LMFDB - Embedded newform 384.4.d.f.193.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,4,Mod(193,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.193"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1534132224.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 $ x^8 + 18x^6 + 107x^4 + 210*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$-2.48330i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.f.193.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +17.4288i q^{5} -2.99032 q^{7} -9.00000 q^{9} -10.6274i q^{11} +43.3156i q^{13} +52.2865 q^{15} -37.8823 q^{17} -79.8823i q^{19} +8.97095i q^{21} -191.204 q^{23} -178.765 q^{25} +27.0000i q^{27} -138.918i q^{29} +212.136 q^{31} -31.8823 q^{33} -52.1177i q^{35} -270.404i q^{37} +129.947 q^{39} -441.411 q^{41} +64.1177i q^{43} -156.860i q^{45} -436.234 q^{47} -334.058 q^{49} +113.647i q^{51} +278.348i q^{53} +185.224 q^{55} -239.647 q^{57} +830.039i q^{59} +724.580i q^{61} +26.9128 q^{63} -754.940 q^{65} -859.529i q^{67} +573.613i q^{69} +681.264 q^{71} -785.058 q^{73} +536.294i q^{75} +31.7793i q^{77} -1018.82 q^{79} +81.0000 q^{81} -467.334i q^{83} -660.244i q^{85} -416.753 q^{87} +510.706 q^{89} -129.527i q^{91} -636.409i q^{93} +1392.26 q^{95} -234.235 q^{97} +95.6468i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) $ 8 * q - 72 * q^9 + 240 * q^17 - 344 * q^25 + 288 * q^33 - 816 * q^41 + 1672 * q^49 - 288 * q^57 - 1152 * q^65 - 1936 * q^73 + 648 * q^81 + 7344 * q^89 - 2960 * q^97

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	17.4288i	1.55888i	0.626475	+	0.779441i	$0.284497\pi$
$5$	17.4288i	1.55888i	−0.626475	+	0.779441i	$0.715503\pi$
$6$	0	0
$7$	−2.99032	−0.161462	−0.0807310	−	0.996736i	$-0.525725\pi$
$7$	−2.99032	−0.161462	−0.0807310	+	0.996736i	$0.525725\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 10.6274i	− 0.291299i	−0.989336	−	0.145649i	$-0.953473\pi$
$11$	− 10.6274i	− 0.291299i	0.989336	−	0.145649i	$-0.0465271\pi$
$12$	0	0
$13$	43.3156i	0.924121i	0.886848	+	0.462061i	$0.152890\pi$
$13$	43.3156i	0.924121i	−0.886848	+	0.462061i	$0.847110\pi$
$14$	0	0
$15$	52.2865	0.900021
$16$	0	0
$17$	−37.8823	−0.540459	−0.270229	−	0.962796i	$-0.587100\pi$
$17$	−37.8823	−0.540459	−0.270229	+	0.962796i	$0.587100\pi$
$18$	0	0
$19$	− 79.8823i	− 0.964539i	−0.876023	−	0.482270i	$-0.839813\pi$
$19$	− 79.8823i	− 0.964539i	0.876023	−	0.482270i	$-0.160187\pi$
$20$	0	0
$21$	8.97095i	0.0932201i
$22$	0	0
$23$	−191.204	−1.73343	−0.866714	−	0.498806i	$-0.833772\pi$
$23$	−191.204	−1.73343	−0.866714	+	0.498806i	$0.833772\pi$
$24$	0	0
$25$	−178.765	−1.43012
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 138.918i	− 0.889530i	−0.895647	−	0.444765i	$-0.853287\pi$
$29$	− 138.918i	− 0.889530i	0.895647	−	0.444765i	$-0.146713\pi$
$30$	0	0
$31$	212.136	1.22906	0.614529	−	0.788894i	$-0.289346\pi$
$31$	212.136	1.22906	0.614529	+	0.788894i	$0.289346\pi$
$32$	0	0
$33$	−31.8823	−0.168181
$34$	0	0
$35$	− 52.1177i	− 0.251700i
$36$	0	0
$37$	− 270.404i	− 1.20146i	−0.799451	−	0.600731i	$-0.794876\pi$
$37$	− 270.404i	− 1.20146i	0.799451	−	0.600731i	$-0.205124\pi$
$38$	0	0
$39$	129.947	0.533542
$40$	0	0
$41$	−441.411	−1.68139	−0.840693	−	0.541511i	$-0.817852\pi$
$41$	−441.411	−1.68139	−0.840693	+	0.541511i	$0.817852\pi$
$42$	0	0
$43$	64.1177i	0.227392i	0.993516	+	0.113696i	$0.0362690\pi$
$43$	64.1177i	0.227392i	−0.993516	+	0.113696i	$0.963731\pi$
$44$	0	0
$45$	− 156.860i	− 0.519628i
$46$	0	0
$47$	−436.234	−1.35386	−0.676929	−	0.736049i	$-0.736689\pi$
$47$	−436.234	−1.35386	−0.676929	+	0.736049i	$0.736689\pi$
$48$	0	0
$49$	−334.058	−0.973930
$50$	0	0
$51$	113.647i	0.312034i
$52$	0	0
$53$	278.348i	0.721398i	0.932682	+	0.360699i	$0.117462\pi$
$53$	278.348i	0.721398i	−0.932682	+	0.360699i	$0.882538\pi$
$54$	0	0
$55$	185.224	0.454101
$56$	0	0
$57$	−239.647	−0.556877
$58$	0	0
$59$	830.039i	1.83156i	0.401684	+	0.915778i	$0.368425\pi$
$59$	830.039i	1.83156i	−0.401684	+	0.915778i	$0.631575\pi$
$60$	0	0
$61$	724.580i	1.52087i	0.649416	+	0.760434i	$0.275014\pi$
$61$	724.580i	1.52087i	−0.649416	+	0.760434i	$0.724986\pi$
$62$	0	0
$63$	26.9128	0.0538206
$64$	0	0
$65$	−754.940	−1.44060
$66$	0	0
$67$	− 859.529i	− 1.56729i	−0.621211	−	0.783643i	$-0.713359\pi$
$67$	− 859.529i	− 1.56729i	0.621211	−	0.783643i	$-0.286641\pi$
$68$	0	0
$69$	573.613i	1.00079i
$70$	0	0
$71$	681.264	1.13875	0.569374	−	0.822078i	$-0.307186\pi$
$71$	681.264	1.13875	0.569374	+	0.822078i	$0.307186\pi$
$72$	0	0
$73$	−785.058	−1.25869	−0.629343	−	0.777128i	$-0.716676\pi$
$73$	−785.058	−1.25869	−0.629343	+	0.777128i	$0.716676\pi$
$74$	0	0
$75$	536.294i	0.825678i
$76$	0	0
$77$	31.7793i	0.0470337i
$78$	0	0
$79$	−1018.82	−1.45096	−0.725481	−	0.688243i	$-0.758382\pi$
$79$	−1018.82	−1.45096	−0.725481	+	0.688243i	$0.758382\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 467.334i	− 0.618031i	−0.951057	−	0.309015i	$-0.900001\pi$
$83$	− 467.334i	− 0.618031i	0.951057	−	0.309015i	$-0.0999995\pi$
$84$	0	0
$85$	− 660.244i	− 0.842512i
$86$	0	0
$87$	−416.753	−0.513570
$88$	0	0
$89$	510.706	0.608256	0.304128	−	0.952631i	$-0.401635\pi$
$89$	510.706	0.608256	0.304128	+	0.952631i	$0.401635\pi$
$90$	0	0
$91$	− 129.527i	− 0.149210i
$92$	0	0
$93$	− 636.409i	− 0.709597i
$94$	0	0
$95$	1392.26	1.50360
$96$	0	0
$97$	−234.235	−0.245186	−0.122593	−	0.992457i	$-0.539121\pi$
$97$	−234.235	−0.245186	−0.122593	+	0.992457i	$0.539121\pi$
$98$	0	0
$99$	95.6468i	0.0970996i
$100$	0	0
$101$	− 205.555i	− 0.202509i	−0.994861	−	0.101255i	$-0.967714\pi$
$101$	− 205.555i	− 0.202509i	0.994861	−	0.101255i	$-0.0322857\pi$
$102$	0	0
$103$	−391.379	−0.374405	−0.187203	−	0.982321i	$-0.559942\pi$
$103$	−391.379	−0.374405	−0.187203	+	0.982321i	$0.559942\pi$
$104$	0	0
$105$	−156.353	−0.145319
$106$	0	0
$107$	934.274i	0.844109i	0.906570	+	0.422055i	$0.138691\pi$
$107$	934.274i	0.844109i	−0.906570	+	0.422055i	$0.861309\pi$
$108$	0	0
$109$	− 584.123i	− 0.513292i	−0.966505	−	0.256646i	$-0.917383\pi$
$109$	− 584.123i	− 0.513292i	0.966505	−	0.256646i	$-0.0826174\pi$
$110$	0	0
$111$	−811.211	−0.693664
$112$	0	0
$113$	−582.706	−0.485101	−0.242551	−	0.970139i	$-0.577984\pi$
$113$	−582.706	−0.485101	−0.242551	+	0.970139i	$0.577984\pi$
$114$	0	0
$115$	− 3332.47i	− 2.70221i
$116$	0	0
$117$	− 389.840i	− 0.308040i
$118$	0	0
$119$	113.280	0.0872635
$120$	0	0
$121$	1218.06	0.915145
$122$	0	0
$123$	1324.23i	0.970749i
$124$	0	0
$125$	− 937.053i	− 0.670501i
$126$	0	0
$127$	1461.03	1.02083	0.510416	−	0.859928i	$-0.329491\pi$
$127$	1461.03	1.02083	0.510416	+	0.859928i	$0.329491\pi$
$128$	0	0
$129$	192.353	0.131285
$130$	0	0
$131$	− 98.4323i	− 0.0656494i	−0.999461	−	0.0328247i	$-0.989550\pi$
$131$	− 98.4323i	− 0.0656494i	0.999461	−	0.0328247i	$-0.0104503\pi$
$132$	0	0
$133$	238.873i	0.155736i
$134$	0	0
$135$	−470.579	−0.300007
$136$	0	0
$137$	2171.06	1.35391	0.676956	−	0.736024i	$-0.263299\pi$
$137$	2171.06	1.35391	0.676956	+	0.736024i	$0.263299\pi$
$138$	0	0
$139$	− 1624.70i	− 0.991407i	−0.868492	−	0.495703i	$-0.834910\pi$
$139$	− 1624.70i	− 0.991407i	0.868492	−	0.495703i	$-0.165090\pi$
$140$	0	0
$141$	1308.70i	0.781650i
$142$	0	0
$143$	460.333	0.269195
$144$	0	0
$145$	2421.17	1.38667
$146$	0	0
$147$	1002.17i	0.562299i
$148$	0	0
$149$	636.658i	0.350047i	0.984564	+	0.175024i	$0.0560002\pi$
$149$	636.658i	0.350047i	−0.984564	+	0.175024i	$0.944000\pi$
$150$	0	0
$151$	−1819.34	−0.980503	−0.490252	−	0.871581i	$-0.663095\pi$
$151$	−1819.34	−0.980503	−0.490252	+	0.871581i	$0.663095\pi$
$152$	0	0
$153$	340.940	0.180153
$154$	0	0
$155$	3697.29i	1.91596i
$156$	0	0
$157$	1656.50i	0.842059i	0.907047	+	0.421029i	$0.138331\pi$
$157$	1656.50i	0.842059i	−0.907047	+	0.421029i	$0.861669\pi$
$158$	0	0
$159$	835.045	0.416499
$160$	0	0
$161$	571.761	0.279882
$162$	0	0
$163$	− 2228.82i	− 1.07101i	−0.844532	−	0.535505i	$-0.820121\pi$
$163$	− 2228.82i	− 1.07101i	0.844532	−	0.535505i	$-0.179879\pi$
$164$	0	0
$165$	− 555.671i	− 0.262175i
$166$	0	0
$167$	−1667.01	−0.772439	−0.386219	−	0.922407i	$-0.626219\pi$
$167$	−1667.01	−0.772439	−0.386219	+	0.922407i	$0.626219\pi$
$168$	0	0
$169$	320.761	0.146000
$170$	0	0
$171$	718.940i	0.321513i
$172$	0	0
$173$	2500.53i	1.09891i	0.835522	+	0.549457i	$0.185165\pi$
$173$	2500.53i	1.09891i	−0.835522	+	0.549457i	$0.814835\pi$
$174$	0	0
$175$	534.562	0.230909
$176$	0	0
$177$	2490.12	1.05745
$178$	0	0
$179$	378.742i	0.158148i	0.996869	+	0.0790740i	$0.0251963\pi$
$179$	378.742i	0.158148i	−0.996869	+	0.0790740i	$0.974804\pi$
$180$	0	0
$181$	3093.88i	1.27053i	0.772294	+	0.635265i	$0.219109\pi$
$181$	3093.88i	1.27053i	−0.772294	+	0.635265i	$0.780891\pi$
$182$	0	0
$183$	2173.74	0.878073
$184$	0	0
$185$	4712.82	1.87294
$186$	0	0
$187$	402.590i	0.157435i
$188$	0	0
$189$	− 80.7385i	− 0.0310734i
$190$	0	0
$191$	−3656.98	−1.38539	−0.692695	−	0.721230i	$-0.743577\pi$
$191$	−3656.98	−1.38539	−0.692695	+	0.721230i	$0.743577\pi$
$192$	0	0
$193$	2788.12	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$193$	2788.12	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$194$	0	0
$195$	2264.82i	0.831729i
$196$	0	0
$197$	− 1147.74i	− 0.415091i	−0.978225	−	0.207546i	$-0.933452\pi$
$197$	− 1147.74i	− 0.415091i	0.978225	−	0.207546i	$-0.0665475\pi$
$198$	0	0
$199$	4842.73	1.72508	0.862542	−	0.505986i	$-0.168871\pi$
$199$	4842.73	1.72508	0.862542	+	0.505986i	$0.168871\pi$
$200$	0	0
$201$	−2578.59	−0.904873
$202$	0	0
$203$	415.408i	0.143625i
$204$	0	0
$205$	− 7693.29i	− 2.62109i
$206$	0	0
$207$	1720.84	0.577809
$208$	0	0
$209$	−848.942	−0.280969
$210$	0	0
$211$	− 3222.35i	− 1.05135i	−0.850684	−	0.525677i	$-0.823812\pi$
$211$	− 3222.35i	− 1.05135i	0.850684	−	0.525677i	$-0.176188\pi$
$212$	0	0
$213$	− 2043.79i	− 0.657457i
$214$	0	0
$215$	−1117.50	−0.354478
$216$	0	0
$217$	−634.355	−0.198446
$218$	0	0
$219$	2355.17i	0.726703i
$220$	0	0
$221$	− 1640.89i	− 0.499449i
$222$	0	0
$223$	−4932.61	−1.48122	−0.740610	−	0.671935i	$-0.765463\pi$
$223$	−4932.61	−1.48122	−0.740610	+	0.671935i	$0.765463\pi$
$224$	0	0
$225$	1608.88	0.476705
$226$	0	0
$227$	3619.49i	1.05830i	0.848529	+	0.529150i	$0.177489\pi$
$227$	3619.49i	1.05830i	−0.848529	+	0.529150i	$0.822511\pi$
$228$	0	0
$229$	− 305.759i	− 0.0882320i	−0.999026	−	0.0441160i	$-0.985953\pi$
$229$	− 305.759i	− 0.0882320i	0.999026	−	0.0441160i	$-0.0140471\pi$
$230$	0	0
$231$	95.3380	0.0271549
$232$	0	0
$233$	639.648	0.179849	0.0899244	−	0.995949i	$-0.471337\pi$
$233$	639.648	0.179849	0.0899244	+	0.995949i	$0.471337\pi$
$234$	0	0
$235$	− 7603.05i	− 2.11050i
$236$	0	0
$237$	3056.45i	0.837713i
$238$	0	0
$239$	−1744.94	−0.472262	−0.236131	−	0.971721i	$-0.575879\pi$
$239$	−1744.94	−0.472262	−0.236131	+	0.971721i	$0.575879\pi$
$240$	0	0
$241$	3357.29	0.897354	0.448677	−	0.893694i	$-0.351895\pi$
$241$	3357.29	0.897354	0.448677	+	0.893694i	$0.351895\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 5822.24i	− 1.51824i
$246$	0	0
$247$	3460.15	0.891351
$248$	0	0
$249$	−1402.00	−0.356820
$250$	0	0
$251$	− 1317.45i	− 0.331301i	−0.986185	−	0.165650i	$-0.947028\pi$
$251$	− 1317.45i	− 0.331301i	0.986185	−	0.165650i	$-0.0529723\pi$
$252$	0	0
$253$	2032.01i	0.504945i
$254$	0	0
$255$	−1980.73	−0.486424
$256$	0	0
$257$	−3036.12	−0.736917	−0.368459	−	0.929644i	$-0.620114\pi$
$257$	−3036.12	−0.736917	−0.368459	+	0.929644i	$0.620114\pi$
$258$	0	0
$259$	808.592i	0.193990i
$260$	0	0
$261$	1250.26i	0.296510i
$262$	0	0
$263$	1655.76	0.388206	0.194103	−	0.980981i	$-0.437820\pi$
$263$	1655.76	0.388206	0.194103	+	0.980981i	$0.437820\pi$
$264$	0	0
$265$	−4851.29	−1.12458
$266$	0	0
$267$	− 1532.12i	− 0.351177i
$268$	0	0
$269$	5292.72i	1.19964i	0.800135	+	0.599820i	$0.204761\pi$
$269$	5292.72i	1.19964i	−0.800135	+	0.599820i	$0.795239\pi$
$270$	0	0
$271$	8010.52	1.79559	0.897795	−	0.440414i	$-0.145168\pi$
$271$	8010.52	1.79559	0.897795	+	0.440414i	$0.145168\pi$
$272$	0	0
$273$	−388.582	−0.0861467
$274$	0	0
$275$	1899.80i	0.416591i
$276$	0	0
$277$	5692.81i	1.23483i	0.786638	+	0.617415i	$0.211820\pi$
$277$	5692.81i	1.23483i	−0.786638	+	0.617415i	$0.788180\pi$
$278$	0	0
$279$	−1909.23	−0.409686
$280$	0	0
$281$	2024.83	0.429861	0.214931	−	0.976629i	$-0.431047\pi$
$281$	2024.83	0.429861	0.214931	+	0.976629i	$0.431047\pi$
$282$	0	0
$283$	− 247.761i	− 0.0520419i	−0.999661	−	0.0260210i	$-0.991716\pi$
$283$	− 247.761i	− 0.0520419i	0.999661	−	0.0260210i	$-0.00828366\pi$
$284$	0	0
$285$	− 4176.77i	− 0.868106i
$286$	0	0
$287$	1319.96	0.271480
$288$	0	0
$289$	−3477.94	−0.707905
$290$	0	0
$291$	702.706i	0.141558i
$292$	0	0
$293$	8133.61i	1.62174i	0.585225	+	0.810871i	$0.301006\pi$
$293$	8133.61i	1.62174i	−0.585225	+	0.810871i	$0.698994\pi$
$294$	0	0
$295$	−14466.6	−2.85518
$296$	0	0
$297$	286.940	0.0560605
$298$	0	0
$299$	− 8282.12i	− 1.60190i
$300$	0	0
$301$	− 191.732i	− 0.0367152i
$302$	0	0
$303$	−616.664	−0.116919
$304$	0	0
$305$	−12628.6	−2.37085
$306$	0	0
$307$	2974.82i	0.553035i	0.961009	+	0.276518i	$0.0891805\pi$
$307$	2974.82i	0.553035i	−0.961009	+	0.276518i	$0.910820\pi$
$308$	0	0
$309$	1174.14i	0.216163i
$310$	0	0
$311$	4451.52	0.811648	0.405824	−	0.913951i	$-0.366985\pi$
$311$	4451.52	0.811648	0.405824	+	0.913951i	$0.366985\pi$
$312$	0	0
$313$	−8273.75	−1.49412	−0.747061	−	0.664755i	$-0.768536\pi$
$313$	−8273.75	−1.49412	−0.747061	+	0.664755i	$0.768536\pi$
$314$	0	0
$315$	469.060i	0.0839001i
$316$	0	0
$317$	− 429.036i	− 0.0760160i	−0.999277	−	0.0380080i	$-0.987899\pi$
$317$	− 429.036i	− 0.0760160i	0.999277	−	0.0380080i	$-0.0121012\pi$
$318$	0	0
$319$	−1476.34	−0.259119
$320$	0	0
$321$	2802.82	0.487347
$322$	0	0
$323$	3026.12i	0.521293i
$324$	0	0
$325$	− 7743.29i	− 1.32160i
$326$	0	0
$327$	−1752.37	−0.296349
$328$	0	0
$329$	1304.48	0.218596
$330$	0	0
$331$	− 8196.71i	− 1.36112i	−0.732691	−	0.680562i	$-0.761736\pi$
$331$	− 8196.71i	− 1.36112i	0.732691	−	0.680562i	$-0.238264\pi$
$332$	0	0
$333$	2433.63i	0.400487i
$334$	0	0
$335$	14980.6	2.44322
$336$	0	0
$337$	−2000.35	−0.323341	−0.161670	−	0.986845i	$-0.551688\pi$
$337$	−2000.35	−0.323341	−0.161670	+	0.986845i	$0.551688\pi$
$338$	0	0
$339$	1748.12i	0.280073i
$340$	0	0
$341$	− 2254.46i	− 0.358023i
$342$	0	0
$343$	2024.62	0.318715
$344$	0	0
$345$	−9997.40	−1.56012
$346$	0	0
$347$	7707.48i	1.19239i	0.802840	+	0.596195i	$0.203321\pi$
$347$	7707.48i	1.19239i	−0.802840	+	0.596195i	$0.796679\pi$
$348$	0	0
$349$	9681.98i	1.48500i	0.669847	+	0.742499i	$0.266360\pi$
$349$	9681.98i	1.48500i	−0.669847	+	0.742499i	$0.733640\pi$
$350$	0	0
$351$	−1169.52	−0.177847
$352$	0	0
$353$	−10540.3	−1.58925	−0.794626	−	0.607099i	$-0.792333\pi$
$353$	−10540.3	−1.58925	−0.794626	+	0.607099i	$0.792333\pi$
$354$	0	0
$355$	11873.6i	1.77518i
$356$	0	0
$357$	− 339.840i	− 0.0503816i
$358$	0	0
$359$	−514.158	−0.0755884	−0.0377942	−	0.999286i	$-0.512033\pi$
$359$	−514.158	−0.0755884	−0.0377942	+	0.999286i	$0.512033\pi$
$360$	0	0
$361$	477.826	0.0696641
$362$	0	0
$363$	− 3654.17i	− 0.528359i
$364$	0	0
$365$	− 13682.7i	− 1.96214i
$366$	0	0
$367$	−11272.4	−1.60331	−0.801657	−	0.597785i	$-0.796048\pi$
$367$	−11272.4	−1.60331	−0.801657	+	0.597785i	$0.796048\pi$
$368$	0	0
$369$	3972.70	0.560462
$370$	0	0
$371$	− 832.350i	− 0.116478i
$372$	0	0
$373$	− 6956.92i	− 0.965726i	−0.875696	−	0.482863i	$-0.839597\pi$
$373$	− 6956.92i	− 0.965726i	0.875696	−	0.482863i	$-0.160403\pi$
$374$	0	0
$375$	−2811.16	−0.387114
$376$	0	0
$377$	6017.30	0.822034
$378$	0	0
$379$	10201.3i	1.38260i	0.722569	+	0.691299i	$0.242961\pi$
$379$	10201.3i	1.38260i	−0.722569	+	0.691299i	$0.757039\pi$
$380$	0	0
$381$	− 4383.10i	− 0.589378i
$382$	0	0
$383$	−2461.56	−0.328406	−0.164203	−	0.986427i	$-0.552505\pi$
$383$	−2461.56	−0.328406	−0.164203	+	0.986427i	$0.552505\pi$
$384$	0	0
$385$	−553.877	−0.0733200
$386$	0	0
$387$	− 577.060i	− 0.0757974i
$388$	0	0
$389$	− 546.451i	− 0.0712240i	−0.999366	−	0.0356120i	$-0.988662\pi$
$389$	− 546.451i	− 0.0712240i	0.999366	−	0.0356120i	$-0.0113381\pi$
$390$	0	0
$391$	7243.25	0.936846
$392$	0	0
$393$	−295.297	−0.0379027
$394$	0	0
$395$	− 17756.8i	− 2.26188i
$396$	0	0
$397$	− 2084.56i	− 0.263529i	−0.991281	−	0.131764i	$-0.957936\pi$
$397$	− 2084.56i	− 0.263529i	0.991281	−	0.131764i	$-0.0420642\pi$
$398$	0	0
$399$	716.620	0.0899144
$400$	0	0
$401$	−9710.59	−1.20929	−0.604643	−	0.796497i	$-0.706684\pi$
$401$	−9710.59	−1.20929	−0.604643	+	0.796497i	$0.706684\pi$
$402$	0	0
$403$	9188.81i	1.13580i
$404$	0	0
$405$	1411.74i	0.173209i
$406$	0	0
$407$	−2873.69	−0.349984
$408$	0	0
$409$	−6659.89	−0.805160	−0.402580	−	0.915385i	$-0.631886\pi$
$409$	−6659.89	−0.805160	−0.402580	+	0.915385i	$0.631886\pi$
$410$	0	0
$411$	− 6513.17i	− 0.781681i
$412$	0	0
$413$	− 2482.08i	− 0.295727i
$414$	0	0
$415$	8145.09	0.963438
$416$	0	0
$417$	−4874.11	−0.572389
$418$	0	0
$419$	10576.7i	1.23318i	0.787283	+	0.616592i	$0.211487\pi$
$419$	10576.7i	1.23318i	−0.787283	+	0.616592i	$0.788513\pi$
$420$	0	0
$421$	− 4871.09i	− 0.563901i	−0.959429	−	0.281951i	$-0.909019\pi$
$421$	− 4871.09i	− 0.563901i	0.959429	−	0.281951i	$-0.0909814\pi$
$422$	0	0
$423$	3926.11	0.451286
$424$	0	0
$425$	6772.00	0.772918
$426$	0	0
$427$	− 2166.72i	− 0.245562i
$428$	0	0
$429$	− 1381.00i	− 0.155420i
$430$	0	0
$431$	−16916.7	−1.89060	−0.945302	−	0.326196i	$-0.894233\pi$
$431$	−16916.7	−1.89060	−0.945302	+	0.326196i	$0.894233\pi$
$432$	0	0
$433$	−1163.88	−0.129174	−0.0645870	−	0.997912i	$-0.520573\pi$
$433$	−1163.88	−0.129174	−0.0645870	+	0.997912i	$0.520573\pi$
$434$	0	0
$435$	− 7263.52i	− 0.800596i
$436$	0	0
$437$	15273.8i	1.67196i
$438$	0	0
$439$	1856.28	0.201812	0.100906	−	0.994896i	$-0.467826\pi$
$439$	1856.28	0.201812	0.100906	+	0.994896i	$0.467826\pi$
$440$	0	0
$441$	3006.52	0.324643
$442$	0	0
$443$	1472.21i	0.157893i	0.996879	+	0.0789465i	$0.0251556\pi$
$443$	1472.21i	0.157893i	−0.996879	+	0.0789465i	$0.974844\pi$
$444$	0	0
$445$	8901.02i	0.948200i
$446$	0	0
$447$	1909.97	0.202100
$448$	0	0
$449$	9620.94	1.01123	0.505613	−	0.862761i	$-0.331267\pi$
$449$	9620.94	1.01123	0.505613	+	0.862761i	$0.331267\pi$
$450$	0	0
$451$	4691.06i	0.489786i
$452$	0	0
$453$	5458.03i	0.566094i
$454$	0	0
$455$	2257.51	0.232602
$456$	0	0
$457$	3613.53	0.369877	0.184938	−	0.982750i	$-0.440791\pi$
$457$	3613.53	0.369877	0.184938	+	0.982750i	$0.440791\pi$
$458$	0	0
$459$	− 1022.82i	− 0.104011i
$460$	0	0
$461$	17710.7i	1.78931i	0.446759	+	0.894654i	$0.352578\pi$
$461$	17710.7i	1.78931i	−0.446759	+	0.894654i	$0.647422\pi$
$462$	0	0
$463$	1674.57	0.168087	0.0840433	−	0.996462i	$-0.473217\pi$
$463$	1674.57	0.168087	0.0840433	+	0.996462i	$0.473217\pi$
$464$	0	0
$465$	11091.9	1.10618
$466$	0	0
$467$	− 15208.3i	− 1.50697i	−0.657466	−	0.753484i	$-0.728372\pi$
$467$	− 15208.3i	− 1.50697i	0.657466	−	0.753484i	$-0.271628\pi$
$468$	0	0
$469$	2570.26i	0.253057i
$470$	0	0
$471$	4969.51	0.486163
$472$	0	0
$473$	681.406	0.0662391
$474$	0	0
$475$	14280.1i	1.37940i
$476$	0	0
$477$	− 2505.14i	− 0.240466i
$478$	0	0
$479$	−6458.37	−0.616056	−0.308028	−	0.951377i	$-0.599669\pi$
$479$	−6458.37	−0.616056	−0.308028	+	0.951377i	$0.599669\pi$
$480$	0	0
$481$	11712.7	1.11030
$482$	0	0
$483$	− 1715.28i	− 0.161590i
$484$	0	0
$485$	− 4082.45i	− 0.382216i
$486$	0	0
$487$	11337.7	1.05495	0.527474	−	0.849571i	$-0.323139\pi$
$487$	11337.7	1.05495	0.527474	+	0.849571i	$0.323139\pi$
$488$	0	0
$489$	−6686.46	−0.618348
$490$	0	0
$491$	− 14946.7i	− 1.37380i	−0.726754	−	0.686898i	$-0.758972\pi$
$491$	− 14946.7i	− 1.37380i	0.726754	−	0.686898i	$-0.241028\pi$
$492$	0	0
$493$	5262.51i	0.480754i
$494$	0	0
$495$	−1667.01	−0.151367
$496$	0	0
$497$	−2037.19	−0.183865
$498$	0	0
$499$	− 2631.77i	− 0.236101i	−0.993008	−	0.118051i	$-0.962336\pi$
$499$	− 2631.77i	− 0.236101i	0.993008	−	0.118051i	$-0.0376645\pi$
$500$	0	0
$501$	5001.04i	0.445968i
$502$	0	0
$503$	−6907.45	−0.612302	−0.306151	−	0.951983i	$-0.599041\pi$
$503$	−6907.45	−0.612302	−0.306151	+	0.951983i	$0.599041\pi$
$504$	0	0
$505$	3582.58	0.315689
$506$	0	0
$507$	− 962.283i	− 0.0842929i
$508$	0	0
$509$	− 12020.3i	− 1.04674i	−0.852107	−	0.523368i	$-0.824675\pi$
$509$	− 12020.3i	− 1.04674i	0.852107	−	0.523368i	$-0.175325\pi$
$510$	0	0
$511$	2347.57	0.203230
$512$	0	0
$513$	2156.82	0.185626
$514$	0	0
$515$	− 6821.29i	− 0.583654i
$516$	0	0
$517$	4636.04i	0.394377i
$518$	0	0
$519$	7501.60	0.634458
$520$	0	0
$521$	15846.1	1.33249	0.666247	−	0.745731i	$-0.267899\pi$
$521$	15846.1	1.33249	0.666247	+	0.745731i	$0.267899\pi$
$522$	0	0
$523$	8891.64i	0.743411i	0.928351	+	0.371706i	$0.121227\pi$
$523$	8891.64i	0.743411i	−0.928351	+	0.371706i	$0.878773\pi$
$524$	0	0
$525$	− 1603.69i	− 0.133316i
$526$	0	0
$527$	−8036.20	−0.664255
$528$	0	0
$529$	24392.0	2.00477
$530$	0	0
$531$	− 7470.35i	− 0.610519i
$532$	0	0
$533$	− 19120.0i	− 1.55381i
$534$	0	0
$535$	−16283.3	−1.31587
$536$	0	0
$537$	1136.23	0.0913068
$538$	0	0
$539$	3550.17i	0.283705i
$540$	0	0
$541$	− 12833.5i	− 1.01988i	−0.860210	−	0.509940i	$-0.829667\pi$
$541$	− 12833.5i	− 1.01988i	0.860210	−	0.509940i	$-0.170333\pi$
$542$	0	0
$543$	9281.63	0.733541
$544$	0	0
$545$	10180.6	0.800162
$546$	0	0
$547$	16257.0i	1.27075i	0.772204	+	0.635375i	$0.219155\pi$
$547$	16257.0i	1.27075i	−0.772204	+	0.635375i	$0.780845\pi$
$548$	0	0
$549$	− 6521.22i	− 0.506956i
$550$	0	0
$551$	−11097.1	−0.857986
$552$	0	0
$553$	3046.59	0.234275
$554$	0	0
$555$	− 14138.5i	− 1.08134i
$556$	0	0
$557$	− 1558.32i	− 0.118542i	−0.998242	−	0.0592712i	$-0.981122\pi$
$557$	− 1558.32i	− 0.118542i	0.998242	−	0.0592712i	$-0.0188777\pi$
$558$	0	0
$559$	−2777.30	−0.210138
$560$	0	0
$561$	1207.77	0.0908951
$562$	0	0
$563$	− 9782.16i	− 0.732272i	−0.930561	−	0.366136i	$-0.880681\pi$
$563$	− 9782.16i	− 0.732272i	0.930561	−	0.366136i	$-0.119319\pi$
$564$	0	0
$565$	− 10155.9i	− 0.756216i
$566$	0	0
$567$	−242.216	−0.0179402
$568$	0	0
$569$	7887.05	0.581094	0.290547	−	0.956861i	$-0.406163\pi$
$569$	7887.05	0.581094	0.290547	+	0.956861i	$0.406163\pi$
$570$	0	0
$571$	21819.3i	1.59914i	0.600573	+	0.799570i	$0.294939\pi$
$571$	21819.3i	1.59914i	−0.600573	+	0.799570i	$0.705061\pi$
$572$	0	0
$573$	10970.9i	0.799856i
$574$	0	0
$575$	34180.5	2.47900
$576$	0	0
$577$	7190.22	0.518774	0.259387	−	0.965773i	$-0.416479\pi$
$577$	7190.22	0.518774	0.259387	+	0.965773i	$0.416479\pi$
$578$	0	0
$579$	− 8364.35i	− 0.600363i
$580$	0	0
$581$	1397.48i	0.0997884i
$582$	0	0
$583$	2958.12	0.210142
$584$	0	0
$585$	6794.46	0.480199
$586$	0	0
$587$	13305.6i	0.935569i	0.883843	+	0.467785i	$0.154948\pi$
$587$	13305.6i	0.935569i	−0.883843	+	0.467785i	$0.845052\pi$
$588$	0	0
$589$	− 16945.9i	− 1.18548i
$590$	0	0
$591$	−3443.21	−0.239653
$592$	0	0
$593$	−8062.23	−0.558307	−0.279153	−	0.960246i	$-0.590054\pi$
$593$	−8062.23	−0.558307	−0.279153	+	0.960246i	$0.590054\pi$
$594$	0	0
$595$	1974.34i	0.136034i
$596$	0	0
$597$	− 14528.2i	− 0.995978i
$598$	0	0
$599$	−2185.21	−0.149058	−0.0745288	−	0.997219i	$-0.523745\pi$
$599$	−2185.21	−0.149058	−0.0745288	+	0.997219i	$0.523745\pi$
$600$	0	0
$601$	−3542.25	−0.240418	−0.120209	−	0.992749i	$-0.538357\pi$
$601$	−3542.25	−0.240418	−0.120209	+	0.992749i	$0.538357\pi$
$602$	0	0
$603$	7735.76i	0.522429i
$604$	0	0
$605$	21229.3i	1.42660i
$606$	0	0
$607$	6050.64	0.404593	0.202296	−	0.979324i	$-0.435160\pi$
$607$	6050.64	0.404593	0.202296	+	0.979324i	$0.435160\pi$
$608$	0	0
$609$	1246.22	0.0829220
$610$	0	0
$611$	− 18895.7i	− 1.25113i
$612$	0	0
$613$	22514.2i	1.48343i	0.670717	+	0.741713i	$0.265986\pi$
$613$	22514.2i	1.48343i	−0.670717	+	0.741713i	$0.734014\pi$
$614$	0	0
$615$	−23079.9	−1.51328
$616$	0	0
$617$	4255.88	0.277691	0.138845	−	0.990314i	$-0.455661\pi$
$617$	4255.88	0.277691	0.138845	+	0.990314i	$0.455661\pi$
$618$	0	0
$619$	− 228.949i	− 0.0148663i	−0.999972	−	0.00743315i	$-0.997634\pi$
$619$	− 228.949i	− 0.0148663i	0.999972	−	0.00743315i	$-0.00236607\pi$
$620$	0	0
$621$	− 5162.51i	− 0.333598i
$622$	0	0
$623$	−1527.17	−0.0982102
$624$	0	0
$625$	−6013.82	−0.384884
$626$	0	0
$627$	2546.83i	0.162218i
$628$	0	0
$629$	10243.5i	0.649340i
$630$	0	0
$631$	−11429.2	−0.721058	−0.360529	−	0.932748i	$-0.617404\pi$
$631$	−11429.2	−0.721058	−0.360529	+	0.932748i	$0.617404\pi$
$632$	0	0
$633$	−9667.04	−0.606999
$634$	0	0
$635$	25464.1i	1.59136i
$636$	0	0
$637$	− 14469.9i	− 0.900030i
$638$	0	0
$639$	−6131.38	−0.379583
$640$	0	0
$641$	29381.4	1.81045	0.905223	−	0.424938i	$-0.139704\pi$
$641$	29381.4	1.81045	0.905223	+	0.424938i	$0.139704\pi$
$642$	0	0
$643$	− 249.316i	− 0.0152909i	−0.999971	−	0.00764546i	$-0.997566\pi$
$643$	− 249.316i	− 0.0152909i	0.999971	−	0.00764546i	$-0.00243365\pi$
$644$	0	0
$645$	3352.49i	0.204658i
$646$	0	0
$647$	16025.8	0.973785	0.486893	−	0.873462i	$-0.338130\pi$
$647$	16025.8	0.973785	0.486893	+	0.873462i	$0.338130\pi$
$648$	0	0
$649$	8821.17	0.533530
$650$	0	0
$651$	1903.06i	0.114573i
$652$	0	0
$653$	− 14008.6i	− 0.839511i	−0.907637	−	0.419755i	$-0.862116\pi$
$653$	− 14008.6i	− 0.839511i	0.907637	−	0.419755i	$-0.137884\pi$
$654$	0	0
$655$	1715.56	0.102340
$656$	0	0
$657$	7065.52	0.419562
$658$	0	0
$659$	1011.91i	0.0598152i	0.999553	+	0.0299076i	$0.00952131\pi$
$659$	1011.91i	0.0598152i	−0.999553	+	0.0299076i	$0.990479\pi$
$660$	0	0
$661$	23619.4i	1.38985i	0.719084	+	0.694923i	$0.244561\pi$
$661$	23619.4i	1.38985i	−0.719084	+	0.694923i	$0.755439\pi$
$662$	0	0
$663$	−4922.67	−0.288357
$664$	0	0
$665$	−4163.28	−0.242775
$666$	0	0
$667$	26561.6i	1.54194i
$668$	0	0
$669$	14797.8i	0.855183i
$670$	0	0
$671$	7700.41	0.443027
$672$	0	0
$673$	−25811.9	−1.47842	−0.739208	−	0.673477i	$-0.764800\pi$
$673$	−25811.9	−1.47842	−0.739208	+	0.673477i	$0.764800\pi$
$674$	0	0
$675$	− 4826.64i	− 0.275226i
$676$	0	0
$677$	− 18255.2i	− 1.03634i	−0.855277	−	0.518172i	$-0.826613\pi$
$677$	− 18255.2i	− 1.03634i	0.855277	−	0.518172i	$-0.173387\pi$
$678$	0	0
$679$	700.438	0.0395881
$680$	0	0
$681$	10858.5	0.611009
$682$	0	0
$683$	− 20090.4i	− 1.12553i	−0.826617	−	0.562765i	$-0.809738\pi$
$683$	− 20090.4i	− 1.12553i	0.826617	−	0.562765i	$-0.190262\pi$
$684$	0	0
$685$	37839.0i	2.11059i
$686$	0	0
$687$	−917.278	−0.0509408
$688$	0	0
$689$	−12056.8	−0.666659
$690$	0	0
$691$	− 16521.5i	− 0.909563i	−0.890603	−	0.454782i	$-0.849717\pi$
$691$	− 16521.5i	− 0.909563i	0.890603	−	0.454782i	$-0.150283\pi$
$692$	0	0
$693$	− 286.014i	− 0.0156779i
$694$	0	0
$695$	28316.7	1.54549
$696$	0	0
$697$	16721.7	0.908720
$698$	0	0
$699$	− 1918.95i	− 0.103836i
$700$	0	0
$701$	12431.4i	0.669795i	0.942255	+	0.334897i	$0.108702\pi$
$701$	12431.4i	0.669795i	−0.942255	+	0.334897i	$0.891298\pi$
$702$	0	0
$703$	−21600.4	−1.15886
$704$	0	0
$705$	−22809.2	−1.21850
$706$	0	0
$707$	614.674i	0.0326976i
$708$	0	0
$709$	980.957i	0.0519614i	0.999662	+	0.0259807i	$0.00827084\pi$
$709$	980.957i	0.0519614i	−0.999662	+	0.0259807i	$0.991729\pi$
$710$	0	0
$711$	9169.36	0.483654
$712$	0	0
$713$	−40561.4	−2.13048
$714$	0	0
$715$	8023.06i	0.419644i
$716$	0	0
$717$	5234.81i	0.272660i
$718$	0	0
$719$	−4115.73	−0.213478	−0.106739	−	0.994287i	$-0.534041\pi$
$719$	−4115.73	−0.213478	−0.106739	+	0.994287i	$0.534041\pi$
$720$	0	0
$721$	1170.35	0.0604522
$722$	0	0
$723$	− 10071.9i	− 0.518088i
$724$	0	0
$725$	24833.5i	1.27213i
$726$	0	0
$727$	20850.1	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$727$	20850.1	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 2428.92i	− 0.122896i
$732$	0	0
$733$	− 31517.2i	− 1.58815i	−0.607821	−	0.794074i	$-0.707956\pi$
$733$	− 31517.2i	− 1.58815i	0.607821	−	0.794074i	$-0.292044\pi$
$734$	0	0
$735$	−17466.7	−0.876558
$736$	0	0
$737$	−9134.57	−0.456549
$738$	0	0
$739$	− 11415.0i	− 0.568213i	−0.958793	−	0.284106i	$-0.908303\pi$
$739$	− 11415.0i	− 0.568213i	0.958793	−	0.284106i	$-0.0916969\pi$
$740$	0	0
$741$	− 10380.4i	− 0.514622i
$742$	0	0
$743$	5732.08	0.283028	0.141514	−	0.989936i	$-0.454803\pi$
$743$	5732.08	0.283028	0.141514	+	0.989936i	$0.454803\pi$
$744$	0	0
$745$	−11096.2	−0.545683
$746$	0	0
$747$	4206.01i	0.206010i
$748$	0	0
$749$	− 2793.78i	− 0.136291i
$750$	0	0
$751$	−7843.07	−0.381089	−0.190544	−	0.981679i	$-0.561025\pi$
$751$	−7843.07	−0.381089	−0.190544	+	0.981679i	$0.561025\pi$
$752$	0	0
$753$	−3952.34	−0.191277
$754$	0	0
$755$	− 31709.0i	− 1.52849i
$756$	0	0
$757$	− 29125.9i	− 1.39841i	−0.714920	−	0.699206i	$-0.753537\pi$
$757$	− 29125.9i	− 1.39841i	0.714920	−	0.699206i	$-0.246463\pi$
$758$	0	0
$759$	6096.02	0.291530
$760$	0	0
$761$	14228.5	0.677768	0.338884	−	0.940828i	$-0.389951\pi$
$761$	14228.5	0.677768	0.338884	+	0.940828i	$0.389951\pi$
$762$	0	0
$763$	1746.71i	0.0828771i
$764$	0	0
$765$	5942.19i	0.280837i
$766$	0	0
$767$	−35953.6	−1.69258
$768$	0	0
$769$	−28133.7	−1.31928	−0.659641	−	0.751581i	$-0.729292\pi$
$769$	−28133.7	−1.31928	−0.659641	+	0.751581i	$0.729292\pi$
$770$	0	0
$771$	9108.35i	0.425459i
$772$	0	0
$773$	− 14686.9i	− 0.683377i	−0.939813	−	0.341689i	$-0.889001\pi$
$773$	− 14686.9i	− 0.683377i	0.939813	−	0.341689i	$-0.110999\pi$
$774$	0	0
$775$	−37922.5	−1.75770
$776$	0	0
$777$	2425.78	0.112000
$778$	0	0
$779$	35260.9i	1.62176i
$780$	0	0
$781$	− 7240.08i	− 0.331716i
$782$	0	0
$783$	3750.78	0.171190
$784$	0	0
$785$	−28870.9	−1.31267
$786$	0	0
$787$	− 21001.0i	− 0.951214i	−0.879658	−	0.475607i	$-0.842228\pi$
$787$	− 21001.0i	− 0.951214i	0.879658	−	0.475607i	$-0.157772\pi$
$788$	0	0
$789$	− 4967.27i	− 0.224131i
$790$	0	0
$791$	1742.48	0.0783253
$792$	0	0
$793$	−31385.6	−1.40547
$794$	0	0
$795$	14553.9i	0.649274i
$796$	0	0
$797$	− 13362.7i	− 0.593892i	−0.954894	−	0.296946i	$-0.904032\pi$
$797$	− 13362.7i	− 0.593892i	0.954894	−	0.296946i	$-0.0959682\pi$
$798$	0	0
$799$	16525.5	0.731704
$800$	0	0
$801$	−4596.36	−0.202752
$802$	0	0
$803$	8343.14i	0.366654i
$804$	0	0
$805$	9965.13i	0.436304i
$806$	0	0
$807$	15878.2	0.692612
$808$	0	0
$809$	−39481.6	−1.71582	−0.857911	−	0.513798i	$-0.828238\pi$
$809$	−39481.6	−1.71582	−0.857911	+	0.513798i	$0.828238\pi$
$810$	0	0
$811$	− 31157.5i	− 1.34906i	−0.738247	−	0.674531i	$-0.764346\pi$
$811$	− 31157.5i	− 1.34906i	0.738247	−	0.674531i	$-0.235654\pi$
$812$	0	0
$813$	− 24031.6i	− 1.03668i
$814$	0	0
$815$	38845.8	1.66958
$816$	0	0
$817$	5121.87	0.219329
$818$	0	0
$819$	1165.75i	0.0497368i
$820$	0	0
$821$	− 21229.9i	− 0.902470i	−0.892405	−	0.451235i	$-0.850984\pi$
$821$	− 21229.9i	− 0.902470i	0.892405	−	0.451235i	$-0.149016\pi$
$822$	0	0
$823$	24603.8	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$823$	24603.8	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$824$	0	0
$825$	5699.41	0.240519
$826$	0	0
$827$	− 13668.1i	− 0.574710i	−0.957824	−	0.287355i	$-0.907224\pi$
$827$	− 13668.1i	− 0.574710i	0.957824	−	0.287355i	$-0.0927759\pi$
$828$	0	0
$829$	− 27518.8i	− 1.15291i	−0.817127	−	0.576457i	$-0.804435\pi$
$829$	− 27518.8i	− 1.15291i	0.817127	−	0.576457i	$-0.195565\pi$
$830$	0	0
$831$	17078.4	0.712929
$832$	0	0
$833$	12654.9	0.526369
$834$	0	0
$835$	− 29054.1i	− 1.20414i
$836$	0	0
$837$	5727.68i	0.236532i
$838$	0	0
$839$	−29951.5	−1.23247	−0.616234	−	0.787563i	$-0.711343\pi$
$839$	−29951.5	−1.23247	−0.616234	+	0.787563i	$0.711343\pi$
$840$	0	0
$841$	5090.88	0.208737
$842$	0	0
$843$	− 6074.48i	− 0.248180i
$844$	0	0
$845$	5590.49i	0.227596i
$846$	0	0
$847$	−3642.38	−0.147761
$848$	0	0
$849$	−743.283	−0.0300464
$850$	0	0
$851$	51702.3i	2.08265i
$852$	0	0
$853$	5174.61i	0.207708i	0.994593	+	0.103854i	$0.0331175\pi$
$853$	5174.61i	0.207708i	−0.994593	+	0.103854i	$0.966882\pi$
$854$	0	0
$855$	−12530.3	−0.501201
$856$	0	0
$857$	9258.34	0.369030	0.184515	−	0.982830i	$-0.440929\pi$
$857$	9258.34	0.369030	0.184515	+	0.982830i	$0.440929\pi$
$858$	0	0
$859$	− 24353.0i	− 0.967304i	−0.875260	−	0.483652i	$-0.839310\pi$
$859$	− 24353.0i	− 0.967304i	0.875260	−	0.483652i	$-0.160690\pi$
$860$	0	0
$861$	− 3959.88i	− 0.156739i
$862$	0	0
$863$	−42283.4	−1.66784	−0.833919	−	0.551887i	$-0.813908\pi$
$863$	−42283.4	−1.66784	−0.833919	+	0.551887i	$0.813908\pi$
$864$	0	0
$865$	−43581.4	−1.71308
$866$	0	0
$867$	10433.8i	0.408709i
$868$	0	0
$869$	10827.4i	0.422663i
$870$	0	0
$871$	37231.0	1.44836
$872$	0	0
$873$	2108.12	0.0817286
$874$	0	0
$875$	2802.08i	0.108260i
$876$	0	0
$877$	49843.1i	1.91914i	0.281476	+	0.959568i	$0.409176\pi$
$877$	49843.1i	1.91914i	−0.281476	+	0.959568i	$0.590824\pi$
$878$	0	0
$879$	24400.8	0.936314
$880$	0	0
$881$	8986.94	0.343675	0.171837	−	0.985125i	$-0.445030\pi$
$881$	8986.94	0.343675	0.171837	+	0.985125i	$0.445030\pi$
$882$	0	0
$883$	3693.99i	0.140784i	0.997519	+	0.0703922i	$0.0224251\pi$
$883$	3693.99i	0.140784i	−0.997519	+	0.0703922i	$0.977575\pi$
$884$	0	0
$885$	43399.8i	1.64844i
$886$	0	0
$887$	−51613.0	−1.95377	−0.976886	−	0.213763i	$-0.931428\pi$
$887$	−51613.0	−1.95377	−0.976886	+	0.213763i	$0.931428\pi$
$888$	0	0
$889$	−4368.95	−0.164825
$890$	0	0
$891$	− 860.821i	− 0.0323665i
$892$	0	0
$893$	34847.4i	1.30585i
$894$	0	0
$895$	−6601.03	−0.246534
$896$	0	0
$897$	−24846.4	−0.924856
$898$	0	0
$899$	− 29469.5i	− 1.09328i
$900$	0	0
$901$	− 10544.5i	− 0.389886i
$902$	0	0
$903$	−575.197	−0.0211975
$904$	0	0
$905$	−53922.7	−1.98061
$906$	0	0
$907$	17016.8i	0.622971i	0.950251	+	0.311486i	$0.100827\pi$
$907$	17016.8i	0.622971i	−0.950251	+	0.311486i	$0.899173\pi$
$908$	0	0
$909$	1849.99i	0.0675032i
$910$	0	0
$911$	2991.72	0.108804	0.0544019	−	0.998519i	$-0.482675\pi$
$911$	2991.72	0.108804	0.0544019	+	0.998519i	$0.482675\pi$
$912$	0	0
$913$	−4966.55	−0.180032
$914$	0	0
$915$	37885.7i	1.36881i
$916$	0	0
$917$	294.344i	0.0105999i
$918$	0	0
$919$	−5174.82	−0.185747	−0.0928736	−	0.995678i	$-0.529605\pi$
$919$	−5174.82	−0.185747	−0.0928736	+	0.995678i	$0.529605\pi$
$920$	0	0
$921$	8924.46	0.319295
$922$	0	0
$923$	29509.3i	1.05234i
$924$	0	0
$925$	48338.6i	1.71823i
$926$	0	0
$927$	3522.41	0.124802
$928$	0	0
$929$	49256.5	1.73956	0.869780	−	0.493439i	$-0.164260\pi$
$929$	49256.5	1.73956	0.869780	+	0.493439i	$0.164260\pi$
$930$	0	0
$931$	26685.3i	0.939394i
$932$	0	0
$933$	− 13354.6i	− 0.468605i
$934$	0	0
$935$	−7016.69	−0.245423
$936$	0	0
$937$	31566.5	1.10057	0.550283	−	0.834978i	$-0.314520\pi$
$937$	31566.5	1.10057	0.550283	+	0.834978i	$0.314520\pi$
$938$	0	0
$939$	24821.3i	0.862632i
$940$	0	0
$941$	− 37575.6i	− 1.30173i	−0.759193	−	0.650866i	$-0.774406\pi$
$941$	− 37575.6i	− 1.30173i	0.759193	−	0.650866i	$-0.225594\pi$
$942$	0	0
$943$	84399.7	2.91456
$944$	0	0
$945$	1407.18	0.0484397
$946$	0	0
$947$	10289.3i	0.353070i	0.984294	+	0.176535i	$0.0564889\pi$
$947$	10289.3i	0.353070i	−0.984294	+	0.176535i	$0.943511\pi$
$948$	0	0
$949$	− 34005.2i	− 1.16318i
$950$	0	0
$951$	−1287.11	−0.0438879
$952$	0	0
$953$	−36779.7	−1.25017	−0.625085	−	0.780557i	$-0.714936\pi$
$953$	−36779.7	−1.25017	−0.625085	+	0.780557i	$0.714936\pi$
$954$	0	0
$955$	− 63736.9i	− 2.15966i
$956$	0	0
$957$	4429.01i	0.149602i
$958$	0	0
$959$	−6492.15	−0.218605
$960$	0	0
$961$	15210.9	0.510586
$962$	0	0
$963$	− 8408.47i	− 0.281370i
$964$	0	0
$965$	48593.6i	1.62102i
$966$	0	0
$967$	−35228.9	−1.17155	−0.585773	−	0.810475i	$-0.699209\pi$
$967$	−35228.9	−1.17155	−0.585773	+	0.810475i	$0.699209\pi$
$968$	0	0
$969$	9078.36	0.300969
$970$	0	0
$971$	− 15314.8i	− 0.506155i	−0.967446	−	0.253077i	$-0.918557\pi$
$971$	− 15314.8i	− 0.506155i	0.967446	−	0.253077i	$-0.0814428\pi$
$972$	0	0
$973$	4858.38i	0.160074i
$974$	0	0
$975$	−23229.9	−0.763027
$976$	0	0
$977$	24074.3	0.788337	0.394168	−	0.919038i	$-0.371033\pi$
$977$	24074.3	0.788337	0.394168	+	0.919038i	$0.371033\pi$
$978$	0	0
$979$	− 5427.49i	− 0.177184i
$980$	0	0
$981$	5257.10i	0.171097i
$982$	0	0
$983$	15244.1	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$983$	15244.1	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$984$	0	0
$985$	20003.7	0.647079
$986$	0	0
$987$	− 3913.43i	− 0.126207i
$988$	0	0
$989$	− 12259.6i	− 0.394168i
$990$	0	0
$991$	−37518.6	−1.20264	−0.601321	−	0.799008i	$-0.705359\pi$
$991$	−37518.6	−1.20264	−0.601321	+	0.799008i	$0.705359\pi$
$992$	0	0
$993$	−24590.1	−0.785845
$994$	0	0
$995$	84403.1i	2.68920i
$996$	0	0
$997$	− 1717.01i	− 0.0545420i	−0.999628	−	0.0272710i	$-0.991318\pi$
$997$	− 1717.01i	− 0.0545420i	0.999628	−	0.0272710i	$-0.00868170\pi$
$998$	0	0
$999$	7300.90	0.231221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.f.193.4	yes	8
3.2	odd	2		1152.4.d.p.577.1		8
4.3	odd	2	inner	384.4.d.f.193.8	yes	8
8.3	odd	2	inner	384.4.d.f.193.1	✓	8
8.5	even	2	inner	384.4.d.f.193.5	yes	8
12.11	even	2		1152.4.d.p.577.2		8
16.3	odd	4		768.4.a.u.1.4		4
16.5	even	4		768.4.a.u.1.1		4
16.11	odd	4		768.4.a.v.1.1		4
16.13	even	4		768.4.a.v.1.4		4
24.5	odd	2		1152.4.d.p.577.7		8
24.11	even	2		1152.4.d.p.577.8		8
48.5	odd	4		2304.4.a.cb.1.4		4
48.11	even	4		2304.4.a.by.1.4		4
48.29	odd	4		2304.4.a.by.1.1		4
48.35	even	4		2304.4.a.cb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.f.193.1	✓	8	8.3	odd	2	inner
384.4.d.f.193.4	yes	8	1.1	even	1	trivial
384.4.d.f.193.5	yes	8	8.5	even	2	inner
384.4.d.f.193.8	yes	8	4.3	odd	2	inner
768.4.a.u.1.1		4	16.5	even	4
768.4.a.u.1.4		4	16.3	odd	4
768.4.a.v.1.1		4	16.11	odd	4
768.4.a.v.1.4		4	16.13	even	4
1152.4.d.p.577.1		8	3.2	odd	2
1152.4.d.p.577.2		8	12.11	even	2
1152.4.d.p.577.7		8	24.5	odd	2
1152.4.d.p.577.8		8	24.11	even	2
2304.4.a.by.1.1		4	48.29	odd	4
2304.4.a.by.1.4		4	48.11	even	4
2304.4.a.cb.1.1		4	48.35	even	4
2304.4.a.cb.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +17.4288i q^{5} -2.99032 q^{7} -9.00000 q^{9} -10.6274i q^{11} +43.3156i q^{13} +52.2865 q^{15} -37.8823 q^{17} -79.8823i q^{19} +8.97095i q^{21} -191.204 q^{23} -178.765 q^{25} +27.0000i q^{27} -138.918i q^{29} +212.136 q^{31} -31.8823 q^{33} -52.1177i q^{35} -270.404i q^{37} +129.947 q^{39} -441.411 q^{41} +64.1177i q^{43} -156.860i q^{45} -436.234 q^{47} -334.058 q^{49} +113.647i q^{51} +278.348i q^{53} +185.224 q^{55} -239.647 q^{57} +830.039i q^{59} +724.580i q^{61} +26.9128 q^{63} -754.940 q^{65} -859.529i q^{67} +573.613i q^{69} +681.264 q^{71} -785.058 q^{73} +536.294i q^{75} +31.7793i q^{77} -1018.82 q^{79} +81.0000 q^{81} -467.334i q^{83} -660.244i q^{85} -416.753 q^{87} +510.706 q^{89} -129.527i q^{91} -636.409i q^{93} +1392.26 q^{95} -234.235 q^{97} +95.6468i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) 8 * q - 72 * q^9 + 240 * q^17 - 344 * q^25 + 288 * q^33 - 816 * q^41 + 1672 * q^49 - 288 * q^57 - 1152 * q^65 - 1936 * q^73 + 648 * q^81 + 7344 * q^89 - 2960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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