Properties

Label 1792.2.m.f.1345.5
Level $1792$
Weight $2$
Character 1792.1345
Analytic conductor $14.309$
Analytic rank $0$
Dimension $16$
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] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 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("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (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: \(14.3091920422\)
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: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1345.5
Root \(0.277956 - 0.213283i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1345
Dual form 1792.2.m.f.449.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.328027 + 0.328027i) q^{3} +(-1.40197 + 1.40197i) q^{5} +1.00000i q^{7} -2.78480i q^{9} +O(q^{10})\) \(q+(0.328027 + 0.328027i) q^{3} +(-1.40197 + 1.40197i) q^{5} +1.00000i q^{7} -2.78480i q^{9} +(0.444087 - 0.444087i) q^{11} +(-2.25590 - 2.25590i) q^{13} -0.919765 q^{15} +4.85578 q^{17} +(-0.114004 - 0.114004i) q^{19} +(-0.328027 + 0.328027i) q^{21} -3.20529i q^{23} +1.06898i q^{25} +(1.89757 - 1.89757i) q^{27} +(0.997091 + 0.997091i) q^{29} +5.34435 q^{31} +0.291345 q^{33} +(-1.40197 - 1.40197i) q^{35} +(-2.03472 + 2.03472i) q^{37} -1.47999i q^{39} -9.57673i q^{41} +(6.86758 - 6.86758i) q^{43} +(3.90419 + 3.90419i) q^{45} +9.70703 q^{47} -1.00000 q^{49} +(1.59282 + 1.59282i) q^{51} +(-7.64426 + 7.64426i) q^{53} +1.24519i q^{55} -0.0747927i q^{57} +(1.50266 - 1.50266i) q^{59} +(1.74157 + 1.74157i) q^{61} +2.78480 q^{63} +6.32541 q^{65} +(8.96491 + 8.96491i) q^{67} +(1.05142 - 1.05142i) q^{69} +7.18356i q^{71} -9.04029i q^{73} +(-0.350652 + 0.350652i) q^{75} +(0.444087 + 0.444087i) q^{77} +9.58806 q^{79} -7.10949 q^{81} +(5.30245 + 5.30245i) q^{83} +(-6.80764 + 6.80764i) q^{85} +0.654145i q^{87} +2.49938i q^{89} +(2.25590 - 2.25590i) q^{91} +(1.75309 + 1.75309i) q^{93} +0.319660 q^{95} +5.89073 q^{97} +(-1.23669 - 1.23669i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 4 q^{5} + 8 q^{11} - 12 q^{13} - 8 q^{17} - 4 q^{19} + 4 q^{21} + 56 q^{27} + 8 q^{31} + 16 q^{33} + 4 q^{35} + 8 q^{37} + 24 q^{43} + 36 q^{45} + 40 q^{47} - 16 q^{49} - 24 q^{51} + 32 q^{53} + 4 q^{59} + 20 q^{61} - 24 q^{63} + 72 q^{65} - 32 q^{67} - 56 q^{69} + 28 q^{75} + 8 q^{77} - 40 q^{81} - 36 q^{83} + 12 q^{91} - 8 q^{93} + 80 q^{95} - 72 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0.328027 + 0.328027i 0.189386 + 0.189386i 0.795431 0.606044i \(-0.207245\pi\)
−0.606044 + 0.795431i \(0.707245\pi\)
\(4\) 0 0
\(5\) −1.40197 + 1.40197i −0.626979 + 0.626979i −0.947307 0.320328i \(-0.896207\pi\)
0.320328 + 0.947307i \(0.396207\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.78480i 0.928266i
\(10\) 0 0
\(11\) 0.444087 0.444087i 0.133897 0.133897i −0.636982 0.770879i \(-0.719817\pi\)
0.770879 + 0.636982i \(0.219817\pi\)
\(12\) 0 0
\(13\) −2.25590 2.25590i −0.625675 0.625675i 0.321302 0.946977i \(-0.395880\pi\)
−0.946977 + 0.321302i \(0.895880\pi\)
\(14\) 0 0
\(15\) −0.919765 −0.237482
\(16\) 0 0
\(17\) 4.85578 1.17770 0.588849 0.808243i \(-0.299581\pi\)
0.588849 + 0.808243i \(0.299581\pi\)
\(18\) 0 0
\(19\) −0.114004 0.114004i −0.0261543 0.0261543i 0.693909 0.720063i \(-0.255887\pi\)
−0.720063 + 0.693909i \(0.755887\pi\)
\(20\) 0 0
\(21\) −0.328027 + 0.328027i −0.0715813 + 0.0715813i
\(22\) 0 0
\(23\) 3.20529i 0.668350i −0.942511 0.334175i \(-0.891542\pi\)
0.942511 0.334175i \(-0.108458\pi\)
\(24\) 0 0
\(25\) 1.06898i 0.213795i
\(26\) 0 0
\(27\) 1.89757 1.89757i 0.365187 0.365187i
\(28\) 0 0
\(29\) 0.997091 + 0.997091i 0.185155 + 0.185155i 0.793598 0.608443i \(-0.208205\pi\)
−0.608443 + 0.793598i \(0.708205\pi\)
\(30\) 0 0
\(31\) 5.34435 0.959874 0.479937 0.877303i \(-0.340660\pi\)
0.479937 + 0.877303i \(0.340660\pi\)
\(32\) 0 0
\(33\) 0.291345 0.0507167
\(34\) 0 0
\(35\) −1.40197 1.40197i −0.236976 0.236976i
\(36\) 0 0
\(37\) −2.03472 + 2.03472i −0.334507 + 0.334507i −0.854295 0.519788i \(-0.826011\pi\)
0.519788 + 0.854295i \(0.326011\pi\)
\(38\) 0 0
\(39\) 1.47999i 0.236989i
\(40\) 0 0
\(41\) 9.57673i 1.49563i −0.663905 0.747817i \(-0.731102\pi\)
0.663905 0.747817i \(-0.268898\pi\)
\(42\) 0 0
\(43\) 6.86758 6.86758i 1.04730 1.04730i 0.0484714 0.998825i \(-0.484565\pi\)
0.998825 0.0484714i \(-0.0154350\pi\)
\(44\) 0 0
\(45\) 3.90419 + 3.90419i 0.582003 + 0.582003i
\(46\) 0 0
\(47\) 9.70703 1.41592 0.707958 0.706255i \(-0.249617\pi\)
0.707958 + 0.706255i \(0.249617\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.59282 + 1.59282i 0.223040 + 0.223040i
\(52\) 0 0
\(53\) −7.64426 + 7.64426i −1.05002 + 1.05002i −0.0513390 + 0.998681i \(0.516349\pi\)
−0.998681 + 0.0513390i \(0.983651\pi\)
\(54\) 0 0
\(55\) 1.24519i 0.167902i
\(56\) 0 0
\(57\) 0.0747927i 0.00990653i
\(58\) 0 0
\(59\) 1.50266 1.50266i 0.195630 0.195630i −0.602494 0.798123i \(-0.705826\pi\)
0.798123 + 0.602494i \(0.205826\pi\)
\(60\) 0 0
\(61\) 1.74157 + 1.74157i 0.222985 + 0.222985i 0.809754 0.586769i \(-0.199601\pi\)
−0.586769 + 0.809754i \(0.699601\pi\)
\(62\) 0 0
\(63\) 2.78480 0.350851
\(64\) 0 0
\(65\) 6.32541 0.784571
\(66\) 0 0
\(67\) 8.96491 + 8.96491i 1.09524 + 1.09524i 0.994959 + 0.100279i \(0.0319735\pi\)
0.100279 + 0.994959i \(0.468027\pi\)
\(68\) 0 0
\(69\) 1.05142 1.05142i 0.126576 0.126576i
\(70\) 0 0
\(71\) 7.18356i 0.852532i 0.904598 + 0.426266i \(0.140171\pi\)
−0.904598 + 0.426266i \(0.859829\pi\)
\(72\) 0 0
\(73\) 9.04029i 1.05809i −0.848595 0.529043i \(-0.822551\pi\)
0.848595 0.529043i \(-0.177449\pi\)
\(74\) 0 0
\(75\) −0.350652 + 0.350652i −0.0404898 + 0.0404898i
\(76\) 0 0
\(77\) 0.444087 + 0.444087i 0.0506085 + 0.0506085i
\(78\) 0 0
\(79\) 9.58806 1.07874 0.539370 0.842069i \(-0.318662\pi\)
0.539370 + 0.842069i \(0.318662\pi\)
\(80\) 0 0
\(81\) −7.10949 −0.789943
\(82\) 0 0
\(83\) 5.30245 + 5.30245i 0.582020 + 0.582020i 0.935458 0.353438i \(-0.114988\pi\)
−0.353438 + 0.935458i \(0.614988\pi\)
\(84\) 0 0
\(85\) −6.80764 + 6.80764i −0.738392 + 0.738392i
\(86\) 0 0
\(87\) 0.654145i 0.0701317i
\(88\) 0 0
\(89\) 2.49938i 0.264934i 0.991187 + 0.132467i \(0.0422898\pi\)
−0.991187 + 0.132467i \(0.957710\pi\)
\(90\) 0 0
\(91\) 2.25590 2.25590i 0.236483 0.236483i
\(92\) 0 0
\(93\) 1.75309 + 1.75309i 0.181787 + 0.181787i
\(94\) 0 0
\(95\) 0.319660 0.0327964
\(96\) 0 0
\(97\) 5.89073 0.598113 0.299057 0.954235i \(-0.403328\pi\)
0.299057 + 0.954235i \(0.403328\pi\)
\(98\) 0 0
\(99\) −1.23669 1.23669i −0.124292 0.124292i
\(100\) 0 0
\(101\) 7.00499 7.00499i 0.697022 0.697022i −0.266745 0.963767i \(-0.585948\pi\)
0.963767 + 0.266745i \(0.0859482\pi\)
\(102\) 0 0
\(103\) 8.10887i 0.798991i −0.916735 0.399496i \(-0.869185\pi\)
0.916735 0.399496i \(-0.130815\pi\)
\(104\) 0 0
\(105\) 0.919765i 0.0897599i
\(106\) 0 0
\(107\) −2.24367 + 2.24367i −0.216904 + 0.216904i −0.807192 0.590288i \(-0.799014\pi\)
0.590288 + 0.807192i \(0.299014\pi\)
\(108\) 0 0
\(109\) −0.299413 0.299413i −0.0286786 0.0286786i 0.692622 0.721301i \(-0.256455\pi\)
−0.721301 + 0.692622i \(0.756455\pi\)
\(110\) 0 0
\(111\) −1.33489 −0.126702
\(112\) 0 0
\(113\) 4.67432 0.439723 0.219862 0.975531i \(-0.429439\pi\)
0.219862 + 0.975531i \(0.429439\pi\)
\(114\) 0 0
\(115\) 4.49372 + 4.49372i 0.419041 + 0.419041i
\(116\) 0 0
\(117\) −6.28224 + 6.28224i −0.580793 + 0.580793i
\(118\) 0 0
\(119\) 4.85578i 0.445128i
\(120\) 0 0
\(121\) 10.6056i 0.964143i
\(122\) 0 0
\(123\) 3.14142 3.14142i 0.283252 0.283252i
\(124\) 0 0
\(125\) −8.50850 8.50850i −0.761024 0.761024i
\(126\) 0 0
\(127\) 13.7602 1.22102 0.610509 0.792009i \(-0.290965\pi\)
0.610509 + 0.792009i \(0.290965\pi\)
\(128\) 0 0
\(129\) 4.50550 0.396687
\(130\) 0 0
\(131\) 6.91851 + 6.91851i 0.604473 + 0.604473i 0.941496 0.337023i \(-0.109420\pi\)
−0.337023 + 0.941496i \(0.609420\pi\)
\(132\) 0 0
\(133\) 0.114004 0.114004i 0.00988539 0.00988539i
\(134\) 0 0
\(135\) 5.32066i 0.457929i
\(136\) 0 0
\(137\) 7.10937i 0.607395i −0.952769 0.303697i \(-0.901779\pi\)
0.952769 0.303697i \(-0.0982211\pi\)
\(138\) 0 0
\(139\) 10.8171 10.8171i 0.917496 0.917496i −0.0793508 0.996847i \(-0.525285\pi\)
0.996847 + 0.0793508i \(0.0252847\pi\)
\(140\) 0 0
\(141\) 3.18416 + 3.18416i 0.268155 + 0.268155i
\(142\) 0 0
\(143\) −2.00364 −0.167553
\(144\) 0 0
\(145\) −2.79578 −0.232177
\(146\) 0 0
\(147\) −0.328027 0.328027i −0.0270552 0.0270552i
\(148\) 0 0
\(149\) −15.3257 + 15.3257i −1.25553 + 1.25553i −0.302325 + 0.953205i \(0.597763\pi\)
−0.953205 + 0.302325i \(0.902237\pi\)
\(150\) 0 0
\(151\) 21.5070i 1.75022i −0.483925 0.875109i \(-0.660789\pi\)
0.483925 0.875109i \(-0.339211\pi\)
\(152\) 0 0
\(153\) 13.5224i 1.09322i
\(154\) 0 0
\(155\) −7.49260 + 7.49260i −0.601821 + 0.601821i
\(156\) 0 0
\(157\) −7.72626 7.72626i −0.616623 0.616623i 0.328041 0.944664i \(-0.393612\pi\)
−0.944664 + 0.328041i \(0.893612\pi\)
\(158\) 0 0
\(159\) −5.01504 −0.397719
\(160\) 0 0
\(161\) 3.20529 0.252612
\(162\) 0 0
\(163\) −16.9336 16.9336i −1.32634 1.32634i −0.908538 0.417802i \(-0.862801\pi\)
−0.417802 0.908538i \(-0.637199\pi\)
\(164\) 0 0
\(165\) −0.408456 + 0.408456i −0.0317983 + 0.0317983i
\(166\) 0 0
\(167\) 8.57678i 0.663691i −0.943334 0.331846i \(-0.892329\pi\)
0.943334 0.331846i \(-0.107671\pi\)
\(168\) 0 0
\(169\) 2.82179i 0.217061i
\(170\) 0 0
\(171\) −0.317478 + 0.317478i −0.0242781 + 0.0242781i
\(172\) 0 0
\(173\) 5.44032 + 5.44032i 0.413620 + 0.413620i 0.882997 0.469378i \(-0.155522\pi\)
−0.469378 + 0.882997i \(0.655522\pi\)
\(174\) 0 0
\(175\) −1.06898 −0.0808069
\(176\) 0 0
\(177\) 0.985825 0.0740991
\(178\) 0 0
\(179\) 8.32748 + 8.32748i 0.622425 + 0.622425i 0.946151 0.323726i \(-0.104936\pi\)
−0.323726 + 0.946151i \(0.604936\pi\)
\(180\) 0 0
\(181\) 6.24148 6.24148i 0.463926 0.463926i −0.436014 0.899940i \(-0.643610\pi\)
0.899940 + 0.436014i \(0.143610\pi\)
\(182\) 0 0
\(183\) 1.14256i 0.0844605i
\(184\) 0 0
\(185\) 5.70524i 0.419457i
\(186\) 0 0
\(187\) 2.15639 2.15639i 0.157691 0.157691i
\(188\) 0 0
\(189\) 1.89757 + 1.89757i 0.138028 + 0.138028i
\(190\) 0 0
\(191\) −1.57521 −0.113979 −0.0569893 0.998375i \(-0.518150\pi\)
−0.0569893 + 0.998375i \(0.518150\pi\)
\(192\) 0 0
\(193\) 0.649369 0.0467426 0.0233713 0.999727i \(-0.492560\pi\)
0.0233713 + 0.999727i \(0.492560\pi\)
\(194\) 0 0
\(195\) 2.07490 + 2.07490i 0.148587 + 0.148587i
\(196\) 0 0
\(197\) 1.16865 1.16865i 0.0832632 0.0832632i −0.664249 0.747512i \(-0.731248\pi\)
0.747512 + 0.664249i \(0.231248\pi\)
\(198\) 0 0
\(199\) 4.09013i 0.289942i −0.989436 0.144971i \(-0.953691\pi\)
0.989436 0.144971i \(-0.0463088\pi\)
\(200\) 0 0
\(201\) 5.88146i 0.414846i
\(202\) 0 0
\(203\) −0.997091 + 0.997091i −0.0699821 + 0.0699821i
\(204\) 0 0
\(205\) 13.4263 + 13.4263i 0.937731 + 0.937731i
\(206\) 0 0
\(207\) −8.92609 −0.620406
\(208\) 0 0
\(209\) −0.101255 −0.00700398
\(210\) 0 0
\(211\) −19.0421 19.0421i −1.31091 1.31091i −0.920745 0.390164i \(-0.872418\pi\)
−0.390164 0.920745i \(-0.627582\pi\)
\(212\) 0 0
\(213\) −2.35640 + 2.35640i −0.161458 + 0.161458i
\(214\) 0 0
\(215\) 19.2562i 1.31326i
\(216\) 0 0
\(217\) 5.34435i 0.362798i
\(218\) 0 0
\(219\) 2.96546 2.96546i 0.200387 0.200387i
\(220\) 0 0
\(221\) −10.9542 10.9542i −0.736857 0.736857i
\(222\) 0 0
\(223\) −26.0121 −1.74190 −0.870950 0.491372i \(-0.836496\pi\)
−0.870950 + 0.491372i \(0.836496\pi\)
\(224\) 0 0
\(225\) 2.97688 0.198459
\(226\) 0 0
\(227\) 0.181612 + 0.181612i 0.0120540 + 0.0120540i 0.713108 0.701054i \(-0.247287\pi\)
−0.701054 + 0.713108i \(0.747287\pi\)
\(228\) 0 0
\(229\) 0.679595 0.679595i 0.0449089 0.0449089i −0.684296 0.729205i \(-0.739890\pi\)
0.729205 + 0.684296i \(0.239890\pi\)
\(230\) 0 0
\(231\) 0.291345i 0.0191691i
\(232\) 0 0
\(233\) 19.6676i 1.28847i 0.764828 + 0.644235i \(0.222824\pi\)
−0.764828 + 0.644235i \(0.777176\pi\)
\(234\) 0 0
\(235\) −13.6089 + 13.6089i −0.887749 + 0.887749i
\(236\) 0 0
\(237\) 3.14514 + 3.14514i 0.204299 + 0.204299i
\(238\) 0 0
\(239\) 9.44846 0.611170 0.305585 0.952165i \(-0.401148\pi\)
0.305585 + 0.952165i \(0.401148\pi\)
\(240\) 0 0
\(241\) −22.8233 −1.47018 −0.735088 0.677971i \(-0.762859\pi\)
−0.735088 + 0.677971i \(0.762859\pi\)
\(242\) 0 0
\(243\) −8.02480 8.02480i −0.514791 0.514791i
\(244\) 0 0
\(245\) 1.40197 1.40197i 0.0895684 0.0895684i
\(246\) 0 0
\(247\) 0.514364i 0.0327282i
\(248\) 0 0
\(249\) 3.47869i 0.220453i
\(250\) 0 0
\(251\) −12.2670 + 12.2670i −0.774287 + 0.774287i −0.978853 0.204566i \(-0.934422\pi\)
0.204566 + 0.978853i \(0.434422\pi\)
\(252\) 0 0
\(253\) −1.42343 1.42343i −0.0894903 0.0894903i
\(254\) 0 0
\(255\) −4.46618 −0.279683
\(256\) 0 0
\(257\) 7.71213 0.481070 0.240535 0.970641i \(-0.422677\pi\)
0.240535 + 0.970641i \(0.422677\pi\)
\(258\) 0 0
\(259\) −2.03472 2.03472i −0.126432 0.126432i
\(260\) 0 0
\(261\) 2.77670 2.77670i 0.171873 0.171873i
\(262\) 0 0
\(263\) 9.24972i 0.570362i 0.958474 + 0.285181i \(0.0920538\pi\)
−0.958474 + 0.285181i \(0.907946\pi\)
\(264\) 0 0
\(265\) 21.4340i 1.31668i
\(266\) 0 0
\(267\) −0.819863 + 0.819863i −0.0501748 + 0.0501748i
\(268\) 0 0
\(269\) −8.36587 8.36587i −0.510076 0.510076i 0.404474 0.914550i \(-0.367455\pi\)
−0.914550 + 0.404474i \(0.867455\pi\)
\(270\) 0 0
\(271\) 1.15740 0.0703071 0.0351535 0.999382i \(-0.488808\pi\)
0.0351535 + 0.999382i \(0.488808\pi\)
\(272\) 0 0
\(273\) 1.47999 0.0895733
\(274\) 0 0
\(275\) 0.474718 + 0.474718i 0.0286266 + 0.0286266i
\(276\) 0 0
\(277\) 16.6925 16.6925i 1.00296 1.00296i 0.00295975 0.999996i \(-0.499058\pi\)
0.999996 0.00295975i \(-0.000942120\pi\)
\(278\) 0 0
\(279\) 14.8829i 0.891018i
\(280\) 0 0
\(281\) 29.3656i 1.75180i 0.482490 + 0.875902i \(0.339732\pi\)
−0.482490 + 0.875902i \(0.660268\pi\)
\(282\) 0 0
\(283\) 13.0502 13.0502i 0.775752 0.775752i −0.203354 0.979105i \(-0.565184\pi\)
0.979105 + 0.203354i \(0.0651842\pi\)
\(284\) 0 0
\(285\) 0.104857 + 0.104857i 0.00621118 + 0.00621118i
\(286\) 0 0
\(287\) 9.57673 0.565296
\(288\) 0 0
\(289\) 6.57857 0.386975
\(290\) 0 0
\(291\) 1.93232 + 1.93232i 0.113274 + 0.113274i
\(292\) 0 0
\(293\) 19.4435 19.4435i 1.13590 1.13590i 0.146723 0.989178i \(-0.453127\pi\)
0.989178 0.146723i \(-0.0468727\pi\)
\(294\) 0 0
\(295\) 4.21336i 0.245311i
\(296\) 0 0
\(297\) 1.68537i 0.0977952i
\(298\) 0 0
\(299\) −7.23084 + 7.23084i −0.418170 + 0.418170i
\(300\) 0 0
\(301\) 6.86758 + 6.86758i 0.395841 + 0.395841i
\(302\) 0 0
\(303\) 4.59564 0.264013
\(304\) 0 0
\(305\) −4.88324 −0.279613
\(306\) 0 0
\(307\) 19.5562 + 19.5562i 1.11613 + 1.11613i 0.992303 + 0.123830i \(0.0395178\pi\)
0.123830 + 0.992303i \(0.460482\pi\)
\(308\) 0 0
\(309\) 2.65993 2.65993i 0.151318 0.151318i
\(310\) 0 0
\(311\) 24.5927i 1.39453i 0.716815 + 0.697263i \(0.245599\pi\)
−0.716815 + 0.697263i \(0.754401\pi\)
\(312\) 0 0
\(313\) 18.0125i 1.01812i 0.860730 + 0.509062i \(0.170008\pi\)
−0.860730 + 0.509062i \(0.829992\pi\)
\(314\) 0 0
\(315\) −3.90419 + 3.90419i −0.219976 + 0.219976i
\(316\) 0 0
\(317\) −9.70483 9.70483i −0.545078 0.545078i 0.379935 0.925013i \(-0.375946\pi\)
−0.925013 + 0.379935i \(0.875946\pi\)
\(318\) 0 0
\(319\) 0.885591 0.0495836
\(320\) 0 0
\(321\) −1.47197 −0.0821573
\(322\) 0 0
\(323\) −0.553578 0.553578i −0.0308019 0.0308019i
\(324\) 0 0
\(325\) 2.41151 2.41151i 0.133766 0.133766i
\(326\) 0 0
\(327\) 0.196431i 0.0108627i
\(328\) 0 0
\(329\) 9.70703i 0.535166i
\(330\) 0 0
\(331\) 1.50341 1.50341i 0.0826349 0.0826349i −0.664581 0.747216i \(-0.731390\pi\)
0.747216 + 0.664581i \(0.231390\pi\)
\(332\) 0 0
\(333\) 5.66630 + 5.66630i 0.310511 + 0.310511i
\(334\) 0 0
\(335\) −25.1370 −1.37338
\(336\) 0 0
\(337\) −17.7244 −0.965512 −0.482756 0.875755i \(-0.660364\pi\)
−0.482756 + 0.875755i \(0.660364\pi\)
\(338\) 0 0
\(339\) 1.53330 + 1.53330i 0.0832776 + 0.0832776i
\(340\) 0 0
\(341\) 2.37336 2.37336i 0.128525 0.128525i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.94812i 0.158721i
\(346\) 0 0
\(347\) 4.92098 4.92098i 0.264172 0.264172i −0.562574 0.826747i \(-0.690189\pi\)
0.826747 + 0.562574i \(0.190189\pi\)
\(348\) 0 0
\(349\) −24.8975 24.8975i −1.33274 1.33274i −0.902914 0.429821i \(-0.858577\pi\)
−0.429821 0.902914i \(-0.641423\pi\)
\(350\) 0 0
\(351\) −8.56146 −0.456977
\(352\) 0 0
\(353\) −5.71306 −0.304075 −0.152038 0.988375i \(-0.548584\pi\)
−0.152038 + 0.988375i \(0.548584\pi\)
\(354\) 0 0
\(355\) −10.0711 10.0711i −0.534519 0.534519i
\(356\) 0 0
\(357\) −1.59282 + 1.59282i −0.0843012 + 0.0843012i
\(358\) 0 0
\(359\) 25.2978i 1.33516i 0.744536 + 0.667582i \(0.232671\pi\)
−0.744536 + 0.667582i \(0.767329\pi\)
\(360\) 0 0
\(361\) 18.9740i 0.998632i
\(362\) 0 0
\(363\) −3.47891 + 3.47891i −0.182595 + 0.182595i
\(364\) 0 0
\(365\) 12.6742 + 12.6742i 0.663398 + 0.663398i
\(366\) 0 0
\(367\) −13.9944 −0.730499 −0.365250 0.930910i \(-0.619016\pi\)
−0.365250 + 0.930910i \(0.619016\pi\)
\(368\) 0 0
\(369\) −26.6692 −1.38835
\(370\) 0 0
\(371\) −7.64426 7.64426i −0.396870 0.396870i
\(372\) 0 0
\(373\) 6.44242 6.44242i 0.333576 0.333576i −0.520367 0.853943i \(-0.674205\pi\)
0.853943 + 0.520367i \(0.174205\pi\)
\(374\) 0 0
\(375\) 5.58203i 0.288255i
\(376\) 0 0
\(377\) 4.49869i 0.231694i
\(378\) 0 0
\(379\) −20.0039 + 20.0039i −1.02753 + 1.02753i −0.0279235 + 0.999610i \(0.508889\pi\)
−0.999610 + 0.0279235i \(0.991111\pi\)
\(380\) 0 0
\(381\) 4.51371 + 4.51371i 0.231244 + 0.231244i
\(382\) 0 0
\(383\) 8.23256 0.420664 0.210332 0.977630i \(-0.432545\pi\)
0.210332 + 0.977630i \(0.432545\pi\)
\(384\) 0 0
\(385\) −1.24519 −0.0634609
\(386\) 0 0
\(387\) −19.1248 19.1248i −0.972169 0.972169i
\(388\) 0 0
\(389\) 22.8204 22.8204i 1.15704 1.15704i 0.171930 0.985109i \(-0.445000\pi\)
0.985109 0.171930i \(-0.0550003\pi\)
\(390\) 0 0
\(391\) 15.5642i 0.787115i
\(392\) 0 0
\(393\) 4.53891i 0.228958i
\(394\) 0 0
\(395\) −13.4421 + 13.4421i −0.676348 + 0.676348i
\(396\) 0 0
\(397\) 20.2540 + 20.2540i 1.01652 + 1.01652i 0.999861 + 0.0166562i \(0.00530208\pi\)
0.0166562 + 0.999861i \(0.494698\pi\)
\(398\) 0 0
\(399\) 0.0747927 0.00374432
\(400\) 0 0
\(401\) −20.9844 −1.04791 −0.523956 0.851745i \(-0.675544\pi\)
−0.523956 + 0.851745i \(0.675544\pi\)
\(402\) 0 0
\(403\) −12.0563 12.0563i −0.600569 0.600569i
\(404\) 0 0
\(405\) 9.96727 9.96727i 0.495277 0.495277i
\(406\) 0 0
\(407\) 1.80719i 0.0895791i
\(408\) 0 0
\(409\) 22.9268i 1.13366i 0.823836 + 0.566828i \(0.191830\pi\)
−0.823836 + 0.566828i \(0.808170\pi\)
\(410\) 0 0
\(411\) 2.33206 2.33206i 0.115032 0.115032i
\(412\) 0 0
\(413\) 1.50266 + 1.50266i 0.0739410 + 0.0739410i
\(414\) 0 0
\(415\) −14.8677 −0.729828
\(416\) 0 0
\(417\) 7.09661 0.347522
\(418\) 0 0
\(419\) 27.8867 + 27.8867i 1.36235 + 1.36235i 0.870909 + 0.491444i \(0.163531\pi\)
0.491444 + 0.870909i \(0.336469\pi\)
\(420\) 0 0
\(421\) 3.93087 3.93087i 0.191579 0.191579i −0.604799 0.796378i \(-0.706747\pi\)
0.796378 + 0.604799i \(0.206747\pi\)
\(422\) 0 0
\(423\) 27.0321i 1.31435i
\(424\) 0 0
\(425\) 5.19071i 0.251786i
\(426\) 0 0
\(427\) −1.74157 + 1.74157i −0.0842803 + 0.0842803i
\(428\) 0 0
\(429\) −0.657247 0.657247i −0.0317322 0.0317322i
\(430\) 0 0
\(431\) −30.8494 −1.48596 −0.742981 0.669313i \(-0.766589\pi\)
−0.742981 + 0.669313i \(0.766589\pi\)
\(432\) 0 0
\(433\) −22.5480 −1.08359 −0.541794 0.840511i \(-0.682255\pi\)
−0.541794 + 0.840511i \(0.682255\pi\)
\(434\) 0 0
\(435\) −0.917090 0.917090i −0.0439711 0.0439711i
\(436\) 0 0
\(437\) −0.365416 + 0.365416i −0.0174802 + 0.0174802i
\(438\) 0 0
\(439\) 20.4983i 0.978329i 0.872192 + 0.489164i \(0.162698\pi\)
−0.872192 + 0.489164i \(0.837302\pi\)
\(440\) 0 0
\(441\) 2.78480i 0.132609i
\(442\) 0 0
\(443\) −15.8801 + 15.8801i −0.754485 + 0.754485i −0.975313 0.220828i \(-0.929124\pi\)
0.220828 + 0.975313i \(0.429124\pi\)
\(444\) 0 0
\(445\) −3.50405 3.50405i −0.166108 0.166108i
\(446\) 0 0
\(447\) −10.0545 −0.475560
\(448\) 0 0
\(449\) 28.3273 1.33685 0.668423 0.743781i \(-0.266969\pi\)
0.668423 + 0.743781i \(0.266969\pi\)
\(450\) 0 0
\(451\) −4.25290 4.25290i −0.200261 0.200261i
\(452\) 0 0
\(453\) 7.05488 7.05488i 0.331467 0.331467i
\(454\) 0 0
\(455\) 6.32541i 0.296540i
\(456\) 0 0
\(457\) 0.979734i 0.0458300i 0.999737 + 0.0229150i \(0.00729472\pi\)
−0.999737 + 0.0229150i \(0.992705\pi\)
\(458\) 0 0
\(459\) 9.21417 9.21417i 0.430080 0.430080i
\(460\) 0 0
\(461\) −25.7845 25.7845i −1.20091 1.20091i −0.973891 0.227014i \(-0.927104\pi\)
−0.227014 0.973891i \(-0.572896\pi\)
\(462\) 0 0
\(463\) 1.56380 0.0726760 0.0363380 0.999340i \(-0.488431\pi\)
0.0363380 + 0.999340i \(0.488431\pi\)
\(464\) 0 0
\(465\) −4.91555 −0.227953
\(466\) 0 0
\(467\) −28.6856 28.6856i −1.32741 1.32741i −0.907620 0.419792i \(-0.862103\pi\)
−0.419792 0.907620i \(-0.637897\pi\)
\(468\) 0 0
\(469\) −8.96491 + 8.96491i −0.413961 + 0.413961i
\(470\) 0 0
\(471\) 5.06884i 0.233560i
\(472\) 0 0
\(473\) 6.09961i 0.280460i
\(474\) 0 0
\(475\) 0.121867 0.121867i 0.00559166 0.00559166i
\(476\) 0 0
\(477\) 21.2877 + 21.2877i 0.974698 + 0.974698i
\(478\) 0 0
\(479\) −25.8040 −1.17902 −0.589508 0.807762i \(-0.700678\pi\)
−0.589508 + 0.807762i \(0.700678\pi\)
\(480\) 0 0
\(481\) 9.18029 0.418585
\(482\) 0 0
\(483\) 1.05142 + 1.05142i 0.0478413 + 0.0478413i
\(484\) 0 0
\(485\) −8.25861 + 8.25861i −0.375004 + 0.375004i
\(486\) 0 0
\(487\) 2.56961i 0.116440i 0.998304 + 0.0582201i \(0.0185425\pi\)
−0.998304 + 0.0582201i \(0.981457\pi\)
\(488\) 0 0
\(489\) 11.1093i 0.502381i
\(490\) 0 0
\(491\) 2.11127 2.11127i 0.0952804 0.0952804i −0.657860 0.753140i \(-0.728538\pi\)
0.753140 + 0.657860i \(0.228538\pi\)
\(492\) 0 0
\(493\) 4.84165 + 4.84165i 0.218057 + 0.218057i
\(494\) 0 0
\(495\) 3.46761 0.155857
\(496\) 0 0
\(497\) −7.18356 −0.322227
\(498\) 0 0
\(499\) −5.30410 5.30410i −0.237444 0.237444i 0.578347 0.815791i \(-0.303698\pi\)
−0.815791 + 0.578347i \(0.803698\pi\)
\(500\) 0 0
\(501\) 2.81341 2.81341i 0.125694 0.125694i
\(502\) 0 0
\(503\) 37.3432i 1.66505i −0.553987 0.832525i \(-0.686894\pi\)
0.553987 0.832525i \(-0.313106\pi\)
\(504\) 0 0
\(505\) 19.6415i 0.874036i
\(506\) 0 0
\(507\) 0.925621 0.925621i 0.0411083 0.0411083i
\(508\) 0 0
\(509\) −21.7303 21.7303i −0.963178 0.963178i 0.0361674 0.999346i \(-0.488485\pi\)
−0.999346 + 0.0361674i \(0.988485\pi\)
\(510\) 0 0
\(511\) 9.04029 0.399919
\(512\) 0 0
\(513\) −0.432660 −0.0191024
\(514\) 0 0
\(515\) 11.3684 + 11.3684i 0.500951 + 0.500951i
\(516\) 0 0
\(517\) 4.31077 4.31077i 0.189587 0.189587i
\(518\) 0 0
\(519\) 3.56914i 0.156668i
\(520\) 0 0
\(521\) 4.96517i 0.217528i 0.994068 + 0.108764i \(0.0346893\pi\)
−0.994068 + 0.108764i \(0.965311\pi\)
\(522\) 0 0
\(523\) 0.922937 0.922937i 0.0403572 0.0403572i −0.686640 0.726997i \(-0.740915\pi\)
0.726997 + 0.686640i \(0.240915\pi\)
\(524\) 0 0
\(525\) −0.350652 0.350652i −0.0153037 0.0153037i
\(526\) 0 0
\(527\) 25.9510 1.13044
\(528\) 0 0
\(529\) 12.7261 0.553309
\(530\) 0 0
\(531\) −4.18460 4.18460i −0.181596 0.181596i
\(532\) 0 0
\(533\) −21.6042 + 21.6042i −0.935781 + 0.935781i
\(534\) 0 0
\(535\) 6.29111i 0.271989i
\(536\) 0 0
\(537\) 5.46327i 0.235757i
\(538\) 0 0
\(539\) −0.444087 + 0.444087i −0.0191282 + 0.0191282i
\(540\) 0 0
\(541\) 20.1027 + 20.1027i 0.864281 + 0.864281i 0.991832 0.127551i \(-0.0407116\pi\)
−0.127551 + 0.991832i \(0.540712\pi\)
\(542\) 0 0
\(543\) 4.09474 0.175722
\(544\) 0 0
\(545\) 0.839535 0.0359617
\(546\) 0 0
\(547\) −16.7415 16.7415i −0.715813 0.715813i 0.251932 0.967745i \(-0.418934\pi\)
−0.967745 + 0.251932i \(0.918934\pi\)
\(548\) 0 0
\(549\) 4.84991 4.84991i 0.206989 0.206989i
\(550\) 0 0
\(551\) 0.227345i 0.00968521i
\(552\) 0 0
\(553\) 9.58806i 0.407726i
\(554\) 0 0
\(555\) 1.87147 1.87147i 0.0794394 0.0794394i
\(556\) 0 0
\(557\) −22.1766 22.1766i −0.939655 0.939655i 0.0586254 0.998280i \(-0.481328\pi\)
−0.998280 + 0.0586254i \(0.981328\pi\)
\(558\) 0 0
\(559\) −30.9852 −1.31053
\(560\) 0 0
\(561\) 1.41471 0.0597290
\(562\) 0 0
\(563\) 24.2669 + 24.2669i 1.02273 + 1.02273i 0.999736 + 0.0229924i \(0.00731934\pi\)
0.0229924 + 0.999736i \(0.492681\pi\)
\(564\) 0 0
\(565\) −6.55325 + 6.55325i −0.275697 + 0.275697i
\(566\) 0 0
\(567\) 7.10949i 0.298570i
\(568\) 0 0
\(569\) 42.8086i 1.79463i 0.441392 + 0.897315i \(0.354485\pi\)
−0.441392 + 0.897315i \(0.645515\pi\)
\(570\) 0 0
\(571\) −6.89908 + 6.89908i −0.288717 + 0.288717i −0.836573 0.547856i \(-0.815444\pi\)
0.547856 + 0.836573i \(0.315444\pi\)
\(572\) 0 0
\(573\) −0.516712 0.516712i −0.0215860 0.0215860i
\(574\) 0 0
\(575\) 3.42638 0.142890
\(576\) 0 0
\(577\) −2.38332 −0.0992189 −0.0496094 0.998769i \(-0.515798\pi\)
−0.0496094 + 0.998769i \(0.515798\pi\)
\(578\) 0 0
\(579\) 0.213010 + 0.213010i 0.00885240 + 0.00885240i
\(580\) 0 0
\(581\) −5.30245 + 5.30245i −0.219983 + 0.219983i
\(582\) 0 0
\(583\) 6.78944i 0.281190i
\(584\) 0 0
\(585\) 17.6150i 0.728290i
\(586\) 0 0
\(587\) −2.70993 + 2.70993i −0.111851 + 0.111851i −0.760817 0.648966i \(-0.775202\pi\)
0.648966 + 0.760817i \(0.275202\pi\)
\(588\) 0 0
\(589\) −0.609277 0.609277i −0.0251048 0.0251048i
\(590\) 0 0
\(591\) 0.766700 0.0315378
\(592\) 0 0
\(593\) 29.3573 1.20556 0.602780 0.797907i \(-0.294060\pi\)
0.602780 + 0.797907i \(0.294060\pi\)
\(594\) 0 0
\(595\) −6.80764 6.80764i −0.279086 0.279086i
\(596\) 0 0
\(597\) 1.34167 1.34167i 0.0549110 0.0549110i
\(598\) 0 0
\(599\) 40.3716i 1.64954i 0.565468 + 0.824770i \(0.308696\pi\)
−0.565468 + 0.824770i \(0.691304\pi\)
\(600\) 0 0
\(601\) 30.9931i 1.26424i −0.774872 0.632118i \(-0.782186\pi\)
0.774872 0.632118i \(-0.217814\pi\)
\(602\) 0 0
\(603\) 24.9655 24.9655i 1.01667 1.01667i
\(604\) 0 0
\(605\) −14.8687 14.8687i −0.604497 0.604497i
\(606\) 0 0
\(607\) 9.81876 0.398531 0.199265 0.979946i \(-0.436144\pi\)
0.199265 + 0.979946i \(0.436144\pi\)
\(608\) 0 0
\(609\) −0.654145 −0.0265073
\(610\) 0 0
\(611\) −21.8981 21.8981i −0.885904 0.885904i
\(612\) 0 0
\(613\) 5.05548 5.05548i 0.204189 0.204189i −0.597603 0.801792i \(-0.703880\pi\)
0.801792 + 0.597603i \(0.203880\pi\)
\(614\) 0 0
\(615\) 8.80834i 0.355187i
\(616\) 0 0
\(617\) 34.8190i 1.40176i 0.713279 + 0.700881i \(0.247209\pi\)
−0.713279 + 0.700881i \(0.752791\pi\)
\(618\) 0 0
\(619\) −34.1722 + 34.1722i −1.37350 + 1.37350i −0.518294 + 0.855203i \(0.673433\pi\)
−0.855203 + 0.518294i \(0.826567\pi\)
\(620\) 0 0
\(621\) −6.08226 6.08226i −0.244073 0.244073i
\(622\) 0 0
\(623\) −2.49938 −0.100135
\(624\) 0 0
\(625\) 18.5124 0.740497
\(626\) 0 0
\(627\) −0.0332145 0.0332145i −0.00132646 0.00132646i
\(628\) 0 0
\(629\) −9.88017 + 9.88017i −0.393948 + 0.393948i
\(630\) 0 0
\(631\) 1.37799i 0.0548569i 0.999624 + 0.0274285i \(0.00873184\pi\)
−0.999624 + 0.0274285i \(0.991268\pi\)
\(632\) 0 0
\(633\) 12.4926i 0.496537i
\(634\) 0 0
\(635\) −19.2913 + 19.2913i −0.765553 + 0.765553i
\(636\) 0 0
\(637\) 2.25590 + 2.25590i 0.0893822 + 0.0893822i
\(638\) 0 0
\(639\) 20.0048 0.791376
\(640\) 0 0
\(641\) −1.24423 −0.0491440 −0.0245720 0.999698i \(-0.507822\pi\)
−0.0245720 + 0.999698i \(0.507822\pi\)
\(642\) 0 0
\(643\) 25.0773 + 25.0773i 0.988954 + 0.988954i 0.999940 0.0109861i \(-0.00349707\pi\)
−0.0109861 + 0.999940i \(0.503497\pi\)
\(644\) 0 0
\(645\) −6.31656 + 6.31656i −0.248714 + 0.248714i
\(646\) 0 0
\(647\) 28.0343i 1.10214i −0.834458 0.551071i \(-0.814219\pi\)
0.834458 0.551071i \(-0.185781\pi\)
\(648\) 0 0
\(649\) 1.33462i 0.0523886i
\(650\) 0 0
\(651\) −1.75309 + 1.75309i −0.0687090 + 0.0687090i
\(652\) 0 0
\(653\) 18.5604 + 18.5604i 0.726324 + 0.726324i 0.969886 0.243561i \(-0.0783157\pi\)
−0.243561 + 0.969886i \(0.578316\pi\)
\(654\) 0 0
\(655\) −19.3991 −0.757984
\(656\) 0 0
\(657\) −25.1754 −0.982185
\(658\) 0 0
\(659\) 2.92378 + 2.92378i 0.113894 + 0.113894i 0.761757 0.647863i \(-0.224337\pi\)
−0.647863 + 0.761757i \(0.724337\pi\)
\(660\) 0 0
\(661\) 1.13234 1.13234i 0.0440428 0.0440428i −0.684742 0.728785i \(-0.740085\pi\)
0.728785 + 0.684742i \(0.240085\pi\)
\(662\) 0 0
\(663\) 7.18652i 0.279101i
\(664\) 0 0
\(665\) 0.319660i 0.0123959i
\(666\) 0 0
\(667\) 3.19597 3.19597i 0.123748 0.123748i
\(668\) 0 0
\(669\) −8.53267 8.53267i −0.329892 0.329892i
\(670\) 0 0
\(671\) 1.54681 0.0597141
\(672\) 0 0
\(673\) −50.3603 −1.94125 −0.970623 0.240607i \(-0.922654\pi\)
−0.970623 + 0.240607i \(0.922654\pi\)
\(674\) 0 0
\(675\) 2.02845 + 2.02845i 0.0780752 + 0.0780752i
\(676\) 0 0
\(677\) 7.37404 7.37404i 0.283407 0.283407i −0.551059 0.834466i \(-0.685776\pi\)
0.834466 + 0.551059i \(0.185776\pi\)
\(678\) 0 0
\(679\) 5.89073i 0.226065i
\(680\) 0 0
\(681\) 0.119147i 0.00456573i
\(682\) 0 0
\(683\) 23.3569 23.3569i 0.893726 0.893726i −0.101146 0.994872i \(-0.532251\pi\)
0.994872 + 0.101146i \(0.0322509\pi\)
\(684\) 0 0
\(685\) 9.96711 + 9.96711i 0.380824 + 0.380824i
\(686\) 0 0
\(687\) 0.445851 0.0170103
\(688\) 0 0
\(689\) 34.4895 1.31394
\(690\) 0 0
\(691\) 12.1302 + 12.1302i 0.461453 + 0.461453i 0.899132 0.437679i \(-0.144199\pi\)
−0.437679 + 0.899132i \(0.644199\pi\)
\(692\) 0 0
\(693\) 1.23669 1.23669i 0.0469781 0.0469781i
\(694\) 0 0
\(695\) 30.3305i 1.15050i
\(696\) 0 0
\(697\) 46.5025i 1.76141i
\(698\) 0 0
\(699\) −6.45151 + 6.45151i −0.244019 + 0.244019i
\(700\) 0 0
\(701\) −4.41607 4.41607i −0.166793 0.166793i 0.618775 0.785568i \(-0.287629\pi\)
−0.785568 + 0.618775i \(0.787629\pi\)
\(702\) 0 0
\(703\) 0.463933 0.0174976
\(704\) 0 0
\(705\) −8.92819 −0.336255
\(706\) 0 0
\(707\) 7.00499 + 7.00499i 0.263450 + 0.263450i
\(708\) 0 0
\(709\) −2.70925 + 2.70925i −0.101748 + 0.101748i −0.756148 0.654400i \(-0.772921\pi\)
0.654400 + 0.756148i \(0.272921\pi\)
\(710\) 0 0
\(711\) 26.7008i 1.00136i
\(712\) 0 0
\(713\) 17.1302i 0.641531i
\(714\) 0 0
\(715\) 2.80903 2.80903i 0.105052 0.105052i
\(716\) 0 0
\(717\) 3.09935 + 3.09935i 0.115747 + 0.115747i
\(718\) 0 0
\(719\) 8.99485 0.335451 0.167726 0.985834i \(-0.446358\pi\)
0.167726 + 0.985834i \(0.446358\pi\)
\(720\) 0 0
\(721\) 8.10887 0.301990
\(722\) 0 0
\(723\) −7.48664 7.48664i −0.278431 0.278431i
\(724\) 0 0
\(725\) −1.06587 + 1.06587i −0.0395853 + 0.0395853i
\(726\) 0 0
\(727\) 33.6636i 1.24851i −0.781219 0.624256i \(-0.785402\pi\)
0.781219 0.624256i \(-0.214598\pi\)
\(728\) 0 0
\(729\) 16.0638i 0.594954i
\(730\) 0 0
\(731\) 33.3474 33.3474i 1.23340 1.23340i
\(732\) 0 0
\(733\) 29.0542 + 29.0542i 1.07314 + 1.07314i 0.997105 + 0.0760379i \(0.0242270\pi\)
0.0760379 + 0.997105i \(0.475773\pi\)
\(734\) 0 0
\(735\) 0.919765 0.0339261
\(736\) 0 0
\(737\) 7.96241 0.293299
\(738\) 0 0
\(739\) 5.63034 + 5.63034i 0.207115 + 0.207115i 0.803040 0.595925i \(-0.203214\pi\)
−0.595925 + 0.803040i \(0.703214\pi\)
\(740\) 0 0
\(741\) −0.168725 + 0.168725i −0.00619827 + 0.00619827i
\(742\) 0 0
\(743\) 27.7688i 1.01874i 0.860548 + 0.509369i \(0.170121\pi\)
−0.860548 + 0.509369i \(0.829879\pi\)
\(744\) 0 0
\(745\) 42.9723i 1.57438i
\(746\) 0 0
\(747\) 14.7663 14.7663i 0.540269 0.540269i
\(748\) 0 0
\(749\) −2.24367 2.24367i −0.0819820 0.0819820i
\(750\) 0 0
\(751\) 45.5138 1.66082 0.830410 0.557153i \(-0.188106\pi\)
0.830410 + 0.557153i \(0.188106\pi\)
\(752\) 0 0
\(753\) −8.04781 −0.293279
\(754\) 0 0
\(755\) 30.1522 + 30.1522i 1.09735 + 1.09735i
\(756\) 0 0
\(757\) −20.7637 + 20.7637i −0.754671 + 0.754671i −0.975347 0.220676i \(-0.929174\pi\)
0.220676 + 0.975347i \(0.429174\pi\)
\(758\) 0 0
\(759\) 0.933846i 0.0338965i
\(760\) 0 0
\(761\) 3.64393i 0.132092i −0.997817 0.0660461i \(-0.978962\pi\)
0.997817 0.0660461i \(-0.0210384\pi\)
\(762\) 0 0
\(763\) 0.299413 0.299413i 0.0108395 0.0108395i
\(764\) 0 0
\(765\) 18.9579 + 18.9579i 0.685424 + 0.685424i
\(766\) 0 0
\(767\) −6.77971 −0.244801
\(768\) 0 0
\(769\) −6.23588 −0.224872 −0.112436 0.993659i \(-0.535865\pi\)
−0.112436 + 0.993659i \(0.535865\pi\)
\(770\) 0 0
\(771\) 2.52979 + 2.52979i 0.0911080 + 0.0911080i
\(772\) 0 0
\(773\) 1.49286 1.49286i 0.0536943 0.0536943i −0.679750 0.733444i \(-0.737912\pi\)
0.733444 + 0.679750i \(0.237912\pi\)
\(774\) 0 0
\(775\) 5.71298i 0.205216i
\(776\) 0 0
\(777\) 1.33489i 0.0478888i
\(778\) 0 0
\(779\) −1.09178 + 1.09178i −0.0391172 + 0.0391172i
\(780\) 0 0
\(781\) 3.19013 + 3.19013i 0.114152 + 0.114152i
\(782\) 0 0
\(783\) 3.78410 0.135233
\(784\) 0 0
\(785\) 21.6639 0.773219
\(786\) 0 0
\(787\) 8.76929 + 8.76929i 0.312591 + 0.312591i 0.845913 0.533321i \(-0.179056\pi\)
−0.533321 + 0.845913i \(0.679056\pi\)
\(788\) 0 0
\(789\) −3.03416 + 3.03416i −0.108019 + 0.108019i
\(790\) 0 0
\(791\) 4.67432i 0.166200i
\(792\) 0 0
\(793\) 7.85761i 0.279032i
\(794\) 0 0
\(795\) 7.03093 7.03093i 0.249361 0.249361i
\(796\) 0 0
\(797\) −22.7473 22.7473i −0.805750 0.805750i 0.178238 0.983987i \(-0.442960\pi\)
−0.983987 + 0.178238i \(0.942960\pi\)
\(798\) 0 0
\(799\) 47.1352 1.66752
\(800\) 0 0
\(801\) 6.96026 0.245929
\(802\) 0 0
\(803\) −4.01468 4.01468i −0.141675 0.141675i
\(804\) 0 0
\(805\) −4.49372 + 4.49372i −0.158383 + 0.158383i
\(806\) 0 0
\(807\) 5.48846i 0.193203i
\(808\) 0 0
\(809\) 2.69445i 0.0947319i −0.998878 0.0473660i \(-0.984917\pi\)
0.998878 0.0473660i \(-0.0150827\pi\)
\(810\) 0 0
\(811\) 30.5311 30.5311i 1.07209 1.07209i 0.0749015 0.997191i \(-0.476136\pi\)
0.997191 0.0749015i \(-0.0238642\pi\)
\(812\) 0 0
\(813\) 0.379658 + 0.379658i 0.0133152 + 0.0133152i
\(814\) 0 0
\(815\) 47.4806 1.66317
\(816\) 0 0
\(817\) −1.56586 −0.0547826
\(818\) 0 0
\(819\) −6.28224 6.28224i −0.219519 0.219519i
\(820\) 0 0
\(821\) −15.4510 + 15.4510i −0.539244 + 0.539244i −0.923307 0.384063i \(-0.874525\pi\)
0.384063 + 0.923307i \(0.374525\pi\)
\(822\) 0 0
\(823\) 22.4104i 0.781178i −0.920565 0.390589i \(-0.872271\pi\)
0.920565 0.390589i \(-0.127729\pi\)
\(824\) 0 0
\(825\) 0.311441i 0.0108430i
\(826\) 0 0
\(827\) −15.9620 + 15.9620i −0.555052 + 0.555052i −0.927895 0.372842i \(-0.878383\pi\)
0.372842 + 0.927895i \(0.378383\pi\)
\(828\) 0 0
\(829\) 6.88152 + 6.88152i 0.239005 + 0.239005i 0.816438 0.577433i \(-0.195946\pi\)
−0.577433 + 0.816438i \(0.695946\pi\)
\(830\) 0 0
\(831\) 10.9512 0.379892
\(832\) 0 0
\(833\) −4.85578 −0.168243
\(834\) 0 0
\(835\) 12.0244 + 12.0244i 0.416120 + 0.416120i
\(836\) 0 0
\(837\) 10.1413 10.1413i 0.350533 0.350533i
\(838\) 0 0
\(839\) 24.1568i 0.833986i 0.908910 + 0.416993i \(0.136916\pi\)
−0.908910 + 0.416993i \(0.863084\pi\)
\(840\) 0 0
\(841\) 27.0116i 0.931435i
\(842\) 0 0
\(843\) −9.63269 + 9.63269i −0.331767 + 0.331767i
\(844\) 0 0
\(845\) 3.95605 + 3.95605i 0.136092 + 0.136092i
\(846\) 0 0
\(847\) −10.6056 −0.364412
\(848\) 0 0
\(849\) 8.56160 0.293833
\(850\) 0 0
\(851\) 6.52189 + 6.52189i 0.223567 + 0.223567i
\(852\) 0 0
\(853\) 21.4719 21.4719i 0.735183 0.735183i −0.236458 0.971642i \(-0.575987\pi\)
0.971642 + 0.236458i \(0.0759867\pi\)
\(854\) 0 0
\(855\) 0.890187i 0.0304438i
\(856\) 0 0
\(857\) 3.52721i 0.120487i −0.998184 0.0602436i \(-0.980812\pi\)
0.998184 0.0602436i \(-0.0191877\pi\)
\(858\) 0 0
\(859\) 38.9597 38.9597i 1.32929 1.32929i 0.423294 0.905992i \(-0.360874\pi\)
0.905992 0.423294i \(-0.139126\pi\)
\(860\) 0 0
\(861\) 3.14142 + 3.14142i 0.107059 + 0.107059i
\(862\) 0 0
\(863\) 18.9798 0.646081 0.323040 0.946385i \(-0.395295\pi\)
0.323040 + 0.946385i \(0.395295\pi\)
\(864\) 0 0
\(865\) −15.2543 −0.518662
\(866\) 0 0
\(867\) 2.15795 + 2.15795i 0.0732877 + 0.0732877i
\(868\) 0 0
\(869\) 4.25794 4.25794i 0.144441 0.144441i
\(870\) 0 0
\(871\) 40.4480i 1.37053i
\(872\) 0 0
\(873\) 16.4045i 0.555208i
\(874\) 0 0
\(875\) 8.50850 8.50850i 0.287640 0.287640i
\(876\) 0 0
\(877\) 6.54582 + 6.54582i 0.221037 + 0.221037i 0.808935 0.587898i \(-0.200044\pi\)
−0.587898 + 0.808935i \(0.700044\pi\)
\(878\) 0 0
\(879\) 12.7560 0.430248
\(880\) 0 0
\(881\) 26.2370 0.883946 0.441973 0.897028i \(-0.354279\pi\)
0.441973 + 0.897028i \(0.354279\pi\)
\(882\) 0 0
\(883\) −19.3060 19.3060i −0.649700 0.649700i 0.303221 0.952920i \(-0.401938\pi\)
−0.952920 + 0.303221i \(0.901938\pi\)
\(884\) 0 0
\(885\) −1.38209 + 1.38209i −0.0464586 + 0.0464586i
\(886\) 0 0
\(887\) 3.40903i 0.114464i 0.998361 + 0.0572321i \(0.0182275\pi\)
−0.998361 + 0.0572321i \(0.981773\pi\)
\(888\) 0 0
\(889\) 13.7602i 0.461502i
\(890\) 0 0
\(891\) −3.15723 + 3.15723i −0.105771 + 0.105771i
\(892\) 0 0
\(893\) −1.10664 1.10664i −0.0370323 0.0370323i
\(894\) 0 0
\(895\) −23.3497 −0.780494
\(896\) 0 0
\(897\) −4.74381 −0.158391
\(898\) 0 0
\(899\) 5.32880 + 5.32880i 0.177726 + 0.177726i
\(900\) 0 0
\(901\) −37.1188 + 37.1188i −1.23661 + 1.23661i
\(902\) 0 0
\(903\) 4.50550i 0.149934i
\(904\) 0 0
\(905\) 17.5007i 0.581743i
\(906\) 0 0
\(907\) 17.9095 17.9095i 0.594675 0.594675i −0.344216 0.938891i \(-0.611855\pi\)
0.938891 + 0.344216i \(0.111855\pi\)
\(908\) 0 0
\(909\) −19.5075 19.5075i −0.647022 0.647022i
\(910\) 0 0
\(911\) 23.6991 0.785186 0.392593 0.919712i \(-0.371578\pi\)
0.392593 + 0.919712i \(0.371578\pi\)
\(912\) 0 0
\(913\) 4.70950 0.155862
\(914\) 0 0
\(915\) −1.60183 1.60183i −0.0529549 0.0529549i
\(916\) 0 0
\(917\) −6.91851 + 6.91851i −0.228469 + 0.228469i
\(918\) 0 0
\(919\) 29.7662i 0.981897i 0.871189 + 0.490949i \(0.163350\pi\)
−0.871189 + 0.490949i \(0.836650\pi\)
\(920\) 0 0
\(921\) 12.8299i 0.422761i
\(922\) 0 0
\(923\) 16.2054 16.2054i 0.533408 0.533408i
\(924\) 0 0
\(925\) −2.17507 2.17507i −0.0715159 0.0715159i
\(926\) 0 0
\(927\) −22.5816 −0.741676
\(928\) 0 0
\(929\) 46.6355 1.53006 0.765031 0.643993i \(-0.222723\pi\)
0.765031 + 0.643993i \(0.222723\pi\)
\(930\) 0 0
\(931\) 0.114004 + 0.114004i 0.00373633 + 0.00373633i
\(932\) 0 0
\(933\) −8.06707 + 8.06707i −0.264104 + 0.264104i
\(934\) 0 0
\(935\) 6.04638i 0.197738i
\(936\) 0 0
\(937\) 13.5471i 0.442564i −0.975210 0.221282i \(-0.928976\pi\)
0.975210 0.221282i \(-0.0710241\pi\)
\(938\) 0 0
\(939\) −5.90856 + 5.90856i −0.192819 + 0.192819i
\(940\) 0 0
\(941\) −13.5931 13.5931i −0.443121 0.443121i 0.449938 0.893060i \(-0.351446\pi\)
−0.893060 + 0.449938i \(0.851446\pi\)
\(942\) 0 0
\(943\) −30.6962 −0.999606
\(944\) 0 0
\(945\) −5.32066 −0.173081
\(946\) 0 0
\(947\) 28.7780 + 28.7780i 0.935159 + 0.935159i 0.998022 0.0628630i \(-0.0200231\pi\)
−0.0628630 + 0.998022i \(0.520023\pi\)
\(948\) 0 0
\(949\) −20.3940 + 20.3940i −0.662019 + 0.662019i
\(950\) 0 0
\(951\) 6.36689i 0.206460i
\(952\) 0 0
\(953\) 30.3026i 0.981598i −0.871273 0.490799i \(-0.836705\pi\)
0.871273 0.490799i \(-0.163295\pi\)
\(954\) 0 0
\(955\) 2.20840 2.20840i 0.0714621 0.0714621i
\(956\) 0 0
\(957\) 0.290498 + 0.290498i 0.00939045 + 0.00939045i
\(958\) 0 0
\(959\) 7.10937 0.229574
\(960\) 0 0
\(961\) −2.43792 −0.0786425
\(962\) 0 0
\(963\) 6.24818 + 6.24818i 0.201345 + 0.201345i
\(964\) 0 0
\(965\) −0.910393 + 0.910393i −0.0293066 + 0.0293066i
\(966\) 0 0
\(967\) 18.8733i 0.606925i 0.952843 + 0.303462i \(0.0981427\pi\)
−0.952843 + 0.303462i \(0.901857\pi\)
\(968\) 0 0
\(969\) 0.363176i 0.0116669i
\(970\) 0 0
\(971\) −28.5680 + 28.5680i −0.916790 + 0.916790i −0.996795 0.0800042i \(-0.974507\pi\)
0.0800042 + 0.996795i \(0.474507\pi\)
\(972\) 0 0
\(973\) 10.8171 + 10.8171i 0.346781 + 0.346781i
\(974\) 0 0
\(975\) 1.58208 0.0506670
\(976\) 0 0
\(977\) 16.4828 0.527331 0.263666 0.964614i \(-0.415068\pi\)
0.263666 + 0.964614i \(0.415068\pi\)
\(978\) 0 0
\(979\) 1.10994 + 1.10994i 0.0354739 + 0.0354739i
\(980\) 0 0
\(981\) −0.833805 + 0.833805i −0.0266213 + 0.0266213i
\(982\) 0 0
\(983\) 33.4039i 1.06542i 0.846298 + 0.532710i \(0.178826\pi\)
−0.846298 + 0.532710i \(0.821174\pi\)
\(984\) 0 0
\(985\) 3.27683i 0.104408i
\(986\) 0 0
\(987\) −3.18416 + 3.18416i −0.101353 + 0.101353i
\(988\) 0 0
\(989\) −22.0126 22.0126i −0.699960 0.699960i
\(990\) 0 0
\(991\) 25.3648 0.805740 0.402870 0.915257i \(-0.368013\pi\)
0.402870 + 0.915257i \(0.368013\pi\)
\(992\) 0 0
\(993\) 0.986318 0.0312998
\(994\) 0 0
\(995\) 5.73423 + 5.73423i 0.181787 + 0.181787i
\(996\) 0 0
\(997\) −11.1405 + 11.1405i −0.352824 + 0.352824i −0.861159 0.508336i \(-0.830261\pi\)
0.508336 + 0.861159i \(0.330261\pi\)
\(998\) 0 0
\(999\) 7.72206i 0.244315i
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 1792.2.m.f.1345.5 yes 16
4.3 odd 2 1792.2.m.h.1345.4 yes 16
8.3 odd 2 1792.2.m.e.1345.5 yes 16
8.5 even 2 1792.2.m.g.1345.4 yes 16
16.3 odd 4 1792.2.m.e.449.5 16
16.5 even 4 inner 1792.2.m.f.449.5 yes 16
16.11 odd 4 1792.2.m.h.449.4 yes 16
16.13 even 4 1792.2.m.g.449.4 yes 16
32.5 even 8 7168.2.a.be.1.4 8
32.11 odd 8 7168.2.a.bb.1.4 8
32.21 even 8 7168.2.a.ba.1.5 8
32.27 odd 8 7168.2.a.bf.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.5 16 16.3 odd 4
1792.2.m.e.1345.5 yes 16 8.3 odd 2
1792.2.m.f.449.5 yes 16 16.5 even 4 inner
1792.2.m.f.1345.5 yes 16 1.1 even 1 trivial
1792.2.m.g.449.4 yes 16 16.13 even 4
1792.2.m.g.1345.4 yes 16 8.5 even 2
1792.2.m.h.449.4 yes 16 16.11 odd 4
1792.2.m.h.1345.4 yes 16 4.3 odd 2
7168.2.a.ba.1.5 8 32.21 even 8
7168.2.a.bb.1.4 8 32.11 odd 8
7168.2.a.be.1.4 8 32.5 even 8
7168.2.a.bf.1.5 8 32.27 odd 8