LMFDB - Embedded newform 4048.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4048,2,Mod(1,4048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4048, 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("4048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4048 = 2^{4} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4048.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:	$32.3234427382$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.10722353.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 12x^{3} + 13x^{2} + 22x - 8 $ x^5 - x^4 - 12x^3 + 13x^2 + 22*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 506)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.67920$ of defining polynomial
Character	$\chi$	$=$	4048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67920 q^{3} -2.64728 q^{5} -1.46915 q^{7} +4.17813 q^{9} +O(q^{10})$	$q-2.67920 q^{3} -2.64728 q^{5} -1.46915 q^{7} +4.17813 q^{9} +1.00000 q^{11} +4.94425 q^{13} +7.09260 q^{15} -1.00809 q^{17} +2.67920 q^{19} +3.93615 q^{21} +1.00000 q^{23} +2.00809 q^{25} -3.15645 q^{27} -2.64728 q^{29} -7.82541 q^{31} -2.67920 q^{33} +3.88925 q^{35} -8.50461 q^{37} -13.2466 q^{39} -10.2329 q^{41} +2.89735 q^{43} -11.0607 q^{45} +3.42149 q^{47} -4.84159 q^{49} +2.70089 q^{51} +4.89735 q^{53} -2.64728 q^{55} -7.17813 q^{57} +11.4748 q^{59} +0.420104 q^{61} -6.13830 q^{63} -13.0888 q^{65} -14.0745 q^{67} -2.67920 q^{69} -10.2455 q^{71} -3.88849 q^{73} -5.38009 q^{75} -1.46915 q^{77} +6.35979 q^{79} -4.07762 q^{81} -0.536535 q^{83} +2.66870 q^{85} +7.09260 q^{87} +3.95234 q^{89} -7.26385 q^{91} +20.9659 q^{93} -7.09260 q^{95} +8.87446 q^{97} +4.17813 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 4 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) $ 5 * q - q^3 + 4 * q^5 - q^7 + 10 * q^9	$ 5 q - q^{3} + 4 q^{5} - q^{7} + 10 q^{9} + 5 q^{11} + 10 q^{13} + 2 q^{15} + q^{19} + 10 q^{21} + 5 q^{23} + 5 q^{25} + 8 q^{27} + 4 q^{29} - 11 q^{31} - q^{33} + q^{35} - 2 q^{37} + 28 q^{39} - 4 q^{41} - 9 q^{43} - 17 q^{45} - 9 q^{47} + 14 q^{49} + 14 q^{51} + q^{53} + 4 q^{55} - 25 q^{57} + 9 q^{59} - 10 q^{61} + 45 q^{63} - 9 q^{65} + 15 q^{67} - q^{69} - 11 q^{71} + 10 q^{73} - 15 q^{75} - q^{77} - 7 q^{79} + 17 q^{81} + 33 q^{83} - 31 q^{85} + 2 q^{87} + 17 q^{91} - 2 q^{93} - 2 q^{95} + 22 q^{97} + 10 q^{99}+O(q^{100}) $ 5 * q - q^3 + 4 * q^5 - q^7 + 10 * q^9 + 5 * q^11 + 10 * q^13 + 2 * q^15 + q^19 + 10 * q^21 + 5 * q^23 + 5 * q^25 + 8 * q^27 + 4 * q^29 - 11 * q^31 - q^33 + q^35 - 2 * q^37 + 28 * q^39 - 4 * q^41 - 9 * q^43 - 17 * q^45 - 9 * q^47 + 14 * q^49 + 14 * q^51 + q^53 + 4 * q^55 - 25 * q^57 + 9 * q^59 - 10 * q^61 + 45 * q^63 - 9 * q^65 + 15 * q^67 - q^69 - 11 * q^71 + 10 * q^73 - 15 * q^75 - q^77 - 7 * q^79 + 17 * q^81 + 33 * q^83 - 31 * q^85 + 2 * q^87 + 17 * q^91 - 2 * q^93 - 2 * q^95 + 22 * q^97 + 10 * 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
$3$	−2.67920	−1.54684	−0.773419	−	0.633895i	$-0.781455\pi$
$3$	−2.67920	−1.54684	−0.773419	+	0.633895i	$0.781455\pi$
$4$	0	0
$5$	−2.64728	−1.18390	−0.591950	−	0.805975i	$-0.701642\pi$
$5$	−2.64728	−1.18390	−0.591950	+	0.805975i	$0.701642\pi$
$6$	0	0
$7$	−1.46915	−0.555287	−0.277643	−	0.960684i	$-0.589553\pi$
$7$	−1.46915	−0.555287	−0.277643	+	0.960684i	$0.589553\pi$
$8$	0	0
$9$	4.17813	1.39271
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.94425	1.37129	0.685644	−	0.727937i	$-0.259521\pi$
$13$	4.94425	1.37129	0.685644	+	0.727937i	$0.259521\pi$
$14$	0	0
$15$	7.09260	1.83130
$16$	0	0
$17$	−1.00809	−0.244498	−0.122249	−	0.992499i	$-0.539011\pi$
$17$	−1.00809	−0.244498	−0.122249	+	0.992499i	$0.539011\pi$
$18$	0	0
$19$	2.67920	0.614651	0.307326	−	0.951604i	$-0.400566\pi$
$19$	2.67920	0.614651	0.307326	+	0.951604i	$0.400566\pi$
$20$	0	0
$21$	3.93615	0.858939
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.00809	0.401619
$26$	0	0
$27$	−3.15645	−0.607459
$28$	0	0
$29$	−2.64728	−0.491588	−0.245794	−	0.969322i	$-0.579049\pi$
$29$	−2.64728	−0.491588	−0.245794	+	0.969322i	$0.579049\pi$
$30$	0	0
$31$	−7.82541	−1.40549	−0.702743	−	0.711444i	$-0.748041\pi$
$31$	−7.82541	−1.40549	−0.702743	+	0.711444i	$0.748041\pi$
$32$	0	0
$33$	−2.67920	−0.466389
$34$	0	0
$35$	3.88925	0.657404
$36$	0	0
$37$	−8.50461	−1.39815	−0.699075	−	0.715049i	$-0.746405\pi$
$37$	−8.50461	−1.39815	−0.699075	+	0.715049i	$0.746405\pi$
$38$	0	0
$39$	−13.2466	−2.12116
$40$	0	0
$41$	−10.2329	−1.59810	−0.799052	−	0.601261i	$-0.794665\pi$
$41$	−10.2329	−1.59810	−0.799052	+	0.601261i	$0.794665\pi$
$42$	0	0
$43$	2.89735	0.441841	0.220921	−	0.975292i	$-0.429094\pi$
$43$	2.89735	0.441841	0.220921	+	0.975292i	$0.429094\pi$
$44$	0	0
$45$	−11.0607	−1.64883
$46$	0	0
$47$	3.42149	0.499076	0.249538	−	0.968365i	$-0.419721\pi$
$47$	3.42149	0.499076	0.249538	+	0.968365i	$0.419721\pi$
$48$	0	0
$49$	−4.84159	−0.691656
$50$	0	0
$51$	2.70089	0.378200
$52$	0	0
$53$	4.89735	0.672702	0.336351	−	0.941737i	$-0.390807\pi$
$53$	4.89735	0.672702	0.336351	+	0.941737i	$0.390807\pi$
$54$	0	0
$55$	−2.64728	−0.356959
$56$	0	0
$57$	−7.17813	−0.950766
$58$	0	0
$59$	11.4748	1.49390	0.746948	−	0.664882i	$-0.231518\pi$
$59$	11.4748	1.49390	0.746948	+	0.664882i	$0.231518\pi$
$60$	0	0
$61$	0.420104	0.0537888	0.0268944	−	0.999638i	$-0.491438\pi$
$61$	0.420104	0.0537888	0.0268944	+	0.999638i	$0.491438\pi$
$62$	0	0
$63$	−6.13830	−0.773354
$64$	0	0
$65$	−13.0888	−1.62347
$66$	0	0
$67$	−14.0745	−1.71947	−0.859734	−	0.510741i	$-0.829371\pi$
$67$	−14.0745	−1.71947	−0.859734	+	0.510741i	$0.829371\pi$
$68$	0	0
$69$	−2.67920	−0.322538
$70$	0	0
$71$	−10.2455	−1.21592	−0.607959	−	0.793968i	$-0.708012\pi$
$71$	−10.2455	−1.21592	−0.607959	+	0.793968i	$0.708012\pi$
$72$	0	0
$73$	−3.88849	−0.455114	−0.227557	−	0.973765i	$-0.573074\pi$
$73$	−3.88849	−0.455114	−0.227557	+	0.973765i	$0.573074\pi$
$74$	0	0
$75$	−5.38009	−0.621239
$76$	0	0
$77$	−1.46915	−0.167425
$78$	0	0
$79$	6.35979	0.715533	0.357766	−	0.933811i	$-0.383538\pi$
$79$	6.35979	0.715533	0.357766	+	0.933811i	$0.383538\pi$
$80$	0	0
$81$	−4.07762	−0.453069
$82$	0	0
$83$	−0.536535	−0.0588924	−0.0294462	−	0.999566i	$-0.509374\pi$
$83$	−0.536535	−0.0588924	−0.0294462	+	0.999566i	$0.509374\pi$
$84$	0	0
$85$	2.66870	0.289462
$86$	0	0
$87$	7.09260	0.760407
$88$	0	0
$89$	3.95234	0.418947	0.209474	−	0.977814i	$-0.432825\pi$
$89$	3.95234	0.418947	0.209474	+	0.977814i	$0.432825\pi$
$90$	0	0
$91$	−7.26385	−0.761458
$92$	0	0
$93$	20.9659	2.17406
$94$	0	0
$95$	−7.09260	−0.727686
$96$	0	0
$97$	8.87446	0.901065	0.450532	−	0.892760i	$-0.351234\pi$
$97$	8.87446	0.901065	0.450532	+	0.892760i	$0.351234\pi$
$98$	0	0
$99$	4.17813	0.419918
$100$	0	0
$101$	12.8013	1.27378	0.636890	−	0.770955i	$-0.280221\pi$
$101$	12.8013	1.27378	0.636890	+	0.770955i	$0.280221\pi$
$102$	0	0
$103$	−13.3633	−1.31673	−0.658364	−	0.752700i	$-0.728751\pi$
$103$	−13.3633	−1.31673	−0.658364	+	0.752700i	$0.728751\pi$
$104$	0	0
$105$	−10.4201	−1.01690
$106$	0	0
$107$	10.6792	1.03240	0.516199	−	0.856469i	$-0.327347\pi$
$107$	10.6792	1.03240	0.516199	+	0.856469i	$0.327347\pi$
$108$	0	0
$109$	7.29456	0.698692	0.349346	−	0.936994i	$-0.386404\pi$
$109$	7.29456	0.698692	0.349346	+	0.936994i	$0.386404\pi$
$110$	0	0
$111$	22.7856	2.16271
$112$	0	0
$113$	5.40607	0.508560	0.254280	−	0.967131i	$-0.418162\pi$
$113$	5.40607	0.508560	0.254280	+	0.967131i	$0.418162\pi$
$114$	0	0
$115$	−2.64728	−0.246860
$116$	0	0
$117$	20.6577	1.90981
$118$	0	0
$119$	1.48104	0.135767
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	27.4159	2.47201
$124$	0	0
$125$	7.92042	0.708424
$126$	0	0
$127$	−12.5677	−1.11520	−0.557601	−	0.830109i	$-0.688278\pi$
$127$	−12.5677	−1.11520	−0.557601	+	0.830109i	$0.688278\pi$
$128$	0	0
$129$	−7.76258	−0.683457
$130$	0	0
$131$	5.46915	0.477842	0.238921	−	0.971039i	$-0.423206\pi$
$131$	5.46915	0.477842	0.238921	+	0.971039i	$0.423206\pi$
$132$	0	0
$133$	−3.93615	−0.341308
$134$	0	0
$135$	8.35600	0.719170
$136$	0	0
$137$	15.5274	1.32660	0.663299	−	0.748355i	$-0.269156\pi$
$137$	15.5274	1.32660	0.663299	+	0.748355i	$0.269156\pi$
$138$	0	0
$139$	5.59127	0.474245	0.237123	−	0.971480i	$-0.423796\pi$
$139$	5.59127	0.474245	0.237123	+	0.971480i	$0.423796\pi$
$140$	0	0
$141$	−9.16687	−0.771990
$142$	0	0
$143$	4.94425	0.413459
$144$	0	0
$145$	7.00809	0.581990
$146$	0	0
$147$	12.9716	1.06988
$148$	0	0
$149$	−19.5913	−1.60498	−0.802490	−	0.596666i	$-0.796492\pi$
$149$	−19.5913	−1.60498	−0.802490	+	0.596666i	$0.796492\pi$
$150$	0	0
$151$	15.6337	1.27225	0.636126	−	0.771585i	$-0.280536\pi$
$151$	15.6337	1.27225	0.636126	+	0.771585i	$0.280536\pi$
$152$	0	0
$153$	−4.21194	−0.340515
$154$	0	0
$155$	20.7161	1.66395
$156$	0	0
$157$	−20.2664	−1.61743	−0.808716	−	0.588199i	$-0.799837\pi$
$157$	−20.2664	−1.61743	−0.808716	+	0.588199i	$0.799837\pi$
$158$	0	0
$159$	−13.1210	−1.04056
$160$	0	0
$161$	−1.46915	−0.115785
$162$	0	0
$163$	8.40884	0.658631	0.329316	−	0.944220i	$-0.393182\pi$
$163$	8.40884	0.658631	0.329316	+	0.944220i	$0.393182\pi$
$164$	0	0
$165$	7.09260	0.552158
$166$	0	0
$167$	−4.29026	−0.331990	−0.165995	−	0.986127i	$-0.553084\pi$
$167$	−4.29026	−0.331990	−0.165995	+	0.986127i	$0.553084\pi$
$168$	0	0
$169$	11.4456	0.880429
$170$	0	0
$171$	11.1941	0.856031
$172$	0	0
$173$	18.6463	1.41765	0.708824	−	0.705385i	$-0.249226\pi$
$173$	18.6463	1.41765	0.708824	+	0.705385i	$0.249226\pi$
$174$	0	0
$175$	−2.95019	−0.223014
$176$	0	0
$177$	−30.7434	−2.31082
$178$	0	0
$179$	−14.6624	−1.09592	−0.547961	−	0.836504i	$-0.684596\pi$
$179$	−14.6624	−1.09592	−0.547961	+	0.836504i	$0.684596\pi$
$180$	0	0
$181$	−0.769656	−0.0572081	−0.0286041	−	0.999591i	$-0.509106\pi$
$181$	−0.769656	−0.0572081	−0.0286041	+	0.999591i	$0.509106\pi$
$182$	0	0
$183$	−1.12554	−0.0832026
$184$	0	0
$185$	22.5141	1.65527
$186$	0	0
$187$	−1.00809	−0.0737190
$188$	0	0
$189$	4.63730	0.337314
$190$	0	0
$191$	1.75709	0.127138	0.0635691	−	0.997977i	$-0.479752\pi$
$191$	1.75709	0.127138	0.0635691	+	0.997977i	$0.479752\pi$
$192$	0	0
$193$	−5.31075	−0.382276	−0.191138	−	0.981563i	$-0.561218\pi$
$193$	−5.31075	−0.382276	−0.191138	+	0.981563i	$0.561218\pi$
$194$	0	0
$195$	35.0676	2.51124
$196$	0	0
$197$	−19.9261	−1.41968	−0.709838	−	0.704365i	$-0.751232\pi$
$197$	−19.9261	−1.41968	−0.709838	+	0.704365i	$0.751232\pi$
$198$	0	0
$199$	−13.0859	−0.927635	−0.463817	−	0.885931i	$-0.653521\pi$
$199$	−13.0859	−0.927635	−0.463817	+	0.885931i	$0.653521\pi$
$200$	0	0
$201$	37.7083	2.65974
$202$	0	0
$203$	3.88925	0.272972
$204$	0	0
$205$	27.0893	1.89200
$206$	0	0
$207$	4.17813	0.290400
$208$	0	0
$209$	2.67920	0.185324
$210$	0	0
$211$	−13.9054	−0.957290	−0.478645	−	0.878009i	$-0.658872\pi$
$211$	−13.9054	−0.957290	−0.478645	+	0.878009i	$0.658872\pi$
$212$	0	0
$213$	27.4498	1.88083
$214$	0	0
$215$	−7.67009	−0.523096
$216$	0	0
$217$	11.4967	0.780447
$218$	0	0
$219$	10.4181	0.703988
$220$	0	0
$221$	−4.98426	−0.335278
$222$	0	0
$223$	−15.9475	−1.06793	−0.533963	−	0.845508i	$-0.679298\pi$
$223$	−15.9475	−1.06793	−0.533963	+	0.845508i	$0.679298\pi$
$224$	0	0
$225$	8.39007	0.559338
$226$	0	0
$227$	24.1874	1.60537	0.802686	−	0.596403i	$-0.203404\pi$
$227$	24.1874	1.60537	0.802686	+	0.596403i	$0.203404\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	3.93615	0.258980
$232$	0	0
$233$	22.8696	1.49824	0.749120	−	0.662434i	$-0.230477\pi$
$233$	22.8696	1.49824	0.749120	+	0.662434i	$0.230477\pi$
$234$	0	0
$235$	−9.05765	−0.590856
$236$	0	0
$237$	−17.0392	−1.10681
$238$	0	0
$239$	18.5662	1.20095	0.600475	−	0.799644i	$-0.294978\pi$
$239$	18.5662	1.20095	0.600475	+	0.799644i	$0.294978\pi$
$240$	0	0
$241$	−13.1624	−0.847864	−0.423932	−	0.905694i	$-0.639350\pi$
$241$	−13.1624	−0.847864	−0.423932	+	0.905694i	$0.639350\pi$
$242$	0	0
$243$	20.3941	1.30828
$244$	0	0
$245$	12.8171	0.818852
$246$	0	0
$247$	13.2466	0.842864
$248$	0	0
$249$	1.43749	0.0910970
$250$	0	0
$251$	−21.4202	−1.35203	−0.676016	−	0.736887i	$-0.736295\pi$
$251$	−21.4202	−1.35203	−0.676016	+	0.736887i	$0.736295\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	−7.15000	−0.447750
$256$	0	0
$257$	30.7982	1.92114	0.960571	−	0.278036i	$-0.0896836\pi$
$257$	30.7982	1.92114	0.960571	+	0.278036i	$0.0896836\pi$
$258$	0	0
$259$	12.4946	0.776374
$260$	0	0
$261$	−11.0607	−0.684639
$262$	0	0
$263$	−19.9645	−1.23106	−0.615531	−	0.788113i	$-0.711058\pi$
$263$	−19.9645	−1.23106	−0.615531	+	0.788113i	$0.711058\pi$
$264$	0	0
$265$	−12.9647	−0.796412
$266$	0	0
$267$	−10.5891	−0.648044
$268$	0	0
$269$	19.6670	1.19912	0.599559	−	0.800330i	$-0.295343\pi$
$269$	19.6670	1.19912	0.599559	+	0.800330i	$0.295343\pi$
$270$	0	0
$271$	10.3676	0.629789	0.314894	−	0.949127i	$-0.398031\pi$
$271$	10.3676	0.629789	0.314894	+	0.949127i	$0.398031\pi$
$272$	0	0
$273$	19.4613	1.17785
$274$	0	0
$275$	2.00809	0.121093
$276$	0	0
$277$	15.3491	0.922239	0.461119	−	0.887338i	$-0.347448\pi$
$277$	15.3491	0.922239	0.461119	+	0.887338i	$0.347448\pi$
$278$	0	0
$279$	−32.6956	−1.95743
$280$	0	0
$281$	−23.2876	−1.38922	−0.694611	−	0.719386i	$-0.744423\pi$
$281$	−23.2876	−1.38922	−0.694611	+	0.719386i	$0.744423\pi$
$282$	0	0
$283$	−14.3676	−0.854067	−0.427034	−	0.904236i	$-0.640441\pi$
$283$	−14.3676	−0.854067	−0.427034	+	0.904236i	$0.640441\pi$
$284$	0	0
$285$	19.0025	1.12561
$286$	0	0
$287$	15.0336	0.887407
$288$	0	0
$289$	−15.9837	−0.940221
$290$	0	0
$291$	−23.7765	−1.39380
$292$	0	0
$293$	15.2008	0.888038	0.444019	−	0.896017i	$-0.353552\pi$
$293$	15.2008	0.888038	0.444019	+	0.896017i	$0.353552\pi$
$294$	0	0
$295$	−30.3771	−1.76862
$296$	0	0
$297$	−3.15645	−0.183156
$298$	0	0
$299$	4.94425	0.285933
$300$	0	0
$301$	−4.25664	−0.245349
$302$	0	0
$303$	−34.2973	−1.97033
$304$	0	0
$305$	−1.11213	−0.0636805
$306$	0	0
$307$	7.37106	0.420688	0.210344	−	0.977627i	$-0.432542\pi$
$307$	7.37106	0.420688	0.210344	+	0.977627i	$0.432542\pi$
$308$	0	0
$309$	35.8031	2.03677
$310$	0	0
$311$	−5.19292	−0.294464	−0.147232	−	0.989102i	$-0.547036\pi$
$311$	−5.19292	−0.294464	−0.147232	+	0.989102i	$0.547036\pi$
$312$	0	0
$313$	3.42225	0.193437	0.0967186	−	0.995312i	$-0.469165\pi$
$313$	3.42225	0.193437	0.0967186	+	0.995312i	$0.469165\pi$
$314$	0	0
$315$	16.2498	0.915573
$316$	0	0
$317$	−11.7813	−0.661703	−0.330851	−	0.943683i	$-0.607336\pi$
$317$	−11.7813	−0.661703	−0.330851	+	0.943683i	$0.607336\pi$
$318$	0	0
$319$	−2.64728	−0.148219
$320$	0	0
$321$	−28.6118	−1.59695
$322$	0	0
$323$	−2.70089	−0.150281
$324$	0	0
$325$	9.92851	0.550734
$326$	0	0
$327$	−19.5436	−1.08076
$328$	0	0
$329$	−5.02669	−0.277130
$330$	0	0
$331$	21.2650	1.16883	0.584414	−	0.811455i	$-0.301324\pi$
$331$	21.2650	1.16883	0.584414	+	0.811455i	$0.301324\pi$
$332$	0	0
$333$	−35.5334	−1.94722
$334$	0	0
$335$	37.2590	2.03568
$336$	0	0
$337$	8.45771	0.460721	0.230360	−	0.973105i	$-0.426009\pi$
$337$	8.45771	0.460721	0.230360	+	0.973105i	$0.426009\pi$
$338$	0	0
$339$	−14.4839	−0.786660
$340$	0	0
$341$	−7.82541	−0.423770
$342$	0	0
$343$	17.3971	0.939355
$344$	0	0
$345$	7.09260	0.381853
$346$	0	0
$347$	21.5444	1.15656	0.578281	−	0.815838i	$-0.303724\pi$
$347$	21.5444	1.15656	0.578281	+	0.815838i	$0.303724\pi$
$348$	0	0
$349$	3.22932	0.172862	0.0864309	−	0.996258i	$-0.472454\pi$
$349$	3.22932	0.172862	0.0864309	+	0.996258i	$0.472454\pi$
$350$	0	0
$351$	−15.6063	−0.833000
$352$	0	0
$353$	27.3623	1.45635	0.728174	−	0.685392i	$-0.240369\pi$
$353$	27.3623	1.45635	0.728174	+	0.685392i	$0.240369\pi$
$354$	0	0
$355$	27.1227	1.43953
$356$	0	0
$357$	−3.96801	−0.210009
$358$	0	0
$359$	30.6180	1.61595	0.807977	−	0.589214i	$-0.200563\pi$
$359$	30.6180	1.61595	0.807977	+	0.589214i	$0.200563\pi$
$360$	0	0
$361$	−11.8219	−0.622204
$362$	0	0
$363$	−2.67920	−0.140622
$364$	0	0
$365$	10.2939	0.538809
$366$	0	0
$367$	−5.72472	−0.298828	−0.149414	−	0.988775i	$-0.547739\pi$
$367$	−5.72472	−0.298828	−0.149414	+	0.988775i	$0.547739\pi$
$368$	0	0
$369$	−42.7542	−2.22570
$370$	0	0
$371$	−7.19494	−0.373543
$372$	0	0
$373$	27.8045	1.43966	0.719832	−	0.694148i	$-0.244219\pi$
$373$	27.8045	1.43966	0.719832	+	0.694148i	$0.244219\pi$
$374$	0	0
$375$	−21.2204	−1.09582
$376$	0	0
$377$	−13.0888	−0.674108
$378$	0	0
$379$	17.3409	0.890744	0.445372	−	0.895346i	$-0.353071\pi$
$379$	17.3409	0.890744	0.445372	+	0.895346i	$0.353071\pi$
$380$	0	0
$381$	33.6714	1.72504
$382$	0	0
$383$	25.5769	1.30692	0.653460	−	0.756961i	$-0.273317\pi$
$383$	25.5769	1.30692	0.653460	+	0.756961i	$0.273317\pi$
$384$	0	0
$385$	3.88925	0.198215
$386$	0	0
$387$	12.1055	0.615357
$388$	0	0
$389$	−16.3736	−0.830173	−0.415087	−	0.909782i	$-0.636249\pi$
$389$	−16.3736	−0.830173	−0.415087	+	0.909782i	$0.636249\pi$
$390$	0	0
$391$	−1.00809	−0.0509814
$392$	0	0
$393$	−14.6530	−0.739144
$394$	0	0
$395$	−16.8362	−0.847119
$396$	0	0
$397$	−16.3248	−0.819317	−0.409659	−	0.912239i	$-0.634352\pi$
$397$	−16.3248	−0.819317	−0.409659	+	0.912239i	$0.634352\pi$
$398$	0	0
$399$	10.5458	0.527948
$400$	0	0
$401$	−10.1824	−0.508486	−0.254243	−	0.967140i	$-0.581826\pi$
$401$	−10.1824	−0.508486	−0.254243	+	0.967140i	$0.581826\pi$
$402$	0	0
$403$	−38.6908	−1.92732
$404$	0	0
$405$	10.7946	0.536389
$406$	0	0
$407$	−8.50461	−0.421558
$408$	0	0
$409$	37.1944	1.83915	0.919573	−	0.392919i	$-0.128535\pi$
$409$	37.1944	1.83915	0.919573	+	0.392919i	$0.128535\pi$
$410$	0	0
$411$	−41.6011	−2.05203
$412$	0	0
$413$	−16.8583	−0.829541
$414$	0	0
$415$	1.42036	0.0697227
$416$	0	0
$417$	−14.9801	−0.733581
$418$	0	0
$419$	−3.52870	−0.172388	−0.0861942	−	0.996278i	$-0.527471\pi$
$419$	−3.52870	−0.172388	−0.0861942	+	0.996278i	$0.527471\pi$
$420$	0	0
$421$	23.3712	1.13904	0.569521	−	0.821977i	$-0.307129\pi$
$421$	23.3712	1.13904	0.569521	+	0.821977i	$0.307129\pi$
$422$	0	0
$423$	14.2954	0.695068
$424$	0	0
$425$	−2.02434	−0.0981951
$426$	0	0
$427$	−0.617196	−0.0298682
$428$	0	0
$429$	−13.2466	−0.639554
$430$	0	0
$431$	9.29103	0.447533	0.223766	−	0.974643i	$-0.428165\pi$
$431$	9.29103	0.447533	0.223766	+	0.974643i	$0.428165\pi$
$432$	0	0
$433$	15.0617	0.723819	0.361909	−	0.932213i	$-0.382125\pi$
$433$	15.0617	0.723819	0.361909	+	0.932213i	$0.382125\pi$
$434$	0	0
$435$	−18.7761	−0.900245
$436$	0	0
$437$	2.67920	0.128164
$438$	0	0
$439$	−36.1755	−1.72656	−0.863280	−	0.504725i	$-0.831594\pi$
$439$	−36.1755	−1.72656	−0.863280	+	0.504725i	$0.831594\pi$
$440$	0	0
$441$	−20.2288	−0.963277
$442$	0	0
$443$	−17.4819	−0.830591	−0.415295	−	0.909687i	$-0.636322\pi$
$443$	−17.4819	−0.830591	−0.415295	+	0.909687i	$0.636322\pi$
$444$	0	0
$445$	−10.4630	−0.495991
$446$	0	0
$447$	52.4890	2.48264
$448$	0	0
$449$	−39.1808	−1.84906	−0.924529	−	0.381111i	$-0.875541\pi$
$449$	−39.1808	−1.84906	−0.924529	+	0.381111i	$0.875541\pi$
$450$	0	0
$451$	−10.2329	−0.481847
$452$	0	0
$453$	−41.8858	−1.96797
$454$	0	0
$455$	19.2294	0.901490
$456$	0	0
$457$	33.1880	1.55247	0.776234	−	0.630445i	$-0.217127\pi$
$457$	33.1880	1.55247	0.776234	+	0.630445i	$0.217127\pi$
$458$	0	0
$459$	3.18199	0.148523
$460$	0	0
$461$	−15.7095	−0.731663	−0.365832	−	0.930681i	$-0.619215\pi$
$461$	−15.7095	−0.731663	−0.365832	+	0.930681i	$0.619215\pi$
$462$	0	0
$463$	21.0128	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$463$	21.0128	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$464$	0	0
$465$	−55.5025	−2.57387
$466$	0	0
$467$	23.7182	1.09755	0.548774	−	0.835971i	$-0.315095\pi$
$467$	23.7182	1.09755	0.548774	+	0.835971i	$0.315095\pi$
$468$	0	0
$469$	20.6775	0.954799
$470$	0	0
$471$	54.2977	2.50191
$472$	0	0
$473$	2.89735	0.133220
$474$	0	0
$475$	5.38009	0.246855
$476$	0	0
$477$	20.4618	0.936879
$478$	0	0
$479$	−27.4757	−1.25540	−0.627699	−	0.778456i	$-0.716003\pi$
$479$	−27.4757	−1.25540	−0.627699	+	0.778456i	$0.716003\pi$
$480$	0	0
$481$	−42.0489	−1.91727
$482$	0	0
$483$	3.93615	0.179101
$484$	0	0
$485$	−23.4932	−1.06677
$486$	0	0
$487$	−16.4657	−0.746133	−0.373067	−	0.927805i	$-0.621694\pi$
$487$	−16.4657	−0.746133	−0.373067	+	0.927805i	$0.621694\pi$
$488$	0	0
$489$	−22.5290	−1.01880
$490$	0	0
$491$	10.2341	0.461860	0.230930	−	0.972970i	$-0.425823\pi$
$491$	10.2341	0.461860	0.230930	+	0.972970i	$0.425823\pi$
$492$	0	0
$493$	2.66870	0.120192
$494$	0	0
$495$	−11.0607	−0.497141
$496$	0	0
$497$	15.0522	0.675184
$498$	0	0
$499$	6.03818	0.270306	0.135153	−	0.990825i	$-0.456847\pi$
$499$	6.03818	0.270306	0.135153	+	0.990825i	$0.456847\pi$
$500$	0	0
$501$	11.4945	0.513536
$502$	0	0
$503$	37.1349	1.65576	0.827881	−	0.560904i	$-0.189546\pi$
$503$	37.1349	1.65576	0.827881	+	0.560904i	$0.189546\pi$
$504$	0	0
$505$	−33.8887	−1.50803
$506$	0	0
$507$	−30.6650	−1.36188
$508$	0	0
$509$	12.2377	0.542425	0.271213	−	0.962519i	$-0.412575\pi$
$509$	12.2377	0.542425	0.271213	+	0.962519i	$0.412575\pi$
$510$	0	0
$511$	5.71279	0.252719
$512$	0	0
$513$	−8.45676	−0.373375
$514$	0	0
$515$	35.3765	1.55887
$516$	0	0
$517$	3.42149	0.150477
$518$	0	0
$519$	−49.9571	−2.19287
$520$	0	0
$521$	3.71896	0.162931	0.0814653	−	0.996676i	$-0.474040\pi$
$521$	3.71896	0.162931	0.0814653	+	0.996676i	$0.474040\pi$
$522$	0	0
$523$	41.9476	1.83424	0.917121	−	0.398609i	$-0.130507\pi$
$523$	41.9476	1.83424	0.917121	+	0.398609i	$0.130507\pi$
$524$	0	0
$525$	7.90416	0.344966
$526$	0	0
$527$	7.88874	0.343639
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	47.9434	2.08056
$532$	0	0
$533$	−50.5938	−2.19146
$534$	0	0
$535$	−28.2708	−1.22226
$536$	0	0
$537$	39.2837	1.69522
$538$	0	0
$539$	−4.84159	−0.208542
$540$	0	0
$541$	14.7799	0.635437	0.317719	−	0.948185i	$-0.397083\pi$
$541$	14.7799	0.635437	0.317719	+	0.948185i	$0.397083\pi$
$542$	0	0
$543$	2.06207	0.0884917
$544$	0	0
$545$	−19.3107	−0.827181
$546$	0	0
$547$	−10.9368	−0.467623	−0.233811	−	0.972282i	$-0.575120\pi$
$547$	−10.9368	−0.467623	−0.233811	+	0.972282i	$0.575120\pi$
$548$	0	0
$549$	1.75525	0.0749121
$550$	0	0
$551$	−7.09260	−0.302155
$552$	0	0
$553$	−9.34350	−0.397326
$554$	0	0
$555$	−60.3198	−2.56043
$556$	0	0
$557$	−19.4223	−0.822947	−0.411474	−	0.911422i	$-0.634986\pi$
$557$	−19.4223	−0.822947	−0.411474	+	0.911422i	$0.634986\pi$
$558$	0	0
$559$	14.3252	0.605892
$560$	0	0
$561$	2.70089	0.114031
$562$	0	0
$563$	−2.75442	−0.116085	−0.0580425	−	0.998314i	$-0.518486\pi$
$563$	−2.75442	−0.116085	−0.0580425	+	0.998314i	$0.518486\pi$
$564$	0	0
$565$	−14.3114	−0.602084
$566$	0	0
$567$	5.99065	0.251584
$568$	0	0
$569$	11.4432	0.479725	0.239863	−	0.970807i	$-0.422898\pi$
$569$	11.4432	0.479725	0.239863	+	0.970807i	$0.422898\pi$
$570$	0	0
$571$	26.7398	1.11903	0.559513	−	0.828822i	$-0.310988\pi$
$571$	26.7398	1.11903	0.559513	+	0.828822i	$0.310988\pi$
$572$	0	0
$573$	−4.70759	−0.196662
$574$	0	0
$575$	2.00809	0.0837433
$576$	0	0
$577$	4.54256	0.189109	0.0945546	−	0.995520i	$-0.469857\pi$
$577$	4.54256	0.189109	0.0945546	+	0.995520i	$0.469857\pi$
$578$	0	0
$579$	14.2286	0.591319
$580$	0	0
$581$	0.788251	0.0327022
$582$	0	0
$583$	4.89735	0.202827
$584$	0	0
$585$	−54.6867	−2.26102
$586$	0	0
$587$	15.7461	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$587$	15.7461	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$588$	0	0
$589$	−20.9659	−0.863883
$590$	0	0
$591$	53.3861	2.19601
$592$	0	0
$593$	20.2609	0.832017	0.416008	−	0.909361i	$-0.363429\pi$
$593$	20.2609	0.832017	0.416008	+	0.909361i	$0.363429\pi$
$594$	0	0
$595$	−3.92073	−0.160734
$596$	0	0
$597$	35.0598	1.43490
$598$	0	0
$599$	−37.6953	−1.54019	−0.770095	−	0.637930i	$-0.779791\pi$
$599$	−37.6953	−1.54019	−0.770095	+	0.637930i	$0.779791\pi$
$600$	0	0
$601$	−15.4321	−0.629489	−0.314744	−	0.949176i	$-0.601919\pi$
$601$	−15.4321	−0.629489	−0.314744	+	0.949176i	$0.601919\pi$
$602$	0	0
$603$	−58.8049	−2.39472
$604$	0	0
$605$	−2.64728	−0.107627
$606$	0	0
$607$	41.4307	1.68162	0.840809	−	0.541332i	$-0.182080\pi$
$607$	41.4307	1.68162	0.840809	+	0.541332i	$0.182080\pi$
$608$	0	0
$609$	−10.4201	−0.422244
$610$	0	0
$611$	16.9167	0.684376
$612$	0	0
$613$	22.8177	0.921598	0.460799	−	0.887505i	$-0.347563\pi$
$613$	22.8177	0.921598	0.460799	+	0.887505i	$0.347563\pi$
$614$	0	0
$615$	−72.5776	−2.92661
$616$	0	0
$617$	7.86750	0.316733	0.158367	−	0.987380i	$-0.449377\pi$
$617$	7.86750	0.316733	0.158367	+	0.987380i	$0.449377\pi$
$618$	0	0
$619$	−36.7732	−1.47804	−0.739020	−	0.673684i	$-0.764711\pi$
$619$	−36.7732	−1.47804	−0.739020	+	0.673684i	$0.764711\pi$
$620$	0	0
$621$	−3.15645	−0.126664
$622$	0	0
$623$	−5.80658	−0.232636
$624$	0	0
$625$	−31.0080	−1.24032
$626$	0	0
$627$	−7.17813	−0.286667
$628$	0	0
$629$	8.57344	0.341845
$630$	0	0
$631$	−28.4752	−1.13358	−0.566790	−	0.823862i	$-0.691815\pi$
$631$	−28.4752	−1.13358	−0.566790	+	0.823862i	$0.691815\pi$
$632$	0	0
$633$	37.2555	1.48077
$634$	0	0
$635$	33.2702	1.32029
$636$	0	0
$637$	−23.9380	−0.948460
$638$	0	0
$639$	−42.8071	−1.69342
$640$	0	0
$641$	33.0893	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$641$	33.0893	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$642$	0	0
$643$	35.0214	1.38111	0.690554	−	0.723281i	$-0.257367\pi$
$643$	35.0214	1.38111	0.690554	+	0.723281i	$0.257367\pi$
$644$	0	0
$645$	20.5497	0.809145
$646$	0	0
$647$	34.7183	1.36492	0.682459	−	0.730924i	$-0.260911\pi$
$647$	34.7183	1.36492	0.682459	+	0.730924i	$0.260911\pi$
$648$	0	0
$649$	11.4748	0.450427
$650$	0	0
$651$	−30.8020	−1.20723
$652$	0	0
$653$	−30.7542	−1.20351	−0.601753	−	0.798682i	$-0.705531\pi$
$653$	−30.7542	−1.20351	−0.601753	+	0.798682i	$0.705531\pi$
$654$	0	0
$655$	−14.4784	−0.565717
$656$	0	0
$657$	−16.2466	−0.633841
$658$	0	0
$659$	−12.6627	−0.493269	−0.246634	−	0.969109i	$-0.579325\pi$
$659$	−12.6627	−0.493269	−0.246634	+	0.969109i	$0.579325\pi$
$660$	0	0
$661$	−24.0616	−0.935889	−0.467945	−	0.883758i	$-0.655005\pi$
$661$	−24.0616	−0.935889	−0.467945	+	0.883758i	$0.655005\pi$
$662$	0	0
$663$	13.3538	0.518620
$664$	0	0
$665$	10.4201	0.404074
$666$	0	0
$667$	−2.64728	−0.102503
$668$	0	0
$669$	42.7267	1.65191
$670$	0	0
$671$	0.420104	0.0162179
$672$	0	0
$673$	−3.28319	−0.126558	−0.0632788	−	0.997996i	$-0.520156\pi$
$673$	−3.28319	−0.126558	−0.0632788	+	0.997996i	$0.520156\pi$
$674$	0	0
$675$	−6.33844	−0.243967
$676$	0	0
$677$	42.7857	1.64439	0.822194	−	0.569207i	$-0.192750\pi$
$677$	42.7857	1.64439	0.822194	+	0.569207i	$0.192750\pi$
$678$	0	0
$679$	−13.0379	−0.500349
$680$	0	0
$681$	−64.8028	−2.48325
$682$	0	0
$683$	4.67491	0.178880	0.0894402	−	0.995992i	$-0.471492\pi$
$683$	4.67491	0.178880	0.0894402	+	0.995992i	$0.471492\pi$
$684$	0	0
$685$	−41.1054	−1.57056
$686$	0	0
$687$	−37.5088	−1.43105
$688$	0	0
$689$	24.2137	0.922468
$690$	0	0
$691$	12.2319	0.465324	0.232662	−	0.972558i	$-0.425256\pi$
$691$	12.2319	0.465324	0.232662	+	0.972558i	$0.425256\pi$
$692$	0	0
$693$	−6.13830	−0.233175
$694$	0	0
$695$	−14.8017	−0.561459
$696$	0	0
$697$	10.3157	0.390734
$698$	0	0
$699$	−61.2724	−2.31754
$700$	0	0
$701$	4.71529	0.178094	0.0890470	−	0.996027i	$-0.471618\pi$
$701$	4.71529	0.178094	0.0890470	+	0.996027i	$0.471618\pi$
$702$	0	0
$703$	−22.7856	−0.859375
$704$	0	0
$705$	24.2673	0.913958
$706$	0	0
$707$	−18.8071	−0.707313
$708$	0	0
$709$	−27.8957	−1.04764	−0.523822	−	0.851828i	$-0.675494\pi$
$709$	−27.8957	−1.04764	−0.523822	+	0.851828i	$0.675494\pi$
$710$	0	0
$711$	26.5720	0.996529
$712$	0	0
$713$	−7.82541	−0.293064
$714$	0	0
$715$	−13.0888	−0.489494
$716$	0	0
$717$	−49.7427	−1.85768
$718$	0	0
$719$	12.9154	0.481664	0.240832	−	0.970567i	$-0.422580\pi$
$719$	12.9154	0.481664	0.240832	+	0.970567i	$0.422580\pi$
$720$	0	0
$721$	19.6328	0.731162
$722$	0	0
$723$	35.2647	1.31151
$724$	0	0
$725$	−5.31598	−0.197431
$726$	0	0
$727$	2.55434	0.0947353	0.0473676	−	0.998878i	$-0.484917\pi$
$727$	2.55434	0.0947353	0.0473676	+	0.998878i	$0.484917\pi$
$728$	0	0
$729$	−42.4071	−1.57063
$730$	0	0
$731$	−2.92080	−0.108030
$732$	0	0
$733$	27.6531	1.02139	0.510695	−	0.859762i	$-0.329388\pi$
$733$	27.6531	1.02139	0.510695	+	0.859762i	$0.329388\pi$
$734$	0	0
$735$	−34.3395	−1.26663
$736$	0	0
$737$	−14.0745	−0.518439
$738$	0	0
$739$	7.85536	0.288964	0.144482	−	0.989507i	$-0.453848\pi$
$739$	7.85536	0.288964	0.144482	+	0.989507i	$0.453848\pi$
$740$	0	0
$741$	−35.4904	−1.30377
$742$	0	0
$743$	−2.46422	−0.0904036	−0.0452018	−	0.998978i	$-0.514393\pi$
$743$	−2.46422	−0.0904036	−0.0452018	+	0.998978i	$0.514393\pi$
$744$	0	0
$745$	51.8636	1.90013
$746$	0	0
$747$	−2.24171	−0.0820200
$748$	0	0
$749$	−15.6894	−0.573277
$750$	0	0
$751$	11.0895	0.404662	0.202331	−	0.979317i	$-0.435148\pi$
$751$	11.0895	0.404662	0.202331	+	0.979317i	$0.435148\pi$
$752$	0	0
$753$	57.3891	2.09138
$754$	0	0
$755$	−41.3868	−1.50622
$756$	0	0
$757$	−43.7690	−1.59081	−0.795406	−	0.606077i	$-0.792742\pi$
$757$	−43.7690	−1.59081	−0.795406	+	0.606077i	$0.792742\pi$
$758$	0	0
$759$	−2.67920	−0.0972489
$760$	0	0
$761$	0.0724438	0.00262609	0.00131304	−	0.999999i	$-0.499582\pi$
$761$	0.0724438	0.00262609	0.00131304	+	0.999999i	$0.499582\pi$
$762$	0	0
$763$	−10.7168	−0.387975
$764$	0	0
$765$	11.1502	0.403136
$766$	0	0
$767$	56.7344	2.04856
$768$	0	0
$769$	25.3286	0.913371	0.456686	−	0.889628i	$-0.349036\pi$
$769$	25.3286	0.913371	0.456686	+	0.889628i	$0.349036\pi$
$770$	0	0
$771$	−82.5147	−2.97170
$772$	0	0
$773$	13.8373	0.497693	0.248847	−	0.968543i	$-0.419949\pi$
$773$	13.8373	0.497693	0.248847	+	0.968543i	$0.419949\pi$
$774$	0	0
$775$	−15.7141	−0.564469
$776$	0	0
$777$	−33.4755	−1.20093
$778$	0	0
$779$	−27.4159	−0.982277
$780$	0	0
$781$	−10.2455	−0.366613
$782$	0	0
$783$	8.35600	0.298619
$784$	0	0
$785$	53.6508	1.91488
$786$	0	0
$787$	44.9348	1.60175	0.800876	−	0.598831i	$-0.204368\pi$
$787$	44.9348	1.60175	0.800876	+	0.598831i	$0.204368\pi$
$788$	0	0
$789$	53.4889	1.90425
$790$	0	0
$791$	−7.94233	−0.282397
$792$	0	0
$793$	2.07710	0.0737599
$794$	0	0
$795$	34.7349	1.23192
$796$	0	0
$797$	51.7910	1.83453	0.917266	−	0.398275i	$-0.130391\pi$
$797$	51.7910	1.83453	0.917266	+	0.398275i	$0.130391\pi$
$798$	0	0
$799$	−3.44918	−0.122023
$800$	0	0
$801$	16.5134	0.583472
$802$	0	0
$803$	−3.88849	−0.137222
$804$	0	0
$805$	3.88925	0.137078
$806$	0	0
$807$	−52.6919	−1.85484
$808$	0	0
$809$	−26.1705	−0.920107	−0.460054	−	0.887891i	$-0.652170\pi$
$809$	−26.1705	−0.920107	−0.460054	+	0.887891i	$0.652170\pi$
$810$	0	0
$811$	46.3621	1.62799	0.813996	−	0.580870i	$-0.197288\pi$
$811$	46.3621	1.62799	0.813996	+	0.580870i	$0.197288\pi$
$812$	0	0
$813$	−27.7770	−0.974182
$814$	0	0
$815$	−22.2606	−0.779754
$816$	0	0
$817$	7.76258	0.271578
$818$	0	0
$819$	−30.3493	−1.06049
$820$	0	0
$821$	−20.5698	−0.717890	−0.358945	−	0.933359i	$-0.616863\pi$
$821$	−20.5698	−0.717890	−0.358945	+	0.933359i	$0.616863\pi$
$822$	0	0
$823$	−16.9104	−0.589458	−0.294729	−	0.955581i	$-0.595229\pi$
$823$	−16.9104	−0.589458	−0.294729	+	0.955581i	$0.595229\pi$
$824$	0	0
$825$	−5.38009	−0.187311
$826$	0	0
$827$	15.9170	0.553488	0.276744	−	0.960944i	$-0.410745\pi$
$827$	15.9170	0.553488	0.276744	+	0.960944i	$0.410745\pi$
$828$	0	0
$829$	−43.1364	−1.49819	−0.749095	−	0.662463i	$-0.769511\pi$
$829$	−43.1364	−1.49819	−0.749095	+	0.662463i	$0.769511\pi$
$830$	0	0
$831$	−41.1234	−1.42655
$832$	0	0
$833$	4.88078	0.169109
$834$	0	0
$835$	11.3575	0.393043
$836$	0	0
$837$	24.7005	0.853774
$838$	0	0
$839$	−23.8722	−0.824159	−0.412079	−	0.911148i	$-0.635197\pi$
$839$	−23.8722	−0.824159	−0.412079	+	0.911148i	$0.635197\pi$
$840$	0	0
$841$	−21.9919	−0.758342
$842$	0	0
$843$	62.3922	2.14890
$844$	0	0
$845$	−30.2997	−1.04234
$846$	0	0
$847$	−1.46915	−0.0504806
$848$	0	0
$849$	38.4938	1.32110
$850$	0	0
$851$	−8.50461	−0.291534
$852$	0	0
$853$	−15.4875	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$853$	−15.4875	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$854$	0	0
$855$	−29.6338	−1.01345
$856$	0	0
$857$	29.7441	1.01604	0.508020	−	0.861346i	$-0.330378\pi$
$857$	29.7441	1.01604	0.508020	+	0.861346i	$0.330378\pi$
$858$	0	0
$859$	−54.9995	−1.87656	−0.938279	−	0.345879i	$-0.887581\pi$
$859$	−54.9995	−1.87656	−0.938279	+	0.345879i	$0.887581\pi$
$860$	0	0
$861$	−40.2781	−1.37267
$862$	0	0
$863$	0.351962	0.0119809	0.00599046	−	0.999982i	$-0.498093\pi$
$863$	0.351962	0.0119809	0.00599046	+	0.999982i	$0.498093\pi$
$864$	0	0
$865$	−49.3619	−1.67835
$866$	0	0
$867$	42.8237	1.45437
$868$	0	0
$869$	6.35979	0.215741
$870$	0	0
$871$	−69.5876	−2.35789
$872$	0	0
$873$	37.0786	1.25492
$874$	0	0
$875$	−11.6363	−0.393378
$876$	0	0
$877$	−33.2688	−1.12341	−0.561703	−	0.827339i	$-0.689854\pi$
$877$	−33.2688	−1.12341	−0.561703	+	0.827339i	$0.689854\pi$
$878$	0	0
$879$	−40.7259	−1.37365
$880$	0	0
$881$	11.2441	0.378824	0.189412	−	0.981898i	$-0.439342\pi$
$881$	11.2441	0.378824	0.189412	+	0.981898i	$0.439342\pi$
$882$	0	0
$883$	30.3285	1.02064	0.510318	−	0.859986i	$-0.329528\pi$
$883$	30.3285	1.02064	0.510318	+	0.859986i	$0.329528\pi$
$884$	0	0
$885$	81.3864	2.73578
$886$	0	0
$887$	34.1571	1.14688	0.573442	−	0.819246i	$-0.305608\pi$
$887$	34.1571	1.14688	0.573442	+	0.819246i	$0.305608\pi$
$888$	0	0
$889$	18.4638	0.619258
$890$	0	0
$891$	−4.07762	−0.136606
$892$	0	0
$893$	9.16687	0.306758
$894$	0	0
$895$	38.8156	1.29746
$896$	0	0
$897$	−13.2466	−0.442293
$898$	0	0
$899$	20.7161	0.690919
$900$	0	0
$901$	−4.93698	−0.164475
$902$	0	0
$903$	11.4044	0.379515
$904$	0	0
$905$	2.03750	0.0677287
$906$	0	0
$907$	16.9714	0.563527	0.281764	−	0.959484i	$-0.409081\pi$
$907$	16.9714	0.563527	0.281764	+	0.959484i	$0.409081\pi$
$908$	0	0
$909$	53.4856	1.77400
$910$	0	0
$911$	23.0071	0.762258	0.381129	−	0.924522i	$-0.375535\pi$
$911$	23.0071	0.762258	0.381129	+	0.924522i	$0.375535\pi$
$912$	0	0
$913$	−0.536535	−0.0177567
$914$	0	0
$915$	2.97963	0.0985035
$916$	0	0
$917$	−8.03501	−0.265339
$918$	0	0
$919$	21.8829	0.721851	0.360925	−	0.932595i	$-0.382461\pi$
$919$	21.8829	0.721851	0.360925	+	0.932595i	$0.382461\pi$
$920$	0	0
$921$	−19.7486	−0.650737
$922$	0	0
$923$	−50.6563	−1.66737
$924$	0	0
$925$	−17.0781	−0.561523
$926$	0	0
$927$	−55.8337	−1.83382
$928$	0	0
$929$	38.3743	1.25902	0.629511	−	0.776992i	$-0.283255\pi$
$929$	38.3743	1.25902	0.629511	+	0.776992i	$0.283255\pi$
$930$	0	0
$931$	−12.9716	−0.425128
$932$	0	0
$933$	13.9129	0.455488
$934$	0	0
$935$	2.66870	0.0872760
$936$	0	0
$937$	0.656502	0.0214470	0.0107235	−	0.999943i	$-0.496587\pi$
$937$	0.656502	0.0214470	0.0107235	+	0.999943i	$0.496587\pi$
$938$	0	0
$939$	−9.16891	−0.299216
$940$	0	0
$941$	−1.65993	−0.0541121	−0.0270560	−	0.999634i	$-0.508613\pi$
$941$	−1.65993	−0.0541121	−0.0270560	+	0.999634i	$0.508613\pi$
$942$	0	0
$943$	−10.2329	−0.333228
$944$	0	0
$945$	−12.2762	−0.399346
$946$	0	0
$947$	−26.5876	−0.863981	−0.431990	−	0.901878i	$-0.642189\pi$
$947$	−26.5876	−0.863981	−0.431990	+	0.901878i	$0.642189\pi$
$948$	0	0
$949$	−19.2257	−0.624092
$950$	0	0
$951$	31.5645	1.02355
$952$	0	0
$953$	−36.5526	−1.18405	−0.592027	−	0.805918i	$-0.701672\pi$
$953$	−36.5526	−1.18405	−0.592027	+	0.805918i	$0.701672\pi$
$954$	0	0
$955$	−4.65150	−0.150519
$956$	0	0
$957$	7.09260	0.229271
$958$	0	0
$959$	−22.8121	−0.736642
$960$	0	0
$961$	30.2370	0.975388
$962$	0	0
$963$	44.6191	1.43783
$964$	0	0
$965$	14.0590	0.452576
$966$	0	0
$967$	52.4407	1.68638	0.843190	−	0.537616i	$-0.180675\pi$
$967$	52.4407	1.68638	0.843190	+	0.537616i	$0.180675\pi$
$968$	0	0
$969$	7.23622	0.232461
$970$	0	0
$971$	−28.0696	−0.900798	−0.450399	−	0.892827i	$-0.648718\pi$
$971$	−28.0696	−0.900798	−0.450399	+	0.892827i	$0.648718\pi$
$972$	0	0
$973$	−8.21442	−0.263342
$974$	0	0
$975$	−26.6005	−0.851897
$976$	0	0
$977$	1.62603	0.0520215	0.0260107	−	0.999662i	$-0.491720\pi$
$977$	1.62603	0.0520215	0.0260107	+	0.999662i	$0.491720\pi$
$978$	0	0
$979$	3.95234	0.126317
$980$	0	0
$981$	30.4776	0.973075
$982$	0	0
$983$	−25.1305	−0.801537	−0.400769	−	0.916179i	$-0.631257\pi$
$983$	−25.1305	−0.801537	−0.400769	+	0.916179i	$0.631257\pi$
$984$	0	0
$985$	52.7500	1.68075
$986$	0	0
$987$	13.4675	0.428676
$988$	0	0
$989$	2.89735	0.0921303
$990$	0	0
$991$	−36.4292	−1.15721	−0.578606	−	0.815607i	$-0.696403\pi$
$991$	−36.4292	−1.15721	−0.578606	+	0.815607i	$0.696403\pi$
$992$	0	0
$993$	−56.9732	−1.80799
$994$	0	0
$995$	34.6420	1.09823
$996$	0	0
$997$	−51.8212	−1.64119	−0.820596	−	0.571508i	$-0.806358\pi$
$997$	−51.8212	−1.64119	−0.820596	+	0.571508i	$0.806358\pi$
$998$	0	0
$999$	26.8444	0.849318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4048.2.a.y.1.1		5
4.3	odd	2		506.2.a.j.1.5	✓	5
12.11	even	2		4554.2.a.by.1.5		5
44.43	even	2		5566.2.a.bi.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
506.2.a.j.1.5	✓	5	4.3	odd	2
4048.2.a.y.1.1		5	1.1	even	1	trivial
4554.2.a.by.1.5		5	12.11	even	2
5566.2.a.bi.1.5		5	44.43	even	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:	\( N \)	\(=\)	\( 4048 = 2^{4} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67920 q^{3} -2.64728 q^{5} -1.46915 q^{7} +4.17813 q^{9} +O(q^{10})\)	\(q-2.67920 q^{3} -2.64728 q^{5} -1.46915 q^{7} +4.17813 q^{9} +1.00000 q^{11} +4.94425 q^{13} +7.09260 q^{15} -1.00809 q^{17} +2.67920 q^{19} +3.93615 q^{21} +1.00000 q^{23} +2.00809 q^{25} -3.15645 q^{27} -2.64728 q^{29} -7.82541 q^{31} -2.67920 q^{33} +3.88925 q^{35} -8.50461 q^{37} -13.2466 q^{39} -10.2329 q^{41} +2.89735 q^{43} -11.0607 q^{45} +3.42149 q^{47} -4.84159 q^{49} +2.70089 q^{51} +4.89735 q^{53} -2.64728 q^{55} -7.17813 q^{57} +11.4748 q^{59} +0.420104 q^{61} -6.13830 q^{63} -13.0888 q^{65} -14.0745 q^{67} -2.67920 q^{69} -10.2455 q^{71} -3.88849 q^{73} -5.38009 q^{75} -1.46915 q^{77} +6.35979 q^{79} -4.07762 q^{81} -0.536535 q^{83} +2.66870 q^{85} +7.09260 q^{87} +3.95234 q^{89} -7.26385 q^{91} +20.9659 q^{93} -7.09260 q^{95} +8.87446 q^{97} +4.17813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 4 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) 5 * q - q^3 + 4 * q^5 - q^7 + 10 * q^9	\( 5 q - q^{3} + 4 q^{5} - q^{7} + 10 q^{9} + 5 q^{11} + 10 q^{13} + 2 q^{15} + q^{19} + 10 q^{21} + 5 q^{23} + 5 q^{25} + 8 q^{27} + 4 q^{29} - 11 q^{31} - q^{33} + q^{35} - 2 q^{37} + 28 q^{39} - 4 q^{41} - 9 q^{43} - 17 q^{45} - 9 q^{47} + 14 q^{49} + 14 q^{51} + q^{53} + 4 q^{55} - 25 q^{57} + 9 q^{59} - 10 q^{61} + 45 q^{63} - 9 q^{65} + 15 q^{67} - q^{69} - 11 q^{71} + 10 q^{73} - 15 q^{75} - q^{77} - 7 q^{79} + 17 q^{81} + 33 q^{83} - 31 q^{85} + 2 q^{87} + 17 q^{91} - 2 q^{93} - 2 q^{95} + 22 q^{97} + 10 q^{99}+O(q^{100}) \) 5 * q - q^3 + 4 * q^5 - q^7 + 10 * q^9 + 5 * q^11 + 10 * q^13 + 2 * q^15 + q^19 + 10 * q^21 + 5 * q^23 + 5 * q^25 + 8 * q^27 + 4 * q^29 - 11 * q^31 - q^33 + q^35 - 2 * q^37 + 28 * q^39 - 4 * q^41 - 9 * q^43 - 17 * q^45 - 9 * q^47 + 14 * q^49 + 14 * q^51 + q^53 + 4 * q^55 - 25 * q^57 + 9 * q^59 - 10 * q^61 + 45 * q^63 - 9 * q^65 + 15 * q^67 - q^69 - 11 * q^71 + 10 * q^73 - 15 * q^75 - q^77 - 7 * q^79 + 17 * q^81 + 33 * q^83 - 31 * q^85 + 2 * q^87 + 17 * q^91 - 2 * q^93 - 2 * q^95 + 22 * q^97 + 10 * q^99

Introduction

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