Properties

Label 460.2.i.b.413.8
Level $460$
Weight $2$
Character 460.413
Analytic conductor $3.673$
Analytic rank $0$
Dimension $16$
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] = [460,2,Mod(137,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), degree \(2\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 413.8
Root \(1.06856 + 1.96422i\) of defining polynomial
Character \(\chi\) \(=\) 460.413
Dual form 460.2.i.b.137.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.03584 + 2.03584i) q^{3} +(1.06856 - 1.96422i) q^{5} +(-0.641104 - 0.641104i) q^{7} +5.28931i q^{9} +O(q^{10})\) \(q+(2.03584 + 2.03584i) q^{3} +(1.06856 - 1.96422i) q^{5} +(-0.641104 - 0.641104i) q^{7} +5.28931i q^{9} +3.68270i q^{11} +(1.51730 + 1.51730i) q^{13} +(6.17428 - 1.82342i) q^{15} +(-0.140805 - 0.140805i) q^{17} +1.03646 q^{19} -2.61037i q^{21} +(4.55009 - 1.51549i) q^{23} +(-2.71634 - 4.19780i) q^{25} +(-4.66067 + 4.66067i) q^{27} -4.36099i q^{29} +3.07168 q^{31} +(-7.49739 + 7.49739i) q^{33} +(-1.94433 + 0.574210i) q^{35} +(-3.28734 - 3.28734i) q^{37} +6.17797i q^{39} -8.10629 q^{41} +(-4.71035 + 4.71035i) q^{43} +(10.3894 + 5.65197i) q^{45} +(-1.77201 + 1.77201i) q^{47} -6.17797i q^{49} -0.573312i q^{51} +(-10.0299 + 10.0299i) q^{53} +(7.23364 + 3.93520i) q^{55} +(2.11007 + 2.11007i) q^{57} -6.32639i q^{59} -13.9583i q^{61} +(3.39100 - 3.39100i) q^{63} +(4.60165 - 1.35898i) q^{65} +(-7.97064 - 7.97064i) q^{67} +(12.3486 + 6.17797i) q^{69} +10.4327 q^{71} +(0.0704445 + 0.0704445i) q^{73} +(3.01602 - 14.0761i) q^{75} +(2.36099 - 2.36099i) q^{77} -9.66645 q^{79} -3.10886 q^{81} +(5.33321 - 5.33321i) q^{83} +(-0.427031 + 0.126113i) q^{85} +(8.87829 - 8.87829i) q^{87} -8.40186 q^{89} -1.94550i q^{91} +(6.25347 + 6.25347i) q^{93} +(1.10753 - 2.03584i) q^{95} +(7.74314 + 7.74314i) q^{97} -19.4789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} + 12 q^{13} + 12 q^{23} + 36 q^{25} + 52 q^{27} - 8 q^{31} + 16 q^{35} - 48 q^{41} + 4 q^{47} + 24 q^{55} + 8 q^{71} - 52 q^{73} - 56 q^{75} - 64 q^{77} - 152 q^{81} + 28 q^{85} + 28 q^{87} + 84 q^{93} - 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 2.03584 + 2.03584i 1.17539 + 1.17539i 0.980905 + 0.194490i \(0.0623050\pi\)
0.194490 + 0.980905i \(0.437695\pi\)
\(4\) 0 0
\(5\) 1.06856 1.96422i 0.477877 0.878427i
\(6\) 0 0
\(7\) −0.641104 0.641104i −0.242315 0.242315i 0.575493 0.817807i \(-0.304810\pi\)
−0.817807 + 0.575493i \(0.804810\pi\)
\(8\) 0 0
\(9\) 5.28931i 1.76310i
\(10\) 0 0
\(11\) 3.68270i 1.11038i 0.831725 + 0.555188i \(0.187354\pi\)
−0.831725 + 0.555188i \(0.812646\pi\)
\(12\) 0 0
\(13\) 1.51730 + 1.51730i 0.420824 + 0.420824i 0.885487 0.464664i \(-0.153825\pi\)
−0.464664 + 0.885487i \(0.653825\pi\)
\(14\) 0 0
\(15\) 6.17428 1.82342i 1.59419 0.470804i
\(16\) 0 0
\(17\) −0.140805 0.140805i −0.0341502 0.0341502i 0.689826 0.723976i \(-0.257687\pi\)
−0.723976 + 0.689826i \(0.757687\pi\)
\(18\) 0 0
\(19\) 1.03646 0.237781 0.118890 0.992907i \(-0.462066\pi\)
0.118890 + 0.992907i \(0.462066\pi\)
\(20\) 0 0
\(21\) 2.61037i 0.569630i
\(22\) 0 0
\(23\) 4.55009 1.51549i 0.948759 0.316001i
\(24\) 0 0
\(25\) −2.71634 4.19780i −0.543268 0.839559i
\(26\) 0 0
\(27\) −4.66067 + 4.66067i −0.896946 + 0.896946i
\(28\) 0 0
\(29\) 4.36099i 0.809816i −0.914357 0.404908i \(-0.867304\pi\)
0.914357 0.404908i \(-0.132696\pi\)
\(30\) 0 0
\(31\) 3.07168 0.551691 0.275845 0.961202i \(-0.411042\pi\)
0.275845 + 0.961202i \(0.411042\pi\)
\(32\) 0 0
\(33\) −7.49739 + 7.49739i −1.30513 + 1.30513i
\(34\) 0 0
\(35\) −1.94433 + 0.574210i −0.328652 + 0.0970591i
\(36\) 0 0
\(37\) −3.28734 3.28734i −0.540435 0.540435i 0.383221 0.923657i \(-0.374815\pi\)
−0.923657 + 0.383221i \(0.874815\pi\)
\(38\) 0 0
\(39\) 6.17797i 0.989267i
\(40\) 0 0
\(41\) −8.10629 −1.26599 −0.632995 0.774156i \(-0.718174\pi\)
−0.632995 + 0.774156i \(0.718174\pi\)
\(42\) 0 0
\(43\) −4.71035 + 4.71035i −0.718322 + 0.718322i −0.968261 0.249939i \(-0.919589\pi\)
0.249939 + 0.968261i \(0.419589\pi\)
\(44\) 0 0
\(45\) 10.3894 + 5.65197i 1.54876 + 0.842546i
\(46\) 0 0
\(47\) −1.77201 + 1.77201i −0.258474 + 0.258474i −0.824433 0.565959i \(-0.808506\pi\)
0.565959 + 0.824433i \(0.308506\pi\)
\(48\) 0 0
\(49\) 6.17797i 0.882567i
\(50\) 0 0
\(51\) 0.573312i 0.0802798i
\(52\) 0 0
\(53\) −10.0299 + 10.0299i −1.37771 + 1.37771i −0.529235 + 0.848475i \(0.677521\pi\)
−0.848475 + 0.529235i \(0.822479\pi\)
\(54\) 0 0
\(55\) 7.23364 + 3.93520i 0.975384 + 0.530623i
\(56\) 0 0
\(57\) 2.11007 + 2.11007i 0.279486 + 0.279486i
\(58\) 0 0
\(59\) 6.32639i 0.823626i −0.911268 0.411813i \(-0.864896\pi\)
0.911268 0.411813i \(-0.135104\pi\)
\(60\) 0 0
\(61\) 13.9583i 1.78718i −0.448883 0.893590i \(-0.648178\pi\)
0.448883 0.893590i \(-0.351822\pi\)
\(62\) 0 0
\(63\) 3.39100 3.39100i 0.427225 0.427225i
\(64\) 0 0
\(65\) 4.60165 1.35898i 0.570765 0.168561i
\(66\) 0 0
\(67\) −7.97064 7.97064i −0.973768 0.973768i 0.0258962 0.999665i \(-0.491756\pi\)
−0.999665 + 0.0258962i \(0.991756\pi\)
\(68\) 0 0
\(69\) 12.3486 + 6.17797i 1.48659 + 0.743740i
\(70\) 0 0
\(71\) 10.4327 1.23813 0.619066 0.785339i \(-0.287512\pi\)
0.619066 + 0.785339i \(0.287512\pi\)
\(72\) 0 0
\(73\) 0.0704445 + 0.0704445i 0.00824490 + 0.00824490i 0.711217 0.702972i \(-0.248144\pi\)
−0.702972 + 0.711217i \(0.748144\pi\)
\(74\) 0 0
\(75\) 3.01602 14.0761i 0.348260 1.62537i
\(76\) 0 0
\(77\) 2.36099 2.36099i 0.269060 0.269060i
\(78\) 0 0
\(79\) −9.66645 −1.08756 −0.543780 0.839228i \(-0.683008\pi\)
−0.543780 + 0.839228i \(0.683008\pi\)
\(80\) 0 0
\(81\) −3.10886 −0.345428
\(82\) 0 0
\(83\) 5.33321 5.33321i 0.585396 0.585396i −0.350985 0.936381i \(-0.614153\pi\)
0.936381 + 0.350985i \(0.114153\pi\)
\(84\) 0 0
\(85\) −0.427031 + 0.126113i −0.0463180 + 0.0136789i
\(86\) 0 0
\(87\) 8.87829 8.87829i 0.951853 0.951853i
\(88\) 0 0
\(89\) −8.40186 −0.890595 −0.445298 0.895383i \(-0.646902\pi\)
−0.445298 + 0.895383i \(0.646902\pi\)
\(90\) 0 0
\(91\) 1.94550i 0.203943i
\(92\) 0 0
\(93\) 6.25347 + 6.25347i 0.648454 + 0.648454i
\(94\) 0 0
\(95\) 1.10753 2.03584i 0.113630 0.208873i
\(96\) 0 0
\(97\) 7.74314 + 7.74314i 0.786197 + 0.786197i 0.980868 0.194672i \(-0.0623641\pi\)
−0.194672 + 0.980868i \(0.562364\pi\)
\(98\) 0 0
\(99\) −19.4789 −1.95771
\(100\) 0 0
\(101\) −16.7912 −1.67079 −0.835393 0.549653i \(-0.814760\pi\)
−0.835393 + 0.549653i \(0.814760\pi\)
\(102\) 0 0
\(103\) −0.859794 + 0.859794i −0.0847180 + 0.0847180i −0.748196 0.663478i \(-0.769080\pi\)
0.663478 + 0.748196i \(0.269080\pi\)
\(104\) 0 0
\(105\) −5.12735 2.78935i −0.500378 0.272213i
\(106\) 0 0
\(107\) 7.32073 + 7.32073i 0.707721 + 0.707721i 0.966056 0.258334i \(-0.0831736\pi\)
−0.258334 + 0.966056i \(0.583174\pi\)
\(108\) 0 0
\(109\) 10.4031 0.996432 0.498216 0.867053i \(-0.333989\pi\)
0.498216 + 0.867053i \(0.333989\pi\)
\(110\) 0 0
\(111\) 13.3850i 1.27045i
\(112\) 0 0
\(113\) −7.51564 + 7.51564i −0.707012 + 0.707012i −0.965906 0.258894i \(-0.916642\pi\)
0.258894 + 0.965906i \(0.416642\pi\)
\(114\) 0 0
\(115\) 1.88531 10.5568i 0.175806 0.984425i
\(116\) 0 0
\(117\) −8.02547 + 8.02547i −0.741955 + 0.741955i
\(118\) 0 0
\(119\) 0.180541i 0.0165502i
\(120\) 0 0
\(121\) −2.56227 −0.232934
\(122\) 0 0
\(123\) −16.5031 16.5031i −1.48804 1.48804i
\(124\) 0 0
\(125\) −11.1480 + 0.849875i −0.997107 + 0.0760151i
\(126\) 0 0
\(127\) 15.7207 15.7207i 1.39499 1.39499i 0.581304 0.813686i \(-0.302543\pi\)
0.813686 0.581304i \(-0.197457\pi\)
\(128\) 0 0
\(129\) −19.1791 −1.68862
\(130\) 0 0
\(131\) −15.4698 −1.35160 −0.675800 0.737085i \(-0.736202\pi\)
−0.675800 + 0.737085i \(0.736202\pi\)
\(132\) 0 0
\(133\) −0.664480 0.664480i −0.0576177 0.0576177i
\(134\) 0 0
\(135\) 4.17436 + 14.1348i 0.359272 + 1.21653i
\(136\) 0 0
\(137\) 12.5584 + 12.5584i 1.07294 + 1.07294i 0.997122 + 0.0758178i \(0.0241567\pi\)
0.0758178 + 0.997122i \(0.475843\pi\)
\(138\) 0 0
\(139\) 4.43268i 0.375975i 0.982171 + 0.187987i \(0.0601964\pi\)
−0.982171 + 0.187987i \(0.939804\pi\)
\(140\) 0 0
\(141\) −7.21505 −0.607617
\(142\) 0 0
\(143\) −5.58776 + 5.58776i −0.467272 + 0.467272i
\(144\) 0 0
\(145\) −8.56596 4.66000i −0.711364 0.386992i
\(146\) 0 0
\(147\) 12.5774 12.5774i 1.03736 1.03736i
\(148\) 0 0
\(149\) 9.63059 0.788969 0.394484 0.918903i \(-0.370923\pi\)
0.394484 + 0.918903i \(0.370923\pi\)
\(150\) 0 0
\(151\) 8.54649 0.695504 0.347752 0.937587i \(-0.386945\pi\)
0.347752 + 0.937587i \(0.386945\pi\)
\(152\) 0 0
\(153\) 0.744759 0.744759i 0.0602102 0.0602102i
\(154\) 0 0
\(155\) 3.28229 6.03347i 0.263640 0.484620i
\(156\) 0 0
\(157\) 12.1116 + 12.1116i 0.966612 + 0.966612i 0.999460 0.0328482i \(-0.0104578\pi\)
−0.0328482 + 0.999460i \(0.510458\pi\)
\(158\) 0 0
\(159\) −40.8385 −3.23871
\(160\) 0 0
\(161\) −3.88866 1.94550i −0.306470 0.153327i
\(162\) 0 0
\(163\) 4.54277 + 4.54277i 0.355818 + 0.355818i 0.862269 0.506451i \(-0.169043\pi\)
−0.506451 + 0.862269i \(0.669043\pi\)
\(164\) 0 0
\(165\) 6.71510 + 22.7380i 0.522770 + 1.77015i
\(166\) 0 0
\(167\) −8.02547 + 8.02547i −0.621030 + 0.621030i −0.945795 0.324765i \(-0.894715\pi\)
0.324765 + 0.945795i \(0.394715\pi\)
\(168\) 0 0
\(169\) 8.39559i 0.645815i
\(170\) 0 0
\(171\) 5.48217i 0.419232i
\(172\) 0 0
\(173\) −10.2405 10.2405i −0.778573 0.778573i 0.201015 0.979588i \(-0.435576\pi\)
−0.979588 + 0.201015i \(0.935576\pi\)
\(174\) 0 0
\(175\) −0.949769 + 4.43268i −0.0717958 + 0.335079i
\(176\) 0 0
\(177\) 12.8795 12.8795i 0.968085 0.968085i
\(178\) 0 0
\(179\) 16.1063i 1.20384i 0.798556 + 0.601920i \(0.205597\pi\)
−0.798556 + 0.601920i \(0.794403\pi\)
\(180\) 0 0
\(181\) 0.801442i 0.0595707i −0.999556 0.0297853i \(-0.990518\pi\)
0.999556 0.0297853i \(-0.00948237\pi\)
\(182\) 0 0
\(183\) 28.4170 28.4170i 2.10064 2.10064i
\(184\) 0 0
\(185\) −9.96980 + 2.94433i −0.732995 + 0.216472i
\(186\) 0 0
\(187\) 0.518541 0.518541i 0.0379195 0.0379195i
\(188\) 0 0
\(189\) 5.97595 0.434686
\(190\) 0 0
\(191\) 20.3788i 1.47456i −0.675587 0.737280i \(-0.736110\pi\)
0.675587 0.737280i \(-0.263890\pi\)
\(192\) 0 0
\(193\) −3.47357 3.47357i −0.250033 0.250033i 0.570951 0.820984i \(-0.306575\pi\)
−0.820984 + 0.570951i \(0.806575\pi\)
\(194\) 0 0
\(195\) 12.1349 + 6.60156i 0.868999 + 0.472748i
\(196\) 0 0
\(197\) −17.9408 + 17.9408i −1.27823 + 1.27823i −0.336576 + 0.941656i \(0.609269\pi\)
−0.941656 + 0.336576i \(0.890731\pi\)
\(198\) 0 0
\(199\) 13.1210 0.930125 0.465062 0.885278i \(-0.346032\pi\)
0.465062 + 0.885278i \(0.346032\pi\)
\(200\) 0 0
\(201\) 32.4539i 2.28912i
\(202\) 0 0
\(203\) −2.79585 + 2.79585i −0.196230 + 0.196230i
\(204\) 0 0
\(205\) −8.66209 + 15.9225i −0.604987 + 1.11208i
\(206\) 0 0
\(207\) 8.01587 + 24.0668i 0.557142 + 1.67276i
\(208\) 0 0
\(209\) 3.81698i 0.264026i
\(210\) 0 0
\(211\) 9.63644 0.663400 0.331700 0.943385i \(-0.392378\pi\)
0.331700 + 0.943385i \(0.392378\pi\)
\(212\) 0 0
\(213\) 21.2393 + 21.2393i 1.45529 + 1.45529i
\(214\) 0 0
\(215\) 4.21886 + 14.2855i 0.287724 + 0.974263i
\(216\) 0 0
\(217\) −1.96927 1.96927i −0.133683 0.133683i
\(218\) 0 0
\(219\) 0.286828i 0.0193820i
\(220\) 0 0
\(221\) 0.427286i 0.0287424i
\(222\) 0 0
\(223\) 13.9257 + 13.9257i 0.932537 + 0.932537i 0.997864 0.0653270i \(-0.0208090\pi\)
−0.0653270 + 0.997864i \(0.520809\pi\)
\(224\) 0 0
\(225\) 22.2034 14.3676i 1.48023 0.957837i
\(226\) 0 0
\(227\) 7.31585 + 7.31585i 0.485570 + 0.485570i 0.906905 0.421335i \(-0.138438\pi\)
−0.421335 + 0.906905i \(0.638438\pi\)
\(228\) 0 0
\(229\) 23.0701 1.52451 0.762256 0.647275i \(-0.224092\pi\)
0.762256 + 0.647275i \(0.224092\pi\)
\(230\) 0 0
\(231\) 9.61322 0.632503
\(232\) 0 0
\(233\) −0.998760 0.998760i −0.0654309 0.0654309i 0.673634 0.739065i \(-0.264732\pi\)
−0.739065 + 0.673634i \(0.764732\pi\)
\(234\) 0 0
\(235\) 1.58711 + 5.37412i 0.103532 + 0.350569i
\(236\) 0 0
\(237\) −19.6794 19.6794i −1.27831 1.27831i
\(238\) 0 0
\(239\) 12.1038i 0.782930i 0.920193 + 0.391465i \(0.128032\pi\)
−0.920193 + 0.391465i \(0.871968\pi\)
\(240\) 0 0
\(241\) 21.1246i 1.36075i −0.732863 0.680377i \(-0.761816\pi\)
0.732863 0.680377i \(-0.238184\pi\)
\(242\) 0 0
\(243\) 7.65287 + 7.65287i 0.490932 + 0.490932i
\(244\) 0 0
\(245\) −12.1349 6.60156i −0.775271 0.421758i
\(246\) 0 0
\(247\) 1.57263 + 1.57263i 0.100064 + 0.100064i
\(248\) 0 0
\(249\) 21.7151 1.37614
\(250\) 0 0
\(251\) 11.4940i 0.725494i 0.931888 + 0.362747i \(0.118161\pi\)
−0.931888 + 0.362747i \(0.881839\pi\)
\(252\) 0 0
\(253\) 5.58108 + 16.7566i 0.350879 + 1.05348i
\(254\) 0 0
\(255\) −1.12611 0.612621i −0.0705199 0.0383638i
\(256\) 0 0
\(257\) 14.2739 14.2739i 0.890381 0.890381i −0.104178 0.994559i \(-0.533221\pi\)
0.994559 + 0.104178i \(0.0332211\pi\)
\(258\) 0 0
\(259\) 4.21505i 0.261911i
\(260\) 0 0
\(261\) 23.0666 1.42779
\(262\) 0 0
\(263\) −5.00958 + 5.00958i −0.308904 + 0.308904i −0.844484 0.535580i \(-0.820093\pi\)
0.535580 + 0.844484i \(0.320093\pi\)
\(264\) 0 0
\(265\) 8.98334 + 30.4185i 0.551842 + 1.86859i
\(266\) 0 0
\(267\) −17.1049 17.1049i −1.04680 1.04680i
\(268\) 0 0
\(269\) 1.51189i 0.0921817i 0.998937 + 0.0460909i \(0.0146764\pi\)
−0.998937 + 0.0460909i \(0.985324\pi\)
\(270\) 0 0
\(271\) −21.2314 −1.28972 −0.644858 0.764303i \(-0.723083\pi\)
−0.644858 + 0.764303i \(0.723083\pi\)
\(272\) 0 0
\(273\) 3.96072 3.96072i 0.239714 0.239714i
\(274\) 0 0
\(275\) 15.4592 10.0035i 0.932226 0.603231i
\(276\) 0 0
\(277\) −10.7299 + 10.7299i −0.644696 + 0.644696i −0.951706 0.307010i \(-0.900671\pi\)
0.307010 + 0.951706i \(0.400671\pi\)
\(278\) 0 0
\(279\) 16.2471i 0.972688i
\(280\) 0 0
\(281\) 23.7522i 1.41694i 0.705742 + 0.708469i \(0.250614\pi\)
−0.705742 + 0.708469i \(0.749386\pi\)
\(282\) 0 0
\(283\) −3.94182 + 3.94182i −0.234317 + 0.234317i −0.814492 0.580175i \(-0.802984\pi\)
0.580175 + 0.814492i \(0.302984\pi\)
\(284\) 0 0
\(285\) 6.39940 1.88990i 0.379068 0.111948i
\(286\) 0 0
\(287\) 5.19697 + 5.19697i 0.306768 + 0.306768i
\(288\) 0 0
\(289\) 16.9603i 0.997668i
\(290\) 0 0
\(291\) 31.5276i 1.84818i
\(292\) 0 0
\(293\) 4.09174 4.09174i 0.239042 0.239042i −0.577411 0.816453i \(-0.695937\pi\)
0.816453 + 0.577411i \(0.195937\pi\)
\(294\) 0 0
\(295\) −12.4264 6.76016i −0.723495 0.393592i
\(296\) 0 0
\(297\) −17.1638 17.1638i −0.995947 0.995947i
\(298\) 0 0
\(299\) 9.20330 + 4.60441i 0.532241 + 0.266280i
\(300\) 0 0
\(301\) 6.03965 0.348120
\(302\) 0 0
\(303\) −34.1842 34.1842i −1.96383 1.96383i
\(304\) 0 0
\(305\) −27.4173 14.9154i −1.56991 0.854052i
\(306\) 0 0
\(307\) −17.3468 + 17.3468i −0.990035 + 0.990035i −0.999951 0.00991588i \(-0.996844\pi\)
0.00991588 + 0.999951i \(0.496844\pi\)
\(308\) 0 0
\(309\) −3.50081 −0.199154
\(310\) 0 0
\(311\) −10.7566 −0.609950 −0.304975 0.952360i \(-0.598648\pi\)
−0.304975 + 0.952360i \(0.598648\pi\)
\(312\) 0 0
\(313\) −9.83073 + 9.83073i −0.555665 + 0.555665i −0.928070 0.372405i \(-0.878533\pi\)
0.372405 + 0.928070i \(0.378533\pi\)
\(314\) 0 0
\(315\) −3.03717 10.2842i −0.171125 0.579447i
\(316\) 0 0
\(317\) 4.99087 4.99087i 0.280315 0.280315i −0.552919 0.833235i \(-0.686486\pi\)
0.833235 + 0.552919i \(0.186486\pi\)
\(318\) 0 0
\(319\) 16.0602 0.899200
\(320\) 0 0
\(321\) 29.8077i 1.66370i
\(322\) 0 0
\(323\) −0.145939 0.145939i −0.00812025 0.00812025i
\(324\) 0 0
\(325\) 2.24782 10.4908i 0.124687 0.581926i
\(326\) 0 0
\(327\) 21.1790 + 21.1790i 1.17120 + 1.17120i
\(328\) 0 0
\(329\) 2.27208 0.125264
\(330\) 0 0
\(331\) −6.00505 −0.330067 −0.165034 0.986288i \(-0.552773\pi\)
−0.165034 + 0.986288i \(0.552773\pi\)
\(332\) 0 0
\(333\) 17.3878 17.3878i 0.952843 0.952843i
\(334\) 0 0
\(335\) −24.1732 + 7.13896i −1.32073 + 0.390043i
\(336\) 0 0
\(337\) −18.1596 18.1596i −0.989215 0.989215i 0.0107275 0.999942i \(-0.496585\pi\)
−0.999942 + 0.0107275i \(0.996585\pi\)
\(338\) 0 0
\(339\) −30.6013 −1.66204
\(340\) 0 0
\(341\) 11.3121i 0.612584i
\(342\) 0 0
\(343\) −8.44845 + 8.44845i −0.456173 + 0.456173i
\(344\) 0 0
\(345\) 25.3301 17.6537i 1.36373 0.950445i
\(346\) 0 0
\(347\) −16.9603 + 16.9603i −0.910479 + 0.910479i −0.996310 0.0858306i \(-0.972646\pi\)
0.0858306 + 0.996310i \(0.472646\pi\)
\(348\) 0 0
\(349\) 17.2818i 0.925072i −0.886600 0.462536i \(-0.846940\pi\)
0.886600 0.462536i \(-0.153060\pi\)
\(350\) 0 0
\(351\) −14.1433 −0.754912
\(352\) 0 0
\(353\) 8.08174 + 8.08174i 0.430148 + 0.430148i 0.888678 0.458531i \(-0.151624\pi\)
−0.458531 + 0.888678i \(0.651624\pi\)
\(354\) 0 0
\(355\) 11.1480 20.4921i 0.591674 1.08761i
\(356\) 0 0
\(357\) −0.367553 + 0.367553i −0.0194530 + 0.0194530i
\(358\) 0 0
\(359\) −19.8505 −1.04767 −0.523835 0.851820i \(-0.675499\pi\)
−0.523835 + 0.851820i \(0.675499\pi\)
\(360\) 0 0
\(361\) −17.9257 −0.943460
\(362\) 0 0
\(363\) −5.21638 5.21638i −0.273789 0.273789i
\(364\) 0 0
\(365\) 0.213643 0.0630941i 0.0111826 0.00330250i
\(366\) 0 0
\(367\) −5.46914 5.46914i −0.285487 0.285487i 0.549806 0.835293i \(-0.314702\pi\)
−0.835293 + 0.549806i \(0.814702\pi\)
\(368\) 0 0
\(369\) 42.8766i 2.23207i
\(370\) 0 0
\(371\) 12.8604 0.667678
\(372\) 0 0
\(373\) −1.09481 + 1.09481i −0.0566873 + 0.0566873i −0.734882 0.678195i \(-0.762763\pi\)
0.678195 + 0.734882i \(0.262763\pi\)
\(374\) 0 0
\(375\) −24.4258 20.9653i −1.26134 1.08265i
\(376\) 0 0
\(377\) 6.61694 6.61694i 0.340790 0.340790i
\(378\) 0 0
\(379\) 19.8147 1.01781 0.508905 0.860823i \(-0.330050\pi\)
0.508905 + 0.860823i \(0.330050\pi\)
\(380\) 0 0
\(381\) 64.0099 3.27933
\(382\) 0 0
\(383\) 6.00265 6.00265i 0.306721 0.306721i −0.536915 0.843636i \(-0.680410\pi\)
0.843636 + 0.536915i \(0.180410\pi\)
\(384\) 0 0
\(385\) −2.11464 7.16039i −0.107772 0.364927i
\(386\) 0 0
\(387\) −24.9145 24.9145i −1.26648 1.26648i
\(388\) 0 0
\(389\) 12.7758 0.647761 0.323880 0.946098i \(-0.395012\pi\)
0.323880 + 0.946098i \(0.395012\pi\)
\(390\) 0 0
\(391\) −0.854061 0.427286i −0.0431917 0.0216088i
\(392\) 0 0
\(393\) −31.4940 31.4940i −1.58866 1.58866i
\(394\) 0 0
\(395\) −10.3292 + 18.9871i −0.519720 + 0.955343i
\(396\) 0 0
\(397\) 12.4084 12.4084i 0.622762 0.622762i −0.323475 0.946237i \(-0.604851\pi\)
0.946237 + 0.323475i \(0.104851\pi\)
\(398\) 0 0
\(399\) 2.70555i 0.135447i
\(400\) 0 0
\(401\) 5.86322i 0.292795i −0.989226 0.146397i \(-0.953232\pi\)
0.989226 0.146397i \(-0.0467679\pi\)
\(402\) 0 0
\(403\) 4.66067 + 4.66067i 0.232165 + 0.232165i
\(404\) 0 0
\(405\) −3.32201 + 6.10648i −0.165072 + 0.303434i
\(406\) 0 0
\(407\) 12.1063 12.1063i 0.600086 0.600086i
\(408\) 0 0
\(409\) 27.4118i 1.35543i 0.735325 + 0.677714i \(0.237029\pi\)
−0.735325 + 0.677714i \(0.762971\pi\)
\(410\) 0 0
\(411\) 51.1340i 2.52225i
\(412\) 0 0
\(413\) −4.05587 + 4.05587i −0.199577 + 0.199577i
\(414\) 0 0
\(415\) −4.77673 16.1745i −0.234480 0.793974i
\(416\) 0 0
\(417\) −9.02423 + 9.02423i −0.441919 + 0.441919i
\(418\) 0 0
\(419\) 26.2153 1.28070 0.640351 0.768082i \(-0.278789\pi\)
0.640351 + 0.768082i \(0.278789\pi\)
\(420\) 0 0
\(421\) 10.7215i 0.522534i 0.965267 + 0.261267i \(0.0841404\pi\)
−0.965267 + 0.261267i \(0.915860\pi\)
\(422\) 0 0
\(423\) −9.37269 9.37269i −0.455716 0.455716i
\(424\) 0 0
\(425\) −0.208596 + 0.973543i −0.0101184 + 0.0472238i
\(426\) 0 0
\(427\) −8.94874 + 8.94874i −0.433060 + 0.433060i
\(428\) 0 0
\(429\) −22.7516 −1.09846
\(430\) 0 0
\(431\) 32.7007i 1.57513i 0.616228 + 0.787567i \(0.288660\pi\)
−0.616228 + 0.787567i \(0.711340\pi\)
\(432\) 0 0
\(433\) −4.99812 + 4.99812i −0.240194 + 0.240194i −0.816930 0.576736i \(-0.804326\pi\)
0.576736 + 0.816930i \(0.304326\pi\)
\(434\) 0 0
\(435\) −7.95191 26.9260i −0.381265 1.29100i
\(436\) 0 0
\(437\) 4.71599 1.57074i 0.225597 0.0751389i
\(438\) 0 0
\(439\) 35.4389i 1.69141i −0.533653 0.845704i \(-0.679181\pi\)
0.533653 0.845704i \(-0.320819\pi\)
\(440\) 0 0
\(441\) 32.6772 1.55606
\(442\) 0 0
\(443\) 15.4430 + 15.4430i 0.733721 + 0.733721i 0.971355 0.237634i \(-0.0763717\pi\)
−0.237634 + 0.971355i \(0.576372\pi\)
\(444\) 0 0
\(445\) −8.97793 + 16.5031i −0.425595 + 0.782323i
\(446\) 0 0
\(447\) 19.6064 + 19.6064i 0.927349 + 0.927349i
\(448\) 0 0
\(449\) 25.6660i 1.21125i −0.795749 0.605626i \(-0.792923\pi\)
0.795749 0.605626i \(-0.207077\pi\)
\(450\) 0 0
\(451\) 29.8530i 1.40572i
\(452\) 0 0
\(453\) 17.3993 + 17.3993i 0.817491 + 0.817491i
\(454\) 0 0
\(455\) −3.82139 2.07889i −0.179149 0.0974598i
\(456\) 0 0
\(457\) −5.76900 5.76900i −0.269862 0.269862i 0.559182 0.829045i \(-0.311115\pi\)
−0.829045 + 0.559182i \(0.811115\pi\)
\(458\) 0 0
\(459\) 1.31249 0.0612617
\(460\) 0 0
\(461\) −9.18302 −0.427696 −0.213848 0.976867i \(-0.568600\pi\)
−0.213848 + 0.976867i \(0.568600\pi\)
\(462\) 0 0
\(463\) 11.5786 + 11.5786i 0.538104 + 0.538104i 0.922972 0.384868i \(-0.125753\pi\)
−0.384868 + 0.922972i \(0.625753\pi\)
\(464\) 0 0
\(465\) 18.9654 5.60096i 0.879501 0.259738i
\(466\) 0 0
\(467\) −9.47482 9.47482i −0.438442 0.438442i 0.453045 0.891488i \(-0.350338\pi\)
−0.891488 + 0.453045i \(0.850338\pi\)
\(468\) 0 0
\(469\) 10.2200i 0.471916i
\(470\) 0 0
\(471\) 49.3147i 2.27230i
\(472\) 0 0
\(473\) −17.3468 17.3468i −0.797607 0.797607i
\(474\) 0 0
\(475\) −2.81538 4.35086i −0.129179 0.199631i
\(476\) 0 0
\(477\) −53.0511 53.0511i −2.42904 2.42904i
\(478\) 0 0
\(479\) 0.427286 0.0195232 0.00976160 0.999952i \(-0.496893\pi\)
0.00976160 + 0.999952i \(0.496893\pi\)
\(480\) 0 0
\(481\) 9.97577i 0.454856i
\(482\) 0 0
\(483\) −3.95598 11.8774i −0.180003 0.540442i
\(484\) 0 0
\(485\) 23.4833 6.93520i 1.06632 0.314911i
\(486\) 0 0
\(487\) −9.15631 + 9.15631i −0.414912 + 0.414912i −0.883446 0.468534i \(-0.844782\pi\)
0.468534 + 0.883446i \(0.344782\pi\)
\(488\) 0 0
\(489\) 18.4967i 0.836452i
\(490\) 0 0
\(491\) 42.2610 1.90721 0.953605 0.301061i \(-0.0973407\pi\)
0.953605 + 0.301061i \(0.0973407\pi\)
\(492\) 0 0
\(493\) −0.614048 + 0.614048i −0.0276553 + 0.0276553i
\(494\) 0 0
\(495\) −20.8145 + 38.2610i −0.935542 + 1.71970i
\(496\) 0 0
\(497\) −6.68843 6.68843i −0.300017 0.300017i
\(498\) 0 0
\(499\) 11.7220i 0.524748i −0.964966 0.262374i \(-0.915495\pi\)
0.964966 0.262374i \(-0.0845054\pi\)
\(500\) 0 0
\(501\) −32.6772 −1.45991
\(502\) 0 0
\(503\) 14.6500 14.6500i 0.653209 0.653209i −0.300555 0.953764i \(-0.597172\pi\)
0.953764 + 0.300555i \(0.0971720\pi\)
\(504\) 0 0
\(505\) −17.9425 + 32.9816i −0.798430 + 1.46766i
\(506\) 0 0
\(507\) 17.0921 17.0921i 0.759087 0.759087i
\(508\) 0 0
\(509\) 14.6453i 0.649139i 0.945862 + 0.324570i \(0.105219\pi\)
−0.945862 + 0.324570i \(0.894781\pi\)
\(510\) 0 0
\(511\) 0.0903245i 0.00399572i
\(512\) 0 0
\(513\) −4.83061 + 4.83061i −0.213277 + 0.213277i
\(514\) 0 0
\(515\) 0.770081 + 2.60757i 0.0339338 + 0.114903i
\(516\) 0 0
\(517\) −6.52577 6.52577i −0.287003 0.287003i
\(518\) 0 0
\(519\) 41.6962i 1.83026i
\(520\) 0 0
\(521\) 19.5415i 0.856130i −0.903748 0.428065i \(-0.859195\pi\)
0.903748 0.428065i \(-0.140805\pi\)
\(522\) 0 0
\(523\) −12.6165 + 12.6165i −0.551680 + 0.551680i −0.926926 0.375245i \(-0.877558\pi\)
0.375245 + 0.926926i \(0.377558\pi\)
\(524\) 0 0
\(525\) −10.9578 + 7.09066i −0.478238 + 0.309462i
\(526\) 0 0
\(527\) −0.432508 0.432508i −0.0188403 0.0188403i
\(528\) 0 0
\(529\) 18.4066 13.7912i 0.800287 0.599617i
\(530\) 0 0
\(531\) 33.4622 1.45214
\(532\) 0 0
\(533\) −12.2997 12.2997i −0.532758 0.532758i
\(534\) 0 0
\(535\) 22.2022 6.55686i 0.959885 0.283478i
\(536\) 0 0
\(537\) −32.7899 + 32.7899i −1.41499 + 1.41499i
\(538\) 0 0
\(539\) 22.7516 0.979981
\(540\) 0 0
\(541\) 5.81193 0.249874 0.124937 0.992165i \(-0.460127\pi\)
0.124937 + 0.992165i \(0.460127\pi\)
\(542\) 0 0
\(543\) 1.63161 1.63161i 0.0700190 0.0700190i
\(544\) 0 0
\(545\) 11.1163 20.4339i 0.476172 0.875293i
\(546\) 0 0
\(547\) 10.8386 10.8386i 0.463427 0.463427i −0.436350 0.899777i \(-0.643729\pi\)
0.899777 + 0.436350i \(0.143729\pi\)
\(548\) 0 0
\(549\) 73.8299 3.15098
\(550\) 0 0
\(551\) 4.52000i 0.192559i
\(552\) 0 0
\(553\) 6.19720 + 6.19720i 0.263532 + 0.263532i
\(554\) 0 0
\(555\) −26.2911 14.3028i −1.11600 0.607118i
\(556\) 0 0
\(557\) 1.86496 + 1.86496i 0.0790209 + 0.0790209i 0.745513 0.666492i \(-0.232205\pi\)
−0.666492 + 0.745513i \(0.732205\pi\)
\(558\) 0 0
\(559\) −14.2940 −0.604574
\(560\) 0 0
\(561\) 2.11134 0.0891407
\(562\) 0 0
\(563\) −24.5429 + 24.5429i −1.03436 + 1.03436i −0.0349727 + 0.999388i \(0.511134\pi\)
−0.999388 + 0.0349727i \(0.988866\pi\)
\(564\) 0 0
\(565\) 6.73144 + 22.7933i 0.283194 + 0.958923i
\(566\) 0 0
\(567\) 1.99310 + 1.99310i 0.0837023 + 0.0837023i
\(568\) 0 0
\(569\) −40.4575 −1.69607 −0.848033 0.529944i \(-0.822213\pi\)
−0.848033 + 0.529944i \(0.822213\pi\)
\(570\) 0 0
\(571\) 14.9032i 0.623681i 0.950135 + 0.311840i \(0.100945\pi\)
−0.950135 + 0.311840i \(0.899055\pi\)
\(572\) 0 0
\(573\) 41.4881 41.4881i 1.73319 1.73319i
\(574\) 0 0
\(575\) −18.7213 14.9838i −0.780732 0.624867i
\(576\) 0 0
\(577\) −9.55190 + 9.55190i −0.397651 + 0.397651i −0.877404 0.479753i \(-0.840726\pi\)
0.479753 + 0.877404i \(0.340726\pi\)
\(578\) 0 0
\(579\) 14.1433i 0.587775i
\(580\) 0 0
\(581\) −6.83828 −0.283700
\(582\) 0 0
\(583\) −36.9370 36.9370i −1.52978 1.52978i
\(584\) 0 0
\(585\) 7.18808 + 24.3396i 0.297190 + 1.00632i
\(586\) 0 0
\(587\) −12.1534 + 12.1534i −0.501625 + 0.501625i −0.911943 0.410317i \(-0.865418\pi\)
0.410317 + 0.911943i \(0.365418\pi\)
\(588\) 0 0
\(589\) 3.18369 0.131181
\(590\) 0 0
\(591\) −73.0495 −3.00485
\(592\) 0 0
\(593\) 13.6477 + 13.6477i 0.560445 + 0.560445i 0.929434 0.368989i \(-0.120296\pi\)
−0.368989 + 0.929434i \(0.620296\pi\)
\(594\) 0 0
\(595\) 0.354622 + 0.192920i 0.0145381 + 0.00790893i
\(596\) 0 0
\(597\) 26.7123 + 26.7123i 1.09326 + 1.09326i
\(598\) 0 0
\(599\) 0.636527i 0.0260078i −0.999915 0.0130039i \(-0.995861\pi\)
0.999915 0.0130039i \(-0.00413938\pi\)
\(600\) 0 0
\(601\) −27.8232 −1.13493 −0.567466 0.823397i \(-0.692076\pi\)
−0.567466 + 0.823397i \(0.692076\pi\)
\(602\) 0 0
\(603\) 42.1592 42.1592i 1.71685 1.71685i
\(604\) 0 0
\(605\) −2.73795 + 5.03287i −0.111314 + 0.204615i
\(606\) 0 0
\(607\) −3.41225 + 3.41225i −0.138499 + 0.138499i −0.772957 0.634458i \(-0.781223\pi\)
0.634458 + 0.772957i \(0.281223\pi\)
\(608\) 0 0
\(609\) −11.3838 −0.461296
\(610\) 0 0
\(611\) −5.37734 −0.217544
\(612\) 0 0
\(613\) 8.13522 8.13522i 0.328578 0.328578i −0.523467 0.852046i \(-0.675362\pi\)
0.852046 + 0.523467i \(0.175362\pi\)
\(614\) 0 0
\(615\) −50.0505 + 14.7811i −2.01823 + 0.596033i
\(616\) 0 0
\(617\) 12.5389 + 12.5389i 0.504797 + 0.504797i 0.912925 0.408128i \(-0.133818\pi\)
−0.408128 + 0.912925i \(0.633818\pi\)
\(618\) 0 0
\(619\) 43.9838 1.76786 0.883928 0.467622i \(-0.154889\pi\)
0.883928 + 0.467622i \(0.154889\pi\)
\(620\) 0 0
\(621\) −14.1433 + 28.2696i −0.567550 + 1.13442i
\(622\) 0 0
\(623\) 5.38647 + 5.38647i 0.215804 + 0.215804i
\(624\) 0 0
\(625\) −10.2430 + 22.8053i −0.409720 + 0.912211i
\(626\) 0 0
\(627\) −7.77077 + 7.77077i −0.310335 + 0.310335i
\(628\) 0 0
\(629\) 0.925746i 0.0369119i
\(630\) 0 0
\(631\) 8.80275i 0.350432i 0.984530 + 0.175216i \(0.0560623\pi\)
−0.984530 + 0.175216i \(0.943938\pi\)
\(632\) 0 0
\(633\) 19.6183 + 19.6183i 0.779756 + 0.779756i
\(634\) 0 0
\(635\) −14.0804 47.6777i −0.558764 1.89203i
\(636\) 0 0
\(637\) 9.37384 9.37384i 0.371405 0.371405i
\(638\) 0 0
\(639\) 55.1816i 2.18295i
\(640\) 0 0
\(641\) 28.4088i 1.12208i −0.827789 0.561040i \(-0.810402\pi\)
0.827789 0.561040i \(-0.189598\pi\)
\(642\) 0 0
\(643\) −31.1486 + 31.1486i −1.22838 + 1.22838i −0.263805 + 0.964576i \(0.584977\pi\)
−0.964576 + 0.263805i \(0.915023\pi\)
\(644\) 0 0
\(645\) −20.4941 + 37.6720i −0.806954 + 1.48333i
\(646\) 0 0
\(647\) 32.5723 32.5723i 1.28055 1.28055i 0.340196 0.940355i \(-0.389507\pi\)
0.940355 0.340196i \(-0.110493\pi\)
\(648\) 0 0
\(649\) 23.2982 0.914534
\(650\) 0 0
\(651\) 8.01824i 0.314260i
\(652\) 0 0
\(653\) −23.0858 23.0858i −0.903418 0.903418i 0.0923122 0.995730i \(-0.470574\pi\)
−0.995730 + 0.0923122i \(0.970574\pi\)
\(654\) 0 0
\(655\) −16.5304 + 30.3860i −0.645898 + 1.18728i
\(656\) 0 0
\(657\) −0.372603 + 0.372603i −0.0145366 + 0.0145366i
\(658\) 0 0
\(659\) −30.7552 −1.19805 −0.599025 0.800730i \(-0.704445\pi\)
−0.599025 + 0.800730i \(0.704445\pi\)
\(660\) 0 0
\(661\) 9.55886i 0.371797i −0.982569 0.185898i \(-0.940481\pi\)
0.982569 0.185898i \(-0.0595195\pi\)
\(662\) 0 0
\(663\) 0.869887 0.869887i 0.0337836 0.0337836i
\(664\) 0 0
\(665\) −2.01523 + 0.595147i −0.0781471 + 0.0230788i
\(666\) 0 0
\(667\) −6.60902 19.8429i −0.255902 0.768320i
\(668\) 0 0
\(669\) 56.7012i 2.19220i
\(670\) 0 0
\(671\) 51.4043 1.98444
\(672\) 0 0
\(673\) 0.330200 + 0.330200i 0.0127283 + 0.0127283i 0.713442 0.700714i \(-0.247135\pi\)
−0.700714 + 0.713442i \(0.747135\pi\)
\(674\) 0 0
\(675\) 32.2245 + 6.90459i 1.24032 + 0.265758i
\(676\) 0 0
\(677\) 9.50253 + 9.50253i 0.365212 + 0.365212i 0.865727 0.500516i \(-0.166856\pi\)
−0.500516 + 0.865727i \(0.666856\pi\)
\(678\) 0 0
\(679\) 9.92832i 0.381014i
\(680\) 0 0
\(681\) 29.7879i 1.14147i
\(682\) 0 0
\(683\) 28.3226 + 28.3226i 1.08373 + 1.08373i 0.996158 + 0.0875756i \(0.0279119\pi\)
0.0875756 + 0.996158i \(0.472088\pi\)
\(684\) 0 0
\(685\) 38.0871 11.2481i 1.45523 0.429766i
\(686\) 0 0
\(687\) 46.9670 + 46.9670i 1.79190 + 1.79190i
\(688\) 0 0
\(689\) −30.4367 −1.15955
\(690\) 0 0
\(691\) 11.6899 0.444703 0.222352 0.974967i \(-0.428627\pi\)
0.222352 + 0.974967i \(0.428627\pi\)
\(692\) 0 0
\(693\) 12.4880 + 12.4880i 0.474381 + 0.474381i
\(694\) 0 0
\(695\) 8.70676 + 4.73660i 0.330266 + 0.179670i
\(696\) 0 0
\(697\) 1.14140 + 1.14140i 0.0432337 + 0.0432337i
\(698\) 0 0
\(699\) 4.06663i 0.153814i
\(700\) 0 0
\(701\) 5.09075i 0.192275i −0.995368 0.0961374i \(-0.969351\pi\)
0.995368 0.0961374i \(-0.0306488\pi\)
\(702\) 0 0
\(703\) −3.40720 3.40720i −0.128505 0.128505i
\(704\) 0 0
\(705\) −7.70975 + 14.1720i −0.290366 + 0.533747i
\(706\) 0 0
\(707\) 10.7649 + 10.7649i 0.404856 + 0.404856i
\(708\) 0 0
\(709\) −33.9271 −1.27416 −0.637080 0.770798i \(-0.719858\pi\)
−0.637080 + 0.770798i \(0.719858\pi\)
\(710\) 0 0
\(711\) 51.1288i 1.91748i
\(712\) 0 0
\(713\) 13.9764 4.65510i 0.523422 0.174335i
\(714\) 0 0
\(715\) 5.00472 + 16.9465i 0.187166 + 0.633763i
\(716\) 0 0
\(717\) −24.6414 + 24.6414i −0.920252 + 0.920252i
\(718\) 0 0
\(719\) 21.1572i 0.789032i 0.918889 + 0.394516i \(0.129088\pi\)
−0.918889 + 0.394516i \(0.870912\pi\)
\(720\) 0 0
\(721\) 1.10243 0.0410568
\(722\) 0 0
\(723\) 43.0063 43.0063i 1.59942 1.59942i
\(724\) 0 0
\(725\) −18.3066 + 11.8459i −0.679889 + 0.439947i
\(726\) 0 0
\(727\) −13.7181 13.7181i −0.508776 0.508776i 0.405375 0.914151i \(-0.367141\pi\)
−0.914151 + 0.405375i \(0.867141\pi\)
\(728\) 0 0
\(729\) 40.4866i 1.49951i
\(730\) 0 0
\(731\) 1.32648 0.0490616
\(732\) 0 0
\(733\) 30.6685 30.6685i 1.13277 1.13277i 0.143050 0.989715i \(-0.454309\pi\)
0.989715 0.143050i \(-0.0456910\pi\)
\(734\) 0 0
\(735\) −11.2650 38.1445i −0.415517 1.40698i
\(736\) 0 0
\(737\) 29.3535 29.3535i 1.08125 1.08125i
\(738\) 0 0
\(739\) 10.5711i 0.388864i −0.980916 0.194432i \(-0.937714\pi\)
0.980916 0.194432i \(-0.0622863\pi\)
\(740\) 0 0
\(741\) 6.40323i 0.235229i
\(742\) 0 0
\(743\) 26.7743 26.7743i 0.982253 0.982253i −0.0175925 0.999845i \(-0.505600\pi\)
0.999845 + 0.0175925i \(0.00560016\pi\)
\(744\) 0 0
\(745\) 10.2909 18.9166i 0.377030 0.693051i
\(746\) 0 0
\(747\) 28.2090 + 28.2090i 1.03211 + 1.03211i
\(748\) 0 0
\(749\) 9.38669i 0.342982i
\(750\) 0 0
\(751\) 37.5857i 1.37152i −0.727827 0.685760i \(-0.759470\pi\)
0.727827 0.685760i \(-0.240530\pi\)
\(752\) 0 0
\(753\) −23.3999 + 23.3999i −0.852741 + 0.852741i
\(754\) 0 0
\(755\) 9.13248 16.7872i 0.332365 0.610949i
\(756\) 0 0
\(757\) 12.8355 + 12.8355i 0.466513 + 0.466513i 0.900783 0.434270i \(-0.142993\pi\)
−0.434270 + 0.900783i \(0.642993\pi\)
\(758\) 0 0
\(759\) −22.7516 + 45.4760i −0.825831 + 1.65067i
\(760\) 0 0
\(761\) 27.5018 0.996939 0.498470 0.866907i \(-0.333896\pi\)
0.498470 + 0.866907i \(0.333896\pi\)
\(762\) 0 0
\(763\) −6.66944 6.66944i −0.241450 0.241450i
\(764\) 0 0
\(765\) −0.667049 2.25870i −0.0241172 0.0816633i
\(766\) 0 0
\(767\) 9.59904 9.59904i 0.346601 0.346601i
\(768\) 0 0
\(769\) 40.2678 1.45209 0.726047 0.687645i \(-0.241356\pi\)
0.726047 + 0.687645i \(0.241356\pi\)
\(770\) 0 0
\(771\) 58.1188 2.09310
\(772\) 0 0
\(773\) 7.63820 7.63820i 0.274727 0.274727i −0.556273 0.831000i \(-0.687769\pi\)
0.831000 + 0.556273i \(0.187769\pi\)
\(774\) 0 0
\(775\) −8.34374 12.8943i −0.299716 0.463177i
\(776\) 0 0
\(777\) −8.58119 + 8.58119i −0.307848 + 0.307848i
\(778\) 0 0
\(779\) −8.40186 −0.301028
\(780\) 0 0
\(781\) 38.4204i 1.37479i
\(782\) 0 0
\(783\) 20.3252 + 20.3252i 0.726362 + 0.726362i
\(784\) 0 0
\(785\) 36.7320 10.8479i 1.31102 0.387177i
\(786\) 0 0
\(787\) 31.8800 + 31.8800i 1.13640 + 1.13640i 0.989091 + 0.147308i \(0.0470608\pi\)
0.147308 + 0.989091i \(0.452939\pi\)
\(788\) 0 0
\(789\) −20.3974 −0.726168
\(790\) 0 0
\(791\) 9.63662 0.342639
\(792\) 0 0
\(793\) 21.1790 21.1790i 0.752088 0.752088i
\(794\) 0 0
\(795\) −43.6386 + 80.2159i −1.54770 + 2.84497i
\(796\) 0 0
\(797\) −15.8582 15.8582i −0.561725 0.561725i 0.368072 0.929797i \(-0.380018\pi\)
−0.929797 + 0.368072i \(0.880018\pi\)
\(798\) 0 0
\(799\) 0.499014 0.0176538
\(800\) 0 0
\(801\) 44.4400i 1.57021i
\(802\) 0 0
\(803\) −0.259426 + 0.259426i −0.00915494 + 0.00915494i
\(804\) 0 0
\(805\) −7.97667 + 5.55931i −0.281141 + 0.195940i
\(806\) 0 0
\(807\) −3.07797 + 3.07797i −0.108350 + 0.108350i
\(808\) 0 0
\(809\) 21.1622i 0.744023i −0.928228 0.372012i \(-0.878668\pi\)
0.928228 0.372012i \(-0.121332\pi\)
\(810\) 0 0
\(811\) −27.4459 −0.963755 −0.481878 0.876238i \(-0.660045\pi\)
−0.481878 + 0.876238i \(0.660045\pi\)
\(812\) 0 0
\(813\) −43.2238 43.2238i −1.51592 1.51592i
\(814\) 0 0
\(815\) 13.7773 4.06877i 0.482597 0.142523i
\(816\) 0 0
\(817\) −4.88210 + 4.88210i −0.170803 + 0.170803i
\(818\) 0 0
\(819\) 10.2903 0.359573
\(820\) 0 0
\(821\) −0.771083 −0.0269110 −0.0134555 0.999909i \(-0.504283\pi\)
−0.0134555 + 0.999909i \(0.504283\pi\)
\(822\) 0 0
\(823\) 8.35975 + 8.35975i 0.291403 + 0.291403i 0.837634 0.546232i \(-0.183938\pi\)
−0.546232 + 0.837634i \(0.683938\pi\)
\(824\) 0 0
\(825\) 51.8380 + 11.1071i 1.80477 + 0.386699i
\(826\) 0 0
\(827\) −2.21958 2.21958i −0.0771824 0.0771824i 0.667462 0.744644i \(-0.267381\pi\)
−0.744644 + 0.667462i \(0.767381\pi\)
\(828\) 0 0
\(829\) 5.94905i 0.206619i −0.994649 0.103310i \(-0.967057\pi\)
0.994649 0.103310i \(-0.0329432\pi\)
\(830\) 0 0
\(831\) −43.6887 −1.51554
\(832\) 0 0
\(833\) −0.869887 + 0.869887i −0.0301398 + 0.0301398i
\(834\) 0 0
\(835\) 7.18808 + 24.3396i 0.248754 + 0.842305i
\(836\) 0 0
\(837\) −14.3161 + 14.3161i −0.494837 + 0.494837i
\(838\) 0 0
\(839\) −17.3157 −0.597804 −0.298902 0.954284i \(-0.596620\pi\)
−0.298902 + 0.954284i \(0.596620\pi\)
\(840\) 0 0
\(841\) 9.98174 0.344198
\(842\) 0 0
\(843\) −48.3558 + 48.3558i −1.66546 + 1.66546i
\(844\) 0 0
\(845\) −16.4908 8.97124i −0.567301 0.308620i
\(846\) 0 0
\(847\) 1.64268 + 1.64268i 0.0564433 + 0.0564433i
\(848\) 0 0
\(849\) −16.0498 −0.550829
\(850\) 0 0
\(851\) −19.9396 9.97577i −0.683521 0.341965i
\(852\) 0 0
\(853\) −33.6940 33.6940i −1.15366 1.15366i −0.985813 0.167849i \(-0.946318\pi\)
−0.167849 0.985813i \(-0.553682\pi\)
\(854\) 0 0
\(855\) 10.7682 + 5.85805i 0.368265 + 0.200341i
\(856\) 0 0
\(857\) 25.8667 25.8667i 0.883589 0.883589i −0.110308 0.993897i \(-0.535184\pi\)
0.993897 + 0.110308i \(0.0351838\pi\)
\(858\) 0 0
\(859\) 25.1384i 0.857711i 0.903373 + 0.428856i \(0.141083\pi\)
−0.903373 + 0.428856i \(0.858917\pi\)
\(860\) 0 0
\(861\) 21.1604i 0.721146i
\(862\) 0 0
\(863\) −17.4031 17.4031i −0.592408 0.592408i 0.345873 0.938281i \(-0.387583\pi\)
−0.938281 + 0.345873i \(0.887583\pi\)
\(864\) 0 0
\(865\) −31.0573 + 9.17201i −1.05598 + 0.311858i
\(866\) 0 0
\(867\) 34.5286 34.5286i 1.17265 1.17265i
\(868\) 0 0
\(869\) 35.5986i 1.20760i
\(870\) 0 0
\(871\) 24.1877i 0.819569i
\(872\) 0 0
\(873\) −40.9559 + 40.9559i −1.38615 + 1.38615i
\(874\) 0 0
\(875\) 7.69188 + 6.60216i 0.260033 + 0.223194i
\(876\) 0 0
\(877\) −19.4764 + 19.4764i −0.657672 + 0.657672i −0.954829 0.297157i \(-0.903962\pi\)
0.297157 + 0.954829i \(0.403962\pi\)
\(878\) 0 0
\(879\) 16.6603 0.561937
\(880\) 0 0
\(881\) 9.42106i 0.317403i −0.987327 0.158702i \(-0.949269\pi\)
0.987327 0.158702i \(-0.0507308\pi\)
\(882\) 0 0
\(883\) 20.4902 + 20.4902i 0.689550 + 0.689550i 0.962132 0.272583i \(-0.0878779\pi\)
−0.272583 + 0.962132i \(0.587878\pi\)
\(884\) 0 0
\(885\) −11.5357 39.0609i −0.387767 1.31302i
\(886\) 0 0
\(887\) 17.5682 17.5682i 0.589884 0.589884i −0.347716 0.937600i \(-0.613043\pi\)
0.937600 + 0.347716i \(0.113043\pi\)
\(888\) 0 0
\(889\) −20.1573 −0.676053
\(890\) 0 0
\(891\) 11.4490i 0.383555i
\(892\) 0 0
\(893\) −1.83662 + 1.83662i −0.0614601 + 0.0614601i
\(894\) 0 0
\(895\) 31.6363 + 17.2106i 1.05749 + 0.575287i
\(896\) 0 0
\(897\) 9.36263 + 28.1103i 0.312609 + 0.938576i
\(898\) 0 0
\(899\) 13.3956i 0.446768i
\(900\) 0 0
\(901\) 2.82451 0.0940980
\(902\) 0 0
\(903\) 12.2958 + 12.2958i 0.409178 + 0.409178i
\(904\) 0 0
\(905\) −1.57421 0.856392i −0.0523285 0.0284674i
\(906\) 0 0
\(907\) −35.9000 35.9000i −1.19204 1.19204i −0.976493 0.215548i \(-0.930846\pi\)
−0.215548 0.976493i \(-0.569154\pi\)
\(908\) 0 0
\(909\) 88.8138i 2.94577i
\(910\) 0 0
\(911\) 12.0034i 0.397689i −0.980031 0.198845i \(-0.936281\pi\)
0.980031 0.198845i \(-0.0637189\pi\)
\(912\) 0 0
\(913\) 19.6406 + 19.6406i 0.650009 + 0.650009i
\(914\) 0 0
\(915\) −25.4519 86.1826i −0.841413 2.84911i
\(916\) 0 0
\(917\) 9.91772 + 9.91772i 0.327512 + 0.327512i
\(918\) 0 0
\(919\) 9.97577 0.329070 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(920\) 0 0
\(921\) −70.6308 −2.32736
\(922\) 0 0
\(923\) 15.8295 + 15.8295i 0.521035 + 0.521035i
\(924\) 0 0
\(925\) −4.87006 + 22.7291i −0.160127 + 0.747329i
\(926\) 0 0
\(927\) −4.54771 4.54771i −0.149367 0.149367i
\(928\) 0 0
\(929\) 5.38430i 0.176653i 0.996092 + 0.0883266i \(0.0281519\pi\)
−0.996092 + 0.0883266i \(0.971848\pi\)
\(930\) 0 0
\(931\) 6.40323i 0.209858i
\(932\) 0 0
\(933\) −21.8987 21.8987i −0.716932 0.716932i
\(934\) 0 0
\(935\) −0.464435 1.57263i −0.0151887 0.0514303i
\(936\) 0 0
\(937\) −35.2785 35.2785i −1.15250 1.15250i −0.986050 0.166447i \(-0.946771\pi\)
−0.166447 0.986050i \(-0.553229\pi\)
\(938\) 0 0
\(939\) −40.0276 −1.30625
\(940\) 0 0
\(941\) 2.05309i 0.0669287i −0.999440 0.0334644i \(-0.989346\pi\)
0.999440 0.0334644i \(-0.0106540\pi\)
\(942\) 0 0
\(943\) −36.8843 + 12.2850i −1.20112 + 0.400053i
\(944\) 0 0
\(945\) 6.38569 11.7381i 0.207726 0.381840i
\(946\) 0 0
\(947\) −28.6627 + 28.6627i −0.931414 + 0.931414i −0.997794 0.0663803i \(-0.978855\pi\)
0.0663803 + 0.997794i \(0.478855\pi\)
\(948\) 0 0
\(949\) 0.213771i 0.00693930i
\(950\) 0 0
\(951\) 20.3213 0.658962
\(952\) 0 0
\(953\) 34.2531 34.2531i 1.10957 1.10957i 0.116360 0.993207i \(-0.462877\pi\)
0.993207 0.116360i \(-0.0371227\pi\)
\(954\) 0 0
\(955\) −40.0286 21.7761i −1.29529 0.704658i
\(956\) 0 0
\(957\) 32.6961 + 32.6961i 1.05691 + 1.05691i
\(958\) 0 0
\(959\) 16.1025i 0.519978i
\(960\) 0 0
\(961\) −21.5648 −0.695637
\(962\) 0 0
\(963\) −38.7216 + 38.7216i −1.24779 + 1.24779i
\(964\) 0 0
\(965\) −10.5346 + 3.11113i −0.339121 + 0.100151i
\(966\) 0 0
\(967\) −17.4403 + 17.4403i −0.560841 + 0.560841i −0.929546 0.368706i \(-0.879801\pi\)
0.368706 + 0.929546i \(0.379801\pi\)
\(968\) 0 0
\(969\) 0.594217i 0.0190890i
\(970\) 0 0
\(971\) 34.8720i 1.11910i 0.828798 + 0.559548i \(0.189025\pi\)
−0.828798 + 0.559548i \(0.810975\pi\)
\(972\) 0 0
\(973\) 2.84181 2.84181i 0.0911041 0.0911041i
\(974\) 0 0
\(975\) 25.9339 16.7815i 0.830549 0.537437i
\(976\) 0 0
\(977\) −15.1727 15.1727i −0.485419 0.485419i 0.421438 0.906857i \(-0.361525\pi\)
−0.906857 + 0.421438i \(0.861525\pi\)
\(978\) 0 0
\(979\) 30.9415i 0.988895i
\(980\) 0 0
\(981\) 55.0250i 1.75681i
\(982\) 0 0
\(983\) −25.9177 + 25.9177i −0.826646 + 0.826646i −0.987051 0.160406i \(-0.948720\pi\)
0.160406 + 0.987051i \(0.448720\pi\)
\(984\) 0 0
\(985\) 16.0689 + 54.4108i 0.511996 + 1.73367i
\(986\) 0 0
\(987\) 4.62560 + 4.62560i 0.147234 + 0.147234i
\(988\) 0 0
\(989\) −14.2940 + 28.5710i −0.454524 + 0.908505i
\(990\) 0 0
\(991\) −32.4736 −1.03156 −0.515779 0.856722i \(-0.672497\pi\)
−0.515779 + 0.856722i \(0.672497\pi\)
\(992\) 0 0
\(993\) −12.2253 12.2253i −0.387959 0.387959i
\(994\) 0 0
\(995\) 14.0207 25.7726i 0.444485 0.817046i
\(996\) 0 0
\(997\) 22.0637 22.0637i 0.698766 0.698766i −0.265378 0.964144i \(-0.585497\pi\)
0.964144 + 0.265378i \(0.0854969\pi\)
\(998\) 0 0
\(999\) 30.6424 0.969483
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 460.2.i.b.413.8 yes 16
5.2 odd 4 inner 460.2.i.b.137.7 16
5.3 odd 4 2300.2.i.d.1057.1 16
5.4 even 2 2300.2.i.d.1793.2 16
23.22 odd 2 inner 460.2.i.b.413.7 yes 16
115.22 even 4 inner 460.2.i.b.137.8 yes 16
115.68 even 4 2300.2.i.d.1057.2 16
115.114 odd 2 2300.2.i.d.1793.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.i.b.137.7 16 5.2 odd 4 inner
460.2.i.b.137.8 yes 16 115.22 even 4 inner
460.2.i.b.413.7 yes 16 23.22 odd 2 inner
460.2.i.b.413.8 yes 16 1.1 even 1 trivial
2300.2.i.d.1057.1 16 5.3 odd 4
2300.2.i.d.1057.2 16 115.68 even 4
2300.2.i.d.1793.1 16 115.114 odd 2
2300.2.i.d.1793.2 16 5.4 even 2