LMFDB - Embedded newform 9200.2.a.cx.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,-4,0,0,0,-9,0,10,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.143376304.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 22x^{2} - 6x - 1 $ x^6 - 12x^4 + 22x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.116918$ 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)

$f(q)$	$=$	$q+0.486391 q^{3} +1.80495 q^{7} -2.76342 q^{9} +2.90652 q^{11} +2.25256 q^{13} -2.14477 q^{17} -0.339824 q^{19} +0.877911 q^{21} -1.00000 q^{23} -2.80328 q^{27} -5.60395 q^{29} -5.92083 q^{31} +1.41371 q^{33} +8.98088 q^{37} +1.09562 q^{39} +1.89222 q^{41} -9.47322 q^{43} -7.83384 q^{47} -3.74216 q^{49} -1.04320 q^{51} -6.47764 q^{53} -0.165287 q^{57} -5.17914 q^{59} -9.12565 q^{61} -4.98784 q^{63} -9.25423 q^{67} -0.486391 q^{69} +4.60255 q^{71} +11.3300 q^{73} +5.24613 q^{77} -7.94972 q^{79} +6.92678 q^{81} +5.37849 q^{83} -2.72571 q^{87} +12.9258 q^{89} +4.06575 q^{91} -2.87984 q^{93} +2.43210 q^{97} -8.03196 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 + 5 * q^17 - 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 - 9 * q^31 - 10 * q^33 + 21 * q^37 + 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 + 12 * q^51 - q^53 + 12 * q^57 + 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 + 17 * q^71 - 14 * q^73 + 20 * q^77 - 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 + 4 * q^91 + 4 * q^97 + 16 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.486391	0.280818	0.140409	−	0.990094i	$-0.455158\pi$
$3$	0.486391	0.280818	0.140409	+	0.990094i	$0.455158\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.80495	0.682207	0.341103	−	0.940026i	$-0.389199\pi$
$7$	1.80495	0.682207	0.341103	+	0.940026i	$0.389199\pi$
$8$	0	0
$9$	−2.76342	−0.921141
$10$	0	0
$11$	2.90652	0.876350	0.438175	−	0.898890i	$-0.355625\pi$
$11$	2.90652	0.876350	0.438175	+	0.898890i	$0.355625\pi$
$12$	0	0
$13$	2.25256	0.624747	0.312373	−	0.949959i	$-0.398876\pi$
$13$	2.25256	0.624747	0.312373	+	0.949959i	$0.398876\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.14477	−0.520184	−0.260092	−	0.965584i	$-0.583753\pi$
$17$	−2.14477	−0.520184	−0.260092	+	0.965584i	$0.583753\pi$
$18$	0	0
$19$	−0.339824	−0.0779610	−0.0389805	−	0.999240i	$-0.512411\pi$
$19$	−0.339824	−0.0779610	−0.0389805	+	0.999240i	$0.512411\pi$
$20$	0	0
$21$	0.877911	0.191576
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.80328	−0.539491
$28$	0	0
$29$	−5.60395	−1.04063	−0.520313	−	0.853975i	$-0.674185\pi$
$29$	−5.60395	−1.04063	−0.520313	+	0.853975i	$0.674185\pi$
$30$	0	0
$31$	−5.92083	−1.06341	−0.531706	−	0.846929i	$-0.678449\pi$
$31$	−5.92083	−1.06341	−0.531706	+	0.846929i	$0.678449\pi$
$32$	0	0
$33$	1.41371	0.246095
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.98088	1.47645	0.738224	−	0.674556i	$-0.235665\pi$
$37$	8.98088	1.47645	0.738224	+	0.674556i	$0.235665\pi$
$38$	0	0
$39$	1.09562	0.175440
$40$	0	0
$41$	1.89222	0.295515	0.147757	−	0.989024i	$-0.452795\pi$
$41$	1.89222	0.295515	0.147757	+	0.989024i	$0.452795\pi$
$42$	0	0
$43$	−9.47322	−1.44465	−0.722327	−	0.691552i	$-0.756927\pi$
$43$	−9.47322	−1.44465	−0.722327	+	0.691552i	$0.756927\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.83384	−1.14268	−0.571342	−	0.820712i	$-0.693577\pi$
$47$	−7.83384	−1.14268	−0.571342	+	0.820712i	$0.693577\pi$
$48$	0	0
$49$	−3.74216	−0.534594
$50$	0	0
$51$	−1.04320	−0.146077
$52$	0	0
$53$	−6.47764	−0.889772	−0.444886	−	0.895587i	$-0.646756\pi$
$53$	−6.47764	−0.889772	−0.444886	+	0.895587i	$0.646756\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.165287	−0.0218928
$58$	0	0
$59$	−5.17914	−0.674267	−0.337134	−	0.941457i	$-0.609457\pi$
$59$	−5.17914	−0.674267	−0.337134	+	0.941457i	$0.609457\pi$
$60$	0	0
$61$	−9.12565	−1.16842	−0.584210	−	0.811602i	$-0.698596\pi$
$61$	−9.12565	−1.16842	−0.584210	+	0.811602i	$0.698596\pi$
$62$	0	0
$63$	−4.98784	−0.628409
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.25423	−1.13058	−0.565292	−	0.824891i	$-0.691236\pi$
$67$	−9.25423	−1.13058	−0.565292	+	0.824891i	$0.691236\pi$
$68$	0	0
$69$	−0.486391	−0.0585546
$70$	0	0
$71$	4.60255	0.546222	0.273111	−	0.961982i	$-0.411947\pi$
$71$	4.60255	0.546222	0.273111	+	0.961982i	$0.411947\pi$
$72$	0	0
$73$	11.3300	1.32608	0.663038	−	0.748585i	$-0.269267\pi$
$73$	11.3300	1.32608	0.663038	+	0.748585i	$0.269267\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.24613	0.597852
$78$	0	0
$79$	−7.94972	−0.894414	−0.447207	−	0.894431i	$-0.647581\pi$
$79$	−7.94972	−0.894414	−0.447207	+	0.894431i	$0.647581\pi$
$80$	0	0
$81$	6.92678	0.769643
$82$	0	0
$83$	5.37849	0.590366	0.295183	−	0.955441i	$-0.404619\pi$
$83$	5.37849	0.590366	0.295183	+	0.955441i	$0.404619\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.72571	−0.292227
$88$	0	0
$89$	12.9258	1.37014	0.685069	−	0.728479i	$-0.259772\pi$
$89$	12.9258	1.37014	0.685069	+	0.728479i	$0.259772\pi$
$90$	0	0
$91$	4.06575	0.426206
$92$	0	0
$93$	−2.87984	−0.298625
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.43210	0.246942	0.123471	−	0.992348i	$-0.460597\pi$
$97$	2.43210	0.246942	0.123471	+	0.992348i	$0.460597\pi$
$98$	0	0
$99$	−8.03196	−0.807242
$100$	0	0
$101$	2.89635	0.288198	0.144099	−	0.989563i	$-0.453972\pi$
$101$	2.89635	0.288198	0.144099	+	0.989563i	$0.453972\pi$
$102$	0	0
$103$	10.3524	1.02005	0.510027	−	0.860159i	$-0.329636\pi$
$103$	10.3524	1.02005	0.510027	+	0.860159i	$0.329636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.2395	−1.47326	−0.736630	−	0.676296i	$-0.763584\pi$
$107$	−15.2395	−1.47326	−0.736630	+	0.676296i	$0.763584\pi$
$108$	0	0
$109$	−7.00555	−0.671010	−0.335505	−	0.942038i	$-0.608907\pi$
$109$	−7.00555	−0.671010	−0.335505	+	0.942038i	$0.608907\pi$
$110$	0	0
$111$	4.36822	0.414613
$112$	0	0
$113$	−6.48460	−0.610020	−0.305010	−	0.952349i	$-0.598660\pi$
$113$	−6.48460	−0.610020	−0.305010	+	0.952349i	$0.598660\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.22477	−0.575480
$118$	0	0
$119$	−3.87121	−0.354873
$120$	0	0
$121$	−2.55212	−0.232011
$122$	0	0
$123$	0.920357	0.0829858
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.28324	−0.557547	−0.278774	−	0.960357i	$-0.589928\pi$
$127$	−6.28324	−0.557547	−0.278774	+	0.960357i	$0.589928\pi$
$128$	0	0
$129$	−4.60769	−0.405684
$130$	0	0
$131$	7.90426	0.690598	0.345299	−	0.938493i	$-0.387777\pi$
$131$	7.90426	0.690598	0.345299	+	0.938493i	$0.387777\pi$
$132$	0	0
$133$	−0.613365	−0.0531855
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.28955	0.708224	0.354112	−	0.935203i	$-0.384783\pi$
$137$	8.28955	0.708224	0.354112	+	0.935203i	$0.384783\pi$
$138$	0	0
$139$	8.97515	0.761262	0.380631	−	0.924727i	$-0.375707\pi$
$139$	8.97515	0.761262	0.380631	+	0.924727i	$0.375707\pi$
$140$	0	0
$141$	−3.81031	−0.320886
$142$	0	0
$143$	6.54711	0.547497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.82015	−0.150124
$148$	0	0
$149$	5.39792	0.442215	0.221107	−	0.975249i	$-0.429033\pi$
$149$	5.39792	0.442215	0.221107	+	0.975249i	$0.429033\pi$
$150$	0	0
$151$	−22.6083	−1.83984	−0.919919	−	0.392108i	$-0.871746\pi$
$151$	−22.6083	−1.83984	−0.919919	+	0.392108i	$0.871746\pi$
$152$	0	0
$153$	5.92692	0.479163
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.00033	−0.638496	−0.319248	−	0.947671i	$-0.603430\pi$
$157$	−8.00033	−0.638496	−0.319248	+	0.947671i	$0.603430\pi$
$158$	0	0
$159$	−3.15066	−0.249864
$160$	0	0
$161$	−1.80495	−0.142250
$162$	0	0
$163$	−18.6504	−1.46081	−0.730404	−	0.683015i	$-0.760668\pi$
$163$	−18.6504	−1.46081	−0.730404	+	0.683015i	$0.760668\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.13547	−0.242630	−0.121315	−	0.992614i	$-0.538711\pi$
$167$	−3.13547	−0.242630	−0.121315	+	0.992614i	$0.538711\pi$
$168$	0	0
$169$	−7.92599	−0.609692
$170$	0	0
$171$	0.939078	0.0718131
$172$	0	0
$173$	−7.56301	−0.575005	−0.287503	−	0.957780i	$-0.592825\pi$
$173$	−7.56301	−0.575005	−0.287503	+	0.957780i	$0.592825\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.51909	−0.189346
$178$	0	0
$179$	−9.73608	−0.727709	−0.363855	−	0.931456i	$-0.618539\pi$
$179$	−9.73608	−0.727709	−0.363855	+	0.931456i	$0.618539\pi$
$180$	0	0
$181$	0.260581	0.0193688	0.00968440	−	0.999953i	$-0.496917\pi$
$181$	0.260581	0.0193688	0.00968440	+	0.999953i	$0.496917\pi$
$182$	0	0
$183$	−4.43863	−0.328113
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.23384	−0.455863
$188$	0	0
$189$	−5.05977	−0.368044
$190$	0	0
$191$	18.1730	1.31495	0.657476	−	0.753475i	$-0.271624\pi$
$191$	18.1730	1.31495	0.657476	+	0.753475i	$0.271624\pi$
$192$	0	0
$193$	2.15073	0.154813	0.0774063	−	0.997000i	$-0.475336\pi$
$193$	2.15073	0.154813	0.0774063	+	0.997000i	$0.475336\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.0886	1.92998	0.964992	−	0.262279i	$-0.0844742\pi$
$197$	27.0886	1.92998	0.964992	+	0.262279i	$0.0844742\pi$
$198$	0	0
$199$	14.2178	1.00787	0.503937	−	0.863741i	$-0.331884\pi$
$199$	14.2178	1.00787	0.503937	+	0.863741i	$0.331884\pi$
$200$	0	0
$201$	−4.50117	−0.317488
$202$	0	0
$203$	−10.1148	−0.709922
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.76342	0.192071
$208$	0	0
$209$	−0.987706	−0.0683211
$210$	0	0
$211$	−1.89855	−0.130701	−0.0653506	−	0.997862i	$-0.520817\pi$
$211$	−1.89855	−0.130701	−0.0653506	+	0.997862i	$0.520817\pi$
$212$	0	0
$213$	2.23864	0.153389
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.6868	−0.725467
$218$	0	0
$219$	5.51081	0.372386
$220$	0	0
$221$	−4.83122	−0.324983
$222$	0	0
$223$	−20.5574	−1.37663	−0.688313	−	0.725414i	$-0.741648\pi$
$223$	−20.5574	−1.37663	−0.688313	+	0.725414i	$0.741648\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.67464	0.509384	0.254692	−	0.967022i	$-0.418026\pi$
$227$	7.67464	0.509384	0.254692	+	0.967022i	$0.418026\pi$
$228$	0	0
$229$	8.50379	0.561946	0.280973	−	0.959716i	$-0.409343\pi$
$229$	8.50379	0.561946	0.280973	+	0.959716i	$0.409343\pi$
$230$	0	0
$231$	2.55167	0.167887
$232$	0	0
$233$	2.37058	0.155302	0.0776511	−	0.996981i	$-0.475258\pi$
$233$	2.37058	0.155302	0.0776511	+	0.996981i	$0.475258\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.86667	−0.251167
$238$	0	0
$239$	0.298079	0.0192811	0.00964057	−	0.999954i	$-0.496931\pi$
$239$	0.298079	0.0192811	0.00964057	+	0.999954i	$0.496931\pi$
$240$	0	0
$241$	29.7779	1.91817	0.959083	−	0.283126i	$-0.0913714\pi$
$241$	29.7779	1.91817	0.959083	+	0.283126i	$0.0913714\pi$
$242$	0	0
$243$	11.7790	0.755620
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.765472	−0.0487058
$248$	0	0
$249$	2.61605	0.165785
$250$	0	0
$251$	−10.1766	−0.642341	−0.321171	−	0.947021i	$-0.604076\pi$
$251$	−10.1766	−0.642341	−0.321171	+	0.947021i	$0.604076\pi$
$252$	0	0
$253$	−2.90652	−0.182732
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.2919	−1.76480	−0.882400	−	0.470501i	$-0.844073\pi$
$257$	−28.2919	−1.76480	−0.882400	+	0.470501i	$0.844073\pi$
$258$	0	0
$259$	16.2100	1.00724
$260$	0	0
$261$	15.4861	0.958564
$262$	0	0
$263$	−24.9182	−1.53652	−0.768260	−	0.640138i	$-0.778877\pi$
$263$	−24.9182	−1.53652	−0.768260	+	0.640138i	$0.778877\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.28701	0.384759
$268$	0	0
$269$	25.8948	1.57883	0.789416	−	0.613859i	$-0.210384\pi$
$269$	25.8948	1.57883	0.789416	+	0.613859i	$0.210384\pi$
$270$	0	0
$271$	−3.49603	−0.212369	−0.106184	−	0.994346i	$-0.533863\pi$
$271$	−3.49603	−0.212369	−0.106184	+	0.994346i	$0.533863\pi$
$272$	0	0
$273$	1.97754	0.119686
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.38317	0.0831064	0.0415532	−	0.999136i	$-0.486769\pi$
$277$	1.38317	0.0831064	0.0415532	+	0.999136i	$0.486769\pi$
$278$	0	0
$279$	16.3618	0.979553
$280$	0	0
$281$	15.3144	0.913582	0.456791	−	0.889574i	$-0.348999\pi$
$281$	15.3144	0.913582	0.456791	+	0.889574i	$0.348999\pi$
$282$	0	0
$283$	−11.8266	−0.703019	−0.351510	−	0.936184i	$-0.614332\pi$
$283$	−11.8266	−0.703019	−0.351510	+	0.936184i	$0.614332\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.41536	0.201602
$288$	0	0
$289$	−12.3999	−0.729409
$290$	0	0
$291$	1.18295	0.0693458
$292$	0	0
$293$	−31.4274	−1.83601	−0.918005	−	0.396569i	$-0.870201\pi$
$293$	−31.4274	−1.83601	−0.918005	+	0.396569i	$0.870201\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.14779	−0.472783
$298$	0	0
$299$	−2.25256	−0.130269
$300$	0	0
$301$	−17.0987	−0.985552
$302$	0	0
$303$	1.40876	0.0809311
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.43504	−0.0819018	−0.0409509	−	0.999161i	$-0.513039\pi$
$307$	−1.43504	−0.0819018	−0.0409509	+	0.999161i	$0.513039\pi$
$308$	0	0
$309$	5.03532	0.286449
$310$	0	0
$311$	−20.9698	−1.18909	−0.594544	−	0.804063i	$-0.702667\pi$
$311$	−20.9698	−1.18909	−0.594544	+	0.804063i	$0.702667\pi$
$312$	0	0
$313$	0.535929	0.0302925	0.0151463	−	0.999885i	$-0.495179\pi$
$313$	0.535929	0.0302925	0.0151463	+	0.999885i	$0.495179\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.61790	−0.371698	−0.185849	−	0.982578i	$-0.559504\pi$
$317$	−6.61790	−0.371698	−0.185849	+	0.982578i	$0.559504\pi$
$318$	0	0
$319$	−16.2880	−0.911953
$320$	0	0
$321$	−7.41236	−0.413718
$322$	0	0
$323$	0.728845	0.0405540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.40744	−0.188432
$328$	0	0
$329$	−14.1397	−0.779546
$330$	0	0
$331$	1.81617	0.0998255	0.0499127	−	0.998754i	$-0.484106\pi$
$331$	1.81617	0.0998255	0.0499127	+	0.998754i	$0.484106\pi$
$332$	0	0
$333$	−24.8180	−1.36002
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.3308	−0.889593	−0.444796	−	0.895632i	$-0.646724\pi$
$337$	−16.3308	−0.889593	−0.444796	+	0.895632i	$0.646724\pi$
$338$	0	0
$339$	−3.15405	−0.171304
$340$	0	0
$341$	−17.2090	−0.931922
$342$	0	0
$343$	−19.3891	−1.04691
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.1771	−0.653700	−0.326850	−	0.945076i	$-0.605987\pi$
$347$	−12.1771	−0.653700	−0.326850	+	0.945076i	$0.605987\pi$
$348$	0	0
$349$	12.0364	0.644293	0.322147	−	0.946690i	$-0.395596\pi$
$349$	12.0364	0.644293	0.322147	+	0.946690i	$0.395596\pi$
$350$	0	0
$351$	−6.31454	−0.337045
$352$	0	0
$353$	−30.0447	−1.59912	−0.799559	−	0.600588i	$-0.794933\pi$
$353$	−30.0447	−1.59912	−0.799559	+	0.600588i	$0.794933\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.88292	−0.0996547
$358$	0	0
$359$	29.6859	1.56676	0.783380	−	0.621543i	$-0.213494\pi$
$359$	29.6859	1.56676	0.783380	+	0.621543i	$0.213494\pi$
$360$	0	0
$361$	−18.8845	−0.993922
$362$	0	0
$363$	−1.24133	−0.0651527
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.4413	0.649428	0.324714	−	0.945812i	$-0.394732\pi$
$367$	12.4413	0.649428	0.324714	+	0.945812i	$0.394732\pi$
$368$	0	0
$369$	−5.22900	−0.272211
$370$	0	0
$371$	−11.6918	−0.607008
$372$	0	0
$373$	−28.5580	−1.47867	−0.739337	−	0.673335i	$-0.764861\pi$
$373$	−28.5580	−1.47867	−0.739337	+	0.673335i	$0.764861\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.6232	−0.650128
$378$	0	0
$379$	−14.8436	−0.762465	−0.381233	−	0.924479i	$-0.624500\pi$
$379$	−14.8436	−0.762465	−0.381233	+	0.924479i	$0.624500\pi$
$380$	0	0
$381$	−3.05611	−0.156569
$382$	0	0
$383$	7.00776	0.358080	0.179040	−	0.983842i	$-0.442701\pi$
$383$	7.00776	0.358080	0.179040	+	0.983842i	$0.442701\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.1785	1.33073
$388$	0	0
$389$	29.2715	1.48413	0.742063	−	0.670330i	$-0.233847\pi$
$389$	29.2715	1.48413	0.742063	+	0.670330i	$0.233847\pi$
$390$	0	0
$391$	2.14477	0.108466
$392$	0	0
$393$	3.84456	0.193932
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.9288	1.65265	0.826324	−	0.563196i	$-0.190428\pi$
$397$	32.9288	1.65265	0.826324	+	0.563196i	$0.190428\pi$
$398$	0	0
$399$	−0.298335	−0.0149354
$400$	0	0
$401$	−24.3403	−1.21550	−0.607749	−	0.794129i	$-0.707927\pi$
$401$	−24.3403	−1.21550	−0.607749	+	0.794129i	$0.707927\pi$
$402$	0	0
$403$	−13.3370	−0.664363
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.1031	1.29389
$408$	0	0
$409$	−5.84661	−0.289096	−0.144548	−	0.989498i	$-0.546173\pi$
$409$	−5.84661	−0.289096	−0.144548	+	0.989498i	$0.546173\pi$
$410$	0	0
$411$	4.03196	0.198882
$412$	0	0
$413$	−9.34809	−0.459990
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.36543	0.213776
$418$	0	0
$419$	28.0557	1.37061	0.685304	−	0.728257i	$-0.259669\pi$
$419$	28.0557	1.37061	0.685304	+	0.728257i	$0.259669\pi$
$420$	0	0
$421$	−30.0719	−1.46561	−0.732807	−	0.680437i	$-0.761790\pi$
$421$	−30.0719	−1.46561	−0.732807	+	0.680437i	$0.761790\pi$
$422$	0	0
$423$	21.6482	1.05257
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.4713	−0.797104
$428$	0	0
$429$	3.18445	0.153747
$430$	0	0
$431$	9.92111	0.477883	0.238941	−	0.971034i	$-0.423200\pi$
$431$	9.92111	0.477883	0.238941	+	0.971034i	$0.423200\pi$
$432$	0	0
$433$	9.82152	0.471992	0.235996	−	0.971754i	$-0.424165\pi$
$433$	9.82152	0.471992	0.235996	+	0.971754i	$0.424165\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.339824	0.0162560
$438$	0	0
$439$	−24.4026	−1.16467	−0.582336	−	0.812948i	$-0.697861\pi$
$439$	−24.4026	−1.16467	−0.582336	+	0.812948i	$0.697861\pi$
$440$	0	0
$441$	10.3412	0.492437
$442$	0	0
$443$	8.35646	0.397027	0.198514	−	0.980098i	$-0.436389\pi$
$443$	8.35646	0.397027	0.198514	+	0.980098i	$0.436389\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.62550	0.124182
$448$	0	0
$449$	15.4109	0.727286	0.363643	−	0.931538i	$-0.381533\pi$
$449$	15.4109	0.727286	0.363643	+	0.931538i	$0.381533\pi$
$450$	0	0
$451$	5.49978	0.258974
$452$	0	0
$453$	−10.9965	−0.516659
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.9314	−1.07269	−0.536343	−	0.844000i	$-0.680195\pi$
$457$	−22.9314	−1.07269	−0.536343	+	0.844000i	$0.680195\pi$
$458$	0	0
$459$	6.01239	0.280634
$460$	0	0
$461$	−9.71207	−0.452336	−0.226168	−	0.974088i	$-0.572620\pi$
$461$	−9.71207	−0.452336	−0.226168	+	0.974088i	$0.572620\pi$
$462$	0	0
$463$	13.6091	0.632469	0.316234	−	0.948681i	$-0.397581\pi$
$463$	13.6091	0.632469	0.316234	+	0.948681i	$0.397581\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.9381	1.61674	0.808371	−	0.588674i	$-0.200350\pi$
$467$	34.9381	1.61674	0.808371	+	0.588674i	$0.200350\pi$
$468$	0	0
$469$	−16.7034	−0.771292
$470$	0	0
$471$	−3.89129	−0.179301
$472$	0	0
$473$	−27.5342	−1.26602
$474$	0	0
$475$	0	0
$476$	0	0
$477$	17.9005	0.819606
$478$	0	0
$479$	6.87962	0.314338	0.157169	−	0.987572i	$-0.449763\pi$
$479$	6.87962	0.314338	0.157169	+	0.987572i	$0.449763\pi$
$480$	0	0
$481$	20.2299	0.922406
$482$	0	0
$483$	−0.877911	−0.0399463
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−31.8903	−1.44509	−0.722544	−	0.691325i	$-0.757027\pi$
$487$	−31.8903	−1.44509	−0.722544	+	0.691325i	$0.757027\pi$
$488$	0	0
$489$	−9.07136	−0.410221
$490$	0	0
$491$	−22.1719	−1.00061	−0.500303	−	0.865851i	$-0.666778\pi$
$491$	−22.1719	−1.00061	−0.500303	+	0.865851i	$0.666778\pi$
$492$	0	0
$493$	12.0192	0.541317
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.30737	0.372636
$498$	0	0
$499$	−11.5721	−0.518037	−0.259019	−	0.965872i	$-0.583399\pi$
$499$	−11.5721	−0.518037	−0.259019	+	0.965872i	$0.583399\pi$
$500$	0	0
$501$	−1.52506	−0.0681349
$502$	0	0
$503$	−39.2532	−1.75021	−0.875106	−	0.483931i	$-0.839208\pi$
$503$	−39.2532	−1.75021	−0.875106	+	0.483931i	$0.839208\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.85513	−0.171212
$508$	0	0
$509$	−3.61391	−0.160184	−0.0800919	−	0.996787i	$-0.525521\pi$
$509$	−3.61391	−0.160184	−0.0800919	+	0.996787i	$0.525521\pi$
$510$	0	0
$511$	20.4501	0.904658
$512$	0	0
$513$	0.952620	0.0420592
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.7692	−1.00139
$518$	0	0
$519$	−3.67858	−0.161472
$520$	0	0
$521$	7.64494	0.334931	0.167465	−	0.985878i	$-0.446442\pi$
$521$	7.64494	0.334931	0.167465	+	0.985878i	$0.446442\pi$
$522$	0	0
$523$	5.62852	0.246118	0.123059	−	0.992399i	$-0.460730\pi$
$523$	5.62852	0.246118	0.123059	+	0.992399i	$0.460730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.6988	0.553170
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.3122	0.621095
$532$	0	0
$533$	4.26233	0.184622
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.73554	−0.204354
$538$	0	0
$539$	−10.8767	−0.468492
$540$	0	0
$541$	0.537263	0.0230987	0.0115494	−	0.999933i	$-0.496324\pi$
$541$	0.537263	0.0230987	0.0115494	+	0.999933i	$0.496324\pi$
$542$	0	0
$543$	0.126744	0.00543911
$544$	0	0
$545$	0	0
$546$	0	0
$547$	41.7590	1.78548	0.892742	−	0.450568i	$-0.148778\pi$
$547$	41.7590	1.78548	0.892742	+	0.450568i	$0.148778\pi$
$548$	0	0
$549$	25.2181	1.07628
$550$	0	0
$551$	1.90435	0.0811282
$552$	0	0
$553$	−14.3488	−0.610175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.5405	0.955071	0.477535	−	0.878613i	$-0.341530\pi$
$557$	22.5405	0.955071	0.477535	+	0.878613i	$0.341530\pi$
$558$	0	0
$559$	−21.3390	−0.902542
$560$	0	0
$561$	−3.03208	−0.128015
$562$	0	0
$563$	−31.5366	−1.32911	−0.664554	−	0.747240i	$-0.731378\pi$
$563$	−31.5366	−1.32911	−0.664554	+	0.747240i	$0.731378\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.5025	0.525055
$568$	0	0
$569$	2.49650	0.104659	0.0523294	−	0.998630i	$-0.483335\pi$
$569$	2.49650	0.104659	0.0523294	+	0.998630i	$0.483335\pi$
$570$	0	0
$571$	43.6658	1.82736	0.913678	−	0.406440i	$-0.133230\pi$
$571$	43.6658	1.82736	0.913678	+	0.406440i	$0.133230\pi$
$572$	0	0
$573$	8.83919	0.369262
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−33.7402	−1.40462	−0.702312	−	0.711869i	$-0.747849\pi$
$577$	−33.7402	−1.40462	−0.702312	+	0.711869i	$0.747849\pi$
$578$	0	0
$579$	1.04609	0.0434742
$580$	0	0
$581$	9.70790	0.402751
$582$	0	0
$583$	−18.8274	−0.779752
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.8050	−0.569795	−0.284898	−	0.958558i	$-0.591960\pi$
$587$	−13.8050	−0.569795	−0.284898	+	0.958558i	$0.591960\pi$
$588$	0	0
$589$	2.01204	0.0829047
$590$	0	0
$591$	13.1757	0.541974
$592$	0	0
$593$	−43.3336	−1.77950	−0.889749	−	0.456450i	$-0.849121\pi$
$593$	−43.3336	−1.77950	−0.889749	+	0.456450i	$0.849121\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.91541	0.283029
$598$	0	0
$599$	27.2077	1.11168	0.555838	−	0.831290i	$-0.312397\pi$
$599$	27.2077	1.11168	0.555838	+	0.831290i	$0.312397\pi$
$600$	0	0
$601$	39.6299	1.61654	0.808269	−	0.588813i	$-0.200405\pi$
$601$	39.6299	1.61654	0.808269	+	0.588813i	$0.200405\pi$
$602$	0	0
$603$	25.5734	1.04143
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.10233	−0.0447421	−0.0223711	−	0.999750i	$-0.507122\pi$
$607$	−1.10233	−0.0447421	−0.0223711	+	0.999750i	$0.507122\pi$
$608$	0	0
$609$	−4.91976	−0.199359
$610$	0	0
$611$	−17.6462	−0.713887
$612$	0	0
$613$	13.6705	0.552145	0.276073	−	0.961137i	$-0.410967\pi$
$613$	13.6705	0.552145	0.276073	+	0.961137i	$0.410967\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.9011	−0.801187	−0.400594	−	0.916256i	$-0.631196\pi$
$617$	−19.9011	−0.801187	−0.400594	+	0.916256i	$0.631196\pi$
$618$	0	0
$619$	32.1024	1.29030	0.645151	−	0.764055i	$-0.276794\pi$
$619$	32.1024	1.29030	0.645151	+	0.764055i	$0.276794\pi$
$620$	0	0
$621$	2.80328	0.112492
$622$	0	0
$623$	23.3305	0.934717
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.480411	−0.0191858
$628$	0	0
$629$	−19.2620	−0.768024
$630$	0	0
$631$	−32.3065	−1.28610	−0.643051	−	0.765823i	$-0.722332\pi$
$631$	−32.3065	−1.28610	−0.643051	+	0.765823i	$0.722332\pi$
$632$	0	0
$633$	−0.923435	−0.0367033
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.42942	−0.333986
$638$	0	0
$639$	−12.7188	−0.503148
$640$	0	0
$641$	−18.6696	−0.737407	−0.368703	−	0.929547i	$-0.620198\pi$
$641$	−18.6696	−0.737407	−0.368703	+	0.929547i	$0.620198\pi$
$642$	0	0
$643$	−19.9459	−0.786590	−0.393295	−	0.919412i	$-0.628665\pi$
$643$	−19.9459	−0.786590	−0.393295	+	0.919412i	$0.628665\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.6920	1.01006	0.505029	−	0.863102i	$-0.331482\pi$
$647$	25.6920	1.01006	0.505029	+	0.863102i	$0.331482\pi$
$648$	0	0
$649$	−15.0533	−0.590894
$650$	0	0
$651$	−5.19796	−0.203724
$652$	0	0
$653$	14.7648	0.577791	0.288896	−	0.957361i	$-0.406712\pi$
$653$	14.7648	0.577791	0.288896	+	0.957361i	$0.406712\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−31.3096	−1.22150
$658$	0	0
$659$	−9.53869	−0.371574	−0.185787	−	0.982590i	$-0.559484\pi$
$659$	−9.53869	−0.371574	−0.185787	+	0.982590i	$0.559484\pi$
$660$	0	0
$661$	13.2912	0.516968	0.258484	−	0.966016i	$-0.416777\pi$
$661$	13.2912	0.516968	0.258484	+	0.966016i	$0.416777\pi$
$662$	0	0
$663$	−2.34986	−0.0912611
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.60395	0.216986
$668$	0	0
$669$	−9.99894	−0.386581
$670$	0	0
$671$	−26.5239	−1.02395
$672$	0	0
$673$	−11.1517	−0.429868	−0.214934	−	0.976629i	$-0.568954\pi$
$673$	−11.1517	−0.429868	−0.214934	+	0.976629i	$0.568954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.0771	−1.04066	−0.520328	−	0.853966i	$-0.674190\pi$
$677$	−27.0771	−1.04066	−0.520328	+	0.853966i	$0.674190\pi$
$678$	0	0
$679$	4.38981	0.168466
$680$	0	0
$681$	3.73287	0.143044
$682$	0	0
$683$	−23.8752	−0.913561	−0.456780	−	0.889579i	$-0.650997\pi$
$683$	−23.8752	−0.913561	−0.456780	+	0.889579i	$0.650997\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.13616	0.157805
$688$	0	0
$689$	−14.5912	−0.555882
$690$	0	0
$691$	−37.5350	−1.42790	−0.713950	−	0.700197i	$-0.753096\pi$
$691$	−37.5350	−1.42790	−0.713950	+	0.700197i	$0.753096\pi$
$692$	0	0
$693$	−14.4973	−0.550706
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.05838	−0.153722
$698$	0	0
$699$	1.15303	0.0436116
$700$	0	0
$701$	45.8587	1.73206	0.866030	−	0.499992i	$-0.166664\pi$
$701$	45.8587	1.73206	0.866030	+	0.499992i	$0.166664\pi$
$702$	0	0
$703$	−3.05192	−0.115105
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.22777	0.196611
$708$	0	0
$709$	10.2917	0.386515	0.193257	−	0.981148i	$-0.438095\pi$
$709$	10.2917	0.386515	0.193257	+	0.981148i	$0.438095\pi$
$710$	0	0
$711$	21.9685	0.823881
$712$	0	0
$713$	5.92083	0.221737
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.144983	0.00541449
$718$	0	0
$719$	15.2862	0.570080	0.285040	−	0.958516i	$-0.407993\pi$
$719$	15.2862	0.570080	0.285040	+	0.958516i	$0.407993\pi$
$720$	0	0
$721$	18.6856	0.695887
$722$	0	0
$723$	14.4837	0.538655
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.5486	−0.687930	−0.343965	−	0.938982i	$-0.611770\pi$
$727$	−18.5486	−0.687930	−0.343965	+	0.938982i	$0.611770\pi$
$728$	0	0
$729$	−15.0512	−0.557451
$730$	0	0
$731$	20.3179	0.751485
$732$	0	0
$733$	29.6045	1.09347	0.546734	−	0.837307i	$-0.315871\pi$
$733$	29.6045	1.09347	0.546734	+	0.837307i	$0.315871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−26.8976	−0.990787
$738$	0	0
$739$	−44.2455	−1.62760	−0.813799	−	0.581147i	$-0.802604\pi$
$739$	−44.2455	−1.62760	−0.813799	+	0.581147i	$0.802604\pi$
$740$	0	0
$741$	−0.372319	−0.0136775
$742$	0	0
$743$	2.33740	0.0857509	0.0428755	−	0.999080i	$-0.486348\pi$
$743$	2.33740	0.0857509	0.0428755	+	0.999080i	$0.486348\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.8630	−0.543810
$748$	0	0
$749$	−27.5066	−1.00507
$750$	0	0
$751$	34.4606	1.25748	0.628742	−	0.777614i	$-0.283570\pi$
$751$	34.4606	1.25748	0.628742	+	0.777614i	$0.283570\pi$
$752$	0	0
$753$	−4.94980	−0.180381
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.5337	0.964384	0.482192	−	0.876066i	$-0.339841\pi$
$757$	26.5337	0.964384	0.482192	+	0.876066i	$0.339841\pi$
$758$	0	0
$759$	−1.41371	−0.0513143
$760$	0	0
$761$	−48.2585	−1.74937	−0.874685	−	0.484691i	$-0.838932\pi$
$761$	−48.2585	−1.74937	−0.874685	+	0.484691i	$0.838932\pi$
$762$	0	0
$763$	−12.6447	−0.457768
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.6663	−0.421246
$768$	0	0
$769$	31.6874	1.14268	0.571338	−	0.820715i	$-0.306424\pi$
$769$	31.6874	1.14268	0.571338	+	0.820715i	$0.306424\pi$
$770$	0	0
$771$	−13.7609	−0.495587
$772$	0	0
$773$	25.7906	0.927623	0.463811	−	0.885934i	$-0.346481\pi$
$773$	25.7906	0.927623	0.463811	+	0.885934i	$0.346481\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.88441	0.282852
$778$	0	0
$779$	−0.643021	−0.0230386
$780$	0	0
$781$	13.3774	0.478682
$782$	0	0
$783$	15.7094	0.561408
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.2714	−0.936475	−0.468237	−	0.883603i	$-0.655111\pi$
$787$	−26.2714	−0.936475	−0.468237	+	0.883603i	$0.655111\pi$
$788$	0	0
$789$	−12.1200	−0.431482
$790$	0	0
$791$	−11.7044	−0.416159
$792$	0	0
$793$	−20.5560	−0.729967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.81284	0.205901	0.102951	−	0.994686i	$-0.467172\pi$
$797$	5.81284	0.205901	0.102951	+	0.994686i	$0.467172\pi$
$798$	0	0
$799$	16.8018	0.594405
$800$	0	0
$801$	−35.7196	−1.26209
$802$	0	0
$803$	32.9309	1.16211
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.5950	0.443364
$808$	0	0
$809$	43.5439	1.53092	0.765462	−	0.643481i	$-0.222511\pi$
$809$	43.5439	1.53092	0.765462	+	0.643481i	$0.222511\pi$
$810$	0	0
$811$	−11.0230	−0.387069	−0.193535	−	0.981093i	$-0.561995\pi$
$811$	−11.0230	−0.387069	−0.193535	+	0.981093i	$0.561995\pi$
$812$	0	0
$813$	−1.70044	−0.0596369
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.21923	0.112627
$818$	0	0
$819$	−11.2354	−0.392596
$820$	0	0
$821$	18.4664	0.644481	0.322240	−	0.946658i	$-0.395564\pi$
$821$	18.4664	0.644481	0.322240	+	0.946658i	$0.395564\pi$
$822$	0	0
$823$	−11.4960	−0.400724	−0.200362	−	0.979722i	$-0.564212\pi$
$823$	−11.4960	−0.400724	−0.200362	+	0.979722i	$0.564212\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.9089	0.483658	0.241829	−	0.970319i	$-0.422253\pi$
$827$	13.9089	0.483658	0.241829	+	0.970319i	$0.422253\pi$
$828$	0	0
$829$	9.63272	0.334558	0.167279	−	0.985910i	$-0.446502\pi$
$829$	9.63272	0.334558	0.167279	+	0.985910i	$0.446502\pi$
$830$	0	0
$831$	0.672759	0.0233378
$832$	0	0
$833$	8.02608	0.278087
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.5977	0.573701
$838$	0	0
$839$	0.697892	0.0240939	0.0120470	−	0.999927i	$-0.496165\pi$
$839$	0.697892	0.0240939	0.0120470	+	0.999927i	$0.496165\pi$
$840$	0	0
$841$	2.40421	0.0829036
$842$	0	0
$843$	7.44880	0.256550
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.60644	−0.158279
$848$	0	0
$849$	−5.75236	−0.197420
$850$	0	0
$851$	−8.98088	−0.307861
$852$	0	0
$853$	40.2978	1.37977	0.689885	−	0.723919i	$-0.257661\pi$
$853$	40.2978	1.37977	0.689885	+	0.723919i	$0.257661\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.1828	−1.64589	−0.822947	−	0.568118i	$-0.807672\pi$
$857$	−48.1828	−1.64589	−0.822947	+	0.568118i	$0.807672\pi$
$858$	0	0
$859$	−55.5268	−1.89455	−0.947276	−	0.320419i	$-0.896176\pi$
$859$	−55.5268	−1.89455	−0.947276	+	0.320419i	$0.896176\pi$
$860$	0	0
$861$	1.66120	0.0566135
$862$	0	0
$863$	39.8794	1.35751	0.678756	−	0.734364i	$-0.262520\pi$
$863$	39.8794	1.35751	0.678756	+	0.734364i	$0.262520\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6.03122	−0.204831
$868$	0	0
$869$	−23.1061	−0.783819
$870$	0	0
$871$	−20.8457	−0.706328
$872$	0	0
$873$	−6.72092	−0.227469
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.8562	1.17701	0.588506	−	0.808493i	$-0.299716\pi$
$877$	34.8562	1.17701	0.588506	+	0.808493i	$0.299716\pi$
$878$	0	0
$879$	−15.2860	−0.515584
$880$	0	0
$881$	−4.94904	−0.166737	−0.0833687	−	0.996519i	$-0.526568\pi$
$881$	−4.94904	−0.166737	−0.0833687	+	0.996519i	$0.526568\pi$
$882$	0	0
$883$	−8.95960	−0.301515	−0.150757	−	0.988571i	$-0.548171\pi$
$883$	−8.95960	−0.301515	−0.150757	+	0.988571i	$0.548171\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.8125	−0.967429	−0.483714	−	0.875226i	$-0.660713\pi$
$887$	−28.8125	−0.967429	−0.483714	+	0.875226i	$0.660713\pi$
$888$	0	0
$889$	−11.3409	−0.380363
$890$	0	0
$891$	20.1329	0.674476
$892$	0	0
$893$	2.66213	0.0890847
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.09562	−0.0365818
$898$	0	0
$899$	33.1800	1.10662
$900$	0	0
$901$	13.8931	0.462845
$902$	0	0
$903$	−8.31665	−0.276761
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.2784	−0.972174	−0.486087	−	0.873910i	$-0.661576\pi$
$907$	−29.2784	−0.972174	−0.486087	+	0.873910i	$0.661576\pi$
$908$	0	0
$909$	−8.00385	−0.265471
$910$	0	0
$911$	8.48369	0.281077	0.140539	−	0.990075i	$-0.455117\pi$
$911$	8.48369	0.281077	0.140539	+	0.990075i	$0.455117\pi$
$912$	0	0
$913$	15.6327	0.517367
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.2668	0.471131
$918$	0	0
$919$	−5.73942	−0.189326	−0.0946630	−	0.995509i	$-0.530177\pi$
$919$	−5.73942	−0.189326	−0.0946630	+	0.995509i	$0.530177\pi$
$920$	0	0
$921$	−0.697988	−0.0229995
$922$	0	0
$923$	10.3675	0.341250
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−28.6081	−0.939613
$928$	0	0
$929$	5.72127	0.187709	0.0938544	−	0.995586i	$-0.470081\pi$
$929$	5.72127	0.187709	0.0938544	+	0.995586i	$0.470081\pi$
$930$	0	0
$931$	1.27167	0.0416775
$932$	0	0
$933$	−10.1995	−0.333917
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.6492	1.16461	0.582304	−	0.812971i	$-0.302151\pi$
$937$	35.6492	1.16461	0.582304	+	0.812971i	$0.302151\pi$
$938$	0	0
$939$	0.260671	0.00850668
$940$	0	0
$941$	−19.5212	−0.636372	−0.318186	−	0.948028i	$-0.603074\pi$
$941$	−19.5212	−0.636372	−0.318186	+	0.948028i	$0.603074\pi$
$942$	0	0
$943$	−1.89222	−0.0616191
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.7799	1.16269	0.581345	−	0.813657i	$-0.302527\pi$
$947$	35.7799	1.16269	0.581345	+	0.813657i	$0.302527\pi$
$948$	0	0
$949$	25.5215	0.828462
$950$	0	0
$951$	−3.21889	−0.104380
$952$	0	0
$953$	9.97808	0.323222	0.161611	−	0.986855i	$-0.448331\pi$
$953$	9.97808	0.323222	0.161611	+	0.986855i	$0.448331\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.92234	−0.256093
$958$	0	0
$959$	14.9622	0.483155
$960$	0	0
$961$	4.05624	0.130847
$962$	0	0
$963$	42.1133	1.35708
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−58.6546	−1.88620	−0.943102	−	0.332505i	$-0.892106\pi$
$967$	−58.6546	−1.88620	−0.943102	+	0.332505i	$0.892106\pi$
$968$	0	0
$969$	0.354504	0.0113883
$970$	0	0
$971$	39.8684	1.27944	0.639719	−	0.768609i	$-0.279051\pi$
$971$	39.8684	1.27944	0.639719	+	0.768609i	$0.279051\pi$
$972$	0	0
$973$	16.1997	0.519338
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.7860	0.856962	0.428481	−	0.903551i	$-0.359049\pi$
$977$	26.7860	0.856962	0.428481	+	0.903551i	$0.359049\pi$
$978$	0	0
$979$	37.5693	1.20072
$980$	0	0
$981$	19.3593	0.618095
$982$	0	0
$983$	−58.5711	−1.86813	−0.934065	−	0.357104i	$-0.883764\pi$
$983$	−58.5711	−1.86813	−0.934065	+	0.357104i	$0.883764\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.87741	−0.218910
$988$	0	0
$989$	9.47322	0.301231
$990$	0	0
$991$	53.9818	1.71479	0.857394	−	0.514661i	$-0.172082\pi$
$991$	53.9818	1.71479	0.857394	+	0.514661i	$0.172082\pi$
$992$	0	0
$993$	0.883366	0.0280328
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−0.592272	−0.0187575	−0.00937873	−	0.999956i	$-0.502985\pi$
$997$	−0.592272	−0.0187575	−0.00937873	+	0.999956i	$0.502985\pi$
$998$	0	0
$999$	−25.1759	−0.796530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cx.1.4		6
4.3	odd	2		2300.2.a.o.1.3		6
5.2	odd	4		1840.2.e.f.369.6		12
5.3	odd	4		1840.2.e.f.369.7		12
5.4	even	2		9200.2.a.cy.1.3		6
20.3	even	4		460.2.c.a.369.6	✓	12
20.7	even	4		460.2.c.a.369.7	yes	12
20.19	odd	2		2300.2.a.n.1.4		6
60.23	odd	4		4140.2.f.b.829.5		12
60.47	odd	4		4140.2.f.b.829.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.6	✓	12	20.3	even	4
460.2.c.a.369.7	yes	12	20.7	even	4
1840.2.e.f.369.6		12	5.2	odd	4
1840.2.e.f.369.7		12	5.3	odd	4
2300.2.a.n.1.4		6	20.19	odd	2
2300.2.a.o.1.3		6	4.3	odd	2
4140.2.f.b.829.5		12	60.23	odd	4
4140.2.f.b.829.6		12	60.47	odd	4
9200.2.a.cx.1.4		6	1.1	even	1	trivial
9200.2.a.cy.1.3		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+0.486391 q^{3} +1.80495 q^{7} -2.76342 q^{9} +2.90652 q^{11} +2.25256 q^{13} -2.14477 q^{17} -0.339824 q^{19} +0.877911 q^{21} -1.00000 q^{23} -2.80328 q^{27} -5.60395 q^{29} -5.92083 q^{31} +1.41371 q^{33} +8.98088 q^{37} +1.09562 q^{39} +1.89222 q^{41} -9.47322 q^{43} -7.83384 q^{47} -3.74216 q^{49} -1.04320 q^{51} -6.47764 q^{53} -0.165287 q^{57} -5.17914 q^{59} -9.12565 q^{61} -4.98784 q^{63} -9.25423 q^{67} -0.486391 q^{69} +4.60255 q^{71} +11.3300 q^{73} +5.24613 q^{77} -7.94972 q^{79} +6.92678 q^{81} +5.37849 q^{83} -2.72571 q^{87} +12.9258 q^{89} +4.06575 q^{91} -2.87984 q^{93} +2.43210 q^{97} -8.03196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 + 5 * q^17 - 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 - 9 * q^31 - 10 * q^33 + 21 * q^37 + 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 + 12 * q^51 - q^53 + 12 * q^57 + 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 + 17 * q^71 - 14 * q^73 + 20 * q^77 - 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 + 4 * q^91 + 4 * q^97 + 16 * q^99

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