Properties

Label 100.4.e.b.43.1
Level $100$
Weight $4$
Character 100.43
Analytic conductor $5.900$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,4,Mod(7,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.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: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 43.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 100.43
Dual form 100.4.e.b.7.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 2.00000i) q^{2} +(7.00000 - 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(-9.00000 - 9.00000i) q^{7} +(16.0000 - 16.0000i) q^{8} -71.0000i q^{9} +O(q^{10})\) \(q+(-2.00000 - 2.00000i) q^{2} +(7.00000 - 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(-9.00000 - 9.00000i) q^{7} +(16.0000 - 16.0000i) q^{8} -71.0000i q^{9} +(56.0000 + 56.0000i) q^{12} +36.0000i q^{14} -64.0000 q^{16} +(-142.000 + 142.000i) q^{18} -126.000 q^{21} +(67.0000 - 67.0000i) q^{23} -224.000i q^{24} +(-308.000 - 308.000i) q^{27} +(72.0000 - 72.0000i) q^{28} +306.000i q^{29} +(128.000 + 128.000i) q^{32} +568.000 q^{36} +252.000 q^{41} +(252.000 + 252.000i) q^{42} +(297.000 - 297.000i) q^{43} -268.000 q^{46} +(301.000 + 301.000i) q^{47} +(-448.000 + 448.000i) q^{48} -181.000i q^{49} +1232.00i q^{54} -288.000 q^{56} +(612.000 - 612.000i) q^{58} +952.000 q^{61} +(-639.000 + 639.000i) q^{63} -512.000i q^{64} +(-549.000 - 549.000i) q^{67} -938.000i q^{69} +(-1136.00 - 1136.00i) q^{72} -2395.00 q^{81} +(-504.000 - 504.000i) q^{82} +(77.0000 - 77.0000i) q^{83} -1008.00i q^{84} -1188.00 q^{86} +(2142.00 + 2142.00i) q^{87} +1386.00i q^{89} +(536.000 + 536.000i) q^{92} -1204.00i q^{94} +1792.00 q^{96} +(-362.000 + 362.000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 14 q^{3} - 56 q^{6} - 18 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 14 q^{3} - 56 q^{6} - 18 q^{7} + 32 q^{8} + 112 q^{12} - 128 q^{16} - 284 q^{18} - 252 q^{21} + 134 q^{23} - 616 q^{27} + 144 q^{28} + 256 q^{32} + 1136 q^{36} + 504 q^{41} + 504 q^{42} + 594 q^{43} - 536 q^{46} + 602 q^{47} - 896 q^{48} - 576 q^{56} + 1224 q^{58} + 1904 q^{61} - 1278 q^{63} - 1098 q^{67} - 2272 q^{72} - 4790 q^{81} - 1008 q^{82} + 154 q^{83} - 2376 q^{86} + 4284 q^{87} + 1072 q^{92} + 3584 q^{96} - 724 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 2.00000i −0.707107 0.707107i
\(3\) 7.00000 7.00000i 1.34715 1.34715i 0.458410 0.888741i \(-0.348419\pi\)
0.888741 0.458410i \(-0.151581\pi\)
\(4\) 8.00000i 1.00000i
\(5\) 0 0
\(6\) −28.0000 −1.90516
\(7\) −9.00000 9.00000i −0.485954 0.485954i 0.421073 0.907027i \(-0.361654\pi\)
−0.907027 + 0.421073i \(0.861654\pi\)
\(8\) 16.0000 16.0000i 0.707107 0.707107i
\(9\) 71.0000i 2.62963i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 56.0000 + 56.0000i 1.34715 + 1.34715i
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 36.0000i 0.687243i
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) −142.000 + 142.000i −1.85943 + 1.85943i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −126.000 −1.30931
\(22\) 0 0
\(23\) 67.0000 67.0000i 0.607412 0.607412i −0.334857 0.942269i \(-0.608688\pi\)
0.942269 + 0.334857i \(0.108688\pi\)
\(24\) 224.000i 1.90516i
\(25\) 0 0
\(26\) 0 0
\(27\) −308.000 308.000i −2.19536 2.19536i
\(28\) 72.0000 72.0000i 0.485954 0.485954i
\(29\) 306.000i 1.95941i 0.200455 + 0.979703i \(0.435758\pi\)
−0.200455 + 0.979703i \(0.564242\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 128.000 + 128.000i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 568.000 2.62963
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 252.000 0.959897 0.479949 0.877297i \(-0.340655\pi\)
0.479949 + 0.877297i \(0.340655\pi\)
\(42\) 252.000 + 252.000i 0.925820 + 0.925820i
\(43\) 297.000 297.000i 1.05330 1.05330i 0.0548071 0.998497i \(-0.482546\pi\)
0.998497 0.0548071i \(-0.0174544\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −268.000 −0.859010
\(47\) 301.000 + 301.000i 0.934157 + 0.934157i 0.997962 0.0638057i \(-0.0203238\pi\)
−0.0638057 + 0.997962i \(0.520324\pi\)
\(48\) −448.000 + 448.000i −1.34715 + 1.34715i
\(49\) 181.000i 0.527697i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 1232.00i 3.10470i
\(55\) 0 0
\(56\) −288.000 −0.687243
\(57\) 0 0
\(58\) 612.000 612.000i 1.38551 1.38551i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 952.000 1.99821 0.999107 0.0422409i \(-0.0134497\pi\)
0.999107 + 0.0422409i \(0.0134497\pi\)
\(62\) 0 0
\(63\) −639.000 + 639.000i −1.27788 + 1.27788i
\(64\) 512.000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −549.000 549.000i −1.00106 1.00106i −0.999999 0.00106064i \(-0.999662\pi\)
−0.00106064 0.999999i \(-0.500338\pi\)
\(68\) 0 0
\(69\) 938.000i 1.63655i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1136.00 1136.00i −1.85943 1.85943i
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2395.00 −3.28532
\(82\) −504.000 504.000i −0.678750 0.678750i
\(83\) 77.0000 77.0000i 0.101829 0.101829i −0.654357 0.756186i \(-0.727060\pi\)
0.756186 + 0.654357i \(0.227060\pi\)
\(84\) 1008.00i 1.30931i
\(85\) 0 0
\(86\) −1188.00 −1.48960
\(87\) 2142.00 + 2142.00i 2.63961 + 2.63961i
\(88\) 0 0
\(89\) 1386.00i 1.65074i 0.564593 + 0.825369i \(0.309033\pi\)
−0.564593 + 0.825369i \(0.690967\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 536.000 + 536.000i 0.607412 + 0.607412i
\(93\) 0 0
\(94\) 1204.00i 1.32110i
\(95\) 0 0
\(96\) 1792.00 1.90516
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −362.000 + 362.000i −0.373138 + 0.373138i
\(99\) 0 0
\(100\) 0 0
\(101\) −378.000 −0.372400 −0.186200 0.982512i \(-0.559617\pi\)
−0.186200 + 0.982512i \(0.559617\pi\)
\(102\) 0 0
\(103\) −1323.00 + 1323.00i −1.26562 + 1.26562i −0.317295 + 0.948327i \(0.602775\pi\)
−0.948327 + 0.317295i \(0.897225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 221.000 + 221.000i 0.199672 + 0.199672i 0.799859 0.600188i \(-0.204907\pi\)
−0.600188 + 0.799859i \(0.704907\pi\)
\(108\) 2464.00 2464.00i 2.19536 2.19536i
\(109\) 1136.00i 0.998248i 0.866530 + 0.499124i \(0.166345\pi\)
−0.866530 + 0.499124i \(0.833655\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 576.000 + 576.000i 0.485954 + 0.485954i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2448.00 −1.95941
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1331.00 1.00000
\(122\) −1904.00 1904.00i −1.41295 1.41295i
\(123\) 1764.00 1764.00i 1.29313 1.29313i
\(124\) 0 0
\(125\) 0 0
\(126\) 2556.00 1.80720
\(127\) −1089.00 1089.00i −0.760891 0.760891i 0.215593 0.976483i \(-0.430832\pi\)
−0.976483 + 0.215593i \(0.930832\pi\)
\(128\) −1024.00 + 1024.00i −0.707107 + 0.707107i
\(129\) 4158.00i 2.83792i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2196.00i 1.41571i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) −1876.00 + 1876.00i −1.15722 + 1.15722i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4214.00 2.51690
\(142\) 0 0
\(143\) 0 0
\(144\) 4544.00i 2.62963i
\(145\) 0 0
\(146\) 0 0
\(147\) −1267.00 1267.00i −0.710887 0.710887i
\(148\) 0 0
\(149\) 3096.00i 1.70224i 0.524969 + 0.851121i \(0.324077\pi\)
−0.524969 + 0.851121i \(0.675923\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1206.00 −0.590349
\(162\) 4790.00 + 4790.00i 2.32307 + 2.32307i
\(163\) −2943.00 + 2943.00i −1.41419 + 1.41419i −0.703336 + 0.710858i \(0.748307\pi\)
−0.710858 + 0.703336i \(0.751693\pi\)
\(164\) 2016.00i 0.959897i
\(165\) 0 0
\(166\) −308.000 −0.144009
\(167\) −1309.00 1309.00i −0.606548 0.606548i 0.335494 0.942042i \(-0.391097\pi\)
−0.942042 + 0.335494i \(0.891097\pi\)
\(168\) −2016.00 + 2016.00i −0.925820 + 0.925820i
\(169\) 2197.00i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 2376.00 + 2376.00i 1.05330 + 1.05330i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 8568.00i 3.73298i
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 2772.00 2772.00i 1.16725 1.16725i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −1078.00 −0.442691 −0.221346 0.975195i \(-0.571045\pi\)
−0.221346 + 0.975195i \(0.571045\pi\)
\(182\) 0 0
\(183\) 6664.00 6664.00i 2.69190 2.69190i
\(184\) 2144.00i 0.859010i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −2408.00 + 2408.00i −0.934157 + 0.934157i
\(189\) 5544.00i 2.13369i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −3584.00 3584.00i −1.34715 1.34715i
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1448.00 0.527697
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −7686.00 −2.69716
\(202\) 756.000 + 756.000i 0.263327 + 0.263327i
\(203\) 2754.00 2754.00i 0.952182 0.952182i
\(204\) 0 0
\(205\) 0 0
\(206\) 5292.00 1.78986
\(207\) −4757.00 4757.00i −1.59727 1.59727i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 884.000i 0.282378i
\(215\) 0 0
\(216\) −9856.00 −3.10470
\(217\) 0 0
\(218\) 2272.00 2272.00i 0.705868 0.705868i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −4473.00 + 4473.00i −1.34320 + 1.34320i −0.450352 + 0.892851i \(0.648701\pi\)
−0.892851 + 0.450352i \(0.851299\pi\)
\(224\) 2304.00i 0.687243i
\(225\) 0 0
\(226\) 0 0
\(227\) 4081.00 + 4081.00i 1.19324 + 1.19324i 0.976152 + 0.217088i \(0.0696558\pi\)
0.217088 + 0.976152i \(0.430344\pi\)
\(228\) 0 0
\(229\) 6874.00i 1.98361i −0.127761 0.991805i \(-0.540779\pi\)
0.127761 0.991805i \(-0.459221\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4896.00 + 4896.00i 1.38551 + 1.38551i
\(233\) 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
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1708.00 −0.456523 −0.228261 0.973600i \(-0.573304\pi\)
−0.228261 + 0.973600i \(0.573304\pi\)
\(242\) −2662.00 2662.00i −0.707107 0.707107i
\(243\) −8449.00 + 8449.00i −2.23047 + 2.23047i
\(244\) 7616.00i 1.99821i
\(245\) 0 0
\(246\) −7056.00 −1.82876
\(247\) 0 0
\(248\) 0 0
\(249\) 1078.00i 0.274359i
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −5112.00 5112.00i −1.27788 1.27788i
\(253\) 0 0
\(254\) 4356.00i 1.07606i
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) −8316.00 + 8316.00i −2.00671 + 2.00671i
\(259\) 0 0
\(260\) 0 0
\(261\) 21726.0 5.15251
\(262\) 0 0
\(263\) 6017.00 6017.00i 1.41074 1.41074i 0.655826 0.754912i \(-0.272321\pi\)
0.754912 0.655826i \(-0.227679\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9702.00 + 9702.00i 2.22379 + 2.22379i
\(268\) 4392.00 4392.00i 1.00106 1.00106i
\(269\) 5544.00i 1.25659i −0.777974 0.628297i \(-0.783752\pi\)
0.777974 0.628297i \(-0.216248\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 7504.00 1.63655
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8388.00 −1.78073 −0.890367 0.455244i \(-0.849552\pi\)
−0.890367 + 0.455244i \(0.849552\pi\)
\(282\) −8428.00 8428.00i −1.77972 1.77972i
\(283\) −693.000 + 693.000i −0.145564 + 0.145564i −0.776133 0.630569i \(-0.782822\pi\)
0.630569 + 0.776133i \(0.282822\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2268.00 2268.00i −0.466466 0.466466i
\(288\) 9088.00 9088.00i 1.85943 1.85943i
\(289\) 4913.00i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 5068.00i 1.00535i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 6192.00 6192.00i 1.20367 1.20367i
\(299\) 0 0
\(300\) 0 0
\(301\) −5346.00 −1.02372
\(302\) 0 0
\(303\) −2646.00 + 2646.00i −0.501679 + 0.501679i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2709.00 2709.00i −0.503618 0.503618i 0.408942 0.912560i \(-0.365898\pi\)
−0.912560 + 0.408942i \(0.865898\pi\)
\(308\) 0 0
\(309\) 18522.0i 3.40997i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(321\) 3094.00 0.537976
\(322\) 2412.00 + 2412.00i 0.417439 + 0.417439i
\(323\) 0 0
\(324\) 19160.0i 3.28532i
\(325\) 0 0
\(326\) 11772.0 1.99997
\(327\) 7952.00 + 7952.00i 1.34479 + 1.34479i
\(328\) 4032.00 4032.00i 0.678750 0.678750i
\(329\) 5418.00i 0.907915i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 616.000 + 616.000i 0.101829 + 0.101829i
\(333\) 0 0
\(334\) 5236.00i 0.857788i
\(335\) 0 0
\(336\) 8064.00 1.30931
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) −4394.00 + 4394.00i −0.707107 + 0.707107i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4716.00 + 4716.00i −0.742391 + 0.742391i
\(344\) 9504.00i 1.48960i
\(345\) 0 0
\(346\) 0 0
\(347\) 7871.00 + 7871.00i 1.21769 + 1.21769i 0.968440 + 0.249247i \(0.0801832\pi\)
0.249247 + 0.968440i \(0.419817\pi\)
\(348\) −17136.0 + 17136.0i −2.63961 + 2.63961i
\(349\) 9646.00i 1.47948i 0.672893 + 0.739740i \(0.265052\pi\)
−0.672893 + 0.739740i \(0.734948\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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
\(356\) −11088.0 −1.65074
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −6859.00 −1.00000
\(362\) 2156.00 + 2156.00i 0.313030 + 0.313030i
\(363\) 9317.00 9317.00i 1.34715 1.34715i
\(364\) 0 0
\(365\) 0 0
\(366\) −26656.0 −3.80692
\(367\) −9639.00 9639.00i −1.37099 1.37099i −0.858985 0.512000i \(-0.828905\pi\)
−0.512000 0.858985i \(-0.671095\pi\)
\(368\) −4288.00 + 4288.00i −0.607412 + 0.607412i
\(369\) 17892.0i 2.52417i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 9632.00 1.32110
\(377\) 0 0
\(378\) 11088.0 11088.0i 1.50874 1.50874i
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −15246.0 −2.05007
\(382\) 0 0
\(383\) −8113.00 + 8113.00i −1.08239 + 1.08239i −0.0861026 + 0.996286i \(0.527441\pi\)
−0.996286 + 0.0861026i \(0.972559\pi\)
\(384\) 14336.0i 1.90516i
\(385\) 0 0
\(386\) 0 0
\(387\) −21087.0 21087.0i −2.76980 2.76980i
\(388\) 0 0
\(389\) 14184.0i 1.84873i −0.381505 0.924367i \(-0.624594\pi\)
0.381505 0.924367i \(-0.375406\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2896.00 2896.00i −0.373138 0.373138i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15822.0 1.97036 0.985178 0.171534i \(-0.0548722\pi\)
0.985178 + 0.171534i \(0.0548722\pi\)
\(402\) 15372.0 + 15372.0i 1.90718 + 1.90718i
\(403\) 0 0
\(404\) 3024.00i 0.372400i
\(405\) 0 0
\(406\) −11016.0 −1.34659
\(407\) 0 0
\(408\) 0 0
\(409\) 4844.00i 0.585624i −0.956170 0.292812i \(-0.905409\pi\)
0.956170 0.292812i \(-0.0945911\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −10584.0 10584.0i −1.26562 1.26562i
\(413\) 0 0
\(414\) 19028.0i 2.25888i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 9592.00 1.11042 0.555208 0.831711i \(-0.312638\pi\)
0.555208 + 0.831711i \(0.312638\pi\)
\(422\) 0 0
\(423\) 21371.0 21371.0i 2.45649 2.45649i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8568.00 8568.00i −0.971041 0.971041i
\(428\) −1768.00 + 1768.00i −0.199672 + 0.199672i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 19712.0 + 19712.0i 2.19536 + 2.19536i
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9088.00 −0.998248
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −12851.0 −1.38765
\(442\) 0 0
\(443\) 10237.0 10237.0i 1.09791 1.09791i 0.103256 0.994655i \(-0.467074\pi\)
0.994655 0.103256i \(-0.0329261\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 17892.0 1.89958
\(447\) 21672.0 + 21672.0i 2.29318 + 2.29318i
\(448\) −4608.00 + 4608.00i −0.485954 + 0.485954i
\(449\) 1836.00i 0.192976i 0.995334 + 0.0964880i \(0.0307609\pi\)
−0.995334 + 0.0964880i \(0.969239\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 16324.0i 1.68750i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) −13748.0 + 13748.0i −1.40262 + 1.40262i
\(459\) 0 0
\(460\) 0 0
\(461\) 16002.0 1.61668 0.808338 0.588719i \(-0.200368\pi\)
0.808338 + 0.588719i \(0.200368\pi\)
\(462\) 0 0
\(463\) −11043.0 + 11043.0i −1.10845 + 1.10845i −0.115094 + 0.993355i \(0.536717\pi\)
−0.993355 + 0.115094i \(0.963283\pi\)
\(464\) 19584.0i 1.95941i
\(465\) 0 0
\(466\) 0 0
\(467\) −7889.00 7889.00i −0.781712 0.781712i 0.198408 0.980120i \(-0.436423\pi\)
−0.980120 + 0.198408i \(0.936423\pi\)
\(468\) 0 0
\(469\) 9882.00i 0.972939i
\(470\) 0 0
\(471\) 0 0
\(472\) 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
\(481\) 0 0
\(482\) 3416.00 + 3416.00i 0.322810 + 0.322810i
\(483\) −8442.00 + 8442.00i −0.795288 + 0.795288i
\(484\) 10648.0i 1.00000i
\(485\) 0 0
\(486\) 33796.0 3.15436
\(487\) 81.0000 + 81.0000i 0.00753688 + 0.00753688i 0.710865 0.703328i \(-0.248304\pi\)
−0.703328 + 0.710865i \(0.748304\pi\)
\(488\) 15232.0 15232.0i 1.41295 1.41295i
\(489\) 41202.0i 3.81026i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 14112.0 + 14112.0i 1.29313 + 1.29313i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −2156.00 + 2156.00i −0.194001 + 0.194001i
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −18326.0 −1.63422
\(502\) 0 0
\(503\) −12803.0 + 12803.0i −1.13491 + 1.13491i −0.145556 + 0.989350i \(0.546497\pi\)
−0.989350 + 0.145556i \(0.953503\pi\)
\(504\) 20448.0i 1.80720i
\(505\) 0 0
\(506\) 0 0
\(507\) −15379.0 15379.0i −1.34715 1.34715i
\(508\) 8712.00 8712.00i 0.760891 0.760891i
\(509\) 8946.00i 0.779026i 0.921021 + 0.389513i \(0.127357\pi\)
−0.921021 + 0.389513i \(0.872643\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8192.00 8192.00i −0.707107 0.707107i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 33264.0 2.83792
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8442.00 0.709886 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(522\) −43452.0 43452.0i −3.64338 3.64338i
\(523\) 14427.0 14427.0i 1.20621 1.20621i 0.233967 0.972245i \(-0.424829\pi\)
0.972245 0.233967i \(-0.0751708\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −24068.0 −1.99508
\(527\) 0 0
\(528\) 0 0
\(529\) 3189.00i 0.262102i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 38808.0i 3.14492i
\(535\) 0 0
\(536\) −17568.0 −1.41571
\(537\) 0 0
\(538\) −11088.0 + 11088.0i −0.888546 + 0.888546i
\(539\) 0 0
\(540\) 0 0
\(541\) 6802.00 0.540556 0.270278 0.962782i \(-0.412884\pi\)
0.270278 + 0.962782i \(0.412884\pi\)
\(542\) 0 0
\(543\) −7546.00 + 7546.00i −0.596372 + 0.596372i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17721.0 + 17721.0i 1.38518 + 1.38518i 0.835129 + 0.550055i \(0.185393\pi\)
0.550055 + 0.835129i \(0.314607\pi\)
\(548\) 0 0
\(549\) 67592.0i 5.25457i
\(550\) 0 0
\(551\) 0 0
\(552\) −15008.0 15008.0i −1.15722 1.15722i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 16776.0 + 16776.0i 1.25917 + 1.25917i
\(563\) 1687.00 1687.00i 0.126285 0.126285i −0.641139 0.767425i \(-0.721538\pi\)
0.767425 + 0.641139i \(0.221538\pi\)
\(564\) 33712.0i 2.51690i
\(565\) 0 0
\(566\) 2772.00 0.205858
\(567\) 21555.0 + 21555.0i 1.59652 + 1.59652i
\(568\) 0 0
\(569\) 14796.0i 1.09012i 0.838396 + 0.545062i \(0.183494\pi\)
−0.838396 + 0.545062i \(0.816506\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9072.00i 0.659683i
\(575\) 0 0
\(576\) −36352.0 −2.62963
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) 9826.00 9826.00i 0.707107 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) −1386.00 −0.0989690
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11641.0 + 11641.0i 0.818527 + 0.818527i 0.985895 0.167367i \(-0.0535266\pi\)
−0.167367 + 0.985895i \(0.553527\pi\)
\(588\) 10136.0 10136.0i 0.710887 0.710887i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24768.0 −1.70224
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 28532.0 1.93651 0.968257 0.249958i \(-0.0804168\pi\)
0.968257 + 0.249958i \(0.0804168\pi\)
\(602\) 10692.0 + 10692.0i 0.723876 + 0.723876i
\(603\) −38979.0 + 38979.0i −2.63242 + 2.63242i
\(604\) 0 0
\(605\) 0 0
\(606\) 10584.0 0.709481
\(607\) 11781.0 + 11781.0i 0.787769 + 0.787769i 0.981128 0.193359i \(-0.0619381\pi\)
−0.193359 + 0.981128i \(0.561938\pi\)
\(608\) 0 0
\(609\) 38556.0i 2.56546i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 10836.0i 0.712224i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 37044.0 37044.0i 2.41121 2.41121i
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −41272.0 −2.66697
\(622\) 0 0
\(623\) 12474.0 12474.0i 0.802183 0.802183i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 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
\(641\) −21348.0 −1.31544 −0.657719 0.753264i \(-0.728478\pi\)
−0.657719 + 0.753264i \(0.728478\pi\)
\(642\) −6188.00 6188.00i −0.380406 0.380406i
\(643\) 21987.0 21987.0i 1.34850 1.34850i 0.461199 0.887297i \(-0.347420\pi\)
0.887297 0.461199i \(-0.152580\pi\)
\(644\) 9648.00i 0.590349i
\(645\) 0 0
\(646\) 0 0
\(647\) −20839.0 20839.0i −1.26625 1.26625i −0.948009 0.318244i \(-0.896907\pi\)
−0.318244 0.948009i \(-0.603093\pi\)
\(648\) −38320.0 + 38320.0i −2.32307 + 2.32307i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −23544.0 23544.0i −1.41419 1.41419i
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 31808.0i 1.90182i
\(655\) 0 0
\(656\) −16128.0 −0.959897
\(657\) 0 0
\(658\) −10836.0 + 10836.0i −0.641993 + 0.641993i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −30688.0 −1.80579 −0.902893 0.429865i \(-0.858561\pi\)
−0.902893 + 0.429865i \(0.858561\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 2464.00i 0.144009i
\(665\) 0 0
\(666\) 0 0
\(667\) 20502.0 + 20502.0i 1.19017 + 1.19017i
\(668\) 10472.0 10472.0i 0.606548 0.606548i
\(669\) 62622.0i 3.61899i
\(670\) 0 0
\(671\) 0 0
\(672\) −16128.0 16128.0i −0.925820 0.925820i
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 17576.0 1.00000
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 57134.0 3.21495
\(682\) 0 0
\(683\) 3937.00 3937.00i 0.220564 0.220564i −0.588172 0.808736i \(-0.700152\pi\)
0.808736 + 0.588172i \(0.200152\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18864.0 1.04990
\(687\) −48118.0 48118.0i −2.67222 2.67222i
\(688\) −19008.0 + 19008.0i −1.05330 + 1.05330i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 31484.0i 1.72207i
\(695\) 0 0
\(696\) 68544.0 3.73298
\(697\) 0 0
\(698\) 19292.0 19292.0i 1.04615 1.04615i
\(699\) 0 0
\(700\) 0 0
\(701\) −9648.00 −0.519829 −0.259914 0.965632i \(-0.583694\pi\)
−0.259914 + 0.965632i \(0.583694\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3402.00 + 3402.00i 0.180969 + 0.180969i
\(708\) 0 0
\(709\) 506.000i 0.0268029i 0.999910 + 0.0134014i \(0.00426594\pi\)
−0.999910 + 0.0134014i \(0.995734\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 22176.0 + 22176.0i 1.16725 + 1.16725i
\(713\) 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
\(721\) 23814.0 1.23007
\(722\) 13718.0 + 13718.0i 0.707107 + 0.707107i
\(723\) −11956.0 + 11956.0i −0.615005 + 0.615005i
\(724\) 8624.00i 0.442691i
\(725\) 0 0
\(726\) −37268.0 −1.90516
\(727\) −6489.00 6489.00i −0.331037 0.331037i 0.521943 0.852980i \(-0.325207\pi\)
−0.852980 + 0.521943i \(0.825207\pi\)
\(728\) 0 0
\(729\) 53621.0i 2.72423i
\(730\) 0 0
\(731\) 0 0
\(732\) 53312.0 + 53312.0i 2.69190 + 2.69190i
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 38556.0i 1.93887i
\(735\) 0 0
\(736\) 17152.0 0.859010
\(737\) 0 0
\(738\) −35784.0 + 35784.0i −1.78486 + 1.78486i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28633.0 + 28633.0i −1.41379 + 1.41379i −0.689504 + 0.724282i \(0.742171\pi\)
−0.724282 + 0.689504i \(0.757829\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5467.00 5467.00i −0.267774 0.267774i
\(748\) 0 0
\(749\) 3978.00i 0.194063i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −19264.0 19264.0i −0.934157 0.934157i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −44352.0 −2.13369
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21798.0 −1.03834 −0.519170 0.854671i \(-0.673759\pi\)
−0.519170 + 0.854671i \(0.673759\pi\)
\(762\) 30492.0 + 30492.0i 1.44962 + 1.44962i
\(763\) 10224.0 10224.0i 0.485103 0.485103i
\(764\) 0 0
\(765\) 0 0
\(766\) 32452.0 1.53073
\(767\) 0 0
\(768\) 28672.0 28672.0i 1.34715 1.34715i
\(769\) 29554.0i 1.38588i −0.720994 0.692942i \(-0.756314\pi\)
0.720994 0.692942i \(-0.243686\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 84348.0i 3.91709i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −28368.0 + 28368.0i −1.30725 + 1.30725i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 94248.0 94248.0i 4.30159 4.30159i
\(784\) 11584.0i 0.527697i
\(785\) 0 0
\(786\) 0 0
\(787\) 24381.0 + 24381.0i 1.10431 + 1.10431i 0.993885 + 0.110421i \(0.0352199\pi\)
0.110421 + 0.993885i \(0.464780\pi\)
\(788\) 0 0
\(789\) 84238.0i 3.80095i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 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
\(801\) 98406.0 4.34083
\(802\) −31644.0 31644.0i −1.39325 1.39325i
\(803\) 0 0
\(804\) 61488.0i 2.69716i
\(805\) 0 0
\(806\) 0 0
\(807\) −38808.0 38808.0i −1.69282 1.69282i
\(808\) −6048.00 + 6048.00i −0.263327 + 0.263327i
\(809\) 26406.0i 1.14757i 0.819005 + 0.573786i \(0.194526\pi\)
−0.819005 + 0.573786i \(0.805474\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 22032.0 + 22032.0i 0.952182 + 0.952182i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −9688.00 + 9688.00i −0.414099 + 0.414099i
\(819\) 0 0
\(820\) 0 0
\(821\) 7632.00 0.324432 0.162216 0.986755i \(-0.448136\pi\)
0.162216 + 0.986755i \(0.448136\pi\)
\(822\) 0 0
\(823\) 22347.0 22347.0i 0.946498 0.946498i −0.0521422 0.998640i \(-0.516605\pi\)
0.998640 + 0.0521422i \(0.0166049\pi\)
\(824\) 42336.0i 1.78986i
\(825\) 0 0
\(826\) 0 0
\(827\) −32729.0 32729.0i −1.37618 1.37618i −0.850981 0.525197i \(-0.823992\pi\)
−0.525197 0.850981i \(-0.676008\pi\)
\(828\) 38056.0 38056.0i 1.59727 1.59727i
\(829\) 36344.0i 1.52265i −0.648369 0.761326i \(-0.724548\pi\)
0.648369 0.761326i \(-0.275452\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −69247.0 −2.83927
\(842\) −19184.0 19184.0i −0.785183 0.785183i
\(843\) −58716.0 + 58716.0i −2.39892 + 2.39892i
\(844\) 0 0
\(845\) 0 0
\(846\) −85484.0 −3.47400
\(847\) −11979.0 11979.0i −0.485954 0.485954i
\(848\) 0 0
\(849\) 9702.00i 0.392193i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 34272.0i 1.37326i
\(855\) 0 0
\(856\) 7072.00 0.282378
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) −31752.0 −1.25680
\(862\) 0 0
\(863\) −31693.0 + 31693.0i −1.25011 + 1.25011i −0.294436 + 0.955671i \(0.595132\pi\)
−0.955671 + 0.294436i \(0.904868\pi\)
\(864\) 78848.0i 3.10470i
\(865\) 0 0
\(866\) 0 0
\(867\) 34391.0 + 34391.0i 1.34715 + 1.34715i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 18176.0 + 18176.0i 0.705868 + 0.705868i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −29988.0 −1.14679 −0.573395 0.819279i \(-0.694374\pi\)
−0.573395 + 0.819279i \(0.694374\pi\)
\(882\) 25702.0 + 25702.0i 0.981215 + 0.981215i
\(883\) 15327.0 15327.0i 0.584139 0.584139i −0.351899 0.936038i \(-0.614464\pi\)
0.936038 + 0.351899i \(0.114464\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −40948.0 −1.55268
\(887\) −2849.00 2849.00i −0.107847 0.107847i 0.651124 0.758971i \(-0.274298\pi\)
−0.758971 + 0.651124i \(0.774298\pi\)
\(888\) 0 0
\(889\) 19602.0i 0.739516i
\(890\) 0 0
\(891\) 0 0
\(892\) −35784.0 35784.0i −1.34320 1.34320i
\(893\) 0 0
\(894\) 86688.0i 3.24304i
\(895\) 0 0
\(896\) 18432.0 0.687243
\(897\) 0 0
\(898\) 3672.00 3672.00i 0.136455 0.136455i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −37422.0 + 37422.0i −1.37910 + 1.37910i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8109.00 8109.00i −0.296863 0.296863i 0.542921 0.839784i \(-0.317318\pi\)
−0.839784 + 0.542921i \(0.817318\pi\)
\(908\) −32648.0 + 32648.0i −1.19324 + 1.19324i
\(909\) 26838.0i 0.979274i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 54992.0 1.98361
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −37926.0 −1.35690
\(922\) −32004.0 32004.0i −1.14316 1.14316i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 44172.0 1.56758
\(927\) 93933.0 + 93933.0i 3.32812 + 3.32812i
\(928\) −39168.0 + 39168.0i −1.38551 + 1.38551i
\(929\) 53676.0i 1.89564i 0.318801 + 0.947822i \(0.396720\pi\)
−0.318801 + 0.947822i \(0.603280\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 31556.0i 1.10551i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 19764.0 19764.0i 0.687972 0.687972i
\(939\) 0 0
\(940\) 0 0
\(941\) −44478.0 −1.54085 −0.770426 0.637530i \(-0.779956\pi\)
−0.770426 + 0.637530i \(0.779956\pi\)
\(942\) 0 0
\(943\) 16884.0 16884.0i 0.583053 0.583053i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36851.0 + 36851.0i 1.26452 + 1.26452i 0.948881 + 0.315635i \(0.102218\pi\)
0.315635 + 0.948881i \(0.397782\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 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
\(961\) 29791.0 1.00000
\(962\) 0 0
\(963\) 15691.0 15691.0i 0.525063 0.525063i
\(964\) 13664.0i 0.456523i
\(965\) 0 0
\(966\) 33768.0 1.12471
\(967\) −23859.0 23859.0i −0.793437 0.793437i 0.188614 0.982051i \(-0.439600\pi\)
−0.982051 + 0.188614i \(0.939600\pi\)
\(968\) 21296.0 21296.0i 0.707107 0.707107i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −67592.0 67592.0i −2.23047 2.23047i
\(973\) 0 0
\(974\) 324.000i 0.0106588i
\(975\) 0 0
\(976\) −60928.0 −1.99821
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 82404.0 82404.0i 2.69426 2.69426i
\(979\) 0 0
\(980\) 0 0
\(981\) 80656.0 2.62502
\(982\) 0 0
\(983\) −36743.0 + 36743.0i −1.19219 + 1.19219i −0.215735 + 0.976452i \(0.569215\pi\)
−0.976452 + 0.215735i \(0.930785\pi\)
\(984\) 56448.0i 1.82876i
\(985\) 0 0
\(986\) 0 0
\(987\) −37926.0 37926.0i −1.22310 1.22310i
\(988\) 0 0
\(989\) 39798.0i 1.27958i
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 8624.00 0.274359
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 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 100.4.e.b.43.1 yes 2
4.3 odd 2 100.4.e.c.43.1 yes 2
5.2 odd 4 100.4.e.c.7.1 yes 2
5.3 odd 4 inner 100.4.e.b.7.1 2
5.4 even 2 100.4.e.c.43.1 yes 2
20.3 even 4 100.4.e.c.7.1 yes 2
20.7 even 4 inner 100.4.e.b.7.1 2
20.19 odd 2 CM 100.4.e.b.43.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.e.b.7.1 2 5.3 odd 4 inner
100.4.e.b.7.1 2 20.7 even 4 inner
100.4.e.b.43.1 yes 2 1.1 even 1 trivial
100.4.e.b.43.1 yes 2 20.19 odd 2 CM
100.4.e.c.7.1 yes 2 5.2 odd 4
100.4.e.c.7.1 yes 2 20.3 even 4
100.4.e.c.43.1 yes 2 4.3 odd 2
100.4.e.c.43.1 yes 2 5.4 even 2