LMFDB - Embedded newform 9200.2.a.cj.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:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.58874$ 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)

$f(q)$	$=$	$q-3.25886 q^{3} -1.44270 q^{7} +7.62018 q^{9} +O(q^{10})$	$q-3.25886 q^{3} -1.44270 q^{7} +7.62018 q^{9} +2.03144 q^{11} +0.557299 q^{13} -3.91861 q^{17} +6.73300 q^{19} +4.70156 q^{21} -1.00000 q^{23} -15.0565 q^{27} -9.69520 q^{29} +3.07502 q^{31} -6.62018 q^{33} -3.65798 q^{37} -1.81616 q^{39} +7.03321 q^{41} +5.06465 q^{43} -0.659753 q^{47} -4.91861 q^{49} +12.7702 q^{51} +11.1275 q^{53} -21.9419 q^{57} -10.7226 q^{59} +1.73937 q^{61} -10.9936 q^{63} -12.0129 q^{67} +3.25886 q^{69} -12.4363 q^{71} -9.26464 q^{73} -2.93076 q^{77} -5.92439 q^{79} +26.2066 q^{81} +11.3775 q^{83} +31.5953 q^{87} +5.25308 q^{89} -0.804016 q^{91} -10.0211 q^{93} -0.0813861 q^{97} +15.4799 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - 3 q^{7} + 7 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - 3 * q^7 + 7 * q^9	$ 4 q - 3 q^{3} - 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} - 5 q^{17} + q^{19} + 6 q^{21} - 4 q^{23} - 6 q^{27} + 2 q^{29} - 5 q^{31} - 3 q^{33} - 6 q^{37} + 23 q^{41} + 2 q^{43} - 2 q^{47} - 9 q^{49} - 7 q^{51} - 28 q^{57} - 16 q^{59} + 9 q^{61} - 16 q^{63} + 2 q^{67} + 3 q^{69} - 19 q^{71} - 8 q^{73} - 10 q^{77} + 6 q^{79} + 16 q^{81} + 14 q^{83} + 38 q^{87} + 30 q^{89} + 13 q^{91} - 26 q^{93} - 11 q^{97} + 27 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - 3 * q^7 + 7 * q^9 - 5 * q^11 + 5 * q^13 - 5 * q^17 + q^19 + 6 * q^21 - 4 * q^23 - 6 * q^27 + 2 * q^29 - 5 * q^31 - 3 * q^33 - 6 * q^37 + 23 * q^41 + 2 * q^43 - 2 * q^47 - 9 * q^49 - 7 * q^51 - 28 * q^57 - 16 * q^59 + 9 * q^61 - 16 * q^63 + 2 * q^67 + 3 * q^69 - 19 * q^71 - 8 * q^73 - 10 * q^77 + 6 * q^79 + 16 * q^81 + 14 * q^83 + 38 * q^87 + 30 * q^89 + 13 * q^91 - 26 * q^93 - 11 * q^97 + 27 * 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
$3$	−3.25886	−1.88150	−0.940752	−	0.339095i	$-0.889879\pi$
$3$	−3.25886	−1.88150	−0.940752	+	0.339095i	$0.889879\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.44270	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$7$	−1.44270	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$8$	0	0
$9$	7.62018	2.54006
$10$	0	0
$11$	2.03144	0.612502	0.306251	−	0.951951i	$-0.400925\pi$
$11$	2.03144	0.612502	0.306251	+	0.951951i	$0.400925\pi$
$12$	0	0
$13$	0.557299	0.154567	0.0772835	−	0.997009i	$-0.475375\pi$
$13$	0.557299	0.154567	0.0772835	+	0.997009i	$0.475375\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.91861	−0.950403	−0.475202	−	0.879877i	$-0.657625\pi$
$17$	−3.91861	−0.950403	−0.475202	+	0.879877i	$0.657625\pi$
$18$	0	0
$19$	6.73300	1.54466	0.772328	−	0.635224i	$-0.219092\pi$
$19$	6.73300	1.54466	0.772328	+	0.635224i	$0.219092\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−15.0565	−2.89763
$28$	0	0
$29$	−9.69520	−1.80035	−0.900176	−	0.435525i	$-0.856563\pi$
$29$	−9.69520	−1.80035	−0.900176	+	0.435525i	$0.856563\pi$
$30$	0	0
$31$	3.07502	0.552290	0.276145	−	0.961116i	$-0.410943\pi$
$31$	3.07502	0.552290	0.276145	+	0.961116i	$0.410943\pi$
$32$	0	0
$33$	−6.62018	−1.15242
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.65798	−0.601368	−0.300684	−	0.953724i	$-0.597215\pi$
$37$	−3.65798	−0.601368	−0.300684	+	0.953724i	$0.597215\pi$
$38$	0	0
$39$	−1.81616	−0.290818
$40$	0	0
$41$	7.03321	1.09840	0.549202	−	0.835690i	$-0.314932\pi$
$41$	7.03321	1.09840	0.549202	+	0.835690i	$0.314932\pi$
$42$	0	0
$43$	5.06465	0.772352	0.386176	−	0.922425i	$-0.373796\pi$
$43$	5.06465	0.772352	0.386176	+	0.922425i	$0.373796\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.659753	−0.0962348	−0.0481174	−	0.998842i	$-0.515322\pi$
$47$	−0.659753	−0.0962348	−0.0481174	+	0.998842i	$0.515322\pi$
$48$	0	0
$49$	−4.91861	−0.702659
$50$	0	0
$51$	12.7702	1.78819
$52$	0	0
$53$	11.1275	1.52848	0.764242	−	0.644930i	$-0.223113\pi$
$53$	11.1275	1.52848	0.764242	+	0.644930i	$0.223113\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−21.9419	−2.90628
$58$	0	0
$59$	−10.7226	−1.39597	−0.697984	−	0.716114i	$-0.745919\pi$
$59$	−10.7226	−1.39597	−0.697984	+	0.716114i	$0.745919\pi$
$60$	0	0
$61$	1.73937	0.222703	0.111351	−	0.993781i	$-0.464482\pi$
$61$	1.73937	0.222703	0.111351	+	0.993781i	$0.464482\pi$
$62$	0	0
$63$	−10.9936	−1.38507
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.0129	−1.46761	−0.733806	−	0.679359i	$-0.762258\pi$
$67$	−12.0129	−1.46761	−0.733806	+	0.679359i	$0.762258\pi$
$68$	0	0
$69$	3.25886	0.392321
$70$	0	0
$71$	−12.4363	−1.47592	−0.737961	−	0.674844i	$-0.764211\pi$
$71$	−12.4363	−1.47592	−0.737961	+	0.674844i	$0.764211\pi$
$72$	0	0
$73$	−9.26464	−1.08434	−0.542172	−	0.840267i	$-0.682398\pi$
$73$	−9.26464	−1.08434	−0.542172	+	0.840267i	$0.682398\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.93076	−0.333991
$78$	0	0
$79$	−5.92439	−0.666546	−0.333273	−	0.942830i	$-0.608153\pi$
$79$	−5.92439	−0.666546	−0.333273	+	0.942830i	$0.608153\pi$
$80$	0	0
$81$	26.2066	2.91184
$82$	0	0
$83$	11.3775	1.24884	0.624420	−	0.781089i	$-0.285336\pi$
$83$	11.3775	1.24884	0.624420	+	0.781089i	$0.285336\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	31.5953	3.38737
$88$	0	0
$89$	5.25308	0.556826	0.278413	−	0.960462i	$-0.410192\pi$
$89$	5.25308	0.556826	0.278413	+	0.960462i	$0.410192\pi$
$90$	0	0
$91$	−0.804016	−0.0842838
$92$	0	0
$93$	−10.0211	−1.03914
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.0813861	−0.00826350	−0.00413175	−	0.999991i	$-0.501315\pi$
$97$	−0.0813861	−0.00826350	−0.00413175	+	0.999991i	$0.501315\pi$
$98$	0	0
$99$	15.4799	1.55579
$100$	0	0
$101$	8.31281	0.827156	0.413578	−	0.910469i	$-0.364279\pi$
$101$	8.31281	0.827156	0.413578	+	0.910469i	$0.364279\pi$
$102$	0	0
$103$	6.36991	0.627646	0.313823	−	0.949481i	$-0.398390\pi$
$103$	6.36991	0.627646	0.313823	+	0.949481i	$0.398390\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.9226	1.63597	0.817986	−	0.575239i	$-0.195091\pi$
$107$	16.9226	1.63597	0.817986	+	0.575239i	$0.195091\pi$
$108$	0	0
$109$	−0.218825	−0.0209597	−0.0104798	−	0.999945i	$-0.503336\pi$
$109$	−0.218825	−0.0209597	−0.0104798	+	0.999945i	$0.503336\pi$
$110$	0	0
$111$	11.9208	1.13148
$112$	0	0
$113$	0.580599	0.0546182	0.0273091	−	0.999627i	$-0.491306\pi$
$113$	0.580599	0.0546182	0.0273091	+	0.999627i	$0.491306\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.24672	0.392609
$118$	0	0
$119$	5.65339	0.518245
$120$	0	0
$121$	−6.87326	−0.624842
$122$	0	0
$123$	−22.9203	−2.06665
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.56944	−0.139266	−0.0696328	−	0.997573i	$-0.522183\pi$
$127$	−1.56944	−0.139266	−0.0696328	+	0.997573i	$0.522183\pi$
$128$	0	0
$129$	−16.5050	−1.45318
$130$	0	0
$131$	−15.9208	−1.39101	−0.695505	−	0.718521i	$-0.744819\pi$
$131$	−15.9208	−1.39101	−0.695505	+	0.718521i	$0.744819\pi$
$132$	0	0
$133$	−9.71371	−0.842285
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.47013	−0.552781	−0.276390	−	0.961045i	$-0.589138\pi$
$137$	−6.47013	−0.552781	−0.276390	+	0.961045i	$0.589138\pi$
$138$	0	0
$139$	0.656206	0.0556586	0.0278293	−	0.999613i	$-0.491141\pi$
$139$	0.656206	0.0556586	0.0278293	+	0.999613i	$0.491141\pi$
$140$	0	0
$141$	2.15004	0.181066
$142$	0	0
$143$	1.13212	0.0946725
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.0291	1.32206
$148$	0	0
$149$	23.8349	1.95263	0.976314	−	0.216357i	$-0.0694176\pi$
$149$	23.8349	1.95263	0.976314	+	0.216357i	$0.0694176\pi$
$150$	0	0
$151$	14.8669	1.20985	0.604925	−	0.796282i	$-0.293203\pi$
$151$	14.8669	1.20985	0.604925	+	0.796282i	$0.293203\pi$
$152$	0	0
$153$	−29.8605	−2.41408
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.24213	0.657793	0.328897	−	0.944366i	$-0.393323\pi$
$157$	8.24213	0.657793	0.328897	+	0.944366i	$0.393323\pi$
$158$	0	0
$159$	−36.2631	−2.87585
$160$	0	0
$161$	1.44270	0.113701
$162$	0	0
$163$	10.3154	0.807962	0.403981	−	0.914767i	$-0.367626\pi$
$163$	10.3154	0.807962	0.403981	+	0.914767i	$0.367626\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.65975	0.360582	0.180291	−	0.983613i	$-0.442296\pi$
$167$	4.65975	0.360582	0.180291	+	0.983613i	$0.442296\pi$
$168$	0	0
$169$	−12.6894	−0.976109
$170$	0	0
$171$	51.3067	3.92352
$172$	0	0
$173$	2.80402	0.213185	0.106593	−	0.994303i	$-0.466006\pi$
$173$	2.80402	0.213185	0.106593	+	0.994303i	$0.466006\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34.9436	2.62652
$178$	0	0
$179$	4.44212	0.332019	0.166010	−	0.986124i	$-0.446912\pi$
$179$	4.44212	0.332019	0.166010	+	0.986124i	$0.446912\pi$
$180$	0	0
$181$	2.42019	0.179891	0.0899455	−	0.995947i	$-0.471331\pi$
$181$	2.42019	0.179891	0.0899455	+	0.995947i	$0.471331\pi$
$182$	0	0
$183$	−5.66835	−0.419016
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.96042	−0.582124
$188$	0	0
$189$	21.7220	1.58005
$190$	0	0
$191$	20.8934	1.51179	0.755897	−	0.654690i	$-0.227201\pi$
$191$	20.8934	1.51179	0.755897	+	0.654690i	$0.227201\pi$
$192$	0	0
$193$	−2.88540	−0.207696	−0.103848	−	0.994593i	$-0.533116\pi$
$193$	−2.88540	−0.207696	−0.103848	+	0.994593i	$0.533116\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.0925	1.36029	0.680144	−	0.733079i	$-0.261917\pi$
$197$	19.0925	1.36029	0.680144	+	0.733079i	$0.261917\pi$
$198$	0	0
$199$	−6.58060	−0.466486	−0.233243	−	0.972418i	$-0.574934\pi$
$199$	−6.58060	−0.466486	−0.233243	+	0.972418i	$0.574934\pi$
$200$	0	0
$201$	39.1485	2.76132
$202$	0	0
$203$	13.9873	0.981714
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.62018	−0.529639
$208$	0	0
$209$	13.6777	0.946105
$210$	0	0
$211$	−6.55986	−0.451599	−0.225800	−	0.974174i	$-0.572499\pi$
$211$	−6.55986	−0.451599	−0.225800	+	0.974174i	$0.572499\pi$
$212$	0	0
$213$	40.5283	2.77695
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.43634	−0.301158
$218$	0	0
$219$	30.1922	2.04020
$220$	0	0
$221$	−2.18384	−0.146901
$222$	0	0
$223$	−20.2094	−1.35332	−0.676660	−	0.736296i	$-0.736573\pi$
$223$	−20.2094	−1.35332	−0.676660	+	0.736296i	$0.736573\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.0098	−0.996234	−0.498117	−	0.867110i	$-0.665975\pi$
$227$	−15.0098	−0.996234	−0.498117	+	0.867110i	$0.665975\pi$
$228$	0	0
$229$	13.7081	0.905859	0.452929	−	0.891546i	$-0.350379\pi$
$229$	13.7081	0.905859	0.452929	+	0.891546i	$0.350379\pi$
$230$	0	0
$231$	9.55093	0.628405
$232$	0	0
$233$	−9.19822	−0.602595	−0.301298	−	0.953530i	$-0.597420\pi$
$233$	−9.19822	−0.602595	−0.301298	+	0.953530i	$0.597420\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	19.3068	1.25411
$238$	0	0
$239$	2.72618	0.176342	0.0881709	−	0.996105i	$-0.471898\pi$
$239$	2.72618	0.176342	0.0881709	+	0.996105i	$0.471898\pi$
$240$	0	0
$241$	3.26109	0.210066	0.105033	−	0.994469i	$-0.466505\pi$
$241$	3.26109	0.210066	0.105033	+	0.994469i	$0.466505\pi$
$242$	0	0
$243$	−40.2340	−2.58101
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.75229	0.238753
$248$	0	0
$249$	−37.0776	−2.34970
$250$	0	0
$251$	−19.9941	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$251$	−19.9941	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$252$	0	0
$253$	−2.03144	−0.127715
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.7935	−1.67133	−0.835667	−	0.549237i	$-0.814919\pi$
$257$	−26.7935	−1.67133	−0.835667	+	0.549237i	$0.814919\pi$
$258$	0	0
$259$	5.27737	0.327920
$260$	0	0
$261$	−73.8791	−4.57300
$262$	0	0
$263$	15.8267	0.975918	0.487959	−	0.872867i	$-0.337742\pi$
$263$	15.8267	0.975918	0.487959	+	0.872867i	$0.337742\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−17.1191	−1.04767
$268$	0	0
$269$	12.0872	0.736967	0.368484	−	0.929634i	$-0.379877\pi$
$269$	12.0872	0.736967	0.368484	+	0.929634i	$0.379877\pi$
$270$	0	0
$271$	−8.40254	−0.510418	−0.255209	−	0.966886i	$-0.582144\pi$
$271$	−8.40254	−0.510418	−0.255209	+	0.966886i	$0.582144\pi$
$272$	0	0
$273$	2.62018	0.158580
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.8488	−0.832093	−0.416046	−	0.909343i	$-0.636585\pi$
$277$	−13.8488	−0.832093	−0.416046	+	0.909343i	$0.636585\pi$
$278$	0	0
$279$	23.4322	1.40285
$280$	0	0
$281$	−8.15004	−0.486191	−0.243095	−	0.970002i	$-0.578163\pi$
$281$	−8.15004	−0.486191	−0.243095	+	0.970002i	$0.578163\pi$
$282$	0	0
$283$	17.3532	1.03154	0.515770	−	0.856727i	$-0.327506\pi$
$283$	17.3532	1.03154	0.515770	+	0.856727i	$0.327506\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.1468	−0.598948
$288$	0	0
$289$	−1.64446	−0.0967332
$290$	0	0
$291$	0.265226	0.0155478
$292$	0	0
$293$	7.02252	0.410260	0.205130	−	0.978735i	$-0.434238\pi$
$293$	7.02252	0.410260	0.205130	+	0.978735i	$0.434238\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−30.5864	−1.77480
$298$	0	0
$299$	−0.557299	−0.0322294
$300$	0	0
$301$	−7.30678	−0.421156
$302$	0	0
$303$	−27.0903	−1.55630
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.4156	−0.936887	−0.468444	−	0.883493i	$-0.655185\pi$
$307$	−16.4156	−0.936887	−0.468444	+	0.883493i	$0.655185\pi$
$308$	0	0
$309$	−20.7587	−1.18092
$310$	0	0
$311$	−7.35181	−0.416883	−0.208441	−	0.978035i	$-0.566839\pi$
$311$	−7.35181	−0.416883	−0.208441	+	0.978035i	$0.566839\pi$
$312$	0	0
$313$	4.08972	0.231165	0.115582	−	0.993298i	$-0.463127\pi$
$313$	4.08972	0.231165	0.115582	+	0.993298i	$0.463127\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.91580	0.500761	0.250381	−	0.968148i	$-0.419444\pi$
$317$	8.91580	0.500761	0.250381	+	0.968148i	$0.419444\pi$
$318$	0	0
$319$	−19.6952	−1.10272
$320$	0	0
$321$	−55.1485	−3.07809
$322$	0	0
$323$	−26.3840	−1.46805
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.713121	0.0394357
$328$	0	0
$329$	0.951826	0.0524759
$330$	0	0
$331$	−21.3240	−1.17207	−0.586036	−	0.810285i	$-0.699312\pi$
$331$	−21.3240	−1.17207	−0.586036	+	0.810285i	$0.699312\pi$
$332$	0	0
$333$	−27.8744	−1.52751
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.15417	−0.171819	−0.0859094	−	0.996303i	$-0.527380\pi$
$337$	−3.15417	−0.171819	−0.0859094	+	0.996303i	$0.527380\pi$
$338$	0	0
$339$	−1.89209	−0.102764
$340$	0	0
$341$	6.24672	0.338279
$342$	0	0
$343$	17.1950	0.928443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.54739	0.190434	0.0952169	−	0.995457i	$-0.469646\pi$
$347$	3.54739	0.190434	0.0952169	+	0.995457i	$0.469646\pi$
$348$	0	0
$349$	−18.9726	−1.01558	−0.507789	−	0.861481i	$-0.669537\pi$
$349$	−18.9726	−1.01558	−0.507789	+	0.861481i	$0.669537\pi$
$350$	0	0
$351$	−8.39098	−0.447877
$352$	0	0
$353$	−33.5710	−1.78680	−0.893402	−	0.449257i	$-0.851689\pi$
$353$	−33.5710	−1.78680	−0.893402	+	0.449257i	$0.851689\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.4236	−0.975081
$358$	0	0
$359$	3.53243	0.186434	0.0932171	−	0.995646i	$-0.470285\pi$
$359$	3.53243	0.186434	0.0932171	+	0.995646i	$0.470285\pi$
$360$	0	0
$361$	26.3333	1.38596
$362$	0	0
$363$	22.3990	1.17564
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.07444	0.421482	0.210741	−	0.977542i	$-0.432412\pi$
$367$	8.07444	0.421482	0.210741	+	0.977542i	$0.432412\pi$
$368$	0	0
$369$	53.5943	2.79001
$370$	0	0
$371$	−16.0537	−0.833466
$372$	0	0
$373$	1.31459	0.0680668	0.0340334	−	0.999421i	$-0.489165\pi$
$373$	1.31459	0.0680668	0.0340334	+	0.999421i	$0.489165\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.40312	−0.278275
$378$	0	0
$379$	−0.311952	−0.0160239	−0.00801196	−	0.999968i	$-0.502550\pi$
$379$	−0.311952	−0.0160239	−0.00801196	+	0.999968i	$0.502550\pi$
$380$	0	0
$381$	5.11460	0.262029
$382$	0	0
$383$	10.6805	0.545748	0.272874	−	0.962050i	$-0.412026\pi$
$383$	10.6805	0.545748	0.272874	+	0.962050i	$0.412026\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	38.5935	1.96182
$388$	0	0
$389$	−7.86867	−0.398957	−0.199479	−	0.979902i	$-0.563925\pi$
$389$	−7.86867	−0.398957	−0.199479	+	0.979902i	$0.563925\pi$
$390$	0	0
$391$	3.91861	0.198173
$392$	0	0
$393$	51.8838	2.61719
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−29.2334	−1.46718	−0.733591	−	0.679591i	$-0.762157\pi$
$397$	−29.2334	−1.46718	−0.733591	+	0.679591i	$0.762157\pi$
$398$	0	0
$399$	31.6556	1.58476
$400$	0	0
$401$	23.7130	1.18417	0.592086	−	0.805874i	$-0.298304\pi$
$401$	23.7130	1.18417	0.592086	+	0.805874i	$0.298304\pi$
$402$	0	0
$403$	1.71371	0.0853658
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.43096	−0.368339
$408$	0	0
$409$	−33.2509	−1.64415	−0.822076	−	0.569378i	$-0.807184\pi$
$409$	−33.2509	−1.64415	−0.822076	+	0.569378i	$0.807184\pi$
$410$	0	0
$411$	21.0853	1.04006
$412$	0	0
$413$	15.4695	0.761207
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.13848	−0.104722
$418$	0	0
$419$	−7.28229	−0.355763	−0.177882	−	0.984052i	$-0.556924\pi$
$419$	−7.28229	−0.355763	−0.177882	+	0.984052i	$0.556924\pi$
$420$	0	0
$421$	−2.71515	−0.132329	−0.0661643	−	0.997809i	$-0.521076\pi$
$421$	−2.71515	−0.132329	−0.0661643	+	0.997809i	$0.521076\pi$
$422$	0	0
$423$	−5.02743	−0.244442
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.50938	−0.121438
$428$	0	0
$429$	−3.68942	−0.178127
$430$	0	0
$431$	−10.3336	−0.497750	−0.248875	−	0.968536i	$-0.580061\pi$
$431$	−10.3336	−0.497750	−0.248875	+	0.968536i	$0.580061\pi$
$432$	0	0
$433$	−28.4153	−1.36555	−0.682775	−	0.730628i	$-0.739227\pi$
$433$	−28.4153	−1.36555	−0.682775	+	0.730628i	$0.739227\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.73300	−0.322083
$438$	0	0
$439$	−6.37313	−0.304173	−0.152087	−	0.988367i	$-0.548599\pi$
$439$	−6.37313	−0.304173	−0.152087	+	0.988367i	$0.548599\pi$
$440$	0	0
$441$	−37.4807	−1.78480
$442$	0	0
$443$	−9.43114	−0.448087	−0.224044	−	0.974579i	$-0.571926\pi$
$443$	−9.43114	−0.448087	−0.224044	+	0.974579i	$0.571926\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−77.6745	−3.67388
$448$	0	0
$449$	11.3185	0.534154	0.267077	−	0.963675i	$-0.413942\pi$
$449$	11.3185	0.534154	0.267077	+	0.963675i	$0.413942\pi$
$450$	0	0
$451$	14.2875	0.672774
$452$	0	0
$453$	−48.4491	−2.27634
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.96456	−0.232232	−0.116116	−	0.993236i	$-0.537044\pi$
$457$	−4.96456	−0.232232	−0.116116	+	0.993236i	$0.537044\pi$
$458$	0	0
$459$	59.0007	2.75391
$460$	0	0
$461$	19.9128	0.927433	0.463717	−	0.885984i	$-0.346516\pi$
$461$	19.9128	0.927433	0.463717	+	0.885984i	$0.346516\pi$
$462$	0	0
$463$	−29.9046	−1.38978	−0.694892	−	0.719114i	$-0.744548\pi$
$463$	−29.9046	−1.38978	−0.694892	+	0.719114i	$0.744548\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.1032	0.698895	0.349447	−	0.936956i	$-0.386369\pi$
$467$	15.1032	0.698895	0.349447	+	0.936956i	$0.386369\pi$
$468$	0	0
$469$	17.3311	0.800274
$470$	0	0
$471$	−26.8599	−1.23764
$472$	0	0
$473$	10.2885	0.473067
$474$	0	0
$475$	0	0
$476$	0	0
$477$	84.7937	3.88244
$478$	0	0
$479$	−28.6678	−1.30986	−0.654932	−	0.755688i	$-0.727303\pi$
$479$	−28.6678	−1.30986	−0.654932	+	0.755688i	$0.727303\pi$
$480$	0	0
$481$	−2.03859	−0.0929516
$482$	0	0
$483$	−4.70156	−0.213928
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.15319	0.414770	0.207385	−	0.978259i	$-0.433505\pi$
$487$	9.15319	0.414770	0.207385	+	0.978259i	$0.433505\pi$
$488$	0	0
$489$	−33.6164	−1.52018
$490$	0	0
$491$	20.8098	0.939133	0.469566	−	0.882897i	$-0.344410\pi$
$491$	20.8098	0.939133	0.469566	+	0.882897i	$0.344410\pi$
$492$	0	0
$493$	37.9917	1.71106
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.9419	0.804805
$498$	0	0
$499$	18.1293	0.811579	0.405789	−	0.913967i	$-0.366997\pi$
$499$	18.1293	0.811579	0.405789	+	0.913967i	$0.366997\pi$
$500$	0	0
$501$	−15.1855	−0.678438
$502$	0	0
$503$	−26.4296	−1.17844	−0.589218	−	0.807974i	$-0.700564\pi$
$503$	−26.4296	−1.17844	−0.589218	+	0.807974i	$0.700564\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	41.3531	1.83655
$508$	0	0
$509$	−12.0501	−0.534113	−0.267057	−	0.963681i	$-0.586051\pi$
$509$	−12.0501	−0.534113	−0.267057	+	0.963681i	$0.586051\pi$
$510$	0	0
$511$	13.3661	0.591282
$512$	0	0
$513$	−101.376	−4.47584
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.34025	−0.0589440
$518$	0	0
$519$	−9.13790	−0.401109
$520$	0	0
$521$	−25.3697	−1.11146	−0.555732	−	0.831361i	$-0.687562\pi$
$521$	−25.3697	−1.11146	−0.555732	+	0.831361i	$0.687562\pi$
$522$	0	0
$523$	−25.0018	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$523$	−25.0018	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0498	−0.524898
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−81.7083	−3.54584
$532$	0	0
$533$	3.91960	0.169777
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−14.4762	−0.624696
$538$	0	0
$539$	−9.99186	−0.430380
$540$	0	0
$541$	24.7900	1.06580	0.532902	−	0.846177i	$-0.321101\pi$
$541$	24.7900	1.06580	0.532902	+	0.846177i	$0.321101\pi$
$542$	0	0
$543$	−7.88705	−0.338466
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−30.9508	−1.32336	−0.661681	−	0.749785i	$-0.730157\pi$
$547$	−30.9508	−1.32336	−0.661681	+	0.749785i	$0.730157\pi$
$548$	0	0
$549$	13.2543	0.565678
$550$	0	0
$551$	−65.2778	−2.78093
$552$	0	0
$553$	8.54713	0.363461
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.45307	0.315797	0.157898	−	0.987455i	$-0.449528\pi$
$557$	7.45307	0.315797	0.157898	+	0.987455i	$0.449528\pi$
$558$	0	0
$559$	2.82252	0.119380
$560$	0	0
$561$	25.9419	1.09527
$562$	0	0
$563$	−7.96161	−0.335542	−0.167771	−	0.985826i	$-0.553657\pi$
$563$	−7.96161	−0.335542	−0.167771	+	0.985826i	$0.553657\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−37.8082	−1.58780
$568$	0	0
$569$	32.3873	1.35774	0.678872	−	0.734257i	$-0.262469\pi$
$569$	32.3873	1.35774	0.678872	+	0.734257i	$0.262469\pi$
$570$	0	0
$571$	6.38994	0.267410	0.133705	−	0.991021i	$-0.457312\pi$
$571$	6.38994	0.267410	0.133705	+	0.991021i	$0.457312\pi$
$572$	0	0
$573$	−68.0887	−2.84445
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.1728	1.63078	0.815392	−	0.578910i	$-0.196522\pi$
$577$	39.1728	1.63078	0.815392	+	0.578910i	$0.196522\pi$
$578$	0	0
$579$	9.40312	0.390781
$580$	0	0
$581$	−16.4143	−0.680979
$582$	0	0
$583$	22.6049	0.936199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.3811	−0.882491	−0.441246	−	0.897386i	$-0.645463\pi$
$587$	−21.3811	−0.882491	−0.441246	+	0.897386i	$0.645463\pi$
$588$	0	0
$589$	20.7041	0.853098
$590$	0	0
$591$	−62.2199	−2.55939
$592$	0	0
$593$	2.45130	0.100663	0.0503314	−	0.998733i	$-0.483972\pi$
$593$	2.45130	0.100663	0.0503314	+	0.998733i	$0.483972\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.4453	0.877696
$598$	0	0
$599$	−38.5828	−1.57645	−0.788226	−	0.615386i	$-0.789000\pi$
$599$	−38.5828	−1.57645	−0.788226	+	0.615386i	$0.789000\pi$
$600$	0	0
$601$	−35.3416	−1.44162	−0.720808	−	0.693135i	$-0.756229\pi$
$601$	−35.3416	−1.44162	−0.720808	+	0.693135i	$0.756229\pi$
$602$	0	0
$603$	−91.5406	−3.72782
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.8129	−0.723005	−0.361502	−	0.932371i	$-0.617736\pi$
$607$	−17.8129	−0.723005	−0.361502	+	0.932371i	$0.617736\pi$
$608$	0	0
$609$	−45.5826	−1.84710
$610$	0	0
$611$	−0.367680	−0.0148747
$612$	0	0
$613$	37.7547	1.52490	0.762450	−	0.647048i	$-0.223997\pi$
$613$	37.7547	1.52490	0.762450	+	0.647048i	$0.223997\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.1248	−0.930971	−0.465486	−	0.885055i	$-0.654120\pi$
$617$	−23.1248	−0.930971	−0.465486	+	0.885055i	$0.654120\pi$
$618$	0	0
$619$	−19.5428	−0.785491	−0.392746	−	0.919647i	$-0.628475\pi$
$619$	−19.5428	−0.785491	−0.392746	+	0.919647i	$0.628475\pi$
$620$	0	0
$621$	15.0565	0.604197
$622$	0	0
$623$	−7.57863	−0.303631
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−44.5736	−1.78010
$628$	0	0
$629$	14.3342	0.571542
$630$	0	0
$631$	26.4807	1.05418	0.527090	−	0.849809i	$-0.323283\pi$
$631$	26.4807	1.05418	0.527090	+	0.849809i	$0.323283\pi$
$632$	0	0
$633$	21.3777	0.849686
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.74114	−0.108608
$638$	0	0
$639$	−94.7671	−3.74893
$640$	0	0
$641$	−35.9594	−1.42031	−0.710156	−	0.704044i	$-0.751376\pi$
$641$	−35.9594	−1.42031	−0.710156	+	0.704044i	$0.751376\pi$
$642$	0	0
$643$	14.6934	0.579452	0.289726	−	0.957110i	$-0.406436\pi$
$643$	14.6934	0.579452	0.289726	+	0.957110i	$0.406436\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.7763	1.05269	0.526343	−	0.850272i	$-0.323563\pi$
$647$	26.7763	1.05269	0.526343	+	0.850272i	$0.323563\pi$
$648$	0	0
$649$	−21.7824	−0.855033
$650$	0	0
$651$	14.4574	0.566630
$652$	0	0
$653$	−18.8331	−0.736996	−0.368498	−	0.929629i	$-0.620128\pi$
$653$	−18.8331	−0.736996	−0.368498	+	0.929629i	$0.620128\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−70.5982	−2.75430
$658$	0	0
$659$	18.0885	0.704629	0.352315	−	0.935882i	$-0.385395\pi$
$659$	18.0885	0.704629	0.352315	+	0.935882i	$0.385395\pi$
$660$	0	0
$661$	30.3235	1.17945	0.589724	−	0.807605i	$-0.299237\pi$
$661$	30.3235	1.17945	0.589724	+	0.807605i	$0.299237\pi$
$662$	0	0
$663$	7.11683	0.276395
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.69520	0.375400
$668$	0	0
$669$	65.8595	2.54628
$670$	0	0
$671$	3.53341	0.136406
$672$	0	0
$673$	−11.1496	−0.429787	−0.214894	−	0.976637i	$-0.568940\pi$
$673$	−11.1496	−0.429787	−0.214894	+	0.976637i	$0.568940\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.5275	1.67290	0.836449	−	0.548045i	$-0.184628\pi$
$677$	43.5275	1.67290	0.836449	+	0.548045i	$0.184628\pi$
$678$	0	0
$679$	0.117416	0.00450600
$680$	0	0
$681$	48.9148	1.87442
$682$	0	0
$683$	24.1375	0.923596	0.461798	−	0.886985i	$-0.347205\pi$
$683$	24.1375	0.923596	0.461798	+	0.886985i	$0.347205\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−44.6729	−1.70438
$688$	0	0
$689$	6.20136	0.236253
$690$	0	0
$691$	20.8241	0.792186	0.396093	−	0.918210i	$-0.370366\pi$
$691$	20.8241	0.792186	0.396093	+	0.918210i	$0.370366\pi$
$692$	0	0
$693$	−22.3329	−0.848356
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−27.5604	−1.04393
$698$	0	0
$699$	29.9757	1.13379
$700$	0	0
$701$	−31.5301	−1.19087	−0.595437	−	0.803402i	$-0.703021\pi$
$701$	−31.5301	−1.19087	−0.595437	+	0.803402i	$0.703021\pi$
$702$	0	0
$703$	−24.6292	−0.928907
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.9929	−0.451040
$708$	0	0
$709$	3.12175	0.117240	0.0586199	−	0.998280i	$-0.481330\pi$
$709$	3.12175	0.117240	0.0586199	+	0.998280i	$0.481330\pi$
$710$	0	0
$711$	−45.1449	−1.69307
$712$	0	0
$713$	−3.07502	−0.115160
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.88423	−0.331788
$718$	0	0
$719$	−20.3945	−0.760588	−0.380294	−	0.924866i	$-0.624177\pi$
$719$	−20.3945	−0.760588	−0.380294	+	0.924866i	$0.624177\pi$
$720$	0	0
$721$	−9.18988	−0.342249
$722$	0	0
$723$	−10.6275	−0.395239
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.8896	1.47942	0.739712	−	0.672924i	$-0.234962\pi$
$727$	39.8896	1.47942	0.739712	+	0.672924i	$0.234962\pi$
$728$	0	0
$729$	52.4973	1.94434
$730$	0	0
$731$	−19.8464	−0.734046
$732$	0	0
$733$	−22.7761	−0.841255	−0.420628	−	0.907233i	$-0.638190\pi$
$733$	−22.7761	−0.841255	−0.420628	+	0.907233i	$0.638190\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.4035	−0.898915
$738$	0	0
$739$	−33.7661	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$739$	−33.7661	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$740$	0	0
$741$	−12.2282	−0.449214
$742$	0	0
$743$	−47.5880	−1.74584	−0.872918	−	0.487868i	$-0.837775\pi$
$743$	−47.5880	−1.74584	−0.872918	+	0.487868i	$0.837775\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	86.6983	3.17212
$748$	0	0
$749$	−24.4143	−0.892078
$750$	0	0
$751$	−51.2698	−1.87086	−0.935430	−	0.353512i	$-0.884987\pi$
$751$	−51.2698	−1.87086	−0.935430	+	0.353512i	$0.884987\pi$
$752$	0	0
$753$	65.1580	2.37449
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−26.2807	−0.955189	−0.477594	−	0.878580i	$-0.658491\pi$
$757$	−26.2807	−0.955189	−0.477594	+	0.878580i	$0.658491\pi$
$758$	0	0
$759$	6.62018	0.240297
$760$	0	0
$761$	−7.18003	−0.260276	−0.130138	−	0.991496i	$-0.541542\pi$
$761$	−7.18003	−0.260276	−0.130138	+	0.991496i	$0.541542\pi$
$762$	0	0
$763$	0.315700	0.0114291
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.97571	−0.215770
$768$	0	0
$769$	−32.2666	−1.16356	−0.581782	−	0.813345i	$-0.697644\pi$
$769$	−32.2666	−1.16356	−0.581782	+	0.813345i	$0.697644\pi$
$770$	0	0
$771$	87.3164	3.14462
$772$	0	0
$773$	3.75316	0.134992	0.0674958	−	0.997720i	$-0.478499\pi$
$773$	3.75316	0.134992	0.0674958	+	0.997720i	$0.478499\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.1982	−0.616983
$778$	0	0
$779$	47.3546	1.69666
$780$	0	0
$781$	−25.2637	−0.904005
$782$	0	0
$783$	145.976	5.21675
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.9373	0.781981	0.390991	−	0.920395i	$-0.372133\pi$
$787$	21.9373	0.781981	0.390991	+	0.920395i	$0.372133\pi$
$788$	0	0
$789$	−51.5771	−1.83619
$790$	0	0
$791$	−0.837631	−0.0297827
$792$	0	0
$793$	0.969347	0.0344225
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.57883	−0.268456	−0.134228	−	0.990950i	$-0.542855\pi$
$797$	−7.57883	−0.268456	−0.134228	+	0.990950i	$0.542855\pi$
$798$	0	0
$799$	2.58532	0.0914619
$800$	0	0
$801$	40.0294	1.41437
$802$	0	0
$803$	−18.8205	−0.664163
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−39.3904	−1.38661
$808$	0	0
$809$	18.9835	0.667423	0.333712	−	0.942675i	$-0.391699\pi$
$809$	18.9835	0.667423	0.333712	+	0.942675i	$0.391699\pi$
$810$	0	0
$811$	−14.1851	−0.498106	−0.249053	−	0.968490i	$-0.580119\pi$
$811$	−14.1851	−0.498106	−0.249053	+	0.968490i	$0.580119\pi$
$812$	0	0
$813$	27.3827	0.960354
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34.1003	1.19302
$818$	0	0
$819$	−6.12674	−0.214086
$820$	0	0
$821$	−44.8294	−1.56456	−0.782278	−	0.622930i	$-0.785942\pi$
$821$	−44.8294	−1.56456	−0.782278	+	0.622930i	$0.785942\pi$
$822$	0	0
$823$	−15.7652	−0.549539	−0.274770	−	0.961510i	$-0.588602\pi$
$823$	−15.7652	−0.549539	−0.274770	+	0.961510i	$0.588602\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.4332	0.397573	0.198786	−	0.980043i	$-0.436300\pi$
$827$	11.4332	0.397573	0.198786	+	0.980043i	$0.436300\pi$
$828$	0	0
$829$	−28.3924	−0.986108	−0.493054	−	0.869999i	$-0.664119\pi$
$829$	−28.3924	−0.986108	−0.493054	+	0.869999i	$0.664119\pi$
$830$	0	0
$831$	45.1313	1.56559
$832$	0	0
$833$	19.2741	0.667810
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−46.2991	−1.60033
$838$	0	0
$839$	−54.3340	−1.87582	−0.937908	−	0.346883i	$-0.887240\pi$
$839$	−54.3340	−1.87582	−0.937908	+	0.346883i	$0.887240\pi$
$840$	0	0
$841$	64.9969	2.24127
$842$	0	0
$843$	26.5599	0.914770
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.91606	0.340720
$848$	0	0
$849$	−56.5516	−1.94085
$850$	0	0
$851$	3.65798	0.125394
$852$	0	0
$853$	−46.8542	−1.60426	−0.802129	−	0.597150i	$-0.796300\pi$
$853$	−46.8542	−1.60426	−0.802129	+	0.597150i	$0.796300\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.5356	0.974756	0.487378	−	0.873191i	$-0.337953\pi$
$857$	28.5356	0.974756	0.487378	+	0.873191i	$0.337953\pi$
$858$	0	0
$859$	−19.4417	−0.663343	−0.331671	−	0.943395i	$-0.607613\pi$
$859$	−19.4417	−0.663343	−0.331671	+	0.943395i	$0.607613\pi$
$860$	0	0
$861$	33.0671	1.12692
$862$	0	0
$863$	−1.94514	−0.0662132	−0.0331066	−	0.999452i	$-0.510540\pi$
$863$	−1.94514	−0.0662132	−0.0331066	+	0.999452i	$0.510540\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.35908	0.182004
$868$	0	0
$869$	−12.0350	−0.408261
$870$	0	0
$871$	−6.69479	−0.226844
$872$	0	0
$873$	−0.620176	−0.0209898
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.6457	−1.30497	−0.652486	−	0.757800i	$-0.726274\pi$
$877$	−38.6457	−1.30497	−0.652486	+	0.757800i	$0.726274\pi$
$878$	0	0
$879$	−22.8854	−0.771905
$880$	0	0
$881$	−54.6889	−1.84252	−0.921258	−	0.388952i	$-0.872837\pi$
$881$	−54.6889	−1.84252	−0.921258	+	0.388952i	$0.872837\pi$
$882$	0	0
$883$	14.6273	0.492247	0.246123	−	0.969238i	$-0.420843\pi$
$883$	14.6273	0.492247	0.246123	+	0.969238i	$0.420843\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.2666	−1.35202	−0.676010	−	0.736892i	$-0.736293\pi$
$887$	−40.2666	−1.35202	−0.676010	+	0.736892i	$0.736293\pi$
$888$	0	0
$889$	2.26424	0.0759401
$890$	0	0
$891$	53.2370	1.78351
$892$	0	0
$893$	−4.44212	−0.148650
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.81616	0.0606398
$898$	0	0
$899$	−29.8129	−0.994317
$900$	0	0
$901$	−43.6045	−1.45268
$902$	0	0
$903$	23.8118	0.792406
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.87090	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$907$	−7.87090	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$908$	0	0
$909$	63.3451	2.10102
$910$	0	0
$911$	−0.586235	−0.0194228	−0.00971141	−	0.999953i	$-0.503091\pi$
$911$	−0.586235	−0.0194228	−0.00971141	+	0.999953i	$0.503091\pi$
$912$	0	0
$913$	23.1126	0.764916
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22.9690	0.758504
$918$	0	0
$919$	27.9422	0.921729	0.460865	−	0.887470i	$-0.347539\pi$
$919$	27.9422	0.921729	0.460865	+	0.887470i	$0.347539\pi$
$920$	0	0
$921$	53.4961	1.76276
$922$	0	0
$923$	−6.93076	−0.228129
$924$	0	0
$925$	0	0
$926$	0	0
$927$	48.5399	1.59426
$928$	0	0
$929$	−4.79352	−0.157270	−0.0786351	−	0.996903i	$-0.525056\pi$
$929$	−4.79352	−0.157270	−0.0786351	+	0.996903i	$0.525056\pi$
$930$	0	0
$931$	−33.1170	−1.08537
$932$	0	0
$933$	23.9585	0.784367
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.110466	0.00360877	0.00180438	−	0.999998i	$-0.499426\pi$
$937$	0.110466	0.00360877	0.00180438	+	0.999998i	$0.499426\pi$
$938$	0	0
$939$	−13.3278	−0.434938
$940$	0	0
$941$	19.2077	0.626155	0.313077	−	0.949728i	$-0.398640\pi$
$941$	19.2077	0.626155	0.313077	+	0.949728i	$0.398640\pi$
$942$	0	0
$943$	−7.03321	−0.229033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.0933	−0.652944	−0.326472	−	0.945207i	$-0.605860\pi$
$947$	−20.0933	−0.652944	−0.326472	+	0.945207i	$0.605860\pi$
$948$	0	0
$949$	−5.16318	−0.167604
$950$	0	0
$951$	−29.0553	−0.942184
$952$	0	0
$953$	55.4762	1.79705	0.898526	−	0.438920i	$-0.144639\pi$
$953$	55.4762	1.79705	0.898526	+	0.438920i	$0.144639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	64.1839	2.07477
$958$	0	0
$959$	9.33447	0.301426
$960$	0	0
$961$	−21.5442	−0.694976
$962$	0	0
$963$	128.953	4.15546
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.86228	0.0920448	0.0460224	−	0.998940i	$-0.485345\pi$
$967$	2.86228	0.0920448	0.0460224	+	0.998940i	$0.485345\pi$
$968$	0	0
$969$	85.9819	2.76214
$970$	0	0
$971$	−42.5901	−1.36678	−0.683391	−	0.730052i	$-0.739496\pi$
$971$	−42.5901	−1.36678	−0.683391	+	0.730052i	$0.739496\pi$
$972$	0	0
$973$	−0.946709	−0.0303501
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.5633	1.36172	0.680861	−	0.732413i	$-0.261606\pi$
$977$	42.5633	1.36172	0.680861	+	0.732413i	$0.261606\pi$
$978$	0	0
$979$	10.6713	0.341057
$980$	0	0
$981$	−1.66749	−0.0532388
$982$	0	0
$983$	−32.6691	−1.04198	−0.520991	−	0.853562i	$-0.674437\pi$
$983$	−32.6691	−1.04198	−0.520991	+	0.853562i	$0.674437\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.10187	−0.0987336
$988$	0	0
$989$	−5.06465	−0.161047
$990$	0	0
$991$	4.39775	0.139699	0.0698495	−	0.997558i	$-0.477748\pi$
$991$	4.39775	0.139699	0.0698495	+	0.997558i	$0.477748\pi$
$992$	0	0
$993$	69.4919	2.20526
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.4135	1.05822	0.529108	−	0.848554i	$-0.322527\pi$
$997$	33.4135	1.05822	0.529108	+	0.848554i	$0.322527\pi$
$998$	0	0
$999$	55.0764	1.74254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cj.1.1		4
4.3	odd	2		1150.2.a.s.1.4		4
5.2	odd	4		1840.2.e.e.369.8		8
5.3	odd	4		1840.2.e.e.369.1		8
5.4	even	2		9200.2.a.cr.1.4		4
20.3	even	4		230.2.b.b.139.4	✓	8
20.7	even	4		230.2.b.b.139.5	yes	8
20.19	odd	2		1150.2.a.r.1.1		4
60.23	odd	4		2070.2.d.f.829.6		8
60.47	odd	4		2070.2.d.f.829.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.b.139.4	✓	8	20.3	even	4
230.2.b.b.139.5	yes	8	20.7	even	4
1150.2.a.r.1.1		4	20.19	odd	2
1150.2.a.s.1.4		4	4.3	odd	2
1840.2.e.e.369.1		8	5.3	odd	4
1840.2.e.e.369.8		8	5.2	odd	4
2070.2.d.f.829.2		8	60.47	odd	4
2070.2.d.f.829.6		8	60.23	odd	4
9200.2.a.cj.1.1		4	1.1	even	1	trivial
9200.2.a.cr.1.4		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.25886 q^{3} -1.44270 q^{7} +7.62018 q^{9} +O(q^{10})\)	\(q-3.25886 q^{3} -1.44270 q^{7} +7.62018 q^{9} +2.03144 q^{11} +0.557299 q^{13} -3.91861 q^{17} +6.73300 q^{19} +4.70156 q^{21} -1.00000 q^{23} -15.0565 q^{27} -9.69520 q^{29} +3.07502 q^{31} -6.62018 q^{33} -3.65798 q^{37} -1.81616 q^{39} +7.03321 q^{41} +5.06465 q^{43} -0.659753 q^{47} -4.91861 q^{49} +12.7702 q^{51} +11.1275 q^{53} -21.9419 q^{57} -10.7226 q^{59} +1.73937 q^{61} -10.9936 q^{63} -12.0129 q^{67} +3.25886 q^{69} -12.4363 q^{71} -9.26464 q^{73} -2.93076 q^{77} -5.92439 q^{79} +26.2066 q^{81} +11.3775 q^{83} +31.5953 q^{87} +5.25308 q^{89} -0.804016 q^{91} -10.0211 q^{93} -0.0813861 q^{97} +15.4799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - 3 q^{7} + 7 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - 3 * q^7 + 7 * q^9	\( 4 q - 3 q^{3} - 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} - 5 q^{17} + q^{19} + 6 q^{21} - 4 q^{23} - 6 q^{27} + 2 q^{29} - 5 q^{31} - 3 q^{33} - 6 q^{37} + 23 q^{41} + 2 q^{43} - 2 q^{47} - 9 q^{49} - 7 q^{51} - 28 q^{57} - 16 q^{59} + 9 q^{61} - 16 q^{63} + 2 q^{67} + 3 q^{69} - 19 q^{71} - 8 q^{73} - 10 q^{77} + 6 q^{79} + 16 q^{81} + 14 q^{83} + 38 q^{87} + 30 q^{89} + 13 q^{91} - 26 q^{93} - 11 q^{97} + 27 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - 3 * q^7 + 7 * q^9 - 5 * q^11 + 5 * q^13 - 5 * q^17 + q^19 + 6 * q^21 - 4 * q^23 - 6 * q^27 + 2 * q^29 - 5 * q^31 - 3 * q^33 - 6 * q^37 + 23 * q^41 + 2 * q^43 - 2 * q^47 - 9 * q^49 - 7 * q^51 - 28 * q^57 - 16 * q^59 + 9 * q^61 - 16 * q^63 + 2 * q^67 + 3 * q^69 - 19 * q^71 - 8 * q^73 - 10 * q^77 + 6 * q^79 + 16 * q^81 + 14 * q^83 + 38 * q^87 + 30 * q^89 + 13 * q^91 - 26 * q^93 - 11 * q^97 + 27 * q^99

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