LMFDB - Embedded newform 1764.3.c.b.197.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(197,1764)] 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("1764.197"); S:= CuspForms(chi, 3); N := Newforms(S);

sage: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")

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:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.b.197.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{5} +1.41421i q^{11} -16.0000 q^{13} -16.9706i q^{17} +8.00000 q^{19} -12.7279i q^{23} -7.00000 q^{25} -24.0416i q^{29} -56.0000 q^{31} +24.0000 q^{37} -50.9117i q^{41} +40.0000 q^{43} -11.3137i q^{47} +43.8406i q^{53} -8.00000 q^{55} -11.3137i q^{59} +40.0000 q^{61} -90.5097i q^{65} -26.0000 q^{67} -134.350i q^{71} +88.0000 q^{73} +82.0000 q^{79} +101.823i q^{83} +96.0000 q^{85} +62.2254i q^{89} +45.2548i q^{95} +40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{13} + 16 q^{19} - 14 q^{25} - 112 q^{31} + 48 q^{37} + 80 q^{43} - 16 q^{55} + 80 q^{61} - 52 q^{67} + 176 q^{73} + 164 q^{79} + 192 q^{85} + 80 q^{97}+O(q^{100}) $ 2 * q - 32 * q^13 + 16 * q^19 - 14 * q^25 - 112 * q^31 + 48 * q^37 + 80 * q^43 - 16 * q^55 + 80 * q^61 - 52 * q^67 + 176 * q^73 + 164 * q^79 + 192 * q^85 + 80 * q^97

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
$5$	5.65685i	1.13137i	0.824621	+	0.565685i	$0.191388\pi$
$5$	5.65685i	1.13137i	−0.824621	+	0.565685i	$0.808612\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421i	0.128565i	0.997932	+	0.0642824i	$0.0204759\pi$
$11$	1.41421i	0.128565i	−0.997932	+	0.0642824i	$0.979524\pi$
$12$	0	0
$13$	−16.0000	−1.23077	−0.615385	−	0.788227i	$-0.710999\pi$
$13$	−16.0000	−1.23077	−0.615385	+	0.788227i	$0.710999\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 16.9706i	− 0.998268i	−0.866525	−	0.499134i	$-0.833651\pi$
$17$	− 16.9706i	− 0.998268i	0.866525	−	0.499134i	$-0.166349\pi$
$18$	0	0
$19$	8.00000	0.421053	0.210526	−	0.977588i	$-0.432482\pi$
$19$	8.00000	0.421053	0.210526	+	0.977588i	$0.432482\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 12.7279i	− 0.553388i	−0.960958	−	0.276694i	$-0.910761\pi$
$23$	− 12.7279i	− 0.553388i	0.960958	−	0.276694i	$-0.0892388\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 24.0416i	− 0.829022i	−0.910044	−	0.414511i	$-0.863953\pi$
$29$	− 24.0416i	− 0.829022i	0.910044	−	0.414511i	$-0.136047\pi$
$30$	0	0
$31$	−56.0000	−1.80645	−0.903226	−	0.429166i	$-0.858808\pi$
$31$	−56.0000	−1.80645	−0.903226	+	0.429166i	$0.858808\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	24.0000	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
$37$	24.0000	0.648649	0.324324	+	0.945946i	$0.394863\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 50.9117i	− 1.24175i	−0.783910	−	0.620874i	$-0.786778\pi$
$41$	− 50.9117i	− 1.24175i	0.783910	−	0.620874i	$-0.213222\pi$
$42$	0	0
$43$	40.0000	0.930233	0.465116	−	0.885250i	$-0.346013\pi$
$43$	40.0000	0.930233	0.465116	+	0.885250i	$0.346013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.3137i	− 0.240717i	−0.992730	−	0.120359i	$-0.961596\pi$
$47$	− 11.3137i	− 0.240717i	0.992730	−	0.120359i	$-0.0384044\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	43.8406i	0.827182i	0.910463	+	0.413591i	$0.135726\pi$
$53$	43.8406i	0.827182i	−0.910463	+	0.413591i	$0.864274\pi$
$54$	0	0
$55$	−8.00000	−0.145455
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.3137i	− 0.191758i	−0.995393	−	0.0958789i	$-0.969434\pi$
$59$	− 11.3137i	− 0.191758i	0.995393	−	0.0958789i	$-0.0305662\pi$
$60$	0	0
$61$	40.0000	0.655738	0.327869	−	0.944723i	$-0.393670\pi$
$61$	40.0000	0.655738	0.327869	+	0.944723i	$0.393670\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 90.5097i	− 1.39246i
$66$	0	0
$67$	−26.0000	−0.388060	−0.194030	−	0.980996i	$-0.562156\pi$
$67$	−26.0000	−0.388060	−0.194030	+	0.980996i	$0.562156\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 134.350i	− 1.89226i	−0.323791	−	0.946129i	$-0.604957\pi$
$71$	− 134.350i	− 1.89226i	0.323791	−	0.946129i	$-0.395043\pi$
$72$	0	0
$73$	88.0000	1.20548	0.602740	−	0.797938i	$-0.294076\pi$
$73$	88.0000	1.20548	0.602740	+	0.797938i	$0.294076\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	82.0000	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$79$	82.0000	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	101.823i	1.22679i	0.789777	+	0.613394i	$0.210196\pi$
$83$	101.823i	1.22679i	−0.789777	+	0.613394i	$0.789804\pi$
$84$	0	0
$85$	96.0000	1.12941
$86$	0	0
$87$	0	0
$88$	0	0
$89$	62.2254i	0.699162i	0.936906	+	0.349581i	$0.113676\pi$
$89$	62.2254i	0.699162i	−0.936906	+	0.349581i	$0.886324\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	45.2548i	0.476367i
$96$	0	0
$97$	40.0000	0.412371	0.206186	−	0.978513i	$-0.433895\pi$
$97$	40.0000	0.412371	0.206186	+	0.978513i	$0.433895\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	107.480i	1.06416i	0.846694	+	0.532080i	$0.178590\pi$
$101$	107.480i	1.06416i	−0.846694	+	0.532080i	$0.821410\pi$
$102$	0	0
$103$	−184.000	−1.78641	−0.893204	−	0.449652i	$-0.851548\pi$
$103$	−184.000	−1.78641	−0.893204	+	0.449652i	$0.851548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 145.664i	− 1.36135i	−0.732587	−	0.680673i	$-0.761688\pi$
$107$	− 145.664i	− 1.36135i	0.732587	−	0.680673i	$-0.238312\pi$
$108$	0	0
$109$	−104.000	−0.954128	−0.477064	−	0.878868i	$-0.658299\pi$
$109$	−104.000	−0.954128	−0.477064	+	0.878868i	$0.658299\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 171.120i	− 1.51433i	−0.653221	−	0.757167i	$-0.726583\pi$
$113$	− 171.120i	− 1.51433i	0.653221	−	0.757167i	$-0.273417\pi$
$114$	0	0
$115$	72.0000	0.626087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	119.000	0.983471
$122$	0	0
$123$	0	0
$124$	0	0
$125$	101.823i	0.814587i
$126$	0	0
$127$	206.000	1.62205	0.811024	−	0.585013i	$-0.198911\pi$
$127$	206.000	1.62205	0.811024	+	0.585013i	$0.198911\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 203.647i	− 1.55456i	−0.629158	−	0.777278i	$-0.716600\pi$
$131$	− 203.647i	− 1.55456i	0.629158	−	0.777278i	$-0.283400\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	179.605i	1.31099i	0.755201	+	0.655493i	$0.227539\pi$
$137$	179.605i	1.31099i	−0.755201	+	0.655493i	$0.772461\pi$
$138$	0	0
$139$	192.000	1.38129	0.690647	−	0.723192i	$-0.257326\pi$
$139$	192.000	1.38129	0.690647	+	0.723192i	$0.257326\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 22.6274i	− 0.158234i
$144$	0	0
$145$	136.000	0.937931
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 236.174i	− 1.58506i	−0.609834	−	0.792529i	$-0.708764\pi$
$149$	− 236.174i	− 1.58506i	0.609834	−	0.792529i	$-0.291236\pi$
$150$	0	0
$151$	16.0000	0.105960	0.0529801	−	0.998596i	$-0.483128\pi$
$151$	16.0000	0.105960	0.0529801	+	0.998596i	$0.483128\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 316.784i	− 2.04377i
$156$	0	0
$157$	56.0000	0.356688	0.178344	−	0.983968i	$-0.442926\pi$
$157$	56.0000	0.356688	0.178344	+	0.983968i	$0.442926\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	102.000	0.625767	0.312883	−	0.949792i	$-0.398705\pi$
$163$	102.000	0.625767	0.312883	+	0.949792i	$0.398705\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 260.215i	− 1.55818i	−0.626915	−	0.779088i	$-0.715683\pi$
$167$	− 260.215i	− 1.55818i	0.626915	−	0.779088i	$-0.284317\pi$
$168$	0	0
$169$	87.0000	0.514793
$170$	0	0
$171$	0	0
$172$	0	0
$173$	96.1665i	0.555876i	0.960599	+	0.277938i	$0.0896510\pi$
$173$	96.1665i	0.555876i	−0.960599	+	0.277938i	$0.910349\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 21.2132i	− 0.118510i	−0.998243	−	0.0592548i	$-0.981128\pi$
$179$	− 21.2132i	− 0.118510i	0.998243	−	0.0592548i	$-0.0188724\pi$
$180$	0	0
$181$	−128.000	−0.707182	−0.353591	−	0.935400i	$-0.615040\pi$
$181$	−128.000	−0.707182	−0.353591	+	0.935400i	$0.615040\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	135.765i	0.733862i
$186$	0	0
$187$	24.0000	0.128342
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 114.551i	− 0.599745i	−0.953979	−	0.299873i	$-0.903056\pi$
$191$	− 114.551i	− 0.599745i	0.953979	−	0.299873i	$-0.0969441\pi$
$192$	0	0
$193$	−208.000	−1.07772	−0.538860	−	0.842395i	$-0.681145\pi$
$193$	−208.000	−1.07772	−0.538860	+	0.842395i	$0.681145\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 123.037i	− 0.624551i	−0.949992	−	0.312276i	$-0.898909\pi$
$197$	− 123.037i	− 0.624551i	0.949992	−	0.312276i	$-0.101091\pi$
$198$	0	0
$199$	−176.000	−0.884422	−0.442211	−	0.896911i	$-0.645806\pi$
$199$	−176.000	−0.884422	−0.442211	+	0.896911i	$0.645806\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	288.000	1.40488
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137i	0.0541326i
$210$	0	0
$211$	248.000	1.17536	0.587678	−	0.809095i	$-0.300042\pi$
$211$	248.000	1.17536	0.587678	+	0.809095i	$0.300042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	226.274i	1.05244i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	271.529i	1.22864i
$222$	0	0
$223$	−272.000	−1.21973	−0.609865	−	0.792505i	$-0.708777\pi$
$223$	−272.000	−1.21973	−0.609865	+	0.792505i	$0.708777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 395.980i	− 1.74440i	−0.489146	−	0.872202i	$-0.662691\pi$
$227$	− 395.980i	− 1.74440i	0.489146	−	0.872202i	$-0.337309\pi$
$228$	0	0
$229$	80.0000	0.349345	0.174672	−	0.984627i	$-0.444113\pi$
$229$	80.0000	0.349345	0.174672	+	0.984627i	$0.444113\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 202.233i	− 0.867951i	−0.900925	−	0.433975i	$-0.857110\pi$
$233$	− 202.233i	− 0.867951i	0.900925	−	0.433975i	$-0.142890\pi$
$234$	0	0
$235$	64.0000	0.272340
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 326.683i	− 1.36688i	−0.730009	−	0.683438i	$-0.760484\pi$
$239$	− 326.683i	− 1.36688i	0.730009	−	0.683438i	$-0.239516\pi$
$240$	0	0
$241$	−248.000	−1.02905	−0.514523	−	0.857477i	$-0.672031\pi$
$241$	−248.000	−1.02905	−0.514523	+	0.857477i	$0.672031\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−128.000	−0.518219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	294.156i	1.17194i	0.810333	+	0.585969i	$0.199286\pi$
$251$	294.156i	1.17194i	−0.810333	+	0.585969i	$0.800714\pi$
$252$	0	0
$253$	18.0000	0.0711462
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 175.362i	− 0.682344i	−0.940001	−	0.341172i	$-0.889176\pi$
$257$	− 175.362i	− 0.682344i	0.940001	−	0.341172i	$-0.110824\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 159.806i	− 0.607628i	−0.952731	−	0.303814i	$-0.901740\pi$
$263$	− 159.806i	− 0.607628i	0.952731	−	0.303814i	$-0.0982601\pi$
$264$	0	0
$265$	−248.000	−0.935849
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 299.813i	− 1.11455i	−0.830329	−	0.557274i	$-0.811847\pi$
$269$	− 299.813i	− 1.11455i	0.830329	−	0.557274i	$-0.188153\pi$
$270$	0	0
$271$	−88.0000	−0.324723	−0.162362	−	0.986731i	$-0.551911\pi$
$271$	−88.0000	−0.324723	−0.162362	+	0.986731i	$0.551911\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9.89949i	− 0.0359982i
$276$	0	0
$277$	−38.0000	−0.137184	−0.0685921	−	0.997645i	$-0.521851\pi$
$277$	−38.0000	−0.137184	−0.0685921	+	0.997645i	$0.521851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	182.434i	0.649230i	0.945846	+	0.324615i	$0.105235\pi$
$281$	182.434i	0.649230i	−0.945846	+	0.324615i	$0.894765\pi$
$282$	0	0
$283$	−152.000	−0.537102	−0.268551	−	0.963265i	$-0.586545\pi$
$283$	−152.000	−0.537102	−0.268551	+	0.963265i	$0.586545\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 458.205i	− 1.56384i	−0.623379	−	0.781920i	$-0.714240\pi$
$293$	− 458.205i	− 1.56384i	0.623379	−	0.781920i	$-0.285760\pi$
$294$	0	0
$295$	64.0000	0.216949
$296$	0	0
$297$	0	0
$298$	0	0
$299$	203.647i	0.681093i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	226.274i	0.741883i
$306$	0	0
$307$	232.000	0.755700	0.377850	−	0.925867i	$-0.376663\pi$
$307$	232.000	0.755700	0.377850	+	0.925867i	$0.376663\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	486.489i	1.56427i	0.623106	+	0.782137i	$0.285871\pi$
$311$	486.489i	1.56427i	−0.623106	+	0.782137i	$0.714129\pi$
$312$	0	0
$313$	−144.000	−0.460064	−0.230032	−	0.973183i	$-0.573883\pi$
$313$	−144.000	−0.460064	−0.230032	+	0.973183i	$0.573883\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 80.6102i	− 0.254291i	−0.991884	−	0.127145i	$-0.959419\pi$
$317$	− 80.6102i	− 0.254291i	0.991884	−	0.127145i	$-0.0405815\pi$
$318$	0	0
$319$	34.0000	0.106583
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 135.765i	− 0.420324i
$324$	0	0
$325$	112.000	0.344615
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	200.000	0.604230	0.302115	−	0.953272i	$-0.402307\pi$
$331$	200.000	0.604230	0.302115	+	0.953272i	$0.402307\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 147.078i	− 0.439039i
$336$	0	0
$337$	−208.000	−0.617211	−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000	−0.617211	−0.308605	+	0.951190i	$0.599862\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 79.1960i	− 0.232246i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 462.448i	− 1.33270i	−0.745638	−	0.666351i	$-0.767855\pi$
$347$	− 462.448i	− 1.33270i	0.745638	−	0.666351i	$-0.232145\pi$
$348$	0	0
$349$	456.000	1.30659	0.653295	−	0.757103i	$-0.273386\pi$
$349$	456.000	1.30659	0.653295	+	0.757103i	$0.273386\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 16.9706i	− 0.0480752i	−0.999711	−	0.0240376i	$-0.992348\pi$
$353$	− 16.9706i	− 0.0480752i	0.999711	−	0.0240376i	$-0.00765215\pi$
$354$	0	0
$355$	760.000	2.14085
$356$	0	0
$357$	0	0
$358$	0	0
$359$	439.820i	1.22513i	0.790422	+	0.612563i	$0.209862\pi$
$359$	439.820i	1.22513i	−0.790422	+	0.612563i	$0.790138\pi$
$360$	0	0
$361$	−297.000	−0.822715
$362$	0	0
$363$	0	0
$364$	0	0
$365$	497.803i	1.36384i
$366$	0	0
$367$	496.000	1.35150	0.675749	−	0.737132i	$-0.263820\pi$
$367$	496.000	1.35150	0.675749	+	0.737132i	$0.263820\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−314.000	−0.841823	−0.420912	−	0.907102i	$-0.638290\pi$
$373$	−314.000	−0.841823	−0.420912	+	0.907102i	$0.638290\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	384.666i	1.02033i
$378$	0	0
$379$	24.0000	0.0633245	0.0316623	−	0.999499i	$-0.489920\pi$
$379$	24.0000	0.0633245	0.0316623	+	0.999499i	$0.489920\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 543.058i	− 1.41791i	−0.705256	−	0.708953i	$-0.749168\pi$
$383$	− 543.058i	− 1.41791i	0.705256	−	0.708953i	$-0.250832\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	250.316i	0.643485i	0.946827	+	0.321743i	$0.104269\pi$
$389$	250.316i	0.643485i	−0.946827	+	0.321743i	$0.895731\pi$
$390$	0	0
$391$	−216.000	−0.552430
$392$	0	0
$393$	0	0
$394$	0	0
$395$	463.862i	1.17433i
$396$	0	0
$397$	−664.000	−1.67254	−0.836272	−	0.548315i	$-0.815270\pi$
$397$	−664.000	−1.67254	−0.836272	+	0.548315i	$0.815270\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	123.037i	0.306824i	0.988162	+	0.153412i	$0.0490262\pi$
$401$	123.037i	0.306824i	−0.988162	+	0.153412i	$0.950974\pi$
$402$	0	0
$403$	896.000	2.22333
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.9411i	0.0833934i
$408$	0	0
$409$	312.000	0.762836	0.381418	−	0.924403i	$-0.375436\pi$
$409$	312.000	0.762836	0.381418	+	0.924403i	$0.375436\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−576.000	−1.38795
$416$	0	0
$417$	0	0
$418$	0	0
$419$	226.274i	0.540034i	0.962856	+	0.270017i	$0.0870293\pi$
$419$	226.274i	0.540034i	−0.962856	+	0.270017i	$0.912971\pi$
$420$	0	0
$421$	−762.000	−1.80998	−0.904988	−	0.425437i	$-0.860120\pi$
$421$	−762.000	−1.80998	−0.904988	+	0.425437i	$0.860120\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	118.794i	0.279515i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 374.767i	− 0.869528i	−0.900544	−	0.434764i	$-0.856832\pi$
$431$	− 374.767i	− 0.869528i	0.900544	−	0.434764i	$-0.143168\pi$
$432$	0	0
$433$	464.000	1.07159	0.535797	−	0.844347i	$-0.320011\pi$
$433$	464.000	1.07159	0.535797	+	0.844347i	$0.320011\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 101.823i	− 0.233005i
$438$	0	0
$439$	−480.000	−1.09339	−0.546697	−	0.837330i	$-0.684115\pi$
$439$	−480.000	−1.09339	−0.546697	+	0.837330i	$0.684115\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	453.963i	1.02475i	0.858763	+	0.512373i	$0.171233\pi$
$443$	453.963i	1.02475i	−0.858763	+	0.512373i	$0.828767\pi$
$444$	0	0
$445$	−352.000	−0.791011
$446$	0	0
$447$	0	0
$448$	0	0
$449$	835.800i	1.86147i	0.365694	+	0.930735i	$0.380832\pi$
$449$	835.800i	1.86147i	−0.365694	+	0.930735i	$0.619168\pi$
$450$	0	0
$451$	72.0000	0.159645
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	416.000	0.910284	0.455142	−	0.890419i	$-0.349588\pi$
$457$	416.000	0.910284	0.455142	+	0.890419i	$0.349588\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	661.852i	1.43569i	0.696204	+	0.717844i	$0.254871\pi$
$461$	661.852i	1.43569i	−0.696204	+	0.717844i	$0.745129\pi$
$462$	0	0
$463$	146.000	0.315335	0.157667	−	0.987492i	$-0.449603\pi$
$463$	146.000	0.315335	0.157667	+	0.987492i	$0.449603\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	588.313i	1.25977i	0.776688	+	0.629885i	$0.216898\pi$
$467$	588.313i	1.25977i	−0.776688	+	0.629885i	$0.783102\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	56.5685i	0.119595i
$474$	0	0
$475$	−56.0000	−0.117895
$476$	0	0
$477$	0	0
$478$	0	0
$479$	67.8823i	0.141717i	0.997486	+	0.0708583i	$0.0225738\pi$
$479$	67.8823i	0.141717i	−0.997486	+	0.0708583i	$0.977426\pi$
$480$	0	0
$481$	−384.000	−0.798337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	226.274i	0.466545i
$486$	0	0
$487$	112.000	0.229979	0.114990	−	0.993367i	$-0.463316\pi$
$487$	112.000	0.229979	0.114990	+	0.993367i	$0.463316\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	428.507i	0.872722i	0.899772	+	0.436361i	$0.143733\pi$
$491$	428.507i	0.872722i	−0.899772	+	0.436361i	$0.856267\pi$
$492$	0	0
$493$	−408.000	−0.827586
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−744.000	−1.49098	−0.745491	−	0.666516i	$-0.767785\pi$
$499$	−744.000	−1.49098	−0.745491	+	0.666516i	$0.767785\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 192.333i	− 0.382372i	−0.981554	−	0.191186i	$-0.938767\pi$
$503$	− 192.333i	− 0.382372i	0.981554	−	0.191186i	$-0.0612333\pi$
$504$	0	0
$505$	−608.000	−1.20396
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 277.186i	− 0.544569i	−0.962217	−	0.272285i	$-0.912221\pi$
$509$	− 277.186i	− 0.544569i	0.962217	−	0.272285i	$-0.0877793\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1040.86i	− 2.02109i
$516$	0	0
$517$	16.0000	0.0309478
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 379.009i	− 0.727465i	−0.931503	−	0.363732i	$-0.881502\pi$
$521$	− 379.009i	− 0.727465i	0.931503	−	0.363732i	$-0.118498\pi$
$522$	0	0
$523$	512.000	0.978967	0.489484	−	0.872012i	$-0.337185\pi$
$523$	512.000	0.978967	0.489484	+	0.872012i	$0.337185\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	950.352i	1.80332i
$528$	0	0
$529$	367.000	0.693762
$530$	0	0
$531$	0	0
$532$	0	0
$533$	814.587i	1.52831i
$534$	0	0
$535$	824.000	1.54019
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−470.000	−0.868762	−0.434381	−	0.900729i	$-0.643033\pi$
$541$	−470.000	−0.868762	−0.434381	+	0.900729i	$0.643033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 588.313i	− 1.07947i
$546$	0	0
$547$	−614.000	−1.12249	−0.561243	−	0.827651i	$-0.689677\pi$
$547$	−614.000	−1.12249	−0.561243	+	0.827651i	$0.689677\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 192.333i	− 0.349062i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	318.198i	0.571271i	0.958338	+	0.285636i	$0.0922047\pi$
$557$	318.198i	0.571271i	−0.958338	+	0.285636i	$0.907795\pi$
$558$	0	0
$559$	−640.000	−1.14490
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 260.215i	− 0.462194i	−0.972931	−	0.231097i	$-0.925769\pi$
$563$	− 260.215i	− 0.462194i	0.972931	−	0.231097i	$-0.0742315\pi$
$564$	0	0
$565$	968.000	1.71327
$566$	0	0
$567$	0	0
$568$	0	0
$569$	284.257i	0.499573i	0.968301	+	0.249786i	$0.0803604\pi$
$569$	284.257i	0.499573i	−0.968301	+	0.249786i	$0.919640\pi$
$570$	0	0
$571$	150.000	0.262697	0.131349	−	0.991336i	$-0.458069\pi$
$571$	150.000	0.262697	0.131349	+	0.991336i	$0.458069\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	89.0955i	0.154949i
$576$	0	0
$577$	−848.000	−1.46967	−0.734835	−	0.678246i	$-0.762741\pi$
$577$	−848.000	−1.46967	−0.734835	+	0.678246i	$0.762741\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−62.0000	−0.106346
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 622.254i	− 1.06006i	−0.847980	−	0.530029i	$-0.822181\pi$
$587$	− 622.254i	− 1.06006i	0.847980	−	0.530029i	$-0.177819\pi$
$588$	0	0
$589$	−448.000	−0.760611
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 288.500i	− 0.486509i	−0.969963	−	0.243254i	$-0.921785\pi$
$593$	− 288.500i	− 0.486509i	0.969963	−	0.243254i	$-0.0782149\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 453.963i	− 0.757867i	−0.925424	−	0.378934i	$-0.876291\pi$
$599$	− 453.963i	− 0.757867i	0.925424	−	0.378934i	$-0.123709\pi$
$600$	0	0
$601$	−208.000	−0.346090	−0.173045	−	0.984914i	$-0.555361\pi$
$601$	−208.000	−0.346090	−0.173045	+	0.984914i	$0.555361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	673.166i	1.11267i
$606$	0	0
$607$	48.0000	0.0790774	0.0395387	−	0.999218i	$-0.487411\pi$
$607$	48.0000	0.0790774	0.0395387	+	0.999218i	$0.487411\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	181.019i	0.296267i
$612$	0	0
$613$	−154.000	−0.251223	−0.125612	−	0.992079i	$-0.540089\pi$
$613$	−154.000	−0.251223	−0.125612	+	0.992079i	$0.540089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1076.22i	1.74427i	0.489263	+	0.872137i	$0.337266\pi$
$617$	1076.22i	1.74427i	−0.489263	+	0.872137i	$0.662734\pi$
$618$	0	0
$619$	−944.000	−1.52504	−0.762520	−	0.646964i	$-0.776038\pi$
$619$	−944.000	−1.52504	−0.762520	+	0.646964i	$0.776038\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−751.000	−1.20160
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 407.294i	− 0.647525i
$630$	0	0
$631$	126.000	0.199683	0.0998415	−	0.995003i	$-0.468166\pi$
$631$	126.000	0.199683	0.0998415	+	0.995003i	$0.468166\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1165.31i	1.83514i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	465.276i	0.725860i	0.931816	+	0.362930i	$0.118224\pi$
$641$	465.276i	0.725860i	−0.931816	+	0.362930i	$0.881776\pi$
$642$	0	0
$643$	552.000	0.858476	0.429238	−	0.903191i	$-0.358782\pi$
$643$	552.000	0.858476	0.429238	+	0.903191i	$0.358782\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 509.117i	− 0.786889i	−0.919348	−	0.393444i	$-0.871283\pi$
$647$	− 509.117i	− 0.786889i	0.919348	−	0.393444i	$-0.128717\pi$
$648$	0	0
$649$	16.0000	0.0246533
$650$	0	0
$651$	0	0
$652$	0	0
$653$	315.370i	0.482955i	0.970406	+	0.241478i	$0.0776320\pi$
$653$	315.370i	0.482955i	−0.970406	+	0.241478i	$0.922368\pi$
$654$	0	0
$655$	1152.00	1.75878
$656$	0	0
$657$	0	0
$658$	0	0
$659$	270.115i	0.409886i	0.978774	+	0.204943i	$0.0657009\pi$
$659$	270.115i	0.409886i	−0.978774	+	0.204943i	$0.934299\pi$
$660$	0	0
$661$	−808.000	−1.22239	−0.611195	−	0.791480i	$-0.709311\pi$
$661$	−808.000	−1.22239	−0.611195	+	0.791480i	$0.709311\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−306.000	−0.458771
$668$	0	0
$669$	0	0
$670$	0	0
$671$	56.5685i	0.0843048i
$672$	0	0
$673$	−832.000	−1.23626	−0.618128	−	0.786078i	$-0.712109\pi$
$673$	−832.000	−1.23626	−0.618128	+	0.786078i	$0.712109\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 424.264i	− 0.626683i	−0.949641	−	0.313341i	$-0.898552\pi$
$677$	− 424.264i	− 0.626683i	0.949641	−	0.313341i	$-0.101448\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1291.18i	− 1.89045i	−0.326420	−	0.945225i	$-0.605842\pi$
$683$	− 1291.18i	− 1.89045i	0.326420	−	0.945225i	$-0.394158\pi$
$684$	0	0
$685$	−1016.00	−1.48321
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 701.450i	− 1.01807i
$690$	0	0
$691$	−272.000	−0.393632	−0.196816	−	0.980440i	$-0.563060\pi$
$691$	−272.000	−0.393632	−0.196816	+	0.980440i	$0.563060\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1086.12i	1.56276i
$696$	0	0
$697$	−864.000	−1.23960
$698$	0	0
$699$	0	0
$700$	0	0
$701$	982.878i	1.40211i	0.713108	+	0.701055i	$0.247287\pi$
$701$	982.878i	1.40211i	−0.713108	+	0.701055i	$0.752713\pi$
$702$	0	0
$703$	192.000	0.273115
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1192.00	1.68124	0.840621	−	0.541624i	$-0.182190\pi$
$709$	1192.00	1.68124	0.840621	+	0.541624i	$0.182190\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	712.764i	0.999668i
$714$	0	0
$715$	128.000	0.179021
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 746.705i	− 1.03853i	−0.854613	−	0.519266i	$-0.826205\pi$
$719$	− 746.705i	− 1.03853i	0.854613	−	0.519266i	$-0.173795\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	168.291i	0.232126i
$726$	0	0
$727$	−184.000	−0.253095	−0.126547	−	0.991961i	$-0.540390\pi$
$727$	−184.000	−0.253095	−0.126547	+	0.991961i	$0.540390\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 678.823i	− 0.928622i
$732$	0	0
$733$	−800.000	−1.09141	−0.545703	−	0.837979i	$-0.683737\pi$
$733$	−800.000	−1.09141	−0.545703	+	0.837979i	$0.683737\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.7696i	− 0.0498908i
$738$	0	0
$739$	346.000	0.468200	0.234100	−	0.972212i	$-0.424786\pi$
$739$	346.000	0.468200	0.234100	+	0.972212i	$0.424786\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 895.197i	− 1.20484i	−0.798179	−	0.602421i	$-0.794203\pi$
$743$	− 895.197i	− 1.20484i	0.798179	−	0.602421i	$-0.205797\pi$
$744$	0	0
$745$	1336.00	1.79329
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1248.00	−1.66178	−0.830892	−	0.556434i	$-0.812169\pi$
$751$	−1248.00	−1.66178	−0.830892	+	0.556434i	$0.812169\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	90.5097i	0.119880i
$756$	0	0
$757$	−392.000	−0.517834	−0.258917	−	0.965900i	$-0.583366\pi$
$757$	−392.000	−0.517834	−0.258917	+	0.965900i	$0.583366\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	956.008i	1.25625i	0.778111	+	0.628126i	$0.216178\pi$
$761$	956.008i	1.25625i	−0.778111	+	0.628126i	$0.783822\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	181.019i	0.236010i
$768$	0	0
$769$	−256.000	−0.332900	−0.166450	−	0.986050i	$-0.553230\pi$
$769$	−256.000	−0.332900	−0.166450	+	0.986050i	$0.553230\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	322.441i	0.417129i	0.978009	+	0.208564i	$0.0668791\pi$
$773$	322.441i	0.417129i	−0.978009	+	0.208564i	$0.933121\pi$
$774$	0	0
$775$	392.000	0.505806
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 407.294i	− 0.522841i
$780$	0	0
$781$	190.000	0.243278
$782$	0	0
$783$	0	0
$784$	0	0
$785$	316.784i	0.403546i
$786$	0	0
$787$	−800.000	−1.01652	−0.508259	−	0.861204i	$-0.669711\pi$
$787$	−800.000	−1.01652	−0.508259	+	0.861204i	$0.669711\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−640.000	−0.807062
$794$	0	0
$795$	0	0
$796$	0	0
$797$	797.616i	1.00077i	0.865802	+	0.500387i	$0.166809\pi$
$797$	797.616i	1.00077i	−0.865802	+	0.500387i	$0.833191\pi$
$798$	0	0
$799$	−192.000	−0.240300
$800$	0	0
$801$	0	0
$802$	0	0
$803$	124.451i	0.154982i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	487.904i	0.603095i	0.953451	+	0.301547i	$0.0975031\pi$
$809$	487.904i	0.603095i	−0.953451	+	0.301547i	$0.902497\pi$
$810$	0	0
$811$	1232.00	1.51911	0.759556	−	0.650442i	$-0.225416\pi$
$811$	1232.00	1.51911	0.759556	+	0.650442i	$0.225416\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	576.999i	0.707974i
$816$	0	0
$817$	320.000	0.391677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 473.762i	− 0.577054i	−0.957472	−	0.288527i	$-0.906834\pi$
$821$	− 473.762i	− 0.577054i	0.957472	−	0.288527i	$-0.0931655\pi$
$822$	0	0
$823$	−254.000	−0.308627	−0.154313	−	0.988022i	$-0.549317\pi$
$823$	−254.000	−0.308627	−0.154313	+	0.988022i	$0.549317\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.7279i	− 0.0153905i	−0.999970	−	0.00769524i	$-0.997551\pi$
$827$	− 12.7279i	− 0.0153905i	0.999970	−	0.00769524i	$-0.00244949\pi$
$828$	0	0
$829$	1504.00	1.81423	0.907117	−	0.420879i	$-0.138278\pi$
$829$	1504.00	1.81423	0.907117	+	0.420879i	$0.138278\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1472.00	1.76287
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 203.647i	− 0.242726i	−0.992608	−	0.121363i	$-0.961274\pi$
$839$	− 203.647i	− 0.242726i	0.992608	−	0.121363i	$-0.0387264\pi$
$840$	0	0
$841$	263.000	0.312723
$842$	0	0
$843$	0	0
$844$	0	0
$845$	492.146i	0.582422i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 305.470i	− 0.358954i
$852$	0	0
$853$	536.000	0.628370	0.314185	−	0.949362i	$-0.398269\pi$
$853$	536.000	0.628370	0.314185	+	0.949362i	$0.398269\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1544.32i	1.80201i	0.433810	+	0.901004i	$0.357169\pi$
$857$	1544.32i	1.80201i	−0.433810	+	0.901004i	$0.642831\pi$
$858$	0	0
$859$	−792.000	−0.922002	−0.461001	−	0.887400i	$-0.652510\pi$
$859$	−792.000	−0.922002	−0.461001	+	0.887400i	$0.652510\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	903.682i	1.04714i	0.851982	+	0.523570i	$0.175400\pi$
$863$	903.682i	1.04714i	−0.851982	+	0.523570i	$0.824600\pi$
$864$	0	0
$865$	−544.000	−0.628902
$866$	0	0
$867$	0	0
$868$	0	0
$869$	115.966i	0.133447i
$870$	0	0
$871$	416.000	0.477612
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−342.000	−0.389966	−0.194983	−	0.980807i	$-0.562465\pi$
$877$	−342.000	−0.389966	−0.194983	+	0.980807i	$0.562465\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1487.75i	− 1.68871i	−0.535785	−	0.844355i	$-0.679984\pi$
$881$	− 1487.75i	− 1.68871i	0.535785	−	0.844355i	$-0.320016\pi$
$882$	0	0
$883$	1096.00	1.24122	0.620612	−	0.784118i	$-0.286884\pi$
$883$	1096.00	1.24122	0.620612	+	0.784118i	$0.286884\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1120.06i	− 1.26275i	−0.775479	−	0.631374i	$-0.782491\pi$
$887$	− 1120.06i	− 1.26275i	0.775479	−	0.631374i	$-0.217509\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 90.5097i	− 0.101355i
$894$	0	0
$895$	120.000	0.134078
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1346.33i	1.49759i
$900$	0	0
$901$	744.000	0.825749
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 724.077i	− 0.800085i
$906$	0	0
$907$	1528.00	1.68467	0.842337	−	0.538951i	$-0.181179\pi$
$907$	1528.00	1.68467	0.842337	+	0.538951i	$0.181179\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 552.958i	− 0.606979i	−0.952835	−	0.303489i	$-0.901848\pi$
$911$	− 552.958i	− 0.606979i	0.952835	−	0.303489i	$-0.0981517\pi$
$912$	0	0
$913$	−144.000	−0.157722
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−672.000	−0.731230	−0.365615	−	0.930766i	$-0.619141\pi$
$919$	−672.000	−0.731230	−0.365615	+	0.930766i	$0.619141\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2149.60i	2.32893i
$924$	0	0
$925$	−168.000	−0.181622
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1035.20i	− 1.11432i	−0.830405	−	0.557161i	$-0.811891\pi$
$929$	− 1035.20i	− 1.11432i	0.830405	−	0.557161i	$-0.188109\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	135.765i	0.145203i
$936$	0	0
$937$	960.000	1.02455	0.512273	−	0.858823i	$-0.328804\pi$
$937$	960.000	1.02455	0.512273	+	0.858823i	$0.328804\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 967.322i	− 1.02797i	−0.857798	−	0.513986i	$-0.828168\pi$
$941$	− 967.322i	− 1.02797i	0.857798	−	0.513986i	$-0.171832\pi$
$942$	0	0
$943$	−648.000	−0.687169
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1356.23i	− 1.43213i	−0.698032	−	0.716067i	$-0.745941\pi$
$947$	− 1356.23i	− 1.43213i	0.698032	−	0.716067i	$-0.254059\pi$
$948$	0	0
$949$	−1408.00	−1.48367
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1107.33i	− 1.16194i	−0.813925	−	0.580970i	$-0.802673\pi$
$953$	− 1107.33i	− 1.16194i	0.813925	−	0.580970i	$-0.197327\pi$
$954$	0	0
$955$	648.000	0.678534
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2175.00	2.26327
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1176.63i	− 1.21930i
$966$	0	0
$967$	994.000	1.02792	0.513961	−	0.857814i	$-0.328178\pi$
$967$	994.000	1.02792	0.513961	+	0.857814i	$0.328178\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1776.25i	− 1.82930i	−0.404244	−	0.914651i	$-0.632465\pi$
$971$	− 1776.25i	− 1.82930i	0.404244	−	0.914651i	$-0.367535\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1087.53i	1.11313i	0.830803	+	0.556566i	$0.187881\pi$
$977$	1087.53i	1.11313i	−0.830803	+	0.556566i	$0.812119\pi$
$978$	0	0
$979$	−88.0000	−0.0898876
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1097.43i	− 1.11641i	−0.829704	−	0.558204i	$-0.811491\pi$
$983$	− 1097.43i	− 1.11641i	0.829704	−	0.558204i	$-0.188509\pi$
$984$	0	0
$985$	696.000	0.706599
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 509.117i	− 0.514779i
$990$	0	0
$991$	−1648.00	−1.66297	−0.831483	−	0.555550i	$-0.812508\pi$
$991$	−1648.00	−1.66297	−0.831483	+	0.555550i	$0.812508\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 995.606i	− 1.00061i
$996$	0	0
$997$	248.000	0.248746	0.124373	−	0.992236i	$-0.460308\pi$
$997$	248.000	0.248746	0.124373	+	0.992236i	$0.460308\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.b.197.2	yes	2
3.2	odd	2	inner	1764.3.c.b.197.1	✓	2
7.2	even	3		1764.3.bk.b.557.2		4
7.3	odd	6		1764.3.bk.c.1745.2		4
7.4	even	3		1764.3.bk.b.1745.1		4
7.5	odd	6		1764.3.bk.c.557.1		4
7.6	odd	2		1764.3.c.c.197.1	yes	2
21.2	odd	6		1764.3.bk.b.557.1		4
21.5	even	6		1764.3.bk.c.557.2		4
21.11	odd	6		1764.3.bk.b.1745.2		4
21.17	even	6		1764.3.bk.c.1745.1		4
21.20	even	2		1764.3.c.c.197.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.b.197.1	✓	2	3.2	odd	2	inner
1764.3.c.b.197.2	yes	2	1.1	even	1	trivial
1764.3.c.c.197.1	yes	2	7.6	odd	2
1764.3.c.c.197.2	yes	2	21.20	even	2
1764.3.bk.b.557.1		4	21.2	odd	6
1764.3.bk.b.557.2		4	7.2	even	3
1764.3.bk.b.1745.1		4	7.4	even	3
1764.3.bk.b.1745.2		4	21.11	odd	6
1764.3.bk.c.557.1		4	7.5	odd	6
1764.3.bk.c.557.2		4	21.5	even	6
1764.3.bk.c.1745.1		4	21.17	even	6
1764.3.bk.c.1745.2		4	7.3	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+5.65685i q^{5} +1.41421i q^{11} -16.0000 q^{13} -16.9706i q^{17} +8.00000 q^{19} -12.7279i q^{23} -7.00000 q^{25} -24.0416i q^{29} -56.0000 q^{31} +24.0000 q^{37} -50.9117i q^{41} +40.0000 q^{43} -11.3137i q^{47} +43.8406i q^{53} -8.00000 q^{55} -11.3137i q^{59} +40.0000 q^{61} -90.5097i q^{65} -26.0000 q^{67} -134.350i q^{71} +88.0000 q^{73} +82.0000 q^{79} +101.823i q^{83} +96.0000 q^{85} +62.2254i q^{89} +45.2548i q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{13} + 16 q^{19} - 14 q^{25} - 112 q^{31} + 48 q^{37} + 80 q^{43} - 16 q^{55} + 80 q^{61} - 52 q^{67} + 176 q^{73} + 164 q^{79} + 192 q^{85} + 80 q^{97}+O(q^{100}) \) 2 * q - 32 * q^13 + 16 * q^19 - 14 * q^25 - 112 * q^31 + 48 * q^37 + 80 * q^43 - 16 * q^55 + 80 * q^61 - 52 * q^67 + 176 * q^73 + 164 * q^79 + 192 * q^85 + 80 * q^97

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