Properties

Label 95.7.d.a.94.1
Level $95$
Weight $7$
Character 95.94
Analytic conductor $21.855$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,7,Mod(94,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.94");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 95.d (of order \(2\), degree \(1\), 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: \(21.8551379439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 94.1
Root \(0.500000 - 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 95.94
Dual form 95.7.d.a.94.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +O(q^{10})\) \(q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +1062.00 q^{11} +4096.00 q^{16} +1952.79i q^{17} -6859.00 q^{19} +(1728.00 + 7811.15i) q^{20} +12937.2i q^{23} +(-14167.0 + 6590.66i) q^{25} +20085.8i q^{28} +(-38304.0 + 8473.70i) q^{35} +46656.0 q^{36} +70300.3i q^{43} -67968.0 q^{44} +(19683.0 + 88973.8i) q^{45} +193570. i q^{47} +19153.0 q^{49} +(-28674.0 - 129616. i) q^{55} +57062.0 q^{61} +228790. i q^{63} -262144. q^{64} -124978. i q^{68} -676641. i q^{73} +438976. q^{76} -333299. i q^{77} +(-110592. - 499913. i) q^{80} +531441. q^{81} -168916. i q^{83} +(238336. - 52725.2i) q^{85} -827982. i q^{92} +(185193. + 837135. i) q^{95} -774198. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9} + 2124 q^{11} + 8192 q^{16} - 13718 q^{19} + 3456 q^{20} - 28334 q^{25} - 76608 q^{35} + 93312 q^{36} - 135936 q^{44} + 39366 q^{45} + 38306 q^{49} - 57348 q^{55} + 114124 q^{61} - 524288 q^{64} + 877952 q^{76} - 221184 q^{80} + 1062882 q^{81} + 476672 q^{85} + 370386 q^{95} - 1548396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −64.0000 −1.00000
\(5\) −27.0000 122.049i −0.216000 0.976393i
\(6\) 0 0
\(7\) 313.841i 0.914988i −0.889213 0.457494i \(-0.848747\pi\)
0.889213 0.457494i \(-0.151253\pi\)
\(8\) 0 0
\(9\) −729.000 −1.00000
\(10\) 0 0
\(11\) 1062.00 0.797896 0.398948 0.916973i \(-0.369375\pi\)
0.398948 + 0.916973i \(0.369375\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 1952.79i 0.397473i 0.980053 + 0.198737i \(0.0636839\pi\)
−0.980053 + 0.198737i \(0.936316\pi\)
\(18\) 0 0
\(19\) −6859.00 −1.00000
\(20\) 1728.00 + 7811.15i 0.216000 + 0.976393i
\(21\) 0 0
\(22\) 0 0
\(23\) 12937.2i 1.06330i 0.846963 + 0.531652i \(0.178428\pi\)
−0.846963 + 0.531652i \(0.821572\pi\)
\(24\) 0 0
\(25\) −14167.0 + 6590.66i −0.906688 + 0.421802i
\(26\) 0 0
\(27\) 0 0
\(28\) 20085.8i 0.914988i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −38304.0 + 8473.70i −0.893388 + 0.197637i
\(36\) 46656.0 1.00000
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 70300.3i 0.884203i 0.896965 + 0.442101i \(0.145767\pi\)
−0.896965 + 0.442101i \(0.854233\pi\)
\(44\) −67968.0 −0.797896
\(45\) 19683.0 + 88973.8i 0.216000 + 0.976393i
\(46\) 0 0
\(47\) 193570.i 1.86442i 0.361914 + 0.932211i \(0.382123\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(48\) 0 0
\(49\) 19153.0 0.162798
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −28674.0 129616.i −0.172346 0.779061i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 57062.0 0.251395 0.125698 0.992069i \(-0.459883\pi\)
0.125698 + 0.992069i \(0.459883\pi\)
\(62\) 0 0
\(63\) 228790.i 0.914988i
\(64\) −262144. −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 124978.i 0.397473i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 676641.i 1.73936i −0.493616 0.869680i \(-0.664325\pi\)
0.493616 0.869680i \(-0.335675\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 438976. 1.00000
\(77\) 333299.i 0.730065i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −110592. 499913.i −0.216000 0.976393i
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 168916.i 0.295418i −0.989031 0.147709i \(-0.952810\pi\)
0.989031 0.147709i \(-0.0471899\pi\)
\(84\) 0 0
\(85\) 238336. 52725.2i 0.388090 0.0858543i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 827982.i 1.06330i
\(93\) 0 0
\(94\) 0 0
\(95\) 185193. + 837135.i 0.216000 + 0.976393i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −774198. −0.797896
\(100\) 906688. 421802.i 0.906688 0.421802i
\(101\) −2.06030e6 −1.99970 −0.999852 0.0171767i \(-0.994532\pi\)
−0.999852 + 0.0171767i \(0.994532\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.28549e6i 0.914988i
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 1.57898e6 349305.i 1.03820 0.229674i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 612864. 0.363683
\(120\) 0 0
\(121\) −643717. −0.363361
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.18689e6 + 1.55112e6i 0.607689 + 0.794175i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.30228e6 0.579283 0.289642 0.957135i \(-0.406464\pi\)
0.289642 + 0.957135i \(0.406464\pi\)
\(132\) 0 0
\(133\) 2.15263e6i 0.914988i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.63738e6i 1.80348i 0.432280 + 0.901739i \(0.357709\pi\)
−0.432280 + 0.901739i \(0.642291\pi\)
\(138\) 0 0
\(139\) −3.77334e6 −1.40502 −0.702508 0.711676i \(-0.747937\pi\)
−0.702508 + 0.711676i \(0.747937\pi\)
\(140\) 2.45146e6 542317.i 0.893388 0.197637i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.98598e6 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.24350e6 −1.88742 −0.943711 0.330770i \(-0.892692\pi\)
−0.943711 + 0.330770i \(0.892692\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.42358e6i 0.397473i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.70448e6i 1.99088i 0.0954122 + 0.995438i \(0.469583\pi\)
−0.0954122 + 0.995438i \(0.530417\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.06022e6 0.972909
\(162\) 0 0
\(163\) 7.51774e6i 1.73590i 0.496652 + 0.867950i \(0.334562\pi\)
−0.496652 + 0.867950i \(0.665438\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −4.82681e6 −1.00000
\(170\) 0 0
\(171\) 5.00021e6 1.00000
\(172\) 4.49922e6i 0.884203i
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.06842e6 + 4.44618e6i 0.385944 + 0.829608i
\(176\) 4.34995e6 0.797896
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.25971e6 5.69433e6i −0.216000 0.976393i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.07386e6i 0.317143i
\(188\) 1.23885e7i 1.86442i
\(189\) 0 0
\(190\) 0 0
\(191\) 9.37384e6 1.34529 0.672647 0.739963i \(-0.265157\pi\)
0.672647 + 0.739963i \(0.265157\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.22579e6 −0.162798
\(197\) 1.17795e7i 1.54073i −0.637603 0.770365i \(-0.720074\pi\)
0.637603 0.770365i \(-0.279926\pi\)
\(198\) 0 0
\(199\) −1.52712e7 −1.93782 −0.968911 0.247410i \(-0.920421\pi\)
−0.968911 + 0.247410i \(0.920421\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.43123e6i 1.06330i
\(208\) 0 0
\(209\) −7.28426e6 −0.797896
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.58010e6 1.89811e6i 0.863330 0.190988i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 1.83514e6 + 8.29544e6i 0.172346 + 0.779061i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 1.03277e7 4.80459e6i 0.906688 0.421802i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −2.66958e6 −0.222298 −0.111149 0.993804i \(-0.535453\pi\)
−0.111149 + 0.993804i \(0.535453\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.94104e6i 0.390616i −0.980742 0.195308i \(-0.937429\pi\)
0.980742 0.195308i \(-0.0625707\pi\)
\(234\) 0 0
\(235\) 2.36251e7 5.22639e6i 1.82041 0.402715i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.53322e7 −1.12308 −0.561541 0.827449i \(-0.689791\pi\)
−0.561541 + 0.827449i \(0.689791\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −3.65197e6 −0.251395
\(245\) −517131. 2.33761e6i −0.0351643 0.158955i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.08340e6 −0.321464 −0.160732 0.986998i \(-0.551386\pi\)
−0.160732 + 0.986998i \(0.551386\pi\)
\(252\) 1.46426e7i 0.914988i
\(253\) 1.37393e7i 0.848406i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.18370e7i 1.75011i −0.484024 0.875054i \(-0.660825\pi\)
0.484024 0.875054i \(-0.339175\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −2.84226e7 −1.42809 −0.714045 0.700100i \(-0.753139\pi\)
−0.714045 + 0.700100i \(0.753139\pi\)
\(272\) 7.99861e6i 0.397473i
\(273\) 0 0
\(274\) 0 0
\(275\) −1.50454e7 + 6.99928e6i −0.723443 + 0.336554i
\(276\) 0 0
\(277\) 3.01347e7i 1.41784i 0.705289 + 0.708920i \(0.250817\pi\)
−0.705289 + 0.708920i \(0.749183\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 3.07828e7i 1.35815i 0.734068 + 0.679076i \(0.237619\pi\)
−0.734068 + 0.679076i \(0.762381\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.03242e7 0.842015
\(290\) 0 0
\(291\) 0 0
\(292\) 4.33050e7i 1.73936i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.20631e7 0.809035
\(302\) 0 0
\(303\) 0 0
\(304\) −2.80945e7 −1.00000
\(305\) −1.54067e6 6.96437e6i −0.0543014 0.245461i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 2.13311e7i 0.730065i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.42879e7 −1.47233 −0.736164 0.676804i \(-0.763365\pi\)
−0.736164 + 0.676804i \(0.763365\pi\)
\(312\) 0 0
\(313\) 5.23342e7i 1.70668i 0.521353 + 0.853341i \(0.325427\pi\)
−0.521353 + 0.853341i \(0.674573\pi\)
\(314\) 0 0
\(315\) 2.79236e7 6.17733e6i 0.893388 0.197637i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.07789e6 + 3.19945e7i 0.216000 + 0.976393i
\(321\) 0 0
\(322\) 0 0
\(323\) 1.33942e7i 0.397473i
\(324\) −3.40122e7 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.07501e7 1.70592
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.08106e7i 0.295418i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.52535e7 + 3.37442e6i −0.388090 + 0.0858543i
\(341\) 0 0
\(342\) 0 0
\(343\) 4.29340e7i 1.06395i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.40634e7i 1.29394i −0.762515 0.646971i \(-0.776035\pi\)
0.762515 0.646971i \(-0.223965\pi\)
\(348\) 0 0
\(349\) −4.62042e7 −1.08694 −0.543469 0.839429i \(-0.682890\pi\)
−0.543469 + 0.839429i \(0.682890\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.61463e7i 1.50377i −0.659295 0.751885i \(-0.729145\pi\)
0.659295 0.751885i \(-0.270855\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.96053e7 1.72052 0.860258 0.509858i \(-0.170302\pi\)
0.860258 + 0.509858i \(0.170302\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.25834e7 + 1.82693e7i −1.69830 + 0.375702i
\(366\) 0 0
\(367\) 9.68013e7i 1.95832i −0.203095 0.979159i \(-0.565100\pi\)
0.203095 0.979159i \(-0.434900\pi\)
\(368\) 5.29908e7i 1.06330i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −1.18524e7 5.35767e7i −0.216000 0.976393i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −4.06788e7 + 8.99907e6i −0.712831 + 0.157694i
\(386\) 0 0
\(387\) 5.12489e7i 0.884203i
\(388\) 0 0
\(389\) 6.58748e7 1.11910 0.559552 0.828795i \(-0.310973\pi\)
0.559552 + 0.828795i \(0.310973\pi\)
\(390\) 0 0
\(391\) −2.52636e7 −0.422635
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 4.95487e7 0.797896
\(397\) 1.09135e8i 1.74418i 0.489346 + 0.872090i \(0.337236\pi\)
−0.489346 + 0.872090i \(0.662764\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.80280e7 + 2.69953e7i −0.906688 + 0.421802i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.31859e8 1.99970
\(405\) −1.43489e7 6.48619e7i −0.216000 0.976393i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.06161e7 + 4.56073e6i −0.288444 + 0.0638102i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.11183e7 0.558976 0.279488 0.960149i \(-0.409835\pi\)
0.279488 + 0.960149i \(0.409835\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 1.41113e8i 1.86442i
\(424\) 0 0
\(425\) −1.28701e7 2.76651e7i −0.167655 0.360384i
\(426\) 0 0
\(427\) 1.79084e7i 0.230024i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.87363e7i 1.06330i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.39625e7 −0.162798
\(442\) 0 0
\(443\) 1.71860e8i 1.97681i −0.151846 0.988404i \(-0.548522\pi\)
0.151846 0.988404i \(-0.451478\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 8.22715e7i 0.914988i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.21229e8i 1.27015i 0.772449 + 0.635077i \(0.219032\pi\)
−0.772449 + 0.635077i \(0.780968\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.01054e8 + 2.23555e7i −1.03820 + 0.229674i
\(461\) −1.95676e8 −1.99726 −0.998631 0.0523154i \(-0.983340\pi\)
−0.998631 + 0.0523154i \(0.983340\pi\)
\(462\) 0 0
\(463\) 1.90683e8i 1.92118i 0.277961 + 0.960592i \(0.410341\pi\)
−0.277961 + 0.960592i \(0.589659\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.16046e8i 1.13941i 0.821849 + 0.569706i \(0.192943\pi\)
−0.821849 + 0.569706i \(0.807057\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.46589e7i 0.705502i
\(474\) 0 0
\(475\) 9.71715e7 4.52053e7i 0.906688 0.421802i
\(476\) −3.92233e7 −0.363683
\(477\) 0 0
\(478\) 0 0
\(479\) 1.87497e8 1.70603 0.853015 0.521886i \(-0.174771\pi\)
0.853015 + 0.521886i \(0.174771\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 4.11979e7 0.363361
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.09684e8 −0.926610 −0.463305 0.886199i \(-0.653337\pi\)
−0.463305 + 0.886199i \(0.653337\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 2.09033e7 + 9.44902e7i 0.172346 + 0.779061i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.47627e8 1.99295 0.996475 0.0838961i \(-0.0267364\pi\)
0.996475 + 0.0838961i \(0.0267364\pi\)
\(500\) −7.59612e7 9.92719e7i −0.607689 0.794175i
\(501\) 0 0
\(502\) 0 0
\(503\) 2.34711e8i 1.84429i 0.386843 + 0.922146i \(0.373566\pi\)
−0.386843 + 0.922146i \(0.626434\pi\)
\(504\) 0 0
\(505\) 5.56280e7 + 2.51458e8i 0.431936 + 1.95250i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.12357e8 −1.59149
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.05571e8i 1.48762i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) −8.33460e7 −0.579283
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.93356e7 −0.130614
\(530\) 0 0
\(531\) 0 0
\(532\) 1.37769e8i 0.914988i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.03405e7 0.129896
\(540\) 0 0
\(541\) −3.05822e8 −1.93142 −0.965709 0.259626i \(-0.916401\pi\)
−0.965709 + 0.259626i \(0.916401\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 2.96792e8i 1.80348i
\(549\) −4.15982e7 −0.251395
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 2.41494e8 1.40502
\(557\) 1.82082e8i 1.05366i −0.849970 0.526831i \(-0.823380\pi\)
0.849970 0.526831i \(-0.176620\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.56893e8 + 3.47083e7i −0.893388 + 0.197637i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.66788e8i 0.914988i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 3.51908e8 1.89026 0.945130 0.326696i \(-0.105935\pi\)
0.945130 + 0.326696i \(0.105935\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.52647e7 1.83281e8i −0.448503 0.964084i
\(576\) 1.91103e8 1.00000
\(577\) 1.40693e8i 0.732394i 0.930537 + 0.366197i \(0.119340\pi\)
−0.930537 + 0.366197i \(0.880660\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.30127e7 −0.270304
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.09139e8i 1.52841i −0.644975 0.764204i \(-0.723132\pi\)
0.644975 0.764204i \(-0.276868\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.15855e8i 1.99424i 0.0758206 + 0.997121i \(0.475842\pi\)
−0.0758206 + 0.997121i \(0.524158\pi\)
\(594\) 0 0
\(595\) −1.65473e7 7.47995e7i −0.0785556 0.355098i
\(596\) 3.99584e8 1.88742
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.73804e7 + 7.85651e7i 0.0784861 + 0.354784i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 9.11092e7i 0.397473i
\(613\) 3.43599e8i 1.49166i −0.666134 0.745832i \(-0.732052\pi\)
0.666134 0.745832i \(-0.267948\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.69696e8i 1.99968i 0.0177783 + 0.999842i \(0.494341\pi\)
−0.0177783 + 0.999842i \(0.505659\pi\)
\(618\) 0 0
\(619\) −4.70840e8 −1.98519 −0.992594 0.121480i \(-0.961236\pi\)
−0.992594 + 0.121480i \(0.961236\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.57267e8 1.86740e8i 0.644166 0.764886i
\(626\) 0 0
\(627\) 0 0
\(628\) 4.93086e8i 1.99088i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.23521e8 −0.491647 −0.245824 0.969315i \(-0.579058\pi\)
−0.245824 + 0.969315i \(0.579058\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 5.31686e8i 1.99996i −0.00603535 0.999982i \(-0.501921\pi\)
0.00603535 0.999982i \(-0.498079\pi\)
\(644\) −2.59854e8 −0.972909
\(645\) 0 0
\(646\) 0 0
\(647\) 4.82072e8i 1.77991i −0.456046 0.889956i \(-0.650735\pi\)
0.456046 0.889956i \(-0.349265\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.81135e8i 1.73590i
\(653\) 3.57517e8i 1.28398i 0.766714 + 0.641988i \(0.221890\pi\)
−0.766714 + 0.641988i \(0.778110\pi\)
\(654\) 0 0
\(655\) −3.51616e7 1.58942e8i −0.125125 0.565609i
\(656\) 0 0
\(657\) 4.93271e8i 1.73936i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.62727e8 5.81211e7i 0.893388 0.197637i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.05998e7 0.200588
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 3.08916e8 1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −3.20014e8 −1.00000
\(685\) 5.65988e8 1.25209e8i 1.76090 0.389551i
\(686\) 0 0
\(687\) 0 0
\(688\) 2.87950e8i 0.884203i
\(689\) 0 0
\(690\) 0 0
\(691\) −2.21024e8 −0.669892 −0.334946 0.942237i \(-0.608718\pi\)
−0.334946 + 0.942237i \(0.608718\pi\)
\(692\) 0 0
\(693\) 2.42975e8i 0.730065i
\(694\) 0 0
\(695\) 1.01880e8 + 4.60533e8i 0.303483 + 1.37185i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.32379e8 2.84556e8i −0.385944 0.829608i
\(701\) 2.94922e8 0.856156 0.428078 0.903742i \(-0.359191\pi\)
0.428078 + 0.903742i \(0.359191\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.78397e8 −0.797896
\(705\) 0 0
\(706\) 0 0
\(707\) 6.46605e8i 1.82971i
\(708\) 0 0
\(709\) −3.01929e8 −0.847161 −0.423580 0.905859i \(-0.639227\pi\)
−0.423580 + 0.905859i \(0.639227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00444e8 1.61542 0.807711 0.589579i \(-0.200706\pi\)
0.807711 + 0.589579i \(0.200706\pi\)
\(720\) 8.06216e7 + 3.64437e8i 0.216000 + 0.976393i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.56680e8i 1.96929i −0.174579 0.984643i \(-0.555856\pi\)
0.174579 0.984643i \(-0.444144\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −1.00000
\(730\) 0 0
\(731\) −1.37282e8 −0.351447
\(732\) 0 0
\(733\) 7.87665e8i 2.00000i 0.00166557 + 0.999999i \(0.499470\pi\)
−0.00166557 + 0.999999i \(0.500530\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −7.32517e8 −1.81503 −0.907517 0.420016i \(-0.862024\pi\)
−0.907517 + 0.420016i \(0.862024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 1.68574e8 + 7.62014e8i 0.407683 + 1.84287i
\(746\) 0 0
\(747\) 1.23140e8i 0.295418i
\(748\) 1.32727e8i 0.317143i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 7.92863e8i 1.86442i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.58953e7i 0.128851i −0.997923 0.0644255i \(-0.979479\pi\)
0.997923 0.0644255i \(-0.0205215\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.84034e8 −1.77902 −0.889510 0.456915i \(-0.848954\pi\)
−0.889510 + 0.456915i \(0.848954\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.99926e8 −1.34529
\(765\) −1.73747e8 + 3.84367e7i −0.388090 + 0.0858543i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.84693e8 1.50563 0.752813 0.658235i \(-0.228697\pi\)
0.752813 + 0.658235i \(0.228697\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 7.84507e7 0.162798
\(785\) 9.40325e8 2.08021e8i 1.94388 0.430029i
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 7.53885e8i 1.54073i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 9.77357e8 1.93782
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −3.78001e8 −0.741058
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.18592e8i 1.38783i
\(804\) 0 0
\(805\) −1.09626e8 4.95547e8i −0.210148 0.949942i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.14710e8 −1.72758 −0.863790 0.503853i \(-0.831915\pi\)
−0.863790 + 0.503853i \(0.831915\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.17534e8 2.02979e8i 1.69492 0.374954i
\(816\) 0 0
\(817\) 4.82190e8i 0.884203i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.70493e8 −1.39232 −0.696160 0.717886i \(-0.745110\pi\)
−0.696160 + 0.717886i \(0.745110\pi\)
\(822\) 0 0
\(823\) 9.03651e8i 1.62107i −0.585691 0.810534i \(-0.699177\pi\)
0.585691 0.810534i \(-0.300823\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 6.03599e8i 1.06330i
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.74017e7i 0.0647078i
\(834\) 0 0
\(835\) 0 0
\(836\) 4.66193e8 0.797896
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.30324e8 + 5.89108e8i 0.216000 + 0.976393i
\(846\) 0 0
\(847\) 2.02025e8i 0.332471i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.53268e8i 0.730311i −0.930947 0.365156i \(-0.881016\pi\)
0.930947 0.365156i \(-0.118984\pi\)
\(854\) 0 0
\(855\) −1.35006e8 6.10272e8i −0.216000 0.976393i
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −2.30012e7 −0.0362886 −0.0181443 0.999835i \(-0.505776\pi\)
−0.0181443 + 0.999835i \(0.505776\pi\)
\(860\) −5.49126e8 + 1.21479e8i −0.863330 + 0.190988i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.86806e8 3.72495e8i 0.726660 0.556028i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.17449e8 5.30908e8i −0.172346 0.779061i
\(881\) −1.09554e9 −1.60214 −0.801071 0.598569i \(-0.795736\pi\)
−0.801071 + 0.598569i \(0.795736\pi\)
\(882\) 0 0
\(883\) 1.28324e8i 0.186392i 0.995648 + 0.0931960i \(0.0297083\pi\)
−0.995648 + 0.0931960i \(0.970292\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.64390e8 0.797896
\(892\) 0 0
\(893\) 1.32770e9i 1.86442i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −6.60976e8 + 3.07494e8i −0.906688 + 0.421802i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 1.50196e9 1.99970
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.79389e8i 0.235713i
\(914\) 0 0
\(915\) 0 0
\(916\) 1.70853e8 0.222298
\(917\) 4.08709e8i 0.530037i
\(918\) 0 0
\(919\) 1.00603e9 1.29618 0.648091 0.761563i \(-0.275568\pi\)
0.648091 + 0.761563i \(0.275568\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.39761e9 −1.74316 −0.871582 0.490250i \(-0.836905\pi\)
−0.871582 + 0.490250i \(0.836905\pi\)
\(930\) 0 0
\(931\) −1.31370e8 −0.162798
\(932\) 3.16226e8i 0.390616i
\(933\) 0 0
\(934\) 0 0
\(935\) 2.53113e8 5.59942e7i 0.309656 0.0685028i
\(936\) 0 0
\(937\) 1.41189e9i 1.71626i −0.513436 0.858128i \(-0.671628\pi\)
0.513436 0.858128i \(-0.328372\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.51200e9 + 3.34489e8i −1.82041 + 0.402715i
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.19710e9i 1.40955i −0.709429 0.704777i \(-0.751047\pi\)
0.709429 0.704777i \(-0.248953\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −2.53094e8 1.14407e9i −0.290584 1.31354i
\(956\) 9.81263e8 1.12308
\(957\) 0 0
\(958\) 0 0
\(959\) 1.45540e9 1.65016
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.66159e9i 1.83757i 0.394761 + 0.918784i \(0.370827\pi\)
−0.394761 + 0.918784i \(0.629173\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1.18423e9i 1.28557i
\(974\) 0 0
\(975\) 0 0
\(976\) 2.33726e8 0.251395
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.30964e7 + 1.49607e8i 0.0351643 + 0.158955i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −1.43767e9 + 3.18045e8i −1.50436 + 0.332798i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.09490e8 −0.940176
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.12322e8 + 1.86384e9i 0.418570 + 1.89208i
\(996\) 0 0
\(997\) 7.98188e8i 0.805415i 0.915329 + 0.402707i \(0.131931\pi\)
−0.915329 + 0.402707i \(0.868069\pi\)
\(998\) 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 95.7.d.a.94.1 2
5.4 even 2 inner 95.7.d.a.94.2 yes 2
19.18 odd 2 CM 95.7.d.a.94.1 2
95.94 odd 2 inner 95.7.d.a.94.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.7.d.a.94.1 2 1.1 even 1 trivial
95.7.d.a.94.1 2 19.18 odd 2 CM
95.7.d.a.94.2 yes 2 5.4 even 2 inner
95.7.d.a.94.2 yes 2 95.94 odd 2 inner