Properties

Label 4225.2.a.t.1.1
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $1$
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] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, 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("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.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: \(33.7367948540\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 4225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} +2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} -1.60555 q^{11} -3.30278 q^{12} +2.30278 q^{14} +0.302776 q^{16} -7.60555 q^{17} +4.60555 q^{18} +5.60555 q^{19} +1.00000 q^{21} +3.69722 q^{22} +3.00000 q^{23} +3.00000 q^{24} +5.00000 q^{27} -3.30278 q^{28} -6.21110 q^{29} +4.00000 q^{31} +5.30278 q^{32} +1.60555 q^{33} +17.5139 q^{34} -6.60555 q^{36} +3.60555 q^{37} -12.9083 q^{38} -3.00000 q^{41} -2.30278 q^{42} +10.2111 q^{43} -5.30278 q^{44} -6.90833 q^{46} +9.21110 q^{47} -0.302776 q^{48} -6.00000 q^{49} +7.60555 q^{51} +3.21110 q^{53} -11.5139 q^{54} +3.00000 q^{56} -5.60555 q^{57} +14.3028 q^{58} +10.8167 q^{59} -1.00000 q^{61} -9.21110 q^{62} +2.00000 q^{63} -12.8167 q^{64} -3.69722 q^{66} -7.00000 q^{67} -25.1194 q^{68} -3.00000 q^{69} -4.81665 q^{71} +6.00000 q^{72} -0.788897 q^{73} -8.30278 q^{74} +18.5139 q^{76} +1.60555 q^{77} +5.21110 q^{79} +1.00000 q^{81} +6.90833 q^{82} -9.21110 q^{83} +3.30278 q^{84} -23.5139 q^{86} +6.21110 q^{87} +4.81665 q^{88} +6.21110 q^{89} +9.90833 q^{92} -4.00000 q^{93} -21.2111 q^{94} -5.30278 q^{96} -8.39445 q^{97} +13.8167 q^{98} +3.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 3 q^{12} + q^{14} - 3 q^{16} - 8 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{21} + 11 q^{22} + 6 q^{23} + 6 q^{24} + 10 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30278 −1.62831 −0.814154 0.580649i \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) 2.30278 0.940104
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −3.00000 −1.06066
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.60555 −0.484092 −0.242046 0.970265i \(-0.577818\pi\)
−0.242046 + 0.970265i \(0.577818\pi\)
\(12\) −3.30278 −0.953429
\(13\) 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 \(0.873700\pi\)
\(18\) 4.60555 1.08554
\(19\) 5.60555 1.28600 0.643001 0.765865i \(-0.277689\pi\)
0.643001 + 0.765865i \(0.277689\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 3.69722 0.788251
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) −3.30278 −0.624166
\(29\) −6.21110 −1.15337 −0.576686 0.816966i \(-0.695655\pi\)
−0.576686 + 0.816966i \(0.695655\pi\)
\(30\) 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
\(33\) 1.60555 0.279491
\(34\) 17.5139 3.00361
\(35\) 0 0
\(36\) −6.60555 −1.10093
\(37\) 3.60555 0.592749 0.296374 0.955072i \(-0.404222\pi\)
0.296374 + 0.955072i \(0.404222\pi\)
\(38\) −12.9083 −2.09401
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) −2.30278 −0.355326
\(43\) 10.2111 1.55718 0.778589 0.627534i \(-0.215936\pi\)
0.778589 + 0.627534i \(0.215936\pi\)
\(44\) −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 \(0.265526\pi\)
\(48\) −0.302776 −0.0437019
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 7.60555 1.06499
\(52\) 0 0
\(53\) 3.21110 0.441079 0.220539 0.975378i \(-0.429218\pi\)
0.220539 + 0.975378i \(0.429218\pi\)
\(54\) −11.5139 −1.56684
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −5.60555 −0.742473
\(58\) 14.3028 1.87805
\(59\) 10.8167 1.40821 0.704104 0.710097i \(-0.251349\pi\)
0.704104 + 0.710097i \(0.251349\pi\)
\(60\) 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
\(63\) 2.00000 0.251976
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) −3.69722 −0.455097
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −25.1194 −3.04618
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −4.81665 −0.571632 −0.285816 0.958285i \(-0.592265\pi\)
−0.285816 + 0.958285i \(0.592265\pi\)
\(72\) 6.00000 0.707107
\(73\) −0.788897 −0.0923335 −0.0461667 0.998934i \(-0.514701\pi\)
−0.0461667 + 0.998934i \(0.514701\pi\)
\(74\) −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 0
\(81\) 1.00000 0.111111
\(82\) 6.90833 0.762897
\(83\) −9.21110 −1.01105 −0.505525 0.862812i \(-0.668701\pi\)
−0.505525 + 0.862812i \(0.668701\pi\)
\(84\) 3.30278 0.360362
\(85\) 0 0
\(86\) −23.5139 −2.53557
\(87\) 6.21110 0.665900
\(88\) 4.81665 0.513457
\(89\) 6.21110 0.658376 0.329188 0.944264i \(-0.393225\pi\)
0.329188 + 0.944264i \(0.393225\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.90833 1.03301
\(93\) −4.00000 −0.414781
\(94\) −21.2111 −2.18776
\(95\) 0 0
\(96\) −5.30278 −0.541212
\(97\) −8.39445 −0.852327 −0.426164 0.904646i \(-0.640135\pi\)
−0.426164 + 0.904646i \(0.640135\pi\)
\(98\) 13.8167 1.39569
\(99\) 3.21110 0.322728
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) −17.5139 −1.73413
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 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 \(0.597062\pi\)
\(108\) 16.5139 1.58905
\(109\) 19.2111 1.84009 0.920045 0.391813i \(-0.128152\pi\)
0.920045 + 0.391813i \(0.128152\pi\)
\(110\) 0 0
\(111\) −3.60555 −0.342224
\(112\) −0.302776 −0.0286096
\(113\) 1.60555 0.151038 0.0755188 0.997144i \(-0.475939\pi\)
0.0755188 + 0.997144i \(0.475939\pi\)
\(114\) 12.9083 1.20898
\(115\) 0 0
\(116\) −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
\(123\) 3.00000 0.270501
\(124\) 13.2111 1.18639
\(125\) 0 0
\(126\) −4.60555 −0.410295
\(127\) 4.21110 0.373675 0.186837 0.982391i \(-0.440176\pi\)
0.186837 + 0.982391i \(0.440176\pi\)
\(128\) 18.9083 1.67128
\(129\) −10.2111 −0.899037
\(130\) 0 0
\(131\) −21.2111 −1.85322 −0.926611 0.376021i \(-0.877292\pi\)
−0.926611 + 0.376021i \(0.877292\pi\)
\(132\) 5.30278 0.461547
\(133\) −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 \(0.521849\pi\)
\(138\) 6.90833 0.588076
\(139\) 6.39445 0.542370 0.271185 0.962527i \(-0.412584\pi\)
0.271185 + 0.962527i \(0.412584\pi\)
\(140\) 0 0
\(141\) −9.21110 −0.775715
\(142\) 11.0917 0.930793
\(143\) 0 0
\(144\) −0.605551 −0.0504626
\(145\) 0 0
\(146\) 1.81665 0.150347
\(147\) 6.00000 0.494872
\(148\) 11.9083 0.978858
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 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
\(153\) 15.2111 1.22974
\(154\) −3.69722 −0.297931
\(155\) 0 0
\(156\) 0 0
\(157\) −11.2111 −0.894743 −0.447372 0.894348i \(-0.647640\pi\)
−0.447372 + 0.894348i \(0.647640\pi\)
\(158\) −12.0000 −0.954669
\(159\) −3.21110 −0.254657
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −2.30278 −0.180923
\(163\) −3.78890 −0.296769 −0.148385 0.988930i \(-0.547407\pi\)
−0.148385 + 0.988930i \(0.547407\pi\)
\(164\) −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 \(0.386786\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 0 0
\(171\) −11.2111 −0.857334
\(172\) 33.7250 2.57151
\(173\) −4.81665 −0.366203 −0.183102 0.983094i \(-0.558614\pi\)
−0.183102 + 0.983094i \(0.558614\pi\)
\(174\) −14.3028 −1.08429
\(175\) 0 0
\(176\) −0.486122 −0.0366428
\(177\) −10.8167 −0.813029
\(178\) −14.3028 −1.07204
\(179\) 22.8167 1.70540 0.852698 0.522404i \(-0.174965\pi\)
0.852698 + 0.522404i \(0.174965\pi\)
\(180\) 0 0
\(181\) 17.6333 1.31067 0.655337 0.755337i \(-0.272527\pi\)
0.655337 + 0.755337i \(0.272527\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) −9.00000 −0.663489
\(185\) 0 0
\(186\) 9.21110 0.675391
\(187\) 12.2111 0.892964
\(188\) 30.4222 2.21877
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 16.8167 1.21681 0.608405 0.793627i \(-0.291810\pi\)
0.608405 + 0.793627i \(0.291810\pi\)
\(192\) 12.8167 0.924962
\(193\) 15.6056 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(194\) 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 \(0.486578\pi\)
\(198\) −7.39445 −0.525501
\(199\) 12.8167 0.908549 0.454274 0.890862i \(-0.349899\pi\)
0.454274 + 0.890862i \(0.349899\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 20.7250 1.45820
\(203\) 6.21110 0.435934
\(204\) 25.1194 1.75871
\(205\) 0 0
\(206\) −9.21110 −0.641768
\(207\) −6.00000 −0.417029
\(208\) 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
\(213\) 4.81665 0.330032
\(214\) 14.3028 0.977718
\(215\) 0 0
\(216\) −15.0000 −1.02062
\(217\) −4.00000 −0.271538
\(218\) −44.2389 −2.99623
\(219\) 0.788897 0.0533087
\(220\) 0 0
\(221\) 0 0
\(222\) 8.30278 0.557246
\(223\) −4.21110 −0.281996 −0.140998 0.990010i \(-0.545031\pi\)
−0.140998 + 0.990010i \(0.545031\pi\)
\(224\) −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 \(0.863948\pi\)
\(228\) −18.5139 −1.22611
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −1.60555 −0.105638
\(232\) 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
\(235\) 0 0
\(236\) 35.7250 2.32550
\(237\) −5.21110 −0.338497
\(238\) −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
\(243\) −16.0000 −1.02640
\(244\) −3.30278 −0.211439
\(245\) 0 0
\(246\) −6.90833 −0.440459
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) 9.21110 0.583730
\(250\) 0 0
\(251\) −7.18335 −0.453409 −0.226704 0.973964i \(-0.572795\pi\)
−0.226704 + 0.973964i \(0.572795\pi\)
\(252\) 6.60555 0.416111
\(253\) −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 \(0.329153\pi\)
\(258\) 23.5139 1.46391
\(259\) −3.60555 −0.224038
\(260\) 0 0
\(261\) 12.4222 0.768915
\(262\) 48.8444 3.01762
\(263\) −11.7889 −0.726935 −0.363467 0.931607i \(-0.618407\pi\)
−0.363467 + 0.931607i \(0.618407\pi\)
\(264\) −4.81665 −0.296445
\(265\) 0 0
\(266\) 12.9083 0.791460
\(267\) −6.21110 −0.380113
\(268\) −23.1194 −1.41224
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 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
\(275\) 0 0
\(276\) −9.90833 −0.596411
\(277\) −27.6056 −1.65866 −0.829328 0.558761i \(-0.811277\pi\)
−0.829328 + 0.558761i \(0.811277\pi\)
\(278\) −14.7250 −0.883146
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 21.2111 1.26310
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) −15.9083 −0.943986
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −10.6056 −0.624938
\(289\) 40.8444 2.40261
\(290\) 0 0
\(291\) 8.39445 0.492091
\(292\) −2.60555 −0.152478
\(293\) 10.3944 0.607250 0.303625 0.952792i \(-0.401803\pi\)
0.303625 + 0.952792i \(0.401803\pi\)
\(294\) −13.8167 −0.805804
\(295\) 0 0
\(296\) −10.8167 −0.628705
\(297\) −8.02776 −0.465818
\(298\) 6.90833 0.400189
\(299\) 0 0
\(300\) 0 0
\(301\) −10.2111 −0.588558
\(302\) −2.78890 −0.160483
\(303\) 9.00000 0.517036
\(304\) 1.69722 0.0973425
\(305\) 0 0
\(306\) −35.0278 −2.00240
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 5.30278 0.302154
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −9.21110 −0.522314 −0.261157 0.965296i \(-0.584104\pi\)
−0.261157 + 0.965296i \(0.584104\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 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 \(0.446109\pi\)
\(318\) 7.39445 0.414660
\(319\) 9.97224 0.558338
\(320\) 0 0
\(321\) 6.21110 0.346670
\(322\) 6.90833 0.384986
\(323\) −42.6333 −2.37218
\(324\) 3.30278 0.183488
\(325\) 0 0
\(326\) 8.72498 0.483232
\(327\) −19.2111 −1.06238
\(328\) 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
\(333\) −7.21110 −0.395166
\(334\) −20.7250 −1.13402
\(335\) 0 0
\(336\) 0.302776 0.0165178
\(337\) 25.6333 1.39634 0.698168 0.715934i \(-0.253999\pi\)
0.698168 + 0.715934i \(0.253999\pi\)
\(338\) 0 0
\(339\) −1.60555 −0.0872016
\(340\) 0 0
\(341\) −6.42221 −0.347782
\(342\) 25.8167 1.39600
\(343\) 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 \(0.549661\pi\)
\(348\) 20.5139 1.09966
\(349\) 3.78890 0.202815 0.101408 0.994845i \(-0.467665\pi\)
0.101408 + 0.994845i \(0.467665\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8.51388 −0.453791
\(353\) 16.8167 0.895060 0.447530 0.894269i \(-0.352304\pi\)
0.447530 + 0.894269i \(0.352304\pi\)
\(354\) 24.9083 1.32386
\(355\) 0 0
\(356\) 20.5139 1.08723
\(357\) −7.60555 −0.402528
\(358\) −52.5416 −2.77691
\(359\) −18.4222 −0.972287 −0.486143 0.873879i \(-0.661597\pi\)
−0.486143 + 0.873879i \(0.661597\pi\)
\(360\) 0 0
\(361\) 12.4222 0.653800
\(362\) −40.6056 −2.13418
\(363\) 8.42221 0.442051
\(364\) 0 0
\(365\) 0 0
\(366\) −2.30278 −0.120368
\(367\) −11.4222 −0.596234 −0.298117 0.954529i \(-0.596359\pi\)
−0.298117 + 0.954529i \(0.596359\pi\)
\(368\) 0.908327 0.0473498
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −3.21110 −0.166712
\(372\) −13.2111 −0.684964
\(373\) 20.3944 1.05598 0.527992 0.849249i \(-0.322945\pi\)
0.527992 + 0.849249i \(0.322945\pi\)
\(374\) −28.1194 −1.45402
\(375\) 0 0
\(376\) −27.6333 −1.42508
\(377\) 0 0
\(378\) 11.5139 0.592210
\(379\) −9.60555 −0.493404 −0.246702 0.969091i \(-0.579347\pi\)
−0.246702 + 0.969091i \(0.579347\pi\)
\(380\) 0 0
\(381\) −4.21110 −0.215741
\(382\) −38.7250 −1.98134
\(383\) −24.6333 −1.25870 −0.629352 0.777121i \(-0.716679\pi\)
−0.629352 + 0.777121i \(0.716679\pi\)
\(384\) −18.9083 −0.964912
\(385\) 0 0
\(386\) −35.9361 −1.82910
\(387\) −20.4222 −1.03812
\(388\) −27.7250 −1.40752
\(389\) −15.2111 −0.771234 −0.385617 0.922659i \(-0.626011\pi\)
−0.385617 + 0.922659i \(0.626011\pi\)
\(390\) 0 0
\(391\) −22.8167 −1.15389
\(392\) 18.0000 0.909137
\(393\) 21.2111 1.06996
\(394\) −2.72498 −0.137283
\(395\) 0 0
\(396\) 10.6056 0.532949
\(397\) 22.0278 1.10554 0.552771 0.833333i \(-0.313571\pi\)
0.552771 + 0.833333i \(0.313571\pi\)
\(398\) −29.5139 −1.47940
\(399\) 5.60555 0.280629
\(400\) 0 0
\(401\) −12.2111 −0.609793 −0.304897 0.952385i \(-0.598622\pi\)
−0.304897 + 0.952385i \(0.598622\pi\)
\(402\) −16.1194 −0.803964
\(403\) 0 0
\(404\) −29.7250 −1.47887
\(405\) 0 0
\(406\) −14.3028 −0.709835
\(407\) −5.78890 −0.286945
\(408\) −22.8167 −1.12959
\(409\) −8.21110 −0.406013 −0.203006 0.979177i \(-0.565071\pi\)
−0.203006 + 0.979177i \(0.565071\pi\)
\(410\) 0 0
\(411\) 1.60555 0.0791960
\(412\) 13.2111 0.650864
\(413\) −10.8167 −0.532253
\(414\) 13.8167 0.679051
\(415\) 0 0
\(416\) 0 0
\(417\) −6.39445 −0.313138
\(418\) 20.7250 1.01369
\(419\) 17.2389 0.842173 0.421087 0.907020i \(-0.361649\pi\)
0.421087 + 0.907020i \(0.361649\pi\)
\(420\) 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
\(423\) −18.4222 −0.895718
\(424\) −9.63331 −0.467835
\(425\) 0 0
\(426\) −11.0917 −0.537393
\(427\) 1.00000 0.0483934
\(428\) −20.5139 −0.991576
\(429\) 0 0
\(430\) 0 0
\(431\) −29.2389 −1.40839 −0.704193 0.710008i \(-0.748691\pi\)
−0.704193 + 0.710008i \(0.748691\pi\)
\(432\) 1.51388 0.0728365
\(433\) −3.60555 −0.173272 −0.0866359 0.996240i \(-0.527612\pi\)
−0.0866359 + 0.996240i \(0.527612\pi\)
\(434\) 9.21110 0.442147
\(435\) 0 0
\(436\) 63.4500 3.03870
\(437\) 16.8167 0.804450
\(438\) −1.81665 −0.0868031
\(439\) −27.2389 −1.30004 −0.650020 0.759917i \(-0.725239\pi\)
−0.650020 + 0.759917i \(0.725239\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) −6.42221 −0.305128 −0.152564 0.988294i \(-0.548753\pi\)
−0.152564 + 0.988294i \(0.548753\pi\)
\(444\) −11.9083 −0.565144
\(445\) 0 0
\(446\) 9.69722 0.459177
\(447\) 3.00000 0.141895
\(448\) 12.8167 0.605530
\(449\) −30.6333 −1.44568 −0.722838 0.691018i \(-0.757163\pi\)
−0.722838 + 0.691018i \(0.757163\pi\)
\(450\) 0 0
\(451\) 4.81665 0.226807
\(452\) 5.30278 0.249422
\(453\) −1.21110 −0.0569026
\(454\) 63.1472 2.96364
\(455\) 0 0
\(456\) 16.8167 0.787512
\(457\) −26.8167 −1.25443 −0.627215 0.778846i \(-0.715805\pi\)
−0.627215 + 0.778846i \(0.715805\pi\)
\(458\) 32.2389 1.50642
\(459\) −38.0278 −1.77498
\(460\) 0 0
\(461\) −36.2111 −1.68652 −0.843260 0.537507i \(-0.819366\pi\)
−0.843260 + 0.537507i \(0.819366\pi\)
\(462\) 3.69722 0.172010
\(463\) −34.4222 −1.59974 −0.799868 0.600176i \(-0.795097\pi\)
−0.799868 + 0.600176i \(0.795097\pi\)
\(464\) −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
\(470\) 0 0
\(471\) 11.2111 0.516580
\(472\) −32.4500 −1.49363
\(473\) −16.3944 −0.753818
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) 25.1194 1.15135
\(477\) −6.42221 −0.294053
\(478\) 0 0
\(479\) 28.8167 1.31667 0.658333 0.752727i \(-0.271262\pi\)
0.658333 + 0.752727i \(0.271262\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4.11943 0.187635
\(483\) 3.00000 0.136505
\(484\) −27.8167 −1.26439
\(485\) 0 0
\(486\) 36.8444 1.67130
\(487\) −1.00000 −0.0453143 −0.0226572 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 3.00000 0.135804
\(489\) 3.78890 0.171340
\(490\) 0 0
\(491\) −16.8167 −0.758925 −0.379462 0.925207i \(-0.623891\pi\)
−0.379462 + 0.925207i \(0.623891\pi\)
\(492\) 9.90833 0.446702
\(493\) 47.2389 2.12753
\(494\) 0 0
\(495\) 0 0
\(496\) 1.21110 0.0543801
\(497\) 4.81665 0.216056
\(498\) −21.2111 −0.950492
\(499\) −2.42221 −0.108433 −0.0542164 0.998529i \(-0.517266\pi\)
−0.0542164 + 0.998529i \(0.517266\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 16.5416 0.738289
\(503\) −3.00000 −0.133763 −0.0668817 0.997761i \(-0.521305\pi\)
−0.0668817 + 0.997761i \(0.521305\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) 11.0917 0.493085
\(507\) 0 0
\(508\) 13.9083 0.617082
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 0.788897 0.0348988
\(512\) 3.42221 0.151242
\(513\) 28.0278 1.23746
\(514\) −37.7527 −1.66520
\(515\) 0 0
\(516\) −33.7250 −1.48466
\(517\) −14.7889 −0.650415
\(518\) 8.30278 0.364803
\(519\) 4.81665 0.211428
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −28.6056 −1.25203
\(523\) 1.42221 0.0621887 0.0310943 0.999516i \(-0.490101\pi\)
0.0310943 + 0.999516i \(0.490101\pi\)
\(524\) −70.0555 −3.06039
\(525\) 0 0
\(526\) 27.1472 1.18367
\(527\) −30.4222 −1.32521
\(528\) 0.486122 0.0211557
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −21.6333 −0.938806
\(532\) −18.5139 −0.802678
\(533\) 0 0
\(534\) 14.3028 0.618942
\(535\) 0 0
\(536\) 21.0000 0.907062
\(537\) −22.8167 −0.984611
\(538\) 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
\(543\) −17.6333 −0.756718
\(544\) −40.3305 −1.72916
\(545\) 0 0
\(546\) 0 0
\(547\) −32.8444 −1.40433 −0.702163 0.712016i \(-0.747782\pi\)
−0.702163 + 0.712016i \(0.747782\pi\)
\(548\) −5.30278 −0.226523
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −34.8167 −1.48324
\(552\) 9.00000 0.383065
\(553\) −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 \(0.510829\pi\)
\(558\) 18.4222 0.779874
\(559\) 0 0
\(560\) 0 0
\(561\) −12.2111 −0.515553
\(562\) −13.8167 −0.582820
\(563\) −9.42221 −0.397099 −0.198549 0.980091i \(-0.563623\pi\)
−0.198549 + 0.980091i \(0.563623\pi\)
\(564\) −30.4222 −1.28101
\(565\) 0 0
\(566\) 11.5139 0.483964
\(567\) −1.00000 −0.0419961
\(568\) 14.4500 0.606307
\(569\) −27.4222 −1.14960 −0.574799 0.818294i \(-0.694920\pi\)
−0.574799 + 0.818294i \(0.694920\pi\)
\(570\) 0 0
\(571\) 20.8444 0.872311 0.436156 0.899871i \(-0.356340\pi\)
0.436156 + 0.899871i \(0.356340\pi\)
\(572\) 0 0
\(573\) −16.8167 −0.702526
\(574\) −6.90833 −0.288348
\(575\) 0 0
\(576\) 25.6333 1.06805
\(577\) −13.6333 −0.567562 −0.283781 0.958889i \(-0.591589\pi\)
−0.283781 + 0.958889i \(0.591589\pi\)
\(578\) −94.0555 −3.91219
\(579\) −15.6056 −0.648545
\(580\) 0 0
\(581\) 9.21110 0.382141
\(582\) −19.3305 −0.801276
\(583\) −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 \(0.742276\pi\)
\(588\) 19.8167 0.817225
\(589\) 22.4222 0.923891
\(590\) 0 0
\(591\) −1.18335 −0.0486764
\(592\) 1.09167 0.0448675
\(593\) 20.7889 0.853698 0.426849 0.904323i \(-0.359624\pi\)
0.426849 + 0.904323i \(0.359624\pi\)
\(594\) 18.4861 0.758495
\(595\) 0 0
\(596\) −9.90833 −0.405861
\(597\) −12.8167 −0.524551
\(598\) 0 0
\(599\) −21.2111 −0.866662 −0.433331 0.901235i \(-0.642662\pi\)
−0.433331 + 0.901235i \(0.642662\pi\)
\(600\) 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
\(603\) 14.0000 0.570124
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) −20.7250 −0.841895
\(607\) 34.2111 1.38859 0.694293 0.719693i \(-0.255717\pi\)
0.694293 + 0.719693i \(0.255717\pi\)
\(608\) 29.7250 1.20551
\(609\) −6.21110 −0.251687
\(610\) 0 0
\(611\) 0 0
\(612\) 50.2389 2.03079
\(613\) −5.60555 −0.226406 −0.113203 0.993572i \(-0.536111\pi\)
−0.113203 + 0.993572i \(0.536111\pi\)
\(614\) 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 \(0.781731\pi\)
\(618\) 9.21110 0.370525
\(619\) −14.4222 −0.579677 −0.289839 0.957076i \(-0.593602\pi\)
−0.289839 + 0.957076i \(0.593602\pi\)
\(620\) 0 0
\(621\) 15.0000 0.601929
\(622\) 21.2111 0.850488
\(623\) −6.21110 −0.248843
\(624\) 0 0
\(625\) 0 0
\(626\) 32.2389 1.28852
\(627\) 9.00000 0.359425
\(628\) −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
\(633\) 23.6056 0.938236
\(634\) −13.8167 −0.548729
\(635\) 0 0
\(636\) −10.6056 −0.420537
\(637\) 0 0
\(638\) −22.9638 −0.909147
\(639\) 9.63331 0.381088
\(640\) 0 0
\(641\) 9.42221 0.372155 0.186077 0.982535i \(-0.440422\pi\)
0.186077 + 0.982535i \(0.440422\pi\)
\(642\) −14.3028 −0.564486
\(643\) 2.63331 0.103848 0.0519238 0.998651i \(-0.483465\pi\)
0.0519238 + 0.998651i \(0.483465\pi\)
\(644\) −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 \(0.782212\pi\)
\(648\) −3.00000 −0.117851
\(649\) −17.3667 −0.681702
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −12.5139 −0.490081
\(653\) 7.18335 0.281106 0.140553 0.990073i \(-0.455112\pi\)
0.140553 + 0.990073i \(0.455112\pi\)
\(654\) 44.2389 1.72988
\(655\) 0 0
\(656\) −0.908327 −0.0354642
\(657\) 1.57779 0.0615556
\(658\) 21.2111 0.826895
\(659\) −34.8167 −1.35626 −0.678132 0.734940i \(-0.737210\pi\)
−0.678132 + 0.734940i \(0.737210\pi\)
\(660\) 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
\(665\) 0 0
\(666\) 16.6056 0.643452
\(667\) −18.6333 −0.721485
\(668\) 29.7250 1.15009
\(669\) 4.21110 0.162811
\(670\) 0 0
\(671\) 1.60555 0.0619816
\(672\) 5.30278 0.204559
\(673\) 17.6056 0.678644 0.339322 0.940670i \(-0.389802\pi\)
0.339322 + 0.940670i \(0.389802\pi\)
\(674\) −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 \(0.440733\pi\)
\(678\) 3.69722 0.141991
\(679\) 8.39445 0.322149
\(680\) 0 0
\(681\) 27.4222 1.05082
\(682\) 14.7889 0.566296
\(683\) 36.2111 1.38558 0.692790 0.721140i \(-0.256381\pi\)
0.692790 + 0.721140i \(0.256381\pi\)
\(684\) −37.0278 −1.41579
\(685\) 0 0
\(686\) −29.9361 −1.14296
\(687\) 14.0000 0.534133
\(688\) 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
\(693\) −3.21110 −0.121980
\(694\) 13.3305 0.506020
\(695\) 0 0
\(696\) −18.6333 −0.706294
\(697\) 22.8167 0.864242
\(698\) −8.72498 −0.330245
\(699\) 15.2111 0.575337
\(700\) 0 0
\(701\) −36.4222 −1.37565 −0.687824 0.725878i \(-0.741434\pi\)
−0.687824 + 0.725878i \(0.741434\pi\)
\(702\) 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
\(708\) −35.7250 −1.34263
\(709\) 13.8444 0.519938 0.259969 0.965617i \(-0.416288\pi\)
0.259969 + 0.965617i \(0.416288\pi\)
\(710\) 0 0
\(711\) −10.4222 −0.390863
\(712\) −18.6333 −0.698313
\(713\) 12.0000 0.449404
\(714\) 17.5139 0.655440
\(715\) 0 0
\(716\) 75.3583 2.81627
\(717\) 0 0
\(718\) 42.4222 1.58318
\(719\) −25.6056 −0.954926 −0.477463 0.878652i \(-0.658444\pi\)
−0.477463 + 0.878652i \(0.658444\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −28.6056 −1.06459
\(723\) 1.78890 0.0665298
\(724\) 58.2389 2.16443
\(725\) 0 0
\(726\) −19.3944 −0.719796
\(727\) −13.5778 −0.503573 −0.251786 0.967783i \(-0.581018\pi\)
−0.251786 + 0.967783i \(0.581018\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −77.6611 −2.87240
\(732\) 3.30278 0.122074
\(733\) −46.8444 −1.73024 −0.865119 0.501567i \(-0.832757\pi\)
−0.865119 + 0.501567i \(0.832757\pi\)
\(734\) 26.3028 0.970853
\(735\) 0 0
\(736\) 15.9083 0.586389
\(737\) 11.2389 0.413989
\(738\) −13.8167 −0.508598
\(739\) 35.6056 1.30977 0.654886 0.755728i \(-0.272717\pi\)
0.654886 + 0.755728i \(0.272717\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7.39445 0.271459
\(743\) 36.6333 1.34395 0.671973 0.740576i \(-0.265447\pi\)
0.671973 + 0.740576i \(0.265447\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) −46.9638 −1.71947
\(747\) 18.4222 0.674033
\(748\) 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
\(753\) 7.18335 0.261776
\(754\) 0 0
\(755\) 0 0
\(756\) −16.5139 −0.600604
\(757\) −0.816654 −0.0296818 −0.0148409 0.999890i \(-0.504724\pi\)
−0.0148409 + 0.999890i \(0.504724\pi\)
\(758\) 22.1194 0.803414
\(759\) 4.81665 0.174833
\(760\) 0 0
\(761\) −18.6333 −0.675457 −0.337728 0.941244i \(-0.609659\pi\)
−0.337728 + 0.941244i \(0.609659\pi\)
\(762\) 9.69722 0.351293
\(763\) −19.2111 −0.695489
\(764\) 55.5416 2.00943
\(765\) 0 0
\(766\) 56.7250 2.04956
\(767\) 0 0
\(768\) 17.9083 0.646211
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) −16.3944 −0.590432
\(772\) 51.5416 1.85502
\(773\) 22.3944 0.805472 0.402736 0.915316i \(-0.368059\pi\)
0.402736 + 0.915316i \(0.368059\pi\)
\(774\) 47.0278 1.69038
\(775\) 0 0
\(776\) 25.1833 0.904029
\(777\) 3.60555 0.129348
\(778\) 35.0278 1.25581
\(779\) −16.8167 −0.602519
\(780\) 0 0
\(781\) 7.73338 0.276722
\(782\) 52.5416 1.87889
\(783\) −31.0555 −1.10983
\(784\) −1.81665 −0.0648805
\(785\) 0 0
\(786\) −48.8444 −1.74222
\(787\) 14.6333 0.521621 0.260811 0.965390i \(-0.416010\pi\)
0.260811 + 0.965390i \(0.416010\pi\)
\(788\) 3.90833 0.139228
\(789\) 11.7889 0.419696
\(790\) 0 0
\(791\) −1.60555 −0.0570868
\(792\) −9.63331 −0.342305
\(793\) 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 \(0.417621\pi\)
\(798\) −12.9083 −0.456950
\(799\) −70.0555 −2.47839
\(800\) 0 0
\(801\) −12.4222 −0.438917
\(802\) 28.1194 0.992932
\(803\) 1.26662 0.0446979
\(804\) 23.1194 0.815359
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 27.0000 0.949857
\(809\) −55.0555 −1.93565 −0.967824 0.251627i \(-0.919034\pi\)
−0.967824 + 0.251627i \(0.919034\pi\)
\(810\) 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
\(813\) −20.8167 −0.730072
\(814\) 13.3305 0.467235
\(815\) 0 0
\(816\) 2.30278 0.0806133
\(817\) 57.2389 2.00253
\(818\) 18.9083 0.661114
\(819\) 0 0
\(820\) 0 0
\(821\) −21.4222 −0.747640 −0.373820 0.927501i \(-0.621952\pi\)
−0.373820 + 0.927501i \(0.621952\pi\)
\(822\) −3.69722 −0.128956
\(823\) 16.6333 0.579801 0.289900 0.957057i \(-0.406378\pi\)
0.289900 + 0.957057i \(0.406378\pi\)
\(824\) −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 \(0.764033\pi\)
\(828\) −19.8167 −0.688676
\(829\) 29.4222 1.02188 0.510938 0.859618i \(-0.329298\pi\)
0.510938 + 0.859618i \(0.329298\pi\)
\(830\) 0 0
\(831\) 27.6056 0.957626
\(832\) 0 0
\(833\) 45.6333 1.58110
\(834\) 14.7250 0.509884
\(835\) 0 0
\(836\) −29.7250 −1.02806
\(837\) 20.0000 0.691301
\(838\) −39.6972 −1.37132
\(839\) 20.0278 0.691435 0.345717 0.938339i \(-0.387636\pi\)
0.345717 + 0.938339i \(0.387636\pi\)
\(840\) 0 0
\(841\) 9.57779 0.330269
\(842\) 74.6611 2.57299
\(843\) −6.00000 −0.206651
\(844\) −77.9638 −2.68363
\(845\) 0 0
\(846\) 42.4222 1.45851
\(847\) 8.42221 0.289390
\(848\) 0.972244 0.0333870
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) 10.8167 0.370790
\(852\) 15.9083 0.545010
\(853\) 47.2111 1.61648 0.808239 0.588855i \(-0.200421\pi\)
0.808239 + 0.588855i \(0.200421\pi\)
\(854\) −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 \(0.441077\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) 67.3305 2.29329
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 26.5139 0.902020
\(865\) 0 0
\(866\) 8.30278 0.282140
\(867\) −40.8444 −1.38715
\(868\) −13.2111 −0.448414
\(869\) −8.36669 −0.283821
\(870\) 0 0
\(871\) 0 0
\(872\) −57.6333 −1.95171
\(873\) 16.7889 0.568218
\(874\) −38.7250 −1.30989
\(875\) 0 0
\(876\) 2.60555 0.0880334
\(877\) −1.97224 −0.0665979 −0.0332990 0.999445i \(-0.510601\pi\)
−0.0332990 + 0.999445i \(0.510601\pi\)
\(878\) 62.7250 2.11687
\(879\) −10.3944 −0.350596
\(880\) 0 0
\(881\) −21.8444 −0.735957 −0.367978 0.929834i \(-0.619950\pi\)
−0.367978 + 0.929834i \(0.619950\pi\)
\(882\) −27.6333 −0.930462
\(883\) −11.6333 −0.391492 −0.195746 0.980655i \(-0.562713\pi\)
−0.195746 + 0.980655i \(0.562713\pi\)
\(884\) 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 \(0.286279\pi\)
\(888\) 10.8167 0.362983
\(889\) −4.21110 −0.141236
\(890\) 0 0
\(891\) −1.60555 −0.0537880
\(892\) −13.9083 −0.465685
\(893\) 51.6333 1.72784
\(894\) −6.90833 −0.231049
\(895\) 0 0
\(896\) −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
\(903\) 10.2111 0.339804
\(904\) −4.81665 −0.160200
\(905\) 0 0
\(906\) 2.78890 0.0926549
\(907\) 38.2666 1.27062 0.635311 0.772256i \(-0.280872\pi\)
0.635311 + 0.772256i \(0.280872\pi\)
\(908\) −90.5694 −3.00565
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −1.69722 −0.0562007
\(913\) 14.7889 0.489441
\(914\) 61.7527 2.04260
\(915\) 0 0
\(916\) −46.2389 −1.52777
\(917\) 21.2111 0.700452
\(918\) 87.5694 2.89022
\(919\) −38.8167 −1.28044 −0.640222 0.768190i \(-0.721157\pi\)
−0.640222 + 0.768190i \(0.721157\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) 83.3860 2.74617
\(923\) 0 0
\(924\) −5.30278 −0.174449
\(925\) 0 0
\(926\) 79.2666 2.60486
\(927\) −8.00000 −0.262754
\(928\) −32.9361 −1.08118
\(929\) 15.4222 0.505986 0.252993 0.967468i \(-0.418585\pi\)
0.252993 + 0.967468i \(0.418585\pi\)
\(930\) 0 0
\(931\) −33.6333 −1.10229
\(932\) −50.2389 −1.64563
\(933\) 9.21110 0.301558
\(934\) 6.42221 0.210141
\(935\) 0 0
\(936\) 0 0
\(937\) −54.4777 −1.77971 −0.889855 0.456244i \(-0.849194\pi\)
−0.889855 + 0.456244i \(0.849194\pi\)
\(938\) −16.1194 −0.526318
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 9.63331 0.314037 0.157018 0.987596i \(-0.449812\pi\)
0.157018 + 0.987596i \(0.449812\pi\)
\(942\) −25.8167 −0.841152
\(943\) −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 \(0.597905\pi\)
\(948\) −17.2111 −0.558991
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) −22.8167 −0.739492
\(953\) 14.4500 0.468080 0.234040 0.972227i \(-0.424805\pi\)
0.234040 + 0.972227i \(0.424805\pi\)
\(954\) 14.7889 0.478808
\(955\) 0 0
\(956\) 0 0
\(957\) −9.97224 −0.322357
\(958\) −66.3583 −2.14394
\(959\) 1.60555 0.0518460
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.4222 0.400300
\(964\) −5.90833 −0.190294
\(965\) 0 0
\(966\) −6.90833 −0.222272
\(967\) −44.4777 −1.43031 −0.715153 0.698967i \(-0.753643\pi\)
−0.715153 + 0.698967i \(0.753643\pi\)
\(968\) 25.2666 0.812100
\(969\) 42.6333 1.36958
\(970\) 0 0
\(971\) −44.0278 −1.41292 −0.706459 0.707754i \(-0.749709\pi\)
−0.706459 + 0.707754i \(0.749709\pi\)
\(972\) −52.8444 −1.69499
\(973\) −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 \(0.347504\pi\)
\(978\) −8.72498 −0.278994
\(979\) −9.97224 −0.318714
\(980\) 0 0
\(981\) −38.4222 −1.22673
\(982\) 38.7250 1.23576
\(983\) 18.4222 0.587577 0.293789 0.955870i \(-0.405084\pi\)
0.293789 + 0.955870i \(0.405084\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) −108.780 −3.46428
\(987\) 9.21110 0.293193
\(988\) 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
\(993\) 10.0278 0.318221
\(994\) −11.0917 −0.351807
\(995\) 0 0
\(996\) 30.4222 0.963964
\(997\) 18.4500 0.584316 0.292158 0.956370i \(-0.405627\pi\)
0.292158 + 0.956370i \(0.405627\pi\)
\(998\) 5.57779 0.176562
\(999\) 18.0278 0.570373
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 4225.2.a.t.1.1 2
5.4 even 2 845.2.a.f.1.2 2
13.4 even 6 325.2.e.a.276.1 4
13.10 even 6 325.2.e.a.126.1 4
13.12 even 2 4225.2.a.x.1.2 2
15.14 odd 2 7605.2.a.bb.1.1 2
65.4 even 6 65.2.e.b.16.2 4
65.9 even 6 845.2.e.d.146.1 4
65.17 odd 12 325.2.o.b.224.4 8
65.19 odd 12 845.2.m.d.361.1 8
65.23 odd 12 325.2.o.b.74.4 8
65.24 odd 12 845.2.m.d.316.1 8
65.29 even 6 845.2.e.d.191.1 4
65.34 odd 4 845.2.c.d.506.4 4
65.43 odd 12 325.2.o.b.224.1 8
65.44 odd 4 845.2.c.d.506.1 4
65.49 even 6 65.2.e.b.61.2 yes 4
65.54 odd 12 845.2.m.d.316.4 8
65.59 odd 12 845.2.m.d.361.4 8
65.62 odd 12 325.2.o.b.74.1 8
65.64 even 2 845.2.a.c.1.1 2
195.134 odd 6 585.2.j.d.406.1 4
195.179 odd 6 585.2.j.d.451.1 4
195.194 odd 2 7605.2.a.bg.1.2 2
260.179 odd 6 1040.2.q.o.321.2 4
260.199 odd 6 1040.2.q.o.81.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.e.b.16.2 4 65.4 even 6
65.2.e.b.61.2 yes 4 65.49 even 6
325.2.e.a.126.1 4 13.10 even 6
325.2.e.a.276.1 4 13.4 even 6
325.2.o.b.74.1 8 65.62 odd 12
325.2.o.b.74.4 8 65.23 odd 12
325.2.o.b.224.1 8 65.43 odd 12
325.2.o.b.224.4 8 65.17 odd 12
585.2.j.d.406.1 4 195.134 odd 6
585.2.j.d.451.1 4 195.179 odd 6
845.2.a.c.1.1 2 65.64 even 2
845.2.a.f.1.2 2 5.4 even 2
845.2.c.d.506.1 4 65.44 odd 4
845.2.c.d.506.4 4 65.34 odd 4
845.2.e.d.146.1 4 65.9 even 6
845.2.e.d.191.1 4 65.29 even 6
845.2.m.d.316.1 8 65.24 odd 12
845.2.m.d.316.4 8 65.54 odd 12
845.2.m.d.361.1 8 65.19 odd 12
845.2.m.d.361.4 8 65.59 odd 12
1040.2.q.o.81.2 4 260.199 odd 6
1040.2.q.o.321.2 4 260.179 odd 6
4225.2.a.t.1.1 2 1.1 even 1 trivial
4225.2.a.x.1.2 2 13.12 even 2
7605.2.a.bb.1.1 2 15.14 odd 2
7605.2.a.bg.1.2 2 195.194 odd 2