LMFDB - Embedded newform 3168.2.f.h.1585.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(1585,3168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3168.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	20.0.74831334220841134637329678336.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2x^{18} + 5x^{16} - 8x^{14} + 28x^{12} - 64x^{10} + 112x^{8} - 128x^{6} + 320x^{4} - 512x^{2} + 1024 $ x^20 - 2x^18 + 5x^16 - 8x^14 + 28x^12 - 64x^10 + 112x^8 - 128x^6 + 320x^4 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{29} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.12
Root			$-0.689696 - 1.23463i$ of defining polynomial
Character	$\chi$	$=$	3168.1585
Dual form			3168.2.f.h.1585.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.752075i q^{5} +3.95740 q^{7} -1.00000i q^{11} +0.420127i q^{13} -3.06410 q^{17} -7.03114i q^{19} +3.63326 q^{23} +4.43438 q^{25} -2.98923i q^{29} +2.93853 q^{31} +2.97626i q^{35} +1.20826i q^{37} -10.2443 q^{41} -10.2443i q^{43} +11.1993 q^{47} +8.66098 q^{49} -8.38101i q^{53} +0.752075 q^{55} +1.21830i q^{59} -3.86288i q^{61} -0.315967 q^{65} +7.79081i q^{67} -6.91633 q^{71} -2.76017 q^{73} -3.95740i q^{77} +6.67984 q^{79} +14.8318i q^{83} -2.30443i q^{85} +11.7579 q^{89} +1.66261i q^{91} +5.28795 q^{95} +3.22439 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.752075i	0.336338i	0.985758	+	0.168169i	$0.0537855\pi$
$5$	0.752075i	0.336338i	−0.985758	+	0.168169i	$0.946215\pi$
$6$	0	0
$7$	3.95740	1.49575	0.747877	−	0.663837i	$-0.231073\pi$
$7$	3.95740	1.49575	0.747877	+	0.663837i	$0.231073\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	0.420127i	0.116522i	0.998301	+	0.0582612i	$0.0185556\pi$
$13$	0.420127i	0.116522i	−0.998301	+	0.0582612i	$0.981444\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.06410	−0.743154	−0.371577	−	0.928402i	$-0.621183\pi$
$17$	−3.06410	−0.743154	−0.371577	+	0.928402i	$0.621183\pi$
$18$	0	0
$19$	− 7.03114i	− 1.61306i	−0.591197	−	0.806528i	$-0.701344\pi$
$19$	− 7.03114i	− 1.61306i	0.591197	−	0.806528i	$-0.298656\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.63326	0.757588	0.378794	−	0.925481i	$-0.376339\pi$
$23$	3.63326	0.757588	0.378794	+	0.925481i	$0.376339\pi$
$24$	0	0
$25$	4.43438	0.886877
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.98923i	− 0.555087i	−0.960713	−	0.277543i	$-0.910480\pi$
$29$	− 2.98923i	− 0.555087i	0.960713	−	0.277543i	$-0.0895202\pi$
$30$	0	0
$31$	2.93853	0.527776	0.263888	−	0.964553i	$-0.414995\pi$
$31$	2.93853	0.527776	0.263888	+	0.964553i	$0.414995\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.97626i	0.503079i
$36$	0	0
$37$	1.20826i	0.198637i	0.995056	+	0.0993183i	$0.0316662\pi$
$37$	1.20826i	0.198637i	−0.995056	+	0.0993183i	$0.968334\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.2443	−1.59989	−0.799944	−	0.600075i	$-0.795137\pi$
$41$	−10.2443	−1.59989	−0.799944	+	0.600075i	$0.795137\pi$
$42$	0	0
$43$	− 10.2443i	− 1.56224i	−0.624382	−	0.781119i	$-0.714649\pi$
$43$	− 10.2443i	− 1.56224i	0.624382	−	0.781119i	$-0.285351\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.1993	1.63359	0.816796	−	0.576926i	$-0.195748\pi$
$47$	11.1993	1.63359	0.816796	+	0.576926i	$0.195748\pi$
$48$	0	0
$49$	8.66098	1.23728
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.38101i	− 1.15122i	−0.817724	−	0.575610i	$-0.804765\pi$
$53$	− 8.38101i	− 1.15122i	0.817724	−	0.575610i	$-0.195235\pi$
$54$	0	0
$55$	0.752075	0.101410
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.21830i	0.158609i	0.996850	+	0.0793044i	$0.0252699\pi$
$59$	1.21830i	0.158609i	−0.996850	+	0.0793044i	$0.974730\pi$
$60$	0	0
$61$	− 3.86288i	− 0.494592i	−0.968940	−	0.247296i	$-0.920458\pi$
$61$	− 3.86288i	− 0.494592i	0.968940	−	0.247296i	$-0.0795419\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.315967	−0.0391909
$66$	0	0
$67$	7.79081i	0.951799i	0.879500	+	0.475900i	$0.157877\pi$
$67$	7.79081i	0.951799i	−0.879500	+	0.475900i	$0.842123\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.91633	−0.820818	−0.410409	−	0.911902i	$-0.634614\pi$
$71$	−6.91633	−0.820818	−0.410409	+	0.911902i	$0.634614\pi$
$72$	0	0
$73$	−2.76017	−0.323054	−0.161527	−	0.986868i	$-0.551642\pi$
$73$	−2.76017	−0.323054	−0.161527	+	0.986868i	$0.551642\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.95740i	− 0.450987i
$78$	0	0
$79$	6.67984	0.751541	0.375770	−	0.926713i	$-0.377378\pi$
$79$	6.67984	0.751541	0.375770	+	0.926713i	$0.377378\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.8318i	1.62800i	0.580864	+	0.814000i	$0.302715\pi$
$83$	14.8318i	1.62800i	−0.580864	+	0.814000i	$0.697285\pi$
$84$	0	0
$85$	− 2.30443i	− 0.249951i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.7579	1.24633	0.623165	−	0.782090i	$-0.285846\pi$
$89$	11.7579	1.24633	0.623165	+	0.782090i	$0.285846\pi$
$90$	0	0
$91$	1.66261i	0.174289i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.28795	0.542532
$96$	0	0
$97$	3.22439	0.327387	0.163693	−	0.986511i	$-0.447659\pi$
$97$	3.22439	0.327387	0.163693	+	0.986511i	$0.447659\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.50662i	0.348921i	0.984664	+	0.174461i	$0.0558182\pi$
$101$	3.50662i	0.348921i	−0.984664	+	0.174461i	$0.944182\pi$
$102$	0	0
$103$	8.90081	0.877023	0.438511	−	0.898726i	$-0.355506\pi$
$103$	8.90081	0.877023	0.438511	+	0.898726i	$0.355506\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.2877i	− 1.57459i	−0.616575	−	0.787296i	$-0.711480\pi$
$107$	− 16.2877i	− 1.57459i	0.616575	−	0.787296i	$-0.288520\pi$
$108$	0	0
$109$	15.3911i	1.47420i	0.675783	+	0.737101i	$0.263806\pi$
$109$	15.3911i	1.47420i	−0.675783	+	0.737101i	$0.736194\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.8277	−1.20673	−0.603366	−	0.797465i	$-0.706174\pi$
$113$	−12.8277	−1.20673	−0.603366	+	0.797465i	$0.706174\pi$
$114$	0	0
$115$	2.73249i	0.254806i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.1259	−1.11158
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.09536i	0.634629i
$126$	0	0
$127$	−11.1542	−0.989772	−0.494886	−	0.868958i	$-0.664790\pi$
$127$	−11.1542	−0.989772	−0.494886	+	0.868958i	$0.664790\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.8450i	1.20965i	0.796360	+	0.604823i	$0.206756\pi$
$131$	13.8450i	1.20965i	−0.796360	+	0.604823i	$0.793244\pi$
$132$	0	0
$133$	− 27.8250i	− 2.41274i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.21313	−0.274517	−0.137258	−	0.990535i	$-0.543829\pi$
$137$	−3.21313	−0.274517	−0.137258	+	0.990535i	$0.543829\pi$
$138$	0	0
$139$	− 2.76943i	− 0.234900i	−0.993079	−	0.117450i	$-0.962528\pi$
$139$	− 2.76943i	− 0.234900i	0.993079	−	0.117450i	$-0.0374720\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.420127	0.0351328
$144$	0	0
$145$	2.24813	0.186697
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.44736i	0.610111i	0.952335	+	0.305056i	$0.0986751\pi$
$149$	7.44736i	0.610111i	−0.952335	+	0.305056i	$0.901325\pi$
$150$	0	0
$151$	6.91137	0.562439	0.281220	−	0.959643i	$-0.409261\pi$
$151$	6.91137	0.562439	0.281220	+	0.959643i	$0.409261\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.21000i	0.177511i
$156$	0	0
$157$	− 19.3502i	− 1.54432i	−0.635430	−	0.772158i	$-0.719177\pi$
$157$	− 19.3502i	− 1.54432i	0.635430	−	0.772158i	$-0.280823\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.3783	1.13317
$162$	0	0
$163$	− 7.49615i	− 0.587143i	−0.955937	−	0.293572i	$-0.905156\pi$
$163$	− 7.49615i	− 0.587143i	0.955937	−	0.293572i	$-0.0948439\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.5371	1.66659	0.833297	−	0.552826i	$-0.186451\pi$
$167$	21.5371	1.66659	0.833297	+	0.552826i	$0.186451\pi$
$168$	0	0
$169$	12.8235	0.986423
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 23.2769i	− 1.76971i	−0.465864	−	0.884857i	$-0.654256\pi$
$173$	− 23.2769i	− 1.76971i	0.465864	−	0.884857i	$-0.345744\pi$
$174$	0	0
$175$	17.5486	1.32655
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 0.559527i	− 0.0418210i	−0.999781	−	0.0209105i	$-0.993343\pi$
$179$	− 0.559527i	− 0.0418210i	0.999781	−	0.0209105i	$-0.00665651\pi$
$180$	0	0
$181$	− 5.51757i	− 0.410118i	−0.978750	−	0.205059i	$-0.934261\pi$
$181$	− 5.51757i	− 0.410118i	0.978750	−	0.205059i	$-0.0657386\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.908701	−0.0668091
$186$	0	0
$187$	3.06410i	0.224069i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.51884	0.688759	0.344379	−	0.938831i	$-0.388089\pi$
$191$	9.51884	0.688759	0.344379	+	0.938831i	$0.388089\pi$
$192$	0	0
$193$	23.4751	1.68978	0.844888	−	0.534943i	$-0.179667\pi$
$193$	23.4751	1.68978	0.844888	+	0.534943i	$0.179667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8565i	1.48597i	0.669310	+	0.742984i	$0.266590\pi$
$197$	20.8565i	1.48597i	−0.669310	+	0.742984i	$0.733410\pi$
$198$	0	0
$199$	5.94683	0.421560	0.210780	−	0.977534i	$-0.432400\pi$
$199$	5.94683	0.421560	0.210780	+	0.977534i	$0.432400\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 11.8296i	− 0.830274i
$204$	0	0
$205$	− 7.70447i	− 0.538103i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.03114	−0.486354
$210$	0	0
$211$	− 12.1108i	− 0.833740i	−0.908966	−	0.416870i	$-0.863127\pi$
$211$	− 12.1108i	− 0.833740i	0.908966	−	0.416870i	$-0.136873\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.70447	0.525440
$216$	0	0
$217$	11.6289	0.789424
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.28731i	− 0.0865940i
$222$	0	0
$223$	18.2816	1.22423	0.612114	−	0.790770i	$-0.290319\pi$
$223$	18.2816	1.22423	0.612114	+	0.790770i	$0.290319\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	13.4647i	0.889770i	0.895588	+	0.444885i	$0.146755\pi$
$229$	13.4647i	0.889770i	−0.895588	+	0.444885i	$0.853245\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.8803	−1.17138	−0.585689	−	0.810536i	$-0.699176\pi$
$233$	−17.8803	−1.17138	−0.585689	+	0.810536i	$0.699176\pi$
$234$	0	0
$235$	8.42275i	0.549439i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.0061	−1.68220	−0.841098	−	0.540882i	$-0.818090\pi$
$239$	−26.0061	−1.68220	−0.841098	+	0.540882i	$0.818090\pi$
$240$	0	0
$241$	0.248128	0.0159833	0.00799167	−	0.999968i	$-0.497456\pi$
$241$	0.248128	0.0159833	0.00799167	+	0.999968i	$0.497456\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.51371i	0.416145i
$246$	0	0
$247$	2.95398	0.187957
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 19.5599i	− 1.23461i	−0.786723	−	0.617306i	$-0.788224\pi$
$251$	− 19.5599i	− 1.23461i	0.786723	−	0.617306i	$-0.211776\pi$
$252$	0	0
$253$	− 3.63326i	− 0.228421i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.80058	0.611343	0.305672	−	0.952137i	$-0.401119\pi$
$257$	9.80058	0.611343	0.305672	+	0.952137i	$0.401119\pi$
$258$	0	0
$259$	4.78156i	0.297112i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.13832	0.0701920	0.0350960	−	0.999384i	$-0.488826\pi$
$263$	1.13832	0.0701920	0.0350960	+	0.999384i	$0.488826\pi$
$264$	0	0
$265$	6.30315	0.387199
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 13.2102i	− 0.805440i	−0.915323	−	0.402720i	$-0.868065\pi$
$269$	− 13.2102i	− 0.805440i	0.915323	−	0.402720i	$-0.131935\pi$
$270$	0	0
$271$	6.44831	0.391707	0.195854	−	0.980633i	$-0.437252\pi$
$271$	6.44831	0.391707	0.195854	+	0.980633i	$0.437252\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.43438i	− 0.267403i
$276$	0	0
$277$	16.4174i	0.986423i	0.869909	+	0.493212i	$0.164177\pi$
$277$	16.4174i	0.986423i	−0.869909	+	0.493212i	$0.835823\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.49037	0.566148	0.283074	−	0.959098i	$-0.408646\pi$
$281$	9.49037	0.566148	0.283074	+	0.959098i	$0.408646\pi$
$282$	0	0
$283$	− 3.54475i	− 0.210714i	−0.994434	−	0.105357i	$-0.966402\pi$
$283$	− 3.54475i	− 0.210714i	0.994434	−	0.105357i	$-0.0335985\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−40.5407	−2.39304
$288$	0	0
$289$	−7.61129	−0.447723
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.7811i	1.44773i	0.689943	+	0.723863i	$0.257635\pi$
$293$	24.7811i	1.44773i	−0.689943	+	0.723863i	$0.742365\pi$
$294$	0	0
$295$	−0.916251	−0.0533462
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.52643i	0.0882759i
$300$	0	0
$301$	− 40.5407i	− 2.33673i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.90518	0.166350
$306$	0	0
$307$	− 18.0530i	− 1.03034i	−0.857088	−	0.515169i	$-0.827729\pi$
$307$	− 18.0530i	− 1.03034i	0.857088	−	0.515169i	$-0.172271\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.21589	0.409176	0.204588	−	0.978848i	$-0.434415\pi$
$311$	7.21589	0.409176	0.204588	+	0.978848i	$0.434415\pi$
$312$	0	0
$313$	10.1625	0.574419	0.287210	−	0.957868i	$-0.407272\pi$
$313$	10.1625	0.574419	0.287210	+	0.957868i	$0.407272\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.2396i	− 0.687446i	−0.939071	−	0.343723i	$-0.888312\pi$
$317$	− 12.2396i	− 0.687446i	0.939071	−	0.343723i	$-0.111688\pi$
$318$	0	0
$319$	−2.98923	−0.167365
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.5441i	1.19875i
$324$	0	0
$325$	1.86301i	0.103341i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	44.3202	2.44345
$330$	0	0
$331$	20.4886i	1.12615i	0.826405	+	0.563077i	$0.190382\pi$
$331$	20.4886i	1.12615i	−0.826405	+	0.563077i	$0.809618\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.85927	−0.320126
$336$	0	0
$337$	−8.06553	−0.439358	−0.219679	−	0.975572i	$-0.570501\pi$
$337$	−8.06553	−0.439358	−0.219679	+	0.975572i	$0.570501\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.93853i	− 0.159130i
$342$	0	0
$343$	6.57315	0.354917
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.3996i	− 0.987743i	−0.869535	−	0.493872i	$-0.835581\pi$
$347$	− 18.3996i	− 0.987743i	0.869535	−	0.493872i	$-0.164419\pi$
$348$	0	0
$349$	22.4165i	1.19993i	0.800027	+	0.599964i	$0.204818\pi$
$349$	22.4165i	1.19993i	−0.800027	+	0.599964i	$0.795182\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.46509	0.184428	0.0922141	−	0.995739i	$-0.470606\pi$
$353$	3.46509	0.184428	0.0922141	+	0.995739i	$0.470606\pi$
$354$	0	0
$355$	− 5.20160i	− 0.276072i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.0676	−1.48135	−0.740677	−	0.671861i	$-0.765495\pi$
$359$	−28.0676	−1.48135	−0.740677	+	0.671861i	$0.765495\pi$
$360$	0	0
$361$	−30.4370	−1.60195
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.07586i	− 0.108655i
$366$	0	0
$367$	−31.5561	−1.64721	−0.823607	−	0.567160i	$-0.808042\pi$
$367$	−31.5561	−1.64721	−0.823607	+	0.567160i	$0.808042\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 33.1670i	− 1.72194i
$372$	0	0
$373$	− 7.28434i	− 0.377169i	−0.982057	−	0.188584i	$-0.939610\pi$
$373$	− 7.28434i	− 0.377169i	0.982057	−	0.188584i	$-0.0603899\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.25586	0.0646800
$378$	0	0
$379$	− 18.1593i	− 0.932781i	−0.884579	−	0.466391i	$-0.845554\pi$
$379$	− 18.1593i	− 0.932781i	0.884579	−	0.466391i	$-0.154446\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.8634	−0.759486	−0.379743	−	0.925092i	$-0.623988\pi$
$383$	−14.8634	−0.759486	−0.379743	+	0.925092i	$0.623988\pi$
$384$	0	0
$385$	2.97626	0.151684
$386$	0	0
$387$	0	0
$388$	0	0
$389$	35.7363i	1.81190i	0.423383	+	0.905951i	$0.360842\pi$
$389$	35.7363i	1.81190i	−0.423383	+	0.905951i	$0.639158\pi$
$390$	0	0
$391$	−11.1327	−0.563004
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.02374i	0.252772i
$396$	0	0
$397$	28.4926i	1.43000i	0.699124	+	0.715001i	$0.253574\pi$
$397$	28.4926i	1.43000i	−0.699124	+	0.715001i	$0.746426\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.1691	1.10707	0.553535	−	0.832826i	$-0.313278\pi$
$401$	22.1691	1.10707	0.553535	+	0.832826i	$0.313278\pi$
$402$	0	0
$403$	1.23456i	0.0614977i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.20826	0.0598912
$408$	0	0
$409$	−21.7865	−1.07727	−0.538637	−	0.842538i	$-0.681061\pi$
$409$	−21.7865	−1.07727	−0.538637	+	0.842538i	$0.681061\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.82128i	0.237240i
$414$	0	0
$415$	−11.1546	−0.547559
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 14.9249i	− 0.729131i	−0.931178	−	0.364566i	$-0.881217\pi$
$419$	− 14.9249i	− 0.729131i	0.931178	−	0.364566i	$-0.118783\pi$
$420$	0	0
$421$	36.7590i	1.79153i	0.444532	+	0.895763i	$0.353370\pi$
$421$	36.7590i	1.79153i	−0.444532	+	0.895763i	$0.646630\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.5874	−0.659086
$426$	0	0
$427$	− 15.2870i	− 0.739788i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.67329	0.273273	0.136636	−	0.990621i	$-0.456371\pi$
$431$	5.67329	0.273273	0.136636	+	0.990621i	$0.456371\pi$
$432$	0	0
$433$	8.16693	0.392478	0.196239	−	0.980556i	$-0.437127\pi$
$433$	8.16693	0.392478	0.196239	+	0.980556i	$0.437127\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 25.5460i	− 1.22203i
$438$	0	0
$439$	−27.0366	−1.29038	−0.645192	−	0.764020i	$-0.723223\pi$
$439$	−27.0366	−1.29038	−0.645192	+	0.764020i	$0.723223\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.6719i	1.36224i	0.732171	+	0.681121i	$0.238507\pi$
$443$	28.6719i	1.36224i	−0.732171	+	0.681121i	$0.761493\pi$
$444$	0	0
$445$	8.84279i	0.419188i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.9657	1.08382	0.541909	−	0.840437i	$-0.317702\pi$
$449$	22.9657	1.08382	0.541909	+	0.840437i	$0.317702\pi$
$450$	0	0
$451$	10.2443i	0.482384i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.25041	−0.0586200
$456$	0	0
$457$	−36.0634	−1.68697	−0.843487	−	0.537150i	$-0.819501\pi$
$457$	−36.0634	−1.68697	−0.843487	+	0.537150i	$0.819501\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 15.0485i	− 0.700878i	−0.936586	−	0.350439i	$-0.886032\pi$
$461$	− 15.0485i	− 0.700878i	0.936586	−	0.350439i	$-0.113968\pi$
$462$	0	0
$463$	0.587148	0.0272871	0.0136435	−	0.999907i	$-0.495657\pi$
$463$	0.587148	0.0272871	0.0136435	+	0.999907i	$0.495657\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.52034i	− 0.255451i	−0.991810	−	0.127726i	$-0.959232\pi$
$467$	− 5.52034i	− 0.255451i	0.991810	−	0.127726i	$-0.0407677\pi$
$468$	0	0
$469$	30.8313i	1.42366i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.2443	−0.471032
$474$	0	0
$475$	− 31.1788i	− 1.43058i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.3911	1.61706	0.808530	−	0.588454i	$-0.200263\pi$
$479$	35.3911	1.61706	0.808530	+	0.588454i	$0.200263\pi$
$480$	0	0
$481$	−0.507623	−0.0231456
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.42498i	0.110113i
$486$	0	0
$487$	−14.2395	−0.645252	−0.322626	−	0.946527i	$-0.604566\pi$
$487$	−14.2395	−0.645252	−0.322626	+	0.946527i	$0.604566\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	23.2206i	1.04793i	0.851740	+	0.523965i	$0.175548\pi$
$491$	23.2206i	1.04793i	−0.851740	+	0.523965i	$0.824452\pi$
$492$	0	0
$493$	9.15932i	0.412515i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.3707	−1.22774
$498$	0	0
$499$	34.4110i	1.54045i	0.637774	+	0.770223i	$0.279855\pi$
$499$	34.4110i	1.54045i	−0.637774	+	0.770223i	$0.720145\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.0641	0.448734	0.224367	−	0.974505i	$-0.427969\pi$
$503$	10.0641	0.448734	0.224367	+	0.974505i	$0.427969\pi$
$504$	0	0
$505$	−2.63724	−0.117356
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.13933i	− 0.0504999i	−0.999681	−	0.0252500i	$-0.991962\pi$
$509$	− 1.13933i	− 0.0504999i	0.999681	−	0.0252500i	$-0.00803817\pi$
$510$	0	0
$511$	−10.9231	−0.483209
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.69407i	0.294976i
$516$	0	0
$517$	− 11.1993i	− 0.492547i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.4828	0.678315	0.339158	−	0.940730i	$-0.389858\pi$
$521$	15.4828	0.678315	0.339158	+	0.940730i	$0.389858\pi$
$522$	0	0
$523$	27.4076i	1.19845i	0.800580	+	0.599226i	$0.204525\pi$
$523$	27.4076i	1.19845i	−0.800580	+	0.599226i	$0.795475\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.00396	−0.392219
$528$	0	0
$529$	−9.79941	−0.426061
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.30390i	− 0.186423i
$534$	0	0
$535$	12.2496	0.529595
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 8.66098i	− 0.373055i
$540$	0	0
$541$	25.6902i	1.10451i	0.833675	+	0.552255i	$0.186232\pi$
$541$	25.6902i	1.10451i	−0.833675	+	0.552255i	$0.813768\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.5753	−0.495830
$546$	0	0
$547$	24.4926i	1.04723i	0.851956	+	0.523613i	$0.175416\pi$
$547$	24.4926i	1.04723i	−0.851956	+	0.523613i	$0.824584\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.0177	−0.895386
$552$	0	0
$553$	26.4348	1.12412
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.22599i	0.263803i	0.991263	+	0.131902i	$0.0421083\pi$
$557$	6.22599i	0.263803i	−0.991263	+	0.131902i	$0.957892\pi$
$558$	0	0
$559$	4.30390	0.182036
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 16.4963i	− 0.695234i	−0.937637	−	0.347617i	$-0.886991\pi$
$563$	− 16.4963i	− 0.695234i	0.937637	−	0.347617i	$-0.113009\pi$
$564$	0	0
$565$	− 9.64741i	− 0.405870i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.2197	−1.47649	−0.738244	−	0.674534i	$-0.764345\pi$
$569$	−35.2197	−1.47649	−0.738244	+	0.674534i	$0.764345\pi$
$570$	0	0
$571$	29.0142i	1.21421i	0.794623	+	0.607104i	$0.207669\pi$
$571$	29.0142i	1.21421i	−0.794623	+	0.607104i	$0.792331\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.1113	0.671887
$576$	0	0
$577$	−43.5942	−1.81485	−0.907426	−	0.420213i	$-0.861955\pi$
$577$	−43.5942	−1.81485	−0.907426	+	0.420213i	$0.861955\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	58.6953i	2.43509i
$582$	0	0
$583$	−8.38101	−0.347106
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.3405i	1.37611i	0.725658	+	0.688055i	$0.241535\pi$
$587$	33.3405i	1.37611i	−0.725658	+	0.688055i	$0.758465\pi$
$588$	0	0
$589$	− 20.6613i	− 0.851332i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.9166	−0.653618	−0.326809	−	0.945090i	$-0.605973\pi$
$593$	−15.9166	−0.653618	−0.326809	+	0.945090i	$0.605973\pi$
$594$	0	0
$595$	− 9.11955i	− 0.373865i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.7388	−1.09252	−0.546259	−	0.837616i	$-0.683949\pi$
$599$	−26.7388	−1.09252	−0.546259	+	0.837616i	$0.683949\pi$
$600$	0	0
$601$	−26.4502	−1.07893	−0.539464	−	0.842009i	$-0.681373\pi$
$601$	−26.4502	−1.07893	−0.539464	+	0.842009i	$0.681373\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.752075i	− 0.0305762i
$606$	0	0
$607$	11.9898	0.486650	0.243325	−	0.969945i	$-0.421762\pi$
$607$	11.9898	0.486650	0.243325	+	0.969945i	$0.421762\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.70515i	0.190350i
$612$	0	0
$613$	− 47.7774i	− 1.92971i	−0.262781	−	0.964856i	$-0.584640\pi$
$613$	− 47.7774i	− 1.92971i	0.262781	−	0.964856i	$-0.415360\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.14149	−0.247247	−0.123624	−	0.992329i	$-0.539452\pi$
$617$	−6.14149	−0.247247	−0.123624	+	0.992329i	$0.539452\pi$
$618$	0	0
$619$	− 16.5082i	− 0.663522i	−0.943364	−	0.331761i	$-0.892357\pi$
$619$	− 16.5082i	− 0.663522i	0.943364	−	0.331761i	$-0.107643\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	46.5305	1.86420
$624$	0	0
$625$	16.8357	0.673427
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.70223i	− 0.147617i
$630$	0	0
$631$	−25.7334	−1.02443	−0.512214	−	0.858858i	$-0.671175\pi$
$631$	−25.7334	−1.02443	−0.512214	+	0.858858i	$0.671175\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 8.38876i	− 0.332898i
$636$	0	0
$637$	3.63871i	0.144171i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.10690	−0.241208	−0.120604	−	0.992701i	$-0.538483\pi$
$641$	−6.10690	−0.241208	−0.120604	+	0.992701i	$0.538483\pi$
$642$	0	0
$643$	− 5.49967i	− 0.216886i	−0.994103	−	0.108443i	$-0.965414\pi$
$643$	− 5.49967i	− 0.216886i	0.994103	−	0.108443i	$-0.0345865\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.3248	1.27082	0.635410	−	0.772175i	$-0.280831\pi$
$647$	32.3248	1.27082	0.635410	+	0.772175i	$0.280831\pi$
$648$	0	0
$649$	1.21830	0.0478223
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 27.5680i	− 1.07882i	−0.842043	−	0.539410i	$-0.818647\pi$
$653$	− 27.5680i	− 1.07882i	0.842043	−	0.539410i	$-0.181353\pi$
$654$	0	0
$655$	−10.4125	−0.406850
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 40.6890i	− 1.58502i	−0.609860	−	0.792509i	$-0.708774\pi$
$659$	− 40.6890i	− 1.58502i	0.609860	−	0.792509i	$-0.291226\pi$
$660$	0	0
$661$	− 21.4900i	− 0.835864i	−0.908478	−	0.417932i	$-0.862755\pi$
$661$	− 21.4900i	− 0.835864i	0.908478	−	0.417932i	$-0.137245\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.9265	0.811495
$666$	0	0
$667$	− 10.8607i	− 0.420527i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.86288	−0.149125
$672$	0	0
$673$	−35.5881	−1.37182	−0.685911	−	0.727686i	$-0.740596\pi$
$673$	−35.5881	−1.37182	−0.685911	+	0.727686i	$0.740596\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.5364i	0.635547i	0.948167	+	0.317774i	$0.102935\pi$
$677$	16.5364i	0.635547i	−0.948167	+	0.317774i	$0.897065\pi$
$678$	0	0
$679$	12.7602	0.489690
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 29.8578i	− 1.14248i	−0.820784	−	0.571238i	$-0.806463\pi$
$683$	− 29.8578i	− 1.14248i	0.820784	−	0.571238i	$-0.193537\pi$
$684$	0	0
$685$	− 2.41652i	− 0.0923304i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.52109	0.134143
$690$	0	0
$691$	− 0.926601i	− 0.0352496i	−0.999845	−	0.0176248i	$-0.994390\pi$
$691$	− 0.926601i	− 0.0352496i	0.999845	−	0.0176248i	$-0.00561043\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.08282	0.0790060
$696$	0	0
$697$	31.3895	1.18896
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 37.2773i	− 1.40795i	−0.710227	−	0.703973i	$-0.751408\pi$
$701$	− 37.2773i	− 1.40795i	0.710227	−	0.703973i	$-0.248592\pi$
$702$	0	0
$703$	8.49544	0.320412
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8771i	0.521901i
$708$	0	0
$709$	− 2.42953i	− 0.0912428i	−0.998959	−	0.0456214i	$-0.985473\pi$
$709$	− 2.42953i	− 0.0912428i	0.998959	−	0.0456214i	$-0.0145268\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.6765	0.399837
$714$	0	0
$715$	0.315967i	0.0118165i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.4107	0.686604	0.343302	−	0.939225i	$-0.388455\pi$
$719$	18.4107	0.686604	0.343302	+	0.939225i	$0.388455\pi$
$720$	0	0
$721$	35.2240	1.31181
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 13.2554i	− 0.492294i
$726$	0	0
$727$	−51.0509	−1.89337	−0.946686	−	0.322157i	$-0.895592\pi$
$727$	−51.0509	−1.89337	−0.946686	+	0.322157i	$0.895592\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31.3895i	1.16098i
$732$	0	0
$733$	19.4009i	0.716587i	0.933609	+	0.358294i	$0.116641\pi$
$733$	19.4009i	0.716587i	−0.933609	+	0.358294i	$0.883359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.79081	0.286978
$738$	0	0
$739$	− 7.20384i	− 0.264997i	−0.991183	−	0.132499i	$-0.957700\pi$
$739$	− 7.20384i	− 0.264997i	0.991183	−	0.132499i	$-0.0423000\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.11702	−0.0409795	−0.0204898	−	0.999790i	$-0.506523\pi$
$743$	−1.11702	−0.0409795	−0.0204898	+	0.999790i	$0.506523\pi$
$744$	0	0
$745$	−5.60097	−0.205204
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 64.4569i	− 2.35520i
$750$	0	0
$751$	−1.59011	−0.0580239	−0.0290120	−	0.999579i	$-0.509236\pi$
$751$	−1.59011	−0.0580239	−0.0290120	+	0.999579i	$0.509236\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.19787i	0.189170i
$756$	0	0
$757$	26.6950i	0.970246i	0.874446	+	0.485123i	$0.161225\pi$
$757$	26.6950i	0.970246i	−0.874446	+	0.485123i	$0.838775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.54477	−0.0559977	−0.0279989	−	0.999608i	$-0.508913\pi$
$761$	−1.54477	−0.0559977	−0.0279989	+	0.999608i	$0.508913\pi$
$762$	0	0
$763$	60.9087i	2.20504i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.511840	−0.0184815
$768$	0	0
$769$	−14.5898	−0.526120	−0.263060	−	0.964779i	$-0.584732\pi$
$769$	−14.5898	−0.526120	−0.263060	+	0.964779i	$0.584732\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.8370i	1.61267i	0.591456	+	0.806337i	$0.298553\pi$
$773$	44.8370i	1.61267i	−0.591456	+	0.806337i	$0.701447\pi$
$774$	0	0
$775$	13.0306	0.468072
$776$	0	0
$777$	0	0
$778$	0	0
$779$	72.0290i	2.58071i
$780$	0	0
$781$	6.91633i	0.247486i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.5528	0.519413
$786$	0	0
$787$	46.6483i	1.66283i	0.555650	+	0.831417i	$0.312470\pi$
$787$	46.6483i	1.66283i	−0.555650	+	0.831417i	$0.687530\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−50.7644	−1.80497
$792$	0	0
$793$	1.62290	0.0576310
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.7608i	0.522853i	0.965223	+	0.261427i	$0.0841930\pi$
$797$	14.7608i	0.522853i	−0.965223	+	0.261427i	$0.915807\pi$
$798$	0	0
$799$	−34.3159	−1.21401
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.76017i	0.0974043i
$804$	0	0
$805$	10.8135i	0.381127i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29.3649	−1.03241	−0.516207	−	0.856464i	$-0.672657\pi$
$809$	−29.3649	−1.03241	−0.516207	+	0.856464i	$0.672657\pi$
$810$	0	0
$811$	17.8918i	0.628267i	0.949379	+	0.314133i	$0.101714\pi$
$811$	17.8918i	0.628267i	−0.949379	+	0.314133i	$0.898286\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.63766	0.197479
$816$	0	0
$817$	−72.0290	−2.51998
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.16499i	0.180259i	0.995930	+	0.0901297i	$0.0287282\pi$
$821$	5.16499i	0.180259i	−0.995930	+	0.0901297i	$0.971272\pi$
$822$	0	0
$823$	10.4520	0.364335	0.182168	−	0.983268i	$-0.441689\pi$
$823$	10.4520	0.364335	0.182168	+	0.983268i	$0.441689\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 15.8284i	− 0.550408i	−0.961386	−	0.275204i	$-0.911255\pi$
$827$	− 15.8284i	− 0.550408i	0.961386	−	0.275204i	$-0.0887454\pi$
$828$	0	0
$829$	− 42.2896i	− 1.46878i	−0.678728	−	0.734390i	$-0.737469\pi$
$829$	− 42.2896i	− 1.46878i	0.678728	−	0.734390i	$-0.262531\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.5381	−0.919491
$834$	0	0
$835$	16.1975i	0.560539i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.3612	0.979137	0.489569	−	0.871965i	$-0.337154\pi$
$839$	28.3612	0.979137	0.489569	+	0.871965i	$0.337154\pi$
$840$	0	0
$841$	20.0645	0.691879
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.64423i	0.331772i
$846$	0	0
$847$	−3.95740	−0.135978
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.38992i	0.150485i
$852$	0	0
$853$	− 23.3645i	− 0.799986i	−0.916518	−	0.399993i	$-0.869013\pi$
$853$	− 23.3645i	− 0.799986i	0.916518	−	0.399993i	$-0.130987\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.6593	−1.76465	−0.882325	−	0.470641i	$-0.844023\pi$
$857$	−51.6593	−1.76465	−0.882325	+	0.470641i	$0.844023\pi$
$858$	0	0
$859$	− 31.1517i	− 1.06288i	−0.847095	−	0.531442i	$-0.821650\pi$
$859$	− 31.1517i	− 1.06288i	0.847095	−	0.531442i	$-0.178350\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.4557	0.832481	0.416241	−	0.909254i	$-0.363347\pi$
$863$	24.4557	0.832481	0.416241	+	0.909254i	$0.363347\pi$
$864$	0	0
$865$	17.5060	0.595222
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 6.67984i	− 0.226598i
$870$	0	0
$871$	−3.27313	−0.110906
$872$	0	0
$873$	0	0
$874$	0	0
$875$	28.0792i	0.949249i
$876$	0	0
$877$	29.1117i	0.983032i	0.870868	+	0.491516i	$0.163557\pi$
$877$	29.1117i	0.983032i	−0.870868	+	0.491516i	$0.836443\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17.8993	−0.603044	−0.301522	−	0.953459i	$-0.597495\pi$
$881$	−17.8993	−0.603044	−0.301522	+	0.953459i	$0.597495\pi$
$882$	0	0
$883$	12.3997i	0.417282i	0.977992	+	0.208641i	$0.0669041\pi$
$883$	12.3997i	0.417282i	−0.977992	+	0.208641i	$0.933096\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.7105	1.19904	0.599520	−	0.800360i	$-0.295358\pi$
$887$	35.7105	1.19904	0.599520	+	0.800360i	$0.295358\pi$
$888$	0	0
$889$	−44.1414	−1.48046
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 78.7442i	− 2.63507i
$894$	0	0
$895$	0.420807	0.0140660
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 8.78396i	− 0.292962i
$900$	0	0
$901$	25.6803i	0.855534i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.14962	0.137938
$906$	0	0
$907$	55.1827i	1.83231i	0.400823	+	0.916155i	$0.368724\pi$
$907$	55.1827i	1.83231i	−0.400823	+	0.916155i	$0.631276\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.99434	−0.198601	−0.0993007	−	0.995057i	$-0.531661\pi$
$911$	−5.99434	−0.198601	−0.0993007	+	0.995057i	$0.531661\pi$
$912$	0	0
$913$	14.8318	0.490861
$914$	0	0
$915$	0	0
$916$	0	0
$917$	54.7902i	1.80933i
$918$	0	0
$919$	−30.7620	−1.01474	−0.507372	−	0.861727i	$-0.669383\pi$
$919$	−30.7620	−1.01474	−0.507372	+	0.861727i	$0.669383\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2.90574i	− 0.0956436i
$924$	0	0
$925$	5.35788i	0.176166i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.6017	−1.00401	−0.502005	−	0.864865i	$-0.667404\pi$
$929$	−30.6017	−1.00401	−0.502005	+	0.864865i	$0.667404\pi$
$930$	0	0
$931$	− 60.8966i	− 1.99581i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.30443	−0.0753630
$936$	0	0
$937$	−0.804683	−0.0262879	−0.0131439	−	0.999914i	$-0.504184\pi$
$937$	−0.804683	−0.0262879	−0.0131439	+	0.999914i	$0.504184\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43.1276i	− 1.40592i	−0.711229	−	0.702961i	$-0.751861\pi$
$941$	− 43.1276i	− 1.40592i	0.711229	−	0.702961i	$-0.248139\pi$
$942$	0	0
$943$	−37.2202	−1.21205
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 40.1863i	− 1.30588i	−0.757410	−	0.652940i	$-0.773535\pi$
$947$	− 40.1863i	− 1.30588i	0.757410	−	0.652940i	$-0.226465\pi$
$948$	0	0
$949$	− 1.15962i	− 0.0376430i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51.9932	1.68423	0.842113	−	0.539302i	$-0.181312\pi$
$953$	51.9932	1.68423	0.842113	+	0.539302i	$0.181312\pi$
$954$	0	0
$955$	7.15888i	0.231656i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.7156	−0.410610
$960$	0	0
$961$	−22.3650	−0.721452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.6550i	0.568336i
$966$	0	0
$967$	20.0086	0.643435	0.321717	−	0.946836i	$-0.395740\pi$
$967$	20.0086	0.643435	0.321717	+	0.946836i	$0.395740\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.4880i	0.432849i	0.976299	+	0.216425i	$0.0694395\pi$
$971$	13.4880i	0.432849i	−0.976299	+	0.216425i	$0.930560\pi$
$972$	0	0
$973$	− 10.9597i	− 0.351353i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.1101	0.899321	0.449661	−	0.893200i	$-0.351545\pi$
$977$	28.1101	0.899321	0.449661	+	0.893200i	$0.351545\pi$
$978$	0	0
$979$	− 11.7579i	− 0.375783i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.3523	1.86115	0.930574	−	0.366104i	$-0.119308\pi$
$983$	58.3523	1.86115	0.930574	+	0.366104i	$0.119308\pi$
$984$	0	0
$985$	−15.6857	−0.499787
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 37.2202i	− 1.18353i
$990$	0	0
$991$	−6.41139	−0.203665	−0.101832	−	0.994802i	$-0.532471\pi$
$991$	−6.41139	−0.203665	−0.101832	+	0.994802i	$0.532471\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.47246i	0.141787i
$996$	0	0
$997$	41.5619i	1.31628i	0.752896	+	0.658140i	$0.228656\pi$
$997$	41.5619i	1.31628i	−0.752896	+	0.658140i	$0.771344\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.f.h.1585.12		20
3.2	odd	2	inner	3168.2.f.h.1585.10		20
4.3	odd	2		792.2.f.h.397.13	yes	20
8.3	odd	2		792.2.f.h.397.14	yes	20
8.5	even	2	inner	3168.2.f.h.1585.9		20
12.11	even	2		792.2.f.h.397.8	yes	20
24.5	odd	2	inner	3168.2.f.h.1585.11		20
24.11	even	2		792.2.f.h.397.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.f.h.397.7	✓	20	24.11	even	2
792.2.f.h.397.8	yes	20	12.11	even	2
792.2.f.h.397.13	yes	20	4.3	odd	2
792.2.f.h.397.14	yes	20	8.3	odd	2
3168.2.f.h.1585.9		20	8.5	even	2	inner
3168.2.f.h.1585.10		20	3.2	odd	2	inner
3168.2.f.h.1585.11		20	24.5	odd	2	inner
3168.2.f.h.1585.12		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.752075i q^{5} +3.95740 q^{7} -1.00000i q^{11} +0.420127i q^{13} -3.06410 q^{17} -7.03114i q^{19} +3.63326 q^{23} +4.43438 q^{25} -2.98923i q^{29} +2.93853 q^{31} +2.97626i q^{35} +1.20826i q^{37} -10.2443 q^{41} -10.2443i q^{43} +11.1993 q^{47} +8.66098 q^{49} -8.38101i q^{53} +0.752075 q^{55} +1.21830i q^{59} -3.86288i q^{61} -0.315967 q^{65} +7.79081i q^{67} -6.91633 q^{71} -2.76017 q^{73} -3.95740i q^{77} +6.67984 q^{79} +14.8318i q^{83} -2.30443i q^{85} +11.7579 q^{89} +1.66261i q^{91} +5.28795 q^{95} +3.22439 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{25} - 40 q^{31} + 36 q^{49} + 16 q^{55} - 56 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 20 * q - 12 * q^25 - 40 * q^31 + 36 * q^49 + 16 * q^55 - 56 * q^73 + 16 * q^79 + 8 * q^97

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