LMFDB - Embedded newform 6800.2.a.bo.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6800,2,Mod(1,6800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("6800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,0,0,0,0,4,0,-2,0,-5] 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:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 680)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68133 q^{3} -0.508203 q^{7} +4.18953 q^{9} -5.87086 q^{11} +1.18953 q^{13} -1.00000 q^{17} -0.810466 q^{19} -1.36266 q^{21} -0.508203 q^{23} +3.18953 q^{27} -0.173127 q^{29} -9.06040 q^{31} -15.7417 q^{33} +4.37907 q^{37} +3.18953 q^{39} +2.00000 q^{41} -1.36266 q^{43} -10.1731 q^{47} -6.74173 q^{49} -2.68133 q^{51} -8.17313 q^{53} -2.17313 q^{57} +9.91486 q^{59} -6.55220 q^{61} -2.12914 q^{63} +2.37907 q^{67} -1.36266 q^{69} +11.6977 q^{71} -10.5522 q^{73} +2.98359 q^{77} +4.85446 q^{79} -4.01641 q^{81} -15.7417 q^{83} -0.464211 q^{87} -1.53579 q^{89} -0.604525 q^{91} -24.2939 q^{93} -8.93126 q^{97} -24.5962 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 4 q^{9} - 2 q^{11} - 5 q^{13} - 3 q^{17} - 11 q^{19} + 10 q^{21} + q^{27} + 5 q^{29} - 3 q^{31} - 16 q^{33} - 4 q^{37} + q^{39} + 6 q^{41} + 10 q^{43} - 25 q^{47} + 11 q^{49} - q^{51}+ \cdots - 30 q^{99}+O(q^{100}) $ 3 * q + q^3 + 4 * q^9 - 2 * q^11 - 5 * q^13 - 3 * q^17 - 11 * q^19 + 10 * q^21 + q^27 + 5 * q^29 - 3 * q^31 - 16 * q^33 - 4 * q^37 + q^39 + 6 * q^41 + 10 * q^43 - 25 * q^47 + 11 * q^49 - q^51 - 19 * q^53 - q^57 - 7 * q^59 + 3 * q^61 - 22 * q^63 - 10 * q^67 + 10 * q^69 + 25 * q^71 - 9 * q^73 + 12 * q^77 + 2 * q^79 - 9 * q^81 - 16 * q^83 - 21 * q^87 + 15 * q^89 - 22 * q^91 - 19 * q^93 + 13 * q^97 - 30 * 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
$3$	2.68133	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$3$	2.68133	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.508203	−0.192083	−0.0960414	−	0.995377i	$-0.530618\pi$
$7$	−0.508203	−0.192083	−0.0960414	+	0.995377i	$0.530618\pi$
$8$	0	0
$9$	4.18953	1.39651
$10$	0	0
$11$	−5.87086	−1.77013	−0.885066	−	0.465465i	$-0.845887\pi$
$11$	−5.87086	−1.77013	−0.885066	+	0.465465i	$0.845887\pi$
$12$	0	0
$13$	1.18953	0.329917	0.164959	−	0.986300i	$-0.447251\pi$
$13$	1.18953	0.329917	0.164959	+	0.986300i	$0.447251\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.810466	−0.185934	−0.0929668	−	0.995669i	$-0.529635\pi$
$19$	−0.810466	−0.185934	−0.0929668	+	0.995669i	$0.529635\pi$
$20$	0	0
$21$	−1.36266	−0.297357
$22$	0	0
$23$	−0.508203	−0.105968	−0.0529839	−	0.998595i	$-0.516873\pi$
$23$	−0.508203	−0.105968	−0.0529839	+	0.998595i	$0.516873\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.18953	0.613826
$28$	0	0
$29$	−0.173127	−0.0321489	−0.0160745	−	0.999871i	$-0.505117\pi$
$29$	−0.173127	−0.0321489	−0.0160745	+	0.999871i	$0.505117\pi$
$30$	0	0
$31$	−9.06040	−1.62730	−0.813648	−	0.581358i	$-0.802522\pi$
$31$	−9.06040	−1.62730	−0.813648	+	0.581358i	$0.802522\pi$
$32$	0	0
$33$	−15.7417	−2.74028
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.37907	0.719914	0.359957	−	0.932969i	$-0.382791\pi$
$37$	4.37907	0.719914	0.359957	+	0.932969i	$0.382791\pi$
$38$	0	0
$39$	3.18953	0.510734
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.36266	−0.207804	−0.103902	−	0.994588i	$-0.533133\pi$
$43$	−1.36266	−0.207804	−0.103902	+	0.994588i	$0.533133\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.1731	−1.48390	−0.741952	−	0.670453i	$-0.766100\pi$
$47$	−10.1731	−1.48390	−0.741952	+	0.670453i	$0.766100\pi$
$48$	0	0
$49$	−6.74173	−0.963104
$50$	0	0
$51$	−2.68133	−0.375461
$52$	0	0
$53$	−8.17313	−1.12267	−0.561333	−	0.827590i	$-0.689711\pi$
$53$	−8.17313	−1.12267	−0.561333	+	0.827590i	$0.689711\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.17313	−0.287838
$58$	0	0
$59$	9.91486	1.29080	0.645402	−	0.763843i	$-0.276690\pi$
$59$	9.91486	1.29080	0.645402	+	0.763843i	$0.276690\pi$
$60$	0	0
$61$	−6.55220	−0.838923	−0.419461	−	0.907773i	$-0.637781\pi$
$61$	−6.55220	−0.838923	−0.419461	+	0.907773i	$0.637781\pi$
$62$	0	0
$63$	−2.12914	−0.268246
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.37907	0.290649	0.145325	−	0.989384i	$-0.453577\pi$
$67$	2.37907	0.290649	0.145325	+	0.989384i	$0.453577\pi$
$68$	0	0
$69$	−1.36266	−0.164045
$70$	0	0
$71$	11.6977	1.38827	0.694133	−	0.719847i	$-0.255788\pi$
$71$	11.6977	1.38827	0.694133	+	0.719847i	$0.255788\pi$
$72$	0	0
$73$	−10.5522	−1.23504	−0.617521	−	0.786555i	$-0.711863\pi$
$73$	−10.5522	−1.23504	−0.617521	+	0.786555i	$0.711863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.98359	0.340012
$78$	0	0
$79$	4.85446	0.546169	0.273085	−	0.961990i	$-0.411956\pi$
$79$	4.85446	0.546169	0.273085	+	0.961990i	$0.411956\pi$
$80$	0	0
$81$	−4.01641	−0.446267
$82$	0	0
$83$	−15.7417	−1.72788	−0.863940	−	0.503595i	$-0.832010\pi$
$83$	−15.7417	−1.72788	−0.863940	+	0.503595i	$0.832010\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.464211	−0.0497687
$88$	0	0
$89$	−1.53579	−0.162793	−0.0813966	−	0.996682i	$-0.525938\pi$
$89$	−1.53579	−0.162793	−0.0813966	+	0.996682i	$0.525938\pi$
$90$	0	0
$91$	−0.604525	−0.0633715
$92$	0	0
$93$	−24.2939	−2.51916
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.93126	−0.906832	−0.453416	−	0.891299i	$-0.649795\pi$
$97$	−8.93126	−0.906832	−0.453416	+	0.891299i	$0.649795\pi$
$98$	0	0
$99$	−24.5962	−2.47201
$100$	0	0
$101$	13.7417	1.36735	0.683677	−	0.729785i	$-0.260380\pi$
$101$	13.7417	1.36735	0.683677	+	0.729785i	$0.260380\pi$
$102$	0	0
$103$	2.03281	0.200299	0.100150	−	0.994972i	$-0.468068\pi$
$103$	2.03281	0.200299	0.100150	+	0.994972i	$0.468068\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8709	−1.34095	−0.670474	−	0.741933i	$-0.733909\pi$
$107$	−13.8709	−1.34095	−0.670474	+	0.741933i	$0.733909\pi$
$108$	0	0
$109$	8.93126	0.855460	0.427730	−	0.903907i	$-0.359313\pi$
$109$	8.93126	0.855460	0.427730	+	0.903907i	$0.359313\pi$
$110$	0	0
$111$	11.7417	1.11448
$112$	0	0
$113$	6.55220	0.616379	0.308189	−	0.951325i	$-0.400277\pi$
$113$	6.55220	0.616379	0.308189	+	0.951325i	$0.400277\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.98359	0.460733
$118$	0	0
$119$	0.508203	0.0465869
$120$	0	0
$121$	23.4671	2.13337
$122$	0	0
$123$	5.36266	0.483535
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.53579	−0.668693	−0.334347	−	0.942450i	$-0.608516\pi$
$127$	−7.53579	−0.668693	−0.334347	+	0.942450i	$0.608516\pi$
$128$	0	0
$129$	−3.65375	−0.321694
$130$	0	0
$131$	9.52461	0.832169	0.416085	−	0.909326i	$-0.363402\pi$
$131$	9.52461	0.832169	0.416085	+	0.909326i	$0.363402\pi$
$132$	0	0
$133$	0.411882	0.0357147
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.7745	1.34771	0.673855	−	0.738864i	$-0.264637\pi$
$137$	15.7745	1.34771	0.673855	+	0.738864i	$0.264637\pi$
$138$	0	0
$139$	−14.8873	−1.26272	−0.631361	−	0.775489i	$-0.717503\pi$
$139$	−14.8873	−1.26272	−0.631361	+	0.775489i	$0.717503\pi$
$140$	0	0
$141$	−27.2775	−2.29718
$142$	0	0
$143$	−6.98359	−0.583997
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.0768	−1.49095
$148$	0	0
$149$	−5.65375	−0.463173	−0.231586	−	0.972814i	$-0.574392\pi$
$149$	−5.65375	−0.463173	−0.231586	+	0.972814i	$0.574392\pi$
$150$	0	0
$151$	−20.8461	−1.69643	−0.848217	−	0.529650i	$-0.822323\pi$
$151$	−20.8461	−1.69643	−0.848217	+	0.529650i	$0.822323\pi$
$152$	0	0
$153$	−4.18953	−0.338704
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.6925	0.853355	0.426678	−	0.904404i	$-0.359684\pi$
$157$	10.6925	0.853355	0.426678	+	0.904404i	$0.359684\pi$
$158$	0	0
$159$	−21.9149	−1.73796
$160$	0	0
$161$	0.258271	0.0203546
$162$	0	0
$163$	−20.3379	−1.59299	−0.796494	−	0.604646i	$-0.793315\pi$
$163$	−20.3379	−1.59299	−0.796494	+	0.604646i	$0.793315\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.2335	0.869276	0.434638	−	0.900605i	$-0.356876\pi$
$167$	11.2335	0.869276	0.434638	+	0.900605i	$0.356876\pi$
$168$	0	0
$169$	−11.5850	−0.891155
$170$	0	0
$171$	−3.39547	−0.259658
$172$	0	0
$173$	−15.4506	−1.17469	−0.587345	−	0.809336i	$-0.699827\pi$
$173$	−15.4506	−1.17469	−0.587345	+	0.809336i	$0.699827\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.5850	1.99825
$178$	0	0
$179$	−25.7745	−1.92648	−0.963240	−	0.268643i	$-0.913425\pi$
$179$	−25.7745	−1.92648	−0.963240	+	0.268643i	$0.913425\pi$
$180$	0	0
$181$	−6.69251	−0.497450	−0.248725	−	0.968574i	$-0.580012\pi$
$181$	−6.69251	−0.497450	−0.248725	+	0.968574i	$0.580012\pi$
$182$	0	0
$183$	−17.5686	−1.29871
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.87086	0.429320
$188$	0	0
$189$	−1.62093	−0.117905
$190$	0	0
$191$	18.7253	1.35492	0.677458	−	0.735561i	$-0.263081\pi$
$191$	18.7253	1.35492	0.677458	+	0.735561i	$0.263081\pi$
$192$	0	0
$193$	0.725323	0.0522099	0.0261049	−	0.999659i	$-0.491690\pi$
$193$	0.725323	0.0522099	0.0261049	+	0.999659i	$0.491690\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.0716	1.50129	0.750644	−	0.660707i	$-0.229743\pi$
$197$	21.0716	1.50129	0.750644	+	0.660707i	$0.229743\pi$
$198$	0	0
$199$	−2.68133	−0.190074	−0.0950372	−	0.995474i	$-0.530297\pi$
$199$	−2.68133	−0.190074	−0.0950372	+	0.995474i	$0.530297\pi$
$200$	0	0
$201$	6.37907	0.449945
$202$	0	0
$203$	0.0879839	0.00617526
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.12914	−0.147985
$208$	0	0
$209$	4.75814	0.329127
$210$	0	0
$211$	−4.76647	−0.328138	−0.164069	−	0.986449i	$-0.552462\pi$
$211$	−4.76647	−0.328138	−0.164069	+	0.986449i	$0.552462\pi$
$212$	0	0
$213$	31.3655	2.14913
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.60453	0.312576
$218$	0	0
$219$	−28.2939	−1.91193
$220$	0	0
$221$	−1.18953	−0.0800167
$222$	0	0
$223$	−10.2611	−0.687135	−0.343567	−	0.939128i	$-0.611635\pi$
$223$	−10.2611	−0.687135	−0.343567	+	0.939128i	$0.611635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.0768	0.668821	0.334411	−	0.942428i	$-0.391463\pi$
$227$	10.0768	0.668821	0.334411	+	0.942428i	$0.391463\pi$
$228$	0	0
$229$	27.7089	1.83106	0.915528	−	0.402253i	$-0.131773\pi$
$229$	27.7089	1.83106	0.915528	+	0.402253i	$0.131773\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	12.5194	0.820172	0.410086	−	0.912047i	$-0.365499\pi$
$233$	12.5194	0.820172	0.410086	+	0.912047i	$0.365499\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.0164	0.845506
$238$	0	0
$239$	21.8625	1.41417	0.707085	−	0.707129i	$-0.250010\pi$
$239$	21.8625	1.41417	0.707085	+	0.707129i	$0.250010\pi$
$240$	0	0
$241$	−3.96719	−0.255549	−0.127774	−	0.991803i	$-0.540783\pi$
$241$	−3.96719	−0.255549	−0.127774	+	0.991803i	$0.540783\pi$
$242$	0	0
$243$	−20.3379	−1.30468
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.964077	−0.0613427
$248$	0	0
$249$	−42.2088	−2.67487
$250$	0	0
$251$	−14.0328	−0.885743	−0.442872	−	0.896585i	$-0.646040\pi$
$251$	−14.0328	−0.885743	−0.442872	+	0.896585i	$0.646040\pi$
$252$	0	0
$253$	2.98359	0.187577
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.7089	−1.47892	−0.739461	−	0.673200i	$-0.764919\pi$
$257$	−23.7089	−1.47892	−0.739461	+	0.673200i	$0.764919\pi$
$258$	0	0
$259$	−2.22546	−0.138283
$260$	0	0
$261$	−0.725323	−0.0448963
$262$	0	0
$263$	−11.1895	−0.689976	−0.344988	−	0.938607i	$-0.612117\pi$
$263$	−11.1895	−0.689976	−0.344988	+	0.938607i	$0.612117\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.11796	−0.252015
$268$	0	0
$269$	10.1403	0.618266	0.309133	−	0.951019i	$-0.399961\pi$
$269$	10.1403	0.618266	0.309133	+	0.951019i	$0.399961\pi$
$270$	0	0
$271$	20.7581	1.26097	0.630483	−	0.776203i	$-0.282857\pi$
$271$	20.7581	1.26097	0.630483	+	0.776203i	$0.282857\pi$
$272$	0	0
$273$	−1.62093	−0.0981033
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.4835	−1.53115	−0.765576	−	0.643345i	$-0.777546\pi$
$277$	−25.4835	−1.53115	−0.765576	+	0.643345i	$0.777546\pi$
$278$	0	0
$279$	−37.9588	−2.27254
$280$	0	0
$281$	4.17313	0.248948	0.124474	−	0.992223i	$-0.460276\pi$
$281$	4.17313	0.248948	0.124474	+	0.992223i	$0.460276\pi$
$282$	0	0
$283$	31.5275	1.87411	0.937056	−	0.349179i	$-0.113539\pi$
$283$	31.5275	1.87411	0.937056	+	0.349179i	$0.113539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.01641	−0.0599966
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−23.9477	−1.40384
$292$	0	0
$293$	−16.1731	−0.944844	−0.472422	−	0.881372i	$-0.656620\pi$
$293$	−16.1731	−0.944844	−0.472422	+	0.881372i	$0.656620\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−18.7253	−1.08655
$298$	0	0
$299$	−0.604525	−0.0349606
$300$	0	0
$301$	0.692509	0.0399156
$302$	0	0
$303$	36.8461	2.11675
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.4835	−1.11198	−0.555990	−	0.831189i	$-0.687661\pi$
$307$	−19.4835	−1.11198	−0.555990	+	0.831189i	$0.687661\pi$
$308$	0	0
$309$	5.45065	0.310076
$310$	0	0
$311$	1.11273	0.0630970	0.0315485	−	0.999502i	$-0.489956\pi$
$311$	1.11273	0.0630970	0.0315485	+	0.999502i	$0.489956\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.8625	1.56492	0.782458	−	0.622704i	$-0.213966\pi$
$317$	27.8625	1.56492	0.782458	+	0.622704i	$0.213966\pi$
$318$	0	0
$319$	1.01641	0.0569079
$320$	0	0
$321$	−37.1924	−2.07588
$322$	0	0
$323$	0.810466	0.0450955
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.9477	1.32431
$328$	0	0
$329$	5.17002	0.285032
$330$	0	0
$331$	−23.1895	−1.27461	−0.637306	−	0.770611i	$-0.719951\pi$
$331$	−23.1895	−1.27461	−0.637306	+	0.770611i	$0.719951\pi$
$332$	0	0
$333$	18.3463	1.00537
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.5850	−1.12134	−0.560668	−	0.828040i	$-0.689456\pi$
$337$	−20.5850	−1.12134	−0.560668	+	0.828040i	$0.689456\pi$
$338$	0	0
$339$	17.5686	0.954195
$340$	0	0
$341$	53.1924	2.88053
$342$	0	0
$343$	6.98359	0.377079
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.4231	1.20373	0.601866	−	0.798597i	$-0.294424\pi$
$347$	22.4231	1.20373	0.601866	+	0.798597i	$0.294424\pi$
$348$	0	0
$349$	10.3463	0.553822	0.276911	−	0.960896i	$-0.410689\pi$
$349$	10.3463	0.553822	0.276911	+	0.960896i	$0.410689\pi$
$350$	0	0
$351$	3.79406	0.202512
$352$	0	0
$353$	19.4506	1.03525	0.517627	−	0.855607i	$-0.326816\pi$
$353$	19.4506	1.03525	0.517627	+	0.855607i	$0.326816\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.36266	0.0721197
$358$	0	0
$359$	−20.1536	−1.06367	−0.531833	−	0.846849i	$-0.678497\pi$
$359$	−20.1536	−1.06367	−0.531833	+	0.846849i	$0.678497\pi$
$360$	0	0
$361$	−18.3431	−0.965429
$362$	0	0
$363$	62.9229	3.30260
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.20071	−0.480273	−0.240137	−	0.970739i	$-0.577192\pi$
$367$	−9.20071	−0.480273	−0.240137	+	0.970739i	$0.577192\pi$
$368$	0	0
$369$	8.37907	0.436197
$370$	0	0
$371$	4.15361	0.215645
$372$	0	0
$373$	−22.4342	−1.16160	−0.580800	−	0.814046i	$-0.697260\pi$
$373$	−22.4342	−1.16160	−0.580800	+	0.814046i	$0.697260\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.205941	−0.0106065
$378$	0	0
$379$	−14.8873	−0.764708	−0.382354	−	0.924016i	$-0.624886\pi$
$379$	−14.8873	−0.764708	−0.382354	+	0.924016i	$0.624886\pi$
$380$	0	0
$381$	−20.2059	−1.03518
$382$	0	0
$383$	1.15672	0.0591057	0.0295528	−	0.999563i	$-0.490592\pi$
$383$	1.15672	0.0591057	0.0295528	+	0.999563i	$0.490592\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70892	−0.290201
$388$	0	0
$389$	12.8133	0.649660	0.324830	−	0.945772i	$-0.394693\pi$
$389$	12.8133	0.649660	0.324830	+	0.945772i	$0.394693\pi$
$390$	0	0
$391$	0.508203	0.0257009
$392$	0	0
$393$	25.5386	1.28825
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.62093	−0.181729	−0.0908647	−	0.995863i	$-0.528963\pi$
$397$	−3.62093	−0.181729	−0.0908647	+	0.995863i	$0.528963\pi$
$398$	0	0
$399$	1.10439	0.0552887
$400$	0	0
$401$	9.07158	0.453013	0.226506	−	0.974010i	$-0.427270\pi$
$401$	9.07158	0.453013	0.226506	+	0.974010i	$0.427270\pi$
$402$	0	0
$403$	−10.7777	−0.536873
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.7089	−1.27434
$408$	0	0
$409$	12.5850	0.622289	0.311144	−	0.950363i	$-0.399288\pi$
$409$	12.5850	0.622289	0.311144	+	0.950363i	$0.399288\pi$
$410$	0	0
$411$	42.2968	2.08635
$412$	0	0
$413$	−5.03876	−0.247941
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−39.9177	−1.95478
$418$	0	0
$419$	6.47539	0.316343	0.158172	−	0.987412i	$-0.449440\pi$
$419$	6.47539	0.316343	0.158172	+	0.987412i	$0.449440\pi$
$420$	0	0
$421$	27.6043	1.34535	0.672675	−	0.739938i	$-0.265145\pi$
$421$	27.6043	1.34535	0.672675	+	0.739938i	$0.265145\pi$
$422$	0	0
$423$	−42.6207	−2.07229
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.32985	0.161143
$428$	0	0
$429$	−18.7253	−0.904067
$430$	0	0
$431$	4.85446	0.233831	0.116916	−	0.993142i	$-0.462699\pi$
$431$	4.85446	0.233831	0.116916	+	0.993142i	$0.462699\pi$
$432$	0	0
$433$	−18.2583	−0.877436	−0.438718	−	0.898625i	$-0.644567\pi$
$433$	−18.2583	−0.877436	−0.438718	+	0.898625i	$0.644567\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.411882	0.0197030
$438$	0	0
$439$	2.62900	0.125475	0.0627377	−	0.998030i	$-0.480017\pi$
$439$	2.62900	0.125475	0.0627377	+	0.998030i	$0.480017\pi$
$440$	0	0
$441$	−28.2447	−1.34499
$442$	0	0
$443$	−38.7253	−1.83990	−0.919948	−	0.392041i	$-0.871769\pi$
$443$	−38.7253	−1.83990	−0.919948	+	0.392041i	$0.871769\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−15.1596	−0.717023
$448$	0	0
$449$	−11.0388	−0.520951	−0.260476	−	0.965480i	$-0.583879\pi$
$449$	−11.0388	−0.520951	−0.260476	+	0.965480i	$0.583879\pi$
$450$	0	0
$451$	−11.7417	−0.552896
$452$	0	0
$453$	−55.8953	−2.62619
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.4999	1.61384	0.806918	−	0.590664i	$-0.201134\pi$
$457$	34.4999	1.61384	0.806918	+	0.590664i	$0.201134\pi$
$458$	0	0
$459$	−3.18953	−0.148875
$460$	0	0
$461$	36.1432	1.68335	0.841677	−	0.539981i	$-0.181569\pi$
$461$	36.1432	1.68335	0.841677	+	0.539981i	$0.181569\pi$
$462$	0	0
$463$	19.2775	0.895902	0.447951	−	0.894058i	$-0.352154\pi$
$463$	19.2775	0.895902	0.447951	+	0.894058i	$0.352154\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.2911	0.476215	0.238107	−	0.971239i	$-0.423473\pi$
$467$	10.2911	0.476215	0.238107	+	0.971239i	$0.423473\pi$
$468$	0	0
$469$	−1.20905	−0.0558288
$470$	0	0
$471$	28.6702	1.32105
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−34.2416	−1.56781
$478$	0	0
$479$	42.7693	1.95418	0.977090	−	0.212827i	$-0.0682670\pi$
$479$	42.7693	1.95418	0.977090	+	0.212827i	$0.0682670\pi$
$480$	0	0
$481$	5.20905	0.237512
$482$	0	0
$483$	0.692509	0.0315103
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.184306	0.00835169	0.00417584	−	0.999991i	$-0.498671\pi$
$487$	0.184306	0.00835169	0.00417584	+	0.999991i	$0.498671\pi$
$488$	0	0
$489$	−54.5327	−2.46605
$490$	0	0
$491$	−34.6074	−1.56181	−0.780904	−	0.624651i	$-0.785241\pi$
$491$	−34.6074	−1.56181	−0.780904	+	0.624651i	$0.785241\pi$
$492$	0	0
$493$	0.173127	0.00779726
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.94483	−0.266662
$498$	0	0
$499$	34.7170	1.55415	0.777073	−	0.629411i	$-0.216704\pi$
$499$	34.7170	1.55415	0.777073	+	0.629411i	$0.216704\pi$
$500$	0	0
$501$	30.1208	1.34570
$502$	0	0
$503$	−5.37100	−0.239481	−0.119741	−	0.992805i	$-0.538206\pi$
$503$	−5.37100	−0.239481	−0.119741	+	0.992805i	$0.538206\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.0632	−1.37957
$508$	0	0
$509$	−15.8625	−0.703094	−0.351547	−	0.936170i	$-0.614344\pi$
$509$	−15.8625	−0.703094	−0.351547	+	0.936170i	$0.614344\pi$
$510$	0	0
$511$	5.36266	0.237230
$512$	0	0
$513$	−2.58501	−0.114131
$514$	0	0
$515$	0	0
$516$	0	0
$517$	59.7251	2.62670
$518$	0	0
$519$	−41.4283	−1.81850
$520$	0	0
$521$	0.895609	0.0392374	0.0196187	−	0.999808i	$-0.493755\pi$
$521$	0.895609	0.0392374	0.0196187	+	0.999808i	$0.493755\pi$
$522$	0	0
$523$	29.7969	1.30293	0.651464	−	0.758680i	$-0.274155\pi$
$523$	29.7969	1.30293	0.651464	+	0.758680i	$0.274155\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.06040	0.394677
$528$	0	0
$529$	−22.7417	−0.988771
$530$	0	0
$531$	41.5386	1.80262
$532$	0	0
$533$	2.37907	0.103049
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−69.1101	−2.98232
$538$	0	0
$539$	39.5798	1.70482
$540$	0	0
$541$	19.5163	0.839070	0.419535	−	0.907739i	$-0.362193\pi$
$541$	19.5163	0.839070	0.419535	+	0.907739i	$0.362193\pi$
$542$	0	0
$543$	−17.9448	−0.770086
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.28586	−0.140493	−0.0702465	−	0.997530i	$-0.522379\pi$
$547$	−3.28586	−0.140493	−0.0702465	+	0.997530i	$0.522379\pi$
$548$	0	0
$549$	−27.4506	−1.17156
$550$	0	0
$551$	0.140314	0.00597757
$552$	0	0
$553$	−2.46705	−0.104910
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.87609	−0.121864	−0.0609320	−	0.998142i	$-0.519407\pi$
$557$	−2.87609	−0.121864	−0.0609320	+	0.998142i	$0.519407\pi$
$558$	0	0
$559$	−1.62093	−0.0685581
$560$	0	0
$561$	15.7417	0.664616
$562$	0	0
$563$	−41.5386	−1.75064	−0.875322	−	0.483540i	$-0.839351\pi$
$563$	−41.5386	−1.75064	−0.875322	+	0.483540i	$0.839351\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.04115	0.0857203
$568$	0	0
$569$	4.17313	0.174947	0.0874733	−	0.996167i	$-0.472121\pi$
$569$	4.17313	0.174947	0.0874733	+	0.996167i	$0.472121\pi$
$570$	0	0
$571$	−31.6678	−1.32525	−0.662627	−	0.748949i	$-0.730559\pi$
$571$	−31.6678	−1.32525	−0.662627	+	0.748949i	$0.730559\pi$
$572$	0	0
$573$	50.2088	2.09750
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.3955	−1.22375	−0.611875	−	0.790955i	$-0.709584\pi$
$577$	−29.3955	−1.22375	−0.611875	+	0.790955i	$0.709584\pi$
$578$	0	0
$579$	1.94483	0.0808244
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	47.9833	1.98727
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0880	0.498924	0.249462	−	0.968385i	$-0.419746\pi$
$587$	12.0880	0.498924	0.249462	+	0.968385i	$0.419746\pi$
$588$	0	0
$589$	7.34314	0.302569
$590$	0	0
$591$	56.4999	2.32409
$592$	0	0
$593$	35.4506	1.45578	0.727892	−	0.685692i	$-0.240500\pi$
$593$	35.4506	1.45578	0.727892	+	0.685692i	$0.240500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.18953	−0.294248
$598$	0	0
$599$	−41.2804	−1.68667	−0.843335	−	0.537388i	$-0.819411\pi$
$599$	−41.2804	−1.68667	−0.843335	+	0.537388i	$0.819411\pi$
$600$	0	0
$601$	−21.0716	−0.859528	−0.429764	−	0.902941i	$-0.641403\pi$
$601$	−21.0716	−0.859528	−0.429764	+	0.902941i	$0.641403\pi$
$602$	0	0
$603$	9.96719	0.405895
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.2992	0.620973	0.310487	−	0.950578i	$-0.399508\pi$
$607$	15.2992	0.620973	0.310487	+	0.950578i	$0.399508\pi$
$608$	0	0
$609$	0.235914	0.00955971
$610$	0	0
$611$	−12.1013	−0.489565
$612$	0	0
$613$	−11.8269	−0.477683	−0.238841	−	0.971059i	$-0.576768\pi$
$613$	−11.8269	−0.477683	−0.238841	+	0.971059i	$0.576768\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.5850	0.667687	0.333844	−	0.942628i	$-0.391654\pi$
$617$	16.5850	0.667687	0.333844	+	0.942628i	$0.391654\pi$
$618$	0	0
$619$	47.9917	1.92895	0.964474	−	0.264178i	$-0.0851007\pi$
$619$	47.9917	1.92895	0.964474	+	0.264178i	$0.0851007\pi$
$620$	0	0
$621$	−1.62093	−0.0650458
$622$	0	0
$623$	0.780493	0.0312698
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.7581	0.509511
$628$	0	0
$629$	−4.37907	−0.174605
$630$	0	0
$631$	−20.4342	−0.813474	−0.406737	−	0.913545i	$-0.633333\pi$
$631$	−20.4342	−0.813474	−0.406737	+	0.913545i	$0.633333\pi$
$632$	0	0
$633$	−12.7805	−0.507979
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.01952	−0.317745
$638$	0	0
$639$	49.0081	1.93873
$640$	0	0
$641$	27.5163	1.08683	0.543414	−	0.839465i	$-0.317132\pi$
$641$	27.5163	1.08683	0.543414	+	0.839465i	$0.317132\pi$
$642$	0	0
$643$	9.61259	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$643$	9.61259	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.3491	1.35040	0.675201	−	0.737634i	$-0.264057\pi$
$647$	34.3491	1.35040	0.675201	+	0.737634i	$0.264057\pi$
$648$	0	0
$649$	−58.2088	−2.28489
$650$	0	0
$651$	12.3463	0.483888
$652$	0	0
$653$	19.9672	0.781376	0.390688	−	0.920523i	$-0.372237\pi$
$653$	19.9672	0.781376	0.390688	+	0.920523i	$0.372237\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−44.2088	−1.72475
$658$	0	0
$659$	−0.898450	−0.0349986	−0.0174993	−	0.999847i	$-0.505570\pi$
$659$	−0.898450	−0.0349986	−0.0174993	+	0.999847i	$0.505570\pi$
$660$	0	0
$661$	−26.4119	−1.02730	−0.513652	−	0.857999i	$-0.671708\pi$
$661$	−26.4119	−1.02730	−0.513652	+	0.857999i	$0.671708\pi$
$662$	0	0
$663$	−3.18953	−0.123871
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.0879839	0.00340675
$668$	0	0
$669$	−27.5134	−1.06373
$670$	0	0
$671$	38.4671	1.48500
$672$	0	0
$673$	−3.76125	−0.144985	−0.0724927	−	0.997369i	$-0.523095\pi$
$673$	−3.76125	−0.144985	−0.0724927	+	0.997369i	$0.523095\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	4.53890	0.174187
$680$	0	0
$681$	27.0192	1.03538
$682$	0	0
$683$	1.34103	0.0513129	0.0256565	−	0.999671i	$-0.491832\pi$
$683$	1.34103	0.0513129	0.0256565	+	0.999671i	$0.491832\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	74.2968	2.83460
$688$	0	0
$689$	−9.72221	−0.370387
$690$	0	0
$691$	4.16195	0.158328	0.0791640	−	0.996862i	$-0.474775\pi$
$691$	4.16195	0.158328	0.0791640	+	0.996862i	$0.474775\pi$
$692$	0	0
$693$	12.4999	0.474831
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	0	0
$699$	33.5686	1.26968
$700$	0	0
$701$	42.0880	1.58964	0.794821	−	0.606844i	$-0.207565\pi$
$701$	42.0880	1.58964	0.794821	+	0.606844i	$0.207565\pi$
$702$	0	0
$703$	−3.54909	−0.133856
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.98359	−0.262645
$708$	0	0
$709$	18.5522	0.696742	0.348371	−	0.937357i	$-0.386735\pi$
$709$	18.5522	0.696742	0.348371	+	0.937357i	$0.386735\pi$
$710$	0	0
$711$	20.3379	0.762731
$712$	0	0
$713$	4.60453	0.172441
$714$	0	0
$715$	0	0
$716$	0	0
$717$	58.6207	2.18923
$718$	0	0
$719$	3.69774	0.137902	0.0689512	−	0.997620i	$-0.478035\pi$
$719$	3.69774	0.137902	0.0689512	+	0.997620i	$0.478035\pi$
$720$	0	0
$721$	−1.03308	−0.0384740
$722$	0	0
$723$	−10.6373	−0.395607
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.2119	0.564178	0.282089	−	0.959388i	$-0.408973\pi$
$727$	15.2119	0.564178	0.282089	+	0.959388i	$0.408973\pi$
$728$	0	0
$729$	−42.4835	−1.57346
$730$	0	0
$731$	1.36266	0.0503998
$732$	0	0
$733$	−13.3075	−0.491523	−0.245762	−	0.969330i	$-0.579038\pi$
$733$	−13.3075	−0.491523	−0.245762	+	0.969330i	$0.579038\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.9672	−0.514488
$738$	0	0
$739$	−3.94767	−0.145217	−0.0726087	−	0.997361i	$-0.523132\pi$
$739$	−3.94767	−0.145217	−0.0726087	+	0.997361i	$0.523132\pi$
$740$	0	0
$741$	−2.58501	−0.0949627
$742$	0	0
$743$	−50.3931	−1.84874	−0.924372	−	0.381493i	$-0.875410\pi$
$743$	−50.3931	−1.84874	−0.924372	+	0.381493i	$0.875410\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−65.9505	−2.41300
$748$	0	0
$749$	7.04922	0.257573
$750$	0	0
$751$	52.9781	1.93320	0.966599	−	0.256293i	$-0.0825013\pi$
$751$	52.9781	1.93320	0.966599	+	0.256293i	$0.0825013\pi$
$752$	0	0
$753$	−37.6266	−1.37119
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.3655	0.921925	0.460962	−	0.887420i	$-0.347504\pi$
$757$	25.3655	0.921925	0.460962	+	0.887420i	$0.347504\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	47.1924	1.71072	0.855361	−	0.518032i	$-0.173335\pi$
$761$	47.1924	1.71072	0.855361	+	0.518032i	$0.173335\pi$
$762$	0	0
$763$	−4.53890	−0.164319
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.7941	0.425859
$768$	0	0
$769$	10.8105	0.389835	0.194918	−	0.980820i	$-0.437556\pi$
$769$	10.8105	0.389835	0.194918	+	0.980820i	$0.437556\pi$
$770$	0	0
$771$	−63.5714	−2.28947
$772$	0	0
$773$	−49.9833	−1.79778	−0.898888	−	0.438179i	$-0.855624\pi$
$773$	−49.9833	−1.79778	−0.898888	+	0.438179i	$0.855624\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.96719	−0.214072
$778$	0	0
$779$	−1.62093	−0.0580759
$780$	0	0
$781$	−68.6758	−2.45741
$782$	0	0
$783$	−0.552195	−0.0197339
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.13198	0.289874	0.144937	−	0.989441i	$-0.453702\pi$
$787$	8.13198	0.289874	0.144937	+	0.989441i	$0.453702\pi$
$788$	0	0
$789$	−30.0028	−1.06813
$790$	0	0
$791$	−3.32985	−0.118396
$792$	0	0
$793$	−7.79406	−0.276775
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.7251	−1.76135	−0.880676	−	0.473719i	$-0.842911\pi$
$797$	−49.7251	−1.76135	−0.880676	+	0.473719i	$0.842911\pi$
$798$	0	0
$799$	10.1731	0.359899
$800$	0	0
$801$	−6.43424	−0.227343
$802$	0	0
$803$	61.9505	2.18619
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.1895	0.957117
$808$	0	0
$809$	13.4178	0.471746	0.235873	−	0.971784i	$-0.424205\pi$
$809$	13.4178	0.471746	0.235873	+	0.971784i	$0.424205\pi$
$810$	0	0
$811$	13.4590	0.472609	0.236304	−	0.971679i	$-0.424064\pi$
$811$	13.4590	0.472609	0.236304	+	0.971679i	$0.424064\pi$
$812$	0	0
$813$	55.6594	1.95206
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.10439	0.0386377
$818$	0	0
$819$	−2.53268	−0.0884990
$820$	0	0
$821$	−38.7282	−1.35162	−0.675811	−	0.737075i	$-0.736206\pi$
$821$	−38.7282	−1.35162	−0.675811	+	0.737075i	$0.736206\pi$
$822$	0	0
$823$	−29.4423	−1.02629	−0.513147	−	0.858301i	$-0.671520\pi$
$823$	−29.4423	−1.02629	−0.513147	+	0.858301i	$0.671520\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.6126	−0.612450	−0.306225	−	0.951959i	$-0.599066\pi$
$827$	−17.6126	−0.612450	−0.306225	+	0.951959i	$0.599066\pi$
$828$	0	0
$829$	−0.895609	−0.0311058	−0.0155529	−	0.999879i	$-0.504951\pi$
$829$	−0.895609	−0.0311058	−0.0155529	+	0.999879i	$0.504951\pi$
$830$	0	0
$831$	−68.3296	−2.37033
$832$	0	0
$833$	6.74173	0.233587
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−28.8984	−0.998877
$838$	0	0
$839$	39.0276	1.34738	0.673691	−	0.739013i	$-0.264708\pi$
$839$	39.0276	1.34738	0.673691	+	0.739013i	$0.264708\pi$
$840$	0	0
$841$	−28.9700	−0.998966
$842$	0	0
$843$	11.1895	0.385388
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.9260	−0.409783
$848$	0	0
$849$	84.5355	2.90125
$850$	0	0
$851$	−2.22546	−0.0762877
$852$	0	0
$853$	−28.2088	−0.965850	−0.482925	−	0.875662i	$-0.660426\pi$
$853$	−28.2088	−0.965850	−0.482925	+	0.875662i	$0.660426\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.9700	1.16039	0.580197	−	0.814476i	$-0.302976\pi$
$857$	33.9700	1.16039	0.580197	+	0.814476i	$0.302976\pi$
$858$	0	0
$859$	2.60737	0.0889622	0.0444811	−	0.999010i	$-0.485837\pi$
$859$	2.60737	0.0889622	0.0444811	+	0.999010i	$0.485837\pi$
$860$	0	0
$861$	−2.72532	−0.0928787
$862$	0	0
$863$	−3.93437	−0.133928	−0.0669638	−	0.997755i	$-0.521331\pi$
$863$	−3.93437	−0.133928	−0.0669638	+	0.997755i	$0.521331\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.68133	0.0910628
$868$	0	0
$869$	−28.4999	−0.966792
$870$	0	0
$871$	2.82998	0.0958903
$872$	0	0
$873$	−37.4178	−1.26640
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.1044	0.510039	0.255020	−	0.966936i	$-0.417918\pi$
$877$	15.1044	0.510039	0.255020	+	0.966936i	$0.417918\pi$
$878$	0	0
$879$	−43.3655	−1.46268
$880$	0	0
$881$	−28.9669	−0.975920	−0.487960	−	0.872866i	$-0.662259\pi$
$881$	−28.9669	−0.975920	−0.487960	+	0.872866i	$0.662259\pi$
$882$	0	0
$883$	−22.3134	−0.750907	−0.375454	−	0.926841i	$-0.622513\pi$
$883$	−22.3134	−0.750907	−0.375454	+	0.926841i	$0.622513\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.4587	−0.485476	−0.242738	−	0.970092i	$-0.578046\pi$
$887$	−14.4587	−0.485476	−0.242738	+	0.970092i	$0.578046\pi$
$888$	0	0
$889$	3.82971	0.128444
$890$	0	0
$891$	23.5798	0.789952
$892$	0	0
$893$	8.24497	0.275908
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.62093	−0.0541213
$898$	0	0
$899$	1.56860	0.0523158
$900$	0	0
$901$	8.17313	0.272286
$902$	0	0
$903$	1.85685	0.0617920
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.1976	−0.936286	−0.468143	−	0.883653i	$-0.655077\pi$
$907$	−28.1976	−0.936286	−0.468143	+	0.883653i	$0.655077\pi$
$908$	0	0
$909$	57.5714	1.90952
$910$	0	0
$911$	−22.0468	−0.730444	−0.365222	−	0.930920i	$-0.619007\pi$
$911$	−22.0468	−0.730444	−0.365222	+	0.930920i	$0.619007\pi$
$912$	0	0
$913$	92.4176	3.05857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.84044	−0.159845
$918$	0	0
$919$	−29.8625	−0.985074	−0.492537	−	0.870292i	$-0.663930\pi$
$919$	−29.8625	−0.985074	−0.492537	+	0.870292i	$0.663930\pi$
$920$	0	0
$921$	−52.2416	−1.72142
$922$	0	0
$923$	13.9149	0.458013
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.51654	0.279720
$928$	0	0
$929$	50.8685	1.66894	0.834470	−	0.551053i	$-0.185774\pi$
$929$	50.8685	1.66894	0.834470	+	0.551053i	$0.185774\pi$
$930$	0	0
$931$	5.46394	0.179073
$932$	0	0
$933$	2.98359	0.0976785
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14.3463	−0.468672	−0.234336	−	0.972156i	$-0.575292\pi$
$937$	−14.3463	−0.468672	−0.234336	+	0.972156i	$0.575292\pi$
$938$	0	0
$939$	−5.36266	−0.175004
$940$	0	0
$941$	−26.3819	−0.860026	−0.430013	−	0.902823i	$-0.641491\pi$
$941$	−26.3819	−0.860026	−0.430013	+	0.902823i	$0.641491\pi$
$942$	0	0
$943$	−1.01641	−0.0330988
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.6649	−1.09396	−0.546982	−	0.837145i	$-0.684223\pi$
$947$	−33.6649	−1.09396	−0.546982	+	0.837145i	$0.684223\pi$
$948$	0	0
$949$	−12.5522	−0.407462
$950$	0	0
$951$	74.7086	2.42259
$952$	0	0
$953$	22.3296	0.723326	0.361663	−	0.932309i	$-0.382209\pi$
$953$	22.3296	0.723326	0.361663	+	0.932309i	$0.382209\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.72532	0.0880972
$958$	0	0
$959$	−8.01668	−0.258872
$960$	0	0
$961$	51.0908	1.64809
$962$	0	0
$963$	−58.1125	−1.87265
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.7745	1.21475	0.607374	−	0.794416i	$-0.292223\pi$
$967$	37.7745	1.21475	0.607374	+	0.794416i	$0.292223\pi$
$968$	0	0
$969$	2.17313	0.0698109
$970$	0	0
$971$	−25.5030	−0.818429	−0.409215	−	0.912438i	$-0.634197\pi$
$971$	−25.5030	−0.818429	−0.409215	+	0.912438i	$0.634197\pi$
$972$	0	0
$973$	7.56576	0.242547
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.17002	−0.229389	−0.114695	−	0.993401i	$-0.536589\pi$
$977$	−7.17002	−0.229389	−0.114695	+	0.993401i	$0.536589\pi$
$978$	0	0
$979$	9.01641	0.288166
$980$	0	0
$981$	37.4178	1.19466
$982$	0	0
$983$	−32.1843	−1.02652	−0.513260	−	0.858233i	$-0.671562\pi$
$983$	−32.1843	−1.02652	−0.513260	+	0.858233i	$0.671562\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.8625	0.441249
$988$	0	0
$989$	0.692509	0.0220205
$990$	0	0
$991$	12.3902	0.393589	0.196795	−	0.980445i	$-0.436947\pi$
$991$	12.3902	0.393589	0.196795	+	0.980445i	$0.436947\pi$
$992$	0	0
$993$	−62.1788	−1.97318
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.0328	−0.381083	−0.190542	−	0.981679i	$-0.561024\pi$
$997$	−12.0328	−0.381083	−0.190542	+	0.981679i	$0.561024\pi$
$998$	0	0
$999$	13.9672	0.441902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bo.1.3		3
4.3	odd	2		3400.2.a.n.1.1		3
5.4	even	2		1360.2.a.r.1.1		3
20.3	even	4		3400.2.e.k.2449.1		6
20.7	even	4		3400.2.e.k.2449.6		6
20.19	odd	2		680.2.a.f.1.3	✓	3
40.19	odd	2		5440.2.a.bo.1.1		3
40.29	even	2		5440.2.a.bt.1.3		3
60.59	even	2		6120.2.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.f.1.3	✓	3	20.19	odd	2
1360.2.a.r.1.1		3	5.4	even	2
3400.2.a.n.1.1		3	4.3	odd	2
3400.2.e.k.2449.1		6	20.3	even	4
3400.2.e.k.2449.6		6	20.7	even	4
5440.2.a.bo.1.1		3	40.19	odd	2
5440.2.a.bt.1.3		3	40.29	even	2
6120.2.a.br.1.2		3	60.59	even	2
6800.2.a.bo.1.3		3	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:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68133 q^{3} -0.508203 q^{7} +4.18953 q^{9} -5.87086 q^{11} +1.18953 q^{13} -1.00000 q^{17} -0.810466 q^{19} -1.36266 q^{21} -0.508203 q^{23} +3.18953 q^{27} -0.173127 q^{29} -9.06040 q^{31} -15.7417 q^{33} +4.37907 q^{37} +3.18953 q^{39} +2.00000 q^{41} -1.36266 q^{43} -10.1731 q^{47} -6.74173 q^{49} -2.68133 q^{51} -8.17313 q^{53} -2.17313 q^{57} +9.91486 q^{59} -6.55220 q^{61} -2.12914 q^{63} +2.37907 q^{67} -1.36266 q^{69} +11.6977 q^{71} -10.5522 q^{73} +2.98359 q^{77} +4.85446 q^{79} -4.01641 q^{81} -15.7417 q^{83} -0.464211 q^{87} -1.53579 q^{89} -0.604525 q^{91} -24.2939 q^{93} -8.93126 q^{97} -24.5962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 4 q^{9} - 2 q^{11} - 5 q^{13} - 3 q^{17} - 11 q^{19} + 10 q^{21} + q^{27} + 5 q^{29} - 3 q^{31} - 16 q^{33} - 4 q^{37} + q^{39} + 6 q^{41} + 10 q^{43} - 25 q^{47} + 11 q^{49} - q^{51}+ \cdots - 30 q^{99}+O(q^{100}) \) 3 * q + q^3 + 4 * q^9 - 2 * q^11 - 5 * q^13 - 3 * q^17 - 11 * q^19 + 10 * q^21 + q^27 + 5 * q^29 - 3 * q^31 - 16 * q^33 - 4 * q^37 + q^39 + 6 * q^41 + 10 * q^43 - 25 * q^47 + 11 * q^49 - q^51 - 19 * q^53 - q^57 - 7 * q^59 + 3 * q^61 - 22 * q^63 - 10 * q^67 + 10 * q^69 + 25 * q^71 - 9 * q^73 + 12 * q^77 + 2 * q^79 - 9 * q^81 - 16 * q^83 - 21 * q^87 + 15 * q^89 - 22 * q^91 - 19 * q^93 + 13 * q^97 - 30 * q^99

Introduction

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