LMFDB - Embedded newform 3484.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3484,2,Mod(1,3484)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3484, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3484.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3484 = 2^{2} \cdot 13 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3484.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:	$27.8198800642$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2444177.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 3x^{2} + 11x - 5 $ x^5 - x^4 - 8x^3 + 3x^2 + 11*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.53147$ of defining polynomial
Character	$\chi$	$=$	3484.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37460 q^{5} +1.40836 q^{7} -3.00000 q^{9} +O(q^{10})$	$q+3.37460 q^{5} +1.40836 q^{7} -3.00000 q^{9} +5.38794 q^{11} -1.00000 q^{13} -3.65459 q^{17} -7.71293 q^{19} -1.13645 q^{23} +6.38794 q^{25} +8.59442 q^{29} +3.53147 q^{31} +4.75267 q^{35} +11.3992 q^{37} +8.32153 q^{41} +6.74920 q^{43} -10.1238 q^{45} +5.95915 q^{47} -5.01651 q^{49} -11.3992 q^{53} +18.1821 q^{55} +10.7901 q^{59} +7.44003 q^{61} -4.22509 q^{63} -3.37460 q^{65} -1.00000 q^{67} -12.5030 q^{71} +13.1530 q^{73} +7.58818 q^{77} +3.72710 q^{79} +9.00000 q^{81} +0.690828 q^{83} -12.3328 q^{85} -2.37708 q^{89} -1.40836 q^{91} -26.0281 q^{95} -2.76463 q^{97} -16.1638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - 5 q^{7} - 15 q^{9}+O(q^{10}) $ 5 * q - 3 * q^5 - 5 * q^7 - 15 * q^9	$ 5 q - 3 q^{5} - 5 q^{7} - 15 q^{9} + 3 q^{11} - 5 q^{13} - 13 q^{17} - 8 q^{19} - 5 q^{23} + 8 q^{25} + 17 q^{29} + 9 q^{31} + 2 q^{35} + 4 q^{37} + 17 q^{41} - 6 q^{43} + 9 q^{45} + 6 q^{47} + 34 q^{49} - 4 q^{53} + 6 q^{55} + 38 q^{59} + 8 q^{61} + 15 q^{63} + 3 q^{65} - 5 q^{67} - 16 q^{71} + 6 q^{73} + 32 q^{77} + 20 q^{79} + 45 q^{81} + 14 q^{83} - 8 q^{85} + 5 q^{91} + 4 q^{95} + 5 q^{97} - 9 q^{99}+O(q^{100}) $ 5 * q - 3 * q^5 - 5 * q^7 - 15 * q^9 + 3 * q^11 - 5 * q^13 - 13 * q^17 - 8 * q^19 - 5 * q^23 + 8 * q^25 + 17 * q^29 + 9 * q^31 + 2 * q^35 + 4 * q^37 + 17 * q^41 - 6 * q^43 + 9 * q^45 + 6 * q^47 + 34 * q^49 - 4 * q^53 + 6 * q^55 + 38 * q^59 + 8 * q^61 + 15 * q^63 + 3 * q^65 - 5 * q^67 - 16 * q^71 + 6 * q^73 + 32 * q^77 + 20 * q^79 + 45 * q^81 + 14 * q^83 - 8 * q^85 + 5 * q^91 + 4 * q^95 + 5 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	3.37460	1.50917	0.754584	−	0.656204i	$-0.227839\pi$
$5$	3.37460	1.50917	0.754584	+	0.656204i	$0.227839\pi$
$6$	0	0
$7$	1.40836	0.532311	0.266156	−	0.963930i	$-0.414246\pi$
$7$	1.40836	0.532311	0.266156	+	0.963930i	$0.414246\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	5.38794	1.62452	0.812262	−	0.583292i	$-0.198236\pi$
$11$	5.38794	1.62452	0.812262	+	0.583292i	$0.198236\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.65459	−0.886367	−0.443184	−	0.896431i	$-0.646151\pi$
$17$	−3.65459	−0.886367	−0.443184	+	0.896431i	$0.646151\pi$
$18$	0	0
$19$	−7.71293	−1.76947	−0.884734	−	0.466097i	$-0.845660\pi$
$19$	−7.71293	−1.76947	−0.884734	+	0.466097i	$0.845660\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.13645	−0.236966	−0.118483	−	0.992956i	$-0.537803\pi$
$23$	−1.13645	−0.236966	−0.118483	+	0.992956i	$0.537803\pi$
$24$	0	0
$25$	6.38794	1.27759
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.59442	1.59594	0.797972	−	0.602694i	$-0.205906\pi$
$29$	8.59442	1.59594	0.797972	+	0.602694i	$0.205906\pi$
$30$	0	0
$31$	3.53147	0.634272	0.317136	−	0.948380i	$-0.397279\pi$
$31$	3.53147	0.634272	0.317136	+	0.948380i	$0.397279\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.75267	0.803347
$36$	0	0
$37$	11.3992	1.87401	0.937007	−	0.349310i	$-0.113584\pi$
$37$	11.3992	1.87401	0.937007	+	0.349310i	$0.113584\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.32153	1.29960	0.649802	−	0.760103i	$-0.274852\pi$
$41$	8.32153	1.29960	0.649802	+	0.760103i	$0.274852\pi$
$42$	0	0
$43$	6.74920	1.02924	0.514622	−	0.857417i	$-0.327932\pi$
$43$	6.74920	1.02924	0.514622	+	0.857417i	$0.327932\pi$
$44$	0	0
$45$	−10.1238	−1.50917
$46$	0	0
$47$	5.95915	0.869231	0.434616	−	0.900616i	$-0.356884\pi$
$47$	5.95915	0.869231	0.434616	+	0.900616i	$0.356884\pi$
$48$	0	0
$49$	−5.01651	−0.716645
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.3992	−1.56580	−0.782899	−	0.622149i	$-0.786260\pi$
$53$	−11.3992	−1.56580	−0.782899	+	0.622149i	$0.786260\pi$
$54$	0	0
$55$	18.1821	2.45168
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.7901	1.40475	0.702373	−	0.711809i	$-0.252124\pi$
$59$	10.7901	1.40475	0.702373	+	0.711809i	$0.252124\pi$
$60$	0	0
$61$	7.44003	0.952599	0.476299	−	0.879283i	$-0.341978\pi$
$61$	7.44003	0.952599	0.476299	+	0.879283i	$0.341978\pi$
$62$	0	0
$63$	−4.22509	−0.532311
$64$	0	0
$65$	−3.37460	−0.418568
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.5030	−1.48383	−0.741915	−	0.670493i	$-0.766082\pi$
$71$	−12.5030	−1.48383	−0.741915	+	0.670493i	$0.766082\pi$
$72$	0	0
$73$	13.1530	1.53944	0.769719	−	0.638383i	$-0.220396\pi$
$73$	13.1530	1.53944	0.769719	+	0.638383i	$0.220396\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.58818	0.864753
$78$	0	0
$79$	3.72710	0.419332	0.209666	−	0.977773i	$-0.432762\pi$
$79$	3.72710	0.419332	0.209666	+	0.977773i	$0.432762\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0.690828	0.0758283	0.0379141	−	0.999281i	$-0.487929\pi$
$83$	0.690828	0.0758283	0.0379141	+	0.999281i	$0.487929\pi$
$84$	0	0
$85$	−12.3328	−1.33768
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.37708	−0.251970	−0.125985	−	0.992032i	$-0.540209\pi$
$89$	−2.37708	−0.251970	−0.125985	+	0.992032i	$0.540209\pi$
$90$	0	0
$91$	−1.40836	−0.147637
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−26.0281	−2.67042
$96$	0	0
$97$	−2.76463	−0.280706	−0.140353	−	0.990102i	$-0.544824\pi$
$97$	−2.76463	−0.280706	−0.140353	+	0.990102i	$0.544824\pi$
$98$	0	0
$99$	−16.1638	−1.62452
$100$	0	0
$101$	−14.4213	−1.43497	−0.717486	−	0.696573i	$-0.754707\pi$
$101$	−14.4213	−1.43497	−0.717486	+	0.696573i	$0.754707\pi$
$102$	0	0
$103$	5.87692	0.579070	0.289535	−	0.957167i	$-0.406499\pi$
$103$	5.87692	0.579070	0.289535	+	0.957167i	$0.406499\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.12838	0.495779	0.247890	−	0.968788i	$-0.420263\pi$
$107$	5.12838	0.495779	0.247890	+	0.968788i	$0.420263\pi$
$108$	0	0
$109$	−13.9849	−1.33951	−0.669754	−	0.742583i	$-0.733601\pi$
$109$	−13.9849	−1.33951	−0.669754	+	0.742583i	$0.733601\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.0266732	0.00250921	0.00125460	−	0.999999i	$-0.499601\pi$
$113$	0.0266732	0.00250921	0.00125460	+	0.999999i	$0.499601\pi$
$114$	0	0
$115$	−3.83506	−0.357621
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	−5.14698	−0.471823
$120$	0	0
$121$	18.0299	1.63908
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.68374	0.418926
$126$	0	0
$127$	16.6612	1.47844	0.739222	−	0.673462i	$-0.235193\pi$
$127$	16.6612	1.47844	0.739222	+	0.673462i	$0.235193\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.37918	0.207870	0.103935	−	0.994584i	$-0.466857\pi$
$131$	2.37918	0.207870	0.103935	+	0.994584i	$0.466857\pi$
$132$	0	0
$133$	−10.8626	−0.941907
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.85898	−0.671438	−0.335719	−	0.941962i	$-0.608979\pi$
$137$	−7.85898	−0.671438	−0.335719	+	0.941962i	$0.608979\pi$
$138$	0	0
$139$	−6.60913	−0.560579	−0.280290	−	0.959916i	$-0.590430\pi$
$139$	−6.60913	−0.560579	−0.280290	+	0.959916i	$0.590430\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.38794	−0.450562
$144$	0	0
$145$	29.0028	2.40855
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.2522	1.41335	0.706677	−	0.707536i	$-0.250193\pi$
$149$	17.2522	1.41335	0.706677	+	0.707536i	$0.250193\pi$
$150$	0	0
$151$	13.0488	1.06189	0.530947	−	0.847405i	$-0.321836\pi$
$151$	13.0488	1.06189	0.530947	+	0.847405i	$0.321836\pi$
$152$	0	0
$153$	10.9638	0.886367
$154$	0	0
$155$	11.9173	0.957222
$156$	0	0
$157$	9.38794	0.749239	0.374620	−	0.927179i	$-0.377773\pi$
$157$	9.38794	0.749239	0.374620	+	0.927179i	$0.377773\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.60053	−0.126140
$162$	0	0
$163$	−23.4576	−1.83734	−0.918669	−	0.395028i	$-0.870735\pi$
$163$	−23.4576	−1.83734	−0.918669	+	0.395028i	$0.870735\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.81215	−0.759287	−0.379644	−	0.925133i	$-0.623953\pi$
$167$	−9.81215	−0.759287	−0.379644	+	0.925133i	$0.623953\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	23.1388	1.76947
$172$	0	0
$173$	−17.3879	−1.32198	−0.660990	−	0.750394i	$-0.729864\pi$
$173$	−17.3879	−1.32198	−0.660990	+	0.750394i	$0.729864\pi$
$174$	0	0
$175$	8.99654	0.680074
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.9954	1.56927	0.784636	−	0.619957i	$-0.212850\pi$
$179$	20.9954	1.56927	0.784636	+	0.619957i	$0.212850\pi$
$180$	0	0
$181$	−18.9933	−1.41176	−0.705882	−	0.708330i	$-0.749449\pi$
$181$	−18.9933	−1.41176	−0.705882	+	0.708330i	$0.749449\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	38.4677	2.82820
$186$	0	0
$187$	−19.6907	−1.43993
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.37669	0.0996141	0.0498070	−	0.998759i	$-0.484139\pi$
$191$	1.37669	0.0996141	0.0498070	+	0.998759i	$0.484139\pi$
$192$	0	0
$193$	−17.1121	−1.23176	−0.615878	−	0.787841i	$-0.711199\pi$
$193$	−17.1121	−1.23176	−0.615878	+	0.787841i	$0.711199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4055	−1.24009	−0.620044	−	0.784567i	$-0.712885\pi$
$197$	−17.4055	−1.24009	−0.620044	+	0.784567i	$0.712885\pi$
$198$	0	0
$199$	14.2398	1.00943	0.504717	−	0.863285i	$-0.331597\pi$
$199$	14.2398	1.00943	0.504717	+	0.863285i	$0.331597\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.1041	0.849539
$204$	0	0
$205$	28.0818	1.96132
$206$	0	0
$207$	3.40934	0.236966
$208$	0	0
$209$	−41.5568	−2.87454
$210$	0	0
$211$	17.3882	1.19706	0.598528	−	0.801102i	$-0.295753\pi$
$211$	17.3882	1.19706	0.598528	+	0.801102i	$0.295753\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	22.7759	1.55330
$216$	0	0
$217$	4.97360	0.337630
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.65459	0.245834
$222$	0	0
$223$	−21.1338	−1.41523	−0.707613	−	0.706601i	$-0.750228\pi$
$223$	−21.1338	−1.41523	−0.707613	+	0.706601i	$0.750228\pi$
$224$	0	0
$225$	−19.1638	−1.27759
$226$	0	0
$227$	11.9163	0.790915	0.395458	−	0.918484i	$-0.370586\pi$
$227$	11.9163	0.790915	0.395458	+	0.918484i	$0.370586\pi$
$228$	0	0
$229$	−2.96774	−0.196114	−0.0980569	−	0.995181i	$-0.531263\pi$
$229$	−2.96774	−0.196114	−0.0980569	+	0.995181i	$0.531263\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.55997	−0.429758	−0.214879	−	0.976641i	$-0.568936\pi$
$233$	−6.55997	−0.429758	−0.214879	+	0.976641i	$0.568936\pi$
$234$	0	0
$235$	20.1098	1.31182
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.12877	−0.137699	−0.0688493	−	0.997627i	$-0.521933\pi$
$239$	−2.12877	−0.137699	−0.0688493	+	0.997627i	$0.521933\pi$
$240$	0	0
$241$	−7.71293	−0.496833	−0.248417	−	0.968653i	$-0.579910\pi$
$241$	−7.71293	−0.496833	−0.248417	+	0.968653i	$0.579910\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.9287	−1.08154
$246$	0	0
$247$	7.71293	0.490762
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.8026	0.808090	0.404045	−	0.914739i	$-0.367604\pi$
$251$	12.8026	0.808090	0.404045	+	0.914739i	$0.367604\pi$
$252$	0	0
$253$	−6.12311	−0.384957
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.95733	−0.309230	−0.154615	−	0.987975i	$-0.549414\pi$
$257$	−4.95733	−0.309230	−0.154615	+	0.987975i	$0.549414\pi$
$258$	0	0
$259$	16.0542	0.997559
$260$	0	0
$261$	−25.7833	−1.59594
$262$	0	0
$263$	4.50798	0.277974	0.138987	−	0.990294i	$-0.455615\pi$
$263$	4.50798	0.277974	0.138987	+	0.990294i	$0.455615\pi$
$264$	0	0
$265$	−38.4677	−2.36305
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.4530	−1.61286	−0.806432	−	0.591326i	$-0.798605\pi$
$269$	−26.4530	−1.61286	−0.806432	+	0.591326i	$0.798605\pi$
$270$	0	0
$271$	14.9547	0.908434	0.454217	−	0.890891i	$-0.349919\pi$
$271$	14.9547	0.908434	0.454217	+	0.890891i	$0.349919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	34.4178	2.07547
$276$	0	0
$277$	8.10477	0.486968	0.243484	−	0.969905i	$-0.421710\pi$
$277$	8.10477	0.486968	0.243484	+	0.969905i	$0.421710\pi$
$278$	0	0
$279$	−10.5944	−0.634272
$280$	0	0
$281$	−16.5366	−0.986493	−0.493247	−	0.869889i	$-0.664190\pi$
$281$	−16.5366	−0.986493	−0.493247	+	0.869889i	$0.664190\pi$
$282$	0	0
$283$	−25.7337	−1.52971	−0.764855	−	0.644203i	$-0.777189\pi$
$283$	−25.7337	−1.52971	−0.764855	+	0.644203i	$0.777189\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.7197	0.691794
$288$	0	0
$289$	−3.64400	−0.214353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.7346	−0.685545	−0.342772	−	0.939418i	$-0.611366\pi$
$293$	−11.7346	−0.685545	−0.342772	+	0.939418i	$0.611366\pi$
$294$	0	0
$295$	36.4121	2.12000
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.13645	0.0657225
$300$	0	0
$301$	9.50533	0.547878
$302$	0	0
$303$	0	0
$304$	0	0
$305$	25.1071	1.43763
$306$	0	0
$307$	15.2008	0.867555	0.433778	−	0.901020i	$-0.357180\pi$
$307$	15.2008	0.867555	0.433778	+	0.901020i	$0.357180\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.8125	0.839942	0.419971	−	0.907537i	$-0.362040\pi$
$311$	14.8125	0.839942	0.419971	+	0.907537i	$0.362040\pi$
$312$	0	0
$313$	0.799267	0.0451772	0.0225886	−	0.999745i	$-0.492809\pi$
$313$	0.799267	0.0451772	0.0225886	+	0.999745i	$0.492809\pi$
$314$	0	0
$315$	−14.2580	−0.803347
$316$	0	0
$317$	−7.25219	−0.407323	−0.203662	−	0.979041i	$-0.565284\pi$
$317$	−7.25219	−0.407323	−0.203662	+	0.979041i	$0.565284\pi$
$318$	0	0
$319$	46.3062	2.59265
$320$	0	0
$321$	0	0
$322$	0	0
$323$	28.1876	1.56840
$324$	0	0
$325$	−6.38794	−0.354339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.39265	0.462702
$330$	0	0
$331$	−19.0039	−1.04455	−0.522274	−	0.852778i	$-0.674916\pi$
$331$	−19.0039	−1.04455	−0.522274	+	0.852778i	$0.674916\pi$
$332$	0	0
$333$	−34.1975	−1.87401
$334$	0	0
$335$	−3.37460	−0.184374
$336$	0	0
$337$	17.2881	0.941741	0.470871	−	0.882202i	$-0.343940\pi$
$337$	17.2881	0.941741	0.470871	+	0.882202i	$0.343940\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.0274	1.03039
$342$	0	0
$343$	−16.9236	−0.913789
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.4038	0.558503	0.279251	−	0.960218i	$-0.409914\pi$
$347$	10.4038	0.558503	0.279251	+	0.960218i	$0.409914\pi$
$348$	0	0
$349$	−18.4093	−0.985429	−0.492714	−	0.870191i	$-0.663995\pi$
$349$	−18.4093	−0.985429	−0.492714	+	0.870191i	$0.663995\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.49248	0.132661	0.0663306	−	0.997798i	$-0.478871\pi$
$353$	2.49248	0.132661	0.0663306	+	0.997798i	$0.478871\pi$
$354$	0	0
$355$	−42.1926	−2.23935
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.08505	−0.215601	−0.107800	−	0.994173i	$-0.534381\pi$
$359$	−4.08505	−0.215601	−0.107800	+	0.994173i	$0.534381\pi$
$360$	0	0
$361$	40.4893	2.13101
$362$	0	0
$363$	0	0
$364$	0	0
$365$	44.3860	2.32327
$366$	0	0
$367$	−5.07294	−0.264805	−0.132403	−	0.991196i	$-0.542269\pi$
$367$	−5.07294	−0.264805	−0.132403	+	0.991196i	$0.542269\pi$
$368$	0	0
$369$	−24.9646	−1.29960
$370$	0	0
$371$	−16.0542	−0.833492
$372$	0	0
$373$	−30.6556	−1.58728	−0.793642	−	0.608385i	$-0.791818\pi$
$373$	−30.6556	−1.58728	−0.793642	+	0.608385i	$0.791818\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.59442	−0.442635
$378$	0	0
$379$	0.616251	0.0316547	0.0158273	−	0.999875i	$-0.494962\pi$
$379$	0.616251	0.0316547	0.0158273	+	0.999875i	$0.494962\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.16382	−0.212761	−0.106381	−	0.994325i	$-0.533926\pi$
$383$	−4.16382	−0.212761	−0.106381	+	0.994325i	$0.533926\pi$
$384$	0	0
$385$	25.6071	1.30506
$386$	0	0
$387$	−20.2476	−1.02924
$388$	0	0
$389$	0.457811	0.0232119	0.0116060	−	0.999933i	$-0.496306\pi$
$389$	0.457811	0.0232119	0.0116060	+	0.999933i	$0.496306\pi$
$390$	0	0
$391$	4.15325	0.210039
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.5775	0.632842
$396$	0	0
$397$	27.1460	1.36242	0.681210	−	0.732088i	$-0.261454\pi$
$397$	27.1460	1.36242	0.681210	+	0.732088i	$0.261454\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.85493	−0.292381	−0.146191	−	0.989256i	$-0.546701\pi$
$401$	−5.85493	−0.292381	−0.146191	+	0.989256i	$0.546701\pi$
$402$	0	0
$403$	−3.53147	−0.175915
$404$	0	0
$405$	30.3714	1.50917
$406$	0	0
$407$	61.4181	3.04438
$408$	0	0
$409$	12.3928	0.612785	0.306393	−	0.951905i	$-0.400878\pi$
$409$	12.3928	0.612785	0.306393	+	0.951905i	$0.400878\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.1963	0.747762
$414$	0	0
$415$	2.33127	0.114438
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.4363	−0.851820	−0.425910	−	0.904766i	$-0.640046\pi$
$419$	−17.4363	−0.851820	−0.425910	+	0.904766i	$0.640046\pi$
$420$	0	0
$421$	−3.25833	−0.158801	−0.0794007	−	0.996843i	$-0.525301\pi$
$421$	−3.25833	−0.158801	−0.0794007	+	0.996843i	$0.525301\pi$
$422$	0	0
$423$	−17.8775	−0.869231
$424$	0	0
$425$	−23.3453	−1.13241
$426$	0	0
$427$	10.4783	0.507079
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.26001	0.446039	0.223019	−	0.974814i	$-0.428409\pi$
$431$	9.26001	0.446039	0.223019	+	0.974814i	$0.428409\pi$
$432$	0	0
$433$	40.0561	1.92497	0.962487	−	0.271327i	$-0.0874624\pi$
$433$	40.0561	1.92497	0.962487	+	0.271327i	$0.0874624\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.76534	0.419303
$438$	0	0
$439$	6.37779	0.304395	0.152198	−	0.988350i	$-0.451365\pi$
$439$	6.37779	0.304395	0.152198	+	0.988350i	$0.451365\pi$
$440$	0	0
$441$	15.0495	0.716645
$442$	0	0
$443$	27.9288	1.32694	0.663470	−	0.748203i	$-0.269083\pi$
$443$	27.9288	1.32694	0.663470	+	0.748203i	$0.269083\pi$
$444$	0	0
$445$	−8.02171	−0.380266
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.0179	0.803125	0.401563	−	0.915832i	$-0.368467\pi$
$449$	17.0179	0.803125	0.401563	+	0.915832i	$0.368467\pi$
$450$	0	0
$451$	44.8359	2.11124
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.75267	−0.222808
$456$	0	0
$457$	−5.97333	−0.279420	−0.139710	−	0.990192i	$-0.544617\pi$
$457$	−5.97333	−0.279420	−0.139710	+	0.990192i	$0.544617\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.1102	−1.35579	−0.677897	−	0.735156i	$-0.737109\pi$
$461$	−29.1102	−1.35579	−0.677897	+	0.735156i	$0.737109\pi$
$462$	0	0
$463$	−24.0421	−1.11733	−0.558666	−	0.829393i	$-0.688687\pi$
$463$	−24.0421	−1.11733	−0.558666	+	0.829393i	$0.688687\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.56535	0.164985	0.0824924	−	0.996592i	$-0.473712\pi$
$467$	3.56535	0.164985	0.0824924	+	0.996592i	$0.473712\pi$
$468$	0	0
$469$	−1.40836	−0.0650322
$470$	0	0
$471$	0	0
$472$	0	0
$473$	36.3643	1.67203
$474$	0	0
$475$	−49.2697	−2.26065
$476$	0	0
$477$	34.1975	1.56580
$478$	0	0
$479$	−30.7640	−1.40564	−0.702821	−	0.711366i	$-0.748077\pi$
$479$	−30.7640	−1.40564	−0.702821	+	0.711366i	$0.748077\pi$
$480$	0	0
$481$	−11.3992	−0.519758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.32954	−0.423632
$486$	0	0
$487$	2.01512	0.0913140	0.0456570	−	0.998957i	$-0.485462\pi$
$487$	2.01512	0.0913140	0.0456570	+	0.998957i	$0.485462\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.1199	−0.772610	−0.386305	−	0.922371i	$-0.626249\pi$
$491$	−17.1199	−0.772610	−0.386305	+	0.922371i	$0.626249\pi$
$492$	0	0
$493$	−31.4091	−1.41459
$494$	0	0
$495$	−54.5464	−2.45168
$496$	0	0
$497$	−17.6087	−0.789860
$498$	0	0
$499$	10.8251	0.484597	0.242298	−	0.970202i	$-0.422099\pi$
$499$	10.8251	0.484597	0.242298	+	0.970202i	$0.422099\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.4319	1.44607	0.723033	−	0.690813i	$-0.242747\pi$
$503$	32.4319	1.44607	0.723033	+	0.690813i	$0.242747\pi$
$504$	0	0
$505$	−48.6661	−2.16561
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.41996	0.195911	0.0979556	−	0.995191i	$-0.468770\pi$
$509$	4.41996	0.195911	0.0979556	+	0.995191i	$0.468770\pi$
$510$	0	0
$511$	18.5241	0.819460
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.8323	0.873914
$516$	0	0
$517$	32.1075	1.41209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.9747	−1.57608	−0.788040	−	0.615623i	$-0.788904\pi$
$521$	−35.9747	−1.57608	−0.788040	+	0.615623i	$0.788904\pi$
$522$	0	0
$523$	10.4426	0.456622	0.228311	−	0.973588i	$-0.426680\pi$
$523$	10.4426	0.456622	0.228311	+	0.973588i	$0.426680\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.9061	−0.562198
$528$	0	0
$529$	−21.7085	−0.943847
$530$	0	0
$531$	−32.3702	−1.40475
$532$	0	0
$533$	−8.32153	−0.360445
$534$	0	0
$535$	17.3062	0.748214
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−27.0287	−1.16421
$540$	0	0
$541$	−36.4226	−1.56593	−0.782966	−	0.622065i	$-0.786294\pi$
$541$	−36.4226	−1.56593	−0.782966	+	0.622065i	$0.786294\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−47.1934	−2.02154
$546$	0	0
$547$	−5.23244	−0.223723	−0.111861	−	0.993724i	$-0.535681\pi$
$547$	−5.23244	−0.223723	−0.111861	+	0.993724i	$0.535681\pi$
$548$	0	0
$549$	−22.3201	−0.952599
$550$	0	0
$551$	−66.2882	−2.82397
$552$	0	0
$553$	5.24912	0.223215
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.3041	0.478972	0.239486	−	0.970900i	$-0.423021\pi$
$557$	11.3041	0.478972	0.239486	+	0.970900i	$0.423021\pi$
$558$	0	0
$559$	−6.74920	−0.285461
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.1075	−1.35317	−0.676586	−	0.736364i	$-0.736541\pi$
$563$	−32.1075	−1.35317	−0.676586	+	0.736364i	$0.736541\pi$
$564$	0	0
$565$	0.0900116	0.00378682
$566$	0	0
$567$	12.6753	0.532311
$568$	0	0
$569$	−7.84875	−0.329037	−0.164518	−	0.986374i	$-0.552607\pi$
$569$	−7.84875	−0.329037	−0.164518	+	0.986374i	$0.552607\pi$
$570$	0	0
$571$	9.25215	0.387191	0.193595	−	0.981081i	$-0.437985\pi$
$571$	9.25215	0.387191	0.193595	+	0.981081i	$0.437985\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.25956	−0.302745
$576$	0	0
$577$	−6.69286	−0.278627	−0.139314	−	0.990248i	$-0.544490\pi$
$577$	−6.69286	−0.278627	−0.139314	+	0.990248i	$0.544490\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.972937	0.0403642
$582$	0	0
$583$	−61.4181	−2.54368
$584$	0	0
$585$	10.1238	0.418568
$586$	0	0
$587$	0.108405	0.00447436	0.00223718	−	0.999997i	$-0.499288\pi$
$587$	0.108405	0.00447436	0.00223718	+	0.999997i	$0.499288\pi$
$588$	0	0
$589$	−27.2380	−1.12232
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.6191	0.436075	0.218038	−	0.975940i	$-0.430034\pi$
$593$	10.6191	0.436075	0.218038	+	0.975940i	$0.430034\pi$
$594$	0	0
$595$	−17.3690	−0.712061
$596$	0	0
$597$	0	0
$598$	0	0
$599$	41.3155	1.68811	0.844053	−	0.536259i	$-0.180163\pi$
$599$	41.3155	1.68811	0.844053	+	0.536259i	$0.180163\pi$
$600$	0	0
$601$	−25.2529	−1.03009	−0.515044	−	0.857164i	$-0.672224\pi$
$601$	−25.2529	−1.03009	−0.515044	+	0.857164i	$0.672224\pi$
$602$	0	0
$603$	3.00000	0.122169
$604$	0	0
$605$	60.8437	2.47365
$606$	0	0
$607$	−24.2069	−0.982529	−0.491265	−	0.871010i	$-0.663465\pi$
$607$	−24.2069	−0.982529	−0.491265	+	0.871010i	$0.663465\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.95915	−0.241081
$612$	0	0
$613$	−36.1517	−1.46015	−0.730076	−	0.683366i	$-0.760515\pi$
$613$	−36.1517	−1.46015	−0.730076	+	0.683366i	$0.760515\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.7479	1.43916	0.719578	−	0.694411i	$-0.244335\pi$
$617$	35.7479	1.43916	0.719578	+	0.694411i	$0.244335\pi$
$618$	0	0
$619$	−16.1190	−0.647876	−0.323938	−	0.946078i	$-0.605007\pi$
$619$	−16.1190	−0.647876	−0.323938	+	0.946078i	$0.605007\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.34780	−0.134127
$624$	0	0
$625$	−16.1339	−0.645357
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−41.6593	−1.66106
$630$	0	0
$631$	21.7263	0.864910	0.432455	−	0.901656i	$-0.357647\pi$
$631$	21.7263	0.864910	0.432455	+	0.901656i	$0.357647\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	56.2250	2.23122
$636$	0	0
$637$	5.01651	0.198761
$638$	0	0
$639$	37.5089	1.48383
$640$	0	0
$641$	32.8884	1.29901	0.649506	−	0.760356i	$-0.274976\pi$
$641$	32.8884	1.29901	0.649506	+	0.760356i	$0.274976\pi$
$642$	0	0
$643$	9.88967	0.390010	0.195005	−	0.980802i	$-0.437528\pi$
$643$	9.88967	0.390010	0.195005	+	0.980802i	$0.437528\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.8039	−1.36828	−0.684142	−	0.729349i	$-0.739823\pi$
$647$	−34.8039	−1.36828	−0.684142	+	0.729349i	$0.739823\pi$
$648$	0	0
$649$	58.1361	2.28204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.04642	0.158349	0.0791744	−	0.996861i	$-0.474772\pi$
$653$	4.04642	0.158349	0.0791744	+	0.996861i	$0.474772\pi$
$654$	0	0
$655$	8.02877	0.313710
$656$	0	0
$657$	−39.4589	−1.53944
$658$	0	0
$659$	−20.1506	−0.784958	−0.392479	−	0.919761i	$-0.628382\pi$
$659$	−20.1506	−0.784958	−0.392479	+	0.919761i	$0.628382\pi$
$660$	0	0
$661$	6.76463	0.263114	0.131557	−	0.991309i	$-0.458002\pi$
$661$	6.76463	0.263114	0.131557	+	0.991309i	$0.458002\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−36.6570	−1.42150
$666$	0	0
$667$	−9.76711	−0.378184
$668$	0	0
$669$	0	0
$670$	0	0
$671$	40.0864	1.54752
$672$	0	0
$673$	−19.8099	−0.763617	−0.381808	−	0.924242i	$-0.624699\pi$
$673$	−19.8099	−0.763617	−0.381808	+	0.924242i	$0.624699\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−46.4493	−1.78519	−0.892597	−	0.450856i	$-0.851119\pi$
$677$	−46.4493	−1.78519	−0.892597	+	0.450856i	$0.851119\pi$
$678$	0	0
$679$	−3.89361	−0.149423
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.1084	−1.84082	−0.920408	−	0.390958i	$-0.872144\pi$
$683$	−48.1084	−1.84082	−0.920408	+	0.390958i	$0.872144\pi$
$684$	0	0
$685$	−26.5209	−1.01331
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.3992	0.434274
$690$	0	0
$691$	−3.34834	−0.127377	−0.0636885	−	0.997970i	$-0.520286\pi$
$691$	−3.34834	−0.127377	−0.0636885	+	0.997970i	$0.520286\pi$
$692$	0	0
$693$	−22.7645	−0.864753
$694$	0	0
$695$	−22.3032	−0.846008
$696$	0	0
$697$	−30.4117	−1.15193
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.0993	1.25014	0.625072	−	0.780567i	$-0.285070\pi$
$701$	33.0993	1.25014	0.625072	+	0.780567i	$0.285070\pi$
$702$	0	0
$703$	−87.9211	−3.31601
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.3104	−0.763851
$708$	0	0
$709$	7.98254	0.299791	0.149895	−	0.988702i	$-0.452106\pi$
$709$	7.98254	0.299791	0.149895	+	0.988702i	$0.452106\pi$
$710$	0	0
$711$	−11.1813	−0.419332
$712$	0	0
$713$	−4.01334	−0.150301
$714$	0	0
$715$	−18.1821	−0.679974
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.8062	−0.738648	−0.369324	−	0.929301i	$-0.620411\pi$
$719$	−19.8062	−0.738648	−0.369324	+	0.929301i	$0.620411\pi$
$720$	0	0
$721$	8.27684	0.308246
$722$	0	0
$723$	0	0
$724$	0	0
$725$	54.9007	2.03896
$726$	0	0
$727$	−23.8788	−0.885615	−0.442807	−	0.896617i	$-0.646017\pi$
$727$	−23.8788	−0.885615	−0.442807	+	0.896617i	$0.646017\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−24.6655	−0.912288
$732$	0	0
$733$	18.3179	0.676587	0.338293	−	0.941041i	$-0.390150\pi$
$733$	18.3179	0.676587	0.338293	+	0.941041i	$0.390150\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.38794	−0.198467
$738$	0	0
$739$	17.2940	0.636169	0.318085	−	0.948062i	$-0.396960\pi$
$739$	17.2940	0.636169	0.318085	+	0.948062i	$0.396960\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.8504	0.911672	0.455836	−	0.890064i	$-0.349340\pi$
$743$	24.8504	0.911672	0.455836	+	0.890064i	$0.349340\pi$
$744$	0	0
$745$	58.2193	2.13299
$746$	0	0
$747$	−2.07249	−0.0758283
$748$	0	0
$749$	7.22262	0.263909
$750$	0	0
$751$	−3.45703	−0.126149	−0.0630744	−	0.998009i	$-0.520091\pi$
$751$	−3.45703	−0.126149	−0.0630744	+	0.998009i	$0.520091\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.0344	1.60258
$756$	0	0
$757$	−27.5321	−1.00067	−0.500335	−	0.865832i	$-0.666790\pi$
$757$	−27.5321	−1.00067	−0.500335	+	0.865832i	$0.666790\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.3634	1.49942	0.749710	−	0.661766i	$-0.230193\pi$
$761$	41.3634	1.49942	0.749710	+	0.661766i	$0.230193\pi$
$762$	0	0
$763$	−19.6958	−0.713035
$764$	0	0
$765$	36.9983	1.33768
$766$	0	0
$767$	−10.7901	−0.389606
$768$	0	0
$769$	−26.5387	−0.957011	−0.478506	−	0.878084i	$-0.658821\pi$
$769$	−26.5387	−0.957011	−0.478506	+	0.878084i	$0.658821\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.1313	1.04778	0.523890	−	0.851786i	$-0.324480\pi$
$773$	29.1313	1.04778	0.523890	+	0.851786i	$0.324480\pi$
$774$	0	0
$775$	22.5588	0.810338
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−64.1833	−2.29961
$780$	0	0
$781$	−67.3653	−2.41052
$782$	0	0
$783$	0	0
$784$	0	0
$785$	31.6806	1.13073
$786$	0	0
$787$	24.4190	0.870444	0.435222	−	0.900323i	$-0.356670\pi$
$787$	24.4190	0.870444	0.435222	+	0.900323i	$0.356670\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.0375656	0.00133568
$792$	0	0
$793$	−7.44003	−0.264203
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.1435	−1.06774	−0.533869	−	0.845567i	$-0.679262\pi$
$797$	−30.1435	−1.06774	−0.533869	+	0.845567i	$0.679262\pi$
$798$	0	0
$799$	−21.7782	−0.770458
$800$	0	0
$801$	7.13125	0.251970
$802$	0	0
$803$	70.8673	2.50085
$804$	0	0
$805$	−5.40116	−0.190366
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.1175	1.69172	0.845861	−	0.533403i	$-0.179087\pi$
$809$	48.1175	1.69172	0.845861	+	0.533403i	$0.179087\pi$
$810$	0	0
$811$	−54.0358	−1.89745	−0.948727	−	0.316096i	$-0.897628\pi$
$811$	−54.0358	−1.89745	−0.948727	+	0.316096i	$0.897628\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−79.1599	−2.77285
$816$	0	0
$817$	−52.0561	−1.82121
$818$	0	0
$819$	4.22509	0.147637
$820$	0	0
$821$	2.85255	0.0995546	0.0497773	−	0.998760i	$-0.484149\pi$
$821$	2.85255	0.0995546	0.0497773	+	0.998760i	$0.484149\pi$
$822$	0	0
$823$	20.9692	0.730939	0.365470	−	0.930823i	$-0.380908\pi$
$823$	20.9692	0.730939	0.365470	+	0.930823i	$0.380908\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.48413	0.155929	0.0779643	−	0.996956i	$-0.475158\pi$
$827$	4.48413	0.155929	0.0779643	+	0.996956i	$0.475158\pi$
$828$	0	0
$829$	36.7239	1.27547	0.637737	−	0.770254i	$-0.279871\pi$
$829$	36.7239	1.27547	0.637737	+	0.770254i	$0.279871\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.3333	0.635210
$834$	0	0
$835$	−33.1121	−1.14589
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.1672	−0.972441	−0.486220	−	0.873836i	$-0.661625\pi$
$839$	−28.1672	−0.972441	−0.486220	+	0.873836i	$0.661625\pi$
$840$	0	0
$841$	44.8641	1.54704
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.37460	0.116090
$846$	0	0
$847$	25.3926	0.872501
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.9546	−0.444077
$852$	0	0
$853$	−21.2625	−0.728013	−0.364006	−	0.931396i	$-0.618591\pi$
$853$	−21.2625	−0.728013	−0.364006	+	0.931396i	$0.618591\pi$
$854$	0	0
$855$	78.0842	2.67042
$856$	0	0
$857$	−19.9182	−0.680394	−0.340197	−	0.940354i	$-0.610494\pi$
$857$	−19.9182	−0.680394	−0.340197	+	0.940354i	$0.610494\pi$
$858$	0	0
$859$	−8.79919	−0.300224	−0.150112	−	0.988669i	$-0.547963\pi$
$859$	−8.79919	−0.300224	−0.150112	+	0.988669i	$0.547963\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.4232	0.559054	0.279527	−	0.960138i	$-0.409822\pi$
$863$	16.4232	0.559054	0.279527	+	0.960138i	$0.409822\pi$
$864$	0	0
$865$	−58.6774	−1.99509
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.0814	0.681215
$870$	0	0
$871$	1.00000	0.0338837
$872$	0	0
$873$	8.29390	0.280706
$874$	0	0
$875$	6.59641	0.222999
$876$	0	0
$877$	39.5412	1.33521	0.667606	−	0.744515i	$-0.267319\pi$
$877$	39.5412	1.33521	0.667606	+	0.744515i	$0.267319\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.5868	−0.356677	−0.178339	−	0.983969i	$-0.557072\pi$
$881$	−10.5868	−0.356677	−0.178339	+	0.983969i	$0.557072\pi$
$882$	0	0
$883$	−33.9411	−1.14221	−0.571104	−	0.820878i	$-0.693485\pi$
$883$	−33.9411	−1.14221	−0.571104	+	0.820878i	$0.693485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.53664	0.219479	0.109740	−	0.993960i	$-0.464998\pi$
$887$	6.53664	0.219479	0.109740	+	0.993960i	$0.464998\pi$
$888$	0	0
$889$	23.4651	0.786993
$890$	0	0
$891$	48.4914	1.62452
$892$	0	0
$893$	−45.9625	−1.53808
$894$	0	0
$895$	70.8512	2.36829
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.3510	1.01226
$900$	0	0
$901$	41.6593	1.38787
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−64.0949	−2.13059
$906$	0	0
$907$	−2.61834	−0.0869407	−0.0434703	−	0.999055i	$-0.513841\pi$
$907$	−2.61834	−0.0869407	−0.0434703	+	0.999055i	$0.513841\pi$
$908$	0	0
$909$	43.2638	1.43497
$910$	0	0
$911$	16.6079	0.550244	0.275122	−	0.961409i	$-0.411282\pi$
$911$	16.6079	0.550244	0.275122	+	0.961409i	$0.411282\pi$
$912$	0	0
$913$	3.72214	0.123185
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.35074	0.110651
$918$	0	0
$919$	−53.9928	−1.78106	−0.890529	−	0.454927i	$-0.849665\pi$
$919$	−53.9928	−1.78106	−0.890529	+	0.454927i	$0.849665\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.5030	0.411541
$924$	0	0
$925$	72.8173	2.39422
$926$	0	0
$927$	−17.6308	−0.579070
$928$	0	0
$929$	−13.7581	−0.451388	−0.225694	−	0.974198i	$-0.572465\pi$
$929$	−13.7581	−0.451388	−0.225694	+	0.974198i	$0.572465\pi$
$930$	0	0
$931$	38.6920	1.26808
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−66.4482	−2.17309
$936$	0	0
$937$	−51.3464	−1.67741	−0.838707	−	0.544584i	$-0.816688\pi$
$937$	−51.3464	−1.67741	−0.838707	+	0.544584i	$0.816688\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.3808	−0.696995	−0.348498	−	0.937310i	$-0.613308\pi$
$941$	−21.3808	−0.696995	−0.348498	+	0.937310i	$0.613308\pi$
$942$	0	0
$943$	−9.45698	−0.307962
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.1856	1.69580	0.847902	−	0.530153i	$-0.177866\pi$
$947$	52.1856	1.69580	0.847902	+	0.530153i	$0.177866\pi$
$948$	0	0
$949$	−13.1530	−0.426963
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.06467	−0.164061	−0.0820305	−	0.996630i	$-0.526140\pi$
$953$	−5.06467	−0.164061	−0.0820305	+	0.996630i	$0.526140\pi$
$954$	0	0
$955$	4.64579	0.150334
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.0683	−0.357414
$960$	0	0
$961$	−18.5287	−0.597700
$962$	0	0
$963$	−15.3851	−0.495779
$964$	0	0
$965$	−57.7466	−1.85893
$966$	0	0
$967$	−15.4921	−0.498191	−0.249095	−	0.968479i	$-0.580133\pi$
$967$	−15.4921	−0.498191	−0.249095	+	0.968479i	$0.580133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.3184	0.427407	0.213704	−	0.976899i	$-0.431447\pi$
$971$	13.3184	0.427407	0.213704	+	0.976899i	$0.431447\pi$
$972$	0	0
$973$	−9.30806	−0.298403
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.6665	−0.757159	−0.378579	−	0.925569i	$-0.623587\pi$
$977$	−23.6665	−0.757159	−0.378579	+	0.925569i	$0.623587\pi$
$978$	0	0
$979$	−12.8076	−0.409332
$980$	0	0
$981$	41.9546	1.33951
$982$	0	0
$983$	4.62699	0.147578	0.0737891	−	0.997274i	$-0.476491\pi$
$983$	4.62699	0.147578	0.0737891	+	0.997274i	$0.476491\pi$
$984$	0	0
$985$	−58.7365	−1.87150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.67012	−0.243896
$990$	0	0
$991$	53.5242	1.70025	0.850126	−	0.526580i	$-0.176526\pi$
$991$	53.5242	1.70025	0.850126	+	0.526580i	$0.176526\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	48.0538	1.52341
$996$	0	0
$997$	−7.19705	−0.227933	−0.113966	−	0.993485i	$-0.536356\pi$
$997$	−7.19705	−0.227933	−0.113966	+	0.993485i	$0.536356\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3484.2.a.b.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3484.2.a.b.1.5	✓	5	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 3484 = 2^{2} \cdot 13 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3484.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37460 q^{5} +1.40836 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+3.37460 q^{5} +1.40836 q^{7} -3.00000 q^{9} +5.38794 q^{11} -1.00000 q^{13} -3.65459 q^{17} -7.71293 q^{19} -1.13645 q^{23} +6.38794 q^{25} +8.59442 q^{29} +3.53147 q^{31} +4.75267 q^{35} +11.3992 q^{37} +8.32153 q^{41} +6.74920 q^{43} -10.1238 q^{45} +5.95915 q^{47} -5.01651 q^{49} -11.3992 q^{53} +18.1821 q^{55} +10.7901 q^{59} +7.44003 q^{61} -4.22509 q^{63} -3.37460 q^{65} -1.00000 q^{67} -12.5030 q^{71} +13.1530 q^{73} +7.58818 q^{77} +3.72710 q^{79} +9.00000 q^{81} +0.690828 q^{83} -12.3328 q^{85} -2.37708 q^{89} -1.40836 q^{91} -26.0281 q^{95} -2.76463 q^{97} -16.1638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - 5 q^{7} - 15 q^{9}+O(q^{10}) \) 5 * q - 3 * q^5 - 5 * q^7 - 15 * q^9	\( 5 q - 3 q^{5} - 5 q^{7} - 15 q^{9} + 3 q^{11} - 5 q^{13} - 13 q^{17} - 8 q^{19} - 5 q^{23} + 8 q^{25} + 17 q^{29} + 9 q^{31} + 2 q^{35} + 4 q^{37} + 17 q^{41} - 6 q^{43} + 9 q^{45} + 6 q^{47} + 34 q^{49} - 4 q^{53} + 6 q^{55} + 38 q^{59} + 8 q^{61} + 15 q^{63} + 3 q^{65} - 5 q^{67} - 16 q^{71} + 6 q^{73} + 32 q^{77} + 20 q^{79} + 45 q^{81} + 14 q^{83} - 8 q^{85} + 5 q^{91} + 4 q^{95} + 5 q^{97} - 9 q^{99}+O(q^{100}) \) 5 * q - 3 * q^5 - 5 * q^7 - 15 * q^9 + 3 * q^11 - 5 * q^13 - 13 * q^17 - 8 * q^19 - 5 * q^23 + 8 * q^25 + 17 * q^29 + 9 * q^31 + 2 * q^35 + 4 * q^37 + 17 * q^41 - 6 * q^43 + 9 * q^45 + 6 * q^47 + 34 * q^49 - 4 * q^53 + 6 * q^55 + 38 * q^59 + 8 * q^61 + 15 * q^63 + 3 * q^65 - 5 * q^67 - 16 * q^71 + 6 * q^73 + 32 * q^77 + 20 * q^79 + 45 * q^81 + 14 * q^83 - 8 * q^85 + 5 * q^91 + 4 * q^95 + 5 * q^97 - 9 * q^99

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