LMFDB - Embedded newform 1764.3.c.h.197.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(197,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.c (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.224054542336.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 199x^{4} + 14641 $ x^8 + 199*x^4 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.7
Root			$-2.67196 - 1.96485i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.h.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.55744i q^{5} +O(q^{10})$	$q+9.55744i q^{5} -9.27362i q^{11} +4.24264 q^{13} +15.5574i q^{17} +34.8919 q^{19} +17.7589i q^{23} -66.3446 q^{25} +26.2442i q^{29} +32.0635 q^{31} +55.3446 q^{37} -38.4426i q^{41} +29.3446 q^{43} +68.2298i q^{47} +1.08929i q^{53} +88.6320 q^{55} +68.2298i q^{59} -47.6198 q^{61} +40.5488i q^{65} -8.65537 q^{67} +95.7031i q^{71} -90.0463 q^{73} -148.689 q^{79} -88.8851i q^{83} -148.689 q^{85} -76.2467i q^{89} +333.477i q^{95} -12.7519 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 216 q^{25} + 128 q^{37} - 80 q^{43} - 384 q^{67} - 560 q^{79} - 560 q^{85}+O(q^{100}) $ 8 * q - 216 * q^25 + 128 * q^37 - 80 * q^43 - 384 * q^67 - 560 * q^79 - 560 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	9.55744i	1.91149i	0.294200	+	0.955744i	$0.404947\pi$
$5$	9.55744i	1.91149i	−0.294200	+	0.955744i	$0.595053\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 9.27362i	− 0.843056i	−0.906815	−	0.421528i	$-0.861494\pi$
$11$	− 9.27362i	− 0.843056i	0.906815	−	0.421528i	$-0.138506\pi$
$12$	0	0
$13$	4.24264	0.326357	0.163178	−	0.986597i	$-0.447825\pi$
$13$	4.24264	0.326357	0.163178	+	0.986597i	$0.447825\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	15.5574i	0.915143i	0.889173	+	0.457572i	$0.151281\pi$
$17$	15.5574i	0.915143i	−0.889173	+	0.457572i	$0.848719\pi$
$18$	0	0
$19$	34.8919	1.83642	0.918209	−	0.396097i	$-0.129636\pi$
$19$	34.8919	1.83642	0.918209	+	0.396097i	$0.129636\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	17.7589i	0.772126i	0.922472	+	0.386063i	$0.126165\pi$
$23$	17.7589i	0.772126i	−0.922472	+	0.386063i	$0.873835\pi$
$24$	0	0
$25$	−66.3446	−2.65379
$26$	0	0
$27$	0	0
$28$	0	0
$29$	26.2442i	0.904972i	0.891772	+	0.452486i	$0.149463\pi$
$29$	26.2442i	0.904972i	−0.891772	+	0.452486i	$0.850537\pi$
$30$	0	0
$31$	32.0635	1.03431	0.517153	−	0.855893i	$-0.326992\pi$
$31$	32.0635	1.03431	0.517153	+	0.855893i	$0.326992\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	55.3446	1.49580	0.747900	−	0.663811i	$-0.231062\pi$
$37$	55.3446	1.49580	0.747900	+	0.663811i	$0.231062\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 38.4426i	− 0.937623i	−0.883298	−	0.468812i	$-0.844682\pi$
$41$	− 38.4426i	− 0.937623i	0.883298	−	0.468812i	$-0.155318\pi$
$42$	0	0
$43$	29.3446	0.682433	0.341217	−	0.939985i	$-0.389161\pi$
$43$	29.3446	0.682433	0.341217	+	0.939985i	$0.389161\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	68.2298i	1.45170i	0.687855	+	0.725848i	$0.258553\pi$
$47$	68.2298i	1.45170i	−0.687855	+	0.725848i	$0.741447\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.08929i	0.0205527i	0.999947	+	0.0102763i	$0.00327112\pi$
$53$	1.08929i	0.0205527i	−0.999947	+	0.0102763i	$0.996729\pi$
$54$	0	0
$55$	88.6320	1.61149
$56$	0	0
$57$	0	0
$58$	0	0
$59$	68.2298i	1.15644i	0.815882	+	0.578218i	$0.196252\pi$
$59$	68.2298i	1.15644i	−0.815882	+	0.578218i	$0.803748\pi$
$60$	0	0
$61$	−47.6198	−0.780653	−0.390327	−	0.920676i	$-0.627638\pi$
$61$	−47.6198	−0.780653	−0.390327	+	0.920676i	$0.627638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	40.5488i	0.623827i
$66$	0	0
$67$	−8.65537	−0.129185	−0.0645923	−	0.997912i	$-0.520575\pi$
$67$	−8.65537	−0.129185	−0.0645923	+	0.997912i	$0.520575\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	95.7031i	1.34793i	0.738763	+	0.673966i	$0.235410\pi$
$71$	95.7031i	1.34793i	−0.738763	+	0.673966i	$0.764590\pi$
$72$	0	0
$73$	−90.0463	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$73$	−90.0463	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−148.689	−1.88214	−0.941071	−	0.338208i	$-0.890179\pi$
$79$	−148.689	−1.88214	−0.941071	+	0.338208i	$0.890179\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 88.8851i	− 1.07091i	−0.844565	−	0.535453i	$-0.820141\pi$
$83$	− 88.8851i	− 1.07091i	0.844565	−	0.535453i	$-0.179859\pi$
$84$	0	0
$85$	−148.689	−1.74929
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 76.2467i	− 0.856705i	−0.903612	−	0.428352i	$-0.859094\pi$
$89$	− 76.2467i	− 0.856705i	0.903612	−	0.428352i	$-0.140906\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	333.477i	3.51029i
$96$	0	0
$97$	−12.7519	−0.131463	−0.0657314	−	0.997837i	$-0.520938\pi$
$97$	−12.7519	−0.131463	−0.0657314	+	0.997837i	$0.520938\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.36158i	− 0.0530849i	−0.999648	−	0.0265425i	$-0.991550\pi$
$101$	− 5.36158i	− 0.0530849i	0.999648	−	0.0265425i	$-0.00844972\pi$
$102$	0	0
$103$	−136.715	−1.32733	−0.663667	−	0.748029i	$-0.731001\pi$
$103$	−136.715	−1.32733	−0.663667	+	0.748029i	$0.731001\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 149.166i	− 1.39408i	−0.717034	−	0.697038i	$-0.754501\pi$
$107$	− 149.166i	− 1.39408i	0.717034	−	0.697038i	$-0.245499\pi$
$108$	0	0
$109$	28.6893	0.263204	0.131602	−	0.991303i	$-0.457988\pi$
$109$	28.6893	0.263204	0.131602	+	0.991303i	$0.457988\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 145.712i	− 1.28949i	−0.764399	−	0.644743i	$-0.776964\pi$
$113$	− 145.712i	− 1.28949i	0.764399	−	0.644743i	$-0.223036\pi$
$114$	0	0
$115$	−169.730	−1.47591
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	35.0000	0.289256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 395.149i	− 3.16119i
$126$	0	0
$127$	−119.345	−0.939722	−0.469861	−	0.882741i	$-0.655696\pi$
$127$	−119.345	−0.939722	−0.469861	+	0.882741i	$0.655696\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	96.0000i	0.732824i	0.930453	+	0.366412i	$0.119414\pi$
$131$	96.0000i	0.732824i	−0.930453	+	0.366412i	$0.880586\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	172.071i	1.25599i	0.778217	+	0.627995i	$0.216124\pi$
$137$	172.071i	1.25599i	−0.778217	+	0.627995i	$0.783876\pi$
$138$	0	0
$139$	211.205	1.51946	0.759731	−	0.650238i	$-0.225331\pi$
$139$	211.205	1.51946	0.759731	+	0.650238i	$0.225331\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 39.3446i	− 0.275137i
$144$	0	0
$145$	−250.827	−1.72984
$146$	0	0
$147$	0	0
$148$	0	0
$149$	151.044i	1.01372i	0.862029	+	0.506859i	$0.169193\pi$
$149$	151.044i	1.01372i	−0.862029	+	0.506859i	$0.830807\pi$
$150$	0	0
$151$	12.0339	0.0796947	0.0398473	−	0.999206i	$-0.487313\pi$
$151$	12.0339	0.0796947	0.0398473	+	0.999206i	$0.487313\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	306.445i	1.97706i
$156$	0	0
$157$	−160.757	−1.02393	−0.511965	−	0.859006i	$-0.671082\pi$
$157$	−160.757	−1.02393	−0.511965	+	0.859006i	$0.671082\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−258.034	−1.58303	−0.791515	−	0.611150i	$-0.790707\pi$
$163$	−258.034	−1.58303	−0.791515	+	0.611150i	$0.790707\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	107.838i	0.645737i	0.946444	+	0.322868i	$0.104647\pi$
$167$	107.838i	0.645737i	−0.946444	+	0.322868i	$0.895353\pi$
$168$	0	0
$169$	−151.000	−0.893491
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.3277i	0.146403i	0.997317	+	0.0732014i	$0.0233216\pi$
$173$	25.3277i	0.146403i	−0.997317	+	0.0732014i	$0.976678\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	56.0572i	0.313169i	0.987665	+	0.156584i	$0.0500483\pi$
$179$	56.0572i	0.313169i	−0.987665	+	0.156584i	$0.949952\pi$
$180$	0	0
$181$	118.307	0.653627	0.326814	−	0.945089i	$-0.394025\pi$
$181$	118.307	0.653627	0.326814	+	0.945089i	$0.394025\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	528.953i	2.85920i
$186$	0	0
$187$	144.274	0.771517
$188$	0	0
$189$	0	0
$190$	0	0
$191$	94.1264i	0.492809i	0.969167	+	0.246404i	$0.0792491\pi$
$191$	94.1264i	0.492809i	−0.969167	+	0.246404i	$0.920751\pi$
$192$	0	0
$193$	−305.379	−1.58227	−0.791136	−	0.611640i	$-0.790510\pi$
$193$	−305.379	−1.58227	−0.791136	+	0.611640i	$0.790510\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	369.483i	1.87555i	0.347249	+	0.937773i	$0.387116\pi$
$197$	369.483i	1.87555i	−0.347249	+	0.937773i	$0.612884\pi$
$198$	0	0
$199$	268.749	1.35050	0.675248	−	0.737591i	$-0.264037\pi$
$199$	268.749	1.35050	0.675248	+	0.737591i	$0.264037\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	367.412	1.79226
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 323.574i	− 1.54820i
$210$	0	0
$211$	296.068	1.40316	0.701582	−	0.712588i	$-0.252477\pi$
$211$	296.068	1.40316	0.701582	+	0.712588i	$0.252477\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	280.460i	1.30446i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	66.0046i	0.298663i
$222$	0	0
$223$	298.934	1.34051	0.670256	−	0.742130i	$-0.266184\pi$
$223$	298.934	1.34051	0.670256	+	0.742130i	$0.266184\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 132.689i	− 0.584534i	−0.956337	−	0.292267i	$-0.905590\pi$
$227$	− 132.689i	− 0.584534i	0.956337	−	0.292267i	$-0.0944096\pi$
$228$	0	0
$229$	235.295	1.02749	0.513744	−	0.857944i	$-0.328258\pi$
$229$	235.295	1.02749	0.513744	+	0.857944i	$0.328258\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 95.7031i	− 0.410743i	−0.978684	−	0.205371i	$-0.934160\pi$
$233$	− 95.7031i	− 0.410743i	0.978684	−	0.205371i	$-0.0658403\pi$
$234$	0	0
$235$	−652.102	−2.77490
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 382.397i	− 1.59999i	−0.600009	−	0.799994i	$-0.704836\pi$
$239$	− 382.397i	− 1.59999i	0.600009	−	0.799994i	$-0.295164\pi$
$240$	0	0
$241$	8.06980	0.0334846	0.0167423	−	0.999860i	$-0.494671\pi$
$241$	8.06980	0.0334846	0.0167423	+	0.999860i	$0.494671\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	148.034	0.599328
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 216.264i	− 0.861608i	−0.902445	−	0.430804i	$-0.858230\pi$
$251$	− 216.264i	− 0.861608i	0.902445	−	0.430804i	$-0.141770\pi$
$252$	0	0
$253$	164.689	0.650946
$254$	0	0
$255$	0	0
$256$	0	0
$257$	288.740i	1.12350i	0.827306	+	0.561751i	$0.189872\pi$
$257$	288.740i	1.12350i	−0.827306	+	0.561751i	$0.810128\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 67.4668i	− 0.256528i	−0.991740	−	0.128264i	$-0.959060\pi$
$263$	− 67.4668i	− 0.256528i	0.991740	−	0.128264i	$-0.0409404\pi$
$264$	0	0
$265$	−10.4108	−0.0392862
$266$	0	0
$267$	0	0
$268$	0	0
$269$	58.4087i	0.217133i	0.994089	+	0.108566i	$0.0346260\pi$
$269$	58.4087i	0.217133i	−0.994089	+	0.108566i	$0.965374\pi$
$270$	0	0
$271$	138.665	0.511678	0.255839	−	0.966719i	$-0.417648\pi$
$271$	138.665	0.511678	0.255839	+	0.966719i	$0.417648\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	615.255i	2.23729i
$276$	0	0
$277$	−128.689	−0.464582	−0.232291	−	0.972646i	$-0.574622\pi$
$277$	−128.689	−0.464582	−0.232291	+	0.972646i	$0.574622\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	225.534i	0.802611i	0.915944	+	0.401306i	$0.131443\pi$
$281$	225.534i	0.802611i	−0.915944	+	0.401306i	$0.868557\pi$
$282$	0	0
$283$	23.5303	0.0831459	0.0415729	−	0.999135i	$-0.486763\pi$
$283$	23.5303	0.0831459	0.0415729	+	0.999135i	$0.486763\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	46.9661	0.162512
$290$	0	0
$291$	0	0
$292$	0	0
$293$	301.855i	1.03022i	0.857124	+	0.515111i	$0.172249\pi$
$293$	301.855i	1.03022i	−0.857124	+	0.515111i	$0.827751\pi$
$294$	0	0
$295$	−652.102	−2.21051
$296$	0	0
$297$	0	0
$298$	0	0
$299$	75.3446i	0.251989i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 455.124i	− 1.49221i
$306$	0	0
$307$	−320.611	−1.04434	−0.522168	−	0.852843i	$-0.674877\pi$
$307$	−320.611	−1.04434	−0.522168	+	0.852843i	$0.674877\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	391.906i	1.26015i	0.776535	+	0.630074i	$0.216975\pi$
$311$	391.906i	1.26015i	−0.776535	+	0.630074i	$0.783025\pi$
$312$	0	0
$313$	399.296	1.27570	0.637852	−	0.770159i	$-0.279823\pi$
$313$	399.296	1.27570	0.637852	+	0.770159i	$0.279823\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 573.129i	− 1.80798i	−0.427555	−	0.903990i	$-0.640625\pi$
$317$	− 573.129i	− 1.80798i	0.427555	−	0.903990i	$-0.359375\pi$
$318$	0	0
$319$	243.379	0.762942
$320$	0	0
$321$	0	0
$322$	0	0
$323$	542.829i	1.68058i
$324$	0	0
$325$	−281.476	−0.866081
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−422.689	−1.27701	−0.638503	−	0.769619i	$-0.720446\pi$
$331$	−422.689	−1.27701	−0.638503	+	0.769619i	$0.720446\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 82.7232i	− 0.246935i
$336$	0	0
$337$	299.345	0.888263	0.444132	−	0.895962i	$-0.353512\pi$
$337$	299.345	0.888263	0.444132	+	0.895962i	$0.353512\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 297.345i	− 0.871978i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	424.823i	1.22427i	0.790751	+	0.612137i	$0.209690\pi$
$347$	424.823i	1.22427i	−0.790751	+	0.612137i	$0.790310\pi$
$348$	0	0
$349$	182.505	0.522938	0.261469	−	0.965212i	$-0.415793\pi$
$349$	182.505	0.522938	0.261469	+	0.965212i	$0.415793\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 211.166i	− 0.598203i	−0.954221	−	0.299102i	$-0.903313\pi$
$353$	− 211.166i	− 0.598203i	0.954221	−	0.299102i	$-0.0966869\pi$
$354$	0	0
$355$	−914.677	−2.57655
$356$	0	0
$357$	0	0
$358$	0	0
$359$	92.5498i	0.257799i	0.991658	+	0.128899i	$0.0411444\pi$
$359$	92.5498i	0.257799i	−0.991658	+	0.128899i	$0.958856\pi$
$360$	0	0
$361$	856.446	2.37243
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 860.612i	− 2.35784i
$366$	0	0
$367$	303.616	0.827293	0.413646	−	0.910438i	$-0.364255\pi$
$367$	303.616	0.827293	0.413646	+	0.910438i	$0.364255\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−347.446	−0.931491	−0.465746	−	0.884919i	$-0.654214\pi$
$373$	−347.446	−0.931491	−0.465746	+	0.884919i	$0.654214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	111.345i	0.295344i
$378$	0	0
$379$	233.379	0.615774	0.307887	−	0.951423i	$-0.400378\pi$
$379$	233.379	0.615774	0.307887	+	0.951423i	$0.400378\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 220.987i	− 0.576989i	−0.957482	−	0.288495i	$-0.906845\pi$
$383$	− 220.987i	− 0.576989i	0.957482	−	0.288495i	$-0.0931547\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	130.992i	0.336740i	0.985724	+	0.168370i	$0.0538503\pi$
$389$	130.992i	0.336740i	−0.985724	+	0.168370i	$0.946150\pi$
$390$	0	0
$391$	−276.283	−0.706606
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1421.09i	− 3.59769i
$396$	0	0
$397$	−196.648	−0.495334	−0.247667	−	0.968845i	$-0.579664\pi$
$397$	−196.648	−0.495334	−0.247667	+	0.968845i	$0.579664\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 635.981i	− 1.58599i	−0.609230	−	0.792993i	$-0.708522\pi$
$401$	− 635.981i	− 1.58599i	0.609230	−	0.792993i	$-0.291478\pi$
$402$	0	0
$403$	136.034	0.337553
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 513.245i	− 1.26104i
$408$	0	0
$409$	−70.2473	−0.171754	−0.0858768	−	0.996306i	$-0.527369\pi$
$409$	−70.2473	−0.171754	−0.0858768	+	0.996306i	$0.527369\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	849.514	2.04702
$416$	0	0
$417$	0	0
$418$	0	0
$419$	436.196i	1.04104i	0.853849	+	0.520520i	$0.174262\pi$
$419$	436.196i	1.04104i	−0.853849	+	0.520520i	$0.825738\pi$
$420$	0	0
$421$	7.37852	0.0175262	0.00876309	−	0.999962i	$-0.497211\pi$
$421$	7.37852	0.0175262	0.00876309	+	0.999962i	$0.497211\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1032.15i	− 2.42859i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	200.451i	0.465083i	0.972586	+	0.232542i	$0.0747041\pi$
$431$	200.451i	0.465083i	−0.972586	+	0.232542i	$0.925296\pi$
$432$	0	0
$433$	454.106	1.04874	0.524372	−	0.851489i	$-0.324300\pi$
$433$	454.106	1.04874	0.524372	+	0.851489i	$0.324300\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	619.642i	1.41795i
$438$	0	0
$439$	−262.213	−0.597296	−0.298648	−	0.954363i	$-0.596536\pi$
$439$	−262.213	−0.597296	−0.298648	+	0.954363i	$0.596536\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	232.671i	0.525218i	0.964902	+	0.262609i	$0.0845829\pi$
$443$	232.671i	0.525218i	−0.964902	+	0.262609i	$0.915417\pi$
$444$	0	0
$445$	728.723	1.63758
$446$	0	0
$447$	0	0
$448$	0	0
$449$	92.0197i	0.204944i	0.994736	+	0.102472i	$0.0326752\pi$
$449$	92.0197i	0.204944i	−0.994736	+	0.102472i	$0.967325\pi$
$450$	0	0
$451$	−356.502	−0.790469
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−694.825	−1.52040	−0.760202	−	0.649687i	$-0.774900\pi$
$457$	−694.825	−1.52040	−0.760202	+	0.649687i	$0.774900\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	432.476i	0.938127i	0.883164	+	0.469063i	$0.155408\pi$
$461$	432.476i	0.938127i	−0.883164	+	0.469063i	$0.844592\pi$
$462$	0	0
$463$	−310.169	−0.669912	−0.334956	−	0.942234i	$-0.608722\pi$
$463$	−310.169	−0.669912	−0.334956	+	0.942234i	$0.608722\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 415.804i	− 0.890373i	−0.895438	−	0.445186i	$-0.853137\pi$
$467$	− 415.804i	− 0.890373i	0.895438	−	0.445186i	$-0.146863\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 272.131i	− 0.575330i
$474$	0	0
$475$	−2314.89	−4.87346
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 286.358i	− 0.597824i	−0.954281	−	0.298912i	$-0.903376\pi$
$479$	− 286.358i	− 0.597824i	0.954281	−	0.298912i	$-0.0966238\pi$
$480$	0	0
$481$	234.807	0.488165
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 121.875i	− 0.251289i
$486$	0	0
$487$	485.412	0.996740	0.498370	−	0.866964i	$-0.333932\pi$
$487$	485.412	0.996740	0.498370	+	0.866964i	$0.333932\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 138.130i	− 0.281323i	−0.990058	−	0.140661i	$-0.955077\pi$
$491$	− 138.130i	− 0.281323i	0.990058	−	0.140661i	$-0.0449229\pi$
$492$	0	0
$493$	−408.292	−0.828179
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−145.345	−0.291272	−0.145636	−	0.989338i	$-0.546523\pi$
$499$	−145.345	−0.291272	−0.145636	+	0.989338i	$0.546523\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	619.115i	1.23084i	0.788198	+	0.615422i	$0.211015\pi$
$503$	619.115i	1.23084i	−0.788198	+	0.615422i	$0.788985\pi$
$504$	0	0
$505$	51.2430	0.101471
$506$	0	0
$507$	0	0
$508$	0	0
$509$	231.923i	0.455644i	0.973703	+	0.227822i	$0.0731604\pi$
$509$	231.923i	0.455644i	−0.973703	+	0.227822i	$0.926840\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1306.65i	− 2.53718i
$516$	0	0
$517$	632.737	1.22386
$518$	0	0
$519$	0	0
$520$	0	0
$521$	52.8343i	0.101409i	0.998714	+	0.0507047i	$0.0161467\pi$
$521$	52.8343i	0.101409i	−0.998714	+	0.0507047i	$0.983853\pi$
$522$	0	0
$523$	56.6165	0.108253	0.0541266	−	0.998534i	$-0.482763\pi$
$523$	56.6165	0.108253	0.0541266	+	0.998534i	$0.482763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	498.826i	0.946539i
$528$	0	0
$529$	213.621	0.403821
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 163.098i	− 0.306000i
$534$	0	0
$535$	1425.65	2.66476
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	91.2430	0.168656	0.0843281	−	0.996438i	$-0.473126\pi$
$541$	91.2430	0.168656	0.0843281	+	0.996438i	$0.473126\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	274.196i	0.503112i
$546$	0	0
$547$	356.825	0.652331	0.326165	−	0.945313i	$-0.394243\pi$
$547$	356.825	0.652331	0.326165	+	0.945313i	$0.394243\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	915.710i	1.66191i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 938.598i	− 1.68510i	−0.538621	−	0.842548i	$-0.681055\pi$
$557$	− 938.598i	− 1.68510i	0.538621	−	0.842548i	$-0.318945\pi$
$558$	0	0
$559$	124.499	0.222717
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1013.15i	− 1.79955i	−0.436350	−	0.899777i	$-0.643729\pi$
$563$	− 1013.15i	− 1.79955i	0.436350	−	0.899777i	$-0.356271\pi$
$564$	0	0
$565$	1392.63	2.46484
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 967.279i	− 1.69996i	−0.526812	−	0.849982i	$-0.676613\pi$
$569$	− 967.279i	− 1.69996i	0.526812	−	0.849982i	$-0.323387\pi$
$570$	0	0
$571$	−631.514	−1.10598	−0.552990	−	0.833188i	$-0.686513\pi$
$571$	−631.514	−1.10598	−0.552990	+	0.833188i	$0.686513\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1178.21i	− 2.04906i
$576$	0	0
$577$	614.352	1.06473	0.532367	−	0.846513i	$-0.321303\pi$
$577$	614.352	1.06473	0.532367	+	0.846513i	$0.321303\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.1017	0.0173271
$584$	0	0
$585$	0	0
$586$	0	0
$587$	411.243i	0.700584i	0.936641	+	0.350292i	$0.113918\pi$
$587$	411.243i	0.700584i	−0.936641	+	0.350292i	$0.886082\pi$
$588$	0	0
$589$	1118.76	1.89942
$590$	0	0
$591$	0	0
$592$	0	0
$593$	381.923i	0.644052i	0.946731	+	0.322026i	$0.104364\pi$
$593$	381.923i	0.644052i	−0.946731	+	0.322026i	$0.895636\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 245.285i	− 0.409491i	−0.978815	−	0.204745i	$-0.934363\pi$
$599$	− 245.285i	− 0.409491i	0.978815	−	0.204745i	$-0.0656366\pi$
$600$	0	0
$601$	−168.243	−0.279939	−0.139970	−	0.990156i	$-0.544700\pi$
$601$	−168.243	−0.279939	−0.139970	+	0.990156i	$0.544700\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	334.510i	0.552910i
$606$	0	0
$607$	558.367	0.919879	0.459940	−	0.887950i	$-0.347871\pi$
$607$	558.367	0.919879	0.459940	+	0.887950i	$0.347871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	289.474i	0.473771i
$612$	0	0
$613$	−260.169	−0.424420	−0.212210	−	0.977224i	$-0.568066\pi$
$613$	−260.169	−0.424420	−0.212210	+	0.977224i	$0.568066\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	351.007i	0.568893i	0.958692	+	0.284447i	$0.0918099\pi$
$617$	351.007i	0.568893i	−0.958692	+	0.284447i	$0.908190\pi$
$618$	0	0
$619$	−317.807	−0.513419	−0.256710	−	0.966489i	$-0.582638\pi$
$619$	−317.807	−0.513419	−0.256710	+	0.966489i	$0.582638\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2117.99	3.38879
$626$	0	0
$627$	0	0
$628$	0	0
$629$	861.021i	1.36887i
$630$	0	0
$631$	297.966	0.472213	0.236106	−	0.971727i	$-0.424129\pi$
$631$	297.966	0.472213	0.236106	+	0.971727i	$0.424129\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1140.63i	− 1.79627i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 761.912i	− 1.18863i	−0.804232	−	0.594315i	$-0.797423\pi$
$641$	− 761.912i	− 1.18863i	0.804232	−	0.594315i	$-0.202577\pi$
$642$	0	0
$643$	−21.5808	−0.0335626	−0.0167813	−	0.999859i	$-0.505342\pi$
$643$	−21.5808	−0.0335626	−0.0167813	+	0.999859i	$0.505342\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 104.554i	− 0.161598i	−0.996730	−	0.0807988i	$-0.974253\pi$
$647$	− 104.554i	− 0.161598i	0.996730	−	0.0807988i	$-0.0257471\pi$
$648$	0	0
$649$	632.737	0.974941
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 359.722i	− 0.550875i	−0.961319	−	0.275438i	$-0.911177\pi$
$653$	− 359.722i	− 0.550875i	0.961319	−	0.275438i	$-0.0888228\pi$
$654$	0	0
$655$	−917.514	−1.40078
$656$	0	0
$657$	0	0
$658$	0	0
$659$	983.046i	1.49172i	0.666100	+	0.745862i	$0.267962\pi$
$659$	983.046i	1.49172i	−0.666100	+	0.745862i	$0.732038\pi$
$660$	0	0
$661$	511.626	0.774018	0.387009	−	0.922076i	$-0.373508\pi$
$661$	511.626	0.774018	0.387009	+	0.922076i	$0.373508\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−466.068	−0.698752
$668$	0	0
$669$	0	0
$670$	0	0
$671$	441.608i	0.658135i
$672$	0	0
$673$	−49.8644	−0.0740928	−0.0370464	−	0.999314i	$-0.511795\pi$
$673$	−49.8644	−0.0740928	−0.0370464	+	0.999314i	$0.511795\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 564.051i	− 0.833162i	−0.909099	−	0.416581i	$-0.863228\pi$
$677$	− 564.051i	− 0.833162i	0.909099	−	0.416581i	$-0.136772\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	132.425i	0.193887i	0.995290	+	0.0969434i	$0.0309066\pi$
$683$	132.425i	0.193887i	−0.995290	+	0.0969434i	$0.969093\pi$
$684$	0	0
$685$	−1644.55	−2.40081
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.62148i	0.00670751i
$690$	0	0
$691$	1102.11	1.59495	0.797476	−	0.603351i	$-0.206168\pi$
$691$	1102.11	1.59495	0.797476	+	0.603351i	$0.206168\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2018.58i	2.90443i
$696$	0	0
$697$	598.068	0.858060
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 302.130i	− 0.430999i	−0.976504	−	0.215500i	$-0.930862\pi$
$701$	− 302.130i	− 0.430999i	0.976504	−	0.215500i	$-0.0691380\pi$
$702$	0	0
$703$	1931.08	2.74691
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.6215	−0.0178018	−0.00890090	−	0.999960i	$-0.502833\pi$
$709$	−12.6215	−0.0178018	−0.00890090	+	0.999960i	$0.502833\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	569.412i	0.798615i
$714$	0	0
$715$	376.034	0.525922
$716$	0	0
$717$	0	0
$718$	0	0
$719$	759.608i	1.05648i	0.849095	+	0.528239i	$0.177148\pi$
$719$	759.608i	1.05648i	−0.849095	+	0.528239i	$0.822852\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1741.16i	− 2.40160i
$726$	0	0
$727$	829.041	1.14036	0.570179	−	0.821520i	$-0.306874\pi$
$727$	829.041	1.14036	0.570179	+	0.821520i	$0.306874\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	456.527i	0.624524i
$732$	0	0
$733$	−1354.52	−1.84791	−0.923957	−	0.382496i	$-0.875064\pi$
$733$	−1354.52	−1.84791	−0.923957	+	0.382496i	$0.875064\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	80.2666i	0.108910i
$738$	0	0
$739$	34.0339	0.0460540	0.0230270	−	0.999735i	$-0.492670\pi$
$739$	34.0339	0.0460540	0.0230270	+	0.999735i	$0.492670\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 61.0163i	− 0.0821215i	−0.999157	−	0.0410607i	$-0.986926\pi$
$743$	− 61.0163i	− 0.0821215i	0.999157	−	0.0410607i	$-0.0130737\pi$
$744$	0	0
$745$	−1443.59	−1.93771
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−140.136	−0.186599	−0.0932993	−	0.995638i	$-0.529741\pi$
$751$	−140.136	−0.186599	−0.0932993	+	0.995638i	$0.529741\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	115.013i	0.152335i
$756$	0	0
$757$	518.000	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$757$	518.000	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 136.145i	− 0.178903i	−0.995991	−	0.0894514i	$-0.971489\pi$
$761$	− 136.145i	− 0.178903i	0.995991	−	0.0894514i	$-0.0285114\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	289.474i	0.377411i
$768$	0	0
$769$	−237.244	−0.308510	−0.154255	−	0.988031i	$-0.549298\pi$
$769$	−237.244	−0.308510	−0.154255	+	0.988031i	$0.549298\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	863.301i	1.11682i	0.829565	+	0.558410i	$0.188588\pi$
$773$	863.301i	1.11682i	−0.829565	+	0.558410i	$0.811412\pi$
$774$	0	0
$775$	−2127.24	−2.74483
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1341.33i	− 1.72187i
$780$	0	0
$781$	887.514	1.13638
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1536.42i	− 1.95723i
$786$	0	0
$787$	−764.842	−0.971845	−0.485922	−	0.874002i	$-0.661516\pi$
$787$	−764.842	−0.971845	−0.485922	+	0.874002i	$0.661516\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−202.034	−0.254772
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1413.60i	− 1.77365i	−0.462106	−	0.886825i	$-0.652906\pi$
$797$	− 1413.60i	− 1.77365i	0.462106	−	0.886825i	$-0.347094\pi$
$798$	0	0
$799$	−1061.48	−1.32851
$800$	0	0
$801$	0	0
$802$	0	0
$803$	835.055i	1.03992i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	721.278i	0.891568i	0.895141	+	0.445784i	$0.147075\pi$
$809$	721.278i	0.891568i	−0.895141	+	0.445784i	$0.852925\pi$
$810$	0	0
$811$	928.795	1.14525	0.572623	−	0.819819i	$-0.305926\pi$
$811$	928.795	1.14525	0.572623	+	0.819819i	$0.305926\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 2466.14i	− 3.02594i
$816$	0	0
$817$	1023.89	1.25323
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1012.19i	− 1.23287i	−0.787406	−	0.616434i	$-0.788577\pi$
$821$	− 1012.19i	− 1.23287i	0.787406	−	0.616434i	$-0.211423\pi$
$822$	0	0
$823$	1331.58	1.61796	0.808980	−	0.587835i	$-0.200020\pi$
$823$	1331.58	1.61796	0.808980	+	0.587835i	$0.200020\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 96.8217i	− 0.117076i	−0.998285	−	0.0585379i	$-0.981356\pi$
$827$	− 96.8217i	− 0.117076i	0.998285	−	0.0585379i	$-0.0186438\pi$
$828$	0	0
$829$	−274.845	−0.331538	−0.165769	−	0.986165i	$-0.553011\pi$
$829$	−274.845	−0.331538	−0.165769	+	0.986165i	$0.553011\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1030.66	−1.23432
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1196.53i	1.42614i	0.701091	+	0.713072i	$0.252697\pi$
$839$	1196.53i	1.42614i	−0.701091	+	0.713072i	$0.747303\pi$
$840$	0	0
$841$	152.243	0.181026
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1443.17i	− 1.70790i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	982.860i	1.15495i
$852$	0	0
$853$	−761.454	−0.892678	−0.446339	−	0.894864i	$-0.647272\pi$
$853$	−761.454	−0.892678	−0.446339	+	0.894864i	$0.647272\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 204.416i	− 0.238525i	−0.992863	−	0.119263i	$-0.961947\pi$
$857$	− 204.416i	− 0.238525i	0.992863	−	0.119263i	$-0.0380531\pi$
$858$	0	0
$859$	−56.4487	−0.0657144	−0.0328572	−	0.999460i	$-0.510461\pi$
$859$	−56.4487	−0.0657144	−0.0328572	+	0.999460i	$0.510461\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 887.673i	− 1.02859i	−0.857613	−	0.514295i	$-0.828054\pi$
$863$	− 887.673i	− 1.02859i	0.857613	−	0.514295i	$-0.171946\pi$
$864$	0	0
$865$	−242.068	−0.279847
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1378.89i	1.58675i
$870$	0	0
$871$	−36.7216	−0.0421603
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	77.1413	0.0879604	0.0439802	−	0.999032i	$-0.485996\pi$
$877$	77.1413	0.0879604	0.0439802	+	0.999032i	$0.485996\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 918.740i	− 1.04284i	−0.853301	−	0.521419i	$-0.825403\pi$
$881$	− 918.740i	− 1.04284i	0.853301	−	0.521419i	$-0.174597\pi$
$882$	0	0
$883$	6.72316	0.00761399	0.00380700	−	0.999993i	$-0.498788\pi$
$883$	6.72316	0.00761399	0.00380700	+	0.999993i	$0.498788\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 157.744i	− 0.177840i	−0.996039	−	0.0889199i	$-0.971658\pi$
$887$	− 157.744i	− 0.177840i	0.996039	−	0.0889199i	$-0.0283415\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2380.67i	2.66592i
$894$	0	0
$895$	−535.763	−0.598618
$896$	0	0
$897$	0	0
$898$	0	0
$899$	841.480i	0.936018i
$900$	0	0
$901$	−16.9466	−0.0188087
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1130.71i	1.24940i
$906$	0	0
$907$	612.169	0.674939	0.337469	−	0.941337i	$-0.390429\pi$
$907$	612.169	0.674939	0.337469	+	0.941337i	$0.390429\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 570.650i	− 0.626399i	−0.949687	−	0.313200i	$-0.898599\pi$
$911$	− 570.650i	− 0.626399i	0.949687	−	0.313200i	$-0.101401\pi$
$912$	0	0
$913$	−824.287	−0.902833
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1266.93	−1.37859	−0.689296	−	0.724480i	$-0.742080\pi$
$919$	−1266.93	−1.37859	−0.689296	+	0.724480i	$0.742080\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	406.034i	0.439907i
$924$	0	0
$925$	−3671.82	−3.96953
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1377.34i	− 1.48260i	−0.671174	−	0.741300i	$-0.734210\pi$
$929$	− 1377.34i	− 1.48260i	0.671174	−	0.741300i	$-0.265790\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1378.89i	1.47475i
$936$	0	0
$937$	754.751	0.805497	0.402748	−	0.915311i	$-0.368055\pi$
$937$	754.751	0.805497	0.402748	+	0.915311i	$0.368055\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 760.126i	− 0.807785i	−0.914806	−	0.403893i	$-0.867657\pi$
$941$	− 760.126i	− 0.807785i	0.914806	−	0.403893i	$-0.132343\pi$
$942$	0	0
$943$	682.698	0.723964
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.1119i	0.0402448i	0.999798	+	0.0201224i	$0.00640560\pi$
$947$	38.1119i	0.0402448i	−0.999798	+	0.0201224i	$0.993594\pi$
$948$	0	0
$949$	−382.034	−0.402565
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1140.96i	− 1.19723i	−0.801039	−	0.598613i	$-0.795719\pi$
$953$	− 1140.96i	− 1.19723i	0.801039	−	0.598613i	$-0.204281\pi$
$954$	0	0
$955$	−899.608	−0.941997
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	67.0678	0.0697896
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2918.64i	− 3.02449i
$966$	0	0
$967$	1212.79	1.25418	0.627089	−	0.778947i	$-0.284246\pi$
$967$	1212.79	1.25418	0.627089	+	0.778947i	$0.284246\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 475.115i	− 0.489305i	−0.969611	−	0.244652i	$-0.921326\pi$
$971$	− 475.115i	− 0.489305i	0.969611	−	0.244652i	$-0.0786738\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1335.82i	− 1.36726i	−0.729827	−	0.683632i	$-0.760399\pi$
$977$	− 1335.82i	− 1.36726i	0.729827	−	0.683632i	$-0.239601\pi$
$978$	0	0
$979$	−707.083	−0.722250
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 108.750i	− 0.110630i	−0.998469	−	0.0553151i	$-0.982384\pi$
$983$	− 108.750i	− 0.110630i	0.998469	−	0.0553151i	$-0.0176163\pi$
$984$	0	0
$985$	−3531.31	−3.58508
$986$	0	0
$987$	0	0
$988$	0	0
$989$	521.128i	0.526925i
$990$	0	0
$991$	550.859	0.555861	0.277931	−	0.960601i	$-0.410351\pi$
$991$	550.859	0.555861	0.277931	+	0.960601i	$0.410351\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2568.55i	2.58145i
$996$	0	0
$997$	71.4617	0.0716767	0.0358384	−	0.999358i	$-0.488590\pi$
$997$	71.4617	0.0716767	0.0358384	+	0.999358i	$0.488590\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.h.197.7	yes	8
3.2	odd	2	inner	1764.3.c.h.197.2	yes	8
7.2	even	3		1764.3.bk.g.557.8		16
7.3	odd	6		1764.3.bk.g.1745.7		16
7.4	even	3		1764.3.bk.g.1745.1		16
7.5	odd	6		1764.3.bk.g.557.2		16
7.6	odd	2	inner	1764.3.c.h.197.1	✓	8
21.2	odd	6		1764.3.bk.g.557.1		16
21.5	even	6		1764.3.bk.g.557.7		16
21.11	odd	6		1764.3.bk.g.1745.8		16
21.17	even	6		1764.3.bk.g.1745.2		16
21.20	even	2	inner	1764.3.c.h.197.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.h.197.1	✓	8	7.6	odd	2	inner
1764.3.c.h.197.2	yes	8	3.2	odd	2	inner
1764.3.c.h.197.7	yes	8	1.1	even	1	trivial
1764.3.c.h.197.8	yes	8	21.20	even	2	inner
1764.3.bk.g.557.1		16	21.2	odd	6
1764.3.bk.g.557.2		16	7.5	odd	6
1764.3.bk.g.557.7		16	21.5	even	6
1764.3.bk.g.557.8		16	7.2	even	3
1764.3.bk.g.1745.1		16	7.4	even	3
1764.3.bk.g.1745.2		16	21.17	even	6
1764.3.bk.g.1745.7		16	7.3	odd	6
1764.3.bk.g.1745.8		16	21.11	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.55744i q^{5} +O(q^{10})\)	\(q+9.55744i q^{5} -9.27362i q^{11} +4.24264 q^{13} +15.5574i q^{17} +34.8919 q^{19} +17.7589i q^{23} -66.3446 q^{25} +26.2442i q^{29} +32.0635 q^{31} +55.3446 q^{37} -38.4426i q^{41} +29.3446 q^{43} +68.2298i q^{47} +1.08929i q^{53} +88.6320 q^{55} +68.2298i q^{59} -47.6198 q^{61} +40.5488i q^{65} -8.65537 q^{67} +95.7031i q^{71} -90.0463 q^{73} -148.689 q^{79} -88.8851i q^{83} -148.689 q^{85} -76.2467i q^{89} +333.477i q^{95} -12.7519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 216 q^{25} + 128 q^{37} - 80 q^{43} - 384 q^{67} - 560 q^{79} - 560 q^{85}+O(q^{100}) \) 8 * q - 216 * q^25 + 128 * q^37 - 80 * q^43 - 384 * q^67 - 560 * q^79 - 560 * q^85

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