LMFDB - Embedded newform 756.2.b.f.55.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(55,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("756.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 4x^{10} - 4x^{8} - 16x^{6} - 16x^{4} + 256 $ x^16 - x^12 - 4x^10 - 4x^8 - 16x^6 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.12
Root			$-0.748450 + 1.19993i$ of defining polynomial
Character	$\chi$	$=$	756.55
Dual form			756.2.b.f.55.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.748450 + 1.19993i) q^{2} +(-0.879646 + 1.79617i) q^{4} -3.90968i q^{5} +(-1.12924 + 2.39266i) q^{7} +(-2.81364 + 0.288831i) q^{8} +O(q^{10})$	\(q+(0.748450 + 1.19993i) q^{2} +(-0.879646 + 1.79617i) q^{4} -3.90968i q^{5} +(-1.12924 + 2.39266i) q^{7} +(-2.81364 + 0.288831i) q^{8} +(4.69133 - 2.92620i) q^{10} -5.01901i q^{11} -5.59667i q^{13} +(-3.71619 + 0.435784i) q^{14} +(-2.45244 - 3.15999i) q^{16} +2.41029i q^{17} -3.11590 q^{19} +(7.02245 + 3.43914i) q^{20} +(6.02245 - 3.75648i) q^{22} -3.45254i q^{23} -10.2856 q^{25} +(6.71560 - 4.18883i) q^{26} +(-3.30429 - 4.13300i) q^{28} +4.76986 q^{29} -0.482412 q^{31} +(1.95622 - 5.30784i) q^{32} +(-2.89217 + 1.80398i) q^{34} +(9.35454 + 4.41496i) q^{35} +5.90489 q^{37} +(-2.33209 - 3.73885i) q^{38} +(1.12924 + 11.0004i) q^{40} -1.51033i q^{41} -1.77561i q^{43} +(9.01499 + 4.41496i) q^{44} +(4.14280 - 2.58406i) q^{46} +2.38630 q^{47} +(-4.44965 - 5.40376i) q^{49} +(-7.69825 - 12.3420i) q^{50} +(10.0526 + 4.92309i) q^{52} -5.62728 q^{53} -19.6227 q^{55} +(2.48619 - 7.05825i) q^{56} +(3.57000 + 5.72348i) q^{58} +6.38071 q^{59} -6.94384i q^{61} +(-0.361061 - 0.578859i) q^{62} +(7.83315 - 1.62533i) q^{64} -21.8812 q^{65} +3.92062i q^{67} +(-4.32929 - 2.12021i) q^{68} +(1.70378 + 14.5291i) q^{70} +8.25225i q^{71} +3.59159i q^{73} +(4.41951 + 7.08543i) q^{74} +(2.74089 - 5.59667i) q^{76} +(12.0088 + 5.66766i) q^{77} +9.32969i q^{79} +(-12.3545 + 9.58827i) q^{80} +(1.81228 - 1.13040i) q^{82} -5.76701 q^{83} +9.42347 q^{85} +(2.13060 - 1.32895i) q^{86} +(1.44965 + 14.1217i) q^{88} -17.6553i q^{89} +(13.3909 + 6.31997i) q^{91} +(6.20135 + 3.03702i) q^{92} +(1.78603 + 2.86339i) q^{94} +12.1822i q^{95} +15.5685i q^{97} +(3.15378 - 9.38369i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + q^{14} + 4 q^{16} + 26 q^{20} + 10 q^{22} - 20 q^{25} + 6 q^{26} - 11 q^{28} + 6 q^{35} + 8 q^{37} + 20 q^{38} - 6 q^{46} + 8 q^{47} - 14 q^{49} - 21 q^{56} + 14 q^{58} + 44 q^{59} - 48 q^{62} + 24 q^{64} + 2 q^{68} - 27 q^{70} - 54 q^{80} - 4 q^{83} + 8 q^{85} - 34 q^{88} + 47 q^{98}+O(q^{100}) $ 16 * q + q^14 + 4 * q^16 + 26 * q^20 + 10 * q^22 - 20 * q^25 + 6 * q^26 - 11 * q^28 + 6 * q^35 + 8 * q^37 + 20 * q^38 - 6 * q^46 + 8 * q^47 - 14 * q^49 - 21 * q^56 + 14 * q^58 + 44 * q^59 - 48 * q^62 + 24 * q^64 + 2 * q^68 - 27 * q^70 - 54 * q^80 - 4 * q^83 + 8 * q^85 - 34 * q^88 + 47 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.748450	+	1.19993i	0.529234	+	0.848476i
$2$	0.748450	+	1.19993i	0.529234	+	0.848476i
$3$		0			0
$3$		0			0
$4$	−0.879646	+	1.79617i	−0.439823	+	0.898084i
$5$		−	3.90968i		−	1.74846i	−0.485510	−	0.874231i	$-0.661366\pi$
$5$		−	3.90968i		−	1.74846i	0.485510	−	0.874231i	$-0.338634\pi$
$6$		0			0
$7$	−1.12924	+	2.39266i	−0.426812	+	0.904341i
$7$	−1.12924	+	2.39266i	−0.426812	+	0.904341i
$8$	−2.81364	+	0.288831i	−0.994772	+	0.102117i
$9$		0			0
$10$	4.69133	−	2.92620i	1.48353	−	0.925345i
$11$		−	5.01901i		−	1.51329i	−0.653826	−	0.756645i	$-0.726837\pi$
$11$		−	5.01901i		−	1.51329i	0.653826	−	0.756645i	$-0.273163\pi$
$12$		0			0
$13$		−	5.59667i		−	1.55224i	−0.630586	−	0.776119i	$-0.717186\pi$
$13$		−	5.59667i		−	1.55224i	0.630586	−	0.776119i	$-0.282814\pi$
$14$	−3.71619	+	0.435784i	−0.993194	+	0.116468i
$15$		0			0
$16$	−2.45244	−	3.15999i	−0.613111	−	0.789997i
$17$			2.41029i			0.584582i	0.956329	+	0.292291i	$0.0944176\pi$
$17$			2.41029i			0.584582i	−0.956329	+	0.292291i	$0.905582\pi$
$18$		0			0
$19$	−3.11590			−0.714836			−0.357418	−	0.933945i	$-0.616343\pi$
$19$	−3.11590			−0.714836			−0.357418	+	0.933945i	$0.616343\pi$
$20$	7.02245	+	3.43914i	1.57027	+	0.769014i
$21$		0			0
$22$	6.02245	−	3.75648i	1.28399	−	0.800884i
$23$		−	3.45254i		−	0.719905i	−0.932971	−	0.359953i	$-0.882793\pi$
$23$		−	3.45254i		−	0.719905i	0.932971	−	0.359953i	$-0.117207\pi$
$24$		0			0
$25$	−10.2856			−2.05712
$26$	6.71560	−	4.18883i	1.31704	−	0.821497i
$27$		0			0
$28$	−3.30429	−	4.13300i	−0.624453	−	0.781063i
$29$	4.76986			0.885741			0.442870	−	0.896586i	$-0.353960\pi$
$29$	4.76986			0.885741			0.442870	+	0.896586i	$0.353960\pi$
$30$		0			0
$31$	−0.482412			−0.0866437			−0.0433219	−	0.999061i	$-0.513794\pi$
$31$	−0.482412			−0.0866437			−0.0433219	+	0.999061i	$0.513794\pi$
$32$	1.95622	−	5.30784i	0.345814	−	0.938303i
$33$		0			0
$34$	−2.89217	+	1.80398i	−0.496004	+	0.309381i
$35$	9.35454	+	4.41496i	1.58121	+	0.746264i
$36$		0			0
$37$	5.90489			0.970758			0.485379	−	0.874304i	$-0.338682\pi$
$37$	5.90489			0.970758			0.485379	+	0.874304i	$0.338682\pi$
$38$	−2.33209	−	3.73885i	−0.378315	−	0.606521i
$39$		0			0
$40$	1.12924	+	11.0004i	0.178548	+	1.73932i
$41$		−	1.51033i		−	0.235873i	−0.993021	−	0.117937i	$-0.962372\pi$
$41$		−	1.51033i		−	0.235873i	0.993021	−	0.117937i	$-0.0376280\pi$
$42$		0			0
$43$		−	1.77561i		−	0.270778i	−0.990793	−	0.135389i	$-0.956772\pi$
$43$		−	1.77561i		−	0.270778i	0.990793	−	0.135389i	$-0.0432284\pi$
$44$	9.01499	+	4.41496i	1.35906	+	0.665580i
$45$		0			0
$46$	4.14280	−	2.58406i	0.610822	−	0.380998i
$47$	2.38630			0.348078			0.174039	−	0.984739i	$-0.444318\pi$
$47$	2.38630			0.348078			0.174039	+	0.984739i	$0.444318\pi$
$48$		0			0
$49$	−4.44965	−	5.40376i	−0.635664	−	0.771966i
$50$	−7.69825	−	12.3420i	−1.08870	−	1.74542i
$51$		0			0
$52$	10.0526	+	4.92309i	1.39404	+	0.682710i
$53$	−5.62728			−0.772967			−0.386483	−	0.922296i	$-0.626310\pi$
$53$	−5.62728			−0.772967			−0.386483	+	0.922296i	$0.626310\pi$
$54$		0			0
$55$	−19.6227			−2.64593
$56$	2.48619	−	7.05825i	0.332232	−	0.943198i
$57$		0			0
$58$	3.57000	+	5.72348i	0.468764	+	0.751530i
$59$	6.38071			0.830697			0.415349	−	0.909662i	$-0.363660\pi$
$59$	6.38071			0.830697			0.415349	+	0.909662i	$0.363660\pi$
$60$		0			0
$61$		−	6.94384i		−	0.889067i	−0.895762	−	0.444534i	$-0.853369\pi$
$61$		−	6.94384i		−	0.889067i	0.895762	−	0.444534i	$-0.146631\pi$
$62$	−0.361061	−	0.578859i	−0.0458548	−	0.0735151i
$63$		0			0
$64$	7.83315	−	1.62533i	0.979144	−	0.203167i
$65$	−21.8812			−2.71403
$66$		0			0
$67$			3.92062i			0.478980i	0.970899	+	0.239490i	$0.0769802\pi$
$67$			3.92062i			0.478980i	−0.970899	+	0.239490i	$0.923020\pi$
$68$	−4.32929	−	2.12021i	−0.525004	−	0.257113i
$69$		0			0
$70$	1.70378	+	14.5291i	0.203640	+	1.73656i
$71$			8.25225i			0.979362i	0.871902	+	0.489681i	$0.162887\pi$
$71$			8.25225i			0.979362i	−0.871902	+	0.489681i	$0.837113\pi$
$72$		0			0
$73$			3.59159i			0.420364i	0.977662	+	0.210182i	$0.0674057\pi$
$73$			3.59159i			0.420364i	−0.977662	+	0.210182i	$0.932594\pi$
$74$	4.41951	+	7.08543i	0.513758	+	0.823665i
$75$		0			0
$76$	2.74089	−	5.59667i	0.314401	−	0.641983i
$77$	12.0088	+	5.66766i	1.36853	+	0.645889i
$78$		0			0
$79$			9.32969i			1.04967i	0.851203	+	0.524836i	$0.175873\pi$
$79$			9.32969i			1.04967i	−0.851203	+	0.524836i	$0.824127\pi$
$80$	−12.3545	+	9.58827i	−1.38128	+	1.07200i
$81$		0			0
$82$	1.81228	−	1.13040i	0.200133	−	0.124832i
$83$	−5.76701			−0.633012			−0.316506	−	0.948591i	$-0.602510\pi$
$83$	−5.76701			−0.633012			−0.316506	+	0.948591i	$0.602510\pi$
$84$		0			0
$85$	9.42347			1.02212
$86$	2.13060	−	1.32895i	0.229748	−	0.143305i
$87$		0			0
$88$	1.44965	+	14.1217i	0.154533	+	1.50538i
$89$		−	17.6553i		−	1.87146i	−0.352722	−	0.935728i	$-0.614744\pi$
$89$		−	17.6553i		−	1.87146i	0.352722	−	0.935728i	$-0.385256\pi$
$90$		0			0
$91$	13.3909	+	6.31997i	1.40375	+	0.662513i
$92$	6.20135	+	3.03702i	0.646536	+	0.316631i
$93$		0			0
$94$	1.78603	+	2.86339i	0.184215	+	0.295336i
$95$			12.1822i			1.24986i
$96$		0			0
$97$			15.5685i			1.58074i	0.612630	+	0.790370i	$0.290112\pi$
$97$			15.5685i			1.58074i	−0.612630	+	0.790370i	$0.709888\pi$
$98$	3.15378	−	9.38369i	0.318580	−	0.947896i
$99$		0			0
$100$	9.04769	−	18.4747i	0.904769	−	1.84747i
$101$			5.42001i			0.539311i	0.962957	+	0.269655i	$0.0869098\pi$
$101$			5.42001i			0.539311i	−0.962957	+	0.269655i	$0.913090\pi$
$102$		0			0
$103$	−4.89196			−0.482019			−0.241009	−	0.970523i	$-0.577478\pi$
$103$	−4.89196			−0.482019			−0.241009	+	0.970523i	$0.577478\pi$
$104$	1.61649	+	15.7470i	0.158510	+	1.54412i
$105$		0			0
$106$	−4.21174	−	6.75232i	−0.409080	−	0.655844i
$107$		−	12.4743i		−	1.20594i	−0.797766	−	0.602968i	$-0.793985\pi$
$107$		−	12.4743i		−	1.20594i	0.797766	−	0.602968i	$-0.206015\pi$
$108$		0			0
$109$	5.90489			0.565586			0.282793	−	0.959181i	$-0.408739\pi$
$109$	5.90489			0.565586			0.282793	+	0.959181i	$0.408739\pi$
$110$	−14.6866	−	23.5458i	−1.40031	−	2.24501i
$111$		0			0
$112$	10.3302	−	2.29949i	0.976109	−	0.217282i
$113$	15.3353			1.44262			0.721311	−	0.692611i	$-0.243540\pi$
$113$	15.3353			1.44262			0.721311	+	0.692611i	$0.243540\pi$
$114$		0			0
$115$	−13.4983			−1.25873
$116$	−4.19579	+	8.56748i	−0.389569	+	0.795470i
$117$		0			0
$118$	4.77564	+	7.65638i	0.439633	+	0.704827i
$119$	−5.76701	−	2.72179i	−0.528661	−	0.249506i
$120$		0			0
$121$	−14.1905			−1.29004
$122$	8.33209	−	5.19711i	0.754352	−	0.470524i
$123$		0			0
$124$	0.424352	−	0.866493i	0.0381079	−	0.0778134i
$125$			20.6650i			1.84833i
$126$		0			0
$127$			0.910906i			0.0808298i	0.999183	+	0.0404149i	$0.0128680\pi$
$127$			0.910906i			0.0808298i	−0.999183	+	0.0404149i	$0.987132\pi$
$128$	7.81300	+	8.18273i	0.690578	+	0.723258i
$129$		0			0
$130$	−16.3770	−	26.2558i	−1.43636	−	2.30279i
$131$	−15.9421			−1.39286			−0.696432	−	0.717623i	$-0.745230\pi$
$131$	−15.9421			−1.39286			−0.696432	+	0.717623i	$0.745230\pi$
$132$		0			0
$133$	3.51859	−	7.45528i	0.305100	−	0.646455i
$134$	−4.70445	+	2.93439i	−0.406403	+	0.253492i
$135$		0			0
$136$	−0.696168	−	6.78170i	−0.0596959	−	0.581526i
$137$	0.627922			0.0536470			0.0268235	−	0.999640i	$-0.491461\pi$
$137$	0.627922			0.0536470			0.0268235	+	0.999640i	$0.491461\pi$
$138$		0			0
$139$	−17.3643			−1.47282			−0.736409	−	0.676537i	$-0.763480\pi$
$139$	−17.3643			−1.47282			−0.736409	+	0.676537i	$0.763480\pi$
$140$	−16.1587	+	12.9187i	−1.36566	+	1.09183i
$141$		0			0
$142$	−9.90209	+	6.17639i	−0.830965	+	0.518311i
$143$	−28.0898			−2.34899
$144$		0			0
$145$		−	18.6486i		−	1.54868i
$146$	−4.30965	+	2.68813i	−0.356669	+	0.222471i
$147$		0			0
$148$	−5.19421	+	10.6062i	−0.426962	+	0.871822i
$149$	0.726607			0.0595259			0.0297630	−	0.999557i	$-0.490525\pi$
$149$	0.726607			0.0595259			0.0297630	+	0.999557i	$0.490525\pi$
$150$		0			0
$151$			6.27377i			0.510552i	0.966868	+	0.255276i	$0.0821663\pi$
$151$			6.27377i			0.510552i	−0.966868	+	0.255276i	$0.917834\pi$
$152$	8.76701	−	0.899968i	0.711099	−	0.0729970i
$153$		0			0
$154$	2.18721	+	18.6516i	0.176250	+	1.50299i
$155$			1.88608i			0.151493i
$156$		0			0
$157$			19.9958i			1.59584i	0.602765	+	0.797919i	$0.294066\pi$
$157$			19.9958i			1.59584i	−0.602765	+	0.797919i	$0.705934\pi$
$158$	−11.1949	+	6.98280i	−0.890621	+	0.555522i
$159$		0			0
$160$	−20.7520	−	7.64819i	−1.64059	−	0.604643i
$161$	8.26077	+	3.89874i	0.651040	+	0.307264i
$162$		0			0
$163$		−	18.0490i		−	1.41371i	−0.707360	−	0.706854i	$-0.750114\pi$
$163$		−	18.0490i		−	1.41371i	0.707360	−	0.706854i	$-0.249886\pi$
$164$	2.71280	+	1.32855i	0.211834	+	0.103743i
$165$		0			0
$166$	−4.31632	−	6.91999i	−0.335011	−	0.537095i
$167$	19.7035			1.52470			0.762350	−	0.647164i	$-0.224045\pi$
$167$	19.7035			1.52470			0.762350	+	0.647164i	$0.224045\pi$
$168$		0			0
$169$	−18.3228			−1.40944
$170$	7.05300	+	11.3075i	0.540940	+	0.867244i
$171$		0			0
$172$	3.18929	+	1.56191i	0.243181	+	0.119094i
$173$			0.621296i			0.0472363i	0.999721	+	0.0236181i	$0.00751859\pi$
$173$			0.621296i			0.0472363i	−0.999721	+	0.0236181i	$0.992481\pi$
$174$		0			0
$175$	11.6149	−	24.6099i	0.878002	−	1.86034i
$176$	−15.8600	+	12.3089i	−1.19549	+	0.927815i
$177$		0			0
$178$	21.1850	−	13.2141i	1.58789	−	0.990438i
$179$		−	12.1294i		−	0.906591i	−0.891360	−	0.453296i	$-0.850248\pi$
$179$		−	12.1294i		−	0.906591i	0.891360	−	0.453296i	$-0.149752\pi$
$180$		0			0
$181$		−	21.5510i		−	1.60187i	−0.598750	−	0.800936i	$-0.704336\pi$
$181$		−	21.5510i		−	1.60187i	0.598750	−	0.800936i	$-0.295664\pi$
$182$	2.43894	+	20.7983i	0.180787	+	1.54167i
$183$		0			0
$184$	0.997202	+	9.71422i	0.0735147	+	0.716142i
$185$		−	23.0862i		−	1.69733i
$186$		0			0
$187$	12.0973			0.884642
$188$	−2.09910	+	4.28620i	−0.153093	+	0.312604i
$189$		0			0
$190$	−14.6177	+	9.11773i	−1.06048	+	0.661470i
$191$		−	12.3487i		−	0.893519i	−0.894654	−	0.446759i	$-0.852578\pi$
$191$		−	12.3487i		−	0.893519i	0.894654	−	0.446759i	$-0.147422\pi$
$192$		0			0
$193$	18.0526			1.29946			0.649728	−	0.760167i	$-0.274883\pi$
$193$	18.0526			1.29946			0.649728	+	0.760167i	$0.274883\pi$
$194$	−18.6810	+	11.6522i	−1.34122	+	0.836581i
$195$		0			0
$196$	13.6202	−	3.23892i	0.972870	−	0.231351i
$197$	−6.84502			−0.487687			−0.243844	−	0.969815i	$-0.578408\pi$
$197$	−6.84502			−0.487687			−0.243844	+	0.969815i	$0.578408\pi$
$198$		0			0
$199$	12.4174			0.880246			0.440123	−	0.897937i	$-0.354935\pi$
$199$	12.4174			0.880246			0.440123	+	0.897937i	$0.354935\pi$
$200$	28.9400	−	2.97080i	2.04637	−	0.210067i
$201$		0			0
$202$	−6.50361	+	4.05660i	−0.457592	+	0.285421i
$203$	−5.38630	+	11.4127i	−0.378044	+	0.801011i
$204$		0			0
$205$	−5.90489			−0.412415
$206$	−3.66138	−	5.86999i	−0.255101	−	0.408982i
$207$		0			0
$208$	−17.6854	+	13.7255i	−1.22626	+	0.951695i
$209$			15.6387i			1.08175i
$210$		0			0
$211$			20.0194i			1.37819i	0.724671	+	0.689095i	$0.241992\pi$
$211$			20.0194i			1.37819i	−0.724671	+	0.689095i	$0.758008\pi$
$212$	4.95002	−	10.1075i	0.339969	−	0.694189i
$213$		0			0
$214$	14.9682	−	9.33638i	1.02321	−	0.638222i
$215$	−6.94206			−0.473445
$216$		0			0
$217$	0.544757	−	1.15425i	0.0369805	−	0.0783555i
$218$	4.41951	+	7.08543i	0.299327	+	0.479886i
$219$		0			0
$220$	17.2611	−	35.2457i	1.16374	−	2.37627i
$221$	13.4896			0.907410
$222$		0			0
$223$	26.3982			1.76775			0.883876	−	0.467722i	$-0.154925\pi$
$223$	26.3982			1.76775			0.883876	+	0.467722i	$0.154925\pi$
$224$	10.4908	+	10.6744i	0.700948	+	0.713212i
$225$		0			0
$226$	11.4777	+	18.4012i	0.763484	+	1.22403i
$227$	2.00560			0.133116			0.0665580	−	0.997783i	$-0.478798\pi$
$227$	2.00560			0.133116			0.0665580	+	0.997783i	$0.478798\pi$
$228$		0			0
$229$		−	14.3991i		−	0.951520i	−0.879575	−	0.475760i	$-0.842173\pi$
$229$		−	14.3991i		−	0.951520i	0.879575	−	0.475760i	$-0.157827\pi$
$230$	−10.1028	−	16.1970i	−0.666161	−	1.06800i
$231$		0			0
$232$	−13.4207	+	1.37768i	−0.881111	+	0.0904494i
$233$	12.6031			0.825658			0.412829	−	0.910809i	$-0.364541\pi$
$233$	12.6031			0.825658			0.412829	+	0.910809i	$0.364541\pi$
$234$		0			0
$235$		−	9.32969i		−	0.608602i
$236$	−5.61277	+	11.4608i	−0.365360	+	0.746036i
$237$		0			0
$238$	−1.05037	−	8.95711i	−0.0680852	−	0.580604i
$239$			2.58274i			0.167064i	0.996505	+	0.0835319i	$0.0266201\pi$
$239$			2.58274i			0.167064i	−0.996505	+	0.0835319i	$0.973380\pi$
$240$		0			0
$241$			9.63816i			0.620848i	0.950598	+	0.310424i	$0.100471\pi$
$241$			9.63816i			0.620848i	−0.950598	+	0.310424i	$0.899529\pi$
$242$	−10.6209	−	17.0275i	−0.682735	−	1.09457i
$243$		0			0
$244$	12.4723	+	6.10812i	0.798457	+	0.391032i
$245$	−21.1270	+	17.3967i	−1.34975	+	1.11143i
$246$		0			0
$247$			17.4387i			1.10960i
$248$	1.35733	−	0.139336i	0.0861908	−	0.00884782i
$249$		0			0
$250$	−24.7965	+	15.4667i	−1.56827	+	0.978201i
$251$	12.9421			0.816896			0.408448	−	0.912782i	$-0.366070\pi$
$251$	12.9421			0.816896			0.408448	+	0.912782i	$0.366070\pi$
$252$		0			0
$253$	−17.3284			−1.08942
$254$	−1.09302	+	0.681767i	−0.0685822	+	0.0427779i
$255$		0			0
$256$	−3.97103	+	15.4994i	−0.248189	+	0.968712i
$257$		−	19.7541i		−	1.23223i	−0.787658	−	0.616113i	$-0.788706\pi$
$257$		−	19.7541i		−	1.23223i	0.787658	−	0.616113i	$-0.211294\pi$
$258$		0			0
$259$	−6.66802	+	14.1284i	−0.414331	+	0.877896i
$260$	19.2477	−	39.3023i	1.19369	−	2.43743i
$261$		0			0
$262$	−11.9318	−	19.1293i	−0.737151	−	1.18181i
$263$			4.77439i			0.294401i	0.989107	+	0.147201i	$0.0470263\pi$
$263$			4.77439i			0.294401i	−0.989107	+	0.147201i	$0.952974\pi$
$264$		0			0
$265$			22.0009i			1.35150i
$266$	11.5793	−	1.35786i	0.709971	−	0.0832557i
$267$		0			0
$268$	−7.04209	−	3.44876i	−0.430164	−	0.210666i
$269$		−	2.19366i		−	0.133750i	−0.997761	−	0.0668750i	$-0.978697\pi$
$269$		−	2.19366i		−	0.133750i	0.997761	−	0.0668750i	$-0.0213029\pi$
$270$		0			0
$271$	12.5797			0.764163			0.382081	−	0.924129i	$-0.375207\pi$
$271$	12.5797			0.764163			0.382081	+	0.924129i	$0.375207\pi$
$272$	7.61649	−	5.91111i	0.461818	−	0.358414i
$273$		0			0
$274$	0.469968	+	0.753460i	0.0283918	+	0.0455182i
$275$			51.6235i			3.11302i
$276$		0			0
$277$	21.7035			1.30404			0.652018	−	0.758204i	$-0.273923\pi$
$277$	21.7035			1.30404			0.652018	+	0.758204i	$0.273923\pi$
$278$	−12.9963	−	20.8358i	−0.779465	−	1.24965i
$279$		0			0
$280$	−27.5955	−	9.72022i	−1.64915	−	0.580894i
$281$	−18.2595			−1.08927			−0.544635	−	0.838673i	$-0.683332\pi$
$281$	−18.2595			−1.08927			−0.544635	+	0.838673i	$0.683332\pi$
$282$		0			0
$283$	4.88324			0.290279			0.145139	−	0.989411i	$-0.453637\pi$
$283$	4.88324			0.290279			0.145139	+	0.989411i	$0.453637\pi$
$284$	−14.8224	−	7.25906i	−0.879550	−	0.430746i
$285$		0			0
$286$	−21.0238	−	33.7057i	−1.24316	−	1.99306i
$287$	3.61370	+	1.70552i	0.213310	+	0.100673i
$288$		0			0
$289$	11.1905			0.658264
$290$	22.3770	−	13.9576i	1.31402	−	0.819616i
$291$		0			0
$292$	−6.45111	−	3.15933i	−0.377522	−	0.184886i
$293$			15.2450i			0.890622i	0.895376	+	0.445311i	$0.146907\pi$
$293$			15.2450i			0.890622i	−0.895376	+	0.445311i	$0.853093\pi$
$294$		0			0
$295$		−	24.9465i		−	1.45244i
$296$	−16.6142	+	1.70552i	−0.965683	+	0.0991311i
$297$		0			0
$298$	0.543828	+	0.871874i	0.0315031	+	0.0505063i
$299$	−19.3228			−1.11746
$300$		0			0
$301$	4.24843	+	2.00508i	0.244875	+	0.115571i
$302$	−7.52806	+	4.69560i	−0.433191	+	0.270201i
$303$		0			0
$304$	7.64156	+	9.84619i	0.438274	+	0.564718i
$305$	−27.1482			−1.55450
$306$		0			0
$307$	−6.30173			−0.359659			−0.179829	−	0.983698i	$-0.557555\pi$
$307$	−6.30173			−0.359659			−0.179829	+	0.983698i	$0.557555\pi$
$308$	−20.7436	+	16.5843i	−1.18197	+	0.944977i
$309$		0			0
$310$	−2.26315	+	1.41163i	−0.128538	+	0.0801754i
$311$	18.4761			1.04768			0.523841	−	0.851816i	$-0.324498\pi$
$311$	18.4761			1.04768			0.523841	+	0.851816i	$0.324498\pi$
$312$		0			0
$313$			18.7951i			1.06236i	0.847258	+	0.531181i	$0.178252\pi$
$313$			18.7951i			1.06236i	−0.847258	+	0.531181i	$0.821748\pi$
$314$	−23.9935	+	14.9658i	−1.35403	+	0.844571i
$315$		0			0
$316$	−16.7577	−	8.20682i	−0.942694	−	0.461670i
$317$	2.06678			0.116082			0.0580410	−	0.998314i	$-0.481515\pi$
$317$	2.06678			0.116082			0.0580410	+	0.998314i	$0.481515\pi$
$318$		0			0
$319$		−	23.9400i		−	1.34038i
$320$	−6.35454	−	30.6251i	−0.355229	−	1.71200i
$321$		0			0
$322$	1.50456	+	12.8303i	0.0838461	+	0.715006i
$323$		−	7.51022i		−	0.417880i
$324$		0			0
$325$			57.5651i			3.19314i
$326$	21.6575	−	13.5088i	1.19950	−	0.748182i
$327$		0			0
$328$	0.436229	+	4.24951i	0.0240867	+	0.234640i
$329$	−2.69470	+	5.70962i	−0.148564	+	0.314781i
$330$		0			0
$331$		−	31.2531i		−	1.71783i	−0.512121	−	0.858913i	$-0.671140\pi$
$331$		−	31.2531i		−	1.71783i	0.512121	−	0.858913i	$-0.328860\pi$
$332$	5.07293	−	10.3585i	0.278413	−	0.568498i
$333$		0			0
$334$	14.7471	+	23.6427i	0.806923	+	1.29367i
$335$	15.3284			0.837478
$336$		0			0
$337$	14.4550			0.787417			0.393708	−	0.919235i	$-0.371192\pi$
$337$	14.4550			0.787417			0.393708	+	0.919235i	$0.371192\pi$
$338$	−13.7137	−	21.9860i	−0.745925	−	1.19588i
$339$		0			0
$340$	−8.28932	+	16.9262i	−0.449552	+	0.917949i
$341$			2.42123i			0.131117i
$342$		0			0
$343$	17.9541	−	4.54436i	0.969429	−	0.245373i
$344$	0.512851	+	4.99592i	0.0276511	+	0.269362i
$345$		0			0
$346$	−0.745510	+	0.465009i	−0.0400789	+	0.0249990i
$347$		−	16.6510i		−	0.893871i	−0.894566	−	0.446936i	$-0.852515\pi$
$347$		−	16.6510i		−	0.893871i	0.894566	−	0.446936i	$-0.147485\pi$
$348$		0			0
$349$			17.0294i			0.911561i	0.890092	+	0.455781i	$0.150640\pi$
$349$			17.0294i			0.911561i	−0.890092	+	0.455781i	$0.849360\pi$
$350$	38.2233	−	4.48230i	2.04312	−	0.239589i
$351$		0			0
$352$	−26.6401	−	9.81829i	−1.41992	−	0.523317i
$353$		−	26.9850i		−	1.43627i	−0.695906	−	0.718133i	$-0.744997\pi$
$353$		−	26.9850i		−	1.43627i	0.695906	−	0.718133i	$-0.255003\pi$
$354$		0			0
$355$	32.2637			1.71238
$356$	31.7119	+	15.5304i	1.68073	+	0.823110i
$357$		0			0
$358$	14.5543	−	9.07822i	0.769221	−	0.479799i
$359$		−	12.9242i		−	0.682112i	−0.940043	−	0.341056i	$-0.889215\pi$
$359$		−	12.9242i		−	0.682112i	0.940043	−	0.341056i	$-0.110785\pi$
$360$		0			0
$361$	−9.29119			−0.489010
$362$	25.8596	−	16.1298i	1.35915	−	0.847765i
$363$		0			0
$364$	−23.1310	+	18.4931i	−1.21240	+	0.969299i
$365$	14.0420			0.734991
$366$		0			0
$367$	−33.8798			−1.76851			−0.884256	−	0.467003i	$-0.845334\pi$
$367$	−33.8798			−1.76851			−0.884256	+	0.467003i	$0.845334\pi$
$368$	−10.9100	+	8.46717i	−0.568723	+	0.441382i
$369$		0			0
$370$	27.7018	−	17.2789i	1.44015	−	0.898286i
$371$	6.35454	−	13.4642i	0.329911	−	0.699025i
$372$		0			0
$373$	14.1849			0.734466			0.367233	−	0.930129i	$-0.380305\pi$
$373$	14.1849			0.734466			0.367233	+	0.930129i	$0.380305\pi$
$374$	9.05421	+	14.5159i	0.468182	+	0.750597i
$375$		0			0
$376$	−6.71420	+	0.689239i	−0.346259	+	0.0355448i
$377$		−	26.6954i		−	1.37488i
$378$		0			0
$379$			10.5772i			0.543313i	0.962394	+	0.271657i	$0.0875715\pi$
$379$			10.5772i			0.543313i	−0.962394	+	0.271657i	$0.912429\pi$
$380$	−21.8812	−	10.7160i	−1.12248	−	0.549719i
$381$		0			0
$382$	14.8175	−	9.24236i	0.758129	−	0.472880i
$383$	15.1477			0.774012			0.387006	−	0.922077i	$-0.373509\pi$
$383$	15.1477			0.774012			0.387006	+	0.922077i	$0.373509\pi$
$384$		0			0
$385$	22.1587	−	46.9505i	1.12931	−	2.39282i
$386$	13.5115	+	21.6618i	0.687716	+	1.10256i
$387$		0			0
$388$	−27.9636	−	13.6948i	−1.41964	−	0.695246i
$389$	28.1976			1.42968			0.714838	−	0.699290i	$-0.246500\pi$
$389$	28.1976			1.42968			0.714838	+	0.699290i	$0.246500\pi$
$390$		0			0
$391$	8.32164			0.420844
$392$	14.0805	+	13.9191i	0.711172	+	0.703018i
$393$		0			0
$394$	−5.12315	−	8.21352i	−0.258101	−	0.413791i
$395$	36.4761			1.83531
$396$		0			0
$397$			7.98758i			0.400885i	0.979705	+	0.200443i	$0.0642380\pi$
$397$			7.98758i			0.400885i	−0.979705	+	0.200443i	$0.935762\pi$
$398$	9.29380	+	14.9000i	0.465856	+	0.746868i
$399$		0			0
$400$	25.2249	+	32.5024i	1.26124	+	1.62512i
$401$	11.8129			0.589908			0.294954	−	0.955512i	$-0.404696\pi$
$401$	11.8129			0.589908			0.294954	+	0.955512i	$0.404696\pi$
$402$		0			0
$403$			2.69990i			0.134492i
$404$	−9.73524	−	4.76769i	−0.484347	−	0.237201i
$405$		0			0
$406$	−17.7257	+	2.07863i	−0.879713	+	0.103161i
$407$		−	29.6367i		−	1.46904i
$408$		0			0
$409$			6.79736i			0.336108i	0.985778	+	0.168054i	$0.0537483\pi$
$409$			6.79736i			0.336108i	−0.985778	+	0.168054i	$0.946252\pi$
$410$	−4.41951	−	7.08543i	−0.218264	−	0.349925i
$411$		0			0
$412$	4.30319	−	8.78678i	0.212003	−	0.432894i
$413$	−7.20533	+	15.2669i	−0.354551	+	0.751233i
$414$		0			0
$415$			22.5472i			1.10680i
$416$	−29.7063	−	10.9483i	−1.45647	−	0.536786i
$417$		0			0
$418$	−18.7653	+	11.7048i	−0.917841	+	0.572500i
$419$	−15.7091			−0.767438			−0.383719	−	0.923450i	$-0.625357\pi$
$419$	−15.7091			−0.767438			−0.383719	+	0.923450i	$0.625357\pi$
$420$		0			0
$421$	12.0154			0.585597			0.292798	−	0.956174i	$-0.405414\pi$
$421$	12.0154			0.585597			0.292798	+	0.956174i	$0.405414\pi$
$422$	−24.0218	+	14.9835i	−1.16936	+	0.729385i
$423$		0			0
$424$	15.8332	−	1.62533i	0.768926	−	0.0789332i
$425$		−	24.7913i		−	1.20255i
$426$		0			0
$427$	16.6142	+	7.84124i	0.804019	+	0.379464i
$428$	22.4059	+	10.9730i	1.08303	+	0.530398i
$429$		0			0
$430$	−5.19578	−	8.32996i	−0.250563	−	0.401706i
$431$			10.0380i			0.483515i	0.970337	+	0.241757i	$0.0777238\pi$
$431$			10.0380i			0.483515i	−0.970337	+	0.241757i	$0.922276\pi$
$432$		0			0
$433$			25.7389i			1.23693i	0.785811	+	0.618467i	$0.212246\pi$
$433$			25.7389i			1.23693i	−0.785811	+	0.618467i	$0.787754\pi$
$434$	1.79274	−	0.210228i	0.0860541	−	0.0100912i
$435$		0			0
$436$	−5.19421	+	10.6062i	−0.248758	+	0.507944i
$437$			10.7578i			0.514614i
$438$		0			0
$439$	28.6653			1.36812			0.684061	−	0.729425i	$-0.260212\pi$
$439$	28.6653			1.36812			0.684061	+	0.729425i	$0.260212\pi$
$440$	55.2113	−	5.66766i	2.63210	−	0.270195i
$441$		0			0
$442$	10.0963	+	16.1866i	0.480232	+	0.769916i
$443$			3.16702i			0.150470i	0.997166	+	0.0752348i	$0.0239706\pi$
$443$			3.16702i			0.150470i	−0.997166	+	0.0752348i	$0.976029\pi$
$444$		0			0
$445$	−69.0265			−3.27217
$446$	19.7577	+	31.6758i	0.935554	+	1.49989i
$447$		0			0
$448$	−4.95662	+	20.5775i	−0.234178	+	0.972194i
$449$	−32.1416			−1.51685			−0.758427	−	0.651758i	$-0.774032\pi$
$449$	−32.1416			−1.51685			−0.758427	+	0.651758i	$0.774032\pi$
$450$		0			0
$451$	−7.58034			−0.356944
$452$	−13.4896	+	27.5448i	−0.634499	+	1.29560i
$453$		0			0
$454$	1.50109	+	2.40657i	0.0704495	+	0.112946i
$455$	24.7091	−	52.3543i	1.15838	−	2.45441i
$456$		0			0
$457$	20.4389			0.956092			0.478046	−	0.878335i	$-0.341345\pi$
$457$	20.4389			0.956092			0.478046	+	0.878335i	$0.341345\pi$
$458$	17.2779	−	10.7770i	0.807342	−	0.503577i
$459$		0			0
$460$	11.8738	−	24.2453i	0.553617	−	1.13044i
$461$			6.14287i			0.286102i	0.989715	+	0.143051i	$0.0456913\pi$
$461$			6.14287i			0.286102i	−0.989715	+	0.143051i	$0.954309\pi$
$462$		0			0
$463$		−	2.13162i		−	0.0990649i	−0.998773	−	0.0495324i	$-0.984227\pi$
$463$		−	2.13162i		−	0.0990649i	0.998773	−	0.0495324i	$-0.0157731\pi$
$464$	−11.6978	−	15.0727i	−0.543058	−	0.699732i
$465$		0			0
$466$	9.43280	+	15.1228i	0.436966	+	0.700551i
$467$	11.4817			0.531309			0.265654	−	0.964068i	$-0.414412\pi$
$467$	11.4817			0.531309			0.265654	+	0.964068i	$0.414412\pi$
$468$		0			0
$469$	−9.38071	−	4.42731i	−0.433161	−	0.204434i
$470$	11.1949	−	6.98280i	0.516384	−	0.322093i
$471$		0			0
$472$	−17.9530	+	1.84295i	−0.826355	+	0.0848285i
$473$	−8.91180			−0.409765
$474$		0			0
$475$	32.0489			1.47050
$476$	9.96173	−	7.96431i	0.456595	−	0.365044i
$477$		0			0
$478$	−3.09910	+	1.93305i	−0.141750	+	0.0884159i
$479$	−14.3863			−0.657327			−0.328664	−	0.944447i	$-0.606598\pi$
$479$	−14.3863			−0.657327			−0.328664	+	0.944447i	$0.606598\pi$
$480$		0			0
$481$		−	33.0477i		−	1.50685i
$482$	−11.5651	+	7.21368i	−0.526775	+	0.328574i
$483$		0			0
$484$	12.4826	−	25.4885i	0.567391	−	1.15857i
$485$	60.8678			2.76386
$486$		0			0
$487$			20.3888i			0.923904i	0.886905	+	0.461952i	$0.152851\pi$
$487$			20.3888i			0.923904i	−0.886905	+	0.461952i	$0.847149\pi$
$488$	2.00560	+	19.5375i	0.0907890	+	0.884419i
$489$		0			0
$490$	−36.6872	−	12.3303i	−1.65736	−	0.557025i
$491$		−	19.9296i		−	0.899409i	−0.893177	−	0.449705i	$-0.851529\pi$
$491$		−	19.9296i		−	0.899409i	0.893177	−	0.449705i	$-0.148471\pi$
$492$		0			0
$493$			11.4968i			0.517788i
$494$	−20.9251	+	13.0520i	−0.941465	+	0.587235i
$495$		0			0
$496$	1.18309	+	1.52442i	0.0531222	+	0.0684483i
$497$	−19.7448	−	9.31875i	−0.885677	−	0.418003i
$498$		0			0
$499$		−	7.62706i		−	0.341434i	−0.985320	−	0.170717i	$-0.945392\pi$
$499$		−	7.62706i		−	0.341434i	0.985320	−	0.170717i	$-0.0546084\pi$
$500$	−37.1178	−	18.1779i	−1.65996	−	0.812940i
$501$		0			0
$502$	9.68648	+	15.5295i	0.432329	+	0.693116i
$503$	−30.4761			−1.35886			−0.679431	−	0.733740i	$-0.737773\pi$
$503$	−30.4761			−1.35886			−0.679431	+	0.733740i	$0.737773\pi$
$504$		0			0
$505$	21.1905			0.942964
$506$	−12.9694	−	20.7928i	−0.576560	−	0.924351i
$507$		0			0
$508$	−1.63614	−	0.801275i	−0.0725920	−	0.0355508i
$509$			6.22512i			0.275923i	0.990438	+	0.137962i	$0.0440551\pi$
$509$			6.22512i			0.275923i	−0.990438	+	0.137962i	$0.955945\pi$
$510$		0			0
$511$	−8.59346	−	4.05576i	−0.380152	−	0.179416i
$512$	−21.5702	+	6.83557i	−0.953279	+	0.302092i
$513$		0			0
$514$	23.7035	−	14.7849i	1.04551	−	0.652136i
$515$			19.1260i			0.842792i
$516$		0			0
$517$		−	11.9769i		−	0.526743i
$518$	−21.9437	+	2.57326i	−0.964151	+	0.113062i
$519$		0			0
$520$	61.5659	−	6.31997i	2.69984	−	0.277149i
$521$			4.99769i			0.218953i	0.993989	+	0.109476i	$0.0349174\pi$
$521$			4.99769i			0.218953i	−0.993989	+	0.109476i	$0.965083\pi$
$522$		0			0
$523$	−4.95318			−0.216587			−0.108294	−	0.994119i	$-0.534539\pi$
$523$	−4.95318			−0.216587			−0.108294	+	0.994119i	$0.534539\pi$
$524$	14.0234	−	28.6346i	0.612614	−	1.25091i
$525$		0			0
$526$	−5.72891	+	3.57339i	−0.249792	+	0.155807i
$527$		−	1.16275i		−	0.0506504i
$528$		0			0
$529$	11.0799			0.481736
$530$	−26.3994	+	16.4665i	−1.14672	+	0.715261i
$531$		0			0
$532$	10.2958	+	12.8780i	0.446381	+	0.558331i
$533$	−8.45280			−0.366131
$534$		0			0
$535$	−48.7705			−2.10853
$536$	−1.13240	−	11.0312i	−0.0489121	−	0.476476i
$537$		0			0
$538$	2.63224	−	1.64185i	0.113484	−	0.0707851i
$539$	−27.1215	+	22.3328i	−1.16821	+	0.961943i
$540$		0			0
$541$	−31.8044			−1.36738			−0.683690	−	0.729773i	$-0.739626\pi$
$541$	−31.8044			−1.36738			−0.683690	+	0.729773i	$0.739626\pi$
$542$	9.41527	+	15.0947i	0.404421	+	0.648374i
$543$		0			0
$544$	12.7935	+	4.71506i	0.548515	+	0.202157i
$545$		−	23.0862i		−	0.988905i
$546$		0			0
$547$		−	40.9363i		−	1.75031i	−0.483844	−	0.875154i	$-0.660760\pi$
$547$		−	40.9363i		−	1.75031i	0.483844	−	0.875154i	$-0.339240\pi$
$548$	−0.552349	+	1.12785i	−0.0235952	+	0.0481795i
$549$		0			0
$550$	−61.9445	+	38.6376i	−2.64132	+	1.64751i
$551$	−14.8624			−0.633159
$552$		0			0
$553$	−22.3228	−	10.5354i	−0.949261	−	0.448012i
$554$	16.2440	+	26.0426i	0.690140	+	1.10644i
$555$		0			0
$556$	15.2744	−	31.1891i	0.647779	−	1.32271i
$557$	13.9112			0.589435			0.294717	−	0.955584i	$-0.404774\pi$
$557$	13.9112			0.589435			0.294717	+	0.955584i	$0.404774\pi$
$558$		0			0
$559$	−9.93750			−0.420312
$560$	−8.99028	−	40.3876i	−0.379909	−	1.70669i
$561$		0			0
$562$	−13.6663	−	21.9100i	−0.576478	−	0.924219i
$563$	−37.5659			−1.58321			−0.791606	−	0.611032i	$-0.790755\pi$
$563$	−37.5659			−1.58321			−0.791606	+	0.611032i	$0.790755\pi$
$564$		0			0
$565$		−	59.9561i		−	2.52237i
$566$	3.65486	+	5.85953i	0.153625	+	0.246294i
$567$		0			0
$568$	−2.38351	−	23.2189i	−0.100010	−	0.974242i
$569$	−4.83108			−0.202529			−0.101265	−	0.994860i	$-0.532289\pi$
$569$	−4.83108			−0.202529			−0.101265	+	0.994860i	$0.532289\pi$
$570$		0			0
$571$		−	35.4877i		−	1.48511i	−0.669784	−	0.742556i	$-0.733613\pi$
$571$		−	35.4877i		−	1.48511i	0.669784	−	0.742556i	$-0.266387\pi$
$572$	24.7091	−	50.4540i	1.03314	−	2.10959i
$573$		0			0
$574$	0.658176	+	5.61266i	0.0274717	+	0.234268i
$575$			35.5115i			1.48093i
$576$		0			0
$577$			10.9331i			0.455153i	0.973760	+	0.227576i	$0.0730801\pi$
$577$			10.9331i			0.455153i	−0.973760	+	0.227576i	$0.926920\pi$
$578$	8.37552	+	13.4278i	0.348376	+	0.558521i
$579$		0			0
$580$	33.4961	+	16.4042i	1.39085	+	0.681147i
$581$	6.51232	−	13.7985i	0.270177	−	0.572458i
$582$		0			0
$583$			28.2434i			1.16972i
$584$	−1.03736	−	10.1055i	−0.0429264	−	0.418167i
$585$		0			0
$586$	−18.2929	+	11.4101i	−0.755671	+	0.471347i
$587$	12.1477			0.501390			0.250695	−	0.968066i	$-0.419341\pi$
$587$	12.1477			0.501390			0.250695	+	0.968066i	$0.419341\pi$
$588$		0			0
$589$	1.50315			0.0619360
$590$	29.9340	−	18.6712i	1.23236	−	0.768682i
$591$		0			0
$592$	−14.4814	−	18.6594i	−0.595182	−	0.766895i
$593$			2.60504i			0.106976i	0.998568	+	0.0534882i	$0.0170339\pi$
$593$			2.60504i			0.106976i	−0.998568	+	0.0534882i	$0.982966\pi$
$594$		0			0
$595$	−10.6413	+	22.5472i	−0.436252	+	0.924344i
$596$	−0.639157	+	1.30511i	−0.0261809	+	0.0534593i
$597$		0			0
$598$	−14.4621	−	23.1859i	−0.591400	−	0.948142i
$599$			34.5554i			1.41190i	0.708263	+	0.705948i	$0.249479\pi$
$599$			34.5554i			1.41190i	−0.708263	+	0.705948i	$0.750521\pi$
$600$		0			0
$601$		−	12.3627i		−	0.504286i	−0.967690	−	0.252143i	$-0.918865\pi$
$601$		−	12.3627i		−	0.504286i	0.967690	−	0.252143i	$-0.0811353\pi$
$602$	0.773782	+	6.59850i	0.0315370	+	0.268935i
$603$		0			0
$604$	−11.2687	−	5.51870i	−0.458519	−	0.224553i
$605$			55.4803i			2.25559i
$606$		0			0
$607$	3.43840			0.139560			0.0697802	−	0.997562i	$-0.477770\pi$
$607$	3.43840			0.139560			0.0697802	+	0.997562i	$0.477770\pi$
$608$	−6.09538	+	16.5387i	−0.247200	+	0.670732i
$609$		0			0
$610$	−20.3190	−	32.5758i	−0.822694	−	1.31896i
$611$		−	13.3554i		−	0.540300i
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$	−4.71653	−	7.56161i	−0.190344	−	0.305162i
$615$		0			0
$616$	−35.4254	−	12.4782i	−1.42733	−	0.502762i
$617$	−38.7921			−1.56171			−0.780856	−	0.624710i	$-0.785217\pi$
$617$	−38.7921			−1.56171			−0.780856	+	0.624710i	$0.785217\pi$
$618$		0			0
$619$	−38.0592			−1.52973			−0.764865	−	0.644191i	$-0.777194\pi$
$619$	−38.0592			−1.52973			−0.764865	+	0.644191i	$0.777194\pi$
$620$	−3.38771	−	1.65908i	−0.136054	−	0.0666303i
$621$		0			0
$622$	13.8284	+	22.1699i	0.554469	+	0.888934i
$623$	42.2431	+	19.9370i	1.69243	+	0.798759i
$624$		0			0
$625$	29.3655			1.17462
$626$	−22.5527	+	14.0672i	−0.901389	+	0.562238i
$627$		0			0
$628$	−35.9158	−	17.5892i	−1.43320	−	0.701886i
$629$			14.2325i			0.567487i
$630$		0			0
$631$		−	41.8076i		−	1.66434i	−0.554524	−	0.832168i	$-0.687100\pi$
$631$		−	41.8076i		−	1.66434i	0.554524	−	0.832168i	$-0.312900\pi$
$632$	−2.69470	−	26.2504i	−0.107190	−	1.04418i
$633$		0			0
$634$	1.54688	+	2.47999i	0.0614345	+	0.0984928i
$635$	3.56135			0.141328
$636$		0			0
$637$	−30.2431	+	24.9032i	−1.19828	+	0.986702i
$638$	28.7262	−	17.9179i	1.13728	−	0.709376i
$639$		0			0
$640$	31.9918	−	30.5463i	1.26459	−	1.20745i
$641$	29.5836			1.16848			0.584242	−	0.811580i	$-0.301392\pi$
$641$	29.5836			1.16848			0.584242	+	0.811580i	$0.301392\pi$
$642$		0			0
$643$	34.8510			1.37439			0.687194	−	0.726474i	$-0.258842\pi$
$643$	34.8510			1.37439			0.687194	+	0.726474i	$0.258842\pi$
$644$	−14.2694	+	11.4082i	−0.562291	+	0.449547i
$645$		0			0
$646$	9.01171	−	5.62102i	0.354561	−	0.221156i
$647$	22.8512			0.898373			0.449187	−	0.893438i	$-0.351714\pi$
$647$	22.8512			0.898373			0.449187	+	0.893438i	$0.351714\pi$
$648$		0			0
$649$		−	32.0249i		−	1.25709i
$650$	−69.0739	+	43.0846i	−2.70930	+	1.68992i
$651$		0			0
$652$	32.4191	+	15.8767i	1.26963	+	0.621781i
$653$	−6.41751			−0.251136			−0.125568	−	0.992085i	$-0.540075\pi$
$653$	−6.41751			−0.251136			−0.125568	+	0.992085i	$0.540075\pi$
$654$		0			0
$655$			62.3284i			2.43537i
$656$	−4.77261	+	3.70399i	−0.186339	+	0.144617i
$657$		0			0
$658$	−8.86797	+	1.03991i	−0.345709	+	0.0405401i
$659$			3.31350i			0.129076i	0.997915	+	0.0645378i	$0.0205573\pi$
$659$			3.31350i			0.129076i	−0.997915	+	0.0645378i	$0.979443\pi$
$660$		0			0
$661$			26.3760i			1.02591i	0.858416	+	0.512954i	$0.171449\pi$
$661$			26.3760i			1.02591i	−0.858416	+	0.512954i	$0.828551\pi$
$662$	37.5014	−	23.3914i	1.45753	−	0.909132i
$663$		0			0
$664$	16.2263	−	1.66569i	0.629703	−	0.0646414i
$665$	−29.1478	−	13.7565i	−1.13030	−	0.533456i
$666$		0			0
$667$		−	16.4682i		−	0.637650i
$668$	−17.3321	+	35.3908i	−0.670599	+	1.36931i
$669$		0			0
$670$	11.4725	+	18.3929i	0.443222	+	0.710580i
$671$	−34.8512			−1.34542
$672$		0			0
$673$	20.4389			0.787862			0.393931	−	0.919140i	$-0.371115\pi$
$673$	20.4389			0.787862			0.393931	+	0.919140i	$0.371115\pi$
$674$	10.8189	+	17.3450i	0.416728	+	0.668104i
$675$		0			0
$676$	16.1176	−	32.9108i	0.619906	−	1.26580i
$677$		−	6.72464i		−	0.258449i	−0.991615	−	0.129224i	$-0.958751\pi$
$677$		−	6.72464i		−	0.258449i	0.991615	−	0.129224i	$-0.0412488\pi$
$678$		0			0
$679$	−37.2501	−	17.5805i	−1.42953	−	0.674678i
$680$	−26.5143	+	2.72179i	−1.01678	+	0.104376i
$681$		0			0
$682$	−2.90530	+	1.81217i	−0.111250	+	0.0693916i
$683$		−	10.7935i		−	0.413001i	−0.978446	−	0.206501i	$-0.933792\pi$
$683$		−	10.7935i		−	0.413001i	0.978446	−	0.206501i	$-0.0662075\pi$
$684$		0			0
$685$		−	2.45497i		−	0.0937997i
$686$	18.8906	+	18.1423i	0.721247	+	0.692678i
$687$		0			0
$688$	−5.61090	+	4.35458i	−0.213913	+	0.166017i
$689$			31.4941i			1.19983i
$690$		0			0
$691$	−21.9337			−0.834398			−0.417199	−	0.908815i	$-0.636988\pi$
$691$	−21.9337			−0.834398			−0.417199	+	0.908815i	$0.636988\pi$
$692$	−1.11595	−	0.546521i	−0.0424222	−	0.0207756i
$693$		0			0
$694$	19.9799	−	12.4624i	0.758428	−	0.473067i
$695$			67.8887i			2.57517i
$696$		0			0
$697$	3.64033			0.137887
$698$	−20.4340	+	12.7456i	−0.773438	+	0.482429i
$699$		0			0
$700$	33.9866	+	42.5103i	1.28457	+	1.60674i
$701$	−18.3612			−0.693493			−0.346747	−	0.937959i	$-0.612714\pi$
$701$	−18.3612			−0.693493			−0.346747	+	0.937959i	$0.612714\pi$
$702$		0			0
$703$	−18.3990			−0.693932
$704$	−8.15757	−	39.3147i	−0.307450	−	1.48173i
$705$		0			0
$706$	32.3800	−	20.1969i	1.21864	−	0.760120i
$707$	−12.9682	−	6.12047i	−0.487721	−	0.230184i
$708$		0			0
$709$	−28.9463			−1.08710			−0.543551	−	0.839376i	$-0.682920\pi$
$709$	−28.9463			−1.08710			−0.543551	+	0.839376i	$0.682920\pi$
$710$	24.1477	+	38.7140i	0.906248	+	1.45291i
$711$		0			0
$712$	5.09940	+	49.6756i	0.191108	+	1.86167i
$713$			1.66555i			0.0623753i
$714$		0			0
$715$			109.822i			4.10711i
$716$	21.7864	+	10.6696i	0.814196	+	0.398740i
$717$		0			0
$718$	15.5081	−	9.67310i	0.578756	−	0.360997i
$719$	−9.14772			−0.341152			−0.170576	−	0.985344i	$-0.554563\pi$
$719$	−9.14772			−0.341152			−0.170576	+	0.985344i	$0.554563\pi$
$720$		0			0
$721$	5.52418	−	11.7048i	0.205731	−	0.435909i
$722$	−6.95399	−	11.1487i	−0.258801	−	0.414913i
$723$		0			0
$724$	38.7092	+	18.9572i	1.43862	+	0.704540i
$725$	−49.0609			−1.82208
$726$		0			0
$727$	−14.7308			−0.546334			−0.273167	−	0.961967i	$-0.588071\pi$
$727$	−14.7308			−0.546334			−0.273167	+	0.961967i	$0.588071\pi$
$728$	−39.5027	−	13.9144i	−1.46407	−	0.515703i
$729$		0			0
$730$	10.5097	+	16.8493i	0.388982	+	0.623622i
$731$	4.27974			0.158292
$732$		0			0
$733$			24.2453i			0.895521i	0.894154	+	0.447760i	$0.147778\pi$
$733$			24.2453i			0.895521i	−0.894154	+	0.447760i	$0.852222\pi$
$734$	−25.3573	−	40.6533i	−0.935956	−	1.50054i
$735$		0			0
$736$	−18.3256	−	6.75394i	−0.675489	−	0.248953i
$737$	19.6776			0.724835
$738$		0			0
$739$			31.2993i			1.15136i	0.817674	+	0.575682i	$0.195263\pi$
$739$			31.2993i			1.15136i	−0.817674	+	0.575682i	$0.804737\pi$
$740$	41.4668	+	20.3077i	1.52435	+	0.746526i
$741$		0			0
$742$	20.9121	−	2.45228i	0.767706	−	0.0900261i
$743$		−	35.0388i		−	1.28545i	−0.766098	−	0.642724i	$-0.777804\pi$
$743$		−	35.0388i		−	1.28545i	0.766098	−	0.642724i	$-0.222196\pi$
$744$		0			0
$745$		−	2.84080i		−	0.104079i
$746$	10.6167	+	17.0208i	0.388704	+	0.623177i
$747$		0			0
$748$	−10.6413	+	21.7288i	−0.389086	+	0.794483i
$749$	29.8467	+	14.0864i	1.09058	+	0.514707i
$750$		0			0
$751$			12.7356i			0.464728i	0.972629	+	0.232364i	$0.0746460\pi$
$751$			12.7356i			0.464728i	−0.972629	+	0.232364i	$0.925354\pi$
$752$	−5.85228	−	7.54069i	−0.213411	−	0.274981i
$753$		0			0
$754$	32.0325	−	19.9801i	1.16655	−	0.727633i
$755$	24.5284			0.892681
$756$		0			0
$757$	−12.8565			−0.467278			−0.233639	−	0.972323i	$-0.575063\pi$
$757$	−12.8565			−0.467278			−0.233639	+	0.972323i	$0.575063\pi$
$758$	−12.6918	+	7.91649i	−0.460988	+	0.287540i
$759$		0			0
$760$	−3.51859	−	34.2762i	−0.127633	−	1.24333i
$761$			1.27182i			0.0461034i	0.999734	+	0.0230517i	$0.00733824\pi$
$761$			1.27182i			0.0461034i	−0.999734	+	0.0230517i	$0.992662\pi$
$762$		0			0
$763$	−6.66802	+	14.1284i	−0.241399	+	0.511482i
$764$	22.1803	+	10.8625i	0.802455	+	0.392990i
$765$		0			0
$766$	11.3373	+	18.1761i	0.409634	+	0.656731i
$767$		−	35.7108i		−	1.28944i
$768$		0			0
$769$		−	10.2961i		−	0.371286i	−0.982617	−	0.185643i	$-0.940563\pi$
$769$		−	10.2961i		−	0.371286i	0.982617	−	0.185643i	$-0.0594368\pi$
$770$	72.9219	−	8.55128i	2.62792	−	0.308167i
$771$		0			0
$772$	−15.8799	+	32.4255i	−0.571531	+	1.16702i
$773$		−	32.7274i		−	1.17712i	−0.808453	−	0.588561i	$-0.799695\pi$
$773$		−	32.7274i		−	1.17712i	0.808453	−	0.588561i	$-0.200305\pi$
$774$		0			0
$775$	4.96190			0.178237
$776$	−4.49666	−	43.8041i	−0.161421	−	1.57248i
$777$		0			0
$778$	21.1045	+	33.8351i	0.756633	+	1.21305i
$779$			4.70602i			0.168611i
$780$		0			0
$781$	41.4181			1.48206
$782$	6.22833	+	9.98536i	0.222725	+	0.357076i
$783$		0			0
$784$	−6.16330	+	27.3133i	−0.220118	+	0.975473i
$785$	78.1771			2.79026
$786$		0			0
$787$	−7.89413			−0.281395			−0.140698	−	0.990053i	$-0.544935\pi$
$787$	−7.89413			−0.281395			−0.140698	+	0.990053i	$0.544935\pi$
$788$	6.02120	−	12.2948i	0.214496	−	0.437984i
$789$		0			0
$790$	27.3005	+	43.7686i	0.971309	+	1.55722i
$791$	−17.3172	+	36.6921i	−0.615728	+	1.30462i
$792$		0			0
$793$	−38.8624			−1.38004
$794$	−9.58451	+	5.97830i	−0.340142	+	0.212162i
$795$		0			0
$796$	−10.9229	+	22.3037i	−0.387153	+	0.790535i
$797$			37.8578i			1.34099i	0.741913	+	0.670496i	$0.233919\pi$
$797$			37.8578i			1.34099i	−0.741913	+	0.670496i	$0.766081\pi$
$798$		0			0
$799$			5.75169i			0.203480i
$800$	−20.1209	+	54.5943i	−0.711381	+	1.93020i
$801$		0			0
$802$	8.84135	+	14.1746i	0.312199	+	0.500522i
$803$	18.0262			0.636132
$804$		0			0
$805$	15.2428	−	32.2970i	0.537239	−	1.13832i
$806$	−3.23968	+	2.02074i	−0.114113	+	0.0711776i
$807$		0			0
$808$	−1.56547	−	15.2500i	−0.0550729	−	0.536491i
$809$	20.6718			0.726783			0.363392	−	0.931636i	$-0.381619\pi$
$809$	20.6718			0.726783			0.363392	+	0.931636i	$0.381619\pi$
$810$		0			0
$811$	3.86591			0.135751			0.0678753	−	0.997694i	$-0.478378\pi$
$811$	3.86591			0.135751			0.0678753	+	0.997694i	$0.478378\pi$
$812$	−15.7610	−	19.7138i	−0.553103	−	0.691819i
$813$		0			0
$814$	35.5619	−	22.1816i	1.24644	−	0.777464i
$815$	−70.5659			−2.47181
$816$		0			0
$817$			5.53261i			0.193562i
$818$	−8.15633	+	5.08748i	−0.285179	+	0.177880i
$819$		0			0
$820$	5.19421	−	10.6062i	0.181390	−	0.370384i
$821$	−29.6449			−1.03461			−0.517306	−	0.855800i	$-0.673065\pi$
$821$	−29.6449			−1.03461			−0.517306	+	0.855800i	$0.673065\pi$
$822$		0			0
$823$		−	27.7188i		−	0.966216i	−0.875561	−	0.483108i	$-0.839508\pi$
$823$		−	27.7188i		−	0.966216i	0.875561	−	0.483108i	$-0.160492\pi$
$824$	13.7642	−	1.41295i	0.479499	−	0.0492224i
$825$		0			0
$826$	−23.7119	+	2.78061i	−0.825044	+	0.0967499i
$827$			42.1357i			1.46520i	0.680658	+	0.732601i	$0.261694\pi$
$827$			42.1357i			1.46520i	−0.680658	+	0.732601i	$0.738306\pi$
$828$		0			0
$829$		−	41.5795i		−	1.44412i	−0.691832	−	0.722058i	$-0.743196\pi$
$829$		−	41.5795i		−	1.44412i	0.691832	−	0.722058i	$-0.256804\pi$
$830$	−27.0549	+	16.8754i	−0.939091	+	0.585755i
$831$		0			0
$832$	−9.09647	−	43.8396i	−0.315363	−	1.51987i
$833$	13.0246	−	10.7250i	0.451277	−	0.371598i
$834$		0			0
$835$		−	77.0343i		−	2.66588i
$836$	−28.0898	−	13.7565i	−0.971505	−	0.475780i
$837$		0			0
$838$	−11.7574	−	18.8497i	−0.406154	−	0.651153i
$839$	−40.2704			−1.39029			−0.695145	−	0.718870i	$-0.744660\pi$
$839$	−40.2704			−1.39029			−0.695145	+	0.718870i	$0.744660\pi$
$840$		0			0
$841$	−6.24843			−0.215463
$842$	8.99295	+	14.4176i	0.309918	+	0.496865i
$843$		0			0
$844$	−35.9582	−	17.6100i	−1.23773	−	0.606160i
$845$			71.6362i			2.46436i
$846$		0			0
$847$	16.0244	−	33.9530i	0.550606	−	1.16664i
$848$	13.8006	+	17.7821i	0.473915	+	0.610641i
$849$		0			0
$850$	29.7477	−	18.5550i	1.02034	−	0.636433i
$851$		−	20.3869i		−	0.698854i
$852$		0			0
$853$			28.2555i			0.967449i	0.875220	+	0.483724i	$0.160716\pi$
$853$			28.2555i			0.967449i	−0.875220	+	0.483724i	$0.839284\pi$
$854$	3.02601	+	25.8046i	0.103548	+	0.883016i
$855$		0			0
$856$	3.60296	+	35.0982i	0.123147	+	1.19963i
$857$		−	51.6638i		−	1.76480i	−0.470499	−	0.882400i	$-0.655926\pi$
$857$		−	51.6638i		−	1.76480i	0.470499	−	0.882400i	$-0.344074\pi$
$858$		0			0
$859$	−40.9603			−1.39755			−0.698774	−	0.715343i	$-0.746271\pi$
$859$	−40.9603			−1.39755			−0.698774	+	0.715343i	$0.746271\pi$
$860$	6.10656	−	12.4691i	0.208232	−	0.425193i
$861$		0			0
$862$	−12.0449	+	7.51296i	−0.410251	+	0.255892i
$863$		−	51.1388i		−	1.74078i	−0.492360	−	0.870392i	$-0.663866\pi$
$863$		−	51.1388i		−	1.74078i	0.492360	−	0.870392i	$-0.336134\pi$
$864$		0			0
$865$	2.42907			0.0825909
$866$	−30.8848	+	19.2643i	−1.04951	+	0.654628i
$867$		0			0
$868$	1.59403	+	1.99381i	0.0541049	+	0.0676742i
$869$	46.8258			1.58846
$870$		0			0
$871$	21.9424			0.743491
$872$	−16.6142	+	1.70552i	−0.562629	+	0.0577560i
$873$		0			0
$874$	−12.9085	+	8.05165i	−0.436638	+	0.272351i
$875$	−49.4443	−	23.3357i	−1.67152	−	0.788890i
$876$		0			0
$877$	15.5244			0.524223			0.262112	−	0.965038i	$-0.415581\pi$
$877$	15.5244			0.524223			0.262112	+	0.965038i	$0.415581\pi$
$878$	21.4546	+	34.3963i	0.724057	+	1.16082i
$879$		0			0
$880$	48.1237	+	62.0076i	1.62225	+	2.09027i
$881$		−	23.2196i		−	0.782287i	−0.920330	−	0.391144i	$-0.872080\pi$
$881$		−	23.2196i		−	0.782287i	0.920330	−	0.391144i	$-0.127920\pi$
$882$		0			0
$883$			5.98339i			0.201357i	0.994919	+	0.100678i	$0.0321013\pi$
$883$			5.98339i			0.201357i	−0.994919	+	0.100678i	$0.967899\pi$
$884$	−11.8661	+	24.2296i	−0.399100	+	0.814931i
$885$		0			0
$886$	−3.80019	+	2.37035i	−0.127670	+	0.0796336i
$887$	39.5090			1.32658			0.663291	−	0.748361i	$-0.269159\pi$
$887$	39.5090			1.32658			0.663291	+	0.748361i	$0.269159\pi$
$888$		0			0
$889$	−2.17949	−	1.02863i	−0.0730977	−	0.0344991i
$890$	−51.6629	−	82.8267i	−1.73174	−	2.77636i
$891$		0			0
$892$	−23.2210	+	47.4155i	−0.777498	+	1.58759i
$893$	−7.43547			−0.248819
$894$		0			0
$895$	−47.4219			−1.58514
$896$	−28.4012	+	9.45362i	−0.948818	+	0.315823i
$897$		0			0
$898$	−24.0563	−	38.5675i	−0.802770	−	1.28701i
$899$	−2.30104			−0.0767439
$900$		0			0
$901$		−	13.5634i		−	0.451862i
$902$	−5.67350	−	9.09585i	−0.188907	−	0.302859i
$903$		0			0
$904$	−43.1480	+	4.42931i	−1.43508	+	0.147317i
$905$	−84.2574			−2.80081
$906$		0			0
$907$		−	20.1580i		−	0.669335i	−0.942336	−	0.334667i	$-0.891376\pi$
$907$		−	20.1580i		−	0.669335i	0.942336	−	0.334667i	$-0.108624\pi$
$908$	−1.76421	+	3.60239i	−0.0585475	+	0.119549i
$909$		0			0
$910$	81.3148	−	9.53549i	2.69556	−	0.316098i
$911$			20.5534i			0.680965i	0.940251	+	0.340482i	$0.110590\pi$
$911$			20.5534i			0.680965i	−0.940251	+	0.340482i	$0.889410\pi$
$912$		0			0
$913$			28.9447i			0.957930i
$914$	15.2975	+	24.5252i	0.505996	+	0.811221i
$915$		0			0
$916$	25.8632	+	12.6661i	0.854546	+	0.418501i
$917$	18.0024	−	38.1439i	0.594491	−	1.25962i
$918$		0			0
$919$			54.4208i			1.79518i	0.440834	+	0.897588i	$0.354683\pi$
$919$			54.4208i			1.79518i	−0.440834	+	0.897588i	$0.645317\pi$
$920$	37.9795	−	3.89874i	1.25215	−	0.128538i
$921$		0			0
$922$	−7.37099	+	4.59763i	−0.242751	+	0.151415i
$923$	46.1852			1.52020
$924$		0			0
$925$	−60.7353			−1.99696
$926$	2.55779	−	1.59541i	0.0840542	−	0.0524285i
$927$		0			0
$928$	9.33090	−	25.3177i	0.306302	−	0.831093i
$929$			26.0631i			0.855103i	0.903991	+	0.427552i	$0.140624\pi$
$929$			26.0631i			0.855103i	−0.903991	+	0.427552i	$0.859376\pi$
$930$		0			0
$931$	13.8646	+	16.8376i	0.454395	+	0.551829i
$932$	−11.0863	+	22.6373i	−0.363143	+	0.741510i
$933$		0			0
$934$	8.59346	+	13.7772i	0.281187	+	0.450803i
$935$		−	47.2965i		−	1.54676i
$936$		0			0
$937$		−	0.657922i		−	0.0214934i	−0.999942	−	0.0107467i	$-0.996579\pi$
$937$		−	0.657922i		−	0.0214934i	0.999942	−	0.0107467i	$-0.00342084\pi$
$938$	−1.70854	−	14.5698i	−0.0557859	−	0.475720i
$939$		0			0
$940$	16.7577	+	8.20682i	0.546576	+	0.267677i
$941$			39.3462i			1.28265i	0.767269	+	0.641325i	$0.221615\pi$
$941$			39.3462i			1.28265i	−0.767269	+	0.641325i	$0.778385\pi$
$942$		0			0
$943$	−5.21447			−0.169806
$944$	−15.6483	−	20.1630i	−0.509310	−	0.656248i
$945$		0			0
$946$	−6.67003	−	10.6935i	−0.216861	−	0.347676i
$947$		−	29.2121i		−	0.949267i	−0.880184	−	0.474634i	$-0.842581\pi$
$947$		−	29.2121i		−	0.949267i	0.880184	−	0.474634i	$-0.157419\pi$
$948$		0			0
$949$	20.1010			0.652505
$950$	23.9870	+	38.4563i	0.778239	+	1.24769i
$951$		0			0
$952$	17.0124	+	5.99245i	0.551376	+	0.194217i
$953$	0.0993268			0.00321751			0.00160876	−	0.999999i	$-0.499488\pi$
$953$	0.0993268			0.00321751			0.00160876	+	0.999999i	$0.499488\pi$
$954$		0			0
$955$	−48.2794			−1.56228
$956$	−4.63905	−	2.27190i	−0.150037	−	0.0734786i
$957$		0			0
$958$	−10.7674	−	17.2625i	−0.347880	−	0.557726i
$959$	−0.709073	+	1.50240i	−0.0228971	+	0.0485151i
$960$		0			0
$961$	−30.7673			−0.992493
$962$	39.6549	−	24.7346i	1.27852	−	0.797475i
$963$		0			0
$964$	−17.3118	−	8.47817i	−0.557574	−	0.273063i
$965$		−	70.5799i		−	2.27205i
$966$		0			0
$967$		−	34.9194i		−	1.12293i	−0.827500	−	0.561466i	$-0.810238\pi$
$967$		−	34.9194i		−	1.12293i	0.827500	−	0.561466i	$-0.189762\pi$
$968$	39.9269	−	4.09865i	1.28330	−	0.131736i
$969$		0			0
$970$	45.5565	+	73.0369i	1.46273	+	2.34507i
$971$	−27.8736			−0.894506			−0.447253	−	0.894408i	$-0.647598\pi$
$971$	−27.8736			−0.894506			−0.447253	+	0.894408i	$0.647598\pi$
$972$		0			0
$973$	19.6084	−	41.5468i	0.628615	−	1.33193i
$974$	−24.4650	+	15.2600i	−0.783910	+	0.488961i
$975$		0			0
$976$	−21.9424	+	17.0294i	−0.702360	+	0.545097i
$977$	−6.42348			−0.205505			−0.102753	−	0.994707i	$-0.532765\pi$
$977$	−6.42348			−0.205505			−0.102753	+	0.994707i	$0.532765\pi$
$978$		0			0
$979$	−88.6121			−2.83205
$980$	−12.6631	−	53.2506i	−0.404509	−	1.70103i
$981$		0			0
$982$	23.9140	−	14.9163i	0.763127	−	0.475998i
$983$	−16.8411			−0.537147			−0.268574	−	0.963259i	$-0.586552\pi$
$983$	−16.8411			−0.537147			−0.268574	+	0.963259i	$0.586552\pi$
$984$		0			0
$985$			26.7618i			0.852703i
$986$	−13.7953	+	8.60475i	−0.439331	+	0.274031i
$987$		0			0
$988$	−31.3228	−	15.3398i	−0.996510	−	0.488026i
$989$	−6.13037			−0.194934
$990$		0			0
$991$		−	14.6792i		−	0.466299i	−0.972441	−	0.233149i	$-0.925097\pi$
$991$		−	14.6792i		−	0.466299i	0.972441	−	0.233149i	$-0.0749031\pi$
$992$	−0.943704	+	2.56057i	−0.0299626	+	0.0812981i
$993$		0			0
$994$	−3.59620	−	30.6670i	−0.114065	−	0.972697i
$995$		−	48.5481i		−	1.53908i
$996$		0			0
$997$			25.0288i			0.792671i	0.918106	+	0.396336i	$0.129718\pi$
$997$			25.0288i			0.792671i	−0.918106	+	0.396336i	$0.870282\pi$
$998$	9.15191	−	5.70847i	0.289699	−	0.180698i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.b.f.55.12	yes	16
3.2	odd	2		756.2.b.e.55.5	✓	16
4.3	odd	2		756.2.b.e.55.11	yes	16
7.6	odd	2		756.2.b.e.55.12	yes	16
12.11	even	2	inner	756.2.b.f.55.6	yes	16
21.20	even	2	inner	756.2.b.f.55.5	yes	16
28.27	even	2	inner	756.2.b.f.55.11	yes	16
84.83	odd	2		756.2.b.e.55.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.b.e.55.5	✓	16	3.2	odd	2
756.2.b.e.55.6	yes	16	84.83	odd	2
756.2.b.e.55.11	yes	16	4.3	odd	2
756.2.b.e.55.12	yes	16	7.6	odd	2
756.2.b.f.55.5	yes	16	21.20	even	2	inner
756.2.b.f.55.6	yes	16	12.11	even	2	inner
756.2.b.f.55.11	yes	16	28.27	even	2	inner
756.2.b.f.55.12	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.748450 + 1.19993i) q^{2} +(-0.879646 + 1.79617i) q^{4} -3.90968i q^{5} +(-1.12924 + 2.39266i) q^{7} +(-2.81364 + 0.288831i) q^{8} +O(q^{10})\)	\(q+(0.748450 + 1.19993i) q^{2} +(-0.879646 + 1.79617i) q^{4} -3.90968i q^{5} +(-1.12924 + 2.39266i) q^{7} +(-2.81364 + 0.288831i) q^{8} +(4.69133 - 2.92620i) q^{10} -5.01901i q^{11} -5.59667i q^{13} +(-3.71619 + 0.435784i) q^{14} +(-2.45244 - 3.15999i) q^{16} +2.41029i q^{17} -3.11590 q^{19} +(7.02245 + 3.43914i) q^{20} +(6.02245 - 3.75648i) q^{22} -3.45254i q^{23} -10.2856 q^{25} +(6.71560 - 4.18883i) q^{26} +(-3.30429 - 4.13300i) q^{28} +4.76986 q^{29} -0.482412 q^{31} +(1.95622 - 5.30784i) q^{32} +(-2.89217 + 1.80398i) q^{34} +(9.35454 + 4.41496i) q^{35} +5.90489 q^{37} +(-2.33209 - 3.73885i) q^{38} +(1.12924 + 11.0004i) q^{40} -1.51033i q^{41} -1.77561i q^{43} +(9.01499 + 4.41496i) q^{44} +(4.14280 - 2.58406i) q^{46} +2.38630 q^{47} +(-4.44965 - 5.40376i) q^{49} +(-7.69825 - 12.3420i) q^{50} +(10.0526 + 4.92309i) q^{52} -5.62728 q^{53} -19.6227 q^{55} +(2.48619 - 7.05825i) q^{56} +(3.57000 + 5.72348i) q^{58} +6.38071 q^{59} -6.94384i q^{61} +(-0.361061 - 0.578859i) q^{62} +(7.83315 - 1.62533i) q^{64} -21.8812 q^{65} +3.92062i q^{67} +(-4.32929 - 2.12021i) q^{68} +(1.70378 + 14.5291i) q^{70} +8.25225i q^{71} +3.59159i q^{73} +(4.41951 + 7.08543i) q^{74} +(2.74089 - 5.59667i) q^{76} +(12.0088 + 5.66766i) q^{77} +9.32969i q^{79} +(-12.3545 + 9.58827i) q^{80} +(1.81228 - 1.13040i) q^{82} -5.76701 q^{83} +9.42347 q^{85} +(2.13060 - 1.32895i) q^{86} +(1.44965 + 14.1217i) q^{88} -17.6553i q^{89} +(13.3909 + 6.31997i) q^{91} +(6.20135 + 3.03702i) q^{92} +(1.78603 + 2.86339i) q^{94} +12.1822i q^{95} +15.5685i q^{97} +(3.15378 - 9.38369i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + q^{14} + 4 q^{16} + 26 q^{20} + 10 q^{22} - 20 q^{25} + 6 q^{26} - 11 q^{28} + 6 q^{35} + 8 q^{37} + 20 q^{38} - 6 q^{46} + 8 q^{47} - 14 q^{49} - 21 q^{56} + 14 q^{58} + 44 q^{59} - 48 q^{62} + 24 q^{64} + 2 q^{68} - 27 q^{70} - 54 q^{80} - 4 q^{83} + 8 q^{85} - 34 q^{88} + 47 q^{98}+O(q^{100}) \) 16 * q + q^14 + 4 * q^16 + 26 * q^20 + 10 * q^22 - 20 * q^25 + 6 * q^26 - 11 * q^28 + 6 * q^35 + 8 * q^37 + 20 * q^38 - 6 * q^46 + 8 * q^47 - 14 * q^49 - 21 * q^56 + 14 * q^58 + 44 * q^59 - 48 * q^62 + 24 * q^64 + 2 * q^68 - 27 * q^70 - 54 * q^80 - 4 * q^83 + 8 * q^85 - 34 * q^88 + 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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