LMFDB - Embedded newform 6800.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6800,2,Mod(1,6800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q-0.585786 q^{3} -3.41421 q^{7} -2.65685 q^{9} +5.41421 q^{11} -2.82843 q^{13} +1.00000 q^{17} -2.82843 q^{19} +2.00000 q^{21} -0.585786 q^{23} +3.31371 q^{27} +0.828427 q^{29} +4.24264 q^{31} -3.17157 q^{33} +10.4853 q^{37} +1.65685 q^{39} +10.4853 q^{41} -3.65685 q^{43} +0.828427 q^{47} +4.65685 q^{49} -0.585786 q^{51} -11.6569 q^{53} +1.65685 q^{57} +14.8284 q^{59} -3.65685 q^{61} +9.07107 q^{63} -8.82843 q^{67} +0.343146 q^{69} -4.24264 q^{71} -0.828427 q^{73} -18.4853 q^{77} -2.58579 q^{79} +6.02944 q^{81} -13.3137 q^{83} -0.485281 q^{87} -13.6569 q^{89} +9.65685 q^{91} -2.48528 q^{93} +7.65685 q^{97} -14.3848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{7} + 6 q^{9} + 8 q^{11} + 2 q^{17} + 4 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 12 q^{33} + 4 q^{37} - 8 q^{39} + 4 q^{41} + 4 q^{43} - 4 q^{47} - 2 q^{49} - 4 q^{51} - 12 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^7 + 6 * q^9 + 8 * q^11 + 2 * q^17 + 4 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 12 * q^33 + 4 * q^37 - 8 * q^39 + 4 * q^41 + 4 * q^43 - 4 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 - 8 * q^57 + 24 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 12 * q^69 + 4 * q^73 - 20 * q^77 - 8 * q^79 + 46 * q^81 - 4 * q^83 + 16 * q^87 - 16 * q^89 + 8 * q^91 + 12 * q^93 + 4 * q^97 + 8 * 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
$3$	−0.585786	−0.338204	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	−0.585786	−0.338204	−0.169102	+	0.985599i	$0.554087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	5.41421	1.63245	0.816223	−	0.577736i	$-0.196064\pi$
$11$	5.41421	1.63245	0.816223	+	0.577736i	$0.196064\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−0.585786	−0.122145	−0.0610725	−	0.998133i	$-0.519452\pi$
$23$	−0.585786	−0.122145	−0.0610725	+	0.998133i	$0.519452\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.31371	0.637723
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	0	0
$33$	−3.17157	−0.552100
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.4853	1.72377	0.861885	−	0.507104i	$-0.169284\pi$
$37$	10.4853	1.72377	0.861885	+	0.507104i	$0.169284\pi$
$38$	0	0
$39$	1.65685	0.265309
$40$	0	0
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	0	0
$43$	−3.65685	−0.557665	−0.278833	−	0.960340i	$-0.589947\pi$
$43$	−3.65685	−0.557665	−0.278833	+	0.960340i	$0.589947\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	−0.585786	−0.0820265
$52$	0	0
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.65685	0.219456
$58$	0	0
$59$	14.8284	1.93050	0.965248	−	0.261334i	$-0.0841625\pi$
$59$	14.8284	1.93050	0.965248	+	0.261334i	$0.0841625\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	0	0
$63$	9.07107	1.14285
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	0	0
$69$	0.343146	0.0413099
$70$	0	0
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	0	0
$73$	−0.828427	−0.0969601	−0.0484800	−	0.998824i	$-0.515438\pi$
$73$	−0.828427	−0.0969601	−0.0484800	+	0.998824i	$0.515438\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.4853	−2.10659
$78$	0	0
$79$	−2.58579	−0.290924	−0.145462	−	0.989364i	$-0.546467\pi$
$79$	−2.58579	−0.290924	−0.145462	+	0.989364i	$0.546467\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	−13.3137	−1.46137	−0.730685	−	0.682715i	$-0.760799\pi$
$83$	−13.3137	−1.46137	−0.730685	+	0.682715i	$0.760799\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.485281	−0.0520276
$88$	0	0
$89$	−13.6569	−1.44762	−0.723812	−	0.689997i	$-0.757612\pi$
$89$	−13.6569	−1.44762	−0.723812	+	0.689997i	$0.757612\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	0	0
$93$	−2.48528	−0.257712
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	−14.3848	−1.44572
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.89949	−0.763673	−0.381837	−	0.924230i	$-0.624708\pi$
$107$	−7.89949	−0.763673	−0.381837	+	0.924230i	$0.624708\pi$
$108$	0	0
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	0	0
$111$	−6.14214	−0.582986
$112$	0	0
$113$	8.82843	0.830509	0.415254	−	0.909705i	$-0.363693\pi$
$113$	8.82843	0.830509	0.415254	+	0.909705i	$0.363693\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.51472	0.694736
$118$	0	0
$119$	−3.41421	−0.312980
$120$	0	0
$121$	18.3137	1.66488
$122$	0	0
$123$	−6.14214	−0.553818
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.31371	0.471515	0.235758	−	0.971812i	$-0.424243\pi$
$127$	5.31371	0.471515	0.235758	+	0.971812i	$0.424243\pi$
$128$	0	0
$129$	2.14214	0.188605
$130$	0	0
$131$	−5.89949	−0.515441	−0.257721	−	0.966219i	$-0.582971\pi$
$131$	−5.89949	−0.515441	−0.257721	+	0.966219i	$0.582971\pi$
$132$	0	0
$133$	9.65685	0.837355
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.82843	−0.583392	−0.291696	−	0.956511i	$-0.594220\pi$
$137$	−6.82843	−0.583392	−0.291696	+	0.956511i	$0.594220\pi$
$138$	0	0
$139$	1.89949	0.161113	0.0805565	−	0.996750i	$-0.474330\pi$
$139$	1.89949	0.161113	0.0805565	+	0.996750i	$0.474330\pi$
$140$	0	0
$141$	−0.485281	−0.0408681
$142$	0	0
$143$	−15.3137	−1.28060
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.72792	−0.224995
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−24.4853	−1.99258	−0.996292	−	0.0860367i	$-0.972580\pi$
$151$	−24.4853	−1.99258	−0.996292	+	0.0860367i	$0.972580\pi$
$152$	0	0
$153$	−2.65685	−0.214794
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.31371	−0.104845	−0.0524227	−	0.998625i	$-0.516694\pi$
$157$	−1.31371	−0.104845	−0.0524227	+	0.998625i	$0.516694\pi$
$158$	0	0
$159$	6.82843	0.541529
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	3.41421	0.267422	0.133711	−	0.991020i	$-0.457311\pi$
$163$	3.41421	0.267422	0.133711	+	0.991020i	$0.457311\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.24264	−0.483070	−0.241535	−	0.970392i	$-0.577651\pi$
$167$	−6.24264	−0.483070	−0.241535	+	0.970392i	$0.577651\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	7.51472	0.574665
$172$	0	0
$173$	−7.17157	−0.545245	−0.272622	−	0.962121i	$-0.587891\pi$
$173$	−7.17157	−0.545245	−0.272622	+	0.962121i	$0.587891\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.68629	−0.652902
$178$	0	0
$179$	1.17157	0.0875675	0.0437837	−	0.999041i	$-0.486059\pi$
$179$	1.17157	0.0875675	0.0437837	+	0.999041i	$0.486059\pi$
$180$	0	0
$181$	−14.4853	−1.07668	−0.538341	−	0.842727i	$-0.680949\pi$
$181$	−14.4853	−1.07668	−0.538341	+	0.842727i	$0.680949\pi$
$182$	0	0
$183$	2.14214	0.158351
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.41421	0.395927
$188$	0	0
$189$	−11.3137	−0.822951
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	15.1716	1.09207	0.546037	−	0.837761i	$-0.316136\pi$
$193$	15.1716	1.09207	0.546037	+	0.837761i	$0.316136\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.17157	−0.510953	−0.255477	−	0.966815i	$-0.582232\pi$
$197$	−7.17157	−0.510953	−0.255477	+	0.966815i	$0.582232\pi$
$198$	0	0
$199$	−15.7574	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$199$	−15.7574	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$200$	0	0
$201$	5.17157	0.364775
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.55635	0.108174
$208$	0	0
$209$	−15.3137	−1.05927
$210$	0	0
$211$	21.8995	1.50762	0.753812	−	0.657090i	$-0.228213\pi$
$211$	21.8995	1.50762	0.753812	+	0.657090i	$0.228213\pi$
$212$	0	0
$213$	2.48528	0.170289
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.4853	−0.983325
$218$	0	0
$219$	0.485281	0.0327923
$220$	0	0
$221$	−2.82843	−0.190261
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.72792	0.181059	0.0905293	−	0.995894i	$-0.471144\pi$
$227$	2.72792	0.181059	0.0905293	+	0.995894i	$0.471144\pi$
$228$	0	0
$229$	−23.3137	−1.54061	−0.770307	−	0.637674i	$-0.779897\pi$
$229$	−23.3137	−1.54061	−0.770307	+	0.637674i	$0.779897\pi$
$230$	0	0
$231$	10.8284	0.712458
$232$	0	0
$233$	13.3137	0.872210	0.436105	−	0.899896i	$-0.356358\pi$
$233$	13.3137	0.872210	0.436105	+	0.899896i	$0.356358\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.51472	0.0983915
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	7.17157	0.461962	0.230981	−	0.972958i	$-0.425807\pi$
$241$	7.17157	0.461962	0.230981	+	0.972958i	$0.425807\pi$
$242$	0	0
$243$	−13.4731	−0.864299
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	7.79899	0.494241
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−3.17157	−0.199395
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.82843	−0.176432	−0.0882162	−	0.996101i	$-0.528117\pi$
$257$	−2.82843	−0.176432	−0.0882162	+	0.996101i	$0.528117\pi$
$258$	0	0
$259$	−35.7990	−2.22444
$260$	0	0
$261$	−2.20101	−0.136239
$262$	0	0
$263$	−13.3137	−0.820958	−0.410479	−	0.911870i	$-0.634639\pi$
$263$	−13.3137	−0.820958	−0.410479	+	0.911870i	$0.634639\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	−14.9706	−0.912771	−0.456386	−	0.889782i	$-0.650856\pi$
$269$	−14.9706	−0.912771	−0.456386	+	0.889782i	$0.650856\pi$
$270$	0	0
$271$	2.34315	0.142336	0.0711680	−	0.997464i	$-0.477327\pi$
$271$	2.34315	0.142336	0.0711680	+	0.997464i	$0.477327\pi$
$272$	0	0
$273$	−5.65685	−0.342368
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−11.2721	−0.674842
$280$	0	0
$281$	−15.6569	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$281$	−15.6569	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$282$	0	0
$283$	5.75736	0.342239	0.171120	−	0.985250i	$-0.445262\pi$
$283$	5.75736	0.342239	0.171120	+	0.985250i	$0.445262\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−35.7990	−2.11315
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−4.48528	−0.262932
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	17.9411	1.04105
$298$	0	0
$299$	1.65685	0.0958184
$300$	0	0
$301$	12.4853	0.719640
$302$	0	0
$303$	4.68629	0.269220
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.1716	−0.637595	−0.318798	−	0.947823i	$-0.603279\pi$
$307$	−11.1716	−0.637595	−0.318798	+	0.947823i	$0.603279\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	0	0
$311$	−0.928932	−0.0526749	−0.0263375	−	0.999653i	$-0.508384\pi$
$311$	−0.928932	−0.0526749	−0.0263375	+	0.999653i	$0.508384\pi$
$312$	0	0
$313$	−33.7990	−1.91043	−0.955216	−	0.295910i	$-0.904377\pi$
$313$	−33.7990	−1.91043	−0.955216	+	0.295910i	$0.904377\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.31371	−0.0737852	−0.0368926	−	0.999319i	$-0.511746\pi$
$317$	−1.31371	−0.0737852	−0.0368926	+	0.999319i	$0.511746\pi$
$318$	0	0
$319$	4.48528	0.251128
$320$	0	0
$321$	4.62742	0.258277
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.11270	−0.172133
$328$	0	0
$329$	−2.82843	−0.155936
$330$	0	0
$331$	−21.1716	−1.16369	−0.581847	−	0.813298i	$-0.697670\pi$
$331$	−21.1716	−1.16369	−0.581847	+	0.813298i	$0.697670\pi$
$332$	0	0
$333$	−27.8579	−1.52660
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.6274	−1.34154	−0.670770	−	0.741665i	$-0.734036\pi$
$337$	−24.6274	−1.34154	−0.670770	+	0.741665i	$0.734036\pi$
$338$	0	0
$339$	−5.17157	−0.280881
$340$	0	0
$341$	22.9706	1.24393
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	35.6985	1.91640	0.958198	−	0.286107i	$-0.0923614\pi$
$347$	35.6985	1.91640	0.958198	+	0.286107i	$0.0923614\pi$
$348$	0	0
$349$	20.3431	1.08894	0.544472	−	0.838779i	$-0.316730\pi$
$349$	20.3431	1.08894	0.544472	+	0.838779i	$0.316730\pi$
$350$	0	0
$351$	−9.37258	−0.500271
$352$	0	0
$353$	16.3431	0.869858	0.434929	−	0.900465i	$-0.356773\pi$
$353$	16.3431	0.869858	0.434929	+	0.900465i	$0.356773\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	−7.79899	−0.411615	−0.205807	−	0.978593i	$-0.565982\pi$
$359$	−7.79899	−0.411615	−0.205807	+	0.978593i	$0.565982\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−10.7279	−0.563070
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.75736	−0.300532	−0.150266	−	0.988646i	$-0.548013\pi$
$367$	−5.75736	−0.300532	−0.150266	+	0.988646i	$0.548013\pi$
$368$	0	0
$369$	−27.8579	−1.45022
$370$	0	0
$371$	39.7990	2.06626
$372$	0	0
$373$	27.7990	1.43938	0.719689	−	0.694297i	$-0.244285\pi$
$373$	27.7990	1.43938	0.719689	+	0.694297i	$0.244285\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.34315	−0.120678
$378$	0	0
$379$	−26.8701	−1.38022	−0.690111	−	0.723703i	$-0.742438\pi$
$379$	−26.8701	−1.38022	−0.690111	+	0.723703i	$0.742438\pi$
$380$	0	0
$381$	−3.11270	−0.159468
$382$	0	0
$383$	22.2843	1.13867	0.569337	−	0.822105i	$-0.307200\pi$
$383$	22.2843	1.13867	0.569337	+	0.822105i	$0.307200\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.71573	0.493878
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	−0.585786	−0.0296245
$392$	0	0
$393$	3.45584	0.174324
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.31371	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$397$	9.31371	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$398$	0	0
$399$	−5.65685	−0.283197
$400$	0	0
$401$	−27.6569	−1.38112	−0.690559	−	0.723276i	$-0.742635\pi$
$401$	−27.6569	−1.38112	−0.690559	+	0.723276i	$0.742635\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	0	0
$406$	0	0
$407$	56.7696	2.81396
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	−50.6274	−2.49121
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.11270	−0.0544891
$418$	0	0
$419$	−3.75736	−0.183559	−0.0917795	−	0.995779i	$-0.529255\pi$
$419$	−3.75736	−0.183559	−0.0917795	+	0.995779i	$0.529255\pi$
$420$	0	0
$421$	4.97056	0.242250	0.121125	−	0.992637i	$-0.461350\pi$
$421$	4.97056	0.242250	0.121125	+	0.992637i	$0.461350\pi$
$422$	0	0
$423$	−2.20101	−0.107017
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.4853	0.604205
$428$	0	0
$429$	8.97056	0.433103
$430$	0	0
$431$	1.41421	0.0681203	0.0340601	−	0.999420i	$-0.489156\pi$
$431$	1.41421	0.0681203	0.0340601	+	0.999420i	$0.489156\pi$
$432$	0	0
$433$	2.82843	0.135926	0.0679628	−	0.997688i	$-0.478350\pi$
$433$	2.82843	0.135926	0.0679628	+	0.997688i	$0.478350\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.65685	0.0792581
$438$	0	0
$439$	−2.58579	−0.123413	−0.0617064	−	0.998094i	$-0.519654\pi$
$439$	−2.58579	−0.123413	−0.0617064	+	0.998094i	$0.519654\pi$
$440$	0	0
$441$	−12.3726	−0.589171
$442$	0	0
$443$	2.48528	0.118079	0.0590396	−	0.998256i	$-0.481196\pi$
$443$	2.48528	0.118079	0.0590396	+	0.998256i	$0.481196\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.17157	−0.0554135
$448$	0	0
$449$	−9.51472	−0.449027	−0.224514	−	0.974471i	$-0.572079\pi$
$449$	−9.51472	−0.449027	−0.224514	+	0.974471i	$0.572079\pi$
$450$	0	0
$451$	56.7696	2.67317
$452$	0	0
$453$	14.3431	0.673900
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.1127	1.08117	0.540583	−	0.841291i	$-0.318204\pi$
$457$	23.1127	1.08117	0.540583	+	0.841291i	$0.318204\pi$
$458$	0	0
$459$	3.31371	0.154671
$460$	0	0
$461$	−37.5980	−1.75111	−0.875556	−	0.483116i	$-0.839505\pi$
$461$	−37.5980	−1.75111	−0.875556	+	0.483116i	$0.839505\pi$
$462$	0	0
$463$	−8.82843	−0.410292	−0.205146	−	0.978731i	$-0.565767\pi$
$463$	−8.82843	−0.410292	−0.205146	+	0.978731i	$0.565767\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.6569	−0.539415	−0.269707	−	0.962942i	$-0.586927\pi$
$467$	−11.6569	−0.539415	−0.269707	+	0.962942i	$0.586927\pi$
$468$	0	0
$469$	30.1421	1.39183
$470$	0	0
$471$	0.769553	0.0354591
$472$	0	0
$473$	−19.7990	−0.910359
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.9706	1.41804
$478$	0	0
$479$	24.2426	1.10767	0.553837	−	0.832625i	$-0.313163\pi$
$479$	24.2426	1.10767	0.553837	+	0.832625i	$0.313163\pi$
$480$	0	0
$481$	−29.6569	−1.35224
$482$	0	0
$483$	−1.17157	−0.0533084
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.8995	0.720475	0.360237	−	0.932861i	$-0.382696\pi$
$487$	15.8995	0.720475	0.360237	+	0.932861i	$0.382696\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−0.485281	−0.0219004	−0.0109502	−	0.999940i	$-0.503486\pi$
$491$	−0.485281	−0.0219004	−0.0109502	+	0.999940i	$0.503486\pi$
$492$	0	0
$493$	0.828427	0.0373105
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.4853	0.649754
$498$	0	0
$499$	−3.75736	−0.168203	−0.0841013	−	0.996457i	$-0.526802\pi$
$499$	−3.75736	−0.168203	−0.0841013	+	0.996457i	$0.526802\pi$
$500$	0	0
$501$	3.65685	0.163376
$502$	0	0
$503$	−27.6985	−1.23501	−0.617507	−	0.786565i	$-0.711857\pi$
$503$	−27.6985	−1.23501	−0.617507	+	0.786565i	$0.711857\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.92893	0.130078
$508$	0	0
$509$	−24.6274	−1.09159	−0.545796	−	0.837918i	$-0.683772\pi$
$509$	−24.6274	−1.09159	−0.545796	+	0.837918i	$0.683772\pi$
$510$	0	0
$511$	2.82843	0.125122
$512$	0	0
$513$	−9.37258	−0.413810
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.48528	0.197262
$518$	0	0
$519$	4.20101	0.184404
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	28.1421	1.23057	0.615285	−	0.788305i	$-0.289041\pi$
$523$	28.1421	1.23057	0.615285	+	0.788305i	$0.289041\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.24264	0.184812
$528$	0	0
$529$	−22.6569	−0.985081
$530$	0	0
$531$	−39.3970	−1.70968
$532$	0	0
$533$	−29.6569	−1.28458
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.686292	−0.0296157
$538$	0	0
$539$	25.2132	1.08601
$540$	0	0
$541$	30.7696	1.32289	0.661443	−	0.749995i	$-0.269944\pi$
$541$	30.7696	1.32289	0.661443	+	0.749995i	$0.269944\pi$
$542$	0	0
$543$	8.48528	0.364138
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−35.2132	−1.50561	−0.752804	−	0.658245i	$-0.771299\pi$
$547$	−35.2132	−1.50561	−0.752804	+	0.658245i	$0.771299\pi$
$548$	0	0
$549$	9.71573	0.414657
$550$	0	0
$551$	−2.34315	−0.0998214
$552$	0	0
$553$	8.82843	0.375423
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.1716	−0.897068	−0.448534	−	0.893766i	$-0.648054\pi$
$557$	−21.1716	−0.897068	−0.448534	+	0.893766i	$0.648054\pi$
$558$	0	0
$559$	10.3431	0.437468
$560$	0	0
$561$	−3.17157	−0.133904
$562$	0	0
$563$	−31.6569	−1.33418	−0.667089	−	0.744978i	$-0.732460\pi$
$563$	−31.6569	−1.33418	−0.667089	+	0.744978i	$0.732460\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−20.5858	−0.864522
$568$	0	0
$569$	30.2843	1.26958	0.634791	−	0.772684i	$-0.281086\pi$
$569$	30.2843	1.26958	0.634791	+	0.772684i	$0.281086\pi$
$570$	0	0
$571$	35.5563	1.48799	0.743993	−	0.668187i	$-0.232929\pi$
$571$	35.5563	1.48799	0.743993	+	0.668187i	$0.232929\pi$
$572$	0	0
$573$	7.02944	0.293659
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17.1716	−0.714862	−0.357431	−	0.933940i	$-0.616347\pi$
$577$	−17.1716	−0.714862	−0.357431	+	0.933940i	$0.616347\pi$
$578$	0	0
$579$	−8.88730	−0.369344
$580$	0	0
$581$	45.4558	1.88583
$582$	0	0
$583$	−63.1127	−2.61386
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.6274	−0.686287	−0.343143	−	0.939283i	$-0.611492\pi$
$587$	−16.6274	−0.686287	−0.343143	+	0.939283i	$0.611492\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	4.20101	0.172806
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.23045	0.377777
$598$	0	0
$599$	21.9411	0.896490	0.448245	−	0.893911i	$-0.352049\pi$
$599$	21.9411	0.896490	0.448245	+	0.893911i	$0.352049\pi$
$600$	0	0
$601$	13.7990	0.562873	0.281436	−	0.959580i	$-0.409189\pi$
$601$	13.7990	0.562873	0.281436	+	0.959580i	$0.409189\pi$
$602$	0	0
$603$	23.4558	0.955196
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.41421	−0.300934	−0.150467	−	0.988615i	$-0.548078\pi$
$607$	−7.41421	−0.300934	−0.150467	+	0.988615i	$0.548078\pi$
$608$	0	0
$609$	1.65685	0.0671391
$610$	0	0
$611$	−2.34315	−0.0947935
$612$	0	0
$613$	13.0294	0.526254	0.263127	−	0.964761i	$-0.415246\pi$
$613$	13.0294	0.526254	0.263127	+	0.964761i	$0.415246\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.4853	−1.71039	−0.855197	−	0.518304i	$-0.826564\pi$
$617$	−42.4853	−1.71039	−0.855197	+	0.518304i	$0.826564\pi$
$618$	0	0
$619$	31.0711	1.24885	0.624426	−	0.781084i	$-0.285333\pi$
$619$	31.0711	1.24885	0.624426	+	0.781084i	$0.285333\pi$
$620$	0	0
$621$	−1.94113	−0.0778947
$622$	0	0
$623$	46.6274	1.86809
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.97056	0.358250
$628$	0	0
$629$	10.4853	0.418076
$630$	0	0
$631$	−23.7990	−0.947423	−0.473711	−	0.880680i	$-0.657086\pi$
$631$	−23.7990	−0.947423	−0.473711	+	0.880680i	$0.657086\pi$
$632$	0	0
$633$	−12.8284	−0.509884
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13.1716	−0.521877
$638$	0	0
$639$	11.2721	0.445917
$640$	0	0
$641$	−0.142136	−0.00561402	−0.00280701	−	0.999996i	$-0.500894\pi$
$641$	−0.142136	−0.00561402	−0.00280701	+	0.999996i	$0.500894\pi$
$642$	0	0
$643$	11.6985	0.461343	0.230672	−	0.973032i	$-0.425908\pi$
$643$	11.6985	0.461343	0.230672	+	0.973032i	$0.425908\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.1716	−0.596456	−0.298228	−	0.954495i	$-0.596396\pi$
$647$	−15.1716	−0.596456	−0.298228	+	0.954495i	$0.596396\pi$
$648$	0	0
$649$	80.2843	3.15143
$650$	0	0
$651$	8.48528	0.332564
$652$	0	0
$653$	−1.51472	−0.0592755	−0.0296378	−	0.999561i	$-0.509435\pi$
$653$	−1.51472	−0.0592755	−0.0296378	+	0.999561i	$0.509435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.20101	0.0858696
$658$	0	0
$659$	27.3137	1.06399	0.531996	−	0.846747i	$-0.321442\pi$
$659$	27.3137	1.06399	0.531996	+	0.846747i	$0.321442\pi$
$660$	0	0
$661$	−9.31371	−0.362261	−0.181131	−	0.983459i	$-0.557976\pi$
$661$	−9.31371	−0.362261	−0.181131	+	0.983459i	$0.557976\pi$
$662$	0	0
$663$	1.65685	0.0643469
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.485281	−0.0187902
$668$	0	0
$669$	−3.51472	−0.135887
$670$	0	0
$671$	−19.7990	−0.764332
$672$	0	0
$673$	−0.142136	−0.00547893	−0.00273946	−	0.999996i	$-0.500872\pi$
$673$	−0.142136	−0.00547893	−0.00273946	+	0.999996i	$0.500872\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.6274	1.56144	0.780719	−	0.624882i	$-0.214853\pi$
$677$	40.6274	1.56144	0.780719	+	0.624882i	$0.214853\pi$
$678$	0	0
$679$	−26.1421	−1.00324
$680$	0	0
$681$	−1.59798	−0.0612347
$682$	0	0
$683$	10.7279	0.410493	0.205246	−	0.978710i	$-0.434200\pi$
$683$	10.7279	0.410493	0.205246	+	0.978710i	$0.434200\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.6569	0.521041
$688$	0	0
$689$	32.9706	1.25608
$690$	0	0
$691$	−25.2132	−0.959155	−0.479578	−	0.877499i	$-0.659210\pi$
$691$	−25.2132	−0.959155	−0.479578	+	0.877499i	$0.659210\pi$
$692$	0	0
$693$	49.1127	1.86564
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.4853	0.397158
$698$	0	0
$699$	−7.79899	−0.294985
$700$	0	0
$701$	41.6569	1.57336	0.786679	−	0.617362i	$-0.211799\pi$
$701$	41.6569	1.57336	0.786679	+	0.617362i	$0.211799\pi$
$702$	0	0
$703$	−29.6569	−1.11853
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.3137	1.02724
$708$	0	0
$709$	−13.7990	−0.518232	−0.259116	−	0.965846i	$-0.583431\pi$
$709$	−13.7990	−0.518232	−0.259116	+	0.965846i	$0.583431\pi$
$710$	0	0
$711$	6.87006	0.257647
$712$	0	0
$713$	−2.48528	−0.0930745
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.7157	0.437532
$718$	0	0
$719$	−45.4975	−1.69677	−0.848385	−	0.529380i	$-0.822425\pi$
$719$	−45.4975	−1.69677	−0.848385	+	0.529380i	$0.822425\pi$
$720$	0	0
$721$	−16.4853	−0.613944
$722$	0	0
$723$	−4.20101	−0.156237
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.4558	−1.46334	−0.731668	−	0.681661i	$-0.761258\pi$
$727$	−39.4558	−1.46334	−0.731668	+	0.681661i	$0.761258\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	−3.65685	−0.135254
$732$	0	0
$733$	10.2843	0.379858	0.189929	−	0.981798i	$-0.439174\pi$
$733$	10.2843	0.379858	0.189929	+	0.981798i	$0.439174\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−47.7990	−1.76070
$738$	0	0
$739$	11.7990	0.434033	0.217016	−	0.976168i	$-0.430367\pi$
$739$	11.7990	0.434033	0.217016	+	0.976168i	$0.430367\pi$
$740$	0	0
$741$	−4.68629	−0.172155
$742$	0	0
$743$	−34.0416	−1.24887	−0.624433	−	0.781078i	$-0.714670\pi$
$743$	−34.0416	−1.24887	−0.624433	+	0.781078i	$0.714670\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	35.3726	1.29422
$748$	0	0
$749$	26.9706	0.985483
$750$	0	0
$751$	−30.1838	−1.10142	−0.550711	−	0.834696i	$-0.685643\pi$
$751$	−30.1838	−1.10142	−0.550711	+	0.834696i	$0.685643\pi$
$752$	0	0
$753$	−7.02944	−0.256167
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.8284	−1.70201	−0.851004	−	0.525159i	$-0.824006\pi$
$757$	−46.8284	−1.70201	−0.851004	+	0.525159i	$0.824006\pi$
$758$	0	0
$759$	1.85786	0.0674362
$760$	0	0
$761$	35.3137	1.28012	0.640060	−	0.768325i	$-0.278909\pi$
$761$	35.3137	1.28012	0.640060	+	0.768325i	$0.278909\pi$
$762$	0	0
$763$	−18.1421	−0.656789
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.9411	−1.51441
$768$	0	0
$769$	−45.6569	−1.64643	−0.823214	−	0.567731i	$-0.807821\pi$
$769$	−45.6569	−1.64643	−0.823214	+	0.567731i	$0.807821\pi$
$770$	0	0
$771$	1.65685	0.0596701
$772$	0	0
$773$	35.1127	1.26292	0.631458	−	0.775410i	$-0.282457\pi$
$773$	35.1127	1.26292	0.631458	+	0.775410i	$0.282457\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.9706	0.752315
$778$	0	0
$779$	−29.6569	−1.06257
$780$	0	0
$781$	−22.9706	−0.821951
$782$	0	0
$783$	2.74517	0.0981042
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.2132	0.399708	0.199854	−	0.979826i	$-0.435953\pi$
$787$	11.2132	0.399708	0.199854	+	0.979826i	$0.435953\pi$
$788$	0	0
$789$	7.79899	0.277651
$790$	0	0
$791$	−30.1421	−1.07173
$792$	0	0
$793$	10.3431	0.367296
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.62742	−0.305599	−0.152799	−	0.988257i	$-0.548829\pi$
$797$	−8.62742	−0.305599	−0.152799	+	0.988257i	$0.548829\pi$
$798$	0	0
$799$	0.828427	0.0293076
$800$	0	0
$801$	36.2843	1.28204
$802$	0	0
$803$	−4.48528	−0.158282
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.76955	0.308703
$808$	0	0
$809$	−20.3431	−0.715227	−0.357613	−	0.933870i	$-0.616409\pi$
$809$	−20.3431	−0.715227	−0.357613	+	0.933870i	$0.616409\pi$
$810$	0	0
$811$	−21.4142	−0.751955	−0.375977	−	0.926629i	$-0.622693\pi$
$811$	−21.4142	−0.751955	−0.375977	+	0.926629i	$0.622693\pi$
$812$	0	0
$813$	−1.37258	−0.0481386
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.3431	0.361861
$818$	0	0
$819$	−25.6569	−0.896523
$820$	0	0
$821$	−8.62742	−0.301099	−0.150549	−	0.988602i	$-0.548104\pi$
$821$	−8.62742	−0.301099	−0.150549	+	0.988602i	$0.548104\pi$
$822$	0	0
$823$	−10.9289	−0.380959	−0.190479	−	0.981691i	$-0.561004\pi$
$823$	−10.9289	−0.380959	−0.190479	+	0.981691i	$0.561004\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.55635	0.0541196	0.0270598	−	0.999634i	$-0.491386\pi$
$827$	1.55635	0.0541196	0.0270598	+	0.999634i	$0.491386\pi$
$828$	0	0
$829$	−27.9411	−0.970435	−0.485218	−	0.874393i	$-0.661260\pi$
$829$	−27.9411	−0.970435	−0.485218	+	0.874393i	$0.661260\pi$
$830$	0	0
$831$	−5.85786	−0.203207
$832$	0	0
$833$	4.65685	0.161350
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.0589	0.485946
$838$	0	0
$839$	34.1838	1.18015	0.590077	−	0.807347i	$-0.299097\pi$
$839$	34.1838	1.18015	0.590077	+	0.807347i	$0.299097\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	9.17157	0.315886
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−62.5269	−2.14845
$848$	0	0
$849$	−3.37258	−0.115747
$850$	0	0
$851$	−6.14214	−0.210550
$852$	0	0
$853$	36.8284	1.26098	0.630491	−	0.776197i	$-0.282854\pi$
$853$	36.8284	1.26098	0.630491	+	0.776197i	$0.282854\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.4853	−1.04136	−0.520679	−	0.853753i	$-0.674321\pi$
$857$	−30.4853	−1.04136	−0.520679	+	0.853753i	$0.674321\pi$
$858$	0	0
$859$	36.7696	1.25456	0.627280	−	0.778793i	$-0.284168\pi$
$859$	36.7696	1.25456	0.627280	+	0.778793i	$0.284168\pi$
$860$	0	0
$861$	20.9706	0.714675
$862$	0	0
$863$	−10.4853	−0.356923	−0.178462	−	0.983947i	$-0.557112\pi$
$863$	−10.4853	−0.356923	−0.178462	+	0.983947i	$0.557112\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.585786	−0.0198944
$868$	0	0
$869$	−14.0000	−0.474917
$870$	0	0
$871$	24.9706	0.846095
$872$	0	0
$873$	−20.3431	−0.688511
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−54.2843	−1.83305	−0.916525	−	0.399978i	$-0.869018\pi$
$877$	−54.2843	−1.83305	−0.916525	+	0.399978i	$0.869018\pi$
$878$	0	0
$879$	10.5442	0.355646
$880$	0	0
$881$	19.8579	0.669028	0.334514	−	0.942391i	$-0.391428\pi$
$881$	19.8579	0.669028	0.334514	+	0.942391i	$0.391428\pi$
$882$	0	0
$883$	20.8284	0.700932	0.350466	−	0.936575i	$-0.386023\pi$
$883$	20.8284	0.700932	0.350466	+	0.936575i	$0.386023\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.0711	−0.976111	−0.488055	−	0.872813i	$-0.662294\pi$
$887$	−29.0711	−0.976111	−0.488055	+	0.872813i	$0.662294\pi$
$888$	0	0
$889$	−18.1421	−0.608468
$890$	0	0
$891$	32.6447	1.09364
$892$	0	0
$893$	−2.34315	−0.0784104
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.970563	−0.0324061
$898$	0	0
$899$	3.51472	0.117222
$900$	0	0
$901$	−11.6569	−0.388346
$902$	0	0
$903$	−7.31371	−0.243385
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.7279	0.621850	0.310925	−	0.950434i	$-0.399361\pi$
$907$	18.7279	0.621850	0.310925	+	0.950434i	$0.399361\pi$
$908$	0	0
$909$	21.2548	0.704978
$910$	0	0
$911$	−3.75736	−0.124487	−0.0622434	−	0.998061i	$-0.519826\pi$
$911$	−3.75736	−0.124487	−0.0622434	+	0.998061i	$0.519826\pi$
$912$	0	0
$913$	−72.0833	−2.38561
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.1421	0.665152
$918$	0	0
$919$	−7.02944	−0.231880	−0.115940	−	0.993256i	$-0.536988\pi$
$919$	−7.02944	−0.231880	−0.115940	+	0.993256i	$0.536988\pi$
$920$	0	0
$921$	6.54416	0.215637
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.8284	−0.421341
$928$	0	0
$929$	−15.4558	−0.507090	−0.253545	−	0.967324i	$-0.581597\pi$
$929$	−15.4558	−0.507090	−0.253545	+	0.967324i	$0.581597\pi$
$930$	0	0
$931$	−13.1716	−0.431681
$932$	0	0
$933$	0.544156	0.0178149
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.2843	0.727995	0.363998	−	0.931400i	$-0.381412\pi$
$937$	22.2843	0.727995	0.363998	+	0.931400i	$0.381412\pi$
$938$	0	0
$939$	19.7990	0.646116
$940$	0	0
$941$	47.4558	1.54702	0.773508	−	0.633786i	$-0.218500\pi$
$941$	47.4558	1.54702	0.773508	+	0.633786i	$0.218500\pi$
$942$	0	0
$943$	−6.14214	−0.200015
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.1005	−0.523196	−0.261598	−	0.965177i	$-0.584249\pi$
$947$	−16.1005	−0.523196	−0.261598	+	0.965177i	$0.584249\pi$
$948$	0	0
$949$	2.34315	0.0760617
$950$	0	0
$951$	0.769553	0.0249545
$952$	0	0
$953$	50.1421	1.62426	0.812132	−	0.583474i	$-0.198307\pi$
$953$	50.1421	1.62426	0.812132	+	0.583474i	$0.198307\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.62742	−0.0849323
$958$	0	0
$959$	23.3137	0.752839
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	20.9878	0.676323
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−34.9706	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$967$	−34.9706	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$968$	0	0
$969$	1.65685	0.0532258
$970$	0	0
$971$	47.7990	1.53394	0.766971	−	0.641681i	$-0.221763\pi$
$971$	47.7990	1.53394	0.766971	+	0.641681i	$0.221763\pi$
$972$	0	0
$973$	−6.48528	−0.207909
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.2548	0.999931	0.499965	−	0.866045i	$-0.333346\pi$
$977$	31.2548	0.999931	0.499965	+	0.866045i	$0.333346\pi$
$978$	0	0
$979$	−73.9411	−2.36317
$980$	0	0
$981$	−14.1177	−0.450745
$982$	0	0
$983$	29.3553	0.936290	0.468145	−	0.883652i	$-0.344923\pi$
$983$	29.3553	0.936290	0.468145	+	0.883652i	$0.344923\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.65685	0.0527383
$988$	0	0
$989$	2.14214	0.0681159
$990$	0	0
$991$	−30.5858	−0.971590	−0.485795	−	0.874073i	$-0.661470\pi$
$991$	−30.5858	−0.971590	−0.485795	+	0.874073i	$0.661470\pi$
$992$	0	0
$993$	12.4020	0.393566
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.8579	−0.375542	−0.187771	−	0.982213i	$-0.560126\pi$
$997$	−11.8579	−0.375542	−0.187771	+	0.982213i	$0.560126\pi$
$998$	0	0
$999$	34.7452	1.09929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.ba.1.2		2
4.3	odd	2		425.2.a.f.1.2		2
5.4	even	2		1360.2.a.o.1.1		2
12.11	even	2		3825.2.a.p.1.1		2
20.3	even	4		425.2.b.e.324.1		4
20.7	even	4		425.2.b.e.324.4		4
20.19	odd	2		85.2.a.b.1.1	✓	2
40.19	odd	2		5440.2.a.bm.1.1		2
40.29	even	2		5440.2.a.ba.1.2		2
60.59	even	2		765.2.a.i.1.2		2
68.67	odd	2		7225.2.a.o.1.2		2
140.139	even	2		4165.2.a.q.1.1		2
340.259	odd	4		1445.2.d.f.866.4		4
340.319	odd	4		1445.2.d.f.866.3		4
340.339	odd	2		1445.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.b.1.1	✓	2	20.19	odd	2
425.2.a.f.1.2		2	4.3	odd	2
425.2.b.e.324.1		4	20.3	even	4
425.2.b.e.324.4		4	20.7	even	4
765.2.a.i.1.2		2	60.59	even	2
1360.2.a.o.1.1		2	5.4	even	2
1445.2.a.f.1.1		2	340.339	odd	2
1445.2.d.f.866.3		4	340.319	odd	4
1445.2.d.f.866.4		4	340.259	odd	4
3825.2.a.p.1.1		2	12.11	even	2
4165.2.a.q.1.1		2	140.139	even	2
5440.2.a.ba.1.2		2	40.29	even	2
5440.2.a.bm.1.1		2	40.19	odd	2
6800.2.a.ba.1.2		2	1.1	even	1	trivial
7225.2.a.o.1.2		2	68.67	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.585786 q^{3} -3.41421 q^{7} -2.65685 q^{9} +5.41421 q^{11} -2.82843 q^{13} +1.00000 q^{17} -2.82843 q^{19} +2.00000 q^{21} -0.585786 q^{23} +3.31371 q^{27} +0.828427 q^{29} +4.24264 q^{31} -3.17157 q^{33} +10.4853 q^{37} +1.65685 q^{39} +10.4853 q^{41} -3.65685 q^{43} +0.828427 q^{47} +4.65685 q^{49} -0.585786 q^{51} -11.6569 q^{53} +1.65685 q^{57} +14.8284 q^{59} -3.65685 q^{61} +9.07107 q^{63} -8.82843 q^{67} +0.343146 q^{69} -4.24264 q^{71} -0.828427 q^{73} -18.4853 q^{77} -2.58579 q^{79} +6.02944 q^{81} -13.3137 q^{83} -0.485281 q^{87} -13.6569 q^{89} +9.65685 q^{91} -2.48528 q^{93} +7.65685 q^{97} -14.3848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{7} + 6 q^{9} + 8 q^{11} + 2 q^{17} + 4 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 12 q^{33} + 4 q^{37} - 8 q^{39} + 4 q^{41} + 4 q^{43} - 4 q^{47} - 2 q^{49} - 4 q^{51} - 12 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^7 + 6 * q^9 + 8 * q^11 + 2 * q^17 + 4 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 12 * q^33 + 4 * q^37 - 8 * q^39 + 4 * q^41 + 4 * q^43 - 4 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 - 8 * q^57 + 24 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 12 * q^69 + 4 * q^73 - 20 * q^77 - 8 * q^79 + 46 * q^81 - 4 * q^83 + 16 * q^87 - 16 * q^89 + 8 * q^91 + 12 * q^93 + 4 * q^97 + 8 * q^99

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