LMFDB - Embedded newform 9200.2.a.cx.1.6

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			1.6
Root			$1.26443$ 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)

$f(q)$	$=$	$q+2.40050 q^{3} -4.41307 q^{7} +2.76241 q^{9} -2.29289 q^{11} +6.92936 q^{13} +1.51387 q^{17} -2.89920 q^{19} -10.5936 q^{21} -1.00000 q^{23} -0.570328 q^{27} -7.68764 q^{29} -3.85746 q^{31} -5.50408 q^{33} +8.62830 q^{37} +16.6340 q^{39} -6.44324 q^{41} +3.48497 q^{43} +6.19747 q^{47} +12.4752 q^{49} +3.63405 q^{51} +2.17710 q^{53} -6.95953 q^{57} +11.7637 q^{59} -5.11443 q^{61} -12.1907 q^{63} -9.94597 q^{67} -2.40050 q^{69} -3.41407 q^{71} -8.95307 q^{73} +10.1187 q^{77} +1.92694 q^{79} -9.65631 q^{81} -8.04131 q^{83} -18.4542 q^{87} -1.09273 q^{89} -30.5798 q^{91} -9.25985 q^{93} -16.9208 q^{97} -6.33390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.40050	1.38593	0.692965	−	0.720971i	$-0.256304\pi$
$3$	2.40050	1.38593	0.692965	+	0.720971i	$0.256304\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.41307	−1.66798	−0.833992	−	0.551777i	$-0.813950\pi$
$7$	−4.41307	−1.66798	−0.833992	+	0.551777i	$0.813950\pi$
$8$	0	0
$9$	2.76241	0.920804
$10$	0	0
$11$	−2.29289	−0.691332	−0.345666	−	0.938358i	$-0.612347\pi$
$11$	−2.29289	−0.691332	−0.345666	+	0.938358i	$0.612347\pi$
$12$	0	0
$13$	6.92936	1.92186	0.960930	−	0.276792i	$-0.0892713\pi$
$13$	6.92936	1.92186	0.960930	+	0.276792i	$0.0892713\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.51387	0.367168	0.183584	−	0.983004i	$-0.441230\pi$
$17$	1.51387	0.367168	0.183584	+	0.983004i	$0.441230\pi$
$18$	0	0
$19$	−2.89920	−0.665121	−0.332561	−	0.943082i	$-0.607913\pi$
$19$	−2.89920	−0.665121	−0.332561	+	0.943082i	$0.607913\pi$
$20$	0	0
$21$	−10.5936	−2.31171
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.570328	−0.109760
$28$	0	0
$29$	−7.68764	−1.42756	−0.713779	−	0.700371i	$-0.753018\pi$
$29$	−7.68764	−1.42756	−0.713779	+	0.700371i	$0.753018\pi$
$30$	0	0
$31$	−3.85746	−0.692821	−0.346410	−	0.938083i	$-0.612599\pi$
$31$	−3.85746	−0.692821	−0.346410	+	0.938083i	$0.612599\pi$
$32$	0	0
$33$	−5.50408	−0.958138
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.62830	1.41848	0.709242	−	0.704965i	$-0.249037\pi$
$37$	8.62830	1.41848	0.709242	+	0.704965i	$0.249037\pi$
$38$	0	0
$39$	16.6340	2.66356
$40$	0	0
$41$	−6.44324	−1.00626	−0.503132	−	0.864209i	$-0.667819\pi$
$41$	−6.44324	−1.00626	−0.503132	+	0.864209i	$0.667819\pi$
$42$	0	0
$43$	3.48497	0.531453	0.265727	−	0.964048i	$-0.414388\pi$
$43$	3.48497	0.531453	0.265727	+	0.964048i	$0.414388\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.19747	0.903994	0.451997	−	0.892019i	$-0.350712\pi$
$47$	6.19747	0.903994	0.451997	+	0.892019i	$0.350712\pi$
$48$	0	0
$49$	12.4752	1.78217
$50$	0	0
$51$	3.63405	0.508869
$52$	0	0
$53$	2.17710	0.299047	0.149524	−	0.988758i	$-0.452226\pi$
$53$	2.17710	0.299047	0.149524	+	0.988758i	$0.452226\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.95953	−0.921812
$58$	0	0
$59$	11.7637	1.53150	0.765750	−	0.643138i	$-0.222368\pi$
$59$	11.7637	1.53150	0.765750	+	0.643138i	$0.222368\pi$
$60$	0	0
$61$	−5.11443	−0.654835	−0.327418	−	0.944880i	$-0.606178\pi$
$61$	−5.11443	−0.654835	−0.327418	+	0.944880i	$0.606178\pi$
$62$	0	0
$63$	−12.1907	−1.53589
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.94597	−1.21509	−0.607547	−	0.794284i	$-0.707846\pi$
$67$	−9.94597	−1.21509	−0.607547	+	0.794284i	$0.707846\pi$
$68$	0	0
$69$	−2.40050	−0.288987
$70$	0	0
$71$	−3.41407	−0.405175	−0.202588	−	0.979264i	$-0.564935\pi$
$71$	−3.41407	−0.405175	−0.202588	+	0.979264i	$0.564935\pi$
$72$	0	0
$73$	−8.95307	−1.04788	−0.523939	−	0.851756i	$-0.675538\pi$
$73$	−8.95307	−1.04788	−0.523939	+	0.851756i	$0.675538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.1187	1.15313
$78$	0	0
$79$	1.92694	0.216798	0.108399	−	0.994107i	$-0.465428\pi$
$79$	1.92694	0.216798	0.108399	+	0.994107i	$0.465428\pi$
$80$	0	0
$81$	−9.65631	−1.07292
$82$	0	0
$83$	−8.04131	−0.882648	−0.441324	−	0.897348i	$-0.645491\pi$
$83$	−8.04131	−0.882648	−0.441324	+	0.897348i	$0.645491\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.4542	−1.97850
$88$	0	0
$89$	−1.09273	−0.115829	−0.0579147	−	0.998322i	$-0.518445\pi$
$89$	−1.09273	−0.115829	−0.0579147	+	0.998322i	$0.518445\pi$
$90$	0	0
$91$	−30.5798	−3.20563
$92$	0	0
$93$	−9.25985	−0.960201
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.9208	−1.71805	−0.859026	−	0.511933i	$-0.828930\pi$
$97$	−16.9208	−1.71805	−0.859026	+	0.511933i	$0.828930\pi$
$98$	0	0
$99$	−6.33390	−0.636581
$100$	0	0
$101$	12.9497	1.28855	0.644274	−	0.764795i	$-0.277160\pi$
$101$	12.9497	1.28855	0.644274	+	0.764795i	$0.277160\pi$
$102$	0	0
$103$	−10.9754	−1.08144	−0.540721	−	0.841202i	$-0.681848\pi$
$103$	−10.9754	−1.08144	−0.540721	+	0.841202i	$0.681848\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.8821	−1.92208	−0.961040	−	0.276411i	$-0.910855\pi$
$107$	−19.8821	−1.92208	−0.961040	+	0.276411i	$0.910855\pi$
$108$	0	0
$109$	0.427249	0.0409231	0.0204615	−	0.999791i	$-0.493486\pi$
$109$	0.427249	0.0409231	0.0204615	+	0.999791i	$0.493486\pi$
$110$	0	0
$111$	20.7123	1.96592
$112$	0	0
$113$	−5.38533	−0.506609	−0.253304	−	0.967387i	$-0.581517\pi$
$113$	−5.38533	−0.506609	−0.253304	+	0.967387i	$0.581517\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19.1418	1.76966
$118$	0	0
$119$	−6.68082	−0.612430
$120$	0	0
$121$	−5.74267	−0.522061
$122$	0	0
$123$	−15.4670	−1.39461
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.16014	−0.546624	−0.273312	−	0.961925i	$-0.588119\pi$
$127$	−6.16014	−0.546624	−0.273312	+	0.961925i	$0.588119\pi$
$128$	0	0
$129$	8.36569	0.736558
$130$	0	0
$131$	−14.6325	−1.27845	−0.639225	−	0.769020i	$-0.720745\pi$
$131$	−14.6325	−1.27845	−0.639225	+	0.769020i	$0.720745\pi$
$132$	0	0
$133$	12.7944	1.10941
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.972256	0.0830654	0.0415327	−	0.999137i	$-0.486776\pi$
$137$	0.972256	0.0830654	0.0415327	+	0.999137i	$0.486776\pi$
$138$	0	0
$139$	−7.74312	−0.656763	−0.328382	−	0.944545i	$-0.606503\pi$
$139$	−7.74312	−0.656763	−0.328382	+	0.944545i	$0.606503\pi$
$140$	0	0
$141$	14.8770	1.25287
$142$	0	0
$143$	−15.8883	−1.32864
$144$	0	0
$145$	0	0
$146$	0	0
$147$	29.9467	2.46996
$148$	0	0
$149$	−17.6823	−1.44859	−0.724293	−	0.689492i	$-0.757834\pi$
$149$	−17.6823	−1.44859	−0.724293	+	0.689492i	$0.757834\pi$
$150$	0	0
$151$	2.01286	0.163804	0.0819022	−	0.996640i	$-0.473900\pi$
$151$	2.01286	0.163804	0.0819022	+	0.996640i	$0.473900\pi$
$152$	0	0
$153$	4.18194	0.338090
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.82703	−0.465048	−0.232524	−	0.972591i	$-0.574698\pi$
$157$	−5.82703	−0.465048	−0.232524	+	0.972591i	$0.574698\pi$
$158$	0	0
$159$	5.22612	0.414459
$160$	0	0
$161$	4.41307	0.347799
$162$	0	0
$163$	6.75147	0.528816	0.264408	−	0.964411i	$-0.414823\pi$
$163$	6.75147	0.528816	0.264408	+	0.964411i	$0.414823\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.4541	1.81493	0.907466	−	0.420125i	$-0.138014\pi$
$167$	23.4541	1.81493	0.907466	+	0.420125i	$0.138014\pi$
$168$	0	0
$169$	35.0161	2.69355
$170$	0	0
$171$	−8.00878	−0.612447
$172$	0	0
$173$	−8.28850	−0.630163	−0.315081	−	0.949065i	$-0.602032\pi$
$173$	−8.28850	−0.630163	−0.315081	+	0.949065i	$0.602032\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	28.2387	2.12255
$178$	0	0
$179$	−1.04002	−0.0777345	−0.0388673	−	0.999244i	$-0.512375\pi$
$179$	−1.04002	−0.0777345	−0.0388673	+	0.999244i	$0.512375\pi$
$180$	0	0
$181$	2.71850	0.202065	0.101032	−	0.994883i	$-0.467785\pi$
$181$	2.71850	0.202065	0.101032	+	0.994883i	$0.467785\pi$
$182$	0	0
$183$	−12.2772	−0.907557
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.47114	−0.253835
$188$	0	0
$189$	2.51690	0.183077
$190$	0	0
$191$	−8.04858	−0.582375	−0.291187	−	0.956666i	$-0.594050\pi$
$191$	−8.04858	−0.582375	−0.291187	+	0.956666i	$0.594050\pi$
$192$	0	0
$193$	−16.0276	−1.15370	−0.576848	−	0.816852i	$-0.695717\pi$
$193$	−16.0276	−1.15370	−0.576848	+	0.816852i	$0.695717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.06316	0.289488	0.144744	−	0.989469i	$-0.453764\pi$
$197$	4.06316	0.289488	0.144744	+	0.989469i	$0.453764\pi$
$198$	0	0
$199$	−18.7042	−1.32590	−0.662952	−	0.748662i	$-0.730697\pi$
$199$	−18.7042	−1.32590	−0.662952	+	0.748662i	$0.730697\pi$
$200$	0	0
$201$	−23.8753	−1.68404
$202$	0	0
$203$	33.9261	2.38114
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.76241	−0.192001
$208$	0	0
$209$	6.64753	0.459819
$210$	0	0
$211$	−7.87839	−0.542371	−0.271185	−	0.962527i	$-0.587416\pi$
$211$	−7.87839	−0.542371	−0.271185	+	0.962527i	$0.587416\pi$
$212$	0	0
$213$	−8.19548	−0.561545
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.0232	1.15561
$218$	0	0
$219$	−21.4919	−1.45229
$220$	0	0
$221$	10.4902	0.705645
$222$	0	0
$223$	2.29263	0.153526	0.0767628	−	0.997049i	$-0.475542\pi$
$223$	2.29263	0.153526	0.0767628	+	0.997049i	$0.475542\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.8325	−0.851725	−0.425862	−	0.904788i	$-0.640029\pi$
$227$	−12.8325	−0.851725	−0.425862	+	0.904788i	$0.640029\pi$
$228$	0	0
$229$	−16.2528	−1.07401	−0.537007	−	0.843578i	$-0.680445\pi$
$229$	−16.2528	−1.07401	−0.537007	+	0.843578i	$0.680445\pi$
$230$	0	0
$231$	24.2899	1.59816
$232$	0	0
$233$	21.5410	1.41120	0.705598	−	0.708613i	$-0.250679\pi$
$233$	21.5410	1.41120	0.705598	+	0.708613i	$0.250679\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.62563	0.300467
$238$	0	0
$239$	11.0622	0.715555	0.357778	−	0.933807i	$-0.383535\pi$
$239$	11.0622	0.715555	0.357778	+	0.933807i	$0.383535\pi$
$240$	0	0
$241$	18.2164	1.17342	0.586710	−	0.809797i	$-0.300423\pi$
$241$	18.2164	1.17342	0.586710	+	0.809797i	$0.300423\pi$
$242$	0	0
$243$	−21.4690	−1.37724
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−20.0896	−1.27827
$248$	0	0
$249$	−19.3032	−1.22329
$250$	0	0
$251$	10.0182	0.632345	0.316172	−	0.948702i	$-0.397602\pi$
$251$	10.0182	0.632345	0.316172	+	0.948702i	$0.397602\pi$
$252$	0	0
$253$	2.29289	0.144153
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.3021	−0.892141	−0.446070	−	0.894998i	$-0.647177\pi$
$257$	−14.3021	−0.892141	−0.446070	+	0.894998i	$0.647177\pi$
$258$	0	0
$259$	−38.0773	−2.36601
$260$	0	0
$261$	−21.2364	−1.31450
$262$	0	0
$263$	9.44361	0.582318	0.291159	−	0.956675i	$-0.405959\pi$
$263$	9.44361	0.582318	0.291159	+	0.956675i	$0.405959\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.62311	−0.160531
$268$	0	0
$269$	23.7630	1.44886	0.724428	−	0.689350i	$-0.242104\pi$
$269$	23.7630	1.44886	0.724428	+	0.689350i	$0.242104\pi$
$270$	0	0
$271$	17.5939	1.06875	0.534375	−	0.845247i	$-0.320547\pi$
$271$	17.5939	1.06875	0.534375	+	0.845247i	$0.320547\pi$
$272$	0	0
$273$	−73.4068	−4.44278
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.6582	−1.78199	−0.890995	−	0.454014i	$-0.849992\pi$
$277$	−29.6582	−1.78199	−0.890995	+	0.454014i	$0.849992\pi$
$278$	0	0
$279$	−10.6559	−0.637952
$280$	0	0
$281$	−8.25484	−0.492442	−0.246221	−	0.969214i	$-0.579189\pi$
$281$	−8.25484	−0.492442	−0.246221	+	0.969214i	$0.579189\pi$
$282$	0	0
$283$	−9.21317	−0.547666	−0.273833	−	0.961777i	$-0.588292\pi$
$283$	−9.21317	−0.547666	−0.273833	+	0.961777i	$0.588292\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.4344	1.67843
$288$	0	0
$289$	−14.7082	−0.865188
$290$	0	0
$291$	−40.6185	−2.38110
$292$	0	0
$293$	11.1138	0.649277	0.324639	−	0.945838i	$-0.394757\pi$
$293$	11.1138	0.649277	0.324639	+	0.945838i	$0.394757\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.30770	0.0758804
$298$	0	0
$299$	−6.92936	−0.400735
$300$	0	0
$301$	−15.3794	−0.886455
$302$	0	0
$303$	31.0859	1.78584
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.7800	−1.07183	−0.535916	−	0.844272i	$-0.680033\pi$
$307$	−18.7800	−1.07183	−0.535916	+	0.844272i	$0.680033\pi$
$308$	0	0
$309$	−26.3465	−1.49880
$310$	0	0
$311$	−2.51258	−0.142475	−0.0712377	−	0.997459i	$-0.522695\pi$
$311$	−2.51258	−0.142475	−0.0712377	+	0.997459i	$0.522695\pi$
$312$	0	0
$313$	19.9278	1.12638	0.563192	−	0.826326i	$-0.309573\pi$
$313$	19.9278	1.12638	0.563192	+	0.826326i	$0.309573\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.8849	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$317$	−27.8849	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$318$	0	0
$319$	17.6269	0.986916
$320$	0	0
$321$	−47.7271	−2.66387
$322$	0	0
$323$	−4.38901	−0.244211
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.02561	0.0567165
$328$	0	0
$329$	−27.3499	−1.50785
$330$	0	0
$331$	28.5409	1.56875	0.784375	−	0.620287i	$-0.212984\pi$
$331$	28.5409	1.56875	0.784375	+	0.620287i	$0.212984\pi$
$332$	0	0
$333$	23.8349	1.30615
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.71371	0.202298	0.101149	−	0.994871i	$-0.467748\pi$
$337$	3.71371	0.202298	0.101149	+	0.994871i	$0.467748\pi$
$338$	0	0
$339$	−12.9275	−0.702125
$340$	0	0
$341$	8.84472	0.478969
$342$	0	0
$343$	−24.1624	−1.30464
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.1459	−0.759393	−0.379696	−	0.925111i	$-0.623972\pi$
$347$	−14.1459	−0.759393	−0.379696	+	0.925111i	$0.623972\pi$
$348$	0	0
$349$	−21.2802	−1.13910	−0.569550	−	0.821957i	$-0.692883\pi$
$349$	−21.2802	−1.13910	−0.569550	+	0.821957i	$0.692883\pi$
$350$	0	0
$351$	−3.95201	−0.210943
$352$	0	0
$353$	24.4641	1.30209	0.651046	−	0.759038i	$-0.274331\pi$
$353$	24.4641	1.30209	0.651046	+	0.759038i	$0.274331\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−16.0373	−0.848786
$358$	0	0
$359$	23.0252	1.21522	0.607612	−	0.794234i	$-0.292128\pi$
$359$	23.0252	1.21522	0.607612	+	0.794234i	$0.292128\pi$
$360$	0	0
$361$	−10.5947	−0.557613
$362$	0	0
$363$	−13.7853	−0.723540
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.478057	−0.0249544	−0.0124772	−	0.999922i	$-0.503972\pi$
$367$	−0.478057	−0.0249544	−0.0124772	+	0.999922i	$0.503972\pi$
$368$	0	0
$369$	−17.7989	−0.926573
$370$	0	0
$371$	−9.60767	−0.498806
$372$	0	0
$373$	27.3508	1.41617	0.708085	−	0.706127i	$-0.249559\pi$
$373$	27.3508	1.41617	0.708085	+	0.706127i	$0.249559\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−53.2704	−2.74357
$378$	0	0
$379$	7.35358	0.377728	0.188864	−	0.982003i	$-0.439519\pi$
$379$	7.35358	0.377728	0.188864	+	0.982003i	$0.439519\pi$
$380$	0	0
$381$	−14.7874	−0.757583
$382$	0	0
$383$	3.34108	0.170721	0.0853606	−	0.996350i	$-0.472796\pi$
$383$	3.34108	0.170721	0.0853606	+	0.996350i	$0.472796\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.62693	0.489364
$388$	0	0
$389$	0.366568	0.0185858	0.00929288	−	0.999957i	$-0.497042\pi$
$389$	0.366568	0.0185858	0.00929288	+	0.999957i	$0.497042\pi$
$390$	0	0
$391$	−1.51387	−0.0765598
$392$	0	0
$393$	−35.1254	−1.77184
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.3222	−0.568245	−0.284122	−	0.958788i	$-0.591702\pi$
$397$	−11.3222	−0.568245	−0.284122	+	0.958788i	$0.591702\pi$
$398$	0	0
$399$	30.7129	1.53757
$400$	0	0
$401$	37.1673	1.85605	0.928024	−	0.372520i	$-0.121506\pi$
$401$	37.1673	1.85605	0.928024	+	0.372520i	$0.121506\pi$
$402$	0	0
$403$	−26.7298	−1.33150
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.7837	−0.980643
$408$	0	0
$409$	−13.7745	−0.681107	−0.340553	−	0.940225i	$-0.610614\pi$
$409$	−13.7745	−0.681107	−0.340553	+	0.940225i	$0.610614\pi$
$410$	0	0
$411$	2.33390	0.115123
$412$	0	0
$413$	−51.9139	−2.55452
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.5874	−0.910228
$418$	0	0
$419$	−7.50726	−0.366753	−0.183377	−	0.983043i	$-0.558703\pi$
$419$	−7.50726	−0.366753	−0.183377	+	0.983043i	$0.558703\pi$
$420$	0	0
$421$	−7.65685	−0.373172	−0.186586	−	0.982439i	$-0.559742\pi$
$421$	−7.65685	−0.373172	−0.186586	+	0.982439i	$0.559742\pi$
$422$	0	0
$423$	17.1200	0.832402
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.5703	1.09225
$428$	0	0
$429$	−38.1398	−1.84141
$430$	0	0
$431$	−6.22764	−0.299975	−0.149987	−	0.988688i	$-0.547923\pi$
$431$	−6.22764	−0.299975	−0.149987	+	0.988688i	$0.547923\pi$
$432$	0	0
$433$	0.704087	0.0338362	0.0169181	−	0.999857i	$-0.494615\pi$
$433$	0.704087	0.0338362	0.0169181	+	0.999857i	$0.494615\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.89920	0.138687
$438$	0	0
$439$	34.8570	1.66363	0.831816	−	0.555052i	$-0.187301\pi$
$439$	34.8570	1.66363	0.831816	+	0.555052i	$0.187301\pi$
$440$	0	0
$441$	34.4616	1.64103
$442$	0	0
$443$	24.8165	1.17907	0.589533	−	0.807745i	$-0.299312\pi$
$443$	24.8165	1.17907	0.589533	+	0.807745i	$0.299312\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−42.4463	−2.00764
$448$	0	0
$449$	6.77738	0.319844	0.159922	−	0.987130i	$-0.448876\pi$
$449$	6.77738	0.319844	0.159922	+	0.987130i	$0.448876\pi$
$450$	0	0
$451$	14.7736	0.695662
$452$	0	0
$453$	4.83188	0.227021
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.47169	0.0688429	0.0344214	−	0.999407i	$-0.489041\pi$
$457$	1.47169	0.0688429	0.0344214	+	0.999407i	$0.489041\pi$
$458$	0	0
$459$	−0.863404	−0.0403003
$460$	0	0
$461$	25.8037	1.20180	0.600899	−	0.799325i	$-0.294809\pi$
$461$	25.8037	1.20180	0.600899	+	0.799325i	$0.294809\pi$
$462$	0	0
$463$	15.3469	0.713231	0.356616	−	0.934251i	$-0.383931\pi$
$463$	15.3469	0.713231	0.356616	+	0.934251i	$0.383931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.2933	−0.476315	−0.238157	−	0.971227i	$-0.576543\pi$
$467$	−10.2933	−0.476315	−0.238157	+	0.971227i	$0.576543\pi$
$468$	0	0
$469$	43.8922	2.02676
$470$	0	0
$471$	−13.9878	−0.644524
$472$	0	0
$473$	−7.99065	−0.367410
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.01404	0.275364
$478$	0	0
$479$	−4.50453	−0.205817	−0.102909	−	0.994691i	$-0.532815\pi$
$479$	−4.50453	−0.205817	−0.102909	+	0.994691i	$0.532815\pi$
$480$	0	0
$481$	59.7886	2.72613
$482$	0	0
$483$	10.5936	0.482025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.2669	1.23558	0.617790	−	0.786343i	$-0.288028\pi$
$487$	27.2669	1.23558	0.617790	+	0.786343i	$0.288028\pi$
$488$	0	0
$489$	16.2069	0.732902
$490$	0	0
$491$	−24.4866	−1.10507	−0.552533	−	0.833491i	$-0.686339\pi$
$491$	−24.4866	−1.10507	−0.552533	+	0.833491i	$0.686339\pi$
$492$	0	0
$493$	−11.6381	−0.524154
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.0665	0.675826
$498$	0	0
$499$	24.1892	1.08286	0.541429	−	0.840746i	$-0.317883\pi$
$499$	24.1892	1.08286	0.541429	+	0.840746i	$0.317883\pi$
$500$	0	0
$501$	56.3016	2.51537
$502$	0	0
$503$	−30.6230	−1.36541	−0.682706	−	0.730693i	$-0.739197\pi$
$503$	−30.6230	−1.36541	−0.682706	+	0.730693i	$0.739197\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	84.0562	3.73307
$508$	0	0
$509$	−2.56826	−0.113836	−0.0569181	−	0.998379i	$-0.518127\pi$
$509$	−2.56826	−0.113836	−0.0569181	+	0.998379i	$0.518127\pi$
$510$	0	0
$511$	39.5105	1.74784
$512$	0	0
$513$	1.65349	0.0730035
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.2101	−0.624960
$518$	0	0
$519$	−19.8966	−0.873362
$520$	0	0
$521$	−44.1355	−1.93361	−0.966807	−	0.255509i	$-0.917757\pi$
$521$	−44.1355	−1.93361	−0.966807	+	0.255509i	$0.917757\pi$
$522$	0	0
$523$	−42.2884	−1.84914	−0.924572	−	0.381008i	$-0.875577\pi$
$523$	−42.2884	−1.84914	−0.924572	+	0.381008i	$0.875577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.83970	−0.254381
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	32.4961	1.41021
$532$	0	0
$533$	−44.6475	−1.93390
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.49656	−0.107735
$538$	0	0
$539$	−28.6042	−1.23207
$540$	0	0
$541$	10.2776	0.441871	0.220935	−	0.975288i	$-0.429089\pi$
$541$	10.2776	0.441871	0.220935	+	0.975288i	$0.429089\pi$
$542$	0	0
$543$	6.52577	0.280048
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.87305	0.165600	0.0827998	−	0.996566i	$-0.473614\pi$
$547$	3.87305	0.165600	0.0827998	+	0.996566i	$0.473614\pi$
$548$	0	0
$549$	−14.1282	−0.602975
$550$	0	0
$551$	22.2880	0.949500
$552$	0	0
$553$	−8.50373	−0.361615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.62364	−0.407767	−0.203883	−	0.978995i	$-0.565356\pi$
$557$	−9.62364	−0.407767	−0.203883	+	0.978995i	$0.565356\pi$
$558$	0	0
$559$	24.1486	1.02138
$560$	0	0
$561$	−8.33248	−0.351797
$562$	0	0
$563$	−3.35865	−0.141550	−0.0707751	−	0.997492i	$-0.522547\pi$
$563$	−3.35865	−0.141550	−0.0707751	+	0.997492i	$0.522547\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	42.6140	1.78962
$568$	0	0
$569$	−4.26939	−0.178982	−0.0894910	−	0.995988i	$-0.528524\pi$
$569$	−4.26939	−0.178982	−0.0894910	+	0.995988i	$0.528524\pi$
$570$	0	0
$571$	−16.4567	−0.688689	−0.344345	−	0.938843i	$-0.611899\pi$
$571$	−16.4567	−0.688689	−0.344345	+	0.938843i	$0.611899\pi$
$572$	0	0
$573$	−19.3206	−0.807131
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.98780	0.0827530	0.0413765	−	0.999144i	$-0.486826\pi$
$577$	1.98780	0.0827530	0.0413765	+	0.999144i	$0.486826\pi$
$578$	0	0
$579$	−38.4744	−1.59894
$580$	0	0
$581$	35.4869	1.47224
$582$	0	0
$583$	−4.99184	−0.206741
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.9720	1.19580	0.597900	−	0.801570i	$-0.296002\pi$
$587$	28.9720	1.19580	0.597900	+	0.801570i	$0.296002\pi$
$588$	0	0
$589$	11.1835	0.460810
$590$	0	0
$591$	9.75363	0.401211
$592$	0	0
$593$	−19.4799	−0.799944	−0.399972	−	0.916527i	$-0.630980\pi$
$593$	−19.4799	−0.799944	−0.399972	+	0.916527i	$0.630980\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−44.8994	−1.83761
$598$	0	0
$599$	−37.8442	−1.54627	−0.773136	−	0.634240i	$-0.781313\pi$
$599$	−37.8442	−1.54627	−0.773136	+	0.634240i	$0.781313\pi$
$600$	0	0
$601$	−20.2347	−0.825389	−0.412695	−	0.910869i	$-0.635412\pi$
$601$	−20.2347	−0.825389	−0.412695	+	0.910869i	$0.635412\pi$
$602$	0	0
$603$	−27.4749	−1.11886
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.4075	−0.990670	−0.495335	−	0.868702i	$-0.664955\pi$
$607$	−24.4075	−0.990670	−0.495335	+	0.868702i	$0.664955\pi$
$608$	0	0
$609$	81.4396	3.30010
$610$	0	0
$611$	42.9445	1.73735
$612$	0	0
$613$	10.6964	0.432025	0.216012	−	0.976391i	$-0.430695\pi$
$613$	10.6964	0.432025	0.216012	+	0.976391i	$0.430695\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.4738	−1.10605	−0.553026	−	0.833164i	$-0.686527\pi$
$617$	−27.4738	−1.10605	−0.553026	+	0.833164i	$0.686527\pi$
$618$	0	0
$619$	0.389330	0.0156485	0.00782425	−	0.999969i	$-0.497509\pi$
$619$	0.389330	0.0156485	0.00782425	+	0.999969i	$0.497509\pi$
$620$	0	0
$621$	0.570328	0.0228865
$622$	0	0
$623$	4.82230	0.193201
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.9574	0.637278
$628$	0	0
$629$	13.0621	0.520822
$630$	0	0
$631$	11.9330	0.475047	0.237523	−	0.971382i	$-0.423664\pi$
$631$	11.9330	0.475047	0.237523	+	0.971382i	$0.423664\pi$
$632$	0	0
$633$	−18.9121	−0.751689
$634$	0	0
$635$	0	0
$636$	0	0
$637$	86.4451	3.42508
$638$	0	0
$639$	−9.43106	−0.373087
$640$	0	0
$641$	22.8525	0.902621	0.451311	−	0.892367i	$-0.350957\pi$
$641$	22.8525	0.902621	0.451311	+	0.892367i	$0.350957\pi$
$642$	0	0
$643$	−34.9279	−1.37742	−0.688711	−	0.725036i	$-0.741823\pi$
$643$	−34.9279	−1.37742	−0.688711	+	0.725036i	$0.741823\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.3738	−1.03686	−0.518430	−	0.855120i	$-0.673483\pi$
$647$	−26.3738	−1.03686	−0.518430	+	0.855120i	$0.673483\pi$
$648$	0	0
$649$	−26.9728	−1.05877
$650$	0	0
$651$	40.8643	1.60160
$652$	0	0
$653$	−2.17022	−0.0849274	−0.0424637	−	0.999098i	$-0.513521\pi$
$653$	−2.17022	−0.0849274	−0.0424637	+	0.999098i	$0.513521\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−24.7321	−0.964891
$658$	0	0
$659$	−19.7820	−0.770596	−0.385298	−	0.922792i	$-0.625901\pi$
$659$	−19.7820	−0.770596	−0.385298	+	0.922792i	$0.625901\pi$
$660$	0	0
$661$	45.7505	1.77949	0.889743	−	0.456461i	$-0.150883\pi$
$661$	45.7505	1.77949	0.889743	+	0.456461i	$0.150883\pi$
$662$	0	0
$663$	25.1817	0.977976
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.68764	0.297666
$668$	0	0
$669$	5.50346	0.212776
$670$	0	0
$671$	11.7268	0.452708
$672$	0	0
$673$	6.99940	0.269807	0.134904	−	0.990859i	$-0.456928\pi$
$673$	6.99940	0.269807	0.134904	+	0.990859i	$0.456928\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.7178	1.02685	0.513425	−	0.858134i	$-0.328376\pi$
$677$	26.7178	1.02685	0.513425	+	0.858134i	$0.328376\pi$
$678$	0	0
$679$	74.6728	2.86568
$680$	0	0
$681$	−30.8045	−1.18043
$682$	0	0
$683$	19.8947	0.761248	0.380624	−	0.924730i	$-0.375709\pi$
$683$	19.8947	0.761248	0.380624	+	0.924730i	$0.375709\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−39.0148	−1.48851
$688$	0	0
$689$	15.0859	0.574727
$690$	0	0
$691$	−1.24438	−0.0473385	−0.0236693	−	0.999720i	$-0.507535\pi$
$691$	−1.24438	−0.0473385	−0.0236693	+	0.999720i	$0.507535\pi$
$692$	0	0
$693$	27.9520	1.06181
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.75424	−0.369468
$698$	0	0
$699$	51.7091	1.95582
$700$	0	0
$701$	−29.8454	−1.12724	−0.563622	−	0.826033i	$-0.690592\pi$
$701$	−29.8454	−1.12724	−0.563622	+	0.826033i	$0.690592\pi$
$702$	0	0
$703$	−25.0151	−0.943464
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−57.1481	−2.14928
$708$	0	0
$709$	25.2255	0.947362	0.473681	−	0.880697i	$-0.342925\pi$
$709$	25.2255	0.947362	0.473681	+	0.880697i	$0.342925\pi$
$710$	0	0
$711$	5.32301	0.199628
$712$	0	0
$713$	3.85746	0.144463
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.5549	0.991710
$718$	0	0
$719$	23.7787	0.886797	0.443398	−	0.896325i	$-0.353773\pi$
$719$	23.7787	0.886797	0.443398	+	0.896325i	$0.353773\pi$
$720$	0	0
$721$	48.4353	1.80383
$722$	0	0
$723$	43.7285	1.62628
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−36.5849	−1.35686	−0.678429	−	0.734666i	$-0.737339\pi$
$727$	−36.5849	−1.35686	−0.678429	+	0.734666i	$0.737339\pi$
$728$	0	0
$729$	−22.5675	−0.835833
$730$	0	0
$731$	5.27580	0.195133
$732$	0	0
$733$	−21.7593	−0.803698	−0.401849	−	0.915706i	$-0.631632\pi$
$733$	−21.7593	−0.803698	−0.401849	+	0.915706i	$0.631632\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.8050	0.840032
$738$	0	0
$739$	−2.72303	−0.100168	−0.0500842	−	0.998745i	$-0.515949\pi$
$739$	−2.72303	−0.100168	−0.0500842	+	0.998745i	$0.515949\pi$
$740$	0	0
$741$	−48.2251	−1.77159
$742$	0	0
$743$	32.0161	1.17456	0.587278	−	0.809385i	$-0.300200\pi$
$743$	32.0161	1.17456	0.587278	+	0.809385i	$0.300200\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.2134	−0.812746
$748$	0	0
$749$	87.7413	3.20600
$750$	0	0
$751$	36.0949	1.31712	0.658561	−	0.752527i	$-0.271165\pi$
$751$	36.0949	1.31712	0.658561	+	0.752527i	$0.271165\pi$
$752$	0	0
$753$	24.0488	0.876386
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.8188	−1.70166	−0.850830	−	0.525442i	$-0.823900\pi$
$757$	−46.8188	−1.70166	−0.850830	+	0.525442i	$0.823900\pi$
$758$	0	0
$759$	5.50408	0.199786
$760$	0	0
$761$	−28.8182	−1.04466	−0.522330	−	0.852744i	$-0.674937\pi$
$761$	−28.8182	−1.04466	−0.522330	+	0.852744i	$0.674937\pi$
$762$	0	0
$763$	−1.88548	−0.0682590
$764$	0	0
$765$	0	0
$766$	0	0
$767$	81.5148	2.94333
$768$	0	0
$769$	51.6243	1.86162	0.930811	−	0.365501i	$-0.119102\pi$
$769$	51.6243	1.86162	0.930811	+	0.365501i	$0.119102\pi$
$770$	0	0
$771$	−34.3322	−1.23645
$772$	0	0
$773$	25.9610	0.933753	0.466877	−	0.884322i	$-0.345379\pi$
$773$	25.9610	0.933753	0.466877	+	0.884322i	$0.345379\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−91.4046	−3.27912
$778$	0	0
$779$	18.6802	0.669288
$780$	0	0
$781$	7.82807	0.280110
$782$	0	0
$783$	4.38448	0.156688
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.2258	−1.89730	−0.948648	−	0.316333i	$-0.897548\pi$
$787$	−53.2258	−1.89730	−0.948648	+	0.316333i	$0.897548\pi$
$788$	0	0
$789$	22.6694	0.807053
$790$	0	0
$791$	23.7658	0.845015
$792$	0	0
$793$	−35.4397	−1.25850
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.1874	1.10471	0.552357	−	0.833608i	$-0.313729\pi$
$797$	31.1874	1.10471	0.552357	+	0.833608i	$0.313729\pi$
$798$	0	0
$799$	9.38218	0.331918
$800$	0	0
$801$	−3.01858	−0.106656
$802$	0	0
$803$	20.5284	0.724431
$804$	0	0
$805$	0	0
$806$	0	0
$807$	57.0432	2.00802
$808$	0	0
$809$	−1.60251	−0.0563414	−0.0281707	−	0.999603i	$-0.508968\pi$
$809$	−1.60251	−0.0563414	−0.0281707	+	0.999603i	$0.508968\pi$
$810$	0	0
$811$	36.9545	1.29765	0.648824	−	0.760938i	$-0.275261\pi$
$811$	36.9545	1.29765	0.648824	+	0.760938i	$0.275261\pi$
$812$	0	0
$813$	42.2341	1.48121
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.1036	−0.353481
$818$	0	0
$819$	−84.4739	−2.95176
$820$	0	0
$821$	−16.5583	−0.577890	−0.288945	−	0.957346i	$-0.593304\pi$
$821$	−16.5583	−0.577890	−0.288945	+	0.957346i	$0.593304\pi$
$822$	0	0
$823$	7.63201	0.266035	0.133018	−	0.991114i	$-0.457533\pi$
$823$	7.63201	0.266035	0.133018	+	0.991114i	$0.457533\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.1969	0.667543	0.333772	−	0.942654i	$-0.391679\pi$
$827$	19.1969	0.667543	0.333772	+	0.942654i	$0.391679\pi$
$828$	0	0
$829$	−6.32413	−0.219646	−0.109823	−	0.993951i	$-0.535028\pi$
$829$	−6.32413	−0.219646	−0.109823	+	0.993951i	$0.535028\pi$
$830$	0	0
$831$	−71.1946	−2.46971
$832$	0	0
$833$	18.8858	0.654355
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.20002	0.0760438
$838$	0	0
$839$	−23.1158	−0.798046	−0.399023	−	0.916941i	$-0.630651\pi$
$839$	−23.1158	−0.798046	−0.399023	+	0.916941i	$0.630651\pi$
$840$	0	0
$841$	30.0997	1.03792
$842$	0	0
$843$	−19.8158	−0.682491
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.3428	0.870789
$848$	0	0
$849$	−22.1163	−0.759028
$850$	0	0
$851$	−8.62830	−0.295774
$852$	0	0
$853$	−37.7049	−1.29099	−0.645496	−	0.763764i	$-0.723349\pi$
$853$	−37.7049	−1.29099	−0.645496	+	0.763764i	$0.723349\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.7133	0.673395	0.336697	−	0.941613i	$-0.390690\pi$
$857$	19.7133	0.673395	0.336697	+	0.941613i	$0.390690\pi$
$858$	0	0
$859$	−2.00609	−0.0684471	−0.0342235	−	0.999414i	$-0.510896\pi$
$859$	−2.00609	−0.0684471	−0.0342235	+	0.999414i	$0.510896\pi$
$860$	0	0
$861$	68.2570	2.32619
$862$	0	0
$863$	−16.8127	−0.572312	−0.286156	−	0.958183i	$-0.592377\pi$
$863$	−16.8127	−0.572312	−0.286156	+	0.958183i	$0.592377\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−35.3071	−1.19909
$868$	0	0
$869$	−4.41826	−0.149879
$870$	0	0
$871$	−68.9192	−2.33524
$872$	0	0
$873$	−46.7424	−1.58199
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.8922	0.502874	0.251437	−	0.967874i	$-0.419097\pi$
$877$	14.8922	0.502874	0.251437	+	0.967874i	$0.419097\pi$
$878$	0	0
$879$	26.6788	0.899853
$880$	0	0
$881$	−16.5218	−0.556632	−0.278316	−	0.960490i	$-0.589776\pi$
$881$	−16.5218	−0.556632	−0.278316	+	0.960490i	$0.589776\pi$
$882$	0	0
$883$	−38.7254	−1.30321	−0.651607	−	0.758557i	$-0.725905\pi$
$883$	−38.7254	−1.30321	−0.651607	+	0.758557i	$0.725905\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.16364	−0.0390711	−0.0195355	−	0.999809i	$-0.506219\pi$
$887$	−1.16364	−0.0390711	−0.0195355	+	0.999809i	$0.506219\pi$
$888$	0	0
$889$	27.1851	0.911760
$890$	0	0
$891$	22.1408	0.741746
$892$	0	0
$893$	−17.9677	−0.601266
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16.6340	−0.555392
$898$	0	0
$899$	29.6548	0.989042
$900$	0	0
$901$	3.29584	0.109800
$902$	0	0
$903$	−36.9183	−1.22857
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.2796	−1.00542	−0.502709	−	0.864456i	$-0.667663\pi$
$907$	−30.2796	−1.00542	−0.502709	+	0.864456i	$0.667663\pi$
$908$	0	0
$909$	35.7725	1.18650
$910$	0	0
$911$	28.2296	0.935290	0.467645	−	0.883916i	$-0.345103\pi$
$911$	28.2296	0.935290	0.467645	+	0.883916i	$0.345103\pi$
$912$	0	0
$913$	18.4378	0.610203
$914$	0	0
$915$	0	0
$916$	0	0
$917$	64.5744	2.13243
$918$	0	0
$919$	−3.28150	−0.108247	−0.0541233	−	0.998534i	$-0.517236\pi$
$919$	−3.28150	−0.108247	−0.0541233	+	0.998534i	$0.517236\pi$
$920$	0	0
$921$	−45.0814	−1.48548
$922$	0	0
$923$	−23.6573	−0.778690
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−30.3187	−0.995796
$928$	0	0
$929$	7.64745	0.250905	0.125452	−	0.992100i	$-0.459962\pi$
$929$	7.64745	0.250905	0.125452	+	0.992100i	$0.459962\pi$
$930$	0	0
$931$	−36.1680	−1.18536
$932$	0	0
$933$	−6.03146	−0.197461
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.95226	−0.161783	−0.0808917	−	0.996723i	$-0.525777\pi$
$937$	−4.95226	−0.161783	−0.0808917	+	0.996723i	$0.525777\pi$
$938$	0	0
$939$	47.8366	1.56109
$940$	0	0
$941$	−8.62677	−0.281225	−0.140612	−	0.990065i	$-0.544907\pi$
$941$	−8.62677	−0.281225	−0.140612	+	0.990065i	$0.544907\pi$
$942$	0	0
$943$	6.44324	0.209821
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6807	1.51692	0.758460	−	0.651720i	$-0.225952\pi$
$947$	46.6807	1.51692	0.758460	+	0.651720i	$0.225952\pi$
$948$	0	0
$949$	−62.0391	−2.01387
$950$	0	0
$951$	−66.9378	−2.17061
$952$	0	0
$953$	−12.2343	−0.396307	−0.198154	−	0.980171i	$-0.563495\pi$
$953$	−12.2343	−0.396307	−0.198154	+	0.980171i	$0.563495\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	42.3134	1.36780
$958$	0	0
$959$	−4.29063	−0.138552
$960$	0	0
$961$	−16.1200	−0.520000
$962$	0	0
$963$	−54.9227	−1.76986
$964$	0	0
$965$	0	0
$966$	0	0
$967$	54.8318	1.76327	0.881635	−	0.471932i	$-0.156443\pi$
$967$	54.8318	1.76327	0.881635	+	0.471932i	$0.156443\pi$
$968$	0	0
$969$	−10.5358	−0.338460
$970$	0	0
$971$	−40.3804	−1.29587	−0.647935	−	0.761696i	$-0.724367\pi$
$971$	−40.3804	−1.29587	−0.647935	+	0.761696i	$0.724367\pi$
$972$	0	0
$973$	34.1709	1.09547
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.6700	−0.341363	−0.170681	−	0.985326i	$-0.554597\pi$
$977$	−10.6700	−0.341363	−0.170681	+	0.985326i	$0.554597\pi$
$978$	0	0
$979$	2.50551	0.0800765
$980$	0	0
$981$	1.18024	0.0376821
$982$	0	0
$983$	−6.27812	−0.200241	−0.100120	−	0.994975i	$-0.531923\pi$
$983$	−6.27812	−0.200241	−0.100120	+	0.994975i	$0.531923\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−65.6535	−2.08977
$988$	0	0
$989$	−3.48497	−0.110816
$990$	0	0
$991$	−61.2864	−1.94683	−0.973413	−	0.229057i	$-0.926436\pi$
$991$	−61.2864	−1.94683	−0.973413	+	0.229057i	$0.926436\pi$
$992$	0	0
$993$	68.5125	2.17418
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.9799	1.26618	0.633088	−	0.774080i	$-0.281787\pi$
$997$	39.9799	1.26618	0.633088	+	0.774080i	$0.281787\pi$
$998$	0	0
$999$	−4.92096	−0.155692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cx.1.6		6
4.3	odd	2		2300.2.a.o.1.1		6
5.2	odd	4		1840.2.e.f.369.3		12
5.3	odd	4		1840.2.e.f.369.10		12
5.4	even	2		9200.2.a.cy.1.1		6
20.3	even	4		460.2.c.a.369.3	✓	12
20.7	even	4		460.2.c.a.369.10	yes	12
20.19	odd	2		2300.2.a.n.1.6		6
60.23	odd	4		4140.2.f.b.829.8		12
60.47	odd	4		4140.2.f.b.829.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.3	✓	12	20.3	even	4
460.2.c.a.369.10	yes	12	20.7	even	4
1840.2.e.f.369.3		12	5.2	odd	4
1840.2.e.f.369.10		12	5.3	odd	4
2300.2.a.n.1.6		6	20.19	odd	2
2300.2.a.o.1.1		6	4.3	odd	2
4140.2.f.b.829.7		12	60.47	odd	4
4140.2.f.b.829.8		12	60.23	odd	4
9200.2.a.cx.1.6		6	1.1	even	1	trivial
9200.2.a.cy.1.1		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)

\(f(q)\)	\(=\)	\(q+2.40050 q^{3} -4.41307 q^{7} +2.76241 q^{9} -2.29289 q^{11} +6.92936 q^{13} +1.51387 q^{17} -2.89920 q^{19} -10.5936 q^{21} -1.00000 q^{23} -0.570328 q^{27} -7.68764 q^{29} -3.85746 q^{31} -5.50408 q^{33} +8.62830 q^{37} +16.6340 q^{39} -6.44324 q^{41} +3.48497 q^{43} +6.19747 q^{47} +12.4752 q^{49} +3.63405 q^{51} +2.17710 q^{53} -6.95953 q^{57} +11.7637 q^{59} -5.11443 q^{61} -12.1907 q^{63} -9.94597 q^{67} -2.40050 q^{69} -3.41407 q^{71} -8.95307 q^{73} +10.1187 q^{77} +1.92694 q^{79} -9.65631 q^{81} -8.04131 q^{83} -18.4542 q^{87} -1.09273 q^{89} -30.5798 q^{91} -9.25985 q^{93} -16.9208 q^{97} -6.33390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49}+ \cdots + 16 q^{99}+O(q^{100}) \) 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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