Properties

Label 896.2.m.g.673.1
Level $896$
Weight $2$
Character 896.673
Analytic conductor $7.155$
Analytic rank $0$
Dimension $12$
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] = [896,2,Mod(225,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, 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("896.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.m (of order \(4\), degree \(2\), not 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.20138089353117696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 673.1
Root \(1.35309 + 0.411286i\) of defining polynomial
Character \(\chi\) \(=\) 896.673
Dual form 896.2.m.g.225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.21570 - 2.21570i) q^{3} +(0.393125 - 0.393125i) q^{5} -1.00000i q^{7} +6.81864i q^{9} +O(q^{10})\) \(q+(-2.21570 - 2.21570i) q^{3} +(0.393125 - 0.393125i) q^{5} -1.00000i q^{7} +6.81864i q^{9} +(-2.22602 + 2.22602i) q^{11} +(3.16316 + 3.16316i) q^{13} -1.74209 q^{15} +0.980951 q^{17} +(5.26429 + 5.26429i) q^{19} +(-2.21570 + 2.21570i) q^{21} -1.25951i q^{23} +4.69090i q^{25} +(8.46094 - 8.46094i) q^{27} +(-3.17349 - 3.17349i) q^{29} -4.43140 q^{31} +9.86440 q^{33} +(-0.393125 - 0.393125i) q^{35} +(-0.645145 + 0.645145i) q^{37} -14.0172i q^{39} +1.21375i q^{41} +(-0.966515 + 0.966515i) q^{43} +(2.68058 + 2.68058i) q^{45} +9.97147 q^{47} -1.00000 q^{49} +(-2.17349 - 2.17349i) q^{51} +(8.07814 - 8.07814i) q^{53} +1.75021i q^{55} -23.3282i q^{57} +(-1.81385 + 1.81385i) q^{59} +(2.58783 + 2.58783i) q^{61} +6.81864 q^{63} +2.48704 q^{65} +(1.59261 + 1.59261i) q^{67} +(-2.79069 + 2.79069i) q^{69} +0.934634i q^{71} +0.710511i q^{73} +(10.3936 - 10.3936i) q^{75} +(2.22602 + 2.22602i) q^{77} +11.8644 q^{79} -17.0379 q^{81} +(6.77482 + 6.77482i) q^{83} +(0.385637 - 0.385637i) q^{85} +14.0630i q^{87} +10.2082i q^{89} +(3.16316 - 3.16316i) q^{91} +(9.81864 + 9.81864i) q^{93} +4.13906 q^{95} -3.03684 q^{97} +(-15.1784 - 15.1784i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 4 q^{5} - 24 q^{15} - 8 q^{17} - 4 q^{21} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{35} + 20 q^{37} - 16 q^{43} - 40 q^{45} + 16 q^{47} - 12 q^{49} + 16 q^{51} - 4 q^{53} + 16 q^{59} + 20 q^{61} + 12 q^{63} + 32 q^{65} - 24 q^{67} + 4 q^{69} + 40 q^{75} + 24 q^{79} - 44 q^{81} + 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −2.21570 2.21570i −1.27923 1.27923i −0.941097 0.338137i \(-0.890203\pi\)
−0.338137 0.941097i \(-0.609797\pi\)
\(4\) 0 0
\(5\) 0.393125 0.393125i 0.175811 0.175811i −0.613716 0.789527i \(-0.710326\pi\)
0.789527 + 0.613716i \(0.210326\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 6.81864i 2.27288i
\(10\) 0 0
\(11\) −2.22602 + 2.22602i −0.671172 + 0.671172i −0.957986 0.286815i \(-0.907404\pi\)
0.286815 + 0.957986i \(0.407404\pi\)
\(12\) 0 0
\(13\) 3.16316 + 3.16316i 0.877304 + 0.877304i 0.993255 0.115951i \(-0.0369916\pi\)
−0.115951 + 0.993255i \(0.536992\pi\)
\(14\) 0 0
\(15\) −1.74209 −0.449807
\(16\) 0 0
\(17\) 0.980951 0.237915 0.118958 0.992899i \(-0.462045\pi\)
0.118958 + 0.992899i \(0.462045\pi\)
\(18\) 0 0
\(19\) 5.26429 + 5.26429i 1.20771 + 1.20771i 0.971766 + 0.235946i \(0.0758188\pi\)
0.235946 + 0.971766i \(0.424181\pi\)
\(20\) 0 0
\(21\) −2.21570 + 2.21570i −0.483505 + 0.483505i
\(22\) 0 0
\(23\) 1.25951i 0.262626i −0.991341 0.131313i \(-0.958081\pi\)
0.991341 0.131313i \(-0.0419192\pi\)
\(24\) 0 0
\(25\) 4.69090i 0.938181i
\(26\) 0 0
\(27\) 8.46094 8.46094i 1.62831 1.62831i
\(28\) 0 0
\(29\) −3.17349 3.17349i −0.589302 0.589302i 0.348140 0.937443i \(-0.386813\pi\)
−0.937443 + 0.348140i \(0.886813\pi\)
\(30\) 0 0
\(31\) −4.43140 −0.795902 −0.397951 0.917407i \(-0.630279\pi\)
−0.397951 + 0.917407i \(0.630279\pi\)
\(32\) 0 0
\(33\) 9.86440 1.71717
\(34\) 0 0
\(35\) −0.393125 0.393125i −0.0664503 0.0664503i
\(36\) 0 0
\(37\) −0.645145 + 0.645145i −0.106061 + 0.106061i −0.758146 0.652085i \(-0.773895\pi\)
0.652085 + 0.758146i \(0.273895\pi\)
\(38\) 0 0
\(39\) 14.0172i 2.24455i
\(40\) 0 0
\(41\) 1.21375i 0.189556i 0.995498 + 0.0947779i \(0.0302141\pi\)
−0.995498 + 0.0947779i \(0.969786\pi\)
\(42\) 0 0
\(43\) −0.966515 + 0.966515i −0.147392 + 0.147392i −0.776952 0.629560i \(-0.783235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(44\) 0 0
\(45\) 2.68058 + 2.68058i 0.399597 + 0.399597i
\(46\) 0 0
\(47\) 9.97147 1.45449 0.727244 0.686379i \(-0.240801\pi\)
0.727244 + 0.686379i \(0.240801\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.17349 2.17349i −0.304350 0.304350i
\(52\) 0 0
\(53\) 8.07814 8.07814i 1.10962 1.10962i 0.116418 0.993200i \(-0.462859\pi\)
0.993200 0.116418i \(-0.0371413\pi\)
\(54\) 0 0
\(55\) 1.75021i 0.235999i
\(56\) 0 0
\(57\) 23.3282i 3.08989i
\(58\) 0 0
\(59\) −1.81385 + 1.81385i −0.236143 + 0.236143i −0.815251 0.579108i \(-0.803401\pi\)
0.579108 + 0.815251i \(0.303401\pi\)
\(60\) 0 0
\(61\) 2.58783 + 2.58783i 0.331337 + 0.331337i 0.853094 0.521757i \(-0.174723\pi\)
−0.521757 + 0.853094i \(0.674723\pi\)
\(62\) 0 0
\(63\) 6.81864 0.859067
\(64\) 0 0
\(65\) 2.48704 0.308479
\(66\) 0 0
\(67\) 1.59261 + 1.59261i 0.194568 + 0.194568i 0.797667 0.603098i \(-0.206067\pi\)
−0.603098 + 0.797667i \(0.706067\pi\)
\(68\) 0 0
\(69\) −2.79069 + 2.79069i −0.335960 + 0.335960i
\(70\) 0 0
\(71\) 0.934634i 0.110921i 0.998461 + 0.0554603i \(0.0176626\pi\)
−0.998461 + 0.0554603i \(0.982337\pi\)
\(72\) 0 0
\(73\) 0.710511i 0.0831591i 0.999135 + 0.0415795i \(0.0132390\pi\)
−0.999135 + 0.0415795i \(0.986761\pi\)
\(74\) 0 0
\(75\) 10.3936 10.3936i 1.20015 1.20015i
\(76\) 0 0
\(77\) 2.22602 + 2.22602i 0.253679 + 0.253679i
\(78\) 0 0
\(79\) 11.8644 1.33485 0.667424 0.744678i \(-0.267397\pi\)
0.667424 + 0.744678i \(0.267397\pi\)
\(80\) 0 0
\(81\) −17.0379 −1.89310
\(82\) 0 0
\(83\) 6.77482 + 6.77482i 0.743634 + 0.743634i 0.973275 0.229642i \(-0.0737554\pi\)
−0.229642 + 0.973275i \(0.573755\pi\)
\(84\) 0 0
\(85\) 0.385637 0.385637i 0.0418282 0.0418282i
\(86\) 0 0
\(87\) 14.0630i 1.50771i
\(88\) 0 0
\(89\) 10.2082i 1.08206i 0.841002 + 0.541032i \(0.181966\pi\)
−0.841002 + 0.541032i \(0.818034\pi\)
\(90\) 0 0
\(91\) 3.16316 3.16316i 0.331590 0.331590i
\(92\) 0 0
\(93\) 9.81864 + 9.81864i 1.01815 + 1.01815i
\(94\) 0 0
\(95\) 4.13906 0.424658
\(96\) 0 0
\(97\) −3.03684 −0.308344 −0.154172 0.988044i \(-0.549271\pi\)
−0.154172 + 0.988044i \(0.549271\pi\)
\(98\) 0 0
\(99\) −15.1784 15.1784i −1.52549 1.52549i
\(100\) 0 0
\(101\) 10.1648 10.1648i 1.01143 1.01143i 0.0114982 0.999934i \(-0.496340\pi\)
0.999934 0.0114982i \(-0.00366008\pi\)
\(102\) 0 0
\(103\) 6.05520i 0.596636i 0.954466 + 0.298318i \(0.0964256\pi\)
−0.954466 + 0.298318i \(0.903574\pi\)
\(104\) 0 0
\(105\) 1.74209i 0.170011i
\(106\) 0 0
\(107\) −12.4973 + 12.4973i −1.20816 + 1.20816i −0.236533 + 0.971624i \(0.576011\pi\)
−0.971624 + 0.236533i \(0.923989\pi\)
\(108\) 0 0
\(109\) −10.0575 10.0575i −0.963333 0.963333i 0.0360181 0.999351i \(-0.488533\pi\)
−0.999351 + 0.0360181i \(0.988533\pi\)
\(110\) 0 0
\(111\) 2.85889 0.271354
\(112\) 0 0
\(113\) 5.01929 0.472175 0.236088 0.971732i \(-0.424135\pi\)
0.236088 + 0.971732i \(0.424135\pi\)
\(114\) 0 0
\(115\) −0.495145 0.495145i −0.0461725 0.0461725i
\(116\) 0 0
\(117\) −21.5685 + 21.5685i −1.99401 + 1.99401i
\(118\) 0 0
\(119\) 0.980951i 0.0899236i
\(120\) 0 0
\(121\) 1.08963i 0.0990575i
\(122\) 0 0
\(123\) 2.68930 2.68930i 0.242486 0.242486i
\(124\) 0 0
\(125\) 3.80974 + 3.80974i 0.340754 + 0.340754i
\(126\) 0 0
\(127\) −19.3869 −1.72031 −0.860156 0.510031i \(-0.829634\pi\)
−0.860156 + 0.510031i \(0.829634\pi\)
\(128\) 0 0
\(129\) 4.28301 0.377098
\(130\) 0 0
\(131\) −8.29224 8.29224i −0.724496 0.724496i 0.245021 0.969518i \(-0.421205\pi\)
−0.969518 + 0.245021i \(0.921205\pi\)
\(132\) 0 0
\(133\) 5.26429 5.26429i 0.456472 0.456472i
\(134\) 0 0
\(135\) 6.65242i 0.572549i
\(136\) 0 0
\(137\) 10.3723i 0.886168i −0.896480 0.443084i \(-0.853884\pi\)
0.896480 0.443084i \(-0.146116\pi\)
\(138\) 0 0
\(139\) 4.83290 4.83290i 0.409921 0.409921i −0.471790 0.881711i \(-0.656392\pi\)
0.881711 + 0.471790i \(0.156392\pi\)
\(140\) 0 0
\(141\) −22.0938 22.0938i −1.86063 1.86063i
\(142\) 0 0
\(143\) −14.0826 −1.17764
\(144\) 0 0
\(145\) −2.49516 −0.207212
\(146\) 0 0
\(147\) 2.21570 + 2.21570i 0.182748 + 0.182748i
\(148\) 0 0
\(149\) −1.74049 + 1.74049i −0.142587 + 0.142587i −0.774797 0.632210i \(-0.782148\pi\)
0.632210 + 0.774797i \(0.282148\pi\)
\(150\) 0 0
\(151\) 20.8131i 1.69374i 0.531798 + 0.846871i \(0.321517\pi\)
−0.531798 + 0.846871i \(0.678483\pi\)
\(152\) 0 0
\(153\) 6.68874i 0.540753i
\(154\) 0 0
\(155\) −1.74209 + 1.74209i −0.139928 + 0.139928i
\(156\) 0 0
\(157\) 10.1441 + 10.1441i 0.809588 + 0.809588i 0.984571 0.174983i \(-0.0559870\pi\)
−0.174983 + 0.984571i \(0.555987\pi\)
\(158\) 0 0
\(159\) −35.7975 −2.83892
\(160\) 0 0
\(161\) −1.25951 −0.0992632
\(162\) 0 0
\(163\) 11.8882 + 11.8882i 0.931152 + 0.931152i 0.997778 0.0666257i \(-0.0212233\pi\)
−0.0666257 + 0.997778i \(0.521223\pi\)
\(164\) 0 0
\(165\) 3.87794 3.87794i 0.301898 0.301898i
\(166\) 0 0
\(167\) 1.10868i 0.0857923i 0.999080 + 0.0428962i \(0.0136585\pi\)
−0.999080 + 0.0428962i \(0.986342\pi\)
\(168\) 0 0
\(169\) 7.01121i 0.539324i
\(170\) 0 0
\(171\) −35.8953 + 35.8953i −2.74498 + 2.74498i
\(172\) 0 0
\(173\) 7.23971 + 7.23971i 0.550425 + 0.550425i 0.926563 0.376139i \(-0.122748\pi\)
−0.376139 + 0.926563i \(0.622748\pi\)
\(174\) 0 0
\(175\) 4.69090 0.354599
\(176\) 0 0
\(177\) 8.03789 0.604164
\(178\) 0 0
\(179\) −9.93959 9.93959i −0.742920 0.742920i 0.230219 0.973139i \(-0.426056\pi\)
−0.973139 + 0.230219i \(0.926056\pi\)
\(180\) 0 0
\(181\) −7.35342 + 7.35342i −0.546576 + 0.546576i −0.925449 0.378873i \(-0.876312\pi\)
0.378873 + 0.925449i \(0.376312\pi\)
\(182\) 0 0
\(183\) 11.4677i 0.847715i
\(184\) 0 0
\(185\) 0.507246i 0.0372935i
\(186\) 0 0
\(187\) −2.18362 + 2.18362i −0.159682 + 0.159682i
\(188\) 0 0
\(189\) −8.46094 8.46094i −0.615443 0.615443i
\(190\) 0 0
\(191\) 3.54684 0.256640 0.128320 0.991733i \(-0.459042\pi\)
0.128320 + 0.991733i \(0.459042\pi\)
\(192\) 0 0
\(193\) 2.99394 0.215509 0.107754 0.994178i \(-0.465634\pi\)
0.107754 + 0.994178i \(0.465634\pi\)
\(194\) 0 0
\(195\) −5.51053 5.51053i −0.394617 0.394617i
\(196\) 0 0
\(197\) −3.55345 + 3.55345i −0.253173 + 0.253173i −0.822270 0.569097i \(-0.807293\pi\)
0.569097 + 0.822270i \(0.307293\pi\)
\(198\) 0 0
\(199\) 1.80109i 0.127676i −0.997960 0.0638380i \(-0.979666\pi\)
0.997960 0.0638380i \(-0.0203341\pi\)
\(200\) 0 0
\(201\) 7.05749i 0.497797i
\(202\) 0 0
\(203\) −3.17349 + 3.17349i −0.222735 + 0.222735i
\(204\) 0 0
\(205\) 0.477156 + 0.477156i 0.0333260 + 0.0333260i
\(206\) 0 0
\(207\) 8.58813 0.596916
\(208\) 0 0
\(209\) −23.4369 −1.62116
\(210\) 0 0
\(211\) 15.1022 + 15.1022i 1.03968 + 1.03968i 0.999180 + 0.0404953i \(0.0128936\pi\)
0.0404953 + 0.999180i \(0.487106\pi\)
\(212\) 0 0
\(213\) 2.07087 2.07087i 0.141893 0.141893i
\(214\) 0 0
\(215\) 0.759924i 0.0518264i
\(216\) 0 0
\(217\) 4.43140i 0.300823i
\(218\) 0 0
\(219\) 1.57428 1.57428i 0.106380 0.106380i
\(220\) 0 0
\(221\) 3.10291 + 3.10291i 0.208724 + 0.208724i
\(222\) 0 0
\(223\) 7.11258 0.476294 0.238147 0.971229i \(-0.423460\pi\)
0.238147 + 0.971229i \(0.423460\pi\)
\(224\) 0 0
\(225\) −31.9856 −2.13237
\(226\) 0 0
\(227\) 12.1687 + 12.1687i 0.807665 + 0.807665i 0.984280 0.176615i \(-0.0565147\pi\)
−0.176615 + 0.984280i \(0.556515\pi\)
\(228\) 0 0
\(229\) 0.621484 0.621484i 0.0410688 0.0410688i −0.686274 0.727343i \(-0.740755\pi\)
0.727343 + 0.686274i \(0.240755\pi\)
\(230\) 0 0
\(231\) 9.86440i 0.649030i
\(232\) 0 0
\(233\) 4.85688i 0.318185i 0.987264 + 0.159093i \(0.0508568\pi\)
−0.987264 + 0.159093i \(0.949143\pi\)
\(234\) 0 0
\(235\) 3.92004 3.92004i 0.255715 0.255715i
\(236\) 0 0
\(237\) −26.2879 26.2879i −1.70758 1.70758i
\(238\) 0 0
\(239\) 4.91033 0.317623 0.158811 0.987309i \(-0.449234\pi\)
0.158811 + 0.987309i \(0.449234\pi\)
\(240\) 0 0
\(241\) 10.8591 0.699498 0.349749 0.936843i \(-0.386267\pi\)
0.349749 + 0.936843i \(0.386267\pi\)
\(242\) 0 0
\(243\) 12.3680 + 12.3680i 0.793406 + 0.793406i
\(244\) 0 0
\(245\) −0.393125 + 0.393125i −0.0251159 + 0.0251159i
\(246\) 0 0
\(247\) 33.3037i 2.11906i
\(248\) 0 0
\(249\) 30.0219i 1.90256i
\(250\) 0 0
\(251\) −8.18516 + 8.18516i −0.516643 + 0.516643i −0.916554 0.399911i \(-0.869041\pi\)
0.399911 + 0.916554i \(0.369041\pi\)
\(252\) 0 0
\(253\) 2.80370 + 2.80370i 0.176267 + 0.176267i
\(254\) 0 0
\(255\) −1.70891 −0.107016
\(256\) 0 0
\(257\) 20.6130 1.28580 0.642902 0.765948i \(-0.277730\pi\)
0.642902 + 0.765948i \(0.277730\pi\)
\(258\) 0 0
\(259\) 0.645145 + 0.645145i 0.0400874 + 0.0400874i
\(260\) 0 0
\(261\) 21.6389 21.6389i 1.33941 1.33941i
\(262\) 0 0
\(263\) 13.6297i 0.840444i −0.907421 0.420222i \(-0.861952\pi\)
0.907421 0.420222i \(-0.138048\pi\)
\(264\) 0 0
\(265\) 6.35145i 0.390166i
\(266\) 0 0
\(267\) 22.6182 22.6182i 1.38421 1.38421i
\(268\) 0 0
\(269\) −4.42999 4.42999i −0.270101 0.270101i 0.559040 0.829141i \(-0.311170\pi\)
−0.829141 + 0.559040i \(0.811170\pi\)
\(270\) 0 0
\(271\) 12.6851 0.770564 0.385282 0.922799i \(-0.374104\pi\)
0.385282 + 0.922799i \(0.374104\pi\)
\(272\) 0 0
\(273\) −14.0172 −0.848362
\(274\) 0 0
\(275\) −10.4421 10.4421i −0.629680 0.629680i
\(276\) 0 0
\(277\) −17.0842 + 17.0842i −1.02649 + 1.02649i −0.0268508 + 0.999639i \(0.508548\pi\)
−0.999639 + 0.0268508i \(0.991452\pi\)
\(278\) 0 0
\(279\) 30.2161i 1.80899i
\(280\) 0 0
\(281\) 12.8628i 0.767330i −0.923472 0.383665i \(-0.874662\pi\)
0.923472 0.383665i \(-0.125338\pi\)
\(282\) 0 0
\(283\) 0.879091 0.879091i 0.0522565 0.0522565i −0.680496 0.732752i \(-0.738235\pi\)
0.732752 + 0.680496i \(0.238235\pi\)
\(284\) 0 0
\(285\) −9.17090 9.17090i −0.543237 0.543237i
\(286\) 0 0
\(287\) 1.21375 0.0716454
\(288\) 0 0
\(289\) −16.0377 −0.943396
\(290\) 0 0
\(291\) 6.72872 + 6.72872i 0.394445 + 0.394445i
\(292\) 0 0
\(293\) 17.9935 17.9935i 1.05119 1.05119i 0.0525765 0.998617i \(-0.483257\pi\)
0.998617 0.0525765i \(-0.0167433\pi\)
\(294\) 0 0
\(295\) 1.42614i 0.0830331i
\(296\) 0 0
\(297\) 37.6685i 2.18575i
\(298\) 0 0
\(299\) 3.98403 3.98403i 0.230403 0.230403i
\(300\) 0 0
\(301\) 0.966515 + 0.966515i 0.0557090 + 0.0557090i
\(302\) 0 0
\(303\) −45.0441 −2.58772
\(304\) 0 0
\(305\) 2.03468 0.116505
\(306\) 0 0
\(307\) 1.74987 + 1.74987i 0.0998702 + 0.0998702i 0.755277 0.655406i \(-0.227503\pi\)
−0.655406 + 0.755277i \(0.727503\pi\)
\(308\) 0 0
\(309\) 13.4165 13.4165i 0.763237 0.763237i
\(310\) 0 0
\(311\) 17.4288i 0.988296i −0.869378 0.494148i \(-0.835480\pi\)
0.869378 0.494148i \(-0.164520\pi\)
\(312\) 0 0
\(313\) 28.3430i 1.60204i −0.598637 0.801021i \(-0.704291\pi\)
0.598637 0.801021i \(-0.295709\pi\)
\(314\) 0 0
\(315\) 2.68058 2.68058i 0.151033 0.151033i
\(316\) 0 0
\(317\) −8.24397 8.24397i −0.463028 0.463028i 0.436619 0.899647i \(-0.356176\pi\)
−0.899647 + 0.436619i \(0.856176\pi\)
\(318\) 0 0
\(319\) 14.1285 0.791046
\(320\) 0 0
\(321\) 55.3803 3.09103
\(322\) 0 0
\(323\) 5.16401 + 5.16401i 0.287333 + 0.287333i
\(324\) 0 0
\(325\) −14.8381 + 14.8381i −0.823070 + 0.823070i
\(326\) 0 0
\(327\) 44.5687i 2.46466i
\(328\) 0 0
\(329\) 9.97147i 0.549745i
\(330\) 0 0
\(331\) −0.235462 + 0.235462i −0.0129422 + 0.0129422i −0.713548 0.700606i \(-0.752913\pi\)
0.700606 + 0.713548i \(0.252913\pi\)
\(332\) 0 0
\(333\) −4.39901 4.39901i −0.241064 0.241064i
\(334\) 0 0
\(335\) 1.25219 0.0684146
\(336\) 0 0
\(337\) 9.22099 0.502299 0.251150 0.967948i \(-0.419191\pi\)
0.251150 + 0.967948i \(0.419191\pi\)
\(338\) 0 0
\(339\) −11.1212 11.1212i −0.604023 0.604023i
\(340\) 0 0
\(341\) 9.86440 9.86440i 0.534187 0.534187i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.19418i 0.118131i
\(346\) 0 0
\(347\) 8.02721 8.02721i 0.430923 0.430923i −0.458019 0.888942i \(-0.651441\pi\)
0.888942 + 0.458019i \(0.151441\pi\)
\(348\) 0 0
\(349\) −11.5879 11.5879i −0.620287 0.620287i 0.325318 0.945605i \(-0.394529\pi\)
−0.945605 + 0.325318i \(0.894529\pi\)
\(350\) 0 0
\(351\) 53.5267 2.85704
\(352\) 0 0
\(353\) 18.9481 1.00850 0.504252 0.863557i \(-0.331768\pi\)
0.504252 + 0.863557i \(0.331768\pi\)
\(354\) 0 0
\(355\) 0.367428 + 0.367428i 0.0195011 + 0.0195011i
\(356\) 0 0
\(357\) −2.17349 + 2.17349i −0.115033 + 0.115033i
\(358\) 0 0
\(359\) 3.54172i 0.186925i 0.995623 + 0.0934624i \(0.0297935\pi\)
−0.995623 + 0.0934624i \(0.970207\pi\)
\(360\) 0 0
\(361\) 36.4256i 1.91714i
\(362\) 0 0
\(363\) 2.41430 2.41430i 0.126718 0.126718i
\(364\) 0 0
\(365\) 0.279320 + 0.279320i 0.0146203 + 0.0146203i
\(366\) 0 0
\(367\) 11.1874 0.583979 0.291989 0.956422i \(-0.405683\pi\)
0.291989 + 0.956422i \(0.405683\pi\)
\(368\) 0 0
\(369\) −8.27611 −0.430837
\(370\) 0 0
\(371\) −8.07814 8.07814i −0.419396 0.419396i
\(372\) 0 0
\(373\) −13.8071 + 13.8071i −0.714907 + 0.714907i −0.967558 0.252651i \(-0.918698\pi\)
0.252651 + 0.967558i \(0.418698\pi\)
\(374\) 0 0
\(375\) 16.8825i 0.871807i
\(376\) 0 0
\(377\) 20.0765i 1.03399i
\(378\) 0 0
\(379\) 10.9875 10.9875i 0.564391 0.564391i −0.366161 0.930552i \(-0.619328\pi\)
0.930552 + 0.366161i \(0.119328\pi\)
\(380\) 0 0
\(381\) 42.9556 + 42.9556i 2.20068 + 2.20068i
\(382\) 0 0
\(383\) −4.83945 −0.247284 −0.123642 0.992327i \(-0.539458\pi\)
−0.123642 + 0.992327i \(0.539458\pi\)
\(384\) 0 0
\(385\) 1.75021 0.0891991
\(386\) 0 0
\(387\) −6.59032 6.59032i −0.335005 0.335005i
\(388\) 0 0
\(389\) −12.4807 + 12.4807i −0.632795 + 0.632795i −0.948768 0.315973i \(-0.897669\pi\)
0.315973 + 0.948768i \(0.397669\pi\)
\(390\) 0 0
\(391\) 1.23552i 0.0624827i
\(392\) 0 0
\(393\) 36.7462i 1.85360i
\(394\) 0 0
\(395\) 4.66419 4.66419i 0.234681 0.234681i
\(396\) 0 0
\(397\) 11.7543 + 11.7543i 0.589931 + 0.589931i 0.937613 0.347682i \(-0.113031\pi\)
−0.347682 + 0.937613i \(0.613031\pi\)
\(398\) 0 0
\(399\) −23.3282 −1.16787
\(400\) 0 0
\(401\) −25.5823 −1.27752 −0.638760 0.769406i \(-0.720552\pi\)
−0.638760 + 0.769406i \(0.720552\pi\)
\(402\) 0 0
\(403\) −14.0172 14.0172i −0.698248 0.698248i
\(404\) 0 0
\(405\) −6.69803 + 6.69803i −0.332828 + 0.332828i
\(406\) 0 0
\(407\) 2.87222i 0.142371i
\(408\) 0 0
\(409\) 19.2126i 0.950001i 0.879985 + 0.475001i \(0.157552\pi\)
−0.879985 + 0.475001i \(0.842448\pi\)
\(410\) 0 0
\(411\) −22.9820 + 22.9820i −1.13362 + 1.13362i
\(412\) 0 0
\(413\) 1.81385 + 1.81385i 0.0892537 + 0.0892537i
\(414\) 0 0
\(415\) 5.32671 0.261478
\(416\) 0 0
\(417\) −21.4165 −1.04877
\(418\) 0 0
\(419\) −11.2764 11.2764i −0.550888 0.550888i 0.375809 0.926697i \(-0.377365\pi\)
−0.926697 + 0.375809i \(0.877365\pi\)
\(420\) 0 0
\(421\) −27.1033 + 27.1033i −1.32094 + 1.32094i −0.407918 + 0.913019i \(0.633745\pi\)
−0.913019 + 0.407918i \(0.866255\pi\)
\(422\) 0 0
\(423\) 67.9918i 3.30588i
\(424\) 0 0
\(425\) 4.60155i 0.223208i
\(426\) 0 0
\(427\) 2.58783 2.58783i 0.125234 0.125234i
\(428\) 0 0
\(429\) 31.2027 + 31.2027i 1.50648 + 1.50648i
\(430\) 0 0
\(431\) −14.7618 −0.711053 −0.355526 0.934666i \(-0.615698\pi\)
−0.355526 + 0.934666i \(0.615698\pi\)
\(432\) 0 0
\(433\) −30.0057 −1.44198 −0.720991 0.692944i \(-0.756313\pi\)
−0.720991 + 0.692944i \(0.756313\pi\)
\(434\) 0 0
\(435\) 5.52852 + 5.52852i 0.265072 + 0.265072i
\(436\) 0 0
\(437\) 6.63043 6.63043i 0.317176 0.317176i
\(438\) 0 0
\(439\) 33.7523i 1.61091i −0.592659 0.805454i \(-0.701922\pi\)
0.592659 0.805454i \(-0.298078\pi\)
\(440\) 0 0
\(441\) 6.81864i 0.324697i
\(442\) 0 0
\(443\) −5.22020 + 5.22020i −0.248019 + 0.248019i −0.820157 0.572138i \(-0.806114\pi\)
0.572138 + 0.820157i \(0.306114\pi\)
\(444\) 0 0
\(445\) 4.01309 + 4.01309i 0.190239 + 0.190239i
\(446\) 0 0
\(447\) 7.71281 0.364803
\(448\) 0 0
\(449\) −1.59006 −0.0750395 −0.0375197 0.999296i \(-0.511946\pi\)
−0.0375197 + 0.999296i \(0.511946\pi\)
\(450\) 0 0
\(451\) −2.70184 2.70184i −0.127224 0.127224i
\(452\) 0 0
\(453\) 46.1154 46.1154i 2.16669 2.16669i
\(454\) 0 0
\(455\) 2.48704i 0.116594i
\(456\) 0 0
\(457\) 2.14917i 0.100534i 0.998736 + 0.0502671i \(0.0160072\pi\)
−0.998736 + 0.0502671i \(0.983993\pi\)
\(458\) 0 0
\(459\) 8.29977 8.29977i 0.387400 0.387400i
\(460\) 0 0
\(461\) 26.7406 + 26.7406i 1.24543 + 1.24543i 0.957715 + 0.287719i \(0.0928968\pi\)
0.287719 + 0.957715i \(0.407103\pi\)
\(462\) 0 0
\(463\) 16.2686 0.756065 0.378033 0.925792i \(-0.376601\pi\)
0.378033 + 0.925792i \(0.376601\pi\)
\(464\) 0 0
\(465\) 7.71991 0.358002
\(466\) 0 0
\(467\) −17.9285 17.9285i −0.829632 0.829632i 0.157834 0.987466i \(-0.449549\pi\)
−0.987466 + 0.157834i \(0.949549\pi\)
\(468\) 0 0
\(469\) 1.59261 1.59261i 0.0735400 0.0735400i
\(470\) 0 0
\(471\) 44.9526i 2.07131i
\(472\) 0 0
\(473\) 4.30297i 0.197851i
\(474\) 0 0
\(475\) −24.6943 + 24.6943i −1.13305 + 1.13305i
\(476\) 0 0
\(477\) 55.0819 + 55.0819i 2.52203 + 2.52203i
\(478\) 0 0
\(479\) −16.9860 −0.776112 −0.388056 0.921636i \(-0.626853\pi\)
−0.388056 + 0.921636i \(0.626853\pi\)
\(480\) 0 0
\(481\) −4.08140 −0.186096
\(482\) 0 0
\(483\) 2.79069 + 2.79069i 0.126981 + 0.126981i
\(484\) 0 0
\(485\) −1.19386 + 1.19386i −0.0542103 + 0.0542103i
\(486\) 0 0
\(487\) 39.0342i 1.76881i −0.466722 0.884404i \(-0.654565\pi\)
0.466722 0.884404i \(-0.345435\pi\)
\(488\) 0 0
\(489\) 52.6811i 2.38232i
\(490\) 0 0
\(491\) 26.8828 26.8828i 1.21320 1.21320i 0.243238 0.969967i \(-0.421790\pi\)
0.969967 0.243238i \(-0.0782097\pi\)
\(492\) 0 0
\(493\) −3.11304 3.11304i −0.140204 0.140204i
\(494\) 0 0
\(495\) −11.9341 −0.536396
\(496\) 0 0
\(497\) 0.934634 0.0419241
\(498\) 0 0
\(499\) −4.52969 4.52969i −0.202777 0.202777i 0.598412 0.801189i \(-0.295799\pi\)
−0.801189 + 0.598412i \(0.795799\pi\)
\(500\) 0 0
\(501\) 2.45650 2.45650i 0.109748 0.109748i
\(502\) 0 0
\(503\) 35.6215i 1.58829i 0.607731 + 0.794143i \(0.292080\pi\)
−0.607731 + 0.794143i \(0.707920\pi\)
\(504\) 0 0
\(505\) 7.99206i 0.355642i
\(506\) 0 0
\(507\) 15.5347 15.5347i 0.689922 0.689922i
\(508\) 0 0
\(509\) −18.0304 18.0304i −0.799183 0.799183i 0.183783 0.982967i \(-0.441165\pi\)
−0.982967 + 0.183783i \(0.941165\pi\)
\(510\) 0 0
\(511\) 0.710511 0.0314312
\(512\) 0 0
\(513\) 89.0818 3.93306
\(514\) 0 0
\(515\) 2.38045 + 2.38045i 0.104895 + 0.104895i
\(516\) 0 0
\(517\) −22.1967 + 22.1967i −0.976212 + 0.976212i
\(518\) 0 0
\(519\) 32.0820i 1.40824i
\(520\) 0 0
\(521\) 15.5300i 0.680380i 0.940357 + 0.340190i \(0.110491\pi\)
−0.940357 + 0.340190i \(0.889509\pi\)
\(522\) 0 0
\(523\) −7.58074 + 7.58074i −0.331483 + 0.331483i −0.853149 0.521667i \(-0.825310\pi\)
0.521667 + 0.853149i \(0.325310\pi\)
\(524\) 0 0
\(525\) −10.3936 10.3936i −0.453615 0.453615i
\(526\) 0 0
\(527\) −4.34698 −0.189357
\(528\) 0 0
\(529\) 21.4136 0.931028
\(530\) 0 0
\(531\) −12.3680 12.3680i −0.536725 0.536725i
\(532\) 0 0
\(533\) −3.83929 + 3.83929i −0.166298 + 0.166298i
\(534\) 0 0
\(535\) 9.82598i 0.424814i
\(536\) 0 0
\(537\) 44.0463i 1.90074i
\(538\) 0 0
\(539\) 2.22602 2.22602i 0.0958817 0.0958817i
\(540\) 0 0
\(541\) −16.7920 16.7920i −0.721944 0.721944i 0.247057 0.969001i \(-0.420537\pi\)
−0.969001 + 0.247057i \(0.920537\pi\)
\(542\) 0 0
\(543\) 32.5859 1.39840
\(544\) 0 0
\(545\) −7.90771 −0.338729
\(546\) 0 0
\(547\) 28.9159 + 28.9159i 1.23636 + 1.23636i 0.961480 + 0.274875i \(0.0886366\pi\)
0.274875 + 0.961480i \(0.411363\pi\)
\(548\) 0 0
\(549\) −17.6454 + 17.6454i −0.753089 + 0.753089i
\(550\) 0 0
\(551\) 33.4124i 1.42342i
\(552\) 0 0
\(553\) 11.8644i 0.504525i
\(554\) 0 0
\(555\) 1.12390 1.12390i 0.0477071 0.0477071i
\(556\) 0 0
\(557\) −11.8756 11.8756i −0.503184 0.503184i 0.409242 0.912426i \(-0.365793\pi\)
−0.912426 + 0.409242i \(0.865793\pi\)
\(558\) 0 0
\(559\) −6.11449 −0.258616
\(560\) 0 0
\(561\) 9.67648 0.408541
\(562\) 0 0
\(563\) 2.51935 + 2.51935i 0.106178 + 0.106178i 0.758200 0.652022i \(-0.226079\pi\)
−0.652022 + 0.758200i \(0.726079\pi\)
\(564\) 0 0
\(565\) 1.97321 1.97321i 0.0830137 0.0830137i
\(566\) 0 0
\(567\) 17.0379i 0.715524i
\(568\) 0 0
\(569\) 1.90395i 0.0798176i 0.999203 + 0.0399088i \(0.0127067\pi\)
−0.999203 + 0.0399088i \(0.987293\pi\)
\(570\) 0 0
\(571\) −1.94546 + 1.94546i −0.0814148 + 0.0814148i −0.746641 0.665227i \(-0.768335\pi\)
0.665227 + 0.746641i \(0.268335\pi\)
\(572\) 0 0
\(573\) −7.85872 7.85872i −0.328303 0.328303i
\(574\) 0 0
\(575\) 5.90824 0.246390
\(576\) 0 0
\(577\) −0.270046 −0.0112422 −0.00562108 0.999984i \(-0.501789\pi\)
−0.00562108 + 0.999984i \(0.501789\pi\)
\(578\) 0 0
\(579\) −6.63367 6.63367i −0.275686 0.275686i
\(580\) 0 0
\(581\) 6.77482 6.77482i 0.281067 0.281067i
\(582\) 0 0
\(583\) 35.9643i 1.48949i
\(584\) 0 0
\(585\) 16.9582i 0.701136i
\(586\) 0 0
\(587\) 22.4556 22.4556i 0.926842 0.926842i −0.0706582 0.997501i \(-0.522510\pi\)
0.997501 + 0.0706582i \(0.0225100\pi\)
\(588\) 0 0
\(589\) −23.3282 23.3282i −0.961221 0.961221i
\(590\) 0 0
\(591\) 15.7468 0.647735
\(592\) 0 0
\(593\) −36.0626 −1.48091 −0.740456 0.672105i \(-0.765390\pi\)
−0.740456 + 0.672105i \(0.765390\pi\)
\(594\) 0 0
\(595\) −0.385637 0.385637i −0.0158096 0.0158096i
\(596\) 0 0
\(597\) −3.99067 + 3.99067i −0.163327 + 0.163327i
\(598\) 0 0
\(599\) 10.7292i 0.438381i 0.975682 + 0.219191i \(0.0703417\pi\)
−0.975682 + 0.219191i \(0.929658\pi\)
\(600\) 0 0
\(601\) 36.0677i 1.47123i 0.677399 + 0.735616i \(0.263107\pi\)
−0.677399 + 0.735616i \(0.736893\pi\)
\(602\) 0 0
\(603\) −10.8594 + 10.8594i −0.442230 + 0.442230i
\(604\) 0 0
\(605\) 0.428362 + 0.428362i 0.0174154 + 0.0174154i
\(606\) 0 0
\(607\) 6.98617 0.283560 0.141780 0.989898i \(-0.454717\pi\)
0.141780 + 0.989898i \(0.454717\pi\)
\(608\) 0 0
\(609\) 14.0630 0.569861
\(610\) 0 0
\(611\) 31.5414 + 31.5414i 1.27603 + 1.27603i
\(612\) 0 0
\(613\) 23.3560 23.3560i 0.943340 0.943340i −0.0551389 0.998479i \(-0.517560\pi\)
0.998479 + 0.0551389i \(0.0175602\pi\)
\(614\) 0 0
\(615\) 2.11447i 0.0852635i
\(616\) 0 0
\(617\) 41.5107i 1.67116i 0.549370 + 0.835579i \(0.314868\pi\)
−0.549370 + 0.835579i \(0.685132\pi\)
\(618\) 0 0
\(619\) 19.3546 19.3546i 0.777929 0.777929i −0.201550 0.979478i \(-0.564598\pi\)
0.979478 + 0.201550i \(0.0645978\pi\)
\(620\) 0 0
\(621\) −10.6566 10.6566i −0.427636 0.427636i
\(622\) 0 0
\(623\) 10.2082 0.408982
\(624\) 0 0
\(625\) −20.4591 −0.818365
\(626\) 0 0
\(627\) 51.9291 + 51.9291i 2.07385 + 2.07385i
\(628\) 0 0
\(629\) −0.632856 + 0.632856i −0.0252336 + 0.0252336i
\(630\) 0 0
\(631\) 2.56032i 0.101925i −0.998701 0.0509624i \(-0.983771\pi\)
0.998701 0.0509624i \(-0.0162289\pi\)
\(632\) 0 0
\(633\) 66.9236i 2.65997i
\(634\) 0 0
\(635\) −7.62149 + 7.62149i −0.302450 + 0.302450i
\(636\) 0 0
\(637\) −3.16316 3.16316i −0.125329 0.125329i
\(638\) 0 0
\(639\) −6.37293 −0.252109
\(640\) 0 0
\(641\) −0.654189 −0.0258389 −0.0129195 0.999917i \(-0.504113\pi\)
−0.0129195 + 0.999917i \(0.504113\pi\)
\(642\) 0 0
\(643\) −3.21708 3.21708i −0.126869 0.126869i 0.640821 0.767690i \(-0.278594\pi\)
−0.767690 + 0.640821i \(0.778594\pi\)
\(644\) 0 0
\(645\) 1.68376 1.68376i 0.0662980 0.0662980i
\(646\) 0 0
\(647\) 39.4069i 1.54925i −0.632423 0.774623i \(-0.717939\pi\)
0.632423 0.774623i \(-0.282061\pi\)
\(648\) 0 0
\(649\) 8.07535i 0.316985i
\(650\) 0 0
\(651\) 9.81864 9.81864i 0.384823 0.384823i
\(652\) 0 0
\(653\) −25.4270 25.4270i −0.995034 0.995034i 0.00495338 0.999988i \(-0.498423\pi\)
−0.999988 + 0.00495338i \(0.998423\pi\)
\(654\) 0 0
\(655\) −6.51978 −0.254749
\(656\) 0 0
\(657\) −4.84472 −0.189010
\(658\) 0 0
\(659\) −31.8006 31.8006i −1.23877 1.23877i −0.960502 0.278272i \(-0.910238\pi\)
−0.278272 0.960502i \(-0.589762\pi\)
\(660\) 0 0
\(661\) 5.45506 5.45506i 0.212177 0.212177i −0.593015 0.805192i \(-0.702062\pi\)
0.805192 + 0.593015i \(0.202062\pi\)
\(662\) 0 0
\(663\) 13.7502i 0.534014i
\(664\) 0 0
\(665\) 4.13906i 0.160506i
\(666\) 0 0
\(667\) −3.99704 + 3.99704i −0.154766 + 0.154766i
\(668\) 0 0
\(669\) −15.7593 15.7593i −0.609291 0.609291i
\(670\) 0 0
\(671\) −11.5211 −0.444768
\(672\) 0 0
\(673\) −28.3929 −1.09446 −0.547232 0.836981i \(-0.684319\pi\)
−0.547232 + 0.836981i \(0.684319\pi\)
\(674\) 0 0
\(675\) 39.6895 + 39.6895i 1.52765 + 1.52765i
\(676\) 0 0
\(677\) −7.58622 + 7.58622i −0.291562 + 0.291562i −0.837697 0.546135i \(-0.816099\pi\)
0.546135 + 0.837697i \(0.316099\pi\)
\(678\) 0 0
\(679\) 3.03684i 0.116543i
\(680\) 0 0
\(681\) 53.9243i 2.06639i
\(682\) 0 0
\(683\) 10.6493 10.6493i 0.407485 0.407485i −0.473376 0.880861i \(-0.656965\pi\)
0.880861 + 0.473376i \(0.156965\pi\)
\(684\) 0 0
\(685\) −4.07763 4.07763i −0.155798 0.155798i
\(686\) 0 0
\(687\) −2.75404 −0.105073
\(688\) 0 0
\(689\) 51.1050 1.94695
\(690\) 0 0
\(691\) −1.31805 1.31805i −0.0501409 0.0501409i 0.681592 0.731733i \(-0.261288\pi\)
−0.731733 + 0.681592i \(0.761288\pi\)
\(692\) 0 0
\(693\) −15.1784 + 15.1784i −0.576582 + 0.576582i
\(694\) 0 0
\(695\) 3.79987i 0.144137i
\(696\) 0 0
\(697\) 1.19063i 0.0450983i
\(698\) 0 0
\(699\) 10.7614 10.7614i 0.407033 0.407033i
\(700\) 0 0
\(701\) −5.77893 5.77893i −0.218267 0.218267i 0.589501 0.807768i \(-0.299325\pi\)
−0.807768 + 0.589501i \(0.799325\pi\)
\(702\) 0 0
\(703\) −6.79247 −0.256183
\(704\) 0 0
\(705\) −17.3712 −0.654239
\(706\) 0 0
\(707\) −10.1648 10.1648i −0.382285 0.382285i
\(708\) 0 0
\(709\) 27.5392 27.5392i 1.03426 1.03426i 0.0348642 0.999392i \(-0.488900\pi\)
0.999392 0.0348642i \(-0.0110999\pi\)
\(710\) 0 0
\(711\) 80.8990i 3.03395i
\(712\) 0 0
\(713\) 5.58138i 0.209024i
\(714\) 0 0
\(715\) −5.53621 + 5.53621i −0.207043 + 0.207043i
\(716\) 0 0
\(717\) −10.8798 10.8798i −0.406314 0.406314i
\(718\) 0 0
\(719\) −3.79808 −0.141645 −0.0708223 0.997489i \(-0.522562\pi\)
−0.0708223 + 0.997489i \(0.522562\pi\)
\(720\) 0 0
\(721\) 6.05520 0.225507
\(722\) 0 0
\(723\) −24.0605 24.0605i −0.894821 0.894821i
\(724\) 0 0
\(725\) 14.8865 14.8865i 0.552872 0.552872i
\(726\) 0 0
\(727\) 25.5053i 0.945937i −0.881079 0.472969i \(-0.843182\pi\)
0.881079 0.472969i \(-0.156818\pi\)
\(728\) 0 0
\(729\) 3.69376i 0.136806i
\(730\) 0 0
\(731\) −0.948104 + 0.948104i −0.0350669 + 0.0350669i
\(732\) 0 0
\(733\) 29.0147 + 29.0147i 1.07168 + 1.07168i 0.997224 + 0.0744582i \(0.0237228\pi\)
0.0744582 + 0.997224i \(0.476277\pi\)
\(734\) 0 0
\(735\) 1.74209 0.0642581
\(736\) 0 0
\(737\) −7.09038 −0.261178
\(738\) 0 0
\(739\) −1.66971 1.66971i −0.0614213 0.0614213i 0.675729 0.737150i \(-0.263829\pi\)
−0.737150 + 0.675729i \(0.763829\pi\)
\(740\) 0 0
\(741\) 73.7909 73.7909i 2.71077 2.71077i
\(742\) 0 0
\(743\) 24.1373i 0.885513i 0.896642 + 0.442756i \(0.145999\pi\)
−0.896642 + 0.442756i \(0.854001\pi\)
\(744\) 0 0
\(745\) 1.36846i 0.0501366i
\(746\) 0 0
\(747\) −46.1951 + 46.1951i −1.69019 + 1.69019i
\(748\) 0 0
\(749\) 12.4973 + 12.4973i 0.456640 + 0.456640i
\(750\) 0 0
\(751\) 17.6703 0.644797 0.322398 0.946604i \(-0.395511\pi\)
0.322398 + 0.946604i \(0.395511\pi\)
\(752\) 0 0
\(753\) 36.2717 1.32181
\(754\) 0 0
\(755\) 8.18214 + 8.18214i 0.297779 + 0.297779i
\(756\) 0 0
\(757\) 14.3436 14.3436i 0.521327 0.521327i −0.396645 0.917972i \(-0.629826\pi\)
0.917972 + 0.396645i \(0.129826\pi\)
\(758\) 0 0
\(759\) 12.4243i 0.450973i
\(760\) 0 0
\(761\) 13.4406i 0.487220i 0.969873 + 0.243610i \(0.0783317\pi\)
−0.969873 + 0.243610i \(0.921668\pi\)
\(762\) 0 0
\(763\) −10.0575 + 10.0575i −0.364106 + 0.364106i
\(764\) 0 0
\(765\) 2.62952 + 2.62952i 0.0950703 + 0.0950703i
\(766\) 0 0
\(767\) −11.4750 −0.414338
\(768\) 0 0
\(769\) −42.7134 −1.54029 −0.770143 0.637871i \(-0.779815\pi\)
−0.770143 + 0.637871i \(0.779815\pi\)
\(770\) 0 0
\(771\) −45.6722 45.6722i −1.64484 1.64484i
\(772\) 0 0
\(773\) −17.6239 + 17.6239i −0.633886 + 0.633886i −0.949040 0.315154i \(-0.897944\pi\)
0.315154 + 0.949040i \(0.397944\pi\)
\(774\) 0 0
\(775\) 20.7873i 0.746700i
\(776\) 0 0
\(777\) 2.85889i 0.102562i
\(778\) 0 0
\(779\) −6.38953 + 6.38953i −0.228929 + 0.228929i
\(780\) 0 0
\(781\) −2.08052 2.08052i −0.0744468 0.0744468i
\(782\) 0 0
\(783\) −53.7014 −1.91913
\(784\) 0 0
\(785\) 7.97582 0.284669
\(786\) 0 0
\(787\) −27.4590 27.4590i −0.978807 0.978807i 0.0209726 0.999780i \(-0.493324\pi\)
−0.999780 + 0.0209726i \(0.993324\pi\)
\(788\) 0 0
\(789\) −30.1993 + 30.1993i −1.07512 + 1.07512i
\(790\) 0 0
\(791\) 5.01929i 0.178466i
\(792\) 0 0
\(793\) 16.3714i 0.581367i
\(794\) 0 0
\(795\) −14.0729 + 14.0729i −0.499114 + 0.499114i
\(796\) 0 0
\(797\) −37.2933 37.2933i −1.32100 1.32100i −0.912971 0.408024i \(-0.866218\pi\)
−0.408024 0.912971i \(-0.633782\pi\)
\(798\) 0 0
\(799\) 9.78152 0.346045
\(800\) 0 0
\(801\) −69.6058 −2.45940
\(802\) 0 0
\(803\) −1.58162 1.58162i −0.0558140 0.0558140i
\(804\) 0 0
\(805\) −0.495145 + 0.495145i −0.0174516 + 0.0174516i
\(806\) 0 0
\(807\) 19.6310i 0.691045i
\(808\) 0 0
\(809\) 44.3124i 1.55794i −0.627061 0.778970i \(-0.715742\pi\)
0.627061 0.778970i \(-0.284258\pi\)
\(810\) 0 0
\(811\) −9.44442 + 9.44442i −0.331639 + 0.331639i −0.853209 0.521570i \(-0.825347\pi\)
0.521570 + 0.853209i \(0.325347\pi\)
\(812\) 0 0
\(813\) −28.1063 28.1063i −0.985731 0.985731i
\(814\) 0 0
\(815\) 9.34707 0.327414
\(816\) 0 0
\(817\) −10.1760 −0.356015
\(818\) 0 0
\(819\) 21.5685 + 21.5685i 0.753663 + 0.753663i
\(820\) 0 0
\(821\) −35.3404 + 35.3404i −1.23339 + 1.23339i −0.270734 + 0.962654i \(0.587266\pi\)
−0.962654 + 0.270734i \(0.912734\pi\)
\(822\) 0 0
\(823\) 23.7136i 0.826605i −0.910594 0.413302i \(-0.864375\pi\)
0.910594 0.413302i \(-0.135625\pi\)
\(824\) 0 0
\(825\) 46.2729i 1.61102i
\(826\) 0 0
\(827\) −34.8453 + 34.8453i −1.21169 + 1.21169i −0.241218 + 0.970471i \(0.577547\pi\)
−0.970471 + 0.241218i \(0.922453\pi\)
\(828\) 0 0
\(829\) −11.9866 11.9866i −0.416313 0.416313i 0.467618 0.883931i \(-0.345112\pi\)
−0.883931 + 0.467618i \(0.845112\pi\)
\(830\) 0 0
\(831\) 75.7069 2.62624
\(832\) 0 0
\(833\) −0.980951 −0.0339879
\(834\) 0 0
\(835\) 0.435851 + 0.435851i 0.0150832 + 0.0150832i
\(836\) 0 0
\(837\) −37.4938 + 37.4938i −1.29598 + 1.29598i
\(838\) 0 0
\(839\) 1.32175i 0.0456320i −0.999740 0.0228160i \(-0.992737\pi\)
0.999740 0.0228160i \(-0.00726318\pi\)
\(840\) 0 0
\(841\) 8.85792i 0.305445i
\(842\) 0 0
\(843\) −28.5001 + 28.5001i −0.981594 + 0.981594i
\(844\) 0 0
\(845\) 2.75629 + 2.75629i 0.0948191 + 0.0948191i
\(846\) 0 0
\(847\) 1.08963 0.0374402
\(848\) 0 0
\(849\) −3.89560 −0.133697
\(850\) 0 0
\(851\) 0.812566 + 0.812566i 0.0278544 + 0.0278544i
\(852\) 0 0
\(853\) −33.7981 + 33.7981i −1.15722 + 1.15722i −0.172153 + 0.985070i \(0.555072\pi\)
−0.985070 + 0.172153i \(0.944928\pi\)
\(854\) 0 0
\(855\) 28.2227i 0.965196i
\(856\) 0 0
\(857\) 12.2024i 0.416826i −0.978041 0.208413i \(-0.933170\pi\)
0.978041 0.208413i \(-0.0668297\pi\)
\(858\) 0 0
\(859\) 9.62063 9.62063i 0.328252 0.328252i −0.523670 0.851921i \(-0.675437\pi\)
0.851921 + 0.523670i \(0.175437\pi\)
\(860\) 0 0
\(861\) −2.68930 2.68930i −0.0916512 0.0916512i
\(862\) 0 0
\(863\) −48.5092 −1.65127 −0.825636 0.564204i \(-0.809183\pi\)
−0.825636 + 0.564204i \(0.809183\pi\)
\(864\) 0 0
\(865\) 5.69222 0.193541
\(866\) 0 0
\(867\) 35.5348 + 35.5348i 1.20682 + 1.20682i
\(868\) 0 0
\(869\) −26.4104 + 26.4104i −0.895913 + 0.895913i
\(870\) 0 0
\(871\) 10.0754i 0.341391i
\(872\) 0 0
\(873\) 20.7071i 0.700829i
\(874\) 0 0
\(875\) 3.80974 3.80974i 0.128793 0.128793i
\(876\) 0 0
\(877\) 15.6211 + 15.6211i 0.527487 + 0.527487i 0.919822 0.392336i \(-0.128333\pi\)
−0.392336 + 0.919822i \(0.628333\pi\)
\(878\) 0 0
\(879\) −79.7365 −2.68944
\(880\) 0 0
\(881\) 22.7269 0.765688 0.382844 0.923813i \(-0.374945\pi\)
0.382844 + 0.923813i \(0.374945\pi\)
\(882\) 0 0
\(883\) 30.2833 + 30.2833i 1.01912 + 1.01912i 0.999814 + 0.0193019i \(0.00614437\pi\)
0.0193019 + 0.999814i \(0.493856\pi\)
\(884\) 0 0
\(885\) 3.15990 3.15990i 0.106219 0.106219i
\(886\) 0 0
\(887\) 38.7161i 1.29996i 0.759951 + 0.649980i \(0.225223\pi\)
−0.759951 + 0.649980i \(0.774777\pi\)
\(888\) 0 0
\(889\) 19.3869i 0.650217i
\(890\) 0 0
\(891\) 37.9267 37.9267i 1.27059 1.27059i
\(892\) 0 0
\(893\) 52.4928 + 52.4928i 1.75660 + 1.75660i
\(894\) 0 0
\(895\) −7.81501 −0.261227
\(896\) 0 0
\(897\) −17.6548 −0.589478
\(898\) 0 0
\(899\) 14.0630 + 14.0630i 0.469027 + 0.469027i
\(900\) 0 0
\(901\) 7.92426 7.92426i 0.263995 0.263995i
\(902\) 0 0
\(903\) 4.28301i 0.142530i
\(904\) 0 0
\(905\) 5.78163i 0.192188i
\(906\) 0 0
\(907\) 25.4724 25.4724i 0.845799 0.845799i −0.143807 0.989606i \(-0.545935\pi\)
0.989606 + 0.143807i \(0.0459345\pi\)
\(908\) 0 0
\(909\) 69.3098 + 69.3098i 2.29886 + 2.29886i
\(910\) 0 0
\(911\) −52.6922 −1.74577 −0.872886 0.487924i \(-0.837754\pi\)
−0.872886 + 0.487924i \(0.837754\pi\)
\(912\) 0 0
\(913\) −30.1618 −0.998212
\(914\) 0 0
\(915\) −4.50824 4.50824i −0.149038 0.149038i
\(916\) 0 0
\(917\) −8.29224 + 8.29224i −0.273834 + 0.273834i
\(918\) 0 0
\(919\) 25.5116i 0.841549i 0.907165 + 0.420775i \(0.138242\pi\)
−0.907165 + 0.420775i \(0.861758\pi\)
\(920\) 0 0
\(921\) 7.75436i 0.255515i
\(922\) 0 0
\(923\) −2.95640 + 2.95640i −0.0973111 + 0.0973111i
\(924\) 0 0
\(925\) −3.02632 3.02632i −0.0995046 0.0995046i
\(926\) 0 0
\(927\) −41.2882 −1.35608
\(928\) 0 0
\(929\) 49.1998 1.61419 0.807097 0.590419i \(-0.201038\pi\)
0.807097 + 0.590419i \(0.201038\pi\)
\(930\) 0 0
\(931\) −5.26429 5.26429i −0.172530 0.172530i
\(932\) 0 0
\(933\) −38.6169 + 38.6169i −1.26426 + 1.26426i
\(934\) 0 0
\(935\) 1.71687i 0.0561477i
\(936\) 0 0
\(937\) 44.3522i 1.44892i 0.689315 + 0.724462i \(0.257912\pi\)
−0.689315 + 0.724462i \(0.742088\pi\)
\(938\) 0 0
\(939\) −62.7995 + 62.7995i −2.04939 + 2.04939i
\(940\) 0 0
\(941\) −14.0676 14.0676i −0.458591 0.458591i 0.439602 0.898193i \(-0.355120\pi\)
−0.898193 + 0.439602i \(0.855120\pi\)
\(942\) 0 0
\(943\) 1.52873 0.0497822
\(944\) 0 0
\(945\) −6.65242 −0.216403
\(946\) 0 0
\(947\) 22.2829 + 22.2829i 0.724098 + 0.724098i 0.969437 0.245340i \(-0.0788995\pi\)
−0.245340 + 0.969437i \(0.578899\pi\)
\(948\) 0 0
\(949\) −2.24746 + 2.24746i −0.0729558 + 0.0729558i
\(950\) 0 0
\(951\) 36.5323i 1.18464i
\(952\) 0 0
\(953\) 52.5006i 1.70066i −0.526250 0.850330i \(-0.676402\pi\)
0.526250 0.850330i \(-0.323598\pi\)
\(954\) 0 0
\(955\) 1.39435 1.39435i 0.0451202 0.0451202i
\(956\) 0 0
\(957\) −31.3046 31.3046i −1.01193 1.01193i
\(958\) 0 0
\(959\) −10.3723 −0.334940
\(960\) 0 0
\(961\) −11.3627 −0.366540
\(962\) 0 0
\(963\) −85.2143 85.2143i −2.74599 2.74599i
\(964\) 0 0
\(965\) 1.17699 1.17699i 0.0378888 0.0378888i
\(966\) 0 0
\(967\) 22.7183i 0.730572i −0.930895 0.365286i \(-0.880971\pi\)
0.930895 0.365286i \(-0.119029\pi\)
\(968\) 0 0
\(969\) 22.8838i 0.735133i
\(970\) 0 0
\(971\) 8.51367 8.51367i 0.273217 0.273217i −0.557177 0.830394i \(-0.688116\pi\)
0.830394 + 0.557177i \(0.188116\pi\)
\(972\) 0 0
\(973\) −4.83290 4.83290i −0.154936 0.154936i
\(974\) 0 0
\(975\) 65.7535 2.10580
\(976\) 0 0
\(977\) −37.7274 −1.20701 −0.603503 0.797361i \(-0.706229\pi\)
−0.603503 + 0.797361i \(0.706229\pi\)
\(978\) 0 0
\(979\) −22.7236 22.7236i −0.726250 0.726250i
\(980\) 0 0
\(981\) 68.5784 68.5784i 2.18954 2.18954i
\(982\) 0 0
\(983\) 45.1569i 1.44028i −0.693828 0.720141i \(-0.744077\pi\)
0.693828 0.720141i \(-0.255923\pi\)
\(984\) 0 0
\(985\) 2.79390i 0.0890212i
\(986\) 0 0
\(987\) −22.0938 + 22.0938i −0.703253 + 0.703253i
\(988\) 0 0
\(989\) 1.21733 + 1.21733i 0.0387090 + 0.0387090i
\(990\) 0 0
\(991\) −46.2280 −1.46848 −0.734240 0.678890i \(-0.762462\pi\)
−0.734240 + 0.678890i \(0.762462\pi\)
\(992\) 0 0
\(993\) 1.04343 0.0331121
\(994\) 0 0
\(995\) −0.708054 0.708054i −0.0224468 0.0224468i
\(996\) 0 0
\(997\) 27.2458 27.2458i 0.862885 0.862885i −0.128788 0.991672i \(-0.541109\pi\)
0.991672 + 0.128788i \(0.0411086\pi\)
\(998\) 0 0
\(999\) 10.9171i 0.345401i
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 896.2.m.g.673.1 12
4.3 odd 2 896.2.m.h.673.6 12
8.3 odd 2 448.2.m.d.337.1 12
8.5 even 2 112.2.m.d.29.2 12
16.3 odd 4 448.2.m.d.113.1 12
16.5 even 4 inner 896.2.m.g.225.1 12
16.11 odd 4 896.2.m.h.225.6 12
16.13 even 4 112.2.m.d.85.2 yes 12
32.5 even 8 7168.2.a.bj.1.12 12
32.11 odd 8 7168.2.a.bi.1.12 12
32.21 even 8 7168.2.a.bj.1.1 12
32.27 odd 8 7168.2.a.bi.1.1 12
56.5 odd 6 784.2.x.m.557.3 24
56.13 odd 2 784.2.m.h.589.2 12
56.37 even 6 784.2.x.l.557.3 24
56.45 odd 6 784.2.x.m.765.6 24
56.53 even 6 784.2.x.l.765.6 24
112.13 odd 4 784.2.m.h.197.2 12
112.45 odd 12 784.2.x.m.373.3 24
112.61 odd 12 784.2.x.m.165.6 24
112.93 even 12 784.2.x.l.165.6 24
112.109 even 12 784.2.x.l.373.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.d.29.2 12 8.5 even 2
112.2.m.d.85.2 yes 12 16.13 even 4
448.2.m.d.113.1 12 16.3 odd 4
448.2.m.d.337.1 12 8.3 odd 2
784.2.m.h.197.2 12 112.13 odd 4
784.2.m.h.589.2 12 56.13 odd 2
784.2.x.l.165.6 24 112.93 even 12
784.2.x.l.373.3 24 112.109 even 12
784.2.x.l.557.3 24 56.37 even 6
784.2.x.l.765.6 24 56.53 even 6
784.2.x.m.165.6 24 112.61 odd 12
784.2.x.m.373.3 24 112.45 odd 12
784.2.x.m.557.3 24 56.5 odd 6
784.2.x.m.765.6 24 56.45 odd 6
896.2.m.g.225.1 12 16.5 even 4 inner
896.2.m.g.673.1 12 1.1 even 1 trivial
896.2.m.h.225.6 12 16.11 odd 4
896.2.m.h.673.6 12 4.3 odd 2
7168.2.a.bi.1.1 12 32.27 odd 8
7168.2.a.bi.1.12 12 32.11 odd 8
7168.2.a.bj.1.1 12 32.21 even 8
7168.2.a.bj.1.12 12 32.5 even 8