LMFDB - Embedded newform 828.3.d.a.737.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [828,3,Mod(737,828)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("828.737"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 828 = 2^{2} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	828.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.5613658890$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 236 x^{14} + 21460 x^{12} + 962776 x^{10} + 22691080 x^{8} + 272328848 x^{6} + 1422566992 x^{4} + \cdots + 390615696 $ x^16 + 236x^14 + 21460x^12 + 962776x^10 + 22691080x^8 + 272328848x^6 + 1422566992x^4 + 1717701408*x^2 + 390615696
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			737.16
Root			$9.02853i$ of defining polynomial
Character	$\chi$	$=$	828.737
Dual form			828.3.d.a.737.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.02853i q^{5} -2.91592 q^{7} -14.2162i q^{11} -12.0916 q^{13} +29.9697i q^{17} +0.454908 q^{19} -4.79583i q^{23} -56.5143 q^{25} +11.9932i q^{29} -2.91889 q^{31} -26.3265i q^{35} -8.71329 q^{37} -48.3972i q^{41} -7.20046 q^{43} -86.6479i q^{47} -40.4974 q^{49} -52.9607i q^{53} +128.352 q^{55} -7.37931i q^{59} -0.539425 q^{61} -109.169i q^{65} -29.5166 q^{67} -79.0353i q^{71} -24.9609 q^{73} +41.4535i q^{77} -147.262 q^{79} +111.470i q^{83} -270.583 q^{85} +42.8828i q^{89} +35.2581 q^{91} +4.10715i q^{95} +59.7681 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{13} + 40 q^{19} - 72 q^{25} - 32 q^{31} - 64 q^{37} + 48 q^{43} + 8 q^{49} + 144 q^{55} - 48 q^{61} - 40 q^{67} + 48 q^{73} - 24 q^{79} - 8 q^{85} - 24 q^{91} + 168 q^{97}+O(q^{100}) $ 16 * q - 8 * q^13 + 40 * q^19 - 72 * q^25 - 32 * q^31 - 64 * q^37 + 48 * q^43 + 8 * q^49 + 144 * q^55 - 48 * q^61 - 40 * q^67 + 48 * q^73 - 24 * q^79 - 8 * q^85 - 24 * q^91 + 168 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/828\mathbb{Z}\right)^\times$.

$n$	$415$	$461$	$649$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	9.02853i	1.80571i	0.429950	+	0.902853i	$0.358531\pi$
$5$	9.02853i	1.80571i	−0.429950	+	0.902853i	$0.641469\pi$
$6$	0	0
$7$	−2.91592	−0.416560	−0.208280	−	0.978069i	$-0.566787\pi$
$7$	−2.91592	−0.416560	−0.208280	+	0.978069i	$0.566787\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.2162i	− 1.29239i	−0.763174	−	0.646193i	$-0.776360\pi$
$11$	− 14.2162i	− 1.29239i	0.763174	−	0.646193i	$-0.223640\pi$
$12$	0	0
$13$	−12.0916	−0.930121	−0.465060	−	0.885279i	$-0.653967\pi$
$13$	−12.0916	−0.930121	−0.465060	+	0.885279i	$0.653967\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	29.9697i	1.76293i	0.472253	+	0.881463i	$0.343441\pi$
$17$	29.9697i	1.76293i	−0.472253	+	0.881463i	$0.656559\pi$
$18$	0	0
$19$	0.454908	0.0239425	0.0119713	−	0.999928i	$-0.496189\pi$
$19$	0.454908	0.0239425	0.0119713	+	0.999928i	$0.496189\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	−56.5143	−2.26057
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11.9932i	0.413558i	0.978388	+	0.206779i	$0.0662981\pi$
$29$	11.9932i	0.413558i	−0.978388	+	0.206779i	$0.933702\pi$
$30$	0	0
$31$	−2.91889	−0.0941576	−0.0470788	−	0.998891i	$-0.514991\pi$
$31$	−2.91889	−0.0941576	−0.0470788	+	0.998891i	$0.514991\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 26.3265i	− 0.752185i
$36$	0	0
$37$	−8.71329	−0.235494	−0.117747	−	0.993044i	$-0.537567\pi$
$37$	−8.71329	−0.235494	−0.117747	+	0.993044i	$0.537567\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 48.3972i	− 1.18042i	−0.807250	−	0.590210i	$-0.799045\pi$
$41$	− 48.3972i	− 1.18042i	0.807250	−	0.590210i	$-0.200955\pi$
$42$	0	0
$43$	−7.20046	−0.167453	−0.0837263	−	0.996489i	$-0.526682\pi$
$43$	−7.20046	−0.167453	−0.0837263	+	0.996489i	$0.526682\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 86.6479i	− 1.84357i	−0.387697	−	0.921787i	$-0.626729\pi$
$47$	− 86.6479i	− 1.84357i	0.387697	−	0.921787i	$-0.373271\pi$
$48$	0	0
$49$	−40.4974	−0.826477
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 52.9607i	− 0.999259i	−0.866239	−	0.499630i	$-0.833469\pi$
$53$	− 52.9607i	− 0.999259i	0.866239	−	0.499630i	$-0.166531\pi$
$54$	0	0
$55$	128.352	2.33367
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.37931i	− 0.125073i	−0.998043	−	0.0625366i	$-0.980081\pi$
$59$	− 7.37931i	− 0.125073i	0.998043	−	0.0625366i	$-0.0199190\pi$
$60$	0	0
$61$	−0.539425	−0.00884304	−0.00442152	−	0.999990i	$-0.501407\pi$
$61$	−0.539425	−0.00884304	−0.00442152	+	0.999990i	$0.501407\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 109.169i	− 1.67952i
$66$	0	0
$67$	−29.5166	−0.440546	−0.220273	−	0.975438i	$-0.570695\pi$
$67$	−29.5166	−0.440546	−0.220273	+	0.975438i	$0.570695\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 79.0353i	− 1.11317i	−0.830790	−	0.556586i	$-0.812111\pi$
$71$	− 79.0353i	− 1.11317i	0.830790	−	0.556586i	$-0.187889\pi$
$72$	0	0
$73$	−24.9609	−0.341930	−0.170965	−	0.985277i	$-0.554689\pi$
$73$	−24.9609	−0.341930	−0.170965	+	0.985277i	$0.554689\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	41.4535i	0.538357i
$78$	0	0
$79$	−147.262	−1.86407	−0.932036	−	0.362366i	$-0.881969\pi$
$79$	−147.262	−1.86407	−0.932036	+	0.362366i	$0.881969\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	111.470i	1.34302i	0.740997	+	0.671508i	$0.234353\pi$
$83$	111.470i	1.34302i	−0.740997	+	0.671508i	$0.765647\pi$
$84$	0	0
$85$	−270.583	−3.18333
$86$	0	0
$87$	0	0
$88$	0	0
$89$	42.8828i	0.481830i	0.970546	+	0.240915i	$0.0774474\pi$
$89$	42.8828i	0.481830i	−0.970546	+	0.240915i	$0.922553\pi$
$90$	0	0
$91$	35.2581	0.387451
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.10715i	0.0432331i
$96$	0	0
$97$	59.7681	0.616166	0.308083	−	0.951360i	$-0.400313\pi$
$97$	59.7681	0.616166	0.308083	+	0.951360i	$0.400313\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.5378i	− 0.153840i	−0.997037	−	0.0769199i	$-0.975491\pi$
$101$	− 15.5378i	− 0.153840i	0.997037	−	0.0769199i	$-0.0245086\pi$
$102$	0	0
$103$	−121.985	−1.18432	−0.592158	−	0.805822i	$-0.701724\pi$
$103$	−121.985	−1.18432	−0.592158	+	0.805822i	$0.701724\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	134.803i	1.25984i	0.776658	+	0.629922i	$0.216913\pi$
$107$	134.803i	1.25984i	−0.776658	+	0.629922i	$0.783087\pi$
$108$	0	0
$109$	56.6131	0.519386	0.259693	−	0.965691i	$-0.416379\pi$
$109$	56.6131	0.519386	0.259693	+	0.965691i	$0.416379\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	213.979i	1.89362i	0.321797	+	0.946809i	$0.395713\pi$
$113$	213.979i	1.89362i	−0.321797	+	0.946809i	$0.604287\pi$
$114$	0	0
$115$	43.2993	0.376516
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 87.3894i	− 0.734365i
$120$	0	0
$121$	−81.1015	−0.670260
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 284.528i	− 2.27622i
$126$	0	0
$127$	37.4051	0.294528	0.147264	−	0.989097i	$-0.452953\pi$
$127$	37.4051	0.294528	0.147264	+	0.989097i	$0.452953\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 161.569i	− 1.23335i	−0.787217	−	0.616677i	$-0.788479\pi$
$131$	− 161.569i	− 1.23335i	0.787217	−	0.616677i	$-0.211521\pi$
$132$	0	0
$133$	−1.32648	−0.00997350
$134$	0	0
$135$	0	0
$136$	0	0
$137$	80.5251i	0.587775i	0.955840	+	0.293887i	$0.0949490\pi$
$137$	80.5251i	0.587775i	−0.955840	+	0.293887i	$0.905051\pi$
$138$	0	0
$139$	−66.1313	−0.475765	−0.237882	−	0.971294i	$-0.576453\pi$
$139$	−66.1313	−0.475765	−0.237882	+	0.971294i	$0.576453\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	171.897i	1.20207i
$144$	0	0
$145$	−108.281	−0.746764
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 161.484i	− 1.08378i	−0.840448	−	0.541891i	$-0.817708\pi$
$149$	− 161.484i	− 1.08378i	0.840448	−	0.541891i	$-0.182292\pi$
$150$	0	0
$151$	266.584	1.76546	0.882730	−	0.469880i	$-0.155703\pi$
$151$	266.584	1.76546	0.882730	+	0.469880i	$0.155703\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 26.3532i	− 0.170021i
$156$	0	0
$157$	−252.887	−1.61075	−0.805374	−	0.592767i	$-0.798036\pi$
$157$	−252.887	−1.61075	−0.805374	+	0.592767i	$0.798036\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.9843i	0.0868588i
$162$	0	0
$163$	−236.018	−1.44796	−0.723982	−	0.689819i	$-0.757690\pi$
$163$	−236.018	−1.44796	−0.723982	+	0.689819i	$0.757690\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	97.8196i	0.585746i	0.956151	+	0.292873i	$0.0946113\pi$
$167$	97.8196i	0.585746i	−0.956151	+	0.292873i	$0.905389\pi$
$168$	0	0
$169$	−22.7939	−0.134875
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 150.303i	− 0.868802i	−0.900719	−	0.434401i	$-0.856960\pi$
$173$	− 150.303i	− 0.868802i	0.900719	−	0.434401i	$-0.143040\pi$
$174$	0	0
$175$	164.791	0.941665
$176$	0	0
$177$	0	0
$178$	0	0
$179$	318.613i	1.77996i	0.455997	+	0.889981i	$0.349283\pi$
$179$	318.613i	1.77996i	−0.455997	+	0.889981i	$0.650717\pi$
$180$	0	0
$181$	222.240	1.22784	0.613921	−	0.789367i	$-0.289591\pi$
$181$	222.240	1.22784	0.613921	+	0.789367i	$0.289591\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 78.6682i	− 0.425233i
$186$	0	0
$187$	426.057	2.27838
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 84.3364i	− 0.441552i	−0.975325	−	0.220776i	$-0.929141\pi$
$191$	− 84.3364i	− 0.441552i	0.975325	−	0.220776i	$-0.0708590\pi$
$192$	0	0
$193$	254.017	1.31615	0.658075	−	0.752953i	$-0.271371\pi$
$193$	254.017	1.31615	0.658075	+	0.752953i	$0.271371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	88.0293i	0.446849i	0.974721	+	0.223425i	$0.0717236\pi$
$197$	88.0293i	0.446849i	−0.974721	+	0.223425i	$0.928276\pi$
$198$	0	0
$199$	−14.1244	−0.0709767	−0.0354884	−	0.999370i	$-0.511299\pi$
$199$	−14.1244	−0.0709767	−0.0354884	+	0.999370i	$0.511299\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 34.9712i	− 0.172272i
$204$	0	0
$205$	436.955	2.13149
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.46708i	− 0.0309430i
$210$	0	0
$211$	−385.707	−1.82799	−0.913997	−	0.405721i	$-0.867020\pi$
$211$	−385.707	−1.82799	−0.913997	+	0.405721i	$0.867020\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 65.0096i	− 0.302370i
$216$	0	0
$217$	8.51124	0.0392223
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 362.381i	− 1.63973i
$222$	0	0
$223$	−42.4422	−0.190324	−0.0951619	−	0.995462i	$-0.530337\pi$
$223$	−42.4422	−0.190324	−0.0951619	+	0.995462i	$0.530337\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 135.404i	− 0.596492i	−0.954489	−	0.298246i	$-0.903598\pi$
$227$	− 135.404i	− 0.596492i	0.954489	−	0.298246i	$-0.0964016\pi$
$228$	0	0
$229$	236.127	1.03112	0.515562	−	0.856852i	$-0.327583\pi$
$229$	236.127	1.03112	0.515562	+	0.856852i	$0.327583\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	388.640i	1.66798i	0.551777	+	0.833991i	$0.313950\pi$
$233$	388.640i	1.66798i	−0.551777	+	0.833991i	$0.686050\pi$
$234$	0	0
$235$	782.303	3.32895
$236$	0	0
$237$	0	0
$238$	0	0
$239$	440.824i	1.84445i	0.386651	+	0.922226i	$0.373632\pi$
$239$	440.824i	1.84445i	−0.386651	+	0.922226i	$0.626368\pi$
$240$	0	0
$241$	−352.103	−1.46101	−0.730504	−	0.682908i	$-0.760715\pi$
$241$	−352.103	−1.46101	−0.730504	+	0.682908i	$0.760715\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 365.632i	− 1.49238i
$246$	0	0
$247$	−5.50055	−0.0222694
$248$	0	0
$249$	0	0
$250$	0	0
$251$	248.282i	0.989173i	0.869128	+	0.494587i	$0.164681\pi$
$251$	248.282i	0.989173i	−0.869128	+	0.494587i	$0.835319\pi$
$252$	0	0
$253$	−68.1787	−0.269481
$254$	0	0
$255$	0	0
$256$	0	0
$257$	348.936i	1.35773i	0.734265	+	0.678863i	$0.237527\pi$
$257$	348.936i	1.35773i	−0.734265	+	0.678863i	$0.762473\pi$
$258$	0	0
$259$	25.4073	0.0980976
$260$	0	0
$261$	0	0
$262$	0	0
$263$	100.481i	0.382057i	0.981585	+	0.191028i	$0.0611822\pi$
$263$	100.481i	0.382057i	−0.981585	+	0.191028i	$0.938818\pi$
$264$	0	0
$265$	478.158	1.80437
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 425.142i	− 1.58045i	−0.612814	−	0.790227i	$-0.709962\pi$
$269$	− 425.142i	− 1.58045i	0.612814	−	0.790227i	$-0.290038\pi$
$270$	0	0
$271$	−108.219	−0.399331	−0.199665	−	0.979864i	$-0.563986\pi$
$271$	−108.219	−0.399331	−0.199665	+	0.979864i	$0.563986\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	803.421i	2.92153i
$276$	0	0
$277$	−322.308	−1.16357	−0.581783	−	0.813344i	$-0.697645\pi$
$277$	−322.308	−1.16357	−0.581783	+	0.813344i	$0.697645\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	222.460i	0.791674i	0.918321	+	0.395837i	$0.129545\pi$
$281$	222.460i	0.791674i	−0.918321	+	0.395837i	$0.870455\pi$
$282$	0	0
$283$	335.466	1.18539	0.592697	−	0.805426i	$-0.298063\pi$
$283$	335.466	1.18539	0.592697	+	0.805426i	$0.298063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	141.122i	0.491716i
$288$	0	0
$289$	−609.185	−2.10791
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 351.122i	− 1.19837i	−0.800611	−	0.599184i	$-0.795492\pi$
$293$	− 351.122i	− 1.19837i	0.800611	−	0.599184i	$-0.204508\pi$
$294$	0	0
$295$	66.6244	0.225845
$296$	0	0
$297$	0	0
$298$	0	0
$299$	57.9891i	0.193944i
$300$	0	0
$301$	20.9960	0.0697541
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.87022i	− 0.0159679i
$306$	0	0
$307$	−356.758	−1.16208	−0.581040	−	0.813875i	$-0.697354\pi$
$307$	−356.758	−1.16208	−0.581040	+	0.813875i	$0.697354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	342.632i	1.10171i	0.834601	+	0.550856i	$0.185699\pi$
$311$	342.632i	1.10171i	−0.834601	+	0.550856i	$0.814301\pi$
$312$	0	0
$313$	−184.635	−0.589889	−0.294944	−	0.955514i	$-0.595301\pi$
$313$	−184.635	−0.589889	−0.294944	+	0.955514i	$0.595301\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	370.627i	1.16917i	0.811332	+	0.584586i	$0.198743\pi$
$317$	370.627i	1.16917i	−0.811332	+	0.584586i	$0.801257\pi$
$318$	0	0
$319$	170.498	0.534477
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.6335i	0.0422089i
$324$	0	0
$325$	683.347	2.10261
$326$	0	0
$327$	0	0
$328$	0	0
$329$	252.659i	0.767960i
$330$	0	0
$331$	44.3212	0.133901	0.0669505	−	0.997756i	$-0.478673\pi$
$331$	44.3212	0.133901	0.0669505	+	0.997756i	$0.478673\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 266.492i	− 0.795497i
$336$	0	0
$337$	−509.936	−1.51316	−0.756581	−	0.653900i	$-0.773132\pi$
$337$	−509.936	−1.51316	−0.756581	+	0.653900i	$0.773132\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	41.4956i	0.121688i
$342$	0	0
$343$	260.967	0.760838
$344$	0	0
$345$	0	0
$346$	0	0
$347$	151.606i	0.436905i	0.975848	+	0.218453i	$0.0701009\pi$
$347$	151.606i	0.436905i	−0.975848	+	0.218453i	$0.929899\pi$
$348$	0	0
$349$	−398.389	−1.14152	−0.570758	−	0.821119i	$-0.693350\pi$
$349$	−398.389	−1.14152	−0.570758	+	0.821119i	$0.693350\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 201.394i	− 0.570522i	−0.958450	−	0.285261i	$-0.907920\pi$
$353$	− 201.394i	− 0.570522i	0.958450	−	0.285261i	$-0.0920804\pi$
$354$	0	0
$355$	713.572	2.01006
$356$	0	0
$357$	0	0
$358$	0	0
$359$	264.556i	0.736925i	0.929643	+	0.368463i	$0.120116\pi$
$359$	264.556i	0.736925i	−0.929643	+	0.368463i	$0.879884\pi$
$360$	0	0
$361$	−360.793	−0.999427
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 225.360i	− 0.617425i
$366$	0	0
$367$	663.386	1.80759	0.903796	−	0.427964i	$-0.140769\pi$
$367$	663.386	1.80759	0.903796	+	0.427964i	$0.140769\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	154.429i	0.416252i
$372$	0	0
$373$	185.416	0.497093	0.248547	−	0.968620i	$-0.420047\pi$
$373$	185.416	0.497093	0.248547	+	0.968620i	$0.420047\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 145.016i	− 0.384659i
$378$	0	0
$379$	445.259	1.17483	0.587413	−	0.809287i	$-0.300146\pi$
$379$	445.259	1.17483	0.587413	+	0.809287i	$0.300146\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	156.103i	0.407581i	0.979015	+	0.203790i	$0.0653261\pi$
$383$	156.103i	0.407581i	−0.979015	+	0.203790i	$0.934674\pi$
$384$	0	0
$385$	−374.264	−0.972113
$386$	0	0
$387$	0	0
$388$	0	0
$389$	507.883i	1.30561i	0.757526	+	0.652805i	$0.226408\pi$
$389$	507.883i	1.30561i	−0.757526	+	0.652805i	$0.773592\pi$
$390$	0	0
$391$	143.730	0.367595
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1329.56i	− 3.36596i
$396$	0	0
$397$	416.146	1.04823	0.524113	−	0.851649i	$-0.324397\pi$
$397$	416.146	1.04823	0.524113	+	0.851649i	$0.324397\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	432.172i	1.07774i	0.842390	+	0.538868i	$0.181148\pi$
$401$	432.172i	1.07774i	−0.842390	+	0.538868i	$0.818852\pi$
$402$	0	0
$403$	35.2939	0.0875779
$404$	0	0
$405$	0	0
$406$	0	0
$407$	123.870i	0.304349i
$408$	0	0
$409$	398.052	0.973232	0.486616	−	0.873616i	$-0.338231\pi$
$409$	398.052	0.973232	0.486616	+	0.873616i	$0.338231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.5175i	0.0521005i
$414$	0	0
$415$	−1006.41	−2.42509
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 339.428i	− 0.810090i	−0.914297	−	0.405045i	$-0.867256\pi$
$419$	− 339.428i	− 0.810090i	0.914297	−	0.405045i	$-0.132744\pi$
$420$	0	0
$421$	−461.411	−1.09599	−0.547994	−	0.836482i	$-0.684609\pi$
$421$	−461.411	−1.09599	−0.547994	+	0.836482i	$0.684609\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1693.72i	− 3.98522i
$426$	0	0
$427$	1.57292	0.00368366
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 390.257i	− 0.905469i	−0.891645	−	0.452734i	$-0.850449\pi$
$431$	− 390.257i	− 0.905469i	0.891645	−	0.452734i	$-0.149551\pi$
$432$	0	0
$433$	362.207	0.836505	0.418253	−	0.908331i	$-0.362643\pi$
$433$	362.207	0.836505	0.418253	+	0.908331i	$0.362643\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.18166i	− 0.00499236i
$438$	0	0
$439$	287.132	0.654058	0.327029	−	0.945014i	$-0.393953\pi$
$439$	287.132	0.654058	0.327029	+	0.945014i	$0.393953\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 675.698i	− 1.52528i	−0.646824	−	0.762639i	$-0.723903\pi$
$443$	− 675.698i	− 1.52528i	0.646824	−	0.762639i	$-0.276097\pi$
$444$	0	0
$445$	−387.169	−0.870042
$446$	0	0
$447$	0	0
$448$	0	0
$449$	369.989i	0.824030i	0.911177	+	0.412015i	$0.135175\pi$
$449$	369.989i	0.824030i	−0.911177	+	0.412015i	$0.864825\pi$
$450$	0	0
$451$	−688.026	−1.52556
$452$	0	0
$453$	0	0
$454$	0	0
$455$	318.329i	0.699623i
$456$	0	0
$457$	236.585	0.517691	0.258846	−	0.965919i	$-0.416658\pi$
$457$	236.585	0.517691	0.258846	+	0.965919i	$0.416658\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 578.951i	− 1.25586i	−0.778270	−	0.627930i	$-0.783902\pi$
$461$	− 578.951i	− 1.25586i	0.778270	−	0.627930i	$-0.216098\pi$
$462$	0	0
$463$	663.374	1.43277	0.716387	−	0.697703i	$-0.245795\pi$
$463$	663.374	1.43277	0.716387	+	0.697703i	$0.245795\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	90.5401i	0.193876i	0.995290	+	0.0969380i	$0.0309049\pi$
$467$	90.5401i	0.193876i	−0.995290	+	0.0969380i	$0.969095\pi$
$468$	0	0
$469$	86.0681	0.183514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	102.363i	0.216413i
$474$	0	0
$475$	−25.7088	−0.0541238
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 658.788i	− 1.37534i	−0.726024	−	0.687670i	$-0.758634\pi$
$479$	− 658.788i	− 1.37534i	0.726024	−	0.687670i	$-0.241366\pi$
$480$	0	0
$481$	105.357	0.219038
$482$	0	0
$483$	0	0
$484$	0	0
$485$	539.618i	1.11261i
$486$	0	0
$487$	−324.307	−0.665928	−0.332964	−	0.942940i	$-0.608049\pi$
$487$	−324.307	−0.665928	−0.332964	+	0.942940i	$0.608049\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	403.920i	0.822647i	0.911489	+	0.411324i	$0.134933\pi$
$491$	403.920i	0.822647i	−0.911489	+	0.411324i	$0.865067\pi$
$492$	0	0
$493$	−359.433	−0.729072
$494$	0	0
$495$	0	0
$496$	0	0
$497$	230.461i	0.463704i
$498$	0	0
$499$	−91.6160	−0.183599	−0.0917996	−	0.995777i	$-0.529262\pi$
$499$	−91.6160	−0.183599	−0.0917996	+	0.995777i	$0.529262\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	599.040i	1.19093i	0.803380	+	0.595467i	$0.203033\pi$
$503$	599.040i	1.19093i	−0.803380	+	0.595467i	$0.796967\pi$
$504$	0	0
$505$	140.284	0.277789
$506$	0	0
$507$	0	0
$508$	0	0
$509$	303.479i	0.596226i	0.954531	+	0.298113i	$0.0963572\pi$
$509$	303.479i	0.596226i	−0.954531	+	0.298113i	$0.903643\pi$
$510$	0	0
$511$	72.7841	0.142435
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1101.34i	− 2.13853i
$516$	0	0
$517$	−1231.81	−2.38261
$518$	0	0
$519$	0	0
$520$	0	0
$521$	333.103i	0.639353i	0.947527	+	0.319677i	$0.103574\pi$
$521$	333.103i	0.639353i	−0.947527	+	0.319677i	$0.896426\pi$
$522$	0	0
$523$	−565.124	−1.08054	−0.540272	−	0.841490i	$-0.681679\pi$
$523$	−565.124	−1.08054	−0.540272	+	0.841490i	$0.681679\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 87.4782i	− 0.165993i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	585.198i	1.09793i
$534$	0	0
$535$	−1217.08	−2.27491
$536$	0	0
$537$	0	0
$538$	0	0
$539$	575.721i	1.06813i
$540$	0	0
$541$	−1000.57	−1.84947	−0.924737	−	0.380606i	$-0.875715\pi$
$541$	−1000.57	−1.84947	−0.924737	+	0.380606i	$0.875715\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	511.133i	0.937858i
$546$	0	0
$547$	188.857	0.345260	0.172630	−	0.984987i	$-0.444774\pi$
$547$	188.857	0.345260	0.172630	+	0.984987i	$0.444774\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.45579i	0.00990162i
$552$	0	0
$553$	429.404	0.776498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 470.641i	− 0.844956i	−0.906373	−	0.422478i	$-0.861160\pi$
$557$	− 470.641i	− 0.844956i	0.906373	−	0.422478i	$-0.138840\pi$
$558$	0	0
$559$	87.0649	0.155751
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 250.163i	− 0.444339i	−0.975008	−	0.222169i	$-0.928686\pi$
$563$	− 250.163i	− 0.444339i	0.975008	−	0.222169i	$-0.0713138\pi$
$564$	0	0
$565$	−1931.91	−3.41932
$566$	0	0
$567$	0	0
$568$	0	0
$569$	331.840i	0.583199i	0.956541	+	0.291599i	$0.0941874\pi$
$569$	331.840i	0.583199i	−0.956541	+	0.291599i	$0.905813\pi$
$570$	0	0
$571$	578.856	1.01376	0.506879	−	0.862017i	$-0.330799\pi$
$571$	578.856	1.01376	0.506879	+	0.862017i	$0.330799\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	271.033i	0.471362i
$576$	0	0
$577$	−118.741	−0.205791	−0.102895	−	0.994692i	$-0.532811\pi$
$577$	−118.741	−0.205791	−0.102895	+	0.994692i	$0.532811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 325.039i	− 0.559447i
$582$	0	0
$583$	−752.903	−1.29143
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 489.679i	− 0.834206i	−0.908859	−	0.417103i	$-0.863045\pi$
$587$	− 489.679i	− 0.834206i	0.908859	−	0.417103i	$-0.136955\pi$
$588$	0	0
$589$	−1.32782	−0.00225437
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 63.2884i	− 0.106726i	−0.998575	−	0.0533629i	$-0.983006\pi$
$593$	− 63.2884i	− 0.106726i	0.998575	−	0.0533629i	$-0.0169940\pi$
$594$	0	0
$595$	788.998	1.32605
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 955.571i	− 1.59528i	−0.603135	−	0.797639i	$-0.706082\pi$
$599$	− 955.571i	− 1.59528i	0.603135	−	0.797639i	$-0.293918\pi$
$600$	0	0
$601$	−498.924	−0.830157	−0.415078	−	0.909786i	$-0.636246\pi$
$601$	−498.924	−0.830157	−0.415078	+	0.909786i	$0.636246\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 732.227i	− 1.21029i
$606$	0	0
$607$	651.861	1.07391	0.536953	−	0.843612i	$-0.319575\pi$
$607$	651.861	1.07391	0.536953	+	0.843612i	$0.319575\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1047.71i	1.71475i
$612$	0	0
$613$	−441.006	−0.719423	−0.359712	−	0.933064i	$-0.617125\pi$
$613$	−441.006	−0.719423	−0.359712	+	0.933064i	$0.617125\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1008.63i	− 1.63474i	−0.576115	−	0.817369i	$-0.695432\pi$
$617$	− 1008.63i	− 1.63474i	0.576115	−	0.817369i	$-0.304568\pi$
$618$	0	0
$619$	−261.344	−0.422203	−0.211102	−	0.977464i	$-0.567705\pi$
$619$	−261.344	−0.422203	−0.211102	+	0.977464i	$0.567705\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 125.043i	− 0.200711i
$624$	0	0
$625$	1156.01	1.84962
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 261.135i	− 0.415159i
$630$	0	0
$631$	−724.141	−1.14761	−0.573805	−	0.818992i	$-0.694533\pi$
$631$	−724.141	−1.14761	−0.573805	+	0.818992i	$0.694533\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	337.713i	0.531831i
$636$	0	0
$637$	489.677	0.768724
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 323.209i	− 0.504226i	−0.967698	−	0.252113i	$-0.918875\pi$
$641$	− 323.209i	− 0.504226i	0.967698	−	0.252113i	$-0.0811255\pi$
$642$	0	0
$643$	−743.023	−1.15556	−0.577779	−	0.816193i	$-0.696080\pi$
$643$	−743.023	−1.15556	−0.577779	+	0.816193i	$0.696080\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 120.104i	− 0.185633i	−0.995683	−	0.0928163i	$-0.970413\pi$
$647$	− 120.104i	− 0.185633i	0.995683	−	0.0928163i	$-0.0295869\pi$
$648$	0	0
$649$	−104.906	−0.161643
$650$	0	0
$651$	0	0
$652$	0	0
$653$	611.706i	0.936762i	0.883526	+	0.468381i	$0.155163\pi$
$653$	611.706i	0.936762i	−0.883526	+	0.468381i	$0.844837\pi$
$654$	0	0
$655$	1458.73	2.22707
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 13.0872i	− 0.0198592i	−0.999951	−	0.00992960i	$-0.996839\pi$
$659$	− 13.0872i	− 0.0198592i	0.999951	−	0.00992960i	$-0.00316074\pi$
$660$	0	0
$661$	−185.252	−0.280261	−0.140130	−	0.990133i	$-0.544752\pi$
$661$	−185.252	−0.280261	−0.140130	+	0.990133i	$0.544752\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.9761i	− 0.0180092i
$666$	0	0
$667$	57.5173	0.0862328
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.66860i	0.0114286i
$672$	0	0
$673$	600.468	0.892227	0.446113	−	0.894977i	$-0.352808\pi$
$673$	600.468	0.892227	0.446113	+	0.894977i	$0.352808\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 333.735i	− 0.492961i	−0.969148	−	0.246481i	$-0.920726\pi$
$677$	− 333.735i	− 0.492961i	0.969148	−	0.246481i	$-0.0792742\pi$
$678$	0	0
$679$	−174.279	−0.256670
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 814.585i	− 1.19266i	−0.802740	−	0.596329i	$-0.796625\pi$
$683$	− 814.585i	− 1.19266i	0.802740	−	0.596329i	$-0.203375\pi$
$684$	0	0
$685$	−727.023	−1.06135
$686$	0	0
$687$	0	0
$688$	0	0
$689$	640.379i	0.929432i
$690$	0	0
$691$	−886.310	−1.28265	−0.641324	−	0.767270i	$-0.721615\pi$
$691$	−886.310	−1.28265	−0.641324	+	0.767270i	$0.721615\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 597.068i	− 0.859091i
$696$	0	0
$697$	1450.45	2.08099
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 595.315i	− 0.849236i	−0.905373	−	0.424618i	$-0.860408\pi$
$701$	− 595.315i	− 0.849236i	0.905373	−	0.424618i	$-0.139592\pi$
$702$	0	0
$703$	−3.96374	−0.00563833
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.3071i	0.0640835i
$708$	0	0
$709$	−439.914	−0.620471	−0.310236	−	0.950660i	$-0.600408\pi$
$709$	−439.914	−0.620471	−0.310236	+	0.950660i	$0.600408\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.9985i	0.0196332i
$714$	0	0
$715$	−1551.97	−2.17059
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 433.159i	− 0.602446i	−0.953554	−	0.301223i	$-0.902605\pi$
$719$	− 433.159i	− 0.602446i	0.953554	−	0.301223i	$-0.0973949\pi$
$720$	0	0
$721$	355.697	0.493339
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 677.787i	− 0.934878i
$726$	0	0
$727$	−97.5476	−0.134178	−0.0670891	−	0.997747i	$-0.521371\pi$
$727$	−97.5476	−0.134178	−0.0670891	+	0.997747i	$0.521371\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 215.796i	− 0.295206i
$732$	0	0
$733$	826.238	1.12720	0.563600	−	0.826048i	$-0.309416\pi$
$733$	826.238	1.12720	0.563600	+	0.826048i	$0.309416\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	419.615i	0.569356i
$738$	0	0
$739$	−779.379	−1.05464	−0.527320	−	0.849667i	$-0.676803\pi$
$739$	−779.379	−1.05464	−0.527320	+	0.849667i	$0.676803\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1053.07i	1.41732i	0.705550	+	0.708660i	$0.250700\pi$
$743$	1053.07i	1.41732i	−0.705550	+	0.708660i	$0.749300\pi$
$744$	0	0
$745$	1457.96	1.95699
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 393.076i	− 0.524801i
$750$	0	0
$751$	−526.121	−0.700561	−0.350280	−	0.936645i	$-0.613914\pi$
$751$	−526.121	−0.700561	−0.350280	+	0.936645i	$0.613914\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2406.87i	3.18790i
$756$	0	0
$757$	877.552	1.15925	0.579625	−	0.814883i	$-0.303199\pi$
$757$	877.552	1.15925	0.579625	+	0.814883i	$0.303199\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	953.180i	1.25254i	0.779608	+	0.626268i	$0.215418\pi$
$761$	953.180i	1.25254i	−0.779608	+	0.626268i	$0.784582\pi$
$762$	0	0
$763$	−165.079	−0.216356
$764$	0	0
$765$	0	0
$766$	0	0
$767$	89.2275i	0.116333i
$768$	0	0
$769$	1048.88	1.36395	0.681976	−	0.731375i	$-0.261121\pi$
$769$	1048.88	1.36395	0.681976	+	0.731375i	$0.261121\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 720.664i	− 0.932295i	−0.884707	−	0.466147i	$-0.845642\pi$
$773$	− 720.664i	− 0.932295i	0.884707	−	0.466147i	$-0.154358\pi$
$774$	0	0
$775$	164.959	0.212850
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 22.0163i	− 0.0282622i
$780$	0	0
$781$	−1123.58	−1.43865
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2283.20i	− 2.90854i
$786$	0	0
$787$	293.303	0.372684	0.186342	−	0.982485i	$-0.440337\pi$
$787$	293.303	0.372684	0.186342	+	0.982485i	$0.440337\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 623.945i	− 0.788806i
$792$	0	0
$793$	6.52250	0.00822510
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1101.43i	1.38197i	0.722870	+	0.690984i	$0.242822\pi$
$797$	1101.43i	1.38197i	−0.722870	+	0.690984i	$0.757178\pi$
$798$	0	0
$799$	2596.82	3.25008
$800$	0	0
$801$	0	0
$802$	0	0
$803$	354.850i	0.441906i
$804$	0	0
$805$	−126.257	−0.156841
$806$	0	0
$807$	0	0
$808$	0	0
$809$	139.592i	0.172548i	0.996271	+	0.0862742i	$0.0274961\pi$
$809$	139.592i	0.172548i	−0.996271	+	0.0862742i	$0.972504\pi$
$810$	0	0
$811$	172.113	0.212223	0.106111	−	0.994354i	$-0.466160\pi$
$811$	172.113	0.212223	0.106111	+	0.994354i	$0.466160\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 2130.90i	− 2.61460i
$816$	0	0
$817$	−3.27555	−0.00400924
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 470.618i	− 0.573226i	−0.958046	−	0.286613i	$-0.907471\pi$
$821$	− 470.618i	− 0.573226i	0.958046	−	0.286613i	$-0.0925293\pi$
$822$	0	0
$823$	−1501.22	−1.82409	−0.912043	−	0.410094i	$-0.865496\pi$
$823$	−1501.22	−1.82409	−0.912043	+	0.410094i	$0.865496\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 665.287i	− 0.804458i	−0.915539	−	0.402229i	$-0.868236\pi$
$827$	− 665.287i	− 0.804458i	0.915539	−	0.402229i	$-0.131764\pi$
$828$	0	0
$829$	120.995	0.145953	0.0729765	−	0.997334i	$-0.476750\pi$
$829$	120.995	0.145953	0.0729765	+	0.997334i	$0.476750\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1213.70i	− 1.45702i
$834$	0	0
$835$	−883.167	−1.05769
$836$	0	0
$837$	0	0
$838$	0	0
$839$	860.528i	1.02566i	0.858491	+	0.512829i	$0.171403\pi$
$839$	860.528i	1.02566i	−0.858491	+	0.512829i	$0.828597\pi$
$840$	0	0
$841$	697.163	0.828970
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 205.796i	− 0.243545i
$846$	0	0
$847$	236.486	0.279204
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41.7875i	0.0491040i
$852$	0	0
$853$	−280.533	−0.328878	−0.164439	−	0.986387i	$-0.552581\pi$
$853$	−280.533	−0.328878	−0.164439	+	0.986387i	$0.552581\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	527.779i	0.615845i	0.951411	+	0.307922i	$0.0996337\pi$
$857$	527.779i	0.615845i	−0.951411	+	0.307922i	$0.900366\pi$
$858$	0	0
$859$	505.353	0.588304	0.294152	−	0.955759i	$-0.404963\pi$
$859$	505.353	0.588304	0.294152	+	0.955759i	$0.404963\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1326.62i	1.53722i	0.639718	+	0.768610i	$0.279051\pi$
$863$	1326.62i	1.53722i	−0.639718	+	0.768610i	$0.720949\pi$
$864$	0	0
$865$	1357.01	1.56880
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2093.51i	2.40910i
$870$	0	0
$871$	356.902	0.409761
$872$	0	0
$873$	0	0
$874$	0	0
$875$	829.661i	0.948184i
$876$	0	0
$877$	−1195.59	−1.36327	−0.681636	−	0.731691i	$-0.738731\pi$
$877$	−1195.59	−1.36327	−0.681636	+	0.731691i	$0.738731\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 571.134i	− 0.648280i	−0.946009	−	0.324140i	$-0.894925\pi$
$881$	− 571.134i	− 0.648280i	0.946009	−	0.324140i	$-0.105075\pi$
$882$	0	0
$883$	−1093.85	−1.23878	−0.619392	−	0.785082i	$-0.712621\pi$
$883$	−1093.85	−1.23878	−0.619392	+	0.785082i	$0.712621\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	866.982i	0.977431i	0.872443	+	0.488716i	$0.162534\pi$
$887$	866.982i	0.977431i	−0.872443	+	0.488716i	$0.837466\pi$
$888$	0	0
$889$	−109.070	−0.122689
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 39.4168i	− 0.0441398i
$894$	0	0
$895$	−2876.61	−3.21409
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 35.0067i	− 0.0389396i
$900$	0	0
$901$	1587.22	1.76162
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2006.50i	2.21712i
$906$	0	0
$907$	439.525	0.484592	0.242296	−	0.970202i	$-0.422100\pi$
$907$	439.525	0.484592	0.242296	+	0.970202i	$0.422100\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	230.522i	0.253043i	0.991964	+	0.126521i	$0.0403812\pi$
$911$	230.522i	0.253043i	−0.991964	+	0.126521i	$0.959619\pi$
$912$	0	0
$913$	1584.69	1.73570
$914$	0	0
$915$	0	0
$916$	0	0
$917$	471.123i	0.513766i
$918$	0	0
$919$	−12.4225	−0.0135174	−0.00675872	−	0.999977i	$-0.502151\pi$
$919$	−12.4225	−0.0135174	−0.00675872	+	0.999977i	$0.502151\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	955.661i	1.03539i
$924$	0	0
$925$	492.426	0.532352
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 979.660i	− 1.05453i	−0.849700	−	0.527266i	$-0.823217\pi$
$929$	− 979.660i	− 1.05453i	0.849700	−	0.527266i	$-0.176783\pi$
$930$	0	0
$931$	−18.4226	−0.0197879
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3846.67i	4.11408i
$936$	0	0
$937$	1142.29	1.21909	0.609544	−	0.792752i	$-0.291352\pi$
$937$	1142.29	1.21909	0.609544	+	0.792752i	$0.291352\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 344.000i	− 0.365569i	−0.983153	−	0.182784i	$-0.941489\pi$
$941$	− 344.000i	− 0.365569i	0.983153	−	0.182784i	$-0.0585110\pi$
$942$	0	0
$943$	−232.105	−0.246134
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1138.07i	1.20176i	0.799339	+	0.600880i	$0.205183\pi$
$947$	1138.07i	1.20176i	−0.799339	+	0.600880i	$0.794817\pi$
$948$	0	0
$949$	301.817	0.318036
$950$	0	0
$951$	0	0
$952$	0	0
$953$	348.129i	0.365298i	0.983178	+	0.182649i	$0.0584671\pi$
$953$	348.129i	0.365298i	−0.983178	+	0.182649i	$0.941533\pi$
$954$	0	0
$955$	761.434	0.797313
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 234.805i	− 0.244844i
$960$	0	0
$961$	−952.480	−0.991134
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2293.40i	2.37658i
$966$	0	0
$967$	1574.89	1.62863	0.814317	−	0.580421i	$-0.197112\pi$
$967$	1574.89	1.62863	0.814317	+	0.580421i	$0.197112\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1804.73i	− 1.85863i	−0.369284	−	0.929317i	$-0.620397\pi$
$971$	− 1804.73i	− 1.85863i	0.369284	−	0.929317i	$-0.379603\pi$
$972$	0	0
$973$	192.834	0.198185
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1604.50i	1.64228i	0.570730	+	0.821138i	$0.306660\pi$
$977$	1604.50i	1.64228i	−0.570730	+	0.821138i	$0.693340\pi$
$978$	0	0
$979$	609.633	0.622710
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1597.25i	− 1.62488i	−0.583048	−	0.812438i	$-0.698140\pi$
$983$	− 1597.25i	− 1.62488i	0.583048	−	0.812438i	$-0.301860\pi$
$984$	0	0
$985$	−794.775	−0.806878
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.5322i	0.0349163i
$990$	0	0
$991$	468.044	0.472294	0.236147	−	0.971717i	$-0.424115\pi$
$991$	468.044	0.472294	0.236147	+	0.971717i	$0.424115\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 127.522i	− 0.128163i
$996$	0	0
$997$	49.8238	0.0499737	0.0249868	−	0.999688i	$-0.492046\pi$
$997$	49.8238	0.0499737	0.0249868	+	0.999688i	$0.492046\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	828.3.d.a.737.16	yes	16
3.2	odd	2	inner	828.3.d.a.737.1	✓	16
4.3	odd	2		3312.3.g.c.737.16		16
12.11	even	2		3312.3.g.c.737.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
828.3.d.a.737.1	✓	16	3.2	odd	2	inner
828.3.d.a.737.16	yes	16	1.1	even	1	trivial
3312.3.g.c.737.1		16	12.11	even	2
3312.3.g.c.737.16		16	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	828.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.02853i q^{5} -2.91592 q^{7} -14.2162i q^{11} -12.0916 q^{13} +29.9697i q^{17} +0.454908 q^{19} -4.79583i q^{23} -56.5143 q^{25} +11.9932i q^{29} -2.91889 q^{31} -26.3265i q^{35} -8.71329 q^{37} -48.3972i q^{41} -7.20046 q^{43} -86.6479i q^{47} -40.4974 q^{49} -52.9607i q^{53} +128.352 q^{55} -7.37931i q^{59} -0.539425 q^{61} -109.169i q^{65} -29.5166 q^{67} -79.0353i q^{71} -24.9609 q^{73} +41.4535i q^{77} -147.262 q^{79} +111.470i q^{83} -270.583 q^{85} +42.8828i q^{89} +35.2581 q^{91} +4.10715i q^{95} +59.7681 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{13} + 40 q^{19} - 72 q^{25} - 32 q^{31} - 64 q^{37} + 48 q^{43} + 8 q^{49} + 144 q^{55} - 48 q^{61} - 40 q^{67} + 48 q^{73} - 24 q^{79} - 8 q^{85} - 24 q^{91} + 168 q^{97}+O(q^{100}) \) 16 * q - 8 * q^13 + 40 * q^19 - 72 * q^25 - 32 * q^31 - 64 * q^37 + 48 * q^43 + 8 * q^49 + 144 * q^55 - 48 * q^61 - 40 * q^67 + 48 * q^73 - 24 * q^79 - 8 * q^85 - 24 * q^91 + 168 * q^97

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