Properties

Label 945.2.cs.a.419.3
Level $945$
Weight $2$
Character 945.419
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(104,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cs (of order \(18\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 419.3
Character \(\chi\) \(=\) 945.419
Dual form 945.2.cs.a.839.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.306808 - 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(-1.43732 + 1.71293i) q^{5} +(-2.48619 - 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +O(q^{10})\) \(q+(0.306808 - 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(-1.43732 + 1.71293i) q^{5} +(-2.48619 - 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +(-2.55941 - 3.05019i) q^{11} +(-0.589446 - 3.41358i) q^{12} +(-1.10071 + 6.24241i) q^{13} +(2.47898 + 2.97568i) q^{15} +(3.06418 - 2.57115i) q^{16} +(-4.17071 - 2.40796i) q^{17} +(-1.52956 + 4.20243i) q^{20} +(-2.30533 + 3.96049i) q^{21} +(-0.868241 - 4.92404i) q^{25} +(-2.64575 + 4.47214i) q^{27} -5.29150 q^{28} +(-10.5889 + 1.86711i) q^{29} +(-5.98479 + 3.42711i) q^{33} +(5.12348 - 2.95804i) q^{35} +(-5.99985 - 0.0425087i) q^{36} +(10.3035 + 3.79155i) q^{39} +(-6.89658 - 3.98174i) q^{44} +(5.83309 - 3.31286i) q^{45} +(4.67425 - 12.8424i) q^{47} +(-3.44283 - 6.01223i) q^{48} +(5.36231 + 4.49951i) q^{49} +(-5.38436 + 6.37086i) q^{51} +(2.20141 + 12.4848i) q^{52} +8.90344 q^{55} +(6.69444 + 3.89672i) q^{60} +(6.04399 + 5.14492i) q^{63} +(4.00000 - 6.92820i) q^{64} +(-9.11074 - 10.8578i) q^{65} +(-9.48551 - 1.67255i) q^{68} +(6.07461 + 3.50718i) q^{71} +(-7.26673 - 12.5863i) q^{73} +(-8.66020 - 0.0306782i) q^{75} +(3.60308 + 9.89937i) q^{77} +(3.00497 + 17.0421i) q^{79} +8.94427i q^{80} +(6.81174 + 5.88220i) q^{81} +(-13.8663 + 2.44500i) q^{83} +(-1.62348 + 9.02022i) q^{84} +(10.1193 - 3.68312i) q^{85} +(-0.0659721 + 18.6234i) q^{87} +(8.38533 - 14.5238i) q^{91} +(11.5742 - 9.71188i) q^{97} +(4.00588 + 11.2535i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(3\) 0.306808 1.70466i 0.177136 0.984186i
\(4\) 1.87939 0.684040i 0.939693 0.342020i
\(5\) −1.43732 + 1.71293i −0.642788 + 0.766044i
\(6\) 0 0
\(7\) −2.48619 0.904900i −0.939693 0.342020i
\(8\) 0 0
\(9\) −2.81174 1.04601i −0.937246 0.348669i
\(10\) 0 0
\(11\) −2.55941 3.05019i −0.771692 0.919667i 0.226834 0.973933i \(-0.427162\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) −0.589446 3.41358i −0.170158 0.985417i
\(13\) −1.10071 + 6.24241i −0.305281 + 1.73133i 0.316898 + 0.948459i \(0.397359\pi\)
−0.622179 + 0.782875i \(0.713753\pi\)
\(14\) 0 0
\(15\) 2.47898 + 2.97568i 0.640070 + 0.768317i
\(16\) 3.06418 2.57115i 0.766044 0.642788i
\(17\) −4.17071 2.40796i −1.01154 0.584016i −0.0999013 0.994997i \(-0.531853\pi\)
−0.911644 + 0.410982i \(0.865186\pi\)
\(18\) 0 0
\(19\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) −1.52956 + 4.20243i −0.342020 + 0.939693i
\(21\) −2.30533 + 3.96049i −0.503065 + 0.864249i
\(22\) 0 0
\(23\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(24\) 0 0
\(25\) −0.868241 4.92404i −0.173648 0.984808i
\(26\) 0 0
\(27\) −2.64575 + 4.47214i −0.509175 + 0.860663i
\(28\) −5.29150 −1.00000
\(29\) −10.5889 + 1.86711i −1.96631 + 0.346714i −0.973104 + 0.230366i \(0.926008\pi\)
−0.993208 + 0.116348i \(0.962881\pi\)
\(30\) 0 0
\(31\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(32\) 0 0
\(33\) −5.98479 + 3.42711i −1.04182 + 0.596583i
\(34\) 0 0
\(35\) 5.12348 2.95804i 0.866025 0.500000i
\(36\) −5.99985 0.0425087i −0.999975 0.00708479i
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 10.3035 + 3.79155i 1.64988 + 0.607134i
\(40\) 0 0
\(41\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(42\) 0 0
\(43\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(44\) −6.89658 3.98174i −1.03970 0.600270i
\(45\) 5.83309 3.31286i 0.869546 0.493852i
\(46\) 0 0
\(47\) 4.67425 12.8424i 0.681810 1.87326i 0.265086 0.964225i \(-0.414600\pi\)
0.416724 0.909033i \(-0.363178\pi\)
\(48\) −3.44283 6.01223i −0.496929 0.867791i
\(49\) 5.36231 + 4.49951i 0.766044 + 0.642788i
\(50\) 0 0
\(51\) −5.38436 + 6.37086i −0.753961 + 0.892099i
\(52\) 2.20141 + 12.4848i 0.305281 + 1.73133i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 8.90344 1.20054
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(60\) 6.69444 + 3.89672i 0.864249 + 0.503065i
\(61\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(62\) 0 0
\(63\) 6.04399 + 5.14492i 0.761471 + 0.648199i
\(64\) 4.00000 6.92820i 0.500000 0.866025i
\(65\) −9.11074 10.8578i −1.13005 1.34674i
\(66\) 0 0
\(67\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(68\) −9.48551 1.67255i −1.15029 0.202827i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.07461 + 3.50718i 0.720923 + 0.416225i 0.815092 0.579331i \(-0.196686\pi\)
−0.0941692 + 0.995556i \(0.530019\pi\)
\(72\) 0 0
\(73\) −7.26673 12.5863i −0.850506 1.47312i −0.880752 0.473577i \(-0.842962\pi\)
0.0302463 0.999542i \(-0.490371\pi\)
\(74\) 0 0
\(75\) −8.66020 0.0306782i −0.999994 0.00354242i
\(76\) 0 0
\(77\) 3.60308 + 9.89937i 0.410609 + 1.12814i
\(78\) 0 0
\(79\) 3.00497 + 17.0421i 0.338086 + 1.91738i 0.394340 + 0.918964i \(0.370973\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 8.94427i 1.00000i
\(81\) 6.81174 + 5.88220i 0.756860 + 0.653577i
\(82\) 0 0
\(83\) −13.8663 + 2.44500i −1.52202 + 0.268373i −0.871227 0.490881i \(-0.836675\pi\)
−0.650794 + 0.759254i \(0.725564\pi\)
\(84\) −1.62348 + 9.02022i −0.177136 + 0.984186i
\(85\) 10.1193 3.68312i 1.09759 0.399490i
\(86\) 0 0
\(87\) −0.0659721 + 18.6234i −0.00707295 + 1.99663i
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 8.38533 14.5238i 0.879021 1.52251i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5742 9.71188i 1.17518 0.986092i 0.175180 0.984536i \(-0.443949\pi\)
0.999999 0.00155612i \(-0.000495328\pi\)
\(98\) 0 0
\(99\) 4.00588 + 11.2535i 0.402606 + 1.13102i
\(100\) −5.00000 8.66025i −0.500000 0.866025i
\(101\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(102\) 0 0
\(103\) −12.3376 10.3524i −1.21566 1.02006i −0.999040 0.0438001i \(-0.986054\pi\)
−0.216616 0.976257i \(-0.569502\pi\)
\(104\) 0 0
\(105\) −3.47053 9.64134i −0.338689 0.940898i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.91326 + 10.2147i −0.184104 + 0.982907i
\(109\) −2.98186 −0.285610 −0.142805 0.989751i \(-0.545612\pi\)
−0.142805 + 0.989751i \(0.545612\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.94477 + 3.61960i −0.939693 + 0.342020i
\(113\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −18.6235 + 10.7523i −1.72915 + 0.998323i
\(117\) 9.62451 16.4007i 0.889786 1.51624i
\(118\) 0 0
\(119\) 8.19022 + 9.76072i 0.750796 + 0.894764i
\(120\) 0 0
\(121\) −0.842932 + 4.78050i −0.0766302 + 0.434591i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.68246 + 5.59017i 0.866025 + 0.500000i
\(126\) 0 0
\(127\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(132\) −8.90344 + 10.5347i −0.774945 + 0.916927i
\(133\) 0 0
\(134\) 0 0
\(135\) −3.85766 10.9599i −0.332015 0.943274i
\(136\) 0 0
\(137\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(138\) 0 0
\(139\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(140\) 7.60557 9.06396i 0.642788 0.766044i
\(141\) −20.4579 11.9082i −1.72286 1.00285i
\(142\) 0 0
\(143\) 21.8577 12.6196i 1.82783 1.05530i
\(144\) −11.3051 + 4.02425i −0.942092 + 0.335354i
\(145\) 12.0214 20.8217i 0.998323 1.72915i
\(146\) 0 0
\(147\) 9.31534 7.76044i 0.768317 0.640070i
\(148\) 0 0
\(149\) −21.6506 3.81759i −1.77369 0.312749i −0.811343 0.584571i \(-0.801263\pi\)
−0.962348 + 0.271821i \(0.912374\pi\)
\(150\) 0 0
\(151\) 15.6508 13.1326i 1.27364 1.06871i 0.279554 0.960130i \(-0.409814\pi\)
0.994088 0.108582i \(-0.0346309\pi\)
\(152\) 0 0
\(153\) 9.20819 + 11.1331i 0.744438 + 0.900061i
\(154\) 0 0
\(155\) 0 0
\(156\) 21.9578 + 0.0777842i 1.75803 + 0.00622772i
\(157\) −7.30588 6.13036i −0.583072 0.489256i 0.302882 0.953028i \(-0.402051\pi\)
−0.885954 + 0.463772i \(0.846496\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 2.73165 15.1773i 0.212658 1.18155i
\(166\) 0 0
\(167\) 2.27900 2.71600i 0.176354 0.210171i −0.670625 0.741796i \(-0.733974\pi\)
0.846979 + 0.531626i \(0.178419\pi\)
\(168\) 0 0
\(169\) −25.5402 9.29586i −1.96463 0.715066i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.66201 + 11.5147i 0.734589 + 0.875449i 0.995961 0.0897890i \(-0.0286193\pi\)
−0.261372 + 0.965238i \(0.584175\pi\)
\(174\) 0 0
\(175\) −2.29715 + 13.0278i −0.173648 + 0.984808i
\(176\) −15.6850 2.76569i −1.18230 0.208472i
\(177\) 0 0
\(178\) 0 0
\(179\) 5.51047 + 3.18147i 0.411872 + 0.237794i 0.691594 0.722287i \(-0.256909\pi\)
−0.279722 + 0.960081i \(0.590242\pi\)
\(180\) 8.69650 10.2162i 0.648199 0.761471i
\(181\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.32983 + 18.8844i 0.243501 + 1.38096i
\(188\) 27.3332i 1.99348i
\(189\) 10.6247 8.72445i 0.772832 0.634611i
\(190\) 0 0
\(191\) 5.82620 1.02732i 0.421569 0.0743340i 0.0411606 0.999153i \(-0.486894\pi\)
0.380409 + 0.924819i \(0.375783\pi\)
\(192\) −10.5830 8.94427i −0.763763 0.645497i
\(193\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(194\) 0 0
\(195\) −21.3040 + 12.1995i −1.52561 + 0.873623i
\(196\) 13.1557 + 4.78828i 0.939693 + 0.342020i
\(197\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0 0
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 28.0156 + 4.93991i 1.96631 + 0.346714i
\(204\) −5.76136 + 15.6564i −0.403376 + 1.09617i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 12.6774 + 21.9579i 0.879021 + 1.52251i
\(209\) 0 0
\(210\) 0 0
\(211\) −21.2066 17.7944i −1.45992 1.22502i −0.924918 0.380166i \(-0.875867\pi\)
−0.535001 0.844851i \(-0.679689\pi\)
\(212\) 0 0
\(213\) 7.84228 9.27911i 0.537344 0.635794i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −23.6849 + 8.52572i −1.60048 + 0.576114i
\(220\) 16.7330 6.09031i 1.12814 0.410609i
\(221\) 19.6222 23.3848i 1.31993 1.57303i
\(222\) 0 0
\(223\) 27.8548 + 10.1383i 1.86530 + 0.678912i 0.974449 + 0.224607i \(0.0721099\pi\)
0.890846 + 0.454305i \(0.150112\pi\)
\(224\) 0 0
\(225\) −2.70931 + 14.7533i −0.180621 + 0.983553i
\(226\) 0 0
\(227\) −11.0341 13.1500i −0.732362 0.872795i 0.263407 0.964685i \(-0.415154\pi\)
−0.995769 + 0.0918895i \(0.970709\pi\)
\(228\) 0 0
\(229\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(230\) 0 0
\(231\) 17.9805 3.10482i 1.18303 0.204282i
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) 15.2797 + 26.4653i 0.996740 + 1.72640i
\(236\) 0 0
\(237\) 29.9729 + 0.106177i 1.94695 + 0.00689694i
\(238\) 0 0
\(239\) 2.39276 + 6.57406i 0.154775 + 0.425241i 0.992710 0.120530i \(-0.0384593\pi\)
−0.837935 + 0.545770i \(0.816237\pi\)
\(240\) 15.2470 + 2.74417i 0.984186 + 0.177136i
\(241\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(242\) 0 0
\(243\) 12.1170 9.80700i 0.777309 0.629119i
\(244\) 0 0
\(245\) −15.4147 + 2.71802i −0.984808 + 0.173648i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.0863910 + 24.3874i −0.00547481 + 1.54549i
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 14.8783 + 5.53495i 0.937246 + 0.348669i
\(253\) 0 0
\(254\) 0 0
\(255\) −3.17379 18.3800i −0.198750 1.15100i
\(256\) 2.77837 15.7569i 0.173648 0.984808i
\(257\) 23.4716 + 4.13868i 1.46412 + 0.258164i 0.848213 0.529655i \(-0.177678\pi\)
0.615907 + 0.787819i \(0.288790\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −24.5497 14.1738i −1.52251 0.879021i
\(261\) 31.7263 + 5.82626i 1.96381 + 0.360636i
\(262\) 0 0
\(263\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −18.9710 + 3.34510i −1.15029 + 0.202827i
\(273\) −22.1855 18.7502i −1.34273 1.13481i
\(274\) 0 0
\(275\) −12.7971 + 15.2509i −0.771692 + 0.919667i
\(276\) 0 0
\(277\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.68148 4.38742i −0.219619 0.261731i 0.644974 0.764204i \(-0.276868\pi\)
−0.864593 + 0.502473i \(0.832424\pi\)
\(282\) 0 0
\(283\) 2.42376 13.7458i 0.144078 0.817106i −0.824025 0.566553i \(-0.808277\pi\)
0.968103 0.250553i \(-0.0806123\pi\)
\(284\) 13.8156 + 2.43606i 0.819804 + 0.144553i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.09653 + 5.36335i 0.182149 + 0.315491i
\(290\) 0 0
\(291\) −13.0044 22.7097i −0.762333 1.33127i
\(292\) −22.2665 18.6838i −1.30305 1.09339i
\(293\) −2.84671 7.82127i −0.166306 0.456923i 0.828344 0.560220i \(-0.189283\pi\)
−0.994651 + 0.103296i \(0.967061\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 20.4124 3.37600i 1.18445 0.195895i
\(298\) 0 0
\(299\) 0 0
\(300\) −16.2968 + 5.86627i −0.940898 + 0.338689i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.316851 + 0.548803i −0.0180837 + 0.0313218i −0.874926 0.484257i \(-0.839090\pi\)
0.856842 + 0.515579i \(0.172423\pi\)
\(308\) 13.5431 + 16.1401i 0.771692 + 0.919667i
\(309\) −21.4327 + 17.8552i −1.21926 + 1.01574i
\(310\) 0 0
\(311\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(312\) 0 0
\(313\) −24.4981 + 20.5564i −1.38472 + 1.16192i −0.417288 + 0.908774i \(0.637019\pi\)
−0.967429 + 0.253141i \(0.918536\pi\)
\(314\) 0 0
\(315\) −17.5000 + 2.95804i −0.986013 + 0.166667i
\(316\) 17.3050 + 29.9731i 0.973480 + 1.68612i
\(317\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(318\) 0 0
\(319\) 32.7965 + 27.5195i 1.83625 + 1.54080i
\(320\) 6.11824 + 16.8097i 0.342020 + 0.939693i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 16.8255 + 6.39541i 0.934752 + 0.355301i
\(325\) 31.6936 1.75804
\(326\) 0 0
\(327\) −0.914858 + 5.08306i −0.0505918 + 0.281094i
\(328\) 0 0
\(329\) −23.2422 + 27.6990i −1.28138 + 1.52709i
\(330\) 0 0
\(331\) −4.34417 1.58115i −0.238777 0.0869078i 0.219860 0.975531i \(-0.429440\pi\)
−0.458637 + 0.888624i \(0.651662\pi\)
\(332\) −24.3876 + 14.0802i −1.33844 + 0.772750i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 3.11906 + 18.0630i 0.170158 + 0.985417i
\(337\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 16.4986 13.8440i 0.894764 0.750796i
\(341\) 0 0
\(342\) 0 0
\(343\) −9.26013 16.0390i −0.500000 0.866025i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(348\) 12.6151 + 35.0456i 0.676243 + 1.87864i
\(349\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(350\) 0 0
\(351\) −25.0047 21.4384i −1.33465 1.14430i
\(352\) 0 0
\(353\) −34.6219 + 6.10478i −1.84274 + 0.324925i −0.982686 0.185279i \(-0.940681\pi\)
−0.860054 + 0.510204i \(0.829570\pi\)
\(354\) 0 0
\(355\) −14.7387 + 5.36444i −0.782247 + 0.284715i
\(356\) 0 0
\(357\) 19.1515 10.9669i 1.01361 0.580429i
\(358\) 0 0
\(359\) 21.8765 12.6304i 1.15460 0.666608i 0.204595 0.978847i \(-0.434412\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −9.50000 + 16.4545i −0.500000 + 0.866025i
\(362\) 0 0
\(363\) 7.89052 + 2.90361i 0.414145 + 0.152400i
\(364\) 5.82439 33.0318i 0.305281 1.73133i
\(365\) 32.0041 + 5.64318i 1.67517 + 0.295378i
\(366\) 0 0
\(367\) −27.7672 + 23.2994i −1.44943 + 1.21622i −0.516443 + 0.856322i \(0.672744\pi\)
−0.932992 + 0.359898i \(0.882811\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(374\) 0 0
\(375\) 12.5000 14.7902i 0.645497 0.763763i
\(376\) 0 0
\(377\) 68.1555i 3.51019i
\(378\) 0 0
\(379\) −37.7356 −1.93835 −0.969173 0.246380i \(-0.920759\pi\)
−0.969173 + 0.246380i \(0.920759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.1202 + 21.5948i −0.925900 + 1.10344i 0.0684880 + 0.997652i \(0.478183\pi\)
−0.994388 + 0.105793i \(0.966262\pi\)
\(384\) 0 0
\(385\) −22.1357 8.05673i −1.12814 0.410609i
\(386\) 0 0
\(387\) 0 0
\(388\) 15.1090 26.1696i 0.767044 1.32856i
\(389\) 13.2008 + 15.7321i 0.669305 + 0.797647i 0.988689 0.149979i \(-0.0479205\pi\)
−0.319384 + 0.947625i \(0.603476\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −33.5109 19.3475i −1.68612 0.973480i
\(396\) 15.2264 + 18.4095i 0.765157 + 0.925111i
\(397\) 0.115521 + 0.200088i 0.00579782 + 0.0100421i 0.868910 0.494971i \(-0.164821\pi\)
−0.863112 + 0.505013i \(0.831488\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −15.3209 12.8558i −0.766044 0.642788i
\(401\) −10.1171 27.7965i −0.505223 1.38809i −0.886113 0.463469i \(-0.846604\pi\)
0.380889 0.924621i \(-0.375618\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −19.8664 + 3.21343i −0.987169 + 0.159677i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −30.2685 11.0168i −1.49122 0.542761i
\(413\) 0 0
\(414\) 0 0
\(415\) 15.7421 27.2662i 0.772750 1.33844i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(420\) −13.1175 15.7458i −0.640070 0.768317i
\(421\) 8.46757 7.10513i 0.412684 0.346283i −0.412688 0.910872i \(-0.635410\pi\)
0.825372 + 0.564590i \(0.190966\pi\)
\(422\) 0 0
\(423\) −26.5760 + 31.2202i −1.29217 + 1.51798i
\(424\) 0 0
\(425\) −8.23570 + 22.6274i −0.399490 + 1.09759i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −14.8059 41.1318i −0.714838 1.98586i
\(430\) 0 0
\(431\) 29.7955i 1.43520i −0.696456 0.717599i \(-0.745241\pi\)
0.696456 0.717599i \(-0.254759\pi\)
\(432\) 3.39148 + 20.5060i 0.163173 + 0.986598i
\(433\) 40.1484 1.92941 0.964703 0.263339i \(-0.0848238\pi\)
0.964703 + 0.263339i \(0.0848238\pi\)
\(434\) 0 0
\(435\) −31.8056 26.8807i −1.52496 1.28883i
\(436\) −5.60406 + 2.03971i −0.268386 + 0.0976845i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(440\) 0 0
\(441\) −10.3709 18.2605i −0.493852 0.869546i
\(442\) 0 0
\(443\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.1503 + 35.7357i −0.621988 + 1.69024i
\(448\) −16.2141 + 13.6052i −0.766044 + 0.642788i
\(449\) −24.6822 14.2503i −1.16482 0.672511i −0.212368 0.977190i \(-0.568118\pi\)
−0.952455 + 0.304679i \(0.901451\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.5848 30.7084i −0.826204 1.44281i
\(454\) 0 0
\(455\) 12.8259 + 35.2388i 0.601286 + 1.65202i
\(456\) 0 0
\(457\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(458\) 0 0
\(459\) 21.8034 12.2811i 1.01769 0.573233i
\(460\) 0 0
\(461\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(462\) 0 0
\(463\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(464\) −27.6457 + 32.9469i −1.28342 + 1.52952i
\(465\) 0 0
\(466\) 0 0
\(467\) 36.9254 21.3189i 1.70871 0.986521i 0.772536 0.634970i \(-0.218988\pi\)
0.936169 0.351551i \(-0.114346\pi\)
\(468\) 6.86943 37.4068i 0.317539 1.72913i
\(469\) 0 0
\(470\) 0 0
\(471\) −12.6917 + 10.5732i −0.584802 + 0.487187i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 22.0693 + 12.7417i 1.01154 + 0.584016i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.68586 + 9.56101i 0.0766302 + 0.434591i
\(485\) 33.7848i 1.53409i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.930698 + 1.10916i −0.0420018 + 0.0500558i −0.786636 0.617417i \(-0.788179\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 48.6592 + 17.7105i 2.19150 + 0.797641i
\(494\) 0 0
\(495\) −25.0341 9.31306i −1.12520 0.418591i
\(496\) 0 0
\(497\) −11.9290 14.2164i −0.535089 0.637694i
\(498\) 0 0
\(499\) −7.74430 + 43.9201i −0.346683 + 1.96613i −0.114816 + 0.993387i \(0.536628\pi\)
−0.231867 + 0.972748i \(0.574483\pi\)
\(500\) 22.0210 + 3.88289i 0.984808 + 0.173648i
\(501\) −3.93065 4.71821i −0.175608 0.210794i
\(502\) 0 0
\(503\) −1.39864 0.807506i −0.0623624 0.0360049i 0.468495 0.883466i \(-0.344797\pi\)
−0.530857 + 0.847461i \(0.678130\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −23.6822 + 40.6853i −1.05176 + 1.80690i
\(508\) 0 0
\(509\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(510\) 0 0
\(511\) 6.67710 + 37.8677i 0.295378 + 1.67517i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 35.4660 6.25361i 1.56282 0.275567i
\(516\) 0 0
\(517\) −51.1351 + 18.6117i −2.24892 + 0.818540i
\(518\) 0 0
\(519\) 22.5931 12.9376i 0.991727 0.567899i
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −5.56344 + 9.63616i −0.243272 + 0.421360i −0.961644 0.274299i \(-0.911554\pi\)
0.718372 + 0.695659i \(0.244888\pi\)
\(524\) 0 0
\(525\) 21.5032 + 7.91289i 0.938475 + 0.345347i
\(526\) 0 0
\(527\) 0 0
\(528\) −9.52684 + 25.8891i −0.414603 + 1.12668i
\(529\) −17.6190 + 14.7841i −0.766044 + 0.642788i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.11399 8.41738i 0.306991 0.363237i
\(538\) 0 0
\(539\) 27.8722i 1.20054i
\(540\) −14.7470 17.9590i −0.634611 0.772832i
\(541\) −44.3483 −1.90668 −0.953340 0.301898i \(-0.902380\pi\)
−0.953340 + 0.301898i \(0.902380\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.28588 5.10771i 0.183587 0.218790i
\(546\) 0 0
\(547\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 7.95041 45.0890i 0.338086 1.91738i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 8.09367 22.2372i 0.342020 0.939693i
\(561\) 33.2131 + 0.117655i 1.40226 + 0.00496742i
\(562\) 0 0
\(563\) 16.2301 + 44.5919i 0.684019 + 1.87933i 0.349503 + 0.936935i \(0.386350\pi\)
0.334515 + 0.942390i \(0.391427\pi\)
\(564\) −46.5939 8.38605i −1.96196 0.353116i
\(565\) 0 0
\(566\) 0 0
\(567\) −11.6125 20.7882i −0.487679 0.873023i
\(568\) 0 0
\(569\) 45.8197 8.07924i 1.92086 0.338699i 0.922032 0.387113i \(-0.126528\pi\)
0.998827 + 0.0484135i \(0.0154165\pi\)
\(570\) 0 0
\(571\) −44.4758 + 16.1879i −1.86125 + 0.677441i −0.883231 + 0.468938i \(0.844637\pi\)
−0.978022 + 0.208502i \(0.933141\pi\)
\(572\) 32.4468 38.6686i 1.35667 1.61681i
\(573\) 0.0362990 10.2469i 0.00151641 0.428070i
\(574\) 0 0
\(575\) 0 0
\(576\) −18.4939 + 15.2963i −0.770579 + 0.637344i
\(577\) 2.39408 4.14668i 0.0996670 0.172628i −0.811880 0.583825i \(-0.801556\pi\)
0.911547 + 0.411196i \(0.134889\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 8.34998 47.3551i 0.346714 1.96631i
\(581\) 36.6867 + 6.46886i 1.52202 + 0.268373i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 14.2597 + 40.0591i 0.589567 + 1.65624i
\(586\) 0 0
\(587\) −12.5764 + 34.5535i −0.519085 + 1.42617i 0.352445 + 0.935833i \(0.385350\pi\)
−0.871530 + 0.490342i \(0.836872\pi\)
\(588\) 12.1987 20.9569i 0.503065 0.864249i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.8642i 1.71916i 0.511003 + 0.859579i \(0.329274\pi\)
−0.511003 + 0.859579i \(0.670726\pi\)
\(594\) 0 0
\(595\) −28.4914 −1.16803
\(596\) −43.3013 + 7.63519i −1.77369 + 0.312749i
\(597\) 0 0
\(598\) 0 0
\(599\) 21.3861 25.4870i 0.873814 1.04137i −0.124975 0.992160i \(-0.539885\pi\)
0.998788 0.0492108i \(-0.0156706\pi\)
\(600\) 0 0
\(601\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 20.4306 35.3869i 0.831310 1.43987i
\(605\) −6.97710 8.31498i −0.283659 0.338052i
\(606\) 0 0
\(607\) −6.93157 + 39.3109i −0.281344 + 1.59558i 0.436717 + 0.899599i \(0.356141\pi\)
−0.718061 + 0.695980i \(0.754970\pi\)
\(608\) 0 0
\(609\) 17.0163 46.2416i 0.689535 1.87380i
\(610\) 0 0
\(611\) 75.0227 + 43.3143i 3.03509 + 1.75231i
\(612\) 24.9213 + 14.6247i 1.00738 + 0.591168i
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(618\) 0 0
\(619\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 41.3204 14.8738i 1.65414 0.595430i
\(625\) −23.4923 + 8.55050i −0.939693 + 0.342020i
\(626\) 0 0
\(627\) 0 0
\(628\) −17.9240 6.52379i −0.715244 0.260328i
\(629\) 0 0
\(630\) 0 0
\(631\) −12.2979 + 21.3005i −0.489570 + 0.847961i −0.999928 0.0120014i \(-0.996180\pi\)
0.510357 + 0.859962i \(0.329513\pi\)
\(632\) 0 0
\(633\) −36.8398 + 30.6905i −1.46425 + 1.21984i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −33.9902 + 28.5211i −1.34674 + 1.13005i
\(638\) 0 0
\(639\) −13.4117 16.2153i −0.530557 0.641469i
\(640\) 0 0
\(641\) −13.4252 + 36.8855i −0.530265 + 1.45689i 0.328491 + 0.944507i \(0.393460\pi\)
−0.858756 + 0.512384i \(0.828762\pi\)
\(642\) 0 0
\(643\) −32.8024 27.5245i −1.29360 1.08546i −0.991213 0.132273i \(-0.957772\pi\)
−0.302386 0.953186i \(-0.597783\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 50.1875i 1.97307i −0.163543 0.986536i \(-0.552292\pi\)
0.163543 0.986536i \(-0.447708\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.26673 + 42.9905i 0.283502 + 1.67722i
\(658\) 0 0
\(659\) 31.6238 + 37.6878i 1.23189 + 1.46811i 0.835009 + 0.550236i \(0.185462\pi\)
0.396878 + 0.917871i \(0.370093\pi\)
\(660\) −5.24810 30.3926i −0.204282 1.18303i
\(661\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(662\) 0 0
\(663\) −33.8429 40.6238i −1.31435 1.57770i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 2.42526 6.66334i 0.0938360 0.257812i
\(669\) 25.8285 44.3725i 0.998587 1.71554i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(674\) 0 0
\(675\) 24.3181 + 9.14489i 0.936005 + 0.351987i
\(676\) −54.3586 −2.09071
\(677\) −43.8233 + 7.72722i −1.68426 + 0.296981i −0.932156 0.362058i \(-0.882074\pi\)
−0.752109 + 0.659039i \(0.770963\pi\)
\(678\) 0 0
\(679\) −37.5639 + 13.6721i −1.44157 + 0.524689i
\(680\) 0 0
\(681\) −25.8016 + 14.7750i −0.988721 + 0.566178i
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(692\) 26.0352 + 15.0314i 0.989709 + 0.571409i
\(693\) 0.223908 31.6033i 0.00850557 1.20051i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 4.59430 + 26.0556i 0.173648 + 0.984808i
\(701\) 25.3074i 0.955849i 0.878401 + 0.477924i \(0.158611\pi\)
−0.878401 + 0.477924i \(0.841389\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −31.3700 + 5.53137i −1.18230 + 0.208472i
\(705\) 49.8023 17.9270i 1.87566 0.675170i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 42.8606 + 15.6000i 1.60966 + 0.585869i 0.981373 0.192110i \(-0.0615331\pi\)
0.628290 + 0.777980i \(0.283755\pi\)
\(710\) 0 0
\(711\) 9.37691 51.0610i 0.351662 1.91494i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −9.80007 + 55.5790i −0.366502 + 2.07854i
\(716\) 12.5326 + 2.20983i 0.468363 + 0.0825851i
\(717\) 11.9407 2.06187i 0.445932 0.0770021i
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 9.35577 25.1489i 0.348669 0.937246i
\(721\) 21.3056 + 36.9024i 0.793463 + 1.37432i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.3875 + 50.5191i 0.682893 + 1.87623i
\(726\) 0 0
\(727\) 2.32382 + 13.1790i 0.0861857 + 0.488783i 0.997094 + 0.0761754i \(0.0242709\pi\)
−0.910909 + 0.412608i \(0.864618\pi\)
\(728\) 0 0
\(729\) −13.0000 23.6643i −0.481481 0.876456i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.24147 + 2.63568i −0.267470 + 0.0973511i −0.472274 0.881452i \(-0.656567\pi\)
0.204804 + 0.978803i \(0.434344\pi\)
\(734\) 0 0
\(735\) −0.0960380 + 27.1107i −0.00354242 + 0.999994i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.8056 44.6967i 0.949276 1.64419i 0.202321 0.979319i \(-0.435152\pi\)
0.746955 0.664875i \(-0.231515\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(744\) 0 0
\(745\) 37.6581 31.5989i 1.37969 1.15769i
\(746\) 0 0
\(747\) 41.5458 + 7.62953i 1.52008 + 0.279150i
\(748\) 19.1757 + 33.2133i 0.701134 + 1.21440i
\(749\) 0 0
\(750\) 0 0
\(751\) −30.3440 25.4616i −1.10727 0.929108i −0.109376 0.994000i \(-0.534885\pi\)
−0.997892 + 0.0648920i \(0.979330\pi\)
\(752\) −18.6970 51.3696i −0.681810 1.87326i
\(753\) 0 0
\(754\) 0 0
\(755\) 45.6843i 1.66262i
\(756\) 14.0000 23.6643i 0.509175 0.860663i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(762\) 0 0
\(763\) 7.41348 + 2.69829i 0.268386 + 0.0976845i
\(764\) 10.2470 5.91608i 0.370722 0.214036i
\(765\) −32.3053 0.228882i −1.16800 0.00827525i
\(766\) 0 0
\(767\) 0 0
\(768\) −26.0078 9.57053i −0.938475 0.345347i
\(769\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(770\) 0 0
\(771\) 14.2563 38.7414i 0.513430 1.39524i
\(772\) 0 0
\(773\) −29.4456 17.0004i −1.05908 0.611463i −0.133906 0.990994i \(-0.542752\pi\)
−0.925179 + 0.379531i \(0.876085\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −31.6936 + 37.5003i −1.13481 + 1.34273i
\(781\) −4.84988 27.5050i −0.173542 0.984207i
\(782\) 0 0
\(783\) 19.6657 52.2950i 0.702794 1.86887i
\(784\) 28.0000 1.00000
\(785\) 21.0017 3.70317i 0.749583 0.132172i
\(786\) 0 0
\(787\) 28.9365 10.5320i 1.03147 0.375426i 0.229832 0.973230i \(-0.426182\pi\)
0.801642 + 0.597804i \(0.203960\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −17.2577 3.04299i −0.611298 0.107788i −0.140576 0.990070i \(-0.544895\pi\)
−0.470722 + 0.882282i \(0.656007\pi\)
\(798\) 0 0
\(799\) −50.4189 + 42.3065i −1.78369 + 1.49670i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.7922 + 54.3785i −0.698450 + 1.91898i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 52.2539i 1.83715i 0.395249 + 0.918574i \(0.370658\pi\)
−0.395249 + 0.918574i \(0.629342\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 56.0313 9.87983i 1.96631 0.346714i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.118195 + 33.3654i −0.00413765 + 1.16802i
\(817\) 0 0
\(818\) 0 0
\(819\) −38.7694 + 32.0660i −1.35471 + 1.12048i
\(820\) 0 0
\(821\) −36.7426 43.7881i −1.28233 1.52822i −0.695592 0.718437i \(-0.744858\pi\)
−0.586734 0.809780i \(-0.699586\pi\)
\(822\) 0 0
\(823\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(824\) 0 0
\(825\) 22.0715 + 26.4938i 0.768429 + 0.922395i
\(826\) 0 0
\(827\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(828\) 0 0
\(829\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 38.8459 + 32.5956i 1.34674 + 1.13005i
\(833\) −11.5300 31.6784i −0.399490 1.09759i
\(834\) 0 0
\(835\) 1.37667 + 7.80751i 0.0476418 + 0.270190i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(840\) 0 0
\(841\) 81.3880 29.6228i 2.80648 1.02148i
\(842\) 0 0
\(843\) −8.60857 + 4.92958i −0.296495 + 0.169784i
\(844\) −52.0274 18.9364i −1.79086 0.651818i
\(845\) 52.6325 30.3874i 1.81061 1.04536i
\(846\) 0 0
\(847\) 6.42157 11.1225i 0.220648 0.382173i
\(848\) 0 0
\(849\) −22.6884 8.34903i −0.778663 0.286538i
\(850\) 0 0
\(851\) 0 0
\(852\) 8.39138 22.8035i 0.287484 0.781234i
\(853\) 42.9160 36.0108i 1.46942 1.23299i 0.552741 0.833353i \(-0.313582\pi\)
0.916675 0.399633i \(-0.130862\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.5244 53.6429i 0.666941 1.83241i 0.124678 0.992197i \(-0.460210\pi\)
0.542263 0.840209i \(-0.317568\pi\)
\(858\) 0 0
\(859\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −33.6113 −1.14282
\(866\) 0 0
\(867\) 10.0927 3.63302i 0.342767 0.123384i
\(868\) 0 0
\(869\) 44.2905 52.7834i 1.50245 1.79055i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −42.7022 + 15.2006i −1.44525 + 0.514462i
\(874\) 0 0
\(875\) −19.0139 22.6599i −0.642788 0.766044i
\(876\) −38.6812 + 32.2246i −1.30692 + 1.08877i
\(877\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(878\) 0 0
\(879\) −14.2060 + 2.45304i −0.479156 + 0.0827391i
\(880\) 27.2817 22.8921i 0.919667 0.771692i
\(881\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(882\) 0 0
\(883\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(884\) 20.8815 57.3715i 0.702321 1.92961i
\(885\) 0 0
\(886\) 0 0
\(887\) 7.98776 + 21.9462i 0.268203 + 0.736881i 0.998551 + 0.0538062i \(0.0171353\pi\)
−0.730349 + 0.683074i \(0.760642\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.507763 35.8321i 0.0170107 1.20042i
\(892\) 59.2849 1.98501
\(893\) 0 0
\(894\) 0 0
\(895\) −13.3699 + 4.86625i −0.446907 + 0.162661i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.00000 + 29.5804i 0.166667 + 0.986013i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(908\) −29.7325 17.1661i −0.986709 0.569677i
\(909\) 0 0
\(910\) 0 0
\(911\) −12.8621 + 35.3384i −0.426141 + 1.17081i 0.521995 + 0.852949i \(0.325188\pi\)
−0.948136 + 0.317865i \(0.897034\pi\)
\(912\) 0 0
\(913\) 42.9472 + 36.0370i 1.42135 + 1.19265i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −60.6113 −1.99938 −0.999691 0.0248659i \(-0.992084\pi\)
−0.999691 + 0.0248659i \(0.992084\pi\)
\(920\) 0 0
\(921\) 0.838310 + 0.708501i 0.0276232 + 0.0233459i
\(922\) 0 0
\(923\) −28.5796 + 34.0598i −0.940709 + 1.12109i
\(924\) 31.6685 18.1346i 1.04182 0.596583i
\(925\) 0 0
\(926\) 0 0
\(927\) 23.8613 + 42.0136i 0.783707 + 1.37991i
\(928\) 0 0
\(929\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −37.1336 21.4391i −1.21440 0.701134i
\(936\) 0 0
\(937\) −8.58659 14.8724i −0.280512 0.485860i 0.690999 0.722856i \(-0.257171\pi\)
−0.971511 + 0.236995i \(0.923837\pi\)
\(938\) 0 0
\(939\) 27.5254 + 48.0679i 0.898259 + 1.56864i
\(940\) 46.8198 + 39.2865i 1.52709 + 1.28138i
\(941\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −0.326685 + 30.7391i −0.0106271 + 0.999944i
\(946\) 0 0
\(947\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(948\) 56.4032 20.3031i 1.83189 0.659414i
\(949\) 86.5677 31.5081i 2.81011 1.02279i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) −6.61438 + 11.4564i −0.214036 + 0.370722i
\(956\) 8.99385 + 10.7184i 0.290882 + 0.346659i
\(957\) 56.9736 47.4636i 1.84170 1.53428i
\(958\) 0 0
\(959\) 0 0
\(960\) 30.5320 5.27217i 0.985417 0.170158i
\(961\) 23.7474 19.9264i 0.766044 0.642788i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 16.0642 26.7197i 0.515260 0.857034i
\(973\) 0 0
\(974\) 0 0
\(975\) 9.72384 54.0268i 0.311412 1.73024i
\(976\) 0 0
\(977\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −27.1109 + 15.6525i −0.866025 + 0.500000i
\(981\) 8.38421 + 3.11905i 0.267687 + 0.0995835i
\(982\) 0 0
\(983\) −38.4158 45.7822i −1.22528 1.46023i −0.844498 0.535559i \(-0.820101\pi\)
−0.380778 0.924667i \(-0.624344\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 40.0865 + 48.1183i 1.27597 + 1.53162i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 21.9935 + 38.0938i 0.698645 + 1.21009i 0.968936 + 0.247310i \(0.0795467\pi\)
−0.270291 + 0.962779i \(0.587120\pi\)
\(992\) 0 0
\(993\) −4.02815 + 6.92023i −0.127829 + 0.219607i
\(994\) 0 0
\(995\) 0 0
\(996\) 16.5196 + 45.8925i 0.523444 + 1.45416i
\(997\) −3.21601 18.2389i −0.101852 0.577632i −0.992431 0.122803i \(-0.960812\pi\)
0.890579 0.454829i \(-0.150299\pi\)
\(998\) 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 945.2.cs.a.419.3 yes 24
5.4 even 2 inner 945.2.cs.a.419.2 24
7.6 odd 2 inner 945.2.cs.a.419.2 24
27.2 odd 18 inner 945.2.cs.a.839.3 yes 24
35.34 odd 2 CM 945.2.cs.a.419.3 yes 24
135.29 odd 18 inner 945.2.cs.a.839.2 yes 24
189.83 even 18 inner 945.2.cs.a.839.2 yes 24
945.839 even 18 inner 945.2.cs.a.839.3 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
945.2.cs.a.419.2 24 5.4 even 2 inner
945.2.cs.a.419.2 24 7.6 odd 2 inner
945.2.cs.a.419.3 yes 24 1.1 even 1 trivial
945.2.cs.a.419.3 yes 24 35.34 odd 2 CM
945.2.cs.a.839.2 yes 24 135.29 odd 18 inner
945.2.cs.a.839.2 yes 24 189.83 even 18 inner
945.2.cs.a.839.3 yes 24 27.2 odd 18 inner
945.2.cs.a.839.3 yes 24 945.839 even 18 inner