LMFDB - Embedded newform 6840.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.38776 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +4.38776 q^{7} -6.38776 q^{11} -1.13536 q^{13} -1.72928 q^{17} +1.00000 q^{19} -1.52311 q^{23} +1.00000 q^{25} -6.65847 q^{29} -3.72928 q^{31} +4.38776 q^{35} -4.59392 q^{37} +9.64015 q^{41} +2.59392 q^{43} +9.52311 q^{47} +12.2524 q^{49} -11.5231 q^{53} -6.38776 q^{55} -14.9817 q^{59} -1.79383 q^{61} -1.13536 q^{65} -7.04623 q^{67} +9.31695 q^{71} -8.50479 q^{73} -28.0279 q^{77} -9.25240 q^{79} -9.04623 q^{83} -1.72928 q^{85} -10.5939 q^{89} -4.98168 q^{91} +1.00000 q^{95} -4.59392 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 2 * q^7	$ 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.38776	1.65842	0.829208	−	0.558940i	$-0.188792\pi$
$7$	4.38776	1.65842	0.829208	+	0.558940i	$0.188792\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.38776	−1.92598	−0.962990	−	0.269536i	$-0.913130\pi$
$11$	−6.38776	−1.92598	−0.962990	+	0.269536i	$0.913130\pi$
$12$	0	0
$13$	−1.13536	−0.314892	−0.157446	−	0.987528i	$-0.550326\pi$
$13$	−1.13536	−0.314892	−0.157446	+	0.987528i	$0.550326\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.72928	−0.419412	−0.209706	−	0.977764i	$-0.567251\pi$
$17$	−1.72928	−0.419412	−0.209706	+	0.977764i	$0.567251\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.52311	−0.317591	−0.158796	−	0.987311i	$-0.550761\pi$
$23$	−1.52311	−0.317591	−0.158796	+	0.987311i	$0.550761\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.65847	−1.23645	−0.618224	−	0.786002i	$-0.712147\pi$
$29$	−6.65847	−1.23645	−0.618224	+	0.786002i	$0.712147\pi$
$30$	0	0
$31$	−3.72928	−0.669799	−0.334899	−	0.942254i	$-0.608702\pi$
$31$	−3.72928	−0.669799	−0.334899	+	0.942254i	$0.608702\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.38776	0.741666
$36$	0	0
$37$	−4.59392	−0.755236	−0.377618	−	0.925961i	$-0.623257\pi$
$37$	−4.59392	−0.755236	−0.377618	+	0.925961i	$0.623257\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.64015	1.50554	0.752769	−	0.658284i	$-0.228718\pi$
$41$	9.64015	1.50554	0.752769	+	0.658284i	$0.228718\pi$
$42$	0	0
$43$	2.59392	0.395569	0.197785	−	0.980245i	$-0.436625\pi$
$43$	2.59392	0.395569	0.197785	+	0.980245i	$0.436625\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.52311	1.38909	0.694544	−	0.719450i	$-0.255606\pi$
$47$	9.52311	1.38909	0.694544	+	0.719450i	$0.255606\pi$
$48$	0	0
$49$	12.2524	1.75034
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.5231	−1.58282	−0.791411	−	0.611285i	$-0.790653\pi$
$53$	−11.5231	−1.58282	−0.791411	+	0.611285i	$0.790653\pi$
$54$	0	0
$55$	−6.38776	−0.861325
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.9817	−1.95045	−0.975224	−	0.221219i	$-0.928996\pi$
$59$	−14.9817	−1.95045	−0.975224	+	0.221219i	$0.928996\pi$
$60$	0	0
$61$	−1.79383	−0.229677	−0.114838	−	0.993384i	$-0.536635\pi$
$61$	−1.79383	−0.229677	−0.114838	+	0.993384i	$0.536635\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.13536	−0.140824
$66$	0	0
$67$	−7.04623	−0.860834	−0.430417	−	0.902630i	$-0.641633\pi$
$67$	−7.04623	−0.860834	−0.430417	+	0.902630i	$0.641633\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.31695	1.10572	0.552859	−	0.833275i	$-0.313537\pi$
$71$	9.31695	1.10572	0.552859	+	0.833275i	$0.313537\pi$
$72$	0	0
$73$	−8.50479	−0.995411	−0.497705	−	0.867346i	$-0.665824\pi$
$73$	−8.50479	−0.995411	−0.497705	+	0.867346i	$0.665824\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−28.0279	−3.19408
$78$	0	0
$79$	−9.25240	−1.04098	−0.520488	−	0.853869i	$-0.674250\pi$
$79$	−9.25240	−1.04098	−0.520488	+	0.853869i	$0.674250\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.04623	−0.992953	−0.496476	−	0.868050i	$-0.665373\pi$
$83$	−9.04623	−0.992953	−0.496476	+	0.868050i	$0.665373\pi$
$84$	0	0
$85$	−1.72928	−0.187567
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.5939	−1.12295	−0.561477	−	0.827492i	$-0.689767\pi$
$89$	−10.5939	−1.12295	−0.561477	+	0.827492i	$0.689767\pi$
$90$	0	0
$91$	−4.98168	−0.522222
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−4.59392	−0.466442	−0.233221	−	0.972424i	$-0.574927\pi$
$97$	−4.59392	−0.466442	−0.233221	+	0.972424i	$0.574927\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.22449	−0.121841	−0.0609206	−	0.998143i	$-0.519404\pi$
$101$	−1.22449	−0.121841	−0.0609206	+	0.998143i	$0.519404\pi$
$102$	0	0
$103$	−7.04623	−0.694286	−0.347143	−	0.937812i	$-0.612848\pi$
$103$	−7.04623	−0.694286	−0.347143	+	0.937812i	$0.612848\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.45856	0.334352	0.167176	−	0.985927i	$-0.446535\pi$
$107$	3.45856	0.334352	0.167176	+	0.985927i	$0.446535\pi$
$108$	0	0
$109$	12.9817	1.24342	0.621710	−	0.783248i	$-0.286438\pi$
$109$	12.9817	1.24342	0.621710	+	0.783248i	$0.286438\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.7938	−1.29761	−0.648807	−	0.760953i	$-0.724732\pi$
$113$	−13.7938	−1.29761	−0.648807	+	0.760953i	$0.724732\pi$
$114$	0	0
$115$	−1.52311	−0.142031
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.58767	−0.695560
$120$	0	0
$121$	29.8034	2.70940
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.04623	0.625252	0.312626	−	0.949876i	$-0.398791\pi$
$127$	7.04623	0.625252	0.312626	+	0.949876i	$0.398791\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.117037	0.0102256	0.00511279	−	0.999987i	$-0.498373\pi$
$131$	0.117037	0.0102256	0.00511279	+	0.999987i	$0.498373\pi$
$132$	0	0
$133$	4.38776	0.380467
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2524	−0.961357	−0.480679	−	0.876897i	$-0.659610\pi$
$137$	−11.2524	−0.961357	−0.480679	+	0.876897i	$0.659610\pi$
$138$	0	0
$139$	13.3169	1.12953	0.564764	−	0.825252i	$-0.308967\pi$
$139$	13.3169	1.12953	0.564764	+	0.825252i	$0.308967\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.25240	0.606476
$144$	0	0
$145$	−6.65847	−0.552956
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	10.2341	0.832837	0.416419	−	0.909173i	$-0.363285\pi$
$151$	10.2341	0.832837	0.416419	+	0.909173i	$0.363285\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.72928	−0.299543
$156$	0	0
$157$	12.0279	0.959931	0.479966	−	0.877287i	$-0.340649\pi$
$157$	12.0279	0.959931	0.479966	+	0.877287i	$0.340649\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.68305	−0.526698
$162$	0	0
$163$	−14.4157	−1.12912	−0.564561	−	0.825391i	$-0.690954\pi$
$163$	−14.4157	−1.12912	−0.564561	+	0.825391i	$0.690954\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.79383	0.293576	0.146788	−	0.989168i	$-0.453107\pi$
$167$	3.79383	0.293576	0.146788	+	0.989168i	$0.453107\pi$
$168$	0	0
$169$	−11.7110	−0.900843
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.2986	0.935047	0.467524	−	0.883981i	$-0.345146\pi$
$173$	12.2986	0.935047	0.467524	+	0.883981i	$0.345146\pi$
$174$	0	0
$175$	4.38776	0.331683
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.02791	0.151573	0.0757864	−	0.997124i	$-0.475853\pi$
$179$	2.02791	0.151573	0.0757864	+	0.997124i	$0.475853\pi$
$180$	0	0
$181$	−17.5231	−1.30248	−0.651241	−	0.758871i	$-0.725751\pi$
$181$	−17.5231	−1.30248	−0.651241	+	0.758871i	$0.725751\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.59392	−0.337752
$186$	0	0
$187$	11.0462	0.807780
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3878	0.751632	0.375816	−	0.926694i	$-0.377363\pi$
$191$	10.3878	0.751632	0.375816	+	0.926694i	$0.377363\pi$
$192$	0	0
$193$	1.13536	0.0817249	0.0408625	−	0.999165i	$-0.486989\pi$
$193$	1.13536	0.0817249	0.0408625	+	0.999165i	$0.486989\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.5693	−1.03802	−0.519011	−	0.854767i	$-0.673700\pi$
$197$	−14.5693	−1.03802	−0.519011	+	0.854767i	$0.673700\pi$
$198$	0	0
$199$	9.18785	0.651309	0.325655	−	0.945489i	$-0.394415\pi$
$199$	9.18785	0.651309	0.325655	+	0.945489i	$0.394415\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−29.2158	−2.05054
$204$	0	0
$205$	9.64015	0.673297
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.38776	−0.441850
$210$	0	0
$211$	−14.5048	−0.998551	−0.499276	−	0.866443i	$-0.666400\pi$
$211$	−14.5048	−0.998551	−0.499276	+	0.866443i	$0.666400\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.59392	0.176904
$216$	0	0
$217$	−16.3632	−1.11080
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.96336	0.132070
$222$	0	0
$223$	−19.4586	−1.30304	−0.651521	−	0.758631i	$-0.725869\pi$
$223$	−19.4586	−1.30304	−0.651521	+	0.758631i	$0.725869\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2524	0.746848	0.373424	−	0.927661i	$-0.378184\pi$
$227$	11.2524	0.746848	0.373424	+	0.927661i	$0.378184\pi$
$228$	0	0
$229$	10.2062	0.674443	0.337221	−	0.941425i	$-0.390513\pi$
$229$	10.2062	0.674443	0.337221	+	0.941425i	$0.390513\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.3449	1.92245	0.961223	−	0.275774i	$-0.0889340\pi$
$233$	29.3449	1.92245	0.961223	+	0.275774i	$0.0889340\pi$
$234$	0	0
$235$	9.52311	0.621219
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.1170	−1.30126	−0.650631	−	0.759394i	$-0.725496\pi$
$239$	−20.1170	−1.30126	−0.650631	+	0.759394i	$0.725496\pi$
$240$	0	0
$241$	3.55102	0.228741	0.114371	−	0.993438i	$-0.463515\pi$
$241$	3.55102	0.228741	0.114371	+	0.993438i	$0.463515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.2524	0.782777
$246$	0	0
$247$	−1.13536	−0.0722412
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.836734	0.0528142	0.0264071	−	0.999651i	$-0.491593\pi$
$251$	0.836734	0.0528142	0.0264071	+	0.999651i	$0.491593\pi$
$252$	0	0
$253$	9.72928	0.611675
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.2986	−1.01668	−0.508340	−	0.861156i	$-0.669741\pi$
$257$	−16.2986	−1.01668	−0.508340	+	0.861156i	$0.669741\pi$
$258$	0	0
$259$	−20.1570	−1.25250
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.206167	0.0127128	0.00635641	−	0.999980i	$-0.497977\pi$
$263$	0.206167	0.0127128	0.00635641	+	0.999980i	$0.497977\pi$
$264$	0	0
$265$	−11.5231	−0.707859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.7509	1.50909	0.754545	−	0.656248i	$-0.227857\pi$
$269$	24.7509	1.50909	0.754545	+	0.656248i	$0.227857\pi$
$270$	0	0
$271$	18.3265	1.11326	0.556629	−	0.830761i	$-0.312095\pi$
$271$	18.3265	1.11326	0.556629	+	0.830761i	$0.312095\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.38776	−0.385196
$276$	0	0
$277$	7.25240	0.435754	0.217877	−	0.975976i	$-0.430087\pi$
$277$	7.25240	0.435754	0.217877	+	0.975976i	$0.430087\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.4061	−1.03836	−0.519180	−	0.854665i	$-0.673762\pi$
$281$	−17.4061	−1.03836	−0.519180	+	0.854665i	$0.673762\pi$
$282$	0	0
$283$	−4.32320	−0.256988	−0.128494	−	0.991710i	$-0.541014\pi$
$283$	−4.32320	−0.256988	−0.128494	+	0.991710i	$0.541014\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	42.2986	2.49681
$288$	0	0
$289$	−14.0096	−0.824093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.2524	−1.00790	−0.503948	−	0.863734i	$-0.668120\pi$
$293$	−17.2524	−1.00790	−0.503948	+	0.863734i	$0.668120\pi$
$294$	0	0
$295$	−14.9817	−0.872267
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.72928	0.100007
$300$	0	0
$301$	11.3815	0.656019
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.79383	−0.102715
$306$	0	0
$307$	−5.31695	−0.303454	−0.151727	−	0.988422i	$-0.548483\pi$
$307$	−5.31695	−0.303454	−0.151727	+	0.988422i	$0.548483\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.65847	0.490977	0.245488	−	0.969400i	$-0.421052\pi$
$311$	8.65847	0.490977	0.245488	+	0.969400i	$0.421052\pi$
$312$	0	0
$313$	−24.7389	−1.39832	−0.699162	−	0.714964i	$-0.746443\pi$
$313$	−24.7389	−1.39832	−0.699162	+	0.714964i	$0.746443\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0279	1.01255	0.506274	−	0.862373i	$-0.331022\pi$
$317$	18.0279	1.01255	0.506274	+	0.862373i	$0.331022\pi$
$318$	0	0
$319$	42.5327	2.38137
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.72928	−0.0962198
$324$	0	0
$325$	−1.13536	−0.0629784
$326$	0	0
$327$	0	0
$328$	0	0
$329$	41.7851	2.30369
$330$	0	0
$331$	12.9817	0.713538	0.356769	−	0.934193i	$-0.383878\pi$
$331$	12.9817	0.713538	0.356769	+	0.934193i	$0.383878\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.04623	−0.384977
$336$	0	0
$337$	−14.8646	−0.809729	−0.404864	−	0.914377i	$-0.632681\pi$
$337$	−14.8646	−0.809729	−0.404864	+	0.914377i	$0.632681\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.8217	1.29002
$342$	0	0
$343$	23.0462	1.24438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.5048	1.53022	0.765109	−	0.643901i	$-0.222685\pi$
$347$	28.5048	1.53022	0.765109	+	0.643901i	$0.222685\pi$
$348$	0	0
$349$	−24.5048	−1.31171	−0.655856	−	0.754886i	$-0.727692\pi$
$349$	−24.5048	−1.31171	−0.655856	+	0.754886i	$0.727692\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.206167	−0.0109732	−0.00548659	−	0.999985i	$-0.501746\pi$
$353$	−0.206167	−0.0109732	−0.00548659	+	0.999985i	$0.501746\pi$
$354$	0	0
$355$	9.31695	0.494492
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.6122	−1.14065	−0.570325	−	0.821419i	$-0.693183\pi$
$359$	−21.6122	−1.14065	−0.570325	+	0.821419i	$0.693183\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.50479	−0.445161
$366$	0	0
$367$	21.3973	1.11693	0.558466	−	0.829527i	$-0.311390\pi$
$367$	21.3973	1.11693	0.558466	+	0.829527i	$0.311390\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−50.5606	−2.62498
$372$	0	0
$373$	−1.67680	−0.0868212	−0.0434106	−	0.999057i	$-0.513822\pi$
$373$	−1.67680	−0.0868212	−0.0434106	+	0.999057i	$0.513822\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.55976	0.389347
$378$	0	0
$379$	7.01832	0.360507	0.180253	−	0.983620i	$-0.442308\pi$
$379$	7.01832	0.360507	0.180253	+	0.983620i	$0.442308\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	28.5972	1.46125	0.730626	−	0.682778i	$-0.239229\pi$
$383$	28.5972	1.46125	0.730626	+	0.682778i	$0.239229\pi$
$384$	0	0
$385$	−28.0279	−1.42843
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.6339	−1.24899	−0.624494	−	0.781030i	$-0.714695\pi$
$389$	−24.6339	−1.24899	−0.624494	+	0.781030i	$0.714695\pi$
$390$	0	0
$391$	2.63389	0.133202
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.25240	−0.465539
$396$	0	0
$397$	−10.8401	−0.544047	−0.272024	−	0.962291i	$-0.587693\pi$
$397$	−10.8401	−0.544047	−0.272024	+	0.962291i	$0.587693\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.6864	−0.833278	−0.416639	−	0.909072i	$-0.636792\pi$
$401$	−16.6864	−0.833278	−0.416639	+	0.909072i	$0.636792\pi$
$402$	0	0
$403$	4.23407	0.210914
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.3449	1.45457
$408$	0	0
$409$	24.3265	1.20287	0.601435	−	0.798922i	$-0.294596\pi$
$409$	24.3265	1.20287	0.601435	+	0.798922i	$0.294596\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−65.7359	−3.23465
$414$	0	0
$415$	−9.04623	−0.444062
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.6681	0.667728	0.333864	−	0.942621i	$-0.391647\pi$
$419$	13.6681	0.667728	0.333864	+	0.942621i	$0.391647\pi$
$420$	0	0
$421$	−15.7938	−0.769744	−0.384872	−	0.922970i	$-0.625754\pi$
$421$	−15.7938	−0.769744	−0.384872	+	0.922970i	$0.625754\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.72928	−0.0838825
$426$	0	0
$427$	−7.87090	−0.380899
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.7389	1.48064	0.740320	−	0.672255i	$-0.234674\pi$
$431$	30.7389	1.48064	0.740320	+	0.672255i	$0.234674\pi$
$432$	0	0
$433$	24.0525	1.15589	0.577944	−	0.816076i	$-0.303855\pi$
$433$	24.0525	1.15589	0.577944	+	0.816076i	$0.303855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.52311	−0.0728604
$438$	0	0
$439$	9.38150	0.447754	0.223877	−	0.974617i	$-0.428129\pi$
$439$	9.38150	0.447754	0.223877	+	0.974617i	$0.428129\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.1974	−1.71979	−0.859896	−	0.510469i	$-0.829472\pi$
$443$	−36.1974	−1.71979	−0.859896	+	0.510469i	$0.829472\pi$
$444$	0	0
$445$	−10.5939	−0.502200
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−29.3328	−1.38430	−0.692150	−	0.721754i	$-0.743336\pi$
$449$	−29.3328	−1.38430	−0.692150	+	0.721754i	$0.743336\pi$
$450$	0	0
$451$	−61.5789	−2.89964
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.98168	−0.233545
$456$	0	0
$457$	−32.1974	−1.50613	−0.753066	−	0.657945i	$-0.771426\pi$
$457$	−32.1974	−1.50613	−0.753066	+	0.657945i	$0.771426\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.6339	0.774718	0.387359	−	0.921929i	$-0.373387\pi$
$461$	16.6339	0.774718	0.387359	+	0.921929i	$0.373387\pi$
$462$	0	0
$463$	−32.2095	−1.49690	−0.748451	−	0.663190i	$-0.769202\pi$
$463$	−32.2095	−1.49690	−0.748451	+	0.663190i	$0.769202\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.81215	−0.407778	−0.203889	−	0.978994i	$-0.565358\pi$
$467$	−8.81215	−0.407778	−0.203889	+	0.978994i	$0.565358\pi$
$468$	0	0
$469$	−30.9171	−1.42762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.5693	−0.761859
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.30488	−0.0596216	−0.0298108	−	0.999556i	$-0.509490\pi$
$479$	−1.30488	−0.0596216	−0.0298108	+	0.999556i	$0.509490\pi$
$480$	0	0
$481$	5.21575	0.237818
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.59392	−0.208599
$486$	0	0
$487$	−24.3632	−1.10400	−0.552000	−	0.833844i	$-0.686135\pi$
$487$	−24.3632	−1.10400	−0.552000	+	0.833844i	$0.686135\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.7605	−1.79437	−0.897183	−	0.441658i	$-0.854390\pi$
$491$	−39.7605	−1.79437	−0.897183	+	0.441658i	$0.854390\pi$
$492$	0	0
$493$	11.5144	0.518581
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.8805	1.83374
$498$	0	0
$499$	42.3265	1.89480	0.947398	−	0.320058i	$-0.103702\pi$
$499$	42.3265	1.89480	0.947398	+	0.320058i	$0.103702\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	38.8959	1.73428	0.867141	−	0.498063i	$-0.165955\pi$
$503$	38.8959	1.73428	0.867141	+	0.498063i	$0.165955\pi$
$504$	0	0
$505$	−1.22449	−0.0544891
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.4340	−0.506802	−0.253401	−	0.967361i	$-0.581549\pi$
$509$	−11.4340	−0.506802	−0.253401	+	0.967361i	$0.581549\pi$
$510$	0	0
$511$	−37.3169	−1.65080
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.04623	−0.310494
$516$	0	0
$517$	−60.8313	−2.67536
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.5910	−1.82213	−0.911067	−	0.412258i	$-0.864740\pi$
$521$	−41.5910	−1.82213	−0.911067	+	0.412258i	$0.864740\pi$
$522$	0	0
$523$	−32.8313	−1.43561	−0.717807	−	0.696242i	$-0.754854\pi$
$523$	−32.8313	−1.43561	−0.717807	+	0.696242i	$0.754854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.44898	0.280922
$528$	0	0
$529$	−20.6801	−0.899136
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.9450	−0.474082
$534$	0	0
$535$	3.45856	0.149527
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−78.2653	−3.37113
$540$	0	0
$541$	−0.0924575	−0.00397506	−0.00198753	−	0.999998i	$-0.500633\pi$
$541$	−0.0924575	−0.00397506	−0.00198753	+	0.999998i	$0.500633\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.9817	0.556074
$546$	0	0
$547$	8.77551	0.375214	0.187607	−	0.982244i	$-0.439927\pi$
$547$	8.77551	0.375214	0.187607	+	0.982244i	$0.439927\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.65847	−0.283661
$552$	0	0
$553$	−40.5972	−1.72637
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.5606	1.88809	0.944047	−	0.329812i	$-0.106985\pi$
$557$	44.5606	1.88809	0.944047	+	0.329812i	$0.106985\pi$
$558$	0	0
$559$	−2.94503	−0.124562
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.6743	1.54564	0.772819	−	0.634626i	$-0.218846\pi$
$563$	36.6743	1.54564	0.772819	+	0.634626i	$0.218846\pi$
$564$	0	0
$565$	−13.7938	−0.580311
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.5939	0.947187	0.473593	−	0.880744i	$-0.342957\pi$
$569$	22.5939	0.947187	0.473593	+	0.880744i	$0.342957\pi$
$570$	0	0
$571$	16.1050	0.673972	0.336986	−	0.941510i	$-0.390592\pi$
$571$	16.1050	0.673972	0.336986	+	0.941510i	$0.390592\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.52311	−0.0635183
$576$	0	0
$577$	8.32653	0.346638	0.173319	−	0.984866i	$-0.444551\pi$
$577$	8.32653	0.346638	0.173319	+	0.984866i	$0.444551\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−39.6926	−1.64673
$582$	0	0
$583$	73.6068	3.04848
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.9508	1.07111	0.535553	−	0.844502i	$-0.320103\pi$
$587$	25.9508	1.07111	0.535553	+	0.844502i	$0.320103\pi$
$588$	0	0
$589$	−3.72928	−0.153662
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.38150	−0.303122	−0.151561	−	0.988448i	$-0.548430\pi$
$593$	−7.38150	−0.303122	−0.151561	+	0.988448i	$0.548430\pi$
$594$	0	0
$595$	−7.58767	−0.311064
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.4219	1.03871	0.519356	−	0.854558i	$-0.326172\pi$
$599$	25.4219	1.03871	0.519356	+	0.854558i	$0.326172\pi$
$600$	0	0
$601$	25.8217	1.05329	0.526645	−	0.850085i	$-0.323450\pi$
$601$	25.8217	1.05329	0.526645	+	0.850085i	$0.323450\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.8034	1.21168
$606$	0	0
$607$	5.31695	0.215808	0.107904	−	0.994161i	$-0.465586\pi$
$607$	5.31695	0.215808	0.107904	+	0.994161i	$0.465586\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.8122	−0.437413
$612$	0	0
$613$	−37.4498	−1.51258	−0.756292	−	0.654234i	$-0.772991\pi$
$613$	−37.4498	−1.51258	−0.756292	+	0.654234i	$0.772991\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.2341	−1.45873	−0.729364	−	0.684125i	$-0.760184\pi$
$617$	−36.2341	−1.45873	−0.729364	+	0.684125i	$0.760184\pi$
$618$	0	0
$619$	37.7293	1.51647	0.758234	−	0.651983i	$-0.226062\pi$
$619$	37.7293	1.51647	0.758234	+	0.651983i	$0.226062\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46.4835	−1.86232
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.94419	0.316755
$630$	0	0
$631$	−19.4586	−0.774633	−0.387317	−	0.921947i	$-0.626598\pi$
$631$	−19.4586	−0.774633	−0.387317	+	0.921947i	$0.626598\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.04623	0.279621
$636$	0	0
$637$	−13.9109	−0.551169
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.1912	−1.70595	−0.852974	−	0.521953i	$-0.825204\pi$
$641$	−43.1912	−1.70595	−0.852974	+	0.521953i	$0.825204\pi$
$642$	0	0
$643$	18.4157	0.726243	0.363121	−	0.931742i	$-0.381711\pi$
$643$	18.4157	0.726243	0.363121	+	0.931742i	$0.381711\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.2986	−0.719393	−0.359697	−	0.933069i	$-0.617120\pi$
$647$	−18.2986	−0.719393	−0.359697	+	0.933069i	$0.617120\pi$
$648$	0	0
$649$	95.6993	3.75653
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.1570	0.554007	0.277003	−	0.960869i	$-0.410659\pi$
$653$	14.1570	0.554007	0.277003	+	0.960869i	$0.410659\pi$
$654$	0	0
$655$	0.117037	0.00457301
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.3544	1.57199	0.785993	−	0.618236i	$-0.212152\pi$
$659$	40.3544	1.57199	0.785993	+	0.618236i	$0.212152\pi$
$660$	0	0
$661$	−48.0837	−1.87024	−0.935120	−	0.354331i	$-0.884709\pi$
$661$	−48.0837	−1.87024	−0.935120	+	0.354331i	$0.884709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.38776	0.170150
$666$	0	0
$667$	10.1416	0.392685
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.4586	0.442353
$672$	0	0
$673$	−2.86464	−0.110424	−0.0552119	−	0.998475i	$-0.517583\pi$
$673$	−2.86464	−0.110424	−0.0552119	+	0.998475i	$0.517583\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.4865	−1.28699	−0.643495	−	0.765450i	$-0.722516\pi$
$677$	−33.4865	−1.28699	−0.643495	+	0.765450i	$0.722516\pi$
$678$	0	0
$679$	−20.1570	−0.773555
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.5972	1.24730	0.623650	−	0.781704i	$-0.285649\pi$
$683$	32.5972	1.24730	0.623650	+	0.781704i	$0.285649\pi$
$684$	0	0
$685$	−11.2524	−0.429932
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.0829	0.498418
$690$	0	0
$691$	−29.3728	−1.11739	−0.558696	−	0.829372i	$-0.688698\pi$
$691$	−29.3728	−1.11739	−0.558696	+	0.829372i	$0.688698\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.3169	0.505141
$696$	0	0
$697$	−16.6705	−0.631442
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.4952	−0.434168	−0.217084	−	0.976153i	$-0.569655\pi$
$701$	−11.4952	−0.434168	−0.217084	+	0.976153i	$0.569655\pi$
$702$	0	0
$703$	−4.59392	−0.173263
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.37276	−0.202063
$708$	0	0
$709$	−31.7572	−1.19267	−0.596333	−	0.802737i	$-0.703376\pi$
$709$	−31.7572	−1.19267	−0.596333	+	0.802737i	$0.703376\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.68012	0.212722
$714$	0	0
$715$	7.25240	0.271224
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.4802	0.912958	0.456479	−	0.889734i	$-0.349110\pi$
$719$	24.4802	0.912958	0.456479	+	0.889734i	$0.349110\pi$
$720$	0	0
$721$	−30.9171	−1.15141
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.65847	−0.247289
$726$	0	0
$727$	−23.4340	−0.869118	−0.434559	−	0.900643i	$-0.643096\pi$
$727$	−23.4340	−0.869118	−0.434559	+	0.900643i	$0.643096\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.48562	−0.165907
$732$	0	0
$733$	45.8863	1.69485	0.847424	−	0.530916i	$-0.178152\pi$
$733$	45.8863	1.69485	0.847424	+	0.530916i	$0.178152\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	45.0096	1.65795
$738$	0	0
$739$	−31.1020	−1.14411	−0.572054	−	0.820216i	$-0.693853\pi$
$739$	−31.1020	−1.14411	−0.572054	+	0.820216i	$0.693853\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.5048	−0.532129	−0.266065	−	0.963955i	$-0.585723\pi$
$743$	−14.5048	−0.532129	−0.266065	+	0.963955i	$0.585723\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.1753	0.554495
$750$	0	0
$751$	−49.4094	−1.80297	−0.901487	−	0.432805i	$-0.857524\pi$
$751$	−49.4094	−1.80297	−0.901487	+	0.432805i	$0.857524\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.2341	0.372456
$756$	0	0
$757$	25.1108	0.912667	0.456333	−	0.889809i	$-0.349162\pi$
$757$	25.1108	0.912667	0.456333	+	0.889809i	$0.349162\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	56.9604	2.06211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.0096	0.614181
$768$	0	0
$769$	15.9634	0.575653	0.287827	−	0.957683i	$-0.407067\pi$
$769$	15.9634	0.575653	0.287827	+	0.957683i	$0.407067\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.6522	−0.562971	−0.281486	−	0.959565i	$-0.590827\pi$
$773$	−15.6522	−0.562971	−0.281486	+	0.959565i	$0.590827\pi$
$774$	0	0
$775$	−3.72928	−0.133960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.64015	0.345394
$780$	0	0
$781$	−59.5144	−2.12959
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0279	0.429294
$786$	0	0
$787$	38.4556	1.37080	0.685398	−	0.728169i	$-0.259628\pi$
$787$	38.4556	1.37080	0.685398	+	0.728169i	$0.259628\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−60.5240	−2.15198
$792$	0	0
$793$	2.03664	0.0723234
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.2986	−1.00239	−0.501194	−	0.865335i	$-0.667106\pi$
$797$	−28.2986	−1.00239	−0.501194	+	0.865335i	$0.667106\pi$
$798$	0	0
$799$	−16.4681	−0.582601
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.3265	1.91714
$804$	0	0
$805$	−6.68305	−0.235547
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.0925	−1.26894	−0.634472	−	0.772946i	$-0.718782\pi$
$809$	−36.0925	−1.26894	−0.634472	+	0.772946i	$0.718782\pi$
$810$	0	0
$811$	−8.05581	−0.282878	−0.141439	−	0.989947i	$-0.545173\pi$
$811$	−8.05581	−0.282878	−0.141439	+	0.989947i	$0.545173\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.4157	−0.504959
$816$	0	0
$817$	2.59392	0.0907499
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.9142	1.67222	0.836108	−	0.548564i	$-0.184825\pi$
$821$	47.9142	1.67222	0.836108	+	0.548564i	$0.184825\pi$
$822$	0	0
$823$	−13.1633	−0.458843	−0.229421	−	0.973327i	$-0.573683\pi$
$823$	−13.1633	−0.458843	−0.229421	+	0.973327i	$0.573683\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.2524	0.808565	0.404283	−	0.914634i	$-0.367521\pi$
$827$	23.2524	0.808565	0.404283	+	0.914634i	$0.367521\pi$
$828$	0	0
$829$	−10.3478	−0.359393	−0.179697	−	0.983722i	$-0.557512\pi$
$829$	−10.3478	−0.359393	−0.179697	+	0.983722i	$0.557512\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21.1878	−0.734115
$834$	0	0
$835$	3.79383	0.131291
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.1512	−1.07546	−0.537729	−	0.843117i	$-0.680718\pi$
$839$	−31.1512	−1.07546	−0.537729	+	0.843117i	$0.680718\pi$
$840$	0	0
$841$	15.3353	0.528802
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.7110	−0.402869
$846$	0	0
$847$	130.770	4.49331
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.99707	0.239856
$852$	0	0
$853$	7.07414	0.242214	0.121107	−	0.992639i	$-0.461356\pi$
$853$	7.07414	0.242214	0.121107	+	0.992639i	$0.461356\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.4769	1.24603	0.623013	−	0.782211i	$-0.285908\pi$
$857$	36.4769	1.24603	0.623013	+	0.782211i	$0.285908\pi$
$858$	0	0
$859$	3.58767	0.122410	0.0612048	−	0.998125i	$-0.480506\pi$
$859$	3.58767	0.122410	0.0612048	+	0.998125i	$0.480506\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.5972	0.973462	0.486731	−	0.873552i	$-0.338189\pi$
$863$	28.5972	0.973462	0.486731	+	0.873552i	$0.338189\pi$
$864$	0	0
$865$	12.2986	0.418166
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.1020	2.00490
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.38776	0.148333
$876$	0	0
$877$	15.6402	0.528130	0.264065	−	0.964505i	$-0.414937\pi$
$877$	15.6402	0.528130	0.264065	+	0.964505i	$0.414937\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.7851	1.60992	0.804960	−	0.593329i	$-0.202187\pi$
$881$	47.7851	1.60992	0.804960	+	0.593329i	$0.202187\pi$
$882$	0	0
$883$	9.09871	0.306196	0.153098	−	0.988211i	$-0.451075\pi$
$883$	9.09871	0.306196	0.153098	+	0.988211i	$0.451075\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.5510	1.39515	0.697573	−	0.716513i	$-0.254263\pi$
$887$	41.5510	1.39515	0.697573	+	0.716513i	$0.254263\pi$
$888$	0	0
$889$	30.9171	1.03693
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.52311	0.318679
$894$	0	0
$895$	2.02791	0.0677854
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.8313	0.828171
$900$	0	0
$901$	19.9267	0.663855
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.5231	−0.582488
$906$	0	0
$907$	0.775511	0.0257504	0.0128752	−	0.999917i	$-0.495902\pi$
$907$	0.775511	0.0257504	0.0128752	+	0.999917i	$0.495902\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.90754	0.195726	0.0978628	−	0.995200i	$-0.468799\pi$
$911$	5.90754	0.195726	0.0978628	+	0.995200i	$0.468799\pi$
$912$	0	0
$913$	57.7851	1.91241
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.513530	0.0169582
$918$	0	0
$919$	−38.1483	−1.25840	−0.629198	−	0.777245i	$-0.716616\pi$
$919$	−38.1483	−1.25840	−0.629198	+	0.777245i	$0.716616\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.5781	−0.348182
$924$	0	0
$925$	−4.59392	−0.151047
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.2341	0.991948	0.495974	−	0.868337i	$-0.334811\pi$
$929$	30.2341	0.991948	0.495974	+	0.868337i	$0.334811\pi$
$930$	0	0
$931$	12.2524	0.401556
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.0462	0.361250
$936$	0	0
$937$	21.6926	0.708668	0.354334	−	0.935119i	$-0.384708\pi$
$937$	21.6926	0.708668	0.354334	+	0.935119i	$0.384708\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.08039	0.133017	0.0665085	−	0.997786i	$-0.478814\pi$
$941$	4.08039	0.133017	0.0665085	+	0.997786i	$0.478814\pi$
$942$	0	0
$943$	−14.6831	−0.478146
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.5606	−0.928095	−0.464047	−	0.885810i	$-0.653603\pi$
$947$	−28.5606	−0.928095	−0.464047	+	0.885810i	$0.653603\pi$
$948$	0	0
$949$	9.65599	0.313447
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.8401	−1.19337	−0.596683	−	0.802477i	$-0.703515\pi$
$953$	−36.8401	−1.19337	−0.596683	+	0.802477i	$0.703515\pi$
$954$	0	0
$955$	10.3878	0.336140
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−49.3728	−1.59433
$960$	0	0
$961$	−17.0925	−0.551370
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.13536	0.0365485
$966$	0	0
$967$	23.1999	0.746059	0.373029	−	0.927820i	$-0.378319\pi$
$967$	23.1999	0.746059	0.373029	+	0.927820i	$0.378319\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.79383	0.0575668	0.0287834	−	0.999586i	$-0.490837\pi$
$971$	1.79383	0.0575668	0.0287834	+	0.999586i	$0.490837\pi$
$972$	0	0
$973$	58.4315	1.87323
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.4961	−1.61551	−0.807756	−	0.589517i	$-0.799318\pi$
$977$	−50.4961	−1.61551	−0.807756	+	0.589517i	$0.799318\pi$
$978$	0	0
$979$	67.6714	2.16279
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.8034	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$983$	−32.8034	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$984$	0	0
$985$	−14.5693	−0.464218
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.95084	−0.125629
$990$	0	0
$991$	32.4277	1.03010	0.515050	−	0.857160i	$-0.327773\pi$
$991$	32.4277	1.03010	0.515050	+	0.857160i	$0.327773\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.18785	0.291274
$996$	0	0
$997$	5.93545	0.187978	0.0939888	−	0.995573i	$-0.470038\pi$
$997$	5.93545	0.187978	0.0939888	+	0.995573i	$0.470038\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bk.1.3		3
3.2	odd	2		2280.2.a.r.1.3	✓	3
12.11	even	2		4560.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.r.1.3	✓	3	3.2	odd	2
4560.2.a.bt.1.1		3	12.11	even	2
6840.2.a.bk.1.3		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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.38776 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +4.38776 q^{7} -6.38776 q^{11} -1.13536 q^{13} -1.72928 q^{17} +1.00000 q^{19} -1.52311 q^{23} +1.00000 q^{25} -6.65847 q^{29} -3.72928 q^{31} +4.38776 q^{35} -4.59392 q^{37} +9.64015 q^{41} +2.59392 q^{43} +9.52311 q^{47} +12.2524 q^{49} -11.5231 q^{53} -6.38776 q^{55} -14.9817 q^{59} -1.79383 q^{61} -1.13536 q^{65} -7.04623 q^{67} +9.31695 q^{71} -8.50479 q^{73} -28.0279 q^{77} -9.25240 q^{79} -9.04623 q^{83} -1.72928 q^{85} -10.5939 q^{89} -4.98168 q^{91} +1.00000 q^{95} -4.59392 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 2 * q^7	\( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

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