LMFDB - Embedded newform 198.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,6,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("198.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	198.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:	$31.7559963230$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	198.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +31.0000 q^{5} -230.000 q^{7} -64.0000 q^{8} +O(q^{10})$

\(q-4.00000 q^{2} +16.0000 q^{4} +31.0000 q^{5} -230.000 q^{7} -64.0000 q^{8} -124.000 q^{10} -121.000 q^{11} +112.000 q^{13} +920.000 q^{14} +256.000 q^{16} +1142.00 q^{17} -612.000 q^{19} +496.000 q^{20} +484.000 q^{22} +1941.00 q^{23} -2164.00 q^{25} -448.000 q^{26} -3680.00 q^{28} -1192.00 q^{29} -1037.00 q^{31} -1024.00 q^{32} -4568.00 q^{34} -7130.00 q^{35} +8083.00 q^{37} +2448.00 q^{38} -1984.00 q^{40} +10444.0 q^{41} +58.0000 q^{43} -1936.00 q^{44} -7764.00 q^{46} -8656.00 q^{47} +36093.0 q^{49} +8656.00 q^{50} +1792.00 q^{52} +20318.0 q^{53} -3751.00 q^{55} +14720.0 q^{56} +4768.00 q^{58} +21351.0 q^{59} +47044.0 q^{61} +4148.00 q^{62} +4096.00 q^{64} +3472.00 q^{65} +48093.0 q^{67} +18272.0 q^{68} +28520.0 q^{70} +24967.0 q^{71} -42288.0 q^{73} -32332.0 q^{74} -9792.00 q^{76} +27830.0 q^{77} -72410.0 q^{79} +7936.00 q^{80} -41776.0 q^{82} +15806.0 q^{83} +35402.0 q^{85} -232.000 q^{86} +7744.00 q^{88} +114761. q^{89} -25760.0 q^{91} +31056.0 q^{92} +34624.0 q^{94} -18972.0 q^{95} -5159.00 q^{97} -144372. q^{98} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	31.0000	0.554545	0.277272	−	0.960791i	$-0.410570\pi$
$5$	31.0000	0.554545	0.277272	+	0.960791i	$0.410570\pi$
$6$	0	0
$7$	−230.000	−1.77412	−0.887059	−	0.461655i	$-0.847256\pi$
$7$	−230.000	−1.77412	−0.887059	+	0.461655i	$0.847256\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−124.000	−0.392122
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	112.000	0.183806	0.0919030	−	0.995768i	$-0.470705\pi$
$13$	112.000	0.183806	0.0919030	+	0.995768i	$0.470705\pi$
$14$	920.000	1.25449
$15$	0	0
$16$	256.000	0.250000
$17$	1142.00	0.958393	0.479197	−	0.877708i	$-0.340928\pi$
$17$	1142.00	0.958393	0.479197	+	0.877708i	$0.340928\pi$
$18$	0	0
$19$	−612.000	−0.388926	−0.194463	−	0.980910i	$-0.562296\pi$
$19$	−612.000	−0.388926	−0.194463	+	0.980910i	$0.562296\pi$
$20$	496.000	0.277272
$21$	0	0
$22$	484.000	0.213201
$23$	1941.00	0.765078	0.382539	−	0.923939i	$-0.375050\pi$
$23$	1941.00	0.765078	0.382539	+	0.923939i	$0.375050\pi$
$24$	0	0
$25$	−2164.00	−0.692480
$26$	−448.000	−0.129970
$27$	0	0
$28$	−3680.00	−0.887059
$29$	−1192.00	−0.263197	−0.131599	−	0.991303i	$-0.542011\pi$
$29$	−1192.00	−0.263197	−0.131599	+	0.991303i	$0.542011\pi$
$30$	0	0
$31$	−1037.00	−0.193809	−0.0969046	−	0.995294i	$-0.530894\pi$
$31$	−1037.00	−0.193809	−0.0969046	+	0.995294i	$0.530894\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−4568.00	−0.677686
$35$	−7130.00	−0.983829
$36$	0	0
$37$	8083.00	0.970663	0.485331	−	0.874330i	$-0.338699\pi$
$37$	8083.00	0.970663	0.485331	+	0.874330i	$0.338699\pi$
$38$	2448.00	0.275012
$39$	0	0
$40$	−1984.00	−0.196061
$41$	10444.0	0.970303	0.485151	−	0.874430i	$-0.338765\pi$
$41$	10444.0	0.970303	0.485151	+	0.874430i	$0.338765\pi$
$42$	0	0
$43$	58.0000	0.00478362	0.00239181	−	0.999997i	$-0.499239\pi$
$43$	58.0000	0.00478362	0.00239181	+	0.999997i	$0.499239\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−7764.00	−0.540992
$47$	−8656.00	−0.571574	−0.285787	−	0.958293i	$-0.592255\pi$
$47$	−8656.00	−0.571574	−0.285787	+	0.958293i	$0.592255\pi$
$48$	0	0
$49$	36093.0	2.14750
$50$	8656.00	0.489657
$51$	0	0
$52$	1792.00	0.0919030
$53$	20318.0	0.993554	0.496777	−	0.867878i	$-0.334517\pi$
$53$	20318.0	0.993554	0.496777	+	0.867878i	$0.334517\pi$
$54$	0	0
$55$	−3751.00	−0.167202
$56$	14720.0	0.627246
$57$	0	0
$58$	4768.00	0.186109
$59$	21351.0	0.798524	0.399262	−	0.916837i	$-0.369266\pi$
$59$	21351.0	0.798524	0.399262	+	0.916837i	$0.369266\pi$
$60$	0	0
$61$	47044.0	1.61875	0.809375	−	0.587293i	$-0.199806\pi$
$61$	47044.0	1.61875	0.809375	+	0.587293i	$0.199806\pi$
$62$	4148.00	0.137044
$63$	0	0
$64$	4096.00	0.125000
$65$	3472.00	0.101929
$66$	0	0
$67$	48093.0	1.30887	0.654433	−	0.756120i	$-0.272908\pi$
$67$	48093.0	1.30887	0.654433	+	0.756120i	$0.272908\pi$
$68$	18272.0	0.479197
$69$	0	0
$70$	28520.0	0.695672
$71$	24967.0	0.587788	0.293894	−	0.955838i	$-0.405049\pi$
$71$	24967.0	0.587788	0.293894	+	0.955838i	$0.405049\pi$
$72$	0	0
$73$	−42288.0	−0.928774	−0.464387	−	0.885632i	$-0.653725\pi$
$73$	−42288.0	−0.928774	−0.464387	+	0.885632i	$0.653725\pi$
$74$	−32332.0	−0.686362
$75$	0	0
$76$	−9792.00	−0.194463
$77$	27830.0	0.534917
$78$	0	0
$79$	−72410.0	−1.30536	−0.652681	−	0.757633i	$-0.726356\pi$
$79$	−72410.0	−1.30536	−0.652681	+	0.757633i	$0.726356\pi$
$80$	7936.00	0.138636
$81$	0	0
$82$	−41776.0	−0.686108
$83$	15806.0	0.251841	0.125921	−	0.992040i	$-0.459812\pi$
$83$	15806.0	0.251841	0.125921	+	0.992040i	$0.459812\pi$
$84$	0	0
$85$	35402.0	0.531472
$86$	−232.000	−0.00338253
$87$	0	0
$88$	7744.00	0.106600
$89$	114761.	1.53575	0.767873	−	0.640602i	$-0.221315\pi$
$89$	114761.	1.53575	0.767873	+	0.640602i	$0.221315\pi$
$90$	0	0
$91$	−25760.0	−0.326094
$92$	31056.0	0.382539
$93$	0	0
$94$	34624.0	0.404164
$95$	−18972.0	−0.215677
$96$	0	0
$97$	−5159.00	−0.0556719	−0.0278360	−	0.999613i	$-0.508862\pi$
$97$	−5159.00	−0.0556719	−0.0278360	+	0.999613i	$0.508862\pi$
$98$	−144372.	−1.51851
$99$	0	0
$100$	−34624.0	−0.346240
$101$	61426.0	0.599168	0.299584	−	0.954070i	$-0.403152\pi$
$101$	61426.0	0.599168	0.299584	+	0.954070i	$0.403152\pi$
$102$	0	0
$103$	185896.	1.72654	0.863271	−	0.504741i	$-0.168412\pi$
$103$	185896.	1.72654	0.863271	+	0.504741i	$0.168412\pi$
$104$	−7168.00	−0.0649852
$105$	0	0
$106$	−81272.0	−0.702548
$107$	23970.0	0.202399	0.101200	−	0.994866i	$-0.467732\pi$
$107$	23970.0	0.202399	0.101200	+	0.994866i	$0.467732\pi$
$108$	0	0
$109$	−56326.0	−0.454091	−0.227045	−	0.973884i	$-0.572907\pi$
$109$	−56326.0	−0.454091	−0.227045	+	0.973884i	$0.572907\pi$
$110$	15004.0	0.118229
$111$	0	0
$112$	−58880.0	−0.443530
$113$	261903.	1.92950	0.964749	−	0.263171i	$-0.0847685\pi$
$113$	261903.	1.92950	0.964749	+	0.263171i	$0.0847685\pi$
$114$	0	0
$115$	60171.0	0.424270
$116$	−19072.0	−0.131599
$117$	0	0
$118$	−85404.0	−0.564642
$119$	−262660.	−1.70030
$120$	0	0
$121$	14641.0	0.0909091
$122$	−188176.	−1.14463
$123$	0	0
$124$	−16592.0	−0.0969046
$125$	−163959.	−0.938556
$126$	0	0
$127$	87404.0	0.480864	0.240432	−	0.970666i	$-0.422711\pi$
$127$	87404.0	0.480864	0.240432	+	0.970666i	$0.422711\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−13888.0	−0.0720744
$131$	−265122.	−1.34979	−0.674897	−	0.737912i	$-0.735812\pi$
$131$	−265122.	−1.34979	−0.674897	+	0.737912i	$0.735812\pi$
$132$	0	0
$133$	140760.	0.690002
$134$	−192372.	−0.925507
$135$	0	0
$136$	−73088.0	−0.338843
$137$	−245857.	−1.11913	−0.559566	−	0.828786i	$-0.689032\pi$
$137$	−245857.	−1.11913	−0.559566	+	0.828786i	$0.689032\pi$
$138$	0	0
$139$	−363594.	−1.59617	−0.798086	−	0.602544i	$-0.794154\pi$
$139$	−363594.	−1.59617	−0.798086	+	0.602544i	$0.794154\pi$
$140$	−114080.	−0.491914
$141$	0	0
$142$	−99868.0	−0.415629
$143$	−13552.0	−0.0554196
$144$	0	0
$145$	−36952.0	−0.145955
$146$	169152.	0.656742
$147$	0	0
$148$	129328.	0.485331
$149$	55750.0	0.205721	0.102861	−	0.994696i	$-0.467200\pi$
$149$	55750.0	0.205721	0.102861	+	0.994696i	$0.467200\pi$
$150$	0	0
$151$	65642.0	0.234282	0.117141	−	0.993115i	$-0.462627\pi$
$151$	65642.0	0.234282	0.117141	+	0.993115i	$0.462627\pi$
$152$	39168.0	0.137506
$153$	0	0
$154$	−111320.	−0.378243
$155$	−32147.0	−0.107476
$156$	0	0
$157$	−275367.	−0.891585	−0.445793	−	0.895136i	$-0.647078\pi$
$157$	−275367.	−0.891585	−0.445793	+	0.895136i	$0.647078\pi$
$158$	289640.	0.923030
$159$	0	0
$160$	−31744.0	−0.0980306
$161$	−446430.	−1.35734
$162$	0	0
$163$	291940.	0.860646	0.430323	−	0.902675i	$-0.358400\pi$
$163$	291940.	0.860646	0.430323	+	0.902675i	$0.358400\pi$
$164$	167104.	0.485151
$165$	0	0
$166$	−63224.0	−0.178079
$167$	337344.	0.936013	0.468006	−	0.883725i	$-0.344972\pi$
$167$	337344.	0.936013	0.468006	+	0.883725i	$0.344972\pi$
$168$	0	0
$169$	−358749.	−0.966215
$170$	−141608.	−0.375808
$171$	0	0
$172$	928.000	0.00239181
$173$	116742.	0.296560	0.148280	−	0.988945i	$-0.452626\pi$
$173$	116742.	0.296560	0.148280	+	0.988945i	$0.452626\pi$
$174$	0	0
$175$	497720.	1.22854
$176$	−30976.0	−0.0753778
$177$	0	0
$178$	−459044.	−1.08594
$179$	19107.0	0.0445718	0.0222859	−	0.999752i	$-0.492906\pi$
$179$	19107.0	0.0445718	0.0222859	+	0.999752i	$0.492906\pi$
$180$	0	0
$181$	−16177.0	−0.0367030	−0.0183515	−	0.999832i	$-0.505842\pi$
$181$	−16177.0	−0.0367030	−0.0183515	+	0.999832i	$0.505842\pi$
$182$	103040.	0.230583
$183$	0	0
$184$	−124224.	−0.270496
$185$	250573.	0.538276
$186$	0	0
$187$	−138182.	−0.288966
$188$	−138496.	−0.285787
$189$	0	0
$190$	75888.0	0.152507
$191$	−685333.	−1.35931	−0.679655	−	0.733532i	$-0.737870\pi$
$191$	−685333.	−1.35931	−0.679655	+	0.733532i	$0.737870\pi$
$192$	0	0
$193$	−309292.	−0.597689	−0.298845	−	0.954302i	$-0.596601\pi$
$193$	−309292.	−0.597689	−0.298845	+	0.954302i	$0.596601\pi$
$194$	20636.0	0.0393660
$195$	0	0
$196$	577488.	1.07375
$197$	120930.	0.222008	0.111004	−	0.993820i	$-0.464593\pi$
$197$	120930.	0.222008	0.111004	+	0.993820i	$0.464593\pi$
$198$	0	0
$199$	915536.	1.63886	0.819432	−	0.573177i	$-0.194289\pi$
$199$	915536.	1.63886	0.819432	+	0.573177i	$0.194289\pi$
$200$	138496.	0.244829
$201$	0	0
$202$	−245704.	−0.423676
$203$	274160.	0.466943
$204$	0	0
$205$	323764.	0.538076
$206$	−743584.	−1.22085
$207$	0	0
$208$	28672.0	0.0459515
$209$	74052.0	0.117266
$210$	0	0
$211$	−134580.	−0.208101	−0.104051	−	0.994572i	$-0.533180\pi$
$211$	−134580.	−0.208101	−0.104051	+	0.994572i	$0.533180\pi$
$212$	325088.	0.496777
$213$	0	0
$214$	−95880.0	−0.143118
$215$	1798.00	0.00265273
$216$	0	0
$217$	238510.	0.343841
$218$	225304.	0.321091
$219$	0	0
$220$	−60016.0	−0.0836008
$221$	127904.	0.176158
$222$	0	0
$223$	468839.	0.631337	0.315669	−	0.948869i	$-0.397771\pi$
$223$	468839.	0.631337	0.315669	+	0.948869i	$0.397771\pi$
$224$	235520.	0.313623
$225$	0	0
$226$	−1.04761e6	−1.36436
$227$	−275022.	−0.354244	−0.177122	−	0.984189i	$-0.556679\pi$
$227$	−275022.	−0.354244	−0.177122	+	0.984189i	$0.556679\pi$
$228$	0	0
$229$	−642281.	−0.809350	−0.404675	−	0.914461i	$-0.632615\pi$
$229$	−642281.	−0.809350	−0.404675	+	0.914461i	$0.632615\pi$
$230$	−240684.	−0.300004
$231$	0	0
$232$	76288.0	0.0930543
$233$	1.50485e6	1.81595	0.907973	−	0.419029i	$-0.137629\pi$
$233$	1.50485e6	1.81595	0.907973	+	0.419029i	$0.137629\pi$
$234$	0	0
$235$	−268336.	−0.316964
$236$	341616.	0.399262
$237$	0	0
$238$	1.05064e6	1.20230
$239$	304694.	0.345040	0.172520	−	0.985006i	$-0.444809\pi$
$239$	304694.	0.345040	0.172520	+	0.985006i	$0.444809\pi$
$240$	0	0
$241$	1.27181e6	1.41052	0.705260	−	0.708949i	$-0.250830\pi$
$241$	1.27181e6	1.41052	0.705260	+	0.708949i	$0.250830\pi$
$242$	−58564.0	−0.0642824
$243$	0	0
$244$	752704.	0.809375
$245$	1.11888e6	1.19088
$246$	0	0
$247$	−68544.0	−0.0714870
$248$	66368.0	0.0685219
$249$	0	0
$250$	655836.	0.663659
$251$	−629965.	−0.631149	−0.315575	−	0.948901i	$-0.602197\pi$
$251$	−629965.	−0.631149	−0.315575	+	0.948901i	$0.602197\pi$
$252$	0	0
$253$	−234861.	−0.230680
$254$	−349616.	−0.340022
$255$	0	0
$256$	65536.0	0.0625000
$257$	−544086.	−0.513848	−0.256924	−	0.966432i	$-0.582709\pi$
$257$	−544086.	−0.513848	−0.256924	+	0.966432i	$0.582709\pi$
$258$	0	0
$259$	−1.85909e6	−1.72207
$260$	55552.0	0.0509643
$261$	0	0
$262$	1.06049e6	0.954449
$263$	1.98933e6	1.77345	0.886724	−	0.462300i	$-0.152976\pi$
$263$	1.98933e6	1.77345	0.886724	+	0.462300i	$0.152976\pi$
$264$	0	0
$265$	629858.	0.550970
$266$	−563040.	−0.487905
$267$	0	0
$268$	769488.	0.654433
$269$	−1.75446e6	−1.47830	−0.739149	−	0.673541i	$-0.764772\pi$
$269$	−1.75446e6	−1.47830	−0.739149	+	0.673541i	$0.764772\pi$
$270$	0	0
$271$	−1.65824e6	−1.37159	−0.685795	−	0.727795i	$-0.740545\pi$
$271$	−1.65824e6	−1.37159	−0.685795	+	0.727795i	$0.740545\pi$
$272$	292352.	0.239598
$273$	0	0
$274$	983428.	0.791346
$275$	261844.	0.208791
$276$	0	0
$277$	−42634.0	−0.0333854	−0.0166927	−	0.999861i	$-0.505314\pi$
$277$	−42634.0	−0.0333854	−0.0166927	+	0.999861i	$0.505314\pi$
$278$	1.45438e6	1.12866
$279$	0	0
$280$	456320.	0.347836
$281$	−319510.	−0.241390	−0.120695	−	0.992690i	$-0.538512\pi$
$281$	−319510.	−0.241390	−0.120695	+	0.992690i	$0.538512\pi$
$282$	0	0
$283$	−2.02735e6	−1.50474	−0.752371	−	0.658739i	$-0.771090\pi$
$283$	−2.02735e6	−1.50474	−0.752371	+	0.658739i	$0.771090\pi$
$284$	399472.	0.293894
$285$	0	0
$286$	54208.0	0.0391876
$287$	−2.40212e6	−1.72143
$288$	0	0
$289$	−115693.	−0.0814821
$290$	147808.	0.103206
$291$	0	0
$292$	−676608.	−0.464387
$293$	−718844.	−0.489177	−0.244588	−	0.969627i	$-0.578653\pi$
$293$	−718844.	−0.489177	−0.244588	+	0.969627i	$0.578653\pi$
$294$	0	0
$295$	661881.	0.442818
$296$	−517312.	−0.343181
$297$	0	0
$298$	−223000.	−0.145467
$299$	217392.	0.140626
$300$	0	0
$301$	−13340.0	−0.00848671
$302$	−262568.	−0.165663
$303$	0	0
$304$	−156672.	−0.0972316
$305$	1.45836e6	0.897669
$306$	0	0
$307$	−1.98142e6	−1.19986	−0.599930	−	0.800052i	$-0.704805\pi$
$307$	−1.98142e6	−1.19986	−0.599930	+	0.800052i	$0.704805\pi$
$308$	445280.	0.267458
$309$	0	0
$310$	128588.	0.0759970
$311$	1.51030e6	0.885446	0.442723	−	0.896658i	$-0.354012\pi$
$311$	1.51030e6	0.885446	0.442723	+	0.896658i	$0.354012\pi$
$312$	0	0
$313$	2.00092e6	1.15443	0.577216	−	0.816591i	$-0.304139\pi$
$313$	2.00092e6	1.15443	0.577216	+	0.816591i	$0.304139\pi$
$314$	1.10147e6	0.630446
$315$	0	0
$316$	−1.15856e6	−0.652681
$317$	259331.	0.144946	0.0724730	−	0.997370i	$-0.476911\pi$
$317$	259331.	0.144946	0.0724730	+	0.997370i	$0.476911\pi$
$318$	0	0
$319$	144232.	0.0793569
$320$	126976.	0.0693181
$321$	0	0
$322$	1.78572e6	0.959784
$323$	−698904.	−0.372744
$324$	0	0
$325$	−242368.	−0.127282
$326$	−1.16776e6	−0.608569
$327$	0	0
$328$	−668416.	−0.343054
$329$	1.99088e6	1.01404
$330$	0	0
$331$	51203.0	0.0256877	0.0128439	−	0.999918i	$-0.495912\pi$
$331$	51203.0	0.0256877	0.0128439	+	0.999918i	$0.495912\pi$
$332$	252896.	0.125921
$333$	0	0
$334$	−1.34938e6	−0.661861
$335$	1.49088e6	0.725824
$336$	0	0
$337$	266870.	0.128004	0.0640022	−	0.997950i	$-0.479614\pi$
$337$	266870.	0.128004	0.0640022	+	0.997950i	$0.479614\pi$
$338$	1.43500e6	0.683217
$339$	0	0
$340$	566432.	0.265736
$341$	125477.	0.0584357
$342$	0	0
$343$	−4.43578e6	−2.03580
$344$	−3712.00	−0.00169127
$345$	0	0
$346$	−466968.	−0.209699
$347$	−622800.	−0.277667	−0.138834	−	0.990316i	$-0.544335\pi$
$347$	−622800.	−0.277667	−0.138834	+	0.990316i	$0.544335\pi$
$348$	0	0
$349$	2.43649e6	1.07078	0.535391	−	0.844604i	$-0.320164\pi$
$349$	2.43649e6	1.07078	0.535391	+	0.844604i	$0.320164\pi$
$350$	−1.99088e6	−0.868710
$351$	0	0
$352$	123904.	0.0533002
$353$	−1.55957e6	−0.666144	−0.333072	−	0.942901i	$-0.608085\pi$
$353$	−1.55957e6	−0.666144	−0.333072	+	0.942901i	$0.608085\pi$
$354$	0	0
$355$	773977.	0.325955
$356$	1.83618e6	0.767873
$357$	0	0
$358$	−76428.0	−0.0315170
$359$	−1.91961e6	−0.786098	−0.393049	−	0.919518i	$-0.628580\pi$
$359$	−1.91961e6	−0.786098	−0.393049	+	0.919518i	$0.628580\pi$
$360$	0	0
$361$	−2.10156e6	−0.848736
$362$	64708.0	0.0259529
$363$	0	0
$364$	−412160.	−0.163047
$365$	−1.31093e6	−0.515047
$366$	0	0
$367$	3.61225e6	1.39995	0.699975	−	0.714167i	$-0.253194\pi$
$367$	3.61225e6	1.39995	0.699975	+	0.714167i	$0.253194\pi$
$368$	496896.	0.191270
$369$	0	0
$370$	−1.00229e6	−0.380619
$371$	−4.67314e6	−1.76268
$372$	0	0
$373$	3.93968e6	1.46619	0.733093	−	0.680128i	$-0.238076\pi$
$373$	3.93968e6	1.46619	0.733093	+	0.680128i	$0.238076\pi$
$374$	552728.	0.204330
$375$	0	0
$376$	553984.	0.202082
$377$	−133504.	−0.0483772
$378$	0	0
$379$	−2.18829e6	−0.782540	−0.391270	−	0.920276i	$-0.627964\pi$
$379$	−2.18829e6	−0.782540	−0.391270	+	0.920276i	$0.627964\pi$
$380$	−303552.	−0.107839
$381$	0	0
$382$	2.74133e6	0.961177
$383$	−768387.	−0.267660	−0.133830	−	0.991004i	$-0.542728\pi$
$383$	−768387.	−0.267660	−0.133830	+	0.991004i	$0.542728\pi$
$384$	0	0
$385$	862730.	0.296635
$386$	1.23717e6	0.422630
$387$	0	0
$388$	−82544.0	−0.0278360
$389$	−324313.	−0.108665	−0.0543326	−	0.998523i	$-0.517303\pi$
$389$	−324313.	−0.108665	−0.0543326	+	0.998523i	$0.517303\pi$
$390$	0	0
$391$	2.21662e6	0.733246
$392$	−2.30995e6	−0.759255
$393$	0	0
$394$	−483720.	−0.156983
$395$	−2.24471e6	−0.723882
$396$	0	0
$397$	334758.	0.106599	0.0532997	−	0.998579i	$-0.483026\pi$
$397$	334758.	0.106599	0.0532997	+	0.998579i	$0.483026\pi$
$398$	−3.66214e6	−1.15885
$399$	0	0
$400$	−553984.	−0.173120
$401$	902022.	0.280128	0.140064	−	0.990142i	$-0.455269\pi$
$401$	902022.	0.280128	0.140064	+	0.990142i	$0.455269\pi$
$402$	0	0
$403$	−116144.	−0.0356233
$404$	982816.	0.299584
$405$	0	0
$406$	−1.09664e6	−0.330179
$407$	−978043.	−0.292666
$408$	0	0
$409$	−5.00457e6	−1.47931	−0.739654	−	0.672987i	$-0.765011\pi$
$409$	−5.00457e6	−1.47931	−0.739654	+	0.672987i	$0.765011\pi$
$410$	−1.29506e6	−0.380477
$411$	0	0
$412$	2.97434e6	0.863271
$413$	−4.91073e6	−1.41668
$414$	0	0
$415$	489986.	0.139657
$416$	−114688.	−0.0324926
$417$	0	0
$418$	−296208.	−0.0829194
$419$	3.00124e6	0.835151	0.417576	−	0.908642i	$-0.362880\pi$
$419$	3.00124e6	0.835151	0.417576	+	0.908642i	$0.362880\pi$
$420$	0	0
$421$	4.56224e6	1.25451	0.627253	−	0.778816i	$-0.284179\pi$
$421$	4.56224e6	1.25451	0.627253	+	0.778816i	$0.284179\pi$
$422$	538320.	0.147150
$423$	0	0
$424$	−1.30035e6	−0.351274
$425$	−2.47129e6	−0.663668
$426$	0	0
$427$	−1.08201e7	−2.87185
$428$	383520.	0.101200
$429$	0	0
$430$	−7192.00	−0.00187577
$431$	−4.89783e6	−1.27002	−0.635009	−	0.772504i	$-0.719004\pi$
$431$	−4.89783e6	−1.27002	−0.635009	+	0.772504i	$0.719004\pi$
$432$	0	0
$433$	6.72876e6	1.72471	0.862353	−	0.506307i	$-0.168990\pi$
$433$	6.72876e6	1.72471	0.862353	+	0.506307i	$0.168990\pi$
$434$	−954040.	−0.243132
$435$	0	0
$436$	−901216.	−0.227045
$437$	−1.18789e6	−0.297559
$438$	0	0
$439$	−3.35034e6	−0.829711	−0.414856	−	0.909887i	$-0.636168\pi$
$439$	−3.35034e6	−0.829711	−0.414856	+	0.909887i	$0.636168\pi$
$440$	240064.	0.0591147
$441$	0	0
$442$	−511616.	−0.124563
$443$	7.12434e6	1.72479	0.862394	−	0.506238i	$-0.168964\pi$
$443$	7.12434e6	1.72479	0.862394	+	0.506238i	$0.168964\pi$
$444$	0	0
$445$	3.55759e6	0.851640
$446$	−1.87536e6	−0.446423
$447$	0	0
$448$	−942080.	−0.221765
$449$	2.70928e6	0.634218	0.317109	−	0.948389i	$-0.397288\pi$
$449$	2.70928e6	0.634218	0.317109	+	0.948389i	$0.397288\pi$
$450$	0	0
$451$	−1.26372e6	−0.292557
$452$	4.19045e6	0.964749
$453$	0	0
$454$	1.10009e6	0.250489
$455$	−798560.	−0.180834
$456$	0	0
$457$	2.41361e6	0.540601	0.270301	−	0.962776i	$-0.412877\pi$
$457$	2.41361e6	0.540601	0.270301	+	0.962776i	$0.412877\pi$
$458$	2.56912e6	0.572297
$459$	0	0
$460$	962736.	0.212135
$461$	6.56065e6	1.43779	0.718894	−	0.695120i	$-0.244649\pi$
$461$	6.56065e6	1.43779	0.718894	+	0.695120i	$0.244649\pi$
$462$	0	0
$463$	−4.72421e6	−1.02418	−0.512090	−	0.858932i	$-0.671129\pi$
$463$	−4.72421e6	−1.02418	−0.512090	+	0.858932i	$0.671129\pi$
$464$	−305152.	−0.0657993
$465$	0	0
$466$	−6.01939e6	−1.28407
$467$	2.28444e6	0.484716	0.242358	−	0.970187i	$-0.422079\pi$
$467$	2.28444e6	0.484716	0.242358	+	0.970187i	$0.422079\pi$
$468$	0	0
$469$	−1.10614e7	−2.32208
$470$	1.07334e6	0.224127
$471$	0	0
$472$	−1.36646e6	−0.282321
$473$	−7018.00	−0.00144232
$474$	0	0
$475$	1.32437e6	0.269324
$476$	−4.20256e6	−0.850152
$477$	0	0
$478$	−1.21878e6	−0.243980
$479$	−951544.	−0.189492	−0.0947458	−	0.995501i	$-0.530204\pi$
$479$	−951544.	−0.189492	−0.0947458	+	0.995501i	$0.530204\pi$
$480$	0	0
$481$	905296.	0.178414
$482$	−5.08723e6	−0.997388
$483$	0	0
$484$	234256.	0.0454545
$485$	−159929.	−0.0308726
$486$	0	0
$487$	3.51484e6	0.671558	0.335779	−	0.941941i	$-0.391000\pi$
$487$	3.51484e6	0.671558	0.335779	+	0.941941i	$0.391000\pi$
$488$	−3.01082e6	−0.572314
$489$	0	0
$490$	−4.47553e6	−0.842082
$491$	5.78719e6	1.08334	0.541669	−	0.840592i	$-0.317793\pi$
$491$	5.78719e6	1.08334	0.541669	+	0.840592i	$0.317793\pi$
$492$	0	0
$493$	−1.36126e6	−0.252246
$494$	274176.	0.0505489
$495$	0	0
$496$	−265472.	−0.0484523
$497$	−5.74241e6	−1.04281
$498$	0	0
$499$	1.02912e6	0.185019	0.0925095	−	0.995712i	$-0.470511\pi$
$499$	1.02912e6	0.185019	0.0925095	+	0.995712i	$0.470511\pi$
$500$	−2.62334e6	−0.469278
$501$	0	0
$502$	2.51986e6	0.446290
$503$	727370.	0.128184	0.0640922	−	0.997944i	$-0.479585\pi$
$503$	727370.	0.128184	0.0640922	+	0.997944i	$0.479585\pi$
$504$	0	0
$505$	1.90421e6	0.332266
$506$	939444.	0.163115
$507$	0	0
$508$	1.39846e6	0.240432
$509$	1.94630e6	0.332977	0.166489	−	0.986043i	$-0.446757\pi$
$509$	1.94630e6	0.332977	0.166489	+	0.986043i	$0.446757\pi$
$510$	0	0
$511$	9.72624e6	1.64776
$512$	−262144.	−0.0441942
$513$	0	0
$514$	2.17634e6	0.363345
$515$	5.76278e6	0.957445
$516$	0	0
$517$	1.04738e6	0.172336
$518$	7.43636e6	1.21769
$519$	0	0
$520$	−222208.	−0.0360372
$521$	1.03133e7	1.66457	0.832286	−	0.554346i	$-0.187032\pi$
$521$	1.03133e7	1.66457	0.832286	+	0.554346i	$0.187032\pi$
$522$	0	0
$523$	−6.86840e6	−1.09800	−0.548998	−	0.835823i	$-0.684991\pi$
$523$	−6.86840e6	−1.09800	−0.548998	+	0.835823i	$0.684991\pi$
$524$	−4.24195e6	−0.674897
$525$	0	0
$526$	−7.95734e6	−1.25402
$527$	−1.18425e6	−0.185746
$528$	0	0
$529$	−2.66886e6	−0.414655
$530$	−2.51943e6	−0.389595
$531$	0	0
$532$	2.25216e6	0.345001
$533$	1.16973e6	0.178347
$534$	0	0
$535$	743070.	0.112239
$536$	−3.07795e6	−0.462754
$537$	0	0
$538$	7.01783e6	1.04532
$539$	−4.36725e6	−0.647495
$540$	0	0
$541$	1.00545e7	1.47695	0.738476	−	0.674280i	$-0.235546\pi$
$541$	1.00545e7	1.47695	0.738476	+	0.674280i	$0.235546\pi$
$542$	6.63296e6	0.969860
$543$	0	0
$544$	−1.16941e6	−0.169422
$545$	−1.74611e6	−0.251814
$546$	0	0
$547$	9.85725e6	1.40860	0.704299	−	0.709903i	$-0.251261\pi$
$547$	9.85725e6	1.40860	0.704299	+	0.709903i	$0.251261\pi$
$548$	−3.93371e6	−0.559566
$549$	0	0
$550$	−1.04738e6	−0.147637
$551$	729504.	0.102364
$552$	0	0
$553$	1.66543e7	2.31587
$554$	170536.	0.0236070
$555$	0	0
$556$	−5.81750e6	−0.798086
$557$	1.45892e7	1.99247	0.996237	−	0.0866757i	$-0.0276244\pi$
$557$	1.45892e7	1.99247	0.996237	+	0.0866757i	$0.0276244\pi$
$558$	0	0
$559$	6496.00	0.000879258 0
$560$	−1.82528e6	−0.245957
$561$	0	0
$562$	1.27804e6	0.170688
$563$	1.02413e7	1.36171	0.680855	−	0.732418i	$-0.261608\pi$
$563$	1.02413e7	1.36171	0.680855	+	0.732418i	$0.261608\pi$
$564$	0	0
$565$	8.11899e6	1.06999
$566$	8.10939e6	1.06401
$567$	0	0
$568$	−1.59789e6	−0.207814
$569$	751816.	0.0973489	0.0486744	−	0.998815i	$-0.484500\pi$
$569$	751816.	0.0973489	0.0486744	+	0.998815i	$0.484500\pi$
$570$	0	0
$571$	−7.01854e6	−0.900858	−0.450429	−	0.892812i	$-0.648729\pi$
$571$	−7.01854e6	−0.900858	−0.450429	+	0.892812i	$0.648729\pi$
$572$	−216832.	−0.0277098
$573$	0	0
$574$	9.60848e6	1.21724
$575$	−4.20032e6	−0.529801
$576$	0	0
$577$	−3.36377e6	−0.420617	−0.210308	−	0.977635i	$-0.567447\pi$
$577$	−3.36377e6	−0.420617	−0.210308	+	0.977635i	$0.567447\pi$
$578$	462772.	0.0576166
$579$	0	0
$580$	−591232.	−0.0729773
$581$	−3.63538e6	−0.446796
$582$	0	0
$583$	−2.45848e6	−0.299568
$584$	2.70643e6	0.328371
$585$	0	0
$586$	2.87538e6	0.345900
$587$	−1.40585e7	−1.68401	−0.842006	−	0.539468i	$-0.818625\pi$
$587$	−1.40585e7	−1.68401	−0.842006	+	0.539468i	$0.818625\pi$
$588$	0	0
$589$	634644.	0.0753775
$590$	−2.64752e6	−0.313119
$591$	0	0
$592$	2.06925e6	0.242666
$593$	5.39420e6	0.629927	0.314963	−	0.949104i	$-0.398008\pi$
$593$	5.39420e6	0.629927	0.314963	+	0.949104i	$0.398008\pi$
$594$	0	0
$595$	−8.14246e6	−0.942895
$596$	892000.	0.102861
$597$	0	0
$598$	−869568.	−0.0994376
$599$	1.20204e7	1.36883	0.684417	−	0.729090i	$-0.260057\pi$
$599$	1.20204e7	1.36883	0.684417	+	0.729090i	$0.260057\pi$
$600$	0	0
$601$	1.64636e6	0.185925	0.0929626	−	0.995670i	$-0.470366\pi$
$601$	1.64636e6	0.185925	0.0929626	+	0.995670i	$0.470366\pi$
$602$	53360.0	0.00600101
$603$	0	0
$604$	1.05027e6	0.117141
$605$	453871.	0.0504132
$606$	0	0
$607$	4.88451e6	0.538083	0.269041	−	0.963129i	$-0.413293\pi$
$607$	4.88451e6	0.538083	0.269041	+	0.963129i	$0.413293\pi$
$608$	626688.	0.0687531
$609$	0	0
$610$	−5.83346e6	−0.634748
$611$	−969472.	−0.105059
$612$	0	0
$613$	3.49011e6	0.375136	0.187568	−	0.982252i	$-0.439940\pi$
$613$	3.49011e6	0.375136	0.187568	+	0.982252i	$0.439940\pi$
$614$	7.92568e6	0.848429
$615$	0	0
$616$	−1.78112e6	−0.189122
$617$	−9.12072e6	−0.964531	−0.482266	−	0.876025i	$-0.660186\pi$
$617$	−9.12072e6	−0.964531	−0.482266	+	0.876025i	$0.660186\pi$
$618$	0	0
$619$	1.46635e7	1.53820	0.769098	−	0.639131i	$-0.220706\pi$
$619$	1.46635e7	1.53820	0.769098	+	0.639131i	$0.220706\pi$
$620$	−514352.	−0.0537380
$621$	0	0
$622$	−6.04120e6	−0.626105
$623$	−2.63950e7	−2.72460
$624$	0	0
$625$	1.67977e6	0.172009
$626$	−8.00368e6	−0.816307
$627$	0	0
$628$	−4.40587e6	−0.445793
$629$	9.23079e6	0.930277
$630$	0	0
$631$	−1.63870e7	−1.63842	−0.819212	−	0.573491i	$-0.805589\pi$
$631$	−1.63870e7	−1.63842	−0.819212	+	0.573491i	$0.805589\pi$
$632$	4.63424e6	0.461515
$633$	0	0
$634$	−1.03732e6	−0.102492
$635$	2.70952e6	0.266660
$636$	0	0
$637$	4.04242e6	0.394723
$638$	−576928.	−0.0561138
$639$	0	0
$640$	−507904.	−0.0490153
$641$	3.26835e6	0.314184	0.157092	−	0.987584i	$-0.449788\pi$
$641$	3.26835e6	0.314184	0.157092	+	0.987584i	$0.449788\pi$
$642$	0	0
$643$	8.32842e6	0.794393	0.397197	−	0.917734i	$-0.369983\pi$
$643$	8.32842e6	0.794393	0.397197	+	0.917734i	$0.369983\pi$
$644$	−7.14288e6	−0.678670
$645$	0	0
$646$	2.79562e6	0.263570
$647$	−2.49694e6	−0.234503	−0.117251	−	0.993102i	$-0.537408\pi$
$647$	−2.49694e6	−0.234503	−0.117251	+	0.993102i	$0.537408\pi$
$648$	0	0
$649$	−2.58347e6	−0.240764
$650$	969472.	0.0900019
$651$	0	0
$652$	4.67104e6	0.430323
$653$	789105.	0.0724189	0.0362094	−	0.999344i	$-0.488472\pi$
$653$	789105.	0.0724189	0.0362094	+	0.999344i	$0.488472\pi$
$654$	0	0
$655$	−8.21878e6	−0.748521
$656$	2.67366e6	0.242576
$657$	0	0
$658$	−7.96352e6	−0.717035
$659$	8.31393e6	0.745749	0.372874	−	0.927882i	$-0.378372\pi$
$659$	8.31393e6	0.745749	0.372874	+	0.927882i	$0.378372\pi$
$660$	0	0
$661$	−4.33517e6	−0.385925	−0.192962	−	0.981206i	$-0.561810\pi$
$661$	−4.33517e6	−0.385925	−0.192962	+	0.981206i	$0.561810\pi$
$662$	−204812.	−0.0181640
$663$	0	0
$664$	−1.01158e6	−0.0890393
$665$	4.36356e6	0.382637
$666$	0	0
$667$	−2.31367e6	−0.201366
$668$	5.39750e6	0.468006
$669$	0	0
$670$	−5.96353e6	−0.513235
$671$	−5.69232e6	−0.488071
$672$	0	0
$673$	7.29313e6	0.620693	0.310346	−	0.950624i	$-0.399555\pi$
$673$	7.29313e6	0.620693	0.310346	+	0.950624i	$0.399555\pi$
$674$	−1.06748e6	−0.0905128
$675$	0	0
$676$	−5.73998e6	−0.483108
$677$	−1.55814e7	−1.30658	−0.653288	−	0.757109i	$-0.726611\pi$
$677$	−1.55814e7	−1.30658	−0.653288	+	0.757109i	$0.726611\pi$
$678$	0	0
$679$	1.18657e6	0.0987686
$680$	−2.26573e6	−0.187904
$681$	0	0
$682$	−501908.	−0.0413203
$683$	2.16930e6	0.177938	0.0889690	−	0.996034i	$-0.471643\pi$
$683$	2.16930e6	0.177938	0.0889690	+	0.996034i	$0.471643\pi$
$684$	0	0
$685$	−7.62157e6	−0.620609
$686$	1.77431e7	1.43953
$687$	0	0
$688$	14848.0	0.00119591
$689$	2.27562e6	0.182621
$690$	0	0
$691$	−1.32195e7	−1.05322	−0.526610	−	0.850107i	$-0.676537\pi$
$691$	−1.32195e7	−1.05322	−0.526610	+	0.850107i	$0.676537\pi$
$692$	1.86787e6	0.148280
$693$	0	0
$694$	2.49120e6	0.196341
$695$	−1.12714e7	−0.885149
$696$	0	0
$697$	1.19270e7	0.929932
$698$	−9.74596e6	−0.757157
$699$	0	0
$700$	7.96352e6	0.614271
$701$	2.59395e7	1.99373	0.996866	−	0.0791122i	$-0.0252085\pi$
$701$	2.59395e7	1.99373	0.996866	+	0.0791122i	$0.0252085\pi$
$702$	0	0
$703$	−4.94680e6	−0.377516
$704$	−495616.	−0.0376889
$705$	0	0
$706$	6.23828e6	0.471035
$707$	−1.41280e7	−1.06300
$708$	0	0
$709$	3.57531e6	0.267115	0.133557	−	0.991041i	$-0.457360\pi$
$709$	3.57531e6	0.267115	0.133557	+	0.991041i	$0.457360\pi$
$710$	−3.09591e6	−0.230485
$711$	0	0
$712$	−7.34470e6	−0.542968
$713$	−2.01282e6	−0.148279
$714$	0	0
$715$	−420112.	−0.0307326
$716$	305712.	0.0222859
$717$	0	0
$718$	7.67843e6	0.555855
$719$	1.95814e6	0.141261	0.0706304	−	0.997503i	$-0.477499\pi$
$719$	1.95814e6	0.141261	0.0706304	+	0.997503i	$0.477499\pi$
$720$	0	0
$721$	−4.27561e7	−3.06309
$722$	8.40622e6	0.600147
$723$	0	0
$724$	−258832.	−0.0183515
$725$	2.57949e6	0.182259
$726$	0	0
$727$	1.55360e7	1.09019	0.545095	−	0.838374i	$-0.316493\pi$
$727$	1.55360e7	1.09019	0.545095	+	0.838374i	$0.316493\pi$
$728$	1.64864e6	0.115292
$729$	0	0
$730$	5.24371e6	0.364193
$731$	66236.0	0.00458459
$732$	0	0
$733$	−1.46002e7	−1.00369	−0.501844	−	0.864958i	$-0.667345\pi$
$733$	−1.46002e7	−1.00369	−0.501844	+	0.864958i	$0.667345\pi$
$734$	−1.44490e7	−0.989915
$735$	0	0
$736$	−1.98758e6	−0.135248
$737$	−5.81925e6	−0.394638
$738$	0	0
$739$	−2.06682e7	−1.39217	−0.696085	−	0.717959i	$-0.745077\pi$
$739$	−2.06682e7	−1.39217	−0.696085	+	0.717959i	$0.745077\pi$
$740$	4.00917e6	0.269138
$741$	0	0
$742$	1.86926e7	1.24640
$743$	−1.17065e7	−0.777953	−0.388976	−	0.921248i	$-0.627171\pi$
$743$	−1.17065e7	−0.777953	−0.388976	+	0.921248i	$0.627171\pi$
$744$	0	0
$745$	1.72825e6	0.114082
$746$	−1.57587e7	−1.03675
$747$	0	0
$748$	−2.21091e6	−0.144483
$749$	−5.51310e6	−0.359080
$750$	0	0
$751$	−1.27607e7	−0.825610	−0.412805	−	0.910819i	$-0.635451\pi$
$751$	−1.27607e7	−0.825610	−0.412805	+	0.910819i	$0.635451\pi$
$752$	−2.21594e6	−0.142894
$753$	0	0
$754$	534016.	0.0342079
$755$	2.03490e6	0.129920
$756$	0	0
$757$	−1.40869e7	−0.893458	−0.446729	−	0.894669i	$-0.647411\pi$
$757$	−1.40869e7	−0.893458	−0.446729	+	0.894669i	$0.647411\pi$
$758$	8.75316e6	0.553339
$759$	0	0
$760$	1.21421e6	0.0762534
$761$	−2.33822e7	−1.46360	−0.731801	−	0.681518i	$-0.761320\pi$
$761$	−2.33822e7	−1.46360	−0.731801	+	0.681518i	$0.761320\pi$
$762$	0	0
$763$	1.29550e7	0.805611
$764$	−1.09653e7	−0.679655
$765$	0	0
$766$	3.07355e6	0.189264
$767$	2.39131e6	0.146774
$768$	0	0
$769$	−1.09575e7	−0.668185	−0.334092	−	0.942540i	$-0.608430\pi$
$769$	−1.09575e7	−0.668185	−0.334092	+	0.942540i	$0.608430\pi$
$770$	−3.45092e6	−0.209753
$771$	0	0
$772$	−4.94867e6	−0.298845
$773$	−1.69336e7	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$773$	−1.69336e7	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$774$	0	0
$775$	2.24407e6	0.134209
$776$	330176.	0.0196830
$777$	0	0
$778$	1.29725e6	0.0768379
$779$	−6.39173e6	−0.377376
$780$	0	0
$781$	−3.02101e6	−0.177225
$782$	−8.86649e6	−0.518483
$783$	0	0
$784$	9.23981e6	0.536875
$785$	−8.53638e6	−0.494424
$786$	0	0
$787$	−7.51655e6	−0.432595	−0.216298	−	0.976327i	$-0.569398\pi$
$787$	−7.51655e6	−0.432595	−0.216298	+	0.976327i	$0.569398\pi$
$788$	1.93488e6	0.111004
$789$	0	0
$790$	8.97884e6	0.511862
$791$	−6.02377e7	−3.42316
$792$	0	0
$793$	5.26893e6	0.297536
$794$	−1.33903e6	−0.0753771
$795$	0	0
$796$	1.46486e7	0.819432
$797$	−3.93788e6	−0.219592	−0.109796	−	0.993954i	$-0.535020\pi$
$797$	−3.93788e6	−0.219592	−0.109796	+	0.993954i	$0.535020\pi$
$798$	0	0
$799$	−9.88515e6	−0.547793
$800$	2.21594e6	0.122414
$801$	0	0
$802$	−3.60809e6	−0.198080
$803$	5.11685e6	0.280036
$804$	0	0
$805$	−1.38393e7	−0.752706
$806$	464576.	0.0251895
$807$	0	0
$808$	−3.93126e6	−0.211838
$809$	1.73609e7	0.932612	0.466306	−	0.884624i	$-0.345585\pi$
$809$	1.73609e7	0.932612	0.466306	+	0.884624i	$0.345585\pi$
$810$	0	0
$811$	2.70850e7	1.44603	0.723014	−	0.690833i	$-0.242756\pi$
$811$	2.70850e7	1.44603	0.723014	+	0.690833i	$0.242756\pi$
$812$	4.38656e6	0.233472
$813$	0	0
$814$	3.91217e6	0.206946
$815$	9.05014e6	0.477267
$816$	0	0
$817$	−35496.0	−0.00186048
$818$	2.00183e7	1.04603
$819$	0	0
$820$	5.18022e6	0.269038
$821$	−3.59384e7	−1.86080	−0.930402	−	0.366540i	$-0.880542\pi$
$821$	−3.59384e7	−1.86080	−0.930402	+	0.366540i	$0.880542\pi$
$822$	0	0
$823$	−505509.	−0.0260153	−0.0130077	−	0.999915i	$-0.504141\pi$
$823$	−505509.	−0.0260153	−0.0130077	+	0.999915i	$0.504141\pi$
$824$	−1.18973e7	−0.610425
$825$	0	0
$826$	1.96429e7	1.00174
$827$	2.99955e7	1.52508	0.762539	−	0.646942i	$-0.223952\pi$
$827$	2.99955e7	1.52508	0.762539	+	0.646942i	$0.223952\pi$
$828$	0	0
$829$	2.96942e7	1.50067	0.750334	−	0.661059i	$-0.229893\pi$
$829$	2.96942e7	1.50067	0.750334	+	0.661059i	$0.229893\pi$
$830$	−1.95994e6	−0.0987526
$831$	0	0
$832$	458752.	0.0229757
$833$	4.12182e7	2.05815
$834$	0	0
$835$	1.04577e7	0.519061
$836$	1.18483e6	0.0586329
$837$	0	0
$838$	−1.20049e7	−0.590541
$839$	1.41371e7	0.693356	0.346678	−	0.937984i	$-0.387310\pi$
$839$	1.41371e7	0.693356	0.346678	+	0.937984i	$0.387310\pi$
$840$	0	0
$841$	−1.90903e7	−0.930727
$842$	−1.82490e7	−0.887070
$843$	0	0
$844$	−2.15328e6	−0.104051
$845$	−1.11212e7	−0.535810
$846$	0	0
$847$	−3.36743e6	−0.161284
$848$	5.20141e6	0.248388
$849$	0	0
$850$	9.88515e6	0.469284
$851$	1.56891e7	0.742633
$852$	0	0
$853$	−4.68539e6	−0.220482	−0.110241	−	0.993905i	$-0.535162\pi$
$853$	−4.68539e6	−0.220482	−0.110241	+	0.993905i	$0.535162\pi$
$854$	4.32805e7	2.03071
$855$	0	0
$856$	−1.53408e6	−0.0715589
$857$	4.12846e7	1.92015	0.960076	−	0.279740i	$-0.0902483\pi$
$857$	4.12846e7	1.92015	0.960076	+	0.279740i	$0.0902483\pi$
$858$	0	0
$859$	−3.54805e6	−0.164062	−0.0820308	−	0.996630i	$-0.526141\pi$
$859$	−3.54805e6	−0.164062	−0.0820308	+	0.996630i	$0.526141\pi$
$860$	28768.0	0.00132637
$861$	0	0
$862$	1.95913e7	0.898039
$863$	3.07605e7	1.40594	0.702970	−	0.711219i	$-0.251857\pi$
$863$	3.07605e7	1.40594	0.702970	+	0.711219i	$0.251857\pi$
$864$	0	0
$865$	3.61900e6	0.164456
$866$	−2.69150e7	−1.21955
$867$	0	0
$868$	3.81616e6	0.171920
$869$	8.76161e6	0.393581
$870$	0	0
$871$	5.38642e6	0.240577
$872$	3.60486e6	0.160545
$873$	0	0
$874$	4.75157e6	0.210406
$875$	3.77106e7	1.66511
$876$	0	0
$877$	3.05535e7	1.34141	0.670706	−	0.741723i	$-0.265991\pi$
$877$	3.05535e7	1.34141	0.670706	+	0.741723i	$0.265991\pi$
$878$	1.34013e7	0.586695
$879$	0	0
$880$	−960256.	−0.0418004
$881$	−4.21018e7	−1.82751	−0.913757	−	0.406262i	$-0.866832\pi$
$881$	−4.21018e7	−1.82751	−0.913757	+	0.406262i	$0.866832\pi$
$882$	0	0
$883$	57164.0	0.00246729	0.00123365	−	0.999999i	$-0.499607\pi$
$883$	57164.0	0.00246729	0.00123365	+	0.999999i	$0.499607\pi$
$884$	2.04646e6	0.0880792
$885$	0	0
$886$	−2.84974e7	−1.21961
$887$	1.16106e7	0.495504	0.247752	−	0.968824i	$-0.420308\pi$
$887$	1.16106e7	0.495504	0.247752	+	0.968824i	$0.420308\pi$
$888$	0	0
$889$	−2.01029e7	−0.853109
$890$	−1.42304e7	−0.602200
$891$	0	0
$892$	7.50142e6	0.315669
$893$	5.29747e6	0.222300
$894$	0	0
$895$	592317.	0.0247170
$896$	3.76832e6	0.156811
$897$	0	0
$898$	−1.08371e7	−0.448460
$899$	1.23610e6	0.0510101
$900$	0	0
$901$	2.32032e7	0.952215
$902$	5.05490e6	0.206869
$903$	0	0
$904$	−1.67618e7	−0.682181
$905$	−501487.	−0.0203535
$906$	0	0
$907$	2.09855e7	0.847034	0.423517	−	0.905888i	$-0.360795\pi$
$907$	2.09855e7	0.847034	0.423517	+	0.905888i	$0.360795\pi$
$908$	−4.40035e6	−0.177122
$909$	0	0
$910$	3.19424e6	0.127869
$911$	−4.74125e7	−1.89277	−0.946383	−	0.323047i	$-0.895293\pi$
$911$	−4.74125e7	−1.89277	−0.946383	+	0.323047i	$0.895293\pi$
$912$	0	0
$913$	−1.91253e6	−0.0759330
$914$	−9.65445e6	−0.382263
$915$	0	0
$916$	−1.02765e7	−0.404675
$917$	6.09781e7	2.39470
$918$	0	0
$919$	4.04326e7	1.57922	0.789610	−	0.613609i	$-0.210283\pi$
$919$	4.04326e7	1.57922	0.789610	+	0.613609i	$0.210283\pi$
$920$	−3.85094e6	−0.150002
$921$	0	0
$922$	−2.62426e7	−1.01667
$923$	2.79630e6	0.108039
$924$	0	0
$925$	−1.74916e7	−0.672164
$926$	1.88968e7	0.724205
$927$	0	0
$928$	1.22061e6	0.0465271
$929$	−3.30757e7	−1.25739	−0.628694	−	0.777652i	$-0.716410\pi$
$929$	−3.30757e7	−1.25739	−0.628694	+	0.777652i	$0.716410\pi$
$930$	0	0
$931$	−2.20889e7	−0.835219
$932$	2.40776e7	0.907973
$933$	0	0
$934$	−9.13776e6	−0.342746
$935$	−4.28364e6	−0.160245
$936$	0	0
$937$	−3.15132e7	−1.17258	−0.586292	−	0.810100i	$-0.699413\pi$
$937$	−3.15132e7	−1.17258	−0.586292	+	0.810100i	$0.699413\pi$
$938$	4.42456e7	1.64196
$939$	0	0
$940$	−4.29338e6	−0.158482
$941$	9.54147e6	0.351270	0.175635	−	0.984455i	$-0.443802\pi$
$941$	9.54147e6	0.351270	0.175635	+	0.984455i	$0.443802\pi$
$942$	0	0
$943$	2.02718e7	0.742358
$944$	5.46586e6	0.199631
$945$	0	0
$946$	28072.0	0.00101987
$947$	−2.24208e7	−0.812410	−0.406205	−	0.913782i	$-0.633148\pi$
$947$	−2.24208e7	−0.812410	−0.406205	+	0.913782i	$0.633148\pi$
$948$	0	0
$949$	−4.73626e6	−0.170714
$950$	−5.29747e6	−0.190441
$951$	0	0
$952$	1.68102e7	0.601148
$953$	−1.68985e7	−0.602720	−0.301360	−	0.953510i	$-0.597441\pi$
$953$	−1.68985e7	−0.602720	−0.301360	+	0.953510i	$0.597441\pi$
$954$	0	0
$955$	−2.12453e7	−0.753798
$956$	4.87510e6	0.172520
$957$	0	0
$958$	3.80618e6	0.133991
$959$	5.65471e7	1.98547
$960$	0	0
$961$	−2.75538e7	−0.962438
$962$	−3.62118e6	−0.126157
$963$	0	0
$964$	2.03489e7	0.705260
$965$	−9.58805e6	−0.331445
$966$	0	0
$967$	−3.06946e7	−1.05559	−0.527796	−	0.849371i	$-0.676982\pi$
$967$	−3.06946e7	−1.05559	−0.527796	+	0.849371i	$0.676982\pi$
$968$	−937024.	−0.0321412
$969$	0	0
$970$	639716.	0.0218302
$971$	−3.35664e7	−1.14250	−0.571251	−	0.820776i	$-0.693542\pi$
$971$	−3.35664e7	−1.14250	−0.571251	+	0.820776i	$0.693542\pi$
$972$	0	0
$973$	8.36266e7	2.83180
$974$	−1.40594e7	−0.474863
$975$	0	0
$976$	1.20433e7	0.404687
$977$	−2.47897e7	−0.830873	−0.415436	−	0.909622i	$-0.636371\pi$
$977$	−2.47897e7	−0.830873	−0.415436	+	0.909622i	$0.636371\pi$
$978$	0	0
$979$	−1.38861e7	−0.463045
$980$	1.79021e7	0.595442
$981$	0	0
$982$	−2.31488e7	−0.766036
$983$	−5.22606e6	−0.172501	−0.0862503	−	0.996274i	$-0.527488\pi$
$983$	−5.22606e6	−0.172501	−0.0862503	+	0.996274i	$0.527488\pi$
$984$	0	0
$985$	3.74883e6	0.123113
$986$	5.44506e6	0.178365
$987$	0	0
$988$	−1.09670e6	−0.0357435
$989$	112578.	0.00365985
$990$	0	0
$991$	2.40826e7	0.778967	0.389484	−	0.921033i	$-0.372653\pi$
$991$	2.40826e7	0.778967	0.389484	+	0.921033i	$0.372653\pi$
$992$	1.06189e6	0.0342610
$993$	0	0
$994$	2.29696e7	0.737375
$995$	2.83816e7	0.908823
$996$	0	0
$997$	−1.32606e7	−0.422499	−0.211249	−	0.977432i	$-0.567753\pi$
$997$	−1.32606e7	−0.422499	−0.211249	+	0.977432i	$0.567753\pi$
$998$	−4.11650e6	−0.130828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.6.a.b.1.1		1
3.2	odd	2		22.6.a.c.1.1	✓	1
12.11	even	2		176.6.a.e.1.1		1
15.2	even	4		550.6.b.a.199.2		2
15.8	even	4		550.6.b.a.199.1		2
15.14	odd	2		550.6.a.c.1.1		1
21.20	even	2		1078.6.a.f.1.1		1
24.5	odd	2		704.6.a.j.1.1		1
24.11	even	2		704.6.a.a.1.1		1
33.32	even	2		242.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.6.a.c.1.1	✓	1	3.2	odd	2
176.6.a.e.1.1		1	12.11	even	2
198.6.a.b.1.1		1	1.1	even	1	trivial
242.6.a.a.1.1		1	33.32	even	2
550.6.a.c.1.1		1	15.14	odd	2
550.6.b.a.199.1		2	15.8	even	4
550.6.b.a.199.2		2	15.2	even	4
704.6.a.a.1.1		1	24.11	even	2
704.6.a.j.1.1		1	24.5	odd	2
1078.6.a.f.1.1		1	21.20	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:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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