Properties

Label 69.7.d.a.22.2
Level $69$
Weight $7$
Character 69.22
Analytic conductor $15.874$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,7,Mod(22,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.22");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(15.8737317698\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 22.2
Character \(\chi\) \(=\) 69.22
Dual form 69.7.d.a.22.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-15.4071 q^{2} +15.5885 q^{3} +173.378 q^{4} +161.383i q^{5} -240.172 q^{6} +585.691i q^{7} -1685.19 q^{8} +243.000 q^{9} +O(q^{10})\) \(q-15.4071 q^{2} +15.5885 q^{3} +173.378 q^{4} +161.383i q^{5} -240.172 q^{6} +585.691i q^{7} -1685.19 q^{8} +243.000 q^{9} -2486.44i q^{10} +2121.45i q^{11} +2702.69 q^{12} +1927.08 q^{13} -9023.79i q^{14} +2515.71i q^{15} +14867.7 q^{16} -1375.26i q^{17} -3743.92 q^{18} -3652.93i q^{19} +27980.2i q^{20} +9130.03i q^{21} -32685.3i q^{22} +(-10900.5 - 5405.04i) q^{23} -26269.5 q^{24} -10419.4 q^{25} -29690.6 q^{26} +3788.00 q^{27} +101546. i q^{28} +6030.06 q^{29} -38759.7i q^{30} +3253.53 q^{31} -121215. q^{32} +33070.1i q^{33} +21188.7i q^{34} -94520.5 q^{35} +42130.8 q^{36} +25548.1i q^{37} +56280.9i q^{38} +30040.1 q^{39} -271961. i q^{40} +101808. q^{41} -140667. i q^{42} -83443.8i q^{43} +367812. i q^{44} +39216.0i q^{45} +(167945. + 83275.8i) q^{46} -119152. q^{47} +231764. q^{48} -225385. q^{49} +160532. q^{50} -21438.2i q^{51} +334112. q^{52} +87012.7i q^{53} -58361.9 q^{54} -342365. q^{55} -987001. i q^{56} -56943.5i q^{57} -92905.5 q^{58} +42117.7 q^{59} +436168. i q^{60} +112804. i q^{61} -50127.4 q^{62} +142323. i q^{63} +916036. q^{64} +310997. i q^{65} -509513. i q^{66} +49872.3i q^{67} -238439. i q^{68} +(-169922. - 84256.3i) q^{69} +1.45628e6 q^{70} +234036. q^{71} -409501. q^{72} -75271.3 q^{73} -393621. i q^{74} -162422. q^{75} -633336. i q^{76} -1.24251e6 q^{77} -462830. q^{78} +539646. i q^{79} +2.39938e6i q^{80} +59049.0 q^{81} -1.56857e6 q^{82} -221270. i q^{83} +1.58294e6i q^{84} +221943. q^{85} +1.28562e6i q^{86} +93999.3 q^{87} -3.57504e6i q^{88} -1.34727e6i q^{89} -604204. i q^{90} +1.12867e6i q^{91} +(-1.88991e6 - 937114. i) q^{92} +50717.6 q^{93} +1.83579e6 q^{94} +589519. q^{95} -1.88955e6 q^{96} +1.22527e6i q^{97} +3.47253e6 q^{98} +515512. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.4071 −1.92588 −0.962942 0.269709i \(-0.913072\pi\)
−0.962942 + 0.269709i \(0.913072\pi\)
\(3\) 15.5885 0.577350
\(4\) 173.378 2.70903
\(5\) 161.383i 1.29106i 0.763734 + 0.645531i \(0.223364\pi\)
−0.763734 + 0.645531i \(0.776636\pi\)
\(6\) −240.172 −1.11191
\(7\) 585.691i 1.70756i 0.520638 + 0.853778i \(0.325694\pi\)
−0.520638 + 0.853778i \(0.674306\pi\)
\(8\) −1685.19 −3.29139
\(9\) 243.000 0.333333
\(10\) 2486.44i 2.48644i
\(11\) 2121.45i 1.59387i 0.604062 + 0.796937i \(0.293548\pi\)
−0.604062 + 0.796937i \(0.706452\pi\)
\(12\) 2702.69 1.56406
\(13\) 1927.08 0.877139 0.438570 0.898697i \(-0.355485\pi\)
0.438570 + 0.898697i \(0.355485\pi\)
\(14\) 9023.79i 3.28855i
\(15\) 2515.71i 0.745395i
\(16\) 14867.7 3.62980
\(17\) 1375.26i 0.279923i −0.990157 0.139961i \(-0.955302\pi\)
0.990157 0.139961i \(-0.0446978\pi\)
\(18\) −3743.92 −0.641961
\(19\) 3652.93i 0.532574i −0.963894 0.266287i \(-0.914203\pi\)
0.963894 0.266287i \(-0.0857969\pi\)
\(20\) 27980.2i 3.49752i
\(21\) 9130.03i 0.985857i
\(22\) 32685.3i 3.06962i
\(23\) −10900.5 5405.04i −0.895909 0.444238i
\(24\) −26269.5 −1.90028
\(25\) −10419.4 −0.666842
\(26\) −29690.6 −1.68927
\(27\) 3788.00 0.192450
\(28\) 101546.i 4.62581i
\(29\) 6030.06 0.247245 0.123623 0.992329i \(-0.460549\pi\)
0.123623 + 0.992329i \(0.460549\pi\)
\(30\) 38759.7i 1.43554i
\(31\) 3253.53 0.109212 0.0546060 0.998508i \(-0.482610\pi\)
0.0546060 + 0.998508i \(0.482610\pi\)
\(32\) −121215. −3.69919
\(33\) 33070.1i 0.920224i
\(34\) 21188.7i 0.539098i
\(35\) −94520.5 −2.20456
\(36\) 42130.8 0.903009
\(37\) 25548.1i 0.504375i 0.967678 + 0.252187i \(0.0811499\pi\)
−0.967678 + 0.252187i \(0.918850\pi\)
\(38\) 56280.9i 1.02568i
\(39\) 30040.1 0.506417
\(40\) 271961.i 4.24939i
\(41\) 101808. 1.47717 0.738587 0.674158i \(-0.235493\pi\)
0.738587 + 0.674158i \(0.235493\pi\)
\(42\) 140667.i 1.89865i
\(43\) 83443.8i 1.04952i −0.851252 0.524758i \(-0.824156\pi\)
0.851252 0.524758i \(-0.175844\pi\)
\(44\) 367812.i 4.31785i
\(45\) 39216.0i 0.430354i
\(46\) 167945. + 83275.8i 1.72542 + 0.855550i
\(47\) −119152. −1.14765 −0.573824 0.818979i \(-0.694541\pi\)
−0.573824 + 0.818979i \(0.694541\pi\)
\(48\) 231764. 2.09567
\(49\) −225385. −1.91574
\(50\) 160532. 1.28426
\(51\) 21438.2i 0.161613i
\(52\) 334112. 2.37619
\(53\) 87012.7i 0.584461i 0.956348 + 0.292230i \(0.0943974\pi\)
−0.956348 + 0.292230i \(0.905603\pi\)
\(54\) −58361.9 −0.370636
\(55\) −342365. −2.05779
\(56\) 987001.i 5.62023i
\(57\) 56943.5i 0.307482i
\(58\) −92905.5 −0.476165
\(59\) 42117.7 0.205073 0.102537 0.994729i \(-0.467304\pi\)
0.102537 + 0.994729i \(0.467304\pi\)
\(60\) 436168.i 2.01930i
\(61\) 112804.i 0.496974i 0.968635 + 0.248487i \(0.0799333\pi\)
−0.968635 + 0.248487i \(0.920067\pi\)
\(62\) −50127.4 −0.210329
\(63\) 142323.i 0.569185i
\(64\) 916036. 3.49440
\(65\) 310997.i 1.13244i
\(66\) 509513.i 1.77224i
\(67\) 49872.3i 0.165819i 0.996557 + 0.0829096i \(0.0264213\pi\)
−0.996557 + 0.0829096i \(0.973579\pi\)
\(68\) 238439.i 0.758318i
\(69\) −169922. 84256.3i −0.517253 0.256481i
\(70\) 1.45628e6 4.24573
\(71\) 234036. 0.653896 0.326948 0.945042i \(-0.393980\pi\)
0.326948 + 0.945042i \(0.393980\pi\)
\(72\) −409501. −1.09713
\(73\) −75271.3 −0.193491 −0.0967455 0.995309i \(-0.530843\pi\)
−0.0967455 + 0.995309i \(0.530843\pi\)
\(74\) 393621.i 0.971367i
\(75\) −162422. −0.385001
\(76\) 633336.i 1.44276i
\(77\) −1.24251e6 −2.72163
\(78\) −462830. −0.975300
\(79\) 539646.i 1.09453i 0.836960 + 0.547265i \(0.184331\pi\)
−0.836960 + 0.547265i \(0.815669\pi\)
\(80\) 2.39938e6i 4.68630i
\(81\) 59049.0 0.111111
\(82\) −1.56857e6 −2.84487
\(83\) 221270.i 0.386981i −0.981102 0.193490i \(-0.938019\pi\)
0.981102 0.193490i \(-0.0619808\pi\)
\(84\) 1.58294e6i 2.67071i
\(85\) 221943. 0.361398
\(86\) 1.28562e6i 2.02124i
\(87\) 93999.3 0.142747
\(88\) 3.57504e6i 5.24606i
\(89\) 1.34727e6i 1.91110i −0.294825 0.955551i \(-0.595261\pi\)
0.294825 0.955551i \(-0.404739\pi\)
\(90\) 604204.i 0.828812i
\(91\) 1.12867e6i 1.49776i
\(92\) −1.88991e6 937114.i −2.42704 1.20345i
\(93\) 50717.6 0.0630535
\(94\) 1.83579e6 2.21024
\(95\) 589519. 0.687586
\(96\) −1.88955e6 −2.13573
\(97\) 1.22527e6i 1.34251i 0.741229 + 0.671253i \(0.234243\pi\)
−0.741229 + 0.671253i \(0.765757\pi\)
\(98\) 3.47253e6 3.68950
\(99\) 515512.i 0.531292i
\(100\) −1.80649e6 −1.80649
\(101\) 1.49877e6 1.45469 0.727346 0.686271i \(-0.240754\pi\)
0.727346 + 0.686271i \(0.240754\pi\)
\(102\) 330300.i 0.311249i
\(103\) 505604.i 0.462700i 0.972871 + 0.231350i \(0.0743142\pi\)
−0.972871 + 0.231350i \(0.925686\pi\)
\(104\) −3.24749e6 −2.88701
\(105\) −1.47343e6 −1.27280
\(106\) 1.34061e6i 1.12560i
\(107\) 1.43394e6i 1.17052i −0.810846 0.585260i \(-0.800993\pi\)
0.810846 0.585260i \(-0.199007\pi\)
\(108\) 656754. 0.521353
\(109\) 2.08905e6i 1.61313i −0.591146 0.806565i \(-0.701324\pi\)
0.591146 0.806565i \(-0.298676\pi\)
\(110\) 5.27484e6 3.96307
\(111\) 398255.i 0.291201i
\(112\) 8.70786e6i 6.19809i
\(113\) 320916.i 0.222411i −0.993797 0.111205i \(-0.964529\pi\)
0.993797 0.111205i \(-0.0354711\pi\)
\(114\) 877332.i 0.592174i
\(115\) 872281. 1.75916e6i 0.573539 1.15667i
\(116\) 1.04548e6 0.669794
\(117\) 468279. 0.292380
\(118\) −648911. −0.394947
\(119\) 805478. 0.477983
\(120\) 4.23945e6i 2.45338i
\(121\) −2.72898e6 −1.54044
\(122\) 1.73797e6i 0.957113i
\(123\) 1.58703e6 0.852847
\(124\) 564090. 0.295858
\(125\) 840094.i 0.430128i
\(126\) 2.19278e6i 1.09618i
\(127\) −2.62996e6 −1.28392 −0.641960 0.766738i \(-0.721878\pi\)
−0.641960 + 0.766738i \(0.721878\pi\)
\(128\) −6.35568e6 −3.03062
\(129\) 1.30076e6i 0.605938i
\(130\) 4.79155e6i 2.18095i
\(131\) 1.69152e6 0.752423 0.376212 0.926534i \(-0.377227\pi\)
0.376212 + 0.926534i \(0.377227\pi\)
\(132\) 5.73362e6i 2.49291i
\(133\) 2.13949e6 0.909400
\(134\) 768386.i 0.319349i
\(135\) 611317.i 0.248465i
\(136\) 2.31757e6i 0.921334i
\(137\) 2.53094e6i 0.984283i 0.870515 + 0.492141i \(0.163786\pi\)
−0.870515 + 0.492141i \(0.836214\pi\)
\(138\) 2.61801e6 + 1.29814e6i 0.996169 + 0.493952i
\(139\) 704489. 0.262319 0.131160 0.991361i \(-0.458130\pi\)
0.131160 + 0.991361i \(0.458130\pi\)
\(140\) −1.63878e7 −5.97221
\(141\) −1.85740e6 −0.662595
\(142\) −3.60582e6 −1.25933
\(143\) 4.08819e6i 1.39805i
\(144\) 3.61284e6 1.20993
\(145\) 973148.i 0.319209i
\(146\) 1.15971e6 0.372641
\(147\) −3.51341e6 −1.10606
\(148\) 4.42947e6i 1.36636i
\(149\) 541350.i 0.163651i −0.996647 0.0818256i \(-0.973925\pi\)
0.996647 0.0818256i \(-0.0260751\pi\)
\(150\) 2.50245e6 0.741467
\(151\) 56757.2 0.0164850 0.00824252 0.999966i \(-0.497376\pi\)
0.00824252 + 0.999966i \(0.497376\pi\)
\(152\) 6.15587e6i 1.75291i
\(153\) 334188.i 0.0933076i
\(154\) 1.91435e7 5.24154
\(155\) 525064.i 0.140999i
\(156\) 5.20829e6 1.37190
\(157\) 1.67461e6i 0.432727i 0.976313 + 0.216363i \(0.0694195\pi\)
−0.976313 + 0.216363i \(0.930580\pi\)
\(158\) 8.31436e6i 2.10794i
\(159\) 1.35639e6i 0.337439i
\(160\) 1.95620e7i 4.77588i
\(161\) 3.16569e6 6.38434e6i 0.758561 1.52981i
\(162\) −909772. −0.213987
\(163\) −5.46518e6 −1.26195 −0.630974 0.775804i \(-0.717345\pi\)
−0.630974 + 0.775804i \(0.717345\pi\)
\(164\) 1.76513e7 4.00171
\(165\) −5.33694e6 −1.18807
\(166\) 3.40913e6i 0.745279i
\(167\) 5.53280e6 1.18794 0.593971 0.804486i \(-0.297559\pi\)
0.593971 + 0.804486i \(0.297559\pi\)
\(168\) 1.53858e7i 3.24484i
\(169\) −1.11319e6 −0.230626
\(170\) −3.41950e6 −0.696010
\(171\) 887661.i 0.177525i
\(172\) 1.44673e7i 2.84317i
\(173\) −254750. −0.0492012 −0.0246006 0.999697i \(-0.507831\pi\)
−0.0246006 + 0.999697i \(0.507831\pi\)
\(174\) −1.44825e6 −0.274914
\(175\) 6.10255e6i 1.13867i
\(176\) 3.15410e7i 5.78545i
\(177\) 656550. 0.118399
\(178\) 2.07574e7i 3.68056i
\(179\) −2.38809e6 −0.416381 −0.208190 0.978088i \(-0.566757\pi\)
−0.208190 + 0.978088i \(0.566757\pi\)
\(180\) 6.79918e6i 1.16584i
\(181\) 1.11133e6i 0.187416i 0.995600 + 0.0937082i \(0.0298721\pi\)
−0.995600 + 0.0937082i \(0.970128\pi\)
\(182\) 1.73895e7i 2.88452i
\(183\) 1.75843e6i 0.286928i
\(184\) 1.83695e7 + 9.10852e6i 2.94878 + 1.46216i
\(185\) −4.12302e6 −0.651179
\(186\) −781409. −0.121434
\(187\) 2.91754e6 0.446162
\(188\) −2.06583e7 −3.10901
\(189\) 2.21860e6i 0.328619i
\(190\) −9.08276e6 −1.32421
\(191\) 1.57559e6i 0.226122i 0.993588 + 0.113061i \(0.0360655\pi\)
−0.993588 + 0.113061i \(0.963934\pi\)
\(192\) 1.42796e7 2.01749
\(193\) 1.05732e7 1.47074 0.735370 0.677665i \(-0.237008\pi\)
0.735370 + 0.677665i \(0.237008\pi\)
\(194\) 1.88778e7i 2.58551i
\(195\) 4.84796e6i 0.653815i
\(196\) −3.90768e7 −5.18980
\(197\) −5.56206e6 −0.727507 −0.363753 0.931495i \(-0.618505\pi\)
−0.363753 + 0.931495i \(0.618505\pi\)
\(198\) 7.94252e6i 1.02321i
\(199\) 1.20492e7i 1.52897i −0.644639 0.764487i \(-0.722992\pi\)
0.644639 0.764487i \(-0.277008\pi\)
\(200\) 1.75587e7 2.19483
\(201\) 777432.i 0.0957358i
\(202\) −2.30916e7 −2.80157
\(203\) 3.53175e6i 0.422185i
\(204\) 3.71690e6i 0.437815i
\(205\) 1.64301e7i 1.90712i
\(206\) 7.78988e6i 0.891105i
\(207\) −2.64883e6 1.31343e6i −0.298636 0.148079i
\(208\) 2.86511e7 3.18384
\(209\) 7.74949e6 0.848856
\(210\) 2.27012e7 2.45127
\(211\) −2.56616e6 −0.273172 −0.136586 0.990628i \(-0.543613\pi\)
−0.136586 + 0.990628i \(0.543613\pi\)
\(212\) 1.50861e7i 1.58332i
\(213\) 3.64827e6 0.377527
\(214\) 2.20928e7i 2.25428i
\(215\) 1.34664e7 1.35499
\(216\) −6.38349e6 −0.633428
\(217\) 1.90557e6i 0.186485i
\(218\) 3.21861e7i 3.10670i
\(219\) −1.17336e6 −0.111712
\(220\) −5.93585e7 −5.57461
\(221\) 2.65023e6i 0.245531i
\(222\) 6.13595e6i 0.560819i
\(223\) −3.21603e6 −0.290005 −0.145002 0.989431i \(-0.546319\pi\)
−0.145002 + 0.989431i \(0.546319\pi\)
\(224\) 7.09946e7i 6.31656i
\(225\) −2.53191e6 −0.222281
\(226\) 4.94437e6i 0.428337i
\(227\) 4.49495e6i 0.384280i −0.981368 0.192140i \(-0.938457\pi\)
0.981368 0.192140i \(-0.0615427\pi\)
\(228\) 9.87273e6i 0.832976i
\(229\) 1.69502e7i 1.41146i 0.708483 + 0.705728i \(0.249380\pi\)
−0.708483 + 0.705728i \(0.750620\pi\)
\(230\) −1.34393e7 + 2.71034e7i −1.10457 + 2.22762i
\(231\) −1.93689e7 −1.57133
\(232\) −1.01618e7 −0.813779
\(233\) 8.99993e6 0.711494 0.355747 0.934582i \(-0.384226\pi\)
0.355747 + 0.934582i \(0.384226\pi\)
\(234\) −7.21481e6 −0.563089
\(235\) 1.92291e7i 1.48168i
\(236\) 7.30228e6 0.555549
\(237\) 8.41224e6i 0.631927i
\(238\) −1.24101e7 −0.920540
\(239\) −2.11370e7 −1.54828 −0.774141 0.633013i \(-0.781818\pi\)
−0.774141 + 0.633013i \(0.781818\pi\)
\(240\) 3.74027e7i 2.70564i
\(241\) 9.50399e6i 0.678977i 0.940610 + 0.339488i \(0.110254\pi\)
−0.940610 + 0.339488i \(0.889746\pi\)
\(242\) 4.20455e7 2.96670
\(243\) 920483. 0.0641500
\(244\) 1.95576e7i 1.34632i
\(245\) 3.63733e7i 2.47335i
\(246\) −2.44516e7 −1.64248
\(247\) 7.03946e6i 0.467142i
\(248\) −5.48282e6 −0.359459
\(249\) 3.44926e6i 0.223423i
\(250\) 1.29434e7i 0.828377i
\(251\) 6.04218e6i 0.382096i 0.981581 + 0.191048i \(0.0611886\pi\)
−0.981581 + 0.191048i \(0.938811\pi\)
\(252\) 2.46756e7i 1.54194i
\(253\) 1.14665e7 2.31249e7i 0.708059 1.42797i
\(254\) 4.05200e7 2.47268
\(255\) 3.45975e6 0.208653
\(256\) 3.92960e7 2.34223
\(257\) −2.02760e7 −1.19449 −0.597246 0.802058i \(-0.703738\pi\)
−0.597246 + 0.802058i \(0.703738\pi\)
\(258\) 2.00409e7i 1.16697i
\(259\) −1.49633e7 −0.861247
\(260\) 5.39199e7i 3.06781i
\(261\) 1.46530e6 0.0824150
\(262\) −2.60613e7 −1.44908
\(263\) 2.16567e7i 1.19049i 0.803544 + 0.595245i \(0.202945\pi\)
−0.803544 + 0.595245i \(0.797055\pi\)
\(264\) 5.57294e7i 3.02881i
\(265\) −1.40424e7 −0.754575
\(266\) −3.29632e7 −1.75140
\(267\) 2.10018e7i 1.10338i
\(268\) 8.64675e6i 0.449209i
\(269\) 3.45160e7 1.77323 0.886613 0.462512i \(-0.153052\pi\)
0.886613 + 0.462512i \(0.153052\pi\)
\(270\) 9.41861e6i 0.478515i
\(271\) 1.67829e7 0.843255 0.421627 0.906769i \(-0.361459\pi\)
0.421627 + 0.906769i \(0.361459\pi\)
\(272\) 2.04469e7i 1.01606i
\(273\) 1.75942e7i 0.864734i
\(274\) 3.89943e7i 1.89561i
\(275\) 2.21042e7i 1.06286i
\(276\) −2.94608e7 1.46082e7i −1.40125 0.694814i
\(277\) 3.71225e7 1.74662 0.873308 0.487168i \(-0.161970\pi\)
0.873308 + 0.487168i \(0.161970\pi\)
\(278\) −1.08541e7 −0.505196
\(279\) 790609. 0.0364040
\(280\) 1.59285e8 7.25606
\(281\) 3.49045e7i 1.57312i −0.617511 0.786562i \(-0.711859\pi\)
0.617511 0.786562i \(-0.288141\pi\)
\(282\) 2.86171e7 1.27608
\(283\) 6.06094e6i 0.267412i 0.991021 + 0.133706i \(0.0426878\pi\)
−0.991021 + 0.133706i \(0.957312\pi\)
\(284\) 4.05767e7 1.77142
\(285\) 9.18970e6 0.396978
\(286\) 6.29870e7i 2.69248i
\(287\) 5.96283e7i 2.52236i
\(288\) −2.94552e7 −1.23306
\(289\) 2.22462e7 0.921643
\(290\) 1.49934e7i 0.614759i
\(291\) 1.91000e7i 0.775096i
\(292\) −1.30504e7 −0.524173
\(293\) 3.32779e7i 1.32298i 0.749954 + 0.661490i \(0.230076\pi\)
−0.749954 + 0.661490i \(0.769924\pi\)
\(294\) 5.41314e7 2.13013
\(295\) 6.79708e6i 0.264762i
\(296\) 4.30534e7i 1.66009i
\(297\) 8.03603e6i 0.306741i
\(298\) 8.34061e6i 0.315173i
\(299\) −2.10061e7 1.04159e7i −0.785837 0.389659i
\(300\) −2.81604e7 −1.04298
\(301\) 4.88723e7 1.79211
\(302\) −874462. −0.0317483
\(303\) 2.33635e7 0.839866
\(304\) 5.43105e7i 1.93314i
\(305\) −1.82046e7 −0.641624
\(306\) 5.14886e6i 0.179699i
\(307\) 8.91735e6 0.308192 0.154096 0.988056i \(-0.450754\pi\)
0.154096 + 0.988056i \(0.450754\pi\)
\(308\) −2.15424e8 −7.37297
\(309\) 7.88159e6i 0.267140i
\(310\) 8.08970e6i 0.271548i
\(311\) 3.92020e7 1.30325 0.651625 0.758542i \(-0.274088\pi\)
0.651625 + 0.758542i \(0.274088\pi\)
\(312\) −5.06233e7 −1.66681
\(313\) 2.03457e7i 0.663497i −0.943368 0.331749i \(-0.892361\pi\)
0.943368 0.331749i \(-0.107639\pi\)
\(314\) 2.58008e7i 0.833381i
\(315\) −2.29685e7 −0.734853
\(316\) 9.35625e7i 2.96511i
\(317\) −1.36669e7 −0.429033 −0.214517 0.976720i \(-0.568818\pi\)
−0.214517 + 0.976720i \(0.568818\pi\)
\(318\) 2.08981e7i 0.649867i
\(319\) 1.27925e7i 0.394078i
\(320\) 1.47832e8i 4.51149i
\(321\) 2.23529e7i 0.675800i
\(322\) −4.87739e7 + 9.83640e7i −1.46090 + 2.94624i
\(323\) −5.02372e6 −0.149080
\(324\) 1.02378e7 0.301003
\(325\) −2.00790e7 −0.584913
\(326\) 8.42024e7 2.43037
\(327\) 3.25651e7i 0.931341i
\(328\) −1.71566e8 −4.86195
\(329\) 6.97864e7i 1.95967i
\(330\) 8.22266e7 2.28808
\(331\) −9.55678e6 −0.263528 −0.131764 0.991281i \(-0.542064\pi\)
−0.131764 + 0.991281i \(0.542064\pi\)
\(332\) 3.83634e7i 1.04834i
\(333\) 6.20818e6i 0.168125i
\(334\) −8.52442e7 −2.28784
\(335\) −8.04853e6 −0.214083
\(336\) 1.35742e8i 3.57847i
\(337\) 3.96943e6i 0.103714i −0.998655 0.0518572i \(-0.983486\pi\)
0.998655 0.0518572i \(-0.0165141\pi\)
\(338\) 1.71510e7 0.444160
\(339\) 5.00258e6i 0.128409i
\(340\) 3.84800e7 0.979036
\(341\) 6.90220e6i 0.174070i
\(342\) 1.36763e7i 0.341892i
\(343\) 6.31003e7i 1.56368i
\(344\) 1.40619e8i 3.45436i
\(345\) 1.35975e7 2.74225e7i 0.331133 0.667806i
\(346\) 3.92495e6 0.0947558
\(347\) 4.00398e6 0.0958304 0.0479152 0.998851i \(-0.484742\pi\)
0.0479152 + 0.998851i \(0.484742\pi\)
\(348\) 1.62974e7 0.386705
\(349\) 6.77965e7 1.59489 0.797446 0.603391i \(-0.206184\pi\)
0.797446 + 0.603391i \(0.206184\pi\)
\(350\) 9.40225e7i 2.19294i
\(351\) 7.29975e6 0.168806
\(352\) 2.57151e8i 5.89604i
\(353\) −3.84370e7 −0.873826 −0.436913 0.899504i \(-0.643928\pi\)
−0.436913 + 0.899504i \(0.643928\pi\)
\(354\) −1.01155e7 −0.228023
\(355\) 3.77695e7i 0.844220i
\(356\) 2.33586e8i 5.17723i
\(357\) 1.25562e7 0.275964
\(358\) 3.67934e7 0.801901
\(359\) 7.06002e7i 1.52589i 0.646465 + 0.762944i \(0.276247\pi\)
−0.646465 + 0.762944i \(0.723753\pi\)
\(360\) 6.60864e7i 1.41646i
\(361\) 3.37020e7 0.716365
\(362\) 1.71223e7i 0.360942i
\(363\) −4.25406e7 −0.889372
\(364\) 1.95687e8i 4.05748i
\(365\) 1.21475e7i 0.249809i
\(366\) 2.70923e7i 0.552590i
\(367\) 7.35519e7i 1.48797i 0.668194 + 0.743987i \(0.267068\pi\)
−0.668194 + 0.743987i \(0.732932\pi\)
\(368\) −1.62065e8 8.03603e7i −3.25197 1.61249i
\(369\) 2.47394e7 0.492391
\(370\) 6.35237e7 1.25409
\(371\) −5.09626e7 −0.997999
\(372\) 8.79330e6 0.170814
\(373\) 4.00245e7i 0.771257i −0.922654 0.385629i \(-0.873985\pi\)
0.922654 0.385629i \(-0.126015\pi\)
\(374\) −4.49508e7 −0.859255
\(375\) 1.30958e7i 0.248335i
\(376\) 2.00794e8 3.77735
\(377\) 1.16204e7 0.216868
\(378\) 3.41821e7i 0.632882i
\(379\) 1.08610e7i 0.199504i −0.995012 0.0997522i \(-0.968195\pi\)
0.995012 0.0997522i \(-0.0318050\pi\)
\(380\) 1.02210e8 1.86269
\(381\) −4.09970e7 −0.741272
\(382\) 2.42752e7i 0.435484i
\(383\) 9.30194e7i 1.65568i 0.560962 + 0.827842i \(0.310431\pi\)
−0.560962 + 0.827842i \(0.689569\pi\)
\(384\) −9.90752e7 −1.74973
\(385\) 2.00520e8i 3.51379i
\(386\) −1.62903e8 −2.83248
\(387\) 2.02768e7i 0.349838i
\(388\) 2.12434e8i 3.63688i
\(389\) 1.08899e7i 0.185001i −0.995713 0.0925004i \(-0.970514\pi\)
0.995713 0.0925004i \(-0.0294859\pi\)
\(390\) 7.46928e7i 1.25917i
\(391\) −7.43334e6 + 1.49911e7i −0.124352 + 0.250785i
\(392\) 3.79817e8 6.30546
\(393\) 2.63681e7 0.434412
\(394\) 8.56950e7 1.40109
\(395\) −8.70895e7 −1.41311
\(396\) 8.93783e7i 1.43928i
\(397\) −2.63976e7 −0.421883 −0.210942 0.977499i \(-0.567653\pi\)
−0.210942 + 0.977499i \(0.567653\pi\)
\(398\) 1.85643e8i 2.94463i
\(399\) 3.33513e7 0.525042
\(400\) −1.54912e8 −2.42050
\(401\) 9.57063e7i 1.48425i 0.670261 + 0.742125i \(0.266182\pi\)
−0.670261 + 0.742125i \(0.733818\pi\)
\(402\) 1.19780e7i 0.184376i
\(403\) 6.26980e6 0.0957941
\(404\) 2.59853e8 3.94080
\(405\) 9.52949e6i 0.143451i
\(406\) 5.44140e7i 0.813078i
\(407\) −5.41989e7 −0.803910
\(408\) 3.61274e7i 0.531932i
\(409\) −7.91380e7 −1.15669 −0.578343 0.815794i \(-0.696300\pi\)
−0.578343 + 0.815794i \(0.696300\pi\)
\(410\) 2.53140e8i 3.67290i
\(411\) 3.94534e7i 0.568276i
\(412\) 8.76605e7i 1.25347i
\(413\) 2.46680e7i 0.350174i
\(414\) 4.08107e7 + 2.02360e7i 0.575139 + 0.285183i
\(415\) 3.57092e7 0.499616
\(416\) −2.33590e8 −3.24470
\(417\) 1.09819e7 0.151450
\(418\) −1.19397e8 −1.63480
\(419\) 2.39477e7i 0.325553i −0.986663 0.162777i \(-0.947955\pi\)
0.986663 0.162777i \(-0.0520450\pi\)
\(420\) −2.55460e8 −3.44806
\(421\) 1.75288e7i 0.234912i 0.993078 + 0.117456i \(0.0374740\pi\)
−0.993078 + 0.117456i \(0.962526\pi\)
\(422\) 3.95371e7 0.526098
\(423\) −2.89540e7 −0.382549
\(424\) 1.46633e8i 1.92369i
\(425\) 1.43294e7i 0.186664i
\(426\) −5.62091e7 −0.727073
\(427\) −6.60681e7 −0.848610
\(428\) 2.48613e8i 3.17097i
\(429\) 6.37286e7i 0.807165i
\(430\) −2.07478e8 −2.60955
\(431\) 1.35583e8i 1.69345i 0.532030 + 0.846725i \(0.321429\pi\)
−0.532030 + 0.846725i \(0.678571\pi\)
\(432\) 5.63186e7 0.698556
\(433\) 3.97273e7i 0.489356i −0.969604 0.244678i \(-0.921318\pi\)
0.969604 0.244678i \(-0.0786822\pi\)
\(434\) 2.93592e7i 0.359149i
\(435\) 1.51699e7i 0.184295i
\(436\) 3.62195e8i 4.37001i
\(437\) −1.97442e7 + 3.98188e7i −0.236590 + 0.477138i
\(438\) 1.80781e7 0.215145
\(439\) −1.21074e7 −0.143106 −0.0715528 0.997437i \(-0.522795\pi\)
−0.0715528 + 0.997437i \(0.522795\pi\)
\(440\) 5.76950e8 6.77299
\(441\) −5.47687e7 −0.638582
\(442\) 4.08323e7i 0.472865i
\(443\) −3.79840e7 −0.436908 −0.218454 0.975847i \(-0.570101\pi\)
−0.218454 + 0.975847i \(0.570101\pi\)
\(444\) 6.90486e7i 0.788871i
\(445\) 2.17426e8 2.46735
\(446\) 4.95495e7 0.558515
\(447\) 8.43881e6i 0.0944841i
\(448\) 5.36515e8i 5.96688i
\(449\) −6.88185e7 −0.760267 −0.380133 0.924932i \(-0.624122\pi\)
−0.380133 + 0.924932i \(0.624122\pi\)
\(450\) 3.90094e7 0.428086
\(451\) 2.15981e8i 2.35443i
\(452\) 5.56396e7i 0.602516i
\(453\) 884757. 0.00951764
\(454\) 6.92540e7i 0.740078i
\(455\) −1.82148e8 −1.93371
\(456\) 9.59606e7i 1.01204i
\(457\) 3.14473e7i 0.329485i −0.986337 0.164742i \(-0.947321\pi\)
0.986337 0.164742i \(-0.0526792\pi\)
\(458\) 2.61152e8i 2.71830i
\(459\) 5.20948e6i 0.0538711i
\(460\) 1.51234e8 3.04999e8i 1.55373 3.13346i
\(461\) 3.45124e7 0.352267 0.176134 0.984366i \(-0.443641\pi\)
0.176134 + 0.984366i \(0.443641\pi\)
\(462\) 2.98417e8 3.02620
\(463\) −1.45561e8 −1.46656 −0.733281 0.679925i \(-0.762012\pi\)
−0.733281 + 0.679925i \(0.762012\pi\)
\(464\) 8.96529e7 0.897450
\(465\) 8.18494e6i 0.0814060i
\(466\) −1.38663e8 −1.37025
\(467\) 1.85068e8i 1.81711i −0.417768 0.908554i \(-0.637188\pi\)
0.417768 0.908554i \(-0.362812\pi\)
\(468\) 8.11892e7 0.792065
\(469\) −2.92098e7 −0.283146
\(470\) 2.96264e8i 2.85355i
\(471\) 2.61045e7i 0.249835i
\(472\) −7.09764e7 −0.674975
\(473\) 1.77022e8 1.67280
\(474\) 1.29608e8i 1.21702i
\(475\) 3.80613e7i 0.355143i
\(476\) 1.39652e8 1.29487
\(477\) 2.11441e7i 0.194820i
\(478\) 3.25660e8 2.98181
\(479\) 8.37663e7i 0.762189i −0.924536 0.381095i \(-0.875547\pi\)
0.924536 0.381095i \(-0.124453\pi\)
\(480\) 3.04941e8i 2.75736i
\(481\) 4.92331e7i 0.442407i
\(482\) 1.46429e8i 1.30763i
\(483\) 4.93482e7 9.95221e7i 0.437955 0.883238i
\(484\) −4.73144e8 −4.17308
\(485\) −1.97737e8 −1.73326
\(486\) −1.41819e7 −0.123545
\(487\) −1.52782e8 −1.32277 −0.661387 0.750045i \(-0.730032\pi\)
−0.661387 + 0.750045i \(0.730032\pi\)
\(488\) 1.90096e8i 1.63573i
\(489\) −8.51937e7 −0.728587
\(490\) 5.60406e8i 4.76338i
\(491\) −1.77832e8 −1.50233 −0.751165 0.660115i \(-0.770508\pi\)
−0.751165 + 0.660115i \(0.770508\pi\)
\(492\) 2.75157e8 2.31039
\(493\) 8.29290e6i 0.0692095i
\(494\) 1.08457e8i 0.899660i
\(495\) −8.31947e7 −0.685930
\(496\) 4.83724e7 0.396418
\(497\) 1.37073e8i 1.11656i
\(498\) 5.31431e7i 0.430287i
\(499\) 1.78735e8 1.43850 0.719248 0.694753i \(-0.244486\pi\)
0.719248 + 0.694753i \(0.244486\pi\)
\(500\) 1.45654e8i 1.16523i
\(501\) 8.62478e7 0.685859
\(502\) 9.30922e7i 0.735872i
\(503\) 1.25380e8i 0.985204i −0.870255 0.492602i \(-0.836046\pi\)
0.870255 0.492602i \(-0.163954\pi\)
\(504\) 2.39841e8i 1.87341i
\(505\) 2.41876e8i 1.87810i
\(506\) −1.76665e8 + 3.56287e8i −1.36364 + 2.75010i
\(507\) −1.73529e7 −0.133152
\(508\) −4.55976e8 −3.47817
\(509\) 3.45688e7 0.262139 0.131069 0.991373i \(-0.458159\pi\)
0.131069 + 0.991373i \(0.458159\pi\)
\(510\) −5.33047e7 −0.401841
\(511\) 4.40858e7i 0.330397i
\(512\) −1.98673e8 −1.48023
\(513\) 1.38373e7i 0.102494i
\(514\) 3.12394e8 2.30045
\(515\) −8.15958e7 −0.597374
\(516\) 2.25523e8i 1.64150i
\(517\) 2.52775e8i 1.82921i
\(518\) 2.30540e8 1.65866
\(519\) −3.97116e6 −0.0284063
\(520\) 5.24089e8i 3.72730i
\(521\) 1.84179e8i 1.30235i 0.758928 + 0.651175i \(0.225724\pi\)
−0.758928 + 0.651175i \(0.774276\pi\)
\(522\) −2.25760e7 −0.158722
\(523\) 1.67317e8i 1.16959i −0.811180 0.584797i \(-0.801174\pi\)
0.811180 0.584797i \(-0.198826\pi\)
\(524\) 2.93271e8 2.03834
\(525\) 9.51294e7i 0.657411i
\(526\) 3.33667e8i 2.29274i
\(527\) 4.47445e6i 0.0305709i
\(528\) 4.91675e8i 3.34023i
\(529\) 8.96069e7 + 1.17836e8i 0.605306 + 0.795993i
\(530\) 2.16352e8 1.45322
\(531\) 1.02346e7 0.0683577
\(532\) 3.70939e8 2.46359
\(533\) 1.96192e8 1.29569
\(534\) 3.23577e8i 2.12497i
\(535\) 2.31413e8 1.51121
\(536\) 8.40443e7i 0.545775i
\(537\) −3.72266e7 −0.240398
\(538\) −5.31791e8 −3.41503
\(539\) 4.78143e8i 3.05346i
\(540\) 1.05989e8i 0.673099i
\(541\) 1.47750e8 0.933115 0.466557 0.884491i \(-0.345494\pi\)
0.466557 + 0.884491i \(0.345494\pi\)
\(542\) −2.58575e8 −1.62401
\(543\) 1.73239e7i 0.108205i
\(544\) 1.66702e8i 1.03549i
\(545\) 3.37137e8 2.08265
\(546\) 2.71076e8i 1.66538i
\(547\) 5.42473e7 0.331449 0.165724 0.986172i \(-0.447004\pi\)
0.165724 + 0.986172i \(0.447004\pi\)
\(548\) 4.38808e8i 2.66645i
\(549\) 2.74113e7i 0.165658i
\(550\) 3.40561e8i 2.04695i
\(551\) 2.20274e7i 0.131676i
\(552\) 2.86351e8 + 1.41988e8i 1.70248 + 0.844178i
\(553\) −3.16066e8 −1.86897
\(554\) −5.71948e8 −3.36378
\(555\) −6.42715e7 −0.375958
\(556\) 1.22143e8 0.710630
\(557\) 1.34098e7i 0.0775991i 0.999247 + 0.0387995i \(0.0123534\pi\)
−0.999247 + 0.0387995i \(0.987647\pi\)
\(558\) −1.21810e7 −0.0701098
\(559\) 1.60802e8i 0.920571i
\(560\) −1.40530e9 −8.00211
\(561\) 4.54800e7 0.257592
\(562\) 5.37777e8i 3.02965i
\(563\) 1.61409e8i 0.904490i 0.891894 + 0.452245i \(0.149377\pi\)
−0.891894 + 0.452245i \(0.850623\pi\)
\(564\) −3.22032e8 −1.79499
\(565\) 5.17902e7 0.287146
\(566\) 9.33813e7i 0.515004i
\(567\) 3.45845e7i 0.189728i
\(568\) −3.94396e8 −2.15222
\(569\) 1.91072e6i 0.0103720i 0.999987 + 0.00518598i \(0.00165076\pi\)
−0.999987 + 0.00518598i \(0.998349\pi\)
\(570\) −1.41586e8 −0.764534
\(571\) 2.08656e8i 1.12078i 0.828228 + 0.560392i \(0.189350\pi\)
−0.828228 + 0.560392i \(0.810650\pi\)
\(572\) 7.08801e8i 3.78736i
\(573\) 2.45610e7i 0.130551i
\(574\) 9.18697e8i 4.85776i
\(575\) 1.13577e8 + 5.63173e7i 0.597429 + 0.296236i
\(576\) 2.22597e8 1.16480
\(577\) 3.82528e8 1.99129 0.995647 0.0932020i \(-0.0297102\pi\)
0.995647 + 0.0932020i \(0.0297102\pi\)
\(578\) −3.42749e8 −1.77498
\(579\) 1.64820e8 0.849133
\(580\) 1.68722e8i 0.864745i
\(581\) 1.29596e8 0.660791
\(582\) 2.94276e8i 1.49274i
\(583\) −1.84593e8 −0.931557
\(584\) 1.26846e8 0.636854
\(585\) 7.55722e7i 0.377481i
\(586\) 5.12715e8i 2.54791i
\(587\) 7.94636e7 0.392874 0.196437 0.980516i \(-0.437063\pi\)
0.196437 + 0.980516i \(0.437063\pi\)
\(588\) −6.09147e8 −2.99633
\(589\) 1.18849e7i 0.0581634i
\(590\) 1.04723e8i 0.509901i
\(591\) −8.67039e7 −0.420026
\(592\) 3.79840e8i 1.83078i
\(593\) 3.55660e7 0.170557 0.0852787 0.996357i \(-0.472822\pi\)
0.0852787 + 0.996357i \(0.472822\pi\)
\(594\) 1.23812e8i 0.590748i
\(595\) 1.29990e8i 0.617106i
\(596\) 9.38580e7i 0.443336i
\(597\) 1.87829e8i 0.882754i
\(598\) 3.23643e8 + 1.60479e8i 1.51343 + 0.750437i
\(599\) 3.23906e8 1.50709 0.753543 0.657398i \(-0.228343\pi\)
0.753543 + 0.657398i \(0.228343\pi\)
\(600\) 2.73713e8 1.26719
\(601\) 1.25196e8 0.576721 0.288360 0.957522i \(-0.406890\pi\)
0.288360 + 0.957522i \(0.406890\pi\)
\(602\) −7.52979e8 −3.45139
\(603\) 1.21190e7i 0.0552731i
\(604\) 9.84043e6 0.0446584
\(605\) 4.40410e8i 1.98880i
\(606\) −3.59963e8 −1.61748
\(607\) −1.10872e8 −0.495743 −0.247871 0.968793i \(-0.579731\pi\)
−0.247871 + 0.968793i \(0.579731\pi\)
\(608\) 4.42789e8i 1.97009i
\(609\) 5.50546e7i 0.243748i
\(610\) 2.80479e8 1.23569
\(611\) −2.29615e8 −1.00665
\(612\) 5.79408e7i 0.252773i
\(613\) 4.30728e8i 1.86992i −0.354758 0.934958i \(-0.615437\pi\)
0.354758 0.934958i \(-0.384563\pi\)
\(614\) −1.37390e8 −0.593541
\(615\) 2.56120e8i 1.10108i
\(616\) 2.09387e9 8.95793
\(617\) 4.44968e7i 0.189441i 0.995504 + 0.0947203i \(0.0301957\pi\)
−0.995504 + 0.0947203i \(0.969804\pi\)
\(618\) 1.21432e8i 0.514480i
\(619\) 3.26072e8i 1.37480i 0.726277 + 0.687402i \(0.241249\pi\)
−0.726277 + 0.687402i \(0.758751\pi\)
\(620\) 9.10344e7i 0.381971i
\(621\) −4.12911e7 2.04743e7i −0.172418 0.0854936i
\(622\) −6.03988e8 −2.50991
\(623\) 7.89083e8 3.26331
\(624\) 4.46627e8 1.83819
\(625\) −2.98380e8 −1.22216
\(626\) 3.13467e8i 1.27782i
\(627\) 1.20803e8 0.490087
\(628\) 2.90339e8i 1.17227i
\(629\) 3.51353e7 0.141186
\(630\) 3.53877e8 1.41524
\(631\) 2.03802e8i 0.811187i −0.914054 0.405594i \(-0.867065\pi\)
0.914054 0.405594i \(-0.132935\pi\)
\(632\) 9.09405e8i 3.60252i
\(633\) −4.00025e7 −0.157716
\(634\) 2.10566e8 0.826268
\(635\) 4.24430e8i 1.65762i
\(636\) 2.35169e8i 0.914130i
\(637\) −4.34335e8 −1.68038
\(638\) 1.97094e8i 0.758948i
\(639\) 5.68709e7 0.217965
\(640\) 1.02570e9i 3.91272i
\(641\) 7.89353e7i 0.299707i −0.988708 0.149854i \(-0.952120\pi\)
0.988708 0.149854i \(-0.0478802\pi\)
\(642\) 3.44392e8i 1.30151i
\(643\) 6.79395e7i 0.255558i 0.991803 + 0.127779i \(0.0407848\pi\)
−0.991803 + 0.127779i \(0.959215\pi\)
\(644\) 5.48860e8 1.10690e9i 2.05496 4.14431i
\(645\) 2.09920e8 0.782303
\(646\) 7.74008e7 0.287110
\(647\) 9.78812e7 0.361399 0.180699 0.983538i \(-0.442164\pi\)
0.180699 + 0.983538i \(0.442164\pi\)
\(648\) −9.95088e7 −0.365710
\(649\) 8.93505e7i 0.326861i
\(650\) 3.09358e8 1.12647
\(651\) 2.97048e7i 0.107667i
\(652\) −9.47541e8 −3.41865
\(653\) −3.62932e7 −0.130342 −0.0651712 0.997874i \(-0.520759\pi\)
−0.0651712 + 0.997874i \(0.520759\pi\)
\(654\) 5.01732e8i 1.79365i
\(655\) 2.72982e8i 0.971425i
\(656\) 1.51365e9 5.36185
\(657\) −1.82909e7 −0.0644970
\(658\) 1.07520e9i 3.77410i
\(659\) 4.37953e8i 1.53028i 0.643863 + 0.765141i \(0.277331\pi\)
−0.643863 + 0.765141i \(0.722669\pi\)
\(660\) −9.25307e8 −3.21850
\(661\) 3.87843e8i 1.34292i −0.741039 0.671462i \(-0.765667\pi\)
0.741039 0.671462i \(-0.234333\pi\)
\(662\) 1.47242e8 0.507525
\(663\) 4.13130e7i 0.141758i
\(664\) 3.72883e8i 1.27370i
\(665\) 3.45276e8i 1.17409i
\(666\) 9.56499e7i 0.323789i
\(667\) −6.57308e7 3.25927e7i −0.221509 0.109836i
\(668\) 9.59264e8 3.21817
\(669\) −5.01329e7 −0.167434
\(670\) 1.24004e8 0.412299
\(671\) −2.39307e8 −0.792114
\(672\) 1.10670e9i 3.64687i
\(673\) 5.10336e8 1.67422 0.837108 0.547038i \(-0.184245\pi\)
0.837108 + 0.547038i \(0.184245\pi\)
\(674\) 6.11573e7i 0.199742i
\(675\) −3.94686e7 −0.128334
\(676\) −1.93002e8 −0.624773
\(677\) 5.73800e8i 1.84924i −0.380886 0.924622i \(-0.624381\pi\)
0.380886 0.924622i \(-0.375619\pi\)
\(678\) 7.70751e7i 0.247300i
\(679\) −7.17629e8 −2.29240
\(680\) −3.74017e8 −1.18950
\(681\) 7.00694e7i 0.221864i
\(682\) 1.06343e8i 0.335239i
\(683\) 4.32001e8 1.35588 0.677942 0.735115i \(-0.262872\pi\)
0.677942 + 0.735115i \(0.262872\pi\)
\(684\) 1.53901e8i 0.480919i
\(685\) −4.08450e8 −1.27077
\(686\) 9.72191e8i 3.01147i
\(687\) 2.64227e8i 0.814905i
\(688\) 1.24061e9i 3.80953i
\(689\) 1.67680e8i 0.512653i
\(690\) −2.09498e8 + 4.22501e8i −0.637723 + 1.28612i
\(691\) −5.16043e7 −0.156405 −0.0782027 0.996937i \(-0.524918\pi\)
−0.0782027 + 0.996937i \(0.524918\pi\)
\(692\) −4.41680e7 −0.133287
\(693\) −3.01931e8 −0.907210
\(694\) −6.16896e7 −0.184558
\(695\) 1.13692e8i 0.338670i
\(696\) −1.58407e8 −0.469836
\(697\) 1.40013e8i 0.413495i
\(698\) −1.04455e9 −3.07158
\(699\) 1.40295e8 0.410781
\(700\) 1.05805e9i 3.08468i
\(701\) 6.70008e8i 1.94503i −0.232847 0.972513i \(-0.574804\pi\)
0.232847 0.972513i \(-0.425196\pi\)
\(702\) −1.12468e8 −0.325100
\(703\) 9.33253e7 0.268617
\(704\) 1.94332e9i 5.56964i
\(705\) 2.99752e8i 0.855451i
\(706\) 5.92201e8 1.68289
\(707\) 8.77817e8i 2.48397i
\(708\) 1.13831e8 0.320746
\(709\) 3.62760e8i 1.01784i 0.860813 + 0.508921i \(0.169955\pi\)
−0.860813 + 0.508921i \(0.830045\pi\)
\(710\) 5.81916e8i 1.62587i
\(711\) 1.31134e8i 0.364843i
\(712\) 2.27040e9i 6.29018i
\(713\) −3.54652e7 1.75855e7i −0.0978440 0.0485161i
\(714\) −1.93454e8 −0.531474
\(715\) −6.59763e8 −1.80497
\(716\) −4.14041e8 −1.12799
\(717\) −3.29494e8 −0.893901
\(718\) 1.08774e9i 2.93868i
\(719\) −1.42205e8 −0.382586 −0.191293 0.981533i \(-0.561268\pi\)
−0.191293 + 0.981533i \(0.561268\pi\)
\(720\) 5.83051e8i 1.56210i
\(721\) −2.96128e8 −0.790085
\(722\) −5.19249e8 −1.37964
\(723\) 1.48153e8i 0.392007i
\(724\) 1.92680e8i 0.507716i
\(725\) −6.28296e7 −0.164873
\(726\) 6.55425e8 1.71283
\(727\) 1.81413e8i 0.472134i −0.971737 0.236067i \(-0.924142\pi\)
0.971737 0.236067i \(-0.0758584\pi\)
\(728\) 1.90203e9i 4.92972i
\(729\) 1.43489e7 0.0370370
\(730\) 1.87157e8i 0.481103i
\(731\) −1.14757e8 −0.293783
\(732\) 3.04873e8i 0.777296i
\(733\) 2.56165e8i 0.650441i 0.945638 + 0.325221i \(0.105439\pi\)
−0.945638 + 0.325221i \(0.894561\pi\)
\(734\) 1.13322e9i 2.86567i
\(735\) 5.67004e8i 1.42799i
\(736\) 1.32131e9 + 6.55172e8i 3.31413 + 1.64332i
\(737\) −1.05801e8 −0.264295
\(738\) −3.81162e8 −0.948289
\(739\) −4.29689e8 −1.06468 −0.532342 0.846529i \(-0.678688\pi\)
−0.532342 + 0.846529i \(0.678688\pi\)
\(740\) −7.14840e8 −1.76406
\(741\) 1.09734e8i 0.269704i
\(742\) 7.85185e8 1.92203
\(743\) 3.88763e8i 0.947804i 0.880578 + 0.473902i \(0.157155\pi\)
−0.880578 + 0.473902i \(0.842845\pi\)
\(744\) −8.54687e7 −0.207534
\(745\) 8.73645e7 0.211284
\(746\) 6.16660e8i 1.48535i
\(747\) 5.37687e7i 0.128994i
\(748\) 5.05837e8 1.20866
\(749\) 8.39845e8 1.99873
\(750\) 2.01767e8i 0.478264i
\(751\) 5.69395e8i 1.34429i −0.740418 0.672146i \(-0.765373\pi\)
0.740418 0.672146i \(-0.234627\pi\)
\(752\) −1.77151e9 −4.16573
\(753\) 9.41882e7i 0.220603i
\(754\) −1.79036e8 −0.417663
\(755\) 9.15963e6i 0.0212832i
\(756\) 3.84655e8i 0.890238i
\(757\) 4.17955e8i 0.963479i 0.876315 + 0.481739i \(0.159995\pi\)
−0.876315 + 0.481739i \(0.840005\pi\)
\(758\) 1.67336e8i 0.384222i
\(759\) 1.78745e8 3.60481e8i 0.408798 0.824437i
\(760\) −9.93452e8 −2.26311
\(761\) −5.53264e7 −0.125539 −0.0627695 0.998028i \(-0.519993\pi\)
−0.0627695 + 0.998028i \(0.519993\pi\)
\(762\) 6.31644e8 1.42760
\(763\) 1.22354e9 2.75451
\(764\) 2.73172e8i 0.612570i
\(765\) 5.39322e7 0.120466
\(766\) 1.43316e9i 3.18865i
\(767\) 8.11640e7 0.179878
\(768\) 6.12564e8 1.35228
\(769\) 3.30904e8i 0.727652i 0.931467 + 0.363826i \(0.118530\pi\)
−0.931467 + 0.363826i \(0.881470\pi\)
\(770\) 3.08943e9i 6.76715i
\(771\) −3.16072e8 −0.689640
\(772\) 1.83316e9 3.98428
\(773\) 7.66293e7i 0.165904i 0.996554 + 0.0829519i \(0.0264348\pi\)
−0.996554 + 0.0829519i \(0.973565\pi\)
\(774\) 3.12407e8i 0.673748i
\(775\) −3.38999e7 −0.0728271
\(776\) 2.06481e9i 4.41870i
\(777\) −2.33255e8 −0.497241
\(778\) 1.67781e8i 0.356290i
\(779\) 3.71898e8i 0.786705i
\(780\) 8.40528e8i 1.77120i
\(781\) 4.96496e8i 1.04223i
\(782\) 1.14526e8 2.30968e8i 0.239488 0.482983i
\(783\) 2.28418e7 0.0475823
\(784\) −3.35096e9 −6.95377
\(785\) −2.70252e8 −0.558677
\(786\) −4.06256e8 −0.836627
\(787\) 6.20091e8i 1.27213i 0.771636 + 0.636065i \(0.219439\pi\)
−0.771636 + 0.636065i \(0.780561\pi\)
\(788\) −9.64337e8 −1.97083
\(789\) 3.37595e8i 0.687329i
\(790\) 1.34179e9 2.72148
\(791\) 1.87957e8 0.379778
\(792\) 8.68735e8i 1.74869i
\(793\) 2.17381e8i 0.435915i
\(794\) 4.06709e8 0.812498
\(795\) −2.18899e8 −0.435654
\(796\) 2.08907e9i 4.14203i
\(797\) 5.15952e8i 1.01914i 0.860429 + 0.509571i \(0.170196\pi\)
−0.860429 + 0.509571i \(0.829804\pi\)
\(798\) −5.13846e8 −1.01117
\(799\) 1.63865e8i 0.321253i
\(800\) 1.26299e9 2.46677
\(801\) 3.27386e8i 0.637034i
\(802\) 1.47455e9i 2.85849i
\(803\) 1.59684e8i 0.308401i
\(804\) 1.34789e8i 0.259351i
\(805\) 1.03032e9 + 5.10887e8i 1.97508 + 0.979349i
\(806\) −9.65993e7 −0.184488
\(807\) 5.38052e8 1.02377
\(808\) −2.52571e9 −4.78795
\(809\) −5.79553e8 −1.09458 −0.547290 0.836943i \(-0.684340\pi\)
−0.547290 + 0.836943i \(0.684340\pi\)
\(810\) 1.46822e8i 0.276271i
\(811\) −8.98110e8 −1.68371 −0.841855 0.539705i \(-0.818536\pi\)
−0.841855 + 0.539705i \(0.818536\pi\)
\(812\) 6.12328e8i 1.14371i
\(813\) 2.61619e8 0.486853
\(814\) 8.35046e8 1.54824
\(815\) 8.81986e8i 1.62925i
\(816\) 3.18736e8i 0.586625i
\(817\) −3.04814e8 −0.558945
\(818\) 1.21928e9 2.22764
\(819\) 2.74267e8i 0.499255i
\(820\) 2.84862e9i 5.16645i
\(821\) −9.78455e8 −1.76812 −0.884059 0.467375i \(-0.845200\pi\)
−0.884059 + 0.467375i \(0.845200\pi\)
\(822\) 6.07862e8i 1.09443i
\(823\) −3.80620e8 −0.682797 −0.341399 0.939919i \(-0.610901\pi\)
−0.341399 + 0.939919i \(0.610901\pi\)
\(824\) 8.52039e8i 1.52292i
\(825\) 3.44571e8i 0.613644i
\(826\) 3.80061e8i 0.674394i
\(827\) 4.38363e8i 0.775029i −0.921864 0.387514i \(-0.873334\pi\)
0.921864 0.387514i \(-0.126666\pi\)
\(828\) −4.59248e8 2.27719e8i −0.809014 0.401151i
\(829\) 2.92030e8 0.512583 0.256291 0.966600i \(-0.417499\pi\)
0.256291 + 0.966600i \(0.417499\pi\)
\(830\) −5.50175e8 −0.962202
\(831\) 5.78682e8 1.00841
\(832\) 1.76527e9 3.06508
\(833\) 3.09964e8i 0.536260i
\(834\) −1.69199e8 −0.291675
\(835\) 8.92898e8i 1.53371i
\(836\) 1.34359e9 2.29957
\(837\) 1.23244e7 0.0210178
\(838\) 3.68964e8i 0.626977i
\(839\) 6.83425e8i 1.15719i −0.815615 0.578595i \(-0.803601\pi\)
0.815615 0.578595i \(-0.196399\pi\)
\(840\) 2.48301e9 4.18929
\(841\) −5.58462e8 −0.938870
\(842\) 2.70067e8i 0.452414i
\(843\) 5.44108e8i 0.908244i
\(844\) −4.44916e8 −0.740032
\(845\) 1.79650e8i 0.297753i
\(846\) 4.46096e8 0.736745
\(847\) 1.59834e9i 2.63038i
\(848\) 1.29368e9i 2.12148i
\(849\) 9.44807e7i 0.154390i
\(850\) 2.20774e8i 0.359493i
\(851\) 1.38088e8 2.78488e8i 0.224062 0.451874i
\(852\) 6.32528e8 1.02273
\(853\) 6.12987e8 0.987652 0.493826 0.869561i \(-0.335598\pi\)
0.493826 + 0.869561i \(0.335598\pi\)
\(854\) 1.01792e9 1.63432
\(855\) 1.43253e8 0.229195
\(856\) 2.41646e9i 3.85263i
\(857\) 3.56047e8 0.565673 0.282836 0.959168i \(-0.408725\pi\)
0.282836 + 0.959168i \(0.408725\pi\)
\(858\) 9.81870e8i 1.55451i
\(859\) −4.90978e8 −0.774609 −0.387304 0.921952i \(-0.626594\pi\)
−0.387304 + 0.921952i \(0.626594\pi\)
\(860\) 2.33477e9 3.67070
\(861\) 9.29513e8i 1.45628i
\(862\) 2.08893e9i 3.26139i
\(863\) −6.95860e8 −1.08265 −0.541327 0.840812i \(-0.682078\pi\)
−0.541327 + 0.840812i \(0.682078\pi\)
\(864\) −4.59162e8 −0.711909
\(865\) 4.11123e7i 0.0635218i
\(866\) 6.12081e8i 0.942443i
\(867\) 3.46784e8 0.532111
\(868\) 3.30383e8i 0.505194i
\(869\) −1.14483e9 −1.74454
\(870\) 2.33723e8i 0.354931i
\(871\) 9.61077e7i 0.145447i
\(872\) 3.52044e9i 5.30943i
\(873\) 2.97740e8i 0.447502i
\(874\) 3.04200e8 6.13491e8i 0.455644 0.918912i
\(875\) −4.92036e8 −0.734468
\(876\) −2.03435e8 −0.302631
\(877\) 8.71349e8 1.29179 0.645897 0.763425i \(-0.276484\pi\)
0.645897 + 0.763425i \(0.276484\pi\)
\(878\) 1.86539e8 0.275605
\(879\) 5.18752e8i 0.763823i
\(880\) −5.09017e9 −7.46937
\(881\) 3.09398e8i 0.452470i 0.974073 + 0.226235i \(0.0726418\pi\)
−0.974073 + 0.226235i \(0.927358\pi\)
\(882\) 8.43824e8 1.22983
\(883\) −7.52724e8 −1.09334 −0.546668 0.837350i \(-0.684104\pi\)
−0.546668 + 0.837350i \(0.684104\pi\)
\(884\) 4.59491e8i 0.665151i
\(885\) 1.05956e8i 0.152861i
\(886\) 5.85222e8 0.841433
\(887\) 8.11020e8 1.16215 0.581073 0.813852i \(-0.302633\pi\)
0.581073 + 0.813852i \(0.302633\pi\)
\(888\) 6.71136e8i 0.958455i
\(889\) 1.54034e9i 2.19236i
\(890\) −3.34989e9 −4.75183
\(891\) 1.25269e8i 0.177097i
\(892\) −5.57587e8 −0.785631
\(893\) 4.35254e8i 0.611207i
\(894\) 1.30017e8i 0.181965i
\(895\) 3.85396e8i 0.537574i
\(896\) 3.72247e9i 5.17496i
\(897\) −3.27453e8 1.62368e8i −0.453703 0.224969i
\(898\) 1.06029e9 1.46419
\(899\) 1.96190e7 0.0270021
\(900\) −4.38978e8 −0.602164
\(901\) 1.19665e8 0.163604
\(902\) 3.32763e9i 4.53436i
\(903\) 7.61844e8 1.03467
\(904\) 5.40804e8i 0.732039i
\(905\) −1.79350e8 −0.241966
\(906\) −1.36315e7 −0.0183299
\(907\) 6.91993e8i 0.927427i 0.885985 + 0.463714i \(0.153483\pi\)
−0.885985 + 0.463714i \(0.846517\pi\)
\(908\) 7.79325e8i 1.04102i
\(909\) 3.64201e8 0.484897
\(910\) 2.80637e9 3.72409
\(911\) 8.23121e8i 1.08870i 0.838858 + 0.544351i \(0.183224\pi\)
−0.838858 + 0.544351i \(0.816776\pi\)
\(912\) 8.46616e8i 1.11610i
\(913\) 4.69414e8 0.616798
\(914\) 4.84511e8i 0.634549i
\(915\) −2.83781e8 −0.370442
\(916\) 2.93878e9i 3.82367i
\(917\) 9.90706e8i 1.28480i
\(918\) 8.02628e7i 0.103750i
\(919\) 5.85375e8i 0.754201i −0.926172 0.377101i \(-0.876921\pi\)
0.926172 0.377101i \(-0.123079\pi\)
\(920\) −1.46996e9 + 2.96451e9i −1.88774 + 3.80706i
\(921\) 1.39008e8 0.177934
\(922\) −5.31735e8 −0.678426
\(923\) 4.51006e8 0.573558
\(924\) −3.35813e9 −4.25678
\(925\) 2.66196e8i 0.336338i
\(926\) 2.24266e9 2.82443
\(927\) 1.22862e8i 0.154233i
\(928\) −7.30933e8 −0.914605
\(929\) 1.03227e9 1.28750 0.643748 0.765237i \(-0.277378\pi\)
0.643748 + 0.765237i \(0.277378\pi\)
\(930\) 1.26106e8i 0.156779i
\(931\) 8.23316e8i 1.02028i
\(932\) 1.56039e9 1.92746
\(933\) 6.11099e8 0.752431
\(934\) 2.85135e9i 3.49954i
\(935\) 4.70841e8i 0.576023i
\(936\) −7.89140e8 −0.962335
\(937\) 8.43729e8i 1.02561i −0.858504 0.512807i \(-0.828606\pi\)
0.858504 0.512807i \(-0.171394\pi\)
\(938\) 4.50037e8 0.545305
\(939\) 3.17158e8i 0.383070i
\(940\) 3.33390e9i 4.01392i
\(941\) 1.50332e9i 1.80420i −0.431530 0.902098i \(-0.642026\pi\)
0.431530 0.902098i \(-0.357974\pi\)
\(942\) 4.02194e8i 0.481153i
\(943\) −1.10976e9 5.50278e8i −1.32341 0.656217i
\(944\) 6.26192e8 0.744375
\(945\) −3.58043e8 −0.424268
\(946\) −2.72738e9 −3.22161
\(947\) 6.65696e8 0.783837 0.391919 0.920000i \(-0.371811\pi\)
0.391919 + 0.920000i \(0.371811\pi\)
\(948\) 1.45850e9i 1.71191i
\(949\) −1.45053e8 −0.169719
\(950\) 5.86413e8i 0.683963i
\(951\) −2.13045e8 −0.247703
\(952\) −1.35738e9 −1.57323
\(953\) 8.53637e7i 0.0986267i −0.998783 0.0493133i \(-0.984297\pi\)
0.998783 0.0493133i \(-0.0157033\pi\)
\(954\) 3.25769e8i 0.375201i
\(955\) −2.54273e8 −0.291937
\(956\) −3.66469e9 −4.19434
\(957\) 1.99415e8i 0.227521i
\(958\) 1.29059e9i 1.46789i
\(959\) −1.48235e9 −1.68072
\(960\) 2.30448e9i 2.60471i
\(961\) −8.76918e8 −0.988073
\(962\) 7.58537e8i 0.852024i
\(963\) 3.48447e8i 0.390173i
\(964\) 1.64778e9i 1.83937i
\(965\) 1.70634e9i 1.89882i
\(966\) −7.60311e8 + 1.53334e9i −0.843451 + 1.70101i
\(967\) 1.10914e9 1.22661 0.613307 0.789845i \(-0.289839\pi\)
0.613307 + 0.789845i \(0.289839\pi\)
\(968\) 4.59885e9 5.07017
\(969\) −7.83121e7 −0.0860711
\(970\) 3.04655e9 3.33805
\(971\) 6.69724e7i 0.0731540i 0.999331 + 0.0365770i \(0.0116454\pi\)
−0.999331 + 0.0365770i \(0.988355\pi\)
\(972\) 1.59591e8 0.173784
\(973\) 4.12613e8i 0.447924i
\(974\) 2.35392e9 2.54751
\(975\) −3.13000e8 −0.337700
\(976\) 1.67713e9i 1.80392i
\(977\) 3.19020e8i 0.342086i −0.985264 0.171043i \(-0.945286\pi\)
0.985264 0.171043i \(-0.0547136\pi\)
\(978\) 1.31259e9 1.40317
\(979\) 2.85816e9 3.04606
\(980\) 6.30633e9i 6.70036i
\(981\) 5.07639e8i 0.537710i
\(982\) 2.73987e9 2.89331
\(983\) 4.41798e7i 0.0465118i 0.999730 + 0.0232559i \(0.00740325\pi\)
−0.999730 + 0.0232559i \(0.992597\pi\)
\(984\) −2.67446e9 −2.80705
\(985\) 8.97621e8i 0.939256i
\(986\) 1.27769e8i 0.133289i
\(987\) 1.08786e9i 1.13142i
\(988\) 1.22049e9i 1.26550i
\(989\) −4.51017e8 + 9.09581e8i −0.466234 + 0.940270i
\(990\) 1.28179e9 1.32102
\(991\) 1.67599e9 1.72207 0.861035 0.508546i \(-0.169817\pi\)
0.861035 + 0.508546i \(0.169817\pi\)
\(992\) −3.94377e8 −0.403995
\(993\) −1.48975e8 −0.152148
\(994\) 2.11190e9i 2.15037i
\(995\) 1.94454e9 1.97400
\(996\) 5.98026e8i 0.605260i
\(997\) −5.36246e8 −0.541102 −0.270551 0.962706i \(-0.587206\pi\)
−0.270551 + 0.962706i \(0.587206\pi\)
\(998\) −2.75379e9 −2.77038
\(999\) 9.67760e7i 0.0970669i
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 69.7.d.a.22.2 yes 24
3.2 odd 2 207.7.d.e.91.23 24
23.22 odd 2 inner 69.7.d.a.22.1 24
69.68 even 2 207.7.d.e.91.24 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.7.d.a.22.1 24 23.22 odd 2 inner
69.7.d.a.22.2 yes 24 1.1 even 1 trivial
207.7.d.e.91.23 24 3.2 odd 2
207.7.d.e.91.24 24 69.68 even 2