Properties

Label 76.7.h.a.69.9
Level $76$
Weight $7$
Character 76.69
Analytic conductor $17.484$
Analytic rank $0$
Dimension $20$
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] = [76,7,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
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("76.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.h (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 19^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 69.9
Root \(-42.7313i\) of defining polynomial
Character \(\chi\) \(=\) 76.69
Dual form 76.7.h.a.65.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(35.5064 - 20.4996i) q^{3} +(20.1615 + 34.9207i) q^{5} +53.8286 q^{7} +(475.968 - 824.401i) q^{9} +O(q^{10})\) \(q+(35.5064 - 20.4996i) q^{3} +(20.1615 + 34.9207i) q^{5} +53.8286 q^{7} +(475.968 - 824.401i) q^{9} +1995.81 q^{11} +(1394.74 + 805.253i) q^{13} +(1431.72 + 826.606i) q^{15} +(-1591.10 - 2755.87i) q^{17} +(-2912.77 - 6209.80i) q^{19} +(1911.26 - 1103.46i) q^{21} +(-3377.10 + 5849.31i) q^{23} +(6999.53 - 12123.5i) q^{25} -9140.20i q^{27} +(454.264 + 262.269i) q^{29} +10098.7i q^{31} +(70864.0 - 40913.4i) q^{33} +(1085.26 + 1879.73i) q^{35} -45019.8i q^{37} +66029.5 q^{39} +(66707.8 - 38513.8i) q^{41} +(24733.2 + 42839.2i) q^{43} +38384.9 q^{45} +(-4692.75 + 8128.08i) q^{47} -114751. q^{49} +(-112989. - 65234.1i) q^{51} +(54739.9 + 31604.1i) q^{53} +(40238.6 + 69695.2i) q^{55} +(-230721. - 160777. i) q^{57} +(-241556. + 139462. i) q^{59} +(-211690. + 366657. i) q^{61} +(25620.7 - 44376.3i) q^{63} +64940.4i q^{65} +(-2972.38 - 1716.11i) q^{67} +276917. i q^{69} +(-173856. + 100376. i) q^{71} +(120923. + 209445. i) q^{73} -573950. i q^{75} +107432. q^{77} +(58299.2 - 33659.0i) q^{79} +(159610. + 276453. i) q^{81} -336237. q^{83} +(64158.1 - 111125. i) q^{85} +21505.7 q^{87} +(345795. + 199645. i) q^{89} +(75076.8 + 43345.6i) q^{91} +(207020. + 358569. i) q^{93} +(158125. - 226915. i) q^{95} +(-1.20292e6 + 694509. i) q^{97} +(949943. - 1.64535e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 35.5064 20.4996i 1.31505 0.759245i 0.332123 0.943236i \(-0.392235\pi\)
0.982928 + 0.183991i \(0.0589019\pi\)
\(4\) 0 0
\(5\) 20.1615 + 34.9207i 0.161292 + 0.279366i 0.935332 0.353770i \(-0.115101\pi\)
−0.774040 + 0.633136i \(0.781767\pi\)
\(6\) 0 0
\(7\) 53.8286 0.156935 0.0784673 0.996917i \(-0.474997\pi\)
0.0784673 + 0.996917i \(0.474997\pi\)
\(8\) 0 0
\(9\) 475.968 824.401i 0.652905 1.13087i
\(10\) 0 0
\(11\) 1995.81 1.49948 0.749742 0.661731i \(-0.230178\pi\)
0.749742 + 0.661731i \(0.230178\pi\)
\(12\) 0 0
\(13\) 1394.74 + 805.253i 0.634838 + 0.366524i 0.782623 0.622496i \(-0.213881\pi\)
−0.147785 + 0.989019i \(0.547214\pi\)
\(14\) 0 0
\(15\) 1431.72 + 826.606i 0.424214 + 0.244920i
\(16\) 0 0
\(17\) −1591.10 2755.87i −0.323856 0.560935i 0.657424 0.753521i \(-0.271646\pi\)
−0.981280 + 0.192586i \(0.938313\pi\)
\(18\) 0 0
\(19\) −2912.77 6209.80i −0.424665 0.905351i
\(20\) 0 0
\(21\) 1911.26 1103.46i 0.206377 0.119152i
\(22\) 0 0
\(23\) −3377.10 + 5849.31i −0.277562 + 0.480752i −0.970778 0.239978i \(-0.922860\pi\)
0.693216 + 0.720730i \(0.256193\pi\)
\(24\) 0 0
\(25\) 6999.53 12123.5i 0.447970 0.775906i
\(26\) 0 0
\(27\) 9140.20i 0.464370i
\(28\) 0 0
\(29\) 454.264 + 262.269i 0.0186258 + 0.0107536i 0.509284 0.860599i \(-0.329910\pi\)
−0.490658 + 0.871352i \(0.663244\pi\)
\(30\) 0 0
\(31\) 10098.7i 0.338986i 0.985531 + 0.169493i \(0.0542130\pi\)
−0.985531 + 0.169493i \(0.945787\pi\)
\(32\) 0 0
\(33\) 70864.0 40913.4i 1.97190 1.13847i
\(34\) 0 0
\(35\) 1085.26 + 1879.73i 0.0253123 + 0.0438422i
\(36\) 0 0
\(37\) 45019.8i 0.888788i −0.895831 0.444394i \(-0.853419\pi\)
0.895831 0.444394i \(-0.146581\pi\)
\(38\) 0 0
\(39\) 66029.5 1.11313
\(40\) 0 0
\(41\) 66707.8 38513.8i 0.967888 0.558811i 0.0692965 0.997596i \(-0.477925\pi\)
0.898592 + 0.438785i \(0.144591\pi\)
\(42\) 0 0
\(43\) 24733.2 + 42839.2i 0.311082 + 0.538810i 0.978597 0.205786i \(-0.0659752\pi\)
−0.667515 + 0.744597i \(0.732642\pi\)
\(44\) 0 0
\(45\) 38384.9 0.421233
\(46\) 0 0
\(47\) −4692.75 + 8128.08i −0.0451995 + 0.0782879i −0.887740 0.460345i \(-0.847726\pi\)
0.842541 + 0.538633i \(0.181059\pi\)
\(48\) 0 0
\(49\) −114751. −0.975372
\(50\) 0 0
\(51\) −112989. 65234.1i −0.851774 0.491772i
\(52\) 0 0
\(53\) 54739.9 + 31604.1i 0.367685 + 0.212283i 0.672447 0.740145i \(-0.265243\pi\)
−0.304761 + 0.952429i \(0.598577\pi\)
\(54\) 0 0
\(55\) 40238.6 + 69695.2i 0.241855 + 0.418904i
\(56\) 0 0
\(57\) −230721. 160777.i −1.24584 0.868158i
\(58\) 0 0
\(59\) −241556. + 139462.i −1.17615 + 0.679049i −0.955120 0.296219i \(-0.904274\pi\)
−0.221027 + 0.975268i \(0.570941\pi\)
\(60\) 0 0
\(61\) −211690. + 366657.i −0.932631 + 1.61536i −0.153827 + 0.988098i \(0.549160\pi\)
−0.778804 + 0.627267i \(0.784174\pi\)
\(62\) 0 0
\(63\) 25620.7 44376.3i 0.102463 0.177472i
\(64\) 0 0
\(65\) 64940.4i 0.236469i
\(66\) 0 0
\(67\) −2972.38 1716.11i −0.00988281 0.00570584i 0.495050 0.868864i \(-0.335150\pi\)
−0.504933 + 0.863158i \(0.668483\pi\)
\(68\) 0 0
\(69\) 276917.i 0.842951i
\(70\) 0 0
\(71\) −173856. + 100376.i −0.485752 + 0.280449i −0.722810 0.691046i \(-0.757150\pi\)
0.237058 + 0.971495i \(0.423817\pi\)
\(72\) 0 0
\(73\) 120923. + 209445.i 0.310843 + 0.538396i 0.978545 0.206032i \(-0.0660553\pi\)
−0.667702 + 0.744429i \(0.732722\pi\)
\(74\) 0 0
\(75\) 573950.i 1.36047i
\(76\) 0 0
\(77\) 107432. 0.235321
\(78\) 0 0
\(79\) 58299.2 33659.0i 0.118245 0.0682685i −0.439711 0.898139i \(-0.644919\pi\)
0.557956 + 0.829871i \(0.311586\pi\)
\(80\) 0 0
\(81\) 159610. + 276453.i 0.300335 + 0.520195i
\(82\) 0 0
\(83\) −336237. −0.588045 −0.294023 0.955798i \(-0.594994\pi\)
−0.294023 + 0.955798i \(0.594994\pi\)
\(84\) 0 0
\(85\) 64158.1 111125.i 0.104471 0.180949i
\(86\) 0 0
\(87\) 21505.7 0.0326584
\(88\) 0 0
\(89\) 345795. + 199645.i 0.490511 + 0.283197i 0.724786 0.688974i \(-0.241938\pi\)
−0.234275 + 0.972170i \(0.575272\pi\)
\(90\) 0 0
\(91\) 75076.8 + 43345.6i 0.0996281 + 0.0575203i
\(92\) 0 0
\(93\) 207020. + 358569.i 0.257373 + 0.445783i
\(94\) 0 0
\(95\) 158125. 226915.i 0.184429 0.264663i
\(96\) 0 0
\(97\) −1.20292e6 + 694509.i −1.31802 + 0.760961i −0.983410 0.181394i \(-0.941939\pi\)
−0.334613 + 0.942356i \(0.608606\pi\)
\(98\) 0 0
\(99\) 949943. 1.64535e6i 0.979021 1.69571i
\(100\) 0 0
\(101\) −667775. + 1.15662e6i −0.648136 + 1.12260i 0.335432 + 0.942065i \(0.391118\pi\)
−0.983568 + 0.180540i \(0.942215\pi\)
\(102\) 0 0
\(103\) 777460.i 0.711486i −0.934584 0.355743i \(-0.884228\pi\)
0.934584 0.355743i \(-0.115772\pi\)
\(104\) 0 0
\(105\) 77067.6 + 44495.0i 0.0665739 + 0.0384365i
\(106\) 0 0
\(107\) 612619.i 0.500080i −0.968236 0.250040i \(-0.919556\pi\)
0.968236 0.250040i \(-0.0804437\pi\)
\(108\) 0 0
\(109\) −565177. + 326305.i −0.436420 + 0.251967i −0.702078 0.712100i \(-0.747744\pi\)
0.265658 + 0.964067i \(0.414411\pi\)
\(110\) 0 0
\(111\) −922888. 1.59849e6i −0.674808 1.16880i
\(112\) 0 0
\(113\) 1.19449e6i 0.827844i 0.910312 + 0.413922i \(0.135841\pi\)
−0.910312 + 0.413922i \(0.864159\pi\)
\(114\) 0 0
\(115\) −272349. −0.179074
\(116\) 0 0
\(117\) 1.32770e6 766549.i 0.828978 0.478611i
\(118\) 0 0
\(119\) −85646.9 148345.i −0.0508242 0.0880302i
\(120\) 0 0
\(121\) 2.21171e6 1.24845
\(122\) 0 0
\(123\) 1.57904e6 2.73497e6i 0.848548 1.46973i
\(124\) 0 0
\(125\) 1.19453e6 0.611600
\(126\) 0 0
\(127\) 2.62269e6 + 1.51421e6i 1.28037 + 0.739221i 0.976916 0.213625i \(-0.0685272\pi\)
0.303453 + 0.952846i \(0.401861\pi\)
\(128\) 0 0
\(129\) 1.75637e6 + 1.01404e6i 0.818178 + 0.472375i
\(130\) 0 0
\(131\) −1.66590e6 2.88542e6i −0.741027 1.28350i −0.952028 0.306011i \(-0.901005\pi\)
0.211000 0.977486i \(-0.432328\pi\)
\(132\) 0 0
\(133\) −156790. 334265.i −0.0666446 0.142081i
\(134\) 0 0
\(135\) 319182. 184280.i 0.129729 0.0748992i
\(136\) 0 0
\(137\) 947050. 1.64034e6i 0.368308 0.637928i −0.620993 0.783816i \(-0.713271\pi\)
0.989301 + 0.145888i \(0.0466038\pi\)
\(138\) 0 0
\(139\) −524600. + 908633.i −0.195337 + 0.338333i −0.947011 0.321202i \(-0.895913\pi\)
0.751674 + 0.659535i \(0.229247\pi\)
\(140\) 0 0
\(141\) 384798.i 0.137270i
\(142\) 0 0
\(143\) 2.78364e6 + 1.60713e6i 0.951929 + 0.549597i
\(144\) 0 0
\(145\) 21151.0i 0.00693787i
\(146\) 0 0
\(147\) −4.07441e6 + 2.35236e6i −1.28266 + 0.740546i
\(148\) 0 0
\(149\) −2.64323e6 4.57822e6i −0.799055 1.38400i −0.920232 0.391373i \(-0.872000\pi\)
0.121177 0.992631i \(-0.461333\pi\)
\(150\) 0 0
\(151\) 4.91871e6i 1.42863i 0.699823 + 0.714316i \(0.253262\pi\)
−0.699823 + 0.714316i \(0.746738\pi\)
\(152\) 0 0
\(153\) −3.02926e6 −0.845789
\(154\) 0 0
\(155\) −352655. + 203605.i −0.0947011 + 0.0546757i
\(156\) 0 0
\(157\) 1.54314e6 + 2.67279e6i 0.398755 + 0.690664i 0.993573 0.113197i \(-0.0361091\pi\)
−0.594818 + 0.803861i \(0.702776\pi\)
\(158\) 0 0
\(159\) 2.59149e6 0.644700
\(160\) 0 0
\(161\) −181784. + 314860.i −0.0435591 + 0.0754466i
\(162\) 0 0
\(163\) −5.84972e6 −1.35074 −0.675371 0.737478i \(-0.736016\pi\)
−0.675371 + 0.737478i \(0.736016\pi\)
\(164\) 0 0
\(165\) 2.85745e6 + 1.64975e6i 0.636102 + 0.367254i
\(166\) 0 0
\(167\) −4.57416e6 2.64089e6i −0.982115 0.567024i −0.0792070 0.996858i \(-0.525239\pi\)
−0.902908 + 0.429834i \(0.858572\pi\)
\(168\) 0 0
\(169\) −1.11654e6 1.93390e6i −0.231320 0.400659i
\(170\) 0 0
\(171\) −6.50575e6 554374.i −1.30110 0.110870i
\(172\) 0 0
\(173\) −7.75647e6 + 4.47820e6i −1.49805 + 0.864898i −0.999997 0.00224984i \(-0.999284\pi\)
−0.498050 + 0.867148i \(0.665951\pi\)
\(174\) 0 0
\(175\) 376775. 652593.i 0.0703020 0.121767i
\(176\) 0 0
\(177\) −5.71785e6 + 9.90360e6i −1.03113 + 1.78597i
\(178\) 0 0
\(179\) 9.92917e6i 1.73123i −0.500713 0.865613i \(-0.666929\pi\)
0.500713 0.865613i \(-0.333071\pi\)
\(180\) 0 0
\(181\) 2.52820e6 + 1.45966e6i 0.426359 + 0.246158i 0.697794 0.716298i \(-0.254165\pi\)
−0.271435 + 0.962457i \(0.587498\pi\)
\(182\) 0 0
\(183\) 1.73582e7i 2.83238i
\(184\) 0 0
\(185\) 1.57212e6 907667.i 0.248297 0.143354i
\(186\) 0 0
\(187\) −3.17555e6 5.50021e6i −0.485617 0.841113i
\(188\) 0 0
\(189\) 492004.i 0.0728757i
\(190\) 0 0
\(191\) 8.15338e6 1.17014 0.585070 0.810983i \(-0.301067\pi\)
0.585070 + 0.810983i \(0.301067\pi\)
\(192\) 0 0
\(193\) 1.47955e6 854219.i 0.205806 0.118822i −0.393555 0.919301i \(-0.628755\pi\)
0.599361 + 0.800479i \(0.295422\pi\)
\(194\) 0 0
\(195\) 1.33125e6 + 2.30580e6i 0.179538 + 0.310969i
\(196\) 0 0
\(197\) 1.30695e7 1.70947 0.854734 0.519066i \(-0.173720\pi\)
0.854734 + 0.519066i \(0.173720\pi\)
\(198\) 0 0
\(199\) 6.93470e6 1.20113e7i 0.879972 1.52416i 0.0286014 0.999591i \(-0.490895\pi\)
0.851370 0.524565i \(-0.175772\pi\)
\(200\) 0 0
\(201\) −140718. −0.0173285
\(202\) 0 0
\(203\) 24452.4 + 14117.6i 0.00292303 + 0.00168761i
\(204\) 0 0
\(205\) 2.68986e6 + 1.55299e6i 0.312225 + 0.180263i
\(206\) 0 0
\(207\) 3.21478e6 + 5.56817e6i 0.362444 + 0.627771i
\(208\) 0 0
\(209\) −5.81335e6 1.23936e7i −0.636777 1.35756i
\(210\) 0 0
\(211\) −4.62038e6 + 2.66758e6i −0.491848 + 0.283968i −0.725341 0.688390i \(-0.758318\pi\)
0.233493 + 0.972359i \(0.424984\pi\)
\(212\) 0 0
\(213\) −4.11533e6 + 7.12796e6i −0.425859 + 0.737609i
\(214\) 0 0
\(215\) −997317. + 1.72740e6i −0.100350 + 0.173811i
\(216\) 0 0
\(217\) 543600.i 0.0531986i
\(218\) 0 0
\(219\) 8.58709e6 + 4.95776e6i 0.817549 + 0.472012i
\(220\) 0 0
\(221\) 5.12497e6i 0.474804i
\(222\) 0 0
\(223\) −9.07806e6 + 5.24122e6i −0.818612 + 0.472626i −0.849938 0.526883i \(-0.823361\pi\)
0.0313254 + 0.999509i \(0.490027\pi\)
\(224\) 0 0
\(225\) −6.66310e6 1.15408e7i −0.584964 1.01319i
\(226\) 0 0
\(227\) 1.93360e7i 1.65306i −0.562889 0.826532i \(-0.690310\pi\)
0.562889 0.826532i \(-0.309690\pi\)
\(228\) 0 0
\(229\) 6.87818e6 0.572753 0.286376 0.958117i \(-0.407549\pi\)
0.286376 + 0.958117i \(0.407549\pi\)
\(230\) 0 0
\(231\) 3.81451e6 2.20231e6i 0.309459 0.178666i
\(232\) 0 0
\(233\) −2.94154e6 5.09490e6i −0.232545 0.402780i 0.726011 0.687683i \(-0.241372\pi\)
−0.958556 + 0.284903i \(0.908039\pi\)
\(234\) 0 0
\(235\) −378452. −0.0291613
\(236\) 0 0
\(237\) 1.37999e6 2.39022e6i 0.103665 0.179553i
\(238\) 0 0
\(239\) −1.89895e6 −0.139098 −0.0695488 0.997579i \(-0.522156\pi\)
−0.0695488 + 0.997579i \(0.522156\pi\)
\(240\) 0 0
\(241\) 1.47225e7 + 8.50001e6i 1.05179 + 0.607251i 0.923150 0.384440i \(-0.125605\pi\)
0.128640 + 0.991691i \(0.458939\pi\)
\(242\) 0 0
\(243\) 1.71049e7 + 9.87549e6i 1.19207 + 0.688240i
\(244\) 0 0
\(245\) −2.31356e6 4.00721e6i −0.157320 0.272486i
\(246\) 0 0
\(247\) 937901. 1.10066e7i 0.0622395 0.730401i
\(248\) 0 0
\(249\) −1.19385e7 + 6.89272e6i −0.773309 + 0.446470i
\(250\) 0 0
\(251\) −789.615 + 1367.65i −4.99338e−5 + 8.64878e-5i −0.866050 0.499957i \(-0.833349\pi\)
0.866000 + 0.500043i \(0.166683\pi\)
\(252\) 0 0
\(253\) −6.74006e6 + 1.16741e7i −0.416200 + 0.720879i
\(254\) 0 0
\(255\) 5.26086e6i 0.317276i
\(256\) 0 0
\(257\) −1.12130e7 6.47380e6i −0.660573 0.381382i 0.131922 0.991260i \(-0.457885\pi\)
−0.792495 + 0.609878i \(0.791218\pi\)
\(258\) 0 0
\(259\) 2.42335e6i 0.139482i
\(260\) 0 0
\(261\) 432430. 249664.i 0.0243217 0.0140422i
\(262\) 0 0
\(263\) −6.92640e6 1.19969e7i −0.380751 0.659479i 0.610419 0.792079i \(-0.291001\pi\)
−0.991170 + 0.132599i \(0.957668\pi\)
\(264\) 0 0
\(265\) 2.54874e6i 0.136958i
\(266\) 0 0
\(267\) 1.63706e7 0.860062
\(268\) 0 0
\(269\) 3.05526e7 1.76396e7i 1.56961 0.906214i 0.573396 0.819279i \(-0.305626\pi\)
0.996214 0.0869358i \(-0.0277075\pi\)
\(270\) 0 0
\(271\) 9.49147e6 + 1.64397e7i 0.476898 + 0.826012i 0.999650 0.0264734i \(-0.00842772\pi\)
−0.522751 + 0.852485i \(0.675094\pi\)
\(272\) 0 0
\(273\) 3.55427e6 0.174688
\(274\) 0 0
\(275\) 1.39697e7 2.41963e7i 0.671723 1.16346i
\(276\) 0 0
\(277\) −3.84807e6 −0.181052 −0.0905261 0.995894i \(-0.528855\pi\)
−0.0905261 + 0.995894i \(0.528855\pi\)
\(278\) 0 0
\(279\) 8.32540e6 + 4.80667e6i 0.383347 + 0.221326i
\(280\) 0 0
\(281\) −1.26128e7 7.28203e6i −0.568453 0.328196i 0.188079 0.982154i \(-0.439774\pi\)
−0.756531 + 0.653958i \(0.773107\pi\)
\(282\) 0 0
\(283\) 1.88520e7 + 3.26526e7i 0.831759 + 1.44065i 0.896642 + 0.442756i \(0.145999\pi\)
−0.0648828 + 0.997893i \(0.520667\pi\)
\(284\) 0 0
\(285\) 962771. 1.12984e7i 0.0415900 0.488072i
\(286\) 0 0
\(287\) 3.59079e6 2.07314e6i 0.151895 0.0876967i
\(288\) 0 0
\(289\) 7.00555e6 1.21340e7i 0.290234 0.502701i
\(290\) 0 0
\(291\) −2.84743e7 + 4.93190e7i −1.15551 + 2.00141i
\(292\) 0 0
\(293\) 6.05616e6i 0.240766i −0.992728 0.120383i \(-0.961588\pi\)
0.992728 0.120383i \(-0.0384122\pi\)
\(294\) 0 0
\(295\) −9.74025e6 5.62354e6i −0.379406 0.219050i
\(296\) 0 0
\(297\) 1.82421e7i 0.696315i
\(298\) 0 0
\(299\) −9.42035e6 + 5.43884e6i −0.352414 + 0.203466i
\(300\) 0 0
\(301\) 1.33135e6 + 2.30597e6i 0.0488196 + 0.0845580i
\(302\) 0 0
\(303\) 5.47565e7i 1.96838i
\(304\) 0 0
\(305\) −1.70719e7 −0.601704
\(306\) 0 0
\(307\) −3.43057e7 + 1.98064e7i −1.18564 + 0.684527i −0.957311 0.289059i \(-0.906658\pi\)
−0.228324 + 0.973585i \(0.573324\pi\)
\(308\) 0 0
\(309\) −1.59376e7 2.76048e7i −0.540192 0.935640i
\(310\) 0 0
\(311\) −5.54908e7 −1.84476 −0.922380 0.386284i \(-0.873759\pi\)
−0.922380 + 0.386284i \(0.873759\pi\)
\(312\) 0 0
\(313\) −8.43988e6 + 1.46183e7i −0.275235 + 0.476720i −0.970194 0.242328i \(-0.922089\pi\)
0.694960 + 0.719049i \(0.255422\pi\)
\(314\) 0 0
\(315\) 2.06620e6 0.0661061
\(316\) 0 0
\(317\) 5.61606e6 + 3.24243e6i 0.176301 + 0.101787i 0.585553 0.810634i \(-0.300877\pi\)
−0.409253 + 0.912421i \(0.634211\pi\)
\(318\) 0 0
\(319\) 906625. + 523440.i 0.0279290 + 0.0161248i
\(320\) 0 0
\(321\) −1.25585e7 2.17519e7i −0.379683 0.657630i
\(322\) 0 0
\(323\) −1.24789e7 + 1.79077e7i −0.370313 + 0.531413i
\(324\) 0 0
\(325\) 1.95250e7 1.12728e7i 0.568777 0.328383i
\(326\) 0 0
\(327\) −1.33782e7 + 2.31718e7i −0.382610 + 0.662699i
\(328\) 0 0
\(329\) −252604. + 437523.i −0.00709337 + 0.0122861i
\(330\) 0 0
\(331\) 2.50727e7i 0.691380i 0.938349 + 0.345690i \(0.112355\pi\)
−0.938349 + 0.345690i \(0.887645\pi\)
\(332\) 0 0
\(333\) −3.71144e7 2.14280e7i −1.00510 0.580295i
\(334\) 0 0
\(335\) 138397.i 0.00368123i
\(336\) 0 0
\(337\) −3.49392e7 + 2.01722e7i −0.912901 + 0.527064i −0.881363 0.472439i \(-0.843374\pi\)
−0.0315376 + 0.999503i \(0.510040\pi\)
\(338\) 0 0
\(339\) 2.44866e7 + 4.24121e7i 0.628536 + 1.08866i
\(340\) 0 0
\(341\) 2.01552e7i 0.508304i
\(342\) 0 0
\(343\) −1.25098e7 −0.310004
\(344\) 0 0
\(345\) −9.67014e6 + 5.58306e6i −0.235492 + 0.135961i
\(346\) 0 0
\(347\) −2.13812e7 3.70333e7i −0.511733 0.886348i −0.999907 0.0136015i \(-0.995670\pi\)
0.488174 0.872746i \(-0.337663\pi\)
\(348\) 0 0
\(349\) 5.87855e7 1.38291 0.691455 0.722420i \(-0.256970\pi\)
0.691455 + 0.722420i \(0.256970\pi\)
\(350\) 0 0
\(351\) 7.36017e6 1.27482e7i 0.170203 0.294800i
\(352\) 0 0
\(353\) 6.57708e7 1.49523 0.747617 0.664130i \(-0.231198\pi\)
0.747617 + 0.664130i \(0.231198\pi\)
\(354\) 0 0
\(355\) −7.01039e6 4.04745e6i −0.156696 0.0904684i
\(356\) 0 0
\(357\) −6.08202e6 3.51146e6i −0.133673 0.0771761i
\(358\) 0 0
\(359\) −1.59674e7 2.76564e7i −0.345105 0.597739i 0.640268 0.768152i \(-0.278823\pi\)
−0.985373 + 0.170413i \(0.945490\pi\)
\(360\) 0 0
\(361\) −3.00774e7 + 3.61755e7i −0.639320 + 0.768941i
\(362\) 0 0
\(363\) 7.85296e7 4.53391e7i 1.64178 0.947879i
\(364\) 0 0
\(365\) −4.87599e6 + 8.44546e6i −0.100273 + 0.173678i
\(366\) 0 0
\(367\) 2.77505e7 4.80653e7i 0.561401 0.972375i −0.435974 0.899959i \(-0.643596\pi\)
0.997375 0.0724154i \(-0.0230707\pi\)
\(368\) 0 0
\(369\) 7.33253e7i 1.45940i
\(370\) 0 0
\(371\) 2.94657e6 + 1.70120e6i 0.0577026 + 0.0333146i
\(372\) 0 0
\(373\) 6.32072e6i 0.121798i −0.998144 0.0608989i \(-0.980603\pi\)
0.998144 0.0608989i \(-0.0193967\pi\)
\(374\) 0 0
\(375\) 4.24134e7 2.44874e7i 0.804284 0.464354i
\(376\) 0 0
\(377\) 422386. + 731595.i 0.00788290 + 0.0136536i
\(378\) 0 0
\(379\) 4.86114e7i 0.892937i 0.894799 + 0.446469i \(0.147319\pi\)
−0.894799 + 0.446469i \(0.852681\pi\)
\(380\) 0 0
\(381\) 1.24163e8 2.24500
\(382\) 0 0
\(383\) 1.61488e7 9.32351e6i 0.287438 0.165952i −0.349348 0.936993i \(-0.613597\pi\)
0.636786 + 0.771041i \(0.280264\pi\)
\(384\) 0 0
\(385\) 2.16598e6 + 3.75160e6i 0.0379554 + 0.0657406i
\(386\) 0 0
\(387\) 4.70889e7 0.812429
\(388\) 0 0
\(389\) −3.32556e7 + 5.76003e7i −0.564957 + 0.978534i 0.432097 + 0.901827i \(0.357774\pi\)
−0.997054 + 0.0767070i \(0.975559\pi\)
\(390\) 0 0
\(391\) 2.14933e7 0.359561
\(392\) 0 0
\(393\) −1.18300e8 6.83005e7i −1.94898 1.12524i
\(394\) 0 0
\(395\) 2.35080e6 + 1.35723e6i 0.0381438 + 0.0220223i
\(396\) 0 0
\(397\) 6.11861e7 + 1.05977e8i 0.977871 + 1.69372i 0.670117 + 0.742255i \(0.266244\pi\)
0.307753 + 0.951466i \(0.400423\pi\)
\(398\) 0 0
\(399\) −1.24194e7 8.65438e6i −0.195515 0.136244i
\(400\) 0 0
\(401\) 8.08056e7 4.66531e7i 1.25317 0.723515i 0.281428 0.959582i \(-0.409192\pi\)
0.971737 + 0.236067i \(0.0758584\pi\)
\(402\) 0 0
\(403\) −8.13203e6 + 1.40851e7i −0.124246 + 0.215201i
\(404\) 0 0
\(405\) −6.43596e6 + 1.11474e7i −0.0968832 + 0.167807i
\(406\) 0 0
\(407\) 8.98511e7i 1.33272i
\(408\) 0 0
\(409\) −9.56148e7 5.52032e7i −1.39751 0.806853i −0.403379 0.915033i \(-0.632164\pi\)
−0.994131 + 0.108180i \(0.965498\pi\)
\(410\) 0 0
\(411\) 7.76566e7i 1.11854i
\(412\) 0 0
\(413\) −1.30026e7 + 7.50706e6i −0.184578 + 0.106566i
\(414\) 0 0
\(415\) −6.77903e6 1.17416e7i −0.0948470 0.164280i
\(416\) 0 0
\(417\) 4.30163e7i 0.593233i
\(418\) 0 0
\(419\) 1.44215e7 0.196051 0.0980256 0.995184i \(-0.468747\pi\)
0.0980256 + 0.995184i \(0.468747\pi\)
\(420\) 0 0
\(421\) −7.16727e7 + 4.13802e7i −0.960522 + 0.554557i −0.896334 0.443380i \(-0.853779\pi\)
−0.0641881 + 0.997938i \(0.520446\pi\)
\(422\) 0 0
\(423\) 4.46720e6 + 7.73741e6i 0.0590220 + 0.102229i
\(424\) 0 0
\(425\) −4.45479e7 −0.580311
\(426\) 0 0
\(427\) −1.13949e7 + 1.97366e7i −0.146362 + 0.253507i
\(428\) 0 0
\(429\) 1.31782e8 1.66911
\(430\) 0 0
\(431\) −1.24661e8 7.19731e7i −1.55704 0.898956i −0.997538 0.0701243i \(-0.977660\pi\)
−0.559499 0.828831i \(-0.689006\pi\)
\(432\) 0 0
\(433\) −5.25908e6 3.03633e6i −0.0647808 0.0374012i 0.467260 0.884120i \(-0.345241\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(434\) 0 0
\(435\) 433587. + 750994.i 0.00526754 + 0.00912365i
\(436\) 0 0
\(437\) 4.61598e7 + 3.93340e6i 0.553120 + 0.0471329i
\(438\) 0 0
\(439\) 8.32621e7 4.80714e7i 0.984133 0.568190i 0.0806179 0.996745i \(-0.474311\pi\)
0.903516 + 0.428555i \(0.140977\pi\)
\(440\) 0 0
\(441\) −5.46180e7 + 9.46012e7i −0.636825 + 1.10301i
\(442\) 0 0
\(443\) −1.76533e6 + 3.05764e6i −0.0203056 + 0.0351703i −0.876000 0.482312i \(-0.839797\pi\)
0.855694 + 0.517482i \(0.173131\pi\)
\(444\) 0 0
\(445\) 1.61006e7i 0.182709i
\(446\) 0 0
\(447\) −1.87703e8 1.08371e8i −2.10160 1.21336i
\(448\) 0 0
\(449\) 3.46471e7i 0.382761i 0.981516 + 0.191380i \(0.0612964\pi\)
−0.981516 + 0.191380i \(0.938704\pi\)
\(450\) 0 0
\(451\) 1.33136e8 7.68663e7i 1.45133 0.837927i
\(452\) 0 0
\(453\) 1.00832e8 + 1.74646e8i 1.08468 + 1.87872i
\(454\) 0 0
\(455\) 3.49565e6i 0.0371102i
\(456\) 0 0
\(457\) 3.35067e7 0.351062 0.175531 0.984474i \(-0.443836\pi\)
0.175531 + 0.984474i \(0.443836\pi\)
\(458\) 0 0
\(459\) −2.51892e7 + 1.45430e7i −0.260481 + 0.150389i
\(460\) 0 0
\(461\) −9.58650e6 1.66043e7i −0.0978492 0.169480i 0.812945 0.582341i \(-0.197863\pi\)
−0.910794 + 0.412861i \(0.864530\pi\)
\(462\) 0 0
\(463\) −8.43784e7 −0.850136 −0.425068 0.905161i \(-0.639750\pi\)
−0.425068 + 0.905161i \(0.639750\pi\)
\(464\) 0 0
\(465\) −8.34766e6 + 1.44586e7i −0.0830245 + 0.143803i
\(466\) 0 0
\(467\) 1.18908e8 1.16751 0.583753 0.811931i \(-0.301584\pi\)
0.583753 + 0.811931i \(0.301584\pi\)
\(468\) 0 0
\(469\) −159999. 92375.5i −0.00155095 0.000895444i
\(470\) 0 0
\(471\) 1.09582e8 + 6.32675e7i 1.04877 + 0.605505i
\(472\) 0 0
\(473\) 4.93628e7 + 8.54990e7i 0.466463 + 0.807937i
\(474\) 0 0
\(475\) −9.56728e7 8.15255e6i −0.892704 0.0760699i
\(476\) 0 0
\(477\) 5.21089e7 3.00851e7i 0.480127 0.277202i
\(478\) 0 0
\(479\) −7.54649e7 + 1.30709e8i −0.686654 + 1.18932i 0.286259 + 0.958152i \(0.407588\pi\)
−0.972914 + 0.231168i \(0.925745\pi\)
\(480\) 0 0
\(481\) 3.62523e7 6.27909e7i 0.325762 0.564237i
\(482\) 0 0
\(483\) 1.49060e7i 0.132288i
\(484\) 0 0
\(485\) −4.85055e7 2.80047e7i −0.425173 0.245474i
\(486\) 0 0
\(487\) 2.09348e8i 1.81252i −0.422721 0.906260i \(-0.638925\pi\)
0.422721 0.906260i \(-0.361075\pi\)
\(488\) 0 0
\(489\) −2.07702e8 + 1.19917e8i −1.77629 + 1.02554i
\(490\) 0 0
\(491\) 1.88325e7 + 3.26189e7i 0.159098 + 0.275565i 0.934544 0.355849i \(-0.115808\pi\)
−0.775446 + 0.631414i \(0.782475\pi\)
\(492\) 0 0
\(493\) 1.66919e6i 0.0139305i
\(494\) 0 0
\(495\) 7.66091e7 0.631633
\(496\) 0 0
\(497\) −9.35842e6 + 5.40309e6i −0.0762313 + 0.0440122i
\(498\) 0 0
\(499\) 3.49677e7 + 6.05659e7i 0.281427 + 0.487446i 0.971736 0.236068i \(-0.0758590\pi\)
−0.690310 + 0.723514i \(0.742526\pi\)
\(500\) 0 0
\(501\) −2.16549e8 −1.72204
\(502\) 0 0
\(503\) −1.48199e7 + 2.56689e7i −0.116451 + 0.201699i −0.918359 0.395749i \(-0.870485\pi\)
0.801908 + 0.597448i \(0.203818\pi\)
\(504\) 0 0
\(505\) −5.38534e7 −0.418157
\(506\) 0 0
\(507\) −7.92885e7 4.57772e7i −0.608396 0.351258i
\(508\) 0 0
\(509\) 1.65604e8 + 9.56113e7i 1.25579 + 0.725030i 0.972253 0.233932i \(-0.0751593\pi\)
0.283535 + 0.958962i \(0.408493\pi\)
\(510\) 0 0
\(511\) 6.50913e6 + 1.12741e7i 0.0487821 + 0.0844930i
\(512\) 0 0
\(513\) −5.67588e7 + 2.66233e7i −0.420418 + 0.197201i
\(514\) 0 0
\(515\) 2.71495e7 1.56748e7i 0.198765 0.114757i
\(516\) 0 0
\(517\) −9.36585e6 + 1.62221e7i −0.0677760 + 0.117391i
\(518\) 0 0
\(519\) −1.83603e8 + 3.18009e8i −1.31334 + 2.27477i
\(520\) 0 0
\(521\) 8.73062e7i 0.617350i −0.951168 0.308675i \(-0.900114\pi\)
0.951168 0.308675i \(-0.0998856\pi\)
\(522\) 0 0
\(523\) −2.23643e8 1.29120e8i −1.56333 0.902588i −0.996917 0.0784647i \(-0.974998\pi\)
−0.566411 0.824123i \(-0.691668\pi\)
\(524\) 0 0
\(525\) 3.08949e7i 0.213506i
\(526\) 0 0
\(527\) 2.78308e7 1.60681e7i 0.190149 0.109783i
\(528\) 0 0
\(529\) 5.12083e7 + 8.86955e7i 0.345918 + 0.599148i
\(530\) 0 0
\(531\) 2.65518e8i 1.77342i
\(532\) 0 0
\(533\) 1.24053e8 0.819270
\(534\) 0 0
\(535\) 2.13931e7 1.23513e7i 0.139705 0.0806588i
\(536\) 0 0
\(537\) −2.03544e8 3.52549e8i −1.31442 2.27665i
\(538\) 0 0
\(539\) −2.29022e8 −1.46255
\(540\) 0 0
\(541\) 1.18637e8 2.05485e8i 0.749251 1.29774i −0.198932 0.980013i \(-0.563747\pi\)
0.948182 0.317727i \(-0.102919\pi\)
\(542\) 0 0
\(543\) 1.19689e8 0.747578
\(544\) 0 0
\(545\) −2.27896e7 1.31576e7i −0.140782 0.0812806i
\(546\) 0 0
\(547\) −1.47631e8 8.52346e7i −0.902017 0.520780i −0.0241628 0.999708i \(-0.507692\pi\)
−0.877854 + 0.478928i \(0.841025\pi\)
\(548\) 0 0
\(549\) 2.01515e8 + 3.49034e8i 1.21784 + 2.10936i
\(550\) 0 0
\(551\) 305473. 3.58482e6i 0.00182607 0.0214295i
\(552\) 0 0
\(553\) 3.13816e6 1.81182e6i 0.0185567 0.0107137i
\(554\) 0 0
\(555\) 3.72136e7 6.44559e7i 0.217682 0.377037i
\(556\) 0 0
\(557\) −2.18734e6 + 3.78859e6i −0.0126576 + 0.0219236i −0.872285 0.488998i \(-0.837362\pi\)
0.859627 + 0.510922i \(0.170696\pi\)
\(558\) 0 0
\(559\) 7.96660e7i 0.456076i
\(560\) 0 0
\(561\) −2.25504e8 1.30195e8i −1.27722 0.737404i
\(562\) 0 0
\(563\) 2.04600e8i 1.14652i −0.819374 0.573259i \(-0.805679\pi\)
0.819374 0.573259i \(-0.194321\pi\)
\(564\) 0 0
\(565\) −4.17126e7 + 2.40828e7i −0.231271 + 0.133525i
\(566\) 0 0
\(567\) 8.59159e6 + 1.48811e7i 0.0471329 + 0.0816366i
\(568\) 0 0
\(569\) 2.15237e8i 1.16837i 0.811621 + 0.584184i \(0.198585\pi\)
−0.811621 + 0.584184i \(0.801415\pi\)
\(570\) 0 0
\(571\) 2.06574e8 1.10960 0.554800 0.831983i \(-0.312795\pi\)
0.554800 + 0.831983i \(0.312795\pi\)
\(572\) 0 0
\(573\) 2.89497e8 1.67141e8i 1.53879 0.888422i
\(574\) 0 0
\(575\) 4.72762e7 + 8.18848e7i 0.248679 + 0.430725i
\(576\) 0 0
\(577\) 3.13530e8 1.63212 0.816058 0.577970i \(-0.196155\pi\)
0.816058 + 0.577970i \(0.196155\pi\)
\(578\) 0 0
\(579\) 3.50223e7 6.06604e7i 0.180430 0.312514i
\(580\) 0 0
\(581\) −1.80991e7 −0.0922847
\(582\) 0 0
\(583\) 1.09251e8 + 6.30758e7i 0.551338 + 0.318315i
\(584\) 0 0
\(585\) 5.35369e7 + 3.09096e7i 0.267415 + 0.154392i
\(586\) 0 0
\(587\) −2.12730e7 3.68458e7i −0.105175 0.182169i 0.808635 0.588311i \(-0.200207\pi\)
−0.913810 + 0.406142i \(0.866874\pi\)
\(588\) 0 0
\(589\) 6.27111e7 2.94153e7i 0.306901 0.143955i
\(590\) 0 0
\(591\) 4.64051e8 2.67920e8i 2.24804 1.29790i
\(592\) 0 0
\(593\) 8.54553e6 1.48013e7i 0.0409803 0.0709799i −0.844808 0.535070i \(-0.820285\pi\)
0.885788 + 0.464090i \(0.153619\pi\)
\(594\) 0 0
\(595\) 3.45354e6 5.98171e6i 0.0163951 0.0283971i
\(596\) 0 0
\(597\) 5.68635e8i 2.67246i
\(598\) 0 0
\(599\) 1.39104e8 + 8.03116e7i 0.647230 + 0.373678i 0.787394 0.616450i \(-0.211430\pi\)
−0.140164 + 0.990128i \(0.544763\pi\)
\(600\) 0 0
\(601\) 1.53421e6i 0.00706740i 0.999994 + 0.00353370i \(0.00112481\pi\)
−0.999994 + 0.00353370i \(0.998875\pi\)
\(602\) 0 0
\(603\) −2.82952e6 + 1.63362e6i −0.0129051 + 0.00745075i
\(604\) 0 0
\(605\) 4.45913e7 + 7.72344e7i 0.201365 + 0.348774i
\(606\) 0 0
\(607\) 6.35483e7i 0.284143i 0.989856 + 0.142072i \(0.0453764\pi\)
−0.989856 + 0.142072i \(0.954624\pi\)
\(608\) 0 0
\(609\) 1.15762e6 0.00512524
\(610\) 0 0
\(611\) −1.30903e7 + 7.55771e6i −0.0573888 + 0.0331334i
\(612\) 0 0
\(613\) −5.29057e7 9.16354e7i −0.229679 0.397815i 0.728034 0.685541i \(-0.240434\pi\)
−0.957713 + 0.287726i \(0.907101\pi\)
\(614\) 0 0
\(615\) 1.27343e8 0.547456
\(616\) 0 0
\(617\) 9.32480e6 1.61510e7i 0.0396994 0.0687614i −0.845493 0.533987i \(-0.820693\pi\)
0.885192 + 0.465225i \(0.154027\pi\)
\(618\) 0 0
\(619\) 2.32518e8 0.980356 0.490178 0.871622i \(-0.336932\pi\)
0.490178 + 0.871622i \(0.336932\pi\)
\(620\) 0 0
\(621\) 5.34638e7 + 3.08673e7i 0.223247 + 0.128892i
\(622\) 0 0
\(623\) 1.86137e7 + 1.07466e7i 0.0769782 + 0.0444434i
\(624\) 0 0
\(625\) −8.52841e7 1.47716e8i −0.349324 0.605046i
\(626\) 0 0
\(627\) −4.60475e8 3.20880e8i −1.86811 1.30179i
\(628\) 0 0
\(629\) −1.24069e8 + 7.16312e7i −0.498553 + 0.287840i
\(630\) 0 0
\(631\) −5.44686e7 + 9.43425e7i −0.216800 + 0.375508i −0.953828 0.300354i \(-0.902895\pi\)
0.737028 + 0.675862i \(0.236228\pi\)
\(632\) 0 0
\(633\) −1.09369e8 + 1.89432e8i −0.431203 + 0.746866i
\(634\) 0 0
\(635\) 1.22115e8i 0.476922i
\(636\) 0 0
\(637\) −1.60048e8 9.24040e7i −0.619203 0.357497i
\(638\) 0 0
\(639\) 1.91103e8i 0.732427i
\(640\) 0 0
\(641\) −2.85786e8 + 1.64999e8i −1.08509 + 0.626478i −0.932266 0.361775i \(-0.882171\pi\)
−0.152827 + 0.988253i \(0.548838\pi\)
\(642\) 0 0
\(643\) −2.05064e8 3.55181e8i −0.771358 1.33603i −0.936819 0.349815i \(-0.886244\pi\)
0.165461 0.986216i \(-0.447089\pi\)
\(644\) 0 0
\(645\) 8.17784e7i 0.304761i
\(646\) 0 0
\(647\) 1.57074e8 0.579949 0.289975 0.957034i \(-0.406353\pi\)
0.289975 + 0.957034i \(0.406353\pi\)
\(648\) 0 0
\(649\) −4.82100e8 + 2.78341e8i −1.76361 + 1.01822i
\(650\) 0 0
\(651\) 1.11436e7 + 1.93013e7i 0.0403908 + 0.0699589i
\(652\) 0 0
\(653\) 5.03130e8 1.80693 0.903463 0.428665i \(-0.141016\pi\)
0.903463 + 0.428665i \(0.141016\pi\)
\(654\) 0 0
\(655\) 6.71740e7 1.16349e8i 0.239044 0.414036i
\(656\) 0 0
\(657\) 2.30222e8 0.811805
\(658\) 0 0
\(659\) 1.96394e8 + 1.13388e8i 0.686232 + 0.396197i 0.802199 0.597057i \(-0.203663\pi\)
−0.115967 + 0.993253i \(0.536997\pi\)
\(660\) 0 0
\(661\) 5.36340e7 + 3.09656e7i 0.185710 + 0.107220i 0.589973 0.807423i \(-0.299138\pi\)
−0.404263 + 0.914643i \(0.632472\pi\)
\(662\) 0 0
\(663\) −1.05060e8 1.81969e8i −0.360492 0.624391i
\(664\) 0 0
\(665\) 8.51164e6 1.22145e7i 0.0289433 0.0415347i
\(666\) 0 0
\(667\) −3.06819e6 + 1.77142e6i −0.0103396 + 0.00596958i
\(668\) 0 0
\(669\) −2.14886e8 + 3.72193e8i −0.717678 + 1.24305i
\(670\) 0 0
\(671\) −4.22493e8 + 7.31779e8i −1.39846 + 2.42221i
\(672\) 0 0
\(673\) 5.12478e8i 1.68124i −0.541624 0.840621i \(-0.682190\pi\)
0.541624 0.840621i \(-0.317810\pi\)
\(674\) 0 0
\(675\) −1.10812e8 6.39771e7i −0.360308 0.208024i
\(676\) 0 0
\(677\) 2.61035e8i 0.841264i 0.907231 + 0.420632i \(0.138192\pi\)
−0.907231 + 0.420632i \(0.861808\pi\)
\(678\) 0 0
\(679\) −6.47517e7 + 3.73844e7i −0.206844 + 0.119421i
\(680\) 0 0
\(681\) −3.96381e8 6.86552e8i −1.25508 2.17386i
\(682\) 0 0
\(683\) 3.68538e8i 1.15670i 0.815789 + 0.578350i \(0.196303\pi\)
−0.815789 + 0.578350i \(0.803697\pi\)
\(684\) 0 0
\(685\) 7.63758e7 0.237621
\(686\) 0 0
\(687\) 2.44219e8 1.41000e8i 0.753199 0.434860i
\(688\) 0 0
\(689\) 5.08986e7 + 8.81589e7i 0.155614 + 0.269531i
\(690\) 0 0
\(691\) 9.17534e7 0.278092 0.139046 0.990286i \(-0.455596\pi\)
0.139046 + 0.990286i \(0.455596\pi\)
\(692\) 0 0
\(693\) 5.11341e7 8.85668e7i 0.153642 0.266116i
\(694\) 0 0
\(695\) −4.23068e7 −0.126025
\(696\) 0 0
\(697\) −2.12278e8 1.22559e8i −0.626913 0.361948i
\(698\) 0 0
\(699\) −2.08887e8 1.20601e8i −0.611617 0.353117i
\(700\) 0 0
\(701\) 1.88928e8 + 3.27233e8i 0.548457 + 0.949955i 0.998381 + 0.0568880i \(0.0181178\pi\)
−0.449924 + 0.893067i \(0.648549\pi\)
\(702\) 0 0
\(703\) −2.79564e8 + 1.31133e8i −0.804665 + 0.377437i
\(704\) 0 0
\(705\) −1.34374e7 + 7.75811e6i −0.0383486 + 0.0221406i
\(706\) 0 0
\(707\) −3.59454e7 + 6.22592e7i −0.101715 + 0.176176i
\(708\) 0 0
\(709\) −2.51715e8 + 4.35983e8i −0.706269 + 1.22329i 0.259963 + 0.965619i \(0.416290\pi\)
−0.966232 + 0.257675i \(0.917044\pi\)
\(710\) 0 0
\(711\) 6.40825e7i 0.178291i
\(712\) 0 0
\(713\) −5.90706e7 3.41044e7i −0.162968 0.0940896i
\(714\) 0 0
\(715\) 1.29609e8i 0.354582i
\(716\) 0 0
\(717\) −6.74248e7 + 3.89277e7i −0.182920 + 0.105609i
\(718\) 0 0
\(719\) −2.59711e6 4.49833e6i −0.00698722 0.0121022i 0.862511 0.506039i \(-0.168891\pi\)
−0.869498 + 0.493937i \(0.835557\pi\)
\(720\) 0 0
\(721\) 4.18496e7i 0.111657i
\(722\) 0 0
\(723\) 6.96988e8 1.84421
\(724\) 0 0
\(725\) 6.35926e6 3.67152e6i 0.0166876 0.00963457i
\(726\) 0 0
\(727\) 6.98308e7 + 1.20951e8i 0.181737 + 0.314778i 0.942472 0.334284i \(-0.108495\pi\)
−0.760735 + 0.649063i \(0.775161\pi\)
\(728\) 0 0
\(729\) 5.77063e8 1.48950
\(730\) 0 0
\(731\) 7.87063e7 1.36323e8i 0.201492 0.348994i
\(732\) 0 0
\(733\) 1.40120e8 0.355785 0.177893 0.984050i \(-0.443072\pi\)
0.177893 + 0.984050i \(0.443072\pi\)
\(734\) 0 0
\(735\) −1.64292e8 9.48542e7i −0.413766 0.238888i
\(736\) 0 0
\(737\) −5.93232e6 3.42503e6i −0.0148191 0.00855581i
\(738\) 0 0
\(739\) 1.14793e8 + 1.98828e8i 0.284435 + 0.492656i 0.972472 0.233020i \(-0.0748607\pi\)
−0.688037 + 0.725676i \(0.741527\pi\)
\(740\) 0 0
\(741\) −1.92329e8 4.10030e8i −0.472705 1.00777i
\(742\) 0 0
\(743\) −2.24945e8 + 1.29872e8i −0.548415 + 0.316627i −0.748482 0.663155i \(-0.769217\pi\)
0.200068 + 0.979782i \(0.435884\pi\)
\(744\) 0 0
\(745\) 1.06583e8 1.84607e8i 0.257762 0.446458i
\(746\) 0 0
\(747\) −1.60038e8 + 2.77194e8i −0.383938 + 0.665000i
\(748\) 0 0
\(749\) 3.29764e7i 0.0784798i
\(750\) 0 0
\(751\) 5.17994e8 + 2.99064e8i 1.22294 + 0.706064i 0.965543 0.260242i \(-0.0838023\pi\)
0.257396 + 0.966306i \(0.417136\pi\)
\(752\) 0 0
\(753\) 64747.2i 0.000151648i
\(754\) 0 0
\(755\) −1.71765e8 + 9.91686e7i −0.399111 + 0.230427i
\(756\) 0 0
\(757\) −3.04726e8 5.27801e8i −0.702460 1.21670i −0.967600 0.252486i \(-0.918752\pi\)
0.265141 0.964210i \(-0.414582\pi\)
\(758\) 0 0
\(759\) 5.52674e8i 1.26399i
\(760\) 0 0
\(761\) 3.84676e7 0.0872853 0.0436427 0.999047i \(-0.486104\pi\)
0.0436427 + 0.999047i \(0.486104\pi\)
\(762\) 0 0
\(763\) −3.04226e7 + 1.75645e7i −0.0684894 + 0.0395424i
\(764\) 0 0
\(765\) −6.10744e7 1.05784e8i −0.136419 0.236285i
\(766\) 0 0
\(767\) −4.49210e8 −0.995550
\(768\) 0 0
\(769\) 4.50076e7 7.79555e7i 0.0989708 0.171423i −0.812288 0.583256i \(-0.801778\pi\)
0.911259 + 0.411834i \(0.135112\pi\)
\(770\) 0 0
\(771\) −5.30842e8 −1.15825
\(772\) 0 0
\(773\) 1.41164e8 + 8.15009e7i 0.305622 + 0.176451i 0.644966 0.764211i \(-0.276872\pi\)
−0.339344 + 0.940662i \(0.610205\pi\)
\(774\) 0 0
\(775\) 1.22432e8 + 7.06863e7i 0.263021 + 0.151855i
\(776\) 0 0
\(777\) −4.96778e7 8.60444e7i −0.105901 0.183425i
\(778\) 0 0
\(779\) −4.33468e8 3.02060e8i −0.916947 0.638971i
\(780\) 0 0
\(781\) −3.46984e8 + 2.00331e8i −0.728377 + 0.420529i
\(782\) 0 0
\(783\) 2.39719e6 4.15206e6i 0.00499365 0.00864925i
\(784\) 0 0
\(785\) −6.22240e7 + 1.07775e8i −0.128632 + 0.222797i
\(786\) 0 0
\(787\) 2.78169e8i 0.570670i 0.958428 + 0.285335i \(0.0921049\pi\)
−0.958428 + 0.285335i \(0.907895\pi\)
\(788\) 0 0
\(789\) −4.91863e8 2.83977e8i −1.00141 0.578166i
\(790\) 0 0
\(791\) 6.42979e7i 0.129917i
\(792\) 0 0
\(793\) −5.90503e8 + 3.40927e8i −1.18414 + 0.683663i
\(794\) 0 0
\(795\) 5.22482e7 + 9.04966e7i 0.103985 + 0.180107i
\(796\) 0 0
\(797\) 6.74509e8i 1.33233i −0.745803 0.666167i \(-0.767934\pi\)
0.745803 0.666167i \(-0.232066\pi\)
\(798\) 0 0
\(799\) 2.98666e7 0.0585526
\(800\) 0 0
\(801\) 3.29175e8 1.90049e8i 0.640514 0.369801i
\(802\) 0 0
\(803\) 2.41340e8 + 4.18014e8i 0.466104 + 0.807316i
\(804\) 0 0
\(805\) −1.46602e7 −0.0281029
\(806\) 0 0
\(807\) 7.23208e8 1.25263e9i 1.37608 2.38344i
\(808\) 0 0
\(809\) 3.15571e8 0.596008 0.298004 0.954565i \(-0.403679\pi\)
0.298004 + 0.954565i \(0.403679\pi\)
\(810\) 0 0
\(811\) 7.08837e8 + 4.09247e8i 1.32887 + 0.767226i 0.985126 0.171836i \(-0.0549699\pi\)
0.343749 + 0.939062i \(0.388303\pi\)
\(812\) 0 0
\(813\) 6.74015e8 + 3.89143e8i 1.25429 + 0.724165i
\(814\) 0 0
\(815\) −1.17939e8 2.04277e8i −0.217864 0.377351i
\(816\) 0 0
\(817\) 1.93981e8 2.78369e8i 0.355707 0.510452i
\(818\) 0 0
\(819\) 7.14683e7 4.12623e7i 0.130095 0.0751106i
\(820\) 0 0
\(821\) 1.16362e8 2.01544e8i 0.210272 0.364201i −0.741528 0.670922i \(-0.765899\pi\)
0.951799 + 0.306721i \(0.0992319\pi\)
\(822\) 0 0
\(823\) −2.52096e8 + 4.36643e8i −0.452237 + 0.783297i −0.998525 0.0543002i \(-0.982707\pi\)
0.546288 + 0.837598i \(0.316041\pi\)
\(824\) 0 0
\(825\) 1.14550e9i 2.04001i
\(826\) 0 0
\(827\) 8.95537e7 + 5.17038e7i 0.158331 + 0.0914126i 0.577072 0.816693i \(-0.304195\pi\)
−0.418741 + 0.908106i \(0.637528\pi\)
\(828\) 0 0
\(829\) 2.55442e8i 0.448362i −0.974547 0.224181i \(-0.928029\pi\)
0.974547 0.224181i \(-0.0719708\pi\)
\(830\) 0 0
\(831\) −1.36631e8 + 7.88839e7i −0.238093 + 0.137463i
\(832\) 0 0
\(833\) 1.82582e8 + 3.16241e8i 0.315880 + 0.547120i
\(834\) 0 0
\(835\) 2.12978e8i 0.365826i
\(836\) 0 0
\(837\) 9.23043e7 0.157415
\(838\) 0 0
\(839\) 6.35755e8 3.67053e8i 1.07647 0.621503i 0.146531 0.989206i \(-0.453189\pi\)
0.929943 + 0.367703i \(0.119856\pi\)
\(840\) 0 0
\(841\) −2.97274e8 5.14894e8i −0.499769 0.865625i
\(842\) 0 0
\(843\) −5.97115e8 −0.996725
\(844\) 0 0
\(845\) 4.50222e7 7.79808e7i 0.0746202 0.129246i
\(846\) 0 0
\(847\) 1.19053e8 0.195925
\(848\) 0 0
\(849\) 1.33873e9 + 7.72916e8i 2.18761 + 1.26302i
\(850\) 0 0
\(851\) 2.63335e8 + 1.52036e8i 0.427287 + 0.246694i
\(852\) 0 0
\(853\) 2.27815e8 + 3.94587e8i 0.367058 + 0.635764i 0.989104 0.147217i \(-0.0470316\pi\)
−0.622046 + 0.782981i \(0.713698\pi\)
\(854\) 0 0
\(855\) −1.11807e8 2.38363e8i −0.178883 0.381364i
\(856\) 0 0
\(857\) −1.13199e8 + 6.53558e7i −0.179847 + 0.103834i −0.587221 0.809427i \(-0.699778\pi\)
0.407374 + 0.913261i \(0.366445\pi\)
\(858\) 0 0
\(859\) 1.48931e8 2.57956e8i 0.234966 0.406974i −0.724297 0.689489i \(-0.757835\pi\)
0.959263 + 0.282515i \(0.0911687\pi\)
\(860\) 0 0
\(861\) 8.49972e7 1.47219e8i 0.133167 0.230651i
\(862\) 0 0
\(863\) 5.27524e7i 0.0820748i 0.999158 + 0.0410374i \(0.0130663\pi\)
−0.999158 + 0.0410374i \(0.986934\pi\)
\(864\) 0 0
\(865\) −3.12764e8 1.80574e8i −0.483246 0.279002i
\(866\) 0 0
\(867\) 5.74445e8i 0.881436i
\(868\) 0 0
\(869\) 1.16354e8 6.71771e7i 0.177306 0.102368i
\(870\) 0 0
\(871\) −2.76380e6 4.78704e6i −0.00418265 0.00724457i
\(872\) 0 0
\(873\) 1.32226e9i 1.98734i
\(874\) 0 0
\(875\) 6.42999e7 0.0959812
\(876\) 0 0
\(877\) 8.37532e8 4.83549e8i 1.24166 0.716873i 0.272228 0.962233i \(-0.412240\pi\)
0.969432 + 0.245360i \(0.0789062\pi\)
\(878\) 0 0
\(879\) −1.24149e8 2.15032e8i −0.182800 0.316619i
\(880\) 0 0
\(881\) 9.00050e8 1.31625 0.658126 0.752908i \(-0.271350\pi\)
0.658126 + 0.752908i \(0.271350\pi\)
\(882\) 0 0
\(883\) 3.37516e8 5.84595e8i 0.490244 0.849127i −0.509693 0.860356i \(-0.670241\pi\)
0.999937 + 0.0112289i \(0.00357435\pi\)
\(884\) 0 0
\(885\) −4.61121e8 −0.665251
\(886\) 0 0
\(887\) −1.06508e9 6.14925e8i −1.52620 0.881152i −0.999517 0.0310913i \(-0.990102\pi\)
−0.526684 0.850061i \(-0.676565\pi\)
\(888\) 0 0
\(889\) 1.41175e8 + 8.15077e7i 0.200934 + 0.116009i
\(890\) 0 0
\(891\) 3.18552e8 + 5.51748e8i 0.450347 + 0.780024i
\(892\) 0 0
\(893\) 6.41427e7 + 5.46578e6i 0.0900726 + 0.00767535i
\(894\) 0 0
\(895\) 3.46734e8 2.00187e8i 0.483646 0.279233i
\(896\) 0 0
\(897\) −2.22988e8 + 3.86227e8i −0.308962 + 0.535137i
\(898\) 0 0
\(899\) −2.64859e6 + 4.58749e6i −0.00364531 + 0.00631387i
\(900\) 0 0
\(901\) 2.01142e8i 0.274997i
\(902\) 0 0
\(903\) 9.45430e7 + 5.45845e7i 0.128400 + 0.0741320i
\(904\) 0 0
\(905\) 1.17715e8i 0.158813i
\(906\) 0 0
\(907\) 7.20201e8 4.15808e8i 0.965232 0.557277i 0.0674529 0.997722i \(-0.478513\pi\)
0.897779 + 0.440445i \(0.145179\pi\)
\(908\) 0 0
\(909\) 6.35679e8 + 1.10103e9i 0.846343 + 1.46591i
\(910\) 0 0
\(911\) 6.31322e8i 0.835017i −0.908673 0.417509i \(-0.862903\pi\)
0.908673 0.417509i \(-0.137097\pi\)
\(912\) 0 0
\(913\) −6.71065e8 −0.881764
\(914\) 0 0
\(915\) −6.06162e8 + 3.49968e8i −0.791271 + 0.456840i
\(916\) 0 0
\(917\) −8.96729e7 1.55318e8i −0.116293 0.201425i
\(918\) 0 0
\(919\) −1.18008e9 −1.52042 −0.760210 0.649677i \(-0.774904\pi\)
−0.760210 + 0.649677i \(0.774904\pi\)
\(920\) 0 0
\(921\) −8.12047e8 + 1.40651e9i −1.03945 + 1.80037i
\(922\) 0 0
\(923\) −3.23312e8 −0.411165
\(924\) 0 0
\(925\) −5.45799e8 3.15117e8i −0.689617 0.398150i
\(926\) 0 0
\(927\) −6.40939e8 3.70046e8i −0.804595 0.464533i
\(928\) 0 0
\(929\) 1.14268e8 + 1.97918e8i 0.142521 + 0.246853i 0.928445 0.371469i \(-0.121146\pi\)
−0.785925 + 0.618322i \(0.787813\pi\)
\(930\) 0 0
\(931\) 3.34245e8 + 7.12584e8i 0.414206 + 0.883053i
\(932\) 0 0
\(933\) −1.97028e9 + 1.13754e9i −2.42595 + 1.40062i
\(934\) 0 0
\(935\) 1.28048e8 2.21785e8i 0.156652 0.271330i
\(936\) 0 0
\(937\) −9.36761e7 + 1.62252e8i −0.113870 + 0.197229i −0.917328 0.398133i \(-0.869658\pi\)
0.803457 + 0.595362i \(0.202991\pi\)
\(938\) 0 0
\(939\) 6.92057e8i 0.835882i
\(940\) 0 0
\(941\) −1.27110e9 7.33870e8i −1.52550 0.880745i −0.999543 0.0302318i \(-0.990375\pi\)
−0.525953 0.850514i \(-0.676291\pi\)
\(942\) 0 0
\(943\) 5.20260e8i 0.620419i
\(944\) 0 0
\(945\) 1.71811e7 9.91953e6i 0.0203590 0.0117543i
\(946\) 0 0
\(947\) −8.11295e8 1.40520e9i −0.955276 1.65459i −0.733736 0.679435i \(-0.762225\pi\)
−0.221540 0.975151i \(-0.571108\pi\)
\(948\) 0 0
\(949\) 3.89495e8i 0.455726i
\(950\) 0 0
\(951\) 2.65874e8 0.309126
\(952\) 0 0
\(953\) −7.24323e8 + 4.18188e8i −0.836861 + 0.483162i −0.856196 0.516651i \(-0.827178\pi\)
0.0193350 + 0.999813i \(0.493845\pi\)
\(954\) 0 0
\(955\) 1.64384e8 + 2.84722e8i 0.188734 + 0.326897i
\(956\) 0 0
\(957\) 4.29213e7 0.0489708
\(958\) 0 0
\(959\) 5.09784e7 8.82971e7i 0.0578003 0.100113i
\(960\) 0 0
\(961\) 7.85519e8 0.885089
\(962\) 0 0
\(963\) −5.05044e8 2.91587e8i −0.565523 0.326505i
\(964\) 0 0
\(965\) 5.96599e7 + 3.44447e7i 0.0663897 + 0.0383301i
\(966\) 0 0
\(967\) −7.70384e8 1.33434e9i −0.851977 1.47567i −0.879421 0.476045i \(-0.842070\pi\)
0.0274439 0.999623i \(-0.491263\pi\)
\(968\) 0 0
\(969\) −7.59800e7 + 8.91649e8i −0.0835080 + 0.979993i
\(970\) 0 0
\(971\) 9.88684e8 5.70817e8i 1.07994 0.623504i 0.149060 0.988828i \(-0.452375\pi\)
0.930880 + 0.365324i \(0.119042\pi\)
\(972\) 0 0
\(973\) −2.82384e7 + 4.89104e7i −0.0306551 + 0.0530961i
\(974\) 0 0
\(975\) 4.62175e8 8.00511e8i 0.498647 0.863681i
\(976\) 0 0
\(977\) 4.74801e8i 0.509129i 0.967056 + 0.254565i \(0.0819322\pi\)
−0.967056 + 0.254565i \(0.918068\pi\)
\(978\) 0 0
\(979\) 6.90142e8 + 3.98454e8i 0.735513 + 0.424649i
\(980\) 0 0
\(981\) 6.21243e8i 0.658043i
\(982\) 0 0
\(983\) −3.92307e8 + 2.26498e8i −0.413014 + 0.238454i −0.692084 0.721817i \(-0.743307\pi\)
0.279070 + 0.960271i \(0.409974\pi\)
\(984\) 0 0
\(985\) 2.63501e8 + 4.56397e8i 0.275723 + 0.477567i
\(986\) 0 0
\(987\) 2.07131e7i 0.0215424i
\(988\) 0 0
\(989\) −3.34106e8 −0.345379
\(990\) 0 0
\(991\) −3.41754e8 + 1.97312e8i −0.351150 + 0.202737i −0.665192 0.746673i \(-0.731650\pi\)
0.314041 + 0.949409i \(0.398317\pi\)
\(992\) 0 0
\(993\) 5.13980e8 + 8.90240e8i 0.524927 + 0.909200i
\(994\) 0 0
\(995\) 5.59256e8 0.567730
\(996\) 0 0
\(997\) −9.07065e8 + 1.57108e9i −0.915278 + 1.58531i −0.108783 + 0.994066i \(0.534695\pi\)
−0.806494 + 0.591242i \(0.798638\pi\)
\(998\) 0 0
\(999\) −4.11490e8 −0.412727
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 76.7.h.a.69.9 yes 20
3.2 odd 2 684.7.y.c.145.4 20
19.8 odd 6 inner 76.7.h.a.65.9 20
57.8 even 6 684.7.y.c.217.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.7.h.a.65.9 20 19.8 odd 6 inner
76.7.h.a.69.9 yes 20 1.1 even 1 trivial
684.7.y.c.145.4 20 3.2 odd 2
684.7.y.c.217.4 20 57.8 even 6