LMFDB - Embedded newform 2352.3.m.o.1471.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,3,Mod(1471,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2352.1471"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2,0,0,0,-18,0,0,0,44,0,0,0,16,0,0,0,0,0,0,0,28,0,0,0, 34,0,0,0,-6,0,0,0,-12,0,0,0,136] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

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

Self dual:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1364138928.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 35x^{4} + 364x^{2} + 972 $ x^6 + 35x^4 + 364x^2 + 972
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.2
Root			$3.78298i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.o.1471.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.31096 q^{5} -3.00000 q^{9} -11.5687i q^{11} +15.7937 q^{13} +4.00270i q^{15} +8.48271 q^{17} -2.46897i q^{19} +45.8352i q^{23} -19.6595 q^{25} +5.19615i q^{27} +35.2814 q^{29} +30.0485i q^{31} -20.0375 q^{33} +39.2468 q^{37} -27.3554i q^{39} -27.0405 q^{41} +1.39177i q^{43} +6.93289 q^{45} +21.2582i q^{47} -14.6925i q^{51} +32.3110 q^{53} +26.7348i q^{55} -4.27639 q^{57} +29.5269i q^{59} -103.394 q^{61} -36.4986 q^{65} -71.5527i q^{67} +79.3890 q^{69} -71.3595i q^{71} -29.2468 q^{73} +34.0512i q^{75} -102.925i q^{79} +9.00000 q^{81} +43.1509i q^{83} -19.6032 q^{85} -61.1091i q^{87} +124.097 q^{89} +52.0455 q^{93} +5.70571i q^{95} +14.7186 q^{97} +34.7060i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 18 q^{9} + 44 q^{13} + 16 q^{17} + 28 q^{25} + 34 q^{29} - 6 q^{33} - 12 q^{37} + 136 q^{41} - 6 q^{45} + 178 q^{53} + 60 q^{57} - 100 q^{61} + 368 q^{65} + 48 q^{69} + 72 q^{73} + 54 q^{81}+ \cdots + 266 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 - 18 * q^9 + 44 * q^13 + 16 * q^17 + 28 * q^25 + 34 * q^29 - 6 * q^33 - 12 * q^37 + 136 * q^41 - 6 * q^45 + 178 * q^53 + 60 * q^57 - 100 * q^61 + 368 * q^65 + 48 * q^69 + 72 * q^73 + 54 * q^81 + 536 * q^85 - 404 * q^89 - 78 * q^93 + 266 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	−2.31096	−0.462193	−0.231096	−	0.972931i	$-0.574231\pi$
$5$	−2.31096	−0.462193	−0.231096	+	0.972931i	$0.574231\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 11.5687i	− 1.05170i	−0.850578	−	0.525849i	$-0.823748\pi$
$11$	− 11.5687i	− 1.05170i	0.850578	−	0.525849i	$-0.176252\pi$
$12$	0	0
$13$	15.7937	1.21490	0.607449	−	0.794359i	$-0.292193\pi$
$13$	15.7937	1.21490	0.607449	+	0.794359i	$0.292193\pi$
$14$	0	0
$15$	4.00270i	0.266847i
$16$	0	0
$17$	8.48271	0.498983	0.249492	−	0.968377i	$-0.419737\pi$
$17$	8.48271	0.498983	0.249492	+	0.968377i	$0.419737\pi$
$18$	0	0
$19$	− 2.46897i	− 0.129946i	−0.997887	−	0.0649730i	$-0.979304\pi$
$19$	− 2.46897i	− 0.129946i	0.997887	−	0.0649730i	$-0.0206961\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	45.8352i	1.99284i	0.0845609	+	0.996418i	$0.473051\pi$
$23$	45.8352i	1.99284i	−0.0845609	+	0.996418i	$0.526949\pi$
$24$	0	0
$25$	−19.6595	−0.786378
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	35.2814	1.21660	0.608300	−	0.793707i	$-0.291852\pi$
$29$	35.2814	1.21660	0.608300	+	0.793707i	$0.291852\pi$
$30$	0	0
$31$	30.0485i	0.969305i	0.874707	+	0.484653i	$0.161054\pi$
$31$	30.0485i	0.969305i	−0.874707	+	0.484653i	$0.838946\pi$
$32$	0	0
$33$	−20.0375	−0.607198
$34$	0	0
$35$	0	0
$36$	0	0
$37$	39.2468	1.06072	0.530362	−	0.847771i	$-0.322056\pi$
$37$	39.2468	1.06072	0.530362	+	0.847771i	$0.322056\pi$
$38$	0	0
$39$	− 27.3554i	− 0.701422i
$40$	0	0
$41$	−27.0405	−0.659524	−0.329762	−	0.944064i	$-0.606968\pi$
$41$	−27.0405	−0.659524	−0.329762	+	0.944064i	$0.606968\pi$
$42$	0	0
$43$	1.39177i	0.0323667i	0.999869	+	0.0161834i	$0.00515155\pi$
$43$	1.39177i	0.0323667i	−0.999869	+	0.0161834i	$0.994848\pi$
$44$	0	0
$45$	6.93289	0.154064
$46$	0	0
$47$	21.2582i	0.452302i	0.974092	+	0.226151i	$0.0726144\pi$
$47$	21.2582i	0.452302i	−0.974092	+	0.226151i	$0.927386\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 14.6925i	− 0.288088i
$52$	0	0
$53$	32.3110	0.609641	0.304820	−	0.952410i	$-0.401404\pi$
$53$	32.3110	0.609641	0.304820	+	0.952410i	$0.401404\pi$
$54$	0	0
$55$	26.7348i	0.486087i
$56$	0	0
$57$	−4.27639	−0.0750244
$58$	0	0
$59$	29.5269i	0.500457i	0.968187	+	0.250228i	$0.0805057\pi$
$59$	29.5269i	0.500457i	−0.968187	+	0.250228i	$0.919494\pi$
$60$	0	0
$61$	−103.394	−1.69498	−0.847491	−	0.530809i	$-0.821888\pi$
$61$	−103.394	−1.69498	−0.847491	+	0.530809i	$0.821888\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−36.4986	−0.561517
$66$	0	0
$67$	− 71.5527i	− 1.06795i	−0.845500	−	0.533975i	$-0.820698\pi$
$67$	− 71.5527i	− 1.06795i	0.845500	−	0.533975i	$-0.179302\pi$
$68$	0	0
$69$	79.3890	1.15056
$70$	0	0
$71$	− 71.3595i	− 1.00506i	−0.864559	−	0.502532i	$-0.832402\pi$
$71$	− 71.3595i	− 1.00506i	0.864559	−	0.502532i	$-0.167598\pi$
$72$	0	0
$73$	−29.2468	−0.400641	−0.200321	−	0.979730i	$-0.564198\pi$
$73$	−29.2468	−0.400641	−0.200321	+	0.979730i	$0.564198\pi$
$74$	0	0
$75$	34.0512i	0.454016i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 102.925i	− 1.30284i	−0.758716	−	0.651421i	$-0.774173\pi$
$79$	− 102.925i	− 1.30284i	0.758716	−	0.651421i	$-0.225827\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	43.1509i	0.519890i	0.965623	+	0.259945i	$0.0837044\pi$
$83$	43.1509i	0.519890i	−0.965623	+	0.259945i	$0.916296\pi$
$84$	0	0
$85$	−19.6032	−0.230626
$86$	0	0
$87$	− 61.1091i	− 0.702404i
$88$	0	0
$89$	124.097	1.39435	0.697173	−	0.716903i	$-0.254441\pi$
$89$	124.097	1.39435	0.697173	+	0.716903i	$0.254441\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	52.0455	0.559629
$94$	0	0
$95$	5.70571i	0.0600601i
$96$	0	0
$97$	14.7186	0.151738	0.0758692	−	0.997118i	$-0.475827\pi$
$97$	14.7186	0.151738	0.0758692	+	0.997118i	$0.475827\pi$
$98$	0	0
$99$	34.7060i	0.350566i
$100$	0	0
$101$	83.5123	0.826854	0.413427	−	0.910537i	$-0.364332\pi$
$101$	83.5123	0.826854	0.413427	+	0.910537i	$0.364332\pi$
$102$	0	0
$103$	124.636i	1.21006i	0.796204	+	0.605028i	$0.206838\pi$
$103$	124.636i	1.21006i	−0.796204	+	0.605028i	$0.793162\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	36.0585i	0.336995i	0.985702	+	0.168498i	$0.0538916\pi$
$107$	36.0585i	0.336995i	−0.985702	+	0.168498i	$0.946108\pi$
$108$	0	0
$109$	184.700	1.69449	0.847247	−	0.531199i	$-0.178258\pi$
$109$	184.700	1.69449	0.847247	+	0.531199i	$0.178258\pi$
$110$	0	0
$111$	− 67.9775i	− 0.612410i
$112$	0	0
$113$	211.813	1.87446	0.937228	−	0.348718i	$-0.113383\pi$
$113$	211.813	1.87446	0.937228	+	0.348718i	$0.113383\pi$
$114$	0	0
$115$	− 105.924i	− 0.921074i
$116$	0	0
$117$	−47.3810	−0.404966
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−12.8342	−0.106067
$122$	0	0
$123$	46.8355i	0.380776i
$124$	0	0
$125$	103.206	0.825651
$126$	0	0
$127$	− 35.1062i	− 0.276427i	−0.990402	−	0.138213i	$-0.955864\pi$
$127$	− 35.1062i	− 0.276427i	0.990402	−	0.138213i	$-0.0441360\pi$
$128$	0	0
$129$	2.41061	0.0186869
$130$	0	0
$131$	157.940i	1.20565i	0.797874	+	0.602824i	$0.205958\pi$
$131$	157.940i	1.20565i	−0.797874	+	0.602824i	$0.794042\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 12.0081i	− 0.0889490i
$136$	0	0
$137$	221.050	1.61351	0.806753	−	0.590888i	$-0.201223\pi$
$137$	221.050	1.61351	0.806753	+	0.590888i	$0.201223\pi$
$138$	0	0
$139$	− 136.854i	− 0.984563i	−0.870436	−	0.492281i	$-0.836163\pi$
$139$	− 136.854i	− 0.984563i	0.870436	−	0.492281i	$-0.163837\pi$
$140$	0	0
$141$	36.8203	0.261137
$142$	0	0
$143$	− 182.712i	− 1.27771i
$144$	0	0
$145$	−81.5339	−0.562303
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−46.2744	−0.310566	−0.155283	−	0.987870i	$-0.549629\pi$
$149$	−46.2744	−0.310566	−0.155283	+	0.987870i	$0.549629\pi$
$150$	0	0
$151$	− 92.7852i	− 0.614471i	−0.951633	−	0.307236i	$-0.900596\pi$
$151$	− 92.7852i	− 0.614471i	0.951633	−	0.307236i	$-0.0994040\pi$
$152$	0	0
$153$	−25.4481	−0.166328
$154$	0	0
$155$	− 69.4409i	− 0.448006i
$156$	0	0
$157$	184.026	1.17214	0.586070	−	0.810261i	$-0.300675\pi$
$157$	184.026	1.17214	0.586070	+	0.810261i	$0.300675\pi$
$158$	0	0
$159$	− 55.9642i	− 0.351976i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	66.5771i	0.408448i	0.978924	+	0.204224i	$0.0654671\pi$
$163$	66.5771i	0.408448i	−0.978924	+	0.204224i	$0.934533\pi$
$164$	0	0
$165$	46.3060	0.280642
$166$	0	0
$167$	118.824i	0.711521i	0.934577	+	0.355761i	$0.115778\pi$
$167$	118.824i	0.711521i	−0.934577	+	0.355761i	$0.884222\pi$
$168$	0	0
$169$	80.4402	0.475978
$170$	0	0
$171$	7.40692i	0.0433153i
$172$	0	0
$173$	−200.419	−1.15849	−0.579247	−	0.815152i	$-0.696653\pi$
$173$	−200.419	−1.15849	−0.579247	+	0.815152i	$0.696653\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	51.1422	0.288939
$178$	0	0
$179$	− 185.220i	− 1.03475i	−0.855759	−	0.517375i	$-0.826909\pi$
$179$	− 185.220i	− 1.03475i	0.855759	−	0.517375i	$-0.173091\pi$
$180$	0	0
$181$	228.372	1.26173	0.630863	−	0.775894i	$-0.282701\pi$
$181$	228.372	1.26173	0.630863	+	0.775894i	$0.282701\pi$
$182$	0	0
$183$	179.084i	0.978599i
$184$	0	0
$185$	−90.6979	−0.490259
$186$	0	0
$187$	− 98.1337i	− 0.524779i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 150.387i	− 0.787369i	−0.919246	−	0.393684i	$-0.871200\pi$
$191$	− 150.387i	− 0.787369i	0.919246	−	0.393684i	$-0.128800\pi$
$192$	0	0
$193$	81.5410	0.422492	0.211246	−	0.977433i	$-0.432248\pi$
$193$	81.5410	0.422492	0.211246	+	0.977433i	$0.432248\pi$
$194$	0	0
$195$	63.2174i	0.324192i
$196$	0	0
$197$	−192.720	−0.978273	−0.489136	−	0.872207i	$-0.662688\pi$
$197$	−192.720	−0.978273	−0.489136	+	0.872207i	$0.662688\pi$
$198$	0	0
$199$	87.5533i	0.439966i	0.975504	+	0.219983i	$0.0706003\pi$
$199$	87.5533i	0.439966i	−0.975504	+	0.219983i	$0.929400\pi$
$200$	0	0
$201$	−123.933	−0.616582
$202$	0	0
$203$	0	0
$204$	0	0
$205$	62.4895	0.304827
$206$	0	0
$207$	− 137.506i	− 0.664279i
$208$	0	0
$209$	−28.5628	−0.136664
$210$	0	0
$211$	− 127.785i	− 0.605618i	−0.953051	−	0.302809i	$-0.902076\pi$
$211$	− 127.785i	− 0.605618i	0.953051	−	0.302809i	$-0.0979244\pi$
$212$	0	0
$213$	−123.598	−0.580273
$214$	0	0
$215$	− 3.21632i	− 0.0149596i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	50.6569i	0.231310i
$220$	0	0
$221$	133.973	0.606214
$222$	0	0
$223$	− 283.951i	− 1.27332i	−0.771144	−	0.636660i	$-0.780315\pi$
$223$	− 283.951i	− 1.27332i	0.771144	−	0.636660i	$-0.219685\pi$
$224$	0	0
$225$	58.9784	0.262126
$226$	0	0
$227$	413.572i	1.82190i	0.412512	+	0.910952i	$0.364651\pi$
$227$	413.572i	1.82190i	−0.412512	+	0.910952i	$0.635349\pi$
$228$	0	0
$229$	−161.310	−0.704411	−0.352205	−	0.935923i	$-0.614568\pi$
$229$	−161.310	−0.704411	−0.352205	+	0.935923i	$0.614568\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	362.730	1.55678	0.778390	−	0.627781i	$-0.216037\pi$
$233$	362.730	1.55678	0.778390	+	0.627781i	$0.216037\pi$
$234$	0	0
$235$	− 49.1269i	− 0.209051i
$236$	0	0
$237$	−178.271	−0.752197
$238$	0	0
$239$	− 342.283i	− 1.43215i	−0.698026	−	0.716073i	$-0.745938\pi$
$239$	− 342.283i	− 1.43215i	0.698026	−	0.716073i	$-0.254062\pi$
$240$	0	0
$241$	235.728	0.978123	0.489061	−	0.872249i	$-0.337339\pi$
$241$	235.728	0.978123	0.489061	+	0.872249i	$0.337339\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 38.9942i	− 0.157871i
$248$	0	0
$249$	74.7395	0.300159
$250$	0	0
$251$	− 409.935i	− 1.63321i	−0.577198	−	0.816604i	$-0.695854\pi$
$251$	− 409.935i	− 1.63321i	0.577198	−	0.816604i	$-0.304146\pi$
$252$	0	0
$253$	530.253	2.09586
$254$	0	0
$255$	33.9538i	0.133152i
$256$	0	0
$257$	−386.268	−1.50299	−0.751494	−	0.659740i	$-0.770666\pi$
$257$	−386.268	−1.50299	−0.751494	+	0.659740i	$0.770666\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−105.844	−0.405533
$262$	0	0
$263$	44.7325i	0.170086i	0.996377	+	0.0850428i	$0.0271027\pi$
$263$	44.7325i	0.170086i	−0.996377	+	0.0850428i	$0.972897\pi$
$264$	0	0
$265$	−74.6694	−0.281771
$266$	0	0
$267$	− 214.942i	− 0.805026i
$268$	0	0
$269$	389.015	1.44615	0.723076	−	0.690769i	$-0.242728\pi$
$269$	389.015	1.44615	0.723076	+	0.690769i	$0.242728\pi$
$270$	0	0
$271$	− 409.479i	− 1.51099i	−0.655154	−	0.755496i	$-0.727396\pi$
$271$	− 409.479i	− 1.51099i	0.655154	−	0.755496i	$-0.272604\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	227.434i	0.827032i
$276$	0	0
$277$	180.056	0.650022	0.325011	−	0.945710i	$-0.394632\pi$
$277$	180.056	0.650022	0.325011	+	0.945710i	$0.394632\pi$
$278$	0	0
$279$	− 90.1454i	− 0.323102i
$280$	0	0
$281$	−210.919	−0.750601	−0.375301	−	0.926903i	$-0.622461\pi$
$281$	−210.919	−0.750601	−0.375301	+	0.926903i	$0.622461\pi$
$282$	0	0
$283$	− 245.018i	− 0.865787i	−0.901445	−	0.432894i	$-0.857493\pi$
$283$	− 245.018i	− 0.865787i	0.901445	−	0.432894i	$-0.142507\pi$
$284$	0	0
$285$	9.88258	0.0346757
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−217.044	−0.751016
$290$	0	0
$291$	− 25.4934i	− 0.0876062i
$292$	0	0
$293$	262.124	0.894622	0.447311	−	0.894378i	$-0.352382\pi$
$293$	262.124	0.894622	0.447311	+	0.894378i	$0.352382\pi$
$294$	0	0
$295$	− 68.2357i	− 0.231307i
$296$	0	0
$297$	60.1126	0.202399
$298$	0	0
$299$	723.907i	2.42109i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 144.648i	− 0.477385i
$304$	0	0
$305$	238.940	0.783408
$306$	0	0
$307$	146.501i	0.477203i	0.971118	+	0.238601i	$0.0766889\pi$
$307$	146.501i	0.477203i	−0.971118	+	0.238601i	$0.923311\pi$
$308$	0	0
$309$	215.876	0.698626
$310$	0	0
$311$	− 15.0566i	− 0.0484134i	−0.999707	−	0.0242067i	$-0.992294\pi$
$311$	− 15.0566i	− 0.0484134i	0.999707	−	0.0242067i	$-0.00770598\pi$
$312$	0	0
$313$	−445.575	−1.42356	−0.711780	−	0.702402i	$-0.752111\pi$
$313$	−445.575	−1.42356	−0.711780	+	0.702402i	$0.752111\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	94.8746	0.299289	0.149645	−	0.988740i	$-0.452187\pi$
$317$	94.8746	0.299289	0.149645	+	0.988740i	$0.452187\pi$
$318$	0	0
$319$	− 408.159i	− 1.27949i
$320$	0	0
$321$	62.4552	0.194564
$322$	0	0
$323$	− 20.9436i	− 0.0648409i
$324$	0	0
$325$	−310.495	−0.955369
$326$	0	0
$327$	− 319.910i	− 0.978317i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 70.0820i	− 0.211728i	−0.994381	−	0.105864i	$-0.966239\pi$
$331$	− 70.0820i	− 0.211728i	0.994381	−	0.105864i	$-0.0337608\pi$
$332$	0	0
$333$	−117.740	−0.353575
$334$	0	0
$335$	165.356i	0.493599i
$336$	0	0
$337$	25.8353	0.0766625	0.0383313	−	0.999265i	$-0.487796\pi$
$337$	25.8353	0.0766625	0.0383313	+	0.999265i	$0.487796\pi$
$338$	0	0
$339$	− 366.872i	− 1.08222i
$340$	0	0
$341$	347.621	1.01942
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−183.465	−0.531782
$346$	0	0
$347$	431.127i	1.24244i	0.783636	+	0.621220i	$0.213363\pi$
$347$	431.127i	1.24244i	−0.783636	+	0.621220i	$0.786637\pi$
$348$	0	0
$349$	−340.822	−0.976566	−0.488283	−	0.872685i	$-0.662377\pi$
$349$	−340.822	−0.976566	−0.488283	+	0.872685i	$0.662377\pi$
$350$	0	0
$351$	82.0663i	0.233807i
$352$	0	0
$353$	89.8708	0.254591	0.127296	−	0.991865i	$-0.459370\pi$
$353$	89.8708	0.254591	0.127296	+	0.991865i	$0.459370\pi$
$354$	0	0
$355$	164.909i	0.464533i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 359.108i	− 1.00030i	−0.865939	−	0.500150i	$-0.833278\pi$
$359$	− 359.108i	− 1.00030i	0.865939	−	0.500150i	$-0.166722\pi$
$360$	0	0
$361$	354.904	0.983114
$362$	0	0
$363$	22.2294i	0.0612380i
$364$	0	0
$365$	67.5883	0.185173
$366$	0	0
$367$	− 496.034i	− 1.35159i	−0.737090	−	0.675795i	$-0.763800\pi$
$367$	− 496.034i	− 1.35159i	0.737090	−	0.675795i	$-0.236200\pi$
$368$	0	0
$369$	81.1214	0.219841
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−409.207	−1.09707	−0.548535	−	0.836128i	$-0.684814\pi$
$373$	−409.207	−1.09707	−0.548535	+	0.836128i	$0.684814\pi$
$374$	0	0
$375$	− 178.759i	− 0.476690i
$376$	0	0
$377$	557.223	1.47804
$378$	0	0
$379$	531.591i	1.40261i	0.712859	+	0.701307i	$0.247400\pi$
$379$	531.591i	1.40261i	−0.712859	+	0.701307i	$0.752600\pi$
$380$	0	0
$381$	−60.8057	−0.159595
$382$	0	0
$383$	746.787i	1.94984i	0.222563	+	0.974918i	$0.428558\pi$
$383$	746.787i	1.94984i	−0.222563	+	0.974918i	$0.571442\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.17530i	− 0.0107889i
$388$	0	0
$389$	331.492	0.852164	0.426082	−	0.904685i	$-0.359894\pi$
$389$	331.492	0.852164	0.426082	+	0.904685i	$0.359894\pi$
$390$	0	0
$391$	388.807i	0.994392i
$392$	0	0
$393$	273.560	0.696081
$394$	0	0
$395$	237.855i	0.602164i
$396$	0	0
$397$	136.540	0.343930	0.171965	−	0.985103i	$-0.444988\pi$
$397$	136.540	0.343930	0.171965	+	0.985103i	$0.444988\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	390.029	0.972640	0.486320	−	0.873781i	$-0.338339\pi$
$401$	390.029	0.972640	0.486320	+	0.873781i	$0.338339\pi$
$402$	0	0
$403$	474.576i	1.17761i
$404$	0	0
$405$	−20.7987	−0.0513547
$406$	0	0
$407$	− 454.033i	− 1.11556i
$408$	0	0
$409$	−769.032	−1.88027	−0.940137	−	0.340797i	$-0.889303\pi$
$409$	−769.032	−1.88027	−0.940137	+	0.340797i	$0.889303\pi$
$410$	0	0
$411$	− 382.871i	− 0.931559i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 99.7200i	− 0.240289i
$416$	0	0
$417$	−237.038	−0.568438
$418$	0	0
$419$	− 664.417i	− 1.58572i	−0.609403	−	0.792861i	$-0.708591\pi$
$419$	− 664.417i	− 1.58572i	0.609403	−	0.792861i	$-0.291409\pi$
$420$	0	0
$421$	83.6243	0.198633	0.0993163	−	0.995056i	$-0.468334\pi$
$421$	83.6243	0.198633	0.0993163	+	0.995056i	$0.468334\pi$
$422$	0	0
$423$	− 63.7746i	− 0.150767i
$424$	0	0
$425$	−166.765	−0.392389
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−316.466	−0.737683
$430$	0	0
$431$	401.205i	0.930870i	0.885082	+	0.465435i	$0.154102\pi$
$431$	401.205i	0.930870i	−0.885082	+	0.465435i	$0.845898\pi$
$432$	0	0
$433$	−96.9743	−0.223959	−0.111980	−	0.993711i	$-0.535719\pi$
$433$	−96.9743	−0.223959	−0.111980	+	0.993711i	$0.535719\pi$
$434$	0	0
$435$	141.221i	0.324646i
$436$	0	0
$437$	113.166	0.258961
$438$	0	0
$439$	437.240i	0.995990i	0.867180	+	0.497995i	$0.165930\pi$
$439$	437.240i	0.995990i	−0.867180	+	0.497995i	$0.834070\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	585.479i	1.32162i	0.750552	+	0.660812i	$0.229788\pi$
$443$	585.479i	1.32162i	−0.750552	+	0.660812i	$0.770212\pi$
$444$	0	0
$445$	−286.783	−0.644457
$446$	0	0
$447$	80.1495i	0.179305i
$448$	0	0
$449$	−430.149	−0.958016	−0.479008	−	0.877811i	$-0.659004\pi$
$449$	−430.149	−0.958016	−0.479008	+	0.877811i	$0.659004\pi$
$450$	0	0
$451$	312.822i	0.693620i
$452$	0	0
$453$	−160.709	−0.354765
$454$	0	0
$455$	0	0
$456$	0	0
$457$	816.499	1.78665	0.893325	−	0.449412i	$-0.148366\pi$
$457$	816.499	1.78665	0.893325	+	0.449412i	$0.148366\pi$
$458$	0	0
$459$	44.0775i	0.0960293i
$460$	0	0
$461$	−280.569	−0.608609	−0.304305	−	0.952575i	$-0.598424\pi$
$461$	−280.569	−0.608609	−0.304305	+	0.952575i	$0.598424\pi$
$462$	0	0
$463$	181.939i	0.392958i	0.980508	+	0.196479i	$0.0629507\pi$
$463$	181.939i	0.392958i	−0.980508	+	0.196479i	$0.937049\pi$
$464$	0	0
$465$	−120.275	−0.258656
$466$	0	0
$467$	88.8370i	0.190229i	0.995466	+	0.0951146i	$0.0303217\pi$
$467$	88.8370i	0.190229i	−0.995466	+	0.0951146i	$0.969678\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 318.742i	− 0.676735i
$472$	0	0
$473$	16.1009	0.0340400
$474$	0	0
$475$	48.5387i	0.102187i
$476$	0	0
$477$	−96.9329	−0.203214
$478$	0	0
$479$	126.963i	0.265058i	0.991179	+	0.132529i	$0.0423097\pi$
$479$	126.963i	0.265058i	−0.991179	+	0.132529i	$0.957690\pi$
$480$	0	0
$481$	619.851	1.28867
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−34.0142	−0.0701323
$486$	0	0
$487$	− 287.582i	− 0.590518i	−0.955417	−	0.295259i	$-0.904594\pi$
$487$	− 287.582i	− 0.590518i	0.955417	−	0.295259i	$-0.0954059\pi$
$488$	0	0
$489$	115.315	0.235818
$490$	0	0
$491$	− 23.7600i	− 0.0483910i	−0.999707	−	0.0241955i	$-0.992298\pi$
$491$	− 23.7600i	− 0.0483910i	0.999707	−	0.0241955i	$-0.00770241\pi$
$492$	0	0
$493$	299.282	0.607062
$494$	0	0
$495$	− 80.2043i	− 0.162029i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 56.2260i	− 0.112677i	−0.998412	−	0.0563387i	$-0.982057\pi$
$499$	− 56.2260i	− 0.112677i	0.998412	−	0.0563387i	$-0.0179427\pi$
$500$	0	0
$501$	205.809	0.410797
$502$	0	0
$503$	913.863i	1.81683i	0.418075	+	0.908413i	$0.362705\pi$
$503$	913.863i	1.81683i	−0.418075	+	0.908413i	$0.637295\pi$
$504$	0	0
$505$	−192.994	−0.382166
$506$	0	0
$507$	− 139.327i	− 0.274806i
$508$	0	0
$509$	898.823	1.76586	0.882930	−	0.469504i	$-0.155567\pi$
$509$	898.823	1.76586	0.882930	+	0.469504i	$0.155567\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.8292	0.0250081
$514$	0	0
$515$	− 288.029i	− 0.559279i
$516$	0	0
$517$	245.929	0.475685
$518$	0	0
$519$	347.137i	0.668857i
$520$	0	0
$521$	311.560	0.598004	0.299002	−	0.954253i	$-0.403346\pi$
$521$	311.560	0.598004	0.299002	+	0.954253i	$0.403346\pi$
$522$	0	0
$523$	− 786.871i	− 1.50453i	−0.658859	−	0.752267i	$-0.728960\pi$
$523$	− 786.871i	− 1.50453i	0.658859	−	0.752267i	$-0.271040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	254.893i	0.483667i
$528$	0	0
$529$	−1571.87	−2.97140
$530$	0	0
$531$	− 88.5808i	− 0.166819i
$532$	0	0
$533$	−427.069	−0.801254
$534$	0	0
$535$	− 83.3299i	− 0.155757i
$536$	0	0
$537$	−320.811	−0.597413
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−280.698	−0.518850	−0.259425	−	0.965763i	$-0.583533\pi$
$541$	−280.698	−0.518850	−0.259425	+	0.965763i	$0.583533\pi$
$542$	0	0
$543$	− 395.552i	− 0.728458i
$544$	0	0
$545$	−426.835	−0.783183
$546$	0	0
$547$	153.761i	0.281099i	0.990074	+	0.140549i	$0.0448869\pi$
$547$	153.761i	0.281099i	−0.990074	+	0.140549i	$0.955113\pi$
$548$	0	0
$549$	310.182	0.564994
$550$	0	0
$551$	− 87.1088i	− 0.158092i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	157.093i	0.283051i
$556$	0	0
$557$	−780.733	−1.40167	−0.700837	−	0.713321i	$-0.747190\pi$
$557$	−780.733	−1.40167	−0.700837	+	0.713321i	$0.747190\pi$
$558$	0	0
$559$	21.9811i	0.0393222i
$560$	0	0
$561$	−169.973	−0.302981
$562$	0	0
$563$	937.153i	1.66457i	0.554348	+	0.832285i	$0.312968\pi$
$563$	937.153i	1.66457i	−0.554348	+	0.832285i	$0.687032\pi$
$564$	0	0
$565$	−489.493	−0.866359
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−786.799	−1.38278	−0.691388	−	0.722484i	$-0.743000\pi$
$569$	−786.799	−1.38278	−0.691388	+	0.722484i	$0.743000\pi$
$570$	0	0
$571$	− 263.128i	− 0.460820i	−0.973094	−	0.230410i	$-0.925993\pi$
$571$	− 263.128i	− 0.460820i	0.973094	−	0.230410i	$-0.0740067\pi$
$572$	0	0
$573$	−260.479	−0.454588
$574$	0	0
$575$	− 901.096i	− 1.56712i
$576$	0	0
$577$	−11.0000	−0.0190641	−0.00953206	−	0.999955i	$-0.503034\pi$
$577$	−11.0000	−0.0190641	−0.00953206	+	0.999955i	$0.503034\pi$
$578$	0	0
$579$	− 141.233i	− 0.243926i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 373.795i	− 0.641158i
$584$	0	0
$585$	109.496	0.187172
$586$	0	0
$587$	461.546i	0.786279i	0.919479	+	0.393139i	$0.128611\pi$
$587$	461.546i	0.786279i	−0.919479	+	0.393139i	$0.871389\pi$
$588$	0	0
$589$	74.1889	0.125957
$590$	0	0
$591$	333.800i	0.564806i
$592$	0	0
$593$	776.206	1.30895	0.654474	−	0.756085i	$-0.272890\pi$
$593$	776.206	1.30895	0.654474	+	0.756085i	$0.272890\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	151.647	0.254015
$598$	0	0
$599$	− 241.933i	− 0.403895i	−0.979396	−	0.201947i	$-0.935273\pi$
$599$	− 241.933i	− 0.403895i	0.979396	−	0.201947i	$-0.0647270\pi$
$600$	0	0
$601$	521.217	0.867250	0.433625	−	0.901093i	$-0.357234\pi$
$601$	521.217	0.867250	0.433625	+	0.901093i	$0.357234\pi$
$602$	0	0
$603$	214.658i	0.355984i
$604$	0	0
$605$	29.6593	0.0490236
$606$	0	0
$607$	− 1139.24i	− 1.87684i	−0.345496	−	0.938420i	$-0.612289\pi$
$607$	− 1139.24i	− 1.87684i	0.345496	−	0.938420i	$-0.387711\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	335.745i	0.549501i
$612$	0	0
$613$	534.066	0.871233	0.435616	−	0.900132i	$-0.356530\pi$
$613$	534.066	0.871233	0.435616	+	0.900132i	$0.356530\pi$
$614$	0	0
$615$	− 108.235i	− 0.175992i
$616$	0	0
$617$	−802.979	−1.30142	−0.650712	−	0.759324i	$-0.725530\pi$
$617$	−802.979	−1.30142	−0.650712	+	0.759324i	$0.725530\pi$
$618$	0	0
$619$	− 317.050i	− 0.512196i	−0.966651	−	0.256098i	$-0.917563\pi$
$619$	− 317.050i	− 0.512196i	0.966651	−	0.256098i	$-0.0824370\pi$
$620$	0	0
$621$	−238.167	−0.383522
$622$	0	0
$623$	0	0
$624$	0	0
$625$	252.980	0.404769
$626$	0	0
$627$	49.4721i	0.0789029i
$628$	0	0
$629$	332.919	0.529284
$630$	0	0
$631$	1168.77i	1.85225i	0.377222	+	0.926123i	$0.376879\pi$
$631$	1168.77i	1.85225i	−0.377222	+	0.926123i	$0.623121\pi$
$632$	0	0
$633$	−221.331	−0.349654
$634$	0	0
$635$	81.1291i	0.127762i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	214.078i	0.335021i
$640$	0	0
$641$	−16.2513	−0.0253530	−0.0126765	−	0.999920i	$-0.504035\pi$
$641$	−16.2513	−0.0253530	−0.0126765	+	0.999920i	$0.504035\pi$
$642$	0	0
$643$	− 956.475i	− 1.48752i	−0.668447	−	0.743760i	$-0.733041\pi$
$643$	− 956.475i	− 1.48752i	0.668447	−	0.743760i	$-0.266959\pi$
$644$	0	0
$645$	−5.57084	−0.00863696
$646$	0	0
$647$	527.324i	0.815029i	0.913199	+	0.407515i	$0.133604\pi$
$647$	527.324i	0.815029i	−0.913199	+	0.407515i	$0.866396\pi$
$648$	0	0
$649$	341.587	0.526329
$650$	0	0
$651$	0	0
$652$	0	0
$653$	319.329	0.489018	0.244509	−	0.969647i	$-0.421373\pi$
$653$	319.329	0.489018	0.244509	+	0.969647i	$0.421373\pi$
$654$	0	0
$655$	− 364.993i	− 0.557241i
$656$	0	0
$657$	87.7404	0.133547
$658$	0	0
$659$	− 289.303i	− 0.439004i	−0.975612	−	0.219502i	$-0.929557\pi$
$659$	− 289.303i	− 0.439004i	0.975612	−	0.219502i	$-0.0704432\pi$
$660$	0	0
$661$	16.4197	0.0248407	0.0124203	−	0.999923i	$-0.496046\pi$
$661$	16.4197	0.0248407	0.0124203	+	0.999923i	$0.496046\pi$
$662$	0	0
$663$	− 232.048i	− 0.349998i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1617.13i	2.42448i
$668$	0	0
$669$	−491.817	−0.735152
$670$	0	0
$671$	1196.13i	1.78261i
$672$	0	0
$673$	−65.5702	−0.0974297	−0.0487149	−	0.998813i	$-0.515513\pi$
$673$	−65.5702	−0.0974297	−0.0487149	+	0.998813i	$0.515513\pi$
$674$	0	0
$675$	− 102.154i	− 0.151339i
$676$	0	0
$677$	1190.46	1.75843	0.879216	−	0.476424i	$-0.158067\pi$
$677$	1190.46	1.75843	0.879216	+	0.476424i	$0.158067\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	716.328	1.05188
$682$	0	0
$683$	890.160i	1.30331i	0.758516	+	0.651654i	$0.225925\pi$
$683$	890.160i	1.30331i	−0.758516	+	0.651654i	$0.774075\pi$
$684$	0	0
$685$	−510.839	−0.745751
$686$	0	0
$687$	279.397i	0.406692i
$688$	0	0
$689$	510.309	0.740651
$690$	0	0
$691$	350.596i	0.507375i	0.967286	+	0.253688i	$0.0816435\pi$
$691$	350.596i	0.507375i	−0.967286	+	0.253688i	$0.918356\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	316.265i	0.455058i
$696$	0	0
$697$	−229.377	−0.329091
$698$	0	0
$699$	− 628.266i	− 0.898807i
$700$	0	0
$701$	998.973	1.42507	0.712534	−	0.701637i	$-0.247547\pi$
$701$	998.973	1.42507	0.712534	+	0.701637i	$0.247547\pi$
$702$	0	0
$703$	− 96.8994i	− 0.137837i
$704$	0	0
$705$	−85.0904	−0.120696
$706$	0	0
$707$	0	0
$708$	0	0
$709$	466.648	0.658177	0.329089	−	0.944299i	$-0.393259\pi$
$709$	466.648	0.658177	0.329089	+	0.944299i	$0.393259\pi$
$710$	0	0
$711$	308.774i	0.434281i
$712$	0	0
$713$	−1377.28	−1.93167
$714$	0	0
$715$	422.240i	0.590546i
$716$	0	0
$717$	−592.851	−0.826850
$718$	0	0
$719$	380.832i	0.529670i	0.964294	+	0.264835i	$0.0853174\pi$
$719$	380.832i	0.529670i	−0.964294	+	0.264835i	$0.914683\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 408.292i	− 0.564720i
$724$	0	0
$725$	−693.613	−0.956707
$726$	0	0
$727$	868.875i	1.19515i	0.801813	+	0.597575i	$0.203869\pi$
$727$	868.875i	1.19515i	−0.801813	+	0.597575i	$0.796131\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	11.8060i	0.0161504i
$732$	0	0
$733$	−271.368	−0.370215	−0.185108	−	0.982718i	$-0.559263\pi$
$733$	−271.368	−0.370215	−0.185108	+	0.982718i	$0.559263\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−827.770	−1.12316
$738$	0	0
$739$	1294.89i	1.75221i	0.482116	+	0.876107i	$0.339868\pi$
$739$	1294.89i	1.75221i	−0.482116	+	0.876107i	$0.660132\pi$
$740$	0	0
$741$	−67.5399	−0.0911470
$742$	0	0
$743$	69.5251i	0.0935735i	0.998905	+	0.0467868i	$0.0148981\pi$
$743$	69.5251i	0.0935735i	−0.998905	+	0.0467868i	$0.985102\pi$
$744$	0	0
$745$	106.938	0.143541
$746$	0	0
$747$	− 129.453i	− 0.173297i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 1080.06i	− 1.43816i	−0.694925	−	0.719082i	$-0.744563\pi$
$751$	− 1080.06i	− 1.43816i	0.694925	−	0.719082i	$-0.255437\pi$
$752$	0	0
$753$	−710.029	−0.942933
$754$	0	0
$755$	214.423i	0.284004i
$756$	0	0
$757$	−1073.01	−1.41745	−0.708727	−	0.705482i	$-0.750730\pi$
$757$	−1073.01	−1.41745	−0.708727	+	0.705482i	$0.750730\pi$
$758$	0	0
$759$	− 918.425i	− 1.21005i
$760$	0	0
$761$	1189.72	1.56336	0.781681	−	0.623678i	$-0.214362\pi$
$761$	1189.72	1.56336	0.781681	+	0.623678i	$0.214362\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	58.8097	0.0768754
$766$	0	0
$767$	466.339i	0.608004i
$768$	0	0
$769$	−497.083	−0.646402	−0.323201	−	0.946330i	$-0.604759\pi$
$769$	−497.083	−0.646402	−0.323201	+	0.946330i	$0.604759\pi$
$770$	0	0
$771$	669.035i	0.867750i
$772$	0	0
$773$	−1221.32	−1.57997	−0.789984	−	0.613127i	$-0.789911\pi$
$773$	−1221.32	−1.57997	−0.789984	+	0.613127i	$0.789911\pi$
$774$	0	0
$775$	− 590.736i	− 0.762240i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	66.7623i	0.0857025i
$780$	0	0
$781$	−825.534	−1.05702
$782$	0	0
$783$	183.327i	0.234135i
$784$	0	0
$785$	−425.277	−0.541754
$786$	0	0
$787$	421.297i	0.535321i	0.963513	+	0.267660i	$0.0862505\pi$
$787$	421.297i	0.535321i	−0.963513	+	0.267660i	$0.913750\pi$
$788$	0	0
$789$	77.4790	0.0981990
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1632.97	−2.05923
$794$	0	0
$795$	129.331i	0.162681i
$796$	0	0
$797$	395.391	0.496099	0.248050	−	0.968747i	$-0.420210\pi$
$797$	395.391	0.496099	0.248050	+	0.968747i	$0.420210\pi$
$798$	0	0
$799$	180.327i	0.225691i
$800$	0	0
$801$	−372.291	−0.464782
$802$	0	0
$803$	338.347i	0.421353i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 673.793i	− 0.834936i
$808$	0	0
$809$	−476.907	−0.589502	−0.294751	−	0.955574i	$-0.595237\pi$
$809$	−476.907	−0.589502	−0.294751	+	0.955574i	$0.595237\pi$
$810$	0	0
$811$	− 268.467i	− 0.331032i	−0.986207	−	0.165516i	$-0.947071\pi$
$811$	− 268.467i	− 0.331032i	0.986207	−	0.165516i	$-0.0529289\pi$
$812$	0	0
$813$	−709.238	−0.872371
$814$	0	0
$815$	− 153.857i	− 0.188782i
$816$	0	0
$817$	3.43624	0.00420592
$818$	0	0
$819$	0	0
$820$	0	0
$821$	646.656	0.787645	0.393822	−	0.919187i	$-0.371153\pi$
$821$	646.656	0.787645	0.393822	+	0.919187i	$0.371153\pi$
$822$	0	0
$823$	210.056i	0.255232i	0.991824	+	0.127616i	$0.0407324\pi$
$823$	210.056i	0.255232i	−0.991824	+	0.127616i	$0.959268\pi$
$824$	0	0
$825$	393.927	0.477487
$826$	0	0
$827$	− 590.763i	− 0.714345i	−0.934038	−	0.357173i	$-0.883741\pi$
$827$	− 590.763i	− 0.714345i	0.934038	−	0.357173i	$-0.116259\pi$
$828$	0	0
$829$	−1493.29	−1.80132	−0.900659	−	0.434527i	$-0.856916\pi$
$829$	−1493.29	−1.80132	−0.900659	+	0.434527i	$0.856916\pi$
$830$	0	0
$831$	− 311.867i	− 0.375291i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 274.598i	− 0.328860i
$836$	0	0
$837$	−156.136	−0.186543
$838$	0	0
$839$	− 529.587i	− 0.631212i	−0.948890	−	0.315606i	$-0.897792\pi$
$839$	− 529.587i	− 0.631212i	0.948890	−	0.315606i	$-0.102208\pi$
$840$	0	0
$841$	403.776	0.480114
$842$	0	0
$843$	365.322i	0.433360i
$844$	0	0
$845$	−185.894	−0.219993
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−424.383	−0.499862
$850$	0	0
$851$	1798.89i	2.11385i
$852$	0	0
$853$	−544.072	−0.637833	−0.318917	−	0.947783i	$-0.603319\pi$
$853$	−544.072	−0.637833	−0.318917	+	0.947783i	$0.603319\pi$
$854$	0	0
$855$	− 17.1171i	− 0.0200200i
$856$	0	0
$857$	−625.051	−0.729348	−0.364674	−	0.931135i	$-0.618820\pi$
$857$	−625.051	−0.729348	−0.364674	+	0.931135i	$0.618820\pi$
$858$	0	0
$859$	748.421i	0.871270i	0.900123	+	0.435635i	$0.143476\pi$
$859$	748.421i	0.871270i	−0.900123	+	0.435635i	$0.856524\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 866.946i	− 1.00457i	−0.864701	−	0.502286i	$-0.832492\pi$
$863$	− 866.946i	− 1.00457i	0.864701	−	0.502286i	$-0.167508\pi$
$864$	0	0
$865$	463.162	0.535447
$866$	0	0
$867$	375.930i	0.433599i
$868$	0	0
$869$	−1190.70	−1.37020
$870$	0	0
$871$	− 1130.08i	− 1.29745i
$872$	0	0
$873$	−44.1559	−0.0505795
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−569.378	−0.649234	−0.324617	−	0.945846i	$-0.605235\pi$
$877$	−569.378	−0.649234	−0.324617	+	0.945846i	$0.605235\pi$
$878$	0	0
$879$	− 454.013i	− 0.516511i
$880$	0	0
$881$	−781.679	−0.887263	−0.443632	−	0.896209i	$-0.646310\pi$
$881$	−781.679	−0.887263	−0.443632	+	0.896209i	$0.646310\pi$
$882$	0	0
$883$	− 1139.96i	− 1.29101i	−0.763757	−	0.645504i	$-0.776647\pi$
$883$	− 1139.96i	− 1.29101i	0.763757	−	0.645504i	$-0.223353\pi$
$884$	0	0
$885$	−118.188	−0.133545
$886$	0	0
$887$	− 152.245i	− 0.171641i	−0.996311	−	0.0858204i	$-0.972649\pi$
$887$	− 152.245i	− 0.171641i	0.996311	−	0.0858204i	$-0.0273511\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 104.118i	− 0.116855i
$892$	0	0
$893$	52.4860	0.0587749
$894$	0	0
$895$	428.037i	0.478253i
$896$	0	0
$897$	1253.84	1.39782
$898$	0	0
$899$	1060.15i	1.17926i
$900$	0	0
$901$	274.085	0.304200
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−527.760	−0.583160
$906$	0	0
$907$	469.617i	0.517769i	0.965908	+	0.258885i	$0.0833549\pi$
$907$	469.617i	0.517769i	−0.965908	+	0.258885i	$0.916645\pi$
$908$	0	0
$909$	−250.537	−0.275618
$910$	0	0
$911$	− 309.559i	− 0.339801i	−0.985461	−	0.169901i	$-0.945655\pi$
$911$	− 309.559i	− 0.339801i	0.985461	−	0.169901i	$-0.0543447\pi$
$912$	0	0
$913$	499.198	0.546767
$914$	0	0
$915$	− 413.855i	− 0.452301i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 1178.27i	− 1.28213i	−0.767488	−	0.641063i	$-0.778494\pi$
$919$	− 1178.27i	− 1.28213i	0.767488	−	0.641063i	$-0.221506\pi$
$920$	0	0
$921$	253.748	0.275513
$922$	0	0
$923$	− 1127.03i	− 1.22105i
$924$	0	0
$925$	−771.571	−0.834130
$926$	0	0
$927$	− 373.907i	− 0.403352i
$928$	0	0
$929$	283.492	0.305158	0.152579	−	0.988291i	$-0.451242\pi$
$929$	283.492	0.305158	0.152579	+	0.988291i	$0.451242\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−26.0787	−0.0279515
$934$	0	0
$935$	226.783i	0.242549i
$936$	0	0
$937$	343.658	0.366764	0.183382	−	0.983042i	$-0.441295\pi$
$937$	343.658	0.366764	0.183382	+	0.983042i	$0.441295\pi$
$938$	0	0
$939$	771.758i	0.821893i
$940$	0	0
$941$	614.597	0.653132	0.326566	−	0.945174i	$-0.394108\pi$
$941$	614.597	0.653132	0.326566	+	0.945174i	$0.394108\pi$
$942$	0	0
$943$	− 1239.41i	− 1.31432i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1563.32i	1.65081i	0.564542	+	0.825404i	$0.309053\pi$
$947$	1563.32i	1.65081i	−0.564542	+	0.825404i	$0.690947\pi$
$948$	0	0
$949$	−461.915	−0.486738
$950$	0	0
$951$	− 164.328i	− 0.172795i
$952$	0	0
$953$	−65.3694	−0.0685933	−0.0342966	−	0.999412i	$-0.510919\pi$
$953$	−65.3694	−0.0685933	−0.0342966	+	0.999412i	$0.510919\pi$
$954$	0	0
$955$	347.540i	0.363916i
$956$	0	0
$957$	−706.952	−0.738716
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58.0898	0.0604472
$962$	0	0
$963$	− 108.176i	− 0.112332i
$964$	0	0
$965$	−188.438	−0.195273
$966$	0	0
$967$	54.5021i	0.0563620i	0.999603	+	0.0281810i	$0.00897148\pi$
$967$	54.5021i	0.0563620i	−0.999603	+	0.0281810i	$0.991029\pi$
$968$	0	0
$969$	−36.2754	−0.0374359
$970$	0	0
$971$	1285.16i	1.32354i	0.749708	+	0.661769i	$0.230194\pi$
$971$	1285.16i	1.32354i	−0.749708	+	0.661769i	$0.769806\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	537.793i	0.551583i
$976$	0	0
$977$	1552.41	1.58895	0.794477	−	0.607294i	$-0.207745\pi$
$977$	1552.41	1.58895	0.794477	+	0.607294i	$0.207745\pi$
$978$	0	0
$979$	− 1435.64i	− 1.46643i
$980$	0	0
$981$	−554.100	−0.564832
$982$	0	0
$983$	− 68.2644i	− 0.0694450i	−0.999397	−	0.0347225i	$-0.988945\pi$
$983$	− 68.2644i	− 0.0694450i	0.999397	−	0.0347225i	$-0.0110547\pi$
$984$	0	0
$985$	445.368	0.452150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63.7920	−0.0645016
$990$	0	0
$991$	− 794.660i	− 0.801877i	−0.916105	−	0.400939i	$-0.868684\pi$
$991$	− 794.660i	− 0.801877i	0.916105	−	0.400939i	$-0.131316\pi$
$992$	0	0
$993$	−121.386	−0.122241
$994$	0	0
$995$	− 202.332i	− 0.203349i
$996$	0	0
$997$	1026.99	1.03008	0.515039	−	0.857167i	$-0.327778\pi$
$997$	1026.99	1.03008	0.515039	+	0.857167i	$0.327778\pi$
$998$	0	0
$999$	203.932i	0.204137i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.o.1471.2		6
4.3	odd	2	inner	2352.3.m.o.1471.5		6
7.3	odd	6		336.3.be.f.79.2	yes	6
7.5	odd	6		336.3.be.d.319.2	yes	6
7.6	odd	2		2352.3.m.n.1471.5		6
21.5	even	6		1008.3.cd.h.991.2		6
21.17	even	6		1008.3.cd.i.415.2		6
28.3	even	6		336.3.be.d.79.2	✓	6
28.19	even	6		336.3.be.f.319.2	yes	6
28.27	even	2		2352.3.m.n.1471.2		6
84.47	odd	6		1008.3.cd.i.991.2		6
84.59	odd	6		1008.3.cd.h.415.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.be.d.79.2	✓	6	28.3	even	6
336.3.be.d.319.2	yes	6	7.5	odd	6
336.3.be.f.79.2	yes	6	7.3	odd	6
336.3.be.f.319.2	yes	6	28.19	even	6
1008.3.cd.h.415.2		6	84.59	odd	6
1008.3.cd.h.991.2		6	21.5	even	6
1008.3.cd.i.415.2		6	21.17	even	6
1008.3.cd.i.991.2		6	84.47	odd	6
2352.3.m.n.1471.2		6	28.27	even	2
2352.3.m.n.1471.5		6	7.6	odd	2
2352.3.m.o.1471.2		6	1.1	even	1	trivial
2352.3.m.o.1471.5		6	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.31096 q^{5} -3.00000 q^{9} -11.5687i q^{11} +15.7937 q^{13} +4.00270i q^{15} +8.48271 q^{17} -2.46897i q^{19} +45.8352i q^{23} -19.6595 q^{25} +5.19615i q^{27} +35.2814 q^{29} +30.0485i q^{31} -20.0375 q^{33} +39.2468 q^{37} -27.3554i q^{39} -27.0405 q^{41} +1.39177i q^{43} +6.93289 q^{45} +21.2582i q^{47} -14.6925i q^{51} +32.3110 q^{53} +26.7348i q^{55} -4.27639 q^{57} +29.5269i q^{59} -103.394 q^{61} -36.4986 q^{65} -71.5527i q^{67} +79.3890 q^{69} -71.3595i q^{71} -29.2468 q^{73} +34.0512i q^{75} -102.925i q^{79} +9.00000 q^{81} +43.1509i q^{83} -19.6032 q^{85} -61.1091i q^{87} +124.097 q^{89} +52.0455 q^{93} +5.70571i q^{95} +14.7186 q^{97} +34.7060i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 18 q^{9} + 44 q^{13} + 16 q^{17} + 28 q^{25} + 34 q^{29} - 6 q^{33} - 12 q^{37} + 136 q^{41} - 6 q^{45} + 178 q^{53} + 60 q^{57} - 100 q^{61} + 368 q^{65} + 48 q^{69} + 72 q^{73} + 54 q^{81}+ \cdots + 266 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 - 18 * q^9 + 44 * q^13 + 16 * q^17 + 28 * q^25 + 34 * q^29 - 6 * q^33 - 12 * q^37 + 136 * q^41 - 6 * q^45 + 178 * q^53 + 60 * q^57 - 100 * q^61 + 368 * q^65 + 48 * q^69 + 72 * q^73 + 54 * q^81 + 536 * q^85 - 404 * q^89 - 78 * q^93 + 266 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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