Properties

Label 315.2.z.a.209.4
Level $315$
Weight $2$
Character 315.209
Analytic conductor $2.515$
Analytic rank $0$
Dimension $8$
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] = [315,2,Mod(104,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.z (of order \(6\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.31116960000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 209.4
Root \(0.306808 - 1.70466i\) of defining polynomial
Character \(\chi\) \(=\) 315.209
Dual form 315.2.z.a.104.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.32288 + 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(1.93649 - 1.11803i) q^{5} +(-1.32288 + 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +O(q^{10})\) \(q+(1.32288 + 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(1.93649 - 1.11803i) q^{5} +(-1.32288 + 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +(-0.311738 - 0.179982i) q^{11} +(-0.613616 + 3.40932i) q^{12} +(-3.56618 - 6.17680i) q^{13} +(3.81174 + 0.686044i) q^{15} +(-2.00000 + 3.46410i) q^{16} -5.75583i q^{17} +(3.87298 + 2.23607i) q^{20} +(-4.31174 + 1.55207i) q^{21} +(2.50000 - 4.33013i) q^{25} +(-2.64575 + 4.47214i) q^{27} -5.29150 q^{28} +(5.12348 + 2.95804i) q^{29} +(-0.211164 - 0.586627i) q^{33} +5.91608i q^{35} +(-4.62348 + 3.82407i) q^{36} +(2.18826 - 12.1582i) q^{39} -0.719927i q^{44} +(4.27543 + 5.16920i) q^{45} +(10.7942 + 6.23202i) q^{47} +(-6.51873 + 2.34651i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(6.43521 - 7.61425i) q^{51} +(7.13235 - 12.3536i) q^{52} -0.804903 q^{55} +(2.62348 + 7.28817i) q^{60} +(-7.43916 - 2.76748i) q^{63} -8.00000 q^{64} +(-13.8117 - 7.97421i) q^{65} +(9.96939 - 5.75583i) q^{68} -16.3084i q^{71} -6.32745 q^{73} +(8.14842 - 2.93313i) q^{75} +(0.824780 - 0.476187i) q^{77} +(-7.93521 + 13.7442i) q^{79} +8.94427i q^{80} +(-8.50000 + 2.95804i) q^{81} +(-15.7789 - 9.10993i) q^{83} +(-7.00000 - 5.91608i) q^{84} +(-6.43521 - 11.1461i) q^{85} +(3.47053 + 9.64134i) q^{87} +18.8704 q^{91} +(-7.53480 + 13.0507i) q^{97} +(0.376525 - 1.01212i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 4 q^{9} + 18 q^{11} + 10 q^{15} - 16 q^{16} - 14 q^{21} + 20 q^{25} + 4 q^{36} + 38 q^{39} - 28 q^{49} - 10 q^{51} - 20 q^{60} - 64 q^{64} - 90 q^{65} - 2 q^{79} - 68 q^{81} - 56 q^{84} + 10 q^{85} + 28 q^{91} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 1.32288 + 1.11803i 0.763763 + 0.645497i
\(4\) 1.00000 + 1.73205i 0.500000 + 0.866025i
\(5\) 1.93649 1.11803i 0.866025 0.500000i
\(6\) 0 0
\(7\) −1.32288 + 2.29129i −0.500000 + 0.866025i
\(8\) 0 0
\(9\) 0.500000 + 2.95804i 0.166667 + 0.986013i
\(10\) 0 0
\(11\) −0.311738 0.179982i −0.0939925 0.0542666i 0.452267 0.891883i \(-0.350615\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) −0.613616 + 3.40932i −0.177136 + 0.984186i
\(13\) −3.56618 6.17680i −0.989079 1.71314i −0.622179 0.782875i \(-0.713753\pi\)
−0.366900 0.930261i \(-0.619581\pi\)
\(14\) 0 0
\(15\) 3.81174 + 0.686044i 0.984186 + 0.177136i
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 5.75583i 1.39599i −0.716101 0.697997i \(-0.754075\pi\)
0.716101 0.697997i \(-0.245925\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 3.87298 + 2.23607i 0.866025 + 0.500000i
\(21\) −4.31174 + 1.55207i −0.940898 + 0.338689i
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) −2.64575 + 4.47214i −0.509175 + 0.860663i
\(28\) −5.29150 −1.00000
\(29\) 5.12348 + 2.95804i 0.951405 + 0.549294i 0.893517 0.449029i \(-0.148230\pi\)
0.0578882 + 0.998323i \(0.481563\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) −0.211164 0.586627i −0.0367590 0.102119i
\(34\) 0 0
\(35\) 5.91608i 1.00000i
\(36\) −4.62348 + 3.82407i −0.770579 + 0.637344i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.18826 12.1582i 0.350402 1.94688i
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0.719927i 0.108533i
\(45\) 4.27543 + 5.16920i 0.637344 + 0.770579i
\(46\) 0 0
\(47\) 10.7942 + 6.23202i 1.57449 + 0.909033i 0.995608 + 0.0936230i \(0.0298448\pi\)
0.578884 + 0.815410i \(0.303489\pi\)
\(48\) −6.51873 + 2.34651i −0.940898 + 0.338689i
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) 0 0
\(51\) 6.43521 7.61425i 0.901110 1.06621i
\(52\) 7.13235 12.3536i 0.989079 1.71314i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −0.804903 −0.108533
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 2.62348 + 7.28817i 0.338689 + 0.940898i
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) −7.43916 2.76748i −0.937246 0.348669i
\(64\) −8.00000 −1.00000
\(65\) −13.8117 7.97421i −1.71314 0.989079i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 9.96939 5.75583i 1.20897 0.697997i
\(69\) 0 0
\(70\) 0 0
\(71\) 16.3084i 1.93545i −0.252010 0.967725i \(-0.581092\pi\)
0.252010 0.967725i \(-0.418908\pi\)
\(72\) 0 0
\(73\) −6.32745 −0.740572 −0.370286 0.928918i \(-0.620740\pi\)
−0.370286 + 0.928918i \(0.620740\pi\)
\(74\) 0 0
\(75\) 8.14842 2.93313i 0.940898 0.338689i
\(76\) 0 0
\(77\) 0.824780 0.476187i 0.0939925 0.0542666i
\(78\) 0 0
\(79\) −7.93521 + 13.7442i −0.892781 + 1.54634i −0.0562544 + 0.998416i \(0.517916\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 8.94427i 1.00000i
\(81\) −8.50000 + 2.95804i −0.944444 + 0.328671i
\(82\) 0 0
\(83\) −15.7789 9.10993i −1.73196 0.999945i −0.871227 0.490881i \(-0.836675\pi\)
−0.860729 0.509064i \(-0.829992\pi\)
\(84\) −7.00000 5.91608i −0.763763 0.645497i
\(85\) −6.43521 11.1461i −0.697997 1.20897i
\(86\) 0 0
\(87\) 3.47053 + 9.64134i 0.372080 + 1.03366i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 18.8704 1.97816
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.53480 + 13.0507i −0.765043 + 1.32509i 0.175180 + 0.984536i \(0.443949\pi\)
−0.940224 + 0.340557i \(0.889384\pi\)
\(98\) 0 0
\(99\) 0.376525 1.01212i 0.0378421 0.101722i
\(100\) 10.0000 1.00000
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 1.32288 + 2.29129i 0.130347 + 0.225767i 0.923810 0.382851i \(-0.125058\pi\)
−0.793463 + 0.608618i \(0.791724\pi\)
\(104\) 0 0
\(105\) −6.61438 + 7.82624i −0.645497 + 0.763763i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −10.3917 0.110440i −0.999944 0.0106271i
\(109\) 9.87043 0.945415 0.472708 0.881219i \(-0.343277\pi\)
0.472708 + 0.881219i \(0.343277\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.29150 9.16515i −0.500000 0.866025i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 11.8322i 1.09859i
\(117\) 16.4881 13.6373i 1.52433 1.26077i
\(118\) 0 0
\(119\) 13.1883 + 7.61425i 1.20897 + 0.697997i
\(120\) 0 0
\(121\) −5.43521 9.41407i −0.494110 0.855824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0.804903 0.952374i 0.0700578 0.0828935i
\(133\) 0 0
\(134\) 0 0
\(135\) −0.123475 + 11.6183i −0.0106271 + 0.999944i
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) −10.2470 + 5.91608i −0.866025 + 0.500000i
\(141\) 7.31174 + 20.3124i 0.615759 + 1.71062i
\(142\) 0 0
\(143\) 2.56739i 0.214696i
\(144\) −11.2470 4.18403i −0.937246 0.348669i
\(145\) 13.2288 1.09859
\(146\) 0 0
\(147\) 2.14766 11.9326i 0.177136 0.984186i
\(148\) 0 0
\(149\) 14.7470 8.51416i 1.20812 0.697507i 0.245770 0.969328i \(-0.420959\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) −3.43521 + 5.94996i −0.279554 + 0.484201i −0.971274 0.237964i \(-0.923520\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) 17.0260 2.87791i 1.37647 0.232666i
\(154\) 0 0
\(155\) 0 0
\(156\) 23.2470 8.36806i 1.86125 0.669981i
\(157\) 11.1010 + 19.2275i 0.885954 + 1.53452i 0.844616 + 0.535373i \(0.179829\pi\)
0.0413387 + 0.999145i \(0.486838\pi\)
\(158\) 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
\(165\) −1.06479 0.899909i −0.0828935 0.0700578i
\(166\) 0 0
\(167\) −16.6036 + 9.58612i −1.28483 + 0.741796i −0.977727 0.209881i \(-0.932692\pi\)
−0.307102 + 0.951677i \(0.599359\pi\)
\(168\) 0 0
\(169\) −18.9352 + 32.7968i −1.45655 + 2.52283i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.68246 5.59017i −0.736144 0.425013i 0.0845218 0.996422i \(-0.473064\pi\)
−0.820666 + 0.571409i \(0.806397\pi\)
\(174\) 0 0
\(175\) 6.61438 + 11.4564i 0.500000 + 0.866025i
\(176\) 1.24695 0.719927i 0.0939925 0.0542666i
\(177\) 0 0
\(178\) 0 0
\(179\) 14.8685i 1.11133i 0.831408 + 0.555663i \(0.187536\pi\)
−0.831408 + 0.555663i \(0.812464\pi\)
\(180\) −4.67789 + 12.5745i −0.348669 + 0.937246i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.03594 + 1.79431i −0.0757558 + 0.131213i
\(188\) 24.9281i 1.81807i
\(189\) −6.74695 11.9783i −0.490768 0.871290i
\(190\) 0 0
\(191\) 5.12348 + 2.95804i 0.370722 + 0.214036i 0.673774 0.738938i \(-0.264672\pi\)
−0.303052 + 0.952974i \(0.598006\pi\)
\(192\) −10.5830 8.94427i −0.763763 0.645497i
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) −9.35577 25.9909i −0.669981 1.86125i
\(196\) 7.00000 12.1244i 0.500000 0.866025i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −13.5554 + 7.82624i −0.951405 + 0.549294i
\(204\) 19.6235 + 3.53187i 1.37392 + 0.247280i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 28.5294 1.97816
\(209\) 0 0
\(210\) 0 0
\(211\) −1.93521 3.35189i −0.133226 0.230753i 0.791693 0.610920i \(-0.209200\pi\)
−0.924918 + 0.380166i \(0.875867\pi\)
\(212\) 0 0
\(213\) 18.2333 21.5740i 1.24933 1.47822i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −8.37043 7.07430i −0.565621 0.478037i
\(220\) −0.804903 1.39413i −0.0542666 0.0939925i
\(221\) −35.5526 + 20.5263i −2.39153 + 1.38075i
\(222\) 0 0
\(223\) 4.37108 7.57093i 0.292709 0.506987i −0.681740 0.731594i \(-0.738777\pi\)
0.974449 + 0.224607i \(0.0721099\pi\)
\(224\) 0 0
\(225\) 14.0587 + 5.23004i 0.937246 + 0.348669i
\(226\) 0 0
\(227\) −6.63426 3.83029i −0.440331 0.254225i 0.263407 0.964685i \(-0.415154\pi\)
−0.703738 + 0.710460i \(0.748487\pi\)
\(228\) 0 0
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 1.62348 + 0.292196i 0.106817 + 0.0192251i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 27.8704 1.81807
\(236\) 0 0
\(237\) −25.8638 + 9.31002i −1.68003 + 0.604751i
\(238\) 0 0
\(239\) 25.6174 14.7902i 1.65705 0.956698i 0.682985 0.730433i \(-0.260682\pi\)
0.974066 0.226266i \(-0.0726518\pi\)
\(240\) −10.0000 + 11.8322i −0.645497 + 0.763763i
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0 0
\(243\) −14.5516 5.59017i −0.933488 0.358610i
\(244\) 0 0
\(245\) −13.5554 7.82624i −0.866025 0.500000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −10.6883 29.6926i −0.677341 1.88169i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −2.64575 15.6525i −0.166667 0.986013i
\(253\) 0 0
\(254\) 0 0
\(255\) 3.94875 21.9397i 0.247280 1.37392i
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) −25.7483 + 14.8658i −1.60613 + 0.927301i −0.615907 + 0.787819i \(0.711210\pi\)
−0.990225 + 0.139482i \(0.955456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 31.8968i 1.97816i
\(261\) −6.18826 + 16.6345i −0.383044 + 1.02965i
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 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 \(\pi\)
\(272\) 19.9388 + 11.5117i 1.20897 + 0.697997i
\(273\) 24.9632 + 21.0978i 1.51084 + 1.27690i
\(274\) 0 0
\(275\) −1.55869 + 0.899909i −0.0939925 + 0.0542666i
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.3117 15.7684i −1.62928 0.940666i −0.984307 0.176462i \(-0.943535\pi\)
−0.644974 0.764204i \(-0.723132\pi\)
\(282\) 0 0
\(283\) 13.8622 + 24.0101i 0.824025 + 1.42725i 0.902662 + 0.430350i \(0.141610\pi\)
−0.0786368 + 0.996903i \(0.525057\pi\)
\(284\) 28.2470 16.3084i 1.67615 0.967725i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.1296 −0.948798
\(290\) 0 0
\(291\) −24.5587 + 8.84024i −1.43966 + 0.518224i
\(292\) −6.32745 10.9595i −0.370286 0.641354i
\(293\) 21.3014 12.2984i 1.24444 0.718479i 0.274446 0.961602i \(-0.411505\pi\)
0.969995 + 0.243124i \(0.0781721\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.62968 0.917946i 0.0945638 0.0532646i
\(298\) 0 0
\(299\) 0 0
\(300\) 13.2288 + 11.1803i 0.763763 + 0.645497i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 23.0069 1.31307 0.656535 0.754295i \(-0.272021\pi\)
0.656535 + 0.754295i \(0.272021\pi\)
\(308\) 1.64956 + 0.952374i 0.0939925 + 0.0542666i
\(309\) −0.811738 + 4.51011i −0.0461781 + 0.256571i
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −14.5516 + 25.2042i −0.822507 + 1.42462i 0.0813030 + 0.996689i \(0.474092\pi\)
−0.903810 + 0.427934i \(0.859241\pi\)
\(314\) 0 0
\(315\) −17.5000 + 2.95804i −0.986013 + 0.166667i
\(316\) −31.7409 −1.78556
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) −1.06479 1.84427i −0.0596166 0.103259i
\(320\) −15.4919 + 8.94427i −0.866025 + 0.500000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −13.6235 11.7644i −0.756860 0.653577i
\(325\) −35.6618 −1.97816
\(326\) 0 0
\(327\) 13.0573 + 11.0355i 0.722073 + 0.610263i
\(328\) 0 0
\(329\) −28.5587 + 16.4884i −1.57449 + 0.909033i
\(330\) 0 0
\(331\) 17.3704 30.0865i 0.954765 1.65370i 0.219860 0.975531i \(-0.429440\pi\)
0.734905 0.678170i \(-0.237227\pi\)
\(332\) 36.4397i 1.99989i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 3.24695 18.0404i 0.177136 0.984186i
\(337\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 12.8704 22.2922i 0.697997 1.20897i
\(341\) 0 0
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) −13.2288 + 15.6525i −0.709136 + 0.839061i
\(349\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 37.0587 + 0.393847i 1.97805 + 0.0210220i
\(352\) 0 0
\(353\) 25.1744 + 14.5344i 1.33990 + 0.773590i 0.986792 0.161993i \(-0.0517922\pi\)
0.353106 + 0.935583i \(0.385126\pi\)
\(354\) 0 0
\(355\) −18.2333 31.5811i −0.967725 1.67615i
\(356\) 0 0
\(357\) 8.93344 + 24.8176i 0.472808 + 1.31349i
\(358\) 0 0
\(359\) 11.8322i 0.624477i 0.950004 + 0.312239i \(0.101079\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 3.33513 18.5304i 0.175049 0.972593i
\(364\) 18.8704 + 32.6845i 0.989079 + 1.71314i
\(365\) −12.2530 + 7.07430i −0.641354 + 0.370286i
\(366\) 0 0
\(367\) 9.89362 17.1363i 0.516443 0.894505i −0.483375 0.875413i \(-0.660589\pi\)
0.999818 0.0190919i \(-0.00607750\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 12.5000 14.7902i 0.645497 0.763763i
\(376\) 0 0
\(377\) 42.1956i 2.17318i
\(378\) 0 0
\(379\) −22.7409 −1.16812 −0.584060 0.811711i \(-0.698537\pi\)
−0.584060 + 0.811711i \(0.698537\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.3978 15.8181i 1.39996 0.808269i 0.405575 0.914062i \(-0.367071\pi\)
0.994388 + 0.105793i \(0.0337381\pi\)
\(384\) 0 0
\(385\) 1.06479 1.84427i 0.0542666 0.0939925i
\(386\) 0 0
\(387\) 0 0
\(388\) −30.1392 −1.53009
\(389\) 26.6883 + 15.4085i 1.35315 + 0.781241i 0.988689 0.149979i \(-0.0479205\pi\)
0.364459 + 0.931219i \(0.381254\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 35.4874i 1.78556i
\(396\) 2.12957 0.359964i 0.107015 0.0180889i
\(397\) −34.3948 −1.72622 −0.863112 0.505013i \(-0.831488\pi\)
−0.863112 + 0.505013i \(0.831488\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 10.0000 + 17.3205i 0.500000 + 0.866025i
\(401\) 25.6174 14.7902i 1.27927 0.738587i 0.302556 0.953131i \(-0.402160\pi\)
0.976714 + 0.214544i \(0.0688266\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −13.1530 + 15.2315i −0.653577 + 0.756860i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.64575 + 4.58258i −0.130347 + 0.225767i
\(413\) 0 0
\(414\) 0 0
\(415\) −40.7409 −1.99989
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(420\) −20.1698 3.63020i −0.984186 0.177136i
\(421\) −16.9352 + 29.3326i −0.825372 + 1.42959i 0.0762630 + 0.997088i \(0.475701\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(422\) 0 0
\(423\) −13.0375 + 35.0456i −0.633903 + 1.70398i
\(424\) 0 0
\(425\) −24.9235 14.3896i −1.20897 0.697997i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.87043 + 3.39633i −0.138585 + 0.163977i
\(430\) 0 0
\(431\) 18.1082i 0.872241i 0.899888 + 0.436121i \(0.143648\pi\)
−0.899888 + 0.436121i \(0.856352\pi\)
\(432\) −10.2004 18.1094i −0.490768 0.871290i
\(433\) −10.5830 −0.508587 −0.254293 0.967127i \(-0.581843\pi\)
−0.254293 + 0.967127i \(0.581843\pi\)
\(434\) 0 0
\(435\) 17.5000 + 14.7902i 0.839061 + 0.709136i
\(436\) 9.87043 + 17.0961i 0.472708 + 0.818754i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 16.1822 13.3842i 0.770579 0.637344i
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 29.0275 + 5.22442i 1.37295 + 0.247107i
\(448\) 10.5830 18.3303i 0.500000 0.866025i
\(449\) 41.4126i 1.95438i −0.212368 0.977190i \(-0.568118\pi\)
0.212368 0.977190i \(-0.431882\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −11.1966 + 4.03038i −0.526063 + 0.189364i
\(454\) 0 0
\(455\) 36.5424 21.0978i 1.71314 0.989079i
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 25.7409 + 15.2285i 1.20148 + 0.710805i
\(460\) 0 0
\(461\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) −20.4939 + 11.8322i −0.951405 + 0.549294i
\(465\) 0 0
\(466\) 0 0
\(467\) 14.3688i 0.664908i 0.943119 + 0.332454i \(0.107877\pi\)
−0.943119 + 0.332454i \(0.892123\pi\)
\(468\) 40.1086 + 14.9210i 1.85402 + 0.689723i
\(469\) 0 0
\(470\) 0 0
\(471\) −6.81174 + 37.8468i −0.313868 + 1.74389i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 30.4570i 1.39599i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 10.8704 18.8281i 0.494110 0.855824i
\(485\) 33.6967i 1.53009i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.9413 6.89432i 0.538904 0.311136i −0.205731 0.978609i \(-0.565957\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 17.0260 29.4899i 0.766811 1.32816i
\(494\) 0 0
\(495\) −0.402452 2.38094i −0.0180889 0.107015i
\(496\) 0 0
\(497\) 37.3672 + 21.5740i 1.67615 + 0.967725i
\(498\) 0 0
\(499\) 2.56479 + 4.44234i 0.114816 + 0.198867i 0.917706 0.397260i \(-0.130039\pi\)
−0.802890 + 0.596127i \(0.796706\pi\)
\(500\) 19.3649 11.1803i 0.866025 0.500000i
\(501\) −32.6822 5.88220i −1.46013 0.262797i
\(502\) 0 0
\(503\) 38.0132i 1.69492i 0.530857 + 0.847461i \(0.321870\pi\)
−0.530857 + 0.847461i \(0.678130\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −61.7168 + 22.2158i −2.74094 + 0.986639i
\(508\) 0 0
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 8.37043 14.4980i 0.370286 0.641354i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.12348 + 2.95804i 0.225767 + 0.130347i
\(516\) 0 0
\(517\) −2.24330 3.88551i −0.0986602 0.170884i
\(518\) 0 0
\(519\) −6.55869 18.2204i −0.287894 0.799788i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 4.71764 0.206288 0.103144 0.994666i \(-0.467110\pi\)
0.103144 + 0.994666i \(0.467110\pi\)
\(524\) 0 0
\(525\) −4.05869 + 22.5505i −0.177136 + 0.984186i
\(526\) 0 0
\(527\) 0 0
\(528\) 2.45446 + 0.441759i 0.106817 + 0.0192251i
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −16.6235 + 19.6692i −0.717358 + 0.848789i
\(538\) 0 0
\(539\) 2.51975i 0.108533i
\(540\) −20.2470 + 11.4044i −0.871290 + 0.490768i
\(541\) 36.8704 1.58518 0.792592 0.609753i \(-0.208731\pi\)
0.792592 + 0.609753i \(0.208731\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.1140 11.0355i 0.818754 0.472708i
\(546\) 0 0
\(547\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.9946 36.3637i −0.892781 1.54634i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −20.4939 11.8322i −0.866025 0.500000i
\(561\) −3.37652 + 1.21543i −0.142557 + 0.0513153i
\(562\) 0 0
\(563\) −7.45904 + 4.30648i −0.314361 + 0.181496i −0.648876 0.760894i \(-0.724761\pi\)
0.334515 + 0.942390i \(0.391427\pi\)
\(564\) −27.8704 + 32.9767i −1.17356 + 1.38857i
\(565\) 0 0
\(566\) 0 0
\(567\) 4.46672 23.3891i 0.187585 0.982248i
\(568\) 0 0
\(569\) 15.9939 + 9.23408i 0.670499 + 0.387113i 0.796266 0.604947i \(-0.206806\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 23.3704 + 40.4788i 0.978022 + 1.69398i 0.669579 + 0.742741i \(0.266474\pi\)
0.308443 + 0.951243i \(0.400192\pi\)
\(572\) −4.44685 + 2.56739i −0.185932 + 0.107348i
\(573\) 3.47053 + 9.64134i 0.144984 + 0.402773i
\(574\) 0 0
\(575\) 0 0
\(576\) −4.00000 23.6643i −0.166667 0.986013i
\(577\) −11.8500 −0.493322 −0.246661 0.969102i \(-0.579333\pi\)
−0.246661 + 0.969102i \(0.579333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 13.2288 + 22.9129i 0.549294 + 0.951405i
\(581\) 41.7470 24.1026i 1.73196 0.999945i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 16.6822 44.8428i 0.689723 1.85402i
\(586\) 0 0
\(587\) 7.74597 + 4.47214i 0.319710 + 0.184585i 0.651263 0.758852i \(-0.274239\pi\)
−0.331553 + 0.943437i \(0.607573\pi\)
\(588\) 22.8156 8.21278i 0.940898 0.338689i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.4853i 1.74466i −0.488916 0.872331i \(-0.662608\pi\)
0.488916 0.872331i \(-0.337392\pi\)
\(594\) 0 0
\(595\) 34.0519 1.39599
\(596\) 29.4939 + 17.0283i 1.20812 + 0.697507i
\(597\) 0 0
\(598\) 0 0
\(599\) −42.0587 + 24.2826i −1.71847 + 0.992160i −0.796748 + 0.604311i \(0.793448\pi\)
−0.921723 + 0.387849i \(0.873218\pi\)
\(600\) 0 0
\(601\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −13.7409 −0.559107
\(605\) −21.0505 12.1535i −0.855824 0.494110i
\(606\) 0 0
\(607\) −22.4889 38.9519i −0.912796 1.58101i −0.810097 0.586296i \(-0.800586\pi\)
−0.102699 0.994712i \(-0.532748\pi\)
\(608\) 0 0
\(609\) −26.6822 4.80230i −1.08122 0.194599i
\(610\) 0 0
\(611\) 88.8979i 3.59642i
\(612\) 22.0107 + 26.6119i 0.889729 + 1.07572i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 37.7409 + 31.8968i 1.51084 + 1.27690i
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −22.2020 + 38.4549i −0.885954 + 1.53452i
\(629\) 0 0
\(630\) 0 0
\(631\) −8.12957 −0.323633 −0.161817 0.986821i \(-0.551735\pi\)
−0.161817 + 0.986821i \(0.551735\pi\)
\(632\) 0 0
\(633\) 1.18748 6.59776i 0.0471980 0.262238i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −24.9632 + 43.2376i −0.989079 + 1.71314i
\(638\) 0 0
\(639\) 48.2409 8.15419i 1.90838 0.322575i
\(640\) 0 0
\(641\) −40.9878 23.6643i −1.61892 0.934684i −0.987200 0.159489i \(-0.949015\pi\)
−0.631721 0.775196i \(-0.717651\pi\)
\(642\) 0 0
\(643\) −15.4721 26.7984i −0.610158 1.05683i −0.991213 0.132273i \(-0.957772\pi\)
0.381055 0.924552i \(-0.375561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8885i 0.703271i 0.936137 + 0.351636i \(0.114374\pi\)
−0.936137 + 0.351636i \(0.885626\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.16372 18.7168i −0.123429 0.730214i
\(658\) 0 0
\(659\) −40.8117 23.5627i −1.58980 0.917871i −0.993339 0.115229i \(-0.963240\pi\)
−0.596461 0.802642i \(-0.703427\pi\)
\(660\) 0.493902 2.74417i 0.0192251 0.106817i
\(661\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 0 0
\(663\) −69.9808 12.5953i −2.71783 0.489160i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −33.2073 19.1722i −1.28483 0.741796i
\(669\) 14.2470 5.12838i 0.550819 0.198275i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 12.7505 + 22.6368i 0.490768 + 0.871290i
\(676\) −75.7409 −2.91311
\(677\) 28.2226 + 16.2943i 1.08468 + 0.626242i 0.932156 0.362058i \(-0.117926\pi\)
0.152527 + 0.988299i \(0.451259\pi\)
\(678\) 0 0
\(679\) −19.9352 34.5288i −0.765043 1.32509i
\(680\) 0 0
\(681\) −4.49390 12.4843i −0.172207 0.478400i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 22.3607i 0.850026i
\(693\) 1.82097 + 2.20164i 0.0691730 + 0.0836334i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −13.2288 + 22.9129i −0.500000 + 0.866025i
\(701\) 49.2851i 1.86147i 0.365690 + 0.930737i \(0.380833\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.49390 + 1.43985i 0.0939925 + 0.0542666i
\(705\) 36.8691 + 31.1601i 1.38857 + 1.17356i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.500000 0.866025i 0.0187779 0.0325243i −0.856484 0.516174i \(-0.827356\pi\)
0.875262 + 0.483650i \(0.160689\pi\)
\(710\) 0 0
\(711\) −44.6235 16.6006i −1.67351 0.622570i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.87043 + 4.97172i 0.107348 + 0.185932i
\(716\) −25.7530 + 14.8685i −0.962437 + 0.555663i
\(717\) 50.4245 + 9.07550i 1.88314 + 0.338931i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −26.4575 + 4.47214i −0.986013 + 0.166667i
\(721\) −7.00000 −0.260694
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 25.6174 14.7902i 0.951405 0.549294i
\(726\) 0 0
\(727\) 24.5608 42.5405i 0.910909 1.57774i 0.0981255 0.995174i \(-0.468715\pi\)
0.812783 0.582566i \(-0.197951\pi\)
\(728\) 0 0
\(729\) −13.0000 23.6643i −0.481481 0.876456i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 25.1346 + 43.5345i 0.928369 + 1.60798i 0.786051 + 0.618161i \(0.212122\pi\)
0.142318 + 0.989821i \(0.454545\pi\)
\(734\) 0 0
\(735\) −9.18216 25.5086i −0.338689 0.940898i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 19.0382 32.9752i 0.697507 1.20812i
\(746\) 0 0
\(747\) 19.0581 51.2295i 0.697300 1.87439i
\(748\) −4.14378 −0.151512
\(749\) 0 0
\(750\) 0 0
\(751\) 6.50000 + 11.2583i 0.237188 + 0.410822i 0.959906 0.280321i \(-0.0904408\pi\)
−0.722718 + 0.691143i \(0.757107\pi\)
\(752\) −43.1767 + 24.9281i −1.57449 + 0.909033i
\(753\) 0 0
\(754\) 0 0
\(755\) 15.3627i 0.559107i
\(756\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) −13.0573 + 22.6160i −0.472708 + 0.818754i
\(764\) 11.8322i 0.428073i
\(765\) 29.7530 24.6087i 1.07572 0.889729i
\(766\) 0 0
\(767\) 0 0
\(768\) 4.90893 27.2746i 0.177136 0.984186i
\(769\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) −50.6822 9.12187i −1.82527 0.328516i
\(772\) 0 0
\(773\) 46.9990i 1.69044i −0.534421 0.845218i \(-0.679470\pi\)
0.534421 0.845218i \(-0.320530\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 35.6618 42.1956i 1.27690 1.51084i
\(781\) −2.93521 + 5.08394i −0.105030 + 0.181918i
\(782\) 0 0
\(783\) −26.7842 + 15.0866i −0.957189 + 0.539153i
\(784\) 28.0000 1.00000
\(785\) 42.9939 + 24.8225i 1.53452 + 0.885954i
\(786\) 0 0
\(787\) 25.7681 + 44.6317i 0.918535 + 1.59095i 0.801642 + 0.597804i \(0.203960\pi\)
0.116892 + 0.993145i \(0.462707\pi\)
\(788\) 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
\(797\) 18.2532 10.5385i 0.646562 0.373293i −0.140576 0.990070i \(-0.544895\pi\)
0.787138 + 0.616777i \(0.211562\pi\)
\(798\) 0 0
\(799\) 35.8704 62.1294i 1.26900 2.19798i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.97250 + 1.13883i 0.0696081 + 0.0401883i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13.0687i 0.459471i −0.973253 0.229736i \(-0.926214\pi\)
0.973253 0.229736i \(-0.0737862\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −27.1109 15.6525i −0.951405 0.549294i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 13.5061 + 37.5207i 0.472808 + 1.31349i
\(817\) 0 0
\(818\) 0 0
\(819\) 9.43521 + 55.8195i 0.329693 + 1.95049i
\(820\) 0 0
\(821\) 40.1883 + 23.2027i 1.40258 + 0.809780i 0.994657 0.103236i \(-0.0329198\pi\)
0.407923 + 0.913016i \(0.366253\pi\)
\(822\) 0 0
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) −3.06808 0.552199i −0.106817 0.0192251i
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 28.5294 + 49.4144i 0.989079 + 1.71314i
\(833\) −34.8929 + 20.1454i −1.20897 + 0.697997i
\(834\) 0 0
\(835\) −21.4352 + 37.1269i −0.741796 + 1.28483i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) 3.00000 + 5.19615i 0.103448 + 0.179178i
\(842\) 0 0
\(843\) −18.5004 51.3951i −0.637187 1.77014i
\(844\) 3.87043 6.70377i 0.133226 0.230753i
\(845\) 84.6808i 2.91311i
\(846\) 0 0
\(847\) 28.7604 0.988221
\(848\) 0 0
\(849\) −8.50610 + 47.2609i −0.291929 + 1.62199i
\(850\) 0 0
\(851\) 0 0
\(852\) 55.6005 + 10.0071i 1.90484 + 0.342837i
\(853\) 21.1660 36.6606i 0.724710 1.25524i −0.234383 0.972144i \(-0.575307\pi\)
0.959093 0.283091i \(-0.0913598\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.51035 + 1.44935i 0.0857520 + 0.0495090i 0.542263 0.840209i \(-0.317568\pi\)
−0.456511 + 0.889718i \(0.650901\pi\)
\(858\) 0 0
\(859\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −25.0000 −0.850026
\(866\) 0 0
\(867\) −21.3374 18.0334i −0.724657 0.612447i
\(868\) 0 0
\(869\) 4.94741 2.85639i 0.167829 0.0968963i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −42.3718 15.7629i −1.43407 0.533494i
\(874\) 0 0
\(875\) 25.6174 + 14.7902i 0.866025 + 0.500000i
\(876\) 3.88262 21.5723i 0.131182 0.728861i
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 41.9291 + 7.54648i 1.41423 + 0.254536i
\(880\) 1.60981 2.78827i 0.0542666 0.0939925i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −71.1052 41.0526i −2.39153 1.38075i
\(885\) 0 0
\(886\) 0 0
\(887\) −30.9839 + 17.8885i −1.04034 + 0.600639i −0.919929 0.392086i \(-0.871754\pi\)
−0.120408 + 0.992725i \(0.538420\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3.18216 + 0.607713i 0.106606 + 0.0203592i
\(892\) 17.4843 0.585418
\(893\) 0 0
\(894\) 0 0
\(895\) 16.6235 + 28.7928i 0.555663 + 0.962437i
\(896\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 15.3212i 0.508450i
\(909\) 0 0
\(910\) 0 0
\(911\) −16.6174 9.59405i −0.550558 0.317865i 0.198789 0.980042i \(-0.436299\pi\)
−0.749347 + 0.662177i \(0.769633\pi\)
\(912\) 0 0
\(913\) 3.27924 + 5.67982i 0.108527 + 0.187975i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 30.4352 + 25.7224i 1.00287 + 0.847584i
\(922\) 0 0
\(923\) −100.734 + 58.1586i −3.31569 + 1.91431i
\(924\) 1.11738 + 3.10414i 0.0367590 + 0.102119i
\(925\) 0 0
\(926\) 0 0
\(927\) −6.11628 + 5.05876i −0.200885 + 0.166152i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.63289i 0.151512i
\(936\) 0 0
\(937\) −24.6167 −0.804191 −0.402096 0.915598i \(-0.631718\pi\)
−0.402096 + 0.915598i \(0.631718\pi\)
\(938\) 0 0
\(939\) −47.4291 + 17.0728i −1.54779 + 0.557148i
\(940\) 27.8704 + 48.2730i 0.909033 + 1.57449i
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −26.4575 15.6525i −0.860663 0.509175i
\(946\) 0 0
\(947\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) −41.9892 35.4874i −1.36375 1.15258i
\(949\) 22.5648 + 39.0834i 0.732484 + 1.26870i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 13.2288 0.428073
\(956\) 51.2348 + 29.5804i 1.65705 + 0.956698i
\(957\) 0.653370 3.63020i 0.0211205 0.117348i
\(958\) 0 0
\(959\) 0 0
\(960\) −30.4939 5.48835i −0.984186 0.177136i
\(961\) −15.5000 + 26.8468i −0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −4.86917 30.7943i −0.156179 0.987729i
\(973\) 0 0
\(974\) 0 0
\(975\) −47.1761 39.8711i −1.51084 1.27690i
\(976\) 0 0
\(977\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 31.3050i 1.00000i
\(981\) 4.93521 + 29.1971i 0.157569 + 0.932192i
\(982\) 0 0
\(983\) 29.0834 + 16.7913i 0.927616 + 0.535559i 0.886057 0.463577i \(-0.153434\pi\)
0.0415592 + 0.999136i \(0.486767\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −56.2141 10.1175i −1.78932 0.322044i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.74085 −0.150598 −0.0752991 0.997161i \(-0.523991\pi\)
−0.0752991 + 0.997161i \(0.523991\pi\)
\(992\) 0 0
\(993\) 56.6166 20.3799i 1.79667 0.646737i
\(994\) 0 0
\(995\) 0 0
\(996\) 40.7409 48.2052i 1.29092 1.52744i
\(997\) 9.26013 16.0390i 0.293271 0.507961i −0.681310 0.731995i \(-0.738589\pi\)
0.974581 + 0.224034i \(0.0719228\pi\)
\(998\) 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 315.2.z.a.209.4 yes 8
3.2 odd 2 945.2.z.a.629.1 8
5.4 even 2 inner 315.2.z.a.209.1 yes 8
7.6 odd 2 inner 315.2.z.a.209.1 yes 8
9.4 even 3 945.2.z.a.314.1 8
9.5 odd 6 inner 315.2.z.a.104.3 yes 8
15.14 odd 2 945.2.z.a.629.4 8
21.20 even 2 945.2.z.a.629.4 8
35.34 odd 2 CM 315.2.z.a.209.4 yes 8
45.4 even 6 945.2.z.a.314.4 8
45.14 odd 6 inner 315.2.z.a.104.2 8
63.13 odd 6 945.2.z.a.314.4 8
63.41 even 6 inner 315.2.z.a.104.2 8
105.104 even 2 945.2.z.a.629.1 8
315.104 even 6 inner 315.2.z.a.104.3 yes 8
315.139 odd 6 945.2.z.a.314.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.z.a.104.2 8 45.14 odd 6 inner
315.2.z.a.104.2 8 63.41 even 6 inner
315.2.z.a.104.3 yes 8 9.5 odd 6 inner
315.2.z.a.104.3 yes 8 315.104 even 6 inner
315.2.z.a.209.1 yes 8 5.4 even 2 inner
315.2.z.a.209.1 yes 8 7.6 odd 2 inner
315.2.z.a.209.4 yes 8 1.1 even 1 trivial
315.2.z.a.209.4 yes 8 35.34 odd 2 CM
945.2.z.a.314.1 8 9.4 even 3
945.2.z.a.314.1 8 315.139 odd 6
945.2.z.a.314.4 8 45.4 even 6
945.2.z.a.314.4 8 63.13 odd 6
945.2.z.a.629.1 8 3.2 odd 2
945.2.z.a.629.1 8 105.104 even 2
945.2.z.a.629.4 8 15.14 odd 2
945.2.z.a.629.4 8 21.20 even 2