LMFDB - Embedded newform 384.3.h.f.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.3.h.f.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23607 - 2.00000i) q^{3} +4.00000 q^{5} -8.94427 q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})$	\(q+(-2.23607 - 2.00000i) q^{3} +4.00000 q^{5} -8.94427 q^{7} +(1.00000 + 8.94427i) q^{9} +4.47214 q^{11} +17.8885i q^{13} +(-8.94427 - 8.00000i) q^{15} -17.8885i q^{17} +20.0000i q^{19} +(20.0000 + 17.8885i) q^{21} +16.0000i q^{23} -9.00000 q^{25} +(15.6525 - 22.0000i) q^{27} +52.0000 q^{29} +26.8328 q^{31} +(-10.0000 - 8.94427i) q^{33} -35.7771 q^{35} +53.6656i q^{37} +(35.7771 - 40.0000i) q^{39} +35.7771i q^{41} -36.0000i q^{43} +(4.00000 + 35.7771i) q^{45} +64.0000i q^{47} +31.0000 q^{49} +(-35.7771 + 40.0000i) q^{51} +20.0000 q^{53} +17.8885 q^{55} +(40.0000 - 44.7214i) q^{57} +102.859 q^{59} +17.8885i q^{61} +(-8.94427 - 80.0000i) q^{63} +71.5542i q^{65} +44.0000i q^{67} +(32.0000 - 35.7771i) q^{69} +80.0000i q^{71} -50.0000 q^{73} +(20.1246 + 18.0000i) q^{75} -40.0000 q^{77} -80.4984 q^{79} +(-79.0000 + 17.8885i) q^{81} -102.859 q^{83} -71.5542i q^{85} +(-116.276 - 104.000i) q^{87} -160.997i q^{89} -160.000i q^{91} +(-60.0000 - 53.6656i) q^{93} +80.0000i q^{95} +50.0000 q^{97} +(4.47214 + 40.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 16 * q^5 + 4 * q^9	$ 4 q + 16 q^{5} + 4 q^{9} + 80 q^{21} - 36 q^{25} + 208 q^{29} - 40 q^{33} + 16 q^{45} + 124 q^{49} + 80 q^{53} + 160 q^{57} + 128 q^{69} - 200 q^{73} - 160 q^{77} - 316 q^{81} - 240 q^{93} + 200 q^{97}+O(q^{100}) $ 4 * q + 16 * q^5 + 4 * q^9 + 80 * q^21 - 36 * q^25 + 208 * q^29 - 40 * q^33 + 16 * q^45 + 124 * q^49 + 80 * q^53 + 160 * q^57 + 128 * q^69 - 200 * q^73 - 160 * q^77 - 316 * q^81 - 240 * q^93 + 200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$3$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$4$		0			0
$5$	4.00000			0.800000			0.400000	−	0.916515i	$-0.369010\pi$
$5$	4.00000			0.800000			0.400000	+	0.916515i	$0.369010\pi$
$6$		0			0
$7$	−8.94427			−1.27775			−0.638877	−	0.769309i	$-0.720601\pi$
$7$	−8.94427			−1.27775			−0.638877	+	0.769309i	$0.720601\pi$
$8$		0			0
$9$	1.00000	+	8.94427i	0.111111	+	0.993808i
$10$		0			0
$11$	4.47214			0.406558			0.203279	−	0.979121i	$-0.434840\pi$
$11$	4.47214			0.406558			0.203279	+	0.979121i	$0.434840\pi$
$12$		0			0
$13$			17.8885i			1.37604i	0.725691	+	0.688021i	$0.241520\pi$
$13$			17.8885i			1.37604i	−0.725691	+	0.688021i	$0.758480\pi$
$14$		0			0
$15$	−8.94427	−	8.00000i	−0.596285	−	0.533333i
$16$		0			0
$17$		−	17.8885i		−	1.05227i	−0.850402	−	0.526134i	$-0.823641\pi$
$17$		−	17.8885i		−	1.05227i	0.850402	−	0.526134i	$-0.176359\pi$
$18$		0			0
$19$			20.0000i			1.05263i	0.850289	+	0.526316i	$0.176427\pi$
$19$			20.0000i			1.05263i	−0.850289	+	0.526316i	$0.823573\pi$
$20$		0			0
$21$	20.0000	+	17.8885i	0.952381	+	0.851835i
$22$		0			0
$23$			16.0000i			0.695652i	0.937559	+	0.347826i	$0.113080\pi$
$23$			16.0000i			0.695652i	−0.937559	+	0.347826i	$0.886920\pi$
$24$		0			0
$25$	−9.00000			−0.360000
$26$		0			0
$27$	15.6525	−	22.0000i	0.579721	−	0.814815i
$28$		0			0
$29$	52.0000			1.79310			0.896552	−	0.442939i	$-0.146064\pi$
$29$	52.0000			1.79310			0.896552	+	0.442939i	$0.146064\pi$
$30$		0			0
$31$	26.8328			0.865575			0.432787	−	0.901496i	$-0.357530\pi$
$31$	26.8328			0.865575			0.432787	+	0.901496i	$0.357530\pi$
$32$		0			0
$33$	−10.0000	−	8.94427i	−0.303030	−	0.271039i
$34$		0			0
$35$	−35.7771			−1.02220
$36$		0			0
$37$			53.6656i			1.45042i	0.688526	+	0.725211i	$0.258258\pi$
$37$			53.6656i			1.45042i	−0.688526	+	0.725211i	$0.741742\pi$
$38$		0			0
$39$	35.7771	−	40.0000i	0.917361	−	1.02564i
$40$		0			0
$41$			35.7771i			0.872612i	0.899798	+	0.436306i	$0.143713\pi$
$41$			35.7771i			0.872612i	−0.899798	+	0.436306i	$0.856287\pi$
$42$		0			0
$43$		−	36.0000i		−	0.837209i	−0.908169	−	0.418605i	$-0.862519\pi$
$43$		−	36.0000i		−	0.837209i	0.908169	−	0.418605i	$-0.137481\pi$
$44$		0			0
$45$	4.00000	+	35.7771i	0.0888889	+	0.795046i
$46$		0			0
$47$			64.0000i			1.36170i	0.732422	+	0.680851i	$0.238390\pi$
$47$			64.0000i			1.36170i	−0.732422	+	0.680851i	$0.761610\pi$
$48$		0			0
$49$	31.0000			0.632653
$50$		0			0
$51$	−35.7771	+	40.0000i	−0.701512	+	0.784314i
$52$		0			0
$53$	20.0000			0.377358			0.188679	−	0.982039i	$-0.439579\pi$
$53$	20.0000			0.377358			0.188679	+	0.982039i	$0.439579\pi$
$54$		0			0
$55$	17.8885			0.325246
$56$		0			0
$57$	40.0000	−	44.7214i	0.701754	−	0.784585i
$58$		0			0
$59$	102.859			1.74338			0.871688	−	0.490062i	$-0.163026\pi$
$59$	102.859			1.74338			0.871688	+	0.490062i	$0.163026\pi$
$60$		0			0
$61$			17.8885i			0.293255i	0.989192	+	0.146627i	$0.0468418\pi$
$61$			17.8885i			0.293255i	−0.989192	+	0.146627i	$0.953158\pi$
$62$		0			0
$63$	−8.94427	−	80.0000i	−0.141973	−	1.26984i
$64$		0			0
$65$			71.5542i			1.10083i
$66$		0			0
$67$			44.0000i			0.656716i	0.944553	+	0.328358i	$0.106495\pi$
$67$			44.0000i			0.656716i	−0.944553	+	0.328358i	$0.893505\pi$
$68$		0			0
$69$	32.0000	−	35.7771i	0.463768	−	0.518509i
$70$		0			0
$71$			80.0000i			1.12676i	0.826198	+	0.563380i	$0.190499\pi$
$71$			80.0000i			1.12676i	−0.826198	+	0.563380i	$0.809501\pi$
$72$		0			0
$73$	−50.0000			−0.684932			−0.342466	−	0.939530i	$-0.611262\pi$
$73$	−50.0000			−0.684932			−0.342466	+	0.939530i	$0.611262\pi$
$74$		0			0
$75$	20.1246	+	18.0000i	0.268328	+	0.240000i
$76$		0			0
$77$	−40.0000			−0.519481
$78$		0			0
$79$	−80.4984			−1.01897			−0.509484	−	0.860480i	$-0.670164\pi$
$79$	−80.4984			−1.01897			−0.509484	+	0.860480i	$0.670164\pi$
$80$		0			0
$81$	−79.0000	+	17.8885i	−0.975309	+	0.220846i
$82$		0			0
$83$	−102.859			−1.23927			−0.619633	−	0.784891i	$-0.712719\pi$
$83$	−102.859			−1.23927			−0.619633	+	0.784891i	$0.712719\pi$
$84$		0			0
$85$		−	71.5542i		−	0.841814i
$86$		0			0
$87$	−116.276	−	104.000i	−1.33650	−	1.19540i
$88$		0			0
$89$		−	160.997i		−	1.80895i	−0.426523	−	0.904477i	$-0.640262\pi$
$89$		−	160.997i		−	1.80895i	0.426523	−	0.904477i	$-0.359738\pi$
$90$		0			0
$91$		−	160.000i		−	1.75824i
$92$		0			0
$93$	−60.0000	−	53.6656i	−0.645161	−	0.577050i
$94$		0			0
$95$			80.0000i			0.842105i
$96$		0			0
$97$	50.0000			0.515464			0.257732	−	0.966216i	$-0.417025\pi$
$97$	50.0000			0.515464			0.257732	+	0.966216i	$0.417025\pi$
$98$		0			0
$99$	4.47214	+	40.0000i	0.0451731	+	0.404040i
$100$		0			0
$101$	−92.0000			−0.910891			−0.455446	−	0.890264i	$-0.650520\pi$
$101$	−92.0000			−0.910891			−0.455446	+	0.890264i	$0.650520\pi$
$102$		0			0
$103$	−44.7214			−0.434188			−0.217094	−	0.976151i	$-0.569658\pi$
$103$	−44.7214			−0.434188			−0.217094	+	0.976151i	$0.569658\pi$
$104$		0			0
$105$	80.0000	+	71.5542i	0.761905	+	0.681468i
$106$		0			0
$107$	−40.2492			−0.376161			−0.188080	−	0.982154i	$-0.560227\pi$
$107$	−40.2492			−0.376161			−0.188080	+	0.982154i	$0.560227\pi$
$108$		0			0
$109$		−	125.220i		−	1.14881i	−0.818573	−	0.574403i	$-0.805234\pi$
$109$		−	125.220i		−	1.14881i	0.818573	−	0.574403i	$-0.194766\pi$
$110$		0			0
$111$	107.331	−	120.000i	0.966948	−	1.08108i
$112$		0			0
$113$		−	35.7771i		−	0.316611i	−0.987390	−	0.158306i	$-0.949397\pi$
$113$		−	35.7771i		−	0.316611i	0.987390	−	0.158306i	$-0.0506031\pi$
$114$		0			0
$115$			64.0000i			0.556522i
$116$		0			0
$117$	−160.000	+	17.8885i	−1.36752	+	0.152894i
$118$		0			0
$119$			160.000i			1.34454i
$120$		0			0
$121$	−101.000			−0.834711
$122$		0			0
$123$	71.5542	−	80.0000i	0.581741	−	0.650407i
$124$		0			0
$125$	−136.000			−1.08800
$126$		0			0
$127$	26.8328			0.211282			0.105641	−	0.994404i	$-0.466311\pi$
$127$	26.8328			0.211282			0.105641	+	0.994404i	$0.466311\pi$
$128$		0			0
$129$	−72.0000	+	80.4984i	−0.558140	+	0.624019i
$130$		0			0
$131$	67.0820			0.512077			0.256038	−	0.966667i	$-0.417583\pi$
$131$	67.0820			0.512077			0.256038	+	0.966667i	$0.417583\pi$
$132$		0			0
$133$		−	178.885i		−	1.34500i
$134$		0			0
$135$	62.6099	−	88.0000i	0.463777	−	0.651852i
$136$		0			0
$137$			143.108i			1.04459i	0.852766	+	0.522293i	$0.174923\pi$
$137$			143.108i			1.04459i	−0.852766	+	0.522293i	$0.825077\pi$
$138$		0			0
$139$			260.000i			1.87050i	0.353983	+	0.935252i	$0.384827\pi$
$139$			260.000i			1.87050i	−0.353983	+	0.935252i	$0.615173\pi$
$140$		0			0
$141$	128.000	−	143.108i	0.907801	−	1.01495i
$142$		0			0
$143$			80.0000i			0.559441i
$144$		0			0
$145$	208.000			1.43448
$146$		0			0
$147$	−69.3181	−	62.0000i	−0.471552	−	0.421769i
$148$		0			0
$149$	−28.0000			−0.187919			−0.0939597	−	0.995576i	$-0.529952\pi$
$149$	−28.0000			−0.187919			−0.0939597	+	0.995576i	$0.529952\pi$
$150$		0			0
$151$	241.495			1.59931			0.799653	−	0.600462i	$-0.205017\pi$
$151$	241.495			1.59931			0.799653	+	0.600462i	$0.205017\pi$
$152$		0			0
$153$	160.000	−	17.8885i	1.04575	−	0.116919i
$154$		0			0
$155$	107.331			0.692460
$156$		0			0
$157$		−	53.6656i		−	0.341819i	−0.985287	−	0.170910i	$-0.945329\pi$
$157$		−	53.6656i		−	0.341819i	0.985287	−	0.170910i	$-0.0546706\pi$
$158$		0			0
$159$	−44.7214	−	40.0000i	−0.281266	−	0.251572i
$160$		0			0
$161$		−	143.108i		−	0.888872i
$162$		0			0
$163$		−	124.000i		−	0.760736i	−0.924835	−	0.380368i	$-0.875797\pi$
$163$		−	124.000i		−	0.760736i	0.924835	−	0.380368i	$-0.124203\pi$
$164$		0			0
$165$	−40.0000	−	35.7771i	−0.242424	−	0.216831i
$166$		0			0
$167$		−	16.0000i		−	0.0958084i	−0.998852	−	0.0479042i	$-0.984746\pi$
$167$		−	16.0000i		−	0.0958084i	0.998852	−	0.0479042i	$-0.0152542\pi$
$168$		0			0
$169$	−151.000			−0.893491
$170$		0			0
$171$	−178.885	+	20.0000i	−1.04611	+	0.116959i
$172$		0			0
$173$	−140.000			−0.809249			−0.404624	−	0.914483i	$-0.632598\pi$
$173$	−140.000			−0.809249			−0.404624	+	0.914483i	$0.632598\pi$
$174$		0			0
$175$	80.4984			0.459991
$176$		0			0
$177$	−230.000	−	205.718i	−1.29944	−	1.16225i
$178$		0			0
$179$	67.0820			0.374760			0.187380	−	0.982288i	$-0.440000\pi$
$179$	67.0820			0.374760			0.187380	+	0.982288i	$0.440000\pi$
$180$		0			0
$181$			125.220i			0.691822i	0.938267	+	0.345911i	$0.112430\pi$
$181$			125.220i			0.691822i	−0.938267	+	0.345911i	$0.887570\pi$
$182$		0			0
$183$	35.7771	−	40.0000i	0.195503	−	0.218579i
$184$		0			0
$185$			214.663i			1.16034i
$186$		0			0
$187$		−	80.0000i		−	0.427807i
$188$		0			0
$189$	−140.000	+	196.774i	−0.740741	+	1.04113i
$190$		0			0
$191$			160.000i			0.837696i	0.908056	+	0.418848i	$0.137566\pi$
$191$			160.000i			0.837696i	−0.908056	+	0.418848i	$0.862434\pi$
$192$		0			0
$193$	30.0000			0.155440			0.0777202	−	0.996975i	$-0.475236\pi$
$193$	30.0000			0.155440			0.0777202	+	0.996975i	$0.475236\pi$
$194$		0			0
$195$	143.108	−	160.000i	0.733889	−	0.820513i
$196$		0			0
$197$	180.000			0.913706			0.456853	−	0.889542i	$-0.348977\pi$
$197$	180.000			0.913706			0.456853	+	0.889542i	$0.348977\pi$
$198$		0			0
$199$	−259.384			−1.30344			−0.651718	−	0.758461i	$-0.725952\pi$
$199$	−259.384			−1.30344			−0.651718	+	0.758461i	$0.725952\pi$
$200$		0			0
$201$	88.0000	−	98.3870i	0.437811	−	0.489488i
$202$		0			0
$203$	−465.102			−2.29114
$204$		0			0
$205$			143.108i			0.698090i
$206$		0			0
$207$	−143.108	+	16.0000i	−0.691345	+	0.0772947i
$208$		0			0
$209$			89.4427i			0.427956i
$210$		0			0
$211$			60.0000i			0.284360i	0.989841	+	0.142180i	$0.0454112\pi$
$211$			60.0000i			0.284360i	−0.989841	+	0.142180i	$0.954589\pi$
$212$		0			0
$213$	160.000	−	178.885i	0.751174	−	0.839838i
$214$		0			0
$215$		−	144.000i		−	0.669767i
$216$		0			0
$217$	−240.000			−1.10599
$218$		0			0
$219$	111.803	+	100.000i	0.510518	+	0.456621i
$220$		0			0
$221$	320.000			1.44796
$222$		0			0
$223$	−152.053			−0.681850			−0.340925	−	0.940090i	$-0.610740\pi$
$223$	−152.053			−0.681850			−0.340925	+	0.940090i	$0.610740\pi$
$224$		0			0
$225$	−9.00000	−	80.4984i	−0.0400000	−	0.357771i
$226$		0			0
$227$	254.912			1.12296			0.561480	−	0.827491i	$-0.310232\pi$
$227$	254.912			1.12296			0.561480	+	0.827491i	$0.310232\pi$
$228$		0			0
$229$			196.774i			0.859275i	0.903001	+	0.429638i	$0.141359\pi$
$229$			196.774i			0.859275i	−0.903001	+	0.429638i	$0.858641\pi$
$230$		0			0
$231$	89.4427	+	80.0000i	0.387198	+	0.346320i
$232$		0			0
$233$		−	160.997i		−	0.690974i	−0.938424	−	0.345487i	$-0.887714\pi$
$233$		−	160.997i		−	0.690974i	0.938424	−	0.345487i	$-0.112286\pi$
$234$		0			0
$235$			256.000i			1.08936i
$236$		0			0
$237$	180.000	+	160.997i	0.759494	+	0.679312i
$238$		0			0
$239$		−	320.000i		−	1.33891i	−0.742852	−	0.669456i	$-0.766527\pi$
$239$		−	320.000i		−	1.33891i	0.742852	−	0.669456i	$-0.233473\pi$
$240$		0			0
$241$	318.000			1.31950			0.659751	−	0.751484i	$-0.270662\pi$
$241$	318.000			1.31950			0.659751	+	0.751484i	$0.270662\pi$
$242$		0			0
$243$	212.426	+	118.000i	0.874183	+	0.485597i
$244$		0			0
$245$	124.000			0.506122
$246$		0			0
$247$	−357.771			−1.44847
$248$		0			0
$249$	230.000	+	205.718i	0.923695	+	0.826178i
$250$		0			0
$251$	147.580			0.587970			0.293985	−	0.955810i	$-0.405018\pi$
$251$	147.580			0.587970			0.293985	+	0.955810i	$0.405018\pi$
$252$		0			0
$253$			71.5542i			0.282823i
$254$		0			0
$255$	−143.108	+	160.000i	−0.561209	+	0.627451i
$256$		0			0
$257$			107.331i			0.417631i	0.977955	+	0.208816i	$0.0669609\pi$
$257$			107.331i			0.417631i	−0.977955	+	0.208816i	$0.933039\pi$
$258$		0			0
$259$		−	480.000i		−	1.85328i
$260$		0			0
$261$	52.0000	+	465.102i	0.199234	+	1.78200i
$262$		0			0
$263$			144.000i			0.547529i	0.961797	+	0.273764i	$0.0882688\pi$
$263$			144.000i			0.547529i	−0.961797	+	0.273764i	$0.911731\pi$
$264$		0			0
$265$	80.0000			0.301887
$266$		0			0
$267$	−321.994	+	360.000i	−1.20597	+	1.34831i
$268$		0			0
$269$	132.000			0.490706			0.245353	−	0.969434i	$-0.421096\pi$
$269$	132.000			0.490706			0.245353	+	0.969434i	$0.421096\pi$
$270$		0			0
$271$	277.272			1.02315			0.511573	−	0.859240i	$-0.329063\pi$
$271$	277.272			1.02315			0.511573	+	0.859240i	$0.329063\pi$
$272$		0			0
$273$	−320.000	+	357.771i	−1.17216	+	1.31052i
$274$		0			0
$275$	−40.2492			−0.146361
$276$		0			0
$277$			268.328i			0.968694i	0.874876	+	0.484347i	$0.160943\pi$
$277$			268.328i			0.968694i	−0.874876	+	0.484347i	$0.839057\pi$
$278$		0			0
$279$	26.8328	+	240.000i	0.0961750	+	0.860215i
$280$		0			0
$281$		−	196.774i		−	0.700263i	−0.936701	−	0.350132i	$-0.886137\pi$
$281$		−	196.774i		−	0.700263i	0.936701	−	0.350132i	$-0.113863\pi$
$282$		0			0
$283$		−	76.0000i		−	0.268551i	−0.990944	−	0.134276i	$-0.957129\pi$
$283$		−	76.0000i		−	0.268551i	0.990944	−	0.134276i	$-0.0428707\pi$
$284$		0			0
$285$	160.000	−	178.885i	0.561404	−	0.627668i
$286$		0			0
$287$		−	320.000i		−	1.11498i
$288$		0			0
$289$	−31.0000			−0.107266
$290$		0			0
$291$	−111.803	−	100.000i	−0.384204	−	0.343643i
$292$		0			0
$293$	−140.000			−0.477816			−0.238908	−	0.971042i	$-0.576789\pi$
$293$	−140.000			−0.477816			−0.238908	+	0.971042i	$0.576789\pi$
$294$		0			0
$295$	411.437			1.39470
$296$		0			0
$297$	70.0000	−	98.3870i	0.235690	−	0.331269i
$298$		0			0
$299$	−286.217			−0.957246
$300$		0			0
$301$			321.994i			1.06975i
$302$		0			0
$303$	205.718	+	184.000i	0.678938	+	0.607261i
$304$		0			0
$305$			71.5542i			0.234604i
$306$		0			0
$307$		−	244.000i		−	0.794788i	−0.917648	−	0.397394i	$-0.869915\pi$
$307$		−	244.000i		−	0.794788i	0.917648	−	0.397394i	$-0.130085\pi$
$308$		0			0
$309$	100.000	+	89.4427i	0.323625	+	0.289459i
$310$		0			0
$311$		−	400.000i		−	1.28617i	−0.765793	−	0.643087i	$-0.777653\pi$
$311$		−	400.000i		−	1.28617i	0.765793	−	0.643087i	$-0.222347\pi$
$312$		0			0
$313$	−290.000			−0.926518			−0.463259	−	0.886223i	$-0.653320\pi$
$313$	−290.000			−0.926518			−0.463259	+	0.886223i	$0.653320\pi$
$314$		0			0
$315$	−35.7771	−	320.000i	−0.113578	−	1.01587i
$316$		0			0
$317$	420.000			1.32492			0.662461	−	0.749097i	$-0.269512\pi$
$317$	420.000			1.32492			0.662461	+	0.749097i	$0.269512\pi$
$318$		0			0
$319$	232.551			0.729000
$320$		0			0
$321$	90.0000	+	80.4984i	0.280374	+	0.250774i
$322$		0			0
$323$	357.771			1.10765
$324$		0			0
$325$		−	160.997i		−	0.495375i
$326$		0			0
$327$	−250.440	+	280.000i	−0.765870	+	0.856269i
$328$		0			0
$329$		−	572.433i		−	1.73992i
$330$		0			0
$331$			340.000i			1.02719i	0.858033	+	0.513595i	$0.171687\pi$
$331$			340.000i			1.02719i	−0.858033	+	0.513595i	$0.828313\pi$
$332$		0			0
$333$	−480.000	+	53.6656i	−1.44144	+	0.161158i
$334$		0			0
$335$			176.000i			0.525373i
$336$		0			0
$337$	110.000			0.326409			0.163205	−	0.986592i	$-0.447817\pi$
$337$	110.000			0.326409			0.163205	+	0.986592i	$0.447817\pi$
$338$		0			0
$339$	−71.5542	+	80.0000i	−0.211074	+	0.235988i
$340$		0			0
$341$	120.000			0.351906
$342$		0			0
$343$	160.997			0.469379
$344$		0			0
$345$	128.000	−	143.108i	0.371014	−	0.414807i
$346$		0			0
$347$	147.580			0.425304			0.212652	−	0.977128i	$-0.431790\pi$
$347$	147.580			0.425304			0.212652	+	0.977128i	$0.431790\pi$
$348$		0			0
$349$			17.8885i			0.0512566i	0.999672	+	0.0256283i	$0.00815863\pi$
$349$			17.8885i			0.0512566i	−0.999672	+	0.0256283i	$0.991841\pi$
$350$		0			0
$351$	393.548	+	280.000i	1.12122	+	0.797721i
$352$		0			0
$353$		−	214.663i		−	0.608109i	−0.952655	−	0.304055i	$-0.901659\pi$
$353$		−	214.663i		−	0.608109i	0.952655	−	0.304055i	$-0.0983405\pi$
$354$		0			0
$355$			320.000i			0.901408i
$356$		0			0
$357$	320.000	−	357.771i	0.896359	−	1.00216i
$358$		0			0
$359$		−	560.000i		−	1.55989i	−0.625849	−	0.779944i	$-0.715247\pi$
$359$		−	560.000i		−	1.55989i	0.625849	−	0.779944i	$-0.284753\pi$
$360$		0			0
$361$	−39.0000			−0.108033
$362$		0			0
$363$	225.843	+	202.000i	0.622157	+	0.556474i
$364$		0			0
$365$	−200.000			−0.547945
$366$		0			0
$367$	−617.155			−1.68162			−0.840810	−	0.541330i	$-0.817921\pi$
$367$	−617.155			−1.68162			−0.840810	+	0.541330i	$0.817921\pi$
$368$		0			0
$369$	−320.000	+	35.7771i	−0.867209	+	0.0969569i
$370$		0			0
$371$	−178.885			−0.482171
$372$		0			0
$373$		−	375.659i		−	1.00713i	−0.863957	−	0.503565i	$-0.832021\pi$
$373$		−	375.659i		−	1.00713i	0.863957	−	0.503565i	$-0.167979\pi$
$374$		0			0
$375$	304.105	+	272.000i	0.810947	+	0.725333i
$376$		0			0
$377$			930.204i			2.46739i
$378$		0			0
$379$		−	260.000i		−	0.686016i	−0.939333	−	0.343008i	$-0.888554\pi$
$379$		−	260.000i		−	0.686016i	0.939333	−	0.343008i	$-0.111446\pi$
$380$		0			0
$381$	−60.0000	−	53.6656i	−0.157480	−	0.140855i
$382$		0			0
$383$		−	544.000i		−	1.42037i	−0.704017	−	0.710183i	$-0.748612\pi$
$383$		−	544.000i		−	1.42037i	0.704017	−	0.710183i	$-0.251388\pi$
$384$		0			0
$385$	−160.000			−0.415584
$386$		0			0
$387$	321.994	−	36.0000i	0.832025	−	0.0930233i
$388$		0			0
$389$	−332.000			−0.853470			−0.426735	−	0.904377i	$-0.640336\pi$
$389$	−332.000			−0.853470			−0.426735	+	0.904377i	$0.640336\pi$
$390$		0			0
$391$	286.217			0.732012
$392$		0			0
$393$	−150.000	−	134.164i	−0.381679	−	0.341384i
$394$		0			0
$395$	−321.994			−0.815174
$396$		0			0
$397$		−	268.328i		−	0.675890i	−0.941166	−	0.337945i	$-0.890268\pi$
$397$		−	268.328i		−	0.675890i	0.941166	−	0.337945i	$-0.109732\pi$
$398$		0			0
$399$	−357.771	+	400.000i	−0.896669	+	1.00251i
$400$		0			0
$401$			160.997i			0.401489i	0.979644	+	0.200744i	$0.0643360\pi$
$401$			160.997i			0.401489i	−0.979644	+	0.200744i	$0.935664\pi$
$402$		0			0
$403$			480.000i			1.19107i
$404$		0			0
$405$	−316.000	+	71.5542i	−0.780247	+	0.176677i
$406$		0			0
$407$			240.000i			0.589681i
$408$		0			0
$409$	178.000			0.435208			0.217604	−	0.976037i	$-0.430176\pi$
$409$	178.000			0.435208			0.217604	+	0.976037i	$0.430176\pi$
$410$		0			0
$411$	286.217	−	320.000i	0.696391	−	0.778589i
$412$		0			0
$413$	−920.000			−2.22760
$414$		0			0
$415$	−411.437			−0.991413
$416$		0			0
$417$	520.000	−	581.378i	1.24700	−	1.39419i
$418$		0			0
$419$	−245.967			−0.587035			−0.293517	−	0.955954i	$-0.594826\pi$
$419$	−245.967			−0.587035			−0.293517	+	0.955954i	$0.594826\pi$
$420$		0			0
$421$			53.6656i			0.127472i	0.997967	+	0.0637359i	$0.0203015\pi$
$421$			53.6656i			0.127472i	−0.997967	+	0.0637359i	$0.979698\pi$
$422$		0			0
$423$	−572.433	+	64.0000i	−1.35327	+	0.151300i
$424$		0			0
$425$			160.997i			0.378816i
$426$		0			0
$427$		−	160.000i		−	0.374707i
$428$		0			0
$429$	160.000	−	178.885i	0.372960	−	0.416982i
$430$		0			0
$431$		−	320.000i		−	0.742459i	−0.928541	−	0.371230i	$-0.878936\pi$
$431$		−	320.000i		−	0.742459i	0.928541	−	0.371230i	$-0.121064\pi$
$432$		0			0
$433$	530.000			1.22402			0.612009	−	0.790851i	$-0.290362\pi$
$433$	530.000			1.22402			0.612009	+	0.790851i	$0.290362\pi$
$434$		0			0
$435$	−465.102	−	416.000i	−1.06920	−	0.956322i
$436$		0			0
$437$	−320.000			−0.732265
$438$		0			0
$439$	−474.046			−1.07983			−0.539916	−	0.841719i	$-0.681544\pi$
$439$	−474.046			−1.07983			−0.539916	+	0.841719i	$0.681544\pi$
$440$		0			0
$441$	31.0000	+	277.272i	0.0702948	+	0.628736i
$442$		0			0
$443$	505.351			1.14075			0.570374	−	0.821385i	$-0.306798\pi$
$443$	505.351			1.14075			0.570374	+	0.821385i	$0.306798\pi$
$444$		0			0
$445$		−	643.988i		−	1.44716i
$446$		0			0
$447$	62.6099	+	56.0000i	0.140067	+	0.125280i
$448$		0			0
$449$			268.328i			0.597613i	0.954314	+	0.298806i	$0.0965885\pi$
$449$			268.328i			0.597613i	−0.954314	+	0.298806i	$0.903412\pi$
$450$		0			0
$451$			160.000i			0.354767i
$452$		0			0
$453$	−540.000	−	482.991i	−1.19205	−	1.06620i
$454$		0			0
$455$		−	640.000i		−	1.40659i
$456$		0			0
$457$	210.000			0.459519			0.229759	−	0.973247i	$-0.426206\pi$
$457$	210.000			0.459519			0.229759	+	0.973247i	$0.426206\pi$
$458$		0			0
$459$	−393.548	−	280.000i	−0.857403	−	0.610022i
$460$		0			0
$461$	372.000			0.806941			0.403471	−	0.914993i	$-0.367804\pi$
$461$	372.000			0.806941			0.403471	+	0.914993i	$0.367804\pi$
$462$		0			0
$463$	169.941			0.367044			0.183522	−	0.983016i	$-0.441250\pi$
$463$	169.941			0.367044			0.183522	+	0.983016i	$0.441250\pi$
$464$		0			0
$465$	−240.000	−	214.663i	−0.516129	−	0.461640i
$466$		0			0
$467$	−460.630			−0.986360			−0.493180	−	0.869927i	$-0.664166\pi$
$467$	−460.630			−0.986360			−0.493180	+	0.869927i	$0.664166\pi$
$468$		0			0
$469$		−	393.548i		−	0.839121i
$470$		0			0
$471$	−107.331	+	120.000i	−0.227880	+	0.254777i
$472$		0			0
$473$		−	160.997i		−	0.340374i
$474$		0			0
$475$		−	180.000i		−	0.378947i
$476$		0			0
$477$	20.0000	+	178.885i	0.0419287	+	0.375022i
$478$		0			0
$479$			320.000i			0.668058i	0.942563	+	0.334029i	$0.108408\pi$
$479$			320.000i			0.668058i	−0.942563	+	0.334029i	$0.891592\pi$
$480$		0			0
$481$	−960.000			−1.99584
$482$		0			0
$483$	−286.217	+	320.000i	−0.592581	+	0.662526i
$484$		0			0
$485$	200.000			0.412371
$486$		0			0
$487$	62.6099			0.128562			0.0642812	−	0.997932i	$-0.479525\pi$
$487$	62.6099			0.128562			0.0642812	+	0.997932i	$0.479525\pi$
$488$		0			0
$489$	−248.000	+	277.272i	−0.507157	+	0.567019i
$490$		0			0
$491$	889.955			1.81254			0.906268	−	0.422704i	$-0.138919\pi$
$491$	889.955			1.81254			0.906268	+	0.422704i	$0.138919\pi$
$492$		0			0
$493$		−	930.204i		−	1.88682i
$494$		0			0
$495$	17.8885	+	160.000i	0.0361385	+	0.323232i
$496$		0			0
$497$		−	715.542i		−	1.43972i
$498$		0			0
$499$		−	100.000i		−	0.200401i	−0.994967	−	0.100200i	$-0.968052\pi$
$499$		−	100.000i		−	0.200401i	0.994967	−	0.100200i	$-0.0319484\pi$
$500$		0			0
$501$	−32.0000	+	35.7771i	−0.0638723	+	0.0714114i
$502$		0			0
$503$		−	16.0000i		−	0.0318091i	−0.999874	−	0.0159046i	$-0.994937\pi$
$503$		−	16.0000i		−	0.0318091i	0.999874	−	0.0159046i	$-0.00506280\pi$
$504$		0			0
$505$	−368.000			−0.728713
$506$		0			0
$507$	337.646	+	302.000i	0.665969	+	0.595661i
$508$		0			0
$509$	−332.000			−0.652259			−0.326130	−	0.945325i	$-0.605745\pi$
$509$	−332.000			−0.652259			−0.326130	+	0.945325i	$0.605745\pi$
$510$		0			0
$511$	447.214			0.875173
$512$		0			0
$513$	440.000	+	313.050i	0.857700	+	0.610233i
$514$		0			0
$515$	−178.885			−0.347350
$516$		0			0
$517$			286.217i			0.553611i
$518$		0			0
$519$	313.050	+	280.000i	0.603178	+	0.539499i
$520$		0			0
$521$			214.663i			0.412020i	0.978550	+	0.206010i	$0.0660480\pi$
$521$			214.663i			0.412020i	−0.978550	+	0.206010i	$0.933952\pi$
$522$		0			0
$523$			76.0000i			0.145315i	0.997357	+	0.0726577i	$0.0231481\pi$
$523$			76.0000i			0.145315i	−0.997357	+	0.0726577i	$0.976852\pi$
$524$		0			0
$525$	−180.000	−	160.997i	−0.342857	−	0.306661i
$526$		0			0
$527$		−	480.000i		−	0.910816i
$528$		0			0
$529$	273.000			0.516068
$530$		0			0
$531$	102.859	+	920.000i	0.193708	+	1.73258i
$532$		0			0
$533$	−640.000			−1.20075
$534$		0			0
$535$	−160.997			−0.300929
$536$		0			0
$537$	−150.000	−	134.164i	−0.279330	−	0.249840i
$538$		0			0
$539$	138.636			0.257210
$540$		0			0
$541$			1019.65i			1.88474i	0.334566	+	0.942372i	$0.391410\pi$
$541$			1019.65i			1.88474i	−0.334566	+	0.942372i	$0.608590\pi$
$542$		0			0
$543$	250.440	−	280.000i	0.461215	−	0.515654i
$544$		0			0
$545$		−	500.879i		−	0.919044i
$546$		0			0
$547$			244.000i			0.446069i	0.974810	+	0.223035i	$0.0715963\pi$
$547$			244.000i			0.446069i	−0.974810	+	0.223035i	$0.928404\pi$
$548$		0			0
$549$	−160.000	+	17.8885i	−0.291439	+	0.0325839i
$550$		0			0
$551$			1040.00i			1.88748i
$552$		0			0
$553$	720.000			1.30199
$554$		0			0
$555$	429.325	−	480.000i	0.773559	−	0.864865i
$556$		0			0
$557$	−60.0000			−0.107720			−0.0538600	−	0.998548i	$-0.517152\pi$
$557$	−60.0000			−0.107720			−0.0538600	+	0.998548i	$0.517152\pi$
$558$		0			0
$559$	643.988			1.15204
$560$		0			0
$561$	−160.000	+	178.885i	−0.285205	+	0.318869i
$562$		0			0
$563$	111.803			0.198585			0.0992925	−	0.995058i	$-0.468342\pi$
$563$	111.803			0.198585			0.0992925	+	0.995058i	$0.468342\pi$
$564$		0			0
$565$		−	143.108i		−	0.253289i
$566$		0			0
$567$	706.597	−	160.000i	1.24620	−	0.282187i
$568$		0			0
$569$		−	858.650i		−	1.50905i	−0.656271	−	0.754526i	$-0.727867\pi$
$569$		−	858.650i		−	1.50905i	0.656271	−	0.754526i	$-0.272133\pi$
$570$		0			0
$571$		−	940.000i		−	1.64623i	−0.567871	−	0.823117i	$-0.692233\pi$
$571$		−	940.000i		−	1.64623i	0.567871	−	0.823117i	$-0.307767\pi$
$572$		0			0
$573$	320.000	−	357.771i	0.558464	−	0.624382i
$574$		0			0
$575$		−	144.000i		−	0.250435i
$576$		0			0
$577$	−370.000			−0.641248			−0.320624	−	0.947207i	$-0.603893\pi$
$577$	−370.000			−0.641248			−0.320624	+	0.947207i	$0.603893\pi$
$578$		0			0
$579$	−67.0820	−	60.0000i	−0.115858	−	0.103627i
$580$		0			0
$581$	920.000			1.58348
$582$		0			0
$583$	89.4427			0.153418
$584$		0			0
$585$	−640.000	+	71.5542i	−1.09402	+	0.122315i
$586$		0			0
$587$	−469.574			−0.799956			−0.399978	−	0.916525i	$-0.630982\pi$
$587$	−469.574			−0.799956			−0.399978	+	0.916525i	$0.630982\pi$
$588$		0			0
$589$			536.656i			0.911131i
$590$		0			0
$591$	−402.492	−	360.000i	−0.681036	−	0.609137i
$592$		0			0
$593$			572.433i			0.965318i	0.875808	+	0.482659i	$0.160329\pi$
$593$			572.433i			0.965318i	−0.875808	+	0.482659i	$0.839671\pi$
$594$		0			0
$595$			640.000i			1.07563i
$596$		0			0
$597$	580.000	+	518.768i	0.971524	+	0.868958i
$598$		0			0
$599$		−	560.000i		−	0.934891i	−0.884022	−	0.467446i	$-0.845174\pi$
$599$		−	560.000i		−	0.934891i	0.884022	−	0.467446i	$-0.154826\pi$
$600$		0			0
$601$	302.000			0.502496			0.251248	−	0.967923i	$-0.419159\pi$
$601$	302.000			0.502496			0.251248	+	0.967923i	$0.419159\pi$
$602$		0			0
$603$	−393.548	+	44.0000i	−0.652650	+	0.0729685i
$604$		0			0
$605$	−404.000			−0.667769
$606$		0			0
$607$	−44.7214			−0.0736760			−0.0368380	−	0.999321i	$-0.511729\pi$
$607$	−44.7214			−0.0736760			−0.0368380	+	0.999321i	$0.511729\pi$
$608$		0			0
$609$	1040.00	+	930.204i	1.70772	+	1.52743i
$610$		0			0
$611$	−1144.87			−1.87376
$612$		0			0
$613$		−	447.214i		−	0.729549i	−0.931096	−	0.364775i	$-0.881146\pi$
$613$		−	447.214i		−	0.729549i	0.931096	−	0.364775i	$-0.118854\pi$
$614$		0			0
$615$	286.217	−	320.000i	0.465393	−	0.520325i
$616$		0			0
$617$			447.214i			0.724819i	0.932019	+	0.362410i	$0.118046\pi$
$617$			447.214i			0.724819i	−0.932019	+	0.362410i	$0.881954\pi$
$618$		0			0
$619$		−	780.000i		−	1.26010i	−0.776556	−	0.630048i	$-0.783035\pi$
$619$		−	780.000i		−	1.26010i	0.776556	−	0.630048i	$-0.216965\pi$
$620$		0			0
$621$	352.000	+	250.440i	0.566828	+	0.403284i
$622$		0			0
$623$			1440.00i			2.31140i
$624$		0			0
$625$	−319.000			−0.510400
$626$		0			0
$627$	178.885	−	200.000i	0.285304	−	0.318979i
$628$		0			0
$629$	960.000			1.52623
$630$		0			0
$631$	−80.4984			−0.127573			−0.0637864	−	0.997964i	$-0.520318\pi$
$631$	−80.4984			−0.127573			−0.0637864	+	0.997964i	$0.520318\pi$
$632$		0			0
$633$	120.000	−	134.164i	0.189573	−	0.211950i
$634$		0			0
$635$	107.331			0.169026
$636$		0			0
$637$			554.545i			0.870557i
$638$		0			0
$639$	−715.542	+	80.0000i	−1.11978	+	0.125196i
$640$		0			0
$641$			661.876i			1.03257i	0.856417	+	0.516284i	$0.172685\pi$
$641$			661.876i			1.03257i	−0.856417	+	0.516284i	$0.827315\pi$
$642$		0			0
$643$		−	844.000i		−	1.31260i	−0.754501	−	0.656299i	$-0.772121\pi$
$643$		−	844.000i		−	1.31260i	0.754501	−	0.656299i	$-0.227879\pi$
$644$		0			0
$645$	−288.000	+	321.994i	−0.446512	+	0.499215i
$646$		0			0
$647$		−	16.0000i		−	0.0247295i	−0.999924	−	0.0123648i	$-0.996064\pi$
$647$		−	16.0000i		−	0.0247295i	0.999924	−	0.0123648i	$-0.00393593\pi$
$648$		0			0
$649$	460.000			0.708783
$650$		0			0
$651$	536.656	+	480.000i	0.824357	+	0.737327i
$652$		0			0
$653$	660.000			1.01072			0.505360	−	0.862909i	$-0.331360\pi$
$653$	660.000			1.01072			0.505360	+	0.862909i	$0.331360\pi$
$654$		0			0
$655$	268.328			0.409661
$656$		0			0
$657$	−50.0000	−	447.214i	−0.0761035	−	0.680690i
$658$		0			0
$659$	210.190			0.318954			0.159477	−	0.987202i	$-0.449019\pi$
$659$	210.190			0.318954			0.159477	+	0.987202i	$0.449019\pi$
$660$		0			0
$661$			769.207i			1.16370i	0.813295	+	0.581851i	$0.197671\pi$
$661$			769.207i			1.16370i	−0.813295	+	0.581851i	$0.802329\pi$
$662$		0			0
$663$	−715.542	−	640.000i	−1.07925	−	0.965309i
$664$		0			0
$665$		−	715.542i		−	1.07600i
$666$		0			0
$667$			832.000i			1.24738i
$668$		0			0
$669$	340.000	+	304.105i	0.508221	+	0.454567i
$670$		0			0
$671$			80.0000i			0.119225i
$672$		0			0
$673$	−910.000			−1.35215			−0.676077	−	0.736831i	$-0.736321\pi$
$673$	−910.000			−1.35215			−0.676077	+	0.736831i	$0.736321\pi$
$674$		0			0
$675$	−140.872	+	198.000i	−0.208700	+	0.293333i
$676$		0			0
$677$	260.000			0.384047			0.192024	−	0.981390i	$-0.438495\pi$
$677$	260.000			0.384047			0.192024	+	0.981390i	$0.438495\pi$
$678$		0			0
$679$	−447.214			−0.658636
$680$		0			0
$681$	−570.000	−	509.823i	−0.837004	−	0.748639i
$682$		0			0
$683$	934.676			1.36849			0.684243	−	0.729254i	$-0.260133\pi$
$683$	934.676			1.36849			0.684243	+	0.729254i	$0.260133\pi$
$684$		0			0
$685$			572.433i			0.835669i
$686$		0			0
$687$	393.548	−	440.000i	0.572850	−	0.640466i
$688$		0			0
$689$			357.771i			0.519261i
$690$		0			0
$691$			100.000i			0.144718i	0.997379	+	0.0723589i	$0.0230527\pi$
$691$			100.000i			0.144718i	−0.997379	+	0.0723589i	$0.976947\pi$
$692$		0			0
$693$	−40.0000	−	357.771i	−0.0577201	−	0.516264i
$694$		0			0
$695$			1040.00i			1.49640i
$696$		0			0
$697$	640.000			0.918221
$698$		0			0
$699$	−321.994	+	360.000i	−0.460649	+	0.515021i
$700$		0			0
$701$	1028.00			1.46648			0.733238	−	0.679972i	$-0.238008\pi$
$701$	1028.00			1.46648			0.733238	+	0.679972i	$0.238008\pi$
$702$		0			0
$703$	−1073.31			−1.52676
$704$		0			0
$705$	512.000	−	572.433i	0.726241	−	0.811962i
$706$		0			0
$707$	822.873			1.16389
$708$		0			0
$709$			912.316i			1.28676i	0.765545	+	0.643382i	$0.222469\pi$
$709$			912.316i			1.28676i	−0.765545	+	0.643382i	$0.777531\pi$
$710$		0			0
$711$	−80.4984	−	720.000i	−0.113219	−	1.01266i
$712$		0			0
$713$			429.325i			0.602139i
$714$		0			0
$715$			320.000i			0.447552i
$716$		0			0
$717$	−640.000	+	715.542i	−0.892608	+	0.997966i
$718$		0			0
$719$		−	480.000i		−	0.667594i	−0.942645	−	0.333797i	$-0.891670\pi$
$719$		−	480.000i		−	0.667594i	0.942645	−	0.333797i	$-0.108330\pi$
$720$		0			0
$721$	400.000			0.554785
$722$		0			0
$723$	−711.070	−	636.000i	−0.983499	−	0.879668i
$724$		0			0
$725$	−468.000			−0.645517
$726$		0			0
$727$	241.495			0.332181			0.166090	−	0.986111i	$-0.446886\pi$
$727$	241.495			0.332181			0.166090	+	0.986111i	$0.446886\pi$
$728$		0			0
$729$	−239.000	−	688.709i	−0.327846	−	0.944731i
$730$		0			0
$731$	−643.988			−0.880968
$732$		0			0
$733$		−	1126.98i		−	1.53749i	−0.639557	−	0.768744i	$-0.720882\pi$
$733$		−	1126.98i		−	1.53749i	0.639557	−	0.768744i	$-0.279118\pi$
$734$		0			0
$735$	−277.272	−	248.000i	−0.377241	−	0.337415i
$736$		0			0
$737$			196.774i			0.266993i
$738$		0			0
$739$			1020.00i			1.38024i	0.723693	+	0.690122i	$0.242443\pi$
$739$			1020.00i			1.38024i	−0.723693	+	0.690122i	$0.757557\pi$
$740$		0			0
$741$	800.000	+	715.542i	1.07962	+	0.965643i
$742$		0			0
$743$			176.000i			0.236878i	0.992961	+	0.118439i	$0.0377889\pi$
$743$			176.000i			0.236878i	−0.992961	+	0.118439i	$0.962211\pi$
$744$		0			0
$745$	−112.000			−0.150336
$746$		0			0
$747$	−102.859	−	920.000i	−0.137696	−	1.23159i
$748$		0			0
$749$	360.000			0.480641
$750$		0			0
$751$	−1296.92			−1.72692			−0.863462	−	0.504414i	$-0.831708\pi$
$751$	−1296.92			−1.72692			−0.863462	+	0.504414i	$0.831708\pi$
$752$		0			0
$753$	−330.000	−	295.161i	−0.438247	−	0.391980i
$754$		0			0
$755$	965.981			1.27945
$756$		0			0
$757$			983.870i			1.29970i	0.760064	+	0.649848i	$0.225167\pi$
$757$			983.870i			1.29970i	−0.760064	+	0.649848i	$0.774833\pi$
$758$		0			0
$759$	143.108	−	160.000i	0.188549	−	0.210804i
$760$		0			0
$761$		−	107.331i		−	0.141040i	−0.997510	−	0.0705199i	$-0.977534\pi$
$761$		−	107.331i		−	0.141040i	0.997510	−	0.0705199i	$-0.0224658\pi$
$762$		0			0
$763$			1120.00i			1.46789i
$764$		0			0
$765$	640.000	−	71.5542i	0.836601	−	0.0935349i
$766$		0			0
$767$			1840.00i			2.39896i
$768$		0			0
$769$	−1378.00			−1.79194			−0.895969	−	0.444117i	$-0.853517\pi$
$769$	−1378.00			−1.79194			−0.895969	+	0.444117i	$0.853517\pi$
$770$		0			0
$771$	214.663	−	240.000i	0.278421	−	0.311284i
$772$		0			0
$773$	−1180.00			−1.52652			−0.763260	−	0.646091i	$-0.776402\pi$
$773$	−1180.00			−1.52652			−0.763260	+	0.646091i	$0.776402\pi$
$774$		0			0
$775$	−241.495			−0.311607
$776$		0			0
$777$	−960.000	+	1073.31i	−1.23552	+	1.38135i
$778$		0			0
$779$	−715.542			−0.918539
$780$		0			0
$781$			357.771i			0.458093i
$782$		0			0
$783$	813.929	−	1144.00i	1.03950	−	1.46105i
$784$		0			0
$785$		−	214.663i		−	0.273455i
$786$		0			0
$787$		−	1244.00i		−	1.58069i	−0.612665	−	0.790343i	$-0.709902\pi$
$787$		−	1244.00i		−	1.58069i	0.612665	−	0.790343i	$-0.290098\pi$
$788$		0			0
$789$	288.000	−	321.994i	0.365019	−	0.408104i
$790$		0			0
$791$			320.000i			0.404551i
$792$		0			0
$793$	−320.000			−0.403531
$794$		0			0
$795$	−178.885	−	160.000i	−0.225013	−	0.201258i
$796$		0			0
$797$	580.000			0.727729			0.363864	−	0.931452i	$-0.381457\pi$
$797$	580.000			0.727729			0.363864	+	0.931452i	$0.381457\pi$
$798$		0			0
$799$	1144.87			1.43287
$800$		0			0
$801$	1440.00	−	160.997i	1.79775	−	0.200995i
$802$		0			0
$803$	−223.607			−0.278464
$804$		0			0
$805$		−	572.433i		−	0.711097i
$806$		0			0
$807$	−295.161	−	264.000i	−0.365751	−	0.327138i
$808$		0			0
$809$			250.440i			0.309567i	0.987948	+	0.154783i	$0.0494680\pi$
$809$			250.440i			0.309567i	−0.987948	+	0.154783i	$0.950532\pi$
$810$		0			0
$811$		−	980.000i		−	1.20838i	−0.796839	−	0.604192i	$-0.793496\pi$
$811$		−	980.000i		−	1.20838i	0.796839	−	0.604192i	$-0.206504\pi$
$812$		0			0
$813$	−620.000	−	554.545i	−0.762608	−	0.682097i
$814$		0			0
$815$		−	496.000i		−	0.608589i
$816$		0			0
$817$	720.000			0.881273
$818$		0			0
$819$	1431.08	−	160.000i	1.74735	−	0.195360i
$820$		0			0
$821$	932.000			1.13520			0.567600	−	0.823304i	$-0.307872\pi$
$821$	932.000			1.13520			0.567600	+	0.823304i	$0.307872\pi$
$822$		0			0
$823$	1565.25			1.90188			0.950940	−	0.309375i	$-0.100120\pi$
$823$	1565.25			1.90188			0.950940	+	0.309375i	$0.100120\pi$
$824$		0			0
$825$	90.0000	+	80.4984i	0.109091	+	0.0975739i
$826$		0			0
$827$	−1542.89			−1.86564			−0.932822	−	0.360339i	$-0.882661\pi$
$827$	−1542.89			−1.86564			−0.932822	+	0.360339i	$0.882661\pi$
$828$		0			0
$829$		−	482.991i		−	0.582618i	−0.956629	−	0.291309i	$-0.905909\pi$
$829$		−	482.991i		−	0.582618i	0.956629	−	0.291309i	$-0.0940909\pi$
$830$		0			0
$831$	536.656	−	600.000i	0.645796	−	0.722022i
$832$		0			0
$833$		−	554.545i		−	0.665720i
$834$		0			0
$835$		−	64.0000i		−	0.0766467i
$836$		0			0
$837$	420.000	−	590.322i	0.501792	−	0.705283i
$838$		0			0
$839$			1360.00i			1.62098i	0.585754	+	0.810489i	$0.300798\pi$
$839$			1360.00i			1.62098i	−0.585754	+	0.810489i	$0.699202\pi$
$840$		0			0
$841$	1863.00			2.21522
$842$		0			0
$843$	−393.548	+	440.000i	−0.466842	+	0.521945i
$844$		0			0
$845$	−604.000			−0.714793
$846$		0			0
$847$	903.371			1.06655
$848$		0			0
$849$	−152.000	+	169.941i	−0.179034	+	0.200166i
$850$		0			0
$851$	−858.650			−1.00899
$852$		0			0
$853$		−	948.093i		−	1.11148i	−0.831356	−	0.555740i	$-0.812435\pi$
$853$		−	948.093i		−	1.11148i	0.831356	−	0.555740i	$-0.187565\pi$
$854$		0			0
$855$	−715.542	+	80.0000i	−0.836891	+	0.0935673i
$856$		0			0
$857$			858.650i			1.00193i	0.865469	+	0.500963i	$0.167021\pi$
$857$			858.650i			1.00193i	−0.865469	+	0.500963i	$0.832979\pi$
$858$		0			0
$859$			1340.00i			1.55995i	0.625809	+	0.779977i	$0.284769\pi$
$859$			1340.00i			1.55995i	−0.625809	+	0.779977i	$0.715231\pi$
$860$		0			0
$861$	−640.000	+	715.542i	−0.743322	+	0.831059i
$862$		0			0
$863$			224.000i			0.259560i	0.991543	+	0.129780i	$0.0414271\pi$
$863$			224.000i			0.259560i	−0.991543	+	0.129780i	$0.958573\pi$
$864$		0			0
$865$	−560.000			−0.647399
$866$		0			0
$867$	69.3181	+	62.0000i	0.0799517	+	0.0715110i
$868$		0			0
$869$	−360.000			−0.414269
$870$		0			0
$871$	−787.096			−0.903669
$872$		0			0
$873$	50.0000	+	447.214i	0.0572738	+	0.512272i
$874$		0			0
$875$	1216.42			1.39020
$876$		0			0
$877$			948.093i			1.08106i	0.841324	+	0.540532i	$0.181777\pi$
$877$			948.093i			1.08106i	−0.841324	+	0.540532i	$0.818223\pi$
$878$		0			0
$879$	313.050	+	280.000i	0.356143	+	0.318544i
$880$		0			0
$881$		−	1538.41i		−	1.74621i	−0.487528	−	0.873107i	$-0.662101\pi$
$881$		−	1538.41i		−	1.74621i	0.487528	−	0.873107i	$-0.337899\pi$
$882$		0			0
$883$		−	1164.00i		−	1.31823i	−0.752041	−	0.659117i	$-0.770930\pi$
$883$		−	1164.00i		−	1.31823i	0.752041	−	0.659117i	$-0.229070\pi$
$884$		0			0
$885$	−920.000	−	822.873i	−1.03955	−	0.929800i
$886$		0			0
$887$		−	336.000i		−	0.378805i	−0.981900	−	0.189402i	$-0.939345\pi$
$887$		−	336.000i		−	0.378805i	0.981900	−	0.189402i	$-0.0606551\pi$
$888$		0			0
$889$	−240.000			−0.269966
$890$		0			0
$891$	−353.299	+	80.0000i	−0.396519	+	0.0897868i
$892$		0			0
$893$	−1280.00			−1.43337
$894$		0			0
$895$	268.328			0.299808
$896$		0			0
$897$	640.000	+	572.433i	0.713489	+	0.638164i
$898$		0			0
$899$	1395.31			1.55206
$900$		0			0
$901$		−	357.771i		−	0.397082i
$902$		0			0
$903$	643.988	−	720.000i	0.713165	−	0.797342i
$904$		0			0
$905$			500.879i			0.553458i
$906$		0			0
$907$			764.000i			0.842337i	0.906982	+	0.421169i	$0.138380\pi$
$907$			764.000i			0.842337i	−0.906982	+	0.421169i	$0.861620\pi$
$908$		0			0
$909$	−92.0000	−	822.873i	−0.101210	−	0.905251i
$910$		0			0
$911$			1280.00i			1.40505i	0.711659	+	0.702525i	$0.247944\pi$
$911$			1280.00i			1.40505i	−0.711659	+	0.702525i	$0.752056\pi$
$912$		0			0
$913$	−460.000			−0.503834
$914$		0			0
$915$	143.108	−	160.000i	0.156403	−	0.174863i
$916$		0			0
$917$	−600.000			−0.654308
$918$		0			0
$919$	849.706			0.924598			0.462299	−	0.886724i	$-0.347025\pi$
$919$	849.706			0.924598			0.462299	+	0.886724i	$0.347025\pi$
$920$		0			0
$921$	−488.000	+	545.601i	−0.529859	+	0.592400i
$922$		0			0
$923$	−1431.08			−1.55047
$924$		0			0
$925$		−	482.991i		−	0.522152i
$926$		0			0
$927$	−44.7214	−	400.000i	−0.0482431	−	0.431499i
$928$		0			0
$929$			375.659i			0.404370i	0.979347	+	0.202185i	$0.0648042\pi$
$929$			375.659i			0.404370i	−0.979347	+	0.202185i	$0.935196\pi$
$930$		0			0
$931$			620.000i			0.665951i
$932$		0			0
$933$	−800.000	+	894.427i	−0.857449	+	0.958657i
$934$		0			0
$935$		−	320.000i		−	0.342246i
$936$		0			0
$937$	−510.000			−0.544290			−0.272145	−	0.962256i	$-0.587733\pi$
$937$	−510.000			−0.544290			−0.272145	+	0.962256i	$0.587733\pi$
$938$		0			0
$939$	648.460	+	580.000i	0.690585	+	0.617678i
$940$		0			0
$941$	−812.000			−0.862912			−0.431456	−	0.902134i	$-0.642000\pi$
$941$	−812.000			−0.862912			−0.431456	+	0.902134i	$0.642000\pi$
$942$		0			0
$943$	−572.433			−0.607034
$944$		0			0
$945$	−560.000	+	787.096i	−0.592593	+	0.832906i
$946$		0			0
$947$	1569.72			1.65757			0.828785	−	0.559566i	$-0.189032\pi$
$947$	1569.72			1.65757			0.828785	+	0.559566i	$0.189032\pi$
$948$		0			0
$949$		−	894.427i		−	0.942494i
$950$		0			0
$951$	−939.149	−	840.000i	−0.987538	−	0.883281i
$952$		0			0
$953$		−	1073.31i		−	1.12625i	−0.826373	−	0.563123i	$-0.809600\pi$
$953$		−	1073.31i		−	1.12625i	0.826373	−	0.563123i	$-0.190400\pi$
$954$		0			0
$955$			640.000i			0.670157i
$956$		0			0
$957$	−520.000	−	465.102i	−0.543365	−	0.486000i
$958$		0			0
$959$		−	1280.00i		−	1.33472i
$960$		0			0
$961$	−241.000			−0.250780
$962$		0			0
$963$	−40.2492	−	360.000i	−0.0417957	−	0.373832i
$964$		0			0
$965$	120.000			0.124352
$966$		0			0
$967$	420.381			0.434727			0.217363	−	0.976091i	$-0.430254\pi$
$967$	420.381			0.434727			0.217363	+	0.976091i	$0.430254\pi$
$968$		0			0
$969$	−800.000	−	715.542i	−0.825593	−	0.738433i
$970$		0			0
$971$	791.568			0.815209			0.407605	−	0.913159i	$-0.366364\pi$
$971$	791.568			0.815209			0.407605	+	0.913159i	$0.366364\pi$
$972$		0			0
$973$		−	2325.51i		−	2.39004i
$974$		0			0
$975$	−321.994	+	360.000i	−0.330250	+	0.369231i
$976$		0			0
$977$		−	661.876i		−	0.677458i	−0.940884	−	0.338729i	$-0.890003\pi$
$977$		−	661.876i		−	0.677458i	0.940884	−	0.338729i	$-0.109997\pi$
$978$		0			0
$979$		−	720.000i		−	0.735444i
$980$		0			0
$981$	1120.00	−	125.220i	1.14169	−	0.127645i
$982$		0			0
$983$			1776.00i			1.80671i	0.428889	+	0.903357i	$0.358905\pi$
$983$			1776.00i			1.80671i	−0.428889	+	0.903357i	$0.641095\pi$
$984$		0			0
$985$	720.000			0.730964
$986$		0			0
$987$	−1144.87	+	1280.00i	−1.15995	+	1.29686i
$988$		0			0
$989$	576.000			0.582406
$990$		0			0
$991$	−1368.47			−1.38090			−0.690451	−	0.723379i	$-0.742588\pi$
$991$	−1368.47			−1.38090			−0.690451	+	0.723379i	$0.742588\pi$
$992$		0			0
$993$	680.000	−	760.263i	0.684794	−	0.765622i
$994$		0			0
$995$	−1037.54			−1.04275
$996$		0			0
$997$		−	1234.31i		−	1.23802i	−0.785382	−	0.619012i	$-0.787533\pi$
$997$		−	1234.31i		−	1.23802i	0.785382	−	0.619012i	$-0.212467\pi$
$998$		0			0
$999$	1180.64	+	840.000i	1.18183	+	0.840841i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.h.f.65.1	yes	4
3.2	odd	2		384.3.h.e.65.3	yes	4
4.3	odd	2	inner	384.3.h.f.65.4	yes	4
8.3	odd	2		384.3.h.e.65.1	✓	4
8.5	even	2		384.3.h.e.65.4	yes	4
12.11	even	2		384.3.h.e.65.2	yes	4
16.3	odd	4		768.3.e.h.257.3		4
16.5	even	4		768.3.e.h.257.4		4
16.11	odd	4		768.3.e.m.257.2		4
16.13	even	4		768.3.e.m.257.1		4
24.5	odd	2	inner	384.3.h.f.65.2	yes	4
24.11	even	2	inner	384.3.h.f.65.3	yes	4
48.5	odd	4		768.3.e.h.257.1		4
48.11	even	4		768.3.e.m.257.3		4
48.29	odd	4		768.3.e.m.257.4		4
48.35	even	4		768.3.e.h.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.h.e.65.1	✓	4	8.3	odd	2
384.3.h.e.65.2	yes	4	12.11	even	2
384.3.h.e.65.3	yes	4	3.2	odd	2
384.3.h.e.65.4	yes	4	8.5	even	2
384.3.h.f.65.1	yes	4	1.1	even	1	trivial
384.3.h.f.65.2	yes	4	24.5	odd	2	inner
384.3.h.f.65.3	yes	4	24.11	even	2	inner
384.3.h.f.65.4	yes	4	4.3	odd	2	inner
768.3.e.h.257.1		4	48.5	odd	4
768.3.e.h.257.2		4	48.35	even	4
768.3.e.h.257.3		4	16.3	odd	4
768.3.e.h.257.4		4	16.5	even	4
768.3.e.m.257.1		4	16.13	even	4
768.3.e.m.257.2		4	16.11	odd	4
768.3.e.m.257.3		4	48.11	even	4
768.3.e.m.257.4		4	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-2.23607 - 2.00000i) q^{3} +4.00000 q^{5} -8.94427 q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})\)	\(q+(-2.23607 - 2.00000i) q^{3} +4.00000 q^{5} -8.94427 q^{7} +(1.00000 + 8.94427i) q^{9} +4.47214 q^{11} +17.8885i q^{13} +(-8.94427 - 8.00000i) q^{15} -17.8885i q^{17} +20.0000i q^{19} +(20.0000 + 17.8885i) q^{21} +16.0000i q^{23} -9.00000 q^{25} +(15.6525 - 22.0000i) q^{27} +52.0000 q^{29} +26.8328 q^{31} +(-10.0000 - 8.94427i) q^{33} -35.7771 q^{35} +53.6656i q^{37} +(35.7771 - 40.0000i) q^{39} +35.7771i q^{41} -36.0000i q^{43} +(4.00000 + 35.7771i) q^{45} +64.0000i q^{47} +31.0000 q^{49} +(-35.7771 + 40.0000i) q^{51} +20.0000 q^{53} +17.8885 q^{55} +(40.0000 - 44.7214i) q^{57} +102.859 q^{59} +17.8885i q^{61} +(-8.94427 - 80.0000i) q^{63} +71.5542i q^{65} +44.0000i q^{67} +(32.0000 - 35.7771i) q^{69} +80.0000i q^{71} -50.0000 q^{73} +(20.1246 + 18.0000i) q^{75} -40.0000 q^{77} -80.4984 q^{79} +(-79.0000 + 17.8885i) q^{81} -102.859 q^{83} -71.5542i q^{85} +(-116.276 - 104.000i) q^{87} -160.997i q^{89} -160.000i q^{91} +(-60.0000 - 53.6656i) q^{93} +80.0000i q^{95} +50.0000 q^{97} +(4.47214 + 40.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 16 * q^5 + 4 * q^9	\( 4 q + 16 q^{5} + 4 q^{9} + 80 q^{21} - 36 q^{25} + 208 q^{29} - 40 q^{33} + 16 q^{45} + 124 q^{49} + 80 q^{53} + 160 q^{57} + 128 q^{69} - 200 q^{73} - 160 q^{77} - 316 q^{81} - 240 q^{93} + 200 q^{97}+O(q^{100}) \) 4 * q + 16 * q^5 + 4 * q^9 + 80 * q^21 - 36 * q^25 + 208 * q^29 - 40 * q^33 + 16 * q^45 + 124 * q^49 + 80 * q^53 + 160 * q^57 + 128 * q^69 - 200 * q^73 - 160 * q^77 - 316 * q^81 - 240 * q^93 + 200 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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