LMFDB - Embedded newform 384.4.f.g.191.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(191,384)]

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, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.f (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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.7
Root			$-1.54779 - 1.54779i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.4.f.g.191.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.44949 + 4.58258i) q^{3} -17.2813 q^{5} -0.269691i q^{7} +(-15.0000 + 22.4499i) q^{9} +O(q^{10})$	\(q+(2.44949 + 4.58258i) q^{3} -17.2813 q^{5} -0.269691i q^{7} +(-15.0000 + 22.4499i) q^{9} -41.1652i q^{11} -64.6606i q^{13} +(-42.3303 - 79.1927i) q^{15} +63.4170i q^{17} +154.026 q^{19} +(1.23588 - 0.660606i) q^{21} -43.3394 q^{23} +173.642 q^{25} +(-139.621 - 13.7477i) q^{27} +240.702 q^{29} +79.1927i q^{31} +(188.642 - 100.834i) q^{33} +4.66061i q^{35} -132.624i q^{37} +(296.312 - 158.385i) q^{39} -101.216i q^{41} +120.317 q^{43} +(259.219 - 387.964i) q^{45} -293.285 q^{47} +342.927 q^{49} +(-290.613 + 155.339i) q^{51} +6.17940 q^{53} +711.386i q^{55} +(377.285 + 705.835i) q^{57} +45.8258i q^{59} -651.230i q^{61} +(6.05455 + 4.04537i) q^{63} +1117.42i q^{65} -685.273 q^{67} +(-106.159 - 198.606i) q^{69} +836.515 q^{71} +285.927 q^{73} +(425.335 + 795.730i) q^{75} -11.1019 q^{77} -940.874i q^{79} +(-279.000 - 673.498i) q^{81} -1011.11i q^{83} -1095.93i q^{85} +(589.597 + 1103.03i) q^{87} -432.503i q^{89} -17.4384 q^{91} +(-362.907 + 193.982i) q^{93} -2661.76 q^{95} +1007.28 q^{97} +(924.155 + 617.477i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 120 q^{9}+O(q^{10}) $ 8 * q - 120 * q^9	$ 8 q - 120 q^{9} - 192 q^{15} - 640 q^{23} + 216 q^{25} + 336 q^{33} + 1344 q^{39} - 776 q^{49} + 672 q^{57} + 2688 q^{63} - 640 q^{71} - 1232 q^{73} - 2232 q^{81} + 1344 q^{87} - 9856 q^{95} + 5712 q^{97}+O(q^{100}) $ 8 * q - 120 * q^9 - 192 * q^15 - 640 * q^23 + 216 * q^25 + 336 * q^33 + 1344 * q^39 - 776 * q^49 + 672 * q^57 + 2688 * q^63 - 640 * q^71 - 1232 * q^73 - 2232 * q^81 + 1344 * q^87 - 9856 * q^95 + 5712 * 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
$3$	2.44949	+	4.58258i	0.471405	+	0.881917i
$3$	2.44949	+	4.58258i	0.471405	+	0.881917i
$4$		0			0
$5$	−17.2813			−1.54568			−0.772842	−	0.634598i	$-0.781166\pi$
$5$	−17.2813			−1.54568			−0.772842	+	0.634598i	$0.781166\pi$
$6$		0			0
$7$		−	0.269691i		−	0.0145620i	−0.999973	−	0.00728098i	$-0.997682\pi$
$7$		−	0.269691i		−	0.0145620i	0.999973	−	0.00728098i	$-0.00231763\pi$
$8$		0			0
$9$	−15.0000	+	22.4499i	−0.555556	+	0.831479i
$10$		0			0
$11$		−	41.1652i		−	1.12834i	−0.825658	−	0.564171i	$-0.809196\pi$
$11$		−	41.1652i		−	1.12834i	0.825658	−	0.564171i	$-0.190804\pi$
$12$		0			0
$13$		−	64.6606i		−	1.37951i	−0.724043	−	0.689755i	$-0.757718\pi$
$13$		−	64.6606i		−	1.37951i	0.724043	−	0.689755i	$-0.242282\pi$
$14$		0			0
$15$	−42.3303	−	79.1927i	−0.728642	−	1.36317i
$16$		0			0
$17$			63.4170i			0.904758i	0.891826	+	0.452379i	$0.149425\pi$
$17$			63.4170i			0.904758i	−0.891826	+	0.452379i	$0.850575\pi$
$18$		0			0
$19$	154.026			1.85979			0.929894	−	0.367828i	$-0.119899\pi$
$19$	154.026			1.85979			0.929894	+	0.367828i	$0.119899\pi$
$20$		0			0
$21$	1.23588	−	0.660606i	0.0128424	−	0.00686457i
$22$		0			0
$23$	−43.3394			−0.392908			−0.196454	−	0.980513i	$-0.562943\pi$
$23$	−43.3394			−0.392908			−0.196454	+	0.980513i	$0.562943\pi$
$24$		0			0
$25$	173.642			1.38914
$26$		0			0
$27$	−139.621	−	13.7477i	−0.995187	−	0.0979908i
$28$		0			0
$29$	240.702			1.54128			0.770642	−	0.637268i	$-0.219936\pi$
$29$	240.702			1.54128			0.770642	+	0.637268i	$0.219936\pi$
$30$		0			0
$31$			79.1927i			0.458821i	0.973330	+	0.229410i	$0.0736797\pi$
$31$			79.1927i			0.458821i	−0.973330	+	0.229410i	$0.926320\pi$
$32$		0			0
$33$	188.642	−	100.834i	0.995104	−	0.531905i
$34$		0			0
$35$			4.66061i			0.0225082i
$36$		0			0
$37$		−	132.624i		−	0.589278i	−0.955609	−	0.294639i	$-0.904801\pi$
$37$		−	132.624i		−	0.589278i	0.955609	−	0.294639i	$-0.0951994\pi$
$38$		0			0
$39$	296.312	−	158.385i	1.21661	−	0.650307i
$40$		0			0
$41$		−	101.216i		−	0.385543i	−0.981244	−	0.192772i	$-0.938252\pi$
$41$		−	101.216i		−	0.385543i	0.981244	−	0.192772i	$-0.0617476\pi$
$42$		0			0
$43$	120.317			0.426701			0.213351	−	0.976976i	$-0.431562\pi$
$43$	120.317			0.426701			0.213351	+	0.976976i	$0.431562\pi$
$44$		0			0
$45$	259.219	−	387.964i	0.858713	−	1.28520i
$46$		0			0
$47$	−293.285			−0.910213			−0.455106	−	0.890437i	$-0.650399\pi$
$47$	−293.285			−0.910213			−0.455106	+	0.890437i	$0.650399\pi$
$48$		0			0
$49$	342.927			0.999788
$50$		0			0
$51$	−290.613	+	155.339i	−0.797922	+	0.426507i
$52$		0			0
$53$	6.17940			0.0160152			0.00800760	−	0.999968i	$-0.497451\pi$
$53$	6.17940			0.0160152			0.00800760	+	0.999968i	$0.497451\pi$
$54$		0			0
$55$			711.386i			1.74406i
$56$		0			0
$57$	377.285	+	705.835i	0.876712	+	1.64018i
$58$		0			0
$59$			45.8258i			0.101119i	0.998721	+	0.0505594i	$0.0161004\pi$
$59$			45.8258i			0.101119i	−0.998721	+	0.0505594i	$0.983900\pi$
$60$		0			0
$61$		−	651.230i		−	1.36691i	−0.729993	−	0.683455i	$-0.760477\pi$
$61$		−	651.230i		−	1.36691i	0.729993	−	0.683455i	$-0.239523\pi$
$62$		0			0
$63$	6.05455	+	4.04537i	0.0121080	+	0.00808997i
$64$		0			0
$65$			1117.42i			2.13229i
$66$		0			0
$67$	−685.273			−1.24954			−0.624772	−	0.780807i	$-0.714808\pi$
$67$	−685.273			−1.24954			−0.624772	+	0.780807i	$0.714808\pi$
$68$		0			0
$69$	−106.159	−	198.606i	−0.185219	−	0.346512i
$70$		0			0
$71$	836.515			1.39825			0.699127	−	0.714997i	$-0.253572\pi$
$71$	836.515			1.39825			0.699127	+	0.714997i	$0.253572\pi$
$72$		0			0
$73$	285.927			0.458428			0.229214	−	0.973376i	$-0.426384\pi$
$73$	285.927			0.458428			0.229214	+	0.973376i	$0.426384\pi$
$74$		0			0
$75$	425.335	+	795.730i	0.654847	+	1.22511i
$76$		0			0
$77$	−11.1019			−0.0164309
$78$		0			0
$79$		−	940.874i		−	1.33996i	−0.742381	−	0.669978i	$-0.766303\pi$
$79$		−	940.874i		−	1.33996i	0.742381	−	0.669978i	$-0.233697\pi$
$80$		0			0
$81$	−279.000	−	673.498i	−0.382716	−	0.923866i
$82$		0			0
$83$		−	1011.11i		−	1.33715i	−0.743643	−	0.668577i	$-0.766904\pi$
$83$		−	1011.11i		−	1.33715i	0.743643	−	0.668577i	$-0.233096\pi$
$84$		0			0
$85$		−	1095.93i		−	1.39847i
$86$		0			0
$87$	589.597	+	1103.03i	0.726568	+	1.35928i
$88$		0			0
$89$		−	432.503i		−	0.515115i	−0.966263	−	0.257558i	$-0.917082\pi$
$89$		−	432.503i		−	0.515115i	0.966263	−	0.257558i	$-0.0829177\pi$
$90$		0			0
$91$	−17.4384			−0.0200884
$92$		0			0
$93$	−362.907	+	193.982i	−0.404642	+	0.216290i
$94$		0			0
$95$	−2661.76			−2.87464
$96$		0			0
$97$	1007.28			1.05437			0.527187	−	0.849749i	$-0.323247\pi$
$97$	1007.28			1.05437			0.527187	+	0.849749i	$0.323247\pi$
$98$		0			0
$99$	924.155	+	617.477i	0.938193	+	0.626857i
$100$		0			0
$101$	1217.08			1.19905			0.599526	−	0.800355i	$-0.295356\pi$
$101$	1217.08			1.19905			0.599526	+	0.800355i	$0.295356\pi$
$102$		0			0
$103$		−	1357.60i		−	1.29872i	−0.760479	−	0.649362i	$-0.775036\pi$
$103$		−	1357.60i		−	1.29872i	0.760479	−	0.649362i	$-0.224964\pi$
$104$		0			0
$105$	−21.3576	+	11.4161i	−0.0198503	+	0.0106105i
$106$		0			0
$107$		−	1490.17i		−	1.34636i	−0.739478	−	0.673180i	$-0.764928\pi$
$107$		−	1490.17i		−	1.34636i	0.739478	−	0.673180i	$-0.235072\pi$
$108$		0			0
$109$			1865.91i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$109$			1865.91i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$110$		0			0
$111$	607.761	−	324.862i	0.519694	−	0.277788i
$112$		0			0
$113$		−	1047.66i		−	0.872177i	−0.899904	−	0.436088i	$-0.856364\pi$
$113$		−	1047.66i		−	0.872177i	0.899904	−	0.436088i	$-0.143636\pi$
$114$		0			0
$115$	748.960			0.607312
$116$		0			0
$117$	1451.63	+	969.909i	1.14703	+	0.766394i
$118$		0			0
$119$	17.1030			0.0131750
$120$		0			0
$121$	−363.570			−0.273155
$122$		0			0
$123$	463.829	−	247.927i	0.340017	−	0.181747i
$124$		0			0
$125$	−840.603			−0.601487
$126$		0			0
$127$		−	1033.28i		−	0.721960i	−0.932574	−	0.360980i	$-0.882442\pi$
$127$		−	1033.28i		−	0.721960i	0.932574	−	0.360980i	$-0.117558\pi$
$128$		0			0
$129$	294.715	+	551.362i	0.201149	+	0.376315i
$130$		0			0
$131$			359.626i			0.239852i	0.992783	+	0.119926i	$0.0382658\pi$
$131$			359.626i			0.239852i	−0.992783	+	0.119926i	$0.961734\pi$
$132$		0			0
$133$		−	41.5394i		−	0.0270821i
$134$		0			0
$135$	2412.83	+	237.578i	1.53825	+	0.151463i
$136$		0			0
$137$		−	379.245i		−	0.236504i	−0.992984	−	0.118252i	$-0.962271\pi$
$137$		−	379.245i		−	0.236504i	0.992984	−	0.118252i	$-0.0377291\pi$
$138$		0			0
$139$	−1984.09			−1.21071			−0.605353	−	0.795957i	$-0.706968\pi$
$139$	−1984.09			−1.21071			−0.605353	+	0.795957i	$0.706968\pi$
$140$		0			0
$141$	−718.398	−	1344.00i	−0.429078	−	0.802732i
$142$		0			0
$143$	−2661.76			−1.55656
$144$		0			0
$145$	−4159.64			−2.38234
$146$		0			0
$147$	839.997	+	1571.49i	0.471305	+	0.881730i
$148$		0			0
$149$	−2323.13			−1.27730			−0.638651	−	0.769497i	$-0.720507\pi$
$149$	−2323.13			−1.27730			−0.638651	+	0.769497i	$0.720507\pi$
$150$		0			0
$151$			2613.36i			1.40843i	0.709989	+	0.704213i	$0.248700\pi$
$151$			2613.36i			1.40843i	−0.709989	+	0.704213i	$0.751300\pi$
$152$		0			0
$153$	−1423.71	−	951.256i	−0.752288	−	0.502644i
$154$		0			0
$155$		−	1368.55i		−	0.709192i
$156$		0			0
$157$			2780.41i			1.41338i	0.707524	+	0.706690i	$0.249812\pi$
$157$			2780.41i			1.41338i	−0.707524	+	0.706690i	$0.750188\pi$
$158$		0			0
$159$	15.1364	+	28.3176i	0.00754964	+	0.0141241i
$160$		0			0
$161$			11.6882i			0.00572151i
$162$		0			0
$163$	635.294			0.305276			0.152638	−	0.988282i	$-0.451223\pi$
$163$	635.294			0.305276			0.152638	+	0.988282i	$0.451223\pi$
$164$		0			0
$165$	−3259.98	+	1742.53i	−1.53812	+	0.822158i
$166$		0			0
$167$	−319.521			−0.148056			−0.0740278	−	0.997256i	$-0.523585\pi$
$167$	−319.521			−0.148056			−0.0740278	+	0.997256i	$0.523585\pi$
$168$		0			0
$169$	−1983.99			−0.903047
$170$		0			0
$171$	−2310.39	+	3457.87i	−1.03322	+	1.54638i
$172$		0			0
$173$	−1862.71			−0.818609			−0.409305	−	0.912398i	$-0.634229\pi$
$173$	−1862.71			−0.818609			−0.409305	+	0.912398i	$0.634229\pi$
$174$		0			0
$175$		−	46.8298i		−	0.0202286i
$176$		0			0
$177$	−210.000	+	112.250i	−0.0891783	+	0.0476678i
$178$		0			0
$179$		−	3946.43i		−	1.64788i	−0.566678	−	0.823940i	$-0.691771\pi$
$179$		−	3946.43i		−	1.64788i	0.566678	−	0.823940i	$-0.308229\pi$
$180$		0			0
$181$			1187.01i			0.487458i	0.969843	+	0.243729i	$0.0783708\pi$
$181$			1187.01i			0.487458i	−0.969843	+	0.243729i	$0.921629\pi$
$182$		0			0
$183$	2984.31	−	1595.18i	1.20550	−	0.644367i
$184$		0			0
$185$			2291.92i			0.910838i
$186$		0			0
$187$	2610.57			1.02088
$188$		0			0
$189$	−3.70764	+	37.6545i	−0.00142694	+	0.0144919i
$190$		0			0
$191$	1147.96			0.434889			0.217444	−	0.976073i	$-0.430228\pi$
$191$	1147.96			0.434889			0.217444	+	0.976073i	$0.430228\pi$
$192$		0			0
$193$	304.933			0.113728			0.0568642	−	0.998382i	$-0.481890\pi$
$193$	304.933			0.113728			0.0568642	+	0.998382i	$0.481890\pi$
$194$		0			0
$195$	−5120.65	+	2737.10i	−1.88050	+	1.00517i
$196$		0			0
$197$	1252.94			0.453140			0.226570	−	0.973995i	$-0.427249\pi$
$197$	1252.94			0.453140			0.226570	+	0.973995i	$0.427249\pi$
$198$		0			0
$199$			4664.28i			1.66152i	0.556632	+	0.830759i	$0.312093\pi$
$199$			4664.28i			1.66152i	−0.556632	+	0.830759i	$0.687907\pi$
$200$		0			0
$201$	−1678.57	−	3140.32i	−0.589041	−	1.10199i
$202$		0			0
$203$		−	64.9152i		−	0.0224441i
$204$		0			0
$205$			1749.14i			0.595928i
$206$		0			0
$207$	650.091	−	972.967i	0.218282	−	0.326695i
$208$		0			0
$209$		−	6340.50i		−	2.09848i
$210$		0			0
$211$	−1845.39			−0.602095			−0.301047	−	0.953609i	$-0.597336\pi$
$211$	−1845.39			−0.602095			−0.301047	+	0.953609i	$0.597336\pi$
$212$		0			0
$213$	2049.04	+	3833.39i	0.659144	+	1.23314i
$214$		0			0
$215$	−2079.23			−0.659546
$216$		0			0
$217$	21.3576			0.00668132
$218$		0			0
$219$	700.376	+	1310.28i	0.216105	+	0.404296i
$220$		0			0
$221$	4100.58			1.24812
$222$		0			0
$223$			2965.36i			0.890473i	0.895413	+	0.445237i	$0.146880\pi$
$223$			2965.36i			0.890473i	−0.895413	+	0.445237i	$0.853120\pi$
$224$		0			0
$225$	−2604.64	+	3898.26i	−0.771744	+	1.15504i
$226$		0			0
$227$			1538.99i			0.449983i	0.974361	+	0.224991i	$0.0722354\pi$
$227$			1538.99i			0.449983i	−0.974361	+	0.224991i	$0.927765\pi$
$228$		0			0
$229$		−	1658.05i		−	0.478460i	−0.970963	−	0.239230i	$-0.923105\pi$
$229$		−	1658.05i		−	0.478460i	0.970963	−	0.239230i	$-0.0768950\pi$
$230$		0			0
$231$	−27.1939	−	50.8752i	−0.00774558	−	0.0144907i
$232$		0			0
$233$		−	1241.69i		−	0.349123i	−0.984646	−	0.174561i	$-0.944149\pi$
$233$		−	1241.69i		−	0.349123i	0.984646	−	0.174561i	$-0.0558507\pi$
$234$		0			0
$235$	5068.34			1.40690
$236$		0			0
$237$	4311.63	−	2304.66i	1.18173	−	0.631662i
$238$		0			0
$239$	−2373.58			−0.642401			−0.321201	−	0.947011i	$-0.604086\pi$
$239$	−2373.58			−0.642401			−0.321201	+	0.947011i	$0.604086\pi$
$240$		0			0
$241$	2592.35			0.692896			0.346448	−	0.938069i	$-0.387388\pi$
$241$	2592.35			0.692896			0.346448	+	0.938069i	$0.387388\pi$
$242$		0			0
$243$	2402.95	−	2928.27i	0.634359	−	0.773038i
$244$		0			0
$245$	−5926.22			−1.54536
$246$		0			0
$247$		−	9959.41i		−	2.56559i
$248$		0			0
$249$	4633.49	−	2476.71i	1.17926	−	0.630341i
$250$		0			0
$251$		−	2750.30i		−	0.691624i	−0.938304	−	0.345812i	$-0.887603\pi$
$251$		−	2750.30i		−	0.691624i	0.938304	−	0.345812i	$-0.112397\pi$
$252$		0			0
$253$			1784.07i			0.443335i
$254$		0			0
$255$	5022.17	−	2684.46i	1.23334	−	0.659245i
$256$		0			0
$257$			218.163i			0.0529519i	0.999649	+	0.0264759i	$0.00842854\pi$
$257$			218.163i			0.0529519i	−0.999649	+	0.0264759i	$0.991571\pi$
$258$		0			0
$259$	−35.7676			−0.00858104
$260$		0			0
$261$	−3610.53	+	5403.75i	−0.856269	+	1.28155i
$262$		0			0
$263$	−7341.25			−1.72122			−0.860611	−	0.509264i	$-0.829918\pi$
$263$	−7341.25			−1.72122			−0.860611	+	0.509264i	$0.829918\pi$
$264$		0			0
$265$	−106.788			−0.0247544
$266$		0			0
$267$	1981.98	−	1059.41i	0.454289	−	0.242828i
$268$		0			0
$269$	54.3577			0.0123206			0.00616031	−	0.999981i	$-0.498039\pi$
$269$	54.3577			0.0123206			0.00616031	+	0.999981i	$0.498039\pi$
$270$		0			0
$271$			3633.43i			0.814446i	0.913329	+	0.407223i	$0.133503\pi$
$271$			3633.43i			0.814446i	−0.913329	+	0.407223i	$0.866497\pi$
$272$		0			0
$273$	−42.7152	−	79.9127i	−0.00946974	−	0.0177163i
$274$		0			0
$275$		−	7148.02i		−	1.56742i
$276$		0			0
$277$		−	6204.12i		−	1.34574i	−0.739762	−	0.672868i	$-0.765062\pi$
$277$		−	6204.12i		−	1.34574i	0.739762	−	0.672868i	$-0.234938\pi$
$278$		0			0
$279$	−1777.87	−	1187.89i	−0.381500	−	0.254900i
$280$		0			0
$281$			5651.49i			1.19979i	0.800081	+	0.599893i	$0.204790\pi$
$281$			5651.49i			1.19979i	−0.800081	+	0.599893i	$0.795210\pi$
$282$		0			0
$283$	−5857.07			−1.23027			−0.615136	−	0.788421i	$-0.710899\pi$
$283$	−5857.07			−1.23027			−0.615136	+	0.788421i	$0.710899\pi$
$284$		0			0
$285$	−6519.96	−	12197.7i	−1.35512	−	2.53520i
$286$		0			0
$287$	−27.2970			−0.00561426
$288$		0			0
$289$	891.279			0.181412
$290$		0			0
$291$	2467.33	+	4615.96i	0.497037	+	0.929870i
$292$		0			0
$293$	5395.42			1.07578			0.537891	−	0.843015i	$-0.319221\pi$
$293$	5395.42			1.07578			0.537891	+	0.843015i	$0.319221\pi$
$294$		0			0
$295$		−	791.927i		−	0.156298i
$296$		0			0
$297$	−565.927	+	5747.52i	−0.110567	+	1.12291i
$298$		0			0
$299$			2802.35i			0.542021i
$300$		0			0
$301$		−	32.4484i		−	0.00621361i
$302$		0			0
$303$	2981.23	+	5577.38i	0.565239	+	1.05747i
$304$		0			0
$305$			11254.1i			2.11281i
$306$		0			0
$307$	8188.99			1.52238			0.761189	−	0.648530i	$-0.224616\pi$
$307$	8188.99			1.52238			0.761189	+	0.648530i	$0.224616\pi$
$308$		0			0
$309$	6221.32	−	3325.44i	1.14537	−	0.612225i
$310$		0			0
$311$	1838.42			0.335200			0.167600	−	0.985855i	$-0.446398\pi$
$311$	1838.42			0.335200			0.167600	+	0.985855i	$0.446398\pi$
$312$		0			0
$313$	−293.079			−0.0529259			−0.0264629	−	0.999650i	$-0.508424\pi$
$313$	−293.079			−0.0529259			−0.0264629	+	0.999650i	$0.508424\pi$
$314$		0			0
$315$	−104.630	−	69.9091i	−0.0187151	−	0.0125045i
$316$		0			0
$317$	1343.02			0.237954			0.118977	−	0.992897i	$-0.462039\pi$
$317$	1343.02			0.237954			0.118977	+	0.992897i	$0.462039\pi$
$318$		0			0
$319$		−	9908.53i		−	1.73909i
$320$		0			0
$321$	6828.84	−	3650.17i	1.18738	−	0.634680i
$322$		0			0
$323$			9767.87i			1.68266i
$324$		0			0
$325$		−	11227.8i		−	1.91633i
$326$		0			0
$327$	−8550.67	+	4570.53i	−1.44603	+	0.772938i
$328$		0			0
$329$			79.0963i			0.0132545i
$330$		0			0
$331$	−8825.95			−1.46561			−0.732807	−	0.680437i	$-0.761790\pi$
$331$	−8825.95			−1.46561			−0.732807	+	0.680437i	$0.761790\pi$
$332$		0			0
$333$	2977.41	+	1989.36i	0.489973	+	0.327377i
$334$		0			0
$335$	11842.4			1.93140
$336$		0			0
$337$	3842.48			0.621108			0.310554	−	0.950556i	$-0.399485\pi$
$337$	3842.48			0.621108			0.310554	+	0.950556i	$0.399485\pi$
$338$		0			0
$339$	4801.00	−	2566.24i	0.769187	−	0.411148i
$340$		0			0
$341$	3259.98			0.517706
$342$		0			0
$343$		−	184.988i		−	0.0291208i
$344$		0			0
$345$	1834.57	+	3432.17i	0.286290	+	0.535599i
$346$		0			0
$347$		−	9565.43i		−	1.47983i	−0.672703	−	0.739913i	$-0.734867\pi$
$347$		−	9565.43i		−	1.47983i	0.672703	−	0.739913i	$-0.265133\pi$
$348$		0			0
$349$			7514.35i			1.15253i	0.817262	+	0.576266i	$0.195491\pi$
$349$			7514.35i			1.15253i	−0.817262	+	0.576266i	$0.804509\pi$
$350$		0			0
$351$	−888.936	+	9027.97i	−0.135179	+	1.37287i
$352$		0			0
$353$		−	4302.30i		−	0.648693i	−0.945938	−	0.324346i	$-0.894856\pi$
$353$		−	4302.30i		−	0.648693i	0.945938	−	0.324346i	$-0.105144\pi$
$354$		0			0
$355$	−14456.0			−2.16126
$356$		0			0
$357$	41.8937	+	78.3758i	0.00621078	+	0.0116193i
$358$		0			0
$359$	−2543.75			−0.373967			−0.186983	−	0.982363i	$-0.559871\pi$
$359$	−2543.75			−0.373967			−0.186983	+	0.982363i	$0.559871\pi$
$360$		0			0
$361$	16865.0			2.45881
$362$		0			0
$363$	−890.560	−	1666.09i	−0.128767	−	0.240900i
$364$		0			0
$365$	−4941.19			−0.708585
$366$		0			0
$367$			5291.35i			0.752606i	0.926497	+	0.376303i	$0.122805\pi$
$367$			5291.35i			0.752606i	−0.926497	+	0.376303i	$0.877195\pi$
$368$		0			0
$369$	2272.29	+	1518.24i	0.320571	+	0.214191i
$370$		0			0
$371$		−	1.66653i		−	0.000233213i
$372$		0			0
$373$		−	7860.77i		−	1.09119i	−0.838048	−	0.545597i	$-0.816303\pi$
$373$		−	7860.77i		−	1.09119i	0.838048	−	0.545597i	$-0.183697\pi$
$374$		0			0
$375$	−2059.05	−	3852.13i	−0.283543	−	0.530461i
$376$		0			0
$377$		−	15563.9i		−	2.12622i
$378$		0			0
$379$	3577.47			0.484861			0.242431	−	0.970169i	$-0.422055\pi$
$379$	3577.47			0.484861			0.242431	+	0.970169i	$0.422055\pi$
$380$		0			0
$381$	4735.09	−	2531.01i	0.636709	−	0.340335i
$382$		0			0
$383$	9144.30			1.21998			0.609990	−	0.792409i	$-0.291174\pi$
$383$	9144.30			1.21998			0.609990	+	0.792409i	$0.291174\pi$
$384$		0			0
$385$	191.855			0.0253969
$386$		0			0
$387$	−1804.75	+	2701.11i	−0.237056	+	0.354794i
$388$		0			0
$389$	12996.7			1.69398			0.846989	−	0.531611i	$-0.178413\pi$
$389$	12996.7			1.69398			0.846989	+	0.531611i	$0.178413\pi$
$390$		0			0
$391$		−	2748.46i		−	0.355487i
$392$		0			0
$393$	−1648.01	+	880.900i	−0.211530	+	0.113067i
$394$		0			0
$395$			16259.5i			2.07115i
$396$		0			0
$397$		−	2309.36i		−	0.291949i	−0.989288	−	0.145974i	$-0.953368\pi$
$397$		−	2309.36i		−	0.291949i	0.989288	−	0.145974i	$-0.0466317\pi$
$398$		0			0
$399$	190.357	−	101.750i	0.0238842	−	0.0127666i
$400$		0			0
$401$			4913.59i			0.611903i	0.952047	+	0.305951i	$0.0989745\pi$
$401$			4913.59i			0.611903i	−0.952047	+	0.305951i	$0.901026\pi$
$402$		0			0
$403$	5120.65			0.632947
$404$		0			0
$405$	4821.48	+	11638.9i	0.591558	+	1.42801i
$406$		0			0
$407$	−5459.50			−0.664907
$408$		0			0
$409$	−3643.58			−0.440497			−0.220248	−	0.975444i	$-0.570687\pi$
$409$	−3643.58			−0.440497			−0.220248	+	0.975444i	$0.570687\pi$
$410$		0			0
$411$	1737.92	−	928.958i	0.208577	−	0.111489i
$412$		0			0
$413$	12.3588			0.00147249
$414$		0			0
$415$			17473.3i			2.06682i
$416$		0			0
$417$	−4860.00	−	9092.23i	−0.570732	−	1.06774i
$418$		0			0
$419$			7200.87i			0.839583i	0.907621	+	0.419792i	$0.137897\pi$
$419$			7200.87i			0.839583i	−0.907621	+	0.419792i	$0.862103\pi$
$420$		0			0
$421$		−	6441.62i		−	0.745713i	−0.927889	−	0.372857i	$-0.878378\pi$
$421$		−	6441.62i		−	0.745713i	0.927889	−	0.372857i	$-0.121622\pi$
$422$		0			0
$423$	4399.27	−	6584.23i	0.505674	−	0.756823i
$424$		0			0
$425$			11011.9i			1.25684i
$426$		0			0
$427$	−175.631			−0.0199049
$428$		0			0
$429$	−6519.96	−	12197.7i	−0.733769	−	1.37276i
$430$		0			0
$431$	5306.42			0.593043			0.296521	−	0.955026i	$-0.404173\pi$
$431$	5306.42			0.593043			0.296521	+	0.955026i	$0.404173\pi$
$432$		0			0
$433$	2336.16			0.259281			0.129640	−	0.991561i	$-0.458618\pi$
$433$	2336.16			0.259281			0.129640	+	0.991561i	$0.458618\pi$
$434$		0			0
$435$	−10189.0	−	19061.8i	−1.12304	−	2.10102i
$436$		0			0
$437$	−6675.39			−0.730726
$438$		0			0
$439$		−	286.123i		−	0.0311068i	−0.999879	−	0.0155534i	$-0.995049\pi$
$439$		−	286.123i		−	0.0311068i	0.999879	−	0.0155534i	$-0.00495100\pi$
$440$		0			0
$441$	−5143.91	+	7698.70i	−0.555438	+	0.831303i
$442$		0			0
$443$			8021.59i			0.860309i	0.902755	+	0.430155i	$0.141541\pi$
$443$			8021.59i			0.860309i	−0.902755	+	0.430155i	$0.858459\pi$
$444$		0			0
$445$			7474.21i			0.796205i
$446$		0			0
$447$	−5690.48	−	10645.9i	−0.602126	−	1.12647i
$448$		0			0
$449$			12605.3i			1.32491i	0.749103	+	0.662453i	$0.230485\pi$
$449$			12605.3i			1.32491i	−0.749103	+	0.662453i	$0.769515\pi$
$450$		0			0
$451$	−4166.57			−0.435024
$452$		0			0
$453$	−11975.9	+	6401.40i	−1.24211	+	0.663938i
$454$		0			0
$455$	301.358			0.0310502
$456$		0			0
$457$	−5652.98			−0.578633			−0.289317	−	0.957233i	$-0.593428\pi$
$457$	−5652.98			−0.578633			−0.289317	+	0.957233i	$0.593428\pi$
$458$		0			0
$459$	871.840	−	8854.35i	0.0886580	−	0.900404i
$460$		0			0
$461$	2094.76			0.211633			0.105817	−	0.994386i	$-0.466254\pi$
$461$	2094.76			0.211633			0.105817	+	0.994386i	$0.466254\pi$
$462$		0			0
$463$		−	7783.64i		−	0.781288i	−0.920542	−	0.390644i	$-0.872252\pi$
$463$		−	7783.64i		−	0.781288i	0.920542	−	0.390644i	$-0.127748\pi$
$464$		0			0
$465$	6271.49	−	3352.25i	0.625448	−	0.334316i
$466$		0			0
$467$		−	12864.3i		−	1.27470i	−0.770573	−	0.637352i	$-0.780030\pi$
$467$		−	12864.3i		−	1.27470i	0.770573	−	0.637352i	$-0.219970\pi$
$468$		0			0
$469$			184.812i			0.0181958i
$470$		0			0
$471$	−12741.4	+	6810.58i	−1.24648	+	0.666273i
$472$		0			0
$473$		−	4952.87i		−	0.481465i
$474$		0			0
$475$	26745.4			2.58350
$476$		0			0
$477$	−92.6910	+	138.727i	−0.00889733	+	0.0133163i
$478$		0			0
$479$	8397.65			0.801040			0.400520	−	0.916288i	$-0.368829\pi$
$479$	8397.65			0.801040			0.400520	+	0.916288i	$0.368829\pi$
$480$		0			0
$481$	−8575.56			−0.812915
$482$		0			0
$483$	−53.5623	+	28.6302i	−0.00504590	+	0.00269715i
$484$		0			0
$485$	−17407.2			−1.62973
$486$		0			0
$487$		−	9622.06i		−	0.895313i	−0.894206	−	0.447656i	$-0.852259\pi$
$487$		−	9622.06i		−	0.895313i	0.894206	−	0.447656i	$-0.147741\pi$
$488$		0			0
$489$	1556.15	+	2911.28i	0.143909	+	0.269228i
$490$		0			0
$491$			3382.57i			0.310902i	0.987844	+	0.155451i	$0.0496831\pi$
$491$			3382.57i			0.310902i	−0.987844	+	0.155451i	$0.950317\pi$
$492$		0			0
$493$			15264.6i			1.39449i
$494$		0			0
$495$	−15970.6	−	10670.8i	−1.45015	−	0.968922i
$496$		0			0
$497$		−	225.601i		−	0.0203613i
$498$		0			0
$499$	12894.7			1.15681			0.578403	−	0.815751i	$-0.303676\pi$
$499$	12894.7			1.15681			0.578403	+	0.815751i	$0.303676\pi$
$500$		0			0
$501$	−782.664	−	1464.23i	−0.0697941	−	0.130573i
$502$		0			0
$503$	1741.55			0.154377			0.0771885	−	0.997017i	$-0.475406\pi$
$503$	1741.55			0.154377			0.0771885	+	0.997017i	$0.475406\pi$
$504$		0			0
$505$	−21032.8			−1.85336
$506$		0			0
$507$	−4859.77	−	9091.80i	−0.425700	−	0.796412i
$508$		0			0
$509$	−5025.21			−0.437600			−0.218800	−	0.975770i	$-0.570214\pi$
$509$	−5025.21			−0.437600			−0.218800	+	0.975770i	$0.570214\pi$
$510$		0			0
$511$		−	77.1120i		−	0.00667561i
$512$		0			0
$513$	−21505.2	−	2117.51i	−1.85084	−	0.182242i
$514$		0			0
$515$			23461.1i			2.00742i
$516$		0			0
$517$			12073.1i			1.02703i
$518$		0			0
$519$	−4562.69	−	8536.02i	−0.385896	−	0.721945i
$520$		0			0
$521$		−	19105.2i		−	1.60656i	−0.595605	−	0.803278i	$-0.703087\pi$
$521$		−	19105.2i		−	1.60656i	0.595605	−	0.803278i	$-0.296913\pi$
$522$		0			0
$523$	−2025.34			−0.169334			−0.0846671	−	0.996409i	$-0.526983\pi$
$523$	−2025.34			−0.169334			−0.0846671	+	0.996409i	$0.526983\pi$
$524$		0			0
$525$	214.601	−	114.709i	0.0178399	−	0.00953584i
$526$		0			0
$527$	−5022.17			−0.415122
$528$		0			0
$529$	−10288.7			−0.845623
$530$		0			0
$531$	−1028.79	−	687.386i	−0.0840781	−	0.0561771i
$532$		0			0
$533$	−6544.68			−0.531860
$534$		0			0
$535$			25752.1i			2.08105i
$536$		0			0
$537$	18084.8	−	9666.75i	1.45329	−	0.776818i
$538$		0			0
$539$		−	14116.7i		−	1.12810i
$540$		0			0
$541$		−	1585.39i		−	0.125991i	−0.998014	−	0.0629955i	$-0.979935\pi$
$541$		−	1585.39i		−	0.125991i	0.998014	−	0.0629955i	$-0.0200654\pi$
$542$		0			0
$543$	−5439.57	+	2907.57i	−0.429898	+	0.229790i
$544$		0			0
$545$		−	32245.3i		−	2.53438i
$546$		0			0
$547$	4480.61			0.350232			0.175116	−	0.984548i	$-0.443970\pi$
$547$	4480.61			0.350232			0.175116	+	0.984548i	$0.443970\pi$
$548$		0			0
$549$	14620.1	+	9768.45i	1.13656	+	0.759394i
$550$		0			0
$551$	37074.3			2.86646
$552$		0			0
$553$	−253.745			−0.0195124
$554$		0			0
$555$	−10502.9	+	5614.02i	−0.803283	+	0.429373i
$556$		0			0
$557$	12069.6			0.918139			0.459070	−	0.888400i	$-0.348183\pi$
$557$	12069.6			0.918139			0.459070	+	0.888400i	$0.348183\pi$
$558$		0			0
$559$		−	7779.77i		−	0.588639i
$560$		0			0
$561$	6394.57	+	11963.1i	0.481246	+	0.900329i
$562$		0			0
$563$		−	6218.25i		−	0.465485i	−0.972538	−	0.232742i	$-0.925230\pi$
$563$		−	6218.25i		−	0.465485i	0.972538	−	0.232742i	$-0.0747699\pi$
$564$		0			0
$565$			18105.0i			1.34811i
$566$		0			0
$567$	−181.636	+	75.2438i	−0.0134533	+	0.00557309i
$568$		0			0
$569$			7681.87i			0.565977i	0.959123	+	0.282988i	$0.0913258\pi$
$569$			7681.87i			0.565977i	−0.959123	+	0.282988i	$0.908674\pi$
$570$		0			0
$571$	−21206.3			−1.55421			−0.777105	−	0.629371i	$-0.783313\pi$
$571$	−21206.3			−1.55421			−0.777105	+	0.629371i	$0.783313\pi$
$572$		0			0
$573$	2811.93	+	5260.63i	0.205008	+	0.383536i
$574$		0			0
$575$	−7525.56			−0.545804
$576$		0			0
$577$	9075.07			0.654766			0.327383	−	0.944892i	$-0.393833\pi$
$577$	9075.07			0.654766			0.327383	+	0.944892i	$0.393833\pi$
$578$		0			0
$579$	746.931	+	1397.38i	0.0536121	+	0.100299i
$580$		0			0
$581$	−272.688			−0.0194716
$582$		0			0
$583$		−	254.376i		−	0.0180706i
$584$		0			0
$585$	−25086.0	−	16761.3i	−1.77295	−	1.18460i
$586$		0			0
$587$			15653.2i			1.10064i	0.834954	+	0.550320i	$0.185494\pi$
$587$			15653.2i			1.10064i	−0.834954	+	0.550320i	$0.814506\pi$
$588$		0			0
$589$			12197.7i			0.853309i
$590$		0			0
$591$	3069.08	+	5741.72i	0.213612	+	0.399632i
$592$		0			0
$593$			1646.23i			0.114001i	0.998374	+	0.0570003i	$0.0181536\pi$
$593$			1646.23i			0.114001i	−0.998374	+	0.0570003i	$0.981846\pi$
$594$		0			0
$595$	−295.562			−0.0203645
$596$		0			0
$597$	−21374.4	+	11425.1i	−1.46532	+	0.783247i
$598$		0			0
$599$	6400.85			0.436614			0.218307	−	0.975880i	$-0.429947\pi$
$599$	6400.85			0.436614			0.218307	+	0.975880i	$0.429947\pi$
$600$		0			0
$601$	1282.48			0.0870443			0.0435222	−	0.999052i	$-0.486142\pi$
$601$	1282.48			0.0870443			0.0435222	+	0.999052i	$0.486142\pi$
$602$		0			0
$603$	10279.1	−	15384.3i	0.694191	−	1.03897i
$604$		0			0
$605$	6282.95			0.422212
$606$		0			0
$607$		−	5242.27i		−	0.350539i	−0.984521	−	0.175269i	$-0.943920\pi$
$607$		−	5242.27i		−	0.350539i	0.984521	−	0.175269i	$-0.0560796\pi$
$608$		0			0
$609$	297.479	−	159.009i	0.0197938	−	0.0105802i
$610$		0			0
$611$			18964.0i			1.25565i
$612$		0			0
$613$			9985.56i			0.657933i	0.944342	+	0.328966i	$0.106700\pi$
$613$			9985.56i			0.657933i	−0.944342	+	0.328966i	$0.893300\pi$
$614$		0			0
$615$	−8015.56	+	4284.50i	−0.525559	+	0.280923i
$616$		0			0
$617$			26504.8i			1.72941i	0.502283	+	0.864703i	$0.332493\pi$
$617$			26504.8i			1.72941i	−0.502283	+	0.864703i	$0.667507\pi$
$618$		0			0
$619$	−22327.0			−1.44976			−0.724878	−	0.688877i	$-0.758104\pi$
$619$	−22327.0			−1.44976			−0.724878	+	0.688877i	$0.758104\pi$
$620$		0			0
$621$	6051.09	+	595.818i	0.391017	+	0.0385014i
$622$		0			0
$623$	−116.642			−0.00750108
$624$		0			0
$625$	−7178.61			−0.459431
$626$		0			0
$627$	29055.8	−	15531.0i	1.85068	−	0.989231i
$628$		0			0
$629$	8410.64			0.533154
$630$		0			0
$631$			5040.02i			0.317971i	0.987281	+	0.158986i	$0.0508224\pi$
$631$			5040.02i			0.317971i	−0.987281	+	0.158986i	$0.949178\pi$
$632$		0			0
$633$	−4520.27	−	8456.64i	−0.283830	−	0.530998i
$634$		0			0
$635$			17856.4i			1.11592i
$636$		0			0
$637$		−	22173.9i		−	1.37922i
$638$		0			0
$639$	−12547.7	+	18779.7i	−0.776808	+	1.16262i
$640$		0			0
$641$		−	8994.13i		−	0.554207i	−0.960840	−	0.277104i	$-0.910626\pi$
$641$		−	8994.13i		−	0.554207i	0.960840	−	0.277104i	$-0.0893745\pi$
$642$		0			0
$643$	−8789.99			−0.539103			−0.269551	−	0.962986i	$-0.586875\pi$
$643$	−8789.99			−0.539103			−0.269551	+	0.962986i	$0.586875\pi$
$644$		0			0
$645$	−5093.05	−	9528.23i	−0.310913	−	0.581665i
$646$		0			0
$647$	14441.6			0.877523			0.438762	−	0.898603i	$-0.355417\pi$
$647$	14441.6			0.877523			0.438762	+	0.898603i	$0.355417\pi$
$648$		0			0
$649$	1886.42			0.114096
$650$		0			0
$651$	52.3152	+	97.8727i	0.00314961	+	0.00589237i
$652$		0			0
$653$	−16798.8			−1.00672			−0.503359	−	0.864078i	$-0.667903\pi$
$653$	−16798.8			−1.00672			−0.503359	+	0.864078i	$0.667903\pi$
$654$		0			0
$655$		−	6214.79i		−	0.370736i
$656$		0			0
$657$	−4288.91	+	6419.05i	−0.254682	+	0.381174i
$658$		0			0
$659$			10301.3i			0.608928i	0.952524	+	0.304464i	$0.0984773\pi$
$659$			10301.3i			0.608928i	−0.952524	+	0.304464i	$0.901523\pi$
$660$		0			0
$661$		−	31766.9i		−	1.86927i	−0.355609	−	0.934635i	$-0.615726\pi$
$661$		−	31766.9i		−	1.86927i	0.355609	−	0.934635i	$-0.384274\pi$
$662$		0			0
$663$	10044.3	+	18791.2i	0.588371	+	1.10074i
$664$		0			0
$665$			717.854i			0.0418604i
$666$		0			0
$667$	−10431.9			−0.605583
$668$		0			0
$669$	−13589.0	+	7263.63i	−0.785323	+	0.419773i
$670$		0			0
$671$	−26808.0			−1.54234
$672$		0			0
$673$	−12707.8			−0.727861			−0.363930	−	0.931426i	$-0.618565\pi$
$673$	−12707.8			−0.727861			−0.363930	+	0.931426i	$0.618565\pi$
$674$		0			0
$675$	−24244.1	−	2387.19i	−1.38245	−	0.136123i
$676$		0			0
$677$	−19127.6			−1.08587			−0.542936	−	0.839774i	$-0.682687\pi$
$677$	−19127.6			−1.08587			−0.542936	+	0.839774i	$0.682687\pi$
$678$		0			0
$679$		−	271.656i		−	0.0153537i
$680$		0			0
$681$	−7052.52	+	3769.73i	−0.396848	+	0.212124i
$682$		0			0
$683$		−	14069.0i		−	0.788191i	−0.919069	−	0.394096i	$-0.871058\pi$
$683$		−	14069.0i		−	0.788191i	0.919069	−	0.394096i	$-0.128942\pi$
$684$		0			0
$685$			6553.84i			0.365561i
$686$		0			0
$687$	7598.16	−	4061.39i	0.421962	−	0.225548i
$688$		0			0
$689$		−	399.564i		−	0.0220931i
$690$		0			0
$691$	−19159.5			−1.05479			−0.527396	−	0.849620i	$-0.676831\pi$
$691$	−19159.5			−1.05479			−0.527396	+	0.849620i	$0.676831\pi$
$692$		0			0
$693$	166.528	−	249.236i	0.00912825	−	0.0136619i
$694$		0			0
$695$	34287.5			1.87137
$696$		0			0
$697$	6418.81			0.348823
$698$		0			0
$699$	5690.12	−	3041.50i	0.307897	−	0.164578i
$700$		0			0
$701$	−24736.8			−1.33281			−0.666403	−	0.745592i	$-0.732167\pi$
$701$	−24736.8			−1.33281			−0.666403	+	0.745592i	$0.732167\pi$
$702$		0			0
$703$		−	20427.6i		−	1.09593i
$704$		0			0
$705$	12414.8	+	23226.0i	0.663220	+	1.24077i
$706$		0			0
$707$		−	328.237i		−	0.0174605i
$708$		0			0
$709$		−	10627.5i		−	0.562937i	−0.959570	−	0.281469i	$-0.909178\pi$
$709$		−	10627.5i		−	0.562937i	0.959570	−	0.281469i	$-0.0908216\pi$
$710$		0			0
$711$	21122.6	+	14113.1i	1.11415	+	0.744420i
$712$		0			0
$713$		−	3432.17i		−	0.180274i
$714$		0			0
$715$	45998.7			2.40595
$716$		0			0
$717$	−5814.05	−	10877.1i	−0.302831	−	0.566544i
$718$		0			0
$719$	−8256.38			−0.428249			−0.214124	−	0.976806i	$-0.568690\pi$
$719$	−8256.38			−0.428249			−0.214124	+	0.976806i	$0.568690\pi$
$720$		0			0
$721$	−366.134			−0.0189120
$722$		0			0
$723$	6349.94	+	11879.6i	0.326635	+	0.611077i
$724$		0			0
$725$	41796.1			2.14106
$726$		0			0
$727$		−	21457.7i		−	1.09466i	−0.836916	−	0.547332i	$-0.815644\pi$
$727$		−	21457.7i		−	1.09466i	0.836916	−	0.547332i	$-0.184356\pi$
$728$		0			0
$729$	19305.0	+	3838.94i	0.980796	+	0.195038i
$730$		0			0
$731$			7630.15i			0.386062i
$732$		0			0
$733$			12687.2i			0.639307i	0.947534	+	0.319654i	$0.103567\pi$
$733$			12687.2i			0.639307i	−0.947534	+	0.319654i	$0.896433\pi$
$734$		0			0
$735$	−14516.2	−	27157.4i	−0.728488	−	1.36288i
$736$		0			0
$737$			28209.4i			1.40991i
$738$		0			0
$739$	−7599.74			−0.378296			−0.189148	−	0.981949i	$-0.560573\pi$
$739$	−7599.74			−0.378296			−0.189148	+	0.981949i	$0.560573\pi$
$740$		0			0
$741$	45639.7	−	24395.5i	2.26264	−	1.20943i
$742$		0			0
$743$	−34061.2			−1.68181			−0.840904	−	0.541185i	$-0.817976\pi$
$743$	−34061.2			−1.68181			−0.840904	+	0.541185i	$0.817976\pi$
$744$		0			0
$745$	40146.6			1.97431
$746$		0			0
$747$	22699.4	+	15166.7i	1.11182	+	0.742864i
$748$		0			0
$749$	−401.887			−0.0196056
$750$		0			0
$751$			7091.67i			0.344579i	0.985046	+	0.172289i	$0.0551164\pi$
$751$			7091.67i			0.344579i	−0.985046	+	0.172289i	$0.944884\pi$
$752$		0			0
$753$	12603.5	−	6736.84i	0.609955	−	0.326035i
$754$		0			0
$755$		−	45162.2i		−	2.17698i
$756$		0			0
$757$			10128.2i			0.486282i	0.969991	+	0.243141i	$0.0781778\pi$
$757$			10128.2i			0.486282i	−0.969991	+	0.243141i	$0.921822\pi$
$758$		0			0
$759$	−8175.65	+	4370.07i	−0.390984	+	0.208990i
$760$		0			0
$761$			22747.8i			1.08358i	0.840513	+	0.541792i	$0.182254\pi$
$761$			22747.8i			1.08358i	−0.840513	+	0.541792i	$0.817746\pi$
$762$		0			0
$763$	503.219			0.0238765
$764$		0			0
$765$	24603.5	+	16438.9i	1.16280	+	0.776928i
$766$		0			0
$767$	2963.12			0.139494
$768$		0			0
$769$	30168.9			1.41472			0.707359	−	0.706855i	$-0.249887\pi$
$769$	30168.9			1.41472			0.707359	+	0.706855i	$0.249887\pi$
$770$		0			0
$771$	−999.748	+	534.388i	−0.0466992	+	0.0249617i
$772$		0			0
$773$	12592.8			0.585940			0.292970	−	0.956122i	$-0.405356\pi$
$773$	12592.8			0.585940			0.292970	+	0.956122i	$0.405356\pi$
$774$		0			0
$775$			13751.2i			0.637366i
$776$		0			0
$777$	−87.6123	−	163.908i	−0.00404514	−	0.00756776i
$778$		0			0
$779$		−	15589.9i		−	0.717028i
$780$		0			0
$781$		−	34435.3i		−	1.57771i
$782$		0			0
$783$	−33607.0	−	3309.10i	−1.53387	−	0.151032i
$784$		0			0
$785$		−	48049.0i		−	2.18464i
$786$		0			0
$787$	14594.4			0.661036			0.330518	−	0.943800i	$-0.392777\pi$
$787$	14594.4			0.661036			0.330518	+	0.943800i	$0.392777\pi$
$788$		0			0
$789$	−17982.3	−	33641.9i	−0.811391	−	1.51797i
$790$		0			0
$791$	−282.546			−0.0127006
$792$		0			0
$793$	−42108.9			−1.88567
$794$		0			0
$795$	−261.576	−	489.364i	−0.0116694	−	0.0218314i
$796$		0			0
$797$	36750.1			1.63332			0.816661	−	0.577118i	$-0.195823\pi$
$797$	36750.1			1.63332			0.816661	+	0.577118i	$0.195823\pi$
$798$		0			0
$799$		−	18599.3i		−	0.823522i
$800$		0			0
$801$	9709.67	+	6487.55i	0.428308	+	0.286175i
$802$		0			0
$803$		−	11770.2i		−	0.517264i
$804$		0			0
$805$		−	201.988i		−	0.00884365i
$806$		0			0
$807$	133.149	+	249.098i	0.00580799	+	0.0108658i
$808$		0			0
$809$		−	15914.1i		−	0.691605i	−0.938307	−	0.345803i	$-0.887607\pi$
$809$		−	15914.1i		−	0.691605i	0.938307	−	0.345803i	$-0.112393\pi$
$810$		0			0
$811$	−1811.41			−0.0784304			−0.0392152	−	0.999231i	$-0.512486\pi$
$811$	−1811.41			−0.0784304			−0.0392152	+	0.999231i	$0.512486\pi$
$812$		0			0
$813$	−16650.5	+	8900.04i	−0.718274	+	0.383934i
$814$		0			0
$815$	−10978.7			−0.471861
$816$		0			0
$817$	18531.9			0.793574
$818$		0			0
$819$	261.576	−	391.491i	0.0111602	−	0.0167031i
$820$		0			0
$821$	15523.7			0.659904			0.329952	−	0.943998i	$-0.392967\pi$
$821$	15523.7			0.659904			0.329952	+	0.943998i	$0.392967\pi$
$822$		0			0
$823$			2879.06i			0.121941i	0.998140	+	0.0609707i	$0.0194196\pi$
$823$			2879.06i			0.121941i	−0.998140	+	0.0609707i	$0.980580\pi$
$824$		0			0
$825$	32756.3	−	17509.0i	1.38234	−	0.738891i
$826$		0			0
$827$			2163.66i			0.0909766i	0.998965	+	0.0454883i	$0.0144844\pi$
$827$			2163.66i			0.0909766i	−0.998965	+	0.0454883i	$0.985516\pi$
$828$		0			0
$829$		−	22363.4i		−	0.936928i	−0.883483	−	0.468464i	$-0.844808\pi$
$829$		−	22363.4i		−	0.936928i	0.883483	−	0.468464i	$-0.155192\pi$
$830$		0			0
$831$	28430.8	−	15196.9i	1.18683	−	0.634387i
$832$		0			0
$833$			21747.4i			0.904566i
$834$		0			0
$835$	5521.73			0.228847
$836$		0			0
$837$	1088.72	−	11057.0i	0.0449602	−	0.456612i
$838$		0			0
$839$	10859.0			0.446833			0.223417	−	0.974723i	$-0.428279\pi$
$839$	10859.0			0.446833			0.223417	+	0.974723i	$0.428279\pi$
$840$		0			0
$841$	33548.4			1.37556
$842$		0			0
$843$	−25898.4	+	13843.3i	−1.05811	+	0.565584i
$844$		0			0
$845$	34285.9			1.39583
$846$		0			0
$847$			98.0515i			0.00397767i
$848$		0			0
$849$	−14346.8	−	26840.5i	−0.579955	−	1.08500i
$850$		0			0
$851$			5747.85i			0.231532i
$852$		0			0
$853$			23518.0i			0.944012i	0.881595	+	0.472006i	$0.156470\pi$
$853$			23518.0i			0.944012i	−0.881595	+	0.472006i	$0.843530\pi$
$854$		0			0
$855$	39926.5	−	59756.4i	1.59702	−	2.39021i
$856$		0			0
$857$			22859.6i			0.911164i	0.890194	+	0.455582i	$0.150569\pi$
$857$			22859.6i			0.911164i	−0.890194	+	0.455582i	$0.849431\pi$
$858$		0			0
$859$	25943.4			1.03048			0.515238	−	0.857047i	$-0.327704\pi$
$859$	25943.4			1.03048			0.515238	+	0.857047i	$0.327704\pi$
$860$		0			0
$861$	−66.8638	−	125.091i	−0.00264659	−	0.00495131i
$862$		0			0
$863$	10444.3			0.411967			0.205983	−	0.978555i	$-0.433961\pi$
$863$	10444.3			0.411967			0.205983	+	0.978555i	$0.433961\pi$
$864$		0			0
$865$	32190.0			1.26531
$866$		0			0
$867$	2183.18	+	4084.35i	0.0855186	+	0.159991i
$868$		0			0
$869$	−38731.2			−1.51193
$870$		0			0
$871$			44310.2i			1.72376i
$872$		0			0
$873$	−15109.3	+	22613.5i	−0.585763	+	0.876690i
$874$		0			0
$875$			226.703i			0.00875882i
$876$		0			0
$877$		−	42209.3i		−	1.62521i	−0.582818	−	0.812603i	$-0.698050\pi$
$877$		−	42209.3i		−	1.62521i	0.582818	−	0.812603i	$-0.301950\pi$
$878$		0			0
$879$	13216.0	+	24724.9i	0.507128	+	0.948750i
$880$		0			0
$881$			42919.3i			1.64130i	0.571429	+	0.820651i	$0.306389\pi$
$881$			42919.3i			1.64130i	−0.571429	+	0.820651i	$0.693611\pi$
$882$		0			0
$883$	9983.97			0.380507			0.190253	−	0.981735i	$-0.439069\pi$
$883$	9983.97			0.380507			0.190253	+	0.981735i	$0.439069\pi$
$884$		0			0
$885$	3629.07	−	1939.82i	0.137842	−	0.0736794i
$886$		0			0
$887$	−42646.7			−1.61436			−0.807180	−	0.590305i	$-0.799007\pi$
$887$	−42646.7			−1.61436			−0.807180	+	0.590305i	$0.799007\pi$
$888$		0			0
$889$	−278.667			−0.0105131
$890$		0			0
$891$	−27724.7	+	11485.1i	−1.04244	+	0.431835i
$892$		0			0
$893$	−45173.5			−1.69280
$894$		0			0
$895$			68199.4i			2.54710i
$896$		0			0
$897$	−12842.0	+	6864.33i	−0.478017	+	0.255511i
$898$		0			0
$899$			19061.8i			0.707173i
$900$		0			0
$901$			391.879i			0.0144899i
$902$		0			0
$903$	148.697	−	79.4821i	0.00547989	−	0.00292912i
$904$		0			0
$905$		−	20513.1i		−	0.753456i
$906$		0			0
$907$	−37924.9			−1.38840			−0.694198	−	0.719784i	$-0.744241\pi$
$907$	−37924.9			−1.38840			−0.694198	+	0.719784i	$0.744241\pi$
$908$		0			0
$909$	−18256.3	+	27323.5i	−0.666140	+	0.996988i
$910$		0			0
$911$	−18358.2			−0.667655			−0.333827	−	0.942634i	$-0.608340\pi$
$911$	−18358.2			−0.667655			−0.333827	+	0.942634i	$0.608340\pi$
$912$		0			0
$913$	−41622.5			−1.50877
$914$		0			0
$915$	−51572.7	+	27566.8i	−1.86332	+	0.995989i
$916$		0			0
$917$	96.9879			0.00349272
$918$		0			0
$919$		−	30049.2i		−	1.07860i	−0.842114	−	0.539300i	$-0.818689\pi$
$919$		−	30049.2i		−	1.07860i	0.842114	−	0.539300i	$-0.181311\pi$
$920$		0			0
$921$	20058.8	+	37526.6i	0.717656	+	1.34261i
$922$		0			0
$923$		−	54089.6i		−	1.92891i
$924$		0			0
$925$		−	23029.2i		−	0.818589i
$926$		0			0
$927$	30478.1	+	20364.1i	1.07986	+	0.721514i
$928$		0			0
$929$			34115.5i			1.20484i	0.798181	+	0.602418i	$0.205796\pi$
$929$			34115.5i			1.20484i	−0.798181	+	0.602418i	$0.794204\pi$
$930$		0			0
$931$	52819.7			1.85939
$932$		0			0
$933$	4503.19	+	8424.69i	0.158015	+	0.295618i
$934$		0			0
$935$	−45114.0			−1.57795
$936$		0			0
$937$	1709.73			0.0596100			0.0298050	−	0.999556i	$-0.490511\pi$
$937$	1709.73			0.0596100			0.0298050	+	0.999556i	$0.490511\pi$
$938$		0			0
$939$	−717.894	−	1343.06i	−0.0249495	−	0.0466762i
$940$		0			0
$941$	6856.94			0.237545			0.118772	−	0.992921i	$-0.462104\pi$
$941$	6856.94			0.237545			0.118772	+	0.992921i	$0.462104\pi$
$942$		0			0
$943$			4386.64i			0.151483i
$944$		0			0
$945$	64.0727	−	650.718i	0.00220559	−	0.0223999i
$946$		0			0
$947$			34558.0i			1.18583i	0.805264	+	0.592917i	$0.202024\pi$
$947$			34558.0i			1.18583i	−0.805264	+	0.592917i	$0.797976\pi$
$948$		0			0
$949$		−	18488.2i		−	0.632406i
$950$		0			0
$951$	3289.71	+	6154.48i	0.112172	+	0.209855i
$952$		0			0
$953$		−	39786.8i		−	1.35238i	−0.736725	−	0.676192i	$-0.763629\pi$
$953$		−	39786.8i		−	1.35238i	0.736725	−	0.676192i	$-0.236371\pi$
$954$		0			0
$955$	−19838.3			−0.672200
$956$		0			0
$957$	45406.6	−	24270.8i	1.53374	−	0.819817i
$958$		0			0
$959$	−102.279			−0.00344397
$960$		0			0
$961$	23519.5			0.789484
$962$		0			0
$963$	33454.3	+	22352.6i	1.11947	+	0.747978i
$964$		0			0
$965$	−5269.64			−0.175788
$966$		0			0
$967$			11168.4i			0.371406i	0.982606	+	0.185703i	$0.0594563\pi$
$967$			11168.4i			0.371406i	−0.982606	+	0.185703i	$0.940544\pi$
$968$		0			0
$969$	−44762.0	+	23926.3i	−1.48397	+	0.793213i
$970$		0			0
$971$		−	13885.1i		−	0.458901i	−0.973320	−	0.229450i	$-0.926307\pi$
$971$		−	13885.1i		−	0.458901i	0.973320	−	0.229450i	$-0.0736929\pi$
$972$		0			0
$973$			535.091i			0.0176302i
$974$		0			0
$975$	51452.4	−	27502.4i	1.69005	−	0.903367i
$976$		0			0
$977$		−	5137.09i		−	0.168219i	−0.996457	−	0.0841096i	$-0.973195\pi$
$977$		−	5137.09i		−	0.168219i	0.996457	−	0.0841096i	$-0.0268046\pi$
$978$		0			0
$979$	−17804.1			−0.581226
$980$		0			0
$981$	−41889.6	−	27988.6i	−1.36333	−	0.910916i
$982$		0			0
$983$	28815.3			0.934959			0.467480	−	0.884004i	$-0.345162\pi$
$983$	28815.3			0.934959			0.467480	+	0.884004i	$0.345162\pi$
$984$		0			0
$985$	−21652.5			−0.700412
$986$		0			0
$987$	−362.465	+	193.746i	−0.0116893	+	0.00624822i
$988$		0			0
$989$	−5214.46			−0.167655
$990$		0			0
$991$		−	52902.8i		−	1.69578i	−0.530175	−	0.847888i	$-0.677874\pi$
$991$		−	52902.8i		−	1.69578i	0.530175	−	0.847888i	$-0.322126\pi$
$992$		0			0
$993$	−21619.1	−	40445.6i	−0.690897	−	1.29255i
$994$		0			0
$995$		−	80604.7i		−	2.56818i
$996$		0			0
$997$			29517.0i			0.937625i	0.883298	+	0.468812i	$0.155318\pi$
$997$			29517.0i			0.937625i	−0.883298	+	0.468812i	$0.844682\pi$
$998$		0			0
$999$	−1823.28	+	18517.1i	−0.0577438	+	0.586442i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.f.g.191.7	yes	8
3.2	odd	2		384.4.f.h.191.6	yes	8
4.3	odd	2		384.4.f.h.191.1	yes	8
8.3	odd	2		384.4.f.h.191.8	yes	8
8.5	even	2	inner	384.4.f.g.191.2	✓	8
12.11	even	2	inner	384.4.f.g.191.4	yes	8
16.3	odd	4		768.4.c.p.767.3		4
16.5	even	4		768.4.c.o.767.3		4
16.11	odd	4		768.4.c.m.767.2		4
16.13	even	4		768.4.c.n.767.2		4
24.5	odd	2		384.4.f.h.191.3	yes	8
24.11	even	2	inner	384.4.f.g.191.5	yes	8
48.5	odd	4		768.4.c.m.767.1		4
48.11	even	4		768.4.c.o.767.4		4
48.29	odd	4		768.4.c.p.767.4		4
48.35	even	4		768.4.c.n.767.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.g.191.2	✓	8	8.5	even	2	inner
384.4.f.g.191.4	yes	8	12.11	even	2	inner
384.4.f.g.191.5	yes	8	24.11	even	2	inner
384.4.f.g.191.7	yes	8	1.1	even	1	trivial
384.4.f.h.191.1	yes	8	4.3	odd	2
384.4.f.h.191.3	yes	8	24.5	odd	2
384.4.f.h.191.6	yes	8	3.2	odd	2
384.4.f.h.191.8	yes	8	8.3	odd	2
768.4.c.m.767.1		4	48.5	odd	4
768.4.c.m.767.2		4	16.11	odd	4
768.4.c.n.767.1		4	48.35	even	4
768.4.c.n.767.2		4	16.13	even	4
768.4.c.o.767.3		4	16.5	even	4
768.4.c.o.767.4		4	48.11	even	4
768.4.c.p.767.3		4	16.3	odd	4
768.4.c.p.767.4		4	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 + 4.58258i) q^{3} -17.2813 q^{5} -0.269691i q^{7} +(-15.0000 + 22.4499i) q^{9} +O(q^{10})\)	\(q+(2.44949 + 4.58258i) q^{3} -17.2813 q^{5} -0.269691i q^{7} +(-15.0000 + 22.4499i) q^{9} -41.1652i q^{11} -64.6606i q^{13} +(-42.3303 - 79.1927i) q^{15} +63.4170i q^{17} +154.026 q^{19} +(1.23588 - 0.660606i) q^{21} -43.3394 q^{23} +173.642 q^{25} +(-139.621 - 13.7477i) q^{27} +240.702 q^{29} +79.1927i q^{31} +(188.642 - 100.834i) q^{33} +4.66061i q^{35} -132.624i q^{37} +(296.312 - 158.385i) q^{39} -101.216i q^{41} +120.317 q^{43} +(259.219 - 387.964i) q^{45} -293.285 q^{47} +342.927 q^{49} +(-290.613 + 155.339i) q^{51} +6.17940 q^{53} +711.386i q^{55} +(377.285 + 705.835i) q^{57} +45.8258i q^{59} -651.230i q^{61} +(6.05455 + 4.04537i) q^{63} +1117.42i q^{65} -685.273 q^{67} +(-106.159 - 198.606i) q^{69} +836.515 q^{71} +285.927 q^{73} +(425.335 + 795.730i) q^{75} -11.1019 q^{77} -940.874i q^{79} +(-279.000 - 673.498i) q^{81} -1011.11i q^{83} -1095.93i q^{85} +(589.597 + 1103.03i) q^{87} -432.503i q^{89} -17.4384 q^{91} +(-362.907 + 193.982i) q^{93} -2661.76 q^{95} +1007.28 q^{97} +(924.155 + 617.477i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 120 q^{9}+O(q^{10}) \) 8 * q - 120 * q^9	\( 8 q - 120 q^{9} - 192 q^{15} - 640 q^{23} + 216 q^{25} + 336 q^{33} + 1344 q^{39} - 776 q^{49} + 672 q^{57} + 2688 q^{63} - 640 q^{71} - 1232 q^{73} - 2232 q^{81} + 1344 q^{87} - 9856 q^{95} + 5712 q^{97}+O(q^{100}) \) 8 * q - 120 * q^9 - 192 * q^15 - 640 * q^23 + 216 * q^25 + 336 * q^33 + 1344 * q^39 - 776 * q^49 + 672 * q^57 + 2688 * q^63 - 640 * q^71 - 1232 * q^73 - 2232 * q^81 + 1344 * q^87 - 9856 * q^95 + 5712 * q^97

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