LMFDB - Embedded newform 8664.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8664,2,Mod(1,8664)] 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("8664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.1823883112$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.06418 q^{5} -2.06418 q^{7} +1.00000 q^{9} -6.45336 q^{11} +1.00000 q^{13} -3.06418 q^{15} +1.38919 q^{17} +2.06418 q^{21} +3.06418 q^{23} +4.38919 q^{25} -1.00000 q^{27} -3.51754 q^{29} +9.45336 q^{31} +6.45336 q^{33} -6.32501 q^{35} -2.38919 q^{37} -1.00000 q^{39} +10.1284 q^{41} +6.06418 q^{43} +3.06418 q^{45} -6.00000 q^{47} -2.73917 q^{49} -1.38919 q^{51} -10.5817 q^{53} -19.7743 q^{55} -11.1925 q^{59} +5.12836 q^{61} -2.06418 q^{63} +3.06418 q^{65} +3.45336 q^{67} -3.06418 q^{69} +6.73917 q^{71} -9.12836 q^{73} -4.38919 q^{75} +13.3209 q^{77} -1.58172 q^{79} +1.00000 q^{81} -17.6459 q^{83} +4.25671 q^{85} +3.51754 q^{87} +10.4534 q^{89} -2.06418 q^{91} -9.45336 q^{93} +6.73917 q^{97} -6.45336 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 3 q^{21} + 9 q^{25} - 3 q^{27} + 12 q^{29} + 15 q^{31} + 6 q^{33} - 24 q^{35} - 3 q^{37} - 3 q^{39} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 3 * q^21 + 9 * q^25 - 3 * q^27 + 12 * q^29 + 15 * q^31 + 6 * q^33 - 24 * q^35 - 3 * q^37 - 3 * q^39 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 - 6 * q^59 - 3 * q^61 + 3 * q^63 - 3 * q^67 + 6 * q^71 - 9 * q^73 - 9 * q^75 - 6 * q^77 + 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 + 18 * q^89 + 3 * q^91 - 15 * q^93 + 6 * q^97 - 6 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.06418	1.37034	0.685171	−	0.728382i	$-0.259727\pi$
$5$	3.06418	1.37034	0.685171	+	0.728382i	$0.259727\pi$
$6$	0	0
$7$	−2.06418	−0.780186	−0.390093	−	0.920775i	$-0.627557\pi$
$7$	−2.06418	−0.780186	−0.390093	+	0.920775i	$0.627557\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.45336	−1.94576	−0.972881	−	0.231306i	$-0.925700\pi$
$11$	−6.45336	−1.94576	−0.972881	+	0.231306i	$0.925700\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−3.06418	−0.791167
$16$	0	0
$17$	1.38919	0.336927	0.168463	−	0.985708i	$-0.446119\pi$
$17$	1.38919	0.336927	0.168463	+	0.985708i	$0.446119\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	2.06418	0.450441
$22$	0	0
$23$	3.06418	0.638925	0.319463	−	0.947599i	$-0.396498\pi$
$23$	3.06418	0.638925	0.319463	+	0.947599i	$0.396498\pi$
$24$	0	0
$25$	4.38919	0.877837
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.51754	−0.653191	−0.326595	−	0.945164i	$-0.605901\pi$
$29$	−3.51754	−0.653191	−0.326595	+	0.945164i	$0.605901\pi$
$30$	0	0
$31$	9.45336	1.69787	0.848937	−	0.528494i	$-0.177243\pi$
$31$	9.45336	1.69787	0.848937	+	0.528494i	$0.177243\pi$
$32$	0	0
$33$	6.45336	1.12339
$34$	0	0
$35$	−6.32501	−1.06912
$36$	0	0
$37$	−2.38919	−0.392780	−0.196390	−	0.980526i	$-0.562922\pi$
$37$	−2.38919	−0.392780	−0.196390	+	0.980526i	$0.562922\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	10.1284	1.58178	0.790892	−	0.611956i	$-0.209617\pi$
$41$	10.1284	1.58178	0.790892	+	0.611956i	$0.209617\pi$
$42$	0	0
$43$	6.06418	0.924778	0.462389	−	0.886677i	$-0.346992\pi$
$43$	6.06418	0.924778	0.462389	+	0.886677i	$0.346992\pi$
$44$	0	0
$45$	3.06418	0.456781
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−2.73917	−0.391310
$50$	0	0
$51$	−1.38919	−0.194525
$52$	0	0
$53$	−10.5817	−1.45351	−0.726755	−	0.686896i	$-0.758973\pi$
$53$	−10.5817	−1.45351	−0.726755	+	0.686896i	$0.758973\pi$
$54$	0	0
$55$	−19.7743	−2.66636
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.1925	−1.45714	−0.728572	−	0.684969i	$-0.759816\pi$
$59$	−11.1925	−1.45714	−0.728572	+	0.684969i	$0.759816\pi$
$60$	0	0
$61$	5.12836	0.656619	0.328309	−	0.944570i	$-0.393521\pi$
$61$	5.12836	0.656619	0.328309	+	0.944570i	$0.393521\pi$
$62$	0	0
$63$	−2.06418	−0.260062
$64$	0	0
$65$	3.06418	0.380064
$66$	0	0
$67$	3.45336	0.421895	0.210948	−	0.977497i	$-0.432345\pi$
$67$	3.45336	0.421895	0.210948	+	0.977497i	$0.432345\pi$
$68$	0	0
$69$	−3.06418	−0.368884
$70$	0	0
$71$	6.73917	0.799792	0.399896	−	0.916560i	$-0.369046\pi$
$71$	6.73917	0.799792	0.399896	+	0.916560i	$0.369046\pi$
$72$	0	0
$73$	−9.12836	−1.06839	−0.534197	−	0.845360i	$-0.679386\pi$
$73$	−9.12836	−1.06839	−0.534197	+	0.845360i	$0.679386\pi$
$74$	0	0
$75$	−4.38919	−0.506819
$76$	0	0
$77$	13.3209	1.51806
$78$	0	0
$79$	−1.58172	−0.177957	−0.0889786	−	0.996034i	$-0.528360\pi$
$79$	−1.58172	−0.177957	−0.0889786	+	0.996034i	$0.528360\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.6459	−1.93689	−0.968444	−	0.249230i	$-0.919823\pi$
$83$	−17.6459	−1.93689	−0.968444	+	0.249230i	$0.919823\pi$
$84$	0	0
$85$	4.25671	0.461705
$86$	0	0
$87$	3.51754	0.377120
$88$	0	0
$89$	10.4534	1.10805	0.554027	−	0.832499i	$-0.313090\pi$
$89$	10.4534	1.10805	0.554027	+	0.832499i	$0.313090\pi$
$90$	0	0
$91$	−2.06418	−0.216385
$92$	0	0
$93$	−9.45336	−0.980268
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.73917	0.684259	0.342130	−	0.939653i	$-0.388852\pi$
$97$	6.73917	0.684259	0.342130	+	0.939653i	$0.388852\pi$
$98$	0	0
$99$	−6.45336	−0.648587
$100$	0	0
$101$	0.610815	0.0607783	0.0303892	−	0.999538i	$-0.490325\pi$
$101$	0.610815	0.0607783	0.0303892	+	0.999538i	$0.490325\pi$
$102$	0	0
$103$	0.0641778	0.00632362	0.00316181	−	0.999995i	$-0.498994\pi$
$103$	0.0641778	0.00632362	0.00316181	+	0.999995i	$0.498994\pi$
$104$	0	0
$105$	6.32501	0.617258
$106$	0	0
$107$	0.610815	0.0590497	0.0295248	−	0.999564i	$-0.490601\pi$
$107$	0.610815	0.0590497	0.0295248	+	0.999564i	$0.490601\pi$
$108$	0	0
$109$	4.61081	0.441636	0.220818	−	0.975315i	$-0.429127\pi$
$109$	4.61081	0.441636	0.220818	+	0.975315i	$0.429127\pi$
$110$	0	0
$111$	2.38919	0.226771
$112$	0	0
$113$	17.4884	1.64517	0.822587	−	0.568639i	$-0.192530\pi$
$113$	17.4884	1.64517	0.822587	+	0.568639i	$0.192530\pi$
$114$	0	0
$115$	9.38919	0.875546
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−2.86753	−0.262866
$120$	0	0
$121$	30.6459	2.78599
$122$	0	0
$123$	−10.1284	−0.913243
$124$	0	0
$125$	−1.87164	−0.167405
$126$	0	0
$127$	14.1284	1.25369	0.626844	−	0.779144i	$-0.284346\pi$
$127$	14.1284	1.25369	0.626844	+	0.779144i	$0.284346\pi$
$128$	0	0
$129$	−6.06418	−0.533921
$130$	0	0
$131$	5.51754	0.482070	0.241035	−	0.970516i	$-0.422513\pi$
$131$	5.51754	0.482070	0.241035	+	0.970516i	$0.422513\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.06418	−0.263722
$136$	0	0
$137$	17.7743	1.51856	0.759278	−	0.650766i	$-0.225552\pi$
$137$	17.7743	1.51856	0.759278	+	0.650766i	$0.225552\pi$
$138$	0	0
$139$	−10.9709	−0.930540	−0.465270	−	0.885169i	$-0.654043\pi$
$139$	−10.9709	−0.930540	−0.465270	+	0.885169i	$0.654043\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−6.45336	−0.539657
$144$	0	0
$145$	−10.7784	−0.895095
$146$	0	0
$147$	2.73917	0.225923
$148$	0	0
$149$	13.0642	1.07026	0.535130	−	0.844770i	$-0.320263\pi$
$149$	13.0642	1.07026	0.535130	+	0.844770i	$0.320263\pi$
$150$	0	0
$151$	4.90673	0.399304	0.199652	−	0.979867i	$-0.436019\pi$
$151$	4.90673	0.399304	0.199652	+	0.979867i	$0.436019\pi$
$152$	0	0
$153$	1.38919	0.112309
$154$	0	0
$155$	28.9668	2.32667
$156$	0	0
$157$	11.7392	0.936888	0.468444	−	0.883493i	$-0.344815\pi$
$157$	11.7392	0.936888	0.468444	+	0.883493i	$0.344815\pi$
$158$	0	0
$159$	10.5817	0.839185
$160$	0	0
$161$	−6.32501	−0.498480
$162$	0	0
$163$	−13.4534	−1.05375	−0.526874	−	0.849943i	$-0.676636\pi$
$163$	−13.4534	−1.05375	−0.526874	+	0.849943i	$0.676636\pi$
$164$	0	0
$165$	19.7743	1.53942
$166$	0	0
$167$	13.9709	1.08110	0.540551	−	0.841312i	$-0.318216\pi$
$167$	13.9709	1.08110	0.540551	+	0.841312i	$0.318216\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.51754	0.267434	0.133717	−	0.991020i	$-0.457309\pi$
$173$	3.51754	0.267434	0.133717	+	0.991020i	$0.457309\pi$
$174$	0	0
$175$	−9.06006	−0.684876
$176$	0	0
$177$	11.1925	0.841282
$178$	0	0
$179$	11.0642	0.826975	0.413488	−	0.910510i	$-0.364310\pi$
$179$	11.0642	0.826975	0.413488	+	0.910510i	$0.364310\pi$
$180$	0	0
$181$	21.5175	1.59939	0.799693	−	0.600409i	$-0.204996\pi$
$181$	21.5175	1.59939	0.799693	+	0.600409i	$0.204996\pi$
$182$	0	0
$183$	−5.12836	−0.379099
$184$	0	0
$185$	−7.32089	−0.538242
$186$	0	0
$187$	−8.96492	−0.655580
$188$	0	0
$189$	2.06418	0.150147
$190$	0	0
$191$	−15.2317	−1.10213	−0.551065	−	0.834462i	$-0.685778\pi$
$191$	−15.2317	−1.10213	−0.551065	+	0.834462i	$0.685778\pi$
$192$	0	0
$193$	6.26083	0.450664	0.225332	−	0.974282i	$-0.427653\pi$
$193$	6.26083	0.450664	0.225332	+	0.974282i	$0.427653\pi$
$194$	0	0
$195$	−3.06418	−0.219430
$196$	0	0
$197$	−6.71007	−0.478073	−0.239036	−	0.971011i	$-0.576832\pi$
$197$	−6.71007	−0.478073	−0.239036	+	0.971011i	$0.576832\pi$
$198$	0	0
$199$	−24.9317	−1.76736	−0.883681	−	0.468090i	$-0.844942\pi$
$199$	−24.9317	−1.76736	−0.883681	+	0.468090i	$0.844942\pi$
$200$	0	0
$201$	−3.45336	−0.243581
$202$	0	0
$203$	7.26083	0.509610
$204$	0	0
$205$	31.0351	2.16758
$206$	0	0
$207$	3.06418	0.212975
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.97090	0.617583	0.308791	−	0.951130i	$-0.400076\pi$
$211$	8.97090	0.617583	0.308791	+	0.951130i	$0.400076\pi$
$212$	0	0
$213$	−6.73917	−0.461760
$214$	0	0
$215$	18.5817	1.26726
$216$	0	0
$217$	−19.5134	−1.32466
$218$	0	0
$219$	9.12836	0.616837
$220$	0	0
$221$	1.38919	0.0934467
$222$	0	0
$223$	25.1385	1.68340	0.841698	−	0.539949i	$-0.181556\pi$
$223$	25.1385	1.68340	0.841698	+	0.539949i	$0.181556\pi$
$224$	0	0
$225$	4.38919	0.292612
$226$	0	0
$227$	−22.8384	−1.51584	−0.757920	−	0.652348i	$-0.773784\pi$
$227$	−22.8384	−1.51584	−0.757920	+	0.652348i	$0.773784\pi$
$228$	0	0
$229$	−6.03508	−0.398809	−0.199405	−	0.979917i	$-0.563901\pi$
$229$	−6.03508	−0.398809	−0.199405	+	0.979917i	$0.563901\pi$
$230$	0	0
$231$	−13.3209	−0.876450
$232$	0	0
$233$	4.61081	0.302065	0.151032	−	0.988529i	$-0.451740\pi$
$233$	4.61081	0.302065	0.151032	+	0.988529i	$0.451740\pi$
$234$	0	0
$235$	−18.3851	−1.19931
$236$	0	0
$237$	1.58172	0.102744
$238$	0	0
$239$	9.36009	0.605454	0.302727	−	0.953077i	$-0.402103\pi$
$239$	9.36009	0.605454	0.302727	+	0.953077i	$0.402103\pi$
$240$	0	0
$241$	−20.1634	−1.29884	−0.649421	−	0.760429i	$-0.724989\pi$
$241$	−20.1634	−1.29884	−0.649421	+	0.760429i	$0.724989\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−8.39330	−0.536229
$246$	0	0
$247$	0	0
$248$	0	0
$249$	17.6459	1.11826
$250$	0	0
$251$	12.1284	0.765535	0.382768	−	0.923845i	$-0.374971\pi$
$251$	12.1284	0.765535	0.382768	+	0.923845i	$0.374971\pi$
$252$	0	0
$253$	−19.7743	−1.24320
$254$	0	0
$255$	−4.25671	−0.266566
$256$	0	0
$257$	30.8384	1.92365	0.961824	−	0.273668i	$-0.0882370\pi$
$257$	30.8384	1.92365	0.961824	+	0.273668i	$0.0882370\pi$
$258$	0	0
$259$	4.93170	0.306441
$260$	0	0
$261$	−3.51754	−0.217730
$262$	0	0
$263$	−10.4825	−0.646376	−0.323188	−	0.946335i	$-0.604755\pi$
$263$	−10.4825	−0.646376	−0.323188	+	0.946335i	$0.604755\pi$
$264$	0	0
$265$	−32.4243	−1.99181
$266$	0	0
$267$	−10.4534	−0.639735
$268$	0	0
$269$	−1.80335	−0.109952	−0.0549760	−	0.998488i	$-0.517508\pi$
$269$	−1.80335	−0.109952	−0.0549760	+	0.998488i	$0.517508\pi$
$270$	0	0
$271$	28.2567	1.71647	0.858236	−	0.513254i	$-0.171560\pi$
$271$	28.2567	1.71647	0.858236	+	0.513254i	$0.171560\pi$
$272$	0	0
$273$	2.06418	0.124930
$274$	0	0
$275$	−28.3250	−1.70806
$276$	0	0
$277$	6.48246	0.389493	0.194747	−	0.980854i	$-0.437612\pi$
$277$	6.48246	0.389493	0.194747	+	0.980854i	$0.437612\pi$
$278$	0	0
$279$	9.45336	0.565958
$280$	0	0
$281$	5.84255	0.348537	0.174269	−	0.984698i	$-0.444244\pi$
$281$	5.84255	0.348537	0.174269	+	0.984698i	$0.444244\pi$
$282$	0	0
$283$	16.2567	0.966361	0.483181	−	0.875521i	$-0.339481\pi$
$283$	16.2567	0.966361	0.483181	+	0.875521i	$0.339481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.9067	−1.23409
$288$	0	0
$289$	−15.0702	−0.886480
$290$	0	0
$291$	−6.73917	−0.395057
$292$	0	0
$293$	26.2567	1.53393	0.766967	−	0.641687i	$-0.221765\pi$
$293$	26.2567	1.53393	0.766967	+	0.641687i	$0.221765\pi$
$294$	0	0
$295$	−34.2959	−1.99679
$296$	0	0
$297$	6.45336	0.374462
$298$	0	0
$299$	3.06418	0.177206
$300$	0	0
$301$	−12.5175	−0.721499
$302$	0	0
$303$	−0.610815	−0.0350904
$304$	0	0
$305$	15.7142	0.899792
$306$	0	0
$307$	14.2959	0.815911	0.407955	−	0.913002i	$-0.366242\pi$
$307$	14.2959	0.815911	0.407955	+	0.913002i	$0.366242\pi$
$308$	0	0
$309$	−0.0641778	−0.00365095
$310$	0	0
$311$	22.2276	1.26041	0.630206	−	0.776428i	$-0.282970\pi$
$311$	22.2276	1.26041	0.630206	+	0.776428i	$0.282970\pi$
$312$	0	0
$313$	3.13247	0.177058	0.0885290	−	0.996074i	$-0.471783\pi$
$313$	3.13247	0.177058	0.0885290	+	0.996074i	$0.471783\pi$
$314$	0	0
$315$	−6.32501	−0.356374
$316$	0	0
$317$	−6.09926	−0.342569	−0.171284	−	0.985222i	$-0.554792\pi$
$317$	−6.09926	−0.342569	−0.171284	+	0.985222i	$0.554792\pi$
$318$	0	0
$319$	22.7000	1.27095
$320$	0	0
$321$	−0.610815	−0.0340923
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.38919	0.243468
$326$	0	0
$327$	−4.61081	−0.254979
$328$	0	0
$329$	12.3851	0.682811
$330$	0	0
$331$	−1.93582	−0.106402	−0.0532012	−	0.998584i	$-0.516942\pi$
$331$	−1.93582	−0.106402	−0.0532012	+	0.998584i	$0.516942\pi$
$332$	0	0
$333$	−2.38919	−0.130927
$334$	0	0
$335$	10.5817	0.578141
$336$	0	0
$337$	−7.16756	−0.390442	−0.195221	−	0.980759i	$-0.562542\pi$
$337$	−7.16756	−0.390442	−0.195221	+	0.980759i	$0.562542\pi$
$338$	0	0
$339$	−17.4884	−0.949842
$340$	0	0
$341$	−61.0060	−3.30366
$342$	0	0
$343$	20.1034	1.08548
$344$	0	0
$345$	−9.38919	−0.505497
$346$	0	0
$347$	−9.61680	−0.516257	−0.258128	−	0.966111i	$-0.583106\pi$
$347$	−9.61680	−0.516257	−0.258128	+	0.966111i	$0.583106\pi$
$348$	0	0
$349$	31.5134	1.68687	0.843437	−	0.537227i	$-0.180528\pi$
$349$	31.5134	1.68687	0.843437	+	0.537227i	$0.180528\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−25.8425	−1.37546	−0.687730	−	0.725967i	$-0.741393\pi$
$353$	−25.8425	−1.37546	−0.687730	+	0.725967i	$0.741393\pi$
$354$	0	0
$355$	20.6500	1.09599
$356$	0	0
$357$	2.86753	0.151766
$358$	0	0
$359$	32.2567	1.70244	0.851222	−	0.524806i	$-0.175862\pi$
$359$	32.2567	1.70244	0.851222	+	0.524806i	$0.175862\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−30.6459	−1.60849
$364$	0	0
$365$	−27.9709	−1.46406
$366$	0	0
$367$	29.2668	1.52772	0.763858	−	0.645385i	$-0.223303\pi$
$367$	29.2668	1.52772	0.763858	+	0.645385i	$0.223303\pi$
$368$	0	0
$369$	10.1284	0.527261
$370$	0	0
$371$	21.8425	1.13401
$372$	0	0
$373$	8.03920	0.416254	0.208127	−	0.978102i	$-0.433263\pi$
$373$	8.03920	0.416254	0.208127	+	0.978102i	$0.433263\pi$
$374$	0	0
$375$	1.87164	0.0966513
$376$	0	0
$377$	−3.51754	−0.181163
$378$	0	0
$379$	8.19253	0.420822	0.210411	−	0.977613i	$-0.432520\pi$
$379$	8.19253	0.420822	0.210411	+	0.977613i	$0.432520\pi$
$380$	0	0
$381$	−14.1284	−0.723818
$382$	0	0
$383$	−5.48845	−0.280446	−0.140223	−	0.990120i	$-0.544782\pi$
$383$	−5.48845	−0.280446	−0.140223	+	0.990120i	$0.544782\pi$
$384$	0	0
$385$	40.8176	2.08026
$386$	0	0
$387$	6.06418	0.308259
$388$	0	0
$389$	8.45336	0.428603	0.214301	−	0.976768i	$-0.431253\pi$
$389$	8.45336	0.428603	0.214301	+	0.976768i	$0.431253\pi$
$390$	0	0
$391$	4.25671	0.215271
$392$	0	0
$393$	−5.51754	−0.278323
$394$	0	0
$395$	−4.84667	−0.243862
$396$	0	0
$397$	25.0702	1.25824	0.629118	−	0.777310i	$-0.283416\pi$
$397$	25.0702	1.25824	0.629118	+	0.777310i	$0.283416\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.7452	−1.28565	−0.642826	−	0.766012i	$-0.722238\pi$
$401$	−25.7452	−1.28565	−0.642826	+	0.766012i	$0.722238\pi$
$402$	0	0
$403$	9.45336	0.470906
$404$	0	0
$405$	3.06418	0.152260
$406$	0	0
$407$	15.4183	0.764256
$408$	0	0
$409$	−6.42427	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$409$	−6.42427	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$410$	0	0
$411$	−17.7743	−0.876739
$412$	0	0
$413$	23.1034	1.13684
$414$	0	0
$415$	−54.0702	−2.65420
$416$	0	0
$417$	10.9709	0.537247
$418$	0	0
$419$	16.2668	0.794686	0.397343	−	0.917670i	$-0.369932\pi$
$419$	16.2668	0.794686	0.397343	+	0.917670i	$0.369932\pi$
$420$	0	0
$421$	−29.4593	−1.43576	−0.717880	−	0.696166i	$-0.754888\pi$
$421$	−29.4593	−1.43576	−0.717880	+	0.696166i	$0.754888\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	6.09739	0.295767
$426$	0	0
$427$	−10.5858	−0.512285
$428$	0	0
$429$	6.45336	0.311571
$430$	0	0
$431$	−17.0351	−0.820551	−0.410276	−	0.911962i	$-0.634567\pi$
$431$	−17.0351	−0.820551	−0.410276	+	0.911962i	$0.634567\pi$
$432$	0	0
$433$	−13.6810	−0.657466	−0.328733	−	0.944423i	$-0.606622\pi$
$433$	−13.6810	−0.657466	−0.328733	+	0.944423i	$0.606622\pi$
$434$	0	0
$435$	10.7784	0.516783
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.41416	0.0674943	0.0337471	−	0.999430i	$-0.489256\pi$
$439$	1.41416	0.0674943	0.0337471	+	0.999430i	$0.489256\pi$
$440$	0	0
$441$	−2.73917	−0.130437
$442$	0	0
$443$	0.739170	0.0351190	0.0175595	−	0.999846i	$-0.494410\pi$
$443$	0.739170	0.0351190	0.0175595	+	0.999846i	$0.494410\pi$
$444$	0	0
$445$	32.0310	1.51841
$446$	0	0
$447$	−13.0642	−0.617914
$448$	0	0
$449$	7.51754	0.354775	0.177387	−	0.984141i	$-0.443235\pi$
$449$	7.51754	0.354775	0.177387	+	0.984141i	$0.443235\pi$
$450$	0	0
$451$	−65.3620	−3.07777
$452$	0	0
$453$	−4.90673	−0.230538
$454$	0	0
$455$	−6.32501	−0.296521
$456$	0	0
$457$	−15.6810	−0.733525	−0.366763	−	0.930315i	$-0.619534\pi$
$457$	−15.6810	−0.733525	−0.366763	+	0.930315i	$0.619534\pi$
$458$	0	0
$459$	−1.38919	−0.0648416
$460$	0	0
$461$	21.4884	1.00082	0.500408	−	0.865790i	$-0.333183\pi$
$461$	21.4884	1.00082	0.500408	+	0.865790i	$0.333183\pi$
$462$	0	0
$463$	−9.62092	−0.447122	−0.223561	−	0.974690i	$-0.571768\pi$
$463$	−9.62092	−0.447122	−0.223561	+	0.974690i	$0.571768\pi$
$464$	0	0
$465$	−28.9668	−1.34330
$466$	0	0
$467$	1.87164	0.0866094	0.0433047	−	0.999062i	$-0.486211\pi$
$467$	1.87164	0.0866094	0.0433047	+	0.999062i	$0.486211\pi$
$468$	0	0
$469$	−7.12836	−0.329157
$470$	0	0
$471$	−11.7392	−0.540912
$472$	0	0
$473$	−39.1343	−1.79940
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.5817	−0.484504
$478$	0	0
$479$	6.73917	0.307921	0.153960	−	0.988077i	$-0.450797\pi$
$479$	6.73917	0.307921	0.153960	+	0.988077i	$0.450797\pi$
$480$	0	0
$481$	−2.38919	−0.108937
$482$	0	0
$483$	6.32501	0.287798
$484$	0	0
$485$	20.6500	0.937669
$486$	0	0
$487$	−2.38507	−0.108078	−0.0540388	−	0.998539i	$-0.517209\pi$
$487$	−2.38507	−0.108078	−0.0540388	+	0.998539i	$0.517209\pi$
$488$	0	0
$489$	13.4534	0.608382
$490$	0	0
$491$	−10.7392	−0.484652	−0.242326	−	0.970195i	$-0.577910\pi$
$491$	−10.7392	−0.484652	−0.242326	+	0.970195i	$0.577910\pi$
$492$	0	0
$493$	−4.88652	−0.220078
$494$	0	0
$495$	−19.7743	−0.888787
$496$	0	0
$497$	−13.9108	−0.623987
$498$	0	0
$499$	28.5466	1.27792	0.638961	−	0.769239i	$-0.279364\pi$
$499$	28.5466	1.27792	0.638961	+	0.769239i	$0.279364\pi$
$500$	0	0
$501$	−13.9709	−0.624174
$502$	0	0
$503$	3.51754	0.156839	0.0784197	−	0.996920i	$-0.475013\pi$
$503$	3.51754	0.156839	0.0784197	+	0.996920i	$0.475013\pi$
$504$	0	0
$505$	1.87164	0.0832871
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	7.22163	0.320093	0.160047	−	0.987109i	$-0.448836\pi$
$509$	7.22163	0.320093	0.160047	+	0.987109i	$0.448836\pi$
$510$	0	0
$511$	18.8425	0.833545
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.196652	0.00866553
$516$	0	0
$517$	38.7202	1.70291
$518$	0	0
$519$	−3.51754	−0.154403
$520$	0	0
$521$	16.8075	0.736348	0.368174	−	0.929757i	$-0.379983\pi$
$521$	16.8075	0.736348	0.368174	+	0.929757i	$0.379983\pi$
$522$	0	0
$523$	−7.76827	−0.339683	−0.169841	−	0.985471i	$-0.554326\pi$
$523$	−7.76827	−0.339683	−0.169841	+	0.985471i	$0.554326\pi$
$524$	0	0
$525$	9.06006	0.395413
$526$	0	0
$527$	13.1325	0.572060
$528$	0	0
$529$	−13.6108	−0.591775
$530$	0	0
$531$	−11.1925	−0.485715
$532$	0	0
$533$	10.1284	0.438708
$534$	0	0
$535$	1.87164	0.0809182
$536$	0	0
$537$	−11.0642	−0.477455
$538$	0	0
$539$	17.6769	0.761396
$540$	0	0
$541$	−36.3310	−1.56199	−0.780996	−	0.624536i	$-0.785288\pi$
$541$	−36.3310	−1.56199	−0.780996	+	0.624536i	$0.785288\pi$
$542$	0	0
$543$	−21.5175	−0.923406
$544$	0	0
$545$	14.1284	0.605192
$546$	0	0
$547$	31.6209	1.35201	0.676006	−	0.736896i	$-0.263709\pi$
$547$	31.6209	1.35201	0.676006	+	0.736896i	$0.263709\pi$
$548$	0	0
$549$	5.12836	0.218873
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.26495	0.138840
$554$	0	0
$555$	7.32089	0.310754
$556$	0	0
$557$	4.03920	0.171146	0.0855732	−	0.996332i	$-0.472728\pi$
$557$	4.03920	0.171146	0.0855732	+	0.996332i	$0.472728\pi$
$558$	0	0
$559$	6.06418	0.256487
$560$	0	0
$561$	8.96492	0.378499
$562$	0	0
$563$	−8.61081	−0.362903	−0.181451	−	0.983400i	$-0.558079\pi$
$563$	−8.61081	−0.362903	−0.181451	+	0.983400i	$0.558079\pi$
$564$	0	0
$565$	53.5877	2.25445
$566$	0	0
$567$	−2.06418	−0.0866873
$568$	0	0
$569$	−27.6851	−1.16062	−0.580310	−	0.814396i	$-0.697069\pi$
$569$	−27.6851	−1.16062	−0.580310	+	0.814396i	$0.697069\pi$
$570$	0	0
$571$	24.1533	1.01079	0.505393	−	0.862889i	$-0.331348\pi$
$571$	24.1533	1.01079	0.505393	+	0.862889i	$0.331348\pi$
$572$	0	0
$573$	15.2317	0.636315
$574$	0	0
$575$	13.4492	0.560872
$576$	0	0
$577$	28.8675	1.20177	0.600885	−	0.799335i	$-0.294815\pi$
$577$	28.8675	1.20177	0.600885	+	0.799335i	$0.294815\pi$
$578$	0	0
$579$	−6.26083	−0.260191
$580$	0	0
$581$	36.4243	1.51113
$582$	0	0
$583$	68.2877	2.82819
$584$	0	0
$585$	3.06418	0.126688
$586$	0	0
$587$	10.2858	0.424541	0.212270	−	0.977211i	$-0.431914\pi$
$587$	10.2858	0.424541	0.212270	+	0.977211i	$0.431914\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.71007	0.276016
$592$	0	0
$593$	−15.9317	−0.654237	−0.327118	−	0.944983i	$-0.606078\pi$
$593$	−15.9317	−0.654237	−0.327118	+	0.944983i	$0.606078\pi$
$594$	0	0
$595$	−8.78661	−0.360216
$596$	0	0
$597$	24.9317	1.02039
$598$	0	0
$599$	−18.0601	−0.737914	−0.368957	−	0.929446i	$-0.620285\pi$
$599$	−18.0601	−0.737914	−0.368957	+	0.929446i	$0.620285\pi$
$600$	0	0
$601$	−7.86753	−0.320923	−0.160462	−	0.987042i	$-0.551298\pi$
$601$	−7.86753	−0.320923	−0.160462	+	0.987042i	$0.551298\pi$
$602$	0	0
$603$	3.45336	0.140632
$604$	0	0
$605$	93.9045	3.81776
$606$	0	0
$607$	−17.2668	−0.700838	−0.350419	−	0.936593i	$-0.613961\pi$
$607$	−17.2668	−0.700838	−0.350419	+	0.936593i	$0.613961\pi$
$608$	0	0
$609$	−7.26083	−0.294224
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−27.5877	−1.11426	−0.557128	−	0.830426i	$-0.688097\pi$
$613$	−27.5877	−1.11426	−0.557128	+	0.830426i	$0.688097\pi$
$614$	0	0
$615$	−31.0351	−1.25146
$616$	0	0
$617$	16.4926	0.663966	0.331983	−	0.943285i	$-0.392282\pi$
$617$	16.4926	0.663966	0.331983	+	0.943285i	$0.392282\pi$
$618$	0	0
$619$	24.4884	0.984274	0.492137	−	0.870518i	$-0.336216\pi$
$619$	24.4884	0.984274	0.492137	+	0.870518i	$0.336216\pi$
$620$	0	0
$621$	−3.06418	−0.122961
$622$	0	0
$623$	−21.5776	−0.864488
$624$	0	0
$625$	−27.6810	−1.10724
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.31902	−0.132338
$630$	0	0
$631$	19.6709	0.783085	0.391543	−	0.920160i	$-0.371942\pi$
$631$	19.6709	0.783085	0.391543	+	0.920160i	$0.371942\pi$
$632$	0	0
$633$	−8.97090	−0.356561
$634$	0	0
$635$	43.2918	1.71798
$636$	0	0
$637$	−2.73917	−0.108530
$638$	0	0
$639$	6.73917	0.266597
$640$	0	0
$641$	8.12836	0.321051	0.160525	−	0.987032i	$-0.448681\pi$
$641$	8.12836	0.321051	0.160525	+	0.987032i	$0.448681\pi$
$642$	0	0
$643$	7.83843	0.309117	0.154559	−	0.987984i	$-0.450604\pi$
$643$	7.83843	0.309117	0.154559	+	0.987984i	$0.450604\pi$
$644$	0	0
$645$	−18.5817	−0.731654
$646$	0	0
$647$	−39.7743	−1.56369	−0.781844	−	0.623475i	$-0.785720\pi$
$647$	−39.7743	−1.56369	−0.781844	+	0.623475i	$0.785720\pi$
$648$	0	0
$649$	72.2295	2.83526
$650$	0	0
$651$	19.5134	0.764791
$652$	0	0
$653$	8.94593	0.350081	0.175041	−	0.984561i	$-0.443994\pi$
$653$	8.94593	0.350081	0.175041	+	0.984561i	$0.443994\pi$
$654$	0	0
$655$	16.9067	0.660600
$656$	0	0
$657$	−9.12836	−0.356131
$658$	0	0
$659$	−11.8735	−0.462526	−0.231263	−	0.972891i	$-0.574286\pi$
$659$	−11.8735	−0.462526	−0.231263	+	0.972891i	$0.574286\pi$
$660$	0	0
$661$	43.8135	1.70415	0.852073	−	0.523423i	$-0.175345\pi$
$661$	43.8135	1.70415	0.852073	+	0.523423i	$0.175345\pi$
$662$	0	0
$663$	−1.38919	−0.0539515
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.7784	−0.417340
$668$	0	0
$669$	−25.1385	−0.971909
$670$	0	0
$671$	−33.0951	−1.27762
$672$	0	0
$673$	−37.6810	−1.45249	−0.726247	−	0.687433i	$-0.758737\pi$
$673$	−37.6810	−1.45249	−0.726247	+	0.687433i	$0.758737\pi$
$674$	0	0
$675$	−4.38919	−0.168940
$676$	0	0
$677$	−35.1052	−1.34920	−0.674602	−	0.738182i	$-0.735685\pi$
$677$	−35.1052	−1.34920	−0.674602	+	0.738182i	$0.735685\pi$
$678$	0	0
$679$	−13.9108	−0.533849
$680$	0	0
$681$	22.8384	0.875171
$682$	0	0
$683$	37.6168	1.43937	0.719683	−	0.694302i	$-0.244287\pi$
$683$	37.6168	1.43937	0.719683	+	0.694302i	$0.244287\pi$
$684$	0	0
$685$	54.4635	2.08094
$686$	0	0
$687$	6.03508	0.230253
$688$	0	0
$689$	−10.5817	−0.403131
$690$	0	0
$691$	−2.69997	−0.102712	−0.0513558	−	0.998680i	$-0.516354\pi$
$691$	−2.69997	−0.102712	−0.0513558	+	0.998680i	$0.516354\pi$
$692$	0	0
$693$	13.3209	0.506019
$694$	0	0
$695$	−33.6168	−1.27516
$696$	0	0
$697$	14.0702	0.532945
$698$	0	0
$699$	−4.61081	−0.174397
$700$	0	0
$701$	−21.7641	−0.822020	−0.411010	−	0.911631i	$-0.634824\pi$
$701$	−21.7641	−0.822020	−0.411010	+	0.911631i	$0.634824\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	18.3851	0.692422
$706$	0	0
$707$	−1.26083	−0.0474184
$708$	0	0
$709$	−26.6067	−0.999235	−0.499618	−	0.866246i	$-0.666526\pi$
$709$	−26.6067	−0.999235	−0.499618	+	0.866246i	$0.666526\pi$
$710$	0	0
$711$	−1.58172	−0.0593191
$712$	0	0
$713$	28.9668	1.08481
$714$	0	0
$715$	−19.7743	−0.739515
$716$	0	0
$717$	−9.36009	−0.349559
$718$	0	0
$719$	−7.63579	−0.284767	−0.142383	−	0.989812i	$-0.545477\pi$
$719$	−7.63579	−0.284767	−0.142383	+	0.989812i	$0.545477\pi$
$720$	0	0
$721$	−0.132474	−0.00493360
$722$	0	0
$723$	20.1634	0.749886
$724$	0	0
$725$	−15.4391	−0.573395
$726$	0	0
$727$	40.7452	1.51115	0.755577	−	0.655060i	$-0.227357\pi$
$727$	40.7452	1.51115	0.755577	+	0.655060i	$0.227357\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.42427	0.311583
$732$	0	0
$733$	48.4742	1.79044	0.895218	−	0.445628i	$-0.147020\pi$
$733$	48.4742	1.79044	0.895218	+	0.445628i	$0.147020\pi$
$734$	0	0
$735$	8.39330	0.309592
$736$	0	0
$737$	−22.2858	−0.820908
$738$	0	0
$739$	−10.6168	−0.390545	−0.195273	−	0.980749i	$-0.562559\pi$
$739$	−10.6168	−0.390545	−0.195273	+	0.980749i	$0.562559\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.5586	1.74476	0.872378	−	0.488832i	$-0.162577\pi$
$743$	47.5586	1.74476	0.872378	+	0.488832i	$0.162577\pi$
$744$	0	0
$745$	40.0310	1.46662
$746$	0	0
$747$	−17.6459	−0.645630
$748$	0	0
$749$	−1.26083	−0.0460697
$750$	0	0
$751$	−15.9358	−0.581506	−0.290753	−	0.956798i	$-0.593906\pi$
$751$	−15.9358	−0.581506	−0.290753	+	0.956798i	$0.593906\pi$
$752$	0	0
$753$	−12.1284	−0.441982
$754$	0	0
$755$	15.0351	0.547183
$756$	0	0
$757$	−18.3500	−0.666942	−0.333471	−	0.942760i	$-0.608220\pi$
$757$	−18.3500	−0.666942	−0.333471	+	0.942760i	$0.608220\pi$
$758$	0	0
$759$	19.7743	0.717760
$760$	0	0
$761$	−36.7802	−1.33328	−0.666641	−	0.745379i	$-0.732269\pi$
$761$	−36.7802	−1.33328	−0.666641	+	0.745379i	$0.732269\pi$
$762$	0	0
$763$	−9.51754	−0.344558
$764$	0	0
$765$	4.25671	0.153902
$766$	0	0
$767$	−11.1925	−0.404139
$768$	0	0
$769$	42.0351	1.51582	0.757912	−	0.652357i	$-0.226220\pi$
$769$	42.0351	1.51582	0.757912	+	0.652357i	$0.226220\pi$
$770$	0	0
$771$	−30.8384	−1.11062
$772$	0	0
$773$	45.9418	1.65241	0.826206	−	0.563368i	$-0.190495\pi$
$773$	45.9418	1.65241	0.826206	+	0.563368i	$0.190495\pi$
$774$	0	0
$775$	41.4926	1.49046
$776$	0	0
$777$	−4.93170	−0.176924
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−43.4903	−1.55621
$782$	0	0
$783$	3.51754	0.125707
$784$	0	0
$785$	35.9709	1.28386
$786$	0	0
$787$	−17.3250	−0.617570	−0.308785	−	0.951132i	$-0.599922\pi$
$787$	−17.3250	−0.617570	−0.308785	+	0.951132i	$0.599922\pi$
$788$	0	0
$789$	10.4825	0.373185
$790$	0	0
$791$	−36.0993	−1.28354
$792$	0	0
$793$	5.12836	0.182113
$794$	0	0
$795$	32.4243	1.14997
$796$	0	0
$797$	−21.8324	−0.773345	−0.386672	−	0.922217i	$-0.626376\pi$
$797$	−21.8324	−0.773345	−0.386672	+	0.922217i	$0.626376\pi$
$798$	0	0
$799$	−8.33511	−0.294875
$800$	0	0
$801$	10.4534	0.369351
$802$	0	0
$803$	58.9086	2.07884
$804$	0	0
$805$	−19.3809	−0.683089
$806$	0	0
$807$	1.80335	0.0634809
$808$	0	0
$809$	−34.6709	−1.21896	−0.609482	−	0.792800i	$-0.708622\pi$
$809$	−34.6709	−1.21896	−0.609482	+	0.792800i	$0.708622\pi$
$810$	0	0
$811$	−28.9067	−1.01505	−0.507526	−	0.861636i	$-0.669440\pi$
$811$	−28.9067	−1.01505	−0.507526	+	0.861636i	$0.669440\pi$
$812$	0	0
$813$	−28.2567	−0.991006
$814$	0	0
$815$	−41.2235	−1.44400
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.06418	−0.0721282
$820$	0	0
$821$	−27.5175	−0.960369	−0.480184	−	0.877168i	$-0.659430\pi$
$821$	−27.5175	−0.960369	−0.480184	+	0.877168i	$0.659430\pi$
$822$	0	0
$823$	−43.9418	−1.53172	−0.765858	−	0.643010i	$-0.777685\pi$
$823$	−43.9418	−1.53172	−0.765858	+	0.643010i	$0.777685\pi$
$824$	0	0
$825$	28.3250	0.986150
$826$	0	0
$827$	1.96080	0.0681837	0.0340918	−	0.999419i	$-0.489146\pi$
$827$	1.96080	0.0681837	0.0340918	+	0.999419i	$0.489146\pi$
$828$	0	0
$829$	−10.4284	−0.362193	−0.181096	−	0.983465i	$-0.557965\pi$
$829$	−10.4284	−0.362193	−0.181096	+	0.983465i	$0.557965\pi$
$830$	0	0
$831$	−6.48246	−0.224874
$832$	0	0
$833$	−3.80522	−0.131843
$834$	0	0
$835$	42.8093	1.48148
$836$	0	0
$837$	−9.45336	−0.326756
$838$	0	0
$839$	32.3952	1.11841	0.559203	−	0.829031i	$-0.311107\pi$
$839$	32.3952	1.11841	0.559203	+	0.829031i	$0.311107\pi$
$840$	0	0
$841$	−16.6269	−0.573342
$842$	0	0
$843$	−5.84255	−0.201228
$844$	0	0
$845$	−36.7701	−1.26493
$846$	0	0
$847$	−63.2586	−2.17359
$848$	0	0
$849$	−16.2567	−0.557929
$850$	0	0
$851$	−7.32089	−0.250957
$852$	0	0
$853$	−29.1676	−0.998678	−0.499339	−	0.866407i	$-0.666424\pi$
$853$	−29.1676	−0.998678	−0.499339	+	0.866407i	$0.666424\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.5567	0.531408	0.265704	−	0.964055i	$-0.414396\pi$
$857$	15.5567	0.531408	0.265704	+	0.964055i	$0.414396\pi$
$858$	0	0
$859$	30.1533	1.02882	0.514409	−	0.857545i	$-0.328011\pi$
$859$	30.1533	1.02882	0.514409	+	0.857545i	$0.328011\pi$
$860$	0	0
$861$	20.9067	0.712499
$862$	0	0
$863$	−19.7351	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$863$	−19.7351	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$864$	0	0
$865$	10.7784	0.366476
$866$	0	0
$867$	15.0702	0.511810
$868$	0	0
$869$	10.2074	0.346263
$870$	0	0
$871$	3.45336	0.117013
$872$	0	0
$873$	6.73917	0.228086
$874$	0	0
$875$	3.86341	0.130607
$876$	0	0
$877$	−10.6459	−0.359486	−0.179743	−	0.983714i	$-0.557527\pi$
$877$	−10.6459	−0.359486	−0.179743	+	0.983714i	$0.557527\pi$
$878$	0	0
$879$	−26.2567	−0.885617
$880$	0	0
$881$	39.9709	1.34665	0.673327	−	0.739345i	$-0.264865\pi$
$881$	39.9709	1.34665	0.673327	+	0.739345i	$0.264865\pi$
$882$	0	0
$883$	29.0993	0.979268	0.489634	−	0.871928i	$-0.337130\pi$
$883$	29.0993	0.979268	0.489634	+	0.871928i	$0.337130\pi$
$884$	0	0
$885$	34.2959	1.15284
$886$	0	0
$887$	19.4492	0.653042	0.326521	−	0.945190i	$-0.394124\pi$
$887$	19.4492	0.653042	0.326521	+	0.945190i	$0.394124\pi$
$888$	0	0
$889$	−29.1634	−0.978110
$890$	0	0
$891$	−6.45336	−0.216196
$892$	0	0
$893$	0	0
$894$	0	0
$895$	33.9026	1.13324
$896$	0	0
$897$	−3.06418	−0.102310
$898$	0	0
$899$	−33.2526	−1.10904
$900$	0	0
$901$	−14.7000	−0.489727
$902$	0	0
$903$	12.5175	0.416558
$904$	0	0
$905$	65.9336	2.19171
$906$	0	0
$907$	1.55674	0.0516908	0.0258454	−	0.999666i	$-0.491772\pi$
$907$	1.55674	0.0516908	0.0258454	+	0.999666i	$0.491772\pi$
$908$	0	0
$909$	0.610815	0.0202594
$910$	0	0
$911$	43.9418	1.45586	0.727929	−	0.685653i	$-0.240483\pi$
$911$	43.9418	1.45586	0.727929	+	0.685653i	$0.240483\pi$
$912$	0	0
$913$	113.875	3.76872
$914$	0	0
$915$	−15.7142	−0.519495
$916$	0	0
$917$	−11.3892	−0.376104
$918$	0	0
$919$	43.4843	1.43442	0.717208	−	0.696859i	$-0.245420\pi$
$919$	43.4843	1.43442	0.717208	+	0.696859i	$0.245420\pi$
$920$	0	0
$921$	−14.2959	−0.471066
$922$	0	0
$923$	6.73917	0.221822
$924$	0	0
$925$	−10.4866	−0.344797
$926$	0	0
$927$	0.0641778	0.00210787
$928$	0	0
$929$	17.2709	0.566641	0.283320	−	0.959025i	$-0.408564\pi$
$929$	17.2709	0.566641	0.283320	+	0.959025i	$0.408564\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22.2276	−0.727699
$934$	0	0
$935$	−27.4701	−0.898368
$936$	0	0
$937$	29.2175	0.954494	0.477247	−	0.878769i	$-0.341635\pi$
$937$	29.2175	0.954494	0.477247	+	0.878769i	$0.341635\pi$
$938$	0	0
$939$	−3.13247	−0.102224
$940$	0	0
$941$	−27.2918	−0.889687	−0.444844	−	0.895608i	$-0.646741\pi$
$941$	−27.2918	−0.889687	−0.444844	+	0.895608i	$0.646741\pi$
$942$	0	0
$943$	31.0351	1.01064
$944$	0	0
$945$	6.32501	0.205753
$946$	0	0
$947$	−51.3310	−1.66803	−0.834017	−	0.551739i	$-0.813964\pi$
$947$	−51.3310	−1.66803	−0.834017	+	0.551739i	$0.813964\pi$
$948$	0	0
$949$	−9.12836	−0.296319
$950$	0	0
$951$	6.09926	0.197782
$952$	0	0
$953$	−55.1343	−1.78598	−0.892988	−	0.450080i	$-0.851395\pi$
$953$	−55.1343	−1.78598	−0.892988	+	0.450080i	$0.851395\pi$
$954$	0	0
$955$	−46.6727	−1.51029
$956$	0	0
$957$	−22.7000	−0.733786
$958$	0	0
$959$	−36.6892	−1.18476
$960$	0	0
$961$	58.3661	1.88278
$962$	0	0
$963$	0.610815	0.0196832
$964$	0	0
$965$	19.1843	0.617564
$966$	0	0
$967$	50.3519	1.61921	0.809603	−	0.586978i	$-0.199682\pi$
$967$	50.3519	1.61921	0.809603	+	0.586978i	$0.199682\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.9918	1.79686	0.898431	−	0.439116i	$-0.144708\pi$
$971$	55.9918	1.79686	0.898431	+	0.439116i	$0.144708\pi$
$972$	0	0
$973$	22.6459	0.725994
$974$	0	0
$975$	−4.38919	−0.140566
$976$	0	0
$977$	−14.8283	−0.474400	−0.237200	−	0.971461i	$-0.576230\pi$
$977$	−14.8283	−0.474400	−0.237200	+	0.971461i	$0.576230\pi$
$978$	0	0
$979$	−67.4593	−2.15601
$980$	0	0
$981$	4.61081	0.147212
$982$	0	0
$983$	16.8384	0.537063	0.268531	−	0.963271i	$-0.413462\pi$
$983$	16.8384	0.537063	0.268531	+	0.963271i	$0.413462\pi$
$984$	0	0
$985$	−20.5609	−0.655123
$986$	0	0
$987$	−12.3851	−0.394221
$988$	0	0
$989$	18.5817	0.590864
$990$	0	0
$991$	54.4100	1.72839	0.864196	−	0.503155i	$-0.167828\pi$
$991$	54.4100	1.72839	0.864196	+	0.503155i	$0.167828\pi$
$992$	0	0
$993$	1.93582	0.0614315
$994$	0	0
$995$	−76.3952	−2.42189
$996$	0	0
$997$	54.3228	1.72042	0.860209	−	0.509941i	$-0.170333\pi$
$997$	54.3228	1.72042	0.860209	+	0.509941i	$0.170333\pi$
$998$	0	0
$999$	2.38919	0.0755905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.x.1.3		3
19.7	even	3		456.2.q.f.49.1	✓	6
19.11	even	3		456.2.q.f.121.1	yes	6
19.18	odd	2		8664.2.a.z.1.3		3
57.11	odd	6		1368.2.s.j.577.3		6
57.26	odd	6		1368.2.s.j.505.3		6
76.7	odd	6		912.2.q.k.49.1		6
76.11	odd	6		912.2.q.k.577.1		6
228.11	even	6		2736.2.s.x.577.3		6
228.83	even	6		2736.2.s.x.1873.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.f.49.1	✓	6	19.7	even	3
456.2.q.f.121.1	yes	6	19.11	even	3
912.2.q.k.49.1		6	76.7	odd	6
912.2.q.k.577.1		6	76.11	odd	6
1368.2.s.j.505.3		6	57.26	odd	6
1368.2.s.j.577.3		6	57.11	odd	6
2736.2.s.x.577.3		6	228.11	even	6
2736.2.s.x.1873.3		6	228.83	even	6
8664.2.a.x.1.3		3	1.1	even	1	trivial
8664.2.a.z.1.3		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.06418 q^{5} -2.06418 q^{7} +1.00000 q^{9} -6.45336 q^{11} +1.00000 q^{13} -3.06418 q^{15} +1.38919 q^{17} +2.06418 q^{21} +3.06418 q^{23} +4.38919 q^{25} -1.00000 q^{27} -3.51754 q^{29} +9.45336 q^{31} +6.45336 q^{33} -6.32501 q^{35} -2.38919 q^{37} -1.00000 q^{39} +10.1284 q^{41} +6.06418 q^{43} +3.06418 q^{45} -6.00000 q^{47} -2.73917 q^{49} -1.38919 q^{51} -10.5817 q^{53} -19.7743 q^{55} -11.1925 q^{59} +5.12836 q^{61} -2.06418 q^{63} +3.06418 q^{65} +3.45336 q^{67} -3.06418 q^{69} +6.73917 q^{71} -9.12836 q^{73} -4.38919 q^{75} +13.3209 q^{77} -1.58172 q^{79} +1.00000 q^{81} -17.6459 q^{83} +4.25671 q^{85} +3.51754 q^{87} +10.4534 q^{89} -2.06418 q^{91} -9.45336 q^{93} +6.73917 q^{97} -6.45336 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 3 q^{21} + 9 q^{25} - 3 q^{27} + 12 q^{29} + 15 q^{31} + 6 q^{33} - 24 q^{35} - 3 q^{37} - 3 q^{39} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 3 * q^21 + 9 * q^25 - 3 * q^27 + 12 * q^29 + 15 * q^31 + 6 * q^33 - 24 * q^35 - 3 * q^37 - 3 * q^39 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 - 6 * q^59 - 3 * q^61 + 3 * q^63 - 3 * q^67 + 6 * q^71 - 9 * q^73 - 9 * q^75 - 6 * q^77 + 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 + 18 * q^89 + 3 * q^91 - 15 * q^93 + 6 * q^97 - 6 * q^99

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