LMFDB - Embedded newform 8464.2.a.ci.1.10

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [15,0,1,0,-10,0,10,0,16,0,-9,0,0] 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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 $ x^15 - x^14 - 30x^13 + 26x^12 + 338x^11 - 238x^10 - 1773x^9 + 894x^8 + 4319x^7 - 1038x^6 - 4236x^5 - 759x^4 + 1320x^3 + 759x^2 + 154*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.881524$ 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)

$f(q)$	$=$	$q+0.881524 q^{3} +2.46842 q^{5} -1.39730 q^{7} -2.22292 q^{9} +0.760030 q^{11} -3.31151 q^{13} +2.17597 q^{15} +2.14059 q^{17} -0.416417 q^{19} -1.23176 q^{21} +1.09309 q^{25} -4.60412 q^{27} -7.29582 q^{29} +7.39924 q^{31} +0.669985 q^{33} -3.44913 q^{35} +7.55976 q^{37} -2.91918 q^{39} +0.0737081 q^{41} -1.34201 q^{43} -5.48709 q^{45} -5.43776 q^{47} -5.04755 q^{49} +1.88698 q^{51} -7.32667 q^{53} +1.87607 q^{55} -0.367081 q^{57} +14.9188 q^{59} -11.5155 q^{61} +3.10609 q^{63} -8.17420 q^{65} -11.2996 q^{67} +4.20758 q^{71} -6.28419 q^{73} +0.963582 q^{75} -1.06199 q^{77} +4.57589 q^{79} +2.61011 q^{81} -2.48083 q^{83} +5.28386 q^{85} -6.43144 q^{87} -6.81924 q^{89} +4.62718 q^{91} +6.52260 q^{93} -1.02789 q^{95} +6.48057 q^{97} -1.68948 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 10 q^{5} + 10 q^{7} + 16 q^{9} - 9 q^{11} + 12 q^{15} - 20 q^{17} - 3 q^{19} - 21 q^{21} + 11 q^{25} + 7 q^{27} - 4 q^{29} + 14 q^{31} - 34 q^{33} - 18 q^{35} - 26 q^{37} - 33 q^{39} - 11 q^{41}+ \cdots - 29 q^{99}+O(q^{100}) $ 15 * q + q^3 - 10 * q^5 + 10 * q^7 + 16 * q^9 - 9 * q^11 + 12 * q^15 - 20 * q^17 - 3 * q^19 - 21 * q^21 + 11 * q^25 + 7 * q^27 - 4 * q^29 + 14 * q^31 - 34 * q^33 - 18 * q^35 - 26 * q^37 - 33 * q^39 - 11 * q^41 + 10 * q^43 - 46 * q^45 - 12 * q^47 + 9 * q^49 - 29 * q^51 - 25 * q^53 + 24 * q^55 - 45 * q^57 - 22 * q^59 - 32 * q^61 + 25 * q^63 - 26 * q^65 + 28 * q^67 + 26 * q^71 - 39 * q^73 - 29 * q^75 - 38 * q^77 + 28 * q^79 + 51 * q^81 - 24 * q^83 + 18 * q^85 + 33 * q^87 - 66 * q^89 - 13 * q^91 + 41 * q^93 + 6 * q^95 - 56 * q^97 - 29 * 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
$3$	0.881524	0.508948	0.254474	−	0.967080i	$-0.418098\pi$
$3$	0.881524	0.508948	0.254474	+	0.967080i	$0.418098\pi$
$4$	0	0
$5$	2.46842	1.10391	0.551955	−	0.833874i	$-0.313882\pi$
$5$	2.46842	1.10391	0.551955	+	0.833874i	$0.313882\pi$
$6$	0	0
$7$	−1.39730	−0.528131	−0.264065	−	0.964505i	$-0.585063\pi$
$7$	−1.39730	−0.528131	−0.264065	+	0.964505i	$0.585063\pi$
$8$	0	0
$9$	−2.22292	−0.740972
$10$	0	0
$11$	0.760030	0.229158	0.114579	−	0.993414i	$-0.463448\pi$
$11$	0.760030	0.229158	0.114579	+	0.993414i	$0.463448\pi$
$12$	0	0
$13$	−3.31151	−0.918448	−0.459224	−	0.888320i	$-0.651873\pi$
$13$	−3.31151	−0.918448	−0.459224	+	0.888320i	$0.651873\pi$
$14$	0	0
$15$	2.17597	0.561833
$16$	0	0
$17$	2.14059	0.519168	0.259584	−	0.965721i	$-0.416415\pi$
$17$	2.14059	0.519168	0.259584	+	0.965721i	$0.416415\pi$
$18$	0	0
$19$	−0.416417	−0.0955325	−0.0477663	−	0.998859i	$-0.515210\pi$
$19$	−0.416417	−0.0955325	−0.0477663	+	0.998859i	$0.515210\pi$
$20$	0	0
$21$	−1.23176	−0.268791
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.09309	0.218617
$26$	0	0
$27$	−4.60412	−0.886064
$28$	0	0
$29$	−7.29582	−1.35480	−0.677400	−	0.735615i	$-0.736893\pi$
$29$	−7.29582	−1.35480	−0.677400	+	0.735615i	$0.736893\pi$
$30$	0	0
$31$	7.39924	1.32894	0.664471	−	0.747314i	$-0.268657\pi$
$31$	7.39924	1.32894	0.664471	+	0.747314i	$0.268657\pi$
$32$	0	0
$33$	0.669985	0.116629
$34$	0	0
$35$	−3.44913	−0.583009
$36$	0	0
$37$	7.55976	1.24282	0.621408	−	0.783487i	$-0.286561\pi$
$37$	7.55976	1.24282	0.621408	+	0.783487i	$0.286561\pi$
$38$	0	0
$39$	−2.91918	−0.467442
$40$	0	0
$41$	0.0737081	0.0115113	0.00575563	−	0.999983i	$-0.498168\pi$
$41$	0.0737081	0.0115113	0.00575563	+	0.999983i	$0.498168\pi$
$42$	0	0
$43$	−1.34201	−0.204655	−0.102327	−	0.994751i	$-0.532629\pi$
$43$	−1.34201	−0.204655	−0.102327	+	0.994751i	$0.532629\pi$
$44$	0	0
$45$	−5.48709	−0.817966
$46$	0	0
$47$	−5.43776	−0.793179	−0.396590	−	0.917996i	$-0.629806\pi$
$47$	−5.43776	−0.793179	−0.396590	+	0.917996i	$0.629806\pi$
$48$	0	0
$49$	−5.04755	−0.721078
$50$	0	0
$51$	1.88698	0.264230
$52$	0	0
$53$	−7.32667	−1.00640	−0.503198	−	0.864171i	$-0.667843\pi$
$53$	−7.32667	−1.00640	−0.503198	+	0.864171i	$0.667843\pi$
$54$	0	0
$55$	1.87607	0.252970
$56$	0	0
$57$	−0.367081	−0.0486211
$58$	0	0
$59$	14.9188	1.94226	0.971130	−	0.238550i	$-0.0766721\pi$
$59$	14.9188	1.94226	0.971130	+	0.238550i	$0.0766721\pi$
$60$	0	0
$61$	−11.5155	−1.47441	−0.737204	−	0.675670i	$-0.763854\pi$
$61$	−11.5155	−1.47441	−0.737204	+	0.675670i	$0.763854\pi$
$62$	0	0
$63$	3.10609	0.391330
$64$	0	0
$65$	−8.17420	−1.01388
$66$	0	0
$67$	−11.2996	−1.38046	−0.690231	−	0.723589i	$-0.742491\pi$
$67$	−11.2996	−1.38046	−0.690231	+	0.723589i	$0.742491\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.20758	0.499347	0.249674	−	0.968330i	$-0.419677\pi$
$71$	4.20758	0.499347	0.249674	+	0.968330i	$0.419677\pi$
$72$	0	0
$73$	−6.28419	−0.735509	−0.367754	−	0.929923i	$-0.619873\pi$
$73$	−6.28419	−0.735509	−0.367754	+	0.929923i	$0.619873\pi$
$74$	0	0
$75$	0.963582	0.111265
$76$	0	0
$77$	−1.06199	−0.121025
$78$	0	0
$79$	4.57589	0.514827	0.257414	−	0.966301i	$-0.417130\pi$
$79$	4.57589	0.514827	0.257414	+	0.966301i	$0.417130\pi$
$80$	0	0
$81$	2.61011	0.290012
$82$	0	0
$83$	−2.48083	−0.272307	−0.136153	−	0.990688i	$-0.543474\pi$
$83$	−2.48083	−0.272307	−0.136153	+	0.990688i	$0.543474\pi$
$84$	0	0
$85$	5.28386	0.573115
$86$	0	0
$87$	−6.43144	−0.689523
$88$	0	0
$89$	−6.81924	−0.722838	−0.361419	−	0.932404i	$-0.617708\pi$
$89$	−6.81924	−0.722838	−0.361419	+	0.932404i	$0.617708\pi$
$90$	0	0
$91$	4.62718	0.485061
$92$	0	0
$93$	6.52260	0.676362
$94$	0	0
$95$	−1.02789	−0.105459
$96$	0	0
$97$	6.48057	0.658003	0.329001	−	0.944329i	$-0.393288\pi$
$97$	6.48057	0.658003	0.329001	+	0.944329i	$0.393288\pi$
$98$	0	0
$99$	−1.68948	−0.169800
$100$	0	0
$101$	6.27270	0.624157	0.312078	−	0.950056i	$-0.398975\pi$
$101$	6.27270	0.624157	0.312078	+	0.950056i	$0.398975\pi$
$102$	0	0
$103$	16.4562	1.62148	0.810738	−	0.585409i	$-0.199066\pi$
$103$	16.4562	1.62148	0.810738	+	0.585409i	$0.199066\pi$
$104$	0	0
$105$	−3.04049	−0.296721
$106$	0	0
$107$	−14.9145	−1.44184	−0.720918	−	0.693020i	$-0.756280\pi$
$107$	−14.9145	−1.44184	−0.720918	+	0.693020i	$0.756280\pi$
$108$	0	0
$109$	−11.7901	−1.12929	−0.564645	−	0.825334i	$-0.690987\pi$
$109$	−11.7901	−1.12929	−0.564645	+	0.825334i	$0.690987\pi$
$110$	0	0
$111$	6.66410	0.632529
$112$	0	0
$113$	6.57582	0.618601	0.309300	−	0.950964i	$-0.399905\pi$
$113$	6.57582	0.618601	0.309300	+	0.950964i	$0.399905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.36121	0.680544
$118$	0	0
$119$	−2.99104	−0.274189
$120$	0	0
$121$	−10.4224	−0.947487
$122$	0	0
$123$	0.0649754	0.00585863
$124$	0	0
$125$	−9.64389	−0.862576
$126$	0	0
$127$	−18.8602	−1.67357	−0.836785	−	0.547532i	$-0.815567\pi$
$127$	−18.8602	−1.67357	−0.836785	+	0.547532i	$0.815567\pi$
$128$	0	0
$129$	−1.18301	−0.104159
$130$	0	0
$131$	−14.5636	−1.27242	−0.636212	−	0.771514i	$-0.719500\pi$
$131$	−14.5636	−1.27242	−0.636212	+	0.771514i	$0.719500\pi$
$132$	0	0
$133$	0.581860	0.0504537
$134$	0	0
$135$	−11.3649	−0.978135
$136$	0	0
$137$	−18.9527	−1.61924	−0.809619	−	0.586956i	$-0.800326\pi$
$137$	−18.9527	−1.61924	−0.809619	+	0.586956i	$0.800326\pi$
$138$	0	0
$139$	1.46141	0.123955	0.0619776	−	0.998078i	$-0.480259\pi$
$139$	1.46141	0.123955	0.0619776	+	0.998078i	$0.480259\pi$
$140$	0	0
$141$	−4.79352	−0.403687
$142$	0	0
$143$	−2.51685	−0.210470
$144$	0	0
$145$	−18.0091	−1.49558
$146$	0	0
$147$	−4.44953	−0.366991
$148$	0	0
$149$	−12.9255	−1.05890	−0.529448	−	0.848343i	$-0.677601\pi$
$149$	−12.9255	−1.05890	−0.529448	+	0.848343i	$0.677601\pi$
$150$	0	0
$151$	−14.5932	−1.18757	−0.593787	−	0.804622i	$-0.702368\pi$
$151$	−14.5932	−1.18757	−0.593787	+	0.804622i	$0.702368\pi$
$152$	0	0
$153$	−4.75834	−0.384689
$154$	0	0
$155$	18.2644	1.46703
$156$	0	0
$157$	1.67568	0.133734	0.0668669	−	0.997762i	$-0.478700\pi$
$157$	1.67568	0.133734	0.0668669	+	0.997762i	$0.478700\pi$
$158$	0	0
$159$	−6.45863	−0.512203
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.53902	0.277197	0.138599	−	0.990349i	$-0.455740\pi$
$163$	3.53902	0.277197	0.138599	+	0.990349i	$0.455740\pi$
$164$	0	0
$165$	1.65380	0.128748
$166$	0	0
$167$	22.0439	1.70581	0.852903	−	0.522070i	$-0.174840\pi$
$167$	22.0439	1.70581	0.852903	+	0.522070i	$0.174840\pi$
$168$	0	0
$169$	−2.03389	−0.156453
$170$	0	0
$171$	0.925659	0.0707869
$172$	0	0
$173$	−13.2523	−1.00755	−0.503776	−	0.863834i	$-0.668056\pi$
$173$	−13.2523	−1.00755	−0.503776	+	0.863834i	$0.668056\pi$
$174$	0	0
$175$	−1.52737	−0.115459
$176$	0	0
$177$	13.1513	0.988509
$178$	0	0
$179$	19.6745	1.47054	0.735272	−	0.677773i	$-0.237055\pi$
$179$	19.6745	1.47054	0.735272	+	0.677773i	$0.237055\pi$
$180$	0	0
$181$	9.41494	0.699807	0.349903	−	0.936786i	$-0.386214\pi$
$181$	9.41494	0.699807	0.349903	+	0.936786i	$0.386214\pi$
$182$	0	0
$183$	−10.1512	−0.750397
$184$	0	0
$185$	18.6606	1.37196
$186$	0	0
$187$	1.62691	0.118971
$188$	0	0
$189$	6.43335	0.467958
$190$	0	0
$191$	−18.9042	−1.36786	−0.683932	−	0.729546i	$-0.739731\pi$
$191$	−18.9042	−1.36786	−0.683932	+	0.729546i	$0.739731\pi$
$192$	0	0
$193$	23.1196	1.66418	0.832092	−	0.554638i	$-0.187143\pi$
$193$	23.1196	1.66418	0.832092	+	0.554638i	$0.187143\pi$
$194$	0	0
$195$	−7.20575	−0.516014
$196$	0	0
$197$	−0.718846	−0.0512157	−0.0256078	−	0.999672i	$-0.508152\pi$
$197$	−0.718846	−0.0512157	−0.0256078	+	0.999672i	$0.508152\pi$
$198$	0	0
$199$	−4.03259	−0.285863	−0.142931	−	0.989733i	$-0.545653\pi$
$199$	−4.03259	−0.285863	−0.142931	+	0.989733i	$0.545653\pi$
$200$	0	0
$201$	−9.96084	−0.702583
$202$	0	0
$203$	10.1945	0.715512
$204$	0	0
$205$	0.181942	0.0127074
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.316489	−0.0218920
$210$	0	0
$211$	−13.2787	−0.914143	−0.457071	−	0.889430i	$-0.651102\pi$
$211$	−13.2787	−0.914143	−0.457071	+	0.889430i	$0.651102\pi$
$212$	0	0
$213$	3.70908	0.254142
$214$	0	0
$215$	−3.31264	−0.225920
$216$	0	0
$217$	−10.3390	−0.701855
$218$	0	0
$219$	−5.53966	−0.374336
$220$	0	0
$221$	−7.08857	−0.476829
$222$	0	0
$223$	−23.8254	−1.59547	−0.797733	−	0.603011i	$-0.793968\pi$
$223$	−23.8254	−1.59547	−0.797733	+	0.603011i	$0.793968\pi$
$224$	0	0
$225$	−2.42984	−0.161989
$226$	0	0
$227$	−3.55519	−0.235967	−0.117983	−	0.993016i	$-0.537643\pi$
$227$	−3.55519	−0.235967	−0.117983	+	0.993016i	$0.537643\pi$
$228$	0	0
$229$	−11.9024	−0.786535	−0.393267	−	0.919424i	$-0.628655\pi$
$229$	−11.9024	−0.786535	−0.393267	+	0.919424i	$0.628655\pi$
$230$	0	0
$231$	−0.936171	−0.0615955
$232$	0	0
$233$	−6.46878	−0.423784	−0.211892	−	0.977293i	$-0.567962\pi$
$233$	−6.46878	−0.423784	−0.211892	+	0.977293i	$0.567962\pi$
$234$	0	0
$235$	−13.4227	−0.875599
$236$	0	0
$237$	4.03375	0.262020
$238$	0	0
$239$	9.24849	0.598235	0.299117	−	0.954216i	$-0.403308\pi$
$239$	9.24849	0.598235	0.299117	+	0.954216i	$0.403308\pi$
$240$	0	0
$241$	−30.3448	−1.95468	−0.977339	−	0.211678i	$-0.932107\pi$
$241$	−30.3448	−1.95468	−0.977339	+	0.211678i	$0.932107\pi$
$242$	0	0
$243$	16.1132	1.03366
$244$	0	0
$245$	−12.4595	−0.796005
$246$	0	0
$247$	1.37897	0.0877417
$248$	0	0
$249$	−2.18691	−0.138590
$250$	0	0
$251$	−21.6362	−1.36566	−0.682831	−	0.730577i	$-0.739251\pi$
$251$	−21.6362	−1.36566	−0.682831	+	0.730577i	$0.739251\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.65785	0.291686
$256$	0	0
$257$	1.24554	0.0776948	0.0388474	−	0.999245i	$-0.487631\pi$
$257$	1.24554	0.0776948	0.0388474	+	0.999245i	$0.487631\pi$
$258$	0	0
$259$	−10.5633	−0.656370
$260$	0	0
$261$	16.2180	1.00387
$262$	0	0
$263$	21.0621	1.29875	0.649374	−	0.760469i	$-0.275031\pi$
$263$	21.0621	1.29875	0.649374	+	0.760469i	$0.275031\pi$
$264$	0	0
$265$	−18.0853	−1.11097
$266$	0	0
$267$	−6.01132	−0.367887
$268$	0	0
$269$	17.9626	1.09520	0.547600	−	0.836740i	$-0.315542\pi$
$269$	17.9626	1.09520	0.547600	+	0.836740i	$0.315542\pi$
$270$	0	0
$271$	7.00843	0.425732	0.212866	−	0.977081i	$-0.431720\pi$
$271$	7.00843	0.425732	0.212866	+	0.977081i	$0.431720\pi$
$272$	0	0
$273$	4.07897	0.246871
$274$	0	0
$275$	0.830779	0.0500979
$276$	0	0
$277$	−13.2905	−0.798546	−0.399273	−	0.916832i	$-0.630737\pi$
$277$	−13.2905	−0.798546	−0.399273	+	0.916832i	$0.630737\pi$
$278$	0	0
$279$	−16.4479	−0.984709
$280$	0	0
$281$	−24.7593	−1.47701	−0.738507	−	0.674245i	$-0.764469\pi$
$281$	−24.7593	−1.47701	−0.738507	+	0.674245i	$0.764469\pi$
$282$	0	0
$283$	28.4689	1.69230	0.846151	−	0.532943i	$-0.178914\pi$
$283$	28.4689	1.69230	0.846151	+	0.532943i	$0.178914\pi$
$284$	0	0
$285$	−0.906109	−0.0536733
$286$	0	0
$287$	−0.102992	−0.00607945
$288$	0	0
$289$	−12.4179	−0.730464
$290$	0	0
$291$	5.71278	0.334889
$292$	0	0
$293$	18.7500	1.09539	0.547694	−	0.836679i	$-0.315506\pi$
$293$	18.7500	1.09539	0.547694	+	0.836679i	$0.315506\pi$
$294$	0	0
$295$	36.8258	2.14408
$296$	0	0
$297$	−3.49927	−0.203048
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.87520	0.108084
$302$	0	0
$303$	5.52953	0.317663
$304$	0	0
$305$	−28.4251	−1.62761
$306$	0	0
$307$	14.9902	0.855534	0.427767	−	0.903889i	$-0.359300\pi$
$307$	14.9902	0.855534	0.427767	+	0.903889i	$0.359300\pi$
$308$	0	0
$309$	14.5065	0.825247
$310$	0	0
$311$	−1.06000	−0.0601070	−0.0300535	−	0.999548i	$-0.509568\pi$
$311$	−1.06000	−0.0601070	−0.0300535	+	0.999548i	$0.509568\pi$
$312$	0	0
$313$	−10.0204	−0.566388	−0.283194	−	0.959063i	$-0.591394\pi$
$313$	−10.0204	−0.566388	−0.283194	+	0.959063i	$0.591394\pi$
$314$	0	0
$315$	7.66712	0.431993
$316$	0	0
$317$	0.0673022	0.00378007	0.00189003	−	0.999998i	$-0.499398\pi$
$317$	0.0673022	0.00378007	0.00189003	+	0.999998i	$0.499398\pi$
$318$	0	0
$319$	−5.54505	−0.310463
$320$	0	0
$321$	−13.1475	−0.733819
$322$	0	0
$323$	−0.891375	−0.0495974
$324$	0	0
$325$	−3.61977	−0.200789
$326$	0	0
$327$	−10.3933	−0.574750
$328$	0	0
$329$	7.59820	0.418902
$330$	0	0
$331$	23.4553	1.28922	0.644611	−	0.764511i	$-0.277019\pi$
$331$	23.4553	1.28922	0.644611	+	0.764511i	$0.277019\pi$
$332$	0	0
$333$	−16.8047	−0.920892
$334$	0	0
$335$	−27.8921	−1.52391
$336$	0	0
$337$	−27.1819	−1.48069	−0.740347	−	0.672225i	$-0.765339\pi$
$337$	−27.1819	−1.48069	−0.740347	+	0.672225i	$0.765339\pi$
$338$	0	0
$339$	5.79674	0.314836
$340$	0	0
$341$	5.62365	0.304537
$342$	0	0
$343$	16.8341	0.908954
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.7174	0.629021	0.314511	−	0.949254i	$-0.398160\pi$
$347$	11.7174	0.629021	0.314511	+	0.949254i	$0.398160\pi$
$348$	0	0
$349$	12.0885	0.647081	0.323540	−	0.946214i	$-0.395127\pi$
$349$	12.0885	0.647081	0.323540	+	0.946214i	$0.395127\pi$
$350$	0	0
$351$	15.2466	0.813804
$352$	0	0
$353$	−7.41442	−0.394630	−0.197315	−	0.980340i	$-0.563222\pi$
$353$	−7.41442	−0.394630	−0.197315	+	0.980340i	$0.563222\pi$
$354$	0	0
$355$	10.3861	0.551235
$356$	0	0
$357$	−2.63668	−0.139548
$358$	0	0
$359$	−7.77344	−0.410267	−0.205133	−	0.978734i	$-0.565763\pi$
$359$	−7.77344	−0.410267	−0.205133	+	0.978734i	$0.565763\pi$
$360$	0	0
$361$	−18.8266	−0.990874
$362$	0	0
$363$	−9.18755	−0.482221
$364$	0	0
$365$	−15.5120	−0.811935
$366$	0	0
$367$	−24.5601	−1.28203	−0.641013	−	0.767530i	$-0.721486\pi$
$367$	−24.5601	−1.28203	−0.641013	+	0.767530i	$0.721486\pi$
$368$	0	0
$369$	−0.163847	−0.00852953
$370$	0	0
$371$	10.2376	0.531508
$372$	0	0
$373$	12.7259	0.658921	0.329461	−	0.944169i	$-0.393133\pi$
$373$	12.7259	0.658921	0.329461	+	0.944169i	$0.393133\pi$
$374$	0	0
$375$	−8.50132	−0.439006
$376$	0	0
$377$	24.1602	1.24431
$378$	0	0
$379$	−15.2724	−0.784490	−0.392245	−	0.919861i	$-0.628302\pi$
$379$	−15.2724	−0.784490	−0.392245	+	0.919861i	$0.628302\pi$
$380$	0	0
$381$	−16.6257	−0.851760
$382$	0	0
$383$	10.0601	0.514047	0.257023	−	0.966405i	$-0.417258\pi$
$383$	10.0601	0.514047	0.257023	+	0.966405i	$0.417258\pi$
$384$	0	0
$385$	−2.62144	−0.133601
$386$	0	0
$387$	2.98318	0.151643
$388$	0	0
$389$	−3.85603	−0.195509	−0.0977543	−	0.995211i	$-0.531166\pi$
$389$	−3.85603	−0.195509	−0.0977543	+	0.995211i	$0.531166\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12.8381	−0.647598
$394$	0	0
$395$	11.2952	0.568323
$396$	0	0
$397$	0.395798	0.0198645	0.00993226	−	0.999951i	$-0.496838\pi$
$397$	0.395798	0.0198645	0.00993226	+	0.999951i	$0.496838\pi$
$398$	0	0
$399$	0.512923	0.0256783
$400$	0	0
$401$	22.4025	1.11873	0.559364	−	0.828922i	$-0.311045\pi$
$401$	22.4025	1.11873	0.559364	+	0.828922i	$0.311045\pi$
$402$	0	0
$403$	−24.5027	−1.22056
$404$	0	0
$405$	6.44283	0.320147
$406$	0	0
$407$	5.74565	0.284801
$408$	0	0
$409$	12.6652	0.626256	0.313128	−	0.949711i	$-0.398623\pi$
$409$	12.6652	0.626256	0.313128	+	0.949711i	$0.398623\pi$
$410$	0	0
$411$	−16.7072	−0.824108
$412$	0	0
$413$	−20.8461	−1.02577
$414$	0	0
$415$	−6.12373	−0.300602
$416$	0	0
$417$	1.28827	0.0630868
$418$	0	0
$419$	28.3843	1.38666	0.693332	−	0.720618i	$-0.256142\pi$
$419$	28.3843	1.38666	0.693332	+	0.720618i	$0.256142\pi$
$420$	0	0
$421$	11.4998	0.560465	0.280232	−	0.959932i	$-0.409588\pi$
$421$	11.4998	0.560465	0.280232	+	0.959932i	$0.409588\pi$
$422$	0	0
$423$	12.0877	0.587724
$424$	0	0
$425$	2.33984	0.113499
$426$	0	0
$427$	16.0906	0.778680
$428$	0	0
$429$	−2.21866	−0.107118
$430$	0	0
$431$	−9.17736	−0.442058	−0.221029	−	0.975267i	$-0.570942\pi$
$431$	−9.17736	−0.442058	−0.221029	+	0.975267i	$0.570942\pi$
$432$	0	0
$433$	16.8170	0.808174	0.404087	−	0.914721i	$-0.367589\pi$
$433$	16.8170	0.808174	0.404087	+	0.914721i	$0.367589\pi$
$434$	0	0
$435$	−15.8755	−0.761171
$436$	0	0
$437$	0	0
$438$	0	0
$439$	32.5131	1.55177	0.775883	−	0.630877i	$-0.217305\pi$
$439$	32.5131	1.55177	0.775883	+	0.630877i	$0.217305\pi$
$440$	0	0
$441$	11.2203	0.534299
$442$	0	0
$443$	1.96531	0.0933745	0.0466873	−	0.998910i	$-0.485134\pi$
$443$	1.96531	0.0933745	0.0466873	+	0.998910i	$0.485134\pi$
$444$	0	0
$445$	−16.8327	−0.797948
$446$	0	0
$447$	−11.3941	−0.538922
$448$	0	0
$449$	13.0311	0.614977	0.307489	−	0.951552i	$-0.400511\pi$
$449$	13.0311	0.614977	0.307489	+	0.951552i	$0.400511\pi$
$450$	0	0
$451$	0.0560204	0.00263790
$452$	0	0
$453$	−12.8642	−0.604413
$454$	0	0
$455$	11.4218	0.535463
$456$	0	0
$457$	−37.7850	−1.76751	−0.883753	−	0.467953i	$-0.844991\pi$
$457$	−37.7850	−1.76751	−0.883753	+	0.467953i	$0.844991\pi$
$458$	0	0
$459$	−9.85552	−0.460016
$460$	0	0
$461$	26.2432	1.22227	0.611133	−	0.791528i	$-0.290714\pi$
$461$	26.2432	1.22227	0.611133	+	0.791528i	$0.290714\pi$
$462$	0	0
$463$	−8.48170	−0.394178	−0.197089	−	0.980386i	$-0.563149\pi$
$463$	−8.48170	−0.394178	−0.197089	+	0.980386i	$0.563149\pi$
$464$	0	0
$465$	16.1005	0.746643
$466$	0	0
$467$	−10.1461	−0.469505	−0.234753	−	0.972055i	$-0.575428\pi$
$467$	−10.1461	−0.469505	−0.234753	+	0.972055i	$0.575428\pi$
$468$	0	0
$469$	15.7889	0.729065
$470$	0	0
$471$	1.47715	0.0680635
$472$	0	0
$473$	−1.01997	−0.0468982
$474$	0	0
$475$	−0.455179	−0.0208851
$476$	0	0
$477$	16.2866	0.745711
$478$	0	0
$479$	23.5492	1.07599	0.537995	−	0.842948i	$-0.319182\pi$
$479$	23.5492	1.07599	0.537995	+	0.842948i	$0.319182\pi$
$480$	0	0
$481$	−25.0342	−1.14146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.9968	0.726376
$486$	0	0
$487$	−10.3092	−0.467153	−0.233577	−	0.972338i	$-0.575043\pi$
$487$	−10.3092	−0.467153	−0.233577	+	0.972338i	$0.575043\pi$
$488$	0	0
$489$	3.11973	0.141079
$490$	0	0
$491$	7.92361	0.357588	0.178794	−	0.983887i	$-0.442780\pi$
$491$	7.92361	0.357588	0.178794	+	0.983887i	$0.442780\pi$
$492$	0	0
$493$	−15.6173	−0.703369
$494$	0	0
$495$	−4.17035	−0.187443
$496$	0	0
$497$	−5.87926	−0.263721
$498$	0	0
$499$	16.8716	0.755274	0.377637	−	0.925954i	$-0.376737\pi$
$499$	16.8716	0.755274	0.377637	+	0.925954i	$0.376737\pi$
$500$	0	0
$501$	19.4322	0.868166
$502$	0	0
$503$	12.8159	0.571431	0.285715	−	0.958315i	$-0.407769\pi$
$503$	12.8159	0.571431	0.285715	+	0.958315i	$0.407769\pi$
$504$	0	0
$505$	15.4836	0.689013
$506$	0	0
$507$	−1.79292	−0.0796264
$508$	0	0
$509$	−17.7782	−0.788005	−0.394003	−	0.919109i	$-0.628910\pi$
$509$	−17.7782	−0.788005	−0.394003	+	0.919109i	$0.628910\pi$
$510$	0	0
$511$	8.78091	0.388445
$512$	0	0
$513$	1.91723	0.0846479
$514$	0	0
$515$	40.6207	1.78996
$516$	0	0
$517$	−4.13287	−0.181763
$518$	0	0
$519$	−11.6822	−0.512792
$520$	0	0
$521$	37.8204	1.65694	0.828470	−	0.560033i	$-0.189211\pi$
$521$	37.8204	1.65694	0.828470	+	0.560033i	$0.189211\pi$
$522$	0	0
$523$	3.22117	0.140852	0.0704260	−	0.997517i	$-0.477564\pi$
$523$	3.22117	0.140852	0.0704260	+	0.997517i	$0.477564\pi$
$524$	0	0
$525$	−1.34641	−0.0587624
$526$	0	0
$527$	15.8387	0.689945
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−33.1632	−1.43916
$532$	0	0
$533$	−0.244085	−0.0105725
$534$	0	0
$535$	−36.8151	−1.59166
$536$	0	0
$537$	17.3436	0.748430
$538$	0	0
$539$	−3.83629	−0.165241
$540$	0	0
$541$	−18.2781	−0.785837	−0.392918	−	0.919573i	$-0.628534\pi$
$541$	−18.2781	−0.785837	−0.392918	+	0.919573i	$0.628534\pi$
$542$	0	0
$543$	8.29949	0.356165
$544$	0	0
$545$	−29.1030	−1.24664
$546$	0	0
$547$	−7.58254	−0.324206	−0.162103	−	0.986774i	$-0.551828\pi$
$547$	−7.58254	−0.324206	−0.162103	+	0.986774i	$0.551828\pi$
$548$	0	0
$549$	25.5980	1.09250
$550$	0	0
$551$	3.03810	0.129427
$552$	0	0
$553$	−6.39390	−0.271896
$554$	0	0
$555$	16.4498	0.698255
$556$	0	0
$557$	5.32598	0.225669	0.112835	−	0.993614i	$-0.464007\pi$
$557$	5.32598	0.225669	0.112835	+	0.993614i	$0.464007\pi$
$558$	0	0
$559$	4.44409	0.187965
$560$	0	0
$561$	1.43416	0.0605502
$562$	0	0
$563$	−28.0589	−1.18254	−0.591272	−	0.806473i	$-0.701374\pi$
$563$	−28.0589	−1.18254	−0.591272	+	0.806473i	$0.701374\pi$
$564$	0	0
$565$	16.2319	0.682880
$566$	0	0
$567$	−3.64711	−0.153164
$568$	0	0
$569$	19.1066	0.800991	0.400495	−	0.916299i	$-0.368838\pi$
$569$	19.1066	0.800991	0.400495	+	0.916299i	$0.368838\pi$
$570$	0	0
$571$	−37.4981	−1.56925	−0.784624	−	0.619972i	$-0.787144\pi$
$571$	−37.4981	−1.56925	−0.784624	+	0.619972i	$0.787144\pi$
$572$	0	0
$573$	−16.6645	−0.696171
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.5252	1.02100	0.510499	−	0.859878i	$-0.329461\pi$
$577$	24.5252	1.02100	0.510499	+	0.859878i	$0.329461\pi$
$578$	0	0
$579$	20.3804	0.846982
$580$	0	0
$581$	3.46647	0.143814
$582$	0	0
$583$	−5.56849	−0.230623
$584$	0	0
$585$	18.1706	0.751260
$586$	0	0
$587$	8.56954	0.353703	0.176851	−	0.984238i	$-0.443409\pi$
$587$	8.56954	0.353703	0.176851	+	0.984238i	$0.443409\pi$
$588$	0	0
$589$	−3.08117	−0.126957
$590$	0	0
$591$	−0.633680	−0.0260661
$592$	0	0
$593$	16.1591	0.663573	0.331787	−	0.943354i	$-0.392349\pi$
$593$	16.1591	0.663573	0.331787	+	0.943354i	$0.392349\pi$
$594$	0	0
$595$	−7.38315	−0.302680
$596$	0	0
$597$	−3.55482	−0.145489
$598$	0	0
$599$	37.1195	1.51666	0.758332	−	0.651869i	$-0.226015\pi$
$599$	37.1195	1.51666	0.758332	+	0.651869i	$0.226015\pi$
$600$	0	0
$601$	−6.03650	−0.246234	−0.123117	−	0.992392i	$-0.539289\pi$
$601$	−6.03650	−0.246234	−0.123117	+	0.992392i	$0.539289\pi$
$602$	0	0
$603$	25.1180	1.02288
$604$	0	0
$605$	−25.7267	−1.04594
$606$	0	0
$607$	−10.7215	−0.435172	−0.217586	−	0.976041i	$-0.569818\pi$
$607$	−10.7215	−0.435172	−0.217586	+	0.976041i	$0.569818\pi$
$608$	0	0
$609$	8.98667	0.364158
$610$	0	0
$611$	18.0072	0.728494
$612$	0	0
$613$	−20.1669	−0.814534	−0.407267	−	0.913309i	$-0.633518\pi$
$613$	−20.1669	−0.814534	−0.407267	+	0.913309i	$0.633518\pi$
$614$	0	0
$615$	0.160386	0.00646741
$616$	0	0
$617$	7.11588	0.286474	0.143237	−	0.989688i	$-0.454249\pi$
$617$	7.11588	0.286474	0.143237	+	0.989688i	$0.454249\pi$
$618$	0	0
$619$	−31.4142	−1.26264	−0.631322	−	0.775521i	$-0.717487\pi$
$619$	−31.4142	−1.26264	−0.631322	+	0.775521i	$0.717487\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.52854	0.381753
$624$	0	0
$625$	−29.2706	−1.17082
$626$	0	0
$627$	−0.278993	−0.0111419
$628$	0	0
$629$	16.1823	0.645231
$630$	0	0
$631$	18.8689	0.751161	0.375580	−	0.926790i	$-0.377443\pi$
$631$	18.8689	0.751161	0.375580	+	0.926790i	$0.377443\pi$
$632$	0	0
$633$	−11.7055	−0.465251
$634$	0	0
$635$	−46.5548	−1.84747
$636$	0	0
$637$	16.7150	0.662273
$638$	0	0
$639$	−9.35309	−0.370003
$640$	0	0
$641$	0.552367	0.0218172	0.0109086	−	0.999940i	$-0.496528\pi$
$641$	0.552367	0.0218172	0.0109086	+	0.999940i	$0.496528\pi$
$642$	0	0
$643$	15.6492	0.617145	0.308572	−	0.951201i	$-0.400149\pi$
$643$	15.6492	0.617145	0.308572	+	0.951201i	$0.400149\pi$
$644$	0	0
$645$	−2.92017	−0.114982
$646$	0	0
$647$	−30.8768	−1.21389	−0.606946	−	0.794743i	$-0.707605\pi$
$647$	−30.8768	−1.21389	−0.606946	+	0.794743i	$0.707605\pi$
$648$	0	0
$649$	11.3387	0.445084
$650$	0	0
$651$	−9.11405	−0.357208
$652$	0	0
$653$	20.9042	0.818045	0.409023	−	0.912524i	$-0.365870\pi$
$653$	20.9042	0.818045	0.409023	+	0.912524i	$0.365870\pi$
$654$	0	0
$655$	−35.9489	−1.40464
$656$	0	0
$657$	13.9692	0.544991
$658$	0	0
$659$	−43.0684	−1.67771	−0.838853	−	0.544359i	$-0.816773\pi$
$659$	−43.0684	−1.67771	−0.838853	+	0.544359i	$0.816773\pi$
$660$	0	0
$661$	−10.3854	−0.403946	−0.201973	−	0.979391i	$-0.564735\pi$
$661$	−10.3854	−0.403946	−0.201973	+	0.979391i	$0.564735\pi$
$662$	0	0
$663$	−6.24874	−0.242681
$664$	0	0
$665$	1.43627	0.0556963
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−21.0026	−0.812009
$670$	0	0
$671$	−8.75213	−0.337872
$672$	0	0
$673$	48.2776	1.86096	0.930482	−	0.366337i	$-0.119388\pi$
$673$	48.2776	1.86096	0.930482	+	0.366337i	$0.119388\pi$
$674$	0	0
$675$	−5.03271	−0.193709
$676$	0	0
$677$	18.6801	0.717936	0.358968	−	0.933350i	$-0.383129\pi$
$677$	18.6801	0.717936	0.358968	+	0.933350i	$0.383129\pi$
$678$	0	0
$679$	−9.05532	−0.347511
$680$	0	0
$681$	−3.13399	−0.120095
$682$	0	0
$683$	−10.2168	−0.390936	−0.195468	−	0.980710i	$-0.562623\pi$
$683$	−10.2168	−0.390936	−0.195468	+	0.980710i	$0.562623\pi$
$684$	0	0
$685$	−46.7832	−1.78749
$686$	0	0
$687$	−10.4923	−0.400305
$688$	0	0
$689$	24.2623	0.924322
$690$	0	0
$691$	−1.65461	−0.0629444	−0.0314722	−	0.999505i	$-0.510020\pi$
$691$	−1.65461	−0.0629444	−0.0314722	+	0.999505i	$0.510020\pi$
$692$	0	0
$693$	2.36072	0.0896763
$694$	0	0
$695$	3.60737	0.136835
$696$	0	0
$697$	0.157778	0.00597628
$698$	0	0
$699$	−5.70238	−0.215684
$700$	0	0
$701$	15.9952	0.604129	0.302064	−	0.953288i	$-0.402324\pi$
$701$	15.9952	0.604129	0.302064	+	0.953288i	$0.402324\pi$
$702$	0	0
$703$	−3.14801	−0.118729
$704$	0	0
$705$	−11.8324	−0.445634
$706$	0	0
$707$	−8.76486	−0.329636
$708$	0	0
$709$	48.0595	1.80491	0.902456	−	0.430783i	$-0.141763\pi$
$709$	48.0595	1.80491	0.902456	+	0.430783i	$0.141763\pi$
$710$	0	0
$711$	−10.1718	−0.381473
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−6.21264	−0.232339
$716$	0	0
$717$	8.15276	0.304470
$718$	0	0
$719$	−12.1273	−0.452271	−0.226136	−	0.974096i	$-0.572609\pi$
$719$	−12.1273	−0.452271	−0.226136	+	0.974096i	$0.572609\pi$
$720$	0	0
$721$	−22.9943	−0.856351
$722$	0	0
$723$	−26.7496	−0.994830
$724$	0	0
$725$	−7.97497	−0.296183
$726$	0	0
$727$	33.5881	1.24571	0.622857	−	0.782336i	$-0.285972\pi$
$727$	33.5881	1.24571	0.622857	+	0.782336i	$0.285972\pi$
$728$	0	0
$729$	6.37388	0.236070
$730$	0	0
$731$	−2.87269	−0.106250
$732$	0	0
$733$	−27.2031	−1.00477	−0.502385	−	0.864644i	$-0.667544\pi$
$733$	−27.2031	−1.00477	−0.502385	+	0.864644i	$0.667544\pi$
$734$	0	0
$735$	−10.9833	−0.405125
$736$	0	0
$737$	−8.58802	−0.316344
$738$	0	0
$739$	−36.9389	−1.35882	−0.679409	−	0.733759i	$-0.737764\pi$
$739$	−36.9389	−1.35882	−0.679409	+	0.733759i	$0.737764\pi$
$740$	0	0
$741$	1.21559	0.0446559
$742$	0	0
$743$	−24.2213	−0.888592	−0.444296	−	0.895880i	$-0.646546\pi$
$743$	−24.2213	−0.888592	−0.444296	+	0.895880i	$0.646546\pi$
$744$	0	0
$745$	−31.9054	−1.16892
$746$	0	0
$747$	5.51468	0.201772
$748$	0	0
$749$	20.8400	0.761478
$750$	0	0
$751$	−2.98502	−0.108925	−0.0544625	−	0.998516i	$-0.517345\pi$
$751$	−2.98502	−0.108925	−0.0544625	+	0.998516i	$0.517345\pi$
$752$	0	0
$753$	−19.0728	−0.695051
$754$	0	0
$755$	−36.0220	−1.31098
$756$	0	0
$757$	−35.4091	−1.28696	−0.643482	−	0.765461i	$-0.722511\pi$
$757$	−35.4091	−1.28696	−0.643482	+	0.765461i	$0.722511\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.5945	1.10905	0.554524	−	0.832168i	$-0.312900\pi$
$761$	30.5945	1.10905	0.554524	+	0.832168i	$0.312900\pi$
$762$	0	0
$763$	16.4744	0.596413
$764$	0	0
$765$	−11.7456	−0.424662
$766$	0	0
$767$	−49.4037	−1.78387
$768$	0	0
$769$	−23.7248	−0.855538	−0.427769	−	0.903888i	$-0.640700\pi$
$769$	−23.7248	−0.855538	−0.427769	+	0.903888i	$0.640700\pi$
$770$	0	0
$771$	1.09797	0.0395426
$772$	0	0
$773$	13.7035	0.492879	0.246440	−	0.969158i	$-0.420739\pi$
$773$	13.7035	0.492879	0.246440	+	0.969158i	$0.420739\pi$
$774$	0	0
$775$	8.08801	0.290530
$776$	0	0
$777$	−9.31177	−0.334058
$778$	0	0
$779$	−0.0306933	−0.00109970
$780$	0	0
$781$	3.19789	0.114429
$782$	0	0
$783$	33.5909	1.20044
$784$	0	0
$785$	4.13628	0.147630
$786$	0	0
$787$	−16.5781	−0.590947	−0.295474	−	0.955351i	$-0.595477\pi$
$787$	−16.5781	−0.590947	−0.295474	+	0.955351i	$0.595477\pi$
$788$	0	0
$789$	18.5668	0.660995
$790$	0	0
$791$	−9.18841	−0.326702
$792$	0	0
$793$	38.1337	1.35417
$794$	0	0
$795$	−15.9426	−0.565426
$796$	0	0
$797$	45.7453	1.62038	0.810190	−	0.586167i	$-0.199364\pi$
$797$	45.7453	1.62038	0.810190	+	0.586167i	$0.199364\pi$
$798$	0	0
$799$	−11.6400	−0.411794
$800$	0	0
$801$	15.1586	0.535603
$802$	0	0
$803$	−4.77617	−0.168548
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.8345	0.557400
$808$	0	0
$809$	−13.8787	−0.487947	−0.243974	−	0.969782i	$-0.578451\pi$
$809$	−13.8787	−0.487947	−0.243974	+	0.969782i	$0.578451\pi$
$810$	0	0
$811$	−52.9204	−1.85829	−0.929144	−	0.369719i	$-0.879454\pi$
$811$	−52.9204	−1.85829	−0.929144	+	0.369719i	$0.879454\pi$
$812$	0	0
$813$	6.17810	0.216675
$814$	0	0
$815$	8.73578	0.306001
$816$	0	0
$817$	0.558836	0.0195512
$818$	0	0
$819$	−10.2858	−0.359416
$820$	0	0
$821$	−43.7368	−1.52643	−0.763213	−	0.646148i	$-0.776379\pi$
$821$	−43.7368	−1.52643	−0.763213	+	0.646148i	$0.776379\pi$
$822$	0	0
$823$	20.0791	0.699915	0.349958	−	0.936766i	$-0.386196\pi$
$823$	20.0791	0.699915	0.349958	+	0.936766i	$0.386196\pi$
$824$	0	0
$825$	0.732351	0.0254972
$826$	0	0
$827$	−30.1446	−1.04823	−0.524115	−	0.851648i	$-0.675604\pi$
$827$	−30.1446	−1.04823	−0.524115	+	0.851648i	$0.675604\pi$
$828$	0	0
$829$	−16.0798	−0.558476	−0.279238	−	0.960222i	$-0.590082\pi$
$829$	−16.0798	−0.558476	−0.279238	+	0.960222i	$0.590082\pi$
$830$	0	0
$831$	−11.7158	−0.406418
$832$	0	0
$833$	−10.8047	−0.374361
$834$	0	0
$835$	54.4135	1.88306
$836$	0	0
$837$	−34.0670	−1.17753
$838$	0	0
$839$	14.7308	0.508564	0.254282	−	0.967130i	$-0.418161\pi$
$839$	14.7308	0.508564	0.254282	+	0.967130i	$0.418161\pi$
$840$	0	0
$841$	24.2290	0.835484
$842$	0	0
$843$	−21.8259	−0.751723
$844$	0	0
$845$	−5.02049	−0.172710
$846$	0	0
$847$	14.5632	0.500397
$848$	0	0
$849$	25.0960	0.861294
$850$	0	0
$851$	0	0
$852$	0	0
$853$	36.3301	1.24392	0.621960	−	0.783049i	$-0.286337\pi$
$853$	36.3301	1.24392	0.621960	+	0.783049i	$0.286337\pi$
$854$	0	0
$855$	2.28491	0.0781424
$856$	0	0
$857$	−6.52818	−0.222998	−0.111499	−	0.993765i	$-0.535565\pi$
$857$	−6.52818	−0.222998	−0.111499	+	0.993765i	$0.535565\pi$
$858$	0	0
$859$	29.8341	1.01793	0.508963	−	0.860788i	$-0.330029\pi$
$859$	29.8341	1.01793	0.508963	+	0.860788i	$0.330029\pi$
$860$	0	0
$861$	−0.0907903	−0.00309412
$862$	0	0
$863$	56.1084	1.90995	0.954976	−	0.296683i	$-0.0958804\pi$
$863$	56.1084	1.90995	0.954976	+	0.296683i	$0.0958804\pi$
$864$	0	0
$865$	−32.7122	−1.11225
$866$	0	0
$867$	−10.9467	−0.371768
$868$	0	0
$869$	3.47781	0.117977
$870$	0	0
$871$	37.4187	1.26788
$872$	0	0
$873$	−14.4058	−0.487562
$874$	0	0
$875$	13.4754	0.455553
$876$	0	0
$877$	−12.5929	−0.425232	−0.212616	−	0.977136i	$-0.568198\pi$
$877$	−12.5929	−0.425232	−0.212616	+	0.977136i	$0.568198\pi$
$878$	0	0
$879$	16.5286	0.557495
$880$	0	0
$881$	−27.6086	−0.930158	−0.465079	−	0.885269i	$-0.653974\pi$
$881$	−27.6086	−0.930158	−0.465079	+	0.885269i	$0.653974\pi$
$882$	0	0
$883$	−36.4366	−1.22619	−0.613095	−	0.790009i	$-0.710076\pi$
$883$	−36.4366	−1.22619	−0.613095	+	0.790009i	$0.710076\pi$
$884$	0	0
$885$	32.4628	1.09123
$886$	0	0
$887$	−8.63911	−0.290073	−0.145036	−	0.989426i	$-0.546330\pi$
$887$	−8.63911	−0.290073	−0.145036	+	0.989426i	$0.546330\pi$
$888$	0	0
$889$	26.3534	0.883863
$890$	0	0
$891$	1.98376	0.0664584
$892$	0	0
$893$	2.26438	0.0757744
$894$	0	0
$895$	48.5650	1.62335
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−53.9835	−1.80045
$900$	0	0
$901$	−15.6834	−0.522488
$902$	0	0
$903$	1.65303	0.0550093
$904$	0	0
$905$	23.2400	0.772524
$906$	0	0
$907$	−26.5131	−0.880352	−0.440176	−	0.897912i	$-0.645084\pi$
$907$	−26.5131	−0.880352	−0.440176	+	0.897912i	$0.645084\pi$
$908$	0	0
$909$	−13.9437	−0.462483
$910$	0	0
$911$	39.2004	1.29877	0.649384	−	0.760461i	$-0.275027\pi$
$911$	39.2004	1.29877	0.649384	+	0.760461i	$0.275027\pi$
$912$	0	0
$913$	−1.88551	−0.0624012
$914$	0	0
$915$	−25.0574	−0.828371
$916$	0	0
$917$	20.3497	0.672006
$918$	0	0
$919$	−27.9225	−0.921079	−0.460539	−	0.887639i	$-0.652344\pi$
$919$	−27.9225	−0.921079	−0.460539	+	0.887639i	$0.652344\pi$
$920$	0	0
$921$	13.2142	0.435422
$922$	0	0
$923$	−13.9334	−0.458625
$924$	0	0
$925$	8.26347	0.271701
$926$	0	0
$927$	−36.5807	−1.20147
$928$	0	0
$929$	−29.0133	−0.951895	−0.475947	−	0.879474i	$-0.657895\pi$
$929$	−29.0133	−0.951895	−0.475947	+	0.879474i	$0.657895\pi$
$930$	0	0
$931$	2.10188	0.0688864
$932$	0	0
$933$	−0.934414	−0.0305913
$934$	0	0
$935$	4.01589	0.131334
$936$	0	0
$937$	15.9119	0.519820	0.259910	−	0.965633i	$-0.416307\pi$
$937$	15.9119	0.519820	0.259910	+	0.965633i	$0.416307\pi$
$938$	0	0
$939$	−8.83324	−0.288262
$940$	0	0
$941$	8.90388	0.290258	0.145129	−	0.989413i	$-0.453640\pi$
$941$	8.90388	0.290258	0.145129	+	0.989413i	$0.453640\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	15.8802	0.516583
$946$	0	0
$947$	5.18152	0.168377	0.0841884	−	0.996450i	$-0.473170\pi$
$947$	5.18152	0.168377	0.0841884	+	0.996450i	$0.473170\pi$
$948$	0	0
$949$	20.8102	0.675526
$950$	0	0
$951$	0.0593285	0.00192386
$952$	0	0
$953$	45.8067	1.48382	0.741912	−	0.670498i	$-0.233919\pi$
$953$	45.8067	1.48382	0.741912	+	0.670498i	$0.233919\pi$
$954$	0	0
$955$	−46.6636	−1.51000
$956$	0	0
$957$	−4.88809	−0.158009
$958$	0	0
$959$	26.4826	0.855169
$960$	0	0
$961$	23.7487	0.766088
$962$	0	0
$963$	33.1536	1.06836
$964$	0	0
$965$	57.0688	1.83711
$966$	0	0
$967$	4.58411	0.147415	0.0737075	−	0.997280i	$-0.476517\pi$
$967$	4.58411	0.147415	0.0737075	+	0.997280i	$0.476517\pi$
$968$	0	0
$969$	−0.785768	−0.0252425
$970$	0	0
$971$	−48.1336	−1.54468	−0.772341	−	0.635208i	$-0.780914\pi$
$971$	−48.1336	−1.54468	−0.772341	+	0.635208i	$0.780914\pi$
$972$	0	0
$973$	−2.04203	−0.0654646
$974$	0	0
$975$	−3.19091	−0.102191
$976$	0	0
$977$	−34.2534	−1.09586	−0.547932	−	0.836523i	$-0.684585\pi$
$977$	−34.2534	−1.09586	−0.547932	+	0.836523i	$0.684585\pi$
$978$	0	0
$979$	−5.18283	−0.165644
$980$	0	0
$981$	26.2085	0.836773
$982$	0	0
$983$	−29.7517	−0.948931	−0.474465	−	0.880274i	$-0.657359\pi$
$983$	−29.7517	−0.948931	−0.474465	+	0.880274i	$0.657359\pi$
$984$	0	0
$985$	−1.77441	−0.0565375
$986$	0	0
$987$	6.69799	0.213200
$988$	0	0
$989$	0	0
$990$	0	0
$991$	49.1617	1.56167	0.780836	−	0.624736i	$-0.214794\pi$
$991$	49.1617	1.56167	0.780836	+	0.624736i	$0.214794\pi$
$992$	0	0
$993$	20.6764	0.656147
$994$	0	0
$995$	−9.95411	−0.315567
$996$	0	0
$997$	−13.1303	−0.415839	−0.207920	−	0.978146i	$-0.566669\pi$
$997$	−13.1303	−0.415839	−0.207920	+	0.978146i	$0.566669\pi$
$998$	0	0
$999$	−34.8061	−1.10122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ci.1.10		15
4.3	odd	2		4232.2.a.y.1.6		15
23.4	even	11		368.2.m.f.177.2		30
23.6	even	11		368.2.m.f.289.2		30
23.22	odd	2		8464.2.a.cj.1.10		15
92.27	odd	22		184.2.i.a.177.2	yes	30
92.75	odd	22		184.2.i.a.105.2	✓	30
92.91	even	2		4232.2.a.z.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.a.105.2	✓	30	92.75	odd	22
184.2.i.a.177.2	yes	30	92.27	odd	22
368.2.m.f.177.2		30	23.4	even	11
368.2.m.f.289.2		30	23.6	even	11
4232.2.a.y.1.6		15	4.3	odd	2
4232.2.a.z.1.6		15	92.91	even	2
8464.2.a.ci.1.10		15	1.1	even	1	trivial
8464.2.a.cj.1.10		15	23.22	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.881524 q^{3} +2.46842 q^{5} -1.39730 q^{7} -2.22292 q^{9} +0.760030 q^{11} -3.31151 q^{13} +2.17597 q^{15} +2.14059 q^{17} -0.416417 q^{19} -1.23176 q^{21} +1.09309 q^{25} -4.60412 q^{27} -7.29582 q^{29} +7.39924 q^{31} +0.669985 q^{33} -3.44913 q^{35} +7.55976 q^{37} -2.91918 q^{39} +0.0737081 q^{41} -1.34201 q^{43} -5.48709 q^{45} -5.43776 q^{47} -5.04755 q^{49} +1.88698 q^{51} -7.32667 q^{53} +1.87607 q^{55} -0.367081 q^{57} +14.9188 q^{59} -11.5155 q^{61} +3.10609 q^{63} -8.17420 q^{65} -11.2996 q^{67} +4.20758 q^{71} -6.28419 q^{73} +0.963582 q^{75} -1.06199 q^{77} +4.57589 q^{79} +2.61011 q^{81} -2.48083 q^{83} +5.28386 q^{85} -6.43144 q^{87} -6.81924 q^{89} +4.62718 q^{91} +6.52260 q^{93} -1.02789 q^{95} +6.48057 q^{97} -1.68948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 10 q^{5} + 10 q^{7} + 16 q^{9} - 9 q^{11} + 12 q^{15} - 20 q^{17} - 3 q^{19} - 21 q^{21} + 11 q^{25} + 7 q^{27} - 4 q^{29} + 14 q^{31} - 34 q^{33} - 18 q^{35} - 26 q^{37} - 33 q^{39} - 11 q^{41}+ \cdots - 29 q^{99}+O(q^{100}) \) 15 * q + q^3 - 10 * q^5 + 10 * q^7 + 16 * q^9 - 9 * q^11 + 12 * q^15 - 20 * q^17 - 3 * q^19 - 21 * q^21 + 11 * q^25 + 7 * q^27 - 4 * q^29 + 14 * q^31 - 34 * q^33 - 18 * q^35 - 26 * q^37 - 33 * q^39 - 11 * q^41 + 10 * q^43 - 46 * q^45 - 12 * q^47 + 9 * q^49 - 29 * q^51 - 25 * q^53 + 24 * q^55 - 45 * q^57 - 22 * q^59 - 32 * q^61 + 25 * q^63 - 26 * q^65 + 28 * q^67 + 26 * q^71 - 39 * q^73 - 29 * q^75 - 38 * q^77 + 28 * q^79 + 51 * q^81 - 24 * q^83 + 18 * q^85 + 33 * q^87 - 66 * q^89 - 13 * q^91 + 41 * q^93 + 6 * q^95 - 56 * q^97 - 29 * q^99

Introduction

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