LMFDB - Embedded newform 192.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.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:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -14.0000 q^{5} +24.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -14.0000 q^{5} +24.0000 q^{7} +9.00000 q^{9} -28.0000 q^{11} +74.0000 q^{13} -42.0000 q^{15} +82.0000 q^{17} +92.0000 q^{19} +72.0000 q^{21} -8.00000 q^{23} +71.0000 q^{25} +27.0000 q^{27} +138.000 q^{29} -80.0000 q^{31} -84.0000 q^{33} -336.000 q^{35} -30.0000 q^{37} +222.000 q^{39} +282.000 q^{41} +4.00000 q^{43} -126.000 q^{45} -240.000 q^{47} +233.000 q^{49} +246.000 q^{51} +130.000 q^{53} +392.000 q^{55} +276.000 q^{57} +596.000 q^{59} +218.000 q^{61} +216.000 q^{63} -1036.00 q^{65} -436.000 q^{67} -24.0000 q^{69} -856.000 q^{71} -998.000 q^{73} +213.000 q^{75} -672.000 q^{77} +32.0000 q^{79} +81.0000 q^{81} -1508.00 q^{83} -1148.00 q^{85} +414.000 q^{87} -246.000 q^{89} +1776.00 q^{91} -240.000 q^{93} -1288.00 q^{95} +866.000 q^{97} -252.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	0	0
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	74.0000	1.57876	0.789381	−	0.613904i	$-0.210402\pi$
$13$	74.0000	1.57876	0.789381	+	0.613904i	$0.210402\pi$
$14$	0	0
$15$	−42.0000	−0.722957
$16$	0	0
$17$	82.0000	1.16988	0.584939	−	0.811077i	$-0.301118\pi$
$17$	82.0000	1.16988	0.584939	+	0.811077i	$0.301118\pi$
$18$	0	0
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	0	0
$21$	72.0000	0.748176
$22$	0	0
$23$	−8.00000	−0.0725268	−0.0362634	−	0.999342i	$-0.511546\pi$
$23$	−8.00000	−0.0725268	−0.0362634	+	0.999342i	$0.511546\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	138.000	0.883654	0.441827	−	0.897100i	$-0.354331\pi$
$29$	138.000	0.883654	0.441827	+	0.897100i	$0.354331\pi$
$30$	0	0
$31$	−80.0000	−0.463498	−0.231749	−	0.972776i	$-0.574445\pi$
$31$	−80.0000	−0.463498	−0.231749	+	0.972776i	$0.574445\pi$
$32$	0	0
$33$	−84.0000	−0.443107
$34$	0	0
$35$	−336.000	−1.62270
$36$	0	0
$37$	−30.0000	−0.133296	−0.0666482	−	0.997777i	$-0.521231\pi$
$37$	−30.0000	−0.133296	−0.0666482	+	0.997777i	$0.521231\pi$
$38$	0	0
$39$	222.000	0.911499
$40$	0	0
$41$	282.000	1.07417	0.537085	−	0.843528i	$-0.319525\pi$
$41$	282.000	1.07417	0.537085	+	0.843528i	$0.319525\pi$
$42$	0	0
$43$	4.00000	0.0141859	0.00709296	−	0.999975i	$-0.497742\pi$
$43$	4.00000	0.0141859	0.00709296	+	0.999975i	$0.497742\pi$
$44$	0	0
$45$	−126.000	−0.417399
$46$	0	0
$47$	−240.000	−0.744843	−0.372421	−	0.928064i	$-0.621472\pi$
$47$	−240.000	−0.744843	−0.372421	+	0.928064i	$0.621472\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	246.000	0.675429
$52$	0	0
$53$	130.000	0.336922	0.168461	−	0.985708i	$-0.446120\pi$
$53$	130.000	0.336922	0.168461	+	0.985708i	$0.446120\pi$
$54$	0	0
$55$	392.000	0.961041
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	596.000	1.31513	0.657564	−	0.753398i	$-0.271587\pi$
$59$	596.000	1.31513	0.657564	+	0.753398i	$0.271587\pi$
$60$	0	0
$61$	218.000	0.457574	0.228787	−	0.973476i	$-0.426524\pi$
$61$	218.000	0.457574	0.228787	+	0.973476i	$0.426524\pi$
$62$	0	0
$63$	216.000	0.431959
$64$	0	0
$65$	−1036.00	−1.97692
$66$	0	0
$67$	−436.000	−0.795013	−0.397507	−	0.917599i	$-0.630124\pi$
$67$	−436.000	−0.795013	−0.397507	+	0.917599i	$0.630124\pi$
$68$	0	0
$69$	−24.0000	−0.0418733
$70$	0	0
$71$	−856.000	−1.43082	−0.715412	−	0.698703i	$-0.753761\pi$
$71$	−856.000	−1.43082	−0.715412	+	0.698703i	$0.753761\pi$
$72$	0	0
$73$	−998.000	−1.60010	−0.800048	−	0.599935i	$-0.795193\pi$
$73$	−998.000	−1.60010	−0.800048	+	0.599935i	$0.795193\pi$
$74$	0	0
$75$	213.000	0.327935
$76$	0	0
$77$	−672.000	−0.994565
$78$	0	0
$79$	32.0000	0.0455732	0.0227866	−	0.999740i	$-0.492746\pi$
$79$	32.0000	0.0455732	0.0227866	+	0.999740i	$0.492746\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1508.00	−1.99427	−0.997136	−	0.0756351i	$-0.975902\pi$
$83$	−1508.00	−1.99427	−0.997136	+	0.0756351i	$0.975902\pi$
$84$	0	0
$85$	−1148.00	−1.46492
$86$	0	0
$87$	414.000	0.510178
$88$	0	0
$89$	−246.000	−0.292988	−0.146494	−	0.989212i	$-0.546799\pi$
$89$	−246.000	−0.292988	−0.146494	+	0.989212i	$0.546799\pi$
$90$	0	0
$91$	1776.00	2.04588
$92$	0	0
$93$	−240.000	−0.267600
$94$	0	0
$95$	−1288.00	−1.39101
$96$	0	0
$97$	866.000	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$97$	866.000	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$98$	0	0
$99$	−252.000	−0.255828
$100$	0	0
$101$	−270.000	−0.266000	−0.133000	−	0.991116i	$-0.542461\pi$
$101$	−270.000	−0.266000	−0.133000	+	0.991116i	$0.542461\pi$
$102$	0	0
$103$	1496.00	1.43112	0.715560	−	0.698552i	$-0.246172\pi$
$103$	1496.00	1.43112	0.715560	+	0.698552i	$0.246172\pi$
$104$	0	0
$105$	−1008.00	−0.936864
$106$	0	0
$107$	−1692.00	−1.52871	−0.764354	−	0.644797i	$-0.776942\pi$
$107$	−1692.00	−1.52871	−0.764354	+	0.644797i	$0.776942\pi$
$108$	0	0
$109$	−406.000	−0.356768	−0.178384	−	0.983961i	$-0.557087\pi$
$109$	−406.000	−0.356768	−0.178384	+	0.983961i	$0.557087\pi$
$110$	0	0
$111$	−90.0000	−0.0769588
$112$	0	0
$113$	786.000	0.654342	0.327171	−	0.944965i	$-0.393905\pi$
$113$	786.000	0.654342	0.327171	+	0.944965i	$0.393905\pi$
$114$	0	0
$115$	112.000	0.0908179
$116$	0	0
$117$	666.000	0.526254
$118$	0	0
$119$	1968.00	1.51602
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	846.000	0.620173
$124$	0	0
$125$	756.000	0.540950
$126$	0	0
$127$	−1744.00	−1.21854	−0.609272	−	0.792962i	$-0.708538\pi$
$127$	−1744.00	−1.21854	−0.609272	+	0.792962i	$0.708538\pi$
$128$	0	0
$129$	12.0000	0.00819024
$130$	0	0
$131$	652.000	0.434851	0.217426	−	0.976077i	$-0.430234\pi$
$131$	652.000	0.434851	0.217426	+	0.976077i	$0.430234\pi$
$132$	0	0
$133$	2208.00	1.43953
$134$	0	0
$135$	−378.000	−0.240986
$136$	0	0
$137$	1530.00	0.954137	0.477068	−	0.878866i	$-0.341699\pi$
$137$	1530.00	0.954137	0.477068	+	0.878866i	$0.341699\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	−720.000	−0.430035
$142$	0	0
$143$	−2072.00	−1.21167
$144$	0	0
$145$	−1932.00	−1.10651
$146$	0	0
$147$	699.000	0.392194
$148$	0	0
$149$	−1342.00	−0.737859	−0.368929	−	0.929457i	$-0.620276\pi$
$149$	−1342.00	−0.737859	−0.368929	+	0.929457i	$0.620276\pi$
$150$	0	0
$151$	424.000	0.228507	0.114254	−	0.993452i	$-0.463552\pi$
$151$	424.000	0.228507	0.114254	+	0.993452i	$0.463552\pi$
$152$	0	0
$153$	738.000	0.389959
$154$	0	0
$155$	1120.00	0.580391
$156$	0	0
$157$	−262.000	−0.133184	−0.0665920	−	0.997780i	$-0.521213\pi$
$157$	−262.000	−0.133184	−0.0665920	+	0.997780i	$0.521213\pi$
$158$	0	0
$159$	390.000	0.194522
$160$	0	0
$161$	−192.000	−0.0939858
$162$	0	0
$163$	−2292.00	−1.10137	−0.550685	−	0.834713i	$-0.685633\pi$
$163$	−2292.00	−1.10137	−0.550685	+	0.834713i	$0.685633\pi$
$164$	0	0
$165$	1176.00	0.554857
$166$	0	0
$167$	1896.00	0.878544	0.439272	−	0.898354i	$-0.355236\pi$
$167$	1896.00	0.878544	0.439272	+	0.898354i	$0.355236\pi$
$168$	0	0
$169$	3279.00	1.49249
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	2874.00	1.26304	0.631521	−	0.775359i	$-0.282431\pi$
$173$	2874.00	1.26304	0.631521	+	0.775359i	$0.282431\pi$
$174$	0	0
$175$	1704.00	0.736059
$176$	0	0
$177$	1788.00	0.759290
$178$	0	0
$179$	−1188.00	−0.496063	−0.248032	−	0.968752i	$-0.579784\pi$
$179$	−1188.00	−0.496063	−0.248032	+	0.968752i	$0.579784\pi$
$180$	0	0
$181$	3474.00	1.42663	0.713316	−	0.700843i	$-0.247192\pi$
$181$	3474.00	1.42663	0.713316	+	0.700843i	$0.247192\pi$
$182$	0	0
$183$	654.000	0.264181
$184$	0	0
$185$	420.000	0.166914
$186$	0	0
$187$	−2296.00	−0.897862
$188$	0	0
$189$	648.000	0.249392
$190$	0	0
$191$	−192.000	−0.0727363	−0.0363681	−	0.999338i	$-0.511579\pi$
$191$	−192.000	−0.0727363	−0.0363681	+	0.999338i	$0.511579\pi$
$192$	0	0
$193$	4802.00	1.79096	0.895481	−	0.445100i	$-0.146832\pi$
$193$	4802.00	1.79096	0.895481	+	0.445100i	$0.146832\pi$
$194$	0	0
$195$	−3108.00	−1.14138
$196$	0	0
$197$	−1518.00	−0.549000	−0.274500	−	0.961587i	$-0.588512\pi$
$197$	−1518.00	−0.549000	−0.274500	+	0.961587i	$0.588512\pi$
$198$	0	0
$199$	−5128.00	−1.82670	−0.913352	−	0.407170i	$-0.866516\pi$
$199$	−5128.00	−1.82670	−0.913352	+	0.407170i	$0.866516\pi$
$200$	0	0
$201$	−1308.00	−0.459001
$202$	0	0
$203$	3312.00	1.14511
$204$	0	0
$205$	−3948.00	−1.34507
$206$	0	0
$207$	−72.0000	−0.0241756
$208$	0	0
$209$	−2576.00	−0.852563
$210$	0	0
$211$	1084.00	0.353676	0.176838	−	0.984240i	$-0.443413\pi$
$211$	1084.00	0.353676	0.176838	+	0.984240i	$0.443413\pi$
$212$	0	0
$213$	−2568.00	−0.826087
$214$	0	0
$215$	−56.0000	−0.0177636
$216$	0	0
$217$	−1920.00	−0.600636
$218$	0	0
$219$	−2994.00	−0.923816
$220$	0	0
$221$	6068.00	1.84696
$222$	0	0
$223$	−688.000	−0.206600	−0.103300	−	0.994650i	$-0.532940\pi$
$223$	−688.000	−0.206600	−0.103300	+	0.994650i	$0.532940\pi$
$224$	0	0
$225$	639.000	0.189333
$226$	0	0
$227$	4812.00	1.40698	0.703488	−	0.710707i	$-0.251625\pi$
$227$	4812.00	1.40698	0.703488	+	0.710707i	$0.251625\pi$
$228$	0	0
$229$	−2494.00	−0.719686	−0.359843	−	0.933013i	$-0.617170\pi$
$229$	−2494.00	−0.719686	−0.359843	+	0.933013i	$0.617170\pi$
$230$	0	0
$231$	−2016.00	−0.574212
$232$	0	0
$233$	698.000	0.196255	0.0981277	−	0.995174i	$-0.468715\pi$
$233$	698.000	0.196255	0.0981277	+	0.995174i	$0.468715\pi$
$234$	0	0
$235$	3360.00	0.932690
$236$	0	0
$237$	96.0000	0.0263117
$238$	0	0
$239$	6320.00	1.71049	0.855244	−	0.518225i	$-0.173407\pi$
$239$	6320.00	1.71049	0.855244	+	0.518225i	$0.173407\pi$
$240$	0	0
$241$	−6510.00	−1.74002	−0.870012	−	0.493030i	$-0.835889\pi$
$241$	−6510.00	−1.74002	−0.870012	+	0.493030i	$0.835889\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−3262.00	−0.850619
$246$	0	0
$247$	6808.00	1.75378
$248$	0	0
$249$	−4524.00	−1.15139
$250$	0	0
$251$	628.000	0.157924	0.0789622	−	0.996878i	$-0.474839\pi$
$251$	628.000	0.157924	0.0789622	+	0.996878i	$0.474839\pi$
$252$	0	0
$253$	224.000	0.0556631
$254$	0	0
$255$	−3444.00	−0.845771
$256$	0	0
$257$	−4862.00	−1.18009	−0.590045	−	0.807370i	$-0.700890\pi$
$257$	−4862.00	−1.18009	−0.590045	+	0.807370i	$0.700890\pi$
$258$	0	0
$259$	−720.000	−0.172736
$260$	0	0
$261$	1242.00	0.294551
$262$	0	0
$263$	−5816.00	−1.36361	−0.681806	−	0.731533i	$-0.738805\pi$
$263$	−5816.00	−1.36361	−0.681806	+	0.731533i	$0.738805\pi$
$264$	0	0
$265$	−1820.00	−0.421893
$266$	0	0
$267$	−738.000	−0.169157
$268$	0	0
$269$	−3526.00	−0.799197	−0.399599	−	0.916690i	$-0.630850\pi$
$269$	−3526.00	−0.799197	−0.399599	+	0.916690i	$0.630850\pi$
$270$	0	0
$271$	256.000	0.0573834	0.0286917	−	0.999588i	$-0.490866\pi$
$271$	256.000	0.0573834	0.0286917	+	0.999588i	$0.490866\pi$
$272$	0	0
$273$	5328.00	1.18119
$274$	0	0
$275$	−1988.00	−0.435931
$276$	0	0
$277$	−142.000	−0.0308013	−0.0154006	−	0.999881i	$-0.504902\pi$
$277$	−142.000	−0.0308013	−0.0154006	+	0.999881i	$0.504902\pi$
$278$	0	0
$279$	−720.000	−0.154499
$280$	0	0
$281$	8842.00	1.87712	0.938558	−	0.345122i	$-0.112162\pi$
$281$	8842.00	1.87712	0.938558	+	0.345122i	$0.112162\pi$
$282$	0	0
$283$	−7180.00	−1.50815	−0.754075	−	0.656788i	$-0.771915\pi$
$283$	−7180.00	−1.50815	−0.754075	+	0.656788i	$0.771915\pi$
$284$	0	0
$285$	−3864.00	−0.803100
$286$	0	0
$287$	6768.00	1.39199
$288$	0	0
$289$	1811.00	0.368614
$290$	0	0
$291$	2598.00	0.523359
$292$	0	0
$293$	−7374.00	−1.47029	−0.735143	−	0.677912i	$-0.762885\pi$
$293$	−7374.00	−1.47029	−0.735143	+	0.677912i	$0.762885\pi$
$294$	0	0
$295$	−8344.00	−1.64680
$296$	0	0
$297$	−756.000	−0.147702
$298$	0	0
$299$	−592.000	−0.114502
$300$	0	0
$301$	96.0000	0.0183832
$302$	0	0
$303$	−810.000	−0.153575
$304$	0	0
$305$	−3052.00	−0.572974
$306$	0	0
$307$	1500.00	0.278858	0.139429	−	0.990232i	$-0.455473\pi$
$307$	1500.00	0.278858	0.139429	+	0.990232i	$0.455473\pi$
$308$	0	0
$309$	4488.00	0.826257
$310$	0	0
$311$	7608.00	1.38717	0.693585	−	0.720374i	$-0.256030\pi$
$311$	7608.00	1.38717	0.693585	+	0.720374i	$0.256030\pi$
$312$	0	0
$313$	−4758.00	−0.859227	−0.429614	−	0.903013i	$-0.641350\pi$
$313$	−4758.00	−0.859227	−0.429614	+	0.903013i	$0.641350\pi$
$314$	0	0
$315$	−3024.00	−0.540899
$316$	0	0
$317$	−4374.00	−0.774979	−0.387489	−	0.921874i	$-0.626658\pi$
$317$	−4374.00	−0.774979	−0.387489	+	0.921874i	$0.626658\pi$
$318$	0	0
$319$	−3864.00	−0.678190
$320$	0	0
$321$	−5076.00	−0.882600
$322$	0	0
$323$	7544.00	1.29956
$324$	0	0
$325$	5254.00	0.896737
$326$	0	0
$327$	−1218.00	−0.205980
$328$	0	0
$329$	−5760.00	−0.965225
$330$	0	0
$331$	−7804.00	−1.29591	−0.647956	−	0.761678i	$-0.724376\pi$
$331$	−7804.00	−1.29591	−0.647956	+	0.761678i	$0.724376\pi$
$332$	0	0
$333$	−270.000	−0.0444322
$334$	0	0
$335$	6104.00	0.995514
$336$	0	0
$337$	5106.00	0.825346	0.412673	−	0.910879i	$-0.364595\pi$
$337$	5106.00	0.825346	0.412673	+	0.910879i	$0.364595\pi$
$338$	0	0
$339$	2358.00	0.377785
$340$	0	0
$341$	2240.00	0.355727
$342$	0	0
$343$	−2640.00	−0.415588
$344$	0	0
$345$	336.000	0.0524337
$346$	0	0
$347$	−4716.00	−0.729591	−0.364796	−	0.931088i	$-0.618861\pi$
$347$	−4716.00	−0.729591	−0.364796	+	0.931088i	$0.618861\pi$
$348$	0	0
$349$	−7302.00	−1.11996	−0.559982	−	0.828505i	$-0.689192\pi$
$349$	−7302.00	−1.11996	−0.559982	+	0.828505i	$0.689192\pi$
$350$	0	0
$351$	1998.00	0.303833
$352$	0	0
$353$	−4382.00	−0.660709	−0.330355	−	0.943857i	$-0.607168\pi$
$353$	−4382.00	−0.660709	−0.330355	+	0.943857i	$0.607168\pi$
$354$	0	0
$355$	11984.0	1.79168
$356$	0	0
$357$	5904.00	0.875274
$358$	0	0
$359$	−7224.00	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$359$	−7224.00	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−1641.00	−0.237273
$364$	0	0
$365$	13972.0	2.00364
$366$	0	0
$367$	−1408.00	−0.200264	−0.100132	−	0.994974i	$-0.531927\pi$
$367$	−1408.00	−0.200264	−0.100132	+	0.994974i	$0.531927\pi$
$368$	0	0
$369$	2538.00	0.358057
$370$	0	0
$371$	3120.00	0.436610
$372$	0	0
$373$	1714.00	0.237929	0.118965	−	0.992899i	$-0.462043\pi$
$373$	1714.00	0.237929	0.118965	+	0.992899i	$0.462043\pi$
$374$	0	0
$375$	2268.00	0.312317
$376$	0	0
$377$	10212.0	1.39508
$378$	0	0
$379$	884.000	0.119810	0.0599051	−	0.998204i	$-0.480920\pi$
$379$	884.000	0.119810	0.0599051	+	0.998204i	$0.480920\pi$
$380$	0	0
$381$	−5232.00	−0.703526
$382$	0	0
$383$	−10368.0	−1.38324	−0.691619	−	0.722263i	$-0.743102\pi$
$383$	−10368.0	−1.38324	−0.691619	+	0.722263i	$0.743102\pi$
$384$	0	0
$385$	9408.00	1.24539
$386$	0	0
$387$	36.0000	0.00472864
$388$	0	0
$389$	−398.000	−0.0518751	−0.0259375	−	0.999664i	$-0.508257\pi$
$389$	−398.000	−0.0518751	−0.0259375	+	0.999664i	$0.508257\pi$
$390$	0	0
$391$	−656.000	−0.0848474
$392$	0	0
$393$	1956.00	0.251061
$394$	0	0
$395$	−448.000	−0.0570666
$396$	0	0
$397$	5098.00	0.644487	0.322243	−	0.946657i	$-0.395563\pi$
$397$	5098.00	0.644487	0.322243	+	0.946657i	$0.395563\pi$
$398$	0	0
$399$	6624.00	0.831115
$400$	0	0
$401$	10002.0	1.24558	0.622788	−	0.782391i	$-0.286000\pi$
$401$	10002.0	1.24558	0.622788	+	0.782391i	$0.286000\pi$
$402$	0	0
$403$	−5920.00	−0.731752
$404$	0	0
$405$	−1134.00	−0.139133
$406$	0	0
$407$	840.000	0.102303
$408$	0	0
$409$	−9270.00	−1.12071	−0.560357	−	0.828251i	$-0.689336\pi$
$409$	−9270.00	−1.12071	−0.560357	+	0.828251i	$0.689336\pi$
$410$	0	0
$411$	4590.00	0.550871
$412$	0	0
$413$	14304.0	1.70425
$414$	0	0
$415$	21112.0	2.49722
$416$	0	0
$417$	1548.00	0.181789
$418$	0	0
$419$	−6516.00	−0.759731	−0.379866	−	0.925042i	$-0.624030\pi$
$419$	−6516.00	−0.759731	−0.379866	+	0.925042i	$0.624030\pi$
$420$	0	0
$421$	2626.00	0.303999	0.151999	−	0.988381i	$-0.451429\pi$
$421$	2626.00	0.303999	0.151999	+	0.988381i	$0.451429\pi$
$422$	0	0
$423$	−2160.00	−0.248281
$424$	0	0
$425$	5822.00	0.664491
$426$	0	0
$427$	5232.00	0.592961
$428$	0	0
$429$	−6216.00	−0.699560
$430$	0	0
$431$	4304.00	0.481012	0.240506	−	0.970648i	$-0.422687\pi$
$431$	4304.00	0.481012	0.240506	+	0.970648i	$0.422687\pi$
$432$	0	0
$433$	11794.0	1.30897	0.654484	−	0.756076i	$-0.272886\pi$
$433$	11794.0	1.30897	0.654484	+	0.756076i	$0.272886\pi$
$434$	0	0
$435$	−5796.00	−0.638844
$436$	0	0
$437$	−736.000	−0.0805667
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	0	0
$443$	−3788.00	−0.406260	−0.203130	−	0.979152i	$-0.565111\pi$
$443$	−3788.00	−0.406260	−0.203130	+	0.979152i	$0.565111\pi$
$444$	0	0
$445$	3444.00	0.366879
$446$	0	0
$447$	−4026.00	−0.426003
$448$	0	0
$449$	−13342.0	−1.40233	−0.701167	−	0.712997i	$-0.747337\pi$
$449$	−13342.0	−1.40233	−0.701167	+	0.712997i	$0.747337\pi$
$450$	0	0
$451$	−7896.00	−0.824408
$452$	0	0
$453$	1272.00	0.131929
$454$	0	0
$455$	−24864.0	−2.56185
$456$	0	0
$457$	−4390.00	−0.449356	−0.224678	−	0.974433i	$-0.572133\pi$
$457$	−4390.00	−0.449356	−0.224678	+	0.974433i	$0.572133\pi$
$458$	0	0
$459$	2214.00	0.225143
$460$	0	0
$461$	−5798.00	−0.585770	−0.292885	−	0.956148i	$-0.594615\pi$
$461$	−5798.00	−0.585770	−0.292885	+	0.956148i	$0.594615\pi$
$462$	0	0
$463$	14656.0	1.47111	0.735553	−	0.677467i	$-0.236922\pi$
$463$	14656.0	1.47111	0.735553	+	0.677467i	$0.236922\pi$
$464$	0	0
$465$	3360.00	0.335089
$466$	0	0
$467$	8412.00	0.833535	0.416768	−	0.909013i	$-0.363163\pi$
$467$	8412.00	0.833535	0.416768	+	0.909013i	$0.363163\pi$
$468$	0	0
$469$	−10464.0	−1.03024
$470$	0	0
$471$	−786.000	−0.0768938
$472$	0	0
$473$	−112.000	−0.0108875
$474$	0	0
$475$	6532.00	0.630966
$476$	0	0
$477$	1170.00	0.112307
$478$	0	0
$479$	−14848.0	−1.41633	−0.708165	−	0.706047i	$-0.750477\pi$
$479$	−14848.0	−1.41633	−0.708165	+	0.706047i	$0.750477\pi$
$480$	0	0
$481$	−2220.00	−0.210443
$482$	0	0
$483$	−576.000	−0.0542627
$484$	0	0
$485$	−12124.0	−1.13510
$486$	0	0
$487$	−18568.0	−1.72771	−0.863857	−	0.503738i	$-0.831958\pi$
$487$	−18568.0	−1.72771	−0.863857	+	0.503738i	$0.831958\pi$
$488$	0	0
$489$	−6876.00	−0.635876
$490$	0	0
$491$	−14364.0	−1.32024	−0.660120	−	0.751160i	$-0.729495\pi$
$491$	−14364.0	−1.32024	−0.660120	+	0.751160i	$0.729495\pi$
$492$	0	0
$493$	11316.0	1.03377
$494$	0	0
$495$	3528.00	0.320347
$496$	0	0
$497$	−20544.0	−1.85417
$498$	0	0
$499$	21660.0	1.94316	0.971578	−	0.236720i	$-0.0760724\pi$
$499$	21660.0	1.94316	0.971578	+	0.236720i	$0.0760724\pi$
$500$	0	0
$501$	5688.00	0.507228
$502$	0	0
$503$	17112.0	1.51687	0.758436	−	0.651748i	$-0.225964\pi$
$503$	17112.0	1.51687	0.758436	+	0.651748i	$0.225964\pi$
$504$	0	0
$505$	3780.00	0.333085
$506$	0	0
$507$	9837.00	0.861689
$508$	0	0
$509$	−11478.0	−0.999516	−0.499758	−	0.866165i	$-0.666578\pi$
$509$	−11478.0	−0.999516	−0.499758	+	0.866165i	$0.666578\pi$
$510$	0	0
$511$	−23952.0	−2.07353
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	−20944.0	−1.79204
$516$	0	0
$517$	6720.00	0.571654
$518$	0	0
$519$	8622.00	0.729217
$520$	0	0
$521$	13114.0	1.10275	0.551377	−	0.834256i	$-0.314103\pi$
$521$	13114.0	1.10275	0.551377	+	0.834256i	$0.314103\pi$
$522$	0	0
$523$	−4508.00	−0.376905	−0.188452	−	0.982082i	$-0.560347\pi$
$523$	−4508.00	−0.376905	−0.188452	+	0.982082i	$0.560347\pi$
$524$	0	0
$525$	5112.00	0.424964
$526$	0	0
$527$	−6560.00	−0.542235
$528$	0	0
$529$	−12103.0	−0.994740
$530$	0	0
$531$	5364.00	0.438376
$532$	0	0
$533$	20868.0	1.69586
$534$	0	0
$535$	23688.0	1.91425
$536$	0	0
$537$	−3564.00	−0.286402
$538$	0	0
$539$	−6524.00	−0.521352
$540$	0	0
$541$	−22950.0	−1.82384	−0.911920	−	0.410368i	$-0.865400\pi$
$541$	−22950.0	−1.82384	−0.911920	+	0.410368i	$0.865400\pi$
$542$	0	0
$543$	10422.0	0.823666
$544$	0	0
$545$	5684.00	0.446745
$546$	0	0
$547$	−6580.00	−0.514334	−0.257167	−	0.966367i	$-0.582789\pi$
$547$	−6580.00	−0.514334	−0.257167	+	0.966367i	$0.582789\pi$
$548$	0	0
$549$	1962.00	0.152525
$550$	0	0
$551$	12696.0	0.981611
$552$	0	0
$553$	768.000	0.0590573
$554$	0	0
$555$	1260.00	0.0963676
$556$	0	0
$557$	−7046.00	−0.535994	−0.267997	−	0.963420i	$-0.586362\pi$
$557$	−7046.00	−0.535994	−0.267997	+	0.963420i	$0.586362\pi$
$558$	0	0
$559$	296.000	0.0223962
$560$	0	0
$561$	−6888.00	−0.518381
$562$	0	0
$563$	8252.00	0.617727	0.308864	−	0.951106i	$-0.400051\pi$
$563$	8252.00	0.617727	0.308864	+	0.951106i	$0.400051\pi$
$564$	0	0
$565$	−11004.0	−0.819366
$566$	0	0
$567$	1944.00	0.143986
$568$	0	0
$569$	−6838.00	−0.503803	−0.251901	−	0.967753i	$-0.581056\pi$
$569$	−6838.00	−0.503803	−0.251901	+	0.967753i	$0.581056\pi$
$570$	0	0
$571$	23316.0	1.70883	0.854417	−	0.519588i	$-0.173915\pi$
$571$	23316.0	1.70883	0.854417	+	0.519588i	$0.173915\pi$
$572$	0	0
$573$	−576.000	−0.0419943
$574$	0	0
$575$	−568.000	−0.0411952
$576$	0	0
$577$	−10558.0	−0.761760	−0.380880	−	0.924625i	$-0.624379\pi$
$577$	−10558.0	−0.761760	−0.380880	+	0.924625i	$0.624379\pi$
$578$	0	0
$579$	14406.0	1.03401
$580$	0	0
$581$	−36192.0	−2.58433
$582$	0	0
$583$	−3640.00	−0.258582
$584$	0	0
$585$	−9324.00	−0.658974
$586$	0	0
$587$	1028.00	0.0722830	0.0361415	−	0.999347i	$-0.488493\pi$
$587$	1028.00	0.0722830	0.0361415	+	0.999347i	$0.488493\pi$
$588$	0	0
$589$	−7360.00	−0.514879
$590$	0	0
$591$	−4554.00	−0.316965
$592$	0	0
$593$	1202.00	0.0832382	0.0416191	−	0.999134i	$-0.486748\pi$
$593$	1202.00	0.0832382	0.0416191	+	0.999134i	$0.486748\pi$
$594$	0	0
$595$	−27552.0	−1.89836
$596$	0	0
$597$	−15384.0	−1.05465
$598$	0	0
$599$	3576.00	0.243926	0.121963	−	0.992535i	$-0.461081\pi$
$599$	3576.00	0.243926	0.121963	+	0.992535i	$0.461081\pi$
$600$	0	0
$601$	8650.00	0.587090	0.293545	−	0.955945i	$-0.405165\pi$
$601$	8650.00	0.587090	0.293545	+	0.955945i	$0.405165\pi$
$602$	0	0
$603$	−3924.00	−0.265004
$604$	0	0
$605$	7658.00	0.514615
$606$	0	0
$607$	−12656.0	−0.846279	−0.423139	−	0.906065i	$-0.639072\pi$
$607$	−12656.0	−0.846279	−0.423139	+	0.906065i	$0.639072\pi$
$608$	0	0
$609$	9936.00	0.661128
$610$	0	0
$611$	−17760.0	−1.17593
$612$	0	0
$613$	3298.00	0.217300	0.108650	−	0.994080i	$-0.465347\pi$
$613$	3298.00	0.217300	0.108650	+	0.994080i	$0.465347\pi$
$614$	0	0
$615$	−11844.0	−0.776579
$616$	0	0
$617$	5370.00	0.350386	0.175193	−	0.984534i	$-0.443945\pi$
$617$	5370.00	0.350386	0.175193	+	0.984534i	$0.443945\pi$
$618$	0	0
$619$	−16220.0	−1.05321	−0.526605	−	0.850110i	$-0.676535\pi$
$619$	−16220.0	−1.05321	−0.526605	+	0.850110i	$0.676535\pi$
$620$	0	0
$621$	−216.000	−0.0139578
$622$	0	0
$623$	−5904.00	−0.379677
$624$	0	0
$625$	−19459.0	−1.24538
$626$	0	0
$627$	−7728.00	−0.492227
$628$	0	0
$629$	−2460.00	−0.155941
$630$	0	0
$631$	20360.0	1.28450	0.642249	−	0.766496i	$-0.278001\pi$
$631$	20360.0	1.28450	0.642249	+	0.766496i	$0.278001\pi$
$632$	0	0
$633$	3252.00	0.204195
$634$	0	0
$635$	24416.0	1.52586
$636$	0	0
$637$	17242.0	1.07245
$638$	0	0
$639$	−7704.00	−0.476941
$640$	0	0
$641$	14498.0	0.893349	0.446674	−	0.894697i	$-0.352608\pi$
$641$	14498.0	0.893349	0.446674	+	0.894697i	$0.352608\pi$
$642$	0	0
$643$	21612.0	1.32550	0.662748	−	0.748842i	$-0.269390\pi$
$643$	21612.0	1.32550	0.662748	+	0.748842i	$0.269390\pi$
$644$	0	0
$645$	−168.000	−0.0102558
$646$	0	0
$647$	−12184.0	−0.740344	−0.370172	−	0.928963i	$-0.620701\pi$
$647$	−12184.0	−0.740344	−0.370172	+	0.928963i	$0.620701\pi$
$648$	0	0
$649$	−16688.0	−1.00934
$650$	0	0
$651$	−5760.00	−0.346778
$652$	0	0
$653$	28122.0	1.68530	0.842648	−	0.538464i	$-0.180995\pi$
$653$	28122.0	1.68530	0.842648	+	0.538464i	$0.180995\pi$
$654$	0	0
$655$	−9128.00	−0.544520
$656$	0	0
$657$	−8982.00	−0.533366
$658$	0	0
$659$	−5700.00	−0.336935	−0.168468	−	0.985707i	$-0.553882\pi$
$659$	−5700.00	−0.336935	−0.168468	+	0.985707i	$0.553882\pi$
$660$	0	0
$661$	29458.0	1.73341	0.866705	−	0.498822i	$-0.166234\pi$
$661$	29458.0	1.73341	0.866705	+	0.498822i	$0.166234\pi$
$662$	0	0
$663$	18204.0	1.06634
$664$	0	0
$665$	−30912.0	−1.80258
$666$	0	0
$667$	−1104.00	−0.0640885
$668$	0	0
$669$	−2064.00	−0.119281
$670$	0	0
$671$	−6104.00	−0.351181
$672$	0	0
$673$	19810.0	1.13465	0.567325	−	0.823494i	$-0.307978\pi$
$673$	19810.0	1.13465	0.567325	+	0.823494i	$0.307978\pi$
$674$	0	0
$675$	1917.00	0.109312
$676$	0	0
$677$	10450.0	0.593244	0.296622	−	0.954995i	$-0.404140\pi$
$677$	10450.0	0.593244	0.296622	+	0.954995i	$0.404140\pi$
$678$	0	0
$679$	20784.0	1.17469
$680$	0	0
$681$	14436.0	0.812318
$682$	0	0
$683$	23300.0	1.30534	0.652672	−	0.757641i	$-0.273648\pi$
$683$	23300.0	1.30534	0.652672	+	0.757641i	$0.273648\pi$
$684$	0	0
$685$	−21420.0	−1.19477
$686$	0	0
$687$	−7482.00	−0.415511
$688$	0	0
$689$	9620.00	0.531920
$690$	0	0
$691$	−14212.0	−0.782417	−0.391208	−	0.920302i	$-0.627943\pi$
$691$	−14212.0	−0.782417	−0.391208	+	0.920302i	$0.627943\pi$
$692$	0	0
$693$	−6048.00	−0.331522
$694$	0	0
$695$	−7224.00	−0.394276
$696$	0	0
$697$	23124.0	1.25665
$698$	0	0
$699$	2094.00	0.113308
$700$	0	0
$701$	15978.0	0.860885	0.430443	−	0.902618i	$-0.358357\pi$
$701$	15978.0	0.860885	0.430443	+	0.902618i	$0.358357\pi$
$702$	0	0
$703$	−2760.00	−0.148073
$704$	0	0
$705$	10080.0	0.538489
$706$	0	0
$707$	−6480.00	−0.344704
$708$	0	0
$709$	8866.00	0.469633	0.234816	−	0.972040i	$-0.424551\pi$
$709$	8866.00	0.469633	0.234816	+	0.972040i	$0.424551\pi$
$710$	0	0
$711$	288.000	0.0151911
$712$	0	0
$713$	640.000	0.0336160
$714$	0	0
$715$	29008.0	1.51726
$716$	0	0
$717$	18960.0	0.987551
$718$	0	0
$719$	−7760.00	−0.402502	−0.201251	−	0.979540i	$-0.564501\pi$
$719$	−7760.00	−0.402502	−0.201251	+	0.979540i	$0.564501\pi$
$720$	0	0
$721$	35904.0	1.85456
$722$	0	0
$723$	−19530.0	−1.00460
$724$	0	0
$725$	9798.00	0.501915
$726$	0	0
$727$	−13080.0	−0.667277	−0.333638	−	0.942701i	$-0.608276\pi$
$727$	−13080.0	−0.667277	−0.333638	+	0.942701i	$0.608276\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	328.000	0.0165958
$732$	0	0
$733$	−16934.0	−0.853304	−0.426652	−	0.904416i	$-0.640307\pi$
$733$	−16934.0	−0.853304	−0.426652	+	0.904416i	$0.640307\pi$
$734$	0	0
$735$	−9786.00	−0.491105
$736$	0	0
$737$	12208.0	0.610159
$738$	0	0
$739$	−7060.00	−0.351429	−0.175715	−	0.984441i	$-0.556224\pi$
$739$	−7060.00	−0.351429	−0.175715	+	0.984441i	$0.556224\pi$
$740$	0	0
$741$	20424.0	1.01254
$742$	0	0
$743$	12520.0	0.618189	0.309094	−	0.951031i	$-0.399974\pi$
$743$	12520.0	0.618189	0.309094	+	0.951031i	$0.399974\pi$
$744$	0	0
$745$	18788.0	0.923945
$746$	0	0
$747$	−13572.0	−0.664757
$748$	0	0
$749$	−40608.0	−1.98102
$750$	0	0
$751$	9792.00	0.475786	0.237893	−	0.971291i	$-0.423543\pi$
$751$	9792.00	0.475786	0.237893	+	0.971291i	$0.423543\pi$
$752$	0	0
$753$	1884.00	0.0911777
$754$	0	0
$755$	−5936.00	−0.286137
$756$	0	0
$757$	−13166.0	−0.632135	−0.316068	−	0.948737i	$-0.602363\pi$
$757$	−13166.0	−0.632135	−0.316068	+	0.948737i	$0.602363\pi$
$758$	0	0
$759$	672.000	0.0321371
$760$	0	0
$761$	−23222.0	−1.10617	−0.553086	−	0.833124i	$-0.686550\pi$
$761$	−23222.0	−1.10617	−0.553086	+	0.833124i	$0.686550\pi$
$762$	0	0
$763$	−9744.00	−0.462328
$764$	0	0
$765$	−10332.0	−0.488306
$766$	0	0
$767$	44104.0	2.07628
$768$	0	0
$769$	−39934.0	−1.87264	−0.936318	−	0.351154i	$-0.885789\pi$
$769$	−39934.0	−1.87264	−0.936318	+	0.351154i	$0.885789\pi$
$770$	0	0
$771$	−14586.0	−0.681325
$772$	0	0
$773$	17106.0	0.795938	0.397969	−	0.917399i	$-0.369715\pi$
$773$	17106.0	0.795938	0.397969	+	0.917399i	$0.369715\pi$
$774$	0	0
$775$	−5680.00	−0.263267
$776$	0	0
$777$	−2160.00	−0.0997292
$778$	0	0
$779$	25944.0	1.19325
$780$	0	0
$781$	23968.0	1.09813
$782$	0	0
$783$	3726.00	0.170059
$784$	0	0
$785$	3668.00	0.166773
$786$	0	0
$787$	−9956.00	−0.450944	−0.225472	−	0.974250i	$-0.572392\pi$
$787$	−9956.00	−0.450944	−0.225472	+	0.974250i	$0.572392\pi$
$788$	0	0
$789$	−17448.0	−0.787282
$790$	0	0
$791$	18864.0	0.847948
$792$	0	0
$793$	16132.0	0.722401
$794$	0	0
$795$	−5460.00	−0.243580
$796$	0	0
$797$	9130.00	0.405773	0.202887	−	0.979202i	$-0.434968\pi$
$797$	9130.00	0.405773	0.202887	+	0.979202i	$0.434968\pi$
$798$	0	0
$799$	−19680.0	−0.871375
$800$	0	0
$801$	−2214.00	−0.0976627
$802$	0	0
$803$	27944.0	1.22805
$804$	0	0
$805$	2688.00	0.117689
$806$	0	0
$807$	−10578.0	−0.461417
$808$	0	0
$809$	11482.0	0.498993	0.249497	−	0.968376i	$-0.419735\pi$
$809$	11482.0	0.498993	0.249497	+	0.968376i	$0.419735\pi$
$810$	0	0
$811$	4612.00	0.199691	0.0998454	−	0.995003i	$-0.468165\pi$
$811$	4612.00	0.199691	0.0998454	+	0.995003i	$0.468165\pi$
$812$	0	0
$813$	768.000	0.0331303
$814$	0	0
$815$	32088.0	1.37913
$816$	0	0
$817$	368.000	0.0157585
$818$	0	0
$819$	15984.0	0.681961
$820$	0	0
$821$	35010.0	1.48826	0.744128	−	0.668038i	$-0.232865\pi$
$821$	35010.0	1.48826	0.744128	+	0.668038i	$0.232865\pi$
$822$	0	0
$823$	−13688.0	−0.579749	−0.289875	−	0.957065i	$-0.593614\pi$
$823$	−13688.0	−0.579749	−0.289875	+	0.957065i	$0.593614\pi$
$824$	0	0
$825$	−5964.00	−0.251685
$826$	0	0
$827$	11668.0	0.490612	0.245306	−	0.969446i	$-0.421112\pi$
$827$	11668.0	0.490612	0.245306	+	0.969446i	$0.421112\pi$
$828$	0	0
$829$	29306.0	1.22779	0.613896	−	0.789387i	$-0.289601\pi$
$829$	29306.0	1.22779	0.613896	+	0.789387i	$0.289601\pi$
$830$	0	0
$831$	−426.000	−0.0177831
$832$	0	0
$833$	19106.0	0.794698
$834$	0	0
$835$	−26544.0	−1.10011
$836$	0	0
$837$	−2160.00	−0.0892001
$838$	0	0
$839$	2664.00	0.109620	0.0548102	−	0.998497i	$-0.482545\pi$
$839$	2664.00	0.109620	0.0548102	+	0.998497i	$0.482545\pi$
$840$	0	0
$841$	−5345.00	−0.219156
$842$	0	0
$843$	26526.0	1.08375
$844$	0	0
$845$	−45906.0	−1.86889
$846$	0	0
$847$	−13128.0	−0.532566
$848$	0	0
$849$	−21540.0	−0.870731
$850$	0	0
$851$	240.000	0.00966756
$852$	0	0
$853$	−26030.0	−1.04484	−0.522421	−	0.852688i	$-0.674971\pi$
$853$	−26030.0	−1.04484	−0.522421	+	0.852688i	$0.674971\pi$
$854$	0	0
$855$	−11592.0	−0.463670
$856$	0	0
$857$	44202.0	1.76186	0.880929	−	0.473249i	$-0.156919\pi$
$857$	44202.0	1.76186	0.880929	+	0.473249i	$0.156919\pi$
$858$	0	0
$859$	−32748.0	−1.30075	−0.650377	−	0.759612i	$-0.725389\pi$
$859$	−32748.0	−1.30075	−0.650377	+	0.759612i	$0.725389\pi$
$860$	0	0
$861$	20304.0	0.803668
$862$	0	0
$863$	−45344.0	−1.78856	−0.894280	−	0.447507i	$-0.852312\pi$
$863$	−45344.0	−1.78856	−0.894280	+	0.447507i	$0.852312\pi$
$864$	0	0
$865$	−40236.0	−1.58158
$866$	0	0
$867$	5433.00	0.212819
$868$	0	0
$869$	−896.000	−0.0349767
$870$	0	0
$871$	−32264.0	−1.25514
$872$	0	0
$873$	7794.00	0.302161
$874$	0	0
$875$	18144.0	0.701005
$876$	0	0
$877$	8778.00	0.337984	0.168992	−	0.985617i	$-0.445949\pi$
$877$	8778.00	0.337984	0.168992	+	0.985617i	$0.445949\pi$
$878$	0	0
$879$	−22122.0	−0.848870
$880$	0	0
$881$	−4142.00	−0.158397	−0.0791984	−	0.996859i	$-0.525236\pi$
$881$	−4142.00	−0.158397	−0.0791984	+	0.996859i	$0.525236\pi$
$882$	0	0
$883$	22076.0	0.841355	0.420678	−	0.907210i	$-0.361792\pi$
$883$	22076.0	0.841355	0.420678	+	0.907210i	$0.361792\pi$
$884$	0	0
$885$	−25032.0	−0.950781
$886$	0	0
$887$	40376.0	1.52840	0.764201	−	0.644978i	$-0.223133\pi$
$887$	40376.0	1.52840	0.764201	+	0.644978i	$0.223133\pi$
$888$	0	0
$889$	−41856.0	−1.57908
$890$	0	0
$891$	−2268.00	−0.0852759
$892$	0	0
$893$	−22080.0	−0.827412
$894$	0	0
$895$	16632.0	0.621169
$896$	0	0
$897$	−1776.00	−0.0661080
$898$	0	0
$899$	−11040.0	−0.409571
$900$	0	0
$901$	10660.0	0.394158
$902$	0	0
$903$	288.000	0.0106136
$904$	0	0
$905$	−48636.0	−1.78643
$906$	0	0
$907$	−26396.0	−0.966334	−0.483167	−	0.875528i	$-0.660514\pi$
$907$	−26396.0	−0.966334	−0.483167	+	0.875528i	$0.660514\pi$
$908$	0	0
$909$	−2430.00	−0.0886667
$910$	0	0
$911$	−24368.0	−0.886222	−0.443111	−	0.896467i	$-0.646125\pi$
$911$	−24368.0	−0.886222	−0.443111	+	0.896467i	$0.646125\pi$
$912$	0	0
$913$	42224.0	1.53057
$914$	0	0
$915$	−9156.00	−0.330807
$916$	0	0
$917$	15648.0	0.563514
$918$	0	0
$919$	5096.00	0.182918	0.0914589	−	0.995809i	$-0.470847\pi$
$919$	5096.00	0.182918	0.0914589	+	0.995809i	$0.470847\pi$
$920$	0	0
$921$	4500.00	0.160999
$922$	0	0
$923$	−63344.0	−2.25893
$924$	0	0
$925$	−2130.00	−0.0757124
$926$	0	0
$927$	13464.0	0.477040
$928$	0	0
$929$	−18494.0	−0.653142	−0.326571	−	0.945173i	$-0.605893\pi$
$929$	−18494.0	−0.653142	−0.326571	+	0.945173i	$0.605893\pi$
$930$	0	0
$931$	21436.0	0.754604
$932$	0	0
$933$	22824.0	0.800883
$934$	0	0
$935$	32144.0	1.12430
$936$	0	0
$937$	−33222.0	−1.15829	−0.579144	−	0.815225i	$-0.696613\pi$
$937$	−33222.0	−1.15829	−0.579144	+	0.815225i	$0.696613\pi$
$938$	0	0
$939$	−14274.0	−0.496075
$940$	0	0
$941$	−27846.0	−0.964669	−0.482335	−	0.875987i	$-0.660211\pi$
$941$	−27846.0	−0.964669	−0.482335	+	0.875987i	$0.660211\pi$
$942$	0	0
$943$	−2256.00	−0.0779061
$944$	0	0
$945$	−9072.00	−0.312288
$946$	0	0
$947$	41052.0	1.40867	0.704335	−	0.709868i	$-0.251245\pi$
$947$	41052.0	1.40867	0.704335	+	0.709868i	$0.251245\pi$
$948$	0	0
$949$	−73852.0	−2.52617
$950$	0	0
$951$	−13122.0	−0.447434
$952$	0	0
$953$	5706.00	0.193951	0.0969756	−	0.995287i	$-0.469083\pi$
$953$	5706.00	0.193951	0.0969756	+	0.995287i	$0.469083\pi$
$954$	0	0
$955$	2688.00	0.0910802
$956$	0	0
$957$	−11592.0	−0.391553
$958$	0	0
$959$	36720.0	1.23644
$960$	0	0
$961$	−23391.0	−0.785170
$962$	0	0
$963$	−15228.0	−0.509570
$964$	0	0
$965$	−67228.0	−2.24264
$966$	0	0
$967$	39352.0	1.30866	0.654330	−	0.756209i	$-0.272951\pi$
$967$	39352.0	1.30866	0.654330	+	0.756209i	$0.272951\pi$
$968$	0	0
$969$	22632.0	0.750304
$970$	0	0
$971$	−33180.0	−1.09660	−0.548299	−	0.836282i	$-0.684724\pi$
$971$	−33180.0	−1.09660	−0.548299	+	0.836282i	$0.684724\pi$
$972$	0	0
$973$	12384.0	0.408030
$974$	0	0
$975$	15762.0	0.517731
$976$	0	0
$977$	−4014.00	−0.131442	−0.0657212	−	0.997838i	$-0.520935\pi$
$977$	−4014.00	−0.131442	−0.0657212	+	0.997838i	$0.520935\pi$
$978$	0	0
$979$	6888.00	0.224864
$980$	0	0
$981$	−3654.00	−0.118923
$982$	0	0
$983$	−20328.0	−0.659575	−0.329788	−	0.944055i	$-0.606977\pi$
$983$	−20328.0	−0.659575	−0.329788	+	0.944055i	$0.606977\pi$
$984$	0	0
$985$	21252.0	0.687457
$986$	0	0
$987$	−17280.0	−0.557273
$988$	0	0
$989$	−32.0000	−0.00102886
$990$	0	0
$991$	−11728.0	−0.375936	−0.187968	−	0.982175i	$-0.560190\pi$
$991$	−11728.0	−0.375936	−0.187968	+	0.982175i	$0.560190\pi$
$992$	0	0
$993$	−23412.0	−0.748195
$994$	0	0
$995$	71792.0	2.28740
$996$	0	0
$997$	−50974.0	−1.61922	−0.809610	−	0.586968i	$-0.800321\pi$
$997$	−50974.0	−1.61922	−0.809610	+	0.586968i	$0.800321\pi$
$998$	0	0
$999$	−810.000	−0.0256529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.4.a.g.1.1		1
3.2	odd	2		576.4.a.v.1.1		1
4.3	odd	2		192.4.a.a.1.1		1
8.3	odd	2		24.4.a.a.1.1	✓	1
8.5	even	2		48.4.a.b.1.1		1
12.11	even	2		576.4.a.u.1.1		1
16.3	odd	4		768.4.d.o.385.1		2
16.5	even	4		768.4.d.b.385.1		2
16.11	odd	4		768.4.d.o.385.2		2
16.13	even	4		768.4.d.b.385.2		2
24.5	odd	2		144.4.a.b.1.1		1
24.11	even	2		72.4.a.b.1.1		1
40.3	even	4		600.4.f.b.49.2		2
40.13	odd	4		1200.4.f.p.49.1		2
40.19	odd	2		600.4.a.h.1.1		1
40.27	even	4		600.4.f.b.49.1		2
40.29	even	2		1200.4.a.u.1.1		1
40.37	odd	4		1200.4.f.p.49.2		2
56.13	odd	2		2352.4.a.w.1.1		1
56.27	even	2		1176.4.a.a.1.1		1
72.11	even	6		648.4.i.k.433.1		2
72.43	odd	6		648.4.i.b.433.1		2
72.59	even	6		648.4.i.k.217.1		2
72.67	odd	6		648.4.i.b.217.1		2
120.59	even	2		1800.4.a.bg.1.1		1
120.83	odd	4		1800.4.f.q.649.2		2
120.107	odd	4		1800.4.f.q.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.a.a.1.1	✓	1	8.3	odd	2
48.4.a.b.1.1		1	8.5	even	2
72.4.a.b.1.1		1	24.11	even	2
144.4.a.b.1.1		1	24.5	odd	2
192.4.a.a.1.1		1	4.3	odd	2
192.4.a.g.1.1		1	1.1	even	1	trivial
576.4.a.u.1.1		1	12.11	even	2
576.4.a.v.1.1		1	3.2	odd	2
600.4.a.h.1.1		1	40.19	odd	2
600.4.f.b.49.1		2	40.27	even	4
600.4.f.b.49.2		2	40.3	even	4
648.4.i.b.217.1		2	72.67	odd	6
648.4.i.b.433.1		2	72.43	odd	6
648.4.i.k.217.1		2	72.59	even	6
648.4.i.k.433.1		2	72.11	even	6
768.4.d.b.385.1		2	16.5	even	4
768.4.d.b.385.2		2	16.13	even	4
768.4.d.o.385.1		2	16.3	odd	4
768.4.d.o.385.2		2	16.11	odd	4
1176.4.a.a.1.1		1	56.27	even	2
1200.4.a.u.1.1		1	40.29	even	2
1200.4.f.p.49.1		2	40.13	odd	4
1200.4.f.p.49.2		2	40.37	odd	4
1800.4.a.bg.1.1		1	120.59	even	2
1800.4.f.q.649.1		2	120.107	odd	4
1800.4.f.q.649.2		2	120.83	odd	4
2352.4.a.w.1.1		1	56.13	odd	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more