Properties

Label 731.2.d.b.560.4
Level $731$
Weight $2$
Character 731.560
Analytic conductor $5.837$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [731,2,Mod(560,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.560");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.d (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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 129x^{4} + 323x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 560.4
Root \(-1.21991i\) of defining polynomial
Character \(\chi\) \(=\) 731.560
Dual form 731.2.d.b.560.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.21991i q^{3} -2.00000 q^{4} +3.81798i q^{5} -4.55222i q^{7} +1.51183 q^{9} +O(q^{10})\) \(q-1.21991i q^{3} -2.00000 q^{4} +3.81798i q^{5} -4.55222i q^{7} +1.51183 q^{9} -0.600184i q^{11} +2.43981i q^{12} -3.14574 q^{13} +4.65757 q^{15} +4.00000 q^{16} +(-0.203767 + 4.11807i) q^{17} +2.00000 q^{19} -7.63595i q^{20} -5.55328 q^{21} -6.06444i q^{23} -9.57694 q^{25} -5.50401i q^{27} +9.10443i q^{28} -7.63595i q^{29} -10.6759i q^{31} -0.732168 q^{33} +17.3803 q^{35} -3.02366 q^{36} -5.44471i q^{37} +3.83751i q^{39} -0.371145i q^{41} +1.00000 q^{43} +1.20037i q^{44} +5.77212i q^{45} +0.958550 q^{47} -4.87963i q^{48} -13.7227 q^{49} +(5.02366 + 0.248577i) q^{51} +6.29149 q^{52} +7.36608 q^{53} +2.29149 q^{55} -2.43981i q^{57} +6.57694 q^{59} -9.31515 q^{60} -6.23825i q^{61} -6.88217i q^{63} -8.00000 q^{64} -12.0104i q^{65} -8.76413 q^{67} +(0.407535 - 8.23613i) q^{68} -7.39805 q^{69} +7.47769i q^{71} -2.86618i q^{73} +11.6830i q^{75} -4.00000 q^{76} -2.73217 q^{77} +12.2475i q^{79} +15.2719i q^{80} -2.17889 q^{81} -5.21085 q^{83} +11.1066 q^{84} +(-15.7227 - 0.777978i) q^{85} -9.31515 q^{87} +10.4217 q^{89} +14.3201i q^{91} +12.1289i q^{92} -13.0237 q^{93} +7.63595i q^{95} -1.57151i q^{97} -0.907375i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 16 q^{9} - 10 q^{13} - 6 q^{15} + 32 q^{16} - 3 q^{17} + 16 q^{19} - 32 q^{21} - 8 q^{25} + 20 q^{33} + 12 q^{35} + 32 q^{36} + 8 q^{43} - 8 q^{47} - 26 q^{49} - 16 q^{51} + 20 q^{52} + 46 q^{53} - 12 q^{55} - 16 q^{59} + 12 q^{60} - 64 q^{64} - 2 q^{67} + 6 q^{68} - 4 q^{69} - 32 q^{76} + 4 q^{77} - 4 q^{81} + 14 q^{83} + 64 q^{84} - 42 q^{85} + 12 q^{87} - 28 q^{89} - 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(562\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.21991i 0.704313i −0.935941 0.352157i \(-0.885448\pi\)
0.935941 0.352157i \(-0.114552\pi\)
\(4\) −2.00000 −1.00000
\(5\) 3.81798i 1.70745i 0.520724 + 0.853725i \(0.325662\pi\)
−0.520724 + 0.853725i \(0.674338\pi\)
\(6\) 0 0
\(7\) 4.55222i 1.72058i −0.509808 0.860288i \(-0.670284\pi\)
0.509808 0.860288i \(-0.329716\pi\)
\(8\) 0 0
\(9\) 1.51183 0.503943
\(10\) 0 0
\(11\) 0.600184i 0.180962i −0.995898 0.0904812i \(-0.971160\pi\)
0.995898 0.0904812i \(-0.0288405\pi\)
\(12\) 2.43981i 0.704313i
\(13\) −3.14574 −0.872472 −0.436236 0.899832i \(-0.643689\pi\)
−0.436236 + 0.899832i \(0.643689\pi\)
\(14\) 0 0
\(15\) 4.65757 1.20258
\(16\) 4.00000 1.00000
\(17\) −0.203767 + 4.11807i −0.0494208 + 0.998778i
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 7.63595i 1.70745i
\(21\) −5.55328 −1.21182
\(22\) 0 0
\(23\) 6.06444i 1.26452i −0.774755 0.632261i \(-0.782127\pi\)
0.774755 0.632261i \(-0.217873\pi\)
\(24\) 0 0
\(25\) −9.57694 −1.91539
\(26\) 0 0
\(27\) 5.50401i 1.05925i
\(28\) 9.10443i 1.72058i
\(29\) 7.63595i 1.41796i −0.705228 0.708980i \(-0.749156\pi\)
0.705228 0.708980i \(-0.250844\pi\)
\(30\) 0 0
\(31\) 10.6759i 1.91746i −0.284324 0.958728i \(-0.591769\pi\)
0.284324 0.958728i \(-0.408231\pi\)
\(32\) 0 0
\(33\) −0.732168 −0.127454
\(34\) 0 0
\(35\) 17.3803 2.93780
\(36\) −3.02366 −0.503943
\(37\) 5.44471i 0.895106i −0.894258 0.447553i \(-0.852296\pi\)
0.894258 0.447553i \(-0.147704\pi\)
\(38\) 0 0
\(39\) 3.83751i 0.614494i
\(40\) 0 0
\(41\) 0.371145i 0.0579632i −0.999580 0.0289816i \(-0.990774\pi\)
0.999580 0.0289816i \(-0.00922641\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) 1.20037i 0.180962i
\(45\) 5.77212i 0.860457i
\(46\) 0 0
\(47\) 0.958550 0.139819 0.0699094 0.997553i \(-0.477729\pi\)
0.0699094 + 0.997553i \(0.477729\pi\)
\(48\) 4.87963i 0.704313i
\(49\) −13.7227 −1.96038
\(50\) 0 0
\(51\) 5.02366 + 0.248577i 0.703453 + 0.0348077i
\(52\) 6.29149 0.872472
\(53\) 7.36608 1.01181 0.505905 0.862589i \(-0.331159\pi\)
0.505905 + 0.862589i \(0.331159\pi\)
\(54\) 0 0
\(55\) 2.29149 0.308984
\(56\) 0 0
\(57\) 2.43981i 0.323161i
\(58\) 0 0
\(59\) 6.57694 0.856244 0.428122 0.903721i \(-0.359175\pi\)
0.428122 + 0.903721i \(0.359175\pi\)
\(60\) −9.31515 −1.20258
\(61\) 6.23825i 0.798726i −0.916793 0.399363i \(-0.869231\pi\)
0.916793 0.399363i \(-0.130769\pi\)
\(62\) 0 0
\(63\) 6.88217i 0.867072i
\(64\) −8.00000 −1.00000
\(65\) 12.0104i 1.48970i
\(66\) 0 0
\(67\) −8.76413 −1.07071 −0.535354 0.844628i \(-0.679822\pi\)
−0.535354 + 0.844628i \(0.679822\pi\)
\(68\) 0.407535 8.23613i 0.0494208 0.998778i
\(69\) −7.39805 −0.890620
\(70\) 0 0
\(71\) 7.47769i 0.887439i 0.896166 + 0.443720i \(0.146341\pi\)
−0.896166 + 0.443720i \(0.853659\pi\)
\(72\) 0 0
\(73\) 2.86618i 0.335461i −0.985833 0.167731i \(-0.946356\pi\)
0.985833 0.167731i \(-0.0536439\pi\)
\(74\) 0 0
\(75\) 11.6830i 1.34903i
\(76\) −4.00000 −0.458831
\(77\) −2.73217 −0.311359
\(78\) 0 0
\(79\) 12.2475i 1.37795i 0.724787 + 0.688974i \(0.241938\pi\)
−0.724787 + 0.688974i \(0.758062\pi\)
\(80\) 15.2719i 1.70745i
\(81\) −2.17889 −0.242099
\(82\) 0 0
\(83\) −5.21085 −0.571965 −0.285983 0.958235i \(-0.592320\pi\)
−0.285983 + 0.958235i \(0.592320\pi\)
\(84\) 11.1066 1.21182
\(85\) −15.7227 0.777978i −1.70536 0.0843836i
\(86\) 0 0
\(87\) −9.31515 −0.998688
\(88\) 0 0
\(89\) 10.4217 1.10470 0.552349 0.833613i \(-0.313732\pi\)
0.552349 + 0.833613i \(0.313732\pi\)
\(90\) 0 0
\(91\) 14.3201i 1.50116i
\(92\) 12.1289i 1.26452i
\(93\) −13.0237 −1.35049
\(94\) 0 0
\(95\) 7.63595i 0.783432i
\(96\) 0 0
\(97\) 1.57151i 0.159563i −0.996812 0.0797815i \(-0.974578\pi\)
0.996812 0.0797815i \(-0.0254223\pi\)
\(98\) 0 0
\(99\) 0.907375i 0.0911947i
\(100\) 19.1539 1.91539
\(101\) 5.73217 0.570372 0.285186 0.958472i \(-0.407945\pi\)
0.285186 + 0.958472i \(0.407945\pi\)
\(102\) 0 0
\(103\) −8.39578 −0.827261 −0.413630 0.910445i \(-0.635739\pi\)
−0.413630 + 0.910445i \(0.635739\pi\)
\(104\) 0 0
\(105\) 21.2023i 2.06913i
\(106\) 0 0
\(107\) 7.86499i 0.760337i −0.924917 0.380169i \(-0.875866\pi\)
0.924917 0.380169i \(-0.124134\pi\)
\(108\) 11.0080i 1.05925i
\(109\) 12.2319i 1.17160i 0.810454 + 0.585802i \(0.199220\pi\)
−0.810454 + 0.585802i \(0.800780\pi\)
\(110\) 0 0
\(111\) −6.64204 −0.630435
\(112\) 18.2089i 1.72058i
\(113\) 16.3140i 1.53469i 0.641233 + 0.767347i \(0.278423\pi\)
−0.641233 + 0.767347i \(0.721577\pi\)
\(114\) 0 0
\(115\) 23.1539 2.15911
\(116\) 15.2719i 1.41796i
\(117\) −4.75583 −0.439676
\(118\) 0 0
\(119\) 18.7463 + 0.927593i 1.71847 + 0.0850323i
\(120\) 0 0
\(121\) 10.6398 0.967253
\(122\) 0 0
\(123\) −0.452762 −0.0408242
\(124\) 21.3519i 1.91746i
\(125\) 17.4746i 1.56298i
\(126\) 0 0
\(127\) −5.81281 −0.515803 −0.257902 0.966171i \(-0.583031\pi\)
−0.257902 + 0.966171i \(0.583031\pi\)
\(128\) 0 0
\(129\) 1.21991i 0.107407i
\(130\) 0 0
\(131\) 15.9468i 1.39328i −0.717420 0.696641i \(-0.754677\pi\)
0.717420 0.696641i \(-0.245323\pi\)
\(132\) 1.46434 0.127454
\(133\) 9.10443i 0.789455i
\(134\) 0 0
\(135\) 21.0142 1.80861
\(136\) 0 0
\(137\) −19.1539 −1.63643 −0.818213 0.574915i \(-0.805035\pi\)
−0.818213 + 0.574915i \(0.805035\pi\)
\(138\) 0 0
\(139\) 7.53292i 0.638934i −0.947597 0.319467i \(-0.896496\pi\)
0.947597 0.319467i \(-0.103504\pi\)
\(140\) −34.7605 −2.93780
\(141\) 1.16934i 0.0984763i
\(142\) 0 0
\(143\) 1.88803i 0.157885i
\(144\) 6.04731 0.503943
\(145\) 29.1539 2.42110
\(146\) 0 0
\(147\) 16.7404i 1.38072i
\(148\) 10.8894i 0.895106i
\(149\) 5.18493 0.424766 0.212383 0.977186i \(-0.431878\pi\)
0.212383 + 0.977186i \(0.431878\pi\)
\(150\) 0 0
\(151\) 15.1539 1.23320 0.616602 0.787275i \(-0.288509\pi\)
0.616602 + 0.787275i \(0.288509\pi\)
\(152\) 0 0
\(153\) −0.308061 + 6.22581i −0.0249053 + 0.503327i
\(154\) 0 0
\(155\) 40.7605 3.27396
\(156\) 7.67503i 0.614494i
\(157\) 19.4454 1.55191 0.775954 0.630789i \(-0.217269\pi\)
0.775954 + 0.630789i \(0.217269\pi\)
\(158\) 0 0
\(159\) 8.98593i 0.712631i
\(160\) 0 0
\(161\) −27.6066 −2.17571
\(162\) 0 0
\(163\) 17.1277i 1.34154i 0.741664 + 0.670772i \(0.234037\pi\)
−0.741664 + 0.670772i \(0.765963\pi\)
\(164\) 0.742291i 0.0579632i
\(165\) 2.79540i 0.217622i
\(166\) 0 0
\(167\) 6.06444i 0.469280i −0.972082 0.234640i \(-0.924609\pi\)
0.972082 0.234640i \(-0.0753912\pi\)
\(168\) 0 0
\(169\) −3.10429 −0.238792
\(170\) 0 0
\(171\) 3.02366 0.231225
\(172\) −2.00000 −0.152499
\(173\) 19.6774i 1.49604i 0.663675 + 0.748021i \(0.268996\pi\)
−0.663675 + 0.748021i \(0.731004\pi\)
\(174\) 0 0
\(175\) 43.5963i 3.29557i
\(176\) 2.40074i 0.180962i
\(177\) 8.02325i 0.603064i
\(178\) 0 0
\(179\) 2.73217 0.204212 0.102106 0.994774i \(-0.467442\pi\)
0.102106 + 0.994774i \(0.467442\pi\)
\(180\) 11.5442i 0.860457i
\(181\) 21.1229i 1.57005i −0.619464 0.785025i \(-0.712650\pi\)
0.619464 0.785025i \(-0.287350\pi\)
\(182\) 0 0
\(183\) −7.61008 −0.562553
\(184\) 0 0
\(185\) 20.7878 1.52835
\(186\) 0 0
\(187\) 2.47160 + 0.122298i 0.180741 + 0.00894331i
\(188\) −1.91710 −0.139819
\(189\) −25.0554 −1.82252
\(190\) 0 0
\(191\) −19.9689 −1.44490 −0.722451 0.691422i \(-0.756984\pi\)
−0.722451 + 0.691422i \(0.756984\pi\)
\(192\) 9.75925i 0.704313i
\(193\) 24.7084i 1.77855i 0.457373 + 0.889275i \(0.348790\pi\)
−0.457373 + 0.889275i \(0.651210\pi\)
\(194\) 0 0
\(195\) −14.6515 −1.04922
\(196\) 27.4454 1.96038
\(197\) 16.2432i 1.15728i −0.815582 0.578641i \(-0.803583\pi\)
0.815582 0.578641i \(-0.196417\pi\)
\(198\) 0 0
\(199\) 3.88876i 0.275667i −0.990455 0.137833i \(-0.955986\pi\)
0.990455 0.137833i \(-0.0440138\pi\)
\(200\) 0 0
\(201\) 10.6914i 0.754114i
\(202\) 0 0
\(203\) −34.7605 −2.43971
\(204\) −10.0473 0.497154i −0.703453 0.0348077i
\(205\) 1.41702 0.0989692
\(206\) 0 0
\(207\) 9.16839i 0.637247i
\(208\) −12.5830 −0.872472
\(209\) 1.20037i 0.0830312i
\(210\) 0 0
\(211\) 4.13449i 0.284630i 0.989821 + 0.142315i \(0.0454546\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(212\) −14.7322 −1.01181
\(213\) 9.12209 0.625035
\(214\) 0 0
\(215\) 3.81798i 0.260384i
\(216\) 0 0
\(217\) −48.5992 −3.29913
\(218\) 0 0
\(219\) −3.49648 −0.236270
\(220\) −4.58298 −0.308984
\(221\) 0.641000 12.9544i 0.0431183 0.871406i
\(222\) 0 0
\(223\) −4.20859 −0.281828 −0.140914 0.990022i \(-0.545004\pi\)
−0.140914 + 0.990022i \(0.545004\pi\)
\(224\) 0 0
\(225\) −14.4787 −0.965246
\(226\) 0 0
\(227\) 23.5627i 1.56391i −0.623335 0.781955i \(-0.714223\pi\)
0.623335 0.781955i \(-0.285777\pi\)
\(228\) 4.87963i 0.323161i
\(229\) 4.18115 0.276298 0.138149 0.990411i \(-0.455885\pi\)
0.138149 + 0.990411i \(0.455885\pi\)
\(230\) 0 0
\(231\) 3.33299i 0.219295i
\(232\) 0 0
\(233\) 13.0018i 0.851778i 0.904775 + 0.425889i \(0.140039\pi\)
−0.904775 + 0.425889i \(0.859961\pi\)
\(234\) 0 0
\(235\) 3.65972i 0.238734i
\(236\) −13.1539 −0.856244
\(237\) 14.9408 0.970506
\(238\) 0 0
\(239\) −8.59247 −0.555800 −0.277900 0.960610i \(-0.589638\pi\)
−0.277900 + 0.960610i \(0.589638\pi\)
\(240\) 18.6303 1.20258
\(241\) 2.53818i 0.163499i 0.996653 + 0.0817494i \(0.0260507\pi\)
−0.996653 + 0.0817494i \(0.973949\pi\)
\(242\) 0 0
\(243\) 13.8540i 0.888733i
\(244\) 12.4765i 0.798726i
\(245\) 52.3929i 3.34726i
\(246\) 0 0
\(247\) −6.29149 −0.400318
\(248\) 0 0
\(249\) 6.35675i 0.402843i
\(250\) 0 0
\(251\) 15.8707 1.00175 0.500874 0.865520i \(-0.333012\pi\)
0.500874 + 0.865520i \(0.333012\pi\)
\(252\) 13.7643i 0.867072i
\(253\) −3.63978 −0.228831
\(254\) 0 0
\(255\) −0.949061 + 19.1802i −0.0594325 + 1.20111i
\(256\) 16.0000 1.00000
\(257\) −7.15387 −0.446246 −0.223123 0.974790i \(-0.571625\pi\)
−0.223123 + 0.974790i \(0.571625\pi\)
\(258\) 0 0
\(259\) −24.7855 −1.54010
\(260\) 24.0207i 1.48970i
\(261\) 11.5442i 0.714571i
\(262\) 0 0
\(263\) 0.338802 0.0208914 0.0104457 0.999945i \(-0.496675\pi\)
0.0104457 + 0.999945i \(0.496675\pi\)
\(264\) 0 0
\(265\) 28.1235i 1.72761i
\(266\) 0 0
\(267\) 12.7135i 0.778054i
\(268\) 17.5283 1.07071
\(269\) 13.9294i 0.849292i 0.905359 + 0.424646i \(0.139601\pi\)
−0.905359 + 0.424646i \(0.860399\pi\)
\(270\) 0 0
\(271\) 24.5911 1.49380 0.746902 0.664935i \(-0.231541\pi\)
0.746902 + 0.664935i \(0.231541\pi\)
\(272\) −0.815069 + 16.4723i −0.0494208 + 0.998778i
\(273\) 17.4692 1.05728
\(274\) 0 0
\(275\) 5.74792i 0.346613i
\(276\) 14.7961 0.890620
\(277\) 5.21568i 0.313380i −0.987648 0.156690i \(-0.949918\pi\)
0.987648 0.156690i \(-0.0500823\pi\)
\(278\) 0 0
\(279\) 16.1402i 0.966289i
\(280\) 0 0
\(281\) −10.2523 −0.611601 −0.305801 0.952096i \(-0.598924\pi\)
−0.305801 + 0.952096i \(0.598924\pi\)
\(282\) 0 0
\(283\) 15.3749i 0.913945i −0.889481 0.456972i \(-0.848934\pi\)
0.889481 0.456972i \(-0.151066\pi\)
\(284\) 14.9554i 0.887439i
\(285\) 9.31515 0.551782
\(286\) 0 0
\(287\) −1.68953 −0.0997300
\(288\) 0 0
\(289\) −16.9170 1.67825i −0.995115 0.0987209i
\(290\) 0 0
\(291\) −1.91710 −0.112382
\(292\) 5.73237i 0.335461i
\(293\) −4.22260 −0.246687 −0.123344 0.992364i \(-0.539362\pi\)
−0.123344 + 0.992364i \(0.539362\pi\)
\(294\) 0 0
\(295\) 25.1106i 1.46199i
\(296\) 0 0
\(297\) −3.30342 −0.191684
\(298\) 0 0
\(299\) 19.0772i 1.10326i
\(300\) 23.3659i 1.34903i
\(301\) 4.55222i 0.262385i
\(302\) 0 0
\(303\) 6.99271i 0.401721i
\(304\) 8.00000 0.458831
\(305\) 23.8175 1.36379
\(306\) 0 0
\(307\) −2.91710 −0.166488 −0.0832438 0.996529i \(-0.526528\pi\)
−0.0832438 + 0.996529i \(0.526528\pi\)
\(308\) 5.46434 0.311359
\(309\) 10.2421i 0.582651i
\(310\) 0 0
\(311\) 18.9276i 1.07329i 0.843809 + 0.536644i \(0.180308\pi\)
−0.843809 + 0.536644i \(0.819692\pi\)
\(312\) 0 0
\(313\) 2.57853i 0.145747i 0.997341 + 0.0728736i \(0.0232170\pi\)
−0.997341 + 0.0728736i \(0.976783\pi\)
\(314\) 0 0
\(315\) 26.2760 1.48048
\(316\) 24.4949i 1.37795i
\(317\) 3.97225i 0.223104i −0.993759 0.111552i \(-0.964418\pi\)
0.993759 0.111552i \(-0.0355821\pi\)
\(318\) 0 0
\(319\) −4.58298 −0.256597
\(320\) 30.5438i 1.70745i
\(321\) −9.59455 −0.535516
\(322\) 0 0
\(323\) −0.407535 + 8.23613i −0.0226758 + 0.458271i
\(324\) 4.35778 0.242099
\(325\) 30.1266 1.67112
\(326\) 0 0
\(327\) 14.9218 0.825176
\(328\) 0 0
\(329\) 4.36353i 0.240569i
\(330\) 0 0
\(331\) 2.20859 0.121395 0.0606975 0.998156i \(-0.480668\pi\)
0.0606975 + 0.998156i \(0.480668\pi\)
\(332\) 10.4217 0.571965
\(333\) 8.23147i 0.451082i
\(334\) 0 0
\(335\) 33.4612i 1.82818i
\(336\) −22.2131 −1.21182
\(337\) 7.76196i 0.422821i 0.977397 + 0.211410i \(0.0678057\pi\)
−0.977397 + 0.211410i \(0.932194\pi\)
\(338\) 0 0
\(339\) 19.9016 1.08090
\(340\) 31.4454 + 1.55596i 1.70536 + 0.0843836i
\(341\) −6.40753 −0.346987
\(342\) 0 0
\(343\) 30.6031i 1.65241i
\(344\) 0 0
\(345\) 28.2456i 1.52069i
\(346\) 0 0
\(347\) 10.5332i 0.565454i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912391\pi\)
\(348\) 18.6303 0.998688
\(349\) 20.8745 1.11738 0.558692 0.829375i \(-0.311303\pi\)
0.558692 + 0.829375i \(0.311303\pi\)
\(350\) 0 0
\(351\) 17.3142i 0.924164i
\(352\) 0 0
\(353\) −1.94302 −0.103417 −0.0517083 0.998662i \(-0.516467\pi\)
−0.0517083 + 0.998662i \(0.516467\pi\)
\(354\) 0 0
\(355\) −28.5497 −1.51526
\(356\) −20.8434 −1.10470
\(357\) 1.13158 22.8688i 0.0598894 1.21034i
\(358\) 0 0
\(359\) −14.8175 −0.782037 −0.391018 0.920383i \(-0.627877\pi\)
−0.391018 + 0.920383i \(0.627877\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 12.9795i 0.681249i
\(364\) 28.6402i 1.50116i
\(365\) 10.9430 0.572784
\(366\) 0 0
\(367\) 28.8693i 1.50696i −0.657469 0.753482i \(-0.728373\pi\)
0.657469 0.753482i \(-0.271627\pi\)
\(368\) 24.2577i 1.26452i
\(369\) 0.561108i 0.0292101i
\(370\) 0 0
\(371\) 33.5320i 1.74090i
\(372\) 26.0473 1.35049
\(373\) 20.1112 1.04132 0.520660 0.853764i \(-0.325686\pi\)
0.520660 + 0.853764i \(0.325686\pi\)
\(374\) 0 0
\(375\) −21.3174 −1.10083
\(376\) 0 0
\(377\) 24.0207i 1.23713i
\(378\) 0 0
\(379\) 28.8848i 1.48371i −0.670559 0.741857i \(-0.733945\pi\)
0.670559 0.741857i \(-0.266055\pi\)
\(380\) 15.2719i 0.783432i
\(381\) 7.09108i 0.363287i
\(382\) 0 0
\(383\) −23.2368 −1.18734 −0.593672 0.804707i \(-0.702322\pi\)
−0.593672 + 0.804707i \(0.702322\pi\)
\(384\) 0 0
\(385\) 10.4314i 0.531631i
\(386\) 0 0
\(387\) 1.51183 0.0768506
\(388\) 3.14303i 0.159563i
\(389\) 32.0283 1.62390 0.811951 0.583726i \(-0.198406\pi\)
0.811951 + 0.583726i \(0.198406\pi\)
\(390\) 0 0
\(391\) 24.9738 + 1.23573i 1.26298 + 0.0624937i
\(392\) 0 0
\(393\) −19.4537 −0.981308
\(394\) 0 0
\(395\) −46.7605 −2.35278
\(396\) 1.81475i 0.0911947i
\(397\) 3.56999i 0.179173i 0.995979 + 0.0895864i \(0.0285545\pi\)
−0.995979 + 0.0895864i \(0.971445\pi\)
\(398\) 0 0
\(399\) −11.1066 −0.556023
\(400\) −38.3077 −1.91539
\(401\) 23.9338i 1.19520i 0.801795 + 0.597599i \(0.203878\pi\)
−0.801795 + 0.597599i \(0.796122\pi\)
\(402\) 0 0
\(403\) 33.5838i 1.67293i
\(404\) −11.4643 −0.570372
\(405\) 8.31895i 0.413372i
\(406\) 0 0
\(407\) −3.26783 −0.161980
\(408\) 0 0
\(409\) 21.8197 1.07892 0.539459 0.842012i \(-0.318629\pi\)
0.539459 + 0.842012i \(0.318629\pi\)
\(410\) 0 0
\(411\) 23.3659i 1.15256i
\(412\) 16.7916 0.827261
\(413\) 29.9396i 1.47323i
\(414\) 0 0
\(415\) 19.8949i 0.976602i
\(416\) 0 0
\(417\) −9.18946 −0.450010
\(418\) 0 0
\(419\) 30.2164i 1.47617i 0.674709 + 0.738084i \(0.264269\pi\)
−0.674709 + 0.738084i \(0.735731\pi\)
\(420\) 42.4046i 2.06913i
\(421\) −22.8745 −1.11483 −0.557417 0.830233i \(-0.688207\pi\)
−0.557417 + 0.830233i \(0.688207\pi\)
\(422\) 0 0
\(423\) 1.44916 0.0704607
\(424\) 0 0
\(425\) 1.95147 39.4385i 0.0946600 1.91305i
\(426\) 0 0
\(427\) −28.3979 −1.37427
\(428\) 15.7300i 0.760337i
\(429\) 2.30321 0.111200
\(430\) 0 0
\(431\) 22.3077i 1.07452i 0.843416 + 0.537261i \(0.180541\pi\)
−0.843416 + 0.537261i \(0.819459\pi\)
\(432\) 22.0160i 1.05925i
\(433\) −33.6539 −1.61731 −0.808653 0.588286i \(-0.799803\pi\)
−0.808653 + 0.588286i \(0.799803\pi\)
\(434\) 0 0
\(435\) 35.5650i 1.70521i
\(436\) 24.4638i 1.17160i
\(437\) 12.1289i 0.580203i
\(438\) 0 0
\(439\) 11.1340i 0.531398i 0.964056 + 0.265699i \(0.0856027\pi\)
−0.964056 + 0.265699i \(0.914397\pi\)
\(440\) 0 0
\(441\) −20.7463 −0.987921
\(442\) 0 0
\(443\) 12.4823 0.593051 0.296526 0.955025i \(-0.404172\pi\)
0.296526 + 0.955025i \(0.404172\pi\)
\(444\) 13.2841 0.630435
\(445\) 39.7898i 1.88622i
\(446\) 0 0
\(447\) 6.32513i 0.299168i
\(448\) 36.4177i 1.72058i
\(449\) 14.6677i 0.692213i −0.938195 0.346107i \(-0.887504\pi\)
0.938195 0.346107i \(-0.112496\pi\)
\(450\) 0 0
\(451\) −0.222756 −0.0104891
\(452\) 32.6280i 1.53469i
\(453\) 18.4863i 0.868562i
\(454\) 0 0
\(455\) −54.6738 −2.56315
\(456\) 0 0
\(457\) 21.9171 1.02524 0.512619 0.858616i \(-0.328675\pi\)
0.512619 + 0.858616i \(0.328675\pi\)
\(458\) 0 0
\(459\) 22.6659 + 1.12154i 1.05795 + 0.0523488i
\(460\) −46.3077 −2.15911
\(461\) 29.6778 1.38223 0.691116 0.722744i \(-0.257119\pi\)
0.691116 + 0.722744i \(0.257119\pi\)
\(462\) 0 0
\(463\) −12.7916 −0.594474 −0.297237 0.954804i \(-0.596065\pi\)
−0.297237 + 0.954804i \(0.596065\pi\)
\(464\) 30.5438i 1.41796i
\(465\) 49.7240i 2.30590i
\(466\) 0 0
\(467\) −30.6776 −1.41959 −0.709795 0.704408i \(-0.751212\pi\)
−0.709795 + 0.704408i \(0.751212\pi\)
\(468\) 9.51165 0.439676
\(469\) 39.8962i 1.84224i
\(470\) 0 0
\(471\) 23.7215i 1.09303i
\(472\) 0 0
\(473\) 0.600184i 0.0275965i
\(474\) 0 0
\(475\) −19.1539 −0.878840
\(476\) −37.4927 1.85519i −1.71847 0.0850323i
\(477\) 11.1363 0.509894
\(478\) 0 0
\(479\) 2.45537i 0.112189i −0.998425 0.0560944i \(-0.982135\pi\)
0.998425 0.0560944i \(-0.0178648\pi\)
\(480\) 0 0
\(481\) 17.1277i 0.780955i
\(482\) 0 0
\(483\) 33.6775i 1.53238i
\(484\) −21.2796 −0.967253
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 15.0658i 0.682698i −0.939937 0.341349i \(-0.889116\pi\)
0.939937 0.341349i \(-0.110884\pi\)
\(488\) 0 0
\(489\) 20.8942 0.944867
\(490\) 0 0
\(491\) 18.8151 0.849112 0.424556 0.905402i \(-0.360430\pi\)
0.424556 + 0.905402i \(0.360430\pi\)
\(492\) 0.905525 0.0408242
\(493\) 31.4454 + 1.55596i 1.41623 + 0.0700768i
\(494\) 0 0
\(495\) 3.46434 0.155710
\(496\) 42.7038i 1.91746i
\(497\) 34.0401 1.52691
\(498\) 0 0
\(499\) 38.5693i 1.72660i 0.504692 + 0.863300i \(0.331606\pi\)
−0.504692 + 0.863300i \(0.668394\pi\)
\(500\) 34.9493i 1.56298i
\(501\) −7.39805 −0.330520
\(502\) 0 0
\(503\) 34.7910i 1.55125i 0.631192 + 0.775627i \(0.282566\pi\)
−0.631192 + 0.775627i \(0.717434\pi\)
\(504\) 0 0
\(505\) 21.8853i 0.973882i
\(506\) 0 0
\(507\) 3.78695i 0.168184i
\(508\) 11.6256 0.515803
\(509\) 43.3233 1.92027 0.960135 0.279536i \(-0.0901807\pi\)
0.960135 + 0.279536i \(0.0901807\pi\)
\(510\) 0 0
\(511\) −13.0475 −0.577187
\(512\) 0 0
\(513\) 11.0080i 0.486016i
\(514\) 0 0
\(515\) 32.0549i 1.41251i
\(516\) 2.43981i 0.107407i
\(517\) 0.575306i 0.0253019i
\(518\) 0 0
\(519\) 24.0045 1.05368
\(520\) 0 0
\(521\) 1.06970i 0.0468644i −0.999725 0.0234322i \(-0.992541\pi\)
0.999725 0.0234322i \(-0.00745938\pi\)
\(522\) 0 0
\(523\) −11.9171 −0.521098 −0.260549 0.965461i \(-0.583904\pi\)
−0.260549 + 0.965461i \(0.583904\pi\)
\(524\) 31.8937i 1.39328i
\(525\) 53.1834 2.32111
\(526\) 0 0
\(527\) 43.9643 + 2.17541i 1.91511 + 0.0947623i
\(528\) −2.92867 −0.127454
\(529\) −13.7774 −0.599017
\(530\) 0 0
\(531\) 9.94320 0.431498
\(532\) 18.2089i 0.789455i
\(533\) 1.16753i 0.0505713i
\(534\) 0 0
\(535\) 30.0283 1.29824
\(536\) 0 0
\(537\) 3.33299i 0.143829i
\(538\) 0 0
\(539\) 8.23613i 0.354755i
\(540\) −42.0283 −1.80861
\(541\) 12.2319i 0.525891i −0.964811 0.262945i \(-0.915306\pi\)
0.964811 0.262945i \(-0.0846939\pi\)
\(542\) 0 0
\(543\) −25.7679 −1.10581
\(544\) 0 0
\(545\) −46.7011 −2.00046
\(546\) 0 0
\(547\) 16.8664i 0.721155i 0.932729 + 0.360577i \(0.117420\pi\)
−0.932729 + 0.360577i \(0.882580\pi\)
\(548\) 38.3077 1.63643
\(549\) 9.43116i 0.402512i
\(550\) 0 0
\(551\) 15.2719i 0.650605i
\(552\) 0 0
\(553\) 55.7531 2.37086
\(554\) 0 0
\(555\) 25.3592i 1.07644i
\(556\) 15.0658i 0.638934i
\(557\) 4.97272 0.210701 0.105350 0.994435i \(-0.466404\pi\)
0.105350 + 0.994435i \(0.466404\pi\)
\(558\) 0 0
\(559\) −3.14574 −0.133051
\(560\) 69.5210 2.93780
\(561\) 0.149192 3.01512i 0.00629889 0.127298i
\(562\) 0 0
\(563\) −29.0283 −1.22340 −0.611699 0.791090i \(-0.709514\pi\)
−0.611699 + 0.791090i \(0.709514\pi\)
\(564\) 2.33868i 0.0984763i
\(565\) −62.2865 −2.62041
\(566\) 0 0
\(567\) 9.91878i 0.416550i
\(568\) 0 0
\(569\) 12.3234 0.516626 0.258313 0.966061i \(-0.416833\pi\)
0.258313 + 0.966061i \(0.416833\pi\)
\(570\) 0 0
\(571\) 18.3867i 0.769461i 0.923029 + 0.384731i \(0.125706\pi\)
−0.923029 + 0.384731i \(0.874294\pi\)
\(572\) 3.77605i 0.157885i
\(573\) 24.3602i 1.01766i
\(574\) 0 0
\(575\) 58.0787i 2.42205i
\(576\) −12.0946 −0.503943
\(577\) −8.50008 −0.353863 −0.176931 0.984223i \(-0.556617\pi\)
−0.176931 + 0.984223i \(0.556617\pi\)
\(578\) 0 0
\(579\) 30.1419 1.25266
\(580\) −58.3077 −2.42110
\(581\) 23.7209i 0.984110i
\(582\) 0 0
\(583\) 4.42101i 0.183099i
\(584\) 0 0
\(585\) 18.1576i 0.750725i
\(586\) 0 0
\(587\) 27.8860 1.15098 0.575490 0.817809i \(-0.304811\pi\)
0.575490 + 0.817809i \(0.304811\pi\)
\(588\) 33.4808i 1.38072i
\(589\) 21.3519i 0.879790i
\(590\) 0 0
\(591\) −19.8152 −0.815089
\(592\) 21.7789i 0.895106i
\(593\) 24.8151 1.01903 0.509516 0.860461i \(-0.329824\pi\)
0.509516 + 0.860461i \(0.329824\pi\)
\(594\) 0 0
\(595\) −3.54153 + 71.5730i −0.145188 + 2.93421i
\(596\) −10.3699 −0.424766
\(597\) −4.74392 −0.194156
\(598\) 0 0
\(599\) 38.1383 1.55829 0.779145 0.626844i \(-0.215653\pi\)
0.779145 + 0.626844i \(0.215653\pi\)
\(600\) 0 0
\(601\) 8.31090i 0.339009i −0.985529 0.169504i \(-0.945783\pi\)
0.985529 0.169504i \(-0.0542167\pi\)
\(602\) 0 0
\(603\) −13.2499 −0.539576
\(604\) −30.3077 −1.23320
\(605\) 40.6224i 1.65154i
\(606\) 0 0
\(607\) 17.2174i 0.698833i 0.936968 + 0.349417i \(0.113620\pi\)
−0.936968 + 0.349417i \(0.886380\pi\)
\(608\) 0 0
\(609\) 42.4046i 1.71832i
\(610\) 0 0
\(611\) −3.01535 −0.121988
\(612\) 0.616122 12.4516i 0.0249053 0.503327i
\(613\) −15.3057 −0.618190 −0.309095 0.951031i \(-0.600026\pi\)
−0.309095 + 0.951031i \(0.600026\pi\)
\(614\) 0 0
\(615\) 1.72864i 0.0697053i
\(616\) 0 0
\(617\) 39.4348i 1.58758i −0.608190 0.793792i \(-0.708104\pi\)
0.608190 0.793792i \(-0.291896\pi\)
\(618\) 0 0
\(619\) 44.9157i 1.80532i 0.430358 + 0.902658i \(0.358387\pi\)
−0.430358 + 0.902658i \(0.641613\pi\)
\(620\) −81.5210 −3.27396
\(621\) −33.3787 −1.33944
\(622\) 0 0
\(623\) 47.4419i 1.90072i
\(624\) 15.3501i 0.614494i
\(625\) 18.8330 0.753321
\(626\) 0 0
\(627\) −1.46434 −0.0584800
\(628\) −38.8907 −1.55191
\(629\) 22.4217 + 1.10945i 0.894012 + 0.0442369i
\(630\) 0 0
\(631\) −36.0283 −1.43427 −0.717133 0.696937i \(-0.754546\pi\)
−0.717133 + 0.696937i \(0.754546\pi\)
\(632\) 0 0
\(633\) 5.04369 0.200469
\(634\) 0 0
\(635\) 22.1931i 0.880708i
\(636\) 17.9719i 0.712631i
\(637\) 43.1680 1.71038
\(638\) 0 0
\(639\) 11.3050i 0.447219i
\(640\) 0 0
\(641\) 7.85626i 0.310304i −0.987891 0.155152i \(-0.950413\pi\)
0.987891 0.155152i \(-0.0495867\pi\)
\(642\) 0 0
\(643\) 6.51509i 0.256930i −0.991714 0.128465i \(-0.958995\pi\)
0.991714 0.128465i \(-0.0410050\pi\)
\(644\) 55.2133 2.17571
\(645\) 4.65757 0.183392
\(646\) 0 0
\(647\) 25.4333 0.999885 0.499943 0.866059i \(-0.333354\pi\)
0.499943 + 0.866059i \(0.333354\pi\)
\(648\) 0 0
\(649\) 3.94737i 0.154948i
\(650\) 0 0
\(651\) 59.2865i 2.32362i
\(652\) 34.2554i 1.34154i
\(653\) 6.65372i 0.260380i −0.991489 0.130190i \(-0.958441\pi\)
0.991489 0.130190i \(-0.0415588\pi\)
\(654\) 0 0
\(655\) 60.8847 2.37896
\(656\) 1.48458i 0.0579632i
\(657\) 4.33318i 0.169053i
\(658\) 0 0
\(659\) −12.1281 −0.472445 −0.236222 0.971699i \(-0.575909\pi\)
−0.236222 + 0.971699i \(0.575909\pi\)
\(660\) 5.59080i 0.217622i
\(661\) −6.68727 −0.260105 −0.130052 0.991507i \(-0.541515\pi\)
−0.130052 + 0.991507i \(0.541515\pi\)
\(662\) 0 0
\(663\) −15.8031 0.781960i −0.613743 0.0303688i
\(664\) 0 0
\(665\) 34.7605 1.34795
\(666\) 0 0
\(667\) −46.3077 −1.79304
\(668\) 12.1289i 0.469280i
\(669\) 5.13408i 0.198495i
\(670\) 0 0
\(671\) −3.74410 −0.144539
\(672\) 0 0
\(673\) 13.0099i 0.501494i 0.968053 + 0.250747i \(0.0806762\pi\)
−0.968053 + 0.250747i \(0.919324\pi\)
\(674\) 0 0
\(675\) 52.7115i 2.02887i
\(676\) 6.20859 0.238792
\(677\) 5.36588i 0.206228i −0.994670 0.103114i \(-0.967119\pi\)
0.994670 0.103114i \(-0.0328806\pi\)
\(678\) 0 0
\(679\) −7.15387 −0.274540
\(680\) 0 0
\(681\) −28.7443 −1.10148
\(682\) 0 0
\(683\) 10.9211i 0.417884i 0.977928 + 0.208942i \(0.0670019\pi\)
−0.977928 + 0.208942i \(0.932998\pi\)
\(684\) −6.04731 −0.231225
\(685\) 73.1290i 2.79412i
\(686\) 0 0
\(687\) 5.10062i 0.194601i
\(688\) 4.00000 0.152499
\(689\) −23.1718 −0.882776
\(690\) 0 0
\(691\) 28.4112i 1.08081i −0.841404 0.540406i \(-0.818271\pi\)
0.841404 0.540406i \(-0.181729\pi\)
\(692\) 39.3547i 1.49604i
\(693\) −4.13057 −0.156907
\(694\) 0 0
\(695\) 28.7605 1.09095
\(696\) 0 0
\(697\) 1.52840 + 0.0756273i 0.0578923 + 0.00286459i
\(698\) 0 0
\(699\) 15.8610 0.599919
\(700\) 87.1926i 3.29557i
\(701\) 5.27958 0.199407 0.0997036 0.995017i \(-0.468211\pi\)
0.0997036 + 0.995017i \(0.468211\pi\)
\(702\) 0 0
\(703\) 10.8894i 0.410703i
\(704\) 4.80147i 0.180962i
\(705\) 4.46451 0.168143
\(706\) 0 0
\(707\) 26.0941i 0.981369i
\(708\) 16.0465i 0.603064i
\(709\) 12.2630i 0.460547i −0.973126 0.230274i \(-0.926038\pi\)
0.973126 0.230274i \(-0.0739621\pi\)
\(710\) 0 0
\(711\) 18.5161i 0.694406i
\(712\) 0 0
\(713\) −64.7436 −2.42467
\(714\) 0 0
\(715\) −7.20844 −0.269580
\(716\) −5.46434 −0.204212
\(717\) 10.4820i 0.391458i
\(718\) 0 0
\(719\) 7.86499i 0.293315i −0.989187 0.146657i \(-0.953149\pi\)
0.989187 0.146657i \(-0.0468515\pi\)
\(720\) 23.0885i 0.860457i
\(721\) 38.2194i 1.42337i
\(722\) 0 0
\(723\) 3.09635 0.115154
\(724\) 42.2457i 1.57005i
\(725\) 73.1290i 2.71594i
\(726\) 0 0
\(727\) 32.8078 1.21677 0.608387 0.793640i \(-0.291817\pi\)
0.608387 + 0.793640i \(0.291817\pi\)
\(728\) 0 0
\(729\) −23.4372 −0.868046
\(730\) 0 0
\(731\) −0.203767 + 4.11807i −0.00753660 + 0.152312i
\(732\) 15.2202 0.562553
\(733\) 45.2651 1.67190 0.835952 0.548802i \(-0.184916\pi\)
0.835952 + 0.548802i \(0.184916\pi\)
\(734\) 0 0
\(735\) −63.9144 −2.35752
\(736\) 0 0
\(737\) 5.26009i 0.193758i
\(738\) 0 0
\(739\) 22.0992 0.812931 0.406465 0.913666i \(-0.366761\pi\)
0.406465 + 0.913666i \(0.366761\pi\)
\(740\) −41.5756 −1.52835
\(741\) 7.67503i 0.281949i
\(742\) 0 0
\(743\) 19.9029i 0.730168i −0.930975 0.365084i \(-0.881040\pi\)
0.930975 0.365084i \(-0.118960\pi\)
\(744\) 0 0
\(745\) 19.7959i 0.725267i
\(746\) 0 0
\(747\) −7.87791 −0.288238
\(748\) −4.94320 0.244596i −0.180741 0.00894331i
\(749\) −35.8031 −1.30822
\(750\) 0 0
\(751\) 51.4728i 1.87827i −0.343551 0.939134i \(-0.611630\pi\)
0.343551 0.939134i \(-0.388370\pi\)
\(752\) 3.83420 0.139819
\(753\) 19.3608i 0.705545i
\(754\) 0 0
\(755\) 57.8571i 2.10564i
\(756\) 50.1109 1.82252
\(757\) 24.7087 0.898052 0.449026 0.893519i \(-0.351771\pi\)
0.449026 + 0.893519i \(0.351771\pi\)
\(758\) 0 0
\(759\) 4.44019i 0.161169i
\(760\) 0 0
\(761\) 19.6895 0.713745 0.356873 0.934153i \(-0.383843\pi\)
0.356873 + 0.934153i \(0.383843\pi\)
\(762\) 0 0
\(763\) 55.6823 2.01583
\(764\) 39.9379 1.44490
\(765\) −23.7700 1.17617i −0.859406 0.0425245i
\(766\) 0 0
\(767\) −20.6894 −0.747049
\(768\) 19.5185i 0.704313i
\(769\) −9.91592 −0.357577 −0.178789 0.983888i \(-0.557218\pi\)
−0.178789 + 0.983888i \(0.557218\pi\)
\(770\) 0 0
\(771\) 8.72705i 0.314297i
\(772\) 49.4168i 1.77855i
\(773\) 9.07097 0.326260 0.163130 0.986605i \(-0.447841\pi\)
0.163130 + 0.986605i \(0.447841\pi\)
\(774\) 0 0
\(775\) 102.243i 3.67267i
\(776\) 0 0
\(777\) 30.2360i 1.08471i
\(778\) 0 0
\(779\) 0.742291i 0.0265953i
\(780\) 29.3031 1.04922
\(781\) 4.48799 0.160593
\(782\) 0 0
\(783\) −42.0283 −1.50197
\(784\) −54.8907 −1.96038
\(785\) 74.2419i 2.64981i
\(786\) 0 0
\(787\) 18.1778i 0.647967i 0.946063 + 0.323983i \(0.105022\pi\)
−0.946063 + 0.323983i \(0.894978\pi\)
\(788\) 32.4865i 1.15728i
\(789\) 0.413306i 0.0147141i
\(790\) 0 0
\(791\) 74.2649 2.64056
\(792\) 0 0
\(793\) 19.6239i 0.696866i
\(794\) 0 0
\(795\) 34.3081 1.21678
\(796\) 7.77752i 0.275667i
\(797\) −8.35055 −0.295792 −0.147896 0.989003i \(-0.547250\pi\)
−0.147896 + 0.989003i \(0.547250\pi\)
\(798\) 0 0
\(799\) −0.195321 + 3.94737i −0.00690996 + 0.139648i
\(800\) 0 0
\(801\) 15.7558 0.556705
\(802\) 0 0
\(803\) −1.72024 −0.0607059
\(804\) 21.3828i 0.754114i
\(805\) 105.401i 3.71491i
\(806\) 0 0
\(807\) 16.9926 0.598168
\(808\) 0 0
\(809\) 12.1209i 0.426149i 0.977036 + 0.213074i \(0.0683476\pi\)
−0.977036 + 0.213074i \(0.931652\pi\)
\(810\) 0 0
\(811\) 15.2126i 0.534187i −0.963671 0.267093i \(-0.913937\pi\)
0.963671 0.267093i \(-0.0860632\pi\)
\(812\) 69.5210 2.43971
\(813\) 29.9988i 1.05211i
\(814\) 0 0
\(815\) −65.3931 −2.29062
\(816\) 20.0946 + 0.994308i 0.703453 + 0.0348077i
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 21.6495i 0.756496i
\(820\) −2.83405 −0.0989692
\(821\) 8.14866i 0.284390i −0.989839 0.142195i \(-0.954584\pi\)
0.989839 0.142195i \(-0.0454160\pi\)
\(822\) 0 0
\(823\) 26.8397i 0.935572i −0.883842 0.467786i \(-0.845052\pi\)
0.883842 0.467786i \(-0.154948\pi\)
\(824\) 0 0
\(825\) 7.01193 0.244124
\(826\) 0 0
\(827\) 30.8349i 1.07223i 0.844144 + 0.536117i \(0.180109\pi\)
−0.844144 + 0.536117i \(0.819891\pi\)
\(828\) 18.3368i 0.637247i
\(829\) −17.4333 −0.605483 −0.302741 0.953073i \(-0.597902\pi\)
−0.302741 + 0.953073i \(0.597902\pi\)
\(830\) 0 0
\(831\) −6.36264 −0.220717
\(832\) 25.1660 0.872472
\(833\) 2.79623 56.5109i 0.0968837 1.95799i
\(834\) 0 0
\(835\) 23.1539 0.801273
\(836\) 2.40074i 0.0830312i
\(837\) −58.7605 −2.03106
\(838\) 0 0
\(839\) 13.1089i 0.452571i −0.974061 0.226285i \(-0.927342\pi\)
0.974061 0.226285i \(-0.0726582\pi\)
\(840\) 0 0
\(841\) −29.3077 −1.01061
\(842\) 0 0
\(843\) 12.5068i 0.430759i
\(844\) 8.26897i 0.284630i
\(845\) 11.8521i 0.407725i
\(846\) 0 0
\(847\) 48.4346i 1.66423i
\(848\) 29.4643 1.01181
\(849\) −18.7560 −0.643703
\(850\) 0 0
\(851\) −33.0191 −1.13188
\(852\) −18.2442 −0.625035
\(853\) 26.8708i 0.920038i −0.887909 0.460019i \(-0.847843\pi\)
0.887909 0.460019i \(-0.152157\pi\)
\(854\) 0 0
\(855\) 11.5442i 0.394805i
\(856\) 0 0
\(857\) 22.1735i 0.757433i −0.925513 0.378717i \(-0.876366\pi\)
0.925513 0.378717i \(-0.123634\pi\)
\(858\) 0 0
\(859\) −11.0831 −0.378149 −0.189074 0.981963i \(-0.560549\pi\)
−0.189074 + 0.981963i \(0.560549\pi\)
\(860\) 7.63595i 0.260384i
\(861\) 2.06107i 0.0702412i
\(862\) 0 0
\(863\) 9.54724 0.324992 0.162496 0.986709i \(-0.448046\pi\)
0.162496 + 0.986709i \(0.448046\pi\)
\(864\) 0 0
\(865\) −75.1276 −2.55442
\(866\) 0 0
\(867\) −2.04731 + 20.6371i −0.0695304 + 0.700873i
\(868\) 97.1985 3.29913
\(869\) 7.35073 0.249357
\(870\) 0 0
\(871\) 27.5697 0.934164
\(872\) 0 0
\(873\) 2.37586i 0.0804106i
\(874\) 0 0
\(875\) −79.5483 −2.68922
\(876\) 6.99295 0.236270
\(877\) 46.0603i 1.55535i −0.628669 0.777673i \(-0.716400\pi\)
0.628669 0.777673i \(-0.283600\pi\)
\(878\) 0 0
\(879\) 5.15118i 0.173745i
\(880\) 9.16595 0.308984
\(881\) 6.30144i 0.212301i −0.994350 0.106150i \(-0.966147\pi\)
0.994350 0.106150i \(-0.0338525\pi\)
\(882\) 0 0
\(883\) 45.6621 1.53665 0.768326 0.640059i \(-0.221090\pi\)
0.768326 + 0.640059i \(0.221090\pi\)
\(884\) −1.28200 + 25.9088i −0.0431183 + 0.871406i
\(885\) 30.6326 1.02970
\(886\) 0 0
\(887\) 19.9830i 0.670962i 0.942047 + 0.335481i \(0.108899\pi\)
−0.942047 + 0.335481i \(0.891101\pi\)
\(888\) 0 0
\(889\) 26.4612i 0.887479i
\(890\) 0 0
\(891\) 1.30774i 0.0438108i
\(892\) 8.41718 0.281828
\(893\) 1.91710 0.0641533
\(894\) 0 0
\(895\) 10.4314i 0.348682i
\(896\) 0 0
\(897\) 23.2724 0.777041
\(898\) 0 0
\(899\) −81.5210 −2.71888
\(900\) 28.9574 0.965246
\(901\) −1.50097 + 30.3340i −0.0500045 + 1.01057i
\(902\) 0 0
\(903\) −5.55328 −0.184802
\(904\) 0 0
\(905\) 80.6465 2.68078
\(906\) 0 0
\(907\) 2.60490i 0.0864942i −0.999064 0.0432471i \(-0.986230\pi\)
0.999064 0.0432471i \(-0.0137703\pi\)
\(908\) 47.1253i 1.56391i
\(909\) 8.66605 0.287435
\(910\) 0 0
\(911\) 6.96100i 0.230628i 0.993329 + 0.115314i \(0.0367875\pi\)
−0.993329 + 0.115314i \(0.963213\pi\)
\(912\) 9.75925i 0.323161i
\(913\) 3.12747i 0.103504i
\(914\) 0 0
\(915\) 29.0551i 0.960532i
\(916\) −8.36231 −0.276298
\(917\) −72.5935 −2.39725
\(918\) 0 0
\(919\) −25.8411 −0.852421 −0.426210 0.904624i \(-0.640152\pi\)
−0.426210 + 0.904624i \(0.640152\pi\)
\(920\) 0 0
\(921\) 3.55859i 0.117259i
\(922\) 0 0
\(923\) 23.5229i 0.774266i
\(924\) 6.66598i 0.219295i
\(925\) 52.1437i 1.71447i
\(926\) 0 0
\(927\) −12.6930 −0.416892
\(928\) 0 0
\(929\) 6.43558i 0.211145i 0.994412 + 0.105572i \(0.0336674\pi\)
−0.994412 + 0.105572i \(0.966333\pi\)
\(930\) 0 0
\(931\) −27.4454 −0.899485
\(932\) 26.0037i 0.851778i
\(933\) 23.0899 0.755931
\(934\) 0 0
\(935\) −0.466930 + 9.43650i −0.0152703 + 0.308607i
\(936\) 0 0
\(937\) 35.2796 1.15253 0.576266 0.817262i \(-0.304509\pi\)
0.576266 + 0.817262i \(0.304509\pi\)
\(938\) 0 0
\(939\) 3.14557 0.102652
\(940\) 7.31944i 0.238734i
\(941\) 37.6416i 1.22708i 0.789663 + 0.613541i \(0.210255\pi\)
−0.789663 + 0.613541i \(0.789745\pi\)
\(942\) 0 0
\(943\) −2.25079 −0.0732957
\(944\) 26.3077 0.856244
\(945\) 95.6611i 3.11185i
\(946\) 0 0
\(947\) 10.8336i 0.352045i −0.984386 0.176023i \(-0.943677\pi\)
0.984386 0.176023i \(-0.0563232\pi\)
\(948\) −29.8815 −0.970506
\(949\) 9.01628i 0.292681i
\(950\) 0 0
\(951\) −4.84577 −0.157135
\(952\) 0 0
\(953\) 9.88604 0.320240 0.160120 0.987098i \(-0.448812\pi\)
0.160120 + 0.987098i \(0.448812\pi\)
\(954\) 0 0
\(955\) 76.2409i 2.46710i
\(956\) 17.1849 0.555800
\(957\) 5.59080i 0.180725i
\(958\) 0 0
\(959\) 87.1926i 2.81560i
\(960\) −37.2606 −1.20258
\(961\) −82.9759 −2.67664
\(962\) 0 0
\(963\) 11.8905i 0.383166i
\(964\) 5.07637i 0.163499i
\(965\) −94.3361 −3.03679
\(966\) 0 0
\(967\) 43.6942 1.40511 0.702556 0.711629i \(-0.252042\pi\)
0.702556 + 0.711629i \(0.252042\pi\)
\(968\) 0 0
\(969\) 10.0473 + 0.497154i 0.322766 + 0.0159709i
\(970\) 0 0
\(971\) 42.9417 1.37806 0.689032 0.724731i \(-0.258036\pi\)
0.689032 + 0.724731i \(0.258036\pi\)
\(972\) 27.7080i 0.888733i
\(973\) −34.2915 −1.09933
\(974\) 0 0
\(975\) 36.7516i 1.17699i
\(976\) 24.9530i 0.798726i
\(977\) −39.2625 −1.25612 −0.628060 0.778165i \(-0.716151\pi\)
−0.628060 + 0.778165i \(0.716151\pi\)
\(978\) 0 0
\(979\) 6.25494i 0.199909i
\(980\) 104.786i 3.34726i
\(981\) 18.4925i 0.590421i
\(982\) 0 0
\(983\) 39.7473i 1.26774i 0.773439 + 0.633871i \(0.218535\pi\)
−0.773439 + 0.633871i \(0.781465\pi\)
\(984\) 0 0
\(985\) 62.0163 1.97600
\(986\) 0 0
\(987\) −5.32309 −0.169436
\(988\) 12.5830 0.400318
\(989\) 6.06444i 0.192838i
\(990\) 0 0
\(991\) 42.1560i 1.33913i 0.742754 + 0.669564i \(0.233519\pi\)
−0.742754 + 0.669564i \(0.766481\pi\)
\(992\) 0 0
\(993\) 2.69427i 0.0855001i
\(994\) 0 0
\(995\) 14.8472 0.470687
\(996\) 12.7135i 0.402843i
\(997\) 47.2560i 1.49661i 0.663353 + 0.748306i \(0.269133\pi\)
−0.663353 + 0.748306i \(0.730867\pi\)
\(998\) 0 0
\(999\) −29.9678 −0.948138
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 731.2.d.b.560.4 8
17.16 even 2 inner 731.2.d.b.560.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.d.b.560.4 8 1.1 even 1 trivial
731.2.d.b.560.5 yes 8 17.16 even 2 inner