LMFDB - Embedded newform 6840.2.a.bb.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -1.41421 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.41421 q^{7} +3.41421 q^{11} +3.41421 q^{13} -2.82843 q^{17} +1.00000 q^{19} -0.828427 q^{23} +1.00000 q^{25} -7.07107 q^{29} +0.828427 q^{31} -1.41421 q^{35} -2.24264 q^{37} +9.89949 q^{41} -3.07107 q^{43} +8.82843 q^{47} -5.00000 q^{49} +8.48528 q^{53} +3.41421 q^{55} +6.82843 q^{59} +9.65685 q^{61} +3.41421 q^{65} +11.3137 q^{67} +9.17157 q^{71} -0.343146 q^{73} -4.82843 q^{77} -2.34315 q^{79} -3.65685 q^{83} -2.82843 q^{85} +3.75736 q^{89} -4.82843 q^{91} +1.00000 q^{95} -7.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{37} + 8 q^{43} + 12 q^{47} - 10 q^{49} + 4 q^{55} + 8 q^{59} + 8 q^{61} + 4 q^{65} + 24 q^{71} - 12 q^{73} - 4 q^{77} - 16 q^{79} + 4 q^{83} + 16 q^{89} - 4 q^{91} + 2 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 4 * q^11 + 4 * q^13 + 2 * q^19 + 4 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^37 + 8 * q^43 + 12 * q^47 - 10 * q^49 + 4 * q^55 + 8 * q^59 + 8 * q^61 + 4 * q^65 + 24 * q^71 - 12 * q^73 - 4 * q^77 - 16 * q^79 + 4 * q^83 + 16 * q^89 - 4 * q^91 + 2 * q^95 + 4 * 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
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.07107	−1.31306	−0.656532	−	0.754298i	$-0.727977\pi$
$29$	−7.07107	−1.31306	−0.656532	+	0.754298i	$0.727977\pi$
$30$	0	0
$31$	0.828427	0.148790	0.0743950	−	0.997229i	$-0.476297\pi$
$31$	0.828427	0.148790	0.0743950	+	0.997229i	$0.476297\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−2.24264	−0.368688	−0.184344	−	0.982862i	$-0.559016\pi$
$37$	−2.24264	−0.368688	−0.184344	+	0.982862i	$0.559016\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.89949	1.54604	0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949	1.54604	0.773021	+	0.634381i	$0.218745\pi$
$42$	0	0
$43$	−3.07107	−0.468333	−0.234167	−	0.972196i	$-0.575236\pi$
$43$	−3.07107	−0.468333	−0.234167	+	0.972196i	$0.575236\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.82843	1.28776	0.643879	−	0.765127i	$-0.277324\pi$
$47$	8.82843	1.28776	0.643879	+	0.765127i	$0.277324\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	0	0
$61$	9.65685	1.23643	0.618217	−	0.786008i	$-0.287855\pi$
$61$	9.65685	1.23643	0.618217	+	0.786008i	$0.287855\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.17157	1.08847	0.544233	−	0.838934i	$-0.316821\pi$
$71$	9.17157	1.08847	0.544233	+	0.838934i	$0.316821\pi$
$72$	0	0
$73$	−0.343146	−0.0401622	−0.0200811	−	0.999798i	$-0.506392\pi$
$73$	−0.343146	−0.0401622	−0.0200811	+	0.999798i	$0.506392\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.82843	−0.550250
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.65685	−0.401392	−0.200696	−	0.979654i	$-0.564320\pi$
$83$	−3.65685	−0.401392	−0.200696	+	0.979654i	$0.564320\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.75736	0.398279	0.199140	−	0.979971i	$-0.436185\pi$
$89$	3.75736	0.398279	0.199140	+	0.979971i	$0.436185\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−7.89949	−0.802072	−0.401036	−	0.916062i	$-0.631350\pi$
$97$	−7.89949	−0.802072	−0.401036	+	0.916062i	$0.631350\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	0	0
$103$	−19.3137	−1.90304	−0.951518	−	0.307593i	$-0.900477\pi$
$103$	−19.3137	−1.90304	−0.951518	+	0.307593i	$0.900477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.34315	0.226520	0.113260	−	0.993565i	$-0.463871\pi$
$107$	2.34315	0.226520	0.113260	+	0.993565i	$0.463871\pi$
$108$	0	0
$109$	−20.1421	−1.92927	−0.964633	−	0.263595i	$-0.915092\pi$
$109$	−20.1421	−1.92927	−0.964633	+	0.263595i	$0.915092\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.31371	−0.688016	−0.344008	−	0.938967i	$-0.611785\pi$
$113$	−7.31371	−0.688016	−0.344008	+	0.938967i	$0.611785\pi$
$114$	0	0
$115$	−0.828427	−0.0772512
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	0.656854	0.0597140
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.7279	1.63627	0.818133	−	0.575029i	$-0.195009\pi$
$131$	18.7279	1.63627	0.818133	+	0.575029i	$0.195009\pi$
$132$	0	0
$133$	−1.41421	−0.122628
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.34315	−0.371060	−0.185530	−	0.982639i	$-0.559400\pi$
$137$	−4.34315	−0.371060	−0.185530	+	0.982639i	$0.559400\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.6569	0.974795
$144$	0	0
$145$	−7.07107	−0.587220
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.6569	1.28266	0.641330	−	0.767265i	$-0.278383\pi$
$149$	15.6569	1.28266	0.641330	+	0.767265i	$0.278383\pi$
$150$	0	0
$151$	−4.82843	−0.392932	−0.196466	−	0.980511i	$-0.562947\pi$
$151$	−4.82843	−0.392932	−0.196466	+	0.980511i	$0.562947\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.828427	0.0665409
$156$	0	0
$157$	24.1421	1.92675	0.963376	−	0.268154i	$-0.0864136\pi$
$157$	24.1421	1.92675	0.963376	+	0.268154i	$0.0864136\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.17157	0.0923329
$162$	0	0
$163$	7.55635	0.591859	0.295929	−	0.955210i	$-0.404371\pi$
$163$	7.55635	0.591859	0.295929	+	0.955210i	$0.404371\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.31371	−0.411187	−0.205594	−	0.978637i	$-0.565912\pi$
$167$	−5.31371	−0.411187	−0.205594	+	0.978637i	$0.565912\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.8284	1.40730	0.703651	−	0.710545i	$-0.251552\pi$
$179$	18.8284	1.40730	0.703651	+	0.710545i	$0.251552\pi$
$180$	0	0
$181$	−8.14214	−0.605200	−0.302600	−	0.953118i	$-0.597855\pi$
$181$	−8.14214	−0.605200	−0.302600	+	0.953118i	$0.597855\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.24264	−0.164882
$186$	0	0
$187$	−9.65685	−0.706179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.2426	1.60942	0.804710	−	0.593667i	$-0.202321\pi$
$191$	22.2426	1.60942	0.804710	+	0.593667i	$0.202321\pi$
$192$	0	0
$193$	−1.07107	−0.0770971	−0.0385486	−	0.999257i	$-0.512273\pi$
$193$	−1.07107	−0.0770971	−0.0385486	+	0.999257i	$0.512273\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.14214	0.437609	0.218805	−	0.975769i	$-0.429784\pi$
$197$	6.14214	0.437609	0.218805	+	0.975769i	$0.429784\pi$
$198$	0	0
$199$	−14.8284	−1.05116	−0.525580	−	0.850744i	$-0.676152\pi$
$199$	−14.8284	−1.05116	−0.525580	+	0.850744i	$0.676152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.0000	0.701862
$204$	0	0
$205$	9.89949	0.691411
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.41421	0.236166
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.07107	−0.209445
$216$	0	0
$217$	−1.17157	−0.0795315
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.65685	−0.649590
$222$	0	0
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.343146	−0.0227754	−0.0113877	−	0.999935i	$-0.503625\pi$
$227$	−0.343146	−0.0227754	−0.0113877	+	0.999935i	$0.503625\pi$
$228$	0	0
$229$	−10.3431	−0.683494	−0.341747	−	0.939792i	$-0.611019\pi$
$229$	−10.3431	−0.683494	−0.341747	+	0.939792i	$0.611019\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.34315	−0.284529	−0.142264	−	0.989829i	$-0.545438\pi$
$233$	−4.34315	−0.284529	−0.142264	+	0.989829i	$0.545438\pi$
$234$	0	0
$235$	8.82843	0.575903
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.2426	1.69750	0.848748	−	0.528798i	$-0.177357\pi$
$239$	26.2426	1.69750	0.848748	+	0.528798i	$0.177357\pi$
$240$	0	0
$241$	23.6569	1.52387	0.761936	−	0.647652i	$-0.224249\pi$
$241$	23.6569	1.52387	0.761936	+	0.647652i	$0.224249\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.00000	−0.319438
$246$	0	0
$247$	3.41421	0.217241
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.2426	0.898988	0.449494	−	0.893283i	$-0.351604\pi$
$251$	14.2426	0.898988	0.449494	+	0.893283i	$0.351604\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	3.17157	0.197072
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.3848	1.36482	0.682412	−	0.730968i	$-0.260931\pi$
$269$	22.3848	1.36482	0.682412	+	0.730968i	$0.260931\pi$
$270$	0	0
$271$	2.82843	0.171815	0.0859074	−	0.996303i	$-0.472621\pi$
$271$	2.82843	0.171815	0.0859074	+	0.996303i	$0.472621\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.41421	0.205885
$276$	0	0
$277$	20.6274	1.23938	0.619691	−	0.784846i	$-0.287258\pi$
$277$	20.6274	1.23938	0.619691	+	0.784846i	$0.287258\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.38478	−0.380884	−0.190442	−	0.981698i	$-0.560992\pi$
$281$	−6.38478	−0.380884	−0.190442	+	0.981698i	$0.560992\pi$
$282$	0	0
$283$	16.2426	0.965525	0.482762	−	0.875751i	$-0.339633\pi$
$283$	16.2426	0.965525	0.482762	+	0.875751i	$0.339633\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.0000	−0.826394
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.65685	0.564159	0.282080	−	0.959391i	$-0.408976\pi$
$293$	9.65685	0.564159	0.282080	+	0.959391i	$0.408976\pi$
$294$	0	0
$295$	6.82843	0.397566
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	4.34315	0.250335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.65685	0.552950
$306$	0	0
$307$	1.85786	0.106034	0.0530170	−	0.998594i	$-0.483116\pi$
$307$	1.85786	0.106034	0.0530170	+	0.998594i	$0.483116\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.2132	−1.08948	−0.544740	−	0.838605i	$-0.683372\pi$
$311$	−19.2132	−1.08948	−0.544740	+	0.838605i	$0.683372\pi$
$312$	0	0
$313$	−13.7990	−0.779965	−0.389983	−	0.920822i	$-0.627519\pi$
$313$	−13.7990	−0.779965	−0.389983	+	0.920822i	$0.627519\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.14214	0.120314	0.0601572	−	0.998189i	$-0.480840\pi$
$317$	2.14214	0.120314	0.0601572	+	0.998189i	$0.480840\pi$
$318$	0	0
$319$	−24.1421	−1.35170
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.4853	−0.688336
$330$	0	0
$331$	16.1421	0.887252	0.443626	−	0.896212i	$-0.353692\pi$
$331$	16.1421	0.887252	0.443626	+	0.896212i	$0.353692\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	26.2426	1.42953	0.714764	−	0.699366i	$-0.246534\pi$
$337$	26.2426	1.42953	0.714764	+	0.699366i	$0.246534\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.82843	0.153168
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.31371	0.0705236	0.0352618	−	0.999378i	$-0.488773\pi$
$347$	1.31371	0.0705236	0.0352618	+	0.999378i	$0.488773\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.3137	1.56021	0.780106	−	0.625648i	$-0.215165\pi$
$353$	29.3137	1.56021	0.780106	+	0.625648i	$0.215165\pi$
$354$	0	0
$355$	9.17157	0.486777
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.5563	0.926589	0.463294	−	0.886204i	$-0.346667\pi$
$359$	17.5563	0.926589	0.463294	+	0.886204i	$0.346667\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.343146	−0.0179611
$366$	0	0
$367$	−0.727922	−0.0379972	−0.0189986	−	0.999820i	$-0.506048\pi$
$367$	−0.727922	−0.0379972	−0.0189986	+	0.999820i	$0.506048\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−2.92893	−0.151654	−0.0758272	−	0.997121i	$-0.524160\pi$
$373$	−2.92893	−0.151654	−0.0758272	+	0.997121i	$0.524160\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.1421	−1.24338
$378$	0	0
$379$	26.4853	1.36046	0.680229	−	0.733000i	$-0.261880\pi$
$379$	26.4853	1.36046	0.680229	+	0.733000i	$0.261880\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.6274	−0.543036	−0.271518	−	0.962433i	$-0.587526\pi$
$383$	−10.6274	−0.543036	−0.271518	+	0.962433i	$0.587526\pi$
$384$	0	0
$385$	−4.82843	−0.246079
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.02944	0.0521945	0.0260973	−	0.999659i	$-0.491692\pi$
$389$	1.02944	0.0521945	0.0260973	+	0.999659i	$0.491692\pi$
$390$	0	0
$391$	2.34315	0.118498
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.34315	−0.117896
$396$	0	0
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.3848	0.918092	0.459046	−	0.888413i	$-0.348191\pi$
$401$	18.3848	0.918092	0.459046	+	0.888413i	$0.348191\pi$
$402$	0	0
$403$	2.82843	0.140894
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.65685	−0.379536
$408$	0	0
$409$	0.828427	0.0409631	0.0204815	−	0.999790i	$-0.493480\pi$
$409$	0.828427	0.0409631	0.0204815	+	0.999790i	$0.493480\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.65685	−0.475183
$414$	0	0
$415$	−3.65685	−0.179508
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.5563	−1.44392	−0.721961	−	0.691934i	$-0.756759\pi$
$419$	−29.5563	−1.44392	−0.721961	+	0.691934i	$0.756759\pi$
$420$	0	0
$421$	28.6274	1.39521	0.697607	−	0.716480i	$-0.254248\pi$
$421$	28.6274	1.39521	0.697607	+	0.716480i	$0.254248\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	−13.6569	−0.660901
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.1421	1.06655	0.533275	−	0.845942i	$-0.320961\pi$
$431$	22.1421	1.06655	0.533275	+	0.845942i	$0.320961\pi$
$432$	0	0
$433$	−5.75736	−0.276681	−0.138341	−	0.990385i	$-0.544177\pi$
$433$	−5.75736	−0.276681	−0.138341	+	0.990385i	$0.544177\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.828427	−0.0396290
$438$	0	0
$439$	27.3137	1.30361	0.651806	−	0.758386i	$-0.274012\pi$
$439$	27.3137	1.30361	0.651806	+	0.758386i	$0.274012\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.4558	1.30447	0.652233	−	0.758018i	$-0.273832\pi$
$443$	27.4558	1.30447	0.652233	+	0.758018i	$0.273832\pi$
$444$	0	0
$445$	3.75736	0.178116
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.3848	−1.05640	−0.528201	−	0.849119i	$-0.677133\pi$
$449$	−22.3848	−1.05640	−0.528201	+	0.849119i	$0.677133\pi$
$450$	0	0
$451$	33.7990	1.59153
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.82843	−0.226360
$456$	0	0
$457$	−12.1421	−0.567985	−0.283993	−	0.958826i	$-0.591659\pi$
$457$	−12.1421	−0.567985	−0.283993	+	0.958826i	$0.591659\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.3137	−1.36528	−0.682638	−	0.730757i	$-0.739167\pi$
$461$	−29.3137	−1.36528	−0.682638	+	0.730757i	$0.739167\pi$
$462$	0	0
$463$	−0.727922	−0.0338294	−0.0169147	−	0.999857i	$-0.505384\pi$
$463$	−0.727922	−0.0338294	−0.0169147	+	0.999857i	$0.505384\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.1716	0.516959	0.258479	−	0.966017i	$-0.416779\pi$
$467$	11.1716	0.516959	0.258479	+	0.966017i	$0.416779\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.4853	−0.482114
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.2426	−0.650763	−0.325381	−	0.945583i	$-0.605493\pi$
$479$	−14.2426	−0.650763	−0.325381	+	0.945583i	$0.605493\pi$
$480$	0	0
$481$	−7.65685	−0.349123
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.89949	−0.358698
$486$	0	0
$487$	−15.7990	−0.715921	−0.357960	−	0.933737i	$-0.616528\pi$
$487$	−15.7990	−0.715921	−0.357960	+	0.933737i	$0.616528\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.3848	−1.10047	−0.550235	−	0.835010i	$-0.685462\pi$
$491$	−24.3848	−1.10047	−0.550235	+	0.835010i	$0.685462\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.9706	−0.581809
$498$	0	0
$499$	30.1421	1.34935	0.674674	−	0.738116i	$-0.264284\pi$
$499$	30.1421	1.34935	0.674674	+	0.738116i	$0.264284\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.3137	1.30703	0.653517	−	0.756912i	$-0.273293\pi$
$503$	29.3137	1.30703	0.653517	+	0.756912i	$0.273293\pi$
$504$	0	0
$505$	−4.82843	−0.214862
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.0711	0.845310	0.422655	−	0.906291i	$-0.361098\pi$
$509$	19.0711	0.845310	0.422655	+	0.906291i	$0.361098\pi$
$510$	0	0
$511$	0.485281	0.0214676
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−19.3137	−0.851064
$516$	0	0
$517$	30.1421	1.32565
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.5563	1.03202	0.516011	−	0.856582i	$-0.327416\pi$
$521$	23.5563	1.03202	0.516011	+	0.856582i	$0.327416\pi$
$522$	0	0
$523$	−15.7990	−0.690842	−0.345421	−	0.938448i	$-0.612264\pi$
$523$	−15.7990	−0.690842	−0.345421	+	0.938448i	$0.612264\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.34315	−0.102069
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	33.7990	1.46400
$534$	0	0
$535$	2.34315	0.101303
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−17.0711	−0.735303
$540$	0	0
$541$	9.31371	0.400428	0.200214	−	0.979752i	$-0.435836\pi$
$541$	9.31371	0.400428	0.200214	+	0.979752i	$0.435836\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.1421	−0.862794
$546$	0	0
$547$	8.48528	0.362804	0.181402	−	0.983409i	$-0.441936\pi$
$547$	8.48528	0.362804	0.181402	+	0.983409i	$0.441936\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.07107	−0.301238
$552$	0	0
$553$	3.31371	0.140913
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.6569	−1.00237	−0.501187	−	0.865339i	$-0.667103\pi$
$557$	−23.6569	−1.00237	−0.501187	+	0.865339i	$0.667103\pi$
$558$	0	0
$559$	−10.4853	−0.443480
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.343146	−0.0144619	−0.00723093	−	0.999974i	$-0.502302\pi$
$563$	−0.343146	−0.0144619	−0.00723093	+	0.999974i	$0.502302\pi$
$564$	0	0
$565$	−7.31371	−0.307690
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.1838	1.76844	0.884218	−	0.467075i	$-0.154692\pi$
$569$	42.1838	1.76844	0.884218	+	0.467075i	$0.154692\pi$
$570$	0	0
$571$	−2.14214	−0.0896456	−0.0448228	−	0.998995i	$-0.514272\pi$
$571$	−2.14214	−0.0896456	−0.0448228	+	0.998995i	$0.514272\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.828427	−0.0345478
$576$	0	0
$577$	−9.51472	−0.396103	−0.198051	−	0.980192i	$-0.563461\pi$
$577$	−9.51472	−0.396103	−0.198051	+	0.980192i	$0.563461\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.17157	0.214553
$582$	0	0
$583$	28.9706	1.19984
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.4558	−1.13322	−0.566612	−	0.823985i	$-0.691746\pi$
$587$	−27.4558	−1.13322	−0.566612	+	0.823985i	$0.691746\pi$
$588$	0	0
$589$	0.828427	0.0341347
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−44.6274	−1.83263	−0.916314	−	0.400460i	$-0.868850\pi$
$593$	−44.6274	−1.83263	−0.916314	+	0.400460i	$0.868850\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.9706	−0.693398	−0.346699	−	0.937976i	$-0.612698\pi$
$599$	−16.9706	−0.693398	−0.346699	+	0.937976i	$0.612698\pi$
$600$	0	0
$601$	6.20101	0.252944	0.126472	−	0.991970i	$-0.459635\pi$
$601$	6.20101	0.252944	0.126472	+	0.991970i	$0.459635\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.656854	0.0267049
$606$	0	0
$607$	33.4558	1.35793	0.678965	−	0.734170i	$-0.262429\pi$
$607$	33.4558	1.35793	0.678965	+	0.734170i	$0.262429\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.1421	1.21942
$612$	0	0
$613$	−12.8284	−0.518135	−0.259068	−	0.965859i	$-0.583415\pi$
$613$	−12.8284	−0.518135	−0.259068	+	0.965859i	$0.583415\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.1127	0.608414	0.304207	−	0.952606i	$-0.401608\pi$
$617$	15.1127	0.608414	0.304207	+	0.952606i	$0.401608\pi$
$618$	0	0
$619$	3.51472	0.141268	0.0706342	−	0.997502i	$-0.477498\pi$
$619$	3.51472	0.141268	0.0706342	+	0.997502i	$0.477498\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.31371	−0.212889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.34315	0.252918
$630$	0	0
$631$	−36.2843	−1.44445	−0.722227	−	0.691656i	$-0.756881\pi$
$631$	−36.2843	−1.44445	−0.722227	+	0.691656i	$0.756881\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.65685	0.224485
$636$	0	0
$637$	−17.0711	−0.676380
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.9289	−0.668653	−0.334326	−	0.942457i	$-0.608509\pi$
$641$	−16.9289	−0.668653	−0.334326	+	0.942457i	$0.608509\pi$
$642$	0	0
$643$	−28.9289	−1.14085	−0.570423	−	0.821351i	$-0.693221\pi$
$643$	−28.9289	−1.14085	−0.570423	+	0.821351i	$0.693221\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.3137	−0.995185	−0.497592	−	0.867411i	$-0.665782\pi$
$647$	−25.3137	−0.995185	−0.497592	+	0.867411i	$0.665782\pi$
$648$	0	0
$649$	23.3137	0.915143
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.1421	−0.396892	−0.198446	−	0.980112i	$-0.563590\pi$
$653$	−10.1421	−0.396892	−0.198446	+	0.980112i	$0.563590\pi$
$654$	0	0
$655$	18.7279	0.731760
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.9706	−1.12853	−0.564266	−	0.825593i	$-0.690841\pi$
$659$	−28.9706	−1.12853	−0.564266	+	0.825593i	$0.690841\pi$
$660$	0	0
$661$	−49.1127	−1.91026	−0.955131	−	0.296183i	$-0.904286\pi$
$661$	−49.1127	−1.91026	−0.955131	+	0.296183i	$0.904286\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	5.85786	0.226817
$668$	0	0
$669$	0	0
$670$	0	0
$671$	32.9706	1.27281
$672$	0	0
$673$	14.2426	0.549013	0.274507	−	0.961585i	$-0.411485\pi$
$673$	14.2426	0.549013	0.274507	+	0.961585i	$0.411485\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.1716	−1.12115	−0.560577	−	0.828102i	$-0.689421\pi$
$677$	−29.1716	−1.12115	−0.560577	+	0.828102i	$0.689421\pi$
$678$	0	0
$679$	11.1716	0.428726
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.3137	−1.35124	−0.675621	−	0.737249i	$-0.736124\pi$
$683$	−35.3137	−1.35124	−0.675621	+	0.737249i	$0.736124\pi$
$684$	0	0
$685$	−4.34315	−0.165943
$686$	0	0
$687$	0	0
$688$	0	0
$689$	28.9706	1.10369
$690$	0	0
$691$	8.20101	0.311981	0.155991	−	0.987759i	$-0.450143\pi$
$691$	8.20101	0.311981	0.155991	+	0.987759i	$0.450143\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.82843	0.107288
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.9706	−0.867586	−0.433793	−	0.901013i	$-0.642825\pi$
$701$	−22.9706	−0.867586	−0.433793	+	0.901013i	$0.642825\pi$
$702$	0	0
$703$	−2.24264	−0.0845828
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.82843	0.256809
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.686292	−0.0257018
$714$	0	0
$715$	11.6569	0.435942
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.75736	−0.0655384	−0.0327692	−	0.999463i	$-0.510433\pi$
$719$	−1.75736	−0.0655384	−0.0327692	+	0.999463i	$0.510433\pi$
$720$	0	0
$721$	27.3137	1.01722
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.07107	−0.262613
$726$	0	0
$727$	15.7574	0.584408	0.292204	−	0.956356i	$-0.405611\pi$
$727$	15.7574	0.584408	0.292204	+	0.956356i	$0.405611\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.68629	0.321274
$732$	0	0
$733$	−33.3137	−1.23047	−0.615235	−	0.788344i	$-0.710939\pi$
$733$	−33.3137	−1.23047	−0.615235	+	0.788344i	$0.710939\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.6274	1.42286
$738$	0	0
$739$	−15.3137	−0.563324	−0.281662	−	0.959514i	$-0.590886\pi$
$739$	−15.3137	−0.563324	−0.281662	+	0.959514i	$0.590886\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.2548	1.80698	0.903492	−	0.428604i	$-0.140995\pi$
$743$	49.2548	1.80698	0.903492	+	0.428604i	$0.140995\pi$
$744$	0	0
$745$	15.6569	0.573623
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.31371	−0.121080
$750$	0	0
$751$	−15.8579	−0.578662	−0.289331	−	0.957229i	$-0.593433\pi$
$751$	−15.8579	−0.578662	−0.289331	+	0.957229i	$0.593433\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.82843	−0.175724
$756$	0	0
$757$	−9.51472	−0.345818	−0.172909	−	0.984938i	$-0.555317\pi$
$757$	−9.51472	−0.345818	−0.172909	+	0.984938i	$0.555317\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.3431	1.02744	0.513719	−	0.857958i	$-0.328267\pi$
$761$	28.3431	1.02744	0.513719	+	0.857958i	$0.328267\pi$
$762$	0	0
$763$	28.4853	1.03124
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.3137	0.841809
$768$	0	0
$769$	22.2843	0.803591	0.401796	−	0.915729i	$-0.368386\pi$
$769$	22.2843	0.803591	0.401796	+	0.915729i	$0.368386\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	45.4558	1.63493	0.817467	−	0.575976i	$-0.195378\pi$
$773$	45.4558	1.63493	0.817467	+	0.575976i	$0.195378\pi$
$774$	0	0
$775$	0.828427	0.0297580
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.89949	0.354686
$780$	0	0
$781$	31.3137	1.12049
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.1421	0.861670
$786$	0	0
$787$	−1.17157	−0.0417621	−0.0208810	−	0.999782i	$-0.506647\pi$
$787$	−1.17157	−0.0417621	−0.0208810	+	0.999782i	$0.506647\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.3431	0.367760
$792$	0	0
$793$	32.9706	1.17082
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.3137	−0.967501	−0.483751	−	0.875206i	$-0.660726\pi$
$797$	−27.3137	−0.967501	−0.483751	+	0.875206i	$0.660726\pi$
$798$	0	0
$799$	−24.9706	−0.883395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.17157	−0.0413439
$804$	0	0
$805$	1.17157	0.0412925
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.6863	1.07887	0.539436	−	0.842026i	$-0.318637\pi$
$809$	30.6863	1.07887	0.539436	+	0.842026i	$0.318637\pi$
$810$	0	0
$811$	19.0294	0.668214	0.334107	−	0.942535i	$-0.391565\pi$
$811$	19.0294	0.668214	0.334107	+	0.942535i	$0.391565\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.55635	0.264687
$816$	0	0
$817$	−3.07107	−0.107443
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.7990	−0.621189	−0.310595	−	0.950543i	$-0.600528\pi$
$821$	−17.7990	−0.621189	−0.310595	+	0.950543i	$0.600528\pi$
$822$	0	0
$823$	2.87006	0.100044	0.0500220	−	0.998748i	$-0.484071\pi$
$823$	2.87006	0.100044	0.0500220	+	0.998748i	$0.484071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.65685	0.127161	0.0635806	−	0.997977i	$-0.479748\pi$
$827$	3.65685	0.127161	0.0635806	+	0.997977i	$0.479748\pi$
$828$	0	0
$829$	−51.4558	−1.78714	−0.893568	−	0.448929i	$-0.851806\pi$
$829$	−51.4558	−1.78714	−0.893568	+	0.448929i	$0.851806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.1421	0.489996
$834$	0	0
$835$	−5.31371	−0.183888
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.85786	−0.202236	−0.101118	−	0.994874i	$-0.532242\pi$
$839$	−5.85786	−0.202236	−0.101118	+	0.994874i	$0.532242\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	−0.928932	−0.0319185
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.85786	0.0636868
$852$	0	0
$853$	8.82843	0.302280	0.151140	−	0.988512i	$-0.451706\pi$
$853$	8.82843	0.302280	0.151140	+	0.988512i	$0.451706\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.4558	−0.869555	−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558	−0.869555	−0.434778	+	0.900538i	$0.643173\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.2548	−1.13201	−0.566004	−	0.824403i	$-0.691511\pi$
$863$	−33.2548	−1.13201	−0.566004	+	0.824403i	$0.691511\pi$
$864$	0	0
$865$	−11.3137	−0.384678
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	38.6274	1.30884
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−46.0416	−1.55472	−0.777358	−	0.629059i	$-0.783440\pi$
$877$	−46.0416	−1.55472	−0.777358	+	0.629059i	$0.783440\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.4853	1.16184	0.580919	−	0.813961i	$-0.302693\pi$
$881$	34.4853	1.16184	0.580919	+	0.813961i	$0.302693\pi$
$882$	0	0
$883$	−35.7574	−1.20333	−0.601665	−	0.798748i	$-0.705496\pi$
$883$	−35.7574	−1.20333	−0.601665	+	0.798748i	$0.705496\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.6569	0.727166	0.363583	−	0.931562i	$-0.381553\pi$
$887$	21.6569	0.727166	0.363583	+	0.931562i	$0.381553\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.82843	0.295432
$894$	0	0
$895$	18.8284	0.629365
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.85786	−0.195371
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.14214	−0.270654
$906$	0	0
$907$	52.7696	1.75218	0.876092	−	0.482144i	$-0.160142\pi$
$907$	52.7696	1.75218	0.876092	+	0.482144i	$0.160142\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.6274	0.352102	0.176051	−	0.984381i	$-0.443668\pi$
$911$	10.6274	0.352102	0.176051	+	0.984381i	$0.443668\pi$
$912$	0	0
$913$	−12.4853	−0.413203
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.4853	−0.874621
$918$	0	0
$919$	27.5980	0.910373	0.455187	−	0.890396i	$-0.349573\pi$
$919$	27.5980	0.910373	0.455187	+	0.890396i	$0.349573\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31.3137	1.03070
$924$	0	0
$925$	−2.24264	−0.0737376
$926$	0	0
$927$	0	0
$928$	0	0
$929$	53.7990	1.76509	0.882544	−	0.470230i	$-0.155829\pi$
$929$	53.7990	1.76509	0.882544	+	0.470230i	$0.155829\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.65685	−0.315813
$936$	0	0
$937$	9.79899	0.320119	0.160060	−	0.987107i	$-0.448831\pi$
$937$	9.79899	0.320119	0.160060	+	0.987107i	$0.448831\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.2426	−0.529495	−0.264747	−	0.964318i	$-0.585289\pi$
$941$	−16.2426	−0.529495	−0.264747	+	0.964318i	$0.585289\pi$
$942$	0	0
$943$	−8.20101	−0.267062
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.2843	0.594159	0.297079	−	0.954853i	$-0.403987\pi$
$947$	18.2843	0.594159	0.297079	+	0.954853i	$0.403987\pi$
$948$	0	0
$949$	−1.17157	−0.0380309
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.02944	−0.0981331	−0.0490665	−	0.998796i	$-0.515625\pi$
$953$	−3.02944	−0.0981331	−0.0490665	+	0.998796i	$0.515625\pi$
$954$	0	0
$955$	22.2426	0.719755
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.14214	0.198340
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.07107	−0.0344789
$966$	0	0
$967$	−14.5858	−0.469047	−0.234524	−	0.972110i	$-0.575353\pi$
$967$	−14.5858	−0.469047	−0.234524	+	0.972110i	$0.575353\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.4853	−0.655382	−0.327691	−	0.944785i	$-0.606271\pi$
$977$	−20.4853	−0.655382	−0.327691	+	0.944785i	$0.606271\pi$
$978$	0	0
$979$	12.8284	0.409998
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15.9411	−0.508443	−0.254221	−	0.967146i	$-0.581819\pi$
$983$	−15.9411	−0.508443	−0.254221	+	0.967146i	$0.581819\pi$
$984$	0	0
$985$	6.14214	0.195705
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.54416	0.0808995
$990$	0	0
$991$	−10.3431	−0.328561	−0.164280	−	0.986414i	$-0.552530\pi$
$991$	−10.3431	−0.328561	−0.164280	+	0.986414i	$0.552530\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.8284	−0.470093
$996$	0	0
$997$	27.4558	0.869535	0.434768	−	0.900543i	$-0.356830\pi$
$997$	27.4558	0.869535	0.434768	+	0.900543i	$0.356830\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bb.1.1		2
3.2	odd	2		2280.2.a.k.1.1	✓	2
12.11	even	2		4560.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.k.1.1	✓	2	3.2	odd	2
4560.2.a.bn.1.2		2	12.11	even	2
6840.2.a.bb.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.41421 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.41421 q^{7} +3.41421 q^{11} +3.41421 q^{13} -2.82843 q^{17} +1.00000 q^{19} -0.828427 q^{23} +1.00000 q^{25} -7.07107 q^{29} +0.828427 q^{31} -1.41421 q^{35} -2.24264 q^{37} +9.89949 q^{41} -3.07107 q^{43} +8.82843 q^{47} -5.00000 q^{49} +8.48528 q^{53} +3.41421 q^{55} +6.82843 q^{59} +9.65685 q^{61} +3.41421 q^{65} +11.3137 q^{67} +9.17157 q^{71} -0.343146 q^{73} -4.82843 q^{77} -2.34315 q^{79} -3.65685 q^{83} -2.82843 q^{85} +3.75736 q^{89} -4.82843 q^{91} +1.00000 q^{95} -7.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{37} + 8 q^{43} + 12 q^{47} - 10 q^{49} + 4 q^{55} + 8 q^{59} + 8 q^{61} + 4 q^{65} + 24 q^{71} - 12 q^{73} - 4 q^{77} - 16 q^{79} + 4 q^{83} + 16 q^{89} - 4 q^{91} + 2 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 4 * q^11 + 4 * q^13 + 2 * q^19 + 4 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^37 + 8 * q^43 + 12 * q^47 - 10 * q^49 + 4 * q^55 + 8 * q^59 + 8 * q^61 + 4 * q^65 + 24 * q^71 - 12 * q^73 - 4 * q^77 - 16 * q^79 + 4 * q^83 + 16 * q^89 - 4 * q^91 + 2 * q^95 + 4 * q^97

Introduction

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