LMFDB - Embedded newform 6840.2.a.bd.1.2

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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +5.23607 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +5.23607 q^{7} +1.23607 q^{11} +0.763932 q^{13} -4.47214 q^{17} +1.00000 q^{19} -2.47214 q^{23} +1.00000 q^{25} -0.763932 q^{29} -8.94427 q^{31} +5.23607 q^{35} +3.23607 q^{37} +9.70820 q^{41} +5.23607 q^{43} -2.47214 q^{47} +20.4164 q^{49} +0.472136 q^{53} +1.23607 q^{55} +10.4721 q^{59} +4.47214 q^{61} +0.763932 q^{65} +1.52786 q^{67} +12.4721 q^{73} +6.47214 q^{77} -4.00000 q^{83} -4.47214 q^{85} +1.70820 q^{89} +4.00000 q^{91} +1.00000 q^{95} -4.76393 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 6 * q^7	$ 2 q + 2 q^{5} + 6 q^{7} - 2 q^{11} + 6 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 6 q^{29} + 6 q^{35} + 2 q^{37} + 6 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 8 q^{53} - 2 q^{55} + 12 q^{59} + 6 q^{65} + 12 q^{67} + 16 q^{73} + 4 q^{77} - 8 q^{83} - 10 q^{89} + 8 q^{91} + 2 q^{95} - 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 6 * q^7 - 2 * q^11 + 6 * q^13 + 2 * q^19 + 4 * q^23 + 2 * q^25 - 6 * q^29 + 6 * q^35 + 2 * q^37 + 6 * q^41 + 6 * q^43 + 4 * q^47 + 14 * q^49 - 8 * q^53 - 2 * q^55 + 12 * q^59 + 6 * q^65 + 12 * q^67 + 16 * q^73 + 4 * q^77 - 8 * q^83 - 10 * q^89 + 8 * q^91 + 2 * q^95 - 14 * 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
$7$	5.23607	1.97905	0.989524	−	0.144370i	$-0.0461154\pi$
$7$	5.23607	1.97905	0.989524	+	0.144370i	$0.0461154\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.763932	−0.141859	−0.0709293	−	0.997481i	$-0.522596\pi$
$29$	−0.763932	−0.141859	−0.0709293	+	0.997481i	$0.522596\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.23607	0.885057
$36$	0	0
$37$	3.23607	0.532006	0.266003	−	0.963972i	$-0.414297\pi$
$37$	3.23607	0.532006	0.266003	+	0.963972i	$0.414297\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.70820	1.51617	0.758083	−	0.652158i	$-0.226136\pi$
$41$	9.70820	1.51617	0.758083	+	0.652158i	$0.226136\pi$
$42$	0	0
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	0	0
$49$	20.4164	2.91663
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.4721	1.36336	0.681678	−	0.731652i	$-0.261251\pi$
$59$	10.4721	1.36336	0.681678	+	0.731652i	$0.261251\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.763932	0.0947541
$66$	0	0
$67$	1.52786	0.186658	0.0933292	−	0.995635i	$-0.470249\pi$
$67$	1.52786	0.186658	0.0933292	+	0.995635i	$0.470249\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.70820	0.181069	0.0905346	−	0.995893i	$-0.471142\pi$
$89$	1.70820	0.181069	0.0905346	+	0.995893i	$0.471142\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−4.76393	−0.483704	−0.241852	−	0.970313i	$-0.577755\pi$
$97$	−4.76393	−0.483704	−0.241852	+	0.970313i	$0.577755\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	9.52786	0.938808	0.469404	−	0.882983i	$-0.344469\pi$
$103$	9.52786	0.938808	0.469404	+	0.882983i	$0.344469\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−4.47214	−0.428353	−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214	−0.428353	−0.214176	+	0.976795i	$0.568707\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−2.47214	−0.230528
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−23.4164	−2.14658
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	0	0
$133$	5.23607	0.454025
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	0	0
$139$	−14.4721	−1.22751	−0.613755	−	0.789496i	$-0.710342\pi$
$139$	−14.4721	−1.22751	−0.613755	+	0.789496i	$0.710342\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.944272	0.0789640
$144$	0	0
$145$	−0.763932	−0.0634411
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.4164	−1.42681	−0.713404	−	0.700753i	$-0.752847\pi$
$149$	−17.4164	−1.42681	−0.713404	+	0.700753i	$0.752847\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.94427	−0.718421
$156$	0	0
$157$	−21.4164	−1.70922	−0.854608	−	0.519274i	$-0.826202\pi$
$157$	−21.4164	−1.70922	−0.854608	+	0.519274i	$0.826202\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.9443	−1.02015
$162$	0	0
$163$	20.6525	1.61763	0.808813	−	0.588065i	$-0.200110\pi$
$163$	20.6525	1.61763	0.808813	+	0.588065i	$0.200110\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8885	−1.07473	−0.537364	−	0.843350i	$-0.680580\pi$
$167$	−13.8885	−1.07473	−0.537364	+	0.843350i	$0.680580\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	5.23607	0.395810
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.4164	1.15228	0.576138	−	0.817352i	$-0.304559\pi$
$179$	15.4164	1.15228	0.576138	+	0.817352i	$0.304559\pi$
$180$	0	0
$181$	8.47214	0.629729	0.314864	−	0.949137i	$-0.398041\pi$
$181$	8.47214	0.629729	0.314864	+	0.949137i	$0.398041\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.23607	0.237920
$186$	0	0
$187$	−5.52786	−0.404237
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.18034	0.447194	0.223597	−	0.974682i	$-0.428220\pi$
$191$	6.18034	0.447194	0.223597	+	0.974682i	$0.428220\pi$
$192$	0	0
$193$	5.70820	0.410886	0.205443	−	0.978669i	$-0.434137\pi$
$193$	5.70820	0.410886	0.205443	+	0.978669i	$0.434137\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.4721	0.888603	0.444301	−	0.895877i	$-0.353452\pi$
$197$	12.4721	0.888603	0.444301	+	0.895877i	$0.353452\pi$
$198$	0	0
$199$	18.4721	1.30945	0.654727	−	0.755865i	$-0.272783\pi$
$199$	18.4721	1.30945	0.654727	+	0.755865i	$0.272783\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	9.70820	0.678050
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.23607	0.0855006
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.23607	0.357097
$216$	0	0
$217$	−46.8328	−3.17922
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.41641	−0.229812
$222$	0	0
$223$	22.4721	1.50485	0.752423	−	0.658680i	$-0.228885\pi$
$223$	22.4721	1.50485	0.752423	+	0.658680i	$0.228885\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−21.4164	−1.41524	−0.707618	−	0.706595i	$-0.750230\pi$
$229$	−21.4164	−1.41524	−0.707618	+	0.706595i	$0.750230\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.8885	−1.56499	−0.782495	−	0.622657i	$-0.786053\pi$
$233$	−23.8885	−1.56499	−0.782495	+	0.622657i	$0.786053\pi$
$234$	0	0
$235$	−2.47214	−0.161264
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.7082	0.757341	0.378670	−	0.925532i	$-0.376381\pi$
$239$	11.7082	0.757341	0.378670	+	0.925532i	$0.376381\pi$
$240$	0	0
$241$	6.94427	0.447320	0.223660	−	0.974667i	$-0.428199\pi$
$241$	6.94427	0.447320	0.223660	+	0.974667i	$0.428199\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.4164	1.30436
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.70820	−0.234060	−0.117030	−	0.993128i	$-0.537337\pi$
$251$	−3.70820	−0.234060	−0.117030	+	0.993128i	$0.537337\pi$
$252$	0	0
$253$	−3.05573	−0.192112
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.9443	1.43122	0.715612	−	0.698498i	$-0.246148\pi$
$257$	22.9443	1.43122	0.715612	+	0.698498i	$0.246148\pi$
$258$	0	0
$259$	16.9443	1.05287
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.05573	−0.188424	−0.0942121	−	0.995552i	$-0.530033\pi$
$263$	−3.05573	−0.188424	−0.0942121	+	0.995552i	$0.530033\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.70820	−0.348035	−0.174018	−	0.984743i	$-0.555675\pi$
$269$	−5.70820	−0.348035	−0.174018	+	0.984743i	$0.555675\pi$
$270$	0	0
$271$	−2.47214	−0.150172	−0.0750858	−	0.997177i	$-0.523923\pi$
$271$	−2.47214	−0.150172	−0.0750858	+	0.997177i	$0.523923\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	−16.4721	−0.989715	−0.494857	−	0.868974i	$-0.664780\pi$
$277$	−16.4721	−0.989715	−0.494857	+	0.868974i	$0.664780\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.2361	0.908908	0.454454	−	0.890770i	$-0.349834\pi$
$281$	15.2361	0.908908	0.454454	+	0.890770i	$0.349834\pi$
$282$	0	0
$283$	−28.0689	−1.66852	−0.834261	−	0.551370i	$-0.814105\pi$
$283$	−28.0689	−1.66852	−0.834261	+	0.551370i	$0.814105\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.8328	3.00057
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.88854	−0.227171	−0.113586	−	0.993528i	$-0.536234\pi$
$293$	−3.88854	−0.227171	−0.113586	+	0.993528i	$0.536234\pi$
$294$	0	0
$295$	10.4721	0.609711
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.88854	−0.109217
$300$	0	0
$301$	27.4164	1.58026
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.47214	0.256074
$306$	0	0
$307$	−17.5279	−1.00037	−0.500184	−	0.865919i	$-0.666734\pi$
$307$	−17.5279	−1.00037	−0.500184	+	0.865919i	$0.666734\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.7082	−1.11755	−0.558775	−	0.829319i	$-0.688728\pi$
$311$	−19.7082	−1.11755	−0.558775	+	0.829319i	$0.688728\pi$
$312$	0	0
$313$	−23.8885	−1.35026	−0.675130	−	0.737699i	$-0.735913\pi$
$313$	−23.8885	−1.35026	−0.675130	+	0.737699i	$0.735913\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.4721	−0.700505	−0.350252	−	0.936655i	$-0.613904\pi$
$317$	−12.4721	−0.700505	−0.350252	+	0.936655i	$0.613904\pi$
$318$	0	0
$319$	−0.944272	−0.0528691
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.47214	−0.248836
$324$	0	0
$325$	0.763932	0.0423753
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.9443	−0.713641
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.52786	0.0834761
$336$	0	0
$337$	−28.7639	−1.56687	−0.783436	−	0.621473i	$-0.786535\pi$
$337$	−28.7639	−1.56687	−0.783436	+	0.621473i	$0.786535\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.0557	−0.598701
$342$	0	0
$343$	70.2492	3.79310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	10.9443	0.585833	0.292917	−	0.956138i	$-0.405374\pi$
$349$	10.9443	0.585833	0.292917	+	0.956138i	$0.405374\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.9443	0.795403	0.397702	−	0.917515i	$-0.369808\pi$
$353$	14.9443	0.795403	0.397702	+	0.917515i	$0.369808\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.1803	1.17063	0.585317	−	0.810805i	$-0.300970\pi$
$359$	22.1803	1.17063	0.585317	+	0.810805i	$0.300970\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.4721	0.652821
$366$	0	0
$367$	−29.2361	−1.52611	−0.763055	−	0.646333i	$-0.776302\pi$
$367$	−29.2361	−1.52611	−0.763055	+	0.646333i	$0.776302\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.47214	0.128347
$372$	0	0
$373$	−7.23607	−0.374669	−0.187335	−	0.982296i	$-0.559985\pi$
$373$	−7.23607	−0.374669	−0.187335	+	0.982296i	$0.559985\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.583592	−0.0300565
$378$	0	0
$379$	1.88854	0.0970080	0.0485040	−	0.998823i	$-0.484555\pi$
$379$	1.88854	0.0970080	0.0485040	+	0.998823i	$0.484555\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.88854	0.505281	0.252640	−	0.967560i	$-0.418701\pi$
$383$	9.88854	0.505281	0.252640	+	0.967560i	$0.418701\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.8328	−1.66469	−0.832345	−	0.554258i	$-0.813002\pi$
$389$	−32.8328	−1.66469	−0.832345	+	0.554258i	$0.813002\pi$
$390$	0	0
$391$	11.0557	0.559112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.58359	−0.330421	−0.165211	−	0.986258i	$-0.552830\pi$
$397$	−6.58359	−0.330421	−0.165211	+	0.986258i	$0.552830\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.2361	1.16035	0.580177	−	0.814490i	$-0.302983\pi$
$401$	23.2361	1.16035	0.580177	+	0.814490i	$0.302983\pi$
$402$	0	0
$403$	−6.83282	−0.340367
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	4.47214	0.221133	0.110566	−	0.993869i	$-0.464734\pi$
$409$	4.47214	0.221133	0.110566	+	0.993869i	$0.464734\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	54.8328	2.69815
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.1803	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$419$	22.1803	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$420$	0	0
$421$	−24.8328	−1.21028	−0.605139	−	0.796120i	$-0.706882\pi$
$421$	−24.8328	−1.21028	−0.605139	+	0.796120i	$0.706882\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	23.4164	1.13320
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.3050	−1.21890	−0.609448	−	0.792826i	$-0.708609\pi$
$431$	−25.3050	−1.21890	−0.609448	+	0.792826i	$0.708609\pi$
$432$	0	0
$433$	0.763932	0.0367122	0.0183561	−	0.999832i	$-0.494157\pi$
$433$	0.763932	0.0367122	0.0183561	+	0.999832i	$0.494157\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.47214	−0.118258
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.4164	−0.922501	−0.461251	−	0.887270i	$-0.652599\pi$
$443$	−19.4164	−0.922501	−0.461251	+	0.887270i	$0.652599\pi$
$444$	0	0
$445$	1.70820	0.0809766
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.70820	0.458158	0.229079	−	0.973408i	$-0.426429\pi$
$449$	9.70820	0.458158	0.229079	+	0.973408i	$0.426429\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	16.8328	0.787406	0.393703	−	0.919238i	$-0.371194\pi$
$457$	16.8328	0.787406	0.393703	+	0.919238i	$0.371194\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.47214	0.394587	0.197293	−	0.980344i	$-0.436785\pi$
$461$	8.47214	0.394587	0.197293	+	0.980344i	$0.436785\pi$
$462$	0	0
$463$	25.5967	1.18958	0.594791	−	0.803880i	$-0.297235\pi$
$463$	25.5967	1.18958	0.594791	+	0.803880i	$0.297235\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.58359	0.212103	0.106052	−	0.994361i	$-0.466179\pi$
$467$	4.58359	0.212103	0.106052	+	0.994361i	$0.466179\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.47214	0.297589
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.1803	0.647916	0.323958	−	0.946071i	$-0.394986\pi$
$479$	14.1803	0.647916	0.323958	+	0.946071i	$0.394986\pi$
$480$	0	0
$481$	2.47214	0.112720
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.76393	−0.216319
$486$	0	0
$487$	4.58359	0.207702	0.103851	−	0.994593i	$-0.466883\pi$
$487$	4.58359	0.207702	0.103851	+	0.994593i	$0.466883\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.6525	1.47359	0.736793	−	0.676119i	$-0.236339\pi$
$491$	32.6525	1.47359	0.736793	+	0.676119i	$0.236339\pi$
$492$	0	0
$493$	3.41641	0.153867
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.4164	−0.511069	−0.255534	−	0.966800i	$-0.582251\pi$
$499$	−11.4164	−0.511069	−0.255534	+	0.966800i	$0.582251\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.9443	0.577157	0.288578	−	0.957456i	$-0.406817\pi$
$503$	12.9443	0.577157	0.288578	+	0.957456i	$0.406817\pi$
$504$	0	0
$505$	13.4164	0.597022
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−30.2918	−1.34266	−0.671330	−	0.741158i	$-0.734277\pi$
$509$	−30.2918	−1.34266	−0.671330	+	0.741158i	$0.734277\pi$
$510$	0	0
$511$	65.3050	2.88892
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.52786	0.419848
$516$	0	0
$517$	−3.05573	−0.134391
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.2918	1.15186	0.575932	−	0.817497i	$-0.304639\pi$
$521$	26.2918	1.15186	0.575932	+	0.817497i	$0.304639\pi$
$522$	0	0
$523$	−0.944272	−0.0412901	−0.0206451	−	0.999787i	$-0.506572\pi$
$523$	−0.944272	−0.0412901	−0.0206451	+	0.999787i	$0.506572\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.41641	0.321240
$534$	0	0
$535$	8.94427	0.386695
$536$	0	0
$537$	0	0
$538$	0	0
$539$	25.2361	1.08699
$540$	0	0
$541$	−29.7771	−1.28022	−0.640108	−	0.768285i	$-0.721111\pi$
$541$	−29.7771	−1.28022	−0.640108	+	0.768285i	$0.721111\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.47214	−0.191565
$546$	0	0
$547$	−37.3050	−1.59504	−0.797522	−	0.603289i	$-0.793856\pi$
$547$	−37.3050	−1.59504	−0.797522	+	0.603289i	$0.793856\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.763932	−0.0325446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.8328	−1.39117	−0.695586	−	0.718443i	$-0.744855\pi$
$557$	−32.8328	−1.39117	−0.695586	+	0.718443i	$0.744855\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.88854	0.416752	0.208376	−	0.978049i	$-0.433182\pi$
$563$	9.88854	0.416752	0.208376	+	0.978049i	$0.433182\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.5410	1.36419	0.682095	−	0.731263i	$-0.261069\pi$
$569$	32.5410	1.36419	0.682095	+	0.731263i	$0.261069\pi$
$570$	0	0
$571$	−21.3050	−0.891584	−0.445792	−	0.895136i	$-0.647078\pi$
$571$	−21.3050	−0.891584	−0.445792	+	0.895136i	$0.647078\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.47214	−0.103095
$576$	0	0
$577$	−9.05573	−0.376995	−0.188497	−	0.982074i	$-0.560362\pi$
$577$	−9.05573	−0.376995	−0.188497	+	0.982074i	$0.560362\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.9443	−0.868915
$582$	0	0
$583$	0.583592	0.0241699
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.4164	0.471206	0.235603	−	0.971849i	$-0.424294\pi$
$587$	11.4164	0.471206	0.235603	+	0.971849i	$0.424294\pi$
$588$	0	0
$589$	−8.94427	−0.368542
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	−23.4164	−0.959979
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−47.4164	−1.93738	−0.968691	−	0.248270i	$-0.920138\pi$
$599$	−47.4164	−1.93738	−0.968691	+	0.248270i	$0.920138\pi$
$600$	0	0
$601$	−10.3607	−0.422621	−0.211310	−	0.977419i	$-0.567773\pi$
$601$	−10.3607	−0.422621	−0.211310	+	0.977419i	$0.567773\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.47214	−0.385097
$606$	0	0
$607$	−33.5279	−1.36085	−0.680427	−	0.732816i	$-0.738206\pi$
$607$	−33.5279	−1.36085	−0.680427	+	0.732816i	$0.738206\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.88854	−0.0764023
$612$	0	0
$613$	−35.5279	−1.43496	−0.717478	−	0.696581i	$-0.754704\pi$
$613$	−35.5279	−1.43496	−0.717478	+	0.696581i	$0.754704\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.3050	1.58236	0.791179	−	0.611585i	$-0.209468\pi$
$617$	39.3050	1.58236	0.791179	+	0.611585i	$0.209468\pi$
$618$	0	0
$619$	32.3607	1.30069	0.650343	−	0.759641i	$-0.274625\pi$
$619$	32.3607	1.30069	0.650343	+	0.759641i	$0.274625\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.94427	0.358345
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.4721	−0.577042
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20.0000	0.793676
$636$	0	0
$637$	15.5967	0.617966
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.7639	1.76807	0.884035	−	0.467422i	$-0.154817\pi$
$641$	44.7639	1.76807	0.884035	+	0.467422i	$0.154817\pi$
$642$	0	0
$643$	17.5967	0.693948	0.346974	−	0.937875i	$-0.387209\pi$
$643$	17.5967	0.693948	0.346974	+	0.937875i	$0.387209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.8328	−0.583138	−0.291569	−	0.956550i	$-0.594177\pi$
$647$	−14.8328	−0.583138	−0.291569	+	0.956550i	$0.594177\pi$
$648$	0	0
$649$	12.9443	0.508107
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.5279	1.23378	0.616890	−	0.787049i	$-0.288392\pi$
$653$	31.5279	1.23378	0.616890	+	0.787049i	$0.288392\pi$
$654$	0	0
$655$	−14.1803	−0.554072
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.8328	−1.20108	−0.600538	−	0.799596i	$-0.705047\pi$
$659$	−30.8328	−1.20108	−0.600538	+	0.799596i	$0.705047\pi$
$660$	0	0
$661$	23.3050	0.906458	0.453229	−	0.891394i	$-0.350272\pi$
$661$	23.3050	0.906458	0.453229	+	0.891394i	$0.350272\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.23607	0.203046
$666$	0	0
$667$	1.88854	0.0731247
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.52786	0.213401
$672$	0	0
$673$	−27.5967	−1.06378	−0.531888	−	0.846815i	$-0.678517\pi$
$673$	−27.5967	−1.06378	−0.531888	+	0.846815i	$0.678517\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.5279	−0.596784	−0.298392	−	0.954443i	$-0.596450\pi$
$677$	−15.5279	−0.596784	−0.298392	+	0.954443i	$0.596450\pi$
$678$	0	0
$679$	−24.9443	−0.957273
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.7771	−1.21592	−0.607958	−	0.793969i	$-0.708011\pi$
$683$	−31.7771	−1.21592	−0.607958	+	0.793969i	$0.708011\pi$
$684$	0	0
$685$	−15.8885	−0.607070
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.360680	0.0137408
$690$	0	0
$691$	−25.5279	−0.971126	−0.485563	−	0.874202i	$-0.661385\pi$
$691$	−25.5279	−0.971126	−0.485563	+	0.874202i	$0.661385\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.4721	−0.548959
$696$	0	0
$697$	−43.4164	−1.64451
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.5836	0.550815	0.275407	−	0.961328i	$-0.411187\pi$
$701$	14.5836	0.550815	0.275407	+	0.961328i	$0.411187\pi$
$702$	0	0
$703$	3.23607	0.122051
$704$	0	0
$705$	0	0
$706$	0	0
$707$	70.2492	2.64199
$708$	0	0
$709$	−31.3050	−1.17568	−0.587841	−	0.808976i	$-0.700022\pi$
$709$	−31.3050	−1.17568	−0.587841	+	0.808976i	$0.700022\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.1115	0.828081
$714$	0	0
$715$	0.944272	0.0353138
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.23607	−0.0460976	−0.0230488	−	0.999734i	$-0.507337\pi$
$719$	−1.23607	−0.0460976	−0.0230488	+	0.999734i	$0.507337\pi$
$720$	0	0
$721$	49.8885	1.85795
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.763932	−0.0283717
$726$	0	0
$727$	20.6525	0.765958	0.382979	−	0.923757i	$-0.374898\pi$
$727$	20.6525	0.765958	0.382979	+	0.923757i	$0.374898\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.4164	−0.866087
$732$	0	0
$733$	−36.8328	−1.36045	−0.680226	−	0.733003i	$-0.738118\pi$
$733$	−36.8328	−1.36045	−0.680226	+	0.733003i	$0.738118\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.88854	0.0695654
$738$	0	0
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.9443	1.64885	0.824423	−	0.565975i	$-0.191500\pi$
$743$	44.9443	1.64885	0.824423	+	0.565975i	$0.191500\pi$
$744$	0	0
$745$	−17.4164	−0.638088
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.8328	1.71123
$750$	0	0
$751$	8.94427	0.326381	0.163191	−	0.986595i	$-0.447821\pi$
$751$	8.94427	0.326381	0.163191	+	0.986595i	$0.447821\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−2.94427	−0.107011	−0.0535057	−	0.998568i	$-0.517040\pi$
$757$	−2.94427	−0.107011	−0.0535057	+	0.998568i	$0.517040\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.41641	−0.341345	−0.170672	−	0.985328i	$-0.554594\pi$
$761$	−9.41641	−0.341345	−0.170672	+	0.985328i	$0.554594\pi$
$762$	0	0
$763$	−23.4164	−0.847731
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	8.11146	0.292507	0.146253	−	0.989247i	$-0.453279\pi$
$769$	8.11146	0.292507	0.146253	+	0.989247i	$0.453279\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.5279	−1.42172	−0.710859	−	0.703334i	$-0.751694\pi$
$773$	−39.5279	−1.42172	−0.710859	+	0.703334i	$0.751694\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.70820	0.347833
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.4164	−0.764384
$786$	0	0
$787$	10.8328	0.386148	0.193074	−	0.981184i	$-0.438154\pi$
$787$	10.8328	0.386148	0.193074	+	0.981184i	$0.438154\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.4721	0.372346
$792$	0	0
$793$	3.41641	0.121320
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.7771	−1.33813	−0.669067	−	0.743202i	$-0.733306\pi$
$797$	−37.7771	−1.33813	−0.669067	+	0.743202i	$0.733306\pi$
$798$	0	0
$799$	11.0557	0.391124
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.4164	0.544033
$804$	0	0
$805$	−12.9443	−0.456226
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	0.944272	0.0331579	0.0165789	−	0.999863i	$-0.494723\pi$
$811$	0.944272	0.0331579	0.0165789	+	0.999863i	$0.494723\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.6525	0.723425
$816$	0	0
$817$	5.23607	0.183187
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.5279	−0.541926	−0.270963	−	0.962590i	$-0.587342\pi$
$821$	−15.5279	−0.541926	−0.270963	+	0.962590i	$0.587342\pi$
$822$	0	0
$823$	−2.18034	−0.0760019	−0.0380009	−	0.999278i	$-0.512099\pi$
$823$	−2.18034	−0.0760019	−0.0380009	+	0.999278i	$0.512099\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	47.3050	1.64297	0.821484	−	0.570231i	$-0.193146\pi$
$829$	47.3050	1.64297	0.821484	+	0.570231i	$0.193146\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−91.3050	−3.16353
$834$	0	0
$835$	−13.8885	−0.480633
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.4164	−0.979876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.4164	−0.427137
$846$	0	0
$847$	−49.5967	−1.70416
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	42.7214	1.46275	0.731376	−	0.681975i	$-0.238879\pi$
$853$	42.7214	1.46275	0.731376	+	0.681975i	$0.238879\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.4721	−0.835952	−0.417976	−	0.908458i	$-0.637260\pi$
$857$	−24.4721	−0.835952	−0.417976	+	0.908458i	$0.637260\pi$
$858$	0	0
$859$	10.8328	0.369611	0.184805	−	0.982775i	$-0.440834\pi$
$859$	10.8328	0.369611	0.184805	+	0.982775i	$0.440834\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.8328	−1.59421	−0.797104	−	0.603842i	$-0.793636\pi$
$863$	−46.8328	−1.59421	−0.797104	+	0.603842i	$0.793636\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.16718	0.0395485
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.23607	0.177011
$876$	0	0
$877$	37.7082	1.27332	0.636658	−	0.771146i	$-0.280316\pi$
$877$	37.7082	1.27332	0.636658	+	0.771146i	$0.280316\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	55.3050	1.86327	0.931636	−	0.363394i	$-0.118382\pi$
$881$	55.3050	1.86327	0.931636	+	0.363394i	$0.118382\pi$
$882$	0	0
$883$	25.5967	0.861399	0.430700	−	0.902495i	$-0.358267\pi$
$883$	25.5967	0.861399	0.430700	+	0.902495i	$0.358267\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.8885	0.869252	0.434626	−	0.900611i	$-0.356881\pi$
$887$	25.8885	0.869252	0.434626	+	0.900611i	$0.356881\pi$
$888$	0	0
$889$	104.721	3.51224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.47214	−0.0827269
$894$	0	0
$895$	15.4164	0.515314
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.83282	0.227887
$900$	0	0
$901$	−2.11146	−0.0703428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.47214	0.281623
$906$	0	0
$907$	10.1115	0.335745	0.167873	−	0.985809i	$-0.446310\pi$
$907$	10.1115	0.335745	0.167873	+	0.985809i	$0.446310\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.7771	1.45040	0.725200	−	0.688538i	$-0.241747\pi$
$911$	43.7771	1.45040	0.725200	+	0.688538i	$0.241747\pi$
$912$	0	0
$913$	−4.94427	−0.163632
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−74.2492	−2.45193
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3.23607	0.106401
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.8885	−0.914993	−0.457497	−	0.889211i	$-0.651254\pi$
$929$	−27.8885	−0.914993	−0.457497	+	0.889211i	$0.651254\pi$
$930$	0	0
$931$	20.4164	0.669121
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.52786	−0.180780
$936$	0	0
$937$	−37.4164	−1.22234	−0.611170	−	0.791499i	$-0.709301\pi$
$937$	−37.4164	−1.22234	−0.611170	+	0.791499i	$0.709301\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.1246	0.558246	0.279123	−	0.960255i	$-0.409956\pi$
$941$	17.1246	0.558246	0.279123	+	0.960255i	$0.409956\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	9.52786	0.309288
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.9443	−1.39110	−0.695551	−	0.718477i	$-0.744840\pi$
$953$	−42.9443	−1.39110	−0.695551	+	0.718477i	$0.744840\pi$
$954$	0	0
$955$	6.18034	0.199991
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−83.1935	−2.68646
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.70820	0.183754
$966$	0	0
$967$	−18.1803	−0.584640	−0.292320	−	0.956321i	$-0.594427\pi$
$967$	−18.1803	−0.584640	−0.292320	+	0.956321i	$0.594427\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.8328	1.24620	0.623102	−	0.782140i	$-0.285872\pi$
$971$	38.8328	1.24620	0.623102	+	0.782140i	$0.285872\pi$
$972$	0	0
$973$	−75.7771	−2.42930
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.5836	−0.978456	−0.489228	−	0.872156i	$-0.662721\pi$
$977$	−30.5836	−0.978456	−0.489228	+	0.872156i	$0.662721\pi$
$978$	0	0
$979$	2.11146	0.0674824
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	12.4721	0.397395
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.9443	−0.411604
$990$	0	0
$991$	28.9443	0.919445	0.459723	−	0.888063i	$-0.347949\pi$
$991$	28.9443	0.919445	0.459723	+	0.888063i	$0.347949\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.4721	0.585606
$996$	0	0
$997$	1.41641	0.0448581	0.0224290	−	0.999748i	$-0.492860\pi$
$997$	1.41641	0.0448581	0.0224290	+	0.999748i	$0.492860\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +5.23607 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +5.23607 q^{7} +1.23607 q^{11} +0.763932 q^{13} -4.47214 q^{17} +1.00000 q^{19} -2.47214 q^{23} +1.00000 q^{25} -0.763932 q^{29} -8.94427 q^{31} +5.23607 q^{35} +3.23607 q^{37} +9.70820 q^{41} +5.23607 q^{43} -2.47214 q^{47} +20.4164 q^{49} +0.472136 q^{53} +1.23607 q^{55} +10.4721 q^{59} +4.47214 q^{61} +0.763932 q^{65} +1.52786 q^{67} +12.4721 q^{73} +6.47214 q^{77} -4.00000 q^{83} -4.47214 q^{85} +1.70820 q^{89} +4.00000 q^{91} +1.00000 q^{95} -4.76393 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 6 * q^7	\( 2 q + 2 q^{5} + 6 q^{7} - 2 q^{11} + 6 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 6 q^{29} + 6 q^{35} + 2 q^{37} + 6 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 8 q^{53} - 2 q^{55} + 12 q^{59} + 6 q^{65} + 12 q^{67} + 16 q^{73} + 4 q^{77} - 8 q^{83} - 10 q^{89} + 8 q^{91} + 2 q^{95} - 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 6 * q^7 - 2 * q^11 + 6 * q^13 + 2 * q^19 + 4 * q^23 + 2 * q^25 - 6 * q^29 + 6 * q^35 + 2 * q^37 + 6 * q^41 + 6 * q^43 + 4 * q^47 + 14 * q^49 - 8 * q^53 - 2 * q^55 + 12 * q^59 + 6 * q^65 + 12 * q^67 + 16 * q^73 + 4 * q^77 - 8 * q^83 - 10 * q^89 + 8 * q^91 + 2 * q^95 - 14 * q^97

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