LMFDB - Embedded newform 6080.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +4.00000 q^{11} -2.56155 q^{13} +2.56155 q^{15} -0.561553 q^{17} +1.00000 q^{19} +11.6847 q^{21} -5.68466 q^{23} +1.00000 q^{25} +1.43845 q^{27} +3.43845 q^{29} -5.12311 q^{31} +10.2462 q^{33} +4.56155 q^{35} +1.12311 q^{37} -6.56155 q^{39} +8.24621 q^{41} +2.00000 q^{43} +3.56155 q^{45} +3.12311 q^{47} +13.8078 q^{49} -1.43845 q^{51} -6.56155 q^{53} +4.00000 q^{55} +2.56155 q^{57} -12.8078 q^{59} +4.87689 q^{61} +16.2462 q^{63} -2.56155 q^{65} +10.5616 q^{67} -14.5616 q^{69} -8.00000 q^{71} +14.8078 q^{73} +2.56155 q^{75} +18.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} +14.0000 q^{83} -0.561553 q^{85} +8.80776 q^{87} -10.0000 q^{89} -11.6847 q^{91} -13.1231 q^{93} +1.00000 q^{95} -1.12311 q^{97} +14.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + 5 q^{7} + 3 q^{9} + 8 q^{11} - q^{13} + q^{15} + 3 q^{17} + 2 q^{19} + 11 q^{21} + q^{23} + 2 q^{25} + 7 q^{27} + 11 q^{29} - 2 q^{31} + 4 q^{33} + 5 q^{35} - 6 q^{37} - 9 q^{39}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + 5 * q^7 + 3 * q^9 + 8 * q^11 - q^13 + q^15 + 3 * q^17 + 2 * q^19 + 11 * q^21 + q^23 + 2 * q^25 + 7 * q^27 + 11 * q^29 - 2 * q^31 + 4 * q^33 + 5 * q^35 - 6 * q^37 - 9 * q^39 + 4 * q^43 + 3 * q^45 - 2 * q^47 + 7 * q^49 - 7 * q^51 - 9 * q^53 + 8 * q^55 + q^57 - 5 * q^59 + 18 * q^61 + 16 * q^63 - q^65 + 17 * q^67 - 25 * q^69 - 16 * q^71 + 9 * q^73 + q^75 + 20 * q^77 + 2 * q^79 - 14 * q^81 + 28 * q^83 + 3 * q^85 - 3 * q^87 - 20 * q^89 - 11 * q^91 - 18 * q^93 + 2 * q^95 + 6 * q^97 + 12 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.56155	1.72410	0.862052	−	0.506819i	$-0.169179\pi$
$7$	4.56155	1.72410	0.862052	+	0.506819i	$0.169179\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.56155	−0.710447	−0.355223	−	0.934781i	$-0.615595\pi$
$13$	−2.56155	−0.710447	−0.355223	+	0.934781i	$0.615595\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	−0.561553	−0.136197	−0.0680983	−	0.997679i	$-0.521693\pi$
$17$	−0.561553	−0.136197	−0.0680983	+	0.997679i	$0.521693\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	11.6847	2.54980
$22$	0	0
$23$	−5.68466	−1.18533	−0.592667	−	0.805448i	$-0.701925\pi$
$23$	−5.68466	−1.18533	−0.592667	+	0.805448i	$0.701925\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	3.43845	0.638504	0.319252	−	0.947670i	$-0.396568\pi$
$29$	3.43845	0.638504	0.319252	+	0.947670i	$0.396568\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	0	0
$33$	10.2462	1.78364
$34$	0	0
$35$	4.56155	0.771043
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	0	0
$39$	−6.56155	−1.05069
$40$	0	0
$41$	8.24621	1.28784	0.643921	−	0.765092i	$-0.277307\pi$
$41$	8.24621	1.28784	0.643921	+	0.765092i	$0.277307\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	3.12311	0.455552	0.227776	−	0.973714i	$-0.426855\pi$
$47$	3.12311	0.455552	0.227776	+	0.973714i	$0.426855\pi$
$48$	0	0
$49$	13.8078	1.97254
$50$	0	0
$51$	−1.43845	−0.201423
$52$	0	0
$53$	−6.56155	−0.901299	−0.450649	−	0.892701i	$-0.648807\pi$
$53$	−6.56155	−0.901299	−0.450649	+	0.892701i	$0.648807\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	2.56155	0.339286
$58$	0	0
$59$	−12.8078	−1.66743	−0.833714	−	0.552196i	$-0.813790\pi$
$59$	−12.8078	−1.66743	−0.833714	+	0.552196i	$0.813790\pi$
$60$	0	0
$61$	4.87689	0.624422	0.312211	−	0.950013i	$-0.398930\pi$
$61$	4.87689	0.624422	0.312211	+	0.950013i	$0.398930\pi$
$62$	0	0
$63$	16.2462	2.04683
$64$	0	0
$65$	−2.56155	−0.317722
$66$	0	0
$67$	10.5616	1.29030	0.645150	−	0.764056i	$-0.276795\pi$
$67$	10.5616	1.29030	0.645150	+	0.764056i	$0.276795\pi$
$68$	0	0
$69$	−14.5616	−1.75300
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	14.8078	1.73312	0.866559	−	0.499075i	$-0.166327\pi$
$73$	14.8078	1.73312	0.866559	+	0.499075i	$0.166327\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	18.2462	2.07935
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	−0.561553	−0.0609090
$86$	0	0
$87$	8.80776	0.944291
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−11.6847	−1.22489
$92$	0	0
$93$	−13.1231	−1.36080
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−1.12311	−0.114034	−0.0570170	−	0.998373i	$-0.518159\pi$
$97$	−1.12311	−0.114034	−0.0570170	+	0.998373i	$0.518159\pi$
$98$	0	0
$99$	14.2462	1.43180
$100$	0	0
$101$	3.12311	0.310761	0.155380	−	0.987855i	$-0.450340\pi$
$101$	3.12311	0.310761	0.155380	+	0.987855i	$0.450340\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	11.6847	1.14031
$106$	0	0
$107$	11.6847	1.12960	0.564799	−	0.825228i	$-0.308954\pi$
$107$	11.6847	1.12960	0.564799	+	0.825228i	$0.308954\pi$
$108$	0	0
$109$	−11.4384	−1.09560	−0.547802	−	0.836608i	$-0.684535\pi$
$109$	−11.4384	−1.09560	−0.547802	+	0.836608i	$0.684535\pi$
$110$	0	0
$111$	2.87689	0.273063
$112$	0	0
$113$	−6.24621	−0.587594	−0.293797	−	0.955868i	$-0.594919\pi$
$113$	−6.24621	−0.587594	−0.293797	+	0.955868i	$0.594919\pi$
$114$	0	0
$115$	−5.68466	−0.530097
$116$	0	0
$117$	−9.12311	−0.843431
$118$	0	0
$119$	−2.56155	−0.234817
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	21.1231	1.90461
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.3693	−1.36381	−0.681903	−	0.731442i	$-0.738847\pi$
$127$	−15.3693	−1.36381	−0.681903	+	0.731442i	$0.738847\pi$
$128$	0	0
$129$	5.12311	0.451064
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	4.56155	0.395537
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−2.80776	−0.239883	−0.119942	−	0.992781i	$-0.538271\pi$
$137$	−2.80776	−0.239883	−0.119942	+	0.992781i	$0.538271\pi$
$138$	0	0
$139$	−2.24621	−0.190521	−0.0952606	−	0.995452i	$-0.530368\pi$
$139$	−2.24621	−0.190521	−0.0952606	+	0.995452i	$0.530368\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−10.2462	−0.856831
$144$	0	0
$145$	3.43845	0.285547
$146$	0	0
$147$	35.3693	2.91721
$148$	0	0
$149$	−3.12311	−0.255855	−0.127927	−	0.991784i	$-0.540832\pi$
$149$	−3.12311	−0.255855	−0.127927	+	0.991784i	$0.540832\pi$
$150$	0	0
$151$	13.1231	1.06794	0.533972	−	0.845502i	$-0.320699\pi$
$151$	13.1231	1.06794	0.533972	+	0.845502i	$0.320699\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−5.12311	−0.411498
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	−16.8078	−1.33294
$160$	0	0
$161$	−25.9309	−2.04364
$162$	0	0
$163$	−3.75379	−0.294019	−0.147010	−	0.989135i	$-0.546965\pi$
$163$	−3.75379	−0.294019	−0.147010	+	0.989135i	$0.546965\pi$
$164$	0	0
$165$	10.2462	0.797666
$166$	0	0
$167$	−19.3693	−1.49884	−0.749421	−	0.662093i	$-0.769668\pi$
$167$	−19.3693	−1.49884	−0.749421	+	0.662093i	$0.769668\pi$
$168$	0	0
$169$	−6.43845	−0.495265
$170$	0	0
$171$	3.56155	0.272359
$172$	0	0
$173$	−2.87689	−0.218726	−0.109363	−	0.994002i	$-0.534881\pi$
$173$	−2.87689	−0.218726	−0.109363	+	0.994002i	$0.534881\pi$
$174$	0	0
$175$	4.56155	0.344821
$176$	0	0
$177$	−32.8078	−2.46598
$178$	0	0
$179$	−14.2462	−1.06481	−0.532406	−	0.846489i	$-0.678712\pi$
$179$	−14.2462	−1.06481	−0.532406	+	0.846489i	$0.678712\pi$
$180$	0	0
$181$	−7.75379	−0.576335	−0.288167	−	0.957580i	$-0.593046\pi$
$181$	−7.75379	−0.576335	−0.288167	+	0.957580i	$0.593046\pi$
$182$	0	0
$183$	12.4924	0.923466
$184$	0	0
$185$	1.12311	0.0825724
$186$	0	0
$187$	−2.24621	−0.164259
$188$	0	0
$189$	6.56155	0.477283
$190$	0	0
$191$	4.31534	0.312247	0.156124	−	0.987738i	$-0.450100\pi$
$191$	4.31534	0.312247	0.156124	+	0.987738i	$0.450100\pi$
$192$	0	0
$193$	9.12311	0.656696	0.328348	−	0.944557i	$-0.393508\pi$
$193$	9.12311	0.656696	0.328348	+	0.944557i	$0.393508\pi$
$194$	0	0
$195$	−6.56155	−0.469883
$196$	0	0
$197$	24.2462	1.72747	0.863736	−	0.503945i	$-0.168119\pi$
$197$	24.2462	1.72747	0.863736	+	0.503945i	$0.168119\pi$
$198$	0	0
$199$	−5.43845	−0.385521	−0.192761	−	0.981246i	$-0.561744\pi$
$199$	−5.43845	−0.385521	−0.192761	+	0.981246i	$0.561744\pi$
$200$	0	0
$201$	27.0540	1.90824
$202$	0	0
$203$	15.6847	1.10085
$204$	0	0
$205$	8.24621	0.575940
$206$	0	0
$207$	−20.2462	−1.40721
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−23.0540	−1.58710	−0.793551	−	0.608504i	$-0.791770\pi$
$211$	−23.0540	−1.58710	−0.793551	+	0.608504i	$0.791770\pi$
$212$	0	0
$213$	−20.4924	−1.40412
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−23.3693	−1.58641
$218$	0	0
$219$	37.9309	2.56313
$220$	0	0
$221$	1.43845	0.0967604
$222$	0	0
$223$	15.3693	1.02921	0.514603	−	0.857429i	$-0.327939\pi$
$223$	15.3693	1.02921	0.514603	+	0.857429i	$0.327939\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	4.31534	0.286419	0.143210	−	0.989692i	$-0.454258\pi$
$227$	4.31534	0.286419	0.143210	+	0.989692i	$0.454258\pi$
$228$	0	0
$229$	28.7386	1.89910	0.949551	−	0.313612i	$-0.101539\pi$
$229$	28.7386	1.89910	0.949551	+	0.313612i	$0.101539\pi$
$230$	0	0
$231$	46.7386	3.07518
$232$	0	0
$233$	−26.4924	−1.73558	−0.867788	−	0.496934i	$-0.834459\pi$
$233$	−26.4924	−1.73558	−0.867788	+	0.496934i	$0.834459\pi$
$234$	0	0
$235$	3.12311	0.203729
$236$	0	0
$237$	13.1231	0.852437
$238$	0	0
$239$	−3.19224	−0.206489	−0.103244	−	0.994656i	$-0.532922\pi$
$239$	−3.19224	−0.206489	−0.103244	+	0.994656i	$0.532922\pi$
$240$	0	0
$241$	−29.8617	−1.92356	−0.961782	−	0.273817i	$-0.911714\pi$
$241$	−29.8617	−1.92356	−0.961782	+	0.273817i	$0.911714\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	13.8078	0.882146
$246$	0	0
$247$	−2.56155	−0.162988
$248$	0	0
$249$	35.8617	2.27265
$250$	0	0
$251$	−5.75379	−0.363176	−0.181588	−	0.983375i	$-0.558124\pi$
$251$	−5.75379	−0.363176	−0.181588	+	0.983375i	$0.558124\pi$
$252$	0	0
$253$	−22.7386	−1.42957
$254$	0	0
$255$	−1.43845	−0.0900791
$256$	0	0
$257$	−31.3693	−1.95676	−0.978382	−	0.206805i	$-0.933693\pi$
$257$	−31.3693	−1.95676	−0.978382	+	0.206805i	$0.933693\pi$
$258$	0	0
$259$	5.12311	0.318334
$260$	0	0
$261$	12.2462	0.758021
$262$	0	0
$263$	−9.36932	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$263$	−9.36932	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$264$	0	0
$265$	−6.56155	−0.403073
$266$	0	0
$267$	−25.6155	−1.56764
$268$	0	0
$269$	24.2462	1.47832	0.739159	−	0.673531i	$-0.235223\pi$
$269$	24.2462	1.47832	0.739159	+	0.673531i	$0.235223\pi$
$270$	0	0
$271$	−20.1771	−1.22567	−0.612835	−	0.790211i	$-0.709971\pi$
$271$	−20.1771	−1.22567	−0.612835	+	0.790211i	$0.709971\pi$
$272$	0	0
$273$	−29.9309	−1.81150
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	19.6155	1.17858	0.589291	−	0.807921i	$-0.299407\pi$
$277$	19.6155	1.17858	0.589291	+	0.807921i	$0.299407\pi$
$278$	0	0
$279$	−18.2462	−1.09237
$280$	0	0
$281$	24.2462	1.44641	0.723204	−	0.690635i	$-0.242669\pi$
$281$	24.2462	1.44641	0.723204	+	0.690635i	$0.242669\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	2.56155	0.151733
$286$	0	0
$287$	37.6155	2.22037
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	−2.87689	−0.168647
$292$	0	0
$293$	−5.93087	−0.346485	−0.173243	−	0.984879i	$-0.555424\pi$
$293$	−5.93087	−0.346485	−0.173243	+	0.984879i	$0.555424\pi$
$294$	0	0
$295$	−12.8078	−0.745697
$296$	0	0
$297$	5.75379	0.333869
$298$	0	0
$299$	14.5616	0.842116
$300$	0	0
$301$	9.12311	0.525847
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	4.87689	0.279250
$306$	0	0
$307$	5.75379	0.328386	0.164193	−	0.986428i	$-0.447498\pi$
$307$	5.75379	0.328386	0.164193	+	0.986428i	$0.447498\pi$
$308$	0	0
$309$	−40.9848	−2.33155
$310$	0	0
$311$	−5.43845	−0.308386	−0.154193	−	0.988041i	$-0.549278\pi$
$311$	−5.43845	−0.308386	−0.154193	+	0.988041i	$0.549278\pi$
$312$	0	0
$313$	−30.1771	−1.70571	−0.852855	−	0.522148i	$-0.825131\pi$
$313$	−30.1771	−1.70571	−0.852855	+	0.522148i	$0.825131\pi$
$314$	0	0
$315$	16.2462	0.915370
$316$	0	0
$317$	−5.43845	−0.305454	−0.152727	−	0.988268i	$-0.548805\pi$
$317$	−5.43845	−0.305454	−0.152727	+	0.988268i	$0.548805\pi$
$318$	0	0
$319$	13.7538	0.770064
$320$	0	0
$321$	29.9309	1.67058
$322$	0	0
$323$	−0.561553	−0.0312456
$324$	0	0
$325$	−2.56155	−0.142089
$326$	0	0
$327$	−29.3002	−1.62030
$328$	0	0
$329$	14.2462	0.785419
$330$	0	0
$331$	12.8078	0.703978	0.351989	−	0.936004i	$-0.385505\pi$
$331$	12.8078	0.703978	0.351989	+	0.936004i	$0.385505\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	10.5616	0.577039
$336$	0	0
$337$	14.8769	0.810396	0.405198	−	0.914229i	$-0.367203\pi$
$337$	14.8769	0.810396	0.405198	+	0.914229i	$0.367203\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	−20.4924	−1.10973
$342$	0	0
$343$	31.0540	1.67676
$344$	0	0
$345$	−14.5616	−0.783968
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	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
$351$	−3.68466	−0.196673
$352$	0	0
$353$	−36.4233	−1.93862	−0.969308	−	0.245849i	$-0.920933\pi$
$353$	−36.4233	−1.93862	−0.969308	+	0.245849i	$0.920933\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−6.56155	−0.347274
$358$	0	0
$359$	5.93087	0.313019	0.156510	−	0.987676i	$-0.449976\pi$
$359$	5.93087	0.313019	0.156510	+	0.987676i	$0.449976\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.8078	0.672233
$364$	0	0
$365$	14.8078	0.775074
$366$	0	0
$367$	35.1231	1.83341	0.916706	−	0.399563i	$-0.130838\pi$
$367$	35.1231	1.83341	0.916706	+	0.399563i	$0.130838\pi$
$368$	0	0
$369$	29.3693	1.52891
$370$	0	0
$371$	−29.9309	−1.55393
$372$	0	0
$373$	24.8078	1.28450	0.642249	−	0.766496i	$-0.278002\pi$
$373$	24.8078	1.28450	0.642249	+	0.766496i	$0.278002\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	−8.80776	−0.453623
$378$	0	0
$379$	−8.31534	−0.427130	−0.213565	−	0.976929i	$-0.568508\pi$
$379$	−8.31534	−0.427130	−0.213565	+	0.976929i	$0.568508\pi$
$380$	0	0
$381$	−39.3693	−2.01695
$382$	0	0
$383$	26.7386	1.36628	0.683140	−	0.730287i	$-0.260614\pi$
$383$	26.7386	1.36628	0.683140	+	0.730287i	$0.260614\pi$
$384$	0	0
$385$	18.2462	0.929913
$386$	0	0
$387$	7.12311	0.362088
$388$	0	0
$389$	−7.12311	−0.361156	−0.180578	−	0.983561i	$-0.557797\pi$
$389$	−7.12311	−0.361156	−0.180578	+	0.983561i	$0.557797\pi$
$390$	0	0
$391$	3.19224	0.161438
$392$	0	0
$393$	40.9848	2.06741
$394$	0	0
$395$	5.12311	0.257771
$396$	0	0
$397$	−25.8617	−1.29796	−0.648982	−	0.760804i	$-0.724805\pi$
$397$	−25.8617	−1.29796	−0.648982	+	0.760804i	$0.724805\pi$
$398$	0	0
$399$	11.6847	0.584965
$400$	0	0
$401$	−20.7386	−1.03564	−0.517819	−	0.855490i	$-0.673256\pi$
$401$	−20.7386	−1.03564	−0.517819	+	0.855490i	$0.673256\pi$
$402$	0	0
$403$	13.1231	0.653708
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	4.49242	0.222681
$408$	0	0
$409$	24.7386	1.22325	0.611623	−	0.791149i	$-0.290517\pi$
$409$	24.7386	1.22325	0.611623	+	0.791149i	$0.290517\pi$
$410$	0	0
$411$	−7.19224	−0.354767
$412$	0	0
$413$	−58.4233	−2.87482
$414$	0	0
$415$	14.0000	0.687233
$416$	0	0
$417$	−5.75379	−0.281764
$418$	0	0
$419$	22.8769	1.11761	0.558805	−	0.829299i	$-0.311260\pi$
$419$	22.8769	1.11761	0.558805	+	0.829299i	$0.311260\pi$
$420$	0	0
$421$	−19.3002	−0.940634	−0.470317	−	0.882498i	$-0.655860\pi$
$421$	−19.3002	−0.940634	−0.470317	+	0.882498i	$0.655860\pi$
$422$	0	0
$423$	11.1231	0.540824
$424$	0	0
$425$	−0.561553	−0.0272393
$426$	0	0
$427$	22.2462	1.07657
$428$	0	0
$429$	−26.2462	−1.26718
$430$	0	0
$431$	20.4924	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$431$	20.4924	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	8.80776	0.422300
$436$	0	0
$437$	−5.68466	−0.271934
$438$	0	0
$439$	31.3693	1.49718	0.748588	−	0.663036i	$-0.230732\pi$
$439$	31.3693	1.49718	0.748588	+	0.663036i	$0.230732\pi$
$440$	0	0
$441$	49.1771	2.34177
$442$	0	0
$443$	−11.6155	−0.551870	−0.275935	−	0.961176i	$-0.588988\pi$
$443$	−11.6155	−0.551870	−0.275935	+	0.961176i	$0.588988\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	16.7386	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$449$	16.7386	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$450$	0	0
$451$	32.9848	1.55320
$452$	0	0
$453$	33.6155	1.57940
$454$	0	0
$455$	−11.6847	−0.547785
$456$	0	0
$457$	16.5616	0.774717	0.387358	−	0.921929i	$-0.373388\pi$
$457$	16.5616	0.774717	0.387358	+	0.921929i	$0.373388\pi$
$458$	0	0
$459$	−0.807764	−0.0377032
$460$	0	0
$461$	−24.2462	−1.12926	−0.564629	−	0.825345i	$-0.690981\pi$
$461$	−24.2462	−1.12926	−0.564629	+	0.825345i	$0.690981\pi$
$462$	0	0
$463$	−21.8617	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$463$	−21.8617	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$464$	0	0
$465$	−13.1231	−0.608569
$466$	0	0
$467$	29.8617	1.38184	0.690918	−	0.722933i	$-0.257206\pi$
$467$	29.8617	1.38184	0.690918	+	0.722933i	$0.257206\pi$
$468$	0	0
$469$	48.1771	2.22461
$470$	0	0
$471$	−15.3693	−0.708181
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−23.3693	−1.07001
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−2.87689	−0.131175
$482$	0	0
$483$	−66.4233	−3.02236
$484$	0	0
$485$	−1.12311	−0.0509976
$486$	0	0
$487$	27.3693	1.24022	0.620111	−	0.784514i	$-0.287088\pi$
$487$	27.3693	1.24022	0.620111	+	0.784514i	$0.287088\pi$
$488$	0	0
$489$	−9.61553	−0.434829
$490$	0	0
$491$	2.87689	0.129832	0.0649162	−	0.997891i	$-0.479322\pi$
$491$	2.87689	0.129832	0.0649162	+	0.997891i	$0.479322\pi$
$492$	0	0
$493$	−1.93087	−0.0869620
$494$	0	0
$495$	14.2462	0.640320
$496$	0	0
$497$	−36.4924	−1.63691
$498$	0	0
$499$	−29.1231	−1.30373	−0.651865	−	0.758335i	$-0.726013\pi$
$499$	−29.1231	−1.30373	−0.651865	+	0.758335i	$0.726013\pi$
$500$	0	0
$501$	−49.6155	−2.21666
$502$	0	0
$503$	−36.4233	−1.62403	−0.812017	−	0.583634i	$-0.801630\pi$
$503$	−36.4233	−1.62403	−0.812017	+	0.583634i	$0.801630\pi$
$504$	0	0
$505$	3.12311	0.138976
$506$	0	0
$507$	−16.4924	−0.732454
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	67.5464	2.98808
$512$	0	0
$513$	1.43845	0.0635090
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	12.4924	0.549416
$518$	0	0
$519$	−7.36932	−0.323477
$520$	0	0
$521$	7.12311	0.312069	0.156034	−	0.987752i	$-0.450129\pi$
$521$	7.12311	0.312069	0.156034	+	0.987752i	$0.450129\pi$
$522$	0	0
$523$	13.9309	0.609154	0.304577	−	0.952488i	$-0.401485\pi$
$523$	13.9309	0.609154	0.304577	+	0.952488i	$0.401485\pi$
$524$	0	0
$525$	11.6847	0.509960
$526$	0	0
$527$	2.87689	0.125319
$528$	0	0
$529$	9.31534	0.405015
$530$	0	0
$531$	−45.6155	−1.97955
$532$	0	0
$533$	−21.1231	−0.914943
$534$	0	0
$535$	11.6847	0.505172
$536$	0	0
$537$	−36.4924	−1.57476
$538$	0	0
$539$	55.2311	2.37897
$540$	0	0
$541$	16.8769	0.725594	0.362797	−	0.931868i	$-0.381822\pi$
$541$	16.8769	0.725594	0.362797	+	0.931868i	$0.381822\pi$
$542$	0	0
$543$	−19.8617	−0.852349
$544$	0	0
$545$	−11.4384	−0.489969
$546$	0	0
$547$	21.7538	0.930125	0.465062	−	0.885278i	$-0.346032\pi$
$547$	21.7538	0.930125	0.465062	+	0.885278i	$0.346032\pi$
$548$	0	0
$549$	17.3693	0.741304
$550$	0	0
$551$	3.43845	0.146483
$552$	0	0
$553$	23.3693	0.993764
$554$	0	0
$555$	2.87689	0.122117
$556$	0	0
$557$	−29.3693	−1.24442	−0.622209	−	0.782851i	$-0.713765\pi$
$557$	−29.3693	−1.24442	−0.622209	+	0.782851i	$0.713765\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	0	0
$561$	−5.75379	−0.242925
$562$	0	0
$563$	−30.7386	−1.29548	−0.647739	−	0.761862i	$-0.724285\pi$
$563$	−30.7386	−1.29548	−0.647739	+	0.761862i	$0.724285\pi$
$564$	0	0
$565$	−6.24621	−0.262780
$566$	0	0
$567$	−31.9309	−1.34097
$568$	0	0
$569$	−0.876894	−0.0367613	−0.0183807	−	0.999831i	$-0.505851\pi$
$569$	−0.876894	−0.0367613	−0.0183807	+	0.999831i	$0.505851\pi$
$570$	0	0
$571$	−27.3693	−1.14537	−0.572685	−	0.819775i	$-0.694098\pi$
$571$	−27.3693	−1.14537	−0.572685	+	0.819775i	$0.694098\pi$
$572$	0	0
$573$	11.0540	0.461786
$574$	0	0
$575$	−5.68466	−0.237067
$576$	0	0
$577$	28.4233	1.18328	0.591639	−	0.806203i	$-0.298481\pi$
$577$	28.4233	1.18328	0.591639	+	0.806203i	$0.298481\pi$
$578$	0	0
$579$	23.3693	0.971196
$580$	0	0
$581$	63.8617	2.64943
$582$	0	0
$583$	−26.2462	−1.08701
$584$	0	0
$585$	−9.12311	−0.377194
$586$	0	0
$587$	15.7538	0.650228	0.325114	−	0.945675i	$-0.394597\pi$
$587$	15.7538	0.650228	0.325114	+	0.945675i	$0.394597\pi$
$588$	0	0
$589$	−5.12311	−0.211094
$590$	0	0
$591$	62.1080	2.55478
$592$	0	0
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	0	0
$595$	−2.56155	−0.105013
$596$	0	0
$597$	−13.9309	−0.570153
$598$	0	0
$599$	9.61553	0.392880	0.196440	−	0.980516i	$-0.437062\pi$
$599$	9.61553	0.392880	0.196440	+	0.980516i	$0.437062\pi$
$600$	0	0
$601$	−48.7386	−1.98809	−0.994045	−	0.108969i	$-0.965245\pi$
$601$	−48.7386	−1.98809	−0.994045	+	0.108969i	$0.965245\pi$
$602$	0	0
$603$	37.6155	1.53182
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−9.12311	−0.370295	−0.185148	−	0.982711i	$-0.559276\pi$
$607$	−9.12311	−0.370295	−0.185148	+	0.982711i	$0.559276\pi$
$608$	0	0
$609$	40.1771	1.62806
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	41.3693	1.67089	0.835445	−	0.549573i	$-0.185210\pi$
$613$	41.3693	1.67089	0.835445	+	0.549573i	$0.185210\pi$
$614$	0	0
$615$	21.1231	0.851766
$616$	0	0
$617$	22.9848	0.925335	0.462668	−	0.886532i	$-0.346892\pi$
$617$	22.9848	0.925335	0.462668	+	0.886532i	$0.346892\pi$
$618$	0	0
$619$	2.24621	0.0902829	0.0451414	−	0.998981i	$-0.485626\pi$
$619$	2.24621	0.0902829	0.0451414	+	0.998981i	$0.485626\pi$
$620$	0	0
$621$	−8.17708	−0.328135
$622$	0	0
$623$	−45.6155	−1.82755
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.2462	0.409194
$628$	0	0
$629$	−0.630683	−0.0251470
$630$	0	0
$631$	−26.7386	−1.06445	−0.532224	−	0.846604i	$-0.678644\pi$
$631$	−26.7386	−1.06445	−0.532224	+	0.846604i	$0.678644\pi$
$632$	0	0
$633$	−59.0540	−2.34718
$634$	0	0
$635$	−15.3693	−0.609913
$636$	0	0
$637$	−35.3693	−1.40138
$638$	0	0
$639$	−28.4924	−1.12714
$640$	0	0
$641$	−7.12311	−0.281346	−0.140673	−	0.990056i	$-0.544927\pi$
$641$	−7.12311	−0.281346	−0.140673	+	0.990056i	$0.544927\pi$
$642$	0	0
$643$	−14.4924	−0.571525	−0.285763	−	0.958300i	$-0.592247\pi$
$643$	−14.4924	−0.571525	−0.285763	+	0.958300i	$0.592247\pi$
$644$	0	0
$645$	5.12311	0.201722
$646$	0	0
$647$	−17.6847	−0.695256	−0.347628	−	0.937633i	$-0.613013\pi$
$647$	−17.6847	−0.695256	−0.347628	+	0.937633i	$0.613013\pi$
$648$	0	0
$649$	−51.2311	−2.01099
$650$	0	0
$651$	−59.8617	−2.34617
$652$	0	0
$653$	−45.3693	−1.77544	−0.887719	−	0.460385i	$-0.847711\pi$
$653$	−45.3693	−1.77544	−0.887719	+	0.460385i	$0.847711\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	52.7386	2.05753
$658$	0	0
$659$	−27.1922	−1.05926	−0.529630	−	0.848229i	$-0.677669\pi$
$659$	−27.1922	−1.05926	−0.529630	+	0.848229i	$0.677669\pi$
$660$	0	0
$661$	−19.9309	−0.775221	−0.387610	−	0.921823i	$-0.626699\pi$
$661$	−19.9309	−0.775221	−0.387610	+	0.921823i	$0.626699\pi$
$662$	0	0
$663$	3.68466	0.143100
$664$	0	0
$665$	4.56155	0.176889
$666$	0	0
$667$	−19.5464	−0.756840
$668$	0	0
$669$	39.3693	1.52211
$670$	0	0
$671$	19.5076	0.753082
$672$	0	0
$673$	−36.4924	−1.40668	−0.703340	−	0.710854i	$-0.748309\pi$
$673$	−36.4924	−1.40668	−0.703340	+	0.710854i	$0.748309\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	−15.1922	−0.583885	−0.291943	−	0.956436i	$-0.594302\pi$
$677$	−15.1922	−0.583885	−0.291943	+	0.956436i	$0.594302\pi$
$678$	0	0
$679$	−5.12311	−0.196607
$680$	0	0
$681$	11.0540	0.423589
$682$	0	0
$683$	−34.2462	−1.31039	−0.655197	−	0.755458i	$-0.727415\pi$
$683$	−34.2462	−1.31039	−0.655197	+	0.755458i	$0.727415\pi$
$684$	0	0
$685$	−2.80776	−0.107279
$686$	0	0
$687$	73.6155	2.80861
$688$	0	0
$689$	16.8078	0.640325
$690$	0	0
$691$	34.1080	1.29753	0.648764	−	0.760990i	$-0.275286\pi$
$691$	34.1080	1.29753	0.648764	+	0.760990i	$0.275286\pi$
$692$	0	0
$693$	64.9848	2.46857
$694$	0	0
$695$	−2.24621	−0.0852036
$696$	0	0
$697$	−4.63068	−0.175400
$698$	0	0
$699$	−67.8617	−2.56677
$700$	0	0
$701$	−11.1231	−0.420114	−0.210057	−	0.977689i	$-0.567365\pi$
$701$	−11.1231	−0.420114	−0.210057	+	0.977689i	$0.567365\pi$
$702$	0	0
$703$	1.12311	0.0423587
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	14.2462	0.535784
$708$	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
$711$	18.2462	0.684286
$712$	0	0
$713$	29.1231	1.09067
$714$	0	0
$715$	−10.2462	−0.383187
$716$	0	0
$717$	−8.17708	−0.305379
$718$	0	0
$719$	−34.4233	−1.28377	−0.641886	−	0.766800i	$-0.721848\pi$
$719$	−34.4233	−1.28377	−0.641886	+	0.766800i	$0.721848\pi$
$720$	0	0
$721$	−72.9848	−2.71810
$722$	0	0
$723$	−76.4924	−2.84478
$724$	0	0
$725$	3.43845	0.127701
$726$	0	0
$727$	−13.6847	−0.507536	−0.253768	−	0.967265i	$-0.581670\pi$
$727$	−13.6847	−0.507536	−0.253768	+	0.967265i	$0.581670\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−1.12311	−0.0415396
$732$	0	0
$733$	4.38447	0.161944	0.0809721	−	0.996716i	$-0.474198\pi$
$733$	4.38447	0.161944	0.0809721	+	0.996716i	$0.474198\pi$
$734$	0	0
$735$	35.3693	1.30462
$736$	0	0
$737$	42.2462	1.55616
$738$	0	0
$739$	−17.1231	−0.629884	−0.314942	−	0.949111i	$-0.601985\pi$
$739$	−17.1231	−0.629884	−0.314942	+	0.949111i	$0.601985\pi$
$740$	0	0
$741$	−6.56155	−0.241045
$742$	0	0
$743$	−35.3693	−1.29757	−0.648787	−	0.760970i	$-0.724723\pi$
$743$	−35.3693	−1.29757	−0.648787	+	0.760970i	$0.724723\pi$
$744$	0	0
$745$	−3.12311	−0.114422
$746$	0	0
$747$	49.8617	1.82435
$748$	0	0
$749$	53.3002	1.94755
$750$	0	0
$751$	31.3693	1.14468	0.572341	−	0.820015i	$-0.306035\pi$
$751$	31.3693	1.14468	0.572341	+	0.820015i	$0.306035\pi$
$752$	0	0
$753$	−14.7386	−0.537106
$754$	0	0
$755$	13.1231	0.477599
$756$	0	0
$757$	12.7386	0.462994	0.231497	−	0.972836i	$-0.425638\pi$
$757$	12.7386	0.462994	0.231497	+	0.972836i	$0.425638\pi$
$758$	0	0
$759$	−58.2462	−2.11420
$760$	0	0
$761$	18.3153	0.663931	0.331965	−	0.943292i	$-0.392288\pi$
$761$	18.3153	0.663931	0.331965	+	0.943292i	$0.392288\pi$
$762$	0	0
$763$	−52.1771	−1.88894
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	32.8078	1.18462
$768$	0	0
$769$	38.1771	1.37670	0.688350	−	0.725378i	$-0.258335\pi$
$769$	38.1771	1.37670	0.688350	+	0.725378i	$0.258335\pi$
$770$	0	0
$771$	−80.3542	−2.89388
$772$	0	0
$773$	−20.1771	−0.725719	−0.362860	−	0.931844i	$-0.618200\pi$
$773$	−20.1771	−0.725719	−0.362860	+	0.931844i	$0.618200\pi$
$774$	0	0
$775$	−5.12311	−0.184027
$776$	0	0
$777$	13.1231	0.470789
$778$	0	0
$779$	8.24621	0.295451
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	4.94602	0.176757
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	35.0540	1.24954	0.624770	−	0.780809i	$-0.285193\pi$
$787$	35.0540	1.24954	0.624770	+	0.780809i	$0.285193\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−28.4924	−1.01307
$792$	0	0
$793$	−12.4924	−0.443619
$794$	0	0
$795$	−16.8078	−0.596110
$796$	0	0
$797$	−24.8078	−0.878736	−0.439368	−	0.898307i	$-0.644798\pi$
$797$	−24.8078	−0.878736	−0.439368	+	0.898307i	$0.644798\pi$
$798$	0	0
$799$	−1.75379	−0.0620446
$800$	0	0
$801$	−35.6155	−1.25841
$802$	0	0
$803$	59.2311	2.09022
$804$	0	0
$805$	−25.9309	−0.913943
$806$	0	0
$807$	62.1080	2.18630
$808$	0	0
$809$	39.9309	1.40389	0.701947	−	0.712229i	$-0.252314\pi$
$809$	39.9309	1.40389	0.701947	+	0.712229i	$0.252314\pi$
$810$	0	0
$811$	48.0388	1.68687	0.843436	−	0.537230i	$-0.180529\pi$
$811$	48.0388	1.68687	0.843436	+	0.537230i	$0.180529\pi$
$812$	0	0
$813$	−51.6847	−1.81266
$814$	0	0
$815$	−3.75379	−0.131489
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	−41.6155	−1.45416
$820$	0	0
$821$	−4.87689	−0.170205	−0.0851024	−	0.996372i	$-0.527122\pi$
$821$	−4.87689	−0.170205	−0.0851024	+	0.996372i	$0.527122\pi$
$822$	0	0
$823$	20.4233	0.711911	0.355956	−	0.934503i	$-0.384155\pi$
$823$	20.4233	0.711911	0.355956	+	0.934503i	$0.384155\pi$
$824$	0	0
$825$	10.2462	0.356727
$826$	0	0
$827$	39.0540	1.35804	0.679020	−	0.734120i	$-0.262405\pi$
$827$	39.0540	1.35804	0.679020	+	0.734120i	$0.262405\pi$
$828$	0	0
$829$	−43.4384	−1.50868	−0.754340	−	0.656484i	$-0.772043\pi$
$829$	−43.4384	−1.50868	−0.754340	+	0.656484i	$0.772043\pi$
$830$	0	0
$831$	50.2462	1.74302
$832$	0	0
$833$	−7.75379	−0.268653
$834$	0	0
$835$	−19.3693	−0.670303
$836$	0	0
$837$	−7.36932	−0.254721
$838$	0	0
$839$	35.2311	1.21631	0.608156	−	0.793818i	$-0.291910\pi$
$839$	35.2311	1.21631	0.608156	+	0.793818i	$0.291910\pi$
$840$	0	0
$841$	−17.1771	−0.592313
$842$	0	0
$843$	62.1080	2.13911
$844$	0	0
$845$	−6.43845	−0.221489
$846$	0	0
$847$	22.8078	0.783684
$848$	0	0
$849$	15.3693	0.527474
$850$	0	0
$851$	−6.38447	−0.218857
$852$	0	0
$853$	44.7386	1.53182	0.765911	−	0.642947i	$-0.222288\pi$
$853$	44.7386	1.53182	0.765911	+	0.642947i	$0.222288\pi$
$854$	0	0
$855$	3.56155	0.121803
$856$	0	0
$857$	−24.9848	−0.853466	−0.426733	−	0.904378i	$-0.640336\pi$
$857$	−24.9848	−0.853466	−0.426733	+	0.904378i	$0.640336\pi$
$858$	0	0
$859$	5.26137	0.179515	0.0897577	−	0.995964i	$-0.471391\pi$
$859$	5.26137	0.179515	0.0897577	+	0.995964i	$0.471391\pi$
$860$	0	0
$861$	96.3542	3.28374
$862$	0	0
$863$	49.4773	1.68423	0.842113	−	0.539301i	$-0.181312\pi$
$863$	49.4773	1.68423	0.842113	+	0.539301i	$0.181312\pi$
$864$	0	0
$865$	−2.87689	−0.0978173
$866$	0	0
$867$	−42.7386	−1.45148
$868$	0	0
$869$	20.4924	0.695158
$870$	0	0
$871$	−27.0540	−0.916689
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	4.56155	0.154209
$876$	0	0
$877$	24.8078	0.837699	0.418849	−	0.908056i	$-0.362434\pi$
$877$	24.8078	0.837699	0.418849	+	0.908056i	$0.362434\pi$
$878$	0	0
$879$	−15.1922	−0.512421
$880$	0	0
$881$	−26.4924	−0.892552	−0.446276	−	0.894895i	$-0.647250\pi$
$881$	−26.4924	−0.892552	−0.446276	+	0.894895i	$0.647250\pi$
$882$	0	0
$883$	−19.7538	−0.664768	−0.332384	−	0.943144i	$-0.607853\pi$
$883$	−19.7538	−0.664768	−0.332384	+	0.943144i	$0.607853\pi$
$884$	0	0
$885$	−32.8078	−1.10282
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−70.1080	−2.35135
$890$	0	0
$891$	−28.0000	−0.938035
$892$	0	0
$893$	3.12311	0.104511
$894$	0	0
$895$	−14.2462	−0.476198
$896$	0	0
$897$	37.3002	1.24542
$898$	0	0
$899$	−17.6155	−0.587511
$900$	0	0
$901$	3.68466	0.122754
$902$	0	0
$903$	23.3693	0.777682
$904$	0	0
$905$	−7.75379	−0.257745
$906$	0	0
$907$	42.4233	1.40864	0.704321	−	0.709881i	$-0.251252\pi$
$907$	42.4233	1.40864	0.704321	+	0.709881i	$0.251252\pi$
$908$	0	0
$909$	11.1231	0.368930
$910$	0	0
$911$	−22.7386	−0.753365	−0.376682	−	0.926343i	$-0.622935\pi$
$911$	−22.7386	−0.753365	−0.376682	+	0.926343i	$0.622935\pi$
$912$	0	0
$913$	56.0000	1.85333
$914$	0	0
$915$	12.4924	0.412987
$916$	0	0
$917$	72.9848	2.41017
$918$	0	0
$919$	−20.8078	−0.686385	−0.343192	−	0.939265i	$-0.611508\pi$
$919$	−20.8078	−0.686385	−0.343192	+	0.939265i	$0.611508\pi$
$920$	0	0
$921$	14.7386	0.485654
$922$	0	0
$923$	20.4924	0.674516
$924$	0	0
$925$	1.12311	0.0369275
$926$	0	0
$927$	−56.9848	−1.87163
$928$	0	0
$929$	−5.05398	−0.165816	−0.0829078	−	0.996557i	$-0.526421\pi$
$929$	−5.05398	−0.165816	−0.0829078	+	0.996557i	$0.526421\pi$
$930$	0	0
$931$	13.8078	0.452531
$932$	0	0
$933$	−13.9309	−0.456076
$934$	0	0
$935$	−2.24621	−0.0734590
$936$	0	0
$937$	17.0540	0.557129	0.278565	−	0.960418i	$-0.410141\pi$
$937$	17.0540	0.557129	0.278565	+	0.960418i	$0.410141\pi$
$938$	0	0
$939$	−77.3002	−2.52260
$940$	0	0
$941$	−14.3153	−0.466667	−0.233333	−	0.972397i	$-0.574963\pi$
$941$	−14.3153	−0.466667	−0.233333	+	0.972397i	$0.574963\pi$
$942$	0	0
$943$	−46.8769	−1.52652
$944$	0	0
$945$	6.56155	0.213447
$946$	0	0
$947$	33.3693	1.08436	0.542179	−	0.840263i	$-0.317600\pi$
$947$	33.3693	1.08436	0.542179	+	0.840263i	$0.317600\pi$
$948$	0	0
$949$	−37.9309	−1.23129
$950$	0	0
$951$	−13.9309	−0.451739
$952$	0	0
$953$	51.2311	1.65954	0.829768	−	0.558108i	$-0.188473\pi$
$953$	51.2311	1.65954	0.829768	+	0.558108i	$0.188473\pi$
$954$	0	0
$955$	4.31534	0.139641
$956$	0	0
$957$	35.2311	1.13886
$958$	0	0
$959$	−12.8078	−0.413584
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	41.6155	1.34104
$964$	0	0
$965$	9.12311	0.293683
$966$	0	0
$967$	39.1231	1.25811	0.629057	−	0.777359i	$-0.283441\pi$
$967$	39.1231	1.25811	0.629057	+	0.777359i	$0.283441\pi$
$968$	0	0
$969$	−1.43845	−0.0462096
$970$	0	0
$971$	−12.9848	−0.416704	−0.208352	−	0.978054i	$-0.566810\pi$
$971$	−12.9848	−0.416704	−0.208352	+	0.978054i	$0.566810\pi$
$972$	0	0
$973$	−10.2462	−0.328478
$974$	0	0
$975$	−6.56155	−0.210138
$976$	0	0
$977$	5.12311	0.163903	0.0819513	−	0.996636i	$-0.473885\pi$
$977$	5.12311	0.163903	0.0819513	+	0.996636i	$0.473885\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	−40.7386	−1.30068
$982$	0	0
$983$	29.6155	0.944589	0.472294	−	0.881441i	$-0.343426\pi$
$983$	29.6155	0.944589	0.472294	+	0.881441i	$0.343426\pi$
$984$	0	0
$985$	24.2462	0.772549
$986$	0	0
$987$	36.4924	1.16157
$988$	0	0
$989$	−11.3693	−0.361523
$990$	0	0
$991$	21.1231	0.670998	0.335499	−	0.942041i	$-0.391095\pi$
$991$	21.1231	0.670998	0.335499	+	0.942041i	$0.391095\pi$
$992$	0	0
$993$	32.8078	1.04112
$994$	0	0
$995$	−5.43845	−0.172410
$996$	0	0
$997$	−8.24621	−0.261160	−0.130580	−	0.991438i	$-0.541684\pi$
$997$	−8.24621	−0.261160	−0.130580	+	0.991438i	$0.541684\pi$
$998$	0	0
$999$	1.61553	0.0511130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bi.1.2		2
4.3	odd	2		6080.2.a.bc.1.1		2
8.3	odd	2		3040.2.a.h.1.2	yes	2
8.5	even	2		3040.2.a.e.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.e.1.1	✓	2	8.5	even	2
3040.2.a.h.1.2	yes	2	8.3	odd	2
6080.2.a.bc.1.1		2	4.3	odd	2
6080.2.a.bi.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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +4.00000 q^{11} -2.56155 q^{13} +2.56155 q^{15} -0.561553 q^{17} +1.00000 q^{19} +11.6847 q^{21} -5.68466 q^{23} +1.00000 q^{25} +1.43845 q^{27} +3.43845 q^{29} -5.12311 q^{31} +10.2462 q^{33} +4.56155 q^{35} +1.12311 q^{37} -6.56155 q^{39} +8.24621 q^{41} +2.00000 q^{43} +3.56155 q^{45} +3.12311 q^{47} +13.8078 q^{49} -1.43845 q^{51} -6.56155 q^{53} +4.00000 q^{55} +2.56155 q^{57} -12.8078 q^{59} +4.87689 q^{61} +16.2462 q^{63} -2.56155 q^{65} +10.5616 q^{67} -14.5616 q^{69} -8.00000 q^{71} +14.8078 q^{73} +2.56155 q^{75} +18.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} +14.0000 q^{83} -0.561553 q^{85} +8.80776 q^{87} -10.0000 q^{89} -11.6847 q^{91} -13.1231 q^{93} +1.00000 q^{95} -1.12311 q^{97} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + 5 q^{7} + 3 q^{9} + 8 q^{11} - q^{13} + q^{15} + 3 q^{17} + 2 q^{19} + 11 q^{21} + q^{23} + 2 q^{25} + 7 q^{27} + 11 q^{29} - 2 q^{31} + 4 q^{33} + 5 q^{35} - 6 q^{37} - 9 q^{39}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + 5 * q^7 + 3 * q^9 + 8 * q^11 - q^13 + q^15 + 3 * q^17 + 2 * q^19 + 11 * q^21 + q^23 + 2 * q^25 + 7 * q^27 + 11 * q^29 - 2 * q^31 + 4 * q^33 + 5 * q^35 - 6 * q^37 - 9 * q^39 + 4 * q^43 + 3 * q^45 - 2 * q^47 + 7 * q^49 - 7 * q^51 - 9 * q^53 + 8 * q^55 + q^57 - 5 * q^59 + 18 * q^61 + 16 * q^63 - q^65 + 17 * q^67 - 25 * q^69 - 16 * q^71 + 9 * q^73 + q^75 + 20 * q^77 + 2 * q^79 - 14 * q^81 + 28 * q^83 + 3 * q^85 - 3 * q^87 - 20 * q^89 - 11 * q^91 - 18 * q^93 + 2 * q^95 + 6 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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