Properties

Label 1400.2.bh.d.849.1
Level $1400$
Weight $2$
Character 1400.849
Analytic conductor $11.179$
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] = [1400,2,Mod(249,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, 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("1400.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (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: \(11.1790562830\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 849.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.849
Dual form 1400.2.bh.d.249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.73205 + 1.00000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.73205 + 1.00000i) q^{3} +(-2.59808 + 0.500000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} +2.00000i q^{13} +(-2.59808 + 1.50000i) q^{17} +(4.00000 - 3.46410i) q^{21} +(2.59808 + 1.50000i) q^{23} -4.00000i q^{27} +6.00000 q^{29} +(-4.50000 - 7.79423i) q^{31} +(6.92820 + 4.00000i) q^{33} +(-2.00000 - 3.46410i) q^{39} +5.00000 q^{41} +6.00000i q^{43} +(-7.79423 - 4.50000i) q^{47} +(6.50000 - 2.59808i) q^{49} +(3.00000 - 5.19615i) q^{51} +(5.19615 - 3.00000i) q^{53} +(4.00000 + 6.92820i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-0.866025 + 2.50000i) q^{63} +(12.1244 - 7.00000i) q^{67} -6.00000 q^{69} +11.0000 q^{71} +(1.73205 - 1.00000i) q^{73} +(6.92820 + 8.00000i) q^{77} +(4.50000 - 7.79423i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.00000i q^{83} +(-10.3923 + 6.00000i) q^{87} +(5.50000 - 9.52628i) q^{89} +(-1.00000 - 5.19615i) q^{91} +(15.5885 + 9.00000i) q^{93} -11.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} - 8 q^{11} + 16 q^{21} + 24 q^{29} - 18 q^{31} - 8 q^{39} + 20 q^{41} + 26 q^{49} + 12 q^{51} + 16 q^{59} - 16 q^{61} - 24 q^{69} + 44 q^{71} + 18 q^{79} + 22 q^{81} + 22 q^{89} - 4 q^{91} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) −1.73205 + 1.00000i −1.00000 + 0.577350i −0.908248 0.418432i \(-0.862580\pi\)
−0.0917517 + 0.995782i \(0.529247\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.59808 + 0.500000i −0.981981 + 0.188982i
\(8\) 0 0
\(9\) 0.500000 0.866025i 0.166667 0.288675i
\(10\) 0 0
\(11\) −2.00000 3.46410i −0.603023 1.04447i −0.992361 0.123371i \(-0.960630\pi\)
0.389338 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.59808 + 1.50000i −0.630126 + 0.363803i −0.780801 0.624780i \(-0.785189\pi\)
0.150675 + 0.988583i \(0.451855\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 4.00000 3.46410i 0.872872 0.755929i
\(22\) 0 0
\(23\) 2.59808 + 1.50000i 0.541736 + 0.312772i 0.745782 0.666190i \(-0.232076\pi\)
−0.204046 + 0.978961i \(0.565409\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.50000 7.79423i −0.808224 1.39988i −0.914093 0.405505i \(-0.867096\pi\)
0.105869 0.994380i \(-0.466238\pi\)
\(32\) 0 0
\(33\) 6.92820 + 4.00000i 1.20605 + 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) −2.00000 3.46410i −0.320256 0.554700i
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.79423 4.50000i −1.13691 0.656392i −0.191243 0.981543i \(-0.561252\pi\)
−0.945662 + 0.325150i \(0.894585\pi\)
\(48\) 0 0
\(49\) 6.50000 2.59808i 0.928571 0.371154i
\(50\) 0 0
\(51\) 3.00000 5.19615i 0.420084 0.727607i
\(52\) 0 0
\(53\) 5.19615 3.00000i 0.713746 0.412082i −0.0987002 0.995117i \(-0.531468\pi\)
0.812447 + 0.583036i \(0.198135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) −0.866025 + 2.50000i −0.109109 + 0.314970i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.1244 7.00000i 1.48123 0.855186i 0.481452 0.876472i \(-0.340109\pi\)
0.999773 + 0.0212861i \(0.00677610\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) 0 0
\(73\) 1.73205 1.00000i 0.202721 0.117041i −0.395203 0.918594i \(-0.629326\pi\)
0.597924 + 0.801553i \(0.295992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 + 8.00000i 0.789542 + 0.911685i
\(78\) 0 0
\(79\) 4.50000 7.79423i 0.506290 0.876919i −0.493684 0.869641i \(-0.664350\pi\)
0.999974 0.00727784i \(-0.00231663\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −10.3923 + 6.00000i −1.11417 + 0.643268i
\(88\) 0 0
\(89\) 5.50000 9.52628i 0.582999 1.00978i −0.412123 0.911128i \(-0.635213\pi\)
0.995122 0.0986553i \(-0.0314541\pi\)
\(90\) 0 0
\(91\) −1.00000 5.19615i −0.104828 0.544705i
\(92\) 0 0
\(93\) 15.5885 + 9.00000i 1.61645 + 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000i 1.11688i −0.829545 0.558440i \(-0.811400\pi\)
0.829545 0.558440i \(-0.188600\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 4.00000 + 6.92820i 0.398015 + 0.689382i 0.993481 0.113998i \(-0.0363659\pi\)
−0.595466 + 0.803380i \(0.703033\pi\)
\(102\) 0 0
\(103\) 12.9904 + 7.50000i 1.27998 + 0.738997i 0.976845 0.213950i \(-0.0686331\pi\)
0.303136 + 0.952947i \(0.401966\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.92820 4.00000i −0.669775 0.386695i 0.126217 0.992003i \(-0.459717\pi\)
−0.795991 + 0.605308i \(0.793050\pi\)
\(108\) 0 0
\(109\) −7.00000 12.1244i −0.670478 1.16130i −0.977769 0.209687i \(-0.932756\pi\)
0.307290 0.951616i \(-0.400578\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.0000i 1.41108i −0.708669 0.705541i \(-0.750704\pi\)
0.708669 0.705541i \(-0.249296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.73205 + 1.00000i 0.160128 + 0.0924500i
\(118\) 0 0
\(119\) 6.00000 5.19615i 0.550019 0.476331i
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) −8.66025 + 5.00000i −0.780869 + 0.450835i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) −6.00000 10.3923i −0.528271 0.914991i
\(130\) 0 0
\(131\) −6.00000 + 10.3923i −0.524222 + 0.907980i 0.475380 + 0.879781i \(0.342311\pi\)
−0.999602 + 0.0281993i \(0.991023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.7224 8.50000i 1.25782 0.726204i 0.285171 0.958477i \(-0.407949\pi\)
0.972651 + 0.232273i \(0.0746161\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) 18.0000 1.51587
\(142\) 0 0
\(143\) 6.92820 4.00000i 0.579365 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −8.66025 + 11.0000i −0.714286 + 0.907265i
\(148\) 0 0
\(149\) −10.0000 + 17.3205i −0.819232 + 1.41895i 0.0870170 + 0.996207i \(0.472267\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(150\) 0 0
\(151\) −10.0000 17.3205i −0.813788 1.40952i −0.910195 0.414181i \(-0.864068\pi\)
0.0964061 0.995342i \(-0.469265\pi\)
\(152\) 0 0
\(153\) 3.00000i 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.66025 5.00000i 0.691164 0.399043i −0.112884 0.993608i \(-0.536009\pi\)
0.804048 + 0.594565i \(0.202676\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) −7.50000 2.59808i −0.591083 0.204757i
\(162\) 0 0
\(163\) −20.7846 12.0000i −1.62798 0.939913i −0.984696 0.174282i \(-0.944240\pi\)
−0.643280 0.765631i \(-0.722427\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.8564 + 8.00000i 1.05348 + 0.608229i 0.923622 0.383304i \(-0.125214\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −13.8564 8.00000i −1.04151 0.601317i
\(178\) 0 0
\(179\) 2.00000 + 3.46410i 0.149487 + 0.258919i 0.931038 0.364922i \(-0.118904\pi\)
−0.781551 + 0.623841i \(0.785571\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 16.0000i 1.18275i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.3923 + 6.00000i 0.759961 + 0.438763i
\(188\) 0 0
\(189\) 2.00000 + 10.3923i 0.145479 + 0.755929i
\(190\) 0 0
\(191\) 6.50000 11.2583i 0.470323 0.814624i −0.529101 0.848559i \(-0.677471\pi\)
0.999424 + 0.0339349i \(0.0108039\pi\)
\(192\) 0 0
\(193\) −4.33013 + 2.50000i −0.311689 + 0.179954i −0.647682 0.761911i \(-0.724262\pi\)
0.335993 + 0.941865i \(0.390928\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 10.5000 + 18.1865i 0.744325 + 1.28921i 0.950509 + 0.310696i \(0.100562\pi\)
−0.206184 + 0.978513i \(0.566105\pi\)
\(200\) 0 0
\(201\) −14.0000 + 24.2487i −0.987484 + 1.71037i
\(202\) 0 0
\(203\) −15.5885 + 3.00000i −1.09410 + 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.59808 1.50000i 0.180579 0.104257i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 0 0
\(213\) −19.0526 + 11.0000i −1.30546 + 0.753708i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.5885 + 18.0000i 1.05821 + 1.22192i
\(218\) 0 0
\(219\) −2.00000 + 3.46410i −0.135147 + 0.234082i
\(220\) 0 0
\(221\) −3.00000 5.19615i −0.201802 0.349531i
\(222\) 0 0
\(223\) 11.0000i 0.736614i −0.929704 0.368307i \(-0.879937\pi\)
0.929704 0.368307i \(-0.120063\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.0526 + 11.0000i −1.26456 + 0.730096i −0.973954 0.226746i \(-0.927191\pi\)
−0.290609 + 0.956842i \(0.593858\pi\)
\(228\) 0 0
\(229\) −4.00000 + 6.92820i −0.264327 + 0.457829i −0.967387 0.253302i \(-0.918483\pi\)
0.703060 + 0.711131i \(0.251817\pi\)
\(230\) 0 0
\(231\) −20.0000 6.92820i −1.31590 0.455842i
\(232\) 0 0
\(233\) −19.0526 11.0000i −1.24817 0.720634i −0.277429 0.960746i \(-0.589482\pi\)
−0.970745 + 0.240112i \(0.922816\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 18.0000i 1.16923i
\(238\) 0 0
\(239\) −11.0000 −0.711531 −0.355765 0.934575i \(-0.615780\pi\)
−0.355765 + 0.934575i \(0.615780\pi\)
\(240\) 0 0
\(241\) −9.00000 15.5885i −0.579741 1.00414i −0.995509 0.0946700i \(-0.969820\pi\)
0.415768 0.909471i \(-0.363513\pi\)
\(242\) 0 0
\(243\) −8.66025 5.00000i −0.555556 0.320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 10.3923i −0.380235 0.658586i
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.5885 9.00000i −0.972381 0.561405i −0.0724199 0.997374i \(-0.523072\pi\)
−0.899961 + 0.435970i \(0.856405\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 5.19615i 0.185695 0.321634i
\(262\) 0 0
\(263\) 2.59808 1.50000i 0.160204 0.0924940i −0.417755 0.908560i \(-0.637183\pi\)
0.577959 + 0.816066i \(0.303849\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 22.0000i 1.34638i
\(268\) 0 0
\(269\) −1.00000 1.73205i −0.0609711 0.105605i 0.833929 0.551872i \(-0.186086\pi\)
−0.894900 + 0.446267i \(0.852753\pi\)
\(270\) 0 0
\(271\) −3.50000 + 6.06218i −0.212610 + 0.368251i −0.952531 0.304443i \(-0.901530\pi\)
0.739921 + 0.672694i \(0.234863\pi\)
\(272\) 0 0
\(273\) 6.92820 + 8.00000i 0.419314 + 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.92820 4.00000i 0.416275 0.240337i −0.277207 0.960810i \(-0.589409\pi\)
0.693482 + 0.720473i \(0.256075\pi\)
\(278\) 0 0
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 0 0
\(283\) −1.73205 + 1.00000i −0.102960 + 0.0594438i −0.550596 0.834772i \(-0.685599\pi\)
0.447636 + 0.894216i \(0.352266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.9904 + 2.50000i −0.766798 + 0.147570i
\(288\) 0 0
\(289\) −4.00000 + 6.92820i −0.235294 + 0.407541i
\(290\) 0 0
\(291\) 11.0000 + 19.0526i 0.644831 + 1.11688i
\(292\) 0 0
\(293\) 14.0000i 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −13.8564 + 8.00000i −0.804030 + 0.464207i
\(298\) 0 0
\(299\) −3.00000 + 5.19615i −0.173494 + 0.300501i
\(300\) 0 0
\(301\) −3.00000 15.5885i −0.172917 0.898504i
\(302\) 0 0
\(303\) −13.8564 8.00000i −0.796030 0.459588i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 24.0000i 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) 0 0
\(309\) −30.0000 −1.70664
\(310\) 0 0
\(311\) −0.500000 0.866025i −0.0283524 0.0491078i 0.851501 0.524353i \(-0.175693\pi\)
−0.879853 + 0.475245i \(0.842359\pi\)
\(312\) 0 0
\(313\) −26.8468 15.5000i −1.51747 0.876112i −0.999789 0.0205381i \(-0.993462\pi\)
−0.517681 0.855574i \(-0.673205\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.3923 + 6.00000i 0.583690 + 0.336994i 0.762598 0.646872i \(-0.223923\pi\)
−0.178908 + 0.983866i \(0.557257\pi\)
\(318\) 0 0
\(319\) −12.0000 20.7846i −0.671871 1.16371i
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.2487 + 14.0000i 1.34096 + 0.774202i
\(328\) 0 0
\(329\) 22.5000 + 7.79423i 1.24047 + 0.429710i
\(330\) 0 0
\(331\) 1.00000 1.73205i 0.0549650 0.0952021i −0.837234 0.546845i \(-0.815829\pi\)
0.892199 + 0.451643i \(0.149162\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.0000i 1.57973i −0.613280 0.789865i \(-0.710150\pi\)
0.613280 0.789865i \(-0.289850\pi\)
\(338\) 0 0
\(339\) 15.0000 + 25.9808i 0.814688 + 1.41108i
\(340\) 0 0
\(341\) −18.0000 + 31.1769i −0.974755 + 1.68832i
\(342\) 0 0
\(343\) −15.5885 + 10.0000i −0.841698 + 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.73205 1.00000i 0.0929814 0.0536828i −0.452788 0.891618i \(-0.649571\pi\)
0.545770 + 0.837935i \(0.316237\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −12.9904 + 7.50000i −0.691408 + 0.399185i −0.804139 0.594441i \(-0.797373\pi\)
0.112731 + 0.993626i \(0.464040\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −5.19615 + 15.0000i −0.275010 + 0.793884i
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.8564 8.00000i 0.723299 0.417597i −0.0926670 0.995697i \(-0.529539\pi\)
0.815966 + 0.578101i \(0.196206\pi\)
\(368\) 0 0
\(369\) 2.50000 4.33013i 0.130145 0.225417i
\(370\) 0 0
\(371\) −12.0000 + 10.3923i −0.623009 + 0.539542i
\(372\) 0 0
\(373\) 5.19615 + 3.00000i 0.269047 + 0.155334i 0.628454 0.777847i \(-0.283688\pi\)
−0.359408 + 0.933181i \(0.617021\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −8.00000 13.8564i −0.409852 0.709885i
\(382\) 0 0
\(383\) 18.1865 + 10.5000i 0.929288 + 0.536525i 0.886586 0.462563i \(-0.153070\pi\)
0.0427020 + 0.999088i \(0.486403\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.19615 + 3.00000i 0.264135 + 0.152499i
\(388\) 0 0
\(389\) 17.0000 + 29.4449i 0.861934 + 1.49291i 0.870059 + 0.492947i \(0.164080\pi\)
−0.00812520 + 0.999967i \(0.502586\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) 24.0000i 1.21064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 32.9090 + 19.0000i 1.65165 + 0.953583i 0.976392 + 0.216004i \(0.0693024\pi\)
0.675261 + 0.737579i \(0.264031\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 + 15.5885i −0.449439 + 0.778450i −0.998350 0.0574304i \(-0.981709\pi\)
0.548911 + 0.835881i \(0.315043\pi\)
\(402\) 0 0
\(403\) 15.5885 9.00000i 0.776516 0.448322i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −15.5000 26.8468i −0.766426 1.32749i −0.939490 0.342578i \(-0.888700\pi\)
0.173064 0.984911i \(-0.444633\pi\)
\(410\) 0 0
\(411\) −17.0000 + 29.4449i −0.838548 + 1.45241i
\(412\) 0 0
\(413\) −13.8564 16.0000i −0.681829 0.787309i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −10.3923 + 6.00000i −0.508913 + 0.293821i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −7.79423 + 4.50000i −0.378968 + 0.218797i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.92820 20.0000i 0.335279 0.967868i
\(428\) 0 0
\(429\) −8.00000 + 13.8564i −0.386244 + 0.668994i
\(430\) 0 0
\(431\) 15.5000 + 26.8468i 0.746609 + 1.29316i 0.949439 + 0.313950i \(0.101653\pi\)
−0.202831 + 0.979214i \(0.565014\pi\)
\(432\) 0 0
\(433\) 21.0000i 1.00920i −0.863355 0.504598i \(-0.831641\pi\)
0.863355 0.504598i \(-0.168359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 17.5000 30.3109i 0.835229 1.44666i −0.0586141 0.998281i \(-0.518668\pi\)
0.893843 0.448379i \(-0.147999\pi\)
\(440\) 0 0
\(441\) 1.00000 6.92820i 0.0476190 0.329914i
\(442\) 0 0
\(443\) 10.3923 + 6.00000i 0.493753 + 0.285069i 0.726130 0.687557i \(-0.241317\pi\)
−0.232377 + 0.972626i \(0.574650\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 40.0000i 1.89194i
\(448\) 0 0
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −10.0000 17.3205i −0.470882 0.815591i
\(452\) 0 0
\(453\) 34.6410 + 20.0000i 1.62758 + 0.939682i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.73205 1.00000i −0.0810219 0.0467780i 0.458942 0.888466i \(-0.348229\pi\)
−0.539964 + 0.841688i \(0.681562\pi\)
\(458\) 0 0
\(459\) 6.00000 + 10.3923i 0.280056 + 0.485071i
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 11.0000i 0.511213i −0.966781 0.255607i \(-0.917725\pi\)
0.966781 0.255607i \(-0.0822752\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.9808 + 15.0000i 1.20225 + 0.694117i 0.961054 0.276360i \(-0.0891283\pi\)
0.241192 + 0.970477i \(0.422462\pi\)
\(468\) 0 0
\(469\) −28.0000 + 24.2487i −1.29292 + 1.11970i
\(470\) 0 0
\(471\) −10.0000 + 17.3205i −0.460776 + 0.798087i
\(472\) 0 0
\(473\) 20.7846 12.0000i 0.955677 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 4.50000 + 7.79423i 0.205610 + 0.356127i 0.950327 0.311253i \(-0.100749\pi\)
−0.744717 + 0.667381i \(0.767415\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 15.5885 3.00000i 0.709299 0.136505i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.9186 11.5000i 0.902597 0.521115i 0.0245553 0.999698i \(-0.492183\pi\)
0.878042 + 0.478584i \(0.158850\pi\)
\(488\) 0 0
\(489\) 48.0000 2.17064
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) −15.5885 + 9.00000i −0.702069 + 0.405340i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −28.5788 + 5.50000i −1.28194 + 0.246709i
\(498\) 0 0
\(499\) 19.0000 32.9090i 0.850557 1.47321i −0.0301498 0.999545i \(-0.509598\pi\)
0.880707 0.473662i \(-0.157068\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.5885 + 9.00000i −0.692308 + 0.399704i
\(508\) 0 0
\(509\) 6.00000 10.3923i 0.265945 0.460631i −0.701866 0.712309i \(-0.747649\pi\)
0.967811 + 0.251679i \(0.0809826\pi\)
\(510\) 0 0
\(511\) −4.00000 + 3.46410i −0.176950 + 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 36.0000i 1.58328i
\(518\) 0 0
\(519\) −32.0000 −1.40464
\(520\) 0 0
\(521\) −22.5000 38.9711i −0.985743 1.70736i −0.638588 0.769549i \(-0.720481\pi\)
−0.347155 0.937808i \(-0.612852\pi\)
\(522\) 0 0
\(523\) 19.0526 + 11.0000i 0.833110 + 0.480996i 0.854916 0.518766i \(-0.173608\pi\)
−0.0218062 + 0.999762i \(0.506942\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23.3827 + 13.5000i 1.01857 + 0.588069i
\(528\) 0 0
\(529\) −7.00000 12.1244i −0.304348 0.527146i
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −6.92820 4.00000i −0.298974 0.172613i
\(538\) 0 0
\(539\) −22.0000 17.3205i −0.947607 0.746047i
\(540\) 0 0
\(541\) 15.0000 25.9808i 0.644900 1.11700i −0.339424 0.940633i \(-0.610232\pi\)
0.984325 0.176367i \(-0.0564345\pi\)
\(542\) 0 0
\(543\) 13.8564 8.00000i 0.594635 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 40.0000i 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) 0 0
\(549\) 4.00000 + 6.92820i 0.170716 + 0.295689i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.79423 + 22.5000i −0.331444 + 0.956797i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.1769 + 18.0000i −1.32101 + 0.762684i −0.983890 0.178778i \(-0.942786\pi\)
−0.337119 + 0.941462i \(0.609452\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 22.5167 13.0000i 0.948964 0.547885i 0.0562051 0.998419i \(-0.482100\pi\)
0.892759 + 0.450535i \(0.148767\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −19.0526 22.0000i −0.800132 0.923913i
\(568\) 0 0
\(569\) −5.50000 + 9.52628i −0.230572 + 0.399362i −0.957977 0.286846i \(-0.907393\pi\)
0.727405 + 0.686209i \(0.240726\pi\)
\(570\) 0 0
\(571\) 6.00000 + 10.3923i 0.251092 + 0.434904i 0.963827 0.266529i \(-0.0858769\pi\)
−0.712735 + 0.701434i \(0.752544\pi\)
\(572\) 0 0
\(573\) 26.0000i 1.08617i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12.1244 7.00000i 0.504744 0.291414i −0.225927 0.974144i \(-0.572541\pi\)
0.730670 + 0.682730i \(0.239208\pi\)
\(578\) 0 0
\(579\) 5.00000 8.66025i 0.207793 0.359908i
\(580\) 0 0
\(581\) −3.00000 15.5885i −0.124461 0.646718i
\(582\) 0 0
\(583\) −20.7846 12.0000i −0.860811 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 + 31.1769i 0.740421 + 1.28245i
\(592\) 0 0
\(593\) −11.2583 6.50000i −0.462324 0.266923i 0.250697 0.968066i \(-0.419340\pi\)
−0.713021 + 0.701143i \(0.752674\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −36.3731 21.0000i −1.48865 0.859473i
\(598\) 0 0
\(599\) 4.50000 + 7.79423i 0.183865 + 0.318464i 0.943193 0.332244i \(-0.107806\pi\)
−0.759328 + 0.650708i \(0.774472\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 14.0000i 0.570124i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11.2583 + 6.50000i 0.456962 + 0.263827i 0.710766 0.703429i \(-0.248349\pi\)
−0.253804 + 0.967256i \(0.581682\pi\)
\(608\) 0 0
\(609\) 24.0000 20.7846i 0.972529 0.842235i
\(610\) 0 0
\(611\) 9.00000 15.5885i 0.364101 0.630641i
\(612\) 0 0
\(613\) −22.5167 + 13.0000i −0.909439 + 0.525065i −0.880251 0.474509i \(-0.842626\pi\)
−0.0291886 + 0.999574i \(0.509292\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.0000i 0.684394i −0.939628 0.342197i \(-0.888829\pi\)
0.939628 0.342197i \(-0.111171\pi\)
\(618\) 0 0
\(619\) −5.00000 8.66025i −0.200967 0.348085i 0.747873 0.663842i \(-0.231075\pi\)
−0.948840 + 0.315757i \(0.897742\pi\)
\(620\) 0 0
\(621\) 6.00000 10.3923i 0.240772 0.417029i
\(622\) 0 0
\(623\) −9.52628 + 27.5000i −0.381662 + 1.10176i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) −38.1051 + 22.0000i −1.51454 + 0.874421i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.19615 + 13.0000i 0.205879 + 0.515079i
\(638\) 0 0
\(639\) 5.50000 9.52628i 0.217577 0.376854i
\(640\) 0 0
\(641\) 17.5000 + 30.3109i 0.691208 + 1.19721i 0.971442 + 0.237276i \(0.0762547\pi\)
−0.280234 + 0.959932i \(0.590412\pi\)
\(642\) 0 0
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.5692 + 24.0000i −1.63425 + 0.943537i −0.651494 + 0.758654i \(0.725858\pi\)
−0.982760 + 0.184884i \(0.940809\pi\)
\(648\) 0 0
\(649\) 16.0000 27.7128i 0.628055 1.08782i
\(650\) 0 0
\(651\) −45.0000 15.5885i −1.76369 0.610960i
\(652\) 0 0
\(653\) 25.9808 + 15.0000i 1.01671 + 0.586995i 0.913148 0.407628i \(-0.133644\pi\)
0.103558 + 0.994623i \(0.466977\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) −22.0000 38.1051i −0.855701 1.48212i −0.875993 0.482323i \(-0.839793\pi\)
0.0202925 0.999794i \(-0.493540\pi\)
\(662\) 0 0
\(663\) 10.3923 + 6.00000i 0.403604 + 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15.5885 + 9.00000i 0.603587 + 0.348481i
\(668\) 0 0
\(669\) 11.0000 + 19.0526i 0.425285 + 0.736614i
\(670\) 0 0
\(671\) 32.0000 1.23535
\(672\) 0 0
\(673\) 43.0000i 1.65753i 0.559598 + 0.828764i \(0.310955\pi\)
−0.559598 + 0.828764i \(0.689045\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.19615 + 3.00000i 0.199704 + 0.115299i 0.596518 0.802600i \(-0.296551\pi\)
−0.396813 + 0.917899i \(0.629884\pi\)
\(678\) 0 0
\(679\) 5.50000 + 28.5788i 0.211071 + 1.09676i
\(680\) 0 0
\(681\) 22.0000 38.1051i 0.843042 1.46019i
\(682\) 0 0
\(683\) 12.1244 7.00000i 0.463926 0.267848i −0.249768 0.968306i \(-0.580354\pi\)
0.713693 + 0.700458i \(0.247021\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16.0000i 0.610438i
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 0 0
\(691\) 11.0000 19.0526i 0.418460 0.724793i −0.577325 0.816514i \(-0.695903\pi\)
0.995785 + 0.0917209i \(0.0292368\pi\)
\(692\) 0 0
\(693\) 10.3923 2.00000i 0.394771 0.0759737i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.9904 + 7.50000i −0.492046 + 0.284083i
\(698\) 0 0
\(699\) 44.0000 1.66423
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.8564 16.0000i −0.521124 0.601742i
\(708\) 0 0
\(709\) −4.00000 + 6.92820i −0.150223 + 0.260194i −0.931309 0.364229i \(-0.881333\pi\)
0.781086 + 0.624423i \(0.214666\pi\)
\(710\) 0 0
\(711\) −4.50000 7.79423i −0.168763 0.292306i
\(712\) 0 0
\(713\) 27.0000i 1.01116i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 19.0526 11.0000i 0.711531 0.410803i
\(718\) 0 0
\(719\) 14.5000 25.1147i 0.540759 0.936622i −0.458102 0.888900i \(-0.651471\pi\)
0.998861 0.0477220i \(-0.0151961\pi\)
\(720\) 0 0
\(721\) −37.5000 12.9904i −1.39657 0.483787i
\(722\) 0 0
\(723\) 31.1769 + 18.0000i 1.15948 + 0.669427i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 29.0000i 1.07555i 0.843088 + 0.537775i \(0.180735\pi\)
−0.843088 + 0.537775i \(0.819265\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −9.00000 15.5885i −0.332877 0.576560i
\(732\) 0 0
\(733\) 12.1244 + 7.00000i 0.447823 + 0.258551i 0.706910 0.707303i \(-0.250088\pi\)
−0.259087 + 0.965854i \(0.583422\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.4974 28.0000i −1.78643 1.03139i
\(738\) 0 0
\(739\) −8.00000 13.8564i −0.294285 0.509716i 0.680534 0.732717i \(-0.261748\pi\)
−0.974818 + 0.223001i \(0.928415\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.0000i 0.990534i 0.868741 + 0.495267i \(0.164930\pi\)
−0.868741 + 0.495267i \(0.835070\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.19615 + 3.00000i 0.190117 + 0.109764i
\(748\) 0 0
\(749\) 20.0000 + 6.92820i 0.730784 + 0.253151i
\(750\) 0 0
\(751\) 16.0000 27.7128i 0.583848 1.01125i −0.411170 0.911559i \(-0.634880\pi\)
0.995018 0.0996961i \(-0.0317870\pi\)
\(752\) 0 0
\(753\) 27.7128 16.0000i 1.00991 0.583072i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000i 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 0 0
\(759\) 12.0000 + 20.7846i 0.435572 + 0.754434i
\(760\) 0 0
\(761\) 21.5000 37.2391i 0.779374 1.34992i −0.152928 0.988237i \(-0.548870\pi\)
0.932303 0.361679i \(-0.117796\pi\)
\(762\) 0 0
\(763\) 24.2487 + 28.0000i 0.877862 + 1.01367i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.8564 + 8.00000i −0.500326 + 0.288863i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) 17.3205 10.0000i 0.622975 0.359675i −0.155051 0.987906i \(-0.549554\pi\)
0.778027 + 0.628231i \(0.216221\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −22.0000 38.1051i −0.787222 1.36351i
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6.92820 4.00000i 0.246964 0.142585i −0.371409 0.928469i \(-0.621125\pi\)
0.618373 + 0.785885i \(0.287792\pi\)
\(788\) 0 0
\(789\) −3.00000 + 5.19615i −0.106803 + 0.184988i
\(790\) 0 0
\(791\) 7.50000 + 38.9711i 0.266669 + 1.38565i
\(792\) 0 0
\(793\) −13.8564 8.00000i −0.492055 0.284088i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000i 0.283375i 0.989911 + 0.141687i \(0.0452527\pi\)
−0.989911 + 0.141687i \(0.954747\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) −5.50000 9.52628i −0.194333 0.336595i
\(802\) 0 0
\(803\) −6.92820 4.00000i −0.244491 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.46410 + 2.00000i 0.121942 + 0.0704033i
\(808\) 0 0
\(809\) 5.00000 + 8.66025i 0.175791 + 0.304478i 0.940435 0.339975i \(-0.110418\pi\)
−0.764644 + 0.644453i \(0.777085\pi\)
\(810\) 0 0
\(811\) −10.0000 −0.351147 −0.175574 0.984466i \(-0.556178\pi\)
−0.175574 + 0.984466i \(0.556178\pi\)
\(812\) 0 0
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −5.00000 1.73205i −0.174714 0.0605228i
\(820\) 0 0
\(821\) −10.0000 + 17.3205i −0.349002 + 0.604490i −0.986073 0.166316i \(-0.946813\pi\)
0.637070 + 0.770806i \(0.280146\pi\)
\(822\) 0 0
\(823\) 3.46410 2.00000i 0.120751 0.0697156i −0.438408 0.898776i \(-0.644457\pi\)
0.559159 + 0.829060i \(0.311124\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.0000i 0.486828i 0.969923 + 0.243414i \(0.0782673\pi\)
−0.969923 + 0.243414i \(0.921733\pi\)
\(828\) 0 0
\(829\) −16.0000 27.7128i −0.555703 0.962506i −0.997848 0.0655624i \(-0.979116\pi\)
0.442145 0.896943i \(-0.354217\pi\)
\(830\) 0 0
\(831\) −8.00000 + 13.8564i −0.277517 + 0.480673i
\(832\) 0 0
\(833\) −12.9904 + 16.5000i −0.450090 + 0.571691i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −31.1769 + 18.0000i −1.07763 + 0.622171i
\(838\) 0 0
\(839\) 29.0000 1.00119 0.500596 0.865681i \(-0.333114\pi\)
0.500596 + 0.865681i \(0.333114\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 22.5167 13.0000i 0.775515 0.447744i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.33013 12.5000i 0.148785 0.429505i
\(848\) 0 0
\(849\) 2.00000 3.46410i 0.0686398 0.118888i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.0526 11.0000i 0.650823 0.375753i −0.137948 0.990439i \(-0.544051\pi\)
0.788771 + 0.614687i \(0.210717\pi\)
\(858\) 0 0
\(859\) 10.0000 17.3205i 0.341196 0.590968i −0.643459 0.765480i \(-0.722501\pi\)
0.984655 + 0.174512i \(0.0558348\pi\)
\(860\) 0 0
\(861\) 20.0000 17.3205i 0.681598 0.590281i
\(862\) 0 0
\(863\) −6.06218 3.50000i −0.206359 0.119141i 0.393259 0.919428i \(-0.371348\pi\)
−0.599618 + 0.800286i \(0.704681\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.0000i 0.543388i
\(868\) 0 0
\(869\) −36.0000 −1.22122
\(870\) 0 0
\(871\) 14.0000 + 24.2487i 0.474372 + 0.821636i
\(872\) 0 0
\(873\) −9.52628 5.50000i −0.322416 0.186147i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.46410 + 2.00000i 0.116974 + 0.0675352i 0.557346 0.830281i \(-0.311820\pi\)
−0.440371 + 0.897816i \(0.645153\pi\)
\(878\) 0 0
\(879\) 14.0000 + 24.2487i 0.472208 + 0.817889i
\(880\) 0 0
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) 0 0
\(883\) 2.00000i 0.0673054i −0.999434 0.0336527i \(-0.989286\pi\)
0.999434 0.0336527i \(-0.0107140\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.3205 + 10.0000i 0.581566 + 0.335767i 0.761755 0.647865i \(-0.224338\pi\)
−0.180190 + 0.983632i \(0.557671\pi\)
\(888\) 0 0
\(889\) −4.00000 20.7846i −0.134156 0.697093i
\(890\) 0 0
\(891\) 22.0000 38.1051i 0.737028 1.27657i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000i 0.400668i
\(898\) 0 0
\(899\) −27.0000 46.7654i −0.900500 1.55971i
\(900\) 0 0
\(901\) −9.00000 + 15.5885i −0.299833 + 0.519327i
\(902\) 0 0
\(903\) 20.7846 + 24.0000i 0.691669 + 0.798670i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −10.3923 + 6.00000i −0.345071 + 0.199227i −0.662512 0.749051i \(-0.730510\pi\)
0.317441 + 0.948278i \(0.397176\pi\)
\(908\) 0 0
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) 47.0000 1.55718 0.778590 0.627533i \(-0.215935\pi\)
0.778590 + 0.627533i \(0.215935\pi\)
\(912\) 0 0
\(913\) 20.7846 12.0000i 0.687870 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.3923 30.0000i 0.343184 0.990687i
\(918\) 0 0
\(919\) 3.50000 6.06218i 0.115454 0.199973i −0.802507 0.596643i \(-0.796501\pi\)
0.917961 + 0.396670i \(0.129834\pi\)
\(920\) 0 0
\(921\) 24.0000 + 41.5692i 0.790827 + 1.36975i
\(922\) 0 0
\(923\) 22.0000i 0.724139i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.9904 7.50000i 0.426660 0.246332i
\(928\) 0 0
\(929\) −3.00000 + 5.19615i −0.0984268 + 0.170480i −0.911034 0.412332i \(-0.864714\pi\)
0.812607 + 0.582812i \(0.198048\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.73205 + 1.00000i 0.0567048 + 0.0327385i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000i 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 0 0
\(939\) 62.0000 2.02329
\(940\) 0 0
\(941\) 13.0000 + 22.5167i 0.423788 + 0.734022i 0.996306 0.0858697i \(-0.0273669\pi\)
−0.572518 + 0.819892i \(0.694034\pi\)
\(942\) 0 0
\(943\) 12.9904 + 7.50000i 0.423025 + 0.244234i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.46410 + 2.00000i 0.112568 + 0.0649913i 0.555227 0.831699i \(-0.312631\pi\)
−0.442659 + 0.896690i \(0.645965\pi\)
\(948\) 0 0
\(949\) 2.00000 + 3.46410i 0.0649227 + 0.112449i
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 10.0000i 0.323932i 0.986796 + 0.161966i \(0.0517835\pi\)
−0.986796 + 0.161966i \(0.948217\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 41.5692 + 24.0000i 1.34374 + 0.775810i
\(958\) 0 0
\(959\) −34.0000 + 29.4449i −1.09792 + 0.950824i
\(960\) 0 0
\(961\) −25.0000 + 43.3013i −0.806452 + 1.39682i
\(962\) 0 0
\(963\) −6.92820 + 4.00000i −0.223258 + 0.128898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.0000i 0.418052i 0.977910 + 0.209026i \(0.0670293\pi\)
−0.977910 + 0.209026i \(0.932971\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −11.0000 + 19.0526i −0.353007 + 0.611426i −0.986775 0.162098i \(-0.948174\pi\)
0.633768 + 0.773523i \(0.281507\pi\)
\(972\) 0 0
\(973\) −15.5885 + 3.00000i −0.499743 + 0.0961756i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.59808 + 1.50000i −0.0831198 + 0.0479893i −0.540984 0.841033i \(-0.681948\pi\)
0.457864 + 0.889022i \(0.348615\pi\)
\(978\) 0 0
\(979\) −44.0000 −1.40625
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) −41.5692 + 24.0000i −1.32585 + 0.765481i −0.984655 0.174511i \(-0.944166\pi\)
−0.341197 + 0.939992i \(0.610832\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −46.7654 + 9.00000i −1.48856 + 0.286473i
\(988\) 0 0
\(989\) −9.00000 + 15.5885i −0.286183 + 0.495684i
\(990\) 0 0
\(991\) 15.5000 + 26.8468i 0.492374 + 0.852816i 0.999961 0.00878379i \(-0.00279600\pi\)
−0.507588 + 0.861600i \(0.669463\pi\)
\(992\) 0 0
\(993\) 4.00000i 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.1051 22.0000i 1.20680 0.696747i 0.244742 0.969588i \(-0.421297\pi\)
0.962059 + 0.272841i \(0.0879634\pi\)
\(998\) 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 1400.2.bh.d.849.1 4
5.2 odd 4 1400.2.q.f.401.1 yes 2
5.3 odd 4 1400.2.q.b.401.1 2
5.4 even 2 inner 1400.2.bh.d.849.2 4
7.4 even 3 inner 1400.2.bh.d.249.2 4
35.2 odd 12 9800.2.a.j.1.1 1
35.4 even 6 inner 1400.2.bh.d.249.1 4
35.12 even 12 9800.2.a.bl.1.1 1
35.18 odd 12 1400.2.q.b.1201.1 yes 2
35.23 odd 12 9800.2.a.bm.1.1 1
35.32 odd 12 1400.2.q.f.1201.1 yes 2
35.33 even 12 9800.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.q.b.401.1 2 5.3 odd 4
1400.2.q.b.1201.1 yes 2 35.18 odd 12
1400.2.q.f.401.1 yes 2 5.2 odd 4
1400.2.q.f.1201.1 yes 2 35.32 odd 12
1400.2.bh.d.249.1 4 35.4 even 6 inner
1400.2.bh.d.249.2 4 7.4 even 3 inner
1400.2.bh.d.849.1 4 1.1 even 1 trivial
1400.2.bh.d.849.2 4 5.4 even 2 inner
9800.2.a.j.1.1 1 35.2 odd 12
9800.2.a.k.1.1 1 35.33 even 12
9800.2.a.bl.1.1 1 35.12 even 12
9800.2.a.bm.1.1 1 35.23 odd 12