Properties

Label 1280.2.l.a.321.2
Level $1280$
Weight $2$
Character 1280.321
Analytic conductor $10.221$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(321,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.l (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 321.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1280.321
Dual form 1280.2.l.a.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.658919 - 0.658919i) q^{3} +(-0.707107 + 0.707107i) q^{5} +2.34607i q^{7} -2.13165i q^{9} +O(q^{10})\) \(q+(-0.658919 - 0.658919i) q^{3} +(-0.707107 + 0.707107i) q^{5} +2.34607i q^{7} -2.13165i q^{9} +(1.19980 - 1.19980i) q^{11} +(-1.51764 - 1.51764i) q^{13} +0.931852 q^{15} +1.11099 q^{17} +(-1.53225 - 1.53225i) q^{19} +(1.54587 - 1.54587i) q^{21} +4.83548i q^{23} -1.00000i q^{25} +(-3.38134 + 3.38134i) q^{27} +(-5.49938 - 5.49938i) q^{29} +6.29253 q^{31} -1.58114 q^{33} +(-1.65892 - 1.65892i) q^{35} +(4.59575 - 4.59575i) q^{37} +2.00000i q^{39} -10.0599i q^{41} +(5.14128 - 5.14128i) q^{43} +(1.50731 + 1.50731i) q^{45} +6.44584 q^{47} +1.49598 q^{49} +(-0.732051 - 0.732051i) q^{51} +(5.02823 - 5.02823i) q^{53} +1.69677i q^{55} +2.01926i q^{57} +(-1.46170 + 1.46170i) q^{59} +(-0.752715 - 0.752715i) q^{61} +5.00100 q^{63} +2.14626 q^{65} +(-11.2975 - 11.2975i) q^{67} +(3.18618 - 3.18618i) q^{69} -0.399602i q^{71} -6.02406i q^{73} +(-0.658919 + 0.658919i) q^{75} +(2.81481 + 2.81481i) q^{77} -15.8637 q^{79} -1.93890 q^{81} +(0.312853 + 0.312853i) q^{83} +(-0.785587 + 0.785587i) q^{85} +7.24728i q^{87} -16.5558i q^{89} +(3.56048 - 3.56048i) q^{91} +(-4.14626 - 4.14626i) q^{93} +2.16693 q^{95} +9.14502 q^{97} +(-2.55756 - 2.55756i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 8 q^{11} - 8 q^{13} - 8 q^{15} - 8 q^{17} - 16 q^{21} + 8 q^{27} - 8 q^{29} + 24 q^{33} - 12 q^{35} - 8 q^{37} + 44 q^{43} + 8 q^{45} - 8 q^{47} + 8 q^{51} + 16 q^{53} - 16 q^{59} + 8 q^{61} + 48 q^{63} - 8 q^{65} - 12 q^{67} + 40 q^{69} - 4 q^{75} + 16 q^{77} - 96 q^{79} - 16 q^{81} + 28 q^{83} - 16 q^{85} - 8 q^{91} - 8 q^{93} - 8 q^{95} - 24 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −0.658919 0.658919i −0.380427 0.380427i 0.490829 0.871256i \(-0.336694\pi\)
−0.871256 + 0.490829i \(0.836694\pi\)
\(4\) 0 0
\(5\) −0.707107 + 0.707107i −0.316228 + 0.316228i
\(6\) 0 0
\(7\) 2.34607i 0.886729i 0.896341 + 0.443365i \(0.146215\pi\)
−0.896341 + 0.443365i \(0.853785\pi\)
\(8\) 0 0
\(9\) 2.13165i 0.710551i
\(10\) 0 0
\(11\) 1.19980 1.19980i 0.361754 0.361754i −0.502705 0.864458i \(-0.667662\pi\)
0.864458 + 0.502705i \(0.167662\pi\)
\(12\) 0 0
\(13\) −1.51764 1.51764i −0.420917 0.420917i 0.464602 0.885519i \(-0.346197\pi\)
−0.885519 + 0.464602i \(0.846197\pi\)
\(14\) 0 0
\(15\) 0.931852 0.240603
\(16\) 0 0
\(17\) 1.11099 0.269454 0.134727 0.990883i \(-0.456984\pi\)
0.134727 + 0.990883i \(0.456984\pi\)
\(18\) 0 0
\(19\) −1.53225 1.53225i −0.351522 0.351522i 0.509153 0.860676i \(-0.329959\pi\)
−0.860676 + 0.509153i \(0.829959\pi\)
\(20\) 0 0
\(21\) 1.54587 1.54587i 0.337336 0.337336i
\(22\) 0 0
\(23\) 4.83548i 1.00827i 0.863626 + 0.504133i \(0.168188\pi\)
−0.863626 + 0.504133i \(0.831812\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) −3.38134 + 3.38134i −0.650739 + 0.650739i
\(28\) 0 0
\(29\) −5.49938 5.49938i −1.02121 1.02121i −0.999770 0.0214387i \(-0.993175\pi\)
−0.0214387 0.999770i \(-0.506825\pi\)
\(30\) 0 0
\(31\) 6.29253 1.13017 0.565086 0.825032i \(-0.308843\pi\)
0.565086 + 0.825032i \(0.308843\pi\)
\(32\) 0 0
\(33\) −1.58114 −0.275242
\(34\) 0 0
\(35\) −1.65892 1.65892i −0.280408 0.280408i
\(36\) 0 0
\(37\) 4.59575 4.59575i 0.755537 0.755537i −0.219969 0.975507i \(-0.570596\pi\)
0.975507 + 0.219969i \(0.0705958\pi\)
\(38\) 0 0
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 10.0599i 1.57109i −0.618807 0.785543i \(-0.712384\pi\)
0.618807 0.785543i \(-0.287616\pi\)
\(42\) 0 0
\(43\) 5.14128 5.14128i 0.784038 0.784038i −0.196472 0.980510i \(-0.562948\pi\)
0.980510 + 0.196472i \(0.0629483\pi\)
\(44\) 0 0
\(45\) 1.50731 + 1.50731i 0.224696 + 0.224696i
\(46\) 0 0
\(47\) 6.44584 0.940223 0.470111 0.882607i \(-0.344214\pi\)
0.470111 + 0.882607i \(0.344214\pi\)
\(48\) 0 0
\(49\) 1.49598 0.213711
\(50\) 0 0
\(51\) −0.732051 0.732051i −0.102508 0.102508i
\(52\) 0 0
\(53\) 5.02823 5.02823i 0.690680 0.690680i −0.271701 0.962382i \(-0.587586\pi\)
0.962382 + 0.271701i \(0.0875863\pi\)
\(54\) 0 0
\(55\) 1.69677i 0.228793i
\(56\) 0 0
\(57\) 2.01926i 0.267457i
\(58\) 0 0
\(59\) −1.46170 + 1.46170i −0.190297 + 0.190297i −0.795824 0.605528i \(-0.792962\pi\)
0.605528 + 0.795824i \(0.292962\pi\)
\(60\) 0 0
\(61\) −0.752715 0.752715i −0.0963753 0.0963753i 0.657275 0.753651i \(-0.271709\pi\)
−0.753651 + 0.657275i \(0.771709\pi\)
\(62\) 0 0
\(63\) 5.00100 0.630066
\(64\) 0 0
\(65\) 2.14626 0.266211
\(66\) 0 0
\(67\) −11.2975 11.2975i −1.38021 1.38021i −0.844230 0.535980i \(-0.819942\pi\)
−0.535980 0.844230i \(-0.680058\pi\)
\(68\) 0 0
\(69\) 3.18618 3.18618i 0.383572 0.383572i
\(70\) 0 0
\(71\) 0.399602i 0.0474240i −0.999719 0.0237120i \(-0.992452\pi\)
0.999719 0.0237120i \(-0.00754847\pi\)
\(72\) 0 0
\(73\) 6.02406i 0.705063i −0.935800 0.352532i \(-0.885321\pi\)
0.935800 0.352532i \(-0.114679\pi\)
\(74\) 0 0
\(75\) −0.658919 + 0.658919i −0.0760854 + 0.0760854i
\(76\) 0 0
\(77\) 2.81481 + 2.81481i 0.320777 + 0.320777i
\(78\) 0 0
\(79\) −15.8637 −1.78481 −0.892403 0.451239i \(-0.850982\pi\)
−0.892403 + 0.451239i \(0.850982\pi\)
\(80\) 0 0
\(81\) −1.93890 −0.215433
\(82\) 0 0
\(83\) 0.312853 + 0.312853i 0.0343401 + 0.0343401i 0.724068 0.689728i \(-0.242270\pi\)
−0.689728 + 0.724068i \(0.742270\pi\)
\(84\) 0 0
\(85\) −0.785587 + 0.785587i −0.0852089 + 0.0852089i
\(86\) 0 0
\(87\) 7.24728i 0.776990i
\(88\) 0 0
\(89\) 16.5558i 1.75491i −0.479655 0.877457i \(-0.659238\pi\)
0.479655 0.877457i \(-0.340762\pi\)
\(90\) 0 0
\(91\) 3.56048 3.56048i 0.373240 0.373240i
\(92\) 0 0
\(93\) −4.14626 4.14626i −0.429948 0.429948i
\(94\) 0 0
\(95\) 2.16693 0.222322
\(96\) 0 0
\(97\) 9.14502 0.928536 0.464268 0.885695i \(-0.346317\pi\)
0.464268 + 0.885695i \(0.346317\pi\)
\(98\) 0 0
\(99\) −2.55756 2.55756i −0.257044 0.257044i
\(100\) 0 0
\(101\) 1.96472 1.96472i 0.195497 0.195497i −0.602569 0.798067i \(-0.705856\pi\)
0.798067 + 0.602569i \(0.205856\pi\)
\(102\) 0 0
\(103\) 4.52885i 0.446241i −0.974791 0.223120i \(-0.928376\pi\)
0.974791 0.223120i \(-0.0716243\pi\)
\(104\) 0 0
\(105\) 2.18618i 0.213350i
\(106\) 0 0
\(107\) −2.87574 + 2.87574i −0.278008 + 0.278008i −0.832313 0.554305i \(-0.812984\pi\)
0.554305 + 0.832313i \(0.312984\pi\)
\(108\) 0 0
\(109\) −2.74202 2.74202i −0.262638 0.262638i 0.563487 0.826125i \(-0.309459\pi\)
−0.826125 + 0.563487i \(0.809459\pi\)
\(110\) 0 0
\(111\) −6.05646 −0.574853
\(112\) 0 0
\(113\) −14.3923 −1.35391 −0.676957 0.736022i \(-0.736702\pi\)
−0.676957 + 0.736022i \(0.736702\pi\)
\(114\) 0 0
\(115\) −3.41920 3.41920i −0.318842 0.318842i
\(116\) 0 0
\(117\) −3.23508 + 3.23508i −0.299083 + 0.299083i
\(118\) 0 0
\(119\) 2.60645i 0.238933i
\(120\) 0 0
\(121\) 8.12096i 0.738269i
\(122\) 0 0
\(123\) −6.62863 + 6.62863i −0.597683 + 0.597683i
\(124\) 0 0
\(125\) 0.707107 + 0.707107i 0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) 18.2611 1.62041 0.810204 0.586148i \(-0.199356\pi\)
0.810204 + 0.586148i \(0.199356\pi\)
\(128\) 0 0
\(129\) −6.77537 −0.596538
\(130\) 0 0
\(131\) −4.61401 4.61401i −0.403128 0.403128i 0.476206 0.879334i \(-0.342012\pi\)
−0.879334 + 0.476206i \(0.842012\pi\)
\(132\) 0 0
\(133\) 3.59476 3.59476i 0.311705 0.311705i
\(134\) 0 0
\(135\) 4.78194i 0.411564i
\(136\) 0 0
\(137\) 14.4195i 1.23194i 0.787768 + 0.615972i \(0.211237\pi\)
−0.787768 + 0.615972i \(0.788763\pi\)
\(138\) 0 0
\(139\) 11.9160 11.9160i 1.01070 1.01070i 0.0107594 0.999942i \(-0.496575\pi\)
0.999942 0.0107594i \(-0.00342490\pi\)
\(140\) 0 0
\(141\) −4.24728 4.24728i −0.357686 0.357686i
\(142\) 0 0
\(143\) −3.64173 −0.304537
\(144\) 0 0
\(145\) 7.77729 0.645869
\(146\) 0 0
\(147\) −0.985728 0.985728i −0.0813015 0.0813015i
\(148\) 0 0
\(149\) −9.47407 + 9.47407i −0.776146 + 0.776146i −0.979173 0.203027i \(-0.934922\pi\)
0.203027 + 0.979173i \(0.434922\pi\)
\(150\) 0 0
\(151\) 10.7062i 0.871260i 0.900126 + 0.435630i \(0.143474\pi\)
−0.900126 + 0.435630i \(0.856526\pi\)
\(152\) 0 0
\(153\) 2.36824i 0.191461i
\(154\) 0 0
\(155\) −4.44949 + 4.44949i −0.357392 + 0.357392i
\(156\) 0 0
\(157\) 16.4986 + 16.4986i 1.31674 + 1.31674i 0.916344 + 0.400391i \(0.131126\pi\)
0.400391 + 0.916344i \(0.368874\pi\)
\(158\) 0 0
\(159\) −6.62639 −0.525507
\(160\) 0 0
\(161\) −11.3443 −0.894059
\(162\) 0 0
\(163\) 10.8152 + 10.8152i 0.847108 + 0.847108i 0.989771 0.142663i \(-0.0455666\pi\)
−0.142663 + 0.989771i \(0.545567\pi\)
\(164\) 0 0
\(165\) 1.11804 1.11804i 0.0870390 0.0870390i
\(166\) 0 0
\(167\) 13.9413i 1.07881i −0.842046 0.539405i \(-0.818649\pi\)
0.842046 0.539405i \(-0.181351\pi\)
\(168\) 0 0
\(169\) 8.39355i 0.645658i
\(170\) 0 0
\(171\) −3.26622 + 3.26622i −0.249774 + 0.249774i
\(172\) 0 0
\(173\) −2.73205 2.73205i −0.207714 0.207714i 0.595581 0.803295i \(-0.296922\pi\)
−0.803295 + 0.595581i \(0.796922\pi\)
\(174\) 0 0
\(175\) 2.34607 0.177346
\(176\) 0 0
\(177\) 1.92628 0.144788
\(178\) 0 0
\(179\) 17.7656 + 17.7656i 1.32786 + 1.32786i 0.907231 + 0.420632i \(0.138192\pi\)
0.420632 + 0.907231i \(0.361808\pi\)
\(180\) 0 0
\(181\) 2.24213 2.24213i 0.166656 0.166656i −0.618852 0.785508i \(-0.712402\pi\)
0.785508 + 0.618852i \(0.212402\pi\)
\(182\) 0 0
\(183\) 0.991956i 0.0733275i
\(184\) 0 0
\(185\) 6.49938i 0.477844i
\(186\) 0 0
\(187\) 1.33296 1.33296i 0.0974760 0.0974760i
\(188\) 0 0
\(189\) −7.93285 7.93285i −0.577030 0.577030i
\(190\) 0 0
\(191\) 2.60164 0.188248 0.0941241 0.995560i \(-0.469995\pi\)
0.0941241 + 0.995560i \(0.469995\pi\)
\(192\) 0 0
\(193\) 17.0452 1.22694 0.613472 0.789717i \(-0.289772\pi\)
0.613472 + 0.789717i \(0.289772\pi\)
\(194\) 0 0
\(195\) −1.41421 1.41421i −0.101274 0.101274i
\(196\) 0 0
\(197\) −10.0535 + 10.0535i −0.716285 + 0.716285i −0.967842 0.251558i \(-0.919057\pi\)
0.251558 + 0.967842i \(0.419057\pi\)
\(198\) 0 0
\(199\) 10.7287i 0.760534i −0.924877 0.380267i \(-0.875832\pi\)
0.924877 0.380267i \(-0.124168\pi\)
\(200\) 0 0
\(201\) 14.8883i 1.05014i
\(202\) 0 0
\(203\) 12.9019 12.9019i 0.905536 0.905536i
\(204\) 0 0
\(205\) 7.11339 + 7.11339i 0.496821 + 0.496821i
\(206\) 0 0
\(207\) 10.3076 0.716424
\(208\) 0 0
\(209\) −3.67679 −0.254329
\(210\) 0 0
\(211\) 17.7350 + 17.7350i 1.22093 + 1.22093i 0.967303 + 0.253622i \(0.0816221\pi\)
0.253622 + 0.967303i \(0.418378\pi\)
\(212\) 0 0
\(213\) −0.263305 + 0.263305i −0.0180414 + 0.0180414i
\(214\) 0 0
\(215\) 7.27087i 0.495869i
\(216\) 0 0
\(217\) 14.7627i 1.00216i
\(218\) 0 0
\(219\) −3.96937 + 3.96937i −0.268225 + 0.268225i
\(220\) 0 0
\(221\) −1.68608 1.68608i −0.113418 0.113418i
\(222\) 0 0
\(223\) −18.2124 −1.21960 −0.609798 0.792557i \(-0.708749\pi\)
−0.609798 + 0.792557i \(0.708749\pi\)
\(224\) 0 0
\(225\) −2.13165 −0.142110
\(226\) 0 0
\(227\) −8.98797 8.98797i −0.596552 0.596552i 0.342841 0.939393i \(-0.388611\pi\)
−0.939393 + 0.342841i \(0.888611\pi\)
\(228\) 0 0
\(229\) 1.60645 1.60645i 0.106157 0.106157i −0.652033 0.758190i \(-0.726084\pi\)
0.758190 + 0.652033i \(0.226084\pi\)
\(230\) 0 0
\(231\) 3.70946i 0.244065i
\(232\) 0 0
\(233\) 0.287369i 0.0188262i 0.999956 + 0.00941309i \(0.00299632\pi\)
−0.999956 + 0.00941309i \(0.997004\pi\)
\(234\) 0 0
\(235\) −4.55790 + 4.55790i −0.297324 + 0.297324i
\(236\) 0 0
\(237\) 10.4529 + 10.4529i 0.678988 + 0.678988i
\(238\) 0 0
\(239\) −17.5274 −1.13375 −0.566875 0.823804i \(-0.691848\pi\)
−0.566875 + 0.823804i \(0.691848\pi\)
\(240\) 0 0
\(241\) 8.39371 0.540687 0.270343 0.962764i \(-0.412863\pi\)
0.270343 + 0.962764i \(0.412863\pi\)
\(242\) 0 0
\(243\) 11.4216 + 11.4216i 0.732696 + 0.732696i
\(244\) 0 0
\(245\) −1.05782 + 1.05782i −0.0675814 + 0.0675814i
\(246\) 0 0
\(247\) 4.65080i 0.295923i
\(248\) 0 0
\(249\) 0.412290i 0.0261278i
\(250\) 0 0
\(251\) −9.83548 + 9.83548i −0.620810 + 0.620810i −0.945738 0.324929i \(-0.894660\pi\)
0.324929 + 0.945738i \(0.394660\pi\)
\(252\) 0 0
\(253\) 5.80161 + 5.80161i 0.364744 + 0.364744i
\(254\) 0 0
\(255\) 1.03528 0.0648315
\(256\) 0 0
\(257\) 18.2054 1.13562 0.567811 0.823159i \(-0.307791\pi\)
0.567811 + 0.823159i \(0.307791\pi\)
\(258\) 0 0
\(259\) 10.7819 + 10.7819i 0.669957 + 0.669957i
\(260\) 0 0
\(261\) −11.7228 + 11.7228i −0.725621 + 0.725621i
\(262\) 0 0
\(263\) 22.5174i 1.38848i −0.719742 0.694242i \(-0.755740\pi\)
0.719742 0.694242i \(-0.244260\pi\)
\(264\) 0 0
\(265\) 7.11099i 0.436825i
\(266\) 0 0
\(267\) −10.9089 + 10.9089i −0.667617 + 0.667617i
\(268\) 0 0
\(269\) −15.0745 15.0745i −0.919107 0.919107i 0.0778580 0.996964i \(-0.475192\pi\)
−0.996964 + 0.0778580i \(0.975192\pi\)
\(270\) 0 0
\(271\) −30.6848 −1.86397 −0.931985 0.362496i \(-0.881925\pi\)
−0.931985 + 0.362496i \(0.881925\pi\)
\(272\) 0 0
\(273\) −4.69213 −0.283981
\(274\) 0 0
\(275\) −1.19980 1.19980i −0.0723507 0.0723507i
\(276\) 0 0
\(277\) −6.85425 + 6.85425i −0.411832 + 0.411832i −0.882376 0.470544i \(-0.844058\pi\)
0.470544 + 0.882376i \(0.344058\pi\)
\(278\) 0 0
\(279\) 13.4135i 0.803044i
\(280\) 0 0
\(281\) 2.44433i 0.145817i 0.997339 + 0.0729083i \(0.0232280\pi\)
−0.997339 + 0.0729083i \(0.976772\pi\)
\(282\) 0 0
\(283\) −2.29171 + 2.29171i −0.136228 + 0.136228i −0.771932 0.635705i \(-0.780710\pi\)
0.635705 + 0.771932i \(0.280710\pi\)
\(284\) 0 0
\(285\) −1.42783 1.42783i −0.0845773 0.0845773i
\(286\) 0 0
\(287\) 23.6011 1.39313
\(288\) 0 0
\(289\) −15.7657 −0.927394
\(290\) 0 0
\(291\) −6.02582 6.02582i −0.353240 0.353240i
\(292\) 0 0
\(293\) −17.9043 + 17.9043i −1.04598 + 1.04598i −0.0470899 + 0.998891i \(0.514995\pi\)
−0.998891 + 0.0470899i \(0.985005\pi\)
\(294\) 0 0
\(295\) 2.06715i 0.120354i
\(296\) 0 0
\(297\) 8.11387i 0.470815i
\(298\) 0 0
\(299\) 7.33850 7.33850i 0.424397 0.424397i
\(300\) 0 0
\(301\) 12.0618 + 12.0618i 0.695229 + 0.695229i
\(302\) 0 0
\(303\) −2.58919 −0.148745
\(304\) 0 0
\(305\) 1.06450 0.0609531
\(306\) 0 0
\(307\) 7.15125 + 7.15125i 0.408143 + 0.408143i 0.881091 0.472947i \(-0.156810\pi\)
−0.472947 + 0.881091i \(0.656810\pi\)
\(308\) 0 0
\(309\) −2.98414 + 2.98414i −0.169762 + 0.169762i
\(310\) 0 0
\(311\) 23.2114i 1.31620i −0.752930 0.658100i \(-0.771360\pi\)
0.752930 0.658100i \(-0.228640\pi\)
\(312\) 0 0
\(313\) 5.10686i 0.288657i 0.989530 + 0.144328i \(0.0461022\pi\)
−0.989530 + 0.144328i \(0.953898\pi\)
\(314\) 0 0
\(315\) −3.53624 + 3.53624i −0.199244 + 0.199244i
\(316\) 0 0
\(317\) 20.6162 + 20.6162i 1.15792 + 1.15792i 0.984922 + 0.172997i \(0.0553451\pi\)
0.172997 + 0.984922i \(0.444655\pi\)
\(318\) 0 0
\(319\) −13.1963 −0.738852
\(320\) 0 0
\(321\) 3.78975 0.211523
\(322\) 0 0
\(323\) −1.70231 1.70231i −0.0947192 0.0947192i
\(324\) 0 0
\(325\) −1.51764 + 1.51764i −0.0841834 + 0.0841834i
\(326\) 0 0
\(327\) 3.61353i 0.199829i
\(328\) 0 0
\(329\) 15.1224i 0.833723i
\(330\) 0 0
\(331\) 10.1686 10.1686i 0.558917 0.558917i −0.370082 0.928999i \(-0.620670\pi\)
0.928999 + 0.370082i \(0.120670\pi\)
\(332\) 0 0
\(333\) −9.79655 9.79655i −0.536848 0.536848i
\(334\) 0 0
\(335\) 15.9771 0.872922
\(336\) 0 0
\(337\) 13.9542 0.760133 0.380067 0.924959i \(-0.375901\pi\)
0.380067 + 0.924959i \(0.375901\pi\)
\(338\) 0 0
\(339\) 9.48336 + 9.48336i 0.515065 + 0.515065i
\(340\) 0 0
\(341\) 7.54978 7.54978i 0.408844 0.408844i
\(342\) 0 0
\(343\) 19.9321i 1.07623i
\(344\) 0 0
\(345\) 4.50595i 0.242592i
\(346\) 0 0
\(347\) −20.1208 + 20.1208i −1.08014 + 1.08014i −0.0836444 + 0.996496i \(0.526656\pi\)
−0.996496 + 0.0836444i \(0.973344\pi\)
\(348\) 0 0
\(349\) 3.33386 + 3.33386i 0.178457 + 0.178457i 0.790683 0.612226i \(-0.209726\pi\)
−0.612226 + 0.790683i \(0.709726\pi\)
\(350\) 0 0
\(351\) 10.2633 0.547815
\(352\) 0 0
\(353\) −35.7352 −1.90199 −0.950997 0.309199i \(-0.899939\pi\)
−0.950997 + 0.309199i \(0.899939\pi\)
\(354\) 0 0
\(355\) 0.282561 + 0.282561i 0.0149968 + 0.0149968i
\(356\) 0 0
\(357\) 1.71744 1.71744i 0.0908965 0.0908965i
\(358\) 0 0
\(359\) 11.2159i 0.591954i −0.955195 0.295977i \(-0.904355\pi\)
0.955195 0.295977i \(-0.0956451\pi\)
\(360\) 0 0
\(361\) 14.3044i 0.752864i
\(362\) 0 0
\(363\) 5.35105 5.35105i 0.280857 0.280857i
\(364\) 0 0
\(365\) 4.25966 + 4.25966i 0.222961 + 0.222961i
\(366\) 0 0
\(367\) −13.2867 −0.693561 −0.346781 0.937946i \(-0.612725\pi\)
−0.346781 + 0.937946i \(0.612725\pi\)
\(368\) 0 0
\(369\) −21.4441 −1.11634
\(370\) 0 0
\(371\) 11.7966 + 11.7966i 0.612446 + 0.612446i
\(372\) 0 0
\(373\) −2.58166 + 2.58166i −0.133673 + 0.133673i −0.770778 0.637104i \(-0.780132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(374\) 0 0
\(375\) 0.931852i 0.0481206i
\(376\) 0 0
\(377\) 16.6921i 0.859688i
\(378\) 0 0
\(379\) 19.2025 19.2025i 0.986365 0.986365i −0.0135436 0.999908i \(-0.504311\pi\)
0.999908 + 0.0135436i \(0.00431118\pi\)
\(380\) 0 0
\(381\) −12.0326 12.0326i −0.616447 0.616447i
\(382\) 0 0
\(383\) −19.5286 −0.997867 −0.498933 0.866640i \(-0.666275\pi\)
−0.498933 + 0.866640i \(0.666275\pi\)
\(384\) 0 0
\(385\) −3.98074 −0.202877
\(386\) 0 0
\(387\) −10.9594 10.9594i −0.557099 0.557099i
\(388\) 0 0
\(389\) 5.67412 5.67412i 0.287689 0.287689i −0.548477 0.836166i \(-0.684792\pi\)
0.836166 + 0.548477i \(0.184792\pi\)
\(390\) 0 0
\(391\) 5.37216i 0.271682i
\(392\) 0 0
\(393\) 6.08052i 0.306722i
\(394\) 0 0
\(395\) 11.2173 11.2173i 0.564405 0.564405i
\(396\) 0 0
\(397\) 13.8102 + 13.8102i 0.693112 + 0.693112i 0.962916 0.269803i \(-0.0869586\pi\)
−0.269803 + 0.962916i \(0.586959\pi\)
\(398\) 0 0
\(399\) −4.73731 −0.237162
\(400\) 0 0
\(401\) 4.06126 0.202810 0.101405 0.994845i \(-0.467666\pi\)
0.101405 + 0.994845i \(0.467666\pi\)
\(402\) 0 0
\(403\) −9.54978 9.54978i −0.475708 0.475708i
\(404\) 0 0
\(405\) 1.37101 1.37101i 0.0681260 0.0681260i
\(406\) 0 0
\(407\) 11.0280i 0.546637i
\(408\) 0 0
\(409\) 15.6651i 0.774587i −0.921956 0.387294i \(-0.873410\pi\)
0.921956 0.387294i \(-0.126590\pi\)
\(410\) 0 0
\(411\) 9.50130 9.50130i 0.468665 0.468665i
\(412\) 0 0
\(413\) −3.42924 3.42924i −0.168742 0.168742i
\(414\) 0 0
\(415\) −0.442442 −0.0217186
\(416\) 0 0
\(417\) −15.7033 −0.768996
\(418\) 0 0
\(419\) 12.6081 + 12.6081i 0.615947 + 0.615947i 0.944489 0.328542i \(-0.106557\pi\)
−0.328542 + 0.944489i \(0.606557\pi\)
\(420\) 0 0
\(421\) −5.77589 + 5.77589i −0.281499 + 0.281499i −0.833707 0.552207i \(-0.813786\pi\)
0.552207 + 0.833707i \(0.313786\pi\)
\(422\) 0 0
\(423\) 13.7403i 0.668076i
\(424\) 0 0
\(425\) 1.11099i 0.0538908i
\(426\) 0 0
\(427\) 1.76592 1.76592i 0.0854588 0.0854588i
\(428\) 0 0
\(429\) 2.39960 + 2.39960i 0.115854 + 0.115854i
\(430\) 0 0
\(431\) 0.928203 0.0447100 0.0223550 0.999750i \(-0.492884\pi\)
0.0223550 + 0.999750i \(0.492884\pi\)
\(432\) 0 0
\(433\) 24.3730 1.17129 0.585647 0.810566i \(-0.300841\pi\)
0.585647 + 0.810566i \(0.300841\pi\)
\(434\) 0 0
\(435\) −5.12460 5.12460i −0.245706 0.245706i
\(436\) 0 0
\(437\) 7.40916 7.40916i 0.354428 0.354428i
\(438\) 0 0
\(439\) 25.1832i 1.20193i 0.799276 + 0.600964i \(0.205216\pi\)
−0.799276 + 0.600964i \(0.794784\pi\)
\(440\) 0 0
\(441\) 3.18891i 0.151853i
\(442\) 0 0
\(443\) −8.38152 + 8.38152i −0.398218 + 0.398218i −0.877604 0.479386i \(-0.840859\pi\)
0.479386 + 0.877604i \(0.340859\pi\)
\(444\) 0 0
\(445\) 11.7067 + 11.7067i 0.554953 + 0.554953i
\(446\) 0 0
\(447\) 12.4853 0.590534
\(448\) 0 0
\(449\) 6.60607 0.311760 0.155880 0.987776i \(-0.450179\pi\)
0.155880 + 0.987776i \(0.450179\pi\)
\(450\) 0 0
\(451\) −12.0698 12.0698i −0.568346 0.568346i
\(452\) 0 0
\(453\) 7.05453 7.05453i 0.331451 0.331451i
\(454\) 0 0
\(455\) 5.03528i 0.236057i
\(456\) 0 0
\(457\) 36.6201i 1.71302i 0.516133 + 0.856508i \(0.327371\pi\)
−0.516133 + 0.856508i \(0.672629\pi\)
\(458\) 0 0
\(459\) −3.75663 + 3.75663i −0.175344 + 0.175344i
\(460\) 0 0
\(461\) 5.96472 + 5.96472i 0.277805 + 0.277805i 0.832232 0.554427i \(-0.187063\pi\)
−0.554427 + 0.832232i \(0.687063\pi\)
\(462\) 0 0
\(463\) −13.8749 −0.644820 −0.322410 0.946600i \(-0.604493\pi\)
−0.322410 + 0.946600i \(0.604493\pi\)
\(464\) 0 0
\(465\) 5.86370 0.271923
\(466\) 0 0
\(467\) −19.8755 19.8755i −0.919729 0.919729i 0.0772800 0.997009i \(-0.475376\pi\)
−0.997009 + 0.0772800i \(0.975376\pi\)
\(468\) 0 0
\(469\) 26.5047 26.5047i 1.22387 1.22387i
\(470\) 0 0
\(471\) 21.7425i 1.00184i
\(472\) 0 0
\(473\) 12.3370i 0.567257i
\(474\) 0 0
\(475\) −1.53225 + 1.53225i −0.0703044 + 0.0703044i
\(476\) 0 0
\(477\) −10.7184 10.7184i −0.490763 0.490763i
\(478\) 0 0
\(479\) 15.5931 0.712467 0.356233 0.934397i \(-0.384061\pi\)
0.356233 + 0.934397i \(0.384061\pi\)
\(480\) 0 0
\(481\) −13.9494 −0.636037
\(482\) 0 0
\(483\) 7.47500 + 7.47500i 0.340124 + 0.340124i
\(484\) 0 0
\(485\) −6.46651 + 6.46651i −0.293629 + 0.293629i
\(486\) 0 0
\(487\) 14.1319i 0.640378i 0.947354 + 0.320189i \(0.103746\pi\)
−0.947354 + 0.320189i \(0.896254\pi\)
\(488\) 0 0
\(489\) 14.2526i 0.644525i
\(490\) 0 0
\(491\) −23.7841 + 23.7841i −1.07336 + 1.07336i −0.0762760 + 0.997087i \(0.524303\pi\)
−0.997087 + 0.0762760i \(0.975697\pi\)
\(492\) 0 0
\(493\) −6.10974 6.10974i −0.275169 0.275169i
\(494\) 0 0
\(495\) 3.61693 0.162569
\(496\) 0 0
\(497\) 0.937492 0.0420522
\(498\) 0 0
\(499\) 1.91119 + 1.91119i 0.0855565 + 0.0855565i 0.748590 0.663033i \(-0.230731\pi\)
−0.663033 + 0.748590i \(0.730731\pi\)
\(500\) 0 0
\(501\) −9.18618 + 9.18618i −0.410408 + 0.410408i
\(502\) 0 0
\(503\) 20.9343i 0.933414i 0.884412 + 0.466707i \(0.154560\pi\)
−0.884412 + 0.466707i \(0.845440\pi\)
\(504\) 0 0
\(505\) 2.77854i 0.123643i
\(506\) 0 0
\(507\) −5.53067 + 5.53067i −0.245625 + 0.245625i
\(508\) 0 0
\(509\) −17.2127 17.2127i −0.762939 0.762939i 0.213914 0.976853i \(-0.431379\pi\)
−0.976853 + 0.213914i \(0.931379\pi\)
\(510\) 0 0
\(511\) 14.1328 0.625200
\(512\) 0 0
\(513\) 10.3621 0.457499
\(514\) 0 0
\(515\) 3.20238 + 3.20238i 0.141114 + 0.141114i
\(516\) 0 0
\(517\) 7.73373 7.73373i 0.340129 0.340129i
\(518\) 0 0
\(519\) 3.60040i 0.158040i
\(520\) 0 0
\(521\) 30.0588i 1.31690i −0.752625 0.658449i \(-0.771213\pi\)
0.752625 0.658449i \(-0.228787\pi\)
\(522\) 0 0
\(523\) 6.17948 6.17948i 0.270210 0.270210i −0.558975 0.829185i \(-0.688805\pi\)
0.829185 + 0.558975i \(0.188805\pi\)
\(524\) 0 0
\(525\) −1.54587 1.54587i −0.0674671 0.0674671i
\(526\) 0 0
\(527\) 6.99093 0.304529
\(528\) 0 0
\(529\) −0.381822 −0.0166009
\(530\) 0 0
\(531\) 3.11583 + 3.11583i 0.135216 + 0.135216i
\(532\) 0 0
\(533\) −15.2672 + 15.2672i −0.661297 + 0.661297i
\(534\) 0 0
\(535\) 4.06690i 0.175828i
\(536\) 0 0
\(537\) 23.4122i 1.01031i
\(538\) 0 0
\(539\) 1.79488 1.79488i 0.0773108 0.0773108i
\(540\) 0 0
\(541\) 11.3198 + 11.3198i 0.486675 + 0.486675i 0.907255 0.420581i \(-0.138174\pi\)
−0.420581 + 0.907255i \(0.638174\pi\)
\(542\) 0 0
\(543\) −2.95476 −0.126801
\(544\) 0 0
\(545\) 3.87780 0.166107
\(546\) 0 0
\(547\) −5.87965 5.87965i −0.251396 0.251396i 0.570147 0.821543i \(-0.306886\pi\)
−0.821543 + 0.570147i \(0.806886\pi\)
\(548\) 0 0
\(549\) −1.60453 + 1.60453i −0.0684795 + 0.0684795i
\(550\) 0 0
\(551\) 16.8528i 0.717955i
\(552\) 0 0
\(553\) 37.2173i 1.58264i
\(554\) 0 0
\(555\) 4.28256 4.28256i 0.181785 0.181785i
\(556\) 0 0
\(557\) −21.0900 21.0900i −0.893611 0.893611i 0.101250 0.994861i \(-0.467716\pi\)
−0.994861 + 0.101250i \(0.967716\pi\)
\(558\) 0 0
\(559\) −15.6052 −0.660030
\(560\) 0 0
\(561\) −1.75663 −0.0741650
\(562\) 0 0
\(563\) −29.9169 29.9169i −1.26084 1.26084i −0.950684 0.310161i \(-0.899617\pi\)
−0.310161 0.950684i \(-0.600383\pi\)
\(564\) 0 0
\(565\) 10.1769 10.1769i 0.428145 0.428145i
\(566\) 0 0
\(567\) 4.54879i 0.191031i
\(568\) 0 0
\(569\) 4.80990i 0.201641i 0.994905 + 0.100821i \(0.0321469\pi\)
−0.994905 + 0.100821i \(0.967853\pi\)
\(570\) 0 0
\(571\) 22.6639 22.6639i 0.948455 0.948455i −0.0502803 0.998735i \(-0.516011\pi\)
0.998735 + 0.0502803i \(0.0160115\pi\)
\(572\) 0 0
\(573\) −1.71427 1.71427i −0.0716147 0.0716147i
\(574\) 0 0
\(575\) 4.83548 0.201653
\(576\) 0 0
\(577\) 45.5463 1.89612 0.948059 0.318094i \(-0.103043\pi\)
0.948059 + 0.318094i \(0.103043\pi\)
\(578\) 0 0
\(579\) −11.2314 11.2314i −0.466762 0.466762i
\(580\) 0 0
\(581\) −0.733974 + 0.733974i −0.0304504 + 0.0304504i
\(582\) 0 0
\(583\) 12.0657i 0.499712i
\(584\) 0 0
\(585\) 4.57509i 0.189157i
\(586\) 0 0
\(587\) −9.27826 + 9.27826i −0.382955 + 0.382955i −0.872166 0.489211i \(-0.837285\pi\)
0.489211 + 0.872166i \(0.337285\pi\)
\(588\) 0 0
\(589\) −9.64173 9.64173i −0.397280 0.397280i
\(590\) 0 0
\(591\) 13.2489 0.544988
\(592\) 0 0
\(593\) −3.32677 −0.136614 −0.0683071 0.997664i \(-0.521760\pi\)
−0.0683071 + 0.997664i \(0.521760\pi\)
\(594\) 0 0
\(595\) −1.84304 1.84304i −0.0755572 0.0755572i
\(596\) 0 0
\(597\) −7.06931 + 7.06931i −0.289328 + 0.289328i
\(598\) 0 0
\(599\) 7.52640i 0.307520i 0.988108 + 0.153760i \(0.0491383\pi\)
−0.988108 + 0.153760i \(0.950862\pi\)
\(600\) 0 0
\(601\) 9.03188i 0.368418i −0.982887 0.184209i \(-0.941028\pi\)
0.982887 0.184209i \(-0.0589723\pi\)
\(602\) 0 0
\(603\) −24.0824 + 24.0824i −0.980710 + 0.980710i
\(604\) 0 0
\(605\) −5.74238 5.74238i −0.233461 0.233461i
\(606\) 0 0
\(607\) 11.7031 0.475014 0.237507 0.971386i \(-0.423670\pi\)
0.237507 + 0.971386i \(0.423670\pi\)
\(608\) 0 0
\(609\) −17.0026 −0.688980
\(610\) 0 0
\(611\) −9.78245 9.78245i −0.395756 0.395756i
\(612\) 0 0
\(613\) 19.2477 19.2477i 0.777408 0.777408i −0.201981 0.979389i \(-0.564738\pi\)
0.979389 + 0.201981i \(0.0647381\pi\)
\(614\) 0 0
\(615\) 9.37429i 0.378008i
\(616\) 0 0
\(617\) 13.5963i 0.547365i −0.961820 0.273683i \(-0.911758\pi\)
0.961820 0.273683i \(-0.0882418\pi\)
\(618\) 0 0
\(619\) −1.50303 + 1.50303i −0.0604117 + 0.0604117i −0.736667 0.676255i \(-0.763602\pi\)
0.676255 + 0.736667i \(0.263602\pi\)
\(620\) 0 0
\(621\) −16.3504 16.3504i −0.656119 0.656119i
\(622\) 0 0
\(623\) 38.8411 1.55613
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 2.42271 + 2.42271i 0.0967535 + 0.0967535i
\(628\) 0 0
\(629\) 5.10583 5.10583i 0.203583 0.203583i
\(630\) 0 0
\(631\) 27.1399i 1.08042i 0.841530 + 0.540210i \(0.181655\pi\)
−0.841530 + 0.540210i \(0.818345\pi\)
\(632\) 0 0
\(633\) 23.3718i 0.928946i
\(634\) 0 0
\(635\) −12.9125 + 12.9125i −0.512418 + 0.512418i
\(636\) 0 0
\(637\) −2.27035 2.27035i −0.0899547 0.0899547i
\(638\) 0 0
\(639\) −0.851812 −0.0336972
\(640\) 0 0
\(641\) 27.5635 1.08869 0.544346 0.838861i \(-0.316778\pi\)
0.544346 + 0.838861i \(0.316778\pi\)
\(642\) 0 0
\(643\) 23.1493 + 23.1493i 0.912918 + 0.912918i 0.996501 0.0835832i \(-0.0266364\pi\)
−0.0835832 + 0.996501i \(0.526636\pi\)
\(644\) 0 0
\(645\) 4.79091 4.79091i 0.188642 0.188642i
\(646\) 0 0
\(647\) 28.4410i 1.11813i 0.829123 + 0.559066i \(0.188840\pi\)
−0.829123 + 0.559066i \(0.811160\pi\)
\(648\) 0 0
\(649\) 3.50749i 0.137681i
\(650\) 0 0
\(651\) 9.72741 9.72741i 0.381247 0.381247i
\(652\) 0 0
\(653\) −0.957676 0.957676i −0.0374767 0.0374767i 0.688120 0.725597i \(-0.258436\pi\)
−0.725597 + 0.688120i \(0.758436\pi\)
\(654\) 0 0
\(655\) 6.52520 0.254961
\(656\) 0 0
\(657\) −12.8412 −0.500983
\(658\) 0 0
\(659\) 0.917240 + 0.917240i 0.0357306 + 0.0357306i 0.724746 0.689016i \(-0.241957\pi\)
−0.689016 + 0.724746i \(0.741957\pi\)
\(660\) 0 0
\(661\) 11.7481 11.7481i 0.456947 0.456947i −0.440705 0.897652i \(-0.645272\pi\)
0.897652 + 0.440705i \(0.145272\pi\)
\(662\) 0 0
\(663\) 2.22198i 0.0862944i
\(664\) 0 0
\(665\) 5.08376i 0.197140i
\(666\) 0 0
\(667\) 26.5921 26.5921i 1.02965 1.02965i
\(668\) 0 0
\(669\) 12.0005 + 12.0005i 0.463967 + 0.463967i
\(670\) 0 0
\(671\) −1.80622 −0.0697282
\(672\) 0 0
\(673\) 7.48817 0.288648 0.144324 0.989531i \(-0.453899\pi\)
0.144324 + 0.989531i \(0.453899\pi\)
\(674\) 0 0
\(675\) 3.38134 + 3.38134i 0.130148 + 0.130148i
\(676\) 0 0
\(677\) 19.3721 19.3721i 0.744528 0.744528i −0.228917 0.973446i \(-0.573519\pi\)
0.973446 + 0.228917i \(0.0735186\pi\)
\(678\) 0 0
\(679\) 21.4548i 0.823360i
\(680\) 0 0
\(681\) 11.8447i 0.453889i
\(682\) 0 0
\(683\) 14.4723 14.4723i 0.553765 0.553765i −0.373760 0.927525i \(-0.621932\pi\)
0.927525 + 0.373760i \(0.121932\pi\)
\(684\) 0 0
\(685\) −10.1962 10.1962i −0.389575 0.389575i
\(686\) 0 0
\(687\) −2.11704 −0.0807701
\(688\) 0 0
\(689\) −15.2621 −0.581438
\(690\) 0 0
\(691\) −18.3482 18.3482i −0.697999 0.697999i 0.265979 0.963979i \(-0.414305\pi\)
−0.963979 + 0.265979i \(0.914305\pi\)
\(692\) 0 0
\(693\) 6.00020 6.00020i 0.227929 0.227929i
\(694\) 0 0
\(695\) 16.8518i 0.639224i
\(696\) 0 0
\(697\) 11.1764i 0.423336i
\(698\) 0 0
\(699\) 0.189353 0.189353i 0.00716198 0.00716198i
\(700\) 0 0
\(701\) −12.9630 12.9630i −0.489604 0.489604i 0.418577 0.908181i \(-0.362529\pi\)
−0.908181 + 0.418577i \(0.862529\pi\)
\(702\) 0 0
\(703\) −14.0837 −0.531176
\(704\) 0 0
\(705\) 6.00657 0.226220
\(706\) 0 0
\(707\) 4.60937 + 4.60937i 0.173353 + 0.173353i
\(708\) 0 0
\(709\) 6.10583 6.10583i 0.229309 0.229309i −0.583095 0.812404i \(-0.698158\pi\)
0.812404 + 0.583095i \(0.198158\pi\)
\(710\) 0 0
\(711\) 33.8159i 1.26820i
\(712\) 0 0
\(713\) 30.4274i 1.13951i
\(714\) 0 0
\(715\) 2.57509 2.57509i 0.0963029 0.0963029i
\(716\) 0 0
\(717\) 11.5491 + 11.5491i 0.431309 + 0.431309i
\(718\) 0 0
\(719\) 49.9396 1.86243 0.931216 0.364467i \(-0.118749\pi\)
0.931216 + 0.364467i \(0.118749\pi\)
\(720\) 0 0
\(721\) 10.6250 0.395695
\(722\) 0 0
\(723\) −5.53077 5.53077i −0.205692 0.205692i
\(724\) 0 0
\(725\) −5.49938 + 5.49938i −0.204242 + 0.204242i
\(726\) 0 0
\(727\) 8.41214i 0.311989i 0.987758 + 0.155994i \(0.0498582\pi\)
−0.987758 + 0.155994i \(0.950142\pi\)
\(728\) 0 0
\(729\) 9.23511i 0.342041i
\(730\) 0 0
\(731\) 5.71190 5.71190i 0.211262 0.211262i
\(732\) 0 0
\(733\) −10.6663 10.6663i −0.393969 0.393969i 0.482130 0.876099i \(-0.339863\pi\)
−0.876099 + 0.482130i \(0.839863\pi\)
\(734\) 0 0
\(735\) 1.39403 0.0514196
\(736\) 0 0
\(737\) −27.1095 −0.998592
\(738\) 0 0
\(739\) −8.63735 8.63735i −0.317730 0.317730i 0.530165 0.847895i \(-0.322130\pi\)
−0.847895 + 0.530165i \(0.822130\pi\)
\(740\) 0 0
\(741\) 3.06450 3.06450i 0.112577 0.112577i
\(742\) 0 0
\(743\) 10.1561i 0.372592i 0.982494 + 0.186296i \(0.0596484\pi\)
−0.982494 + 0.186296i \(0.940352\pi\)
\(744\) 0 0
\(745\) 13.3984i 0.490878i
\(746\) 0 0
\(747\) 0.666895 0.666895i 0.0244004 0.0244004i
\(748\) 0 0
\(749\) −6.74666 6.74666i −0.246518 0.246518i
\(750\) 0 0
\(751\) −24.3700 −0.889272 −0.444636 0.895711i \(-0.646667\pi\)
−0.444636 + 0.895711i \(0.646667\pi\)
\(752\) 0 0
\(753\) 12.9616 0.472345
\(754\) 0 0
\(755\) −7.57045 7.57045i −0.275517 0.275517i
\(756\) 0 0
\(757\) −8.98325 + 8.98325i −0.326502 + 0.326502i −0.851255 0.524753i \(-0.824158\pi\)
0.524753 + 0.851255i \(0.324158\pi\)
\(758\) 0 0
\(759\) 7.64557i 0.277517i
\(760\) 0 0
\(761\) 4.00929i 0.145337i 0.997356 + 0.0726683i \(0.0231514\pi\)
−0.997356 + 0.0726683i \(0.976849\pi\)
\(762\) 0 0
\(763\) 6.43295 6.43295i 0.232889 0.232889i
\(764\) 0 0
\(765\) 1.67460 + 1.67460i 0.0605453 + 0.0605453i
\(766\) 0 0
\(767\) 4.43666 0.160198
\(768\) 0 0
\(769\) 46.6663 1.68283 0.841414 0.540391i \(-0.181724\pi\)
0.841414 + 0.540391i \(0.181724\pi\)
\(770\) 0 0
\(771\) −11.9959 11.9959i −0.432021 0.432021i
\(772\) 0 0
\(773\) 15.8133 15.8133i 0.568766 0.568766i −0.363017 0.931783i \(-0.618253\pi\)
0.931783 + 0.363017i \(0.118253\pi\)
\(774\) 0 0
\(775\) 6.29253i 0.226034i
\(776\) 0 0
\(777\) 14.2088i 0.509739i
\(778\) 0 0
\(779\) −15.4142 + 15.4142i −0.552272 + 0.552272i
\(780\) 0 0
\(781\) −0.479442 0.479442i −0.0171558 0.0171558i
\(782\) 0 0
\(783\) 37.1905 1.32908
\(784\) 0 0
\(785\) −23.3326 −0.832777
\(786\) 0 0
\(787\) −20.1108 20.1108i −0.716873 0.716873i 0.251091 0.967964i \(-0.419211\pi\)
−0.967964 + 0.251091i \(0.919211\pi\)
\(788\) 0 0
\(789\) −14.8372 + 14.8372i −0.528217 + 0.528217i
\(790\) 0 0
\(791\) 33.7653i 1.20056i
\(792\) 0 0
\(793\) 2.28470i 0.0811320i
\(794\) 0 0
\(795\) 4.68556 4.68556i 0.166180 0.166180i
\(796\) 0 0
\(797\) −2.79415 2.79415i −0.0989738 0.0989738i 0.655886 0.754860i \(-0.272295\pi\)
−0.754860 + 0.655886i \(0.772295\pi\)
\(798\) 0 0
\(799\) 7.16125 0.253347
\(800\) 0 0
\(801\) −35.2913 −1.24696
\(802\) 0 0
\(803\) −7.22768 7.22768i −0.255059 0.255059i
\(804\) 0 0
\(805\) 8.02166 8.02166i 0.282726 0.282726i
\(806\) 0 0
\(807\) 19.8657i 0.699306i
\(808\) 0 0
\(809\) 39.9600i 1.40492i −0.711723 0.702460i \(-0.752085\pi\)
0.711723 0.702460i \(-0.247915\pi\)
\(810\) 0 0
\(811\) −29.8685 + 29.8685i −1.04883 + 1.04883i −0.0500814 + 0.998745i \(0.515948\pi\)
−0.998745 + 0.0500814i \(0.984052\pi\)
\(812\) 0 0
\(813\) 20.2188 + 20.2188i 0.709105 + 0.709105i
\(814\) 0 0
\(815\) −15.2949 −0.535758
\(816\) 0 0
\(817\) −15.7555 −0.551214
\(818\) 0 0
\(819\) −7.58970 7.58970i −0.265206 0.265206i
\(820\) 0 0
\(821\) 11.2658 11.2658i 0.393180 0.393180i −0.482640 0.875819i \(-0.660322\pi\)
0.875819 + 0.482640i \(0.160322\pi\)
\(822\) 0 0
\(823\) 11.8670i 0.413657i 0.978377 + 0.206828i \(0.0663142\pi\)
−0.978377 + 0.206828i \(0.933686\pi\)
\(824\) 0 0
\(825\) 1.58114i 0.0550483i
\(826\) 0 0
\(827\) 13.5457 13.5457i 0.471030 0.471030i −0.431218 0.902248i \(-0.641916\pi\)
0.902248 + 0.431218i \(0.141916\pi\)
\(828\) 0 0
\(829\) 10.3975 + 10.3975i 0.361119 + 0.361119i 0.864225 0.503106i \(-0.167809\pi\)
−0.503106 + 0.864225i \(0.667809\pi\)
\(830\) 0 0
\(831\) 9.03279 0.313344
\(832\) 0 0
\(833\) 1.66201 0.0575854
\(834\) 0 0
\(835\) 9.85799 + 9.85799i 0.341150 + 0.341150i
\(836\) 0 0
\(837\) −21.2772 + 21.2772i −0.735447 + 0.735447i
\(838\) 0 0
\(839\) 2.89168i 0.0998320i −0.998753 0.0499160i \(-0.984105\pi\)
0.998753 0.0499160i \(-0.0158954\pi\)
\(840\) 0 0
\(841\) 31.4863i 1.08573i
\(842\) 0 0
\(843\) 1.61061 1.61061i 0.0554725 0.0554725i
\(844\) 0 0
\(845\) 5.93514 + 5.93514i 0.204175 + 0.204175i
\(846\) 0 0
\(847\) −19.0523 −0.654645
\(848\) 0 0
\(849\) 3.02010 0.103650
\(850\) 0 0
\(851\) 22.2227 + 22.2227i 0.761783 + 0.761783i
\(852\) 0 0
\(853\) 13.6325 13.6325i 0.466769 0.466769i −0.434097 0.900866i \(-0.642932\pi\)
0.900866 + 0.434097i \(0.142932\pi\)
\(854\) 0 0
\(855\) 4.61914i 0.157971i
\(856\) 0 0
\(857\) 37.4062i 1.27777i −0.769302 0.638886i \(-0.779396\pi\)
0.769302 0.638886i \(-0.220604\pi\)
\(858\) 0 0
\(859\) 13.8754 13.8754i 0.473422 0.473422i −0.429598 0.903020i \(-0.641345\pi\)
0.903020 + 0.429598i \(0.141345\pi\)
\(860\) 0 0
\(861\) −15.5512 15.5512i −0.529983 0.529983i
\(862\) 0 0
\(863\) −47.5152 −1.61744 −0.808718 0.588196i \(-0.799838\pi\)
−0.808718 + 0.588196i \(0.799838\pi\)
\(864\) 0 0
\(865\) 3.86370 0.131370
\(866\) 0 0
\(867\) 10.3883 + 10.3883i 0.352806 + 0.352806i
\(868\) 0 0
\(869\) −19.0333 + 19.0333i −0.645660 + 0.645660i
\(870\) 0 0
\(871\) 34.2911i 1.16191i
\(872\) 0 0
\(873\) 19.4940i 0.659772i
\(874\) 0 0
\(875\) −1.65892 + 1.65892i −0.0560817 + 0.0560817i
\(876\) 0 0
\(877\) −22.0217 22.0217i −0.743620 0.743620i 0.229653 0.973273i \(-0.426241\pi\)
−0.973273 + 0.229653i \(0.926241\pi\)
\(878\) 0 0
\(879\) 23.5950 0.795838
\(880\) 0 0
\(881\) −19.8379 −0.668355 −0.334178 0.942510i \(-0.608459\pi\)
−0.334178 + 0.942510i \(0.608459\pi\)
\(882\) 0 0
\(883\) −10.7398 10.7398i −0.361423 0.361423i 0.502914 0.864337i \(-0.332261\pi\)
−0.864337 + 0.502914i \(0.832261\pi\)
\(884\) 0 0
\(885\) −1.36209 + 1.36209i −0.0457860 + 0.0457860i
\(886\) 0 0
\(887\) 57.5726i 1.93310i −0.256478 0.966550i \(-0.582562\pi\)
0.256478 0.966550i \(-0.417438\pi\)
\(888\) 0 0
\(889\) 42.8416i 1.43686i
\(890\) 0 0
\(891\) −2.32629 + 2.32629i −0.0779338 + 0.0779338i
\(892\) 0 0
\(893\) −9.87664 9.87664i −0.330509 0.330509i
\(894\) 0 0
\(895\) −25.1244 −0.839815
\(896\) 0 0
\(897\) −9.67095 −0.322904
\(898\) 0 0
\(899\) −34.6050 34.6050i −1.15414 1.15414i
\(900\) 0 0
\(901\) 5.58630 5.58630i 0.186107 0.186107i
\(902\) 0 0
\(903\) 15.8955i 0.528968i
\(904\) 0 0
\(905\) 3.17084i 0.105402i
\(906\) 0 0
\(907\) −3.73175 + 3.73175i −0.123911 + 0.123911i −0.766343 0.642432i \(-0.777925\pi\)
0.642432 + 0.766343i \(0.277925\pi\)
\(908\) 0 0
\(909\) −4.18811 4.18811i −0.138911 0.138911i
\(910\) 0 0
\(911\) −11.9590 −0.396219 −0.198110 0.980180i \(-0.563480\pi\)
−0.198110 + 0.980180i \(0.563480\pi\)
\(912\) 0 0
\(913\) 0.750724 0.0248453
\(914\) 0 0
\(915\) −0.701419 0.701419i −0.0231882 0.0231882i
\(916\) 0 0
\(917\) 10.8248 10.8248i 0.357466 0.357466i
\(918\) 0 0
\(919\) 53.6932i 1.77118i −0.464472 0.885588i \(-0.653756\pi\)
0.464472 0.885588i \(-0.346244\pi\)
\(920\) 0 0
\(921\) 9.42418i 0.310537i
\(922\) 0 0
\(923\) −0.606451 + 0.606451i −0.0199616 + 0.0199616i
\(924\) 0 0
\(925\) −4.59575 4.59575i −0.151107 0.151107i
\(926\) 0 0
\(927\) −9.65393 −0.317077
\(928\) 0 0
\(929\) 7.25033 0.237876 0.118938 0.992902i \(-0.462051\pi\)
0.118938 + 0.992902i \(0.462051\pi\)
\(930\) 0 0
\(931\) −2.29221 2.29221i −0.0751242 0.0751242i
\(932\) 0 0
\(933\) −15.2945 + 15.2945i −0.500718 + 0.500718i
\(934\) 0 0
\(935\) 1.88510i 0.0616493i
\(936\) 0 0
\(937\) 31.3398i 1.02383i −0.859037 0.511913i \(-0.828937\pi\)
0.859037 0.511913i \(-0.171063\pi\)
\(938\) 0 0
\(939\) 3.36500 3.36500i 0.109813 0.109813i
\(940\) 0 0
\(941\) 25.5972 + 25.5972i 0.834444 + 0.834444i 0.988121 0.153677i \(-0.0491117\pi\)
−0.153677 + 0.988121i \(0.549112\pi\)
\(942\) 0 0
\(943\) 48.6442 1.58407
\(944\) 0 0
\(945\) 11.2187 0.364946
\(946\) 0 0
\(947\) 37.1739 + 37.1739i 1.20799 + 1.20799i 0.971678 + 0.236310i \(0.0759382\pi\)
0.236310 + 0.971678i \(0.424062\pi\)
\(948\) 0 0
\(949\) −9.14235 + 9.14235i −0.296773 + 0.296773i
\(950\) 0 0
\(951\) 27.1688i 0.881007i
\(952\) 0 0
\(953\) 33.0538i 1.07072i 0.844625 + 0.535358i \(0.179823\pi\)
−0.844625 + 0.535358i \(0.820177\pi\)
\(954\) 0 0
\(955\) −1.83964 + 1.83964i −0.0595293 + 0.0595293i
\(956\) 0 0
\(957\) 8.69530 + 8.69530i 0.281079 + 0.281079i
\(958\) 0 0
\(959\) −33.8292 −1.09240
\(960\) 0 0
\(961\) 8.59592 0.277288
\(962\) 0 0
\(963\) 6.13007 + 6.13007i 0.197539 + 0.197539i
\(964\) 0 0
\(965\) −12.0528 + 12.0528i −0.387994 + 0.387994i
\(966\) 0 0
\(967\) 19.8952i 0.639785i −0.947454 0.319893i \(-0.896353\pi\)
0.947454 0.319893i \(-0.103647\pi\)
\(968\) 0 0
\(969\) 2.24337i 0.0720674i
\(970\) 0 0
\(971\) −36.6690 + 36.6690i −1.17676 + 1.17676i −0.196201 + 0.980564i \(0.562860\pi\)
−0.980564 + 0.196201i \(0.937140\pi\)
\(972\) 0 0
\(973\) 27.9557 + 27.9557i 0.896219 + 0.896219i
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) 14.8822 0.476124 0.238062 0.971250i \(-0.423488\pi\)
0.238062 + 0.971250i \(0.423488\pi\)
\(978\) 0 0
\(979\) −19.8637 19.8637i −0.634847 0.634847i
\(980\) 0 0
\(981\) −5.84503 + 5.84503i −0.186617 + 0.186617i
\(982\) 0 0
\(983\) 39.9254i 1.27342i 0.771102 + 0.636711i \(0.219706\pi\)
−0.771102 + 0.636711i \(0.780294\pi\)
\(984\) 0 0
\(985\) 14.2178i 0.453018i
\(986\) 0 0
\(987\) 9.96441 9.96441i 0.317171 0.317171i
\(988\) 0 0
\(989\) 24.8605 + 24.8605i 0.790519 + 0.790519i
\(990\) 0 0
\(991\) 2.22646 0.0707257 0.0353629 0.999375i \(-0.488741\pi\)
0.0353629 + 0.999375i \(0.488741\pi\)
\(992\) 0 0
\(993\) −13.4006 −0.425254
\(994\) 0 0
\(995\) 7.58630 + 7.58630i 0.240502 + 0.240502i
\(996\) 0 0
\(997\) 17.9730 17.9730i 0.569211 0.569211i −0.362696 0.931907i \(-0.618144\pi\)
0.931907 + 0.362696i \(0.118144\pi\)
\(998\) 0 0
\(999\) 31.0796i 0.983316i
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 1280.2.l.a.321.2 8
4.3 odd 2 1280.2.l.f.321.3 yes 8
8.3 odd 2 1280.2.l.d.321.2 yes 8
8.5 even 2 1280.2.l.g.321.3 yes 8
16.3 odd 4 1280.2.l.d.961.2 yes 8
16.5 even 4 inner 1280.2.l.a.961.2 yes 8
16.11 odd 4 1280.2.l.f.961.3 yes 8
16.13 even 4 1280.2.l.g.961.3 yes 8
32.5 even 8 5120.2.a.a.1.4 4
32.11 odd 8 5120.2.a.b.1.4 4
32.21 even 8 5120.2.a.r.1.1 4
32.27 odd 8 5120.2.a.q.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1280.2.l.a.321.2 8 1.1 even 1 trivial
1280.2.l.a.961.2 yes 8 16.5 even 4 inner
1280.2.l.d.321.2 yes 8 8.3 odd 2
1280.2.l.d.961.2 yes 8 16.3 odd 4
1280.2.l.f.321.3 yes 8 4.3 odd 2
1280.2.l.f.961.3 yes 8 16.11 odd 4
1280.2.l.g.321.3 yes 8 8.5 even 2
1280.2.l.g.961.3 yes 8 16.13 even 4
5120.2.a.a.1.4 4 32.5 even 8
5120.2.a.b.1.4 4 32.11 odd 8
5120.2.a.q.1.1 4 32.27 odd 8
5120.2.a.r.1.1 4 32.21 even 8