LMFDB - Embedded newform 6800.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+2.73205 q^{3} -2.73205 q^{7} +4.46410 q^{9} -4.73205 q^{11} +4.00000 q^{13} +1.00000 q^{17} +1.46410 q^{19} -7.46410 q^{21} -8.19615 q^{23} +4.00000 q^{27} -3.46410 q^{29} -3.26795 q^{31} -12.9282 q^{33} +0.535898 q^{37} +10.9282 q^{39} -3.46410 q^{41} -0.535898 q^{43} +12.9282 q^{47} +0.464102 q^{49} +2.73205 q^{51} -6.00000 q^{53} +4.00000 q^{57} -2.53590 q^{59} -4.92820 q^{61} -12.1962 q^{63} -10.0000 q^{67} -22.3923 q^{69} -11.6603 q^{71} -6.39230 q^{73} +12.9282 q^{77} -14.5885 q^{79} -2.46410 q^{81} +8.53590 q^{83} -9.46410 q^{87} +4.39230 q^{89} -10.9282 q^{91} -8.92820 q^{93} +4.92820 q^{97} -21.1244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 8 q^{13} + 2 q^{17} - 4 q^{19} - 8 q^{21} - 6 q^{23} + 8 q^{27} - 10 q^{31} - 12 q^{33} + 8 q^{37} + 8 q^{39} - 8 q^{43} + 12 q^{47} - 6 q^{49} + 2 q^{51}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 8 * q^13 + 2 * q^17 - 4 * q^19 - 8 * q^21 - 6 * q^23 + 8 * q^27 - 10 * q^31 - 12 * q^33 + 8 * q^37 + 8 * q^39 - 8 * q^43 + 12 * q^47 - 6 * q^49 + 2 * q^51 - 12 * q^53 + 8 * q^57 - 12 * q^59 + 4 * q^61 - 14 * q^63 - 20 * q^67 - 24 * q^69 - 6 * q^71 + 8 * q^73 + 12 * q^77 + 2 * q^79 + 2 * q^81 + 24 * q^83 - 12 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^93 - 4 * q^97 - 18 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	−7.46410	−1.62880
$22$	0	0
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−3.26795	−0.586941	−0.293471	−	0.955968i	$-0.594810\pi$
$31$	−3.26795	−0.586941	−0.293471	+	0.955968i	$0.594810\pi$
$32$	0	0
$33$	−12.9282	−2.25051
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	10.9282	1.74991
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−0.535898	−0.0817237	−0.0408619	−	0.999165i	$-0.513010\pi$
$43$	−0.535898	−0.0817237	−0.0408619	+	0.999165i	$0.513010\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9282	1.88577	0.942886	−	0.333115i	$-0.108100\pi$
$47$	12.9282	1.88577	0.942886	+	0.333115i	$0.108100\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.73205	0.382564
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	−12.1962	−1.53657
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−22.3923	−2.69572
$70$	0	0
$71$	−11.6603	−1.38382	−0.691909	−	0.721985i	$-0.743230\pi$
$71$	−11.6603	−1.38382	−0.691909	+	0.721985i	$0.743230\pi$
$72$	0	0
$73$	−6.39230	−0.748163	−0.374081	−	0.927396i	$-0.622042\pi$
$73$	−6.39230	−0.748163	−0.374081	+	0.927396i	$0.622042\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.9282	1.47331
$78$	0	0
$79$	−14.5885	−1.64133	−0.820665	−	0.571410i	$-0.806397\pi$
$79$	−14.5885	−1.64133	−0.820665	+	0.571410i	$0.806397\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	8.53590	0.936937	0.468468	−	0.883480i	$-0.344806\pi$
$83$	8.53590	0.936937	0.468468	+	0.883480i	$0.344806\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.46410	−1.01466
$88$	0	0
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	0	0
$91$	−10.9282	−1.14559
$92$	0	0
$93$	−8.92820	−0.925812
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.92820	0.500383	0.250192	−	0.968196i	$-0.419506\pi$
$97$	4.92820	0.500383	0.250192	+	0.968196i	$0.419506\pi$
$98$	0	0
$99$	−21.1244	−2.12308
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	0	0
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.6603	1.70728	0.853641	−	0.520862i	$-0.174390\pi$
$107$	17.6603	1.70728	0.853641	+	0.520862i	$0.174390\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	1.46410	0.138966
$112$	0	0
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	17.8564	1.65083
$118$	0	0
$119$	−2.73205	−0.250447
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	−9.46410	−0.853349
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.3923	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$127$	−14.3923	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$128$	0	0
$129$	−1.46410	−0.128907
$130$	0	0
$131$	2.19615	0.191879	0.0959394	−	0.995387i	$-0.469415\pi$
$131$	2.19615	0.191879	0.0959394	+	0.995387i	$0.469415\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	0	0
$141$	35.3205	2.97452
$142$	0	0
$143$	−18.9282	−1.58286
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.26795	0.104579
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	0	0
$153$	4.46410	0.360901
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.92820	−0.712548	−0.356274	−	0.934381i	$-0.615953\pi$
$157$	−8.92820	−0.712548	−0.356274	+	0.934381i	$0.615953\pi$
$158$	0	0
$159$	−16.3923	−1.29999
$160$	0	0
$161$	22.3923	1.76476
$162$	0	0
$163$	−0.196152	−0.0153638	−0.00768192	−	0.999970i	$-0.502445\pi$
$163$	−0.196152	−0.0153638	−0.00768192	+	0.999970i	$0.502445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.5885	0.974124	0.487062	−	0.873367i	$-0.338069\pi$
$167$	12.5885	0.974124	0.487062	+	0.873367i	$0.338069\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	6.53590	0.499813
$172$	0	0
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.92820	−0.520756
$178$	0	0
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	0	0
$183$	−13.4641	−0.995295
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.73205	−0.346042
$188$	0	0
$189$	−10.9282	−0.794910
$190$	0	0
$191$	−1.85641	−0.134325	−0.0671624	−	0.997742i	$-0.521395\pi$
$191$	−1.85641	−0.134325	−0.0671624	+	0.997742i	$0.521395\pi$
$192$	0	0
$193$	−16.5359	−1.19028	−0.595140	−	0.803622i	$-0.702903\pi$
$193$	−16.5359	−1.19028	−0.595140	+	0.803622i	$0.702903\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.3205	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$198$	0	0
$199$	−10.1962	−0.722786	−0.361393	−	0.932414i	$-0.617699\pi$
$199$	−10.1962	−0.722786	−0.361393	+	0.932414i	$0.617699\pi$
$200$	0	0
$201$	−27.3205	−1.92704
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−36.5885	−2.54307
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	−10.1962	−0.701932	−0.350966	−	0.936388i	$-0.614147\pi$
$211$	−10.1962	−0.701932	−0.350966	+	0.936388i	$0.614147\pi$
$212$	0	0
$213$	−31.8564	−2.18277
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.92820	0.606086
$218$	0	0
$219$	−17.4641	−1.18011
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−26.3923	−1.76736	−0.883680	−	0.468092i	$-0.844942\pi$
$223$	−26.3923	−1.76736	−0.883680	+	0.468092i	$0.844942\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.7321	1.50878	0.754390	−	0.656427i	$-0.227933\pi$
$227$	22.7321	1.50878	0.754390	+	0.656427i	$0.227933\pi$
$228$	0	0
$229$	−8.39230	−0.554579	−0.277290	−	0.960786i	$-0.589436\pi$
$229$	−8.39230	−0.554579	−0.277290	+	0.960786i	$0.589436\pi$
$230$	0	0
$231$	35.3205	2.32392
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−39.8564	−2.58895
$238$	0	0
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.85641	0.372634
$248$	0	0
$249$	23.3205	1.47788
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	38.7846	2.43837
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	0	0
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	−15.4641	−0.957204
$262$	0	0
$263$	−1.60770	−0.0991347	−0.0495674	−	0.998771i	$-0.515784\pi$
$263$	−1.60770	−0.0991347	−0.0495674	+	0.998771i	$0.515784\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	0.928203	0.0565935	0.0282968	−	0.999600i	$-0.490992\pi$
$269$	0.928203	0.0565935	0.0282968	+	0.999600i	$0.490992\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	0	0
$273$	−29.8564	−1.80699
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.9282	−1.25745	−0.628727	−	0.777626i	$-0.716424\pi$
$277$	−20.9282	−1.25745	−0.628727	+	0.777626i	$0.716424\pi$
$278$	0	0
$279$	−14.5885	−0.873388
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	0	0
$283$	−5.26795	−0.313147	−0.156574	−	0.987666i	$-0.550045\pi$
$283$	−5.26795	−0.313147	−0.156574	+	0.987666i	$0.550045\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.46410	0.558648
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.4641	0.789280
$292$	0	0
$293$	0.928203	0.0542262	0.0271131	−	0.999632i	$-0.491369\pi$
$293$	0.928203	0.0542262	0.0271131	+	0.999632i	$0.491369\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−18.9282	−1.09833
$298$	0	0
$299$	−32.7846	−1.89598
$300$	0	0
$301$	1.46410	0.0843894
$302$	0	0
$303$	−25.8564	−1.48541
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	24.3923	1.38763
$310$	0	0
$311$	16.0526	0.910257	0.455129	−	0.890426i	$-0.349593\pi$
$311$	16.0526	0.910257	0.455129	+	0.890426i	$0.349593\pi$
$312$	0	0
$313$	26.3923	1.49178	0.745891	−	0.666068i	$-0.232024\pi$
$313$	26.3923	1.49178	0.745891	+	0.666068i	$0.232024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	0	0
$319$	16.3923	0.917793
$320$	0	0
$321$	48.2487	2.69298
$322$	0	0
$323$	1.46410	0.0814648
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−27.3205	−1.51083
$328$	0	0
$329$	−35.3205	−1.94728
$330$	0	0
$331$	6.53590	0.359245	0.179623	−	0.983736i	$-0.442512\pi$
$331$	6.53590	0.359245	0.179623	+	0.983736i	$0.442512\pi$
$332$	0	0
$333$	2.39230	0.131097
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.78461	0.369581	0.184791	−	0.982778i	$-0.440839\pi$
$337$	6.78461	0.369581	0.184791	+	0.982778i	$0.440839\pi$
$338$	0	0
$339$	47.3205	2.57010
$340$	0	0
$341$	15.4641	0.837428
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.80385	0.204201	0.102101	−	0.994774i	$-0.467444\pi$
$347$	3.80385	0.204201	0.102101	+	0.994774i	$0.467444\pi$
$348$	0	0
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.46410	−0.395042
$358$	0	0
$359$	21.4641	1.13283	0.566416	−	0.824119i	$-0.308330\pi$
$359$	21.4641	1.13283	0.566416	+	0.824119i	$0.308330\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	31.1244	1.63361
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.80385	−0.407358	−0.203679	−	0.979038i	$-0.565290\pi$
$367$	−7.80385	−0.407358	−0.203679	+	0.979038i	$0.565290\pi$
$368$	0	0
$369$	−15.4641	−0.805029
$370$	0	0
$371$	16.3923	0.851046
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.8564	−0.713641
$378$	0	0
$379$	−17.8038	−0.914522	−0.457261	−	0.889332i	$-0.651170\pi$
$379$	−17.8038	−0.914522	−0.457261	+	0.889332i	$0.651170\pi$
$380$	0	0
$381$	−39.3205	−2.01445
$382$	0	0
$383$	15.4641	0.790179	0.395089	−	0.918643i	$-0.370714\pi$
$383$	15.4641	0.790179	0.395089	+	0.918643i	$0.370714\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.39230	−0.121608
$388$	0	0
$389$	−4.39230	−0.222699	−0.111349	−	0.993781i	$-0.535517\pi$
$389$	−4.39230	−0.222699	−0.111349	+	0.993781i	$0.535517\pi$
$390$	0	0
$391$	−8.19615	−0.414497
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	−10.9282	−0.547094
$400$	0	0
$401$	12.9282	0.645604	0.322802	−	0.946467i	$-0.395375\pi$
$401$	12.9282	0.645604	0.322802	+	0.946467i	$0.395375\pi$
$402$	0	0
$403$	−13.0718	−0.651153
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.53590	−0.125700
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	−38.1962	−1.86600	−0.933002	−	0.359871i	$-0.882821\pi$
$419$	−38.1962	−1.86600	−0.933002	+	0.359871i	$0.882821\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	0	0
$423$	57.7128	2.80609
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4641	0.651574
$428$	0	0
$429$	−51.7128	−2.49672
$430$	0	0
$431$	−9.80385	−0.472235	−0.236117	−	0.971725i	$-0.575875\pi$
$431$	−9.80385	−0.472235	−0.236117	+	0.971725i	$0.575875\pi$
$432$	0	0
$433$	17.8564	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$433$	17.8564	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	20.0526	0.957056	0.478528	−	0.878072i	$-0.341170\pi$
$439$	20.0526	0.957056	0.478528	+	0.878072i	$0.341170\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	0	0
$443$	−12.9282	−0.614237	−0.307119	−	0.951671i	$-0.599365\pi$
$443$	−12.9282	−0.614237	−0.307119	+	0.951671i	$0.599365\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.3923	−0.775329
$448$	0	0
$449$	−34.3923	−1.62307	−0.811537	−	0.584302i	$-0.801369\pi$
$449$	−34.3923	−1.62307	−0.811537	+	0.584302i	$0.801369\pi$
$450$	0	0
$451$	16.3923	0.771883
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.7846	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$457$	36.7846	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−24.9282	−1.16102	−0.580511	−	0.814252i	$-0.697147\pi$
$461$	−24.9282	−1.16102	−0.580511	+	0.814252i	$0.697147\pi$
$462$	0	0
$463$	−23.8564	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$463$	−23.8564	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.60770	0.0743953	0.0371976	−	0.999308i	$-0.488157\pi$
$467$	1.60770	0.0743953	0.0371976	+	0.999308i	$0.488157\pi$
$468$	0	0
$469$	27.3205	1.26154
$470$	0	0
$471$	−24.3923	−1.12394
$472$	0	0
$473$	2.53590	0.116601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−26.7846	−1.22638
$478$	0	0
$479$	11.6603	0.532771	0.266385	−	0.963867i	$-0.414171\pi$
$479$	11.6603	0.532771	0.266385	+	0.963867i	$0.414171\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	0	0
$483$	61.1769	2.78365
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.9808	1.13199	0.565993	−	0.824410i	$-0.308493\pi$
$487$	24.9808	1.13199	0.565993	+	0.824410i	$0.308493\pi$
$488$	0	0
$489$	−0.535898	−0.0242342
$490$	0	0
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.8564	1.42896
$498$	0	0
$499$	15.6603	0.701049	0.350525	−	0.936554i	$-0.386003\pi$
$499$	15.6603	0.701049	0.350525	+	0.936554i	$0.386003\pi$
$500$	0	0
$501$	34.3923	1.53653
$502$	0	0
$503$	15.1244	0.674362	0.337181	−	0.941440i	$-0.390527\pi$
$503$	15.1244	0.674362	0.337181	+	0.941440i	$0.390527\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.19615	0.364004
$508$	0	0
$509$	19.8564	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$509$	19.8564	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$510$	0	0
$511$	17.4641	0.772566
$512$	0	0
$513$	5.85641	0.258567
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−61.1769	−2.69056
$518$	0	0
$519$	−9.46410	−0.415428
$520$	0	0
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.26795	−0.142354
$528$	0	0
$529$	44.1769	1.92074
$530$	0	0
$531$	−11.3205	−0.491268
$532$	0	0
$533$	−13.8564	−0.600188
$534$	0	0
$535$	0	0
$536$	0	0
$537$	30.9282	1.33465
$538$	0	0
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	39.1769	1.68435	0.842174	−	0.539207i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539207i	$0.181276\pi$
$542$	0	0
$543$	−6.53590	−0.280482
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.9090	−1.70638	−0.853192	−	0.521597i	$-0.825337\pi$
$547$	−39.9090	−1.70638	−0.853192	+	0.521597i	$0.825337\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	−5.07180	−0.216066
$552$	0	0
$553$	39.8564	1.69487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	−2.14359	−0.0906643
$560$	0	0
$561$	−12.9282	−0.545829
$562$	0	0
$563$	27.4641	1.15747	0.578737	−	0.815514i	$-0.303546\pi$
$563$	27.4641	1.15747	0.578737	+	0.815514i	$0.303546\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.73205	0.282720
$568$	0	0
$569$	40.6410	1.70376	0.851880	−	0.523737i	$-0.175463\pi$
$569$	40.6410	1.70376	0.851880	+	0.523737i	$0.175463\pi$
$570$	0	0
$571$	36.4449	1.52517	0.762585	−	0.646888i	$-0.223930\pi$
$571$	36.4449	1.52517	0.762585	+	0.646888i	$0.223930\pi$
$572$	0	0
$573$	−5.07180	−0.211877
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.6410	1.60865	0.804323	−	0.594192i	$-0.202528\pi$
$577$	38.6410	1.60865	0.804323	+	0.594192i	$0.202528\pi$
$578$	0	0
$579$	−45.1769	−1.87749
$580$	0	0
$581$	−23.3205	−0.967498
$582$	0	0
$583$	28.3923	1.17589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46.3923	1.91482	0.957408	−	0.288740i	$-0.0932362\pi$
$587$	46.3923	1.91482	0.957408	+	0.288740i	$0.0932362\pi$
$588$	0	0
$589$	−4.78461	−0.197146
$590$	0	0
$591$	−47.3205	−1.94651
$592$	0	0
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−27.8564	−1.14009
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	0	0
$603$	−44.6410	−1.81792
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−21.6603	−0.879163	−0.439581	−	0.898203i	$-0.644873\pi$
$607$	−21.6603	−0.879163	−0.439581	+	0.898203i	$0.644873\pi$
$608$	0	0
$609$	25.8564	1.04775
$610$	0	0
$611$	51.7128	2.09208
$612$	0	0
$613$	−15.8564	−0.640434	−0.320217	−	0.947344i	$-0.603756\pi$
$613$	−15.8564	−0.640434	−0.320217	+	0.947344i	$0.603756\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.4641	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$617$	27.4641	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$618$	0	0
$619$	−38.5885	−1.55100	−0.775501	−	0.631347i	$-0.782502\pi$
$619$	−38.5885	−1.55100	−0.775501	+	0.631347i	$0.782502\pi$
$620$	0	0
$621$	−32.7846	−1.31560
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18.9282	−0.755920
$628$	0	0
$629$	0.535898	0.0213677
$630$	0	0
$631$	32.3923	1.28952	0.644759	−	0.764386i	$-0.276958\pi$
$631$	32.3923	1.28952	0.644759	+	0.764386i	$0.276958\pi$
$632$	0	0
$633$	−27.8564	−1.10719
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.85641	0.0735535
$638$	0	0
$639$	−52.0526	−2.05917
$640$	0	0
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	0	0
$643$	−24.1962	−0.954203	−0.477102	−	0.878848i	$-0.658313\pi$
$643$	−24.1962	−0.954203	−0.477102	+	0.878848i	$0.658313\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.7846	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$647$	−38.7846	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	24.3923	0.956010
$652$	0	0
$653$	25.6077	1.00211	0.501053	−	0.865416i	$-0.332946\pi$
$653$	25.6077	1.00211	0.501053	+	0.865416i	$0.332946\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−28.5359	−1.11329
$658$	0	0
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	0	0
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	0	0
$663$	10.9282	0.424416
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.3923	1.09935
$668$	0	0
$669$	−72.1051	−2.78774
$670$	0	0
$671$	23.3205	0.900278
$672$	0	0
$673$	−23.4641	−0.904475	−0.452237	−	0.891898i	$-0.649374\pi$
$673$	−23.4641	−0.904475	−0.452237	+	0.891898i	$0.649374\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.78461	−0.107021	−0.0535106	−	0.998567i	$-0.517041\pi$
$677$	−2.78461	−0.107021	−0.0535106	+	0.998567i	$0.517041\pi$
$678$	0	0
$679$	−13.4641	−0.516705
$680$	0	0
$681$	62.1051	2.37987
$682$	0	0
$683$	−30.8372	−1.17995	−0.589976	−	0.807421i	$-0.700863\pi$
$683$	−30.8372	−1.17995	−0.589976	+	0.807421i	$0.700863\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.9282	−0.874766
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	38.9808	1.48290	0.741449	−	0.671009i	$-0.234139\pi$
$691$	38.9808	1.48290	0.741449	+	0.671009i	$0.234139\pi$
$692$	0	0
$693$	57.7128	2.19233
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.46410	−0.131212
$698$	0	0
$699$	−16.3923	−0.620014
$700$	0	0
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	0	0
$703$	0.784610	0.0295921
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.8564	0.972430
$708$	0	0
$709$	4.53590	0.170349	0.0851746	−	0.996366i	$-0.472855\pi$
$709$	4.53590	0.170349	0.0851746	+	0.996366i	$0.472855\pi$
$710$	0	0
$711$	−65.1244	−2.44235
$712$	0	0
$713$	26.7846	1.00309
$714$	0	0
$715$	0	0
$716$	0	0
$717$	56.7846	2.12066
$718$	0	0
$719$	5.41154	0.201816	0.100908	−	0.994896i	$-0.467825\pi$
$719$	5.41154	0.201816	0.100908	+	0.994896i	$0.467825\pi$
$720$	0	0
$721$	−24.3923	−0.908417
$722$	0	0
$723$	−15.3205	−0.569776
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−0.535898	−0.0198209
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	47.3205	1.74307
$738$	0	0
$739$	11.6077	0.426996	0.213498	−	0.976944i	$-0.431514\pi$
$739$	11.6077	0.426996	0.213498	+	0.976944i	$0.431514\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−22.7321	−0.833958	−0.416979	−	0.908916i	$-0.636911\pi$
$743$	−22.7321	−0.833958	−0.416979	+	0.908916i	$0.636911\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	38.1051	1.39419
$748$	0	0
$749$	−48.2487	−1.76297
$750$	0	0
$751$	−43.6603	−1.59319	−0.796593	−	0.604516i	$-0.793366\pi$
$751$	−43.6603	−1.59319	−0.796593	+	0.604516i	$0.793366\pi$
$752$	0	0
$753$	−18.9282	−0.689782
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.6410	−1.11367	−0.556833	−	0.830624i	$-0.687984\pi$
$757$	−30.6410	−1.11367	−0.556833	+	0.830624i	$0.687984\pi$
$758$	0	0
$759$	105.962	3.84616
$760$	0	0
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	0	0
$763$	27.3205	0.989069
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.1436	−0.366264
$768$	0	0
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	0	0
$771$	18.9282	0.681683
$772$	0	0
$773$	46.6410	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$773$	46.6410	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−5.07180	−0.181716
$780$	0	0
$781$	55.1769	1.97439
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.9090	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$787$	43.9090	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$788$	0	0
$789$	−4.39230	−0.156370
$790$	0	0
$791$	−47.3205	−1.68252
$792$	0	0
$793$	−19.7128	−0.700023
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.9282	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$797$	24.9282	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$798$	0	0
$799$	12.9282	0.457367
$800$	0	0
$801$	19.6077	0.692804
$802$	0	0
$803$	30.2487	1.06745
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.53590	0.0892679
$808$	0	0
$809$	−55.8564	−1.96381	−0.981903	−	0.189383i	$-0.939351\pi$
$809$	−55.8564	−1.96381	−0.981903	+	0.189383i	$0.939351\pi$
$810$	0	0
$811$	−44.8372	−1.57445	−0.787223	−	0.616668i	$-0.788482\pi$
$811$	−44.8372	−1.57445	−0.787223	+	0.616668i	$0.788482\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.784610	−0.0274500
$818$	0	0
$819$	−48.7846	−1.70467
$820$	0	0
$821$	−24.9282	−0.870000	−0.435000	−	0.900430i	$-0.643252\pi$
$821$	−24.9282	−0.870000	−0.435000	+	0.900430i	$0.643252\pi$
$822$	0	0
$823$	−8.98076	−0.313050	−0.156525	−	0.987674i	$-0.550029\pi$
$823$	−8.98076	−0.313050	−0.156525	+	0.987674i	$0.550029\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.8038	−0.549554	−0.274777	−	0.961508i	$-0.588604\pi$
$827$	−15.8038	−0.549554	−0.274777	+	0.961508i	$0.588604\pi$
$828$	0	0
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	0	0
$831$	−57.1769	−1.98345
$832$	0	0
$833$	0.464102	0.0160802
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−13.0718	−0.451827
$838$	0	0
$839$	−19.2679	−0.665203	−0.332602	−	0.943067i	$-0.607926\pi$
$839$	−19.2679	−0.665203	−0.332602	+	0.943067i	$0.607926\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−35.3205	−1.21650
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−31.1244	−1.06945
$848$	0	0
$849$	−14.3923	−0.493943
$850$	0	0
$851$	−4.39230	−0.150566
$852$	0	0
$853$	23.1769	0.793562	0.396781	−	0.917913i	$-0.370127\pi$
$853$	23.1769	0.793562	0.396781	+	0.917913i	$0.370127\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.1769	−1.06498	−0.532492	−	0.846435i	$-0.678744\pi$
$857$	−31.1769	−1.06498	−0.532492	+	0.846435i	$0.678744\pi$
$858$	0	0
$859$	25.4641	0.868824	0.434412	−	0.900714i	$-0.356956\pi$
$859$	25.4641	0.868824	0.434412	+	0.900714i	$0.356956\pi$
$860$	0	0
$861$	25.8564	0.881184
$862$	0	0
$863$	23.0718	0.785373	0.392687	−	0.919672i	$-0.371546\pi$
$863$	23.0718	0.785373	0.392687	+	0.919672i	$0.371546\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.73205	0.0927853
$868$	0	0
$869$	69.0333	2.34180
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	22.0000	0.744587
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.21539	0.0410408	0.0205204	−	0.999789i	$-0.493468\pi$
$877$	1.21539	0.0410408	0.0205204	+	0.999789i	$0.493468\pi$
$878$	0	0
$879$	2.53590	0.0855337
$880$	0	0
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	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
$887$	3.12436	0.104906	0.0524528	−	0.998623i	$-0.483296\pi$
$887$	3.12436	0.104906	0.0524528	+	0.998623i	$0.483296\pi$
$888$	0	0
$889$	39.3205	1.31877
$890$	0	0
$891$	11.6603	0.390633
$892$	0	0
$893$	18.9282	0.633408
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−89.5692	−2.99063
$898$	0	0
$899$	11.3205	0.377560
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.7321	1.02044	0.510220	−	0.860044i	$-0.329564\pi$
$907$	30.7321	1.02044	0.510220	+	0.860044i	$0.329564\pi$
$908$	0	0
$909$	−42.2487	−1.40130
$910$	0	0
$911$	−36.3397	−1.20399	−0.601995	−	0.798500i	$-0.705627\pi$
$911$	−36.3397	−1.20399	−0.601995	+	0.798500i	$0.705627\pi$
$912$	0	0
$913$	−40.3923	−1.33679
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−27.3205	−0.900241
$922$	0	0
$923$	−46.6410	−1.53521
$924$	0	0
$925$	0	0
$926$	0	0
$927$	39.8564	1.30906
$928$	0	0
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	0.679492	0.0222694
$932$	0	0
$933$	43.8564	1.43579
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.6410	1.06634	0.533168	−	0.846010i	$-0.321001\pi$
$937$	32.6410	1.06634	0.533168	+	0.846010i	$0.321001\pi$
$938$	0	0
$939$	72.1051	2.35306
$940$	0	0
$941$	48.2487	1.57286	0.786432	−	0.617677i	$-0.211926\pi$
$941$	48.2487	1.57286	0.786432	+	0.617677i	$0.211926\pi$
$942$	0	0
$943$	28.3923	0.924581
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.19615	0.266339	0.133170	−	0.991093i	$-0.457485\pi$
$947$	8.19615	0.266339	0.133170	+	0.991093i	$0.457485\pi$
$948$	0	0
$949$	−25.5692	−0.830012
$950$	0	0
$951$	68.1051	2.20846
$952$	0	0
$953$	8.78461	0.284561	0.142281	−	0.989826i	$-0.454556\pi$
$953$	8.78461	0.284561	0.142281	+	0.989826i	$0.454556\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	44.7846	1.44768
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.3205	−0.655500
$962$	0	0
$963$	78.8372	2.54049
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.1769	1.25984	0.629922	−	0.776658i	$-0.283087\pi$
$967$	39.1769	1.25984	0.629922	+	0.776658i	$0.283087\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−54.2487	−1.74092	−0.870462	−	0.492236i	$-0.836180\pi$
$971$	−54.2487	−1.74092	−0.870462	+	0.492236i	$0.836180\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−20.7846	−0.664279
$980$	0	0
$981$	−44.6410	−1.42528
$982$	0	0
$983$	58.7321	1.87326	0.936631	−	0.350318i	$-0.113927\pi$
$983$	58.7321	1.87326	0.936631	+	0.350318i	$0.113927\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−96.4974	−3.07155
$988$	0	0
$989$	4.39230	0.139667
$990$	0	0
$991$	2.98076	0.0946870	0.0473435	−	0.998879i	$-0.484924\pi$
$991$	2.98076	0.0946870	0.0473435	+	0.998879i	$0.484924\pi$
$992$	0	0
$993$	17.8564	0.566656
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.39230	0.0757651	0.0378825	−	0.999282i	$-0.487939\pi$
$997$	2.39230	0.0757651	0.0378825	+	0.999282i	$0.487939\pi$
$998$	0	0
$999$	2.14359	0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bg.1.2		2
4.3	odd	2		425.2.a.e.1.2		2
5.4	even	2		1360.2.a.k.1.1		2
12.11	even	2		3825.2.a.v.1.1		2
20.3	even	4		425.2.b.d.324.1		4
20.7	even	4		425.2.b.d.324.4		4
20.19	odd	2		85.2.a.c.1.1	✓	2
40.19	odd	2		5440.2.a.bb.1.1		2
40.29	even	2		5440.2.a.bl.1.2		2
60.59	even	2		765.2.a.g.1.2		2
68.67	odd	2		7225.2.a.l.1.2		2
140.139	even	2		4165.2.a.t.1.1		2
340.259	odd	4		1445.2.d.e.866.3		4
340.319	odd	4		1445.2.d.e.866.4		4
340.339	odd	2		1445.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	20.19	odd	2
425.2.a.e.1.2		2	4.3	odd	2
425.2.b.d.324.1		4	20.3	even	4
425.2.b.d.324.4		4	20.7	even	4
765.2.a.g.1.2		2	60.59	even	2
1360.2.a.k.1.1		2	5.4	even	2
1445.2.a.g.1.1		2	340.339	odd	2
1445.2.d.e.866.3		4	340.259	odd	4
1445.2.d.e.866.4		4	340.319	odd	4
3825.2.a.v.1.1		2	12.11	even	2
4165.2.a.t.1.1		2	140.139	even	2
5440.2.a.bb.1.1		2	40.19	odd	2
5440.2.a.bl.1.2		2	40.29	even	2
6800.2.a.bg.1.2		2	1.1	even	1	trivial
7225.2.a.l.1.2		2	68.67	odd	2

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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)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} -2.73205 q^{7} +4.46410 q^{9} -4.73205 q^{11} +4.00000 q^{13} +1.00000 q^{17} +1.46410 q^{19} -7.46410 q^{21} -8.19615 q^{23} +4.00000 q^{27} -3.46410 q^{29} -3.26795 q^{31} -12.9282 q^{33} +0.535898 q^{37} +10.9282 q^{39} -3.46410 q^{41} -0.535898 q^{43} +12.9282 q^{47} +0.464102 q^{49} +2.73205 q^{51} -6.00000 q^{53} +4.00000 q^{57} -2.53590 q^{59} -4.92820 q^{61} -12.1962 q^{63} -10.0000 q^{67} -22.3923 q^{69} -11.6603 q^{71} -6.39230 q^{73} +12.9282 q^{77} -14.5885 q^{79} -2.46410 q^{81} +8.53590 q^{83} -9.46410 q^{87} +4.39230 q^{89} -10.9282 q^{91} -8.92820 q^{93} +4.92820 q^{97} -21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 8 q^{13} + 2 q^{17} - 4 q^{19} - 8 q^{21} - 6 q^{23} + 8 q^{27} - 10 q^{31} - 12 q^{33} + 8 q^{37} + 8 q^{39} - 8 q^{43} + 12 q^{47} - 6 q^{49} + 2 q^{51}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 8 * q^13 + 2 * q^17 - 4 * q^19 - 8 * q^21 - 6 * q^23 + 8 * q^27 - 10 * q^31 - 12 * q^33 + 8 * q^37 + 8 * q^39 - 8 * q^43 + 12 * q^47 - 6 * q^49 + 2 * q^51 - 12 * q^53 + 8 * q^57 - 12 * q^59 + 4 * q^61 - 14 * q^63 - 20 * q^67 - 24 * q^69 - 6 * q^71 + 8 * q^73 + 12 * q^77 + 2 * q^79 + 2 * q^81 + 24 * q^83 - 12 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^93 - 4 * q^97 - 18 * q^99

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