LMFDB - Embedded newform 384.3.g.a.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(127,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	384.127
Dual form			384.3.g.a.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} -5.79796i q^{11} +8.78710 q^{13} -4.56512i q^{15} -30.1810 q^{17} -17.4125i q^{19} -21.7473 q^{21} +2.48425i q^{23} -18.0532 q^{25} -5.19615i q^{27} -26.4175 q^{29} +38.0082i q^{31} +10.0424 q^{33} -33.0931i q^{35} -47.7930 q^{37} +15.2197i q^{39} +53.3294 q^{41} +30.7044i q^{43} +7.90702 q^{45} +16.2238i q^{47} -108.649 q^{49} -52.2750i q^{51} -49.8432 q^{53} +15.2815i q^{55} +30.1593 q^{57} -107.412i q^{59} +62.6220 q^{61} -37.6675i q^{63} -23.1599 q^{65} +60.9540i q^{67} -4.30285 q^{69} -19.9465i q^{71} +5.13929 q^{73} -31.2691i q^{75} +72.7982 q^{77} -6.83349i q^{79} +9.00000 q^{81} +159.213i q^{83} +79.5472 q^{85} -45.7565i q^{87} +39.4473 q^{89} +110.329i q^{91} -65.8321 q^{93} +45.8936i q^{95} -60.5944 q^{97} +17.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{5} - 24 q^{9}+O(q^{10}) $ 8 * q - 16 * q^5 - 24 * q^9	$ 8 q - 16 q^{5} - 24 q^{9} + 48 q^{13} + 16 q^{17} - 8 q^{25} - 80 q^{29} - 16 q^{37} + 80 q^{41} + 48 q^{45} - 88 q^{49} + 176 q^{53} + 96 q^{57} - 272 q^{61} - 160 q^{65} - 16 q^{73} + 320 q^{77} + 72 q^{81} + 32 q^{85} - 240 q^{89} - 192 q^{93} + 400 q^{97}+O(q^{100}) $ 8 * q - 16 * q^5 - 24 * q^9 + 48 * q^13 + 16 * q^17 - 8 * q^25 - 80 * q^29 - 16 * q^37 + 80 * q^41 + 48 * q^45 - 88 * q^49 + 176 * q^53 + 96 * q^57 - 272 * q^61 - 160 * q^65 - 16 * q^73 + 320 * q^77 + 72 * q^81 + 32 * q^85 - 240 * q^89 - 192 * q^93 + 400 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	−2.63567	−0.527135	−0.263567	−	0.964641i	$-0.584899\pi$
$5$	−2.63567	−0.527135	−0.263567	+	0.964641i	$0.584899\pi$
$6$	0	0
$7$	12.5558i	1.79369i	0.442344	+	0.896845i	$0.354147\pi$
$7$	12.5558i	1.79369i	−0.442344	+	0.896845i	$0.645853\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 5.79796i	− 0.527087i	−0.964647	−	0.263544i	$-0.915109\pi$
$11$	− 5.79796i	− 0.527087i	0.964647	−	0.263544i	$-0.0848913\pi$
$12$	0	0
$13$	8.78710	0.675931	0.337965	−	0.941159i	$-0.390261\pi$
$13$	8.78710	0.675931	0.337965	+	0.941159i	$0.390261\pi$
$14$	0	0
$15$	− 4.56512i	− 0.304341i
$16$	0	0
$17$	−30.1810	−1.77535	−0.887676	−	0.460469i	$-0.847681\pi$
$17$	−30.1810	−1.77535	−0.887676	+	0.460469i	$0.847681\pi$
$18$	0	0
$19$	− 17.4125i	− 0.916445i	−0.888838	−	0.458222i	$-0.848486\pi$
$19$	− 17.4125i	− 0.916445i	0.888838	−	0.458222i	$-0.151514\pi$
$20$	0	0
$21$	−21.7473	−1.03559
$22$	0	0
$23$	2.48425i	0.108011i	0.998541	+	0.0540054i	$0.0171988\pi$
$23$	2.48425i	0.108011i	−0.998541	+	0.0540054i	$0.982801\pi$
$24$	0	0
$25$	−18.0532	−0.722129
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−26.4175	−0.910950	−0.455475	−	0.890249i	$-0.650531\pi$
$29$	−26.4175	−0.910950	−0.455475	+	0.890249i	$0.650531\pi$
$30$	0	0
$31$	38.0082i	1.22607i	0.790056	+	0.613035i	$0.210051\pi$
$31$	38.0082i	1.22607i	−0.790056	+	0.613035i	$0.789949\pi$
$32$	0	0
$33$	10.0424	0.304314
$34$	0	0
$35$	− 33.0931i	− 0.945517i
$36$	0	0
$37$	−47.7930	−1.29170	−0.645851	−	0.763463i	$-0.723497\pi$
$37$	−47.7930	−1.29170	−0.645851	+	0.763463i	$0.723497\pi$
$38$	0	0
$39$	15.2197i	0.390249i
$40$	0	0
$41$	53.3294	1.30072	0.650358	−	0.759628i	$-0.274619\pi$
$41$	53.3294	1.30072	0.650358	+	0.759628i	$0.274619\pi$
$42$	0	0
$43$	30.7044i	0.714057i	0.934094	+	0.357028i	$0.116210\pi$
$43$	30.7044i	0.714057i	−0.934094	+	0.357028i	$0.883790\pi$
$44$	0	0
$45$	7.90702	0.175712
$46$	0	0
$47$	16.2238i	0.345186i	0.984993	+	0.172593i	$0.0552146\pi$
$47$	16.2238i	0.345186i	−0.984993	+	0.172593i	$0.944785\pi$
$48$	0	0
$49$	−108.649	−2.21733
$50$	0	0
$51$	− 52.2750i	− 1.02500i
$52$	0	0
$53$	−49.8432	−0.940437	−0.470219	−	0.882550i	$-0.655825\pi$
$53$	−49.8432	−0.940437	−0.470219	+	0.882550i	$0.655825\pi$
$54$	0	0
$55$	15.2815i	0.277846i
$56$	0	0
$57$	30.1593	0.529110
$58$	0	0
$59$	− 107.412i	− 1.82054i	−0.414012	−	0.910271i	$-0.635873\pi$
$59$	− 107.412i	− 1.82054i	0.414012	−	0.910271i	$-0.364127\pi$
$60$	0	0
$61$	62.6220	1.02659	0.513295	−	0.858212i	$-0.328425\pi$
$61$	62.6220	1.02659	0.513295	+	0.858212i	$0.328425\pi$
$62$	0	0
$63$	− 37.6675i	− 0.597897i
$64$	0	0
$65$	−23.1599	−0.356307
$66$	0	0
$67$	60.9540i	0.909762i	0.890552	+	0.454881i	$0.150318\pi$
$67$	60.9540i	0.909762i	−0.890552	+	0.454881i	$0.849682\pi$
$68$	0	0
$69$	−4.30285	−0.0623601
$70$	0	0
$71$	− 19.9465i	− 0.280937i	−0.990085	−	0.140469i	$-0.955139\pi$
$71$	− 19.9465i	− 0.280937i	0.990085	−	0.140469i	$-0.0448609\pi$
$72$	0	0
$73$	5.13929	0.0704013	0.0352006	−	0.999380i	$-0.488793\pi$
$73$	5.13929	0.0704013	0.0352006	+	0.999380i	$0.488793\pi$
$74$	0	0
$75$	− 31.2691i	− 0.416921i
$76$	0	0
$77$	72.7982	0.945431
$78$	0	0
$79$	− 6.83349i	− 0.0864999i	−0.999064	−	0.0432500i	$-0.986229\pi$
$79$	− 6.83349i	− 0.0864999i	0.999064	−	0.0432500i	$-0.0137712\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	159.213i	1.91823i	0.283016	+	0.959115i	$0.408665\pi$
$83$	159.213i	1.91823i	−0.283016	+	0.959115i	$0.591335\pi$
$84$	0	0
$85$	79.5472	0.935850
$86$	0	0
$87$	− 45.7565i	− 0.525937i
$88$	0	0
$89$	39.4473	0.443229	0.221614	−	0.975134i	$-0.428867\pi$
$89$	39.4473	0.443229	0.221614	+	0.975134i	$0.428867\pi$
$90$	0	0
$91$	110.329i	1.21241i
$92$	0	0
$93$	−65.8321	−0.707872
$94$	0	0
$95$	45.8936i	0.483090i
$96$	0	0
$97$	−60.5944	−0.624684	−0.312342	−	0.949970i	$-0.601114\pi$
$97$	−60.5944	−0.624684	−0.312342	+	0.949970i	$0.601114\pi$
$98$	0	0
$99$	17.3939i	0.175696i
$100$	0	0
$101$	−169.076	−1.67402	−0.837011	−	0.547186i	$-0.815699\pi$
$101$	−169.076	−1.67402	−0.837011	+	0.547186i	$0.815699\pi$
$102$	0	0
$103$	163.390i	1.58631i	0.609020	+	0.793155i	$0.291563\pi$
$103$	163.390i	1.58631i	−0.609020	+	0.793155i	$0.708437\pi$
$104$	0	0
$105$	57.3189	0.545894
$106$	0	0
$107$	37.4288i	0.349802i	0.984586	+	0.174901i	$0.0559605\pi$
$107$	37.4288i	0.349802i	−0.984586	+	0.174901i	$0.944040\pi$
$108$	0	0
$109$	210.117	1.92768	0.963838	−	0.266489i	$-0.0858635\pi$
$109$	210.117	1.92768	0.963838	+	0.266489i	$0.0858635\pi$
$110$	0	0
$111$	− 82.7799i	− 0.745765i
$112$	0	0
$113$	190.809	1.68857	0.844286	−	0.535893i	$-0.180025\pi$
$113$	190.809	1.68857	0.844286	+	0.535893i	$0.180025\pi$
$114$	0	0
$115$	− 6.54768i	− 0.0569363i
$116$	0	0
$117$	−26.3613	−0.225310
$118$	0	0
$119$	− 378.947i	− 3.18443i
$120$	0	0
$121$	87.3837	0.722179
$122$	0	0
$123$	92.3692i	0.750969i
$124$	0	0
$125$	113.474	0.907794
$126$	0	0
$127$	34.4377i	0.271163i	0.990766	+	0.135581i	$0.0432902\pi$
$127$	34.4377i	0.271163i	−0.990766	+	0.135581i	$0.956710\pi$
$128$	0	0
$129$	−53.1817	−0.412261
$130$	0	0
$131$	− 92.5491i	− 0.706482i	−0.935532	−	0.353241i	$-0.885080\pi$
$131$	− 92.5491i	− 0.706482i	0.935532	−	0.353241i	$-0.114920\pi$
$132$	0	0
$133$	218.628	1.64382
$134$	0	0
$135$	13.6954i	0.101447i
$136$	0	0
$137$	−20.0745	−0.146529	−0.0732647	−	0.997313i	$-0.523342\pi$
$137$	−20.0745	−0.146529	−0.0732647	+	0.997313i	$0.523342\pi$
$138$	0	0
$139$	70.0557i	0.503998i	0.967728	+	0.251999i	$0.0810879\pi$
$139$	70.0557i	0.503998i	−0.967728	+	0.251999i	$0.918912\pi$
$140$	0	0
$141$	−28.1004	−0.199293
$142$	0	0
$143$	− 50.9472i	− 0.356274i
$144$	0	0
$145$	69.6281	0.480193
$146$	0	0
$147$	− 188.186i	− 1.28017i
$148$	0	0
$149$	165.919	1.11355	0.556774	−	0.830664i	$-0.312039\pi$
$149$	165.919	1.11355	0.556774	+	0.830664i	$0.312039\pi$
$150$	0	0
$151$	− 59.7106i	− 0.395434i	−0.980259	−	0.197717i	$-0.936647\pi$
$151$	− 59.7106i	− 0.395434i	0.980259	−	0.197717i	$-0.0633528\pi$
$152$	0	0
$153$	90.5429	0.591784
$154$	0	0
$155$	− 100.177i	− 0.646304i
$156$	0	0
$157$	−138.133	−0.879829	−0.439915	−	0.898040i	$-0.644991\pi$
$157$	−138.133	−0.879829	−0.439915	+	0.898040i	$0.644991\pi$
$158$	0	0
$159$	− 86.3309i	− 0.542962i
$160$	0	0
$161$	−31.1918	−0.193738
$162$	0	0
$163$	178.296i	1.09384i	0.837185	+	0.546920i	$0.184200\pi$
$163$	178.296i	1.09384i	−0.837185	+	0.546920i	$0.815800\pi$
$164$	0	0
$165$	−26.4684	−0.160414
$166$	0	0
$167$	98.6284i	0.590589i	0.955406	+	0.295295i	$0.0954178\pi$
$167$	98.6284i	0.590589i	−0.955406	+	0.295295i	$0.904582\pi$
$168$	0	0
$169$	−91.7869	−0.543118
$170$	0	0
$171$	52.2374i	0.305482i
$172$	0	0
$173$	−166.380	−0.961733	−0.480867	−	0.876794i	$-0.659678\pi$
$173$	−166.380	−0.961733	−0.480867	+	0.876794i	$0.659678\pi$
$174$	0	0
$175$	− 226.673i	− 1.29528i
$176$	0	0
$177$	186.043	1.05109
$178$	0	0
$179$	206.496i	1.15361i	0.816882	+	0.576805i	$0.195701\pi$
$179$	206.496i	1.15361i	−0.816882	+	0.576805i	$0.804299\pi$
$180$	0	0
$181$	−181.330	−1.00182	−0.500911	−	0.865499i	$-0.667002\pi$
$181$	−181.330	−1.00182	−0.500911	+	0.865499i	$0.667002\pi$
$182$	0	0
$183$	108.465i	0.592702i
$184$	0	0
$185$	125.967	0.680901
$186$	0	0
$187$	174.988i	0.935765i
$188$	0	0
$189$	65.2420	0.345196
$190$	0	0
$191$	185.390i	0.970627i	0.874340	+	0.485314i	$0.161295\pi$
$191$	185.390i	0.970627i	−0.874340	+	0.485314i	$0.838705\pi$
$192$	0	0
$193$	15.5439	0.0805382	0.0402691	−	0.999189i	$-0.487178\pi$
$193$	15.5439	0.0805382	0.0402691	+	0.999189i	$0.487178\pi$
$194$	0	0
$195$	− 40.1142i	− 0.205714i
$196$	0	0
$197$	114.464	0.581034	0.290517	−	0.956870i	$-0.406173\pi$
$197$	114.464	0.581034	0.290517	+	0.956870i	$0.406173\pi$
$198$	0	0
$199$	299.009i	1.50256i	0.659984	+	0.751280i	$0.270563\pi$
$199$	299.009i	1.50256i	−0.659984	+	0.751280i	$0.729437\pi$
$200$	0	0
$201$	−105.576	−0.525251
$202$	0	0
$203$	− 331.694i	− 1.63396i
$204$	0	0
$205$	−140.559	−0.685653
$206$	0	0
$207$	− 7.45275i	− 0.0360036i
$208$	0	0
$209$	−100.957	−0.483046
$210$	0	0
$211$	− 139.706i	− 0.662111i	−0.943611	−	0.331056i	$-0.892595\pi$
$211$	− 139.706i	− 0.662111i	0.943611	−	0.331056i	$-0.107405\pi$
$212$	0	0
$213$	34.5484	0.162199
$214$	0	0
$215$	− 80.9269i	− 0.376404i
$216$	0	0
$217$	−477.224	−2.19919
$218$	0	0
$219$	8.90152i	0.0406462i
$220$	0	0
$221$	−265.203	−1.20001
$222$	0	0
$223$	− 66.1141i	− 0.296476i	−0.988952	−	0.148238i	$-0.952640\pi$
$223$	− 66.1141i	− 0.296476i	0.988952	−	0.148238i	$-0.0473601\pi$
$224$	0	0
$225$	54.1597	0.240710
$226$	0	0
$227$	49.9611i	0.220093i	0.993926	+	0.110046i	$0.0351000\pi$
$227$	49.9611i	0.220093i	−0.993926	+	0.110046i	$0.964900\pi$
$228$	0	0
$229$	129.255	0.564431	0.282215	−	0.959351i	$-0.408931\pi$
$229$	129.255	0.564431	0.282215	+	0.959351i	$0.408931\pi$
$230$	0	0
$231$	126.090i	0.545845i
$232$	0	0
$233$	−196.318	−0.842567	−0.421284	−	0.906929i	$-0.638420\pi$
$233$	−196.318	−0.842567	−0.421284	+	0.906929i	$0.638420\pi$
$234$	0	0
$235$	− 42.7606i	− 0.181960i
$236$	0	0
$237$	11.8360	0.0499407
$238$	0	0
$239$	153.654i	0.642905i	0.946926	+	0.321452i	$0.104171\pi$
$239$	153.654i	0.642905i	−0.946926	+	0.321452i	$0.895829\pi$
$240$	0	0
$241$	−41.3543	−0.171595	−0.0857974	−	0.996313i	$-0.527344\pi$
$241$	−41.3543	−0.171595	−0.0857974	+	0.996313i	$0.527344\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	286.363	1.16883
$246$	0	0
$247$	− 153.005i	− 0.619453i
$248$	0	0
$249$	−275.765	−1.10749
$250$	0	0
$251$	− 35.4495i	− 0.141233i	−0.997504	−	0.0706165i	$-0.977503\pi$
$251$	− 35.4495i	− 0.141233i	0.997504	−	0.0706165i	$-0.0224967\pi$
$252$	0	0
$253$	14.4036	0.0569312
$254$	0	0
$255$	137.780i	0.540313i
$256$	0	0
$257$	66.0626	0.257053	0.128527	−	0.991706i	$-0.458975\pi$
$257$	66.0626	0.257053	0.128527	+	0.991706i	$0.458975\pi$
$258$	0	0
$259$	− 600.081i	− 2.31691i
$260$	0	0
$261$	79.2526	0.303650
$262$	0	0
$263$	93.5910i	0.355859i	0.984043	+	0.177930i	$0.0569400\pi$
$263$	93.5910i	0.355859i	−0.984043	+	0.177930i	$0.943060\pi$
$264$	0	0
$265$	131.370	0.495737
$266$	0	0
$267$	68.3248i	0.255898i
$268$	0	0
$269$	−305.548	−1.13587	−0.567933	−	0.823074i	$-0.692257\pi$
$269$	−305.548	−1.13587	−0.567933	+	0.823074i	$0.692257\pi$
$270$	0	0
$271$	191.602i	0.707017i	0.935431	+	0.353509i	$0.115012\pi$
$271$	191.602i	0.707017i	−0.935431	+	0.353509i	$0.884988\pi$
$272$	0	0
$273$	−191.096	−0.699985
$274$	0	0
$275$	104.672i	0.380625i
$276$	0	0
$277$	−188.436	−0.680276	−0.340138	−	0.940376i	$-0.610474\pi$
$277$	−188.436	−0.680276	−0.340138	+	0.940376i	$0.610474\pi$
$278$	0	0
$279$	− 114.024i	− 0.408690i
$280$	0	0
$281$	−296.087	−1.05369	−0.526844	−	0.849962i	$-0.676625\pi$
$281$	−296.087	−1.05369	−0.526844	+	0.849962i	$0.676625\pi$
$282$	0	0
$283$	− 260.681i	− 0.921136i	−0.887625	−	0.460568i	$-0.847646\pi$
$283$	− 260.681i	− 0.921136i	0.887625	−	0.460568i	$-0.152354\pi$
$284$	0	0
$285$	−79.4900	−0.278912
$286$	0	0
$287$	669.595i	2.33308i
$288$	0	0
$289$	621.891	2.15187
$290$	0	0
$291$	− 104.953i	− 0.360662i
$292$	0	0
$293$	−314.756	−1.07425	−0.537127	−	0.843501i	$-0.680490\pi$
$293$	−314.756	−1.07425	−0.537127	+	0.843501i	$0.680490\pi$
$294$	0	0
$295$	283.103i	0.959672i
$296$	0	0
$297$	−30.1271	−0.101438
$298$	0	0
$299$	21.8294i	0.0730079i
$300$	0	0
$301$	−385.520	−1.28080
$302$	0	0
$303$	− 292.849i	− 0.966497i
$304$	0	0
$305$	−165.051	−0.541152
$306$	0	0
$307$	− 168.120i	− 0.547621i	−0.961784	−	0.273811i	$-0.911716\pi$
$307$	− 168.120i	− 0.547621i	0.961784	−	0.273811i	$-0.0882841\pi$
$308$	0	0
$309$	−283.000	−0.915856
$310$	0	0
$311$	− 468.033i	− 1.50493i	−0.658633	−	0.752464i	$-0.728865\pi$
$311$	− 468.033i	− 1.50493i	0.658633	−	0.752464i	$-0.271135\pi$
$312$	0	0
$313$	225.370	0.720033	0.360017	−	0.932946i	$-0.382771\pi$
$313$	225.370	0.720033	0.360017	+	0.932946i	$0.382771\pi$
$314$	0	0
$315$	99.2793i	0.315172i
$316$	0	0
$317$	162.613	0.512976	0.256488	−	0.966547i	$-0.417435\pi$
$317$	162.613	0.512976	0.256488	+	0.966547i	$0.417435\pi$
$318$	0	0
$319$	153.168i	0.480150i
$320$	0	0
$321$	−64.8285	−0.201958
$322$	0	0
$323$	525.525i	1.62701i
$324$	0	0
$325$	−158.635	−0.488109
$326$	0	0
$327$	363.933i	1.11294i
$328$	0	0
$329$	−203.703	−0.619158
$330$	0	0
$331$	− 254.527i	− 0.768964i	−0.923133	−	0.384482i	$-0.874380\pi$
$331$	− 254.527i	− 0.768964i	0.923133	−	0.384482i	$-0.125620\pi$
$332$	0	0
$333$	143.379	0.430567
$334$	0	0
$335$	− 160.655i	− 0.479567i
$336$	0	0
$337$	−94.7347	−0.281112	−0.140556	−	0.990073i	$-0.544889\pi$
$337$	−94.7347	−0.281112	−0.140556	+	0.990073i	$0.544889\pi$
$338$	0	0
$339$	330.490i	0.974897i
$340$	0	0
$341$	220.370	0.646246
$342$	0	0
$343$	− 748.942i	− 2.18351i
$344$	0	0
$345$	11.3409	0.0328722
$346$	0	0
$347$	− 173.315i	− 0.499467i	−0.968315	−	0.249734i	$-0.919657\pi$
$347$	− 173.315i	− 0.499467i	0.968315	−	0.249734i	$-0.0803431\pi$
$348$	0	0
$349$	−195.247	−0.559448	−0.279724	−	0.960081i	$-0.590243\pi$
$349$	−195.247	−0.559448	−0.279724	+	0.960081i	$0.590243\pi$
$350$	0	0
$351$	− 45.6591i	− 0.130083i
$352$	0	0
$353$	296.107	0.838830	0.419415	−	0.907795i	$-0.362235\pi$
$353$	296.107	0.838830	0.419415	+	0.907795i	$0.362235\pi$
$354$	0	0
$355$	52.5726i	0.148092i
$356$	0	0
$357$	656.356	1.83853
$358$	0	0
$359$	448.964i	1.25060i	0.780386	+	0.625298i	$0.215023\pi$
$359$	448.964i	1.25060i	−0.780386	+	0.625298i	$0.784977\pi$
$360$	0	0
$361$	57.8065	0.160129
$362$	0	0
$363$	151.353i	0.416950i
$364$	0	0
$365$	−13.5455	−0.0371110
$366$	0	0
$367$	− 623.564i	− 1.69908i	−0.527521	−	0.849542i	$-0.676878\pi$
$367$	− 623.564i	− 1.69908i	0.527521	−	0.849542i	$-0.323122\pi$
$368$	0	0
$369$	−159.988	−0.433572
$370$	0	0
$371$	− 625.823i	− 1.68685i
$372$	0	0
$373$	297.116	0.796558	0.398279	−	0.917264i	$-0.369608\pi$
$373$	297.116	0.796558	0.398279	+	0.917264i	$0.369608\pi$
$374$	0	0
$375$	196.543i	0.524115i
$376$	0	0
$377$	−232.134	−0.615739
$378$	0	0
$379$	− 224.021i	− 0.591084i	−0.955330	−	0.295542i	$-0.904500\pi$
$379$	− 224.021i	− 0.591084i	0.955330	−	0.295542i	$-0.0955003\pi$
$380$	0	0
$381$	−59.6478	−0.156556
$382$	0	0
$383$	− 573.847i	− 1.49830i	−0.662403	−	0.749148i	$-0.730463\pi$
$383$	− 573.847i	− 1.49830i	0.662403	−	0.749148i	$-0.269537\pi$
$384$	0	0
$385$	−191.872	−0.498370
$386$	0	0
$387$	− 92.1133i	− 0.238019i
$388$	0	0
$389$	−148.849	−0.382646	−0.191323	−	0.981527i	$-0.561278\pi$
$389$	−148.849	−0.382646	−0.191323	+	0.981527i	$0.561278\pi$
$390$	0	0
$391$	− 74.9771i	− 0.191757i
$392$	0	0
$393$	160.300	0.407887
$394$	0	0
$395$	18.0109i	0.0455971i
$396$	0	0
$397$	−245.952	−0.619526	−0.309763	−	0.950814i	$-0.600250\pi$
$397$	−245.952	−0.619526	−0.309763	+	0.950814i	$0.600250\pi$
$398$	0	0
$399$	378.675i	0.949059i
$400$	0	0
$401$	64.0509	0.159728	0.0798640	−	0.996806i	$-0.474551\pi$
$401$	64.0509	0.159728	0.0798640	+	0.996806i	$0.474551\pi$
$402$	0	0
$403$	333.981i	0.828738i
$404$	0	0
$405$	−23.7211	−0.0585705
$406$	0	0
$407$	277.102i	0.680840i
$408$	0	0
$409$	555.403	1.35795	0.678977	−	0.734160i	$-0.262424\pi$
$409$	555.403	1.35795	0.678977	+	0.734160i	$0.262424\pi$
$410$	0	0
$411$	− 34.7701i	− 0.0845988i
$412$	0	0
$413$	1348.65	3.26549
$414$	0	0
$415$	− 419.634i	− 1.01117i
$416$	0	0
$417$	−121.340	−0.290983
$418$	0	0
$419$	793.486i	1.89376i	0.321583	+	0.946881i	$0.395785\pi$
$419$	793.486i	1.89376i	−0.321583	+	0.946881i	$0.604215\pi$
$420$	0	0
$421$	−87.1175	−0.206930	−0.103465	−	0.994633i	$-0.532993\pi$
$421$	−87.1175	−0.206930	−0.103465	+	0.994633i	$0.532993\pi$
$422$	0	0
$423$	− 48.6713i	− 0.115062i
$424$	0	0
$425$	544.864	1.28203
$426$	0	0
$427$	786.272i	1.84139i
$428$	0	0
$429$	88.2432	0.205695
$430$	0	0
$431$	432.932i	1.00448i	0.864727	+	0.502242i	$0.167491\pi$
$431$	432.932i	1.00448i	−0.864727	+	0.502242i	$0.832509\pi$
$432$	0	0
$433$	391.344	0.903798	0.451899	−	0.892069i	$-0.350747\pi$
$433$	391.344	0.903798	0.451899	+	0.892069i	$0.350747\pi$
$434$	0	0
$435$	120.599i	0.277240i
$436$	0	0
$437$	43.2569	0.0989860
$438$	0	0
$439$	12.3066i	0.0280332i	0.999902	+	0.0140166i	$0.00446177\pi$
$439$	12.3066i	0.0280332i	−0.999902	+	0.0140166i	$0.995538\pi$
$440$	0	0
$441$	325.947	0.739109
$442$	0	0
$443$	567.554i	1.28116i	0.767892	+	0.640580i	$0.221306\pi$
$443$	567.554i	1.28116i	−0.767892	+	0.640580i	$0.778694\pi$
$444$	0	0
$445$	−103.970	−0.233641
$446$	0	0
$447$	287.380i	0.642908i
$448$	0	0
$449$	−407.897	−0.908457	−0.454229	−	0.890885i	$-0.650085\pi$
$449$	−407.897	−0.908457	−0.454229	+	0.890885i	$0.650085\pi$
$450$	0	0
$451$	− 309.201i	− 0.685591i
$452$	0	0
$453$	103.422	0.228304
$454$	0	0
$455$	− 290.792i	− 0.639104i
$456$	0	0
$457$	231.789	0.507197	0.253598	−	0.967310i	$-0.418386\pi$
$457$	231.789	0.507197	0.253598	+	0.967310i	$0.418386\pi$
$458$	0	0
$459$	156.825i	0.341667i
$460$	0	0
$461$	438.217	0.950578	0.475289	−	0.879830i	$-0.342343\pi$
$461$	438.217	0.950578	0.475289	+	0.879830i	$0.342343\pi$
$462$	0	0
$463$	− 185.141i	− 0.399873i	−0.979809	−	0.199936i	$-0.935926\pi$
$463$	− 185.141i	− 0.399873i	0.979809	−	0.199936i	$-0.0640735\pi$
$464$	0	0
$465$	173.512	0.373144
$466$	0	0
$467$	− 82.9765i	− 0.177680i	−0.996046	−	0.0888399i	$-0.971684\pi$
$467$	− 82.9765i	− 0.177680i	0.996046	−	0.0888399i	$-0.0283159\pi$
$468$	0	0
$469$	−765.329	−1.63183
$470$	0	0
$471$	− 239.254i	− 0.507970i
$472$	0	0
$473$	178.023	0.376370
$474$	0	0
$475$	314.351i	0.661791i
$476$	0	0
$477$	149.530	0.313479
$478$	0	0
$479$	173.847i	0.362936i	0.983397	+	0.181468i	$0.0580849\pi$
$479$	173.847i	0.362936i	−0.983397	+	0.181468i	$0.941915\pi$
$480$	0	0
$481$	−419.962	−0.873101
$482$	0	0
$483$	− 54.0258i	− 0.111855i
$484$	0	0
$485$	159.707	0.329293
$486$	0	0
$487$	− 250.667i	− 0.514716i	−0.966316	−	0.257358i	$-0.917148\pi$
$487$	− 250.667i	− 0.514716i	0.966316	−	0.257358i	$-0.0828520\pi$
$488$	0	0
$489$	−308.817	−0.631529
$490$	0	0
$491$	283.478i	0.577349i	0.957427	+	0.288674i	$0.0932145\pi$
$491$	283.478i	0.577349i	−0.957427	+	0.288674i	$0.906786\pi$
$492$	0	0
$493$	797.307	1.61726
$494$	0	0
$495$	− 45.8446i	− 0.0926153i
$496$	0	0
$497$	250.445	0.503914
$498$	0	0
$499$	− 192.802i	− 0.386377i	−0.981162	−	0.193188i	$-0.938117\pi$
$499$	− 192.802i	− 0.386377i	0.981162	−	0.193188i	$-0.0618829\pi$
$500$	0	0
$501$	−170.829	−0.340977
$502$	0	0
$503$	− 398.251i	− 0.791752i	−0.918304	−	0.395876i	$-0.870441\pi$
$503$	− 398.251i	− 0.791752i	0.918304	−	0.395876i	$-0.129559\pi$
$504$	0	0
$505$	445.630	0.882436
$506$	0	0
$507$	− 158.980i	− 0.313569i
$508$	0	0
$509$	−550.212	−1.08097	−0.540483	−	0.841355i	$-0.681758\pi$
$509$	−550.212	−1.08097	−0.540483	+	0.841355i	$0.681758\pi$
$510$	0	0
$511$	64.5281i	0.126278i
$512$	0	0
$513$	−90.4778	−0.176370
$514$	0	0
$515$	− 430.643i	− 0.836199i
$516$	0	0
$517$	94.0647	0.181943
$518$	0	0
$519$	− 288.178i	− 0.555257i
$520$	0	0
$521$	−164.181	−0.315126	−0.157563	−	0.987509i	$-0.550364\pi$
$521$	−164.181	−0.315126	−0.157563	+	0.987509i	$0.550364\pi$
$522$	0	0
$523$	− 643.556i	− 1.23051i	−0.788328	−	0.615255i	$-0.789053\pi$
$523$	− 643.556i	− 1.23051i	0.788328	−	0.615255i	$-0.210947\pi$
$524$	0	0
$525$	392.610	0.747828
$526$	0	0
$527$	− 1147.12i	− 2.17670i
$528$	0	0
$529$	522.828	0.988334
$530$	0	0
$531$	322.236i	0.606848i
$532$	0	0
$533$	468.610	0.879194
$534$	0	0
$535$	− 98.6501i	− 0.184393i
$536$	0	0
$537$	−357.662	−0.666037
$538$	0	0
$539$	629.942i	1.16872i
$540$	0	0
$541$	727.713	1.34513	0.672563	−	0.740040i	$-0.265193\pi$
$541$	727.713	1.34513	0.672563	+	0.740040i	$0.265193\pi$
$542$	0	0
$543$	− 314.072i	− 0.578402i
$544$	0	0
$545$	−553.799	−1.01615
$546$	0	0
$547$	97.2958i	0.177872i	0.996037	+	0.0889359i	$0.0283466\pi$
$547$	97.2958i	0.177872i	−0.996037	+	0.0889359i	$0.971653\pi$
$548$	0	0
$549$	−187.866	−0.342197
$550$	0	0
$551$	459.994i	0.834835i
$552$	0	0
$553$	85.8002	0.155154
$554$	0	0
$555$	218.181i	0.393119i
$556$	0	0
$557$	−520.548	−0.934556	−0.467278	−	0.884110i	$-0.654765\pi$
$557$	−520.548	−0.934556	−0.467278	+	0.884110i	$0.654765\pi$
$558$	0	0
$559$	269.803i	0.482653i
$560$	0	0
$561$	−303.088	−0.540264
$562$	0	0
$563$	− 525.371i	− 0.933163i	−0.884478	−	0.466581i	$-0.845485\pi$
$563$	− 525.371i	− 0.933163i	0.884478	−	0.466581i	$-0.154515\pi$
$564$	0	0
$565$	−502.909	−0.890105
$566$	0	0
$567$	113.003i	0.199299i
$568$	0	0
$569$	824.774	1.44951	0.724757	−	0.689004i	$-0.241952\pi$
$569$	824.774	1.44951	0.724757	+	0.689004i	$0.241952\pi$
$570$	0	0
$571$	− 1088.90i	− 1.90701i	−0.301381	−	0.953504i	$-0.597448\pi$
$571$	− 1088.90i	− 1.90701i	0.301381	−	0.953504i	$-0.402552\pi$
$572$	0	0
$573$	−321.105	−0.560392
$574$	0	0
$575$	− 44.8487i	− 0.0779978i
$576$	0	0
$577$	−688.463	−1.19318	−0.596589	−	0.802547i	$-0.703478\pi$
$577$	−688.463	−1.19318	−0.596589	+	0.802547i	$0.703478\pi$
$578$	0	0
$579$	26.9228i	0.0464988i
$580$	0	0
$581$	−1999.05	−3.44071
$582$	0	0
$583$	288.989i	0.495692i
$584$	0	0
$585$	69.4798	0.118769
$586$	0	0
$587$	− 441.108i	− 0.751462i	−0.926729	−	0.375731i	$-0.877392\pi$
$587$	− 441.108i	− 0.751462i	0.926729	−	0.375731i	$-0.122608\pi$
$588$	0	0
$589$	661.815	1.12363
$590$	0	0
$591$	198.257i	0.335460i
$592$	0	0
$593$	−481.317	−0.811664	−0.405832	−	0.913948i	$-0.633018\pi$
$593$	−481.317	−0.811664	−0.405832	+	0.913948i	$0.633018\pi$
$594$	0	0
$595$	998.782i	1.67862i
$596$	0	0
$597$	−517.899	−0.867503
$598$	0	0
$599$	541.322i	0.903709i	0.892092	+	0.451854i	$0.149237\pi$
$599$	541.322i	0.903709i	−0.892092	+	0.451854i	$0.850763\pi$
$600$	0	0
$601$	64.7772	0.107782	0.0538912	−	0.998547i	$-0.482838\pi$
$601$	64.7772	0.107782	0.0538912	+	0.998547i	$0.482838\pi$
$602$	0	0
$603$	− 182.862i	− 0.303254i
$604$	0	0
$605$	−230.315	−0.380686
$606$	0	0
$607$	608.260i	1.00208i	0.865426	+	0.501038i	$0.167048\pi$
$607$	608.260i	1.00208i	−0.865426	+	0.501038i	$0.832952\pi$
$608$	0	0
$609$	574.511	0.943368
$610$	0	0
$611$	142.560i	0.233322i
$612$	0	0
$613$	924.407	1.50800	0.754002	−	0.656872i	$-0.228121\pi$
$613$	924.407	1.50800	0.754002	+	0.656872i	$0.228121\pi$
$614$	0	0
$615$	− 243.455i	− 0.395862i
$616$	0	0
$617$	652.781	1.05799	0.528996	−	0.848624i	$-0.322569\pi$
$617$	652.781	1.05799	0.528996	+	0.848624i	$0.322569\pi$
$618$	0	0
$619$	− 246.098i	− 0.397574i	−0.980043	−	0.198787i	$-0.936300\pi$
$619$	− 246.098i	− 0.397574i	0.980043	−	0.198787i	$-0.0637001\pi$
$620$	0	0
$621$	12.9085	0.0207867
$622$	0	0
$623$	495.294i	0.795015i
$624$	0	0
$625$	152.249	0.243599
$626$	0	0
$627$	− 174.862i	− 0.278887i
$628$	0	0
$629$	1442.44	2.29323
$630$	0	0
$631$	563.715i	0.893368i	0.894692	+	0.446684i	$0.147395\pi$
$631$	563.715i	0.893368i	−0.894692	+	0.446684i	$0.852605\pi$
$632$	0	0
$633$	241.977	0.382270
$634$	0	0
$635$	− 90.7665i	− 0.142939i
$636$	0	0
$637$	−954.709	−1.49876
$638$	0	0
$639$	59.8396i	0.0936457i
$640$	0	0
$641$	−163.485	−0.255046	−0.127523	−	0.991836i	$-0.540703\pi$
$641$	−163.485	−0.255046	−0.127523	+	0.991836i	$0.540703\pi$
$642$	0	0
$643$	307.908i	0.478862i	0.970913	+	0.239431i	$0.0769609\pi$
$643$	307.908i	0.478862i	−0.970913	+	0.239431i	$0.923039\pi$
$644$	0	0
$645$	140.170	0.217317
$646$	0	0
$647$	909.931i	1.40638i	0.711000	+	0.703192i	$0.248243\pi$
$647$	909.931i	1.40638i	−0.711000	+	0.703192i	$0.751757\pi$
$648$	0	0
$649$	−622.771	−0.959585
$650$	0	0
$651$	− 826.576i	− 1.26970i
$652$	0	0
$653$	−299.487	−0.458632	−0.229316	−	0.973352i	$-0.573649\pi$
$653$	−299.487	−0.458632	−0.229316	+	0.973352i	$0.573649\pi$
$654$	0	0
$655$	243.929i	0.372411i
$656$	0	0
$657$	−15.4179	−0.0234671
$658$	0	0
$659$	− 141.105i	− 0.214120i	−0.994253	−	0.107060i	$-0.965856\pi$
$659$	− 141.105i	− 0.214120i	0.994253	−	0.107060i	$-0.0341437\pi$
$660$	0	0
$661$	−1176.74	−1.78024	−0.890121	−	0.455725i	$-0.849380\pi$
$661$	−1176.74	−1.78024	−0.890121	+	0.455725i	$0.849380\pi$
$662$	0	0
$663$	− 459.345i	− 0.692829i
$664$	0	0
$665$	−576.232	−0.866514
$666$	0	0
$667$	− 65.6278i	− 0.0983925i
$668$	0	0
$669$	114.513	0.171170
$670$	0	0
$671$	− 363.080i	− 0.541103i
$672$	0	0
$673$	−230.780	−0.342912	−0.171456	−	0.985192i	$-0.554847\pi$
$673$	−230.780	−0.342912	−0.171456	+	0.985192i	$0.554847\pi$
$674$	0	0
$675$	93.8073i	0.138974i
$676$	0	0
$677$	−768.499	−1.13515	−0.567577	−	0.823320i	$-0.692119\pi$
$677$	−768.499	−1.13515	−0.567577	+	0.823320i	$0.692119\pi$
$678$	0	0
$679$	− 760.813i	− 1.12049i
$680$	0	0
$681$	−86.5352	−0.127071
$682$	0	0
$683$	963.083i	1.41008i	0.709169	+	0.705039i	$0.249070\pi$
$683$	963.083i	1.41008i	−0.709169	+	0.705039i	$0.750930\pi$
$684$	0	0
$685$	52.9099	0.0772408
$686$	0	0
$687$	223.876i	0.325874i
$688$	0	0
$689$	−437.977	−0.635670
$690$	0	0
$691$	− 376.440i	− 0.544776i	−0.962188	−	0.272388i	$-0.912187\pi$
$691$	− 376.440i	− 0.544776i	0.962188	−	0.272388i	$-0.0878134\pi$
$692$	0	0
$693$	−218.395	−0.315144
$694$	0	0
$695$	− 184.644i	− 0.265675i
$696$	0	0
$697$	−1609.53	−2.30923
$698$	0	0
$699$	− 340.033i	− 0.486456i
$700$	0	0
$701$	543.266	0.774988	0.387494	−	0.921872i	$-0.373341\pi$
$701$	543.266	0.774988	0.387494	+	0.921872i	$0.373341\pi$
$702$	0	0
$703$	832.193i	1.18377i
$704$	0	0
$705$	74.0635	0.105055
$706$	0	0
$707$	− 2122.89i	− 3.00268i
$708$	0	0
$709$	403.852	0.569608	0.284804	−	0.958586i	$-0.408071\pi$
$709$	403.852	0.569608	0.284804	+	0.958586i	$0.408071\pi$
$710$	0	0
$711$	20.5005i	0.0288333i
$712$	0	0
$713$	−94.4218	−0.132429
$714$	0	0
$715$	134.280i	0.187805i
$716$	0	0
$717$	−266.137	−0.371181
$718$	0	0
$719$	183.791i	0.255620i	0.991799	+	0.127810i	$0.0407948\pi$
$719$	183.791i	0.255620i	−0.991799	+	0.127810i	$0.959205\pi$
$720$	0	0
$721$	−2051.50	−2.84535
$722$	0	0
$723$	− 71.6278i	− 0.0990703i
$724$	0	0
$725$	476.922	0.657823
$726$	0	0
$727$	843.226i	1.15987i	0.814663	+	0.579935i	$0.196922\pi$
$727$	843.226i	1.15987i	−0.814663	+	0.579935i	$0.803078\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 926.690i	− 1.26770i
$732$	0	0
$733$	−286.176	−0.390418	−0.195209	−	0.980762i	$-0.562539\pi$
$733$	−286.176	−0.390418	−0.195209	+	0.980762i	$0.562539\pi$
$734$	0	0
$735$	495.996i	0.674824i
$736$	0	0
$737$	353.409	0.479524
$738$	0	0
$739$	821.243i	1.11129i	0.831420	+	0.555645i	$0.187529\pi$
$739$	821.243i	1.11129i	−0.831420	+	0.555645i	$0.812471\pi$
$740$	0	0
$741$	265.012	0.357641
$742$	0	0
$743$	− 1227.08i	− 1.65153i	−0.564018	−	0.825763i	$-0.690745\pi$
$743$	− 1227.08i	− 1.65153i	0.564018	−	0.825763i	$-0.309255\pi$
$744$	0	0
$745$	−437.308	−0.586990
$746$	0	0
$747$	− 477.639i	− 0.639410i
$748$	0	0
$749$	−469.949	−0.627436
$750$	0	0
$751$	966.507i	1.28696i	0.765463	+	0.643480i	$0.222510\pi$
$751$	966.507i	1.28696i	−0.765463	+	0.643480i	$0.777490\pi$
$752$	0	0
$753$	61.4003	0.0815409
$754$	0	0
$755$	157.378i	0.208447i
$756$	0	0
$757$	876.359	1.15767	0.578837	−	0.815443i	$-0.303507\pi$
$757$	876.359	1.15767	0.578837	+	0.815443i	$0.303507\pi$
$758$	0	0
$759$	24.9477i	0.0328692i
$760$	0	0
$761$	217.575	0.285907	0.142953	−	0.989729i	$-0.454340\pi$
$761$	217.575	0.285907	0.142953	+	0.989729i	$0.454340\pi$
$762$	0	0
$763$	2638.19i	3.45765i
$764$	0	0
$765$	−238.642	−0.311950
$766$	0	0
$767$	− 943.840i	− 1.23056i
$768$	0	0
$769$	497.583	0.647053	0.323526	−	0.946219i	$-0.395132\pi$
$769$	497.583	0.647053	0.323526	+	0.946219i	$0.395132\pi$
$770$	0	0
$771$	114.424i	0.148410i
$772$	0	0
$773$	−664.490	−0.859624	−0.429812	−	0.902918i	$-0.641420\pi$
$773$	−664.490	−0.859624	−0.429812	+	0.902918i	$0.641420\pi$
$774$	0	0
$775$	− 686.170i	− 0.885380i
$776$	0	0
$777$	1039.37	1.33767
$778$	0	0
$779$	− 928.595i	− 1.19203i
$780$	0	0
$781$	−115.649	−0.148078
$782$	0	0
$783$	137.270i	0.175312i
$784$	0	0
$785$	364.074	0.463789
$786$	0	0
$787$	− 214.321i	− 0.272326i	−0.990686	−	0.136163i	$-0.956523\pi$
$787$	− 214.321i	− 0.272326i	0.990686	−	0.136163i	$-0.0434772\pi$
$788$	0	0
$789$	−162.104	−0.205456
$790$	0	0
$791$	2395.76i	3.02878i
$792$	0	0
$793$	550.266	0.693904
$794$	0	0
$795$	227.540i	0.286214i
$796$	0	0
$797$	−487.535	−0.611713	−0.305857	−	0.952078i	$-0.598943\pi$
$797$	−487.535	−0.611713	−0.305857	+	0.952078i	$0.598943\pi$
$798$	0	0
$799$	− 489.649i	− 0.612827i
$800$	0	0
$801$	−118.342	−0.147743
$802$	0	0
$803$	− 29.7974i	− 0.0371076i
$804$	0	0
$805$	82.2115	0.102126
$806$	0	0
$807$	− 529.225i	− 0.655793i
$808$	0	0
$809$	−1232.31	−1.52325	−0.761623	−	0.648021i	$-0.775597\pi$
$809$	−1232.31	−1.52325	−0.761623	+	0.648021i	$0.775597\pi$
$810$	0	0
$811$	− 38.1721i	− 0.0470679i	−0.999723	−	0.0235340i	$-0.992508\pi$
$811$	− 38.1721i	− 0.0470679i	0.999723	−	0.0235340i	$-0.00749178\pi$
$812$	0	0
$813$	−331.864	−0.408197
$814$	0	0
$815$	− 469.930i	− 0.576601i
$816$	0	0
$817$	534.640	0.654394
$818$	0	0
$819$	− 330.988i	− 0.404137i
$820$	0	0
$821$	−490.147	−0.597012	−0.298506	−	0.954408i	$-0.596488\pi$
$821$	−490.147	−0.597012	−0.298506	+	0.954408i	$0.596488\pi$
$822$	0	0
$823$	582.258i	0.707482i	0.935343	+	0.353741i	$0.115091\pi$
$823$	582.258i	0.707482i	−0.935343	+	0.353741i	$0.884909\pi$
$824$	0	0
$825$	−181.297	−0.219754
$826$	0	0
$827$	852.873i	1.03128i	0.856804	+	0.515642i	$0.172447\pi$
$827$	852.873i	1.03128i	−0.856804	+	0.515642i	$0.827553\pi$
$828$	0	0
$829$	−692.476	−0.835315	−0.417657	−	0.908605i	$-0.637149\pi$
$829$	−692.476	−0.835315	−0.417657	+	0.908605i	$0.637149\pi$
$830$	0	0
$831$	− 326.381i	− 0.392757i
$832$	0	0
$833$	3279.13	3.93653
$834$	0	0
$835$	− 259.952i	− 0.311320i
$836$	0	0
$837$	197.496	0.235957
$838$	0	0
$839$	− 761.022i	− 0.907059i	−0.891241	−	0.453529i	$-0.850165\pi$
$839$	− 761.022i	− 0.907059i	0.891241	−	0.453529i	$-0.149835\pi$
$840$	0	0
$841$	−143.113	−0.170170
$842$	0	0
$843$	− 512.837i	− 0.608347i
$844$	0	0
$845$	241.920	0.286296
$846$	0	0
$847$	1097.17i	1.29537i
$848$	0	0
$849$	451.513	0.531818
$850$	0	0
$851$	− 118.730i	− 0.139518i
$852$	0	0
$853$	1245.38	1.46000	0.729999	−	0.683448i	$-0.239520\pi$
$853$	1245.38	1.46000	0.729999	+	0.683448i	$0.239520\pi$
$854$	0	0
$855$	− 137.681i	− 0.161030i
$856$	0	0
$857$	−1517.00	−1.77013	−0.885064	−	0.465469i	$-0.845886\pi$
$857$	−1517.00	−1.77013	−0.885064	+	0.465469i	$0.845886\pi$
$858$	0	0
$859$	− 1311.75i	− 1.52706i	−0.645770	−	0.763532i	$-0.723463\pi$
$859$	− 1311.75i	− 1.52706i	0.645770	−	0.763532i	$-0.276537\pi$
$860$	0	0
$861$	−1159.77	−1.34701
$862$	0	0
$863$	− 1304.34i	− 1.51140i	−0.654920	−	0.755698i	$-0.727298\pi$
$863$	− 1304.34i	− 1.51140i	0.654920	−	0.755698i	$-0.272702\pi$
$864$	0	0
$865$	438.523	0.506963
$866$	0	0
$867$	1077.15i	1.24238i
$868$	0	0
$869$	−39.6203	−0.0455930
$870$	0	0
$871$	535.609i	0.614936i
$872$	0	0
$873$	181.783	0.208228
$874$	0	0
$875$	1424.76i	1.62830i
$876$	0	0
$877$	1207.07	1.37637	0.688183	−	0.725537i	$-0.258409\pi$
$877$	1207.07	1.37637	0.688183	+	0.725537i	$0.258409\pi$
$878$	0	0
$879$	− 545.174i	− 0.620221i
$880$	0	0
$881$	−375.974	−0.426758	−0.213379	−	0.976970i	$-0.568447\pi$
$881$	−375.974	−0.426758	−0.213379	+	0.976970i	$0.568447\pi$
$882$	0	0
$883$	395.955i	0.448420i	0.974541	+	0.224210i	$0.0719802\pi$
$883$	395.955i	0.448420i	−0.974541	+	0.224210i	$0.928020\pi$
$884$	0	0
$885$	−490.349	−0.554067
$886$	0	0
$887$	960.050i	1.08236i	0.840908	+	0.541178i	$0.182021\pi$
$887$	960.050i	1.08236i	−0.840908	+	0.541178i	$0.817979\pi$
$888$	0	0
$889$	−432.394	−0.486382
$890$	0	0
$891$	− 52.1816i	− 0.0585652i
$892$	0	0
$893$	282.496	0.316344
$894$	0	0
$895$	− 544.256i	− 0.608108i
$896$	0	0
$897$	−37.8095	−0.0421511
$898$	0	0
$899$	− 1004.08i	− 1.11689i
$900$	0	0
$901$	1504.32	1.66961
$902$	0	0
$903$	− 667.740i	− 0.739468i
$904$	0	0
$905$	477.926	0.528095
$906$	0	0
$907$	− 1301.18i	− 1.43460i	−0.696766	−	0.717298i	$-0.745378\pi$
$907$	− 1301.18i	− 1.43460i	0.696766	−	0.717298i	$-0.254622\pi$
$908$	0	0
$909$	507.229	0.558008
$910$	0	0
$911$	− 1274.60i	− 1.39912i	−0.714573	−	0.699561i	$-0.753379\pi$
$911$	− 1274.60i	− 1.39912i	0.714573	−	0.699561i	$-0.246621\pi$
$912$	0	0
$913$	923.111	1.01107
$914$	0	0
$915$	− 285.877i	− 0.312434i
$916$	0	0
$917$	1162.03	1.26721
$918$	0	0
$919$	− 957.109i	− 1.04147i	−0.853719	−	0.520734i	$-0.825658\pi$
$919$	− 957.109i	− 1.04147i	0.853719	−	0.520734i	$-0.174342\pi$
$920$	0	0
$921$	291.192	0.316169
$922$	0	0
$923$	− 175.272i	− 0.189894i
$924$	0	0
$925$	862.817	0.932775
$926$	0	0
$927$	− 490.170i	− 0.528770i
$928$	0	0
$929$	−912.989	−0.982765	−0.491383	−	0.870944i	$-0.663508\pi$
$929$	−912.989	−0.982765	−0.491383	+	0.870944i	$0.663508\pi$
$930$	0	0
$931$	1891.84i	2.03206i
$932$	0	0
$933$	810.657	0.868871
$934$	0	0
$935$	− 461.212i	− 0.493274i
$936$	0	0
$937$	751.958	0.802516	0.401258	−	0.915965i	$-0.368573\pi$
$937$	751.958	0.802516	0.401258	+	0.915965i	$0.368573\pi$
$938$	0	0
$939$	390.353i	0.415711i
$940$	0	0
$941$	−417.550	−0.443730	−0.221865	−	0.975077i	$-0.571214\pi$
$941$	−417.550	−0.443730	−0.221865	+	0.975077i	$0.571214\pi$
$942$	0	0
$943$	132.484i	0.140492i
$944$	0	0
$945$	−171.957	−0.181965
$946$	0	0
$947$	928.909i	0.980897i	0.871470	+	0.490448i	$0.163167\pi$
$947$	928.909i	0.980897i	−0.871470	+	0.490448i	$0.836833\pi$
$948$	0	0
$949$	45.1595	0.0475864
$950$	0	0
$951$	281.655i	0.296167i
$952$	0	0
$953$	676.685	0.710058	0.355029	−	0.934855i	$-0.384471\pi$
$953$	676.685	0.710058	0.355029	+	0.934855i	$0.384471\pi$
$954$	0	0
$955$	− 488.627i	− 0.511652i
$956$	0	0
$957$	−265.294	−0.277215
$958$	0	0
$959$	− 252.053i	− 0.262828i
$960$	0	0
$961$	−483.620	−0.503247
$962$	0	0
$963$	− 112.286i	− 0.116601i
$964$	0	0
$965$	−40.9686	−0.0424545
$966$	0	0
$967$	− 208.530i	− 0.215646i	−0.994170	−	0.107823i	$-0.965612\pi$
$967$	− 208.530i	− 0.215646i	0.994170	−	0.107823i	$-0.0343880\pi$
$968$	0	0
$969$	−910.236	−0.939356
$970$	0	0
$971$	932.159i	0.959999i	0.877269	+	0.480000i	$0.159363\pi$
$971$	932.159i	0.959999i	−0.877269	+	0.480000i	$0.840637\pi$
$972$	0	0
$973$	−879.608	−0.904016
$974$	0	0
$975$	− 274.765i	− 0.281810i
$976$	0	0
$977$	1567.22	1.60412	0.802059	−	0.597245i	$-0.203738\pi$
$977$	1567.22	1.60412	0.802059	+	0.597245i	$0.203738\pi$
$978$	0	0
$979$	− 228.714i	− 0.233620i
$980$	0	0
$981$	−630.350	−0.642559
$982$	0	0
$983$	− 1201.11i	− 1.22188i	−0.791676	−	0.610941i	$-0.790791\pi$
$983$	− 1201.11i	− 1.22188i	0.791676	−	0.610941i	$-0.209209\pi$
$984$	0	0
$985$	−301.689	−0.306283
$986$	0	0
$987$	− 352.824i	− 0.357471i
$988$	0	0
$989$	−76.2775	−0.0771259
$990$	0	0
$991$	568.813i	0.573979i	0.957934	+	0.286989i	$0.0926545\pi$
$991$	568.813i	0.573979i	−0.957934	+	0.286989i	$0.907346\pi$
$992$	0	0
$993$	440.854	0.443961
$994$	0	0
$995$	− 788.091i	− 0.792052i
$996$	0	0
$997$	−66.4659	−0.0666659	−0.0333330	−	0.999444i	$-0.510612\pi$
$997$	−66.4659	−0.0666659	−0.0333330	+	0.999444i	$0.510612\pi$
$998$	0	0
$999$	248.340i	0.248588i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.g.a.127.6	yes	8
3.2	odd	2		1152.3.g.f.127.6		8
4.3	odd	2	inner	384.3.g.a.127.2	✓	8
8.3	odd	2		384.3.g.b.127.7	yes	8
8.5	even	2		384.3.g.b.127.3	yes	8
12.11	even	2		1152.3.g.f.127.5		8
16.3	odd	4		768.3.b.e.127.7		8
16.5	even	4		768.3.b.e.127.6		8
16.11	odd	4		768.3.b.f.127.2		8
16.13	even	4		768.3.b.f.127.3		8
24.5	odd	2		1152.3.g.c.127.4		8
24.11	even	2		1152.3.g.c.127.3		8
48.5	odd	4		2304.3.b.t.127.6		8
48.11	even	4		2304.3.b.q.127.6		8
48.29	odd	4		2304.3.b.q.127.3		8
48.35	even	4		2304.3.b.t.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.2	✓	8	4.3	odd	2	inner
384.3.g.a.127.6	yes	8	1.1	even	1	trivial
384.3.g.b.127.3	yes	8	8.5	even	2
384.3.g.b.127.7	yes	8	8.3	odd	2
768.3.b.e.127.6		8	16.5	even	4
768.3.b.e.127.7		8	16.3	odd	4
768.3.b.f.127.2		8	16.11	odd	4
768.3.b.f.127.3		8	16.13	even	4
1152.3.g.c.127.3		8	24.11	even	2
1152.3.g.c.127.4		8	24.5	odd	2
1152.3.g.f.127.5		8	12.11	even	2
1152.3.g.f.127.6		8	3.2	odd	2
2304.3.b.q.127.3		8	48.29	odd	4
2304.3.b.q.127.6		8	48.11	even	4
2304.3.b.t.127.3		8	48.35	even	4
2304.3.b.t.127.6		8	48.5	odd	4

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -2.63567 q^{5} +12.5558i q^{7} -3.00000 q^{9} -5.79796i q^{11} +8.78710 q^{13} -4.56512i q^{15} -30.1810 q^{17} -17.4125i q^{19} -21.7473 q^{21} +2.48425i q^{23} -18.0532 q^{25} -5.19615i q^{27} -26.4175 q^{29} +38.0082i q^{31} +10.0424 q^{33} -33.0931i q^{35} -47.7930 q^{37} +15.2197i q^{39} +53.3294 q^{41} +30.7044i q^{43} +7.90702 q^{45} +16.2238i q^{47} -108.649 q^{49} -52.2750i q^{51} -49.8432 q^{53} +15.2815i q^{55} +30.1593 q^{57} -107.412i q^{59} +62.6220 q^{61} -37.6675i q^{63} -23.1599 q^{65} +60.9540i q^{67} -4.30285 q^{69} -19.9465i q^{71} +5.13929 q^{73} -31.2691i q^{75} +72.7982 q^{77} -6.83349i q^{79} +9.00000 q^{81} +159.213i q^{83} +79.5472 q^{85} -45.7565i q^{87} +39.4473 q^{89} +110.329i q^{91} -65.8321 q^{93} +45.8936i q^{95} -60.5944 q^{97} +17.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{5} - 24 q^{9}+O(q^{10}) \) 8 * q - 16 * q^5 - 24 * q^9	\( 8 q - 16 q^{5} - 24 q^{9} + 48 q^{13} + 16 q^{17} - 8 q^{25} - 80 q^{29} - 16 q^{37} + 80 q^{41} + 48 q^{45} - 88 q^{49} + 176 q^{53} + 96 q^{57} - 272 q^{61} - 160 q^{65} - 16 q^{73} + 320 q^{77} + 72 q^{81} + 32 q^{85} - 240 q^{89} - 192 q^{93} + 400 q^{97}+O(q^{100}) \) 8 * q - 16 * q^5 - 24 * q^9 + 48 * q^13 + 16 * q^17 - 8 * q^25 - 80 * q^29 - 16 * q^37 + 80 * q^41 + 48 * q^45 - 88 * q^49 + 176 * q^53 + 96 * q^57 - 272 * q^61 - 160 * q^65 - 16 * q^73 + 320 * q^77 + 72 * q^81 + 32 * q^85 - 240 * q^89 - 192 * q^93 + 400 * q^97

Introduction

L-functions

Modular forms

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