LMFDB - Embedded newform 400.2.bi.b.223.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(47,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 0, 17]))

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

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

chi := DirichletCharacter("400.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	400.bi (of order $20$, degree $8$, 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:	$3.19401608085$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
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} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 806 x^{12} - 2288 x^{11} + 5530 x^{10} - 11062 x^{9} + \cdots + 521 $ x^16 - 8x^15 + 52x^14 - 224x^13 + 806x^12 - 2288x^11 + 5530x^10 - 11062x^9 + 18924x^8 - 27056x^7 + 32750x^6 - 32636x^5 + 26711x^4 - 17122x^3 + 8368x^2 - 2746*x + 521
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			223.1
Root			$0.500000 - 0.509304i$ of defining polynomial
Character	$\chi$	$=$	400.223
Dual form			400.2.bi.b.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.648013 - 1.27180i) q^{3} +(0.166977 - 2.22982i) q^{5} +(2.92911 + 2.92911i) q^{7} +(0.565808 - 0.778768i) q^{9} +O(q^{10})$	\(q+(-0.648013 - 1.27180i) q^{3} +(0.166977 - 2.22982i) q^{5} +(2.92911 + 2.92911i) q^{7} +(0.565808 - 0.778768i) q^{9} +(2.25687 + 3.10632i) q^{11} +(-0.645322 - 4.07440i) q^{13} +(-2.94409 + 1.23259i) q^{15} +(-0.260074 - 0.132514i) q^{17} +(-2.41910 - 7.44523i) q^{19} +(1.82714 - 5.62334i) q^{21} +(0.240636 - 1.51932i) q^{23} +(-4.94424 - 0.744661i) q^{25} +(-5.58648 - 0.884812i) q^{27} +(5.30751 + 1.72451i) q^{29} +(1.44900 - 0.470809i) q^{31} +(2.48812 - 4.88322i) q^{33} +(7.02051 - 6.04232i) q^{35} +(-2.98459 + 0.472712i) q^{37} +(-4.76364 + 3.46098i) q^{39} +(5.49614 + 3.99318i) q^{41} +(-7.55900 + 7.55900i) q^{43} +(-1.64204 - 1.39169i) q^{45} +(2.05781 - 1.04851i) q^{47} +10.1594i q^{49} +0.416632i q^{51} +(7.08097 - 3.60793i) q^{53} +(7.30339 - 4.51375i) q^{55} +(-7.90122 + 7.90122i) q^{57} +(10.1729 + 7.39105i) q^{59} +(5.63043 - 4.09075i) q^{61} +(3.93842 - 0.623784i) q^{63} +(-9.19296 + 0.758621i) q^{65} +(-4.04794 + 7.94452i) q^{67} +(-2.08820 + 0.678497i) q^{69} +(-6.63736 - 2.15661i) q^{71} +(-14.8333 - 2.34937i) q^{73} +(2.25687 + 6.77062i) q^{75} +(-2.48812 + 15.7094i) q^{77} +(0.815963 - 2.51127i) q^{79} +(1.60242 + 4.93175i) q^{81} +(-2.28110 - 1.16228i) q^{83} +(-0.338910 + 0.557791i) q^{85} +(-1.24610 - 7.86758i) q^{87} +(6.85369 + 9.43329i) q^{89} +(10.0442 - 13.8246i) q^{91} +(-1.53775 - 1.53775i) q^{93} +(-17.0055 + 4.15099i) q^{95} +(1.71702 + 3.36984i) q^{97} +3.69606 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{9}+O(q^{10}) $ 16 * q - 20 * q^9	$ 16 q - 20 q^{9} + 20 q^{17} - 20 q^{21} - 20 q^{25} + 20 q^{29} + 60 q^{33} - 20 q^{37} + 28 q^{41} - 20 q^{45} + 60 q^{53} - 20 q^{57} - 12 q^{61} - 20 q^{65} - 80 q^{69} - 40 q^{73} - 60 q^{77} + 56 q^{81} - 60 q^{85} + 40 q^{97}+O(q^{100}) $ 16 * q - 20 * q^9 + 20 * q^17 - 20 * q^21 - 20 * q^25 + 20 * q^29 + 60 * q^33 - 20 * q^37 + 28 * q^41 - 20 * q^45 + 60 * q^53 - 20 * q^57 - 12 * q^61 - 20 * q^65 - 80 * q^69 - 40 * q^73 - 60 * q^77 + 56 * q^81 - 60 * q^85 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{11}{20}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.648013	−	1.27180i	−0.374131	−	0.734273i	0.624787	−	0.780795i	$-0.285186\pi$
$3$	−0.648013	−	1.27180i	−0.374131	−	0.734273i	−0.998917	+	0.0465230i	$0.985186\pi$
$4$		0			0
$5$	0.166977	−	2.22982i	0.0746746	−	0.997208i
$5$	0.166977	−	2.22982i	0.0746746	−	0.997208i
$6$		0			0
$7$	2.92911	+	2.92911i	1.10710	+	1.10710i	0.993530	+	0.113571i	$0.0362291\pi$
$7$	2.92911	+	2.92911i	1.10710	+	1.10710i	0.113571	+	0.993530i	$0.463771\pi$
$8$		0			0
$9$	0.565808	−	0.778768i	0.188603	−	0.259589i
$10$		0			0
$11$	2.25687	+	3.10632i	0.680473	+	0.936590i	0.999939	−	0.0110072i	$-0.00350377\pi$
$11$	2.25687	+	3.10632i	0.680473	+	0.936590i	−0.319467	+	0.947597i	$0.603504\pi$
$12$		0			0
$13$	−0.645322	−	4.07440i	−0.178980	−	1.13004i	−0.899603	−	0.436708i	$-0.856144\pi$
$13$	−0.645322	−	4.07440i	−0.178980	−	1.13004i	0.720623	−	0.693327i	$-0.243856\pi$
$14$		0			0
$15$	−2.94409	+	1.23259i	−0.760160	+	0.318254i
$16$		0			0
$17$	−0.260074	−	0.132514i	−0.0630771	−	0.0321394i	0.422167	−	0.906518i	$-0.361269\pi$
$17$	−0.260074	−	0.132514i	−0.0630771	−	0.0321394i	−0.485245	+	0.874378i	$0.661269\pi$
$18$		0			0
$19$	−2.41910	−	7.44523i	−0.554980	−	1.70805i	−0.695996	−	0.718046i	$-0.745037\pi$
$19$	−2.41910	−	7.44523i	−0.554980	−	1.70805i	0.141016	−	0.990007i	$-0.454963\pi$
$20$		0			0
$21$	1.82714	−	5.62334i	0.398714	−	1.22711i
$22$		0			0
$23$	0.240636	−	1.51932i	0.0501761	−	0.316799i	−0.949816	−	0.312809i	$-0.898730\pi$
$23$	0.240636	−	1.51932i	0.0501761	−	0.316799i	0.999992	−	0.00399005i	$-0.00127008\pi$
$24$		0			0
$25$	−4.94424	−	0.744661i	−0.988847	−	0.148932i
$26$		0			0
$27$	−5.58648	−	0.884812i	−1.07512	−	0.170282i
$28$		0			0
$29$	5.30751	+	1.72451i	0.985580	+	0.320234i	0.757089	−	0.653312i	$-0.226621\pi$
$29$	5.30751	+	1.72451i	0.985580	+	0.320234i	0.228491	+	0.973546i	$0.426621\pi$
$30$		0			0
$31$	1.44900	−	0.470809i	0.260248	−	0.0845598i	−0.175987	−	0.984393i	$-0.556312\pi$
$31$	1.44900	−	0.470809i	0.260248	−	0.0845598i	0.436235	+	0.899833i	$0.356312\pi$
$32$		0			0
$33$	2.48812	−	4.88322i	0.433127	−	0.850059i
$34$		0			0
$35$	7.02051	−	6.04232i	1.18668	−	1.02134i
$36$		0			0
$37$	−2.98459	+	0.472712i	−0.490663	+	0.0777134i	−0.396863	−	0.917878i	$-0.629901\pi$
$37$	−2.98459	+	0.472712i	−0.490663	+	0.0777134i	−0.0937997	+	0.995591i	$0.529901\pi$
$38$		0			0
$39$	−4.76364	+	3.46098i	−0.762792	+	0.554201i
$40$		0			0
$41$	5.49614	+	3.99318i	0.858353	+	0.623630i	0.927436	−	0.373981i	$-0.122007\pi$
$41$	5.49614	+	3.99318i	0.858353	+	0.623630i	−0.0690833	+	0.997611i	$0.522007\pi$
$42$		0			0
$43$	−7.55900	+	7.55900i	−1.15274	+	1.15274i	−0.166736	+	0.986002i	$0.553323\pi$
$43$	−7.55900	+	7.55900i	−1.15274	+	1.15274i	−0.986002	+	0.166736i	$0.946677\pi$
$44$		0			0
$45$	−1.64204	−	1.39169i	−0.244781	−	0.207461i
$46$		0			0
$47$	2.05781	−	1.04851i	0.300163	−	0.152941i	−0.297423	−	0.954746i	$-0.596127\pi$
$47$	2.05781	−	1.04851i	0.300163	−	0.152941i	0.597585	+	0.801805i	$0.296127\pi$
$48$		0			0
$49$			10.1594i			1.45135i
$50$		0			0
$51$			0.416632i			0.0583401i
$52$		0			0
$53$	7.08097	−	3.60793i	0.972645	−	0.495588i	0.105921	−	0.994375i	$-0.466221\pi$
$53$	7.08097	−	3.60793i	0.972645	−	0.495588i	0.866725	+	0.498787i	$0.166221\pi$
$54$		0			0
$55$	7.30339	−	4.51375i	0.984789	−	0.608633i
$56$		0			0
$57$	−7.90122	+	7.90122i	−1.04654	+	1.04654i
$58$		0			0
$59$	10.1729	+	7.39105i	1.32440	+	0.962233i	0.999866	+	0.0163593i	$0.00520757\pi$
$59$	10.1729	+	7.39105i	1.32440	+	0.962233i	0.324534	+	0.945874i	$0.394792\pi$
$60$		0			0
$61$	5.63043	−	4.09075i	0.720903	−	0.523766i	−0.165770	−	0.986164i	$-0.553011\pi$
$61$	5.63043	−	4.09075i	0.720903	−	0.523766i	0.886672	+	0.462398i	$0.153011\pi$
$62$		0			0
$63$	3.93842	−	0.623784i	0.496194	−	0.0785894i
$64$		0			0
$65$	−9.19296	+	0.758621i	−1.14025	+	0.0940954i
$66$		0			0
$67$	−4.04794	+	7.94452i	−0.494534	+	0.970578i	0.499986	+	0.866033i	$0.333338\pi$
$67$	−4.04794	+	7.94452i	−0.494534	+	0.970578i	−0.994520	+	0.104545i	$0.966662\pi$
$68$		0			0
$69$	−2.08820	+	0.678497i	−0.251389	+	0.0816814i
$70$		0			0
$71$	−6.63736	−	2.15661i	−0.787710	−	0.255942i	−0.112581	−	0.993643i	$-0.535912\pi$
$71$	−6.63736	−	2.15661i	−0.787710	−	0.255942i	−0.675129	+	0.737700i	$0.735912\pi$
$72$		0			0
$73$	−14.8333	−	2.34937i	−1.73611	−	0.274973i	−0.793427	−	0.608665i	$-0.791705\pi$
$73$	−14.8333	−	2.34937i	−1.73611	−	0.274973i	−0.942682	+	0.333693i	$0.891705\pi$
$74$		0			0
$75$	2.25687	+	6.77062i	0.260601	+	0.781804i
$76$		0			0
$77$	−2.48812	+	15.7094i	−0.283548	+	1.79025i
$78$		0			0
$79$	0.815963	−	2.51127i	0.0918030	−	0.282540i	−0.894604	−	0.446859i	$-0.852543\pi$
$79$	0.815963	−	2.51127i	0.0918030	−	0.282540i	0.986407	+	0.164319i	$0.0525425\pi$
$80$		0			0
$81$	1.60242	+	4.93175i	0.178047	+	0.547972i
$82$		0			0
$83$	−2.28110	−	1.16228i	−0.250383	−	0.127577i	0.324295	−	0.945956i	$-0.394873\pi$
$83$	−2.28110	−	1.16228i	−0.250383	−	0.127577i	−0.574679	+	0.818379i	$0.694873\pi$
$84$		0			0
$85$	−0.338910	+	0.557791i	−0.0367599	+	0.0605010i
$86$		0			0
$87$	−1.24610	−	7.86758i	−0.133596	−	0.843493i
$88$		0			0
$89$	6.85369	+	9.43329i	0.726490	+	0.999927i	0.999283	+	0.0378546i	$0.0120524\pi$
$89$	6.85369	+	9.43329i	0.726490	+	0.999927i	−0.272794	+	0.962073i	$0.587948\pi$
$90$		0			0
$91$	10.0442	−	13.8246i	1.05291	−	1.44921i
$92$		0			0
$93$	−1.53775	−	1.53775i	−0.159457	−	0.159457i
$94$		0			0
$95$	−17.0055	+	4.15099i	−1.74473	+	0.425882i
$96$		0			0
$97$	1.71702	+	3.36984i	0.174337	+	0.342155i	0.961597	−	0.274466i	$-0.0885011\pi$
$97$	1.71702	+	3.36984i	0.174337	+	0.342155i	−0.787260	+	0.616621i	$0.788501\pi$
$98$		0			0
$99$	3.69606			0.371468
$100$		0			0
$101$	11.8096			1.17510			0.587551	−	0.809187i	$-0.300092\pi$
$101$	11.8096			1.17510			0.587551	+	0.809187i	$0.300092\pi$
$102$		0			0
$103$	2.75616	+	5.40928i	0.271573	+	0.532992i	0.986006	−	0.166712i	$-0.0533151\pi$
$103$	2.75616	+	5.40928i	0.271573	+	0.532992i	−0.714433	+	0.699704i	$0.753315\pi$
$104$		0			0
$105$	−12.2340	−	5.01316i	−1.19391	−	0.489235i
$106$		0			0
$107$	−2.65974	−	2.65974i	−0.257127	−	0.257127i	0.566758	−	0.823884i	$-0.308198\pi$
$107$	−2.65974	−	2.65974i	−0.257127	−	0.257127i	−0.823884	+	0.566758i	$0.808198\pi$
$108$		0			0
$109$	−2.79215	+	3.84307i	−0.267440	+	0.368099i	−0.921523	−	0.388323i	$-0.873054\pi$
$109$	−2.79215	+	3.84307i	−0.267440	+	0.368099i	0.654084	+	0.756422i	$0.273054\pi$
$110$		0			0
$111$	2.53525	+	3.48947i	0.240635	+	0.331206i
$112$		0			0
$113$	1.61719	+	10.2106i	0.152133	+	0.960528i	0.939128	+	0.343569i	$0.111636\pi$
$113$	1.61719	+	10.2106i	0.152133	+	0.960528i	−0.786995	+	0.616960i	$0.788364\pi$
$114$		0			0
$115$	−3.34763	−	0.790268i	−0.312168	−	0.0736929i
$116$		0			0
$117$	−3.53814	−	1.80277i	−0.327101	−	0.166666i
$118$		0			0
$119$	−0.373636	−	1.14993i	−0.0342512	−	0.105414i
$120$		0			0
$121$	−1.15655	+	3.55951i	−0.105141	+	0.323592i
$122$		0			0
$123$	1.51694	−	9.57761i	0.136778	−	0.863584i
$124$		0			0
$125$	−2.48604	+	10.9004i	−0.222358	+	0.974965i
$126$		0			0
$127$	9.12441	+	1.44516i	0.809660	+	0.128238i	0.547520	−	0.836792i	$-0.315572\pi$
$127$	9.12441	+	1.44516i	0.809660	+	0.128238i	0.262140	+	0.965030i	$0.415572\pi$
$128$		0			0
$129$	14.5119	+	4.71519i	1.27770	+	0.415149i
$130$		0			0
$131$	−20.0702	+	6.52122i	−1.75355	+	0.569761i	−0.996499	−	0.0836006i	$-0.973358\pi$
$131$	−20.0702	+	6.52122i	−1.75355	+	0.569761i	−0.757046	+	0.653362i	$0.773358\pi$
$132$		0			0
$133$	14.7221	−	28.8938i	1.27657	−	2.50541i
$134$		0			0
$135$	−2.90579	+	12.3091i	−0.250091	+	1.05940i
$136$		0			0
$137$	−7.25033	+	1.14834i	−0.619437	+	0.0981092i	−0.458263	−	0.888817i	$-0.651528\pi$
$137$	−7.25033	+	1.14834i	−0.619437	+	0.0981092i	−0.161174	+	0.986926i	$0.551528\pi$
$138$		0			0
$139$	−8.32767	+	6.05041i	−0.706344	+	0.513189i	−0.881992	−	0.471264i	$-0.843798\pi$
$139$	−8.32767	+	6.05041i	−0.706344	+	0.513189i	0.175648	+	0.984453i	$0.443798\pi$
$140$		0			0
$141$	−2.66698	−	1.93767i	−0.224600	−	0.163181i
$142$		0			0
$143$	11.2000	−	11.2000i	0.936589	−	0.936589i
$144$		0			0
$145$	4.73160	−	11.5469i	0.392938	−	0.958914i
$146$		0			0
$147$	12.9207	−	6.58344i	1.06568	−	0.542993i
$148$		0			0
$149$			11.8576i			0.971416i	0.874121	+	0.485708i	$0.161438\pi$
$149$			11.8576i			0.971416i	−0.874121	+	0.485708i	$0.838562\pi$
$150$		0			0
$151$			13.0162i			1.05925i	0.848233	+	0.529624i	$0.177667\pi$
$151$			13.0162i			1.05925i	−0.848233	+	0.529624i	$0.822333\pi$
$152$		0			0
$153$	−0.250349	+	0.127559i	−0.0202396	+	0.0103126i
$154$		0			0
$155$	−0.807871	−	3.30963i	−0.0648898	−	0.265836i
$156$		0			0
$157$	−2.52932	+	2.52932i	−0.201861	+	0.201861i	−0.800797	−	0.598936i	$-0.795590\pi$
$157$	−2.52932	+	2.52932i	−0.201861	+	0.201861i	0.598936	+	0.800797i	$0.295590\pi$
$158$		0			0
$159$	−9.17712	−	6.66757i	−0.727793	−	0.528772i
$160$		0			0
$161$	5.15510	−	3.74540i	0.406279	−	0.295179i
$162$		0			0
$163$	14.3738	−	2.27659i	1.12584	−	0.178316i	0.434382	−	0.900729i	$-0.356967\pi$
$163$	14.3738	−	2.27659i	1.12584	−	0.178316i	0.691463	+	0.722412i	$0.256967\pi$
$164$		0			0
$165$	−10.4733	−	6.36347i	−0.815342	−	0.495395i
$166$		0			0
$167$	3.25120	−	6.38084i	0.251586	−	0.493765i	−0.730327	−	0.683097i	$-0.760632\pi$
$167$	3.25120	−	6.38084i	0.251586	−	0.493765i	0.981913	+	0.189333i	$0.0606325\pi$
$168$		0			0
$169$	−3.82057	+	1.24138i	−0.293890	+	0.0954907i
$170$		0			0
$171$	−7.16686	−	2.32865i	−0.548063	−	0.178077i
$172$		0			0
$173$	−7.30401	−	1.15684i	−0.555313	−	0.0879530i	−0.127531	−	0.991835i	$-0.540705\pi$
$173$	−7.30401	−	1.15684i	−0.555313	−	0.0879530i	−0.427783	+	0.903882i	$0.640705\pi$
$174$		0			0
$175$	−12.3010	−	16.6634i	−0.929871	−	1.25964i
$176$		0			0
$177$	2.80774	−	17.7274i	0.211043	−	1.33247i
$178$		0			0
$179$	−1.32026	+	4.06333i	−0.0986805	+	0.303707i	−0.988195	−	0.153199i	$-0.951042\pi$
$179$	−1.32026	+	4.06333i	−0.0986805	+	0.303707i	0.889515	+	0.456906i	$0.151042\pi$
$180$		0			0
$181$	−5.03463	−	15.4950i	−0.374221	−	1.15173i	−0.944003	−	0.329938i	$-0.892972\pi$
$181$	−5.03463	−	15.4950i	−0.374221	−	1.15173i	0.569782	−	0.821796i	$-0.307028\pi$
$182$		0			0
$183$	−8.85119	−	4.50991i	−0.654299	−	0.333382i
$184$		0			0
$185$	0.555707	+	6.73404i	0.0408564	+	0.495096i
$186$		0			0
$187$	−0.175322	−	1.10694i	−0.0128208	−	0.0809474i
$188$		0			0
$189$	−13.7717	−	18.9552i	−1.00175	−	1.37878i
$190$		0			0
$191$	−0.396240	+	0.545378i	−0.0286709	+	0.0394622i	−0.823112	−	0.567879i	$-0.807764\pi$
$191$	−0.396240	+	0.545378i	−0.0286709	+	0.0394622i	0.794441	+	0.607341i	$0.207764\pi$
$192$		0			0
$193$	−1.94252	−	1.94252i	−0.139826	−	0.139826i	0.633729	−	0.773555i	$-0.281523\pi$
$193$	−1.94252	−	1.94252i	−0.139826	−	0.139826i	−0.773555	+	0.633729i	$0.781523\pi$
$194$		0			0
$195$	6.92197	+	11.2000i	0.495692	+	0.802047i
$196$		0			0
$197$	3.35301	+	6.58064i	0.238892	+	0.468852i	0.979060	−	0.203570i	$-0.0652544\pi$
$197$	3.35301	+	6.58064i	0.238892	+	0.468852i	−0.740169	+	0.672421i	$0.765254\pi$
$198$		0			0
$199$	5.25484			0.372505			0.186253	−	0.982502i	$-0.440366\pi$
$199$	5.25484			0.372505			0.186253	+	0.982502i	$0.440366\pi$
$200$		0			0
$201$	12.7269			0.897689
$202$		0			0
$203$	10.4950	+	20.5976i	0.736605	+	1.44567i
$204$		0			0
$205$	9.82182	−	11.5887i	0.685986	−	0.809387i
$206$		0			0
$207$	−1.04704	−	1.04704i	−0.0727744	−	0.0727744i
$208$		0			0
$209$	17.6677	−	24.3174i	1.22210	−	1.68207i
$210$		0			0
$211$	4.76815	+	6.56279i	0.328253	+	0.451801i	0.940964	−	0.338505i	$-0.109921\pi$
$211$	4.76815	+	6.56279i	0.328253	+	0.451801i	−0.612712	+	0.790307i	$0.709921\pi$
$212$		0			0
$213$	1.55833	+	9.83889i	0.106775	+	0.674149i
$214$		0			0
$215$	15.5931	+	18.1174i	1.06344	+	1.23560i
$216$		0			0
$217$	5.62334	+	2.86524i	0.381738	+	0.194505i
$218$		0			0
$219$	6.62427	+	20.3874i	0.447627	+	1.37765i
$220$		0			0
$221$	−0.372084	+	1.14516i	−0.0250291	+	0.0770317i
$222$		0			0
$223$	−0.586047	+	3.70015i	−0.0392446	+	0.247781i	−0.999510	−	0.0313086i	$-0.990033\pi$
$223$	−0.586047	+	3.70015i	−0.0392446	+	0.247781i	0.960265	+	0.279089i	$0.0900325\pi$
$224$		0			0
$225$	−3.37741	+	3.42908i	−0.225161	+	0.228605i
$226$		0			0
$227$	−9.19209	−	1.45588i	−0.610101	−	0.0966304i	−0.156264	−	0.987715i	$-0.549945\pi$
$227$	−9.19209	−	1.45588i	−0.610101	−	0.0966304i	−0.453837	+	0.891085i	$0.649945\pi$
$228$		0			0
$229$	5.57028	+	1.80989i	0.368094	+	0.119601i	0.487223	−	0.873278i	$-0.338010\pi$
$229$	5.57028	+	1.80989i	0.368094	+	0.119601i	−0.119128	+	0.992879i	$0.538010\pi$
$230$		0			0
$231$	21.5915	−	7.01551i	1.42062	−	0.461587i
$232$		0			0
$233$	3.76580	−	7.39079i	0.246706	−	0.484187i	−0.734134	−	0.679005i	$-0.762412\pi$
$233$	3.76580	−	7.39079i	0.246706	−	0.484187i	0.980839	+	0.194818i	$0.0624116\pi$
$234$		0			0
$235$	−1.99438	−	4.76364i	−0.130099	−	0.310745i
$236$		0			0
$237$	−3.72259	+	0.589600i	−0.241808	+	0.0382986i
$238$		0			0
$239$	20.0914	−	14.5973i	1.29961	−	0.944219i	0.299654	−	0.954048i	$-0.403129\pi$
$239$	20.0914	−	14.5973i	1.29961	−	0.944219i	0.999952	+	0.00982862i	$0.00312860\pi$
$240$		0			0
$241$	−5.83422	−	4.23881i	−0.375815	−	0.273046i	0.383803	−	0.923415i	$-0.374614\pi$
$241$	−5.83422	−	4.23881i	−0.375815	−	0.273046i	−0.759618	+	0.650369i	$0.774614\pi$
$242$		0			0
$243$	−6.76464	+	6.76464i	−0.433952	+	0.433952i
$244$		0			0
$245$	22.6537	+	1.69639i	1.44729	+	0.108379i
$246$		0			0
$247$	−28.7738	+	14.6610i	−1.83083	+	0.932855i
$248$		0			0
$249$			3.65427i			0.231580i
$250$		0			0
$251$		−	29.3481i		−	1.85244i	−0.376988	−	0.926218i	$-0.623040\pi$
$251$		−	29.3481i		−	1.85244i	0.376988	−	0.926218i	$-0.376960\pi$
$252$		0			0
$253$	5.26257	−	2.68141i	0.330855	−	0.168579i
$254$		0			0
$255$	0.929016	+	0.0695681i	0.0581772	+	0.00435652i
$256$		0			0
$257$	16.1302	−	16.1302i	1.00617	−	1.00617i	0.00619398	−	0.999981i	$-0.498028\pi$
$257$	16.1302	−	16.1302i	1.00617	−	1.00617i	0.999981	−	0.00619398i	$-0.00197162\pi$
$258$		0			0
$259$	−10.1268	−	7.35757i	−0.629250	−	0.457177i
$260$		0			0
$261$	4.34603	−	3.15757i	0.269012	−	0.195449i
$262$		0			0
$263$	−21.4105	+	3.39109i	−1.32023	+	0.209103i	−0.776478	−	0.630145i	$-0.782995\pi$
$263$	−21.4105	+	3.39109i	−1.32023	+	0.209103i	−0.543749	+	0.839248i	$0.682995\pi$
$264$		0			0
$265$	−6.86269	−	16.3918i	−0.421572	−	1.00694i
$266$		0			0
$267$	7.55596	−	14.8294i	0.462417	−	0.907545i
$268$		0			0
$269$	−28.4092	+	9.23070i	−1.73214	+	0.562806i	−0.993756	−	0.111577i	$-0.964410\pi$
$269$	−28.4092	+	9.23070i	−1.73214	+	0.562806i	−0.738382	+	0.674382i	$0.764410\pi$
$270$		0			0
$271$	−16.2853	−	5.29141i	−0.989261	−	0.321430i	−0.230694	−	0.973026i	$-0.574100\pi$
$271$	−16.2853	−	5.29141i	−0.989261	−	0.321430i	−0.758566	+	0.651596i	$0.774100\pi$
$272$		0			0
$273$	−24.0909	−	3.81562i	−1.45804	−	0.230932i
$274$		0			0
$275$	−8.84536	−	17.0390i	−0.533395	−	1.02749i
$276$		0			0
$277$	−1.85546	+	11.7149i	−0.111484	+	0.703882i	0.867115	+	0.498107i	$0.165971\pi$
$277$	−1.85546	+	11.7149i	−0.111484	+	0.703882i	−0.978599	+	0.205775i	$0.934029\pi$
$278$		0			0
$279$	0.453206	−	1.39482i	0.0271327	−	0.0835059i
$280$		0			0
$281$	5.76571	+	17.7450i	0.343953	+	1.05858i	0.962142	+	0.272549i	$0.0878668\pi$
$281$	5.76571	+	17.7450i	0.343953	+	1.05858i	−0.618189	+	0.786030i	$0.712133\pi$
$282$		0			0
$283$	−14.0251	−	7.14616i	−0.833708	−	0.424795i	−0.0156126	−	0.999878i	$-0.504970\pi$
$283$	−14.0251	−	7.14616i	−0.833708	−	0.424795i	−0.818095	+	0.575083i	$0.804970\pi$
$284$		0			0
$285$	16.2990	+	18.9377i	0.965469	+	1.12177i
$286$		0			0
$287$	4.40234	+	27.7953i	0.259862	+	1.64071i
$288$		0			0
$289$	−9.94227	−	13.6844i	−0.584839	−	0.804962i
$290$		0			0
$291$	3.17310	−	4.36740i	0.186010	−	0.256021i
$292$		0			0
$293$	23.5937	+	23.5937i	1.37836	+	1.37836i	0.847398	+	0.530959i	$0.178168\pi$
$293$	23.5937	+	23.5937i	1.37836	+	1.37836i	0.530959	+	0.847398i	$0.321832\pi$
$294$		0			0
$295$	18.1794	−	21.4497i	1.05845	−	1.24885i
$296$		0			0
$297$	−9.85947	−	19.3503i	−0.572104	−	1.12282i
$298$		0			0
$299$	−6.34559			−0.366975
$300$		0			0
$301$	−44.2824			−2.55239
$302$		0			0
$303$	−7.65280	−	15.0195i	−0.439642	−	0.862846i
$304$		0			0
$305$	−8.18149	−	13.2379i	−0.468471	−	0.758002i
$306$		0			0
$307$	−0.100680	−	0.100680i	−0.00574611	−	0.00574611i	0.704228	−	0.709974i	$-0.251293\pi$
$307$	−0.100680	−	0.100680i	−0.00574611	−	0.00574611i	−0.709974	+	0.704228i	$0.751293\pi$
$308$		0			0
$309$	5.09347	−	7.01057i	0.289758	−	0.398817i
$310$		0			0
$311$	6.25063	+	8.60326i	0.354441	+	0.487846i	0.948589	−	0.316509i	$-0.102511\pi$
$311$	6.25063	+	8.60326i	0.354441	+	0.487846i	−0.594149	+	0.804355i	$0.702511\pi$
$312$		0			0
$313$	3.65646	+	23.0860i	0.206676	+	1.30490i	0.844849	+	0.535005i	$0.179690\pi$
$313$	3.65646	+	23.0860i	0.206676	+	1.30490i	−0.638173	+	0.769893i	$0.720310\pi$
$314$		0			0
$315$	−0.733302	−	8.88614i	−0.0413169	−	0.500677i
$316$		0			0
$317$	−8.59220	−	4.37794i	−0.482586	−	0.245890i	0.195733	−	0.980657i	$-0.437291\pi$
$317$	−8.59220	−	4.37794i	−0.482586	−	0.245890i	−0.678319	+	0.734767i	$0.737291\pi$
$318$		0			0
$319$	6.62148	+	20.3788i	0.370732	+	1.14099i
$320$		0			0
$321$	−1.65910	+	5.10619i	−0.0926021	+	0.285000i
$322$		0			0
$323$	−0.357453	+	2.25687i	−0.0198892	+	0.125576i
$324$		0			0
$325$	0.156576	+	20.6254i	0.00868527	+	1.14409i
$326$		0			0
$327$	6.69695	+	1.06069i	0.370342	+	0.0586565i
$328$		0			0
$329$	9.09876	+	2.95637i	0.501631	+	0.162990i
$330$		0			0
$331$	−9.37813	+	3.04714i	−0.515469	+	0.167486i	−0.555188	−	0.831725i	$-0.687354\pi$
$331$	−9.37813	+	3.04714i	−0.515469	+	0.167486i	0.0397191	+	0.999211i	$0.487354\pi$
$332$		0			0
$333$	−1.32057	+	2.59177i	−0.0723668	+	0.142028i
$334$		0			0
$335$	17.0390	+	10.3527i	0.930939	+	0.565631i
$336$		0			0
$337$	32.0835	−	5.08152i	1.74770	−	0.276808i	0.800937	−	0.598748i	$-0.204335\pi$
$337$	32.0835	−	5.08152i	1.74770	−	0.276808i	0.946760	+	0.321940i	$0.104335\pi$
$338$		0			0
$339$	11.9378	−	8.67331i	0.648372	−	0.471070i
$340$		0			0
$341$	4.73269	+	3.43850i	0.256290	+	0.186205i
$342$		0			0
$343$	−9.25432	+	9.25432i	−0.499686	+	0.499686i
$344$		0			0
$345$	1.16425	+	4.76961i	0.0626809	+	0.256787i
$346$		0			0
$347$	−10.8276	+	5.51691i	−0.581253	+	0.296163i	−0.719788	−	0.694194i	$-0.755761\pi$
$347$	−10.8276	+	5.51691i	−0.581253	+	0.296163i	0.138534	+	0.990358i	$0.455761\pi$
$348$		0			0
$349$			13.9855i			0.748629i	0.927302	+	0.374314i	$0.122122\pi$
$349$			13.9855i			0.748629i	−0.927302	+	0.374314i	$0.877878\pi$
$350$		0			0
$351$			23.3326i			1.24540i
$352$		0			0
$353$	−10.8959	+	5.55174i	−0.579930	+	0.295489i	−0.719242	−	0.694759i	$-0.755511\pi$
$353$	−10.8959	+	5.55174i	−0.579930	+	0.295489i	0.139312	+	0.990249i	$0.455511\pi$
$354$		0			0
$355$	−5.91715	+	14.4400i	−0.314050	+	0.766398i
$356$		0			0
$357$	−1.22036	+	1.22036i	−0.0645884	+	0.0645884i
$358$		0			0
$359$	−16.3725	−	11.8953i	−0.864105	−	0.627809i	0.0648934	−	0.997892i	$-0.479329\pi$
$359$	−16.3725	−	11.8953i	−0.864105	−	0.627809i	−0.928999	+	0.370083i	$0.879329\pi$
$360$		0			0
$361$	−34.2081	+	24.8536i	−1.80043	+	1.30809i
$362$		0			0
$363$	5.27644	−	0.835705i	0.276941	−	0.0438632i
$364$		0			0
$365$	−7.71551	+	32.6834i	−0.403848	+	1.71073i
$366$		0			0
$367$	4.25004	−	8.34118i	0.221850	−	0.435406i	−0.753076	−	0.657934i	$-0.771431\pi$
$367$	4.25004	−	8.34118i	0.221850	−	0.435406i	0.974926	+	0.222528i	$0.0714307\pi$
$368$		0			0
$369$	6.21952	−	2.02085i	0.323776	−	0.105201i
$370$		0			0
$371$	31.3090	+	10.1729i	1.62548	+	0.528151i
$372$		0			0
$373$	−32.0548	−	5.07698i	−1.65974	−	0.262876i	−0.745040	−	0.667020i	$-0.767570\pi$
$373$	−32.0548	−	5.07698i	−1.65974	−	0.262876i	−0.914696	+	0.404143i	$0.867570\pi$
$374$		0			0
$375$	15.4741	−	3.90189i	0.799081	−	0.201493i
$376$		0			0
$377$	3.60131	−	22.7378i	0.185477	−	1.17106i
$378$		0			0
$379$	3.21609	−	9.89812i	0.165200	−	0.508432i	−0.833851	−	0.551989i	$-0.813869\pi$
$379$	3.21609	−	9.89812i	0.165200	−	0.508432i	0.999051	+	0.0435569i	$0.0138690\pi$
$380$		0			0
$381$	−4.07478	−	12.5409i	−0.208757	−	0.642489i
$382$		0			0
$383$	26.0530	+	13.2746i	1.33124	+	0.678303i	0.967423	−	0.253166i	$-0.0814719\pi$
$383$	26.0530	+	13.2746i	1.33124	+	0.678303i	0.363821	+	0.931469i	$0.381472\pi$
$384$		0			0
$385$	34.6138	+	8.17120i	1.76408	+	0.416443i
$386$		0			0
$387$	1.60977	+	10.1637i	0.0818290	+	0.516648i
$388$		0			0
$389$	−15.6435	−	21.5314i	−0.793155	−	1.09168i	−0.993708	−	0.112002i	$-0.964274\pi$
$389$	−15.6435	−	21.5314i	−0.793155	−	1.09168i	0.200553	−	0.979683i	$-0.435726\pi$
$390$		0			0
$391$	−0.263914	+	0.363246i	−0.0133467	+	0.0183702i
$392$		0			0
$393$	21.2994	+	21.2994i	1.07441	+	1.07441i
$394$		0			0
$395$	−5.46345	−	2.23878i	−0.274896	−	0.112645i
$396$		0			0
$397$	10.2475	+	20.1119i	0.514308	+	1.00939i	0.991441	+	0.130556i	$0.0416761\pi$
$397$	10.2475	+	20.1119i	0.514308	+	1.00939i	−0.477133	+	0.878831i	$0.658324\pi$
$398$		0			0
$399$	−46.2871			−2.31725
$400$		0			0
$401$	−6.67802			−0.333484			−0.166742	−	0.986001i	$-0.553325\pi$
$401$	−6.67802			−0.333484			−0.166742	+	0.986001i	$0.553325\pi$
$402$		0			0
$403$	−2.85334	−	5.59999i	−0.142135	−	0.278955i
$404$		0			0
$405$	11.2645	−	2.74963i	0.559738	−	0.136630i
$406$		0			0
$407$	−8.20423	−	8.20423i	−0.406669	−	0.406669i
$408$		0			0
$409$	−2.62566	+	3.61391i	−0.129831	+	0.178696i	−0.868983	−	0.494841i	$-0.835226\pi$
$409$	−2.62566	+	3.61391i	−0.129831	+	0.178696i	0.739153	+	0.673538i	$0.235226\pi$
$410$		0			0
$411$	6.15876	+	8.47681i	0.303789	+	0.418130i
$412$		0			0
$413$	8.14838	+	51.4469i	0.400956	+	2.53154i
$414$		0			0
$415$	−2.97257	+	4.89238i	−0.145918	+	0.240158i
$416$		0			0
$417$	13.0913	+	6.67037i	0.641085	+	0.326649i
$418$		0			0
$419$	−10.9171	−	33.5994i	−0.533335	−	1.64144i	−0.747219	−	0.664577i	$-0.768612\pi$
$419$	−10.9171	−	33.5994i	−0.533335	−	1.64144i	0.213885	−	0.976859i	$-0.431388\pi$
$420$		0			0
$421$	−2.30880	+	7.10575i	−0.112524	+	0.346313i	−0.991423	−	0.130696i	$-0.958279\pi$
$421$	−2.30880	+	7.10575i	−0.112524	+	0.346313i	0.878899	+	0.477009i	$0.158279\pi$
$422$		0			0
$423$	0.347782	−	2.19581i	0.0169098	−	0.106764i
$424$		0			0
$425$	1.18719	+	0.848848i	0.0575870	+	0.0411752i
$426$		0			0
$427$	28.4744	+	4.50991i	1.37797	+	0.218250i
$428$		0			0
$429$	−21.5018	−	6.98637i	−1.03812	−	0.337305i
$430$		0			0
$431$	1.66992	−	0.542590i	0.0804372	−	0.0261356i	−0.268522	−	0.963274i	$-0.586535\pi$
$431$	1.66992	−	0.542590i	0.0804372	−	0.0261356i	0.348959	+	0.937138i	$0.386535\pi$
$432$		0			0
$433$	1.36660	−	2.68210i	0.0656745	−	0.128894i	−0.855835	−	0.517248i	$-0.826956\pi$
$433$	1.36660	−	2.68210i	0.0656745	−	0.128894i	0.921510	+	0.388355i	$0.126956\pi$
$434$		0			0
$435$	−17.7514	+	1.46488i	−0.851115	+	0.0702357i
$436$		0			0
$437$	−11.8938	+	1.88379i	−0.568957	+	0.0901139i
$438$		0			0
$439$	20.5365	−	14.9207i	0.980156	−	0.712125i	0.0224121	−	0.999749i	$-0.492865\pi$
$439$	20.5365	−	14.9207i	0.980156	−	0.712125i	0.957743	+	0.287624i	$0.0928654\pi$
$440$		0			0
$441$	7.91184	+	5.74829i	0.376754	+	0.273728i
$442$		0			0
$443$	6.77536	−	6.77536i	0.321907	−	0.321907i	−0.527591	−	0.849498i	$-0.676905\pi$
$443$	6.77536	−	6.77536i	0.321907	−	0.321907i	0.849498	+	0.527591i	$0.176905\pi$
$444$		0			0
$445$	22.1790	−	13.7074i	1.05139	−	0.649792i
$446$		0			0
$447$	15.0805	−	7.68391i	0.713284	−	0.363437i
$448$		0			0
$449$		−	20.3687i		−	0.961260i	−0.876924	−	0.480630i	$-0.840408\pi$
$449$		−	20.3687i		−	0.961260i	0.876924	−	0.480630i	$-0.159592\pi$
$450$		0			0
$451$			26.0849i			1.22829i
$452$		0			0
$453$	16.5540	−	8.43470i	0.777776	−	0.396297i
$454$		0			0
$455$	−29.1493	−	24.7051i	−1.36654	−	1.15819i
$456$		0			0
$457$	5.54512	−	5.54512i	0.259390	−	0.259390i	−0.565416	−	0.824806i	$-0.691284\pi$
$457$	5.54512	−	5.54512i	0.259390	−	0.259390i	0.824806	+	0.565416i	$0.191284\pi$
$458$		0			0
$459$	1.33565	+	0.970403i	0.0623426	+	0.0452946i
$460$		0			0
$461$	−10.0726	+	7.31820i	−0.469130	+	0.340843i	−0.797102	−	0.603845i	$-0.793635\pi$
$461$	−10.0726	+	7.31820i	−0.469130	+	0.340843i	0.327972	+	0.944687i	$0.393635\pi$
$462$		0			0
$463$	14.8396	−	2.35037i	0.689656	−	0.109231i	0.198237	−	0.980154i	$-0.436478\pi$
$463$	14.8396	−	2.35037i	0.689656	−	0.109231i	0.491419	+	0.870923i	$0.336478\pi$
$464$		0			0
$465$	−3.68567	+	3.17213i	−0.170919	+	0.147104i
$466$		0			0
$467$	−12.4383	+	24.4115i	−0.575576	+	1.12963i	0.401325	+	0.915936i	$0.368550\pi$
$467$	−12.4383	+	24.4115i	−0.575576	+	1.12963i	−0.976900	+	0.213695i	$0.931450\pi$
$468$		0			0
$469$	−35.1273	+	11.4135i	−1.62203	+	0.527029i
$470$		0			0
$471$	4.85581	+	1.57775i	0.223744	+	0.0726988i
$472$		0			0
$473$	−40.5404	−	6.42097i	−1.86405	−	0.295236i
$474$		0			0
$475$	6.41644	+	38.6124i	0.294407	+	1.77166i
$476$		0			0
$477$	1.19673	−	7.55583i	0.0547943	−	0.345958i
$478$		0			0
$479$	−7.53740	+	23.1977i	−0.344393	+	1.05993i	0.617515	+	0.786559i	$0.288139\pi$
$479$	−7.53740	+	23.1977i	−0.344393	+	1.05993i	−0.961908	+	0.273373i	$0.911861\pi$
$480$		0			0
$481$	3.85204	+	11.8554i	0.175638	+	0.540558i
$482$		0			0
$483$	−8.10396	−	4.12918i	−0.368743	−	0.187884i
$484$		0			0
$485$	7.80085	−	3.26596i	0.354218	−	0.148300i
$486$		0			0
$487$	−2.80650	−	17.7196i	−0.127175	−	0.802950i	−0.965999	−	0.258547i	$-0.916756\pi$
$487$	−2.80650	−	17.7196i	−0.127175	−	0.802950i	0.838824	−	0.544403i	$-0.183244\pi$
$488$		0			0
$489$	−12.2098	−	16.8053i	−0.552146	−	0.759963i
$490$		0			0
$491$	−16.3602	+	22.5178i	−0.738324	+	1.01622i	0.260390	+	0.965504i	$0.416149\pi$
$491$	−16.3602	+	22.5178i	−0.738324	+	1.01622i	−0.998713	+	0.0507117i	$0.983851\pi$
$492$		0			0
$493$	−1.15182	−	1.15182i	−0.0518754	−	0.0518754i
$494$		0			0
$495$	0.617159	−	8.24156i	0.0277392	−	0.370431i
$496$		0			0
$497$	−13.1246	−	25.7585i	−0.588720	−	1.15543i
$498$		0			0
$499$	27.2018			1.21772			0.608860	−	0.793278i	$-0.291627\pi$
$499$	27.2018			1.21772			0.608860	+	0.793278i	$0.291627\pi$
$500$		0			0
$501$	−10.2220			−0.456684
$502$		0			0
$503$	−14.0273	−	27.5300i	−0.625444	−	1.22750i	−0.958633	−	0.284643i	$-0.908125\pi$
$503$	−14.0273	−	27.5300i	−0.625444	−	1.22750i	0.333189	−	0.942860i	$-0.391875\pi$
$504$		0			0
$505$	1.97194	−	26.3334i	0.0877503	−	1.17182i
$506$		0			0
$507$	4.05456	+	4.05456i	0.180069	+	0.180069i
$508$		0			0
$509$	19.5761	−	26.9441i	0.867693	−	1.19428i	−0.111987	−	0.993710i	$-0.535721\pi$
$509$	19.5761	−	26.9441i	0.867693	−	1.19428i	0.979680	−	0.200568i	$-0.0642786\pi$
$510$		0			0
$511$	−36.5669	−	50.3301i	−1.61763	−	2.22647i
$512$		0			0
$513$	6.92664	+	43.7331i	0.305819	+	1.93086i
$514$		0			0
$515$	12.5220	−	5.24254i	0.551783	−	0.231014i
$516$		0			0
$517$	7.90122	+	4.02587i	0.347495	+	0.177058i
$518$		0			0
$519$	3.26182	+	10.0389i	0.143178	+	0.440657i
$520$		0			0
$521$	5.65254	−	17.3967i	0.247642	−	0.762164i	−0.747548	−	0.664207i	$-0.768769\pi$
$521$	5.65254	−	17.3967i	0.247642	−	0.762164i	0.995191	−	0.0979570i	$-0.0312308\pi$
$522$		0			0
$523$	2.44996	−	15.4684i	0.107129	−	0.676387i	−0.874419	−	0.485172i	$-0.838757\pi$
$523$	2.44996	−	15.4684i	0.107129	−	0.676387i	0.981548	−	0.191215i	$-0.0612428\pi$
$524$		0			0
$525$	−13.2213	+	26.4426i	−0.577024	+	1.15405i
$526$		0			0
$527$	−0.439236	−	0.0695681i	−0.0191334	−	0.00303043i
$528$		0			0
$529$	19.6239	+	6.37619i	0.853212	+	0.277225i
$530$		0			0
$531$	11.5118	−	3.74042i	0.499571	−	0.162321i
$532$		0			0
$533$	12.7230	−	24.9704i	0.551096	−	1.08159i
$534$		0			0
$535$	−6.37487	+	5.48663i	−0.275610	+	0.237208i
$536$		0			0
$537$	6.02327	−	0.953993i	0.259923	−	0.0411678i
$538$		0			0
$539$	−31.5584	+	22.9285i	−1.35932	+	0.987602i
$540$		0			0
$541$	−27.1227	−	19.7058i	−1.16610	−	0.847220i	−0.175561	−	0.984468i	$-0.556174\pi$
$541$	−27.1227	−	19.7058i	−1.16610	−	0.847220i	−0.990537	+	0.137249i	$0.956174\pi$
$542$		0			0
$543$	−16.4440	+	16.4440i	−0.705679	+	0.705679i
$544$		0			0
$545$	8.10314	+	6.86772i	0.347100	+	0.294181i
$546$		0			0
$547$	14.4364	−	7.35569i	0.617254	−	0.314507i	−0.117259	−	0.993101i	$-0.537411\pi$
$547$	14.4364	−	7.35569i	0.617254	−	0.314507i	0.734513	+	0.678595i	$0.237411\pi$
$548$		0			0
$549$		−	6.69938i		−	0.285922i
$550$		0			0
$551$		−	43.6874i		−	1.86115i
$552$		0			0
$553$	9.74586	−	4.96576i	0.414436	−	0.211166i
$554$		0			0
$555$	8.20423	−	5.07049i	0.348250	−	0.215230i
$556$		0			0
$557$	23.4966	−	23.4966i	0.995583	−	0.995583i	−0.00440744	−	0.999990i	$-0.501403\pi$
$557$	23.4966	−	23.4966i	0.995583	−	0.995583i	0.999990	+	0.00440744i	$0.00140294\pi$
$558$		0			0
$559$	35.6764	+	25.9204i	1.50895	+	1.09632i
$560$		0			0
$561$	−1.29419	+	0.940285i	−0.0546408	+	0.0396988i
$562$		0			0
$563$	−18.2001	+	2.88261i	−0.767043	+	0.121488i	−0.527681	−	0.849443i	$-0.676938\pi$
$563$	−18.2001	+	2.88261i	−0.767043	+	0.121488i	−0.239362	+	0.970930i	$0.576938\pi$
$564$		0			0
$565$	23.0378	−	1.90112i	0.969207	−	0.0799809i
$566$		0			0
$567$	−9.75198	+	19.1393i	−0.409545	+	0.803777i
$568$		0			0
$569$	1.24601	−	0.404854i	0.0522355	−	0.0169724i	−0.282783	−	0.959184i	$-0.591258\pi$
$569$	1.24601	−	0.404854i	0.0522355	−	0.0169724i	0.335018	+	0.942212i	$0.391258\pi$
$570$		0			0
$571$	−4.68049	−	1.52078i	−0.195872	−	0.0636428i	0.209438	−	0.977822i	$-0.432837\pi$
$571$	−4.68049	−	1.52078i	−0.195872	−	0.0636428i	−0.405310	+	0.914179i	$0.632837\pi$
$572$		0			0
$573$	0.950379	+	0.150525i	0.0397027	+	0.00628828i
$574$		0			0
$575$	−2.32114	+	7.33267i	−0.0967981	+	0.305793i
$576$		0			0
$577$	3.10016	−	19.5737i	0.129062	−	0.814862i	−0.835207	−	0.549936i	$-0.814652\pi$
$577$	3.10016	−	19.5737i	0.129062	−	0.814862i	0.964269	−	0.264927i	$-0.0853478\pi$
$578$		0			0
$579$	−1.21171	+	3.72927i	−0.0503571	+	0.154983i
$580$		0			0
$581$	−3.27716	−	10.0861i	−0.135959	−	0.418440i
$582$		0			0
$583$	27.1882	+	13.8531i	1.12602	+	0.573736i
$584$		0			0
$585$	−4.61066	+	7.58842i	−0.190627	+	0.313742i
$586$		0			0
$587$	−4.17496	−	26.3597i	−0.172319	−	1.08798i	−0.910540	−	0.413421i	$-0.864334\pi$
$587$	−4.17496	−	26.3597i	−0.172319	−	1.08798i	0.738221	−	0.674559i	$-0.235666\pi$
$588$		0			0
$589$	−7.01057	−	9.64922i	−0.288865	−	0.397589i
$590$		0			0
$591$	6.19645	−	8.52869i	0.254888	−	0.350823i
$592$		0			0
$593$	7.10899	+	7.10899i	0.291931	+	0.291931i	0.837843	−	0.545911i	$-0.183817\pi$
$593$	7.10899	+	7.10899i	0.291931	+	0.291931i	−0.545911	+	0.837843i	$0.683817\pi$
$594$		0			0
$595$	−2.62654	+	0.641130i	−0.107678	+	0.0262838i
$596$		0			0
$597$	−3.40520	−	6.68309i	−0.139366	−	0.273521i
$598$		0			0
$599$	39.7238			1.62307			0.811536	−	0.584302i	$-0.198632\pi$
$599$	39.7238			1.62307			0.811536	+	0.584302i	$0.198632\pi$
$600$		0			0
$601$	7.44239			0.303582			0.151791	−	0.988413i	$-0.451496\pi$
$601$	7.44239			0.303582			0.151791	+	0.988413i	$0.451496\pi$
$602$		0			0
$603$	3.89659	+	7.64748i	0.158681	+	0.311430i
$604$		0			0
$605$	7.74396	+	3.17327i	0.314837	+	0.129012i
$606$		0			0
$607$	−22.8498	−	22.8498i	−0.927445	−	0.927445i	0.0700955	−	0.997540i	$-0.477670\pi$
$607$	−22.8498	−	22.8498i	−0.927445	−	0.927445i	−0.997540	+	0.0700955i	$0.977670\pi$
$608$		0			0
$609$	19.3951	−	26.6950i	0.785928	−	1.08174i
$610$		0			0
$611$	−5.59999	−	7.70772i	−0.226551	−	0.311821i
$612$		0			0
$613$	−4.46309	−	28.1788i	−0.180262	−	1.13813i	−0.897406	−	0.441207i	$-0.854551\pi$
$613$	−4.46309	−	28.1788i	−0.180262	−	1.13813i	0.717143	−	0.696926i	$-0.245449\pi$
$614$		0			0
$615$	−21.1031	−	4.98177i	−0.850959	−	0.200884i
$616$		0			0
$617$	−7.60111	−	3.87296i	−0.306009	−	0.155920i	0.294242	−	0.955731i	$-0.404933\pi$
$617$	−7.60111	−	3.87296i	−0.306009	−	0.155920i	−0.600251	+	0.799811i	$0.704933\pi$
$618$		0			0
$619$	−6.83480	−	21.0353i	−0.274714	−	0.845482i	−0.989295	−	0.145929i	$-0.953383\pi$
$619$	−6.83480	−	21.0353i	−0.274714	−	0.845482i	0.714581	−	0.699552i	$-0.246617\pi$
$620$		0			0
$621$	−2.68862	+	8.27471i	−0.107891	+	0.332053i
$622$		0			0
$623$	−7.55596	+	47.7064i	−0.302723	+	1.91132i
$624$		0			0
$625$	23.8910	+	7.36356i	0.955638	+	0.294542i
$626$		0			0
$627$	−42.3757	−	6.71166i	−1.69232	−	0.268038i
$628$		0			0
$629$	0.838853	+	0.272560i	0.0334473	+	0.0108677i
$630$		0			0
$631$	−14.3275	+	4.65528i	−0.570368	+	0.185324i	−0.579981	−	0.814630i	$-0.696940\pi$
$631$	−14.3275	+	4.65528i	−0.570368	+	0.185324i	0.00961299	+	0.999954i	$0.496940\pi$
$632$		0			0
$633$	5.25672	−	10.3169i	0.208936	−	0.410060i
$634$		0			0
$635$	4.74603	−	20.1045i	0.188341	−	0.797823i
$636$		0			0
$637$	41.3936	−	6.55610i	1.64007	−	0.259762i
$638$		0			0
$639$	−5.43497	+	3.94874i	−0.215004	+	0.156210i
$640$		0			0
$641$	−26.9426	−	19.5749i	−1.06417	−	0.773163i	−0.0893123	−	0.996004i	$-0.528467\pi$
$641$	−26.9426	−	19.5749i	−1.06417	−	0.773163i	−0.974855	+	0.222841i	$0.928467\pi$
$642$		0			0
$643$	26.5370	−	26.5370i	1.04652	−	1.04652i	0.0476541	−	0.998864i	$-0.484825\pi$
$643$	26.5370	−	26.5370i	1.04652	−	1.04652i	0.998864	−	0.0476541i	$-0.0151745\pi$
$644$		0			0
$645$	12.9372	−	31.5716i	0.509402	−	1.24313i
$646$		0			0
$647$	14.8905	−	7.58707i	0.585404	−	0.298278i	−0.136092	−	0.990696i	$-0.543454\pi$
$647$	14.8905	−	7.58707i	0.585404	−	0.298278i	0.721497	+	0.692418i	$0.243454\pi$
$648$		0			0
$649$			48.2810i			1.89519i
$650$		0			0
$651$		−	9.00847i		−	0.353070i
$652$		0			0
$653$	−8.39806	+	4.27902i	−0.328641	+	0.167451i	−0.610524	−	0.791998i	$-0.709041\pi$
$653$	−8.39806	+	4.27902i	−0.328641	+	0.167451i	0.281883	+	0.959449i	$0.409041\pi$
$654$		0			0
$655$	11.1899	+	45.8420i	0.437225	+	1.79120i
$656$		0			0
$657$	−10.2224	+	10.2224i	−0.398815	+	0.398815i
$658$		0			0
$659$	21.3774	+	15.5316i	0.832745	+	0.605025i	0.920335	−	0.391132i	$-0.127916\pi$
$659$	21.3774	+	15.5316i	0.832745	+	0.605025i	−0.0875893	+	0.996157i	$0.527916\pi$
$660$		0			0
$661$	17.7042	−	12.8628i	0.688612	−	0.500306i	−0.187591	−	0.982247i	$-0.560068\pi$
$661$	17.7042	−	12.8628i	0.688612	−	0.500306i	0.876204	+	0.481941i	$0.160068\pi$
$662$		0			0
$663$	1.69752	−	0.268861i	0.0659264	−	0.0104417i
$664$		0			0
$665$	−61.9698	−	37.6523i	−2.40308	−	1.46009i
$666$		0			0
$667$	3.89726	−	7.64880i	0.150903	−	0.296163i
$668$		0			0
$669$	5.08561	−	1.65242i	0.196621	−	0.0638861i
$670$		0			0
$671$	25.4143	+	8.25761i	0.981109	+	0.318782i
$672$		0			0
$673$	−7.36687	−	1.16680i	−0.283972	−	0.0449768i	0.0128235	−	0.999918i	$-0.495918\pi$
$673$	−7.36687	−	1.16680i	−0.283972	−	0.0449768i	−0.296796	+	0.954941i	$0.595918\pi$
$674$		0			0
$675$	26.9620	+	8.53475i	1.03777	+	0.328503i
$676$		0			0
$677$	−2.11287	+	13.3401i	−0.0812042	+	0.512703i	0.913239	+	0.407425i	$0.133573\pi$
$677$	−2.11287	+	13.3401i	−0.0812042	+	0.512703i	−0.994443	+	0.105278i	$0.966427\pi$
$678$		0			0
$679$	−4.84130	+	14.9000i	−0.185792	+	0.571809i
$680$		0			0
$681$	4.10501	+	12.6339i	0.157304	+	0.484132i
$682$		0			0
$683$	28.3996	+	14.4703i	1.08668	+	0.553692i	0.903151	−	0.429323i	$-0.141248\pi$
$683$	28.3996	+	14.4703i	1.08668	+	0.553692i	0.183530	+	0.983014i	$0.441248\pi$
$684$		0			0
$685$	1.34995	+	16.3587i	0.0515791	+	0.625034i
$686$		0			0
$687$	−1.30780	−	8.25710i	−0.0498955	−	0.315028i
$688$		0			0
$689$	−19.2697	−	26.5224i	−0.734116	−	1.01042i
$690$		0			0
$691$	−15.1560	+	20.8605i	−0.576563	+	0.793571i	−0.993313	−	0.115450i	$-0.963169\pi$
$691$	−15.1560	+	20.8605i	−0.576563	+	0.793571i	0.416750	+	0.909021i	$0.363169\pi$
$692$		0			0
$693$	10.8262	+	10.8262i	0.411253	+	0.411253i
$694$		0			0
$695$	12.1008	+	19.5795i	0.459010	+	0.742694i
$696$		0			0
$697$	−0.900248	−	1.76684i	−0.0340993	−	0.0669237i
$698$		0			0
$699$	−11.8399			−0.447825
$700$		0			0
$701$	20.3302			0.767860			0.383930	−	0.923362i	$-0.374570\pi$
$701$	20.3302			0.767860			0.383930	+	0.923362i	$0.374570\pi$
$702$		0			0
$703$	10.7395	+	21.0774i	0.405047	+	0.794949i
$704$		0			0
$705$	−4.76600	+	5.62334i	−0.179498	+	0.211787i
$706$		0			0
$707$	34.5918	+	34.5918i	1.30096	+	1.30096i
$708$		0			0
$709$	−7.51229	+	10.3398i	−0.282130	+	0.388319i	−0.926438	−	0.376447i	$-0.877146\pi$
$709$	−7.51229	+	10.3398i	−0.282130	+	0.388319i	0.644308	+	0.764766i	$0.277146\pi$
$710$		0			0
$711$	−1.49402	−	2.05635i	−0.0560302	−	0.0771190i
$712$		0			0
$713$	−0.366626	−	2.31479i	−0.0137302	−	0.0866894i
$714$		0			0
$715$	−23.1038	−	26.8441i	−0.864035	−	1.00391i
$716$		0			0
$717$	−31.5843	−	16.0930i	−1.17954	−	0.601004i
$718$		0			0
$719$	3.29703	+	10.1472i	0.122959	+	0.378427i	0.993524	−	0.113626i	$-0.0362465\pi$
$719$	3.29703	+	10.1472i	0.122959	+	0.378427i	−0.870565	+	0.492053i	$0.836247\pi$
$720$		0			0
$721$	−7.77127	+	23.9175i	−0.289417	+	0.890735i
$722$		0			0
$723$	−1.61025	+	10.1667i	−0.0598860	+	0.378105i
$724$		0			0
$725$	−24.9574	−	12.4787i	−0.926895	−	0.463447i
$726$		0			0
$727$	−27.4276	−	4.34410i	−1.01723	−	0.161114i	−0.374518	−	0.927220i	$-0.622192\pi$
$727$	−27.4276	−	4.34410i	−1.01723	−	0.161114i	−0.642714	+	0.766106i	$0.722192\pi$
$728$		0			0
$729$	27.7821	+	9.02694i	1.02897	+	0.334331i
$730$		0			0
$731$	2.96757	−	0.964222i	0.109760	−	0.0356630i
$732$		0			0
$733$	14.4667	−	28.3925i	0.534339	−	1.04870i	−0.453212	−	0.891403i	$-0.649722\pi$
$733$	14.4667	−	28.3925i	0.534339	−	1.04870i	0.987551	−	0.157297i	$-0.0502779\pi$
$734$		0			0
$735$	−12.5224	−	29.9102i	−0.461897	−	1.10326i
$736$		0			0
$737$	−33.8139	+	5.35560i	−1.24555	+	0.197276i
$738$		0			0
$739$	−37.2176	+	27.0402i	−1.36907	+	0.994688i	−0.371262	+	0.928528i	$0.621075\pi$
$739$	−37.2176	+	27.0402i	−1.36907	+	0.994688i	−0.997809	+	0.0661598i	$0.978925\pi$
$740$		0			0
$741$	37.2915	+	27.0939i	1.36994	+	0.995319i
$742$		0			0
$743$	18.2973	−	18.2973i	0.671262	−	0.671262i	−0.286745	−	0.958007i	$-0.592573\pi$
$743$	18.2973	−	18.2973i	0.671262	−	0.671262i	0.958007	+	0.286745i	$0.0925733\pi$
$744$		0			0
$745$	26.4405	+	1.97996i	0.968704	+	0.0725401i
$746$		0			0
$747$	−2.19581	+	1.11882i	−0.0803406	+	0.0409356i
$748$		0			0
$749$		−	15.5814i		−	0.569330i
$750$		0			0
$751$		−	3.66430i		−	0.133712i	−0.997763	−	0.0668561i	$-0.978703\pi$
$751$		−	3.66430i		−	0.133712i	0.997763	−	0.0668561i	$-0.0212968\pi$
$752$		0			0
$753$	−37.3248	+	19.0180i	−1.36019	+	0.693053i
$754$		0			0
$755$	29.0240	+	2.17342i	1.05629	+	0.0790989i
$756$		0			0
$757$	−16.0349	+	16.0349i	−0.582797	+	0.582797i	−0.935671	−	0.352874i	$-0.885204\pi$
$757$	−16.0349	+	16.0349i	−0.582797	+	0.582797i	0.352874	+	0.935671i	$0.385204\pi$
$758$		0			0
$759$	−6.82042	−	4.95533i	−0.247566	−	0.179867i
$760$		0			0
$761$	2.89867	−	2.10600i	0.105077	−	0.0763426i	−0.534006	−	0.845481i	$-0.679314\pi$
$761$	2.89867	−	2.10600i	0.105077	−	0.0763426i	0.639083	+	0.769138i	$0.279314\pi$
$762$		0			0
$763$	−19.4353	+	3.07825i	−0.703606	+	0.111440i
$764$		0			0
$765$	0.242632	+	0.579535i	0.00877240	+	0.0209531i
$766$		0			0
$767$	23.5493	−	46.2181i	0.850317	−	1.66884i
$768$		0			0
$769$	−32.4148	+	10.5322i	−1.16891	+	0.379802i	−0.828236	−	0.560380i	$-0.810655\pi$
$769$	−32.4148	+	10.5322i	−1.16891	+	0.379802i	−0.340674	+	0.940182i	$0.610655\pi$
$770$		0			0
$771$	−30.9669	−	10.0618i	−1.11525	−	0.362366i
$772$		0			0
$773$	7.62309	+	1.20738i	0.274183	+	0.0434264i	0.292012	−	0.956415i	$-0.405675\pi$
$773$	7.62309	+	1.20738i	0.274183	+	0.0434264i	−0.0178285	+	0.999841i	$0.505675\pi$
$774$		0			0
$775$	−7.51480	+	1.24878i	−0.269940	+	0.0448574i
$776$		0			0
$777$	−2.79502	+	17.6471i	−0.100271	+	0.633085i
$778$		0			0
$779$	16.4344	−	50.5800i	0.588824	−	1.81222i
$780$		0			0
$781$	−8.28056	−	25.4849i	−0.296302	−	0.911923i
$782$		0			0
$783$	−28.1244	−	14.3301i	−1.00508	−	0.512116i
$784$		0			0
$785$	5.21759	+	6.06227i	0.186224	+	0.216372i
$786$		0			0
$787$	−3.28182	−	20.7206i	−0.116984	−	0.738611i	−0.974539	−	0.224218i	$-0.928017\pi$
$787$	−3.28182	−	20.7206i	−0.116984	−	0.738611i	0.857555	−	0.514393i	$-0.171983\pi$
$788$		0			0
$789$	18.1870	+	25.0323i	0.647476	+	0.891174i
$790$		0			0
$791$	−25.1709	+	34.6448i	−0.894976	+	1.23183i
$792$		0			0
$793$	−20.3008	−	20.3008i	−0.720902	−	0.720902i
$794$		0			0
$795$	−16.3999	+	19.3500i	−0.581644	+	0.686275i
$796$		0			0
$797$	8.94048	+	17.5467i	0.316688	+	0.621535i	0.993399	−	0.114714i	$-0.0365951\pi$
$797$	8.94048	+	17.5467i	0.316688	+	0.621535i	−0.676711	+	0.736249i	$0.736595\pi$
$798$		0			0
$799$	−0.674124			−0.0238488
$800$		0			0
$801$	11.2242			0.396588
$802$		0			0
$803$	−26.1790	−	51.3792i	−0.923838	−	1.81313i
$804$		0			0
$805$	−7.49080	−	12.1204i	−0.264016	−	0.427187i
$806$		0			0
$807$	30.1491	+	30.1491i	1.06130	+	1.06130i
$808$		0			0
$809$	−10.5652	+	14.5418i	−0.371454	+	0.511263i	−0.953295	−	0.302040i	$-0.902333\pi$
$809$	−10.5652	+	14.5418i	−0.371454	+	0.511263i	0.581841	+	0.813302i	$0.302333\pi$
$810$		0			0
$811$	3.88707	+	5.35010i	0.136494	+	0.187867i	0.871792	−	0.489876i	$-0.162958\pi$
$811$	3.88707	+	5.35010i	0.136494	+	0.187867i	−0.735298	+	0.677743i	$0.762958\pi$
$812$		0			0
$813$	3.82348	+	24.1405i	0.134095	+	0.846644i
$814$		0			0
$815$	−2.67629	−	32.4313i	−0.0937464	−	1.13602i
$816$		0			0
$817$	74.5645	+	37.9925i	2.60868	+	1.32919i
$818$		0			0
$819$	−5.08309	−	15.6442i	−0.177618	−	0.546651i
$820$		0			0
$821$	−0.360464	+	1.10939i	−0.0125803	+	0.0387181i	−0.957150	−	0.289594i	$-0.906480\pi$
$821$	−0.360464	+	1.10939i	−0.0125803	+	0.0387181i	0.944569	+	0.328312i	$0.106480\pi$
$822$		0			0
$823$	5.62489	−	35.5142i	0.196071	−	1.23795i	−0.671640	−	0.740878i	$-0.734410\pi$
$823$	5.62489	−	35.5142i	0.196071	−	1.23795i	0.867711	−	0.497068i	$-0.165590\pi$
$824$		0			0
$825$	−15.9382	+	22.2910i	−0.554898	+	0.776073i
$826$		0			0
$827$	17.3046	+	2.74077i	0.601739	+	0.0953060i	0.449869	−	0.893095i	$-0.351471\pi$
$827$	17.3046	+	2.74077i	0.601739	+	0.0953060i	0.151870	+	0.988401i	$0.451471\pi$
$828$		0			0
$829$	15.2346	+	4.95002i	0.529120	+	0.171921i	0.561380	−	0.827558i	$-0.310271\pi$
$829$	15.2346	+	4.95002i	0.529120	+	0.171921i	−0.0322604	+	0.999479i	$0.510271\pi$
$830$		0			0
$831$	16.1014	−	5.23166i	0.558551	−	0.181484i
$832$		0			0
$833$	1.34627	−	2.64220i	0.0466454	−	0.0915467i
$834$		0			0
$835$	−13.6853	−	8.31507i	−0.473599	−	0.287755i
$836$		0			0
$837$	−8.51140	+	1.34807i	−0.294197	+	0.0465962i
$838$		0			0
$839$	−31.4048	+	22.8169i	−1.08421	+	0.787727i	−0.978413	−	0.206661i	$-0.933740\pi$
$839$	−31.4048	+	22.8169i	−1.08421	+	0.787727i	−0.105800	+	0.994387i	$0.533740\pi$
$840$		0			0
$841$	1.73420	+	1.25997i	0.0598001	+	0.0434473i
$842$		0			0
$843$	18.8318	−	18.8318i	0.648602	−	0.648602i
$844$		0			0
$845$	2.13011	+	8.72649i	0.0732779	+	0.300200i
$846$		0			0
$847$	−13.8139	+	7.03853i	−0.474651	+	0.241847i
$848$		0			0
$849$			22.4679i			0.771098i
$850$		0			0
$851$			4.64828i			0.159341i
$852$		0			0
$853$	19.3009	−	9.83431i	0.660851	−	0.336720i	−0.0911931	−	0.995833i	$-0.529068\pi$
$853$	19.3009	−	9.83431i	0.660851	−	0.336720i	0.752044	+	0.659113i	$0.229068\pi$
$854$		0			0
$855$	−6.38919	+	15.5920i	−0.218506	+	0.533235i
$856$		0			0
$857$	−30.7036	+	30.7036i	−1.04881	+	1.04881i	−0.0500684	+	0.998746i	$0.515944\pi$
$857$	−30.7036	+	30.7036i	−1.04881	+	1.04881i	−0.998746	+	0.0500684i	$0.984056\pi$
$858$		0			0
$859$	−0.0460861	−	0.0334835i	−0.00157244	−	0.00114244i	0.586999	−	0.809588i	$-0.300309\pi$
$859$	−0.0460861	−	0.0334835i	−0.00157244	−	0.00114244i	−0.588571	+	0.808445i	$0.700309\pi$
$860$		0			0
$861$	32.4972	−	23.6106i	1.10750	−	0.804648i
$862$		0			0
$863$	−9.17549	+	1.45326i	−0.312337	+	0.0494694i	−0.310635	−	0.950529i	$-0.600542\pi$
$863$	−9.17549	+	1.45326i	−0.312337	+	0.0494694i	−0.00170208	+	0.999999i	$0.500542\pi$
$864$		0			0
$865$	−3.79916	+	16.0935i	−0.129175	+	0.547195i
$866$		0			0
$867$	−10.9610	+	21.5122i	−0.372256	+	0.730593i
$868$		0			0
$869$	9.64234	−	3.13299i	0.327094	−	0.106279i
$870$		0			0
$871$	34.9814	+	11.3661i	1.18530	+	0.385127i
$872$		0			0
$873$	3.59582	+	0.569523i	0.121700	+	0.0192754i
$874$		0			0
$875$	−39.2105	+	24.6467i	−1.32556	+	0.833212i
$876$		0			0
$877$	−7.07858	+	44.6924i	−0.239027	+	1.50916i	0.517779	+	0.855515i	$0.326759\pi$
$877$	−7.07858	+	44.6924i	−0.239027	+	1.50916i	−0.756805	+	0.653640i	$0.773241\pi$
$878$		0			0
$879$	14.7174	−	45.2954i	0.496404	−	1.52777i
$880$		0			0
$881$	5.21481	+	16.0495i	0.175691	+	0.540723i	0.999664	−	0.0259066i	$-0.00824725\pi$
$881$	5.21481	+	16.0495i	0.175691	+	0.540723i	−0.823973	+	0.566629i	$0.808247\pi$
$882$		0			0
$883$	−11.4537	−	5.83596i	−0.385448	−	0.196396i	0.250519	−	0.968112i	$-0.419399\pi$
$883$	−11.4537	−	5.83596i	−0.385448	−	0.196396i	−0.635967	+	0.771716i	$0.719399\pi$
$884$		0			0
$885$	−39.0601	−	9.22085i	−1.31299	−	0.309955i
$886$		0			0
$887$	7.92120	+	50.0125i	0.265968	+	1.67926i	0.653132	+	0.757244i	$0.273455\pi$
$887$	7.92120	+	50.0125i	0.265968	+	1.67926i	−0.387164	+	0.922011i	$0.626545\pi$
$888$		0			0
$889$	22.4934	+	30.9595i	0.754404	+	1.03835i
$890$		0			0
$891$	−11.7031	+	16.1080i	−0.392069	+	0.539637i
$892$		0			0
$893$	−12.7844	−	12.7844i	−0.427815	−	0.427815i
$894$		0			0
$895$	8.84006	+	3.62242i	0.295490	+	0.121084i
$896$		0			0
$897$	4.11203	+	8.07031i	0.137297	+	0.269460i
$898$		0			0
$899$	8.50250			0.283574
$900$		0			0
$901$	−2.31967			−0.0772795
$902$		0			0
$903$	28.6956	+	56.3182i	0.954928	+	1.87415i
$904$		0			0
$905$	−35.3918	+	8.63902i	−1.17646	+	0.287171i
$906$		0			0
$907$	18.8330	+	18.8330i	0.625339	+	0.625339i	0.946892	−	0.321553i	$-0.104205\pi$
$907$	18.8330	+	18.8330i	0.625339	+	0.625339i	−0.321553	+	0.946892i	$0.604205\pi$
$908$		0			0
$909$	6.68199	−	9.19697i	0.221628	−	0.305044i
$910$		0			0
$911$	6.27133	+	8.63174i	0.207778	+	0.285982i	0.900169	−	0.435540i	$-0.143443\pi$
$911$	6.27133	+	8.63174i	0.207778	+	0.285982i	−0.692391	+	0.721523i	$0.743443\pi$
$912$		0			0
$913$	−1.53775	−	9.70894i	−0.0508920	−	0.321319i
$914$		0			0
$915$	−11.5343	+	18.9836i	−0.381311	+	0.627577i
$916$		0			0
$917$	−77.8894	−	39.6866i	−2.57214	−	1.31057i
$918$		0			0
$919$	−13.7726	−	42.3878i	−0.454318	−	1.39825i	−0.871934	−	0.489623i	$-0.837135\pi$
$919$	−13.7726	−	42.3878i	−0.454318	−	1.39825i	0.417617	−	0.908623i	$-0.362865\pi$
$920$		0			0
$921$	−0.0628025	+	0.193286i	−0.00206941	+	0.00636900i
$922$		0			0
$923$	−4.50366	+	28.4350i	−0.148240	+	0.935949i
$924$		0			0
$925$	15.1085	−	0.114695i	0.496765	−	0.00377116i
$926$		0			0
$927$	5.77203	+	0.914200i	0.189578	+	0.0300263i
$928$		0			0
$929$	3.04696	+	0.990017i	0.0999675	+	0.0324814i	0.358574	−	0.933501i	$-0.383263\pi$
$929$	3.04696	+	0.990017i	0.0999675	+	0.0324814i	−0.258606	+	0.965983i	$0.583263\pi$
$930$		0			0
$931$	75.6393	−	24.5767i	2.47898	−	0.805468i
$932$		0			0
$933$	6.89111	−	13.5246i	0.225605	−	0.442774i
$934$		0			0
$935$	−2.49755	+	0.206103i	−0.0816787	+	0.00674029i
$936$		0			0
$937$	−45.1439	+	7.15010i	−1.47479	+	0.233583i	−0.841468	−	0.540307i	$-0.818308\pi$
$937$	−45.1439	+	7.15010i	−1.47479	+	0.233583i	−0.633320	+	0.773890i	$0.718308\pi$
$938$		0			0
$939$	26.9913	−	19.6103i	0.880827	−	0.639958i
$940$		0			0
$941$	−1.31101	−	0.952505i	−0.0427377	−	0.0310508i	0.566211	−	0.824260i	$-0.308409\pi$
$941$	−1.31101	−	0.952505i	−0.0427377	−	0.0310508i	−0.608949	+	0.793209i	$0.708409\pi$
$942$		0			0
$943$	7.38947	−	7.38947i	0.240634	−	0.240634i
$944$		0			0
$945$	−44.5663	+	27.5435i	−1.44974	+	0.895989i
$946$		0			0
$947$	6.11091	−	3.11366i	0.198578	−	0.101180i	−0.351874	−	0.936047i	$-0.614455\pi$
$947$	6.11091	−	3.11366i	0.198578	−	0.101180i	0.550452	+	0.834867i	$0.314455\pi$
$948$		0			0
$949$			61.9530i			2.01108i
$950$		0			0
$951$			13.7645i			0.446345i
$952$		0			0
$953$	45.7444	−	23.3080i	1.48181	−	0.755019i	0.488725	−	0.872438i	$-0.337462\pi$
$953$	45.7444	−	23.3080i	1.48181	−	0.755019i	0.993082	+	0.117419i	$0.0374621\pi$
$954$		0			0
$955$	1.14993	+	0.974612i	0.0372110	+	0.0315377i
$956$		0			0
$957$	21.6269	−	21.6269i	0.699099	−	0.699099i
$958$		0			0
$959$	−24.6007	−	17.8734i	−0.794397	−	0.577163i
$960$		0			0
$961$	−23.2016	+	16.8569i	−0.748438	+	0.543772i
$962$		0			0
$963$	−3.57622	+	0.566418i	−0.115242	+	0.0182526i
$964$		0			0
$965$	−4.65584	+	4.00712i	−0.149877	+	0.128994i
$966$		0			0
$967$	−19.2666	+	37.8129i	−0.619573	+	1.21598i	0.341551	+	0.939863i	$0.389048\pi$
$967$	−19.2666	+	37.8129i	−0.619573	+	1.21598i	−0.961124	+	0.276117i	$0.910952\pi$
$968$		0			0
$969$	3.10192	−	1.00787i	0.0996480	−	0.0323776i
$970$		0			0
$971$	28.6325	+	9.30326i	0.918861	+	0.298556i	0.729999	−	0.683448i	$-0.239520\pi$
$971$	28.6325	+	9.30326i	0.918861	+	0.298556i	0.188861	+	0.982004i	$0.439520\pi$
$972$		0			0
$973$	−42.1150	−	6.67037i	−1.35015	−	0.213842i
$974$		0			0
$975$	26.1298	−	13.5646i	0.836823	−	0.434416i
$976$		0			0
$977$	3.59037	−	22.6687i	0.114866	−	0.725235i	−0.861282	−	0.508128i	$-0.830338\pi$
$977$	3.59037	−	22.6687i	0.114866	−	0.725235i	0.976148	−	0.217107i	$-0.0696622\pi$
$978$		0			0
$979$	−13.8349	+	42.5795i	−0.442166	+	1.36085i
$980$		0			0
$981$	1.41304	+	4.34888i	0.0451148	+	0.138849i
$982$		0			0
$983$	−45.8856	−	23.3799i	−1.46352	−	0.745702i	−0.472745	−	0.881199i	$-0.656737\pi$
$983$	−45.8856	−	23.3799i	−1.46352	−	0.745702i	−0.990778	+	0.135497i	$0.956737\pi$
$984$		0			0
$985$	15.2336	−	6.37780i	0.485382	−	0.203213i
$986$		0			0
$987$	−2.13622	−	13.4875i	−0.0679966	−	0.429313i
$988$		0			0
$989$	9.66555	+	13.3035i	0.307347	+	0.423026i
$990$		0			0
$991$	7.43042	−	10.2271i	0.236035	−	0.324874i	−0.674524	−	0.738253i	$-0.735651\pi$
$991$	7.43042	−	10.2271i	0.236035	−	0.324874i	0.910559	+	0.413378i	$0.135651\pi$
$992$		0			0
$993$	9.95250	+	9.95250i	0.315833	+	0.315833i
$994$		0			0
$995$	0.877439	−	11.7174i	0.0278167	−	0.371465i
$996$		0			0
$997$	10.5746	+	20.7537i	0.334900	+	0.657277i	0.995634	−	0.0933423i	$-0.0297551\pi$
$997$	10.5746	+	20.7537i	0.334900	+	0.657277i	−0.660735	+	0.750620i	$0.729755\pi$
$998$		0			0
$999$	17.0916			0.540754

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bi.b.223.1	✓	16
4.3	odd	2	inner	400.2.bi.b.223.2	yes	16
25.12	odd	20	inner	400.2.bi.b.287.2	yes	16
100.87	even	20	inner	400.2.bi.b.287.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bi.b.223.1	✓	16	1.1	even	1	trivial
400.2.bi.b.223.2	yes	16	4.3	odd	2	inner
400.2.bi.b.287.1	yes	16	100.87	even	20	inner
400.2.bi.b.287.2	yes	16	25.12	odd	20	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bi (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.648013 - 1.27180i) q^{3} +(0.166977 - 2.22982i) q^{5} +(2.92911 + 2.92911i) q^{7} +(0.565808 - 0.778768i) q^{9} +O(q^{10})\)	\(q+(-0.648013 - 1.27180i) q^{3} +(0.166977 - 2.22982i) q^{5} +(2.92911 + 2.92911i) q^{7} +(0.565808 - 0.778768i) q^{9} +(2.25687 + 3.10632i) q^{11} +(-0.645322 - 4.07440i) q^{13} +(-2.94409 + 1.23259i) q^{15} +(-0.260074 - 0.132514i) q^{17} +(-2.41910 - 7.44523i) q^{19} +(1.82714 - 5.62334i) q^{21} +(0.240636 - 1.51932i) q^{23} +(-4.94424 - 0.744661i) q^{25} +(-5.58648 - 0.884812i) q^{27} +(5.30751 + 1.72451i) q^{29} +(1.44900 - 0.470809i) q^{31} +(2.48812 - 4.88322i) q^{33} +(7.02051 - 6.04232i) q^{35} +(-2.98459 + 0.472712i) q^{37} +(-4.76364 + 3.46098i) q^{39} +(5.49614 + 3.99318i) q^{41} +(-7.55900 + 7.55900i) q^{43} +(-1.64204 - 1.39169i) q^{45} +(2.05781 - 1.04851i) q^{47} +10.1594i q^{49} +0.416632i q^{51} +(7.08097 - 3.60793i) q^{53} +(7.30339 - 4.51375i) q^{55} +(-7.90122 + 7.90122i) q^{57} +(10.1729 + 7.39105i) q^{59} +(5.63043 - 4.09075i) q^{61} +(3.93842 - 0.623784i) q^{63} +(-9.19296 + 0.758621i) q^{65} +(-4.04794 + 7.94452i) q^{67} +(-2.08820 + 0.678497i) q^{69} +(-6.63736 - 2.15661i) q^{71} +(-14.8333 - 2.34937i) q^{73} +(2.25687 + 6.77062i) q^{75} +(-2.48812 + 15.7094i) q^{77} +(0.815963 - 2.51127i) q^{79} +(1.60242 + 4.93175i) q^{81} +(-2.28110 - 1.16228i) q^{83} +(-0.338910 + 0.557791i) q^{85} +(-1.24610 - 7.86758i) q^{87} +(6.85369 + 9.43329i) q^{89} +(10.0442 - 13.8246i) q^{91} +(-1.53775 - 1.53775i) q^{93} +(-17.0055 + 4.15099i) q^{95} +(1.71702 + 3.36984i) q^{97} +3.69606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{9}+O(q^{10}) \) 16 * q - 20 * q^9	\( 16 q - 20 q^{9} + 20 q^{17} - 20 q^{21} - 20 q^{25} + 20 q^{29} + 60 q^{33} - 20 q^{37} + 28 q^{41} - 20 q^{45} + 60 q^{53} - 20 q^{57} - 12 q^{61} - 20 q^{65} - 80 q^{69} - 40 q^{73} - 60 q^{77} + 56 q^{81} - 60 q^{85} + 40 q^{97}+O(q^{100}) \) 16 * q - 20 * q^9 + 20 * q^17 - 20 * q^21 - 20 * q^25 + 20 * q^29 + 60 * q^33 - 20 * q^37 + 28 * q^41 - 20 * q^45 + 60 * q^53 - 20 * q^57 - 12 * q^61 - 20 * q^65 - 80 * q^69 - 40 * q^73 - 60 * q^77 + 56 * q^81 - 60 * q^85 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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