Properties

Label 912.2.bn.m.65.2
Level $912$
Weight $2$
Character 912.65
Analytic conductor $7.282$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(65,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.2
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 912.65
Dual form 912.2.bn.m.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.158919 - 1.72474i) q^{3} +(-1.22474 - 0.707107i) q^{5} +3.73205 q^{7} +(-2.94949 + 0.548188i) q^{9} +O(q^{10})\) \(q+(-0.158919 - 1.72474i) q^{3} +(-1.22474 - 0.707107i) q^{5} +3.73205 q^{7} +(-2.94949 + 0.548188i) q^{9} -0.378937i q^{11} +(3.23205 - 1.86603i) q^{13} +(-1.02494 + 2.22474i) q^{15} +(4.24264 + 2.44949i) q^{17} +(1.73205 - 4.00000i) q^{19} +(-0.593092 - 6.43684i) q^{21} +(-0.328169 + 0.189469i) q^{23} +(-1.50000 - 2.59808i) q^{25} +(1.41421 + 5.00000i) q^{27} +(-3.86370 - 6.69213i) q^{29} -4.46410i q^{31} +(-0.653570 + 0.0602202i) q^{33} +(-4.57081 - 2.63896i) q^{35} +4.26795i q^{37} +(-3.73205 - 5.27792i) q^{39} +(-2.82843 + 4.89898i) q^{41} +(1.13397 - 1.96410i) q^{43} +(4.00000 + 1.41421i) q^{45} +(-9.14162 + 5.27792i) q^{47} +6.92820 q^{49} +(3.55051 - 7.70674i) q^{51} +(-3.01790 - 5.22715i) q^{53} +(-0.267949 + 0.464102i) q^{55} +(-7.17423 - 2.35167i) q^{57} +(4.19187 - 7.26054i) q^{59} +(1.76795 + 3.06218i) q^{61} +(-11.0076 + 2.04587i) q^{63} -5.27792 q^{65} +(-0.866025 + 0.500000i) q^{67} +(0.378937 + 0.535898i) q^{69} +(1.79315 - 3.10583i) q^{71} +(1.50000 - 2.59808i) q^{73} +(-4.24264 + 3.00000i) q^{75} -1.41421i q^{77} +(3.06218 + 1.76795i) q^{79} +(8.39898 - 3.23375i) q^{81} -7.72741i q^{83} +(-3.46410 - 6.00000i) q^{85} +(-10.9282 + 7.72741i) q^{87} +(-3.67423 - 6.36396i) q^{89} +(12.0622 - 6.96410i) q^{91} +(-7.69944 + 0.709429i) q^{93} +(-4.94975 + 3.67423i) q^{95} +(6.46410 + 3.73205i) q^{97} +(0.207729 + 1.11767i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.158919 1.72474i −0.0917517 0.995782i
\(4\) 0 0
\(5\) −1.22474 0.707107i −0.547723 0.316228i 0.200480 0.979698i \(-0.435750\pi\)
−0.748203 + 0.663470i \(0.769083\pi\)
\(6\) 0 0
\(7\) 3.73205 1.41058 0.705291 0.708918i \(-0.250816\pi\)
0.705291 + 0.708918i \(0.250816\pi\)
\(8\) 0 0
\(9\) −2.94949 + 0.548188i −0.983163 + 0.182729i
\(10\) 0 0
\(11\) 0.378937i 0.114254i −0.998367 0.0571270i \(-0.981806\pi\)
0.998367 0.0571270i \(-0.0181940\pi\)
\(12\) 0 0
\(13\) 3.23205 1.86603i 0.896410 0.517542i 0.0203760 0.999792i \(-0.493514\pi\)
0.876034 + 0.482250i \(0.160180\pi\)
\(14\) 0 0
\(15\) −1.02494 + 2.22474i −0.264639 + 0.574427i
\(16\) 0 0
\(17\) 4.24264 + 2.44949i 1.02899 + 0.594089i 0.916696 0.399586i \(-0.130846\pi\)
0.112296 + 0.993675i \(0.464180\pi\)
\(18\) 0 0
\(19\) 1.73205 4.00000i 0.397360 0.917663i
\(20\) 0 0
\(21\) −0.593092 6.43684i −0.129423 1.40463i
\(22\) 0 0
\(23\) −0.328169 + 0.189469i −0.0684280 + 0.0395070i −0.533824 0.845596i \(-0.679245\pi\)
0.465396 + 0.885103i \(0.345912\pi\)
\(24\) 0 0
\(25\) −1.50000 2.59808i −0.300000 0.519615i
\(26\) 0 0
\(27\) 1.41421 + 5.00000i 0.272166 + 0.962250i
\(28\) 0 0
\(29\) −3.86370 6.69213i −0.717472 1.24270i −0.961998 0.273055i \(-0.911966\pi\)
0.244527 0.969643i \(-0.421367\pi\)
\(30\) 0 0
\(31\) 4.46410i 0.801776i −0.916127 0.400888i \(-0.868702\pi\)
0.916127 0.400888i \(-0.131298\pi\)
\(32\) 0 0
\(33\) −0.653570 + 0.0602202i −0.113772 + 0.0104830i
\(34\) 0 0
\(35\) −4.57081 2.63896i −0.772608 0.446065i
\(36\) 0 0
\(37\) 4.26795i 0.701647i 0.936442 + 0.350823i \(0.114098\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 0 0
\(39\) −3.73205 5.27792i −0.597606 0.845143i
\(40\) 0 0
\(41\) −2.82843 + 4.89898i −0.441726 + 0.765092i −0.997818 0.0660290i \(-0.978967\pi\)
0.556092 + 0.831121i \(0.312300\pi\)
\(42\) 0 0
\(43\) 1.13397 1.96410i 0.172930 0.299523i −0.766513 0.642228i \(-0.778010\pi\)
0.939443 + 0.342706i \(0.111343\pi\)
\(44\) 0 0
\(45\) 4.00000 + 1.41421i 0.596285 + 0.210819i
\(46\) 0 0
\(47\) −9.14162 + 5.27792i −1.33344 + 0.769863i −0.985826 0.167773i \(-0.946342\pi\)
−0.347617 + 0.937637i \(0.613009\pi\)
\(48\) 0 0
\(49\) 6.92820 0.989743
\(50\) 0 0
\(51\) 3.55051 7.70674i 0.497171 1.07916i
\(52\) 0 0
\(53\) −3.01790 5.22715i −0.414540 0.718004i 0.580840 0.814018i \(-0.302724\pi\)
−0.995380 + 0.0960135i \(0.969391\pi\)
\(54\) 0 0
\(55\) −0.267949 + 0.464102i −0.0361303 + 0.0625794i
\(56\) 0 0
\(57\) −7.17423 2.35167i −0.950251 0.311486i
\(58\) 0 0
\(59\) 4.19187 7.26054i 0.545735 0.945241i −0.452825 0.891599i \(-0.649584\pi\)
0.998560 0.0536419i \(-0.0170829\pi\)
\(60\) 0 0
\(61\) 1.76795 + 3.06218i 0.226363 + 0.392072i 0.956727 0.290986i \(-0.0939832\pi\)
−0.730365 + 0.683057i \(0.760650\pi\)
\(62\) 0 0
\(63\) −11.0076 + 2.04587i −1.38683 + 0.257755i
\(64\) 0 0
\(65\) −5.27792 −0.654645
\(66\) 0 0
\(67\) −0.866025 + 0.500000i −0.105802 + 0.0610847i −0.551967 0.833866i \(-0.686123\pi\)
0.446165 + 0.894951i \(0.352789\pi\)
\(68\) 0 0
\(69\) 0.378937 + 0.535898i 0.0456187 + 0.0645146i
\(70\) 0 0
\(71\) 1.79315 3.10583i 0.212808 0.368594i −0.739784 0.672844i \(-0.765072\pi\)
0.952592 + 0.304250i \(0.0984058\pi\)
\(72\) 0 0
\(73\) 1.50000 2.59808i 0.175562 0.304082i −0.764794 0.644275i \(-0.777159\pi\)
0.940356 + 0.340193i \(0.110493\pi\)
\(74\) 0 0
\(75\) −4.24264 + 3.00000i −0.489898 + 0.346410i
\(76\) 0 0
\(77\) 1.41421i 0.161165i
\(78\) 0 0
\(79\) 3.06218 + 1.76795i 0.344522 + 0.198910i 0.662270 0.749265i \(-0.269593\pi\)
−0.317748 + 0.948175i \(0.602927\pi\)
\(80\) 0 0
\(81\) 8.39898 3.23375i 0.933220 0.359306i
\(82\) 0 0
\(83\) 7.72741i 0.848193i −0.905617 0.424097i \(-0.860592\pi\)
0.905617 0.424097i \(-0.139408\pi\)
\(84\) 0 0
\(85\) −3.46410 6.00000i −0.375735 0.650791i
\(86\) 0 0
\(87\) −10.9282 + 7.72741i −1.17163 + 0.828465i
\(88\) 0 0
\(89\) −3.67423 6.36396i −0.389468 0.674579i 0.602910 0.797809i \(-0.294008\pi\)
−0.992378 + 0.123231i \(0.960674\pi\)
\(90\) 0 0
\(91\) 12.0622 6.96410i 1.26446 0.730036i
\(92\) 0 0
\(93\) −7.69944 + 0.709429i −0.798394 + 0.0735643i
\(94\) 0 0
\(95\) −4.94975 + 3.67423i −0.507833 + 0.376969i
\(96\) 0 0
\(97\) 6.46410 + 3.73205i 0.656330 + 0.378932i 0.790877 0.611975i \(-0.209625\pi\)
−0.134547 + 0.990907i \(0.542958\pi\)
\(98\) 0 0
\(99\) 0.207729 + 1.11767i 0.0208775 + 0.112330i
\(100\) 0 0
\(101\) −6.69213 + 3.86370i −0.665892 + 0.384453i −0.794518 0.607240i \(-0.792277\pi\)
0.128626 + 0.991693i \(0.458943\pi\)
\(102\) 0 0
\(103\) 10.4641i 1.03106i 0.856872 + 0.515529i \(0.172405\pi\)
−0.856872 + 0.515529i \(0.827595\pi\)
\(104\) 0 0
\(105\) −3.82514 + 8.30286i −0.373296 + 0.810276i
\(106\) 0 0
\(107\) 4.89898 0.473602 0.236801 0.971558i \(-0.423901\pi\)
0.236801 + 0.971558i \(0.423901\pi\)
\(108\) 0 0
\(109\) −12.4641 7.19615i −1.19384 0.689266i −0.234668 0.972076i \(-0.575400\pi\)
−0.959176 + 0.282809i \(0.908734\pi\)
\(110\) 0 0
\(111\) 7.36112 0.678257i 0.698687 0.0643773i
\(112\) 0 0
\(113\) 18.6622 1.75559 0.877795 0.479036i \(-0.159014\pi\)
0.877795 + 0.479036i \(0.159014\pi\)
\(114\) 0 0
\(115\) 0.535898 0.0499728
\(116\) 0 0
\(117\) −8.50997 + 7.27559i −0.786747 + 0.672629i
\(118\) 0 0
\(119\) 15.8338 + 9.14162i 1.45148 + 0.838011i
\(120\) 0 0
\(121\) 10.8564 0.986946
\(122\) 0 0
\(123\) 8.89898 + 4.09978i 0.802394 + 0.369664i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) −6.80385 + 3.92820i −0.603744 + 0.348572i −0.770513 0.637424i \(-0.780000\pi\)
0.166769 + 0.985996i \(0.446667\pi\)
\(128\) 0 0
\(129\) −3.56778 1.64368i −0.314126 0.144718i
\(130\) 0 0
\(131\) 6.03579 + 3.48477i 0.527350 + 0.304465i 0.739936 0.672677i \(-0.234856\pi\)
−0.212587 + 0.977142i \(0.568189\pi\)
\(132\) 0 0
\(133\) 6.46410 14.9282i 0.560509 1.29444i
\(134\) 0 0
\(135\) 1.80348 7.12372i 0.155219 0.613113i
\(136\) 0 0
\(137\) −0.656339 + 0.378937i −0.0560748 + 0.0323748i −0.527775 0.849384i \(-0.676974\pi\)
0.471700 + 0.881759i \(0.343640\pi\)
\(138\) 0 0
\(139\) 6.59808 + 11.4282i 0.559642 + 0.969328i 0.997526 + 0.0702964i \(0.0223945\pi\)
−0.437885 + 0.899031i \(0.644272\pi\)
\(140\) 0 0
\(141\) 10.5558 + 14.9282i 0.888962 + 1.25718i
\(142\) 0 0
\(143\) −0.707107 1.22474i −0.0591312 0.102418i
\(144\) 0 0
\(145\) 10.9282i 0.907538i
\(146\) 0 0
\(147\) −1.10102 11.9494i −0.0908106 0.985568i
\(148\) 0 0
\(149\) 14.6090 + 8.43451i 1.19682 + 0.690982i 0.959844 0.280534i \(-0.0905116\pi\)
0.236972 + 0.971516i \(0.423845\pi\)
\(150\) 0 0
\(151\) 2.00000i 0.162758i −0.996683 0.0813788i \(-0.974068\pi\)
0.996683 0.0813788i \(-0.0259324\pi\)
\(152\) 0 0
\(153\) −13.8564 4.89898i −1.12022 0.396059i
\(154\) 0 0
\(155\) −3.15660 + 5.46739i −0.253544 + 0.439151i
\(156\) 0 0
\(157\) −1.76795 + 3.06218i −0.141098 + 0.244388i −0.927910 0.372804i \(-0.878397\pi\)
0.786813 + 0.617192i \(0.211730\pi\)
\(158\) 0 0
\(159\) −8.53590 + 6.03579i −0.676941 + 0.478669i
\(160\) 0 0
\(161\) −1.22474 + 0.707107i −0.0965234 + 0.0557278i
\(162\) 0 0
\(163\) −5.19615 −0.406994 −0.203497 0.979076i \(-0.565231\pi\)
−0.203497 + 0.979076i \(0.565231\pi\)
\(164\) 0 0
\(165\) 0.843039 + 0.388390i 0.0656305 + 0.0302361i
\(166\) 0 0
\(167\) 3.81294 + 6.60420i 0.295054 + 0.511048i 0.974997 0.222216i \(-0.0713291\pi\)
−0.679944 + 0.733264i \(0.737996\pi\)
\(168\) 0 0
\(169\) 0.464102 0.803848i 0.0357001 0.0618344i
\(170\) 0 0
\(171\) −2.91591 + 12.7474i −0.222985 + 0.974822i
\(172\) 0 0
\(173\) 5.93426 10.2784i 0.451173 0.781455i −0.547286 0.836946i \(-0.684339\pi\)
0.998459 + 0.0554909i \(0.0176724\pi\)
\(174\) 0 0
\(175\) −5.59808 9.69615i −0.423175 0.732960i
\(176\) 0 0
\(177\) −13.1887 6.07608i −0.991326 0.456706i
\(178\) 0 0
\(179\) −1.41421 −0.105703 −0.0528516 0.998602i \(-0.516831\pi\)
−0.0528516 + 0.998602i \(0.516831\pi\)
\(180\) 0 0
\(181\) −3.00000 + 1.73205i −0.222988 + 0.128742i −0.607333 0.794447i \(-0.707761\pi\)
0.384345 + 0.923190i \(0.374427\pi\)
\(182\) 0 0
\(183\) 5.00052 3.53590i 0.369649 0.261381i
\(184\) 0 0
\(185\) 3.01790 5.22715i 0.221880 0.384308i
\(186\) 0 0
\(187\) 0.928203 1.60770i 0.0678769 0.117566i
\(188\) 0 0
\(189\) 5.27792 + 18.6603i 0.383912 + 1.35733i
\(190\) 0 0
\(191\) 1.69161i 0.122401i 0.998125 + 0.0612005i \(0.0194929\pi\)
−0.998125 + 0.0612005i \(0.980507\pi\)
\(192\) 0 0
\(193\) 11.4282 + 6.59808i 0.822620 + 0.474940i 0.851319 0.524648i \(-0.175803\pi\)
−0.0286991 + 0.999588i \(0.509136\pi\)
\(194\) 0 0
\(195\) 0.838759 + 9.10306i 0.0600648 + 0.651884i
\(196\) 0 0
\(197\) 9.14162i 0.651313i 0.945488 + 0.325657i \(0.105585\pi\)
−0.945488 + 0.325657i \(0.894415\pi\)
\(198\) 0 0
\(199\) −6.40192 11.0885i −0.453820 0.786040i 0.544799 0.838567i \(-0.316606\pi\)
−0.998620 + 0.0525267i \(0.983273\pi\)
\(200\) 0 0
\(201\) 1.00000 + 1.41421i 0.0705346 + 0.0997509i
\(202\) 0 0
\(203\) −14.4195 24.9754i −1.01205 1.75293i
\(204\) 0 0
\(205\) 6.92820 4.00000i 0.483887 0.279372i
\(206\) 0 0
\(207\) 0.864068 0.738735i 0.0600569 0.0513456i
\(208\) 0 0
\(209\) −1.51575 0.656339i −0.104847 0.0453999i
\(210\) 0 0
\(211\) 9.52628 + 5.50000i 0.655816 + 0.378636i 0.790681 0.612228i \(-0.209727\pi\)
−0.134865 + 0.990864i \(0.543060\pi\)
\(212\) 0 0
\(213\) −5.64173 2.59915i −0.386565 0.178091i
\(214\) 0 0
\(215\) −2.77766 + 1.60368i −0.189435 + 0.109370i
\(216\) 0 0
\(217\) 16.6603i 1.13097i
\(218\) 0 0
\(219\) −4.71940 2.17423i −0.318907 0.146921i
\(220\) 0 0
\(221\) 18.2832 1.22986
\(222\) 0 0
\(223\) 8.13397 + 4.69615i 0.544691 + 0.314478i 0.746978 0.664849i \(-0.231504\pi\)
−0.202287 + 0.979326i \(0.564837\pi\)
\(224\) 0 0
\(225\) 5.84847 + 6.84072i 0.389898 + 0.456048i
\(226\) 0 0
\(227\) 24.5964 1.63252 0.816261 0.577683i \(-0.196043\pi\)
0.816261 + 0.577683i \(0.196043\pi\)
\(228\) 0 0
\(229\) 9.39230 0.620661 0.310330 0.950629i \(-0.399560\pi\)
0.310330 + 0.950629i \(0.399560\pi\)
\(230\) 0 0
\(231\) −2.43916 + 0.224745i −0.160485 + 0.0147871i
\(232\) 0 0
\(233\) 3.10583 + 1.79315i 0.203470 + 0.117473i 0.598273 0.801292i \(-0.295854\pi\)
−0.394803 + 0.918766i \(0.629187\pi\)
\(234\) 0 0
\(235\) 14.9282 0.973809
\(236\) 0 0
\(237\) 2.56262 5.56244i 0.166460 0.361319i
\(238\) 0 0
\(239\) 29.9759i 1.93898i 0.245133 + 0.969489i \(0.421168\pi\)
−0.245133 + 0.969489i \(0.578832\pi\)
\(240\) 0 0
\(241\) −24.8205 + 14.3301i −1.59883 + 0.923085i −0.607116 + 0.794613i \(0.707674\pi\)
−0.991713 + 0.128472i \(0.958993\pi\)
\(242\) 0 0
\(243\) −6.91215 13.9722i −0.443415 0.896317i
\(244\) 0 0
\(245\) −8.48528 4.89898i −0.542105 0.312984i
\(246\) 0 0
\(247\) −1.86603 16.1603i −0.118732 1.02825i
\(248\) 0 0
\(249\) −13.3278 + 1.22803i −0.844615 + 0.0778232i
\(250\) 0 0
\(251\) −12.7279 + 7.34847i −0.803379 + 0.463831i −0.844651 0.535317i \(-0.820192\pi\)
0.0412721 + 0.999148i \(0.486859\pi\)
\(252\) 0 0
\(253\) 0.0717968 + 0.124356i 0.00451382 + 0.00781817i
\(254\) 0 0
\(255\) −9.79796 + 6.92820i −0.613572 + 0.433861i
\(256\) 0 0
\(257\) −1.60368 2.77766i −0.100035 0.173266i 0.811664 0.584125i \(-0.198562\pi\)
−0.911699 + 0.410859i \(0.865229\pi\)
\(258\) 0 0
\(259\) 15.9282i 0.989730i
\(260\) 0 0
\(261\) 15.0645 + 17.6203i 0.932469 + 1.09067i
\(262\) 0 0
\(263\) −0.480473 0.277401i −0.0296273 0.0171053i 0.485113 0.874451i \(-0.338778\pi\)
−0.514740 + 0.857346i \(0.672112\pi\)
\(264\) 0 0
\(265\) 8.53590i 0.524356i
\(266\) 0 0
\(267\) −10.3923 + 7.34847i −0.635999 + 0.449719i
\(268\) 0 0
\(269\) −3.01790 + 5.22715i −0.184004 + 0.318705i −0.943241 0.332110i \(-0.892239\pi\)
0.759236 + 0.650815i \(0.225573\pi\)
\(270\) 0 0
\(271\) −3.46410 + 6.00000i −0.210429 + 0.364474i −0.951849 0.306568i \(-0.900819\pi\)
0.741420 + 0.671042i \(0.234153\pi\)
\(272\) 0 0
\(273\) −13.9282 19.6975i −0.842973 1.19214i
\(274\) 0 0
\(275\) −0.984508 + 0.568406i −0.0593681 + 0.0342762i
\(276\) 0 0
\(277\) −9.85641 −0.592214 −0.296107 0.955155i \(-0.595689\pi\)
−0.296107 + 0.955155i \(0.595689\pi\)
\(278\) 0 0
\(279\) 2.44717 + 13.1668i 0.146508 + 0.788277i
\(280\) 0 0
\(281\) 7.53794 + 13.0561i 0.449676 + 0.778861i 0.998365 0.0571654i \(-0.0182062\pi\)
−0.548689 + 0.836027i \(0.684873\pi\)
\(282\) 0 0
\(283\) −10.9282 + 18.9282i −0.649614 + 1.12516i 0.333601 + 0.942714i \(0.391736\pi\)
−0.983215 + 0.182450i \(0.941597\pi\)
\(284\) 0 0
\(285\) 7.12372 + 7.95315i 0.421973 + 0.471104i
\(286\) 0 0
\(287\) −10.5558 + 18.2832i −0.623091 + 1.07923i
\(288\) 0 0
\(289\) 3.50000 + 6.06218i 0.205882 + 0.356599i
\(290\) 0 0
\(291\) 5.40957 11.7420i 0.317115 0.688329i
\(292\) 0 0
\(293\) −25.2528 −1.47528 −0.737641 0.675193i \(-0.764060\pi\)
−0.737641 + 0.675193i \(0.764060\pi\)
\(294\) 0 0
\(295\) −10.2679 + 5.92820i −0.597823 + 0.345153i
\(296\) 0 0
\(297\) 1.89469 0.535898i 0.109941 0.0310960i
\(298\) 0 0
\(299\) −0.707107 + 1.22474i −0.0408930 + 0.0708288i
\(300\) 0 0
\(301\) 4.23205 7.33013i 0.243931 0.422501i
\(302\) 0 0
\(303\) 7.72741 + 10.9282i 0.443928 + 0.627809i
\(304\) 0 0
\(305\) 5.00052i 0.286329i
\(306\) 0 0
\(307\) 12.0000 + 6.92820i 0.684876 + 0.395413i 0.801690 0.597740i \(-0.203935\pi\)
−0.116814 + 0.993154i \(0.537268\pi\)
\(308\) 0 0
\(309\) 18.0479 1.66294i 1.02671 0.0946014i
\(310\) 0 0
\(311\) 27.1475i 1.53939i −0.638411 0.769696i \(-0.720408\pi\)
0.638411 0.769696i \(-0.279592\pi\)
\(312\) 0 0
\(313\) 12.3923 + 21.4641i 0.700454 + 1.21322i 0.968307 + 0.249763i \(0.0803527\pi\)
−0.267853 + 0.963460i \(0.586314\pi\)
\(314\) 0 0
\(315\) 14.9282 + 5.27792i 0.841109 + 0.297377i
\(316\) 0 0
\(317\) 6.88160 + 11.9193i 0.386509 + 0.669453i 0.991977 0.126416i \(-0.0403474\pi\)
−0.605468 + 0.795870i \(0.707014\pi\)
\(318\) 0 0
\(319\) −2.53590 + 1.46410i −0.141983 + 0.0819740i
\(320\) 0 0
\(321\) −0.778539 8.44949i −0.0434538 0.471605i
\(322\) 0 0
\(323\) 17.1464 12.7279i 0.954053 0.708201i
\(324\) 0 0
\(325\) −9.69615 5.59808i −0.537846 0.310525i
\(326\) 0 0
\(327\) −10.4307 + 22.6410i −0.576822 + 1.25205i
\(328\) 0 0
\(329\) −34.1170 + 19.6975i −1.88093 + 1.08596i
\(330\) 0 0
\(331\) 31.9282i 1.75493i −0.479638 0.877466i \(-0.659232\pi\)
0.479638 0.877466i \(-0.340768\pi\)
\(332\) 0 0
\(333\) −2.33964 12.5883i −0.128211 0.689833i
\(334\) 0 0
\(335\) 1.41421 0.0772667
\(336\) 0 0
\(337\) 6.35641 + 3.66987i 0.346256 + 0.199911i 0.663035 0.748589i \(-0.269268\pi\)
−0.316779 + 0.948499i \(0.602601\pi\)
\(338\) 0 0
\(339\) −2.96577 32.1875i −0.161078 1.74818i
\(340\) 0 0
\(341\) −1.69161 −0.0916061
\(342\) 0 0
\(343\) −0.267949 −0.0144679
\(344\) 0 0
\(345\) −0.0851642 0.924288i −0.00458509 0.0497620i
\(346\) 0 0
\(347\) 21.5414 + 12.4369i 1.15640 + 0.667649i 0.950439 0.310911i \(-0.100634\pi\)
0.205963 + 0.978560i \(0.433968\pi\)
\(348\) 0 0
\(349\) −11.3923 −0.609816 −0.304908 0.952382i \(-0.598626\pi\)
−0.304908 + 0.952382i \(0.598626\pi\)
\(350\) 0 0
\(351\) 13.9009 + 13.5213i 0.741977 + 0.721713i
\(352\) 0 0
\(353\) 11.9700i 0.637101i −0.947906 0.318551i \(-0.896804\pi\)
0.947906 0.318551i \(-0.103196\pi\)
\(354\) 0 0
\(355\) −4.39230 + 2.53590i −0.233119 + 0.134592i
\(356\) 0 0
\(357\) 13.2507 28.7620i 0.701301 1.52224i
\(358\) 0 0
\(359\) −23.1822 13.3843i −1.22351 0.706394i −0.257846 0.966186i \(-0.583013\pi\)
−0.965665 + 0.259792i \(0.916346\pi\)
\(360\) 0 0
\(361\) −13.0000 13.8564i −0.684211 0.729285i
\(362\) 0 0
\(363\) −1.72529 18.7245i −0.0905540 0.982783i
\(364\) 0 0
\(365\) −3.67423 + 2.12132i −0.192318 + 0.111035i
\(366\) 0 0
\(367\) −8.52628 14.7679i −0.445068 0.770881i 0.552989 0.833189i \(-0.313487\pi\)
−0.998057 + 0.0623080i \(0.980154\pi\)
\(368\) 0 0
\(369\) 5.65685 16.0000i 0.294484 0.832927i
\(370\) 0 0
\(371\) −11.2629 19.5080i −0.584743 1.01280i
\(372\) 0 0
\(373\) 12.5359i 0.649084i 0.945871 + 0.324542i \(0.105210\pi\)
−0.945871 + 0.324542i \(0.894790\pi\)
\(374\) 0 0
\(375\) 19.5133 1.79796i 1.00766 0.0928462i
\(376\) 0 0
\(377\) −24.9754 14.4195i −1.28630 0.742644i
\(378\) 0 0
\(379\) 17.7846i 0.913534i −0.889586 0.456767i \(-0.849007\pi\)
0.889586 0.456767i \(-0.150993\pi\)
\(380\) 0 0
\(381\) 7.85641 + 11.1106i 0.402496 + 0.569215i
\(382\) 0 0
\(383\) 13.3335 23.0943i 0.681310 1.18006i −0.293272 0.956029i \(-0.594744\pi\)
0.974581 0.224034i \(-0.0719226\pi\)
\(384\) 0 0
\(385\) −1.00000 + 1.73205i −0.0509647 + 0.0882735i
\(386\) 0 0
\(387\) −2.26795 + 6.41473i −0.115286 + 0.326079i
\(388\) 0 0
\(389\) −4.33057 + 2.50026i −0.219569 + 0.126768i −0.605751 0.795655i \(-0.707127\pi\)
0.386182 + 0.922423i \(0.373794\pi\)
\(390\) 0 0
\(391\) −1.85641 −0.0938825
\(392\) 0 0
\(393\) 5.05113 10.9640i 0.254796 0.553060i
\(394\) 0 0
\(395\) −2.50026 4.33057i −0.125802 0.217895i
\(396\) 0 0
\(397\) −4.16025 + 7.20577i −0.208797 + 0.361647i −0.951336 0.308156i \(-0.900288\pi\)
0.742539 + 0.669803i \(0.233622\pi\)
\(398\) 0 0
\(399\) −26.7746 8.77656i −1.34041 0.439377i
\(400\) 0 0
\(401\) 5.74479 9.95026i 0.286881 0.496892i −0.686183 0.727429i \(-0.740715\pi\)
0.973064 + 0.230537i \(0.0740482\pi\)
\(402\) 0 0
\(403\) −8.33013 14.4282i −0.414953 0.718720i
\(404\) 0 0
\(405\) −12.5732 1.97846i −0.624768 0.0983103i
\(406\) 0 0
\(407\) 1.61729 0.0801659
\(408\) 0 0
\(409\) 24.4641 14.1244i 1.20967 0.698404i 0.246984 0.969019i \(-0.420560\pi\)
0.962688 + 0.270615i \(0.0872270\pi\)
\(410\) 0 0
\(411\) 0.757875 + 1.07180i 0.0373832 + 0.0528678i
\(412\) 0 0
\(413\) 15.6443 27.0967i 0.769805 1.33334i
\(414\) 0 0
\(415\) −5.46410 + 9.46410i −0.268222 + 0.464574i
\(416\) 0 0
\(417\) 18.6622 13.1962i 0.913891 0.646218i
\(418\) 0 0
\(419\) 32.0464i 1.56557i 0.622292 + 0.782785i \(0.286202\pi\)
−0.622292 + 0.782785i \(0.713798\pi\)
\(420\) 0 0
\(421\) −30.7128 17.7321i −1.49685 0.864207i −0.496857 0.867832i \(-0.665513\pi\)
−0.999993 + 0.00362487i \(0.998846\pi\)
\(422\) 0 0
\(423\) 24.0698 20.5785i 1.17031 1.00056i
\(424\) 0 0
\(425\) 14.6969i 0.712906i
\(426\) 0 0
\(427\) 6.59808 + 11.4282i 0.319303 + 0.553050i
\(428\) 0 0
\(429\) −2.00000 + 1.41421i −0.0965609 + 0.0682789i
\(430\) 0 0
\(431\) 12.7279 + 22.0454i 0.613082 + 1.06189i 0.990718 + 0.135935i \(0.0434040\pi\)
−0.377635 + 0.925954i \(0.623263\pi\)
\(432\) 0 0
\(433\) 2.89230 1.66987i 0.138995 0.0802490i −0.428890 0.903357i \(-0.641095\pi\)
0.567885 + 0.823108i \(0.307762\pi\)
\(434\) 0 0
\(435\) 18.8484 1.73670i 0.903710 0.0832682i
\(436\) 0 0
\(437\) 0.189469 + 1.64085i 0.00906352 + 0.0784924i
\(438\) 0 0
\(439\) −19.4545 11.2321i −0.928512 0.536077i −0.0421712 0.999110i \(-0.513427\pi\)
−0.886341 + 0.463034i \(0.846761\pi\)
\(440\) 0 0
\(441\) −20.4347 + 3.79796i −0.973079 + 0.180855i
\(442\) 0 0
\(443\) −33.4607 + 19.3185i −1.58976 + 0.917850i −0.596419 + 0.802674i \(0.703410\pi\)
−0.993345 + 0.115177i \(0.963257\pi\)
\(444\) 0 0
\(445\) 10.3923i 0.492642i
\(446\) 0 0
\(447\) 12.2257 26.5372i 0.578258 1.25517i
\(448\) 0 0
\(449\) 34.4959 1.62796 0.813982 0.580890i \(-0.197296\pi\)
0.813982 + 0.580890i \(0.197296\pi\)
\(450\) 0 0
\(451\) 1.85641 + 1.07180i 0.0874148 + 0.0504689i
\(452\) 0 0
\(453\) −3.44949 + 0.317837i −0.162071 + 0.0149333i
\(454\) 0 0
\(455\) −19.6975 −0.923431
\(456\) 0 0
\(457\) −20.7128 −0.968905 −0.484452 0.874818i \(-0.660981\pi\)
−0.484452 + 0.874818i \(0.660981\pi\)
\(458\) 0 0
\(459\) −6.24745 + 24.6773i −0.291606 + 1.15184i
\(460\) 0 0
\(461\) 28.4737 + 16.4393i 1.32615 + 0.765656i 0.984703 0.174244i \(-0.0557481\pi\)
0.341452 + 0.939899i \(0.389081\pi\)
\(462\) 0 0
\(463\) 29.5885 1.37509 0.687546 0.726141i \(-0.258688\pi\)
0.687546 + 0.726141i \(0.258688\pi\)
\(464\) 0 0
\(465\) 9.93149 + 4.57545i 0.460562 + 0.212182i
\(466\) 0 0
\(467\) 22.6274i 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) 0 0
\(469\) −3.23205 + 1.86603i −0.149242 + 0.0861650i
\(470\) 0 0
\(471\) 5.56244 + 2.56262i 0.256304 + 0.118079i
\(472\) 0 0
\(473\) −0.744272 0.429705i −0.0342216 0.0197579i
\(474\) 0 0
\(475\) −12.9904 + 1.50000i −0.596040 + 0.0688247i
\(476\) 0 0
\(477\) 11.7667 + 13.7630i 0.538761 + 0.630167i
\(478\) 0 0
\(479\) 10.2784 5.93426i 0.469634 0.271143i −0.246453 0.969155i \(-0.579265\pi\)
0.716086 + 0.698012i \(0.245932\pi\)
\(480\) 0 0
\(481\) 7.96410 + 13.7942i 0.363132 + 0.628963i
\(482\) 0 0
\(483\) 1.41421 + 2.00000i 0.0643489 + 0.0910032i
\(484\) 0 0
\(485\) −5.27792 9.14162i −0.239658 0.415100i
\(486\) 0 0
\(487\) 24.9282i 1.12960i 0.825226 + 0.564802i \(0.191048\pi\)
−0.825226 + 0.564802i \(0.808952\pi\)
\(488\) 0 0
\(489\) 0.825765 + 8.96204i 0.0373424 + 0.405277i
\(490\) 0 0
\(491\) −8.66115 5.00052i −0.390872 0.225670i 0.291666 0.956520i \(-0.405790\pi\)
−0.682538 + 0.730850i \(0.739124\pi\)
\(492\) 0 0
\(493\) 37.8564i 1.70497i
\(494\) 0 0
\(495\) 0.535898 1.51575i 0.0240868 0.0681279i
\(496\) 0 0
\(497\) 6.69213 11.5911i 0.300183 0.519932i
\(498\) 0 0
\(499\) −8.06218 + 13.9641i −0.360913 + 0.625119i −0.988111 0.153740i \(-0.950868\pi\)
0.627199 + 0.778859i \(0.284201\pi\)
\(500\) 0 0
\(501\) 10.7846 7.62587i 0.481821 0.340699i
\(502\) 0 0
\(503\) 15.8338 9.14162i 0.705992 0.407605i −0.103583 0.994621i \(-0.533031\pi\)
0.809575 + 0.587016i \(0.199697\pi\)
\(504\) 0 0
\(505\) 10.9282 0.486299
\(506\) 0 0
\(507\) −1.46019 0.672711i −0.0648492 0.0298761i
\(508\) 0 0
\(509\) −6.79367 11.7670i −0.301124 0.521562i 0.675267 0.737574i \(-0.264028\pi\)
−0.976391 + 0.216011i \(0.930695\pi\)
\(510\) 0 0
\(511\) 5.59808 9.69615i 0.247644 0.428933i
\(512\) 0 0
\(513\) 22.4495 + 3.00340i 0.991169 + 0.132603i
\(514\) 0 0
\(515\) 7.39924 12.8159i 0.326049 0.564734i
\(516\) 0 0
\(517\) 2.00000 + 3.46410i 0.0879599 + 0.152351i
\(518\) 0 0
\(519\) −18.6707 8.60164i −0.819554 0.377570i
\(520\) 0 0
\(521\) −4.52004 −0.198027 −0.0990133 0.995086i \(-0.531569\pi\)
−0.0990133 + 0.995086i \(0.531569\pi\)
\(522\) 0 0
\(523\) 14.5981 8.42820i 0.638329 0.368540i −0.145641 0.989337i \(-0.546525\pi\)
0.783971 + 0.620798i \(0.213191\pi\)
\(524\) 0 0
\(525\) −15.8338 + 11.1962i −0.691042 + 0.488640i
\(526\) 0 0
\(527\) 10.9348 18.9396i 0.476326 0.825021i
\(528\) 0 0
\(529\) −11.4282 + 19.7942i −0.496878 + 0.860619i
\(530\) 0 0
\(531\) −8.38375 + 23.7128i −0.363824 + 1.02905i
\(532\) 0 0
\(533\) 21.1117i 0.914448i
\(534\) 0 0
\(535\) −6.00000 3.46410i −0.259403 0.149766i
\(536\) 0 0
\(537\) 0.224745 + 2.43916i 0.00969846 + 0.105257i
\(538\) 0 0
\(539\) 2.62536i 0.113082i
\(540\) 0 0
\(541\) 8.76795 + 15.1865i 0.376964 + 0.652920i 0.990619 0.136654i \(-0.0436349\pi\)
−0.613655 + 0.789574i \(0.710302\pi\)
\(542\) 0 0
\(543\) 3.46410 + 4.89898i 0.148659 + 0.210235i
\(544\) 0 0
\(545\) 10.1769 + 17.6269i 0.435930 + 0.755054i
\(546\) 0 0
\(547\) 11.1340 6.42820i 0.476054 0.274850i −0.242716 0.970097i \(-0.578038\pi\)
0.718771 + 0.695247i \(0.244705\pi\)
\(548\) 0 0
\(549\) −6.89320 8.06269i −0.294195 0.344107i
\(550\) 0 0
\(551\) −33.4607 + 3.86370i −1.42547 + 0.164599i
\(552\) 0 0
\(553\) 11.4282 + 6.59808i 0.485977 + 0.280579i
\(554\) 0 0
\(555\) −9.49510 4.37441i −0.403044 0.185683i
\(556\) 0 0
\(557\) −2.92996 + 1.69161i −0.124147 + 0.0716760i −0.560787 0.827960i \(-0.689501\pi\)
0.436641 + 0.899636i \(0.356168\pi\)
\(558\) 0 0
\(559\) 8.46410i 0.357993i
\(560\) 0 0
\(561\) −2.92037 1.34542i −0.123298 0.0568037i
\(562\) 0 0
\(563\) −3.38323 −0.142586 −0.0712931 0.997455i \(-0.522713\pi\)
−0.0712931 + 0.997455i \(0.522713\pi\)
\(564\) 0 0
\(565\) −22.8564 13.1962i −0.961576 0.555166i
\(566\) 0 0
\(567\) 31.3454 12.0685i 1.31638 0.506830i
\(568\) 0 0
\(569\) −13.9391 −0.584356 −0.292178 0.956364i \(-0.594380\pi\)
−0.292178 + 0.956364i \(0.594380\pi\)
\(570\) 0 0
\(571\) −25.1962 −1.05443 −0.527213 0.849733i \(-0.676763\pi\)
−0.527213 + 0.849733i \(0.676763\pi\)
\(572\) 0 0
\(573\) 2.91760 0.268829i 0.121885 0.0112305i
\(574\) 0 0
\(575\) 0.984508 + 0.568406i 0.0410568 + 0.0237042i
\(576\) 0 0
\(577\) 42.9282 1.78712 0.893562 0.448939i \(-0.148198\pi\)
0.893562 + 0.448939i \(0.148198\pi\)
\(578\) 0 0
\(579\) 9.56384 20.7593i 0.397460 0.862727i
\(580\) 0 0
\(581\) 28.8391i 1.19645i
\(582\) 0 0
\(583\) −1.98076 + 1.14359i −0.0820348 + 0.0473628i
\(584\) 0 0
\(585\) 15.5672 2.89329i 0.643623 0.119623i
\(586\) 0 0
\(587\) −9.29392 5.36585i −0.383601 0.221472i 0.295783 0.955255i \(-0.404420\pi\)
−0.679384 + 0.733783i \(0.737753\pi\)
\(588\) 0 0
\(589\) −17.8564 7.73205i −0.735760 0.318594i
\(590\) 0 0
\(591\) 15.7670 1.45277i 0.648566 0.0597591i
\(592\) 0 0
\(593\) −21.7816 + 12.5756i −0.894463 + 0.516419i −0.875400 0.483400i \(-0.839402\pi\)
−0.0190636 + 0.999818i \(0.506069\pi\)
\(594\) 0 0
\(595\) −12.9282 22.3923i −0.530005 0.917995i
\(596\) 0 0
\(597\) −18.1074 + 12.8038i −0.741086 + 0.524027i
\(598\) 0 0
\(599\) 8.43451 + 14.6090i 0.344625 + 0.596908i 0.985286 0.170916i \(-0.0546728\pi\)
−0.640661 + 0.767824i \(0.721339\pi\)
\(600\) 0 0
\(601\) 44.3731i 1.81002i −0.425395 0.905008i \(-0.639865\pi\)
0.425395 0.905008i \(-0.360135\pi\)
\(602\) 0 0
\(603\) 2.28024 1.94949i 0.0928585 0.0793894i
\(604\) 0 0
\(605\) −13.2963 7.67664i −0.540573 0.312100i
\(606\) 0 0
\(607\) 13.2487i 0.537749i −0.963175 0.268874i \(-0.913348\pi\)
0.963175 0.268874i \(-0.0866516\pi\)
\(608\) 0 0
\(609\) −40.7846 + 28.8391i −1.65268 + 1.16862i
\(610\) 0 0
\(611\) −19.6975 + 34.1170i −0.796874 + 1.38023i
\(612\) 0 0
\(613\) −11.3205 + 19.6077i −0.457231 + 0.791947i −0.998813 0.0487003i \(-0.984492\pi\)
0.541582 + 0.840648i \(0.317825\pi\)
\(614\) 0 0
\(615\) −8.00000 11.3137i −0.322591 0.456213i
\(616\) 0 0
\(617\) 22.4379 12.9546i 0.903318 0.521531i 0.0250427 0.999686i \(-0.492028\pi\)
0.878275 + 0.478156i \(0.158695\pi\)
\(618\) 0 0
\(619\) −2.80385 −0.112696 −0.0563481 0.998411i \(-0.517946\pi\)
−0.0563481 + 0.998411i \(0.517946\pi\)
\(620\) 0 0
\(621\) −1.41145 1.37290i −0.0566393 0.0550925i
\(622\) 0 0
\(623\) −13.7124 23.7506i −0.549377 0.951549i
\(624\) 0 0
\(625\) 0.500000 0.866025i 0.0200000 0.0346410i
\(626\) 0 0
\(627\) −0.891136 + 2.71859i −0.0355885 + 0.108570i
\(628\) 0 0
\(629\) −10.4543 + 18.1074i −0.416840 + 0.721988i
\(630\) 0 0
\(631\) 24.5263 + 42.4808i 0.976376 + 1.69113i 0.675318 + 0.737526i \(0.264006\pi\)
0.301058 + 0.953606i \(0.402660\pi\)
\(632\) 0 0
\(633\) 7.97219 17.3045i 0.316866 0.687790i
\(634\) 0 0
\(635\) 11.1106 0.440912
\(636\) 0 0
\(637\) 22.3923 12.9282i 0.887215 0.512234i
\(638\) 0 0
\(639\) −3.58630 + 10.1436i −0.141872 + 0.401274i
\(640\) 0 0
\(641\) −6.41473 + 11.1106i −0.253367 + 0.438844i −0.964451 0.264263i \(-0.914871\pi\)
0.711084 + 0.703107i \(0.248205\pi\)
\(642\) 0 0
\(643\) −21.7942 + 37.7487i −0.859480 + 1.48866i 0.0129447 + 0.999916i \(0.495879\pi\)
−0.872425 + 0.488748i \(0.837454\pi\)
\(644\) 0 0
\(645\) 3.20736 + 4.53590i 0.126290 + 0.178601i
\(646\) 0 0
\(647\) 20.5569i 0.808174i −0.914721 0.404087i \(-0.867589\pi\)
0.914721 0.404087i \(-0.132411\pi\)
\(648\) 0 0
\(649\) −2.75129 1.58846i −0.107998 0.0623524i
\(650\) 0 0
\(651\) −28.7347 + 2.64762i −1.12620 + 0.103769i
\(652\) 0 0
\(653\) 47.1223i 1.84404i −0.387144 0.922019i \(-0.626538\pi\)
0.387144 0.922019i \(-0.373462\pi\)
\(654\) 0 0
\(655\) −4.92820 8.53590i −0.192561 0.333525i
\(656\) 0 0
\(657\) −3.00000 + 8.48528i −0.117041 + 0.331042i
\(658\) 0 0
\(659\) 16.5409 + 28.6496i 0.644340 + 1.11603i 0.984453 + 0.175646i \(0.0562013\pi\)
−0.340113 + 0.940385i \(0.610465\pi\)
\(660\) 0 0
\(661\) 5.32051 3.07180i 0.206944 0.119479i −0.392946 0.919561i \(-0.628544\pi\)
0.599890 + 0.800082i \(0.295211\pi\)
\(662\) 0 0
\(663\) −2.90555 31.5339i −0.112842 1.22468i
\(664\) 0 0
\(665\) −18.4727 + 13.7124i −0.716341 + 0.531745i
\(666\) 0 0
\(667\) 2.53590 + 1.46410i 0.0981904 + 0.0566902i
\(668\) 0 0
\(669\) 6.80702 14.7753i 0.263175 0.571248i
\(670\) 0 0
\(671\) 1.16037 0.669942i 0.0447957 0.0258628i
\(672\) 0 0
\(673\) 21.9808i 0.847296i 0.905827 + 0.423648i \(0.139251\pi\)
−0.905827 + 0.423648i \(0.860749\pi\)
\(674\) 0 0
\(675\) 10.8691 11.1742i 0.418350 0.430096i
\(676\) 0 0
\(677\) −18.8380 −0.724005 −0.362002 0.932177i \(-0.617907\pi\)
−0.362002 + 0.932177i \(0.617907\pi\)
\(678\) 0 0
\(679\) 24.1244 + 13.9282i 0.925808 + 0.534515i
\(680\) 0 0
\(681\) −3.90883 42.4226i −0.149787 1.62564i
\(682\) 0 0
\(683\) 16.3142 0.624246 0.312123 0.950042i \(-0.398960\pi\)
0.312123 + 0.950042i \(0.398960\pi\)
\(684\) 0 0
\(685\) 1.07180 0.0409512
\(686\) 0 0
\(687\) −1.49261 16.1993i −0.0569467 0.618043i
\(688\) 0 0
\(689\) −19.5080 11.2629i −0.743195 0.429084i
\(690\) 0 0
\(691\) −9.85641 −0.374955 −0.187478 0.982269i \(-0.560031\pi\)
−0.187478 + 0.982269i \(0.560031\pi\)
\(692\) 0 0
\(693\) 0.775255 + 4.17121i 0.0294495 + 0.158451i
\(694\) 0 0
\(695\) 18.6622i 0.707897i
\(696\) 0 0
\(697\) −24.0000 + 13.8564i −0.909065 + 0.524849i
\(698\) 0 0
\(699\) 2.59915 5.64173i 0.0983090 0.213390i
\(700\) 0 0
\(701\) 17.2344 + 9.95026i 0.650933 + 0.375816i 0.788814 0.614633i \(-0.210696\pi\)
−0.137881 + 0.990449i \(0.544029\pi\)
\(702\) 0 0
\(703\) 17.0718 + 7.39230i 0.643875 + 0.278806i
\(704\) 0 0
\(705\) −2.37237 25.7473i −0.0893486 0.969701i
\(706\) 0 0
\(707\) −24.9754 + 14.4195i −0.939295 + 0.542303i
\(708\) 0 0
\(709\) −21.1603 36.6506i −0.794690 1.37644i −0.923036 0.384714i \(-0.874300\pi\)
0.128346 0.991729i \(-0.459033\pi\)
\(710\) 0 0
\(711\) −10.0010 3.53590i −0.375068 0.132607i
\(712\) 0 0
\(713\) 0.845807 + 1.46498i 0.0316757 + 0.0548640i
\(714\) 0 0
\(715\) 2.00000i 0.0747958i
\(716\) 0 0
\(717\) 51.7008 4.76373i 1.93080 0.177905i
\(718\) 0 0
\(719\) 5.22715 + 3.01790i 0.194940 + 0.112549i 0.594293 0.804249i \(-0.297432\pi\)
−0.399353 + 0.916797i \(0.630765\pi\)
\(720\) 0 0
\(721\) 39.0526i 1.45439i
\(722\) 0 0
\(723\) 28.6603 + 40.5317i 1.06589 + 1.50739i
\(724\) 0 0
\(725\) −11.5911 + 20.0764i −0.430483 + 0.745618i
\(726\) 0 0
\(727\) −6.79423 + 11.7679i −0.251984 + 0.436449i −0.964072 0.265641i \(-0.914416\pi\)
0.712088 + 0.702090i \(0.247750\pi\)
\(728\) 0 0
\(729\) −23.0000 + 14.1421i −0.851852 + 0.523783i
\(730\) 0 0
\(731\) 9.62209 5.55532i 0.355886 0.205471i
\(732\) 0 0
\(733\) −35.7128 −1.31908 −0.659541 0.751668i \(-0.729249\pi\)
−0.659541 + 0.751668i \(0.729249\pi\)
\(734\) 0 0
\(735\) −7.10102 + 15.4135i −0.261925 + 0.568535i
\(736\) 0 0
\(737\) 0.189469 + 0.328169i 0.00697917 + 0.0120883i
\(738\) 0 0
\(739\) −17.4019 + 30.1410i −0.640140 + 1.10876i 0.345261 + 0.938507i \(0.387790\pi\)
−0.985401 + 0.170249i \(0.945543\pi\)
\(740\) 0 0
\(741\) −27.5758 + 5.78658i −1.01302 + 0.212575i
\(742\) 0 0
\(743\) −3.53553 + 6.12372i −0.129706 + 0.224658i −0.923563 0.383447i \(-0.874737\pi\)
0.793857 + 0.608105i \(0.208070\pi\)
\(744\) 0 0
\(745\) −11.9282 20.6603i −0.437016 0.756933i
\(746\) 0 0
\(747\) 4.23607 + 22.7919i 0.154990 + 0.833912i
\(748\) 0 0
\(749\) 18.2832 0.668055
\(750\) 0 0
\(751\) 7.45448 4.30385i 0.272018 0.157050i −0.357786 0.933803i \(-0.616468\pi\)
0.629804 + 0.776754i \(0.283135\pi\)
\(752\) 0 0
\(753\) 14.6969 + 20.7846i 0.535586 + 0.757433i
\(754\) 0 0
\(755\) −1.41421 + 2.44949i −0.0514685 + 0.0891461i
\(756\) 0 0
\(757\) −2.30385 + 3.99038i −0.0837348 + 0.145033i −0.904851 0.425728i \(-0.860018\pi\)
0.821117 + 0.570760i \(0.193352\pi\)
\(758\) 0 0
\(759\) 0.203072 0.143594i 0.00737104 0.00521212i
\(760\) 0 0
\(761\) 46.2629i 1.67703i 0.544879 + 0.838514i \(0.316575\pi\)
−0.544879 + 0.838514i \(0.683425\pi\)
\(762\) 0 0
\(763\) −46.5167 26.8564i −1.68402 0.972267i
\(764\) 0 0
\(765\) 13.5065 + 15.7980i 0.488327 + 0.571176i
\(766\) 0 0
\(767\) 31.2886i 1.12976i
\(768\) 0 0
\(769\) 0.428203 + 0.741670i 0.0154414 + 0.0267453i 0.873643 0.486568i \(-0.161751\pi\)
−0.858201 + 0.513313i \(0.828418\pi\)
\(770\) 0 0
\(771\) −4.53590 + 3.20736i −0.163356 + 0.115510i
\(772\) 0 0
\(773\) −9.04008 15.6579i −0.325149 0.563175i 0.656393 0.754419i \(-0.272081\pi\)
−0.981543 + 0.191244i \(0.938748\pi\)
\(774\) 0 0
\(775\) −11.5981 + 6.69615i −0.416615 + 0.240533i
\(776\) 0 0
\(777\) 27.4721 2.53129i 0.985556 0.0908095i
\(778\) 0 0
\(779\) 14.6969 + 19.7990i 0.526572 + 0.709372i
\(780\) 0 0
\(781\) −1.17691 0.679492i −0.0421133 0.0243141i
\(782\) 0 0
\(783\) 27.9966 28.7826i 1.00052 1.02861i
\(784\) 0 0
\(785\) 4.33057 2.50026i 0.154565 0.0892380i
\(786\) 0 0
\(787\) 24.0718i 0.858067i 0.903289 + 0.429033i \(0.141146\pi\)
−0.903289 + 0.429033i \(0.858854\pi\)
\(788\) 0 0
\(789\) −0.402091 + 0.872778i −0.0143148 + 0.0310717i
\(790\) 0 0
\(791\) 69.6482 2.47640
\(792\) 0 0
\(793\) 11.4282 + 6.59808i 0.405827 + 0.234305i
\(794\) 0 0
\(795\) 14.7222 1.35651i 0.522144 0.0481106i
\(796\) 0 0
\(797\) 14.4939 0.513399 0.256700 0.966491i \(-0.417365\pi\)
0.256700 + 0.966491i \(0.417365\pi\)
\(798\) 0 0
\(799\) −51.7128 −1.82947
\(800\) 0 0
\(801\) 14.3258 + 16.7563i 0.506176 + 0.592054i
\(802\) 0 0
\(803\) −0.984508 0.568406i −0.0347425 0.0200586i
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 9.49510 + 4.37441i 0.334243 + 0.153986i
\(808\) 0 0
\(809\) 51.1619i 1.79876i 0.437172 + 0.899378i \(0.355980\pi\)
−0.437172 + 0.899378i \(0.644020\pi\)
\(810\) 0 0
\(811\) −18.8038 + 10.8564i −0.660292 + 0.381220i −0.792388 0.610017i \(-0.791163\pi\)
0.132096 + 0.991237i \(0.457829\pi\)
\(812\) 0 0
\(813\) 10.8990 + 5.02118i 0.382244 + 0.176100i
\(814\) 0 0
\(815\) 6.36396 + 3.67423i 0.222920 + 0.128703i
\(816\) 0 0
\(817\) −5.89230 7.93782i −0.206146 0.277709i
\(818\) 0 0
\(819\) −31.7596 + 27.1529i −1.10977 + 0.948799i
\(820\) 0 0
\(821\) 29.8744 17.2480i 1.04262 0.601958i 0.122047 0.992524i \(-0.461054\pi\)
0.920575 + 0.390566i \(0.127721\pi\)
\(822\) 0 0
\(823\) −4.00000 6.92820i −0.139431 0.241502i 0.787850 0.615867i \(-0.211194\pi\)
−0.927281 + 0.374365i \(0.877861\pi\)
\(824\) 0 0
\(825\) 1.13681 + 1.60770i 0.0395787 + 0.0559728i
\(826\) 0 0
\(827\) 12.2474 + 21.2132i 0.425886 + 0.737655i 0.996503 0.0835608i \(-0.0266293\pi\)
−0.570617 + 0.821216i \(0.693296\pi\)
\(828\) 0 0
\(829\) 38.6603i 1.34273i 0.741129 + 0.671363i \(0.234291\pi\)
−0.741129 + 0.671363i \(0.765709\pi\)
\(830\) 0 0
\(831\) 1.56637 + 16.9998i 0.0543367 + 0.589716i
\(832\) 0 0
\(833\) 29.3939 + 16.9706i 1.01844 + 0.587995i
\(834\) 0 0
\(835\) 10.7846i 0.373217i
\(836\) 0 0
\(837\) 22.3205 6.31319i 0.771510 0.218216i
\(838\) 0 0
\(839\) −2.87920 + 4.98691i −0.0994009 + 0.172167i −0.911437 0.411440i \(-0.865026\pi\)
0.812036 + 0.583607i \(0.198359\pi\)
\(840\) 0 0
\(841\) −15.3564 + 26.5981i −0.529531 + 0.917175i
\(842\) 0 0
\(843\) 21.3205 15.0759i 0.734317 0.519241i
\(844\) 0 0
\(845\) −1.13681 + 0.656339i −0.0391075 + 0.0225787i
\(846\) 0 0
\(847\) 40.5167 1.39217
\(848\) 0 0
\(849\) 34.3830 + 15.8403i 1.18002 + 0.543638i
\(850\) 0 0
\(851\) −0.808643 1.40061i −0.0277199 0.0480123i
\(852\) 0 0
\(853\) −9.83975 + 17.0429i −0.336906 + 0.583539i −0.983849 0.178999i \(-0.942714\pi\)
0.646943 + 0.762539i \(0.276047\pi\)
\(854\) 0 0
\(855\) 12.5851 13.5505i 0.430400 0.463418i
\(856\) 0 0
\(857\) 1.51575 2.62536i 0.0517770 0.0896804i −0.838975 0.544170i \(-0.816845\pi\)
0.890752 + 0.454489i \(0.150178\pi\)
\(858\) 0 0
\(859\) −2.33013 4.03590i −0.0795029 0.137703i 0.823533 0.567269i \(-0.192000\pi\)
−0.903036 + 0.429566i \(0.858667\pi\)
\(860\) 0 0
\(861\) 33.2114 + 15.3006i 1.13184 + 0.521442i
\(862\) 0 0
\(863\) −18.2832 −0.622369 −0.311184 0.950350i \(-0.600726\pi\)
−0.311184 + 0.950350i \(0.600726\pi\)
\(864\) 0 0
\(865\) −14.5359 + 8.39230i −0.494235 + 0.285347i
\(866\) 0 0
\(867\) 9.89949 7.00000i 0.336204 0.237732i
\(868\) 0 0
\(869\) 0.669942 1.16037i 0.0227262 0.0393630i
\(870\) 0 0
\(871\) −1.86603 + 3.23205i −0.0632279 + 0.109514i
\(872\) 0 0
\(873\) −21.1117 7.46410i −0.714522 0.252622i
\(874\) 0 0
\(875\) 42.2233i 1.42741i
\(876\) 0 0
\(877\) −3.23205 1.86603i −0.109139 0.0630112i 0.444437 0.895810i \(-0.353404\pi\)
−0.553576 + 0.832799i \(0.686737\pi\)
\(878\) 0 0
\(879\) 4.01314 + 43.5546i 0.135360 + 1.46906i
\(880\) 0 0
\(881\) 39.8482i 1.34252i −0.741222 0.671260i \(-0.765754\pi\)
0.741222 0.671260i \(-0.234246\pi\)
\(882\) 0 0
\(883\) −11.1340 19.2846i −0.374688 0.648979i 0.615592 0.788065i \(-0.288917\pi\)
−0.990280 + 0.139086i \(0.955584\pi\)
\(884\) 0 0
\(885\) 11.8564 + 16.7675i 0.398549 + 0.563633i
\(886\) 0 0
\(887\) 5.98502 + 10.3664i 0.200957 + 0.348068i 0.948837 0.315766i \(-0.102261\pi\)
−0.747880 + 0.663834i \(0.768928\pi\)
\(888\) 0 0
\(889\) −25.3923 + 14.6603i −0.851631 + 0.491689i
\(890\) 0 0
\(891\) −1.22539 3.18269i −0.0410521 0.106624i
\(892\) 0 0
\(893\) 5.27792 + 45.7081i 0.176619 + 1.52956i
\(894\) 0 0
\(895\) 1.73205 + 1.00000i 0.0578961 + 0.0334263i
\(896\) 0 0
\(897\) 2.22474 + 1.02494i 0.0742821 + 0.0342219i
\(898\) 0 0
\(899\) −29.8744 + 17.2480i −0.996365 + 0.575252i
\(900\) 0 0
\(901\) 29.5692i 0.985094i
\(902\) 0 0
\(903\) −13.3152 6.13431i −0.443100 0.204137i
\(904\) 0 0
\(905\) 4.89898 0.162848
\(906\) 0 0
\(907\) −35.4449 20.4641i −1.17693 0.679499i −0.221626 0.975132i \(-0.571136\pi\)
−0.955302 + 0.295632i \(0.904470\pi\)
\(908\) 0 0
\(909\) 17.6203 15.0645i 0.584430 0.499658i
\(910\) 0 0
\(911\) 40.9107 1.35543 0.677715 0.735324i \(-0.262970\pi\)
0.677715 + 0.735324i \(0.262970\pi\)
\(912\) 0 0
\(913\) −2.92820 −0.0969094
\(914\) 0 0
\(915\) −8.62461 + 0.794675i −0.285121 + 0.0262711i
\(916\) 0 0
\(917\) 22.5259 + 13.0053i 0.743870 + 0.429474i
\(918\) 0 0
\(919\) 31.9808 1.05495 0.527474 0.849571i \(-0.323139\pi\)
0.527474 + 0.849571i \(0.323139\pi\)
\(920\) 0 0
\(921\) 10.0424 21.7980i 0.330907 0.718267i
\(922\) 0 0
\(923\) 13.3843i 0.440548i
\(924\) 0 0
\(925\) 11.0885 6.40192i 0.364586 0.210494i
\(926\) 0 0
\(927\) −5.73630 30.8638i −0.188405 1.01370i
\(928\) 0 0
\(929\) 17.7148 + 10.2277i 0.581205 + 0.335559i 0.761612 0.648033i \(-0.224408\pi\)
−0.180407 + 0.983592i \(0.557742\pi\)
\(930\) 0 0
\(931\) 12.0000 27.7128i 0.393284 0.908251i
\(932\) 0 0
\(933\) −46.8224 + 4.31424i −1.53290 + 0.141242i
\(934\) 0 0
\(935\) −2.27362 + 1.31268i −0.0743555 + 0.0429291i
\(936\) 0 0
\(937\) −2.89230 5.00962i −0.0944875 0.163657i 0.814907 0.579592i \(-0.196788\pi\)
−0.909395 + 0.415934i \(0.863455\pi\)
\(938\) 0 0
\(939\) 35.0507 24.7846i 1.14384 0.808815i
\(940\) 0 0
\(941\) −6.96953 12.0716i −0.227200 0.393522i 0.729777 0.683685i \(-0.239624\pi\)
−0.956977 + 0.290163i \(0.906291\pi\)
\(942\) 0 0
\(943\) 2.14359i 0.0698050i
\(944\) 0 0
\(945\) 6.73069 26.5861i 0.218949 0.864846i
\(946\) 0 0
\(947\) −38.3596 22.1469i −1.24652 0.719679i −0.276107 0.961127i \(-0.589044\pi\)
−0.970414 + 0.241448i \(0.922378\pi\)
\(948\) 0 0
\(949\) 11.1962i 0.363442i
\(950\) 0 0
\(951\) 19.4641 13.7632i 0.631167 0.446302i
\(952\) 0 0
\(953\) 21.3011 36.8947i 0.690011 1.19513i −0.281822 0.959467i \(-0.590939\pi\)
0.971834 0.235668i \(-0.0757279\pi\)
\(954\) 0 0
\(955\) 1.19615 2.07180i 0.0387066 0.0670418i
\(956\) 0 0
\(957\) 2.92820 + 4.14110i 0.0946554 + 0.133863i
\(958\) 0 0
\(959\) −2.44949 + 1.41421i −0.0790981 + 0.0456673i
\(960\) 0 0
\(961\) 11.0718 0.357155
\(962\) 0 0
\(963\) −14.4495 + 2.68556i −0.465628 + 0.0865410i
\(964\) 0 0
\(965\) −9.33109 16.1619i −0.300378 0.520271i
\(966\) 0 0
\(967\) −4.40192 + 7.62436i −0.141556 + 0.245183i −0.928083 0.372374i \(-0.878544\pi\)
0.786527 + 0.617556i \(0.211877\pi\)
\(968\) 0 0
\(969\) −24.6773 27.5505i −0.792749 0.885050i
\(970\) 0 0
\(971\) 7.17260 12.4233i 0.230180 0.398683i −0.727681 0.685916i \(-0.759402\pi\)
0.957861 + 0.287232i \(0.0927352\pi\)
\(972\) 0 0
\(973\) 24.6244 + 42.6506i 0.789421 + 1.36732i
\(974\) 0 0
\(975\) −8.11435 + 17.6130i −0.259867 + 0.564068i
\(976\) 0 0
\(977\) 20.3538 0.651176 0.325588 0.945512i \(-0.394438\pi\)
0.325588 + 0.945512i \(0.394438\pi\)
\(978\) 0 0
\(979\) −2.41154 + 1.39230i −0.0770732 + 0.0444983i
\(980\) 0 0
\(981\) 40.7076 + 14.3923i 1.29969 + 0.459511i
\(982\) 0 0
\(983\) −16.2635 + 28.1691i −0.518724 + 0.898456i 0.481040 + 0.876699i \(0.340259\pi\)
−0.999763 + 0.0217569i \(0.993074\pi\)
\(984\) 0 0
\(985\) 6.46410 11.1962i 0.205963 0.356739i
\(986\) 0 0
\(987\) 39.3949 + 55.7128i 1.25395 + 1.77336i
\(988\) 0 0
\(989\) 0.859411i 0.0273277i
\(990\) 0 0
\(991\) 3.61731 + 2.08846i 0.114908 + 0.0663420i 0.556352 0.830946i \(-0.312200\pi\)
−0.441445 + 0.897289i \(0.645534\pi\)
\(992\) 0 0
\(993\) −55.0680 + 5.07399i −1.74753 + 0.161018i
\(994\) 0 0
\(995\) 18.1074i 0.574042i
\(996\) 0 0
\(997\) −25.0885 43.4545i −0.794559 1.37622i −0.923119 0.384515i \(-0.874369\pi\)
0.128559 0.991702i \(-0.458965\pi\)
\(998\) 0 0
\(999\) −21.3397 + 6.03579i −0.675160 + 0.190964i
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 912.2.bn.m.65.2 8
3.2 odd 2 inner 912.2.bn.m.65.1 8
4.3 odd 2 57.2.f.a.8.4 yes 8
12.11 even 2 57.2.f.a.8.1 8
19.12 odd 6 inner 912.2.bn.m.449.1 8
57.50 even 6 inner 912.2.bn.m.449.2 8
76.11 odd 6 1083.2.d.b.1082.1 8
76.27 even 6 1083.2.d.b.1082.8 8
76.31 even 6 57.2.f.a.50.1 yes 8
228.11 even 6 1083.2.d.b.1082.7 8
228.107 odd 6 57.2.f.a.50.4 yes 8
228.179 odd 6 1083.2.d.b.1082.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.f.a.8.1 8 12.11 even 2
57.2.f.a.8.4 yes 8 4.3 odd 2
57.2.f.a.50.1 yes 8 76.31 even 6
57.2.f.a.50.4 yes 8 228.107 odd 6
912.2.bn.m.65.1 8 3.2 odd 2 inner
912.2.bn.m.65.2 8 1.1 even 1 trivial
912.2.bn.m.449.1 8 19.12 odd 6 inner
912.2.bn.m.449.2 8 57.50 even 6 inner
1083.2.d.b.1082.1 8 76.11 odd 6
1083.2.d.b.1082.2 8 228.179 odd 6
1083.2.d.b.1082.7 8 228.11 even 6
1083.2.d.b.1082.8 8 76.27 even 6