Properties

Label 1155.2.i.a.76.2
Level $1155$
Weight $2$
Character 1155.76
Analytic conductor $9.223$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1155,2,Mod(76,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.i (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 76.2
Root \(0.396143 + 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 1155.76
Dual form 1155.2.i.a.76.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52434i q^{2} +1.00000i q^{3} -4.37228 q^{4} -1.00000i q^{5} +2.52434 q^{6} +(2.52434 + 0.792287i) q^{7} +5.98844i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.52434i q^{2} +1.00000i q^{3} -4.37228 q^{4} -1.00000i q^{5} +2.52434 q^{6} +(2.52434 + 0.792287i) q^{7} +5.98844i q^{8} -1.00000 q^{9} -2.52434 q^{10} +3.31662i q^{11} -4.37228i q^{12} +(2.00000 - 6.37228i) q^{14} +1.00000 q^{15} +6.37228 q^{16} -5.84096 q^{17} +2.52434i q^{18} -7.72049 q^{19} +4.37228i q^{20} +(-0.792287 + 2.52434i) q^{21} +8.37228 q^{22} -8.11684 q^{23} -5.98844 q^{24} -1.00000 q^{25} -1.00000i q^{27} +(-11.0371 - 3.46410i) q^{28} -2.67181i q^{29} -2.52434i q^{30} -4.74456i q^{31} -4.10891i q^{32} -3.31662 q^{33} +14.7446i q^{34} +(0.792287 - 2.52434i) q^{35} +4.37228 q^{36} -0.744563 q^{37} +19.4891i q^{38} +5.98844 q^{40} -8.21782 q^{41} +(6.37228 + 2.00000i) q^{42} +5.84096i q^{43} -14.5012i q^{44} +1.00000i q^{45} +20.4897i q^{46} -1.25544i q^{47} +6.37228i q^{48} +(5.74456 + 4.00000i) q^{49} +2.52434i q^{50} -5.84096i q^{51} -7.37228 q^{53} -2.52434 q^{54} +3.31662 q^{55} +(-4.74456 + 15.1168i) q^{56} -7.72049i q^{57} -6.74456 q^{58} -7.37228i q^{59} -4.37228 q^{60} -1.08724 q^{61} -11.9769 q^{62} +(-2.52434 - 0.792287i) q^{63} +2.37228 q^{64} +8.37228i q^{66} +6.74456 q^{67} +25.5383 q^{68} -8.11684i q^{69} +(-6.37228 - 2.00000i) q^{70} +14.7446 q^{71} -5.98844i q^{72} +6.92820 q^{73} +1.87953i q^{74} -1.00000i q^{75} +33.7562 q^{76} +(-2.62772 + 8.37228i) q^{77} +13.5615i q^{79} -6.37228i q^{80} +1.00000 q^{81} +20.7446i q^{82} -6.13592 q^{83} +(3.46410 - 11.0371i) q^{84} +5.84096i q^{85} +14.7446 q^{86} +2.67181 q^{87} -19.8614 q^{88} +2.62772i q^{89} +2.52434 q^{90} +35.4891 q^{92} +4.74456 q^{93} -3.16915 q^{94} +7.72049i q^{95} +4.10891 q^{96} -2.62772i q^{97} +(10.0974 - 14.5012i) q^{98} -3.31662i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{4} - 8 q^{9} + 16 q^{14} + 8 q^{15} + 28 q^{16} + 44 q^{22} + 4 q^{23} - 8 q^{25} + 12 q^{36} + 40 q^{37} + 28 q^{42} - 36 q^{53} + 8 q^{56} - 8 q^{58} - 12 q^{60} - 4 q^{64} + 8 q^{67} - 28 q^{70} + 72 q^{71} - 44 q^{77} + 8 q^{81} + 72 q^{86} - 44 q^{88} + 192 q^{92} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.52434i 1.78498i −0.451071 0.892488i \(-0.648958\pi\)
0.451071 0.892488i \(-0.351042\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −4.37228 −2.18614
\(5\) 1.00000i 0.447214i
\(6\) 2.52434 1.03056
\(7\) 2.52434 + 0.792287i 0.954110 + 0.299456i
\(8\) 5.98844i 2.11723i
\(9\) −1.00000 −0.333333
\(10\) −2.52434 −0.798266
\(11\) 3.31662i 1.00000i
\(12\) 4.37228i 1.26217i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 6.37228i 0.534522 1.70306i
\(15\) 1.00000 0.258199
\(16\) 6.37228 1.59307
\(17\) −5.84096 −1.41664 −0.708321 0.705891i \(-0.750547\pi\)
−0.708321 + 0.705891i \(0.750547\pi\)
\(18\) 2.52434i 0.594992i
\(19\) −7.72049 −1.77120 −0.885601 0.464447i \(-0.846253\pi\)
−0.885601 + 0.464447i \(0.846253\pi\)
\(20\) 4.37228i 0.977672i
\(21\) −0.792287 + 2.52434i −0.172891 + 0.550856i
\(22\) 8.37228 1.78498
\(23\) −8.11684 −1.69248 −0.846239 0.532803i \(-0.821139\pi\)
−0.846239 + 0.532803i \(0.821139\pi\)
\(24\) −5.98844 −1.22239
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) −11.0371 3.46410i −2.08582 0.654654i
\(29\) 2.67181i 0.496144i −0.968742 0.248072i \(-0.920203\pi\)
0.968742 0.248072i \(-0.0797969\pi\)
\(30\) 2.52434i 0.460879i
\(31\) 4.74456i 0.852149i −0.904688 0.426074i \(-0.859896\pi\)
0.904688 0.426074i \(-0.140104\pi\)
\(32\) 4.10891i 0.726360i
\(33\) −3.31662 −0.577350
\(34\) 14.7446i 2.52867i
\(35\) 0.792287 2.52434i 0.133921 0.426691i
\(36\) 4.37228 0.728714
\(37\) −0.744563 −0.122405 −0.0612027 0.998125i \(-0.519494\pi\)
−0.0612027 + 0.998125i \(0.519494\pi\)
\(38\) 19.4891i 3.16155i
\(39\) 0 0
\(40\) 5.98844 0.946855
\(41\) −8.21782 −1.28341 −0.641704 0.766952i \(-0.721772\pi\)
−0.641704 + 0.766952i \(0.721772\pi\)
\(42\) 6.37228 + 2.00000i 0.983264 + 0.308607i
\(43\) 5.84096i 0.890738i 0.895347 + 0.445369i \(0.146928\pi\)
−0.895347 + 0.445369i \(0.853072\pi\)
\(44\) 14.5012i 2.18614i
\(45\) 1.00000i 0.149071i
\(46\) 20.4897i 3.02103i
\(47\) 1.25544i 0.183124i −0.995799 0.0915622i \(-0.970814\pi\)
0.995799 0.0915622i \(-0.0291860\pi\)
\(48\) 6.37228i 0.919760i
\(49\) 5.74456 + 4.00000i 0.820652 + 0.571429i
\(50\) 2.52434i 0.356995i
\(51\) 5.84096i 0.817898i
\(52\) 0 0
\(53\) −7.37228 −1.01266 −0.506330 0.862340i \(-0.668998\pi\)
−0.506330 + 0.862340i \(0.668998\pi\)
\(54\) −2.52434 −0.343519
\(55\) 3.31662 0.447214
\(56\) −4.74456 + 15.1168i −0.634019 + 2.02007i
\(57\) 7.72049i 1.02260i
\(58\) −6.74456 −0.885604
\(59\) 7.37228i 0.959789i −0.877326 0.479895i \(-0.840675\pi\)
0.877326 0.479895i \(-0.159325\pi\)
\(60\) −4.37228 −0.564459
\(61\) −1.08724 −0.139207 −0.0696034 0.997575i \(-0.522173\pi\)
−0.0696034 + 0.997575i \(0.522173\pi\)
\(62\) −11.9769 −1.52107
\(63\) −2.52434 0.792287i −0.318037 0.0998188i
\(64\) 2.37228 0.296535
\(65\) 0 0
\(66\) 8.37228i 1.03056i
\(67\) 6.74456 0.823979 0.411990 0.911188i \(-0.364834\pi\)
0.411990 + 0.911188i \(0.364834\pi\)
\(68\) 25.5383 3.09698
\(69\) 8.11684i 0.977153i
\(70\) −6.37228 2.00000i −0.761633 0.239046i
\(71\) 14.7446 1.74986 0.874929 0.484252i \(-0.160908\pi\)
0.874929 + 0.484252i \(0.160908\pi\)
\(72\) 5.98844i 0.705744i
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 1.87953i 0.218491i
\(75\) 1.00000i 0.115470i
\(76\) 33.7562 3.87210
\(77\) −2.62772 + 8.37228i −0.299456 + 0.954110i
\(78\) 0 0
\(79\) 13.5615i 1.52578i 0.646527 + 0.762891i \(0.276221\pi\)
−0.646527 + 0.762891i \(0.723779\pi\)
\(80\) 6.37228i 0.712443i
\(81\) 1.00000 0.111111
\(82\) 20.7446i 2.29085i
\(83\) −6.13592 −0.673504 −0.336752 0.941593i \(-0.609328\pi\)
−0.336752 + 0.941593i \(0.609328\pi\)
\(84\) 3.46410 11.0371i 0.377964 1.20425i
\(85\) 5.84096i 0.633541i
\(86\) 14.7446 1.58995
\(87\) 2.67181 0.286449
\(88\) −19.8614 −2.11723
\(89\) 2.62772i 0.278538i 0.990255 + 0.139269i \(0.0444752\pi\)
−0.990255 + 0.139269i \(0.955525\pi\)
\(90\) 2.52434 0.266089
\(91\) 0 0
\(92\) 35.4891 3.70000
\(93\) 4.74456 0.491988
\(94\) −3.16915 −0.326873
\(95\) 7.72049i 0.792106i
\(96\) 4.10891 0.419364
\(97\) 2.62772i 0.266804i −0.991062 0.133402i \(-0.957410\pi\)
0.991062 0.133402i \(-0.0425902\pi\)
\(98\) 10.0974 14.5012i 1.01999 1.46484i
\(99\) 3.31662i 0.333333i
\(100\) 4.37228 0.437228
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) −14.7446 −1.45993
\(103\) 6.11684i 0.602711i −0.953512 0.301355i \(-0.902561\pi\)
0.953512 0.301355i \(-0.0974390\pi\)
\(104\) 0 0
\(105\) 2.52434 + 0.792287i 0.246350 + 0.0773193i
\(106\) 18.6101i 1.80758i
\(107\) 15.1460i 1.46422i −0.681186 0.732111i \(-0.738535\pi\)
0.681186 0.732111i \(-0.261465\pi\)
\(108\) 4.37228i 0.420723i
\(109\) 8.51278i 0.815376i 0.913121 + 0.407688i \(0.133665\pi\)
−0.913121 + 0.407688i \(0.866335\pi\)
\(110\) 8.37228i 0.798266i
\(111\) 0.744563i 0.0706708i
\(112\) 16.0858 + 5.04868i 1.51996 + 0.477055i
\(113\) −7.37228 −0.693526 −0.346763 0.937953i \(-0.612719\pi\)
−0.346763 + 0.937953i \(0.612719\pi\)
\(114\) −19.4891 −1.82532
\(115\) 8.11684i 0.756900i
\(116\) 11.6819i 1.08464i
\(117\) 0 0
\(118\) −18.6101 −1.71320
\(119\) −14.7446 4.62772i −1.35163 0.424222i
\(120\) 5.98844i 0.546667i
\(121\) −11.0000 −1.00000
\(122\) 2.74456i 0.248481i
\(123\) 8.21782i 0.740976i
\(124\) 20.7446i 1.86292i
\(125\) 1.00000i 0.0894427i
\(126\) −2.00000 + 6.37228i −0.178174 + 0.567688i
\(127\) 19.6974i 1.74786i 0.486053 + 0.873929i \(0.338436\pi\)
−0.486053 + 0.873929i \(0.661564\pi\)
\(128\) 14.2063i 1.25567i
\(129\) −5.84096 −0.514268
\(130\) 0 0
\(131\) 18.6101 1.62597 0.812987 0.582282i \(-0.197840\pi\)
0.812987 + 0.582282i \(0.197840\pi\)
\(132\) 14.5012 1.26217
\(133\) −19.4891 6.11684i −1.68992 0.530398i
\(134\) 17.0256i 1.47078i
\(135\) −1.00000 −0.0860663
\(136\) 34.9783i 2.99936i
\(137\) 1.25544 0.107259 0.0536296 0.998561i \(-0.482921\pi\)
0.0536296 + 0.998561i \(0.482921\pi\)
\(138\) −20.4897 −1.74420
\(139\) 3.46410 0.293821 0.146911 0.989150i \(-0.453067\pi\)
0.146911 + 0.989150i \(0.453067\pi\)
\(140\) −3.46410 + 11.0371i −0.292770 + 0.932806i
\(141\) 1.25544 0.105727
\(142\) 37.2203i 3.12345i
\(143\) 0 0
\(144\) −6.37228 −0.531023
\(145\) −2.67181 −0.221882
\(146\) 17.4891i 1.44741i
\(147\) −4.00000 + 5.74456i −0.329914 + 0.473804i
\(148\) 3.25544 0.267595
\(149\) 13.8564i 1.13516i 0.823318 + 0.567581i \(0.192120\pi\)
−0.823318 + 0.567581i \(0.807880\pi\)
\(150\) −2.52434 −0.206111
\(151\) 1.87953i 0.152954i 0.997071 + 0.0764769i \(0.0243671\pi\)
−0.997071 + 0.0764769i \(0.975633\pi\)
\(152\) 46.2337i 3.75005i
\(153\) 5.84096 0.472214
\(154\) 21.1345 + 6.63325i 1.70306 + 0.534522i
\(155\) −4.74456 −0.381092
\(156\) 0 0
\(157\) 20.1168i 1.60550i −0.596316 0.802749i \(-0.703370\pi\)
0.596316 0.802749i \(-0.296630\pi\)
\(158\) 34.2337 2.72349
\(159\) 7.37228i 0.584660i
\(160\) −4.10891 −0.324838
\(161\) −20.4897 6.43087i −1.61481 0.506824i
\(162\) 2.52434i 0.198331i
\(163\) 16.2337 1.27152 0.635760 0.771887i \(-0.280687\pi\)
0.635760 + 0.771887i \(0.280687\pi\)
\(164\) 35.9306 2.80571
\(165\) 3.31662i 0.258199i
\(166\) 15.4891i 1.20219i
\(167\) −23.6588 −1.83077 −0.915387 0.402576i \(-0.868115\pi\)
−0.915387 + 0.402576i \(0.868115\pi\)
\(168\) −15.1168 4.74456i −1.16629 0.366051i
\(169\) −13.0000 −1.00000
\(170\) 14.7446 1.13086
\(171\) 7.72049 0.590401
\(172\) 25.5383i 1.94728i
\(173\) 6.92820 0.526742 0.263371 0.964695i \(-0.415166\pi\)
0.263371 + 0.964695i \(0.415166\pi\)
\(174\) 6.74456i 0.511304i
\(175\) −2.52434 0.792287i −0.190822 0.0598913i
\(176\) 21.1345i 1.59307i
\(177\) 7.37228 0.554135
\(178\) 6.63325 0.497183
\(179\) 1.48913 0.111302 0.0556512 0.998450i \(-0.482277\pi\)
0.0556512 + 0.998450i \(0.482277\pi\)
\(180\) 4.37228i 0.325891i
\(181\) 5.25544i 0.390634i −0.980740 0.195317i \(-0.937426\pi\)
0.980740 0.195317i \(-0.0625735\pi\)
\(182\) 0 0
\(183\) 1.08724i 0.0803711i
\(184\) 48.6072i 3.58337i
\(185\) 0.744563i 0.0547413i
\(186\) 11.9769i 0.878187i
\(187\) 19.3723i 1.41664i
\(188\) 5.48913i 0.400336i
\(189\) 0.792287 2.52434i 0.0576304 0.183619i
\(190\) 19.4891 1.41389
\(191\) 2.74456 0.198590 0.0992948 0.995058i \(-0.468341\pi\)
0.0992948 + 0.995058i \(0.468341\pi\)
\(192\) 2.37228i 0.171205i
\(193\) 15.1460i 1.09023i −0.838360 0.545117i \(-0.816485\pi\)
0.838360 0.545117i \(-0.183515\pi\)
\(194\) −6.63325 −0.476240
\(195\) 0 0
\(196\) −25.1168 17.4891i −1.79406 1.24922i
\(197\) 10.0974i 0.719406i −0.933067 0.359703i \(-0.882878\pi\)
0.933067 0.359703i \(-0.117122\pi\)
\(198\) −8.37228 −0.594992
\(199\) 4.74456i 0.336333i −0.985759 0.168167i \(-0.946215\pi\)
0.985759 0.168167i \(-0.0537846\pi\)
\(200\) 5.98844i 0.423447i
\(201\) 6.74456i 0.475725i
\(202\) 26.2337i 1.84580i
\(203\) 2.11684 6.74456i 0.148573 0.473375i
\(204\) 25.5383i 1.78804i
\(205\) 8.21782i 0.573958i
\(206\) −15.4410 −1.07582
\(207\) 8.11684 0.564160
\(208\) 0 0
\(209\) 25.6060i 1.77120i
\(210\) 2.00000 6.37228i 0.138013 0.439729i
\(211\) 3.46410i 0.238479i −0.992866 0.119239i \(-0.961954\pi\)
0.992866 0.119239i \(-0.0380456\pi\)
\(212\) 32.2337 2.21382
\(213\) 14.7446i 1.01028i
\(214\) −38.2337 −2.61360
\(215\) 5.84096 0.398350
\(216\) 5.98844 0.407462
\(217\) 3.75906 11.9769i 0.255181 0.813044i
\(218\) 21.4891 1.45543
\(219\) 6.92820i 0.468165i
\(220\) −14.5012 −0.977672
\(221\) 0 0
\(222\) −1.87953 −0.126146
\(223\) 8.62772i 0.577755i −0.957366 0.288877i \(-0.906718\pi\)
0.957366 0.288877i \(-0.0932820\pi\)
\(224\) 3.25544 10.3723i 0.217513 0.693027i
\(225\) 1.00000 0.0666667
\(226\) 18.6101i 1.23793i
\(227\) 1.38219 0.0917395 0.0458697 0.998947i \(-0.485394\pi\)
0.0458697 + 0.998947i \(0.485394\pi\)
\(228\) 33.7562i 2.23556i
\(229\) 22.7446i 1.50300i 0.659731 + 0.751502i \(0.270670\pi\)
−0.659731 + 0.751502i \(0.729330\pi\)
\(230\) 20.4897 1.35105
\(231\) −8.37228 2.62772i −0.550856 0.172891i
\(232\) 16.0000 1.05045
\(233\) 23.3639i 1.53062i −0.643664 0.765308i \(-0.722586\pi\)
0.643664 0.765308i \(-0.277414\pi\)
\(234\) 0 0
\(235\) −1.25544 −0.0818957
\(236\) 32.2337i 2.09823i
\(237\) −13.5615 −0.880911
\(238\) −11.6819 + 37.2203i −0.757227 + 2.41263i
\(239\) 15.6434i 1.01188i 0.862567 + 0.505942i \(0.168855\pi\)
−0.862567 + 0.505942i \(0.831145\pi\)
\(240\) 6.37228 0.411329
\(241\) −13.8564 −0.892570 −0.446285 0.894891i \(-0.647253\pi\)
−0.446285 + 0.894891i \(0.647253\pi\)
\(242\) 27.7677i 1.78498i
\(243\) 1.00000i 0.0641500i
\(244\) 4.75372 0.304326
\(245\) 4.00000 5.74456i 0.255551 0.367007i
\(246\) −20.7446 −1.32263
\(247\) 0 0
\(248\) 28.4125 1.80420
\(249\) 6.13592i 0.388848i
\(250\) 2.52434 0.159653
\(251\) 5.48913i 0.346471i 0.984880 + 0.173235i \(0.0554221\pi\)
−0.984880 + 0.173235i \(0.944578\pi\)
\(252\) 11.0371 + 3.46410i 0.695273 + 0.218218i
\(253\) 26.9205i 1.69248i
\(254\) 49.7228 3.11989
\(255\) −5.84096 −0.365775
\(256\) −31.1168 −1.94480
\(257\) 10.2337i 0.638360i 0.947694 + 0.319180i \(0.103407\pi\)
−0.947694 + 0.319180i \(0.896593\pi\)
\(258\) 14.7446i 0.917956i
\(259\) −1.87953 0.589907i −0.116788 0.0366551i
\(260\) 0 0
\(261\) 2.67181i 0.165381i
\(262\) 46.9783i 2.90233i
\(263\) 9.80240i 0.604442i −0.953238 0.302221i \(-0.902272\pi\)
0.953238 0.302221i \(-0.0977280\pi\)
\(264\) 19.8614i 1.22239i
\(265\) 7.37228i 0.452876i
\(266\) −15.4410 + 49.1971i −0.946747 + 3.01647i
\(267\) −2.62772 −0.160814
\(268\) −29.4891 −1.80134
\(269\) 25.3723i 1.54698i 0.633811 + 0.773488i \(0.281490\pi\)
−0.633811 + 0.773488i \(0.718510\pi\)
\(270\) 2.52434i 0.153626i
\(271\) 0.792287 0.0481280 0.0240640 0.999710i \(-0.492339\pi\)
0.0240640 + 0.999710i \(0.492339\pi\)
\(272\) −37.2203 −2.25681
\(273\) 0 0
\(274\) 3.16915i 0.191455i
\(275\) 3.31662i 0.200000i
\(276\) 35.4891i 2.13619i
\(277\) 15.7359i 0.945481i 0.881202 + 0.472740i \(0.156735\pi\)
−0.881202 + 0.472740i \(0.843265\pi\)
\(278\) 8.74456i 0.524464i
\(279\) 4.74456i 0.284050i
\(280\) 15.1168 + 4.74456i 0.903404 + 0.283542i
\(281\) 27.1229i 1.61802i 0.587797 + 0.809008i \(0.299995\pi\)
−0.587797 + 0.809008i \(0.700005\pi\)
\(282\) 3.16915i 0.188720i
\(283\) −22.0742 −1.31218 −0.656088 0.754684i \(-0.727790\pi\)
−0.656088 + 0.754684i \(0.727790\pi\)
\(284\) −64.4674 −3.82543
\(285\) −7.72049 −0.457322
\(286\) 0 0
\(287\) −20.7446 6.51087i −1.22451 0.384325i
\(288\) 4.10891i 0.242120i
\(289\) 17.1168 1.00687
\(290\) 6.74456i 0.396054i
\(291\) 2.62772 0.154040
\(292\) −30.2921 −1.77271
\(293\) 25.0410 1.46291 0.731455 0.681889i \(-0.238841\pi\)
0.731455 + 0.681889i \(0.238841\pi\)
\(294\) 14.5012 + 10.0974i 0.845728 + 0.588889i
\(295\) −7.37228 −0.429231
\(296\) 4.45877i 0.259161i
\(297\) 3.31662 0.192450
\(298\) 34.9783 2.02624
\(299\) 0 0
\(300\) 4.37228i 0.252434i
\(301\) −4.62772 + 14.7446i −0.266737 + 0.849862i
\(302\) 4.74456 0.273019
\(303\) 10.3923i 0.597022i
\(304\) −49.1971 −2.82165
\(305\) 1.08724i 0.0622552i
\(306\) 14.7446i 0.842891i
\(307\) −8.80773 −0.502684 −0.251342 0.967898i \(-0.580872\pi\)
−0.251342 + 0.967898i \(0.580872\pi\)
\(308\) 11.4891 36.6060i 0.654654 2.08582i
\(309\) 6.11684 0.347975
\(310\) 11.9769i 0.680241i
\(311\) 16.0000i 0.907277i −0.891186 0.453638i \(-0.850126\pi\)
0.891186 0.453638i \(-0.149874\pi\)
\(312\) 0 0
\(313\) 6.86141i 0.387830i −0.981018 0.193915i \(-0.937881\pi\)
0.981018 0.193915i \(-0.0621185\pi\)
\(314\) −50.7817 −2.86578
\(315\) −0.792287 + 2.52434i −0.0446403 + 0.142230i
\(316\) 59.2945i 3.33558i
\(317\) −13.2554 −0.744500 −0.372250 0.928133i \(-0.621414\pi\)
−0.372250 + 0.928133i \(0.621414\pi\)
\(318\) −18.6101 −1.04360
\(319\) 8.86141 0.496144
\(320\) 2.37228i 0.132615i
\(321\) 15.1460 0.845369
\(322\) −16.2337 + 51.7228i −0.904668 + 2.88240i
\(323\) 45.0951 2.50916
\(324\) −4.37228 −0.242905
\(325\) 0 0
\(326\) 40.9793i 2.26963i
\(327\) −8.51278 −0.470758
\(328\) 49.2119i 2.71727i
\(329\) 0.994667 3.16915i 0.0548377 0.174721i
\(330\) 8.37228 0.460879
\(331\) −15.3723 −0.844937 −0.422468 0.906378i \(-0.638836\pi\)
−0.422468 + 0.906378i \(0.638836\pi\)
\(332\) 26.8280 1.47238
\(333\) 0.744563 0.0408018
\(334\) 59.7228i 3.26789i
\(335\) 6.74456i 0.368495i
\(336\) −5.04868 + 16.0858i −0.275428 + 0.877552i
\(337\) 22.5716i 1.22955i 0.788702 + 0.614776i \(0.210753\pi\)
−0.788702 + 0.614776i \(0.789247\pi\)
\(338\) 32.8164i 1.78498i
\(339\) 7.37228i 0.400407i
\(340\) 25.5383i 1.38501i
\(341\) 15.7359 0.852149
\(342\) 19.4891i 1.05385i
\(343\) 11.3321 + 14.6487i 0.611874 + 0.790955i
\(344\) −34.9783 −1.88590
\(345\) −8.11684 −0.436996
\(346\) 17.4891i 0.940221i
\(347\) 24.2487i 1.30174i −0.759190 0.650870i \(-0.774404\pi\)
0.759190 0.650870i \(-0.225596\pi\)
\(348\) −11.6819 −0.626217
\(349\) −17.5229 −0.937979 −0.468989 0.883204i \(-0.655382\pi\)
−0.468989 + 0.883204i \(0.655382\pi\)
\(350\) −2.00000 + 6.37228i −0.106904 + 0.340613i
\(351\) 0 0
\(352\) 13.6277 0.726360
\(353\) 28.7446i 1.52992i 0.644079 + 0.764959i \(0.277241\pi\)
−0.644079 + 0.764959i \(0.722759\pi\)
\(354\) 18.6101i 0.989117i
\(355\) 14.7446i 0.782560i
\(356\) 11.4891i 0.608922i
\(357\) 4.62772 14.7446i 0.244925 0.780365i
\(358\) 3.75906i 0.198672i
\(359\) 22.5716i 1.19128i 0.803251 + 0.595641i \(0.203102\pi\)
−0.803251 + 0.595641i \(0.796898\pi\)
\(360\) −5.98844 −0.315618
\(361\) 40.6060 2.13716
\(362\) −13.2665 −0.697272
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 6.92820i 0.362639i
\(366\) −2.74456 −0.143461
\(367\) 6.11684i 0.319297i 0.987174 + 0.159648i \(0.0510360\pi\)
−0.987174 + 0.159648i \(0.948964\pi\)
\(368\) −51.7228 −2.69624
\(369\) 8.21782 0.427803
\(370\) 1.87953 0.0977120
\(371\) −18.6101 5.84096i −0.966190 0.303248i
\(372\) −20.7446 −1.07556
\(373\) 18.4077i 0.953117i 0.879143 + 0.476559i \(0.158116\pi\)
−0.879143 + 0.476559i \(0.841884\pi\)
\(374\) −48.9022 −2.52867
\(375\) −1.00000 −0.0516398
\(376\) 7.51811 0.387717
\(377\) 0 0
\(378\) −6.37228 2.00000i −0.327755 0.102869i
\(379\) 24.6277 1.26504 0.632520 0.774544i \(-0.282020\pi\)
0.632520 + 0.774544i \(0.282020\pi\)
\(380\) 33.7562i 1.73165i
\(381\) −19.6974 −1.00913
\(382\) 6.92820i 0.354478i
\(383\) 4.23369i 0.216331i −0.994133 0.108166i \(-0.965502\pi\)
0.994133 0.108166i \(-0.0344977\pi\)
\(384\) 14.2063 0.724960
\(385\) 8.37228 + 2.62772i 0.426691 + 0.133921i
\(386\) −38.2337 −1.94604
\(387\) 5.84096i 0.296913i
\(388\) 11.4891i 0.583272i
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 47.4102 2.39764
\(392\) −23.9538 + 34.4010i −1.20985 + 1.73751i
\(393\) 18.6101i 0.938757i
\(394\) −25.4891 −1.28412
\(395\) 13.5615 0.682351
\(396\) 14.5012i 0.728714i
\(397\) 4.51087i 0.226394i −0.993573 0.113197i \(-0.963891\pi\)
0.993573 0.113197i \(-0.0361092\pi\)
\(398\) −11.9769 −0.600347
\(399\) 6.11684 19.4891i 0.306225 0.975677i
\(400\) −6.37228 −0.318614
\(401\) −27.7228 −1.38441 −0.692206 0.721700i \(-0.743361\pi\)
−0.692206 + 0.721700i \(0.743361\pi\)
\(402\) 17.0256 0.849157
\(403\) 0 0
\(404\) 45.4381 2.26063
\(405\) 1.00000i 0.0496904i
\(406\) −17.0256 5.34363i −0.844964 0.265200i
\(407\) 2.46943i 0.122405i
\(408\) 34.9783 1.73168
\(409\) 34.6410 1.71289 0.856444 0.516240i \(-0.172669\pi\)
0.856444 + 0.516240i \(0.172669\pi\)
\(410\) 20.7446 1.02450
\(411\) 1.25544i 0.0619262i
\(412\) 26.7446i 1.31761i
\(413\) 5.84096 18.6101i 0.287415 0.915745i
\(414\) 20.4897i 1.00701i
\(415\) 6.13592i 0.301200i
\(416\) 0 0
\(417\) 3.46410i 0.169638i
\(418\) −64.6381 −3.16155
\(419\) 4.62772i 0.226079i −0.993591 0.113039i \(-0.963941\pi\)
0.993591 0.113039i \(-0.0360586\pi\)
\(420\) −11.0371 3.46410i −0.538556 0.169031i
\(421\) 18.8614 0.919249 0.459624 0.888113i \(-0.347984\pi\)
0.459624 + 0.888113i \(0.347984\pi\)
\(422\) −8.74456 −0.425679
\(423\) 1.25544i 0.0610415i
\(424\) 44.1485i 2.14404i
\(425\) 5.84096 0.283328
\(426\) 37.2203 1.80333
\(427\) −2.74456 0.861407i −0.132819 0.0416864i
\(428\) 66.2227i 3.20099i
\(429\) 0 0
\(430\) 14.7446i 0.711046i
\(431\) 26.8280i 1.29226i −0.763229 0.646128i \(-0.776387\pi\)
0.763229 0.646128i \(-0.223613\pi\)
\(432\) 6.37228i 0.306587i
\(433\) 32.9783i 1.58483i −0.609980 0.792417i \(-0.708823\pi\)
0.609980 0.792417i \(-0.291177\pi\)
\(434\) −30.2337 9.48913i −1.45126 0.455493i
\(435\) 2.67181i 0.128104i
\(436\) 37.2203i 1.78253i
\(437\) 62.6660 2.99772
\(438\) 17.4891 0.835663
\(439\) 18.8125 0.897872 0.448936 0.893564i \(-0.351803\pi\)
0.448936 + 0.893564i \(0.351803\pi\)
\(440\) 19.8614i 0.946855i
\(441\) −5.74456 4.00000i −0.273551 0.190476i
\(442\) 0 0
\(443\) −34.2337 −1.62649 −0.813246 0.581920i \(-0.802302\pi\)
−0.813246 + 0.581920i \(0.802302\pi\)
\(444\) 3.25544i 0.154496i
\(445\) 2.62772 0.124566
\(446\) −21.7793 −1.03128
\(447\) −13.8564 −0.655386
\(448\) 5.98844 + 1.87953i 0.282927 + 0.0887993i
\(449\) −31.7228 −1.49709 −0.748546 0.663083i \(-0.769248\pi\)
−0.748546 + 0.663083i \(0.769248\pi\)
\(450\) 2.52434i 0.118998i
\(451\) 27.2554i 1.28341i
\(452\) 32.2337 1.51615
\(453\) −1.87953 −0.0883079
\(454\) 3.48913i 0.163753i
\(455\) 0 0
\(456\) 46.2337 2.16509
\(457\) 8.31040i 0.388744i 0.980928 + 0.194372i \(0.0622669\pi\)
−0.980928 + 0.194372i \(0.937733\pi\)
\(458\) 57.4150 2.68282
\(459\) 5.84096i 0.272633i
\(460\) 35.4891i 1.65469i
\(461\) −9.80240 −0.456543 −0.228272 0.973597i \(-0.573307\pi\)
−0.228272 + 0.973597i \(0.573307\pi\)
\(462\) −6.63325 + 21.1345i −0.308607 + 0.983264i
\(463\) −25.4891 −1.18458 −0.592290 0.805725i \(-0.701776\pi\)
−0.592290 + 0.805725i \(0.701776\pi\)
\(464\) 17.0256i 0.790392i
\(465\) 4.74456i 0.220024i
\(466\) −58.9783 −2.73211
\(467\) 38.9783i 1.80370i −0.432051 0.901849i \(-0.642210\pi\)
0.432051 0.901849i \(-0.357790\pi\)
\(468\) 0 0
\(469\) 17.0256 + 5.34363i 0.786167 + 0.246746i
\(470\) 3.16915i 0.146182i
\(471\) 20.1168 0.926935
\(472\) 44.1485 2.03210
\(473\) −19.3723 −0.890738
\(474\) 34.2337i 1.57241i
\(475\) 7.72049 0.354240
\(476\) 64.4674 + 20.2337i 2.95486 + 0.927410i
\(477\) 7.37228 0.337554
\(478\) 39.4891 1.80619
\(479\) −7.51811 −0.343511 −0.171756 0.985140i \(-0.554944\pi\)
−0.171756 + 0.985140i \(0.554944\pi\)
\(480\) 4.10891i 0.187545i
\(481\) 0 0
\(482\) 34.9783i 1.59322i
\(483\) 6.43087 20.4897i 0.292615 0.932312i
\(484\) 48.0951 2.18614
\(485\) −2.62772 −0.119319
\(486\) 2.52434 0.114506
\(487\) −14.9783 −0.678729 −0.339365 0.940655i \(-0.610212\pi\)
−0.339365 + 0.940655i \(0.610212\pi\)
\(488\) 6.51087i 0.294733i
\(489\) 16.2337i 0.734113i
\(490\) −14.5012 10.0974i −0.655098 0.456152i
\(491\) 19.4024i 0.875619i −0.899068 0.437809i \(-0.855754\pi\)
0.899068 0.437809i \(-0.144246\pi\)
\(492\) 35.9306i 1.61988i
\(493\) 15.6060i 0.702858i
\(494\) 0 0
\(495\) −3.31662 −0.149071
\(496\) 30.2337i 1.35753i
\(497\) 37.2203 + 11.6819i 1.66956 + 0.524006i
\(498\) −15.4891 −0.694084
\(499\) −34.3505 −1.53774 −0.768870 0.639405i \(-0.779181\pi\)
−0.768870 + 0.639405i \(0.779181\pi\)
\(500\) 4.37228i 0.195534i
\(501\) 23.6588i 1.05700i
\(502\) 13.8564 0.618442
\(503\) −10.8896 −0.485545 −0.242772 0.970083i \(-0.578057\pi\)
−0.242772 + 0.970083i \(0.578057\pi\)
\(504\) 4.74456 15.1168i 0.211340 0.673358i
\(505\) 10.3923i 0.462451i
\(506\) −67.9565 −3.02103
\(507\) 13.0000i 0.577350i
\(508\) 86.1224i 3.82107i
\(509\) 10.6277i 0.471065i 0.971866 + 0.235533i \(0.0756835\pi\)
−0.971866 + 0.235533i \(0.924317\pi\)
\(510\) 14.7446i 0.652900i
\(511\) 17.4891 + 5.48913i 0.773673 + 0.242825i
\(512\) 50.1369i 2.21576i
\(513\) 7.72049i 0.340868i
\(514\) 25.8333 1.13946
\(515\) −6.11684 −0.269540
\(516\) 25.5383 1.12426
\(517\) 4.16381 0.183124
\(518\) −1.48913 + 4.74456i −0.0654284 + 0.208464i
\(519\) 6.92820i 0.304114i
\(520\) 0 0
\(521\) 22.6277i 0.991338i 0.868511 + 0.495669i \(0.165077\pi\)
−0.868511 + 0.495669i \(0.834923\pi\)
\(522\) 6.74456 0.295201
\(523\) 10.3923 0.454424 0.227212 0.973845i \(-0.427039\pi\)
0.227212 + 0.973845i \(0.427039\pi\)
\(524\) −81.3687 −3.55461
\(525\) 0.792287 2.52434i 0.0345782 0.110171i
\(526\) −24.7446 −1.07891
\(527\) 27.7128i 1.20719i
\(528\) −21.1345 −0.919760
\(529\) 42.8832 1.86449
\(530\) 18.6101 0.808372
\(531\) 7.37228i 0.319930i
\(532\) 85.2119 + 26.7446i 3.69441 + 1.15952i
\(533\) 0 0
\(534\) 6.63325i 0.287049i
\(535\) −15.1460 −0.654820
\(536\) 40.3894i 1.74456i
\(537\) 1.48913i 0.0642605i
\(538\) 64.0482 2.76131
\(539\) −13.2665 + 19.0526i −0.571429 + 0.820652i
\(540\) 4.37228 0.188153
\(541\) 9.50744i 0.408757i −0.978892 0.204378i \(-0.934483\pi\)
0.978892 0.204378i \(-0.0655173\pi\)
\(542\) 2.00000i 0.0859074i
\(543\) 5.25544 0.225532
\(544\) 24.0000i 1.02899i
\(545\) 8.51278 0.364647
\(546\) 0 0
\(547\) 24.4511i 1.04545i −0.852500 0.522727i \(-0.824915\pi\)
0.852500 0.522727i \(-0.175085\pi\)
\(548\) −5.48913 −0.234484
\(549\) 1.08724 0.0464023
\(550\) −8.37228 −0.356995
\(551\) 20.6277i 0.878770i
\(552\) 48.6072 2.06886
\(553\) −10.7446 + 34.2337i −0.456905 + 1.45576i
\(554\) 39.7228 1.68766
\(555\) −0.744563 −0.0316049
\(556\) −15.1460 −0.642335
\(557\) 33.4612i 1.41780i −0.705311 0.708898i \(-0.749193\pi\)
0.705311 0.708898i \(-0.250807\pi\)
\(558\) 11.9769 0.507022
\(559\) 0 0
\(560\) 5.04868 16.0858i 0.213345 0.679749i
\(561\) 19.3723 0.817898
\(562\) 68.4674 2.88812
\(563\) −17.9104 −0.754834 −0.377417 0.926043i \(-0.623188\pi\)
−0.377417 + 0.926043i \(0.623188\pi\)
\(564\) −5.48913 −0.231134
\(565\) 7.37228i 0.310154i
\(566\) 55.7228i 2.34220i
\(567\) 2.52434 + 0.792287i 0.106012 + 0.0332729i
\(568\) 88.2969i 3.70486i
\(569\) 5.25106i 0.220136i −0.993924 0.110068i \(-0.964893\pi\)
0.993924 0.110068i \(-0.0351068\pi\)
\(570\) 19.4891i 0.816310i
\(571\) 10.3923i 0.434904i −0.976071 0.217452i \(-0.930225\pi\)
0.976071 0.217452i \(-0.0697746\pi\)
\(572\) 0 0
\(573\) 2.74456i 0.114656i
\(574\) −16.4356 + 52.3663i −0.686011 + 2.18573i
\(575\) 8.11684 0.338496
\(576\) −2.37228 −0.0988451
\(577\) 42.4674i 1.76794i 0.467544 + 0.883970i \(0.345139\pi\)
−0.467544 + 0.883970i \(0.654861\pi\)
\(578\) 43.2087i 1.79724i
\(579\) 15.1460 0.629447
\(580\) 11.6819 0.485066
\(581\) −15.4891 4.86141i −0.642597 0.201685i
\(582\) 6.63325i 0.274957i
\(583\) 24.4511i 1.01266i
\(584\) 41.4891i 1.71683i
\(585\) 0 0
\(586\) 63.2119i 2.61126i
\(587\) 4.00000i 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) 17.4891 25.1168i 0.721239 1.03580i
\(589\) 36.6303i 1.50933i
\(590\) 18.6101i 0.766167i
\(591\) 10.0974 0.415350
\(592\) −4.74456 −0.195000
\(593\) −40.3894 −1.65859 −0.829297 0.558808i \(-0.811259\pi\)
−0.829297 + 0.558808i \(0.811259\pi\)
\(594\) 8.37228i 0.343519i
\(595\) −4.62772 + 14.7446i −0.189718 + 0.604468i
\(596\) 60.5841i 2.48162i
\(597\) 4.74456 0.194182
\(598\) 0 0
\(599\) 13.7228 0.560699 0.280349 0.959898i \(-0.409550\pi\)
0.280349 + 0.959898i \(0.409550\pi\)
\(600\) 5.98844 0.244477
\(601\) −16.5282 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(602\) 37.2203 + 11.6819i 1.51698 + 0.476120i
\(603\) −6.74456 −0.274660
\(604\) 8.21782i 0.334378i
\(605\) 11.0000i 0.447214i
\(606\) −26.2337 −1.06567
\(607\) 6.04334 0.245292 0.122646 0.992451i \(-0.460862\pi\)
0.122646 + 0.992451i \(0.460862\pi\)
\(608\) 31.7228i 1.28653i
\(609\) 6.74456 + 2.11684i 0.273303 + 0.0857788i
\(610\) 2.74456 0.111124
\(611\) 0 0
\(612\) −25.5383 −1.03233
\(613\) 5.63858i 0.227740i −0.993496 0.113870i \(-0.963675\pi\)
0.993496 0.113870i \(-0.0363248\pi\)
\(614\) 22.2337i 0.897279i
\(615\) −8.21782 −0.331375
\(616\) −50.1369 15.7359i −2.02007 0.634019i
\(617\) 14.7446 0.593594 0.296797 0.954941i \(-0.404082\pi\)
0.296797 + 0.954941i \(0.404082\pi\)
\(618\) 15.4410i 0.621127i
\(619\) 19.4891i 0.783334i −0.920107 0.391667i \(-0.871899\pi\)
0.920107 0.391667i \(-0.128101\pi\)
\(620\) 20.7446 0.833122
\(621\) 8.11684i 0.325718i
\(622\) −40.3894 −1.61947
\(623\) −2.08191 + 6.63325i −0.0834099 + 0.265756i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −17.3205 −0.692267
\(627\) 25.6060 1.02260
\(628\) 87.9565i 3.50985i
\(629\) 4.34896 0.173404
\(630\) 6.37228 + 2.00000i 0.253878 + 0.0796819i
\(631\) 23.6060 0.939739 0.469869 0.882736i \(-0.344301\pi\)
0.469869 + 0.882736i \(0.344301\pi\)
\(632\) −81.2119 −3.23044
\(633\) 3.46410 0.137686
\(634\) 33.4612i 1.32891i
\(635\) 19.6974 0.781666
\(636\) 32.2337i 1.27815i
\(637\) 0 0
\(638\) 22.3692i 0.885604i
\(639\) −14.7446 −0.583286
\(640\) −14.2063 −0.561552
\(641\) 44.7446 1.76730 0.883652 0.468144i \(-0.155077\pi\)
0.883652 + 0.468144i \(0.155077\pi\)
\(642\) 38.2337i 1.50896i
\(643\) 0.627719i 0.0247548i −0.999923 0.0123774i \(-0.996060\pi\)
0.999923 0.0123774i \(-0.00393995\pi\)
\(644\) 89.5865 + 28.1176i 3.53020 + 1.10799i
\(645\) 5.84096i 0.229988i
\(646\) 113.835i 4.47879i
\(647\) 3.76631i 0.148069i 0.997256 + 0.0740345i \(0.0235875\pi\)
−0.997256 + 0.0740345i \(0.976413\pi\)
\(648\) 5.98844i 0.235248i
\(649\) 24.4511 0.959789
\(650\) 0 0
\(651\) 11.9769 + 3.75906i 0.469411 + 0.147329i
\(652\) −70.9783 −2.77972
\(653\) −38.3505 −1.50077 −0.750386 0.661000i \(-0.770132\pi\)
−0.750386 + 0.661000i \(0.770132\pi\)
\(654\) 21.4891i 0.840291i
\(655\) 18.6101i 0.727158i
\(656\) −52.3663 −2.04456
\(657\) −6.92820 −0.270295
\(658\) −8.00000 2.51087i −0.311872 0.0978841i
\(659\) 19.9923i 0.778790i 0.921071 + 0.389395i \(0.127316\pi\)
−0.921071 + 0.389395i \(0.872684\pi\)
\(660\) 14.5012i 0.564459i
\(661\) 20.2337i 0.786999i −0.919325 0.393500i \(-0.871264\pi\)
0.919325 0.393500i \(-0.128736\pi\)
\(662\) 38.8048i 1.50819i
\(663\) 0 0
\(664\) 36.7446i 1.42597i
\(665\) −6.11684 + 19.4891i −0.237201 + 0.755756i
\(666\) 1.87953i 0.0728302i
\(667\) 21.6867i 0.839712i
\(668\) 103.443 4.00233
\(669\) 8.62772 0.333567
\(670\) −17.0256 −0.657755
\(671\) 3.60597i 0.139207i
\(672\) 10.3723 + 3.25544i 0.400119 + 0.125581i
\(673\) 24.7460i 0.953890i −0.878933 0.476945i \(-0.841744\pi\)
0.878933 0.476945i \(-0.158256\pi\)
\(674\) 56.9783 2.19472
\(675\) 1.00000i 0.0384900i
\(676\) 56.8397 2.18614
\(677\) 25.6309 0.985076 0.492538 0.870291i \(-0.336069\pi\)
0.492538 + 0.870291i \(0.336069\pi\)
\(678\) −18.6101 −0.714718
\(679\) 2.08191 6.63325i 0.0798963 0.254561i
\(680\) −34.9783 −1.34135
\(681\) 1.38219i 0.0529658i
\(682\) 39.7228i 1.52107i
\(683\) −8.74456 −0.334601 −0.167301 0.985906i \(-0.553505\pi\)
−0.167301 + 0.985906i \(0.553505\pi\)
\(684\) −33.7562 −1.29070
\(685\) 1.25544i 0.0479678i
\(686\) 36.9783 28.6060i 1.41184 1.09218i
\(687\) −22.7446 −0.867759
\(688\) 37.2203i 1.41901i
\(689\) 0 0
\(690\) 20.4897i 0.780028i
\(691\) 26.4674i 1.00687i 0.864034 + 0.503433i \(0.167930\pi\)
−0.864034 + 0.503433i \(0.832070\pi\)
\(692\) −30.2921 −1.15153
\(693\) 2.62772 8.37228i 0.0998188 0.318037i
\(694\) −61.2119 −2.32357
\(695\) 3.46410i 0.131401i
\(696\) 16.0000i 0.606478i
\(697\) 48.0000 1.81813
\(698\) 44.2337i 1.67427i
\(699\) 23.3639 0.883702
\(700\) 11.0371 + 3.46410i 0.417164 + 0.130931i
\(701\) 18.7027i 0.706391i −0.935549 0.353196i \(-0.885095\pi\)
0.935549 0.353196i \(-0.114905\pi\)
\(702\) 0 0
\(703\) 5.74839 0.216805
\(704\) 7.86797i 0.296535i
\(705\) 1.25544i 0.0472825i
\(706\) 72.5610 2.73087
\(707\) −26.2337 8.23369i −0.986619 0.309660i
\(708\) −32.2337 −1.21142
\(709\) −43.0951 −1.61847 −0.809235 0.587485i \(-0.800118\pi\)
−0.809235 + 0.587485i \(0.800118\pi\)
\(710\) −37.2203 −1.39685
\(711\) 13.5615i 0.508594i
\(712\) −15.7359 −0.589729
\(713\) 38.5109i 1.44224i
\(714\) −37.2203 11.6819i −1.39293 0.437185i
\(715\) 0 0
\(716\) −6.51087 −0.243323
\(717\) −15.6434 −0.584212
\(718\) 56.9783 2.12641
\(719\) 11.3723i 0.424115i −0.977257 0.212057i \(-0.931984\pi\)
0.977257 0.212057i \(-0.0680163\pi\)
\(720\) 6.37228i 0.237481i
\(721\) 4.84630 15.4410i 0.180485 0.575052i
\(722\) 102.503i 3.81477i
\(723\) 13.8564i 0.515325i
\(724\) 22.9783i 0.853980i
\(725\) 2.67181i 0.0992287i
\(726\) −27.7677 −1.03056
\(727\) 42.1168i 1.56203i 0.624514 + 0.781014i \(0.285297\pi\)
−0.624514 + 0.781014i \(0.714703\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −17.4891 −0.647302
\(731\) 34.1168i 1.26186i
\(732\) 4.75372i 0.175703i
\(733\) −45.7330 −1.68919 −0.844594 0.535407i \(-0.820158\pi\)
−0.844594 + 0.535407i \(0.820158\pi\)
\(734\) 15.4410 0.569937
\(735\) 5.74456 + 4.00000i 0.211891 + 0.147542i
\(736\) 33.3514i 1.22935i
\(737\) 22.3692i 0.823979i
\(738\) 20.7446i 0.763618i
\(739\) 16.3258i 0.600556i 0.953852 + 0.300278i \(0.0970794\pi\)
−0.953852 + 0.300278i \(0.902921\pi\)
\(740\) 3.25544i 0.119672i
\(741\) 0 0
\(742\) −14.7446 + 46.9783i −0.541290 + 1.72463i
\(743\) 22.0742i 0.809825i −0.914355 0.404912i \(-0.867302\pi\)
0.914355 0.404912i \(-0.132698\pi\)
\(744\) 28.4125i 1.04165i
\(745\) 13.8564 0.507659
\(746\) 46.4674 1.70129
\(747\) 6.13592 0.224501
\(748\) 84.7011i 3.09698i
\(749\) 12.0000 38.2337i 0.438470 1.39703i
\(750\) 2.52434i 0.0921758i
\(751\) 0.861407 0.0314332 0.0157166 0.999876i \(-0.494997\pi\)
0.0157166 + 0.999876i \(0.494997\pi\)
\(752\) 8.00000i 0.291730i
\(753\) −5.48913 −0.200035
\(754\) 0 0
\(755\) 1.87953 0.0684030
\(756\) −3.46410 + 11.0371i −0.125988 + 0.401416i
\(757\) −0.978251 −0.0355551 −0.0177776 0.999842i \(-0.505659\pi\)
−0.0177776 + 0.999842i \(0.505659\pi\)
\(758\) 62.1687i 2.25807i
\(759\) 26.9205 0.977153
\(760\) −46.2337 −1.67707
\(761\) 1.87953 0.0681328 0.0340664 0.999420i \(-0.489154\pi\)
0.0340664 + 0.999420i \(0.489154\pi\)
\(762\) 49.7228i 1.80127i
\(763\) −6.74456 + 21.4891i −0.244170 + 0.777959i
\(764\) −12.0000 −0.434145
\(765\) 5.84096i 0.211180i
\(766\) −10.6873 −0.386146
\(767\) 0 0
\(768\) 31.1168i 1.12283i
\(769\) −8.60535 −0.310317 −0.155158 0.987890i \(-0.549589\pi\)
−0.155158 + 0.987890i \(0.549589\pi\)
\(770\) 6.63325 21.1345i 0.239046 0.761633i
\(771\) −10.2337 −0.368557
\(772\) 66.2227i 2.38341i
\(773\) 34.4674i 1.23971i −0.784718 0.619853i \(-0.787192\pi\)
0.784718 0.619853i \(-0.212808\pi\)
\(774\) −14.7446 −0.529982
\(775\) 4.74456i 0.170430i
\(776\) 15.7359 0.564887
\(777\) 0.589907 1.87953i 0.0211628 0.0674277i
\(778\) 5.04868i 0.181004i
\(779\) 63.4456 2.27318
\(780\) 0 0
\(781\) 48.9022i 1.74986i
\(782\) 119.679i 4.27972i
\(783\) −2.67181 −0.0954829
\(784\) 36.6060 + 25.4891i 1.30736 + 0.910326i
\(785\) −20.1168 −0.718001
\(786\) 46.9783 1.67566
\(787\) 34.9360 1.24533 0.622666 0.782487i \(-0.286049\pi\)
0.622666 + 0.782487i \(0.286049\pi\)
\(788\) 44.1485i 1.57272i
\(789\) 9.80240 0.348975
\(790\) 34.2337i 1.21798i
\(791\) −18.6101 5.84096i −0.661700 0.207681i
\(792\) 19.8614 0.705744
\(793\) 0 0
\(794\) −11.3870 −0.404108
\(795\) −7.37228 −0.261468
\(796\) 20.7446i 0.735272i
\(797\) 46.0000i 1.62940i 0.579880 + 0.814702i \(0.303099\pi\)
−0.579880 + 0.814702i \(0.696901\pi\)
\(798\) −49.1971 15.4410i −1.74156 0.546605i
\(799\) 7.33296i 0.259422i
\(800\) 4.10891i 0.145272i
\(801\) 2.62772i 0.0928459i
\(802\) 69.9817i 2.47114i
\(803\) 22.9783i 0.810885i
\(804\) 29.4891i 1.04000i
\(805\) −6.43087 + 20.4897i −0.226658 + 0.722165i
\(806\) 0 0
\(807\) −25.3723 −0.893147
\(808\) 62.2337i 2.18937i
\(809\) 13.2665i 0.466425i 0.972426 + 0.233213i \(0.0749238\pi\)
−0.972426 + 0.233213i \(0.925076\pi\)
\(810\) −2.52434 −0.0886962
\(811\) −6.63325 −0.232925 −0.116462 0.993195i \(-0.537155\pi\)
−0.116462 + 0.993195i \(0.537155\pi\)
\(812\) −9.25544 + 29.4891i −0.324802 + 1.03487i
\(813\) 0.792287i 0.0277867i
\(814\) −6.23369 −0.218491
\(815\) 16.2337i 0.568641i
\(816\) 37.2203i 1.30297i
\(817\) 45.0951i 1.57768i
\(818\) 87.4456i 3.05746i
\(819\) 0 0
\(820\) 35.9306i 1.25475i
\(821\) 16.5282i 0.576839i −0.957504 0.288419i \(-0.906870\pi\)
0.957504 0.288419i \(-0.0931297\pi\)
\(822\) 3.16915 0.110537
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 36.6303 1.27608
\(825\) 3.31662 0.115470
\(826\) −46.9783 14.7446i −1.63458 0.513029i
\(827\) 11.9769i 0.416477i 0.978078 + 0.208238i \(0.0667730\pi\)
−0.978078 + 0.208238i \(0.933227\pi\)
\(828\) −35.4891 −1.23333
\(829\) 21.4891i 0.746348i −0.927761 0.373174i \(-0.878269\pi\)
0.927761 0.373174i \(-0.121731\pi\)
\(830\) 15.4891 0.537635
\(831\) −15.7359 −0.545874
\(832\) 0 0
\(833\) −33.5538 23.3639i −1.16257 0.809509i
\(834\) 8.74456 0.302799
\(835\) 23.6588i 0.818747i
\(836\) 111.957i 3.87210i
\(837\) −4.74456 −0.163996
\(838\) −11.6819 −0.403545
\(839\) 6.11684i 0.211177i 0.994410 + 0.105588i \(0.0336726\pi\)
−0.994410 + 0.105588i \(0.966327\pi\)
\(840\) −4.74456 + 15.1168i −0.163703 + 0.521581i
\(841\) 21.8614 0.753842
\(842\) 47.6126i 1.64084i
\(843\) −27.1229 −0.934162
\(844\) 15.1460i 0.521348i
\(845\) 13.0000i 0.447214i
\(846\) 3.16915 0.108958
\(847\) −27.7677 8.71516i −0.954110 0.299456i
\(848\) −46.9783 −1.61324
\(849\) 22.0742i 0.757586i
\(850\) 14.7446i 0.505734i
\(851\) 6.04350 0.207168
\(852\) 64.4674i 2.20862i
\(853\) −38.4001 −1.31479 −0.657397 0.753545i \(-0.728342\pi\)
−0.657397 + 0.753545i \(0.728342\pi\)
\(854\) −2.17448 + 6.92820i −0.0744092 + 0.237078i
\(855\) 7.72049i 0.264035i
\(856\) 90.7011 3.10010
\(857\) −20.7846 −0.709989 −0.354994 0.934868i \(-0.615517\pi\)
−0.354994 + 0.934868i \(0.615517\pi\)
\(858\) 0 0
\(859\) 47.7228i 1.62828i −0.580668 0.814141i \(-0.697208\pi\)
0.580668 0.814141i \(-0.302792\pi\)
\(860\) −25.5383 −0.870850
\(861\) 6.51087 20.7446i 0.221890 0.706973i
\(862\) −67.7228 −2.30665
\(863\) −44.1168 −1.50175 −0.750877 0.660442i \(-0.770369\pi\)
−0.750877 + 0.660442i \(0.770369\pi\)
\(864\) −4.10891 −0.139788
\(865\) 6.92820i 0.235566i
\(866\) −83.2482 −2.82889
\(867\) 17.1168i 0.581318i
\(868\) −16.4356 + 52.3663i −0.557862 + 1.77743i
\(869\) −44.9783 −1.52578
\(870\) −6.74456 −0.228662
\(871\) 0 0
\(872\) −50.9783 −1.72634
\(873\) 2.62772i 0.0889348i
\(874\) 158.190i 5.35086i
\(875\) −0.792287 + 2.52434i −0.0267842 + 0.0853382i
\(876\) 30.2921i 1.02347i
\(877\) 13.6540i 0.461064i −0.973065 0.230532i \(-0.925953\pi\)
0.973065 0.230532i \(-0.0740466\pi\)
\(878\) 47.4891i 1.60268i
\(879\) 25.0410i 0.844612i
\(880\) 21.1345 0.712443
\(881\) 1.37228i 0.0462333i 0.999733 + 0.0231167i \(0.00735892\pi\)
−0.999733 + 0.0231167i \(0.992641\pi\)
\(882\) −10.0974 + 14.5012i −0.339996 + 0.488281i
\(883\) −49.7228 −1.67331 −0.836653 0.547733i \(-0.815491\pi\)
−0.836653 + 0.547733i \(0.815491\pi\)
\(884\) 0 0
\(885\) 7.37228i 0.247817i
\(886\) 86.4174i 2.90325i
\(887\) −40.7769 −1.36916 −0.684578 0.728940i \(-0.740013\pi\)
−0.684578 + 0.728940i \(0.740013\pi\)
\(888\) 4.45877 0.149626
\(889\) −15.6060 + 49.7228i −0.523407 + 1.66765i
\(890\) 6.63325i 0.222347i
\(891\) 3.31662i 0.111111i
\(892\) 37.7228i 1.26305i
\(893\) 9.69259i 0.324350i
\(894\) 34.9783i 1.16985i
\(895\) 1.48913i 0.0497760i
\(896\) 11.2554 35.8614i 0.376018 1.19805i
\(897\) 0 0
\(898\) 80.0791i 2.67227i
\(899\) −12.6766 −0.422788
\(900\) −4.37228 −0.145743
\(901\) 43.0612 1.43458
\(902\) −68.8019 −2.29085
\(903\) −14.7446 4.62772i −0.490668 0.154001i
\(904\) 44.1485i 1.46836i
\(905\) −5.25544 −0.174697
\(906\) 4.74456i 0.157628i
\(907\) 24.4674 0.812426 0.406213 0.913778i \(-0.366849\pi\)
0.406213 + 0.913778i \(0.366849\pi\)
\(908\) −6.04334 −0.200555
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) −13.2554 −0.439172 −0.219586 0.975593i \(-0.570471\pi\)
−0.219586 + 0.975593i \(0.570471\pi\)
\(912\) 49.1971i 1.62908i
\(913\) 20.3505i 0.673504i
\(914\) 20.9783 0.693899
\(915\) −1.08724 −0.0359431
\(916\) 99.4456i 3.28578i
\(917\) 46.9783 + 14.7446i 1.55136 + 0.486908i
\(918\) 14.7446 0.486643
\(919\) 29.9971i 0.989513i −0.869032 0.494757i \(-0.835257\pi\)
0.869032 0.494757i \(-0.164743\pi\)
\(920\) −48.6072 −1.60253
\(921\) 8.80773i 0.290225i
\(922\) 24.7446i 0.814919i
\(923\) 0 0
\(924\) 36.6060 + 11.4891i 1.20425 + 0.377964i
\(925\) 0.744563 0.0244811
\(926\) 64.3432i 2.11445i
\(927\) 6.11684i 0.200904i
\(928\) −10.9783 −0.360379
\(929\) 59.4891i 1.95177i 0.218274 + 0.975887i \(0.429957\pi\)
−0.218274 + 0.975887i \(0.570043\pi\)
\(930\) −11.9769 −0.392737
\(931\) −44.3508 30.8820i −1.45354 1.01212i
\(932\) 102.153i 3.34614i
\(933\) 16.0000 0.523816
\(934\) −98.3943 −3.21956
\(935\) −19.3723 −0.633541
\(936\) 0 0
\(937\) −7.92287 −0.258829 −0.129414 0.991591i \(-0.541310\pi\)
−0.129414 + 0.991591i \(0.541310\pi\)
\(938\) 13.4891 42.9783i 0.440436 1.40329i
\(939\) 6.86141 0.223914
\(940\) 5.48913 0.179036
\(941\) −26.4232 −0.861371 −0.430686 0.902502i \(-0.641728\pi\)
−0.430686 + 0.902502i \(0.641728\pi\)
\(942\) 50.7817i 1.65456i
\(943\) 66.7028 2.17214
\(944\) 46.9783i 1.52901i
\(945\) −2.52434 0.792287i −0.0821167 0.0257731i
\(946\) 48.9022i 1.58995i
\(947\) 48.3505 1.57118 0.785591 0.618747i \(-0.212359\pi\)
0.785591 + 0.618747i \(0.212359\pi\)
\(948\) 59.2945 1.92580
\(949\) 0 0
\(950\) 19.4891i 0.632311i
\(951\) 13.2554i 0.429837i
\(952\) 27.7128 88.2969i 0.898177 2.86172i
\(953\) 41.3841i 1.34056i 0.742108 + 0.670281i \(0.233826\pi\)
−0.742108 + 0.670281i \(0.766174\pi\)
\(954\) 18.6101i 0.602525i
\(955\) 2.74456i 0.0888120i
\(956\) 68.3972i 2.21212i
\(957\) 8.86141i 0.286449i
\(958\) 18.9783i 0.613159i
\(959\) 3.16915 + 0.994667i 0.102337 + 0.0321195i
\(960\) 2.37228 0.0765651
\(961\) 8.48913 0.273843
\(962\) 0 0
\(963\) 15.1460i 0.488074i
\(964\) 60.5841 1.95128
\(965\) −15.1460 −0.487568
\(966\) −51.7228 16.2337i −1.66415 0.522310i
\(967\) 8.01544i 0.257759i 0.991660 + 0.128880i \(0.0411381\pi\)
−0.991660 + 0.128880i \(0.958862\pi\)
\(968\) 65.8728i 2.11723i
\(969\) 45.0951i 1.44866i
\(970\) 6.63325i 0.212981i
\(971\) 7.37228i 0.236588i −0.992979 0.118294i \(-0.962258\pi\)
0.992979 0.118294i \(-0.0377425\pi\)
\(972\) 4.37228i 0.140241i
\(973\) 8.74456 + 2.74456i 0.280338 + 0.0879866i
\(974\) 37.8102i 1.21152i
\(975\) 0 0
\(976\) −6.92820 −0.221766
\(977\) 33.0951 1.05881 0.529403 0.848371i \(-0.322416\pi\)
0.529403 + 0.848371i \(0.322416\pi\)
\(978\) 40.9793 1.31037
\(979\) −8.71516 −0.278538
\(980\) −17.4891 + 25.1168i −0.558670 + 0.802328i
\(981\) 8.51278i 0.271792i
\(982\) −48.9783 −1.56296
\(983\) 40.4674i 1.29071i 0.763883 + 0.645354i \(0.223290\pi\)
−0.763883 + 0.645354i \(0.776710\pi\)
\(984\) 49.2119 1.56882
\(985\) −10.0974 −0.321728
\(986\) 39.3947 1.25458
\(987\) 3.16915 + 0.994667i 0.100875 + 0.0316606i
\(988\) 0 0
\(989\) 47.4102i 1.50756i
\(990\) 8.37228i 0.266089i
\(991\) −44.6277 −1.41765 −0.708823 0.705386i \(-0.750774\pi\)
−0.708823 + 0.705386i \(0.750774\pi\)
\(992\) −19.4950 −0.618967
\(993\) 15.3723i 0.487825i
\(994\) 29.4891 93.9565i 0.935338 2.98012i
\(995\) −4.74456 −0.150413
\(996\) 26.8280i 0.850076i
\(997\) −39.3947 −1.24764 −0.623822 0.781566i \(-0.714421\pi\)
−0.623822 + 0.781566i \(0.714421\pi\)
\(998\) 86.7123i 2.74483i
\(999\) 0.744563i 0.0235569i
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 1155.2.i.a.76.2 yes 8
7.6 odd 2 inner 1155.2.i.a.76.1 8
11.10 odd 2 inner 1155.2.i.a.76.8 yes 8
77.76 even 2 inner 1155.2.i.a.76.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.i.a.76.1 8 7.6 odd 2 inner
1155.2.i.a.76.2 yes 8 1.1 even 1 trivial
1155.2.i.a.76.7 yes 8 77.76 even 2 inner
1155.2.i.a.76.8 yes 8 11.10 odd 2 inner