Properties

Label 4225.2.a.w.1.2
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: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 1.2
Root \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.26795 q^{6} +2.00000 q^{7} -1.73205 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.26795 q^{6} +2.00000 q^{7} -1.73205 q^{8} -2.46410 q^{9} +1.26795 q^{11} +0.732051 q^{12} +3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} -4.26795 q^{18} -4.19615 q^{19} +1.46410 q^{21} +2.19615 q^{22} -4.73205 q^{23} -1.26795 q^{24} -4.00000 q^{27} +2.00000 q^{28} -9.46410 q^{29} +0.196152 q^{31} -5.19615 q^{32} +0.928203 q^{33} -6.00000 q^{34} -2.46410 q^{36} -4.00000 q^{37} -7.26795 q^{38} +3.46410 q^{41} +2.53590 q^{42} -10.1962 q^{43} +1.26795 q^{44} -8.19615 q^{46} +6.00000 q^{47} -3.66025 q^{48} -3.00000 q^{49} -2.53590 q^{51} +10.3923 q^{53} -6.92820 q^{54} -3.46410 q^{56} -3.07180 q^{57} -16.3923 q^{58} +15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} -4.92820 q^{63} +1.00000 q^{64} +1.60770 q^{66} -14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{69} -1.26795 q^{71} +4.26795 q^{72} -4.00000 q^{73} -6.92820 q^{74} -4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} +4.46410 q^{81} +6.00000 q^{82} -6.00000 q^{83} +1.46410 q^{84} -17.6603 q^{86} -6.92820 q^{87} -2.19615 q^{88} -0.928203 q^{89} -4.73205 q^{92} +0.143594 q^{93} +10.3923 q^{94} -3.80385 q^{96} +2.00000 q^{97} -5.19615 q^{98} -3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 4 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 10 q^{16} - 12 q^{18} + 2 q^{19} - 4 q^{21} - 6 q^{22} - 6 q^{23} - 6 q^{24} - 8 q^{27} + 4 q^{28} - 12 q^{29} - 10 q^{31} - 12 q^{33} - 12 q^{34} + 2 q^{36} - 8 q^{37} - 18 q^{38} + 12 q^{42} - 10 q^{43} + 6 q^{44} - 6 q^{46} + 12 q^{47} + 10 q^{48} - 6 q^{49} - 12 q^{51} - 20 q^{57} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 4 q^{63} + 2 q^{64} + 24 q^{66} - 8 q^{67} - 6 q^{71} + 12 q^{72} - 8 q^{73} + 2 q^{76} + 12 q^{77} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 12 q^{83} - 4 q^{84} - 18 q^{86} + 6 q^{88} + 12 q^{89} - 6 q^{92} + 28 q^{93} - 18 q^{96} + 4 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.26795 0.517638
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.73205 −0.612372
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0.732051 0.211325
\(13\) 0 0
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) −4.26795 −1.00597
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) 0 0
\(21\) 1.46410 0.319493
\(22\) 2.19615 0.468221
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) −1.26795 −0.258819
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 2.00000 0.377964
\(29\) −9.46410 −1.75744 −0.878720 0.477338i \(-0.841602\pi\)
−0.878720 + 0.477338i \(0.841602\pi\)
\(30\) 0 0
\(31\) 0.196152 0.0352300 0.0176150 0.999845i \(-0.494393\pi\)
0.0176150 + 0.999845i \(0.494393\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0.928203 0.161579
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −7.26795 −1.17902
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 2.53590 0.391298
\(43\) −10.1962 −1.55490 −0.777449 0.628946i \(-0.783487\pi\)
−0.777449 + 0.628946i \(0.783487\pi\)
\(44\) 1.26795 0.191151
\(45\) 0 0
\(46\) −8.19615 −1.20846
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −3.66025 −0.528312
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) −6.92820 −0.942809
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) −3.07180 −0.406869
\(58\) −16.3923 −2.15242
\(59\) 15.1244 1.96902 0.984512 0.175319i \(-0.0560957\pi\)
0.984512 + 0.175319i \(0.0560957\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0.339746 0.0431478
\(63\) −4.92820 −0.620895
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.60770 0.197894
\(67\) −14.3923 −1.75830 −0.879150 0.476545i \(-0.841889\pi\)
−0.879150 + 0.476545i \(0.841889\pi\)
\(68\) −3.46410 −0.420084
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) −1.26795 −0.150478 −0.0752389 0.997166i \(-0.523972\pi\)
−0.0752389 + 0.997166i \(0.523972\pi\)
\(72\) 4.26795 0.502983
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0 0
\(76\) −4.19615 −0.481332
\(77\) 2.53590 0.288992
\(78\) 0 0
\(79\) 12.3923 1.39424 0.697122 0.716953i \(-0.254464\pi\)
0.697122 + 0.716953i \(0.254464\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 1.46410 0.159747
\(85\) 0 0
\(86\) −17.6603 −1.90435
\(87\) −6.92820 −0.742781
\(88\) −2.19615 −0.234111
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.73205 −0.493350
\(93\) 0.143594 0.0148900
\(94\) 10.3923 1.07188
\(95\) 0 0
\(96\) −3.80385 −0.388229
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −5.19615 −0.524891
\(99\) −3.12436 −0.314010
\(100\) 0 0
\(101\) 12.9282 1.28640 0.643202 0.765696i \(-0.277605\pi\)
0.643202 + 0.765696i \(0.277605\pi\)
\(102\) −4.39230 −0.434903
\(103\) −10.1962 −1.00466 −0.502328 0.864677i \(-0.667523\pi\)
−0.502328 + 0.864677i \(0.667523\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) −0.339746 −0.0328445 −0.0164222 0.999865i \(-0.505228\pi\)
−0.0164222 + 0.999865i \(0.505228\pi\)
\(108\) −4.00000 −0.384900
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −2.92820 −0.277933
\(112\) −10.0000 −0.944911
\(113\) −15.4641 −1.45474 −0.727370 0.686245i \(-0.759258\pi\)
−0.727370 + 0.686245i \(0.759258\pi\)
\(114\) −5.32051 −0.498311
\(115\) 0 0
\(116\) −9.46410 −0.878720
\(117\) 0 0
\(118\) 26.1962 2.41155
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 21.4641 1.94327
\(123\) 2.53590 0.228654
\(124\) 0.196152 0.0176150
\(125\) 0 0
\(126\) −8.53590 −0.760438
\(127\) −5.80385 −0.515008 −0.257504 0.966277i \(-0.582900\pi\)
−0.257504 + 0.966277i \(0.582900\pi\)
\(128\) 12.1244 1.07165
\(129\) −7.46410 −0.657178
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0.928203 0.0807897
\(133\) −8.39230 −0.727705
\(134\) −24.9282 −2.15347
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −12.9282 −1.10453 −0.552265 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(138\) −6.00000 −0.510754
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 0 0
\(141\) 4.39230 0.369899
\(142\) −2.19615 −0.184297
\(143\) 0 0
\(144\) 12.3205 1.02671
\(145\) 0 0
\(146\) −6.92820 −0.573382
\(147\) −2.19615 −0.181136
\(148\) −4.00000 −0.328798
\(149\) −19.8564 −1.62670 −0.813350 0.581775i \(-0.802359\pi\)
−0.813350 + 0.581775i \(0.802359\pi\)
\(150\) 0 0
\(151\) 12.1962 0.992509 0.496254 0.868177i \(-0.334708\pi\)
0.496254 + 0.868177i \(0.334708\pi\)
\(152\) 7.26795 0.589509
\(153\) 8.53590 0.690086
\(154\) 4.39230 0.353942
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 21.4641 1.70759
\(159\) 7.60770 0.603329
\(160\) 0 0
\(161\) −9.46410 −0.745876
\(162\) 7.73205 0.607487
\(163\) 6.39230 0.500684 0.250342 0.968157i \(-0.419457\pi\)
0.250342 + 0.968157i \(0.419457\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) −10.3923 −0.806599
\(167\) 12.9282 1.00041 0.500207 0.865906i \(-0.333257\pi\)
0.500207 + 0.865906i \(0.333257\pi\)
\(168\) −2.53590 −0.195649
\(169\) 0 0
\(170\) 0 0
\(171\) 10.3397 0.790700
\(172\) −10.1962 −0.777449
\(173\) −15.4641 −1.17571 −0.587857 0.808965i \(-0.700028\pi\)
−0.587857 + 0.808965i \(0.700028\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) −6.33975 −0.477876
\(177\) 11.0718 0.832207
\(178\) −1.60770 −0.120502
\(179\) −5.07180 −0.379084 −0.189542 0.981873i \(-0.560700\pi\)
−0.189542 + 0.981873i \(0.560700\pi\)
\(180\) 0 0
\(181\) −20.3923 −1.51575 −0.757874 0.652401i \(-0.773762\pi\)
−0.757874 + 0.652401i \(0.773762\pi\)
\(182\) 0 0
\(183\) 9.07180 0.670607
\(184\) 8.19615 0.604228
\(185\) 0 0
\(186\) 0.248711 0.0182364
\(187\) −4.39230 −0.321197
\(188\) 6.00000 0.437595
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −18.9282 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(192\) 0.732051 0.0528312
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 3.46410 0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 0.928203 0.0661317 0.0330659 0.999453i \(-0.489473\pi\)
0.0330659 + 0.999453i \(0.489473\pi\)
\(198\) −5.41154 −0.384582
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −10.5359 −0.743145
\(202\) 22.3923 1.57552
\(203\) −18.9282 −1.32850
\(204\) −2.53590 −0.177548
\(205\) 0 0
\(206\) −17.6603 −1.23045
\(207\) 11.6603 0.810444
\(208\) 0 0
\(209\) −5.32051 −0.368027
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.3923 0.713746
\(213\) −0.928203 −0.0635994
\(214\) −0.588457 −0.0402261
\(215\) 0 0
\(216\) 6.92820 0.471405
\(217\) 0.392305 0.0266314
\(218\) −3.46410 −0.234619
\(219\) −2.92820 −0.197870
\(220\) 0 0
\(221\) 0 0
\(222\) −5.07180 −0.340397
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) −26.7846 −1.78169
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) −3.07180 −0.203435
\(229\) 14.3923 0.951070 0.475535 0.879697i \(-0.342254\pi\)
0.475535 + 0.879697i \(0.342254\pi\)
\(230\) 0 0
\(231\) 1.85641 0.122143
\(232\) 16.3923 1.07621
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 15.1244 0.984512
\(237\) 9.07180 0.589277
\(238\) −12.0000 −0.777844
\(239\) 3.80385 0.246050 0.123025 0.992404i \(-0.460740\pi\)
0.123025 + 0.992404i \(0.460740\pi\)
\(240\) 0 0
\(241\) −18.3923 −1.18475 −0.592376 0.805661i \(-0.701810\pi\)
−0.592376 + 0.805661i \(0.701810\pi\)
\(242\) −16.2679 −1.04574
\(243\) 15.2679 0.979439
\(244\) 12.3923 0.793336
\(245\) 0 0
\(246\) 4.39230 0.280043
\(247\) 0 0
\(248\) −0.339746 −0.0215739
\(249\) −4.39230 −0.278351
\(250\) 0 0
\(251\) −14.5359 −0.917498 −0.458749 0.888566i \(-0.651702\pi\)
−0.458749 + 0.888566i \(0.651702\pi\)
\(252\) −4.92820 −0.310448
\(253\) −6.00000 −0.377217
\(254\) −10.0526 −0.630754
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 7.85641 0.490069 0.245035 0.969514i \(-0.421201\pi\)
0.245035 + 0.969514i \(0.421201\pi\)
\(258\) −12.9282 −0.804875
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 23.3205 1.44350
\(262\) 0 0
\(263\) −4.73205 −0.291791 −0.145895 0.989300i \(-0.546606\pi\)
−0.145895 + 0.989300i \(0.546606\pi\)
\(264\) −1.60770 −0.0989468
\(265\) 0 0
\(266\) −14.5359 −0.891253
\(267\) −0.679492 −0.0415842
\(268\) −14.3923 −0.879150
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 0 0
\(271\) 20.9808 1.27449 0.637245 0.770661i \(-0.280074\pi\)
0.637245 + 0.770661i \(0.280074\pi\)
\(272\) 17.3205 1.05021
\(273\) 0 0
\(274\) −22.3923 −1.35277
\(275\) 0 0
\(276\) −3.46410 −0.208514
\(277\) 5.60770 0.336934 0.168467 0.985707i \(-0.446118\pi\)
0.168467 + 0.985707i \(0.446118\pi\)
\(278\) −14.5359 −0.871805
\(279\) −0.483340 −0.0289368
\(280\) 0 0
\(281\) −1.60770 −0.0959071 −0.0479535 0.998850i \(-0.515270\pi\)
−0.0479535 + 0.998850i \(0.515270\pi\)
\(282\) 7.60770 0.453032
\(283\) −1.41154 −0.0839075 −0.0419538 0.999120i \(-0.513358\pi\)
−0.0419538 + 0.999120i \(0.513358\pi\)
\(284\) −1.26795 −0.0752389
\(285\) 0 0
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) 12.8038 0.754474
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 1.46410 0.0858272
\(292\) −4.00000 −0.234082
\(293\) −18.9282 −1.10580 −0.552899 0.833248i \(-0.686478\pi\)
−0.552899 + 0.833248i \(0.686478\pi\)
\(294\) −3.80385 −0.221845
\(295\) 0 0
\(296\) 6.92820 0.402694
\(297\) −5.07180 −0.294295
\(298\) −34.3923 −1.99229
\(299\) 0 0
\(300\) 0 0
\(301\) −20.3923 −1.17539
\(302\) 21.1244 1.21557
\(303\) 9.46410 0.543698
\(304\) 20.9808 1.20333
\(305\) 0 0
\(306\) 14.7846 0.845180
\(307\) 22.7846 1.30039 0.650193 0.759769i \(-0.274688\pi\)
0.650193 + 0.759769i \(0.274688\pi\)
\(308\) 2.53590 0.144496
\(309\) −7.46410 −0.424618
\(310\) 0 0
\(311\) 4.39230 0.249065 0.124532 0.992216i \(-0.460257\pi\)
0.124532 + 0.992216i \(0.460257\pi\)
\(312\) 0 0
\(313\) −6.39230 −0.361314 −0.180657 0.983546i \(-0.557822\pi\)
−0.180657 + 0.983546i \(0.557822\pi\)
\(314\) 17.3205 0.977453
\(315\) 0 0
\(316\) 12.3923 0.697122
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 13.1769 0.738925
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −0.248711 −0.0138817
\(322\) −16.3923 −0.913507
\(323\) 14.5359 0.808799
\(324\) 4.46410 0.248006
\(325\) 0 0
\(326\) 11.0718 0.613210
\(327\) −1.46410 −0.0809650
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 28.5885 1.57136 0.785682 0.618631i \(-0.212312\pi\)
0.785682 + 0.618631i \(0.212312\pi\)
\(332\) −6.00000 −0.329293
\(333\) 9.85641 0.540128
\(334\) 22.3923 1.22525
\(335\) 0 0
\(336\) −7.32051 −0.399366
\(337\) 5.60770 0.305471 0.152735 0.988267i \(-0.451192\pi\)
0.152735 + 0.988267i \(0.451192\pi\)
\(338\) 0 0
\(339\) −11.3205 −0.614846
\(340\) 0 0
\(341\) 0.248711 0.0134685
\(342\) 17.9090 0.968406
\(343\) −20.0000 −1.07990
\(344\) 17.6603 0.952177
\(345\) 0 0
\(346\) −26.7846 −1.43995
\(347\) −11.6603 −0.625955 −0.312978 0.949761i \(-0.601326\pi\)
−0.312978 + 0.949761i \(0.601326\pi\)
\(348\) −6.92820 −0.371391
\(349\) −6.39230 −0.342172 −0.171086 0.985256i \(-0.554728\pi\)
−0.171086 + 0.985256i \(0.554728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.58846 −0.351166
\(353\) 27.7128 1.47500 0.737502 0.675345i \(-0.236005\pi\)
0.737502 + 0.675345i \(0.236005\pi\)
\(354\) 19.1769 1.01924
\(355\) 0 0
\(356\) −0.928203 −0.0491947
\(357\) −5.07180 −0.268428
\(358\) −8.78461 −0.464281
\(359\) −8.19615 −0.432576 −0.216288 0.976330i \(-0.569395\pi\)
−0.216288 + 0.976330i \(0.569395\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) −35.3205 −1.85640
\(363\) −6.87564 −0.360878
\(364\) 0 0
\(365\) 0 0
\(366\) 15.7128 0.821322
\(367\) −22.1962 −1.15863 −0.579315 0.815104i \(-0.696680\pi\)
−0.579315 + 0.815104i \(0.696680\pi\)
\(368\) 23.6603 1.23338
\(369\) −8.53590 −0.444361
\(370\) 0 0
\(371\) 20.7846 1.07908
\(372\) 0.143594 0.00744498
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −7.60770 −0.393385
\(375\) 0 0
\(376\) −10.3923 −0.535942
\(377\) 0 0
\(378\) −13.8564 −0.712697
\(379\) 32.9808 1.69411 0.847054 0.531507i \(-0.178374\pi\)
0.847054 + 0.531507i \(0.178374\pi\)
\(380\) 0 0
\(381\) −4.24871 −0.217668
\(382\) −32.7846 −1.67741
\(383\) −0.928203 −0.0474290 −0.0237145 0.999719i \(-0.507549\pi\)
−0.0237145 + 0.999719i \(0.507549\pi\)
\(384\) 8.87564 0.452933
\(385\) 0 0
\(386\) −17.3205 −0.881591
\(387\) 25.1244 1.27714
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 16.3923 0.828994
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) 1.60770 0.0809945
\(395\) 0 0
\(396\) −3.12436 −0.157005
\(397\) −12.7846 −0.641641 −0.320821 0.947140i \(-0.603959\pi\)
−0.320821 + 0.947140i \(0.603959\pi\)
\(398\) 34.6410 1.73640
\(399\) −6.14359 −0.307564
\(400\) 0 0
\(401\) 23.0718 1.15215 0.576075 0.817397i \(-0.304584\pi\)
0.576075 + 0.817397i \(0.304584\pi\)
\(402\) −18.2487 −0.910163
\(403\) 0 0
\(404\) 12.9282 0.643202
\(405\) 0 0
\(406\) −32.7846 −1.62707
\(407\) −5.07180 −0.251400
\(408\) 4.39230 0.217451
\(409\) 38.3923 1.89838 0.949189 0.314708i \(-0.101906\pi\)
0.949189 + 0.314708i \(0.101906\pi\)
\(410\) 0 0
\(411\) −9.46410 −0.466830
\(412\) −10.1962 −0.502328
\(413\) 30.2487 1.48844
\(414\) 20.1962 0.992587
\(415\) 0 0
\(416\) 0 0
\(417\) −6.14359 −0.300853
\(418\) −9.21539 −0.450739
\(419\) −9.46410 −0.462352 −0.231176 0.972912i \(-0.574257\pi\)
−0.231176 + 0.972912i \(0.574257\pi\)
\(420\) 0 0
\(421\) −10.7846 −0.525610 −0.262805 0.964849i \(-0.584648\pi\)
−0.262805 + 0.964849i \(0.584648\pi\)
\(422\) 13.8564 0.674519
\(423\) −14.7846 −0.718852
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) −1.60770 −0.0778931
\(427\) 24.7846 1.19941
\(428\) −0.339746 −0.0164222
\(429\) 0 0
\(430\) 0 0
\(431\) −19.5167 −0.940084 −0.470042 0.882644i \(-0.655761\pi\)
−0.470042 + 0.882644i \(0.655761\pi\)
\(432\) 20.0000 0.962250
\(433\) 6.78461 0.326048 0.163024 0.986622i \(-0.447875\pi\)
0.163024 + 0.986622i \(0.447875\pi\)
\(434\) 0.679492 0.0326167
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 19.8564 0.949861
\(438\) −5.07180 −0.242340
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 7.39230 0.352015
\(442\) 0 0
\(443\) −34.9808 −1.66199 −0.830993 0.556283i \(-0.812227\pi\)
−0.830993 + 0.556283i \(0.812227\pi\)
\(444\) −2.92820 −0.138966
\(445\) 0 0
\(446\) 3.46410 0.164030
\(447\) −14.5359 −0.687524
\(448\) 2.00000 0.0944911
\(449\) −27.4641 −1.29611 −0.648056 0.761593i \(-0.724418\pi\)
−0.648056 + 0.761593i \(0.724418\pi\)
\(450\) 0 0
\(451\) 4.39230 0.206826
\(452\) −15.4641 −0.727370
\(453\) 8.92820 0.419484
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 5.32051 0.249156
\(457\) −30.7846 −1.44004 −0.720022 0.693952i \(-0.755868\pi\)
−0.720022 + 0.693952i \(0.755868\pi\)
\(458\) 24.9282 1.16482
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) 3.21539 0.149593
\(463\) 18.3923 0.854763 0.427381 0.904071i \(-0.359436\pi\)
0.427381 + 0.904071i \(0.359436\pi\)
\(464\) 47.3205 2.19680
\(465\) 0 0
\(466\) 10.3923 0.481414
\(467\) −38.1962 −1.76751 −0.883754 0.467953i \(-0.844992\pi\)
−0.883754 + 0.467953i \(0.844992\pi\)
\(468\) 0 0
\(469\) −28.7846 −1.32915
\(470\) 0 0
\(471\) 7.32051 0.337311
\(472\) −26.1962 −1.20578
\(473\) −12.9282 −0.594439
\(474\) 15.7128 0.721713
\(475\) 0 0
\(476\) −6.92820 −0.317554
\(477\) −25.6077 −1.17250
\(478\) 6.58846 0.301349
\(479\) −18.3397 −0.837964 −0.418982 0.907994i \(-0.637613\pi\)
−0.418982 + 0.907994i \(0.637613\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −31.8564 −1.45102
\(483\) −6.92820 −0.315244
\(484\) −9.39230 −0.426923
\(485\) 0 0
\(486\) 26.4449 1.19956
\(487\) −5.60770 −0.254109 −0.127054 0.991896i \(-0.540552\pi\)
−0.127054 + 0.991896i \(0.540552\pi\)
\(488\) −21.4641 −0.971634
\(489\) 4.67949 0.211614
\(490\) 0 0
\(491\) −9.46410 −0.427109 −0.213554 0.976931i \(-0.568504\pi\)
−0.213554 + 0.976931i \(0.568504\pi\)
\(492\) 2.53590 0.114327
\(493\) 32.7846 1.47654
\(494\) 0 0
\(495\) 0 0
\(496\) −0.980762 −0.0440375
\(497\) −2.53590 −0.113751
\(498\) −7.60770 −0.340909
\(499\) −12.9808 −0.581099 −0.290549 0.956860i \(-0.593838\pi\)
−0.290549 + 0.956860i \(0.593838\pi\)
\(500\) 0 0
\(501\) 9.46410 0.422825
\(502\) −25.1769 −1.12370
\(503\) 25.5167 1.13773 0.568866 0.822430i \(-0.307382\pi\)
0.568866 + 0.822430i \(0.307382\pi\)
\(504\) 8.53590 0.380219
\(505\) 0 0
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) −5.80385 −0.257504
\(509\) −32.5359 −1.44213 −0.721064 0.692868i \(-0.756347\pi\)
−0.721064 + 0.692868i \(0.756347\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 8.66025 0.382733
\(513\) 16.7846 0.741059
\(514\) 13.6077 0.600210
\(515\) 0 0
\(516\) −7.46410 −0.328589
\(517\) 7.60770 0.334586
\(518\) −13.8564 −0.608816
\(519\) −11.3205 −0.496915
\(520\) 0 0
\(521\) −7.60770 −0.333299 −0.166650 0.986016i \(-0.553295\pi\)
−0.166650 + 0.986016i \(0.553295\pi\)
\(522\) 40.3923 1.76792
\(523\) 13.8038 0.603600 0.301800 0.953371i \(-0.402412\pi\)
0.301800 + 0.953371i \(0.402412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.19615 −0.357369
\(527\) −0.679492 −0.0295991
\(528\) −4.64102 −0.201974
\(529\) −0.607695 −0.0264215
\(530\) 0 0
\(531\) −37.2679 −1.61729
\(532\) −8.39230 −0.363853
\(533\) 0 0
\(534\) −1.17691 −0.0509301
\(535\) 0 0
\(536\) 24.9282 1.07673
\(537\) −3.71281 −0.160220
\(538\) 13.6077 0.586669
\(539\) −3.80385 −0.163843
\(540\) 0 0
\(541\) 5.60770 0.241094 0.120547 0.992708i \(-0.461535\pi\)
0.120547 + 0.992708i \(0.461535\pi\)
\(542\) 36.3397 1.56093
\(543\) −14.9282 −0.640631
\(544\) 18.0000 0.771744
\(545\) 0 0
\(546\) 0 0
\(547\) 1.80385 0.0771270 0.0385635 0.999256i \(-0.487722\pi\)
0.0385635 + 0.999256i \(0.487722\pi\)
\(548\) −12.9282 −0.552265
\(549\) −30.5359 −1.30324
\(550\) 0 0
\(551\) 39.7128 1.69182
\(552\) 6.00000 0.255377
\(553\) 24.7846 1.05395
\(554\) 9.71281 0.412658
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) −25.8564 −1.09557 −0.547786 0.836619i \(-0.684529\pi\)
−0.547786 + 0.836619i \(0.684529\pi\)
\(558\) −0.837169 −0.0354402
\(559\) 0 0
\(560\) 0 0
\(561\) −3.21539 −0.135754
\(562\) −2.78461 −0.117462
\(563\) −16.0526 −0.676535 −0.338267 0.941050i \(-0.609841\pi\)
−0.338267 + 0.941050i \(0.609841\pi\)
\(564\) 4.39230 0.184949
\(565\) 0 0
\(566\) −2.44486 −0.102765
\(567\) 8.92820 0.374949
\(568\) 2.19615 0.0921485
\(569\) −9.46410 −0.396756 −0.198378 0.980126i \(-0.563567\pi\)
−0.198378 + 0.980126i \(0.563567\pi\)
\(570\) 0 0
\(571\) 15.6077 0.653162 0.326581 0.945169i \(-0.394103\pi\)
0.326581 + 0.945169i \(0.394103\pi\)
\(572\) 0 0
\(573\) −13.8564 −0.578860
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) −2.46410 −0.102671
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −8.66025 −0.360219
\(579\) −7.32051 −0.304230
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 2.53590 0.105116
\(583\) 13.1769 0.545732
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) −32.7846 −1.35432
\(587\) −15.4641 −0.638272 −0.319136 0.947709i \(-0.603393\pi\)
−0.319136 + 0.947709i \(0.603393\pi\)
\(588\) −2.19615 −0.0905678
\(589\) −0.823085 −0.0339146
\(590\) 0 0
\(591\) 0.679492 0.0279506
\(592\) 20.0000 0.821995
\(593\) −14.7846 −0.607131 −0.303566 0.952811i \(-0.598177\pi\)
−0.303566 + 0.952811i \(0.598177\pi\)
\(594\) −8.78461 −0.360437
\(595\) 0 0
\(596\) −19.8564 −0.813350
\(597\) 14.6410 0.599217
\(598\) 0 0
\(599\) −28.3923 −1.16008 −0.580039 0.814589i \(-0.696963\pi\)
−0.580039 + 0.814589i \(0.696963\pi\)
\(600\) 0 0
\(601\) −39.5692 −1.61406 −0.807031 0.590509i \(-0.798927\pi\)
−0.807031 + 0.590509i \(0.798927\pi\)
\(602\) −35.3205 −1.43956
\(603\) 35.4641 1.44421
\(604\) 12.1962 0.496254
\(605\) 0 0
\(606\) 16.3923 0.665892
\(607\) 26.9808 1.09512 0.547558 0.836768i \(-0.315558\pi\)
0.547558 + 0.836768i \(0.315558\pi\)
\(608\) 21.8038 0.884263
\(609\) −13.8564 −0.561490
\(610\) 0 0
\(611\) 0 0
\(612\) 8.53590 0.345043
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 39.4641 1.59264
\(615\) 0 0
\(616\) −4.39230 −0.176971
\(617\) −21.7128 −0.874125 −0.437062 0.899431i \(-0.643981\pi\)
−0.437062 + 0.899431i \(0.643981\pi\)
\(618\) −12.9282 −0.520049
\(619\) 44.9808 1.80793 0.903965 0.427607i \(-0.140643\pi\)
0.903965 + 0.427607i \(0.140643\pi\)
\(620\) 0 0
\(621\) 18.9282 0.759563
\(622\) 7.60770 0.305041
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 0 0
\(626\) −11.0718 −0.442518
\(627\) −3.89488 −0.155547
\(628\) 10.0000 0.399043
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) −16.1962 −0.644759 −0.322379 0.946611i \(-0.604483\pi\)
−0.322379 + 0.946611i \(0.604483\pi\)
\(632\) −21.4641 −0.853796
\(633\) 5.85641 0.232771
\(634\) 41.5692 1.65092
\(635\) 0 0
\(636\) 7.60770 0.301665
\(637\) 0 0
\(638\) −20.7846 −0.822871
\(639\) 3.12436 0.123598
\(640\) 0 0
\(641\) −0.928203 −0.0366618 −0.0183309 0.999832i \(-0.505835\pi\)
−0.0183309 + 0.999832i \(0.505835\pi\)
\(642\) −0.430781 −0.0170016
\(643\) 34.7846 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(644\) −9.46410 −0.372938
\(645\) 0 0
\(646\) 25.1769 0.990572
\(647\) 16.0526 0.631091 0.315546 0.948910i \(-0.397812\pi\)
0.315546 + 0.948910i \(0.397812\pi\)
\(648\) −7.73205 −0.303744
\(649\) 19.1769 0.752760
\(650\) 0 0
\(651\) 0.287187 0.0112557
\(652\) 6.39230 0.250342
\(653\) 19.8564 0.777041 0.388521 0.921440i \(-0.372986\pi\)
0.388521 + 0.921440i \(0.372986\pi\)
\(654\) −2.53590 −0.0991615
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 9.85641 0.384535
\(658\) 20.7846 0.810268
\(659\) −14.5359 −0.566238 −0.283119 0.959085i \(-0.591369\pi\)
−0.283119 + 0.959085i \(0.591369\pi\)
\(660\) 0 0
\(661\) 30.7846 1.19738 0.598691 0.800980i \(-0.295688\pi\)
0.598691 + 0.800980i \(0.295688\pi\)
\(662\) 49.5167 1.92452
\(663\) 0 0
\(664\) 10.3923 0.403300
\(665\) 0 0
\(666\) 17.0718 0.661519
\(667\) 44.7846 1.73407
\(668\) 12.9282 0.500207
\(669\) 1.46410 0.0566054
\(670\) 0 0
\(671\) 15.7128 0.606586
\(672\) −7.60770 −0.293473
\(673\) −6.39230 −0.246405 −0.123203 0.992382i \(-0.539317\pi\)
−0.123203 + 0.992382i \(0.539317\pi\)
\(674\) 9.71281 0.374124
\(675\) 0 0
\(676\) 0 0
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) −19.6077 −0.753029
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 2.53590 0.0971758
\(682\) 0.430781 0.0164954
\(683\) 39.4641 1.51005 0.755026 0.655695i \(-0.227624\pi\)
0.755026 + 0.655695i \(0.227624\pi\)
\(684\) 10.3397 0.395350
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) 10.5359 0.401970
\(688\) 50.9808 1.94362
\(689\) 0 0
\(690\) 0 0
\(691\) −45.7654 −1.74100 −0.870498 0.492171i \(-0.836203\pi\)
−0.870498 + 0.492171i \(0.836203\pi\)
\(692\) −15.4641 −0.587857
\(693\) −6.24871 −0.237369
\(694\) −20.1962 −0.766635
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) −12.0000 −0.454532
\(698\) −11.0718 −0.419074
\(699\) 4.39230 0.166132
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 16.7846 0.633044
\(704\) 1.26795 0.0477876
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) 25.8564 0.972430
\(708\) 11.0718 0.416104
\(709\) −9.60770 −0.360825 −0.180412 0.983591i \(-0.557743\pi\)
−0.180412 + 0.983591i \(0.557743\pi\)
\(710\) 0 0
\(711\) −30.5359 −1.14519
\(712\) 1.60770 0.0602509
\(713\) −0.928203 −0.0347615
\(714\) −8.78461 −0.328756
\(715\) 0 0
\(716\) −5.07180 −0.189542
\(717\) 2.78461 0.103993
\(718\) −14.1962 −0.529796
\(719\) 1.85641 0.0692323 0.0346161 0.999401i \(-0.488979\pi\)
0.0346161 + 0.999401i \(0.488979\pi\)
\(720\) 0 0
\(721\) −20.3923 −0.759449
\(722\) −2.41154 −0.0897483
\(723\) −13.4641 −0.500735
\(724\) −20.3923 −0.757874
\(725\) 0 0
\(726\) −11.9090 −0.441983
\(727\) −13.4115 −0.497407 −0.248703 0.968580i \(-0.580004\pi\)
−0.248703 + 0.968580i \(0.580004\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 35.3205 1.30638
\(732\) 9.07180 0.335303
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) −38.4449 −1.41903
\(735\) 0 0
\(736\) 24.5885 0.906343
\(737\) −18.2487 −0.672200
\(738\) −14.7846 −0.544229
\(739\) 7.80385 0.287069 0.143535 0.989645i \(-0.454153\pi\)
0.143535 + 0.989645i \(0.454153\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) −43.8564 −1.60894 −0.804468 0.593996i \(-0.797549\pi\)
−0.804468 + 0.593996i \(0.797549\pi\)
\(744\) −0.248711 −0.00911820
\(745\) 0 0
\(746\) 17.3205 0.634149
\(747\) 14.7846 0.540941
\(748\) −4.39230 −0.160599
\(749\) −0.679492 −0.0248281
\(750\) 0 0
\(751\) 15.6077 0.569533 0.284766 0.958597i \(-0.408084\pi\)
0.284766 + 0.958597i \(0.408084\pi\)
\(752\) −30.0000 −1.09399
\(753\) −10.6410 −0.387780
\(754\) 0 0
\(755\) 0 0
\(756\) −8.00000 −0.290957
\(757\) −18.3923 −0.668480 −0.334240 0.942488i \(-0.608480\pi\)
−0.334240 + 0.942488i \(0.608480\pi\)
\(758\) 57.1244 2.07485
\(759\) −4.39230 −0.159431
\(760\) 0 0
\(761\) −7.85641 −0.284795 −0.142397 0.989810i \(-0.545481\pi\)
−0.142397 + 0.989810i \(0.545481\pi\)
\(762\) −7.35898 −0.266588
\(763\) −4.00000 −0.144810
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) −1.60770 −0.0580884
\(767\) 0 0
\(768\) 13.9090 0.501897
\(769\) 6.78461 0.244659 0.122330 0.992490i \(-0.460963\pi\)
0.122330 + 0.992490i \(0.460963\pi\)
\(770\) 0 0
\(771\) 5.75129 0.207128
\(772\) −10.0000 −0.359908
\(773\) −6.92820 −0.249190 −0.124595 0.992208i \(-0.539763\pi\)
−0.124595 + 0.992208i \(0.539763\pi\)
\(774\) 43.5167 1.56417
\(775\) 0 0
\(776\) −3.46410 −0.124354
\(777\) −5.85641 −0.210097
\(778\) 10.3923 0.372582
\(779\) −14.5359 −0.520803
\(780\) 0 0
\(781\) −1.60770 −0.0575279
\(782\) 28.3923 1.01531
\(783\) 37.8564 1.35288
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) −51.5692 −1.83824 −0.919122 0.393973i \(-0.871100\pi\)
−0.919122 + 0.393973i \(0.871100\pi\)
\(788\) 0.928203 0.0330659
\(789\) −3.46410 −0.123325
\(790\) 0 0
\(791\) −30.9282 −1.09968
\(792\) 5.41154 0.192291
\(793\) 0 0
\(794\) −22.1436 −0.785847
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −28.6410 −1.01452 −0.507258 0.861794i \(-0.669341\pi\)
−0.507258 + 0.861794i \(0.669341\pi\)
\(798\) −10.6410 −0.376688
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) 2.28719 0.0808138
\(802\) 39.9615 1.41109
\(803\) −5.07180 −0.178980
\(804\) −10.5359 −0.371572
\(805\) 0 0
\(806\) 0 0
\(807\) 5.75129 0.202455
\(808\) −22.3923 −0.787759
\(809\) −9.46410 −0.332740 −0.166370 0.986063i \(-0.553205\pi\)
−0.166370 + 0.986063i \(0.553205\pi\)
\(810\) 0 0
\(811\) −28.1962 −0.990101 −0.495050 0.868864i \(-0.664850\pi\)
−0.495050 + 0.868864i \(0.664850\pi\)
\(812\) −18.9282 −0.664250
\(813\) 15.3590 0.538663
\(814\) −8.78461 −0.307900
\(815\) 0 0
\(816\) 12.6795 0.443871
\(817\) 42.7846 1.49684
\(818\) 66.4974 2.32503
\(819\) 0 0
\(820\) 0 0
\(821\) 40.6410 1.41838 0.709191 0.705017i \(-0.249061\pi\)
0.709191 + 0.705017i \(0.249061\pi\)
\(822\) −16.3923 −0.571747
\(823\) 46.5885 1.62397 0.811986 0.583677i \(-0.198387\pi\)
0.811986 + 0.583677i \(0.198387\pi\)
\(824\) 17.6603 0.615224
\(825\) 0 0
\(826\) 52.3923 1.82296
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 11.6603 0.405222
\(829\) −20.3923 −0.708254 −0.354127 0.935197i \(-0.615222\pi\)
−0.354127 + 0.935197i \(0.615222\pi\)
\(830\) 0 0
\(831\) 4.10512 0.142405
\(832\) 0 0
\(833\) 10.3923 0.360072
\(834\) −10.6410 −0.368468
\(835\) 0 0
\(836\) −5.32051 −0.184014
\(837\) −0.784610 −0.0271201
\(838\) −16.3923 −0.566263
\(839\) −17.6603 −0.609700 −0.304850 0.952400i \(-0.598606\pi\)
−0.304850 + 0.952400i \(0.598606\pi\)
\(840\) 0 0
\(841\) 60.5692 2.08859
\(842\) −18.6795 −0.643738
\(843\) −1.17691 −0.0405351
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) −25.6077 −0.880411
\(847\) −18.7846 −0.645447
\(848\) −51.9615 −1.78437
\(849\) −1.03332 −0.0354635
\(850\) 0 0
\(851\) 18.9282 0.648850
\(852\) −0.928203 −0.0317997
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 42.9282 1.46897
\(855\) 0 0
\(856\) 0.588457 0.0201131
\(857\) −47.5692 −1.62493 −0.812467 0.583007i \(-0.801876\pi\)
−0.812467 + 0.583007i \(0.801876\pi\)
\(858\) 0 0
\(859\) 45.1769 1.54142 0.770708 0.637188i \(-0.219903\pi\)
0.770708 + 0.637188i \(0.219903\pi\)
\(860\) 0 0
\(861\) 5.07180 0.172846
\(862\) −33.8038 −1.15136
\(863\) 2.78461 0.0947892 0.0473946 0.998876i \(-0.484908\pi\)
0.0473946 + 0.998876i \(0.484908\pi\)
\(864\) 20.7846 0.707107
\(865\) 0 0
\(866\) 11.7513 0.399325
\(867\) −3.66025 −0.124309
\(868\) 0.392305 0.0133157
\(869\) 15.7128 0.533021
\(870\) 0 0
\(871\) 0 0
\(872\) 3.46410 0.117309
\(873\) −4.92820 −0.166794
\(874\) 34.3923 1.16334
\(875\) 0 0
\(876\) −2.92820 −0.0989348
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 55.4256 1.87052
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) −12.6795 −0.427183 −0.213591 0.976923i \(-0.568516\pi\)
−0.213591 + 0.976923i \(0.568516\pi\)
\(882\) 12.8038 0.431128
\(883\) −34.1962 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −60.5885 −2.03551
\(887\) −17.9090 −0.601324 −0.300662 0.953731i \(-0.597208\pi\)
−0.300662 + 0.953731i \(0.597208\pi\)
\(888\) 5.07180 0.170198
\(889\) −11.6077 −0.389310
\(890\) 0 0
\(891\) 5.66025 0.189626
\(892\) 2.00000 0.0669650
\(893\) −25.1769 −0.842513
\(894\) −25.1769 −0.842042
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) −47.5692 −1.58741
\(899\) −1.85641 −0.0619146
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 7.60770 0.253309
\(903\) −14.9282 −0.496779
\(904\) 26.7846 0.890843
\(905\) 0 0
\(906\) 15.4641 0.513760
\(907\) −39.7654 −1.32039 −0.660194 0.751095i \(-0.729526\pi\)
−0.660194 + 0.751095i \(0.729526\pi\)
\(908\) 3.46410 0.114960
\(909\) −31.8564 −1.05661
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 15.3590 0.508587
\(913\) −7.60770 −0.251778
\(914\) −53.3205 −1.76369
\(915\) 0 0
\(916\) 14.3923 0.475535
\(917\) 0 0
\(918\) 24.0000 0.792118
\(919\) −53.1769 −1.75414 −0.877072 0.480358i \(-0.840507\pi\)
−0.877072 + 0.480358i \(0.840507\pi\)
\(920\) 0 0
\(921\) 16.6795 0.549608
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 1.85641 0.0610713
\(925\) 0 0
\(926\) 31.8564 1.04687
\(927\) 25.1244 0.825192
\(928\) 49.1769 1.61431
\(929\) −51.4641 −1.68848 −0.844241 0.535963i \(-0.819948\pi\)
−0.844241 + 0.535963i \(0.819948\pi\)
\(930\) 0 0
\(931\) 12.5885 0.412570
\(932\) 6.00000 0.196537
\(933\) 3.21539 0.105267
\(934\) −66.1577 −2.16475
\(935\) 0 0
\(936\) 0 0
\(937\) 6.78461 0.221644 0.110822 0.993840i \(-0.464652\pi\)
0.110822 + 0.993840i \(0.464652\pi\)
\(938\) −49.8564 −1.62787
\(939\) −4.67949 −0.152709
\(940\) 0 0
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) 12.6795 0.413120
\(943\) −16.3923 −0.533807
\(944\) −75.6218 −2.46128
\(945\) 0 0
\(946\) −22.3923 −0.728037
\(947\) −28.6410 −0.930708 −0.465354 0.885125i \(-0.654073\pi\)
−0.465354 + 0.885125i \(0.654073\pi\)
\(948\) 9.07180 0.294638
\(949\) 0 0
\(950\) 0 0
\(951\) 17.5692 0.569721
\(952\) 12.0000 0.388922
\(953\) 12.9282 0.418786 0.209393 0.977832i \(-0.432851\pi\)
0.209393 + 0.977832i \(0.432851\pi\)
\(954\) −44.3538 −1.43601
\(955\) 0 0
\(956\) 3.80385 0.123025
\(957\) −8.78461 −0.283966
\(958\) −31.7654 −1.02629
\(959\) −25.8564 −0.834947
\(960\) 0 0
\(961\) −30.9615 −0.998759
\(962\) 0 0
\(963\) 0.837169 0.0269774
\(964\) −18.3923 −0.592376
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) −29.6077 −0.952119 −0.476060 0.879413i \(-0.657935\pi\)
−0.476060 + 0.879413i \(0.657935\pi\)
\(968\) 16.2679 0.522872
\(969\) 10.6410 0.341839
\(970\) 0 0
\(971\) 5.07180 0.162762 0.0813809 0.996683i \(-0.474067\pi\)
0.0813809 + 0.996683i \(0.474067\pi\)
\(972\) 15.2679 0.489720
\(973\) −16.7846 −0.538090
\(974\) −9.71281 −0.311219
\(975\) 0 0
\(976\) −61.9615 −1.98334
\(977\) 39.7128 1.27053 0.635263 0.772296i \(-0.280892\pi\)
0.635263 + 0.772296i \(0.280892\pi\)
\(978\) 8.10512 0.259173
\(979\) −1.17691 −0.0376144
\(980\) 0 0
\(981\) 4.92820 0.157345
\(982\) −16.3923 −0.523099
\(983\) −13.6077 −0.434018 −0.217009 0.976170i \(-0.569630\pi\)
−0.217009 + 0.976170i \(0.569630\pi\)
\(984\) −4.39230 −0.140022
\(985\) 0 0
\(986\) 56.7846 1.80839
\(987\) 8.78461 0.279617
\(988\) 0 0
\(989\) 48.2487 1.53422
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −1.01924 −0.0323608
\(993\) 20.9282 0.664136
\(994\) −4.39230 −0.139315
\(995\) 0 0
\(996\) −4.39230 −0.139176
\(997\) −54.3923 −1.72262 −0.861311 0.508078i \(-0.830356\pi\)
−0.861311 + 0.508078i \(0.830356\pi\)
\(998\) −22.4833 −0.711698
\(999\) 16.0000 0.506218
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.w.1.2 2
5.4 even 2 845.2.a.d.1.1 2
13.12 even 2 325.2.a.g.1.1 2
15.14 odd 2 7605.2.a.be.1.2 2
39.38 odd 2 2925.2.a.z.1.2 2
52.51 odd 2 5200.2.a.ca.1.1 2
65.4 even 6 845.2.e.e.146.1 4
65.9 even 6 845.2.e.f.146.2 4
65.12 odd 4 325.2.b.e.274.2 4
65.19 odd 12 845.2.m.c.361.2 4
65.24 odd 12 845.2.m.c.316.2 4
65.29 even 6 845.2.e.f.191.2 4
65.34 odd 4 845.2.c.e.506.1 4
65.38 odd 4 325.2.b.e.274.3 4
65.44 odd 4 845.2.c.e.506.3 4
65.49 even 6 845.2.e.e.191.1 4
65.54 odd 12 845.2.m.a.316.2 4
65.59 odd 12 845.2.m.a.361.2 4
65.64 even 2 65.2.a.c.1.2 2
195.38 even 4 2925.2.c.v.2224.2 4
195.77 even 4 2925.2.c.v.2224.3 4
195.194 odd 2 585.2.a.k.1.1 2
260.259 odd 2 1040.2.a.h.1.2 2
455.454 odd 2 3185.2.a.k.1.2 2
520.259 odd 2 4160.2.a.bj.1.1 2
520.389 even 2 4160.2.a.y.1.2 2
715.714 odd 2 7865.2.a.h.1.1 2
780.779 even 2 9360.2.a.cm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.2 2 65.64 even 2
325.2.a.g.1.1 2 13.12 even 2
325.2.b.e.274.2 4 65.12 odd 4
325.2.b.e.274.3 4 65.38 odd 4
585.2.a.k.1.1 2 195.194 odd 2
845.2.a.d.1.1 2 5.4 even 2
845.2.c.e.506.1 4 65.34 odd 4
845.2.c.e.506.3 4 65.44 odd 4
845.2.e.e.146.1 4 65.4 even 6
845.2.e.e.191.1 4 65.49 even 6
845.2.e.f.146.2 4 65.9 even 6
845.2.e.f.191.2 4 65.29 even 6
845.2.m.a.316.2 4 65.54 odd 12
845.2.m.a.361.2 4 65.59 odd 12
845.2.m.c.316.2 4 65.24 odd 12
845.2.m.c.361.2 4 65.19 odd 12
1040.2.a.h.1.2 2 260.259 odd 2
2925.2.a.z.1.2 2 39.38 odd 2
2925.2.c.v.2224.2 4 195.38 even 4
2925.2.c.v.2224.3 4 195.77 even 4
3185.2.a.k.1.2 2 455.454 odd 2
4160.2.a.y.1.2 2 520.389 even 2
4160.2.a.bj.1.1 2 520.259 odd 2
4225.2.a.w.1.2 2 1.1 even 1 trivial
5200.2.a.ca.1.1 2 52.51 odd 2
7605.2.a.be.1.2 2 15.14 odd 2
7865.2.a.h.1.1 2 715.714 odd 2
9360.2.a.cm.1.2 2 780.779 even 2