LMFDB - Embedded newform 6800.2.a.cd.1.1

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)

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);

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")

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 = [5,0,1,0,0,0,1,0,6,0,-4,0,3] 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:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.48887$ 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)

$f(q)$	$=$	$q-2.48887 q^{3} +3.05725 q^{7} +3.19447 q^{9} +5.36180 q^{11} +4.59895 q^{13} -1.00000 q^{17} -4.57325 q^{19} -7.60910 q^{21} +1.24730 q^{23} -0.483999 q^{27} -5.93018 q^{29} -9.84580 q^{31} -13.3448 q^{33} -4.20461 q^{37} -11.4462 q^{39} +0.404485 q^{41} -5.76142 q^{43} -3.35693 q^{47} +2.34678 q^{49} +2.48887 q^{51} -4.81568 q^{53} +11.3822 q^{57} -12.7392 q^{59} +4.97774 q^{61} +9.76628 q^{63} +6.82926 q^{67} -3.10436 q^{69} -11.9408 q^{71} -10.8876 q^{73} +16.3924 q^{77} -16.7139 q^{79} -8.37879 q^{81} +4.11450 q^{83} +14.7594 q^{87} -10.4142 q^{89} +14.0601 q^{91} +24.5049 q^{93} +2.27443 q^{97} +17.1281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + q^{7} + 6 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{17} - 6 q^{19} - 5 q^{21} + 4 q^{23} - 5 q^{27} + 2 q^{29} - 21 q^{31} - 12 q^{33} + 2 q^{37} - 23 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q + q^3 + q^7 + 6 * q^9 - 4 * q^11 + 3 * q^13 - 5 * q^17 - 6 * q^19 - 5 * q^21 + 4 * q^23 - 5 * q^27 + 2 * q^29 - 21 * q^31 - 12 * q^33 + 2 * q^37 - 23 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 + 10 * q^49 - q^51 - 21 * q^53 + 20 * q^57 - 12 * q^59 - 2 * q^61 + 8 * q^63 - 12 * q^67 - 21 * q^71 - 22 * q^73 + 32 * q^77 - 41 * q^79 - 35 * q^81 - 8 * q^83 - 4 * q^87 - 10 * q^89 - 13 * q^91 - 9 * q^93 + 20 * q^97 + 14 * 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
$3$	−2.48887	−1.43695	−0.718474	−	0.695553i	$-0.755159\pi$
$3$	−2.48887	−1.43695	−0.718474	+	0.695553i	$0.755159\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.05725	1.15553	0.577766	−	0.816202i	$-0.303925\pi$
$7$	3.05725	1.15553	0.577766	+	0.816202i	$0.303925\pi$
$8$	0	0
$9$	3.19447	1.06482
$10$	0	0
$11$	5.36180	1.61664	0.808322	−	0.588741i	$-0.200376\pi$
$11$	5.36180	1.61664	0.808322	+	0.588741i	$0.200376\pi$
$12$	0	0
$13$	4.59895	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$13$	4.59895	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.57325	−1.04918	−0.524588	−	0.851356i	$-0.675781\pi$
$19$	−4.57325	−1.04918	−0.524588	+	0.851356i	$0.675781\pi$
$20$	0	0
$21$	−7.60910	−1.66044
$22$	0	0
$23$	1.24730	0.260079	0.130040	−	0.991509i	$-0.458490\pi$
$23$	1.24730	0.260079	0.130040	+	0.991509i	$0.458490\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.483999	−0.0931457
$28$	0	0
$29$	−5.93018	−1.10121	−0.550604	−	0.834767i	$-0.685602\pi$
$29$	−5.93018	−1.10121	−0.550604	+	0.834767i	$0.685602\pi$
$30$	0	0
$31$	−9.84580	−1.76836	−0.884179	−	0.467149i	$-0.845281\pi$
$31$	−9.84580	−1.76836	−0.884179	+	0.467149i	$0.845281\pi$
$32$	0	0
$33$	−13.3448	−2.32303
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.20461	−0.691234	−0.345617	−	0.938376i	$-0.612330\pi$
$37$	−4.20461	−0.691234	−0.345617	+	0.938376i	$0.612330\pi$
$38$	0	0
$39$	−11.4462	−1.83286
$40$	0	0
$41$	0.404485	0.0631699	0.0315850	−	0.999501i	$-0.489945\pi$
$41$	0.404485	0.0631699	0.0315850	+	0.999501i	$0.489945\pi$
$42$	0	0
$43$	−5.76142	−0.878608	−0.439304	−	0.898339i	$-0.644775\pi$
$43$	−5.76142	−0.878608	−0.439304	+	0.898339i	$0.644775\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.35693	−0.489659	−0.244829	−	0.969566i	$-0.578732\pi$
$47$	−3.35693	−0.489659	−0.244829	+	0.969566i	$0.578732\pi$
$48$	0	0
$49$	2.34678	0.335255
$50$	0	0
$51$	2.48887	0.348511
$52$	0	0
$53$	−4.81568	−0.661484	−0.330742	−	0.943721i	$-0.607299\pi$
$53$	−4.81568	−0.661484	−0.330742	+	0.943721i	$0.607299\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.3822	1.50761
$58$	0	0
$59$	−12.7392	−1.65850	−0.829248	−	0.558881i	$-0.811231\pi$
$59$	−12.7392	−1.65850	−0.829248	+	0.558881i	$0.811231\pi$
$60$	0	0
$61$	4.97774	0.637334	0.318667	−	0.947867i	$-0.396765\pi$
$61$	4.97774	0.637334	0.318667	+	0.947867i	$0.396765\pi$
$62$	0	0
$63$	9.76628	1.23044
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.82926	0.834327	0.417163	−	0.908831i	$-0.363024\pi$
$67$	6.82926	0.834327	0.417163	+	0.908831i	$0.363024\pi$
$68$	0	0
$69$	−3.10436	−0.373721
$70$	0	0
$71$	−11.9408	−1.41711	−0.708555	−	0.705656i	$-0.750652\pi$
$71$	−11.9408	−1.41711	−0.708555	+	0.705656i	$0.750652\pi$
$72$	0	0
$73$	−10.8876	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$73$	−10.8876	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.3924	1.86808
$78$	0	0
$79$	−16.7139	−1.88046	−0.940230	−	0.340539i	$-0.889391\pi$
$79$	−16.7139	−1.88046	−0.940230	+	0.340539i	$0.889391\pi$
$80$	0	0
$81$	−8.37879	−0.930976
$82$	0	0
$83$	4.11450	0.451625	0.225813	−	0.974171i	$-0.427496\pi$
$83$	4.11450	0.451625	0.225813	+	0.974171i	$0.427496\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	14.7594	1.58238
$88$	0	0
$89$	−10.4142	−1.10391	−0.551953	−	0.833875i	$-0.686117\pi$
$89$	−10.4142	−1.10391	−0.551953	+	0.833875i	$0.686117\pi$
$90$	0	0
$91$	14.0601	1.47390
$92$	0	0
$93$	24.5049	2.54104
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.27443	0.230933	0.115467	−	0.993311i	$-0.463164\pi$
$97$	2.27443	0.230933	0.115467	+	0.993311i	$0.463164\pi$
$98$	0	0
$99$	17.1281	1.72144
$100$	0	0
$101$	8.12465	0.808433	0.404216	−	0.914663i	$-0.367544\pi$
$101$	8.12465	0.808433	0.404216	+	0.914663i	$0.367544\pi$
$102$	0	0
$103$	−5.15706	−0.508140	−0.254070	−	0.967186i	$-0.581769\pi$
$103$	−5.15706	−0.508140	−0.254070	+	0.967186i	$0.581769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.61021	0.542360	0.271180	−	0.962529i	$-0.412586\pi$
$107$	5.61021	0.542360	0.271180	+	0.962529i	$0.412586\pi$
$108$	0	0
$109$	−3.98158	−0.381366	−0.190683	−	0.981652i	$-0.561070\pi$
$109$	−3.98158	−0.381366	−0.190683	+	0.981652i	$0.561070\pi$
$110$	0	0
$111$	10.4647	0.993268
$112$	0	0
$113$	0.913734	0.0859569	0.0429785	−	0.999076i	$-0.486315\pi$
$113$	0.913734	0.0859569	0.0429785	+	0.999076i	$0.486315\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.6912	1.35820
$118$	0	0
$119$	−3.05725	−0.280258
$120$	0	0
$121$	17.7489	1.61354
$122$	0	0
$123$	−1.00671	−0.0907720
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.41422	0.746642	0.373321	−	0.927702i	$-0.378219\pi$
$127$	8.41422	0.746642	0.373321	+	0.927702i	$0.378219\pi$
$128$	0	0
$129$	14.3394	1.26251
$130$	0	0
$131$	1.34723	0.117708	0.0588542	−	0.998267i	$-0.481255\pi$
$131$	1.34723	0.117708	0.0588542	+	0.998267i	$0.481255\pi$
$132$	0	0
$133$	−13.9816	−1.21236
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.5650	−1.07350	−0.536749	−	0.843742i	$-0.680348\pi$
$137$	−12.5650	−1.07350	−0.536749	+	0.843742i	$0.680348\pi$
$138$	0	0
$139$	9.83024	0.833790	0.416895	−	0.908955i	$-0.363118\pi$
$139$	9.83024	0.833790	0.416895	+	0.908955i	$0.363118\pi$
$140$	0	0
$141$	8.35496	0.703614
$142$	0	0
$143$	24.6586	2.06206
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.84084	−0.481744
$148$	0	0
$149$	−9.03003	−0.739769	−0.369884	−	0.929078i	$-0.620603\pi$
$149$	−9.03003	−0.739769	−0.369884	+	0.929078i	$0.620603\pi$
$150$	0	0
$151$	5.97576	0.486301	0.243150	−	0.969989i	$-0.421819\pi$
$151$	5.97576	0.486301	0.243150	+	0.969989i	$0.421819\pi$
$152$	0	0
$153$	−3.19447	−0.258257
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.71002	0.695135	0.347568	−	0.937655i	$-0.387008\pi$
$157$	8.71002	0.695135	0.347568	+	0.937655i	$0.387008\pi$
$158$	0	0
$159$	11.9856	0.950519
$160$	0	0
$161$	3.81330	0.300530
$162$	0	0
$163$	0.852508	0.0667736	0.0333868	−	0.999443i	$-0.489371\pi$
$163$	0.852508	0.0667736	0.0333868	+	0.999443i	$0.489371\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.32122	0.102239	0.0511195	−	0.998693i	$-0.483721\pi$
$167$	1.32122	0.102239	0.0511195	+	0.998693i	$0.483721\pi$
$168$	0	0
$169$	8.15035	0.626950
$170$	0	0
$171$	−14.6091	−1.11719
$172$	0	0
$173$	7.49672	0.569965	0.284983	−	0.958533i	$-0.408012\pi$
$173$	7.49672	0.569965	0.284983	+	0.958533i	$0.408012\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	31.7061	2.38317
$178$	0	0
$179$	−15.9913	−1.19525	−0.597624	−	0.801777i	$-0.703888\pi$
$179$	−15.9913	−1.19525	−0.597624	+	0.801777i	$0.703888\pi$
$180$	0	0
$181$	−11.0136	−0.818633	−0.409317	−	0.912392i	$-0.634233\pi$
$181$	−11.0136	−0.818633	−0.409317	+	0.912392i	$0.634233\pi$
$182$	0	0
$183$	−12.3889	−0.915816
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.36180	−0.392094
$188$	0	0
$189$	−1.47971	−0.107633
$190$	0	0
$191$	−8.49870	−0.614944	−0.307472	−	0.951557i	$-0.599483\pi$
$191$	−8.49870	−0.614944	−0.307472	+	0.951557i	$0.599483\pi$
$192$	0	0
$193$	−18.8634	−1.35782	−0.678908	−	0.734223i	$-0.737547\pi$
$193$	−18.8634	−1.35782	−0.678908	+	0.734223i	$0.737547\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.7750	−0.767687	−0.383843	−	0.923398i	$-0.625400\pi$
$197$	−10.7750	−0.767687	−0.383843	+	0.923398i	$0.625400\pi$
$198$	0	0
$199$	12.7488	0.903735	0.451868	−	0.892085i	$-0.350758\pi$
$199$	12.7488	0.903735	0.451868	+	0.892085i	$0.350758\pi$
$200$	0	0
$201$	−16.9971	−1.19889
$202$	0	0
$203$	−18.1301	−1.27248
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.98445	0.276938
$208$	0	0
$209$	−24.5209	−1.69614
$210$	0	0
$211$	−26.3541	−1.81429	−0.907146	−	0.420817i	$-0.861743\pi$
$211$	−26.3541	−1.81429	−0.907146	+	0.420817i	$0.861743\pi$
$212$	0	0
$213$	29.7190	2.03631
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−30.1011	−2.04339
$218$	0	0
$219$	27.0979	1.83110
$220$	0	0
$221$	−4.59895	−0.309359
$222$	0	0
$223$	−20.0300	−1.34131	−0.670655	−	0.741769i	$-0.733987\pi$
$223$	−20.0300	−1.34131	−0.670655	+	0.741769i	$0.733987\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.2928	0.882277	0.441138	−	0.897439i	$-0.354575\pi$
$227$	13.2928	0.882277	0.441138	+	0.897439i	$0.354575\pi$
$228$	0	0
$229$	10.0240	0.662404	0.331202	−	0.943560i	$-0.392546\pi$
$229$	10.0240	0.662404	0.331202	+	0.943560i	$0.392546\pi$
$230$	0	0
$231$	−40.7984	−2.68434
$232$	0	0
$233$	5.12121	0.335502	0.167751	−	0.985829i	$-0.446350\pi$
$233$	5.12121	0.335502	0.167751	+	0.985829i	$0.446350\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	41.5987	2.70213
$238$	0	0
$239$	13.3667	0.864618	0.432309	−	0.901726i	$-0.357699\pi$
$239$	13.3667	0.864618	0.432309	+	0.901726i	$0.357699\pi$
$240$	0	0
$241$	−4.12595	−0.265776	−0.132888	−	0.991131i	$-0.542425\pi$
$241$	−4.12595	−0.265776	−0.132888	+	0.991131i	$0.542425\pi$
$242$	0	0
$243$	22.3057	1.43091
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−21.0322	−1.33824
$248$	0	0
$249$	−10.2405	−0.648962
$250$	0	0
$251$	7.71502	0.486968	0.243484	−	0.969905i	$-0.421710\pi$
$251$	7.71502	0.486968	0.243484	+	0.969905i	$0.421710\pi$
$252$	0	0
$253$	6.68775	0.420456
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.4651	1.15182	0.575912	−	0.817512i	$-0.304647\pi$
$257$	18.4651	1.15182	0.575912	+	0.817512i	$0.304647\pi$
$258$	0	0
$259$	−12.8546	−0.798743
$260$	0	0
$261$	−18.9438	−1.17259
$262$	0	0
$263$	−18.7644	−1.15707	−0.578533	−	0.815659i	$-0.696374\pi$
$263$	−18.7644	−1.15707	−0.578533	+	0.815659i	$0.696374\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	25.9196	1.58626
$268$	0	0
$269$	14.7847	0.901441	0.450721	−	0.892665i	$-0.351167\pi$
$269$	14.7847	0.901441	0.450721	+	0.892665i	$0.351167\pi$
$270$	0	0
$271$	24.2920	1.47563	0.737817	−	0.675001i	$-0.235857\pi$
$271$	24.2920	1.47563	0.737817	+	0.675001i	$0.235857\pi$
$272$	0	0
$273$	−34.9939	−2.11792
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.84481	0.351181	0.175590	−	0.984463i	$-0.443817\pi$
$277$	5.84481	0.351181	0.175590	+	0.984463i	$0.443817\pi$
$278$	0	0
$279$	−31.4521	−1.88299
$280$	0	0
$281$	−0.651914	−0.0388899	−0.0194450	−	0.999811i	$-0.506190\pi$
$281$	−0.651914	−0.0388899	−0.0194450	+	0.999811i	$0.506190\pi$
$282$	0	0
$283$	4.62452	0.274899	0.137450	−	0.990509i	$-0.456110\pi$
$283$	4.62452	0.274899	0.137450	+	0.990509i	$0.456110\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.23661	0.0729949
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.66075	−0.331839
$292$	0	0
$293$	−3.46169	−0.202234	−0.101117	−	0.994875i	$-0.532242\pi$
$293$	−3.46169	−0.202234	−0.101117	+	0.994875i	$0.532242\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.59511	−0.150583
$298$	0	0
$299$	5.73626	0.331736
$300$	0	0
$301$	−17.6141	−1.01526
$302$	0	0
$303$	−20.2212	−1.16168
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.7625	−1.12790	−0.563952	−	0.825808i	$-0.690720\pi$
$307$	−19.7625	−1.12790	−0.563952	+	0.825808i	$0.690720\pi$
$308$	0	0
$309$	12.8352	0.730171
$310$	0	0
$311$	−26.2366	−1.48774	−0.743871	−	0.668324i	$-0.767012\pi$
$311$	−26.2366	−1.48774	−0.743871	+	0.668324i	$0.767012\pi$
$312$	0	0
$313$	22.9825	1.29905	0.649523	−	0.760342i	$-0.274969\pi$
$313$	22.9825	1.29905	0.649523	+	0.760342i	$0.274969\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.9419	−0.614558	−0.307279	−	0.951619i	$-0.599418\pi$
$317$	−10.9419	−0.614558	−0.307279	+	0.951619i	$0.599418\pi$
$318$	0	0
$319$	−31.7964	−1.78026
$320$	0	0
$321$	−13.9631	−0.779343
$322$	0	0
$323$	4.57325	0.254463
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.90963	0.548004
$328$	0	0
$329$	−10.2630	−0.565816
$330$	0	0
$331$	−11.1037	−0.610314	−0.305157	−	0.952302i	$-0.598709\pi$
$331$	−11.1037	−0.610314	−0.305157	+	0.952302i	$0.598709\pi$
$332$	0	0
$333$	−13.4315	−0.736041
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.0360	−1.63617	−0.818083	−	0.575101i	$-0.804963\pi$
$337$	−30.0360	−1.63617	−0.818083	+	0.575101i	$0.804963\pi$
$338$	0	0
$339$	−2.27416	−0.123516
$340$	0	0
$341$	−52.7912	−2.85880
$342$	0	0
$343$	−14.2260	−0.768134
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.4956	1.42236	0.711179	−	0.703011i	$-0.248161\pi$
$347$	26.4956	1.42236	0.711179	+	0.703011i	$0.248161\pi$
$348$	0	0
$349$	11.8225	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$349$	11.8225	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$350$	0	0
$351$	−2.22589	−0.118809
$352$	0	0
$353$	−11.3987	−0.606690	−0.303345	−	0.952881i	$-0.598103\pi$
$353$	−11.3987	−0.606690	−0.303345	+	0.952881i	$0.598103\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.60910	0.402716
$358$	0	0
$359$	−12.2135	−0.644601	−0.322301	−	0.946637i	$-0.604456\pi$
$359$	−12.2135	−0.644601	−0.322301	+	0.946637i	$0.604456\pi$
$360$	0	0
$361$	1.91463	0.100770
$362$	0	0
$363$	−44.1746	−2.31857
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.91651	−0.361039	−0.180519	−	0.983571i	$-0.557778\pi$
$367$	−6.91651	−0.361039	−0.180519	+	0.983571i	$0.557778\pi$
$368$	0	0
$369$	1.29211	0.0672647
$370$	0	0
$371$	−14.7227	−0.764367
$372$	0	0
$373$	27.3487	1.41606	0.708030	−	0.706183i	$-0.249584\pi$
$373$	27.3487	1.41606	0.708030	+	0.706183i	$0.249584\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−27.2726	−1.40461
$378$	0	0
$379$	−14.9563	−0.768255	−0.384127	−	0.923280i	$-0.625498\pi$
$379$	−14.9563	−0.768255	−0.384127	+	0.923280i	$0.625498\pi$
$380$	0	0
$381$	−20.9419	−1.07289
$382$	0	0
$383$	28.4094	1.45165	0.725826	−	0.687879i	$-0.241458\pi$
$383$	28.4094	1.45165	0.725826	+	0.687879i	$0.241458\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.4046	−0.935561
$388$	0	0
$389$	−22.2240	−1.12680	−0.563401	−	0.826184i	$-0.690507\pi$
$389$	−22.2240	−1.12680	−0.563401	+	0.826184i	$0.690507\pi$
$390$	0	0
$391$	−1.24730	−0.0630785
$392$	0	0
$393$	−3.35309	−0.169141
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.76615	−0.0886406	−0.0443203	−	0.999017i	$-0.514112\pi$
$397$	−1.76615	−0.0886406	−0.0443203	+	0.999017i	$0.514112\pi$
$398$	0	0
$399$	34.7983	1.74209
$400$	0	0
$401$	−4.79423	−0.239412	−0.119706	−	0.992809i	$-0.538195\pi$
$401$	−4.79423	−0.239412	−0.119706	+	0.992809i	$0.538195\pi$
$402$	0	0
$403$	−45.2803	−2.25557
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.5443	−1.11748
$408$	0	0
$409$	−36.2834	−1.79410	−0.897050	−	0.441929i	$-0.854294\pi$
$409$	−36.2834	−1.79410	−0.897050	+	0.441929i	$0.854294\pi$
$410$	0	0
$411$	31.2726	1.54256
$412$	0	0
$413$	−38.9468	−1.91645
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−24.4662	−1.19811
$418$	0	0
$419$	24.4452	1.19423	0.597113	−	0.802157i	$-0.296314\pi$
$419$	24.4452	1.19423	0.597113	+	0.802157i	$0.296314\pi$
$420$	0	0
$421$	−14.0909	−0.686750	−0.343375	−	0.939198i	$-0.611570\pi$
$421$	−14.0909	−0.686750	−0.343375	+	0.939198i	$0.611570\pi$
$422$	0	0
$423$	−10.7236	−0.521399
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.2182	0.736460
$428$	0	0
$429$	−61.3721	−2.96307
$430$	0	0
$431$	17.5277	0.844282	0.422141	−	0.906530i	$-0.361279\pi$
$431$	17.5277	0.844282	0.422141	+	0.906530i	$0.361279\pi$
$432$	0	0
$433$	11.1755	0.537059	0.268530	−	0.963271i	$-0.413462\pi$
$433$	11.1755	0.537059	0.268530	+	0.963271i	$0.413462\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.70420	−0.272869
$438$	0	0
$439$	12.0419	0.574727	0.287363	−	0.957822i	$-0.407221\pi$
$439$	12.0419	0.574727	0.287363	+	0.957822i	$0.407221\pi$
$440$	0	0
$441$	7.49672	0.356987
$442$	0	0
$443$	39.5984	1.88138	0.940689	−	0.339269i	$-0.110180\pi$
$443$	39.5984	1.88138	0.940689	+	0.339269i	$0.110180\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	22.4746	1.06301
$448$	0	0
$449$	11.6778	0.551107	0.275554	−	0.961286i	$-0.411139\pi$
$449$	11.6778	0.551107	0.275554	+	0.961286i	$0.411139\pi$
$450$	0	0
$451$	2.16877	0.102123
$452$	0	0
$453$	−14.8729	−0.698789
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.2148	−0.945606	−0.472803	−	0.881168i	$-0.656758\pi$
$457$	−20.2148	−0.945606	−0.472803	+	0.881168i	$0.656758\pi$
$458$	0	0
$459$	0.483999	0.0225912
$460$	0	0
$461$	2.77786	0.129378	0.0646890	−	0.997905i	$-0.479394\pi$
$461$	2.77786	0.129378	0.0646890	+	0.997905i	$0.479394\pi$
$462$	0	0
$463$	33.8186	1.57168	0.785842	−	0.618427i	$-0.212230\pi$
$463$	33.8186	1.57168	0.785842	+	0.618427i	$0.212230\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.1417	1.90381	0.951904	−	0.306395i	$-0.0991228\pi$
$467$	41.1417	1.90381	0.951904	+	0.306395i	$0.0991228\pi$
$468$	0	0
$469$	20.8788	0.964092
$470$	0	0
$471$	−21.6781	−0.998874
$472$	0	0
$473$	−30.8916	−1.42039
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.3835	−0.704363
$478$	0	0
$479$	10.8273	0.494710	0.247355	−	0.968925i	$-0.420439\pi$
$479$	10.8273	0.494710	0.247355	+	0.968925i	$0.420439\pi$
$480$	0	0
$481$	−19.3368	−0.881682
$482$	0	0
$483$	−9.49080	−0.431846
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.2533	−0.781820	−0.390910	−	0.920429i	$-0.627840\pi$
$487$	−17.2533	−0.781820	−0.390910	+	0.920429i	$0.627840\pi$
$488$	0	0
$489$	−2.12178	−0.0959502
$490$	0	0
$491$	28.6442	1.29269	0.646347	−	0.763043i	$-0.276296\pi$
$491$	28.6442	1.29269	0.646347	+	0.763043i	$0.276296\pi$
$492$	0	0
$493$	5.93018	0.267082
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−36.5060	−1.63752
$498$	0	0
$499$	10.5083	0.470418	0.235209	−	0.971945i	$-0.424423\pi$
$499$	10.5083	0.470418	0.235209	+	0.971945i	$0.424423\pi$
$500$	0	0
$501$	−3.28834	−0.146912
$502$	0	0
$503$	14.7844	0.659206	0.329603	−	0.944120i	$-0.393085\pi$
$503$	14.7844	0.659206	0.329603	+	0.944120i	$0.393085\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−20.2851	−0.900895
$508$	0	0
$509$	32.5112	1.44103	0.720517	−	0.693438i	$-0.243905\pi$
$509$	32.5112	1.44103	0.720517	+	0.693438i	$0.243905\pi$
$510$	0	0
$511$	−33.2862	−1.47250
$512$	0	0
$513$	2.21345	0.0977263
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−17.9992	−0.791603
$518$	0	0
$519$	−18.6584	−0.819011
$520$	0	0
$521$	40.4949	1.77411	0.887057	−	0.461660i	$-0.152746\pi$
$521$	40.4949	1.77411	0.887057	+	0.461660i	$0.152746\pi$
$522$	0	0
$523$	−19.7050	−0.861640	−0.430820	−	0.902438i	$-0.641776\pi$
$523$	−19.7050	−0.861640	−0.430820	+	0.902438i	$0.641776\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.84580	0.428890
$528$	0	0
$529$	−21.4443	−0.932359
$530$	0	0
$531$	−40.6948	−1.76600
$532$	0	0
$533$	1.86021	0.0805745
$534$	0	0
$535$	0	0
$536$	0	0
$537$	39.8003	1.71751
$538$	0	0
$539$	12.5830	0.541988
$540$	0	0
$541$	21.0224	0.903824	0.451912	−	0.892062i	$-0.350742\pi$
$541$	21.0224	0.903824	0.451912	+	0.892062i	$0.350742\pi$
$542$	0	0
$543$	27.4114	1.17633
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.1379	−1.07482	−0.537410	−	0.843321i	$-0.680597\pi$
$547$	−25.1379	−1.07482	−0.537410	+	0.843321i	$0.680597\pi$
$548$	0	0
$549$	15.9012	0.678647
$550$	0	0
$551$	27.1202	1.15536
$552$	0	0
$553$	−51.0986	−2.17293
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.9470	0.972297	0.486149	−	0.873876i	$-0.338401\pi$
$557$	22.9470	0.972297	0.486149	+	0.873876i	$0.338401\pi$
$558$	0	0
$559$	−26.4965	−1.12068
$560$	0	0
$561$	13.3448	0.563418
$562$	0	0
$563$	−34.8974	−1.47075	−0.735374	−	0.677661i	$-0.762994\pi$
$563$	−34.8974	−1.47075	−0.735374	+	0.677661i	$0.762994\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−25.6161	−1.07577
$568$	0	0
$569$	6.87405	0.288175	0.144088	−	0.989565i	$-0.453975\pi$
$569$	6.87405	0.288175	0.144088	+	0.989565i	$0.453975\pi$
$570$	0	0
$571$	−24.7188	−1.03445	−0.517224	−	0.855850i	$-0.673035\pi$
$571$	−24.7188	−1.03445	−0.517224	+	0.855850i	$0.673035\pi$
$572$	0	0
$573$	21.1521	0.883643
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.89908	0.203951	0.101976	−	0.994787i	$-0.467484\pi$
$577$	4.89908	0.203951	0.101976	+	0.994787i	$0.467484\pi$
$578$	0	0
$579$	46.9485	1.95111
$580$	0	0
$581$	12.5791	0.521868
$582$	0	0
$583$	−25.8207	−1.06938
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.8761	−0.985471	−0.492736	−	0.870179i	$-0.664003\pi$
$587$	−23.8761	−0.985471	−0.492736	+	0.870179i	$0.664003\pi$
$588$	0	0
$589$	45.0273	1.85532
$590$	0	0
$591$	26.8175	1.10313
$592$	0	0
$593$	−37.8742	−1.55531	−0.777654	−	0.628693i	$-0.783590\pi$
$593$	−37.8742	−1.55531	−0.777654	+	0.628693i	$0.783590\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−31.7300	−1.29862
$598$	0	0
$599$	−43.9867	−1.79725	−0.898625	−	0.438718i	$-0.855433\pi$
$599$	−43.9867	−1.79725	−0.898625	+	0.438718i	$0.855433\pi$
$600$	0	0
$601$	11.4083	0.465355	0.232678	−	0.972554i	$-0.425251\pi$
$601$	11.4083	0.465355	0.232678	+	0.972554i	$0.425251\pi$
$602$	0	0
$603$	21.8158	0.888410
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.4364	0.707721	0.353861	−	0.935298i	$-0.384869\pi$
$607$	17.4364	0.707721	0.353861	+	0.935298i	$0.384869\pi$
$608$	0	0
$609$	45.1233	1.82849
$610$	0	0
$611$	−15.4384	−0.624569
$612$	0	0
$613$	−0.620807	−0.0250741	−0.0125371	−	0.999921i	$-0.503991\pi$
$613$	−0.620807	−0.0250741	−0.0125371	+	0.999921i	$0.503991\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.0359	1.81308	0.906539	−	0.422123i	$-0.138715\pi$
$617$	45.0359	1.81308	0.906539	+	0.422123i	$0.138715\pi$
$618$	0	0
$619$	11.4780	0.461340	0.230670	−	0.973032i	$-0.425908\pi$
$619$	11.4780	0.461340	0.230670	+	0.973032i	$0.425908\pi$
$620$	0	0
$621$	−0.603691	−0.0242253
$622$	0	0
$623$	−31.8389	−1.27560
$624$	0	0
$625$	0	0
$626$	0	0
$627$	61.0292	2.43727
$628$	0	0
$629$	4.20461	0.167649
$630$	0	0
$631$	10.9485	0.435853	0.217926	−	0.975965i	$-0.430071\pi$
$631$	10.9485	0.435853	0.217926	+	0.975965i	$0.430071\pi$
$632$	0	0
$633$	65.5919	2.60704
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.7927	0.427624
$638$	0	0
$639$	−38.1444	−1.50897
$640$	0	0
$641$	−1.51186	−0.0597147	−0.0298574	−	0.999554i	$-0.509505\pi$
$641$	−1.51186	−0.0597147	−0.0298574	+	0.999554i	$0.509505\pi$
$642$	0	0
$643$	−17.7547	−0.700179	−0.350089	−	0.936716i	$-0.613849\pi$
$643$	−17.7547	−0.700179	−0.350089	+	0.936716i	$0.613849\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.1787	−1.38302	−0.691508	−	0.722369i	$-0.743053\pi$
$647$	−35.1787	−1.38302	−0.691508	+	0.722369i	$0.743053\pi$
$648$	0	0
$649$	−68.3048	−2.68120
$650$	0	0
$651$	74.9176	2.93625
$652$	0	0
$653$	−9.99410	−0.391099	−0.195550	−	0.980694i	$-0.562649\pi$
$653$	−9.99410	−0.391099	−0.195550	+	0.980694i	$0.562649\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−34.7802	−1.35690
$658$	0	0
$659$	23.9165	0.931655	0.465827	−	0.884876i	$-0.345757\pi$
$659$	23.9165	0.931655	0.465827	+	0.884876i	$0.345757\pi$
$660$	0	0
$661$	28.6063	1.11266	0.556328	−	0.830963i	$-0.312210\pi$
$661$	28.6063	1.11266	0.556328	+	0.830963i	$0.312210\pi$
$662$	0	0
$663$	11.4462	0.444533
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.39670	−0.286401
$668$	0	0
$669$	49.8521	1.92739
$670$	0	0
$671$	26.6896	1.03034
$672$	0	0
$673$	−12.9564	−0.499431	−0.249716	−	0.968319i	$-0.580337\pi$
$673$	−12.9564	−0.499431	−0.249716	+	0.968319i	$0.580337\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.5326	0.904430	0.452215	−	0.891909i	$-0.350634\pi$
$677$	23.5326	0.904430	0.452215	+	0.891909i	$0.350634\pi$
$678$	0	0
$679$	6.95350	0.266851
$680$	0	0
$681$	−33.0841	−1.26779
$682$	0	0
$683$	25.4677	0.974495	0.487247	−	0.873264i	$-0.338001\pi$
$683$	25.4677	0.974495	0.487247	+	0.873264i	$0.338001\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.9484	−0.951840
$688$	0	0
$689$	−22.1471	−0.843736
$690$	0	0
$691$	−47.8923	−1.82191	−0.910955	−	0.412507i	$-0.864653\pi$
$691$	−47.8923	−1.82191	−0.910955	+	0.412507i	$0.864653\pi$
$692$	0	0
$693$	52.3649	1.98918
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.404485	−0.0153210
$698$	0	0
$699$	−12.7460	−0.482099
$700$	0	0
$701$	−5.52783	−0.208783	−0.104392	−	0.994536i	$-0.533290\pi$
$701$	−5.52783	−0.208783	−0.104392	+	0.994536i	$0.533290\pi$
$702$	0	0
$703$	19.2287	0.725226
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.8391	0.934170
$708$	0	0
$709$	22.8895	0.859633	0.429817	−	0.902916i	$-0.358578\pi$
$709$	22.8895	0.859633	0.429817	+	0.902916i	$0.358578\pi$
$710$	0	0
$711$	−53.3920	−2.00236
$712$	0	0
$713$	−12.2806	−0.459913
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−33.2679	−1.24241
$718$	0	0
$719$	−0.748823	−0.0279264	−0.0139632	−	0.999903i	$-0.504445\pi$
$719$	−0.748823	−0.0279264	−0.0139632	+	0.999903i	$0.504445\pi$
$720$	0	0
$721$	−15.7664	−0.587172
$722$	0	0
$723$	10.2690	0.381906
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−51.8312	−1.92231	−0.961157	−	0.276004i	$-0.910990\pi$
$727$	−51.8312	−1.92231	−0.961157	+	0.276004i	$0.910990\pi$
$728$	0	0
$729$	−30.3796	−1.12517
$730$	0	0
$731$	5.76142	0.213094
$732$	0	0
$733$	0.440590	0.0162735	0.00813677	−	0.999967i	$-0.497410\pi$
$733$	0.440590	0.0162735	0.00813677	+	0.999967i	$0.497410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.6171	1.34881
$738$	0	0
$739$	30.4658	1.12070	0.560351	−	0.828255i	$-0.310666\pi$
$739$	30.4658	1.12070	0.560351	+	0.828255i	$0.310666\pi$
$740$	0	0
$741$	52.3463	1.92299
$742$	0	0
$743$	15.1434	0.555557	0.277779	−	0.960645i	$-0.410402\pi$
$743$	15.1434	0.555557	0.277779	+	0.960645i	$0.410402\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.1436	0.480901
$748$	0	0
$749$	17.1518	0.626714
$750$	0	0
$751$	−34.1610	−1.24655	−0.623277	−	0.782001i	$-0.714199\pi$
$751$	−34.1610	−1.24655	−0.623277	+	0.782001i	$0.714199\pi$
$752$	0	0
$753$	−19.2017	−0.699748
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.5792	−0.747965	−0.373982	−	0.927436i	$-0.622008\pi$
$757$	−20.5792	−0.747965	−0.373982	+	0.927436i	$0.622008\pi$
$758$	0	0
$759$	−16.6449	−0.604173
$760$	0	0
$761$	−18.8045	−0.681661	−0.340831	−	0.940125i	$-0.610708\pi$
$761$	−18.8045	−0.681661	−0.340831	+	0.940125i	$0.610708\pi$
$762$	0	0
$763$	−12.1727	−0.440681
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−58.5867	−2.11544
$768$	0	0
$769$	−36.5974	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$769$	−36.5974	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$770$	0	0
$771$	−45.9573	−1.65511
$772$	0	0
$773$	−19.4040	−0.697912	−0.348956	−	0.937139i	$-0.613464\pi$
$773$	−19.4040	−0.697912	−0.348956	+	0.937139i	$0.613464\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	31.9933	1.14775
$778$	0	0
$779$	−1.84981	−0.0662764
$780$	0	0
$781$	−64.0240	−2.29096
$782$	0	0
$783$	2.87020	0.102573
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.7760	1.02575	0.512876	−	0.858462i	$-0.328580\pi$
$787$	28.7760	1.02575	0.512876	+	0.858462i	$0.328580\pi$
$788$	0	0
$789$	46.7022	1.66264
$790$	0	0
$791$	2.79352	0.0993260
$792$	0	0
$793$	22.8924	0.812932
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.55967	−0.161512	−0.0807559	−	0.996734i	$-0.525733\pi$
$797$	−4.55967	−0.161512	−0.0807559	+	0.996734i	$0.525733\pi$
$798$	0	0
$799$	3.35693	0.118760
$800$	0	0
$801$	−33.2679	−1.17546
$802$	0	0
$803$	−58.3773	−2.06009
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.7973	−1.29532
$808$	0	0
$809$	37.0052	1.30103	0.650516	−	0.759493i	$-0.274553\pi$
$809$	37.0052	1.30103	0.650516	+	0.759493i	$0.274553\pi$
$810$	0	0
$811$	30.4268	1.06843	0.534215	−	0.845349i	$-0.320607\pi$
$811$	30.4268	1.06843	0.534215	+	0.845349i	$0.320607\pi$
$812$	0	0
$813$	−60.4596	−2.12041
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.3484	0.921814
$818$	0	0
$819$	44.9147	1.56945
$820$	0	0
$821$	2.43436	0.0849596	0.0424798	−	0.999097i	$-0.486474\pi$
$821$	2.43436	0.0849596	0.0424798	+	0.999097i	$0.486474\pi$
$822$	0	0
$823$	−30.1177	−1.04984	−0.524918	−	0.851153i	$-0.675904\pi$
$823$	−30.1177	−1.04984	−0.524918	+	0.851153i	$0.675904\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.7944	−0.618771	−0.309385	−	0.950937i	$-0.600123\pi$
$827$	−17.7944	−0.618771	−0.309385	+	0.950937i	$0.600123\pi$
$828$	0	0
$829$	27.9359	0.970254	0.485127	−	0.874444i	$-0.338773\pi$
$829$	27.9359	0.970254	0.485127	+	0.874444i	$0.338773\pi$
$830$	0	0
$831$	−14.5470	−0.504629
$832$	0	0
$833$	−2.34678	−0.0813113
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.76536	0.164715
$838$	0	0
$839$	40.4717	1.39724	0.698618	−	0.715494i	$-0.253799\pi$
$839$	40.4717	1.39724	0.698618	+	0.715494i	$0.253799\pi$
$840$	0	0
$841$	6.16706	0.212657
$842$	0	0
$843$	1.62253	0.0558828
$844$	0	0
$845$	0	0
$846$	0	0
$847$	54.2628	1.86449
$848$	0	0
$849$	−11.5098	−0.395016
$850$	0	0
$851$	−5.24440	−0.179776
$852$	0	0
$853$	−7.32217	−0.250706	−0.125353	−	0.992112i	$-0.540006\pi$
$853$	−7.32217	−0.250706	−0.125353	+	0.992112i	$0.540006\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.74086	−0.161945	−0.0809724	−	0.996716i	$-0.525803\pi$
$857$	−4.74086	−0.161945	−0.0809724	+	0.996716i	$0.525803\pi$
$858$	0	0
$859$	−8.14829	−0.278016	−0.139008	−	0.990291i	$-0.544391\pi$
$859$	−8.14829	−0.278016	−0.139008	+	0.990291i	$0.544391\pi$
$860$	0	0
$861$	−3.07776	−0.104890
$862$	0	0
$863$	15.8044	0.537988	0.268994	−	0.963142i	$-0.413309\pi$
$863$	15.8044	0.537988	0.268994	+	0.963142i	$0.413309\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.48887	−0.0845264
$868$	0	0
$869$	−89.6166	−3.04003
$870$	0	0
$871$	31.4074	1.06420
$872$	0	0
$873$	7.26559	0.245903
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.3063	0.753231	0.376616	−	0.926370i	$-0.377088\pi$
$877$	22.3063	0.753231	0.376616	+	0.926370i	$0.377088\pi$
$878$	0	0
$879$	8.61570	0.290600
$880$	0	0
$881$	9.67865	0.326082	0.163041	−	0.986619i	$-0.447870\pi$
$881$	9.67865	0.326082	0.163041	+	0.986619i	$0.447870\pi$
$882$	0	0
$883$	21.7255	0.731120	0.365560	−	0.930788i	$-0.380877\pi$
$883$	21.7255	0.731120	0.365560	+	0.930788i	$0.380877\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.17244	0.0729434	0.0364717	−	0.999335i	$-0.488388\pi$
$887$	2.17244	0.0729434	0.0364717	+	0.999335i	$0.488388\pi$
$888$	0	0
$889$	25.7244	0.862768
$890$	0	0
$891$	−44.9254	−1.50506
$892$	0	0
$893$	15.3521	0.513738
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−14.2768	−0.476688
$898$	0	0
$899$	58.3874	1.94733
$900$	0	0
$901$	4.81568	0.160434
$902$	0	0
$903$	43.8392	1.45888
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.8704	0.859012	0.429506	−	0.903064i	$-0.358688\pi$
$907$	25.8704	0.859012	0.429506	+	0.903064i	$0.358688\pi$
$908$	0	0
$909$	25.9539	0.860837
$910$	0	0
$911$	36.1769	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$911$	36.1769	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$912$	0	0
$913$	22.0611	0.730117
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.11883	0.136016
$918$	0	0
$919$	−52.2475	−1.72349	−0.861743	−	0.507346i	$-0.830627\pi$
$919$	−52.2475	−1.72349	−0.861743	+	0.507346i	$0.830627\pi$
$920$	0	0
$921$	49.1862	1.62074
$922$	0	0
$923$	−54.9150	−1.80755
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.4740	−0.541078
$928$	0	0
$929$	19.6884	0.645955	0.322978	−	0.946407i	$-0.395316\pi$
$929$	19.6884	0.645955	0.322978	+	0.946407i	$0.395316\pi$
$930$	0	0
$931$	−10.7324	−0.351741
$932$	0	0
$933$	65.2994	2.13781
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.0805	−0.492657	−0.246328	−	0.969186i	$-0.579224\pi$
$937$	−15.0805	−0.492657	−0.246328	+	0.969186i	$0.579224\pi$
$938$	0	0
$939$	−57.2004	−1.86666
$940$	0	0
$941$	−12.2054	−0.397886	−0.198943	−	0.980011i	$-0.563751\pi$
$941$	−12.2054	−0.397886	−0.198943	+	0.980011i	$0.563751\pi$
$942$	0	0
$943$	0.504513	0.0164292
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.4424	−0.761776	−0.380888	−	0.924621i	$-0.624382\pi$
$947$	−23.4424	−0.761776	−0.380888	+	0.924621i	$0.624382\pi$
$948$	0	0
$949$	−50.0717	−1.62540
$950$	0	0
$951$	27.2329	0.883088
$952$	0	0
$953$	50.7165	1.64287	0.821435	−	0.570302i	$-0.193174\pi$
$953$	50.7165	1.64287	0.821435	+	0.570302i	$0.193174\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	79.1372	2.55814
$958$	0	0
$959$	−38.4143	−1.24046
$960$	0	0
$961$	65.9397	2.12709
$962$	0	0
$963$	17.9216	0.577517
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.6170	1.33831	0.669157	−	0.743121i	$-0.266656\pi$
$967$	41.6170	1.33831	0.669157	+	0.743121i	$0.266656\pi$
$968$	0	0
$969$	−11.3822	−0.365650
$970$	0	0
$971$	−19.2045	−0.616302	−0.308151	−	0.951337i	$-0.599710\pi$
$971$	−19.2045	−0.616302	−0.308151	+	0.951337i	$0.599710\pi$
$972$	0	0
$973$	30.0535	0.963472
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.0264	0.672696	0.336348	−	0.941738i	$-0.390808\pi$
$977$	21.0264	0.672696	0.336348	+	0.941738i	$0.390808\pi$
$978$	0	0
$979$	−55.8390	−1.78462
$980$	0	0
$981$	−12.7190	−0.406087
$982$	0	0
$983$	3.77964	0.120552	0.0602758	−	0.998182i	$-0.480802\pi$
$983$	3.77964	0.120552	0.0602758	+	0.998182i	$0.480802\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	25.5432	0.813049
$988$	0	0
$989$	−7.18619	−0.228508
$990$	0	0
$991$	−35.7886	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$991$	−35.7886	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$992$	0	0
$993$	27.6356	0.876990
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.92887	−0.156099	−0.0780494	−	0.996949i	$-0.524869\pi$
$997$	−4.92887	−0.156099	−0.0780494	+	0.996949i	$0.524869\pi$
$998$	0	0
$999$	2.03503	0.0643855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.cd.1.1		5
4.3	odd	2		425.2.a.j.1.4	yes	5
5.4	even	2		6800.2.a.bz.1.5		5
12.11	even	2		3825.2.a.bl.1.2		5
20.3	even	4		425.2.b.f.324.3		10
20.7	even	4		425.2.b.f.324.8		10
20.19	odd	2		425.2.a.i.1.2	✓	5
60.59	even	2		3825.2.a.bq.1.4		5
68.67	odd	2		7225.2.a.y.1.4		5
340.339	odd	2		7225.2.a.x.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.2	✓	5	20.19	odd	2
425.2.a.j.1.4	yes	5	4.3	odd	2
425.2.b.f.324.3		10	20.3	even	4
425.2.b.f.324.8		10	20.7	even	4
3825.2.a.bl.1.2		5	12.11	even	2
3825.2.a.bq.1.4		5	60.59	even	2
6800.2.a.bz.1.5		5	5.4	even	2
6800.2.a.cd.1.1		5	1.1	even	1	trivial
7225.2.a.x.1.2		5	340.339	odd	2
7225.2.a.y.1.4		5	68.67	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:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48887 q^{3} +3.05725 q^{7} +3.19447 q^{9} +5.36180 q^{11} +4.59895 q^{13} -1.00000 q^{17} -4.57325 q^{19} -7.60910 q^{21} +1.24730 q^{23} -0.483999 q^{27} -5.93018 q^{29} -9.84580 q^{31} -13.3448 q^{33} -4.20461 q^{37} -11.4462 q^{39} +0.404485 q^{41} -5.76142 q^{43} -3.35693 q^{47} +2.34678 q^{49} +2.48887 q^{51} -4.81568 q^{53} +11.3822 q^{57} -12.7392 q^{59} +4.97774 q^{61} +9.76628 q^{63} +6.82926 q^{67} -3.10436 q^{69} -11.9408 q^{71} -10.8876 q^{73} +16.3924 q^{77} -16.7139 q^{79} -8.37879 q^{81} +4.11450 q^{83} +14.7594 q^{87} -10.4142 q^{89} +14.0601 q^{91} +24.5049 q^{93} +2.27443 q^{97} +17.1281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + q^{7} + 6 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{17} - 6 q^{19} - 5 q^{21} + 4 q^{23} - 5 q^{27} + 2 q^{29} - 21 q^{31} - 12 q^{33} + 2 q^{37} - 23 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q + q^3 + q^7 + 6 * q^9 - 4 * q^11 + 3 * q^13 - 5 * q^17 - 6 * q^19 - 5 * q^21 + 4 * q^23 - 5 * q^27 + 2 * q^29 - 21 * q^31 - 12 * q^33 + 2 * q^37 - 23 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 + 10 * q^49 - q^51 - 21 * q^53 + 20 * q^57 - 12 * q^59 - 2 * q^61 + 8 * q^63 - 12 * q^67 - 21 * q^71 - 22 * q^73 + 32 * q^77 - 41 * q^79 - 35 * q^81 - 8 * q^83 - 4 * q^87 - 10 * q^89 - 13 * q^91 - 9 * q^93 + 20 * q^97 + 14 * q^99

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