LMFDB - Embedded newform 6080.2.a.bm.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			$0.311108$ 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)

$f(q)$	$=$	$q-1.31111 q^{3} -1.00000 q^{5} +1.52543 q^{7} -1.28100 q^{9} -1.09679 q^{11} -0.836535 q^{13} +1.31111 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -1.52543 q^{23} +1.00000 q^{25} +5.61285 q^{27} -4.23506 q^{29} +5.18421 q^{31} +1.43801 q^{33} -1.52543 q^{35} -2.83654 q^{37} +1.09679 q^{39} +10.2351 q^{41} -11.4652 q^{43} +1.28100 q^{45} +1.52543 q^{47} -4.67307 q^{49} -2.62222 q^{51} +6.83654 q^{53} +1.09679 q^{55} -1.31111 q^{57} -7.80642 q^{59} +12.1891 q^{61} -1.95407 q^{63} +0.836535 q^{65} -6.92396 q^{67} +2.00000 q^{69} +5.18421 q^{71} +12.2351 q^{73} -1.31111 q^{75} -1.67307 q^{77} -8.66370 q^{79} -3.51606 q^{81} +7.19850 q^{83} -2.00000 q^{85} +5.55262 q^{87} -8.23506 q^{89} -1.27607 q^{91} -6.79706 q^{93} -1.00000 q^{95} -3.39853 q^{97} +1.40498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 10 q^{11} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 2 q^{23} + 3 q^{25} - 10 q^{27} + 14 q^{29} + 2 q^{31} + 18 q^{33} + 2 q^{35} - 2 q^{37}+ \cdots - 30 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 10 * q^11 + 4 * q^13 + 4 * q^15 + 6 * q^17 + 3 * q^19 - 6 * q^21 + 2 * q^23 + 3 * q^25 - 10 * q^27 + 14 * q^29 + 2 * q^31 + 18 * q^33 + 2 * q^35 - 2 * q^37 + 10 * q^39 + 4 * q^41 - 14 * q^43 - 3 * q^45 - 2 * q^47 - q^49 - 8 * q^51 + 14 * q^53 + 10 * q^55 - 4 * q^57 - 10 * q^59 - 10 * q^61 + 14 * q^63 - 4 * q^65 + 6 * q^67 + 6 * q^69 + 2 * q^71 + 10 * q^73 - 4 * q^75 + 8 * q^77 + 14 * q^79 + 23 * q^81 + 2 * q^83 - 6 * q^85 - 24 * q^87 + 2 * q^89 - 30 * q^91 + 20 * q^93 - 3 * q^95 + 10 * q^97 - 30 * 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
$3$	−1.31111	−0.756968	−0.378484	−	0.925608i	$-0.623555\pi$
$3$	−1.31111	−0.756968	−0.378484	+	0.925608i	$0.623555\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.52543	0.576557	0.288279	−	0.957547i	$-0.406917\pi$
$7$	1.52543	0.576557	0.288279	+	0.957547i	$0.406917\pi$
$8$	0	0
$9$	−1.28100	−0.426999
$10$	0	0
$11$	−1.09679	−0.330694	−0.165347	−	0.986235i	$-0.552874\pi$
$11$	−1.09679	−0.330694	−0.165347	+	0.986235i	$0.552874\pi$
$12$	0	0
$13$	−0.836535	−0.232013	−0.116007	−	0.993248i	$-0.537009\pi$
$13$	−0.836535	−0.232013	−0.116007	+	0.993248i	$0.537009\pi$
$14$	0	0
$15$	1.31111	0.338527
$16$	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
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−1.52543	−0.318074	−0.159037	−	0.987273i	$-0.550839\pi$
$23$	−1.52543	−0.318074	−0.159037	+	0.987273i	$0.550839\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.61285	1.08019
$28$	0	0
$29$	−4.23506	−0.786432	−0.393216	−	0.919446i	$-0.628637\pi$
$29$	−4.23506	−0.786432	−0.393216	+	0.919446i	$0.628637\pi$
$30$	0	0
$31$	5.18421	0.931111	0.465556	−	0.885019i	$-0.345855\pi$
$31$	5.18421	0.931111	0.465556	+	0.885019i	$0.345855\pi$
$32$	0	0
$33$	1.43801	0.250325
$34$	0	0
$35$	−1.52543	−0.257844
$36$	0	0
$37$	−2.83654	−0.466324	−0.233162	−	0.972438i	$-0.574907\pi$
$37$	−2.83654	−0.466324	−0.233162	+	0.972438i	$0.574907\pi$
$38$	0	0
$39$	1.09679	0.175627
$40$	0	0
$41$	10.2351	1.59845	0.799224	−	0.601033i	$-0.205244\pi$
$41$	10.2351	1.59845	0.799224	+	0.601033i	$0.205244\pi$
$42$	0	0
$43$	−11.4652	−1.74843	−0.874214	−	0.485541i	$-0.838622\pi$
$43$	−11.4652	−1.74843	−0.874214	+	0.485541i	$0.838622\pi$
$44$	0	0
$45$	1.28100	0.190960
$46$	0	0
$47$	1.52543	0.222506	0.111253	−	0.993792i	$-0.464514\pi$
$47$	1.52543	0.222506	0.111253	+	0.993792i	$0.464514\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	−2.62222	−0.367184
$52$	0	0
$53$	6.83654	0.939070	0.469535	−	0.882914i	$-0.344422\pi$
$53$	6.83654	0.939070	0.469535	+	0.882914i	$0.344422\pi$
$54$	0	0
$55$	1.09679	0.147891
$56$	0	0
$57$	−1.31111	−0.173660
$58$	0	0
$59$	−7.80642	−1.01631	−0.508155	−	0.861266i	$-0.669672\pi$
$59$	−7.80642	−1.01631	−0.508155	+	0.861266i	$0.669672\pi$
$60$	0	0
$61$	12.1891	1.56066	0.780329	−	0.625369i	$-0.215052\pi$
$61$	12.1891	1.56066	0.780329	+	0.625369i	$0.215052\pi$
$62$	0	0
$63$	−1.95407	−0.246189
$64$	0	0
$65$	0.836535	0.103759
$66$	0	0
$67$	−6.92396	−0.845896	−0.422948	−	0.906154i	$-0.639005\pi$
$67$	−6.92396	−0.845896	−0.422948	+	0.906154i	$0.639005\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	5.18421	0.615252	0.307626	−	0.951507i	$-0.400465\pi$
$71$	5.18421	0.615252	0.307626	+	0.951507i	$0.400465\pi$
$72$	0	0
$73$	12.2351	1.43201	0.716003	−	0.698097i	$-0.245970\pi$
$73$	12.2351	1.43201	0.716003	+	0.698097i	$0.245970\pi$
$74$	0	0
$75$	−1.31111	−0.151394
$76$	0	0
$77$	−1.67307	−0.190664
$78$	0	0
$79$	−8.66370	−0.974743	−0.487371	−	0.873195i	$-0.662044\pi$
$79$	−8.66370	−0.974743	−0.487371	+	0.873195i	$0.662044\pi$
$80$	0	0
$81$	−3.51606	−0.390673
$82$	0	0
$83$	7.19850	0.790138	0.395069	−	0.918651i	$-0.370721\pi$
$83$	7.19850	0.790138	0.395069	+	0.918651i	$0.370721\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	5.55262	0.595304
$88$	0	0
$89$	−8.23506	−0.872915	−0.436457	−	0.899725i	$-0.643767\pi$
$89$	−8.23506	−0.872915	−0.436457	+	0.899725i	$0.643767\pi$
$90$	0	0
$91$	−1.27607	−0.133769
$92$	0	0
$93$	−6.79706	−0.704822
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−3.39853	−0.345068	−0.172534	−	0.985004i	$-0.555196\pi$
$97$	−3.39853	−0.345068	−0.172534	+	0.985004i	$0.555196\pi$
$98$	0	0
$99$	1.40498	0.141206
$100$	0	0
$101$	1.39207	0.138517	0.0692583	−	0.997599i	$-0.477937\pi$
$101$	1.39207	0.138517	0.0692583	+	0.997599i	$0.477937\pi$
$102$	0	0
$103$	13.0257	1.28346	0.641728	−	0.766932i	$-0.278218\pi$
$103$	13.0257	1.28346	0.641728	+	0.766932i	$0.278218\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−14.7906	−1.42986	−0.714931	−	0.699195i	$-0.753542\pi$
$107$	−14.7906	−1.42986	−0.714931	+	0.699195i	$0.753542\pi$
$108$	0	0
$109$	−4.56199	−0.436960	−0.218480	−	0.975841i	$-0.570110\pi$
$109$	−4.56199	−0.436960	−0.218480	+	0.975841i	$0.570110\pi$
$110$	0	0
$111$	3.71900	0.352992
$112$	0	0
$113$	3.07160	0.288952	0.144476	−	0.989508i	$-0.453850\pi$
$113$	3.07160	0.288952	0.144476	+	0.989508i	$0.453850\pi$
$114$	0	0
$115$	1.52543	0.142247
$116$	0	0
$117$	1.07160	0.0990693
$118$	0	0
$119$	3.05086	0.279671
$120$	0	0
$121$	−9.79706	−0.890641
$122$	0	0
$123$	−13.4193	−1.20997
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.1082	−1.07443	−0.537213	−	0.843447i	$-0.680523\pi$
$127$	−12.1082	−1.07443	−0.537213	+	0.843447i	$0.680523\pi$
$128$	0	0
$129$	15.0321	1.32350
$130$	0	0
$131$	−13.9684	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$131$	−13.9684	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$132$	0	0
$133$	1.52543	0.132271
$134$	0	0
$135$	−5.61285	−0.483077
$136$	0	0
$137$	1.43801	0.122857	0.0614286	−	0.998111i	$-0.480434\pi$
$137$	1.43801	0.122857	0.0614286	+	0.998111i	$0.480434\pi$
$138$	0	0
$139$	−3.29036	−0.279085	−0.139543	−	0.990216i	$-0.544563\pi$
$139$	−3.29036	−0.279085	−0.139543	+	0.990216i	$0.544563\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0.917502	0.0767254
$144$	0	0
$145$	4.23506	0.351703
$146$	0	0
$147$	6.12690	0.505338
$148$	0	0
$149$	20.1891	1.65396	0.826979	−	0.562233i	$-0.190058\pi$
$149$	20.1891	1.65396	0.826979	+	0.562233i	$0.190058\pi$
$150$	0	0
$151$	−0.428639	−0.0348822	−0.0174411	−	0.999848i	$-0.505552\pi$
$151$	−0.428639	−0.0348822	−0.0174411	+	0.999848i	$0.505552\pi$
$152$	0	0
$153$	−2.56199	−0.207125
$154$	0	0
$155$	−5.18421	−0.416406
$156$	0	0
$157$	−14.5620	−1.16217	−0.581087	−	0.813842i	$-0.697372\pi$
$157$	−14.5620	−1.16217	−0.581087	+	0.813842i	$0.697372\pi$
$158$	0	0
$159$	−8.96343	−0.710847
$160$	0	0
$161$	−2.32693	−0.183388
$162$	0	0
$163$	1.95407	0.153054	0.0765272	−	0.997067i	$-0.475617\pi$
$163$	1.95407	0.153054	0.0765272	+	0.997067i	$0.475617\pi$
$164$	0	0
$165$	−1.43801	−0.111949
$166$	0	0
$167$	−15.5877	−1.20621	−0.603105	−	0.797662i	$-0.706070\pi$
$167$	−15.5877	−1.20621	−0.603105	+	0.797662i	$0.706070\pi$
$168$	0	0
$169$	−12.3002	−0.946170
$170$	0	0
$171$	−1.28100	−0.0979602
$172$	0	0
$173$	17.3985	1.32279	0.661393	−	0.750040i	$-0.269966\pi$
$173$	17.3985	1.32279	0.661393	+	0.750040i	$0.269966\pi$
$174$	0	0
$175$	1.52543	0.115311
$176$	0	0
$177$	10.2351	0.769314
$178$	0	0
$179$	−7.80642	−0.583480	−0.291740	−	0.956498i	$-0.594234\pi$
$179$	−7.80642	−0.583480	−0.291740	+	0.956498i	$0.594234\pi$
$180$	0	0
$181$	17.3461	1.28933	0.644664	−	0.764466i	$-0.276997\pi$
$181$	17.3461	1.28933	0.644664	+	0.764466i	$0.276997\pi$
$182$	0	0
$183$	−15.9813	−1.18137
$184$	0	0
$185$	2.83654	0.208546
$186$	0	0
$187$	−2.19358	−0.160410
$188$	0	0
$189$	8.56199	0.622793
$190$	0	0
$191$	18.6637	1.35046	0.675229	−	0.737608i	$-0.264045\pi$
$191$	18.6637	1.35046	0.675229	+	0.737608i	$0.264045\pi$
$192$	0	0
$193$	−18.7447	−1.34927	−0.674635	−	0.738151i	$-0.735699\pi$
$193$	−18.7447	−1.34927	−0.674635	+	0.738151i	$0.735699\pi$
$194$	0	0
$195$	−1.09679	−0.0785426
$196$	0	0
$197$	−4.23506	−0.301736	−0.150868	−	0.988554i	$-0.548207\pi$
$197$	−4.23506	−0.301736	−0.150868	+	0.988554i	$0.548207\pi$
$198$	0	0
$199$	0.857279	0.0607709	0.0303854	−	0.999538i	$-0.490327\pi$
$199$	0.857279	0.0607709	0.0303854	+	0.999538i	$0.490327\pi$
$200$	0	0
$201$	9.07805	0.640316
$202$	0	0
$203$	−6.46028	−0.453423
$204$	0	0
$205$	−10.2351	−0.714848
$206$	0	0
$207$	1.95407	0.135817
$208$	0	0
$209$	−1.09679	−0.0758664
$210$	0	0
$211$	−19.5714	−1.34735	−0.673674	−	0.739029i	$-0.735285\pi$
$211$	−19.5714	−1.34735	−0.673674	+	0.739029i	$0.735285\pi$
$212$	0	0
$213$	−6.79706	−0.465727
$214$	0	0
$215$	11.4652	0.781920
$216$	0	0
$217$	7.90813	0.536839
$218$	0	0
$219$	−16.0415	−1.08398
$220$	0	0
$221$	−1.67307	−0.112543
$222$	0	0
$223$	−10.8222	−0.724711	−0.362356	−	0.932040i	$-0.618027\pi$
$223$	−10.8222	−0.724711	−0.362356	+	0.932040i	$0.618027\pi$
$224$	0	0
$225$	−1.28100	−0.0853998
$226$	0	0
$227$	16.0765	1.06704	0.533518	−	0.845789i	$-0.320870\pi$
$227$	16.0765	1.06704	0.533518	+	0.845789i	$0.320870\pi$
$228$	0	0
$229$	6.60793	0.436664	0.218332	−	0.975875i	$-0.429938\pi$
$229$	6.60793	0.436664	0.218332	+	0.975875i	$0.429938\pi$
$230$	0	0
$231$	2.19358	0.144327
$232$	0	0
$233$	8.23506	0.539497	0.269748	−	0.962931i	$-0.413059\pi$
$233$	8.23506	0.539497	0.269748	+	0.962931i	$0.413059\pi$
$234$	0	0
$235$	−1.52543	−0.0995079
$236$	0	0
$237$	11.3590	0.737849
$238$	0	0
$239$	−16.0415	−1.03764	−0.518819	−	0.854884i	$-0.673628\pi$
$239$	−16.0415	−1.03764	−0.518819	+	0.854884i	$0.673628\pi$
$240$	0	0
$241$	6.79706	0.437837	0.218918	−	0.975743i	$-0.429747\pi$
$241$	6.79706	0.437837	0.218918	+	0.975743i	$0.429747\pi$
$242$	0	0
$243$	−12.2286	−0.784466
$244$	0	0
$245$	4.67307	0.298552
$246$	0	0
$247$	−0.836535	−0.0532275
$248$	0	0
$249$	−9.43801	−0.598109
$250$	0	0
$251$	−28.1748	−1.77838	−0.889190	−	0.457538i	$-0.848731\pi$
$251$	−28.1748	−1.77838	−0.889190	+	0.457538i	$0.848731\pi$
$252$	0	0
$253$	1.67307	0.105185
$254$	0	0
$255$	2.62222	0.164210
$256$	0	0
$257$	3.96052	0.247050	0.123525	−	0.992341i	$-0.460580\pi$
$257$	3.96052	0.247050	0.123525	+	0.992341i	$0.460580\pi$
$258$	0	0
$259$	−4.32693	−0.268862
$260$	0	0
$261$	5.42510	0.335805
$262$	0	0
$263$	−4.57628	−0.282186	−0.141093	−	0.989996i	$-0.545062\pi$
$263$	−4.57628	−0.282186	−0.141093	+	0.989996i	$0.545062\pi$
$264$	0	0
$265$	−6.83654	−0.419965
$266$	0	0
$267$	10.7971	0.660769
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−19.7605	−1.20036	−0.600182	−	0.799863i	$-0.704905\pi$
$271$	−19.7605	−1.20036	−0.600182	+	0.799863i	$0.704905\pi$
$272$	0	0
$273$	1.67307	0.101259
$274$	0	0
$275$	−1.09679	−0.0661388
$276$	0	0
$277$	−8.78415	−0.527788	−0.263894	−	0.964552i	$-0.585007\pi$
$277$	−8.78415	−0.527788	−0.263894	+	0.964552i	$0.585007\pi$
$278$	0	0
$279$	−6.64095	−0.397583
$280$	0	0
$281$	−6.88892	−0.410959	−0.205479	−	0.978661i	$-0.565875\pi$
$281$	−6.88892	−0.410959	−0.205479	+	0.978661i	$0.565875\pi$
$282$	0	0
$283$	−6.34122	−0.376946	−0.188473	−	0.982078i	$-0.560354\pi$
$283$	−6.34122	−0.376946	−0.188473	+	0.982078i	$0.560354\pi$
$284$	0	0
$285$	1.31111	0.0776633
$286$	0	0
$287$	15.6128	0.921597
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	4.45584	0.261206
$292$	0	0
$293$	10.2745	0.600245	0.300123	−	0.953901i	$-0.402972\pi$
$293$	10.2745	0.600245	0.300123	+	0.953901i	$0.402972\pi$
$294$	0	0
$295$	7.80642	0.454508
$296$	0	0
$297$	−6.15610	−0.357213
$298$	0	0
$299$	1.27607	0.0737973
$300$	0	0
$301$	−17.4893	−1.00807
$302$	0	0
$303$	−1.82516	−0.104853
$304$	0	0
$305$	−12.1891	−0.697948
$306$	0	0
$307$	−5.15902	−0.294441	−0.147220	−	0.989104i	$-0.547033\pi$
$307$	−5.15902	−0.294441	−0.147220	+	0.989104i	$0.547033\pi$
$308$	0	0
$309$	−17.0781	−0.971536
$310$	0	0
$311$	10.6079	0.601520	0.300760	−	0.953700i	$-0.402760\pi$
$311$	10.6079	0.601520	0.300760	+	0.953700i	$0.402760\pi$
$312$	0	0
$313$	−4.23506	−0.239380	−0.119690	−	0.992811i	$-0.538190\pi$
$313$	−4.23506	−0.239380	−0.119690	+	0.992811i	$0.538190\pi$
$314$	0	0
$315$	1.95407	0.110099
$316$	0	0
$317$	−0.274543	−0.0154199	−0.00770993	−	0.999970i	$-0.502454\pi$
$317$	−0.274543	−0.0154199	−0.00770993	+	0.999970i	$0.502454\pi$
$318$	0	0
$319$	4.64497	0.260068
$320$	0	0
$321$	19.3921	1.08236
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−0.836535	−0.0464026
$326$	0	0
$327$	5.98126	0.330765
$328$	0	0
$329$	2.32693	0.128288
$330$	0	0
$331$	−22.6222	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$331$	−22.6222	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$332$	0	0
$333$	3.63359	0.199120
$334$	0	0
$335$	6.92396	0.378296
$336$	0	0
$337$	−16.1827	−0.881527	−0.440763	−	0.897623i	$-0.645292\pi$
$337$	−16.1827	−0.881527	−0.440763	+	0.897623i	$0.645292\pi$
$338$	0	0
$339$	−4.02720	−0.218727
$340$	0	0
$341$	−5.68598	−0.307913
$342$	0	0
$343$	−17.8064	−0.961457
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	17.1383	0.920031	0.460015	−	0.887911i	$-0.347844\pi$
$347$	17.1383	0.920031	0.460015	+	0.887911i	$0.347844\pi$
$348$	0	0
$349$	−0.326929	−0.0175001	−0.00875006	−	0.999962i	$-0.502785\pi$
$349$	−0.326929	−0.0175001	−0.00875006	+	0.999962i	$0.502785\pi$
$350$	0	0
$351$	−4.69535	−0.250619
$352$	0	0
$353$	−1.34614	−0.0716479	−0.0358239	−	0.999358i	$-0.511406\pi$
$353$	−1.34614	−0.0716479	−0.0358239	+	0.999358i	$0.511406\pi$
$354$	0	0
$355$	−5.18421	−0.275149
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−0.119063	−0.00628390	−0.00314195	−	0.999995i	$-0.501000\pi$
$359$	−0.119063	−0.00628390	−0.00314195	+	0.999995i	$0.501000\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.8450	0.674187
$364$	0	0
$365$	−12.2351	−0.640412
$366$	0	0
$367$	−6.34122	−0.331009	−0.165504	−	0.986209i	$-0.552925\pi$
$367$	−6.34122	−0.331009	−0.165504	+	0.986209i	$0.552925\pi$
$368$	0	0
$369$	−13.1111	−0.682535
$370$	0	0
$371$	10.4286	0.541428
$372$	0	0
$373$	31.2148	1.61624	0.808120	−	0.589017i	$-0.200485\pi$
$373$	31.2148	1.61624	0.808120	+	0.589017i	$0.200485\pi$
$374$	0	0
$375$	1.31111	0.0677053
$376$	0	0
$377$	3.54278	0.182462
$378$	0	0
$379$	−25.9813	−1.33457	−0.667284	−	0.744803i	$-0.732543\pi$
$379$	−25.9813	−1.33457	−0.667284	+	0.744803i	$0.732543\pi$
$380$	0	0
$381$	15.8751	0.813306
$382$	0	0
$383$	2.58718	0.132199	0.0660994	−	0.997813i	$-0.478945\pi$
$383$	2.58718	0.132199	0.0660994	+	0.997813i	$0.478945\pi$
$384$	0	0
$385$	1.67307	0.0852676
$386$	0	0
$387$	14.6869	0.746576
$388$	0	0
$389$	−23.0192	−1.16712	−0.583560	−	0.812070i	$-0.698341\pi$
$389$	−23.0192	−1.16712	−0.583560	+	0.812070i	$0.698341\pi$
$390$	0	0
$391$	−3.05086	−0.154288
$392$	0	0
$393$	18.3140	0.923820
$394$	0	0
$395$	8.66370	0.435918
$396$	0	0
$397$	−11.6731	−0.585855	−0.292927	−	0.956135i	$-0.594629\pi$
$397$	−11.6731	−0.585855	−0.292927	+	0.956135i	$0.594629\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−21.9081	−1.09404	−0.547020	−	0.837120i	$-0.684238\pi$
$401$	−21.9081	−1.09404	−0.547020	+	0.837120i	$0.684238\pi$
$402$	0	0
$403$	−4.33677	−0.216030
$404$	0	0
$405$	3.51606	0.174714
$406$	0	0
$407$	3.11108	0.154210
$408$	0	0
$409$	−2.87601	−0.142210	−0.0711049	−	0.997469i	$-0.522653\pi$
$409$	−2.87601	−0.142210	−0.0711049	+	0.997469i	$0.522653\pi$
$410$	0	0
$411$	−1.88538	−0.0929991
$412$	0	0
$413$	−11.9081	−0.585961
$414$	0	0
$415$	−7.19850	−0.353360
$416$	0	0
$417$	4.31402	0.211259
$418$	0	0
$419$	−9.08250	−0.443709	−0.221855	−	0.975080i	$-0.571211\pi$
$419$	−9.08250	−0.443709	−0.221855	+	0.975080i	$0.571211\pi$
$420$	0	0
$421$	11.4509	0.558083	0.279042	−	0.960279i	$-0.409983\pi$
$421$	11.4509	0.558083	0.279042	+	0.960279i	$0.409983\pi$
$422$	0	0
$423$	−1.95407	−0.0950100
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	18.5936	0.899809
$428$	0	0
$429$	−1.20294	−0.0580787
$430$	0	0
$431$	−2.62222	−0.126308	−0.0631538	−	0.998004i	$-0.520116\pi$
$431$	−2.62222	−0.126308	−0.0631538	+	0.998004i	$0.520116\pi$
$432$	0	0
$433$	−15.6207	−0.750682	−0.375341	−	0.926887i	$-0.622474\pi$
$433$	−15.6207	−0.750682	−0.375341	+	0.926887i	$0.622474\pi$
$434$	0	0
$435$	−5.55262	−0.266228
$436$	0	0
$437$	−1.52543	−0.0729711
$438$	0	0
$439$	−32.4514	−1.54882	−0.774410	−	0.632684i	$-0.781953\pi$
$439$	−32.4514	−1.54882	−0.774410	+	0.632684i	$0.781953\pi$
$440$	0	0
$441$	5.98619	0.285056
$442$	0	0
$443$	−26.2908	−1.24912	−0.624558	−	0.780979i	$-0.714721\pi$
$443$	−26.2908	−1.24912	−0.624558	+	0.780979i	$0.714721\pi$
$444$	0	0
$445$	8.23506	0.390379
$446$	0	0
$447$	−26.4701	−1.25199
$448$	0	0
$449$	−16.5620	−0.781609	−0.390804	−	0.920474i	$-0.627803\pi$
$449$	−16.5620	−0.781609	−0.390804	+	0.920474i	$0.627803\pi$
$450$	0	0
$451$	−11.2257	−0.528597
$452$	0	0
$453$	0.561993	0.0264047
$454$	0	0
$455$	1.27607	0.0598233
$456$	0	0
$457$	25.8163	1.20763	0.603817	−	0.797123i	$-0.293646\pi$
$457$	25.8163	1.20763	0.603817	+	0.797123i	$0.293646\pi$
$458$	0	0
$459$	11.2257	0.523971
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	−31.8938	−1.48223	−0.741116	−	0.671377i	$-0.765703\pi$
$463$	−31.8938	−1.48223	−0.741116	+	0.671377i	$0.765703\pi$
$464$	0	0
$465$	6.79706	0.315206
$466$	0	0
$467$	32.7511	1.51554	0.757771	−	0.652521i	$-0.226289\pi$
$467$	32.7511	1.51554	0.757771	+	0.652521i	$0.226289\pi$
$468$	0	0
$469$	−10.5620	−0.487708
$470$	0	0
$471$	19.0923	0.879729
$472$	0	0
$473$	12.5749	0.578194
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−8.75758	−0.400982
$478$	0	0
$479$	−34.1575	−1.56070	−0.780348	−	0.625346i	$-0.784958\pi$
$479$	−34.1575	−1.56070	−0.780348	+	0.625346i	$0.784958\pi$
$480$	0	0
$481$	2.37286	0.108193
$482$	0	0
$483$	3.05086	0.138819
$484$	0	0
$485$	3.39853	0.154319
$486$	0	0
$487$	−16.1269	−0.730780	−0.365390	−	0.930855i	$-0.619064\pi$
$487$	−16.1269	−0.730780	−0.365390	+	0.930855i	$0.619064\pi$
$488$	0	0
$489$	−2.56199	−0.115857
$490$	0	0
$491$	−29.1526	−1.31564	−0.657818	−	0.753177i	$-0.728521\pi$
$491$	−29.1526	−1.31564	−0.657818	+	0.753177i	$0.728521\pi$
$492$	0	0
$493$	−8.47013	−0.381475
$494$	0	0
$495$	−1.40498	−0.0631492
$496$	0	0
$497$	7.90813	0.354728
$498$	0	0
$499$	30.9862	1.38713	0.693566	−	0.720393i	$-0.256039\pi$
$499$	30.9862	1.38713	0.693566	+	0.720393i	$0.256039\pi$
$500$	0	0
$501$	20.4371	0.913062
$502$	0	0
$503$	24.5763	1.09580	0.547901	−	0.836543i	$-0.315427\pi$
$503$	24.5763	1.09580	0.547901	+	0.836543i	$0.315427\pi$
$504$	0	0
$505$	−1.39207	−0.0619465
$506$	0	0
$507$	16.1269	0.716221
$508$	0	0
$509$	−3.20294	−0.141968	−0.0709840	−	0.997477i	$-0.522614\pi$
$509$	−3.20294	−0.141968	−0.0709840	+	0.997477i	$0.522614\pi$
$510$	0	0
$511$	18.6637	0.825634
$512$	0	0
$513$	5.61285	0.247813
$514$	0	0
$515$	−13.0257	−0.573979
$516$	0	0
$517$	−1.67307	−0.0735816
$518$	0	0
$519$	−22.8113	−1.00131
$520$	0	0
$521$	−7.68598	−0.336729	−0.168364	−	0.985725i	$-0.553849\pi$
$521$	−7.68598	−0.336729	−0.168364	+	0.985725i	$0.553849\pi$
$522$	0	0
$523$	15.9462	0.697280	0.348640	−	0.937257i	$-0.386644\pi$
$523$	15.9462	0.697280	0.348640	+	0.937257i	$0.386644\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	10.3684	0.451655
$528$	0	0
$529$	−20.6731	−0.898829
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	−8.56199	−0.370861
$534$	0	0
$535$	14.7906	0.639454
$536$	0	0
$537$	10.2351	0.441676
$538$	0	0
$539$	5.12537	0.220765
$540$	0	0
$541$	−29.3921	−1.26366	−0.631832	−	0.775105i	$-0.717697\pi$
$541$	−29.3921	−1.26366	−0.631832	+	0.775105i	$0.717697\pi$
$542$	0	0
$543$	−22.7427	−0.975981
$544$	0	0
$545$	4.56199	0.195414
$546$	0	0
$547$	36.1367	1.54510	0.772548	−	0.634957i	$-0.218982\pi$
$547$	36.1367	1.54510	0.772548	+	0.634957i	$0.218982\pi$
$548$	0	0
$549$	−15.6142	−0.666399
$550$	0	0
$551$	−4.23506	−0.180420
$552$	0	0
$553$	−13.2159	−0.561995
$554$	0	0
$555$	−3.71900	−0.157863
$556$	0	0
$557$	−10.0919	−0.427606	−0.213803	−	0.976877i	$-0.568585\pi$
$557$	−10.0919	−0.427606	−0.213803	+	0.976877i	$0.568585\pi$
$558$	0	0
$559$	9.59105	0.405658
$560$	0	0
$561$	2.87601	0.121425
$562$	0	0
$563$	−15.2795	−0.643953	−0.321976	−	0.946748i	$-0.604347\pi$
$563$	−15.2795	−0.643953	−0.321976	+	0.946748i	$0.604347\pi$
$564$	0	0
$565$	−3.07160	−0.129223
$566$	0	0
$567$	−5.36349	−0.225246
$568$	0	0
$569$	−12.4701	−0.522775	−0.261387	−	0.965234i	$-0.584180\pi$
$569$	−12.4701	−0.522775	−0.261387	+	0.965234i	$0.584180\pi$
$570$	0	0
$571$	−10.6079	−0.443928	−0.221964	−	0.975055i	$-0.571247\pi$
$571$	−10.6079	−0.443928	−0.221964	+	0.975055i	$0.571247\pi$
$572$	0	0
$573$	−24.4701	−1.02225
$574$	0	0
$575$	−1.52543	−0.0636147
$576$	0	0
$577$	−28.7052	−1.19501	−0.597506	−	0.801864i	$-0.703842\pi$
$577$	−28.7052	−1.19501	−0.597506	+	0.801864i	$0.703842\pi$
$578$	0	0
$579$	24.5763	1.02136
$580$	0	0
$581$	10.9808	0.455560
$582$	0	0
$583$	−7.49823	−0.310545
$584$	0	0
$585$	−1.07160	−0.0443052
$586$	0	0
$587$	−8.05578	−0.332498	−0.166249	−	0.986084i	$-0.553165\pi$
$587$	−8.05578	−0.332498	−0.166249	+	0.986084i	$0.553165\pi$
$588$	0	0
$589$	5.18421	0.213612
$590$	0	0
$591$	5.55262	0.228404
$592$	0	0
$593$	−43.5941	−1.79020	−0.895098	−	0.445870i	$-0.852894\pi$
$593$	−43.5941	−1.79020	−0.895098	+	0.445870i	$0.852894\pi$
$594$	0	0
$595$	−3.05086	−0.125073
$596$	0	0
$597$	−1.12399	−0.0460017
$598$	0	0
$599$	−4.75557	−0.194307	−0.0971536	−	0.995269i	$-0.530974\pi$
$599$	−4.75557	−0.194307	−0.0971536	+	0.995269i	$0.530974\pi$
$600$	0	0
$601$	19.2672	0.785925	0.392962	−	0.919555i	$-0.371450\pi$
$601$	19.2672	0.785925	0.392962	+	0.919555i	$0.371450\pi$
$602$	0	0
$603$	8.86956	0.361196
$604$	0	0
$605$	9.79706	0.398307
$606$	0	0
$607$	−39.7941	−1.61519	−0.807597	−	0.589735i	$-0.799232\pi$
$607$	−39.7941	−1.61519	−0.807597	+	0.589735i	$0.799232\pi$
$608$	0	0
$609$	8.47013	0.343227
$610$	0	0
$611$	−1.27607	−0.0516244
$612$	0	0
$613$	−20.8889	−0.843696	−0.421848	−	0.906667i	$-0.638618\pi$
$613$	−20.8889	−0.843696	−0.421848	+	0.906667i	$0.638618\pi$
$614$	0	0
$615$	13.4193	0.541117
$616$	0	0
$617$	3.11108	0.125247	0.0626236	−	0.998037i	$-0.480053\pi$
$617$	3.11108	0.125247	0.0626236	+	0.998037i	$0.480053\pi$
$618$	0	0
$619$	−8.41435	−0.338201	−0.169101	−	0.985599i	$-0.554086\pi$
$619$	−8.41435	−0.338201	−0.169101	+	0.985599i	$0.554086\pi$
$620$	0	0
$621$	−8.56199	−0.343581
$622$	0	0
$623$	−12.5620	−0.503286
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.43801	0.0574285
$628$	0	0
$629$	−5.67307	−0.226200
$630$	0	0
$631$	−29.2716	−1.16529	−0.582643	−	0.812729i	$-0.697981\pi$
$631$	−29.2716	−1.16529	−0.582643	+	0.812729i	$0.697981\pi$
$632$	0	0
$633$	25.6602	1.01990
$634$	0	0
$635$	12.1082	0.480498
$636$	0	0
$637$	3.90919	0.154888
$638$	0	0
$639$	−6.64095	−0.262712
$640$	0	0
$641$	22.1432	0.874604	0.437302	−	0.899315i	$-0.355934\pi$
$641$	22.1432	0.874604	0.437302	+	0.899315i	$0.355934\pi$
$642$	0	0
$643$	31.4148	1.23888	0.619440	−	0.785044i	$-0.287360\pi$
$643$	31.4148	1.23888	0.619440	+	0.785044i	$0.287360\pi$
$644$	0	0
$645$	−15.0321	−0.591889
$646$	0	0
$647$	20.7382	0.815303	0.407652	−	0.913137i	$-0.366348\pi$
$647$	20.7382	0.815303	0.407652	+	0.913137i	$0.366348\pi$
$648$	0	0
$649$	8.56199	0.336088
$650$	0	0
$651$	−10.3684	−0.406370
$652$	0	0
$653$	27.7649	1.08653	0.543263	−	0.839563i	$-0.317189\pi$
$653$	27.7649	1.08653	0.543263	+	0.839563i	$0.317189\pi$
$654$	0	0
$655$	13.9684	0.545789
$656$	0	0
$657$	−15.6731	−0.611465
$658$	0	0
$659$	34.7654	1.35427	0.677134	−	0.735860i	$-0.263222\pi$
$659$	34.7654	1.35427	0.677134	+	0.735860i	$0.263222\pi$
$660$	0	0
$661$	−2.79706	−0.108793	−0.0543964	−	0.998519i	$-0.517323\pi$
$661$	−2.79706	−0.108793	−0.0543964	+	0.998519i	$0.517323\pi$
$662$	0	0
$663$	2.19358	0.0851914
$664$	0	0
$665$	−1.52543	−0.0591535
$666$	0	0
$667$	6.46028	0.250143
$668$	0	0
$669$	14.1891	0.548583
$670$	0	0
$671$	−13.3689	−0.516100
$672$	0	0
$673$	21.3067	0.821311	0.410656	−	0.911790i	$-0.365300\pi$
$673$	21.3067	0.821311	0.410656	+	0.911790i	$0.365300\pi$
$674$	0	0
$675$	5.61285	0.216039
$676$	0	0
$677$	10.9284	0.420013	0.210006	−	0.977700i	$-0.432652\pi$
$677$	10.9284	0.420013	0.210006	+	0.977700i	$0.432652\pi$
$678$	0	0
$679$	−5.18421	−0.198952
$680$	0	0
$681$	−21.0781	−0.807713
$682$	0	0
$683$	34.3718	1.31520	0.657600	−	0.753367i	$-0.271572\pi$
$683$	34.3718	1.31520	0.657600	+	0.753367i	$0.271572\pi$
$684$	0	0
$685$	−1.43801	−0.0549434
$686$	0	0
$687$	−8.66370	−0.330541
$688$	0	0
$689$	−5.71900	−0.217877
$690$	0	0
$691$	29.3921	1.11813	0.559064	−	0.829125i	$-0.311161\pi$
$691$	29.3921	1.11813	0.559064	+	0.829125i	$0.311161\pi$
$692$	0	0
$693$	2.14320	0.0814133
$694$	0	0
$695$	3.29036	0.124811
$696$	0	0
$697$	20.4701	0.775361
$698$	0	0
$699$	−10.7971	−0.408382
$700$	0	0
$701$	−0.372862	−0.0140828	−0.00704141	−	0.999975i	$-0.502241\pi$
$701$	−0.372862	−0.0140828	−0.00704141	+	0.999975i	$0.502241\pi$
$702$	0	0
$703$	−2.83654	−0.106982
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	2.12351	0.0798628
$708$	0	0
$709$	−32.6133	−1.22482	−0.612410	−	0.790541i	$-0.709800\pi$
$709$	−32.6133	−1.22482	−0.612410	+	0.790541i	$0.709800\pi$
$710$	0	0
$711$	11.0982	0.416214
$712$	0	0
$713$	−7.90813	−0.296162
$714$	0	0
$715$	−0.917502	−0.0343126
$716$	0	0
$717$	21.0321	0.785459
$718$	0	0
$719$	−28.4143	−1.05968	−0.529838	−	0.848099i	$-0.677747\pi$
$719$	−28.4143	−1.05968	−0.529838	+	0.848099i	$0.677747\pi$
$720$	0	0
$721$	19.8697	0.739987
$722$	0	0
$723$	−8.91167	−0.331429
$724$	0	0
$725$	−4.23506	−0.157286
$726$	0	0
$727$	−2.07451	−0.0769394	−0.0384697	−	0.999260i	$-0.512248\pi$
$727$	−2.07451	−0.0769394	−0.0384697	+	0.999260i	$0.512248\pi$
$728$	0	0
$729$	26.5812	0.984489
$730$	0	0
$731$	−22.9304	−0.848112
$732$	0	0
$733$	11.0192	0.407004	0.203502	−	0.979075i	$-0.434768\pi$
$733$	11.0192	0.407004	0.203502	+	0.979075i	$0.434768\pi$
$734$	0	0
$735$	−6.12690	−0.225994
$736$	0	0
$737$	7.59411	0.279733
$738$	0	0
$739$	−33.9180	−1.24769	−0.623847	−	0.781547i	$-0.714431\pi$
$739$	−33.9180	−1.24769	−0.623847	+	0.781547i	$0.714431\pi$
$740$	0	0
$741$	1.09679	0.0402915
$742$	0	0
$743$	39.1175	1.43508	0.717542	−	0.696516i	$-0.245267\pi$
$743$	39.1175	1.43508	0.717542	+	0.696516i	$0.245267\pi$
$744$	0	0
$745$	−20.1891	−0.739673
$746$	0	0
$747$	−9.22125	−0.337388
$748$	0	0
$749$	−22.5620	−0.824397
$750$	0	0
$751$	25.0736	0.914949	0.457474	−	0.889223i	$-0.348754\pi$
$751$	25.0736	0.914949	0.457474	+	0.889223i	$0.348754\pi$
$752$	0	0
$753$	36.9403	1.34618
$754$	0	0
$755$	0.428639	0.0155998
$756$	0	0
$757$	11.7649	0.427604	0.213802	−	0.976877i	$-0.431415\pi$
$757$	11.7649	0.427604	0.213802	+	0.976877i	$0.431415\pi$
$758$	0	0
$759$	−2.19358	−0.0796218
$760$	0	0
$761$	−13.3921	−0.485462	−0.242731	−	0.970094i	$-0.578043\pi$
$761$	−13.3921	−0.485462	−0.242731	+	0.970094i	$0.578043\pi$
$762$	0	0
$763$	−6.95899	−0.251932
$764$	0	0
$765$	2.56199	0.0926290
$766$	0	0
$767$	6.53035	0.235797
$768$	0	0
$769$	7.26178	0.261867	0.130933	−	0.991391i	$-0.458203\pi$
$769$	7.26178	0.261867	0.130933	+	0.991391i	$0.458203\pi$
$770$	0	0
$771$	−5.19267	−0.187009
$772$	0	0
$773$	6.41774	0.230830	0.115415	−	0.993317i	$-0.463180\pi$
$773$	6.41774	0.230830	0.115415	+	0.993317i	$0.463180\pi$
$774$	0	0
$775$	5.18421	0.186222
$776$	0	0
$777$	5.67307	0.203520
$778$	0	0
$779$	10.2351	0.366709
$780$	0	0
$781$	−5.68598	−0.203460
$782$	0	0
$783$	−23.7708	−0.849498
$784$	0	0
$785$	14.5620	0.519740
$786$	0	0
$787$	25.6479	0.914248	0.457124	−	0.889403i	$-0.348880\pi$
$787$	25.6479	0.914248	0.457124	+	0.889403i	$0.348880\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	4.68550	0.166597
$792$	0	0
$793$	−10.1966	−0.362093
$794$	0	0
$795$	8.96343	0.317900
$796$	0	0
$797$	47.1229	1.66918	0.834590	−	0.550872i	$-0.185705\pi$
$797$	47.1229	1.66918	0.834590	+	0.550872i	$0.185705\pi$
$798$	0	0
$799$	3.05086	0.107931
$800$	0	0
$801$	10.5491	0.372734
$802$	0	0
$803$	−13.4193	−0.473556
$804$	0	0
$805$	2.32693	0.0820135
$806$	0	0
$807$	15.7333	0.553838
$808$	0	0
$809$	20.6133	0.724726	0.362363	−	0.932037i	$-0.381970\pi$
$809$	20.6133	0.724726	0.362363	+	0.932037i	$0.381970\pi$
$810$	0	0
$811$	7.87649	0.276581	0.138291	−	0.990392i	$-0.455839\pi$
$811$	7.87649	0.276581	0.138291	+	0.990392i	$0.455839\pi$
$812$	0	0
$813$	25.9081	0.908638
$814$	0	0
$815$	−1.95407	−0.0684480
$816$	0	0
$817$	−11.4652	−0.401117
$818$	0	0
$819$	1.63465	0.0571192
$820$	0	0
$821$	−4.32693	−0.151011	−0.0755054	−	0.997145i	$-0.524057\pi$
$821$	−4.32693	−0.151011	−0.0755054	+	0.997145i	$0.524057\pi$
$822$	0	0
$823$	−40.6178	−1.41585	−0.707923	−	0.706289i	$-0.750368\pi$
$823$	−40.6178	−1.41585	−0.707923	+	0.706289i	$0.750368\pi$
$824$	0	0
$825$	1.43801	0.0500650
$826$	0	0
$827$	23.1461	0.804869	0.402435	−	0.915449i	$-0.368164\pi$
$827$	23.1461	0.804869	0.402435	+	0.915449i	$0.368164\pi$
$828$	0	0
$829$	37.5022	1.30251	0.651253	−	0.758860i	$-0.274243\pi$
$829$	37.5022	1.30251	0.651253	+	0.758860i	$0.274243\pi$
$830$	0	0
$831$	11.5170	0.399519
$832$	0	0
$833$	−9.34614	−0.323825
$834$	0	0
$835$	15.5877	0.539433
$836$	0	0
$837$	29.0982	1.00578
$838$	0	0
$839$	26.5906	0.918009	0.459004	−	0.888434i	$-0.348206\pi$
$839$	26.5906	0.918009	0.459004	+	0.888434i	$0.348206\pi$
$840$	0	0
$841$	−11.0642	−0.381525
$842$	0	0
$843$	9.03212	0.311083
$844$	0	0
$845$	12.3002	0.423140
$846$	0	0
$847$	−14.9447	−0.513506
$848$	0	0
$849$	8.31402	0.285337
$850$	0	0
$851$	4.32693	0.148325
$852$	0	0
$853$	32.1561	1.10100	0.550502	−	0.834834i	$-0.314436\pi$
$853$	32.1561	1.10100	0.550502	+	0.834834i	$0.314436\pi$
$854$	0	0
$855$	1.28100	0.0438091
$856$	0	0
$857$	−47.8816	−1.63560	−0.817801	−	0.575500i	$-0.804807\pi$
$857$	−47.8816	−1.63560	−0.817801	+	0.575500i	$0.804807\pi$
$858$	0	0
$859$	1.28592	0.0438750	0.0219375	−	0.999759i	$-0.493017\pi$
$859$	1.28592	0.0438750	0.0219375	+	0.999759i	$0.493017\pi$
$860$	0	0
$861$	−20.4701	−0.697620
$862$	0	0
$863$	−1.26073	−0.0429157	−0.0214579	−	0.999770i	$-0.506831\pi$
$863$	−1.26073	−0.0429157	−0.0214579	+	0.999770i	$0.506831\pi$
$864$	0	0
$865$	−17.3985	−0.591568
$866$	0	0
$867$	17.0444	0.578858
$868$	0	0
$869$	9.50225	0.322342
$870$	0	0
$871$	5.79213	0.196259
$872$	0	0
$873$	4.35350	0.147344
$874$	0	0
$875$	−1.52543	−0.0515689
$876$	0	0
$877$	24.9668	0.843070	0.421535	−	0.906812i	$-0.361491\pi$
$877$	24.9668	0.843070	0.421535	+	0.906812i	$0.361491\pi$
$878$	0	0
$879$	−13.4710	−0.454367
$880$	0	0
$881$	−4.28100	−0.144230	−0.0721152	−	0.997396i	$-0.522975\pi$
$881$	−4.28100	−0.144230	−0.0721152	+	0.997396i	$0.522975\pi$
$882$	0	0
$883$	6.64941	0.223771	0.111885	−	0.993721i	$-0.464311\pi$
$883$	6.64941	0.223771	0.111885	+	0.993721i	$0.464311\pi$
$884$	0	0
$885$	−10.2351	−0.344048
$886$	0	0
$887$	34.1911	1.14803	0.574013	−	0.818846i	$-0.305386\pi$
$887$	34.1911	1.14803	0.574013	+	0.818846i	$0.305386\pi$
$888$	0	0
$889$	−18.4701	−0.619468
$890$	0	0
$891$	3.85637	0.129193
$892$	0	0
$893$	1.52543	0.0510465
$894$	0	0
$895$	7.80642	0.260940
$896$	0	0
$897$	−1.67307	−0.0558622
$898$	0	0
$899$	−21.9555	−0.732255
$900$	0	0
$901$	13.6731	0.455516
$902$	0	0
$903$	22.9304	0.763076
$904$	0	0
$905$	−17.3461	−0.576605
$906$	0	0
$907$	54.7402	1.81762	0.908810	−	0.417211i	$-0.136992\pi$
$907$	54.7402	1.81762	0.908810	+	0.417211i	$0.136992\pi$
$908$	0	0
$909$	−1.78324	−0.0591464
$910$	0	0
$911$	46.9086	1.55415	0.777076	−	0.629407i	$-0.216702\pi$
$911$	46.9086	1.55415	0.777076	+	0.629407i	$0.216702\pi$
$912$	0	0
$913$	−7.89523	−0.261294
$914$	0	0
$915$	15.9813	0.528324
$916$	0	0
$917$	−21.3077	−0.703643
$918$	0	0
$919$	−45.5723	−1.50329	−0.751646	−	0.659567i	$-0.770740\pi$
$919$	−45.5723	−1.50329	−0.751646	+	0.659567i	$0.770740\pi$
$920$	0	0
$921$	6.76403	0.222882
$922$	0	0
$923$	−4.33677	−0.142747
$924$	0	0
$925$	−2.83654	−0.0932647
$926$	0	0
$927$	−16.6858	−0.548035
$928$	0	0
$929$	44.0642	1.44570	0.722850	−	0.691005i	$-0.242832\pi$
$929$	44.0642	1.44570	0.722850	+	0.691005i	$0.242832\pi$
$930$	0	0
$931$	−4.67307	−0.153154
$932$	0	0
$933$	−13.9081	−0.455332
$934$	0	0
$935$	2.19358	0.0717376
$936$	0	0
$937$	41.1753	1.34514	0.672569	−	0.740034i	$-0.265191\pi$
$937$	41.1753	1.34514	0.672569	+	0.740034i	$0.265191\pi$
$938$	0	0
$939$	5.55262	0.181203
$940$	0	0
$941$	3.21585	0.104834	0.0524169	−	0.998625i	$-0.483308\pi$
$941$	3.21585	0.104834	0.0524169	+	0.998625i	$0.483308\pi$
$942$	0	0
$943$	−15.6128	−0.508424
$944$	0	0
$945$	−8.56199	−0.278522
$946$	0	0
$947$	23.7190	0.770764	0.385382	−	0.922757i	$-0.374070\pi$
$947$	23.7190	0.770764	0.385382	+	0.922757i	$0.374070\pi$
$948$	0	0
$949$	−10.2351	−0.332244
$950$	0	0
$951$	0.359955	0.0116723
$952$	0	0
$953$	19.3985	0.628380	0.314190	−	0.949360i	$-0.398267\pi$
$953$	19.3985	0.628380	0.314190	+	0.949360i	$0.398267\pi$
$954$	0	0
$955$	−18.6637	−0.603943
$956$	0	0
$957$	−6.09005	−0.196863
$958$	0	0
$959$	2.19358	0.0708343
$960$	0	0
$961$	−4.12399	−0.133032
$962$	0	0
$963$	18.9467	0.610549
$964$	0	0
$965$	18.7447	0.603412
$966$	0	0
$967$	14.1378	0.454641	0.227320	−	0.973820i	$-0.427004\pi$
$967$	14.1378	0.454641	0.227320	+	0.973820i	$0.427004\pi$
$968$	0	0
$969$	−2.62222	−0.0842377
$970$	0	0
$971$	−23.1111	−0.741670	−0.370835	−	0.928699i	$-0.620928\pi$
$971$	−23.1111	−0.741670	−0.370835	+	0.928699i	$0.620928\pi$
$972$	0	0
$973$	−5.01921	−0.160909
$974$	0	0
$975$	1.09679	0.0351253
$976$	0	0
$977$	−33.2019	−1.06222	−0.531111	−	0.847302i	$-0.678225\pi$
$977$	−33.2019	−1.06222	−0.531111	+	0.847302i	$0.678225\pi$
$978$	0	0
$979$	9.03212	0.288668
$980$	0	0
$981$	5.84390	0.186581
$982$	0	0
$983$	39.6765	1.26548	0.632741	−	0.774363i	$-0.281930\pi$
$983$	39.6765	1.26548	0.632741	+	0.774363i	$0.281930\pi$
$984$	0	0
$985$	4.23506	0.134940
$986$	0	0
$987$	−3.05086	−0.0971098
$988$	0	0
$989$	17.4893	0.556129
$990$	0	0
$991$	5.79352	0.184037	0.0920186	−	0.995757i	$-0.470668\pi$
$991$	5.79352	0.184037	0.0920186	+	0.995757i	$0.470668\pi$
$992$	0	0
$993$	29.6602	0.941237
$994$	0	0
$995$	−0.857279	−0.0271776
$996$	0	0
$997$	−35.1240	−1.11239	−0.556194	−	0.831053i	$-0.687739\pi$
$997$	−35.1240	−1.11239	−0.556194	+	0.831053i	$0.687739\pi$
$998$	0	0
$999$	−15.9210	−0.503719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bm.1.2		3
4.3	odd	2		6080.2.a.ca.1.2		3
8.3	odd	2		3040.2.a.j.1.2	✓	3
8.5	even	2		3040.2.a.p.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.j.1.2	✓	3	8.3	odd	2
3040.2.a.p.1.2	yes	3	8.5	even	2
6080.2.a.bm.1.2		3	1.1	even	1	trivial
6080.2.a.ca.1.2		3	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.31111 q^{3} -1.00000 q^{5} +1.52543 q^{7} -1.28100 q^{9} -1.09679 q^{11} -0.836535 q^{13} +1.31111 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -1.52543 q^{23} +1.00000 q^{25} +5.61285 q^{27} -4.23506 q^{29} +5.18421 q^{31} +1.43801 q^{33} -1.52543 q^{35} -2.83654 q^{37} +1.09679 q^{39} +10.2351 q^{41} -11.4652 q^{43} +1.28100 q^{45} +1.52543 q^{47} -4.67307 q^{49} -2.62222 q^{51} +6.83654 q^{53} +1.09679 q^{55} -1.31111 q^{57} -7.80642 q^{59} +12.1891 q^{61} -1.95407 q^{63} +0.836535 q^{65} -6.92396 q^{67} +2.00000 q^{69} +5.18421 q^{71} +12.2351 q^{73} -1.31111 q^{75} -1.67307 q^{77} -8.66370 q^{79} -3.51606 q^{81} +7.19850 q^{83} -2.00000 q^{85} +5.55262 q^{87} -8.23506 q^{89} -1.27607 q^{91} -6.79706 q^{93} -1.00000 q^{95} -3.39853 q^{97} +1.40498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 10 q^{11} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 2 q^{23} + 3 q^{25} - 10 q^{27} + 14 q^{29} + 2 q^{31} + 18 q^{33} + 2 q^{35} - 2 q^{37}+ \cdots - 30 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 10 * q^11 + 4 * q^13 + 4 * q^15 + 6 * q^17 + 3 * q^19 - 6 * q^21 + 2 * q^23 + 3 * q^25 - 10 * q^27 + 14 * q^29 + 2 * q^31 + 18 * q^33 + 2 * q^35 - 2 * q^37 + 10 * q^39 + 4 * q^41 - 14 * q^43 - 3 * q^45 - 2 * q^47 - q^49 - 8 * q^51 + 14 * q^53 + 10 * q^55 - 4 * q^57 - 10 * q^59 - 10 * q^61 + 14 * q^63 - 4 * q^65 + 6 * q^67 + 6 * q^69 + 2 * q^71 + 10 * q^73 - 4 * q^75 + 8 * q^77 + 14 * q^79 + 23 * q^81 + 2 * q^83 - 6 * q^85 - 24 * q^87 + 2 * q^89 - 30 * q^91 + 20 * q^93 - 3 * q^95 + 10 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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