LMFDB - Embedded newform 384.3.g.a.127.8

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			127.8
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	384.127
Dual form			384.3.g.a.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} -13.7980i q^{11} +14.5266 q^{13} +7.43488i q^{15} +22.8673 q^{17} -16.0399i q^{19} +4.77678 q^{21} +17.1117i q^{23} -6.57420 q^{25} -5.19615i q^{27} -21.8667 q^{29} +38.6944i q^{31} +23.8988 q^{33} -11.8383i q^{35} +66.4204 q^{37} +25.1608i q^{39} +23.2392 q^{41} -47.9230i q^{43} -12.8776 q^{45} +14.8512i q^{47} +41.3941 q^{49} +39.6073i q^{51} +65.5589 q^{53} -59.2281i q^{55} +27.7819 q^{57} +65.8428i q^{59} -40.1123 q^{61} +8.27362i q^{63} +62.3559 q^{65} -74.8105i q^{67} -29.6383 q^{69} +122.681i q^{71} -144.904 q^{73} -11.3868i q^{75} -38.0530 q^{77} -128.657i q^{79} +9.00000 q^{81} -22.0417i q^{83} +98.1584 q^{85} -37.8743i q^{87} -122.075 q^{89} -40.0626i q^{91} -67.0208 q^{93} -68.8516i q^{95} -88.3072 q^{97} +41.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.29253	0.858506	0.429253	−	0.903184i	$-0.358777\pi$
$5$	4.29253	0.858506	0.429253	+	0.903184i	$0.358777\pi$
$6$	0	0
$7$	− 2.75787i	− 0.393982i	−0.980405	−	0.196991i	$-0.936883\pi$
$7$	− 2.75787i	− 0.393982i	0.980405	−	0.196991i	$-0.0631170\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 13.7980i	− 1.25436i	−0.778874	−	0.627180i	$-0.784209\pi$
$11$	− 13.7980i	− 1.25436i	0.778874	−	0.627180i	$-0.215791\pi$
$12$	0	0
$13$	14.5266	1.11743	0.558716	−	0.829359i	$-0.311294\pi$
$13$	14.5266	1.11743	0.558716	+	0.829359i	$0.311294\pi$
$14$	0	0
$15$	7.43488i	0.495659i
$16$	0	0
$17$	22.8673	1.34513	0.672567	−	0.740037i	$-0.265192\pi$
$17$	22.8673	1.34513	0.672567	+	0.740037i	$0.265192\pi$
$18$	0	0
$19$	− 16.0399i	− 0.844204i	−0.906548	−	0.422102i	$-0.861292\pi$
$19$	− 16.0399i	− 0.844204i	0.906548	−	0.422102i	$-0.138708\pi$
$20$	0	0
$21$	4.77678	0.227466
$22$	0	0
$23$	17.1117i	0.743986i	0.928236	+	0.371993i	$0.121325\pi$
$23$	17.1117i	0.743986i	−0.928236	+	0.371993i	$0.878675\pi$
$24$	0	0
$25$	−6.57420	−0.262968
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−21.8667	−0.754025	−0.377013	−	0.926208i	$-0.623049\pi$
$29$	−21.8667	−0.754025	−0.377013	+	0.926208i	$0.623049\pi$
$30$	0	0
$31$	38.6944i	1.24821i	0.781341	+	0.624104i	$0.214536\pi$
$31$	38.6944i	1.24821i	−0.781341	+	0.624104i	$0.785464\pi$
$32$	0	0
$33$	23.8988	0.724205
$34$	0	0
$35$	− 11.8383i	− 0.338236i
$36$	0	0
$37$	66.4204	1.79515	0.897573	−	0.440866i	$-0.145329\pi$
$37$	66.4204	1.79515	0.897573	+	0.440866i	$0.145329\pi$
$38$	0	0
$39$	25.1608i	0.645149i
$40$	0	0
$41$	23.2392	0.566809	0.283405	−	0.959000i	$-0.408536\pi$
$41$	23.2392	0.566809	0.283405	+	0.959000i	$0.408536\pi$
$42$	0	0
$43$	− 47.9230i	− 1.11449i	−0.830349	−	0.557244i	$-0.811859\pi$
$43$	− 47.9230i	− 1.11449i	0.830349	−	0.557244i	$-0.188141\pi$
$44$	0	0
$45$	−12.8776	−0.286169
$46$	0	0
$47$	14.8512i	0.315983i	0.987441	+	0.157991i	$0.0505018\pi$
$47$	14.8512i	0.315983i	−0.987441	+	0.157991i	$0.949498\pi$
$48$	0	0
$49$	41.3941	0.844778
$50$	0	0
$51$	39.6073i	0.776613i
$52$	0	0
$53$	65.5589	1.23696	0.618480	−	0.785800i	$-0.287749\pi$
$53$	65.5589	1.23696	0.618480	+	0.785800i	$0.287749\pi$
$54$	0	0
$55$	− 59.2281i	− 1.07688i
$56$	0	0
$57$	27.7819	0.487401
$58$	0	0
$59$	65.8428i	1.11598i	0.829848	+	0.557990i	$0.188427\pi$
$59$	65.8428i	1.11598i	−0.829848	+	0.557990i	$0.811573\pi$
$60$	0	0
$61$	−40.1123	−0.657579	−0.328790	−	0.944403i	$-0.606641\pi$
$61$	−40.1123	−0.657579	−0.328790	+	0.944403i	$0.606641\pi$
$62$	0	0
$63$	8.27362i	0.131327i
$64$	0	0
$65$	62.3559	0.959321
$66$	0	0
$67$	− 74.8105i	− 1.11657i	−0.829648	−	0.558287i	$-0.811459\pi$
$67$	− 74.8105i	− 1.11657i	0.829648	−	0.558287i	$-0.188541\pi$
$68$	0	0
$69$	−29.6383	−0.429540
$70$	0	0
$71$	122.681i	1.72790i	0.503578	+	0.863950i	$0.332017\pi$
$71$	122.681i	1.72790i	−0.503578	+	0.863950i	$0.667983\pi$
$72$	0	0
$73$	−144.904	−1.98498	−0.992492	−	0.122312i	$-0.960969\pi$
$73$	−144.904	−1.98498	−0.992492	+	0.122312i	$0.960969\pi$
$74$	0	0
$75$	− 11.3868i	− 0.151825i
$76$	0	0
$77$	−38.0530	−0.494195
$78$	0	0
$79$	− 128.657i	− 1.62857i	−0.580467	−	0.814284i	$-0.697130\pi$
$79$	− 128.657i	− 1.62857i	0.580467	−	0.814284i	$-0.302870\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 22.0417i	− 0.265563i	−0.991145	−	0.132781i	$-0.957609\pi$
$83$	− 22.0417i	− 0.265563i	0.991145	−	0.132781i	$-0.0423908\pi$
$84$	0	0
$85$	98.1584	1.15480
$86$	0	0
$87$	− 37.8743i	− 0.435337i
$88$	0	0
$89$	−122.075	−1.37163	−0.685813	−	0.727778i	$-0.740553\pi$
$89$	−122.075	−1.37163	−0.685813	+	0.727778i	$0.740553\pi$
$90$	0	0
$91$	− 40.0626i	− 0.440248i
$92$	0	0
$93$	−67.0208	−0.720653
$94$	0	0
$95$	− 68.8516i	− 0.724754i
$96$	0	0
$97$	−88.3072	−0.910384	−0.455192	−	0.890393i	$-0.650429\pi$
$97$	−88.3072	−0.910384	−0.455192	+	0.890393i	$0.650429\pi$
$98$	0	0
$99$	41.3939i	0.418120i
$100$	0	0
$101$	−104.345	−1.03312	−0.516560	−	0.856251i	$-0.672788\pi$
$101$	−104.345	−1.03312	−0.516560	+	0.856251i	$0.672788\pi$
$102$	0	0
$103$	− 69.0609i	− 0.670494i	−0.942130	−	0.335247i	$-0.891180\pi$
$103$	− 69.0609i	− 0.670494i	0.942130	−	0.335247i	$-0.108820\pi$
$104$	0	0
$105$	20.5045	0.195281
$106$	0	0
$107$	149.429i	1.39653i	0.715839	+	0.698265i	$0.246044\pi$
$107$	149.429i	1.39653i	−0.715839	+	0.698265i	$0.753956\pi$
$108$	0	0
$109$	54.3341	0.498478	0.249239	−	0.968442i	$-0.419820\pi$
$109$	54.3341	0.498478	0.249239	+	0.968442i	$0.419820\pi$
$110$	0	0
$111$	115.044i	1.03643i
$112$	0	0
$113$	6.32846	0.0560041	0.0280021	−	0.999608i	$-0.491086\pi$
$113$	6.32846	0.0560041	0.0280021	+	0.999608i	$0.491086\pi$
$114$	0	0
$115$	73.4523i	0.638716i
$116$	0	0
$117$	−43.5798	−0.372477
$118$	0	0
$119$	− 63.0651i	− 0.529958i
$120$	0	0
$121$	−69.3837	−0.573419
$122$	0	0
$123$	40.2514i	0.327247i
$124$	0	0
$125$	−135.533	−1.08427
$126$	0	0
$127$	− 16.5228i	− 0.130101i	−0.997882	−	0.0650504i	$-0.979279\pi$
$127$	− 16.5228i	− 0.130101i	0.997882	−	0.0650504i	$-0.0207208\pi$
$128$	0	0
$129$	83.0050	0.643450
$130$	0	0
$131$	50.9799i	0.389159i	0.980887	+	0.194580i	$0.0623343\pi$
$131$	50.9799i	0.389159i	−0.980887	+	0.194580i	$0.937666\pi$
$132$	0	0
$133$	−44.2360	−0.332601
$134$	0	0
$135$	− 22.3046i	− 0.165220i
$136$	0	0
$137$	10.0157	0.0731070	0.0365535	−	0.999332i	$-0.488362\pi$
$137$	10.0157	0.0731070	0.0365535	+	0.999332i	$0.488362\pi$
$138$	0	0
$139$	− 65.7088i	− 0.472725i	−0.971665	−	0.236363i	$-0.924045\pi$
$139$	− 65.7088i	− 0.472725i	0.971665	−	0.236363i	$-0.0759553\pi$
$140$	0	0
$141$	−25.7230	−0.182433
$142$	0	0
$143$	− 200.438i	− 1.40166i
$144$	0	0
$145$	−93.8635	−0.647335
$146$	0	0
$147$	71.6967i	0.487733i
$148$	0	0
$149$	−94.7716	−0.636051	−0.318026	−	0.948082i	$-0.603020\pi$
$149$	−94.7716	−0.636051	−0.318026	+	0.948082i	$0.603020\pi$
$150$	0	0
$151$	− 269.769i	− 1.78655i	−0.449508	−	0.893276i	$-0.648401\pi$
$151$	− 269.769i	− 1.78655i	0.449508	−	0.893276i	$-0.351599\pi$
$152$	0	0
$153$	−68.6018	−0.448378
$154$	0	0
$155$	166.097i	1.07159i
$156$	0	0
$157$	31.5058	0.200674	0.100337	−	0.994954i	$-0.468008\pi$
$157$	31.5058	0.200674	0.100337	+	0.994954i	$0.468008\pi$
$158$	0	0
$159$	113.551i	0.714159i
$160$	0	0
$161$	47.1918	0.293117
$162$	0	0
$163$	201.159i	1.23410i	0.786923	+	0.617051i	$0.211673\pi$
$163$	201.159i	1.23410i	−0.786923	+	0.617051i	$0.788327\pi$
$164$	0	0
$165$	102.586	0.621734
$166$	0	0
$167$	266.393i	1.59517i	0.603208	+	0.797584i	$0.293889\pi$
$167$	266.393i	1.59517i	−0.603208	+	0.797584i	$0.706111\pi$
$168$	0	0
$169$	42.0224	0.248653
$170$	0	0
$171$	48.1196i	0.281401i
$172$	0	0
$173$	−43.8456	−0.253443	−0.126721	−	0.991938i	$-0.540445\pi$
$173$	−43.8456	−0.253443	−0.126721	+	0.991938i	$0.540445\pi$
$174$	0	0
$175$	18.1308i	0.103605i
$176$	0	0
$177$	−114.043	−0.644311
$178$	0	0
$179$	− 275.778i	− 1.54066i	−0.637646	−	0.770330i	$-0.720092\pi$
$179$	− 275.778i	− 1.54066i	0.637646	−	0.770330i	$-0.279908\pi$
$180$	0	0
$181$	−79.0033	−0.436482	−0.218241	−	0.975895i	$-0.570032\pi$
$181$	−79.0033	−0.436482	−0.218241	+	0.975895i	$0.570032\pi$
$182$	0	0
$183$	− 69.4766i	− 0.379654i
$184$	0	0
$185$	285.111	1.54114
$186$	0	0
$187$	− 315.522i	− 1.68728i
$188$	0	0
$189$	−14.3303	−0.0758219
$190$	0	0
$191$	142.409i	0.745598i	0.927912	+	0.372799i	$0.121602\pi$
$191$	142.409i	0.745598i	−0.927912	+	0.372799i	$0.878398\pi$
$192$	0	0
$193$	−277.818	−1.43947	−0.719736	−	0.694248i	$-0.755737\pi$
$193$	−277.818	−1.43947	−0.719736	+	0.694248i	$0.755737\pi$
$194$	0	0
$195$	108.004i	0.553864i
$196$	0	0
$197$	3.40848	0.0173019	0.00865095	−	0.999963i	$-0.497246\pi$
$197$	3.40848	0.0173019	0.00865095	+	0.999963i	$0.497246\pi$
$198$	0	0
$199$	314.323i	1.57951i	0.613421	+	0.789756i	$0.289793\pi$
$199$	314.323i	1.57951i	−0.613421	+	0.789756i	$0.710207\pi$
$200$	0	0
$201$	129.576	0.644654
$202$	0	0
$203$	60.3057i	0.297072i
$204$	0	0
$205$	99.7548	0.486609
$206$	0	0
$207$	− 51.3350i	− 0.247995i
$208$	0	0
$209$	−221.317	−1.05894
$210$	0	0
$211$	1.54933i	0.00734281i	0.999993	+	0.00367140i	$0.00116865\pi$
$211$	1.54933i	0.00734281i	−0.999993	+	0.00367140i	$0.998831\pi$
$212$	0	0
$213$	−212.490	−0.997603
$214$	0	0
$215$	− 205.711i	− 0.956794i
$216$	0	0
$217$	106.714	0.491772
$218$	0	0
$219$	− 250.981i	− 1.14603i
$220$	0	0
$221$	332.184	1.50309
$222$	0	0
$223$	148.964i	0.668001i	0.942573	+	0.334000i	$0.108399\pi$
$223$	148.964i	0.668001i	−0.942573	+	0.334000i	$0.891601\pi$
$224$	0	0
$225$	19.7226	0.0876560
$226$	0	0
$227$	− 237.803i	− 1.04759i	−0.851844	−	0.523796i	$-0.824515\pi$
$227$	− 237.803i	− 1.04759i	0.851844	−	0.523796i	$-0.175485\pi$
$228$	0	0
$229$	70.0590	0.305935	0.152967	−	0.988231i	$-0.451117\pi$
$229$	70.0590	0.305935	0.152967	+	0.988231i	$0.451117\pi$
$230$	0	0
$231$	− 65.9098i	− 0.285324i
$232$	0	0
$233$	7.18107	0.0308200	0.0154100	−	0.999881i	$-0.495095\pi$
$233$	7.18107	0.0308200	0.0154100	+	0.999881i	$0.495095\pi$
$234$	0	0
$235$	63.7491i	0.271273i
$236$	0	0
$237$	222.840	0.940254
$238$	0	0
$239$	− 216.620i	− 0.906359i	−0.891419	−	0.453180i	$-0.850290\pi$
$239$	− 216.620i	− 0.906359i	0.891419	−	0.453180i	$-0.149710\pi$
$240$	0	0
$241$	385.001	1.59751	0.798757	−	0.601653i	$-0.205491\pi$
$241$	385.001	1.59751	0.798757	+	0.601653i	$0.205491\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	177.685	0.725247
$246$	0	0
$247$	− 233.005i	− 0.943340i
$248$	0	0
$249$	38.1774	0.153323
$250$	0	0
$251$	− 85.4882i	− 0.340590i	−0.985393	−	0.170295i	$-0.945528\pi$
$251$	− 85.4882i	− 0.340590i	0.985393	−	0.170295i	$-0.0544721\pi$
$252$	0	0
$253$	236.106	0.933226
$254$	0	0
$255$	170.015i	0.666727i
$256$	0	0
$257$	−54.2981	−0.211277	−0.105638	−	0.994405i	$-0.533689\pi$
$257$	−54.2981	−0.211277	−0.105638	+	0.994405i	$0.533689\pi$
$258$	0	0
$259$	− 183.179i	− 0.707255i
$260$	0	0
$261$	65.6002	0.251342
$262$	0	0
$263$	501.827i	1.90809i	0.299670	+	0.954043i	$0.403123\pi$
$263$	501.827i	1.90809i	−0.299670	+	0.954043i	$0.596877\pi$
$264$	0	0
$265$	281.413	1.06194
$266$	0	0
$267$	− 211.440i	− 0.791909i
$268$	0	0
$269$	284.911	1.05915	0.529574	−	0.848264i	$-0.322352\pi$
$269$	284.911	1.05915	0.529574	+	0.848264i	$0.322352\pi$
$270$	0	0
$271$	241.190i	0.889998i	0.895531	+	0.444999i	$0.146796\pi$
$271$	241.190i	0.889998i	−0.895531	+	0.444999i	$0.853204\pi$
$272$	0	0
$273$	69.3904	0.254177
$274$	0	0
$275$	90.7105i	0.329856i
$276$	0	0
$277$	−242.877	−0.876813	−0.438407	−	0.898777i	$-0.644457\pi$
$277$	−242.877	−0.876813	−0.438407	+	0.898777i	$0.644457\pi$
$278$	0	0
$279$	− 116.083i	− 0.416069i
$280$	0	0
$281$	105.341	0.374880	0.187440	−	0.982276i	$-0.439981\pi$
$281$	105.341	0.374880	0.187440	+	0.982276i	$0.439981\pi$
$282$	0	0
$283$	223.867i	0.791049i	0.918455	+	0.395525i	$0.129437\pi$
$283$	223.867i	0.791049i	−0.918455	+	0.395525i	$0.870563\pi$
$284$	0	0
$285$	119.254	0.418437
$286$	0	0
$287$	− 64.0907i	− 0.223313i
$288$	0	0
$289$	233.912	0.809384
$290$	0	0
$291$	− 152.953i	− 0.525610i
$292$	0	0
$293$	−421.057	−1.43705	−0.718527	−	0.695499i	$-0.755183\pi$
$293$	−421.057	−1.43705	−0.718527	+	0.695499i	$0.755183\pi$
$294$	0	0
$295$	282.632i	0.958075i
$296$	0	0
$297$	−71.6963	−0.241402
$298$	0	0
$299$	248.575i	0.831353i
$300$	0	0
$301$	−132.166	−0.439088
$302$	0	0
$303$	− 180.731i	− 0.596472i
$304$	0	0
$305$	−172.183	−0.564536
$306$	0	0
$307$	− 122.865i	− 0.400211i	−0.979774	−	0.200106i	$-0.935871\pi$
$307$	− 122.865i	− 0.400211i	0.979774	−	0.200106i	$-0.0641285\pi$
$308$	0	0
$309$	119.617	0.387110
$310$	0	0
$311$	437.535i	1.40686i	0.710762	+	0.703432i	$0.248350\pi$
$311$	437.535i	1.40686i	−0.710762	+	0.703432i	$0.751650\pi$
$312$	0	0
$313$	375.413	1.19940	0.599702	−	0.800223i	$-0.295286\pi$
$313$	375.413	1.19940	0.599702	+	0.800223i	$0.295286\pi$
$314$	0	0
$315$	35.5148i	0.112745i
$316$	0	0
$317$	388.867	1.22671	0.613355	−	0.789808i	$-0.289820\pi$
$317$	388.867	1.22671	0.613355	+	0.789808i	$0.289820\pi$
$318$	0	0
$319$	301.716i	0.945819i
$320$	0	0
$321$	−258.818	−0.806287
$322$	0	0
$323$	− 366.788i	− 1.13557i
$324$	0	0
$325$	−95.5008	−0.293849
$326$	0	0
$327$	94.1094i	0.287796i
$328$	0	0
$329$	40.9577	0.124491
$330$	0	0
$331$	− 15.4690i	− 0.0467341i	−0.999727	−	0.0233670i	$-0.992561\pi$
$331$	− 15.4690i	− 0.0467341i	0.999727	−	0.0233670i	$-0.00743864\pi$
$332$	0	0
$333$	−199.261	−0.598382
$334$	0	0
$335$	− 321.126i	− 0.958585i
$336$	0	0
$337$	−88.0105	−0.261159	−0.130579	−	0.991438i	$-0.541684\pi$
$337$	−88.0105	−0.261159	−0.130579	+	0.991438i	$0.541684\pi$
$338$	0	0
$339$	10.9612i	0.0323340i
$340$	0	0
$341$	533.904	1.56570
$342$	0	0
$343$	− 249.296i	− 0.726810i
$344$	0	0
$345$	−127.223	−0.368763
$346$	0	0
$347$	− 183.589i	− 0.529076i	−0.964375	−	0.264538i	$-0.914781\pi$
$347$	− 183.589i	− 0.529076i	0.964375	−	0.264538i	$-0.0852194\pi$
$348$	0	0
$349$	−242.556	−0.695003	−0.347501	−	0.937679i	$-0.612970\pi$
$349$	−242.556	−0.695003	−0.347501	+	0.937679i	$0.612970\pi$
$350$	0	0
$351$	− 75.4825i	− 0.215050i
$352$	0	0
$353$	10.2850	0.0291360	0.0145680	−	0.999894i	$-0.495363\pi$
$353$	10.2850	0.0291360	0.0145680	+	0.999894i	$0.495363\pi$
$354$	0	0
$355$	526.611i	1.48341i
$356$	0	0
$357$	109.232	0.305972
$358$	0	0
$359$	− 137.976i	− 0.384334i	−0.981362	−	0.192167i	$-0.938448\pi$
$359$	− 137.976i	− 0.384334i	0.981362	−	0.192167i	$-0.0615515\pi$
$360$	0	0
$361$	103.723	0.287320
$362$	0	0
$363$	− 120.176i	− 0.331063i
$364$	0	0
$365$	−622.004	−1.70412
$366$	0	0
$367$	400.180i	1.09041i	0.838303	+	0.545205i	$0.183548\pi$
$367$	400.180i	1.09041i	−0.838303	+	0.545205i	$0.816452\pi$
$368$	0	0
$369$	−69.7175	−0.188936
$370$	0	0
$371$	− 180.803i	− 0.487340i
$372$	0	0
$373$	−644.881	−1.72890	−0.864451	−	0.502717i	$-0.832334\pi$
$373$	−644.881	−1.72890	−0.864451	+	0.502717i	$0.832334\pi$
$374$	0	0
$375$	− 234.750i	− 0.626001i
$376$	0	0
$377$	−317.649	−0.842571
$378$	0	0
$379$	485.900i	1.28206i	0.767517	+	0.641029i	$0.221492\pi$
$379$	485.900i	1.28206i	−0.767517	+	0.641029i	$0.778508\pi$
$380$	0	0
$381$	28.6183	0.0751137
$382$	0	0
$383$	− 168.828i	− 0.440804i	−0.975409	−	0.220402i	$-0.929263\pi$
$383$	− 168.828i	− 0.440804i	0.975409	−	0.220402i	$-0.0707370\pi$
$384$	0	0
$385$	−163.344	−0.424270
$386$	0	0
$387$	143.769i	0.371496i
$388$	0	0
$389$	−192.592	−0.495095	−0.247548	−	0.968876i	$-0.579625\pi$
$389$	−192.592	−0.495095	−0.247548	+	0.968876i	$0.579625\pi$
$390$	0	0
$391$	391.297i	1.00076i
$392$	0	0
$393$	−88.2998	−0.224681
$394$	0	0
$395$	− 552.263i	− 1.39813i
$396$	0	0
$397$	−62.0483	−0.156293	−0.0781465	−	0.996942i	$-0.524900\pi$
$397$	−62.0483	−0.156293	−0.0781465	+	0.996942i	$0.524900\pi$
$398$	0	0
$399$	− 76.6189i	− 0.192027i
$400$	0	0
$401$	353.066	0.880464	0.440232	−	0.897884i	$-0.354896\pi$
$401$	353.066	0.880464	0.440232	+	0.897884i	$0.354896\pi$
$402$	0	0
$403$	562.099i	1.39479i
$404$	0	0
$405$	38.6328	0.0953895
$406$	0	0
$407$	− 916.466i	− 2.25176i
$408$	0	0
$409$	−62.9725	−0.153967	−0.0769835	−	0.997032i	$-0.524529\pi$
$409$	−62.9725	−0.153967	−0.0769835	+	0.997032i	$0.524529\pi$
$410$	0	0
$411$	17.3476i	0.0422084i
$412$	0	0
$413$	181.586	0.439676
$414$	0	0
$415$	− 94.6146i	− 0.227987i
$416$	0	0
$417$	113.811	0.272928
$418$	0	0
$419$	398.741i	0.951650i	0.879540	+	0.475825i	$0.157850\pi$
$419$	398.741i	0.951650i	−0.879540	+	0.475825i	$0.842150\pi$
$420$	0	0
$421$	533.059	1.26617	0.633086	−	0.774081i	$-0.281788\pi$
$421$	533.059	1.26617	0.633086	+	0.774081i	$0.281788\pi$
$422$	0	0
$423$	− 44.5535i	− 0.105328i
$424$	0	0
$425$	−150.334	−0.353727
$426$	0	0
$427$	110.625i	0.259075i
$428$	0	0
$429$	347.168	0.809250
$430$	0	0
$431$	− 284.282i	− 0.659587i	−0.944053	−	0.329793i	$-0.893021\pi$
$431$	− 284.282i	− 0.659587i	0.944053	−	0.329793i	$-0.106979\pi$
$432$	0	0
$433$	−304.599	−0.703462	−0.351731	−	0.936101i	$-0.614407\pi$
$433$	−304.599	−0.703462	−0.351731	+	0.936101i	$0.614407\pi$
$434$	0	0
$435$	− 162.576i	− 0.373739i
$436$	0	0
$437$	274.469	0.628075
$438$	0	0
$439$	− 204.615i	− 0.466094i	−0.972465	−	0.233047i	$-0.925130\pi$
$439$	− 204.615i	− 0.466094i	0.972465	−	0.233047i	$-0.0748696\pi$
$440$	0	0
$441$	−124.182	−0.281593
$442$	0	0
$443$	625.828i	1.41270i	0.707861	+	0.706352i	$0.249660\pi$
$443$	625.828i	1.41270i	−0.707861	+	0.706352i	$0.750340\pi$
$444$	0	0
$445$	−524.009	−1.17755
$446$	0	0
$447$	− 164.149i	− 0.367224i
$448$	0	0
$449$	−557.532	−1.24172	−0.620860	−	0.783921i	$-0.713217\pi$
$449$	−557.532	−1.24172	−0.620860	+	0.783921i	$0.713217\pi$
$450$	0	0
$451$	− 320.653i	− 0.710983i
$452$	0	0
$453$	467.254	1.03147
$454$	0	0
$455$	− 171.970i	− 0.377955i
$456$	0	0
$457$	−312.142	−0.683024	−0.341512	−	0.939877i	$-0.610939\pi$
$457$	−312.142	−0.683024	−0.341512	+	0.939877i	$0.610939\pi$
$458$	0	0
$459$	− 118.822i	− 0.258871i
$460$	0	0
$461$	−603.187	−1.30843	−0.654216	−	0.756308i	$-0.727001\pi$
$461$	−603.187	−1.30843	−0.654216	+	0.756308i	$0.727001\pi$
$462$	0	0
$463$	− 707.866i	− 1.52887i	−0.644702	−	0.764434i	$-0.723019\pi$
$463$	− 707.866i	− 1.52887i	0.644702	−	0.764434i	$-0.276981\pi$
$464$	0	0
$465$	−287.689	−0.618685
$466$	0	0
$467$	195.220i	0.418031i	0.977912	+	0.209015i	$0.0670259\pi$
$467$	195.220i	0.418031i	−0.977912	+	0.209015i	$0.932974\pi$
$468$	0	0
$469$	−206.318	−0.439910
$470$	0	0
$471$	54.5696i	0.115859i
$472$	0	0
$473$	−661.239	−1.39797
$474$	0	0
$475$	105.449i	0.221998i
$476$	0	0
$477$	−196.677	−0.412320
$478$	0	0
$479$	233.729i	0.487952i	0.969781	+	0.243976i	$0.0784517\pi$
$479$	233.729i	0.487952i	−0.969781	+	0.243976i	$0.921548\pi$
$480$	0	0
$481$	964.863	2.00595
$482$	0	0
$483$	81.7387i	0.169231i
$484$	0	0
$485$	−379.061	−0.781569
$486$	0	0
$487$	224.058i	0.460078i	0.973181	+	0.230039i	$0.0738854\pi$
$487$	224.058i	0.460078i	−0.973181	+	0.230039i	$0.926115\pi$
$488$	0	0
$489$	−348.417	−0.712510
$490$	0	0
$491$	194.772i	0.396684i	0.980133	+	0.198342i	$0.0635556\pi$
$491$	194.772i	0.396684i	−0.980133	+	0.198342i	$0.936444\pi$
$492$	0	0
$493$	−500.032	−1.01426
$494$	0	0
$495$	177.684i	0.358958i
$496$	0	0
$497$	338.339	0.680762
$498$	0	0
$499$	99.7462i	0.199892i	0.994993	+	0.0999461i	$0.0318670\pi$
$499$	99.7462i	0.199892i	−0.994993	+	0.0999461i	$0.968133\pi$
$500$	0	0
$501$	−461.406	−0.920970
$502$	0	0
$503$	− 772.172i	− 1.53513i	−0.640969	−	0.767567i	$-0.721467\pi$
$503$	− 772.172i	− 1.53513i	0.640969	−	0.767567i	$-0.278533\pi$
$504$	0	0
$505$	−447.904	−0.886939
$506$	0	0
$507$	72.7849i	0.143560i
$508$	0	0
$509$	−292.306	−0.574276	−0.287138	−	0.957889i	$-0.592704\pi$
$509$	−292.306	−0.574276	−0.287138	+	0.957889i	$0.592704\pi$
$510$	0	0
$511$	399.627i	0.782048i
$512$	0	0
$513$	−83.3456	−0.162467
$514$	0	0
$515$	− 296.446i	− 0.575623i
$516$	0	0
$517$	204.916	0.396356
$518$	0	0
$519$	− 75.9428i	− 0.146325i
$520$	0	0
$521$	−412.035	−0.790853	−0.395427	−	0.918498i	$-0.629403\pi$
$521$	−412.035	−0.790853	−0.395427	+	0.918498i	$0.629403\pi$
$522$	0	0
$523$	− 124.693i	− 0.238420i	−0.992869	−	0.119210i	$-0.961964\pi$
$523$	− 124.693i	− 0.238420i	0.992869	−	0.119210i	$-0.0380361\pi$
$524$	0	0
$525$	−31.4035	−0.0598162
$526$	0	0
$527$	884.836i	1.67901i
$528$	0	0
$529$	236.191	0.446486
$530$	0	0
$531$	− 197.528i	− 0.371993i
$532$	0	0
$533$	337.586	0.633370
$534$	0	0
$535$	641.427i	1.19893i
$536$	0	0
$537$	477.662	0.889500
$538$	0	0
$539$	− 571.154i	− 1.05966i
$540$	0	0
$541$	−572.988	−1.05913	−0.529564	−	0.848270i	$-0.677644\pi$
$541$	−572.988	−1.05913	−0.529564	+	0.848270i	$0.677644\pi$
$542$	0	0
$543$	− 136.838i	− 0.252003i
$544$	0	0
$545$	233.231	0.427946
$546$	0	0
$547$	682.433i	1.24759i	0.781587	+	0.623796i	$0.214410\pi$
$547$	682.433i	1.24759i	−0.781587	+	0.623796i	$0.785590\pi$
$548$	0	0
$549$	120.337	0.219193
$550$	0	0
$551$	350.739i	0.636551i
$552$	0	0
$553$	−354.820	−0.641627
$554$	0	0
$555$	493.828i	0.889779i
$556$	0	0
$557$	856.125	1.53703	0.768515	−	0.639832i	$-0.220996\pi$
$557$	856.125	1.53703	0.768515	+	0.639832i	$0.220996\pi$
$558$	0	0
$559$	− 696.158i	− 1.24536i
$560$	0	0
$561$	546.499	0.974152
$562$	0	0
$563$	680.745i	1.20914i	0.796552	+	0.604570i	$0.206655\pi$
$563$	680.745i	1.20914i	−0.796552	+	0.604570i	$0.793345\pi$
$564$	0	0
$565$	27.1651	0.0480798
$566$	0	0
$567$	− 24.8209i	− 0.0437758i
$568$	0	0
$569$	343.401	0.603516	0.301758	−	0.953384i	$-0.402426\pi$
$569$	343.401	0.603516	0.301758	+	0.953384i	$0.402426\pi$
$570$	0	0
$571$	− 329.137i	− 0.576422i	−0.957567	−	0.288211i	$-0.906940\pi$
$571$	− 329.137i	− 0.576422i	0.957567	−	0.288211i	$-0.0930604\pi$
$572$	0	0
$573$	−246.660	−0.430471
$574$	0	0
$575$	− 112.495i	− 0.195644i
$576$	0	0
$577$	734.738	1.27338	0.636688	−	0.771122i	$-0.280304\pi$
$577$	734.738	1.27338	0.636688	+	0.771122i	$0.280304\pi$
$578$	0	0
$579$	− 481.195i	− 0.831079i
$580$	0	0
$581$	−60.7883	−0.104627
$582$	0	0
$583$	− 904.579i	− 1.55159i
$584$	0	0
$585$	−187.068	−0.319774
$586$	0	0
$587$	− 433.815i	− 0.739037i	−0.929223	−	0.369519i	$-0.879523\pi$
$587$	− 433.815i	− 0.739037i	0.929223	−	0.369519i	$-0.120477\pi$
$588$	0	0
$589$	620.654	1.05374
$590$	0	0
$591$	5.90365i	0.00998926i
$592$	0	0
$593$	−425.075	−0.716822	−0.358411	−	0.933564i	$-0.616681\pi$
$593$	−425.075	−0.716822	−0.358411	+	0.933564i	$0.616681\pi$
$594$	0	0
$595$	− 270.709i	− 0.454972i
$596$	0	0
$597$	−544.424	−0.911932
$598$	0	0
$599$	1048.73i	1.75081i	0.483394	+	0.875403i	$0.339404\pi$
$599$	1048.73i	1.75081i	−0.483394	+	0.875403i	$0.660596\pi$
$600$	0	0
$601$	−478.816	−0.796699	−0.398349	−	0.917234i	$-0.630417\pi$
$601$	−478.816	−0.796699	−0.398349	+	0.917234i	$0.630417\pi$
$602$	0	0
$603$	224.431i	0.372191i
$604$	0	0
$605$	−297.831	−0.492283
$606$	0	0
$607$	− 18.7009i	− 0.0308088i	−0.999881	−	0.0154044i	$-0.995096\pi$
$607$	− 18.7009i	− 0.0308088i	0.999881	−	0.0154044i	$-0.00490356\pi$
$608$	0	0
$609$	−104.453	−0.171515
$610$	0	0
$611$	215.737i	0.353089i
$612$	0	0
$613$	314.964	0.513807	0.256904	−	0.966437i	$-0.417298\pi$
$613$	314.964	0.513807	0.256904	+	0.966437i	$0.417298\pi$
$614$	0	0
$615$	172.780i	0.280944i
$616$	0	0
$617$	786.590	1.27486	0.637431	−	0.770507i	$-0.279997\pi$
$617$	786.590	1.27486	0.637431	+	0.770507i	$0.279997\pi$
$618$	0	0
$619$	505.431i	0.816528i	0.912864	+	0.408264i	$0.133866\pi$
$619$	505.431i	0.816528i	−0.912864	+	0.408264i	$0.866134\pi$
$620$	0	0
$621$	88.9148	0.143180
$622$	0	0
$623$	336.667i	0.540396i
$624$	0	0
$625$	−417.425	−0.667880
$626$	0	0
$627$	− 383.333i	− 0.611377i
$628$	0	0
$629$	1518.85	2.41471
$630$	0	0
$631$	682.834i	1.08215i	0.840976	+	0.541073i	$0.181982\pi$
$631$	682.834i	1.08215i	−0.840976	+	0.541073i	$0.818018\pi$
$632$	0	0
$633$	−2.68352	−0.00423937
$634$	0	0
$635$	− 70.9246i	− 0.111692i
$636$	0	0
$637$	601.316	0.943982
$638$	0	0
$639$	− 368.043i	− 0.575967i
$640$	0	0
$641$	816.719	1.27413	0.637066	−	0.770809i	$-0.280148\pi$
$641$	816.719	1.27413	0.637066	+	0.770809i	$0.280148\pi$
$642$	0	0
$643$	− 1164.05i	− 1.81034i	−0.425045	−	0.905172i	$-0.639742\pi$
$643$	− 1164.05i	− 1.81034i	0.425045	−	0.905172i	$-0.360258\pi$
$644$	0	0
$645$	356.301	0.552405
$646$	0	0
$647$	− 471.519i	− 0.728778i	−0.931247	−	0.364389i	$-0.881278\pi$
$647$	− 471.519i	− 0.728778i	0.931247	−	0.364389i	$-0.118722\pi$
$648$	0	0
$649$	908.496	1.39984
$650$	0	0
$651$	184.835i	0.283924i
$652$	0	0
$653$	−1187.25	−1.81814	−0.909071	−	0.416642i	$-0.863207\pi$
$653$	−1187.25	−1.81814	−0.909071	+	0.416642i	$0.863207\pi$
$654$	0	0
$655$	218.833i	0.334096i
$656$	0	0
$657$	434.711	0.661661
$658$	0	0
$659$	245.640i	0.372747i	0.982479	+	0.186373i	$0.0596734\pi$
$659$	245.640i	0.372747i	−0.982479	+	0.186373i	$0.940327\pi$
$660$	0	0
$661$	1244.70	1.88305	0.941527	−	0.336936i	$-0.109391\pi$
$661$	1244.70	1.88305	0.941527	+	0.336936i	$0.109391\pi$
$662$	0	0
$663$	575.359i	0.867812i
$664$	0	0
$665$	−189.884	−0.285540
$666$	0	0
$667$	− 374.176i	− 0.560984i
$668$	0	0
$669$	−258.013	−0.385670
$670$	0	0
$671$	553.468i	0.824841i
$672$	0	0
$673$	−852.278	−1.26639	−0.633193	−	0.773994i	$-0.718256\pi$
$673$	−852.278	−1.26639	−0.633193	+	0.773994i	$0.718256\pi$
$674$	0	0
$675$	34.1605i	0.0506082i
$676$	0	0
$677$	−555.765	−0.820923	−0.410461	−	0.911878i	$-0.634632\pi$
$677$	−555.765	−0.820923	−0.410461	+	0.911878i	$0.634632\pi$
$678$	0	0
$679$	243.540i	0.358675i
$680$	0	0
$681$	411.888	0.604828
$682$	0	0
$683$	335.241i	0.490836i	0.969417	+	0.245418i	$0.0789253\pi$
$683$	335.241i	0.490836i	−0.969417	+	0.245418i	$0.921075\pi$
$684$	0	0
$685$	42.9925	0.0627628
$686$	0	0
$687$	121.346i	0.176631i
$688$	0	0
$689$	952.349	1.38222
$690$	0	0
$691$	− 1274.20i	− 1.84400i	−0.387192	−	0.921999i	$-0.626555\pi$
$691$	− 1274.20i	− 1.84400i	0.387192	−	0.921999i	$-0.373445\pi$
$692$	0	0
$693$	114.159	0.164732
$694$	0	0
$695$	− 282.057i	− 0.405837i
$696$	0	0
$697$	531.416	0.762434
$698$	0	0
$699$	12.4380i	0.0177940i
$700$	0	0
$701$	676.056	0.964416	0.482208	−	0.876057i	$-0.339835\pi$
$701$	676.056	0.964416	0.482208	+	0.876057i	$0.339835\pi$
$702$	0	0
$703$	− 1065.37i	− 1.51547i
$704$	0	0
$705$	−110.417	−0.156619
$706$	0	0
$707$	287.771i	0.407031i
$708$	0	0
$709$	−62.3418	−0.0879292	−0.0439646	−	0.999033i	$-0.513999\pi$
$709$	−62.3418	−0.0879292	−0.0439646	+	0.999033i	$0.513999\pi$
$710$	0	0
$711$	385.971i	0.542856i
$712$	0	0
$713$	−662.127	−0.928649
$714$	0	0
$715$	− 860.384i	− 1.20333i
$716$	0	0
$717$	375.197	0.523287
$718$	0	0
$719$	322.341i	0.448318i	0.974553	+	0.224159i	$0.0719636\pi$
$719$	322.341i	0.448318i	−0.974553	+	0.224159i	$0.928036\pi$
$720$	0	0
$721$	−190.461	−0.264163
$722$	0	0
$723$	666.841i	0.922326i
$724$	0	0
$725$	143.756	0.198284
$726$	0	0
$727$	368.070i	0.506287i	0.967429	+	0.253143i	$0.0814644\pi$
$727$	368.070i	0.506287i	−0.967429	+	0.253143i	$0.918536\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1095.87i	− 1.49913i
$732$	0	0
$733$	430.587	0.587431	0.293715	−	0.955893i	$-0.405108\pi$
$733$	430.587	0.587431	0.293715	+	0.955893i	$0.405108\pi$
$734$	0	0
$735$	307.760i	0.418721i
$736$	0	0
$737$	−1032.23	−1.40059
$738$	0	0
$739$	− 818.050i	− 1.10697i	−0.832859	−	0.553485i	$-0.813298\pi$
$739$	− 818.050i	− 1.10697i	0.832859	−	0.553485i	$-0.186702\pi$
$740$	0	0
$741$	403.576	0.544638
$742$	0	0
$743$	− 685.438i	− 0.922528i	−0.887263	−	0.461264i	$-0.847396\pi$
$743$	− 685.438i	− 0.922528i	0.887263	−	0.461264i	$-0.152604\pi$
$744$	0	0
$745$	−406.810	−0.546053
$746$	0	0
$747$	66.1251i	0.0885209i
$748$	0	0
$749$	412.106	0.550208
$750$	0	0
$751$	− 104.336i	− 0.138929i	−0.997584	−	0.0694647i	$-0.977871\pi$
$751$	− 104.336i	− 0.138929i	0.997584	−	0.0694647i	$-0.0221291\pi$
$752$	0	0
$753$	148.070	0.196640
$754$	0	0
$755$	− 1157.99i	− 1.53377i
$756$	0	0
$757$	−539.949	−0.713274	−0.356637	−	0.934243i	$-0.616077\pi$
$757$	−539.949	−0.713274	−0.356637	+	0.934243i	$0.616077\pi$
$758$	0	0
$759$	408.948i	0.538798i
$760$	0	0
$761$	1192.21	1.56663	0.783317	−	0.621623i	$-0.213526\pi$
$761$	1192.21	1.56663	0.783317	+	0.621623i	$0.213526\pi$
$762$	0	0
$763$	− 149.847i	− 0.196391i
$764$	0	0
$765$	−294.475	−0.384935
$766$	0	0
$767$	956.473i	1.24703i
$768$	0	0
$769$	−321.506	−0.418083	−0.209042	−	0.977907i	$-0.567034\pi$
$769$	−321.506	−0.418083	−0.209042	+	0.977907i	$0.567034\pi$
$770$	0	0
$771$	− 94.0471i	− 0.121981i
$772$	0	0
$773$	−1453.29	−1.88006	−0.940029	−	0.341094i	$-0.889202\pi$
$773$	−1453.29	−1.88006	−0.940029	+	0.341094i	$0.889202\pi$
$774$	0	0
$775$	− 254.385i	− 0.328239i
$776$	0	0
$777$	317.276	0.408334
$778$	0	0
$779$	− 372.753i	− 0.478502i
$780$	0	0
$781$	1692.75	2.16741
$782$	0	0
$783$	113.623i	0.145112i
$784$	0	0
$785$	135.240	0.172280
$786$	0	0
$787$	− 231.223i	− 0.293802i	−0.989151	−	0.146901i	$-0.953070\pi$
$787$	− 231.223i	− 0.293802i	0.989151	−	0.146901i	$-0.0469299\pi$
$788$	0	0
$789$	−869.189	−1.10163
$790$	0	0
$791$	− 17.4531i	− 0.0220646i
$792$	0	0
$793$	−582.696	−0.734800
$794$	0	0
$795$	487.422i	0.613110i
$796$	0	0
$797$	51.5068	0.0646259	0.0323129	−	0.999478i	$-0.489713\pi$
$797$	51.5068	0.0646259	0.0323129	+	0.999478i	$0.489713\pi$
$798$	0	0
$799$	339.606i	0.425039i
$800$	0	0
$801$	366.224	0.457209
$802$	0	0
$803$	1999.38i	2.48988i
$804$	0	0
$805$	202.572	0.251643
$806$	0	0
$807$	493.480i	0.611499i
$808$	0	0
$809$	580.914	0.718065	0.359032	−	0.933325i	$-0.383107\pi$
$809$	580.914	0.718065	0.359032	+	0.933325i	$0.383107\pi$
$810$	0	0
$811$	1351.98i	1.66706i	0.552475	+	0.833529i	$0.313683\pi$
$811$	1351.98i	1.66706i	−0.552475	+	0.833529i	$0.686317\pi$
$812$	0	0
$813$	−417.753	−0.513841
$814$	0	0
$815$	863.480i	1.05948i
$816$	0	0
$817$	−768.678	−0.940855
$818$	0	0
$819$	120.188i	0.146749i
$820$	0	0
$821$	179.511	0.218650	0.109325	−	0.994006i	$-0.465131\pi$
$821$	179.511	0.218650	0.109325	+	0.994006i	$0.465131\pi$
$822$	0	0
$823$	− 665.446i	− 0.808561i	−0.914635	−	0.404281i	$-0.867522\pi$
$823$	− 665.446i	− 0.808561i	0.914635	−	0.404281i	$-0.132478\pi$
$824$	0	0
$825$	−157.115	−0.190443
$826$	0	0
$827$	− 901.166i	− 1.08968i	−0.838540	−	0.544840i	$-0.816590\pi$
$827$	− 901.166i	− 1.08968i	0.838540	−	0.544840i	$-0.183410\pi$
$828$	0	0
$829$	−183.151	−0.220930	−0.110465	−	0.993880i	$-0.535234\pi$
$829$	−183.151	−0.220930	−0.110465	+	0.993880i	$0.535234\pi$
$830$	0	0
$831$	− 420.676i	− 0.506228i
$832$	0	0
$833$	946.571	1.13634
$834$	0	0
$835$	1143.50i	1.36946i
$836$	0	0
$837$	201.062	0.240218
$838$	0	0
$839$	− 846.082i	− 1.00844i	−0.863575	−	0.504220i	$-0.831780\pi$
$839$	− 846.082i	− 1.00844i	0.863575	−	0.504220i	$-0.168220\pi$
$840$	0	0
$841$	−362.846	−0.431446
$842$	0	0
$843$	182.457i	0.216437i
$844$	0	0
$845$	180.382	0.213470
$846$	0	0
$847$	191.351i	0.225917i
$848$	0	0
$849$	−387.749	−0.456713
$850$	0	0
$851$	1136.56i	1.33556i
$852$	0	0
$853$	−540.635	−0.633804	−0.316902	−	0.948458i	$-0.602643\pi$
$853$	−540.635	−0.633804	−0.316902	+	0.948458i	$0.602643\pi$
$854$	0	0
$855$	206.555i	0.241585i
$856$	0	0
$857$	1333.41	1.55590	0.777952	−	0.628323i	$-0.216258\pi$
$857$	1333.41	1.55590	0.777952	+	0.628323i	$0.216258\pi$
$858$	0	0
$859$	439.034i	0.511099i	0.966796	+	0.255549i	$0.0822563\pi$
$859$	439.034i	0.511099i	−0.966796	+	0.255549i	$0.917744\pi$
$860$	0	0
$861$	111.008	0.128930
$862$	0	0
$863$	− 1164.41i	− 1.34926i	−0.738155	−	0.674631i	$-0.764303\pi$
$863$	− 1164.41i	− 1.34926i	0.738155	−	0.674631i	$-0.235697\pi$
$864$	0	0
$865$	−188.208	−0.217582
$866$	0	0
$867$	405.147i	0.467298i
$868$	0	0
$869$	−1775.20	−2.04281
$870$	0	0
$871$	− 1086.74i	− 1.24769i
$872$	0	0
$873$	264.922	0.303461
$874$	0	0
$875$	373.783i	0.427181i
$876$	0	0
$877$	404.456	0.461181	0.230591	−	0.973051i	$-0.425934\pi$
$877$	404.456	0.461181	0.230591	+	0.973051i	$0.425934\pi$
$878$	0	0
$879$	− 729.292i	− 0.829684i
$880$	0	0
$881$	−227.084	−0.257757	−0.128879	−	0.991660i	$-0.541138\pi$
$881$	−227.084	−0.257757	−0.128879	+	0.991660i	$0.541138\pi$
$882$	0	0
$883$	− 673.730i	− 0.763001i	−0.924369	−	0.381501i	$-0.875407\pi$
$883$	− 673.730i	− 0.763001i	0.924369	−	0.381501i	$-0.124593\pi$
$884$	0	0
$885$	−489.533	−0.553145
$886$	0	0
$887$	1370.95i	1.54560i	0.634648	+	0.772801i	$0.281145\pi$
$887$	1370.95i	1.54560i	−0.634648	+	0.772801i	$0.718855\pi$
$888$	0	0
$889$	−45.5678	−0.0512574
$890$	0	0
$891$	− 124.182i	− 0.139373i
$892$	0	0
$893$	238.211	0.266754
$894$	0	0
$895$	− 1183.79i	− 1.32267i
$896$	0	0
$897$	−430.544	−0.479982
$898$	0	0
$899$	− 846.121i	− 0.941180i
$900$	0	0
$901$	1499.15	1.66388
$902$	0	0
$903$	− 228.917i	− 0.253508i
$904$	0	0
$905$	−339.124	−0.374723
$906$	0	0
$907$	− 1469.46i	− 1.62013i	−0.586342	−	0.810063i	$-0.699433\pi$
$907$	− 1469.46i	− 1.62013i	0.586342	−	0.810063i	$-0.300567\pi$
$908$	0	0
$909$	313.035	0.344373
$910$	0	0
$911$	871.203i	0.956315i	0.878274	+	0.478157i	$0.158695\pi$
$911$	871.203i	0.956315i	−0.878274	+	0.478157i	$0.841305\pi$
$912$	0	0
$913$	−304.131	−0.333111
$914$	0	0
$915$	− 298.230i	− 0.325935i
$916$	0	0
$917$	140.596	0.153322
$918$	0	0
$919$	256.361i	0.278957i	0.990225	+	0.139478i	$0.0445426\pi$
$919$	256.361i	0.278957i	−0.990225	+	0.139478i	$0.955457\pi$
$920$	0	0
$921$	212.808	0.231062
$922$	0	0
$923$	1782.14i	1.93081i
$924$	0	0
$925$	−436.661	−0.472066
$926$	0	0
$927$	207.183i	0.223498i
$928$	0	0
$929$	−1399.93	−1.50692	−0.753462	−	0.657491i	$-0.771618\pi$
$929$	−1399.93	−1.50692	−0.753462	+	0.657491i	$0.771618\pi$
$930$	0	0
$931$	− 663.956i	− 0.713165i
$932$	0	0
$933$	−757.832	−0.812253
$934$	0	0
$935$	− 1354.39i	− 1.44854i
$936$	0	0
$937$	365.610	0.390192	0.195096	−	0.980784i	$-0.437498\pi$
$937$	365.610	0.390192	0.195096	+	0.980784i	$0.437498\pi$
$938$	0	0
$939$	650.235i	0.692476i
$940$	0	0
$941$	−407.498	−0.433048	−0.216524	−	0.976277i	$-0.569472\pi$
$941$	−407.498	−0.433048	−0.216524	+	0.976277i	$0.569472\pi$
$942$	0	0
$943$	397.661i	0.421698i
$944$	0	0
$945$	−61.5134	−0.0650935
$946$	0	0
$947$	1150.64i	1.21503i	0.794307	+	0.607516i	$0.207834\pi$
$947$	1150.64i	1.21503i	−0.794307	+	0.607516i	$0.792166\pi$
$948$	0	0
$949$	−2104.96	−2.21808
$950$	0	0
$951$	673.537i	0.708241i
$952$	0	0
$953$	−1341.53	−1.40769	−0.703844	−	0.710355i	$-0.748534\pi$
$953$	−1341.53	−1.40769	−0.703844	+	0.710355i	$0.748534\pi$
$954$	0	0
$955$	611.295i	0.640100i
$956$	0	0
$957$	−522.588	−0.546069
$958$	0	0
$959$	− 27.6219i	− 0.0288029i
$960$	0	0
$961$	−536.260	−0.558023
$962$	0	0
$963$	− 448.286i	− 0.465510i
$964$	0	0
$965$	−1192.54	−1.23579
$966$	0	0
$967$	− 1227.25i	− 1.26914i	−0.772867	−	0.634568i	$-0.781178\pi$
$967$	− 1227.25i	− 1.26914i	0.772867	−	0.634568i	$-0.218822\pi$
$968$	0	0
$969$	635.295	0.655620
$970$	0	0
$971$	− 1864.31i	− 1.91999i	−0.280020	−	0.959994i	$-0.590341\pi$
$971$	− 1864.31i	− 1.91999i	0.280020	−	0.959994i	$-0.409659\pi$
$972$	0	0
$973$	−181.217	−0.186245
$974$	0	0
$975$	− 165.412i	− 0.169654i
$976$	0	0
$977$	−1209.56	−1.23803	−0.619017	−	0.785378i	$-0.712469\pi$
$977$	−1209.56	−1.23803	−0.619017	+	0.785378i	$0.712469\pi$
$978$	0	0
$979$	1684.38i	1.72051i
$980$	0	0
$981$	−163.002	−0.166159
$982$	0	0
$983$	1018.26i	1.03587i	0.855420	+	0.517935i	$0.173299\pi$
$983$	1018.26i	1.03587i	−0.855420	+	0.517935i	$0.826701\pi$
$984$	0	0
$985$	14.6310	0.0148538
$986$	0	0
$987$	70.9408i	0.0718752i
$988$	0	0
$989$	820.042	0.829163
$990$	0	0
$991$	− 1627.52i	− 1.64230i	−0.570712	−	0.821150i	$-0.693333\pi$
$991$	− 1627.52i	− 1.64230i	0.570712	−	0.821150i	$-0.306667\pi$
$992$	0	0
$993$	26.7931	0.0269819
$994$	0	0
$995$	1349.24i	1.35602i
$996$	0	0
$997$	540.192	0.541817	0.270909	−	0.962605i	$-0.412676\pi$
$997$	540.192	0.541817	0.270909	+	0.962605i	$0.412676\pi$
$998$	0	0
$999$	− 345.131i	− 0.345476i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.g.a.127.8	yes	8
3.2	odd	2		1152.3.g.f.127.1		8
4.3	odd	2	inner	384.3.g.a.127.4	✓	8
8.3	odd	2		384.3.g.b.127.5	yes	8
8.5	even	2		384.3.g.b.127.1	yes	8
12.11	even	2		1152.3.g.f.127.2		8
16.3	odd	4		768.3.b.f.127.5		8
16.5	even	4		768.3.b.f.127.8		8
16.11	odd	4		768.3.b.e.127.4		8
16.13	even	4		768.3.b.e.127.1		8
24.5	odd	2		1152.3.g.c.127.7		8
24.11	even	2		1152.3.g.c.127.8		8
48.5	odd	4		2304.3.b.q.127.2		8
48.11	even	4		2304.3.b.t.127.2		8
48.29	odd	4		2304.3.b.t.127.7		8
48.35	even	4		2304.3.b.q.127.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.4	✓	8	4.3	odd	2	inner
384.3.g.a.127.8	yes	8	1.1	even	1	trivial
384.3.g.b.127.1	yes	8	8.5	even	2
384.3.g.b.127.5	yes	8	8.3	odd	2
768.3.b.e.127.1		8	16.13	even	4
768.3.b.e.127.4		8	16.11	odd	4
768.3.b.f.127.5		8	16.3	odd	4
768.3.b.f.127.8		8	16.5	even	4
1152.3.g.c.127.7		8	24.5	odd	2
1152.3.g.c.127.8		8	24.11	even	2
1152.3.g.f.127.1		8	3.2	odd	2
1152.3.g.f.127.2		8	12.11	even	2
2304.3.b.q.127.2		8	48.5	odd	4
2304.3.b.q.127.7		8	48.35	even	4
2304.3.b.t.127.2		8	48.11	even	4
2304.3.b.t.127.7		8	48.29	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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} -13.7980i q^{11} +14.5266 q^{13} +7.43488i q^{15} +22.8673 q^{17} -16.0399i q^{19} +4.77678 q^{21} +17.1117i q^{23} -6.57420 q^{25} -5.19615i q^{27} -21.8667 q^{29} +38.6944i q^{31} +23.8988 q^{33} -11.8383i q^{35} +66.4204 q^{37} +25.1608i q^{39} +23.2392 q^{41} -47.9230i q^{43} -12.8776 q^{45} +14.8512i q^{47} +41.3941 q^{49} +39.6073i q^{51} +65.5589 q^{53} -59.2281i q^{55} +27.7819 q^{57} +65.8428i q^{59} -40.1123 q^{61} +8.27362i q^{63} +62.3559 q^{65} -74.8105i q^{67} -29.6383 q^{69} +122.681i q^{71} -144.904 q^{73} -11.3868i q^{75} -38.0530 q^{77} -128.657i q^{79} +9.00000 q^{81} -22.0417i q^{83} +98.1584 q^{85} -37.8743i q^{87} -122.075 q^{89} -40.0626i q^{91} -67.0208 q^{93} -68.8516i q^{95} -88.3072 q^{97} +41.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{5} - 24 q^{9}+O(q^{10}) \) 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

Varieties

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Representations

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Database

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