Properties

Label 7605.2.a.bb.1.1
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $2$
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] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +2.30278 q^{10} +1.60555 q^{11} -2.30278 q^{14} +0.302776 q^{16} -7.60555 q^{17} +5.60555 q^{19} -3.30278 q^{20} -3.69722 q^{22} +3.00000 q^{23} +1.00000 q^{25} +3.30278 q^{28} +6.21110 q^{29} +4.00000 q^{31} +5.30278 q^{32} +17.5139 q^{34} -1.00000 q^{35} -3.60555 q^{37} -12.9083 q^{38} +3.00000 q^{40} +3.00000 q^{41} -10.2111 q^{43} +5.30278 q^{44} -6.90833 q^{46} +9.21110 q^{47} -6.00000 q^{49} -2.30278 q^{50} +3.21110 q^{53} -1.60555 q^{55} -3.00000 q^{56} -14.3028 q^{58} -10.8167 q^{59} -1.00000 q^{61} -9.21110 q^{62} -12.8167 q^{64} +7.00000 q^{67} -25.1194 q^{68} +2.30278 q^{70} +4.81665 q^{71} +0.788897 q^{73} +8.30278 q^{74} +18.5139 q^{76} +1.60555 q^{77} +5.21110 q^{79} -0.302776 q^{80} -6.90833 q^{82} -9.21110 q^{83} +7.60555 q^{85} +23.5139 q^{86} -4.81665 q^{88} -6.21110 q^{89} +9.90833 q^{92} -21.2111 q^{94} -5.60555 q^{95} +8.39445 q^{97} +13.8167 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - 4 q^{11} - q^{14} - 3 q^{16} - 8 q^{17} + 4 q^{19} - 3 q^{20} - 11 q^{22} + 6 q^{23} + 2 q^{25} + 3 q^{28} - 2 q^{29} + 8 q^{31} + 7 q^{32} + 17 q^{34} - 2 q^{35} - 15 q^{38} + 6 q^{40} + 6 q^{41} - 6 q^{43} + 7 q^{44} - 3 q^{46} + 4 q^{47} - 12 q^{49} - q^{50} - 8 q^{53} + 4 q^{55} - 6 q^{56} - 25 q^{58} - 2 q^{61} - 4 q^{62} - 4 q^{64} + 14 q^{67} - 25 q^{68} + q^{70} - 12 q^{71} + 16 q^{73} + 13 q^{74} + 19 q^{76} - 4 q^{77} - 4 q^{79} + 3 q^{80} - 3 q^{82} - 4 q^{83} + 8 q^{85} + 29 q^{86} + 12 q^{88} + 2 q^{89} + 9 q^{92} - 28 q^{94} - 4 q^{95} + 24 q^{97} + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30278 −1.62831 −0.814154 0.580649i \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) 0 0
\(4\) 3.30278 1.65139
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 2.30278 0.728202
\(11\) 1.60555 0.484092 0.242046 0.970265i \(-0.422182\pi\)
0.242046 + 0.970265i \(0.422182\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −2.30278 −0.615443
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) −7.60555 −1.84462 −0.922309 0.386454i \(-0.873700\pi\)
−0.922309 + 0.386454i \(0.873700\pi\)
\(18\) 0 0
\(19\) 5.60555 1.28600 0.643001 0.765865i \(-0.277689\pi\)
0.643001 + 0.765865i \(0.277689\pi\)
\(20\) −3.30278 −0.738523
\(21\) 0 0
\(22\) −3.69722 −0.788251
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 3.30278 0.624166
\(29\) 6.21110 1.15337 0.576686 0.816966i \(-0.304345\pi\)
0.576686 + 0.816966i \(0.304345\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.30278 0.937407
\(33\) 0 0
\(34\) 17.5139 3.00361
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −3.60555 −0.592749 −0.296374 0.955072i \(-0.595778\pi\)
−0.296374 + 0.955072i \(0.595778\pi\)
\(38\) −12.9083 −2.09401
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −10.2111 −1.55718 −0.778589 0.627534i \(-0.784064\pi\)
−0.778589 + 0.627534i \(0.784064\pi\)
\(44\) 5.30278 0.799424
\(45\) 0 0
\(46\) −6.90833 −1.01858
\(47\) 9.21110 1.34358 0.671789 0.740743i \(-0.265526\pi\)
0.671789 + 0.740743i \(0.265526\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −2.30278 −0.325662
\(51\) 0 0
\(52\) 0 0
\(53\) 3.21110 0.441079 0.220539 0.975378i \(-0.429218\pi\)
0.220539 + 0.975378i \(0.429218\pi\)
\(54\) 0 0
\(55\) −1.60555 −0.216492
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −14.3028 −1.87805
\(59\) −10.8167 −1.40821 −0.704104 0.710097i \(-0.748651\pi\)
−0.704104 + 0.710097i \(0.748651\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −9.21110 −1.16981
\(63\) 0 0
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −25.1194 −3.04618
\(69\) 0 0
\(70\) 2.30278 0.275234
\(71\) 4.81665 0.571632 0.285816 0.958285i \(-0.407735\pi\)
0.285816 + 0.958285i \(0.407735\pi\)
\(72\) 0 0
\(73\) 0.788897 0.0923335 0.0461667 0.998934i \(-0.485299\pi\)
0.0461667 + 0.998934i \(0.485299\pi\)
\(74\) 8.30278 0.965178
\(75\) 0 0
\(76\) 18.5139 2.12369
\(77\) 1.60555 0.182970
\(78\) 0 0
\(79\) 5.21110 0.586295 0.293147 0.956067i \(-0.405297\pi\)
0.293147 + 0.956067i \(0.405297\pi\)
\(80\) −0.302776 −0.0338513
\(81\) 0 0
\(82\) −6.90833 −0.762897
\(83\) −9.21110 −1.01105 −0.505525 0.862812i \(-0.668701\pi\)
−0.505525 + 0.862812i \(0.668701\pi\)
\(84\) 0 0
\(85\) 7.60555 0.824938
\(86\) 23.5139 2.53557
\(87\) 0 0
\(88\) −4.81665 −0.513457
\(89\) −6.21110 −0.658376 −0.329188 0.944264i \(-0.606775\pi\)
−0.329188 + 0.944264i \(0.606775\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.90833 1.03301
\(93\) 0 0
\(94\) −21.2111 −2.18776
\(95\) −5.60555 −0.575117
\(96\) 0 0
\(97\) 8.39445 0.852327 0.426164 0.904646i \(-0.359865\pi\)
0.426164 + 0.904646i \(0.359865\pi\)
\(98\) 13.8167 1.39569
\(99\) 0 0
\(100\) 3.30278 0.330278
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −7.39445 −0.718212
\(107\) −6.21110 −0.600450 −0.300225 0.953868i \(-0.597062\pi\)
−0.300225 + 0.953868i \(0.597062\pi\)
\(108\) 0 0
\(109\) 19.2111 1.84009 0.920045 0.391813i \(-0.128152\pi\)
0.920045 + 0.391813i \(0.128152\pi\)
\(110\) 3.69722 0.352517
\(111\) 0 0
\(112\) 0.302776 0.0286096
\(113\) 1.60555 0.151038 0.0755188 0.997144i \(-0.475939\pi\)
0.0755188 + 0.997144i \(0.475939\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 20.5139 1.90467
\(117\) 0 0
\(118\) 24.9083 2.29300
\(119\) −7.60555 −0.697200
\(120\) 0 0
\(121\) −8.42221 −0.765655
\(122\) 2.30278 0.208484
\(123\) 0 0
\(124\) 13.2111 1.18639
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.21110 −0.373675 −0.186837 0.982391i \(-0.559824\pi\)
−0.186837 + 0.982391i \(0.559824\pi\)
\(128\) 18.9083 1.67128
\(129\) 0 0
\(130\) 0 0
\(131\) 21.2111 1.85322 0.926611 0.376021i \(-0.122708\pi\)
0.926611 + 0.376021i \(0.122708\pi\)
\(132\) 0 0
\(133\) 5.60555 0.486063
\(134\) −16.1194 −1.39251
\(135\) 0 0
\(136\) 22.8167 1.95651
\(137\) −1.60555 −0.137172 −0.0685858 0.997645i \(-0.521849\pi\)
−0.0685858 + 0.997645i \(0.521849\pi\)
\(138\) 0 0
\(139\) 6.39445 0.542370 0.271185 0.962527i \(-0.412584\pi\)
0.271185 + 0.962527i \(0.412584\pi\)
\(140\) −3.30278 −0.279135
\(141\) 0 0
\(142\) −11.0917 −0.930793
\(143\) 0 0
\(144\) 0 0
\(145\) −6.21110 −0.515804
\(146\) −1.81665 −0.150347
\(147\) 0 0
\(148\) −11.9083 −0.978858
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 1.21110 0.0985581 0.0492791 0.998785i \(-0.484308\pi\)
0.0492791 + 0.998785i \(0.484308\pi\)
\(152\) −16.8167 −1.36401
\(153\) 0 0
\(154\) −3.69722 −0.297931
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 11.2111 0.894743 0.447372 0.894348i \(-0.352360\pi\)
0.447372 + 0.894348i \(0.352360\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) −5.30278 −0.419221
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 3.78890 0.296769 0.148385 0.988930i \(-0.452593\pi\)
0.148385 + 0.988930i \(0.452593\pi\)
\(164\) 9.90833 0.773710
\(165\) 0 0
\(166\) 21.2111 1.64630
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −17.5139 −1.34325
\(171\) 0 0
\(172\) −33.7250 −2.57151
\(173\) −4.81665 −0.366203 −0.183102 0.983094i \(-0.558614\pi\)
−0.183102 + 0.983094i \(0.558614\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0.486122 0.0366428
\(177\) 0 0
\(178\) 14.3028 1.07204
\(179\) −22.8167 −1.70540 −0.852698 0.522404i \(-0.825035\pi\)
−0.852698 + 0.522404i \(0.825035\pi\)
\(180\) 0 0
\(181\) 17.6333 1.31067 0.655337 0.755337i \(-0.272527\pi\)
0.655337 + 0.755337i \(0.272527\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −9.00000 −0.663489
\(185\) 3.60555 0.265085
\(186\) 0 0
\(187\) −12.2111 −0.892964
\(188\) 30.4222 2.21877
\(189\) 0 0
\(190\) 12.9083 0.936468
\(191\) −16.8167 −1.21681 −0.608405 0.793627i \(-0.708190\pi\)
−0.608405 + 0.793627i \(0.708190\pi\)
\(192\) 0 0
\(193\) −15.6056 −1.12331 −0.561656 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(194\) −19.3305 −1.38785
\(195\) 0 0
\(196\) −19.8167 −1.41548
\(197\) 1.18335 0.0843099 0.0421550 0.999111i \(-0.486578\pi\)
0.0421550 + 0.999111i \(0.486578\pi\)
\(198\) 0 0
\(199\) 12.8167 0.908549 0.454274 0.890862i \(-0.349899\pi\)
0.454274 + 0.890862i \(0.349899\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −20.7250 −1.45820
\(203\) 6.21110 0.435934
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 9.21110 0.641768
\(207\) 0 0
\(208\) 0 0
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) −23.6056 −1.62507 −0.812537 0.582910i \(-0.801914\pi\)
−0.812537 + 0.582910i \(0.801914\pi\)
\(212\) 10.6056 0.728392
\(213\) 0 0
\(214\) 14.3028 0.977718
\(215\) 10.2111 0.696391
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −44.2389 −2.99623
\(219\) 0 0
\(220\) −5.30278 −0.357513
\(221\) 0 0
\(222\) 0 0
\(223\) 4.21110 0.281996 0.140998 0.990010i \(-0.454969\pi\)
0.140998 + 0.990010i \(0.454969\pi\)
\(224\) 5.30278 0.354307
\(225\) 0 0
\(226\) −3.69722 −0.245936
\(227\) −27.4222 −1.82008 −0.910038 0.414525i \(-0.863948\pi\)
−0.910038 + 0.414525i \(0.863948\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 6.90833 0.455522
\(231\) 0 0
\(232\) −18.6333 −1.22334
\(233\) −15.2111 −0.996512 −0.498256 0.867030i \(-0.666026\pi\)
−0.498256 + 0.867030i \(0.666026\pi\)
\(234\) 0 0
\(235\) −9.21110 −0.600866
\(236\) −35.7250 −2.32550
\(237\) 0 0
\(238\) 17.5139 1.13526
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.78890 −0.115233 −0.0576165 0.998339i \(-0.518350\pi\)
−0.0576165 + 0.998339i \(0.518350\pi\)
\(242\) 19.3944 1.24672
\(243\) 0 0
\(244\) −3.30278 −0.211439
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) 2.30278 0.145640
\(251\) 7.18335 0.453409 0.226704 0.973964i \(-0.427205\pi\)
0.226704 + 0.973964i \(0.427205\pi\)
\(252\) 0 0
\(253\) 4.81665 0.302820
\(254\) 9.69722 0.608458
\(255\) 0 0
\(256\) −17.9083 −1.11927
\(257\) 16.3944 1.02266 0.511329 0.859385i \(-0.329153\pi\)
0.511329 + 0.859385i \(0.329153\pi\)
\(258\) 0 0
\(259\) −3.60555 −0.224038
\(260\) 0 0
\(261\) 0 0
\(262\) −48.8444 −3.01762
\(263\) −11.7889 −0.726935 −0.363467 0.931607i \(-0.618407\pi\)
−0.363467 + 0.931607i \(0.618407\pi\)
\(264\) 0 0
\(265\) −3.21110 −0.197256
\(266\) −12.9083 −0.791460
\(267\) 0 0
\(268\) 23.1194 1.41224
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) 20.8167 1.26452 0.632261 0.774756i \(-0.282127\pi\)
0.632261 + 0.774756i \(0.282127\pi\)
\(272\) −2.30278 −0.139626
\(273\) 0 0
\(274\) 3.69722 0.223357
\(275\) 1.60555 0.0968184
\(276\) 0 0
\(277\) 27.6056 1.65866 0.829328 0.558761i \(-0.188723\pi\)
0.829328 + 0.558761i \(0.188723\pi\)
\(278\) −14.7250 −0.883146
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) 15.9083 0.943986
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) 0 0
\(289\) 40.8444 2.40261
\(290\) 14.3028 0.839888
\(291\) 0 0
\(292\) 2.60555 0.152478
\(293\) 10.3944 0.607250 0.303625 0.952792i \(-0.401803\pi\)
0.303625 + 0.952792i \(0.401803\pi\)
\(294\) 0 0
\(295\) 10.8167 0.629770
\(296\) 10.8167 0.628705
\(297\) 0 0
\(298\) −6.90833 −0.400189
\(299\) 0 0
\(300\) 0 0
\(301\) −10.2111 −0.588558
\(302\) −2.78890 −0.160483
\(303\) 0 0
\(304\) 1.69722 0.0973425
\(305\) 1.00000 0.0572598
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 5.30278 0.302154
\(309\) 0 0
\(310\) 9.21110 0.523155
\(311\) 9.21110 0.522314 0.261157 0.965296i \(-0.415896\pi\)
0.261157 + 0.965296i \(0.415896\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −25.8167 −1.45692
\(315\) 0 0
\(316\) 17.2111 0.968200
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 9.97224 0.558338
\(320\) 12.8167 0.716473
\(321\) 0 0
\(322\) −6.90833 −0.384986
\(323\) −42.6333 −2.37218
\(324\) 0 0
\(325\) 0 0
\(326\) −8.72498 −0.483232
\(327\) 0 0
\(328\) −9.00000 −0.496942
\(329\) 9.21110 0.507825
\(330\) 0 0
\(331\) −10.0278 −0.551175 −0.275588 0.961276i \(-0.588872\pi\)
−0.275588 + 0.961276i \(0.588872\pi\)
\(332\) −30.4222 −1.66964
\(333\) 0 0
\(334\) −20.7250 −1.13402
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) −25.6333 −1.39634 −0.698168 0.715934i \(-0.746001\pi\)
−0.698168 + 0.715934i \(0.746001\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 25.1194 1.36229
\(341\) 6.42221 0.347782
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 30.6333 1.65164
\(345\) 0 0
\(346\) 11.0917 0.596292
\(347\) −5.78890 −0.310764 −0.155382 0.987854i \(-0.549661\pi\)
−0.155382 + 0.987854i \(0.549661\pi\)
\(348\) 0 0
\(349\) 3.78890 0.202815 0.101408 0.994845i \(-0.467665\pi\)
0.101408 + 0.994845i \(0.467665\pi\)
\(350\) −2.30278 −0.123089
\(351\) 0 0
\(352\) 8.51388 0.453791
\(353\) 16.8167 0.895060 0.447530 0.894269i \(-0.352304\pi\)
0.447530 + 0.894269i \(0.352304\pi\)
\(354\) 0 0
\(355\) −4.81665 −0.255641
\(356\) −20.5139 −1.08723
\(357\) 0 0
\(358\) 52.5416 2.77691
\(359\) 18.4222 0.972287 0.486143 0.873879i \(-0.338403\pi\)
0.486143 + 0.873879i \(0.338403\pi\)
\(360\) 0 0
\(361\) 12.4222 0.653800
\(362\) −40.6056 −2.13418
\(363\) 0 0
\(364\) 0 0
\(365\) −0.788897 −0.0412928
\(366\) 0 0
\(367\) 11.4222 0.596234 0.298117 0.954529i \(-0.403641\pi\)
0.298117 + 0.954529i \(0.403641\pi\)
\(368\) 0.908327 0.0473498
\(369\) 0 0
\(370\) −8.30278 −0.431641
\(371\) 3.21110 0.166712
\(372\) 0 0
\(373\) −20.3944 −1.05598 −0.527992 0.849249i \(-0.677055\pi\)
−0.527992 + 0.849249i \(0.677055\pi\)
\(374\) 28.1194 1.45402
\(375\) 0 0
\(376\) −27.6333 −1.42508
\(377\) 0 0
\(378\) 0 0
\(379\) −9.60555 −0.493404 −0.246702 0.969091i \(-0.579347\pi\)
−0.246702 + 0.969091i \(0.579347\pi\)
\(380\) −18.5139 −0.949742
\(381\) 0 0
\(382\) 38.7250 1.98134
\(383\) −24.6333 −1.25870 −0.629352 0.777121i \(-0.716679\pi\)
−0.629352 + 0.777121i \(0.716679\pi\)
\(384\) 0 0
\(385\) −1.60555 −0.0818265
\(386\) 35.9361 1.82910
\(387\) 0 0
\(388\) 27.7250 1.40752
\(389\) 15.2111 0.771234 0.385617 0.922659i \(-0.373989\pi\)
0.385617 + 0.922659i \(0.373989\pi\)
\(390\) 0 0
\(391\) −22.8167 −1.15389
\(392\) 18.0000 0.909137
\(393\) 0 0
\(394\) −2.72498 −0.137283
\(395\) −5.21110 −0.262199
\(396\) 0 0
\(397\) −22.0278 −1.10554 −0.552771 0.833333i \(-0.686429\pi\)
−0.552771 + 0.833333i \(0.686429\pi\)
\(398\) −29.5139 −1.47940
\(399\) 0 0
\(400\) 0.302776 0.0151388
\(401\) 12.2111 0.609793 0.304897 0.952385i \(-0.401378\pi\)
0.304897 + 0.952385i \(0.401378\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 29.7250 1.47887
\(405\) 0 0
\(406\) −14.3028 −0.709835
\(407\) −5.78890 −0.286945
\(408\) 0 0
\(409\) −8.21110 −0.406013 −0.203006 0.979177i \(-0.565071\pi\)
−0.203006 + 0.979177i \(0.565071\pi\)
\(410\) 6.90833 0.341178
\(411\) 0 0
\(412\) −13.2111 −0.650864
\(413\) −10.8167 −0.532253
\(414\) 0 0
\(415\) 9.21110 0.452155
\(416\) 0 0
\(417\) 0 0
\(418\) −20.7250 −1.01369
\(419\) −17.2389 −0.842173 −0.421087 0.907020i \(-0.638351\pi\)
−0.421087 + 0.907020i \(0.638351\pi\)
\(420\) 0 0
\(421\) −32.4222 −1.58016 −0.790081 0.613003i \(-0.789961\pi\)
−0.790081 + 0.613003i \(0.789961\pi\)
\(422\) 54.3583 2.64612
\(423\) 0 0
\(424\) −9.63331 −0.467835
\(425\) −7.60555 −0.368923
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) −20.5139 −0.991576
\(429\) 0 0
\(430\) −23.5139 −1.13394
\(431\) 29.2389 1.40839 0.704193 0.710008i \(-0.251309\pi\)
0.704193 + 0.710008i \(0.251309\pi\)
\(432\) 0 0
\(433\) 3.60555 0.173272 0.0866359 0.996240i \(-0.472388\pi\)
0.0866359 + 0.996240i \(0.472388\pi\)
\(434\) −9.21110 −0.442147
\(435\) 0 0
\(436\) 63.4500 3.03870
\(437\) 16.8167 0.804450
\(438\) 0 0
\(439\) −27.2389 −1.30004 −0.650020 0.759917i \(-0.725239\pi\)
−0.650020 + 0.759917i \(0.725239\pi\)
\(440\) 4.81665 0.229625
\(441\) 0 0
\(442\) 0 0
\(443\) −6.42221 −0.305128 −0.152564 0.988294i \(-0.548753\pi\)
−0.152564 + 0.988294i \(0.548753\pi\)
\(444\) 0 0
\(445\) 6.21110 0.294434
\(446\) −9.69722 −0.459177
\(447\) 0 0
\(448\) −12.8167 −0.605530
\(449\) 30.6333 1.44568 0.722838 0.691018i \(-0.242837\pi\)
0.722838 + 0.691018i \(0.242837\pi\)
\(450\) 0 0
\(451\) 4.81665 0.226807
\(452\) 5.30278 0.249422
\(453\) 0 0
\(454\) 63.1472 2.96364
\(455\) 0 0
\(456\) 0 0
\(457\) 26.8167 1.25443 0.627215 0.778846i \(-0.284195\pi\)
0.627215 + 0.778846i \(0.284195\pi\)
\(458\) 32.2389 1.50642
\(459\) 0 0
\(460\) −9.90833 −0.461978
\(461\) 36.2111 1.68652 0.843260 0.537507i \(-0.180634\pi\)
0.843260 + 0.537507i \(0.180634\pi\)
\(462\) 0 0
\(463\) 34.4222 1.59974 0.799868 0.600176i \(-0.204903\pi\)
0.799868 + 0.600176i \(0.204903\pi\)
\(464\) 1.88057 0.0873033
\(465\) 0 0
\(466\) 35.0278 1.62263
\(467\) −2.78890 −0.129055 −0.0645274 0.997916i \(-0.520554\pi\)
−0.0645274 + 0.997916i \(0.520554\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 21.2111 0.978395
\(471\) 0 0
\(472\) 32.4500 1.49363
\(473\) −16.3944 −0.753818
\(474\) 0 0
\(475\) 5.60555 0.257200
\(476\) −25.1194 −1.15135
\(477\) 0 0
\(478\) 0 0
\(479\) −28.8167 −1.31667 −0.658333 0.752727i \(-0.728738\pi\)
−0.658333 + 0.752727i \(0.728738\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4.11943 0.187635
\(483\) 0 0
\(484\) −27.8167 −1.26439
\(485\) −8.39445 −0.381172
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 3.00000 0.135804
\(489\) 0 0
\(490\) −13.8167 −0.624173
\(491\) 16.8167 0.758925 0.379462 0.925207i \(-0.376109\pi\)
0.379462 + 0.925207i \(0.376109\pi\)
\(492\) 0 0
\(493\) −47.2389 −2.12753
\(494\) 0 0
\(495\) 0 0
\(496\) 1.21110 0.0543801
\(497\) 4.81665 0.216056
\(498\) 0 0
\(499\) −2.42221 −0.108433 −0.0542164 0.998529i \(-0.517266\pi\)
−0.0542164 + 0.998529i \(0.517266\pi\)
\(500\) −3.30278 −0.147705
\(501\) 0 0
\(502\) −16.5416 −0.738289
\(503\) −3.00000 −0.133763 −0.0668817 0.997761i \(-0.521305\pi\)
−0.0668817 + 0.997761i \(0.521305\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) −11.0917 −0.493085
\(507\) 0 0
\(508\) −13.9083 −0.617082
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 0.788897 0.0348988
\(512\) 3.42221 0.151242
\(513\) 0 0
\(514\) −37.7527 −1.66520
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) 14.7889 0.650415
\(518\) 8.30278 0.364803
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −1.42221 −0.0621887 −0.0310943 0.999516i \(-0.509899\pi\)
−0.0310943 + 0.999516i \(0.509899\pi\)
\(524\) 70.0555 3.06039
\(525\) 0 0
\(526\) 27.1472 1.18367
\(527\) −30.4222 −1.32521
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 7.39445 0.321194
\(531\) 0 0
\(532\) 18.5139 0.802678
\(533\) 0 0
\(534\) 0 0
\(535\) 6.21110 0.268529
\(536\) −21.0000 −0.907062
\(537\) 0 0
\(538\) −20.7250 −0.893517
\(539\) −9.63331 −0.414936
\(540\) 0 0
\(541\) 25.6333 1.10206 0.551031 0.834485i \(-0.314235\pi\)
0.551031 + 0.834485i \(0.314235\pi\)
\(542\) −47.9361 −2.05903
\(543\) 0 0
\(544\) −40.3305 −1.72916
\(545\) −19.2111 −0.822913
\(546\) 0 0
\(547\) 32.8444 1.40433 0.702163 0.712016i \(-0.252218\pi\)
0.702163 + 0.712016i \(0.252218\pi\)
\(548\) −5.30278 −0.226523
\(549\) 0 0
\(550\) −3.69722 −0.157650
\(551\) 34.8167 1.48324
\(552\) 0 0
\(553\) 5.21110 0.221599
\(554\) −63.5694 −2.70080
\(555\) 0 0
\(556\) 21.1194 0.895663
\(557\) −1.60555 −0.0680294 −0.0340147 0.999421i \(-0.510829\pi\)
−0.0340147 + 0.999421i \(0.510829\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.302776 −0.0127946
\(561\) 0 0
\(562\) 13.8167 0.582820
\(563\) −9.42221 −0.397099 −0.198549 0.980091i \(-0.563623\pi\)
−0.198549 + 0.980091i \(0.563623\pi\)
\(564\) 0 0
\(565\) −1.60555 −0.0675460
\(566\) −11.5139 −0.483964
\(567\) 0 0
\(568\) −14.4500 −0.606307
\(569\) 27.4222 1.14960 0.574799 0.818294i \(-0.305080\pi\)
0.574799 + 0.818294i \(0.305080\pi\)
\(570\) 0 0
\(571\) 20.8444 0.872311 0.436156 0.899871i \(-0.356340\pi\)
0.436156 + 0.899871i \(0.356340\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.90833 −0.288348
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 13.6333 0.567562 0.283781 0.958889i \(-0.408411\pi\)
0.283781 + 0.958889i \(0.408411\pi\)
\(578\) −94.0555 −3.91219
\(579\) 0 0
\(580\) −20.5139 −0.851792
\(581\) −9.21110 −0.382141
\(582\) 0 0
\(583\) 5.15559 0.213523
\(584\) −2.36669 −0.0979344
\(585\) 0 0
\(586\) −23.9361 −0.988790
\(587\) −33.4222 −1.37948 −0.689741 0.724056i \(-0.742276\pi\)
−0.689741 + 0.724056i \(0.742276\pi\)
\(588\) 0 0
\(589\) 22.4222 0.923891
\(590\) −24.9083 −1.02546
\(591\) 0 0
\(592\) −1.09167 −0.0448675
\(593\) 20.7889 0.853698 0.426849 0.904323i \(-0.359624\pi\)
0.426849 + 0.904323i \(0.359624\pi\)
\(594\) 0 0
\(595\) 7.60555 0.311797
\(596\) 9.90833 0.405861
\(597\) 0 0
\(598\) 0 0
\(599\) 21.2111 0.866662 0.433331 0.901235i \(-0.357338\pi\)
0.433331 + 0.901235i \(0.357338\pi\)
\(600\) 0 0
\(601\) 13.7889 0.562461 0.281230 0.959640i \(-0.409257\pi\)
0.281230 + 0.959640i \(0.409257\pi\)
\(602\) 23.5139 0.958354
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 8.42221 0.342411
\(606\) 0 0
\(607\) −34.2111 −1.38859 −0.694293 0.719693i \(-0.744283\pi\)
−0.694293 + 0.719693i \(0.744283\pi\)
\(608\) 29.7250 1.20551
\(609\) 0 0
\(610\) −2.30278 −0.0932367
\(611\) 0 0
\(612\) 0 0
\(613\) 5.60555 0.226406 0.113203 0.993572i \(-0.463889\pi\)
0.113203 + 0.993572i \(0.463889\pi\)
\(614\) −36.8444 −1.48692
\(615\) 0 0
\(616\) −4.81665 −0.194069
\(617\) −38.4500 −1.54794 −0.773969 0.633224i \(-0.781731\pi\)
−0.773969 + 0.633224i \(0.781731\pi\)
\(618\) 0 0
\(619\) −14.4222 −0.579677 −0.289839 0.957076i \(-0.593602\pi\)
−0.289839 + 0.957076i \(0.593602\pi\)
\(620\) −13.2111 −0.530571
\(621\) 0 0
\(622\) −21.2111 −0.850488
\(623\) −6.21110 −0.248843
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −32.2389 −1.28852
\(627\) 0 0
\(628\) 37.0278 1.47757
\(629\) 27.4222 1.09339
\(630\) 0 0
\(631\) 36.0278 1.43424 0.717121 0.696949i \(-0.245459\pi\)
0.717121 + 0.696949i \(0.245459\pi\)
\(632\) −15.6333 −0.621860
\(633\) 0 0
\(634\) −13.8167 −0.548729
\(635\) 4.21110 0.167113
\(636\) 0 0
\(637\) 0 0
\(638\) −22.9638 −0.909147
\(639\) 0 0
\(640\) −18.9083 −0.747417
\(641\) −9.42221 −0.372155 −0.186077 0.982535i \(-0.559578\pi\)
−0.186077 + 0.982535i \(0.559578\pi\)
\(642\) 0 0
\(643\) −2.63331 −0.103848 −0.0519238 0.998651i \(-0.516535\pi\)
−0.0519238 + 0.998651i \(0.516535\pi\)
\(644\) 9.90833 0.390443
\(645\) 0 0
\(646\) 98.1749 3.86264
\(647\) −39.4222 −1.54985 −0.774923 0.632055i \(-0.782212\pi\)
−0.774923 + 0.632055i \(0.782212\pi\)
\(648\) 0 0
\(649\) −17.3667 −0.681702
\(650\) 0 0
\(651\) 0 0
\(652\) 12.5139 0.490081
\(653\) 7.18335 0.281106 0.140553 0.990073i \(-0.455112\pi\)
0.140553 + 0.990073i \(0.455112\pi\)
\(654\) 0 0
\(655\) −21.2111 −0.828786
\(656\) 0.908327 0.0354642
\(657\) 0 0
\(658\) −21.2111 −0.826895
\(659\) 34.8167 1.35626 0.678132 0.734940i \(-0.262790\pi\)
0.678132 + 0.734940i \(0.262790\pi\)
\(660\) 0 0
\(661\) 4.63331 0.180215 0.0901074 0.995932i \(-0.471279\pi\)
0.0901074 + 0.995932i \(0.471279\pi\)
\(662\) 23.0917 0.897483
\(663\) 0 0
\(664\) 27.6333 1.07238
\(665\) −5.60555 −0.217374
\(666\) 0 0
\(667\) 18.6333 0.721485
\(668\) 29.7250 1.15009
\(669\) 0 0
\(670\) 16.1194 0.622748
\(671\) −1.60555 −0.0619816
\(672\) 0 0
\(673\) −17.6056 −0.678644 −0.339322 0.940670i \(-0.610198\pi\)
−0.339322 + 0.940670i \(0.610198\pi\)
\(674\) 59.0278 2.27366
\(675\) 0 0
\(676\) 0 0
\(677\) 9.63331 0.370238 0.185119 0.982716i \(-0.440733\pi\)
0.185119 + 0.982716i \(0.440733\pi\)
\(678\) 0 0
\(679\) 8.39445 0.322149
\(680\) −22.8167 −0.874979
\(681\) 0 0
\(682\) −14.7889 −0.566296
\(683\) 36.2111 1.38558 0.692790 0.721140i \(-0.256381\pi\)
0.692790 + 0.721140i \(0.256381\pi\)
\(684\) 0 0
\(685\) 1.60555 0.0613450
\(686\) 29.9361 1.14296
\(687\) 0 0
\(688\) −3.09167 −0.117869
\(689\) 0 0
\(690\) 0 0
\(691\) 30.0278 1.14231 0.571155 0.820842i \(-0.306496\pi\)
0.571155 + 0.820842i \(0.306496\pi\)
\(692\) −15.9083 −0.604744
\(693\) 0 0
\(694\) 13.3305 0.506020
\(695\) −6.39445 −0.242555
\(696\) 0 0
\(697\) −22.8167 −0.864242
\(698\) −8.72498 −0.330245
\(699\) 0 0
\(700\) 3.30278 0.124833
\(701\) 36.4222 1.37565 0.687824 0.725878i \(-0.258566\pi\)
0.687824 + 0.725878i \(0.258566\pi\)
\(702\) 0 0
\(703\) −20.2111 −0.762276
\(704\) −20.5778 −0.775555
\(705\) 0 0
\(706\) −38.7250 −1.45743
\(707\) 9.00000 0.338480
\(708\) 0 0
\(709\) 13.8444 0.519938 0.259969 0.965617i \(-0.416288\pi\)
0.259969 + 0.965617i \(0.416288\pi\)
\(710\) 11.0917 0.416263
\(711\) 0 0
\(712\) 18.6333 0.698313
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −75.3583 −2.81627
\(717\) 0 0
\(718\) −42.4222 −1.58318
\(719\) 25.6056 0.954926 0.477463 0.878652i \(-0.341556\pi\)
0.477463 + 0.878652i \(0.341556\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −28.6056 −1.06459
\(723\) 0 0
\(724\) 58.2389 2.16443
\(725\) 6.21110 0.230675
\(726\) 0 0
\(727\) 13.5778 0.503573 0.251786 0.967783i \(-0.418982\pi\)
0.251786 + 0.967783i \(0.418982\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.81665 0.0672374
\(731\) 77.6611 2.87240
\(732\) 0 0
\(733\) 46.8444 1.73024 0.865119 0.501567i \(-0.167243\pi\)
0.865119 + 0.501567i \(0.167243\pi\)
\(734\) −26.3028 −0.970853
\(735\) 0 0
\(736\) 15.9083 0.586389
\(737\) 11.2389 0.413989
\(738\) 0 0
\(739\) 35.6056 1.30977 0.654886 0.755728i \(-0.272717\pi\)
0.654886 + 0.755728i \(0.272717\pi\)
\(740\) 11.9083 0.437759
\(741\) 0 0
\(742\) −7.39445 −0.271459
\(743\) 36.6333 1.34395 0.671973 0.740576i \(-0.265447\pi\)
0.671973 + 0.740576i \(0.265447\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 46.9638 1.71947
\(747\) 0 0
\(748\) −40.3305 −1.47463
\(749\) −6.21110 −0.226949
\(750\) 0 0
\(751\) 46.4500 1.69498 0.847492 0.530809i \(-0.178112\pi\)
0.847492 + 0.530809i \(0.178112\pi\)
\(752\) 2.78890 0.101701
\(753\) 0 0
\(754\) 0 0
\(755\) −1.21110 −0.0440765
\(756\) 0 0
\(757\) 0.816654 0.0296818 0.0148409 0.999890i \(-0.495276\pi\)
0.0148409 + 0.999890i \(0.495276\pi\)
\(758\) 22.1194 0.803414
\(759\) 0 0
\(760\) 16.8167 0.610004
\(761\) 18.6333 0.675457 0.337728 0.941244i \(-0.390341\pi\)
0.337728 + 0.941244i \(0.390341\pi\)
\(762\) 0 0
\(763\) 19.2111 0.695489
\(764\) −55.5416 −2.00943
\(765\) 0 0
\(766\) 56.7250 2.04956
\(767\) 0 0
\(768\) 0 0
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 3.69722 0.133239
\(771\) 0 0
\(772\) −51.5416 −1.85502
\(773\) 22.3944 0.805472 0.402736 0.915316i \(-0.368059\pi\)
0.402736 + 0.915316i \(0.368059\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −25.1833 −0.904029
\(777\) 0 0
\(778\) −35.0278 −1.25581
\(779\) 16.8167 0.602519
\(780\) 0 0
\(781\) 7.73338 0.276722
\(782\) 52.5416 1.87889
\(783\) 0 0
\(784\) −1.81665 −0.0648805
\(785\) −11.2111 −0.400141
\(786\) 0 0
\(787\) −14.6333 −0.521621 −0.260811 0.965390i \(-0.583990\pi\)
−0.260811 + 0.965390i \(0.583990\pi\)
\(788\) 3.90833 0.139228
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 1.60555 0.0570868
\(792\) 0 0
\(793\) 0 0
\(794\) 50.7250 1.80016
\(795\) 0 0
\(796\) 42.3305 1.50037
\(797\) 14.4500 0.511844 0.255922 0.966697i \(-0.417621\pi\)
0.255922 + 0.966697i \(0.417621\pi\)
\(798\) 0 0
\(799\) −70.0555 −2.47839
\(800\) 5.30278 0.187481
\(801\) 0 0
\(802\) −28.1194 −0.992932
\(803\) 1.26662 0.0446979
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 0 0
\(808\) −27.0000 −0.949857
\(809\) 55.0555 1.93565 0.967824 0.251627i \(-0.0809655\pi\)
0.967824 + 0.251627i \(0.0809655\pi\)
\(810\) 0 0
\(811\) 46.4222 1.63010 0.815052 0.579388i \(-0.196708\pi\)
0.815052 + 0.579388i \(0.196708\pi\)
\(812\) 20.5139 0.719896
\(813\) 0 0
\(814\) 13.3305 0.467235
\(815\) −3.78890 −0.132719
\(816\) 0 0
\(817\) −57.2389 −2.00253
\(818\) 18.9083 0.661114
\(819\) 0 0
\(820\) −9.90833 −0.346014
\(821\) 21.4222 0.747640 0.373820 0.927501i \(-0.378048\pi\)
0.373820 + 0.927501i \(0.378048\pi\)
\(822\) 0 0
\(823\) −16.6333 −0.579801 −0.289900 0.957057i \(-0.593622\pi\)
−0.289900 + 0.957057i \(0.593622\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 24.9083 0.866672
\(827\) −42.4222 −1.47516 −0.737582 0.675257i \(-0.764033\pi\)
−0.737582 + 0.675257i \(0.764033\pi\)
\(828\) 0 0
\(829\) 29.4222 1.02188 0.510938 0.859618i \(-0.329298\pi\)
0.510938 + 0.859618i \(0.329298\pi\)
\(830\) −21.2111 −0.736248
\(831\) 0 0
\(832\) 0 0
\(833\) 45.6333 1.58110
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) 29.7250 1.02806
\(837\) 0 0
\(838\) 39.6972 1.37132
\(839\) −20.0278 −0.691435 −0.345717 0.938339i \(-0.612364\pi\)
−0.345717 + 0.938339i \(0.612364\pi\)
\(840\) 0 0
\(841\) 9.57779 0.330269
\(842\) 74.6611 2.57299
\(843\) 0 0
\(844\) −77.9638 −2.68363
\(845\) 0 0
\(846\) 0 0
\(847\) −8.42221 −0.289390
\(848\) 0.972244 0.0333870
\(849\) 0 0
\(850\) 17.5139 0.600721
\(851\) −10.8167 −0.370790
\(852\) 0 0
\(853\) −47.2111 −1.61648 −0.808239 0.588855i \(-0.799579\pi\)
−0.808239 + 0.588855i \(0.799579\pi\)
\(854\) 2.30278 0.0787994
\(855\) 0 0
\(856\) 18.6333 0.636873
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 10.7889 0.368112 0.184056 0.982916i \(-0.441077\pi\)
0.184056 + 0.982916i \(0.441077\pi\)
\(860\) 33.7250 1.15001
\(861\) 0 0
\(862\) −67.3305 −2.29329
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 4.81665 0.163771
\(866\) −8.30278 −0.282140
\(867\) 0 0
\(868\) 13.2111 0.448414
\(869\) 8.36669 0.283821
\(870\) 0 0
\(871\) 0 0
\(872\) −57.6333 −1.95171
\(873\) 0 0
\(874\) −38.7250 −1.30989
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 1.97224 0.0665979 0.0332990 0.999445i \(-0.489399\pi\)
0.0332990 + 0.999445i \(0.489399\pi\)
\(878\) 62.7250 2.11687
\(879\) 0 0
\(880\) −0.486122 −0.0163872
\(881\) 21.8444 0.735957 0.367978 0.929834i \(-0.380050\pi\)
0.367978 + 0.929834i \(0.380050\pi\)
\(882\) 0 0
\(883\) 11.6333 0.391492 0.195746 0.980655i \(-0.437287\pi\)
0.195746 + 0.980655i \(0.437287\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 14.7889 0.496843
\(887\) 37.0555 1.24420 0.622101 0.782937i \(-0.286279\pi\)
0.622101 + 0.782937i \(0.286279\pi\)
\(888\) 0 0
\(889\) −4.21110 −0.141236
\(890\) −14.3028 −0.479430
\(891\) 0 0
\(892\) 13.9083 0.465685
\(893\) 51.6333 1.72784
\(894\) 0 0
\(895\) 22.8167 0.762677
\(896\) 18.9083 0.631683
\(897\) 0 0
\(898\) −70.5416 −2.35400
\(899\) 24.8444 0.828607
\(900\) 0 0
\(901\) −24.4222 −0.813622
\(902\) −11.0917 −0.369312
\(903\) 0 0
\(904\) −4.81665 −0.160200
\(905\) −17.6333 −0.586151
\(906\) 0 0
\(907\) −38.2666 −1.27062 −0.635311 0.772256i \(-0.719128\pi\)
−0.635311 + 0.772256i \(0.719128\pi\)
\(908\) −90.5694 −3.00565
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −14.7889 −0.489441
\(914\) −61.7527 −2.04260
\(915\) 0 0
\(916\) −46.2389 −1.52777
\(917\) 21.2111 0.700452
\(918\) 0 0
\(919\) −38.8167 −1.28044 −0.640222 0.768190i \(-0.721157\pi\)
−0.640222 + 0.768190i \(0.721157\pi\)
\(920\) 9.00000 0.296721
\(921\) 0 0
\(922\) −83.3860 −2.74617
\(923\) 0 0
\(924\) 0 0
\(925\) −3.60555 −0.118550
\(926\) −79.2666 −2.60486
\(927\) 0 0
\(928\) 32.9361 1.08118
\(929\) −15.4222 −0.505986 −0.252993 0.967468i \(-0.581415\pi\)
−0.252993 + 0.967468i \(0.581415\pi\)
\(930\) 0 0
\(931\) −33.6333 −1.10229
\(932\) −50.2389 −1.64563
\(933\) 0 0
\(934\) 6.42221 0.210141
\(935\) 12.2111 0.399346
\(936\) 0 0
\(937\) 54.4777 1.77971 0.889855 0.456244i \(-0.150806\pi\)
0.889855 + 0.456244i \(0.150806\pi\)
\(938\) −16.1194 −0.526318
\(939\) 0 0
\(940\) −30.4222 −0.992263
\(941\) −9.63331 −0.314037 −0.157018 0.987596i \(-0.550188\pi\)
−0.157018 + 0.987596i \(0.550188\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) −3.27502 −0.106593
\(945\) 0 0
\(946\) 37.7527 1.22745
\(947\) −18.6333 −0.605501 −0.302751 0.953070i \(-0.597905\pi\)
−0.302751 + 0.953070i \(0.597905\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −12.9083 −0.418801
\(951\) 0 0
\(952\) 22.8167 0.739492
\(953\) 14.4500 0.468080 0.234040 0.972227i \(-0.424805\pi\)
0.234040 + 0.972227i \(0.424805\pi\)
\(954\) 0 0
\(955\) 16.8167 0.544174
\(956\) 0 0
\(957\) 0 0
\(958\) 66.3583 2.14394
\(959\) −1.60555 −0.0518460
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) −5.90833 −0.190294
\(965\) 15.6056 0.502360
\(966\) 0 0
\(967\) 44.4777 1.43031 0.715153 0.698967i \(-0.246357\pi\)
0.715153 + 0.698967i \(0.246357\pi\)
\(968\) 25.2666 0.812100
\(969\) 0 0
\(970\) 19.3305 0.620666
\(971\) 44.0278 1.41292 0.706459 0.707754i \(-0.250291\pi\)
0.706459 + 0.707754i \(0.250291\pi\)
\(972\) 0 0
\(973\) 6.39445 0.204997
\(974\) −2.30278 −0.0737857
\(975\) 0 0
\(976\) −0.302776 −0.00969161
\(977\) 28.8167 0.921926 0.460963 0.887419i \(-0.347504\pi\)
0.460963 + 0.887419i \(0.347504\pi\)
\(978\) 0 0
\(979\) −9.97224 −0.318714
\(980\) 19.8167 0.633020
\(981\) 0 0
\(982\) −38.7250 −1.23576
\(983\) 18.4222 0.587577 0.293789 0.955870i \(-0.405084\pi\)
0.293789 + 0.955870i \(0.405084\pi\)
\(984\) 0 0
\(985\) −1.18335 −0.0377045
\(986\) 108.780 3.46428
\(987\) 0 0
\(988\) 0 0
\(989\) −30.6333 −0.974083
\(990\) 0 0
\(991\) 40.0278 1.27152 0.635762 0.771885i \(-0.280686\pi\)
0.635762 + 0.771885i \(0.280686\pi\)
\(992\) 21.2111 0.673453
\(993\) 0 0
\(994\) −11.0917 −0.351807
\(995\) −12.8167 −0.406315
\(996\) 0 0
\(997\) −18.4500 −0.584316 −0.292158 0.956370i \(-0.594373\pi\)
−0.292158 + 0.956370i \(0.594373\pi\)
\(998\) 5.57779 0.176562
\(999\) 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 7605.2.a.bb.1.1 2
3.2 odd 2 845.2.a.f.1.2 2
13.4 even 6 585.2.j.d.406.1 4
13.10 even 6 585.2.j.d.451.1 4
13.12 even 2 7605.2.a.bg.1.2 2
15.14 odd 2 4225.2.a.t.1.1 2
39.2 even 12 845.2.m.d.316.4 8
39.5 even 4 845.2.c.d.506.1 4
39.8 even 4 845.2.c.d.506.4 4
39.11 even 12 845.2.m.d.316.1 8
39.17 odd 6 65.2.e.b.16.2 4
39.20 even 12 845.2.m.d.361.4 8
39.23 odd 6 65.2.e.b.61.2 yes 4
39.29 odd 6 845.2.e.d.191.1 4
39.32 even 12 845.2.m.d.361.1 8
39.35 odd 6 845.2.e.d.146.1 4
39.38 odd 2 845.2.a.c.1.1 2
156.23 even 6 1040.2.q.o.321.2 4
156.95 even 6 1040.2.q.o.81.2 4
195.17 even 12 325.2.o.b.224.1 8
195.23 even 12 325.2.o.b.74.1 8
195.62 even 12 325.2.o.b.74.4 8
195.134 odd 6 325.2.e.a.276.1 4
195.173 even 12 325.2.o.b.224.4 8
195.179 odd 6 325.2.e.a.126.1 4
195.194 odd 2 4225.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.e.b.16.2 4 39.17 odd 6
65.2.e.b.61.2 yes 4 39.23 odd 6
325.2.e.a.126.1 4 195.179 odd 6
325.2.e.a.276.1 4 195.134 odd 6
325.2.o.b.74.1 8 195.23 even 12
325.2.o.b.74.4 8 195.62 even 12
325.2.o.b.224.1 8 195.17 even 12
325.2.o.b.224.4 8 195.173 even 12
585.2.j.d.406.1 4 13.4 even 6
585.2.j.d.451.1 4 13.10 even 6
845.2.a.c.1.1 2 39.38 odd 2
845.2.a.f.1.2 2 3.2 odd 2
845.2.c.d.506.1 4 39.5 even 4
845.2.c.d.506.4 4 39.8 even 4
845.2.e.d.146.1 4 39.35 odd 6
845.2.e.d.191.1 4 39.29 odd 6
845.2.m.d.316.1 8 39.11 even 12
845.2.m.d.316.4 8 39.2 even 12
845.2.m.d.361.1 8 39.32 even 12
845.2.m.d.361.4 8 39.20 even 12
1040.2.q.o.81.2 4 156.95 even 6
1040.2.q.o.321.2 4 156.23 even 6
4225.2.a.t.1.1 2 15.14 odd 2
4225.2.a.x.1.2 2 195.194 odd 2
7605.2.a.bb.1.1 2 1.1 even 1 trivial
7605.2.a.bg.1.2 2 13.12 even 2