Properties

Label 1400.2.q.e.1201.1
Level $1400$
Weight $2$
Character 1400.1201
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1400,2,Mod(401,1400)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1201.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1400.1201
Dual form 1400.2.q.e.401.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(-1.00000 + 1.73205i) q^{11} +(2.00000 - 3.46410i) q^{17} +(1.00000 + 1.73205i) q^{19} +(0.500000 - 2.59808i) q^{21} +(0.500000 + 0.866025i) q^{23} +5.00000 q^{27} +9.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(1.00000 + 1.73205i) q^{33} +(2.00000 + 3.46410i) q^{37} +1.00000 q^{41} -9.00000 q^{43} +(5.50000 - 4.33013i) q^{49} +(-2.00000 - 3.46410i) q^{51} +(-5.00000 + 8.66025i) q^{53} +2.00000 q^{57} +(5.00000 - 8.66025i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(4.00000 + 3.46410i) q^{63} +(2.50000 - 4.33013i) q^{67} +1.00000 q^{69} +14.0000 q^{71} +(6.00000 - 10.3923i) q^{73} +(-1.00000 + 5.19615i) q^{77} +(-7.00000 - 12.1244i) q^{79} +(-0.500000 + 0.866025i) q^{81} -11.0000 q^{83} +(4.50000 - 7.79423i) q^{87} +(7.50000 + 12.9904i) q^{89} +(2.00000 + 3.46410i) q^{93} +18.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 5 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{21} + q^{23} + 10 q^{27} + 18 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{41} - 18 q^{43} + 11 q^{49} - 4 q^{51} - 10 q^{53} + 4 q^{57} + 10 q^{59} - 9 q^{61} + 8 q^{63} + 5 q^{67} + 2 q^{69} + 28 q^{71} + 12 q^{73} - 2 q^{77} - 14 q^{79} - q^{81} - 22 q^{83} + 9 q^{87} + 15 q^{89} + 4 q^{93} + 36 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i −0.684819 0.728714i \(-0.740119\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.50000 0.866025i 0.944911 0.327327i
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) −1.00000 + 1.73205i −0.301511 + 0.522233i −0.976478 0.215615i \(-0.930824\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 3.46410i 0.485071 0.840168i −0.514782 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 0 0
\(21\) 0.500000 2.59808i 0.109109 0.566947i
\(22\) 0 0
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 1.00000 + 1.73205i 0.174078 + 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 + 3.46410i 0.328798 + 0.569495i 0.982274 0.187453i \(-0.0600231\pi\)
−0.653476 + 0.756948i \(0.726690\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) −2.00000 3.46410i −0.280056 0.485071i
\(52\) 0 0
\(53\) −5.00000 + 8.66025i −0.686803 + 1.18958i 0.286064 + 0.958211i \(0.407653\pi\)
−0.972867 + 0.231367i \(0.925680\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 5.00000 8.66025i 0.650945 1.12747i −0.331949 0.943297i \(-0.607706\pi\)
0.982894 0.184172i \(-0.0589603\pi\)
\(60\) 0 0
\(61\) −4.50000 7.79423i −0.576166 0.997949i −0.995914 0.0903080i \(-0.971215\pi\)
0.419748 0.907641i \(-0.362118\pi\)
\(62\) 0 0
\(63\) 4.00000 + 3.46410i 0.503953 + 0.436436i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.50000 4.33013i 0.305424 0.529009i −0.671932 0.740613i \(-0.734535\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 6.00000 10.3923i 0.702247 1.21633i −0.265429 0.964130i \(-0.585514\pi\)
0.967676 0.252197i \(-0.0811531\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 + 5.19615i −0.113961 + 0.592157i
\(78\) 0 0
\(79\) −7.00000 12.1244i −0.787562 1.36410i −0.927457 0.373930i \(-0.878010\pi\)
0.139895 0.990166i \(-0.455323\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.50000 7.79423i 0.482451 0.835629i
\(88\) 0 0
\(89\) 7.50000 + 12.9904i 0.794998 + 1.37698i 0.922840 + 0.385183i \(0.125862\pi\)
−0.127842 + 0.991795i \(0.540805\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 + 3.46410i 0.207390 + 0.359211i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −1.50000 + 2.59808i −0.149256 + 0.258518i −0.930953 0.365140i \(-0.881021\pi\)
0.781697 + 0.623658i \(0.214354\pi\)
\(102\) 0 0
\(103\) −6.50000 11.2583i −0.640464 1.10932i −0.985329 0.170664i \(-0.945409\pi\)
0.344865 0.938652i \(-0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.50000 + 7.79423i 0.435031 + 0.753497i 0.997298 0.0734594i \(-0.0234039\pi\)
−0.562267 + 0.826956i \(0.690071\pi\)
\(108\) 0 0
\(109\) 0.500000 0.866025i 0.0478913 0.0829502i −0.841086 0.540901i \(-0.818083\pi\)
0.888977 + 0.457951i \(0.151417\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 10.3923i 0.183340 0.952661i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0.500000 0.866025i 0.0450835 0.0780869i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.50000 + 7.79423i −0.396203 + 0.686244i
\(130\) 0 0
\(131\) 4.00000 + 6.92820i 0.349482 + 0.605320i 0.986157 0.165812i \(-0.0530244\pi\)
−0.636676 + 0.771132i \(0.719691\pi\)
\(132\) 0 0
\(133\) 4.00000 + 3.46410i 0.346844 + 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 10.3923i 0.512615 0.887875i −0.487278 0.873247i \(-0.662010\pi\)
0.999893 0.0146279i \(-0.00465636\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 6.92820i −0.0824786 0.571429i
\(148\) 0 0
\(149\) 2.50000 + 4.33013i 0.204808 + 0.354738i 0.950072 0.312032i \(-0.101010\pi\)
−0.745264 + 0.666770i \(0.767676\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) 8.00000 0.646762
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.00000 1.73205i 0.0798087 0.138233i −0.823359 0.567521i \(-0.807902\pi\)
0.903167 + 0.429289i \(0.141236\pi\)
\(158\) 0 0
\(159\) 5.00000 + 8.66025i 0.396526 + 0.686803i
\(160\) 0 0
\(161\) 2.00000 + 1.73205i 0.157622 + 0.136505i
\(162\) 0 0
\(163\) −10.0000 17.3205i −0.783260 1.35665i −0.930033 0.367477i \(-0.880222\pi\)
0.146772 0.989170i \(-0.453112\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.0000 −1.31550 −0.657750 0.753237i \(-0.728492\pi\)
−0.657750 + 0.753237i \(0.728492\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −2.00000 + 3.46410i −0.152944 + 0.264906i
\(172\) 0 0
\(173\) 8.00000 + 13.8564i 0.608229 + 1.05348i 0.991532 + 0.129861i \(0.0414530\pi\)
−0.383304 + 0.923622i \(0.625214\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.00000 8.66025i −0.375823 0.650945i
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) −9.00000 −0.665299
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 + 6.92820i 0.292509 + 0.506640i
\(188\) 0 0
\(189\) 12.5000 4.33013i 0.909241 0.314970i
\(190\) 0 0
\(191\) −9.00000 15.5885i −0.651217 1.12794i −0.982828 0.184525i \(-0.940925\pi\)
0.331611 0.943416i \(-0.392408\pi\)
\(192\) 0 0
\(193\) −7.00000 + 12.1244i −0.503871 + 0.872730i 0.496119 + 0.868255i \(0.334758\pi\)
−0.999990 + 0.00447566i \(0.998575\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 2.00000 3.46410i 0.141776 0.245564i −0.786389 0.617731i \(-0.788052\pi\)
0.928166 + 0.372168i \(0.121385\pi\)
\(200\) 0 0
\(201\) −2.50000 4.33013i −0.176336 0.305424i
\(202\) 0 0
\(203\) 22.5000 7.79423i 1.57919 0.547048i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 + 1.73205i −0.0695048 + 0.120386i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) 7.00000 12.1244i 0.479632 0.830747i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 + 10.3923i −0.135769 + 0.705476i
\(218\) 0 0
\(219\) −6.00000 10.3923i −0.405442 0.702247i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.0000 + 17.3205i −0.663723 + 1.14960i 0.315906 + 0.948790i \(0.397691\pi\)
−0.979630 + 0.200812i \(0.935642\pi\)
\(228\) 0 0
\(229\) 5.00000 + 8.66025i 0.330409 + 0.572286i 0.982592 0.185776i \(-0.0594799\pi\)
−0.652183 + 0.758062i \(0.726147\pi\)
\(230\) 0 0
\(231\) 4.00000 + 3.46410i 0.263181 + 0.227921i
\(232\) 0 0
\(233\) 4.00000 + 6.92820i 0.262049 + 0.453882i 0.966786 0.255586i \(-0.0822686\pi\)
−0.704737 + 0.709468i \(0.748935\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 11.0000 19.0526i 0.708572 1.22728i −0.256814 0.966461i \(-0.582673\pi\)
0.965387 0.260822i \(-0.0839937\pi\)
\(242\) 0 0
\(243\) 8.00000 + 13.8564i 0.513200 + 0.888889i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.50000 + 9.52628i −0.348548 + 0.603703i
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 8.00000 + 6.92820i 0.497096 + 0.430498i
\(260\) 0 0
\(261\) 9.00000 + 15.5885i 0.557086 + 0.964901i
\(262\) 0 0
\(263\) −0.500000 + 0.866025i −0.0308313 + 0.0534014i −0.881029 0.473062i \(-0.843149\pi\)
0.850198 + 0.526463i \(0.176482\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.0000 0.917985
\(268\) 0 0
\(269\) 10.5000 18.1865i 0.640196 1.10885i −0.345192 0.938532i \(-0.612186\pi\)
0.985389 0.170321i \(-0.0544803\pi\)
\(270\) 0 0
\(271\) −11.0000 19.0526i −0.668202 1.15736i −0.978406 0.206691i \(-0.933731\pi\)
0.310204 0.950670i \(-0.399603\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 + 24.2487i −0.841178 + 1.45696i 0.0477206 + 0.998861i \(0.484804\pi\)
−0.888899 + 0.458103i \(0.848529\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −2.00000 + 3.46410i −0.118888 + 0.205919i −0.919327 0.393494i \(-0.871266\pi\)
0.800439 + 0.599414i \(0.204600\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.50000 0.866025i 0.147570 0.0511199i
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 9.00000 15.5885i 0.527589 0.913812i
\(292\) 0 0
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.00000 + 8.66025i −0.290129 + 0.502519i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −22.5000 + 7.79423i −1.29688 + 0.449252i
\(302\) 0 0
\(303\) 1.50000 + 2.59808i 0.0861727 + 0.149256i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.0000 −1.19853 −0.599267 0.800549i \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) −13.0000 + 22.5167i −0.737162 + 1.27680i 0.216606 + 0.976259i \(0.430501\pi\)
−0.953768 + 0.300544i \(0.902832\pi\)
\(312\) 0 0
\(313\) 8.00000 + 13.8564i 0.452187 + 0.783210i 0.998522 0.0543564i \(-0.0173107\pi\)
−0.546335 + 0.837567i \(0.683977\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.00000 13.8564i −0.449325 0.778253i 0.549017 0.835811i \(-0.315002\pi\)
−0.998342 + 0.0575576i \(0.981669\pi\)
\(318\) 0 0
\(319\) −9.00000 + 15.5885i −0.503903 + 0.872786i
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.500000 0.866025i −0.0276501 0.0478913i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) −4.00000 + 6.92820i −0.219199 + 0.379663i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −1.00000 + 1.73205i −0.0543125 + 0.0940721i
\(340\) 0 0
\(341\) −4.00000 6.92820i −0.216612 0.375183i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.50000 + 6.06218i −0.187890 + 0.325435i −0.944547 0.328378i \(-0.893498\pi\)
0.756657 + 0.653812i \(0.226831\pi\)
\(348\) 0 0
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.00000 12.1244i 0.372572 0.645314i −0.617388 0.786659i \(-0.711809\pi\)
0.989960 + 0.141344i \(0.0451425\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.00000 6.92820i −0.423405 0.366679i
\(358\) 0 0
\(359\) −6.00000 10.3923i −0.316668 0.548485i 0.663123 0.748511i \(-0.269231\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(360\) 0 0
\(361\) 7.50000 12.9904i 0.394737 0.683704i
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.50000 6.06218i 0.182699 0.316443i −0.760100 0.649806i \(-0.774850\pi\)
0.942799 + 0.333363i \(0.108183\pi\)
\(368\) 0 0
\(369\) 1.00000 + 1.73205i 0.0520579 + 0.0901670i
\(370\) 0 0
\(371\) −5.00000 + 25.9808i −0.259587 + 1.34885i
\(372\) 0 0
\(373\) −14.0000 24.2487i −0.724893 1.25555i −0.959018 0.283344i \(-0.908556\pi\)
0.234126 0.972206i \(-0.424777\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) −4.00000 + 6.92820i −0.204926 + 0.354943i
\(382\) 0 0
\(383\) 10.5000 + 18.1865i 0.536525 + 0.929288i 0.999088 + 0.0427020i \(0.0135966\pi\)
−0.462563 + 0.886586i \(0.653070\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.00000 15.5885i −0.457496 0.792406i
\(388\) 0 0
\(389\) −13.0000 + 22.5167i −0.659126 + 1.14164i 0.321716 + 0.946836i \(0.395740\pi\)
−0.980842 + 0.194804i \(0.937593\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.00000 + 5.19615i 0.150566 + 0.260787i 0.931436 0.363906i \(-0.118557\pi\)
−0.780870 + 0.624694i \(0.785224\pi\)
\(398\) 0 0
\(399\) 5.00000 1.73205i 0.250313 0.0867110i
\(400\) 0 0
\(401\) 8.50000 + 14.7224i 0.424470 + 0.735203i 0.996371 0.0851195i \(-0.0271272\pi\)
−0.571901 + 0.820323i \(0.693794\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 10.5000 18.1865i 0.519192 0.899266i −0.480560 0.876962i \(-0.659566\pi\)
0.999751 0.0223042i \(-0.00710022\pi\)
\(410\) 0 0
\(411\) −6.00000 10.3923i −0.295958 0.512615i
\(412\) 0 0
\(413\) 5.00000 25.9808i 0.246034 1.27843i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00000 + 1.73205i −0.0489702 + 0.0848189i
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18.0000 15.5885i −0.871081 0.754378i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.00000 + 12.1244i −0.337178 + 0.584010i −0.983901 0.178716i \(-0.942806\pi\)
0.646723 + 0.762725i \(0.276139\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.00000 + 1.73205i −0.0478365 + 0.0828552i
\(438\) 0 0
\(439\) 14.0000 + 24.2487i 0.668184 + 1.15733i 0.978412 + 0.206666i \(0.0662612\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 13.0000 + 5.19615i 0.619048 + 0.247436i
\(442\) 0 0
\(443\) −5.50000 9.52628i −0.261313 0.452607i 0.705278 0.708931i \(-0.250822\pi\)
−0.966591 + 0.256323i \(0.917489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.00000 0.236492
\(448\) 0 0
\(449\) −41.0000 −1.93491 −0.967455 0.253044i \(-0.918568\pi\)
−0.967455 + 0.253044i \(0.918568\pi\)
\(450\) 0 0
\(451\) −1.00000 + 1.73205i −0.0470882 + 0.0815591i
\(452\) 0 0
\(453\) 4.00000 + 6.92820i 0.187936 + 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.00000 + 6.92820i 0.187112 + 0.324088i 0.944286 0.329125i \(-0.106754\pi\)
−0.757174 + 0.653213i \(0.773421\pi\)
\(458\) 0 0
\(459\) 10.0000 17.3205i 0.466760 0.808452i
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −39.0000 −1.81248 −0.906242 0.422760i \(-0.861061\pi\)
−0.906242 + 0.422760i \(0.861061\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.50000 6.06218i −0.161961 0.280524i 0.773611 0.633661i \(-0.218448\pi\)
−0.935572 + 0.353137i \(0.885115\pi\)
\(468\) 0 0
\(469\) 2.50000 12.9904i 0.115439 0.599840i
\(470\) 0 0
\(471\) −1.00000 1.73205i −0.0460776 0.0798087i
\(472\) 0 0
\(473\) 9.00000 15.5885i 0.413820 0.716758i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −20.0000 −0.915737
\(478\) 0 0
\(479\) 8.00000 13.8564i 0.365529 0.633115i −0.623332 0.781958i \(-0.714221\pi\)
0.988861 + 0.148842i \(0.0475547\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 2.50000 0.866025i 0.113754 0.0394055i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −4.00000 + 6.92820i −0.181257 + 0.313947i −0.942309 0.334744i \(-0.891350\pi\)
0.761052 + 0.648691i \(0.224683\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 18.0000 31.1769i 0.810679 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 35.0000 12.1244i 1.56996 0.543852i
\(498\) 0 0
\(499\) 19.0000 + 32.9090i 0.850557 + 1.47321i 0.880707 + 0.473662i \(0.157068\pi\)
−0.0301498 + 0.999545i \(0.509598\pi\)
\(500\) 0 0
\(501\) −8.50000 + 14.7224i −0.379752 + 0.657750i
\(502\) 0 0
\(503\) 23.0000 1.02552 0.512760 0.858532i \(-0.328623\pi\)
0.512760 + 0.858532i \(0.328623\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.50000 + 11.2583i −0.288675 + 0.500000i
\(508\) 0 0
\(509\) 7.50000 + 12.9904i 0.332432 + 0.575789i 0.982988 0.183669i \(-0.0587976\pi\)
−0.650556 + 0.759458i \(0.725464\pi\)
\(510\) 0 0
\(511\) 6.00000 31.1769i 0.265424 1.37919i
\(512\) 0 0
\(513\) 5.00000 + 8.66025i 0.220755 + 0.382360i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 21.0000 36.3731i 0.920027 1.59353i 0.120656 0.992694i \(-0.461500\pi\)
0.799370 0.600839i \(-0.205167\pi\)
\(522\) 0 0
\(523\) 14.0000 + 24.2487i 0.612177 + 1.06032i 0.990873 + 0.134801i \(0.0430394\pi\)
−0.378695 + 0.925521i \(0.623627\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 + 13.8564i 0.348485 + 0.603595i
\(528\) 0 0
\(529\) 11.0000 19.0526i 0.478261 0.828372i
\(530\) 0 0
\(531\) 20.0000 0.867926
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000 + 10.3923i 0.258919 + 0.448461i
\(538\) 0 0
\(539\) 2.00000 + 13.8564i 0.0861461 + 0.596838i
\(540\) 0 0
\(541\) 6.50000 + 11.2583i 0.279457 + 0.484033i 0.971250 0.238062i \(-0.0765123\pi\)
−0.691793 + 0.722096i \(0.743179\pi\)
\(542\) 0 0
\(543\) −12.5000 + 21.6506i −0.536426 + 0.929118i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.0000 1.49649 0.748246 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 0 0
\(549\) 9.00000 15.5885i 0.384111 0.665299i
\(550\) 0 0
\(551\) 9.00000 + 15.5885i 0.383413 + 0.664091i
\(552\) 0 0
\(553\) −28.0000 24.2487i −1.19068 1.03116i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 25.9808i 0.635570 1.10084i −0.350824 0.936442i \(-0.614098\pi\)
0.986394 0.164399i \(-0.0525683\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) −22.5000 + 38.9711i −0.948262 + 1.64244i −0.199177 + 0.979963i \(0.563827\pi\)
−0.749085 + 0.662474i \(0.769506\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.500000 + 2.59808i −0.0209980 + 0.109109i
\(568\) 0 0
\(569\) −23.0000 39.8372i −0.964210 1.67006i −0.711722 0.702461i \(-0.752085\pi\)
−0.252488 0.967600i \(-0.581249\pi\)
\(570\) 0 0
\(571\) 13.0000 22.5167i 0.544033 0.942293i −0.454634 0.890678i \(-0.650230\pi\)
0.998667 0.0516146i \(-0.0164367\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.00000 1.73205i 0.0416305 0.0721062i −0.844459 0.535620i \(-0.820078\pi\)
0.886090 + 0.463513i \(0.153411\pi\)
\(578\) 0 0
\(579\) 7.00000 + 12.1244i 0.290910 + 0.503871i
\(580\) 0 0
\(581\) −27.5000 + 9.52628i −1.14089 + 0.395217i
\(582\) 0 0
\(583\) −10.0000 17.3205i −0.414158 0.717342i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −9.00000 + 15.5885i −0.370211 + 0.641223i
\(592\) 0 0
\(593\) 9.00000 + 15.5885i 0.369586 + 0.640141i 0.989501 0.144528i \(-0.0461663\pi\)
−0.619915 + 0.784669i \(0.712833\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.00000 3.46410i −0.0818546 0.141776i
\(598\) 0 0
\(599\) −2.00000 + 3.46410i −0.0817178 + 0.141539i −0.903988 0.427558i \(-0.859374\pi\)
0.822270 + 0.569097i \(0.192707\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13.5000 + 23.3827i 0.547948 + 0.949074i 0.998415 + 0.0562808i \(0.0179242\pi\)
−0.450467 + 0.892793i \(0.648742\pi\)
\(608\) 0 0
\(609\) 4.50000 23.3827i 0.182349 0.947514i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 + 17.3205i −0.403896 + 0.699569i −0.994192 0.107618i \(-0.965678\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 17.0000 29.4449i 0.683288 1.18349i −0.290684 0.956819i \(-0.593883\pi\)
0.973972 0.226670i \(-0.0727838\pi\)
\(620\) 0 0
\(621\) 2.50000 + 4.33013i 0.100322 + 0.173762i
\(622\) 0 0
\(623\) 30.0000 + 25.9808i 1.20192 + 1.04090i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.00000 + 3.46410i −0.0798723 + 0.138343i
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 0 0
\(633\) 1.00000 1.73205i 0.0397464 0.0688428i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.0000 + 24.2487i 0.553831 + 0.959264i
\(640\) 0 0
\(641\) 9.50000 16.4545i 0.375227 0.649913i −0.615134 0.788423i \(-0.710898\pi\)
0.990361 + 0.138510i \(0.0442313\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.5000 19.9186i 0.452112 0.783080i −0.546405 0.837521i \(-0.684004\pi\)
0.998517 + 0.0544405i \(0.0173375\pi\)
\(648\) 0 0
\(649\) 10.0000 + 17.3205i 0.392534 + 0.679889i
\(650\) 0 0
\(651\) 8.00000 + 6.92820i 0.313545 + 0.271538i
\(652\) 0 0
\(653\) 18.0000 + 31.1769i 0.704394 + 1.22005i 0.966910 + 0.255119i \(0.0821147\pi\)
−0.262515 + 0.964928i \(0.584552\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 24.0000 0.936329
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 1.50000 2.59808i 0.0583432 0.101053i −0.835379 0.549675i \(-0.814752\pi\)
0.893722 + 0.448622i \(0.148085\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.50000 + 7.79423i 0.174241 + 0.301794i
\(668\) 0 0
\(669\) 2.00000 3.46410i 0.0773245 0.133930i
\(670\) 0 0
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.0000 20.7846i −0.461197 0.798817i 0.537823 0.843057i \(-0.319247\pi\)
−0.999021 + 0.0442400i \(0.985913\pi\)
\(678\) 0 0
\(679\) 45.0000 15.5885i 1.72694 0.598230i
\(680\) 0 0
\(681\) 10.0000 + 17.3205i 0.383201 + 0.663723i
\(682\) 0 0
\(683\) 4.50000 7.79423i 0.172188 0.298238i −0.766997 0.641651i \(-0.778250\pi\)
0.939184 + 0.343413i \(0.111583\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −25.0000 43.3013i −0.951045 1.64726i −0.743170 0.669102i \(-0.766679\pi\)
−0.207875 0.978155i \(-0.566655\pi\)
\(692\) 0 0
\(693\) −10.0000 + 3.46410i −0.379869 + 0.131590i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.00000 3.46410i 0.0757554 0.131212i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) −4.00000 + 6.92820i −0.150863 + 0.261302i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.50000 + 7.79423i −0.0564133 + 0.293132i
\(708\) 0 0
\(709\) −10.5000 18.1865i −0.394336 0.683010i 0.598680 0.800988i \(-0.295692\pi\)
−0.993016 + 0.117978i \(0.962359\pi\)
\(710\) 0 0
\(711\) 14.0000 24.2487i 0.525041 0.909398i
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 + 20.7846i −0.448148 + 0.776215i
\(718\) 0 0
\(719\) 21.0000 + 36.3731i 0.783168 + 1.35649i 0.930087 + 0.367338i \(0.119731\pi\)
−0.146920 + 0.989148i \(0.546936\pi\)
\(720\) 0 0
\(721\) −26.0000 22.5167i −0.968291 0.838564i
\(722\) 0 0
\(723\) −11.0000 19.0526i −0.409094 0.708572i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 31.0000 1.14973 0.574863 0.818250i \(-0.305055\pi\)
0.574863 + 0.818250i \(0.305055\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −18.0000 + 31.1769i −0.665754 + 1.15312i
\(732\) 0 0
\(733\) −13.0000 22.5167i −0.480166 0.831672i 0.519575 0.854425i \(-0.326090\pi\)
−0.999741 + 0.0227529i \(0.992757\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.00000 + 8.66025i 0.184177 + 0.319005i
\(738\) 0 0
\(739\) −5.00000 + 8.66025i −0.183928 + 0.318573i −0.943215 0.332184i \(-0.892215\pi\)
0.759287 + 0.650756i \(0.225548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −27.0000 −0.990534 −0.495267 0.868741i \(-0.664930\pi\)
−0.495267 + 0.868741i \(0.664930\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.0000 19.0526i −0.402469 0.697097i
\(748\) 0 0
\(749\) 18.0000 + 15.5885i 0.657706 + 0.569590i
\(750\) 0 0
\(751\) −6.00000 10.3923i −0.218943 0.379221i 0.735542 0.677479i \(-0.236928\pi\)
−0.954485 + 0.298259i \(0.903594\pi\)
\(752\) 0 0
\(753\) 4.00000 6.92820i 0.145768 0.252478i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 24.0000 0.872295 0.436147 0.899875i \(-0.356343\pi\)
0.436147 + 0.899875i \(0.356343\pi\)
\(758\) 0 0
\(759\) −1.00000 + 1.73205i −0.0362977 + 0.0628695i
\(760\) 0 0
\(761\) 3.00000 + 5.19615i 0.108750 + 0.188360i 0.915264 0.402854i \(-0.131982\pi\)
−0.806514 + 0.591215i \(0.798649\pi\)
\(762\) 0 0
\(763\) 0.500000 2.59808i 0.0181012 0.0940567i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.0000 3.46410i 0.358748 0.124274i
\(778\) 0 0
\(779\) 1.00000 + 1.73205i 0.0358287 + 0.0620572i
\(780\) 0 0
\(781\) −14.0000 + 24.2487i −0.500959 + 0.867687i
\(782\) 0 0
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 22.5000 38.9711i 0.802038 1.38917i −0.116234 0.993222i \(-0.537082\pi\)
0.918272 0.395949i \(-0.129584\pi\)
\(788\) 0 0
\(789\) 0.500000 + 0.866025i 0.0178005 + 0.0308313i
\(790\) 0 0
\(791\) −5.00000 + 1.73205i −0.177780 + 0.0615846i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −15.0000 + 25.9808i −0.529999 + 0.917985i
\(802\) 0 0
\(803\) 12.0000 + 20.7846i 0.423471 + 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.5000 18.1865i −0.369618 0.640196i
\(808\) 0 0
\(809\) −6.50000 + 11.2583i −0.228528 + 0.395822i −0.957372 0.288858i \(-0.906725\pi\)
0.728844 + 0.684680i \(0.240058\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.00000 15.5885i −0.314870 0.545371i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0000 25.9808i −0.523504 0.906735i −0.999626 0.0273557i \(-0.991291\pi\)
0.476122 0.879379i \(-0.342042\pi\)
\(822\) 0 0
\(823\) 24.5000 42.4352i 0.854016 1.47920i −0.0235383 0.999723i \(-0.507493\pi\)
0.877555 0.479477i \(-0.159174\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.0000 0.591148 0.295574 0.955320i \(-0.404489\pi\)
0.295574 + 0.955320i \(0.404489\pi\)
\(828\) 0 0
\(829\) −1.00000 + 1.73205i −0.0347314 + 0.0601566i −0.882869 0.469620i \(-0.844391\pi\)
0.848137 + 0.529777i \(0.177724\pi\)
\(830\) 0 0
\(831\) 14.0000 + 24.2487i 0.485655 + 0.841178i
\(832\) 0 0
\(833\) −4.00000 27.7128i −0.138592 0.960192i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 + 17.3205i −0.345651 + 0.598684i
\(838\) 0 0
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 9.00000 15.5885i 0.309976 0.536895i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 + 12.1244i 0.481046 + 0.416598i
\(848\) 0 0
\(849\) 2.00000 + 3.46410i 0.0686398 + 0.118888i
\(850\) 0 0
\(851\) −2.00000 + 3.46410i −0.0685591 + 0.118748i
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.0000 25.9808i 0.512390 0.887486i −0.487507 0.873119i \(-0.662093\pi\)
0.999897 0.0143666i \(-0.00457319\pi\)
\(858\) 0 0
\(859\) −20.0000 34.6410i −0.682391 1.18194i −0.974249 0.225475i \(-0.927607\pi\)
0.291858 0.956462i \(-0.405727\pi\)
\(860\) 0 0
\(861\) 0.500000 2.59808i 0.0170400 0.0885422i
\(862\) 0 0
\(863\) −0.500000 0.866025i −0.0170202 0.0294798i 0.857390 0.514667i \(-0.172085\pi\)
−0.874410 + 0.485188i \(0.838751\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 18.0000 + 31.1769i 0.609208 + 1.05518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.00000 + 5.19615i 0.101303 + 0.175462i 0.912222 0.409697i \(-0.134366\pi\)
−0.810919 + 0.585159i \(0.801032\pi\)
\(878\) 0 0
\(879\) −6.00000 + 10.3923i −0.202375 + 0.350524i
\(880\) 0 0
\(881\) 17.0000 0.572745 0.286372 0.958118i \(-0.407551\pi\)
0.286372 + 0.958118i \(0.407551\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.50000 + 12.9904i 0.251825 + 0.436174i 0.964028 0.265799i \(-0.0856358\pi\)
−0.712203 + 0.701974i \(0.752302\pi\)
\(888\) 0 0
\(889\) −20.0000 + 6.92820i −0.670778 + 0.232364i
\(890\) 0 0
\(891\) −1.00000 1.73205i −0.0335013 0.0580259i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.0000 + 31.1769i −0.600334 + 1.03981i
\(900\) 0 0
\(901\) 20.0000 + 34.6410i 0.666297 + 1.15406i
\(902\) 0 0
\(903\) −4.50000 + 23.3827i −0.149751 + 0.778127i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.50000 6.06218i 0.116216 0.201291i −0.802049 0.597258i \(-0.796257\pi\)
0.918265 + 0.395966i \(0.129590\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) 11.0000 19.0526i 0.364047 0.630548i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 + 13.8564i 0.528367 + 0.457579i
\(918\) 0 0
\(919\) −21.0000 36.3731i −0.692726 1.19984i −0.970941 0.239318i \(-0.923076\pi\)
0.278215 0.960519i \(-0.410257\pi\)
\(920\) 0 0
\(921\) −10.5000 + 18.1865i −0.345987 + 0.599267i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 13.0000 22.5167i 0.426976 0.739544i
\(928\) 0 0
\(929\) 6.50000 + 11.2583i 0.213258 + 0.369374i 0.952732 0.303811i \(-0.0982592\pi\)
−0.739474 + 0.673185i \(0.764926\pi\)
\(930\) 0 0
\(931\) 13.0000 + 5.19615i 0.426058 + 0.170297i
\(932\) 0 0
\(933\) 13.0000 + 22.5167i 0.425601 + 0.737162i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.00000 −0.130674 −0.0653372 0.997863i \(-0.520812\pi\)
−0.0653372 + 0.997863i \(0.520812\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) 15.0000 25.9808i 0.488986 0.846949i −0.510934 0.859620i \(-0.670700\pi\)
0.999920 + 0.0126715i \(0.00403357\pi\)
\(942\) 0 0
\(943\) 0.500000 + 0.866025i 0.0162822 + 0.0282017i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.50000 + 4.33013i 0.0812391 + 0.140710i 0.903782 0.427992i \(-0.140779\pi\)
−0.822543 + 0.568702i \(0.807446\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −16.0000 −0.518836
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.00000 + 15.5885i 0.290929 + 0.503903i
\(958\) 0 0
\(959\) 6.00000 31.1769i 0.193750 1.00676i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) −9.00000 + 15.5885i −0.290021 + 0.502331i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −55.0000 −1.76868 −0.884340 0.466843i \(-0.845391\pi\)
−0.884340 + 0.466843i \(0.845391\pi\)
\(968\) 0 0
\(969\) 4.00000 6.92820i 0.128499 0.222566i
\(970\) 0 0
\(971\) 4.00000 + 6.92820i 0.128366 + 0.222337i 0.923044 0.384695i \(-0.125693\pi\)
−0.794678 + 0.607032i \(0.792360\pi\)
\(972\) 0 0
\(973\) −5.00000 + 1.73205i −0.160293 + 0.0555270i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.0000 36.3731i 0.671850 1.16368i −0.305530 0.952183i \(-0.598833\pi\)
0.977379 0.211495i \(-0.0678332\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 1.50000 2.59808i 0.0478426 0.0828658i −0.841112 0.540860i \(-0.818099\pi\)
0.888955 + 0.457995i \(0.151432\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.50000 7.79423i −0.143092 0.247842i
\(990\) 0 0
\(991\) 20.0000 34.6410i 0.635321 1.10041i −0.351126 0.936328i \(-0.614201\pi\)
0.986447 0.164080i \(-0.0524655\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.00000 8.66025i 0.158352 0.274273i −0.775923 0.630828i \(-0.782715\pi\)
0.934274 + 0.356555i \(0.116049\pi\)
\(998\) 0 0
\(999\) 10.0000 + 17.3205i 0.316386 + 0.547997i
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.q.e.1201.1 2
5.2 odd 4 1400.2.bh.b.249.1 4
5.3 odd 4 1400.2.bh.b.249.2 4
5.4 even 2 280.2.q.a.81.1 2
7.2 even 3 inner 1400.2.q.e.401.1 2
7.3 odd 6 9800.2.a.bc.1.1 1
7.4 even 3 9800.2.a.r.1.1 1
15.14 odd 2 2520.2.bi.a.361.1 2
20.19 odd 2 560.2.q.h.81.1 2
35.2 odd 12 1400.2.bh.b.849.2 4
35.4 even 6 1960.2.a.i.1.1 1
35.9 even 6 280.2.q.a.121.1 yes 2
35.19 odd 6 1960.2.q.k.961.1 2
35.23 odd 12 1400.2.bh.b.849.1 4
35.24 odd 6 1960.2.a.e.1.1 1
35.34 odd 2 1960.2.q.k.361.1 2
105.44 odd 6 2520.2.bi.a.1801.1 2
140.39 odd 6 3920.2.a.m.1.1 1
140.59 even 6 3920.2.a.y.1.1 1
140.79 odd 6 560.2.q.h.401.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.a.81.1 2 5.4 even 2
280.2.q.a.121.1 yes 2 35.9 even 6
560.2.q.h.81.1 2 20.19 odd 2
560.2.q.h.401.1 2 140.79 odd 6
1400.2.q.e.401.1 2 7.2 even 3 inner
1400.2.q.e.1201.1 2 1.1 even 1 trivial
1400.2.bh.b.249.1 4 5.2 odd 4
1400.2.bh.b.249.2 4 5.3 odd 4
1400.2.bh.b.849.1 4 35.23 odd 12
1400.2.bh.b.849.2 4 35.2 odd 12
1960.2.a.e.1.1 1 35.24 odd 6
1960.2.a.i.1.1 1 35.4 even 6
1960.2.q.k.361.1 2 35.34 odd 2
1960.2.q.k.961.1 2 35.19 odd 6
2520.2.bi.a.361.1 2 15.14 odd 2
2520.2.bi.a.1801.1 2 105.44 odd 6
3920.2.a.m.1.1 1 140.39 odd 6
3920.2.a.y.1.1 1 140.59 even 6
9800.2.a.r.1.1 1 7.4 even 3
9800.2.a.bc.1.1 1 7.3 odd 6