LMFDB - Embedded newform 4160.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} -1.00000 q^{5} +5.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{3} -1.00000 q^{5} +5.00000 q^{9} -2.82843 q^{11} -1.00000 q^{13} +2.82843 q^{15} +2.00000 q^{17} -2.82843 q^{19} -2.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -2.82843 q^{31} +8.00000 q^{33} +2.00000 q^{37} +2.82843 q^{39} -6.00000 q^{41} +2.82843 q^{43} -5.00000 q^{45} -7.00000 q^{49} -5.65685 q^{51} -6.00000 q^{53} +2.82843 q^{55} +8.00000 q^{57} -8.48528 q^{59} +2.00000 q^{61} +1.00000 q^{65} -5.65685 q^{67} +8.00000 q^{69} -14.1421 q^{71} +10.0000 q^{73} -2.82843 q^{75} +5.65685 q^{79} +1.00000 q^{81} +11.3137 q^{83} -2.00000 q^{85} -5.65685 q^{87} +10.0000 q^{89} +8.00000 q^{93} +2.82843 q^{95} +10.0000 q^{97} -14.1421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 10 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 10 * q^9	$ 2 q - 2 q^{5} + 10 q^{9} - 2 q^{13} + 4 q^{17} + 2 q^{25} + 4 q^{29} + 16 q^{33} + 4 q^{37} - 12 q^{41} - 10 q^{45} - 14 q^{49} - 12 q^{53} + 16 q^{57} + 4 q^{61} + 2 q^{65} + 16 q^{69} + 20 q^{73} + 2 q^{81} - 4 q^{85} + 20 q^{89} + 16 q^{93} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 10 * q^9 - 2 * q^13 + 4 * q^17 + 2 * q^25 + 4 * q^29 + 16 * q^33 + 4 * q^37 - 12 * q^41 - 10 * q^45 - 14 * q^49 - 12 * q^53 + 16 * q^57 + 4 * q^61 + 2 * q^65 + 16 * q^69 + 20 * q^73 + 2 * q^81 - 4 * q^85 + 20 * q^89 + 16 * q^93 + 20 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	−5.00000	−0.745356
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−5.65685	−0.792118
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−14.1421	−1.67836	−0.839181	−	0.543852i	$-0.816965\pi$
$71$	−14.1421	−1.67836	−0.839181	+	0.543852i	$0.816965\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−2.82843	−0.326599
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−5.65685	−0.606478
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−14.1421	−1.42134
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	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
$115$	2.82843	0.263752
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	16.9706	1.53018
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	5.65685	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	19.7990	1.63299
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	10.0000	0.808452
$154$	0	0
$155$	2.82843	0.227185
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	16.9706	1.34585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.9706	1.32924	0.664619	−	0.747183i	$-0.268594\pi$
$163$	16.9706	1.32924	0.664619	+	0.747183i	$0.268594\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−14.1421	−1.08148
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.0000	1.80395
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−5.65685	−0.418167
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	−2.82843	−0.202548
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−14.1421	−0.982946
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	40.0000	2.74075
$214$	0	0
$215$	−2.82843	−0.192897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−28.2843	−1.91127
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	11.3137	0.757622	0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137	0.757622	0.378811	+	0.925474i	$0.376333\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	28.2843	1.87729	0.938647	−	0.344881i	$-0.112081\pi$
$227$	28.2843	1.87729	0.938647	+	0.344881i	$0.112081\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	14.1421	0.914779	0.457389	−	0.889267i	$-0.348785\pi$
$239$	14.1421	0.914779	0.457389	+	0.889267i	$0.348785\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	7.00000	0.447214
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	−32.0000	−2.02792
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	5.65685	0.354246
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	14.1421	0.872041	0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421	0.872041	0.436021	+	0.899937i	$0.356387\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−28.2843	−1.73097
$268$	0	0
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	14.1421	0.859074	0.429537	−	0.903049i	$-0.358677\pi$
$271$	14.1421	0.859074	0.429537	+	0.903049i	$0.358677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−14.1421	−0.846668
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	25.4558	1.51319	0.756596	−	0.653882i	$-0.226861\pi$
$283$	25.4558	1.51319	0.756596	+	0.653882i	$0.226861\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−28.2843	−1.65805
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	8.48528	0.494032
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	2.82843	0.163572
$300$	0	0
$301$	0	0
$302$	0	0
$303$	39.5980	2.27484
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	0	0
$309$	40.0000	2.27552
$310$	0	0
$311$	−28.2843	−1.60385	−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843	−1.60385	−0.801927	+	0.597422i	$0.796192\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−5.65685	−0.316723
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	39.5980	2.18977
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.4558	1.39918	0.699590	−	0.714545i	$-0.253366\pi$
$331$	25.4558	1.39918	0.699590	+	0.714545i	$0.253366\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	5.65685	0.309067
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−19.7990	−1.06287	−0.531433	−	0.847100i	$-0.678346\pi$
$347$	−19.7990	−1.06287	−0.531433	+	0.847100i	$0.678346\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	14.1421	0.750587
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	8.48528	0.445362
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	31.1127	1.62407	0.812035	−	0.583609i	$-0.198360\pi$
$367$	31.1127	1.62407	0.812035	+	0.583609i	$0.198360\pi$
$368$	0	0
$369$	−30.0000	−1.56174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	2.82843	0.146059
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−2.82843	−0.145287	−0.0726433	−	0.997358i	$-0.523143\pi$
$379$	−2.82843	−0.145287	−0.0726433	+	0.997358i	$0.523143\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	22.6274	1.15621	0.578103	−	0.815963i	$-0.303793\pi$
$383$	22.6274	1.15621	0.578103	+	0.815963i	$0.303793\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.1421	0.718885
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	−32.0000	−1.61419
$394$	0	0
$395$	−5.65685	−0.284627
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	2.82843	0.140894
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−5.65685	−0.280400
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−50.9117	−2.51129
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.3137	−0.555368
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	5.65685	0.276355	0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685	0.276355	0.138178	+	0.990407i	$0.455875\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	19.7990	0.953684	0.476842	−	0.878989i	$-0.341781\pi$
$431$	19.7990	0.953684	0.476842	+	0.878989i	$0.341781\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	5.65685	0.271225
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	−35.0000	−1.66667
$442$	0	0
$443$	19.7990	0.940678	0.470339	−	0.882486i	$-0.344132\pi$
$443$	19.7990	0.940678	0.470339	+	0.882486i	$0.344132\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−28.2843	−1.33780
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	16.9706	0.799113
$452$	0	0
$453$	24.0000	1.12762
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	−11.3137	−0.528079
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	16.9706	0.788689	0.394344	−	0.918963i	$-0.370972\pi$
$463$	16.9706	0.788689	0.394344	+	0.918963i	$0.370972\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−14.1421	−0.654420	−0.327210	−	0.944952i	$-0.606108\pi$
$467$	−14.1421	−0.654420	−0.327210	+	0.944952i	$0.606108\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.65685	−0.260654
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−30.0000	−1.37361
$478$	0	0
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−5.65685	−0.256337	−0.128168	−	0.991752i	$-0.540910\pi$
$487$	−5.65685	−0.256337	−0.128168	+	0.991752i	$0.540910\pi$
$488$	0	0
$489$	−48.0000	−2.17064
$490$	0	0
$491$	28.2843	1.27645	0.638226	−	0.769849i	$-0.279669\pi$
$491$	28.2843	1.27645	0.638226	+	0.769849i	$0.279669\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	14.1421	0.635642
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.48528	0.379853	0.189927	−	0.981798i	$-0.439175\pi$
$499$	8.48528	0.379853	0.189927	+	0.981798i	$0.439175\pi$
$500$	0	0
$501$	32.0000	1.42965
$502$	0	0
$503$	14.1421	0.630567	0.315283	−	0.948998i	$-0.397900\pi$
$503$	14.1421	0.630567	0.315283	+	0.948998i	$0.397900\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	−2.82843	−0.125615
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	14.1421	0.623177
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−50.9117	−2.23478
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−14.1421	−0.618392	−0.309196	−	0.950998i	$-0.600060\pi$
$523$	−14.1421	−0.618392	−0.309196	+	0.950998i	$0.600060\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.65685	−0.246416
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−42.4264	−1.84115
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	8.48528	0.366851
$536$	0	0
$537$	−32.0000	−1.38090
$538$	0	0
$539$	19.7990	0.852803
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	−28.2843	−1.21379
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−8.48528	−0.362804	−0.181402	−	0.983409i	$-0.558064\pi$
$547$	−8.48528	−0.362804	−0.181402	+	0.983409i	$0.558064\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−5.65685	−0.240990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.65685	0.240120
$556$	0	0
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−2.82843	−0.119630
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	14.1421	0.596020	0.298010	−	0.954563i	$-0.403677\pi$
$563$	14.1421	0.596020	0.298010	+	0.954563i	$0.403677\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	0	0
$571$	−16.9706	−0.710196	−0.355098	−	0.934829i	$-0.615552\pi$
$571$	−16.9706	−0.710196	−0.355098	+	0.934829i	$0.615552\pi$
$572$	0	0
$573$	32.0000	1.33682
$574$	0	0
$575$	−2.82843	−0.117954
$576$	0	0
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	0	0
$579$	−28.2843	−1.17545
$580$	0	0
$581$	0	0
$582$	0	0
$583$	16.9706	0.702849
$584$	0	0
$585$	5.00000	0.206725
$586$	0	0
$587$	−28.2843	−1.16742	−0.583708	−	0.811963i	$-0.698399\pi$
$587$	−28.2843	−1.16742	−0.583708	+	0.811963i	$0.698399\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	16.9706	0.698076
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.0000	−1.30967
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	−28.2843	−1.15182
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	−14.1421	−0.574012	−0.287006	−	0.957929i	$-0.592660\pi$
$607$	−14.1421	−0.574012	−0.287006	+	0.957929i	$0.592660\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	−16.9706	−0.684319
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	25.4558	1.02316	0.511578	−	0.859237i	$-0.329061\pi$
$619$	25.4558	1.02316	0.511578	+	0.859237i	$0.329061\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−22.6274	−0.903652
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−31.1127	−1.23858	−0.619288	−	0.785164i	$-0.712579\pi$
$631$	−31.1127	−1.23858	−0.619288	+	0.785164i	$0.712579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.48528	0.336728
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	−70.7107	−2.79727
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	33.9411	1.33851	0.669254	−	0.743034i	$-0.266614\pi$
$643$	33.9411	1.33851	0.669254	+	0.743034i	$0.266614\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	2.82843	0.111197	0.0555985	−	0.998453i	$-0.482293\pi$
$647$	2.82843	0.111197	0.0555985	+	0.998453i	$0.482293\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	0	0
$657$	50.0000	1.95069
$658$	0	0
$659$	39.5980	1.54252	0.771259	−	0.636521i	$-0.219627\pi$
$659$	39.5980	1.54252	0.771259	+	0.636521i	$0.219627\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	0	0
$663$	5.65685	0.219694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.65685	−0.219034
$668$	0	0
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−80.0000	−3.06561
$682$	0	0
$683$	−28.2843	−1.08227	−0.541134	−	0.840937i	$-0.682005\pi$
$683$	−28.2843	−1.08227	−0.541134	+	0.840937i	$0.682005\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	−73.5391	−2.80569
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	2.82843	0.107598	0.0537992	−	0.998552i	$-0.482867\pi$
$691$	2.82843	0.107598	0.0537992	+	0.998552i	$0.482867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.65685	−0.214577
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	62.2254	2.35358
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−5.65685	−0.213352
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	28.2843	1.06074
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−2.82843	−0.105777
$716$	0	0
$717$	−40.0000	−1.49383
$718$	0	0
$719$	45.2548	1.68772	0.843860	−	0.536563i	$-0.180278\pi$
$719$	45.2548	1.68772	0.843860	+	0.536563i	$0.180278\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−50.9117	−1.89343
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−8.48528	−0.314702	−0.157351	−	0.987543i	$-0.550295\pi$
$727$	−8.48528	−0.314702	−0.157351	+	0.987543i	$0.550295\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	5.65685	0.209226
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	−19.7990	−0.730297
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−42.4264	−1.56068	−0.780340	−	0.625355i	$-0.784954\pi$
$739$	−42.4264	−1.56068	−0.780340	+	0.625355i	$0.784954\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	56.5685	2.06973
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5.65685	−0.206422	−0.103211	−	0.994660i	$-0.532912\pi$
$751$	−5.65685	−0.206422	−0.103211	+	0.994660i	$0.532912\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	8.48528	0.308811
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	8.48528	0.306386
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−5.65685	−0.203727
$772$	0	0
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	0	0
$775$	−2.82843	−0.101600
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.9706	0.608034
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−11.3137	−0.403290	−0.201645	−	0.979459i	$-0.564629\pi$
$787$	−11.3137	−0.403290	−0.201645	+	0.979459i	$0.564629\pi$
$788$	0	0
$789$	−40.0000	−1.42404
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−16.9706	−0.601884
$796$	0	0
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	50.0000	1.76666
$802$	0	0
$803$	−28.2843	−0.998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	62.2254	2.19044
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	2.82843	0.0993195	0.0496598	−	0.998766i	$-0.484186\pi$
$811$	2.82843	0.0993195	0.0496598	+	0.998766i	$0.484186\pi$
$812$	0	0
$813$	−40.0000	−1.40286
$814$	0	0
$815$	−16.9706	−0.594453
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	42.4264	1.47889	0.739446	−	0.673216i	$-0.235088\pi$
$823$	42.4264	1.47889	0.739446	+	0.673216i	$0.235088\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	11.3137	0.393416	0.196708	−	0.980462i	$-0.436975\pi$
$827$	11.3137	0.393416	0.196708	+	0.980462i	$0.436975\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−28.2843	−0.981170
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	11.3137	0.391527
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	−36.7696	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$839$	−36.7696	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−28.2843	−0.974162
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−72.0000	−2.47103
$850$	0	0
$851$	−5.65685	−0.193914
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	14.1421	0.483651
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	16.9706	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$859$	16.9706	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.6274	−0.770246	−0.385123	−	0.922865i	$-0.625841\pi$
$863$	−22.6274	−0.770246	−0.385123	+	0.922865i	$0.625841\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	36.7696	1.24876
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	5.65685	0.191675
$872$	0	0
$873$	50.0000	1.69224
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	39.5980	1.33561
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−53.7401	−1.80850	−0.904249	−	0.427005i	$-0.859569\pi$
$883$	−53.7401	−1.80850	−0.904249	+	0.427005i	$0.859569\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−5.65685	−0.188667
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−59.3970	−1.97224	−0.986122	−	0.166022i	$-0.946908\pi$
$907$	−59.3970	−1.97224	−0.986122	+	0.166022i	$0.946908\pi$
$908$	0	0
$909$	−70.0000	−2.32175
$910$	0	0
$911$	−11.3137	−0.374840	−0.187420	−	0.982280i	$-0.560013\pi$
$911$	−11.3137	−0.374840	−0.187420	+	0.982280i	$0.560013\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	0	0
$915$	5.65685	0.187010
$916$	0	0
$917$	0	0
$918$	0	0
$919$	50.9117	1.67942	0.839711	−	0.543034i	$-0.182724\pi$
$919$	50.9117	1.67942	0.839711	+	0.543034i	$0.182724\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	14.1421	0.465494
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	−70.7107	−2.32244
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	19.7990	0.648886
$932$	0	0
$933$	80.0000	2.61908
$934$	0	0
$935$	5.65685	0.184999
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	62.2254	2.03065
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.3137	0.367646	0.183823	−	0.982959i	$-0.441153\pi$
$947$	11.3137	0.367646	0.183823	+	0.982959i	$0.441153\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	16.9706	0.550308
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	−42.4264	−1.36717
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	39.5980	1.27339	0.636693	−	0.771118i	$-0.280302\pi$
$967$	39.5980	1.27339	0.636693	+	0.771118i	$0.280302\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.82843	0.0905822
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−28.2843	−0.903969
$980$	0	0
$981$	−70.0000	−2.23493
$982$	0	0
$983$	−16.9706	−0.541277	−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706	−0.541277	−0.270638	+	0.962681i	$0.587235\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	33.9411	1.07818	0.539088	−	0.842250i	$-0.318769\pi$
$991$	33.9411	1.07818	0.539088	+	0.842250i	$0.318769\pi$
$992$	0	0
$993$	−72.0000	−2.28485
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.be.1.1		2
4.3	odd	2	inner	4160.2.a.be.1.2		2
8.3	odd	2		2080.2.a.l.1.1	✓	2
8.5	even	2		2080.2.a.l.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.l.1.1	✓	2	8.3	odd	2
2080.2.a.l.1.2	yes	2	8.5	even	2
4160.2.a.be.1.1		2	1.1	even	1	trivial
4160.2.a.be.1.2		2	4.3	odd	2	inner

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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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} -1.00000 q^{5} +5.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{3} -1.00000 q^{5} +5.00000 q^{9} -2.82843 q^{11} -1.00000 q^{13} +2.82843 q^{15} +2.00000 q^{17} -2.82843 q^{19} -2.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -2.82843 q^{31} +8.00000 q^{33} +2.00000 q^{37} +2.82843 q^{39} -6.00000 q^{41} +2.82843 q^{43} -5.00000 q^{45} -7.00000 q^{49} -5.65685 q^{51} -6.00000 q^{53} +2.82843 q^{55} +8.00000 q^{57} -8.48528 q^{59} +2.00000 q^{61} +1.00000 q^{65} -5.65685 q^{67} +8.00000 q^{69} -14.1421 q^{71} +10.0000 q^{73} -2.82843 q^{75} +5.65685 q^{79} +1.00000 q^{81} +11.3137 q^{83} -2.00000 q^{85} -5.65685 q^{87} +10.0000 q^{89} +8.00000 q^{93} +2.82843 q^{95} +10.0000 q^{97} -14.1421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 10 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 10 * q^9	\( 2 q - 2 q^{5} + 10 q^{9} - 2 q^{13} + 4 q^{17} + 2 q^{25} + 4 q^{29} + 16 q^{33} + 4 q^{37} - 12 q^{41} - 10 q^{45} - 14 q^{49} - 12 q^{53} + 16 q^{57} + 4 q^{61} + 2 q^{65} + 16 q^{69} + 20 q^{73} + 2 q^{81} - 4 q^{85} + 20 q^{89} + 16 q^{93} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 10 * q^9 - 2 * q^13 + 4 * q^17 + 2 * q^25 + 4 * q^29 + 16 * q^33 + 4 * q^37 - 12 * q^41 - 10 * q^45 - 14 * q^49 - 12 * q^53 + 16 * q^57 + 4 * q^61 + 2 * q^65 + 16 * q^69 + 20 * q^73 + 2 * q^81 - 4 * q^85 + 20 * q^89 + 16 * q^93 + 20 * q^97

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