LMFDB - Embedded newform 384.3.e.c.257.2

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			257.2
Root			$-1.32750i$ of defining polynomial
Character	$\chi$	$=$	384.257
Dual form			384.3.e.c.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.69031 + 1.32750i) q^{3} -0.640013i q^{5} -2.72077 q^{7} +(5.47550 - 7.14275i) q^{9} +O(q^{10})$	\(q+(-2.69031 + 1.32750i) q^{3} -0.640013i q^{5} -2.72077 q^{7} +(5.47550 - 7.14275i) q^{9} +11.2836i q^{11} -5.25176 q^{13} +(0.849616 + 1.72183i) q^{15} -14.8718i q^{17} -15.0798 q^{19} +(7.31969 - 3.61181i) q^{21} -36.4411i q^{23} +24.5904 q^{25} +(-5.24877 + 26.4849i) q^{27} -51.7310i q^{29} +36.5009 q^{31} +(-14.9790 - 30.3565i) q^{33} +1.74133i q^{35} +63.6951 q^{37} +(14.1289 - 6.97170i) q^{39} +12.1500i q^{41} +11.8032 q^{43} +(-4.57146 - 3.50439i) q^{45} -61.1247i q^{47} -41.5974 q^{49} +(19.7423 + 40.0097i) q^{51} -59.1695i q^{53} +7.22168 q^{55} +(40.5694 - 20.0185i) q^{57} -37.2898i q^{59} -58.1987 q^{61} +(-14.8975 + 19.4338i) q^{63} +3.36120i q^{65} -23.0991 q^{67} +(48.3754 + 98.0376i) q^{69} -7.29656i q^{71} +73.4504 q^{73} +(-66.1557 + 32.6437i) q^{75} -30.7002i q^{77} -58.5098 q^{79} +(-21.0379 - 78.2203i) q^{81} -32.3939i q^{83} -9.51815 q^{85} +(68.6728 + 139.172i) q^{87} +112.260i q^{89} +14.2888 q^{91} +(-98.1987 + 48.4549i) q^{93} +9.65130i q^{95} -80.0338 q^{97} +(80.5963 + 61.7836i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.69031	+	1.32750i	−0.896769	+	0.442499i
$3$	−2.69031	+	1.32750i	−0.896769	+	0.442499i
$4$		0			0
$5$		−	0.640013i		−	0.128003i	−0.997950	−	0.0640013i	$-0.979614\pi$
$5$		−	0.640013i		−	0.128003i	0.997950	−	0.0640013i	$-0.0203862\pi$
$6$		0			0
$7$	−2.72077			−0.388681			−0.194340	−	0.980934i	$-0.562257\pi$
$7$	−2.72077			−0.388681			−0.194340	+	0.980934i	$0.562257\pi$
$8$		0			0
$9$	5.47550	−	7.14275i	0.608389	−	0.793639i
$10$		0			0
$11$			11.2836i			1.02579i	0.858452	+	0.512893i	$0.171426\pi$
$11$			11.2836i			1.02579i	−0.858452	+	0.512893i	$0.828574\pi$
$12$		0			0
$13$	−5.25176			−0.403982			−0.201991	−	0.979387i	$-0.564741\pi$
$13$	−5.25176			−0.403982			−0.201991	+	0.979387i	$0.564741\pi$
$14$		0			0
$15$	0.849616	+	1.72183i	0.0566411	+	0.114789i
$16$		0			0
$17$		−	14.8718i		−	0.874812i	−0.899264	−	0.437406i	$-0.855897\pi$
$17$		−	14.8718i		−	0.874812i	0.899264	−	0.437406i	$-0.144103\pi$
$18$		0			0
$19$	−15.0798			−0.793676			−0.396838	−	0.917889i	$-0.629893\pi$
$19$	−15.0798			−0.793676			−0.396838	+	0.917889i	$0.629893\pi$
$20$		0			0
$21$	7.31969	−	3.61181i	0.348557	−	0.171991i
$22$		0			0
$23$		−	36.4411i		−	1.58439i	−0.610266	−	0.792197i	$-0.708937\pi$
$23$		−	36.4411i		−	1.58439i	0.610266	−	0.792197i	$-0.291063\pi$
$24$		0			0
$25$	24.5904			0.983615
$26$		0			0
$27$	−5.24877	+	26.4849i	−0.194399	+	0.980923i
$28$		0			0
$29$		−	51.7310i		−	1.78383i	−0.452205	−	0.891914i	$-0.649362\pi$
$29$		−	51.7310i		−	1.78383i	0.452205	−	0.891914i	$-0.350638\pi$
$30$		0			0
$31$	36.5009			1.17745			0.588724	−	0.808334i	$-0.299630\pi$
$31$	36.5009			1.17745			0.588724	+	0.808334i	$0.299630\pi$
$32$		0			0
$33$	−14.9790	−	30.3565i	−0.453910	−	0.919893i
$34$		0			0
$35$			1.74133i			0.0497522i
$36$		0			0
$37$	63.6951			1.72149			0.860745	−	0.509036i	$-0.169998\pi$
$37$	63.6951			1.72149			0.860745	+	0.509036i	$0.169998\pi$
$38$		0			0
$39$	14.1289	−	6.97170i	0.362278	−	0.178762i
$40$		0			0
$41$			12.1500i			0.296342i	0.988962	+	0.148171i	$0.0473386\pi$
$41$			12.1500i			0.296342i	−0.988962	+	0.148171i	$0.952661\pi$
$42$		0			0
$43$	11.8032			0.274493			0.137247	−	0.990537i	$-0.456175\pi$
$43$	11.8032			0.274493			0.137247	+	0.990537i	$0.456175\pi$
$44$		0			0
$45$	−4.57146	−	3.50439i	−0.101588	−	0.0778753i
$46$		0			0
$47$		−	61.1247i		−	1.30053i	−0.759709	−	0.650263i	$-0.774659\pi$
$47$		−	61.1247i		−	1.30053i	0.759709	−	0.650263i	$-0.225341\pi$
$48$		0			0
$49$	−41.5974			−0.848927
$50$		0			0
$51$	19.7423	+	40.0097i	0.387104	+	0.784504i
$52$		0			0
$53$		−	59.1695i		−	1.11641i	−0.829705	−	0.558203i	$-0.811491\pi$
$53$		−	59.1695i		−	1.11641i	0.829705	−	0.558203i	$-0.188509\pi$
$54$		0			0
$55$	7.22168			0.131303
$56$		0			0
$57$	40.5694	−	20.0185i	0.711744	−	0.351201i
$58$		0			0
$59$		−	37.2898i		−	0.632031i	−0.948754	−	0.316016i	$-0.897655\pi$
$59$		−	37.2898i		−	0.632031i	0.948754	−	0.316016i	$-0.102345\pi$
$60$		0			0
$61$	−58.1987			−0.954076			−0.477038	−	0.878883i	$-0.658290\pi$
$61$	−58.1987			−0.954076			−0.477038	+	0.878883i	$0.658290\pi$
$62$		0			0
$63$	−14.8975	+	19.4338i	−0.236469	+	0.308472i
$64$		0			0
$65$			3.36120i			0.0517107i
$66$		0			0
$67$	−23.0991			−0.344762			−0.172381	−	0.985030i	$-0.555146\pi$
$67$	−23.0991			−0.344762			−0.172381	+	0.985030i	$0.555146\pi$
$68$		0			0
$69$	48.3754	+	98.0376i	0.701093	+	1.42084i
$70$		0			0
$71$		−	7.29656i		−	0.102768i	−0.998679	−	0.0513842i	$-0.983637\pi$
$71$		−	7.29656i		−	0.102768i	0.998679	−	0.0513842i	$-0.0163633\pi$
$72$		0			0
$73$	73.4504			1.00617			0.503085	−	0.864237i	$-0.332198\pi$
$73$	73.4504			1.00617			0.503085	+	0.864237i	$0.332198\pi$
$74$		0			0
$75$	−66.1557	+	32.6437i	−0.882076	+	0.435249i
$76$		0			0
$77$		−	30.7002i		−	0.398703i
$78$		0			0
$79$	−58.5098			−0.740630			−0.370315	−	0.928906i	$-0.620750\pi$
$79$	−58.5098			−0.740630			−0.370315	+	0.928906i	$0.620750\pi$
$80$		0			0
$81$	−21.0379	−	78.2203i	−0.259727	−	0.965682i
$82$		0			0
$83$		−	32.3939i		−	0.390287i	−0.980775	−	0.195144i	$-0.937483\pi$
$83$		−	32.3939i		−	0.390287i	0.980775	−	0.195144i	$-0.0625173\pi$
$84$		0			0
$85$	−9.51815			−0.111978
$86$		0			0
$87$	68.6728	+	139.172i	0.789343	+	1.59968i
$88$		0			0
$89$			112.260i			1.26135i	0.776046	+	0.630677i	$0.217223\pi$
$89$			112.260i			1.26135i	−0.776046	+	0.630677i	$0.782777\pi$
$90$		0			0
$91$	14.2888			0.157020
$92$		0			0
$93$	−98.1987	+	48.4549i	−1.05590	+	0.521020i
$94$		0			0
$95$			9.65130i			0.101593i
$96$		0			0
$97$	−80.0338			−0.825090			−0.412545	−	0.910937i	$-0.635360\pi$
$97$	−80.0338			−0.825090			−0.412545	+	0.910937i	$0.635360\pi$
$98$		0			0
$99$	80.5963	+	61.7836i	0.814104	+	0.624077i
$100$		0			0
$101$			119.373i			1.18191i	0.806705	+	0.590954i	$0.201249\pi$
$101$			119.373i			1.18191i	−0.806705	+	0.590954i	$0.798751\pi$
$102$		0			0
$103$	−125.522			−1.21866			−0.609330	−	0.792917i	$-0.708562\pi$
$103$	−125.522			−1.21866			−0.609330	+	0.792917i	$0.708562\pi$
$104$		0			0
$105$	−2.31161	−	4.68470i	−0.0220153	−	0.0446162i
$106$		0			0
$107$		−	148.669i		−	1.38943i	−0.719284	−	0.694716i	$-0.755530\pi$
$107$		−	148.669i		−	1.38943i	0.719284	−	0.694716i	$-0.244470\pi$
$108$		0			0
$109$	−70.6664			−0.648316			−0.324158	−	0.946003i	$-0.605081\pi$
$109$	−70.6664			−0.648316			−0.324158	+	0.946003i	$0.605081\pi$
$110$		0			0
$111$	−171.359	+	84.5552i	−1.54378	+	0.761758i
$112$		0			0
$113$			130.968i			1.15901i	0.814969	+	0.579504i	$0.196754\pi$
$113$			130.968i			1.15901i	−0.814969	+	0.579504i	$0.803246\pi$
$114$		0			0
$115$	−23.3228			−0.202807
$116$		0			0
$117$	−28.7560	+	37.5120i	−0.245778	+	0.320616i
$118$		0			0
$119$			40.4627i			0.340023i
$120$		0			0
$121$	−6.32073			−0.0522374
$122$		0			0
$123$	−16.1291	−	32.6873i	−0.131131	−	0.265750i
$124$		0			0
$125$		−	31.7385i		−	0.253908i
$126$		0			0
$127$	209.206			1.64729			0.823647	−	0.567103i	$-0.191936\pi$
$127$	209.206			1.64729			0.823647	+	0.567103i	$0.191936\pi$
$128$		0			0
$129$	−31.7543	+	15.6687i	−0.246157	+	0.121463i
$130$		0			0
$131$		−	174.839i		−	1.33465i	−0.744766	−	0.667326i	$-0.767439\pi$
$131$		−	174.839i		−	1.33465i	0.744766	−	0.667326i	$-0.232561\pi$
$132$		0			0
$133$	41.0287			0.308487
$134$		0			0
$135$	16.9507	+	3.35929i	0.125561	+	0.0248836i
$136$		0			0
$137$			162.945i			1.18938i	0.803956	+	0.594689i	$0.202725\pi$
$137$			162.945i			1.18938i	−0.803956	+	0.594689i	$0.797275\pi$
$138$		0			0
$139$	194.227			1.39732			0.698660	−	0.715454i	$-0.253780\pi$
$139$	194.227			1.39732			0.698660	+	0.715454i	$0.253780\pi$
$140$		0			0
$141$	81.1429	+	164.444i	0.575482	+	1.16627i
$142$		0			0
$143$		−	59.2590i		−	0.414399i
$144$		0			0
$145$	−33.1085			−0.228335
$146$		0			0
$147$	111.910	−	55.2205i	0.761291	−	0.375650i
$148$		0			0
$149$		−	83.4695i		−	0.560198i	−0.959971	−	0.280099i	$-0.909633\pi$
$149$		−	83.4695i		−	0.560198i	0.959971	−	0.280099i	$-0.0903673\pi$
$150$		0			0
$151$	−60.3488			−0.399661			−0.199830	−	0.979830i	$-0.564039\pi$
$151$	−60.3488			−0.399661			−0.199830	+	0.979830i	$0.564039\pi$
$152$		0			0
$153$	−106.226	−	81.4306i	−0.694285	−	0.532226i
$154$		0			0
$155$		−	23.3611i		−	0.150717i
$156$		0			0
$157$	100.762			0.641798			0.320899	−	0.947113i	$-0.396015\pi$
$157$	100.762			0.641798			0.320899	+	0.947113i	$0.396015\pi$
$158$		0			0
$159$	78.5474	+	159.184i	0.494009	+	1.00116i
$160$		0			0
$161$			99.1476i			0.615824i
$162$		0			0
$163$	−69.7149			−0.427699			−0.213849	−	0.976867i	$-0.568600\pi$
$163$	−69.7149			−0.427699			−0.213849	+	0.976867i	$0.568600\pi$
$164$		0			0
$165$	−19.4285	+	9.58677i	−0.117749	+	0.0581016i
$166$		0			0
$167$		−	120.823i		−	0.723491i	−0.932277	−	0.361745i	$-0.882181\pi$
$167$		−	120.823i		−	0.723491i	0.932277	−	0.361745i	$-0.117819\pi$
$168$		0			0
$169$	−141.419			−0.836799
$170$		0			0
$171$	−82.5697	+	107.712i	−0.482863	+	0.629893i
$172$		0			0
$173$			30.0602i			0.173758i	0.996219	+	0.0868791i	$0.0276894\pi$
$173$			30.0602i			0.173758i	−0.996219	+	0.0868791i	$0.972311\pi$
$174$		0			0
$175$	−66.9047			−0.382312
$176$		0			0
$177$	49.5022	+	100.321i	0.279673	+	0.566786i
$178$		0			0
$179$		−	114.784i		−	0.641250i	−0.947206	−	0.320625i	$-0.896107\pi$
$179$		−	114.784i		−	0.641250i	0.947206	−	0.320625i	$-0.103893\pi$
$180$		0			0
$181$	−181.627			−1.00347			−0.501733	−	0.865022i	$-0.667304\pi$
$181$	−181.627			−1.00347			−0.501733	+	0.865022i	$0.667304\pi$
$182$		0			0
$183$	156.572	−	77.2586i	0.855586	−	0.422178i
$184$		0			0
$185$		−	40.7657i		−	0.220355i
$186$		0			0
$187$	167.808			0.897370
$188$		0			0
$189$	14.2807	−	72.0592i	0.0755592	−	0.381266i
$190$		0			0
$191$			209.226i			1.09543i	0.836666	+	0.547713i	$0.184502\pi$
$191$			209.226i			1.09543i	−0.836666	+	0.547713i	$0.815498\pi$
$192$		0			0
$193$	161.330			0.835907			0.417954	−	0.908468i	$-0.362747\pi$
$193$	161.330			0.835907			0.417954	+	0.908468i	$0.362747\pi$
$194$		0			0
$195$	−4.46198	−	9.04265i	−0.0228820	−	0.0463726i
$196$		0			0
$197$		−	144.953i		−	0.735801i	−0.929865	−	0.367901i	$-0.880077\pi$
$197$		−	144.953i		−	0.735801i	0.929865	−	0.367901i	$-0.119923\pi$
$198$		0			0
$199$	−237.264			−1.19228			−0.596140	−	0.802880i	$-0.703300\pi$
$199$	−237.264			−1.19228			−0.596140	+	0.802880i	$0.703300\pi$
$200$		0			0
$201$	62.1436	−	30.6640i	0.309172	−	0.152557i
$202$		0			0
$203$			140.748i			0.693340i
$204$		0			0
$205$	7.77617			0.0379325
$206$		0			0
$207$	−260.289	−	199.533i	−1.25744	−	0.963927i
$208$		0			0
$209$		−	170.156i		−	0.814142i
$210$		0			0
$211$	−307.117			−1.45553			−0.727765	−	0.685827i	$-0.759441\pi$
$211$	−307.117			−1.45553			−0.727765	+	0.685827i	$0.759441\pi$
$212$		0			0
$213$	9.68617	+	19.6300i	0.0454750	+	0.0921595i
$214$		0			0
$215$		−	7.55422i		−	0.0351359i
$216$		0			0
$217$	−99.3105			−0.457652
$218$		0			0
$219$	−197.604	+	97.5053i	−0.902302	+	0.445230i
$220$		0			0
$221$			78.1032i			0.353408i
$222$		0			0
$223$	438.001			1.96413			0.982065	−	0.188544i	$-0.0603770\pi$
$223$	438.001			1.96413			0.982065	+	0.188544i	$0.0603770\pi$
$224$		0			0
$225$	134.645	−	175.643i	0.598420	−	0.780636i
$226$		0			0
$227$			270.755i			1.19275i	0.802705	+	0.596376i	$0.203393\pi$
$227$			270.755i			1.19275i	−0.802705	+	0.596376i	$0.796607\pi$
$228$		0			0
$229$	−98.6778			−0.430907			−0.215454	−	0.976514i	$-0.569123\pi$
$229$	−98.6778			−0.430907			−0.215454	+	0.976514i	$0.569123\pi$
$230$		0			0
$231$	40.7544	+	82.5929i	0.176426	+	0.357545i
$232$		0			0
$233$		−	432.405i		−	1.85581i	−0.372812	−	0.927907i	$-0.621606\pi$
$233$		−	432.405i		−	1.85581i	0.372812	−	0.927907i	$-0.378394\pi$
$234$		0			0
$235$	−39.1206			−0.166471
$236$		0			0
$237$	157.409	−	77.6717i	0.664174	−	0.327729i
$238$		0			0
$239$		−	275.799i		−	1.15397i	−0.816754	−	0.576986i	$-0.804229\pi$
$239$		−	275.799i		−	1.15397i	0.816754	−	0.576986i	$-0.195771\pi$
$240$		0			0
$241$	416.770			1.72934			0.864669	−	0.502343i	$-0.167528\pi$
$241$	416.770			1.72934			0.864669	+	0.502343i	$0.167528\pi$
$242$		0			0
$243$	160.436	+	182.509i	0.660228	+	0.751065i
$244$		0			0
$245$			26.6229i			0.108665i
$246$		0			0
$247$	79.1958			0.320631
$248$		0			0
$249$	43.0028	+	87.1494i	0.172702	+	0.349998i
$250$		0			0
$251$			165.903i			0.660966i	0.943812	+	0.330483i	$0.107212\pi$
$251$			165.903i			0.660966i	−0.943812	+	0.330483i	$0.892788\pi$
$252$		0			0
$253$	411.188			1.62525
$254$		0			0
$255$	25.6068	−	12.6353i	0.100419	−	0.0495503i
$256$		0			0
$257$			59.6381i			0.232055i	0.993246	+	0.116028i	$0.0370161\pi$
$257$			59.6381i			0.232055i	−0.993246	+	0.116028i	$0.962984\pi$
$258$		0			0
$259$	−173.300			−0.669110
$260$		0			0
$261$	−369.502	−	283.253i	−1.41572	−	1.08526i
$262$		0			0
$263$		−	67.2606i		−	0.255744i	−0.991791	−	0.127872i	$-0.959185\pi$
$263$		−	67.2606i		−	0.255744i	0.991791	−	0.127872i	$-0.0408146\pi$
$264$		0			0
$265$	−37.8693			−0.142903
$266$		0			0
$267$	−149.026	−	302.015i	−0.558148	−	1.13114i
$268$		0			0
$269$		−	36.1495i		−	0.134385i	−0.997740	−	0.0671923i	$-0.978596\pi$
$269$		−	36.1495i		−	0.134385i	0.997740	−	0.0671923i	$-0.0214041\pi$
$270$		0			0
$271$	−328.311			−1.21148			−0.605741	−	0.795662i	$-0.707123\pi$
$271$	−328.311			−1.21148			−0.605741	+	0.795662i	$0.707123\pi$
$272$		0			0
$273$	−38.4413	+	18.9684i	−0.140811	+	0.0694812i
$274$		0			0
$275$			277.469i			1.00898i
$276$		0			0
$277$	60.7455			0.219298			0.109649	−	0.993970i	$-0.465027\pi$
$277$	60.7455			0.219298			0.109649	+	0.993970i	$0.465027\pi$
$278$		0			0
$279$	199.861	−	260.717i	0.716347	−	0.934470i
$280$		0			0
$281$		−	366.827i		−	1.30543i	−0.757602	−	0.652717i	$-0.773629\pi$
$281$		−	366.827i		−	1.30543i	0.757602	−	0.652717i	$-0.226371\pi$
$282$		0			0
$283$	−104.032			−0.367603			−0.183801	−	0.982963i	$-0.558840\pi$
$283$	−104.032			−0.367603			−0.183801	+	0.982963i	$0.558840\pi$
$284$		0			0
$285$	−12.8121	−	25.9650i	−0.0449547	−	0.0911051i
$286$		0			0
$287$		−	33.0574i		−	0.115182i
$288$		0			0
$289$	67.8293			0.234703
$290$		0			0
$291$	215.315	−	106.245i	0.739915	−	0.365102i
$292$		0			0
$293$		−	410.579i		−	1.40129i	−0.713508	−	0.700647i	$-0.752895\pi$
$293$		−	410.579i		−	1.40129i	0.713508	−	0.700647i	$-0.247105\pi$
$294$		0			0
$295$	−23.8660			−0.0809017
$296$		0			0
$297$	−298.846	−	59.2253i	−1.00622	−	0.199412i
$298$		0			0
$299$			191.380i			0.640066i
$300$		0			0
$301$	−32.1138			−0.106690
$302$		0			0
$303$	−158.467	−	321.149i	−0.522994	−	1.05990i
$304$		0			0
$305$			37.2479i			0.122124i
$306$		0			0
$307$	270.025			0.879561			0.439781	−	0.898105i	$-0.355056\pi$
$307$	270.025			0.879561			0.439781	+	0.898105i	$0.355056\pi$
$308$		0			0
$309$	337.693	−	166.630i	1.09286	−	0.539256i
$310$		0			0
$311$			251.555i			0.808858i	0.914569	+	0.404429i	$0.132530\pi$
$311$			251.555i			0.808858i	−0.914569	+	0.404429i	$0.867470\pi$
$312$		0			0
$313$	−83.9560			−0.268230			−0.134115	−	0.990966i	$-0.542819\pi$
$313$	−83.9560			−0.268230			−0.134115	+	0.990966i	$0.542819\pi$
$314$		0			0
$315$	12.4379	+	9.53463i	0.0394853	+	0.0302687i
$316$		0			0
$317$			270.737i			0.854060i	0.904238	+	0.427030i	$0.140440\pi$
$317$			270.737i			0.854060i	−0.904238	+	0.427030i	$0.859560\pi$
$318$		0			0
$319$	583.715			1.82983
$320$		0			0
$321$	197.358	+	399.966i	0.614823	+	1.24600i
$322$		0			0
$323$			224.265i			0.694318i
$324$		0			0
$325$	−129.143			−0.397363
$326$		0			0
$327$	190.114	−	93.8095i	0.581389	−	0.286879i
$328$		0			0
$329$			166.306i			0.505489i
$330$		0			0
$331$	−9.91562			−0.0299566			−0.0149783	−	0.999888i	$-0.504768\pi$
$331$	−9.91562			−0.0299566			−0.0149783	+	0.999888i	$0.504768\pi$
$332$		0			0
$333$	348.763	−	454.959i	1.04733	−	1.36624i
$334$		0			0
$335$			14.7837i			0.0441305i
$336$		0			0
$337$	−289.542			−0.859174			−0.429587	−	0.903025i	$-0.641341\pi$
$337$	−289.542			−0.859174			−0.429587	+	0.903025i	$0.641341\pi$
$338$		0			0
$339$	−173.860	−	352.344i	−0.512860	−	1.03936i
$340$		0			0
$341$			411.864i			1.20781i
$342$		0			0
$343$	246.494			0.718643
$344$		0			0
$345$	62.7454	−	30.9609i	0.181871	−	0.0897418i
$346$		0			0
$347$		−	331.079i		−	0.954119i	−0.878871	−	0.477060i	$-0.841703\pi$
$347$		−	331.079i		−	0.954119i	0.878871	−	0.477060i	$-0.158297\pi$
$348$		0			0
$349$	302.280			0.866133			0.433067	−	0.901362i	$-0.357431\pi$
$349$	302.280			0.866133			0.433067	+	0.901362i	$0.357431\pi$
$350$		0			0
$351$	27.5653	−	139.092i	0.0785337	−	0.396275i
$352$		0			0
$353$		−	528.345i		−	1.49673i	−0.663289	−	0.748364i	$-0.730840\pi$
$353$		−	528.345i		−	1.49673i	0.663289	−	0.748364i	$-0.269160\pi$
$354$		0			0
$355$	−4.66989			−0.0131546
$356$		0			0
$357$	−53.7142	−	108.857i	−0.150460	−	0.304922i
$358$		0			0
$359$		−	59.5167i		−	0.165785i	−0.996559	−	0.0828923i	$-0.973584\pi$
$359$		−	59.5167i		−	0.165785i	0.996559	−	0.0828923i	$-0.0264158\pi$
$360$		0			0
$361$	−133.598			−0.370078
$362$		0			0
$363$	17.0047	−	8.39076i	0.0468449	−	0.0231150i
$364$		0			0
$365$		−	47.0092i		−	0.128792i
$366$		0			0
$367$	−423.762			−1.15466			−0.577332	−	0.816509i	$-0.695906\pi$
$367$	−423.762			−1.15466			−0.577332	+	0.816509i	$0.695906\pi$
$368$		0			0
$369$	86.7846	+	66.5274i	0.235189	+	0.180291i
$370$		0			0
$371$			160.986i			0.433925i
$372$		0			0
$373$	26.5076			0.0710660			0.0355330	−	0.999369i	$-0.488687\pi$
$373$	26.5076			0.0710660			0.0355330	+	0.999369i	$0.488687\pi$
$374$		0			0
$375$	42.1328	+	85.3863i	0.112354	+	0.227697i
$376$		0			0
$377$			271.679i			0.720634i
$378$		0			0
$379$	383.169			1.01100			0.505500	−	0.862827i	$-0.331308\pi$
$379$	383.169			1.01100			0.505500	+	0.862827i	$0.331308\pi$
$380$		0			0
$381$	−562.829	+	277.721i	−1.47724	+	0.728927i
$382$		0			0
$383$			299.496i			0.781973i	0.920396	+	0.390987i	$0.127866\pi$
$383$			299.496i			0.781973i	−0.920396	+	0.390987i	$0.872134\pi$
$384$		0			0
$385$	−19.6485			−0.0510351
$386$		0			0
$387$	64.6285	−	84.3075i	0.166999	−	0.217849i
$388$		0			0
$389$			437.687i			1.12516i	0.826743	+	0.562580i	$0.190191\pi$
$389$			437.687i			1.12516i	−0.826743	+	0.562580i	$0.809809\pi$
$390$		0			0
$391$	−541.944			−1.38605
$392$		0			0
$393$	232.099	+	470.372i	0.590583	+	1.19687i
$394$		0			0
$395$			37.4470i			0.0948027i
$396$		0			0
$397$	−669.283			−1.68585			−0.842926	−	0.538029i	$-0.819169\pi$
$397$	−669.283			−1.68585			−0.842926	+	0.538029i	$0.819169\pi$
$398$		0			0
$399$	−110.380	+	54.4656i	−0.276641	+	0.136505i
$400$		0			0
$401$		−	56.1062i		−	0.139916i	−0.997550	−	0.0699578i	$-0.977714\pi$
$401$		−	56.1062i		−	0.139916i	0.997550	−	0.0699578i	$-0.0222865\pi$
$402$		0			0
$403$	−191.694			−0.475668
$404$		0			0
$405$	−50.0620	+	13.4645i	−0.123610	+	0.0332457i
$406$		0			0
$407$			718.713i			1.76588i
$408$		0			0
$409$	248.582			0.607779			0.303889	−	0.952707i	$-0.401715\pi$
$409$	248.582			0.607779			0.303889	+	0.952707i	$0.401715\pi$
$410$		0			0
$411$	−216.309	−	438.371i	−0.526299	−	1.06660i
$412$		0			0
$413$			101.457i			0.245658i
$414$		0			0
$415$	−20.7325			−0.0499578
$416$		0			0
$417$	−522.531	+	257.836i	−1.25307	+	0.618313i
$418$		0			0
$419$		−	459.935i		−	1.09770i	−0.835922	−	0.548849i	$-0.815066\pi$
$419$		−	459.935i		−	1.09770i	0.835922	−	0.548849i	$-0.184934\pi$
$420$		0			0
$421$	−97.4789			−0.231541			−0.115771	−	0.993276i	$-0.536934\pi$
$421$	−97.4789			−0.231541			−0.115771	+	0.993276i	$0.536934\pi$
$422$		0			0
$423$	−436.599	−	334.688i	−1.03215	−	0.791225i
$424$		0			0
$425$		−	365.703i		−	0.860479i
$426$		0			0
$427$	158.345			0.370831
$428$		0			0
$429$	78.6663	+	159.425i	0.183371	+	0.371620i
$430$		0			0
$431$			545.207i			1.26498i	0.774568	+	0.632490i	$0.217967\pi$
$431$			545.207i			1.26498i	−0.774568	+	0.632490i	$0.782033\pi$
$432$		0			0
$433$	−24.5297			−0.0566506			−0.0283253	−	0.999599i	$-0.509017\pi$
$433$	−24.5297			−0.0566506			−0.0283253	+	0.999599i	$0.509017\pi$
$434$		0			0
$435$	89.0721	−	43.9515i	0.204763	−	0.101038i
$436$		0			0
$437$			549.526i			1.25750i
$438$		0			0
$439$	−81.8743			−0.186502			−0.0932509	−	0.995643i	$-0.529726\pi$
$439$	−81.8743			−0.186502			−0.0932509	+	0.995643i	$0.529726\pi$
$440$		0			0
$441$	−227.767	+	297.120i	−0.516478	+	0.673742i
$442$		0			0
$443$			831.356i			1.87665i	0.345753	+	0.938325i	$0.387623\pi$
$443$			831.356i			1.87665i	−0.345753	+	0.938325i	$0.612377\pi$
$444$		0			0
$445$	71.8482			0.161457
$446$		0			0
$447$	110.806	+	224.559i	0.247887	+	0.502368i
$448$		0			0
$449$			259.553i			0.578070i	0.957319	+	0.289035i	$0.0933344\pi$
$449$			259.553i			0.578070i	−0.957319	+	0.289035i	$0.906666\pi$
$450$		0			0
$451$	−137.097			−0.303983
$452$		0			0
$453$	162.357	−	80.1129i	0.358403	−	0.176850i
$454$		0			0
$455$		−	9.14503i		−	0.0200990i
$456$		0			0
$457$	373.184			0.816596			0.408298	−	0.912849i	$-0.366122\pi$
$457$	373.184			0.816596			0.408298	+	0.912849i	$0.366122\pi$
$458$		0			0
$459$	393.879	+	78.0588i	0.858123	+	0.170063i
$460$		0			0
$461$			672.996i			1.45986i	0.683522	+	0.729930i	$0.260447\pi$
$461$			672.996i			1.45986i	−0.683522	+	0.729930i	$0.739553\pi$
$462$		0			0
$463$	28.0435			0.0605690			0.0302845	−	0.999541i	$-0.490359\pi$
$463$	28.0435			0.0605690			0.0302845	+	0.999541i	$0.490359\pi$
$464$		0			0
$465$	31.0118	+	62.8484i	0.0666920	+	0.135158i
$466$		0			0
$467$		−	373.758i		−	0.800339i	−0.916441	−	0.400169i	$-0.868951\pi$
$467$		−	373.758i		−	0.800339i	0.916441	−	0.400169i	$-0.131049\pi$
$468$		0			0
$469$	62.8472			0.134003
$470$		0			0
$471$	−271.082	+	133.762i	−0.575545	+	0.283995i
$472$		0			0
$473$			133.183i			0.281572i
$474$		0			0
$475$	−370.819			−0.780672
$476$		0			0
$477$	−422.633	−	323.982i	−0.886023	−	0.679208i
$478$		0			0
$479$			915.154i			1.91055i	0.295718	+	0.955275i	$0.404441\pi$
$479$			915.154i			1.91055i	−0.295718	+	0.955275i	$0.595559\pi$
$480$		0			0
$481$	−334.512			−0.695451
$482$		0			0
$483$	−131.618	−	266.737i	−0.272502	−	0.552251i
$484$		0			0
$485$			51.2227i			0.105614i
$486$		0			0
$487$	238.560			0.489856			0.244928	−	0.969541i	$-0.421236\pi$
$487$	238.560			0.489856			0.244928	+	0.969541i	$0.421236\pi$
$488$		0			0
$489$	187.555	−	92.5464i	0.383547	−	0.189257i
$490$		0			0
$491$		−	435.657i		−	0.887286i	−0.896204	−	0.443643i	$-0.853686\pi$
$491$		−	435.657i		−	0.887286i	0.896204	−	0.443643i	$-0.146314\pi$
$492$		0			0
$493$	−769.334			−1.56052
$494$		0			0
$495$	39.5423	−	51.5827i	0.0798835	−	0.104207i
$496$		0			0
$497$			19.8522i			0.0399441i
$498$		0			0
$499$	209.499			0.419838			0.209919	−	0.977719i	$-0.432680\pi$
$499$	209.499			0.419838			0.209919	+	0.977719i	$0.432680\pi$
$500$		0			0
$501$	160.392	+	325.051i	0.320144	+	0.648804i
$502$		0			0
$503$		−	558.314i		−	1.10997i	−0.831861	−	0.554984i	$-0.812724\pi$
$503$		−	558.314i		−	1.10997i	0.831861	−	0.554984i	$-0.187276\pi$
$504$		0			0
$505$	76.4002			0.151287
$506$		0			0
$507$	380.460	−	187.733i	0.750415	−	0.370283i
$508$		0			0
$509$		−	319.621i		−	0.627939i	−0.949433	−	0.313969i	$-0.898341\pi$
$509$		−	319.621i		−	0.627939i	0.949433	−	0.313969i	$-0.101659\pi$
$510$		0			0
$511$	−199.841			−0.391079
$512$		0			0
$513$	79.1507	−	399.388i	0.154290	−	0.778535i
$514$		0			0
$515$			80.3357i			0.155992i
$516$		0			0
$517$	689.710			1.33406
$518$		0			0
$519$	−39.9048	−	80.8710i	−0.0768879	−	0.155821i
$520$		0			0
$521$			771.602i			1.48100i	0.672055	+	0.740501i	$0.265412\pi$
$521$			771.602i			1.48100i	−0.672055	+	0.740501i	$0.734588\pi$
$522$		0			0
$523$	−288.856			−0.552306			−0.276153	−	0.961114i	$-0.589060\pi$
$523$	−288.856			−0.552306			−0.276153	+	0.961114i	$0.589060\pi$
$524$		0			0
$525$	179.994	−	88.8158i	0.342846	−	0.169173i
$526$		0			0
$527$		−	542.835i		−	1.03005i
$528$		0			0
$529$	−798.951			−1.51030
$530$		0			0
$531$	−266.352	−	204.180i	−0.501605	−	0.384521i
$532$		0			0
$533$		−	63.8090i		−	0.119717i
$534$		0			0
$535$	−95.1503			−0.177851
$536$		0			0
$537$	152.375	+	308.803i	0.283753	+	0.575053i
$538$		0			0
$539$		−	469.371i		−	0.870818i
$540$		0			0
$541$	232.871			0.430446			0.215223	−	0.976565i	$-0.430952\pi$
$541$	232.871			0.430446			0.215223	+	0.976565i	$0.430952\pi$
$542$		0			0
$543$	488.633	−	241.110i	0.899877	−	0.444033i
$544$		0			0
$545$			45.2274i			0.0829861i
$546$		0			0
$547$	910.003			1.66363			0.831813	−	0.555056i	$-0.187303\pi$
$547$	910.003			1.66363			0.831813	+	0.555056i	$0.187303\pi$
$548$		0			0
$549$	−318.667	+	415.699i	−0.580449	+	0.757192i
$550$		0			0
$551$			780.096i			1.41578i
$552$		0			0
$553$	159.191			0.287869
$554$		0			0
$555$	54.1164	+	109.672i	0.0975071	+	0.197608i
$556$		0			0
$557$		−	297.809i		−	0.534666i	−0.963604	−	0.267333i	$-0.913858\pi$
$557$		−	297.809i		−	0.534666i	0.963604	−	0.267333i	$-0.0861424\pi$
$558$		0			0
$559$	−61.9877			−0.110890
$560$		0			0
$561$	−451.456	+	222.765i	−0.804734	+	0.397086i
$562$		0			0
$563$			184.465i			0.327647i	0.986490	+	0.163824i	$0.0523828\pi$
$563$			184.465i			0.327647i	−0.986490	+	0.163824i	$0.947617\pi$
$564$		0			0
$565$	83.8212			0.148356
$566$		0			0
$567$	57.2391	+	212.819i	0.100951	+	0.375342i
$568$		0			0
$569$		−	404.137i		−	0.710258i	−0.934817	−	0.355129i	$-0.884437\pi$
$569$		−	404.137i		−	0.710258i	0.934817	−	0.355129i	$-0.115563\pi$
$570$		0			0
$571$	762.365			1.33514			0.667570	−	0.744547i	$-0.267334\pi$
$571$	762.365			1.33514			0.667570	+	0.744547i	$0.267334\pi$
$572$		0			0
$573$	−277.748	−	562.883i	−0.484726	−	0.982344i
$574$		0			0
$575$		−	896.100i		−	1.55843i
$576$		0			0
$577$	−290.766			−0.503927			−0.251964	−	0.967737i	$-0.581076\pi$
$577$	−290.766			−0.503927			−0.251964	+	0.967737i	$0.581076\pi$
$578$		0			0
$579$	−434.027	+	214.165i	−0.749616	+	0.369888i
$580$		0			0
$581$			88.1361i			0.151697i
$582$		0			0
$583$	667.648			1.14519
$584$		0			0
$585$	24.0082	+	18.4042i	0.0410397	+	0.0314602i
$586$		0			0
$587$			224.506i			0.382464i	0.981545	+	0.191232i	$0.0612482\pi$
$587$			224.506i			0.382464i	−0.981545	+	0.191232i	$0.938752\pi$
$588$		0			0
$589$	−550.428			−0.934513
$590$		0			0
$591$	192.425	+	389.968i	0.325592	+	0.659844i
$592$		0			0
$593$		−	482.620i		−	0.813862i	−0.913459	−	0.406931i	$-0.866599\pi$
$593$		−	482.620i		−	0.813862i	0.913459	−	0.406931i	$-0.133401\pi$
$594$		0			0
$595$	25.8967			0.0435238
$596$		0			0
$597$	638.312	−	314.967i	1.06920	−	0.527583i
$598$		0			0
$599$		−	803.277i		−	1.34103i	−0.741896	−	0.670515i	$-0.766073\pi$
$599$		−	803.277i		−	1.34103i	0.741896	−	0.670515i	$-0.233927\pi$
$600$		0			0
$601$	−126.365			−0.210258			−0.105129	−	0.994459i	$-0.533526\pi$
$601$	−126.365			−0.210258			−0.105129	+	0.994459i	$0.533526\pi$
$602$		0			0
$603$	−126.479	+	164.991i	−0.209750	+	0.273617i
$604$		0			0
$605$			4.04535i			0.00668653i
$606$		0			0
$607$	−397.356			−0.654623			−0.327312	−	0.944916i	$-0.606143\pi$
$607$	−397.356			−0.654623			−0.327312	+	0.944916i	$0.606143\pi$
$608$		0			0
$609$	−186.843	−	378.655i	−0.306802	−	0.621766i
$610$		0			0
$611$			321.012i			0.525388i
$612$		0			0
$613$	1056.55			1.72357			0.861785	−	0.507275i	$-0.169347\pi$
$613$	1056.55			1.72357			0.861785	+	0.507275i	$0.169347\pi$
$614$		0			0
$615$	−20.9203	+	10.3229i	−0.0340167	+	0.0167851i
$616$		0			0
$617$			411.284i			0.666586i	0.942823	+	0.333293i	$0.108160\pi$
$617$			411.284i			0.666586i	−0.942823	+	0.333293i	$0.891840\pi$
$618$		0			0
$619$	−691.383			−1.11693			−0.558467	−	0.829526i	$-0.688610\pi$
$619$	−691.383			−1.11693			−0.558467	+	0.829526i	$0.688610\pi$
$620$		0			0
$621$	965.138	+	191.271i	1.55417	+	0.308005i
$622$		0			0
$623$		−	305.435i		−	0.490264i
$624$		0			0
$625$	594.447			0.951114
$626$		0			0
$627$	225.881	+	457.771i	0.360257	+	0.730097i
$628$		0			0
$629$		−	947.262i		−	1.50598i
$630$		0			0
$631$	528.057			0.836858			0.418429	−	0.908250i	$-0.362581\pi$
$631$	528.057			0.836858			0.418429	+	0.908250i	$0.362581\pi$
$632$		0			0
$633$	826.238	−	407.697i	1.30527	−	0.644071i
$634$		0			0
$635$		−	133.895i		−	0.210858i
$636$		0			0
$637$	218.460			0.342951
$638$		0			0
$639$	−52.1175	−	39.9523i	−0.0815611	−	0.0625232i
$640$		0			0
$641$		−	134.165i		−	0.209306i	−0.994509	−	0.104653i	$-0.966627\pi$
$641$		−	134.165i		−	0.209306i	0.994509	−	0.104653i	$-0.0333731\pi$
$642$		0			0
$643$	−633.985			−0.985979			−0.492990	−	0.870035i	$-0.664096\pi$
$643$	−633.985			−0.985979			−0.492990	+	0.870035i	$0.664096\pi$
$644$		0			0
$645$	10.0282	+	20.3232i	0.0155476	+	0.0315088i
$646$		0			0
$647$			227.758i			0.352021i	0.984388	+	0.176011i	$0.0563193\pi$
$647$			227.758i			0.352021i	−0.984388	+	0.176011i	$0.943681\pi$
$648$		0			0
$649$	420.766			0.648329
$650$		0			0
$651$	267.176	−	131.834i	0.410408	−	0.202511i
$652$		0			0
$653$		−	1190.65i		−	1.82336i	−0.410906	−	0.911678i	$-0.634788\pi$
$653$		−	1190.65i		−	1.82336i	0.410906	−	0.911678i	$-0.365212\pi$
$654$		0			0
$655$	−111.900			−0.170839
$656$		0			0
$657$	402.178	−	524.638i	0.612142	−	0.798536i
$658$		0			0
$659$			266.873i			0.404967i	0.979286	+	0.202484i	$0.0649013\pi$
$659$			266.873i			0.404967i	−0.979286	+	0.202484i	$0.935099\pi$
$660$		0			0
$661$	−1045.10			−1.58109			−0.790544	−	0.612405i	$-0.790202\pi$
$661$	−1045.10			−1.58109			−0.790544	+	0.612405i	$0.790202\pi$
$662$		0			0
$663$	−103.682	−	210.122i	−0.156383	−	0.316925i
$664$		0			0
$665$		−	26.2589i		−	0.0394871i
$666$		0			0
$667$	−1885.13			−2.82629
$668$		0			0
$669$	−1178.36	+	581.445i	−1.76137	+	0.869126i
$670$		0			0
$671$		−	656.693i		−	0.978678i
$672$		0			0
$673$	408.300			0.606687			0.303343	−	0.952881i	$-0.401897\pi$
$673$	408.300			0.606687			0.303343	+	0.952881i	$0.401897\pi$
$674$		0			0
$675$	−129.069	+	651.274i	−0.191214	+	0.964850i
$676$		0			0
$677$			749.557i			1.10717i	0.832791	+	0.553587i	$0.186741\pi$
$677$			749.557i			1.10717i	−0.832791	+	0.553587i	$0.813259\pi$
$678$		0			0
$679$	217.753			0.320697
$680$		0			0
$681$	−359.426	−	728.413i	−0.527792	−	1.06962i
$682$		0			0
$683$			258.242i			0.378099i	0.981968	+	0.189050i	$0.0605407\pi$
$683$			258.242i			0.378099i	−0.981968	+	0.189050i	$0.939459\pi$
$684$		0			0
$685$	104.287			0.152244
$686$		0			0
$687$	265.474	−	130.995i	0.386424	−	0.190676i
$688$		0			0
$689$			310.744i			0.451007i
$690$		0			0
$691$	929.714			1.34546			0.672731	−	0.739887i	$-0.265121\pi$
$691$	929.714			1.34546			0.672731	+	0.739887i	$0.265121\pi$
$692$		0			0
$693$	−219.284	−	168.099i	−0.316427	−	0.242567i
$694$		0			0
$695$		−	124.308i		−	0.178861i
$696$		0			0
$697$	180.693			0.259244
$698$		0			0
$699$	574.016	+	1163.30i	0.821197	+	1.66424i
$700$		0			0
$701$		−	335.731i		−	0.478932i	−0.970905	−	0.239466i	$-0.923028\pi$
$701$		−	335.731i		−	0.478932i	0.970905	−	0.239466i	$-0.0769724\pi$
$702$		0			0
$703$	−960.513			−1.36631
$704$		0			0
$705$	105.246	−	51.9325i	0.149286	−	0.0736632i
$706$		0			0
$707$		−	324.785i		−	0.459385i
$708$		0			0
$709$	356.121			0.502287			0.251143	−	0.967950i	$-0.419193\pi$
$709$	356.121			0.502287			0.251143	+	0.967950i	$0.419193\pi$
$710$		0			0
$711$	−320.370	+	417.921i	−0.450591	+	0.587793i
$712$		0			0
$713$		−	1330.13i		−	1.86554i
$714$		0			0
$715$	−37.9266			−0.0530442
$716$		0			0
$717$	366.123	+	741.985i	0.510632	+	1.03485i
$718$		0			0
$719$		−	31.7234i		−	0.0441216i	−0.999757	−	0.0220608i	$-0.992977\pi$
$719$		−	31.7234i		−	0.0441216i	0.999757	−	0.0220608i	$-0.00702274\pi$
$720$		0			0
$721$	341.516			0.473670
$722$		0			0
$723$	−1121.24	+	553.262i	−1.55082	+	0.765231i
$724$		0			0
$725$		−	1272.09i		−	1.75460i
$726$		0			0
$727$	234.202			0.322148			0.161074	−	0.986942i	$-0.448504\pi$
$727$	234.202			0.322148			0.161074	+	0.986942i	$0.448504\pi$
$728$		0			0
$729$	−673.901	−	278.027i	−0.924418	−	0.381381i
$730$		0			0
$731$		−	175.535i		−	0.240130i
$732$		0			0
$733$	616.813			0.841491			0.420745	−	0.907179i	$-0.361769\pi$
$733$	616.813			0.841491			0.420745	+	0.907179i	$0.361769\pi$
$734$		0			0
$735$	−35.3419	−	71.6238i	−0.0480842	−	0.0974473i
$736$		0			0
$737$		−	260.642i		−	0.353653i
$738$		0			0
$739$	605.806			0.819764			0.409882	−	0.912138i	$-0.365570\pi$
$739$	605.806			0.819764			0.409882	+	0.912138i	$0.365570\pi$
$740$		0			0
$741$	−213.061	+	105.132i	−0.287532	+	0.141879i
$742$		0			0
$743$		−	1033.33i		−	1.39076i	−0.718642	−	0.695380i	$-0.755236\pi$
$743$		−	1033.33i		−	1.39076i	0.718642	−	0.695380i	$-0.244764\pi$
$744$		0			0
$745$	−53.4216			−0.0717068
$746$		0			0
$747$	−231.381	−	177.372i	−0.309747	−	0.237446i
$748$		0			0
$749$			404.494i			0.540046i
$750$		0			0
$751$	969.301			1.29068			0.645340	−	0.763896i	$-0.276716\pi$
$751$	969.301			1.29068			0.645340	+	0.763896i	$0.276716\pi$
$752$		0			0
$753$	−220.235	−	446.329i	−0.292477	−	0.592734i
$754$		0			0
$755$			38.6240i			0.0511577i
$756$		0			0
$757$	−533.594			−0.704880			−0.352440	−	0.935835i	$-0.614648\pi$
$757$	−533.594			−0.704880			−0.352440	+	0.935835i	$0.614648\pi$
$758$		0			0
$759$	−1106.22	+	545.851i	−1.45747	+	0.719172i
$760$		0			0
$761$			867.717i			1.14023i	0.821564	+	0.570116i	$0.193102\pi$
$761$			867.717i			1.14023i	−0.821564	+	0.570116i	$0.806898\pi$
$762$		0			0
$763$	192.267			0.251988
$764$		0			0
$765$	−52.1166	+	67.9858i	−0.0681263	+	0.0888704i
$766$		0			0
$767$			195.837i			0.255329i
$768$		0			0
$769$	194.555			0.252997			0.126498	−	0.991967i	$-0.459626\pi$
$769$	194.555			0.252997			0.126498	+	0.991967i	$0.459626\pi$
$770$		0			0
$771$	−79.1695	−	160.445i	−0.102684	−	0.208100i
$772$		0			0
$773$		−	420.140i		−	0.543519i	−0.962365	−	0.271760i	$-0.912394\pi$
$773$		−	420.140i		−	0.543519i	0.962365	−	0.271760i	$-0.0876056\pi$
$774$		0			0
$775$	897.572			1.15816
$776$		0			0
$777$	466.229	−	230.055i	0.600037	−	0.296081i
$778$		0			0
$779$		−	183.220i		−	0.235199i
$780$		0			0
$781$	82.3318			0.105418
$782$		0			0
$783$	1370.09	+	271.524i	1.74980	+	0.346775i
$784$		0			0
$785$		−	64.4892i		−	0.0821519i
$786$		0			0
$787$	−849.081			−1.07888			−0.539441	−	0.842023i	$-0.681365\pi$
$787$	−849.081			−1.07888			−0.539441	+	0.842023i	$0.681365\pi$
$788$		0			0
$789$	89.2883	+	180.952i	0.113166	+	0.229343i
$790$		0			0
$791$		−	356.333i		−	0.450484i
$792$		0			0
$793$	305.646			0.385429
$794$		0			0
$795$	101.880	−	50.2714i	0.128151	−	0.0632344i
$796$		0			0
$797$		−	779.186i		−	0.977648i	−0.872382	−	0.488824i	$-0.837426\pi$
$797$		−	779.186i		−	0.977648i	0.872382	−	0.488824i	$-0.162574\pi$
$798$		0			0
$799$	−909.035			−1.13772
$800$		0			0
$801$	801.849	+	614.682i	1.00106	+	0.767393i
$802$		0			0
$803$			828.789i			1.03212i
$804$		0			0
$805$	63.4558			0.0788270
$806$		0			0
$807$	47.9884	+	97.2532i	0.0594651	+	0.120512i
$808$		0			0
$809$			865.140i			1.06939i	0.845044	+	0.534697i	$0.179574\pi$
$809$			865.140i			1.06939i	−0.845044	+	0.534697i	$0.820426\pi$
$810$		0			0
$811$	251.360			0.309938			0.154969	−	0.987919i	$-0.450472\pi$
$811$	251.360			0.309938			0.154969	+	0.987919i	$0.450472\pi$
$812$		0			0
$813$	883.258	−	435.833i	1.08642	−	0.536080i
$814$		0			0
$815$			44.6185i			0.0547466i
$816$		0			0
$817$	−177.991			−0.217859
$818$		0			0
$819$	78.2384	−	102.061i	0.0955292	−	0.124617i
$820$		0			0
$821$		−	103.359i		−	0.125894i	−0.998017	−	0.0629470i	$-0.979950\pi$
$821$		−	103.359i		−	0.125894i	0.998017	−	0.0629470i	$-0.0200499\pi$
$822$		0			0
$823$	−474.728			−0.576826			−0.288413	−	0.957506i	$-0.593128\pi$
$823$	−474.728			−0.576826			−0.288413	+	0.957506i	$0.593128\pi$
$824$		0			0
$825$	−368.340	−	746.477i	−0.446473	−	0.904821i
$826$		0			0
$827$		−	471.776i		−	0.570467i	−0.958458	−	0.285234i	$-0.907929\pi$
$827$		−	471.776i		−	0.570467i	0.958458	−	0.285234i	$-0.0920712\pi$
$828$		0			0
$829$	771.883			0.931102			0.465551	−	0.885021i	$-0.345856\pi$
$829$	771.883			0.931102			0.465551	+	0.885021i	$0.345856\pi$
$830$		0			0
$831$	−163.424	+	80.6396i	−0.196660	+	0.0970392i
$832$		0			0
$833$			618.629i			0.742652i
$834$		0			0
$835$	−77.3283			−0.0926087
$836$		0			0
$837$	−191.585	+	966.723i	−0.228895	+	1.15499i
$838$		0			0
$839$		−	510.944i		−	0.608992i	−0.952514	−	0.304496i	$-0.901512\pi$
$839$		−	510.944i		−	0.608992i	0.952514	−	0.304496i	$-0.0984881\pi$
$840$		0			0
$841$	−1835.10			−2.18204
$842$		0			0
$843$	486.962	+	986.877i	0.577654	+	1.17067i
$844$		0			0
$845$			90.5100i			0.107112i
$846$		0			0
$847$	17.1972			0.0203037
$848$		0			0
$849$	279.877	−	138.102i	0.329655	−	0.162664i
$850$		0			0
$851$		−	2321.12i		−	2.72752i
$852$		0			0
$853$	−581.090			−0.681231			−0.340616	−	0.940203i	$-0.610635\pi$
$853$	−581.090			−0.681231			−0.340616	+	0.940203i	$0.610635\pi$
$854$		0			0
$855$	68.9369	+	52.8457i	0.0806279	+	0.0618078i
$856$		0			0
$857$			7.21760i			0.00842193i	0.999991	+	0.00421097i	$0.00134040\pi$
$857$			7.21760i			0.00842193i	−0.999991	+	0.00421097i	$0.998660\pi$
$858$		0			0
$859$	104.856			0.122068			0.0610339	−	0.998136i	$-0.480560\pi$
$859$	104.856			0.122068			0.0610339	+	0.998136i	$0.480560\pi$
$860$		0			0
$861$	43.8836	+	88.9344i	0.0509681	+	0.103292i
$862$		0			0
$863$		−	1213.72i		−	1.40640i	−0.710992	−	0.703200i	$-0.751754\pi$
$863$		−	1213.72i		−	1.40640i	0.710992	−	0.703200i	$-0.248246\pi$
$864$		0			0
$865$	19.2389			0.0222415
$866$		0			0
$867$	−182.482	+	90.0432i	−0.210475	+	0.103856i
$868$		0			0
$869$		−	660.204i		−	0.759729i
$870$		0			0
$871$	121.311			0.139278
$872$		0			0
$873$	−438.225	+	571.661i	−0.501975	+	0.654824i
$874$		0			0
$875$			86.3530i			0.0986892i
$876$		0			0
$877$	−706.591			−0.805691			−0.402846	−	0.915268i	$-0.631979\pi$
$877$	−706.591			−0.805691			−0.402846	+	0.915268i	$0.631979\pi$
$878$		0			0
$879$	545.043	+	1104.58i	0.620072	+	1.25664i
$880$		0			0
$881$			1715.95i			1.94773i	0.227134	+	0.973864i	$0.427065\pi$
$881$			1715.95i			1.94773i	−0.227134	+	0.973864i	$0.572935\pi$
$882$		0			0
$883$	1472.11			1.66717			0.833586	−	0.552390i	$-0.186284\pi$
$883$	1472.11			1.66717			0.833586	+	0.552390i	$0.186284\pi$
$884$		0			0
$885$	64.2068	−	31.6821i	0.0725501	−	0.0357989i
$886$		0			0
$887$			277.658i			0.313030i	0.987676	+	0.156515i	$0.0500259\pi$
$887$			277.658i			0.313030i	−0.987676	+	0.156515i	$0.949974\pi$
$888$		0			0
$889$	−569.202			−0.640272
$890$		0			0
$891$	882.610	−	237.384i	0.990584	−	0.266424i
$892$		0			0
$893$			921.751i			1.03220i
$894$		0			0
$895$	−73.4631			−0.0820817
$896$		0			0
$897$	−254.056	−	514.870i	−0.283229	−	0.573991i
$898$		0			0
$899$		−	1888.23i		−	2.10037i
$900$		0			0
$901$	−879.957			−0.976645
$902$		0			0
$903$	86.3959	−	42.6310i	0.0956766	−	0.0472104i
$904$		0			0
$905$			116.244i			0.128446i
$906$		0			0
$907$	365.591			0.403078			0.201539	−	0.979481i	$-0.435406\pi$
$907$	365.591			0.403078			0.201539	+	0.979481i	$0.435406\pi$
$908$		0			0
$909$	852.650	+	653.625i	0.938009	+	0.719060i
$910$		0			0
$911$			1664.90i			1.82755i	0.406223	+	0.913774i	$0.366846\pi$
$911$			1664.90i			1.82755i	−0.406223	+	0.913774i	$0.633154\pi$
$912$		0			0
$913$	365.521			0.400351
$914$		0			0
$915$	−49.4465	−	100.208i	−0.0540399	−	0.109517i
$916$		0			0
$917$			475.697i			0.518754i
$918$		0			0
$919$	−195.745			−0.212998			−0.106499	−	0.994313i	$-0.533964\pi$
$919$	−195.745			−0.212998			−0.106499	+	0.994313i	$0.533964\pi$
$920$		0			0
$921$	−726.451	+	358.458i	−0.788763	+	0.389205i
$922$		0			0
$923$			38.3198i			0.0415166i
$924$		0			0
$925$	1566.29			1.69328
$926$		0			0
$927$	−687.295	+	896.573i	−0.741419	+	0.967177i
$928$		0			0
$929$			769.046i			0.827822i	0.910317	+	0.413911i	$0.135837\pi$
$929$			769.046i			0.827822i	−0.910317	+	0.413911i	$0.864163\pi$
$930$		0			0
$931$	627.283			0.673773
$932$		0			0
$933$	−333.939	−	676.760i	−0.357919	−	0.725359i
$934$		0			0
$935$		−	107.400i		−	0.114866i
$936$		0			0
$937$	−0.0296961			−3.16928e−5			−1.58464e−5	−	1.00000i	$-0.500005\pi$
$937$	−0.0296961			−3.16928e−5			−1.58464e−5		1.00000i	$0.500005\pi$
$938$		0			0
$939$	225.867	−	111.451i	0.240540	−	0.118692i
$940$		0			0
$941$			1283.68i			1.36416i	0.731277	+	0.682081i	$0.238925\pi$
$941$			1283.68i			1.36416i	−0.731277	+	0.682081i	$0.761075\pi$
$942$		0			0
$943$	442.760			0.469522
$944$		0			0
$945$	−46.1189	−	9.13983i	−0.0488030	−	0.00967178i
$946$		0			0
$947$			1514.62i			1.59938i	0.600410	+	0.799692i	$0.295004\pi$
$947$			1514.62i			1.59938i	−0.600410	+	0.799692i	$0.704996\pi$
$948$		0			0
$949$	−385.744			−0.406474
$950$		0			0
$951$	−359.403	−	728.365i	−0.377921	−	0.765894i
$952$		0			0
$953$		−	937.870i		−	0.984124i	−0.870560	−	0.492062i	$-0.836243\pi$
$953$		−	937.870i		−	0.984124i	0.870560	−	0.492062i	$-0.163757\pi$
$954$		0			0
$955$	133.908			0.140218
$956$		0			0
$957$	−1570.37	+	774.880i	−1.64093	+	0.809697i
$958$		0			0
$959$		−	443.335i		−	0.462288i
$960$		0			0
$961$	371.317			0.386386
$962$		0			0
$963$	−1061.91	−	814.038i	−1.10271	−	0.845315i
$964$		0			0
$965$		−	103.253i		−	0.106998i
$966$		0			0
$967$	−461.673			−0.477428			−0.238714	−	0.971090i	$-0.576726\pi$
$967$	−461.673			−0.477428			−0.238714	+	0.971090i	$0.576726\pi$
$968$		0			0
$969$	−297.711	−	603.340i	−0.307235	−	0.622642i
$970$		0			0
$971$		−	1360.95i		−	1.40160i	−0.713360	−	0.700798i	$-0.752827\pi$
$971$		−	1360.95i		−	1.40160i	0.713360	−	0.700798i	$-0.247173\pi$
$972$		0			0
$973$	−528.447			−0.543111
$974$		0			0
$975$	347.434	−	171.437i	0.356342	−	0.175833i
$976$		0			0
$977$			1162.17i			1.18953i	0.803901	+	0.594763i	$0.202754\pi$
$977$			1162.17i			1.18953i	−0.803901	+	0.594763i	$0.797246\pi$
$978$		0			0
$979$	−1266.71			−1.29388
$980$		0			0
$981$	−386.934	+	504.753i	−0.394428	+	0.514529i
$982$		0			0
$983$			969.961i			0.986735i	0.869821	+	0.493368i	$0.164234\pi$
$983$			969.961i			0.986735i	−0.869821	+	0.493368i	$0.835766\pi$
$984$		0			0
$985$	−92.7717			−0.0941845
$986$		0			0
$987$	−220.771	−	447.414i	−0.223679	−	0.453307i
$988$		0			0
$989$		−	430.122i		−	0.434906i
$990$		0			0
$991$	−1602.81			−1.61737			−0.808684	−	0.588243i	$-0.799820\pi$
$991$	−1602.81			−1.61737			−0.808684	+	0.588243i	$0.799820\pi$
$992$		0			0
$993$	26.6761	−	13.1630i	0.0268641	−	0.0132558i
$994$		0			0
$995$			151.852i			0.152615i
$996$		0			0
$997$	−742.855			−0.745090			−0.372545	−	0.928014i	$-0.621515\pi$
$997$	−742.855			−0.745090			−0.372545	+	0.928014i	$0.621515\pi$
$998$		0			0
$999$	−334.321	+	1686.96i	−0.334656	+	1.68865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.e.c.257.2	yes	8
3.2	odd	2	inner	384.3.e.c.257.1	yes	8
4.3	odd	2		384.3.e.b.257.7	yes	8
8.3	odd	2		384.3.e.d.257.2	yes	8
8.5	even	2		384.3.e.a.257.7	✓	8
12.11	even	2		384.3.e.b.257.8	yes	8
16.3	odd	4		768.3.h.g.641.12		16
16.5	even	4		768.3.h.h.641.12		16
16.11	odd	4		768.3.h.g.641.5		16
16.13	even	4		768.3.h.h.641.5		16
24.5	odd	2		384.3.e.a.257.8	yes	8
24.11	even	2		384.3.e.d.257.1	yes	8
48.5	odd	4		768.3.h.h.641.6		16
48.11	even	4		768.3.h.g.641.11		16
48.29	odd	4		768.3.h.h.641.11		16
48.35	even	4		768.3.h.g.641.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.e.a.257.7	✓	8	8.5	even	2
384.3.e.a.257.8	yes	8	24.5	odd	2
384.3.e.b.257.7	yes	8	4.3	odd	2
384.3.e.b.257.8	yes	8	12.11	even	2
384.3.e.c.257.1	yes	8	3.2	odd	2	inner
384.3.e.c.257.2	yes	8	1.1	even	1	trivial
384.3.e.d.257.1	yes	8	24.11	even	2
384.3.e.d.257.2	yes	8	8.3	odd	2
768.3.h.g.641.5		16	16.11	odd	4
768.3.h.g.641.6		16	48.35	even	4
768.3.h.g.641.11		16	48.11	even	4
768.3.h.g.641.12		16	16.3	odd	4
768.3.h.h.641.5		16	16.13	even	4
768.3.h.h.641.6		16	48.5	odd	4
768.3.h.h.641.11		16	48.29	odd	4
768.3.h.h.641.12		16	16.5	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)

\(f(q)\)	\(=\)	\(q+(-2.69031 + 1.32750i) q^{3} -0.640013i q^{5} -2.72077 q^{7} +(5.47550 - 7.14275i) q^{9} +O(q^{10})\)	\(q+(-2.69031 + 1.32750i) q^{3} -0.640013i q^{5} -2.72077 q^{7} +(5.47550 - 7.14275i) q^{9} +11.2836i q^{11} -5.25176 q^{13} +(0.849616 + 1.72183i) q^{15} -14.8718i q^{17} -15.0798 q^{19} +(7.31969 - 3.61181i) q^{21} -36.4411i q^{23} +24.5904 q^{25} +(-5.24877 + 26.4849i) q^{27} -51.7310i q^{29} +36.5009 q^{31} +(-14.9790 - 30.3565i) q^{33} +1.74133i q^{35} +63.6951 q^{37} +(14.1289 - 6.97170i) q^{39} +12.1500i q^{41} +11.8032 q^{43} +(-4.57146 - 3.50439i) q^{45} -61.1247i q^{47} -41.5974 q^{49} +(19.7423 + 40.0097i) q^{51} -59.1695i q^{53} +7.22168 q^{55} +(40.5694 - 20.0185i) q^{57} -37.2898i q^{59} -58.1987 q^{61} +(-14.8975 + 19.4338i) q^{63} +3.36120i q^{65} -23.0991 q^{67} +(48.3754 + 98.0376i) q^{69} -7.29656i q^{71} +73.4504 q^{73} +(-66.1557 + 32.6437i) q^{75} -30.7002i q^{77} -58.5098 q^{79} +(-21.0379 - 78.2203i) q^{81} -32.3939i q^{83} -9.51815 q^{85} +(68.6728 + 139.172i) q^{87} +112.260i q^{89} +14.2888 q^{91} +(-98.1987 + 48.4549i) q^{93} +9.65130i q^{95} -80.0338 q^{97} +(80.5963 + 61.7836i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{7}+O(q^{10}) \) 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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