LMFDB - Embedded newform 816.2.e.b.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(239,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.e (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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40282095616.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.1
Root			$-1.03179 + 1.39119i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.b.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39119 - 1.03179i) q^{3} -2.00000i q^{5} -2.78238i q^{7} +(0.870829 + 2.87083i) q^{9} +O(q^{10})$	\(q+(-1.39119 - 1.03179i) q^{3} -2.00000i q^{5} -2.78238i q^{7} +(0.870829 + 2.87083i) q^{9} +3.50119 q^{11} +5.74166 q^{13} +(-2.06358 + 2.78238i) q^{15} -1.00000i q^{17} -5.56477i q^{19} +(-2.87083 + 3.87083i) q^{21} -6.90953 q^{23} +1.00000 q^{25} +(1.75060 - 4.89238i) q^{27} +5.48331i q^{29} -3.50119i q^{31} +(-4.87083 - 3.61249i) q^{33} -5.56477 q^{35} -9.48331 q^{37} +(-7.98775 - 5.92417i) q^{39} -9.48331i q^{41} +7.00238i q^{43} +(5.74166 - 1.74166i) q^{45} -9.69192 q^{47} -0.741657 q^{49} +(-1.03179 + 1.39119i) q^{51} -2.00000i q^{53} -7.00238i q^{55} +(-5.74166 + 7.74166i) q^{57} +5.56477 q^{59} +5.48331 q^{61} +(7.98775 - 2.42298i) q^{63} -11.4833i q^{65} +11.1295i q^{67} +(9.61249 + 7.12917i) q^{69} +4.93881 q^{71} -6.00000 q^{73} +(-1.39119 - 1.03179i) q^{75} -9.74166i q^{77} -11.7555i q^{79} +(-7.48331 + 5.00000i) q^{81} +2.68953 q^{83} -2.00000 q^{85} +(5.65762 - 7.62834i) q^{87} +13.2250i q^{89} -15.9755i q^{91} +(-3.61249 + 4.87083i) q^{93} -11.1295 q^{95} -9.48331 q^{97} +(3.04894 + 10.0513i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{13} - 8 q^{21} + 8 q^{25} - 24 q^{33} - 16 q^{37} + 16 q^{45} + 24 q^{49} - 16 q^{57} - 16 q^{61} + 32 q^{69} - 48 q^{73} - 16 q^{85} + 16 q^{93} - 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^13 - 8 * q^21 + 8 * q^25 - 24 * q^33 - 16 * q^37 + 16 * q^45 + 24 * q^49 - 16 * q^57 - 16 * q^61 + 32 * q^69 - 48 * q^73 - 16 * q^85 + 16 * q^93 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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			0
$2$		0			0
$3$	−1.39119	−	1.03179i	−0.803205	−	0.595703i
$3$	−1.39119	−	1.03179i	−0.803205	−	0.595703i
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		−	2.78238i		−	1.05164i	−0.850595	−	0.525821i	$-0.823758\pi$
$7$		−	2.78238i		−	1.05164i	0.850595	−	0.525821i	$-0.176242\pi$
$8$		0			0
$9$	0.870829	+	2.87083i	0.290276	+	0.956943i
$10$		0			0
$11$	3.50119			1.05565			0.527824	−	0.849353i	$-0.323008\pi$
$11$	3.50119			1.05565			0.527824	+	0.849353i	$0.323008\pi$
$12$		0			0
$13$	5.74166			1.59245			0.796225	−	0.605001i	$-0.206827\pi$
$13$	5.74166			1.59245			0.796225	+	0.605001i	$0.206827\pi$
$14$		0			0
$15$	−2.06358	+	2.78238i	−0.532813	+	0.718408i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$		−	5.56477i		−	1.27665i	−0.769769	−	0.638323i	$-0.779629\pi$
$19$		−	5.56477i		−	1.27665i	0.769769	−	0.638323i	$-0.220371\pi$
$20$		0			0
$21$	−2.87083	+	3.87083i	−0.626466	+	0.844684i
$22$		0			0
$23$	−6.90953			−1.44074			−0.720369	−	0.693591i	$-0.756027\pi$
$23$	−6.90953			−1.44074			−0.720369	+	0.693591i	$0.756027\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.75060	−	4.89238i	0.336902	−	0.941540i
$28$		0			0
$29$			5.48331i			1.01823i	0.860700	+	0.509113i	$0.170026\pi$
$29$			5.48331i			1.01823i	−0.860700	+	0.509113i	$0.829974\pi$
$30$		0			0
$31$		−	3.50119i		−	0.628833i	−0.949285	−	0.314416i	$-0.898191\pi$
$31$		−	3.50119i		−	0.628833i	0.949285	−	0.314416i	$-0.101809\pi$
$32$		0			0
$33$	−4.87083	−	3.61249i	−0.847902	−	0.628853i
$34$		0			0
$35$	−5.56477			−0.940617
$36$		0			0
$37$	−9.48331			−1.55905			−0.779524	−	0.626373i	$-0.784539\pi$
$37$	−9.48331			−1.55905			−0.779524	+	0.626373i	$0.784539\pi$
$38$		0			0
$39$	−7.98775	−	5.92417i	−1.27906	−	0.948627i
$40$		0			0
$41$		−	9.48331i		−	1.48104i	−0.672032	−	0.740522i	$-0.734578\pi$
$41$		−	9.48331i		−	1.48104i	0.672032	−	0.740522i	$-0.265422\pi$
$42$		0			0
$43$			7.00238i			1.06785i	0.845531	+	0.533927i	$0.179284\pi$
$43$			7.00238i			1.06785i	−0.845531	+	0.533927i	$0.820716\pi$
$44$		0			0
$45$	5.74166	−	1.74166i	0.855916	−	0.259631i
$46$		0			0
$47$	−9.69192			−1.41371			−0.706856	−	0.707358i	$-0.749887\pi$
$47$	−9.69192			−1.41371			−0.706856	+	0.707358i	$0.749887\pi$
$48$		0			0
$49$	−0.741657			−0.105951
$50$		0			0
$51$	−1.03179	+	1.39119i	−0.144479	+	0.194806i
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$		−	7.00238i		−	0.944201i
$56$		0			0
$57$	−5.74166	+	7.74166i	−0.760501	+	1.02541i
$58$		0			0
$59$	5.56477			0.724471			0.362235	−	0.932087i	$-0.382014\pi$
$59$	5.56477			0.724471			0.362235	+	0.932087i	$0.382014\pi$
$60$		0			0
$61$	5.48331			0.702067			0.351033	−	0.936363i	$-0.385830\pi$
$61$	5.48331			0.702067			0.351033	+	0.936363i	$0.385830\pi$
$62$		0			0
$63$	7.98775	−	2.42298i	1.00636	−	0.305267i
$64$		0			0
$65$		−	11.4833i		−	1.42433i
$66$		0			0
$67$			11.1295i			1.35969i	0.733356	+	0.679844i	$0.237953\pi$
$67$			11.1295i			1.35969i	−0.733356	+	0.679844i	$0.762047\pi$
$68$		0			0
$69$	9.61249	+	7.12917i	1.15721	+	0.858251i
$70$		0			0
$71$	4.93881			0.586129			0.293064	−	0.956093i	$-0.405325\pi$
$71$	4.93881			0.586129			0.293064	+	0.956093i	$0.405325\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$	−1.39119	−	1.03179i	−0.160641	−	0.119141i
$76$		0			0
$77$		−	9.74166i		−	1.11016i
$78$		0			0
$79$		−	11.7555i		−	1.32260i	−0.750123	−	0.661298i	$-0.770006\pi$
$79$		−	11.7555i		−	1.32260i	0.750123	−	0.661298i	$-0.229994\pi$
$80$		0			0
$81$	−7.48331	+	5.00000i	−0.831479	+	0.555556i
$82$		0			0
$83$	2.68953			0.295215			0.147607	−	0.989046i	$-0.452843\pi$
$83$	2.68953			0.295215			0.147607	+	0.989046i	$0.452843\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	5.65762	−	7.62834i	0.606560	−	0.817844i
$88$		0			0
$89$			13.2250i			1.40184i	0.713238	+	0.700922i	$0.247228\pi$
$89$			13.2250i			1.40184i	−0.713238	+	0.700922i	$0.752772\pi$
$90$		0			0
$91$		−	15.9755i		−	1.67469i
$92$		0			0
$93$	−3.61249	+	4.87083i	−0.374597	+	0.505081i
$94$		0			0
$95$	−11.1295			−1.14187
$96$		0			0
$97$	−9.48331			−0.962885			−0.481442	−	0.876478i	$-0.659887\pi$
$97$	−9.48331			−0.962885			−0.481442	+	0.876478i	$0.659887\pi$
$98$		0			0
$99$	3.04894	+	10.0513i	0.306430	+	1.01020i
$100$		0			0
$101$		−	17.7417i		−	1.76536i	−0.469974	−	0.882680i	$-0.655737\pi$
$101$		−	17.7417i		−	1.76536i	0.469974	−	0.882680i	$-0.344263\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	7.74166	+	5.74166i	0.755508	+	0.560328i
$106$		0			0
$107$	8.34715			0.806949			0.403475	−	0.914991i	$-0.367802\pi$
$107$	8.34715			0.806949			0.403475	+	0.914991i	$0.367802\pi$
$108$		0			0
$109$	5.48331			0.525206			0.262603	−	0.964904i	$-0.415419\pi$
$109$	5.48331			0.525206			0.262603	+	0.964904i	$0.415419\pi$
$110$		0			0
$111$	13.1931	+	9.78477i	1.25223	+	0.928729i
$112$		0			0
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$			13.8191i			1.28863i
$116$		0			0
$117$	5.00000	+	16.4833i	0.462250	+	1.52388i
$118$		0			0
$119$	−2.78238			−0.255061
$120$		0			0
$121$	1.25834			0.114395
$122$		0			0
$123$	−9.78477	+	13.1931i	−0.882263	+	1.18958i
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$			5.56477i			0.493793i	0.969042	+	0.246897i	$0.0794108\pi$
$127$			5.56477i			0.493793i	−0.969042	+	0.246897i	$0.920589\pi$
$128$		0			0
$129$	7.22497	−	9.74166i	0.636123	−	0.857705i
$130$		0			0
$131$	−0.0928494			−0.00811229			−0.00405614	−	0.999992i	$-0.501291\pi$
$131$	−0.0928494			−0.00811229			−0.00405614	+	0.999992i	$0.501291\pi$
$132$		0			0
$133$	−15.4833			−1.34257
$134$		0			0
$135$	−9.78477	−	3.50119i	−0.842139	−	0.301335i
$136$		0			0
$137$		−	6.25834i		−	0.534686i	−0.963601	−	0.267343i	$-0.913854\pi$
$137$		−	6.25834i		−	0.534686i	0.963601	−	0.267343i	$-0.0861457\pi$
$138$		0			0
$139$			9.78477i			0.829933i	0.909837	+	0.414967i	$0.136207\pi$
$139$			9.78477i			0.829933i	−0.909837	+	0.414967i	$0.863793\pi$
$140$		0			0
$141$	13.4833	+	10.0000i	1.13550	+	0.842152i
$142$		0			0
$143$	20.1026			1.68107
$144$		0			0
$145$	10.9666			0.910729
$146$		0			0
$147$	1.03179	+	0.765233i	0.0851004	+	0.0631153i
$148$		0			0
$149$		−	4.00000i		−	0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$		−	4.00000i		−	0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$		0			0
$151$			15.2567i			1.24157i	0.783980	+	0.620786i	$0.213186\pi$
$151$			15.2567i			1.24157i	−0.783980	+	0.620786i	$0.786814\pi$
$152$		0			0
$153$	2.87083	−	0.870829i	0.232093	−	0.0704023i
$154$		0			0
$155$	−7.00238			−0.562445
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$	−2.06358	+	2.78238i	−0.163652	+	0.220657i
$160$		0			0
$161$			19.2250i			1.51514i
$162$		0			0
$163$			15.1638i			1.18772i	0.804567	+	0.593862i	$0.202397\pi$
$163$			15.1638i			1.18772i	−0.804567	+	0.593862i	$0.797603\pi$
$164$		0			0
$165$	−7.22497	+	9.74166i	−0.562463	+	0.758387i
$166$		0			0
$167$	−9.06596			−0.701545			−0.350772	−	0.936461i	$-0.614081\pi$
$167$	−9.06596			−0.701545			−0.350772	+	0.936461i	$0.614081\pi$
$168$		0			0
$169$	19.9666			1.53589
$170$		0			0
$171$	15.9755	−	4.84596i	1.22168	−	0.370580i
$172$		0			0
$173$			5.48331i			0.416889i	0.978034	+	0.208444i	$0.0668400\pi$
$173$			5.48331i			0.416889i	−0.978034	+	0.208444i	$0.933160\pi$
$174$		0			0
$175$		−	2.78238i		−	0.210328i
$176$		0			0
$177$	−7.74166	−	5.74166i	−0.581899	−	0.431569i
$178$		0			0
$179$	23.5110			1.75729			0.878647	−	0.477472i	$-0.158447\pi$
$179$	23.5110			1.75729			0.878647	+	0.477472i	$0.158447\pi$
$180$		0			0
$181$	1.48331			0.110254			0.0551270	−	0.998479i	$-0.482444\pi$
$181$	1.48331			0.110254			0.0551270	+	0.998479i	$0.482444\pi$
$182$		0			0
$183$	−7.62834	−	5.65762i	−0.563903	−	0.418223i
$184$		0			0
$185$			18.9666i			1.39445i
$186$		0			0
$187$		−	3.50119i		−	0.256032i
$188$		0			0
$189$	−13.6125	−	4.87083i	−0.990163	−	0.354301i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$	−11.8483	+	15.9755i	−0.848477	+	1.14403i
$196$		0			0
$197$		−	24.4499i		−	1.74199i	−0.491295	−	0.870993i	$-0.663476\pi$
$197$		−	24.4499i		−	1.74199i	0.491295	−	0.870993i	$-0.336524\pi$
$198$		0			0
$199$			17.5060i			1.24096i	0.784221	+	0.620482i	$0.213063\pi$
$199$			17.5060i			1.24096i	−0.784221	+	0.620482i	$0.786937\pi$
$200$		0			0
$201$	11.4833	−	15.4833i	0.809971	−	1.09211i
$202$		0			0
$203$	15.2567			1.07081
$204$		0			0
$205$	−18.9666			−1.32469
$206$		0			0
$207$	−6.01702	−	19.8361i	−0.418212	−	1.37870i
$208$		0			0
$209$		−	19.4833i		−	1.34769i
$210$		0			0
$211$			0.811658i			0.0558768i	0.999610	+	0.0279384i	$0.00889423\pi$
$211$			0.811658i			0.0558768i	−0.999610	+	0.0279384i	$0.991106\pi$
$212$		0			0
$213$	−6.87083	−	5.09580i	−0.470781	−	0.349159i
$214$		0			0
$215$	14.0048			0.955117
$216$		0			0
$217$	−9.74166			−0.661307
$218$		0			0
$219$	8.34715	+	6.19073i	0.564048	+	0.418330i
$220$		0			0
$221$		−	5.74166i		−	0.386226i
$222$		0			0
$223$		−	10.9438i		−	0.732853i	−0.930447	−	0.366427i	$-0.880581\pi$
$223$		−	10.9438i		−	0.732853i	0.930447	−	0.366427i	$-0.119419\pi$
$224$		0			0
$225$	0.870829	+	2.87083i	0.0580552	+	0.191389i
$226$		0			0
$227$	21.6331			1.43584			0.717920	−	0.696126i	$-0.245094\pi$
$227$	21.6331			1.43584			0.717920	+	0.696126i	$0.245094\pi$
$228$		0			0
$229$	−1.74166			−0.115092			−0.0575460	−	0.998343i	$-0.518328\pi$
$229$	−1.74166			−0.115092			−0.0575460	+	0.998343i	$0.518328\pi$
$230$		0			0
$231$	−10.0513	+	13.5525i	−0.661328	+	0.891690i
$232$		0			0
$233$		−	4.96663i		−	0.325375i	−0.986678	−	0.162687i	$-0.947984\pi$
$233$		−	4.96663i		−	0.325375i	0.986678	−	0.162687i	$-0.0520162\pi$
$234$		0			0
$235$			19.3838i			1.26446i
$236$		0			0
$237$	−12.1292	+	16.3541i	−0.787874	+	1.06232i
$238$		0			0
$239$	19.3838			1.25384			0.626918	−	0.779085i	$-0.284316\pi$
$239$	19.3838			1.25384			0.626918	+	0.779085i	$0.284316\pi$
$240$		0			0
$241$	−4.96663			−0.319929			−0.159964	−	0.987123i	$-0.551138\pi$
$241$	−4.96663			−0.319929			−0.159964	+	0.987123i	$0.551138\pi$
$242$		0			0
$243$	15.5697	+	0.765233i	0.998794	+	0.0490897i
$244$		0			0
$245$			1.48331i			0.0947655i
$246$		0			0
$247$		−	31.9510i		−	2.03299i
$248$		0			0
$249$	−3.74166	−	2.77503i	−0.237118	−	0.175860i
$250$		0			0
$251$	−25.1343			−1.58646			−0.793231	−	0.608920i	$-0.791603\pi$
$251$	−25.1343			−1.58646			−0.793231	+	0.608920i	$0.791603\pi$
$252$		0			0
$253$	−24.1916			−1.52091
$254$		0			0
$255$	2.78238	+	2.06358i	0.174240	+	0.129226i
$256$		0			0
$257$			21.7417i			1.35621i	0.734966	+	0.678104i	$0.237198\pi$
$257$			21.7417i			1.35621i	−0.734966	+	0.678104i	$0.762802\pi$
$258$		0			0
$259$			26.3862i			1.63956i
$260$		0			0
$261$	−15.7417	+	4.77503i	−0.974384	+	0.295567i
$262$		0			0
$263$	−18.1319			−1.11806			−0.559031	−	0.829147i	$-0.688827\pi$
$263$	−18.1319			−1.11806			−0.559031	+	0.829147i	$0.688827\pi$
$264$		0			0
$265$	−4.00000			−0.245718
$266$		0			0
$267$	13.6454	−	18.3985i	0.835083	−	1.12597i
$268$		0			0
$269$			20.9666i			1.27836i	0.769058	+	0.639179i	$0.220726\pi$
$269$			20.9666i			1.27836i	−0.769058	+	0.639179i	$0.779274\pi$
$270$		0			0
$271$			6.81668i			0.414084i	0.978332	+	0.207042i	$0.0663837\pi$
$271$			6.81668i			0.414084i	−0.978332	+	0.207042i	$0.933616\pi$
$272$		0			0
$273$	−16.4833	+	22.2250i	−0.997616	+	1.34512i
$274$		0			0
$275$	3.50119			0.211130
$276$		0			0
$277$	4.96663			0.298416			0.149208	−	0.988806i	$-0.452328\pi$
$277$	4.96663			0.298416			0.149208	+	0.988806i	$0.452328\pi$
$278$		0			0
$279$	10.0513	−	3.04894i	0.601757	−	0.182535i
$280$		0			0
$281$		−	26.9666i		−	1.60869i	−0.594160	−	0.804347i	$-0.702515\pi$
$281$		−	26.9666i		−	1.60869i	0.594160	−	0.804347i	$-0.297485\pi$
$282$		0			0
$283$		−	9.59907i		−	0.570605i	−0.958437	−	0.285303i	$-0.907906\pi$
$283$		−	9.59907i		−	0.570605i	0.958437	−	0.285303i	$-0.0920941\pi$
$284$		0			0
$285$	15.4833	+	11.4833i	0.917152	+	0.680213i
$286$		0			0
$287$	−26.3862			−1.55753
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	13.1931	+	9.78477i	0.773394	+	0.573593i
$292$		0			0
$293$		−	2.96663i		−	0.173312i	−0.996238	−	0.0866562i	$-0.972382\pi$
$293$		−	2.96663i		−	0.173312i	0.996238	−	0.0866562i	$-0.0276182\pi$
$294$		0			0
$295$		−	11.1295i		−	0.647986i
$296$		0			0
$297$	6.12917	−	17.1292i	0.355651	−	0.993935i
$298$		0			0
$299$	−39.6722			−2.29430
$300$		0			0
$301$	19.4833			1.12300
$302$		0			0
$303$	−18.3056	+	24.6820i	−1.05163	+	1.41795i
$304$		0			0
$305$		−	10.9666i		−	0.627947i
$306$		0			0
$307$		−	17.9462i		−	1.02424i	−0.858912	−	0.512122i	$-0.828859\pi$
$307$		−	17.9462i		−	1.02424i	0.858912	−	0.512122i	$-0.171141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.811658			0.0460249			0.0230124	−	0.999735i	$-0.492674\pi$
$311$	0.811658			0.0460249			0.0230124	+	0.999735i	$0.492674\pi$
$312$		0			0
$313$	8.96663			0.506824			0.253412	−	0.967358i	$-0.418447\pi$
$313$	8.96663			0.506824			0.253412	+	0.967358i	$0.418447\pi$
$314$		0			0
$315$	−4.84596	−	15.9755i	−0.273039	−	0.900117i
$316$		0			0
$317$		−	9.48331i		−	0.532636i	−0.963885	−	0.266318i	$-0.914193\pi$
$317$		−	9.48331i		−	0.532636i	0.963885	−	0.266318i	$-0.0858071\pi$
$318$		0			0
$319$			19.1981i			1.07489i
$320$		0			0
$321$	−11.6125	−	8.61249i	−0.648146	−	0.480702i
$322$		0			0
$323$	−5.56477			−0.309632
$324$		0			0
$325$	5.74166			0.318490
$326$		0			0
$327$	−7.62834	−	5.65762i	−0.421848	−	0.312867i
$328$		0			0
$329$			26.9666i			1.48672i
$330$		0			0
$331$		−	32.1367i		−	1.76639i	−0.469004	−	0.883196i	$-0.655387\pi$
$331$		−	32.1367i		−	1.76639i	0.469004	−	0.883196i	$-0.344613\pi$
$332$		0			0
$333$	−8.25834	−	27.2250i	−0.452554	−	1.49192i
$334$		0			0
$335$	22.2591			1.21614
$336$		0			0
$337$	12.9666			0.706337			0.353169	−	0.935560i	$-0.385104\pi$
$337$	12.9666			0.706337			0.353169	+	0.935560i	$0.385104\pi$
$338$		0			0
$339$	2.06358	−	2.78238i	0.112078	−	0.151118i
$340$		0			0
$341$		−	12.2583i		−	0.663826i
$342$		0			0
$343$		−	17.4131i		−	0.940219i
$344$		0			0
$345$	14.2583	−	19.2250i	0.767643	−	1.03504i
$346$		0			0
$347$	−13.0074			−0.698274			−0.349137	−	0.937072i	$-0.613525\pi$
$347$	−13.0074			−0.698274			−0.349137	+	0.937072i	$0.613525\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$		0			0
$351$	10.0513	−	28.0904i	0.536500	−	1.49935i
$352$		0			0
$353$			6.96663i			0.370796i	0.982663	+	0.185398i	$0.0593575\pi$
$353$			6.96663i			0.370796i	−0.982663	+	0.185398i	$0.940643\pi$
$354$		0			0
$355$		−	9.87762i		−	0.524249i
$356$		0			0
$357$	3.87083	+	2.87083i	0.204866	+	0.151940i
$358$		0			0
$359$	19.5695			1.03284			0.516420	−	0.856335i	$-0.327264\pi$
$359$	19.5695			1.03284			0.516420	+	0.856335i	$0.327264\pi$
$360$		0			0
$361$	−11.9666			−0.629823
$362$		0			0
$363$	−1.75060	−	1.29834i	−0.0918825	−	0.0681453i
$364$		0			0
$365$			12.0000i			0.628109i
$366$		0			0
$367$			22.1662i			1.15707i	0.815659	+	0.578534i	$0.196375\pi$
$367$			22.1662i			1.15707i	−0.815659	+	0.578534i	$0.803625\pi$
$368$		0			0
$369$	27.2250	−	8.25834i	1.41728	−	0.429912i
$370$		0			0
$371$	−5.56477			−0.288908
$372$		0			0
$373$	−1.22497			−0.0634267			−0.0317133	−	0.999497i	$-0.510096\pi$
$373$	−1.22497			−0.0634267			−0.0317133	+	0.999497i	$0.510096\pi$
$374$		0			0
$375$	−12.3815	+	16.6943i	−0.639375	+	0.862090i
$376$		0			0
$377$			31.4833i			1.62147i
$378$		0			0
$379$			7.81404i			0.401380i	0.979655	+	0.200690i	$0.0643184\pi$
$379$			7.81404i			0.401380i	−0.979655	+	0.200690i	$0.935682\pi$
$380$		0			0
$381$	5.74166	−	7.74166i	0.294154	−	0.396617i
$382$		0			0
$383$	−12.3815			−0.632663			−0.316331	−	0.948649i	$-0.602451\pi$
$383$	−12.3815			−0.632663			−0.316331	+	0.948649i	$0.602451\pi$
$384$		0			0
$385$	−19.4833			−0.992962
$386$		0			0
$387$	−20.1026	+	6.09788i	−1.02187	+	0.309972i
$388$		0			0
$389$			1.22497i			0.0621086i	0.999518	+	0.0310543i	$0.00988647\pi$
$389$			1.22497i			0.0621086i	−0.999518	+	0.0310543i	$0.990114\pi$
$390$		0			0
$391$			6.90953i			0.349430i
$392$		0			0
$393$	0.129171	+	0.0958009i	0.00651583	+	0.00483251i
$394$		0			0
$395$	−23.5110			−1.18297
$396$		0			0
$397$	13.4833			0.676708			0.338354	−	0.941019i	$-0.390130\pi$
$397$	13.4833			0.676708			0.338354	+	0.941019i	$0.390130\pi$
$398$		0			0
$399$	21.5403	+	15.9755i	1.07836	+	0.799775i
$400$		0			0
$401$			29.4833i			1.47233i	0.676804	+	0.736163i	$0.263364\pi$
$401$			29.4833i			1.47233i	−0.676804	+	0.736163i	$0.736636\pi$
$402$		0			0
$403$		−	20.1026i		−	1.00138i
$404$		0			0
$405$	10.0000	+	14.9666i	0.496904	+	0.743698i
$406$		0			0
$407$	−33.2029			−1.64581
$408$		0			0
$409$	22.9666			1.13563			0.567813	−	0.823157i	$-0.307790\pi$
$409$	22.9666			1.13563			0.567813	+	0.823157i	$0.307790\pi$
$410$		0			0
$411$	−6.45728	+	8.70655i	−0.318514	+	0.429463i
$412$		0			0
$413$		−	15.4833i		−	0.761884i
$414$		0			0
$415$		−	5.37907i		−	0.264048i
$416$		0			0
$417$	10.0958	−	13.6125i	0.494393	−	0.666606i
$418$		0			0
$419$	35.9853			1.75800			0.878998	−	0.476825i	$-0.158213\pi$
$419$	35.9853			1.75800			0.878998	+	0.476825i	$0.158213\pi$
$420$		0			0
$421$	−21.7417			−1.05962			−0.529812	−	0.848115i	$-0.677737\pi$
$421$	−21.7417			−1.05962			−0.529812	+	0.848115i	$0.677737\pi$
$422$		0			0
$423$	−8.44000	−	27.8238i	−0.410367	−	1.35284i
$424$		0			0
$425$		−	1.00000i		−	0.0485071i
$426$		0			0
$427$		−	15.2567i		−	0.738323i
$428$		0			0
$429$	−27.9666	−	20.7417i	−1.35024	−	1.00142i
$430$		0			0
$431$	−4.40570			−0.212215			−0.106108	−	0.994355i	$-0.533839\pi$
$431$	−4.40570			−0.212215			−0.106108	+	0.994355i	$0.533839\pi$
$432$		0			0
$433$	−29.2250			−1.40446			−0.702231	−	0.711949i	$-0.747813\pi$
$433$	−29.2250			−1.40446			−0.702231	+	0.711949i	$0.747813\pi$
$434$		0			0
$435$	−15.2567	−	11.3152i	−0.731502	−	0.542524i
$436$		0			0
$437$			38.4499i			1.83931i
$438$		0			0
$439$		−	27.9167i		−	1.33239i	−0.745777	−	0.666195i	$-0.767922\pi$
$439$		−	27.9167i		−	1.33239i	0.745777	−	0.666195i	$-0.232078\pi$
$440$		0			0
$441$	−0.645857	−	2.12917i	−0.0307551	−	0.101389i
$442$		0			0
$443$	15.4424			0.733690			0.366845	−	0.930282i	$-0.380438\pi$
$443$	15.4424			0.733690			0.366845	+	0.930282i	$0.380438\pi$
$444$		0			0
$445$	26.4499			1.25385
$446$		0			0
$447$	−4.12715	+	5.56477i	−0.195208	+	0.263204i
$448$		0			0
$449$			10.0000i			0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$			10.0000i			0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$		0			0
$451$		−	33.2029i		−	1.56346i
$452$		0			0
$453$	15.7417	−	21.2250i	0.739608	−	0.997236i
$454$		0			0
$455$	−31.9510			−1.49789
$456$		0			0
$457$	24.7083			1.15580			0.577902	−	0.816106i	$-0.303871\pi$
$457$	24.7083			1.15580			0.577902	+	0.816106i	$0.303871\pi$
$458$		0			0
$459$	−4.89238	−	1.75060i	−0.228357	−	0.0817108i
$460$		0			0
$461$			2.00000i			0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$			2.00000i			0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$		0			0
$463$			8.25430i			0.383610i	0.981433	+	0.191805i	$0.0614341\pi$
$463$			8.25430i			0.383610i	−0.981433	+	0.191805i	$0.938566\pi$
$464$		0			0
$465$	9.74166	+	7.22497i	0.451759	+	0.335050i
$466$		0			0
$467$	−9.50622			−0.439895			−0.219948	−	0.975512i	$-0.570589\pi$
$467$	−9.50622			−0.439895			−0.219948	+	0.975512i	$0.570589\pi$
$468$		0			0
$469$	30.9666			1.42991
$470$		0			0
$471$	−16.6943	−	12.3815i	−0.769233	−	0.570507i
$472$		0			0
$473$			24.5167i			1.12728i
$474$		0			0
$475$		−	5.56477i		−	0.255329i
$476$		0			0
$477$	5.74166	−	1.74166i	0.262892	−	0.0797450i
$478$		0			0
$479$	15.1638			0.692853			0.346427	−	0.938077i	$-0.387395\pi$
$479$	15.1638			0.692853			0.346427	+	0.938077i	$0.387395\pi$
$480$		0			0
$481$	−54.4499			−2.48270
$482$		0			0
$483$	19.8361	−	26.7456i	0.902573	−	1.21697i
$484$		0			0
$485$			18.9666i			0.861230i
$486$		0			0
$487$		−	16.4158i		−	0.743869i	−0.928259	−	0.371934i	$-0.878695\pi$
$487$		−	16.4158i		−	0.743869i	0.928259	−	0.371934i	$-0.121305\pi$
$488$		0			0
$489$	15.6459	−	21.0958i	0.707530	−	0.953985i
$490$		0			0
$491$	−33.2029			−1.49843			−0.749213	−	0.662329i	$-0.769568\pi$
$491$	−33.2029			−1.49843			−0.749213	+	0.662329i	$0.769568\pi$
$492$		0			0
$493$	5.48331			0.246956
$494$		0			0
$495$	20.1026	−	6.09788i	0.903547	−	0.274079i
$496$		0			0
$497$		−	13.7417i		−	0.616398i
$498$		0			0
$499$			18.9436i			0.848031i	0.905655	+	0.424015i	$0.139380\pi$
$499$			18.9436i			0.848031i	−0.905655	+	0.424015i	$0.860620\pi$
$500$		0			0
$501$	12.6125	+	9.35414i	0.563484	+	0.417912i
$502$		0			0
$503$	35.4522			1.58073			0.790367	−	0.612633i	$-0.209890\pi$
$503$	35.4522			1.58073			0.790367	+	0.612633i	$0.209890\pi$
$504$		0			0
$505$	−35.4833			−1.57899
$506$		0			0
$507$	−27.7774	−	20.6013i	−1.23364	−	0.914937i
$508$		0			0
$509$			18.9666i			0.840681i	0.907366	+	0.420341i	$0.138089\pi$
$509$			18.9666i			0.840681i	−0.907366	+	0.420341i	$0.861911\pi$
$510$		0			0
$511$			16.6943i			0.738512i
$512$		0			0
$513$	−27.2250	−	9.74166i	−1.20201	−	0.430105i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−33.9333			−1.49238
$518$		0			0
$519$	5.65762	−	7.62834i	0.248342	−	0.334847i
$520$		0			0
$521$			18.5167i			0.811231i	0.914044	+	0.405615i	$0.132943\pi$
$521$			18.5167i			0.811231i	−0.914044	+	0.405615i	$0.867057\pi$
$522$		0			0
$523$			37.3301i			1.63233i	0.577819	+	0.816165i	$0.303904\pi$
$523$			37.3301i			1.63233i	−0.577819	+	0.816165i	$0.696096\pi$
$524$		0			0
$525$	−2.87083	+	3.87083i	−0.125293	+	0.168937i
$526$		0			0
$527$	−3.50119			−0.152514
$528$		0			0
$529$	24.7417			1.07572
$530$		0			0
$531$	4.84596	+	15.9755i	0.210297	+	0.693277i
$532$		0			0
$533$		−	54.4499i		−	2.35849i
$534$		0			0
$535$		−	16.6943i		−	0.721758i
$536$		0			0
$537$	−32.7083	−	24.2583i	−1.41147	−	1.04682i
$538$		0			0
$539$	−2.59668			−0.111847
$540$		0			0
$541$	13.4833			0.579693			0.289846	−	0.957073i	$-0.406396\pi$
$541$	13.4833			0.579693			0.289846	+	0.957073i	$0.406396\pi$
$542$		0			0
$543$	−2.06358	−	1.53047i	−0.0885565	−	0.0656786i
$544$		0			0
$545$		−	10.9666i		−	0.469759i
$546$		0			0
$547$		−	13.1931i		−	0.564097i	−0.959400	−	0.282048i	$-0.908986\pi$
$547$		−	13.1931i		−	0.564097i	0.959400	−	0.282048i	$-0.0910138\pi$
$548$		0			0
$549$	4.77503	+	15.7417i	0.203793	+	0.671838i
$550$		0			0
$551$	30.5134			1.29991
$552$		0			0
$553$	−32.7083			−1.39090
$554$		0			0
$555$	19.5695	−	26.3862i	0.830680	−	1.12003i
$556$		0			0
$557$		−	0.708287i		−	0.0300111i	−0.999887	−	0.0150055i	$-0.995223\pi$
$557$		−	0.708287i		−	0.0300111i	0.999887	−	0.0150055i	$-0.00477659\pi$
$558$		0			0
$559$			40.2053i			1.70050i
$560$		0			0
$561$	−3.61249	+	4.87083i	−0.152519	+	0.205647i
$562$		0			0
$563$	45.7701			1.92898			0.964489	−	0.264124i	$-0.0850827\pi$
$563$	45.7701			1.92898			0.964489	+	0.264124i	$0.0850827\pi$
$564$		0			0
$565$	4.00000			0.168281
$566$		0			0
$567$	13.9119	+	20.8215i	0.584246	+	0.874419i
$568$		0			0
$569$		−	19.9333i		−	0.835646i	−0.908529	−	0.417823i	$-0.862793\pi$
$569$		−	19.9333i		−	0.835646i	0.908529	−	0.417823i	$-0.137207\pi$
$570$		0			0
$571$			15.6969i			0.656897i	0.944522	+	0.328448i	$0.106526\pi$
$571$			15.6969i			0.656897i	−0.944522	+	0.328448i	$0.893474\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.90953			−0.288147
$576$		0			0
$577$	33.2250			1.38317			0.691587	−	0.722293i	$-0.256912\pi$
$577$	33.2250			1.38317			0.691587	+	0.722293i	$0.256912\pi$
$578$		0			0
$579$	−19.4767	−	14.4450i	−0.809423	−	0.600315i
$580$		0			0
$581$		−	7.48331i		−	0.310460i
$582$		0			0
$583$		−	7.00238i		−	0.290009i
$584$		0			0
$585$	32.9666	−	10.0000i	1.36300	−	0.413449i
$586$		0			0
$587$	32.1367			1.32642			0.663211	−	0.748432i	$-0.269193\pi$
$587$	32.1367			1.32642			0.663211	+	0.748432i	$0.269193\pi$
$588$		0			0
$589$	−19.4833			−0.802796
$590$		0			0
$591$	−25.2271	+	34.0146i	−1.03771	+	1.39917i
$592$		0			0
$593$		−	15.9333i		−	0.654301i	−0.944972	−	0.327150i	$-0.893912\pi$
$593$		−	15.9333i		−	0.654301i	0.944972	−	0.327150i	$-0.106088\pi$
$594$		0			0
$595$			5.56477i			0.228133i
$596$		0			0
$597$	18.0624	−	24.3541i	0.739246	−	0.996749i
$598$		0			0
$599$	−26.5719			−1.08570			−0.542850	−	0.839830i	$-0.682655\pi$
$599$	−26.5719			−1.08570			−0.542850	+	0.839830i	$0.682655\pi$
$600$		0			0
$601$	−27.9333			−1.13942			−0.569711	−	0.821845i	$-0.692945\pi$
$601$	−27.9333			−1.13942			−0.569711	+	0.821845i	$0.692945\pi$
$602$		0			0
$603$	−31.9510	+	9.69192i	−1.30114	+	0.394685i
$604$		0			0
$605$		−	2.51669i		−	0.102318i
$606$		0			0
$607$		−	27.0122i		−	1.09639i	−0.836350	−	0.548195i	$-0.815315\pi$
$607$		−	27.0122i		−	1.09639i	0.836350	−	0.548195i	$-0.184685\pi$
$608$		0			0
$609$	−21.2250	−	15.7417i	−0.860079	−	0.637884i
$610$		0			0
$611$	−55.6477			−2.25126
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$	26.3862	+	19.5695i	1.06399	+	0.789120i
$616$		0			0
$617$			44.9666i			1.81029i	0.425105	+	0.905144i	$0.360237\pi$
$617$			44.9666i			1.81029i	−0.425105	+	0.905144i	$0.639763\pi$
$618$		0			0
$619$		−	34.7334i		−	1.39605i	−0.716072	−	0.698026i	$-0.754062\pi$
$619$		−	34.7334i		−	1.39605i	0.716072	−	0.698026i	$-0.245938\pi$
$620$		0			0
$621$	−12.0958	+	33.8041i	−0.485388	+	1.35651i
$622$		0			0
$623$	36.7969			1.47424
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−20.1026	+	27.1050i	−0.802822	+	1.08247i
$628$		0			0
$629$			9.48331i			0.378125i
$630$		0			0
$631$		−	20.6358i		−	0.821496i	−0.911749	−	0.410748i	$-0.865268\pi$
$631$		−	20.6358i		−	0.821496i	0.911749	−	0.410748i	$-0.134732\pi$
$632$		0			0
$633$	0.837458	−	1.12917i	0.0332860	−	0.0448805i
$634$		0			0
$635$	11.1295			0.441662
$636$		0			0
$637$	−4.25834			−0.168722
$638$		0			0
$639$	4.30086	+	14.1785i	0.170139	+	0.560892i
$640$		0			0
$641$		−	3.03337i		−	0.119811i	−0.998204	−	0.0599055i	$-0.980920\pi$
$641$		−	3.03337i		−	0.119811i	0.998204	−	0.0599055i	$-0.0190799\pi$
$642$		0			0
$643$			40.8312i			1.61023i	0.593121	+	0.805114i	$0.297896\pi$
$643$			40.8312i			1.61023i	−0.593121	+	0.805114i	$0.702104\pi$
$644$		0			0
$645$	−19.4833	−	14.4499i	−0.767155	−	0.568966i
$646$		0			0
$647$	−47.3934			−1.86323			−0.931613	−	0.363452i	$-0.881598\pi$
$647$	−47.3934			−1.86323			−0.931613	+	0.363452i	$0.881598\pi$
$648$		0			0
$649$	19.4833			0.764787
$650$		0			0
$651$	13.5525	+	10.0513i	0.531165	+	0.393942i
$652$		0			0
$653$			30.5167i			1.19421i	0.802163	+	0.597105i	$0.203682\pi$
$653$			30.5167i			1.19421i	−0.802163	+	0.597105i	$0.796318\pi$
$654$		0			0
$655$			0.185699i			0.00725585i
$656$		0			0
$657$	−5.22497	−	17.2250i	−0.203846	−	0.672010i
$658$		0			0
$659$	−4.31285			−0.168005			−0.0840024	−	0.996466i	$-0.526770\pi$
$659$	−4.31285			−0.168005			−0.0840024	+	0.996466i	$0.526770\pi$
$660$		0			0
$661$	43.9333			1.70881			0.854403	−	0.519611i	$-0.173923\pi$
$661$	43.9333			1.70881			0.854403	+	0.519611i	$0.173923\pi$
$662$		0			0
$663$	−5.92417	+	7.98775i	−0.230076	+	0.310218i
$664$		0			0
$665$			30.9666i			1.20083i
$666$		0			0
$667$		−	37.8871i		−	1.46700i
$668$		0			0
$669$	−11.2917	+	15.2250i	−0.436563	+	0.588631i
$670$		0			0
$671$	19.1981			0.741136
$672$		0			0
$673$	30.5167			1.17633			0.588166	−	0.808740i	$-0.299850\pi$
$673$	30.5167			1.17633			0.588166	+	0.808740i	$0.299850\pi$
$674$		0			0
$675$	1.75060	−	4.89238i	0.0673805	−	0.188308i
$676$		0			0
$677$		−	13.4833i		−	0.518206i	−0.965850	−	0.259103i	$-0.916573\pi$
$677$		−	13.4833i		−	0.518206i	0.965850	−	0.259103i	$-0.0834269\pi$
$678$		0			0
$679$			26.3862i			1.01261i
$680$		0			0
$681$	−30.0958	−	22.3208i	−1.15327	−	0.855334i
$682$		0			0
$683$	2.59668			0.0993594			0.0496797	−	0.998765i	$-0.484180\pi$
$683$	2.59668			0.0993594			0.0496797	+	0.998765i	$0.484180\pi$
$684$		0			0
$685$	−12.5167			−0.478238
$686$		0			0
$687$	2.42298	+	1.79702i	0.0924424	+	0.0685606i
$688$		0			0
$689$		−	11.4833i		−	0.437479i
$690$		0			0
$691$			29.3543i			1.11669i	0.829609	+	0.558345i	$0.188563\pi$
$691$			29.3543i			1.11669i	−0.829609	+	0.558345i	$0.811437\pi$
$692$		0			0
$693$	27.9666	−	8.48331i	1.06236	−	0.322254i
$694$		0			0
$695$	19.5695			0.742315
$696$		0			0
$697$	−9.48331			−0.359206
$698$		0			0
$699$	−5.12451	+	6.90953i	−0.193827	+	0.261343i
$700$		0			0
$701$			28.1916i			1.06478i	0.846499	+	0.532391i	$0.178706\pi$
$701$			28.1916i			1.06478i	−0.846499	+	0.532391i	$0.821294\pi$
$702$		0			0
$703$			52.7724i			1.99035i
$704$		0			0
$705$	20.0000	−	26.9666i	0.753244	−	1.01562i
$706$		0			0
$707$	−49.3641			−1.85653
$708$		0			0
$709$	−40.9666			−1.53853			−0.769267	−	0.638927i	$-0.779378\pi$
$709$	−40.9666			−1.53853			−0.769267	+	0.638927i	$0.779378\pi$
$710$		0			0
$711$	33.7480	−	10.2370i	1.26565	−	0.383918i
$712$		0			0
$713$			24.1916i			0.905983i
$714$		0			0
$715$		−	40.2053i		−	1.50359i
$716$		0			0
$717$	−26.9666	−	20.0000i	−1.00709	−	0.746914i
$718$		0			0
$719$	−6.56212			−0.244726			−0.122363	−	0.992485i	$-0.539047\pi$
$719$	−6.56212			−0.244726			−0.122363	+	0.992485i	$0.539047\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	6.90953	+	5.12451i	0.256968	+	0.190582i
$724$		0			0
$725$			5.48331i			0.203645i
$726$		0			0
$727$			35.8924i			1.33118i	0.746319	+	0.665588i	$0.231819\pi$
$727$			35.8924i			1.33118i	−0.746319	+	0.665588i	$0.768181\pi$
$728$		0			0
$729$	−20.8708	−	17.1292i	−0.772994	−	0.634414i
$730$		0			0
$731$	7.00238			0.258992
$732$		0			0
$733$	2.00000			0.0738717			0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000			0.0738717			0.0369358	+	0.999318i	$0.488240\pi$
$734$		0			0
$735$	1.53047	−	2.06358i	0.0564521	−	0.0761161i
$736$		0			0
$737$			38.9666i			1.43535i
$738$		0			0
$739$			37.5158i			1.38004i	0.723791	+	0.690020i	$0.242398\pi$
$739$			37.5158i			1.38004i	−0.723791	+	0.690020i	$0.757602\pi$
$740$		0			0
$741$	−32.9666	+	44.4499i	−1.21106	+	1.63291i
$742$		0			0
$743$	54.3029			1.99218			0.996090	−	0.0883430i	$-0.0281572\pi$
$743$	54.3029			1.99218			0.996090	+	0.0883430i	$0.0281572\pi$
$744$		0			0
$745$	−8.00000			−0.293097
$746$		0			0
$747$	2.34212	+	7.72119i	0.0856938	+	0.282504i
$748$		0			0
$749$		−	23.2250i		−	0.848622i
$750$		0			0
$751$			41.9214i			1.52974i	0.644187	+	0.764868i	$0.277196\pi$
$751$			41.9214i			1.52974i	−0.644187	+	0.764868i	$0.722804\pi$
$752$		0			0
$753$	34.9666	+	25.9333i	1.27425	+	0.945060i
$754$		0			0
$755$	30.5134			1.11050
$756$		0			0
$757$	10.7750			0.391625			0.195813	−	0.980641i	$-0.437266\pi$
$757$	10.7750			0.391625			0.195813	+	0.980641i	$0.437266\pi$
$758$		0			0
$759$	33.6552	+	24.9606i	1.22160	+	0.906012i
$760$		0			0
$761$		−	13.2250i		−	0.479405i	−0.970846	−	0.239702i	$-0.922950\pi$
$761$		−	13.2250i		−	0.479405i	0.970846	−	0.239702i	$-0.0770499\pi$
$762$		0			0
$763$		−	15.2567i		−	0.552329i
$764$		0			0
$765$	−1.74166	−	5.74166i	−0.0629698	−	0.207590i
$766$		0			0
$767$	31.9510			1.15368
$768$		0			0
$769$	28.7083			1.03525			0.517624	−	0.855608i	$-0.326817\pi$
$769$	28.7083			1.03525			0.517624	+	0.855608i	$0.326817\pi$
$770$		0			0
$771$	22.4328	−	30.2468i	0.807897	−	1.08931i
$772$		0			0
$773$			16.1916i			0.582371i	0.956667	+	0.291186i	$0.0940498\pi$
$773$			16.1916i			0.582371i	−0.956667	+	0.291186i	$0.905950\pi$
$774$		0			0
$775$		−	3.50119i		−	0.125767i
$776$		0			0
$777$	27.2250	−	36.7083i	0.976690	−	1.31690i
$778$		0			0
$779$	−52.7724			−1.89077
$780$		0			0
$781$	17.2917			0.618746
$782$		0			0
$783$	26.8265	+	9.59907i	0.958700	+	0.343043i
$784$		0			0
$785$		−	24.0000i		−	0.856597i
$786$		0			0
$787$		−	29.1686i		−	1.03975i	−0.854243	−	0.519874i	$-0.825979\pi$
$787$		−	29.1686i		−	1.03975i	0.854243	−	0.519874i	$-0.174021\pi$
$788$		0			0
$789$	25.2250	+	18.7083i	0.898033	+	0.666033i
$790$		0			0
$791$	5.56477			0.197860
$792$		0			0
$793$	31.4833			1.11801
$794$		0			0
$795$	5.56477	+	4.12715i	0.197362	+	0.146375i
$796$		0			0
$797$		−	14.9666i		−	0.530145i	−0.964228	−	0.265073i	$-0.914604\pi$
$797$		−	14.9666i		−	0.530145i	0.964228	−	0.265073i	$-0.0853959\pi$
$798$		0			0
$799$			9.69192i			0.342875i
$800$		0			0
$801$	−37.9666	+	11.5167i	−1.34148	+	0.406922i
$802$		0			0
$803$	−21.0071			−0.741326
$804$		0			0
$805$	38.4499			1.35518
$806$		0			0
$807$	21.6331	−	29.1686i	0.761521	−	1.02678i
$808$		0			0
$809$			52.9666i			1.86221i	0.364754	+	0.931104i	$0.381153\pi$
$809$			52.9666i			1.86221i	−0.364754	+	0.931104i	$0.618847\pi$
$810$		0			0
$811$		−	4.03430i		−	0.141663i	−0.997488	−	0.0708317i	$-0.977435\pi$
$811$		−	4.03430i		−	0.141663i	0.997488	−	0.0708317i	$-0.0225653\pi$
$812$		0			0
$813$	7.03337	−	9.48331i	0.246671	−	0.332594i
$814$		0			0
$815$	30.3277			1.06233
$816$		0			0
$817$	38.9666			1.36327
$818$		0			0
$819$	45.8629	−	13.9119i	1.60258	−	0.486122i
$820$		0			0
$821$		−	7.55006i		−	0.263499i	−0.991283	−	0.131749i	$-0.957941\pi$
$821$		−	7.55006i		−	0.263499i	0.991283	−	0.131749i	$-0.0420594\pi$
$822$		0			0
$823$			10.5036i			0.366132i	0.983101	+	0.183066i	$0.0586021\pi$
$823$			10.5036i			0.366132i	−0.983101	+	0.183066i	$0.941398\pi$
$824$		0			0
$825$	−4.87083	−	3.61249i	−0.169580	−	0.125771i
$826$		0			0
$827$	0.0928494			0.00322869			0.00161434	−	0.999999i	$-0.499486\pi$
$827$	0.0928494			0.00322869			0.00161434	+	0.999999i	$0.499486\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$	−6.90953	−	5.12451i	−0.239689	−	0.177767i
$832$		0			0
$833$			0.741657i			0.0256969i
$834$		0			0
$835$			18.1319i			0.627481i
$836$		0			0
$837$	−17.1292	−	6.12917i	−0.592071	−	0.211855i
$838$		0			0
$839$	25.5746			0.882932			0.441466	−	0.897278i	$-0.354459\pi$
$839$	25.5746			0.882932			0.441466	+	0.897278i	$0.354459\pi$
$840$		0			0
$841$	−1.06674			−0.0367842
$842$		0			0
$843$	−27.8238	+	37.5158i	−0.958304	+	1.29211i
$844$		0			0
$845$		−	39.9333i		−	1.37375i
$846$		0			0
$847$		−	3.50119i		−	0.120302i
$848$		0			0
$849$	−9.90420	+	13.3541i	−0.339911	+	0.458313i
$850$		0			0
$851$	65.5253			2.24618
$852$		0			0
$853$	20.4499			0.700193			0.350096	−	0.936714i	$-0.386149\pi$
$853$	20.4499			0.700193			0.350096	+	0.936714i	$0.386149\pi$
$854$		0			0
$855$	−9.69192	−	31.9510i	−0.331457	−	1.09270i
$856$		0			0
$857$		−	37.4833i		−	1.28041i	−0.768206	−	0.640203i	$-0.778850\pi$
$857$		−	37.4833i		−	1.28041i	0.768206	−	0.640203i	$-0.221150\pi$
$858$		0			0
$859$			11.3152i			0.386071i	0.981192	+	0.193035i	$0.0618332\pi$
$859$			11.3152i			0.386071i	−0.981192	+	0.193035i	$0.938167\pi$
$860$		0			0
$861$	36.7083	+	27.2250i	1.25102	+	0.927825i
$862$		0			0
$863$	−20.8215			−0.708770			−0.354385	−	0.935100i	$-0.615310\pi$
$863$	−20.8215			−0.708770			−0.354385	+	0.935100i	$0.615310\pi$
$864$		0			0
$865$	10.9666			0.372877
$866$		0			0
$867$	1.39119	+	1.03179i	0.0472473	+	0.0350413i
$868$		0			0
$869$		−	41.1582i		−	1.39620i
$870$		0			0
$871$			63.9020i			2.16524i
$872$		0			0
$873$	−8.25834	−	27.2250i	−0.279503	−	0.921426i
$874$		0			0
$875$	−33.3886			−1.12874
$876$		0			0
$877$	−54.3832			−1.83639			−0.918195	−	0.396128i	$-0.870354\pi$
$877$	−54.3832			−1.83639			−0.918195	+	0.396128i	$0.870354\pi$
$878$		0			0
$879$	−3.06093	+	4.12715i	−0.103243	+	0.139205i
$880$		0			0
$881$		−	18.5167i		−	0.623843i	−0.950108	−	0.311921i	$-0.899027\pi$
$881$		−	18.5167i		−	0.623843i	0.950108	−	0.311921i	$-0.100973\pi$
$882$		0			0
$883$		−	12.5671i		−	0.422918i	−0.977387	−	0.211459i	$-0.932178\pi$
$883$		−	12.5671i		−	0.422918i	0.977387	−	0.211459i	$-0.0678215\pi$
$884$		0			0
$885$	−11.4833	+	15.4833i	−0.386007	+	0.520466i
$886$		0			0
$887$	−26.2934			−0.882845			−0.441422	−	0.897299i	$-0.645526\pi$
$887$	−26.2934			−0.882845			−0.441422	+	0.897299i	$0.645526\pi$
$888$		0			0
$889$	15.4833			0.519294
$890$		0			0
$891$	−26.2005	+	17.5060i	−0.877750	+	0.586472i
$892$		0			0
$893$			53.9333i			1.80481i
$894$		0			0
$895$		−	47.0220i		−	1.57177i
$896$		0			0
$897$	55.1916	+	40.9333i	1.84279	+	1.36672i
$898$		0			0
$899$	19.1981			0.640294
$900$		0			0
$901$	−2.00000			−0.0666297
$902$		0			0
$903$	−27.1050	−	20.1026i	−0.901999	−	0.668974i
$904$		0			0
$905$		−	2.96663i		−	0.0986141i
$906$		0			0
$907$		−	45.6772i		−	1.51669i	−0.651855	−	0.758343i	$-0.726009\pi$
$907$		−	45.6772i		−	1.51669i	0.651855	−	0.758343i	$-0.273991\pi$
$908$		0			0
$909$	50.9333	−	15.4499i	1.68935	−	0.512442i
$910$		0			0
$911$	−16.0683			−0.532368			−0.266184	−	0.963922i	$-0.585763\pi$
$911$	−16.0683			−0.532368			−0.266184	+	0.963922i	$0.585763\pi$
$912$		0			0
$913$	9.41657			0.311643
$914$		0			0
$915$	−11.3152	+	15.2567i	−0.374070	+	0.504370i
$916$		0			0
$917$			0.258343i			0.00853122i
$918$		0			0
$919$		−	8.62570i		−	0.284536i	−0.989828	−	0.142268i	$-0.954561\pi$
$919$		−	8.62570i		−	0.284536i	0.989828	−	0.142268i	$-0.0454394\pi$
$920$		0			0
$921$	−18.5167	+	24.9666i	−0.610146	+	0.822678i
$922$		0			0
$923$	28.3569			0.933380
$924$		0			0
$925$	−9.48331			−0.311809
$926$		0			0
$927$		0			0
$928$		0			0
$929$			27.4166i			0.899509i	0.893152	+	0.449754i	$0.148488\pi$
$929$			27.4166i			0.899509i	−0.893152	+	0.449754i	$0.851512\pi$
$930$		0			0
$931$			4.12715i			0.135262i
$932$		0			0
$933$	−1.12917	−	0.837458i	−0.0369674	−	0.0274172i
$934$		0			0
$935$	−7.00238			−0.229002
$936$		0			0
$937$	−25.9333			−0.847203			−0.423601	−	0.905849i	$-0.639234\pi$
$937$	−25.9333			−0.847203			−0.423601	+	0.905849i	$0.639234\pi$
$938$		0			0
$939$	−12.4743	−	9.25166i	−0.407083	−	0.301916i
$940$		0			0
$941$		−	28.4499i		−	0.927442i	−0.885981	−	0.463721i	$-0.846514\pi$
$941$		−	28.4499i		−	0.927442i	0.885981	−	0.463721i	$-0.153486\pi$
$942$		0			0
$943$			65.5253i			2.13380i
$944$		0			0
$945$	−9.74166	+	27.2250i	−0.316896	+	0.885628i
$946$		0			0
$947$	−22.8850			−0.743663			−0.371832	−	0.928300i	$-0.621270\pi$
$947$	−22.8850			−0.743663			−0.371832	+	0.928300i	$0.621270\pi$
$948$		0			0
$949$	−34.4499			−1.11829
$950$		0			0
$951$	−9.78477	+	13.1931i	−0.317293	+	0.427816i
$952$		0			0
$953$			24.7083i			0.800380i	0.916432	+	0.400190i	$0.131056\pi$
$953$			24.7083i			0.800380i	−0.916432	+	0.400190i	$0.868944\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	19.8084	−	26.7083i	0.640315	−	0.863356i
$958$		0			0
$959$	−17.4131			−0.562299
$960$		0			0
$961$	18.7417			0.604570
$962$		0			0
$963$	7.26894	+	23.9632i	0.234238	+	0.772205i
$964$		0			0
$965$		−	28.0000i		−	0.901352i
$966$		0			0
$967$			4.12715i			0.132720i	0.997796	+	0.0663601i	$0.0211386\pi$
$967$			4.12715i			0.132720i	−0.997796	+	0.0663601i	$0.978861\pi$
$968$		0			0
$969$	7.74166	+	5.74166i	0.248698	+	0.184449i
$970$		0			0
$971$	35.0119			1.12359			0.561793	−	0.827278i	$-0.310112\pi$
$971$	35.0119			1.12359			0.561793	+	0.827278i	$0.310112\pi$
$972$		0			0
$973$	27.2250			0.872793
$974$		0			0
$975$	−7.98775	−	5.92417i	−0.255813	−	0.189725i
$976$		0			0
$977$		−	6.00000i		−	0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$		−	6.00000i		−	0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$		0			0
$979$			46.3032i			1.47986i
$980$		0			0
$981$	4.77503	+	15.7417i	0.152455	+	0.502592i
$982$		0			0
$983$	−49.4569			−1.57743			−0.788716	−	0.614758i	$-0.789254\pi$
$983$	−49.4569			−1.57743			−0.788716	+	0.614758i	$0.789254\pi$
$984$		0			0
$985$	−48.8999			−1.55808
$986$		0			0
$987$	27.8238	−	37.5158i	0.885642	−	1.19414i
$988$		0			0
$989$		−	48.3832i		−	1.53850i
$990$		0			0
$991$			19.2910i			0.612798i	0.951903	+	0.306399i	$0.0991242\pi$
$991$			19.2910i			0.612798i	−0.951903	+	0.306399i	$0.900876\pi$
$992$		0			0
$993$	−33.1582	+	44.7083i	−1.05224	+	1.41877i
$994$		0			0
$995$	35.0119			1.10995
$996$		0			0
$997$	−19.0334			−0.602793			−0.301396	−	0.953499i	$-0.597453\pi$
$997$	−19.0334			−0.602793			−0.301396	+	0.953499i	$0.597453\pi$
$998$		0			0
$999$	−16.6015	+	46.3960i	−0.525247	+	1.46790i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.e.b.239.1	✓	8
3.2	odd	2	inner	816.2.e.b.239.7	yes	8
4.3	odd	2	inner	816.2.e.b.239.8	yes	8
12.11	even	2	inner	816.2.e.b.239.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.e.b.239.1	✓	8	1.1	even	1	trivial
816.2.e.b.239.2	yes	8	12.11	even	2	inner
816.2.e.b.239.7	yes	8	3.2	odd	2	inner
816.2.e.b.239.8	yes	8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.39119 - 1.03179i) q^{3} -2.00000i q^{5} -2.78238i q^{7} +(0.870829 + 2.87083i) q^{9} +O(q^{10})\)	\(q+(-1.39119 - 1.03179i) q^{3} -2.00000i q^{5} -2.78238i q^{7} +(0.870829 + 2.87083i) q^{9} +3.50119 q^{11} +5.74166 q^{13} +(-2.06358 + 2.78238i) q^{15} -1.00000i q^{17} -5.56477i q^{19} +(-2.87083 + 3.87083i) q^{21} -6.90953 q^{23} +1.00000 q^{25} +(1.75060 - 4.89238i) q^{27} +5.48331i q^{29} -3.50119i q^{31} +(-4.87083 - 3.61249i) q^{33} -5.56477 q^{35} -9.48331 q^{37} +(-7.98775 - 5.92417i) q^{39} -9.48331i q^{41} +7.00238i q^{43} +(5.74166 - 1.74166i) q^{45} -9.69192 q^{47} -0.741657 q^{49} +(-1.03179 + 1.39119i) q^{51} -2.00000i q^{53} -7.00238i q^{55} +(-5.74166 + 7.74166i) q^{57} +5.56477 q^{59} +5.48331 q^{61} +(7.98775 - 2.42298i) q^{63} -11.4833i q^{65} +11.1295i q^{67} +(9.61249 + 7.12917i) q^{69} +4.93881 q^{71} -6.00000 q^{73} +(-1.39119 - 1.03179i) q^{75} -9.74166i q^{77} -11.7555i q^{79} +(-7.48331 + 5.00000i) q^{81} +2.68953 q^{83} -2.00000 q^{85} +(5.65762 - 7.62834i) q^{87} +13.2250i q^{89} -15.9755i q^{91} +(-3.61249 + 4.87083i) q^{93} -11.1295 q^{95} -9.48331 q^{97} +(3.04894 + 10.0513i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{13} - 8 q^{21} + 8 q^{25} - 24 q^{33} - 16 q^{37} + 16 q^{45} + 24 q^{49} - 16 q^{57} - 16 q^{61} + 32 q^{69} - 48 q^{73} - 16 q^{85} + 16 q^{93} - 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^13 - 8 * q^21 + 8 * q^25 - 24 * q^33 - 16 * q^37 + 16 * q^45 + 24 * q^49 - 16 * q^57 - 16 * q^61 + 32 * q^69 - 48 * q^73 - 16 * q^85 + 16 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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