LMFDB - Embedded newform 384.3.e.c.257.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(257,384)]

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, 0, 1]))

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

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

chi := DirichletCharacter("384.257");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 $ x^8 + 18x^6 + 99x^4 + 170*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.7
Root			$0.888828i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.3.e.c.257.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.86531 - 0.888828i) q^{3} +8.59176i q^{5} -10.9340 q^{7} +(7.41997 - 5.09353i) q^{9} +O(q^{10})$	\(q+(2.86531 - 0.888828i) q^{3} +8.59176i q^{5} -10.9340 q^{7} +(7.41997 - 5.09353i) q^{9} +2.75255i q^{11} +4.43326 q^{13} +(7.63660 + 24.6180i) q^{15} +25.4208i q^{17} -17.5426 q^{19} +(-31.3291 + 9.71841i) q^{21} +17.5482i q^{23} -48.8183 q^{25} +(16.7332 - 21.1896i) q^{27} +19.6224i q^{29} -2.58322 q^{31} +(2.44655 + 7.88691i) q^{33} -93.9419i q^{35} -7.73178 q^{37} +(12.7026 - 3.94040i) q^{39} +58.0069i q^{41} +42.1932 q^{43} +(43.7624 + 63.7506i) q^{45} -17.4666i q^{47} +70.5514 q^{49} +(22.5947 + 72.8384i) q^{51} -69.0052i q^{53} -23.6493 q^{55} +(-50.2649 + 15.5923i) q^{57} -50.5878i q^{59} +32.5983 q^{61} +(-81.1296 + 55.6924i) q^{63} +38.0895i q^{65} +48.0128 q^{67} +(15.5974 + 50.2811i) q^{69} +22.1021i q^{71} -27.0316 q^{73} +(-139.880 + 43.3911i) q^{75} -30.0963i q^{77} +97.4827 q^{79} +(29.1119 - 75.5877i) q^{81} +59.5252i q^{83} -218.409 q^{85} +(17.4409 + 56.2242i) q^{87} -110.469i q^{89} -48.4730 q^{91} +(-7.40171 + 2.29603i) q^{93} -150.722i q^{95} +55.1169 q^{97} +(14.0202 + 20.4238i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{7}+O(q^{10}) $ 8 * q + 4 * q^3 - 8 * q^7	$ 8 q + 4 q^{3} - 8 q^{7} - 16 q^{15} - 24 q^{19} - 16 q^{21} - 40 q^{25} - 44 q^{27} + 56 q^{31} + 8 q^{33} - 32 q^{37} + 104 q^{39} + 136 q^{43} + 80 q^{45} + 72 q^{49} + 176 q^{51} - 192 q^{55} - 40 q^{57} + 160 q^{61} - 264 q^{63} - 280 q^{67} - 80 q^{69} - 80 q^{73} - 348 q^{75} + 408 q^{79} + 72 q^{81} - 192 q^{85} + 368 q^{87} + 336 q^{91} - 160 q^{93} + 96 q^{97} + 432 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^7 - 16 * q^15 - 24 * q^19 - 16 * q^21 - 40 * q^25 - 44 * q^27 + 56 * q^31 + 8 * q^33 - 32 * q^37 + 104 * q^39 + 136 * q^43 + 80 * q^45 + 72 * q^49 + 176 * q^51 - 192 * q^55 - 40 * q^57 + 160 * q^61 - 264 * q^63 - 280 * q^67 - 80 * q^69 - 80 * q^73 - 348 * q^75 + 408 * q^79 + 72 * q^81 - 192 * q^85 + 368 * q^87 + 336 * q^91 - 160 * q^93 + 96 * q^97 + 432 * q^99

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
$3$	2.86531	−	0.888828i	0.955102	−	0.296276i
$3$	2.86531	−	0.888828i	0.955102	−	0.296276i
$4$		0			0
$5$			8.59176i			1.71835i	0.511680	+	0.859176i	$0.329023\pi$
$5$			8.59176i			1.71835i	−0.511680	+	0.859176i	$0.670977\pi$
$6$		0			0
$7$	−10.9340			−1.56199			−0.780997	−	0.624535i	$-0.785288\pi$
$7$	−10.9340			−1.56199			−0.780997	+	0.624535i	$0.785288\pi$
$8$		0			0
$9$	7.41997	−	5.09353i	0.824441	−	0.565948i
$10$		0			0
$11$			2.75255i			0.250232i	0.992142	+	0.125116i	$0.0399303\pi$
$11$			2.75255i			0.250232i	−0.992142	+	0.125116i	$0.960070\pi$
$12$		0			0
$13$	4.43326			0.341020			0.170510	−	0.985356i	$-0.445459\pi$
$13$	4.43326			0.341020			0.170510	+	0.985356i	$0.445459\pi$
$14$		0			0
$15$	7.63660	+	24.6180i	0.509107	+	1.64120i
$16$		0			0
$17$			25.4208i			1.49534i	0.664070	+	0.747670i	$0.268828\pi$
$17$			25.4208i			1.49534i	−0.664070	+	0.747670i	$0.731172\pi$
$18$		0			0
$19$	−17.5426			−0.923294			−0.461647	−	0.887064i	$-0.652741\pi$
$19$	−17.5426			−0.923294			−0.461647	+	0.887064i	$0.652741\pi$
$20$		0			0
$21$	−31.3291	+	9.71841i	−1.49186	+	0.462781i
$22$		0			0
$23$			17.5482i			0.762966i	0.924376	+	0.381483i	$0.124587\pi$
$23$			17.5482i			0.762966i	−0.924376	+	0.381483i	$0.875413\pi$
$24$		0			0
$25$	−48.8183			−1.95273
$26$		0			0
$27$	16.7332	−	21.1896i	0.619749	−	0.784800i
$28$		0			0
$29$			19.6224i			0.676635i	0.941032	+	0.338317i	$0.109858\pi$
$29$			19.6224i			0.676635i	−0.941032	+	0.338317i	$0.890142\pi$
$30$		0			0
$31$	−2.58322			−0.0833295			−0.0416648	−	0.999132i	$-0.513266\pi$
$31$	−2.58322			−0.0833295			−0.0416648	+	0.999132i	$0.513266\pi$
$32$		0			0
$33$	2.44655	+	7.88691i	0.0741377	+	0.238997i
$34$		0			0
$35$		−	93.9419i		−	2.68405i
$36$		0			0
$37$	−7.73178			−0.208967			−0.104484	−	0.994527i	$-0.533319\pi$
$37$	−7.73178			−0.208967			−0.104484	+	0.994527i	$0.533319\pi$
$38$		0			0
$39$	12.7026	−	3.94040i	0.325709	−	0.101036i
$40$		0			0
$41$			58.0069i			1.41480i	0.706812	+	0.707402i	$0.250133\pi$
$41$			58.0069i			1.41480i	−0.706812	+	0.707402i	$0.749867\pi$
$42$		0			0
$43$	42.1932			0.981238			0.490619	−	0.871374i	$-0.336771\pi$
$43$	42.1932			0.981238			0.490619	+	0.871374i	$0.336771\pi$
$44$		0			0
$45$	43.7624	+	63.7506i	0.972498	+	1.41668i
$46$		0			0
$47$		−	17.4666i		−	0.371629i	−0.982585	−	0.185815i	$-0.940508\pi$
$47$		−	17.4666i		−	0.371629i	0.982585	−	0.185815i	$-0.0594924\pi$
$48$		0			0
$49$	70.5514			1.43982
$50$		0			0
$51$	22.5947	+	72.8384i	0.443034	+	1.42820i
$52$		0			0
$53$		−	69.0052i		−	1.30198i	−0.759085	−	0.650992i	$-0.774353\pi$
$53$		−	69.0052i		−	1.30198i	0.759085	−	0.650992i	$-0.225647\pi$
$54$		0			0
$55$	−23.6493			−0.429987
$56$		0			0
$57$	−50.2649	+	15.5923i	−0.881840	+	0.273550i
$58$		0			0
$59$		−	50.5878i		−	0.857420i	−0.903442	−	0.428710i	$-0.858968\pi$
$59$		−	50.5878i		−	0.857420i	0.903442	−	0.428710i	$-0.141032\pi$
$60$		0			0
$61$	32.5983			0.534398			0.267199	−	0.963641i	$-0.413902\pi$
$61$	32.5983			0.534398			0.267199	+	0.963641i	$0.413902\pi$
$62$		0			0
$63$	−81.1296	+	55.6924i	−1.28777	+	0.884007i
$64$		0			0
$65$			38.0895i			0.585992i
$66$		0			0
$67$	48.0128			0.716609			0.358305	−	0.933605i	$-0.383355\pi$
$67$	48.0128			0.716609			0.358305	+	0.933605i	$0.383355\pi$
$68$		0			0
$69$	15.5974	+	50.2811i	0.226049	+	0.728711i
$70$		0			0
$71$			22.1021i			0.311297i	0.987813	+	0.155648i	$0.0497466\pi$
$71$			22.1021i			0.311297i	−0.987813	+	0.155648i	$0.950253\pi$
$72$		0			0
$73$	−27.0316			−0.370295			−0.185148	−	0.982711i	$-0.559276\pi$
$73$	−27.0316			−0.370295			−0.185148	+	0.982711i	$0.559276\pi$
$74$		0			0
$75$	−139.880	+	43.3911i	−1.86506	+	0.578548i
$76$		0			0
$77$		−	30.0963i		−	0.390861i
$78$		0			0
$79$	97.4827			1.23396			0.616979	−	0.786980i	$-0.288356\pi$
$79$	97.4827			1.23396			0.616979	+	0.786980i	$0.288356\pi$
$80$		0			0
$81$	29.1119	−	75.5877i	0.359406	−	0.933181i
$82$		0			0
$83$			59.5252i			0.717170i	0.933497	+	0.358585i	$0.116741\pi$
$83$			59.5252i			0.717170i	−0.933497	+	0.358585i	$0.883259\pi$
$84$		0			0
$85$	−218.409			−2.56952
$86$		0			0
$87$	17.4409	+	56.2242i	0.200471	+	0.646255i
$88$		0			0
$89$		−	110.469i		−	1.24122i	−0.784119	−	0.620611i	$-0.786885\pi$
$89$		−	110.469i		−	1.24122i	0.784119	−	0.620611i	$-0.213115\pi$
$90$		0			0
$91$	−48.4730			−0.532671
$92$		0			0
$93$	−7.40171	+	2.29603i	−0.0795882	+	0.0246885i
$94$		0			0
$95$		−	150.722i		−	1.58654i
$96$		0			0
$97$	55.1169			0.568215			0.284108	−	0.958792i	$-0.408303\pi$
$97$	55.1169			0.568215			0.284108	+	0.958792i	$0.408303\pi$
$98$		0			0
$99$	14.0202	+	20.4238i	0.141618	+	0.206301i
$100$		0			0
$101$		−	20.2294i		−	0.200291i	−0.994973	−	0.100146i	$-0.968069\pi$
$101$		−	20.2294i		−	0.200291i	0.994973	−	0.100146i	$-0.0319309\pi$
$102$		0			0
$103$	65.3051			0.634030			0.317015	−	0.948421i	$-0.397320\pi$
$103$	65.3051			0.634030			0.317015	+	0.948421i	$0.397320\pi$
$104$		0			0
$105$	−83.4982	−	269.172i	−0.795221	−	2.56355i
$106$		0			0
$107$		−	10.3305i		−	0.0965468i	−0.998834	−	0.0482734i	$-0.984628\pi$
$107$		−	10.3305i		−	0.0965468i	0.998834	−	0.0482734i	$-0.0153719\pi$
$108$		0			0
$109$	151.542			1.39029			0.695145	−	0.718870i	$-0.255340\pi$
$109$	151.542			1.39029			0.695145	+	0.718870i	$0.255340\pi$
$110$		0			0
$111$	−22.1539	+	6.87222i	−0.199585	+	0.0619119i
$112$		0			0
$113$			106.377i			0.941391i	0.882296	+	0.470696i	$0.155997\pi$
$113$			106.377i			0.941391i	−0.882296	+	0.470696i	$0.844003\pi$
$114$		0			0
$115$	−150.770			−1.31104
$116$		0			0
$117$	32.8946	−	22.5809i	0.281151	−	0.192999i
$118$		0			0
$119$		−	277.950i		−	2.33571i
$120$		0			0
$121$	113.423			0.937384
$122$		0			0
$123$	51.5582	+	166.208i	0.419172	+	1.35128i
$124$		0			0
$125$		−	204.641i		−	1.63713i
$126$		0			0
$127$	−112.285			−0.884138			−0.442069	−	0.896981i	$-0.645755\pi$
$127$	−112.285			−0.884138			−0.442069	+	0.896981i	$0.645755\pi$
$128$		0			0
$129$	120.897	−	37.5025i	0.937183	−	0.290717i
$130$		0			0
$131$			137.804i			1.05194i	0.850504	+	0.525968i	$0.176297\pi$
$131$			137.804i			1.05194i	−0.850504	+	0.525968i	$0.823703\pi$
$132$		0			0
$133$	191.810			1.44218
$134$		0			0
$135$	182.056	+	143.768i	1.34856	+	1.06495i
$136$		0			0
$137$			152.889i			1.11598i	0.829847	+	0.557990i	$0.188427\pi$
$137$			152.889i			1.11598i	−0.829847	+	0.557990i	$0.811573\pi$
$138$		0			0
$139$	236.450			1.70108			0.850540	−	0.525910i	$-0.176275\pi$
$139$	236.450			1.70108			0.850540	+	0.525910i	$0.176275\pi$
$140$		0			0
$141$	−15.5248	−	50.0471i	−0.110105	−	0.354944i
$142$		0			0
$143$			12.2028i			0.0853340i
$144$		0			0
$145$	−168.591			−1.16270
$146$		0			0
$147$	202.151	−	62.7080i	1.37518	−	0.426585i
$148$		0			0
$149$		−	185.019i		−	1.24174i	−0.783914	−	0.620869i	$-0.786780\pi$
$149$		−	185.019i		−	1.24174i	0.783914	−	0.620869i	$-0.213220\pi$
$150$		0			0
$151$	−283.340			−1.87643			−0.938214	−	0.346057i	$-0.887520\pi$
$151$	−283.340			−1.87643			−0.938214	+	0.346057i	$0.887520\pi$
$152$		0			0
$153$	129.482	+	188.621i	0.846285	+	1.23282i
$154$		0			0
$155$		−	22.1944i		−	0.143189i
$156$		0			0
$157$	32.9672			0.209982			0.104991	−	0.994473i	$-0.466519\pi$
$157$	32.9672			0.209982			0.104991	+	0.994473i	$0.466519\pi$
$158$		0			0
$159$	−61.3337	−	197.721i	−0.385747	−	1.24353i
$160$		0			0
$161$		−	191.872i		−	1.19175i
$162$		0			0
$163$	−248.216			−1.52280			−0.761399	−	0.648283i	$-0.775487\pi$
$163$	−248.216			−1.52280			−0.761399	+	0.648283i	$0.775487\pi$
$164$		0			0
$165$	−67.7624	+	21.0201i	−0.410681	+	0.127395i
$166$		0			0
$167$		−	235.204i		−	1.40841i	−0.709997	−	0.704205i	$-0.751304\pi$
$167$		−	235.204i		−	1.40841i	0.709997	−	0.704205i	$-0.248696\pi$
$168$		0			0
$169$	−149.346			−0.883706
$170$		0			0
$171$	−130.165	+	89.3537i	−0.761201	+	0.522536i
$172$		0			0
$173$			38.6880i			0.223630i	0.993729	+	0.111815i	$0.0356664\pi$
$173$			38.6880i			0.223630i	−0.993729	+	0.111815i	$0.964334\pi$
$174$		0			0
$175$	533.777			3.05016
$176$		0			0
$177$	−44.9639	−	144.950i	−0.254033	−	0.818924i
$178$		0			0
$179$		−	287.329i		−	1.60519i	−0.596524	−	0.802595i	$-0.703452\pi$
$179$		−	287.329i		−	1.60519i	0.596524	−	0.802595i	$-0.296548\pi$
$180$		0			0
$181$	199.173			1.10040			0.550201	−	0.835032i	$-0.314551\pi$
$181$	199.173			1.10040			0.550201	+	0.835032i	$0.314551\pi$
$182$		0			0
$183$	93.4041	−	28.9743i	0.510405	−	0.158329i
$184$		0			0
$185$		−	66.4296i		−	0.359079i
$186$		0			0
$187$	−69.9720			−0.374182
$188$		0			0
$189$	−182.960	+	231.686i	−0.968044	+	1.22585i
$190$		0			0
$191$			227.555i			1.19139i	0.803211	+	0.595695i	$0.203123\pi$
$191$			227.555i			1.19139i	−0.803211	+	0.595695i	$0.796877\pi$
$192$		0			0
$193$	−23.8956			−0.123811			−0.0619056	−	0.998082i	$-0.519718\pi$
$193$	−23.8956			−0.123811			−0.0619056	+	0.998082i	$0.519718\pi$
$194$		0			0
$195$	33.8550	+	109.138i	0.173615	+	0.559682i
$196$		0			0
$197$		−	78.6009i		−	0.398990i	−0.979899	−	0.199495i	$-0.936070\pi$
$197$		−	78.6009i		−	0.398990i	0.979899	−	0.199495i	$-0.0639301\pi$
$198$		0			0
$199$	−181.811			−0.913625			−0.456812	−	0.889563i	$-0.651009\pi$
$199$	−181.811			−0.913625			−0.456812	+	0.889563i	$0.651009\pi$
$200$		0			0
$201$	137.571	−	42.6751i	0.684435	−	0.212314i
$202$		0			0
$203$		−	214.550i		−	1.05690i
$204$		0			0
$205$	−498.382			−2.43113
$206$		0			0
$207$	89.3824	+	130.207i	0.431799	+	0.629021i
$208$		0			0
$209$		−	48.2869i		−	0.231038i
$210$		0			0
$211$	−2.18826			−0.0103709			−0.00518544	−	0.999987i	$-0.501651\pi$
$211$	−2.18826			−0.0103709			−0.00518544	+	0.999987i	$0.501651\pi$
$212$		0			0
$213$	19.6449	+	63.3292i	0.0922297	+	0.297320i
$214$		0			0
$215$			362.514i			1.68611i
$216$		0			0
$217$	28.2448			0.130160
$218$		0			0
$219$	−77.4537	+	24.0264i	−0.353670	+	0.109710i
$220$		0			0
$221$			112.697i			0.509941i
$222$		0			0
$223$	−317.724			−1.42477			−0.712386	−	0.701788i	$-0.752385\pi$
$223$	−317.724			−1.42477			−0.712386	+	0.701788i	$0.752385\pi$
$224$		0			0
$225$	−362.231	+	248.658i	−1.60991	+	1.10515i
$226$		0			0
$227$			241.233i			1.06270i	0.847152	+	0.531350i	$0.178315\pi$
$227$			241.233i			1.06270i	−0.847152	+	0.531350i	$0.821685\pi$
$228$		0			0
$229$	−58.1799			−0.254061			−0.127030	−	0.991899i	$-0.540545\pi$
$229$	−58.1799			−0.254061			−0.127030	+	0.991899i	$0.540545\pi$
$230$		0			0
$231$	−26.7504	−	86.2351i	−0.115803	−	0.373312i
$232$		0			0
$233$			242.432i			1.04048i	0.854020	+	0.520240i	$0.174158\pi$
$233$			242.432i			1.04048i	−0.854020	+	0.520240i	$0.825842\pi$
$234$		0			0
$235$	150.069			0.638590
$236$		0			0
$237$	279.318	−	86.6453i	1.17856	−	0.365592i
$238$		0			0
$239$			215.651i			0.902304i	0.892447	+	0.451152i	$0.148987\pi$
$239$			215.651i			0.902304i	−0.892447	+	0.451152i	$0.851013\pi$
$240$		0			0
$241$	123.252			0.511417			0.255709	−	0.966754i	$-0.417691\pi$
$241$	123.252			0.511417			0.255709	+	0.966754i	$0.417691\pi$
$242$		0			0
$243$	16.2300	−	242.457i	0.0667902	−	0.997767i
$244$		0			0
$245$			606.160i			2.47412i
$246$		0			0
$247$	−77.7708			−0.314861
$248$		0			0
$249$	52.9076	+	170.558i	0.212480	+	0.684971i
$250$		0			0
$251$			194.768i			0.775967i	0.921666	+	0.387984i	$0.126828\pi$
$251$			194.768i			0.775967i	−0.921666	+	0.387984i	$0.873172\pi$
$252$		0			0
$253$	−48.3024			−0.190919
$254$		0			0
$255$	−625.810	+	194.128i	−2.45416	+	0.761288i
$256$		0			0
$257$			200.615i			0.780605i	0.920687	+	0.390303i	$0.127630\pi$
$257$			200.615i			0.780605i	−0.920687	+	0.390303i	$0.872370\pi$
$258$		0			0
$259$	84.5389			0.326405
$260$		0			0
$261$	99.9473	+	145.598i	0.382940	+	0.557845i
$262$		0			0
$263$			440.044i			1.67317i	0.547837	+	0.836585i	$0.315451\pi$
$263$			440.044i			1.67317i	−0.547837	+	0.836585i	$0.684549\pi$
$264$		0			0
$265$	592.876			2.23727
$266$		0			0
$267$	−98.1877	−	316.527i	−0.367744	−	1.18549i
$268$		0			0
$269$			64.1922i			0.238633i	0.992856	+	0.119316i	$0.0380703\pi$
$269$			64.1922i			0.238633i	−0.992856	+	0.119316i	$0.961930\pi$
$270$		0			0
$271$	229.609			0.847265			0.423633	−	0.905834i	$-0.360755\pi$
$271$	229.609			0.847265			0.423633	+	0.905834i	$0.360755\pi$
$272$		0			0
$273$	−138.890	+	43.0842i	−0.508755	+	0.157818i
$274$		0			0
$275$		−	134.375i		−	0.488636i
$276$		0			0
$277$	329.621			1.18997			0.594983	−	0.803738i	$-0.297159\pi$
$277$	329.621			1.18997			0.594983	+	0.803738i	$0.297159\pi$
$278$		0			0
$279$	−19.1674	+	13.1577i	−0.0687003	+	0.0471602i
$280$		0			0
$281$		−	222.349i		−	0.791279i	−0.918406	−	0.395640i	$-0.870523\pi$
$281$		−	222.349i		−	0.791279i	0.918406	−	0.395640i	$-0.129477\pi$
$282$		0			0
$283$	550.386			1.94483			0.972414	−	0.233261i	$-0.0749397\pi$
$283$	550.386			1.94483			0.972414	+	0.233261i	$0.0749397\pi$
$284$		0			0
$285$	−133.966	−	431.864i	−0.470055	−	1.51531i
$286$		0			0
$287$		−	634.245i		−	2.20991i
$288$		0			0
$289$	−357.217			−1.23604
$290$		0			0
$291$	157.927	−	48.9894i	0.542704	−	0.168349i
$292$		0			0
$293$			322.712i			1.10141i	0.834701	+	0.550703i	$0.185640\pi$
$293$			322.712i			1.10141i	−0.834701	+	0.550703i	$0.814360\pi$
$294$		0			0
$295$	434.638			1.47335
$296$		0			0
$297$	58.3255	+	46.0590i	0.196382	+	0.155081i
$298$		0			0
$299$			77.7958i			0.260187i
$300$		0			0
$301$	−461.339			−1.53269
$302$		0			0
$303$	−17.9805	−	57.9635i	−0.0593415	−	0.191299i
$304$		0			0
$305$			280.077i			0.918284i
$306$		0			0
$307$	66.6568			0.217123			0.108562	−	0.994090i	$-0.465376\pi$
$307$	66.6568			0.217123			0.108562	+	0.994090i	$0.465376\pi$
$308$		0			0
$309$	187.119	−	58.0450i	0.605563	−	0.187848i
$310$		0			0
$311$		−	20.7250i		−	0.0666398i	−0.999445	−	0.0333199i	$-0.989392\pi$
$311$		−	20.7250i		−	0.0666398i	0.999445	−	0.0333199i	$-0.0106080\pi$
$312$		0			0
$313$	−211.296			−0.675066			−0.337533	−	0.941314i	$-0.609592\pi$
$313$	−211.296			−0.675066			−0.337533	+	0.941314i	$0.609592\pi$
$314$		0			0
$315$	−478.496	−	697.046i	−1.51904	−	2.21284i
$316$		0			0
$317$			238.040i			0.750915i	0.926840	+	0.375457i	$0.122514\pi$
$317$			238.040i			0.750915i	−0.926840	+	0.375457i	$0.877486\pi$
$318$		0			0
$319$	−54.0117			−0.169316
$320$		0			0
$321$	−9.18205	−	29.6001i	−0.0286045	−	0.0922121i
$322$		0			0
$323$		−	445.946i		−	1.38064i
$324$		0			0
$325$	−216.424			−0.665921
$326$		0			0
$327$	434.213	−	134.694i	1.32787	−	0.411910i
$328$		0			0
$329$			190.979i			0.580483i
$330$		0			0
$331$	−344.811			−1.04173			−0.520863	−	0.853640i	$-0.674390\pi$
$331$	−344.811			−1.04173			−0.520863	+	0.853640i	$0.674390\pi$
$332$		0			0
$333$	−57.3696	+	39.3821i	−0.172281	+	0.118264i
$334$		0			0
$335$			412.515i			1.23139i
$336$		0			0
$337$	−355.471			−1.05481			−0.527405	−	0.849614i	$-0.676835\pi$
$337$	−355.471			−1.05481			−0.527405	+	0.849614i	$0.676835\pi$
$338$		0			0
$339$	94.5511	+	304.803i	0.278912	+	0.899125i
$340$		0			0
$341$		−	7.11043i		−	0.0208517i
$342$		0			0
$343$	−235.642			−0.687002
$344$		0			0
$345$	−432.003	+	134.009i	−1.25218	+	0.388431i
$346$		0			0
$347$			73.9748i			0.213184i	0.994303	+	0.106592i	$0.0339939\pi$
$347$			73.9748i			0.213184i	−0.994303	+	0.106592i	$0.966006\pi$
$348$		0			0
$349$	443.377			1.27042			0.635210	−	0.772340i	$-0.280914\pi$
$349$	443.377			1.27042			0.635210	+	0.772340i	$0.280914\pi$
$350$		0			0
$351$	74.1827	−	93.9390i	0.211347	−	0.267632i
$352$		0			0
$353$		−	553.017i		−	1.56662i	−0.621630	−	0.783311i	$-0.713529\pi$
$353$		−	553.017i		−	1.56662i	0.621630	−	0.783311i	$-0.286471\pi$
$354$		0			0
$355$	−189.896			−0.534917
$356$		0			0
$357$	−247.050	−	796.411i	−0.692016	−	2.23084i
$358$		0			0
$359$			352.615i			0.982214i	0.871099	+	0.491107i	$0.163408\pi$
$359$			352.615i			0.982214i	−0.871099	+	0.491107i	$0.836592\pi$
$360$		0			0
$361$	−53.2578			−0.147529
$362$		0			0
$363$	324.993	−	100.814i	0.895298	−	0.277724i
$364$		0			0
$365$		−	232.249i		−	0.636298i
$366$		0			0
$367$	305.686			0.832932			0.416466	−	0.909151i	$-0.363269\pi$
$367$	305.686			0.832932			0.416466	+	0.909151i	$0.363269\pi$
$368$		0			0
$369$	295.460	+	430.410i	0.800705	+	1.16642i
$370$		0			0
$371$			754.499i			2.03369i
$372$		0			0
$373$	−133.295			−0.357359			−0.178679	−	0.983907i	$-0.557182\pi$
$373$	−133.295			−0.357359			−0.178679	+	0.983907i	$0.557182\pi$
$374$		0			0
$375$	−181.891	−	586.361i	−0.485043	−	1.56363i
$376$		0			0
$377$			86.9912i			0.230746i
$378$		0			0
$379$	−239.300			−0.631399			−0.315699	−	0.948859i	$-0.602239\pi$
$379$	−239.300			−0.631399			−0.315699	+	0.948859i	$0.602239\pi$
$380$		0			0
$381$	−321.732	+	99.8025i	−0.844442	+	0.261949i
$382$		0			0
$383$			249.576i			0.651634i	0.945433	+	0.325817i	$0.105639\pi$
$383$			249.576i			0.651634i	−0.945433	+	0.325817i	$0.894361\pi$
$384$		0			0
$385$	258.580			0.671636
$386$		0			0
$387$	313.073	−	214.913i	0.808973	−	0.555330i
$388$		0			0
$389$		−	499.992i		−	1.28533i	−0.766148	−	0.642664i	$-0.777829\pi$
$389$		−	499.992i		−	1.28533i	0.766148	−	0.642664i	$-0.222171\pi$
$390$		0			0
$391$	−446.090			−1.14089
$392$		0			0
$393$	122.484	+	394.850i	0.311663	+	1.00471i
$394$		0			0
$395$			837.548i			2.12037i
$396$		0			0
$397$	302.418			0.761759			0.380880	−	0.924625i	$-0.375621\pi$
$397$	302.418			0.761759			0.380880	+	0.924625i	$0.375621\pi$
$398$		0			0
$399$	549.594	−	170.486i	1.37743	−	0.427283i
$400$		0			0
$401$		−	799.221i		−	1.99307i	−0.0831706	−	0.996535i	$-0.526505\pi$
$401$		−	799.221i		−	1.99307i	0.0831706	−	0.996535i	$-0.473495\pi$
$402$		0			0
$403$	−11.4521			−0.0284170
$404$		0			0
$405$	649.431	+	250.122i	1.60353	+	0.617586i
$406$		0			0
$407$		−	21.2821i		−	0.0522902i
$408$		0			0
$409$	108.812			0.266044			0.133022	−	0.991113i	$-0.457532\pi$
$409$	108.812			0.266044			0.133022	+	0.991113i	$0.457532\pi$
$410$		0			0
$411$	135.892	+	438.075i	0.330638	+	1.06588i
$412$		0			0
$413$			553.125i			1.33929i
$414$		0			0
$415$	−511.426			−1.23235
$416$		0			0
$417$	677.502	−	210.164i	1.62471	−	0.503989i
$418$		0			0
$419$		−	183.076i		−	0.436936i	−0.975844	−	0.218468i	$-0.929894\pi$
$419$		−	183.076i		−	0.436936i	0.975844	−	0.218468i	$-0.0701059\pi$
$420$		0			0
$421$	213.105			0.506187			0.253093	−	0.967442i	$-0.418552\pi$
$421$	213.105			0.506187			0.253093	+	0.967442i	$0.418552\pi$
$422$		0			0
$423$	−88.9666	−	129.602i	−0.210323	−	0.306387i
$424$		0			0
$425$		−	1241.00i		−	2.92000i
$426$		0			0
$427$	−356.428			−0.834727
$428$		0			0
$429$	10.8462	+	34.9647i	0.0252824	+	0.0815027i
$430$		0			0
$431$		−	471.854i		−	1.09479i	−0.836875	−	0.547394i	$-0.815620\pi$
$431$		−	471.854i		−	1.09479i	0.836875	−	0.547394i	$-0.184380\pi$
$432$		0			0
$433$	396.992			0.916841			0.458421	−	0.888735i	$-0.348415\pi$
$433$	396.992			0.916841			0.458421	+	0.888735i	$0.348415\pi$
$434$		0			0
$435$	−483.065	+	149.848i	−1.11049	+	0.344479i
$436$		0			0
$437$		−	307.841i		−	0.704442i
$438$		0			0
$439$	−201.084			−0.458050			−0.229025	−	0.973420i	$-0.573554\pi$
$439$	−201.084			−0.458050			−0.229025	+	0.973420i	$0.573554\pi$
$440$		0			0
$441$	523.489	−	359.356i	1.18705	−	0.814865i
$442$		0			0
$443$		−	189.532i		−	0.427838i	−0.976851	−	0.213919i	$-0.931377\pi$
$443$		−	189.532i		−	0.427838i	0.976851	−	0.213919i	$-0.0686229\pi$
$444$		0			0
$445$	949.121			2.13286
$446$		0			0
$447$	−164.450	−	530.136i	−0.367897	−	1.18599i
$448$		0			0
$449$			49.3773i			0.109972i	0.998487	+	0.0549859i	$0.0175114\pi$
$449$			49.3773i			0.109972i	−0.998487	+	0.0549859i	$0.982489\pi$
$450$		0			0
$451$	−159.667			−0.354029
$452$		0			0
$453$	−811.858	+	251.841i	−1.79218	+	0.555940i
$454$		0			0
$455$		−	416.469i		−	0.915316i
$456$		0			0
$457$	−438.599			−0.959734			−0.479867	−	0.877341i	$-0.659315\pi$
$457$	−438.599			−0.959734			−0.479867	+	0.877341i	$0.659315\pi$
$458$		0			0
$459$	538.657	+	425.372i	1.17354	+	0.926736i
$460$		0			0
$461$			27.0211i			0.0586142i	0.999570	+	0.0293071i	$0.00933007\pi$
$461$			27.0211i			0.0586142i	−0.999570	+	0.0293071i	$0.990670\pi$
$462$		0			0
$463$	128.181			0.276850			0.138425	−	0.990373i	$-0.455796\pi$
$463$	128.181			0.276850			0.138425	+	0.990373i	$0.455796\pi$
$464$		0			0
$465$	−19.7270	−	63.5937i	−0.0424236	−	0.136761i
$466$		0			0
$467$			624.873i			1.33806i	0.743236	+	0.669029i	$0.233290\pi$
$467$			624.873i			1.33806i	−0.743236	+	0.669029i	$0.766710\pi$
$468$		0			0
$469$	−524.970			−1.11934
$470$		0			0
$471$	94.4612	−	29.3022i	0.200554	−	0.0622127i
$472$		0			0
$473$			116.139i			0.245537i
$474$		0			0
$475$	856.400			1.80295
$476$		0			0
$477$	−351.480	−	512.016i	−0.736855	−	1.07341i
$478$		0			0
$479$			782.010i			1.63259i	0.577636	+	0.816295i	$0.303975\pi$
$479$			782.010i			1.63259i	−0.577636	+	0.816295i	$0.696025\pi$
$480$		0			0
$481$	−34.2770			−0.0712619
$482$		0			0
$483$	−170.541	−	549.771i	−0.353087	−	1.13824i
$484$		0			0
$485$			473.551i			0.976393i
$486$		0			0
$487$	732.325			1.50375			0.751874	−	0.659307i	$-0.229150\pi$
$487$	732.325			1.50375			0.751874	+	0.659307i	$0.229150\pi$
$488$		0			0
$489$	−711.215	+	220.621i	−1.45443	+	0.451169i
$490$		0			0
$491$		−	65.5662i		−	0.133536i	−0.997769	−	0.0667680i	$-0.978731\pi$
$491$		−	65.5662i		−	0.133536i	0.997769	−	0.0667680i	$-0.0212687\pi$
$492$		0			0
$493$	−498.817			−1.01180
$494$		0			0
$495$	−175.477	+	120.458i	−0.354499	+	0.243350i
$496$		0			0
$497$		−	241.663i		−	0.486243i
$498$		0			0
$499$	−846.549			−1.69649			−0.848245	−	0.529604i	$-0.822341\pi$
$499$	−846.549			−1.69649			−0.848245	+	0.529604i	$0.822341\pi$
$500$		0			0
$501$	−209.056	−	673.933i	−0.417278	−	1.34518i
$502$		0			0
$503$		−	186.077i		−	0.369934i	−0.982745	−	0.184967i	$-0.940782\pi$
$503$		−	186.077i		−	0.369934i	0.982745	−	0.184967i	$-0.0592179\pi$
$504$		0			0
$505$	173.806			0.344171
$506$		0			0
$507$	−427.923	+	132.743i	−0.844029	+	0.261821i
$508$		0			0
$509$		−	996.730i		−	1.95821i	−0.203350	−	0.979106i	$-0.565183\pi$
$509$		−	996.730i		−	1.95821i	0.203350	−	0.979106i	$-0.434817\pi$
$510$		0			0
$511$	295.562			0.578399
$512$		0			0
$513$	−293.544	+	371.720i	−0.572210	+	0.724601i
$514$		0			0
$515$			561.085i			1.08949i
$516$		0			0
$517$	48.0777			0.0929936
$518$		0			0
$519$	34.3870	+	110.853i	0.0662563	+	0.213590i
$520$		0			0
$521$		−	471.553i		−	0.905092i	−0.891741	−	0.452546i	$-0.850516\pi$
$521$		−	471.553i		−	0.905092i	0.891741	−	0.452546i	$-0.149484\pi$
$522$		0			0
$523$	−364.836			−0.697582			−0.348791	−	0.937200i	$-0.613408\pi$
$523$	−364.836			−0.697582			−0.348791	+	0.937200i	$0.613408\pi$
$524$		0			0
$525$	1529.44	−	474.436i	2.91321	−	0.903688i
$526$		0			0
$527$		−	65.6674i		−	0.124606i
$528$		0			0
$529$	221.060			0.417882
$530$		0			0
$531$	−257.671	−	375.360i	−0.485255	−	0.706893i
$532$		0			0
$533$			257.160i			0.482476i
$534$		0			0
$535$	88.7573			0.165901
$536$		0			0
$537$	−255.386	−	823.286i	−0.475579	−	1.53312i
$538$		0			0
$539$			194.196i			0.360290i
$540$		0			0
$541$	68.1097			0.125896			0.0629480	−	0.998017i	$-0.479950\pi$
$541$	68.1097			0.125896			0.0629480	+	0.998017i	$0.479950\pi$
$542$		0			0
$543$	570.691	−	177.030i	1.05100	−	0.326023i
$544$		0			0
$545$			1302.01i			2.38901i
$546$		0			0
$547$	459.725			0.840448			0.420224	−	0.907420i	$-0.361951\pi$
$547$	459.725			0.840448			0.420224	+	0.907420i	$0.361951\pi$
$548$		0			0
$549$	241.878	−	166.040i	0.440580	−	0.302442i
$550$		0			0
$551$		−	344.228i		−	0.624733i
$552$		0			0
$553$	−1065.87			−1.92743
$554$		0			0
$555$	−59.0445	−	190.341i	−0.106386	−	0.342957i
$556$		0			0
$557$			95.1108i			0.170756i	0.996349	+	0.0853778i	$0.0272097\pi$
$557$			95.1108i			0.170756i	−0.996349	+	0.0853778i	$0.972790\pi$
$558$		0			0
$559$	187.053			0.334622
$560$		0			0
$561$	−200.491	+	62.1931i	−0.357382	+	0.110861i
$562$		0			0
$563$		−	422.228i		−	0.749960i	−0.927033	−	0.374980i	$-0.877650\pi$
$563$		−	422.228i		−	0.749960i	0.927033	−	0.374980i	$-0.122350\pi$
$564$		0			0
$565$	−913.967			−1.61764
$566$		0			0
$567$	−318.308	+	826.472i	−0.561390	+	1.45762i
$568$		0			0
$569$			328.583i			0.577474i	0.957408	+	0.288737i	$0.0932354\pi$
$569$			328.583i			0.577474i	−0.957408	+	0.288737i	$0.906765\pi$
$570$		0			0
$571$	−503.834			−0.882371			−0.441186	−	0.897416i	$-0.645442\pi$
$571$	−503.834			−0.882371			−0.441186	+	0.897416i	$0.645442\pi$
$572$		0			0
$573$	202.258	+	652.016i	0.352980	+	1.13790i
$574$		0			0
$575$		−	856.675i		−	1.48987i
$576$		0			0
$577$	1035.33			1.79433			0.897163	−	0.441700i	$-0.145624\pi$
$577$	1035.33			1.79433			0.897163	+	0.441700i	$0.145624\pi$
$578$		0			0
$579$	−68.4681	+	21.2391i	−0.118252	+	0.0366823i
$580$		0			0
$581$		−	650.845i		−	1.12022i
$582$		0			0
$583$	189.940			0.325798
$584$		0			0
$585$	194.010	+	282.623i	0.331641	+	0.483116i
$586$		0			0
$587$		−	501.148i		−	0.853745i	−0.904312	−	0.426872i	$-0.859615\pi$
$587$		−	501.148i		−	0.853745i	0.904312	−	0.426872i	$-0.140385\pi$
$588$		0			0
$589$	45.3163			0.0769376
$590$		0			0
$591$	−69.8627	−	225.216i	−0.118211	−	0.381076i
$592$		0			0
$593$			737.286i			1.24332i	0.783289	+	0.621658i	$0.213541\pi$
$593$			737.286i			1.24332i	−0.783289	+	0.621658i	$0.786459\pi$
$594$		0			0
$595$	2388.08			4.01358
$596$		0			0
$597$	−520.945	+	161.599i	−0.872605	+	0.270685i
$598$		0			0
$599$		−	801.823i		−	1.33860i	−0.742991	−	0.669301i	$-0.766593\pi$
$599$		−	801.823i		−	1.33860i	0.742991	−	0.669301i	$-0.233407\pi$
$600$		0			0
$601$	252.561			0.420234			0.210117	−	0.977676i	$-0.432615\pi$
$601$	252.561			0.420234			0.210117	+	0.977676i	$0.432615\pi$
$602$		0			0
$603$	356.254	−	244.555i	0.590802	−	0.405564i
$604$		0			0
$605$			974.507i			1.61076i
$606$		0			0
$607$	−627.073			−1.03307			−0.516535	−	0.856266i	$-0.672778\pi$
$607$	−627.073			−1.03307			−0.516535	+	0.856266i	$0.672778\pi$
$608$		0			0
$609$	−190.699	−	614.753i	−0.313134	−	1.00945i
$610$		0			0
$611$		−	77.4339i		−	0.126733i
$612$		0			0
$613$	375.629			0.612772			0.306386	−	0.951907i	$-0.400880\pi$
$613$	375.629			0.612772			0.306386	+	0.951907i	$0.400880\pi$
$614$		0			0
$615$	−1428.02	+	442.976i	−2.32198	+	0.720286i
$616$		0			0
$617$			161.548i			0.261829i	0.991394	+	0.130914i	$0.0417913\pi$
$617$			161.548i			0.261829i	−0.991394	+	0.130914i	$0.958209\pi$
$618$		0			0
$619$	−9.45164			−0.0152692			−0.00763460	−	0.999971i	$-0.502430\pi$
$619$	−9.45164			−0.0152692			−0.00763460	+	0.999971i	$0.502430\pi$
$620$		0			0
$621$	371.840	+	293.638i	0.598776	+	0.472847i
$622$		0			0
$623$			1207.86i			1.93878i
$624$		0			0
$625$	537.772			0.860435
$626$		0			0
$627$	−42.9187	−	138.357i	−0.0684509	−	0.220665i
$628$		0			0
$629$		−	196.548i		−	0.312477i
$630$		0			0
$631$	1073.00			1.70048			0.850239	−	0.526398i	$-0.176458\pi$
$631$	1073.00			1.70048			0.850239	+	0.526398i	$0.176458\pi$
$632$		0			0
$633$	−6.27002	+	1.94498i	−0.00990525	+	0.00307264i
$634$		0			0
$635$		−	964.730i		−	1.51926i
$636$		0			0
$637$	312.772			0.491008
$638$		0			0
$639$	112.578	+	163.997i	0.176178	+	0.256646i
$640$		0			0
$641$		−	713.963i		−	1.11383i	−0.830570	−	0.556914i	$-0.811985\pi$
$641$		−	713.963i		−	1.11383i	0.830570	−	0.556914i	$-0.188015\pi$
$642$		0			0
$643$	339.764			0.528405			0.264202	−	0.964467i	$-0.414891\pi$
$643$	339.764			0.528405			0.264202	+	0.964467i	$0.414891\pi$
$644$		0			0
$645$	322.213	+	1038.71i	0.499555	+	1.61041i
$646$		0			0
$647$			108.874i			0.168275i	0.996454	+	0.0841375i	$0.0268135\pi$
$647$			108.874i			0.168275i	−0.996454	+	0.0841375i	$0.973187\pi$
$648$		0			0
$649$	139.246			0.214554
$650$		0			0
$651$	80.9299	−	25.1047i	0.124316	−	0.0385633i
$652$		0			0
$653$			970.043i			1.48552i	0.669559	+	0.742759i	$0.266483\pi$
$653$			970.043i			1.48552i	−0.669559	+	0.742759i	$0.733517\pi$
$654$		0			0
$655$	−1183.98			−1.80760
$656$		0			0
$657$	−200.573	+	137.686i	−0.305287	+	0.209568i
$658$		0			0
$659$		−	302.641i		−	0.459243i	−0.973280	−	0.229622i	$-0.926251\pi$
$659$		−	302.641i		−	0.459243i	0.973280	−	0.229622i	$-0.0737489\pi$
$660$		0			0
$661$	406.011			0.614237			0.307118	−	0.951671i	$-0.400635\pi$
$661$	406.011			0.614237			0.307118	+	0.951671i	$0.400635\pi$
$662$		0			0
$663$	100.168	+	322.911i	0.151083	+	0.487046i
$664$		0			0
$665$			1647.98i			2.47817i
$666$		0			0
$667$	−344.338			−0.516250
$668$		0			0
$669$	−910.377	+	282.402i	−1.36080	+	0.422126i
$670$		0			0
$671$			89.7285i			0.133724i
$672$		0			0
$673$	−143.090			−0.212615			−0.106307	−	0.994333i	$-0.533903\pi$
$673$	−143.090			−0.212615			−0.106307	+	0.994333i	$0.533903\pi$
$674$		0			0
$675$	−816.888	+	1034.44i	−1.21020	+	1.53251i
$676$		0			0
$677$			791.131i			1.16858i	0.811544	+	0.584292i	$0.198628\pi$
$677$			791.131i			1.16858i	−0.811544	+	0.584292i	$0.801372\pi$
$678$		0			0
$679$	−602.645			−0.887548
$680$		0			0
$681$	214.415	+	691.207i	0.314853	+	1.01499i
$682$		0			0
$683$		−	925.330i		−	1.35480i	−0.735614	−	0.677401i	$-0.763106\pi$
$683$		−	925.330i		−	1.35480i	0.735614	−	0.677401i	$-0.236894\pi$
$684$		0			0
$685$	−1313.59			−1.91765
$686$		0			0
$687$	−166.703	+	51.7120i	−0.242654	+	0.0752722i
$688$		0			0
$689$		−	305.918i		−	0.444002i
$690$		0			0
$691$	−666.330			−0.964299			−0.482149	−	0.876089i	$-0.660144\pi$
$691$	−666.330			−0.964299			−0.482149	+	0.876089i	$0.660144\pi$
$692$		0			0
$693$	−153.296	−	223.313i	−0.221207	−	0.322242i
$694$		0			0
$695$			2031.52i			2.92305i
$696$		0			0
$697$	−1474.58			−2.11561
$698$		0			0
$699$	215.480	+	694.642i	0.308269	+	0.993765i
$700$		0			0
$701$		−	1238.38i		−	1.76659i	−0.468820	−	0.883294i	$-0.655321\pi$
$701$		−	1238.38i		−	1.76659i	0.468820	−	0.883294i	$-0.344679\pi$
$702$		0			0
$703$	135.635			0.192938
$704$		0			0
$705$	429.993	−	133.385i	0.609919	−	0.189199i
$706$		0			0
$707$			221.187i			0.312854i
$708$		0			0
$709$	−162.142			−0.228692			−0.114346	−	0.993441i	$-0.536477\pi$
$709$	−162.142			−0.228692			−0.114346	+	0.993441i	$0.536477\pi$
$710$		0			0
$711$	723.318	−	496.531i	1.01733	−	0.698356i
$712$		0			0
$713$		−	45.3309i		−	0.0635776i
$714$		0			0
$715$	−104.843			−0.146634
$716$		0			0
$717$	191.676	+	617.906i	0.267331	+	0.861793i
$718$		0			0
$719$		−	534.958i		−	0.744031i	−0.928226	−	0.372016i	$-0.878667\pi$
$719$		−	534.958i		−	0.744031i	0.928226	−	0.372016i	$-0.121333\pi$
$720$		0			0
$721$	−714.042			−0.990350
$722$		0			0
$723$	353.154	−	109.549i	0.488456	−	0.151521i
$724$		0			0
$725$		−	957.933i		−	1.32129i
$726$		0			0
$727$	723.175			0.994738			0.497369	−	0.867539i	$-0.334299\pi$
$727$	723.175			0.994738			0.497369	+	0.867539i	$0.334299\pi$
$728$		0			0
$729$	−168.999	−	709.141i	−0.231823	−	0.972758i
$730$		0			0
$731$			1072.59i			1.46729i
$732$		0			0
$733$	473.296			0.645697			0.322849	−	0.946451i	$-0.395360\pi$
$733$	473.296			0.645697			0.322849	+	0.946451i	$0.395360\pi$
$734$		0			0
$735$	538.772	+	1736.84i	0.733024	+	2.36304i
$736$		0			0
$737$			132.158i			0.179319i
$738$		0			0
$739$	−724.955			−0.980994			−0.490497	−	0.871443i	$-0.663185\pi$
$739$	−724.955			−0.980994			−0.490497	+	0.871443i	$0.663185\pi$
$740$		0			0
$741$	−222.837	+	69.1249i	−0.300725	+	0.0932859i
$742$		0			0
$743$			905.636i			1.21889i	0.792828	+	0.609445i	$0.208608\pi$
$743$			905.636i			1.21889i	−0.792828	+	0.609445i	$0.791392\pi$
$744$		0			0
$745$	1589.64			2.13374
$746$		0			0
$747$	303.193	+	441.675i	0.405881	+	0.591265i
$748$		0			0
$749$			112.953i			0.150806i
$750$		0			0
$751$	−1039.66			−1.38436			−0.692181	−	0.721724i	$-0.743350\pi$
$751$	−1039.66			−1.38436			−0.692181	+	0.721724i	$0.743350\pi$
$752$		0			0
$753$	173.115	+	558.070i	0.229901	+	0.741128i
$754$		0			0
$755$		−	2434.39i		−	3.22436i
$756$		0			0
$757$	−826.135			−1.09133			−0.545664	−	0.838004i	$-0.683722\pi$
$757$	−826.135			−1.09133			−0.545664	+	0.838004i	$0.683722\pi$
$758$		0			0
$759$	−138.401	+	42.9325i	−0.182347	+	0.0565646i
$760$		0			0
$761$			159.752i			0.209924i	0.994476	+	0.104962i	$0.0334721\pi$
$761$			159.752i			0.209924i	−0.994476	+	0.104962i	$0.966528\pi$
$762$		0			0
$763$	−1656.95			−2.17162
$764$		0			0
$765$	−1620.59	+	1112.47i	−2.11842	+	1.45422i
$766$		0			0
$767$		−	224.269i		−	0.292397i
$768$		0			0
$769$	−1382.69			−1.79804			−0.899020	−	0.437908i	$-0.855720\pi$
$769$	−1382.69			−1.79804			−0.899020	+	0.437908i	$0.855720\pi$
$770$		0			0
$771$	178.313	+	574.825i	0.231275	+	0.745558i
$772$		0			0
$773$		−	33.4297i		−	0.0432466i	−0.999766	−	0.0216233i	$-0.993117\pi$
$773$		−	33.4297i		−	0.0432466i	0.999766	−	0.0216233i	$-0.00688345\pi$
$774$		0			0
$775$	126.108			0.162720
$776$		0			0
$777$	242.230	−	75.1406i	0.311750	−	0.0967060i
$778$		0			0
$779$		−	1017.59i		−	1.30628i
$780$		0			0
$781$	−60.8370			−0.0778964
$782$		0			0
$783$	415.791	+	328.346i	0.531023	+	0.419343i
$784$		0			0
$785$			283.246i			0.360823i
$786$		0			0
$787$	−516.814			−0.656688			−0.328344	−	0.944558i	$-0.606491\pi$
$787$	−516.814			−0.656688			−0.328344	+	0.944558i	$0.606491\pi$
$788$		0			0
$789$	391.123	+	1260.86i	0.495720	+	1.59805i
$790$		0			0
$791$		−	1163.12i		−	1.47045i
$792$		0			0
$793$	144.517			0.182240
$794$		0			0
$795$	1698.77	−	526.965i	2.13682	−	0.662849i
$796$		0			0
$797$		−	593.861i		−	0.745120i	−0.928008	−	0.372560i	$-0.878480\pi$
$797$		−	593.861i		−	0.745120i	0.928008	−	0.372560i	$-0.121520\pi$
$798$		0			0
$799$	444.014			0.555713
$800$		0			0
$801$	−562.676	−	819.675i	−0.702467	−	1.02331i
$802$		0			0
$803$		−	74.4057i		−	0.0926597i
$804$		0			0
$805$	1648.51			2.04784
$806$		0			0
$807$	57.0559	+	183.930i	0.0707012	+	0.227919i
$808$		0			0
$809$		−	163.348i		−	0.201913i	−0.994891	−	0.100956i	$-0.967810\pi$
$809$		−	163.348i		−	0.201913i	0.994891	−	0.100956i	$-0.0321903\pi$
$810$		0			0
$811$	911.243			1.12360			0.561802	−	0.827271i	$-0.310108\pi$
$811$	911.243			1.12360			0.561802	+	0.827271i	$0.310108\pi$
$812$		0			0
$813$	657.900	−	204.083i	0.809225	−	0.251024i
$814$		0			0
$815$		−	2132.61i		−	2.61670i
$816$		0			0
$817$	−740.178			−0.905971
$818$		0			0
$819$	−359.668	+	246.899i	−0.439156	+	0.301464i
$820$		0			0
$821$			803.790i			0.979038i	0.871993	+	0.489519i	$0.162828\pi$
$821$			803.790i			0.979038i	−0.871993	+	0.489519i	$0.837172\pi$
$822$		0			0
$823$	−1214.84			−1.47611			−0.738056	−	0.674740i	$-0.764256\pi$
$823$	−1214.84			−1.47611			−0.738056	+	0.674740i	$0.764256\pi$
$824$		0			0
$825$	−119.436	−	385.026i	−0.144771	−	0.466698i
$826$		0			0
$827$		−	54.7192i		−	0.0661658i	−0.999453	−	0.0330829i	$-0.989467\pi$
$827$		−	54.7192i		−	0.0661658i	0.999453	−	0.0330829i	$-0.0105325\pi$
$828$		0			0
$829$	656.990			0.792508			0.396254	−	0.918141i	$-0.370310\pi$
$829$	656.990			0.792508			0.396254	+	0.918141i	$0.370310\pi$
$830$		0			0
$831$	944.465	−	292.976i	1.13654	−	0.352559i
$832$		0			0
$833$			1793.47i			2.15303i
$834$		0			0
$835$	2020.82			2.42014
$836$		0			0
$837$	−43.2255	+	54.7373i	−0.0516434	+	0.0653970i
$838$		0			0
$839$		−	1104.30i		−	1.31621i	−0.752927	−	0.658104i	$-0.771359\pi$
$839$		−	1104.30i		−	1.31621i	0.752927	−	0.658104i	$-0.228641\pi$
$840$		0			0
$841$	455.961			0.542166
$842$		0			0
$843$	−197.630	−	637.099i	−0.234437	−	0.755753i
$844$		0			0
$845$		−	1283.15i		−	1.51852i
$846$		0			0
$847$	−1240.17			−1.46419
$848$		0			0
$849$	1577.03	−	489.199i	1.85751	−	0.576206i
$850$		0			0
$851$		−	135.679i		−	0.159435i
$852$		0			0
$853$	−455.418			−0.533901			−0.266950	−	0.963710i	$-0.586016\pi$
$853$	−455.418			−0.533901			−0.266950	+	0.963710i	$0.586016\pi$
$854$		0			0
$855$	−767.705	−	1118.35i	−0.897901	−	1.30801i
$856$		0			0
$857$		−	909.199i		−	1.06091i	−0.847713	−	0.530454i	$-0.822021\pi$
$857$		−	909.199i		−	1.06091i	0.847713	−	0.530454i	$-0.177979\pi$
$858$		0			0
$859$	653.942			0.761283			0.380641	−	0.924723i	$-0.375703\pi$
$859$	653.942			0.761283			0.380641	+	0.924723i	$0.375703\pi$
$860$		0			0
$861$	−563.735	−	1817.31i	−0.654744	−	2.11069i
$862$		0			0
$863$			1592.91i			1.84578i	0.385062	+	0.922891i	$0.374180\pi$
$863$			1592.91i			1.84578i	−0.385062	+	0.922891i	$0.625820\pi$
$864$		0			0
$865$	−332.398			−0.384275
$866$		0			0
$867$	−1023.54	+	317.504i	−1.18055	+	0.366210i
$868$		0			0
$869$			268.326i			0.308776i
$870$		0			0
$871$	212.853			0.244378
$872$		0			0
$873$	408.965	−	280.739i	0.468460	−	0.321580i
$874$		0			0
$875$			2237.54i			2.55719i
$876$		0			0
$877$	91.9913			0.104893			0.0524466	−	0.998624i	$-0.483298\pi$
$877$	91.9913			0.104893			0.0524466	+	0.998624i	$0.483298\pi$
$878$		0			0
$879$	286.835	+	924.668i	0.326320	+	1.05195i
$880$		0			0
$881$			278.420i			0.316027i	0.987437	+	0.158014i	$0.0505090\pi$
$881$			278.420i			0.316027i	−0.987437	+	0.158014i	$0.949491\pi$
$882$		0			0
$883$	487.284			0.551851			0.275925	−	0.961179i	$-0.411016\pi$
$883$	487.284			0.551851			0.275925	+	0.961179i	$0.411016\pi$
$884$		0			0
$885$	1245.37	−	386.319i	1.40720	−	0.436518i
$886$		0			0
$887$		−	910.822i		−	1.02686i	−0.858132	−	0.513428i	$-0.828375\pi$
$887$		−	910.822i		−	1.02686i	0.858132	−	0.513428i	$-0.171625\pi$
$888$		0			0
$889$	1227.72			1.38102
$890$		0			0
$891$	208.059	+	80.1320i	0.233512	+	0.0899349i
$892$		0			0
$893$			306.409i			0.343123i
$894$		0			0
$895$	2468.66			2.75828
$896$		0			0
$897$	69.1471	+	222.909i	0.0770871	+	0.248505i
$898$		0			0
$899$		−	50.6889i		−	0.0563837i
$900$		0			0
$901$	1754.17			1.94691
$902$		0			0
$903$	−1321.88	+	410.051i	−1.46387	+	0.454099i
$904$		0			0
$905$			1711.24i			1.89088i
$906$		0			0
$907$	−476.332			−0.525173			−0.262587	−	0.964908i	$-0.584576\pi$
$907$	−476.332			−0.525173			−0.262587	+	0.964908i	$0.584576\pi$
$908$		0			0
$909$	−103.039	−	150.102i	−0.113354	−	0.165128i
$910$		0			0
$911$		−	1146.69i		−	1.25871i	−0.777117	−	0.629356i	$-0.783319\pi$
$911$		−	1146.69i		−	1.25871i	0.777117	−	0.629356i	$-0.216681\pi$
$912$		0			0
$913$	−163.846			−0.179459
$914$		0			0
$915$	248.940	+	802.506i	0.272066	+	0.877055i
$916$		0			0
$917$		−	1506.74i		−	1.64312i
$918$		0			0
$919$	1156.90			1.25887			0.629435	−	0.777053i	$-0.283286\pi$
$919$	1156.90			1.25887			0.629435	+	0.777053i	$0.283286\pi$
$920$		0			0
$921$	190.992	−	59.2464i	0.207375	−	0.0643284i
$922$		0			0
$923$			97.9841i			0.106158i
$924$		0			0
$925$	377.453			0.408057
$926$		0			0
$927$	484.561	−	332.633i	0.522720	−	0.358828i
$928$		0			0
$929$		−	721.346i		−	0.776476i	−0.921559	−	0.388238i	$-0.873084\pi$
$929$		−	721.346i		−	0.776476i	0.921559	−	0.388238i	$-0.126916\pi$
$930$		0			0
$931$	−1237.65			−1.32938
$932$		0			0
$933$	−18.4209	−	59.3834i	−0.0197438	−	0.0636478i
$934$		0			0
$935$		−	601.183i		−	0.642976i
$936$		0			0
$937$	629.504			0.671829			0.335915	−	0.941892i	$-0.390955\pi$
$937$	629.504			0.671829			0.335915	+	0.941892i	$0.390955\pi$
$938$		0			0
$939$	−605.427	+	187.805i	−0.644757	+	0.200006i
$940$		0			0
$941$		−	1735.34i		−	1.84414i	−0.387017	−	0.922072i	$-0.626495\pi$
$941$		−	1735.34i		−	1.84414i	0.387017	−	0.922072i	$-0.373505\pi$
$942$		0			0
$943$	−1017.92			−1.07945
$944$		0			0
$945$	−1990.59	−	1571.95i	−2.10645	−	1.66344i
$946$		0			0
$947$			1363.10i			1.43939i	0.694293	+	0.719693i	$0.255717\pi$
$947$			1363.10i			1.43939i	−0.694293	+	0.719693i	$0.744283\pi$
$948$		0			0
$949$	−119.838			−0.126278
$950$		0			0
$951$	211.577	+	682.058i	0.222478	+	0.717201i
$952$		0			0
$953$		−	83.9917i		−	0.0881340i	−0.999029	−	0.0440670i	$-0.985968\pi$
$953$		−	83.9917i		−	0.0881340i	0.999029	−	0.0440670i	$-0.0140315\pi$
$954$		0			0
$955$	−1955.10			−2.04723
$956$		0			0
$957$	−154.760	+	48.0071i	−0.161714	+	0.0501642i
$958$		0			0
$959$		−	1671.69i		−	1.74315i
$960$		0			0
$961$	−954.327			−0.993056
$962$		0			0
$963$	−52.6188	−	76.6521i	−0.0546405	−	0.0795972i
$964$		0			0
$965$		−	205.305i		−	0.212751i
$966$		0			0
$967$	−450.068			−0.465427			−0.232714	−	0.972545i	$-0.574761\pi$
$967$	−450.068			−0.465427			−0.232714	+	0.972545i	$0.574761\pi$
$968$		0			0
$969$	−396.370	−	1277.77i	−0.409050	−	1.31865i
$970$		0			0
$971$		−	129.710i		−	0.133584i	−0.997767	−	0.0667921i	$-0.978724\pi$
$971$		−	129.710i		−	0.133584i	0.997767	−	0.0667921i	$-0.0212764\pi$
$972$		0			0
$973$	−2585.33			−2.65708
$974$		0			0
$975$	−620.122	+	192.364i	−0.636022	+	0.197296i
$976$		0			0
$977$		−	1234.40i		−	1.26346i	−0.775189	−	0.631729i	$-0.782345\pi$
$977$		−	1234.40i		−	1.26346i	0.775189	−	0.631729i	$-0.217655\pi$
$978$		0			0
$979$	304.071			0.310593
$980$		0			0
$981$	1124.43	−	771.882i	1.14621	−	0.786831i
$982$		0			0
$983$			686.635i			0.698509i	0.937028	+	0.349255i	$0.113565\pi$
$983$			686.635i			0.698509i	−0.937028	+	0.349255i	$0.886435\pi$
$984$		0			0
$985$	675.320			0.685604
$986$		0			0
$987$	169.747	+	547.213i	0.171983	+	0.554420i
$988$		0			0
$989$			740.417i			0.748652i
$990$		0			0
$991$	564.593			0.569721			0.284860	−	0.958569i	$-0.408053\pi$
$991$	564.593			0.569721			0.284860	+	0.958569i	$0.408053\pi$
$992$		0			0
$993$	−987.991	+	306.478i	−0.994955	+	0.308639i
$994$		0			0
$995$		−	1562.08i		−	1.56993i
$996$		0			0
$997$	−1411.47			−1.41572			−0.707858	−	0.706354i	$-0.750338\pi$
$997$	−1411.47			−1.41572			−0.707858	+	0.706354i	$0.750338\pi$
$998$		0			0
$999$	−129.378	+	163.833i	−0.129507	+	0.163997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.e.c.257.7	yes	8
3.2	odd	2	inner	384.3.e.c.257.8	yes	8
4.3	odd	2		384.3.e.b.257.2	yes	8
8.3	odd	2		384.3.e.d.257.7	yes	8
8.5	even	2		384.3.e.a.257.2	yes	8
12.11	even	2		384.3.e.b.257.1	yes	8
16.3	odd	4		768.3.h.g.641.7		16
16.5	even	4		768.3.h.h.641.7		16
16.11	odd	4		768.3.h.g.641.10		16
16.13	even	4		768.3.h.h.641.10		16
24.5	odd	2		384.3.e.a.257.1	✓	8
24.11	even	2		384.3.e.d.257.8	yes	8
48.5	odd	4		768.3.h.h.641.9		16
48.11	even	4		768.3.h.g.641.8		16
48.29	odd	4		768.3.h.h.641.8		16
48.35	even	4		768.3.h.g.641.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.1	✓	8	24.5	odd	2
384.3.e.a.257.2	yes	8	8.5	even	2
384.3.e.b.257.1	yes	8	12.11	even	2
384.3.e.b.257.2	yes	8	4.3	odd	2
384.3.e.c.257.7	yes	8	1.1	even	1	trivial
384.3.e.c.257.8	yes	8	3.2	odd	2	inner
384.3.e.d.257.7	yes	8	8.3	odd	2
384.3.e.d.257.8	yes	8	24.11	even	2
768.3.h.g.641.7		16	16.3	odd	4
768.3.h.g.641.8		16	48.11	even	4
768.3.h.g.641.9		16	48.35	even	4
768.3.h.g.641.10		16	16.11	odd	4
768.3.h.h.641.7		16	16.5	even	4
768.3.h.h.641.8		16	48.29	odd	4
768.3.h.h.641.9		16	48.5	odd	4
768.3.h.h.641.10		16	16.13	even	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.86531 - 0.888828i) q^{3} +8.59176i q^{5} -10.9340 q^{7} +(7.41997 - 5.09353i) q^{9} +O(q^{10})\)	\(q+(2.86531 - 0.888828i) q^{3} +8.59176i q^{5} -10.9340 q^{7} +(7.41997 - 5.09353i) q^{9} +2.75255i q^{11} +4.43326 q^{13} +(7.63660 + 24.6180i) q^{15} +25.4208i q^{17} -17.5426 q^{19} +(-31.3291 + 9.71841i) q^{21} +17.5482i q^{23} -48.8183 q^{25} +(16.7332 - 21.1896i) q^{27} +19.6224i q^{29} -2.58322 q^{31} +(2.44655 + 7.88691i) q^{33} -93.9419i q^{35} -7.73178 q^{37} +(12.7026 - 3.94040i) q^{39} +58.0069i q^{41} +42.1932 q^{43} +(43.7624 + 63.7506i) q^{45} -17.4666i q^{47} +70.5514 q^{49} +(22.5947 + 72.8384i) q^{51} -69.0052i q^{53} -23.6493 q^{55} +(-50.2649 + 15.5923i) q^{57} -50.5878i q^{59} +32.5983 q^{61} +(-81.1296 + 55.6924i) q^{63} +38.0895i q^{65} +48.0128 q^{67} +(15.5974 + 50.2811i) q^{69} +22.1021i q^{71} -27.0316 q^{73} +(-139.880 + 43.3911i) q^{75} -30.0963i q^{77} +97.4827 q^{79} +(29.1119 - 75.5877i) q^{81} +59.5252i q^{83} -218.409 q^{85} +(17.4409 + 56.2242i) q^{87} -110.469i q^{89} -48.4730 q^{91} +(-7.40171 + 2.29603i) q^{93} -150.722i q^{95} +55.1169 q^{97} +(14.0202 + 20.4238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{7}+O(q^{10}) \) 8 * q + 4 * q^3 - 8 * q^7	\( 8 q + 4 q^{3} - 8 q^{7} - 16 q^{15} - 24 q^{19} - 16 q^{21} - 40 q^{25} - 44 q^{27} + 56 q^{31} + 8 q^{33} - 32 q^{37} + 104 q^{39} + 136 q^{43} + 80 q^{45} + 72 q^{49} + 176 q^{51} - 192 q^{55} - 40 q^{57} + 160 q^{61} - 264 q^{63} - 280 q^{67} - 80 q^{69} - 80 q^{73} - 348 q^{75} + 408 q^{79} + 72 q^{81} - 192 q^{85} + 368 q^{87} + 336 q^{91} - 160 q^{93} + 96 q^{97} + 432 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^7 - 16 * q^15 - 24 * q^19 - 16 * q^21 - 40 * q^25 - 44 * q^27 + 56 * q^31 + 8 * q^33 - 32 * q^37 + 104 * q^39 + 136 * q^43 + 80 * q^45 + 72 * q^49 + 176 * q^51 - 192 * q^55 - 40 * q^57 + 160 * q^61 - 264 * q^63 - 280 * q^67 - 80 * q^69 - 80 * q^73 - 348 * q^75 + 408 * q^79 + 72 * q^81 - 192 * q^85 + 368 * q^87 + 336 * q^91 - 160 * q^93 + 96 * q^97 + 432 * q^99

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