Properties

Label 1815.2.c.g.364.1
Level $1815$
Weight $2$
Character 1815.364
Analytic conductor $14.493$
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] = [1815,2,Mod(364,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
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("1815.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 364.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1815.364
Dual form 1815.2.c.g.364.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.00000i q^{3} +(-2.22474 - 0.224745i) q^{5} -1.41421 q^{6} +1.73205i q^{7} -2.82843i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{2} -1.00000i q^{3} +(-2.22474 - 0.224745i) q^{5} -1.41421 q^{6} +1.73205i q^{7} -2.82843i q^{8} -1.00000 q^{9} +(-0.317837 + 3.14626i) q^{10} +0.635674i q^{13} +2.44949 q^{14} +(-0.224745 + 2.22474i) q^{15} -4.00000 q^{16} -4.87832i q^{17} +1.41421i q^{18} -1.73205 q^{19} +1.73205 q^{21} -2.00000i q^{23} -2.82843 q^{24} +(4.89898 + 1.00000i) q^{25} +0.898979 q^{26} +1.00000i q^{27} -8.34242 q^{29} +(3.14626 + 0.317837i) q^{30} -7.89898 q^{31} -6.89898 q^{34} +(0.389270 - 3.85337i) q^{35} +5.89898i q^{37} +2.44949i q^{38} +0.635674 q^{39} +(-0.635674 + 6.29253i) q^{40} -9.12096 q^{41} -2.44949i q^{42} +5.65685i q^{43} +(2.22474 + 0.224745i) q^{45} -2.82843 q^{46} +7.34847i q^{47} +4.00000i q^{48} +4.00000 q^{49} +(1.41421 - 6.92820i) q^{50} -4.87832 q^{51} +10.0000i q^{53} +1.41421 q^{54} +4.89898 q^{56} +1.73205i q^{57} +11.7980i q^{58} +12.2474 q^{59} -3.28913 q^{61} +11.1708i q^{62} -1.73205i q^{63} -8.00000 q^{64} +(0.142865 - 1.41421i) q^{65} -4.10102i q^{67} -2.00000 q^{69} +(-5.44949 - 0.550510i) q^{70} -3.55051 q^{71} +2.82843i q^{72} -3.28913i q^{73} +8.34242 q^{74} +(1.00000 - 4.89898i) q^{75} -0.898979i q^{78} -6.75323 q^{79} +(8.89898 + 0.898979i) q^{80} +1.00000 q^{81} +12.8990i q^{82} -12.7279i q^{83} +(-1.09638 + 10.8530i) q^{85} +8.00000 q^{86} +8.34242i q^{87} -1.34847 q^{89} +(0.317837 - 3.14626i) q^{90} -1.10102 q^{91} +7.89898i q^{93} +10.3923 q^{94} +(3.85337 + 0.389270i) q^{95} -5.00000i q^{97} -5.65685i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{9} + 8 q^{15} - 32 q^{16} - 32 q^{26} - 24 q^{31} - 16 q^{34} + 8 q^{45} + 32 q^{49} - 64 q^{64} - 16 q^{69} - 24 q^{70} - 48 q^{71} + 8 q^{75} + 32 q^{80} + 8 q^{81} + 64 q^{86} + 48 q^{89} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.22474 0.224745i −0.994936 0.100509i
\(6\) −1.41421 −0.577350
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −1.00000 −0.333333
\(10\) −0.317837 + 3.14626i −0.100509 + 0.994936i
\(11\) 0 0
\(12\) 0 0
\(13\) 0.635674i 0.176304i 0.996107 + 0.0881522i \(0.0280962\pi\)
−0.996107 + 0.0881522i \(0.971904\pi\)
\(14\) 2.44949 0.654654
\(15\) −0.224745 + 2.22474i −0.0580289 + 0.574427i
\(16\) −4.00000 −1.00000
\(17\) 4.87832i 1.18317i −0.806244 0.591583i \(-0.798503\pi\)
0.806244 0.591583i \(-0.201497\pi\)
\(18\) 1.41421i 0.333333i
\(19\) −1.73205 −0.397360 −0.198680 0.980064i \(-0.563665\pi\)
−0.198680 + 0.980064i \(0.563665\pi\)
\(20\) 0 0
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) −2.82843 −0.577350
\(25\) 4.89898 + 1.00000i 0.979796 + 0.200000i
\(26\) 0.898979 0.176304
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −8.34242 −1.54915 −0.774574 0.632483i \(-0.782036\pi\)
−0.774574 + 0.632483i \(0.782036\pi\)
\(30\) 3.14626 + 0.317837i 0.574427 + 0.0580289i
\(31\) −7.89898 −1.41870 −0.709349 0.704857i \(-0.751011\pi\)
−0.709349 + 0.704857i \(0.751011\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) −6.89898 −1.18317
\(35\) 0.389270 3.85337i 0.0657986 0.651339i
\(36\) 0 0
\(37\) 5.89898i 0.969786i 0.874573 + 0.484893i \(0.161142\pi\)
−0.874573 + 0.484893i \(0.838858\pi\)
\(38\) 2.44949i 0.397360i
\(39\) 0.635674 0.101789
\(40\) −0.635674 + 6.29253i −0.100509 + 0.994936i
\(41\) −9.12096 −1.42445 −0.712227 0.701949i \(-0.752313\pi\)
−0.712227 + 0.701949i \(0.752313\pi\)
\(42\) 2.44949i 0.377964i
\(43\) 5.65685i 0.862662i 0.902194 + 0.431331i \(0.141956\pi\)
−0.902194 + 0.431331i \(0.858044\pi\)
\(44\) 0 0
\(45\) 2.22474 + 0.224745i 0.331645 + 0.0335030i
\(46\) −2.82843 −0.417029
\(47\) 7.34847i 1.07188i 0.844255 + 0.535942i \(0.180044\pi\)
−0.844255 + 0.535942i \(0.819956\pi\)
\(48\) 4.00000i 0.577350i
\(49\) 4.00000 0.571429
\(50\) 1.41421 6.92820i 0.200000 0.979796i
\(51\) −4.87832 −0.683101
\(52\) 0 0
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 1.41421 0.192450
\(55\) 0 0
\(56\) 4.89898 0.654654
\(57\) 1.73205i 0.229416i
\(58\) 11.7980i 1.54915i
\(59\) 12.2474 1.59448 0.797241 0.603661i \(-0.206292\pi\)
0.797241 + 0.603661i \(0.206292\pi\)
\(60\) 0 0
\(61\) −3.28913 −0.421130 −0.210565 0.977580i \(-0.567530\pi\)
−0.210565 + 0.977580i \(0.567530\pi\)
\(62\) 11.1708i 1.41870i
\(63\) 1.73205i 0.218218i
\(64\) −8.00000 −1.00000
\(65\) 0.142865 1.41421i 0.0177202 0.175412i
\(66\) 0 0
\(67\) 4.10102i 0.501019i −0.968114 0.250510i \(-0.919402\pi\)
0.968114 0.250510i \(-0.0805982\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) −5.44949 0.550510i −0.651339 0.0657986i
\(71\) −3.55051 −0.421368 −0.210684 0.977554i \(-0.567569\pi\)
−0.210684 + 0.977554i \(0.567569\pi\)
\(72\) 2.82843i 0.333333i
\(73\) 3.28913i 0.384963i −0.981301 0.192482i \(-0.938346\pi\)
0.981301 0.192482i \(-0.0616536\pi\)
\(74\) 8.34242 0.969786
\(75\) 1.00000 4.89898i 0.115470 0.565685i
\(76\) 0 0
\(77\) 0 0
\(78\) 0.898979i 0.101789i
\(79\) −6.75323 −0.759798 −0.379899 0.925028i \(-0.624041\pi\)
−0.379899 + 0.925028i \(0.624041\pi\)
\(80\) 8.89898 + 0.898979i 0.994936 + 0.100509i
\(81\) 1.00000 0.111111
\(82\) 12.8990i 1.42445i
\(83\) 12.7279i 1.39707i −0.715575 0.698535i \(-0.753835\pi\)
0.715575 0.698535i \(-0.246165\pi\)
\(84\) 0 0
\(85\) −1.09638 + 10.8530i −0.118919 + 1.17717i
\(86\) 8.00000 0.862662
\(87\) 8.34242i 0.894401i
\(88\) 0 0
\(89\) −1.34847 −0.142937 −0.0714687 0.997443i \(-0.522769\pi\)
−0.0714687 + 0.997443i \(0.522769\pi\)
\(90\) 0.317837 3.14626i 0.0335030 0.331645i
\(91\) −1.10102 −0.115418
\(92\) 0 0
\(93\) 7.89898i 0.819086i
\(94\) 10.3923 1.07188
\(95\) 3.85337 + 0.389270i 0.395348 + 0.0399382i
\(96\) 0 0
\(97\) 5.00000i 0.507673i −0.967247 0.253837i \(-0.918307\pi\)
0.967247 0.253837i \(-0.0816925\pi\)
\(98\) 5.65685i 0.571429i
\(99\) 0 0
\(100\) 0 0
\(101\) 0.142865 0.0142156 0.00710778 0.999975i \(-0.497738\pi\)
0.00710778 + 0.999975i \(0.497738\pi\)
\(102\) 6.89898i 0.683101i
\(103\) 17.8990i 1.76364i 0.471587 + 0.881819i \(0.343681\pi\)
−0.471587 + 0.881819i \(0.656319\pi\)
\(104\) 1.79796 0.176304
\(105\) −3.85337 0.389270i −0.376051 0.0379888i
\(106\) 14.1421 1.37361
\(107\) 8.34242i 0.806492i 0.915092 + 0.403246i \(0.132118\pi\)
−0.915092 + 0.403246i \(0.867882\pi\)
\(108\) 0 0
\(109\) −19.3383 −1.85227 −0.926136 0.377190i \(-0.876890\pi\)
−0.926136 + 0.377190i \(0.876890\pi\)
\(110\) 0 0
\(111\) 5.89898 0.559906
\(112\) 6.92820i 0.654654i
\(113\) 13.5505i 1.27472i 0.770564 + 0.637362i \(0.219975\pi\)
−0.770564 + 0.637362i \(0.780025\pi\)
\(114\) 2.44949 0.229416
\(115\) −0.449490 + 4.44949i −0.0419151 + 0.414917i
\(116\) 0 0
\(117\) 0.635674i 0.0587681i
\(118\) 17.3205i 1.59448i
\(119\) 8.44949 0.774563
\(120\) 6.29253 + 0.635674i 0.574427 + 0.0580289i
\(121\) 0 0
\(122\) 4.65153i 0.421130i
\(123\) 9.12096i 0.822409i
\(124\) 0 0
\(125\) −10.6742 3.32577i −0.954733 0.297465i
\(126\) −2.44949 −0.218218
\(127\) 18.7026i 1.65959i −0.558069 0.829794i \(-0.688458\pi\)
0.558069 0.829794i \(-0.311542\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 5.65685 0.498058
\(130\) −2.00000 0.202041i −0.175412 0.0177202i
\(131\) 10.6780 0.932944 0.466472 0.884536i \(-0.345525\pi\)
0.466472 + 0.884536i \(0.345525\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) −5.79972 −0.501019
\(135\) 0.224745 2.22474i 0.0193430 0.191476i
\(136\) −13.7980 −1.18317
\(137\) 21.7980i 1.86233i −0.364604 0.931163i \(-0.618796\pi\)
0.364604 0.931163i \(-0.381204\pi\)
\(138\) 2.82843i 0.240772i
\(139\) 12.5851 1.06745 0.533725 0.845658i \(-0.320792\pi\)
0.533725 + 0.845658i \(0.320792\pi\)
\(140\) 0 0
\(141\) 7.34847 0.618853
\(142\) 5.02118i 0.421368i
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 18.5597 + 1.87492i 1.54130 + 0.155703i
\(146\) −4.65153 −0.384963
\(147\) 4.00000i 0.329914i
\(148\) 0 0
\(149\) −11.8065 −0.967228 −0.483614 0.875281i \(-0.660676\pi\)
−0.483614 + 0.875281i \(0.660676\pi\)
\(150\) −6.92820 1.41421i −0.565685 0.115470i
\(151\) −16.0492 −1.30606 −0.653031 0.757331i \(-0.726503\pi\)
−0.653031 + 0.757331i \(0.726503\pi\)
\(152\) 4.89898i 0.397360i
\(153\) 4.87832i 0.394388i
\(154\) 0 0
\(155\) 17.5732 + 1.77526i 1.41151 + 0.142592i
\(156\) 0 0
\(157\) 18.5959i 1.48412i −0.670336 0.742058i \(-0.733850\pi\)
0.670336 0.742058i \(-0.266150\pi\)
\(158\) 9.55051i 0.759798i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 3.46410 0.273009
\(162\) 1.41421i 0.111111i
\(163\) 1.00000i 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −18.0000 −1.39707
\(167\) 13.2207i 1.02305i −0.859268 0.511525i \(-0.829081\pi\)
0.859268 0.511525i \(-0.170919\pi\)
\(168\) 4.89898i 0.377964i
\(169\) 12.5959 0.968917
\(170\) 15.3485 + 1.55051i 1.17717 + 0.118919i
\(171\) 1.73205 0.132453
\(172\) 0 0
\(173\) 14.6349i 1.11267i −0.830957 0.556337i \(-0.812206\pi\)
0.830957 0.556337i \(-0.187794\pi\)
\(174\) 11.7980 0.894401
\(175\) −1.73205 + 8.48528i −0.130931 + 0.641427i
\(176\) 0 0
\(177\) 12.2474i 0.920575i
\(178\) 1.90702i 0.142937i
\(179\) −26.0454 −1.94673 −0.973363 0.229271i \(-0.926366\pi\)
−0.973363 + 0.229271i \(0.926366\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 1.55708i 0.115418i
\(183\) 3.28913i 0.243139i
\(184\) −5.65685 −0.417029
\(185\) 1.32577 13.1237i 0.0974722 0.964875i
\(186\) 11.1708 0.819086
\(187\) 0 0
\(188\) 0 0
\(189\) −1.73205 −0.125988
\(190\) 0.550510 5.44949i 0.0399382 0.395348i
\(191\) −5.79796 −0.419526 −0.209763 0.977752i \(-0.567269\pi\)
−0.209763 + 0.977752i \(0.567269\pi\)
\(192\) 8.00000i 0.577350i
\(193\) 17.7812i 1.27992i 0.768408 + 0.639960i \(0.221049\pi\)
−0.768408 + 0.639960i \(0.778951\pi\)
\(194\) −7.07107 −0.507673
\(195\) −1.41421 0.142865i −0.101274 0.0102307i
\(196\) 0 0
\(197\) 2.82843i 0.201517i −0.994911 0.100759i \(-0.967873\pi\)
0.994911 0.100759i \(-0.0321270\pi\)
\(198\) 0 0
\(199\) −16.5959 −1.17645 −0.588227 0.808696i \(-0.700174\pi\)
−0.588227 + 0.808696i \(0.700174\pi\)
\(200\) 2.82843 13.8564i 0.200000 0.979796i
\(201\) −4.10102 −0.289264
\(202\) 0.202041i 0.0142156i
\(203\) 14.4495i 1.01416i
\(204\) 0 0
\(205\) 20.2918 + 2.04989i 1.41724 + 0.143170i
\(206\) 25.3130 1.76364
\(207\) 2.00000i 0.139010i
\(208\) 2.54270i 0.176304i
\(209\) 0 0
\(210\) −0.550510 + 5.44949i −0.0379888 + 0.376051i
\(211\) 12.4101 0.854345 0.427173 0.904170i \(-0.359510\pi\)
0.427173 + 0.904170i \(0.359510\pi\)
\(212\) 0 0
\(213\) 3.55051i 0.243277i
\(214\) 11.7980 0.806492
\(215\) 1.27135 12.5851i 0.0867053 0.858294i
\(216\) 2.82843 0.192450
\(217\) 13.6814i 0.928756i
\(218\) 27.3485i 1.85227i
\(219\) −3.28913 −0.222259
\(220\) 0 0
\(221\) 3.10102 0.208597
\(222\) 8.34242i 0.559906i
\(223\) 11.0000i 0.736614i 0.929704 + 0.368307i \(0.120063\pi\)
−0.929704 + 0.368307i \(0.879937\pi\)
\(224\) 0 0
\(225\) −4.89898 1.00000i −0.326599 0.0666667i
\(226\) 19.1633 1.27472
\(227\) 4.09978i 0.272112i −0.990701 0.136056i \(-0.956557\pi\)
0.990701 0.136056i \(-0.0434427\pi\)
\(228\) 0 0
\(229\) 3.10102 0.204921 0.102461 0.994737i \(-0.467328\pi\)
0.102461 + 0.994737i \(0.467328\pi\)
\(230\) 6.29253 + 0.635674i 0.414917 + 0.0419151i
\(231\) 0 0
\(232\) 23.5959i 1.54915i
\(233\) 23.7559i 1.55630i 0.628078 + 0.778150i \(0.283842\pi\)
−0.628078 + 0.778150i \(0.716158\pi\)
\(234\) −0.898979 −0.0587681
\(235\) 1.65153 16.3485i 0.107734 1.06646i
\(236\) 0 0
\(237\) 6.75323i 0.438669i
\(238\) 11.9494i 0.774563i
\(239\) 9.89949 0.640345 0.320173 0.947359i \(-0.396259\pi\)
0.320173 + 0.947359i \(0.396259\pi\)
\(240\) 0.898979 8.89898i 0.0580289 0.574427i
\(241\) 0.635674 0.0409474 0.0204737 0.999790i \(-0.493483\pi\)
0.0204737 + 0.999790i \(0.493483\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −8.89898 0.898979i −0.568535 0.0574337i
\(246\) 12.8990 0.822409
\(247\) 1.10102i 0.0700563i
\(248\) 22.3417i 1.41870i
\(249\) −12.7279 −0.806599
\(250\) −4.70334 + 15.0956i −0.297465 + 0.954733i
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −26.4495 −1.65959
\(255\) 10.8530 + 1.09638i 0.679642 + 0.0686577i
\(256\) 0 0
\(257\) 8.69694i 0.542500i −0.962509 0.271250i \(-0.912563\pi\)
0.962509 0.271250i \(-0.0874370\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −10.2173 −0.634874
\(260\) 0 0
\(261\) 8.34242 0.516383
\(262\) 15.1010i 0.932944i
\(263\) 16.0492i 0.989634i −0.868997 0.494817i \(-0.835235\pi\)
0.868997 0.494817i \(-0.164765\pi\)
\(264\) 0 0
\(265\) 2.24745 22.2474i 0.138060 1.36665i
\(266\) −4.24264 −0.260133
\(267\) 1.34847i 0.0825250i
\(268\) 0 0
\(269\) −10.4495 −0.637117 −0.318558 0.947903i \(-0.603199\pi\)
−0.318558 + 0.947903i \(0.603199\pi\)
\(270\) −3.14626 0.317837i −0.191476 0.0193430i
\(271\) 8.19955 0.498087 0.249044 0.968492i \(-0.419884\pi\)
0.249044 + 0.968492i \(0.419884\pi\)
\(272\) 19.5133i 1.18317i
\(273\) 1.10102i 0.0666368i
\(274\) −30.8270 −1.86233
\(275\) 0 0
\(276\) 0 0
\(277\) 22.1667i 1.33187i 0.746010 + 0.665934i \(0.231967\pi\)
−0.746010 + 0.665934i \(0.768033\pi\)
\(278\) 17.7980i 1.06745i
\(279\) 7.89898 0.472900
\(280\) −10.8990 1.10102i −0.651339 0.0657986i
\(281\) 2.04989 0.122286 0.0611430 0.998129i \(-0.480525\pi\)
0.0611430 + 0.998129i \(0.480525\pi\)
\(282\) 10.3923i 0.618853i
\(283\) 11.1387i 0.662129i 0.943608 + 0.331065i \(0.107408\pi\)
−0.943608 + 0.331065i \(0.892592\pi\)
\(284\) 0 0
\(285\) 0.389270 3.85337i 0.0230583 0.228254i
\(286\) 0 0
\(287\) 15.7980i 0.932524i
\(288\) 0 0
\(289\) −6.79796 −0.399880
\(290\) 2.65153 26.2474i 0.155703 1.54130i
\(291\) −5.00000 −0.293105
\(292\) 0 0
\(293\) 22.8345i 1.33401i −0.745055 0.667003i \(-0.767577\pi\)
0.745055 0.667003i \(-0.232423\pi\)
\(294\) −5.65685 −0.329914
\(295\) −27.2474 2.75255i −1.58641 0.160260i
\(296\) 16.6848 0.969786
\(297\) 0 0
\(298\) 16.6969i 0.967228i
\(299\) 1.27135 0.0735240
\(300\) 0 0
\(301\) −9.79796 −0.564745
\(302\) 22.6969i 1.30606i
\(303\) 0.142865i 0.00820736i
\(304\) 6.92820 0.397360
\(305\) 7.31747 + 0.739215i 0.418997 + 0.0423273i
\(306\) 6.89898 0.394388
\(307\) 1.44632i 0.0825459i 0.999148 + 0.0412730i \(0.0131413\pi\)
−0.999148 + 0.0412730i \(0.986859\pi\)
\(308\) 0 0
\(309\) 17.8990 1.01824
\(310\) 2.51059 24.8523i 0.142592 1.41151i
\(311\) 5.75255 0.326197 0.163099 0.986610i \(-0.447851\pi\)
0.163099 + 0.986610i \(0.447851\pi\)
\(312\) 1.79796i 0.101789i
\(313\) 22.6969i 1.28291i −0.767162 0.641453i \(-0.778332\pi\)
0.767162 0.641453i \(-0.221668\pi\)
\(314\) −26.2986 −1.48412
\(315\) −0.389270 + 3.85337i −0.0219329 + 0.217113i
\(316\) 0 0
\(317\) 8.44949i 0.474571i −0.971440 0.237285i \(-0.923742\pi\)
0.971440 0.237285i \(-0.0762576\pi\)
\(318\) 14.1421i 0.793052i
\(319\) 0 0
\(320\) 17.7980 + 1.79796i 0.994936 + 0.100509i
\(321\) 8.34242 0.465628
\(322\) 4.89898i 0.273009i
\(323\) 8.44949i 0.470142i
\(324\) 0 0
\(325\) −0.635674 + 3.11416i −0.0352609 + 0.172742i
\(326\) −1.41421 −0.0783260
\(327\) 19.3383i 1.06941i
\(328\) 25.7980i 1.42445i
\(329\) −12.7279 −0.701713
\(330\) 0 0
\(331\) −18.5959 −1.02212 −0.511062 0.859544i \(-0.670748\pi\)
−0.511062 + 0.859544i \(0.670748\pi\)
\(332\) 0 0
\(333\) 5.89898i 0.323262i
\(334\) −18.6969 −1.02305
\(335\) −0.921683 + 9.12372i −0.0503569 + 0.498482i
\(336\) −6.92820 −0.377964
\(337\) 11.1387i 0.606766i −0.952869 0.303383i \(-0.901884\pi\)
0.952869 0.303383i \(-0.0981161\pi\)
\(338\) 17.8133i 0.968917i
\(339\) 13.5505 0.735963
\(340\) 0 0
\(341\) 0 0
\(342\) 2.44949i 0.132453i
\(343\) 19.0526i 1.02874i
\(344\) 16.0000 0.862662
\(345\) 4.44949 + 0.449490i 0.239552 + 0.0241997i
\(346\) −20.6969 −1.11267
\(347\) 13.2207i 0.709726i 0.934918 + 0.354863i \(0.115473\pi\)
−0.934918 + 0.354863i \(0.884527\pi\)
\(348\) 0 0
\(349\) −5.19615 −0.278144 −0.139072 0.990282i \(-0.544412\pi\)
−0.139072 + 0.990282i \(0.544412\pi\)
\(350\) 12.0000 + 2.44949i 0.641427 + 0.130931i
\(351\) −0.635674 −0.0339298
\(352\) 0 0
\(353\) 7.55051i 0.401873i 0.979604 + 0.200937i \(0.0643985\pi\)
−0.979604 + 0.200937i \(0.935601\pi\)
\(354\) −17.3205 −0.920575
\(355\) 7.89898 + 0.797959i 0.419234 + 0.0423513i
\(356\) 0 0
\(357\) 8.44949i 0.447194i
\(358\) 36.8338i 1.94673i
\(359\) −25.9487 −1.36952 −0.684759 0.728770i \(-0.740092\pi\)
−0.684759 + 0.728770i \(0.740092\pi\)
\(360\) 0.635674 6.29253i 0.0335030 0.331645i
\(361\) −16.0000 −0.842105
\(362\) 7.07107i 0.371647i
\(363\) 0 0
\(364\) 0 0
\(365\) −0.739215 + 7.31747i −0.0386923 + 0.383014i
\(366\) 4.65153 0.243139
\(367\) 35.7980i 1.86864i −0.356438 0.934319i \(-0.616009\pi\)
0.356438 0.934319i \(-0.383991\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 9.12096 0.474818
\(370\) −18.5597 1.87492i −0.964875 0.0974722i
\(371\) −17.3205 −0.899236
\(372\) 0 0
\(373\) 26.9022i 1.39294i −0.717585 0.696471i \(-0.754753\pi\)
0.717585 0.696471i \(-0.245247\pi\)
\(374\) 0 0
\(375\) −3.32577 + 10.6742i −0.171742 + 0.551215i
\(376\) 20.7846 1.07188
\(377\) 5.30306i 0.273122i
\(378\) 2.44949i 0.125988i
\(379\) −1.79796 −0.0923549 −0.0461775 0.998933i \(-0.514704\pi\)
−0.0461775 + 0.998933i \(0.514704\pi\)
\(380\) 0 0
\(381\) −18.7026 −0.958164
\(382\) 8.19955i 0.419526i
\(383\) 22.4949i 1.14944i −0.818352 0.574718i \(-0.805112\pi\)
0.818352 0.574718i \(-0.194888\pi\)
\(384\) 11.3137 0.577350
\(385\) 0 0
\(386\) 25.1464 1.27992
\(387\) 5.65685i 0.287554i
\(388\) 0 0
\(389\) 10.8990 0.552600 0.276300 0.961071i \(-0.410892\pi\)
0.276300 + 0.961071i \(0.410892\pi\)
\(390\) −0.202041 + 2.00000i −0.0102307 + 0.101274i
\(391\) −9.75663 −0.493414
\(392\) 11.3137i 0.571429i
\(393\) 10.6780i 0.538636i
\(394\) −4.00000 −0.201517
\(395\) 15.0242 + 1.51775i 0.755950 + 0.0763665i
\(396\) 0 0
\(397\) 24.3939i 1.22429i 0.790744 + 0.612147i \(0.209694\pi\)
−0.790744 + 0.612147i \(0.790306\pi\)
\(398\) 23.4702i 1.17645i
\(399\) −3.00000 −0.150188
\(400\) −19.5959 4.00000i −0.979796 0.200000i
\(401\) 10.6515 0.531912 0.265956 0.963985i \(-0.414312\pi\)
0.265956 + 0.963985i \(0.414312\pi\)
\(402\) 5.79972i 0.289264i
\(403\) 5.02118i 0.250123i
\(404\) 0 0
\(405\) −2.22474 0.224745i −0.110548 0.0111677i
\(406\) −20.4347 −1.01416
\(407\) 0 0
\(408\) 13.7980i 0.683101i
\(409\) −28.8092 −1.42452 −0.712261 0.701914i \(-0.752329\pi\)
−0.712261 + 0.701914i \(0.752329\pi\)
\(410\) 2.89898 28.6969i 0.143170 1.41724i
\(411\) −21.7980 −1.07521
\(412\) 0 0
\(413\) 21.2132i 1.04383i
\(414\) 2.82843 0.139010
\(415\) −2.86054 + 28.3164i −0.140418 + 1.39000i
\(416\) 0 0
\(417\) 12.5851i 0.616293i
\(418\) 0 0
\(419\) −9.55051 −0.466573 −0.233286 0.972408i \(-0.574948\pi\)
−0.233286 + 0.972408i \(0.574948\pi\)
\(420\) 0 0
\(421\) 34.6969 1.69103 0.845513 0.533955i \(-0.179295\pi\)
0.845513 + 0.533955i \(0.179295\pi\)
\(422\) 17.5505i 0.854345i
\(423\) 7.34847i 0.357295i
\(424\) 28.2843 1.37361
\(425\) 4.87832 23.8988i 0.236633 1.15926i
\(426\) 5.02118 0.243277
\(427\) 5.69694i 0.275694i
\(428\) 0 0
\(429\) 0 0
\(430\) −17.7980 1.79796i −0.858294 0.0867053i
\(431\) −22.9774 −1.10678 −0.553390 0.832922i \(-0.686666\pi\)
−0.553390 + 0.832922i \(0.686666\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) −19.3485 −0.928756
\(435\) 1.87492 18.5597i 0.0898953 0.889872i
\(436\) 0 0
\(437\) 3.46410i 0.165710i
\(438\) 4.65153i 0.222259i
\(439\) 21.2453 1.01398 0.506992 0.861951i \(-0.330757\pi\)
0.506992 + 0.861951i \(0.330757\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 4.38551i 0.208597i
\(443\) 1.34847i 0.0640677i 0.999487 + 0.0320339i \(0.0101984\pi\)
−0.999487 + 0.0320339i \(0.989802\pi\)
\(444\) 0 0
\(445\) 3.00000 + 0.303062i 0.142214 + 0.0143665i
\(446\) 15.5563 0.736614
\(447\) 11.8065i 0.558429i
\(448\) 13.8564i 0.654654i
\(449\) −31.5505 −1.48896 −0.744480 0.667644i \(-0.767303\pi\)
−0.744480 + 0.667644i \(0.767303\pi\)
\(450\) −1.41421 + 6.92820i −0.0666667 + 0.326599i
\(451\) 0 0
\(452\) 0 0
\(453\) 16.0492i 0.754055i
\(454\) −5.79796 −0.272112
\(455\) 2.44949 + 0.247449i 0.114834 + 0.0116006i
\(456\) 4.89898 0.229416
\(457\) 23.8988i 1.11794i −0.829189 0.558969i \(-0.811197\pi\)
0.829189 0.558969i \(-0.188803\pi\)
\(458\) 4.38551i 0.204921i
\(459\) 4.87832 0.227700
\(460\) 0 0
\(461\) −6.57826 −0.306380 −0.153190 0.988197i \(-0.548955\pi\)
−0.153190 + 0.988197i \(0.548955\pi\)
\(462\) 0 0
\(463\) 36.6969i 1.70545i −0.522359 0.852726i \(-0.674948\pi\)
0.522359 0.852726i \(-0.325052\pi\)
\(464\) 33.3697 1.54915
\(465\) 1.77526 17.5732i 0.0823255 0.814938i
\(466\) 33.5959 1.55630
\(467\) 10.2474i 0.474195i −0.971486 0.237098i \(-0.923804\pi\)
0.971486 0.237098i \(-0.0761961\pi\)
\(468\) 0 0
\(469\) 7.10318 0.327994
\(470\) −23.1202 2.33562i −1.06646 0.107734i
\(471\) −18.5959 −0.856855
\(472\) 34.6410i 1.59448i
\(473\) 0 0
\(474\) 9.55051 0.438669
\(475\) −8.48528 1.73205i −0.389331 0.0794719i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 14.0000i 0.640345i
\(479\) 8.34242 0.381175 0.190587 0.981670i \(-0.438961\pi\)
0.190587 + 0.981670i \(0.438961\pi\)
\(480\) 0 0
\(481\) −3.74983 −0.170978
\(482\) 0.898979i 0.0409474i
\(483\) 3.46410i 0.157622i
\(484\) 0 0
\(485\) −1.12372 + 11.1237i −0.0510257 + 0.505102i
\(486\) −1.41421 −0.0641500
\(487\) 38.4949i 1.74437i −0.489176 0.872185i \(-0.662702\pi\)
0.489176 0.872185i \(-0.337298\pi\)
\(488\) 9.30306i 0.421130i
\(489\) −1.00000 −0.0452216
\(490\) −1.27135 + 12.5851i −0.0574337 + 0.568535i
\(491\) −30.1913 −1.36251 −0.681257 0.732044i \(-0.738566\pi\)
−0.681257 + 0.732044i \(0.738566\pi\)
\(492\) 0 0
\(493\) 40.6969i 1.83290i
\(494\) −1.55708 −0.0700563
\(495\) 0 0
\(496\) 31.5959 1.41870
\(497\) 6.14966i 0.275850i
\(498\) 18.0000i 0.806599i
\(499\) 10.3031 0.461228 0.230614 0.973045i \(-0.425926\pi\)
0.230614 + 0.973045i \(0.425926\pi\)
\(500\) 0 0
\(501\) −13.2207 −0.590659
\(502\) 2.82843i 0.126239i
\(503\) 15.4135i 0.687253i −0.939106 0.343627i \(-0.888345\pi\)
0.939106 0.343627i \(-0.111655\pi\)
\(504\) −4.89898 −0.218218
\(505\) −0.317837 0.0321081i −0.0141436 0.00142879i
\(506\) 0 0
\(507\) 12.5959i 0.559404i
\(508\) 0 0
\(509\) 13.1464 0.582705 0.291353 0.956616i \(-0.405895\pi\)
0.291353 + 0.956616i \(0.405895\pi\)
\(510\) 1.55051 15.3485i 0.0686577 0.679642i
\(511\) 5.69694 0.252018
\(512\) 22.6274i 1.00000i
\(513\) 1.73205i 0.0764719i
\(514\) −12.2993 −0.542500
\(515\) 4.02270 39.8207i 0.177262 1.75471i
\(516\) 0 0
\(517\) 0 0
\(518\) 14.4495i 0.634874i
\(519\) −14.6349 −0.642403
\(520\) −4.00000 0.404082i −0.175412 0.0177202i
\(521\) −9.14643 −0.400712 −0.200356 0.979723i \(-0.564210\pi\)
−0.200356 + 0.979723i \(0.564210\pi\)
\(522\) 11.7980i 0.516383i
\(523\) 0.810647i 0.0354471i −0.999843 0.0177236i \(-0.994358\pi\)
0.999843 0.0177236i \(-0.00564188\pi\)
\(524\) 0 0
\(525\) 8.48528 + 1.73205i 0.370328 + 0.0755929i
\(526\) −22.6969 −0.989634
\(527\) 38.5337i 1.67855i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) −31.4626 3.17837i −1.36665 0.138060i
\(531\) −12.2474 −0.531494
\(532\) 0 0
\(533\) 5.79796i 0.251137i
\(534\) 1.90702 0.0825250
\(535\) 1.87492 18.5597i 0.0810597 0.802408i
\(536\) −11.5994 −0.501019
\(537\) 26.0454i 1.12394i
\(538\) 14.7778i 0.637117i
\(539\) 0 0
\(540\) 0 0
\(541\) −20.1489 −0.866270 −0.433135 0.901329i \(-0.642593\pi\)
−0.433135 + 0.901329i \(0.642593\pi\)
\(542\) 11.5959i 0.498087i
\(543\) 5.00000i 0.214571i
\(544\) 0 0
\(545\) 43.0228 + 4.34618i 1.84289 + 0.186170i
\(546\) 1.55708 0.0666368
\(547\) 29.5556i 1.26371i −0.775088 0.631854i \(-0.782294\pi\)
0.775088 0.631854i \(-0.217706\pi\)
\(548\) 0 0
\(549\) 3.28913 0.140377
\(550\) 0 0
\(551\) 14.4495 0.615569
\(552\) 5.65685i 0.240772i
\(553\) 11.6969i 0.497404i
\(554\) 31.3485 1.33187
\(555\) −13.1237 1.32577i −0.557071 0.0562756i
\(556\) 0 0
\(557\) 7.70674i 0.326545i −0.986581 0.163273i \(-0.947795\pi\)
0.986581 0.163273i \(-0.0522050\pi\)
\(558\) 11.1708i 0.472900i
\(559\) −3.59592 −0.152091
\(560\) −1.55708 + 15.4135i −0.0657986 + 0.651339i
\(561\) 0 0
\(562\) 2.89898i 0.122286i
\(563\) 2.54270i 0.107162i 0.998564 + 0.0535810i \(0.0170635\pi\)
−0.998564 + 0.0535810i \(0.982936\pi\)
\(564\) 0 0
\(565\) 3.04541 30.1464i 0.128121 1.26827i
\(566\) 15.7526 0.662129
\(567\) 1.73205i 0.0727393i
\(568\) 10.0424i 0.421368i
\(569\) 17.9562 0.752762 0.376381 0.926465i \(-0.377168\pi\)
0.376381 + 0.926465i \(0.377168\pi\)
\(570\) −5.44949 0.550510i −0.228254 0.0230583i
\(571\) −15.3027 −0.640399 −0.320200 0.947350i \(-0.603750\pi\)
−0.320200 + 0.947350i \(0.603750\pi\)
\(572\) 0 0
\(573\) 5.79796i 0.242213i
\(574\) −22.3417 −0.932524
\(575\) 2.00000 9.79796i 0.0834058 0.408603i
\(576\) 8.00000 0.333333
\(577\) 23.4949i 0.978105i −0.872254 0.489053i \(-0.837343\pi\)
0.872254 0.489053i \(-0.162657\pi\)
\(578\) 9.61377i 0.399880i
\(579\) 17.7812 0.738962
\(580\) 0 0
\(581\) 22.0454 0.914598
\(582\) 7.07107i 0.293105i
\(583\) 0 0
\(584\) −9.30306 −0.384963
\(585\) −0.142865 + 1.41421i −0.00590672 + 0.0584705i
\(586\) −32.2929 −1.33401
\(587\) 10.6969i 0.441510i 0.975329 + 0.220755i \(0.0708521\pi\)
−0.975329 + 0.220755i \(0.929148\pi\)
\(588\) 0 0
\(589\) 13.6814 0.563734
\(590\) −3.89270 + 38.5337i −0.160260 + 1.58641i
\(591\) −2.82843 −0.116346
\(592\) 23.5959i 0.969786i
\(593\) 10.6780i 0.438494i 0.975669 + 0.219247i \(0.0703601\pi\)
−0.975669 + 0.219247i \(0.929640\pi\)
\(594\) 0 0
\(595\) −18.7980 1.89898i −0.770641 0.0778506i
\(596\) 0 0
\(597\) 16.5959i 0.679226i
\(598\) 1.79796i 0.0735240i
\(599\) −29.3485 −1.19915 −0.599573 0.800320i \(-0.704663\pi\)
−0.599573 + 0.800320i \(0.704663\pi\)
\(600\) −13.8564 2.82843i −0.565685 0.115470i
\(601\) −4.56048 −0.186026 −0.0930129 0.995665i \(-0.529650\pi\)
−0.0930129 + 0.995665i \(0.529650\pi\)
\(602\) 13.8564i 0.564745i
\(603\) 4.10102i 0.167006i
\(604\) 0 0
\(605\) 0 0
\(606\) −0.202041 −0.00820736
\(607\) 30.5412i 1.23963i −0.784748 0.619815i \(-0.787207\pi\)
0.784748 0.619815i \(-0.212793\pi\)
\(608\) 0 0
\(609\) −14.4495 −0.585523
\(610\) 1.04541 10.3485i 0.0423273 0.418997i
\(611\) −4.67123 −0.188978
\(612\) 0 0
\(613\) 0.174973i 0.00706708i −0.999994 0.00353354i \(-0.998875\pi\)
0.999994 0.00353354i \(-0.00112476\pi\)
\(614\) 2.04541 0.0825459
\(615\) 2.04989 20.2918i 0.0826595 0.818244i
\(616\) 0 0
\(617\) 40.0454i 1.61217i −0.591802 0.806084i \(-0.701583\pi\)
0.591802 0.806084i \(-0.298417\pi\)
\(618\) 25.3130i 1.01824i
\(619\) 28.6969 1.15343 0.576714 0.816946i \(-0.304335\pi\)
0.576714 + 0.816946i \(0.304335\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 8.13534i 0.326197i
\(623\) 2.33562i 0.0935745i
\(624\) −2.54270 −0.101789
\(625\) 23.0000 + 9.79796i 0.920000 + 0.391918i
\(626\) −32.0983 −1.28291
\(627\) 0 0
\(628\) 0 0
\(629\) 28.7771 1.14742
\(630\) 5.44949 + 0.550510i 0.217113 + 0.0219329i
\(631\) −18.8990 −0.752356 −0.376178 0.926547i \(-0.622762\pi\)
−0.376178 + 0.926547i \(0.622762\pi\)
\(632\) 19.1010i 0.759798i
\(633\) 12.4101i 0.493257i
\(634\) −11.9494 −0.474571
\(635\) −4.20332 + 41.6085i −0.166804 + 1.65118i
\(636\) 0 0
\(637\) 2.54270i 0.100745i
\(638\) 0 0
\(639\) 3.55051 0.140456
\(640\) 2.54270 25.1701i 0.100509 0.994936i
\(641\) 8.44949 0.333735 0.166867 0.985979i \(-0.446635\pi\)
0.166867 + 0.985979i \(0.446635\pi\)
\(642\) 11.7980i 0.465628i
\(643\) 29.6969i 1.17113i 0.810624 + 0.585566i \(0.199128\pi\)
−0.810624 + 0.585566i \(0.800872\pi\)
\(644\) 0 0
\(645\) −12.5851 1.27135i −0.495536 0.0500593i
\(646\) 11.9494 0.470142
\(647\) 22.2020i 0.872852i −0.899740 0.436426i \(-0.856244\pi\)
0.899740 0.436426i \(-0.143756\pi\)
\(648\) 2.82843i 0.111111i
\(649\) 0 0
\(650\) 4.40408 + 0.898979i 0.172742 + 0.0352609i
\(651\) −13.6814 −0.536218
\(652\) 0 0
\(653\) 36.6969i 1.43606i 0.696011 + 0.718031i \(0.254956\pi\)
−0.696011 + 0.718031i \(0.745044\pi\)
\(654\) 27.3485 1.06941
\(655\) −23.7559 2.39983i −0.928220 0.0937692i
\(656\) 36.4838 1.42445
\(657\) 3.28913i 0.128321i
\(658\) 18.0000i 0.701713i
\(659\) 36.8338 1.43484 0.717420 0.696641i \(-0.245323\pi\)
0.717420 + 0.696641i \(0.245323\pi\)
\(660\) 0 0
\(661\) −15.2020 −0.591291 −0.295645 0.955298i \(-0.595535\pi\)
−0.295645 + 0.955298i \(0.595535\pi\)
\(662\) 26.2986i 1.02212i
\(663\) 3.10102i 0.120434i
\(664\) −36.0000 −1.39707
\(665\) −0.674235 + 6.67423i −0.0261457 + 0.258816i
\(666\) −8.34242 −0.323262
\(667\) 16.6848i 0.646039i
\(668\) 0 0
\(669\) 11.0000 0.425285
\(670\) 12.9029 + 1.30346i 0.498482 + 0.0503569i
\(671\) 0 0
\(672\) 0 0
\(673\) 45.1441i 1.74018i 0.492896 + 0.870088i \(0.335938\pi\)
−0.492896 + 0.870088i \(0.664062\pi\)
\(674\) −15.7526 −0.606766
\(675\) −1.00000 + 4.89898i −0.0384900 + 0.188562i
\(676\) 0 0
\(677\) 29.5556i 1.13591i 0.823058 + 0.567957i \(0.192266\pi\)
−0.823058 + 0.567957i \(0.807734\pi\)
\(678\) 19.1633i 0.735963i
\(679\) 8.66025 0.332350
\(680\) 30.6969 + 3.10102i 1.17717 + 0.118919i
\(681\) −4.09978 −0.157104
\(682\) 0 0
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) 0 0
\(685\) −4.89898 + 48.4949i −0.187180 + 1.85289i
\(686\) 26.9444 1.02874
\(687\) 3.10102i 0.118311i
\(688\) 22.6274i 0.862662i
\(689\) −6.35674 −0.242173
\(690\) 0.635674 6.29253i 0.0241997 0.239552i
\(691\) 28.1010 1.06901 0.534507 0.845164i \(-0.320497\pi\)
0.534507 + 0.845164i \(0.320497\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 18.6969 0.709726
\(695\) −27.9985 2.82843i −1.06205 0.107288i
\(696\) 23.5959 0.894401
\(697\) 44.4949i 1.68536i
\(698\) 7.34847i 0.278144i
\(699\) 23.7559 0.898531
\(700\) 0 0
\(701\) 4.73545 0.178856 0.0894278 0.995993i \(-0.471496\pi\)
0.0894278 + 0.995993i \(0.471496\pi\)
\(702\) 0.898979i 0.0339298i
\(703\) 10.2173i 0.385354i
\(704\) 0 0
\(705\) −16.3485 1.65153i −0.615719 0.0622002i
\(706\) 10.6780 0.401873
\(707\) 0.247449i 0.00930627i
\(708\) 0 0
\(709\) 24.0000 0.901339 0.450669 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 1.12848 11.1708i 0.0423513 0.419234i
\(711\) 6.75323 0.253266
\(712\) 3.81405i 0.142937i
\(713\) 15.7980i 0.591638i
\(714\) −11.9494 −0.447194
\(715\) 0 0
\(716\) 0 0
\(717\) 9.89949i 0.369703i
\(718\) 36.6969i 1.36952i
\(719\) −46.5403 −1.73566 −0.867830 0.496861i \(-0.834486\pi\)
−0.867830 + 0.496861i \(0.834486\pi\)
\(720\) −8.89898 0.898979i −0.331645 0.0335030i
\(721\) −31.0019 −1.15457
\(722\) 22.6274i 0.842105i
\(723\) 0.635674i 0.0236410i
\(724\) 0 0
\(725\) −40.8693 8.34242i −1.51785 0.309830i
\(726\) 0 0
\(727\) 43.3939i 1.60939i 0.593689 + 0.804695i \(0.297671\pi\)
−0.593689 + 0.804695i \(0.702329\pi\)
\(728\) 3.11416i 0.115418i
\(729\) −1.00000 −0.0370370
\(730\) 10.3485 + 1.04541i 0.383014 + 0.0386923i
\(731\) 27.5959 1.02067
\(732\) 0 0
\(733\) 4.73545i 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278731\pi\)
\(734\) −50.6260 −1.86864
\(735\) −0.898979 + 8.89898i −0.0331594 + 0.328244i
\(736\) 0 0
\(737\) 0 0
\(738\) 12.8990i 0.474818i
\(739\) 24.9951 0.919461 0.459731 0.888058i \(-0.347946\pi\)
0.459731 + 0.888058i \(0.347946\pi\)
\(740\) 0 0
\(741\) −1.10102 −0.0404470
\(742\) 24.4949i 0.899236i
\(743\) 3.81405i 0.139924i 0.997550 + 0.0699619i \(0.0222878\pi\)
−0.997550 + 0.0699619i \(0.977712\pi\)
\(744\) 22.3417 0.819086
\(745\) 26.2665 + 2.65345i 0.962330 + 0.0972150i
\(746\) −38.0454 −1.39294
\(747\) 12.7279i 0.465690i
\(748\) 0 0
\(749\) −14.4495 −0.527973
\(750\) 15.0956 + 4.70334i 0.551215 + 0.171742i
\(751\) −4.10102 −0.149648 −0.0748242 0.997197i \(-0.523840\pi\)
−0.0748242 + 0.997197i \(0.523840\pi\)
\(752\) 29.3939i 1.07188i
\(753\) 2.00000i 0.0728841i
\(754\) −7.49966 −0.273122
\(755\) 35.7053 + 3.60697i 1.29945 + 0.131271i
\(756\) 0 0
\(757\) 13.8990i 0.505167i −0.967575 0.252584i \(-0.918720\pi\)
0.967575 0.252584i \(-0.0812802\pi\)
\(758\) 2.54270i 0.0923549i
\(759\) 0 0
\(760\) 1.10102 10.8990i 0.0399382 0.395348i
\(761\) 48.2903 1.75052 0.875262 0.483650i \(-0.160689\pi\)
0.875262 + 0.483650i \(0.160689\pi\)
\(762\) 26.4495i 0.958164i
\(763\) 33.4949i 1.21260i
\(764\) 0 0
\(765\) 1.09638 10.8530i 0.0396396 0.392391i
\(766\) −31.8126 −1.14944
\(767\) 7.78539i 0.281114i
\(768\) 0 0
\(769\) 49.8153 1.79639 0.898193 0.439601i \(-0.144880\pi\)
0.898193 + 0.439601i \(0.144880\pi\)
\(770\) 0 0
\(771\) −8.69694 −0.313213
\(772\) 0 0
\(773\) 31.5959i 1.13643i 0.822881 + 0.568213i \(0.192365\pi\)
−0.822881 + 0.568213i \(0.807635\pi\)
\(774\) −8.00000 −0.287554
\(775\) −38.6969 7.89898i −1.39004 0.283740i
\(776\) −14.1421 −0.507673
\(777\) 10.2173i 0.366545i
\(778\) 15.4135i 0.552600i
\(779\) 15.7980 0.566021
\(780\) 0 0
\(781\) 0 0
\(782\) 13.7980i 0.493414i
\(783\) 8.34242i 0.298134i
\(784\) −16.0000 −0.571429
\(785\) −4.17934 + 41.3712i −0.149167 + 1.47660i
\(786\) −15.1010 −0.538636
\(787\) 50.0545i 1.78425i −0.451788 0.892125i \(-0.649214\pi\)
0.451788 0.892125i \(-0.350786\pi\)
\(788\) 0 0
\(789\) −16.0492 −0.571365
\(790\) 2.14643 21.2474i 0.0763665 0.755950i
\(791\) −23.4702 −0.834503
\(792\) 0 0
\(793\) 2.09082i 0.0742470i
\(794\) 34.4982 1.22429
\(795\) −22.2474 2.24745i −0.789036 0.0797088i
\(796\) 0 0
\(797\) 23.3939i 0.828654i 0.910128 + 0.414327i \(0.135983\pi\)
−0.910128 + 0.414327i \(0.864017\pi\)
\(798\) 4.24264i 0.150188i
\(799\) 35.8481 1.26822
\(800\) 0 0
\(801\) 1.34847 0.0476458
\(802\) 15.0635i 0.531912i
\(803\) 0 0
\(804\) 0 0
\(805\) −7.70674 0.778539i −0.271627 0.0274399i
\(806\) −7.10102 −0.250123
\(807\) 10.4495i 0.367839i
\(808\) 0.404082i 0.0142156i
\(809\) −35.4196 −1.24529 −0.622643 0.782506i \(-0.713941\pi\)
−0.622643 + 0.782506i \(0.713941\pi\)
\(810\) −0.317837 + 3.14626i −0.0111677 + 0.110548i
\(811\) 29.4449 1.03395 0.516975 0.856001i \(-0.327058\pi\)
0.516975 + 0.856001i \(0.327058\pi\)
\(812\) 0 0
\(813\) 8.19955i 0.287571i
\(814\) 0 0
\(815\) −0.224745 + 2.22474i −0.00787247 + 0.0779294i
\(816\) 19.5133 0.683101
\(817\) 9.79796i 0.342787i
\(818\) 40.7423i 1.42452i
\(819\) 1.10102 0.0384728
\(820\) 0 0
\(821\) 6.00680 0.209639 0.104819 0.994491i \(-0.466574\pi\)
0.104819 + 0.994491i \(0.466574\pi\)
\(822\) 30.8270i 1.07521i
\(823\) 1.49490i 0.0521088i 0.999661 + 0.0260544i \(0.00829432\pi\)
−0.999661 + 0.0260544i \(0.991706\pi\)
\(824\) 50.6260 1.76364
\(825\) 0 0
\(826\) 30.0000 1.04383
\(827\) 27.5699i 0.958701i −0.877624 0.479351i \(-0.840872\pi\)
0.877624 0.479351i \(-0.159128\pi\)
\(828\) 0 0
\(829\) 10.7980 0.375029 0.187514 0.982262i \(-0.439957\pi\)
0.187514 + 0.982262i \(0.439957\pi\)
\(830\) 40.0454 + 4.04541i 1.39000 + 0.140418i
\(831\) 22.1667 0.768955
\(832\) 5.08540i 0.176304i
\(833\) 19.5133i 0.676094i
\(834\) −17.7980 −0.616293
\(835\) −2.97129 + 29.4128i −0.102826 + 1.01787i
\(836\) 0 0
\(837\) 7.89898i 0.273029i
\(838\) 13.5065i 0.466573i
\(839\) 0.898979 0.0310362 0.0155181 0.999880i \(-0.495060\pi\)
0.0155181 + 0.999880i \(0.495060\pi\)
\(840\) −1.10102 + 10.8990i −0.0379888 + 0.376051i
\(841\) 40.5959 1.39986
\(842\) 49.0689i 1.69103i
\(843\) 2.04989i 0.0706019i
\(844\) 0 0
\(845\) −28.0227 2.83087i −0.964010 0.0973848i
\(846\) −10.3923 −0.357295
\(847\) 0 0
\(848\) 40.0000i 1.37361i
\(849\) 11.1387 0.382280
\(850\) −33.7980 6.89898i −1.15926 0.236633i
\(851\) 11.7980 0.404429
\(852\) 0 0
\(853\) 1.09638i 0.0375392i 0.999824 + 0.0187696i \(0.00597490\pi\)
−0.999824 + 0.0187696i \(0.994025\pi\)
\(854\) −8.05669 −0.275694
\(855\) −3.85337 0.389270i −0.131783 0.0133127i
\(856\) 23.5959 0.806492
\(857\) 38.5337i 1.31629i 0.752893 + 0.658143i \(0.228658\pi\)
−0.752893 + 0.658143i \(0.771342\pi\)
\(858\) 0 0
\(859\) 35.2929 1.20418 0.602088 0.798429i \(-0.294335\pi\)
0.602088 + 0.798429i \(0.294335\pi\)
\(860\) 0 0
\(861\) −15.7980 −0.538393
\(862\) 32.4949i 1.10678i
\(863\) 15.8434i 0.539314i 0.962956 + 0.269657i \(0.0869104\pi\)
−0.962956 + 0.269657i \(0.913090\pi\)
\(864\) 0 0
\(865\) −3.28913 + 32.5590i −0.111834 + 1.10704i
\(866\) 26.8701 0.913082
\(867\) 6.79796i 0.230871i
\(868\) 0 0
\(869\) 0 0
\(870\) −26.2474 2.65153i −0.889872 0.0898953i
\(871\) 2.60691 0.0883319
\(872\) 54.6969i 1.85227i
\(873\) 5.00000i 0.169224i
\(874\) 4.89898 0.165710
\(875\) 5.76039 18.4883i 0.194737 0.625019i
\(876\) 0 0
\(877\) 31.2877i 1.05651i −0.849086 0.528255i \(-0.822847\pi\)
0.849086 0.528255i \(-0.177153\pi\)
\(878\) 30.0454i 1.01398i
\(879\) −22.8345 −0.770188
\(880\) 0 0
\(881\) −45.6413 −1.53770 −0.768848 0.639432i \(-0.779170\pi\)
−0.768848 + 0.639432i \(0.779170\pi\)
\(882\) 5.65685i 0.190476i
\(883\) 6.10102i 0.205316i 0.994717 + 0.102658i \(0.0327347\pi\)
−0.994717 + 0.102658i \(0.967265\pi\)
\(884\) 0 0
\(885\) −2.75255 + 27.2474i −0.0925260 + 0.915913i
\(886\) 1.90702 0.0640677
\(887\) 13.2207i 0.443909i −0.975057 0.221954i \(-0.928756\pi\)
0.975057 0.221954i \(-0.0712436\pi\)
\(888\) 16.6848i 0.559906i
\(889\) 32.3939 1.08646
\(890\) 0.428594 4.24264i 0.0143665 0.142214i
\(891\) 0 0
\(892\) 0 0
\(893\) 12.7279i 0.425924i
\(894\) 16.6969 0.558429
\(895\) 57.9444 + 5.85357i 1.93687 + 0.195663i
\(896\) −19.5959 −0.654654
\(897\) 1.27135i 0.0424491i
\(898\) 44.6192i 1.48896i
\(899\) 65.8966 2.19777
\(900\) 0 0
\(901\) 48.7832 1.62520
\(902\) 0 0
\(903\) 9.79796i 0.326056i
\(904\) 38.3266 1.27472
\(905\) −11.1237 1.12372i −0.369765 0.0373539i
\(906\) 22.6969 0.754055
\(907\) 12.3031i 0.408516i −0.978917 0.204258i \(-0.934522\pi\)
0.978917 0.204258i \(-0.0654782\pi\)
\(908\) 0 0
\(909\) −0.142865 −0.00473852
\(910\) 0.349945 3.46410i 0.0116006 0.114834i
\(911\) 25.7980 0.854725 0.427362 0.904080i \(-0.359443\pi\)
0.427362 + 0.904080i \(0.359443\pi\)
\(912\) 6.92820i 0.229416i
\(913\) 0 0
\(914\) −33.7980 −1.11794
\(915\) 0.739215 7.31747i 0.0244377 0.241908i
\(916\) 0 0
\(917\) 18.4949i 0.610755i
\(918\) 6.89898i 0.227700i
\(919\) 7.45312 0.245856 0.122928 0.992416i \(-0.460772\pi\)
0.122928 + 0.992416i \(0.460772\pi\)
\(920\) 12.5851 + 1.27135i 0.414917 + 0.0419151i
\(921\) 1.44632 0.0476579
\(922\) 9.30306i 0.306380i
\(923\) 2.25697i 0.0742890i
\(924\) 0 0
\(925\) −5.89898 + 28.8990i −0.193957 + 0.950193i
\(926\) −51.8973 −1.70545
\(927\) 17.8990i 0.587880i
\(928\) 0 0
\(929\) −38.8990 −1.27623 −0.638117 0.769939i \(-0.720286\pi\)
−0.638117 + 0.769939i \(0.720286\pi\)
\(930\) −24.8523 2.51059i −0.814938 0.0823255i
\(931\) −6.92820 −0.227063
\(932\) 0 0
\(933\) 5.75255i 0.188330i
\(934\) −14.4921 −0.474195
\(935\) 0 0
\(936\) −1.79796 −0.0587681
\(937\) 7.67463i 0.250719i 0.992111 + 0.125360i \(0.0400085\pi\)
−0.992111 + 0.125360i \(0.959992\pi\)
\(938\) 10.0454i 0.327994i
\(939\) −22.6969 −0.740687
\(940\) 0 0
\(941\) 49.4188 1.61101 0.805504 0.592591i \(-0.201895\pi\)
0.805504 + 0.592591i \(0.201895\pi\)
\(942\) 26.2986i 0.856855i
\(943\) 18.2419i 0.594038i
\(944\) −48.9898 −1.59448
\(945\) 3.85337 + 0.389270i 0.125350 + 0.0126629i
\(946\) 0 0
\(947\) 5.55051i 0.180367i 0.995925 + 0.0901837i \(0.0287454\pi\)
−0.995925 + 0.0901837i \(0.971255\pi\)
\(948\) 0 0
\(949\) 2.09082 0.0678707
\(950\) −2.44949 + 12.0000i −0.0794719 + 0.389331i
\(951\) −8.44949 −0.273993
\(952\) 23.8988i 0.774563i
\(953\) 8.83523i 0.286201i −0.989708 0.143101i \(-0.954293\pi\)
0.989708 0.143101i \(-0.0457072\pi\)
\(954\) −14.1421 −0.457869
\(955\) 12.8990 + 1.30306i 0.417401 + 0.0421661i
\(956\) 0 0
\(957\) 0 0
\(958\) 11.7980i 0.381175i
\(959\) 37.7552 1.21918
\(960\) 1.79796 17.7980i 0.0580289 0.574427i
\(961\) 31.3939 1.01271
\(962\) 5.30306i 0.170978i
\(963\) 8.34242i 0.268831i
\(964\) 0 0
\(965\) 3.99624 39.5587i 0.128643 1.27344i
\(966\) −4.89898 −0.157622
\(967\) 6.11756i 0.196727i −0.995151 0.0983637i \(-0.968639\pi\)
0.995151 0.0983637i \(-0.0313608\pi\)
\(968\) 0 0
\(969\) 8.44949 0.271437
\(970\) 15.7313 + 1.58919i 0.505102 + 0.0510257i
\(971\) −13.1010 −0.420432 −0.210216 0.977655i \(-0.567417\pi\)
−0.210216 + 0.977655i \(0.567417\pi\)
\(972\) 0 0
\(973\) 21.7980i 0.698810i
\(974\) −54.4400 −1.74437
\(975\) 3.11416 + 0.635674i 0.0997328 + 0.0203579i
\(976\) 13.1565 0.421130
\(977\) 54.4949i 1.74345i 0.489999 + 0.871723i \(0.336997\pi\)
−0.489999 + 0.871723i \(0.663003\pi\)
\(978\) 1.41421i 0.0452216i
\(979\) 0 0
\(980\) 0 0
\(981\) 19.3383 0.617424
\(982\) 42.6969i 1.36251i
\(983\) 52.2929i 1.66788i −0.551853 0.833942i \(-0.686079\pi\)
0.551853 0.833942i \(-0.313921\pi\)
\(984\) 25.7980 0.822409
\(985\) −0.635674 + 6.29253i −0.0202543 + 0.200497i
\(986\) 57.5542 1.83290
\(987\) 12.7279i 0.405134i
\(988\) 0 0
\(989\) 11.3137 0.359755
\(990\) 0 0
\(991\) −49.1918 −1.56263 −0.781315 0.624137i \(-0.785451\pi\)
−0.781315 + 0.624137i \(0.785451\pi\)
\(992\) 0 0
\(993\) 18.5959i 0.590124i
\(994\) −8.69694 −0.275850
\(995\) 36.9217 + 3.72985i 1.17050 + 0.118244i
\(996\) 0 0
\(997\) 32.5590i 1.03115i 0.856843 + 0.515577i \(0.172422\pi\)
−0.856843 + 0.515577i \(0.827578\pi\)
\(998\) 14.5707i 0.461228i
\(999\) −5.89898 −0.186635
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 1815.2.c.g.364.1 8
5.2 odd 4 9075.2.a.cr.1.3 4
5.3 odd 4 9075.2.a.cy.1.2 4
5.4 even 2 inner 1815.2.c.g.364.7 yes 8
11.10 odd 2 inner 1815.2.c.g.364.5 yes 8
55.32 even 4 9075.2.a.cr.1.2 4
55.43 even 4 9075.2.a.cy.1.3 4
55.54 odd 2 inner 1815.2.c.g.364.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.c.g.364.1 8 1.1 even 1 trivial
1815.2.c.g.364.3 yes 8 55.54 odd 2 inner
1815.2.c.g.364.5 yes 8 11.10 odd 2 inner
1815.2.c.g.364.7 yes 8 5.4 even 2 inner
9075.2.a.cr.1.2 4 55.32 even 4
9075.2.a.cr.1.3 4 5.2 odd 4
9075.2.a.cy.1.2 4 5.3 odd 4
9075.2.a.cy.1.3 4 55.43 even 4