Properties

Label 75.12.e.a.32.1
Level $75$
Weight $12$
Character 75.32
Analytic conductor $57.626$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,12,Mod(32,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.32");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), 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: \(57.6257385420\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 32.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.32
Dual form 75.12.e.a.68.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-297.613 + 297.613i) q^{3} +2048.00i q^{4} +(31607.0 + 31607.0i) q^{7} -177147. i q^{9} +O(q^{10})\) \(q+(-297.613 + 297.613i) q^{3} +2048.00i q^{4} +(31607.0 + 31607.0i) q^{7} -177147. i q^{9} +(-609511. - 609511. i) q^{12} +(-1.02771e6 + 1.02771e6i) q^{13} -4.19430e6 q^{16} +2.05819e7i q^{19} -1.88133e7 q^{21} +(5.27213e7 + 5.27213e7i) q^{27} +(-6.47311e7 + 6.47311e7i) q^{28} -2.96477e8 q^{31} +3.62797e8 q^{36} +(2.22157e8 + 2.22157e8i) q^{37} -6.11719e8i q^{39} +(5.43178e8 - 5.43178e8i) q^{43} +(1.24828e9 - 1.24828e9i) q^{48} +2.06770e7i q^{49} +(-2.10475e9 - 2.10475e9i) q^{52} +(-6.12544e9 - 6.12544e9i) q^{57} -1.29773e10 q^{61} +(5.59908e9 - 5.59908e9i) q^{63} -8.58993e9i q^{64} +(1.50516e10 + 1.50516e10i) q^{67} +(2.07683e10 - 2.07683e10i) q^{73} -4.21517e10 q^{76} -3.28858e10i q^{79} -3.13811e10 q^{81} -3.85296e10i q^{84} -6.49656e10 q^{91} +(8.82354e10 - 8.82354e10i) q^{93} +(-1.13592e11 - 1.13592e11i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16777216 q^{16} - 75253212 q^{21} - 1185907772 q^{31} + 1451188224 q^{36} - 51909171652 q^{61} - 168606932992 q^{76} - 125524238436 q^{81} - 259862347764 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −297.613 + 297.613i −0.707107 + 0.707107i
\(4\) 2048.00i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 31607.0 + 31607.0i 0.710794 + 0.710794i 0.966701 0.255907i \(-0.0823741\pi\)
−0.255907 + 0.966701i \(0.582374\pi\)
\(8\) 0 0
\(9\) 177147.i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −609511. 609511.i −0.707107 0.707107i
\(13\) −1.02771e6 + 1.02771e6i −0.767683 + 0.767683i −0.977698 0.210015i \(-0.932649\pi\)
0.210015 + 0.977698i \(0.432649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.19430e6 −1.00000
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 2.05819e7i 1.90696i 0.301463 + 0.953478i \(0.402525\pi\)
−0.301463 + 0.953478i \(0.597475\pi\)
\(20\) 0 0
\(21\) −1.88133e7 −1.00521
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.27213e7 + 5.27213e7i 0.707107 + 0.707107i
\(28\) −6.47311e7 + 6.47311e7i −0.710794 + 0.710794i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.96477e8 −1.85995 −0.929976 0.367621i \(-0.880172\pi\)
−0.929976 + 0.367621i \(0.880172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.62797e8 1.00000
\(37\) 2.22157e8 + 2.22157e8i 0.526684 + 0.526684i 0.919582 0.392898i \(-0.128528\pi\)
−0.392898 + 0.919582i \(0.628528\pi\)
\(38\) 0 0
\(39\) 6.11719e8i 1.08567i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.43178e8 5.43178e8i 0.563463 0.563463i −0.366826 0.930289i \(-0.619556\pi\)
0.930289 + 0.366826i \(0.119556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 1.24828e9 1.24828e9i 0.707107 0.707107i
\(49\) 2.06770e7i 0.0104570i
\(50\) 0 0
\(51\) 0 0
\(52\) −2.10475e9 2.10475e9i −0.767683 0.767683i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.12544e9 6.12544e9i −1.34842 1.34842i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.29773e10 −1.96730 −0.983649 0.180098i \(-0.942359\pi\)
−0.983649 + 0.180098i \(0.942359\pi\)
\(62\) 0 0
\(63\) 5.59908e9 5.59908e9i 0.710794 0.710794i
\(64\) 8.58993e9i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.50516e10 + 1.50516e10i 1.36198 + 1.36198i 0.871387 + 0.490597i \(0.163221\pi\)
0.490597 + 0.871387i \(0.336779\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.07683e10 2.07683e10i 1.17254 1.17254i 0.190933 0.981603i \(-0.438849\pi\)
0.981603 0.190933i \(-0.0611512\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.21517e10 −1.90696
\(77\) 0 0
\(78\) 0 0
\(79\) 3.28858e10i 1.20243i −0.799087 0.601215i \(-0.794684\pi\)
0.799087 0.601215i \(-0.205316\pi\)
\(80\) 0 0
\(81\) −3.13811e10 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 3.85296e10i 1.00521i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −6.49656e10 −1.09133
\(92\) 0 0
\(93\) 8.82354e10 8.82354e10i 1.31518 1.31518i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.13592e11 1.13592e11i −1.34309 1.34309i −0.892969 0.450117i \(-0.851382\pi\)
−0.450117 0.892969i \(-0.648618\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.58162e11 1.58162e11i 1.34430 1.34430i 0.452579 0.891724i \(-0.350504\pi\)
0.891724 0.452579i \(-0.149496\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) −1.07973e11 + 1.07973e11i −0.707107 + 0.707107i
\(109\) 2.57951e9i 0.0160580i −0.999968 0.00802899i \(-0.997444\pi\)
0.999968 0.00802899i \(-0.00255573\pi\)
\(110\) 0 0
\(111\) −1.32233e11 −0.744843
\(112\) −1.32569e11 1.32569e11i −0.710794 0.710794i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.82056e11 + 1.82056e11i 0.767683 + 0.767683i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.85312e11 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 6.07185e11i 1.85995i
\(125\) 0 0
\(126\) 0 0
\(127\) 4.46400e11 + 4.46400e11i 1.19896 + 1.19896i 0.974479 + 0.224478i \(0.0720676\pi\)
0.224478 + 0.974479i \(0.427932\pi\)
\(128\) 0 0
\(129\) 3.23314e11i 0.796857i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −6.50532e11 + 6.50532e11i −1.35545 + 1.35545i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 1.14947e12i 1.87896i 0.342607 + 0.939479i \(0.388690\pi\)
−0.342607 + 0.939479i \(0.611310\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 7.43008e11i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) −6.15375e9 6.15375e9i −0.00739425 0.00739425i
\(148\) −4.54977e11 + 4.54977e11i −0.526684 + 0.526684i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 1.31670e12 1.36494 0.682470 0.730913i \(-0.260906\pi\)
0.682470 + 0.730913i \(0.260906\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.25280e12 1.08567
\(157\) −1.16557e12 1.16557e12i −0.975190 0.975190i 0.0245092 0.999700i \(-0.492198\pi\)
−0.999700 + 0.0245092i \(0.992198\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.61096e12 + 1.61096e12i −1.09661 + 1.09661i −0.101806 + 0.994804i \(0.532462\pi\)
−0.994804 + 0.101806i \(0.967538\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 3.20212e11i 0.178674i
\(170\) 0 0
\(171\) 3.64602e12 1.90696
\(172\) 1.11243e12 + 1.11243e12i 0.563463 + 0.563463i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −1.27601e12 −0.488228 −0.244114 0.969747i \(-0.578497\pi\)
−0.244114 + 0.969747i \(0.578497\pi\)
\(182\) 0 0
\(183\) 3.86221e12 3.86221e12i 1.39109 1.39109i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.33272e12i 1.00521i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2.55648e12 + 2.55648e12i 0.707107 + 0.707107i
\(193\) −5.09139e12 + 5.09139e12i −1.36858 + 1.36858i −0.506123 + 0.862461i \(0.668922\pi\)
−0.862461 + 0.506123i \(0.831078\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.23465e10 −0.0104570
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 8.39365e12i 1.90660i −0.302028 0.953299i \(-0.597664\pi\)
0.302028 0.953299i \(-0.402336\pi\)
\(200\) 0 0
\(201\) −8.95912e12 −1.92614
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 4.31052e12 4.31052e12i 0.767683 0.767683i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.12332e13 −1.84906 −0.924531 0.381106i \(-0.875543\pi\)
−0.924531 + 0.381106i \(0.875543\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.37074e12 9.37074e12i −1.32204 1.32204i
\(218\) 0 0
\(219\) 1.23619e13i 1.65822i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.21242e12 3.21242e12i 0.390081 0.390081i −0.484635 0.874716i \(-0.661048\pi\)
0.874716 + 0.484635i \(0.161048\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 1.25449e13 1.25449e13i 1.34842 1.34842i
\(229\) 1.90357e13i 1.99744i −0.0505740 0.998720i \(-0.516105\pi\)
0.0505740 0.998720i \(-0.483895\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.78725e12 + 9.78725e12i 0.850246 + 0.850246i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −2.27780e12 −0.180477 −0.0902386 0.995920i \(-0.528763\pi\)
−0.0902386 + 0.995920i \(0.528763\pi\)
\(242\) 0 0
\(243\) 9.33941e12 9.33941e12i 0.707107 0.707107i
\(244\) 2.65775e13i 1.96730i
\(245\) 0 0
\(246\) 0 0
\(247\) −2.11522e13 2.11522e13i −1.46394 1.46394i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.14669e13 + 1.14669e13i 0.710794 + 0.710794i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.75922e13 1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 1.40434e13i 0.748728i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3.08257e13 + 3.08257e13i −1.36198 + 1.36198i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −4.72629e13 −1.96422 −0.982109 0.188315i \(-0.939697\pi\)
−0.982109 + 0.188315i \(0.939697\pi\)
\(272\) 0 0
\(273\) 1.93346e13 1.93346e13i 0.771686 0.771686i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.36566e12 + 3.36566e12i 0.124003 + 0.124003i 0.766385 0.642382i \(-0.222054\pi\)
−0.642382 + 0.766385i \(0.722054\pi\)
\(278\) 0 0
\(279\) 5.25200e13i 1.85995i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −4.22277e13 + 4.22277e13i −1.38284 + 1.38284i −0.543302 + 0.839537i \(0.682826\pi\)
−0.839537 + 0.543302i \(0.817174\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.42719e13i 1.00000i
\(290\) 0 0
\(291\) 6.76130e13 1.89941
\(292\) 4.25336e13 + 4.25336e13i 1.17254 + 1.17254i
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.43364e13 0.801013
\(302\) 0 0
\(303\) 0 0
\(304\) 8.63267e13i 1.90696i
\(305\) 0 0
\(306\) 0 0
\(307\) −3.49686e13 3.49686e13i −0.731841 0.731841i 0.239143 0.970984i \(-0.423134\pi\)
−0.970984 + 0.239143i \(0.923134\pi\)
\(308\) 0 0
\(309\) 9.41421e13i 1.90113i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −7.50682e13 + 7.50682e13i −1.41242 + 1.41242i −0.670557 + 0.741858i \(0.733945\pi\)
−0.741858 + 0.670557i \(0.766055\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 6.73502e13 1.20243
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 6.42684e13i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) 7.67694e11 + 7.67694e11i 0.0113547 + 0.0113547i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.80000e13 1.35573 0.677863 0.735188i \(-0.262906\pi\)
0.677863 + 0.735188i \(0.262906\pi\)
\(332\) 0 0
\(333\) 3.93544e13 3.93544e13i 0.526684 0.526684i
\(334\) 0 0
\(335\) 0 0
\(336\) 7.89087e13 1.00521
\(337\) 7.67948e12 + 7.67948e12i 0.0962426 + 0.0962426i 0.753589 0.657346i \(-0.228321\pi\)
−0.657346 + 0.753589i \(0.728321\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.18438e13 6.18438e13i 0.703361 0.703361i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 1.70909e14i 1.76695i 0.468478 + 0.883475i \(0.344803\pi\)
−0.468478 + 0.883475i \(0.655197\pi\)
\(350\) 0 0
\(351\) −1.08364e14 −1.08567
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.07124e14 −2.63648
\(362\) 0 0
\(363\) −8.49125e13 + 8.49125e13i −0.707107 + 0.707107i
\(364\) 1.33050e14i 1.09133i
\(365\) 0 0
\(366\) 0 0
\(367\) −1.79414e14 1.79414e14i −1.40668 1.40668i −0.776245 0.630431i \(-0.782878\pi\)
−0.630431 0.776245i \(-0.717122\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.80706e14 + 1.80706e14i 1.31518 + 1.31518i
\(373\) −2.17763e13 + 2.17763e13i −0.156166 + 0.156166i −0.780865 0.624699i \(-0.785221\pi\)
0.624699 + 0.780865i \(0.285221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.50919e13i 0.361886i −0.983494 0.180943i \(-0.942085\pi\)
0.983494 0.180943i \(-0.0579150\pi\)
\(380\) 0 0
\(381\) −2.65709e14 −1.69558
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.62223e13 9.62223e13i −0.563463 0.563463i
\(388\) 2.32637e14 2.32637e14i 1.34309 1.34309i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.99277e14 + 1.99277e14i 1.01416 + 1.01416i 0.999898 + 0.0142662i \(0.00454122\pi\)
0.0142662 + 0.999898i \(0.495459\pi\)
\(398\) 0 0
\(399\) 3.87214e14i 1.91690i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 3.04692e14 3.04692e14i 1.42785 1.42785i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.33592e13i 0.360143i −0.983654 0.180072i \(-0.942367\pi\)
0.983654 0.180072i \(-0.0576329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.23916e14 + 3.23916e14i 1.34430 + 1.34430i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.42098e14 3.42098e14i −1.32862 1.32862i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 3.20990e13 0.118288 0.0591438 0.998249i \(-0.481163\pi\)
0.0591438 + 0.998249i \(0.481163\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.10173e14 4.10173e14i −1.39834 1.39834i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −2.21129e14 2.21129e14i −0.707107 0.707107i
\(433\) −2.34313e12 + 2.34313e12i −0.00739796 + 0.00739796i −0.710796 0.703398i \(-0.751665\pi\)
0.703398 + 0.710796i \(0.251665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.28283e12 0.0160580
\(437\) 0 0
\(438\) 0 0
\(439\) 4.50030e14i 1.31731i 0.752446 + 0.658654i \(0.228874\pi\)
−0.752446 + 0.658654i \(0.771126\pi\)
\(440\) 0 0
\(441\) 3.66287e12 0.0104570
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 2.70814e14i 0.744843i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 2.71502e14 2.71502e14i 0.710794 0.710794i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −3.91867e14 + 3.91867e14i −0.965159 + 0.965159i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.15408e14 + 5.15408e14i 1.20952 + 1.20952i 0.971183 + 0.238335i \(0.0766017\pi\)
0.238335 + 0.971183i \(0.423398\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −3.67905e14 + 3.67905e14i −0.803600 + 0.803600i −0.983656 0.180057i \(-0.942372\pi\)
0.180057 + 0.983656i \(0.442372\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) −3.72850e14 + 3.72850e14i −0.767683 + 0.767683i
\(469\) 9.51473e14i 1.93618i
\(470\) 0 0
\(471\) 6.93776e14 1.37913
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −4.56625e14 −0.808652
\(482\) 0 0
\(483\) 0 0
\(484\) 5.84318e14i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 8.05794e14 + 8.05794e14i 1.33295 + 1.33295i 0.902716 + 0.430236i \(0.141570\pi\)
0.430236 + 0.902716i \(0.358430\pi\)
\(488\) 0 0
\(489\) 9.58884e14i 1.55084i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.24351e15 1.85995
\(497\) 0 0
\(498\) 0 0
\(499\) 5.44758e14i 0.788225i 0.919062 + 0.394113i \(0.128948\pi\)
−0.919062 + 0.394113i \(0.871052\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.52992e13 + 9.52992e13i 0.126341 + 0.126341i
\(508\) −9.14227e14 + 9.14227e14i −1.19896 + 1.19896i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 1.31285e15 1.66686
\(512\) 0 0
\(513\) −1.08510e15 + 1.08510e15i −1.34842 + 1.34842i
\(514\) 0 0
\(515\) 0 0
\(516\) −6.62146e14 −0.796857
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.24928e15 1.24928e15i 1.39605 1.39605i 0.585046 0.811000i \(-0.301077\pi\)
0.811000 0.585046i \(-0.198923\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 9.52810e14i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) −1.33229e15 1.33229e15i −1.35545 1.35545i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.78270e15 1.65384 0.826921 0.562319i \(-0.190091\pi\)
0.826921 + 0.562319i \(0.190091\pi\)
\(542\) 0 0
\(543\) 3.79758e14 3.79758e14i 0.345229 0.345229i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.60712e14 4.60712e14i −0.402253 0.402253i 0.476773 0.879026i \(-0.341806\pi\)
−0.879026 + 0.476773i \(0.841806\pi\)
\(548\) 0 0
\(549\) 2.29889e15i 1.96730i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.03942e15 1.03942e15i 0.854680 0.854680i
\(554\) 0 0
\(555\) 0 0
\(556\) −2.35412e15 −1.87896
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 1.11646e15i 0.865122i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −9.91861e14 9.91861e14i −0.710794 0.710794i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −1.21874e15 −0.840261 −0.420131 0.907464i \(-0.638016\pi\)
−0.420131 + 0.907464i \(0.638016\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −1.52168e15 −1.00000
\(577\) 1.95934e15 + 1.95934e15i 1.27539 + 1.27539i 0.943220 + 0.332169i \(0.107780\pi\)
0.332169 + 0.943220i \(0.392220\pi\)
\(578\) 0 0
\(579\) 3.03053e15i 1.93547i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 1.26029e13 1.26029e13i 0.00739425 0.00739425i
\(589\) 6.10206e15i 3.54685i
\(590\) 0 0
\(591\) 0 0
\(592\) −9.31793e14 9.31793e14i −0.526684 0.526684i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.49806e15 + 2.49806e15i 1.34817 + 1.34817i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −3.04058e15 −1.58178 −0.790892 0.611956i \(-0.790383\pi\)
−0.790892 + 0.611956i \(0.790383\pi\)
\(602\) 0 0
\(603\) 2.66635e15 2.66635e15i 1.36198 1.36198i
\(604\) 2.69660e15i 1.36494i
\(605\) 0 0
\(606\) 0 0
\(607\) −3.29288e14 3.29288e14i −0.162195 0.162195i 0.621343 0.783539i \(-0.286587\pi\)
−0.783539 + 0.621343i \(0.786587\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.87256e15 + 1.87256e15i −0.873784 + 0.873784i −0.992882 0.119099i \(-0.962000\pi\)
0.119099 + 0.992882i \(0.462000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 1.56531e15i 0.692312i −0.938177 0.346156i \(-0.887487\pi\)
0.938177 0.346156i \(-0.112513\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.56574e15i 1.08567i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 2.38708e15 2.38708e15i 0.975190 0.975190i
\(629\) 0 0
\(630\) 0 0
\(631\) −4.79425e15 −1.90791 −0.953957 0.299943i \(-0.903032\pi\)
−0.953957 + 0.299943i \(0.903032\pi\)
\(632\) 0 0
\(633\) 3.34316e15 3.34316e15i 1.30748 1.30748i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.12499e13 2.12499e13i −0.00802770 0.00802770i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −3.31582e15 + 3.31582e15i −1.18968 + 1.18968i −0.212524 + 0.977156i \(0.568168\pi\)
−0.977156 + 0.212524i \(0.931832\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5.57771e15 1.86965
\(652\) −3.29924e15 3.29924e15i −1.09661 1.09661i
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.67905e15 3.67905e15i −1.17254 1.17254i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.77508e15 0.547155 0.273578 0.961850i \(-0.411793\pi\)
0.273578 + 0.961850i \(0.411793\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.91211e15i 0.551658i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.46023e14 + 8.46023e14i −0.236211 + 0.236211i −0.815279 0.579068i \(-0.803416\pi\)
0.579068 + 0.815279i \(0.303416\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.55794e14 0.178674
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 7.18062e15i 1.90932i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 7.46705e15i 1.90696i
\(685\) 0 0
\(686\) 0 0
\(687\) 5.66527e15 + 5.66527e15i 1.41240 + 1.41240i
\(688\) −2.27825e15 + 2.27825e15i −0.563463 + 0.563463i
\(689\) 0 0
\(690\) 0 0
\(691\) 8.23279e15 1.98801 0.994003 0.109351i \(-0.0348773\pi\)
0.994003 + 0.109351i \(0.0348773\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −4.57241e15 + 4.57241e15i −1.00436 + 1.00436i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.99833e15i 1.04778i 0.851785 + 0.523891i \(0.175520\pi\)
−0.851785 + 0.523891i \(0.824480\pi\)
\(710\) 0 0
\(711\) −5.82563e15 −1.20243
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 9.99805e15 1.91105
\(722\) 0 0
\(723\) 6.77904e14 6.77904e14i 0.127617 0.127617i
\(724\) 2.61327e15i 0.488228i
\(725\) 0 0
\(726\) 0 0
\(727\) 4.04370e15 + 4.04370e15i 0.738481 + 0.738481i 0.972284 0.233803i \(-0.0751171\pi\)
−0.233803 + 0.972284i \(0.575117\pi\)
\(728\) 0 0
\(729\) 5.55906e15i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 7.90981e15 + 7.90981e15i 1.39109 + 1.39109i
\(733\) −7.52748e15 + 7.52748e15i −1.31395 + 1.31395i −0.395466 + 0.918481i \(0.629417\pi\)
−0.918481 + 0.395466i \(0.870583\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.72182e15i 0.287372i −0.989623 0.143686i \(-0.954105\pi\)
0.989623 0.143686i \(-0.0458955\pi\)
\(740\) 0 0
\(741\) 1.25903e16 2.07032
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.20840e16 −1.84582 −0.922911 0.385013i \(-0.874197\pi\)
−0.922911 + 0.385013i \(0.874197\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −6.82541e15 −1.00521
\(757\) −3.28303e15 3.28303e15i −0.480006 0.480006i 0.425127 0.905134i \(-0.360229\pi\)
−0.905134 + 0.425127i \(0.860229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 8.15304e13 8.15304e13i 0.0114139 0.0114139i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −5.23566e15 + 5.23566e15i −0.707107 + 0.707107i
\(769\) 9.03163e15i 1.21108i 0.795817 + 0.605538i \(0.207042\pi\)
−0.795817 + 0.605538i \(0.792958\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.04272e16 1.04272e16i −1.36858 1.36858i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.17950e15 4.17950e15i −0.529430 0.529430i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.67256e13i 0.0104570i
\(785\) 0 0
\(786\) 0 0
\(787\) −4.30470e15 4.30470e15i −0.508255 0.508255i 0.405735 0.913991i \(-0.367015\pi\)
−0.913991 + 0.405735i \(0.867015\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.33369e16 1.33369e16i 1.51026 1.51026i
\(794\) 0 0
\(795\) 0 0
\(796\) 1.71902e16 1.90660
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.83483e16i 1.92614i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −1.99246e16 −1.99423 −0.997113 0.0759375i \(-0.975805\pi\)
−0.997113 + 0.0759375i \(0.975805\pi\)
\(812\) 0 0
\(813\) 1.40661e16 1.40661e16i 1.38891 1.38891i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.11796e16 + 1.11796e16i 1.07450 + 1.07450i
\(818\) 0 0
\(819\) 1.15085e16i 1.09133i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 8.78784e15 8.78784e15i 0.811303 0.811303i −0.173526 0.984829i \(-0.555516\pi\)
0.984829 + 0.173526i \(0.0555161\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 1.54331e16i 1.36900i −0.729012 0.684501i \(-0.760020\pi\)
0.729012 0.684501i \(-0.239980\pi\)
\(830\) 0 0
\(831\) −2.00333e15 −0.175367
\(832\) 8.82795e15 + 8.82795e15i 0.767683 + 0.767683i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.56306e16 1.56306e16i −1.31518 1.31518i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −1.22005e16 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.30057e16i 1.84906i
\(845\) 0 0
\(846\) 0 0
\(847\) 9.01784e15 + 9.01784e15i 0.710794 + 0.710794i
\(848\) 0 0
\(849\) 2.51350e16i 1.95563i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.32054e16 + 1.32054e16i −1.00122 + 1.00122i −0.00122395 + 0.999999i \(0.500390\pi\)
−0.999999 + 0.00122395i \(0.999610\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 2.70738e16i 1.97509i 0.157335 + 0.987545i \(0.449710\pi\)
−0.157335 + 0.987545i \(0.550290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.01998e16 1.01998e16i −0.707107 0.707107i
\(868\) 1.91913e16 1.91913e16i 1.32204 1.32204i
\(869\) 0 0
\(870\) 0 0
\(871\) −3.09374e16 −2.09114
\(872\) 0 0
\(873\) −2.01225e16 + 2.01225e16i −1.34309 + 1.34309i
\(874\) 0 0
\(875\) 0 0
\(876\) −2.53171e16 −1.65822
\(877\) 1.75621e15 + 1.75621e15i 0.114308 + 0.114308i 0.761947 0.647639i \(-0.224243\pi\)
−0.647639 + 0.761947i \(0.724243\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.04282e16 2.04282e16i 1.28070 1.28070i 0.340425 0.940272i \(-0.389429\pi\)
0.940272 0.340425i \(-0.110571\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) 2.82187e16i 1.70442i
\(890\) 0 0
\(891\) 0 0
\(892\) 6.57903e15 + 6.57903e15i 0.390081 + 0.390081i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −1.02190e16 + 1.02190e16i −0.566402 + 0.566402i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.51966e15 3.51966e15i −0.190397 0.190397i 0.605470 0.795868i \(-0.292985\pi\)
−0.795868 + 0.605470i \(0.792985\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 2.56920e16 + 2.56920e16i 1.34842 + 1.34842i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.89851e16 1.99744
\(917\) 0 0
\(918\) 0 0
\(919\) 7.17141e15i 0.360885i 0.983586 + 0.180443i \(0.0577530\pi\)
−0.983586 + 0.180443i \(0.942247\pi\)
\(920\) 0 0
\(921\) 2.08142e16 1.03498
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.80179e16 2.80179e16i −1.34430 1.34430i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −4.25572e14 −0.0199411
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.86610e16 2.86610e16i −1.29635 1.29635i −0.930785 0.365567i \(-0.880875\pi\)
−0.365567 0.930785i \(-0.619125\pi\)
\(938\) 0 0
\(939\) 4.46826e16i 1.99746i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) −2.00443e16 + 2.00443e16i −0.850246 + 0.850246i
\(949\) 4.26876e16i 1.80027i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.24901e16 2.45942
\(962\) 0 0
\(963\) 0 0
\(964\) 4.66494e15i 0.180477i
\(965\) 0 0
\(966\) 0 0
\(967\) 3.33231e16 + 3.33231e16i 1.26736 + 1.26736i 0.947447 + 0.319912i \(0.103653\pi\)
0.319912 + 0.947447i \(0.396347\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 1.91271e16 + 1.91271e16i 0.707107 + 0.707107i
\(973\) −3.63313e16 + 3.63313e16i −1.33555 + 1.33555i
\(974\) 0 0
\(975\) 0 0
\(976\) 5.44307e16 1.96730
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.56952e14 −0.0160580
\(982\) 0 0
\(983\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.33197e16 4.33197e16i 1.46394 1.46394i
\(989\) 0 0
\(990\) 0 0
\(991\) −8.04799e15 −0.267474 −0.133737 0.991017i \(-0.542698\pi\)
−0.133737 + 0.991017i \(0.542698\pi\)
\(992\) 0 0
\(993\) −2.91661e16 + 2.91661e16i −0.958643 + 0.958643i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.80745e15 7.80745e15i −0.251007 0.251007i 0.570376 0.821383i \(-0.306797\pi\)
−0.821383 + 0.570376i \(0.806797\pi\)
\(998\) 0 0
\(999\) 2.34248e16i 0.744843i
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 75.12.e.a.32.1 4
3.2 odd 2 CM 75.12.e.a.32.1 4
5.2 odd 4 inner 75.12.e.a.68.2 yes 4
5.3 odd 4 inner 75.12.e.a.68.1 yes 4
5.4 even 2 inner 75.12.e.a.32.2 yes 4
15.2 even 4 inner 75.12.e.a.68.2 yes 4
15.8 even 4 inner 75.12.e.a.68.1 yes 4
15.14 odd 2 inner 75.12.e.a.32.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.e.a.32.1 4 1.1 even 1 trivial
75.12.e.a.32.1 4 3.2 odd 2 CM
75.12.e.a.32.2 yes 4 5.4 even 2 inner
75.12.e.a.32.2 yes 4 15.14 odd 2 inner
75.12.e.a.68.1 yes 4 5.3 odd 4 inner
75.12.e.a.68.1 yes 4 15.8 even 4 inner
75.12.e.a.68.2 yes 4 5.2 odd 4 inner
75.12.e.a.68.2 yes 4 15.2 even 4 inner