LMFDB - Embedded newform 6800.2.a.bw.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.6224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 2x + 5 $ x^4 - 6x^2 - 2x + 5
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			1.1
Root			$2.44579$ 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)

$f(q)$	$=$	$q-1.44579 q^{3} +2.86537 q^{7} -0.909700 q^{9} +O(q^{10})$	$q-1.44579 q^{3} +2.86537 q^{7} -0.909700 q^{9} +3.84724 q^{11} -6.22085 q^{13} +1.00000 q^{17} -6.62231 q^{19} -4.14271 q^{21} +4.51796 q^{23} +5.65259 q^{27} -0.658562 q^{29} -3.49984 q^{31} -5.56229 q^{33} -3.34144 q^{37} +8.99403 q^{39} +2.04189 q^{41} -1.29303 q^{43} +6.22085 q^{47} +1.21033 q^{49} -1.44579 q^{51} +9.92750 q^{53} +9.57445 q^{57} +2.00000 q^{59} -7.61634 q^{61} -2.60662 q^{63} +0.257106 q^{67} -6.53201 q^{69} -1.18868 q^{71} -3.26330 q^{73} +11.0238 q^{77} +4.99756 q^{79} -5.44335 q^{81} +7.91534 q^{83} +0.952140 q^{87} +12.8013 q^{89} -17.8250 q^{91} +5.06002 q^{93} +4.08866 q^{97} -3.49984 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 10 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 10 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 10 q^{7} + 4 q^{9} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 4 q^{19} + 12 q^{21} + 4 q^{23} + 10 q^{27} - 4 q^{29} + 12 q^{31} - 2 q^{33} - 12 q^{37} + 22 q^{39} - 6 q^{41} + 18 q^{43} + 6 q^{47} + 8 q^{49} + 4 q^{51} - 8 q^{53} + 16 q^{57} + 8 q^{59} + 6 q^{61} + 16 q^{63} + 6 q^{67} + 4 q^{69} + 10 q^{71} - 2 q^{73} + 18 q^{77} + 12 q^{79} - 4 q^{81} - 14 q^{83} + 4 q^{87} + 24 q^{89} - 18 q^{91} + 18 q^{93} + 4 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 10 * q^7 + 4 * q^9 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 4 * q^19 + 12 * q^21 + 4 * q^23 + 10 * q^27 - 4 * q^29 + 12 * q^31 - 2 * q^33 - 12 * q^37 + 22 * q^39 - 6 * q^41 + 18 * q^43 + 6 * q^47 + 8 * q^49 + 4 * q^51 - 8 * q^53 + 16 * q^57 + 8 * q^59 + 6 * q^61 + 16 * q^63 + 6 * q^67 + 4 * q^69 + 10 * q^71 - 2 * q^73 + 18 * q^77 + 12 * q^79 - 4 * q^81 - 14 * q^83 + 4 * q^87 + 24 * q^89 - 18 * q^91 + 18 * q^93 + 4 * q^97 + 12 * 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
$3$	−1.44579	−0.834726	−0.417363	−	0.908740i	$-0.637046\pi$
$3$	−1.44579	−0.834726	−0.417363	+	0.908740i	$0.637046\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.86537	1.08301	0.541504	−	0.840698i	$-0.317855\pi$
$7$	2.86537	1.08301	0.541504	+	0.840698i	$0.317855\pi$
$8$	0	0
$9$	−0.909700	−0.303233
$10$	0	0
$11$	3.84724	1.15999	0.579994	−	0.814621i	$-0.303055\pi$
$11$	3.84724	1.15999	0.579994	+	0.814621i	$0.303055\pi$
$12$	0	0
$13$	−6.22085	−1.72535	−0.862677	−	0.505755i	$-0.831214\pi$
$13$	−6.22085	−1.72535	−0.862677	+	0.505755i	$0.831214\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.62231	−1.51926	−0.759631	−	0.650354i	$-0.774620\pi$
$19$	−6.62231	−1.51926	−0.759631	+	0.650354i	$0.774620\pi$
$20$	0	0
$21$	−4.14271	−0.904014
$22$	0	0
$23$	4.51796	0.942060	0.471030	−	0.882117i	$-0.343882\pi$
$23$	4.51796	0.942060	0.471030	+	0.882117i	$0.343882\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65259	1.08784
$28$	0	0
$29$	−0.658562	−0.122292	−0.0611459	−	0.998129i	$-0.519476\pi$
$29$	−0.658562	−0.122292	−0.0611459	+	0.998129i	$0.519476\pi$
$30$	0	0
$31$	−3.49984	−0.628589	−0.314295	−	0.949326i	$-0.601768\pi$
$31$	−3.49984	−0.628589	−0.314295	+	0.949326i	$0.601768\pi$
$32$	0	0
$33$	−5.56229	−0.968271
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.34144	−0.549329	−0.274665	−	0.961540i	$-0.588567\pi$
$37$	−3.34144	−0.549329	−0.274665	+	0.961540i	$0.588567\pi$
$38$	0	0
$39$	8.99403	1.44020
$40$	0	0
$41$	2.04189	0.318890	0.159445	−	0.987207i	$-0.449030\pi$
$41$	2.04189	0.318890	0.159445	+	0.987207i	$0.449030\pi$
$42$	0	0
$43$	−1.29303	−0.197185	−0.0985926	−	0.995128i	$-0.531434\pi$
$43$	−1.29303	−0.197185	−0.0985926	+	0.995128i	$0.531434\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.22085	0.907405	0.453702	−	0.891153i	$-0.350103\pi$
$47$	6.22085	0.907405	0.453702	+	0.891153i	$0.350103\pi$
$48$	0	0
$49$	1.21033	0.172905
$50$	0	0
$51$	−1.44579	−0.202451
$52$	0	0
$53$	9.92750	1.36365	0.681823	−	0.731517i	$-0.261187\pi$
$53$	9.92750	1.36365	0.681823	+	0.731517i	$0.261187\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.57445	1.26817
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−7.61634	−0.975173	−0.487586	−	0.873075i	$-0.662123\pi$
$61$	−7.61634	−0.975173	−0.487586	+	0.873075i	$0.662123\pi$
$62$	0	0
$63$	−2.60662	−0.328404
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.257106	0.0314106	0.0157053	−	0.999877i	$-0.495001\pi$
$67$	0.257106	0.0314106	0.0157053	+	0.999877i	$0.495001\pi$
$68$	0	0
$69$	−6.53201	−0.786362
$70$	0	0
$71$	−1.18868	−0.141070	−0.0705352	−	0.997509i	$-0.522471\pi$
$71$	−1.18868	−0.141070	−0.0705352	+	0.997509i	$0.522471\pi$
$72$	0	0
$73$	−3.26330	−0.381940	−0.190970	−	0.981596i	$-0.561163\pi$
$73$	−3.26330	−0.381940	−0.190970	+	0.981596i	$0.561163\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.0238	1.25627
$78$	0	0
$79$	4.99756	0.562269	0.281135	−	0.959668i	$-0.409289\pi$
$79$	4.99756	0.562269	0.281135	+	0.959668i	$0.409289\pi$
$80$	0	0
$81$	−5.44335	−0.604816
$82$	0	0
$83$	7.91534	0.868821	0.434411	−	0.900715i	$-0.356957\pi$
$83$	7.91534	0.868821	0.434411	+	0.900715i	$0.356957\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.952140	0.102080
$88$	0	0
$89$	12.8013	1.35693	0.678466	−	0.734632i	$-0.262645\pi$
$89$	12.8013	1.35693	0.678466	+	0.734632i	$0.262645\pi$
$90$	0	0
$91$	−17.8250	−1.86857
$92$	0	0
$93$	5.06002	0.524699
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.08866	0.415141	0.207570	−	0.978220i	$-0.433444\pi$
$97$	4.08866	0.415141	0.207570	+	0.978220i	$0.433444\pi$
$98$	0	0
$99$	−3.49984	−0.351747
$100$	0	0
$101$	12.2871	1.22261	0.611304	−	0.791396i	$-0.290645\pi$
$101$	12.2871	1.22261	0.611304	+	0.791396i	$0.290645\pi$
$102$	0	0
$103$	13.5623	1.33633	0.668166	−	0.744012i	$-0.267079\pi$
$103$	13.5623	1.33633	0.668166	+	0.744012i	$0.267079\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.63800	−0.641719	−0.320860	−	0.947127i	$-0.603972\pi$
$107$	−6.63800	−0.641719	−0.320860	+	0.947127i	$0.603972\pi$
$108$	0	0
$109$	−2.58042	−0.247159	−0.123580	−	0.992335i	$-0.539437\pi$
$109$	−2.58042	−0.247159	−0.123580	+	0.992335i	$0.539437\pi$
$110$	0	0
$111$	4.83101	0.458539
$112$	0	0
$113$	8.63283	0.812108	0.406054	−	0.913849i	$-0.366904\pi$
$113$	8.63283	0.812108	0.406054	+	0.913849i	$0.366904\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.65911	0.523185
$118$	0	0
$119$	2.86537	0.262668
$120$	0	0
$121$	3.80127	0.345570
$122$	0	0
$123$	−2.95214	−0.266186
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.24550	0.287991	0.143996	−	0.989578i	$-0.454005\pi$
$127$	3.24550	0.287991	0.143996	+	0.989578i	$0.454005\pi$
$128$	0	0
$129$	1.86945	0.164595
$130$	0	0
$131$	3.44742	0.301203	0.150601	−	0.988595i	$-0.451879\pi$
$131$	3.44742	0.301203	0.150601	+	0.988595i	$0.451879\pi$
$132$	0	0
$133$	−18.9754	−1.64537
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.6869	−1.25478	−0.627392	−	0.778703i	$-0.715878\pi$
$137$	−14.6869	−1.25478	−0.627392	+	0.778703i	$0.715878\pi$
$138$	0	0
$139$	−13.9883	−1.18647	−0.593237	−	0.805028i	$-0.702150\pi$
$139$	−13.9883	−1.18647	−0.593237	+	0.805028i	$0.702150\pi$
$140$	0	0
$141$	−8.99403	−0.757434
$142$	0	0
$143$	−23.9331	−2.00139
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.74989	−0.144328
$148$	0	0
$149$	−17.8388	−1.46141	−0.730707	−	0.682691i	$-0.760809\pi$
$149$	−17.8388	−1.46141	−0.730707	+	0.682691i	$0.760809\pi$
$150$	0	0
$151$	2.80291	0.228098	0.114049	−	0.993475i	$-0.463618\pi$
$151$	2.80291	0.228098	0.114049	+	0.993475i	$0.463618\pi$
$152$	0	0
$153$	−0.909700	−0.0735449
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.60582	−0.287776	−0.143888	−	0.989594i	$-0.545960\pi$
$157$	−3.60582	−0.287776	−0.143888	+	0.989594i	$0.545960\pi$
$158$	0	0
$159$	−14.3530	−1.13827
$160$	0	0
$161$	12.9456	1.02026
$162$	0	0
$163$	13.7256	1.07507	0.537535	−	0.843242i	$-0.319356\pi$
$163$	13.7256	1.07507	0.537535	+	0.843242i	$0.319356\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.32684	0.489586	0.244793	−	0.969575i	$-0.421280\pi$
$167$	6.32684	0.489586	0.244793	+	0.969575i	$0.421280\pi$
$168$	0	0
$169$	25.6990	1.97685
$170$	0	0
$171$	6.02431	0.460691
$172$	0	0
$173$	1.76699	0.134342	0.0671708	−	0.997741i	$-0.478603\pi$
$173$	1.76699	0.134342	0.0671708	+	0.997741i	$0.478603\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.89157	−0.217344
$178$	0	0
$179$	−1.10843	−0.0828476	−0.0414238	−	0.999142i	$-0.513189\pi$
$179$	−1.10843	−0.0828476	−0.0414238	+	0.999142i	$0.513189\pi$
$180$	0	0
$181$	−10.8310	−0.805062	−0.402531	−	0.915406i	$-0.631870\pi$
$181$	−10.8310	−0.805062	−0.402531	+	0.915406i	$0.631870\pi$
$182$	0	0
$183$	11.0116	0.814002
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.84724	0.281338
$188$	0	0
$189$	16.1968	1.17814
$190$	0	0
$191$	−3.19221	−0.230980	−0.115490	−	0.993309i	$-0.536844\pi$
$191$	−3.19221	−0.230980	−0.115490	+	0.993309i	$0.536844\pi$
$192$	0	0
$193$	−7.11407	−0.512082	−0.256041	−	0.966666i	$-0.582418\pi$
$193$	−7.11407	−0.512082	−0.256041	+	0.966666i	$0.582418\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.1721	1.65095	0.825473	−	0.564442i	$-0.190909\pi$
$197$	23.1721	1.65095	0.825473	+	0.564442i	$0.190909\pi$
$198$	0	0
$199$	24.7131	1.75186	0.875932	−	0.482434i	$-0.160247\pi$
$199$	24.7131	1.75186	0.875932	+	0.482434i	$0.160247\pi$
$200$	0	0
$201$	−0.371721	−0.0262192
$202$	0	0
$203$	−1.88702	−0.132443
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.10999	−0.285664
$208$	0	0
$209$	−25.4776	−1.76232
$210$	0	0
$211$	4.27138	0.294054	0.147027	−	0.989133i	$-0.453030\pi$
$211$	4.27138	0.294054	0.147027	+	0.989133i	$0.453030\pi$
$212$	0	0
$213$	1.71858	0.117755
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0283	−0.680767
$218$	0	0
$219$	4.71803	0.318815
$220$	0	0
$221$	−6.22085	−0.418460
$222$	0	0
$223$	3.65423	0.244705	0.122353	−	0.992487i	$-0.460956\pi$
$223$	3.65423	0.244705	0.122353	+	0.992487i	$0.460956\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.0016	1.26118	0.630589	−	0.776117i	$-0.282813\pi$
$227$	19.0016	1.26118	0.630589	+	0.776117i	$0.282813\pi$
$228$	0	0
$229$	−20.8851	−1.38012	−0.690062	−	0.723751i	$-0.742417\pi$
$229$	−20.8851	−1.38012	−0.690062	+	0.723751i	$0.742417\pi$
$230$	0	0
$231$	−15.9380	−1.04864
$232$	0	0
$233$	0.682876	0.0447367	0.0223684	−	0.999750i	$-0.492879\pi$
$233$	0.682876	0.0447367	0.0223684	+	0.999750i	$0.492879\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.22541	−0.469341
$238$	0	0
$239$	−28.9915	−1.87531	−0.937653	−	0.347574i	$-0.887006\pi$
$239$	−28.9915	−1.87531	−0.937653	+	0.347574i	$0.887006\pi$
$240$	0	0
$241$	−18.0943	−1.16556	−0.582778	−	0.812631i	$-0.698034\pi$
$241$	−18.0943	−1.16556	−0.582778	+	0.812631i	$0.698034\pi$
$242$	0	0
$243$	−9.08786	−0.582986
$244$	0	0
$245$	0	0
$246$	0	0
$247$	41.1964	2.62127
$248$	0	0
$249$	−11.4439	−0.725227
$250$	0	0
$251$	7.21325	0.455296	0.227648	−	0.973743i	$-0.426896\pi$
$251$	7.21325	0.455296	0.227648	+	0.973743i	$0.426896\pi$
$252$	0	0
$253$	17.3817	1.09278
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.87942	0.304370	0.152185	−	0.988352i	$-0.451369\pi$
$257$	4.87942	0.304370	0.152185	+	0.988352i	$0.451369\pi$
$258$	0	0
$259$	−9.57445	−0.594927
$260$	0	0
$261$	0.599094	0.0370830
$262$	0	0
$263$	30.0234	1.85132	0.925662	−	0.378351i	$-0.123509\pi$
$263$	30.0234	1.85132	0.925662	+	0.378351i	$0.123509\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.5079	−1.13267
$268$	0	0
$269$	−29.6568	−1.80821	−0.904104	−	0.427312i	$-0.859460\pi$
$269$	−29.6568	−1.80821	−0.904104	+	0.427312i	$0.859460\pi$
$270$	0	0
$271$	−3.91622	−0.237893	−0.118947	−	0.992901i	$-0.537952\pi$
$271$	−3.91622	−0.237893	−0.118947	+	0.992901i	$0.537952\pi$
$272$	0	0
$273$	25.7712	1.55974
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.455504	0.0273686	0.0136843	−	0.999906i	$-0.495644\pi$
$277$	0.455504	0.0273686	0.0136843	+	0.999906i	$0.495644\pi$
$278$	0	0
$279$	3.18380	0.190609
$280$	0	0
$281$	16.4941	0.983957	0.491978	−	0.870607i	$-0.336274\pi$
$281$	16.4941	0.983957	0.491978	+	0.870607i	$0.336274\pi$
$282$	0	0
$283$	8.79775	0.522972	0.261486	−	0.965207i	$-0.415788\pi$
$283$	8.79775	0.522972	0.261486	+	0.965207i	$0.415788\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.85077	0.345360
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.91134	−0.346529
$292$	0	0
$293$	22.5023	1.31460	0.657299	−	0.753630i	$-0.271699\pi$
$293$	22.5023	1.31460	0.657299	+	0.753630i	$0.271699\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	21.7469	1.26188
$298$	0	0
$299$	−28.1056	−1.62539
$300$	0	0
$301$	−3.70501	−0.213553
$302$	0	0
$303$	−17.7645	−1.02054
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.7034	0.667947	0.333973	−	0.942583i	$-0.391610\pi$
$307$	11.7034	0.667947	0.333973	+	0.942583i	$0.391610\pi$
$308$	0	0
$309$	−19.6082	−1.11547
$310$	0	0
$311$	21.0605	1.19423	0.597115	−	0.802155i	$-0.296313\pi$
$311$	21.0605	1.19423	0.597115	+	0.802155i	$0.296313\pi$
$312$	0	0
$313$	−22.2911	−1.25997	−0.629983	−	0.776609i	$-0.716938\pi$
$313$	−22.2911	−1.25997	−0.629983	+	0.776609i	$0.716938\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.8613	1.56485	0.782423	−	0.622747i	$-0.213984\pi$
$317$	27.8613	1.56485	0.782423	+	0.622747i	$0.213984\pi$
$318$	0	0
$319$	−2.53365	−0.141857
$320$	0	0
$321$	9.59713	0.535659
$322$	0	0
$323$	−6.62231	−0.368475
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.73074	0.206310
$328$	0	0
$329$	17.8250	0.982726
$330$	0	0
$331$	13.0067	0.714914	0.357457	−	0.933930i	$-0.383644\pi$
$331$	13.0067	0.714914	0.357457	+	0.933930i	$0.383644\pi$
$332$	0	0
$333$	3.03971	0.166575
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.1126	0.823238	0.411619	−	0.911356i	$-0.364963\pi$
$337$	15.1126	0.823238	0.411619	+	0.911356i	$0.364963\pi$
$338$	0	0
$339$	−12.4812	−0.677888
$340$	0	0
$341$	−13.4647	−0.729155
$342$	0	0
$343$	−16.5895	−0.895750
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.58828	−0.192629	−0.0963144	−	0.995351i	$-0.530705\pi$
$347$	−3.58828	−0.192629	−0.0963144	+	0.995351i	$0.530705\pi$
$348$	0	0
$349$	31.2851	1.67465	0.837326	−	0.546703i	$-0.184117\pi$
$349$	31.2851	1.67465	0.837326	+	0.546703i	$0.184117\pi$
$350$	0	0
$351$	−35.1640	−1.87691
$352$	0	0
$353$	24.4417	1.30090	0.650450	−	0.759549i	$-0.274580\pi$
$353$	24.4417	1.30090	0.650450	+	0.759549i	$0.274580\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.14271	−0.219256
$358$	0	0
$359$	0.136195	0.00718808	0.00359404	−	0.999994i	$-0.498856\pi$
$359$	0.136195	0.00718808	0.00359404	+	0.999994i	$0.498856\pi$
$360$	0	0
$361$	24.8550	1.30816
$362$	0	0
$363$	−5.49583	−0.288456
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.9896	−0.886851	−0.443426	−	0.896311i	$-0.646237\pi$
$367$	−16.9896	−0.886851	−0.443426	+	0.896311i	$0.646237\pi$
$368$	0	0
$369$	−1.85751	−0.0966980
$370$	0	0
$371$	28.4459	1.47684
$372$	0	0
$373$	18.3533	0.950296	0.475148	−	0.879906i	$-0.342394\pi$
$373$	18.3533	0.950296	0.475148	+	0.879906i	$0.342394\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.09682	0.210997
$378$	0	0
$379$	35.1943	1.80781	0.903905	−	0.427732i	$-0.140687\pi$
$379$	35.1943	1.80781	0.903905	+	0.427732i	$0.140687\pi$
$380$	0	0
$381$	−4.69230	−0.240394
$382$	0	0
$383$	10.8237	0.553067	0.276533	−	0.961004i	$-0.410814\pi$
$383$	10.8237	0.553067	0.276533	+	0.961004i	$0.410814\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.17627	0.0597931
$388$	0	0
$389$	0.174083	0.00882636	0.00441318	−	0.999990i	$-0.498595\pi$
$389$	0.174083	0.00882636	0.00441318	+	0.999990i	$0.498595\pi$
$390$	0	0
$391$	4.51796	0.228483
$392$	0	0
$393$	−4.98424	−0.251422
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.75287	−0.138162	−0.0690812	−	0.997611i	$-0.522007\pi$
$397$	−2.75287	−0.138162	−0.0690812	+	0.997611i	$0.522007\pi$
$398$	0	0
$399$	27.4343	1.37343
$400$	0	0
$401$	33.9912	1.69744	0.848719	−	0.528843i	$-0.177374\pi$
$401$	33.9912	1.69744	0.848719	+	0.528843i	$0.177374\pi$
$402$	0	0
$403$	21.7720	1.08454
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.8553	−0.637215
$408$	0	0
$409$	−4.68288	−0.231553	−0.115777	−	0.993275i	$-0.536936\pi$
$409$	−4.68288	−0.231553	−0.115777	+	0.993275i	$0.536936\pi$
$410$	0	0
$411$	21.2341	1.04740
$412$	0	0
$413$	5.73074	0.281991
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.2241	0.990380
$418$	0	0
$419$	29.0991	1.42158	0.710792	−	0.703402i	$-0.248337\pi$
$419$	29.0991	1.42158	0.710792	+	0.703402i	$0.248337\pi$
$420$	0	0
$421$	7.19057	0.350447	0.175224	−	0.984529i	$-0.443935\pi$
$421$	7.19057	0.350447	0.175224	+	0.984529i	$0.443935\pi$
$422$	0	0
$423$	−5.65911	−0.275155
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−21.8236	−1.05612
$428$	0	0
$429$	34.6022	1.67061
$430$	0	0
$431$	−7.63181	−0.367611	−0.183806	−	0.982963i	$-0.558842\pi$
$431$	−7.63181	−0.367611	−0.183806	+	0.982963i	$0.558842\pi$
$432$	0	0
$433$	32.2894	1.55173	0.775865	−	0.630898i	$-0.217314\pi$
$433$	32.2894	1.55173	0.775865	+	0.630898i	$0.217314\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.9193	−1.43124
$438$	0	0
$439$	13.3974	0.639422	0.319711	−	0.947515i	$-0.396414\pi$
$439$	13.3974	0.639422	0.319711	+	0.947515i	$0.396414\pi$
$440$	0	0
$441$	−1.10104	−0.0524305
$442$	0	0
$443$	6.89070	0.327387	0.163693	−	0.986511i	$-0.447659\pi$
$443$	6.89070	0.327387	0.163693	+	0.986511i	$0.447659\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	25.7912	1.21988
$448$	0	0
$449$	21.1778	0.999440	0.499720	−	0.866187i	$-0.333436\pi$
$449$	21.1778	0.999440	0.499720	+	0.866187i	$0.333436\pi$
$450$	0	0
$451$	7.85565	0.369908
$452$	0	0
$453$	−4.05241	−0.190399
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.2321	1.41420	0.707100	−	0.707114i	$-0.250003\pi$
$457$	30.2321	1.41420	0.707100	+	0.707114i	$0.250003\pi$
$458$	0	0
$459$	5.65259	0.263840
$460$	0	0
$461$	3.74267	0.174314	0.0871568	−	0.996195i	$-0.472222\pi$
$461$	3.74267	0.174314	0.0871568	+	0.996195i	$0.472222\pi$
$462$	0	0
$463$	2.53452	0.117789	0.0588947	−	0.998264i	$-0.481242\pi$
$463$	2.53452	0.117789	0.0588947	+	0.998264i	$0.481242\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.1170	−1.39365	−0.696824	−	0.717242i	$-0.745404\pi$
$467$	−30.1170	−1.39365	−0.696824	+	0.717242i	$0.745404\pi$
$468$	0	0
$469$	0.736705	0.0340179
$470$	0	0
$471$	5.21325	0.240214
$472$	0	0
$473$	−4.97460	−0.228732
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.03104	−0.413503
$478$	0	0
$479$	−14.4995	−0.662499	−0.331250	−	0.943543i	$-0.607470\pi$
$479$	−14.4995	−0.662499	−0.331250	+	0.943543i	$0.607470\pi$
$480$	0	0
$481$	20.7866	0.947787
$482$	0	0
$483$	−18.7166	−0.851635
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.9424	1.08493	0.542467	−	0.840077i	$-0.317490\pi$
$487$	23.9424	1.08493	0.542467	+	0.840077i	$0.317490\pi$
$488$	0	0
$489$	−19.8443	−0.897388
$490$	0	0
$491$	21.1295	0.953559	0.476780	−	0.879023i	$-0.341804\pi$
$491$	21.1295	0.953559	0.476780	+	0.879023i	$0.341804\pi$
$492$	0	0
$493$	−0.658562	−0.0296601
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.40601	−0.152780
$498$	0	0
$499$	−4.01027	−0.179524	−0.0897621	−	0.995963i	$-0.528611\pi$
$499$	−4.01027	−0.179524	−0.0897621	+	0.995963i	$0.528611\pi$
$500$	0	0
$501$	−9.14726	−0.408670
$502$	0	0
$503$	1.44015	0.0642130	0.0321065	−	0.999484i	$-0.489778\pi$
$503$	1.44015	0.0642130	0.0321065	+	0.999484i	$0.489778\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−37.1553	−1.65013
$508$	0	0
$509$	31.1478	1.38060	0.690301	−	0.723522i	$-0.257478\pi$
$509$	31.1478	1.38060	0.690301	+	0.723522i	$0.257478\pi$
$510$	0	0
$511$	−9.35054	−0.413644
$512$	0	0
$513$	−37.4332	−1.65272
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.9331	1.05258
$518$	0	0
$519$	−2.55469	−0.112138
$520$	0	0
$521$	−39.1197	−1.71387	−0.856933	−	0.515428i	$-0.827633\pi$
$521$	−39.1197	−1.71387	−0.856933	+	0.515428i	$0.827633\pi$
$522$	0	0
$523$	−40.4813	−1.77012	−0.885062	−	0.465473i	$-0.845884\pi$
$523$	−40.4813	−1.77012	−0.885062	+	0.465473i	$0.845884\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.49984	−0.152455
$528$	0	0
$529$	−2.58802	−0.112523
$530$	0	0
$531$	−1.81940	−0.0789552
$532$	0	0
$533$	−12.7023	−0.550198
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.60255	0.0691550
$538$	0	0
$539$	4.65645	0.200568
$540$	0	0
$541$	−18.5288	−0.796614	−0.398307	−	0.917252i	$-0.630402\pi$
$541$	−18.5288	−0.796614	−0.398307	+	0.917252i	$0.630402\pi$
$542$	0	0
$543$	15.6593	0.672006
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.2959	1.85120	0.925600	−	0.378504i	$-0.123561\pi$
$547$	43.2959	1.85120	0.925600	+	0.378504i	$0.123561\pi$
$548$	0	0
$549$	6.92858	0.295705
$550$	0	0
$551$	4.36120	0.185793
$552$	0	0
$553$	14.3198	0.608942
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.3698	1.07495	0.537476	−	0.843279i	$-0.319378\pi$
$557$	25.3698	1.07495	0.537476	+	0.843279i	$0.319378\pi$
$558$	0	0
$559$	8.04375	0.340214
$560$	0	0
$561$	−5.56229	−0.234840
$562$	0	0
$563$	−35.6622	−1.50298	−0.751492	−	0.659742i	$-0.770665\pi$
$563$	−35.6622	−1.50298	−0.751492	+	0.659742i	$0.770665\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−15.5972	−0.655021
$568$	0	0
$569$	27.7944	1.16520	0.582602	−	0.812758i	$-0.302035\pi$
$569$	27.7944	1.16520	0.582602	+	0.812758i	$0.302035\pi$
$570$	0	0
$571$	15.8578	0.663627	0.331813	−	0.943345i	$-0.392340\pi$
$571$	15.8578	0.663627	0.331813	+	0.943345i	$0.392340\pi$
$572$	0	0
$573$	4.61525	0.192805
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.2762	−1.05226	−0.526131	−	0.850403i	$-0.676358\pi$
$577$	−25.2762	−1.05226	−0.526131	+	0.850403i	$0.676358\pi$
$578$	0	0
$579$	10.2854	0.427448
$580$	0	0
$581$	22.6804	0.940940
$582$	0	0
$583$	38.1935	1.58181
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.7829	0.651431	0.325716	−	0.945468i	$-0.394395\pi$
$587$	15.7829	0.651431	0.325716	+	0.945468i	$0.394395\pi$
$588$	0	0
$589$	23.1770	0.954992
$590$	0	0
$591$	−33.5019	−1.37809
$592$	0	0
$593$	−19.3560	−0.794855	−0.397428	−	0.917634i	$-0.630097\pi$
$593$	−19.3560	−0.794855	−0.397428	+	0.917634i	$0.630097\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−35.7299	−1.46233
$598$	0	0
$599$	15.8356	0.647023	0.323512	−	0.946224i	$-0.395136\pi$
$599$	15.8356	0.647023	0.323512	+	0.946224i	$0.395136\pi$
$600$	0	0
$601$	9.30443	0.379536	0.189768	−	0.981829i	$-0.439226\pi$
$601$	9.30443	0.379536	0.189768	+	0.981829i	$0.439226\pi$
$602$	0	0
$603$	−0.233890	−0.00952473
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.6661	1.08234	0.541172	−	0.840912i	$-0.317981\pi$
$607$	26.6661	1.08234	0.541172	+	0.840912i	$0.317981\pi$
$608$	0	0
$609$	2.72823	0.110554
$610$	0	0
$611$	−38.6990	−1.56560
$612$	0	0
$613$	−26.0929	−1.05388	−0.526941	−	0.849902i	$-0.676661\pi$
$613$	−26.0929	−1.05388	−0.526941	+	0.849902i	$0.676661\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.4622	−0.944554	−0.472277	−	0.881450i	$-0.656568\pi$
$617$	−23.4622	−0.944554	−0.472277	+	0.881450i	$0.656568\pi$
$618$	0	0
$619$	−3.50613	−0.140923	−0.0704617	−	0.997514i	$-0.522447\pi$
$619$	−3.50613	−0.140923	−0.0704617	+	0.997514i	$0.522447\pi$
$620$	0	0
$621$	25.5382	1.02481
$622$	0	0
$623$	36.6804	1.46957
$624$	0	0
$625$	0	0
$626$	0	0
$627$	36.8352	1.47106
$628$	0	0
$629$	−3.34144	−0.133232
$630$	0	0
$631$	7.57823	0.301685	0.150842	−	0.988558i	$-0.451801\pi$
$631$	7.57823	0.301685	0.150842	+	0.988558i	$0.451801\pi$
$632$	0	0
$633$	−6.17550	−0.245454
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.52932	−0.298322
$638$	0	0
$639$	1.08134	0.0427772
$640$	0	0
$641$	4.21055	0.166307	0.0831535	−	0.996537i	$-0.473501\pi$
$641$	4.21055	0.166307	0.0831535	+	0.996537i	$0.473501\pi$
$642$	0	0
$643$	−39.0726	−1.54087	−0.770437	−	0.637516i	$-0.779962\pi$
$643$	−39.0726	−1.54087	−0.770437	+	0.637516i	$0.779962\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.0481	−1.06337	−0.531685	−	0.846942i	$-0.678441\pi$
$647$	−27.0481	−1.06337	−0.531685	+	0.846942i	$0.678441\pi$
$648$	0	0
$649$	7.69448	0.302035
$650$	0	0
$651$	14.4988	0.568253
$652$	0	0
$653$	−35.6550	−1.39529	−0.697643	−	0.716445i	$-0.745768\pi$
$653$	−35.6550	−1.39529	−0.697643	+	0.716445i	$0.745768\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.96862	0.115817
$658$	0	0
$659$	−0.586716	−0.0228552	−0.0114276	−	0.999935i	$-0.503638\pi$
$659$	−0.586716	−0.0228552	−0.0114276	+	0.999935i	$0.503638\pi$
$660$	0	0
$661$	18.8785	0.734290	0.367145	−	0.930164i	$-0.380335\pi$
$661$	18.8785	0.734290	0.367145	+	0.930164i	$0.380335\pi$
$662$	0	0
$663$	8.99403	0.349299
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.97536	−0.115206
$668$	0	0
$669$	−5.28324	−0.204262
$670$	0	0
$671$	−29.3019	−1.13119
$672$	0	0
$673$	44.9680	1.73339	0.866694	−	0.498841i	$-0.166241\pi$
$673$	44.9680	1.73339	0.866694	+	0.498841i	$0.166241\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.3876	−1.28319	−0.641594	−	0.767044i	$-0.721727\pi$
$677$	−33.3876	−1.28319	−0.641594	+	0.767044i	$0.721727\pi$
$678$	0	0
$679$	11.7155	0.449601
$680$	0	0
$681$	−27.4722	−1.05274
$682$	0	0
$683$	−22.0237	−0.842713	−0.421357	−	0.906895i	$-0.638446\pi$
$683$	−22.0237	−0.842713	−0.421357	+	0.906895i	$0.638446\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.1953	1.15202
$688$	0	0
$689$	−61.7575	−2.35277
$690$	0	0
$691$	−4.51396	−0.171719	−0.0858595	−	0.996307i	$-0.527364\pi$
$691$	−4.51396	−0.171719	−0.0858595	+	0.996307i	$0.527364\pi$
$692$	0	0
$693$	−10.0283	−0.380944
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.04189	0.0773422
$698$	0	0
$699$	−0.987294	−0.0373429
$700$	0	0
$701$	−29.3948	−1.11023	−0.555114	−	0.831774i	$-0.687325\pi$
$701$	−29.3948	−1.11023	−0.555114	+	0.831774i	$0.687325\pi$
$702$	0	0
$703$	22.1280	0.834575
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.2070	1.32409
$708$	0	0
$709$	−1.86205	−0.0699308	−0.0349654	−	0.999389i	$-0.511132\pi$
$709$	−1.86205	−0.0699308	−0.0349654	+	0.999389i	$0.511132\pi$
$710$	0	0
$711$	−4.54628	−0.170499
$712$	0	0
$713$	−15.8121	−0.592169
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.9156	1.56537
$718$	0	0
$719$	15.8028	0.589346	0.294673	−	0.955598i	$-0.404789\pi$
$719$	15.8028	0.589346	0.294673	+	0.955598i	$0.404789\pi$
$720$	0	0
$721$	38.8610	1.44726
$722$	0	0
$723$	26.1605	0.972920
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.0930	0.782296	0.391148	−	0.920328i	$-0.372078\pi$
$727$	21.0930	0.782296	0.391148	+	0.920328i	$0.372078\pi$
$728$	0	0
$729$	29.4692	1.09145
$730$	0	0
$731$	−1.29303	−0.0478244
$732$	0	0
$733$	−40.8012	−1.50703	−0.753513	−	0.657434i	$-0.771642\pi$
$733$	−40.8012	−1.50703	−0.753513	+	0.657434i	$0.771642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.989151	0.0364358
$738$	0	0
$739$	−13.8161	−0.508234	−0.254117	−	0.967173i	$-0.581785\pi$
$739$	−13.8161	−0.508234	−0.254117	+	0.967173i	$0.581785\pi$
$740$	0	0
$741$	−59.5613	−2.18804
$742$	0	0
$743$	3.01696	0.110682	0.0553408	−	0.998468i	$-0.482375\pi$
$743$	3.01696	0.110682	0.0553408	+	0.998468i	$0.482375\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.20058	−0.263456
$748$	0	0
$749$	−19.0203	−0.694987
$750$	0	0
$751$	44.6641	1.62982	0.814909	−	0.579589i	$-0.196787\pi$
$751$	44.6641	1.62982	0.814909	+	0.579589i	$0.196787\pi$
$752$	0	0
$753$	−10.4288	−0.380047
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.3652	0.449421	0.224710	−	0.974426i	$-0.427856\pi$
$757$	12.3652	0.449421	0.224710	+	0.974426i	$0.427856\pi$
$758$	0	0
$759$	−25.1302	−0.912169
$760$	0	0
$761$	16.7175	0.606009	0.303004	−	0.952989i	$-0.402010\pi$
$761$	16.7175	0.606009	0.303004	+	0.952989i	$0.402010\pi$
$762$	0	0
$763$	−7.39385	−0.267675
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.4417	−0.449244
$768$	0	0
$769$	13.4025	0.483308	0.241654	−	0.970362i	$-0.422310\pi$
$769$	13.4025	0.483308	0.241654	+	0.970362i	$0.422310\pi$
$770$	0	0
$771$	−7.05460	−0.254065
$772$	0	0
$773$	25.8358	0.929248	0.464624	−	0.885508i	$-0.346189\pi$
$773$	25.8358	0.929248	0.464624	+	0.885508i	$0.346189\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	13.8426	0.496601
$778$	0	0
$779$	−13.5220	−0.484477
$780$	0	0
$781$	−4.57314	−0.163640
$782$	0	0
$783$	−3.72258	−0.133034
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−45.2989	−1.61473	−0.807366	−	0.590051i	$-0.799108\pi$
$787$	−45.2989	−1.61473	−0.807366	+	0.590051i	$0.799108\pi$
$788$	0	0
$789$	−43.4075	−1.54535
$790$	0	0
$791$	24.7362	0.879519
$792$	0	0
$793$	47.3802	1.68252
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.3607	1.04001	0.520005	−	0.854163i	$-0.325930\pi$
$797$	29.3607	1.04001	0.520005	+	0.854163i	$0.325930\pi$
$798$	0	0
$799$	6.22085	0.220078
$800$	0	0
$801$	−11.6453	−0.411467
$802$	0	0
$803$	−12.5547	−0.443045
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.8774	1.50936
$808$	0	0
$809$	5.04830	0.177489	0.0887444	−	0.996054i	$-0.471715\pi$
$809$	5.04830	0.177489	0.0887444	+	0.996054i	$0.471715\pi$
$810$	0	0
$811$	−54.3374	−1.90805	−0.954023	−	0.299734i	$-0.903102\pi$
$811$	−54.3374	−1.90805	−0.954023	+	0.299734i	$0.903102\pi$
$812$	0	0
$813$	5.66202	0.198576
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.56284	0.299576
$818$	0	0
$819$	16.2154	0.566613
$820$	0	0
$821$	3.05948	0.106777	0.0533883	−	0.998574i	$-0.482998\pi$
$821$	3.05948	0.106777	0.0533883	+	0.998574i	$0.482998\pi$
$822$	0	0
$823$	−6.87523	−0.239656	−0.119828	−	0.992795i	$-0.538234\pi$
$823$	−6.87523	−0.239656	−0.119828	+	0.992795i	$0.538234\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.2438	−0.843040	−0.421520	−	0.906819i	$-0.638503\pi$
$827$	−24.2438	−0.843040	−0.421520	+	0.906819i	$0.638503\pi$
$828$	0	0
$829$	15.7734	0.547832	0.273916	−	0.961754i	$-0.411681\pi$
$829$	15.7734	0.547832	0.273916	+	0.961754i	$0.411681\pi$
$830$	0	0
$831$	−0.658562	−0.0228453
$832$	0	0
$833$	1.21033	0.0419356
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−19.7831	−0.683806
$838$	0	0
$839$	−9.98832	−0.344835	−0.172418	−	0.985024i	$-0.555158\pi$
$839$	−9.98832	−0.344835	−0.172418	+	0.985024i	$0.555158\pi$
$840$	0	0
$841$	−28.5663	−0.985045
$842$	0	0
$843$	−23.8470	−0.821334
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.8920	0.374255
$848$	0	0
$849$	−12.7197	−0.436538
$850$	0	0
$851$	−15.0965	−0.517501
$852$	0	0
$853$	31.1852	1.06776	0.533880	−	0.845560i	$-0.320734\pi$
$853$	31.1852	1.06776	0.533880	+	0.845560i	$0.320734\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.4712	−0.904240	−0.452120	−	0.891957i	$-0.649332\pi$
$857$	−26.4712	−0.904240	−0.452120	+	0.891957i	$0.649332\pi$
$858$	0	0
$859$	−22.7223	−0.775273	−0.387637	−	0.921812i	$-0.626708\pi$
$859$	−22.7223	−0.775273	−0.387637	+	0.921812i	$0.626708\pi$
$860$	0	0
$861$	−8.45897	−0.288281
$862$	0	0
$863$	35.3962	1.20490	0.602451	−	0.798156i	$-0.294191\pi$
$863$	35.3962	1.20490	0.602451	+	0.798156i	$0.294191\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.44579	−0.0491015
$868$	0	0
$869$	19.2268	0.652225
$870$	0	0
$871$	−1.59942	−0.0541944
$872$	0	0
$873$	−3.71946	−0.125885
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.09791	−0.0370736	−0.0185368	−	0.999828i	$-0.505901\pi$
$877$	−1.09791	−0.0370736	−0.0185368	+	0.999828i	$0.505901\pi$
$878$	0	0
$879$	−32.5335	−1.09733
$880$	0	0
$881$	−16.5399	−0.557245	−0.278622	−	0.960401i	$-0.589878\pi$
$881$	−16.5399	−0.557245	−0.278622	+	0.960401i	$0.589878\pi$
$882$	0	0
$883$	22.1505	0.745425	0.372712	−	0.927947i	$-0.378428\pi$
$883$	22.1505	0.745425	0.372712	+	0.927947i	$0.378428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.5836	−0.657553	−0.328776	−	0.944408i	$-0.606636\pi$
$887$	−19.5836	−0.657553	−0.328776	+	0.944408i	$0.606636\pi$
$888$	0	0
$889$	9.29955	0.311897
$890$	0	0
$891$	−20.9419	−0.701579
$892$	0	0
$893$	−41.1964	−1.37859
$894$	0	0
$895$	0	0
$896$	0	0
$897$	40.6347	1.35675
$898$	0	0
$899$	2.30486	0.0768713
$900$	0	0
$901$	9.92750	0.330733
$902$	0	0
$903$	5.35665	0.178258
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.3180	1.27233	0.636165	−	0.771553i	$-0.280520\pi$
$907$	38.3180	1.27233	0.636165	+	0.771553i	$0.280520\pi$
$908$	0	0
$909$	−11.1775	−0.370736
$910$	0	0
$911$	37.8395	1.25368	0.626840	−	0.779148i	$-0.284348\pi$
$911$	37.8395	1.25368	0.626840	+	0.779148i	$0.284348\pi$
$912$	0	0
$913$	30.4522	1.00782
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.87814	0.326205
$918$	0	0
$919$	28.8996	0.953309	0.476655	−	0.879091i	$-0.341849\pi$
$919$	28.8996	0.953309	0.476655	+	0.879091i	$0.341849\pi$
$920$	0	0
$921$	−16.9206	−0.557552
$922$	0	0
$923$	7.39461	0.243397
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.3376	−0.405220
$928$	0	0
$929$	−47.2978	−1.55179	−0.775895	−	0.630862i	$-0.782701\pi$
$929$	−47.2978	−1.55179	−0.775895	+	0.630862i	$0.782701\pi$
$930$	0	0
$931$	−8.01521	−0.262688
$932$	0	0
$933$	−30.4490	−0.996855
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.24572	0.0406958	0.0203479	−	0.999793i	$-0.493523\pi$
$937$	1.24572	0.0406958	0.0203479	+	0.999793i	$0.493523\pi$
$938$	0	0
$939$	32.2281	1.05173
$940$	0	0
$941$	37.9458	1.23700	0.618499	−	0.785785i	$-0.287741\pi$
$941$	37.9458	1.23700	0.618499	+	0.785785i	$0.287741\pi$
$942$	0	0
$943$	9.22519	0.300413
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.2986	−1.69948	−0.849738	−	0.527205i	$-0.823240\pi$
$947$	−52.2986	−1.69948	−0.849738	+	0.527205i	$0.823240\pi$
$948$	0	0
$949$	20.3005	0.658982
$950$	0	0
$951$	−40.2815	−1.30622
$952$	0	0
$953$	−0.806914	−0.0261385	−0.0130693	−	0.999915i	$-0.504160\pi$
$953$	−0.806914	−0.0261385	−0.0130693	+	0.999915i	$0.504160\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.66311	0.118412
$958$	0	0
$959$	−42.0833	−1.35894
$960$	0	0
$961$	−18.7511	−0.604876
$962$	0	0
$963$	6.03858	0.194591
$964$	0	0
$965$	0	0
$966$	0	0
$967$	27.3892	0.880777	0.440388	−	0.897807i	$-0.354841\pi$
$967$	27.3892	0.880777	0.440388	+	0.897807i	$0.354841\pi$
$968$	0	0
$969$	9.57445	0.307576
$970$	0	0
$971$	−26.7596	−0.858757	−0.429378	−	0.903125i	$-0.641267\pi$
$971$	−26.7596	−0.858757	−0.429378	+	0.903125i	$0.641267\pi$
$972$	0	0
$973$	−40.0817	−1.28496
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.64662	−0.0846730	−0.0423365	−	0.999103i	$-0.513480\pi$
$977$	−2.64662	−0.0846730	−0.0423365	+	0.999103i	$0.513480\pi$
$978$	0	0
$979$	49.2496	1.57402
$980$	0	0
$981$	2.34741	0.0749469
$982$	0	0
$983$	33.0275	1.05341	0.526707	−	0.850047i	$-0.323427\pi$
$983$	33.0275	1.05341	0.526707	+	0.850047i	$0.323427\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−25.7712	−0.820307
$988$	0	0
$989$	−5.84186	−0.185760
$990$	0	0
$991$	32.2132	1.02329	0.511643	−	0.859198i	$-0.329037\pi$
$991$	32.2132	1.02329	0.511643	+	0.859198i	$0.329037\pi$
$992$	0	0
$993$	−18.8050	−0.596757
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.7919	−1.67194	−0.835969	−	0.548777i	$-0.815093\pi$
$997$	−52.7919	−1.67194	−0.835969	+	0.548777i	$0.815093\pi$
$998$	0	0
$999$	−18.8878	−0.597583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bw.1.1		4
4.3	odd	2		425.2.a.g.1.3		4
5.2	odd	4		1360.2.e.d.1089.6		8
5.3	odd	4		1360.2.e.d.1089.3		8
5.4	even	2		6800.2.a.bt.1.4		4
12.11	even	2		3825.2.a.bj.1.2		4
20.3	even	4		85.2.b.a.69.5	yes	8
20.7	even	4		85.2.b.a.69.4	✓	8
20.19	odd	2		425.2.a.h.1.2		4
60.23	odd	4		765.2.b.c.154.4		8
60.47	odd	4		765.2.b.c.154.5		8
60.59	even	2		3825.2.a.bh.1.3		4
68.67	odd	2		7225.2.a.v.1.3		4
340.67	even	4		1445.2.b.e.579.4		8
340.203	even	4		1445.2.b.e.579.5		8
340.339	odd	2		7225.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.4	✓	8	20.7	even	4
85.2.b.a.69.5	yes	8	20.3	even	4
425.2.a.g.1.3		4	4.3	odd	2
425.2.a.h.1.2		4	20.19	odd	2
765.2.b.c.154.4		8	60.23	odd	4
765.2.b.c.154.5		8	60.47	odd	4
1360.2.e.d.1089.3		8	5.3	odd	4
1360.2.e.d.1089.6		8	5.2	odd	4
1445.2.b.e.579.4		8	340.67	even	4
1445.2.b.e.579.5		8	340.203	even	4
3825.2.a.bh.1.3		4	60.59	even	2
3825.2.a.bj.1.2		4	12.11	even	2
6800.2.a.bt.1.4		4	5.4	even	2
6800.2.a.bw.1.1		4	1.1	even	1	trivial
7225.2.a.v.1.3		4	68.67	odd	2
7225.2.a.w.1.2		4	340.339	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.44579 q^{3} +2.86537 q^{7} -0.909700 q^{9} +O(q^{10})\)	\(q-1.44579 q^{3} +2.86537 q^{7} -0.909700 q^{9} +3.84724 q^{11} -6.22085 q^{13} +1.00000 q^{17} -6.62231 q^{19} -4.14271 q^{21} +4.51796 q^{23} +5.65259 q^{27} -0.658562 q^{29} -3.49984 q^{31} -5.56229 q^{33} -3.34144 q^{37} +8.99403 q^{39} +2.04189 q^{41} -1.29303 q^{43} +6.22085 q^{47} +1.21033 q^{49} -1.44579 q^{51} +9.92750 q^{53} +9.57445 q^{57} +2.00000 q^{59} -7.61634 q^{61} -2.60662 q^{63} +0.257106 q^{67} -6.53201 q^{69} -1.18868 q^{71} -3.26330 q^{73} +11.0238 q^{77} +4.99756 q^{79} -5.44335 q^{81} +7.91534 q^{83} +0.952140 q^{87} +12.8013 q^{89} -17.8250 q^{91} +5.06002 q^{93} +4.08866 q^{97} -3.49984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 10 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 10 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 10 q^{7} + 4 q^{9} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 4 q^{19} + 12 q^{21} + 4 q^{23} + 10 q^{27} - 4 q^{29} + 12 q^{31} - 2 q^{33} - 12 q^{37} + 22 q^{39} - 6 q^{41} + 18 q^{43} + 6 q^{47} + 8 q^{49} + 4 q^{51} - 8 q^{53} + 16 q^{57} + 8 q^{59} + 6 q^{61} + 16 q^{63} + 6 q^{67} + 4 q^{69} + 10 q^{71} - 2 q^{73} + 18 q^{77} + 12 q^{79} - 4 q^{81} - 14 q^{83} + 4 q^{87} + 24 q^{89} - 18 q^{91} + 18 q^{93} + 4 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 10 * q^7 + 4 * q^9 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 4 * q^19 + 12 * q^21 + 4 * q^23 + 10 * q^27 - 4 * q^29 + 12 * q^31 - 2 * q^33 - 12 * q^37 + 22 * q^39 - 6 * q^41 + 18 * q^43 + 6 * q^47 + 8 * q^49 + 4 * q^51 - 8 * q^53 + 16 * q^57 + 8 * q^59 + 6 * q^61 + 16 * q^63 + 6 * q^67 + 4 * q^69 + 10 * q^71 - 2 * q^73 + 18 * q^77 + 12 * q^79 - 4 * q^81 - 14 * q^83 + 4 * q^87 + 24 * q^89 - 18 * q^91 + 18 * q^93 + 4 * q^97 + 12 * q^99

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