LMFDB - Embedded newform 384.3.h.g.65.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(65,384)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.h (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 $ x^16 + 28x^14 + 274x^12 + 1236x^10 + 2703x^8 + 2676x^6 + 946x^4 + 64*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.8
Root			$-1.26301i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.3.h.g.65.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.628052 + 2.93352i) q^{3} +6.51323 q^{5} +7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +O(q^{10})$	\(q+(-0.628052 + 2.93352i) q^{3} +6.51323 q^{5} +7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +17.8506 q^{11} -14.4222i q^{13} +(-4.09065 + 19.1067i) q^{15} -18.6833i q^{17} -12.9726i q^{19} +(-4.80048 + 22.4222i) q^{21} +28.1646i q^{23} +17.4222 q^{25} +(15.9665 - 21.7732i) q^{27} +3.08772 q^{29} -32.1873 q^{31} +(-11.2111 + 52.3652i) q^{33} +49.7835 q^{35} -1.57779i q^{37} +(42.3079 + 9.05789i) q^{39} +18.1647i q^{41} +22.2296i q^{43} +(-53.4808 - 24.0000i) q^{45} +86.4757i q^{47} +9.42221 q^{49} +(54.8079 + 11.7341i) q^{51} +25.7151 q^{53} +116.265 q^{55} +(38.0555 + 8.14748i) q^{57} -11.3049 q^{59} +91.2666i q^{61} +(-62.7611 - 28.1646i) q^{63} -93.9352i q^{65} -17.6011i q^{67} +(-82.6215 - 17.6888i) q^{69} -58.3111i q^{71} -104.533 q^{73} +(-10.9420 + 51.1084i) q^{75} +136.440 q^{77} -123.909 q^{79} +(53.8444 + 60.5126i) q^{81} -81.7164 q^{83} -121.689i q^{85} +(-1.93925 + 9.05789i) q^{87} -142.097i q^{89} -110.235i q^{91} +(20.2153 - 94.4222i) q^{93} -84.4938i q^{95} -58.1110 q^{97} +(-146.573 - 65.7760i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) $ 16 * q - 16 * q^9 + 48 * q^25 - 64 * q^33 - 80 * q^49 + 32 * q^57 - 288 * q^73 + 400 * q^81 + 224 * 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
$3$	−0.628052	+	2.93352i	−0.209351	+	0.977841i
$3$	−0.628052	+	2.93352i	−0.209351	+	0.977841i
$4$		0			0
$5$	6.51323			1.30265			0.651323	−	0.758800i	$-0.274214\pi$
$5$	6.51323			1.30265			0.651323	+	0.758800i	$0.274214\pi$
$6$		0			0
$7$	7.64344			1.09192			0.545960	−	0.837811i	$-0.316165\pi$
$7$	7.64344			1.09192			0.545960	+	0.837811i	$0.316165\pi$
$8$		0			0
$9$	−8.21110	−	3.68481i	−0.912345	−	0.409423i
$10$		0			0
$11$	17.8506			1.62278			0.811391	−	0.584503i	$-0.198711\pi$
$11$	17.8506			1.62278			0.811391	+	0.584503i	$0.198711\pi$
$12$		0			0
$13$		−	14.4222i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	14.4222i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$	−4.09065	+	19.1067i	−0.272710	+	1.27378i
$16$		0			0
$17$		−	18.6833i		−	1.09902i	−0.835488	−	0.549509i	$-0.814815\pi$
$17$		−	18.6833i		−	1.09902i	0.835488	−	0.549509i	$-0.185185\pi$
$18$		0			0
$19$		−	12.9726i		−	0.682770i	−0.939923	−	0.341385i	$-0.889104\pi$
$19$		−	12.9726i		−	0.682770i	0.939923	−	0.341385i	$-0.110896\pi$
$20$		0			0
$21$	−4.80048	+	22.4222i	−0.228594	+	1.06772i
$22$		0			0
$23$			28.1646i			1.22455i	0.790646	+	0.612274i	$0.209745\pi$
$23$			28.1646i			1.22455i	−0.790646	+	0.612274i	$0.790255\pi$
$24$		0			0
$25$	17.4222			0.696888
$26$		0			0
$27$	15.9665	−	21.7732i	0.591350	−	0.806415i
$28$		0			0
$29$	3.08772			0.106473			0.0532365	−	0.998582i	$-0.483046\pi$
$29$	3.08772			0.106473			0.0532365	+	0.998582i	$0.483046\pi$
$30$		0			0
$31$	−32.1873			−1.03830			−0.519150	−	0.854683i	$-0.673752\pi$
$31$	−32.1873			−1.03830			−0.519150	+	0.854683i	$0.673752\pi$
$32$		0			0
$33$	−11.2111	+	52.3652i	−0.339730	+	1.58682i
$34$		0			0
$35$	49.7835			1.42239
$36$		0			0
$37$		−	1.57779i		−	0.0426431i	−0.999773	−	0.0213216i	$-0.993213\pi$
$37$		−	1.57779i		−	0.0426431i	0.999773	−	0.0213216i	$-0.00678738\pi$
$38$		0			0
$39$	42.3079	+	9.05789i	1.08482	+	0.232254i
$40$		0			0
$41$			18.1647i			0.443042i	0.975156	+	0.221521i	$0.0711022\pi$
$41$			18.1647i			0.443042i	−0.975156	+	0.221521i	$0.928898\pi$
$42$		0			0
$43$			22.2296i			0.516968i	0.966016	+	0.258484i	$0.0832229\pi$
$43$			22.2296i			0.516968i	−0.966016	+	0.258484i	$0.916777\pi$
$44$		0			0
$45$	−53.4808	−	24.0000i	−1.18846	−	0.533333i
$46$		0			0
$47$			86.4757i			1.83991i	0.392026	+	0.919954i	$0.371774\pi$
$47$			86.4757i			1.83991i	−0.392026	+	0.919954i	$0.628226\pi$
$48$		0			0
$49$	9.42221			0.192290
$50$		0			0
$51$	54.8079	+	11.7341i	1.07467	+	0.230080i
$52$		0			0
$53$	25.7151			0.485191			0.242596	−	0.970128i	$-0.422001\pi$
$53$	25.7151			0.485191			0.242596	+	0.970128i	$0.422001\pi$
$54$		0			0
$55$	116.265			2.11391
$56$		0			0
$57$	38.0555	+	8.14748i	0.667641	+	0.142938i
$58$		0			0
$59$	−11.3049			−0.191609			−0.0958045	−	0.995400i	$-0.530542\pi$
$59$	−11.3049			−0.191609			−0.0958045	+	0.995400i	$0.530542\pi$
$60$		0			0
$61$			91.2666i			1.49617i	0.663601	+	0.748087i	$0.269027\pi$
$61$			91.2666i			1.49617i	−0.663601	+	0.748087i	$0.730973\pi$
$62$		0			0
$63$	−62.7611	−	28.1646i	−0.996208	−	0.447057i
$64$		0			0
$65$		−	93.9352i		−	1.44516i
$66$		0			0
$67$		−	17.6011i		−	0.262703i	−0.991336	−	0.131352i	$-0.958068\pi$
$67$		−	17.6011i		−	0.262703i	0.991336	−	0.131352i	$-0.0419317\pi$
$68$		0			0
$69$	−82.6215	−	17.6888i	−1.19741	−	0.256360i
$70$		0			0
$71$		−	58.3111i		−	0.821283i	−0.911797	−	0.410641i	$-0.865305\pi$
$71$		−	58.3111i		−	0.821283i	0.911797	−	0.410641i	$-0.134695\pi$
$72$		0			0
$73$	−104.533			−1.43196			−0.715981	−	0.698120i	$-0.754020\pi$
$73$	−104.533			−1.43196			−0.715981	+	0.698120i	$0.754020\pi$
$74$		0			0
$75$	−10.9420	+	51.1084i	−0.145894	+	0.681446i
$76$		0			0
$77$	136.440			1.77195
$78$		0			0
$79$	−123.909			−1.56846			−0.784232	−	0.620468i	$-0.786943\pi$
$79$	−123.909			−1.56846			−0.784232	+	0.620468i	$0.786943\pi$
$80$		0			0
$81$	53.8444	+	60.5126i	0.664746	+	0.747070i
$82$		0			0
$83$	−81.7164			−0.984535			−0.492268	−	0.870444i	$-0.663832\pi$
$83$	−81.7164			−0.984535			−0.492268	+	0.870444i	$0.663832\pi$
$84$		0			0
$85$		−	121.689i		−	1.43163i
$86$		0			0
$87$	−1.93925	+	9.05789i	−0.0222902	+	0.104114i
$88$		0			0
$89$		−	142.097i		−	1.59659i	−0.602263	−	0.798297i	$-0.705734\pi$
$89$		−	142.097i		−	1.59659i	0.602263	−	0.798297i	$-0.294266\pi$
$90$		0			0
$91$		−	110.235i		−	1.21138i
$92$		0			0
$93$	20.2153	−	94.4222i	0.217369	−	1.01529i
$94$		0			0
$95$		−	84.4938i		−	0.889408i
$96$		0			0
$97$	−58.1110			−0.599083			−0.299541	−	0.954083i	$-0.596834\pi$
$97$	−58.1110			−0.599083			−0.299541	+	0.954083i	$0.596834\pi$
$98$		0			0
$99$	−146.573	−	65.7760i	−1.48054	−	0.664404i
$100$		0			0
$101$	119.650			1.18466			0.592328	−	0.805697i	$-0.298209\pi$
$101$	119.650			1.18466			0.592328	+	0.805697i	$0.298209\pi$
$102$		0			0
$103$	−4.41634			−0.0428771			−0.0214386	−	0.999770i	$-0.506825\pi$
$103$	−4.41634			−0.0428771			−0.0214386	+	0.999770i	$0.506825\pi$
$104$		0			0
$105$	−31.2666	+	146.041i	−0.297777	+	1.39087i
$106$		0			0
$107$	5.75018			0.0537400			0.0268700	−	0.999639i	$-0.491446\pi$
$107$	5.75018			0.0537400			0.0268700	+	0.999639i	$0.491446\pi$
$108$		0			0
$109$			155.267i			1.42446i	0.701944	+	0.712232i	$0.252316\pi$
$109$			155.267i			1.42446i	−0.701944	+	0.712232i	$0.747684\pi$
$110$		0			0
$111$	4.62850	+	0.990937i	0.0416982	+	0.00892736i
$112$		0			0
$113$		−	30.5156i		−	0.270050i	−0.990842	−	0.135025i	$-0.956889\pi$
$113$		−	30.5156i		−	0.270050i	0.990842	−	0.135025i	$-0.0431114\pi$
$114$		0			0
$115$			183.443i			1.59515i
$116$		0			0
$117$	−53.1430	+	118.422i	−0.454214	+	1.01216i
$118$		0			0
$119$		−	142.805i		−	1.20004i
$120$		0			0
$121$	197.644			1.63342
$122$		0			0
$123$	−53.2867	−	11.4084i	−0.433225	−	0.0927512i
$124$		0			0
$125$	−49.3559			−0.394848
$126$		0			0
$127$	−13.6733			−0.107664			−0.0538320	−	0.998550i	$-0.517144\pi$
$127$	−13.6733			−0.107664			−0.0538320	+	0.998550i	$0.517144\pi$
$128$		0			0
$129$	−65.2111	−	13.9614i	−0.505512	−	0.108228i
$130$		0			0
$131$	−110.872			−0.846351			−0.423175	−	0.906048i	$-0.639085\pi$
$131$	−110.872			−0.846351			−0.423175	+	0.906048i	$0.639085\pi$
$132$		0			0
$133$		−	99.1556i		−	0.745531i
$134$		0			0
$135$	103.993	−	141.814i	0.770320	−	1.05047i
$136$		0			0
$137$		−	109.712i		−	0.800814i	−0.916337	−	0.400407i	$-0.868869\pi$
$137$		−	109.712i		−	0.800814i	0.916337	−	0.400407i	$-0.131131\pi$
$138$		0			0
$139$			210.301i			1.51295i	0.654020	+	0.756477i	$0.273081\pi$
$139$			210.301i			1.51295i	−0.654020	+	0.756477i	$0.726919\pi$
$140$		0			0
$141$	−253.678	−	54.3112i	−1.79914	−	0.385186i
$142$		0			0
$143$		−	257.445i		−	1.80032i
$144$		0			0
$145$	20.1110			0.138697
$146$		0			0
$147$	−5.91763	+	27.6402i	−0.0402560	+	0.188029i
$148$		0			0
$149$	−45.5926			−0.305991			−0.152995	−	0.988227i	$-0.548892\pi$
$149$	−45.5926			−0.305991			−0.152995	+	0.988227i	$0.548892\pi$
$150$		0			0
$151$	−22.9303			−0.151856			−0.0759282	−	0.997113i	$-0.524192\pi$
$151$	−22.9303			−0.151856			−0.0759282	+	0.997113i	$0.524192\pi$
$152$		0			0
$153$	−68.8444	+	153.411i	−0.449963	+	1.00268i
$154$		0			0
$155$	−209.644			−1.35254
$156$		0			0
$157$		−	123.489i		−	0.786552i	−0.919420	−	0.393276i	$-0.871342\pi$
$157$		−	123.489i		−	0.786552i	0.919420	−	0.393276i	$-0.128658\pi$
$158$		0			0
$159$	−16.1504	+	75.4359i	−0.101575	+	0.474440i
$160$		0			0
$161$			215.274i			1.33711i
$162$		0			0
$163$		−	163.039i		−	1.00024i	−0.865957	−	0.500119i	$-0.833290\pi$
$163$		−	163.039i		−	1.00024i	0.865957	−	0.500119i	$-0.166710\pi$
$164$		0			0
$165$	−73.0205	+	341.066i	−0.442549	+	2.06707i
$166$		0			0
$167$			144.787i			0.866987i	0.901157	+	0.433493i	$0.142719\pi$
$167$			144.787i			0.866987i	−0.901157	+	0.433493i	$0.857281\pi$
$168$		0			0
$169$	−39.0000			−0.230769
$170$		0			0
$171$	−47.8016	+	106.520i	−0.279542	+	0.622922i
$172$		0			0
$173$	168.331			0.973010			0.486505	−	0.873678i	$-0.338272\pi$
$173$	168.331			0.973010			0.486505	+	0.873678i	$0.338272\pi$
$174$		0			0
$175$	133.166			0.760946
$176$		0			0
$177$	7.10008	−	33.1633i	0.0401134	−	0.187363i
$178$		0			0
$179$	105.317			0.588364			0.294182	−	0.955749i	$-0.404953\pi$
$179$	105.317			0.588364			0.294182	+	0.955749i	$0.404953\pi$
$180$		0			0
$181$		−	30.2002i		−	0.166852i	−0.996514	−	0.0834258i	$-0.973414\pi$
$181$		−	30.2002i		−	0.166852i	0.996514	−	0.0834258i	$-0.0265862\pi$
$182$		0			0
$183$	−267.733	−	57.3201i	−1.46302	−	0.313225i
$184$		0			0
$185$		−	10.2765i		−	0.0555489i
$186$		0			0
$187$		−	333.509i		−	1.78347i
$188$		0			0
$189$	122.039	−	166.422i	0.645707	−	0.880541i
$190$		0			0
$191$			255.463i			1.33750i	0.743486	+	0.668752i	$0.233171\pi$
$191$			255.463i			1.33750i	−0.743486	+	0.668752i	$0.766829\pi$
$192$		0			0
$193$	286.222			1.48302			0.741508	−	0.670944i	$-0.234111\pi$
$193$	286.222			1.48302			0.741508	+	0.670944i	$0.234111\pi$
$194$		0			0
$195$	275.561	+	58.9961i	1.41313	+	0.302544i
$196$		0			0
$197$	121.002			0.614221			0.307110	−	0.951674i	$-0.400638\pi$
$197$	121.002			0.614221			0.307110	+	0.951674i	$0.400638\pi$
$198$		0			0
$199$	−243.401			−1.22312			−0.611560	−	0.791198i	$-0.709458\pi$
$199$	−243.401			−1.22312			−0.611560	+	0.791198i	$0.709458\pi$
$200$		0			0
$201$	51.6333	+	11.0544i	0.256882	+	0.0549971i
$202$		0			0
$203$	23.6008			0.116260
$204$		0			0
$205$			118.311i			0.577128i
$206$		0			0
$207$	103.781	−	231.262i	0.501358	−	1.11721i
$208$		0			0
$209$		−	231.569i		−	1.10799i
$210$		0			0
$211$		−	112.125i		−	0.531399i	−0.964056	−	0.265700i	$-0.914397\pi$
$211$		−	112.125i		−	0.531399i	0.964056	−	0.265700i	$-0.0856029\pi$
$212$		0			0
$213$	171.057	+	36.6224i	0.803084	+	0.171936i
$214$		0			0
$215$			144.787i			0.673427i
$216$		0			0
$217$	−246.022			−1.13374
$218$		0			0
$219$	65.6523	−	306.651i	0.299782	−	1.40023i
$220$		0			0
$221$	−269.455			−1.21925
$222$		0			0
$223$	341.152			1.52983			0.764915	−	0.644131i	$-0.222781\pi$
$223$	341.152			1.52983			0.764915	+	0.644131i	$0.222781\pi$
$224$		0			0
$225$	−143.056	−	64.1974i	−0.635802	−	0.285322i
$226$		0			0
$227$	57.1247			0.251651			0.125825	−	0.992052i	$-0.459842\pi$
$227$	57.1247			0.251651			0.125825	+	0.992052i	$0.459842\pi$
$228$		0			0
$229$		−	254.644i		−	1.11198i	−0.831188	−	0.555992i	$-0.812339\pi$
$229$		−	254.644i		−	1.11198i	0.831188	−	0.555992i	$-0.187661\pi$
$230$		0			0
$231$	−85.6914	+	400.250i	−0.370958	+	1.73268i
$232$		0			0
$233$		−	117.395i		−	0.503842i	−0.967748	−	0.251921i	$-0.918938\pi$
$233$		−	117.395i		−	0.503842i	0.967748	−	0.251921i	$-0.0810623\pi$
$234$		0			0
$235$			563.236i			2.39675i
$236$		0			0
$237$	77.8210	−	363.489i	0.328359	−	1.53371i
$238$		0			0
$239$			138.841i			0.580925i	0.956886	+	0.290463i	$0.0938092\pi$
$239$			138.841i			0.580925i	−0.956886	+	0.290463i	$0.906191\pi$
$240$		0			0
$241$	−379.911			−1.57639			−0.788197	−	0.615423i	$-0.788985\pi$
$241$	−379.911			−1.57639			−0.788197	+	0.615423i	$0.788985\pi$
$242$		0			0
$243$	−211.332	+	119.949i	−0.869680	+	0.493616i
$244$		0			0
$245$	61.3690			0.250486
$246$		0			0
$247$	−187.094			−0.757466
$248$		0			0
$249$	51.3221	−	239.717i	0.206113	−	0.962719i
$250$		0			0
$251$	399.064			1.58990			0.794948	−	0.606678i	$-0.207498\pi$
$251$	399.064			1.58990			0.794948	+	0.606678i	$0.207498\pi$
$252$		0			0
$253$			502.755i			1.98718i
$254$		0			0
$255$	356.977	+	76.4268i	1.39991	+	0.299713i
$256$		0			0
$257$		−	78.4728i		−	0.305342i	−0.988277	−	0.152671i	$-0.951213\pi$
$257$		−	78.4728i		−	0.305342i	0.988277	−	0.152671i	$-0.0487874\pi$
$258$		0			0
$259$		−	12.0598i		−	0.0465629i
$260$		0			0
$261$	−25.3536	−	11.3776i	−0.0971401	−	0.0435925i
$262$		0			0
$263$			1.98187i			0.00753564i	0.999993	+	0.00376782i	$0.00119934\pi$
$263$			1.98187i			0.00753564i	−0.999993	+	0.00376782i	$0.998801\pi$
$264$		0			0
$265$	167.489			0.632033
$266$		0			0
$267$	416.845	+	89.2442i	1.56122	+	0.334248i
$268$		0			0
$269$	47.6194			0.177024			0.0885119	−	0.996075i	$-0.471789\pi$
$269$	47.6194			0.177024			0.0885119	+	0.996075i	$0.471789\pi$
$270$		0			0
$271$	181.829			0.670956			0.335478	−	0.942048i	$-0.391102\pi$
$271$	181.829			0.670956			0.335478	+	0.942048i	$0.391102\pi$
$272$		0			0
$273$	323.378	+	69.2334i	1.18453	+	0.253602i
$274$		0			0
$275$	310.997			1.13090
$276$		0			0
$277$		−	71.8890i		−	0.259527i	−0.991545	−	0.129763i	$-0.958578\pi$
$277$		−	71.8890i		−	0.259527i	0.991545	−	0.129763i	$-0.0414218\pi$
$278$		0			0
$279$	264.293	+	118.604i	0.947288	+	0.425104i
$280$		0			0
$281$			459.509i			1.63526i	0.575742	+	0.817631i	$0.304713\pi$
$281$			459.509i			1.63526i	−0.575742	+	0.817631i	$0.695287\pi$
$282$		0			0
$283$		−	343.678i		−	1.21441i	−0.794545	−	0.607206i	$-0.792290\pi$
$283$		−	343.678i		−	1.21441i	0.794545	−	0.607206i	$-0.207710\pi$
$284$		0			0
$285$	247.864	+	53.0665i	0.869700	+	0.186198i
$286$		0			0
$287$			138.841i			0.483767i
$288$		0			0
$289$	−60.0665			−0.207842
$290$		0			0
$291$	36.4967	−	170.470i	0.125418	−	0.585807i
$292$		0			0
$293$	−431.610			−1.47307			−0.736536	−	0.676399i	$-0.763540\pi$
$293$	−431.610			−1.47307			−0.736536	+	0.676399i	$0.763540\pi$
$294$		0			0
$295$	−73.6316			−0.249599
$296$		0			0
$297$	285.011	−	388.665i	0.959633	−	1.30864i
$298$		0			0
$299$	406.196			1.35851
$300$		0			0
$301$			169.911i			0.564488i
$302$		0			0
$303$	−75.1466	+	350.997i	−0.248008	+	1.15841i
$304$		0			0
$305$			594.441i			1.94899i
$306$		0			0
$307$			221.383i			0.721119i	0.932736	+	0.360559i	$0.117414\pi$
$307$			221.383i			0.721119i	−0.932736	+	0.360559i	$0.882586\pi$
$308$		0			0
$309$	2.77369	−	12.9554i	0.00897634	−	0.0419270i
$310$		0			0
$311$			170.969i			0.549741i	0.961481	+	0.274871i	$0.0886350\pi$
$311$			170.969i			0.549741i	−0.961481	+	0.274871i	$0.911365\pi$
$312$		0			0
$313$	−17.2666			−0.0551649			−0.0275825	−	0.999620i	$-0.508781\pi$
$313$	−17.2666			−0.0551649			−0.0275825	+	0.999620i	$0.508781\pi$
$314$		0			0
$315$	−408.778	−	183.443i	−1.29771	−	0.582357i
$316$		0			0
$317$	103.874			0.327678			0.163839	−	0.986487i	$-0.447612\pi$
$317$	103.874			0.327678			0.163839	+	0.986487i	$0.447612\pi$
$318$		0			0
$319$	55.1176			0.172783
$320$		0			0
$321$	−3.61141	+	16.8683i	−0.0112505	+	0.0525492i
$322$		0			0
$323$	−242.372			−0.750377
$324$		0			0
$325$		−	251.267i		−	0.773128i
$326$		0			0
$327$	−455.478	−	97.5154i	−1.39290	−	0.298212i
$328$		0			0
$329$			660.972i			2.00903i
$330$		0			0
$331$		−	189.961i		−	0.573900i	−0.957945	−	0.286950i	$-0.907359\pi$
$331$		−	189.961i		−	0.573900i	0.957945	−	0.286950i	$-0.0926414\pi$
$332$		0			0
$333$	−5.81387	+	12.9554i	−0.0174591	+	0.0389052i
$334$		0			0
$335$		−	114.640i		−	0.342210i
$336$		0			0
$337$	−157.889			−0.468513			−0.234257	−	0.972175i	$-0.575266\pi$
$337$	−157.889			−0.468513			−0.234257	+	0.972175i	$0.575266\pi$
$338$		0			0
$339$	89.5182	+	19.1654i	0.264066	+	0.0565350i
$340$		0			0
$341$	−574.563			−1.68494
$342$		0			0
$343$	−302.511			−0.881955
$344$		0			0
$345$	−538.133	−	115.211i	−1.55981	−	0.333946i
$346$		0			0
$347$	−258.632			−0.745336			−0.372668	−	0.927965i	$-0.621557\pi$
$347$	−258.632			−0.745336			−0.372668	+	0.927965i	$0.621557\pi$
$348$		0			0
$349$			228.955i			0.656033i	0.944672	+	0.328016i	$0.106380\pi$
$349$			228.955i			0.656033i	−0.944672	+	0.328016i	$0.893620\pi$
$350$		0			0
$351$	−314.018	−	230.271i	−0.894637	−	0.656044i
$352$		0			0
$353$		−	523.133i		−	1.48196i	−0.671525	−	0.740982i	$-0.734361\pi$
$353$		−	523.133i		−	1.48196i	0.671525	−	0.740982i	$-0.265639\pi$
$354$		0			0
$355$		−	379.794i		−	1.06984i
$356$		0			0
$357$	418.921	+	89.6888i	1.17345	+	0.251229i
$358$		0			0
$359$			114.640i			0.319332i	0.987171	+	0.159666i	$0.0510418\pi$
$359$			114.640i			0.319332i	−0.987171	+	0.159666i	$0.948958\pi$
$360$		0			0
$361$	192.711			0.533825
$362$		0			0
$363$	−124.131	+	579.794i	−0.341958	+	1.59723i
$364$		0			0
$365$	−680.849			−1.86534
$366$		0			0
$367$	188.283			0.513033			0.256517	−	0.966540i	$-0.417425\pi$
$367$	188.283			0.513033			0.256517	+	0.966540i	$0.417425\pi$
$368$		0			0
$369$	66.9335	−	149.153i	0.181392	−	0.404207i
$370$		0			0
$371$	196.552			0.529790
$372$		0			0
$373$		−	30.6443i		−	0.0821562i	−0.999156	−	0.0410781i	$-0.986921\pi$
$373$		−	30.6443i		−	0.0821562i	0.999156	−	0.0410781i	$-0.0130792\pi$
$374$		0			0
$375$	30.9981	−	144.787i	0.0826615	−	0.386098i
$376$		0			0
$377$		−	44.5317i		−	0.118121i
$378$		0			0
$379$		−	216.755i		−	0.571913i	−0.958243	−	0.285956i	$-0.907689\pi$
$379$		−	216.755i		−	0.571913i	0.958243	−	0.285956i	$-0.0923112\pi$
$380$		0			0
$381$	8.58756	−	40.1110i	0.0225395	−	0.105278i
$382$		0			0
$383$		−	653.731i		−	1.70687i	−0.521199	−	0.853435i	$-0.674515\pi$
$383$		−	653.731i		−	1.70687i	0.521199	−	0.853435i	$-0.325485\pi$
$384$		0			0
$385$	888.666			2.30822
$386$		0			0
$387$	81.9119	−	182.530i	0.211659	−	0.471653i
$388$		0			0
$389$	227.288			0.584287			0.292143	−	0.956375i	$-0.405632\pi$
$389$	227.288			0.584287			0.292143	+	0.956375i	$0.405632\pi$
$390$		0			0
$391$	526.208			1.34580
$392$		0			0
$393$	69.6333	−	325.245i	0.177184	−	0.827596i
$394$		0			0
$395$	−807.046			−2.04315
$396$		0			0
$397$		−	338.244i		−	0.852000i	−0.904723	−	0.426000i	$-0.859922\pi$
$397$		−	338.244i		−	0.852000i	0.904723	−	0.426000i	$-0.140078\pi$
$398$		0			0
$399$	290.875	+	62.2748i	0.729010	+	0.156077i
$400$		0			0
$401$			217.553i			0.542527i	0.962505	+	0.271264i	$0.0874415\pi$
$401$			217.553i			0.542527i	−0.962505	+	0.271264i	$0.912559\pi$
$402$		0			0
$403$			464.212i			1.15189i
$404$		0			0
$405$	350.701	+	394.133i	0.865929	+	0.973168i
$406$		0			0
$407$		−	28.1646i		−	0.0692005i
$408$		0			0
$409$	−225.822			−0.552131			−0.276066	−	0.961139i	$-0.589031\pi$
$409$	−225.822			−0.552131			−0.276066	+	0.961139i	$0.589031\pi$
$410$		0			0
$411$	321.841	+	68.9045i	0.783069	+	0.167651i
$412$		0			0
$413$	−86.4086			−0.209222
$414$		0			0
$415$	−532.238			−1.28250
$416$		0			0
$417$	−616.922	−	132.080i	−1.47943	−	0.316738i
$418$		0			0
$419$	−185.247			−0.442117			−0.221059	−	0.975260i	$-0.570951\pi$
$419$	−185.247			−0.442117			−0.221059	+	0.975260i	$0.570951\pi$
$420$		0			0
$421$			72.1110i			0.171285i	0.996326	+	0.0856425i	$0.0272943\pi$
$421$			72.1110i			0.171285i	−0.996326	+	0.0856425i	$0.972706\pi$
$422$		0			0
$423$	318.646	−	710.061i	0.753300	−	1.67863i
$424$		0			0
$425$		−	325.505i		−	0.765893i
$426$		0			0
$427$			697.591i			1.63370i
$428$		0			0
$429$	755.221	+	161.689i	1.76042	+	0.376897i
$430$		0			0
$431$			380.013i			0.881701i	0.897581	+	0.440850i	$0.145323\pi$
$431$			380.013i			0.881701i	−0.897581	+	0.440850i	$0.854677\pi$
$432$		0			0
$433$	−451.800			−1.04342			−0.521709	−	0.853124i	$-0.674705\pi$
$433$	−451.800			−1.04342			−0.521709	+	0.853124i	$0.674705\pi$
$434$		0			0
$435$	−12.6308	+	58.9961i	−0.0290362	+	0.135623i
$436$		0			0
$437$	365.369			0.836085
$438$		0			0
$439$	173.421			0.395036			0.197518	−	0.980299i	$-0.436712\pi$
$439$	173.421			0.395036			0.197518	+	0.980299i	$0.436712\pi$
$440$		0			0
$441$	−77.3667	−	34.7190i	−0.175435	−	0.0787279i
$442$		0			0
$443$	−466.893			−1.05394			−0.526968	−	0.849885i	$-0.676671\pi$
$443$	−466.893			−1.05394			−0.526968	+	0.849885i	$0.676671\pi$
$444$		0			0
$445$		−	925.511i		−	2.07980i
$446$		0			0
$447$	28.6345	−	133.747i	0.0640593	−	0.299210i
$448$		0			0
$449$			694.394i			1.54654i	0.634080	+	0.773268i	$0.281379\pi$
$449$			694.394i			1.54654i	−0.634080	+	0.773268i	$0.718621\pi$
$450$		0			0
$451$			324.252i			0.718962i
$452$		0			0
$453$	14.4014	−	67.2666i	0.0317912	−	0.148491i
$454$		0			0
$455$		−	717.988i		−	1.57800i
$456$		0			0
$457$	−113.378			−0.248091			−0.124046	−	0.992277i	$-0.539587\pi$
$457$	−113.378			−0.248091			−0.124046	+	0.992277i	$0.539587\pi$
$458$		0			0
$459$	−406.796	−	298.306i	−0.886265	−	0.649905i
$460$		0			0
$461$	247.213			0.536253			0.268126	−	0.963384i	$-0.413595\pi$
$461$	247.213			0.536253			0.268126	+	0.963384i	$0.413595\pi$
$462$		0			0
$463$	−625.149			−1.35021			−0.675107	−	0.737720i	$-0.735902\pi$
$463$	−625.149			−1.35021			−0.675107	+	0.737720i	$0.735902\pi$
$464$		0			0
$465$	131.667	−	614.994i	0.283155	−	1.32257i
$466$		0			0
$467$	−756.467			−1.61984			−0.809922	−	0.586538i	$-0.800490\pi$
$467$	−756.467			−1.61984			−0.809922	+	0.586538i	$0.800490\pi$
$468$		0			0
$469$		−	134.533i		−	0.286851i
$470$		0			0
$471$	362.257	+	77.5572i	0.769123	+	0.164665i
$472$		0			0
$473$			396.812i			0.838927i
$474$		0			0
$475$		−	226.012i		−	0.475815i
$476$		0			0
$477$	−211.150	−	94.7553i	−0.442662	−	0.198648i
$478$		0			0
$479$		−	510.927i		−	1.06665i	−0.845910	−	0.533326i	$-0.820942\pi$
$479$		−	510.927i		−	1.06665i	0.845910	−	0.533326i	$-0.179058\pi$
$480$		0			0
$481$	−22.7553			−0.0473083
$482$		0			0
$483$	−631.512	−	135.203i	−1.30748	−	0.279924i
$484$		0			0
$485$	−378.491			−0.780393
$486$		0			0
$487$	368.923			0.757542			0.378771	−	0.925490i	$-0.376347\pi$
$487$	368.923			0.757542			0.378771	+	0.925490i	$0.376347\pi$
$488$		0			0
$489$	478.278	+	102.397i	0.978073	+	0.209400i
$490$		0			0
$491$	567.842			1.15650			0.578251	−	0.815859i	$-0.303736\pi$
$491$	567.842			1.15650			0.578251	+	0.815859i	$0.303736\pi$
$492$		0			0
$493$		−	57.6888i		−	0.117016i
$494$		0			0
$495$	−954.665	−	428.415i	−1.92862	−	0.865484i
$496$		0			0
$497$		−	445.697i		−	0.896775i
$498$		0			0
$499$			28.6838i			0.0574826i	0.999587	+	0.0287413i	$0.00914990\pi$
$499$			28.6838i			0.0574826i	−0.999587	+	0.0287413i	$0.990850\pi$
$500$		0			0
$501$	−424.735	−	90.9335i	−0.847775	−	0.181504i
$502$		0			0
$503$		−	573.201i		−	1.13957i	−0.821795	−	0.569783i	$-0.807027\pi$
$503$		−	573.201i		−	1.13957i	0.821795	−	0.569783i	$-0.192973\pi$
$504$		0			0
$505$	779.310			1.54319
$506$		0			0
$507$	24.4940	−	114.407i	0.0483117	−	0.225656i
$508$		0			0
$509$	−197.857			−0.388716			−0.194358	−	0.980931i	$-0.562262\pi$
$509$	−197.857			−0.388716			−0.194358	+	0.980931i	$0.562262\pi$
$510$		0			0
$511$	−798.994			−1.56359
$512$		0			0
$513$	−282.456	−	207.127i	−0.550596	−	0.403756i
$514$		0			0
$515$	−28.7647			−0.0558537
$516$		0			0
$517$			1543.64i			2.98577i
$518$		0			0
$519$	−105.720	+	493.802i	−0.203700	+	0.951448i
$520$		0			0
$521$		−	375.018i		−	0.719803i	−0.932990	−	0.359902i	$-0.882810\pi$
$521$		−	375.018i		−	0.719803i	0.932990	−	0.359902i	$-0.117190\pi$
$522$		0			0
$523$		−	422.363i		−	0.807577i	−0.914852	−	0.403789i	$-0.867693\pi$
$523$		−	422.363i		−	0.807577i	0.914852	−	0.403789i	$-0.132307\pi$
$524$		0			0
$525$	−83.6349	+	390.644i	−0.159305	+	0.744084i
$526$		0			0
$527$			601.366i			1.14111i
$528$		0			0
$529$	−264.245			−0.499517
$530$		0			0
$531$	92.8259	+	41.6565i	0.174813	+	0.0784491i
$532$		0			0
$533$	261.976			0.491511
$534$		0			0
$535$	37.4523			0.0700043
$536$		0			0
$537$	−66.1446	+	308.950i	−0.123174	+	0.575327i
$538$		0			0
$539$	168.192			0.312045
$540$		0			0
$541$			226.022i			0.417785i	0.977939	+	0.208893i	$0.0669859\pi$
$541$			226.022i			0.417785i	−0.977939	+	0.208893i	$0.933014\pi$
$542$		0			0
$543$	88.5928	+	18.9673i	0.163154	+	0.0349305i
$544$		0			0
$545$			1011.29i			1.85557i
$546$		0			0
$547$		−	581.814i		−	1.06365i	−0.846855	−	0.531823i	$-0.821507\pi$
$547$		−	581.814i		−	1.06365i	0.846855	−	0.531823i	$-0.178493\pi$
$548$		0			0
$549$	336.300	−	749.400i	0.612568	−	1.36503i
$550$		0			0
$551$		−	40.0558i		−	0.0726966i
$552$		0			0
$553$	−947.088			−1.71264
$554$		0			0
$555$	30.1465	+	6.45420i	0.0543180	+	0.0116292i
$556$		0			0
$557$	−960.871			−1.72508			−0.862541	−	0.505987i	$-0.831128\pi$
$557$	−960.871			−1.72508			−0.862541	+	0.505987i	$0.831128\pi$
$558$		0			0
$559$	320.600			0.573525
$560$		0			0
$561$	978.355	+	209.461i	1.74395	+	0.373370i
$562$		0			0
$563$	−376.454			−0.668657			−0.334328	−	0.942457i	$-0.608509\pi$
$563$	−376.454			−0.668657			−0.334328	+	0.942457i	$0.608509\pi$
$564$		0			0
$565$		−	198.755i		−	0.351779i
$566$		0			0
$567$	411.557	+	462.525i	0.725849	+	0.815740i
$568$		0			0
$569$		−	596.515i		−	1.04836i	−0.851608	−	0.524178i	$-0.824372\pi$
$569$		−	596.515i		−	1.04836i	0.851608	−	0.524178i	$-0.175628\pi$
$570$		0			0
$571$			275.100i			0.481786i	0.970552	+	0.240893i	$0.0774403\pi$
$571$			275.100i			0.481786i	−0.970552	+	0.240893i	$0.922560\pi$
$572$		0			0
$573$	−749.407	−	160.444i	−1.30787	−	0.280007i
$574$		0			0
$575$			490.689i			0.853373i
$576$		0			0
$577$	475.356			0.823840			0.411920	−	0.911220i	$-0.364858\pi$
$577$	475.356			0.823840			0.411920	+	0.911220i	$0.364858\pi$
$578$		0			0
$579$	−179.762	+	839.639i	−0.310470	+	1.45015i
$580$		0			0
$581$	−624.595			−1.07503
$582$		0			0
$583$	459.031			0.787360
$584$		0			0
$585$	−346.133	+	771.311i	−0.591680	+	1.31848i
$586$		0			0
$587$	621.408			1.05862			0.529308	−	0.848430i	$-0.322452\pi$
$587$	621.408			1.05862			0.529308	+	0.848430i	$0.322452\pi$
$588$		0			0
$589$			417.554i			0.708921i
$590$		0			0
$591$	−75.9952	+	354.961i	−0.128587	+	0.600610i
$592$		0			0
$593$			260.529i			0.439341i	0.975574	+	0.219671i	$0.0704983\pi$
$593$			260.529i			0.439341i	−0.975574	+	0.219671i	$0.929502\pi$
$594$		0			0
$595$		−	930.121i		−	1.56323i
$596$		0			0
$597$	152.868	−	714.022i	0.256061	−	1.19602i
$598$		0			0
$599$		−	205.080i		−	0.342370i	−0.985239	−	0.171185i	$-0.945240\pi$
$599$		−	205.080i		−	0.342370i	0.985239	−	0.171185i	$-0.0547596\pi$
$600$		0			0
$601$	202.844			0.337511			0.168756	−	0.985658i	$-0.446025\pi$
$601$	202.844			0.337511			0.168756	+	0.985658i	$0.446025\pi$
$602$		0			0
$603$	−64.8568	+	144.525i	−0.107557	+	0.239676i
$604$		0			0
$605$	1287.30			2.12777
$606$		0			0
$607$	377.331			0.621633			0.310817	−	0.950470i	$-0.399397\pi$
$607$	377.331			0.621633			0.310817	+	0.950470i	$0.399397\pi$
$608$		0			0
$609$	−14.8225	+	69.2334i	−0.0243391	+	0.113684i
$610$		0			0
$611$	1247.17			2.04119
$612$		0			0
$613$		−	533.400i		−	0.870146i	−0.900395	−	0.435073i	$-0.856723\pi$
$613$		−	533.400i		−	0.870146i	0.900395	−	0.435073i	$-0.143277\pi$
$614$		0			0
$615$	−347.068	−	74.3055i	−0.564339	−	0.120822i
$616$		0			0
$617$			545.242i			0.883698i	0.897089	+	0.441849i	$0.145677\pi$
$617$			545.242i			0.883698i	−0.897089	+	0.441849i	$0.854323\pi$
$618$		0			0
$619$			843.941i			1.36339i	0.731634	+	0.681697i	$0.238758\pi$
$619$			843.941i			1.36339i	−0.731634	+	0.681697i	$0.761242\pi$
$620$		0			0
$621$	613.234	+	449.689i	0.987494	+	0.724137i
$622$		0			0
$623$		−	1086.11i		−	1.74335i
$624$		0			0
$625$	−757.022			−1.21124
$626$		0			0
$627$	679.314	+	145.438i	1.08344	+	0.231958i
$628$		0			0
$629$	−29.4784			−0.0468656
$630$		0			0
$631$	1150.93			1.82398			0.911991	−	0.410210i	$-0.134545\pi$
$631$	1150.93			1.82398			0.911991	+	0.410210i	$0.134545\pi$
$632$		0			0
$633$	328.922	+	70.4204i	0.519624	+	0.111249i
$634$		0			0
$635$	−89.0576			−0.140248
$636$		0			0
$637$		−	135.889i		−	0.213326i
$638$		0			0
$639$	−214.865	+	478.798i	−0.336252	+	0.749293i
$640$		0			0
$641$		−	1266.93i		−	1.97649i	−0.152875	−	0.988246i	$-0.548853\pi$
$641$		−	1266.93i		−	1.97649i	0.152875	−	0.988246i	$-0.451147\pi$
$642$		0			0
$643$			996.810i			1.55025i	0.631808	+	0.775125i	$0.282313\pi$
$643$			996.810i			1.55025i	−0.631808	+	0.775125i	$0.717687\pi$
$644$		0			0
$645$	−424.735	−	90.9335i	−0.658504	−	0.140982i
$646$		0			0
$647$			1061.91i			1.64128i	0.571445	+	0.820641i	$0.306383\pi$
$647$			1061.91i			1.64128i	−0.571445	+	0.820641i	$0.693617\pi$
$648$		0			0
$649$	−201.800			−0.310940
$650$		0			0
$651$	154.514	−	721.711i	0.237349	−	1.10862i
$652$		0			0
$653$	513.147			0.785830			0.392915	−	0.919575i	$-0.371467\pi$
$653$	513.147			0.785830			0.392915	+	0.919575i	$0.371467\pi$
$654$		0			0
$655$	−722.135			−1.10250
$656$		0			0
$657$	858.333	+	385.185i	1.30644	+	0.586278i
$658$		0			0
$659$	495.658			0.752136			0.376068	−	0.926592i	$-0.377276\pi$
$659$	495.658			0.752136			0.376068	+	0.926592i	$0.377276\pi$
$660$		0			0
$661$			1150.42i			1.74043i	0.492675	+	0.870213i	$0.336019\pi$
$661$			1150.42i			1.74043i	−0.492675	+	0.870213i	$0.663981\pi$
$662$		0			0
$663$	169.231	−	790.451i	0.255251	−	1.19223i
$664$		0			0
$665$		−	645.823i		−	0.971163i
$666$		0			0
$667$			86.9643i			0.130381i
$668$		0			0
$669$	−214.261	+	1000.78i	−0.320271	+	1.49593i
$670$		0			0
$671$			1629.16i			2.42797i
$672$		0			0
$673$	72.8225			0.108206			0.0541029	−	0.998535i	$-0.482770\pi$
$673$	72.8225			0.108206			0.0541029	+	0.998535i	$0.482770\pi$
$674$		0			0
$675$	278.171	−	379.337i	0.412105	−	0.561981i
$676$		0			0
$677$	927.386			1.36985			0.684923	−	0.728615i	$-0.259836\pi$
$677$	927.386			1.36985			0.684923	+	0.728615i	$0.259836\pi$
$678$		0			0
$679$	−444.168			−0.654151
$680$		0			0
$681$	−35.8773	+	167.577i	−0.0526832	+	0.246074i
$682$		0			0
$683$	−709.265			−1.03846			−0.519228	−	0.854636i	$-0.673780\pi$
$683$	−709.265			−1.03846			−0.519228	+	0.854636i	$0.673780\pi$
$684$		0			0
$685$		−	714.577i		−	1.04318i
$686$		0			0
$687$	747.005	+	159.930i	1.08734	+	0.232794i
$688$		0			0
$689$		−	370.869i		−	0.538271i
$690$		0			0
$691$			27.7067i			0.0400966i	0.999799	+	0.0200483i	$0.00638200\pi$
$691$			27.7067i			0.0400966i	−0.999799	+	0.0200483i	$0.993618\pi$
$692$		0			0
$693$	−1120.32	−	502.755i	−1.61663	−	0.725477i
$694$		0			0
$695$			1369.74i			1.97085i
$696$		0			0
$697$	339.378			0.486912
$698$		0			0
$699$	344.381	+	73.7302i	0.492677	+	0.105480i
$700$		0			0
$701$	765.474			1.09197			0.545987	−	0.837794i	$-0.316155\pi$
$701$	765.474			1.09197			0.545987	+	0.837794i	$0.316155\pi$
$702$		0			0
$703$	−20.4682			−0.0291154
$704$		0			0
$705$	−1652.27	−	353.741i	−2.34364	−	0.501761i
$706$		0			0
$707$	914.540			1.29355
$708$		0			0
$709$		−	796.599i		−	1.12355i	−0.827289	−	0.561776i	$-0.810118\pi$
$709$		−	796.599i		−	1.12355i	0.827289	−	0.561776i	$-0.189882\pi$
$710$		0			0
$711$	1017.43	+	456.579i	1.43098	+	0.642165i
$712$		0			0
$713$		−	906.543i		−	1.27145i
$714$		0			0
$715$		−	1676.80i		−	2.34518i
$716$		0			0
$717$	−407.294	−	87.1994i	−0.568052	−	0.121617i
$718$		0			0
$719$			108.695i			0.151175i	0.997139	+	0.0755874i	$0.0240832\pi$
$719$			108.695i			0.151175i	−0.997139	+	0.0755874i	$0.975917\pi$
$720$		0			0
$721$	−33.7561			−0.0468184
$722$		0			0
$723$	238.604	−	1114.48i	0.330019	−	1.54146i
$724$		0			0
$725$	53.7949			0.0741998
$726$		0			0
$727$	−960.355			−1.32098			−0.660491	−	0.750834i	$-0.729652\pi$
$727$	−960.355			−1.32098			−0.660491	+	0.750834i	$0.729652\pi$
$728$		0			0
$729$	−219.145	−	695.282i	−0.300610	−	0.953747i
$730$		0			0
$731$	415.323			0.568158
$732$		0			0
$733$		−	519.667i		−	0.708959i	−0.935064	−	0.354479i	$-0.884658\pi$
$733$		−	519.667i		−	0.708959i	0.935064	−	0.354479i	$-0.115342\pi$
$734$		0			0
$735$	−38.5429	+	180.027i	−0.0524393	+	0.244935i
$736$		0			0
$737$		−	314.191i		−	0.426311i
$738$		0			0
$739$		−	614.214i		−	0.831142i	−0.909561	−	0.415571i	$-0.863582\pi$
$739$		−	614.214i		−	0.831142i	0.909561	−	0.415571i	$-0.136418\pi$
$740$		0			0
$741$	117.505	−	548.844i	0.158576	−	0.740681i
$742$		0			0
$743$		−	20.2371i		−	0.0272370i	−0.999907	−	0.0136185i	$-0.995665\pi$
$743$		−	20.2371i		−	0.0272370i	0.999907	−	0.0136185i	$-0.00433504\pi$
$744$		0			0
$745$	−296.955			−0.398598
$746$		0			0
$747$	670.982	+	301.109i	0.898236	+	0.403091i
$748$		0			0
$749$	43.9512			0.0586798
$750$		0			0
$751$	−107.092			−0.142599			−0.0712995	−	0.997455i	$-0.522715\pi$
$751$	−107.092			−0.142599			−0.0712995	+	0.997455i	$0.522715\pi$
$752$		0			0
$753$	−250.633	+	1170.66i	−0.332845	+	1.55466i
$754$		0			0
$755$	−149.351			−0.197815
$756$		0			0
$757$			181.177i			0.239336i	0.992814	+	0.119668i	$0.0381830\pi$
$757$			181.177i			0.239336i	−0.992814	+	0.119668i	$0.961817\pi$
$758$		0			0
$759$	−1474.84	−	315.756i	−1.94314	−	0.416016i
$760$		0			0
$761$			74.4193i			0.0977914i	0.998804	+	0.0488957i	$0.0155702\pi$
$761$			74.4193i			0.0977914i	−0.998804	+	0.0488957i	$0.984430\pi$
$762$		0			0
$763$			1186.77i			1.55540i
$764$		0			0
$765$	−448.400	+	999.199i	−0.586143	+	1.30614i
$766$		0			0
$767$			163.042i			0.212571i
$768$		0			0
$769$	631.467			0.821153			0.410577	−	0.911826i	$-0.365327\pi$
$769$	631.467			0.821153			0.410577	+	0.911826i	$0.365327\pi$
$770$		0			0
$771$	230.202	+	49.2850i	0.298576	+	0.0639234i
$772$		0			0
$773$	1030.15			1.33267			0.666334	−	0.745654i	$-0.267863\pi$
$773$	1030.15			1.33267			0.666334	+	0.745654i	$0.267863\pi$
$774$		0			0
$775$	−560.774			−0.723579
$776$		0			0
$777$	35.3776	+	7.57417i	0.0455311	+	0.00974796i
$778$		0			0
$779$	235.645			0.302496
$780$		0			0
$781$		−	1040.89i		−	1.33276i
$782$		0			0
$783$	49.2999	−	67.2295i	0.0629628	−	0.0858614i
$784$		0			0
$785$		−	804.310i		−	1.02460i
$786$		0			0
$787$		−	157.433i		−	0.200042i	−0.994985	−	0.100021i	$-0.968109\pi$
$787$		−	157.433i		−	0.200042i	0.994985	−	0.100021i	$-0.0318910\pi$
$788$		0			0
$789$	−5.81387	−	1.24472i	−0.00736865	−	0.00157759i
$790$		0			0
$791$		−	233.244i		−	0.294873i
$792$		0			0
$793$	1316.27			1.65986
$794$		0			0
$795$	−105.192	+	491.332i	−0.132316	+	0.618027i
$796$		0			0
$797$	−703.044			−0.882113			−0.441056	−	0.897479i	$-0.645396\pi$
$797$	−703.044			−0.882113			−0.441056	+	0.897479i	$0.645396\pi$
$798$		0			0
$799$	1615.65			2.02209
$800$		0			0
$801$	−523.600	+	1166.77i	−0.653683	+	1.45664i
$802$		0			0
$803$	−1865.98			−2.32376
$804$		0			0
$805$			1402.13i			1.74178i
$806$		0			0
$807$	−29.9074	+	139.693i	−0.0370600	+	0.173101i
$808$		0			0
$809$			1374.99i			1.69962i	0.527090	+	0.849809i	$0.323283\pi$
$809$			1374.99i			1.69962i	−0.527090	+	0.849809i	$0.676717\pi$
$810$		0			0
$811$		−	867.212i		−	1.06931i	−0.845070	−	0.534656i	$-0.820441\pi$
$811$		−	867.212i		−	1.06931i	0.845070	−	0.534656i	$-0.179559\pi$
$812$		0			0
$813$	−114.198	+	533.400i	−0.140465	+	0.656088i
$814$		0			0
$815$		−	1061.91i		−	1.30296i
$816$		0			0
$817$	288.377			0.352970
$818$		0			0
$819$	−406.196	+	905.153i	−0.495965	+	1.10519i
$820$		0			0
$821$	1577.36			1.92126			0.960632	−	0.277823i	$-0.0896127\pi$
$821$	1577.36			1.92126			0.960632	+	0.277823i	$0.0896127\pi$
$822$		0			0
$823$	−303.700			−0.369016			−0.184508	−	0.982831i	$-0.559069\pi$
$823$	−303.700			−0.369016			−0.184508	+	0.982831i	$0.559069\pi$
$824$		0			0
$825$	−195.322	+	912.316i	−0.236754	+	1.10584i
$826$		0			0
$827$	1302.89			1.57544			0.787718	−	0.616036i	$-0.211263\pi$
$827$	1302.89			1.57544			0.787718	+	0.616036i	$0.211263\pi$
$828$		0			0
$829$		−	238.866i		−	0.288138i	−0.989568	−	0.144069i	$-0.953981\pi$
$829$		−	238.866i		−	0.288138i	0.989568	−	0.144069i	$-0.0460187\pi$
$830$		0			0
$831$	210.888	+	45.1500i	0.253776	+	0.0543321i
$832$		0			0
$833$		−	176.038i		−	0.211330i
$834$		0			0
$835$			943.030i			1.12938i
$836$		0			0
$837$	−513.917	+	700.821i	−0.613999	+	0.837301i
$838$		0			0
$839$			355.812i			0.424091i	0.977260	+	0.212045i	$0.0680124\pi$
$839$			355.812i			0.424091i	−0.977260	+	0.212045i	$0.931988\pi$
$840$		0			0
$841$	−831.466			−0.988663
$842$		0			0
$843$	−1347.98	−	288.595i	−1.59903	−	0.342343i
$844$		0			0
$845$	−254.016			−0.300611
$846$		0			0
$847$	1510.68			1.78357
$848$		0			0
$849$	1008.19	+	215.848i	1.18750	+	0.254238i
$850$		0			0
$851$	44.4380			0.0522185
$852$		0			0
$853$			719.755i			0.843792i	0.906644	+	0.421896i	$0.138635\pi$
$853$			719.755i			0.843792i	−0.906644	+	0.421896i	$0.861365\pi$
$854$		0			0
$855$	−311.343	+	693.787i	−0.364144	+	0.811447i
$856$		0			0
$857$		−	604.812i		−	0.705732i	−0.935674	−	0.352866i	$-0.885207\pi$
$857$		−	604.812i		−	0.705732i	0.935674	−	0.352866i	$-0.114793\pi$
$858$		0			0
$859$			309.453i			0.360248i	0.983644	+	0.180124i	$0.0576500\pi$
$859$			309.453i			0.360248i	−0.983644	+	0.180124i	$0.942350\pi$
$860$		0			0
$861$	−407.294	−	87.1994i	−0.473047	−	0.101277i
$862$		0			0
$863$		−	1120.22i		−	1.29805i	−0.760766	−	0.649027i	$-0.775176\pi$
$863$		−	1120.22i		−	1.29805i	0.760766	−	0.649027i	$-0.224824\pi$
$864$		0			0
$865$	1096.38			1.26749
$866$		0			0
$867$	37.7248	−	176.206i	0.0435119	−	0.203237i
$868$		0			0
$869$	−2211.84			−2.54528
$870$		0			0
$871$	−253.847			−0.291443
$872$		0			0
$873$	477.156	+	214.128i	0.546570	+	0.245278i
$874$		0			0
$875$	−377.249			−0.431142
$876$		0			0
$877$		−	1384.11i		−	1.57823i	−0.614243	−	0.789117i	$-0.710539\pi$
$877$		−	1384.11i		−	1.57823i	0.614243	−	0.789117i	$-0.289461\pi$
$878$		0			0
$879$	271.073	−	1266.14i	0.308388	−	1.44043i
$880$		0			0
$881$		−	807.327i		−	0.916376i	−0.888855	−	0.458188i	$-0.848499\pi$
$881$		−	807.327i		−	0.916376i	0.888855	−	0.458188i	$-0.151501\pi$
$882$		0			0
$883$			1276.73i			1.44590i	0.690899	+	0.722951i	$0.257215\pi$
$883$			1276.73i			1.44590i	−0.690899	+	0.722951i	$0.742785\pi$
$884$		0			0
$885$	46.2445	−	216.000i	0.0522536	−	0.244068i
$886$		0			0
$887$		−	670.005i		−	0.755361i	−0.925936	−	0.377680i	$-0.876722\pi$
$887$		−	670.005i		−	0.755361i	0.925936	−	0.377680i	$-0.123278\pi$
$888$		0			0
$889$	−104.511			−0.117561
$890$		0			0
$891$	961.156	+	1080.19i	1.07874	+	1.21233i
$892$		0			0
$893$	1121.82			1.25623
$894$		0			0
$895$	685.956			0.766431
$896$		0			0
$897$	−255.112	+	1191.58i	−0.284406	+	1.32841i
$898$		0			0
$899$	−99.3854			−0.110551
$900$		0			0
$901$		−	480.444i		−	0.533234i
$902$		0			0
$903$	−498.437	−	106.713i	−0.551979	−	0.118176i
$904$		0			0
$905$		−	196.701i		−	0.217349i
$906$		0			0
$907$		−	1637.75i		−	1.80568i	−0.429974	−	0.902841i	$-0.641477\pi$
$907$		−	1637.75i		−	1.80568i	0.429974	−	0.902841i	$-0.358523\pi$
$908$		0			0
$909$	−982.461	−	440.888i	−1.08082	−	0.485026i
$910$		0			0
$911$			1116.26i			1.22531i	0.790351	+	0.612654i	$0.209898\pi$
$911$			1116.26i			1.22531i	−0.790351	+	0.612654i	$0.790102\pi$
$912$		0			0
$913$	−1458.69			−1.59769
$914$		0			0
$915$	−1743.80	−	373.339i	−1.90580	−	0.408021i
$916$		0			0
$917$	−847.443			−0.924148
$918$		0			0
$919$	−366.545			−0.398852			−0.199426	−	0.979913i	$-0.563908\pi$
$919$	−366.545			−0.398852			−0.199426	+	0.979913i	$0.563908\pi$
$920$		0			0
$921$	−649.433	−	139.040i	−0.705139	−	0.150967i
$922$		0			0
$923$	−840.974			−0.911131
$924$		0			0
$925$		−	27.4887i		−	0.0297175i
$926$		0			0
$927$	36.2630	+	16.2734i	0.0391187	+	0.0175549i
$928$		0			0
$929$		−	488.045i		−	0.525345i	−0.964885	−	0.262672i	$-0.915396\pi$
$929$		−	488.045i		−	0.525345i	0.964885	−	0.262672i	$-0.0846038\pi$
$930$		0			0
$931$		−	122.231i		−	0.131290i
$932$		0			0
$933$	−501.543	−	107.378i	−0.537559	−	0.115089i
$934$		0			0
$935$		−	2172.22i		−	2.32323i
$936$		0			0
$937$	−973.489			−1.03894			−0.519471	−	0.854488i	$-0.673871\pi$
$937$	−973.489			−1.03894			−0.519471	+	0.854488i	$0.673871\pi$
$938$		0			0
$939$	10.8443	−	50.6520i	0.0115488	−	0.0539425i
$940$		0			0
$941$	232.064			0.246615			0.123307	−	0.992369i	$-0.460650\pi$
$941$	232.064			0.246615			0.123307	+	0.992369i	$0.460650\pi$
$942$		0			0
$943$	−511.603			−0.542527
$944$		0			0
$945$	794.866	−	1083.95i	0.841128	−	1.14703i
$946$		0			0
$947$	1005.38			1.06165			0.530826	−	0.847481i	$-0.321882\pi$
$947$	1005.38			1.06165			0.530826	+	0.847481i	$0.321882\pi$
$948$		0			0
$949$			1507.60i			1.58862i
$950$		0			0
$951$	−65.2382	+	304.716i	−0.0685996	+	0.320417i
$952$		0			0
$953$			937.182i			0.983402i	0.870764	+	0.491701i	$0.163625\pi$
$953$			937.182i			0.983402i	−0.870764	+	0.491701i	$0.836375\pi$
$954$		0			0
$955$			1663.89i			1.74230i
$956$		0			0
$957$	−34.6167	+	161.689i	−0.0361721	+	0.168954i
$958$		0			0
$959$		−	838.574i		−	0.874425i
$960$		0			0
$961$	75.0234			0.0780681
$962$		0			0
$963$	−47.2153	−	21.1883i	−0.0490294	−	0.0220024i
$964$		0			0
$965$	1864.23			1.93185
$966$		0			0
$967$	−1748.82			−1.80850			−0.904249	−	0.427005i	$-0.859569\pi$
$967$	−1748.82			−1.80850			−0.904249	+	0.427005i	$0.859569\pi$
$968$		0			0
$969$	152.222	−	711.003i	0.157092	−	0.733750i
$970$		0			0
$971$	−845.706			−0.870964			−0.435482	−	0.900197i	$-0.643422\pi$
$971$	−845.706			−0.870964			−0.435482	+	0.900197i	$0.643422\pi$
$972$		0			0
$973$			1607.42i			1.65203i
$974$		0			0
$975$	737.096	+	157.808i	0.755996	+	0.161855i
$976$		0			0
$977$			880.190i			0.900911i	0.892799	+	0.450456i	$0.148738\pi$
$977$			880.190i			0.900911i	−0.892799	+	0.450456i	$0.851262\pi$
$978$		0			0
$979$		−	2536.52i		−	2.59093i
$980$		0			0
$981$	572.127	−	1274.91i	0.583208	−	1.29960i
$982$		0			0
$983$		−	1353.88i		−	1.37730i	−0.725095	−	0.688648i	$-0.758204\pi$
$983$		−	1353.88i		−	1.37730i	0.725095	−	0.688648i	$-0.241796\pi$
$984$		0			0
$985$	788.111			0.800113
$986$		0			0
$987$	−1938.98	−	415.124i	−1.96451	−	0.420592i
$988$		0			0
$989$	−626.089			−0.633052
$990$		0			0
$991$	−1880.37			−1.89745			−0.948724	−	0.316106i	$-0.897624\pi$
$991$	−1880.37			−1.89745			−0.948724	+	0.316106i	$0.897624\pi$
$992$		0			0
$993$	557.255	+	119.305i	0.561183	+	0.120146i
$994$		0			0
$995$	−1585.33			−1.59329
$996$		0			0
$997$			786.688i			0.789055i	0.918884	+	0.394528i	$0.129092\pi$
$997$			786.688i			0.789055i	−0.918884	+	0.394528i	$0.870908\pi$
$998$		0			0
$999$	−34.3536	−	25.1918i	−0.0343880	−	0.0252170i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.h.g.65.8	yes	16
3.2	odd	2	inner	384.3.h.g.65.11	yes	16
4.3	odd	2	inner	384.3.h.g.65.10	yes	16
8.3	odd	2	inner	384.3.h.g.65.7	yes	16
8.5	even	2	inner	384.3.h.g.65.9	yes	16
12.11	even	2	inner	384.3.h.g.65.5	✓	16
16.3	odd	4		768.3.e.p.257.8		8
16.5	even	4		768.3.e.o.257.8		8
16.11	odd	4		768.3.e.o.257.1		8
16.13	even	4		768.3.e.p.257.1		8
24.5	odd	2	inner	384.3.h.g.65.6	yes	16
24.11	even	2	inner	384.3.h.g.65.12	yes	16
48.5	odd	4		768.3.e.o.257.7		8
48.11	even	4		768.3.e.o.257.2		8
48.29	odd	4		768.3.e.p.257.2		8
48.35	even	4		768.3.e.p.257.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.h.g.65.5	✓	16	12.11	even	2	inner
384.3.h.g.65.6	yes	16	24.5	odd	2	inner
384.3.h.g.65.7	yes	16	8.3	odd	2	inner
384.3.h.g.65.8	yes	16	1.1	even	1	trivial
384.3.h.g.65.9	yes	16	8.5	even	2	inner
384.3.h.g.65.10	yes	16	4.3	odd	2	inner
384.3.h.g.65.11	yes	16	3.2	odd	2	inner
384.3.h.g.65.12	yes	16	24.11	even	2	inner
768.3.e.o.257.1		8	16.11	odd	4
768.3.e.o.257.2		8	48.11	even	4
768.3.e.o.257.7		8	48.5	odd	4
768.3.e.o.257.8		8	16.5	even	4
768.3.e.p.257.1		8	16.13	even	4
768.3.e.p.257.2		8	48.29	odd	4
768.3.e.p.257.7		8	48.35	even	4
768.3.e.p.257.8		8	16.3	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.628052 + 2.93352i) q^{3} +6.51323 q^{5} +7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +O(q^{10})\)	\(q+(-0.628052 + 2.93352i) q^{3} +6.51323 q^{5} +7.64344 q^{7} +(-8.21110 - 3.68481i) q^{9} +17.8506 q^{11} -14.4222i q^{13} +(-4.09065 + 19.1067i) q^{15} -18.6833i q^{17} -12.9726i q^{19} +(-4.80048 + 22.4222i) q^{21} +28.1646i q^{23} +17.4222 q^{25} +(15.9665 - 21.7732i) q^{27} +3.08772 q^{29} -32.1873 q^{31} +(-11.2111 + 52.3652i) q^{33} +49.7835 q^{35} -1.57779i q^{37} +(42.3079 + 9.05789i) q^{39} +18.1647i q^{41} +22.2296i q^{43} +(-53.4808 - 24.0000i) q^{45} +86.4757i q^{47} +9.42221 q^{49} +(54.8079 + 11.7341i) q^{51} +25.7151 q^{53} +116.265 q^{55} +(38.0555 + 8.14748i) q^{57} -11.3049 q^{59} +91.2666i q^{61} +(-62.7611 - 28.1646i) q^{63} -93.9352i q^{65} -17.6011i q^{67} +(-82.6215 - 17.6888i) q^{69} -58.3111i q^{71} -104.533 q^{73} +(-10.9420 + 51.1084i) q^{75} +136.440 q^{77} -123.909 q^{79} +(53.8444 + 60.5126i) q^{81} -81.7164 q^{83} -121.689i q^{85} +(-1.93925 + 9.05789i) q^{87} -142.097i q^{89} -110.235i q^{91} +(20.2153 - 94.4222i) q^{93} -84.4938i q^{95} -58.1110 q^{97} +(-146.573 - 65.7760i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) \) 16 * q - 16 * q^9 + 48 * q^25 - 64 * q^33 - 80 * q^49 + 32 * q^57 - 288 * q^73 + 400 * q^81 + 224 * q^97

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