Properties

Label 124.6.f.a.109.2
Level $124$
Weight $6$
Character 124.109
Analytic conductor $19.888$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [124,6,Mod(33,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.33");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 124.f (of order \(5\), degree \(4\), 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: \(19.8875936568\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 109.2
Character \(\chi\) \(=\) 124.109
Dual form 124.6.f.a.33.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-21.3000 - 15.4753i) q^{3} +92.0001 q^{5} +(26.8893 - 82.7567i) q^{7} +(139.112 + 428.142i) q^{9} +O(q^{10})\) \(q+(-21.3000 - 15.4753i) q^{3} +92.0001 q^{5} +(26.8893 - 82.7567i) q^{7} +(139.112 + 428.142i) q^{9} +(-196.836 + 605.799i) q^{11} +(225.634 + 163.933i) q^{13} +(-1959.60 - 1423.73i) q^{15} +(382.219 + 1176.35i) q^{17} +(1587.34 - 1153.27i) q^{19} +(-1853.43 + 1346.60i) q^{21} +(726.521 + 2236.00i) q^{23} +5339.02 q^{25} +(1685.54 - 5187.57i) q^{27} +(3261.85 - 2369.87i) q^{29} +(851.215 - 5282.48i) q^{31} +(13567.5 - 9857.39i) q^{33} +(2473.82 - 7613.63i) q^{35} +3614.50 q^{37} +(-2269.08 - 6983.52i) q^{39} +(-6762.45 + 4913.21i) q^{41} +(6386.36 - 4639.96i) q^{43} +(12798.3 + 39389.1i) q^{45} +(-12861.0 - 9344.07i) q^{47} +(7471.51 + 5428.37i) q^{49} +(10063.1 - 30971.1i) q^{51} +(5266.64 + 16209.1i) q^{53} +(-18108.9 + 55733.5i) q^{55} -51657.5 q^{57} +(-19214.1 - 13959.9i) q^{59} -746.505 q^{61} +39172.2 q^{63} +(20758.3 + 15081.8i) q^{65} +52388.5 q^{67} +(19128.0 - 58869.9i) q^{69} +(-2993.42 - 9212.79i) q^{71} +(22224.9 - 68401.1i) q^{73} +(-113721. - 82623.1i) q^{75} +(44841.1 + 32579.0i) q^{77} +(23898.1 + 73550.7i) q^{79} +(-27681.1 + 20111.5i) q^{81} +(-87903.8 + 63865.8i) q^{83} +(35164.2 + 108224. i) q^{85} -106152. q^{87} +(42405.2 - 130510. i) q^{89} +(19633.7 - 14264.7i) q^{91} +(-99879.0 + 99343.8i) q^{93} +(146035. - 106101. i) q^{95} +(-25537.7 + 78596.9i) q^{97} -286750. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 2 q^{3} - 58 q^{5} + 104 q^{7} - 1234 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 2 q^{3} - 58 q^{5} + 104 q^{7} - 1234 q^{9} - 509 q^{11} - 117 q^{13} + 89 q^{15} - 3504 q^{17} + 262 q^{19} + 352 q^{21} - 2448 q^{23} + 49618 q^{25} + 14324 q^{27} - 9888 q^{29} - 12771 q^{31} + 27699 q^{33} + 13840 q^{35} + 76096 q^{37} + 33520 q^{39} - 4843 q^{41} - 40778 q^{43} + 56692 q^{45} + 38922 q^{47} - 17126 q^{49} - 69292 q^{51} - 41728 q^{53} - 172096 q^{55} + 57066 q^{57} - 58198 q^{59} + 176328 q^{61} - 37444 q^{63} + 143863 q^{65} + 9812 q^{67} - 9250 q^{69} - 67356 q^{71} - 63512 q^{73} - 198012 q^{75} - 74257 q^{77} + 137651 q^{79} + 196077 q^{81} + 156427 q^{83} + 238828 q^{85} - 558144 q^{87} - 99292 q^{89} - 243609 q^{91} - 325925 q^{93} - 75077 q^{95} - 476340 q^{97} + 745812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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
\(3\) −21.3000 15.4753i −1.36639 0.992743i −0.998009 0.0630696i \(-0.979911\pi\)
−0.368385 0.929674i \(-0.620089\pi\)
\(4\) 0 0
\(5\) 92.0001 1.64575 0.822874 0.568224i \(-0.192369\pi\)
0.822874 + 0.568224i \(0.192369\pi\)
\(6\) 0 0
\(7\) 26.8893 82.7567i 0.207412 0.638349i −0.792194 0.610270i \(-0.791061\pi\)
0.999606 0.0280791i \(-0.00893904\pi\)
\(8\) 0 0
\(9\) 139.112 + 428.142i 0.572476 + 1.76190i
\(10\) 0 0
\(11\) −196.836 + 605.799i −0.490482 + 1.50955i 0.333400 + 0.942785i \(0.391804\pi\)
−0.823882 + 0.566762i \(0.808196\pi\)
\(12\) 0 0
\(13\) 225.634 + 163.933i 0.370293 + 0.269034i 0.757333 0.653029i \(-0.226502\pi\)
−0.387039 + 0.922063i \(0.626502\pi\)
\(14\) 0 0
\(15\) −1959.60 1423.73i −2.24874 1.63380i
\(16\) 0 0
\(17\) 382.219 + 1176.35i 0.320767 + 0.987219i 0.973315 + 0.229472i \(0.0737000\pi\)
−0.652548 + 0.757747i \(0.726300\pi\)
\(18\) 0 0
\(19\) 1587.34 1153.27i 1.00876 0.732904i 0.0448077 0.998996i \(-0.485732\pi\)
0.963948 + 0.266092i \(0.0857325\pi\)
\(20\) 0 0
\(21\) −1853.43 + 1346.60i −0.917123 + 0.666329i
\(22\) 0 0
\(23\) 726.521 + 2236.00i 0.286371 + 0.881358i 0.985984 + 0.166837i \(0.0533554\pi\)
−0.699614 + 0.714521i \(0.746645\pi\)
\(24\) 0 0
\(25\) 5339.02 1.70849
\(26\) 0 0
\(27\) 1685.54 5187.57i 0.444970 1.36948i
\(28\) 0 0
\(29\) 3261.85 2369.87i 0.720227 0.523275i −0.166230 0.986087i \(-0.553159\pi\)
0.886457 + 0.462812i \(0.153159\pi\)
\(30\) 0 0
\(31\) 851.215 5282.48i 0.159087 0.987265i
\(32\) 0 0
\(33\) 13567.5 9857.39i 2.16878 1.57571i
\(34\) 0 0
\(35\) 2473.82 7613.63i 0.341348 1.05056i
\(36\) 0 0
\(37\) 3614.50 0.434054 0.217027 0.976166i \(-0.430364\pi\)
0.217027 + 0.976166i \(0.430364\pi\)
\(38\) 0 0
\(39\) −2269.08 6983.52i −0.238885 0.735213i
\(40\) 0 0
\(41\) −6762.45 + 4913.21i −0.628267 + 0.456463i −0.855800 0.517308i \(-0.826934\pi\)
0.227532 + 0.973771i \(0.426934\pi\)
\(42\) 0 0
\(43\) 6386.36 4639.96i 0.526723 0.382686i −0.292408 0.956294i \(-0.594456\pi\)
0.819130 + 0.573607i \(0.194456\pi\)
\(44\) 0 0
\(45\) 12798.3 + 39389.1i 0.942151 + 2.89964i
\(46\) 0 0
\(47\) −12861.0 9344.07i −0.849240 0.617009i 0.0756961 0.997131i \(-0.475882\pi\)
−0.924936 + 0.380122i \(0.875882\pi\)
\(48\) 0 0
\(49\) 7471.51 + 5428.37i 0.444547 + 0.322982i
\(50\) 0 0
\(51\) 10063.1 30971.1i 0.541761 1.66737i
\(52\) 0 0
\(53\) 5266.64 + 16209.1i 0.257540 + 0.792625i 0.993319 + 0.115404i \(0.0368162\pi\)
−0.735779 + 0.677222i \(0.763184\pi\)
\(54\) 0 0
\(55\) −18108.9 + 55733.5i −0.807209 + 2.48433i
\(56\) 0 0
\(57\) −51657.5 −2.10594
\(58\) 0 0
\(59\) −19214.1 13959.9i −0.718606 0.522098i 0.167333 0.985900i \(-0.446485\pi\)
−0.885938 + 0.463803i \(0.846485\pi\)
\(60\) 0 0
\(61\) −746.505 −0.0256867 −0.0128433 0.999918i \(-0.504088\pi\)
−0.0128433 + 0.999918i \(0.504088\pi\)
\(62\) 0 0
\(63\) 39172.2 1.24345
\(64\) 0 0
\(65\) 20758.3 + 15081.8i 0.609410 + 0.442762i
\(66\) 0 0
\(67\) 52388.5 1.42577 0.712884 0.701282i \(-0.247389\pi\)
0.712884 + 0.701282i \(0.247389\pi\)
\(68\) 0 0
\(69\) 19128.0 58869.9i 0.483667 1.48858i
\(70\) 0 0
\(71\) −2993.42 9212.79i −0.0704727 0.216893i 0.909617 0.415448i \(-0.136375\pi\)
−0.980090 + 0.198555i \(0.936375\pi\)
\(72\) 0 0
\(73\) 22224.9 68401.1i 0.488126 1.50230i −0.339276 0.940687i \(-0.610182\pi\)
0.827402 0.561610i \(-0.189818\pi\)
\(74\) 0 0
\(75\) −113721. 82623.1i −2.33446 1.69609i
\(76\) 0 0
\(77\) 44841.1 + 32579.0i 0.861886 + 0.626197i
\(78\) 0 0
\(79\) 23898.1 + 73550.7i 0.430819 + 1.32592i 0.897311 + 0.441399i \(0.145518\pi\)
−0.466492 + 0.884526i \(0.654482\pi\)
\(80\) 0 0
\(81\) −27681.1 + 20111.5i −0.468783 + 0.340591i
\(82\) 0 0
\(83\) −87903.8 + 63865.8i −1.40059 + 1.01759i −0.405987 + 0.913879i \(0.633072\pi\)
−0.994607 + 0.103713i \(0.966928\pi\)
\(84\) 0 0
\(85\) 35164.2 + 108224.i 0.527902 + 1.62471i
\(86\) 0 0
\(87\) −106152. −1.50359
\(88\) 0 0
\(89\) 42405.2 130510.i 0.567471 1.74650i −0.0930219 0.995664i \(-0.529653\pi\)
0.660493 0.750832i \(-0.270347\pi\)
\(90\) 0 0
\(91\) 19633.7 14264.7i 0.248541 0.180576i
\(92\) 0 0
\(93\) −99879.0 + 99343.8i −1.19748 + 1.19106i
\(94\) 0 0
\(95\) 146035. 106101.i 1.66016 1.20617i
\(96\) 0 0
\(97\) −25537.7 + 78596.9i −0.275583 + 0.848157i 0.713482 + 0.700674i \(0.247117\pi\)
−0.989065 + 0.147483i \(0.952883\pi\)
\(98\) 0 0
\(99\) −286750. −2.94046
\(100\) 0 0
\(101\) −39449.7 121414.i −0.384805 1.18431i −0.936622 0.350342i \(-0.886065\pi\)
0.551817 0.833965i \(-0.313935\pi\)
\(102\) 0 0
\(103\) 151684. 110205.i 1.40879 1.02354i 0.415290 0.909689i \(-0.363680\pi\)
0.993497 0.113855i \(-0.0363199\pi\)
\(104\) 0 0
\(105\) −170516. + 123887.i −1.50935 + 1.09661i
\(106\) 0 0
\(107\) 59717.1 + 183790.i 0.504242 + 1.55190i 0.802041 + 0.597269i \(0.203747\pi\)
−0.297799 + 0.954628i \(0.596253\pi\)
\(108\) 0 0
\(109\) −35378.8 25704.2i −0.285218 0.207223i 0.435972 0.899960i \(-0.356405\pi\)
−0.721190 + 0.692737i \(0.756405\pi\)
\(110\) 0 0
\(111\) −76988.7 55935.5i −0.593088 0.430904i
\(112\) 0 0
\(113\) −60304.0 + 185597.i −0.444273 + 1.36733i 0.439006 + 0.898484i \(0.355331\pi\)
−0.883279 + 0.468847i \(0.844669\pi\)
\(114\) 0 0
\(115\) 66840.0 + 205712.i 0.471294 + 1.45049i
\(116\) 0 0
\(117\) −38798.1 + 119408.i −0.262027 + 0.806435i
\(118\) 0 0
\(119\) 107628. 0.696722
\(120\) 0 0
\(121\) −197955. 143823.i −1.22914 0.893025i
\(122\) 0 0
\(123\) 220074. 1.31161
\(124\) 0 0
\(125\) 203690. 1.16599
\(126\) 0 0
\(127\) 133806. + 97215.9i 0.736151 + 0.534845i 0.891503 0.453014i \(-0.149651\pi\)
−0.155352 + 0.987859i \(0.549651\pi\)
\(128\) 0 0
\(129\) −207834. −1.09962
\(130\) 0 0
\(131\) −928.745 + 2858.38i −0.00472845 + 0.0145527i −0.953393 0.301732i \(-0.902435\pi\)
0.948664 + 0.316284i \(0.102435\pi\)
\(132\) 0 0
\(133\) −52758.4 162374.i −0.258620 0.795951i
\(134\) 0 0
\(135\) 155070. 477257.i 0.732309 2.25381i
\(136\) 0 0
\(137\) 247431. + 179769.i 1.12630 + 0.818301i 0.985151 0.171688i \(-0.0549220\pi\)
0.141144 + 0.989989i \(0.454922\pi\)
\(138\) 0 0
\(139\) −260998. 189626.i −1.14578 0.832455i −0.157862 0.987461i \(-0.550460\pi\)
−0.987913 + 0.155006i \(0.950460\pi\)
\(140\) 0 0
\(141\) 129337. + 398057.i 0.547865 + 1.68616i
\(142\) 0 0
\(143\) −143723. + 104421.i −0.587741 + 0.427019i
\(144\) 0 0
\(145\) 300091. 218029.i 1.18531 0.861179i
\(146\) 0 0
\(147\) −75137.1 231248.i −0.286788 0.882642i
\(148\) 0 0
\(149\) 81148.0 0.299442 0.149721 0.988728i \(-0.452163\pi\)
0.149721 + 0.988728i \(0.452163\pi\)
\(150\) 0 0
\(151\) 84746.8 260824.i 0.302469 0.930904i −0.678140 0.734932i \(-0.737214\pi\)
0.980610 0.195972i \(-0.0627861\pi\)
\(152\) 0 0
\(153\) −450472. + 327287.i −1.55575 + 1.13032i
\(154\) 0 0
\(155\) 78311.8 485989.i 0.261817 1.62479i
\(156\) 0 0
\(157\) −285793. + 207641.i −0.925342 + 0.672300i −0.944848 0.327510i \(-0.893791\pi\)
0.0195060 + 0.999810i \(0.493791\pi\)
\(158\) 0 0
\(159\) 138661. 426755.i 0.434973 1.33871i
\(160\) 0 0
\(161\) 204580. 0.622011
\(162\) 0 0
\(163\) −5140.88 15822.0i −0.0151554 0.0466436i 0.943193 0.332246i \(-0.107806\pi\)
−0.958348 + 0.285602i \(0.907806\pi\)
\(164\) 0 0
\(165\) 1.24821e6 906881.i 3.56927 2.59323i
\(166\) 0 0
\(167\) −335277. + 243593.i −0.930279 + 0.675887i −0.946061 0.323988i \(-0.894976\pi\)
0.0157823 + 0.999875i \(0.494976\pi\)
\(168\) 0 0
\(169\) −90699.1 279143.i −0.244279 0.751814i
\(170\) 0 0
\(171\) 714580. + 519173.i 1.86879 + 1.35776i
\(172\) 0 0
\(173\) 316373. + 229858.i 0.803681 + 0.583908i 0.911992 0.410209i \(-0.134544\pi\)
−0.108311 + 0.994117i \(0.534544\pi\)
\(174\) 0 0
\(175\) 143562. 441840.i 0.354361 1.09061i
\(176\) 0 0
\(177\) 193226. + 594690.i 0.463589 + 1.42678i
\(178\) 0 0
\(179\) −211623. + 651310.i −0.493664 + 1.51934i 0.325366 + 0.945588i \(0.394513\pi\)
−0.819029 + 0.573752i \(0.805487\pi\)
\(180\) 0 0
\(181\) 160308. 0.363712 0.181856 0.983325i \(-0.441789\pi\)
0.181856 + 0.983325i \(0.441789\pi\)
\(182\) 0 0
\(183\) 15900.5 + 11552.4i 0.0350981 + 0.0255003i
\(184\) 0 0
\(185\) 332534. 0.714343
\(186\) 0 0
\(187\) −787865. −1.64758
\(188\) 0 0
\(189\) −383984. 278980.i −0.781912 0.568093i
\(190\) 0 0
\(191\) 425784. 0.844512 0.422256 0.906477i \(-0.361238\pi\)
0.422256 + 0.906477i \(0.361238\pi\)
\(192\) 0 0
\(193\) −23069.6 + 71001.0i −0.0445807 + 0.137205i −0.970869 0.239609i \(-0.922981\pi\)
0.926289 + 0.376815i \(0.122981\pi\)
\(194\) 0 0
\(195\) −208756. 642485.i −0.393144 1.20997i
\(196\) 0 0
\(197\) −58729.3 + 180750.i −0.107818 + 0.331828i −0.990381 0.138364i \(-0.955816\pi\)
0.882564 + 0.470193i \(0.155816\pi\)
\(198\) 0 0
\(199\) 493567. + 358597.i 0.883513 + 0.641910i 0.934179 0.356806i \(-0.116134\pi\)
−0.0506653 + 0.998716i \(0.516134\pi\)
\(200\) 0 0
\(201\) −1.11587e6 810729.i −1.94816 1.41542i
\(202\) 0 0
\(203\) −108414. 333664.i −0.184649 0.568290i
\(204\) 0 0
\(205\) −622146. + 452016.i −1.03397 + 0.751223i
\(206\) 0 0
\(207\) −856258. + 622108.i −1.38892 + 1.00911i
\(208\) 0 0
\(209\) 386204. + 1.18861e6i 0.611577 + 1.88224i
\(210\) 0 0
\(211\) 433626. 0.670515 0.335258 0.942126i \(-0.391177\pi\)
0.335258 + 0.942126i \(0.391177\pi\)
\(212\) 0 0
\(213\) −78811.3 + 242556.i −0.119025 + 0.366322i
\(214\) 0 0
\(215\) 587545. 426877.i 0.866852 0.629805i
\(216\) 0 0
\(217\) −414272. 212486.i −0.597223 0.306324i
\(218\) 0 0
\(219\) −1.53192e6 + 1.11300e6i −2.15837 + 1.56815i
\(220\) 0 0
\(221\) −106600. + 328082.i −0.146818 + 0.451858i
\(222\) 0 0
\(223\) 403593. 0.543478 0.271739 0.962371i \(-0.412401\pi\)
0.271739 + 0.962371i \(0.412401\pi\)
\(224\) 0 0
\(225\) 742719. + 2.28585e6i 0.978067 + 3.01018i
\(226\) 0 0
\(227\) 475055. 345147.i 0.611898 0.444570i −0.238184 0.971220i \(-0.576552\pi\)
0.850082 + 0.526650i \(0.176552\pi\)
\(228\) 0 0
\(229\) −930234. + 675854.i −1.17220 + 0.851657i −0.991271 0.131839i \(-0.957912\pi\)
−0.180933 + 0.983495i \(0.557912\pi\)
\(230\) 0 0
\(231\) −450944. 1.38786e6i −0.556023 1.71126i
\(232\) 0 0
\(233\) −703630. 511217.i −0.849092 0.616901i 0.0758036 0.997123i \(-0.475848\pi\)
−0.924896 + 0.380221i \(0.875848\pi\)
\(234\) 0 0
\(235\) −1.18321e6 859656.i −1.39764 1.01544i
\(236\) 0 0
\(237\) 629193. 1.93646e6i 0.727634 2.23943i
\(238\) 0 0
\(239\) 235545. + 724932.i 0.266734 + 0.820923i 0.991289 + 0.131706i \(0.0420454\pi\)
−0.724555 + 0.689217i \(0.757955\pi\)
\(240\) 0 0
\(241\) 502696. 1.54714e6i 0.557523 1.71588i −0.131662 0.991295i \(-0.542031\pi\)
0.689185 0.724585i \(-0.257969\pi\)
\(242\) 0 0
\(243\) −424612. −0.461293
\(244\) 0 0
\(245\) 687379. + 499410.i 0.731613 + 0.531548i
\(246\) 0 0
\(247\) 547216. 0.570711
\(248\) 0 0
\(249\) 2.86069e6 2.92397
\(250\) 0 0
\(251\) −1.11268e6 808410.i −1.11477 0.809930i −0.131364 0.991334i \(-0.541936\pi\)
−0.983409 + 0.181404i \(0.941936\pi\)
\(252\) 0 0
\(253\) −1.49757e6 −1.47091
\(254\) 0 0
\(255\) 925809. 2.84935e6i 0.891602 2.74407i
\(256\) 0 0
\(257\) −172755. 531684.i −0.163154 0.502135i 0.835742 0.549122i \(-0.185038\pi\)
−0.998895 + 0.0469874i \(0.985038\pi\)
\(258\) 0 0
\(259\) 97191.3 299124.i 0.0900280 0.277078i
\(260\) 0 0
\(261\) 1.46840e6 + 1.06686e6i 1.33427 + 0.969405i
\(262\) 0 0
\(263\) −485571. 352788.i −0.432876 0.314503i 0.349922 0.936779i \(-0.386208\pi\)
−0.782798 + 0.622276i \(0.786208\pi\)
\(264\) 0 0
\(265\) 484531. + 1.49123e6i 0.423845 + 1.30446i
\(266\) 0 0
\(267\) −2.92291e6 + 2.12362e6i −2.50921 + 1.82305i
\(268\) 0 0
\(269\) −1.29546e6 + 941209.i −1.09155 + 0.793059i −0.979661 0.200661i \(-0.935691\pi\)
−0.111892 + 0.993720i \(0.535691\pi\)
\(270\) 0 0
\(271\) 340093. + 1.04670e6i 0.281303 + 0.865762i 0.987482 + 0.157729i \(0.0504172\pi\)
−0.706179 + 0.708033i \(0.749583\pi\)
\(272\) 0 0
\(273\) −638947. −0.518870
\(274\) 0 0
\(275\) −1.05091e6 + 3.23437e6i −0.837981 + 2.57904i
\(276\) 0 0
\(277\) −370053. + 268859.i −0.289778 + 0.210536i −0.723171 0.690669i \(-0.757316\pi\)
0.433393 + 0.901205i \(0.357316\pi\)
\(278\) 0 0
\(279\) 2.38006e6 370414.i 1.83053 0.284890i
\(280\) 0 0
\(281\) 1.21900e6 885652.i 0.920951 0.669110i −0.0228099 0.999740i \(-0.507261\pi\)
0.943760 + 0.330630i \(0.107261\pi\)
\(282\) 0 0
\(283\) 379910. 1.16924e6i 0.281977 0.867837i −0.705311 0.708898i \(-0.749193\pi\)
0.987288 0.158939i \(-0.0508073\pi\)
\(284\) 0 0
\(285\) −4.75250e6 −3.46585
\(286\) 0 0
\(287\) 224764. + 691751.i 0.161072 + 0.495730i
\(288\) 0 0
\(289\) −89015.7 + 64673.7i −0.0626934 + 0.0455494i
\(290\) 0 0
\(291\) 1.76027e6 1.27891e6i 1.21856 0.885333i
\(292\) 0 0
\(293\) −256614. 789777.i −0.174627 0.537447i 0.824989 0.565148i \(-0.191181\pi\)
−0.999616 + 0.0277019i \(0.991181\pi\)
\(294\) 0 0
\(295\) −1.76770e6 1.28431e6i −1.18264 0.859241i
\(296\) 0 0
\(297\) 2.81085e6 + 2.04220e6i 1.84904 + 1.34341i
\(298\) 0 0
\(299\) −202626. + 623618.i −0.131074 + 0.403405i
\(300\) 0 0
\(301\) −212263. 653279.i −0.135039 0.415607i
\(302\) 0 0
\(303\) −1.03864e6 + 3.19661e6i −0.649918 + 2.00024i
\(304\) 0 0
\(305\) −68678.5 −0.0422738
\(306\) 0 0
\(307\) 402908. + 292730.i 0.243983 + 0.177264i 0.703056 0.711135i \(-0.251818\pi\)
−0.459073 + 0.888399i \(0.651818\pi\)
\(308\) 0 0
\(309\) −4.93631e6 −2.94107
\(310\) 0 0
\(311\) −1.84521e6 −1.08180 −0.540899 0.841088i \(-0.681916\pi\)
−0.540899 + 0.841088i \(0.681916\pi\)
\(312\) 0 0
\(313\) −1.19380e6 867344.i −0.688763 0.500415i 0.187491 0.982266i \(-0.439965\pi\)
−0.876253 + 0.481851i \(0.839965\pi\)
\(314\) 0 0
\(315\) 3.60385e6 2.04640
\(316\) 0 0
\(317\) 749684. 2.30729e6i 0.419016 1.28960i −0.489594 0.871951i \(-0.662855\pi\)
0.908609 0.417647i \(-0.137145\pi\)
\(318\) 0 0
\(319\) 793617. + 2.44250e6i 0.436651 + 1.34387i
\(320\) 0 0
\(321\) 1.57224e6 4.83887e6i 0.851642 2.62108i
\(322\) 0 0
\(323\) 1.96336e6 + 1.42646e6i 1.04711 + 0.760771i
\(324\) 0 0
\(325\) 1.20466e6 + 875239.i 0.632641 + 0.459641i
\(326\) 0 0
\(327\) 355786. + 1.09500e6i 0.184001 + 0.566297i
\(328\) 0 0
\(329\) −1.11911e6 + 813080.i −0.570010 + 0.414137i
\(330\) 0 0
\(331\) −298376. + 216783.i −0.149691 + 0.108757i −0.660110 0.751169i \(-0.729490\pi\)
0.510419 + 0.859926i \(0.329490\pi\)
\(332\) 0 0
\(333\) 502818. + 1.54752e6i 0.248485 + 0.764759i
\(334\) 0 0
\(335\) 4.81975e6 2.34645
\(336\) 0 0
\(337\) 124044. 381769.i 0.0594979 0.183116i −0.916890 0.399140i \(-0.869309\pi\)
0.976388 + 0.216024i \(0.0693089\pi\)
\(338\) 0 0
\(339\) 4.15664e6 3.01998e6i 1.96446 1.42726i
\(340\) 0 0
\(341\) 3.03257e6 + 1.55545e6i 1.41229 + 0.724384i
\(342\) 0 0
\(343\) 1.83330e6 1.33197e6i 0.841392 0.611307i
\(344\) 0 0
\(345\) 1.75978e6 5.41604e6i 0.795994 2.44982i
\(346\) 0 0
\(347\) −4.31110e6 −1.92205 −0.961025 0.276463i \(-0.910838\pi\)
−0.961025 + 0.276463i \(0.910838\pi\)
\(348\) 0 0
\(349\) −181554. 558765.i −0.0797888 0.245565i 0.903203 0.429213i \(-0.141209\pi\)
−0.982992 + 0.183649i \(0.941209\pi\)
\(350\) 0 0
\(351\) 1.23073e6 894177.i 0.533205 0.387396i
\(352\) 0 0
\(353\) 1.47323e6 1.07036e6i 0.629263 0.457187i −0.226882 0.973922i \(-0.572853\pi\)
0.856145 + 0.516736i \(0.172853\pi\)
\(354\) 0 0
\(355\) −275394. 847577.i −0.115980 0.356951i
\(356\) 0 0
\(357\) −2.29248e6 1.66558e6i −0.951996 0.691665i
\(358\) 0 0
\(359\) 227973. + 165632.i 0.0933570 + 0.0678278i 0.633485 0.773755i \(-0.281624\pi\)
−0.540128 + 0.841583i \(0.681624\pi\)
\(360\) 0 0
\(361\) 424459. 1.30635e6i 0.171423 0.527584i
\(362\) 0 0
\(363\) 1.99073e6 + 6.12683e6i 0.792949 + 2.44045i
\(364\) 0 0
\(365\) 2.04469e6 6.29291e6i 0.803332 2.47240i
\(366\) 0 0
\(367\) −1.30343e6 −0.505152 −0.252576 0.967577i \(-0.581278\pi\)
−0.252576 + 0.967577i \(0.581278\pi\)
\(368\) 0 0
\(369\) −3.04428e6 2.21180e6i −1.16391 0.845630i
\(370\) 0 0
\(371\) 1.48302e6 0.559388
\(372\) 0 0
\(373\) −1.89977e6 −0.707015 −0.353507 0.935432i \(-0.615011\pi\)
−0.353507 + 0.935432i \(0.615011\pi\)
\(374\) 0 0
\(375\) −4.33859e6 3.15217e6i −1.59320 1.15753i
\(376\) 0 0
\(377\) 1.12448e6 0.407474
\(378\) 0 0
\(379\) 534798. 1.64594e6i 0.191246 0.588594i −0.808754 0.588147i \(-0.799858\pi\)
1.00000 0.000447235i \(-0.000142359\pi\)
\(380\) 0 0
\(381\) −1.34562e6 4.14139e6i −0.474908 1.46162i
\(382\) 0 0
\(383\) −1.10330e6 + 3.39562e6i −0.384324 + 1.18283i 0.552645 + 0.833417i \(0.313619\pi\)
−0.936969 + 0.349412i \(0.886381\pi\)
\(384\) 0 0
\(385\) 4.12539e6 + 2.99727e6i 1.41845 + 1.03056i
\(386\) 0 0
\(387\) 2.87498e6 + 2.08879e6i 0.975791 + 0.708954i
\(388\) 0 0
\(389\) 190079. + 585003.i 0.0636883 + 0.196013i 0.977837 0.209366i \(-0.0671399\pi\)
−0.914149 + 0.405378i \(0.867140\pi\)
\(390\) 0 0
\(391\) −2.35263e6 + 1.70928e6i −0.778236 + 0.565421i
\(392\) 0 0
\(393\) 64016.7 46510.9i 0.0209080 0.0151905i
\(394\) 0 0
\(395\) 2.19862e6 + 6.76667e6i 0.709020 + 2.18214i
\(396\) 0 0
\(397\) −1.74680e6 −0.556245 −0.278123 0.960546i \(-0.589712\pi\)
−0.278123 + 0.960546i \(0.589712\pi\)
\(398\) 0 0
\(399\) −1.38903e6 + 4.27501e6i −0.436798 + 1.34433i
\(400\) 0 0
\(401\) 1.90297e6 1.38259e6i 0.590978 0.429371i −0.251687 0.967809i \(-0.580985\pi\)
0.842665 + 0.538438i \(0.180985\pi\)
\(402\) 0 0
\(403\) 1.05803e6 1.05236e6i 0.324517 0.322778i
\(404\) 0 0
\(405\) −2.54667e6 + 1.85026e6i −0.771498 + 0.560526i
\(406\) 0 0
\(407\) −711463. + 2.18966e6i −0.212895 + 0.655224i
\(408\) 0 0
\(409\) 243721. 0.0720419 0.0360210 0.999351i \(-0.488532\pi\)
0.0360210 + 0.999351i \(0.488532\pi\)
\(410\) 0 0
\(411\) −2.48828e6 7.65815e6i −0.726600 2.23624i
\(412\) 0 0
\(413\) −1.67193e6 + 1.21473e6i −0.482328 + 0.350432i
\(414\) 0 0
\(415\) −8.08716e6 + 5.87566e6i −2.30502 + 1.67470i
\(416\) 0 0
\(417\) 2.62472e6 + 8.07805e6i 0.739167 + 2.27492i
\(418\) 0 0
\(419\) 1.04588e6 + 759873.i 0.291035 + 0.211449i 0.723716 0.690098i \(-0.242432\pi\)
−0.432681 + 0.901547i \(0.642432\pi\)
\(420\) 0 0
\(421\) −3.67580e6 2.67063e6i −1.01076 0.734358i −0.0463897 0.998923i \(-0.514772\pi\)
−0.964368 + 0.264565i \(0.914772\pi\)
\(422\) 0 0
\(423\) 2.21147e6 6.80620e6i 0.600939 1.84950i
\(424\) 0 0
\(425\) 2.04067e6 + 6.28054e6i 0.548026 + 1.68665i
\(426\) 0 0
\(427\) −20073.0 + 61778.3i −0.00532773 + 0.0163971i
\(428\) 0 0
\(429\) 4.67724e6 1.22701
\(430\) 0 0
\(431\) 5.08870e6 + 3.69715e6i 1.31951 + 0.958682i 0.999938 + 0.0111204i \(0.00353981\pi\)
0.319574 + 0.947561i \(0.396460\pi\)
\(432\) 0 0
\(433\) −4.24187e6 −1.08727 −0.543636 0.839321i \(-0.682953\pi\)
−0.543636 + 0.839321i \(0.682953\pi\)
\(434\) 0 0
\(435\) −9.76599e6 −2.47453
\(436\) 0 0
\(437\) 3.73195e6 + 2.71142e6i 0.934829 + 0.679193i
\(438\) 0 0
\(439\) −5.51917e6 −1.36682 −0.683412 0.730033i \(-0.739505\pi\)
−0.683412 + 0.730033i \(0.739505\pi\)
\(440\) 0 0
\(441\) −1.28474e6 + 3.95401e6i −0.314570 + 0.968147i
\(442\) 0 0
\(443\) −1.54539e6 4.75621e6i −0.374134 1.15147i −0.944061 0.329772i \(-0.893028\pi\)
0.569926 0.821696i \(-0.306972\pi\)
\(444\) 0 0
\(445\) 3.90128e6 1.20069e7i 0.933914 2.87429i
\(446\) 0 0
\(447\) −1.72845e6 1.25579e6i −0.409155 0.297269i
\(448\) 0 0
\(449\) −1.90141e6 1.38146e6i −0.445103 0.323387i 0.342556 0.939497i \(-0.388707\pi\)
−0.787659 + 0.616111i \(0.788707\pi\)
\(450\) 0 0
\(451\) −1.64532e6 5.06378e6i −0.380899 1.17229i
\(452\) 0 0
\(453\) −5.84144e6 + 4.24406e6i −1.33744 + 0.971707i
\(454\) 0 0
\(455\) 1.80630e6 1.31235e6i 0.409036 0.297182i
\(456\) 0 0
\(457\) 435833. + 1.34135e6i 0.0976179 + 0.300437i 0.987927 0.154920i \(-0.0495118\pi\)
−0.890309 + 0.455356i \(0.849512\pi\)
\(458\) 0 0
\(459\) 6.74664e6 1.49471
\(460\) 0 0
\(461\) 681148. 2.09636e6i 0.149276 0.459423i −0.848260 0.529579i \(-0.822350\pi\)
0.997536 + 0.0701562i \(0.0223498\pi\)
\(462\) 0 0
\(463\) 3.30779e6 2.40325e6i 0.717109 0.521010i −0.168350 0.985727i \(-0.553844\pi\)
0.885459 + 0.464717i \(0.153844\pi\)
\(464\) 0 0
\(465\) −9.18887e6 + 9.13964e6i −1.97074 + 1.96018i
\(466\) 0 0
\(467\) −4.62498e6 + 3.36025e6i −0.981336 + 0.712983i −0.958007 0.286745i \(-0.907427\pi\)
−0.0233296 + 0.999728i \(0.507427\pi\)
\(468\) 0 0
\(469\) 1.40869e6 4.33550e6i 0.295722 0.910138i
\(470\) 0 0
\(471\) 9.30069e6 1.93180
\(472\) 0 0
\(473\) 1.55382e6 + 4.78216e6i 0.319335 + 0.982813i
\(474\) 0 0
\(475\) 8.47483e6 6.15733e6i 1.72344 1.25216i
\(476\) 0 0
\(477\) −6.20712e6 + 4.50973e6i −1.24909 + 0.907518i
\(478\) 0 0
\(479\) 845373. + 2.60179e6i 0.168349 + 0.518124i 0.999267 0.0382699i \(-0.0121847\pi\)
−0.830919 + 0.556394i \(0.812185\pi\)
\(480\) 0 0
\(481\) 815553. + 592534.i 0.160727 + 0.116775i
\(482\) 0 0
\(483\) −4.35754e6 3.16594e6i −0.849912 0.617497i
\(484\) 0 0
\(485\) −2.34947e6 + 7.23092e6i −0.453540 + 1.39585i
\(486\) 0 0
\(487\) 474428. + 1.46014e6i 0.0906458 + 0.278979i 0.986094 0.166186i \(-0.0531451\pi\)
−0.895449 + 0.445165i \(0.853145\pi\)
\(488\) 0 0
\(489\) −135350. + 416565.i −0.0255968 + 0.0787790i
\(490\) 0 0
\(491\) −2.38069e6 −0.445655 −0.222827 0.974858i \(-0.571529\pi\)
−0.222827 + 0.974858i \(0.571529\pi\)
\(492\) 0 0
\(493\) 4.03454e6 + 2.93126e6i 0.747612 + 0.543172i
\(494\) 0 0
\(495\) −2.63810e7 −4.83925
\(496\) 0 0
\(497\) −842911. −0.153070
\(498\) 0 0
\(499\) 3.08363e6 + 2.24039e6i 0.554384 + 0.402784i 0.829399 0.558656i \(-0.188683\pi\)
−0.275015 + 0.961440i \(0.588683\pi\)
\(500\) 0 0
\(501\) 1.09111e7 1.94211
\(502\) 0 0
\(503\) −216949. + 667701.i −0.0382329 + 0.117669i −0.968351 0.249591i \(-0.919704\pi\)
0.930118 + 0.367260i \(0.119704\pi\)
\(504\) 0 0
\(505\) −3.62938e6 1.11701e7i −0.633292 1.94907i
\(506\) 0 0
\(507\) −2.38794e6 + 7.34934e6i −0.412576 + 1.26978i
\(508\) 0 0
\(509\) 596994. + 433742.i 0.102135 + 0.0742056i 0.637681 0.770301i \(-0.279894\pi\)
−0.535545 + 0.844506i \(0.679894\pi\)
\(510\) 0 0
\(511\) −5.06304e6 3.67851e6i −0.857747 0.623190i
\(512\) 0 0
\(513\) −3.30714e6 1.01783e7i −0.554829 1.70759i
\(514\) 0 0
\(515\) 1.39549e7 1.01388e7i 2.31851 1.68450i
\(516\) 0 0
\(517\) 8.19214e6 5.95194e6i 1.34794 0.979337i
\(518\) 0 0
\(519\) −3.18160e6 9.79195e6i −0.518474 1.59570i
\(520\) 0 0
\(521\) 758303. 0.122391 0.0611954 0.998126i \(-0.480509\pi\)
0.0611954 + 0.998126i \(0.480509\pi\)
\(522\) 0 0
\(523\) −2.40674e6 + 7.40719e6i −0.384747 + 1.18413i 0.551917 + 0.833899i \(0.313897\pi\)
−0.936664 + 0.350230i \(0.886103\pi\)
\(524\) 0 0
\(525\) −9.89549e6 + 7.18949e6i −1.56689 + 1.13841i
\(526\) 0 0
\(527\) 6.53939e6 1.01774e6i 1.02568 0.159628i
\(528\) 0 0
\(529\) 735240. 534183.i 0.114233 0.0829948i
\(530\) 0 0
\(531\) 3.30390e6 1.01683e7i 0.508499 1.56500i
\(532\) 0 0
\(533\) −2.33127e6 −0.355447
\(534\) 0 0
\(535\) 5.49397e6 + 1.69087e7i 0.829855 + 2.55403i
\(536\) 0 0
\(537\) 1.45868e7 1.05979e7i 2.18285 1.58594i
\(538\) 0 0
\(539\) −4.75916e6 + 3.45773e6i −0.705599 + 0.512648i
\(540\) 0 0
\(541\) 2.41122e6 + 7.42096e6i 0.354196 + 1.09010i 0.956474 + 0.291816i \(0.0942596\pi\)
−0.602279 + 0.798286i \(0.705740\pi\)
\(542\) 0 0
\(543\) −3.41455e6 2.48082e6i −0.496974 0.361073i
\(544\) 0 0
\(545\) −3.25485e6 2.36479e6i −0.469397 0.341037i
\(546\) 0 0
\(547\) −2.82431e6 + 8.69234e6i −0.403594 + 1.24213i 0.518470 + 0.855096i \(0.326502\pi\)
−0.922064 + 0.387038i \(0.873498\pi\)
\(548\) 0 0
\(549\) −103847. 319610.i −0.0147050 0.0452573i
\(550\) 0 0
\(551\) 2.44456e6 7.52359e6i 0.343022 1.05571i
\(552\) 0 0
\(553\) 6.72942e6 0.935760
\(554\) 0 0
\(555\) −7.08296e6 5.14607e6i −0.976074 0.709159i
\(556\) 0 0
\(557\) 812063. 0.110905 0.0554526 0.998461i \(-0.482340\pi\)
0.0554526 + 0.998461i \(0.482340\pi\)
\(558\) 0 0
\(559\) 2.20162e6 0.297998
\(560\) 0 0
\(561\) 1.67815e7 + 1.21925e7i 2.25125 + 1.63563i
\(562\) 0 0
\(563\) 3.50634e6 0.466211 0.233105 0.972451i \(-0.425111\pi\)
0.233105 + 0.972451i \(0.425111\pi\)
\(564\) 0 0
\(565\) −5.54797e6 + 1.70749e7i −0.731161 + 2.25028i
\(566\) 0 0
\(567\) 920038. + 2.83159e6i 0.120184 + 0.369890i
\(568\) 0 0
\(569\) 2.32677e6 7.16107e6i 0.301282 0.927251i −0.679756 0.733438i \(-0.737914\pi\)
0.981038 0.193813i \(-0.0620855\pi\)
\(570\) 0 0
\(571\) 2.60569e6 + 1.89314e6i 0.334450 + 0.242992i 0.742317 0.670049i \(-0.233727\pi\)
−0.407866 + 0.913042i \(0.633727\pi\)
\(572\) 0 0
\(573\) −9.06919e6 6.58915e6i −1.15394 0.838384i
\(574\) 0 0
\(575\) 3.87891e6 + 1.19381e7i 0.489260 + 1.50579i
\(576\) 0 0
\(577\) 2.84653e6 2.06813e6i 0.355940 0.258605i −0.395417 0.918502i \(-0.629400\pi\)
0.751357 + 0.659896i \(0.229400\pi\)
\(578\) 0 0
\(579\) 1.59015e6 1.15531e6i 0.197125 0.143219i
\(580\) 0 0
\(581\) 2.92166e6 + 8.99194e6i 0.359078 + 1.10513i
\(582\) 0 0
\(583\) −1.08561e7 −1.32282
\(584\) 0 0
\(585\) −3.56943e6 + 1.09856e7i −0.431230 + 1.32719i
\(586\) 0 0
\(587\) 9.66206e6 7.01990e6i 1.15738 0.840883i 0.167933 0.985798i \(-0.446291\pi\)
0.989444 + 0.144915i \(0.0462909\pi\)
\(588\) 0 0
\(589\) −4.74096e6 9.36677e6i −0.563090 1.11250i
\(590\) 0 0
\(591\) 4.04810e6 2.94112e6i 0.476741 0.346373i
\(592\) 0 0
\(593\) 1.27919e6 3.93693e6i 0.149382 0.459749i −0.848167 0.529729i \(-0.822294\pi\)
0.997548 + 0.0699799i \(0.0222935\pi\)
\(594\) 0 0
\(595\) 9.90182e6 1.14663
\(596\) 0 0
\(597\) −4.96354e6 1.52762e7i −0.569975 1.75420i
\(598\) 0 0
\(599\) −2.51390e6 + 1.82645e6i −0.286273 + 0.207990i −0.721649 0.692259i \(-0.756616\pi\)
0.435376 + 0.900249i \(0.356616\pi\)
\(600\) 0 0
\(601\) 1.92636e6 1.39959e6i 0.217547 0.158057i −0.473675 0.880700i \(-0.657073\pi\)
0.691222 + 0.722643i \(0.257073\pi\)
\(602\) 0 0
\(603\) 7.28785e6 + 2.24297e7i 0.816218 + 2.51206i
\(604\) 0 0
\(605\) −1.82118e7 1.32317e7i −2.02286 1.46969i
\(606\) 0 0
\(607\) −1.05726e7 7.68141e6i −1.16468 0.846193i −0.174322 0.984689i \(-0.555773\pi\)
−0.990363 + 0.138496i \(0.955773\pi\)
\(608\) 0 0
\(609\) −2.85435e6 + 8.78479e6i −0.311863 + 0.959816i
\(610\) 0 0
\(611\) −1.37008e6 4.21668e6i −0.148472 0.456949i
\(612\) 0 0
\(613\) −50272.2 + 154722.i −0.00540352 + 0.0166303i −0.953722 0.300690i \(-0.902783\pi\)
0.948318 + 0.317320i \(0.102783\pi\)
\(614\) 0 0
\(615\) 2.02468e7 2.15858
\(616\) 0 0
\(617\) −4.33384e6 3.14872e6i −0.458310 0.332982i 0.334558 0.942375i \(-0.391413\pi\)
−0.792868 + 0.609393i \(0.791413\pi\)
\(618\) 0 0
\(619\) −1.41681e6 −0.148623 −0.0743115 0.997235i \(-0.523676\pi\)
−0.0743115 + 0.997235i \(0.523676\pi\)
\(620\) 0 0
\(621\) 1.28240e7 1.33443
\(622\) 0 0
\(623\) −9.66031e6 7.01863e6i −0.997174 0.724489i
\(624\) 0 0
\(625\) 2.05505e6 0.210437
\(626\) 0 0
\(627\) 1.01681e7 3.12941e7i 1.03293 3.17902i
\(628\) 0 0
\(629\) 1.38153e6 + 4.25191e6i 0.139230 + 0.428506i
\(630\) 0 0
\(631\) −1.85188e6 + 5.69950e6i −0.185157 + 0.569853i −0.999951 0.00989658i \(-0.996850\pi\)
0.814794 + 0.579750i \(0.196850\pi\)
\(632\) 0 0
\(633\) −9.23621e6 6.71050e6i −0.916188 0.665650i
\(634\) 0 0
\(635\) 1.23102e7 + 8.94387e6i 1.21152 + 0.880220i
\(636\) 0 0
\(637\) 795939. + 2.44965e6i 0.0777197 + 0.239197i
\(638\) 0 0
\(639\) 3.52796e6 2.56321e6i 0.341799 0.248332i
\(640\) 0 0
\(641\) −2.20751e6 + 1.60385e6i −0.212206 + 0.154176i −0.688811 0.724941i \(-0.741867\pi\)
0.476606 + 0.879117i \(0.341867\pi\)
\(642\) 0 0
\(643\) 299728. + 922467.i 0.0285890 + 0.0879880i 0.964333 0.264692i \(-0.0852704\pi\)
−0.935744 + 0.352680i \(0.885270\pi\)
\(644\) 0 0
\(645\) −1.91208e7 −1.80970
\(646\) 0 0
\(647\) −139635. + 429753.i −0.0131140 + 0.0403606i −0.957399 0.288767i \(-0.906755\pi\)
0.944285 + 0.329128i \(0.106755\pi\)
\(648\) 0 0
\(649\) 1.22389e7 8.89209e6i 1.14059 0.828690i
\(650\) 0 0
\(651\) 5.53569e6 + 1.09369e7i 0.511941 + 1.01145i
\(652\) 0 0
\(653\) 4.77836e6 3.47168e6i 0.438527 0.318608i −0.346523 0.938042i \(-0.612638\pi\)
0.785049 + 0.619433i \(0.212638\pi\)
\(654\) 0 0
\(655\) −85444.6 + 262972.i −0.00778183 + 0.0239500i
\(656\) 0 0
\(657\) 3.23771e7 2.92634
\(658\) 0 0
\(659\) −5.15107e6 1.58534e7i −0.462045 1.42203i −0.862661 0.505782i \(-0.831204\pi\)
0.400617 0.916246i \(-0.368796\pi\)
\(660\) 0 0
\(661\) −1.21323e7 + 8.81464e6i −1.08004 + 0.784695i −0.977690 0.210054i \(-0.932636\pi\)
−0.102350 + 0.994748i \(0.532636\pi\)
\(662\) 0 0
\(663\) 7.34777e6 5.33846e6i 0.649190 0.471664i
\(664\) 0 0
\(665\) −4.85378e6 1.49384e7i −0.425624 1.30993i
\(666\) 0 0
\(667\) 7.66885e6 + 5.57174e6i 0.667445 + 0.484927i
\(668\) 0 0
\(669\) −8.59653e6 6.24574e6i −0.742605 0.539534i
\(670\) 0 0
\(671\) 146939. 452232.i 0.0125988 0.0387752i
\(672\) 0 0
\(673\) 3.13440e6 + 9.64669e6i 0.266758 + 0.820995i 0.991283 + 0.131748i \(0.0420588\pi\)
−0.724526 + 0.689248i \(0.757941\pi\)
\(674\) 0 0
\(675\) 8.99915e6 2.76965e7i 0.760225 2.33973i
\(676\) 0 0
\(677\) −1.00296e7 −0.841031 −0.420515 0.907285i \(-0.638151\pi\)
−0.420515 + 0.907285i \(0.638151\pi\)
\(678\) 0 0
\(679\) 5.81773e6 + 4.22683e6i 0.484261 + 0.351836i
\(680\) 0 0
\(681\) −1.54599e7 −1.27744
\(682\) 0 0
\(683\) −1.72101e7 −1.41167 −0.705833 0.708378i \(-0.749427\pi\)
−0.705833 + 0.708378i \(0.749427\pi\)
\(684\) 0 0
\(685\) 2.27637e7 + 1.65388e7i 1.85360 + 1.34672i
\(686\) 0 0
\(687\) 3.02730e7 2.44717
\(688\) 0 0
\(689\) −1.46886e6 + 4.52069e6i −0.117878 + 0.362791i
\(690\) 0 0
\(691\) −659314. 2.02916e6i −0.0525288 0.161667i 0.921351 0.388732i \(-0.127087\pi\)
−0.973880 + 0.227065i \(0.927087\pi\)
\(692\) 0 0
\(693\) −7.71050e6 + 2.37305e7i −0.609887 + 1.87704i
\(694\) 0 0
\(695\) −2.40118e7 1.74456e7i −1.88566 1.37001i
\(696\) 0 0
\(697\) −8.36438e6 6.07708e6i −0.652156 0.473819i
\(698\) 0 0
\(699\) 7.07604e6 + 2.17778e7i 0.547769 + 1.68586i
\(700\) 0 0
\(701\) −1.93968e7 + 1.40926e7i −1.49086 + 1.08317i −0.517005 + 0.855982i \(0.672953\pi\)
−0.973851 + 0.227188i \(0.927047\pi\)
\(702\) 0 0
\(703\) 5.73743e6 4.16849e6i 0.437854 0.318120i
\(704\) 0 0
\(705\) 1.18990e7 + 3.66213e7i 0.901648 + 2.77499i
\(706\) 0 0
\(707\) −1.11086e7 −0.835815
\(708\) 0 0
\(709\) 3.13045e6 9.63453e6i 0.233879 0.719805i −0.763389 0.645939i \(-0.776466\pi\)
0.997268 0.0738663i \(-0.0235338\pi\)
\(710\) 0 0
\(711\) −2.81656e7 + 2.04635e7i −2.08951 + 1.51812i
\(712\) 0 0
\(713\) 1.24301e7 1.93451e6i 0.915692 0.142511i
\(714\) 0 0
\(715\) −1.32225e7 + 9.60673e6i −0.967274 + 0.702766i
\(716\) 0 0
\(717\) 6.20147e6 1.90862e7i 0.450502 1.38650i
\(718\) 0 0
\(719\) −2.18439e7 −1.57583 −0.787913 0.615787i \(-0.788838\pi\)
−0.787913 + 0.615787i \(0.788838\pi\)
\(720\) 0 0
\(721\) −5.04151e6 1.55162e7i −0.361179 1.11159i
\(722\) 0 0
\(723\) −3.46499e7 + 2.51746e7i −2.46522 + 1.79109i
\(724\) 0 0
\(725\) 1.74151e7 1.26528e7i 1.23050 0.894008i
\(726\) 0 0
\(727\) −5.69539e6 1.75286e7i −0.399657 1.23002i −0.925275 0.379297i \(-0.876166\pi\)
0.525618 0.850721i \(-0.323834\pi\)
\(728\) 0 0
\(729\) 1.57708e7 + 1.14581e7i 1.09909 + 0.798536i
\(730\) 0 0
\(731\) 7.89919e6 + 5.73910e6i 0.546751 + 0.397238i
\(732\) 0 0
\(733\) −6.70308e6 + 2.06300e7i −0.460802 + 1.41820i 0.403384 + 0.915031i \(0.367834\pi\)
−0.864186 + 0.503173i \(0.832166\pi\)
\(734\) 0 0
\(735\) −6.91262e6 2.12748e7i −0.471981 1.45261i
\(736\) 0 0
\(737\) −1.03119e7 + 3.17369e7i −0.699313 + 2.15226i
\(738\) 0 0
\(739\) −2.08603e7 −1.40511 −0.702555 0.711630i \(-0.747957\pi\)
−0.702555 + 0.711630i \(0.747957\pi\)
\(740\) 0 0
\(741\) −1.16557e7 8.46835e6i −0.779817 0.566570i
\(742\) 0 0
\(743\) −1.24362e7 −0.826447 −0.413223 0.910630i \(-0.635597\pi\)
−0.413223 + 0.910630i \(0.635597\pi\)
\(744\) 0 0
\(745\) 7.46562e6 0.492805
\(746\) 0 0
\(747\) −3.95721e7 2.87508e7i −2.59470 1.88516i
\(748\) 0 0
\(749\) 1.68156e7 1.09524
\(750\) 0 0
\(751\) −1.98666e6 + 6.11430e6i −0.128536 + 0.395592i −0.994529 0.104464i \(-0.966687\pi\)
0.865993 + 0.500056i \(0.166687\pi\)
\(752\) 0 0
\(753\) 1.11897e7 + 3.44382e7i 0.719166 + 2.21337i
\(754\) 0 0
\(755\) 7.79671e6 2.39958e7i 0.497788 1.53203i
\(756\) 0 0
\(757\) −1.18273e6 859304.i −0.0750147 0.0545014i 0.549646 0.835398i \(-0.314763\pi\)
−0.624661 + 0.780896i \(0.714763\pi\)
\(758\) 0 0
\(759\) 3.18982e7 + 2.31754e7i 2.00984 + 1.46024i
\(760\) 0 0
\(761\) 690805. + 2.12608e6i 0.0432408 + 0.133081i 0.970346 0.241719i \(-0.0777112\pi\)
−0.927105 + 0.374800i \(0.877711\pi\)
\(762\) 0 0
\(763\) −3.07851e6 + 2.23667e6i −0.191438 + 0.139088i
\(764\) 0 0
\(765\) −4.14435e7 + 3.01105e7i −2.56037 + 1.86022i
\(766\) 0 0
\(767\) −2.04688e6 6.29964e6i −0.125633 0.386659i
\(768\) 0 0
\(769\) 2.30960e6 0.140838 0.0704191 0.997517i \(-0.477566\pi\)
0.0704191 + 0.997517i \(0.477566\pi\)
\(770\) 0 0
\(771\) −4.54832e6 + 1.39983e7i −0.275559 + 0.848084i
\(772\) 0 0
\(773\) −5.24266e6 + 3.80901e6i −0.315575 + 0.229279i −0.734285 0.678841i \(-0.762482\pi\)
0.418710 + 0.908120i \(0.362482\pi\)
\(774\) 0 0
\(775\) 4.54465e6 2.82032e7i 0.271798 1.68673i
\(776\) 0 0
\(777\) −6.69921e6 + 4.86726e6i −0.398081 + 0.289223i
\(778\) 0 0
\(779\) −5.06805e6 + 1.55979e7i −0.299225 + 0.920919i
\(780\) 0 0
\(781\) 6.17031e6 0.361975
\(782\) 0 0
\(783\) −6.79590e6 2.09156e7i −0.396134 1.21918i
\(784\) 0 0
\(785\) −2.62930e7 + 1.91030e7i −1.52288 + 1.10644i
\(786\) 0 0
\(787\) −1.31585e7 + 9.56018e6i −0.757301 + 0.550211i −0.898081 0.439830i \(-0.855039\pi\)
0.140781 + 0.990041i \(0.455039\pi\)
\(788\) 0 0
\(789\) 4.88314e6 + 1.50288e7i 0.279259 + 0.859470i
\(790\) 0 0
\(791\) 1.37378e7 + 9.98112e6i 0.780687 + 0.567202i
\(792\) 0 0
\(793\) −168437. 122376.i −0.00951161 0.00691059i
\(794\) 0 0
\(795\) 1.27568e7 3.92615e7i 0.715856 2.20318i
\(796\) 0 0
\(797\) −8.73726e6 2.68905e7i −0.487225 1.49952i −0.828733 0.559644i \(-0.810938\pi\)
0.341508 0.939879i \(-0.389062\pi\)
\(798\) 0 0
\(799\) 6.07617e6 1.87005e7i 0.336715 1.03630i
\(800\) 0 0
\(801\) 6.17757e7 3.40201
\(802\) 0 0
\(803\) 3.70626e7 + 2.69276e7i 2.02837 + 1.47370i
\(804\) 0 0
\(805\) 1.88214e7 1.02367
\(806\) 0 0
\(807\) 4.21589e7 2.27879
\(808\) 0 0
\(809\) −1.17431e7 8.53183e6i −0.630826 0.458322i 0.225860 0.974160i \(-0.427481\pi\)
−0.856686 + 0.515838i \(0.827481\pi\)
\(810\) 0 0
\(811\) −1.18842e7 −0.634482 −0.317241 0.948345i \(-0.602756\pi\)
−0.317241 + 0.948345i \(0.602756\pi\)
\(812\) 0 0
\(813\) 8.95404e6 2.75577e7i 0.475108 1.46223i
\(814\) 0 0
\(815\) −472961. 1.45562e6i −0.0249420 0.0767636i
\(816\) 0 0
\(817\) 4.78619e6 1.47304e7i 0.250862 0.772074i
\(818\) 0 0
\(819\) 8.83858e6 + 6.42160e6i 0.460440 + 0.334529i
\(820\) 0 0
\(821\) −2.93998e7 2.13602e7i −1.52225 1.10598i −0.960359 0.278768i \(-0.910074\pi\)
−0.561891 0.827211i \(-0.689926\pi\)
\(822\) 0 0
\(823\) −4.30367e6 1.32453e7i −0.221483 0.681654i −0.998630 0.0523344i \(-0.983334\pi\)
0.777147 0.629319i \(-0.216666\pi\)
\(824\) 0 0
\(825\) 7.24373e7 5.26288e7i 3.70533 2.69208i
\(826\) 0 0
\(827\) 1.51525e7 1.10089e7i 0.770408 0.559734i −0.131677 0.991293i \(-0.542036\pi\)
0.902085 + 0.431558i \(0.142036\pi\)
\(828\) 0 0
\(829\) −1.06015e7 3.26279e7i −0.535771 1.64893i −0.741977 0.670426i \(-0.766112\pi\)
0.206205 0.978509i \(-0.433888\pi\)
\(830\) 0 0
\(831\) 1.20428e7 0.604958
\(832\) 0 0
\(833\) −3.52990e6 + 1.08639e7i −0.176258 + 0.542468i
\(834\) 0 0
\(835\) −3.08456e7 + 2.24106e7i −1.53100 + 1.11234i
\(836\) 0 0
\(837\) −2.59685e7 1.33196e7i −1.28125 0.657169i
\(838\) 0 0
\(839\) 1.21668e7 8.83969e6i 0.596721 0.433543i −0.247993 0.968762i \(-0.579771\pi\)
0.844713 + 0.535219i \(0.179771\pi\)
\(840\) 0 0
\(841\) −1.31492e6 + 4.04691e6i −0.0641076 + 0.197303i
\(842\) 0 0
\(843\) −3.96703e7 −1.92264
\(844\) 0 0
\(845\) −8.34432e6 2.56812e7i −0.402022 1.23730i
\(846\) 0 0
\(847\) −1.72251e7 + 1.25148e7i −0.825001 + 0.599398i
\(848\) 0 0
\(849\) −2.61865e7 + 1.90256e7i −1.24683 + 0.905876i
\(850\) 0 0
\(851\) 2.62601e6 + 8.08202e6i 0.124300 + 0.382557i
\(852\) 0 0
\(853\) 5.58440e6 + 4.05730e6i 0.262787 + 0.190926i 0.711375 0.702813i \(-0.248073\pi\)
−0.448588 + 0.893739i \(0.648073\pi\)
\(854\) 0 0
\(855\) 6.57414e7 + 4.77639e7i 3.07556 + 2.23452i
\(856\) 0 0
\(857\) 2.78372e6 8.56742e6i 0.129472 0.398472i −0.865218 0.501396i \(-0.832820\pi\)
0.994689 + 0.102924i \(0.0328198\pi\)
\(858\) 0 0
\(859\) −5.04893e6 1.55390e7i −0.233462 0.718523i −0.997322 0.0731403i \(-0.976698\pi\)
0.763859 0.645383i \(-0.223302\pi\)
\(860\) 0 0
\(861\) 5.91762e6 1.82126e7i 0.272044 0.837266i
\(862\) 0 0
\(863\) −2.77704e6 −0.126927 −0.0634636 0.997984i \(-0.520215\pi\)
−0.0634636 + 0.997984i \(0.520215\pi\)
\(864\) 0 0
\(865\) 2.91063e7 + 2.11470e7i 1.32266 + 0.960966i
\(866\) 0 0
\(867\) 2.89688e6 0.130883
\(868\) 0 0
\(869\) −4.92609e7 −2.21285
\(870\) 0 0
\(871\) 1.18206e7 + 8.58818e6i 0.527952 + 0.383580i
\(872\) 0 0
\(873\) −3.72032e7 −1.65213
\(874\) 0 0
\(875\) 5.47707e6 1.68567e7i 0.241840 0.744307i
\(876\) 0 0
\(877\) 1.14838e7 + 3.53436e7i 0.504182 + 1.55171i 0.802141 + 0.597135i \(0.203694\pi\)
−0.297958 + 0.954579i \(0.596306\pi\)
\(878\) 0 0
\(879\) −6.75619e6 + 2.07934e7i −0.294937 + 0.907723i
\(880\) 0 0
\(881\) 9.41929e6 + 6.84351e6i 0.408863 + 0.297057i 0.773141 0.634234i \(-0.218684\pi\)
−0.364278 + 0.931290i \(0.618684\pi\)
\(882\) 0 0
\(883\) −1.35344e7 9.83334e6i −0.584168 0.424423i 0.256056 0.966662i \(-0.417577\pi\)
−0.840224 + 0.542239i \(0.817577\pi\)
\(884\) 0 0
\(885\) 1.77769e7 + 5.47115e7i 0.762951 + 2.34812i
\(886\) 0 0
\(887\) 1.93918e7 1.40890e7i 0.827580 0.601272i −0.0912935 0.995824i \(-0.529100\pi\)
0.918874 + 0.394552i \(0.129100\pi\)
\(888\) 0 0
\(889\) 1.16432e7 8.45930e6i 0.494105 0.358988i
\(890\) 0 0
\(891\) −6.73490e6 2.07279e7i −0.284208 0.874703i
\(892\) 0 0
\(893\) −3.11910e7 −1.30888
\(894\) 0 0
\(895\) −1.94694e7 + 5.99206e7i −0.812446 + 2.50045i
\(896\) 0 0
\(897\) 1.39666e7 1.01473e7i 0.579576 0.421087i
\(898\) 0 0
\(899\) −9.74227e6 1.92479e7i −0.402032 0.794301i
\(900\) 0 0
\(901\) −1.70545e7 + 1.23908e7i −0.699885 + 0.508496i
\(902\) 0 0
\(903\) −5.58851e6 + 1.71997e7i −0.228075 + 0.701941i
\(904\) 0 0
\(905\) 1.47483e7 0.598579
\(906\) 0 0
\(907\) 6.64526e6 + 2.04520e7i 0.268221 + 0.825501i 0.990934 + 0.134352i \(0.0428952\pi\)
−0.722712 + 0.691149i \(0.757105\pi\)
\(908\) 0 0
\(909\) 4.64944e7 3.37801e7i 1.86634 1.35597i
\(910\) 0 0
\(911\) 2.94967e7 2.14306e7i 1.17755 0.855537i 0.185653 0.982615i \(-0.440560\pi\)
0.991893 + 0.127078i \(0.0405599\pi\)
\(912\) 0 0
\(913\) −2.13872e7 6.58231e7i −0.849136 2.61337i
\(914\) 0 0
\(915\) 1.46285e6 + 1.06282e6i 0.0577626 + 0.0419670i
\(916\) 0 0
\(917\) 211577. + 153720.i 0.00830894 + 0.00603680i
\(918\) 0 0
\(919\) 1.35041e6 4.15613e6i 0.0527444 0.162330i −0.921215 0.389055i \(-0.872802\pi\)
0.973959 + 0.226724i \(0.0728017\pi\)
\(920\) 0 0
\(921\) −4.05183e6 1.24703e7i −0.157399 0.484425i
\(922\) 0 0
\(923\) 834860. 2.56944e6i 0.0322559 0.0992735i
\(924\) 0 0
\(925\) 1.92979e7 0.741574
\(926\) 0 0
\(927\) 6.82841e7 + 4.96113e7i 2.60988 + 1.89619i
\(928\) 0 0
\(929\) 1.31305e7 0.499163 0.249582 0.968354i \(-0.419707\pi\)
0.249582 + 0.968354i \(0.419707\pi\)
\(930\) 0 0
\(931\) 1.81202e7 0.685154
\(932\) 0 0
\(933\) 3.93030e7 + 2.85553e7i 1.47816 + 1.07395i
\(934\) 0 0
\(935\) −7.24836e7 −2.71151
\(936\) 0 0
\(937\) −7.70490e6 + 2.37132e7i −0.286694 + 0.882352i 0.699192 + 0.714934i \(0.253543\pi\)
−0.985886 + 0.167419i \(0.946457\pi\)
\(938\) 0 0
\(939\) 1.20054e7 + 3.69488e7i 0.444337 + 1.36753i
\(940\) 0 0
\(941\) 3.90480e6 1.20177e7i 0.143756 0.442434i −0.853093 0.521759i \(-0.825276\pi\)
0.996849 + 0.0793241i \(0.0252762\pi\)
\(942\) 0 0
\(943\) −1.58990e7 1.15513e7i −0.582225 0.423011i
\(944\) 0 0
\(945\) −3.53265e7 2.56662e7i −1.28683 0.934937i
\(946\) 0 0
\(947\) −1.01647e7 3.12836e7i −0.368314 1.13355i −0.947880 0.318628i \(-0.896778\pi\)
0.579566 0.814926i \(-0.303222\pi\)
\(948\) 0 0
\(949\) 1.62279e7 1.17902e7i 0.584919 0.424968i
\(950\) 0 0
\(951\) −5.16743e7 + 3.75436e7i −1.85278 + 1.34612i
\(952\) 0 0
\(953\) 1.05996e6 + 3.26222e6i 0.0378057 + 0.116354i 0.968178 0.250261i \(-0.0805165\pi\)
−0.930373 + 0.366615i \(0.880517\pi\)
\(954\) 0 0
\(955\) 3.91722e7 1.38985
\(956\) 0 0
\(957\) 2.08945e7 6.43067e7i 0.737484 2.26974i
\(958\) 0 0
\(959\) 2.15303e7 1.56427e7i 0.755969 0.549244i
\(960\) 0 0
\(961\) −2.71800e7 8.99305e6i −0.949383 0.314122i
\(962\) 0 0
\(963\) −7.03809e7 + 5.11347e7i −2.44562 + 1.77685i
\(964\) 0 0
\(965\) −2.12241e6 + 6.53210e6i −0.0733686 + 0.225805i
\(966\) 0 0
\(967\) 2.68178e6 0.0922266 0.0461133 0.998936i \(-0.485316\pi\)
0.0461133 + 0.998936i \(0.485316\pi\)
\(968\) 0 0
\(969\) −1.97445e7 6.07672e7i −0.675517 2.07903i
\(970\) 0 0
\(971\) 2.76124e6 2.00616e6i 0.0939845 0.0682837i −0.539801 0.841793i \(-0.681500\pi\)
0.633785 + 0.773509i \(0.281500\pi\)
\(972\) 0 0
\(973\) −2.27109e7 + 1.65004e7i −0.769045 + 0.558744i
\(974\) 0 0
\(975\) −1.21147e7 3.72851e7i −0.408132 1.25610i
\(976\) 0 0
\(977\) 4.00465e7 + 2.90955e7i 1.34224 + 0.975191i 0.999359 + 0.0358106i \(0.0114013\pi\)
0.342877 + 0.939380i \(0.388599\pi\)
\(978\) 0 0
\(979\) 7.07157e7 + 5.13780e7i 2.35808 + 1.71325i
\(980\) 0 0
\(981\) 6.08344e6 1.87229e7i 0.201826 0.621156i
\(982\) 0 0
\(983\) −7.25715e6 2.23352e7i −0.239542 0.737235i −0.996486 0.0837555i \(-0.973309\pi\)
0.756944 0.653480i \(-0.226691\pi\)
\(984\) 0 0
\(985\) −5.40310e6 + 1.66290e7i −0.177440 + 0.546106i
\(986\) 0 0
\(987\) 3.64197e7 1.18999
\(988\) 0 0
\(989\) 1.50148e7 + 1.09089e7i 0.488122 + 0.354641i
\(990\) 0 0
\(991\) 4.34729e7 1.40616 0.703079 0.711111i \(-0.251808\pi\)
0.703079 + 0.711111i \(0.251808\pi\)
\(992\) 0 0
\(993\) 9.71020e6 0.312503
\(994\) 0 0
\(995\) 4.54082e7 + 3.29910e7i 1.45404 + 1.05642i
\(996\) 0 0
\(997\) 6.17903e7 1.96871 0.984357 0.176188i \(-0.0563766\pi\)
0.984357 + 0.176188i \(0.0563766\pi\)
\(998\) 0 0
\(999\) 6.09240e6 1.87505e7i 0.193141 0.594427i
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 124.6.f.a.109.2 yes 56
31.2 even 5 inner 124.6.f.a.33.2 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.6.f.a.33.2 56 31.2 even 5 inner
124.6.f.a.109.2 yes 56 1.1 even 1 trivial