Properties

Label 3042.2.b.l.1351.3
Level $3042$
Weight $2$
Character 3042.1351
Analytic conductor $24.290$
Analytic rank $0$
Dimension $4$
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] = [3042,2,Mod(1351,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (of order \(2\), degree \(1\), 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: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3042.1351
Dual form 3042.2.b.l.1351.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.73205i q^{5} -1.26795i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.73205i q^{5} -1.26795i q^{7} -1.00000i q^{8} +1.73205 q^{10} +1.26795i q^{11} +1.26795 q^{14} +1.00000 q^{16} +5.19615 q^{17} +4.73205i q^{19} +1.73205i q^{20} -1.26795 q^{22} -8.19615 q^{23} +2.00000 q^{25} +1.26795i q^{28} +3.00000 q^{29} +9.46410i q^{31} +1.00000i q^{32} +5.19615i q^{34} -2.19615 q^{35} -3.00000i q^{37} -4.73205 q^{38} -1.73205 q^{40} -6.46410i q^{41} +4.19615 q^{43} -1.26795i q^{44} -8.19615i q^{46} -4.73205i q^{47} +5.39230 q^{49} +2.00000i q^{50} -3.00000 q^{53} +2.19615 q^{55} -1.26795 q^{56} +3.00000i q^{58} -13.8564i q^{59} +15.1962 q^{61} -9.46410 q^{62} -1.00000 q^{64} +7.26795i q^{67} -5.19615 q^{68} -2.19615i q^{70} +2.19615i q^{71} -12.1244i q^{73} +3.00000 q^{74} -4.73205i q^{76} +1.60770 q^{77} +8.39230 q^{79} -1.73205i q^{80} +6.46410 q^{82} -5.66025i q^{83} -9.00000i q^{85} +4.19615i q^{86} +1.26795 q^{88} +9.46410i q^{89} +8.19615 q^{92} +4.73205 q^{94} +8.19615 q^{95} +6.00000i q^{97} +5.39230i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64} + 12 q^{74} + 48 q^{77} - 8 q^{79} + 12 q^{82} + 12 q^{88} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) − 1.73205i − 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) − 1.26795i − 0.479240i −0.970867 0.239620i \(-0.922977\pi\)
0.970867 0.239620i \(-0.0770228\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 1.73205 0.547723
\(11\) 1.26795i 0.382301i 0.981561 + 0.191151i \(0.0612219\pi\)
−0.981561 + 0.191151i \(0.938778\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.26795 0.338874
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.19615 1.26025 0.630126 0.776493i \(-0.283003\pi\)
0.630126 + 0.776493i \(0.283003\pi\)
\(18\) 0 0
\(19\) 4.73205i 1.08561i 0.839860 + 0.542803i \(0.182637\pi\)
−0.839860 + 0.542803i \(0.817363\pi\)
\(20\) 1.73205i 0.387298i
\(21\) 0 0
\(22\) −1.26795 −0.270328
\(23\) −8.19615 −1.70902 −0.854508 0.519438i \(-0.826141\pi\)
−0.854508 + 0.519438i \(0.826141\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.26795i 0.239620i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 9.46410i 1.69980i 0.526942 + 0.849901i \(0.323339\pi\)
−0.526942 + 0.849901i \(0.676661\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) −2.19615 −0.371218
\(36\) 0 0
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) −4.73205 −0.767640
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) − 6.46410i − 1.00952i −0.863259 0.504762i \(-0.831580\pi\)
0.863259 0.504762i \(-0.168420\pi\)
\(42\) 0 0
\(43\) 4.19615 0.639907 0.319954 0.947433i \(-0.396333\pi\)
0.319954 + 0.947433i \(0.396333\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) 0 0
\(46\) − 8.19615i − 1.20846i
\(47\) − 4.73205i − 0.690241i −0.938558 0.345120i \(-0.887838\pi\)
0.938558 0.345120i \(-0.112162\pi\)
\(48\) 0 0
\(49\) 5.39230 0.770329
\(50\) 2.00000i 0.282843i
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 2.19615 0.296129
\(56\) −1.26795 −0.169437
\(57\) 0 0
\(58\) 3.00000i 0.393919i
\(59\) − 13.8564i − 1.80395i −0.431788 0.901975i \(-0.642117\pi\)
0.431788 0.901975i \(-0.357883\pi\)
\(60\) 0 0
\(61\) 15.1962 1.94567 0.972834 0.231504i \(-0.0743646\pi\)
0.972834 + 0.231504i \(0.0743646\pi\)
\(62\) −9.46410 −1.20194
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.26795i 0.887921i 0.896046 + 0.443961i \(0.146427\pi\)
−0.896046 + 0.443961i \(0.853573\pi\)
\(68\) −5.19615 −0.630126
\(69\) 0 0
\(70\) − 2.19615i − 0.262490i
\(71\) 2.19615i 0.260635i 0.991472 + 0.130318i \(0.0415997\pi\)
−0.991472 + 0.130318i \(0.958400\pi\)
\(72\) 0 0
\(73\) − 12.1244i − 1.41905i −0.704681 0.709524i \(-0.748910\pi\)
0.704681 0.709524i \(-0.251090\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) − 4.73205i − 0.542803i
\(77\) 1.60770 0.183214
\(78\) 0 0
\(79\) 8.39230 0.944208 0.472104 0.881543i \(-0.343495\pi\)
0.472104 + 0.881543i \(0.343495\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) 6.46410 0.713841
\(83\) − 5.66025i − 0.621294i −0.950525 0.310647i \(-0.899454\pi\)
0.950525 0.310647i \(-0.100546\pi\)
\(84\) 0 0
\(85\) − 9.00000i − 0.976187i
\(86\) 4.19615i 0.452483i
\(87\) 0 0
\(88\) 1.26795 0.135164
\(89\) 9.46410i 1.00319i 0.865102 + 0.501596i \(0.167254\pi\)
−0.865102 + 0.501596i \(0.832746\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.19615 0.854508
\(93\) 0 0
\(94\) 4.73205 0.488074
\(95\) 8.19615 0.840907
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 5.39230i 0.544705i
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) 19.3923 1.92961 0.964803 0.262973i \(-0.0847030\pi\)
0.964803 + 0.262973i \(0.0847030\pi\)
\(102\) 0 0
\(103\) −6.19615 −0.610525 −0.305263 0.952268i \(-0.598744\pi\)
−0.305263 + 0.952268i \(0.598744\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 3.00000i − 0.291386i
\(107\) 2.19615 0.212310 0.106155 0.994350i \(-0.466146\pi\)
0.106155 + 0.994350i \(0.466146\pi\)
\(108\) 0 0
\(109\) − 4.39230i − 0.420707i −0.977625 0.210353i \(-0.932539\pi\)
0.977625 0.210353i \(-0.0674614\pi\)
\(110\) 2.19615i 0.209395i
\(111\) 0 0
\(112\) − 1.26795i − 0.119810i
\(113\) −0.803848 −0.0756196 −0.0378098 0.999285i \(-0.512038\pi\)
−0.0378098 + 0.999285i \(0.512038\pi\)
\(114\) 0 0
\(115\) 14.1962i 1.32380i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 13.8564 1.27559
\(119\) − 6.58846i − 0.603963i
\(120\) 0 0
\(121\) 9.39230 0.853846
\(122\) 15.1962i 1.37579i
\(123\) 0 0
\(124\) − 9.46410i − 0.849901i
\(125\) − 12.1244i − 1.08444i
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 4.39230 0.383757 0.191879 0.981419i \(-0.438542\pi\)
0.191879 + 0.981419i \(0.438542\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −7.26795 −0.627855
\(135\) 0 0
\(136\) − 5.19615i − 0.445566i
\(137\) − 9.00000i − 0.768922i −0.923141 0.384461i \(-0.874387\pi\)
0.923141 0.384461i \(-0.125613\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.19615 0.185609
\(141\) 0 0
\(142\) −2.19615 −0.184297
\(143\) 0 0
\(144\) 0 0
\(145\) − 5.19615i − 0.431517i
\(146\) 12.1244 1.00342
\(147\) 0 0
\(148\) 3.00000i 0.246598i
\(149\) − 6.12436i − 0.501727i −0.968023 0.250863i \(-0.919286\pi\)
0.968023 0.250863i \(-0.0807145\pi\)
\(150\) 0 0
\(151\) − 10.7321i − 0.873362i −0.899616 0.436681i \(-0.856154\pi\)
0.899616 0.436681i \(-0.143846\pi\)
\(152\) 4.73205 0.383820
\(153\) 0 0
\(154\) 1.60770i 0.129552i
\(155\) 16.3923 1.31666
\(156\) 0 0
\(157\) 7.19615 0.574315 0.287158 0.957883i \(-0.407290\pi\)
0.287158 + 0.957883i \(0.407290\pi\)
\(158\) 8.39230i 0.667656i
\(159\) 0 0
\(160\) 1.73205 0.136931
\(161\) 10.3923i 0.819028i
\(162\) 0 0
\(163\) − 2.53590i − 0.198627i −0.995056 0.0993134i \(-0.968335\pi\)
0.995056 0.0993134i \(-0.0316646\pi\)
\(164\) 6.46410i 0.504762i
\(165\) 0 0
\(166\) 5.66025 0.439321
\(167\) − 9.46410i − 0.732354i −0.930545 0.366177i \(-0.880666\pi\)
0.930545 0.366177i \(-0.119334\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) 0 0
\(172\) −4.19615 −0.319954
\(173\) −4.39230 −0.333941 −0.166970 0.985962i \(-0.553398\pi\)
−0.166970 + 0.985962i \(0.553398\pi\)
\(174\) 0 0
\(175\) − 2.53590i − 0.191696i
\(176\) 1.26795i 0.0955753i
\(177\) 0 0
\(178\) −9.46410 −0.709364
\(179\) 2.19615 0.164148 0.0820741 0.996626i \(-0.473846\pi\)
0.0820741 + 0.996626i \(0.473846\pi\)
\(180\) 0 0
\(181\) −19.5885 −1.45600 −0.727999 0.685578i \(-0.759550\pi\)
−0.727999 + 0.685578i \(0.759550\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.19615i 0.604228i
\(185\) −5.19615 −0.382029
\(186\) 0 0
\(187\) 6.58846i 0.481796i
\(188\) 4.73205i 0.345120i
\(189\) 0 0
\(190\) 8.19615i 0.594611i
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 23.1962i 1.66970i 0.550481 + 0.834848i \(0.314444\pi\)
−0.550481 + 0.834848i \(0.685556\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −5.39230 −0.385165
\(197\) 6.92820i 0.493614i 0.969065 + 0.246807i \(0.0793814\pi\)
−0.969065 + 0.246807i \(0.920619\pi\)
\(198\) 0 0
\(199\) 22.5885 1.60125 0.800627 0.599164i \(-0.204500\pi\)
0.800627 + 0.599164i \(0.204500\pi\)
\(200\) − 2.00000i − 0.141421i
\(201\) 0 0
\(202\) 19.3923i 1.36444i
\(203\) − 3.80385i − 0.266978i
\(204\) 0 0
\(205\) −11.1962 −0.781973
\(206\) − 6.19615i − 0.431706i
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −24.3923 −1.67924 −0.839618 0.543178i \(-0.817221\pi\)
−0.839618 + 0.543178i \(0.817221\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) 2.19615i 0.150126i
\(215\) − 7.26795i − 0.495670i
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 4.39230 0.297484
\(219\) 0 0
\(220\) −2.19615 −0.148065
\(221\) 0 0
\(222\) 0 0
\(223\) − 5.07180i − 0.339633i −0.985476 0.169816i \(-0.945683\pi\)
0.985476 0.169816i \(-0.0543174\pi\)
\(224\) 1.26795 0.0847184
\(225\) 0 0
\(226\) − 0.803848i − 0.0534711i
\(227\) 20.1962i 1.34047i 0.742151 + 0.670233i \(0.233806\pi\)
−0.742151 + 0.670233i \(0.766194\pi\)
\(228\) 0 0
\(229\) 7.85641i 0.519166i 0.965721 + 0.259583i \(0.0835851\pi\)
−0.965721 + 0.259583i \(0.916415\pi\)
\(230\) −14.1962 −0.936067
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −8.19615 −0.534658
\(236\) 13.8564i 0.901975i
\(237\) 0 0
\(238\) 6.58846 0.427066
\(239\) − 6.58846i − 0.426172i −0.977033 0.213086i \(-0.931649\pi\)
0.977033 0.213086i \(-0.0683514\pi\)
\(240\) 0 0
\(241\) − 11.1962i − 0.721208i −0.932719 0.360604i \(-0.882571\pi\)
0.932719 0.360604i \(-0.117429\pi\)
\(242\) 9.39230i 0.603760i
\(243\) 0 0
\(244\) −15.1962 −0.972834
\(245\) − 9.33975i − 0.596694i
\(246\) 0 0
\(247\) 0 0
\(248\) 9.46410 0.600971
\(249\) 0 0
\(250\) 12.1244 0.766812
\(251\) 16.3923 1.03467 0.517337 0.855782i \(-0.326924\pi\)
0.517337 + 0.855782i \(0.326924\pi\)
\(252\) 0 0
\(253\) − 10.3923i − 0.653359i
\(254\) 4.00000i 0.250982i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.1962 1.44694 0.723468 0.690358i \(-0.242547\pi\)
0.723468 + 0.690358i \(0.242547\pi\)
\(258\) 0 0
\(259\) −3.80385 −0.236360
\(260\) 0 0
\(261\) 0 0
\(262\) 4.39230i 0.271357i
\(263\) −8.19615 −0.505396 −0.252698 0.967545i \(-0.581318\pi\)
−0.252698 + 0.967545i \(0.581318\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 6.00000i 0.367884i
\(267\) 0 0
\(268\) − 7.26795i − 0.443961i
\(269\) 7.60770 0.463849 0.231925 0.972734i \(-0.425498\pi\)
0.231925 + 0.972734i \(0.425498\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 5.19615 0.315063
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 2.53590i 0.152920i
\(276\) 0 0
\(277\) −4.80385 −0.288635 −0.144318 0.989531i \(-0.546099\pi\)
−0.144318 + 0.989531i \(0.546099\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 2.19615i 0.131245i
\(281\) 17.5359i 1.04610i 0.852301 + 0.523052i \(0.175207\pi\)
−0.852301 + 0.523052i \(0.824793\pi\)
\(282\) 0 0
\(283\) 19.8038 1.17722 0.588608 0.808418i \(-0.299676\pi\)
0.588608 + 0.808418i \(0.299676\pi\)
\(284\) − 2.19615i − 0.130318i
\(285\) 0 0
\(286\) 0 0
\(287\) −8.19615 −0.483804
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) 5.19615 0.305129
\(291\) 0 0
\(292\) 12.1244i 0.709524i
\(293\) 2.66025i 0.155414i 0.996976 + 0.0777069i \(0.0247598\pi\)
−0.996976 + 0.0777069i \(0.975240\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −3.00000 −0.174371
\(297\) 0 0
\(298\) 6.12436 0.354774
\(299\) 0 0
\(300\) 0 0
\(301\) − 5.32051i − 0.306669i
\(302\) 10.7321 0.617560
\(303\) 0 0
\(304\) 4.73205i 0.271402i
\(305\) − 26.3205i − 1.50711i
\(306\) 0 0
\(307\) − 7.26795i − 0.414804i −0.978256 0.207402i \(-0.933499\pi\)
0.978256 0.207402i \(-0.0665008\pi\)
\(308\) −1.60770 −0.0916069
\(309\) 0 0
\(310\) 16.3923i 0.931020i
\(311\) −8.19615 −0.464761 −0.232381 0.972625i \(-0.574651\pi\)
−0.232381 + 0.972625i \(0.574651\pi\)
\(312\) 0 0
\(313\) −3.60770 −0.203919 −0.101959 0.994789i \(-0.532511\pi\)
−0.101959 + 0.994789i \(0.532511\pi\)
\(314\) 7.19615i 0.406102i
\(315\) 0 0
\(316\) −8.39230 −0.472104
\(317\) 18.1244i 1.01797i 0.860777 + 0.508983i \(0.169978\pi\)
−0.860777 + 0.508983i \(0.830022\pi\)
\(318\) 0 0
\(319\) 3.80385i 0.212975i
\(320\) 1.73205i 0.0968246i
\(321\) 0 0
\(322\) −10.3923 −0.579141
\(323\) 24.5885i 1.36814i
\(324\) 0 0
\(325\) 0 0
\(326\) 2.53590 0.140450
\(327\) 0 0
\(328\) −6.46410 −0.356920
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 12.0000i 0.659580i 0.944054 + 0.329790i \(0.106978\pi\)
−0.944054 + 0.329790i \(0.893022\pi\)
\(332\) 5.66025i 0.310647i
\(333\) 0 0
\(334\) 9.46410 0.517853
\(335\) 12.5885 0.687781
\(336\) 0 0
\(337\) 31.0000 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9.00000i 0.488094i
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) − 15.7128i − 0.848412i
\(344\) − 4.19615i − 0.226241i
\(345\) 0 0
\(346\) − 4.39230i − 0.236132i
\(347\) −18.5885 −0.997881 −0.498940 0.866636i \(-0.666277\pi\)
−0.498940 + 0.866636i \(0.666277\pi\)
\(348\) 0 0
\(349\) − 9.46410i − 0.506602i −0.967388 0.253301i \(-0.918484\pi\)
0.967388 0.253301i \(-0.0815163\pi\)
\(350\) 2.53590 0.135549
\(351\) 0 0
\(352\) −1.26795 −0.0675819
\(353\) 35.7846i 1.90462i 0.305128 + 0.952311i \(0.401301\pi\)
−0.305128 + 0.952311i \(0.598699\pi\)
\(354\) 0 0
\(355\) 3.80385 0.201887
\(356\) − 9.46410i − 0.501596i
\(357\) 0 0
\(358\) 2.19615i 0.116070i
\(359\) − 16.0526i − 0.847222i −0.905844 0.423611i \(-0.860762\pi\)
0.905844 0.423611i \(-0.139238\pi\)
\(360\) 0 0
\(361\) −3.39230 −0.178542
\(362\) − 19.5885i − 1.02955i
\(363\) 0 0
\(364\) 0 0
\(365\) −21.0000 −1.09919
\(366\) 0 0
\(367\) −13.8038 −0.720555 −0.360277 0.932845i \(-0.617318\pi\)
−0.360277 + 0.932845i \(0.617318\pi\)
\(368\) −8.19615 −0.427254
\(369\) 0 0
\(370\) − 5.19615i − 0.270135i
\(371\) 3.80385i 0.197486i
\(372\) 0 0
\(373\) −27.9808 −1.44879 −0.724394 0.689386i \(-0.757881\pi\)
−0.724394 + 0.689386i \(0.757881\pi\)
\(374\) −6.58846 −0.340681
\(375\) 0 0
\(376\) −4.73205 −0.244037
\(377\) 0 0
\(378\) 0 0
\(379\) 30.2487i 1.55377i 0.629641 + 0.776886i \(0.283202\pi\)
−0.629641 + 0.776886i \(0.716798\pi\)
\(380\) −8.19615 −0.420454
\(381\) 0 0
\(382\) 20.7846i 1.06343i
\(383\) 23.3205i 1.19162i 0.803125 + 0.595811i \(0.203169\pi\)
−0.803125 + 0.595811i \(0.796831\pi\)
\(384\) 0 0
\(385\) − 2.78461i − 0.141917i
\(386\) −23.1962 −1.18065
\(387\) 0 0
\(388\) − 6.00000i − 0.304604i
\(389\) 7.39230 0.374805 0.187402 0.982283i \(-0.439993\pi\)
0.187402 + 0.982283i \(0.439993\pi\)
\(390\) 0 0
\(391\) −42.5885 −2.15379
\(392\) − 5.39230i − 0.272353i
\(393\) 0 0
\(394\) −6.92820 −0.349038
\(395\) − 14.5359i − 0.731380i
\(396\) 0 0
\(397\) − 4.39230i − 0.220443i −0.993907 0.110222i \(-0.964844\pi\)
0.993907 0.110222i \(-0.0351561\pi\)
\(398\) 22.5885i 1.13226i
\(399\) 0 0
\(400\) 2.00000 0.100000
\(401\) 21.0000i 1.04869i 0.851506 + 0.524345i \(0.175690\pi\)
−0.851506 + 0.524345i \(0.824310\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −19.3923 −0.964803
\(405\) 0 0
\(406\) 3.80385 0.188782
\(407\) 3.80385 0.188550
\(408\) 0 0
\(409\) − 20.6603i − 1.02158i −0.859704 0.510792i \(-0.829352\pi\)
0.859704 0.510792i \(-0.170648\pi\)
\(410\) − 11.1962i − 0.552939i
\(411\) 0 0
\(412\) 6.19615 0.305263
\(413\) −17.5692 −0.864525
\(414\) 0 0
\(415\) −9.80385 −0.481252
\(416\) 0 0
\(417\) 0 0
\(418\) − 6.00000i − 0.293470i
\(419\) 4.39230 0.214578 0.107289 0.994228i \(-0.465783\pi\)
0.107289 + 0.994228i \(0.465783\pi\)
\(420\) 0 0
\(421\) − 6.46410i − 0.315041i −0.987516 0.157521i \(-0.949650\pi\)
0.987516 0.157521i \(-0.0503500\pi\)
\(422\) − 24.3923i − 1.18740i
\(423\) 0 0
\(424\) 3.00000i 0.145693i
\(425\) 10.3923 0.504101
\(426\) 0 0
\(427\) − 19.2679i − 0.932441i
\(428\) −2.19615 −0.106155
\(429\) 0 0
\(430\) 7.26795 0.350492
\(431\) − 38.1962i − 1.83984i −0.392101 0.919922i \(-0.628252\pi\)
0.392101 0.919922i \(-0.371748\pi\)
\(432\) 0 0
\(433\) −7.78461 −0.374104 −0.187052 0.982350i \(-0.559893\pi\)
−0.187052 + 0.982350i \(0.559893\pi\)
\(434\) 12.0000i 0.576018i
\(435\) 0 0
\(436\) 4.39230i 0.210353i
\(437\) − 38.7846i − 1.85532i
\(438\) 0 0
\(439\) 14.5885 0.696269 0.348135 0.937445i \(-0.386815\pi\)
0.348135 + 0.937445i \(0.386815\pi\)
\(440\) − 2.19615i − 0.104697i
\(441\) 0 0
\(442\) 0 0
\(443\) −16.3923 −0.778822 −0.389411 0.921064i \(-0.627321\pi\)
−0.389411 + 0.921064i \(0.627321\pi\)
\(444\) 0 0
\(445\) 16.3923 0.777070
\(446\) 5.07180 0.240157
\(447\) 0 0
\(448\) 1.26795i 0.0599050i
\(449\) 26.5359i 1.25231i 0.779700 + 0.626153i \(0.215372\pi\)
−0.779700 + 0.626153i \(0.784628\pi\)
\(450\) 0 0
\(451\) 8.19615 0.385942
\(452\) 0.803848 0.0378098
\(453\) 0 0
\(454\) −20.1962 −0.947852
\(455\) 0 0
\(456\) 0 0
\(457\) − 31.9808i − 1.49600i −0.663700 0.747998i \(-0.731015\pi\)
0.663700 0.747998i \(-0.268985\pi\)
\(458\) −7.85641 −0.367106
\(459\) 0 0
\(460\) − 14.1962i − 0.661899i
\(461\) − 31.9808i − 1.48949i −0.667348 0.744746i \(-0.732570\pi\)
0.667348 0.744746i \(-0.267430\pi\)
\(462\) 0 0
\(463\) 15.8038i 0.734467i 0.930129 + 0.367234i \(0.119695\pi\)
−0.930129 + 0.367234i \(0.880305\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) − 18.0000i − 0.833834i
\(467\) 5.41154 0.250416 0.125208 0.992130i \(-0.460040\pi\)
0.125208 + 0.992130i \(0.460040\pi\)
\(468\) 0 0
\(469\) 9.21539 0.425527
\(470\) − 8.19615i − 0.378060i
\(471\) 0 0
\(472\) −13.8564 −0.637793
\(473\) 5.32051i 0.244637i
\(474\) 0 0
\(475\) 9.46410i 0.434243i
\(476\) 6.58846i 0.301981i
\(477\) 0 0
\(478\) 6.58846 0.301349
\(479\) − 0.679492i − 0.0310468i −0.999880 0.0155234i \(-0.995059\pi\)
0.999880 0.0155234i \(-0.00494145\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 11.1962 0.509971
\(483\) 0 0
\(484\) −9.39230 −0.426923
\(485\) 10.3923 0.471890
\(486\) 0 0
\(487\) − 15.1244i − 0.685350i −0.939454 0.342675i \(-0.888667\pi\)
0.939454 0.342675i \(-0.111333\pi\)
\(488\) − 15.1962i − 0.687897i
\(489\) 0 0
\(490\) 9.33975 0.421927
\(491\) −30.5885 −1.38044 −0.690219 0.723601i \(-0.742486\pi\)
−0.690219 + 0.723601i \(0.742486\pi\)
\(492\) 0 0
\(493\) 15.5885 0.702069
\(494\) 0 0
\(495\) 0 0
\(496\) 9.46410i 0.424951i
\(497\) 2.78461 0.124907
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 12.1244i 0.542218i
\(501\) 0 0
\(502\) 16.3923i 0.731624i
\(503\) −12.5885 −0.561292 −0.280646 0.959811i \(-0.590549\pi\)
−0.280646 + 0.959811i \(0.590549\pi\)
\(504\) 0 0
\(505\) − 33.5885i − 1.49467i
\(506\) 10.3923 0.461994
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) − 26.6603i − 1.18169i −0.806783 0.590847i \(-0.798793\pi\)
0.806783 0.590847i \(-0.201207\pi\)
\(510\) 0 0
\(511\) −15.3731 −0.680064
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 23.1962i 1.02314i
\(515\) 10.7321i 0.472911i
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) − 3.80385i − 0.167131i
\(519\) 0 0
\(520\) 0 0
\(521\) 29.1962 1.27911 0.639553 0.768747i \(-0.279119\pi\)
0.639553 + 0.768747i \(0.279119\pi\)
\(522\) 0 0
\(523\) 32.5885 1.42499 0.712497 0.701675i \(-0.247564\pi\)
0.712497 + 0.701675i \(0.247564\pi\)
\(524\) −4.39230 −0.191879
\(525\) 0 0
\(526\) − 8.19615i − 0.357369i
\(527\) 49.1769i 2.14218i
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) −5.19615 −0.225706
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) − 3.80385i − 0.164455i
\(536\) 7.26795 0.313928
\(537\) 0 0
\(538\) 7.60770i 0.327991i
\(539\) 6.83717i 0.294498i
\(540\) 0 0
\(541\) 10.8564i 0.466753i 0.972386 + 0.233377i \(0.0749775\pi\)
−0.972386 + 0.233377i \(0.925022\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 5.19615i 0.222783i
\(545\) −7.60770 −0.325878
\(546\) 0 0
\(547\) 4.19615 0.179415 0.0897073 0.995968i \(-0.471407\pi\)
0.0897073 + 0.995968i \(0.471407\pi\)
\(548\) 9.00000i 0.384461i
\(549\) 0 0
\(550\) −2.53590 −0.108131
\(551\) 14.1962i 0.604776i
\(552\) 0 0
\(553\) − 10.6410i − 0.452502i
\(554\) − 4.80385i − 0.204096i
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 25.7321i 1.09030i 0.838338 + 0.545151i \(0.183528\pi\)
−0.838338 + 0.545151i \(0.816472\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.19615 −0.0928044
\(561\) 0 0
\(562\) −17.5359 −0.739707
\(563\) −32.7846 −1.38171 −0.690853 0.722995i \(-0.742765\pi\)
−0.690853 + 0.722995i \(0.742765\pi\)
\(564\) 0 0
\(565\) 1.39230i 0.0585747i
\(566\) 19.8038i 0.832418i
\(567\) 0 0
\(568\) 2.19615 0.0921485
\(569\) 8.78461 0.368270 0.184135 0.982901i \(-0.441052\pi\)
0.184135 + 0.982901i \(0.441052\pi\)
\(570\) 0 0
\(571\) 24.1962 1.01258 0.506289 0.862364i \(-0.331017\pi\)
0.506289 + 0.862364i \(0.331017\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 8.19615i − 0.342101i
\(575\) −16.3923 −0.683606
\(576\) 0 0
\(577\) − 19.7321i − 0.821456i −0.911758 0.410728i \(-0.865275\pi\)
0.911758 0.410728i \(-0.134725\pi\)
\(578\) 10.0000i 0.415945i
\(579\) 0 0
\(580\) 5.19615i 0.215758i
\(581\) −7.17691 −0.297749
\(582\) 0 0
\(583\) − 3.80385i − 0.157539i
\(584\) −12.1244 −0.501709
\(585\) 0 0
\(586\) −2.66025 −0.109894
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −44.7846 −1.84532
\(590\) − 24.0000i − 0.988064i
\(591\) 0 0
\(592\) − 3.00000i − 0.123299i
\(593\) − 19.1436i − 0.786133i −0.919510 0.393067i \(-0.871414\pi\)
0.919510 0.393067i \(-0.128586\pi\)
\(594\) 0 0
\(595\) −11.4115 −0.467828
\(596\) 6.12436i 0.250863i
\(597\) 0 0
\(598\) 0 0
\(599\) −16.3923 −0.669771 −0.334886 0.942259i \(-0.608698\pi\)
−0.334886 + 0.942259i \(0.608698\pi\)
\(600\) 0 0
\(601\) −19.7846 −0.807031 −0.403516 0.914973i \(-0.632212\pi\)
−0.403516 + 0.914973i \(0.632212\pi\)
\(602\) 5.32051 0.216848
\(603\) 0 0
\(604\) 10.7321i 0.436681i
\(605\) − 16.2679i − 0.661386i
\(606\) 0 0
\(607\) −7.21539 −0.292864 −0.146432 0.989221i \(-0.546779\pi\)
−0.146432 + 0.989221i \(0.546779\pi\)
\(608\) −4.73205 −0.191910
\(609\) 0 0
\(610\) 26.3205 1.06569
\(611\) 0 0
\(612\) 0 0
\(613\) − 13.1436i − 0.530865i −0.964129 0.265432i \(-0.914485\pi\)
0.964129 0.265432i \(-0.0855147\pi\)
\(614\) 7.26795 0.293311
\(615\) 0 0
\(616\) − 1.60770i − 0.0647759i
\(617\) − 31.3923i − 1.26381i −0.775047 0.631903i \(-0.782274\pi\)
0.775047 0.631903i \(-0.217726\pi\)
\(618\) 0 0
\(619\) 28.3923i 1.14118i 0.821234 + 0.570592i \(0.193286\pi\)
−0.821234 + 0.570592i \(0.806714\pi\)
\(620\) −16.3923 −0.658331
\(621\) 0 0
\(622\) − 8.19615i − 0.328636i
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) − 3.60770i − 0.144192i
\(627\) 0 0
\(628\) −7.19615 −0.287158
\(629\) − 15.5885i − 0.621552i
\(630\) 0 0
\(631\) 1.85641i 0.0739024i 0.999317 + 0.0369512i \(0.0117646\pi\)
−0.999317 + 0.0369512i \(0.988235\pi\)
\(632\) − 8.39230i − 0.333828i
\(633\) 0 0
\(634\) −18.1244 −0.719810
\(635\) − 6.92820i − 0.274937i
\(636\) 0 0
\(637\) 0 0
\(638\) −3.80385 −0.150596
\(639\) 0 0
\(640\) −1.73205 −0.0684653
\(641\) −41.1962 −1.62715 −0.813575 0.581460i \(-0.802482\pi\)
−0.813575 + 0.581460i \(0.802482\pi\)
\(642\) 0 0
\(643\) 27.7128i 1.09289i 0.837496 + 0.546443i \(0.184019\pi\)
−0.837496 + 0.546443i \(0.815981\pi\)
\(644\) − 10.3923i − 0.409514i
\(645\) 0 0
\(646\) −24.5885 −0.967420
\(647\) −49.1769 −1.93334 −0.966672 0.256018i \(-0.917589\pi\)
−0.966672 + 0.256018i \(0.917589\pi\)
\(648\) 0 0
\(649\) 17.5692 0.689652
\(650\) 0 0
\(651\) 0 0
\(652\) 2.53590i 0.0993134i
\(653\) −13.1769 −0.515653 −0.257826 0.966191i \(-0.583006\pi\)
−0.257826 + 0.966191i \(0.583006\pi\)
\(654\) 0 0
\(655\) − 7.60770i − 0.297257i
\(656\) − 6.46410i − 0.252381i
\(657\) 0 0
\(658\) − 6.00000i − 0.233904i
\(659\) 37.1769 1.44821 0.724103 0.689691i \(-0.242254\pi\)
0.724103 + 0.689691i \(0.242254\pi\)
\(660\) 0 0
\(661\) − 9.00000i − 0.350059i −0.984563 0.175030i \(-0.943998\pi\)
0.984563 0.175030i \(-0.0560022\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −5.66025 −0.219660
\(665\) − 10.3923i − 0.402996i
\(666\) 0 0
\(667\) −24.5885 −0.952069
\(668\) 9.46410i 0.366177i
\(669\) 0 0
\(670\) 12.5885i 0.486335i
\(671\) 19.2679i 0.743831i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 31.0000i 1.19408i
\(675\) 0 0
\(676\) 0 0
\(677\) 16.3923 0.630007 0.315004 0.949090i \(-0.397994\pi\)
0.315004 + 0.949090i \(0.397994\pi\)
\(678\) 0 0
\(679\) 7.60770 0.291957
\(680\) −9.00000 −0.345134
\(681\) 0 0
\(682\) − 12.0000i − 0.459504i
\(683\) − 27.7128i − 1.06040i −0.847872 0.530201i \(-0.822117\pi\)
0.847872 0.530201i \(-0.177883\pi\)
\(684\) 0 0
\(685\) −15.5885 −0.595604
\(686\) 15.7128 0.599918
\(687\) 0 0
\(688\) 4.19615 0.159977
\(689\) 0 0
\(690\) 0 0
\(691\) 25.5167i 0.970700i 0.874320 + 0.485350i \(0.161308\pi\)
−0.874320 + 0.485350i \(0.838692\pi\)
\(692\) 4.39230 0.166970
\(693\) 0 0
\(694\) − 18.5885i − 0.705608i
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) − 33.5885i − 1.27225i
\(698\) 9.46410 0.358222
\(699\) 0 0
\(700\) 2.53590i 0.0958479i
\(701\) −16.3923 −0.619129 −0.309564 0.950878i \(-0.600183\pi\)
−0.309564 + 0.950878i \(0.600183\pi\)
\(702\) 0 0
\(703\) 14.1962 0.535418
\(704\) − 1.26795i − 0.0477876i
\(705\) 0 0
\(706\) −35.7846 −1.34677
\(707\) − 24.5885i − 0.924744i
\(708\) 0 0
\(709\) 45.2487i 1.69935i 0.527306 + 0.849676i \(0.323202\pi\)
−0.527306 + 0.849676i \(0.676798\pi\)
\(710\) 3.80385i 0.142756i
\(711\) 0 0
\(712\) 9.46410 0.354682
\(713\) − 77.5692i − 2.90499i
\(714\) 0 0
\(715\) 0 0
\(716\) −2.19615 −0.0820741
\(717\) 0 0
\(718\) 16.0526 0.599076
\(719\) 31.6077 1.17877 0.589384 0.807853i \(-0.299370\pi\)
0.589384 + 0.807853i \(0.299370\pi\)
\(720\) 0 0
\(721\) 7.85641i 0.292588i
\(722\) − 3.39230i − 0.126249i
\(723\) 0 0
\(724\) 19.5885 0.727999
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 13.8038 0.511956 0.255978 0.966683i \(-0.417602\pi\)
0.255978 + 0.966683i \(0.417602\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 21.0000i − 0.777245i
\(731\) 21.8038 0.806444
\(732\) 0 0
\(733\) 20.3205i 0.750555i 0.926912 + 0.375278i \(0.122453\pi\)
−0.926912 + 0.375278i \(0.877547\pi\)
\(734\) − 13.8038i − 0.509509i
\(735\) 0 0
\(736\) − 8.19615i − 0.302114i
\(737\) −9.21539 −0.339453
\(738\) 0 0
\(739\) 5.07180i 0.186569i 0.995639 + 0.0932845i \(0.0297366\pi\)
−0.995639 + 0.0932845i \(0.970263\pi\)
\(740\) 5.19615 0.191014
\(741\) 0 0
\(742\) −3.80385 −0.139644
\(743\) 16.3923i 0.601375i 0.953723 + 0.300688i \(0.0972162\pi\)
−0.953723 + 0.300688i \(0.902784\pi\)
\(744\) 0 0
\(745\) −10.6077 −0.388636
\(746\) − 27.9808i − 1.02445i
\(747\) 0 0
\(748\) − 6.58846i − 0.240898i
\(749\) − 2.78461i − 0.101747i
\(750\) 0 0
\(751\) −26.9808 −0.984542 −0.492271 0.870442i \(-0.663833\pi\)
−0.492271 + 0.870442i \(0.663833\pi\)
\(752\) − 4.73205i − 0.172560i
\(753\) 0 0
\(754\) 0 0
\(755\) −18.5885 −0.676503
\(756\) 0 0
\(757\) −22.7846 −0.828121 −0.414060 0.910249i \(-0.635890\pi\)
−0.414060 + 0.910249i \(0.635890\pi\)
\(758\) −30.2487 −1.09868
\(759\) 0 0
\(760\) − 8.19615i − 0.297306i
\(761\) 16.3923i 0.594221i 0.954843 + 0.297110i \(0.0960229\pi\)
−0.954843 + 0.297110i \(0.903977\pi\)
\(762\) 0 0
\(763\) −5.56922 −0.201619
\(764\) −20.7846 −0.751961
\(765\) 0 0
\(766\) −23.3205 −0.842604
\(767\) 0 0
\(768\) 0 0
\(769\) − 21.7128i − 0.782984i −0.920181 0.391492i \(-0.871959\pi\)
0.920181 0.391492i \(-0.128041\pi\)
\(770\) 2.78461 0.100350
\(771\) 0 0
\(772\) − 23.1962i − 0.834848i
\(773\) 9.21539i 0.331455i 0.986172 + 0.165727i \(0.0529971\pi\)
−0.986172 + 0.165727i \(0.947003\pi\)
\(774\) 0 0
\(775\) 18.9282i 0.679921i
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 7.39230i 0.265027i
\(779\) 30.5885 1.09595
\(780\) 0 0
\(781\) −2.78461 −0.0996412
\(782\) − 42.5885i − 1.52296i
\(783\) 0 0
\(784\) 5.39230 0.192582
\(785\) − 12.4641i − 0.444863i
\(786\) 0 0
\(787\) 21.4641i 0.765113i 0.923932 + 0.382556i \(0.124956\pi\)
−0.923932 + 0.382556i \(0.875044\pi\)
\(788\) − 6.92820i − 0.246807i
\(789\) 0 0
\(790\) 14.5359 0.517164
\(791\) 1.01924i 0.0362399i
\(792\) 0 0
\(793\) 0 0
\(794\) 4.39230 0.155877
\(795\) 0 0
\(796\) −22.5885 −0.800627
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) − 24.5885i − 0.869877i
\(800\) 2.00000i 0.0707107i
\(801\) 0 0
\(802\) −21.0000 −0.741536
\(803\) 15.3731 0.542504
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 0 0
\(807\) 0 0
\(808\) − 19.3923i − 0.682219i
\(809\) 36.8038 1.29395 0.646977 0.762509i \(-0.276033\pi\)
0.646977 + 0.762509i \(0.276033\pi\)
\(810\) 0 0
\(811\) − 16.3923i − 0.575612i −0.957689 0.287806i \(-0.907074\pi\)
0.957689 0.287806i \(-0.0929258\pi\)
\(812\) 3.80385i 0.133489i
\(813\) 0 0
\(814\) 3.80385i 0.133325i
\(815\) −4.39230 −0.153856
\(816\) 0 0
\(817\) 19.8564i 0.694688i
\(818\) 20.6603 0.722369
\(819\) 0 0
\(820\) 11.1962 0.390987
\(821\) − 28.6410i − 0.999578i −0.866147 0.499789i \(-0.833411\pi\)
0.866147 0.499789i \(-0.166589\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 6.19615i 0.215853i
\(825\) 0 0
\(826\) − 17.5692i − 0.611311i
\(827\) 44.1051i 1.53369i 0.641835 + 0.766843i \(0.278173\pi\)
−0.641835 + 0.766843i \(0.721827\pi\)
\(828\) 0 0
\(829\) −39.9808 −1.38859 −0.694295 0.719691i \(-0.744284\pi\)
−0.694295 + 0.719691i \(0.744284\pi\)
\(830\) − 9.80385i − 0.340297i
\(831\) 0 0
\(832\) 0 0
\(833\) 28.0192 0.970809
\(834\) 0 0
\(835\) −16.3923 −0.567279
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 4.39230i 0.151730i
\(839\) − 12.0000i − 0.414286i −0.978311 0.207143i \(-0.933583\pi\)
0.978311 0.207143i \(-0.0664165\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 6.46410 0.222768
\(843\) 0 0
\(844\) 24.3923 0.839618
\(845\) 0 0
\(846\) 0 0
\(847\) − 11.9090i − 0.409197i
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) 10.3923i 0.356453i
\(851\) 24.5885i 0.842881i
\(852\) 0 0
\(853\) − 9.00000i − 0.308154i −0.988059 0.154077i \(-0.950760\pi\)
0.988059 0.154077i \(-0.0492404\pi\)
\(854\) 19.2679 0.659336
\(855\) 0 0
\(856\) − 2.19615i − 0.0750629i
\(857\) 18.3731 0.627612 0.313806 0.949487i \(-0.398396\pi\)
0.313806 + 0.949487i \(0.398396\pi\)
\(858\) 0 0
\(859\) −20.5885 −0.702469 −0.351235 0.936288i \(-0.614238\pi\)
−0.351235 + 0.936288i \(0.614238\pi\)
\(860\) 7.26795i 0.247835i
\(861\) 0 0
\(862\) 38.1962 1.30097
\(863\) − 49.5167i − 1.68557i −0.538253 0.842783i \(-0.680915\pi\)
0.538253 0.842783i \(-0.319085\pi\)
\(864\) 0 0
\(865\) 7.60770i 0.258669i
\(866\) − 7.78461i − 0.264532i
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 10.6410i 0.360972i
\(870\) 0 0
\(871\) 0 0
\(872\) −4.39230 −0.148742
\(873\) 0 0
\(874\) 38.7846 1.31191
\(875\) −15.3731 −0.519705
\(876\) 0 0
\(877\) 22.6077i 0.763408i 0.924285 + 0.381704i \(0.124663\pi\)
−0.924285 + 0.381704i \(0.875337\pi\)
\(878\) 14.5885i 0.492337i
\(879\) 0 0
\(880\) 2.19615 0.0740323
\(881\) −13.9808 −0.471024 −0.235512 0.971871i \(-0.575677\pi\)
−0.235512 + 0.971871i \(0.575677\pi\)
\(882\) 0 0
\(883\) −16.7846 −0.564847 −0.282424 0.959290i \(-0.591138\pi\)
−0.282424 + 0.959290i \(0.591138\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 16.3923i − 0.550710i
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) − 5.07180i − 0.170103i
\(890\) 16.3923i 0.549471i
\(891\) 0 0
\(892\) 5.07180i 0.169816i
\(893\) 22.3923 0.749330
\(894\) 0 0
\(895\) − 3.80385i − 0.127149i
\(896\) −1.26795 −0.0423592
\(897\) 0 0
\(898\) −26.5359 −0.885514
\(899\) 28.3923i 0.946936i
\(900\) 0 0
\(901\) −15.5885 −0.519327
\(902\) 8.19615i 0.272902i
\(903\) 0 0
\(904\) 0.803848i 0.0267356i
\(905\) 33.9282i 1.12781i
\(906\) 0 0
\(907\) −21.1769 −0.703168 −0.351584 0.936156i \(-0.614357\pi\)
−0.351584 + 0.936156i \(0.614357\pi\)
\(908\) − 20.1962i − 0.670233i
\(909\) 0 0
\(910\) 0 0
\(911\) 25.1769 0.834148 0.417074 0.908872i \(-0.363056\pi\)
0.417074 + 0.908872i \(0.363056\pi\)
\(912\) 0 0
\(913\) 7.17691 0.237521
\(914\) 31.9808 1.05783
\(915\) 0 0
\(916\) − 7.85641i − 0.259583i
\(917\) − 5.56922i − 0.183912i
\(918\) 0 0
\(919\) 11.6077 0.382903 0.191451 0.981502i \(-0.438681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(920\) 14.1962 0.468033
\(921\) 0 0
\(922\) 31.9808 1.05323
\(923\) 0 0
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) −15.8038 −0.519347
\(927\) 0 0
\(928\) 3.00000i 0.0984798i
\(929\) 55.3923i 1.81736i 0.417491 + 0.908681i \(0.362910\pi\)
−0.417491 + 0.908681i \(0.637090\pi\)
\(930\) 0 0
\(931\) 25.5167i 0.836275i
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 5.41154i 0.177071i
\(935\) 11.4115 0.373197
\(936\) 0 0
\(937\) −15.3923 −0.502845 −0.251422 0.967877i \(-0.580898\pi\)
−0.251422 + 0.967877i \(0.580898\pi\)
\(938\) 9.21539i 0.300893i
\(939\) 0 0
\(940\) 8.19615 0.267329
\(941\) 38.7846i 1.26434i 0.774829 + 0.632171i \(0.217836\pi\)
−0.774829 + 0.632171i \(0.782164\pi\)
\(942\) 0 0
\(943\) 52.9808i 1.72529i
\(944\) − 13.8564i − 0.450988i
\(945\) 0 0
\(946\) −5.32051 −0.172985
\(947\) − 29.0718i − 0.944706i −0.881409 0.472353i \(-0.843405\pi\)
0.881409 0.472353i \(-0.156595\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −9.46410 −0.307056
\(951\) 0 0
\(952\) −6.58846 −0.213533
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) − 36.0000i − 1.16493i
\(956\) 6.58846i 0.213086i
\(957\) 0 0
\(958\) 0.679492 0.0219534
\(959\) −11.4115 −0.368498
\(960\) 0 0
\(961\) −58.5692 −1.88933
\(962\) 0 0
\(963\) 0 0
\(964\) 11.1962i 0.360604i
\(965\) 40.1769 1.29334
\(966\) 0 0
\(967\) − 39.1244i − 1.25815i −0.777343 0.629077i \(-0.783433\pi\)
0.777343 0.629077i \(-0.216567\pi\)
\(968\) − 9.39230i − 0.301880i
\(969\) 0 0
\(970\) 10.3923i 0.333677i
\(971\) −49.1769 −1.57816 −0.789081 0.614289i \(-0.789443\pi\)
−0.789081 + 0.614289i \(0.789443\pi\)
\(972\) 0 0
\(973\) 5.07180i 0.162594i
\(974\) 15.1244 0.484616
\(975\) 0 0
\(976\) 15.1962 0.486417
\(977\) − 10.8564i − 0.347327i −0.984805 0.173664i \(-0.944439\pi\)
0.984805 0.173664i \(-0.0555605\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 9.33975i 0.298347i
\(981\) 0 0
\(982\) − 30.5885i − 0.976117i
\(983\) 20.7846i 0.662926i 0.943468 + 0.331463i \(0.107542\pi\)
−0.943468 + 0.331463i \(0.892458\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 15.5885i 0.496438i
\(987\) 0 0
\(988\) 0 0
\(989\) −34.3923 −1.09361
\(990\) 0 0
\(991\) −43.3731 −1.37779 −0.688895 0.724861i \(-0.741904\pi\)
−0.688895 + 0.724861i \(0.741904\pi\)
\(992\) −9.46410 −0.300486
\(993\) 0 0
\(994\) 2.78461i 0.0883225i
\(995\) − 39.1244i − 1.24033i
\(996\) 0 0
\(997\) 2.80385 0.0887987 0.0443994 0.999014i \(-0.485863\pi\)
0.0443994 + 0.999014i \(0.485863\pi\)
\(998\) 0 0
\(999\) 0 0
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 3042.2.b.l.1351.3 4
3.2 odd 2 1014.2.b.d.337.2 4
13.5 odd 4 3042.2.a.v.1.1 2
13.8 odd 4 3042.2.a.s.1.2 2
13.9 even 3 234.2.l.a.127.2 4
13.10 even 6 234.2.l.a.199.2 4
13.12 even 2 inner 3042.2.b.l.1351.2 4
39.2 even 12 1014.2.e.j.529.2 4
39.5 even 4 1014.2.a.h.1.2 2
39.8 even 4 1014.2.a.j.1.1 2
39.11 even 12 1014.2.e.h.529.1 4
39.17 odd 6 1014.2.i.f.361.2 4
39.20 even 12 1014.2.e.h.991.1 4
39.23 odd 6 78.2.i.b.43.1 4
39.29 odd 6 1014.2.i.f.823.2 4
39.32 even 12 1014.2.e.j.991.2 4
39.35 odd 6 78.2.i.b.49.1 yes 4
39.38 odd 2 1014.2.b.d.337.3 4
52.23 odd 6 1872.2.by.k.433.1 4
52.35 odd 6 1872.2.by.k.1297.1 4
156.23 even 6 624.2.bv.d.433.1 4
156.35 even 6 624.2.bv.d.49.1 4
156.47 odd 4 8112.2.a.bx.1.1 2
156.83 odd 4 8112.2.a.bq.1.2 2
195.23 even 12 1950.2.y.a.199.2 4
195.62 even 12 1950.2.y.h.199.1 4
195.74 odd 6 1950.2.bc.c.751.2 4
195.113 even 12 1950.2.y.h.49.1 4
195.152 even 12 1950.2.y.a.49.2 4
195.179 odd 6 1950.2.bc.c.901.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.b.43.1 4 39.23 odd 6
78.2.i.b.49.1 yes 4 39.35 odd 6
234.2.l.a.127.2 4 13.9 even 3
234.2.l.a.199.2 4 13.10 even 6
624.2.bv.d.49.1 4 156.35 even 6
624.2.bv.d.433.1 4 156.23 even 6
1014.2.a.h.1.2 2 39.5 even 4
1014.2.a.j.1.1 2 39.8 even 4
1014.2.b.d.337.2 4 3.2 odd 2
1014.2.b.d.337.3 4 39.38 odd 2
1014.2.e.h.529.1 4 39.11 even 12
1014.2.e.h.991.1 4 39.20 even 12
1014.2.e.j.529.2 4 39.2 even 12
1014.2.e.j.991.2 4 39.32 even 12
1014.2.i.f.361.2 4 39.17 odd 6
1014.2.i.f.823.2 4 39.29 odd 6
1872.2.by.k.433.1 4 52.23 odd 6
1872.2.by.k.1297.1 4 52.35 odd 6
1950.2.y.a.49.2 4 195.152 even 12
1950.2.y.a.199.2 4 195.23 even 12
1950.2.y.h.49.1 4 195.113 even 12
1950.2.y.h.199.1 4 195.62 even 12
1950.2.bc.c.751.2 4 195.74 odd 6
1950.2.bc.c.901.2 4 195.179 odd 6
3042.2.a.s.1.2 2 13.8 odd 4
3042.2.a.v.1.1 2 13.5 odd 4
3042.2.b.l.1351.2 4 13.12 even 2 inner
3042.2.b.l.1351.3 4 1.1 even 1 trivial
8112.2.a.bq.1.2 2 156.83 odd 4
8112.2.a.bx.1.1 2 156.47 odd 4