LMFDB - Embedded newform 6800.2.a.br.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.321637$ 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)

$f(q)$	$=$	$q+0.321637 q^{3} -2.21819 q^{7} -2.89655 q^{9} +O(q^{10})$	$q+0.321637 q^{3} -2.21819 q^{7} -2.89655 q^{9} -4.21819 q^{11} -6.53982 q^{13} +1.00000 q^{17} -5.25328 q^{19} -0.713451 q^{21} +6.21819 q^{23} -1.89655 q^{27} +7.25328 q^{29} -5.67836 q^{31} -1.35673 q^{33} +1.35673 q^{37} -2.10345 q^{39} -10.4364 q^{41} -2.43637 q^{43} +0.103450 q^{47} -2.07965 q^{49} +0.321637 q^{51} +11.8965 q^{53} -1.68965 q^{57} -1.25328 q^{59} +3.89655 q^{61} +6.42509 q^{63} -3.28655 q^{67} +2.00000 q^{69} -12.8282 q^{71} +4.53982 q^{73} +9.35673 q^{77} -9.50474 q^{79} +8.07965 q^{81} -0.0701770 q^{83} +2.33292 q^{87} +9.68965 q^{89} +14.5066 q^{91} -1.82637 q^{93} -11.6896 q^{97} +12.2182 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q + q^3 + 6 * q^7 + 4 * q^9	$ 3 q + q^{3} + 6 q^{7} + 4 q^{9} - 7 q^{13} + 3 q^{17} - 3 q^{19} - 2 q^{21} + 6 q^{23} + 7 q^{27} + 9 q^{29} - 17 q^{31} - 4 q^{33} + 4 q^{37} - 19 q^{39} - 6 q^{41} + 18 q^{43} + 13 q^{47} + 19 q^{49} + q^{51} + 23 q^{53} + 33 q^{57} + 9 q^{59} - q^{61} + 32 q^{63} - 10 q^{67} + 6 q^{69} - 13 q^{71} + q^{73} + 28 q^{77} - 16 q^{79} - q^{81} - 31 q^{87} - 9 q^{89} + 18 q^{91} + 7 q^{93} + 3 q^{97} + 24 q^{99}+O(q^{100}) $ 3 * q + q^3 + 6 * q^7 + 4 * q^9 - 7 * q^13 + 3 * q^17 - 3 * q^19 - 2 * q^21 + 6 * q^23 + 7 * q^27 + 9 * q^29 - 17 * q^31 - 4 * q^33 + 4 * q^37 - 19 * q^39 - 6 * q^41 + 18 * q^43 + 13 * q^47 + 19 * q^49 + q^51 + 23 * q^53 + 33 * q^57 + 9 * q^59 - q^61 + 32 * q^63 - 10 * q^67 + 6 * q^69 - 13 * q^71 + q^73 + 28 * q^77 - 16 * q^79 - q^81 - 31 * q^87 - 9 * q^89 + 18 * q^91 + 7 * q^93 + 3 * q^97 + 24 * 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
$3$	0.321637	0.185697	0.0928487	−	0.995680i	$-0.470403\pi$
$3$	0.321637	0.185697	0.0928487	+	0.995680i	$0.470403\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.21819	−0.838396	−0.419198	−	0.907895i	$-0.637689\pi$
$7$	−2.21819	−0.838396	−0.419198	+	0.907895i	$0.637689\pi$
$8$	0	0
$9$	−2.89655	−0.965517
$10$	0	0
$11$	−4.21819	−1.27183	−0.635916	−	0.771759i	$-0.719377\pi$
$11$	−4.21819	−1.27183	−0.635916	+	0.771759i	$0.719377\pi$
$12$	0	0
$13$	−6.53982	−1.81382	−0.906910	−	0.421324i	$-0.861566\pi$
$13$	−6.53982	−1.81382	−0.906910	+	0.421324i	$0.861566\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−5.25328	−1.20518	−0.602592	−	0.798049i	$-0.705865\pi$
$19$	−5.25328	−1.20518	−0.602592	+	0.798049i	$0.705865\pi$
$20$	0	0
$21$	−0.713451	−0.155688
$22$	0	0
$23$	6.21819	1.29658	0.648291	−	0.761393i	$-0.275484\pi$
$23$	6.21819	1.29658	0.648291	+	0.761393i	$0.275484\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.89655	−0.364991
$28$	0	0
$29$	7.25328	1.34690	0.673450	−	0.739233i	$-0.264812\pi$
$29$	7.25328	1.34690	0.673450	+	0.739233i	$0.264812\pi$
$30$	0	0
$31$	−5.67836	−1.01986	−0.509932	−	0.860215i	$-0.670329\pi$
$31$	−5.67836	−1.01986	−0.509932	+	0.860215i	$0.670329\pi$
$32$	0	0
$33$	−1.35673	−0.236176
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.35673	0.223044	0.111522	−	0.993762i	$-0.464427\pi$
$37$	1.35673	0.223044	0.111522	+	0.993762i	$0.464427\pi$
$38$	0	0
$39$	−2.10345	−0.336822
$40$	0	0
$41$	−10.4364	−1.62989	−0.814944	−	0.579540i	$-0.803232\pi$
$41$	−10.4364	−1.62989	−0.814944	+	0.579540i	$0.803232\pi$
$42$	0	0
$43$	−2.43637	−0.371543	−0.185772	−	0.982593i	$-0.559478\pi$
$43$	−2.43637	−0.371543	−0.185772	+	0.982593i	$0.559478\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.103450	0.0150898	0.00754490	−	0.999972i	$-0.497598\pi$
$47$	0.103450	0.0150898	0.00754490	+	0.999972i	$0.497598\pi$
$48$	0	0
$49$	−2.07965	−0.297093
$50$	0	0
$51$	0.321637	0.0450382
$52$	0	0
$53$	11.8965	1.63412	0.817058	−	0.576555i	$-0.195603\pi$
$53$	11.8965	1.63412	0.817058	+	0.576555i	$0.195603\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.68965	−0.223799
$58$	0	0
$59$	−1.25328	−0.163163	−0.0815813	−	0.996667i	$-0.525997\pi$
$59$	−1.25328	−0.163163	−0.0815813	+	0.996667i	$0.525997\pi$
$60$	0	0
$61$	3.89655	0.498902	0.249451	−	0.968387i	$-0.419750\pi$
$61$	3.89655	0.498902	0.249451	+	0.968387i	$0.419750\pi$
$62$	0	0
$63$	6.42509	0.809485
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.28655	−0.401516	−0.200758	−	0.979641i	$-0.564340\pi$
$67$	−3.28655	−0.401516	−0.200758	+	0.979641i	$0.564340\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−12.8282	−1.52243	−0.761213	−	0.648502i	$-0.775396\pi$
$71$	−12.8282	−1.52243	−0.761213	+	0.648502i	$0.775396\pi$
$72$	0	0
$73$	4.53982	0.531346	0.265673	−	0.964063i	$-0.414406\pi$
$73$	4.53982	0.531346	0.265673	+	0.964063i	$0.414406\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.35673	1.06630
$78$	0	0
$79$	−9.50474	−1.06937	−0.534683	−	0.845053i	$-0.679569\pi$
$79$	−9.50474	−1.06937	−0.534683	+	0.845053i	$0.679569\pi$
$80$	0	0
$81$	8.07965	0.897739
$82$	0	0
$83$	−0.0701770	−0.00770292	−0.00385146	−	0.999993i	$-0.501226\pi$
$83$	−0.0701770	−0.00770292	−0.00385146	+	0.999993i	$0.501226\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.33292	0.250116
$88$	0	0
$89$	9.68965	1.02710	0.513550	−	0.858059i	$-0.328330\pi$
$89$	9.68965	1.02710	0.513550	+	0.858059i	$0.328330\pi$
$90$	0	0
$91$	14.5066	1.52070
$92$	0	0
$93$	−1.82637	−0.189386
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.6896	−1.18690	−0.593452	−	0.804869i	$-0.702235\pi$
$97$	−11.6896	−1.18690	−0.593452	+	0.804869i	$0.702235\pi$
$98$	0	0
$99$	12.2182	1.22797
$100$	0	0
$101$	−9.07965	−0.903459	−0.451729	−	0.892155i	$-0.649193\pi$
$101$	−9.07965	−0.903459	−0.451729	+	0.892155i	$0.649193\pi$
$102$	0	0
$103$	12.3662	1.21848	0.609239	−	0.792987i	$-0.291475\pi$
$103$	12.3662	1.21848	0.609239	+	0.792987i	$0.291475\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.0113	0.967828	0.483914	−	0.875116i	$-0.339215\pi$
$107$	10.0113	0.967828	0.483914	+	0.875116i	$0.339215\pi$
$108$	0	0
$109$	−4.97620	−0.476633	−0.238317	−	0.971188i	$-0.576596\pi$
$109$	−4.97620	−0.476633	−0.238317	+	0.971188i	$0.576596\pi$
$110$	0	0
$111$	0.436373	0.0414187
$112$	0	0
$113$	13.6195	1.28121	0.640606	−	0.767870i	$-0.278683\pi$
$113$	13.6195	1.28121	0.640606	+	0.767870i	$0.278683\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	18.9429	1.75127
$118$	0	0
$119$	−2.21819	−0.203341
$120$	0	0
$121$	6.79310	0.617554
$122$	0	0
$123$	−3.35673	−0.302666
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.5398	−1.46767	−0.733836	−	0.679327i	$-0.762272\pi$
$127$	−16.5398	−1.46767	−0.733836	+	0.679327i	$0.762272\pi$
$128$	0	0
$129$	−0.783628	−0.0689946
$130$	0	0
$131$	16.4477	1.43704	0.718519	−	0.695507i	$-0.244820\pi$
$131$	16.4477	1.43704	0.718519	+	0.695507i	$0.244820\pi$
$132$	0	0
$133$	11.6527	1.01042
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.9429	1.96015	0.980073	−	0.198637i	$-0.0636515\pi$
$137$	22.9429	1.96015	0.980073	+	0.198637i	$0.0636515\pi$
$138$	0	0
$139$	1.50474	0.127630	0.0638150	−	0.997962i	$-0.479673\pi$
$139$	1.50474	0.127630	0.0638150	+	0.997962i	$0.479673\pi$
$140$	0	0
$141$	0.0332735	0.00280214
$142$	0	0
$143$	27.5862	2.30687
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.668892	−0.0551693
$148$	0	0
$149$	13.7931	1.12997	0.564987	−	0.825100i	$-0.308881\pi$
$149$	13.7931	1.12997	0.564987	+	0.825100i	$0.308881\pi$
$150$	0	0
$151$	−20.2295	−1.64625	−0.823126	−	0.567859i	$-0.807772\pi$
$151$	−20.2295	−1.64625	−0.823126	+	0.567859i	$0.807772\pi$
$152$	0	0
$153$	−2.89655	−0.234172
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	3.82637	0.303451
$160$	0	0
$161$	−13.7931	−1.08705
$162$	0	0
$163$	19.2978	1.51152	0.755762	−	0.654847i	$-0.227267\pi$
$163$	19.2978	1.51152	0.755762	+	0.654847i	$0.227267\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.9411	−1.54309	−0.771545	−	0.636175i	$-0.780516\pi$
$167$	−19.9411	−1.54309	−0.771545	+	0.636175i	$0.780516\pi$
$168$	0	0
$169$	29.7693	2.28995
$170$	0	0
$171$	15.2164	1.16363
$172$	0	0
$173$	−5.14982	−0.391534	−0.195767	−	0.980650i	$-0.562720\pi$
$173$	−5.14982	−0.391534	−0.195767	+	0.980650i	$0.562720\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.403100	−0.0302988
$178$	0	0
$179$	17.8633	1.33516	0.667582	−	0.744536i	$-0.267329\pi$
$179$	17.8633	1.33516	0.667582	+	0.744536i	$0.267329\pi$
$180$	0	0
$181$	14.4364	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$181$	14.4364	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$182$	0	0
$183$	1.25328	0.0926448
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.21819	−0.308464
$188$	0	0
$189$	4.20690	0.306007
$190$	0	0
$191$	18.2996	1.32412	0.662058	−	0.749453i	$-0.269683\pi$
$191$	18.2996	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$192$	0	0
$193$	5.56363	0.400479	0.200239	−	0.979747i	$-0.435828\pi$
$193$	5.56363	0.400479	0.200239	+	0.979747i	$0.435828\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.64327	0.473314	0.236657	−	0.971593i	$-0.423948\pi$
$197$	6.64327	0.473314	0.236657	+	0.971593i	$0.423948\pi$
$198$	0	0
$199$	3.40128	0.241111	0.120555	−	0.992707i	$-0.461532\pi$
$199$	3.40128	0.241111	0.120555	+	0.992707i	$0.461532\pi$
$200$	0	0
$201$	−1.05708	−0.0745604
$202$	0	0
$203$	−16.0891	−1.12923
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.0113	−1.25187
$208$	0	0
$209$	22.1593	1.53279
$210$	0	0
$211$	−16.4477	−1.13230	−0.566152	−	0.824301i	$-0.691568\pi$
$211$	−16.4477	−1.13230	−0.566152	+	0.824301i	$0.691568\pi$
$212$	0	0
$213$	−4.12602	−0.282710
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.5957	0.855050
$218$	0	0
$219$	1.46018	0.0986696
$220$	0	0
$221$	−6.53982	−0.439916
$222$	0	0
$223$	−21.4126	−1.43389	−0.716946	−	0.697129i	$-0.754461\pi$
$223$	−21.4126	−1.43389	−0.716946	+	0.697129i	$0.754461\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.7580	−0.846779	−0.423389	−	0.905948i	$-0.639160\pi$
$227$	−12.7580	−0.846779	−0.423389	+	0.905948i	$0.639160\pi$
$228$	0	0
$229$	−2.50655	−0.165638	−0.0828188	−	0.996565i	$-0.526392\pi$
$229$	−2.50655	−0.165638	−0.0828188	+	0.996565i	$0.526392\pi$
$230$	0	0
$231$	3.00947	0.198009
$232$	0	0
$233$	−5.39000	−0.353111	−0.176555	−	0.984291i	$-0.556495\pi$
$233$	−5.39000	−0.353111	−0.176555	+	0.984291i	$0.556495\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.05708	−0.198578
$238$	0	0
$239$	14.5066	0.938351	0.469175	−	0.883105i	$-0.344551\pi$
$239$	14.5066	0.938351	0.469175	+	0.883105i	$0.344551\pi$
$240$	0	0
$241$	−7.86328	−0.506518	−0.253259	−	0.967398i	$-0.581503\pi$
$241$	−7.86328	−0.506518	−0.253259	+	0.967398i	$0.581503\pi$
$242$	0	0
$243$	8.28836	0.531699
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.3555	2.18599
$248$	0	0
$249$	−0.0225715	−0.00143041
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−26.2295	−1.64903
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.07018	−0.129134	−0.0645670	−	0.997913i	$-0.520567\pi$
$257$	−2.07018	−0.129134	−0.0645670	+	0.997913i	$0.520567\pi$
$258$	0	0
$259$	−3.00947	−0.186999
$260$	0	0
$261$	−21.0095	−1.30045
$262$	0	0
$263$	−17.6195	−1.08646	−0.543232	−	0.839583i	$-0.682799\pi$
$263$	−17.6195	−1.08646	−0.543232	+	0.839583i	$0.682799\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.11655	0.190730
$268$	0	0
$269$	11.4827	0.700115	0.350058	−	0.936728i	$-0.386162\pi$
$269$	11.4827	0.700115	0.350058	+	0.936728i	$0.386162\pi$
$270$	0	0
$271$	−27.4458	−1.66722	−0.833608	−	0.552356i	$-0.813729\pi$
$271$	−27.4458	−1.66722	−0.833608	+	0.552356i	$0.813729\pi$
$272$	0	0
$273$	4.66585	0.282390
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.573097	−0.0344341	−0.0172170	−	0.999852i	$-0.505481\pi$
$277$	−0.573097	−0.0344341	−0.0172170	+	0.999852i	$0.505481\pi$
$278$	0	0
$279$	16.4477	0.984696
$280$	0	0
$281$	18.0558	1.07712	0.538561	−	0.842587i	$-0.318968\pi$
$281$	18.0558	1.07712	0.538561	+	0.842587i	$0.318968\pi$
$282$	0	0
$283$	−16.3442	−0.971562	−0.485781	−	0.874080i	$-0.661465\pi$
$283$	−16.3442	−0.971562	−0.485781	+	0.874080i	$0.661465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.1498	1.36649
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.75983	−0.220405
$292$	0	0
$293$	17.1831	1.00385	0.501924	−	0.864912i	$-0.332626\pi$
$293$	17.1831	1.00385	0.501924	+	0.864912i	$0.332626\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	−40.6658	−2.35177
$300$	0	0
$301$	5.40433	0.311500
$302$	0	0
$303$	−2.92035	−0.167770
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.7931	−0.787214	−0.393607	−	0.919279i	$-0.628773\pi$
$307$	−13.7931	−0.787214	−0.393607	+	0.919279i	$0.628773\pi$
$308$	0	0
$309$	3.97743	0.226268
$310$	0	0
$311$	−20.2182	−1.14647	−0.573234	−	0.819392i	$-0.694311\pi$
$311$	−20.2182	−1.14647	−0.573234	+	0.819392i	$0.694311\pi$
$312$	0	0
$313$	−19.7229	−1.11481	−0.557403	−	0.830242i	$-0.688202\pi$
$313$	−19.7229	−1.11481	−0.557403	+	0.830242i	$0.688202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.79310	−0.550035	−0.275018	−	0.961439i	$-0.588684\pi$
$317$	−9.79310	−0.550035	−0.275018	+	0.961439i	$0.588684\pi$
$318$	0	0
$319$	−30.5957	−1.71303
$320$	0	0
$321$	3.22000	0.179723
$322$	0	0
$323$	−5.25328	−0.292300
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.60053	−0.0885095
$328$	0	0
$329$	−0.229472	−0.0126512
$330$	0	0
$331$	29.2533	1.60791	0.803953	−	0.594693i	$-0.202726\pi$
$331$	29.2533	1.60791	0.803953	+	0.594693i	$0.202726\pi$
$332$	0	0
$333$	−3.92982	−0.215353
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.4031	0.566693	0.283346	−	0.959018i	$-0.408555\pi$
$337$	10.4031	0.566693	0.283346	+	0.959018i	$0.408555\pi$
$338$	0	0
$339$	4.38053	0.237918
$340$	0	0
$341$	23.9524	1.29709
$342$	0	0
$343$	20.1404	1.08748
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.89473	0.155397	0.0776987	−	0.996977i	$-0.475243\pi$
$347$	2.89473	0.155397	0.0776987	+	0.996977i	$0.475243\pi$
$348$	0	0
$349$	−13.6527	−0.730815	−0.365407	−	0.930848i	$-0.619070\pi$
$349$	−13.6527	−0.730815	−0.365407	+	0.930848i	$0.619070\pi$
$350$	0	0
$351$	12.4031	0.662028
$352$	0	0
$353$	28.2996	1.50624	0.753119	−	0.657884i	$-0.228548\pi$
$353$	28.2996	1.50624	0.753119	+	0.657884i	$0.228548\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.713451	−0.0377598
$358$	0	0
$359$	26.3888	1.39275	0.696373	−	0.717680i	$-0.254796\pi$
$359$	26.3888	1.39275	0.696373	+	0.717680i	$0.254796\pi$
$360$	0	0
$361$	8.59690	0.452468
$362$	0	0
$363$	2.18491	0.114678
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.09093	−0.161345	−0.0806727	−	0.996741i	$-0.525707\pi$
$367$	−3.09093	−0.161345	−0.0806727	+	0.996741i	$0.525707\pi$
$368$	0	0
$369$	30.2295	1.57368
$370$	0	0
$371$	−26.3888	−1.37004
$372$	0	0
$373$	−19.3567	−1.00225	−0.501127	−	0.865374i	$-0.667081\pi$
$373$	−19.3567	−1.00225	−0.501127	+	0.865374i	$0.667081\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−47.4351	−2.44303
$378$	0	0
$379$	34.8139	1.78827	0.894134	−	0.447800i	$-0.147792\pi$
$379$	34.8139	1.78827	0.894134	+	0.447800i	$0.147792\pi$
$380$	0	0
$381$	−5.31982	−0.272543
$382$	0	0
$383$	16.7467	0.855718	0.427859	−	0.903846i	$-0.359268\pi$
$383$	16.7467	0.855718	0.427859	+	0.903846i	$0.359268\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.05708	0.358731
$388$	0	0
$389$	−3.12725	−0.158558	−0.0792790	−	0.996852i	$-0.525262\pi$
$389$	−3.12725	−0.158558	−0.0792790	+	0.996852i	$0.525262\pi$
$390$	0	0
$391$	6.21819	0.314467
$392$	0	0
$393$	5.29018	0.266854
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.7931	−0.692256	−0.346128	−	0.938187i	$-0.612504\pi$
$397$	−13.7931	−0.692256	−0.346128	+	0.938187i	$0.612504\pi$
$398$	0	0
$399$	3.74796	0.187632
$400$	0	0
$401$	−0.506550	−0.0252959	−0.0126480	−	0.999920i	$-0.504026\pi$
$401$	−0.506550	−0.0252959	−0.0126480	+	0.999920i	$0.504026\pi$
$402$	0	0
$403$	37.1355	1.84985
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.72292	−0.283675
$408$	0	0
$409$	9.39000	0.464306	0.232153	−	0.972679i	$-0.425423\pi$
$409$	9.39000	0.464306	0.232153	+	0.972679i	$0.425423\pi$
$410$	0	0
$411$	7.37930	0.363994
$412$	0	0
$413$	2.78000	0.136795
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.483979	0.0237005
$418$	0	0
$419$	13.2978	0.649642	0.324821	−	0.945776i	$-0.394696\pi$
$419$	13.2978	0.649642	0.324821	+	0.945776i	$0.394696\pi$
$420$	0	0
$421$	7.58620	0.369729	0.184864	−	0.982764i	$-0.440815\pi$
$421$	7.58620	0.369729	0.184864	+	0.982764i	$0.440815\pi$
$422$	0	0
$423$	−0.299649	−0.0145695
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.64327	−0.418277
$428$	0	0
$429$	8.87275	0.428380
$430$	0	0
$431$	−22.7913	−1.09782	−0.548909	−	0.835882i	$-0.684957\pi$
$431$	−22.7913	−1.09782	−0.548909	+	0.835882i	$0.684957\pi$
$432$	0	0
$433$	−10.9429	−0.525883	−0.262942	−	0.964812i	$-0.584693\pi$
$433$	−10.9429	−0.525883	−0.262942	+	0.964812i	$0.584693\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−32.6658	−1.56262
$438$	0	0
$439$	23.3680	1.11529	0.557647	−	0.830078i	$-0.311704\pi$
$439$	23.3680	1.11529	0.557647	+	0.830078i	$0.311704\pi$
$440$	0	0
$441$	6.02380	0.286848
$442$	0	0
$443$	−20.9204	−0.993956	−0.496978	−	0.867763i	$-0.665557\pi$
$443$	−20.9204	−0.993956	−0.496978	+	0.867763i	$0.665557\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.43637	0.209833
$448$	0	0
$449$	−34.2295	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$449$	−34.2295	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$450$	0	0
$451$	44.0226	2.07294
$452$	0	0
$453$	−6.50655	−0.305704
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.0891	0.752617	0.376309	−	0.926494i	$-0.377193\pi$
$457$	16.0891	0.752617	0.376309	+	0.926494i	$0.377193\pi$
$458$	0	0
$459$	−1.89655	−0.0885234
$460$	0	0
$461$	−33.0320	−1.53846	−0.769228	−	0.638975i	$-0.779359\pi$
$461$	−33.0320	−1.53846	−0.769228	+	0.638975i	$0.779359\pi$
$462$	0	0
$463$	17.1166	0.795474	0.397737	−	0.917500i	$-0.369796\pi$
$463$	17.1166	0.795474	0.397737	+	0.917500i	$0.369796\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.8157	−1.00951	−0.504754	−	0.863263i	$-0.668417\pi$
$467$	−21.8157	−1.00951	−0.504754	+	0.863263i	$0.668417\pi$
$468$	0	0
$469$	7.29018	0.336629
$470$	0	0
$471$	0.643274	0.0296405
$472$	0	0
$473$	10.2771	0.472541
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−34.4589	−1.57777
$478$	0	0
$479$	−22.3442	−1.02093	−0.510466	−	0.859898i	$-0.670527\pi$
$479$	−22.3442	−1.02093	−0.510466	+	0.859898i	$0.670527\pi$
$480$	0	0
$481$	−8.87275	−0.404562
$482$	0	0
$483$	−4.43637	−0.201862
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.21819	−0.100516	−0.0502578	−	0.998736i	$-0.516004\pi$
$487$	−2.21819	−0.100516	−0.0502578	+	0.998736i	$0.516004\pi$
$488$	0	0
$489$	6.20690	0.280686
$490$	0	0
$491$	−10.7467	−0.484993	−0.242496	−	0.970152i	$-0.577966\pi$
$491$	−10.7467	−0.484993	−0.242496	+	0.970152i	$0.577966\pi$
$492$	0	0
$493$	7.25328	0.326671
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.4553	1.27640
$498$	0	0
$499$	−20.8615	−0.933887	−0.466944	−	0.884287i	$-0.654645\pi$
$499$	−20.8615	−0.933887	−0.466944	+	0.884287i	$0.654645\pi$
$500$	0	0
$501$	−6.41380	−0.286548
$502$	0	0
$503$	18.3073	0.816282	0.408141	−	0.912919i	$-0.366177\pi$
$503$	18.3073	0.816282	0.408141	+	0.912919i	$0.366177\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.57491	0.425237
$508$	0	0
$509$	7.63380	0.338362	0.169181	−	0.985585i	$-0.445888\pi$
$509$	7.63380	0.338362	0.169181	+	0.985585i	$0.445888\pi$
$510$	0	0
$511$	−10.0702	−0.445478
$512$	0	0
$513$	9.96310	0.439881
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.436373	−0.0191917
$518$	0	0
$519$	−1.65638	−0.0727068
$520$	0	0
$521$	−2.20690	−0.0966861	−0.0483430	−	0.998831i	$-0.515394\pi$
$521$	−2.20690	−0.0966861	−0.0483430	+	0.998831i	$0.515394\pi$
$522$	0	0
$523$	34.8727	1.52488	0.762439	−	0.647060i	$-0.224002\pi$
$523$	34.8727	1.52488	0.762439	+	0.647060i	$0.224002\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.67836	−0.247353
$528$	0	0
$529$	15.6658	0.681124
$530$	0	0
$531$	3.63017	0.157536
$532$	0	0
$533$	68.2520	2.95632
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.74549	0.247936
$538$	0	0
$539$	8.77234	0.377852
$540$	0	0
$541$	−42.8953	−1.84421	−0.922107	−	0.386935i	$-0.873534\pi$
$541$	−42.8953	−1.84421	−0.922107	+	0.386935i	$0.873534\pi$
$542$	0	0
$543$	4.64327	0.199262
$544$	0	0
$545$	0	0
$546$	0	0
$547$	27.0351	1.15594	0.577968	−	0.816059i	$-0.303846\pi$
$547$	27.0351	1.15594	0.577968	+	0.816059i	$0.303846\pi$
$548$	0	0
$549$	−11.2865	−0.481698
$550$	0	0
$551$	−38.1035	−1.62326
$552$	0	0
$553$	21.0833	0.896552
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.3329	0.437820	0.218910	−	0.975745i	$-0.429750\pi$
$557$	10.3329	0.437820	0.218910	+	0.975745i	$0.429750\pi$
$558$	0	0
$559$	15.9335	0.673913
$560$	0	0
$561$	−1.35673	−0.0572810
$562$	0	0
$563$	−10.9905	−0.463196	−0.231598	−	0.972812i	$-0.574395\pi$
$563$	−10.9905	−0.463196	−0.231598	+	0.972812i	$0.574395\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.9222	−0.752660
$568$	0	0
$569$	−43.2758	−1.81422	−0.907109	−	0.420896i	$-0.861716\pi$
$569$	−43.2758	−1.81422	−0.907109	+	0.420896i	$0.861716\pi$
$570$	0	0
$571$	21.4382	0.897160	0.448580	−	0.893743i	$-0.351930\pi$
$571$	21.4382	0.897160	0.448580	+	0.893743i	$0.351930\pi$
$572$	0	0
$573$	5.88585	0.245885
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.8822	−1.32727	−0.663637	−	0.748055i	$-0.730988\pi$
$577$	−31.8822	−1.32727	−0.663637	+	0.748055i	$0.730988\pi$
$578$	0	0
$579$	1.78947	0.0743678
$580$	0	0
$581$	0.155666	0.00645810
$582$	0	0
$583$	−50.1819	−2.07832
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.57673	0.271451	0.135725	−	0.990747i	$-0.456664\pi$
$587$	6.57673	0.271451	0.135725	+	0.990747i	$0.456664\pi$
$588$	0	0
$589$	29.8300	1.22912
$590$	0	0
$591$	2.13672	0.0878931
$592$	0	0
$593$	−15.7455	−0.646590	−0.323295	−	0.946298i	$-0.604791\pi$
$593$	−15.7455	−0.646590	−0.323295	+	0.946298i	$0.604791\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.09398	0.0447736
$598$	0	0
$599$	33.0796	1.35160	0.675799	−	0.737086i	$-0.263799\pi$
$599$	33.0796	1.35160	0.675799	+	0.737086i	$0.263799\pi$
$600$	0	0
$601$	−22.6433	−0.923638	−0.461819	−	0.886974i	$-0.652803\pi$
$601$	−22.6433	−0.923638	−0.461819	+	0.886974i	$0.652803\pi$
$602$	0	0
$603$	9.51965	0.387670
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.5142	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$607$	22.5142	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$608$	0	0
$609$	−5.17486	−0.209696
$610$	0	0
$611$	−0.676548	−0.0273702
$612$	0	0
$613$	17.1166	0.691331	0.345665	−	0.938358i	$-0.387653\pi$
$613$	17.1166	0.691331	0.345665	+	0.938358i	$0.387653\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.0464	0.444710	0.222355	−	0.974966i	$-0.428626\pi$
$617$	11.0464	0.444710	0.222355	+	0.974966i	$0.428626\pi$
$618$	0	0
$619$	−9.85199	−0.395985	−0.197992	−	0.980204i	$-0.563442\pi$
$619$	−9.85199	−0.395985	−0.197992	+	0.980204i	$0.563442\pi$
$620$	0	0
$621$	−11.7931	−0.473241
$622$	0	0
$623$	−21.4934	−0.861117
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.12725	0.284635
$628$	0	0
$629$	1.35673	0.0540962
$630$	0	0
$631$	−0.576727	−0.0229592	−0.0114796	−	0.999934i	$-0.503654\pi$
$631$	−0.576727	−0.0229592	−0.0114796	+	0.999934i	$0.503654\pi$
$632$	0	0
$633$	−5.29018	−0.210266
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.6005	0.538873
$638$	0	0
$639$	37.1575	1.46993
$640$	0	0
$641$	30.1629	1.19136	0.595682	−	0.803220i	$-0.296882\pi$
$641$	30.1629	1.19136	0.595682	+	0.803220i	$0.296882\pi$
$642$	0	0
$643$	32.1706	1.26868	0.634342	−	0.773053i	$-0.281271\pi$
$643$	32.1706	1.26868	0.634342	+	0.773053i	$0.281271\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.3235	0.681055	0.340528	−	0.940235i	$-0.389394\pi$
$647$	17.3235	0.681055	0.340528	+	0.940235i	$0.389394\pi$
$648$	0	0
$649$	5.28655	0.207515
$650$	0	0
$651$	4.05124	0.158780
$652$	0	0
$653$	35.3091	1.38175	0.690876	−	0.722973i	$-0.257225\pi$
$653$	35.3091	1.38175	0.690876	+	0.722973i	$0.257225\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.1498	−0.513024
$658$	0	0
$659$	0.262746	0.0102351	0.00511757	−	0.999987i	$-0.498371\pi$
$659$	0.262746	0.0102351	0.00511757	+	0.999987i	$0.498371\pi$
$660$	0	0
$661$	−4.78000	−0.185920	−0.0929602	−	0.995670i	$-0.529633\pi$
$661$	−4.78000	−0.185920	−0.0929602	+	0.995670i	$0.529633\pi$
$662$	0	0
$663$	−2.10345	−0.0816912
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.1022	1.74636
$668$	0	0
$669$	−6.88708	−0.266270
$670$	0	0
$671$	−16.4364	−0.634519
$672$	0	0
$673$	19.3936	0.747569	0.373785	−	0.927515i	$-0.378060\pi$
$673$	19.3936	0.747569	0.373785	+	0.927515i	$0.378060\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.0131	0.884465	0.442233	−	0.896900i	$-0.354187\pi$
$677$	23.0131	0.884465	0.442233	+	0.896900i	$0.354187\pi$
$678$	0	0
$679$	25.9298	0.995095
$680$	0	0
$681$	−4.10345	−0.157245
$682$	0	0
$683$	−25.8377	−0.988651	−0.494325	−	0.869277i	$-0.664585\pi$
$683$	−25.8377	−0.988651	−0.494325	+	0.869277i	$0.664585\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.806200	−0.0307584
$688$	0	0
$689$	−77.8013	−2.96399
$690$	0	0
$691$	27.3680	1.04113	0.520564	−	0.853823i	$-0.325722\pi$
$691$	27.3680	1.04113	0.520564	+	0.853823i	$0.325722\pi$
$692$	0	0
$693$	−27.1022	−1.02953
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.4364	−0.395306
$698$	0	0
$699$	−1.73362	−0.0655717
$700$	0	0
$701$	12.7324	0.480896	0.240448	−	0.970662i	$-0.422706\pi$
$701$	12.7324	0.480896	0.240448	+	0.970662i	$0.422706\pi$
$702$	0	0
$703$	−7.12725	−0.268809
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.1404	0.757456
$708$	0	0
$709$	−47.5053	−1.78410	−0.892050	−	0.451937i	$-0.850733\pi$
$709$	−47.5053	−1.78410	−0.892050	+	0.451937i	$0.850733\pi$
$710$	0	0
$711$	27.5309	1.03249
$712$	0	0
$713$	−35.3091	−1.32234
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.66585	0.174249
$718$	0	0
$719$	−42.1147	−1.57061	−0.785307	−	0.619106i	$-0.787495\pi$
$719$	−42.1147	−1.57061	−0.785307	+	0.619106i	$0.787495\pi$
$720$	0	0
$721$	−27.4305	−1.02157
$722$	0	0
$723$	−2.52912	−0.0940591
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.0558	−0.818006	−0.409003	−	0.912533i	$-0.634123\pi$
$727$	−22.0558	−0.818006	−0.409003	+	0.912533i	$0.634123\pi$
$728$	0	0
$729$	−21.5731	−0.799004
$730$	0	0
$731$	−2.43637	−0.0901125
$732$	0	0
$733$	−45.0320	−1.66330	−0.831648	−	0.555303i	$-0.812602\pi$
$733$	−45.0320	−1.66330	−0.831648	+	0.555303i	$0.812602\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.8633	0.510660
$738$	0	0
$739$	35.9667	1.32306	0.661529	−	0.749920i	$-0.269908\pi$
$739$	35.9667	1.32306	0.661529	+	0.749920i	$0.269908\pi$
$740$	0	0
$741$	11.0500	0.405932
$742$	0	0
$743$	20.4953	0.751898	0.375949	−	0.926640i	$-0.377317\pi$
$743$	20.4953	0.751898	0.375949	+	0.926640i	$0.377317\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.203271	0.00743730
$748$	0	0
$749$	−22.2069	−0.811423
$750$	0	0
$751$	−6.18128	−0.225558	−0.112779	−	0.993620i	$-0.535975\pi$
$751$	−6.18128	−0.225558	−0.112779	+	0.993620i	$0.535975\pi$
$752$	0	0
$753$	−3.85965	−0.140653
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.7788	0.936945	0.468473	−	0.883478i	$-0.344804\pi$
$757$	25.7788	0.936945	0.468473	+	0.883478i	$0.344804\pi$
$758$	0	0
$759$	−8.43637	−0.306221
$760$	0	0
$761$	13.2200	0.479225	0.239612	−	0.970869i	$-0.422980\pi$
$761$	13.2200	0.479225	0.239612	+	0.970869i	$0.422980\pi$
$762$	0	0
$763$	11.0381	0.399607
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.19620	0.295948
$768$	0	0
$769$	−36.2627	−1.30767	−0.653834	−	0.756638i	$-0.726841\pi$
$769$	−36.2627	−1.30767	−0.653834	+	0.756638i	$0.726841\pi$
$770$	0	0
$771$	−0.665846	−0.0239799
$772$	0	0
$773$	37.6564	1.35441	0.677203	−	0.735796i	$-0.263192\pi$
$773$	37.6564	1.35441	0.677203	+	0.735796i	$0.263192\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.967958	−0.0347253
$778$	0	0
$779$	54.8251	1.96431
$780$	0	0
$781$	54.1117	1.93627
$782$	0	0
$783$	−13.7562	−0.491606
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.4846	1.30053	0.650267	−	0.759706i	$-0.274657\pi$
$787$	36.4846	1.30053	0.650267	+	0.759706i	$0.274657\pi$
$788$	0	0
$789$	−5.66708	−0.201753
$790$	0	0
$791$	−30.2105	−1.07416
$792$	0	0
$793$	−25.4827	−0.904919
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.1593	−1.28083	−0.640414	−	0.768030i	$-0.721237\pi$
$797$	−36.1593	−1.28083	−0.640414	+	0.768030i	$0.721237\pi$
$798$	0	0
$799$	0.103450	0.00365981
$800$	0	0
$801$	−28.0665	−0.991683
$802$	0	0
$803$	−19.1498	−0.675783
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.69328	0.130010
$808$	0	0
$809$	28.7135	1.00951	0.504756	−	0.863262i	$-0.331583\pi$
$809$	28.7135	1.00951	0.504756	+	0.863262i	$0.331583\pi$
$810$	0	0
$811$	−22.7473	−0.798766	−0.399383	−	0.916784i	$-0.630776\pi$
$811$	−22.7473	−0.798766	−0.399383	+	0.916784i	$0.630776\pi$
$812$	0	0
$813$	−8.82760	−0.309598
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.7989	0.447778
$818$	0	0
$819$	−42.0189	−1.46826
$820$	0	0
$821$	−19.5493	−0.682275	−0.341138	−	0.940013i	$-0.610812\pi$
$821$	−19.5493	−0.682275	−0.341138	+	0.940013i	$0.610812\pi$
$822$	0	0
$823$	−31.6677	−1.10387	−0.551933	−	0.833889i	$-0.686109\pi$
$823$	−31.6677	−1.10387	−0.551933	+	0.833889i	$0.686109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.3204	1.50640	0.753199	−	0.657793i	$-0.228510\pi$
$827$	43.3204	1.50640	0.753199	+	0.657793i	$0.228510\pi$
$828$	0	0
$829$	−11.6338	−0.404059	−0.202029	−	0.979379i	$-0.564754\pi$
$829$	−11.6338	−0.404059	−0.202029	+	0.979379i	$0.564754\pi$
$830$	0	0
$831$	−0.184329	−0.00639432
$832$	0	0
$833$	−2.07965	−0.0720555
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.7693	0.372241
$838$	0	0
$839$	−21.4715	−0.741277	−0.370639	−	0.928777i	$-0.620861\pi$
$839$	−21.4715	−0.741277	−0.370639	+	0.928777i	$0.620861\pi$
$840$	0	0
$841$	23.6100	0.814138
$842$	0	0
$843$	5.80743	0.200019
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.0684	−0.517755
$848$	0	0
$849$	−5.25691	−0.180417
$850$	0	0
$851$	8.43637	0.289195
$852$	0	0
$853$	30.4364	1.04212	0.521061	−	0.853520i	$-0.325536\pi$
$853$	30.4364	1.04212	0.521061	+	0.853520i	$0.325536\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.11292	−0.106335	−0.0531677	−	0.998586i	$-0.516932\pi$
$857$	−3.11292	−0.106335	−0.0531677	+	0.998586i	$0.516932\pi$
$858$	0	0
$859$	1.87398	0.0639393	0.0319697	−	0.999489i	$-0.489822\pi$
$859$	1.87398	0.0639393	0.0319697	+	0.999489i	$0.489822\pi$
$860$	0	0
$861$	7.44584	0.253754
$862$	0	0
$863$	−48.0927	−1.63710	−0.818548	−	0.574438i	$-0.805221\pi$
$863$	−48.0927	−1.63710	−0.818548	+	0.574438i	$0.805221\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.321637	0.0109234
$868$	0	0
$869$	40.0927	1.36005
$870$	0	0
$871$	21.4934	0.728278
$872$	0	0
$873$	33.8596	1.14598
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.2520	−1.02154	−0.510769	−	0.859718i	$-0.670639\pi$
$877$	−30.2520	−1.02154	−0.510769	+	0.859718i	$0.670639\pi$
$878$	0	0
$879$	5.52672	0.186412
$880$	0	0
$881$	15.7229	0.529719	0.264859	−	0.964287i	$-0.414674\pi$
$881$	15.7229	0.529719	0.264859	+	0.964287i	$0.414674\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.3870	1.05387	0.526935	−	0.849905i	$-0.323341\pi$
$887$	31.3870	1.05387	0.526935	+	0.849905i	$0.323341\pi$
$888$	0	0
$889$	36.6884	1.23049
$890$	0	0
$891$	−34.0815	−1.14177
$892$	0	0
$893$	−0.543454	−0.0181860
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−13.0796	−0.436717
$898$	0	0
$899$	−41.1867	−1.37365
$900$	0	0
$901$	11.8965	0.396332
$902$	0	0
$903$	1.73823	0.0578448
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.2551	1.07101	0.535506	−	0.844531i	$-0.320121\pi$
$907$	32.2551	1.07101	0.535506	+	0.844531i	$0.320121\pi$
$908$	0	0
$909$	26.2996	0.872304
$910$	0	0
$911$	−29.5273	−0.978283	−0.489142	−	0.872204i	$-0.662690\pi$
$911$	−29.5273	−0.978283	−0.489142	+	0.872204i	$0.662690\pi$
$912$	0	0
$913$	0.296019	0.00979682
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−36.4840	−1.20481
$918$	0	0
$919$	26.5066	0.874370	0.437185	−	0.899371i	$-0.355975\pi$
$919$	26.5066	0.874370	0.437185	+	0.899371i	$0.355975\pi$
$920$	0	0
$921$	−4.43637	−0.146183
$922$	0	0
$923$	83.8941	2.76141
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−35.8193	−1.17646
$928$	0	0
$929$	−18.0226	−0.591301	−0.295651	−	0.955296i	$-0.595536\pi$
$929$	−18.0226	−0.591301	−0.295651	+	0.955296i	$0.595536\pi$
$930$	0	0
$931$	10.9250	0.358051
$932$	0	0
$933$	−6.50292	−0.212896
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.2389	0.824520	0.412260	−	0.911066i	$-0.364739\pi$
$937$	25.2389	0.824520	0.412260	+	0.911066i	$0.364739\pi$
$938$	0	0
$939$	−6.34362	−0.207016
$940$	0	0
$941$	25.4126	0.828426	0.414213	−	0.910180i	$-0.364057\pi$
$941$	25.4126	0.828426	0.414213	+	0.910180i	$0.364057\pi$
$942$	0	0
$943$	−64.8953	−2.11328
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.6402	−0.800700	−0.400350	−	0.916362i	$-0.631111\pi$
$947$	−24.6402	−0.800700	−0.400350	+	0.916362i	$0.631111\pi$
$948$	0	0
$949$	−29.6896	−0.963767
$950$	0	0
$951$	−3.14982	−0.102140
$952$	0	0
$953$	−40.5767	−1.31441	−0.657205	−	0.753712i	$-0.728261\pi$
$953$	−40.5767	−1.31441	−0.657205	+	0.753712i	$0.728261\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.84070	−0.318105
$958$	0	0
$959$	−50.8917	−1.64338
$960$	0	0
$961$	1.24380	0.0401227
$962$	0	0
$963$	−28.9982	−0.934453
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.5029	−0.466382	−0.233191	−	0.972431i	$-0.574917\pi$
$967$	−14.5029	−0.466382	−0.233191	+	0.972431i	$0.574917\pi$
$968$	0	0
$969$	−1.68965	−0.0542793
$970$	0	0
$971$	26.1926	0.840560	0.420280	−	0.907395i	$-0.361932\pi$
$971$	26.1926	0.840560	0.420280	+	0.907395i	$0.361932\pi$
$972$	0	0
$973$	−3.33778	−0.107004
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.9524	−1.40616	−0.703081	−	0.711110i	$-0.748193\pi$
$977$	−43.9524	−1.40616	−0.703081	+	0.711110i	$0.748193\pi$
$978$	0	0
$979$	−40.8727	−1.30630
$980$	0	0
$981$	14.4138	0.460197
$982$	0	0
$983$	15.6451	0.499001	0.249500	−	0.968375i	$-0.419734\pi$
$983$	15.6451	0.499001	0.249500	+	0.968375i	$0.419734\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.0738069	−0.00234930
$988$	0	0
$989$	−15.1498	−0.481736
$990$	0	0
$991$	25.8853	0.822273	0.411136	−	0.911574i	$-0.365132\pi$
$991$	25.8853	0.822273	0.411136	+	0.911574i	$0.365132\pi$
$992$	0	0
$993$	9.40894	0.298584
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.4495	1.50274	0.751370	−	0.659881i	$-0.229393\pi$
$997$	47.4495	1.50274	0.751370	+	0.659881i	$0.229393\pi$
$998$	0	0
$999$	−2.57310	−0.0814092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.br.1.2		3
4.3	odd	2		3400.2.a.l.1.2		3
5.4	even	2		1360.2.a.q.1.2		3
20.3	even	4		3400.2.e.j.2449.3		6
20.7	even	4		3400.2.e.j.2449.4		6
20.19	odd	2		680.2.a.g.1.2	✓	3
40.19	odd	2		5440.2.a.bp.1.2		3
40.29	even	2		5440.2.a.bs.1.2		3
60.59	even	2		6120.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.g.1.2	✓	3	20.19	odd	2
1360.2.a.q.1.2		3	5.4	even	2
3400.2.a.l.1.2		3	4.3	odd	2
3400.2.e.j.2449.3		6	20.3	even	4
3400.2.e.j.2449.4		6	20.7	even	4
5440.2.a.bp.1.2		3	40.19	odd	2
5440.2.a.bs.1.2		3	40.29	even	2
6120.2.a.bt.1.1		3	60.59	even	2
6800.2.a.br.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.321637 q^{3} -2.21819 q^{7} -2.89655 q^{9} +O(q^{10})\)	\(q+0.321637 q^{3} -2.21819 q^{7} -2.89655 q^{9} -4.21819 q^{11} -6.53982 q^{13} +1.00000 q^{17} -5.25328 q^{19} -0.713451 q^{21} +6.21819 q^{23} -1.89655 q^{27} +7.25328 q^{29} -5.67836 q^{31} -1.35673 q^{33} +1.35673 q^{37} -2.10345 q^{39} -10.4364 q^{41} -2.43637 q^{43} +0.103450 q^{47} -2.07965 q^{49} +0.321637 q^{51} +11.8965 q^{53} -1.68965 q^{57} -1.25328 q^{59} +3.89655 q^{61} +6.42509 q^{63} -3.28655 q^{67} +2.00000 q^{69} -12.8282 q^{71} +4.53982 q^{73} +9.35673 q^{77} -9.50474 q^{79} +8.07965 q^{81} -0.0701770 q^{83} +2.33292 q^{87} +9.68965 q^{89} +14.5066 q^{91} -1.82637 q^{93} -11.6896 q^{97} +12.2182 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q + q^3 + 6 * q^7 + 4 * q^9	\( 3 q + q^{3} + 6 q^{7} + 4 q^{9} - 7 q^{13} + 3 q^{17} - 3 q^{19} - 2 q^{21} + 6 q^{23} + 7 q^{27} + 9 q^{29} - 17 q^{31} - 4 q^{33} + 4 q^{37} - 19 q^{39} - 6 q^{41} + 18 q^{43} + 13 q^{47} + 19 q^{49} + q^{51} + 23 q^{53} + 33 q^{57} + 9 q^{59} - q^{61} + 32 q^{63} - 10 q^{67} + 6 q^{69} - 13 q^{71} + q^{73} + 28 q^{77} - 16 q^{79} - q^{81} - 31 q^{87} - 9 q^{89} + 18 q^{91} + 7 q^{93} + 3 q^{97} + 24 q^{99}+O(q^{100}) \) 3 * q + q^3 + 6 * q^7 + 4 * q^9 - 7 * q^13 + 3 * q^17 - 3 * q^19 - 2 * q^21 + 6 * q^23 + 7 * q^27 + 9 * q^29 - 17 * q^31 - 4 * q^33 + 4 * q^37 - 19 * q^39 - 6 * q^41 + 18 * q^43 + 13 * q^47 + 19 * q^49 + q^51 + 23 * q^53 + 33 * q^57 + 9 * q^59 - q^61 + 32 * q^63 - 10 * q^67 + 6 * q^69 - 13 * q^71 + q^73 + 28 * q^77 - 16 * q^79 - q^81 - 31 * q^87 - 9 * q^89 + 18 * q^91 + 7 * q^93 + 3 * q^97 + 24 * q^99

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