999640
\(341\) 35.1506 1.90351
\(342\) −16.0989 −0.870529
\(343\) −1.00000 −0.0539949
\(344\) 19.5174 1.05231
\(345\) 3.41855 0.184049
\(346\) −16.4969 −0.886880
\(347\) 21.7321 1.16664 0.583319 0.812243i \(-0.301754\pi\)
0.583319 + 0.812243i \(0.301754\pi\)
\(348\) 18.0989 0.970203
\(349\) −5.65983 −0.302964 −0.151482 0.988460i \(-0.548404\pi\)
−0.151482 + 0.988460i \(0.548404\pi\)
\(350\) 9.98667 0.533810
\(351\) 0 0
\(352\) 43.3256 2.30926
\(353\) −33.7237 −1.79493 −0.897464 0.441087i \(-0.854593\pi\)
−0.897464 + 0.441087i \(0.854593\pi\)
\(354\) −2.97107 −0.157911
\(355\) −3.03734 −0.161205
\(356\) 16.8143 0.891157
\(357\) 1.07838 0.0570738
\(358\) −19.7587 −1.04428
\(359\) −18.7031 −0.987114 −0.493557 0.869714i \(-0.664303\pi\)
−0.493557 + 0.869714i \(0.664303\pi\)
\(360\) −0.971071 −0.0511799
\(361\) 36.0349 1.89657
\(362\) −33.3340 −1.75200
\(363\) 21.5958 1.13349
\(364\) 0 0
\(365\) 2.40787 0.126034
\(366\) 27.4596 1.43534
\(367\) 10.0722 0.525766 0.262883 0.964828i \(-0.415327\pi\)
0.262883 + 0.964828i \(0.415327\pi\)
\(368\) 11.2618 0.587062
\(369\) −1.21235 −0.0631123
\(370\) −4.68035 −0.243320
\(371\) 6.00000 0.311504
\(372\) −16.6803 −0.864836
\(373\) 4.24128 0.219605 0.109802 0.993953i \(-0.464978\pi\)
0.109802 + 0.993953i \(0.464978\pi\)
\(374\) 13.3607 0.690865
\(375\) 6.05786 0.312826
\(376\) 9.31124 0.480191
\(377\) 0 0
\(378\) −2.17009 −0.111617
\(379\) 20.1978 1.03749 0.518745 0.854929i \(-0.326399\pi\)
0.518745 + 0.854929i \(0.326399\pi\)
\(380\) 12.6803 0.650488
\(381\) −6.34017 −0.324817
\(382\) −53.3172 −2.72795
\(383\) −7.04331 −0.359896 −0.179948 0.983676i \(-0.557593\pi\)
−0.179948 + 0.983676i \(0.557593\pi\)
\(384\) −11.5392 −0.588857
\(385\) −3.60197 −0.183573
\(386\) −29.4596 −1.49945
\(387\) 12.6803 0.644578
\(388\) 2.92162 0.148323
\(389\) 3.84324 0.194860 0.0974301 0.995242i \(-0.468938\pi\)
0.0974301 + 0.995242i \(0.468938\pi\)
\(390\) 0 0
\(391\) 5.84324 0.295506
\(392\) 1.53919 0.0777408
\(393\) −7.51745 −0.379205
\(394\) 6.38962 0.321904
\(395\) 1.04718 0.0526894
\(396\) −15.4680 −0.777296
\(397\) −14.4391 −0.724676 −0.362338 0.932047i \(-0.618021\pi\)
−0.362338 + 0.932047i \(0.618021\pi\)
\(398\) 22.8371 1.14472
\(399\) 7.41855 0.371392
\(400\) 9.56463 0.478231
\(401\) 9.83483 0.491128 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(402\) −1.26180 −0.0629326
\(403\) 0 0
\(404\) 19.6020 0.975234
\(405\) −0.630898 −0.0313496
\(406\) −14.4969 −0.719470
\(407\) −19.5174 −0.967444
\(408\) −1.65983 −0.0821737
\(409\) −7.54864 −0.373256 −0.186628 0.982431i \(-0.559756\pi\)
−0.186628 + 0.982431i \(0.559756\pi\)
\(410\) 1.65983 0.0819730
\(411\) −4.47414 −0.220693
\(412\) −37.2762 −1.83647
\(413\) 1.36910 0.0673691
\(414\) −11.7587 −0.577910
\(415\) −8.13009 −0.399091
\(416\) 0 0
\(417\) −9.75872 −0.477887
\(418\) 91.9130 4.49561
\(419\) −35.1506 −1.71722 −0.858610 0.512630i \(-0.828671\pi\)
−0.858610 + 0.512630i \(0.828671\pi\)
\(420\) 1.70928 0.0834041
\(421\) 25.6742 1.25128 0.625642 0.780110i \(-0.284837\pi\)
0.625642 + 0.780110i \(0.284837\pi\)
\(422\) −6.55479 −0.319082
\(423\) 6.04945 0.294134
\(424\) −9.23513 −0.448498
\(425\) 4.96266 0.240724
\(426\) 10.4475 0.506182
\(427\) −12.6537 −0.612355
\(428\) −27.3607 −1.32253
\(429\) 0 0
\(430\) −17.3607 −0.837207
\(431\) 16.2329 0.781910 0.390955 0.920410i \(-0.372145\pi\)
0.390955 + 0.920410i \(0.372145\pi\)
\(432\) −2.07838 −0.0999960
\(433\) 5.38735 0.258900 0.129450 0.991586i \(-0.458679\pi\)
0.129450 + 0.991586i \(0.458679\pi\)
\(434\) 13.3607 0.641334
\(435\) −4.21461 −0.202075
\(436\) −48.6102 −2.32801
\(437\) 40.1978 1.92292
\(438\) −8.28231 −0.395744
\(439\) 10.2413 0.488789 0.244395 0.969676i \(-0.421411\pi\)
0.244395 + 0.969676i \(0.421411\pi\)
\(440\) 5.54411 0.264305
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 15.9421 0.757434 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(444\) 9.26180 0.439545
\(445\) −3.91548 −0.185612
\(446\) −49.0928 −2.32461
\(447\) −1.46800 −0.0694340
\(448\) 12.3112 0.581652
\(449\) −13.2001 −0.622949 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(450\) −9.98667 −0.470776
\(451\) 6.92162 0.325926
\(452\) 28.9360 1.36103
\(453\) −13.3607 −0.627740
\(454\) 49.7914 2.33682
\(455\) 0 0
\(456\) −11.4186 −0.534723
\(457\) −18.3090 −0.856458 −0.428229 0.903670i \(-0.640862\pi\)
−0.428229 + 0.903670i \(0.640862\pi\)
\(458\) 46.7526 2.18460
\(459\) −1.07838 −0.0503344
\(460\) 9.26180 0.431833
\(461\) 18.9399 0.882118 0.441059 0.897478i \(-0.354603\pi\)
0.441059 + 0.897478i \(0.354603\pi\)
\(462\) 12.3896 0.576417
\(463\) −1.74435 −0.0810667 −0.0405334 0.999178i \(-0.512906\pi\)
−0.0405334 + 0.999178i \(0.512906\pi\)
\(464\) −13.8843 −0.644562
\(465\) 3.88428 0.180129
\(466\) −15.9733 −0.739951
\(467\) 18.1568 0.840194 0.420097 0.907479i \(-0.361996\pi\)
0.420097 + 0.907479i \(0.361996\pi\)
\(468\) 0 0
\(469\) 0.581449 0.0268488
\(470\) −8.28231 −0.382035
\(471\) −8.15676 −0.375843
\(472\) −2.10731 −0.0969967
\(473\) −72.3956 −3.32875
\(474\) −3.60197 −0.165444
\(475\) 34.1399 1.56645
\(476\) 2.92162 0.133912
\(477\) −6.00000 −0.274721
\(478\) −33.7503 −1.54370
\(479\) 30.4619 1.39184 0.695919 0.718120i \(-0.254997\pi\)
0.695919 + 0.718120i \(0.254997\pi\)
\(480\) 4.78765 0.218525
\(481\) 0 0
\(482\) 15.9155 0.724930
\(483\) 5.41855 0.246553
\(484\) 58.5090 2.65950
\(485\) −0.680346 −0.0308929
\(486\) 2.17009 0.0984371
\(487\) 0.313511 0.0142065 0.00710327 0.999975i \(-0.497739\pi\)
0.00710327 + 0.999975i \(0.497739\pi\)
\(488\) 19.4764 0.881656
\(489\) −4.58145 −0.207180
\(490\) −1.36910 −0.0618497
\(491\) 8.93600 0.403276 0.201638 0.979460i \(-0.435374\pi\)
0.201638 + 0.979460i \(0.435374\pi\)
\(492\) −3.28458 −0.148080
\(493\) −7.20394 −0.324449
\(494\) 0 0
\(495\) 3.60197 0.161896
\(496\) 12.7961 0.574560
\(497\) −4.81432 −0.215952
\(498\) 27.9649 1.25314
\(499\) −33.9877 −1.52150 −0.760750 0.649046i \(-0.775168\pi\)
−0.760750 + 0.649046i \(0.775168\pi\)
\(500\) 16.4124 0.733985
\(501\) −7.31124 −0.326642
\(502\) 29.6742 1.32442
\(503\) 33.5585 1.49630 0.748149 0.663530i \(-0.230943\pi\)
0.748149 + 0.663530i \(0.230943\pi\)
\(504\) −1.53919 −0.0685609
\(505\) −4.56463 −0.203123
\(506\) 67.1338 2.98446
\(507\) 0 0
\(508\) −17.1773 −0.762118
\(509\) −24.8287 −1.10051 −0.550256 0.834996i \(-0.685470\pi\)
−0.550256 + 0.834996i \(0.685470\pi\)
\(510\) 1.47641 0.0653765
\(511\) 3.81658 0.168836
\(512\) 22.1701 0.979789
\(513\) −7.41855 −0.327537
\(514\) 56.6947 2.50070
\(515\) 8.68035 0.382502
\(516\) 34.3545 1.51237
\(517\) −34.5380 −1.51898
\(518\) −7.41855 −0.325952
\(519\) −7.60197 −0.333689
\(520\) 0 0
\(521\) 20.2290 0.886248 0.443124 0.896460i \(-0.353870\pi\)
0.443124 + 0.896460i \(0.353870\pi\)
\(522\) 14.4969 0.634513
\(523\) −40.1666 −1.75636 −0.878181 0.478328i \(-0.841243\pi\)
−0.878181 + 0.478328i \(0.841243\pi\)
\(524\) −20.3668 −0.889729
\(525\) 4.60197 0.200846
\(526\) −66.7936 −2.91234
\(527\) 6.63931 0.289213
\(528\) 11.8660 0.516403
\(529\) 6.36069 0.276552
\(530\) 8.21461 0.356820
\(531\) −1.36910 −0.0594140
\(532\) 20.0989 0.871398
\(533\) 0 0
\(534\) 13.4680 0.582817
\(535\) 6.37137 0.275458
\(536\) −0.894960 −0.0386564
\(537\) −9.10504 −0.392911
\(538\) 11.7009 0.504460
\(539\) −5.70928 −0.245916
\(540\) −1.70928 −0.0735555
\(541\) −18.2101 −0.782912 −0.391456 0.920197i \(-0.628029\pi\)
−0.391456 + 0.920197i \(0.628029\pi\)
\(542\) 24.7792 1.06436
\(543\) −15.3607 −0.659190
\(544\) 8.18342 0.350861
\(545\) 11.3197 0.484881
\(546\) 0 0
\(547\) −6.32580 −0.270472 −0.135236 0.990813i \(-0.543179\pi\)
−0.135236 + 0.990813i \(0.543179\pi\)
\(548\) −12.1217 −0.517813
\(549\) 12.6537 0.540046
\(550\) 57.0166 2.43120
\(551\) −49.5585 −2.11126
\(552\) −8.34017 −0.354981
\(553\) 1.65983 0.0705830
\(554\) −28.5380 −1.21246
\(555\) −2.15676 −0.0915492
\(556\) −26.4391 −1.12127
\(557\) −10.6309 −0.450446 −0.225223 0.974307i \(-0.572311\pi\)
−0.225223 + 0.974307i \(0.572311\pi\)
\(558\) −13.3607 −0.565603
\(559\) 0 0
\(560\) −1.31124 −0.0554102
\(561\) 6.15676 0.259938
\(562\) −4.66209 −0.196659
\(563\) 37.0472 1.56135 0.780676 0.624936i \(-0.214875\pi\)
0.780676 + 0.624936i \(0.214875\pi\)
\(564\) 16.3896 0.690128
\(565\) −6.73820 −0.283478
\(566\) 26.4703 1.11263
\(567\) −1.00000 −0.0419961
\(568\) 7.41014 0.310923
\(569\) −25.7152 −1.07804 −0.539019 0.842293i \(-0.681205\pi\)
−0.539019 + 0.842293i \(0.681205\pi\)
\(570\) 10.1568 0.425420
\(571\) 12.4969 0.522980 0.261490 0.965206i \(-0.415786\pi\)
0.261490 + 0.965206i \(0.415786\pi\)
\(572\) 0 0
\(573\) −24.5692 −1.02639
\(574\) 2.63090 0.109812
\(575\) 24.9360 1.03990
\(576\) −12.3112 −0.512968
\(577\) 3.91548 0.163004 0.0815018 0.996673i \(-0.474028\pi\)
0.0815018 + 0.996673i \(0.474028\pi\)
\(578\) −34.3679 −1.42952
\(579\) −13.5753 −0.564170
\(580\) −11.4186 −0.474130
\(581\) −12.8865 −0.534624
\(582\) 2.34017 0.0970033
\(583\) 34.2557 1.41872
\(584\) −5.87444 −0.243086
\(585\) 0 0
\(586\) 15.5259 0.641367
\(587\) −6.94441 −0.286626 −0.143313 0.989677i \(-0.545776\pi\)
−0.143313 + 0.989677i \(0.545776\pi\)
\(588\) 2.70928 0.111729
\(589\) 45.6742 1.88197
\(590\) 1.87444 0.0771695
\(591\) 2.94441 0.121117
\(592\) −7.10504 −0.292015
\(593\) −29.9383 −1.22942 −0.614709 0.788754i \(-0.710726\pi\)
−0.614709 + 0.788754i \(0.710726\pi\)
\(594\) −12.3896 −0.508352
\(595\) −0.680346 −0.0278915
\(596\) −3.97721 −0.162913
\(597\) 10.5236 0.430702
\(598\) 0 0
\(599\) 43.2905 1.76880 0.884402 0.466726i \(-0.154567\pi\)
0.884402 + 0.466726i \(0.154567\pi\)
\(600\) −7.08330 −0.289174
\(601\) −31.2306 −1.27392 −0.636961 0.770896i \(-0.719809\pi\)
−0.636961 + 0.770896i \(0.719809\pi\)
\(602\) −27.5174 −1.12153
\(603\) −0.581449 −0.0236784
\(604\) −36.1978 −1.47287
\(605\) −13.6248 −0.553925
\(606\) 15.7009 0.637804
\(607\) 4.96266 0.201428 0.100714 0.994915i \(-0.467887\pi\)
0.100714 + 0.994915i \(0.467887\pi\)
\(608\) 56.2967 2.28313
\(609\) −6.68035 −0.270701
\(610\) −17.3242 −0.701436
\(611\) 0 0
\(612\) −2.92162 −0.118100
\(613\) 0.366835 0.0148163 0.00740816 0.999973i \(-0.497642\pi\)
0.00740816 + 0.999973i \(0.497642\pi\)
\(614\) −47.4017 −1.91298
\(615\) 0.764867 0.0308424
\(616\) 8.78765 0.354065
\(617\) 47.7237 1.92128 0.960641 0.277793i \(-0.0896030\pi\)
0.960641 + 0.277793i \(0.0896030\pi\)
\(618\) −29.8576 −1.20105
\(619\) −19.7854 −0.795242 −0.397621 0.917550i \(-0.630164\pi\)
−0.397621 + 0.917550i \(0.630164\pi\)
\(620\) 10.5236 0.422638
\(621\) −5.41855 −0.217439
\(622\) 16.3135 0.654112
\(623\) −6.20620 −0.248646
\(624\) 0 0
\(625\) 19.1880 0.767518
\(626\) 55.7731 2.22914
\(627\) 42.3545 1.69148
\(628\) −22.0989 −0.881842
\(629\) −3.68649 −0.146990
\(630\) 1.36910 0.0545463
\(631\) 20.1522 0.802247 0.401124 0.916024i \(-0.368620\pi\)
0.401124 + 0.916024i \(0.368620\pi\)
\(632\) −2.55479 −0.101624
\(633\) −3.02052 −0.120055
\(634\) −20.3318 −0.807477
\(635\) 4.00000 0.158735
\(636\) −16.2557 −0.644579
\(637\) 0 0
\(638\) −82.7670 −3.27678
\(639\) 4.81432 0.190451
\(640\) 7.28005 0.287769
\(641\) 12.1034 0.478057 0.239028 0.971013i \(-0.423171\pi\)
0.239028 + 0.971013i \(0.423171\pi\)
\(642\) −21.9155 −0.864935
\(643\) −20.6803 −0.815553 −0.407777 0.913082i \(-0.633696\pi\)
−0.407777 + 0.913082i \(0.633696\pi\)
\(644\) 14.6803 0.578487
\(645\) −8.00000 −0.315000
\(646\) 17.3607 0.683047
\(647\) −40.8781 −1.60709 −0.803543 0.595247i \(-0.797054\pi\)
−0.803543 + 0.595247i \(0.797054\pi\)
\(648\) 1.53919 0.0604650
\(649\) 7.81658 0.306828
\(650\) 0 0
\(651\) 6.15676 0.241302
\(652\) −12.4124 −0.486107
\(653\) 6.31351 0.247067 0.123533 0.992340i \(-0.460577\pi\)
0.123533 + 0.992340i \(0.460577\pi\)
\(654\) −38.9360 −1.52252
\(655\) 4.74274 0.185314
\(656\) 2.51971 0.0983783
\(657\) −3.81658 −0.148899
\(658\) −13.1278 −0.511776
\(659\) −20.9360 −0.815551 −0.407775 0.913082i \(-0.633695\pi\)
−0.407775 + 0.913082i \(0.633695\pi\)
\(660\) 9.75872 0.379858
\(661\) 22.2245 0.864431 0.432216 0.901770i \(-0.357732\pi\)
0.432216 + 0.901770i \(0.357732\pi\)
\(662\) −45.6742 −1.77518
\(663\) 0 0
\(664\) 19.8348 0.769741
\(665\) −4.68035 −0.181496
\(666\) 7.41855 0.287463
\(667\) −36.1978 −1.40158
\(668\) −19.8082 −0.766401
\(669\) −22.6225 −0.874636
\(670\) 0.796064 0.0307546
\(671\) −72.2434 −2.78892
\(672\) 7.58864 0.292738
\(673\) 1.15061 0.0443528 0.0221764 0.999754i \(-0.492940\pi\)
0.0221764 + 0.999754i \(0.492940\pi\)
\(674\) −33.5174 −1.29104
\(675\) −4.60197 −0.177130
\(676\) 0 0
\(677\) −33.5897 −1.29096 −0.645478 0.763779i \(-0.723342\pi\)
−0.645478 + 0.763779i \(0.723342\pi\)
\(678\) 23.1773 0.890118
\(679\) −1.07838 −0.0413843
\(680\) 1.04718 0.0401576
\(681\) 22.9444 0.879232
\(682\) 76.2799 2.92091
\(683\) −25.1110 −0.960846 −0.480423 0.877037i \(-0.659517\pi\)
−0.480423 + 0.877037i \(0.659517\pi\)
\(684\) −20.0989 −0.768501
\(685\) 2.82273 0.107851
\(686\) −2.17009 −0.0828543
\(687\) 21.5441 0.821959
\(688\) −26.3545 −1.00476
\(689\) 0 0
\(690\) 7.41855 0.282419
\(691\) 6.30898 0.240005 0.120002 0.992774i \(-0.461710\pi\)
0.120002 + 0.992774i \(0.461710\pi\)
\(692\) −20.5958 −0.782936
\(693\) 5.70928 0.216877
\(694\) 47.1605 1.79019
\(695\) 6.15676 0.233539
\(696\) 10.2823 0.389750
\(697\) 1.30737 0.0495201
\(698\) −12.2823 −0.464892
\(699\) −7.36069 −0.278407
\(700\) 12.4680 0.471246
\(701\) −45.8843 −1.73303 −0.866513 0.499155i \(-0.833644\pi\)
−0.866513 + 0.499155i \(0.833644\pi\)
\(702\) 0 0
\(703\) −25.3607 −0.956497
\(704\) 70.2883 2.64909
\(705\) −3.81658 −0.143741
\(706\) −73.1832 −2.75429
\(707\) −7.23513 −0.272105
\(708\) −3.70928 −0.139403
\(709\) 46.4534 1.74460 0.872298 0.488975i \(-0.162629\pi\)
0.872298 + 0.488975i \(0.162629\pi\)
\(710\) −6.59129 −0.247367
\(711\) −1.65983 −0.0622484
\(712\) 9.55252 0.357996
\(713\) 33.3607 1.24937
\(714\) 2.34017 0.0875788
\(715\) 0 0
\(716\) −24.6681 −0.921889
\(717\) −15.5525 −0.580819
\(718\) −40.5874 −1.51471
\(719\) 7.20394 0.268661 0.134331 0.990937i \(-0.457112\pi\)
0.134331 + 0.990937i \(0.457112\pi\)
\(720\) 1.31124 0.0488672
\(721\) 13.7587 0.512402
\(722\) 78.1988 2.91026
\(723\) 7.33403 0.272756
\(724\) −41.6163 −1.54666
\(725\) −30.7427 −1.14176
\(726\) 46.8648 1.73932
\(727\) −13.6742 −0.507148 −0.253574 0.967316i \(-0.581606\pi\)
−0.253574 + 0.967316i \(0.581606\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 5.22529 0.193397
\(731\) −13.6742 −0.505759
\(732\) 34.2823 1.26711
\(733\) 7.70086 0.284438 0.142219 0.989835i \(-0.454576\pi\)
0.142219 + 0.989835i \(0.454576\pi\)
\(734\) 21.8576 0.806779
\(735\) −0.630898 −0.0232710
\(736\) 41.1194 1.51568
\(737\) 3.31965 0.122281
\(738\) −2.63090 −0.0968447
\(739\) 24.5814 0.904243 0.452122 0.891956i \(-0.350667\pi\)
0.452122 + 0.891956i \(0.350667\pi\)
\(740\) −5.84324 −0.214802
\(741\) 0 0
\(742\) 13.0205 0.477998
\(743\) −24.3773 −0.894318 −0.447159 0.894455i \(-0.647564\pi\)
−0.447159 + 0.894455i \(0.647564\pi\)
\(744\) −9.47641 −0.347422
\(745\) 0.926157 0.0339318
\(746\) 9.20394 0.336980
\(747\) 12.8865 0.471494
\(748\) 16.6803 0.609894
\(749\) 10.0989 0.369006
\(750\) 13.1461 0.480027
\(751\) 24.6803 0.900599 0.450299 0.892878i \(-0.351317\pi\)
0.450299 + 0.892878i \(0.351317\pi\)
\(752\) −12.5730 −0.458492
\(753\) 13.6742 0.498316
\(754\) 0 0
\(755\) 8.42923 0.306771
\(756\) −2.70928 −0.0985354
\(757\) −41.3172 −1.50170 −0.750850 0.660473i \(-0.770356\pi\)
−0.750850 + 0.660473i \(0.770356\pi\)
\(758\) 43.8310 1.59201
\(759\) 30.9360 1.12291
\(760\) 7.20394 0.261314
\(761\) 15.9506 0.578207 0.289104 0.957298i \(-0.406643\pi\)
0.289104 + 0.957298i \(0.406643\pi\)
\(762\) −13.7587 −0.498426
\(763\) 17.9421 0.649549
\(764\) −66.5646 −2.40822
\(765\) 0.680346 0.0245980
\(766\) −15.2846 −0.552254
\(767\) 0 0
\(768\) −0.418551 −0.0151031
\(769\) 6.22446 0.224460 0.112230 0.993682i \(-0.464201\pi\)
0.112230 + 0.993682i \(0.464201\pi\)
\(770\) −7.81658 −0.281690
\(771\) 26.1256 0.940889
\(772\) −36.7792 −1.32371
\(773\) 39.1545 1.40829 0.704145 0.710057i \(-0.251331\pi\)
0.704145 + 0.710057i \(0.251331\pi\)
\(774\) 27.5174 0.989094
\(775\) 28.3332 1.01776
\(776\) 1.65983 0.0595843
\(777\) −3.41855 −0.122640
\(778\) 8.34017 0.299010
\(779\) 8.99386 0.322238
\(780\) 0 0
\(781\) −27.4863 −0.983535
\(782\) 12.6803 0.453448
\(783\) 6.68035 0.238736
\(784\) −2.07838 −0.0742278
\(785\) 5.14608 0.183671
\(786\) −16.3135 −0.581884
\(787\) −32.2967 −1.15125 −0.575626 0.817713i \(-0.695242\pi\)
−0.575626 + 0.817713i \(0.695242\pi\)
\(788\) 7.97721 0.284176
\(789\) −30.7792 −1.09577
\(790\) 2.27247 0.0808510
\(791\) −10.6803 −0.379749
\(792\) −8.78765 −0.312256
\(793\) 0 0
\(794\) −31.3340 −1.11200
\(795\) 3.78539 0.134254
\(796\) 28.5113 1.01056
\(797\) 30.2700 1.07222 0.536110 0.844148i \(-0.319893\pi\)
0.536110 + 0.844148i \(0.319893\pi\)
\(798\) 16.0989 0.569895
\(799\) −6.52359 −0.230788
\(800\) 34.9227 1.23470
\(801\) 6.20620 0.219285
\(802\) 21.3424 0.753628
\(803\) 21.7899 0.768950
\(804\) −1.57531 −0.0555568
\(805\) −3.41855 −0.120488
\(806\) 0 0
\(807\) 5.39189 0.189803
\(808\) 11.1362 0.391771
\(809\) 39.5585 1.39080 0.695401 0.718622i \(-0.255227\pi\)
0.695401 + 0.718622i \(0.255227\pi\)
\(810\) −1.36910 −0.0481054
\(811\) 34.4079 1.20822 0.604112 0.796899i \(-0.293528\pi\)
0.604112 + 0.796899i \(0.293528\pi\)
\(812\) −18.0989 −0.635147
\(813\) 11.4186 0.400466
\(814\) −42.3545 −1.48453
\(815\) 2.89043 0.101247
\(816\) 2.24128 0.0784604
\(817\) −94.0698 −3.29109
\(818\) −16.3812 −0.572756
\(819\) 0 0
\(820\) 2.07223 0.0723656
\(821\) 7.89269 0.275457 0.137728 0.990470i \(-0.456020\pi\)
0.137728 + 0.990470i \(0.456020\pi\)
\(822\) −9.70928 −0.338650
\(823\) −29.6742 −1.03438 −0.517189 0.855871i \(-0.673022\pi\)
−0.517189 + 0.855871i \(0.673022\pi\)
\(824\) −21.1773 −0.737745
\(825\) 26.2739 0.914740
\(826\) 2.97107 0.103377
\(827\) −14.1918 −0.493498 −0.246749 0.969079i \(-0.579362\pi\)
−0.246749 + 0.969079i \(0.579362\pi\)
\(828\) −14.6803 −0.510177
\(829\) −3.71315 −0.128963 −0.0644815 0.997919i \(-0.520539\pi\)
−0.0644815 + 0.997919i \(0.520539\pi\)
\(830\) −17.6430 −0.612398
\(831\) −13.1506 −0.456190
\(832\) 0 0
\(833\) −1.07838 −0.0373636
\(834\) −21.1773 −0.733309
\(835\) 4.61265 0.159627
\(836\) 114.750 3.96872
\(837\) −6.15676 −0.212809
\(838\) −76.2799 −2.63504
\(839\) 48.6719 1.68034 0.840171 0.542322i \(-0.182455\pi\)
0.840171 + 0.542322i \(0.182455\pi\)
\(840\) 0.971071 0.0335051
\(841\) 15.6270 0.538863
\(842\) 55.7152 1.92007
\(843\) −2.14834 −0.0739929
\(844\) −8.18342 −0.281685
\(845\) 0 0
\(846\) 13.1278 0.451344
\(847\) −21.5958 −0.742041
\(848\) 12.4703 0.428231
\(849\) 12.1978 0.418627
\(850\) 10.7694 0.369387
\(851\) −18.5236 −0.634981
\(852\) 13.0433 0.446856
\(853\) 1.92777 0.0660054 0.0330027 0.999455i \(-0.489493\pi\)
0.0330027 + 0.999455i \(0.489493\pi\)
\(854\) −27.4596 −0.939648
\(855\) 4.68035 0.160064
\(856\) −15.5441 −0.531287
\(857\) −24.3980 −0.833421 −0.416710 0.909039i \(-0.636817\pi\)
−0.416710 + 0.909039i \(0.636817\pi\)
\(858\) 0 0
\(859\) −15.1506 −0.516932 −0.258466 0.966020i \(-0.583217\pi\)
−0.258466 + 0.966020i \(0.583217\pi\)
\(860\) −21.6742 −0.739084
\(861\) 1.21235 0.0413167
\(862\) 35.2267 1.19983
\(863\) 49.7380 1.69310 0.846551 0.532308i \(-0.178675\pi\)
0.846551 + 0.532308i \(0.178675\pi\)
\(864\) −7.58864 −0.258171
\(865\) 4.79606 0.163071
\(866\) 11.6910 0.397277
\(867\) −15.8371 −0.537856
\(868\) 16.6803 0.566168
\(869\) 9.47641 0.321465
\(870\) −9.14608 −0.310081
\(871\) 0 0
\(872\) −27.6163 −0.935207
\(873\) 1.07838 0.0364976
\(874\) 87.2327 2.95069
\(875\) −6.05786 −0.204793
\(876\) −10.3402 −0.349362
\(877\) −9.72753 −0.328475 −0.164238 0.986421i \(-0.552516\pi\)
−0.164238 + 0.986421i \(0.552516\pi\)
\(878\) 22.2245 0.750039
\(879\) 7.15449 0.241315
\(880\) −7.48625 −0.252361
\(881\) −6.12556 −0.206375 −0.103188 0.994662i \(-0.532904\pi\)
−0.103188 + 0.994662i \(0.532904\pi\)
\(882\) 2.17009 0.0730706
\(883\) −1.03281 −0.0347567 −0.0173783 0.999849i \(-0.505532\pi\)
−0.0173783 + 0.999849i \(0.505532\pi\)
\(884\) 0 0
\(885\) 0.863763 0.0290351
\(886\) 34.5958 1.16227
\(887\) 39.0882 1.31245 0.656227 0.754564i \(-0.272151\pi\)
0.656227 + 0.754564i \(0.272151\pi\)
\(888\) 5.26180 0.176574
\(889\) 6.34017 0.212643
\(890\) −8.49693 −0.284818
\(891\) −5.70928 −0.191268
\(892\) −61.2905 −2.05216
\(893\) −44.8781 −1.50179
\(894\) −3.18568 −0.106545
\(895\) 5.74435 0.192012
\(896\) 11.5392 0.385497
\(897\) 0 0
\(898\) −28.6453 −0.955905
\(899\) −41.1293 −1.37174
\(900\) −12.4680 −0.415600
\(901\) 6.47027 0.215556
\(902\) 15.0205 0.500129
\(903\) −12.6803 −0.421975
\(904\) 16.4391 0.546755
\(905\) 9.69102 0.322141
\(906\) −28.9939 −0.963256
\(907\) −45.3751 −1.50665 −0.753327 0.657646i \(-0.771552\pi\)
−0.753327 + 0.657646i \(0.771552\pi\)
\(908\) 62.1627 2.06294
\(909\) 7.23513 0.239974
\(910\) 0 0
\(911\) −24.3090 −0.805392 −0.402696 0.915334i \(-0.631927\pi\)
−0.402696 + 0.915334i \(0.631927\pi\)
\(912\) 15.4186 0.510559
\(913\) −73.5729 −2.43491
\(914\) −39.7321 −1.31422
\(915\) −7.98318 −0.263916
\(916\) 58.3689 1.92856
\(917\) 7.51745 0.248248
\(918\) −2.34017 −0.0772372
\(919\) 55.8453 1.84217 0.921084 0.389364i \(-0.127305\pi\)
0.921084 + 0.389364i \(0.127305\pi\)
\(920\) 5.26180 0.173476
\(921\) −21.8432 −0.719759
\(922\) 41.1012 1.35359
\(923\) 0 0
\(924\) 15.4680 0.508860
\(925\) −15.7321 −0.517267
\(926\) −3.78539 −0.124395
\(927\) −13.7587 −0.451896
\(928\) −50.6947 −1.66414
\(929\) 39.4680 1.29490 0.647452 0.762107i \(-0.275835\pi\)
0.647452 + 0.762107i \(0.275835\pi\)
\(930\) 8.42923 0.276405
\(931\) −7.41855 −0.243133
\(932\) −19.9421 −0.653227
\(933\) 7.51745 0.246110
\(934\) 39.4017 1.28926
\(935\) −3.88428 −0.127030
\(936\) 0 0
\(937\) −22.2122 −0.725640 −0.362820 0.931859i \(-0.618186\pi\)
−0.362820 + 0.931859i \(0.618186\pi\)
\(938\) 1.26180 0.0411991
\(939\) 25.7009 0.838716
\(940\) −10.3402 −0.337259
\(941\) 32.1483 1.04801 0.524003 0.851716i \(-0.324438\pi\)
0.524003 + 0.851716i \(0.324438\pi\)
\(942\) −17.7009 −0.576725
\(943\) 6.56916 0.213921
\(944\) 2.84551 0.0926135
\(945\) 0.630898 0.0205231
\(946\) −157.105 −5.10791
\(947\) 19.2390 0.625184 0.312592 0.949888i \(-0.398803\pi\)
0.312592 + 0.949888i \(0.398803\pi\)
\(948\) −4.49693 −0.146053
\(949\) 0 0
\(950\) 74.0866 2.40369
\(951\) −9.36910 −0.303814
\(952\) 1.65983 0.0537953
\(953\) −54.3423 −1.76032 −0.880159 0.474678i \(-0.842564\pi\)
−0.880159 + 0.474678i \(0.842564\pi\)
\(954\) −13.0205 −0.421555
\(955\) 15.5006 0.501588
\(956\) −42.1361 −1.36278
\(957\) −38.1399 −1.23289
\(958\) 66.1049 2.13575
\(959\) 4.47414 0.144478
\(960\) 7.76713 0.250683
\(961\) 6.90564 0.222763
\(962\) 0 0
\(963\) −10.0989 −0.325432
\(964\) 19.8699 0.639967
\(965\) 8.56463 0.275705
\(966\) 11.7587 0.378331
\(967\) 1.43084 0.0460126 0.0230063 0.999735i \(-0.492676\pi\)
0.0230063 + 0.999735i \(0.492676\pi\)
\(968\) 33.2401 1.06838
\(969\) 8.00000 0.256997
\(970\) −1.47641 −0.0474047
\(971\) −25.2450 −0.810150 −0.405075 0.914284i \(-0.632755\pi\)
−0.405075 + 0.914284i \(0.632755\pi\)
\(972\) 2.70928 0.0869000
\(973\) 9.75872 0.312850
\(974\) 0.680346 0.0217997
\(975\) 0 0
\(976\) −26.2991 −0.841815
\(977\) 7.31124 0.233907 0.116954 0.993137i \(-0.462687\pi\)
0.116954 + 0.993137i \(0.462687\pi\)
\(978\) −9.94214 −0.317915
\(979\) −35.4329 −1.13244
\(980\) −1.70928 −0.0546008
\(981\) −17.9421 −0.572848
\(982\) 19.3919 0.618820
\(983\) −21.0556 −0.671569 −0.335785 0.941939i \(-0.609001\pi\)
−0.335785 + 0.941939i \(0.609001\pi\)
\(984\) −1.86603 −0.0594869
\(985\) −1.85762 −0.0591887
\(986\) −15.6332 −0.497861
\(987\) −6.04945 −0.192556
\(988\) 0 0
\(989\) −68.7091 −2.18482
\(990\) 7.81658 0.248427
\(991\) 42.0410 1.33548 0.667739 0.744396i \(-0.267262\pi\)
0.667739 + 0.744396i \(0.267262\pi\)
\(992\) 46.7214 1.48341
\(993\) −21.0472 −0.667912
\(994\) −10.4475 −0.331374
\(995\) −6.63931 −0.210480
\(996\) 34.9132 1.10627
\(997\) −9.20394 −0.291492 −0.145746 0.989322i \(-0.546558\pi\)
−0.145746 + 0.989322i \(0.546558\pi\)
\(998\) −73.7563 −2.33471
\(999\) 3.41855 0.108158
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 3549.2.a.q.1.3 3
13.5 odd 4 273.2.c.b.64.1 6
13.8 odd 4 273.2.c.b.64.6 yes 6
13.12 even 2 3549.2.a.k.1.1 3
39.5 even 4 819.2.c.c.64.6 6
39.8 even 4 819.2.c.c.64.1 6
52.31 even 4 4368.2.h.o.337.4 6
52.47 even 4 4368.2.h.o.337.3 6
91.34 even 4 1911.2.c.h.883.6 6
91.83 even 4 1911.2.c.h.883.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.c.b.64.1 6 13.5 odd 4
273.2.c.b.64.6 yes 6 13.8 odd 4
819.2.c.c.64.1 6 39.8 even 4
819.2.c.c.64.6 6 39.5 even 4
1911.2.c.h.883.1 6 91.83 even 4
1911.2.c.h.883.6 6 91.34 even 4
3549.2.a.k.1.1 3 13.12 even 2
3549.2.a.q.1.3 3 1.1 even 1 trivial
4368.2.h.o.337.3 6 52.47 even 4
4368.2.h.o.337.4 6 52.31 even 4