Properties

Label 2940.2.bb.a.949.1
Level $2940$
Weight $2$
Character 2940.949
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
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] = [2940,2,Mod(949,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.bb (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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2940.949
Dual form 2940.2.bb.a.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 - 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(-1.86603 - 1.23205i) q^{5} +(0.500000 + 0.866025i) q^{9} +(-2.00000 + 3.46410i) q^{11} -2.00000i q^{13} +(1.00000 + 2.00000i) q^{15} +(-1.73205 - 1.00000i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(5.19615 - 3.00000i) q^{23} +(1.96410 + 4.59808i) q^{25} -1.00000i q^{27} -6.00000 q^{29} +(-3.00000 + 5.19615i) q^{31} +(3.46410 - 2.00000i) q^{33} +(-3.46410 + 2.00000i) q^{37} +(-1.00000 + 1.73205i) q^{39} +4.00000i q^{43} +(0.133975 - 2.23205i) q^{45} +(3.46410 - 2.00000i) q^{47} +(1.00000 + 1.73205i) q^{51} +(-1.73205 - 1.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} +2.00000i q^{57} +(2.00000 - 3.46410i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.46410 + 3.73205i) q^{65} +(10.3923 + 6.00000i) q^{67} -6.00000 q^{69} -8.00000 q^{71} +(12.1244 + 7.00000i) q^{73} +(0.598076 - 4.96410i) q^{75} +(8.00000 + 13.8564i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +(5.19615 + 3.00000i) q^{87} +(8.00000 + 13.8564i) q^{89} +(5.19615 - 3.00000i) q^{93} +(-0.267949 + 4.46410i) q^{95} -14.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 2 q^{9} - 8 q^{11} + 4 q^{15} - 4 q^{19} - 6 q^{25} - 24 q^{29} - 12 q^{31} - 4 q^{39} + 4 q^{45} + 4 q^{51} + 32 q^{55} + 8 q^{59} + 4 q^{61} + 4 q^{65} - 24 q^{69} - 32 q^{71} - 8 q^{75} + 32 q^{79} - 2 q^{81} + 8 q^{85} + 32 q^{89} - 8 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(1\)

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
\(3\) −0.866025 0.500000i −0.500000 0.288675i
\(4\) 0 0
\(5\) −1.86603 1.23205i −0.834512 0.550990i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) −1.73205 1.00000i −0.420084 0.242536i 0.275029 0.961436i \(-0.411312\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) −1.00000 1.73205i −0.229416 0.397360i 0.728219 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615 3.00000i 1.08347 0.625543i 0.151642 0.988436i \(-0.451544\pi\)
0.931831 + 0.362892i \(0.118211\pi\)
\(24\) 0 0
\(25\) 1.96410 + 4.59808i 0.392820 + 0.919615i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −3.00000 + 5.19615i −0.538816 + 0.933257i 0.460152 + 0.887840i \(0.347795\pi\)
−0.998968 + 0.0454165i \(0.985539\pi\)
\(32\) 0 0
\(33\) 3.46410 2.00000i 0.603023 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.46410 + 2.00000i −0.569495 + 0.328798i −0.756948 0.653476i \(-0.773310\pi\)
0.187453 + 0.982274i \(0.439977\pi\)
\(38\) 0 0
\(39\) −1.00000 + 1.73205i −0.160128 + 0.277350i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0.133975 2.23205i 0.0199718 0.332734i
\(46\) 0 0
\(47\) 3.46410 2.00000i 0.505291 0.291730i −0.225605 0.974219i \(-0.572436\pi\)
0.730896 + 0.682489i \(0.239102\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 + 1.73205i 0.140028 + 0.242536i
\(52\) 0 0
\(53\) −1.73205 1.00000i −0.237915 0.137361i 0.376303 0.926497i \(-0.377195\pi\)
−0.614218 + 0.789136i \(0.710529\pi\)
\(54\) 0 0
\(55\) 8.00000 4.00000i 1.07872 0.539360i
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 2.00000 3.46410i 0.260378 0.450988i −0.705965 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.46410 + 3.73205i −0.305634 + 0.462904i
\(66\) 0 0
\(67\) 10.3923 + 6.00000i 1.26962 + 0.733017i 0.974916 0.222571i \(-0.0714450\pi\)
0.294706 + 0.955588i \(0.404778\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 12.1244 + 7.00000i 1.41905 + 0.819288i 0.996215 0.0869195i \(-0.0277023\pi\)
0.422833 + 0.906208i \(0.361036\pi\)
\(74\) 0 0
\(75\) 0.598076 4.96410i 0.0690599 0.573205i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 + 13.8564i 0.900070 + 1.55897i 0.827401 + 0.561611i \(0.189818\pi\)
0.0726692 + 0.997356i \(0.476848\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 16.0000i 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 5.19615 + 3.00000i 0.557086 + 0.321634i
\(88\) 0 0
\(89\) 8.00000 + 13.8564i 0.847998 + 1.46878i 0.882992 + 0.469389i \(0.155526\pi\)
−0.0349934 + 0.999388i \(0.511141\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.19615 3.00000i 0.538816 0.311086i
\(94\) 0 0
\(95\) −0.267949 + 4.46410i −0.0274910 + 0.458007i
\(96\) 0 0
\(97\) 14.0000i 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −4.00000 + 6.92820i −0.398015 + 0.689382i −0.993481 0.113998i \(-0.963634\pi\)
0.595466 + 0.803380i \(0.296967\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.5885 9.00000i 1.50699 0.870063i 0.507026 0.861931i \(-0.330745\pi\)
0.999967 0.00813215i \(-0.00258857\pi\)
\(108\) 0 0
\(109\) 9.00000 15.5885i 0.862044 1.49310i −0.00790932 0.999969i \(-0.502518\pi\)
0.869953 0.493135i \(-0.164149\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) −13.3923 0.803848i −1.24884 0.0749592i
\(116\) 0 0
\(117\) 1.73205 1.00000i 0.160128 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 2.00000 3.46410i 0.176090 0.304997i
\(130\) 0 0
\(131\) −2.00000 3.46410i −0.174741 0.302660i 0.765331 0.643637i \(-0.222575\pi\)
−0.940072 + 0.340977i \(0.889242\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.23205 + 1.86603i −0.106038 + 0.160602i
\(136\) 0 0
\(137\) −8.66025 5.00000i −0.739895 0.427179i 0.0821359 0.996621i \(-0.473826\pi\)
−0.822031 + 0.569442i \(0.807159\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 6.92820 + 4.00000i 0.579365 + 0.334497i
\(144\) 0 0
\(145\) 11.1962 + 7.39230i 0.929790 + 0.613898i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) 8.00000 13.8564i 0.651031 1.12762i −0.331842 0.943335i \(-0.607670\pi\)
0.982873 0.184284i \(-0.0589965\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 12.0000 6.00000i 0.963863 0.481932i
\(156\) 0 0
\(157\) −1.73205 1.00000i −0.138233 0.0798087i 0.429289 0.903167i \(-0.358764\pi\)
−0.567521 + 0.823359i \(0.692098\pi\)
\(158\) 0 0
\(159\) 1.00000 + 1.73205i 0.0793052 + 0.137361i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.7846 12.0000i 1.62798 0.939913i 0.643280 0.765631i \(-0.277573\pi\)
0.984696 0.174282i \(-0.0557604\pi\)
\(164\) 0 0
\(165\) −8.92820 0.535898i −0.695060 0.0417196i
\(166\) 0 0
\(167\) 8.00000i 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 1.00000 1.73205i 0.0764719 0.132453i
\(172\) 0 0
\(173\) −12.1244 + 7.00000i −0.921798 + 0.532200i −0.884208 0.467093i \(-0.845301\pi\)
−0.0375896 + 0.999293i \(0.511968\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.46410 + 2.00000i −0.260378 + 0.150329i
\(178\) 0 0
\(179\) −10.0000 + 17.3205i −0.747435 + 1.29460i 0.201613 + 0.979465i \(0.435382\pi\)
−0.949048 + 0.315130i \(0.897952\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 8.92820 + 0.535898i 0.656415 + 0.0394000i
\(186\) 0 0
\(187\) 6.92820 4.00000i 0.506640 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 6.92820i −0.289430 0.501307i 0.684244 0.729253i \(-0.260132\pi\)
−0.973674 + 0.227946i \(0.926799\pi\)
\(192\) 0 0
\(193\) 20.7846 + 12.0000i 1.49611 + 0.863779i 0.999990 0.00447566i \(-0.00142465\pi\)
0.496119 + 0.868255i \(0.334758\pi\)
\(194\) 0 0
\(195\) 4.00000 2.00000i 0.286446 0.143223i
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 5.00000 8.66025i 0.354441 0.613909i −0.632581 0.774494i \(-0.718005\pi\)
0.987022 + 0.160585i \(0.0513380\pi\)
\(200\) 0 0
\(201\) −6.00000 10.3923i −0.423207 0.733017i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.19615 + 3.00000i 0.361158 + 0.208514i
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 6.92820 + 4.00000i 0.474713 + 0.274075i
\(214\) 0 0
\(215\) 4.92820 7.46410i 0.336101 0.509048i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −7.00000 12.1244i −0.473016 0.819288i
\(220\) 0 0
\(221\) −2.00000 + 3.46410i −0.134535 + 0.233021i
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) −3.46410 2.00000i −0.229920 0.132745i 0.380615 0.924734i \(-0.375712\pi\)
−0.610535 + 0.791989i \(0.709046\pi\)
\(228\) 0 0
\(229\) 1.00000 + 1.73205i 0.0660819 + 0.114457i 0.897173 0.441679i \(-0.145617\pi\)
−0.831092 + 0.556136i \(0.812283\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.5885 9.00000i 1.02123 0.589610i 0.106773 0.994283i \(-0.465948\pi\)
0.914461 + 0.404674i \(0.132615\pi\)
\(234\) 0 0
\(235\) −8.92820 0.535898i −0.582412 0.0349582i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −13.0000 + 22.5167i −0.837404 + 1.45043i 0.0546547 + 0.998505i \(0.482594\pi\)
−0.892058 + 0.451920i \(0.850739\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.46410 + 2.00000i −0.220416 + 0.127257i
\(248\) 0 0
\(249\) −8.00000 + 13.8564i −0.506979 + 0.878114i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) 0 0
\(255\) 0.267949 4.46410i 0.0167796 0.279553i
\(256\) 0 0
\(257\) 19.0526 11.0000i 1.18847 0.686161i 0.230508 0.973070i \(-0.425961\pi\)
0.957958 + 0.286909i \(0.0926278\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 5.19615i −0.185695 0.321634i
\(262\) 0 0
\(263\) 25.9808 + 15.0000i 1.60204 + 0.924940i 0.991078 + 0.133281i \(0.0425514\pi\)
0.610964 + 0.791658i \(0.290782\pi\)
\(264\) 0 0
\(265\) 2.00000 + 4.00000i 0.122859 + 0.245718i
\(266\) 0 0
\(267\) 16.0000i 0.979184i
\(268\) 0 0
\(269\) −2.00000 + 3.46410i −0.121942 + 0.211210i −0.920534 0.390664i \(-0.872246\pi\)
0.798591 + 0.601874i \(0.205579\pi\)
\(270\) 0 0
\(271\) 11.0000 + 19.0526i 0.668202 + 1.15736i 0.978406 + 0.206691i \(0.0662693\pi\)
−0.310204 + 0.950670i \(0.600397\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −19.8564 2.39230i −1.19739 0.144261i
\(276\) 0 0
\(277\) −3.46410 2.00000i −0.208138 0.120168i 0.392308 0.919834i \(-0.371677\pi\)
−0.600446 + 0.799666i \(0.705010\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −17.3205 10.0000i −1.02960 0.594438i −0.112728 0.993626i \(-0.535959\pi\)
−0.916869 + 0.399188i \(0.869292\pi\)
\(284\) 0 0
\(285\) 2.46410 3.73205i 0.145961 0.221068i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.50000 11.2583i −0.382353 0.662255i
\(290\) 0 0
\(291\) −7.00000 + 12.1244i −0.410347 + 0.710742i
\(292\) 0 0
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 0 0
\(295\) −8.00000 + 4.00000i −0.465778 + 0.232889i
\(296\) 0 0
\(297\) 3.46410 + 2.00000i 0.201008 + 0.116052i
\(298\) 0 0
\(299\) −6.00000 10.3923i −0.346989 0.601003i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.92820 4.00000i 0.398015 0.229794i
\(304\) 0 0
\(305\) 0.267949 4.46410i 0.0153427 0.255614i
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.0000 17.3205i 0.567048 0.982156i −0.429808 0.902920i \(-0.641419\pi\)
0.996856 0.0792356i \(-0.0252479\pi\)
\(312\) 0 0
\(313\) −19.0526 + 11.0000i −1.07691 + 0.621757i −0.930062 0.367402i \(-0.880247\pi\)
−0.146852 + 0.989158i \(0.546914\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.5167 + 13.0000i −1.26466 + 0.730153i −0.973973 0.226665i \(-0.927218\pi\)
−0.290689 + 0.956818i \(0.593884\pi\)
\(318\) 0 0
\(319\) 12.0000 20.7846i 0.671871 1.16371i
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) 0 0
\(325\) 9.19615 3.92820i 0.510111 0.217898i
\(326\) 0 0
\(327\) −15.5885 + 9.00000i −0.862044 + 0.497701i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 + 20.7846i 0.659580 + 1.14243i 0.980725 + 0.195395i \(0.0625990\pi\)
−0.321145 + 0.947030i \(0.604068\pi\)
\(332\) 0 0
\(333\) −3.46410 2.00000i −0.189832 0.109599i
\(334\) 0 0
\(335\) −12.0000 24.0000i −0.655630 1.31126i
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 3.00000 5.19615i 0.162938 0.282216i
\(340\) 0 0
\(341\) −12.0000 20.7846i −0.649836 1.12555i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 11.1962 + 7.39230i 0.602781 + 0.397988i
\(346\) 0 0
\(347\) −8.66025 5.00000i −0.464907 0.268414i 0.249198 0.968452i \(-0.419833\pi\)
−0.714105 + 0.700038i \(0.753166\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 25.9808 + 15.0000i 1.38282 + 0.798369i 0.992492 0.122308i \(-0.0390296\pi\)
0.390324 + 0.920677i \(0.372363\pi\)
\(354\) 0 0
\(355\) 14.9282 + 9.85641i 0.792307 + 0.523124i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.00000 + 6.92820i 0.211112 + 0.365657i 0.952063 0.305903i \(-0.0989582\pi\)
−0.740951 + 0.671559i \(0.765625\pi\)
\(360\) 0 0
\(361\) 7.50000 12.9904i 0.394737 0.683704i
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) −14.0000 28.0000i −0.732793 1.46559i
\(366\) 0 0
\(367\) −13.8564 8.00000i −0.723299 0.417597i 0.0926670 0.995697i \(-0.470461\pi\)
−0.815966 + 0.578101i \(0.803794\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 20.7846 12.0000i 1.07619 0.621336i 0.146321 0.989237i \(-0.453257\pi\)
0.929865 + 0.367901i \(0.119923\pi\)
\(374\) 0 0
\(375\) −7.23205 + 8.52628i −0.373461 + 0.440295i
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.3923 + 6.00000i −0.531022 + 0.306586i −0.741433 0.671027i \(-0.765853\pi\)
0.210411 + 0.977613i \(0.432520\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.46410 + 2.00000i −0.176090 + 0.101666i
\(388\) 0 0
\(389\) −15.0000 + 25.9808i −0.760530 + 1.31728i 0.182047 + 0.983290i \(0.441728\pi\)
−0.942578 + 0.333987i \(0.891606\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 4.00000i 0.201773i
\(394\) 0 0
\(395\) 2.14359 35.7128i 0.107856 1.79691i
\(396\) 0 0
\(397\) −29.4449 + 17.0000i −1.47780 + 0.853206i −0.999685 0.0250943i \(-0.992011\pi\)
−0.478110 + 0.878300i \(0.658678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 5.19615i −0.149813 0.259483i 0.781345 0.624099i \(-0.214534\pi\)
−0.931158 + 0.364615i \(0.881200\pi\)
\(402\) 0 0
\(403\) 10.3923 + 6.00000i 0.517678 + 0.298881i
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) −13.0000 + 22.5167i −0.642809 + 1.11338i 0.341994 + 0.939702i \(0.388898\pi\)
−0.984803 + 0.173675i \(0.944436\pi\)
\(410\) 0 0
\(411\) 5.00000 + 8.66025i 0.246632 + 0.427179i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −19.7128 + 29.8564i −0.967664 + 1.46559i
\(416\) 0 0
\(417\) −12.1244 7.00000i −0.593732 0.342791i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 3.46410 + 2.00000i 0.168430 + 0.0972433i
\(424\) 0 0
\(425\) 1.19615 9.92820i 0.0580219 0.481589i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 6.92820i −0.193122 0.334497i
\(430\) 0 0
\(431\) 12.0000 20.7846i 0.578020 1.00116i −0.417687 0.908591i \(-0.637159\pi\)
0.995706 0.0925683i \(-0.0295076\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) −6.00000 12.0000i −0.287678 0.575356i
\(436\) 0 0
\(437\) −10.3923 6.00000i −0.497131 0.287019i
\(438\) 0 0
\(439\) −3.00000 5.19615i −0.143182 0.247999i 0.785511 0.618848i \(-0.212400\pi\)
−0.928693 + 0.370849i \(0.879067\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.19615 3.00000i 0.246877 0.142534i −0.371457 0.928450i \(-0.621142\pi\)
0.618333 + 0.785916i \(0.287808\pi\)
\(444\) 0 0
\(445\) 2.14359 35.7128i 0.101616 1.69295i
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.8564 + 8.00000i −0.651031 + 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.2487 + 14.0000i −1.13431 + 0.654892i −0.945015 0.327028i \(-0.893953\pi\)
−0.189292 + 0.981921i \(0.560619\pi\)
\(458\) 0 0
\(459\) −1.00000 + 1.73205i −0.0466760 + 0.0808452i
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) −13.3923 0.803848i −0.621053 0.0372775i
\(466\) 0 0
\(467\) 24.2487 14.0000i 1.12210 0.647843i 0.180161 0.983637i \(-0.442338\pi\)
0.941935 + 0.335794i \(0.109005\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.00000 + 1.73205i 0.0460776 + 0.0798087i
\(472\) 0 0
\(473\) −13.8564 8.00000i −0.637118 0.367840i
\(474\) 0 0
\(475\) 6.00000 8.00000i 0.275299 0.367065i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) −2.00000 + 3.46410i −0.0913823 + 0.158279i −0.908093 0.418769i \(-0.862462\pi\)
0.816711 + 0.577047i \(0.195795\pi\)
\(480\) 0 0
\(481\) 4.00000 + 6.92820i 0.182384 + 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.2487 + 26.1244i −0.783224 + 1.18625i
\(486\) 0 0
\(487\) 10.3923 + 6.00000i 0.470920 + 0.271886i 0.716625 0.697459i \(-0.245686\pi\)
−0.245705 + 0.969345i \(0.579019\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 10.3923 + 6.00000i 0.468046 + 0.270226i
\(494\) 0 0
\(495\) 7.46410 + 4.92820i 0.335486 + 0.221506i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.00000 3.46410i −0.0895323 0.155074i 0.817781 0.575529i \(-0.195204\pi\)
−0.907314 + 0.420455i \(0.861871\pi\)
\(500\) 0 0
\(501\) −4.00000 + 6.92820i −0.178707 + 0.309529i
\(502\) 0 0
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) 0 0
\(505\) 16.0000 8.00000i 0.711991 0.355995i
\(506\) 0 0
\(507\) −7.79423 4.50000i −0.346154 0.199852i
\(508\) 0 0
\(509\) −14.0000 24.2487i −0.620539 1.07481i −0.989385 0.145315i \(-0.953580\pi\)
0.368846 0.929490i \(-0.379753\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.73205 + 1.00000i −0.0764719 + 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −6.00000 + 10.3923i −0.262865 + 0.455295i −0.967002 0.254769i \(-0.918001\pi\)
0.704137 + 0.710064i \(0.251334\pi\)
\(522\) 0 0
\(523\) −10.3923 + 6.00000i −0.454424 + 0.262362i −0.709697 0.704507i \(-0.751168\pi\)
0.255273 + 0.966869i \(0.417835\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.3923 6.00000i 0.452696 0.261364i
\(528\) 0 0
\(529\) 6.50000 11.2583i 0.282609 0.489493i
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −40.1769 2.41154i −1.73700 0.104260i
\(536\) 0 0
\(537\) 17.3205 10.0000i 0.747435 0.431532i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 8.66025i −0.214967 0.372333i 0.738296 0.674477i \(-0.235631\pi\)
−0.953262 + 0.302144i \(0.902298\pi\)
\(542\) 0 0
\(543\) −19.0526 11.0000i −0.817624 0.472055i
\(544\) 0 0
\(545\) −36.0000 + 18.0000i −1.54207 + 0.771035i
\(546\) 0 0
\(547\) 28.0000i 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) −1.00000 + 1.73205i −0.0426790 + 0.0739221i
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −7.46410 4.92820i −0.316833 0.209191i
\(556\) 0 0
\(557\) −25.9808 15.0000i −1.10084 0.635570i −0.164399 0.986394i \(-0.552568\pi\)
−0.936442 + 0.350824i \(0.885902\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 20.7846 + 12.0000i 0.875967 + 0.505740i 0.869326 0.494238i \(-0.164553\pi\)
0.00664037 + 0.999978i \(0.497886\pi\)
\(564\) 0 0
\(565\) 7.39230 11.1962i 0.310997 0.471026i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.00000 + 5.19615i 0.125767 + 0.217834i 0.922032 0.387113i \(-0.126528\pi\)
−0.796266 + 0.604947i \(0.793194\pi\)
\(570\) 0 0
\(571\) 4.00000 6.92820i 0.167395 0.289936i −0.770108 0.637913i \(-0.779798\pi\)
0.937503 + 0.347977i \(0.113131\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) 24.0000 + 18.0000i 1.00087 + 0.750652i
\(576\) 0 0
\(577\) 15.5885 + 9.00000i 0.648956 + 0.374675i 0.788056 0.615603i \(-0.211088\pi\)
−0.139100 + 0.990278i \(0.544421\pi\)
\(578\) 0 0
\(579\) −12.0000 20.7846i −0.498703 0.863779i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.92820 4.00000i 0.286937 0.165663i
\(584\) 0 0
\(585\) −4.46410 0.267949i −0.184568 0.0110783i
\(586\) 0 0
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) 12.1244 7.00000i 0.497888 0.287456i −0.229953 0.973202i \(-0.573857\pi\)
0.727841 + 0.685746i \(0.240524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.66025 + 5.00000i −0.354441 + 0.204636i
\(598\) 0 0
\(599\) 4.00000 6.92820i 0.163436 0.283079i −0.772663 0.634816i \(-0.781076\pi\)
0.936099 + 0.351738i \(0.114409\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) −0.669873 + 11.1603i −0.0272342 + 0.453729i
\(606\) 0 0
\(607\) 6.92820 4.00000i 0.281207 0.162355i −0.352763 0.935713i \(-0.614758\pi\)
0.633970 + 0.773358i \(0.281424\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.00000 6.92820i −0.161823 0.280285i
\(612\) 0 0
\(613\) −3.46410 2.00000i −0.139914 0.0807792i 0.428409 0.903585i \(-0.359074\pi\)
−0.568323 + 0.822806i \(0.692408\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) −15.0000 + 25.9808i −0.602901 + 1.04425i 0.389479 + 0.921036i \(0.372655\pi\)
−0.992379 + 0.123219i \(0.960678\pi\)
\(620\) 0 0
\(621\) −3.00000 5.19615i −0.120386 0.208514i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17.2846 + 18.0622i −0.691384 + 0.722487i
\(626\) 0 0
\(627\) −6.92820 4.00000i −0.276686 0.159745i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 6.92820 + 4.00000i 0.275371 + 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 6.92820i −0.158238 0.274075i
\(640\) 0 0
\(641\) −9.00000 + 15.5885i −0.355479 + 0.615707i −0.987200 0.159489i \(-0.949015\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) −8.00000 + 4.00000i −0.315000 + 0.157500i
\(646\) 0 0
\(647\) −34.6410 20.0000i −1.36188 0.786281i −0.372005 0.928231i \(-0.621330\pi\)
−0.989874 + 0.141950i \(0.954663\pi\)
\(648\) 0 0
\(649\) 8.00000 + 13.8564i 0.314027 + 0.543912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.5167 13.0000i 0.881145 0.508729i 0.0101092 0.999949i \(-0.496782\pi\)
0.871036 + 0.491220i \(0.163449\pi\)
\(654\) 0 0
\(655\) −0.535898 + 8.92820i −0.0209393 + 0.348854i
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 1.00000 1.73205i 0.0388955 0.0673690i −0.845922 0.533306i \(-0.820949\pi\)
0.884818 + 0.465937i \(0.154283\pi\)
\(662\) 0 0
\(663\) 3.46410 2.00000i 0.134535 0.0776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −31.1769 + 18.0000i −1.20717 + 0.696963i
\(668\) 0 0
\(669\) 8.00000 13.8564i 0.309298 0.535720i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 12.0000i 0.462566i −0.972887 0.231283i \(-0.925708\pi\)
0.972887 0.231283i \(-0.0742923\pi\)
\(674\) 0 0
\(675\) 4.59808 1.96410i 0.176980 0.0755983i
\(676\) 0 0
\(677\) −36.3731 + 21.0000i −1.39793 + 0.807096i −0.994176 0.107772i \(-0.965628\pi\)
−0.403755 + 0.914867i \(0.632295\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 2.00000 + 3.46410i 0.0766402 + 0.132745i
\(682\) 0 0
\(683\) −12.1244 7.00000i −0.463926 0.267848i 0.249768 0.968306i \(-0.419646\pi\)
−0.713693 + 0.700458i \(0.752979\pi\)
\(684\) 0 0
\(685\) 10.0000 + 20.0000i 0.382080 + 0.764161i
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) −2.00000 + 3.46410i −0.0761939 + 0.131972i
\(690\) 0 0
\(691\) −11.0000 19.0526i −0.418460 0.724793i 0.577325 0.816514i \(-0.304097\pi\)
−0.995785 + 0.0917209i \(0.970763\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −26.1244 17.2487i −0.990953 0.654281i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 6.92820 + 4.00000i 0.261302 + 0.150863i
\(704\) 0 0
\(705\) 7.46410 + 4.92820i 0.281114 + 0.185607i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.0000 32.9090i −0.713560 1.23592i −0.963512 0.267664i \(-0.913748\pi\)
0.249952 0.968258i \(-0.419585\pi\)
\(710\) 0 0
\(711\) −8.00000 + 13.8564i −0.300023 + 0.519656i
\(712\) 0 0
\(713\) 36.0000i 1.34821i
\(714\) 0 0
\(715\) −8.00000 16.0000i −0.299183 0.598366i
\(716\) 0 0
\(717\) 6.92820 + 4.00000i 0.258738 + 0.149383i
\(718\) 0 0
\(719\) 14.0000 + 24.2487i 0.522112 + 0.904324i 0.999669 + 0.0257237i \(0.00818900\pi\)
−0.477557 + 0.878601i \(0.658478\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 22.5167 13.0000i 0.837404 0.483475i
\(724\) 0 0
\(725\) −11.7846 27.5885i −0.437669 1.02461i
\(726\) 0 0
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 4.00000 6.92820i 0.147945 0.256249i
\(732\) 0 0
\(733\) −36.3731 + 21.0000i −1.34347 + 0.775653i −0.987315 0.158774i \(-0.949246\pi\)
−0.356155 + 0.934427i \(0.615912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −41.5692 + 24.0000i −1.53122 + 0.884051i
\(738\) 0 0
\(739\) −8.00000 + 13.8564i −0.294285 + 0.509716i −0.974818 0.223001i \(-0.928415\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 34.0000i 1.24734i −0.781688 0.623670i \(-0.785641\pi\)
0.781688 0.623670i \(-0.214359\pi\)
\(744\) 0 0
\(745\) 0.803848 13.3923i 0.0294507 0.490656i
\(746\) 0 0
\(747\) 13.8564 8.00000i 0.506979 0.292705i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.00000 3.46410i −0.0729810 0.126407i 0.827225 0.561870i \(-0.189918\pi\)
−0.900207 + 0.435463i \(0.856585\pi\)
\(752\) 0 0
\(753\) 10.3923 + 6.00000i 0.378717 + 0.218652i
\(754\) 0 0
\(755\) −32.0000 + 16.0000i −1.16460 + 0.582300i
\(756\) 0 0
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 0 0
\(759\) 12.0000 20.7846i 0.435572 0.754434i
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.46410 + 3.73205i −0.0890898 + 0.134933i
\(766\) 0 0
\(767\) −6.92820 4.00000i −0.250163 0.144432i
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 0 0
\(773\) 8.66025 + 5.00000i 0.311488 + 0.179838i 0.647592 0.761987i \(-0.275776\pi\)
−0.336104 + 0.941825i \(0.609109\pi\)
\(774\) 0 0
\(775\) −29.7846 3.58846i −1.06989 0.128901i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 16.0000 27.7128i 0.572525 0.991642i
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 2.00000 + 4.00000i 0.0713831 + 0.142766i
\(786\) 0 0
\(787\) −24.2487 14.0000i −0.864373 0.499046i 0.00110111 0.999999i \(-0.499650\pi\)
−0.865474 + 0.500953i \(0.832983\pi\)
\(788\) 0 0
\(789\) −15.0000 25.9808i −0.534014 0.924940i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.46410 2.00000i 0.123014 0.0710221i
\(794\) 0 0
\(795\) 0.267949 4.46410i 0.00950318 0.158325i
\(796\) 0 0
\(797\) 18.0000i 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) −8.00000 + 13.8564i −0.282666 + 0.489592i
\(802\) 0 0
\(803\) −48.4974 + 28.0000i −1.71144 + 0.988099i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.46410 2.00000i 0.121942 0.0704033i
\(808\) 0 0
\(809\) −9.00000 + 15.5885i −0.316423 + 0.548061i −0.979739 0.200279i \(-0.935815\pi\)
0.663316 + 0.748340i \(0.269149\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) 22.0000i 0.771574i
\(814\) 0 0
\(815\) −53.5692 3.21539i −1.87645 0.112630i
\(816\) 0 0
\(817\) 6.92820 4.00000i 0.242387 0.139942i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 + 5.19615i 0.104701 + 0.181347i 0.913616 0.406578i \(-0.133278\pi\)
−0.808915 + 0.587925i \(0.799945\pi\)
\(822\) 0 0
\(823\) 6.92820 + 4.00000i 0.241502 + 0.139431i 0.615867 0.787850i \(-0.288806\pi\)
−0.374365 + 0.927281i \(0.622139\pi\)
\(824\) 0 0
\(825\) 16.0000 + 12.0000i 0.557048 + 0.417786i
\(826\) 0 0
\(827\) 30.0000i 1.04320i −0.853189 0.521601i \(-0.825335\pi\)
0.853189 0.521601i \(-0.174665\pi\)
\(828\) 0 0
\(829\) 19.0000 32.9090i 0.659897 1.14298i −0.320745 0.947166i \(-0.603933\pi\)
0.980642 0.195810i \(-0.0627335\pi\)
\(830\) 0 0
\(831\) 2.00000 + 3.46410i 0.0693792 + 0.120168i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.85641 + 14.9282i −0.341095 + 0.516612i
\(836\) 0 0
\(837\) 5.19615 + 3.00000i 0.179605 + 0.103695i
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −8.66025 5.00000i −0.298275 0.172209i
\(844\) 0 0
\(845\) −16.7942 11.0885i −0.577739 0.381455i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 10.0000 + 17.3205i 0.343199 + 0.594438i
\(850\) 0 0
\(851\) −12.0000 + 20.7846i −0.411355 + 0.712487i
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) −4.00000 + 2.00000i −0.136797 + 0.0683986i
\(856\) 0 0
\(857\) 8.66025 + 5.00000i 0.295829 + 0.170797i 0.640567 0.767902i \(-0.278699\pi\)
−0.344739 + 0.938699i \(0.612033\pi\)
\(858\) 0 0
\(859\) −3.00000 5.19615i −0.102359 0.177290i 0.810297 0.586019i \(-0.199306\pi\)
−0.912656 + 0.408729i \(0.865972\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.7654 27.0000i 1.59191 0.919091i 0.598933 0.800799i \(-0.295592\pi\)
0.992979 0.118291i \(-0.0377417\pi\)
\(864\) 0 0
\(865\) 31.2487 + 1.87564i 1.06249 + 0.0637738i
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) 12.0000 20.7846i 0.406604 0.704260i
\(872\) 0 0
\(873\) 12.1244 7.00000i 0.410347 0.236914i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.1769 18.0000i 1.05277 0.607817i 0.129346 0.991600i \(-0.458712\pi\)
0.923423 + 0.383783i \(0.125379\pi\)
\(878\) 0 0
\(879\) 11.0000 19.0526i 0.371021 0.642627i
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 56.0000i 1.88455i 0.334840 + 0.942275i \(0.391318\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(884\) 0 0
\(885\) 8.92820 + 0.535898i 0.300118 + 0.0180140i
\(886\) 0 0
\(887\) 10.3923 6.00000i 0.348939 0.201460i −0.315279 0.948999i \(-0.602098\pi\)
0.664218 + 0.747539i \(0.268765\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 3.46410i −0.0670025 0.116052i
\(892\) 0 0
\(893\) −6.92820 4.00000i −0.231843 0.133855i
\(894\) 0 0
\(895\) 40.0000 20.0000i 1.33705 0.668526i
\(896\) 0 0
\(897\) 12.0000i 0.400668i
\(898\) 0 0
\(899\) 18.0000 31.1769i 0.600334 1.03981i
\(900\) 0 0
\(901\) 2.00000 + 3.46410i 0.0666297 + 0.115406i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −41.0526 27.1051i −1.36463 0.901005i
\(906\) 0 0
\(907\) 6.92820 + 4.00000i 0.230047 + 0.132818i 0.610594 0.791944i \(-0.290931\pi\)
−0.380547 + 0.924762i \(0.624264\pi\)
\(908\) 0 0
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 55.4256 + 32.0000i 1.83432 + 1.05905i
\(914\) 0 0
\(915\) −2.46410 + 3.73205i −0.0814607 + 0.123378i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 10.0000 + 17.3205i 0.329870 + 0.571351i 0.982486 0.186338i \(-0.0596619\pi\)
−0.652616 + 0.757689i \(0.726329\pi\)
\(920\) 0 0
\(921\) 2.00000 3.46410i 0.0659022 0.114146i
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) −16.0000 12.0000i −0.526077 0.394558i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.0000 + 41.5692i 0.787414 + 1.36384i 0.927546 + 0.373709i \(0.121914\pi\)
−0.140132 + 0.990133i \(0.544753\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −17.3205 + 10.0000i −0.567048 + 0.327385i
\(934\) 0 0
\(935\) −17.8564 1.07180i −0.583967 0.0350515i
\(936\) 0 0
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) 30.0000 51.9615i 0.977972 1.69390i 0.308215 0.951317i \(-0.400268\pi\)
0.669757 0.742581i \(-0.266398\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.5885 + 9.00000i −0.506557 + 0.292461i −0.731417 0.681930i \(-0.761141\pi\)
0.224860 + 0.974391i \(0.427807\pi\)
\(948\) 0 0
\(949\) 14.0000 24.2487i 0.454459 0.787146i
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 0 0
\(955\) −1.07180 + 17.8564i −0.0346825 + 0.577820i
\(956\) 0 0
\(957\) −20.7846 + 12.0000i −0.671871 + 0.387905i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.50000 4.33013i −0.0806452 0.139682i
\(962\) 0 0
\(963\) 15.5885 + 9.00000i 0.502331 + 0.290021i
\(964\) 0 0
\(965\) −24.0000 48.0000i −0.772587 1.54517i
\(966\) 0 0
\(967\) 36.0000i 1.15768i −0.815440 0.578841i \(-0.803505\pi\)
0.815440 0.578841i \(-0.196495\pi\)
\(968\) 0 0
\(969\) 2.00000 3.46410i 0.0642493 0.111283i
\(970\) 0 0
\(971\) 12.0000 + 20.7846i 0.385098 + 0.667010i 0.991783 0.127933i \(-0.0408342\pi\)
−0.606685 + 0.794943i \(0.707501\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −9.92820 1.19615i −0.317957 0.0383075i
\(976\) 0 0
\(977\) 36.3731 + 21.0000i 1.16368 + 0.671850i 0.952183 0.305530i \(-0.0988335\pi\)
0.211495 + 0.977379i \(0.432167\pi\)
\(978\) 0 0
\(979\) −64.0000 −2.04545
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) 0 0
\(983\) −20.7846 12.0000i −0.662926 0.382741i 0.130465 0.991453i \(-0.458353\pi\)
−0.793391 + 0.608712i \(0.791686\pi\)
\(984\) 0 0
\(985\) 22.1769 33.5885i 0.706615 1.07022i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 + 20.7846i 0.381578 + 0.660912i
\(990\) 0 0
\(991\) −18.0000 + 31.1769i −0.571789 + 0.990367i 0.424594 + 0.905384i \(0.360417\pi\)
−0.996382 + 0.0849833i \(0.972916\pi\)
\(992\) 0 0
\(993\) 24.0000i 0.761617i
\(994\) 0 0
\(995\) −20.0000 + 10.0000i −0.634043 + 0.317021i
\(996\) 0 0
\(997\) −19.0526 11.0000i −0.603401 0.348373i 0.166978 0.985961i \(-0.446599\pi\)
−0.770378 + 0.637587i \(0.779933\pi\)
\(998\) 0 0
\(999\) 2.00000 + 3.46410i 0.0632772 + 0.109599i
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 2940.2.bb.a.949.1 4
5.4 even 2 inner 2940.2.bb.a.949.2 4
7.2 even 3 inner 2940.2.bb.a.1549.2 4
7.3 odd 6 2940.2.k.b.589.1 2
7.4 even 3 420.2.k.b.169.2 yes 2
7.5 odd 6 2940.2.bb.f.1549.1 4
7.6 odd 2 2940.2.bb.f.949.2 4
21.11 odd 6 1260.2.k.a.1009.2 2
28.11 odd 6 1680.2.t.g.1009.1 2
35.4 even 6 420.2.k.b.169.1 2
35.9 even 6 inner 2940.2.bb.a.1549.1 4
35.18 odd 12 2100.2.a.i.1.1 1
35.19 odd 6 2940.2.bb.f.1549.2 4
35.24 odd 6 2940.2.k.b.589.2 2
35.32 odd 12 2100.2.a.n.1.1 1
35.34 odd 2 2940.2.bb.f.949.1 4
84.11 even 6 5040.2.t.d.1009.2 2
105.32 even 12 6300.2.a.b.1.1 1
105.53 even 12 6300.2.a.r.1.1 1
105.74 odd 6 1260.2.k.a.1009.1 2
140.39 odd 6 1680.2.t.g.1009.2 2
140.67 even 12 8400.2.a.o.1.1 1
140.123 even 12 8400.2.a.bm.1.1 1
420.179 even 6 5040.2.t.d.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.b.169.1 2 35.4 even 6
420.2.k.b.169.2 yes 2 7.4 even 3
1260.2.k.a.1009.1 2 105.74 odd 6
1260.2.k.a.1009.2 2 21.11 odd 6
1680.2.t.g.1009.1 2 28.11 odd 6
1680.2.t.g.1009.2 2 140.39 odd 6
2100.2.a.i.1.1 1 35.18 odd 12
2100.2.a.n.1.1 1 35.32 odd 12
2940.2.k.b.589.1 2 7.3 odd 6
2940.2.k.b.589.2 2 35.24 odd 6
2940.2.bb.a.949.1 4 1.1 even 1 trivial
2940.2.bb.a.949.2 4 5.4 even 2 inner
2940.2.bb.a.1549.1 4 35.9 even 6 inner
2940.2.bb.a.1549.2 4 7.2 even 3 inner
2940.2.bb.f.949.1 4 35.34 odd 2
2940.2.bb.f.949.2 4 7.6 odd 2
2940.2.bb.f.1549.1 4 7.5 odd 6
2940.2.bb.f.1549.2 4 35.19 odd 6
5040.2.t.d.1009.1 2 420.179 even 6
5040.2.t.d.1009.2 2 84.11 even 6
6300.2.a.b.1.1 1 105.32 even 12
6300.2.a.r.1.1 1 105.53 even 12
8400.2.a.o.1.1 1 140.67 even 12
8400.2.a.bm.1.1 1 140.123 even 12