Properties

Label 560.2.bj.b.97.1
Level $560$
Weight $2$
Character 560.97
Analytic conductor $4.472$
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] = [560,2,Mod(97,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bj (of order \(4\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 97.1
Root \(-1.83051 - 1.83051i\) of defining polynomial
Character \(\chi\) \(=\) 560.97
Dual form 560.2.bj.b.433.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.83051 - 1.83051i) q^{3} +(1.83051 - 1.28422i) q^{5} +(2.12393 + 1.57763i) q^{7} +3.70156i q^{9} +O(q^{10})\) \(q+(-1.83051 - 1.83051i) q^{3} +(1.83051 - 1.28422i) q^{5} +(2.12393 + 1.57763i) q^{7} +3.70156i q^{9} +4.70156 q^{11} +(1.83051 + 1.83051i) q^{13} +(-5.70156 - 1.00000i) q^{15} +(-0.737925 + 0.737925i) q^{17} +4.75362 q^{19} +(-1.00000 - 6.77576i) q^{21} +(-3.70156 + 3.70156i) q^{23} +(1.70156 - 4.70156i) q^{25} +(1.28422 - 1.28422i) q^{27} +0.701562i q^{29} -8.79790i q^{31} +(-8.60627 - 8.60627i) q^{33} +(5.91391 + 0.160291i) q^{35} +(-3.70156 - 3.70156i) q^{37} -6.70156i q^{39} -4.75362i q^{41} +(5.00000 - 5.00000i) q^{43} +(4.75362 + 6.77576i) q^{45} +(-8.05998 + 8.05998i) q^{47} +(2.02214 + 6.70156i) q^{49} +2.70156 q^{51} +(5.00000 - 5.00000i) q^{53} +(8.60627 - 6.03784i) q^{55} +(-8.70156 - 8.70156i) q^{57} -4.75362 q^{59} +9.50723i q^{61} +(-5.83971 + 7.86185i) q^{63} +(5.70156 + 1.00000i) q^{65} +(-5.00000 - 5.00000i) q^{67} +13.5515 q^{69} +5.40312 q^{71} +(3.11473 + 3.11473i) q^{73} +(-11.7210 + 5.49154i) q^{75} +(9.98578 + 7.41734i) q^{77} +6.70156i q^{79} +6.40312 q^{81} +(-7.86835 - 7.86835i) q^{83} +(-0.403124 + 2.29844i) q^{85} +(1.28422 - 1.28422i) q^{87} -13.5515 q^{89} +(1.00000 + 6.77576i) q^{91} +(-16.1047 + 16.1047i) q^{93} +(8.70156 - 6.10469i) q^{95} +(-5.49154 + 5.49154i) q^{97} +17.4031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{7} + 12 q^{11} - 20 q^{15} - 8 q^{21} - 4 q^{23} - 12 q^{25} + 14 q^{35} - 4 q^{37} + 40 q^{43} - 4 q^{51} + 40 q^{53} - 44 q^{57} - 42 q^{63} + 20 q^{65} - 40 q^{67} - 8 q^{71} + 44 q^{77} + 48 q^{85} + 8 q^{91} - 52 q^{93} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) −1.83051 1.83051i −1.05685 1.05685i −0.998284 0.0585640i \(-0.981348\pi\)
−0.0585640 0.998284i \(-0.518652\pi\)
\(4\) 0 0
\(5\) 1.83051 1.28422i 0.818631 0.574320i
\(6\) 0 0
\(7\) 2.12393 + 1.57763i 0.802769 + 0.596289i
\(8\) 0 0
\(9\) 3.70156i 1.23385i
\(10\) 0 0
\(11\) 4.70156 1.41757 0.708787 0.705422i \(-0.249243\pi\)
0.708787 + 0.705422i \(0.249243\pi\)
\(12\) 0 0
\(13\) 1.83051 + 1.83051i 0.507693 + 0.507693i 0.913818 0.406125i \(-0.133120\pi\)
−0.406125 + 0.913818i \(0.633120\pi\)
\(14\) 0 0
\(15\) −5.70156 1.00000i −1.47214 0.258199i
\(16\) 0 0
\(17\) −0.737925 + 0.737925i −0.178973 + 0.178973i −0.790908 0.611935i \(-0.790391\pi\)
0.611935 + 0.790908i \(0.290391\pi\)
\(18\) 0 0
\(19\) 4.75362 1.09055 0.545277 0.838256i \(-0.316424\pi\)
0.545277 + 0.838256i \(0.316424\pi\)
\(20\) 0 0
\(21\) −1.00000 6.77576i −0.218218 1.47859i
\(22\) 0 0
\(23\) −3.70156 + 3.70156i −0.771829 + 0.771829i −0.978426 0.206597i \(-0.933761\pi\)
0.206597 + 0.978426i \(0.433761\pi\)
\(24\) 0 0
\(25\) 1.70156 4.70156i 0.340312 0.940312i
\(26\) 0 0
\(27\) 1.28422 1.28422i 0.247148 0.247148i
\(28\) 0 0
\(29\) 0.701562i 0.130277i 0.997876 + 0.0651384i \(0.0207489\pi\)
−0.997876 + 0.0651384i \(0.979251\pi\)
\(30\) 0 0
\(31\) 8.79790i 1.58015i −0.613010 0.790075i \(-0.710041\pi\)
0.613010 0.790075i \(-0.289959\pi\)
\(32\) 0 0
\(33\) −8.60627 8.60627i −1.49816 1.49816i
\(34\) 0 0
\(35\) 5.91391 + 0.160291i 0.999633 + 0.0270941i
\(36\) 0 0
\(37\) −3.70156 3.70156i −0.608533 0.608533i 0.334030 0.942563i \(-0.391591\pi\)
−0.942563 + 0.334030i \(0.891591\pi\)
\(38\) 0 0
\(39\) 6.70156i 1.07311i
\(40\) 0 0
\(41\) 4.75362i 0.742390i −0.928555 0.371195i \(-0.878948\pi\)
0.928555 0.371195i \(-0.121052\pi\)
\(42\) 0 0
\(43\) 5.00000 5.00000i 0.762493 0.762493i −0.214280 0.976772i \(-0.568740\pi\)
0.976772 + 0.214280i \(0.0687403\pi\)
\(44\) 0 0
\(45\) 4.75362 + 6.77576i 0.708627 + 1.01007i
\(46\) 0 0
\(47\) −8.05998 + 8.05998i −1.17567 + 1.17567i −0.194832 + 0.980837i \(0.562416\pi\)
−0.980837 + 0.194832i \(0.937584\pi\)
\(48\) 0 0
\(49\) 2.02214 + 6.70156i 0.288878 + 0.957366i
\(50\) 0 0
\(51\) 2.70156 0.378294
\(52\) 0 0
\(53\) 5.00000 5.00000i 0.686803 0.686803i −0.274721 0.961524i \(-0.588586\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 8.60627 6.03784i 1.16047 0.814142i
\(56\) 0 0
\(57\) −8.70156 8.70156i −1.15255 1.15255i
\(58\) 0 0
\(59\) −4.75362 −0.618868 −0.309434 0.950921i \(-0.600140\pi\)
−0.309434 + 0.950921i \(0.600140\pi\)
\(60\) 0 0
\(61\) 9.50723i 1.21728i 0.793448 + 0.608638i \(0.208284\pi\)
−0.793448 + 0.608638i \(0.791716\pi\)
\(62\) 0 0
\(63\) −5.83971 + 7.86185i −0.735734 + 0.990500i
\(64\) 0 0
\(65\) 5.70156 + 1.00000i 0.707192 + 0.124035i
\(66\) 0 0
\(67\) −5.00000 5.00000i −0.610847 0.610847i 0.332320 0.943167i \(-0.392169\pi\)
−0.943167 + 0.332320i \(0.892169\pi\)
\(68\) 0 0
\(69\) 13.5515 1.63141
\(70\) 0 0
\(71\) 5.40312 0.641233 0.320616 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(72\) 0 0
\(73\) 3.11473 + 3.11473i 0.364552 + 0.364552i 0.865486 0.500934i \(-0.167010\pi\)
−0.500934 + 0.865486i \(0.667010\pi\)
\(74\) 0 0
\(75\) −11.7210 + 5.49154i −1.35343 + 0.634109i
\(76\) 0 0
\(77\) 9.98578 + 7.41734i 1.13799 + 0.845285i
\(78\) 0 0
\(79\) 6.70156i 0.753985i 0.926216 + 0.376992i \(0.123042\pi\)
−0.926216 + 0.376992i \(0.876958\pi\)
\(80\) 0 0
\(81\) 6.40312 0.711458
\(82\) 0 0
\(83\) −7.86835 7.86835i −0.863664 0.863664i 0.128098 0.991762i \(-0.459113\pi\)
−0.991762 + 0.128098i \(0.959113\pi\)
\(84\) 0 0
\(85\) −0.403124 + 2.29844i −0.0437250 + 0.249301i
\(86\) 0 0
\(87\) 1.28422 1.28422i 0.137683 0.137683i
\(88\) 0 0
\(89\) −13.5515 −1.43646 −0.718229 0.695807i \(-0.755047\pi\)
−0.718229 + 0.695807i \(0.755047\pi\)
\(90\) 0 0
\(91\) 1.00000 + 6.77576i 0.104828 + 0.710293i
\(92\) 0 0
\(93\) −16.1047 + 16.1047i −1.66998 + 1.66998i
\(94\) 0 0
\(95\) 8.70156 6.10469i 0.892761 0.626328i
\(96\) 0 0
\(97\) −5.49154 + 5.49154i −0.557582 + 0.557582i −0.928618 0.371037i \(-0.879002\pi\)
0.371037 + 0.928618i \(0.379002\pi\)
\(98\) 0 0
\(99\) 17.4031i 1.74908i
\(100\) 0 0
\(101\) 4.75362i 0.473003i 0.971631 + 0.236501i \(0.0760008\pi\)
−0.971631 + 0.236501i \(0.923999\pi\)
\(102\) 0 0
\(103\) 2.92310 + 2.92310i 0.288022 + 0.288022i 0.836298 0.548276i \(-0.184716\pi\)
−0.548276 + 0.836298i \(0.684716\pi\)
\(104\) 0 0
\(105\) −10.5321 11.1189i −1.02783 1.08509i
\(106\) 0 0
\(107\) −2.40312 2.40312i −0.232319 0.232319i 0.581341 0.813660i \(-0.302528\pi\)
−0.813660 + 0.581341i \(0.802528\pi\)
\(108\) 0 0
\(109\) 3.29844i 0.315933i 0.987444 + 0.157967i \(0.0504938\pi\)
−0.987444 + 0.157967i \(0.949506\pi\)
\(110\) 0 0
\(111\) 13.5515i 1.28625i
\(112\) 0 0
\(113\) −6.29844 + 6.29844i −0.592507 + 0.592507i −0.938308 0.345801i \(-0.887607\pi\)
0.345801 + 0.938308i \(0.387607\pi\)
\(114\) 0 0
\(115\) −2.02214 + 11.5294i −0.188566 + 1.07512i
\(116\) 0 0
\(117\) −6.77576 + 6.77576i −0.626419 + 0.626419i
\(118\) 0 0
\(119\) −2.73147 + 0.403124i −0.250394 + 0.0369543i
\(120\) 0 0
\(121\) 11.1047 1.00952
\(122\) 0 0
\(123\) −8.70156 + 8.70156i −0.784593 + 0.784593i
\(124\) 0 0
\(125\) −2.92310 10.7915i −0.261450 0.965217i
\(126\) 0 0
\(127\) −2.40312 2.40312i −0.213243 0.213243i 0.592401 0.805643i \(-0.298180\pi\)
−0.805643 + 0.592401i \(0.798180\pi\)
\(128\) 0 0
\(129\) −18.3051 −1.61168
\(130\) 0 0
\(131\) 4.04429i 0.353351i −0.984269 0.176676i \(-0.943466\pi\)
0.984269 0.176676i \(-0.0565343\pi\)
\(132\) 0 0
\(133\) 10.0963 + 7.49947i 0.875464 + 0.650286i
\(134\) 0 0
\(135\) 0.701562 4.00000i 0.0603809 0.344265i
\(136\) 0 0
\(137\) −3.70156 3.70156i −0.316246 0.316246i 0.531077 0.847323i \(-0.321787\pi\)
−0.847323 + 0.531077i \(0.821787\pi\)
\(138\) 0 0
\(139\) 17.5958 1.49246 0.746229 0.665690i \(-0.231863\pi\)
0.746229 + 0.665690i \(0.231863\pi\)
\(140\) 0 0
\(141\) 29.5078 2.48501
\(142\) 0 0
\(143\) 8.60627 + 8.60627i 0.719693 + 0.719693i
\(144\) 0 0
\(145\) 0.900960 + 1.28422i 0.0748206 + 0.106649i
\(146\) 0 0
\(147\) 8.56574 15.9689i 0.706490 1.31709i
\(148\) 0 0
\(149\) 18.8062i 1.54067i −0.637641 0.770334i \(-0.720090\pi\)
0.637641 0.770334i \(-0.279910\pi\)
\(150\) 0 0
\(151\) 7.29844 0.593938 0.296969 0.954887i \(-0.404024\pi\)
0.296969 + 0.954887i \(0.404024\pi\)
\(152\) 0 0
\(153\) −2.73147 2.73147i −0.220827 0.220827i
\(154\) 0 0
\(155\) −11.2984 16.1047i −0.907512 1.29356i
\(156\) 0 0
\(157\) 8.25161 8.25161i 0.658550 0.658550i −0.296487 0.955037i \(-0.595815\pi\)
0.955037 + 0.296487i \(0.0958151\pi\)
\(158\) 0 0
\(159\) −18.3051 −1.45169
\(160\) 0 0
\(161\) −13.7016 + 2.02214i −1.07983 + 0.159367i
\(162\) 0 0
\(163\) −3.70156 + 3.70156i −0.289929 + 0.289929i −0.837052 0.547123i \(-0.815723\pi\)
0.547123 + 0.837052i \(0.315723\pi\)
\(164\) 0 0
\(165\) −26.8062 4.70156i −2.08686 0.366016i
\(166\) 0 0
\(167\) −3.30636 + 3.30636i −0.255854 + 0.255854i −0.823365 0.567511i \(-0.807906\pi\)
0.567511 + 0.823365i \(0.307906\pi\)
\(168\) 0 0
\(169\) 6.29844i 0.484495i
\(170\) 0 0
\(171\) 17.5958i 1.34559i
\(172\) 0 0
\(173\) 6.58413 + 6.58413i 0.500582 + 0.500582i 0.911619 0.411037i \(-0.134833\pi\)
−0.411037 + 0.911619i \(0.634833\pi\)
\(174\) 0 0
\(175\) 11.0313 7.30134i 0.833891 0.551929i
\(176\) 0 0
\(177\) 8.70156 + 8.70156i 0.654049 + 0.654049i
\(178\) 0 0
\(179\) 8.80625i 0.658210i 0.944293 + 0.329105i \(0.106747\pi\)
−0.944293 + 0.329105i \(0.893253\pi\)
\(180\) 0 0
\(181\) 14.2609i 1.06000i 0.847997 + 0.530000i \(0.177808\pi\)
−0.847997 + 0.530000i \(0.822192\pi\)
\(182\) 0 0
\(183\) 17.4031 17.4031i 1.28648 1.28648i
\(184\) 0 0
\(185\) −11.5294 2.02214i −0.847657 0.148671i
\(186\) 0 0
\(187\) −3.46940 + 3.46940i −0.253708 + 0.253708i
\(188\) 0 0
\(189\) 4.75362 0.701562i 0.345775 0.0510311i
\(190\) 0 0
\(191\) 2.10469 0.152290 0.0761449 0.997097i \(-0.475739\pi\)
0.0761449 + 0.997097i \(0.475739\pi\)
\(192\) 0 0
\(193\) −1.10469 + 1.10469i −0.0795171 + 0.0795171i −0.745747 0.666230i \(-0.767907\pi\)
0.666230 + 0.745747i \(0.267907\pi\)
\(194\) 0 0
\(195\) −8.60627 12.2673i −0.616308 0.878480i
\(196\) 0 0
\(197\) 2.40312 + 2.40312i 0.171216 + 0.171216i 0.787513 0.616298i \(-0.211368\pi\)
−0.616298 + 0.787513i \(0.711368\pi\)
\(198\) 0 0
\(199\) −22.3494 −1.58431 −0.792154 0.610321i \(-0.791041\pi\)
−0.792154 + 0.610321i \(0.791041\pi\)
\(200\) 0 0
\(201\) 18.3051i 1.29114i
\(202\) 0 0
\(203\) −1.10681 + 1.49007i −0.0776827 + 0.104582i
\(204\) 0 0
\(205\) −6.10469 8.70156i −0.426370 0.607743i
\(206\) 0 0
\(207\) −13.7016 13.7016i −0.952324 0.952324i
\(208\) 0 0
\(209\) 22.3494 1.54594
\(210\) 0 0
\(211\) −12.7016 −0.874412 −0.437206 0.899361i \(-0.644032\pi\)
−0.437206 + 0.899361i \(0.644032\pi\)
\(212\) 0 0
\(213\) −9.89049 9.89049i −0.677685 0.677685i
\(214\) 0 0
\(215\) 2.73147 15.5737i 0.186285 1.06212i
\(216\) 0 0
\(217\) 13.8799 18.6861i 0.942227 1.26850i
\(218\) 0 0
\(219\) 11.4031i 0.770552i
\(220\) 0 0
\(221\) −2.70156 −0.181727
\(222\) 0 0
\(223\) 15.7653 + 15.7653i 1.05572 + 1.05572i 0.998353 + 0.0573692i \(0.0182712\pi\)
0.0573692 + 0.998353i \(0.481729\pi\)
\(224\) 0 0
\(225\) 17.4031 + 6.29844i 1.16021 + 0.419896i
\(226\) 0 0
\(227\) 6.20087 6.20087i 0.411566 0.411566i −0.470718 0.882284i \(-0.656005\pi\)
0.882284 + 0.470718i \(0.156005\pi\)
\(228\) 0 0
\(229\) 0.709330 0.0468738 0.0234369 0.999725i \(-0.492539\pi\)
0.0234369 + 0.999725i \(0.492539\pi\)
\(230\) 0 0
\(231\) −4.70156 31.8567i −0.309340 2.09601i
\(232\) 0 0
\(233\) 11.1047 11.1047i 0.727492 0.727492i −0.242627 0.970120i \(-0.578009\pi\)
0.970120 + 0.242627i \(0.0780092\pi\)
\(234\) 0 0
\(235\) −4.40312 + 25.1047i −0.287228 + 1.63765i
\(236\) 0 0
\(237\) 12.2673 12.2673i 0.796847 0.796847i
\(238\) 0 0
\(239\) 24.1047i 1.55920i 0.626276 + 0.779601i \(0.284578\pi\)
−0.626276 + 0.779601i \(0.715422\pi\)
\(240\) 0 0
\(241\) 27.1030i 1.74586i −0.487845 0.872930i \(-0.662217\pi\)
0.487845 0.872930i \(-0.337783\pi\)
\(242\) 0 0
\(243\) −15.5737 15.5737i −0.999051 0.999051i
\(244\) 0 0
\(245\) 12.3078 + 9.67043i 0.786319 + 0.617821i
\(246\) 0 0
\(247\) 8.70156 + 8.70156i 0.553667 + 0.553667i
\(248\) 0 0
\(249\) 28.8062i 1.82552i
\(250\) 0 0
\(251\) 4.04429i 0.255273i −0.991821 0.127637i \(-0.959261\pi\)
0.991821 0.127637i \(-0.0407391\pi\)
\(252\) 0 0
\(253\) −17.4031 + 17.4031i −1.09413 + 1.09413i
\(254\) 0 0
\(255\) 4.94525 3.46940i 0.309683 0.217262i
\(256\) 0 0
\(257\) −9.34420 + 9.34420i −0.582875 + 0.582875i −0.935692 0.352817i \(-0.885224\pi\)
0.352817 + 0.935692i \(0.385224\pi\)
\(258\) 0 0
\(259\) −2.02214 13.7016i −0.125650 0.851374i
\(260\) 0 0
\(261\) −2.59688 −0.160743
\(262\) 0 0
\(263\) −12.4031 + 12.4031i −0.764809 + 0.764809i −0.977188 0.212378i \(-0.931879\pi\)
0.212378 + 0.977188i \(0.431879\pi\)
\(264\) 0 0
\(265\) 2.73147 15.5737i 0.167793 0.956683i
\(266\) 0 0
\(267\) 24.8062 + 24.8062i 1.51812 + 1.51812i
\(268\) 0 0
\(269\) −13.5515 −0.826251 −0.413125 0.910674i \(-0.635563\pi\)
−0.413125 + 0.910674i \(0.635563\pi\)
\(270\) 0 0
\(271\) 23.0588i 1.40072i 0.713790 + 0.700360i \(0.246977\pi\)
−0.713790 + 0.700360i \(0.753023\pi\)
\(272\) 0 0
\(273\) 10.5726 14.2336i 0.639883 0.861459i
\(274\) 0 0
\(275\) 8.00000 22.1047i 0.482418 1.33296i
\(276\) 0 0
\(277\) 11.1047 + 11.1047i 0.667216 + 0.667216i 0.957071 0.289855i \(-0.0936070\pi\)
−0.289855 + 0.957071i \(0.593607\pi\)
\(278\) 0 0
\(279\) 32.5660 1.94967
\(280\) 0 0
\(281\) −26.9109 −1.60537 −0.802686 0.596402i \(-0.796596\pi\)
−0.802686 + 0.596402i \(0.796596\pi\)
\(282\) 0 0
\(283\) 7.67672 + 7.67672i 0.456334 + 0.456334i 0.897450 0.441116i \(-0.145417\pi\)
−0.441116 + 0.897450i \(0.645417\pi\)
\(284\) 0 0
\(285\) −27.1030 4.75362i −1.60545 0.281580i
\(286\) 0 0
\(287\) 7.49947 10.0963i 0.442680 0.595968i
\(288\) 0 0
\(289\) 15.9109i 0.935937i
\(290\) 0 0
\(291\) 20.1047 1.17856
\(292\) 0 0
\(293\) −7.67672 7.67672i −0.448479 0.448479i 0.446370 0.894849i \(-0.352717\pi\)
−0.894849 + 0.446370i \(0.852717\pi\)
\(294\) 0 0
\(295\) −8.70156 + 6.10469i −0.506625 + 0.355429i
\(296\) 0 0
\(297\) 6.03784 6.03784i 0.350351 0.350351i
\(298\) 0 0
\(299\) −13.5515 −0.783705
\(300\) 0 0
\(301\) 18.5078 2.73147i 1.06677 0.157440i
\(302\) 0 0
\(303\) 8.70156 8.70156i 0.499892 0.499892i
\(304\) 0 0
\(305\) 12.2094 + 17.4031i 0.699107 + 0.996500i
\(306\) 0 0
\(307\) 9.53583 9.53583i 0.544239 0.544239i −0.380530 0.924769i \(-0.624258\pi\)
0.924769 + 0.380530i \(0.124258\pi\)
\(308\) 0 0
\(309\) 10.7016i 0.608791i
\(310\) 0 0
\(311\) 8.79790i 0.498883i −0.968390 0.249442i \(-0.919753\pi\)
0.968390 0.249442i \(-0.0802471\pi\)
\(312\) 0 0
\(313\) 11.3377 + 11.3377i 0.640847 + 0.640847i 0.950764 0.309916i \(-0.100301\pi\)
−0.309916 + 0.950764i \(0.600301\pi\)
\(314\) 0 0
\(315\) −0.593326 + 21.8907i −0.0334302 + 1.23340i
\(316\) 0 0
\(317\) −12.4031 12.4031i −0.696629 0.696629i 0.267053 0.963682i \(-0.413950\pi\)
−0.963682 + 0.267053i \(0.913950\pi\)
\(318\) 0 0
\(319\) 3.29844i 0.184677i
\(320\) 0 0
\(321\) 8.79790i 0.491051i
\(322\) 0 0
\(323\) −3.50781 + 3.50781i −0.195180 + 0.195180i
\(324\) 0 0
\(325\) 11.7210 5.49154i 0.650165 0.304616i
\(326\) 0 0
\(327\) 6.03784 6.03784i 0.333893 0.333893i
\(328\) 0 0
\(329\) −29.8345 + 4.40312i −1.64483 + 0.242752i
\(330\) 0 0
\(331\) −9.40312 −0.516842 −0.258421 0.966032i \(-0.583202\pi\)
−0.258421 + 0.966032i \(0.583202\pi\)
\(332\) 0 0
\(333\) 13.7016 13.7016i 0.750841 0.750841i
\(334\) 0 0
\(335\) −15.5737 2.73147i −0.850880 0.149236i
\(336\) 0 0
\(337\) −21.1047 21.1047i −1.14965 1.14965i −0.986622 0.163023i \(-0.947875\pi\)
−0.163023 0.986622i \(-0.552125\pi\)
\(338\) 0 0
\(339\) 23.0588 1.25238
\(340\) 0 0
\(341\) 41.3639i 2.23998i
\(342\) 0 0
\(343\) −6.27772 + 17.4238i −0.338965 + 0.940799i
\(344\) 0 0
\(345\) 24.8062 17.4031i 1.33552 0.936953i
\(346\) 0 0
\(347\) −13.7016 13.7016i −0.735538 0.735538i 0.236173 0.971711i \(-0.424107\pi\)
−0.971711 + 0.236173i \(0.924107\pi\)
\(348\) 0 0
\(349\) −8.79790 −0.470941 −0.235471 0.971881i \(-0.575663\pi\)
−0.235471 + 0.971881i \(0.575663\pi\)
\(350\) 0 0
\(351\) 4.70156 0.250951
\(352\) 0 0
\(353\) 16.0914 + 16.0914i 0.856457 + 0.856457i 0.990919 0.134462i \(-0.0429306\pi\)
−0.134462 + 0.990919i \(0.542931\pi\)
\(354\) 0 0
\(355\) 9.89049 6.93880i 0.524933 0.368273i
\(356\) 0 0
\(357\) 5.73792 + 4.26208i 0.303683 + 0.225573i
\(358\) 0 0
\(359\) 6.00000i 0.316668i −0.987386 0.158334i \(-0.949388\pi\)
0.987386 0.158334i \(-0.0506123\pi\)
\(360\) 0 0
\(361\) 3.59688 0.189309
\(362\) 0 0
\(363\) −20.3273 20.3273i −1.06691 1.06691i
\(364\) 0 0
\(365\) 9.70156 + 1.70156i 0.507803 + 0.0890638i
\(366\) 0 0
\(367\) 0.0285948 0.0285948i 0.00149264 0.00149264i −0.706360 0.707853i \(-0.749664\pi\)
0.707853 + 0.706360i \(0.249664\pi\)
\(368\) 0 0
\(369\) 17.5958 0.916001
\(370\) 0 0
\(371\) 18.5078 2.73147i 0.960878 0.141811i
\(372\) 0 0
\(373\) −6.29844 + 6.29844i −0.326121 + 0.326121i −0.851109 0.524989i \(-0.824070\pi\)
0.524989 + 0.851109i \(0.324070\pi\)
\(374\) 0 0
\(375\) −14.4031 + 25.1047i −0.743774 + 1.29640i
\(376\) 0 0
\(377\) −1.28422 + 1.28422i −0.0661407 + 0.0661407i
\(378\) 0 0
\(379\) 33.0156i 1.69590i −0.530077 0.847949i \(-0.677837\pi\)
0.530077 0.847949i \(-0.322163\pi\)
\(380\) 0 0
\(381\) 8.79790i 0.450730i
\(382\) 0 0
\(383\) −7.86835 7.86835i −0.402054 0.402054i 0.476902 0.878956i \(-0.341760\pi\)
−0.878956 + 0.476902i \(0.841760\pi\)
\(384\) 0 0
\(385\) 27.8046 + 0.753617i 1.41705 + 0.0384079i
\(386\) 0 0
\(387\) 18.5078 + 18.5078i 0.940805 + 0.940805i
\(388\) 0 0
\(389\) 20.7016i 1.04961i 0.851222 + 0.524805i \(0.175862\pi\)
−0.851222 + 0.524805i \(0.824138\pi\)
\(390\) 0 0
\(391\) 5.46295i 0.276273i
\(392\) 0 0
\(393\) −7.40312 + 7.40312i −0.373438 + 0.373438i
\(394\) 0 0
\(395\) 8.60627 + 12.2673i 0.433029 + 0.617235i
\(396\) 0 0
\(397\) −10.2452 + 10.2452i −0.514190 + 0.514190i −0.915807 0.401618i \(-0.868448\pi\)
0.401618 + 0.915807i \(0.368448\pi\)
\(398\) 0 0
\(399\) −4.75362 32.2094i −0.237979 1.61249i
\(400\) 0 0
\(401\) −24.3141 −1.21419 −0.607093 0.794631i \(-0.707664\pi\)
−0.607093 + 0.794631i \(0.707664\pi\)
\(402\) 0 0
\(403\) 16.1047 16.1047i 0.802232 0.802232i
\(404\) 0 0
\(405\) 11.7210 8.22302i 0.582422 0.408605i
\(406\) 0 0
\(407\) −17.4031 17.4031i −0.862641 0.862641i
\(408\) 0 0
\(409\) 35.9009 1.77519 0.887594 0.460627i \(-0.152375\pi\)
0.887594 + 0.460627i \(0.152375\pi\)
\(410\) 0 0
\(411\) 13.5515i 0.668447i
\(412\) 0 0
\(413\) −10.0963 7.49947i −0.496809 0.369025i
\(414\) 0 0
\(415\) −24.5078 4.29844i −1.20304 0.211002i
\(416\) 0 0
\(417\) −32.2094 32.2094i −1.57730 1.57730i
\(418\) 0 0
\(419\) −4.75362 −0.232229 −0.116115 0.993236i \(-0.537044\pi\)
−0.116115 + 0.993236i \(0.537044\pi\)
\(420\) 0 0
\(421\) 5.29844 0.258230 0.129115 0.991630i \(-0.458786\pi\)
0.129115 + 0.991630i \(0.458786\pi\)
\(422\) 0 0
\(423\) −29.8345 29.8345i −1.45060 1.45060i
\(424\) 0 0
\(425\) 2.21377 + 4.72502i 0.107384 + 0.229197i
\(426\) 0 0
\(427\) −14.9989 + 20.1927i −0.725849 + 0.977193i
\(428\) 0 0
\(429\) 31.5078i 1.52121i
\(430\) 0 0
\(431\) 7.29844 0.351553 0.175777 0.984430i \(-0.443756\pi\)
0.175777 + 0.984430i \(0.443756\pi\)
\(432\) 0 0
\(433\) −9.72746 9.72746i −0.467472 0.467472i 0.433623 0.901095i \(-0.357235\pi\)
−0.901095 + 0.433623i \(0.857235\pi\)
\(434\) 0 0
\(435\) 0.701562 4.00000i 0.0336373 0.191785i
\(436\) 0 0
\(437\) −17.5958 + 17.5958i −0.841722 + 0.841722i
\(438\) 0 0
\(439\) 19.0145 0.907511 0.453756 0.891126i \(-0.350084\pi\)
0.453756 + 0.891126i \(0.350084\pi\)
\(440\) 0 0
\(441\) −24.8062 + 7.48509i −1.18125 + 0.356433i
\(442\) 0 0
\(443\) −0.193752 + 0.193752i −0.00920541 + 0.00920541i −0.711695 0.702489i \(-0.752072\pi\)
0.702489 + 0.711695i \(0.252072\pi\)
\(444\) 0 0
\(445\) −24.8062 + 17.4031i −1.17593 + 0.824987i
\(446\) 0 0
\(447\) −34.4251 + 34.4251i −1.62825 + 1.62825i
\(448\) 0 0
\(449\) 0.701562i 0.0331088i 0.999863 + 0.0165544i \(0.00526966\pi\)
−0.999863 + 0.0165544i \(0.994730\pi\)
\(450\) 0 0
\(451\) 22.3494i 1.05239i
\(452\) 0 0
\(453\) −13.3599 13.3599i −0.627702 0.627702i
\(454\) 0 0
\(455\) 10.5321 + 11.1189i 0.493751 + 0.521262i
\(456\) 0 0
\(457\) 22.4031 + 22.4031i 1.04797 + 1.04797i 0.998790 + 0.0491842i \(0.0156621\pi\)
0.0491842 + 0.998790i \(0.484338\pi\)
\(458\) 0 0
\(459\) 1.89531i 0.0884657i
\(460\) 0 0
\(461\) 17.5958i 0.819519i 0.912194 + 0.409759i \(0.134387\pi\)
−0.912194 + 0.409759i \(0.865613\pi\)
\(462\) 0 0
\(463\) −21.1047 + 21.1047i −0.980818 + 0.980818i −0.999819 0.0190015i \(-0.993951\pi\)
0.0190015 + 0.999819i \(0.493951\pi\)
\(464\) 0 0
\(465\) −8.79790 + 50.1618i −0.407993 + 2.32620i
\(466\) 0 0
\(467\) −8.05998 + 8.05998i −0.372971 + 0.372971i −0.868558 0.495587i \(-0.834953\pi\)
0.495587 + 0.868558i \(0.334953\pi\)
\(468\) 0 0
\(469\) −2.73147 18.5078i −0.126128 0.854611i
\(470\) 0 0
\(471\) −30.2094 −1.39197
\(472\) 0 0
\(473\) 23.5078 23.5078i 1.08089 1.08089i
\(474\) 0 0
\(475\) 8.08857 22.3494i 0.371129 1.02546i
\(476\) 0 0
\(477\) 18.5078 + 18.5078i 0.847414 + 0.847414i
\(478\) 0 0
\(479\) −28.5217 −1.30319 −0.651595 0.758567i \(-0.725900\pi\)
−0.651595 + 0.758567i \(0.725900\pi\)
\(480\) 0 0
\(481\) 13.5515i 0.617896i
\(482\) 0 0
\(483\) 28.7825 + 21.3793i 1.30965 + 0.972794i
\(484\) 0 0
\(485\) −3.00000 + 17.1047i −0.136223 + 0.776684i
\(486\) 0 0
\(487\) 15.9109 + 15.9109i 0.720993 + 0.720993i 0.968808 0.247814i \(-0.0797123\pi\)
−0.247814 + 0.968808i \(0.579712\pi\)
\(488\) 0 0
\(489\) 13.5515 0.612821
\(490\) 0 0
\(491\) −24.9109 −1.12421 −0.562107 0.827064i \(-0.690009\pi\)
−0.562107 + 0.827064i \(0.690009\pi\)
\(492\) 0 0
\(493\) −0.517700 0.517700i −0.0233160 0.0233160i
\(494\) 0 0
\(495\) 22.3494 + 31.8567i 1.00453 + 1.43185i
\(496\) 0 0
\(497\) 11.4758 + 8.52415i 0.514762 + 0.382360i
\(498\) 0 0
\(499\) 4.10469i 0.183751i 0.995771 + 0.0918755i \(0.0292862\pi\)
−0.995771 + 0.0918755i \(0.970714\pi\)
\(500\) 0 0
\(501\) 12.1047 0.540798
\(502\) 0 0
\(503\) 17.1840 + 17.1840i 0.766195 + 0.766195i 0.977434 0.211240i \(-0.0677500\pi\)
−0.211240 + 0.977434i \(0.567750\pi\)
\(504\) 0 0
\(505\) 6.10469 + 8.70156i 0.271655 + 0.387214i
\(506\) 0 0
\(507\) −11.5294 + 11.5294i −0.512038 + 0.512038i
\(508\) 0 0
\(509\) 13.5515 0.600661 0.300330 0.953835i \(-0.402903\pi\)
0.300330 + 0.953835i \(0.402903\pi\)
\(510\) 0 0
\(511\) 1.70156 + 11.5294i 0.0752727 + 0.510030i
\(512\) 0 0
\(513\) 6.10469 6.10469i 0.269528 0.269528i
\(514\) 0 0
\(515\) 9.10469 + 1.59688i 0.401200 + 0.0703668i
\(516\) 0 0
\(517\) −37.8945 + 37.8945i −1.66660 + 1.66660i
\(518\) 0 0
\(519\) 24.1047i 1.05808i
\(520\) 0 0
\(521\) 3.33496i 0.146107i 0.997328 + 0.0730536i \(0.0232744\pi\)
−0.997328 + 0.0730536i \(0.976726\pi\)
\(522\) 0 0
\(523\) 14.4811 + 14.4811i 0.633213 + 0.633213i 0.948873 0.315659i \(-0.102226\pi\)
−0.315659 + 0.948873i \(0.602226\pi\)
\(524\) 0 0
\(525\) −33.5582 6.82782i −1.46460 0.297990i
\(526\) 0 0
\(527\) 6.49219 + 6.49219i 0.282804 + 0.282804i
\(528\) 0 0
\(529\) 4.40312i 0.191440i
\(530\) 0 0
\(531\) 17.5958i 0.763593i
\(532\) 0 0
\(533\) 8.70156 8.70156i 0.376906 0.376906i
\(534\) 0 0
\(535\) −7.48509 1.31281i −0.323609 0.0567579i
\(536\) 0 0
\(537\) 16.1200 16.1200i 0.695628 0.695628i
\(538\) 0 0
\(539\) 9.50723 + 31.5078i 0.409506 + 1.35714i
\(540\) 0 0
\(541\) −32.1047 −1.38029 −0.690144 0.723672i \(-0.742453\pi\)
−0.690144 + 0.723672i \(0.742453\pi\)
\(542\) 0 0
\(543\) 26.1047 26.1047i 1.12026 1.12026i
\(544\) 0 0
\(545\) 4.23592 + 6.03784i 0.181447 + 0.258632i
\(546\) 0 0
\(547\) 3.70156 + 3.70156i 0.158267 + 0.158267i 0.781799 0.623531i \(-0.214303\pi\)
−0.623531 + 0.781799i \(0.714303\pi\)
\(548\) 0 0
\(549\) −35.1916 −1.50194
\(550\) 0 0
\(551\) 3.33496i 0.142074i
\(552\) 0 0
\(553\) −10.5726 + 14.2336i −0.449593 + 0.605276i
\(554\) 0 0
\(555\) 17.4031 + 24.8062i 0.738721 + 1.05297i
\(556\) 0 0
\(557\) −8.89531 8.89531i −0.376907 0.376907i 0.493078 0.869985i \(-0.335872\pi\)
−0.869985 + 0.493078i \(0.835872\pi\)
\(558\) 0 0
\(559\) 18.3051 0.774225
\(560\) 0 0
\(561\) 12.7016 0.536260
\(562\) 0 0
\(563\) 19.2347 + 19.2347i 0.810646 + 0.810646i 0.984731 0.174085i \(-0.0556967\pi\)
−0.174085 + 0.984731i \(0.555697\pi\)
\(564\) 0 0
\(565\) −3.44080 + 19.6180i −0.144756 + 0.825333i
\(566\) 0 0
\(567\) 13.5998 + 10.1018i 0.571137 + 0.424235i
\(568\) 0 0
\(569\) 30.8062i 1.29147i 0.763564 + 0.645733i \(0.223448\pi\)
−0.763564 + 0.645733i \(0.776552\pi\)
\(570\) 0 0
\(571\) −4.20937 −0.176157 −0.0880784 0.996114i \(-0.528073\pi\)
−0.0880784 + 0.996114i \(0.528073\pi\)
\(572\) 0 0
\(573\) −3.85266 3.85266i −0.160947 0.160947i
\(574\) 0 0
\(575\) 11.1047 + 23.7016i 0.463097 + 0.988423i
\(576\) 0 0
\(577\) −10.2452 + 10.2452i −0.426512 + 0.426512i −0.887438 0.460927i \(-0.847517\pi\)
0.460927 + 0.887438i \(0.347517\pi\)
\(578\) 0 0
\(579\) 4.04429 0.168075
\(580\) 0 0
\(581\) −4.29844 29.1252i −0.178329 1.20832i
\(582\) 0 0
\(583\) 23.5078 23.5078i 0.973594 0.973594i
\(584\) 0 0
\(585\) −3.70156 + 21.1047i −0.153041 + 0.872571i
\(586\) 0 0
\(587\) −18.4682 + 18.4682i −0.762263 + 0.762263i −0.976731 0.214468i \(-0.931198\pi\)
0.214468 + 0.976731i \(0.431198\pi\)
\(588\) 0 0
\(589\) 41.8219i 1.72324i
\(590\) 0 0
\(591\) 8.79790i 0.361897i
\(592\) 0 0
\(593\) −12.4303 12.4303i −0.510453 0.510453i 0.404212 0.914665i \(-0.367546\pi\)
−0.914665 + 0.404212i \(0.867546\pi\)
\(594\) 0 0
\(595\) −4.48230 + 4.24574i −0.183756 + 0.174058i
\(596\) 0 0
\(597\) 40.9109 + 40.9109i 1.67437 + 1.67437i
\(598\) 0 0
\(599\) 6.70156i 0.273818i 0.990584 + 0.136909i \(0.0437169\pi\)
−0.990584 + 0.136909i \(0.956283\pi\)
\(600\) 0 0
\(601\) 4.75362i 0.193904i 0.995289 + 0.0969520i \(0.0309093\pi\)
−0.995289 + 0.0969520i \(0.969091\pi\)
\(602\) 0 0
\(603\) 18.5078 18.5078i 0.753696 0.753696i
\(604\) 0 0
\(605\) 20.3273 14.2609i 0.826422 0.579786i
\(606\) 0 0
\(607\) 1.44725 1.44725i 0.0587422 0.0587422i −0.677125 0.735868i \(-0.736775\pi\)
0.735868 + 0.677125i \(0.236775\pi\)
\(608\) 0 0
\(609\) 4.75362 0.701562i 0.192626 0.0284287i
\(610\) 0 0
\(611\) −29.5078 −1.19376
\(612\) 0 0
\(613\) 19.8062 19.8062i 0.799967 0.799967i −0.183123 0.983090i \(-0.558621\pi\)
0.983090 + 0.183123i \(0.0586207\pi\)
\(614\) 0 0
\(615\) −4.75362 + 27.1030i −0.191684 + 1.09290i
\(616\) 0 0
\(617\) −1.10469 1.10469i −0.0444730 0.0444730i 0.684521 0.728994i \(-0.260012\pi\)
−0.728994 + 0.684521i \(0.760012\pi\)
\(618\) 0 0
\(619\) −27.1030 −1.08936 −0.544682 0.838643i \(-0.683350\pi\)
−0.544682 + 0.838643i \(0.683350\pi\)
\(620\) 0 0
\(621\) 9.50723i 0.381512i
\(622\) 0 0
\(623\) −28.7825 21.3793i −1.15314 0.856545i
\(624\) 0 0
\(625\) −19.2094 16.0000i −0.768375 0.640000i
\(626\) 0 0
\(627\) −40.9109 40.9109i −1.63383 1.63383i
\(628\) 0 0
\(629\) 5.46295 0.217822
\(630\) 0 0
\(631\) 7.29844 0.290546 0.145273 0.989392i \(-0.453594\pi\)
0.145273 + 0.989392i \(0.453594\pi\)
\(632\) 0 0
\(633\) 23.2504 + 23.2504i 0.924120 + 0.924120i
\(634\) 0 0
\(635\) −7.48509 1.31281i −0.297037 0.0520974i
\(636\) 0 0
\(637\) −8.56574 + 15.9689i −0.339387 + 0.632709i
\(638\) 0 0
\(639\) 20.0000i 0.791188i
\(640\) 0 0
\(641\) 9.79063 0.386707 0.193353 0.981129i \(-0.438064\pi\)
0.193353 + 0.981129i \(0.438064\pi\)
\(642\) 0 0
\(643\) −24.1799 24.1799i −0.953564 0.953564i 0.0454049 0.998969i \(-0.485542\pi\)
−0.998969 + 0.0454049i \(0.985542\pi\)
\(644\) 0 0
\(645\) −33.5078 + 23.5078i −1.31937 + 0.925619i
\(646\) 0 0
\(647\) 32.4030 32.4030i 1.27389 1.27389i 0.329863 0.944029i \(-0.392998\pi\)
0.944029 0.329863i \(-0.107002\pi\)
\(648\) 0 0
\(649\) −22.3494 −0.877292
\(650\) 0 0
\(651\) −59.6125 + 8.79790i −2.33640 + 0.344817i
\(652\) 0 0
\(653\) 5.00000 5.00000i 0.195665 0.195665i −0.602474 0.798139i \(-0.705818\pi\)
0.798139 + 0.602474i \(0.205818\pi\)
\(654\) 0 0
\(655\) −5.19375 7.40312i −0.202937 0.289264i
\(656\) 0 0
\(657\) −11.5294 + 11.5294i −0.449804 + 0.449804i
\(658\) 0 0
\(659\) 38.9109i 1.51575i 0.652397 + 0.757877i \(0.273764\pi\)
−0.652397 + 0.757877i \(0.726236\pi\)
\(660\) 0 0
\(661\) 19.0145i 0.739577i 0.929116 + 0.369789i \(0.120570\pi\)
−0.929116 + 0.369789i \(0.879430\pi\)
\(662\) 0 0
\(663\) 4.94525 + 4.94525i 0.192057 + 0.192057i
\(664\) 0 0
\(665\) 28.1125 + 0.761961i 1.09015 + 0.0295476i
\(666\) 0 0
\(667\) −2.59688 2.59688i −0.100551 0.100551i
\(668\) 0 0
\(669\) 57.7172i 2.23148i
\(670\) 0 0
\(671\) 44.6989i 1.72558i
\(672\) 0 0
\(673\) 13.7016 13.7016i 0.528156 0.528156i −0.391866 0.920022i \(-0.628170\pi\)
0.920022 + 0.391866i \(0.128170\pi\)
\(674\) 0 0
\(675\) −3.85266 8.22302i −0.148289 0.316504i
\(676\) 0 0
\(677\) 12.1043 12.1043i 0.465205 0.465205i −0.435152 0.900357i \(-0.643305\pi\)
0.900357 + 0.435152i \(0.143305\pi\)
\(678\) 0 0
\(679\) −20.3273 + 3.00000i −0.780090 + 0.115129i
\(680\) 0 0
\(681\) −22.7016 −0.869926
\(682\) 0 0
\(683\) 28.5078 28.5078i 1.09082 1.09082i 0.0953801 0.995441i \(-0.469593\pi\)
0.995441 0.0953801i \(-0.0304067\pi\)
\(684\) 0 0
\(685\) −11.5294 2.02214i −0.440515 0.0772621i
\(686\) 0 0
\(687\) −1.29844 1.29844i −0.0495385 0.0495385i
\(688\) 0 0
\(689\) 18.3051 0.697370
\(690\) 0 0
\(691\) 32.5660i 1.23887i −0.785048 0.619434i \(-0.787362\pi\)
0.785048 0.619434i \(-0.212638\pi\)
\(692\) 0 0
\(693\) −27.4558 + 36.9630i −1.04296 + 1.40411i
\(694\) 0 0
\(695\) 32.2094 22.5969i 1.22177 0.857148i
\(696\) 0 0
\(697\) 3.50781 + 3.50781i 0.132868 + 0.132868i
\(698\) 0 0
\(699\) −40.6546 −1.53770
\(700\) 0 0
\(701\) −34.7016 −1.31066 −0.655330 0.755343i \(-0.727470\pi\)
−0.655330 + 0.755343i \(0.727470\pi\)
\(702\) 0 0
\(703\) −17.5958 17.5958i −0.663639 0.663639i
\(704\) 0 0
\(705\) 54.0145 37.8945i 2.03430 1.42719i
\(706\) 0 0
\(707\) −7.49947 + 10.0963i −0.282046 + 0.379712i
\(708\) 0 0
\(709\) 11.5078i 0.432185i −0.976373 0.216092i \(-0.930669\pi\)
0.976373 0.216092i \(-0.0693313\pi\)
\(710\) 0 0
\(711\) −24.8062 −0.930307
\(712\) 0 0
\(713\) 32.5660 + 32.5660i 1.21961 + 1.21961i
\(714\) 0 0
\(715\) 26.8062 + 4.70156i 1.00250 + 0.175828i
\(716\) 0 0
\(717\) 44.1240 44.1240i 1.64784 1.64784i
\(718\) 0 0
\(719\) 44.6989 1.66699 0.833493 0.552530i \(-0.186338\pi\)
0.833493 + 0.552530i \(0.186338\pi\)
\(720\) 0 0
\(721\) 1.59688 + 10.8200i 0.0594708 + 0.402960i
\(722\) 0 0
\(723\) −49.6125 + 49.6125i −1.84511 + 1.84511i
\(724\) 0 0
\(725\) 3.29844 + 1.19375i 0.122501 + 0.0443348i
\(726\) 0 0
\(727\) 19.5608 19.5608i 0.725469 0.725469i −0.244245 0.969714i \(-0.578540\pi\)
0.969714 + 0.244245i \(0.0785401\pi\)
\(728\) 0 0
\(729\) 37.8062i 1.40023i
\(730\) 0 0
\(731\) 7.37925i 0.272931i
\(732\) 0 0
\(733\) 19.4263 + 19.4263i 0.717528 + 0.717528i 0.968098 0.250571i \(-0.0806183\pi\)
−0.250571 + 0.968098i \(0.580618\pi\)
\(734\) 0 0
\(735\) −4.82782 40.2315i −0.178077 1.48396i
\(736\) 0 0
\(737\) −23.5078 23.5078i −0.865921 0.865921i
\(738\) 0 0
\(739\) 25.5078i 0.938320i −0.883113 0.469160i \(-0.844557\pi\)
0.883113 0.469160i \(-0.155443\pi\)
\(740\) 0 0
\(741\) 31.8567i 1.17028i
\(742\) 0 0
\(743\) 32.0156 32.0156i 1.17454 1.17454i 0.193424 0.981115i \(-0.438041\pi\)
0.981115 0.193424i \(-0.0619593\pi\)
\(744\) 0 0
\(745\) −24.1513 34.4251i −0.884837 1.26124i
\(746\) 0 0
\(747\) 29.1252 29.1252i 1.06563 1.06563i
\(748\) 0 0
\(749\) −1.31281 8.89531i −0.0479692 0.325028i
\(750\) 0 0
\(751\) 2.10469 0.0768011 0.0384005 0.999262i \(-0.487774\pi\)
0.0384005 + 0.999262i \(0.487774\pi\)
\(752\) 0 0
\(753\) −7.40312 + 7.40312i −0.269785 + 0.269785i
\(754\) 0 0
\(755\) 13.3599 9.37279i 0.486216 0.341111i
\(756\) 0 0
\(757\) −6.29844 6.29844i −0.228921 0.228921i 0.583321 0.812242i \(-0.301753\pi\)
−0.812242 + 0.583321i \(0.801753\pi\)
\(758\) 0 0
\(759\) 63.7133 2.31265
\(760\) 0 0
\(761\) 35.1916i 1.27570i 0.770163 + 0.637848i \(0.220175\pi\)
−0.770163 + 0.637848i \(0.779825\pi\)
\(762\) 0 0
\(763\) −5.20373 + 7.00565i −0.188388 + 0.253621i
\(764\) 0 0
\(765\) −8.50781 1.49219i −0.307601 0.0539502i
\(766\) 0 0
\(767\) −8.70156 8.70156i −0.314195 0.314195i
\(768\) 0 0
\(769\) −26.3937 −0.951782 −0.475891 0.879504i \(-0.657874\pi\)
−0.475891 + 0.879504i \(0.657874\pi\)
\(770\) 0 0
\(771\) 34.2094 1.23202
\(772\) 0 0
\(773\) −20.5189 20.5189i −0.738014 0.738014i 0.234179 0.972193i \(-0.424760\pi\)
−0.972193 + 0.234179i \(0.924760\pi\)
\(774\) 0 0
\(775\) −41.3639 14.9702i −1.48583 0.537745i
\(776\) 0 0
\(777\) −21.3793 + 28.7825i −0.766979 + 1.03257i
\(778\) 0 0
\(779\) 22.5969i 0.809617i
\(780\) 0 0
\(781\) 25.4031 0.908995
\(782\) 0 0
\(783\) 0.900960 + 0.900960i 0.0321977 + 0.0321977i
\(784\) 0 0
\(785\) 4.50781 25.7016i 0.160891 0.917328i
\(786\) 0 0
\(787\) −3.30636 + 3.30636i −0.117859 + 0.117859i −0.763576 0.645717i \(-0.776558\pi\)
0.645717 + 0.763576i \(0.276558\pi\)
\(788\) 0 0
\(789\) 45.4082 1.61657
\(790\) 0 0
\(791\) −23.3141 + 3.44080i −0.828953 + 0.122341i
\(792\) 0 0
\(793\) −17.4031 + 17.4031i −0.618003 + 0.618003i
\(794\) 0 0
\(795\) −33.5078 + 23.5078i −1.18840 + 0.833736i
\(796\) 0 0
\(797\) 16.8579 16.8579i 0.597137 0.597137i −0.342413 0.939550i \(-0.611244\pi\)
0.939550 + 0.342413i \(0.111244\pi\)
\(798\) 0 0
\(799\) 11.8953i 0.420826i
\(800\) 0 0
\(801\) 50.1618i 1.77238i
\(802\) 0 0
\(803\) 14.6441 + 14.6441i 0.516779 + 0.516779i
\(804\) 0 0
\(805\) −22.4840 + 21.2974i −0.792458 + 0.750634i
\(806\) 0 0
\(807\) 24.8062 + 24.8062i 0.873221 + 0.873221i
\(808\) 0 0
\(809\) 19.2984i 0.678497i −0.940697 0.339248i \(-0.889827\pi\)
0.940697 0.339248i \(-0.110173\pi\)
\(810\) 0 0
\(811\) 4.04429i 0.142014i −0.997476 0.0710071i \(-0.977379\pi\)
0.997476 0.0710071i \(-0.0226213\pi\)
\(812\) 0 0
\(813\) 42.2094 42.2094i 1.48035 1.48035i
\(814\) 0 0
\(815\) −2.02214 + 11.5294i −0.0708326 + 0.403856i
\(816\) 0 0
\(817\) 23.7681 23.7681i 0.831540 0.831540i
\(818\) 0 0
\(819\) −25.0809 + 3.70156i −0.876398 + 0.129343i
\(820\) 0 0
\(821\) 52.3141 1.82577 0.912887 0.408213i \(-0.133848\pi\)
0.912887 + 0.408213i \(0.133848\pi\)
\(822\) 0 0
\(823\) 10.1938 10.1938i 0.355332 0.355332i −0.506757 0.862089i \(-0.669156\pi\)
0.862089 + 0.506757i \(0.169156\pi\)
\(824\) 0 0
\(825\) −55.1070 + 25.8188i −1.91858 + 0.898896i
\(826\) 0 0
\(827\) −31.1047 31.1047i −1.08162 1.08162i −0.996359 0.0852569i \(-0.972829\pi\)
−0.0852569 0.996359i \(-0.527171\pi\)
\(828\) 0 0
\(829\) −10.2166 −0.354836 −0.177418 0.984136i \(-0.556774\pi\)
−0.177418 + 0.984136i \(0.556774\pi\)
\(830\) 0 0
\(831\) 40.6546i 1.41029i
\(832\) 0 0
\(833\) −6.43744 3.45306i −0.223044 0.119641i
\(834\) 0 0
\(835\) −1.80625 + 10.2984i −0.0625078 + 0.356392i
\(836\) 0 0
\(837\) −11.2984 11.2984i −0.390531 0.390531i
\(838\) 0 0
\(839\) −35.1916 −1.21495 −0.607475 0.794339i \(-0.707817\pi\)
−0.607475 + 0.794339i \(0.707817\pi\)
\(840\) 0 0
\(841\) 28.5078 0.983028
\(842\) 0 0
\(843\) 49.2608 + 49.2608i 1.69663 + 1.69663i
\(844\) 0 0
\(845\) −8.08857 11.5294i −0.278255 0.396623i
\(846\) 0 0
\(847\) 23.5856 + 17.5191i 0.810409 + 0.601964i
\(848\) 0 0
\(849\) 28.1047i 0.964550i
\(850\) 0 0
\(851\) 27.4031 0.939367
\(852\) 0 0
\(853\) −23.9883 23.9883i −0.821344 0.821344i 0.164957 0.986301i \(-0.447252\pi\)
−0.986301 + 0.164957i \(0.947252\pi\)
\(854\) 0 0
\(855\) 22.5969 + 32.2094i 0.772797 + 1.10154i
\(856\) 0 0
\(857\) 8.25161 8.25161i 0.281870 0.281870i −0.551985 0.833854i \(-0.686129\pi\)
0.833854 + 0.551985i \(0.186129\pi\)
\(858\) 0 0
\(859\) 27.1030 0.924744 0.462372 0.886686i \(-0.346998\pi\)
0.462372 + 0.886686i \(0.346998\pi\)
\(860\) 0 0
\(861\) −32.2094 + 4.75362i −1.09769 + 0.162003i
\(862\) 0 0
\(863\) −12.4031 + 12.4031i −0.422207 + 0.422207i −0.885963 0.463756i \(-0.846502\pi\)
0.463756 + 0.885963i \(0.346502\pi\)
\(864\) 0 0
\(865\) 20.5078 + 3.59688i 0.697286 + 0.122297i
\(866\) 0 0
\(867\) 29.1252 29.1252i 0.989143 0.989143i
\(868\) 0 0
\(869\) 31.5078i 1.06883i
\(870\) 0 0
\(871\) 18.3051i 0.620246i
\(872\) 0 0
\(873\) −20.3273 20.3273i −0.687974 0.687974i
\(874\) 0 0
\(875\) 10.8165 27.5319i 0.365664 0.930747i
\(876\) 0 0
\(877\) 39.8062 + 39.8062i 1.34416 + 1.34416i 0.891870 + 0.452291i \(0.149393\pi\)
0.452291 + 0.891870i \(0.350607\pi\)
\(878\) 0 0
\(879\) 28.1047i 0.947948i
\(880\) 0 0
\(881\) 27.1030i 0.913125i −0.889691 0.456562i \(-0.849081\pi\)
0.889691 0.456562i \(-0.150919\pi\)
\(882\) 0 0
\(883\) −23.7016 + 23.7016i −0.797621 + 0.797621i −0.982720 0.185099i \(-0.940739\pi\)
0.185099 + 0.982720i \(0.440739\pi\)
\(884\) 0 0
\(885\) 27.1030 + 4.75362i 0.911059 + 0.159791i
\(886\) 0 0
\(887\) −0.872365 + 0.872365i −0.0292911 + 0.0292911i −0.721601 0.692310i \(-0.756593\pi\)
0.692310 + 0.721601i \(0.256593\pi\)
\(888\) 0 0
\(889\) −1.31281 8.89531i −0.0440304 0.298339i
\(890\) 0 0
\(891\) 30.1047 1.00854
\(892\) 0 0
\(893\) −38.3141 + 38.3141i −1.28213 + 1.28213i
\(894\) 0 0
\(895\) 11.3092 + 16.1200i 0.378023 + 0.538831i
\(896\) 0 0
\(897\) 24.8062 + 24.8062i 0.828257 + 0.828257i
\(898\) 0 0
\(899\) 6.17228 0.205857
\(900\) 0 0
\(901\) 7.37925i 0.245838i
\(902\) 0 0
\(903\) −38.8788 28.8788i −1.29381 0.961026i
\(904\) 0 0
\(905\) 18.3141 + 26.1047i 0.608780 + 0.867749i
\(906\) 0 0
\(907\) 18.5078 + 18.5078i 0.614542 + 0.614542i 0.944126 0.329584i \(-0.106909\pi\)
−0.329584 + 0.944126i \(0.606909\pi\)
\(908\) 0 0
\(909\) −17.5958 −0.583616
\(910\) 0 0
\(911\) 10.5969 0.351090 0.175545 0.984471i \(-0.443831\pi\)
0.175545 + 0.984471i \(0.443831\pi\)
\(912\) 0 0
\(913\) −36.9935 36.9935i −1.22431 1.22431i
\(914\) 0 0
\(915\) 9.50723 54.2061i 0.314299 1.79200i
\(916\) 0 0
\(917\) 6.38040 8.58978i 0.210700 0.283659i
\(918\) 0 0
\(919\) 22.9109i 0.755762i −0.925854 0.377881i \(-0.876653\pi\)
0.925854 0.377881i \(-0.123347\pi\)
\(920\) 0 0
\(921\) −34.9109 −1.15035
\(922\) 0 0
\(923\) 9.89049 + 9.89049i 0.325550 + 0.325550i
\(924\) 0 0
\(925\) −23.7016 + 11.1047i −0.779303 + 0.365120i
\(926\) 0 0
\(927\) −10.8200 + 10.8200i −0.355377 + 0.355377i
\(928\) 0 0
\(929\) −2.12799 −0.0698171 −0.0349085 0.999391i \(-0.511114\pi\)
−0.0349085 + 0.999391i \(0.511114\pi\)
\(930\) 0 0
\(931\) 9.61250 + 31.8567i 0.315037 + 1.04406i
\(932\) 0 0
\(933\) −16.1047 + 16.1047i −0.527244 + 0.527244i
\(934\) 0 0
\(935\) −1.89531 + 10.8062i −0.0619834 + 0.353402i
\(936\) 0 0
\(937\) 40.6260 40.6260i 1.32719 1.32719i 0.419383 0.907809i \(-0.362246\pi\)
0.907809 0.419383i \(-0.137754\pi\)
\(938\) 0 0
\(939\) 41.5078i 1.35456i
\(940\) 0 0
\(941\) 46.1175i 1.50339i 0.659512 + 0.751694i \(0.270763\pi\)
−0.659512 + 0.751694i \(0.729237\pi\)
\(942\) 0 0
\(943\) 17.5958 + 17.5958i 0.572998 + 0.572998i
\(944\) 0 0
\(945\) 7.80060 7.38891i 0.253754 0.240361i
\(946\) 0 0
\(947\) 15.0000 + 15.0000i 0.487435 + 0.487435i 0.907496 0.420061i \(-0.137991\pi\)
−0.420061 + 0.907496i \(0.637991\pi\)
\(948\) 0 0
\(949\) 11.4031i 0.370161i
\(950\) 0 0
\(951\) 45.4082i 1.47246i
\(952\) 0 0
\(953\) 39.8062 39.8062i 1.28945 1.28945i 0.354331 0.935120i \(-0.384709\pi\)
0.935120 0.354331i \(-0.115291\pi\)
\(954\) 0 0
\(955\) 3.85266 2.70288i 0.124669 0.0874631i
\(956\) 0 0
\(957\) 6.03784 6.03784i 0.195176 0.195176i
\(958\) 0 0
\(959\) −2.02214 13.7016i −0.0652984 0.442447i
\(960\) 0 0
\(961\) −46.4031 −1.49687
\(962\) 0 0
\(963\) 8.89531 8.89531i 0.286647 0.286647i
\(964\) 0 0
\(965\) −0.603484 + 3.44080i −0.0194268 + 0.110763i
\(966\) 0 0
\(967\) −25.0000 25.0000i −0.803946 0.803946i 0.179764 0.983710i \(-0.442467\pi\)
−0.983710 + 0.179764i \(0.942467\pi\)
\(968\) 0 0
\(969\) 12.8422 0.412551
\(970\) 0 0
\(971\) 54.9154i 1.76232i −0.472819 0.881160i \(-0.656763\pi\)
0.472819 0.881160i \(-0.343237\pi\)
\(972\) 0 0
\(973\) 37.3722 + 27.7597i 1.19810 + 0.889936i
\(974\) 0 0
\(975\) −31.5078 11.4031i −1.00906 0.365192i
\(976\) 0 0
\(977\) −32.4031 32.4031i −1.03667 1.03667i −0.999302 0.0373661i \(-0.988103\pi\)
−0.0373661 0.999302i \(-0.511897\pi\)
\(978\) 0 0
\(979\) −63.7133 −2.03629
\(980\) 0 0
\(981\) −12.2094 −0.389815
\(982\) 0 0
\(983\) 12.4303 + 12.4303i 0.396466 + 0.396466i 0.876985 0.480518i \(-0.159551\pi\)
−0.480518 + 0.876985i \(0.659551\pi\)
\(984\) 0 0
\(985\) 7.48509 + 1.31281i 0.238495 + 0.0418297i
\(986\) 0 0
\(987\) 62.6725 + 46.5525i 1.99489 + 1.48178i
\(988\) 0 0
\(989\) 37.0156i 1.17703i
\(990\) 0 0
\(991\) 17.6125 0.559479 0.279740 0.960076i \(-0.409752\pi\)
0.279740 + 0.960076i \(0.409752\pi\)
\(992\) 0 0
\(993\) 17.2125 + 17.2125i 0.546224 + 0.546224i
\(994\) 0 0
\(995\) −40.9109 + 28.7016i −1.29696 + 0.909901i
\(996\) 0 0
\(997\) −2.15658 + 2.15658i −0.0682997 + 0.0682997i −0.740432 0.672132i \(-0.765379\pi\)
0.672132 + 0.740432i \(0.265379\pi\)
\(998\) 0 0
\(999\) −9.50723 −0.300796
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 560.2.bj.b.97.1 8
4.3 odd 2 140.2.m.a.97.4 yes 8
5.3 odd 4 inner 560.2.bj.b.433.4 8
7.6 odd 2 inner 560.2.bj.b.97.4 8
12.11 even 2 1260.2.ba.a.937.1 8
20.3 even 4 140.2.m.a.13.1 8
20.7 even 4 700.2.m.c.293.4 8
20.19 odd 2 700.2.m.c.657.1 8
28.3 even 6 980.2.v.b.117.1 16
28.11 odd 6 980.2.v.b.117.4 16
28.19 even 6 980.2.v.b.717.4 16
28.23 odd 6 980.2.v.b.717.1 16
28.27 even 2 140.2.m.a.97.1 yes 8
35.13 even 4 inner 560.2.bj.b.433.1 8
60.23 odd 4 1260.2.ba.a.433.4 8
84.83 odd 2 1260.2.ba.a.937.4 8
140.3 odd 12 980.2.v.b.313.1 16
140.23 even 12 980.2.v.b.913.1 16
140.27 odd 4 700.2.m.c.293.1 8
140.83 odd 4 140.2.m.a.13.4 yes 8
140.103 odd 12 980.2.v.b.913.4 16
140.123 even 12 980.2.v.b.313.4 16
140.139 even 2 700.2.m.c.657.4 8
420.83 even 4 1260.2.ba.a.433.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.m.a.13.1 8 20.3 even 4
140.2.m.a.13.4 yes 8 140.83 odd 4
140.2.m.a.97.1 yes 8 28.27 even 2
140.2.m.a.97.4 yes 8 4.3 odd 2
560.2.bj.b.97.1 8 1.1 even 1 trivial
560.2.bj.b.97.4 8 7.6 odd 2 inner
560.2.bj.b.433.1 8 35.13 even 4 inner
560.2.bj.b.433.4 8 5.3 odd 4 inner
700.2.m.c.293.1 8 140.27 odd 4
700.2.m.c.293.4 8 20.7 even 4
700.2.m.c.657.1 8 20.19 odd 2
700.2.m.c.657.4 8 140.139 even 2
980.2.v.b.117.1 16 28.3 even 6
980.2.v.b.117.4 16 28.11 odd 6
980.2.v.b.313.1 16 140.3 odd 12
980.2.v.b.313.4 16 140.123 even 12
980.2.v.b.717.1 16 28.23 odd 6
980.2.v.b.717.4 16 28.19 even 6
980.2.v.b.913.1 16 140.23 even 12
980.2.v.b.913.4 16 140.103 odd 12
1260.2.ba.a.433.1 8 420.83 even 4
1260.2.ba.a.433.4 8 60.23 odd 4
1260.2.ba.a.937.1 8 12.11 even 2
1260.2.ba.a.937.4 8 84.83 odd 2