LMFDB - Embedded newform 9200.2.a.ck.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.92022$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39945 q^{3} -4.60747 q^{7} -1.04155 q^{9} +O(q^{10})$	$q-1.39945 q^{3} -4.60747 q^{7} -1.04155 q^{9} -1.56592 q^{11} +2.60055 q^{13} +0.559006 q^{17} +1.16647 q^{19} +6.44791 q^{21} +1.00000 q^{23} +5.65593 q^{27} -3.17339 q^{29} -10.0554 q^{31} +2.19143 q^{33} +5.07341 q^{37} -3.63934 q^{39} -11.8127 q^{41} +2.76426 q^{43} -9.32298 q^{47} +14.2288 q^{49} -0.782299 q^{51} -5.54789 q^{53} -1.63242 q^{57} -3.84044 q^{59} -4.29832 q^{61} +4.79889 q^{63} -2.60747 q^{67} -1.39945 q^{69} -7.89582 q^{71} +9.90579 q^{73} +7.21494 q^{77} -12.0485 q^{79} -4.79054 q^{81} -13.3856 q^{83} +4.44099 q^{87} -7.71551 q^{89} -11.9820 q^{91} +14.0720 q^{93} -2.62130 q^{97} +1.63098 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9	$ 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9} - 2 q^{11} + 14 q^{13} + 14 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} - 14 q^{27} + 4 q^{29} + 14 q^{33} + 2 q^{37} + 8 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} - 10 q^{51} + 4 q^{53} - 8 q^{61} + 12 q^{63} + 2 q^{67} - 2 q^{69} + 24 q^{71} + 18 q^{73} + 4 q^{77} - 24 q^{79} + 8 q^{81} + 6 q^{83} + 6 q^{87} - 8 q^{89} - 26 q^{91} + 2 q^{93} + 34 q^{97} - 36 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9 - 2 * q^11 + 14 * q^13 + 14 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 - 14 * q^27 + 4 * q^29 + 14 * q^33 + 2 * q^37 + 8 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 - 10 * q^51 + 4 * q^53 - 8 * q^61 + 12 * q^63 + 2 * q^67 - 2 * q^69 + 24 * q^71 + 18 * q^73 + 4 * q^77 - 24 * q^79 + 8 * q^81 + 6 * q^83 + 6 * q^87 - 8 * q^89 - 26 * q^91 + 2 * q^93 + 34 * q^97 - 36 * 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
$3$	−1.39945	−0.807971	−0.403986	−	0.914765i	$-0.632375\pi$
$3$	−1.39945	−0.807971	−0.403986	+	0.914765i	$0.632375\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.60747	−1.74146	−0.870730	−	0.491762i	$-0.836353\pi$
$7$	−4.60747	−1.74146	−0.870730	+	0.491762i	$0.836353\pi$
$8$	0	0
$9$	−1.04155	−0.347182
$10$	0	0
$11$	−1.56592	−0.472143	−0.236072	−	0.971736i	$-0.575860\pi$
$11$	−1.56592	−0.472143	−0.236072	+	0.971736i	$0.575860\pi$
$12$	0	0
$13$	2.60055	0.721264	0.360632	−	0.932708i	$-0.382561\pi$
$13$	2.60055	0.721264	0.360632	+	0.932708i	$0.382561\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.559006	0.135579	0.0677894	−	0.997700i	$-0.478405\pi$
$17$	0.559006	0.135579	0.0677894	+	0.997700i	$0.478405\pi$
$18$	0	0
$19$	1.16647	0.267608	0.133804	−	0.991008i	$-0.457281\pi$
$19$	1.16647	0.267608	0.133804	+	0.991008i	$0.457281\pi$
$20$	0	0
$21$	6.44791	1.40705
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65593	1.08848
$28$	0	0
$29$	−3.17339	−0.589284	−0.294642	−	0.955608i	$-0.595200\pi$
$29$	−3.17339	−0.589284	−0.294642	+	0.955608i	$0.595200\pi$
$30$	0	0
$31$	−10.0554	−1.80600	−0.903000	−	0.429641i	$-0.858640\pi$
$31$	−10.0554	−1.80600	−0.903000	+	0.429641i	$0.858640\pi$
$32$	0	0
$33$	2.19143	0.381478
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.07341	0.834064	0.417032	−	0.908892i	$-0.363070\pi$
$37$	5.07341	0.834064	0.417032	+	0.908892i	$0.363070\pi$
$38$	0	0
$39$	−3.63934	−0.582760
$40$	0	0
$41$	−11.8127	−1.84484	−0.922419	−	0.386190i	$-0.873791\pi$
$41$	−11.8127	−1.84484	−0.922419	+	0.386190i	$0.873791\pi$
$42$	0	0
$43$	2.76426	0.421546	0.210773	−	0.977535i	$-0.432402\pi$
$43$	2.76426	0.421546	0.210773	+	0.977535i	$0.432402\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.32298	−1.35990	−0.679948	−	0.733260i	$-0.737998\pi$
$47$	−9.32298	−1.35990	−0.679948	+	0.733260i	$0.737998\pi$
$48$	0	0
$49$	14.2288	2.03268
$50$	0	0
$51$	−0.782299	−0.109544
$52$	0	0
$53$	−5.54789	−0.762061	−0.381030	−	0.924562i	$-0.624431\pi$
$53$	−5.54789	−0.762061	−0.381030	+	0.924562i	$0.624431\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.63242	−0.216219
$58$	0	0
$59$	−3.84044	−0.499983	−0.249991	−	0.968248i	$-0.580428\pi$
$59$	−3.84044	−0.499983	−0.249991	+	0.968248i	$0.580428\pi$
$60$	0	0
$61$	−4.29832	−0.550343	−0.275172	−	0.961395i	$-0.588735\pi$
$61$	−4.29832	−0.550343	−0.275172	+	0.961395i	$0.588735\pi$
$62$	0	0
$63$	4.79889	0.604604
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.60747	−0.318553	−0.159277	−	0.987234i	$-0.550916\pi$
$67$	−2.60747	−0.318553	−0.159277	+	0.987234i	$0.550916\pi$
$68$	0	0
$69$	−1.39945	−0.168474
$70$	0	0
$71$	−7.89582	−0.937062	−0.468531	−	0.883447i	$-0.655217\pi$
$71$	−7.89582	−0.937062	−0.468531	+	0.883447i	$0.655217\pi$
$72$	0	0
$73$	9.90579	1.15938	0.579692	−	0.814835i	$-0.303173\pi$
$73$	9.90579	1.15938	0.579692	+	0.814835i	$0.303173\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.21494	0.822219
$78$	0	0
$79$	−12.0485	−1.35556	−0.677779	−	0.735266i	$-0.737057\pi$
$79$	−12.0485	−1.35556	−0.677779	+	0.735266i	$0.737057\pi$
$80$	0	0
$81$	−4.79054	−0.532282
$82$	0	0
$83$	−13.3856	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$83$	−13.3856	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.44099	0.476125
$88$	0	0
$89$	−7.71551	−0.817843	−0.408921	−	0.912570i	$-0.634095\pi$
$89$	−7.71551	−0.817843	−0.408921	+	0.912570i	$0.634095\pi$
$90$	0	0
$91$	−11.9820	−1.25605
$92$	0	0
$93$	14.0720	1.45920
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.62130	−0.266153	−0.133076	−	0.991106i	$-0.542486\pi$
$97$	−2.62130	−0.266153	−0.133076	+	0.991106i	$0.542486\pi$
$98$	0	0
$99$	1.63098	0.163920
$100$	0	0
$101$	−8.88199	−0.883791	−0.441895	−	0.897067i	$-0.645694\pi$
$101$	−8.88199	−0.883791	−0.441895	+	0.897067i	$0.645694\pi$
$102$	0	0
$103$	8.65593	0.852894	0.426447	−	0.904512i	$-0.359765\pi$
$103$	8.65593	0.852894	0.426447	+	0.904512i	$0.359765\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.91662	0.668655	0.334327	−	0.942457i	$-0.391491\pi$
$107$	6.91662	0.668655	0.334327	+	0.942457i	$0.391491\pi$
$108$	0	0
$109$	−6.84736	−0.655858	−0.327929	−	0.944702i	$-0.606351\pi$
$109$	−6.84736	−0.655858	−0.327929	+	0.944702i	$0.606351\pi$
$110$	0	0
$111$	−7.09998	−0.673900
$112$	0	0
$113$	−4.60747	−0.433434	−0.216717	−	0.976234i	$-0.569535\pi$
$113$	−4.60747	−0.433434	−0.216717	+	0.976234i	$0.569535\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.70860	−0.250410
$118$	0	0
$119$	−2.57560	−0.236105
$120$	0	0
$121$	−8.54789	−0.777081
$122$	0	0
$123$	16.5313	1.49058
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.5324	−1.02334	−0.511669	−	0.859182i	$-0.670973\pi$
$127$	−11.5324	−1.02334	−0.511669	+	0.859182i	$0.670973\pi$
$128$	0	0
$129$	−3.86844	−0.340597
$130$	0	0
$131$	12.9639	1.13266	0.566332	−	0.824177i	$-0.308362\pi$
$131$	12.9639	1.13266	0.566332	+	0.824177i	$0.308362\pi$
$132$	0	0
$133$	−5.37450	−0.466028
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.1457	−1.46485	−0.732427	−	0.680846i	$-0.761612\pi$
$137$	−17.1457	−1.46485	−0.732427	+	0.680846i	$0.761612\pi$
$138$	0	0
$139$	19.4798	1.65225	0.826127	−	0.563485i	$-0.190540\pi$
$139$	19.4798	1.65225	0.826127	+	0.563485i	$0.190540\pi$
$140$	0	0
$141$	13.0470	1.09876
$142$	0	0
$143$	−4.07226	−0.340540
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.9124	−1.64235
$148$	0	0
$149$	13.2288	1.08374	0.541872	−	0.840461i	$-0.317716\pi$
$149$	13.2288	1.08374	0.541872	+	0.840461i	$0.317716\pi$
$150$	0	0
$151$	2.50749	0.204057	0.102028	−	0.994781i	$-0.467467\pi$
$151$	2.50749	0.204057	0.102028	+	0.994781i	$0.467467\pi$
$152$	0	0
$153$	−0.582231	−0.0470706
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.52824	−0.680628	−0.340314	−	0.940312i	$-0.610533\pi$
$157$	−8.52824	−0.680628	−0.340314	+	0.940312i	$0.610533\pi$
$158$	0	0
$159$	7.76398	0.615723
$160$	0	0
$161$	−4.60747	−0.363119
$162$	0	0
$163$	7.73240	0.605648	0.302824	−	0.953046i	$-0.402071\pi$
$163$	7.73240	0.605648	0.302824	+	0.953046i	$0.402071\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.3064	1.10706	0.553531	−	0.832829i	$-0.313280\pi$
$167$	14.3064	1.10706	0.553531	+	0.832829i	$0.313280\pi$
$168$	0	0
$169$	−6.23713	−0.479779
$170$	0	0
$171$	−1.21494	−0.0929086
$172$	0	0
$173$	15.4714	1.17627	0.588135	−	0.808763i	$-0.299862\pi$
$173$	15.4714	1.17627	0.588135	+	0.808763i	$0.299862\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.37450	0.403972
$178$	0	0
$179$	−5.98612	−0.447424	−0.223712	−	0.974655i	$-0.571817\pi$
$179$	−5.98612	−0.447424	−0.223712	+	0.974655i	$0.571817\pi$
$180$	0	0
$181$	−5.69892	−0.423597	−0.211799	−	0.977313i	$-0.567932\pi$
$181$	−5.69892	−0.423597	−0.211799	+	0.977313i	$0.567932\pi$
$182$	0	0
$183$	6.01527	0.444662
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.875359	−0.0640126
$188$	0	0
$189$	−26.0595	−1.89555
$190$	0	0
$191$	8.34402	0.603752	0.301876	−	0.953347i	$-0.402387\pi$
$191$	8.34402	0.603752	0.301876	+	0.953347i	$0.402387\pi$
$192$	0	0
$193$	16.0250	1.15350	0.576751	−	0.816920i	$-0.304320\pi$
$193$	16.0250	1.15350	0.576751	+	0.816920i	$0.304320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.78893	0.412444	0.206222	−	0.978505i	$-0.433883\pi$
$197$	5.78893	0.412444	0.206222	+	0.978505i	$0.433883\pi$
$198$	0	0
$199$	18.8085	1.33330	0.666651	−	0.745370i	$-0.267727\pi$
$199$	18.8085	1.33330	0.666651	+	0.745370i	$0.267727\pi$
$200$	0	0
$201$	3.64902	0.257382
$202$	0	0
$203$	14.6213	1.02621
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.04155	−0.0723925
$208$	0	0
$209$	−1.82661	−0.126349
$210$	0	0
$211$	4.88199	0.336090	0.168045	−	0.985779i	$-0.446255\pi$
$211$	4.88199	0.336090	0.168045	+	0.985779i	$0.446255\pi$
$212$	0	0
$213$	11.0498	0.757119
$214$	0	0
$215$	0	0
$216$	0	0
$217$	46.3299	3.14508
$218$	0	0
$219$	−13.8626	−0.936750
$220$	0	0
$221$	1.45372	0.0977880
$222$	0	0
$223$	13.4786	0.902596	0.451298	−	0.892373i	$-0.350961\pi$
$223$	13.4786	0.902596	0.451298	+	0.892373i	$0.350961\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.6894	−0.908598	−0.454299	−	0.890849i	$-0.650110\pi$
$227$	−13.6894	−0.908598	−0.454299	+	0.890849i	$0.650110\pi$
$228$	0	0
$229$	27.3964	1.81040	0.905202	−	0.424981i	$-0.139719\pi$
$229$	27.3964	1.81040	0.905202	+	0.424981i	$0.139719\pi$
$230$	0	0
$231$	−10.0969	−0.664329
$232$	0	0
$233$	−0.0387841	−0.00254083	−0.00127041	−	0.999999i	$-0.500404\pi$
$233$	−0.0387841	−0.00254083	−0.00127041	+	0.999999i	$0.500404\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.8612	1.09525
$238$	0	0
$239$	13.9030	0.899312	0.449656	−	0.893202i	$-0.351547\pi$
$239$	13.9030	0.899312	0.449656	+	0.893202i	$0.351547\pi$
$240$	0	0
$241$	−11.9972	−0.772810	−0.386405	−	0.922329i	$-0.626283\pi$
$241$	−11.9972	−0.772810	−0.386405	+	0.922329i	$0.626283\pi$
$242$	0	0
$243$	−10.2637	−0.658416
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.03348	0.193016
$248$	0	0
$249$	18.7324	1.18712
$250$	0	0
$251$	17.7806	1.12230	0.561150	−	0.827714i	$-0.310359\pi$
$251$	17.7806	1.12230	0.561150	+	0.827714i	$0.310359\pi$
$252$	0	0
$253$	−1.56592	−0.0984487
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.2315	0.638226	0.319113	−	0.947717i	$-0.396615\pi$
$257$	10.2315	0.638226	0.319113	+	0.947717i	$0.396615\pi$
$258$	0	0
$259$	−23.3756	−1.45249
$260$	0	0
$261$	3.30524	0.204589
$262$	0	0
$263$	3.99856	0.246562	0.123281	−	0.992372i	$-0.460658\pi$
$263$	3.99856	0.246562	0.123281	+	0.992372i	$0.460658\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.7975	0.660794
$268$	0	0
$269$	−2.88084	−0.175648	−0.0878239	−	0.996136i	$-0.527991\pi$
$269$	−2.88084	−0.175648	−0.0878239	+	0.996136i	$0.527991\pi$
$270$	0	0
$271$	−14.1944	−0.862250	−0.431125	−	0.902292i	$-0.641883\pi$
$271$	−14.1944	−0.862250	−0.431125	+	0.902292i	$0.641883\pi$
$272$	0	0
$273$	16.7681	1.01485
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.13576	0.548915	0.274457	−	0.961599i	$-0.411502\pi$
$277$	9.13576	0.548915	0.274457	+	0.961599i	$0.411502\pi$
$278$	0	0
$279$	10.4731	0.627011
$280$	0	0
$281$	14.8640	0.886709	0.443355	−	0.896346i	$-0.353788\pi$
$281$	14.8640	0.886709	0.443355	+	0.896346i	$0.353788\pi$
$282$	0	0
$283$	−3.03878	−0.180637	−0.0903185	−	0.995913i	$-0.528788\pi$
$283$	−3.03878	−0.180637	−0.0903185	+	0.995913i	$0.528788\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	54.4268	3.21271
$288$	0	0
$289$	−16.6875	−0.981618
$290$	0	0
$291$	3.66837	0.215044
$292$	0	0
$293$	−19.2288	−1.12336	−0.561678	−	0.827356i	$-0.689844\pi$
$293$	−19.2288	−1.12336	−0.561678	+	0.827356i	$0.689844\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.85675	−0.513921
$298$	0	0
$299$	2.60055	0.150394
$300$	0	0
$301$	−12.7363	−0.734106
$302$	0	0
$303$	12.4299	0.714078
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.5395	1.17225	0.586127	−	0.810220i	$-0.300652\pi$
$307$	20.5395	1.17225	0.586127	+	0.810220i	$0.300652\pi$
$308$	0	0
$309$	−12.1135	−0.689114
$310$	0	0
$311$	9.83929	0.557935	0.278967	−	0.960301i	$-0.410008\pi$
$311$	9.83929	0.557935	0.278967	+	0.960301i	$0.410008\pi$
$312$	0	0
$313$	22.4659	1.26985	0.634925	−	0.772574i	$-0.281031\pi$
$313$	22.4659	1.26985	0.634925	+	0.772574i	$0.281031\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.8404	−1.00202	−0.501010	−	0.865442i	$-0.667038\pi$
$317$	−17.8404	−1.00202	−0.501010	+	0.865442i	$0.667038\pi$
$318$	0	0
$319$	4.96928	0.278226
$320$	0	0
$321$	−9.67944	−0.540254
$322$	0	0
$323$	0.652066	0.0362819
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.58252	0.529914
$328$	0	0
$329$	42.9554	2.36821
$330$	0	0
$331$	−13.3130	−0.731750	−0.365875	−	0.930664i	$-0.619230\pi$
$331$	−13.3130	−0.731750	−0.365875	+	0.930664i	$0.619230\pi$
$332$	0	0
$333$	−5.28420	−0.289572
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.08338	0.167962	0.0839812	−	0.996467i	$-0.473236\pi$
$337$	3.08338	0.167962	0.0839812	+	0.996467i	$0.473236\pi$
$338$	0	0
$339$	6.44791	0.350202
$340$	0	0
$341$	15.7459	0.852691
$342$	0	0
$343$	−33.3063	−1.79837
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.5617	−0.620666	−0.310333	−	0.950628i	$-0.600441\pi$
$347$	−11.5617	−0.620666	−0.310333	+	0.950628i	$0.600441\pi$
$348$	0	0
$349$	−16.4980	−0.883117	−0.441558	−	0.897232i	$-0.645574\pi$
$349$	−16.4980	−0.883117	−0.441558	+	0.897232i	$0.645574\pi$
$350$	0	0
$351$	14.7085	0.785084
$352$	0	0
$353$	13.7252	0.730518	0.365259	−	0.930906i	$-0.380980\pi$
$353$	13.7252	0.730518	0.365259	+	0.930906i	$0.380980\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.60442	0.190766
$358$	0	0
$359$	−15.1442	−0.799282	−0.399641	−	0.916672i	$-0.630865\pi$
$359$	−15.1442	−0.799282	−0.399641	+	0.916672i	$0.630865\pi$
$360$	0	0
$361$	−17.6393	−0.928386
$362$	0	0
$363$	11.9623	0.627859
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.2719	−0.588390	−0.294195	−	0.955745i	$-0.595051\pi$
$367$	−11.2719	−0.588390	−0.294195	+	0.955745i	$0.595051\pi$
$368$	0	0
$369$	12.3035	0.640495
$370$	0	0
$371$	25.5617	1.32710
$372$	0	0
$373$	−35.3296	−1.82930	−0.914648	−	0.404252i	$-0.867532\pi$
$373$	−35.3296	−1.82930	−0.914648	+	0.404252i	$0.867532\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.25257	−0.425029
$378$	0	0
$379$	36.8598	1.89336	0.946679	−	0.322178i	$-0.104415\pi$
$379$	36.8598	1.89336	0.946679	+	0.322178i	$0.104415\pi$
$380$	0	0
$381$	16.1390	0.826828
$382$	0	0
$383$	4.27065	0.218220	0.109110	−	0.994030i	$-0.465200\pi$
$383$	4.27065	0.218220	0.109110	+	0.994030i	$0.465200\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.87911	−0.146353
$388$	0	0
$389$	−6.42988	−0.326008	−0.163004	−	0.986625i	$-0.552118\pi$
$389$	−6.42988	−0.326008	−0.163004	+	0.986625i	$0.552118\pi$
$390$	0	0
$391$	0.559006	0.0282701
$392$	0	0
$393$	−18.1423	−0.915160
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−33.9539	−1.70410	−0.852049	−	0.523462i	$-0.824640\pi$
$397$	−33.9539	−1.70410	−0.852049	+	0.523462i	$0.824640\pi$
$398$	0	0
$399$	7.52133	0.376537
$400$	0	0
$401$	27.6421	1.38038	0.690189	−	0.723629i	$-0.257527\pi$
$401$	27.6421	1.38038	0.690189	+	0.723629i	$0.257527\pi$
$402$	0	0
$403$	−26.1495	−1.30260
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.94457	−0.393798
$408$	0	0
$409$	−13.2161	−0.653494	−0.326747	−	0.945112i	$-0.605952\pi$
$409$	−13.2161	−0.653494	−0.326747	+	0.945112i	$0.605952\pi$
$410$	0	0
$411$	23.9945	1.18356
$412$	0	0
$413$	17.6947	0.870700
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−27.2609	−1.33497
$418$	0	0
$419$	−23.4772	−1.14694	−0.573468	−	0.819228i	$-0.694402\pi$
$419$	−23.4772	−1.14694	−0.573468	+	0.819228i	$0.694402\pi$
$420$	0	0
$421$	23.1138	1.12650	0.563249	−	0.826287i	$-0.309551\pi$
$421$	23.1138	1.12650	0.563249	+	0.826287i	$0.309551\pi$
$422$	0	0
$423$	9.71032	0.472132
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.8044	0.958401
$428$	0	0
$429$	5.69892	0.275146
$430$	0	0
$431$	9.52709	0.458904	0.229452	−	0.973320i	$-0.426307\pi$
$431$	9.52709	0.458904	0.229452	+	0.973320i	$0.426307\pi$
$432$	0	0
$433$	2.79445	0.134293	0.0671464	−	0.997743i	$-0.478611\pi$
$433$	2.79445	0.134293	0.0671464	+	0.997743i	$0.478611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.16647	0.0558001
$438$	0	0
$439$	−2.09578	−0.100026	−0.0500129	−	0.998749i	$-0.515926\pi$
$439$	−2.09578	−0.100026	−0.0500129	+	0.998749i	$0.515926\pi$
$440$	0	0
$441$	−14.8199	−0.705711
$442$	0	0
$443$	27.0870	1.28694	0.643470	−	0.765471i	$-0.277494\pi$
$443$	27.0870	1.28694	0.643470	+	0.765471i	$0.277494\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.5130	−0.875633
$448$	0	0
$449$	−0.470272	−0.0221935	−0.0110967	−	0.999938i	$-0.503532\pi$
$449$	−0.470272	−0.0221935	−0.0110967	+	0.999938i	$0.503532\pi$
$450$	0	0
$451$	18.4978	0.871028
$452$	0	0
$453$	−3.50910	−0.164872
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.83768	0.273075	0.136538	−	0.990635i	$-0.456403\pi$
$457$	5.83768	0.273075	0.136538	+	0.990635i	$0.456403\pi$
$458$	0	0
$459$	3.16170	0.147575
$460$	0	0
$461$	10.8487	0.505277	0.252638	−	0.967561i	$-0.418702\pi$
$461$	10.8487	0.505277	0.252638	+	0.967561i	$0.418702\pi$
$462$	0	0
$463$	−1.06258	−0.0493825	−0.0246912	−	0.999695i	$-0.507860\pi$
$463$	−1.06258	−0.0493825	−0.0246912	+	0.999695i	$0.507860\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.99885	0.231319	0.115660	−	0.993289i	$-0.463102\pi$
$467$	4.99885	0.231319	0.115660	+	0.993289i	$0.463102\pi$
$468$	0	0
$469$	12.0138	0.554747
$470$	0	0
$471$	11.9348	0.549928
$472$	0	0
$473$	−4.32862	−0.199030
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.77838	0.264574
$478$	0	0
$479$	−33.4283	−1.52738	−0.763688	−	0.645585i	$-0.776613\pi$
$479$	−33.4283	−1.52738	−0.763688	+	0.645585i	$0.776613\pi$
$480$	0	0
$481$	13.1937	0.601580
$482$	0	0
$483$	6.44791	0.293390
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.7901	0.488945	0.244473	−	0.969656i	$-0.421385\pi$
$487$	10.7901	0.488945	0.244473	+	0.969656i	$0.421385\pi$
$488$	0	0
$489$	−10.8211	−0.489346
$490$	0	0
$491$	−15.8127	−0.713618	−0.356809	−	0.934177i	$-0.616135\pi$
$491$	−15.8127	−0.713618	−0.356809	+	0.934177i	$0.616135\pi$
$492$	0	0
$493$	−1.77394	−0.0798944
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.3798	1.63185
$498$	0	0
$499$	−12.9711	−0.580668	−0.290334	−	0.956925i	$-0.593766\pi$
$499$	−12.9711	−0.580668	−0.290334	+	0.956925i	$0.593766\pi$
$500$	0	0
$501$	−20.0210	−0.894474
$502$	0	0
$503$	0.998849	0.0445365	0.0222682	−	0.999752i	$-0.492911\pi$
$503$	0.998849	0.0445365	0.0222682	+	0.999752i	$0.492911\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.72853	0.387648
$508$	0	0
$509$	−20.6936	−0.917228	−0.458614	−	0.888636i	$-0.651654\pi$
$509$	−20.6936	−0.917228	−0.458614	+	0.888636i	$0.651654\pi$
$510$	0	0
$511$	−45.6406	−2.01902
$512$	0	0
$513$	6.59750	0.291287
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.5991	0.642066
$518$	0	0
$519$	−21.6514	−0.950393
$520$	0	0
$521$	39.2620	1.72010	0.860049	−	0.510212i	$-0.170433\pi$
$521$	39.2620	1.72010	0.860049	+	0.510212i	$0.170433\pi$
$522$	0	0
$523$	1.74162	0.0761556	0.0380778	−	0.999275i	$-0.487877\pi$
$523$	1.74162	0.0761556	0.0380778	+	0.999275i	$0.487877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.62101	−0.244855
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−30.7196	−1.33061
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.37726	0.361505
$538$	0	0
$539$	−22.2811	−0.959717
$540$	0	0
$541$	−8.18117	−0.351736	−0.175868	−	0.984414i	$-0.556273\pi$
$541$	−8.18117	−0.351736	−0.175868	+	0.984414i	$0.556273\pi$
$542$	0	0
$543$	7.97534	0.342254
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.2470	1.03673	0.518364	−	0.855160i	$-0.326541\pi$
$547$	24.2470	1.03673	0.518364	+	0.855160i	$0.326541\pi$
$548$	0	0
$549$	4.47690	0.191069
$550$	0	0
$551$	−3.70168	−0.157697
$552$	0	0
$553$	55.5129	2.36065
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.0966	−1.48709	−0.743546	−	0.668685i	$-0.766858\pi$
$557$	−35.0966	−1.48709	−0.743546	+	0.668685i	$0.766858\pi$
$558$	0	0
$559$	7.18862	0.304046
$560$	0	0
$561$	1.22502	0.0517204
$562$	0	0
$563$	6.98179	0.294247	0.147124	−	0.989118i	$-0.452998\pi$
$563$	6.98179	0.294247	0.147124	+	0.989118i	$0.452998\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	22.0723	0.926948
$568$	0	0
$569$	21.7280	0.910886	0.455443	−	0.890265i	$-0.349481\pi$
$569$	21.7280	0.910886	0.455443	+	0.890265i	$0.349481\pi$
$570$	0	0
$571$	4.71016	0.197114	0.0985570	−	0.995131i	$-0.468577\pi$
$571$	4.71016	0.197114	0.0985570	+	0.995131i	$0.468577\pi$
$572$	0	0
$573$	−11.6770	−0.487814
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9.49085	−0.395109	−0.197555	−	0.980292i	$-0.563300\pi$
$577$	−9.49085	−0.395109	−0.197555	+	0.980292i	$0.563300\pi$
$578$	0	0
$579$	−22.4261	−0.931996
$580$	0	0
$581$	61.6736	2.55865
$582$	0	0
$583$	8.68756	0.359802
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.4952	−0.433184	−0.216592	−	0.976262i	$-0.569494\pi$
$587$	−10.4952	−0.433184	−0.216592	+	0.976262i	$0.569494\pi$
$588$	0	0
$589$	−11.7293	−0.483299
$590$	0	0
$591$	−8.10130	−0.333243
$592$	0	0
$593$	−18.2138	−0.747951	−0.373975	−	0.927439i	$-0.622006\pi$
$593$	−18.2138	−0.747951	−0.373975	+	0.927439i	$0.622006\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−26.3215	−1.07727
$598$	0	0
$599$	20.8055	0.850091	0.425045	−	0.905172i	$-0.360258\pi$
$599$	20.8055	0.850091	0.425045	+	0.905172i	$0.360258\pi$
$600$	0	0
$601$	−47.4004	−1.93350	−0.966752	−	0.255716i	$-0.917689\pi$
$601$	−47.4004	−1.93350	−0.966752	+	0.255716i	$0.917689\pi$
$602$	0	0
$603$	2.71580	0.110596
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.77118	0.234245	0.117123	−	0.993117i	$-0.462633\pi$
$607$	5.77118	0.234245	0.117123	+	0.993117i	$0.462633\pi$
$608$	0	0
$609$	−20.4617	−0.829152
$610$	0	0
$611$	−24.2449	−0.980844
$612$	0	0
$613$	38.9329	1.57248	0.786242	−	0.617919i	$-0.212024\pi$
$613$	38.9329	1.57248	0.786242	+	0.617919i	$0.212024\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.1653	0.409241	0.204620	−	0.978841i	$-0.434404\pi$
$617$	10.1653	0.409241	0.204620	+	0.978841i	$0.434404\pi$
$618$	0	0
$619$	−2.09923	−0.0843752	−0.0421876	−	0.999110i	$-0.513433\pi$
$619$	−2.09923	−0.0843752	−0.0421876	+	0.999110i	$0.513433\pi$
$620$	0	0
$621$	5.65593	0.226965
$622$	0	0
$623$	35.5490	1.42424
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.55624	0.102087
$628$	0	0
$629$	2.83607	0.113081
$630$	0	0
$631$	32.3066	1.28611	0.643053	−	0.765821i	$-0.277667\pi$
$631$	32.3066	1.28611	0.643053	+	0.765821i	$0.277667\pi$
$632$	0	0
$633$	−6.83209	−0.271551
$634$	0	0
$635$	0	0
$636$	0	0
$637$	37.0027	1.46610
$638$	0	0
$639$	8.22387	0.325331
$640$	0	0
$641$	14.5758	0.575711	0.287856	−	0.957674i	$-0.407058\pi$
$641$	14.5758	0.575711	0.287856	+	0.957674i	$0.407058\pi$
$642$	0	0
$643$	42.5090	1.67639	0.838196	−	0.545369i	$-0.183611\pi$
$643$	42.5090	1.67639	0.838196	+	0.545369i	$0.183611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.04979	0.119900	0.0599498	−	0.998201i	$-0.480906\pi$
$647$	3.04979	0.119900	0.0599498	+	0.998201i	$0.480906\pi$
$648$	0	0
$649$	6.01383	0.236064
$650$	0	0
$651$	−64.8362	−2.54113
$652$	0	0
$653$	17.9600	0.702830	0.351415	−	0.936220i	$-0.385701\pi$
$653$	17.9600	0.702830	0.351415	+	0.936220i	$0.385701\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.3173	−0.402518
$658$	0	0
$659$	−46.3411	−1.80519	−0.902597	−	0.430486i	$-0.858342\pi$
$659$	−46.3411	−1.80519	−0.902597	+	0.430486i	$0.858342\pi$
$660$	0	0
$661$	1.23165	0.0479056	0.0239528	−	0.999713i	$-0.492375\pi$
$661$	1.23165	0.0479056	0.0239528	+	0.999713i	$0.492375\pi$
$662$	0	0
$663$	−2.03441	−0.0790099
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.17339	−0.122874
$668$	0	0
$669$	−18.8626	−0.729271
$670$	0	0
$671$	6.73083	0.259841
$672$	0	0
$673$	42.4664	1.63696	0.818479	−	0.574537i	$-0.194818\pi$
$673$	42.4664	1.63696	0.818479	+	0.574537i	$0.194818\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.41777	0.0544893	0.0272447	−	0.999629i	$-0.491327\pi$
$677$	1.41777	0.0544893	0.0272447	+	0.999629i	$0.491327\pi$
$678$	0	0
$679$	12.0776	0.463495
$680$	0	0
$681$	19.1576	0.734121
$682$	0	0
$683$	−4.60886	−0.176353	−0.0881766	−	0.996105i	$-0.528104\pi$
$683$	−4.60886	−0.176353	−0.0881766	+	0.996105i	$0.528104\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−38.3398	−1.46276
$688$	0	0
$689$	−14.4276	−0.549647
$690$	0	0
$691$	16.8903	0.642537	0.321269	−	0.946988i	$-0.395891\pi$
$691$	16.8903	0.642537	0.321269	+	0.946988i	$0.395891\pi$
$692$	0	0
$693$	−7.51470	−0.285460
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.60338	−0.250121
$698$	0	0
$699$	0.0542763	0.00205292
$700$	0	0
$701$	−11.9543	−0.451508	−0.225754	−	0.974184i	$-0.572484\pi$
$701$	−11.9543	−0.451508	−0.225754	+	0.974184i	$0.572484\pi$
$702$	0	0
$703$	5.91801	0.223202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.9235	1.53909
$708$	0	0
$709$	−12.6755	−0.476040	−0.238020	−	0.971260i	$-0.576498\pi$
$709$	−12.6755	−0.476040	−0.238020	+	0.971260i	$0.576498\pi$
$710$	0	0
$711$	12.5490	0.470626
$712$	0	0
$713$	−10.0554	−0.376577
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.4566	−0.726618
$718$	0	0
$719$	34.7351	1.29540	0.647700	−	0.761896i	$-0.275731\pi$
$719$	34.7351	1.29540	0.647700	+	0.761896i	$0.275731\pi$
$720$	0	0
$721$	−39.8819	−1.48528
$722$	0	0
$723$	16.7895	0.624408
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27.6631	−1.02597	−0.512984	−	0.858398i	$-0.671460\pi$
$727$	−27.6631	−1.02597	−0.512984	+	0.858398i	$0.671460\pi$
$728$	0	0
$729$	28.7351	1.06426
$730$	0	0
$731$	1.54524	0.0571528
$732$	0	0
$733$	−2.14239	−0.0791309	−0.0395654	−	0.999217i	$-0.512597\pi$
$733$	−2.14239	−0.0791309	−0.0395654	+	0.999217i	$0.512597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.08309	0.150403
$738$	0	0
$739$	42.6249	1.56798	0.783991	−	0.620772i	$-0.213181\pi$
$739$	42.6249	1.56798	0.783991	+	0.620772i	$0.213181\pi$
$740$	0	0
$741$	−4.24519	−0.155951
$742$	0	0
$743$	−23.4687	−0.860982	−0.430491	−	0.902595i	$-0.641660\pi$
$743$	−23.4687	−0.860982	−0.430491	+	0.902595i	$0.641660\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.9417	0.510100
$748$	0	0
$749$	−31.8681	−1.16444
$750$	0	0
$751$	27.2579	0.994654	0.497327	−	0.867563i	$-0.334315\pi$
$751$	27.2579	0.994654	0.497327	+	0.867563i	$0.334315\pi$
$752$	0	0
$753$	−24.8830	−0.906786
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.7382	−0.462976	−0.231488	−	0.972838i	$-0.574359\pi$
$757$	−12.7382	−0.462976	−0.231488	+	0.972838i	$0.574359\pi$
$758$	0	0
$759$	2.19143	0.0795437
$760$	0	0
$761$	6.61715	0.239871	0.119936	−	0.992782i	$-0.461731\pi$
$761$	6.61715	0.239871	0.119936	+	0.992782i	$0.461731\pi$
$762$	0	0
$763$	31.5490	1.14215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.98727	−0.360619
$768$	0	0
$769$	−27.2116	−0.981275	−0.490638	−	0.871364i	$-0.663236\pi$
$769$	−27.2116	−0.981275	−0.490638	+	0.871364i	$0.663236\pi$
$770$	0	0
$771$	−14.3185	−0.515668
$772$	0	0
$773$	22.9249	0.824552	0.412276	−	0.911059i	$-0.364734\pi$
$773$	22.9249	0.824552	0.412276	+	0.911059i	$0.364734\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	32.7129	1.17357
$778$	0	0
$779$	−13.7792	−0.493693
$780$	0	0
$781$	12.3642	0.442427
$782$	0	0
$783$	−17.9485	−0.641427
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.0620	−0.465610	−0.232805	−	0.972523i	$-0.574790\pi$
$787$	−13.0620	−0.465610	−0.232805	+	0.972523i	$0.574790\pi$
$788$	0	0
$789$	−5.59578	−0.199215
$790$	0	0
$791$	21.2288	0.754808
$792$	0	0
$793$	−11.1780	−0.396943
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.9305	0.741395	0.370697	−	0.928754i	$-0.379119\pi$
$797$	20.9305	0.741395	0.370697	+	0.928754i	$0.379119\pi$
$798$	0	0
$799$	−5.21160	−0.184373
$800$	0	0
$801$	8.03607	0.283941
$802$	0	0
$803$	−15.5117	−0.547396
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.03158	0.141918
$808$	0	0
$809$	18.0664	0.635180	0.317590	−	0.948228i	$-0.397126\pi$
$809$	18.0664	0.635180	0.317590	+	0.948228i	$0.397126\pi$
$810$	0	0
$811$	−34.8658	−1.22430	−0.612152	−	0.790740i	$-0.709696\pi$
$811$	−34.8658	−1.22430	−0.612152	+	0.790740i	$0.709696\pi$
$812$	0	0
$813$	19.8644	0.696673
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.22444	0.112809
$818$	0	0
$819$	12.4798	0.436079
$820$	0	0
$821$	14.5989	0.509507	0.254753	−	0.967006i	$-0.418006\pi$
$821$	14.5989	0.509507	0.254753	+	0.967006i	$0.418006\pi$
$822$	0	0
$823$	−25.8126	−0.899771	−0.449886	−	0.893086i	$-0.648535\pi$
$823$	−25.8126	−0.899771	−0.449886	+	0.893086i	$0.648535\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.9097	−1.90940	−0.954698	−	0.297577i	$-0.903822\pi$
$827$	−54.9097	−1.90940	−0.954698	+	0.297577i	$0.903822\pi$
$828$	0	0
$829$	−27.4809	−0.954452	−0.477226	−	0.878781i	$-0.658358\pi$
$829$	−27.4809	−0.954452	−0.477226	+	0.878781i	$0.658358\pi$
$830$	0	0
$831$	−12.7850	−0.443507
$832$	0	0
$833$	7.95396	0.275589
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−56.8725	−1.96580
$838$	0	0
$839$	1.46577	0.0506041	0.0253020	−	0.999680i	$-0.491945\pi$
$839$	1.46577	0.0506041	0.0253020	+	0.999680i	$0.491945\pi$
$840$	0	0
$841$	−18.9296	−0.652744
$842$	0	0
$843$	−20.8013	−0.716436
$844$	0	0
$845$	0	0
$846$	0	0
$847$	39.3841	1.35325
$848$	0	0
$849$	4.25262	0.145949
$850$	0	0
$851$	5.07341	0.173914
$852$	0	0
$853$	35.8044	1.22592	0.612959	−	0.790115i	$-0.289979\pi$
$853$	35.8044	1.22592	0.612959	+	0.790115i	$0.289979\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.1663	0.962141	0.481070	−	0.876682i	$-0.340248\pi$
$857$	28.1663	0.962141	0.481070	+	0.876682i	$0.340248\pi$
$858$	0	0
$859$	−20.3185	−0.693258	−0.346629	−	0.938002i	$-0.612674\pi$
$859$	−20.3185	−0.693258	−0.346629	+	0.938002i	$0.612674\pi$
$860$	0	0
$861$	−76.1674	−2.59578
$862$	0	0
$863$	15.1357	0.515224	0.257612	−	0.966248i	$-0.417064\pi$
$863$	15.1357	0.515224	0.257612	+	0.966248i	$0.417064\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23.3533	0.793120
$868$	0	0
$869$	18.8670	0.640018
$870$	0	0
$871$	−6.78086	−0.229761
$872$	0	0
$873$	2.73021	0.0924036
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.9668	0.437856	0.218928	−	0.975741i	$-0.429744\pi$
$877$	12.9668	0.437856	0.218928	+	0.975741i	$0.429744\pi$
$878$	0	0
$879$	26.9097	0.907640
$880$	0	0
$881$	−34.3573	−1.15753	−0.578763	−	0.815496i	$-0.696464\pi$
$881$	−34.3573	−1.15753	−0.578763	+	0.815496i	$0.696464\pi$
$882$	0	0
$883$	14.7201	0.495372	0.247686	−	0.968840i	$-0.420330\pi$
$883$	14.7201	0.495372	0.247686	+	0.968840i	$0.420330\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.9401	−0.703101	−0.351550	−	0.936169i	$-0.614345\pi$
$887$	−20.9401	−0.703101	−0.351550	+	0.936169i	$0.614345\pi$
$888$	0	0
$889$	53.1354	1.78210
$890$	0	0
$891$	7.50161	0.251313
$892$	0	0
$893$	−10.8750	−0.363919
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.63934	−0.121514
$898$	0	0
$899$	31.9097	1.06425
$900$	0	0
$901$	−3.10130	−0.103319
$902$	0	0
$903$	17.8237	0.593137
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.2008	1.33485	0.667423	−	0.744679i	$-0.267397\pi$
$907$	40.2008	1.33485	0.667423	+	0.744679i	$0.267397\pi$
$908$	0	0
$909$	9.25101	0.306837
$910$	0	0
$911$	−54.6058	−1.80917	−0.904586	−	0.426292i	$-0.859820\pi$
$911$	−54.6058	−1.80917	−0.904586	+	0.426292i	$0.859820\pi$
$912$	0	0
$913$	20.9608	0.693700
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.7309	−1.97249
$918$	0	0
$919$	−13.8019	−0.455284	−0.227642	−	0.973745i	$-0.573102\pi$
$919$	−13.8019	−0.455284	−0.227642	+	0.973745i	$0.573102\pi$
$920$	0	0
$921$	−28.7440	−0.947147
$922$	0	0
$923$	−20.5335	−0.675868
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9.01556	−0.296110
$928$	0	0
$929$	19.2775	0.632475	0.316237	−	0.948680i	$-0.397580\pi$
$929$	19.2775	0.632475	0.316237	+	0.948680i	$0.397580\pi$
$930$	0	0
$931$	16.5975	0.543961
$932$	0	0
$933$	−13.7696	−0.450795
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.0855	1.57089	0.785443	−	0.618934i	$-0.212435\pi$
$937$	48.0855	1.57089	0.785443	+	0.618934i	$0.212435\pi$
$938$	0	0
$939$	−31.4399	−1.02600
$940$	0	0
$941$	−30.6853	−1.00031	−0.500156	−	0.865935i	$-0.666724\pi$
$941$	−30.6853	−1.00031	−0.500156	+	0.865935i	$0.666724\pi$
$942$	0	0
$943$	−11.8127	−0.384675
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.9280	0.420103	0.210051	−	0.977690i	$-0.432637\pi$
$947$	12.9280	0.420103	0.210051	+	0.977690i	$0.432637\pi$
$948$	0	0
$949$	25.7605	0.836222
$950$	0	0
$951$	24.9668	0.809603
$952$	0	0
$953$	−6.38285	−0.206761	−0.103380	−	0.994642i	$-0.532966\pi$
$953$	−6.38285	−0.206761	−0.103380	+	0.994642i	$0.532966\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.95425	−0.224799
$958$	0	0
$959$	78.9982	2.55098
$960$	0	0
$961$	70.1107	2.26163
$962$	0	0
$963$	−7.20398	−0.232145
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.6028	1.24138	0.620691	−	0.784056i	$-0.286852\pi$
$967$	38.6028	1.24138	0.620691	+	0.784056i	$0.286852\pi$
$968$	0	0
$969$	−0.912532	−0.0293148
$970$	0	0
$971$	24.0216	0.770889	0.385444	−	0.922731i	$-0.374048\pi$
$971$	24.0216	0.770889	0.385444	+	0.922731i	$0.374048\pi$
$972$	0	0
$973$	−89.7525	−2.87733
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.5558	1.45746	0.728730	−	0.684801i	$-0.240111\pi$
$977$	45.5558	1.45746	0.728730	+	0.684801i	$0.240111\pi$
$978$	0	0
$979$	12.0819	0.386139
$980$	0	0
$981$	7.13184	0.227702
$982$	0	0
$983$	29.1396	0.929408	0.464704	−	0.885466i	$-0.346161\pi$
$983$	29.1396	0.929408	0.464704	+	0.885466i	$0.346161\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−60.1138	−1.91344
$988$	0	0
$989$	2.76426	0.0878985
$990$	0	0
$991$	20.0415	0.636639	0.318320	−	0.947983i	$-0.396881\pi$
$991$	20.0415	0.636639	0.318320	+	0.947983i	$0.396881\pi$
$992$	0	0
$993$	18.6309	0.591233
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.340153	0.0107728	0.00538638	−	0.999985i	$-0.498285\pi$
$997$	0.340153	0.0107728	0.00538638	+	0.999985i	$0.498285\pi$
$998$	0	0
$999$	28.6949	0.907866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ck.1.2		4
4.3	odd	2		575.2.a.j.1.1		4
5.2	odd	4		1840.2.e.d.369.6		8
5.3	odd	4		1840.2.e.d.369.3		8
5.4	even	2		9200.2.a.cq.1.3		4
12.11	even	2		5175.2.a.bw.1.4		4
20.3	even	4		115.2.b.b.24.7	yes	8
20.7	even	4		115.2.b.b.24.2	✓	8
20.19	odd	2		575.2.a.i.1.4		4
60.23	odd	4		1035.2.b.e.829.2		8
60.47	odd	4		1035.2.b.e.829.7		8
60.59	even	2		5175.2.a.bv.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.2	✓	8	20.7	even	4
115.2.b.b.24.7	yes	8	20.3	even	4
575.2.a.i.1.4		4	20.19	odd	2
575.2.a.j.1.1		4	4.3	odd	2
1035.2.b.e.829.2		8	60.23	odd	4
1035.2.b.e.829.7		8	60.47	odd	4
1840.2.e.d.369.3		8	5.3	odd	4
1840.2.e.d.369.6		8	5.2	odd	4
5175.2.a.bv.1.1		4	60.59	even	2
5175.2.a.bw.1.4		4	12.11	even	2
9200.2.a.ck.1.2		4	1.1	even	1	trivial
9200.2.a.cq.1.3		4	5.4	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39945 q^{3} -4.60747 q^{7} -1.04155 q^{9} +O(q^{10})\)	\(q-1.39945 q^{3} -4.60747 q^{7} -1.04155 q^{9} -1.56592 q^{11} +2.60055 q^{13} +0.559006 q^{17} +1.16647 q^{19} +6.44791 q^{21} +1.00000 q^{23} +5.65593 q^{27} -3.17339 q^{29} -10.0554 q^{31} +2.19143 q^{33} +5.07341 q^{37} -3.63934 q^{39} -11.8127 q^{41} +2.76426 q^{43} -9.32298 q^{47} +14.2288 q^{49} -0.782299 q^{51} -5.54789 q^{53} -1.63242 q^{57} -3.84044 q^{59} -4.29832 q^{61} +4.79889 q^{63} -2.60747 q^{67} -1.39945 q^{69} -7.89582 q^{71} +9.90579 q^{73} +7.21494 q^{77} -12.0485 q^{79} -4.79054 q^{81} -13.3856 q^{83} +4.44099 q^{87} -7.71551 q^{89} -11.9820 q^{91} +14.0720 q^{93} -2.62130 q^{97} +1.63098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9	\( 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9} - 2 q^{11} + 14 q^{13} + 14 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} - 14 q^{27} + 4 q^{29} + 14 q^{33} + 2 q^{37} + 8 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} - 10 q^{51} + 4 q^{53} - 8 q^{61} + 12 q^{63} + 2 q^{67} - 2 q^{69} + 24 q^{71} + 18 q^{73} + 4 q^{77} - 24 q^{79} + 8 q^{81} + 6 q^{83} + 6 q^{87} - 8 q^{89} - 26 q^{91} + 2 q^{93} + 34 q^{97} - 36 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9 - 2 * q^11 + 14 * q^13 + 14 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 - 14 * q^27 + 4 * q^29 + 14 * q^33 + 2 * q^37 + 8 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 - 10 * q^51 + 4 * q^53 - 8 * q^61 + 12 * q^63 + 2 * q^67 - 2 * q^69 + 24 * q^71 + 18 * q^73 + 4 * q^77 - 24 * q^79 + 8 * q^81 + 6 * q^83 + 6 * q^87 - 8 * q^89 - 26 * q^91 + 2 * q^93 + 34 * q^97 - 36 * q^99

Introduction

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