Properties

Label 315.2.z.a.104.2
Level $315$
Weight $2$
Character 315.104
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 104.2
Root \(-0.306808 - 1.70466i\) of defining polynomial
Character \(\chi\) \(=\) 315.104
Dual form 315.2.z.a.209.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32288 + 1.11803i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(1.32288 + 2.29129i) q^{7} +(0.500000 - 2.95804i) q^{9} +O(q^{10})\) \(q+(-1.32288 + 1.11803i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(1.32288 + 2.29129i) q^{7} +(0.500000 - 2.95804i) q^{9} +(-0.311738 + 0.179982i) q^{11} +(0.613616 + 3.40932i) q^{12} +(3.56618 - 6.17680i) q^{13} +(3.81174 - 0.686044i) q^{15} +(-2.00000 - 3.46410i) q^{16} -5.75583i q^{17} +(-3.87298 + 2.23607i) q^{20} +(-4.31174 - 1.55207i) q^{21} +(2.50000 + 4.33013i) q^{25} +(2.64575 + 4.47214i) q^{27} +5.29150 q^{28} +(5.12348 - 2.95804i) q^{29} +(0.211164 - 0.586627i) q^{33} -5.91608i q^{35} +(-4.62348 - 3.82407i) q^{36} +(2.18826 + 12.1582i) q^{39} +0.719927i q^{44} +(-4.27543 + 5.16920i) q^{45} +(-10.7942 + 6.23202i) q^{47} +(6.51873 + 2.34651i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(6.43521 + 7.61425i) q^{51} +(-7.13235 - 12.3536i) q^{52} +0.804903 q^{55} +(2.62348 - 7.28817i) q^{60} +(7.43916 - 2.76748i) q^{63} -8.00000 q^{64} +(-13.8117 + 7.97421i) q^{65} +(-9.96939 - 5.75583i) q^{68} +16.3084i q^{71} +6.32745 q^{73} +(-8.14842 - 2.93313i) q^{75} +(-0.824780 - 0.476187i) q^{77} +(-7.93521 - 13.7442i) q^{79} +8.94427i q^{80} +(-8.50000 - 2.95804i) q^{81} +(15.7789 - 9.10993i) q^{83} +(-7.00000 + 5.91608i) q^{84} +(-6.43521 + 11.1461i) q^{85} +(-3.47053 + 9.64134i) q^{87} +18.8704 q^{91} +(7.53480 + 13.0507i) q^{97} +(0.376525 + 1.01212i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 4 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) −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 −0.546259 0.837616i \(-0.683949\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0.613616 + 3.40932i 0.177136 + 0.984186i
\(13\) 3.56618 6.17680i 0.989079 1.71314i 0.366900 0.930261i \(-0.380419\pi\)
0.622179 0.782875i \(-0.286247\pi\)
\(14\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 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 0.0578882 0.998323i \(-0.481563\pi\)
0.893517 + 0.449029i \(0.148230\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 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 −0.578884 + 0.815410i \(0.696511\pi\)
−0.995608 + 0.0936230i \(0.970155\pi\)
\(48\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 2.62348 7.28817i 0.338689 0.940898i
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −9.96939 5.75583i −1.20897 0.697997i
\(69\) 0 0
\(70\) 0 0
\(71\) 16.3084i 1.93545i 0.252010 + 0.967725i \(0.418908\pi\)
−0.252010 + 0.967725i \(0.581092\pi\)
\(72\) 0 0
\(73\) 6.32745 0.740572 0.370286 0.928918i \(-0.379260\pi\)
0.370286 + 0.928918i \(0.379260\pi\)
\(74\) 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 −0.836527 0.547926i \(-0.815418\pi\)
−0.0562544 0.998416i \(-0.517916\pi\)
\(80\) 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 0.860729 0.509064i \(-0.170008\pi\)
0.871227 0.490881i \(-0.163325\pi\)
\(84\) −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 0.940224 + 0.340557i \(0.110616\pi\)
−0.175180 + 0.984536i \(0.556051\pi\)
\(98\) 0 0
\(99\) 0.376525 + 1.01212i 0.0378421 + 0.101722i
\(100\) 10.0000 1.00000
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −1.32288 + 2.29129i −0.130347 + 0.225767i −0.923810 0.382851i \(-0.874942\pi\)
0.793463 + 0.608618i \(0.208276\pi\)
\(104\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(114\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) −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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) −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 0.962348 0.271821i \(-0.0876260\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −3.43521 5.94996i −0.279554 0.484201i 0.691720 0.722166i \(-0.256853\pi\)
−0.971274 + 0.237964i \(0.923520\pi\)
\(152\) 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 −0.0413387 + 0.999145i \(0.513162\pi\)
−0.844616 + 0.535373i \(0.820171\pi\)
\(158\) 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 \(-0.0673075\pi\)
0.307102 + 0.951677i \(0.400641\pi\)
\(168\) 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 \(-0.526936\pi\)
0.820666 + 0.571409i \(0.193603\pi\)
\(174\) 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 \(-0.812464\pi\)
0.831408 0.555663i \(-0.187536\pi\)
\(180\) 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 −0.303052 0.952974i \(-0.598006\pi\)
0.673774 + 0.738938i \(0.264672\pi\)
\(192\) 10.5830 8.94427i 0.763763 0.645497i
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 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 −0.924918 0.380166i \(-0.875867\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 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 \(-0.261223\pi\)
−0.974449 + 0.224607i \(0.927890\pi\)
\(224\) 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 \(-0.584846\pi\)
0.703738 + 0.710460i \(0.251513\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 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 0.974066 + 0.226266i \(0.0726518\pi\)
0.682985 + 0.730433i \(0.260682\pi\)
\(240\) −10.0000 11.8322i −0.645497 0.763763i
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 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 0.990225 + 0.139482i \(0.0445438\pi\)
0.615907 + 0.787819i \(0.288790\pi\)
\(258\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.3117 + 15.7684i −1.62928 + 0.940666i −0.644974 + 0.764204i \(0.723132\pi\)
−0.984307 + 0.176462i \(0.943535\pi\)
\(282\) 0 0
\(283\) −13.8622 + 24.0101i −0.824025 + 1.42725i 0.0786368 + 0.996903i \(0.474943\pi\)
−0.902662 + 0.430350i \(0.858390\pi\)
\(284\) 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 \(-0.588495\pi\)
−0.969995 + 0.243124i \(0.921828\pi\)
\(294\) 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 \(-0.727979\pi\)
−0.656535 + 0.754295i \(0.727979\pi\)
\(308\) −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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 14.5516 + 25.2042i 0.822507 + 1.42462i 0.903810 + 0.427934i \(0.140759\pi\)
−0.0813030 + 0.996689i \(0.525908\pi\)
\(314\) 0 0
\(315\) −17.5000 2.95804i −0.986013 0.166667i
\(316\) −31.7409 −1.78556
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 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 0.734905 + 0.678170i \(0.237227\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) 13.2288 + 15.6525i 0.709136 + 0.839061i
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 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 \(-0.948208\pi\)
−0.353106 + 0.935583i \(0.614874\pi\)
\(354\) 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 \(-0.898921\pi\)
0.950004 0.312239i \(-0.101079\pi\)
\(360\) 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 −0.999818 0.0190919i \(-0.993923\pi\)
0.483375 0.875413i \(-0.339411\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 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 \(-0.632929\pi\)
−0.994388 + 0.105793i \(0.966262\pi\)
\(384\) 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 0.364459 0.931219i \(-0.381254\pi\)
0.988689 + 0.149979i \(0.0479205\pi\)
\(390\) 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 \(-0.168512\pi\)
0.863112 + 0.505013i \(0.168512\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 10.0000 17.3205i 0.500000 0.866025i
\(401\) 25.6174 + 14.7902i 1.27927 + 0.738587i 0.976714 0.214544i \(-0.0688266\pi\)
0.302556 + 0.953131i \(0.402160\pi\)
\(402\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 20.1698 3.63020i 0.984186 0.177136i
\(421\) −16.9352 29.3326i −0.825372 1.42959i −0.901635 0.432498i \(-0.857632\pi\)
0.0762630 0.997088i \(-0.475701\pi\)
\(422\) 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 \(-0.856352\pi\)
0.899888 0.436121i \(-0.143648\pi\)
\(432\) 10.2004 18.1094i 0.490768 0.871290i
\(433\) 10.5830 0.508587 0.254293 0.967127i \(-0.418157\pi\)
0.254293 + 0.967127i \(0.418157\pi\)
\(434\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 16.1822 + 13.3842i 0.770579 + 0.637344i
\(442\) 0 0
\(443\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 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 \(0.431882\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 25.7409 15.2285i 1.20148 0.710805i
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) −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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 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 0.744635 0.667472i \(-0.232624\pi\)
−0.205731 + 0.978609i \(0.565957\pi\)
\(492\) 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 −0.802890 0.596127i \(-0.796706\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) −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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 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 \(-0.532890\pi\)
−0.103144 + 0.994666i \(0.532890\pi\)
\(524\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 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 \(-0.275239\pi\)
−0.334515 + 0.942390i \(0.608573\pi\)
\(564\) −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 −0.125767 0.992060i \(-0.540139\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 23.3704 40.4788i 0.978022 1.69398i 0.308443 0.951243i \(-0.400192\pi\)
0.669579 0.742741i \(-0.266474\pi\)
\(572\) 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 \(-0.420667\pi\)
0.246661 + 0.969102i \(0.420667\pi\)
\(578\) 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 \(-0.725761\pi\)
0.331553 + 0.943437i \(0.392427\pi\)
\(588\) −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 −0.921723 0.387849i \(-0.873218\pi\)
−0.796748 0.604311i \(-0.793448\pi\)
\(600\) 0 0
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 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 0.102699 0.994712i \(-0.467252\pi\)
0.810097 0.586296i \(-0.199414\pi\)
\(608\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) 0 0
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 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 −0.631721 + 0.775196i \(0.717651\pi\)
−0.987200 + 0.159489i \(0.949015\pi\)
\(642\) 0 0
\(643\) 15.4721 26.7984i 0.610158 1.05683i −0.381055 0.924552i \(-0.624439\pi\)
0.991213 0.132273i \(-0.0422275\pi\)
\(644\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 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 −0.596461 + 0.802642i \(0.703427\pi\)
−0.993339 + 0.115229i \(0.963240\pi\)
\(660\) 0.493902 + 2.74417i 0.0192251 + 0.106817i
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 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 \(-0.882074\pi\)
−0.152527 + 0.988299i \(0.548741\pi\)
\(678\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 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 \(-0.619167\pi\)
0.365690 0.930737i \(-0.380833\pi\)
\(702\) 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 0.875262 0.483650i \(-0.160689\pi\)
−0.856484 + 0.516174i \(0.827356\pi\)
\(710\) 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 −0.812783 0.582566i \(-0.802049\pi\)
−0.0981255 0.995174i \(-0.531285\pi\)
\(728\) 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 −0.142318 + 0.989821i \(0.545455\pi\)
−0.786051 + 0.618161i \(0.787878\pi\)
\(734\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 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 −0.722718 0.691143i \(-0.757107\pi\)
0.959906 + 0.280321i \(0.0904408\pi\)
\(752\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 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 −0.116892 + 0.993145i \(0.537293\pi\)
−0.801642 + 0.597804i \(0.796040\pi\)
\(788\) 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 \(-0.455105\pi\)
−0.787138 + 0.616777i \(0.788438\pi\)
\(798\) 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 \(0.0737862\pi\)
−0.973253 + 0.229736i \(0.926214\pi\)
\(810\) 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 0.407923 0.913016i \(-0.366253\pi\)
0.994657 + 0.103236i \(0.0329198\pi\)
\(822\) 0 0
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 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 −0.959093 0.283091i \(-0.908640\pi\)
0.234383 0.972144i \(-0.424693\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.51035 + 1.44935i −0.0857520 + 0.0495090i −0.542263 0.840209i \(-0.682432\pi\)
0.456511 + 0.889718i \(0.349099\pi\)
\(858\) 0 0
\(859\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 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 \(-0.128246\pi\)
0.120408 + 0.992725i \(0.461580\pi\)
\(888\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(908\) 15.3212i 0.508450i
\(909\) 0 0
\(910\) 0 0
\(911\) −16.6174 + 9.59405i −0.550558 + 0.317865i −0.749347 0.662177i \(-0.769633\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 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 \(-0.368282\pi\)
0.402096 + 0.915598i \(0.368282\pi\)
\(938\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(942\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(948\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 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 \(-0.846566\pi\)
−0.0415592 + 0.999136i \(0.513233\pi\)
\(984\) 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 \(-0.261411\pi\)
−0.974581 + 0.224034i \(0.928077\pi\)
\(998\) 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.104.2 8
3.2 odd 2 945.2.z.a.314.4 8
5.4 even 2 inner 315.2.z.a.104.3 yes 8
7.6 odd 2 inner 315.2.z.a.104.3 yes 8
9.2 odd 6 inner 315.2.z.a.209.1 yes 8
9.7 even 3 945.2.z.a.629.4 8
15.14 odd 2 945.2.z.a.314.1 8
21.20 even 2 945.2.z.a.314.1 8
35.34 odd 2 CM 315.2.z.a.104.2 8
45.29 odd 6 inner 315.2.z.a.209.4 yes 8
45.34 even 6 945.2.z.a.629.1 8
63.20 even 6 inner 315.2.z.a.209.4 yes 8
63.34 odd 6 945.2.z.a.629.1 8
105.104 even 2 945.2.z.a.314.4 8
315.34 odd 6 945.2.z.a.629.4 8
315.209 even 6 inner 315.2.z.a.209.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.z.a.104.2 8 1.1 even 1 trivial
315.2.z.a.104.2 8 35.34 odd 2 CM
315.2.z.a.104.3 yes 8 5.4 even 2 inner
315.2.z.a.104.3 yes 8 7.6 odd 2 inner
315.2.z.a.209.1 yes 8 9.2 odd 6 inner
315.2.z.a.209.1 yes 8 315.209 even 6 inner
315.2.z.a.209.4 yes 8 45.29 odd 6 inner
315.2.z.a.209.4 yes 8 63.20 even 6 inner
945.2.z.a.314.1 8 15.14 odd 2
945.2.z.a.314.1 8 21.20 even 2
945.2.z.a.314.4 8 3.2 odd 2
945.2.z.a.314.4 8 105.104 even 2
945.2.z.a.629.1 8 45.34 even 6
945.2.z.a.629.1 8 63.34 odd 6
945.2.z.a.629.4 8 9.7 even 3
945.2.z.a.629.4 8 315.34 odd 6