LMFDB - Embedded newform 74.8.b.a.73.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,8,Mod(73,74)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("74.73");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	74.b (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:	$23.1164918858$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.12
Character	$\chi$	$=$	74.73
Dual form			74.8.b.a.73.24

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} +86.4953 q^{3} -64.0000 q^{4} +509.039i q^{5} -691.963i q^{6} -1106.69 q^{7} +512.000i q^{8} +5294.44 q^{9} +4072.31 q^{10} -6321.65 q^{11} -5535.70 q^{12} +5645.60i q^{13} +8853.49i q^{14} +44029.5i q^{15} +4096.00 q^{16} +20217.2i q^{17} -42355.5i q^{18} -2675.00i q^{19} -32578.5i q^{20} -95723.2 q^{21} +50573.2i q^{22} -61864.7i q^{23} +44285.6i q^{24} -180996. q^{25} +45164.8 q^{26} +268779. q^{27} +70827.9 q^{28} +113185. i q^{29} +352236. q^{30} +171767. i q^{31} -32768.0i q^{32} -546794. q^{33} +161737. q^{34} -563347. i q^{35} -338844. q^{36} +(298951. + 74568.2i) q^{37} -21400.0 q^{38} +488318. i q^{39} -260628. q^{40} +392684. q^{41} +765785. i q^{42} -322691. i q^{43} +404586. q^{44} +2.69508e6i q^{45} -494918. q^{46} +661976. q^{47} +354285. q^{48} +401211. q^{49} +1.44797e6i q^{50} +1.74869e6i q^{51} -361318. i q^{52} +1.18393e6 q^{53} -2.15023e6i q^{54} -3.21797e6i q^{55} -566623. i q^{56} -231375. i q^{57} +905482. q^{58} +284535. i q^{59} -2.81789e6i q^{60} +220678. i q^{61} +1.37413e6 q^{62} -5.85928e6 q^{63} -262144. q^{64} -2.87383e6 q^{65} +4.37435e6i q^{66} -1.34002e6 q^{67} -1.29390e6i q^{68} -5.35101e6i q^{69} -4.50677e6 q^{70} +4.05819e6 q^{71} +2.71075e6i q^{72} +1.17183e6 q^{73} +(596545. - 2.39160e6i) q^{74} -1.56553e7 q^{75} +171200. i q^{76} +6.99609e6 q^{77} +3.90654e6 q^{78} -5.12044e6i q^{79} +2.08502e6i q^{80} +1.16692e7 q^{81} -3.14148e6i q^{82} -208229. q^{83} +6.12628e6 q^{84} -1.02913e7 q^{85} -2.58152e6 q^{86} +9.78999e6i q^{87} -3.23669e6i q^{88} -9.07881e6i q^{89} +2.15606e7 q^{90} -6.24791e6i q^{91} +3.95934e6i q^{92} +1.48570e7i q^{93} -5.29581e6i q^{94} +1.36168e6 q^{95} -2.83428e6i q^{96} +1.55814e7i q^{97} -3.20969e6i q^{98} -3.34696e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9} + 1136 q^{10} + 366 q^{11} + 6784 q^{12} + 98304 q^{16} - 239820 q^{21} - 675570 q^{25} + 97008 q^{26} + 338780 q^{27} - 6656 q^{28} + 350400 q^{30}+ \cdots - 53279900 q^{99}+O(q^{100}) $ 24 * q - 106 * q^3 - 1536 * q^4 + 104 * q^7 + 17554 * q^9 + 1136 * q^10 + 366 * q^11 + 6784 * q^12 + 98304 * q^16 - 239820 * q^21 - 675570 * q^25 + 97008 * q^26 + 338780 * q^27 - 6656 * q^28 + 350400 * q^30 - 763792 * q^33 + 713632 * q^34 - 1123456 * q^36 + 41652 * q^37 - 30816 * q^38 - 72704 * q^40 + 1729722 * q^41 - 23424 * q^44 - 488496 * q^46 + 114756 * q^47 - 434176 * q^48 + 4003056 * q^49 - 1115964 * q^53 - 2075632 * q^58 + 3248208 * q^62 - 2350900 * q^63 - 6291456 * q^64 - 5246556 * q^65 - 2717994 * q^67 - 5649440 * q^70 + 10643280 * q^71 - 10450370 * q^73 - 1064064 * q^74 - 17737980 * q^75 + 11665236 * q^77 + 8431856 * q^78 + 47300176 * q^81 + 555912 * q^83 + 15348480 * q^84 - 4853096 * q^85 - 12070464 * q^86 + 32064160 * q^90 - 58516956 * q^95 - 53279900 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$.

$n$	$39$
$\chi(n)$	$-1$

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 $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	8.00000i		−	0.707107i
$2$		−	8.00000i		−	0.707107i
$3$	86.4953			1.84956			0.924780	−	0.380503i	$-0.124249\pi$
$3$	86.4953			1.84956			0.924780	+	0.380503i	$0.124249\pi$
$4$	−64.0000			−0.500000
$5$			509.039i			1.82119i	0.413296	+	0.910597i	$0.364378\pi$
$5$			509.039i			1.82119i	−0.413296	+	0.910597i	$0.635622\pi$
$6$		−	691.963i		−	1.30784i
$7$	−1106.69			−1.21950			−0.609749	−	0.792594i	$-0.708730\pi$
$7$	−1106.69			−1.21950			−0.609749	+	0.792594i	$0.708730\pi$
$8$			512.000i			0.353553i
$9$	5294.44			2.42087
$10$	4072.31			1.28778
$11$	−6321.65			−1.43204			−0.716022	−	0.698078i	$-0.754039\pi$
$11$	−6321.65			−1.43204			−0.716022	+	0.698078i	$0.754039\pi$
$12$	−5535.70			−0.924780
$13$			5645.60i			0.712703i	0.934352	+	0.356351i	$0.115979\pi$
$13$			5645.60i			0.712703i	−0.934352	+	0.356351i	$0.884021\pi$
$14$			8853.49i			0.862316i
$15$			44029.5i			3.36841i
$16$	4096.00			0.250000
$17$			20217.2i			0.998042i	0.866590	+	0.499021i	$0.166307\pi$
$17$			20217.2i			0.998042i	−0.866590	+	0.499021i	$0.833693\pi$
$18$		−	42355.5i		−	1.71181i
$19$		−	2675.00i		−	0.0894717i	−0.998999	−	0.0447358i	$-0.985755\pi$
$19$		−	2675.00i		−	0.0894717i	0.998999	−	0.0447358i	$-0.0142446\pi$
$20$		−	32578.5i		−	0.910597i
$21$	−95723.2			−2.25553
$22$			50573.2i			1.01261i
$23$		−	61864.7i		−	1.06022i	−0.847929	−	0.530110i	$-0.822151\pi$
$23$		−	61864.7i		−	1.06022i	0.847929	−	0.530110i	$-0.177849\pi$
$24$			44285.6i			0.653918i
$25$	−180996.			−2.31675
$26$	45164.8			0.503957
$27$	268779.			2.62798
$28$	70827.9			0.609749
$29$			113185.i			0.861781i	0.902404	+	0.430890i	$0.141800\pi$
$29$			113185.i			0.861781i	−0.902404	+	0.430890i	$0.858200\pi$
$30$	352236.			2.38182
$31$			171767.i			1.03555i	0.855515	+	0.517777i	$0.173240\pi$
$31$			171767.i			1.03555i	−0.855515	+	0.517777i	$0.826760\pi$
$32$		−	32768.0i		−	0.176777i
$33$	−546794.			−2.64865
$34$	161737.			0.705722
$35$		−	563347.i		−	2.22094i
$36$	−338844.			−1.21043
$37$	298951.	+	74568.2i	0.970272	+	0.242018i
$37$	298951.	+	74568.2i	0.970272	+	0.242018i
$38$	−21400.0			−0.0632660
$39$			488318.i			1.31819i
$40$	−260628.			−0.643889
$41$	392684.			0.889816			0.444908	−	0.895576i	$-0.353236\pi$
$41$	392684.			0.889816			0.444908	+	0.895576i	$0.353236\pi$
$42$			765785.i			1.59490i
$43$		−	322691.i		−	0.618937i	−0.950910	−	0.309469i	$-0.899849\pi$
$43$		−	322691.i		−	0.618937i	0.950910	−	0.309469i	$-0.100151\pi$
$44$	404586.			0.716022
$45$			2.69508e6i			4.40887i
$46$	−494918.			−0.749688
$47$	661976.			0.930036			0.465018	−	0.885301i	$-0.346048\pi$
$47$	661976.			0.930036			0.465018	+	0.885301i	$0.346048\pi$
$48$	354285.			0.462390
$49$	401211.			0.487177
$50$			1.44797e6i			1.63819i
$51$			1.74869e6i			1.84594i
$52$		−	361318.i		−	0.356351i
$53$	1.18393e6			1.09235			0.546173	−	0.837672i	$-0.316084\pi$
$53$	1.18393e6			1.09235			0.546173	+	0.837672i	$0.316084\pi$
$54$		−	2.15023e6i		−	1.85826i
$55$		−	3.21797e6i		−	2.60803i
$56$		−	566623.i		−	0.431158i
$57$		−	231375.i		−	0.165483i
$58$	905482.			0.609371
$59$			284535.i			0.180366i	0.995925	+	0.0901829i	$0.0287452\pi$
$59$			284535.i			0.180366i	−0.995925	+	0.0901829i	$0.971255\pi$
$60$		−	2.81789e6i		−	1.68420i
$61$			220678.i			0.124482i	0.998061	+	0.0622408i	$0.0198247\pi$
$61$			220678.i			0.124482i	−0.998061	+	0.0622408i	$0.980175\pi$
$62$	1.37413e6			0.732248
$63$	−5.85928e6			−2.95225
$64$	−262144.			−0.125000
$65$	−2.87383e6			−1.29797
$66$			4.37435e6i			1.87288i
$67$	−1.34002e6			−0.544313			−0.272156	−	0.962253i	$-0.587737\pi$
$67$	−1.34002e6			−0.544313			−0.272156	+	0.962253i	$0.587737\pi$
$68$		−	1.29390e6i		−	0.499021i
$69$		−	5.35101e6i		−	1.96094i
$70$	−4.50677e6			−1.57044
$71$	4.05819e6			1.34564			0.672820	−	0.739807i	$-0.265083\pi$
$71$	4.05819e6			1.34564			0.672820	+	0.739807i	$0.265083\pi$
$72$			2.71075e6i			0.855907i
$73$	1.17183e6			0.352561			0.176281	−	0.984340i	$-0.443593\pi$
$73$	1.17183e6			0.352561			0.176281	+	0.984340i	$0.443593\pi$
$74$	596545.	−	2.39160e6i	0.171133	−	0.686086i
$75$	−1.56553e7			−4.28496
$76$			171200.i			0.0447358i
$77$	6.99609e6			1.74638
$78$	3.90654e6			0.932098
$79$		−	5.12044e6i		−	1.16846i	−0.811589	−	0.584228i	$-0.801397\pi$
$79$		−	5.12044e6i		−	1.16846i	0.811589	−	0.584228i	$-0.198603\pi$
$80$			2.08502e6i			0.455298i
$81$	1.16692e7			2.43974
$82$		−	3.14148e6i		−	0.629195i
$83$	−208229.			−0.0399731			−0.0199866	−	0.999800i	$-0.506362\pi$
$83$	−208229.			−0.0399731			−0.0199866	+	0.999800i	$0.506362\pi$
$84$	6.12628e6			1.12777
$85$	−1.02913e7			−1.81763
$86$	−2.58152e6			−0.437655
$87$			9.78999e6i			1.59391i
$88$		−	3.23669e6i		−	0.506304i
$89$		−	9.07881e6i		−	1.36510i	−0.730839	−	0.682549i	$-0.760871\pi$
$89$		−	9.07881e6i		−	1.36510i	0.730839	−	0.682549i	$-0.239129\pi$
$90$	2.15606e7			3.11754
$91$		−	6.24791e6i		−	0.869140i
$92$			3.95934e6i			0.530110i
$93$			1.48570e7i			1.91532i
$94$		−	5.29581e6i		−	0.657635i
$95$	1.36168e6			0.162945
$96$		−	2.83428e6i		−	0.326959i
$97$			1.55814e7i			1.73343i	0.498808	+	0.866713i	$0.333771\pi$
$97$			1.55814e7i			1.73343i	−0.498808	+	0.866713i	$0.666229\pi$
$98$		−	3.20969e6i		−	0.344486i
$99$	−3.34696e7			−3.46679
$100$	1.15837e7			1.15837
$101$	1.06003e6			0.102375			0.0511873	−	0.998689i	$-0.483699\pi$
$101$	1.06003e6			0.102375			0.0511873	+	0.998689i	$0.483699\pi$
$102$	1.39895e7			1.30527
$103$		−	1.49250e6i		−	0.134581i	−0.997733	−	0.0672907i	$-0.978565\pi$
$103$		−	1.49250e6i		−	0.134581i	0.997733	−	0.0672907i	$-0.0214355\pi$
$104$	−2.89055e6			−0.251978
$105$		−	4.87268e7i		−	4.10777i
$106$		−	9.47144e6i		−	0.772406i
$107$	−2.23088e7			−1.76049			−0.880244	−	0.474522i	$-0.842621\pi$
$107$	−2.23088e7			−1.76049			−0.880244	+	0.474522i	$0.842621\pi$
$108$	−1.72019e7			−1.31399
$109$			2.52596e7i			1.86825i	0.356952	+	0.934123i	$0.383816\pi$
$109$			2.52596e7i			1.86825i	−0.356952	+	0.934123i	$0.616184\pi$
$110$	−2.57438e7			−1.84416
$111$	2.58578e7	+	6.44980e6i	1.79458	+	0.447626i
$112$	−4.53299e6			−0.304875
$113$		−	787004.i		−	0.0513101i	−0.999671	−	0.0256550i	$-0.991833\pi$
$113$		−	787004.i		−	0.0513101i	0.999671	−	0.0256550i	$-0.00816715\pi$
$114$	−1.85100e6			−0.117014
$115$	3.14916e7			1.93086
$116$		−	7.24385e6i		−	0.430890i
$117$			2.98903e7i			1.72536i
$118$	2.27628e6			0.127538
$119$		−	2.23740e7i		−	1.21711i
$120$	−2.25431e7			−1.19091
$121$	2.04761e7			1.05075
$122$	1.76543e6			0.0880218
$123$	3.39654e7			1.64577
$124$		−	1.09931e7i		−	0.517777i
$125$		−	5.23653e7i		−	2.39805i
$126$			4.68743e7i			2.08755i
$127$	−1.48462e7			−0.643134			−0.321567	−	0.946887i	$-0.604210\pi$
$127$	−1.48462e7			−0.643134			−0.321567	+	0.946887i	$0.604210\pi$
$128$			2.09715e6i			0.0883883i
$129$		−	2.79112e7i		−	1.14476i
$130$			2.29907e7i			0.917803i
$131$			3.31929e7i			1.29002i	0.764175	+	0.645008i	$0.223146\pi$
$131$			3.31929e7i			1.29002i	−0.764175	+	0.645008i	$0.776854\pi$
$132$	3.49948e7			1.32432
$133$			2.96038e6i			0.109111i
$134$			1.07201e7i			0.384887i
$135$			1.36819e8i			4.78607i
$136$	−1.03512e7			−0.352861
$137$	−3.05428e7			−1.01482			−0.507408	−	0.861706i	$-0.669396\pi$
$137$	−3.05428e7			−1.01482			−0.507408	+	0.861706i	$0.669396\pi$
$138$	−4.28081e7			−1.38659
$139$	−9.75469e6			−0.308079			−0.154039	−	0.988065i	$-0.549228\pi$
$139$	−9.75469e6			−0.308079			−0.154039	+	0.988065i	$0.549228\pi$
$140$			3.60542e7i			1.11047i
$141$	5.72578e7			1.72016
$142$		−	3.24656e7i		−	0.951511i
$143$		−	3.56895e7i		−	1.02062i
$144$	2.16860e7			0.605217
$145$	−5.76157e7			−1.56947
$146$		−	9.37464e6i		−	0.249298i
$147$	3.47029e7			0.901062
$148$	−1.91328e7	−	4.77236e6i	−0.485136	−	0.121009i
$149$	−6.86988e6			−0.170136			−0.0850682	−	0.996375i	$-0.527111\pi$
$149$	−6.86988e6			−0.170136			−0.0850682	+	0.996375i	$0.527111\pi$
$150$			1.25242e8i			3.02992i
$151$	−3.09039e7			−0.730455			−0.365227	−	0.930918i	$-0.619009\pi$
$151$	−3.09039e7			−0.730455			−0.365227	+	0.930918i	$0.619009\pi$
$152$	1.36960e6			0.0316330
$153$			1.07039e8i			2.41613i
$154$		−	5.59687e7i		−	1.23487i
$155$	−8.74360e7			−1.88595
$156$		−	3.12524e7i		−	0.659093i
$157$	7.14654e7			1.47383			0.736914	−	0.675987i	$-0.236282\pi$
$157$	7.14654e7			1.47383			0.736914	+	0.675987i	$0.236282\pi$
$158$	−4.09635e7			−0.826224
$159$	1.02404e8			2.02036
$160$	1.66802e7			0.321945
$161$			6.84648e7i			1.29294i
$162$		−	9.33536e7i		−	1.72516i
$163$		−	3.69665e7i		−	0.668577i	−0.942471	−	0.334288i	$-0.891504\pi$
$163$		−	3.69665e7i		−	0.668577i	0.942471	−	0.334288i	$-0.108496\pi$
$164$	−2.51318e7			−0.444908
$165$		−	2.78339e8i		−	4.82370i
$166$			1.66583e6i			0.0282653i
$167$			2.60407e6i			0.0432658i	0.999766	+	0.0216329i	$0.00688650\pi$
$167$			2.60407e6i			0.0432658i	−0.999766	+	0.0216329i	$0.993113\pi$
$168$		−	4.90103e7i		−	0.797452i
$169$	3.08757e7			0.492055
$170$			8.23306e7i			1.28526i
$171$		−	1.41626e7i		−	0.216599i
$172$			2.06522e7i			0.309469i
$173$	1.26966e7			0.186434			0.0932170	−	0.995646i	$-0.470285\pi$
$173$	1.26966e7			0.186434			0.0932170	+	0.995646i	$0.470285\pi$
$174$	7.83199e7			1.12707
$175$	2.00306e8			2.82527
$176$	−2.58935e7			−0.358011
$177$			2.46110e7i			0.333597i
$178$	−7.26305e7			−0.965271
$179$		−	1.22581e8i		−	1.59748i	−0.601675	−	0.798741i	$-0.705500\pi$
$179$		−	1.22581e8i		−	1.59748i	0.601675	−	0.798741i	$-0.294500\pi$
$180$		−	1.72485e8i		−	2.20444i
$181$	14169.2			0.000177611			8.88056e−5	−	1.00000i	$-0.499972\pi$
$181$	14169.2			0.000177611			8.88056e−5		1.00000i	$0.499972\pi$
$182$	−4.99833e7			−0.614575
$183$			1.90877e7i			0.230236i
$184$	3.16747e7			0.374844
$185$	−3.79581e7	+	1.52178e8i	−0.440762	+	1.76705i
$186$	1.18856e8			1.35434
$187$		−	1.27806e8i		−	1.42924i
$188$	−4.23665e7			−0.465018
$189$	−2.97454e8			−3.20482
$190$		−	1.08934e7i		−	0.115220i
$191$		−	6.31477e7i		−	0.655753i	−0.944721	−	0.327877i	$-0.893667\pi$
$191$		−	6.31477e7i		−	0.655753i	0.944721	−	0.327877i	$-0.106333\pi$
$192$	−2.26742e7			−0.231195
$193$			5.98484e7i			0.599241i	0.954058	+	0.299621i	$0.0968601\pi$
$193$			5.98484e7i			0.599241i	−0.954058	+	0.299621i	$0.903140\pi$
$194$	1.24651e8			1.22572
$195$	−2.48573e8			−2.40067
$196$	−2.56775e7			−0.243588
$197$	6.91995e7			0.644868			0.322434	−	0.946592i	$-0.395499\pi$
$197$	6.91995e7			0.644868			0.322434	+	0.946592i	$0.395499\pi$
$198$			2.67757e8i			2.45139i
$199$			1.63722e8i			1.47273i	0.676586	+	0.736364i	$0.263459\pi$
$199$			1.63722e8i			1.47273i	−0.676586	+	0.736364i	$0.736541\pi$
$200$		−	9.26699e7i		−	0.819094i
$201$	−1.15905e8			−1.00674
$202$		−	8.48022e6i		−	0.0723898i
$203$		−	1.25260e8i		−	1.05094i
$204$		−	1.11916e8i		−	0.922969i
$205$			1.99892e8i			1.62053i
$206$	−1.19400e7			−0.0951634
$207$		−	3.27539e8i		−	2.56665i
$208$			2.31244e7i			0.178176i
$209$			1.69104e7i			0.128127i
$210$	−3.89815e8			−2.90463
$211$	2.46475e8			1.80627			0.903137	−	0.429352i	$-0.141258\pi$
$211$	2.46475e8			1.80627			0.903137	+	0.429352i	$0.141258\pi$
$212$	−7.57715e7			−0.546173
$213$	3.51015e8			2.48884
$214$			1.78470e8i			1.24485i
$215$	1.64262e8			1.12720
$216$			1.37615e8i			0.929132i
$217$		−	1.90092e8i		−	1.26286i
$218$	2.02077e8			1.32105
$219$	1.01358e8			0.652083
$220$			2.05950e8i			1.30401i
$221$	−1.14138e8			−0.711307
$222$	5.15984e7	−	2.06863e8i	0.316520	−	1.26896i
$223$	914059.			0.00551959			0.00275980	−	0.999996i	$-0.499122\pi$
$223$	914059.			0.00551959			0.00275980	+	0.999996i	$0.499122\pi$
$224$			3.62639e7i			0.215579i
$225$	−9.58272e8			−5.60854
$226$	−6.29603e6			−0.0362817
$227$			4.60910e7i			0.261533i	0.991413	+	0.130766i	$0.0417438\pi$
$227$			4.60910e7i			0.261533i	−0.991413	+	0.130766i	$0.958256\pi$
$228$			1.48080e7i			0.0827416i
$229$	1.44463e8			0.794936			0.397468	−	0.917616i	$-0.369889\pi$
$229$	1.44463e8			0.794936			0.397468	+	0.917616i	$0.369889\pi$
$230$		−	2.51933e8i		−	1.36533i
$231$	6.05129e8			3.23002
$232$	−5.79508e7			−0.304685
$233$	1.14251e6			0.00591718			0.00295859	−	0.999996i	$-0.499058\pi$
$233$	1.14251e6			0.00591718			0.00295859	+	0.999996i	$0.499058\pi$
$234$	2.39122e8			1.22001
$235$			3.36972e8i			1.69378i
$236$		−	1.82103e7i		−	0.0901829i
$237$		−	4.42894e8i		−	2.16113i
$238$	−1.78992e8			−0.860627
$239$			2.39631e8i			1.13540i	0.823234	+	0.567702i	$0.192167\pi$
$239$			2.39631e8i			1.13540i	−0.823234	+	0.567702i	$0.807833\pi$
$240$			1.80345e8i			0.842101i
$241$			1.93481e8i			0.890386i	0.895435	+	0.445193i	$0.146865\pi$
$241$			1.93481e8i			0.890386i	−0.895435	+	0.445193i	$0.853135\pi$
$242$		−	1.63809e8i		−	0.742992i
$243$	4.21511e8			1.88446
$244$		−	1.41234e7i		−	0.0622408i
$245$			2.04232e8i			0.887243i
$246$		−	2.71723e8i		−	1.16373i
$247$	1.51020e7			0.0637667
$248$	−8.79445e7			−0.366124
$249$	−1.80109e7			−0.0739327
$250$	−4.18922e8			−1.69568
$251$		−	3.24978e8i		−	1.29717i	−0.761143	−	0.648585i	$-0.775361\pi$
$251$		−	3.24978e8i		−	1.29717i	0.761143	−	0.648585i	$-0.224639\pi$
$252$	3.74994e8			1.47612
$253$			3.91087e8i			1.51828i
$254$			1.18769e8i			0.454764i
$255$	−8.90151e8			−3.36181
$256$	1.67772e7			0.0625000
$257$		−	1.82037e8i		−	0.668951i	−0.942404	−	0.334475i	$-0.891441\pi$
$257$		−	1.82037e8i		−	0.668951i	0.942404	−	0.334475i	$-0.108559\pi$
$258$	−2.23290e8			−0.809468
$259$	−3.30844e8	−	8.25236e7i	−1.18325	−	0.295140i
$260$	1.83925e8			0.648985
$261$			5.99252e8i			2.08626i
$262$	2.65543e8			0.912180
$263$	−2.22855e8			−0.755399			−0.377699	−	0.925928i	$-0.623285\pi$
$263$	−2.22855e8			−0.755399			−0.377699	+	0.925928i	$0.623285\pi$
$264$		−	2.79958e8i		−	0.936439i
$265$			6.02667e8i			1.98937i
$266$	2.36830e7			0.0771528
$267$		−	7.85275e8i		−	2.52483i
$268$	8.57611e7			0.272156
$269$	−4.20621e8			−1.31752			−0.658761	−	0.752352i	$-0.728919\pi$
$269$	−4.20621e8			−1.31752			−0.658761	+	0.752352i	$0.728919\pi$
$270$	1.09455e9			3.38426
$271$	4.37173e8			1.33432			0.667161	−	0.744913i	$-0.267509\pi$
$271$	4.37173e8			1.33432			0.667161	+	0.744913i	$0.267509\pi$
$272$			8.28095e7i			0.249510i
$273$		−	5.40415e8i		−	1.60753i
$274$			2.44343e8i			0.717583i
$275$	1.14419e9			3.31768
$276$			3.42465e8i			0.980469i
$277$			5.94338e7i			0.168018i	0.996465	+	0.0840088i	$0.0267724\pi$
$277$			5.94338e7i			0.168018i	−0.996465	+	0.0840088i	$0.973228\pi$
$278$			7.80375e7i			0.217845i
$279$			9.09409e8i			2.50694i
$280$	2.88433e8			0.785222
$281$			3.66736e7i			0.0986010i	0.998784	+	0.0493005i	$0.0156992\pi$
$281$			3.66736e7i			0.0986010i	−0.998784	+	0.0493005i	$0.984301\pi$
$282$		−	4.58063e8i		−	1.21633i
$283$		−	1.96753e8i		−	0.516024i	−0.966142	−	0.258012i	$-0.916933\pi$
$283$		−	1.96753e8i		−	0.516024i	0.966142	−	0.258012i	$-0.0830674\pi$
$284$	−2.59724e8			−0.672820
$285$	1.17779e8			0.301377
$286$	−2.85516e8			−0.721688
$287$	−4.34578e8			−1.08513
$288$		−	1.73488e8i		−	0.427953i
$289$	1.60542e6			0.00391243
$290$			4.60926e8i			1.10978i
$291$			1.34772e9i			3.20607i
$292$	−7.49971e7			−0.176281
$293$	2.35350e8			0.546610			0.273305	−	0.961927i	$-0.411883\pi$
$293$	2.35350e8			0.546610			0.273305	+	0.961927i	$0.411883\pi$
$294$		−	2.77623e8i		−	0.637147i
$295$	−1.44840e8			−0.328481
$296$	−3.81789e7	+	1.53063e8i	−0.0855663	+	0.343043i
$297$	−1.69913e9			−3.76339
$298$			5.49591e7i			0.120305i
$299$	3.49264e8			0.755621
$300$	1.00194e9			2.14248
$301$			3.57117e8i			0.754793i
$302$			2.47231e8i			0.516509i
$303$	9.16874e7			0.189348
$304$		−	1.09568e7i		−	0.0223679i
$305$	−1.12334e8			−0.226705
$306$	8.56308e8			1.70846
$307$	−4.48461e8			−0.884587			−0.442293	−	0.896870i	$-0.645835\pi$
$307$	−4.48461e8			−0.884587			−0.442293	+	0.896870i	$0.645835\pi$
$308$	−4.47749e8			−0.873188
$309$		−	1.29094e8i		−	0.248916i
$310$			6.99488e8i			1.33356i
$311$			8.61677e8i			1.62436i	0.583405	+	0.812182i	$0.301720\pi$
$311$			8.61677e8i			1.62436i	−0.583405	+	0.812182i	$0.698280\pi$
$312$	−2.50019e8			−0.466049
$313$		−	1.00288e9i		−	1.84861i	−0.381657	−	0.924304i	$-0.624646\pi$
$313$		−	1.00288e9i		−	1.84861i	0.381657	−	0.924304i	$-0.375354\pi$
$314$		−	5.71723e8i		−	1.04215i
$315$		−	2.98261e9i		−	5.37661i
$316$			3.27708e8i			0.584228i
$317$	7.93266e8			1.39866			0.699329	−	0.714800i	$-0.253482\pi$
$317$	7.93266e8			1.39866			0.699329	+	0.714800i	$0.253482\pi$
$318$		−	8.19235e8i		−	1.42861i
$319$		−	7.15518e8i		−	1.23411i
$320$		−	1.33442e8i		−	0.227649i
$321$	−1.92961e9			−3.25613
$322$	5.47719e8			0.914244
$323$	5.40808e7			0.0892965
$324$	−7.46829e8			−1.21987
$325$		−	1.02183e9i		−	1.65115i
$326$	−2.95732e8			−0.472755
$327$			2.18484e9i			3.45543i
$328$			2.01054e8i			0.314597i
$329$	−7.32600e8			−1.13418
$330$	−2.22671e9			−3.41087
$331$		−	2.68525e8i		−	0.406993i	−0.979076	−	0.203496i	$-0.934769\pi$
$331$		−	2.68525e8i		−	0.406993i	0.979076	−	0.203496i	$-0.0652305\pi$
$332$	1.33267e7			0.0199866
$333$	1.58278e9	+	3.94797e8i	2.34890	+	0.585894i
$334$	2.08325e7			0.0305935
$335$		−	6.82121e8i		−	0.991299i
$336$	−3.92082e8			−0.563884
$337$	−6.44046e7			−0.0916669			−0.0458334	−	0.998949i	$-0.514594\pi$
$337$	−6.44046e7			−0.0916669			−0.0458334	+	0.998949i	$0.514594\pi$
$338$		−	2.47006e8i		−	0.347935i
$339$		−	6.80722e7i		−	0.0949010i
$340$	6.58645e8			0.908814
$341$		−	1.08585e9i		−	1.48296i
$342$	−1.13301e8			−0.153159
$343$	4.67389e8			0.625387
$344$	1.65218e8			0.218827
$345$	2.72387e9			3.57125
$346$		−	1.01573e8i		−	0.131829i
$347$			3.89685e8i			0.500681i	0.968158	+	0.250340i	$0.0805425\pi$
$347$			3.89685e8i			0.500681i	−0.968158	+	0.250340i	$0.919457\pi$
$348$		−	6.26559e8i		−	0.796957i
$349$	−5.64662e8			−0.711050			−0.355525	−	0.934667i	$-0.615698\pi$
$349$	−5.64662e8			−0.711050			−0.355525	+	0.934667i	$0.615698\pi$
$350$		−	1.60244e9i		−	1.99777i
$351$			1.51742e9i			1.87297i
$352$			2.07148e8i			0.253152i
$353$		−	1.18515e9i		−	1.43404i	−0.697052	−	0.717020i	$-0.745505\pi$
$353$		−	1.18515e9i		−	1.43404i	0.697052	−	0.717020i	$-0.254495\pi$
$354$	1.96888e8			0.235889
$355$			2.06578e9i			2.45067i
$356$			5.81044e8i			0.682549i
$357$		−	1.93525e9i		−	2.25112i
$358$	−9.80644e8			−1.12959
$359$	−6.40809e8			−0.730968			−0.365484	−	0.930818i	$-0.619096\pi$
$359$	−6.40809e8			−0.730968			−0.365484	+	0.930818i	$0.619096\pi$
$360$	−1.37988e9			−1.55877
$361$	8.86716e8			0.991995
$362$		−	113354.i		−	0.000125590i
$363$	1.77109e9			1.94342
$364$			3.99866e8i			0.434570i
$365$			5.96507e8i			0.642082i
$366$	1.52701e8			0.162802
$367$	3.02555e8			0.319502			0.159751	−	0.987157i	$-0.448931\pi$
$367$	3.02555e8			0.319502			0.159751	+	0.987157i	$0.448931\pi$
$368$		−	2.53398e8i		−	0.265055i
$369$	2.07904e9			2.15413
$370$	1.21742e9	+	3.03665e8i	1.24950	+	0.311665i
$371$	−1.31024e9			−1.33212
$372$		−	9.50849e8i		−	0.957660i
$373$	−1.43704e9			−1.43380			−0.716900	−	0.697176i	$-0.754440\pi$
$373$	−1.43704e9			−1.43380			−0.716900	+	0.697176i	$0.754440\pi$
$374$	−1.02245e9			−1.01063
$375$		−	4.52935e9i		−	4.43534i
$376$			3.38932e8i			0.328817i
$377$	−6.38998e8			−0.614193
$378$			2.37963e9i			2.26615i
$379$	4.27432e8			0.403302			0.201651	−	0.979457i	$-0.435369\pi$
$379$	4.27432e8			0.403302			0.201651	+	0.979457i	$0.435369\pi$
$380$	−8.71474e7			−0.0814726
$381$	−1.28412e9			−1.18951
$382$	−5.05181e8			−0.463688
$383$		−	2.08447e8i		−	0.189583i	−0.995497	−	0.0947915i	$-0.969782\pi$
$383$		−	2.08447e8i		−	0.189583i	0.995497	−	0.0947915i	$-0.0302184\pi$
$384$			1.81394e8i			0.163479i
$385$			3.56128e9i			3.18049i
$386$	4.78787e8			0.423727
$387$		−	1.70847e9i		−	1.49837i
$388$		−	9.97209e8i		−	0.866713i
$389$		−	1.20130e9i		−	1.03473i	−0.855765	−	0.517365i	$-0.826913\pi$
$389$		−	1.20130e9i		−	1.03473i	0.855765	−	0.517365i	$-0.173087\pi$
$390$			1.98858e9i			1.69753i
$391$	1.25073e9			1.05814
$392$			2.05420e8i			0.172243i
$393$			2.87103e9i			2.38596i
$394$		−	5.53596e8i		−	0.455991i
$395$	2.60651e9			2.12799
$396$	2.14206e9			1.73340
$397$	1.33907e9			1.07408			0.537040	−	0.843557i	$-0.319542\pi$
$397$	1.33907e9			1.07408			0.537040	+	0.843557i	$0.319542\pi$
$398$	1.30978e9			1.04138
$399$			2.56059e8i			0.201806i
$400$	−7.41359e8			−0.579187
$401$			1.69970e9i			1.31634i	0.752870	+	0.658169i	$0.228669\pi$
$401$			1.69970e9i			1.31634i	−0.752870	+	0.658169i	$0.771331\pi$
$402$			9.27241e8i			0.711872i
$403$	−9.69726e8			−0.738042
$404$	−6.78417e7			−0.0511873
$405$			5.94008e9i			4.44324i
$406$	−1.00208e9			−0.743127
$407$	−1.88986e9	−	4.71394e8i	−1.38947	−	0.346580i
$408$	−8.95329e8			−0.652637
$409$		−	3.92630e8i		−	0.283760i	−0.989884	−	0.141880i	$-0.954685\pi$
$409$		−	3.92630e8i		−	0.283760i	0.989884	−	0.141880i	$-0.0453148\pi$
$410$	1.59913e9			1.14589
$411$	−2.64181e9			−1.87696
$412$			9.55202e7i			0.0672907i
$413$		−	3.14891e8i		−	0.219956i
$414$	−2.62031e9			−1.81490
$415$		−	1.05997e8i		−	0.0727988i
$416$	1.84995e8			0.125989
$417$	−8.43735e8			−0.569810
$418$	1.35283e8			0.0905997
$419$	−1.13765e9			−0.755546			−0.377773	−	0.925898i	$-0.623310\pi$
$419$	−1.13765e9			−0.755546			−0.377773	+	0.925898i	$0.623310\pi$
$420$			3.11852e9i			2.05388i
$421$		−	9.38142e7i		−	0.0612747i	−0.999531	−	0.0306374i	$-0.990246\pi$
$421$		−	9.38142e7i		−	0.0612747i	0.999531	−	0.0306374i	$-0.00975370\pi$
$422$		−	1.97180e9i		−	1.27723i
$423$	3.50479e9			2.25150
$424$			6.06172e8i			0.386203i
$425$		−	3.65922e9i		−	2.31221i
$426$		−	2.80812e9i		−	1.75988i
$427$		−	2.44222e8i		−	0.151805i
$428$	1.42776e9			0.880244
$429$		−	3.08698e9i		−	1.88770i
$430$		−	1.31410e9i		−	0.797054i
$431$			2.22183e8i			0.133672i	0.997764	+	0.0668359i	$0.0212904\pi$
$431$			2.22183e8i			0.133672i	−0.997764	+	0.0668359i	$0.978710\pi$
$432$	1.10092e9			0.656996
$433$	−1.04808e9			−0.620423			−0.310211	−	0.950668i	$-0.600400\pi$
$433$	−1.04808e9			−0.620423			−0.310211	+	0.950668i	$0.600400\pi$
$434$	−1.52073e9			−0.892975
$435$	−4.98349e9			−2.90283
$436$		−	1.61661e9i		−	0.934123i
$437$	−1.65488e8			−0.0948596
$438$		−	8.10863e8i		−	0.461092i
$439$		−	9.02209e8i		−	0.508957i	−0.967078	−	0.254479i	$-0.918096\pi$
$439$		−	9.02209e8i		−	0.508957i	0.967078	−	0.254479i	$-0.0819038\pi$
$440$	1.64760e9			0.922078
$441$	2.12419e9			1.17939
$442$			9.13104e8i			0.502970i
$443$	1.88343e9			1.02929			0.514644	−	0.857404i	$-0.327924\pi$
$443$	1.88343e9			1.02929			0.514644	+	0.857404i	$0.327924\pi$
$444$	−1.65490e9	−	4.12787e8i	−0.897288	−	0.223813i
$445$	4.62147e9			2.48611
$446$		−	7.31247e6i		−	0.00390294i
$447$	−5.94213e8			−0.314678
$448$	2.90111e8			0.152437
$449$		−	8.86247e8i		−	0.462054i	−0.972947	−	0.231027i	$-0.925791\pi$
$449$		−	8.86247e8i		−	0.462054i	0.972947	−	0.231027i	$-0.0742085\pi$
$450$			7.66618e9i			3.96584i
$451$	−2.48241e9			−1.27426
$452$			5.03683e7i			0.0256550i
$453$	−2.67304e9			−1.35102
$454$	3.68728e8			0.184932
$455$	3.18043e9			1.58287
$456$	1.18464e8			0.0585071
$457$			3.74122e9i			1.83361i	0.399335	+	0.916805i	$0.369241\pi$
$457$			3.74122e9i			1.83361i	−0.399335	+	0.916805i	$0.630759\pi$
$458$		−	1.15570e9i		−	0.562105i
$459$			5.43395e9i			2.62284i
$460$	−2.01546e9			−0.965432
$461$		−	2.26010e9i		−	1.07442i	−0.843449	−	0.537209i	$-0.819479\pi$
$461$		−	2.26010e9i		−	1.07442i	0.843449	−	0.537209i	$-0.180521\pi$
$462$		−	4.84103e9i		−	2.28397i
$463$			2.96960e9i			1.39048i	0.718777	+	0.695240i	$0.244702\pi$
$463$			2.96960e9i			1.39048i	−0.718777	+	0.695240i	$0.755298\pi$
$464$			4.63607e8i			0.215445i
$465$	−7.56280e9			−3.48817
$466$		−	9.14010e6i		−	0.00418408i
$467$			2.77850e7i			0.0126241i	0.999980	+	0.00631205i	$0.00200920\pi$
$467$			2.77850e7i			0.0126241i	−0.999980	+	0.00631205i	$0.997991\pi$
$468$		−	1.91298e9i		−	0.862680i
$469$	1.48298e9			0.663788
$470$	2.69577e9			1.19768
$471$	6.18142e9			2.72593
$472$	−1.45682e8			−0.0637689
$473$			2.03994e9i			0.886345i
$474$	−3.54315e9			−1.52815
$475$			4.84163e8i			0.207283i
$476$			1.43194e9i			0.608555i
$477$	6.26825e9			2.64443
$478$	1.91705e9			0.802852
$479$		−	4.02671e8i		−	0.167408i	−0.996491	−	0.0837039i	$-0.973325\pi$
$479$		−	4.02671e8i		−	0.167408i	0.996491	−	0.0837039i	$-0.0266750\pi$
$480$	1.44276e9			0.595456
$481$	−4.20982e8	+	1.68776e9i	−0.172487	+	0.691515i
$482$	1.54785e9			0.629598
$483$			5.92189e9i			2.39136i
$484$	−1.31047e9			−0.525375
$485$	−7.93154e9			−3.15690
$486$		−	3.37209e9i		−	1.33252i
$487$			4.70840e9i			1.84724i	0.383315	+	0.923618i	$0.374782\pi$
$487$			4.70840e9i			1.84724i	−0.383315	+	0.923618i	$0.625218\pi$
$488$	−1.12987e8			−0.0440109
$489$		−	3.19743e9i		−	1.23657i
$490$	1.63386e9			0.627376
$491$	4.56487e9			1.74038			0.870189	−	0.492718i	$-0.163997\pi$
$491$	4.56487e9			1.74038			0.870189	+	0.492718i	$0.163997\pi$
$492$	−2.17378e9			−0.822884
$493$	−2.28828e9			−0.860093
$494$		−	1.20816e8i		−	0.0450899i
$495$		−	1.70374e10i		−	6.31370i
$496$			7.03556e8i			0.258889i
$497$	−4.49115e9			−1.64101
$498$			1.44087e8i			0.0522783i
$499$		−	2.35941e9i		−	0.850063i	−0.905178	−	0.425032i	$-0.860263\pi$
$499$		−	2.35941e9i		−	0.850063i	0.905178	−	0.425032i	$-0.139737\pi$
$500$			3.35138e9i			1.19903i
$501$			2.25240e8i			0.0800226i
$502$	−2.59983e9			−0.917237
$503$			4.88647e8i			0.171201i	0.996330	+	0.0856007i	$0.0272809\pi$
$503$			4.88647e8i			0.171201i	−0.996330	+	0.0856007i	$0.972719\pi$
$504$		−	2.99995e9i		−	1.04378i
$505$			5.39595e8i			0.186444i
$506$	3.12870e9			1.07359
$507$	2.67061e9			0.910085
$508$	9.50155e8			0.321567
$509$	2.01370e9			0.676834			0.338417	−	0.940996i	$-0.390109\pi$
$509$	2.01370e9			0.676834			0.338417	+	0.940996i	$0.390109\pi$
$510$			7.12121e9i			2.37716i
$511$	−1.29685e9			−0.429948
$512$		−	1.34218e8i		−	0.0441942i
$513$		−	7.18983e8i		−	0.235130i
$514$	−1.45630e9			−0.473020
$515$	7.59742e8			0.245099
$516$			1.78632e9i			0.572381i
$517$	−4.18478e9			−1.33185
$518$	−6.60189e8	+	2.64676e9i	−0.208696	+	0.836681i
$519$	1.09819e9			0.344821
$520$		−	1.47140e9i		−	0.458902i
$521$	−2.55729e9			−0.792224			−0.396112	−	0.918202i	$-0.629641\pi$
$521$	−2.55729e9			−0.792224			−0.396112	+	0.918202i	$0.629641\pi$
$522$	4.79402e9			1.47521
$523$		−	3.10477e9i		−	0.949015i	−0.880251	−	0.474508i	$-0.842626\pi$
$523$		−	3.10477e9i		−	0.949015i	0.880251	−	0.474508i	$-0.157374\pi$
$524$		−	2.12434e9i		−	0.645008i
$525$	1.73255e10			5.22550
$526$			1.78284e9i			0.534148i
$527$	−3.47263e9			−1.03353
$528$	−2.23967e9			−0.662162
$529$	−4.22419e8			−0.124065
$530$	4.82133e9			1.40670
$531$			1.50646e9i			0.436642i
$532$		−	1.89464e8i		−	0.0545553i
$533$			2.21694e9i			0.634174i
$534$	−6.28220e9			−1.78533
$535$		−	1.13560e10i		−	3.20619i
$536$		−	6.86088e8i		−	0.192444i
$537$		−	1.06026e10i		−	2.95464i
$538$			3.36497e9i			0.931629i
$539$	−2.53632e9			−0.697658
$540$		−	8.75642e9i		−	2.39303i
$541$			1.28625e9i			0.349249i	0.984635	+	0.174625i	$0.0558712\pi$
$541$			1.28625e9i			0.349249i	−0.984635	+	0.174625i	$0.944129\pi$
$542$		−	3.49738e9i		−	0.943508i
$543$	1.22557e6			0.000328503
$544$	6.62476e8			0.176431
$545$	−1.28581e10			−3.40244
$546$	−4.32332e9			−1.13669
$547$			2.39589e8i			0.0625908i	0.999510	+	0.0312954i	$0.00996326\pi$
$547$			2.39589e8i			0.0625908i	−0.999510	+	0.0312954i	$0.990037\pi$
$548$	1.95474e9			0.507408
$549$			1.16837e9i			0.301354i
$550$		−	9.15354e9i		−	2.34596i
$551$	3.02770e8			0.0771049
$552$	2.73972e9			0.693296
$553$			5.66672e9i			1.42493i
$554$	4.75471e8			0.118806
$555$	−3.28320e9	+	1.31626e10i	−0.815215	+	3.26827i
$556$	6.24300e8			0.154039
$557$		−	6.89749e9i		−	1.69121i	−0.533808	−	0.845606i	$-0.679239\pi$
$557$		−	6.89749e9i		−	1.69121i	0.533808	−	0.845606i	$-0.320761\pi$
$558$	7.27527e9			1.77268
$559$	1.82178e9			0.441118
$560$		−	2.30747e9i		−	0.555236i
$561$		−	1.10546e10i		−	2.64346i
$562$	2.93389e8			0.0697215
$563$		−	5.84475e9i		−	1.38034i	−0.723647	−	0.690171i	$-0.757535\pi$
$563$		−	5.84475e9i		−	1.38034i	0.723647	−	0.690171i	$-0.242465\pi$
$564$	−3.66450e9			−0.860079
$565$	4.00616e8			0.0934456
$566$	−1.57403e9			−0.364884
$567$	−1.29141e10			−2.97526
$568$			2.07780e9i			0.475755i
$569$			8.17699e9i			1.86080i	0.366542	+	0.930401i	$0.380542\pi$
$569$			8.17699e9i			1.86080i	−0.366542	+	0.930401i	$0.619458\pi$
$570$		−	9.42230e8i		−	0.213106i
$571$	1.60114e9			0.359917			0.179958	−	0.983674i	$-0.442404\pi$
$571$	1.60114e9			0.359917			0.179958	+	0.983674i	$0.442404\pi$
$572$			2.28413e9i			0.510311i
$573$		−	5.46198e9i		−	1.21285i
$574$			3.47663e9i			0.767302i
$575$			1.11973e10i			2.45626i
$576$	−1.38791e9			−0.302609
$577$			5.93214e9i			1.28557i	0.766046	+	0.642786i	$0.222221\pi$
$577$			5.93214e9i			1.28557i	−0.766046	+	0.642786i	$0.777779\pi$
$578$		−	1.28434e7i		−	0.00276651i
$579$			5.17660e9i			1.10833i
$580$	3.68740e9			0.784735
$581$	2.30444e8			0.0487472
$582$	1.07817e10			2.26704
$583$	−7.48439e9			−1.56429
$584$			5.99977e8i			0.124649i
$585$	−1.52153e10			−3.14222
$586$		−	1.88280e9i		−	0.386512i
$587$			3.75795e9i			0.766862i	0.923570	+	0.383431i	$0.125258\pi$
$587$			3.75795e9i			0.766862i	−0.923570	+	0.383431i	$0.874742\pi$
$588$	−2.22098e9			−0.450531
$589$	4.59475e8			0.0926528
$590$			1.15872e9i			0.232271i
$591$	5.98543e9			1.19272
$592$	1.22450e9	+	3.05431e8i	0.242568	+	0.0605045i
$593$	−2.57978e9			−0.508032			−0.254016	−	0.967200i	$-0.581752\pi$
$593$	−2.57978e9			−0.508032			−0.254016	+	0.967200i	$0.581752\pi$
$594$			1.35930e10i			2.66112i
$595$	1.13893e10			2.21659
$596$	4.39673e8			0.0850682
$597$			1.41612e10i			2.72390i
$598$		−	2.79411e9i		−	0.534305i
$599$	7.72604e8			0.146880			0.0734401	−	0.997300i	$-0.476602\pi$
$599$	7.72604e8			0.146880			0.0734401	+	0.997300i	$0.476602\pi$
$600$		−	8.01551e9i		−	1.51496i
$601$	3.33777e9			0.627186			0.313593	−	0.949558i	$-0.398467\pi$
$601$	3.33777e9			0.627186			0.313593	+	0.949558i	$0.398467\pi$
$602$	2.85694e9			0.533719
$603$	−7.09464e9			−1.31771
$604$	1.97785e9			0.365227
$605$			1.04232e10i			1.91362i
$606$		−	7.33499e8i		−	0.133889i
$607$		−	6.72918e9i		−	1.22124i	−0.791923	−	0.610621i	$-0.790920\pi$
$607$		−	6.72918e9i		−	1.22124i	0.791923	−	0.610621i	$-0.209080\pi$
$608$	−8.76543e7			−0.0158165
$609$		−	1.08344e10i		−	1.94378i
$610$			8.98672e8i			0.160305i
$611$			3.73725e9i			0.662839i
$612$		−	6.85047e9i		−	1.20806i
$613$	7.52818e9			1.32001			0.660006	−	0.751260i	$-0.270554\pi$
$613$	7.52818e9			1.32001			0.660006	+	0.751260i	$0.270554\pi$
$614$			3.58769e9i			0.625497i
$615$			1.72897e10i			2.99726i
$616$			3.58200e9i			0.617437i
$617$	3.37298e9			0.578117			0.289058	−	0.957311i	$-0.406658\pi$
$617$	3.37298e9			0.578117			0.289058	+	0.957311i	$0.406658\pi$
$618$	−1.03276e9			−0.176010
$619$	−4.27712e9			−0.724827			−0.362413	−	0.932017i	$-0.618047\pi$
$619$	−4.27712e9			−0.724827			−0.362413	+	0.932017i	$0.618047\pi$
$620$	5.59590e9			0.942973
$621$		−	1.66280e10i		−	2.78624i
$622$	6.89342e9			1.14860
$623$			1.00474e10i			1.66474i
$624$			2.00015e9i			0.329546i
$625$	1.25157e10			2.05057
$626$	−8.02306e9			−1.30716
$627$			1.46267e9i			0.236979i
$628$	−4.57378e9			−0.736914
$629$	−1.50756e9	+	6.04393e9i	−0.241544	+	0.968372i
$630$	−2.38608e10			−3.80184
$631$		−	2.58317e9i		−	0.409308i	−0.978834	−	0.204654i	$-0.934393\pi$
$631$		−	2.58317e9i		−	0.409308i	0.978834	−	0.204654i	$-0.0656069\pi$
$632$	2.62167e9			0.413112
$633$	2.13189e10			3.34081
$634$		−	6.34613e9i		−	0.989000i
$635$		−	7.55728e9i		−	1.17127i
$636$	−6.55388e9			−1.01018
$637$			2.26508e9i			0.347212i
$638$	−5.72414e9			−0.872646
$639$	2.14859e10			3.25762
$640$	−1.06753e9			−0.160972
$641$	1.55018e9			0.232476			0.116238	−	0.993221i	$-0.462916\pi$
$641$	1.55018e9			0.232476			0.116238	+	0.993221i	$0.462916\pi$
$642$			1.54369e10i			2.30243i
$643$		−	3.45874e9i		−	0.513074i	−0.966534	−	0.256537i	$-0.917418\pi$
$643$		−	3.45874e9i		−	0.513074i	0.966534	−	0.256537i	$-0.0825815\pi$
$644$		−	4.38175e9i		−	0.646468i
$645$	1.42079e10			2.08483
$646$		−	4.32646e8i		−	0.0631421i
$647$		−	1.76132e8i		−	0.0255666i	−0.999918	−	0.0127833i	$-0.995931\pi$
$647$		−	1.76132e8i		−	0.0255666i	0.999918	−	0.0127833i	$-0.00406915\pi$
$648$			5.97463e9i			0.862578i
$649$		−	1.79873e9i		−	0.258292i
$650$	−8.17464e9			−1.16754
$651$		−	1.64421e10i		−	2.33573i
$652$			2.36585e9i			0.334288i
$653$		−	1.53927e9i		−	0.216331i	−0.994133	−	0.108166i	$-0.965502\pi$
$653$		−	1.53927e9i		−	0.216331i	0.994133	−	0.108166i	$-0.0344977\pi$
$654$	1.74787e10			2.44336
$655$	−1.68965e10			−2.34937
$656$	1.60844e9			0.222454
$657$	6.20419e9			0.853505
$658$			5.86080e9i			0.801985i
$659$	−3.11702e9			−0.424268			−0.212134	−	0.977241i	$-0.568041\pi$
$659$	−3.11702e9			−0.424268			−0.212134	+	0.977241i	$0.568041\pi$
$660$			1.78137e10i			2.41185i
$661$		−	7.82563e9i		−	1.05394i	−0.849885	−	0.526968i	$-0.823329\pi$
$661$		−	7.82563e9i		−	1.05394i	0.849885	−	0.526968i	$-0.176671\pi$
$662$	−2.14820e9			−0.287787
$663$	−9.87240e9			−1.31560
$664$		−	1.06613e8i		−	0.0141326i
$665$	−1.50695e9			−0.198711
$666$	3.15838e9	−	1.26622e10i	0.414289	−	1.66092i
$667$	7.00217e9			0.913676
$668$		−	1.66660e8i		−	0.0216329i
$669$	7.90618e7			0.0102088
$670$	−5.45697e9			−0.700954
$671$		−	1.39505e9i		−	0.178263i
$672$			3.13666e9i			0.398726i
$673$	7.99847e9			1.01147			0.505736	−	0.862688i	$-0.331221\pi$
$673$	7.99847e9			1.01147			0.505736	+	0.862688i	$0.331221\pi$
$674$			5.15237e8i			0.0648183i
$675$	−4.86479e10			−6.08837
$676$	−1.97605e9			−0.246027
$677$	−1.06929e10			−1.32445			−0.662227	−	0.749303i	$-0.730389\pi$
$677$	−1.06929e10			−1.32445			−0.662227	+	0.749303i	$0.730389\pi$
$678$	−5.44578e8			−0.0671051
$679$		−	1.72437e10i		−	2.11391i
$680$		−	5.26916e9i		−	0.642628i
$681$			3.98666e9i			0.483720i
$682$	−8.68680e9			−1.04861
$683$			8.53778e9i			1.02535i	0.858583	+	0.512675i	$0.171346\pi$
$683$			8.53778e9i			1.02535i	−0.858583	+	0.512675i	$0.828654\pi$
$684$			9.06407e8i			0.108300i
$685$		−	1.55475e10i		−	1.84818i
$686$		−	3.73911e9i		−	0.442216i
$687$	1.24954e10			1.47028
$688$		−	1.32174e9i		−	0.154734i
$689$			6.68399e9i			0.778518i
$690$		−	2.17910e10i		−	2.52525i
$691$	−5.88446e9			−0.678475			−0.339237	−	0.940701i	$-0.610169\pi$
$691$	−5.88446e9			−0.678475			−0.339237	+	0.940701i	$0.610169\pi$
$692$	−8.12581e8			−0.0932170
$693$	3.70404e10			4.22775
$694$	3.11748e9			0.354035
$695$		−	4.96552e9i		−	0.561071i
$696$	−5.01248e9			−0.563534
$697$			7.93896e9i			0.888074i
$698$			4.51730e9i			0.502788i
$699$	9.88220e7			0.0109442
$700$	−1.28196e10			−1.41263
$701$		−	6.66380e9i		−	0.730649i	−0.930880	−	0.365325i	$-0.880958\pi$
$701$		−	6.66380e9i		−	0.730649i	0.930880	−	0.365325i	$-0.119042\pi$
$702$	1.21394e10			1.32439
$703$	1.99470e8	−	7.99692e8i	0.0216537	−	0.0868118i
$704$	1.65718e9			0.179005
$705$			2.91465e10i			3.13274i
$706$	−9.48119e9			−1.01402
$707$	−1.17312e9			−0.124846
$708$		−	1.57510e9i		−	0.166799i
$709$			1.49120e10i			1.57136i	0.618635	+	0.785679i	$0.287686\pi$
$709$			1.49120e10i			1.57136i	−0.618635	+	0.785679i	$0.712314\pi$
$710$	1.65262e10			1.73289
$711$		−	2.71099e10i		−	2.82868i
$712$	4.64835e9			0.482635
$713$	1.06263e10			1.09791
$714$	−1.54820e10			−1.59178
$715$	1.81674e10			1.85875
$716$			7.84516e9i			0.798741i
$717$			2.07270e10i			2.10000i
$718$			5.12647e9i			0.516872i
$719$	−1.94302e10			−1.94952			−0.974758	−	0.223264i	$-0.928329\pi$
$719$	−1.94302e10			−1.94952			−0.974758	+	0.223264i	$0.928329\pi$
$720$			1.10390e10i			1.10222i
$721$			1.65173e9i			0.164122i
$722$		−	7.09373e9i		−	0.701446i
$723$			1.67352e10i			1.64682i
$724$	−906829.			−8.88056e−5
$725$		−	2.04861e10i		−	1.99653i
$726$		−	1.41687e10i		−	1.37421i
$727$			9.32238e9i			0.899822i	0.893073	+	0.449911i	$0.148544\pi$
$727$			9.32238e9i			0.899822i	−0.893073	+	0.449911i	$0.851456\pi$
$728$	3.19893e9			0.307287
$729$	1.09382e10			1.04568
$730$	4.77206e9			0.454021
$731$	6.52388e9			0.617725
$732$		−	1.22161e9i		−	0.115118i
$733$	−1.90022e10			−1.78213			−0.891067	−	0.453873i	$-0.850042\pi$
$733$	−1.90022e10			−1.78213			−0.891067	+	0.453873i	$0.850042\pi$
$734$		−	2.42044e9i		−	0.225922i
$735$			1.76651e10i			1.64101i
$736$	−2.02718e9			−0.187422
$737$	8.47112e9			0.779479
$738$		−	1.66324e10i		−	1.52320i
$739$	−6.18911e9			−0.564121			−0.282061	−	0.959397i	$-0.591018\pi$
$739$	−6.18911e9			−0.564121			−0.282061	+	0.959397i	$0.591018\pi$
$740$	2.42932e9	−	9.73936e9i	0.220381	−	0.883526i
$741$	1.30625e9			0.117940
$742$			1.04819e10i			0.941948i
$743$	−1.21001e10			−1.08225			−0.541127	−	0.840941i	$-0.682002\pi$
$743$	−1.21001e10			−1.08225			−0.541127	+	0.840941i	$0.682002\pi$
$744$	−7.60679e9			−0.677168
$745$		−	3.49704e9i		−	0.309852i
$746$			1.14963e10i			1.01385i
$747$	−1.10246e9			−0.0967698
$748$			8.17957e9i			0.714620i
$749$	2.46888e10			2.14691
$750$	−3.62348e10			−3.13626
$751$	5.46345e9			0.470682			0.235341	−	0.971913i	$-0.424379\pi$
$751$	5.46345e9			0.470682			0.235341	+	0.971913i	$0.424379\pi$
$752$	2.71145e9			0.232509
$753$		−	2.81091e10i		−	2.39919i
$754$			5.11199e9i			0.434300i
$755$		−	1.57313e10i		−	1.33030i
$756$	1.90371e10			1.60241
$757$		−	1.56833e10i		−	1.31402i	−0.753883	−	0.657009i	$-0.771821\pi$
$757$		−	1.56833e10i		−	1.31402i	0.753883	−	0.657009i	$-0.228179\pi$
$758$		−	3.41946e9i		−	0.285177i
$759$			3.38272e10i			2.80815i
$760$			6.97179e8i			0.0576098i
$761$	2.91220e9			0.239538			0.119769	−	0.992802i	$-0.461785\pi$
$761$	2.91220e9			0.239538			0.119769	+	0.992802i	$0.461785\pi$
$762$			1.02730e10i			0.841114i
$763$		−	2.79545e10i		−	2.27832i
$764$			4.04145e9i			0.327877i
$765$	−5.44868e10			−4.40024
$766$	−1.66757e9			−0.134055
$767$	−1.60637e9			−0.128547
$768$	1.45115e9			0.115597
$769$		−	2.25058e10i		−	1.78465i	−0.451398	−	0.892323i	$-0.649074\pi$
$769$		−	2.25058e10i		−	1.78465i	0.451398	−	0.892323i	$-0.350926\pi$
$770$	2.84903e10			2.24894
$771$		−	1.57454e10i		−	1.23726i
$772$		−	3.83029e9i		−	0.299621i
$773$	9.32957e9			0.726496			0.363248	−	0.931692i	$-0.381668\pi$
$773$	9.32957e9			0.726496			0.363248	+	0.931692i	$0.381668\pi$
$774$	−1.36677e10			−1.05951
$775$		−	3.10891e10i		−	2.39912i
$776$	−7.97767e9			−0.612858
$777$	−2.86165e10	−	7.13790e9i	−2.18848	−	0.545880i
$778$	−9.61037e9			−0.731664
$779$		−	1.05043e9i		−	0.0796133i
$780$	1.59087e10			1.20034
$781$	−2.56545e10			−1.92701
$782$		−	1.00058e10i		−	0.748220i
$783$			3.04218e10i			2.26474i
$784$	1.64336e9			0.121794
$785$			3.63787e10i			2.68413i
$786$	2.29682e10			1.68713
$787$	1.54820e10			1.13218			0.566090	−	0.824344i	$-0.308456\pi$
$787$	1.54820e10			1.13218			0.566090	+	0.824344i	$0.308456\pi$
$788$	−4.42877e9			−0.322434
$789$	−1.92759e10			−1.39716
$790$		−	2.08520e10i		−	1.50471i
$791$			8.70967e8i			0.0625726i
$792$		−	1.71364e10i		−	1.22570i
$793$	−1.24586e9			−0.0887184
$794$		−	1.07126e10i		−	0.759490i
$795$			5.21278e10i			3.67947i
$796$		−	1.04782e10i		−	0.736364i
$797$		−	1.75062e10i		−	1.22487i	−0.790523	−	0.612433i	$-0.790191\pi$
$797$		−	1.75062e10i		−	1.22487i	0.790523	−	0.612433i	$-0.209809\pi$
$798$	2.04847e9			0.142699
$799$			1.33833e10i			0.928215i
$800$			5.93087e9i			0.409547i
$801$		−	4.80672e10i		−	3.30473i
$802$	1.35976e10			0.930791
$803$	−7.40790e9			−0.504883
$804$	7.41793e9			0.503369
$805$	−3.48513e10			−2.35469
$806$			7.75781e9i			0.521875i
$807$	−3.63818e10			−2.43684
$808$			5.42734e8i			0.0361949i
$809$		−	2.28769e9i		−	0.151907i	−0.997111	−	0.0759533i	$-0.975800\pi$
$809$		−	2.28769e9i		−	0.151907i	0.997111	−	0.0759533i	$-0.0242000\pi$
$810$	4.75206e10			3.14184
$811$	6.64278e9			0.437298			0.218649	−	0.975804i	$-0.429835\pi$
$811$	6.64278e9			0.437298			0.218649	+	0.975804i	$0.429835\pi$
$812$			8.01667e9i			0.525470i
$813$	3.78134e10			2.46791
$814$	−3.77115e9	+	1.51189e10i	−0.245069	+	0.982505i
$815$	1.88174e10			1.21761
$816$			7.16263e9i			0.461484i
$817$	−8.63196e8			−0.0553773
$818$	−3.14104e9			−0.200649
$819$		−	3.30792e10i		−	2.10407i
$820$		−	1.27931e10i		−	0.810264i
$821$	−1.98367e10			−1.25103			−0.625515	−	0.780212i	$-0.715111\pi$
$821$	−1.98367e10			−1.25103			−0.625515	+	0.780212i	$0.715111\pi$
$822$			2.11345e10i			1.32721i
$823$	2.91932e10			1.82550			0.912750	−	0.408519i	$-0.133955\pi$
$823$	2.91932e10			1.82550			0.912750	+	0.408519i	$0.133955\pi$
$824$	7.64161e8			0.0475817
$825$	9.89674e10			6.13625
$826$	−2.51913e9			−0.155532
$827$		−	2.37846e9i		−	0.146227i	−0.997324	−	0.0731134i	$-0.976706\pi$
$827$		−	2.37846e9i		−	0.146227i	0.997324	−	0.0731134i	$-0.0232935\pi$
$828$			2.09625e10i			1.28333i
$829$		−	3.51164e9i		−	0.214077i	−0.994255	−	0.107038i	$-0.965863\pi$
$829$		−	3.51164e9i		−	0.214077i	0.994255	−	0.107038i	$-0.0341367\pi$
$830$	−8.47974e8			−0.0514766
$831$			5.14075e9i			0.310758i
$832$		−	1.47996e9i		−	0.0890878i
$833$			8.11134e9i			0.486223i
$834$			6.74988e9i			0.402916i
$835$	−1.32557e9			−0.0787954
$836$		−	1.08227e9i		−	0.0640637i
$837$			4.61673e10i			2.72142i
$838$			9.10123e9i			0.534252i
$839$	1.62760e10			0.951437			0.475719	−	0.879598i	$-0.342188\pi$
$839$	1.62760e10			0.951437			0.475719	+	0.879598i	$0.342188\pi$
$840$	2.49481e10			1.45231
$841$	4.43899e9			0.257334
$842$	−7.50514e8			−0.0433278
$843$			3.17209e9i			0.182368i
$844$	−1.57744e10			−0.903137
$845$			1.57169e10i			0.896127i
$846$		−	2.80384e10i		−	1.59205i
$847$	−2.26607e10			−1.28139
$848$	4.84938e9			0.273087
$849$		−	1.70183e10i		−	0.954417i
$850$	−2.92738e10			−1.63498
$851$	4.61314e9	−	1.84945e10i	0.256592	−	1.02870i
$852$	−2.24650e10			−1.24442
$853$		−	9.46428e9i		−	0.522115i	−0.965323	−	0.261057i	$-0.915929\pi$
$853$		−	9.46428e9i		−	0.522115i	0.965323	−	0.261057i	$-0.0840712\pi$
$854$	−1.95377e9			−0.107343
$855$	7.20932e9			0.394469
$856$		−	1.14221e10i		−	0.622426i
$857$		−	3.12460e10i		−	1.69575i	−0.530198	−	0.847874i	$-0.677882\pi$
$857$		−	3.12460e10i		−	1.69575i	0.530198	−	0.847874i	$-0.322118\pi$
$858$	−2.46958e10			−1.33481
$859$			8.13485e9i			0.437899i	0.975736	+	0.218949i	$0.0702629\pi$
$859$			8.13485e9i			0.437899i	−0.975736	+	0.218949i	$0.929737\pi$
$860$	−1.05128e10			−0.563602
$861$	−3.75890e10			−2.00701
$862$	1.77746e9			0.0945203
$863$	6.77890e8			0.0359022			0.0179511	−	0.999839i	$-0.494286\pi$
$863$	6.77890e8			0.0359022			0.0179511	+	0.999839i	$0.494286\pi$
$864$		−	8.80736e9i		−	0.464566i
$865$			6.46305e9i			0.339533i
$866$			8.38466e9i			0.438705i
$867$	1.38861e8			0.00723627
$868$			1.21659e10i			0.631429i
$869$			3.23697e10i			1.67328i
$870$			3.98679e10i			2.05261i
$871$		−	7.56520e9i		−	0.387933i
$872$	−1.29329e10			−0.660524
$873$			8.24948e10i			4.19640i
$874$			1.32390e9i			0.0670759i
$875$			5.79519e10i			2.92442i
$876$	−6.48690e9			−0.326041
$877$	1.32215e9			0.0661883			0.0330941	−	0.999452i	$-0.489464\pi$
$877$	1.32215e9			0.0661883			0.0330941	+	0.999452i	$0.489464\pi$
$878$	−7.21767e9			−0.359887
$879$	2.03567e10			1.01099
$880$		−	1.31808e10i		−	0.652007i
$881$	−1.58731e10			−0.782070			−0.391035	−	0.920376i	$-0.627883\pi$
$881$	−1.58731e10			−0.782070			−0.391035	+	0.920376i	$0.627883\pi$
$882$		−	1.69935e10i		−	0.833956i
$883$			3.13504e10i			1.53243i	0.642585	+	0.766214i	$0.277862\pi$
$883$			3.13504e10i			1.53243i	−0.642585	+	0.766214i	$0.722138\pi$
$884$	7.30483e9			0.355654
$885$	−1.25279e10			−0.607545
$886$		−	1.50675e10i		−	0.727817i
$887$	−1.31138e10			−0.630951			−0.315476	−	0.948934i	$-0.602164\pi$
$887$	−1.31138e10			−0.630951			−0.315476	+	0.948934i	$0.602164\pi$
$888$	−3.30230e9	+	1.32392e10i	−0.158260	+	0.634478i
$889$	1.64300e10			0.784301
$890$		−	3.69718e10i		−	1.75794i
$891$	−7.37686e10			−3.49381
$892$	−5.84997e7			−0.00275980
$893$		−	1.77078e9i		−	0.0832119i
$894$			4.75370e9i			0.222511i
$895$	6.23983e10			2.90932
$896$		−	2.32089e9i		−	0.107789i
$897$	3.02097e10			1.39757
$898$	−7.08997e9			−0.326721
$899$	−1.94414e10			−0.892421
$900$	6.13294e10			2.80427
$901$			2.39357e10i			1.09021i
$902$			1.98593e10i			0.901035i
$903$			3.08890e10i			1.39603i
$904$	4.02946e8			0.0181408
$905$			7.21268e6i			0.000323465i
$906$			2.13843e10i			0.955315i
$907$		−	2.96277e10i		−	1.31848i	−0.751934	−	0.659238i	$-0.770879\pi$
$907$		−	2.96277e10i		−	1.31848i	0.751934	−	0.659238i	$-0.229121\pi$
$908$		−	2.94983e9i		−	0.130766i
$909$	5.61225e9			0.247836
$910$		−	2.54434e10i		−	1.11926i
$911$			4.62330e7i			0.00202599i	0.999999	+	0.00101300i	$0.000322446\pi$
$911$			4.62330e7i			0.00202599i	−0.999999	+	0.00101300i	$0.999678\pi$
$912$		−	9.47711e8i		−	0.0413708i
$913$	1.31635e9			0.0572433
$914$	2.99298e10			1.29656
$915$	−9.71636e9			−0.419305
$916$	−9.24563e9			−0.397468
$917$		−	3.67341e10i		−	1.57317i
$918$	4.34716e10			1.85463
$919$		−	1.43473e10i		−	0.609771i	−0.952389	−	0.304885i	$-0.901382\pi$
$919$		−	1.43473e10i		−	0.609771i	0.952389	−	0.304885i	$-0.0986182\pi$
$920$			1.61237e10i			0.682664i
$921$	−3.87898e10			−1.63610
$922$	−1.80808e10			−0.759729
$923$			2.29109e10i			0.959041i
$924$	−3.87282e10			−1.61501
$925$	−5.41088e10	−	1.34965e10i	−2.24787	−	0.560694i
$926$	2.37568e10			0.983218
$927$		−	7.90197e9i		−	0.325804i
$928$	3.70885e9			0.152343
$929$	−4.56460e10			−1.86788			−0.933938	−	0.357435i	$-0.883651\pi$
$929$	−4.56460e10			−1.86788			−0.933938	+	0.357435i	$0.883651\pi$
$930$			6.05024e10i			2.46651i
$931$		−	1.07324e9i		−	0.0435885i
$932$	−7.31208e7			−0.00295859
$933$			7.45310e10i			3.00436i
$934$	2.22280e8			0.00892659
$935$	6.50582e10			2.60292
$936$	−1.53038e10			−0.610007
$937$	1.04750e10			0.415973			0.207987	−	0.978132i	$-0.433309\pi$
$937$	1.04750e10			0.415973			0.207987	+	0.978132i	$0.433309\pi$
$938$		−	1.18638e10i		−	0.469369i
$939$		−	8.67446e10i		−	3.41911i
$940$		−	2.15662e10i		−	0.846888i
$941$	−3.11657e10			−1.21931			−0.609653	−	0.792669i	$-0.708691\pi$
$941$	−3.11657e10			−1.21931			−0.609653	+	0.792669i	$0.708691\pi$
$942$		−	4.94514e10i		−	1.92753i
$943$		−	2.42933e10i		−	0.943400i
$944$			1.16546e9i			0.0450914i
$945$		−	1.51416e11i		−	5.83660i
$946$	1.63195e10			0.626741
$947$		−	3.55045e10i		−	1.35850i	−0.733909	−	0.679248i	$-0.762306\pi$
$947$		−	3.55045e10i		−	1.35850i	0.733909	−	0.679248i	$-0.237694\pi$
$948$			2.83452e10i			1.08056i
$949$			6.61568e9i			0.251271i
$950$	3.87331e9			0.146571
$951$	6.86138e10			2.58690
$952$	1.14555e10			0.430314
$953$	2.20749e10			0.826177			0.413088	−	0.910691i	$-0.364450\pi$
$953$	2.20749e10			0.826177			0.413088	+	0.910691i	$0.364450\pi$
$954$		−	5.01460e10i		−	1.86989i
$955$	3.21446e10			1.19425
$956$		−	1.53364e10i		−	0.567702i
$957$		−	6.18889e10i		−	2.28255i
$958$	−3.22137e9			−0.118375
$959$	3.38013e10			1.23757
$960$		−	1.15421e10i		−	0.421051i
$961$	−1.99118e9			−0.0723733
$962$	1.35020e10	+	3.36786e9i	0.488975	+	0.121967i
$963$	−1.18113e11			−4.26191
$964$		−	1.23828e10i		−	0.445193i
$965$	−3.04652e10			−1.09133
$966$	4.73751e10			1.69095
$967$			5.07804e10i			1.80594i	0.429704	+	0.902970i	$0.358618\pi$
$967$			5.07804e10i			1.80594i	−0.429704	+	0.902970i	$0.641382\pi$
$968$			1.04838e10i			0.371496i
$969$	4.67774e9			0.165159
$970$			6.34523e10i			2.23227i
$971$	1.46222e10			0.512561			0.256281	−	0.966602i	$-0.417503\pi$
$971$	1.46222e10			0.512561			0.256281	+	0.966602i	$0.417503\pi$
$972$	−2.69767e10			−0.942230
$973$	1.07954e10			0.375702
$974$	3.76672e10			1.30619
$975$		−	8.83835e10i		−	3.05390i
$976$			9.03899e8i			0.0311204i
$977$		−	7.79041e9i		−	0.267257i	−0.991031	−	0.133629i	$-0.957337\pi$
$977$		−	7.79041e9i		−	0.267257i	0.991031	−	0.133629i	$-0.0426629\pi$
$978$	−2.55794e10			−0.874388
$979$			5.73931e10i			1.95488i
$980$		−	1.30709e10i		−	0.443622i
$981$			1.33736e11i			4.52278i
$982$		−	3.65190e10i		−	1.23063i
$983$	5.16803e10			1.73535			0.867677	−	0.497128i	$-0.165612\pi$
$983$	5.16803e10			1.73535			0.867677	+	0.497128i	$0.165612\pi$
$984$			1.73903e10i			0.581867i
$985$			3.52252e10i			1.17443i
$986$			1.83063e10i			0.608178i
$987$	−6.33665e10			−2.09773
$988$	−9.66525e8			−0.0318833
$989$	−1.99632e10			−0.656209
$990$	−1.36299e11			−4.46446
$991$			4.03250e10i			1.31618i	0.752938	+	0.658091i	$0.228636\pi$
$991$			4.03250e10i			1.31618i	−0.752938	+	0.658091i	$0.771364\pi$
$992$	5.62845e9			0.183062
$993$		−	2.32262e10i		−	0.752757i
$994$			3.59292e10i			1.16037i
$995$	−8.33411e10			−2.68212
$996$	1.15269e9			0.0369663
$997$			4.85850e9i			0.155263i	0.996982	+	0.0776317i	$0.0247358\pi$
$997$			4.85850e9i			0.155263i	−0.996982	+	0.0776317i	$0.975264\pi$
$998$	−1.88753e10			−0.601086
$999$	8.03517e10	+	2.00424e10i	2.54986	+	0.636019i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.8.b.a.73.12	✓	24
37.36	even	2	inner	74.8.b.a.73.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.b.a.73.12	✓	24	1.1	even	1	trivial
74.8.b.a.73.24	yes	24	37.36	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	74.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} +86.4953 q^{3} -64.0000 q^{4} +509.039i q^{5} -691.963i q^{6} -1106.69 q^{7} +512.000i q^{8} +5294.44 q^{9} +4072.31 q^{10} -6321.65 q^{11} -5535.70 q^{12} +5645.60i q^{13} +8853.49i q^{14} +44029.5i q^{15} +4096.00 q^{16} +20217.2i q^{17} -42355.5i q^{18} -2675.00i q^{19} -32578.5i q^{20} -95723.2 q^{21} +50573.2i q^{22} -61864.7i q^{23} +44285.6i q^{24} -180996. q^{25} +45164.8 q^{26} +268779. q^{27} +70827.9 q^{28} +113185. i q^{29} +352236. q^{30} +171767. i q^{31} -32768.0i q^{32} -546794. q^{33} +161737. q^{34} -563347. i q^{35} -338844. q^{36} +(298951. + 74568.2i) q^{37} -21400.0 q^{38} +488318. i q^{39} -260628. q^{40} +392684. q^{41} +765785. i q^{42} -322691. i q^{43} +404586. q^{44} +2.69508e6i q^{45} -494918. q^{46} +661976. q^{47} +354285. q^{48} +401211. q^{49} +1.44797e6i q^{50} +1.74869e6i q^{51} -361318. i q^{52} +1.18393e6 q^{53} -2.15023e6i q^{54} -3.21797e6i q^{55} -566623. i q^{56} -231375. i q^{57} +905482. q^{58} +284535. i q^{59} -2.81789e6i q^{60} +220678. i q^{61} +1.37413e6 q^{62} -5.85928e6 q^{63} -262144. q^{64} -2.87383e6 q^{65} +4.37435e6i q^{66} -1.34002e6 q^{67} -1.29390e6i q^{68} -5.35101e6i q^{69} -4.50677e6 q^{70} +4.05819e6 q^{71} +2.71075e6i q^{72} +1.17183e6 q^{73} +(596545. - 2.39160e6i) q^{74} -1.56553e7 q^{75} +171200. i q^{76} +6.99609e6 q^{77} +3.90654e6 q^{78} -5.12044e6i q^{79} +2.08502e6i q^{80} +1.16692e7 q^{81} -3.14148e6i q^{82} -208229. q^{83} +6.12628e6 q^{84} -1.02913e7 q^{85} -2.58152e6 q^{86} +9.78999e6i q^{87} -3.23669e6i q^{88} -9.07881e6i q^{89} +2.15606e7 q^{90} -6.24791e6i q^{91} +3.95934e6i q^{92} +1.48570e7i q^{93} -5.29581e6i q^{94} +1.36168e6 q^{95} -2.83428e6i q^{96} +1.55814e7i q^{97} -3.20969e6i q^{98} -3.34696e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 106 q^{3} - 1536 q^{4} + 104 q^{7} + 17554 q^{9} + 1136 q^{10} + 366 q^{11} + 6784 q^{12} + 98304 q^{16} - 239820 q^{21} - 675570 q^{25} + 97008 q^{26} + 338780 q^{27} - 6656 q^{28} + 350400 q^{30}+ \cdots - 53279900 q^{99}+O(q^{100}) \) 24 * q - 106 * q^3 - 1536 * q^4 + 104 * q^7 + 17554 * q^9 + 1136 * q^10 + 366 * q^11 + 6784 * q^12 + 98304 * q^16 - 239820 * q^21 - 675570 * q^25 + 97008 * q^26 + 338780 * q^27 - 6656 * q^28 + 350400 * q^30 - 763792 * q^33 + 713632 * q^34 - 1123456 * q^36 + 41652 * q^37 - 30816 * q^38 - 72704 * q^40 + 1729722 * q^41 - 23424 * q^44 - 488496 * q^46 + 114756 * q^47 - 434176 * q^48 + 4003056 * q^49 - 1115964 * q^53 - 2075632 * q^58 + 3248208 * q^62 - 2350900 * q^63 - 6291456 * q^64 - 5246556 * q^65 - 2717994 * q^67 - 5649440 * q^70 + 10643280 * q^71 - 10450370 * q^73 - 1064064 * q^74 - 17737980 * q^75 + 11665236 * q^77 + 8431856 * q^78 + 47300176 * q^81 + 555912 * q^83 + 15348480 * q^84 - 4853096 * q^85 - 12070464 * q^86 + 32064160 * q^90 - 58516956 * q^95 - 53279900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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