Properties

Label 1305.2.a.q.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.39138\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} -1.32733 q^{7} -4.11009 q^{8} +O(q^{10})\) \(q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} -1.32733 q^{7} -4.11009 q^{8} +2.39138 q^{10} -3.00000 q^{11} +6.11009 q^{13} +3.17415 q^{14} +2.39138 q^{16} -0.672673 q^{17} -5.43742 q^{19} -3.71871 q^{20} +7.17415 q^{22} +7.89286 q^{23} +1.00000 q^{25} -14.6116 q^{26} -4.93594 q^{28} +1.00000 q^{29} +2.50147 q^{32} +1.60862 q^{34} +1.32733 q^{35} -3.89286 q^{37} +13.0029 q^{38} +4.11009 q^{40} -4.32733 q^{41} -8.45544 q^{43} -11.1561 q^{44} -18.8748 q^{46} -4.76475 q^{47} -5.23820 q^{49} -2.39138 q^{50} +22.7217 q^{52} +13.3303 q^{53} +3.00000 q^{55} +5.45544 q^{56} -2.39138 q^{58} -3.21724 q^{59} +7.43742 q^{61} -10.7647 q^{64} -6.11009 q^{65} -1.10714 q^{67} -2.50147 q^{68} -3.17415 q^{70} -12.3483 q^{71} +9.20217 q^{73} +9.30931 q^{74} -20.2202 q^{76} +3.98198 q^{77} +11.6576 q^{79} -2.39138 q^{80} +10.3483 q^{82} +0.889908 q^{83} +0.672673 q^{85} +20.2202 q^{86} +12.3303 q^{88} +2.33028 q^{89} -8.11009 q^{91} +29.3512 q^{92} +11.3943 q^{94} +5.43742 q^{95} -12.6756 q^{97} +12.5265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7} + q^{10} - 9 q^{11} + 6 q^{13} - 9 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{22} - q^{23} + 3 q^{25} - 13 q^{26} - 21 q^{28} + 3 q^{29} - 11 q^{32} + 11 q^{34} + 4 q^{35} + 13 q^{37} + 2 q^{38} - 13 q^{41} - 13 q^{43} - 15 q^{44} - 32 q^{46} - 2 q^{47} + 9 q^{49} - q^{50} + 25 q^{52} + 3 q^{53} + 9 q^{55} + 4 q^{56} - q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} - 6 q^{65} - 28 q^{67} + 11 q^{68} + 9 q^{70} + 3 q^{73} + 28 q^{74} - 36 q^{76} + 12 q^{77} - 2 q^{79} - q^{80} - 6 q^{82} + 15 q^{83} + 2 q^{85} + 36 q^{86} - 30 q^{89} - 12 q^{91} + 14 q^{92} - 9 q^{94} + 4 q^{95} - q^{97} + 50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39138 −1.69096 −0.845481 0.534005i \(-0.820686\pi\)
−0.845481 + 0.534005i \(0.820686\pi\)
\(3\) 0 0
\(4\) 3.71871 1.85935
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.32733 −0.501683 −0.250841 0.968028i \(-0.580707\pi\)
−0.250841 + 0.968028i \(0.580707\pi\)
\(8\) −4.11009 −1.45314
\(9\) 0 0
\(10\) 2.39138 0.756222
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 6.11009 1.69463 0.847317 0.531087i \(-0.178216\pi\)
0.847317 + 0.531087i \(0.178216\pi\)
\(14\) 3.17415 0.848327
\(15\) 0 0
\(16\) 2.39138 0.597846
\(17\) −0.672673 −0.163147 −0.0815735 0.996667i \(-0.525995\pi\)
−0.0815735 + 0.996667i \(0.525995\pi\)
\(18\) 0 0
\(19\) −5.43742 −1.24743 −0.623715 0.781652i \(-0.714377\pi\)
−0.623715 + 0.781652i \(0.714377\pi\)
\(20\) −3.71871 −0.831529
\(21\) 0 0
\(22\) 7.17415 1.52953
\(23\) 7.89286 1.64577 0.822887 0.568205i \(-0.192362\pi\)
0.822887 + 0.568205i \(0.192362\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −14.6116 −2.86556
\(27\) 0 0
\(28\) −4.93594 −0.932806
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 2.50147 0.442202
\(33\) 0 0
\(34\) 1.60862 0.275876
\(35\) 1.32733 0.224359
\(36\) 0 0
\(37\) −3.89286 −0.639982 −0.319991 0.947421i \(-0.603680\pi\)
−0.319991 + 0.947421i \(0.603680\pi\)
\(38\) 13.0029 2.10936
\(39\) 0 0
\(40\) 4.11009 0.649863
\(41\) −4.32733 −0.675815 −0.337907 0.941179i \(-0.609719\pi\)
−0.337907 + 0.941179i \(0.609719\pi\)
\(42\) 0 0
\(43\) −8.45544 −1.28944 −0.644721 0.764418i \(-0.723026\pi\)
−0.644721 + 0.764418i \(0.723026\pi\)
\(44\) −11.1561 −1.68185
\(45\) 0 0
\(46\) −18.8748 −2.78294
\(47\) −4.76475 −0.695010 −0.347505 0.937678i \(-0.612971\pi\)
−0.347505 + 0.937678i \(0.612971\pi\)
\(48\) 0 0
\(49\) −5.23820 −0.748315
\(50\) −2.39138 −0.338193
\(51\) 0 0
\(52\) 22.7217 3.15093
\(53\) 13.3303 1.83105 0.915527 0.402256i \(-0.131774\pi\)
0.915527 + 0.402256i \(0.131774\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 5.45544 0.729013
\(57\) 0 0
\(58\) −2.39138 −0.314004
\(59\) −3.21724 −0.418848 −0.209424 0.977825i \(-0.567159\pi\)
−0.209424 + 0.977825i \(0.567159\pi\)
\(60\) 0 0
\(61\) 7.43742 0.952264 0.476132 0.879374i \(-0.342038\pi\)
0.476132 + 0.879374i \(0.342038\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −10.7647 −1.34559
\(65\) −6.11009 −0.757864
\(66\) 0 0
\(67\) −1.10714 −0.135259 −0.0676295 0.997711i \(-0.521544\pi\)
−0.0676295 + 0.997711i \(0.521544\pi\)
\(68\) −2.50147 −0.303348
\(69\) 0 0
\(70\) −3.17415 −0.379383
\(71\) −12.3483 −1.46547 −0.732736 0.680513i \(-0.761757\pi\)
−0.732736 + 0.680513i \(0.761757\pi\)
\(72\) 0 0
\(73\) 9.20217 1.07703 0.538516 0.842615i \(-0.318985\pi\)
0.538516 + 0.842615i \(0.318985\pi\)
\(74\) 9.30931 1.08219
\(75\) 0 0
\(76\) −20.2202 −2.31941
\(77\) 3.98198 0.453789
\(78\) 0 0
\(79\) 11.6576 1.31158 0.655791 0.754942i \(-0.272335\pi\)
0.655791 + 0.754942i \(0.272335\pi\)
\(80\) −2.39138 −0.267365
\(81\) 0 0
\(82\) 10.3483 1.14278
\(83\) 0.889908 0.0976801 0.0488400 0.998807i \(-0.484448\pi\)
0.0488400 + 0.998807i \(0.484448\pi\)
\(84\) 0 0
\(85\) 0.672673 0.0729616
\(86\) 20.2202 2.18040
\(87\) 0 0
\(88\) 12.3303 1.31441
\(89\) 2.33028 0.247009 0.123504 0.992344i \(-0.460587\pi\)
0.123504 + 0.992344i \(0.460587\pi\)
\(90\) 0 0
\(91\) −8.11009 −0.850169
\(92\) 29.3512 3.06008
\(93\) 0 0
\(94\) 11.3943 1.17524
\(95\) 5.43742 0.557867
\(96\) 0 0
\(97\) −12.6756 −1.28701 −0.643507 0.765440i \(-0.722521\pi\)
−0.643507 + 0.765440i \(0.722521\pi\)
\(98\) 12.5265 1.26537
\(99\) 0 0
\(100\) 3.71871 0.371871
\(101\) −3.56258 −0.354490 −0.177245 0.984167i \(-0.556719\pi\)
−0.177245 + 0.984167i \(0.556719\pi\)
\(102\) 0 0
\(103\) −12.9109 −1.27215 −0.636073 0.771629i \(-0.719442\pi\)
−0.636073 + 0.771629i \(0.719442\pi\)
\(104\) −25.1130 −2.46254
\(105\) 0 0
\(106\) −31.8778 −3.09624
\(107\) −4.22018 −0.407981 −0.203990 0.978973i \(-0.565391\pi\)
−0.203990 + 0.978973i \(0.565391\pi\)
\(108\) 0 0
\(109\) −11.8748 −1.13740 −0.568702 0.822544i \(-0.692554\pi\)
−0.568702 + 0.822544i \(0.692554\pi\)
\(110\) −7.17415 −0.684028
\(111\) 0 0
\(112\) −3.17415 −0.299929
\(113\) −14.2382 −1.33942 −0.669709 0.742624i \(-0.733581\pi\)
−0.669709 + 0.742624i \(0.733581\pi\)
\(114\) 0 0
\(115\) −7.89286 −0.736013
\(116\) 3.71871 0.345274
\(117\) 0 0
\(118\) 7.69364 0.708257
\(119\) 0.892857 0.0818481
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −17.7857 −1.61024
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.20217 −0.639089 −0.319544 0.947571i \(-0.603530\pi\)
−0.319544 + 0.947571i \(0.603530\pi\)
\(128\) 20.7397 1.83315
\(129\) 0 0
\(130\) 14.6116 1.28152
\(131\) −14.2382 −1.24400 −0.621999 0.783018i \(-0.713679\pi\)
−0.621999 + 0.783018i \(0.713679\pi\)
\(132\) 0 0
\(133\) 7.21724 0.625814
\(134\) 2.64760 0.228718
\(135\) 0 0
\(136\) 2.76475 0.237075
\(137\) −15.0950 −1.28965 −0.644827 0.764328i \(-0.723071\pi\)
−0.644827 + 0.764328i \(0.723071\pi\)
\(138\) 0 0
\(139\) −6.43742 −0.546015 −0.273007 0.962012i \(-0.588018\pi\)
−0.273007 + 0.962012i \(0.588018\pi\)
\(140\) 4.93594 0.417163
\(141\) 0 0
\(142\) 29.5295 2.47806
\(143\) −18.3303 −1.53285
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) −22.0059 −1.82122
\(147\) 0 0
\(148\) −14.4764 −1.18995
\(149\) −1.21724 −0.0997198 −0.0498599 0.998756i \(-0.515877\pi\)
−0.0498599 + 0.998756i \(0.515877\pi\)
\(150\) 0 0
\(151\) −2.32733 −0.189395 −0.0946976 0.995506i \(-0.530188\pi\)
−0.0946976 + 0.995506i \(0.530188\pi\)
\(152\) 22.3483 1.81269
\(153\) 0 0
\(154\) −9.52244 −0.767340
\(155\) 0 0
\(156\) 0 0
\(157\) 9.43742 0.753188 0.376594 0.926378i \(-0.377095\pi\)
0.376594 + 0.926378i \(0.377095\pi\)
\(158\) −27.8778 −2.21784
\(159\) 0 0
\(160\) −2.50147 −0.197759
\(161\) −10.4764 −0.825656
\(162\) 0 0
\(163\) −21.3303 −1.67072 −0.835358 0.549706i \(-0.814740\pi\)
−0.835358 + 0.549706i \(0.814740\pi\)
\(164\) −16.0921 −1.25658
\(165\) 0 0
\(166\) −2.12811 −0.165173
\(167\) −5.30931 −0.410847 −0.205423 0.978673i \(-0.565857\pi\)
−0.205423 + 0.978673i \(0.565857\pi\)
\(168\) 0 0
\(169\) 24.3332 1.87179
\(170\) −1.60862 −0.123375
\(171\) 0 0
\(172\) −31.4433 −2.39753
\(173\) −22.7677 −1.73100 −0.865498 0.500913i \(-0.832998\pi\)
−0.865498 + 0.500913i \(0.832998\pi\)
\(174\) 0 0
\(175\) −1.32733 −0.100337
\(176\) −7.17415 −0.540772
\(177\) 0 0
\(178\) −5.57258 −0.417683
\(179\) −17.6576 −1.31979 −0.659896 0.751357i \(-0.729400\pi\)
−0.659896 + 0.751357i \(0.729400\pi\)
\(180\) 0 0
\(181\) 10.5655 0.785330 0.392665 0.919682i \(-0.371553\pi\)
0.392665 + 0.919682i \(0.371553\pi\)
\(182\) 19.3943 1.43760
\(183\) 0 0
\(184\) −32.4404 −2.39154
\(185\) 3.89286 0.286209
\(186\) 0 0
\(187\) 2.01802 0.147572
\(188\) −17.7187 −1.29227
\(189\) 0 0
\(190\) −13.0029 −0.943333
\(191\) 3.23820 0.234308 0.117154 0.993114i \(-0.462623\pi\)
0.117154 + 0.993114i \(0.462623\pi\)
\(192\) 0 0
\(193\) −3.30931 −0.238209 −0.119105 0.992882i \(-0.538002\pi\)
−0.119105 + 0.992882i \(0.538002\pi\)
\(194\) 30.3123 2.17629
\(195\) 0 0
\(196\) −19.4794 −1.39138
\(197\) 1.89286 0.134860 0.0674302 0.997724i \(-0.478520\pi\)
0.0674302 + 0.997724i \(0.478520\pi\)
\(198\) 0 0
\(199\) −5.56258 −0.394321 −0.197160 0.980371i \(-0.563172\pi\)
−0.197160 + 0.980371i \(0.563172\pi\)
\(200\) −4.11009 −0.290627
\(201\) 0 0
\(202\) 8.51949 0.599429
\(203\) −1.32733 −0.0931601
\(204\) 0 0
\(205\) 4.32733 0.302234
\(206\) 30.8748 2.15115
\(207\) 0 0
\(208\) 14.6116 1.01313
\(209\) 16.3123 1.12834
\(210\) 0 0
\(211\) 15.7497 1.08425 0.542126 0.840297i \(-0.317619\pi\)
0.542126 + 0.840297i \(0.317619\pi\)
\(212\) 49.5714 3.40458
\(213\) 0 0
\(214\) 10.0921 0.689880
\(215\) 8.45544 0.576656
\(216\) 0 0
\(217\) 0 0
\(218\) 28.3973 1.92331
\(219\) 0 0
\(220\) 11.1561 0.752146
\(221\) −4.11009 −0.276475
\(222\) 0 0
\(223\) −2.01802 −0.135136 −0.0675682 0.997715i \(-0.521524\pi\)
−0.0675682 + 0.997715i \(0.521524\pi\)
\(224\) −3.32028 −0.221845
\(225\) 0 0
\(226\) 34.0490 2.26490
\(227\) 20.6756 1.37229 0.686145 0.727465i \(-0.259302\pi\)
0.686145 + 0.727465i \(0.259302\pi\)
\(228\) 0 0
\(229\) 14.2562 0.942078 0.471039 0.882113i \(-0.343879\pi\)
0.471039 + 0.882113i \(0.343879\pi\)
\(230\) 18.8748 1.24457
\(231\) 0 0
\(232\) −4.11009 −0.269841
\(233\) −20.7677 −1.36054 −0.680268 0.732963i \(-0.738137\pi\)
−0.680268 + 0.732963i \(0.738137\pi\)
\(234\) 0 0
\(235\) 4.76475 0.310818
\(236\) −11.9640 −0.778788
\(237\) 0 0
\(238\) −2.13516 −0.138402
\(239\) −3.30931 −0.214061 −0.107031 0.994256i \(-0.534134\pi\)
−0.107031 + 0.994256i \(0.534134\pi\)
\(240\) 0 0
\(241\) −8.78571 −0.565938 −0.282969 0.959129i \(-0.591319\pi\)
−0.282969 + 0.959129i \(0.591319\pi\)
\(242\) 4.78276 0.307448
\(243\) 0 0
\(244\) 27.6576 1.77060
\(245\) 5.23820 0.334656
\(246\) 0 0
\(247\) −33.2231 −2.11394
\(248\) 0 0
\(249\) 0 0
\(250\) 2.39138 0.151244
\(251\) −23.8037 −1.50248 −0.751239 0.660030i \(-0.770543\pi\)
−0.751239 + 0.660030i \(0.770543\pi\)
\(252\) 0 0
\(253\) −23.6786 −1.48866
\(254\) 17.2231 1.08068
\(255\) 0 0
\(256\) −28.0670 −1.75419
\(257\) 27.3303 1.70482 0.852408 0.522877i \(-0.175141\pi\)
0.852408 + 0.522877i \(0.175141\pi\)
\(258\) 0 0
\(259\) 5.16710 0.321068
\(260\) −22.7217 −1.40914
\(261\) 0 0
\(262\) 34.0490 2.10355
\(263\) 21.7497 1.34114 0.670571 0.741845i \(-0.266049\pi\)
0.670571 + 0.741845i \(0.266049\pi\)
\(264\) 0 0
\(265\) −13.3303 −0.818872
\(266\) −17.2592 −1.05823
\(267\) 0 0
\(268\) −4.11714 −0.251495
\(269\) −10.3303 −0.629848 −0.314924 0.949117i \(-0.601979\pi\)
−0.314924 + 0.949117i \(0.601979\pi\)
\(270\) 0 0
\(271\) −6.25622 −0.380038 −0.190019 0.981780i \(-0.560855\pi\)
−0.190019 + 0.981780i \(0.560855\pi\)
\(272\) −1.60862 −0.0975368
\(273\) 0 0
\(274\) 36.0980 2.18076
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 31.6815 1.90356 0.951779 0.306784i \(-0.0992531\pi\)
0.951779 + 0.306784i \(0.0992531\pi\)
\(278\) 15.3943 0.923291
\(279\) 0 0
\(280\) −5.45544 −0.326025
\(281\) −27.9138 −1.66520 −0.832600 0.553875i \(-0.813148\pi\)
−0.832600 + 0.553875i \(0.813148\pi\)
\(282\) 0 0
\(283\) 23.5295 1.39868 0.699342 0.714788i \(-0.253477\pi\)
0.699342 + 0.714788i \(0.253477\pi\)
\(284\) −45.9197 −2.72483
\(285\) 0 0
\(286\) 43.8347 2.59200
\(287\) 5.74378 0.339045
\(288\) 0 0
\(289\) −16.5475 −0.973383
\(290\) 2.39138 0.140427
\(291\) 0 0
\(292\) 34.2202 2.00258
\(293\) 6.23820 0.364440 0.182220 0.983258i \(-0.441672\pi\)
0.182220 + 0.983258i \(0.441672\pi\)
\(294\) 0 0
\(295\) 3.21724 0.187315
\(296\) 16.0000 0.929981
\(297\) 0 0
\(298\) 2.91087 0.168622
\(299\) 48.2261 2.78899
\(300\) 0 0
\(301\) 11.2231 0.646891
\(302\) 5.56553 0.320260
\(303\) 0 0
\(304\) −13.0029 −0.745770
\(305\) −7.43742 −0.425865
\(306\) 0 0
\(307\) 3.45249 0.197044 0.0985220 0.995135i \(-0.468589\pi\)
0.0985220 + 0.995135i \(0.468589\pi\)
\(308\) 14.8078 0.843755
\(309\) 0 0
\(310\) 0 0
\(311\) −11.0000 −0.623753 −0.311876 0.950123i \(-0.600957\pi\)
−0.311876 + 0.950123i \(0.600957\pi\)
\(312\) 0 0
\(313\) −32.9908 −1.86475 −0.932376 0.361490i \(-0.882268\pi\)
−0.932376 + 0.361490i \(0.882268\pi\)
\(314\) −22.5685 −1.27361
\(315\) 0 0
\(316\) 43.3512 2.43870
\(317\) −4.63664 −0.260419 −0.130210 0.991486i \(-0.541565\pi\)
−0.130210 + 0.991486i \(0.541565\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 10.7647 0.601768
\(321\) 0 0
\(322\) 25.0531 1.39615
\(323\) 3.65760 0.203515
\(324\) 0 0
\(325\) 6.11009 0.338927
\(326\) 51.0088 2.82512
\(327\) 0 0
\(328\) 17.7857 0.982052
\(329\) 6.32438 0.348674
\(330\) 0 0
\(331\) −22.2261 −1.22166 −0.610828 0.791763i \(-0.709163\pi\)
−0.610828 + 0.791763i \(0.709163\pi\)
\(332\) 3.30931 0.181622
\(333\) 0 0
\(334\) 12.6966 0.694726
\(335\) 1.10714 0.0604897
\(336\) 0 0
\(337\) −27.5655 −1.50159 −0.750795 0.660535i \(-0.770329\pi\)
−0.750795 + 0.660535i \(0.770329\pi\)
\(338\) −58.1900 −3.16512
\(339\) 0 0
\(340\) 2.50147 0.135661
\(341\) 0 0
\(342\) 0 0
\(343\) 16.2441 0.877099
\(344\) 34.7526 1.87374
\(345\) 0 0
\(346\) 54.4463 2.92705
\(347\) 22.3273 1.19859 0.599297 0.800527i \(-0.295447\pi\)
0.599297 + 0.800527i \(0.295447\pi\)
\(348\) 0 0
\(349\) −20.5115 −1.09795 −0.548977 0.835837i \(-0.684983\pi\)
−0.548977 + 0.835837i \(0.684983\pi\)
\(350\) 3.17415 0.169665
\(351\) 0 0
\(352\) −7.50442 −0.399987
\(353\) 28.8748 1.53685 0.768426 0.639938i \(-0.221040\pi\)
0.768426 + 0.639938i \(0.221040\pi\)
\(354\) 0 0
\(355\) 12.3483 0.655379
\(356\) 8.66562 0.459277
\(357\) 0 0
\(358\) 42.2261 2.23172
\(359\) −2.76770 −0.146073 −0.0730367 0.997329i \(-0.523269\pi\)
−0.0730367 + 0.997329i \(0.523269\pi\)
\(360\) 0 0
\(361\) 10.5655 0.556081
\(362\) −25.2662 −1.32796
\(363\) 0 0
\(364\) −30.1591 −1.58077
\(365\) −9.20217 −0.481663
\(366\) 0 0
\(367\) 25.4584 1.32892 0.664458 0.747325i \(-0.268662\pi\)
0.664458 + 0.747325i \(0.268662\pi\)
\(368\) 18.8748 0.983919
\(369\) 0 0
\(370\) −9.30931 −0.483968
\(371\) −17.6936 −0.918608
\(372\) 0 0
\(373\) −4.03604 −0.208978 −0.104489 0.994526i \(-0.533321\pi\)
−0.104489 + 0.994526i \(0.533321\pi\)
\(374\) −4.82585 −0.249539
\(375\) 0 0
\(376\) 19.5835 1.00994
\(377\) 6.11009 0.314686
\(378\) 0 0
\(379\) 20.4043 1.04810 0.524050 0.851687i \(-0.324420\pi\)
0.524050 + 0.851687i \(0.324420\pi\)
\(380\) 20.2202 1.03727
\(381\) 0 0
\(382\) −7.74378 −0.396206
\(383\) 5.80078 0.296406 0.148203 0.988957i \(-0.452651\pi\)
0.148203 + 0.988957i \(0.452651\pi\)
\(384\) 0 0
\(385\) −3.98198 −0.202941
\(386\) 7.91382 0.402803
\(387\) 0 0
\(388\) −47.1370 −2.39302
\(389\) 32.2592 1.63560 0.817802 0.575499i \(-0.195192\pi\)
0.817802 + 0.575499i \(0.195192\pi\)
\(390\) 0 0
\(391\) −5.30931 −0.268503
\(392\) 21.5295 1.08740
\(393\) 0 0
\(394\) −4.52654 −0.228044
\(395\) −11.6576 −0.586558
\(396\) 0 0
\(397\) 22.1841 1.11339 0.556695 0.830717i \(-0.312069\pi\)
0.556695 + 0.830717i \(0.312069\pi\)
\(398\) 13.3023 0.666782
\(399\) 0 0
\(400\) 2.39138 0.119569
\(401\) 38.6045 1.92782 0.963909 0.266233i \(-0.0857790\pi\)
0.963909 + 0.266233i \(0.0857790\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −13.2482 −0.659123
\(405\) 0 0
\(406\) 3.17415 0.157530
\(407\) 11.6786 0.578885
\(408\) 0 0
\(409\) 28.0980 1.38936 0.694678 0.719321i \(-0.255547\pi\)
0.694678 + 0.719321i \(0.255547\pi\)
\(410\) −10.3483 −0.511066
\(411\) 0 0
\(412\) −48.0118 −2.36537
\(413\) 4.27032 0.210129
\(414\) 0 0
\(415\) −0.889908 −0.0436839
\(416\) 15.2842 0.749371
\(417\) 0 0
\(418\) −39.0088 −1.90799
\(419\) −3.56553 −0.174188 −0.0870938 0.996200i \(-0.527758\pi\)
−0.0870938 + 0.996200i \(0.527758\pi\)
\(420\) 0 0
\(421\) −17.6216 −0.858823 −0.429411 0.903109i \(-0.641279\pi\)
−0.429411 + 0.903109i \(0.641279\pi\)
\(422\) −37.6635 −1.83343
\(423\) 0 0
\(424\) −54.7887 −2.66077
\(425\) −0.672673 −0.0326294
\(426\) 0 0
\(427\) −9.87189 −0.477734
\(428\) −15.6936 −0.758581
\(429\) 0 0
\(430\) −20.2202 −0.975104
\(431\) 33.7916 1.62768 0.813842 0.581086i \(-0.197372\pi\)
0.813842 + 0.581086i \(0.197372\pi\)
\(432\) 0 0
\(433\) 23.3663 1.12291 0.561457 0.827506i \(-0.310241\pi\)
0.561457 + 0.827506i \(0.310241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −44.1591 −2.11484
\(437\) −42.9168 −2.05299
\(438\) 0 0
\(439\) −16.1461 −0.770613 −0.385306 0.922789i \(-0.625904\pi\)
−0.385306 + 0.922789i \(0.625904\pi\)
\(440\) −12.3303 −0.587823
\(441\) 0 0
\(442\) 9.82880 0.467508
\(443\) −9.38337 −0.445817 −0.222909 0.974839i \(-0.571555\pi\)
−0.222909 + 0.974839i \(0.571555\pi\)
\(444\) 0 0
\(445\) −2.33028 −0.110466
\(446\) 4.82585 0.228511
\(447\) 0 0
\(448\) 14.2883 0.675061
\(449\) −17.5685 −0.829108 −0.414554 0.910025i \(-0.636062\pi\)
−0.414554 + 0.910025i \(0.636062\pi\)
\(450\) 0 0
\(451\) 12.9820 0.611298
\(452\) −52.9477 −2.49045
\(453\) 0 0
\(454\) −49.4433 −2.32049
\(455\) 8.11009 0.380207
\(456\) 0 0
\(457\) −17.4194 −0.814845 −0.407423 0.913240i \(-0.633572\pi\)
−0.407423 + 0.913240i \(0.633572\pi\)
\(458\) −34.0921 −1.59302
\(459\) 0 0
\(460\) −29.3512 −1.36851
\(461\) 0.113041 0.00526484 0.00263242 0.999997i \(-0.499162\pi\)
0.00263242 + 0.999997i \(0.499162\pi\)
\(462\) 0 0
\(463\) −26.8568 −1.24814 −0.624071 0.781368i \(-0.714522\pi\)
−0.624071 + 0.781368i \(0.714522\pi\)
\(464\) 2.39138 0.111017
\(465\) 0 0
\(466\) 49.6635 2.30062
\(467\) −0.726727 −0.0336289 −0.0168145 0.999859i \(-0.505352\pi\)
−0.0168145 + 0.999859i \(0.505352\pi\)
\(468\) 0 0
\(469\) 1.46954 0.0678571
\(470\) −11.3943 −0.525581
\(471\) 0 0
\(472\) 13.2231 0.608644
\(473\) 25.3663 1.16634
\(474\) 0 0
\(475\) −5.43742 −0.249486
\(476\) 3.32028 0.152185
\(477\) 0 0
\(478\) 7.91382 0.361970
\(479\) 9.12516 0.416939 0.208470 0.978029i \(-0.433152\pi\)
0.208470 + 0.978029i \(0.433152\pi\)
\(480\) 0 0
\(481\) −23.7857 −1.08454
\(482\) 21.0100 0.956979
\(483\) 0 0
\(484\) −7.43742 −0.338065
\(485\) 12.6756 0.575570
\(486\) 0 0
\(487\) 35.5714 1.61190 0.805948 0.591987i \(-0.201656\pi\)
0.805948 + 0.591987i \(0.201656\pi\)
\(488\) −30.5685 −1.38377
\(489\) 0 0
\(490\) −12.5265 −0.565892
\(491\) 4.44037 0.200391 0.100196 0.994968i \(-0.468053\pi\)
0.100196 + 0.994968i \(0.468053\pi\)
\(492\) 0 0
\(493\) −0.672673 −0.0302957
\(494\) 79.4492 3.57459
\(495\) 0 0
\(496\) 0 0
\(497\) 16.3902 0.735202
\(498\) 0 0
\(499\) −40.3663 −1.80704 −0.903522 0.428541i \(-0.859028\pi\)
−0.903522 + 0.428541i \(0.859028\pi\)
\(500\) −3.71871 −0.166306
\(501\) 0 0
\(502\) 56.9238 2.54063
\(503\) −23.7117 −1.05725 −0.528625 0.848855i \(-0.677292\pi\)
−0.528625 + 0.848855i \(0.677292\pi\)
\(504\) 0 0
\(505\) 3.56258 0.158533
\(506\) 56.6245 2.51727
\(507\) 0 0
\(508\) −26.7828 −1.18829
\(509\) 0.568478 0.0251974 0.0125987 0.999921i \(-0.495990\pi\)
0.0125987 + 0.999921i \(0.495990\pi\)
\(510\) 0 0
\(511\) −12.2143 −0.540328
\(512\) 25.6396 1.13312
\(513\) 0 0
\(514\) −65.3571 −2.88278
\(515\) 12.9109 0.568921
\(516\) 0 0
\(517\) 14.2942 0.628660
\(518\) −12.3565 −0.542913
\(519\) 0 0
\(520\) 25.1130 1.10128
\(521\) −21.6576 −0.948837 −0.474418 0.880299i \(-0.657342\pi\)
−0.474418 + 0.880299i \(0.657342\pi\)
\(522\) 0 0
\(523\) 14.0180 0.612965 0.306483 0.951876i \(-0.400848\pi\)
0.306483 + 0.951876i \(0.400848\pi\)
\(524\) −52.9477 −2.31303
\(525\) 0 0
\(526\) −52.0118 −2.26782
\(527\) 0 0
\(528\) 0 0
\(529\) 39.2972 1.70857
\(530\) 31.8778 1.38468
\(531\) 0 0
\(532\) 26.8388 1.16361
\(533\) −26.4404 −1.14526
\(534\) 0 0
\(535\) 4.22018 0.182454
\(536\) 4.55046 0.196550
\(537\) 0 0
\(538\) 24.7036 1.06505
\(539\) 15.7146 0.676876
\(540\) 0 0
\(541\) 20.8748 0.897479 0.448740 0.893663i \(-0.351873\pi\)
0.448740 + 0.893663i \(0.351873\pi\)
\(542\) 14.9610 0.642631
\(543\) 0 0
\(544\) −1.68267 −0.0721440
\(545\) 11.8748 0.508662
\(546\) 0 0
\(547\) 4.45249 0.190375 0.0951873 0.995459i \(-0.469655\pi\)
0.0951873 + 0.995459i \(0.469655\pi\)
\(548\) −56.1340 −2.39793
\(549\) 0 0
\(550\) 7.17415 0.305907
\(551\) −5.43742 −0.231642
\(552\) 0 0
\(553\) −15.4735 −0.657998
\(554\) −75.7626 −3.21885
\(555\) 0 0
\(556\) −23.9389 −1.01524
\(557\) −34.4253 −1.45865 −0.729323 0.684169i \(-0.760165\pi\)
−0.729323 + 0.684169i \(0.760165\pi\)
\(558\) 0 0
\(559\) −51.6635 −2.18513
\(560\) 3.17415 0.134132
\(561\) 0 0
\(562\) 66.7526 2.81579
\(563\) −19.6756 −0.829229 −0.414614 0.909997i \(-0.636083\pi\)
−0.414614 + 0.909997i \(0.636083\pi\)
\(564\) 0 0
\(565\) 14.2382 0.599006
\(566\) −56.2680 −2.36512
\(567\) 0 0
\(568\) 50.7526 2.12953
\(569\) −26.2942 −1.10231 −0.551156 0.834402i \(-0.685813\pi\)
−0.551156 + 0.834402i \(0.685813\pi\)
\(570\) 0 0
\(571\) 37.0600 1.55091 0.775455 0.631402i \(-0.217520\pi\)
0.775455 + 0.631402i \(0.217520\pi\)
\(572\) −68.1650 −2.85012
\(573\) 0 0
\(574\) −13.7356 −0.573312
\(575\) 7.89286 0.329155
\(576\) 0 0
\(577\) −14.2202 −0.591994 −0.295997 0.955189i \(-0.595652\pi\)
−0.295997 + 0.955189i \(0.595652\pi\)
\(578\) 39.5714 1.64595
\(579\) 0 0
\(580\) −3.71871 −0.154411
\(581\) −1.18120 −0.0490044
\(582\) 0 0
\(583\) −39.9908 −1.65625
\(584\) −37.8217 −1.56508
\(585\) 0 0
\(586\) −14.9179 −0.616254
\(587\) −32.0980 −1.32483 −0.662413 0.749139i \(-0.730467\pi\)
−0.662413 + 0.749139i \(0.730467\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −7.69364 −0.316742
\(591\) 0 0
\(592\) −9.30931 −0.382610
\(593\) 27.1871 1.11644 0.558220 0.829693i \(-0.311484\pi\)
0.558220 + 0.829693i \(0.311484\pi\)
\(594\) 0 0
\(595\) −0.892857 −0.0366036
\(596\) −4.52654 −0.185414
\(597\) 0 0
\(598\) −115.327 −4.71607
\(599\) −31.8037 −1.29947 −0.649733 0.760163i \(-0.725119\pi\)
−0.649733 + 0.760163i \(0.725119\pi\)
\(600\) 0 0
\(601\) −13.3453 −0.544368 −0.272184 0.962245i \(-0.587746\pi\)
−0.272184 + 0.962245i \(0.587746\pi\)
\(602\) −26.8388 −1.09387
\(603\) 0 0
\(604\) −8.65465 −0.352153
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −37.6635 −1.52871 −0.764357 0.644793i \(-0.776944\pi\)
−0.764357 + 0.644793i \(0.776944\pi\)
\(608\) −13.6016 −0.551616
\(609\) 0 0
\(610\) 17.7857 0.720122
\(611\) −29.1130 −1.17779
\(612\) 0 0
\(613\) 38.3663 1.54960 0.774800 0.632206i \(-0.217850\pi\)
0.774800 + 0.632206i \(0.217850\pi\)
\(614\) −8.25622 −0.333194
\(615\) 0 0
\(616\) −16.3663 −0.659418
\(617\) 34.6245 1.39393 0.696965 0.717105i \(-0.254533\pi\)
0.696965 + 0.717105i \(0.254533\pi\)
\(618\) 0 0
\(619\) 0.00589781 0.000237053 0 0.000118526 1.00000i \(-0.499962\pi\)
0.000118526 1.00000i \(0.499962\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 26.3052 1.05474
\(623\) −3.09304 −0.123920
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 78.8937 3.15323
\(627\) 0 0
\(628\) 35.0950 1.40044
\(629\) 2.61862 0.104411
\(630\) 0 0
\(631\) 42.3001 1.68394 0.841971 0.539523i \(-0.181395\pi\)
0.841971 + 0.539523i \(0.181395\pi\)
\(632\) −47.9138 −1.90591
\(633\) 0 0
\(634\) 11.0880 0.440360
\(635\) 7.20217 0.285809
\(636\) 0 0
\(637\) −32.0059 −1.26812
\(638\) 7.17415 0.284027
\(639\) 0 0
\(640\) −20.7397 −0.819808
\(641\) −14.7297 −0.581787 −0.290894 0.956755i \(-0.593953\pi\)
−0.290894 + 0.956755i \(0.593953\pi\)
\(642\) 0 0
\(643\) 37.9820 1.49786 0.748932 0.662647i \(-0.230567\pi\)
0.748932 + 0.662647i \(0.230567\pi\)
\(644\) −38.9587 −1.53519
\(645\) 0 0
\(646\) −8.74673 −0.344135
\(647\) −31.5144 −1.23896 −0.619480 0.785012i \(-0.712656\pi\)
−0.619480 + 0.785012i \(0.712656\pi\)
\(648\) 0 0
\(649\) 9.65171 0.378863
\(650\) −14.6116 −0.573113
\(651\) 0 0
\(652\) −79.3211 −3.10645
\(653\) 38.8929 1.52200 0.760998 0.648755i \(-0.224710\pi\)
0.760998 + 0.648755i \(0.224710\pi\)
\(654\) 0 0
\(655\) 14.2382 0.556333
\(656\) −10.3483 −0.404033
\(657\) 0 0
\(658\) −15.1240 −0.589595
\(659\) 32.2792 1.25742 0.628709 0.777641i \(-0.283584\pi\)
0.628709 + 0.777641i \(0.283584\pi\)
\(660\) 0 0
\(661\) −29.8447 −1.16082 −0.580412 0.814323i \(-0.697109\pi\)
−0.580412 + 0.814323i \(0.697109\pi\)
\(662\) 53.1511 2.06577
\(663\) 0 0
\(664\) −3.65760 −0.141943
\(665\) −7.21724 −0.279872
\(666\) 0 0
\(667\) 7.89286 0.305613
\(668\) −19.7438 −0.763910
\(669\) 0 0
\(670\) −2.64760 −0.102286
\(671\) −22.3123 −0.861355
\(672\) 0 0
\(673\) 10.5505 0.406690 0.203345 0.979107i \(-0.434819\pi\)
0.203345 + 0.979107i \(0.434819\pi\)
\(674\) 65.9197 2.53913
\(675\) 0 0
\(676\) 90.4882 3.48032
\(677\) −40.0600 −1.53963 −0.769815 0.638268i \(-0.779651\pi\)
−0.769815 + 0.638268i \(0.779651\pi\)
\(678\) 0 0
\(679\) 16.8247 0.645673
\(680\) −2.76475 −0.106023
\(681\) 0 0
\(682\) 0 0
\(683\) −24.2913 −0.929480 −0.464740 0.885447i \(-0.653852\pi\)
−0.464740 + 0.885447i \(0.653852\pi\)
\(684\) 0 0
\(685\) 15.0950 0.576751
\(686\) −38.8459 −1.48314
\(687\) 0 0
\(688\) −20.2202 −0.770887
\(689\) 81.4492 3.10297
\(690\) 0 0
\(691\) −43.9017 −1.67010 −0.835050 0.550174i \(-0.814561\pi\)
−0.835050 + 0.550174i \(0.814561\pi\)
\(692\) −84.6665 −3.21854
\(693\) 0 0
\(694\) −53.3932 −2.02678
\(695\) 6.43742 0.244185
\(696\) 0 0
\(697\) 2.91087 0.110257
\(698\) 49.0508 1.85660
\(699\) 0 0
\(700\) −4.93594 −0.186561
\(701\) 39.9197 1.50775 0.753874 0.657020i \(-0.228183\pi\)
0.753874 + 0.657020i \(0.228183\pi\)
\(702\) 0 0
\(703\) 21.1671 0.798332
\(704\) 32.2942 1.21713
\(705\) 0 0
\(706\) −69.0508 −2.59876
\(707\) 4.72871 0.177841
\(708\) 0 0
\(709\) 11.4885 0.431461 0.215730 0.976453i \(-0.430787\pi\)
0.215730 + 0.976453i \(0.430787\pi\)
\(710\) −29.5295 −1.10822
\(711\) 0 0
\(712\) −9.57765 −0.358938
\(713\) 0 0
\(714\) 0 0
\(715\) 18.3303 0.685513
\(716\) −65.6635 −2.45396
\(717\) 0 0
\(718\) 6.61862 0.247005
\(719\) 1.00295 0.0374037 0.0187018 0.999825i \(-0.494047\pi\)
0.0187018 + 0.999825i \(0.494047\pi\)
\(720\) 0 0
\(721\) 17.1370 0.638214
\(722\) −25.2662 −0.940311
\(723\) 0 0
\(724\) 39.2901 1.46021
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −20.4404 −0.758091 −0.379046 0.925378i \(-0.623748\pi\)
−0.379046 + 0.925378i \(0.623748\pi\)
\(728\) 33.3332 1.23541
\(729\) 0 0
\(730\) 22.0059 0.814475
\(731\) 5.68774 0.210369
\(732\) 0 0
\(733\) 30.9669 1.14379 0.571895 0.820327i \(-0.306209\pi\)
0.571895 + 0.820327i \(0.306209\pi\)
\(734\) −60.8807 −2.24715
\(735\) 0 0
\(736\) 19.7438 0.727765
\(737\) 3.32143 0.122346
\(738\) 0 0
\(739\) 3.62157 0.133222 0.0666108 0.997779i \(-0.478781\pi\)
0.0666108 + 0.997779i \(0.478781\pi\)
\(740\) 14.4764 0.532163
\(741\) 0 0
\(742\) 42.3123 1.55333
\(743\) 25.1691 0.923364 0.461682 0.887046i \(-0.347246\pi\)
0.461682 + 0.887046i \(0.347246\pi\)
\(744\) 0 0
\(745\) 1.21724 0.0445960
\(746\) 9.65171 0.353374
\(747\) 0 0
\(748\) 7.50442 0.274389
\(749\) 5.60157 0.204677
\(750\) 0 0
\(751\) −9.63760 −0.351681 −0.175841 0.984419i \(-0.556264\pi\)
−0.175841 + 0.984419i \(0.556264\pi\)
\(752\) −11.3943 −0.415509
\(753\) 0 0
\(754\) −14.6116 −0.532122
\(755\) 2.32733 0.0847001
\(756\) 0 0
\(757\) −17.4282 −0.633440 −0.316720 0.948519i \(-0.602582\pi\)
−0.316720 + 0.948519i \(0.602582\pi\)
\(758\) −48.7946 −1.77230
\(759\) 0 0
\(760\) −22.3483 −0.810658
\(761\) −2.27622 −0.0825130 −0.0412565 0.999149i \(-0.513136\pi\)
−0.0412565 + 0.999149i \(0.513136\pi\)
\(762\) 0 0
\(763\) 15.7618 0.570615
\(764\) 12.0419 0.435662
\(765\) 0 0
\(766\) −13.8719 −0.501212
\(767\) −19.6576 −0.709795
\(768\) 0 0
\(769\) 29.4433 1.06175 0.530877 0.847449i \(-0.321863\pi\)
0.530877 + 0.847449i \(0.321863\pi\)
\(770\) 9.52244 0.343165
\(771\) 0 0
\(772\) −12.3064 −0.442916
\(773\) −14.2562 −0.512761 −0.256380 0.966576i \(-0.582530\pi\)
−0.256380 + 0.966576i \(0.582530\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 52.0980 1.87021
\(777\) 0 0
\(778\) −77.1440 −2.76575
\(779\) 23.5295 0.843032
\(780\) 0 0
\(781\) 37.0449 1.32557
\(782\) 12.6966 0.454029
\(783\) 0 0
\(784\) −12.5265 −0.447377
\(785\) −9.43742 −0.336836
\(786\) 0 0
\(787\) −27.1311 −0.967118 −0.483559 0.875312i \(-0.660656\pi\)
−0.483559 + 0.875312i \(0.660656\pi\)
\(788\) 7.03899 0.250753
\(789\) 0 0
\(790\) 27.8778 0.991847
\(791\) 18.8988 0.671962
\(792\) 0 0
\(793\) 45.4433 1.61374
\(794\) −53.0508 −1.88270
\(795\) 0 0
\(796\) −20.6856 −0.733182
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 3.20511 0.113389
\(800\) 2.50147 0.0884405
\(801\) 0 0
\(802\) −92.3182 −3.25987
\(803\) −27.6065 −0.974212
\(804\) 0 0
\(805\) 10.4764 0.369245
\(806\) 0 0
\(807\) 0 0
\(808\) 14.6425 0.515123
\(809\) 13.5324 0.475775 0.237888 0.971293i \(-0.423545\pi\)
0.237888 + 0.971293i \(0.423545\pi\)
\(810\) 0 0
\(811\) −40.4492 −1.42036 −0.710182 0.704018i \(-0.751387\pi\)
−0.710182 + 0.704018i \(0.751387\pi\)
\(812\) −4.93594 −0.173218
\(813\) 0 0
\(814\) −27.9279 −0.978873
\(815\) 21.3303 0.747167
\(816\) 0 0
\(817\) 45.9758 1.60849
\(818\) −67.1930 −2.34935
\(819\) 0 0
\(820\) 16.0921 0.561960
\(821\) 21.6016 0.753900 0.376950 0.926234i \(-0.376973\pi\)
0.376950 + 0.926234i \(0.376973\pi\)
\(822\) 0 0
\(823\) 25.6995 0.895830 0.447915 0.894076i \(-0.352167\pi\)
0.447915 + 0.894076i \(0.352167\pi\)
\(824\) 53.0649 1.84860
\(825\) 0 0
\(826\) −10.2120 −0.355320
\(827\) 19.6016 0.681613 0.340807 0.940133i \(-0.389300\pi\)
0.340807 + 0.940133i \(0.389300\pi\)
\(828\) 0 0
\(829\) −4.22608 −0.146778 −0.0733889 0.997303i \(-0.523381\pi\)
−0.0733889 + 0.997303i \(0.523381\pi\)
\(830\) 2.12811 0.0738678
\(831\) 0 0
\(832\) −65.7736 −2.28029
\(833\) 3.52360 0.122085
\(834\) 0 0
\(835\) 5.30931 0.183736
\(836\) 60.6606 2.09799
\(837\) 0 0
\(838\) 8.52654 0.294545
\(839\) 2.45839 0.0848729 0.0424365 0.999099i \(-0.486488\pi\)
0.0424365 + 0.999099i \(0.486488\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 42.1399 1.45224
\(843\) 0 0
\(844\) 58.5685 2.01601
\(845\) −24.3332 −0.837088
\(846\) 0 0
\(847\) 2.65465 0.0912150
\(848\) 31.8778 1.09469
\(849\) 0 0
\(850\) 1.60862 0.0551751
\(851\) −30.7258 −1.05327
\(852\) 0 0
\(853\) 39.8427 1.36419 0.682094 0.731264i \(-0.261069\pi\)
0.682094 + 0.731264i \(0.261069\pi\)
\(854\) 23.6075 0.807831
\(855\) 0 0
\(856\) 17.3453 0.592852
\(857\) 16.9518 0.579064 0.289532 0.957168i \(-0.406500\pi\)
0.289532 + 0.957168i \(0.406500\pi\)
\(858\) 0 0
\(859\) 38.3902 1.30986 0.654929 0.755691i \(-0.272699\pi\)
0.654929 + 0.755691i \(0.272699\pi\)
\(860\) 31.4433 1.07221
\(861\) 0 0
\(862\) −80.8087 −2.75235
\(863\) −49.6995 −1.69179 −0.845896 0.533348i \(-0.820934\pi\)
−0.845896 + 0.533348i \(0.820934\pi\)
\(864\) 0 0
\(865\) 22.7677 0.774125
\(866\) −55.8778 −1.89880
\(867\) 0 0
\(868\) 0 0
\(869\) −34.9728 −1.18637
\(870\) 0 0
\(871\) −6.76475 −0.229215
\(872\) 48.8067 1.65280
\(873\) 0 0
\(874\) 102.630 3.47153
\(875\) 1.32733 0.0448719
\(876\) 0 0
\(877\) −40.9168 −1.38166 −0.690831 0.723017i \(-0.742755\pi\)
−0.690831 + 0.723017i \(0.742755\pi\)
\(878\) 38.6116 1.30308
\(879\) 0 0
\(880\) 7.17415 0.241840
\(881\) 26.9079 0.906551 0.453276 0.891370i \(-0.350255\pi\)
0.453276 + 0.891370i \(0.350255\pi\)
\(882\) 0 0
\(883\) −6.69069 −0.225160 −0.112580 0.993643i \(-0.535911\pi\)
−0.112580 + 0.993643i \(0.535911\pi\)
\(884\) −15.2842 −0.514065
\(885\) 0 0
\(886\) 22.4392 0.753860
\(887\) 46.0859 1.54741 0.773706 0.633545i \(-0.218401\pi\)
0.773706 + 0.633545i \(0.218401\pi\)
\(888\) 0 0
\(889\) 9.55963 0.320620
\(890\) 5.57258 0.186793
\(891\) 0 0
\(892\) −7.50442 −0.251267
\(893\) 25.9079 0.866976
\(894\) 0 0
\(895\) 17.6576 0.590229
\(896\) −27.5283 −0.919657
\(897\) 0 0
\(898\) 42.0129 1.40199
\(899\) 0 0
\(900\) 0 0
\(901\) −8.96691 −0.298731
\(902\) −31.0449 −1.03368
\(903\) 0 0
\(904\) 58.5203 1.94636
\(905\) −10.5655 −0.351210
\(906\) 0 0
\(907\) −18.7117 −0.621310 −0.310655 0.950523i \(-0.600548\pi\)
−0.310655 + 0.950523i \(0.600548\pi\)
\(908\) 76.8866 2.55157
\(909\) 0 0
\(910\) −19.3943 −0.642916
\(911\) −25.2923 −0.837970 −0.418985 0.907993i \(-0.637614\pi\)
−0.418985 + 0.907993i \(0.637614\pi\)
\(912\) 0 0
\(913\) −2.66972 −0.0883550
\(914\) 41.6564 1.37787
\(915\) 0 0
\(916\) 53.0147 1.75166
\(917\) 18.8988 0.624092
\(918\) 0 0
\(919\) −45.6396 −1.50551 −0.752756 0.658300i \(-0.771276\pi\)
−0.752756 + 0.658300i \(0.771276\pi\)
\(920\) 32.4404 1.06953
\(921\) 0 0
\(922\) −0.270324 −0.00890265
\(923\) −75.4492 −2.48344
\(924\) 0 0
\(925\) −3.89286 −0.127996
\(926\) 64.2249 2.11056
\(927\) 0 0
\(928\) 2.50147 0.0821149
\(929\) 21.2733 0.697953 0.348977 0.937131i \(-0.386529\pi\)
0.348977 + 0.937131i \(0.386529\pi\)
\(930\) 0 0
\(931\) 28.4823 0.933470
\(932\) −77.2290 −2.52972
\(933\) 0 0
\(934\) 1.73788 0.0568652
\(935\) −2.01802 −0.0659962
\(936\) 0 0
\(937\) −46.9489 −1.53375 −0.766877 0.641794i \(-0.778190\pi\)
−0.766877 + 0.641794i \(0.778190\pi\)
\(938\) −3.51424 −0.114744
\(939\) 0 0
\(940\) 17.7187 0.577921
\(941\) 18.5986 0.606298 0.303149 0.952943i \(-0.401962\pi\)
0.303149 + 0.952943i \(0.401962\pi\)
\(942\) 0 0
\(943\) −34.1550 −1.11224
\(944\) −7.69364 −0.250407
\(945\) 0 0
\(946\) −60.6606 −1.97224
\(947\) −40.0800 −1.30242 −0.651212 0.758896i \(-0.725739\pi\)
−0.651212 + 0.758896i \(0.725739\pi\)
\(948\) 0 0
\(949\) 56.2261 1.82518
\(950\) 13.0029 0.421871
\(951\) 0 0
\(952\) −3.66972 −0.118936
\(953\) 5.03309 0.163038 0.0815188 0.996672i \(-0.474023\pi\)
0.0815188 + 0.996672i \(0.474023\pi\)
\(954\) 0 0
\(955\) −3.23820 −0.104786
\(956\) −12.3064 −0.398016
\(957\) 0 0
\(958\) −21.8217 −0.705029
\(959\) 20.0360 0.646997
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 56.8807 1.83391
\(963\) 0 0
\(964\) −32.6715 −1.05228
\(965\) 3.30931 0.106530
\(966\) 0 0
\(967\) 1.72871 0.0555916 0.0277958 0.999614i \(-0.491151\pi\)
0.0277958 + 0.999614i \(0.491151\pi\)
\(968\) 8.22018 0.264207
\(969\) 0 0
\(970\) −30.3123 −0.973268
\(971\) 16.1071 0.516903 0.258451 0.966024i \(-0.416788\pi\)
0.258451 + 0.966024i \(0.416788\pi\)
\(972\) 0 0
\(973\) 8.54456 0.273926
\(974\) −85.0649 −2.72565
\(975\) 0 0
\(976\) 17.7857 0.569307
\(977\) 13.2323 0.423339 0.211669 0.977341i \(-0.432110\pi\)
0.211669 + 0.977341i \(0.432110\pi\)
\(978\) 0 0
\(979\) −6.99083 −0.223428
\(980\) 19.4794 0.622245
\(981\) 0 0
\(982\) −10.6186 −0.338854
\(983\) −3.08913 −0.0985278 −0.0492639 0.998786i \(-0.515688\pi\)
−0.0492639 + 0.998786i \(0.515688\pi\)
\(984\) 0 0
\(985\) −1.89286 −0.0603114
\(986\) 1.60862 0.0512288
\(987\) 0 0
\(988\) −123.547 −3.93056
\(989\) −66.7376 −2.12213
\(990\) 0 0
\(991\) 30.9138 0.982010 0.491005 0.871157i \(-0.336630\pi\)
0.491005 + 0.871157i \(0.336630\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −39.1953 −1.24320
\(995\) 5.56258 0.176346
\(996\) 0 0
\(997\) −25.3863 −0.803993 −0.401996 0.915641i \(-0.631684\pi\)
−0.401996 + 0.915641i \(0.631684\pi\)
\(998\) 96.5313 3.05564
\(999\) 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 1305.2.a.q.1.1 3
3.2 odd 2 435.2.a.i.1.3 3
5.4 even 2 6525.2.a.bf.1.3 3
12.11 even 2 6960.2.a.cl.1.2 3
15.2 even 4 2175.2.c.m.349.6 6
15.8 even 4 2175.2.c.m.349.1 6
15.14 odd 2 2175.2.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.i.1.3 3 3.2 odd 2
1305.2.a.q.1.1 3 1.1 even 1 trivial
2175.2.a.u.1.1 3 15.14 odd 2
2175.2.c.m.349.1 6 15.8 even 4
2175.2.c.m.349.6 6 15.2 even 4
6525.2.a.bf.1.3 3 5.4 even 2
6960.2.a.cl.1.2 3 12.11 even 2