Properties

Label 676.2.l.c.587.1
Level $676$
Weight $2$
Character 676.587
Analytic conductor $5.398$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [676,2,Mod(19,676)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("676.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.l (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,0,6,0,0,-8,-6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 587.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 676.587
Dual form 676.2.l.c.319.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(0.633975 - 0.633975i) q^{5} +(-2.00000 - 2.00000i) q^{8} +(-1.50000 + 2.59808i) q^{9} +(-1.09808 + 0.633975i) q^{10} +(2.00000 + 3.46410i) q^{16} +(6.86603 + 3.96410i) q^{17} +(3.00000 - 3.00000i) q^{18} +(1.73205 - 0.464102i) q^{20} +4.19615i q^{25} +(-3.33013 - 5.76795i) q^{29} +(-1.46410 - 5.46410i) q^{32} +(-7.92820 - 7.92820i) q^{34} +(-5.19615 + 3.00000i) q^{36} +(-3.13397 + 11.6962i) q^{37} -2.53590 q^{40} +(9.96410 + 2.66987i) q^{41} +(0.696152 + 2.59808i) q^{45} +(6.06218 - 3.50000i) q^{49} +(1.53590 - 5.73205i) q^{50} +10.4641 q^{53} +(2.43782 + 9.09808i) q^{58} +(-2.69615 + 4.66987i) q^{61} +8.00000i q^{64} +(7.92820 + 13.7321i) q^{68} +(8.19615 - 2.19615i) q^{72} +(9.83013 + 9.83013i) q^{73} +(8.56218 - 14.8301i) q^{74} +(3.46410 + 0.928203i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-12.6340 - 7.29423i) q^{82} +(6.86603 - 1.83975i) q^{85} +(1.09808 - 4.09808i) q^{89} -3.80385i q^{90} +(1.83013 + 6.83013i) q^{97} +(-9.56218 + 2.56218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9} + 6 q^{10} + 8 q^{16} + 24 q^{17} + 12 q^{18} + 4 q^{29} + 8 q^{32} - 4 q^{34} - 16 q^{37} - 24 q^{40} + 26 q^{41} - 18 q^{45} + 20 q^{50} + 28 q^{53} + 34 q^{58}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{12}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36603 0.366025i −0.965926 0.258819i
\(3\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) 0.633975 0.633975i 0.283522 0.283522i −0.550990 0.834512i \(-0.685750\pi\)
0.834512 + 0.550990i \(0.185750\pi\)
\(6\) 0 0
\(7\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(8\) −2.00000 2.00000i −0.707107 0.707107i
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) −1.09808 + 0.633975i −0.347242 + 0.200480i
\(11\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 6.86603 + 3.96410i 1.66526 + 0.961436i 0.970143 + 0.242536i \(0.0779791\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 3.00000 3.00000i 0.707107 0.707107i
\(19\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(20\) 1.73205 0.464102i 0.387298 0.103776i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 4.19615i 0.839230i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.33013 5.76795i −0.618389 1.07108i −0.989780 0.142605i \(-0.954452\pi\)
0.371391 0.928477i \(-0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) −1.46410 5.46410i −0.258819 0.965926i
\(33\) 0 0
\(34\) −7.92820 7.92820i −1.35968 1.35968i
\(35\) 0 0
\(36\) −5.19615 + 3.00000i −0.866025 + 0.500000i
\(37\) −3.13397 + 11.6962i −0.515222 + 1.92284i −0.164399 + 0.986394i \(0.552568\pi\)
−0.350823 + 0.936442i \(0.614098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.53590 −0.400961
\(41\) 9.96410 + 2.66987i 1.55613 + 0.416964i 0.931436 0.363905i \(-0.118557\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(44\) 0 0
\(45\) 0.696152 + 2.59808i 0.103776 + 0.387298i
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 6.06218 3.50000i 0.866025 0.500000i
\(50\) 1.53590 5.73205i 0.217209 0.810634i
\(51\) 0 0
\(52\) 0 0
\(53\) 10.4641 1.43735 0.718677 0.695344i \(-0.244748\pi\)
0.718677 + 0.695344i \(0.244748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.43782 + 9.09808i 0.320102 + 1.19464i
\(59\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(60\) 0 0
\(61\) −2.69615 + 4.66987i −0.345207 + 0.597916i −0.985391 0.170305i \(-0.945525\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(68\) 7.92820 + 13.7321i 0.961436 + 1.66526i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(72\) 8.19615 2.19615i 0.965926 0.258819i
\(73\) 9.83013 + 9.83013i 1.15053 + 1.15053i 0.986447 + 0.164083i \(0.0524664\pi\)
0.164083 + 0.986447i \(0.447534\pi\)
\(74\) 8.56218 14.8301i 0.995333 1.72397i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 3.46410 + 0.928203i 0.387298 + 0.103776i
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) −12.6340 7.29423i −1.39519 0.805513i
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 6.86603 1.83975i 0.744725 0.199548i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.09808 4.09808i 0.116396 0.434395i −0.882992 0.469389i \(-0.844474\pi\)
0.999388 + 0.0349934i \(0.0111410\pi\)
\(90\) 3.80385i 0.400961i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.83013 + 6.83013i 0.185821 + 0.693494i 0.994453 + 0.105180i \(0.0335417\pi\)
−0.808632 + 0.588315i \(0.799792\pi\)
\(98\) −9.56218 + 2.56218i −0.965926 + 0.258819i
\(99\) 0 0
\(100\) −4.19615 + 7.26795i −0.419615 + 0.726795i
\(101\) 10.1603 5.86603i 1.01098 0.583691i 0.0995037 0.995037i \(-0.468274\pi\)
0.911479 + 0.411346i \(0.134941\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −14.2942 3.83013i −1.38838 0.372015i
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) −7.00000 + 7.00000i −0.670478 + 0.670478i −0.957826 0.287348i \(-0.907226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.06218 3.57180i 0.193993 0.336006i −0.752577 0.658505i \(-0.771189\pi\)
0.946570 + 0.322498i \(0.104523\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 13.3205i 1.23678i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.52628 5.50000i −0.866025 0.500000i
\(122\) 5.39230 5.39230i 0.488196 0.488196i
\(123\) 0 0
\(124\) 0 0
\(125\) 5.83013 + 5.83013i 0.521462 + 0.521462i
\(126\) 0 0
\(127\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(128\) 2.92820 10.9282i 0.258819 0.965926i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.80385 21.6603i −0.497676 1.85735i
\(137\) −22.5263 + 6.03590i −1.92455 + 0.515682i −0.939793 + 0.341743i \(0.888983\pi\)
−0.984757 + 0.173939i \(0.944351\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) −5.76795 1.54552i −0.479002 0.128348i
\(146\) −9.83013 17.0263i −0.813547 1.40910i
\(147\) 0 0
\(148\) −17.1244 + 17.1244i −1.40761 + 1.40761i
\(149\) −4.06218 15.1603i −0.332787 1.24198i −0.906249 0.422744i \(-0.861067\pi\)
0.573462 0.819232i \(-0.305600\pi\)
\(150\) 0 0
\(151\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(152\) 0 0
\(153\) −20.5981 + 11.8923i −1.66526 + 0.961436i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −25.0526 −1.99941 −0.999706 0.0242497i \(-0.992280\pi\)
−0.999706 + 0.0242497i \(0.992280\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −4.39230 2.53590i −0.347242 0.200480i
\(161\) 0 0
\(162\) 3.29423 + 12.2942i 0.258819 + 0.965926i
\(163\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(164\) 14.5885 + 14.5885i 1.13917 + 1.13917i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −10.0526 −0.770996
\(171\) 0 0
\(172\) 0 0
\(173\) 3.46410 + 2.00000i 0.263371 + 0.152057i 0.625871 0.779926i \(-0.284744\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −3.00000 + 5.19615i −0.224860 + 0.389468i
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) −1.39230 + 5.19615i −0.103776 + 0.387298i
\(181\) 26.3205i 1.95639i −0.207693 0.978194i \(-0.566596\pi\)
0.207693 0.978194i \(-0.433404\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.42820 + 9.40192i 0.399089 + 0.691243i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 5.10770 19.0622i 0.367660 1.37213i −0.496119 0.868255i \(-0.665242\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 10.0000i 0.717958i
\(195\) 0 0
\(196\) 14.0000 1.00000
\(197\) −20.4904 5.49038i −1.45988 0.391173i −0.560431 0.828201i \(-0.689365\pi\)
−0.899448 + 0.437028i \(0.856031\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 8.39230 8.39230i 0.593426 0.593426i
\(201\) 0 0
\(202\) −16.0263 + 4.29423i −1.12761 + 0.302141i
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00962 4.62436i 0.559416 0.322979i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) 18.1244 + 10.4641i 1.24479 + 0.718677i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 12.1244 7.00000i 0.821165 0.474100i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(224\) 0 0
\(225\) −10.9019 6.29423i −0.726795 0.419615i
\(226\) −4.12436 + 4.12436i −0.274348 + 0.274348i
\(227\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(228\) 0 0
\(229\) −17.0000 17.0000i −1.12339 1.12339i −0.991228 0.132164i \(-0.957808\pi\)
−0.132164 0.991228i \(-0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.87564 + 18.1962i −0.320102 + 1.19464i
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) 25.9904 6.96410i 1.67419 0.448597i 0.707953 0.706260i \(-0.249619\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) 11.0000 + 11.0000i 0.707107 + 0.707107i
\(243\) 0 0
\(244\) −9.33975 + 5.39230i −0.597916 + 0.345207i
\(245\) 1.62436 6.06218i 0.103776 0.387298i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −5.83013 10.0981i −0.368730 0.638658i
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) −12.3564 + 7.13397i −0.770771 + 0.445005i −0.833150 0.553047i \(-0.813465\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 19.9808 1.23678
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 6.63397 6.63397i 0.407522 0.407522i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 17.3205i 0.609711 1.05605i −0.381577 0.924337i \(-0.624619\pi\)
0.991288 0.131713i \(-0.0420477\pi\)
\(270\) 0 0
\(271\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(272\) 31.7128i 1.92287i
\(273\) 0 0
\(274\) 32.9808 1.99244
\(275\) 0 0
\(276\) 0 0
\(277\) −1.37564 0.794229i −0.0826545 0.0477206i 0.458103 0.888899i \(-0.348529\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.6865 12.6865i −0.756815 0.756815i 0.218926 0.975741i \(-0.429745\pi\)
−0.975741 + 0.218926i \(0.929745\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 16.3923 + 4.39230i 0.965926 + 0.258819i
\(289\) 22.9282 + 39.7128i 1.34872 + 2.33605i
\(290\) 7.31347 + 4.22243i 0.429462 + 0.247950i
\(291\) 0 0
\(292\) 7.19615 + 26.8564i 0.421123 + 1.57165i
\(293\) −4.76795 + 1.27757i −0.278547 + 0.0746363i −0.395388 0.918514i \(-0.629390\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 29.6603 17.1244i 1.72397 0.995333i
\(297\) 0 0
\(298\) 22.1962i 1.28579i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.25129 + 4.66987i 0.0716486 + 0.267396i
\(306\) 32.4904 8.70577i 1.85735 0.497676i
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 34.2224 + 9.16987i 1.93128 + 0.517486i
\(315\) 0 0
\(316\) 0 0
\(317\) 23.1506 23.1506i 1.30027 1.30027i 0.372061 0.928208i \(-0.378651\pi\)
0.928208 0.372061i \(-0.121349\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.07180 + 5.07180i 0.283522 + 0.283522i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −14.5885 25.2679i −0.805513 1.39519i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(332\) 0 0
\(333\) −25.6865 25.6865i −1.40761 1.40761i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.7128i 1.01935i 0.860366 + 0.509676i \(0.170235\pi\)
−0.860366 + 0.509676i \(0.829765\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 13.7321 + 3.67949i 0.744725 + 0.199548i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −4.00000 4.00000i −0.215041 0.215041i
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −8.41858 + 31.4186i −0.450636 + 1.68180i 0.249973 + 0.968253i \(0.419578\pi\)
−0.700609 + 0.713545i \(0.747088\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.7224 + 7.42820i 1.47552 + 0.395363i 0.904819 0.425797i \(-0.140006\pi\)
0.570697 + 0.821160i \(0.306673\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 6.00000i 0.317999 0.317999i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 3.80385 6.58846i 0.200480 0.347242i
\(361\) −16.4545 + 9.50000i −0.866025 + 0.500000i
\(362\) −9.63397 + 35.9545i −0.506350 + 1.88973i
\(363\) 0 0
\(364\) 0 0
\(365\) 12.4641 0.652401
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 0 0
\(369\) −21.8827 + 21.8827i −1.13917 + 1.13917i
\(370\) −3.97372 14.8301i −0.206584 0.770982i
\(371\) 0 0
\(372\) 0 0
\(373\) 15.0622 26.0885i 0.779890 1.35081i −0.152115 0.988363i \(-0.548608\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13.9545 + 24.1699i −0.710264 + 1.23021i
\(387\) 0 0
\(388\) −3.66025 + 13.6603i −0.185821 + 0.693494i
\(389\) 0.320508i 0.0162504i −0.999967 0.00812520i \(-0.997414\pi\)
0.999967 0.00812520i \(-0.00258636\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −19.1244 5.12436i −0.965926 0.258819i
\(393\) 0 0
\(394\) 25.9808 + 15.0000i 1.30889 + 0.755689i
\(395\) 0 0
\(396\) 0 0
\(397\) 34.1506 9.15064i 1.71397 0.459257i 0.737579 0.675261i \(-0.235969\pi\)
0.976392 + 0.216004i \(0.0693024\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −14.5359 + 8.39230i −0.726795 + 0.419615i
\(401\) 9.86603 36.8205i 0.492686 1.83873i −0.0499376 0.998752i \(-0.515902\pi\)
0.542623 0.839976i \(-0.317431\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 23.4641 1.16738
\(405\) −7.79423 2.08846i −0.387298 0.103776i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.17949 + 15.5981i 0.206663 + 0.771275i 0.988936 + 0.148340i \(0.0473931\pi\)
−0.782274 + 0.622935i \(0.785940\pi\)
\(410\) −12.6340 + 3.38526i −0.623948 + 0.167186i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 13.6340 13.6340i 0.664479 0.664479i −0.291953 0.956433i \(-0.594305\pi\)
0.956433 + 0.291953i \(0.0943052\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −20.9282 20.9282i −1.01636 1.01636i
\(425\) −16.6340 + 28.8109i −0.806866 + 1.39753i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(432\) 0 0
\(433\) 15.1077 + 8.72243i 0.726029 + 0.419173i 0.816968 0.576683i \(-0.195653\pi\)
−0.0909384 + 0.995857i \(0.528987\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −19.1244 + 5.12436i −0.915891 + 0.245412i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −1.90192 3.29423i −0.0901598 0.156161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −36.8827 + 9.88269i −1.74060 + 0.466393i −0.982581 0.185837i \(-0.940500\pi\)
−0.758021 + 0.652230i \(0.773834\pi\)
\(450\) 12.5885 + 12.5885i 0.593426 + 0.593426i
\(451\) 0 0
\(452\) 7.14359 4.12436i 0.336006 0.193993i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.03590 0.813467i −0.142013 0.0380524i 0.187112 0.982339i \(-0.440087\pi\)
−0.329125 + 0.944286i \(0.606754\pi\)
\(458\) 17.0000 + 29.4449i 0.794358 + 1.37587i
\(459\) 0 0
\(460\) 0 0
\(461\) −7.54552 28.1603i −0.351430 1.31155i −0.884918 0.465746i \(-0.845786\pi\)
0.533488 0.845807i \(-0.320881\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 13.3205 23.0718i 0.618389 1.07108i
\(465\) 0 0
\(466\) 5.85641 21.8564i 0.271293 1.01248i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.6962 + 27.1865i −0.718677 + 1.24479i
\(478\) 0 0
\(479\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −38.0526 −1.73325
\(483\) 0 0
\(484\) −11.0000 19.0526i −0.500000 0.866025i
\(485\) 5.49038 + 3.16987i 0.249305 + 0.143937i
\(486\) 0 0
\(487\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(488\) 14.7321 3.94744i 0.666889 0.178692i
\(489\) 0 0
\(490\) −4.43782 + 7.68653i −0.200480 + 0.347242i
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 52.8038i 2.37817i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 4.26795 + 15.9282i 0.190868 + 0.712331i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 2.72243 10.1603i 0.121147 0.452125i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.5526 10.3301i −1.70881 0.457875i −0.733679 0.679496i \(-0.762199\pi\)
−0.975133 + 0.221621i \(0.928865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 16.0000 16.0000i 0.707107 0.707107i
\(513\) 0 0
\(514\) 19.4904 5.22243i 0.859684 0.230352i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.947441 0.0415081 0.0207541 0.999785i \(-0.493393\pi\)
0.0207541 + 0.999785i \(0.493393\pi\)
\(522\) −27.2942 7.31347i −1.19464 0.320102i
\(523\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) −11.4904 + 6.63397i −0.499110 + 0.288161i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −20.0000 + 20.0000i −0.862261 + 0.862261i
\(539\) 0 0
\(540\) 0 0
\(541\) 32.3468 + 32.3468i 1.39070 + 1.39070i 0.823764 + 0.566933i \(0.191870\pi\)
0.566933 + 0.823764i \(0.308130\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 11.6077 43.3205i 0.497676 1.85735i
\(545\) 8.87564i 0.380191i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −45.0526 12.0718i −1.92455 0.515682i
\(549\) −8.08846 14.0096i −0.345207 0.597916i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.58846 + 1.58846i 0.0674871 + 0.0674871i
\(555\) 0 0
\(556\) 0 0
\(557\) −11.3756 + 42.4545i −0.482002 + 1.79885i 0.111198 + 0.993798i \(0.464531\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 12.6865 + 21.9737i 0.535149 + 0.926905i
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) −0.957060 3.57180i −0.0402638 0.150267i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.6410 20.0000i 1.45223 0.838444i 0.453619 0.891196i \(-0.350133\pi\)
0.998608 + 0.0527519i \(0.0167993\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −20.7846 12.0000i −0.866025 0.500000i
\(577\) 10.1506 10.1506i 0.422576 0.422576i −0.463513 0.886090i \(-0.653411\pi\)
0.886090 + 0.463513i \(0.153411\pi\)
\(578\) −16.7846 62.6410i −0.698148 2.60552i
\(579\) 0 0
\(580\) −8.44486 8.44486i −0.350654 0.350654i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 39.3205i 1.62709i
\(585\) 0 0
\(586\) 6.98076 0.288373
\(587\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −46.7846 + 12.5359i −1.92284 + 0.515222i
\(593\) 19.3468 + 19.3468i 0.794477 + 0.794477i 0.982219 0.187741i \(-0.0601166\pi\)
−0.187741 + 0.982219i \(0.560117\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.12436 30.3205i 0.332787 1.24198i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −16.3301 28.2846i −0.666120 1.15375i −0.978980 0.203954i \(-0.934621\pi\)
0.312861 0.949799i \(-0.398713\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.52628 + 2.55256i −0.387298 + 0.103776i
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 6.83717i 0.276829i
\(611\) 0 0
\(612\) −47.5692 −1.92287
\(613\) 40.7224 + 10.9115i 1.64476 + 0.440713i 0.958140 0.286300i \(-0.0924254\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.45448 + 20.3564i 0.219589 + 0.819518i 0.984500 + 0.175382i \(0.0561162\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −13.5885 −0.543538
\(626\) 32.7846 + 8.78461i 1.31034 + 0.351104i
\(627\) 0 0
\(628\) −43.3923 25.0526i −1.73154 0.999706i
\(629\) −67.8827 + 67.8827i −2.70666 + 2.70666i
\(630\) 0 0
\(631\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −40.0981 + 23.1506i −1.59250 + 0.919429i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −5.07180 8.78461i −0.200480 0.347242i
\(641\) −15.6506 9.03590i −0.618163 0.356897i 0.157991 0.987441i \(-0.449498\pi\)
−0.776153 + 0.630544i \(0.782832\pi\)
\(642\) 0 0
\(643\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) −6.58846 + 24.5885i −0.258819 + 0.965926i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.0000 38.1051i −0.860927 1.49117i −0.871036 0.491220i \(-0.836551\pi\)
0.0101092 0.999949i \(-0.496782\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10.6795 + 39.8564i 0.416964 + 1.55613i
\(657\) −40.2846 + 10.7942i −1.57165 + 0.421123i
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) 0.349365 1.30385i 0.0135887 0.0507138i −0.958799 0.284087i \(-0.908310\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 25.6865 + 44.4904i 0.995333 + 1.72397i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.91858 1.10770i 0.0739560 0.0426985i −0.462566 0.886585i \(-0.653071\pi\)
0.536522 + 0.843886i \(0.319738\pi\)
\(674\) 6.84936 25.5622i 0.263828 0.984618i
\(675\) 0 0
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −17.4115 10.0526i −0.667702 0.385498i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(684\) 0 0
\(685\) −10.4545 + 18.1077i −0.399445 + 0.691859i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(692\) 4.00000 + 6.92820i 0.152057 + 0.263371i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 57.8301 + 57.8301i 2.19047 + 2.19047i
\(698\) 23.0000 39.8372i 0.870563 1.50786i
\(699\) 0 0
\(700\) 0 0
\(701\) 10.0000i 0.377695i −0.982006 0.188847i \(-0.939525\pi\)
0.982006 0.188847i \(-0.0604752\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −35.1506 20.2942i −1.32291 0.763783i
\(707\) 0 0
\(708\) 0 0
\(709\) 38.9904 10.4474i 1.46431 0.392362i 0.563337 0.826227i \(-0.309517\pi\)
0.900978 + 0.433865i \(0.142851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.3923 + 6.00000i −0.389468 + 0.224860i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) −7.60770 + 7.60770i −0.283522 + 0.283522i
\(721\) 0 0
\(722\) 25.9545 6.95448i 0.965926 0.258819i
\(723\) 0 0
\(724\) 26.3205 45.5885i 0.978194 1.69428i
\(725\) 24.2032 13.9737i 0.898884 0.518971i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) −17.0263 4.56218i −0.630171 0.168854i
\(731\) 0 0
\(732\) 0 0
\(733\) 36.1506 36.1506i 1.33525 1.33525i 0.434659 0.900595i \(-0.356869\pi\)
0.900595 0.434659i \(-0.143131\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 37.9019 21.8827i 1.39519 0.805513i
\(739\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(740\) 21.7128i 0.798179i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(744\) 0 0
\(745\) −12.1865 7.03590i −0.446480 0.257775i
\(746\) −30.1244 + 30.1244i −1.10293 + 1.10293i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −9.00000 15.5885i −0.327111 0.566572i 0.654827 0.755779i \(-0.272742\pi\)
−0.981937 + 0.189207i \(0.939408\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.36603 + 0.366025i −0.0495184 + 0.0132684i −0.283493 0.958974i \(-0.591493\pi\)
0.233975 + 0.972243i \(0.424827\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.51924 + 20.5981i −0.199548 + 0.744725i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 50.5429 + 13.5429i 1.82263 + 0.488371i 0.997107 0.0760054i \(-0.0242166\pi\)
0.825518 + 0.564376i \(0.190883\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 27.9090 27.9090i 1.00447 1.00447i
\(773\) 1.83013 + 6.83013i 0.0658251 + 0.245663i 0.990997 0.133887i \(-0.0427458\pi\)
−0.925172 + 0.379549i \(0.876079\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 17.3205i 0.358979 0.621770i
\(777\) 0 0
\(778\) −0.117314 + 0.437822i −0.00420591 + 0.0156967i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 24.2487 + 14.0000i 0.866025 + 0.500000i
\(785\) −15.8827 + 15.8827i −0.566877 + 0.566877i
\(786\) 0 0
\(787\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(788\) −30.0000 30.0000i −1.06871 1.06871i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −50.0000 −1.77443
\(795\) 0 0
\(796\) 0 0
\(797\) −19.0526 11.0000i −0.674876 0.389640i 0.123045 0.992401i \(-0.460734\pi\)
−0.797922 + 0.602761i \(0.794067\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 22.9282 6.14359i 0.810634 0.217209i
\(801\) 9.00000 + 9.00000i 0.317999 + 0.317999i
\(802\) −26.9545 + 46.6865i −0.951796 + 1.64856i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −32.0526 8.58846i −1.12761 0.302141i
\(809\) 9.66987 + 16.7487i 0.339975 + 0.588853i 0.984428 0.175791i \(-0.0562482\pi\)
−0.644453 + 0.764644i \(0.722915\pi\)
\(810\) 9.88269 + 5.70577i 0.347242 + 0.200480i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 22.8372i 0.798483i
\(819\) 0 0
\(820\) 18.4974 0.645958
\(821\) 15.0263 + 4.02628i 0.524421 + 0.140518i 0.511311 0.859396i \(-0.329160\pi\)
0.0131101 + 0.999914i \(0.495827\pi\)
\(822\) 0 0
\(823\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 49.1603 28.3827i 1.70741 0.985771i 0.769657 0.638457i \(-0.220427\pi\)
0.937749 0.347314i \(-0.112906\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 55.4974 1.92287
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(840\) 0 0
\(841\) −7.67949 + 13.3013i −0.264810 + 0.458664i
\(842\) −23.6147 + 13.6340i −0.813818 + 0.469858i
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 20.9282 + 36.2487i 0.718677 + 1.24479i
\(849\) 0 0
\(850\) 33.2679 33.2679i 1.14108 1.14108i
\(851\) 0 0
\(852\) 0 0
\(853\) −16.1699 16.1699i −0.553646 0.553646i 0.373845 0.927491i \(-0.378039\pi\)
−0.927491 + 0.373845i \(0.878039\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.2295i 1.85244i 0.376979 + 0.926222i \(0.376963\pi\)
−0.376979 + 0.926222i \(0.623037\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 3.46410 0.928203i 0.117783 0.0315599i
\(866\) −17.4449 17.4449i −0.592801 0.592801i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 28.0000 0.948200
\(873\) −20.4904 5.49038i −0.693494 0.185821i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.6962 + 51.1147i 0.462486 + 1.72602i 0.665092 + 0.746762i \(0.268392\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −51.3564 + 29.6506i −1.73024 + 0.998955i −0.842271 + 0.539054i \(0.818782\pi\)
−0.887970 + 0.459902i \(0.847885\pi\)
\(882\) 7.68653 28.6865i 0.258819 0.965926i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.39230 + 5.19615i 0.0466702 + 0.174175i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 54.0000 1.80200
\(899\) 0 0
\(900\) −12.5885 21.8038i −0.419615 0.726795i
\(901\) 71.8468 + 41.4808i 2.39356 + 1.38192i
\(902\) 0 0
\(903\) 0 0
\(904\) −11.2679 + 3.01924i −0.374766 + 0.100418i
\(905\) −16.6865 16.6865i −0.554679 0.554679i
\(906\) 0 0
\(907\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(908\) 0 0
\(909\) 35.1962i 1.16738i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.84936 + 2.22243i 0.127326 + 0.0735115i
\(915\) 0 0
\(916\) −12.4449 46.4449i −0.411190 1.53458i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 41.2295i 1.35782i
\(923\) 0 0
\(924\) 0 0
\(925\) −49.0788 13.1506i −1.61370 0.432390i
\(926\) 0 0
\(927\) 0 0
\(928\) −26.6410 + 26.6410i −0.874534 + 0.874534i
\(929\) −8.82051 32.9186i −0.289391 1.08002i −0.945570 0.325418i \(-0.894495\pi\)
0.656179 0.754606i \(-0.272172\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −16.0000 + 27.7128i −0.524097 + 0.907763i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −60.5692 −1.97871 −0.989355 0.145522i \(-0.953514\pi\)
−0.989355 + 0.145522i \(0.953514\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.0000 19.0000i 0.619382 0.619382i −0.325991 0.945373i \(-0.605698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.4974 + 28.0000i 1.57099 + 0.907009i 0.996048 + 0.0888114i \(0.0283068\pi\)
0.574937 + 0.818198i \(0.305026\pi\)
\(954\) 31.3923 31.3923i 1.01636 1.01636i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000i 1.00000i
\(962\) 0 0
\(963\) 0 0
\(964\) 51.9808 + 13.9282i 1.67419 + 0.448597i
\(965\) −8.84679 15.3231i −0.284788 0.493268i
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) 8.05256 + 30.0526i 0.258819 + 0.965926i
\(969\) 0 0
\(970\) −6.33975 6.33975i −0.203557 0.203557i
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −21.5692 −0.690414
\(977\) 22.9641 + 6.15321i 0.734687 + 0.196859i 0.606715 0.794919i \(-0.292487\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 8.87564 8.87564i 0.283522 0.283522i
\(981\) −7.68653 28.6865i −0.245412 0.915891i
\(982\) 0 0
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 0 0
\(985\) −16.4711 + 9.50962i −0.524814 + 0.303002i
\(986\) −19.3275 + 72.1314i −0.615515 + 2.29713i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.3038 17.8468i 0.326326 0.565213i −0.655454 0.755235i \(-0.727523\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) 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 676.2.l.c.587.1 4
4.3 odd 2 CM 676.2.l.c.587.1 4
13.2 odd 12 676.2.f.d.99.2 4
13.3 even 3 676.2.f.e.239.1 4
13.4 even 6 676.2.l.d.19.1 4
13.5 odd 4 676.2.l.d.427.1 4
13.6 odd 12 676.2.l.e.319.1 4
13.7 odd 12 inner 676.2.l.c.319.1 4
13.8 odd 4 52.2.l.a.11.1 4
13.9 even 3 52.2.l.a.19.1 yes 4
13.10 even 6 676.2.f.d.239.2 4
13.11 odd 12 676.2.f.e.99.1 4
13.12 even 2 676.2.l.e.587.1 4
39.8 even 4 468.2.cb.d.271.1 4
39.35 odd 6 468.2.cb.d.19.1 4
52.3 odd 6 676.2.f.e.239.1 4
52.7 even 12 inner 676.2.l.c.319.1 4
52.11 even 12 676.2.f.e.99.1 4
52.15 even 12 676.2.f.d.99.2 4
52.19 even 12 676.2.l.e.319.1 4
52.23 odd 6 676.2.f.d.239.2 4
52.31 even 4 676.2.l.d.427.1 4
52.35 odd 6 52.2.l.a.19.1 yes 4
52.43 odd 6 676.2.l.d.19.1 4
52.47 even 4 52.2.l.a.11.1 4
52.51 odd 2 676.2.l.e.587.1 4
104.21 odd 4 832.2.bu.d.63.1 4
104.35 odd 6 832.2.bu.d.383.1 4
104.61 even 6 832.2.bu.d.383.1 4
104.99 even 4 832.2.bu.d.63.1 4
156.35 even 6 468.2.cb.d.19.1 4
156.47 odd 4 468.2.cb.d.271.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
52.2.l.a.11.1 4 13.8 odd 4
52.2.l.a.11.1 4 52.47 even 4
52.2.l.a.19.1 yes 4 13.9 even 3
52.2.l.a.19.1 yes 4 52.35 odd 6
468.2.cb.d.19.1 4 39.35 odd 6
468.2.cb.d.19.1 4 156.35 even 6
468.2.cb.d.271.1 4 39.8 even 4
468.2.cb.d.271.1 4 156.47 odd 4
676.2.f.d.99.2 4 13.2 odd 12
676.2.f.d.99.2 4 52.15 even 12
676.2.f.d.239.2 4 13.10 even 6
676.2.f.d.239.2 4 52.23 odd 6
676.2.f.e.99.1 4 13.11 odd 12
676.2.f.e.99.1 4 52.11 even 12
676.2.f.e.239.1 4 13.3 even 3
676.2.f.e.239.1 4 52.3 odd 6
676.2.l.c.319.1 4 13.7 odd 12 inner
676.2.l.c.319.1 4 52.7 even 12 inner
676.2.l.c.587.1 4 1.1 even 1 trivial
676.2.l.c.587.1 4 4.3 odd 2 CM
676.2.l.d.19.1 4 13.4 even 6
676.2.l.d.19.1 4 52.43 odd 6
676.2.l.d.427.1 4 13.5 odd 4
676.2.l.d.427.1 4 52.31 even 4
676.2.l.e.319.1 4 13.6 odd 12
676.2.l.e.319.1 4 52.19 even 12
676.2.l.e.587.1 4 13.12 even 2
676.2.l.e.587.1 4 52.51 odd 2
832.2.bu.d.63.1 4 104.21 odd 4
832.2.bu.d.63.1 4 104.99 even 4
832.2.bu.d.383.1 4 104.35 odd 6
832.2.bu.d.383.1 4 104.61 even 6