Properties

Label 2790.2.a.bf.1.2
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2790,2,Mod(1,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, 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("2790.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.53113 q^{11} +6.00000 q^{13} -3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.53113 q^{19} +1.00000 q^{20} -1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} +3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +3.53113 q^{35} +9.06226 q^{37} +3.53113 q^{38} -1.00000 q^{40} +9.06226 q^{41} -0.468871 q^{43} +1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} +5.46887 q^{49} -1.00000 q^{50} +6.00000 q^{52} -5.53113 q^{53} +1.53113 q^{55} -3.53113 q^{56} -7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -11.0623 q^{67} +4.00000 q^{68} -3.53113 q^{70} +4.46887 q^{71} -0.468871 q^{73} -9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -0.468871 q^{79} +1.00000 q^{80} -9.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +0.468871 q^{86} -1.53113 q^{88} -1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} +11.0623 q^{94} -3.53113 q^{95} -16.1245 q^{97} -5.46887 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 5 q^{11} + 12 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 5 q^{22} - 5 q^{23} + 2 q^{25} - 12 q^{26} - q^{28} + 2 q^{31} - 2 q^{32} - 8 q^{34} - q^{35} + 2 q^{37} - q^{38} - 2 q^{40} + 2 q^{41} - 9 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 19 q^{49} - 2 q^{50} + 12 q^{52} - 3 q^{53} - 5 q^{55} + q^{56} + 2 q^{59} - 6 q^{61} - 2 q^{62} + 2 q^{64} + 12 q^{65} - 6 q^{67} + 8 q^{68} + q^{70} + 17 q^{71} - 9 q^{73} - 2 q^{74} + q^{76} + 35 q^{77} - 9 q^{79} + 2 q^{80} - 2 q^{82} + 16 q^{83} + 8 q^{85} + 9 q^{86} + 5 q^{88} + 5 q^{89} - 6 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} - 19 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.53113 1.33464 0.667321 0.744771i \(-0.267441\pi\)
0.667321 + 0.744771i \(0.267441\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.53113 0.461653 0.230826 0.972995i \(-0.425857\pi\)
0.230826 + 0.972995i \(0.425857\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −3.53113 −0.943734
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −3.53113 −0.810097 −0.405048 0.914295i \(-0.632745\pi\)
−0.405048 + 0.914295i \(0.632745\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.53113 −0.326438
\(23\) 1.53113 0.319262 0.159631 0.987177i \(-0.448969\pi\)
0.159631 + 0.987177i \(0.448969\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 3.53113 0.667321
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 3.53113 0.596870
\(36\) 0 0
\(37\) 9.06226 1.48983 0.744913 0.667162i \(-0.232491\pi\)
0.744913 + 0.667162i \(0.232491\pi\)
\(38\) 3.53113 0.572825
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.06226 1.41529 0.707643 0.706570i \(-0.249758\pi\)
0.707643 + 0.706570i \(0.249758\pi\)
\(42\) 0 0
\(43\) −0.468871 −0.0715022 −0.0357511 0.999361i \(-0.511382\pi\)
−0.0357511 + 0.999361i \(0.511382\pi\)
\(44\) 1.53113 0.230826
\(45\) 0 0
\(46\) −1.53113 −0.225753
\(47\) −11.0623 −1.61360 −0.806798 0.590827i \(-0.798801\pi\)
−0.806798 + 0.590827i \(0.798801\pi\)
\(48\) 0 0
\(49\) 5.46887 0.781267
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −5.53113 −0.759759 −0.379879 0.925036i \(-0.624035\pi\)
−0.379879 + 0.925036i \(0.624035\pi\)
\(54\) 0 0
\(55\) 1.53113 0.206457
\(56\) −3.53113 −0.471867
\(57\) 0 0
\(58\) 0 0
\(59\) −7.06226 −0.919428 −0.459714 0.888067i \(-0.652048\pi\)
−0.459714 + 0.888067i \(0.652048\pi\)
\(60\) 0 0
\(61\) −11.0623 −1.41638 −0.708188 0.706023i \(-0.750487\pi\)
−0.708188 + 0.706023i \(0.750487\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −11.0623 −1.35147 −0.675735 0.737145i \(-0.736174\pi\)
−0.675735 + 0.737145i \(0.736174\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −3.53113 −0.422051
\(71\) 4.46887 0.530357 0.265179 0.964199i \(-0.414569\pi\)
0.265179 + 0.964199i \(0.414569\pi\)
\(72\) 0 0
\(73\) −0.468871 −0.0548772 −0.0274386 0.999623i \(-0.508735\pi\)
−0.0274386 + 0.999623i \(0.508735\pi\)
\(74\) −9.06226 −1.05347
\(75\) 0 0
\(76\) −3.53113 −0.405048
\(77\) 5.40661 0.616141
\(78\) 0 0
\(79\) −0.468871 −0.0527521 −0.0263761 0.999652i \(-0.508397\pi\)
−0.0263761 + 0.999652i \(0.508397\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −9.06226 −1.00076
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0.468871 0.0505597
\(87\) 0 0
\(88\) −1.53113 −0.163219
\(89\) −1.53113 −0.162299 −0.0811497 0.996702i \(-0.525859\pi\)
−0.0811497 + 0.996702i \(0.525859\pi\)
\(90\) 0 0
\(91\) 21.1868 2.22098
\(92\) 1.53113 0.159631
\(93\) 0 0
\(94\) 11.0623 1.14098
\(95\) −3.53113 −0.362286
\(96\) 0 0
\(97\) −16.1245 −1.63720 −0.818598 0.574367i \(-0.805248\pi\)
−0.818598 + 0.574367i \(0.805248\pi\)
\(98\) −5.46887 −0.552439
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 17.5311 1.74441 0.872206 0.489138i \(-0.162689\pi\)
0.872206 + 0.489138i \(0.162689\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 5.53113 0.537231
\(107\) 14.5934 1.41080 0.705398 0.708811i \(-0.250768\pi\)
0.705398 + 0.708811i \(0.250768\pi\)
\(108\) 0 0
\(109\) −1.06226 −0.101746 −0.0508729 0.998705i \(-0.516200\pi\)
−0.0508729 + 0.998705i \(0.516200\pi\)
\(110\) −1.53113 −0.145987
\(111\) 0 0
\(112\) 3.53113 0.333660
\(113\) −16.5934 −1.56097 −0.780487 0.625172i \(-0.785029\pi\)
−0.780487 + 0.625172i \(0.785029\pi\)
\(114\) 0 0
\(115\) 1.53113 0.142779
\(116\) 0 0
\(117\) 0 0
\(118\) 7.06226 0.650134
\(119\) 14.1245 1.29479
\(120\) 0 0
\(121\) −8.65564 −0.786877
\(122\) 11.0623 1.00153
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.06226 −0.804145 −0.402073 0.915608i \(-0.631710\pi\)
−0.402073 + 0.915608i \(0.631710\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) −7.06226 −0.617032 −0.308516 0.951219i \(-0.599832\pi\)
−0.308516 + 0.951219i \(0.599832\pi\)
\(132\) 0 0
\(133\) −12.4689 −1.08119
\(134\) 11.0623 0.955634
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −0.937742 −0.0801167 −0.0400584 0.999197i \(-0.512754\pi\)
−0.0400584 + 0.999197i \(0.512754\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 3.53113 0.298435
\(141\) 0 0
\(142\) −4.46887 −0.375019
\(143\) 9.18677 0.768237
\(144\) 0 0
\(145\) 0 0
\(146\) 0.468871 0.0388041
\(147\) 0 0
\(148\) 9.06226 0.744913
\(149\) 10.4689 0.857643 0.428822 0.903389i \(-0.358929\pi\)
0.428822 + 0.903389i \(0.358929\pi\)
\(150\) 0 0
\(151\) −18.1245 −1.47495 −0.737476 0.675373i \(-0.763983\pi\)
−0.737476 + 0.675373i \(0.763983\pi\)
\(152\) 3.53113 0.286412
\(153\) 0 0
\(154\) −5.40661 −0.435677
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −14.4689 −1.15474 −0.577371 0.816482i \(-0.695921\pi\)
−0.577371 + 0.816482i \(0.695921\pi\)
\(158\) 0.468871 0.0373014
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 5.40661 0.426101
\(162\) 0 0
\(163\) −11.0623 −0.866463 −0.433231 0.901283i \(-0.642627\pi\)
−0.433231 + 0.901283i \(0.642627\pi\)
\(164\) 9.06226 0.707643
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −0.593387 −0.0459176 −0.0229588 0.999736i \(-0.507309\pi\)
−0.0229588 + 0.999736i \(0.507309\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −0.468871 −0.0357511
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 3.53113 0.266928
\(176\) 1.53113 0.115413
\(177\) 0 0
\(178\) 1.53113 0.114763
\(179\) −13.0623 −0.976319 −0.488159 0.872754i \(-0.662332\pi\)
−0.488159 + 0.872754i \(0.662332\pi\)
\(180\) 0 0
\(181\) 26.5934 1.97667 0.988335 0.152293i \(-0.0486657\pi\)
0.988335 + 0.152293i \(0.0486657\pi\)
\(182\) −21.1868 −1.57047
\(183\) 0 0
\(184\) −1.53113 −0.112876
\(185\) 9.06226 0.666270
\(186\) 0 0
\(187\) 6.12452 0.447869
\(188\) −11.0623 −0.806798
\(189\) 0 0
\(190\) 3.53113 0.256175
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 16.1245 1.15767
\(195\) 0 0
\(196\) 5.46887 0.390634
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −0.468871 −0.0332374 −0.0166187 0.999862i \(-0.505290\pi\)
−0.0166187 + 0.999862i \(0.505290\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −17.5311 −1.23349
\(203\) 0 0
\(204\) 0 0
\(205\) 9.06226 0.632936
\(206\) 0 0
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −5.40661 −0.373983
\(210\) 0 0
\(211\) 22.5934 1.55539 0.777696 0.628640i \(-0.216388\pi\)
0.777696 + 0.628640i \(0.216388\pi\)
\(212\) −5.53113 −0.379879
\(213\) 0 0
\(214\) −14.5934 −0.997583
\(215\) −0.468871 −0.0319767
\(216\) 0 0
\(217\) 3.53113 0.239709
\(218\) 1.06226 0.0719452
\(219\) 0 0
\(220\) 1.53113 0.103229
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −3.53113 −0.235933
\(225\) 0 0
\(226\) 16.5934 1.10378
\(227\) 16.4689 1.09308 0.546539 0.837434i \(-0.315945\pi\)
0.546539 + 0.837434i \(0.315945\pi\)
\(228\) 0 0
\(229\) 18.5934 1.22869 0.614343 0.789039i \(-0.289421\pi\)
0.614343 + 0.789039i \(0.289421\pi\)
\(230\) −1.53113 −0.100960
\(231\) 0 0
\(232\) 0 0
\(233\) −9.53113 −0.624405 −0.312203 0.950016i \(-0.601067\pi\)
−0.312203 + 0.950016i \(0.601067\pi\)
\(234\) 0 0
\(235\) −11.0623 −0.721622
\(236\) −7.06226 −0.459714
\(237\) 0 0
\(238\) −14.1245 −0.915556
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 8.65564 0.556406
\(243\) 0 0
\(244\) −11.0623 −0.708188
\(245\) 5.46887 0.349393
\(246\) 0 0
\(247\) −21.1868 −1.34808
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 13.0623 0.824482 0.412241 0.911075i \(-0.364746\pi\)
0.412241 + 0.911075i \(0.364746\pi\)
\(252\) 0 0
\(253\) 2.34436 0.147388
\(254\) 9.06226 0.568617
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.6556 1.72511 0.862556 0.505962i \(-0.168862\pi\)
0.862556 + 0.505962i \(0.168862\pi\)
\(258\) 0 0
\(259\) 32.0000 1.98838
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 7.06226 0.436308
\(263\) 6.93774 0.427800 0.213900 0.976856i \(-0.431383\pi\)
0.213900 + 0.976856i \(0.431383\pi\)
\(264\) 0 0
\(265\) −5.53113 −0.339775
\(266\) 12.4689 0.764516
\(267\) 0 0
\(268\) −11.0623 −0.675735
\(269\) −29.1868 −1.77955 −0.889774 0.456400i \(-0.849138\pi\)
−0.889774 + 0.456400i \(0.849138\pi\)
\(270\) 0 0
\(271\) −19.5311 −1.18643 −0.593216 0.805043i \(-0.702142\pi\)
−0.593216 + 0.805043i \(0.702142\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 0.937742 0.0566511
\(275\) 1.53113 0.0923305
\(276\) 0 0
\(277\) 1.06226 0.0638249 0.0319124 0.999491i \(-0.489840\pi\)
0.0319124 + 0.999491i \(0.489840\pi\)
\(278\) −6.00000 −0.359856
\(279\) 0 0
\(280\) −3.53113 −0.211025
\(281\) 1.06226 0.0633690 0.0316845 0.999498i \(-0.489913\pi\)
0.0316845 + 0.999498i \(0.489913\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 4.46887 0.265179
\(285\) 0 0
\(286\) −9.18677 −0.543225
\(287\) 32.0000 1.88890
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −0.468871 −0.0274386
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −7.06226 −0.411181
\(296\) −9.06226 −0.526733
\(297\) 0 0
\(298\) −10.4689 −0.606445
\(299\) 9.18677 0.531285
\(300\) 0 0
\(301\) −1.65564 −0.0954298
\(302\) 18.1245 1.04295
\(303\) 0 0
\(304\) −3.53113 −0.202524
\(305\) −11.0623 −0.633423
\(306\) 0 0
\(307\) 2.12452 0.121253 0.0606263 0.998161i \(-0.480690\pi\)
0.0606263 + 0.998161i \(0.480690\pi\)
\(308\) 5.40661 0.308070
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 27.0623 1.52965 0.764825 0.644239i \(-0.222826\pi\)
0.764825 + 0.644239i \(0.222826\pi\)
\(314\) 14.4689 0.816526
\(315\) 0 0
\(316\) −0.468871 −0.0263761
\(317\) −29.0623 −1.63230 −0.816150 0.577841i \(-0.803895\pi\)
−0.816150 + 0.577841i \(0.803895\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −5.40661 −0.301299
\(323\) −14.1245 −0.785909
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 11.0623 0.612682
\(327\) 0 0
\(328\) −9.06226 −0.500379
\(329\) −39.0623 −2.15357
\(330\) 0 0
\(331\) 29.0623 1.59741 0.798703 0.601725i \(-0.205520\pi\)
0.798703 + 0.601725i \(0.205520\pi\)
\(332\) 8.00000 0.439057
\(333\) 0 0
\(334\) 0.593387 0.0324687
\(335\) −11.0623 −0.604396
\(336\) 0 0
\(337\) 29.1868 1.58990 0.794952 0.606672i \(-0.207496\pi\)
0.794952 + 0.606672i \(0.207496\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 1.53113 0.0829153
\(342\) 0 0
\(343\) −5.40661 −0.291930
\(344\) 0.468871 0.0252798
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 22.1245 1.18771 0.593853 0.804573i \(-0.297606\pi\)
0.593853 + 0.804573i \(0.297606\pi\)
\(348\) 0 0
\(349\) 25.0623 1.34155 0.670776 0.741660i \(-0.265961\pi\)
0.670776 + 0.741660i \(0.265961\pi\)
\(350\) −3.53113 −0.188747
\(351\) 0 0
\(352\) −1.53113 −0.0816094
\(353\) 23.0623 1.22748 0.613740 0.789508i \(-0.289664\pi\)
0.613740 + 0.789508i \(0.289664\pi\)
\(354\) 0 0
\(355\) 4.46887 0.237183
\(356\) −1.53113 −0.0811497
\(357\) 0 0
\(358\) 13.0623 0.690362
\(359\) −11.5311 −0.608590 −0.304295 0.952578i \(-0.598421\pi\)
−0.304295 + 0.952578i \(0.598421\pi\)
\(360\) 0 0
\(361\) −6.53113 −0.343744
\(362\) −26.5934 −1.39772
\(363\) 0 0
\(364\) 21.1868 1.11049
\(365\) −0.468871 −0.0245418
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 1.53113 0.0798156
\(369\) 0 0
\(370\) −9.06226 −0.471124
\(371\) −19.5311 −1.01401
\(372\) 0 0
\(373\) −10.4689 −0.542058 −0.271029 0.962571i \(-0.587364\pi\)
−0.271029 + 0.962571i \(0.587364\pi\)
\(374\) −6.12452 −0.316691
\(375\) 0 0
\(376\) 11.0623 0.570492
\(377\) 0 0
\(378\) 0 0
\(379\) −2.59339 −0.133213 −0.0666067 0.997779i \(-0.521217\pi\)
−0.0666067 + 0.997779i \(0.521217\pi\)
\(380\) −3.53113 −0.181143
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −13.0623 −0.667450 −0.333725 0.942670i \(-0.608306\pi\)
−0.333725 + 0.942670i \(0.608306\pi\)
\(384\) 0 0
\(385\) 5.40661 0.275547
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −16.1245 −0.818598
\(389\) −27.0623 −1.37211 −0.686055 0.727549i \(-0.740659\pi\)
−0.686055 + 0.727549i \(0.740659\pi\)
\(390\) 0 0
\(391\) 6.12452 0.309730
\(392\) −5.46887 −0.276220
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −0.468871 −0.0235915
\(396\) 0 0
\(397\) −18.7179 −0.939425 −0.469712 0.882820i \(-0.655642\pi\)
−0.469712 + 0.882820i \(0.655642\pi\)
\(398\) 0.468871 0.0235024
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.7179 1.73373 0.866865 0.498544i \(-0.166132\pi\)
0.866865 + 0.498544i \(0.166132\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 17.5311 0.872206
\(405\) 0 0
\(406\) 0 0
\(407\) 13.8755 0.687782
\(408\) 0 0
\(409\) −9.06226 −0.448100 −0.224050 0.974578i \(-0.571928\pi\)
−0.224050 + 0.974578i \(0.571928\pi\)
\(410\) −9.06226 −0.447553
\(411\) 0 0
\(412\) 0 0
\(413\) −24.9377 −1.22711
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 5.40661 0.264446
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 38.2490 1.86414 0.932072 0.362273i \(-0.117999\pi\)
0.932072 + 0.362273i \(0.117999\pi\)
\(422\) −22.5934 −1.09983
\(423\) 0 0
\(424\) 5.53113 0.268615
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −39.0623 −1.89036
\(428\) 14.5934 0.705398
\(429\) 0 0
\(430\) 0.468871 0.0226110
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −1.40661 −0.0675975 −0.0337988 0.999429i \(-0.510761\pi\)
−0.0337988 + 0.999429i \(0.510761\pi\)
\(434\) −3.53113 −0.169500
\(435\) 0 0
\(436\) −1.06226 −0.0508729
\(437\) −5.40661 −0.258633
\(438\) 0 0
\(439\) −8.93774 −0.426575 −0.213288 0.976989i \(-0.568417\pi\)
−0.213288 + 0.976989i \(0.568417\pi\)
\(440\) −1.53113 −0.0729937
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) −38.5934 −1.83363 −0.916814 0.399316i \(-0.869248\pi\)
−0.916814 + 0.399316i \(0.869248\pi\)
\(444\) 0 0
\(445\) −1.53113 −0.0725825
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) 3.53113 0.166830
\(449\) −28.1245 −1.32728 −0.663639 0.748053i \(-0.730989\pi\)
−0.663639 + 0.748053i \(0.730989\pi\)
\(450\) 0 0
\(451\) 13.8755 0.653371
\(452\) −16.5934 −0.780487
\(453\) 0 0
\(454\) −16.4689 −0.772922
\(455\) 21.1868 0.993251
\(456\) 0 0
\(457\) 31.0623 1.45303 0.726516 0.687150i \(-0.241138\pi\)
0.726516 + 0.687150i \(0.241138\pi\)
\(458\) −18.5934 −0.868812
\(459\) 0 0
\(460\) 1.53113 0.0713893
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) −35.1868 −1.63527 −0.817634 0.575738i \(-0.804715\pi\)
−0.817634 + 0.575738i \(0.804715\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 9.53113 0.441521
\(467\) −10.1245 −0.468507 −0.234253 0.972176i \(-0.575265\pi\)
−0.234253 + 0.972176i \(0.575265\pi\)
\(468\) 0 0
\(469\) −39.0623 −1.80373
\(470\) 11.0623 0.510264
\(471\) 0 0
\(472\) 7.06226 0.325067
\(473\) −0.717902 −0.0330092
\(474\) 0 0
\(475\) −3.53113 −0.162019
\(476\) 14.1245 0.647396
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) −41.6556 −1.90329 −0.951647 0.307192i \(-0.900611\pi\)
−0.951647 + 0.307192i \(0.900611\pi\)
\(480\) 0 0
\(481\) 54.3735 2.47922
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −8.65564 −0.393438
\(485\) −16.1245 −0.732177
\(486\) 0 0
\(487\) 6.93774 0.314379 0.157190 0.987568i \(-0.449757\pi\)
0.157190 + 0.987568i \(0.449757\pi\)
\(488\) 11.0623 0.500765
\(489\) 0 0
\(490\) −5.46887 −0.247058
\(491\) −16.5934 −0.748849 −0.374425 0.927257i \(-0.622160\pi\)
−0.374425 + 0.927257i \(0.622160\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 21.1868 0.953238
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 15.7802 0.707837
\(498\) 0 0
\(499\) 9.06226 0.405682 0.202841 0.979212i \(-0.434982\pi\)
0.202841 + 0.979212i \(0.434982\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −13.0623 −0.582997
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 17.5311 0.780125
\(506\) −2.34436 −0.104219
\(507\) 0 0
\(508\) −9.06226 −0.402073
\(509\) −5.87548 −0.260426 −0.130213 0.991486i \(-0.541566\pi\)
−0.130213 + 0.991486i \(0.541566\pi\)
\(510\) 0 0
\(511\) −1.65564 −0.0732414
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −27.6556 −1.21984
\(515\) 0 0
\(516\) 0 0
\(517\) −16.9377 −0.744921
\(518\) −32.0000 −1.40600
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −20.1245 −0.881671 −0.440836 0.897588i \(-0.645318\pi\)
−0.440836 + 0.897588i \(0.645318\pi\)
\(522\) 0 0
\(523\) −14.5934 −0.638124 −0.319062 0.947734i \(-0.603368\pi\)
−0.319062 + 0.947734i \(0.603368\pi\)
\(524\) −7.06226 −0.308516
\(525\) 0 0
\(526\) −6.93774 −0.302500
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −20.6556 −0.898071
\(530\) 5.53113 0.240257
\(531\) 0 0
\(532\) −12.4689 −0.540594
\(533\) 54.3735 2.35518
\(534\) 0 0
\(535\) 14.5934 0.630927
\(536\) 11.0623 0.477817
\(537\) 0 0
\(538\) 29.1868 1.25833
\(539\) 8.37355 0.360674
\(540\) 0 0
\(541\) 28.1245 1.20917 0.604584 0.796542i \(-0.293340\pi\)
0.604584 + 0.796542i \(0.293340\pi\)
\(542\) 19.5311 0.838934
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −1.06226 −0.0455021
\(546\) 0 0
\(547\) 41.1868 1.76102 0.880510 0.474028i \(-0.157201\pi\)
0.880510 + 0.474028i \(0.157201\pi\)
\(548\) −0.937742 −0.0400584
\(549\) 0 0
\(550\) −1.53113 −0.0652876
\(551\) 0 0
\(552\) 0 0
\(553\) −1.65564 −0.0704052
\(554\) −1.06226 −0.0451310
\(555\) 0 0
\(556\) 6.00000 0.254457
\(557\) −45.7802 −1.93977 −0.969884 0.243568i \(-0.921682\pi\)
−0.969884 + 0.243568i \(0.921682\pi\)
\(558\) 0 0
\(559\) −2.81323 −0.118987
\(560\) 3.53113 0.149217
\(561\) 0 0
\(562\) −1.06226 −0.0448086
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −16.5934 −0.698089
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) −4.46887 −0.187510
\(569\) −17.5311 −0.734943 −0.367472 0.930035i \(-0.619776\pi\)
−0.367472 + 0.930035i \(0.619776\pi\)
\(570\) 0 0
\(571\) 3.87548 0.162184 0.0810920 0.996707i \(-0.474159\pi\)
0.0810920 + 0.996707i \(0.474159\pi\)
\(572\) 9.18677 0.384118
\(573\) 0 0
\(574\) −32.0000 −1.33565
\(575\) 1.53113 0.0638525
\(576\) 0 0
\(577\) −11.1868 −0.465711 −0.232856 0.972511i \(-0.574807\pi\)
−0.232856 + 0.972511i \(0.574807\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 28.2490 1.17197
\(582\) 0 0
\(583\) −8.46887 −0.350745
\(584\) 0.468871 0.0194020
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −20.2490 −0.835767 −0.417883 0.908501i \(-0.637228\pi\)
−0.417883 + 0.908501i \(0.637228\pi\)
\(588\) 0 0
\(589\) −3.53113 −0.145498
\(590\) 7.06226 0.290749
\(591\) 0 0
\(592\) 9.06226 0.372456
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 14.1245 0.579049
\(596\) 10.4689 0.428822
\(597\) 0 0
\(598\) −9.18677 −0.375675
\(599\) −10.5934 −0.432834 −0.216417 0.976301i \(-0.569437\pi\)
−0.216417 + 0.976301i \(0.569437\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 1.65564 0.0674790
\(603\) 0 0
\(604\) −18.1245 −0.737476
\(605\) −8.65564 −0.351902
\(606\) 0 0
\(607\) −38.5934 −1.56646 −0.783229 0.621734i \(-0.786429\pi\)
−0.783229 + 0.621734i \(0.786429\pi\)
\(608\) 3.53113 0.143206
\(609\) 0 0
\(610\) 11.0623 0.447898
\(611\) −66.3735 −2.68519
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) −2.12452 −0.0857385
\(615\) 0 0
\(616\) −5.40661 −0.217839
\(617\) −20.5934 −0.829059 −0.414529 0.910036i \(-0.636054\pi\)
−0.414529 + 0.910036i \(0.636054\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) −5.40661 −0.216611
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −27.0623 −1.08163
\(627\) 0 0
\(628\) −14.4689 −0.577371
\(629\) 36.2490 1.44534
\(630\) 0 0
\(631\) −9.65564 −0.384385 −0.192193 0.981357i \(-0.561560\pi\)
−0.192193 + 0.981357i \(0.561560\pi\)
\(632\) 0.468871 0.0186507
\(633\) 0 0
\(634\) 29.0623 1.15421
\(635\) −9.06226 −0.359625
\(636\) 0 0
\(637\) 32.8132 1.30011
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) 6.59339 0.260018 0.130009 0.991513i \(-0.458499\pi\)
0.130009 + 0.991513i \(0.458499\pi\)
\(644\) 5.40661 0.213050
\(645\) 0 0
\(646\) 14.1245 0.555722
\(647\) −1.53113 −0.0601949 −0.0300974 0.999547i \(-0.509582\pi\)
−0.0300974 + 0.999547i \(0.509582\pi\)
\(648\) 0 0
\(649\) −10.8132 −0.424456
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −11.0623 −0.433231
\(653\) 26.2490 1.02720 0.513602 0.858029i \(-0.328311\pi\)
0.513602 + 0.858029i \(0.328311\pi\)
\(654\) 0 0
\(655\) −7.06226 −0.275945
\(656\) 9.06226 0.353822
\(657\) 0 0
\(658\) 39.0623 1.52281
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −29.0623 −1.12954
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) −12.4689 −0.483522
\(666\) 0 0
\(667\) 0 0
\(668\) −0.593387 −0.0229588
\(669\) 0 0
\(670\) 11.0623 0.427372
\(671\) −16.9377 −0.653874
\(672\) 0 0
\(673\) 7.06226 0.272230 0.136115 0.990693i \(-0.456538\pi\)
0.136115 + 0.990693i \(0.456538\pi\)
\(674\) −29.1868 −1.12423
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −3.40661 −0.130927 −0.0654634 0.997855i \(-0.520853\pi\)
−0.0654634 + 0.997855i \(0.520853\pi\)
\(678\) 0 0
\(679\) −56.9377 −2.18507
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) −1.53113 −0.0586300
\(683\) −15.5311 −0.594282 −0.297141 0.954834i \(-0.596033\pi\)
−0.297141 + 0.954834i \(0.596033\pi\)
\(684\) 0 0
\(685\) −0.937742 −0.0358293
\(686\) 5.40661 0.206425
\(687\) 0 0
\(688\) −0.468871 −0.0178755
\(689\) −33.1868 −1.26432
\(690\) 0 0
\(691\) −7.53113 −0.286498 −0.143249 0.989687i \(-0.545755\pi\)
−0.143249 + 0.989687i \(0.545755\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −22.1245 −0.839835
\(695\) 6.00000 0.227593
\(696\) 0 0
\(697\) 36.2490 1.37303
\(698\) −25.0623 −0.948620
\(699\) 0 0
\(700\) 3.53113 0.133464
\(701\) 21.5311 0.813220 0.406610 0.913602i \(-0.366711\pi\)
0.406610 + 0.913602i \(0.366711\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 1.53113 0.0577066
\(705\) 0 0
\(706\) −23.0623 −0.867960
\(707\) 61.9047 2.32816
\(708\) 0 0
\(709\) 25.4066 0.954165 0.477083 0.878858i \(-0.341694\pi\)
0.477083 + 0.878858i \(0.341694\pi\)
\(710\) −4.46887 −0.167714
\(711\) 0 0
\(712\) 1.53113 0.0573815
\(713\) 1.53113 0.0573412
\(714\) 0 0
\(715\) 9.18677 0.343566
\(716\) −13.0623 −0.488159
\(717\) 0 0
\(718\) 11.5311 0.430338
\(719\) 33.1868 1.23766 0.618829 0.785526i \(-0.287607\pi\)
0.618829 + 0.785526i \(0.287607\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.53113 0.243063
\(723\) 0 0
\(724\) 26.5934 0.988335
\(725\) 0 0
\(726\) 0 0
\(727\) −50.8424 −1.88564 −0.942820 0.333301i \(-0.891837\pi\)
−0.942820 + 0.333301i \(0.891837\pi\)
\(728\) −21.1868 −0.785234
\(729\) 0 0
\(730\) 0.468871 0.0173537
\(731\) −1.87548 −0.0693673
\(732\) 0 0
\(733\) 4.12452 0.152342 0.0761712 0.997095i \(-0.475730\pi\)
0.0761712 + 0.997095i \(0.475730\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −1.53113 −0.0564382
\(737\) −16.9377 −0.623910
\(738\) 0 0
\(739\) 27.1868 1.00008 0.500041 0.866002i \(-0.333318\pi\)
0.500041 + 0.866002i \(0.333318\pi\)
\(740\) 9.06226 0.333135
\(741\) 0 0
\(742\) 19.5311 0.717010
\(743\) −11.6556 −0.427604 −0.213802 0.976877i \(-0.568585\pi\)
−0.213802 + 0.976877i \(0.568585\pi\)
\(744\) 0 0
\(745\) 10.4689 0.383550
\(746\) 10.4689 0.383293
\(747\) 0 0
\(748\) 6.12452 0.223934
\(749\) 51.5311 1.88291
\(750\) 0 0
\(751\) 47.0623 1.71733 0.858663 0.512540i \(-0.171296\pi\)
0.858663 + 0.512540i \(0.171296\pi\)
\(752\) −11.0623 −0.403399
\(753\) 0 0
\(754\) 0 0
\(755\) −18.1245 −0.659619
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 2.59339 0.0941960
\(759\) 0 0
\(760\) 3.53113 0.128088
\(761\) 12.5934 0.456510 0.228255 0.973601i \(-0.426698\pi\)
0.228255 + 0.973601i \(0.426698\pi\)
\(762\) 0 0
\(763\) −3.75097 −0.135794
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 13.0623 0.471959
\(767\) −42.3735 −1.53002
\(768\) 0 0
\(769\) −28.5934 −1.03110 −0.515552 0.856858i \(-0.672413\pi\)
−0.515552 + 0.856858i \(0.672413\pi\)
\(770\) −5.40661 −0.194841
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −14.4689 −0.520409 −0.260205 0.965554i \(-0.583790\pi\)
−0.260205 + 0.965554i \(0.583790\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 16.1245 0.578836
\(777\) 0 0
\(778\) 27.0623 0.970229
\(779\) −32.0000 −1.14652
\(780\) 0 0
\(781\) 6.84242 0.244841
\(782\) −6.12452 −0.219012
\(783\) 0 0
\(784\) 5.46887 0.195317
\(785\) −14.4689 −0.516416
\(786\) 0 0
\(787\) −28.4689 −1.01481 −0.507403 0.861709i \(-0.669394\pi\)
−0.507403 + 0.861709i \(0.669394\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0.468871 0.0166817
\(791\) −58.5934 −2.08334
\(792\) 0 0
\(793\) −66.3735 −2.35699
\(794\) 18.7179 0.664273
\(795\) 0 0
\(796\) −0.468871 −0.0166187
\(797\) 20.1245 0.712847 0.356423 0.934325i \(-0.383996\pi\)
0.356423 + 0.934325i \(0.383996\pi\)
\(798\) 0 0
\(799\) −44.2490 −1.56542
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −34.7179 −1.22593
\(803\) −0.717902 −0.0253342
\(804\) 0 0
\(805\) 5.40661 0.190558
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) −17.5311 −0.616743
\(809\) 8.59339 0.302127 0.151064 0.988524i \(-0.451730\pi\)
0.151064 + 0.988524i \(0.451730\pi\)
\(810\) 0 0
\(811\) −39.7802 −1.39687 −0.698435 0.715673i \(-0.746120\pi\)
−0.698435 + 0.715673i \(0.746120\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −13.8755 −0.486335
\(815\) −11.0623 −0.387494
\(816\) 0 0
\(817\) 1.65564 0.0579237
\(818\) 9.06226 0.316854
\(819\) 0 0
\(820\) 9.06226 0.316468
\(821\) −9.87548 −0.344657 −0.172328 0.985040i \(-0.555129\pi\)
−0.172328 + 0.985040i \(0.555129\pi\)
\(822\) 0 0
\(823\) 7.87548 0.274522 0.137261 0.990535i \(-0.456170\pi\)
0.137261 + 0.990535i \(0.456170\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 24.9377 0.867695
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) −9.65564 −0.335354 −0.167677 0.985842i \(-0.553627\pi\)
−0.167677 + 0.985842i \(0.553627\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 6.00000 0.208013
\(833\) 21.8755 0.757941
\(834\) 0 0
\(835\) −0.593387 −0.0205350
\(836\) −5.40661 −0.186992
\(837\) 0 0
\(838\) 0 0
\(839\) 54.8424 1.89337 0.946685 0.322160i \(-0.104409\pi\)
0.946685 + 0.322160i \(0.104409\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −38.2490 −1.31815
\(843\) 0 0
\(844\) 22.5934 0.777696
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −30.5642 −1.05020
\(848\) −5.53113 −0.189940
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 13.8755 0.475645
\(852\) 0 0
\(853\) 1.28210 0.0438982 0.0219491 0.999759i \(-0.493013\pi\)
0.0219491 + 0.999759i \(0.493013\pi\)
\(854\) 39.0623 1.33668
\(855\) 0 0
\(856\) −14.5934 −0.498792
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 0 0
\(859\) −6.93774 −0.236713 −0.118356 0.992971i \(-0.537763\pi\)
−0.118356 + 0.992971i \(0.537763\pi\)
\(860\) −0.468871 −0.0159884
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −41.5311 −1.41374 −0.706868 0.707345i \(-0.749893\pi\)
−0.706868 + 0.707345i \(0.749893\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 1.40661 0.0477987
\(867\) 0 0
\(868\) 3.53113 0.119854
\(869\) −0.717902 −0.0243532
\(870\) 0 0
\(871\) −66.3735 −2.24898
\(872\) 1.06226 0.0359726
\(873\) 0 0
\(874\) 5.40661 0.182881
\(875\) 3.53113 0.119374
\(876\) 0 0
\(877\) 44.1245 1.48998 0.744990 0.667076i \(-0.232454\pi\)
0.744990 + 0.667076i \(0.232454\pi\)
\(878\) 8.93774 0.301634
\(879\) 0 0
\(880\) 1.53113 0.0516143
\(881\) −36.1245 −1.21707 −0.608533 0.793529i \(-0.708242\pi\)
−0.608533 + 0.793529i \(0.708242\pi\)
\(882\) 0 0
\(883\) −53.6556 −1.80566 −0.902828 0.430002i \(-0.858513\pi\)
−0.902828 + 0.430002i \(0.858513\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 38.5934 1.29657
\(887\) 25.1868 0.845689 0.422845 0.906202i \(-0.361032\pi\)
0.422845 + 0.906202i \(0.361032\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 1.53113 0.0513236
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 39.0623 1.30717
\(894\) 0 0
\(895\) −13.0623 −0.436623
\(896\) −3.53113 −0.117967
\(897\) 0 0
\(898\) 28.1245 0.938527
\(899\) 0 0
\(900\) 0 0
\(901\) −22.1245 −0.737074
\(902\) −13.8755 −0.462003
\(903\) 0 0
\(904\) 16.5934 0.551888
\(905\) 26.5934 0.883994
\(906\) 0 0
\(907\) −32.2490 −1.07081 −0.535406 0.844595i \(-0.679841\pi\)
−0.535406 + 0.844595i \(0.679841\pi\)
\(908\) 16.4689 0.546539
\(909\) 0 0
\(910\) −21.1868 −0.702335
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 12.2490 0.405384
\(914\) −31.0623 −1.02745
\(915\) 0 0
\(916\) 18.5934 0.614343
\(917\) −24.9377 −0.823517
\(918\) 0 0
\(919\) −8.93774 −0.294829 −0.147414 0.989075i \(-0.547095\pi\)
−0.147414 + 0.989075i \(0.547095\pi\)
\(920\) −1.53113 −0.0504798
\(921\) 0 0
\(922\) 24.0000 0.790398
\(923\) 26.8132 0.882568
\(924\) 0 0
\(925\) 9.06226 0.297965
\(926\) 35.1868 1.15631
\(927\) 0 0
\(928\) 0 0
\(929\) −36.8424 −1.20876 −0.604380 0.796696i \(-0.706579\pi\)
−0.604380 + 0.796696i \(0.706579\pi\)
\(930\) 0 0
\(931\) −19.3113 −0.632902
\(932\) −9.53113 −0.312203
\(933\) 0 0
\(934\) 10.1245 0.331284
\(935\) 6.12452 0.200293
\(936\) 0 0
\(937\) −25.0623 −0.818748 −0.409374 0.912367i \(-0.634253\pi\)
−0.409374 + 0.912367i \(0.634253\pi\)
\(938\) 39.0623 1.27543
\(939\) 0 0
\(940\) −11.0623 −0.360811
\(941\) 21.8755 0.713120 0.356560 0.934272i \(-0.383949\pi\)
0.356560 + 0.934272i \(0.383949\pi\)
\(942\) 0 0
\(943\) 13.8755 0.451848
\(944\) −7.06226 −0.229857
\(945\) 0 0
\(946\) 0.717902 0.0233410
\(947\) −35.0623 −1.13937 −0.569685 0.821863i \(-0.692935\pi\)
−0.569685 + 0.821863i \(0.692935\pi\)
\(948\) 0 0
\(949\) −2.81323 −0.0913212
\(950\) 3.53113 0.114565
\(951\) 0 0
\(952\) −14.1245 −0.457778
\(953\) −54.1245 −1.75327 −0.876633 0.481161i \(-0.840215\pi\)
−0.876633 + 0.481161i \(0.840215\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 41.6556 1.34583
\(959\) −3.31129 −0.106927
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −54.3735 −1.75307
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −17.0623 −0.548685 −0.274343 0.961632i \(-0.588460\pi\)
−0.274343 + 0.961632i \(0.588460\pi\)
\(968\) 8.65564 0.278203
\(969\) 0 0
\(970\) 16.1245 0.517727
\(971\) 33.1868 1.06501 0.532507 0.846426i \(-0.321250\pi\)
0.532507 + 0.846426i \(0.321250\pi\)
\(972\) 0 0
\(973\) 21.1868 0.679217
\(974\) −6.93774 −0.222300
\(975\) 0 0
\(976\) −11.0623 −0.354094
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) −2.34436 −0.0749259
\(980\) 5.46887 0.174697
\(981\) 0 0
\(982\) 16.5934 0.529516
\(983\) −17.0623 −0.544202 −0.272101 0.962269i \(-0.587718\pi\)
−0.272101 + 0.962269i \(0.587718\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) −21.1868 −0.674041
\(989\) −0.717902 −0.0228280
\(990\) 0 0
\(991\) −24.4689 −0.777279 −0.388640 0.921390i \(-0.627055\pi\)
−0.388640 + 0.921390i \(0.627055\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) −15.7802 −0.500516
\(995\) −0.468871 −0.0148642
\(996\) 0 0
\(997\) 62.2490 1.97145 0.985723 0.168373i \(-0.0538514\pi\)
0.985723 + 0.168373i \(0.0538514\pi\)
\(998\) −9.06226 −0.286861
\(999\) 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 2790.2.a.bf.1.2 2
3.2 odd 2 930.2.a.q.1.2 2
12.11 even 2 7440.2.a.bd.1.1 2
15.2 even 4 4650.2.d.bg.3349.4 4
15.8 even 4 4650.2.d.bg.3349.1 4
15.14 odd 2 4650.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.2 2 3.2 odd 2
2790.2.a.bf.1.2 2 1.1 even 1 trivial
4650.2.a.bz.1.1 2 15.14 odd 2
4650.2.d.bg.3349.1 4 15.8 even 4
4650.2.d.bg.3349.4 4 15.2 even 4
7440.2.a.bd.1.1 2 12.11 even 2