LMFDB - Embedded newform 9200.2.a.cs.1.1

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.791953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 $ x^5 - 2x^4 - 7x^3 + 7x^2 + 9x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08344$ 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)

$f(q)$	$=$	$q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} +O(q^{10})$	$q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} -4.65190 q^{11} +5.02256 q^{13} +1.32867 q^{17} +0.196402 q^{19} -1.71138 q^{21} -1.00000 q^{23} -10.8154 q^{27} -0.812298 q^{29} +2.11145 q^{31} +14.3439 q^{33} -5.64564 q^{37} -15.4868 q^{39} -4.89714 q^{41} -1.66507 q^{43} +9.89310 q^{47} -6.69195 q^{49} -4.09688 q^{51} +2.23261 q^{53} -0.605593 q^{57} -2.43488 q^{59} -5.71138 q^{61} +3.61185 q^{63} -6.91830 q^{67} +3.08344 q^{69} -0.0120411 q^{71} +15.2989 q^{73} -2.58191 q^{77} -10.6351 q^{79} +13.8260 q^{81} +5.64020 q^{83} +2.50467 q^{87} +9.06693 q^{89} +2.78763 q^{91} -6.51051 q^{93} -14.0720 q^{97} -30.2727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + q^{7} + 4 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 + q^7 + 4 * q^9	$ 5 q - 3 q^{3} + q^{7} + 4 q^{9} + 4 q^{11} - q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} - 4 q^{31} + 13 q^{33} + 6 q^{37} - 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49} + 5 q^{51} + 18 q^{53} + 27 q^{57} - 23 q^{59} - 26 q^{61} - 5 q^{63} - 3 q^{67} + 3 q^{69} + 2 q^{71} + 4 q^{73} + 15 q^{77} - 43 q^{79} - 3 q^{81} - 30 q^{83} - 27 q^{87} + 15 q^{89} + 19 q^{91} - 15 q^{93} + 8 q^{97} - 37 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + q^7 + 4 * q^9 + 4 * q^11 - q^13 + 5 * q^17 - 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 - 4 * q^31 + 13 * q^33 + 6 * q^37 - 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 + 5 * q^51 + 18 * q^53 + 27 * q^57 - 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 + 2 * q^71 + 4 * q^73 + 15 * q^77 - 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 + 19 * q^91 - 15 * q^93 + 8 * q^97 - 37 * 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
$3$	−3.08344	−1.78022	−0.890112	−	0.455742i	$-0.849374\pi$
$3$	−3.08344	−1.78022	−0.890112	+	0.455742i	$0.849374\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.555022	0.209779	0.104889	−	0.994484i	$-0.466551\pi$
$7$	0.555022	0.209779	0.104889	+	0.994484i	$0.466551\pi$
$8$	0	0
$9$	6.50759	2.16920
$10$	0	0
$11$	−4.65190	−1.40260	−0.701301	−	0.712866i	$-0.747397\pi$
$11$	−4.65190	−1.40260	−0.701301	+	0.712866i	$0.747397\pi$
$12$	0	0
$13$	5.02256	1.39301	0.696504	−	0.717553i	$-0.254738\pi$
$13$	5.02256	1.39301	0.696504	+	0.717553i	$0.254738\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.32867	0.322251	0.161125	−	0.986934i	$-0.448488\pi$
$17$	1.32867	0.322251	0.161125	+	0.986934i	$0.448488\pi$
$18$	0	0
$19$	0.196402	0.0450577	0.0225288	−	0.999746i	$-0.492828\pi$
$19$	0.196402	0.0450577	0.0225288	+	0.999746i	$0.492828\pi$
$20$	0	0
$21$	−1.71138	−0.373453
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−10.8154	−2.08143
$28$	0	0
$29$	−0.812298	−0.150840	−0.0754200	−	0.997152i	$-0.524030\pi$
$29$	−0.812298	−0.150840	−0.0754200	+	0.997152i	$0.524030\pi$
$30$	0	0
$31$	2.11145	0.379227	0.189613	−	0.981859i	$-0.439277\pi$
$31$	2.11145	0.379227	0.189613	+	0.981859i	$0.439277\pi$
$32$	0	0
$33$	14.3439	2.49694
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.64564	−0.928138	−0.464069	−	0.885799i	$-0.653611\pi$
$37$	−5.64564	−0.928138	−0.464069	+	0.885799i	$0.653611\pi$
$38$	0	0
$39$	−15.4868	−2.47987
$40$	0	0
$41$	−4.89714	−0.764804	−0.382402	−	0.923996i	$-0.624903\pi$
$41$	−4.89714	−0.764804	−0.382402	+	0.923996i	$0.624903\pi$
$42$	0	0
$43$	−1.66507	−0.253920	−0.126960	−	0.991908i	$-0.540522\pi$
$43$	−1.66507	−0.253920	−0.126960	+	0.991908i	$0.540522\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.89310	1.44306	0.721528	−	0.692385i	$-0.243440\pi$
$47$	9.89310	1.44306	0.721528	+	0.692385i	$0.243440\pi$
$48$	0	0
$49$	−6.69195	−0.955993
$50$	0	0
$51$	−4.09688	−0.573678
$52$	0	0
$53$	2.23261	0.306673	0.153336	−	0.988174i	$-0.450998\pi$
$53$	2.23261	0.306673	0.153336	+	0.988174i	$0.450998\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.605593	−0.0802127
$58$	0	0
$59$	−2.43488	−0.316994	−0.158497	−	0.987359i	$-0.550665\pi$
$59$	−2.43488	−0.316994	−0.158497	+	0.987359i	$0.550665\pi$
$60$	0	0
$61$	−5.71138	−0.731267	−0.365633	−	0.930759i	$-0.619148\pi$
$61$	−5.71138	−0.731267	−0.365633	+	0.930759i	$0.619148\pi$
$62$	0	0
$63$	3.61185	0.455051
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.91830	−0.845205	−0.422602	−	0.906315i	$-0.638883\pi$
$67$	−6.91830	−0.845205	−0.422602	+	0.906315i	$0.638883\pi$
$68$	0	0
$69$	3.08344	0.371202
$70$	0	0
$71$	−0.0120411	−0.00142902	−0.000714510	−	1.00000i	$-0.500227\pi$
$71$	−0.0120411	−0.00142902	−0.000714510		1.00000i	$0.500227\pi$
$72$	0	0
$73$	15.2989	1.79061	0.895303	−	0.445458i	$-0.146959\pi$
$73$	15.2989	1.79061	0.895303	+	0.445458i	$0.146959\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.58191	−0.294236
$78$	0	0
$79$	−10.6351	−1.19654	−0.598272	−	0.801293i	$-0.704146\pi$
$79$	−10.6351	−1.19654	−0.598272	+	0.801293i	$0.704146\pi$
$80$	0	0
$81$	13.8260	1.53622
$82$	0	0
$83$	5.64020	0.619092	0.309546	−	0.950884i	$-0.399823\pi$
$83$	5.64020	0.619092	0.309546	+	0.950884i	$0.399823\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.50467	0.268529
$88$	0	0
$89$	9.06693	0.961093	0.480547	−	0.876969i	$-0.340438\pi$
$89$	9.06693	0.961093	0.480547	+	0.876969i	$0.340438\pi$
$90$	0	0
$91$	2.78763	0.292223
$92$	0	0
$93$	−6.51051	−0.675108
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0720	−1.42880	−0.714398	−	0.699739i	$-0.753299\pi$
$97$	−14.0720	−1.42880	−0.714398	+	0.699739i	$0.753299\pi$
$98$	0	0
$99$	−30.2727	−3.04252
$100$	0	0
$101$	7.53015	0.749278	0.374639	−	0.927171i	$-0.377767\pi$
$101$	7.53015	0.749278	0.374639	+	0.927171i	$0.377767\pi$
$102$	0	0
$103$	14.4476	1.42357	0.711784	−	0.702399i	$-0.247888\pi$
$103$	14.4476	1.42357	0.711784	+	0.702399i	$0.247888\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.35862	−0.228016	−0.114008	−	0.993480i	$-0.536369\pi$
$107$	−2.35862	−0.228016	−0.114008	+	0.993480i	$0.536369\pi$
$108$	0	0
$109$	−10.3578	−0.992101	−0.496050	−	0.868294i	$-0.665217\pi$
$109$	−10.3578	−0.992101	−0.496050	+	0.868294i	$0.665217\pi$
$110$	0	0
$111$	17.4080	1.65229
$112$	0	0
$113$	−19.2723	−1.81298	−0.906491	−	0.422226i	$-0.861249\pi$
$113$	−19.2723	−1.81298	−0.906491	+	0.422226i	$0.861249\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	32.6848	3.02171
$118$	0	0
$119$	0.737442	0.0676012
$120$	0	0
$121$	10.6402	0.967291
$122$	0	0
$123$	15.1000	1.36152
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.4934	1.64102	0.820510	−	0.571632i	$-0.193689\pi$
$127$	18.4934	1.64102	0.820510	+	0.571632i	$0.193689\pi$
$128$	0	0
$129$	5.13413	0.452035
$130$	0	0
$131$	14.7727	1.29070	0.645351	−	0.763887i	$-0.276711\pi$
$131$	14.7727	1.29070	0.645351	+	0.763887i	$0.276711\pi$
$132$	0	0
$133$	0.109007	0.00945213
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.6173	0.992533	0.496266	−	0.868170i	$-0.334704\pi$
$137$	11.6173	0.992533	0.496266	+	0.868170i	$0.334704\pi$
$138$	0	0
$139$	−11.1389	−0.944787	−0.472393	−	0.881388i	$-0.656610\pi$
$139$	−11.1389	−0.944787	−0.472393	+	0.881388i	$0.656610\pi$
$140$	0	0
$141$	−30.5047	−2.56896
$142$	0	0
$143$	−23.3645	−1.95384
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.6342	1.70188
$148$	0	0
$149$	−12.4796	−1.02237	−0.511184	−	0.859472i	$-0.670793\pi$
$149$	−12.4796	−1.02237	−0.511184	+	0.859472i	$0.670793\pi$
$150$	0	0
$151$	−1.32673	−0.107968	−0.0539840	−	0.998542i	$-0.517192\pi$
$151$	−1.32673	−0.107968	−0.0539840	+	0.998542i	$0.517192\pi$
$152$	0	0
$153$	8.64646	0.699025
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.1754	1.84960	0.924798	−	0.380458i	$-0.124234\pi$
$157$	23.1754	1.84960	0.924798	+	0.380458i	$0.124234\pi$
$158$	0	0
$159$	−6.88412	−0.545946
$160$	0	0
$161$	−0.555022	−0.0437418
$162$	0	0
$163$	−25.0682	−1.96349	−0.981746	−	0.190196i	$-0.939088\pi$
$163$	−25.0682	−1.96349	−0.981746	+	0.190196i	$0.939088\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.39603	0.340175	0.170087	−	0.985429i	$-0.445595\pi$
$167$	4.39603	0.340175	0.170087	+	0.985429i	$0.445595\pi$
$168$	0	0
$169$	12.2261	0.940473
$170$	0	0
$171$	1.27810	0.0977389
$172$	0	0
$173$	19.4211	1.47656	0.738281	−	0.674493i	$-0.235638\pi$
$173$	19.4211	1.47656	0.738281	+	0.674493i	$0.235638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.50779	0.564320
$178$	0	0
$179$	−10.4169	−0.778592	−0.389296	−	0.921113i	$-0.627282\pi$
$179$	−10.4169	−0.778592	−0.389296	+	0.921113i	$0.627282\pi$
$180$	0	0
$181$	−9.13693	−0.679143	−0.339571	−	0.940580i	$-0.610282\pi$
$181$	−9.13693	−0.679143	−0.339571	+	0.940580i	$0.610282\pi$
$182$	0	0
$183$	17.6107	1.30182
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.18086	−0.451989
$188$	0	0
$189$	−6.00280	−0.436640
$190$	0	0
$191$	1.49819	0.108405	0.0542026	−	0.998530i	$-0.482738\pi$
$191$	1.49819	0.108405	0.0542026	+	0.998530i	$0.482738\pi$
$192$	0	0
$193$	14.4113	1.03735	0.518675	−	0.854972i	$-0.326425\pi$
$193$	14.4113	1.03735	0.518675	+	0.854972i	$0.326425\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.4686	1.74331	0.871657	−	0.490116i	$-0.163046\pi$
$197$	24.4686	1.74331	0.871657	+	0.490116i	$0.163046\pi$
$198$	0	0
$199$	1.61185	0.114261	0.0571307	−	0.998367i	$-0.481805\pi$
$199$	1.61185	0.114261	0.0571307	+	0.998367i	$0.481805\pi$
$200$	0	0
$201$	21.3321	1.50465
$202$	0	0
$203$	−0.450843	−0.0316430
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.50759	−0.452309
$208$	0	0
$209$	−0.913642	−0.0631979
$210$	0	0
$211$	−6.25081	−0.430324	−0.215162	−	0.976578i	$-0.569028\pi$
$211$	−6.25081	−0.430324	−0.215162	+	0.976578i	$0.569028\pi$
$212$	0	0
$213$	0.0371281	0.00254398
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.17190	0.0795536
$218$	0	0
$219$	−47.1733	−3.18768
$220$	0	0
$221$	6.67334	0.448898
$222$	0	0
$223$	−21.2953	−1.42604	−0.713019	−	0.701144i	$-0.752673\pi$
$223$	−21.2953	−1.42604	−0.713019	+	0.701144i	$0.752673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.5174	−0.698061	−0.349031	−	0.937111i	$-0.613489\pi$
$227$	−10.5174	−0.698061	−0.349031	+	0.937111i	$0.613489\pi$
$228$	0	0
$229$	−24.0063	−1.58638	−0.793190	−	0.608975i	$-0.791581\pi$
$229$	−24.0063	−1.58638	−0.793190	+	0.608975i	$0.791581\pi$
$230$	0	0
$231$	7.96115	0.523805
$232$	0	0
$233$	−11.3112	−0.741021	−0.370510	−	0.928828i	$-0.620817\pi$
$233$	−11.3112	−0.741021	−0.370510	+	0.928828i	$0.620817\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.7927	2.13012
$238$	0	0
$239$	25.5592	1.65329	0.826644	−	0.562726i	$-0.190247\pi$
$239$	25.5592	1.65329	0.826644	+	0.562726i	$0.190247\pi$
$240$	0	0
$241$	−16.1617	−1.04106	−0.520532	−	0.853842i	$-0.674266\pi$
$241$	−16.1617	−1.04106	−0.520532	+	0.853842i	$0.674266\pi$
$242$	0	0
$243$	−10.1852	−0.653380
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.986440	0.0627657
$248$	0	0
$249$	−17.3912	−1.10212
$250$	0	0
$251$	13.3497	0.842627	0.421313	−	0.906915i	$-0.361569\pi$
$251$	13.3497	0.842627	0.421313	+	0.906915i	$0.361569\pi$
$252$	0	0
$253$	4.65190	0.292463
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.1834	0.635222	0.317611	−	0.948221i	$-0.397119\pi$
$257$	10.1834	0.635222	0.317611	+	0.948221i	$0.397119\pi$
$258$	0	0
$259$	−3.13345	−0.194703
$260$	0	0
$261$	−5.28610	−0.327201
$262$	0	0
$263$	23.3073	1.43719	0.718594	−	0.695430i	$-0.244786\pi$
$263$	23.3073	1.43719	0.718594	+	0.695430i	$0.244786\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−27.9573	−1.71096
$268$	0	0
$269$	9.80817	0.598015	0.299007	−	0.954251i	$-0.403344\pi$
$269$	9.80817	0.598015	0.299007	+	0.954251i	$0.403344\pi$
$270$	0	0
$271$	6.08332	0.369535	0.184768	−	0.982782i	$-0.440847\pi$
$271$	6.08332	0.369535	0.184768	+	0.982782i	$0.440847\pi$
$272$	0	0
$273$	−8.59549	−0.520223
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.0445722	0.00267809	0.00133904	−	0.999999i	$-0.499574\pi$
$277$	0.0445722	0.00267809	0.00133904	+	0.999999i	$0.499574\pi$
$278$	0	0
$279$	13.7404	0.822617
$280$	0	0
$281$	27.9996	1.67032	0.835158	−	0.550010i	$-0.185376\pi$
$281$	27.9996	1.67032	0.835158	+	0.550010i	$0.185376\pi$
$282$	0	0
$283$	12.1798	0.724013	0.362006	−	0.932176i	$-0.382092\pi$
$283$	12.1798	0.724013	0.362006	+	0.932176i	$0.382092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.71802	−0.160440
$288$	0	0
$289$	−15.2346	−0.896155
$290$	0	0
$291$	43.3902	2.54358
$292$	0	0
$293$	2.71056	0.158352	0.0791762	−	0.996861i	$-0.474771\pi$
$293$	2.71056	0.158352	0.0791762	+	0.996861i	$0.474771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	50.3124	2.91942
$298$	0	0
$299$	−5.02256	−0.290462
$300$	0	0
$301$	−0.924148	−0.0532670
$302$	0	0
$303$	−23.2188	−1.33388
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.1631	−1.83564	−0.917821	−	0.396994i	$-0.870053\pi$
$307$	−32.1631	−1.83564	−0.917821	+	0.396994i	$0.870053\pi$
$308$	0	0
$309$	−44.5484	−2.53427
$310$	0	0
$311$	4.07951	0.231328	0.115664	−	0.993288i	$-0.463100\pi$
$311$	4.07951	0.231328	0.115664	+	0.993288i	$0.463100\pi$
$312$	0	0
$313$	−13.7806	−0.778925	−0.389462	−	0.921042i	$-0.627339\pi$
$313$	−13.7806	−0.778925	−0.389462	+	0.921042i	$0.627339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.4630	−0.587658	−0.293829	−	0.955858i	$-0.594930\pi$
$317$	−10.4630	−0.587658	−0.293829	+	0.955858i	$0.594930\pi$
$318$	0	0
$319$	3.77873	0.211568
$320$	0	0
$321$	7.27266	0.405920
$322$	0	0
$323$	0.260954	0.0145199
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.9377	1.76616
$328$	0	0
$329$	5.49088	0.302722
$330$	0	0
$331$	12.6451	0.695038	0.347519	−	0.937673i	$-0.387024\pi$
$331$	12.6451	0.695038	0.347519	+	0.937673i	$0.387024\pi$
$332$	0	0
$333$	−36.7395	−2.01331
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.5733	−1.33859	−0.669295	−	0.742997i	$-0.733404\pi$
$337$	−24.5733	−1.33859	−0.669295	+	0.742997i	$0.733404\pi$
$338$	0	0
$339$	59.4248	3.22751
$340$	0	0
$341$	−9.82224	−0.531904
$342$	0	0
$343$	−7.59933	−0.410325
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.50569	−0.188195	−0.0940977	−	0.995563i	$-0.529997\pi$
$347$	−3.50569	−0.188195	−0.0940977	+	0.995563i	$0.529997\pi$
$348$	0	0
$349$	−18.1085	−0.969327	−0.484664	−	0.874701i	$-0.661058\pi$
$349$	−18.1085	−0.969327	−0.484664	+	0.874701i	$0.661058\pi$
$350$	0	0
$351$	−54.3212	−2.89945
$352$	0	0
$353$	20.1915	1.07469	0.537343	−	0.843364i	$-0.319428\pi$
$353$	20.1915	1.07469	0.537343	+	0.843364i	$0.319428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.27386	−0.120345
$358$	0	0
$359$	9.43409	0.497912	0.248956	−	0.968515i	$-0.419912\pi$
$359$	9.43409	0.497912	0.248956	+	0.968515i	$0.419912\pi$
$360$	0	0
$361$	−18.9614	−0.997970
$362$	0	0
$363$	−32.8084	−1.72199
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−30.3412	−1.58380	−0.791899	−	0.610652i	$-0.790908\pi$
$367$	−30.3412	−1.58380	−0.791899	+	0.610652i	$0.790908\pi$
$368$	0	0
$369$	−31.8686	−1.65901
$370$	0	0
$371$	1.23915	0.0643333
$372$	0	0
$373$	−14.7893	−0.765758	−0.382879	−	0.923798i	$-0.625067\pi$
$373$	−14.7893	−0.765758	−0.382879	+	0.923798i	$0.625067\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.07982	−0.210121
$378$	0	0
$379$	11.2405	0.577385	0.288693	−	0.957422i	$-0.406779\pi$
$379$	11.2405	0.577385	0.288693	+	0.957422i	$0.406779\pi$
$380$	0	0
$381$	−57.0231	−2.92138
$382$	0	0
$383$	−30.4602	−1.55644	−0.778221	−	0.627991i	$-0.783878\pi$
$383$	−30.4602	−1.55644	−0.778221	+	0.627991i	$0.783878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.8356	−0.550803
$388$	0	0
$389$	1.94011	0.0983673	0.0491836	−	0.998790i	$-0.484338\pi$
$389$	1.94011	0.0983673	0.0491836	+	0.998790i	$0.484338\pi$
$390$	0	0
$391$	−1.32867	−0.0671939
$392$	0	0
$393$	−45.5509	−2.29774
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.3206	−0.919487	−0.459743	−	0.888052i	$-0.652059\pi$
$397$	−18.3206	−0.919487	−0.459743	+	0.888052i	$0.652059\pi$
$398$	0	0
$399$	−0.336117	−0.0168269
$400$	0	0
$401$	−1.32199	−0.0660170	−0.0330085	−	0.999455i	$-0.510509\pi$
$401$	−1.32199	−0.0660170	−0.0330085	+	0.999455i	$0.510509\pi$
$402$	0	0
$403$	10.6049	0.528266
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.2630	1.30181
$408$	0	0
$409$	21.9963	1.08765	0.543823	−	0.839200i	$-0.316976\pi$
$409$	21.9963	1.08765	0.543823	+	0.839200i	$0.316976\pi$
$410$	0	0
$411$	−35.8212	−1.76693
$412$	0	0
$413$	−1.35141	−0.0664985
$414$	0	0
$415$	0	0
$416$	0	0
$417$	34.3460	1.68193
$418$	0	0
$419$	−28.3194	−1.38349	−0.691746	−	0.722141i	$-0.743158\pi$
$419$	−28.3194	−1.38349	−0.691746	+	0.722141i	$0.743158\pi$
$420$	0	0
$421$	14.7650	0.719602	0.359801	−	0.933029i	$-0.382845\pi$
$421$	14.7650	0.719602	0.359801	+	0.933029i	$0.382845\pi$
$422$	0	0
$423$	64.3802	3.13027
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.16994	−0.153404
$428$	0	0
$429$	72.0429	3.47826
$430$	0	0
$431$	27.0514	1.30302	0.651510	−	0.758640i	$-0.274136\pi$
$431$	27.0514	1.30302	0.651510	+	0.758640i	$0.274136\pi$
$432$	0	0
$433$	−38.2700	−1.83914	−0.919570	−	0.392926i	$-0.871463\pi$
$433$	−38.2700	−1.83914	−0.919570	+	0.392926i	$0.871463\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.196402	−0.00939517
$438$	0	0
$439$	2.39959	0.114526	0.0572630	−	0.998359i	$-0.481763\pi$
$439$	2.39959	0.114526	0.0572630	+	0.998359i	$0.481763\pi$
$440$	0	0
$441$	−43.5485	−2.07374
$442$	0	0
$443$	−10.6542	−0.506198	−0.253099	−	0.967440i	$-0.581450\pi$
$443$	−10.6542	−0.506198	−0.253099	+	0.967440i	$0.581450\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	38.4800	1.82004
$448$	0	0
$449$	−28.9533	−1.36639	−0.683195	−	0.730236i	$-0.739410\pi$
$449$	−28.9533	−1.36639	−0.683195	+	0.730236i	$0.739410\pi$
$450$	0	0
$451$	22.7810	1.07272
$452$	0	0
$453$	4.09090	0.192207
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.5855	−1.80496	−0.902478	−	0.430736i	$-0.858254\pi$
$457$	−38.5855	−1.80496	−0.902478	+	0.430736i	$0.858254\pi$
$458$	0	0
$459$	−14.3702	−0.670743
$460$	0	0
$461$	−8.07659	−0.376164	−0.188082	−	0.982153i	$-0.560227\pi$
$461$	−8.07659	−0.376164	−0.188082	+	0.982153i	$0.560227\pi$
$462$	0	0
$463$	−6.07771	−0.282455	−0.141228	−	0.989977i	$-0.545105\pi$
$463$	−6.07771	−0.282455	−0.141228	+	0.989977i	$0.545105\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.2961	0.476446	0.238223	−	0.971210i	$-0.423435\pi$
$467$	10.2961	0.476446	0.238223	+	0.971210i	$0.423435\pi$
$468$	0	0
$469$	−3.83981	−0.177306
$470$	0	0
$471$	−71.4598	−3.29270
$472$	0	0
$473$	7.74572	0.356149
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14.5289	0.665233
$478$	0	0
$479$	33.9760	1.55240	0.776201	−	0.630486i	$-0.217144\pi$
$479$	33.9760	1.55240	0.776201	+	0.630486i	$0.217144\pi$
$480$	0	0
$481$	−28.3556	−1.29290
$482$	0	0
$483$	1.71138	0.0778703
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.4951	−0.747464	−0.373732	−	0.927537i	$-0.621922\pi$
$487$	−16.4951	−0.747464	−0.373732	+	0.927537i	$0.621922\pi$
$488$	0	0
$489$	77.2962	3.49546
$490$	0	0
$491$	−27.4876	−1.24050	−0.620249	−	0.784405i	$-0.712968\pi$
$491$	−27.4876	−1.24050	−0.620249	+	0.784405i	$0.712968\pi$
$492$	0	0
$493$	−1.07928	−0.0486082
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.00668310	−0.000299778 0
$498$	0	0
$499$	−18.2961	−0.819045	−0.409523	−	0.912300i	$-0.634305\pi$
$499$	−18.2961	−0.819045	−0.409523	+	0.912300i	$0.634305\pi$
$500$	0	0
$501$	−13.5549	−0.605587
$502$	0	0
$503$	1.97389	0.0880115	0.0440058	−	0.999031i	$-0.485988\pi$
$503$	1.97389	0.0880115	0.0440058	+	0.999031i	$0.485988\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−37.6986	−1.67425
$508$	0	0
$509$	−38.3441	−1.69957	−0.849787	−	0.527126i	$-0.823269\pi$
$509$	−38.3441	−1.69957	−0.849787	+	0.527126i	$0.823269\pi$
$510$	0	0
$511$	8.49125	0.375631
$512$	0	0
$513$	−2.12417	−0.0937844
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−46.0217	−2.02403
$518$	0	0
$519$	−59.8839	−2.62861
$520$	0	0
$521$	−23.1814	−1.01560	−0.507799	−	0.861476i	$-0.669541\pi$
$521$	−23.1814	−1.01560	−0.507799	+	0.861476i	$0.669541\pi$
$522$	0	0
$523$	43.5123	1.90266	0.951331	−	0.308172i	$-0.0997173\pi$
$523$	43.5123	1.90266	0.951331	+	0.308172i	$0.0997173\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.80542	0.122206
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−15.8452	−0.687622
$532$	0	0
$533$	−24.5962	−1.06538
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.1197	1.38607
$538$	0	0
$539$	31.1303	1.34088
$540$	0	0
$541$	18.6112	0.800158	0.400079	−	0.916481i	$-0.368983\pi$
$541$	18.6112	0.800158	0.400079	+	0.916481i	$0.368983\pi$
$542$	0	0
$543$	28.1732	1.20903
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.78956	−0.290300	−0.145150	−	0.989410i	$-0.546367\pi$
$547$	−6.78956	−0.290300	−0.145150	+	0.989410i	$0.546367\pi$
$548$	0	0
$549$	−37.1673	−1.58626
$550$	0	0
$551$	−0.159537	−0.00679649
$552$	0	0
$553$	−5.90272	−0.251009
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.9869	−1.05873	−0.529364	−	0.848395i	$-0.677570\pi$
$557$	−24.9869	−1.05873	−0.529364	+	0.848395i	$0.677570\pi$
$558$	0	0
$559$	−8.36290	−0.353713
$560$	0	0
$561$	19.0583	0.804642
$562$	0	0
$563$	−44.7551	−1.88620	−0.943100	−	0.332508i	$-0.892105\pi$
$563$	−44.7551	−1.88620	−0.943100	+	0.332508i	$0.892105\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.67371	0.322265
$568$	0	0
$569$	−27.4895	−1.15242	−0.576211	−	0.817301i	$-0.695469\pi$
$569$	−27.4895	−1.15242	−0.576211	+	0.817301i	$0.695469\pi$
$570$	0	0
$571$	1.40495	0.0587952	0.0293976	−	0.999568i	$-0.490641\pi$
$571$	1.40495	0.0587952	0.0293976	+	0.999568i	$0.490641\pi$
$572$	0	0
$573$	−4.61957	−0.192985
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.1987	0.799253	0.399627	−	0.916678i	$-0.369140\pi$
$577$	19.1987	0.799253	0.399627	+	0.916678i	$0.369140\pi$
$578$	0	0
$579$	−44.4364	−1.84671
$580$	0	0
$581$	3.13043	0.129872
$582$	0	0
$583$	−10.3859	−0.430139
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.8565	−0.943390	−0.471695	−	0.881762i	$-0.656358\pi$
$587$	−22.8565	−0.943390	−0.471695	+	0.881762i	$0.656358\pi$
$588$	0	0
$589$	0.414692	0.0170871
$590$	0	0
$591$	−75.4473	−3.10349
$592$	0	0
$593$	26.5387	1.08981	0.544906	−	0.838497i	$-0.316565\pi$
$593$	26.5387	1.08981	0.544906	+	0.838497i	$0.316565\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.97005	−0.203411
$598$	0	0
$599$	8.82768	0.360689	0.180345	−	0.983603i	$-0.442279\pi$
$599$	8.82768	0.360689	0.180345	+	0.983603i	$0.442279\pi$
$600$	0	0
$601$	−30.3072	−1.23626	−0.618129	−	0.786077i	$-0.712109\pi$
$601$	−30.3072	−1.23626	−0.618129	+	0.786077i	$0.712109\pi$
$602$	0	0
$603$	−45.0215	−1.83342
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.0102	1.54278	0.771392	−	0.636360i	$-0.219561\pi$
$607$	38.0102	1.54278	0.771392	+	0.636360i	$0.219561\pi$
$608$	0	0
$609$	1.39015	0.0563316
$610$	0	0
$611$	49.6887	2.01019
$612$	0	0
$613$	31.7417	1.28204	0.641018	−	0.767526i	$-0.278512\pi$
$613$	31.7417	1.28204	0.641018	+	0.767526i	$0.278512\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.6699	−1.87886	−0.939430	−	0.342741i	$-0.888645\pi$
$617$	−46.6699	−1.87886	−0.939430	+	0.342741i	$0.888645\pi$
$618$	0	0
$619$	−23.2223	−0.933383	−0.466692	−	0.884420i	$-0.654554\pi$
$619$	−23.2223	−0.933383	−0.466692	+	0.884420i	$0.654554\pi$
$620$	0	0
$621$	10.8154	0.434009
$622$	0	0
$623$	5.03235	0.201617
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.81716	0.112506
$628$	0	0
$629$	−7.50121	−0.299093
$630$	0	0
$631$	13.0906	0.521129	0.260565	−	0.965456i	$-0.416091\pi$
$631$	13.0906	0.521129	0.260565	+	0.965456i	$0.416091\pi$
$632$	0	0
$633$	19.2740	0.766072
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−33.6107	−1.33171
$638$	0	0
$639$	−0.0783588	−0.00309983
$640$	0	0
$641$	22.8746	0.903494	0.451747	−	0.892146i	$-0.350801\pi$
$641$	22.8746	0.903494	0.451747	+	0.892146i	$0.350801\pi$
$642$	0	0
$643$	−30.6333	−1.20806	−0.604030	−	0.796962i	$-0.706439\pi$
$643$	−30.6333	−1.20806	−0.604030	+	0.796962i	$0.706439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.74454	−0.343783	−0.171892	−	0.985116i	$-0.554988\pi$
$647$	−8.74454	−0.343783	−0.171892	+	0.985116i	$0.554988\pi$
$648$	0	0
$649$	11.3268	0.444616
$650$	0	0
$651$	−3.61348	−0.141623
$652$	0	0
$653$	24.7443	0.968320	0.484160	−	0.874979i	$-0.339125\pi$
$653$	24.7443	0.968320	0.484160	+	0.874979i	$0.339125\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	99.5593	3.88418
$658$	0	0
$659$	−8.30726	−0.323605	−0.161803	−	0.986823i	$-0.551731\pi$
$659$	−8.30726	−0.323605	−0.161803	+	0.986823i	$0.551731\pi$
$660$	0	0
$661$	−14.1755	−0.551364	−0.275682	−	0.961249i	$-0.588904\pi$
$661$	−14.1755	−0.551364	−0.275682	+	0.961249i	$0.588904\pi$
$662$	0	0
$663$	−20.5768	−0.799138
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.812298	0.0314523
$668$	0	0
$669$	65.6627	2.53867
$670$	0	0
$671$	26.5688	1.02568
$672$	0	0
$673$	−7.30229	−0.281482	−0.140741	−	0.990046i	$-0.544949\pi$
$673$	−7.30229	−0.281482	−0.140741	+	0.990046i	$0.544949\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.7090	−1.52614	−0.763071	−	0.646315i	$-0.776309\pi$
$677$	−39.7090	−1.52614	−0.763071	+	0.646315i	$0.776309\pi$
$678$	0	0
$679$	−7.81027	−0.299731
$680$	0	0
$681$	32.4296	1.24271
$682$	0	0
$683$	−38.5993	−1.47696	−0.738481	−	0.674274i	$-0.764457\pi$
$683$	−38.5993	−1.47696	−0.738481	+	0.674274i	$0.764457\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	74.0219	2.82411
$688$	0	0
$689$	11.2134	0.427198
$690$	0	0
$691$	21.4163	0.814713	0.407357	−	0.913269i	$-0.366451\pi$
$691$	21.4163	0.814713	0.407357	+	0.913269i	$0.366451\pi$
$692$	0	0
$693$	−16.8020	−0.638255
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.50669	−0.246459
$698$	0	0
$699$	34.8773	1.31918
$700$	0	0
$701$	−5.40147	−0.204011	−0.102005	−	0.994784i	$-0.532526\pi$
$701$	−5.40147	−0.204011	−0.102005	+	0.994784i	$0.532526\pi$
$702$	0	0
$703$	−1.10881	−0.0418197
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.17940	0.157182
$708$	0	0
$709$	−15.3161	−0.575208	−0.287604	−	0.957749i	$-0.592859\pi$
$709$	−15.3161	−0.575208	−0.287604	+	0.957749i	$0.592859\pi$
$710$	0	0
$711$	−69.2090	−2.59554
$712$	0	0
$713$	−2.11145	−0.0790742
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−78.8102	−2.94322
$718$	0	0
$719$	−11.6505	−0.434491	−0.217245	−	0.976117i	$-0.569707\pi$
$719$	−11.6505	−0.434491	−0.217245	+	0.976117i	$0.569707\pi$
$720$	0	0
$721$	8.01875	0.298634
$722$	0	0
$723$	49.8334	1.85333
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.8188	−1.06883	−0.534415	−	0.845223i	$-0.679468\pi$
$727$	−28.8188	−1.06883	−0.534415	+	0.845223i	$0.679468\pi$
$728$	0	0
$729$	−10.0725	−0.373056
$730$	0	0
$731$	−2.21233	−0.0818259
$732$	0	0
$733$	−25.2304	−0.931906	−0.465953	−	0.884810i	$-0.654288\pi$
$733$	−25.2304	−0.931906	−0.465953	+	0.884810i	$0.654288\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.1833	1.18549
$738$	0	0
$739$	1.22800	0.0451728	0.0225864	−	0.999745i	$-0.492810\pi$
$739$	1.22800	0.0451728	0.0225864	+	0.999745i	$0.492810\pi$
$740$	0	0
$741$	−3.04163	−0.111737
$742$	0	0
$743$	−35.1117	−1.28812	−0.644061	−	0.764974i	$-0.722752\pi$
$743$	−35.1117	−1.28812	−0.644061	+	0.764974i	$0.722752\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	36.7041	1.34293
$748$	0	0
$749$	−1.30909	−0.0478329
$750$	0	0
$751$	−31.7367	−1.15809	−0.579044	−	0.815296i	$-0.696574\pi$
$751$	−31.7367	−1.15809	−0.579044	+	0.815296i	$0.696574\pi$
$752$	0	0
$753$	−41.1630	−1.50006
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.22287	−0.117137	−0.0585685	−	0.998283i	$-0.518654\pi$
$757$	−3.22287	−0.117137	−0.0585685	+	0.998283i	$0.518654\pi$
$758$	0	0
$759$	−14.3439	−0.520649
$760$	0	0
$761$	−9.60077	−0.348027	−0.174014	−	0.984743i	$-0.555674\pi$
$761$	−9.60077	−0.348027	−0.174014	+	0.984743i	$0.555674\pi$
$762$	0	0
$763$	−5.74882	−0.208121
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.2293	−0.441575
$768$	0	0
$769$	14.9778	0.540113	0.270056	−	0.962844i	$-0.412958\pi$
$769$	14.9778	0.540113	0.270056	+	0.962844i	$0.412958\pi$
$770$	0	0
$771$	−31.3998	−1.13084
$772$	0	0
$773$	28.7430	1.03381	0.516906	−	0.856042i	$-0.327084\pi$
$773$	28.7430	1.03381	0.516906	+	0.856042i	$0.327084\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.66181	0.346616
$778$	0	0
$779$	−0.961806	−0.0344603
$780$	0	0
$781$	0.0560142	0.00200435
$782$	0	0
$783$	8.78536	0.313963
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.1965	1.21897	0.609486	−	0.792797i	$-0.291376\pi$
$787$	34.1965	1.21897	0.609486	+	0.792797i	$0.291376\pi$
$788$	0	0
$789$	−71.8666	−2.55852
$790$	0	0
$791$	−10.6965	−0.380324
$792$	0	0
$793$	−28.6857	−1.01866
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.303391	−0.0107467	−0.00537334	−	0.999986i	$-0.501710\pi$
$797$	−0.303391	−0.0107467	−0.00537334	+	0.999986i	$0.501710\pi$
$798$	0	0
$799$	13.1447	0.465026
$800$	0	0
$801$	59.0039	2.08480
$802$	0	0
$803$	−71.1692	−2.51151
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.2429	−1.06460
$808$	0	0
$809$	−13.1865	−0.463611	−0.231806	−	0.972762i	$-0.574463\pi$
$809$	−13.1865	−0.463611	−0.231806	+	0.972762i	$0.574463\pi$
$810$	0	0
$811$	−14.7284	−0.517185	−0.258593	−	0.965986i	$-0.583259\pi$
$811$	−14.7284	−0.517185	−0.258593	+	0.965986i	$0.583259\pi$
$812$	0	0
$813$	−18.7575	−0.657856
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.327022	−0.0114410
$818$	0	0
$819$	18.1408	0.633890
$820$	0	0
$821$	36.6177	1.27797	0.638984	−	0.769220i	$-0.279355\pi$
$821$	36.6177	1.27797	0.638984	+	0.769220i	$0.279355\pi$
$822$	0	0
$823$	−0.420867	−0.0146705	−0.00733525	−	0.999973i	$-0.502335\pi$
$823$	−0.420867	−0.0146705	−0.00733525	+	0.999973i	$0.502335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.1118	−1.22096	−0.610479	−	0.792032i	$-0.709023\pi$
$827$	−35.1118	−1.22096	−0.610479	+	0.792032i	$0.709023\pi$
$828$	0	0
$829$	−35.6234	−1.23725	−0.618626	−	0.785685i	$-0.712311\pi$
$829$	−35.6234	−1.23725	−0.618626	+	0.785685i	$0.712311\pi$
$830$	0	0
$831$	−0.137436	−0.00476759
$832$	0	0
$833$	−8.89141	−0.308069
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−22.8362	−0.789335
$838$	0	0
$839$	19.9998	0.690469	0.345235	−	0.938516i	$-0.387799\pi$
$839$	19.9998	0.690469	0.345235	+	0.938516i	$0.387799\pi$
$840$	0	0
$841$	−28.3402	−0.977247
$842$	0	0
$843$	−86.3350	−2.97354
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.90554	0.202917
$848$	0	0
$849$	−37.5556	−1.28890
$850$	0	0
$851$	5.64564	0.193530
$852$	0	0
$853$	42.4994	1.45515	0.727577	−	0.686026i	$-0.240647\pi$
$853$	42.4994	1.45515	0.727577	+	0.686026i	$0.240647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0100	−0.615208	−0.307604	−	0.951514i	$-0.599527\pi$
$857$	−18.0100	−0.615208	−0.307604	+	0.951514i	$0.599527\pi$
$858$	0	0
$859$	−22.3869	−0.763832	−0.381916	−	0.924197i	$-0.624736\pi$
$859$	−22.3869	−0.763832	−0.381916	+	0.924197i	$0.624736\pi$
$860$	0	0
$861$	8.38084	0.285618
$862$	0	0
$863$	−18.7613	−0.638642	−0.319321	−	0.947647i	$-0.603455\pi$
$863$	−18.7613	−0.638642	−0.319321	+	0.947647i	$0.603455\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	46.9750	1.59536
$868$	0	0
$869$	49.4735	1.67827
$870$	0	0
$871$	−34.7476	−1.17738
$872$	0	0
$873$	−91.5749	−3.09934
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.6345	−1.06822	−0.534111	−	0.845415i	$-0.679353\pi$
$877$	−31.6345	−1.06822	−0.534111	+	0.845415i	$0.679353\pi$
$878$	0	0
$879$	−8.35783	−0.281903
$880$	0	0
$881$	39.6850	1.33702	0.668512	−	0.743702i	$-0.266932\pi$
$881$	39.6850	1.33702	0.668512	+	0.743702i	$0.266932\pi$
$882$	0	0
$883$	10.3778	0.349242	0.174621	−	0.984636i	$-0.444130\pi$
$883$	10.3778	0.349242	0.174621	+	0.984636i	$0.444130\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0199	−0.806508	−0.403254	−	0.915088i	$-0.632121\pi$
$887$	−24.0199	−0.806508	−0.403254	+	0.915088i	$0.632121\pi$
$888$	0	0
$889$	10.2642	0.344251
$890$	0	0
$891$	−64.3170	−2.15470
$892$	0	0
$893$	1.94302	0.0650207
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.4868	0.517088
$898$	0	0
$899$	−1.71512	−0.0572025
$900$	0	0
$901$	2.96641	0.0988254
$902$	0	0
$903$	2.84955	0.0948271
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.5181	0.614885	0.307442	−	0.951567i	$-0.400527\pi$
$907$	18.5181	0.614885	0.307442	+	0.951567i	$0.400527\pi$
$908$	0	0
$909$	49.0032	1.62533
$910$	0	0
$911$	3.34956	0.110976	0.0554879	−	0.998459i	$-0.482329\pi$
$911$	3.34956	0.110976	0.0554879	+	0.998459i	$0.482329\pi$
$912$	0	0
$913$	−26.2376	−0.868339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.19920	0.270761
$918$	0	0
$919$	32.9754	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$919$	32.9754	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$920$	0	0
$921$	99.1728	3.26785
$922$	0	0
$923$	−0.0604774	−0.00199064
$924$	0	0
$925$	0	0
$926$	0	0
$927$	94.0193	3.08800
$928$	0	0
$929$	−36.5008	−1.19755	−0.598777	−	0.800916i	$-0.704346\pi$
$929$	−36.5008	−1.19755	−0.598777	+	0.800916i	$0.704346\pi$
$930$	0	0
$931$	−1.31431	−0.0430748
$932$	0	0
$933$	−12.5789	−0.411816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.4598	−0.407043	−0.203521	−	0.979071i	$-0.565239\pi$
$937$	−12.4598	−0.407043	−0.203521	+	0.979071i	$0.565239\pi$
$938$	0	0
$939$	42.4916	1.38666
$940$	0	0
$941$	−45.2938	−1.47653	−0.738267	−	0.674508i	$-0.764356\pi$
$941$	−45.2938	−1.47653	−0.738267	+	0.674508i	$0.764356\pi$
$942$	0	0
$943$	4.89714	0.159473
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.4016	1.34537	0.672685	−	0.739929i	$-0.265141\pi$
$947$	41.4016	1.34537	0.672685	+	0.739929i	$0.265141\pi$
$948$	0	0
$949$	76.8399	2.49433
$950$	0	0
$951$	32.2619	1.04616
$952$	0	0
$953$	15.3738	0.498006	0.249003	−	0.968503i	$-0.419897\pi$
$953$	15.3738	0.498006	0.249003	+	0.968503i	$0.419897\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−11.6515	−0.376639
$958$	0	0
$959$	6.44785	0.208212
$960$	0	0
$961$	−26.5418	−0.856187
$962$	0	0
$963$	−15.3489	−0.494612
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53.4925	1.72020	0.860101	−	0.510124i	$-0.170400\pi$
$967$	53.4925	1.72020	0.860101	+	0.510124i	$0.170400\pi$
$968$	0	0
$969$	−0.804635	−0.0258486
$970$	0	0
$971$	−19.8881	−0.638241	−0.319120	−	0.947714i	$-0.603387\pi$
$971$	−19.8881	−0.638241	−0.319120	+	0.947714i	$0.603387\pi$
$972$	0	0
$973$	−6.18231	−0.198196
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.8758	0.475919	0.237960	−	0.971275i	$-0.423521\pi$
$977$	14.8758	0.475919	0.237960	+	0.971275i	$0.423521\pi$
$978$	0	0
$979$	−42.1785	−1.34803
$980$	0	0
$981$	−67.4045	−2.15206
$982$	0	0
$983$	−1.16403	−0.0371269	−0.0185634	−	0.999828i	$-0.505909\pi$
$983$	−1.16403	−0.0371269	−0.0185634	+	0.999828i	$0.505909\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−16.9308	−0.538913
$988$	0	0
$989$	1.66507	0.0529460
$990$	0	0
$991$	47.8693	1.52062	0.760310	−	0.649561i	$-0.225047\pi$
$991$	47.8693	1.52062	0.760310	+	0.649561i	$0.225047\pi$
$992$	0	0
$993$	−38.9904	−1.23732
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.4353	0.362159	0.181079	−	0.983468i	$-0.442041\pi$
$997$	11.4353	0.362159	0.181079	+	0.983468i	$0.442041\pi$
$998$	0	0
$999$	61.0601	1.93186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cs.1.1		5
4.3	odd	2		4600.2.a.bg.1.5	yes	5
5.4	even	2		9200.2.a.cw.1.5		5
20.3	even	4		4600.2.e.v.4049.10		10
20.7	even	4		4600.2.e.v.4049.1		10
20.19	odd	2		4600.2.a.bc.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.1	✓	5	20.19	odd	2
4600.2.a.bg.1.5	yes	5	4.3	odd	2
4600.2.e.v.4049.1		10	20.7	even	4
4600.2.e.v.4049.10		10	20.3	even	4
9200.2.a.cs.1.1		5	1.1	even	1	trivial
9200.2.a.cw.1.5		5	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)

\(f(q)\)	\(=\)	\(q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} +O(q^{10})\)	\(q-3.08344 q^{3} +0.555022 q^{7} +6.50759 q^{9} -4.65190 q^{11} +5.02256 q^{13} +1.32867 q^{17} +0.196402 q^{19} -1.71138 q^{21} -1.00000 q^{23} -10.8154 q^{27} -0.812298 q^{29} +2.11145 q^{31} +14.3439 q^{33} -5.64564 q^{37} -15.4868 q^{39} -4.89714 q^{41} -1.66507 q^{43} +9.89310 q^{47} -6.69195 q^{49} -4.09688 q^{51} +2.23261 q^{53} -0.605593 q^{57} -2.43488 q^{59} -5.71138 q^{61} +3.61185 q^{63} -6.91830 q^{67} +3.08344 q^{69} -0.0120411 q^{71} +15.2989 q^{73} -2.58191 q^{77} -10.6351 q^{79} +13.8260 q^{81} +5.64020 q^{83} +2.50467 q^{87} +9.06693 q^{89} +2.78763 q^{91} -6.51051 q^{93} -14.0720 q^{97} -30.2727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 + q^7 + 4 * q^9	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9} + 4 q^{11} - q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} - 4 q^{31} + 13 q^{33} + 6 q^{37} - 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49} + 5 q^{51} + 18 q^{53} + 27 q^{57} - 23 q^{59} - 26 q^{61} - 5 q^{63} - 3 q^{67} + 3 q^{69} + 2 q^{71} + 4 q^{73} + 15 q^{77} - 43 q^{79} - 3 q^{81} - 30 q^{83} - 27 q^{87} + 15 q^{89} + 19 q^{91} - 15 q^{93} + 8 q^{97} - 37 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + q^7 + 4 * q^9 + 4 * q^11 - q^13 + 5 * q^17 - 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 - 4 * q^31 + 13 * q^33 + 6 * q^37 - 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 + 5 * q^51 + 18 * q^53 + 27 * q^57 - 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 + 2 * q^71 + 4 * q^73 + 15 * q^77 - 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 + 19 * q^91 - 15 * q^93 + 8 * q^97 - 37 * q^99

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