Properties

Label 3174.2.a.ba.1.4
Level $3174$
Weight $2$
Character 3174.1
Self dual yes
Analytic conductor $25.345$
Analytic rank $0$
Dimension $5$
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] = [3174,2,Mod(1,3174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3174, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.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: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.830830\) of defining polynomial
Character \(\chi\) \(=\) 3174.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.324635 q^{5} -1.00000 q^{6} -2.43232 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.324635 q^{5} -1.00000 q^{6} -2.43232 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.324635 q^{10} +3.40009 q^{11} -1.00000 q^{12} +0.269342 q^{13} -2.43232 q^{14} -0.324635 q^{15} +1.00000 q^{16} +4.84815 q^{17} +1.00000 q^{18} +4.02509 q^{19} +0.324635 q^{20} +2.43232 q^{21} +3.40009 q^{22} -1.00000 q^{24} -4.89461 q^{25} +0.269342 q^{26} -1.00000 q^{27} -2.43232 q^{28} +2.00935 q^{29} -0.324635 q^{30} -9.50557 q^{31} +1.00000 q^{32} -3.40009 q^{33} +4.84815 q^{34} -0.789617 q^{35} +1.00000 q^{36} -2.01732 q^{37} +4.02509 q^{38} -0.269342 q^{39} +0.324635 q^{40} -2.60502 q^{41} +2.43232 q^{42} +3.93559 q^{43} +3.40009 q^{44} +0.324635 q^{45} +5.19874 q^{47} -1.00000 q^{48} -1.08380 q^{49} -4.89461 q^{50} -4.84815 q^{51} +0.269342 q^{52} -1.08001 q^{53} -1.00000 q^{54} +1.10379 q^{55} -2.43232 q^{56} -4.02509 q^{57} +2.00935 q^{58} +9.90548 q^{59} -0.324635 q^{60} +4.35062 q^{61} -9.50557 q^{62} -2.43232 q^{63} +1.00000 q^{64} +0.0874378 q^{65} -3.40009 q^{66} +10.2730 q^{67} +4.84815 q^{68} -0.789617 q^{70} +11.4304 q^{71} +1.00000 q^{72} -1.13935 q^{73} -2.01732 q^{74} +4.89461 q^{75} +4.02509 q^{76} -8.27012 q^{77} -0.269342 q^{78} +16.0547 q^{79} +0.324635 q^{80} +1.00000 q^{81} -2.60502 q^{82} -1.05529 q^{83} +2.43232 q^{84} +1.57388 q^{85} +3.93559 q^{86} -2.00935 q^{87} +3.40009 q^{88} -2.74085 q^{89} +0.324635 q^{90} -0.655127 q^{91} +9.50557 q^{93} +5.19874 q^{94} +1.30668 q^{95} -1.00000 q^{96} -13.6079 q^{97} -1.08380 q^{98} +3.40009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} + 11 q^{7} + 5 q^{8} + 5 q^{9} - q^{10} + 11 q^{11} - 5 q^{12} + 12 q^{13} + 11 q^{14} + q^{15} + 5 q^{16} - q^{17} + 5 q^{18} + 15 q^{19} - q^{20} - 11 q^{21} + 11 q^{22} - 5 q^{24} + 6 q^{25} + 12 q^{26} - 5 q^{27} + 11 q^{28} + q^{29} + q^{30} - 18 q^{31} + 5 q^{32} - 11 q^{33} - q^{34} - 11 q^{35} + 5 q^{36} + 10 q^{37} + 15 q^{38} - 12 q^{39} - q^{40} - 16 q^{41} - 11 q^{42} + 18 q^{43} + 11 q^{44} - q^{45} - 4 q^{47} - 5 q^{48} + 20 q^{49} + 6 q^{50} + q^{51} + 12 q^{52} - q^{53} - 5 q^{54} - 22 q^{55} + 11 q^{56} - 15 q^{57} + q^{58} + 2 q^{59} + q^{60} - q^{61} - 18 q^{62} + 11 q^{63} + 5 q^{64} + 24 q^{65} - 11 q^{66} + 29 q^{67} - q^{68} - 11 q^{70} - 11 q^{71} + 5 q^{72} - 8 q^{73} + 10 q^{74} - 6 q^{75} + 15 q^{76} + 11 q^{77} - 12 q^{78} + 40 q^{79} - q^{80} + 5 q^{81} - 16 q^{82} + 8 q^{83} - 11 q^{84} - 13 q^{85} + 18 q^{86} - q^{87} + 11 q^{88} + 2 q^{89} - q^{90} + 33 q^{91} + 18 q^{93} - 4 q^{94} - 3 q^{95} - 5 q^{96} + 17 q^{97} + 20 q^{98} + 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0.324635 0.145181 0.0725905 0.997362i \(-0.476873\pi\)
0.0725905 + 0.997362i \(0.476873\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.43232 −0.919332 −0.459666 0.888092i \(-0.652031\pi\)
−0.459666 + 0.888092i \(0.652031\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.324635 0.102659
\(11\) 3.40009 1.02517 0.512583 0.858638i \(-0.328689\pi\)
0.512583 + 0.858638i \(0.328689\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0.269342 0.0747021 0.0373510 0.999302i \(-0.488108\pi\)
0.0373510 + 0.999302i \(0.488108\pi\)
\(14\) −2.43232 −0.650066
\(15\) −0.324635 −0.0838203
\(16\) 1.00000 0.250000
\(17\) 4.84815 1.17585 0.587925 0.808916i \(-0.299945\pi\)
0.587925 + 0.808916i \(0.299945\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.02509 0.923419 0.461710 0.887031i \(-0.347236\pi\)
0.461710 + 0.887031i \(0.347236\pi\)
\(20\) 0.324635 0.0725905
\(21\) 2.43232 0.530776
\(22\) 3.40009 0.724901
\(23\) 0 0
\(24\) −1.00000 −0.204124
\(25\) −4.89461 −0.978922
\(26\) 0.269342 0.0528223
\(27\) −1.00000 −0.192450
\(28\) −2.43232 −0.459666
\(29\) 2.00935 0.373128 0.186564 0.982443i \(-0.440265\pi\)
0.186564 + 0.982443i \(0.440265\pi\)
\(30\) −0.324635 −0.0592699
\(31\) −9.50557 −1.70725 −0.853625 0.520888i \(-0.825601\pi\)
−0.853625 + 0.520888i \(0.825601\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.40009 −0.591880
\(34\) 4.84815 0.831451
\(35\) −0.789617 −0.133470
\(36\) 1.00000 0.166667
\(37\) −2.01732 −0.331645 −0.165823 0.986156i \(-0.553028\pi\)
−0.165823 + 0.986156i \(0.553028\pi\)
\(38\) 4.02509 0.652956
\(39\) −0.269342 −0.0431293
\(40\) 0.324635 0.0513293
\(41\) −2.60502 −0.406836 −0.203418 0.979092i \(-0.565205\pi\)
−0.203418 + 0.979092i \(0.565205\pi\)
\(42\) 2.43232 0.375316
\(43\) 3.93559 0.600172 0.300086 0.953912i \(-0.402985\pi\)
0.300086 + 0.953912i \(0.402985\pi\)
\(44\) 3.40009 0.512583
\(45\) 0.324635 0.0483937
\(46\) 0 0
\(47\) 5.19874 0.758314 0.379157 0.925332i \(-0.376214\pi\)
0.379157 + 0.925332i \(0.376214\pi\)
\(48\) −1.00000 −0.144338
\(49\) −1.08380 −0.154829
\(50\) −4.89461 −0.692203
\(51\) −4.84815 −0.678877
\(52\) 0.269342 0.0373510
\(53\) −1.08001 −0.148351 −0.0741754 0.997245i \(-0.523632\pi\)
−0.0741754 + 0.997245i \(0.523632\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.10379 0.148835
\(56\) −2.43232 −0.325033
\(57\) −4.02509 −0.533136
\(58\) 2.00935 0.263841
\(59\) 9.90548 1.28958 0.644792 0.764358i \(-0.276944\pi\)
0.644792 + 0.764358i \(0.276944\pi\)
\(60\) −0.324635 −0.0419102
\(61\) 4.35062 0.557040 0.278520 0.960430i \(-0.410156\pi\)
0.278520 + 0.960430i \(0.410156\pi\)
\(62\) −9.50557 −1.20721
\(63\) −2.43232 −0.306444
\(64\) 1.00000 0.125000
\(65\) 0.0874378 0.0108453
\(66\) −3.40009 −0.418522
\(67\) 10.2730 1.25504 0.627521 0.778600i \(-0.284070\pi\)
0.627521 + 0.778600i \(0.284070\pi\)
\(68\) 4.84815 0.587925
\(69\) 0 0
\(70\) −0.789617 −0.0943772
\(71\) 11.4304 1.35654 0.678268 0.734814i \(-0.262731\pi\)
0.678268 + 0.734814i \(0.262731\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.13935 −0.133351 −0.0666756 0.997775i \(-0.521239\pi\)
−0.0666756 + 0.997775i \(0.521239\pi\)
\(74\) −2.01732 −0.234509
\(75\) 4.89461 0.565181
\(76\) 4.02509 0.461710
\(77\) −8.27012 −0.942467
\(78\) −0.269342 −0.0304970
\(79\) 16.0547 1.80629 0.903145 0.429335i \(-0.141252\pi\)
0.903145 + 0.429335i \(0.141252\pi\)
\(80\) 0.324635 0.0362953
\(81\) 1.00000 0.111111
\(82\) −2.60502 −0.287676
\(83\) −1.05529 −0.115833 −0.0579167 0.998321i \(-0.518446\pi\)
−0.0579167 + 0.998321i \(0.518446\pi\)
\(84\) 2.43232 0.265388
\(85\) 1.57388 0.170711
\(86\) 3.93559 0.424385
\(87\) −2.00935 −0.215425
\(88\) 3.40009 0.362451
\(89\) −2.74085 −0.290529 −0.145265 0.989393i \(-0.546403\pi\)
−0.145265 + 0.989393i \(0.546403\pi\)
\(90\) 0.324635 0.0342195
\(91\) −0.655127 −0.0686760
\(92\) 0 0
\(93\) 9.50557 0.985681
\(94\) 5.19874 0.536209
\(95\) 1.30668 0.134063
\(96\) −1.00000 −0.102062
\(97\) −13.6079 −1.38167 −0.690836 0.723011i \(-0.742758\pi\)
−0.690836 + 0.723011i \(0.742758\pi\)
\(98\) −1.08380 −0.109481
\(99\) 3.40009 0.341722
\(100\) −4.89461 −0.489461
\(101\) 15.9029 1.58240 0.791198 0.611560i \(-0.209458\pi\)
0.791198 + 0.611560i \(0.209458\pi\)
\(102\) −4.84815 −0.480038
\(103\) 13.9721 1.37672 0.688358 0.725371i \(-0.258332\pi\)
0.688358 + 0.725371i \(0.258332\pi\)
\(104\) 0.269342 0.0264112
\(105\) 0.789617 0.0770587
\(106\) −1.08001 −0.104900
\(107\) 10.7919 1.04329 0.521644 0.853163i \(-0.325319\pi\)
0.521644 + 0.853163i \(0.325319\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.3580 1.56681 0.783404 0.621513i \(-0.213482\pi\)
0.783404 + 0.621513i \(0.213482\pi\)
\(110\) 1.10379 0.105242
\(111\) 2.01732 0.191476
\(112\) −2.43232 −0.229833
\(113\) 19.9598 1.87766 0.938830 0.344382i \(-0.111911\pi\)
0.938830 + 0.344382i \(0.111911\pi\)
\(114\) −4.02509 −0.376984
\(115\) 0 0
\(116\) 2.00935 0.186564
\(117\) 0.269342 0.0249007
\(118\) 9.90548 0.911873
\(119\) −11.7923 −1.08100
\(120\) −0.324635 −0.0296350
\(121\) 0.560608 0.0509643
\(122\) 4.35062 0.393887
\(123\) 2.60502 0.234887
\(124\) −9.50557 −0.853625
\(125\) −3.21214 −0.287302
\(126\) −2.43232 −0.216689
\(127\) −5.44618 −0.483270 −0.241635 0.970367i \(-0.577684\pi\)
−0.241635 + 0.970367i \(0.577684\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.93559 −0.346509
\(130\) 0.0874378 0.00766881
\(131\) 15.3071 1.33739 0.668694 0.743537i \(-0.266853\pi\)
0.668694 + 0.743537i \(0.266853\pi\)
\(132\) −3.40009 −0.295940
\(133\) −9.79032 −0.848929
\(134\) 10.2730 0.887448
\(135\) −0.324635 −0.0279401
\(136\) 4.84815 0.415725
\(137\) −17.0753 −1.45884 −0.729419 0.684067i \(-0.760209\pi\)
−0.729419 + 0.684067i \(0.760209\pi\)
\(138\) 0 0
\(139\) 16.2779 1.38067 0.690337 0.723488i \(-0.257462\pi\)
0.690337 + 0.723488i \(0.257462\pi\)
\(140\) −0.789617 −0.0667348
\(141\) −5.19874 −0.437813
\(142\) 11.4304 0.959216
\(143\) 0.915787 0.0765820
\(144\) 1.00000 0.0833333
\(145\) 0.652306 0.0541711
\(146\) −1.13935 −0.0942935
\(147\) 1.08380 0.0893907
\(148\) −2.01732 −0.165823
\(149\) −1.63045 −0.133571 −0.0667857 0.997767i \(-0.521274\pi\)
−0.0667857 + 0.997767i \(0.521274\pi\)
\(150\) 4.89461 0.399643
\(151\) −11.7663 −0.957530 −0.478765 0.877943i \(-0.658915\pi\)
−0.478765 + 0.877943i \(0.658915\pi\)
\(152\) 4.02509 0.326478
\(153\) 4.84815 0.391950
\(154\) −8.27012 −0.666425
\(155\) −3.08584 −0.247860
\(156\) −0.269342 −0.0215646
\(157\) 10.0065 0.798607 0.399303 0.916819i \(-0.369252\pi\)
0.399303 + 0.916819i \(0.369252\pi\)
\(158\) 16.0547 1.27724
\(159\) 1.08001 0.0856504
\(160\) 0.324635 0.0256646
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −15.9393 −1.24846 −0.624230 0.781241i \(-0.714587\pi\)
−0.624230 + 0.781241i \(0.714587\pi\)
\(164\) −2.60502 −0.203418
\(165\) −1.10379 −0.0859297
\(166\) −1.05529 −0.0819066
\(167\) −9.53697 −0.737993 −0.368997 0.929431i \(-0.620299\pi\)
−0.368997 + 0.929431i \(0.620299\pi\)
\(168\) 2.43232 0.187658
\(169\) −12.9275 −0.994420
\(170\) 1.57388 0.120711
\(171\) 4.02509 0.307806
\(172\) 3.93559 0.300086
\(173\) −24.7305 −1.88022 −0.940112 0.340865i \(-0.889280\pi\)
−0.940112 + 0.340865i \(0.889280\pi\)
\(174\) −2.00935 −0.152329
\(175\) 11.9053 0.899954
\(176\) 3.40009 0.256291
\(177\) −9.90548 −0.744541
\(178\) −2.74085 −0.205435
\(179\) −26.3484 −1.96937 −0.984684 0.174348i \(-0.944218\pi\)
−0.984684 + 0.174348i \(0.944218\pi\)
\(180\) 0.324635 0.0241968
\(181\) −18.1768 −1.35107 −0.675535 0.737328i \(-0.736087\pi\)
−0.675535 + 0.737328i \(0.736087\pi\)
\(182\) −0.655127 −0.0485613
\(183\) −4.35062 −0.321607
\(184\) 0 0
\(185\) −0.654892 −0.0481486
\(186\) 9.50557 0.696982
\(187\) 16.4841 1.20544
\(188\) 5.19874 0.379157
\(189\) 2.43232 0.176925
\(190\) 1.30668 0.0947969
\(191\) 7.85270 0.568201 0.284101 0.958794i \(-0.408305\pi\)
0.284101 + 0.958794i \(0.408305\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 23.5784 1.69721 0.848604 0.529028i \(-0.177443\pi\)
0.848604 + 0.529028i \(0.177443\pi\)
\(194\) −13.6079 −0.976990
\(195\) −0.0874378 −0.00626155
\(196\) −1.08380 −0.0774146
\(197\) 9.26049 0.659783 0.329891 0.944019i \(-0.392988\pi\)
0.329891 + 0.944019i \(0.392988\pi\)
\(198\) 3.40009 0.241634
\(199\) −25.7180 −1.82310 −0.911552 0.411185i \(-0.865115\pi\)
−0.911552 + 0.411185i \(0.865115\pi\)
\(200\) −4.89461 −0.346101
\(201\) −10.2730 −0.724599
\(202\) 15.9029 1.11892
\(203\) −4.88740 −0.343028
\(204\) −4.84815 −0.339438
\(205\) −0.845680 −0.0590649
\(206\) 13.9721 0.973485
\(207\) 0 0
\(208\) 0.269342 0.0186755
\(209\) 13.6857 0.946658
\(210\) 0.789617 0.0544887
\(211\) 19.6782 1.35470 0.677350 0.735660i \(-0.263128\pi\)
0.677350 + 0.735660i \(0.263128\pi\)
\(212\) −1.08001 −0.0741754
\(213\) −11.4304 −0.783197
\(214\) 10.7919 0.737716
\(215\) 1.27763 0.0871336
\(216\) −1.00000 −0.0680414
\(217\) 23.1206 1.56953
\(218\) 16.3580 1.10790
\(219\) 1.13935 0.0769903
\(220\) 1.10379 0.0744173
\(221\) 1.30581 0.0878384
\(222\) 2.01732 0.135394
\(223\) −4.79039 −0.320788 −0.160394 0.987053i \(-0.551277\pi\)
−0.160394 + 0.987053i \(0.551277\pi\)
\(224\) −2.43232 −0.162516
\(225\) −4.89461 −0.326307
\(226\) 19.9598 1.32771
\(227\) 6.03416 0.400501 0.200251 0.979745i \(-0.435824\pi\)
0.200251 + 0.979745i \(0.435824\pi\)
\(228\) −4.02509 −0.266568
\(229\) 7.27420 0.480692 0.240346 0.970687i \(-0.422739\pi\)
0.240346 + 0.970687i \(0.422739\pi\)
\(230\) 0 0
\(231\) 8.27012 0.544134
\(232\) 2.00935 0.131921
\(233\) 8.41855 0.551517 0.275759 0.961227i \(-0.411071\pi\)
0.275759 + 0.961227i \(0.411071\pi\)
\(234\) 0.269342 0.0176074
\(235\) 1.68769 0.110093
\(236\) 9.90548 0.644792
\(237\) −16.0547 −1.04286
\(238\) −11.7923 −0.764379
\(239\) −5.16012 −0.333781 −0.166890 0.985975i \(-0.553373\pi\)
−0.166890 + 0.985975i \(0.553373\pi\)
\(240\) −0.324635 −0.0209551
\(241\) 15.6301 1.00682 0.503411 0.864047i \(-0.332078\pi\)
0.503411 + 0.864047i \(0.332078\pi\)
\(242\) 0.560608 0.0360372
\(243\) −1.00000 −0.0641500
\(244\) 4.35062 0.278520
\(245\) −0.351841 −0.0224783
\(246\) 2.60502 0.166090
\(247\) 1.08413 0.0689813
\(248\) −9.50557 −0.603604
\(249\) 1.05529 0.0668765
\(250\) −3.21214 −0.203153
\(251\) −13.8839 −0.876345 −0.438172 0.898891i \(-0.644374\pi\)
−0.438172 + 0.898891i \(0.644374\pi\)
\(252\) −2.43232 −0.153222
\(253\) 0 0
\(254\) −5.44618 −0.341724
\(255\) −1.57388 −0.0985601
\(256\) 1.00000 0.0625000
\(257\) −9.84808 −0.614307 −0.307153 0.951660i \(-0.599376\pi\)
−0.307153 + 0.951660i \(0.599376\pi\)
\(258\) −3.93559 −0.245019
\(259\) 4.90677 0.304892
\(260\) 0.0874378 0.00542266
\(261\) 2.00935 0.124376
\(262\) 15.3071 0.945677
\(263\) 0.229051 0.0141239 0.00706195 0.999975i \(-0.497752\pi\)
0.00706195 + 0.999975i \(0.497752\pi\)
\(264\) −3.40009 −0.209261
\(265\) −0.350609 −0.0215377
\(266\) −9.79032 −0.600283
\(267\) 2.74085 0.167737
\(268\) 10.2730 0.627521
\(269\) −0.432758 −0.0263857 −0.0131928 0.999913i \(-0.504200\pi\)
−0.0131928 + 0.999913i \(0.504200\pi\)
\(270\) −0.324635 −0.0197566
\(271\) −7.28255 −0.442383 −0.221192 0.975230i \(-0.570995\pi\)
−0.221192 + 0.975230i \(0.570995\pi\)
\(272\) 4.84815 0.293962
\(273\) 0.655127 0.0396501
\(274\) −17.0753 −1.03155
\(275\) −16.6421 −1.00356
\(276\) 0 0
\(277\) −24.9693 −1.50026 −0.750130 0.661291i \(-0.770009\pi\)
−0.750130 + 0.661291i \(0.770009\pi\)
\(278\) 16.2779 0.976285
\(279\) −9.50557 −0.569083
\(280\) −0.789617 −0.0471886
\(281\) −7.32093 −0.436730 −0.218365 0.975867i \(-0.570072\pi\)
−0.218365 + 0.975867i \(0.570072\pi\)
\(282\) −5.19874 −0.309581
\(283\) 10.4946 0.623837 0.311918 0.950109i \(-0.399028\pi\)
0.311918 + 0.950109i \(0.399028\pi\)
\(284\) 11.4304 0.678268
\(285\) −1.30668 −0.0774013
\(286\) 0.915787 0.0541516
\(287\) 6.33625 0.374017
\(288\) 1.00000 0.0589256
\(289\) 6.50456 0.382621
\(290\) 0.652306 0.0383047
\(291\) 13.6079 0.797709
\(292\) −1.13935 −0.0666756
\(293\) −16.1874 −0.945677 −0.472839 0.881149i \(-0.656771\pi\)
−0.472839 + 0.881149i \(0.656771\pi\)
\(294\) 1.08380 0.0632088
\(295\) 3.21566 0.187223
\(296\) −2.01732 −0.117254
\(297\) −3.40009 −0.197293
\(298\) −1.63045 −0.0944492
\(299\) 0 0
\(300\) 4.89461 0.282591
\(301\) −9.57262 −0.551757
\(302\) −11.7663 −0.677076
\(303\) −15.9029 −0.913597
\(304\) 4.02509 0.230855
\(305\) 1.41236 0.0808717
\(306\) 4.84815 0.277150
\(307\) 17.8501 1.01876 0.509380 0.860541i \(-0.329875\pi\)
0.509380 + 0.860541i \(0.329875\pi\)
\(308\) −8.27012 −0.471234
\(309\) −13.9721 −0.794847
\(310\) −3.08584 −0.175264
\(311\) −14.7357 −0.835586 −0.417793 0.908542i \(-0.637196\pi\)
−0.417793 + 0.908542i \(0.637196\pi\)
\(312\) −0.269342 −0.0152485
\(313\) −5.71682 −0.323134 −0.161567 0.986862i \(-0.551655\pi\)
−0.161567 + 0.986862i \(0.551655\pi\)
\(314\) 10.0065 0.564700
\(315\) −0.789617 −0.0444899
\(316\) 16.0547 0.903145
\(317\) −7.00248 −0.393298 −0.196649 0.980474i \(-0.563006\pi\)
−0.196649 + 0.980474i \(0.563006\pi\)
\(318\) 1.08001 0.0605640
\(319\) 6.83198 0.382518
\(320\) 0.324635 0.0181476
\(321\) −10.7919 −0.602343
\(322\) 0 0
\(323\) 19.5143 1.08580
\(324\) 1.00000 0.0555556
\(325\) −1.31833 −0.0731275
\(326\) −15.9393 −0.882794
\(327\) −16.3580 −0.904597
\(328\) −2.60502 −0.143838
\(329\) −12.6450 −0.697142
\(330\) −1.10379 −0.0607615
\(331\) 19.5035 1.07201 0.536004 0.844215i \(-0.319933\pi\)
0.536004 + 0.844215i \(0.319933\pi\)
\(332\) −1.05529 −0.0579167
\(333\) −2.01732 −0.110548
\(334\) −9.53697 −0.521840
\(335\) 3.33496 0.182208
\(336\) 2.43232 0.132694
\(337\) −20.3735 −1.10981 −0.554906 0.831913i \(-0.687246\pi\)
−0.554906 + 0.831913i \(0.687246\pi\)
\(338\) −12.9275 −0.703161
\(339\) −19.9598 −1.08407
\(340\) 1.57388 0.0853555
\(341\) −32.3198 −1.75021
\(342\) 4.02509 0.217652
\(343\) 19.6624 1.06167
\(344\) 3.93559 0.212193
\(345\) 0 0
\(346\) −24.7305 −1.32952
\(347\) −1.87358 −0.100579 −0.0502896 0.998735i \(-0.516014\pi\)
−0.0502896 + 0.998735i \(0.516014\pi\)
\(348\) −2.00935 −0.107713
\(349\) 16.4616 0.881172 0.440586 0.897710i \(-0.354771\pi\)
0.440586 + 0.897710i \(0.354771\pi\)
\(350\) 11.9053 0.636364
\(351\) −0.269342 −0.0143764
\(352\) 3.40009 0.181225
\(353\) −8.08237 −0.430181 −0.215090 0.976594i \(-0.569005\pi\)
−0.215090 + 0.976594i \(0.569005\pi\)
\(354\) −9.90548 −0.526470
\(355\) 3.71070 0.196943
\(356\) −2.74085 −0.145265
\(357\) 11.7923 0.624113
\(358\) −26.3484 −1.39255
\(359\) 17.8120 0.940083 0.470042 0.882644i \(-0.344239\pi\)
0.470042 + 0.882644i \(0.344239\pi\)
\(360\) 0.324635 0.0171098
\(361\) −2.79864 −0.147297
\(362\) −18.1768 −0.955350
\(363\) −0.560608 −0.0294243
\(364\) −0.655127 −0.0343380
\(365\) −0.369874 −0.0193601
\(366\) −4.35062 −0.227411
\(367\) −4.22445 −0.220515 −0.110257 0.993903i \(-0.535167\pi\)
−0.110257 + 0.993903i \(0.535167\pi\)
\(368\) 0 0
\(369\) −2.60502 −0.135612
\(370\) −0.654892 −0.0340462
\(371\) 2.62693 0.136384
\(372\) 9.50557 0.492841
\(373\) −6.77016 −0.350545 −0.175273 0.984520i \(-0.556081\pi\)
−0.175273 + 0.984520i \(0.556081\pi\)
\(374\) 16.4841 0.852375
\(375\) 3.21214 0.165874
\(376\) 5.19874 0.268105
\(377\) 0.541204 0.0278734
\(378\) 2.43232 0.125105
\(379\) 8.78615 0.451314 0.225657 0.974207i \(-0.427547\pi\)
0.225657 + 0.974207i \(0.427547\pi\)
\(380\) 1.30668 0.0670315
\(381\) 5.44618 0.279016
\(382\) 7.85270 0.401779
\(383\) −23.0045 −1.17547 −0.587737 0.809052i \(-0.699981\pi\)
−0.587737 + 0.809052i \(0.699981\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.68477 −0.136828
\(386\) 23.5784 1.20011
\(387\) 3.93559 0.200057
\(388\) −13.6079 −0.690836
\(389\) −6.52670 −0.330917 −0.165458 0.986217i \(-0.552910\pi\)
−0.165458 + 0.986217i \(0.552910\pi\)
\(390\) −0.0874378 −0.00442759
\(391\) 0 0
\(392\) −1.08380 −0.0547404
\(393\) −15.3071 −0.772142
\(394\) 9.26049 0.466537
\(395\) 5.21190 0.262239
\(396\) 3.40009 0.170861
\(397\) 34.7651 1.74481 0.872404 0.488785i \(-0.162560\pi\)
0.872404 + 0.488785i \(0.162560\pi\)
\(398\) −25.7180 −1.28913
\(399\) 9.79032 0.490129
\(400\) −4.89461 −0.244731
\(401\) −4.23343 −0.211408 −0.105704 0.994398i \(-0.533710\pi\)
−0.105704 + 0.994398i \(0.533710\pi\)
\(402\) −10.2730 −0.512369
\(403\) −2.56025 −0.127535
\(404\) 15.9029 0.791198
\(405\) 0.324635 0.0161312
\(406\) −4.88740 −0.242557
\(407\) −6.85907 −0.339991
\(408\) −4.84815 −0.240019
\(409\) 23.7882 1.17625 0.588126 0.808769i \(-0.299866\pi\)
0.588126 + 0.808769i \(0.299866\pi\)
\(410\) −0.845680 −0.0417652
\(411\) 17.0753 0.842260
\(412\) 13.9721 0.688358
\(413\) −24.0933 −1.18555
\(414\) 0 0
\(415\) −0.342585 −0.0168168
\(416\) 0.269342 0.0132056
\(417\) −16.2779 −0.797133
\(418\) 13.6857 0.669388
\(419\) 22.3001 1.08943 0.544717 0.838620i \(-0.316637\pi\)
0.544717 + 0.838620i \(0.316637\pi\)
\(420\) 0.789617 0.0385294
\(421\) −23.9034 −1.16498 −0.582491 0.812837i \(-0.697922\pi\)
−0.582491 + 0.812837i \(0.697922\pi\)
\(422\) 19.6782 0.957918
\(423\) 5.19874 0.252771
\(424\) −1.08001 −0.0524499
\(425\) −23.7298 −1.15107
\(426\) −11.4304 −0.553804
\(427\) −10.5821 −0.512105
\(428\) 10.7919 0.521644
\(429\) −0.915787 −0.0442146
\(430\) 1.27763 0.0616127
\(431\) 10.0745 0.485269 0.242635 0.970118i \(-0.421988\pi\)
0.242635 + 0.970118i \(0.421988\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −32.9245 −1.58225 −0.791125 0.611655i \(-0.790504\pi\)
−0.791125 + 0.611655i \(0.790504\pi\)
\(434\) 23.1206 1.10982
\(435\) −0.652306 −0.0312757
\(436\) 16.3580 0.783404
\(437\) 0 0
\(438\) 1.13935 0.0544404
\(439\) −17.6807 −0.843852 −0.421926 0.906630i \(-0.638646\pi\)
−0.421926 + 0.906630i \(0.638646\pi\)
\(440\) 1.10379 0.0526210
\(441\) −1.08380 −0.0516097
\(442\) 1.30581 0.0621111
\(443\) −32.6096 −1.54933 −0.774665 0.632371i \(-0.782082\pi\)
−0.774665 + 0.632371i \(0.782082\pi\)
\(444\) 2.01732 0.0957378
\(445\) −0.889774 −0.0421793
\(446\) −4.79039 −0.226832
\(447\) 1.63045 0.0771174
\(448\) −2.43232 −0.114916
\(449\) 35.3685 1.66914 0.834572 0.550898i \(-0.185715\pi\)
0.834572 + 0.550898i \(0.185715\pi\)
\(450\) −4.89461 −0.230734
\(451\) −8.85730 −0.417074
\(452\) 19.9598 0.938830
\(453\) 11.7663 0.552830
\(454\) 6.03416 0.283197
\(455\) −0.212677 −0.00997045
\(456\) −4.02509 −0.188492
\(457\) −8.77948 −0.410687 −0.205343 0.978690i \(-0.565831\pi\)
−0.205343 + 0.978690i \(0.565831\pi\)
\(458\) 7.27420 0.339901
\(459\) −4.84815 −0.226292
\(460\) 0 0
\(461\) 12.7701 0.594765 0.297382 0.954758i \(-0.403886\pi\)
0.297382 + 0.954758i \(0.403886\pi\)
\(462\) 8.27012 0.384761
\(463\) 15.8063 0.734579 0.367290 0.930107i \(-0.380286\pi\)
0.367290 + 0.930107i \(0.380286\pi\)
\(464\) 2.00935 0.0932819
\(465\) 3.08584 0.143102
\(466\) 8.41855 0.389982
\(467\) 3.47362 0.160740 0.0803700 0.996765i \(-0.474390\pi\)
0.0803700 + 0.996765i \(0.474390\pi\)
\(468\) 0.269342 0.0124503
\(469\) −24.9872 −1.15380
\(470\) 1.68769 0.0778474
\(471\) −10.0065 −0.461076
\(472\) 9.90548 0.455936
\(473\) 13.3814 0.615275
\(474\) −16.0547 −0.737415
\(475\) −19.7013 −0.903956
\(476\) −11.7923 −0.540498
\(477\) −1.08001 −0.0494503
\(478\) −5.16012 −0.236019
\(479\) 0.885748 0.0404709 0.0202354 0.999795i \(-0.493558\pi\)
0.0202354 + 0.999795i \(0.493558\pi\)
\(480\) −0.324635 −0.0148175
\(481\) −0.543349 −0.0247746
\(482\) 15.6301 0.711931
\(483\) 0 0
\(484\) 0.560608 0.0254822
\(485\) −4.41760 −0.200593
\(486\) −1.00000 −0.0453609
\(487\) −3.28852 −0.149017 −0.0745085 0.997220i \(-0.523739\pi\)
−0.0745085 + 0.997220i \(0.523739\pi\)
\(488\) 4.35062 0.196943
\(489\) 15.9393 0.720798
\(490\) −0.351841 −0.0158945
\(491\) 24.1044 1.08781 0.543907 0.839145i \(-0.316944\pi\)
0.543907 + 0.839145i \(0.316944\pi\)
\(492\) 2.60502 0.117443
\(493\) 9.74165 0.438742
\(494\) 1.08413 0.0487772
\(495\) 1.10379 0.0496116
\(496\) −9.50557 −0.426812
\(497\) −27.8024 −1.24711
\(498\) 1.05529 0.0472888
\(499\) 13.5901 0.608375 0.304187 0.952612i \(-0.401615\pi\)
0.304187 + 0.952612i \(0.401615\pi\)
\(500\) −3.21214 −0.143651
\(501\) 9.53697 0.426081
\(502\) −13.8839 −0.619669
\(503\) −16.4428 −0.733149 −0.366575 0.930389i \(-0.619470\pi\)
−0.366575 + 0.930389i \(0.619470\pi\)
\(504\) −2.43232 −0.108344
\(505\) 5.16263 0.229734
\(506\) 0 0
\(507\) 12.9275 0.574128
\(508\) −5.44618 −0.241635
\(509\) −25.3556 −1.12387 −0.561934 0.827182i \(-0.689943\pi\)
−0.561934 + 0.827182i \(0.689943\pi\)
\(510\) −1.57388 −0.0696925
\(511\) 2.77127 0.122594
\(512\) 1.00000 0.0441942
\(513\) −4.02509 −0.177712
\(514\) −9.84808 −0.434380
\(515\) 4.53584 0.199873
\(516\) −3.93559 −0.173255
\(517\) 17.6762 0.777398
\(518\) 4.90677 0.215591
\(519\) 24.7305 1.08555
\(520\) 0.0874378 0.00383440
\(521\) 33.3911 1.46289 0.731445 0.681900i \(-0.238846\pi\)
0.731445 + 0.681900i \(0.238846\pi\)
\(522\) 2.00935 0.0879470
\(523\) 8.13303 0.355633 0.177816 0.984064i \(-0.443097\pi\)
0.177816 + 0.984064i \(0.443097\pi\)
\(524\) 15.3071 0.668694
\(525\) −11.9053 −0.519589
\(526\) 0.229051 0.00998711
\(527\) −46.0844 −2.00747
\(528\) −3.40009 −0.147970
\(529\) 0 0
\(530\) −0.350609 −0.0152295
\(531\) 9.90548 0.429861
\(532\) −9.79032 −0.424464
\(533\) −0.701642 −0.0303915
\(534\) 2.74085 0.118608
\(535\) 3.50341 0.151466
\(536\) 10.2730 0.443724
\(537\) 26.3484 1.13702
\(538\) −0.432758 −0.0186575
\(539\) −3.68503 −0.158726
\(540\) −0.324635 −0.0139701
\(541\) 8.92929 0.383900 0.191950 0.981405i \(-0.438519\pi\)
0.191950 + 0.981405i \(0.438519\pi\)
\(542\) −7.28255 −0.312812
\(543\) 18.1768 0.780040
\(544\) 4.84815 0.207863
\(545\) 5.31036 0.227471
\(546\) 0.655127 0.0280369
\(547\) 18.3908 0.786335 0.393168 0.919467i \(-0.371379\pi\)
0.393168 + 0.919467i \(0.371379\pi\)
\(548\) −17.0753 −0.729419
\(549\) 4.35062 0.185680
\(550\) −16.6421 −0.709622
\(551\) 8.08783 0.344553
\(552\) 0 0
\(553\) −39.0501 −1.66058
\(554\) −24.9693 −1.06084
\(555\) 0.654892 0.0277986
\(556\) 16.2779 0.690337
\(557\) −15.0330 −0.636968 −0.318484 0.947928i \(-0.603174\pi\)
−0.318484 + 0.947928i \(0.603174\pi\)
\(558\) −9.50557 −0.402403
\(559\) 1.06002 0.0448341
\(560\) −0.789617 −0.0333674
\(561\) −16.4841 −0.695961
\(562\) −7.32093 −0.308815
\(563\) −45.8315 −1.93157 −0.965785 0.259346i \(-0.916493\pi\)
−0.965785 + 0.259346i \(0.916493\pi\)
\(564\) −5.19874 −0.218906
\(565\) 6.47964 0.272601
\(566\) 10.4946 0.441119
\(567\) −2.43232 −0.102148
\(568\) 11.4304 0.479608
\(569\) 35.3665 1.48264 0.741321 0.671150i \(-0.234199\pi\)
0.741321 + 0.671150i \(0.234199\pi\)
\(570\) −1.30668 −0.0547310
\(571\) 21.0408 0.880528 0.440264 0.897868i \(-0.354885\pi\)
0.440264 + 0.897868i \(0.354885\pi\)
\(572\) 0.915787 0.0382910
\(573\) −7.85270 −0.328051
\(574\) 6.33625 0.264470
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −17.0656 −0.710452 −0.355226 0.934780i \(-0.615596\pi\)
−0.355226 + 0.934780i \(0.615596\pi\)
\(578\) 6.50456 0.270554
\(579\) −23.5784 −0.979884
\(580\) 0.652306 0.0270855
\(581\) 2.56681 0.106489
\(582\) 13.6079 0.564066
\(583\) −3.67213 −0.152084
\(584\) −1.13935 −0.0471467
\(585\) 0.0874378 0.00361511
\(586\) −16.1874 −0.668695
\(587\) −3.20673 −0.132356 −0.0661780 0.997808i \(-0.521081\pi\)
−0.0661780 + 0.997808i \(0.521081\pi\)
\(588\) 1.08380 0.0446953
\(589\) −38.2608 −1.57651
\(590\) 3.21566 0.132387
\(591\) −9.26049 −0.380926
\(592\) −2.01732 −0.0829114
\(593\) −8.15422 −0.334854 −0.167427 0.985884i \(-0.553546\pi\)
−0.167427 + 0.985884i \(0.553546\pi\)
\(594\) −3.40009 −0.139507
\(595\) −3.82818 −0.156940
\(596\) −1.63045 −0.0667857
\(597\) 25.7180 1.05257
\(598\) 0 0
\(599\) −44.6481 −1.82427 −0.912136 0.409888i \(-0.865568\pi\)
−0.912136 + 0.409888i \(0.865568\pi\)
\(600\) 4.89461 0.199822
\(601\) −27.7579 −1.13227 −0.566135 0.824312i \(-0.691562\pi\)
−0.566135 + 0.824312i \(0.691562\pi\)
\(602\) −9.57262 −0.390151
\(603\) 10.2730 0.418347
\(604\) −11.7663 −0.478765
\(605\) 0.181993 0.00739906
\(606\) −15.9029 −0.646011
\(607\) −23.5817 −0.957153 −0.478577 0.878046i \(-0.658847\pi\)
−0.478577 + 0.878046i \(0.658847\pi\)
\(608\) 4.02509 0.163239
\(609\) 4.88740 0.198047
\(610\) 1.41236 0.0571849
\(611\) 1.40024 0.0566477
\(612\) 4.84815 0.195975
\(613\) −32.8468 −1.32667 −0.663336 0.748322i \(-0.730860\pi\)
−0.663336 + 0.748322i \(0.730860\pi\)
\(614\) 17.8501 0.720373
\(615\) 0.845680 0.0341011
\(616\) −8.27012 −0.333212
\(617\) −23.3881 −0.941571 −0.470786 0.882248i \(-0.656029\pi\)
−0.470786 + 0.882248i \(0.656029\pi\)
\(618\) −13.9721 −0.562042
\(619\) −12.4486 −0.500350 −0.250175 0.968201i \(-0.580488\pi\)
−0.250175 + 0.968201i \(0.580488\pi\)
\(620\) −3.08584 −0.123930
\(621\) 0 0
\(622\) −14.7357 −0.590848
\(623\) 6.66662 0.267093
\(624\) −0.269342 −0.0107823
\(625\) 23.4303 0.937212
\(626\) −5.71682 −0.228490
\(627\) −13.6857 −0.546553
\(628\) 10.0065 0.399303
\(629\) −9.78027 −0.389965
\(630\) −0.789617 −0.0314591
\(631\) 32.4757 1.29284 0.646419 0.762982i \(-0.276266\pi\)
0.646419 + 0.762982i \(0.276266\pi\)
\(632\) 16.0547 0.638620
\(633\) −19.6782 −0.782137
\(634\) −7.00248 −0.278104
\(635\) −1.76802 −0.0701617
\(636\) 1.08001 0.0428252
\(637\) −0.291914 −0.0115661
\(638\) 6.83198 0.270481
\(639\) 11.4304 0.452179
\(640\) 0.324635 0.0128323
\(641\) 3.78968 0.149683 0.0748416 0.997195i \(-0.476155\pi\)
0.0748416 + 0.997195i \(0.476155\pi\)
\(642\) −10.7919 −0.425920
\(643\) 16.4223 0.647634 0.323817 0.946120i \(-0.395034\pi\)
0.323817 + 0.946120i \(0.395034\pi\)
\(644\) 0 0
\(645\) −1.27763 −0.0503066
\(646\) 19.5143 0.767778
\(647\) −5.18571 −0.203871 −0.101936 0.994791i \(-0.532504\pi\)
−0.101936 + 0.994791i \(0.532504\pi\)
\(648\) 1.00000 0.0392837
\(649\) 33.6795 1.32204
\(650\) −1.31833 −0.0517090
\(651\) −23.1206 −0.906168
\(652\) −15.9393 −0.624230
\(653\) −25.7770 −1.00873 −0.504366 0.863490i \(-0.668274\pi\)
−0.504366 + 0.863490i \(0.668274\pi\)
\(654\) −16.3580 −0.639647
\(655\) 4.96922 0.194164
\(656\) −2.60502 −0.101709
\(657\) −1.13935 −0.0444504
\(658\) −12.6450 −0.492954
\(659\) −43.8640 −1.70870 −0.854349 0.519700i \(-0.826044\pi\)
−0.854349 + 0.519700i \(0.826044\pi\)
\(660\) −1.10379 −0.0429649
\(661\) 9.43101 0.366824 0.183412 0.983036i \(-0.441286\pi\)
0.183412 + 0.983036i \(0.441286\pi\)
\(662\) 19.5035 0.758024
\(663\) −1.30581 −0.0507135
\(664\) −1.05529 −0.0409533
\(665\) −3.17828 −0.123248
\(666\) −2.01732 −0.0781696
\(667\) 0 0
\(668\) −9.53697 −0.368997
\(669\) 4.79039 0.185207
\(670\) 3.33496 0.128841
\(671\) 14.7925 0.571059
\(672\) 2.43232 0.0938289
\(673\) −8.94031 −0.344624 −0.172312 0.985042i \(-0.555124\pi\)
−0.172312 + 0.985042i \(0.555124\pi\)
\(674\) −20.3735 −0.784756
\(675\) 4.89461 0.188394
\(676\) −12.9275 −0.497210
\(677\) −35.3761 −1.35961 −0.679807 0.733391i \(-0.737937\pi\)
−0.679807 + 0.733391i \(0.737937\pi\)
\(678\) −19.9598 −0.766551
\(679\) 33.0988 1.27022
\(680\) 1.57388 0.0603555
\(681\) −6.03416 −0.231229
\(682\) −32.3198 −1.23759
\(683\) 29.2895 1.12073 0.560366 0.828245i \(-0.310661\pi\)
0.560366 + 0.828245i \(0.310661\pi\)
\(684\) 4.02509 0.153903
\(685\) −5.54322 −0.211796
\(686\) 19.6624 0.750715
\(687\) −7.27420 −0.277528
\(688\) 3.93559 0.150043
\(689\) −0.290892 −0.0110821
\(690\) 0 0
\(691\) −20.9634 −0.797487 −0.398744 0.917062i \(-0.630554\pi\)
−0.398744 + 0.917062i \(0.630554\pi\)
\(692\) −24.7305 −0.940112
\(693\) −8.27012 −0.314156
\(694\) −1.87358 −0.0711203
\(695\) 5.28438 0.200448
\(696\) −2.00935 −0.0761644
\(697\) −12.6295 −0.478377
\(698\) 16.4616 0.623083
\(699\) −8.41855 −0.318419
\(700\) 11.9053 0.449977
\(701\) −32.1001 −1.21240 −0.606202 0.795311i \(-0.707308\pi\)
−0.606202 + 0.795311i \(0.707308\pi\)
\(702\) −0.269342 −0.0101657
\(703\) −8.11990 −0.306248
\(704\) 3.40009 0.128146
\(705\) −1.68769 −0.0635622
\(706\) −8.08237 −0.304184
\(707\) −38.6810 −1.45475
\(708\) −9.90548 −0.372271
\(709\) −33.7537 −1.26765 −0.633824 0.773477i \(-0.718516\pi\)
−0.633824 + 0.773477i \(0.718516\pi\)
\(710\) 3.71070 0.139260
\(711\) 16.0547 0.602097
\(712\) −2.74085 −0.102718
\(713\) 0 0
\(714\) 11.7923 0.441315
\(715\) 0.297296 0.0111183
\(716\) −26.3484 −0.984684
\(717\) 5.16012 0.192708
\(718\) 17.8120 0.664739
\(719\) −46.9884 −1.75237 −0.876185 0.481974i \(-0.839920\pi\)
−0.876185 + 0.481974i \(0.839920\pi\)
\(720\) 0.324635 0.0120984
\(721\) −33.9848 −1.26566
\(722\) −2.79864 −0.104154
\(723\) −15.6301 −0.581289
\(724\) −18.1768 −0.675535
\(725\) −9.83501 −0.365263
\(726\) −0.560608 −0.0208061
\(727\) 16.8502 0.624939 0.312470 0.949928i \(-0.398844\pi\)
0.312470 + 0.949928i \(0.398844\pi\)
\(728\) −0.655127 −0.0242806
\(729\) 1.00000 0.0370370
\(730\) −0.369874 −0.0136896
\(731\) 19.0803 0.705711
\(732\) −4.35062 −0.160804
\(733\) 10.6895 0.394825 0.197413 0.980321i \(-0.436746\pi\)
0.197413 + 0.980321i \(0.436746\pi\)
\(734\) −4.22445 −0.155927
\(735\) 0.351841 0.0129778
\(736\) 0 0
\(737\) 34.9290 1.28663
\(738\) −2.60502 −0.0958921
\(739\) 24.8920 0.915668 0.457834 0.889038i \(-0.348625\pi\)
0.457834 + 0.889038i \(0.348625\pi\)
\(740\) −0.654892 −0.0240743
\(741\) −1.08413 −0.0398264
\(742\) 2.62693 0.0964378
\(743\) 38.1691 1.40029 0.700145 0.714001i \(-0.253119\pi\)
0.700145 + 0.714001i \(0.253119\pi\)
\(744\) 9.50557 0.348491
\(745\) −0.529300 −0.0193920
\(746\) −6.77016 −0.247873
\(747\) −1.05529 −0.0386111
\(748\) 16.4841 0.602720
\(749\) −26.2493 −0.959128
\(750\) 3.21214 0.117291
\(751\) 44.2589 1.61503 0.807514 0.589848i \(-0.200812\pi\)
0.807514 + 0.589848i \(0.200812\pi\)
\(752\) 5.19874 0.189579
\(753\) 13.8839 0.505958
\(754\) 0.541204 0.0197095
\(755\) −3.81976 −0.139015
\(756\) 2.43232 0.0884627
\(757\) −9.78484 −0.355636 −0.177818 0.984063i \(-0.556904\pi\)
−0.177818 + 0.984063i \(0.556904\pi\)
\(758\) 8.78615 0.319128
\(759\) 0 0
\(760\) 1.30668 0.0473984
\(761\) 23.9490 0.868150 0.434075 0.900877i \(-0.357075\pi\)
0.434075 + 0.900877i \(0.357075\pi\)
\(762\) 5.44618 0.197294
\(763\) −39.7878 −1.44042
\(764\) 7.85270 0.284101
\(765\) 1.57388 0.0569037
\(766\) −23.0045 −0.831186
\(767\) 2.66796 0.0963345
\(768\) −1.00000 −0.0360844
\(769\) −27.6827 −0.998265 −0.499132 0.866526i \(-0.666348\pi\)
−0.499132 + 0.866526i \(0.666348\pi\)
\(770\) −2.68477 −0.0967523
\(771\) 9.84808 0.354670
\(772\) 23.5784 0.848604
\(773\) −33.8343 −1.21693 −0.608467 0.793579i \(-0.708215\pi\)
−0.608467 + 0.793579i \(0.708215\pi\)
\(774\) 3.93559 0.141462
\(775\) 46.5261 1.67127
\(776\) −13.6079 −0.488495
\(777\) −4.90677 −0.176030
\(778\) −6.52670 −0.233993
\(779\) −10.4854 −0.375680
\(780\) −0.0874378 −0.00313078
\(781\) 38.8643 1.39067
\(782\) 0 0
\(783\) −2.00935 −0.0718084
\(784\) −1.08380 −0.0387073
\(785\) 3.24846 0.115943
\(786\) −15.3071 −0.545987
\(787\) 24.4836 0.872745 0.436372 0.899766i \(-0.356263\pi\)
0.436372 + 0.899766i \(0.356263\pi\)
\(788\) 9.26049 0.329891
\(789\) −0.229051 −0.00815444
\(790\) 5.21190 0.185431
\(791\) −48.5487 −1.72619
\(792\) 3.40009 0.120817
\(793\) 1.17181 0.0416121
\(794\) 34.7651 1.23377
\(795\) 0.350609 0.0124348
\(796\) −25.7180 −0.911552
\(797\) −8.69614 −0.308033 −0.154017 0.988068i \(-0.549221\pi\)
−0.154017 + 0.988068i \(0.549221\pi\)
\(798\) 9.79032 0.346574
\(799\) 25.2043 0.891663
\(800\) −4.89461 −0.173051
\(801\) −2.74085 −0.0968430
\(802\) −4.23343 −0.149488
\(803\) −3.87390 −0.136707
\(804\) −10.2730 −0.362299
\(805\) 0 0
\(806\) −2.56025 −0.0901809
\(807\) 0.432758 0.0152338
\(808\) 15.9029 0.559462
\(809\) 18.5536 0.652308 0.326154 0.945317i \(-0.394247\pi\)
0.326154 + 0.945317i \(0.394247\pi\)
\(810\) 0.324635 0.0114065
\(811\) −21.5690 −0.757390 −0.378695 0.925521i \(-0.623627\pi\)
−0.378695 + 0.925521i \(0.623627\pi\)
\(812\) −4.88740 −0.171514
\(813\) 7.28255 0.255410
\(814\) −6.85907 −0.240410
\(815\) −5.17444 −0.181253
\(816\) −4.84815 −0.169719
\(817\) 15.8411 0.554210
\(818\) 23.7882 0.831736
\(819\) −0.655127 −0.0228920
\(820\) −0.845680 −0.0295324
\(821\) 6.92328 0.241624 0.120812 0.992675i \(-0.461450\pi\)
0.120812 + 0.992675i \(0.461450\pi\)
\(822\) 17.0753 0.595568
\(823\) −31.3176 −1.09166 −0.545831 0.837895i \(-0.683786\pi\)
−0.545831 + 0.837895i \(0.683786\pi\)
\(824\) 13.9721 0.486743
\(825\) 16.6421 0.579404
\(826\) −24.0933 −0.838314
\(827\) −4.77095 −0.165902 −0.0829511 0.996554i \(-0.526435\pi\)
−0.0829511 + 0.996554i \(0.526435\pi\)
\(828\) 0 0
\(829\) −1.94168 −0.0674372 −0.0337186 0.999431i \(-0.510735\pi\)
−0.0337186 + 0.999431i \(0.510735\pi\)
\(830\) −0.342585 −0.0118913
\(831\) 24.9693 0.866175
\(832\) 0.269342 0.00933776
\(833\) −5.25445 −0.182056
\(834\) −16.2779 −0.563658
\(835\) −3.09603 −0.107143
\(836\) 13.6857 0.473329
\(837\) 9.50557 0.328560
\(838\) 22.3001 0.770346
\(839\) −45.8225 −1.58197 −0.790985 0.611836i \(-0.790431\pi\)
−0.790985 + 0.611836i \(0.790431\pi\)
\(840\) 0.789617 0.0272444
\(841\) −24.9625 −0.860776
\(842\) −23.9034 −0.823766
\(843\) 7.32093 0.252146
\(844\) 19.6782 0.677350
\(845\) −4.19670 −0.144371
\(846\) 5.19874 0.178736
\(847\) −1.36358 −0.0468531
\(848\) −1.08001 −0.0370877
\(849\) −10.4946 −0.360172
\(850\) −23.7298 −0.813926
\(851\) 0 0
\(852\) −11.4304 −0.391598
\(853\) −19.1208 −0.654685 −0.327342 0.944906i \(-0.606153\pi\)
−0.327342 + 0.944906i \(0.606153\pi\)
\(854\) −10.5821 −0.362113
\(855\) 1.30668 0.0446877
\(856\) 10.7919 0.368858
\(857\) 9.48763 0.324091 0.162046 0.986783i \(-0.448191\pi\)
0.162046 + 0.986783i \(0.448191\pi\)
\(858\) −0.915787 −0.0312645
\(859\) 39.2043 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(860\) 1.27763 0.0435668
\(861\) −6.33625 −0.215939
\(862\) 10.0745 0.343137
\(863\) 12.9692 0.441477 0.220739 0.975333i \(-0.429153\pi\)
0.220739 + 0.975333i \(0.429153\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −8.02838 −0.272973
\(866\) −32.9245 −1.11882
\(867\) −6.50456 −0.220907
\(868\) 23.1206 0.784765
\(869\) 54.5873 1.85175
\(870\) −0.652306 −0.0221153
\(871\) 2.76694 0.0937542
\(872\) 16.3580 0.553950
\(873\) −13.6079 −0.460558
\(874\) 0 0
\(875\) 7.81295 0.264126
\(876\) 1.13935 0.0384952
\(877\) 28.5809 0.965108 0.482554 0.875866i \(-0.339709\pi\)
0.482554 + 0.875866i \(0.339709\pi\)
\(878\) −17.6807 −0.596694
\(879\) 16.1874 0.545987
\(880\) 1.10379 0.0372087
\(881\) 33.5556 1.13052 0.565259 0.824913i \(-0.308776\pi\)
0.565259 + 0.824913i \(0.308776\pi\)
\(882\) −1.08380 −0.0364936
\(883\) 21.6186 0.727524 0.363762 0.931492i \(-0.381492\pi\)
0.363762 + 0.931492i \(0.381492\pi\)
\(884\) 1.30581 0.0439192
\(885\) −3.21566 −0.108093
\(886\) −32.6096 −1.09554
\(887\) −39.1158 −1.31338 −0.656690 0.754161i \(-0.728044\pi\)
−0.656690 + 0.754161i \(0.728044\pi\)
\(888\) 2.01732 0.0676968
\(889\) 13.2469 0.444285
\(890\) −0.889774 −0.0298253
\(891\) 3.40009 0.113907
\(892\) −4.79039 −0.160394
\(893\) 20.9254 0.700242
\(894\) 1.63045 0.0545303
\(895\) −8.55359 −0.285915
\(896\) −2.43232 −0.0812582
\(897\) 0 0
\(898\) 35.3685 1.18026
\(899\) −19.1000 −0.637022
\(900\) −4.89461 −0.163154
\(901\) −5.23605 −0.174438
\(902\) −8.85730 −0.294916
\(903\) 9.57262 0.318557
\(904\) 19.9598 0.663853
\(905\) −5.90081 −0.196150
\(906\) 11.7663 0.390910
\(907\) −18.6215 −0.618317 −0.309158 0.951011i \(-0.600047\pi\)
−0.309158 + 0.951011i \(0.600047\pi\)
\(908\) 6.03416 0.200251
\(909\) 15.9029 0.527465
\(910\) −0.212677 −0.00705018
\(911\) 16.2164 0.537273 0.268637 0.963242i \(-0.413427\pi\)
0.268637 + 0.963242i \(0.413427\pi\)
\(912\) −4.02509 −0.133284
\(913\) −3.58809 −0.118748
\(914\) −8.77948 −0.290400
\(915\) −1.41236 −0.0466913
\(916\) 7.27420 0.240346
\(917\) −37.2318 −1.22950
\(918\) −4.84815 −0.160013
\(919\) 7.51602 0.247930 0.123965 0.992287i \(-0.460439\pi\)
0.123965 + 0.992287i \(0.460439\pi\)
\(920\) 0 0
\(921\) −17.8501 −0.588182
\(922\) 12.7701 0.420562
\(923\) 3.07868 0.101336
\(924\) 8.27012 0.272067
\(925\) 9.87400 0.324655
\(926\) 15.8063 0.519426
\(927\) 13.9721 0.458905
\(928\) 2.00935 0.0659603
\(929\) 16.8540 0.552962 0.276481 0.961019i \(-0.410832\pi\)
0.276481 + 0.961019i \(0.410832\pi\)
\(930\) 3.08584 0.101189
\(931\) −4.36241 −0.142972
\(932\) 8.41855 0.275759
\(933\) 14.7357 0.482426
\(934\) 3.47362 0.113660
\(935\) 5.35133 0.175007
\(936\) 0.269342 0.00880372
\(937\) −0.877608 −0.0286702 −0.0143351 0.999897i \(-0.504563\pi\)
−0.0143351 + 0.999897i \(0.504563\pi\)
\(938\) −24.9872 −0.815859
\(939\) 5.71682 0.186561
\(940\) 1.68769 0.0550465
\(941\) 13.9836 0.455851 0.227926 0.973679i \(-0.426806\pi\)
0.227926 + 0.973679i \(0.426806\pi\)
\(942\) −10.0065 −0.326030
\(943\) 0 0
\(944\) 9.90548 0.322396
\(945\) 0.789617 0.0256862
\(946\) 13.3814 0.435065
\(947\) −24.6457 −0.800876 −0.400438 0.916324i \(-0.631142\pi\)
−0.400438 + 0.916324i \(0.631142\pi\)
\(948\) −16.0547 −0.521431
\(949\) −0.306876 −0.00996161
\(950\) −19.7013 −0.639193
\(951\) 7.00248 0.227071
\(952\) −11.7923 −0.382190
\(953\) 56.1029 1.81735 0.908675 0.417504i \(-0.137095\pi\)
0.908675 + 0.417504i \(0.137095\pi\)
\(954\) −1.08001 −0.0349666
\(955\) 2.54926 0.0824921
\(956\) −5.16012 −0.166890
\(957\) −6.83198 −0.220847
\(958\) 0.885748 0.0286172
\(959\) 41.5325 1.34116
\(960\) −0.324635 −0.0104775
\(961\) 59.3558 1.91470
\(962\) −0.543349 −0.0175183
\(963\) 10.7919 0.347763
\(964\) 15.6301 0.503411
\(965\) 7.65436 0.246403
\(966\) 0 0
\(967\) −38.0371 −1.22319 −0.611596 0.791170i \(-0.709472\pi\)
−0.611596 + 0.791170i \(0.709472\pi\)
\(968\) 0.560608 0.0180186
\(969\) −19.5143 −0.626888
\(970\) −4.41760 −0.141841
\(971\) 46.8643 1.50395 0.751974 0.659193i \(-0.229102\pi\)
0.751974 + 0.659193i \(0.229102\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −39.5932 −1.26930
\(974\) −3.28852 −0.105371
\(975\) 1.31833 0.0422202
\(976\) 4.35062 0.139260
\(977\) −6.32527 −0.202363 −0.101182 0.994868i \(-0.532262\pi\)
−0.101182 + 0.994868i \(0.532262\pi\)
\(978\) 15.9393 0.509681
\(979\) −9.31912 −0.297840
\(980\) −0.351841 −0.0112391
\(981\) 16.3580 0.522269
\(982\) 24.1044 0.769201
\(983\) −29.8592 −0.952361 −0.476181 0.879348i \(-0.657979\pi\)
−0.476181 + 0.879348i \(0.657979\pi\)
\(984\) 2.60502 0.0830450
\(985\) 3.00628 0.0957880
\(986\) 9.74165 0.310237
\(987\) 12.6450 0.402495
\(988\) 1.08413 0.0344907
\(989\) 0 0
\(990\) 1.10379 0.0350807
\(991\) 8.95060 0.284325 0.142163 0.989843i \(-0.454594\pi\)
0.142163 + 0.989843i \(0.454594\pi\)
\(992\) −9.50557 −0.301802
\(993\) −19.5035 −0.618924
\(994\) −27.8024 −0.881838
\(995\) −8.34897 −0.264680
\(996\) 1.05529 0.0334382
\(997\) 16.7521 0.530545 0.265272 0.964174i \(-0.414538\pi\)
0.265272 + 0.964174i \(0.414538\pi\)
\(998\) 13.5901 0.430186
\(999\) 2.01732 0.0638252
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 3174.2.a.ba.1.4 5
3.2 odd 2 9522.2.a.bs.1.2 5
23.2 even 11 138.2.e.b.73.1 10
23.12 even 11 138.2.e.b.121.1 yes 10
23.22 odd 2 3174.2.a.bb.1.2 5
69.2 odd 22 414.2.i.e.73.1 10
69.35 odd 22 414.2.i.e.397.1 10
69.68 even 2 9522.2.a.br.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.73.1 10 23.2 even 11
138.2.e.b.121.1 yes 10 23.12 even 11
414.2.i.e.73.1 10 69.2 odd 22
414.2.i.e.397.1 10 69.35 odd 22
3174.2.a.ba.1.4 5 1.1 even 1 trivial
3174.2.a.bb.1.2 5 23.22 odd 2
9522.2.a.br.1.4 5 69.68 even 2
9522.2.a.bs.1.2 5 3.2 odd 2