LMFDB - Embedded newform 8464.2.a.by.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,0,0,0,0,10,0,0,0,-4] 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:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.26849792.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 $ x^6 - 14x^4 - 2x^3 + 28x^2 - 4x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.52912$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.357926 q^{3} +1.57672 q^{5} -2.08290 q^{7} -2.87189 q^{9} +0.668688 q^{11} -0.357926 q^{13} -0.564349 q^{15} +2.99093 q^{17} -5.58002 q^{19} +0.745525 q^{21} -2.51396 q^{25} +2.10170 q^{27} +5.53341 q^{29} +3.87189 q^{31} -0.239341 q^{33} -3.28415 q^{35} +2.58909 q^{37} +0.128111 q^{39} +10.8176 q^{41} +9.71832 q^{43} -4.52816 q^{45} -8.58774 q^{47} -2.66152 q^{49} -1.07053 q^{51} +3.33461 q^{53} +1.05433 q^{55} +1.99724 q^{57} -1.77018 q^{59} -4.13830 q^{61} +5.98186 q^{63} -0.564349 q^{65} -7.66292 q^{67} -5.64207 q^{71} +10.2493 q^{73} +0.899813 q^{75} -1.39281 q^{77} +1.99724 q^{79} +7.86341 q^{81} -10.7889 q^{83} +4.71585 q^{85} -1.98055 q^{87} -2.72409 q^{89} +0.745525 q^{91} -1.38585 q^{93} -8.79811 q^{95} -10.3102 q^{97} -1.92040 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} - 40 q^{27} - 12 q^{29} - 4 q^{31} - 16 q^{35} + 28 q^{39} + 16 q^{41} - 28 q^{47} + 2 q^{49} + 28 q^{55} - 36 q^{59} - 32 q^{71} + 20 q^{73} - 80 q^{75}+ \cdots - 20 q^{95}+O(q^{100}) $ 6 * q - 4 * q^3 + 10 * q^9 - 4 * q^13 + 14 * q^25 - 40 * q^27 - 12 * q^29 - 4 * q^31 - 16 * q^35 + 28 * q^39 + 16 * q^41 - 28 * q^47 + 2 * q^49 + 28 * q^55 - 36 * q^59 - 32 * q^71 + 20 * q^73 - 80 * q^75 - 48 * q^77 + 62 * q^81 + 32 * q^85 - 28 * q^87 + 48 * q^93 - 20 * q^95

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
$3$	−0.357926	−0.206649	−0.103324	−	0.994648i	$-0.532948\pi$
$3$	−0.357926	−0.206649	−0.103324	+	0.994648i	$0.532948\pi$
$4$	0	0
$5$	1.57672	0.705129	0.352565	−	0.935787i	$-0.385310\pi$
$5$	1.57672	0.705129	0.352565	+	0.935787i	$0.385310\pi$
$6$	0	0
$7$	−2.08290	−0.787263	−0.393631	−	0.919268i	$-0.628781\pi$
$7$	−2.08290	−0.787263	−0.393631	+	0.919268i	$0.628781\pi$
$8$	0	0
$9$	−2.87189	−0.957296
$10$	0	0
$11$	0.668688	0.201617	0.100809	−	0.994906i	$-0.467857\pi$
$11$	0.668688	0.201617	0.100809	+	0.994906i	$0.467857\pi$
$12$	0	0
$13$	−0.357926	−0.0992709	−0.0496355	−	0.998767i	$-0.515806\pi$
$13$	−0.357926	−0.0992709	−0.0496355	+	0.998767i	$0.515806\pi$
$14$	0	0
$15$	−0.564349	−0.145714
$16$	0	0
$17$	2.99093	0.725407	0.362704	−	0.931905i	$-0.381854\pi$
$17$	2.99093	0.725407	0.362704	+	0.931905i	$0.381854\pi$
$18$	0	0
$19$	−5.58002	−1.28014	−0.640072	−	0.768315i	$-0.721095\pi$
$19$	−5.58002	−1.28014	−0.640072	+	0.768315i	$0.721095\pi$
$20$	0	0
$21$	0.745525	0.162687
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.51396	−0.502792
$26$	0	0
$27$	2.10170	0.404473
$28$	0	0
$29$	5.53341	1.02753	0.513764	−	0.857931i	$-0.328251\pi$
$29$	5.53341	1.02753	0.513764	+	0.857931i	$0.328251\pi$
$30$	0	0
$31$	3.87189	0.695412	0.347706	−	0.937604i	$-0.386961\pi$
$31$	3.87189	0.695412	0.347706	+	0.937604i	$0.386961\pi$
$32$	0	0
$33$	−0.239341	−0.0416639
$34$	0	0
$35$	−3.28415	−0.555122
$36$	0	0
$37$	2.58909	0.425643	0.212822	−	0.977091i	$-0.431735\pi$
$37$	2.58909	0.425643	0.212822	+	0.977091i	$0.431735\pi$
$38$	0	0
$39$	0.128111	0.0205142
$40$	0	0
$41$	10.8176	1.68942	0.844709	−	0.535225i	$-0.179773\pi$
$41$	10.8176	1.68942	0.844709	+	0.535225i	$0.179773\pi$
$42$	0	0
$43$	9.71832	1.48203	0.741015	−	0.671489i	$-0.234345\pi$
$43$	9.71832	1.48203	0.741015	+	0.671489i	$0.234345\pi$
$44$	0	0
$45$	−4.52816	−0.675018
$46$	0	0
$47$	−8.58774	−1.25265	−0.626325	−	0.779562i	$-0.715442\pi$
$47$	−8.58774	−1.25265	−0.626325	+	0.779562i	$0.715442\pi$
$48$	0	0
$49$	−2.66152	−0.380217
$50$	0	0
$51$	−1.07053	−0.149905
$52$	0	0
$53$	3.33461	0.458044	0.229022	−	0.973421i	$-0.426447\pi$
$53$	3.33461	0.458044	0.229022	+	0.973421i	$0.426447\pi$
$54$	0	0
$55$	1.05433	0.142166
$56$	0	0
$57$	1.99724	0.264540
$58$	0	0
$59$	−1.77018	−0.230458	−0.115229	−	0.993339i	$-0.536760\pi$
$59$	−1.77018	−0.230458	−0.115229	+	0.993339i	$0.536760\pi$
$60$	0	0
$61$	−4.13830	−0.529855	−0.264928	−	0.964268i	$-0.585348\pi$
$61$	−4.13830	−0.529855	−0.264928	+	0.964268i	$0.585348\pi$
$62$	0	0
$63$	5.98186	0.753644
$64$	0	0
$65$	−0.564349	−0.0699988
$66$	0	0
$67$	−7.66292	−0.936175	−0.468087	−	0.883682i	$-0.655057\pi$
$67$	−7.66292	−0.936175	−0.468087	+	0.883682i	$0.655057\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.64207	−0.669591	−0.334795	−	0.942291i	$-0.608667\pi$
$71$	−5.64207	−0.669591	−0.334795	+	0.942291i	$0.608667\pi$
$72$	0	0
$73$	10.2493	1.19959	0.599793	−	0.800155i	$-0.295250\pi$
$73$	10.2493	1.19959	0.599793	+	0.800155i	$0.295250\pi$
$74$	0	0
$75$	0.899813	0.103902
$76$	0	0
$77$	−1.39281	−0.158726
$78$	0	0
$79$	1.99724	0.224707	0.112353	−	0.993668i	$-0.464161\pi$
$79$	1.99724	0.224707	0.112353	+	0.993668i	$0.464161\pi$
$80$	0	0
$81$	7.86341	0.873712
$82$	0	0
$83$	−10.7889	−1.18423	−0.592115	−	0.805853i	$-0.701707\pi$
$83$	−10.7889	−1.18423	−0.592115	+	0.805853i	$0.701707\pi$
$84$	0	0
$85$	4.71585	0.511506
$86$	0	0
$87$	−1.98055	−0.212338
$88$	0	0
$89$	−2.72409	−0.288753	−0.144376	−	0.989523i	$-0.546118\pi$
$89$	−2.72409	−0.288753	−0.144376	+	0.989523i	$0.546118\pi$
$90$	0	0
$91$	0.745525	0.0781523
$92$	0	0
$93$	−1.38585	−0.143706
$94$	0	0
$95$	−8.79811	−0.902667
$96$	0	0
$97$	−10.3102	−1.04684	−0.523420	−	0.852075i	$-0.675344\pi$
$97$	−10.3102	−1.04684	−0.523420	+	0.852075i	$0.675344\pi$
$98$	0	0
$99$	−1.92040	−0.193007
$100$	0	0
$101$	−8.94567	−0.890127	−0.445064	−	0.895499i	$-0.646819\pi$
$101$	−8.94567	−0.890127	−0.445064	+	0.895499i	$0.646819\pi$
$102$	0	0
$103$	−18.3661	−1.80967	−0.904833	−	0.425767i	$-0.860004\pi$
$103$	−18.3661	−1.80967	−0.904833	+	0.425767i	$0.860004\pi$
$104$	0	0
$105$	1.17548	0.114715
$106$	0	0
$107$	15.4027	1.48903	0.744516	−	0.667604i	$-0.232680\pi$
$107$	15.4027	1.48903	0.744516	+	0.667604i	$0.232680\pi$
$108$	0	0
$109$	9.49765	0.909710	0.454855	−	0.890566i	$-0.349691\pi$
$109$	9.49765	0.909710	0.454855	+	0.890566i	$0.349691\pi$
$110$	0	0
$111$	−0.926702	−0.0879587
$112$	0	0
$113$	10.1388	0.953781	0.476891	−	0.878963i	$-0.341764\pi$
$113$	10.1388	0.953781	0.476891	+	0.878963i	$0.341764\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.02792	0.0950317
$118$	0	0
$119$	−6.22982	−0.571086
$120$	0	0
$121$	−10.5529	−0.959351
$122$	0	0
$123$	−3.87189	−0.349116
$124$	0	0
$125$	−11.8474	−1.05966
$126$	0	0
$127$	−9.38585	−0.832860	−0.416430	−	0.909168i	$-0.636719\pi$
$127$	−9.38585	−0.832860	−0.416430	+	0.909168i	$0.636719\pi$
$128$	0	0
$129$	−3.47844	−0.306260
$130$	0	0
$131$	5.64207	0.492950	0.246475	−	0.969149i	$-0.420728\pi$
$131$	5.64207	0.492950	0.246475	+	0.969149i	$0.420728\pi$
$132$	0	0
$133$	11.6226	1.00781
$134$	0	0
$135$	3.31379	0.285206
$136$	0	0
$137$	−18.2036	−1.55524	−0.777619	−	0.628735i	$-0.783573\pi$
$137$	−18.2036	−1.55524	−0.777619	+	0.628735i	$0.783573\pi$
$138$	0	0
$139$	−18.3579	−1.55710	−0.778549	−	0.627583i	$-0.784044\pi$
$139$	−18.3579	−1.55710	−0.778549	+	0.627583i	$0.784044\pi$
$140$	0	0
$141$	3.07378	0.258859
$142$	0	0
$143$	−0.239341	−0.0200147
$144$	0	0
$145$	8.72462	0.724540
$146$	0	0
$147$	0.952628	0.0785715
$148$	0	0
$149$	21.6052	1.76997	0.884984	−	0.465622i	$-0.154169\pi$
$149$	21.6052	1.76997	0.884984	+	0.465622i	$0.154169\pi$
$150$	0	0
$151$	−23.2772	−1.89427	−0.947135	−	0.320834i	$-0.896037\pi$
$151$	−23.2772	−1.89427	−0.947135	+	0.320834i	$0.896037\pi$
$152$	0	0
$153$	−8.58962	−0.694430
$154$	0	0
$155$	6.10487	0.490355
$156$	0	0
$157$	−8.81029	−0.703138	−0.351569	−	0.936162i	$-0.614352\pi$
$157$	−8.81029	−0.703138	−0.351569	+	0.936162i	$0.614352\pi$
$158$	0	0
$159$	−1.19355	−0.0946543
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.9262	−0.855808	−0.427904	−	0.903824i	$-0.640748\pi$
$163$	−10.9262	−0.855808	−0.427904	+	0.903824i	$0.640748\pi$
$164$	0	0
$165$	−0.377373	−0.0293785
$166$	0	0
$167$	−11.8649	−0.918136	−0.459068	−	0.888401i	$-0.651817\pi$
$167$	−11.8649	−0.918136	−0.459068	+	0.888401i	$0.651817\pi$
$168$	0	0
$169$	−12.8719	−0.990145
$170$	0	0
$171$	16.0252	1.22548
$172$	0	0
$173$	4.10866	0.312376	0.156188	−	0.987727i	$-0.450079\pi$
$173$	4.10866	0.312376	0.156188	+	0.987727i	$0.450079\pi$
$174$	0	0
$175$	5.23634	0.395830
$176$	0	0
$177$	0.633596	0.0476240
$178$	0	0
$179$	3.04737	0.227771	0.113886	−	0.993494i	$-0.463670\pi$
$179$	3.04737	0.227771	0.113886	+	0.993494i	$0.463670\pi$
$180$	0	0
$181$	−10.0620	−0.747902	−0.373951	−	0.927448i	$-0.621997\pi$
$181$	−10.0620	−0.747902	−0.373951	+	0.927448i	$0.621997\pi$
$182$	0	0
$183$	1.48121	0.109494
$184$	0	0
$185$	4.08226	0.300133
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−4.37764	−0.318427
$190$	0	0
$191$	15.3565	1.11116	0.555579	−	0.831464i	$-0.312497\pi$
$191$	15.3565	1.11116	0.555579	+	0.831464i	$0.312497\pi$
$192$	0	0
$193$	1.30359	0.0938348	0.0469174	−	0.998899i	$-0.485060\pi$
$193$	1.30359	0.0938348	0.0469174	+	0.998899i	$0.485060\pi$
$194$	0	0
$195$	0.201995	0.0144652
$196$	0	0
$197$	−26.1142	−1.86056	−0.930280	−	0.366852i	$-0.880436\pi$
$197$	−26.1142	−1.86056	−0.930280	+	0.366852i	$0.880436\pi$
$198$	0	0
$199$	−11.7244	−0.831119	−0.415560	−	0.909566i	$-0.636414\pi$
$199$	−11.7244	−0.831119	−0.415560	+	0.909566i	$0.636414\pi$
$200$	0	0
$201$	2.74276	0.193459
$202$	0	0
$203$	−11.5255	−0.808935
$204$	0	0
$205$	17.0562	1.19126
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.73129	−0.258099
$210$	0	0
$211$	−26.4332	−1.81974	−0.909869	−	0.414896i	$-0.863818\pi$
$211$	−26.4332	−1.81974	−0.909869	+	0.414896i	$0.863818\pi$
$212$	0	0
$213$	2.01945	0.138370
$214$	0	0
$215$	15.3230	1.04502
$216$	0	0
$217$	−8.06476	−0.547472
$218$	0	0
$219$	−3.66848	−0.247893
$220$	0	0
$221$	−1.07053	−0.0720118
$222$	0	0
$223$	16.4985	1.10482	0.552411	−	0.833572i	$-0.313708\pi$
$223$	16.4985	1.10482	0.552411	+	0.833572i	$0.313708\pi$
$224$	0	0
$225$	7.21982	0.481321
$226$	0	0
$227$	22.7526	1.51014	0.755071	−	0.655643i	$-0.227603\pi$
$227$	22.7526	1.51014	0.755071	+	0.655643i	$0.227603\pi$
$228$	0	0
$229$	−24.0811	−1.59133	−0.795663	−	0.605740i	$-0.792877\pi$
$229$	−24.0811	−1.59133	−0.795663	+	0.605740i	$0.792877\pi$
$230$	0	0
$231$	0.498524	0.0328005
$232$	0	0
$233$	6.24926	0.409403	0.204701	−	0.978824i	$-0.434378\pi$
$233$	6.24926	0.409403	0.204701	+	0.978824i	$0.434378\pi$
$234$	0	0
$235$	−13.5404	−0.883281
$236$	0	0
$237$	−0.714863	−0.0464354
$238$	0	0
$239$	24.3704	1.57639	0.788195	−	0.615425i	$-0.211016\pi$
$239$	24.3704	1.57639	0.788195	+	0.615425i	$0.211016\pi$
$240$	0	0
$241$	1.50972	0.0972499	0.0486249	−	0.998817i	$-0.484516\pi$
$241$	1.50972	0.0972499	0.0486249	+	0.998817i	$0.484516\pi$
$242$	0	0
$243$	−9.11963	−0.585025
$244$	0	0
$245$	−4.19647	−0.268102
$246$	0	0
$247$	1.99724	0.127081
$248$	0	0
$249$	3.86161	0.244720
$250$	0	0
$251$	4.53699	0.286372	0.143186	−	0.989696i	$-0.454265\pi$
$251$	4.53699	0.286372	0.143186	+	0.989696i	$0.454265\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.68793	−0.105702
$256$	0	0
$257$	−11.2772	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$257$	−11.2772	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$258$	0	0
$259$	−5.39281	−0.335093
$260$	0	0
$261$	−15.8913	−0.983649
$262$	0	0
$263$	3.81329	0.235138	0.117569	−	0.993065i	$-0.462490\pi$
$263$	3.81329	0.235138	0.117569	+	0.993065i	$0.462490\pi$
$264$	0	0
$265$	5.25774	0.322980
$266$	0	0
$267$	0.975023	0.0596704
$268$	0	0
$269$	2.92622	0.178415	0.0892074	−	0.996013i	$-0.471567\pi$
$269$	2.92622	0.178415	0.0892074	+	0.996013i	$0.471567\pi$
$270$	0	0
$271$	−20.2423	−1.22963	−0.614816	−	0.788670i	$-0.710770\pi$
$271$	−20.2423	−1.22963	−0.614816	+	0.788670i	$0.710770\pi$
$272$	0	0
$273$	−0.266843	−0.0161501
$274$	0	0
$275$	−1.68106	−0.101372
$276$	0	0
$277$	26.8300	1.61206	0.806031	−	0.591874i	$-0.201612\pi$
$277$	26.8300	1.61206	0.806031	+	0.591874i	$0.201612\pi$
$278$	0	0
$279$	−11.1196	−0.665715
$280$	0	0
$281$	6.46937	0.385930	0.192965	−	0.981206i	$-0.438190\pi$
$281$	6.46937	0.385930	0.192965	+	0.981206i	$0.438190\pi$
$282$	0	0
$283$	−6.00053	−0.356695	−0.178347	−	0.983968i	$-0.557075\pi$
$283$	−6.00053	−0.356695	−0.178347	+	0.983968i	$0.557075\pi$
$284$	0	0
$285$	3.14908	0.186535
$286$	0	0
$287$	−22.5319	−1.33002
$288$	0	0
$289$	−8.05433	−0.473784
$290$	0	0
$291$	3.69028	0.216328
$292$	0	0
$293$	22.7987	1.33192	0.665959	−	0.745989i	$-0.268023\pi$
$293$	22.7987	1.33192	0.665959	+	0.745989i	$0.268023\pi$
$294$	0	0
$295$	−2.79108	−0.162503
$296$	0	0
$297$	1.40538	0.0815487
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−20.2423	−1.16675
$302$	0	0
$303$	3.20189	0.183944
$304$	0	0
$305$	−6.52493	−0.373616
$306$	0	0
$307$	−14.0823	−0.803717	−0.401858	−	0.915702i	$-0.631636\pi$
$307$	−14.0823	−0.803717	−0.401858	+	0.915702i	$0.631636\pi$
$308$	0	0
$309$	6.57371	0.373965
$310$	0	0
$311$	−2.61415	−0.148235	−0.0741174	−	0.997250i	$-0.523614\pi$
$311$	−2.61415	−0.148235	−0.0741174	+	0.997250i	$0.523614\pi$
$312$	0	0
$313$	−5.55251	−0.313847	−0.156923	−	0.987611i	$-0.550158\pi$
$313$	−5.55251	−0.313847	−0.156923	+	0.987611i	$0.550158\pi$
$314$	0	0
$315$	9.43171	0.531416
$316$	0	0
$317$	3.77018	0.211755	0.105877	−	0.994379i	$-0.466235\pi$
$317$	3.77018	0.211755	0.105877	+	0.994379i	$0.466235\pi$
$318$	0	0
$319$	3.70013	0.207167
$320$	0	0
$321$	−5.51302	−0.307707
$322$	0	0
$323$	−16.6894	−0.928626
$324$	0	0
$325$	0.899813	0.0499127
$326$	0	0
$327$	−3.39946	−0.187991
$328$	0	0
$329$	17.8874	0.986165
$330$	0	0
$331$	0.249262	0.0137007	0.00685034	−	0.999977i	$-0.497819\pi$
$331$	0.249262	0.0137007	0.00685034	+	0.999977i	$0.497819\pi$
$332$	0	0
$333$	−7.43557	−0.407467
$334$	0	0
$335$	−12.0823	−0.660124
$336$	0	0
$337$	−8.58962	−0.467906	−0.233953	−	0.972248i	$-0.575166\pi$
$337$	−8.58962	−0.467906	−0.233953	+	0.972248i	$0.575166\pi$
$338$	0	0
$339$	−3.62896	−0.197098
$340$	0	0
$341$	2.58909	0.140207
$342$	0	0
$343$	20.1240	1.08659
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.0279	−1.12884	−0.564419	−	0.825488i	$-0.690900\pi$
$347$	−21.0279	−1.12884	−0.564419	+	0.825488i	$0.690900\pi$
$348$	0	0
$349$	−1.87189	−0.100200	−0.0501000	−	0.998744i	$-0.515954\pi$
$349$	−1.87189	−0.100200	−0.0501000	+	0.998744i	$0.515954\pi$
$350$	0	0
$351$	−0.752255	−0.0401524
$352$	0	0
$353$	−13.6810	−0.728164	−0.364082	−	0.931367i	$-0.618617\pi$
$353$	−13.6810	−0.728164	−0.364082	+	0.931367i	$0.618617\pi$
$354$	0	0
$355$	−8.89596	−0.472148
$356$	0	0
$357$	2.22982	0.118014
$358$	0	0
$359$	−25.4460	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$359$	−25.4460	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$360$	0	0
$361$	12.1366	0.638768
$362$	0	0
$363$	3.77715	0.198249
$364$	0	0
$365$	16.1602	0.845863
$366$	0	0
$367$	−34.7043	−1.81155	−0.905775	−	0.423759i	$-0.860710\pi$
$367$	−34.7043	−1.81155	−0.905775	+	0.423759i	$0.860710\pi$
$368$	0	0
$369$	−31.0668	−1.61727
$370$	0	0
$371$	−6.94567	−0.360601
$372$	0	0
$373$	6.10487	0.316098	0.158049	−	0.987431i	$-0.449480\pi$
$373$	6.10487	0.316098	0.158049	+	0.987431i	$0.449480\pi$
$374$	0	0
$375$	4.24050	0.218978
$376$	0	0
$377$	−1.98055	−0.102004
$378$	0	0
$379$	0.908029	0.0466423	0.0233212	−	0.999728i	$-0.492576\pi$
$379$	0.908029	0.0466423	0.0233212	+	0.999728i	$0.492576\pi$
$380$	0	0
$381$	3.35944	0.172110
$382$	0	0
$383$	−12.9761	−0.663047	−0.331524	−	0.943447i	$-0.607563\pi$
$383$	−12.9761	−0.663047	−0.331524	+	0.943447i	$0.607563\pi$
$384$	0	0
$385$	−2.19607	−0.111922
$386$	0	0
$387$	−27.9099	−1.41874
$388$	0	0
$389$	30.9217	1.56779	0.783896	−	0.620893i	$-0.213230\pi$
$389$	30.9217	1.56779	0.783896	+	0.620893i	$0.213230\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.01945	−0.101868
$394$	0	0
$395$	3.14908	0.158447
$396$	0	0
$397$	−19.5070	−0.979028	−0.489514	−	0.871995i	$-0.662826\pi$
$397$	−19.5070	−0.979028	−0.489514	+	0.871995i	$0.662826\pi$
$398$	0	0
$399$	−4.16004	−0.208263
$400$	0	0
$401$	−5.67553	−0.283422	−0.141711	−	0.989908i	$-0.545260\pi$
$401$	−5.67553	−0.283422	−0.141711	+	0.989908i	$0.545260\pi$
$402$	0	0
$403$	−1.38585	−0.0690342
$404$	0	0
$405$	12.3984	0.616080
$406$	0	0
$407$	1.73129	0.0858169
$408$	0	0
$409$	36.1964	1.78980	0.894900	−	0.446268i	$-0.147247\pi$
$409$	36.1964	1.78980	0.894900	+	0.446268i	$0.147247\pi$
$410$	0	0
$411$	6.51555	0.321388
$412$	0	0
$413$	3.68712	0.181431
$414$	0	0
$415$	−17.0110	−0.835036
$416$	0	0
$417$	6.57079	0.321773
$418$	0	0
$419$	27.6639	1.35147	0.675735	−	0.737145i	$-0.263826\pi$
$419$	27.6639	1.35147	0.675735	+	0.737145i	$0.263826\pi$
$420$	0	0
$421$	13.5503	0.660400	0.330200	−	0.943911i	$-0.392884\pi$
$421$	13.5503	0.660400	0.330200	+	0.943911i	$0.392884\pi$
$422$	0	0
$423$	24.6630	1.19916
$424$	0	0
$425$	−7.51909	−0.364729
$426$	0	0
$427$	8.61967	0.417135
$428$	0	0
$429$	0.0856665	0.00413602
$430$	0	0
$431$	17.6579	0.850552	0.425276	−	0.905064i	$-0.360177\pi$
$431$	17.6579	0.850552	0.425276	+	0.905064i	$0.360177\pi$
$432$	0	0
$433$	−25.1672	−1.20946	−0.604728	−	0.796432i	$-0.706718\pi$
$433$	−25.1672	−1.20946	−0.604728	+	0.796432i	$0.706718\pi$
$434$	0	0
$435$	−3.12277	−0.149725
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−17.2508	−0.823335	−0.411667	−	0.911334i	$-0.635053\pi$
$439$	−17.2508	−0.823335	−0.411667	+	0.911334i	$0.635053\pi$
$440$	0	0
$441$	7.64359	0.363981
$442$	0	0
$443$	−21.1949	−1.00700	−0.503501	−	0.863995i	$-0.667955\pi$
$443$	−21.1949	−1.00700	−0.503501	+	0.863995i	$0.667955\pi$
$444$	0	0
$445$	−4.29512	−0.203608
$446$	0	0
$447$	−7.73307	−0.365762
$448$	0	0
$449$	−7.32304	−0.345596	−0.172798	−	0.984957i	$-0.555281\pi$
$449$	−7.32304	−0.345596	−0.172798	+	0.984957i	$0.555281\pi$
$450$	0	0
$451$	7.23357	0.340616
$452$	0	0
$453$	8.33152	0.391449
$454$	0	0
$455$	1.17548	0.0551075
$456$	0	0
$457$	−5.66568	−0.265029	−0.132515	−	0.991181i	$-0.542305\pi$
$457$	−5.66568	−0.265029	−0.132515	+	0.991181i	$0.542305\pi$
$458$	0	0
$459$	6.28605	0.293408
$460$	0	0
$461$	−24.8998	−1.15970	−0.579850	−	0.814723i	$-0.696889\pi$
$461$	−24.8998	−1.15970	−0.579850	+	0.814723i	$0.696889\pi$
$462$	0	0
$463$	21.4178	0.995369	0.497684	−	0.867358i	$-0.334184\pi$
$463$	21.4178	0.995369	0.497684	+	0.867358i	$0.334184\pi$
$464$	0	0
$465$	−2.18510	−0.101331
$466$	0	0
$467$	8.61712	0.398753	0.199376	−	0.979923i	$-0.436108\pi$
$467$	8.61712	0.398753	0.199376	+	0.979923i	$0.436108\pi$
$468$	0	0
$469$	15.9611	0.737015
$470$	0	0
$471$	3.15343	0.145303
$472$	0	0
$473$	6.49852	0.298802
$474$	0	0
$475$	14.0280	0.643647
$476$	0	0
$477$	−9.57663	−0.438484
$478$	0	0
$479$	31.9922	1.46176	0.730881	−	0.682505i	$-0.239110\pi$
$479$	31.9922	1.46176	0.730881	+	0.682505i	$0.239110\pi$
$480$	0	0
$481$	−0.926702	−0.0422540
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.2562	−0.738157
$486$	0	0
$487$	20.7911	0.942137	0.471068	−	0.882097i	$-0.343868\pi$
$487$	20.7911	0.942137	0.471068	+	0.882097i	$0.343868\pi$
$488$	0	0
$489$	3.91078	0.176852
$490$	0	0
$491$	−31.8455	−1.43717	−0.718583	−	0.695441i	$-0.755209\pi$
$491$	−31.8455	−1.43717	−0.718583	+	0.695441i	$0.755209\pi$
$492$	0	0
$493$	16.5500	0.745377
$494$	0	0
$495$	−3.02792	−0.136095
$496$	0	0
$497$	11.7519	0.527144
$498$	0	0
$499$	−17.9108	−0.801797	−0.400898	−	0.916123i	$-0.631302\pi$
$499$	−17.9108	−0.801797	−0.400898	+	0.916123i	$0.631302\pi$
$500$	0	0
$501$	4.24677	0.189732
$502$	0	0
$503$	19.7891	0.882354	0.441177	−	0.897420i	$-0.354561\pi$
$503$	19.7891	0.882354	0.441177	+	0.897420i	$0.354561\pi$
$504$	0	0
$505$	−14.1048	−0.627655
$506$	0	0
$507$	4.60719	0.204612
$508$	0	0
$509$	−21.9497	−0.972902	−0.486451	−	0.873708i	$-0.661709\pi$
$509$	−21.9497	−0.972902	−0.486451	+	0.873708i	$0.661709\pi$
$510$	0	0
$511$	−21.3482	−0.944389
$512$	0	0
$513$	−11.7275	−0.517784
$514$	0	0
$515$	−28.9582	−1.27605
$516$	0	0
$517$	−5.74252	−0.252556
$518$	0	0
$519$	−1.47060	−0.0645521
$520$	0	0
$521$	−4.59515	−0.201317	−0.100659	−	0.994921i	$-0.532095\pi$
$521$	−4.59515	−0.201317	−0.100659	+	0.994921i	$0.532095\pi$
$522$	0	0
$523$	24.9860	1.09256	0.546281	−	0.837602i	$-0.316043\pi$
$523$	24.9860	1.09256	0.546281	+	0.837602i	$0.316043\pi$
$524$	0	0
$525$	−1.87422	−0.0817978
$526$	0	0
$527$	11.5806	0.504457
$528$	0	0
$529$	0	0
$530$	0	0
$531$	5.08377	0.220617
$532$	0	0
$533$	−3.87189	−0.167710
$534$	0	0
$535$	24.2857	1.04996
$536$	0	0
$537$	−1.09073	−0.0470687
$538$	0	0
$539$	−1.77973	−0.0766583
$540$	0	0
$541$	−16.8609	−0.724908	−0.362454	−	0.932002i	$-0.618061\pi$
$541$	−16.8609	−0.724908	−0.362454	+	0.932002i	$0.618061\pi$
$542$	0	0
$543$	3.60145	0.154553
$544$	0	0
$545$	14.9751	0.641463
$546$	0	0
$547$	33.8455	1.44713	0.723564	−	0.690257i	$-0.242503\pi$
$547$	33.8455	1.44713	0.723564	+	0.690257i	$0.242503\pi$
$548$	0	0
$549$	11.8847	0.507228
$550$	0	0
$551$	−30.8765	−1.31538
$552$	0	0
$553$	−4.16004	−0.176903
$554$	0	0
$555$	−1.46115	−0.0620223
$556$	0	0
$557$	−9.61398	−0.407357	−0.203679	−	0.979038i	$-0.565290\pi$
$557$	−9.61398	−0.407357	−0.203679	+	0.979038i	$0.565290\pi$
$558$	0	0
$559$	−3.47844	−0.147122
$560$	0	0
$561$	−0.715853	−0.0302233
$562$	0	0
$563$	11.0930	0.467516	0.233758	−	0.972295i	$-0.424898\pi$
$563$	11.0930	0.467516	0.233758	+	0.972295i	$0.424898\pi$
$564$	0	0
$565$	15.9861	0.672539
$566$	0	0
$567$	−16.3787	−0.687841
$568$	0	0
$569$	17.7556	0.744353	0.372176	−	0.928162i	$-0.378612\pi$
$569$	17.7556	0.744353	0.372176	+	0.928162i	$0.378612\pi$
$570$	0	0
$571$	−19.9241	−0.833800	−0.416900	−	0.908952i	$-0.636883\pi$
$571$	−19.9241	−0.833800	−0.416900	+	0.908952i	$0.636883\pi$
$572$	0	0
$573$	−5.49650	−0.229619
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.0753	−0.585962	−0.292981	−	0.956118i	$-0.594647\pi$
$577$	−14.0753	−0.585962	−0.292981	+	0.956118i	$0.594647\pi$
$578$	0	0
$579$	−0.466591	−0.0193908
$580$	0	0
$581$	22.4721	0.932300
$582$	0	0
$583$	2.22982	0.0923495
$584$	0	0
$585$	1.62075	0.0670096
$586$	0	0
$587$	−12.1670	−0.502186	−0.251093	−	0.967963i	$-0.580790\pi$
$587$	−12.1670	−0.502186	−0.251093	+	0.967963i	$0.580790\pi$
$588$	0	0
$589$	−21.6052	−0.890227
$590$	0	0
$591$	9.34696	0.384482
$592$	0	0
$593$	−40.9582	−1.68195	−0.840975	−	0.541074i	$-0.818018\pi$
$593$	−40.9582	−1.68195	−0.840975	+	0.541074i	$0.818018\pi$
$594$	0	0
$595$	−9.82266	−0.402690
$596$	0	0
$597$	4.19647	0.171750
$598$	0	0
$599$	29.4317	1.20255	0.601273	−	0.799043i	$-0.294660\pi$
$599$	29.4317	1.20255	0.601273	+	0.799043i	$0.294660\pi$
$600$	0	0
$601$	−16.0194	−0.653447	−0.326723	−	0.945120i	$-0.605945\pi$
$601$	−16.0194	−0.653447	−0.326723	+	0.945120i	$0.605945\pi$
$602$	0	0
$603$	22.0070	0.896196
$604$	0	0
$605$	−16.6389	−0.676466
$606$	0	0
$607$	−13.0932	−0.531438	−0.265719	−	0.964051i	$-0.585609\pi$
$607$	−13.0932	−0.531438	−0.265719	+	0.964051i	$0.585609\pi$
$608$	0	0
$609$	4.12530	0.167165
$610$	0	0
$611$	3.07378	0.124352
$612$	0	0
$613$	9.34714	0.377527	0.188764	−	0.982023i	$-0.439552\pi$
$613$	9.34714	0.377527	0.188764	+	0.982023i	$0.439552\pi$
$614$	0	0
$615$	−6.10487	−0.246172
$616$	0	0
$617$	−0.285516	−0.0114944	−0.00574722	−	0.999983i	$-0.501829\pi$
$617$	−0.285516	−0.0114944	−0.00574722	+	0.999983i	$0.501829\pi$
$618$	0	0
$619$	30.9097	1.24237	0.621183	−	0.783666i	$-0.286653\pi$
$619$	30.9097	1.24237	0.621183	+	0.783666i	$0.286653\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.67401	0.227324
$624$	0	0
$625$	−6.11018	−0.244407
$626$	0	0
$627$	1.33553	0.0533358
$628$	0	0
$629$	7.74378	0.308765
$630$	0	0
$631$	17.7919	0.708285	0.354142	−	0.935192i	$-0.384773\pi$
$631$	17.7919	0.708285	0.354142	+	0.935192i	$0.384773\pi$
$632$	0	0
$633$	9.46115	0.376047
$634$	0	0
$635$	−14.7988	−0.587274
$636$	0	0
$637$	0.952628	0.0377445
$638$	0	0
$639$	16.2034	0.640997
$640$	0	0
$641$	49.1803	1.94250	0.971252	−	0.238053i	$-0.0765092\pi$
$641$	49.1803	1.94250	0.971252	+	0.238053i	$0.0765092\pi$
$642$	0	0
$643$	−6.44187	−0.254043	−0.127021	−	0.991900i	$-0.540542\pi$
$643$	−6.44187	−0.254043	−0.127021	+	0.991900i	$0.540542\pi$
$644$	0	0
$645$	−5.48452	−0.215953
$646$	0	0
$647$	−26.9387	−1.05907	−0.529535	−	0.848288i	$-0.677634\pi$
$647$	−26.9387	−1.05907	−0.529535	+	0.848288i	$0.677634\pi$
$648$	0	0
$649$	−1.18370	−0.0464644
$650$	0	0
$651$	2.88659	0.113134
$652$	0	0
$653$	−29.2338	−1.14401	−0.572004	−	0.820251i	$-0.693834\pi$
$653$	−29.2338	−1.14401	−0.572004	+	0.820251i	$0.693834\pi$
$654$	0	0
$655$	8.89596	0.347594
$656$	0	0
$657$	−29.4347	−1.14836
$658$	0	0
$659$	3.83197	0.149272	0.0746361	−	0.997211i	$-0.476220\pi$
$659$	3.83197	0.149272	0.0746361	+	0.997211i	$0.476220\pi$
$660$	0	0
$661$	18.7768	0.730332	0.365166	−	0.930942i	$-0.381012\pi$
$661$	18.7768	0.730332	0.365166	+	0.930942i	$0.381012\pi$
$662$	0	0
$663$	0.383172	0.0148812
$664$	0	0
$665$	18.3256	0.710636
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−5.90526	−0.228310
$670$	0	0
$671$	−2.76723	−0.106828
$672$	0	0
$673$	14.9776	0.577344	0.288672	−	0.957428i	$-0.406786\pi$
$673$	14.9776	0.577344	0.288672	+	0.957428i	$0.406786\pi$
$674$	0	0
$675$	−5.28360	−0.203366
$676$	0	0
$677$	−13.8073	−0.530657	−0.265329	−	0.964158i	$-0.585480\pi$
$677$	−13.8073	−0.530657	−0.265329	+	0.964158i	$0.585480\pi$
$678$	0	0
$679$	21.4751	0.824137
$680$	0	0
$681$	−8.14375	−0.312069
$682$	0	0
$683$	0.128111	0.00490204	0.00245102	−	0.999997i	$-0.499220\pi$
$683$	0.128111	0.00490204	0.00245102	+	0.999997i	$0.499220\pi$
$684$	0	0
$685$	−28.7019	−1.09664
$686$	0	0
$687$	8.61927	0.328846
$688$	0	0
$689$	−1.19355	−0.0454705
$690$	0	0
$691$	−27.3230	−1.03942	−0.519709	−	0.854344i	$-0.673960\pi$
$691$	−27.3230	−1.03942	−0.519709	+	0.854344i	$0.673960\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−28.9453	−1.09796
$696$	0	0
$697$	32.3546	1.22552
$698$	0	0
$699$	−2.23678	−0.0846026
$700$	0	0
$701$	−33.0419	−1.24798	−0.623988	−	0.781434i	$-0.714489\pi$
$701$	−33.0419	−1.24798	−0.623988	+	0.781434i	$0.714489\pi$
$702$	0	0
$703$	−14.4471	−0.544884
$704$	0	0
$705$	4.84648	0.182529
$706$	0	0
$707$	18.6329	0.700764
$708$	0	0
$709$	−42.1222	−1.58193	−0.790967	−	0.611859i	$-0.790422\pi$
$709$	−42.1222	−1.58193	−0.790967	+	0.611859i	$0.790422\pi$
$710$	0	0
$711$	−5.73584	−0.215111
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−0.377373	−0.0141130
$716$	0	0
$717$	−8.72281	−0.325759
$718$	0	0
$719$	40.9193	1.52603	0.763015	−	0.646380i	$-0.223718\pi$
$719$	40.9193	1.52603	0.763015	+	0.646380i	$0.223718\pi$
$720$	0	0
$721$	38.2548	1.42468
$722$	0	0
$723$	−0.540370	−0.0200966
$724$	0	0
$725$	−13.9108	−0.516633
$726$	0	0
$727$	48.9353	1.81491	0.907454	−	0.420151i	$-0.138023\pi$
$727$	48.9353	1.81491	0.907454	+	0.420151i	$0.138023\pi$
$728$	0	0
$729$	−20.3261	−0.752818
$730$	0	0
$731$	29.0668	1.07508
$732$	0	0
$733$	11.6969	0.432034	0.216017	−	0.976390i	$-0.430693\pi$
$733$	11.6969	0.432034	0.216017	+	0.976390i	$0.430693\pi$
$734$	0	0
$735$	1.50203	0.0554031
$736$	0	0
$737$	−5.12410	−0.188749
$738$	0	0
$739$	32.4402	1.19333	0.596666	−	0.802490i	$-0.296492\pi$
$739$	32.4402	1.19333	0.596666	+	0.802490i	$0.296492\pi$
$740$	0	0
$741$	−0.714863	−0.0262612
$742$	0	0
$743$	38.1277	1.39877	0.699386	−	0.714744i	$-0.253457\pi$
$743$	38.1277	1.39877	0.699386	+	0.714744i	$0.253457\pi$
$744$	0	0
$745$	34.0653	1.24806
$746$	0	0
$747$	30.9844	1.13366
$748$	0	0
$749$	−32.0823	−1.17226
$750$	0	0
$751$	−31.8001	−1.16040	−0.580200	−	0.814474i	$-0.697026\pi$
$751$	−31.8001	−1.16040	−0.580200	+	0.814474i	$0.697026\pi$
$752$	0	0
$753$	−1.62391	−0.0591785
$754$	0	0
$755$	−36.7015	−1.33571
$756$	0	0
$757$	−49.4690	−1.79798	−0.898990	−	0.437969i	$-0.855698\pi$
$757$	−49.4690	−1.79798	−0.898990	+	0.437969i	$0.855698\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.28816	0.155446	0.0777228	−	0.996975i	$-0.475235\pi$
$761$	4.28816	0.155446	0.0777228	+	0.996975i	$0.475235\pi$
$762$	0	0
$763$	−19.7827	−0.716181
$764$	0	0
$765$	−13.5434	−0.489663
$766$	0	0
$767$	0.633596	0.0228778
$768$	0	0
$769$	38.4133	1.38522	0.692609	−	0.721313i	$-0.256461\pi$
$769$	38.4133	1.38522	0.692609	+	0.721313i	$0.256461\pi$
$770$	0	0
$771$	4.03640	0.145367
$772$	0	0
$773$	−38.6033	−1.38846	−0.694232	−	0.719752i	$-0.744256\pi$
$773$	−38.6033	−1.38846	−0.694232	+	0.719752i	$0.744256\pi$
$774$	0	0
$775$	−9.73378	−0.349648
$776$	0	0
$777$	1.93023	0.0692466
$778$	0	0
$779$	−60.3621	−2.16270
$780$	0	0
$781$	−3.77279	−0.135001
$782$	0	0
$783$	11.6296	0.415608
$784$	0	0
$785$	−13.8913	−0.495803
$786$	0	0
$787$	−8.01859	−0.285832	−0.142916	−	0.989735i	$-0.545648\pi$
$787$	−8.01859	−0.285832	−0.142916	+	0.989735i	$0.545648\pi$
$788$	0	0
$789$	−1.36488	−0.0485910
$790$	0	0
$791$	−21.1182	−0.750877
$792$	0	0
$793$	1.48121	0.0525992
$794$	0	0
$795$	−1.88188	−0.0667435
$796$	0	0
$797$	17.5931	0.623179	0.311589	−	0.950217i	$-0.399139\pi$
$797$	17.5931	0.623179	0.311589	+	0.950217i	$0.399139\pi$
$798$	0	0
$799$	−25.6853	−0.908682
$800$	0	0
$801$	7.82328	0.276422
$802$	0	0
$803$	6.85356	0.241857
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.04737	−0.0368692
$808$	0	0
$809$	−15.2841	−0.537362	−0.268681	−	0.963229i	$-0.586588\pi$
$809$	−15.2841	−0.537362	−0.268681	+	0.963229i	$0.586588\pi$
$810$	0	0
$811$	−53.1157	−1.86514	−0.932572	−	0.360983i	$-0.882441\pi$
$811$	−53.1157	−1.86514	−0.932572	+	0.360983i	$0.882441\pi$
$812$	0	0
$813$	7.24525	0.254102
$814$	0	0
$815$	−17.2276	−0.603455
$816$	0	0
$817$	−54.2284	−1.89721
$818$	0	0
$819$	−2.14107	−0.0748149
$820$	0	0
$821$	−20.8106	−0.726295	−0.363147	−	0.931732i	$-0.618298\pi$
$821$	−20.8106	−0.726295	−0.363147	+	0.931732i	$0.618298\pi$
$822$	0	0
$823$	15.5334	0.541461	0.270730	−	0.962655i	$-0.412735\pi$
$823$	15.5334	0.541461	0.270730	+	0.962655i	$0.412735\pi$
$824$	0	0
$825$	0.601695	0.0209483
$826$	0	0
$827$	15.3378	0.533348	0.266674	−	0.963787i	$-0.414075\pi$
$827$	15.3378	0.533348	0.266674	+	0.963787i	$0.414075\pi$
$828$	0	0
$829$	−44.6366	−1.55029	−0.775147	−	0.631781i	$-0.782324\pi$
$829$	−44.6366	−1.55029	−0.775147	+	0.631781i	$0.782324\pi$
$830$	0	0
$831$	−9.60318	−0.333131
$832$	0	0
$833$	−7.96042	−0.275812
$834$	0	0
$835$	−18.7076	−0.647405
$836$	0	0
$837$	8.13756	0.281275
$838$	0	0
$839$	18.0037	0.621558	0.310779	−	0.950482i	$-0.399410\pi$
$839$	18.0037	0.621558	0.310779	+	0.950482i	$0.399410\pi$
$840$	0	0
$841$	1.61862	0.0558144
$842$	0	0
$843$	−2.31556	−0.0797521
$844$	0	0
$845$	−20.2953	−0.698181
$846$	0	0
$847$	21.9806	0.755261
$848$	0	0
$849$	2.14775	0.0737106
$850$	0	0
$851$	0	0
$852$	0	0
$853$	31.2577	1.07024	0.535122	−	0.844775i	$-0.320266\pi$
$853$	31.2577	1.07024	0.535122	+	0.844775i	$0.320266\pi$
$854$	0	0
$855$	25.2672	0.864120
$856$	0	0
$857$	−22.2353	−0.759545	−0.379772	−	0.925080i	$-0.623998\pi$
$857$	−22.2353	−0.759545	−0.379772	+	0.925080i	$0.623998\pi$
$858$	0	0
$859$	2.37041	0.0808774	0.0404387	−	0.999182i	$-0.487124\pi$
$859$	2.37041	0.0808774	0.0404387	+	0.999182i	$0.487124\pi$
$860$	0	0
$861$	8.06476	0.274846
$862$	0	0
$863$	33.0474	1.12495	0.562473	−	0.826816i	$-0.309850\pi$
$863$	33.0474	1.12495	0.562473	+	0.826816i	$0.309850\pi$
$864$	0	0
$865$	6.47820	0.220265
$866$	0	0
$867$	2.88286	0.0979070
$868$	0	0
$869$	1.33553	0.0453047
$870$	0	0
$871$	2.74276	0.0929349
$872$	0	0
$873$	29.6097	1.00214
$874$	0	0
$875$	24.6770	0.834233
$876$	0	0
$877$	−2.13507	−0.0720963	−0.0360481	−	0.999350i	$-0.511477\pi$
$877$	−2.13507	−0.0720963	−0.0360481	+	0.999350i	$0.511477\pi$
$878$	0	0
$879$	−8.16027	−0.275239
$880$	0	0
$881$	−56.4237	−1.90096	−0.950481	−	0.310782i	$-0.899409\pi$
$881$	−56.4237	−1.90096	−0.950481	+	0.310782i	$0.899409\pi$
$882$	0	0
$883$	−52.4068	−1.76363	−0.881815	−	0.471596i	$-0.843678\pi$
$883$	−52.4068	−1.76363	−0.881815	+	0.471596i	$0.843678\pi$
$884$	0	0
$885$	0.999002	0.0335811
$886$	0	0
$887$	18.3968	0.617705	0.308852	−	0.951110i	$-0.400055\pi$
$887$	18.3968	0.617705	0.308852	+	0.951110i	$0.400055\pi$
$888$	0	0
$889$	19.5498	0.655679
$890$	0	0
$891$	5.25817	0.176155
$892$	0	0
$893$	47.9197	1.60357
$894$	0	0
$895$	4.80484	0.160608
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.4247	0.714555
$900$	0	0
$901$	9.97359	0.332269
$902$	0	0
$903$	7.24525	0.241107
$904$	0	0
$905$	−15.8649	−0.527368
$906$	0	0
$907$	−26.0903	−0.866316	−0.433158	−	0.901318i	$-0.642601\pi$
$907$	−26.0903	−0.866316	−0.433158	+	0.901318i	$0.642601\pi$
$908$	0	0
$909$	25.6910	0.852115
$910$	0	0
$911$	−36.2810	−1.20204	−0.601022	−	0.799232i	$-0.705240\pi$
$911$	−36.2810	−1.20204	−0.601022	+	0.799232i	$0.705240\pi$
$912$	0	0
$913$	−7.21438	−0.238761
$914$	0	0
$915$	2.33545	0.0772074
$916$	0	0
$917$	−11.7519	−0.388081
$918$	0	0
$919$	−8.33477	−0.274939	−0.137469	−	0.990506i	$-0.543897\pi$
$919$	−8.33477	−0.274939	−0.137469	+	0.990506i	$0.543897\pi$
$920$	0	0
$921$	5.04041	0.166087
$922$	0	0
$923$	2.01945	0.0664709
$924$	0	0
$925$	−6.50886	−0.214010
$926$	0	0
$927$	52.7454	1.73239
$928$	0	0
$929$	−4.80507	−0.157649	−0.0788246	−	0.996888i	$-0.525117\pi$
$929$	−4.80507	−0.157649	−0.0788246	+	0.996888i	$0.525117\pi$
$930$	0	0
$931$	14.8513	0.486733
$932$	0	0
$933$	0.935673	0.0306326
$934$	0	0
$935$	3.15343	0.103128
$936$	0	0
$937$	−2.12239	−0.0693356	−0.0346678	−	0.999399i	$-0.511037\pi$
$937$	−2.12239	−0.0693356	−0.0346678	+	0.999399i	$0.511037\pi$
$938$	0	0
$939$	1.98739	0.0648561
$940$	0	0
$941$	−46.6307	−1.52012	−0.760058	−	0.649855i	$-0.774830\pi$
$941$	−46.6307	−1.52012	−0.760058	+	0.649855i	$0.774830\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−6.90230	−0.224532
$946$	0	0
$947$	−6.12811	−0.199137	−0.0995684	−	0.995031i	$-0.531746\pi$
$947$	−6.12811	−0.199137	−0.0995684	+	0.995031i	$0.531746\pi$
$948$	0	0
$949$	−3.66848	−0.119084
$950$	0	0
$951$	−1.34945	−0.0437589
$952$	0	0
$953$	27.2127	0.881507	0.440753	−	0.897628i	$-0.354711\pi$
$953$	27.2127	0.881507	0.440753	+	0.897628i	$0.354711\pi$
$954$	0	0
$955$	24.2129	0.783510
$956$	0	0
$957$	−1.32437	−0.0428109
$958$	0	0
$959$	37.9163	1.22438
$960$	0	0
$961$	−16.0085	−0.516403
$962$	0	0
$963$	−44.2348	−1.42545
$964$	0	0
$965$	2.05540	0.0661657
$966$	0	0
$967$	24.2353	0.779356	0.389678	−	0.920951i	$-0.372586\pi$
$967$	24.2353	0.779356	0.389678	+	0.920951i	$0.372586\pi$
$968$	0	0
$969$	5.97359	0.191899
$970$	0	0
$971$	18.8635	0.605357	0.302679	−	0.953093i	$-0.402119\pi$
$971$	18.8635	0.605357	0.302679	+	0.953093i	$0.402119\pi$
$972$	0	0
$973$	38.2378	1.22585
$974$	0	0
$975$	−0.322067	−0.0103144
$976$	0	0
$977$	−32.1064	−1.02717	−0.513587	−	0.858037i	$-0.671684\pi$
$977$	−32.1064	−1.02717	−0.513587	+	0.858037i	$0.671684\pi$
$978$	0	0
$979$	−1.82157	−0.0582175
$980$	0	0
$981$	−27.2762	−0.870862
$982$	0	0
$983$	26.4584	0.843891	0.421945	−	0.906621i	$-0.361347\pi$
$983$	26.4584	0.843891	0.421945	+	0.906621i	$0.361347\pi$
$984$	0	0
$985$	−41.1747	−1.31193
$986$	0	0
$987$	−6.40238	−0.203790
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.7019	−0.530554	−0.265277	−	0.964172i	$-0.585463\pi$
$991$	−16.7019	−0.530554	−0.265277	+	0.964172i	$0.585463\pi$
$992$	0	0
$993$	−0.0892174	−0.00283123
$994$	0	0
$995$	−18.4860	−0.586047
$996$	0	0
$997$	0.621110	0.0196707	0.00983537	−	0.999952i	$-0.496869\pi$
$997$	0.621110	0.0196707	0.00983537	+	0.999952i	$0.496869\pi$
$998$	0	0
$999$	5.44149	0.172161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.by.1.4		6
4.3	odd	2		4232.2.a.v.1.4	yes	6
23.22	odd	2	inner	8464.2.a.by.1.3		6
92.91	even	2		4232.2.a.v.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.v.1.3	✓	6	92.91	even	2
4232.2.a.v.1.4	yes	6	4.3	odd	2
8464.2.a.by.1.3		6	23.22	odd	2	inner
8464.2.a.by.1.4		6	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.357926 q^{3} +1.57672 q^{5} -2.08290 q^{7} -2.87189 q^{9} +0.668688 q^{11} -0.357926 q^{13} -0.564349 q^{15} +2.99093 q^{17} -5.58002 q^{19} +0.745525 q^{21} -2.51396 q^{25} +2.10170 q^{27} +5.53341 q^{29} +3.87189 q^{31} -0.239341 q^{33} -3.28415 q^{35} +2.58909 q^{37} +0.128111 q^{39} +10.8176 q^{41} +9.71832 q^{43} -4.52816 q^{45} -8.58774 q^{47} -2.66152 q^{49} -1.07053 q^{51} +3.33461 q^{53} +1.05433 q^{55} +1.99724 q^{57} -1.77018 q^{59} -4.13830 q^{61} +5.98186 q^{63} -0.564349 q^{65} -7.66292 q^{67} -5.64207 q^{71} +10.2493 q^{73} +0.899813 q^{75} -1.39281 q^{77} +1.99724 q^{79} +7.86341 q^{81} -10.7889 q^{83} +4.71585 q^{85} -1.98055 q^{87} -2.72409 q^{89} +0.745525 q^{91} -1.38585 q^{93} -8.79811 q^{95} -10.3102 q^{97} -1.92040 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} - 40 q^{27} - 12 q^{29} - 4 q^{31} - 16 q^{35} + 28 q^{39} + 16 q^{41} - 28 q^{47} + 2 q^{49} + 28 q^{55} - 36 q^{59} - 32 q^{71} + 20 q^{73} - 80 q^{75}+ \cdots - 20 q^{95}+O(q^{100}) \) 6 * q - 4 * q^3 + 10 * q^9 - 4 * q^13 + 14 * q^25 - 40 * q^27 - 12 * q^29 - 4 * q^31 - 16 * q^35 + 28 * q^39 + 16 * q^41 - 28 * q^47 + 2 * q^49 + 28 * q^55 - 36 * q^59 - 32 * q^71 + 20 * q^73 - 80 * q^75 - 48 * q^77 + 62 * q^81 + 32 * q^85 - 28 * q^87 + 48 * q^93 - 20 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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