LMFDB - Embedded newform 6800.2.a.bj.1.3

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:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3400)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ 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)

$f(q)$	$=$	$q+2.08613 q^{3} -3.43807 q^{7} +1.35194 q^{9} +O(q^{10})$	$q+2.08613 q^{3} -3.43807 q^{7} +1.35194 q^{9} -1.35194 q^{11} +2.35194 q^{13} +1.00000 q^{17} -2.00000 q^{19} -7.17226 q^{21} -2.64806 q^{23} -3.43807 q^{27} +6.17226 q^{29} +4.08613 q^{31} -2.82032 q^{33} +6.17226 q^{37} +4.90645 q^{39} -8.17226 q^{41} +4.87614 q^{43} -4.17226 q^{47} +4.82032 q^{49} +2.08613 q^{51} +3.00000 q^{53} -4.17226 q^{57} +2.17226 q^{59} -14.3445 q^{61} -4.64806 q^{63} -9.64064 q^{67} -5.52420 q^{69} -8.90645 q^{71} -4.87614 q^{73} +4.64806 q^{77} +0.141948 q^{79} -11.2281 q^{81} -4.00000 q^{83} +12.8761 q^{87} +1.23550 q^{89} -8.08613 q^{91} +8.52420 q^{93} -8.34452 q^{97} -1.82774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q - q^3 - q^7 + 2 * q^9	$ 3 q - q^{3} - q^{7} + 2 q^{9} - 2 q^{11} + 5 q^{13} + 3 q^{17} - 6 q^{19} - 7 q^{21} - 10 q^{23} - q^{27} + 4 q^{29} + 5 q^{31} + 4 q^{33} + 4 q^{37} - 5 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{47} + 2 q^{49} - q^{51} + 9 q^{53} + 2 q^{57} - 8 q^{59} - 14 q^{61} - 16 q^{63} - 4 q^{67} - 7 q^{71} + 4 q^{73} + 16 q^{77} - 13 q^{79} - 13 q^{81} - 12 q^{83} + 20 q^{87} + 10 q^{89} - 17 q^{91} + 9 q^{93} + 4 q^{97} - 20 q^{99}+O(q^{100}) $ 3 * q - q^3 - q^7 + 2 * q^9 - 2 * q^11 + 5 * q^13 + 3 * q^17 - 6 * q^19 - 7 * q^21 - 10 * q^23 - q^27 + 4 * q^29 + 5 * q^31 + 4 * q^33 + 4 * q^37 - 5 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^47 + 2 * q^49 - q^51 + 9 * q^53 + 2 * q^57 - 8 * q^59 - 14 * q^61 - 16 * q^63 - 4 * q^67 - 7 * q^71 + 4 * q^73 + 16 * q^77 - 13 * q^79 - 13 * q^81 - 12 * q^83 + 20 * q^87 + 10 * q^89 - 17 * q^91 + 9 * q^93 + 4 * q^97 - 20 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.08613	1.20443	0.602214	−	0.798335i	$-0.294285\pi$
$3$	2.08613	1.20443	0.602214	+	0.798335i	$0.294285\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.43807	−1.29947	−0.649734	−	0.760162i	$-0.725120\pi$
$7$	−3.43807	−1.29947	−0.649734	+	0.760162i	$0.725120\pi$
$8$	0	0
$9$	1.35194	0.450646
$10$	0	0
$11$	−1.35194	−0.407625	−0.203813	−	0.979010i	$-0.565333\pi$
$11$	−1.35194	−0.407625	−0.203813	+	0.979010i	$0.565333\pi$
$12$	0	0
$13$	2.35194	0.652311	0.326155	−	0.945316i	$-0.394247\pi$
$13$	2.35194	0.652311	0.326155	+	0.945316i	$0.394247\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−7.17226	−1.56512
$22$	0	0
$23$	−2.64806	−0.552159	−0.276079	−	0.961135i	$-0.589035\pi$
$23$	−2.64806	−0.552159	−0.276079	+	0.961135i	$0.589035\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.43807	−0.661657
$28$	0	0
$29$	6.17226	1.14616	0.573080	−	0.819499i	$-0.305748\pi$
$29$	6.17226	1.14616	0.573080	+	0.819499i	$0.305748\pi$
$30$	0	0
$31$	4.08613	0.733891	0.366945	−	0.930243i	$-0.380404\pi$
$31$	4.08613	0.733891	0.366945	+	0.930243i	$0.380404\pi$
$32$	0	0
$33$	−2.82032	−0.490955
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.17226	1.01471	0.507357	−	0.861736i	$-0.330623\pi$
$37$	6.17226	1.01471	0.507357	+	0.861736i	$0.330623\pi$
$38$	0	0
$39$	4.90645	0.785661
$40$	0	0
$41$	−8.17226	−1.27629	−0.638146	−	0.769915i	$-0.720299\pi$
$41$	−8.17226	−1.27629	−0.638146	+	0.769915i	$0.720299\pi$
$42$	0	0
$43$	4.87614	0.743604	0.371802	−	0.928312i	$-0.378740\pi$
$43$	4.87614	0.743604	0.371802	+	0.928312i	$0.378740\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.17226	−0.608587	−0.304293	−	0.952578i	$-0.598420\pi$
$47$	−4.17226	−0.608587	−0.304293	+	0.952578i	$0.598420\pi$
$48$	0	0
$49$	4.82032	0.688617
$50$	0	0
$51$	2.08613	0.292117
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.17226	−0.552629
$58$	0	0
$59$	2.17226	0.282804	0.141402	−	0.989952i	$-0.454839\pi$
$59$	2.17226	0.282804	0.141402	+	0.989952i	$0.454839\pi$
$60$	0	0
$61$	−14.3445	−1.83663	−0.918314	−	0.395853i	$-0.870449\pi$
$61$	−14.3445	−1.83663	−0.918314	+	0.395853i	$0.870449\pi$
$62$	0	0
$63$	−4.64806	−0.585601
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.64064	−1.17779	−0.588896	−	0.808209i	$-0.700437\pi$
$67$	−9.64064	−1.17779	−0.588896	+	0.808209i	$0.700437\pi$
$68$	0	0
$69$	−5.52420	−0.665035
$70$	0	0
$71$	−8.90645	−1.05700	−0.528501	−	0.848933i	$-0.677246\pi$
$71$	−8.90645	−1.05700	−0.528501	+	0.848933i	$0.677246\pi$
$72$	0	0
$73$	−4.87614	−0.570709	−0.285354	−	0.958422i	$-0.592111\pi$
$73$	−4.87614	−0.570709	−0.285354	+	0.958422i	$0.592111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.64806	0.529696
$78$	0	0
$79$	0.141948	0.0159704	0.00798519	−	0.999968i	$-0.497458\pi$
$79$	0.141948	0.0159704	0.00798519	+	0.999968i	$0.497458\pi$
$80$	0	0
$81$	−11.2281	−1.24756
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.8761	1.38047
$88$	0	0
$89$	1.23550	0.130962	0.0654812	−	0.997854i	$-0.479142\pi$
$89$	1.23550	0.130962	0.0654812	+	0.997854i	$0.479142\pi$
$90$	0	0
$91$	−8.08613	−0.847657
$92$	0	0
$93$	8.52420	0.883918
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.34452	−0.847258	−0.423629	−	0.905836i	$-0.639244\pi$
$97$	−8.34452	−0.847258	−0.423629	+	0.905836i	$0.639244\pi$
$98$	0	0
$99$	−1.82774	−0.183695
$100$	0	0
$101$	−5.64806	−0.562003	−0.281002	−	0.959707i	$-0.590667\pi$
$101$	−5.64806	−0.562003	−0.281002	+	0.959707i	$0.590667\pi$
$102$	0	0
$103$	−14.8761	−1.46579	−0.732895	−	0.680342i	$-0.761831\pi$
$103$	−14.8761	−1.46579	−0.732895	+	0.680342i	$0.761831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.03031	−0.196278	−0.0981389	−	0.995173i	$-0.531289\pi$
$107$	−2.03031	−0.196278	−0.0981389	+	0.995173i	$0.531289\pi$
$108$	0	0
$109$	−14.2839	−1.36815	−0.684075	−	0.729412i	$-0.739794\pi$
$109$	−14.2839	−1.36815	−0.684075	+	0.729412i	$0.739794\pi$
$110$	0	0
$111$	12.8761	1.22215
$112$	0	0
$113$	−4.34452	−0.408698	−0.204349	−	0.978898i	$-0.565508\pi$
$113$	−4.34452	−0.408698	−0.204349	+	0.978898i	$0.565508\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.17968	0.293961
$118$	0	0
$119$	−3.43807	−0.315167
$120$	0	0
$121$	−9.17226	−0.833842
$122$	0	0
$123$	−17.0484	−1.53720
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.87614	−0.432687	−0.216344	−	0.976317i	$-0.569413\pi$
$127$	−4.87614	−0.432687	−0.216344	+	0.976317i	$0.569413\pi$
$128$	0	0
$129$	10.1723	0.895618
$130$	0	0
$131$	−8.84583	−0.772863	−0.386432	−	0.922318i	$-0.626293\pi$
$131$	−8.84583	−0.772863	−0.386432	+	0.922318i	$0.626293\pi$
$132$	0	0
$133$	6.87614	0.596237
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.77192	0.407693	0.203846	−	0.979003i	$-0.434656\pi$
$137$	4.77192	0.407693	0.203846	+	0.979003i	$0.434656\pi$
$138$	0	0
$139$	−16.0861	−1.36441	−0.682204	−	0.731162i	$-0.738978\pi$
$139$	−16.0861	−1.36441	−0.682204	+	0.731162i	$0.738978\pi$
$140$	0	0
$141$	−8.70388	−0.732999
$142$	0	0
$143$	−3.17968	−0.265898
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.0558	0.829390
$148$	0	0
$149$	18.2207	1.49269	0.746347	−	0.665557i	$-0.231806\pi$
$149$	18.2207	1.49269	0.746347	+	0.665557i	$0.231806\pi$
$150$	0	0
$151$	13.5800	1.10513	0.552563	−	0.833471i	$-0.313650\pi$
$151$	13.5800	1.10513	0.552563	+	0.833471i	$0.313650\pi$
$152$	0	0
$153$	1.35194	0.109298
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.40776	0.192160	0.0960800	−	0.995374i	$-0.469370\pi$
$157$	2.40776	0.192160	0.0960800	+	0.995374i	$0.469370\pi$
$158$	0	0
$159$	6.25839	0.496323
$160$	0	0
$161$	9.10422	0.717513
$162$	0	0
$163$	−5.32163	−0.416822	−0.208411	−	0.978041i	$-0.566829\pi$
$163$	−5.32163	−0.416822	−0.208411	+	0.978041i	$0.566829\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.35194	−0.104616	−0.0523081	−	0.998631i	$-0.516658\pi$
$167$	−1.35194	−0.104616	−0.0523081	+	0.998631i	$0.516658\pi$
$168$	0	0
$169$	−7.46838	−0.574491
$170$	0	0
$171$	−2.70388	−0.206771
$172$	0	0
$173$	17.8129	1.35429	0.677145	−	0.735850i	$-0.263217\pi$
$173$	17.8129	1.35429	0.677145	+	0.735850i	$0.263217\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.53162	0.340617
$178$	0	0
$179$	0.703878	0.0526103	0.0263052	−	0.999654i	$-0.491626\pi$
$179$	0.703878	0.0526103	0.0263052	+	0.999654i	$0.491626\pi$
$180$	0	0
$181$	13.9245	1.03500	0.517501	−	0.855682i	$-0.326862\pi$
$181$	13.9245	1.03500	0.517501	+	0.855682i	$0.326862\pi$
$182$	0	0
$183$	−29.9245	−2.21209
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.35194	−0.0988636
$188$	0	0
$189$	11.8203	0.859802
$190$	0	0
$191$	−22.6890	−1.64172	−0.820861	−	0.571128i	$-0.806506\pi$
$191$	−22.6890	−1.64172	−0.820861	+	0.571128i	$0.806506\pi$
$192$	0	0
$193$	0.419983	0.0302310	0.0151155	−	0.999886i	$-0.495188\pi$
$193$	0.419983	0.0302310	0.0151155	+	0.999886i	$0.495188\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4078	0.812769	0.406385	−	0.913702i	$-0.366789\pi$
$197$	11.4078	0.812769	0.406385	+	0.913702i	$0.366789\pi$
$198$	0	0
$199$	−2.75970	−0.195630	−0.0978148	−	0.995205i	$-0.531185\pi$
$199$	−2.75970	−0.195630	−0.0978148	+	0.995205i	$0.531185\pi$
$200$	0	0
$201$	−20.1116	−1.41857
$202$	0	0
$203$	−21.2207	−1.48940
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.58002	−0.248828
$208$	0	0
$209$	2.70388	0.187031
$210$	0	0
$211$	−13.2510	−0.912235	−0.456117	−	0.889920i	$-0.650760\pi$
$211$	−13.2510	−0.912235	−0.456117	+	0.889920i	$0.650760\pi$
$212$	0	0
$213$	−18.5800	−1.27308
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.0484	−0.953667
$218$	0	0
$219$	−10.1723	−0.687378
$220$	0	0
$221$	2.35194	0.158209
$222$	0	0
$223$	12.6284	0.845661	0.422831	−	0.906209i	$-0.361037\pi$
$223$	12.6284	0.845661	0.422831	+	0.906209i	$0.361037\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.6661	−0.907054	−0.453527	−	0.891243i	$-0.649834\pi$
$227$	−13.6661	−0.907054	−0.453527	+	0.891243i	$0.649834\pi$
$228$	0	0
$229$	9.46096	0.625198	0.312599	−	0.949885i	$-0.398800\pi$
$229$	9.46096	0.625198	0.312599	+	0.949885i	$0.398800\pi$
$230$	0	0
$231$	9.69646	0.637980
$232$	0	0
$233$	−23.9245	−1.56735	−0.783674	−	0.621172i	$-0.786657\pi$
$233$	−23.9245	−1.56735	−0.783674	+	0.621172i	$0.786657\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.296122	0.0192352
$238$	0	0
$239$	−15.2813	−0.988464	−0.494232	−	0.869330i	$-0.664551\pi$
$239$	−15.2813	−0.988464	−0.494232	+	0.869330i	$0.664551\pi$
$240$	0	0
$241$	17.3929	1.12038	0.560188	−	0.828365i	$-0.310729\pi$
$241$	17.3929	1.12038	0.560188	+	0.828365i	$0.310729\pi$
$242$	0	0
$243$	−13.1090	−0.840944
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.70388	−0.299301
$248$	0	0
$249$	−8.34452	−0.528813
$250$	0	0
$251$	−3.88836	−0.245431	−0.122716	−	0.992442i	$-0.539160\pi$
$251$	−3.88836	−0.245431	−0.122716	+	0.992442i	$0.539160\pi$
$252$	0	0
$253$	3.58002	0.225074
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.0894	−1.75217	−0.876084	−	0.482159i	$-0.839853\pi$
$257$	−28.0894	−1.75217	−0.876084	+	0.482159i	$0.839853\pi$
$258$	0	0
$259$	−21.2207	−1.31859
$260$	0	0
$261$	8.34452	0.516513
$262$	0	0
$263$	−13.4078	−0.826758	−0.413379	−	0.910559i	$-0.635651\pi$
$263$	−13.4078	−0.826758	−0.413379	+	0.910559i	$0.635651\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.57741	0.157735
$268$	0	0
$269$	5.04840	0.307806	0.153903	−	0.988086i	$-0.450816\pi$
$269$	5.04840	0.307806	0.153903	+	0.988086i	$0.450816\pi$
$270$	0	0
$271$	−0.703878	−0.0427576	−0.0213788	−	0.999771i	$-0.506806\pi$
$271$	−0.703878	−0.0427576	−0.0213788	+	0.999771i	$0.506806\pi$
$272$	0	0
$273$	−16.8687	−1.02094
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.9729	1.74082	0.870408	−	0.492332i	$-0.163855\pi$
$277$	28.9729	1.74082	0.870408	+	0.492332i	$0.163855\pi$
$278$	0	0
$279$	5.52420	0.330725
$280$	0	0
$281$	−5.24772	−0.313053	−0.156526	−	0.987674i	$-0.550030\pi$
$281$	−5.24772	−0.313053	−0.156526	+	0.987674i	$0.550030\pi$
$282$	0	0
$283$	12.4003	0.737124	0.368562	−	0.929603i	$-0.379850\pi$
$283$	12.4003	0.737124	0.368562	+	0.929603i	$0.379850\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.0968	1.65850
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−17.4078	−1.02046
$292$	0	0
$293$	14.6890	0.858143	0.429071	−	0.903271i	$-0.358841\pi$
$293$	14.6890	0.858143	0.429071	+	0.903271i	$0.358841\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.64806	0.269708
$298$	0	0
$299$	−6.22808	−0.360179
$300$	0	0
$301$	−16.7645	−0.966290
$302$	0	0
$303$	−11.7826	−0.676892
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.0968	1.71772	0.858858	−	0.512215i	$-0.171175\pi$
$307$	30.0968	1.71772	0.858858	+	0.512215i	$0.171175\pi$
$308$	0	0
$309$	−31.0336	−1.76544
$310$	0	0
$311$	−23.8384	−1.35175	−0.675876	−	0.737015i	$-0.736235\pi$
$311$	−23.8384	−1.35175	−0.675876	+	0.737015i	$0.736235\pi$
$312$	0	0
$313$	−14.9878	−0.847159	−0.423580	−	0.905859i	$-0.639227\pi$
$313$	−14.9878	−0.847159	−0.423580	+	0.905859i	$0.639227\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.7013	0.769540	0.384770	−	0.923013i	$-0.374281\pi$
$317$	13.7013	0.769540	0.384770	+	0.923013i	$0.374281\pi$
$318$	0	0
$319$	−8.34452	−0.467203
$320$	0	0
$321$	−4.23550	−0.236402
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−29.7981	−1.64784
$328$	0	0
$329$	14.3445	0.790839
$330$	0	0
$331$	8.28390	0.455324	0.227662	−	0.973740i	$-0.426892\pi$
$331$	8.28390	0.455324	0.227662	+	0.973740i	$0.426892\pi$
$332$	0	0
$333$	8.34452	0.457277
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.5774	0.903029	0.451514	−	0.892264i	$-0.350884\pi$
$337$	16.5774	0.903029	0.451514	+	0.892264i	$0.350884\pi$
$338$	0	0
$339$	−9.06324	−0.492247
$340$	0	0
$341$	−5.52420	−0.299152
$342$	0	0
$343$	7.49389	0.404632
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.73419	−0.361510	−0.180755	−	0.983528i	$-0.557854\pi$
$347$	−6.73419	−0.361510	−0.180755	+	0.983528i	$0.557854\pi$
$348$	0	0
$349$	0.407757	0.0218267	0.0109134	−	0.999940i	$-0.496526\pi$
$349$	0.407757	0.0218267	0.0109134	+	0.999940i	$0.496526\pi$
$350$	0	0
$351$	−8.08613	−0.431606
$352$	0	0
$353$	30.5046	1.62359	0.811797	−	0.583940i	$-0.198490\pi$
$353$	30.5046	1.62359	0.811797	+	0.583940i	$0.198490\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.17226	−0.379596
$358$	0	0
$359$	−10.6890	−0.564146	−0.282073	−	0.959393i	$-0.591022\pi$
$359$	−10.6890	−0.564146	−0.282073	+	0.959393i	$0.591022\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−19.1345	−1.00430
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.7119	1.75975	0.879874	−	0.475206i	$-0.157627\pi$
$367$	33.7119	1.75975	0.879874	+	0.475206i	$0.157627\pi$
$368$	0	0
$369$	−11.0484	−0.575157
$370$	0	0
$371$	−10.3142	−0.535487
$372$	0	0
$373$	25.3249	1.31127	0.655636	−	0.755077i	$-0.272401\pi$
$373$	25.3249	1.31127	0.655636	+	0.755077i	$0.272401\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5168	0.747652
$378$	0	0
$379$	−19.4381	−0.998467	−0.499233	−	0.866468i	$-0.666385\pi$
$379$	−19.4381	−0.998467	−0.499233	+	0.866468i	$0.666385\pi$
$380$	0	0
$381$	−10.1723	−0.521141
$382$	0	0
$383$	24.2084	1.23699	0.618497	−	0.785787i	$-0.287742\pi$
$383$	24.2084	1.23699	0.618497	+	0.785787i	$0.287742\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.59224	0.335103
$388$	0	0
$389$	0.516781	0.0262018	0.0131009	−	0.999914i	$-0.495830\pi$
$389$	0.516781	0.0262018	0.0131009	+	0.999914i	$0.495830\pi$
$390$	0	0
$391$	−2.64806	−0.133918
$392$	0	0
$393$	−18.4535	−0.930858
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.689042	0.0345820	0.0172910	−	0.999850i	$-0.494496\pi$
$397$	0.689042	0.0345820	0.0172910	+	0.999850i	$0.494496\pi$
$398$	0	0
$399$	14.3445	0.718124
$400$	0	0
$401$	14.7645	0.737304	0.368652	−	0.929567i	$-0.379819\pi$
$401$	14.7645	0.737304	0.368652	+	0.929567i	$0.379819\pi$
$402$	0	0
$403$	9.61033	0.478725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.34452	−0.413623
$408$	0	0
$409$	−19.2839	−0.953527	−0.476764	−	0.879032i	$-0.658190\pi$
$409$	−19.2839	−0.953527	−0.476764	+	0.879032i	$0.658190\pi$
$410$	0	0
$411$	9.95485	0.491037
$412$	0	0
$413$	−7.46838	−0.367495
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−33.5578	−1.64333
$418$	0	0
$419$	34.4971	1.68530	0.842648	−	0.538465i	$-0.180996\pi$
$419$	34.4971	1.68530	0.842648	+	0.538465i	$0.180996\pi$
$420$	0	0
$421$	−19.7449	−0.962306	−0.481153	−	0.876637i	$-0.659782\pi$
$421$	−19.7449	−0.962306	−0.481153	+	0.876637i	$0.659782\pi$
$422$	0	0
$423$	−5.64064	−0.274257
$424$	0	0
$425$	0	0
$426$	0	0
$427$	49.3175	2.38664
$428$	0	0
$429$	−6.63322	−0.320255
$430$	0	0
$431$	−15.7416	−0.758247	−0.379123	−	0.925346i	$-0.623774\pi$
$431$	−15.7416	−0.758247	−0.379123	+	0.925346i	$0.623774\pi$
$432$	0	0
$433$	−10.7645	−0.517309	−0.258655	−	0.965970i	$-0.583279\pi$
$433$	−10.7645	−0.517309	−0.258655	+	0.965970i	$0.583279\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.29612	0.253348
$438$	0	0
$439$	−18.6587	−0.890533	−0.445266	−	0.895398i	$-0.646891\pi$
$439$	−18.6587	−0.890533	−0.445266	+	0.895398i	$0.646891\pi$
$440$	0	0
$441$	6.51678	0.310323
$442$	0	0
$443$	32.2180	1.53073	0.765363	−	0.643599i	$-0.222560\pi$
$443$	32.2180	1.53073	0.765363	+	0.643599i	$0.222560\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	38.0107	1.79784
$448$	0	0
$449$	−33.3323	−1.57305	−0.786524	−	0.617560i	$-0.788121\pi$
$449$	−33.3323	−1.57305	−0.786524	+	0.617560i	$0.788121\pi$
$450$	0	0
$451$	11.0484	0.520249
$452$	0	0
$453$	28.3297	1.33104
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.1042	1.12755	0.563774	−	0.825929i	$-0.309349\pi$
$457$	24.1042	1.12755	0.563774	+	0.825929i	$0.309349\pi$
$458$	0	0
$459$	−3.43807	−0.160475
$460$	0	0
$461$	−4.09680	−0.190807	−0.0954035	−	0.995439i	$-0.530414\pi$
$461$	−4.09680	−0.190807	−0.0954035	+	0.995439i	$0.530414\pi$
$462$	0	0
$463$	11.8129	0.548992	0.274496	−	0.961588i	$-0.411489\pi$
$463$	11.8129	0.548992	0.274496	+	0.961588i	$0.411489\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.1723	−0.748363	−0.374181	−	0.927355i	$-0.622076\pi$
$467$	−16.1723	−0.748363	−0.374181	+	0.927355i	$0.622076\pi$
$468$	0	0
$469$	33.1452	1.53050
$470$	0	0
$471$	5.02289	0.231443
$472$	0	0
$473$	−6.59224	−0.303112
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.05582	0.185703
$478$	0	0
$479$	36.7449	1.67892	0.839458	−	0.543425i	$-0.182873\pi$
$479$	36.7449	1.67892	0.839458	+	0.543425i	$0.182873\pi$
$480$	0	0
$481$	14.5168	0.661908
$482$	0	0
$483$	18.9926	0.864192
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.4152	0.653214	0.326607	−	0.945160i	$-0.394095\pi$
$487$	14.4152	0.653214	0.326607	+	0.945160i	$0.394095\pi$
$488$	0	0
$489$	−11.1016	−0.502032
$490$	0	0
$491$	34.4413	1.55431	0.777157	−	0.629306i	$-0.216661\pi$
$491$	34.4413	1.55431	0.777157	+	0.629306i	$0.216661\pi$
$492$	0	0
$493$	6.17226	0.277985
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.6210	1.37354
$498$	0	0
$499$	2.60291	0.116522	0.0582612	−	0.998301i	$-0.481444\pi$
$499$	2.60291	0.116522	0.0582612	+	0.998301i	$0.481444\pi$
$500$	0	0
$501$	−2.82032	−0.126003
$502$	0	0
$503$	−42.1526	−1.87949	−0.939746	−	0.341873i	$-0.888939\pi$
$503$	−42.1526	−1.87949	−0.939746	+	0.341873i	$0.888939\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−15.5800	−0.691933
$508$	0	0
$509$	−12.9394	−0.573528	−0.286764	−	0.958001i	$-0.592580\pi$
$509$	−12.9394	−0.573528	−0.286764	+	0.958001i	$0.592580\pi$
$510$	0	0
$511$	16.7645	0.741618
$512$	0	0
$513$	6.87614	0.303589
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.64064	0.248075
$518$	0	0
$519$	37.1600	1.63114
$520$	0	0
$521$	29.0484	1.27263	0.636317	−	0.771428i	$-0.280457\pi$
$521$	29.0484	1.27263	0.636317	+	0.771428i	$0.280457\pi$
$522$	0	0
$523$	35.3929	1.54762	0.773812	−	0.633415i	$-0.218347\pi$
$523$	35.3929	1.54762	0.773812	+	0.633415i	$0.218347\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.08613	0.177995
$528$	0	0
$529$	−15.9878	−0.695121
$530$	0	0
$531$	2.93676	0.127445
$532$	0	0
$533$	−19.2207	−0.832539
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.46838	0.0633654
$538$	0	0
$539$	−6.51678	−0.280698
$540$	0	0
$541$	−28.0362	−1.20537	−0.602685	−	0.797979i	$-0.705902\pi$
$541$	−28.0362	−1.20537	−0.602685	+	0.797979i	$0.705902\pi$
$542$	0	0
$543$	29.0484	1.24659
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−40.8162	−1.74517	−0.872586	−	0.488460i	$-0.837559\pi$
$547$	−40.8162	−1.74517	−0.872586	+	0.488460i	$0.837559\pi$
$548$	0	0
$549$	−19.3929	−0.827670
$550$	0	0
$551$	−12.3445	−0.525894
$552$	0	0
$553$	−0.488026	−0.0207530
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00742	0.0850570	0.0425285	−	0.999095i	$-0.486459\pi$
$557$	2.00742	0.0850570	0.0425285	+	0.999095i	$0.486459\pi$
$558$	0	0
$559$	11.4684	0.485061
$560$	0	0
$561$	−2.82032	−0.119074
$562$	0	0
$563$	33.8491	1.42657	0.713284	−	0.700875i	$-0.247207\pi$
$563$	33.8491	1.42657	0.713284	+	0.700875i	$0.247207\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	38.6029	1.62117
$568$	0	0
$569$	−3.18449	−0.133501	−0.0667503	−	0.997770i	$-0.521263\pi$
$569$	−3.18449	−0.133501	−0.0667503	+	0.997770i	$0.521263\pi$
$570$	0	0
$571$	−7.98933	−0.334343	−0.167172	−	0.985928i	$-0.553463\pi$
$571$	−7.98933	−0.334343	−0.167172	+	0.985928i	$0.553463\pi$
$572$	0	0
$573$	−47.3323	−1.97733
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.9581	1.45532	0.727662	−	0.685935i	$-0.240607\pi$
$577$	34.9581	1.45532	0.727662	+	0.685935i	$0.240607\pi$
$578$	0	0
$579$	0.876139	0.0364111
$580$	0	0
$581$	13.7523	0.570541
$582$	0	0
$583$	−4.05582	−0.167975
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.7039	−0.937089	−0.468545	−	0.883440i	$-0.655221\pi$
$587$	−22.7039	−0.937089	−0.468545	+	0.883440i	$0.655221\pi$
$588$	0	0
$589$	−8.17226	−0.336732
$590$	0	0
$591$	23.7981	0.978922
$592$	0	0
$593$	−9.40515	−0.386223	−0.193112	−	0.981177i	$-0.561858\pi$
$593$	−9.40515	−0.386223	−0.193112	+	0.981177i	$0.561858\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.75709	−0.235622
$598$	0	0
$599$	28.7645	1.17529	0.587643	−	0.809120i	$-0.300056\pi$
$599$	28.7645	1.17529	0.587643	+	0.809120i	$0.300056\pi$
$600$	0	0
$601$	5.29612	0.216033	0.108017	−	0.994149i	$-0.465550\pi$
$601$	5.29612	0.216033	0.108017	+	0.994149i	$0.465550\pi$
$602$	0	0
$603$	−13.0336	−0.530768
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.5274	0.752006	0.376003	−	0.926618i	$-0.377298\pi$
$607$	18.5274	0.752006	0.376003	+	0.926618i	$0.377298\pi$
$608$	0	0
$609$	−44.2691	−1.79387
$610$	0	0
$611$	−9.81290	−0.396988
$612$	0	0
$613$	−14.3471	−0.579475	−0.289738	−	0.957106i	$-0.593568\pi$
$613$	−14.3471	−0.579475	−0.289738	+	0.957106i	$0.593568\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.6768	0.872676	0.436338	−	0.899783i	$-0.356275\pi$
$617$	21.6768	0.872676	0.436338	+	0.899783i	$0.356275\pi$
$618$	0	0
$619$	−46.3855	−1.86439	−0.932195	−	0.361956i	$-0.882109\pi$
$619$	−46.3855	−1.86439	−0.932195	+	0.361956i	$0.882109\pi$
$620$	0	0
$621$	9.10422	0.365340
$622$	0	0
$623$	−4.24772	−0.170181
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.64064	0.225266
$628$	0	0
$629$	6.17226	0.246104
$630$	0	0
$631$	−9.59485	−0.381965	−0.190983	−	0.981593i	$-0.561167\pi$
$631$	−9.59485	−0.381965	−0.190983	+	0.981593i	$0.561167\pi$
$632$	0	0
$633$	−27.6433	−1.09872
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11.3371	0.449192
$638$	0	0
$639$	−12.0410	−0.476334
$640$	0	0
$641$	−40.6136	−1.60414	−0.802070	−	0.597230i	$-0.796268\pi$
$641$	−40.6136	−1.60414	−0.802070	+	0.597230i	$0.796268\pi$
$642$	0	0
$643$	12.5619	0.495394	0.247697	−	0.968838i	$-0.420326\pi$
$643$	12.5619	0.495394	0.247697	+	0.968838i	$0.420326\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.0606	0.552780	0.276390	−	0.961046i	$-0.410862\pi$
$647$	14.0606	0.552780	0.276390	+	0.961046i	$0.410862\pi$
$648$	0	0
$649$	−2.93676	−0.115278
$650$	0	0
$651$	−29.3068	−1.14862
$652$	0	0
$653$	−8.53162	−0.333868	−0.166934	−	0.985968i	$-0.553387\pi$
$653$	−8.53162	−0.333868	−0.166934	+	0.985968i	$0.553387\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.59224	−0.257188
$658$	0	0
$659$	−28.2180	−1.09922	−0.549610	−	0.835422i	$-0.685224\pi$
$659$	−28.2180	−1.09922	−0.549610	+	0.835422i	$0.685224\pi$
$660$	0	0
$661$	−30.3419	−1.18016	−0.590082	−	0.807343i	$-0.700905\pi$
$661$	−30.3419	−1.18016	−0.590082	+	0.807343i	$0.700905\pi$
$662$	0	0
$663$	4.90645	0.190551
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.3445	−0.632862
$668$	0	0
$669$	26.3445	1.01854
$670$	0	0
$671$	19.3929	0.748655
$672$	0	0
$673$	−17.3929	−0.670448	−0.335224	−	0.942138i	$-0.608812\pi$
$673$	−17.3929	−0.670448	−0.335224	+	0.942138i	$0.608812\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.8007	1.56810	0.784049	−	0.620699i	$-0.213151\pi$
$677$	40.8007	1.56810	0.784049	+	0.620699i	$0.213151\pi$
$678$	0	0
$679$	28.6890	1.10098
$680$	0	0
$681$	−28.5094	−1.09248
$682$	0	0
$683$	−1.68579	−0.0645050	−0.0322525	−	0.999480i	$-0.510268\pi$
$683$	−1.68579	−0.0645050	−0.0322525	+	0.999480i	$0.510268\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19.7368	0.753006
$688$	0	0
$689$	7.05582	0.268805
$690$	0	0
$691$	−43.4232	−1.65190	−0.825949	−	0.563745i	$-0.809360\pi$
$691$	−43.4232	−1.65190	−0.825949	+	0.563745i	$0.809360\pi$
$692$	0	0
$693$	6.28390	0.238705
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.17226	−0.309546
$698$	0	0
$699$	−49.9097	−1.88776
$700$	0	0
$701$	−2.98777	−0.112847	−0.0564233	−	0.998407i	$-0.517970\pi$
$701$	−2.98777	−0.112847	−0.0564233	+	0.998407i	$0.517970\pi$
$702$	0	0
$703$	−12.3445	−0.465582
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.4184	0.730305
$708$	0	0
$709$	45.2420	1.69910	0.849549	−	0.527509i	$-0.176874\pi$
$709$	45.2420	1.69910	0.849549	+	0.527509i	$0.176874\pi$
$710$	0	0
$711$	0.191905	0.00719699
$712$	0	0
$713$	−10.8203	−0.405224
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−31.8787	−1.19053
$718$	0	0
$719$	34.2584	1.27762	0.638811	−	0.769364i	$-0.279426\pi$
$719$	34.2584	1.27762	0.638811	+	0.769364i	$0.279426\pi$
$720$	0	0
$721$	51.1452	1.90475
$722$	0	0
$723$	36.2839	1.34941
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−9.73744	−0.361142	−0.180571	−	0.983562i	$-0.557795\pi$
$727$	−9.73744	−0.361142	−0.180571	+	0.983562i	$0.557795\pi$
$728$	0	0
$729$	6.33710	0.234707
$730$	0	0
$731$	4.87614	0.180351
$732$	0	0
$733$	10.1239	0.373933	0.186967	−	0.982366i	$-0.440134\pi$
$733$	10.1239	0.373933	0.186967	+	0.982366i	$0.440134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.0336	0.480097
$738$	0	0
$739$	22.5019	0.827747	0.413874	−	0.910334i	$-0.364176\pi$
$739$	22.5019	0.827747	0.413874	+	0.910334i	$0.364176\pi$
$740$	0	0
$741$	−9.81290	−0.360486
$742$	0	0
$743$	39.7119	1.45689	0.728445	−	0.685104i	$-0.240243\pi$
$743$	39.7119	1.45689	0.728445	+	0.685104i	$0.240243\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.40776	−0.197859
$748$	0	0
$749$	6.98036	0.255057
$750$	0	0
$751$	−35.5700	−1.29797	−0.648984	−	0.760802i	$-0.724806\pi$
$751$	−35.5700	−1.29797	−0.648984	+	0.760802i	$0.724806\pi$
$752$	0	0
$753$	−8.11164	−0.295604
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.2691	−1.39091	−0.695456	−	0.718568i	$-0.744798\pi$
$757$	−38.2691	−1.39091	−0.695456	+	0.718568i	$0.744798\pi$
$758$	0	0
$759$	7.46838	0.271085
$760$	0	0
$761$	46.3371	1.67972	0.839859	−	0.542804i	$-0.182637\pi$
$761$	46.3371	1.67972	0.839859	+	0.542804i	$0.182637\pi$
$762$	0	0
$763$	49.1090	1.77787
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.10902	0.184476
$768$	0	0
$769$	−48.0745	−1.73361	−0.866806	−	0.498645i	$-0.833831\pi$
$769$	−48.0745	−1.73361	−0.866806	+	0.498645i	$0.833831\pi$
$770$	0	0
$771$	−58.5981	−2.11036
$772$	0	0
$773$	−22.5752	−0.811974	−0.405987	−	0.913879i	$-0.633072\pi$
$773$	−22.5752	−0.811974	−0.405987	+	0.913879i	$0.633072\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−44.2691	−1.58814
$778$	0	0
$779$	16.3445	0.585603
$780$	0	0
$781$	12.0410	0.430860
$782$	0	0
$783$	−21.2207	−0.758365
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.7119	−1.27299	−0.636497	−	0.771280i	$-0.719617\pi$
$787$	−35.7119	−1.27299	−0.636497	+	0.771280i	$0.719617\pi$
$788$	0	0
$789$	−27.9703	−0.995770
$790$	0	0
$791$	14.9368	0.531090
$792$	0	0
$793$	−33.7374	−1.19805
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.77934	0.0630275	0.0315137	−	0.999503i	$-0.489967\pi$
$797$	1.77934	0.0630275	0.0315137	+	0.999503i	$0.489967\pi$
$798$	0	0
$799$	−4.17226	−0.147604
$800$	0	0
$801$	1.67032	0.0590177
$802$	0	0
$803$	6.59224	0.232635
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.5316	0.370731
$808$	0	0
$809$	−12.3594	−0.434532	−0.217266	−	0.976112i	$-0.569714\pi$
$809$	−12.3594	−0.434532	−0.217266	+	0.976112i	$0.569714\pi$
$810$	0	0
$811$	49.1707	1.72662	0.863308	−	0.504677i	$-0.168388\pi$
$811$	49.1707	1.72662	0.863308	+	0.504677i	$0.168388\pi$
$812$	0	0
$813$	−1.46838	−0.0514984
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.75228	−0.341189
$818$	0	0
$819$	−10.9320	−0.381993
$820$	0	0
$821$	−3.86391	−0.134851	−0.0674257	−	0.997724i	$-0.521479\pi$
$821$	−3.86391	−0.134851	−0.0674257	+	0.997724i	$0.521479\pi$
$822$	0	0
$823$	−11.3626	−0.396076	−0.198038	−	0.980194i	$-0.563457\pi$
$823$	−11.3626	−0.396076	−0.198038	+	0.980194i	$0.563457\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.6720	−1.24044	−0.620219	−	0.784429i	$-0.712956\pi$
$827$	−35.6720	−1.24044	−0.620219	+	0.784429i	$0.712956\pi$
$828$	0	0
$829$	2.75228	0.0955906	0.0477953	−	0.998857i	$-0.484780\pi$
$829$	2.75228	0.0955906	0.0477953	+	0.998857i	$0.484780\pi$
$830$	0	0
$831$	60.4413	2.09669
$832$	0	0
$833$	4.82032	0.167014
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−14.0484	−0.485584
$838$	0	0
$839$	33.7119	1.16387	0.581933	−	0.813237i	$-0.302297\pi$
$839$	33.7119	1.16387	0.581933	+	0.813237i	$0.302297\pi$
$840$	0	0
$841$	9.09680	0.313683
$842$	0	0
$843$	−10.9474	−0.377050
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.5349	1.08355
$848$	0	0
$849$	25.8687	0.887812
$850$	0	0
$851$	−16.3445	−0.560283
$852$	0	0
$853$	−2.07546	−0.0710625	−0.0355312	−	0.999369i	$-0.511312\pi$
$853$	−2.07546	−0.0710625	−0.0355312	+	0.999369i	$0.511312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.56518	−0.190103	−0.0950515	−	0.995472i	$-0.530302\pi$
$857$	−5.56518	−0.190103	−0.0950515	+	0.995472i	$0.530302\pi$
$858$	0	0
$859$	−16.8613	−0.575300	−0.287650	−	0.957736i	$-0.592874\pi$
$859$	−16.8613	−0.575300	−0.287650	+	0.957736i	$0.592874\pi$
$860$	0	0
$861$	58.6136	1.99755
$862$	0	0
$863$	1.98516	0.0675757	0.0337879	−	0.999429i	$-0.489243\pi$
$863$	1.98516	0.0675757	0.0337879	+	0.999429i	$0.489243\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.08613	0.0708487
$868$	0	0
$869$	−0.191905	−0.00650992
$870$	0	0
$871$	−22.6742	−0.768286
$872$	0	0
$873$	−11.2813	−0.381814
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−43.9245	−1.48323	−0.741613	−	0.670828i	$-0.765939\pi$
$877$	−43.9245	−1.48323	−0.741613	+	0.670828i	$0.765939\pi$
$878$	0	0
$879$	30.6433	1.03357
$880$	0	0
$881$	17.9245	0.603893	0.301947	−	0.953325i	$-0.402364\pi$
$881$	17.9245	0.603893	0.301947	+	0.953325i	$0.402364\pi$
$882$	0	0
$883$	5.42259	0.182485	0.0912424	−	0.995829i	$-0.470916\pi$
$883$	5.42259	0.182485	0.0912424	+	0.995829i	$0.470916\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.3888	−1.59116	−0.795579	−	0.605850i	$-0.792833\pi$
$887$	−47.3888	−1.59116	−0.795579	+	0.605850i	$0.792833\pi$
$888$	0	0
$889$	16.7645	0.562263
$890$	0	0
$891$	15.1797	0.508538
$892$	0	0
$893$	8.34452	0.279239
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−12.9926	−0.433810
$898$	0	0
$899$	25.2207	0.841156
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	−34.9729	−1.16383
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.0894	1.09871	0.549357	−	0.835587i	$-0.314873\pi$
$907$	33.0894	1.09871	0.549357	+	0.835587i	$0.314873\pi$
$908$	0	0
$909$	−7.63583	−0.253265
$910$	0	0
$911$	14.3142	0.474251	0.237125	−	0.971479i	$-0.423795\pi$
$911$	14.3142	0.474251	0.237125	+	0.971479i	$0.423795\pi$
$912$	0	0
$913$	5.40776	0.178971
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.4126	1.00431
$918$	0	0
$919$	−15.9245	−0.525302	−0.262651	−	0.964891i	$-0.584597\pi$
$919$	−15.9245	−0.525302	−0.262651	+	0.964891i	$0.584597\pi$
$920$	0	0
$921$	62.7858	2.06886
$922$	0	0
$923$	−20.9474	−0.689493
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−20.1116	−0.660553
$928$	0	0
$929$	21.2207	0.696227	0.348114	−	0.937452i	$-0.386822\pi$
$929$	21.2207	0.696227	0.348114	+	0.937452i	$0.386822\pi$
$930$	0	0
$931$	−9.64064	−0.315959
$932$	0	0
$933$	−49.7300	−1.62809
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−54.4387	−1.77844	−0.889218	−	0.457485i	$-0.848750\pi$
$937$	−54.4387	−1.77844	−0.889218	+	0.457485i	$0.848750\pi$
$938$	0	0
$939$	−31.2664	−1.02034
$940$	0	0
$941$	28.7793	0.938180	0.469090	−	0.883150i	$-0.344582\pi$
$941$	28.7793	0.938180	0.469090	+	0.883150i	$0.344582\pi$
$942$	0	0
$943$	21.6406	0.704716
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.2403	−1.34013	−0.670065	−	0.742303i	$-0.733734\pi$
$947$	−41.2403	−1.34013	−0.670065	+	0.742303i	$0.733734\pi$
$948$	0	0
$949$	−11.4684	−0.372279
$950$	0	0
$951$	28.5826	0.926855
$952$	0	0
$953$	29.2739	0.948274	0.474137	−	0.880451i	$-0.342760\pi$
$953$	29.2739	0.948274	0.474137	+	0.880451i	$0.342760\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.4078	−0.562713
$958$	0	0
$959$	−16.4062	−0.529784
$960$	0	0
$961$	−14.3035	−0.461405
$962$	0	0
$963$	−2.74486	−0.0884519
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.82774	−0.187407	−0.0937037	−	0.995600i	$-0.529871\pi$
$967$	−5.82774	−0.187407	−0.0937037	+	0.995600i	$0.529871\pi$
$968$	0	0
$969$	−4.17226	−0.134032
$970$	0	0
$971$	−4.60708	−0.147848	−0.0739241	−	0.997264i	$-0.523552\pi$
$971$	−4.60708	−0.147848	−0.0739241	+	0.997264i	$0.523552\pi$
$972$	0	0
$973$	55.3052	1.77300
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.5019	1.71168	0.855839	−	0.517242i	$-0.173041\pi$
$977$	53.5019	1.71168	0.855839	+	0.517242i	$0.173041\pi$
$978$	0	0
$979$	−1.67032	−0.0533835
$980$	0	0
$981$	−19.3110	−0.616551
$982$	0	0
$983$	−5.59549	−0.178469	−0.0892343	−	0.996011i	$-0.528442\pi$
$983$	−5.59549	−0.178469	−0.0892343	+	0.996011i	$0.528442\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29.9245	0.952509
$988$	0	0
$989$	−12.9123	−0.410588
$990$	0	0
$991$	−56.7507	−1.80275	−0.901373	−	0.433044i	$-0.857440\pi$
$991$	−56.7507	−1.80275	−0.901373	+	0.433044i	$0.857440\pi$
$992$	0	0
$993$	17.2813	0.548405
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.9852	−0.949640	−0.474820	−	0.880083i	$-0.657487\pi$
$997$	−29.9852	−0.949640	−0.474820	+	0.880083i	$0.657487\pi$
$998$	0	0
$999$	−21.2207	−0.671392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bj.1.3		3
4.3	odd	2		3400.2.a.q.1.1	yes	3
5.4	even	2		6800.2.a.bq.1.1		3
20.3	even	4		3400.2.e.l.2449.2		6
20.7	even	4		3400.2.e.l.2449.5		6
20.19	odd	2		3400.2.a.m.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3400.2.a.m.1.3	✓	3	20.19	odd	2
3400.2.a.q.1.1	yes	3	4.3	odd	2
3400.2.e.l.2449.2		6	20.3	even	4
3400.2.e.l.2449.5		6	20.7	even	4
6800.2.a.bj.1.3		3	1.1	even	1	trivial
6800.2.a.bq.1.1		3	5.4	even	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)

\(f(q)\)	\(=\)	\(q+2.08613 q^{3} -3.43807 q^{7} +1.35194 q^{9} +O(q^{10})\)	\(q+2.08613 q^{3} -3.43807 q^{7} +1.35194 q^{9} -1.35194 q^{11} +2.35194 q^{13} +1.00000 q^{17} -2.00000 q^{19} -7.17226 q^{21} -2.64806 q^{23} -3.43807 q^{27} +6.17226 q^{29} +4.08613 q^{31} -2.82032 q^{33} +6.17226 q^{37} +4.90645 q^{39} -8.17226 q^{41} +4.87614 q^{43} -4.17226 q^{47} +4.82032 q^{49} +2.08613 q^{51} +3.00000 q^{53} -4.17226 q^{57} +2.17226 q^{59} -14.3445 q^{61} -4.64806 q^{63} -9.64064 q^{67} -5.52420 q^{69} -8.90645 q^{71} -4.87614 q^{73} +4.64806 q^{77} +0.141948 q^{79} -11.2281 q^{81} -4.00000 q^{83} +12.8761 q^{87} +1.23550 q^{89} -8.08613 q^{91} +8.52420 q^{93} -8.34452 q^{97} -1.82774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q - q^3 - q^7 + 2 * q^9	\( 3 q - q^{3} - q^{7} + 2 q^{9} - 2 q^{11} + 5 q^{13} + 3 q^{17} - 6 q^{19} - 7 q^{21} - 10 q^{23} - q^{27} + 4 q^{29} + 5 q^{31} + 4 q^{33} + 4 q^{37} - 5 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{47} + 2 q^{49} - q^{51} + 9 q^{53} + 2 q^{57} - 8 q^{59} - 14 q^{61} - 16 q^{63} - 4 q^{67} - 7 q^{71} + 4 q^{73} + 16 q^{77} - 13 q^{79} - 13 q^{81} - 12 q^{83} + 20 q^{87} + 10 q^{89} - 17 q^{91} + 9 q^{93} + 4 q^{97} - 20 q^{99}+O(q^{100}) \) 3 * q - q^3 - q^7 + 2 * q^9 - 2 * q^11 + 5 * q^13 + 3 * q^17 - 6 * q^19 - 7 * q^21 - 10 * q^23 - q^27 + 4 * q^29 + 5 * q^31 + 4 * q^33 + 4 * q^37 - 5 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^47 + 2 * q^49 - q^51 + 9 * q^53 + 2 * q^57 - 8 * q^59 - 14 * q^61 - 16 * q^63 - 4 * q^67 - 7 * q^71 + 4 * q^73 + 16 * q^77 - 13 * q^79 - 13 * q^81 - 12 * q^83 + 20 * q^87 + 10 * q^89 - 17 * q^91 + 9 * q^93 + 4 * q^97 - 20 * q^99

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