LMFDB - Embedded newform 384.3.h.g.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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 $ x^16 + 28x^14 + 274x^12 + 1236x^10 + 2703x^8 + 2676x^6 + 946x^4 + 64*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$0.242964i$ of defining polynomial
Character	$\chi$	$=$	384.65
Dual form			384.3.h.g.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.75782 - 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} +O(q^{10})$	\(q+(-2.75782 - 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} -1.16436 q^{11} +14.4222i q^{13} +(10.1620 + 4.35127i) q^{15} +1.71276i q^{17} -28.8394i q^{19} +(-14.9985 - 6.42221i) q^{21} -35.4225i q^{23} -11.4222 q^{25} +(-9.43781 - 25.2968i) q^{27} +33.6818 q^{29} +55.5336 q^{31} +(3.21110 + 1.37496i) q^{33} -20.0400 q^{35} -30.4222i q^{37} +(17.0307 - 39.7738i) q^{39} -63.4196i q^{41} -43.0098i q^{43} +(-22.8867 - 24.0000i) q^{45} +61.5301i q^{47} -19.4222 q^{49} +(2.02254 - 4.72347i) q^{51} +56.3093 q^{53} +4.29046 q^{55} +(-34.0555 + 79.5338i) q^{57} -49.6407 q^{59} +4.73338i q^{61} +(33.7794 + 35.4225i) q^{63} -53.1430i q^{65} +7.08521i q^{67} +(-41.8293 + 97.6888i) q^{69} -96.9527i q^{71} +68.5332 q^{73} +(31.5003 + 13.4881i) q^{75} -6.33245 q^{77} -9.72900 q^{79} +(-3.84441 + 80.9087i) q^{81} +38.9156 q^{83} -6.31118i q^{85} +(-92.8883 - 39.7738i) q^{87} +0.675595i q^{89} +78.4358i q^{91} +(-153.151 - 65.5778i) q^{93} +106.268i q^{95} +86.1110 q^{97} +(-7.23199 - 7.58378i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) $ 16 * q - 16 * q^9 + 48 * q^25 - 64 * q^33 - 80 * q^49 + 32 * q^57 - 288 * q^73 + 400 * q^81 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.75782	−	1.18087i	−0.919272	−	0.393623i
$3$	−2.75782	−	1.18087i	−0.919272	−	0.393623i
$4$		0			0
$5$	−3.68481			−0.736961			−0.368481	−	0.929635i	$-0.620122\pi$
$5$	−3.68481			−0.736961			−0.368481	+	0.929635i	$0.620122\pi$
$6$		0			0
$7$	5.43855			0.776935			0.388468	−	0.921462i	$-0.373005\pi$
$7$	5.43855			0.776935			0.388468	+	0.921462i	$0.373005\pi$
$8$		0			0
$9$	6.21110	+	6.51323i	0.690123	+	0.723693i
$10$		0			0
$11$	−1.16436			−0.105851			−0.0529256	−	0.998598i	$-0.516855\pi$
$11$	−1.16436			−0.105851			−0.0529256	+	0.998598i	$0.516855\pi$
$12$		0			0
$13$			14.4222i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			14.4222i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	10.1620	+	4.35127i	0.677468	+	0.290085i
$16$		0			0
$17$			1.71276i			0.100750i	0.998730	+	0.0503752i	$0.0160417\pi$
$17$			1.71276i			0.100750i	−0.998730	+	0.0503752i	$0.983958\pi$
$18$		0			0
$19$		−	28.8394i		−	1.51786i	−0.651171	−	0.758931i	$-0.725722\pi$
$19$		−	28.8394i		−	1.51786i	0.651171	−	0.758931i	$-0.274278\pi$
$20$		0			0
$21$	−14.9985	−	6.42221i	−0.714215	−	0.305819i
$22$		0			0
$23$		−	35.4225i		−	1.54011i	−0.637978	−	0.770055i	$-0.720229\pi$
$23$		−	35.4225i		−	1.54011i	0.637978	−	0.770055i	$-0.279771\pi$
$24$		0			0
$25$	−11.4222			−0.456888
$26$		0			0
$27$	−9.43781	−	25.2968i	−0.349549	−	0.936918i
$28$		0			0
$29$	33.6818			1.16144			0.580721	−	0.814102i	$-0.302771\pi$
$29$	33.6818			1.16144			0.580721	+	0.814102i	$0.302771\pi$
$30$		0			0
$31$	55.5336			1.79141			0.895703	−	0.444654i	$-0.146673\pi$
$31$	55.5336			1.79141			0.895703	+	0.444654i	$0.146673\pi$
$32$		0			0
$33$	3.21110	+	1.37496i	0.0973061	+	0.0416655i
$34$		0			0
$35$	−20.0400			−0.572571
$36$		0			0
$37$		−	30.4222i		−	0.822222i	−0.911585	−	0.411111i	$-0.865141\pi$
$37$		−	30.4222i		−	0.822222i	0.911585	−	0.411111i	$-0.134859\pi$
$38$		0			0
$39$	17.0307	−	39.7738i	0.436685	−	1.01984i
$40$		0			0
$41$		−	63.4196i		−	1.54682i	−0.633907	−	0.773409i	$-0.718550\pi$
$41$		−	63.4196i		−	1.54682i	0.633907	−	0.773409i	$-0.281450\pi$
$42$		0			0
$43$		−	43.0098i		−	1.00023i	−0.865960	−	0.500114i	$-0.833291\pi$
$43$		−	43.0098i		−	1.00023i	0.865960	−	0.500114i	$-0.166709\pi$
$44$		0			0
$45$	−22.8867	−	24.0000i	−0.508594	−	0.533333i
$46$		0			0
$47$			61.5301i			1.30915i	0.755996	+	0.654576i	$0.227153\pi$
$47$			61.5301i			1.30915i	−0.755996	+	0.654576i	$0.772847\pi$
$48$		0			0
$49$	−19.4222			−0.396372
$50$		0			0
$51$	2.02254	−	4.72347i	0.0396577	−	0.0926171i
$52$		0			0
$53$	56.3093			1.06244			0.531219	−	0.847234i	$-0.321734\pi$
$53$	56.3093			1.06244			0.531219	+	0.847234i	$0.321734\pi$
$54$		0			0
$55$	4.29046			0.0780083
$56$		0			0
$57$	−34.0555	+	79.5338i	−0.597465	+	1.39533i
$58$		0			0
$59$	−49.6407			−0.841368			−0.420684	−	0.907207i	$-0.638210\pi$
$59$	−49.6407			−0.841368			−0.420684	+	0.907207i	$0.638210\pi$
$60$		0			0
$61$			4.73338i			0.0775965i	0.999247	+	0.0387982i	$0.0123530\pi$
$61$			4.73338i			0.0775965i	−0.999247	+	0.0387982i	$0.987647\pi$
$62$		0			0
$63$	33.7794	+	35.4225i	0.536181	+	0.562262i
$64$		0			0
$65$		−	53.1430i		−	0.817585i
$66$		0			0
$67$			7.08521i			0.105749i	0.998601	+	0.0528747i	$0.0168384\pi$
$67$			7.08521i			0.105749i	−0.998601	+	0.0528747i	$0.983162\pi$
$68$		0			0
$69$	−41.8293	+	97.6888i	−0.606222	+	1.41578i
$70$		0			0
$71$		−	96.9527i		−	1.36553i	−0.730638	−	0.682765i	$-0.760777\pi$
$71$		−	96.9527i		−	1.36553i	0.730638	−	0.682765i	$-0.239223\pi$
$72$		0			0
$73$	68.5332			0.938811			0.469406	−	0.882983i	$-0.344468\pi$
$73$	68.5332			0.938811			0.469406	+	0.882983i	$0.344468\pi$
$74$		0			0
$75$	31.5003	+	13.4881i	0.420005	+	0.179842i
$76$		0			0
$77$	−6.33245			−0.0822396
$78$		0			0
$79$	−9.72900			−0.123152			−0.0615760	−	0.998102i	$-0.519613\pi$
$79$	−9.72900			−0.123152			−0.0615760	+	0.998102i	$0.519613\pi$
$80$		0			0
$81$	−3.84441	+	80.9087i	−0.0474619	+	0.998873i
$82$		0			0
$83$	38.9156			0.468863			0.234431	−	0.972133i	$-0.424677\pi$
$83$	38.9156			0.468863			0.234431	+	0.972133i	$0.424677\pi$
$84$		0			0
$85$		−	6.31118i		−	0.0742492i
$86$		0			0
$87$	−92.8883	−	39.7738i	−1.06768	−	0.457170i
$88$		0			0
$89$			0.675595i			0.00759096i	0.999993	+	0.00379548i	$0.00120814\pi$
$89$			0.675595i			0.00759096i	−0.999993	+	0.00379548i	$0.998792\pi$
$90$		0			0
$91$			78.4358i			0.861932i
$92$		0			0
$93$	−153.151	−	65.5778i	−1.64679	−	0.705138i
$94$		0			0
$95$			106.268i			1.11861i
$96$		0			0
$97$	86.1110			0.887743			0.443871	−	0.896091i	$-0.353605\pi$
$97$	86.1110			0.887743			0.443871	+	0.896091i	$0.353605\pi$
$98$		0			0
$99$	−7.23199	−	7.58378i	−0.0730504	−	0.0766038i
$100$		0			0
$101$	109.452			1.08369			0.541843	−	0.840480i	$-0.317727\pi$
$101$	109.452			1.08369			0.541843	+	0.840480i	$0.317727\pi$
$102$		0			0
$103$	−160.014			−1.55353			−0.776767	−	0.629788i	$-0.783142\pi$
$103$	−160.014			−1.55353			−0.776767	+	0.629788i	$0.783142\pi$
$104$		0			0
$105$	55.2666	+	23.6646i	0.526349	+	0.225377i
$106$		0			0
$107$	184.345			1.72285			0.861423	−	0.507887i	$-0.169573\pi$
$107$	184.345			1.72285			0.861423	+	0.507887i	$0.169573\pi$
$108$		0			0
$109$			68.7334i			0.630582i	0.948995	+	0.315291i	$0.102102\pi$
$109$			68.7334i			0.630582i	−0.948995	+	0.315291i	$0.897898\pi$
$110$		0			0
$111$	−35.9246	+	83.8989i	−0.323645	+	0.755846i
$112$		0			0
$113$		−	71.3078i		−	0.631042i	−0.948919	−	0.315521i	$-0.897821\pi$
$113$		−	71.3078i		−	0.631042i	0.948919	−	0.315521i	$-0.102179\pi$
$114$		0			0
$115$			130.525i			1.13500i
$116$		0			0
$117$	−93.9352	+	89.5778i	−0.802865	+	0.765622i
$118$		0			0
$119$			9.31491i			0.0782766i
$120$		0			0
$121$	−119.644			−0.988796
$122$		0			0
$123$	−74.8901	+	174.900i	−0.608863	+	1.42195i
$124$		0			0
$125$	134.209			1.07367
$126$		0			0
$127$	−88.1648			−0.694211			−0.347106	−	0.937826i	$-0.612835\pi$
$127$	−88.1648			−0.694211			−0.347106	+	0.937826i	$0.612835\pi$
$128$		0			0
$129$	−50.7889	+	118.613i	−0.393712	+	0.919482i
$130$		0			0
$131$	−9.56071			−0.0729825			−0.0364913	−	0.999334i	$-0.511618\pi$
$131$	−9.56071			−0.0729825			−0.0364913	+	0.999334i	$0.511618\pi$
$132$		0			0
$133$		−	156.844i		−	1.17928i
$134$		0			0
$135$	34.7765	+	93.2138i	0.257604	+	0.690472i
$136$		0			0
$137$		−	150.504i		−	1.09857i	−0.835636	−	0.549284i	$-0.814901\pi$
$137$		−	150.504i		−	1.09857i	0.835636	−	0.549284i	$-0.185099\pi$
$138$		0			0
$139$			51.5907i			0.371156i	0.982630	+	0.185578i	$0.0594158\pi$
$139$			51.5907i			0.371156i	−0.982630	+	0.185578i	$0.940584\pi$
$140$		0			0
$141$	72.6590	−	169.689i	0.515312	−	1.20347i
$142$		0			0
$143$		−	16.7927i		−	0.117431i
$144$		0			0
$145$	−124.111			−0.855938
$146$		0			0
$147$	53.5629	+	22.9351i	0.364373	+	0.156021i
$148$		0			0
$149$	25.7936			0.173112			0.0865558	−	0.996247i	$-0.472414\pi$
$149$	25.7936			0.173112			0.0865558	+	0.996247i	$0.472414\pi$
$150$		0			0
$151$	−16.3156			−0.108051			−0.0540253	−	0.998540i	$-0.517205\pi$
$151$	−16.3156			−0.108051			−0.0540253	+	0.998540i	$0.517205\pi$
$152$		0			0
$153$	−11.1556	+	10.6381i	−0.0729124	+	0.0695302i
$154$		0			0
$155$	−204.630			−1.32020
$156$		0			0
$157$			251.489i			1.60184i	0.598772	+	0.800919i	$0.295655\pi$
$157$			251.489i			1.60184i	−0.598772	+	0.800919i	$0.704345\pi$
$158$		0			0
$159$	−155.291	−	66.4938i	−0.976670	−	0.418200i
$160$		0			0
$161$		−	192.647i		−	1.19657i
$162$		0			0
$163$			99.6915i			0.611604i	0.952095	+	0.305802i	$0.0989246\pi$
$163$			99.6915i			0.611604i	−0.952095	+	0.305802i	$0.901075\pi$
$164$		0			0
$165$	−11.8323	−	5.06646i	−0.0717108	−	0.0307058i
$166$		0			0
$167$			158.483i			0.948999i	0.880256	+	0.474499i	$0.157371\pi$
$167$			158.483i			0.948999i	−0.880256	+	0.474499i	$0.842629\pi$
$168$		0			0
$169$	−39.0000			−0.230769
$170$		0			0
$171$	187.838	−	179.124i	1.09847	−	1.04751i
$172$		0			0
$173$	117.340			0.678269			0.339134	−	0.940738i	$-0.389866\pi$
$173$	117.340			0.678269			0.339134	+	0.940738i	$0.389866\pi$
$174$		0			0
$175$	−62.1202			−0.354973
$176$		0			0
$177$	136.900	+	58.6191i	0.773446	+	0.331181i
$178$		0			0
$179$	144.265			0.805948			0.402974	−	0.915212i	$-0.367977\pi$
$179$	144.265			0.805948			0.402974	+	0.915212i	$0.367977\pi$
$180$		0			0
$181$		−	289.800i		−	1.60110i	−0.599263	−	0.800552i	$-0.704540\pi$
$181$		−	289.800i		−	1.60110i	0.599263	−	0.800552i	$-0.295460\pi$
$182$		0			0
$183$	5.58950	−	13.0538i	0.0305437	−	0.0713323i
$184$		0			0
$185$			112.100i			0.605946i
$186$		0			0
$187$		−	1.99427i		−	0.0106646i
$188$		0			0
$189$	−51.3280	−	137.578i	−0.271577	−	0.727925i
$190$		0			0
$191$		−	151.005i		−	0.790602i	−0.918552	−	0.395301i	$-0.870640\pi$
$191$		−	151.005i		−	0.790602i	0.918552	−	0.395301i	$-0.129360\pi$
$192$		0			0
$193$	−2.22205			−0.0115132			−0.00575661	−	0.999983i	$-0.501832\pi$
$193$	−2.22205			−0.0115132			−0.00575661	+	0.999983i	$0.501832\pi$
$194$		0			0
$195$	−62.7549	+	146.559i	−0.321820	+	0.751583i
$196$		0			0
$197$	−174.742			−0.887013			−0.443507	−	0.896271i	$-0.646266\pi$
$197$	−174.742			−0.887013			−0.443507	+	0.896271i	$0.646266\pi$
$198$		0			0
$199$	140.556			0.706312			0.353156	−	0.935564i	$-0.385109\pi$
$199$	140.556			0.706312			0.353156	+	0.935564i	$0.385109\pi$
$200$		0			0
$201$	8.36669	−	19.5397i	0.0416253	−	0.0972124i
$202$		0			0
$203$	183.180			0.902366
$204$		0			0
$205$			233.689i			1.13995i
$206$		0			0
$207$	230.715	−	220.013i	1.11457	−	1.06286i
$208$		0			0
$209$			33.5796i			0.160668i
$210$		0			0
$211$		−	295.479i		−	1.40038i	−0.713959	−	0.700188i	$-0.753100\pi$
$211$		−	295.479i		−	1.40038i	0.713959	−	0.700188i	$-0.246900\pi$
$212$		0			0
$213$	−114.488	+	267.378i	−0.537504	+	1.25529i
$214$		0			0
$215$			158.483i			0.737129i
$216$		0			0
$217$	302.022			1.39181
$218$		0			0
$219$	−189.002	−	80.9287i	−0.863023	−	0.369537i
$220$		0			0
$221$	−24.7017			−0.111773
$222$		0			0
$223$	7.43282			0.0333310			0.0166655	−	0.999861i	$-0.494695\pi$
$223$	7.43282			0.0333310			0.0166655	+	0.999861i	$0.494695\pi$
$224$		0			0
$225$	−70.9445	−	74.3955i	−0.315309	−	0.330647i
$226$		0			0
$227$	−305.995			−1.34799			−0.673997	−	0.738734i	$-0.735424\pi$
$227$	−305.995			−1.34799			−0.673997	+	0.738734i	$0.735424\pi$
$228$		0			0
$229$			62.6443i			0.273556i	0.990602	+	0.136778i	$0.0436747\pi$
$229$			62.6443i			0.273556i	−0.990602	+	0.136778i	$0.956325\pi$
$230$		0			0
$231$	17.4637	+	7.47779i	0.0756006	+	0.0323714i
$232$		0			0
$233$			270.130i			1.15936i	0.814845	+	0.579679i	$0.196822\pi$
$233$			270.130i			1.15936i	−0.814845	+	0.579679i	$0.803178\pi$
$234$		0			0
$235$		−	226.727i		−	0.964794i
$236$		0			0
$237$	26.8308	+	11.4887i	0.113210	+	0.0484754i
$238$		0			0
$239$		−	344.910i		−	1.44314i	−0.692342	−	0.721570i	$-0.743421\pi$
$239$		−	344.910i		−	1.44314i	0.692342	−	0.721570i	$-0.256579\pi$
$240$		0			0
$241$	23.9109			0.0992152			0.0496076	−	0.998769i	$-0.484203\pi$
$241$	23.9109			0.0992152			0.0496076	+	0.998769i	$0.484203\pi$
$242$		0			0
$243$	106.145	−	218.592i	0.436809	−	0.899554i
$244$		0			0
$245$	71.5671			0.292110
$246$		0			0
$247$	415.928			1.68392
$248$		0			0
$249$	−107.322	−	45.9542i	−0.431013	−	0.184555i
$250$		0			0
$251$	−395.470			−1.57558			−0.787788	−	0.615946i	$-0.788774\pi$
$251$	−395.470			−1.57558			−0.787788	+	0.615946i	$0.788774\pi$
$252$		0			0
$253$			41.2447i			0.163023i
$254$		0			0
$255$	−7.45267	+	17.4051i	−0.0292262	+	0.0682552i
$256$		0			0
$257$			451.825i			1.75807i	0.476753	+	0.879037i	$0.341814\pi$
$257$			451.825i			1.75807i	−0.476753	+	0.879037i	$0.658186\pi$
$258$		0			0
$259$		−	165.453i		−	0.638813i
$260$		0			0
$261$	209.201	+	219.378i	0.801538	+	0.840527i
$262$		0			0
$263$			167.798i			0.638014i	0.947752	+	0.319007i	$0.103349\pi$
$263$			167.798i			0.638014i	−0.947752	+	0.319007i	$0.896651\pi$
$264$		0			0
$265$	−207.489			−0.782976
$266$		0			0
$267$	0.797788	−	1.86317i	0.00298797	−	0.00697815i
$268$		0			0
$269$	−452.084			−1.68061			−0.840306	−	0.542113i	$-0.817625\pi$
$269$	−452.084			−1.68061			−0.840306	+	0.542113i	$0.817625\pi$
$270$		0			0
$271$	207.813			0.766837			0.383419	−	0.923575i	$-0.374747\pi$
$271$	207.813			0.766837			0.383419	+	0.923575i	$0.374747\pi$
$272$		0			0
$273$	92.6224	−	216.312i	0.339276	−	0.792350i
$274$		0			0
$275$	13.2996			0.0483622
$276$		0			0
$277$		−	216.111i		−	0.780184i	−0.920776	−	0.390092i	$-0.872443\pi$
$277$		−	216.111i		−	0.780184i	0.920776	−	0.390092i	$-0.127557\pi$
$278$		0			0
$279$	344.925	+	361.703i	1.23629	+	1.29643i
$280$		0			0
$281$		−	499.107i		−	1.77618i	−0.459669	−	0.888090i	$-0.652032\pi$
$281$		−	499.107i		−	1.77618i	0.459669	−	0.888090i	$-0.347968\pi$
$282$		0			0
$283$			206.468i			0.729569i	0.931092	+	0.364785i	$0.118857\pi$
$283$			206.468i			0.729569i	−0.931092	+	0.364785i	$0.881143\pi$
$284$		0			0
$285$	125.488	−	293.066i	0.440309	−	1.02830i
$286$		0			0
$287$		−	344.910i		−	1.20178i
$288$		0			0
$289$	286.066			0.989849
$290$		0			0
$291$	−237.478	−	101.686i	−0.816077	−	0.349436i
$292$		0			0
$293$	−74.6785			−0.254876			−0.127438	−	0.991847i	$-0.540675\pi$
$293$	−74.6785			−0.254876			−0.127438	+	0.991847i	$0.540675\pi$
$294$		0			0
$295$	182.916			0.620055
$296$		0			0
$297$	10.9891	+	29.4547i	0.0370002	+	0.0991740i
$298$		0			0
$299$	510.871			1.70860
$300$		0			0
$301$		−	233.911i		−	0.777113i
$302$		0			0
$303$	−301.849	−	129.249i	−0.996202	−	0.426563i
$304$		0			0
$305$		−	17.4416i		−	0.0571856i
$306$		0			0
$307$		−	293.485i		−	0.955977i	−0.878366	−	0.477988i	$-0.841366\pi$
$307$		−	293.485i		−	0.955977i	0.878366	−	0.477988i	$-0.158634\pi$
$308$		0			0
$309$	441.289	+	188.955i	1.42812	+	0.611506i
$310$		0			0
$311$		−	44.7374i		−	0.143850i	−0.997410	−	0.0719251i	$-0.977086\pi$
$311$		−	44.7374i		−	0.143850i	0.997410	−	0.0719251i	$-0.0229143\pi$
$312$		0			0
$313$	69.2666			0.221299			0.110650	−	0.993859i	$-0.464707\pi$
$313$	69.2666			0.221299			0.110650	+	0.993859i	$0.464707\pi$
$314$		0			0
$315$	−124.470	−	130.525i	−0.395144	−	0.414365i
$316$		0			0
$317$	12.0916			0.0381438			0.0190719	−	0.999818i	$-0.493929\pi$
$317$	12.0916			0.0381438			0.0190719	+	0.999818i	$0.493929\pi$
$318$		0			0
$319$	−39.2179			−0.122940
$320$		0			0
$321$	−508.389	−	217.687i	−1.58377	−	0.678151i
$322$		0			0
$323$	49.3949			0.152925
$324$		0			0
$325$		−	164.733i		−	0.506872i
$326$		0			0
$327$	81.1650	−	189.554i	0.248211	−	0.579676i
$328$		0			0
$329$			334.635i			1.01713i
$330$		0			0
$331$		−	468.516i		−	1.41545i	−0.706486	−	0.707727i	$-0.749721\pi$
$331$		−	468.516i		−	1.41545i	0.706486	−	0.707727i	$-0.250279\pi$
$332$		0			0
$333$	198.147	−	188.955i	0.595036	−	0.567434i
$334$		0			0
$335$		−	26.1076i		−	0.0779332i
$336$		0			0
$337$	−302.111			−0.896472			−0.448236	−	0.893915i	$-0.647948\pi$
$337$	−302.111			−0.896472			−0.448236	+	0.893915i	$0.647948\pi$
$338$		0			0
$339$	−84.2050	+	196.654i	−0.248392	+	0.580099i
$340$		0			0
$341$	−64.6613			−0.189623
$342$		0			0
$343$	−372.117			−1.08489
$344$		0			0
$345$	154.133	−	359.964i	0.446762	−	1.04337i
$346$		0			0
$347$	−419.740			−1.20963			−0.604813	−	0.796368i	$-0.706752\pi$
$347$	−419.740			−1.20963			−0.604813	+	0.796368i	$0.706752\pi$
$348$		0			0
$349$			27.0446i			0.0774916i	0.999249	+	0.0387458i	$0.0123363\pi$
$349$			27.0446i			0.0774916i	−0.999249	+	0.0387458i	$0.987664\pi$
$350$		0			0
$351$	364.836	−	136.114i	1.03942	−	0.387789i
$352$		0			0
$353$			47.9572i			0.135856i	0.997690	+	0.0679281i	$0.0216388\pi$
$353$			47.9572i			0.135856i	−0.997690	+	0.0679281i	$0.978361\pi$
$354$		0			0
$355$			357.252i			1.00634i
$356$		0			0
$357$	10.9997	−	25.6888i	0.0308114	−	0.0719575i
$358$		0			0
$359$			26.1076i			0.0727232i	0.999339	+	0.0363616i	$0.0115768\pi$
$359$			26.1076i			0.0727232i	−0.999339	+	0.0363616i	$0.988423\pi$
$360$		0			0
$361$	−470.711			−1.30391
$362$		0			0
$363$	329.957	+	141.284i	0.908972	+	0.389212i
$364$		0			0
$365$	−252.532			−0.691868
$366$		0			0
$367$	−101.338			−0.276126			−0.138063	−	0.990423i	$-0.544088\pi$
$367$	−101.338			−0.276126			−0.138063	+	0.990423i	$0.544088\pi$
$368$		0			0
$369$	413.066	−	393.905i	1.11942	−	1.06749i
$370$		0			0
$371$	306.241			0.825446
$372$		0			0
$373$			286.644i			0.768483i	0.923233	+	0.384242i	$0.125537\pi$
$373$			286.644i			0.768483i	−0.923233	+	0.384242i	$0.874463\pi$
$374$		0			0
$375$	−370.123	−	158.483i	−0.986995	−	0.422621i
$376$		0			0
$377$			485.766i			1.28850i
$378$		0			0
$379$			257.560i			0.679579i	0.940502	+	0.339789i	$0.110356\pi$
$379$			257.560i			0.679579i	−0.940502	+	0.339789i	$0.889644\pi$
$380$		0			0
$381$	243.142	+	104.111i	0.638169	+	0.273257i
$382$		0			0
$383$			311.325i			0.812859i	0.913682	+	0.406429i	$0.133226\pi$
$383$			311.325i			0.812859i	−0.913682	+	0.406429i	$0.866774\pi$
$384$		0			0
$385$	23.3338			0.0606074
$386$		0			0
$387$	280.133	−	267.138i	0.723858	−	0.690280i
$388$		0			0
$389$	13.1287			0.0337500			0.0168750	−	0.999858i	$-0.494628\pi$
$389$	13.1287			0.0337500			0.0168750	+	0.999858i	$0.494628\pi$
$390$		0			0
$391$	60.6702			0.155167
$392$		0			0
$393$	26.3667	+	11.2899i	0.0670908	+	0.0287276i
$394$		0			0
$395$	35.8495			0.0907582
$396$		0			0
$397$			498.244i			1.25502i	0.778607	+	0.627511i	$0.215926\pi$
$397$			498.244i			1.25502i	−0.778607	+	0.627511i	$0.784074\pi$
$398$		0			0
$399$	−185.213	+	432.548i	−0.464192	+	1.08408i
$400$		0			0
$401$			523.495i			1.30547i	0.757585	+	0.652736i	$0.226379\pi$
$401$			523.495i			1.30547i	−0.757585	+	0.652736i	$0.773621\pi$
$402$		0			0
$403$			800.916i			1.98739i
$404$		0			0
$405$	14.1659	−	298.133i	0.0349775	−	0.736131i
$406$		0			0
$407$			35.4225i			0.0870332i
$408$		0			0
$409$	581.822			1.42255			0.711274	−	0.702915i	$-0.248119\pi$
$409$	581.822			1.42255			0.711274	+	0.702915i	$0.248119\pi$
$410$		0			0
$411$	−177.725	+	415.062i	−0.432421	+	1.00988i
$412$		0			0
$413$	−269.973			−0.653688
$414$		0			0
$415$	−143.396			−0.345534
$416$		0			0
$417$	60.9218	−	142.278i	0.146095	−	0.341194i
$418$		0			0
$419$	−256.600			−0.612410			−0.306205	−	0.951966i	$-0.599059\pi$
$419$	−256.600			−0.612410			−0.306205	+	0.951966i	$0.599059\pi$
$420$		0			0
$421$		−	72.1110i		−	0.171285i	−0.996326	−	0.0856425i	$-0.972706\pi$
$421$		−	72.1110i		−	0.171285i	0.996326	−	0.0856425i	$-0.0272943\pi$
$422$		0			0
$423$	−400.760	+	382.170i	−0.947423	+	0.903475i
$424$		0			0
$425$		−	19.5635i		−	0.0460317i
$426$		0			0
$427$			25.7427i			0.0602874i
$428$		0			0
$429$	−19.8300	+	46.3112i	−0.0462237	+	0.107951i
$430$		0			0
$431$			714.091i			1.65682i	0.560119	+	0.828412i	$0.310755\pi$
$431$			714.091i			1.65682i	−0.560119	+	0.828412i	$0.689245\pi$
$432$		0			0
$433$	−192.200			−0.443880			−0.221940	−	0.975060i	$-0.571239\pi$
$433$	−192.200			−0.443880			−0.221940	+	0.975060i	$0.571239\pi$
$434$		0			0
$435$	342.275	+	146.559i	0.786840	+	0.336917i
$436$		0			0
$437$	−1021.56			−2.33768
$438$		0			0
$439$	−504.093			−1.14827			−0.574137	−	0.818759i	$-0.694662\pi$
$439$	−504.093			−1.14827			−0.574137	+	0.818759i	$0.694662\pi$
$440$		0			0
$441$	−120.633	−	126.501i	−0.273545	−	0.286851i
$442$		0			0
$443$	97.6254			0.220373			0.110187	−	0.993911i	$-0.464855\pi$
$443$	97.6254			0.220373			0.110187	+	0.993911i	$0.464855\pi$
$444$		0			0
$445$		−	2.48944i		−	0.00559424i
$446$		0			0
$447$	−71.1341	−	30.4589i	−0.159137	−	0.0681407i
$448$		0			0
$449$			306.869i			0.683450i	0.939800	+	0.341725i	$0.111011\pi$
$449$			306.869i			0.683450i	−0.939800	+	0.341725i	$0.888989\pi$
$450$		0			0
$451$			73.8435i			0.163733i
$452$		0			0
$453$	44.9955	+	19.2666i	0.0993279	+	0.0425312i
$454$		0			0
$455$		−	289.021i		−	0.635211i
$456$		0			0
$457$	117.378			0.256844			0.128422	−	0.991720i	$-0.459009\pi$
$457$	117.378			0.256844			0.128422	+	0.991720i	$0.459009\pi$
$458$		0			0
$459$	43.3273	−	16.1647i	0.0943949	−	0.0352172i
$460$		0			0
$461$	604.144			1.31051			0.655254	−	0.755409i	$-0.272562\pi$
$461$	604.144			1.31051			0.655254	+	0.755409i	$0.272562\pi$
$462$		0			0
$463$	−523.249			−1.13013			−0.565063	−	0.825048i	$-0.691148\pi$
$463$	−523.249			−1.13013			−0.565063	+	0.825048i	$0.691148\pi$
$464$		0			0
$465$	564.333	+	241.641i	1.21362	+	0.519659i
$466$		0			0
$467$	−219.340			−0.469679			−0.234840	−	0.972034i	$-0.575456\pi$
$467$	−219.340			−0.469679			−0.234840	+	0.972034i	$0.575456\pi$
$468$		0			0
$469$			38.5332i			0.0821604i
$470$		0			0
$471$	296.975	−	693.560i	0.630520	−	1.47253i
$472$		0			0
$473$			50.0791i			0.105875i
$474$		0			0
$475$			329.409i			0.693494i
$476$		0			0
$477$	349.743	+	366.755i	0.733213	+	0.768879i
$478$		0			0
$479$			302.010i			0.630501i	0.949008	+	0.315251i	$0.102089\pi$
$479$			302.010i			0.630501i	−0.949008	+	0.315251i	$0.897911\pi$
$480$		0			0
$481$	438.755			0.912173
$482$		0			0
$483$	−227.491	+	531.285i	−0.470995	+	1.09997i
$484$		0			0
$485$	−317.302			−0.654232
$486$		0			0
$487$	−208.115			−0.427340			−0.213670	−	0.976906i	$-0.568542\pi$
$487$	−208.115			−0.427340			−0.213670	+	0.976906i	$0.568542\pi$
$488$		0			0
$489$	117.722	−	274.931i	0.240741	−	0.562231i
$490$		0			0
$491$	584.291			1.19000			0.595001	−	0.803725i	$-0.297152\pi$
$491$	584.291			1.19000			0.595001	+	0.803725i	$0.297152\pi$
$492$		0			0
$493$			57.6888i			0.117016i
$494$		0			0
$495$	26.6485	+	27.9447i	0.0538353	+	0.0564540i
$496$		0			0
$497$		−	527.282i		−	1.06093i
$498$		0			0
$499$		−	352.161i		−	0.705733i	−0.935674	−	0.352867i	$-0.885207\pi$
$499$		−	352.161i		−	0.705733i	0.935674	−	0.352867i	$-0.114793\pi$
$500$		0			0
$501$	187.147	−	437.066i	0.373547	−	0.872388i
$502$		0			0
$503$		−	130.538i		−	0.259519i	−0.991545	−	0.129760i	$-0.958579\pi$
$503$		−	130.538i		−	0.259519i	0.991545	−	0.129760i	$-0.0414205\pi$
$504$		0			0
$505$	−403.310			−0.798634
$506$		0			0
$507$	107.555	+	46.0538i	0.212140	+	0.0908360i
$508$		0			0
$509$	−738.353			−1.45059			−0.725297	−	0.688436i	$-0.758298\pi$
$509$	−738.353			−1.45059			−0.725297	+	0.688436i	$0.758298\pi$
$510$		0			0
$511$	372.721			0.729396
$512$		0			0
$513$	−729.544	+	272.181i	−1.42211	+	0.530567i
$514$		0			0
$515$	589.621			1.14489
$516$		0			0
$517$		−	71.6435i		−	0.138575i
$518$		0			0
$519$	−323.603	−	138.564i	−0.623513	−	0.266982i
$520$		0			0
$521$			318.449i			0.611227i	0.952156	+	0.305613i	$0.0988615\pi$
$521$			318.449i			0.611227i	−0.952156	+	0.305613i	$0.901139\pi$
$522$		0			0
$523$			817.186i			1.56250i	0.624220	+	0.781249i	$0.285417\pi$
$523$			817.186i			1.56250i	−0.624220	+	0.781249i	$0.714583\pi$
$524$		0			0
$525$	171.316	+	73.3557i	0.326316	+	0.139725i
$526$		0			0
$527$			95.1155i			0.180485i
$528$		0			0
$529$	−725.755			−1.37194
$530$		0			0
$531$	−308.323	−	323.321i	−0.580647	−	0.608892i
$532$		0			0
$533$	914.650			1.71604
$534$		0			0
$535$	−679.274			−1.26967
$536$		0			0
$537$	−397.855	−	170.357i	−0.740885	−	0.317239i
$538$		0			0
$539$	22.6145			0.0419564
$540$		0			0
$541$		−	322.022i		−	0.595235i	−0.954685	−	0.297617i	$-0.903808\pi$
$541$		−	322.022i		−	0.595235i	0.954685	−	0.297617i	$-0.0961919\pi$
$542$		0			0
$543$	−342.215	+	799.215i	−0.630231	+	1.47185i
$544$		0			0
$545$		−	253.269i		−	0.464714i
$546$		0			0
$547$		−	276.716i		−	0.505880i	−0.967482	−	0.252940i	$-0.918602\pi$
$547$		−	276.716i		−	0.505880i	0.967482	−	0.252940i	$-0.0813975\pi$
$548$		0			0
$549$	−30.8296	+	29.3995i	−0.0561560	+	0.0535511i
$550$		0			0
$551$		−	971.364i		−	1.76291i
$552$		0			0
$553$	−52.9116			−0.0956811
$554$		0			0
$555$	132.375	−	309.151i	0.238514	−	0.557029i
$556$		0			0
$557$	579.033			1.03956			0.519778	−	0.854301i	$-0.326015\pi$
$557$	579.033			1.03956			0.519778	+	0.854301i	$0.326015\pi$
$558$		0			0
$559$	620.296			1.10965
$560$		0			0
$561$	−2.35497	+	5.49984i	−0.00419781	+	0.00980364i
$562$		0			0
$563$	494.751			0.878776			0.439388	−	0.898297i	$-0.355195\pi$
$563$	494.751			0.878776			0.439388	+	0.898297i	$0.355195\pi$
$564$		0			0
$565$			262.755i			0.465054i
$566$		0			0
$567$	−20.9080	+	440.026i	−0.0368748	+	0.776060i
$568$		0			0
$569$		−	229.386i		−	0.403138i	−0.979474	−	0.201569i	$-0.935396\pi$
$569$		−	229.386i		−	0.403138i	0.979474	−	0.201569i	$-0.0646041\pi$
$570$		0			0
$571$		−	451.354i		−	0.790462i	−0.918582	−	0.395231i	$-0.870665\pi$
$571$		−	451.354i		−	0.790462i	0.918582	−	0.395231i	$-0.129335\pi$
$572$		0			0
$573$	−178.317	+	416.444i	−0.311199	+	0.726779i
$574$		0			0
$575$			404.603i			0.703658i
$576$		0			0
$577$	792.644			1.37373			0.686867	−	0.726783i	$-0.258986\pi$
$577$	792.644			1.37373			0.686867	+	0.726783i	$0.258986\pi$
$578$		0			0
$579$	6.12801	+	2.62395i	0.0105838	+	0.00453186i
$580$		0			0
$581$	211.644			0.364276
$582$		0			0
$583$	−65.5645			−0.112461
$584$		0			0
$585$	346.133	−	330.077i	0.591680	−	0.564234i
$586$		0			0
$587$	−930.546			−1.58526			−0.792629	−	0.609704i	$-0.791288\pi$
$587$	−930.546			−1.58526			−0.792629	+	0.609704i	$0.791288\pi$
$588$		0			0
$589$		−	1601.55i		−	2.71911i
$590$		0			0
$591$	481.905	+	206.347i	0.815407	+	0.349148i
$592$		0			0
$593$		−	147.392i		−	0.248554i	−0.992248	−	0.124277i	$-0.960339\pi$
$593$		−	147.392i		−	0.248554i	0.992248	−	0.124277i	$-0.0396611\pi$
$594$		0			0
$595$		−	34.3236i		−	0.0576868i
$596$		0			0
$597$	−387.628	−	165.978i	−0.649293	−	0.278020i
$598$		0			0
$599$		−	423.233i		−	0.706566i	−0.935516	−	0.353283i	$-0.885065\pi$
$599$		−	423.233i		−	0.706566i	0.935516	−	0.353283i	$-0.114935\pi$
$600$		0			0
$601$	145.156			0.241523			0.120762	−	0.992682i	$-0.461466\pi$
$601$	145.156			0.241523			0.120762	+	0.992682i	$0.461466\pi$
$602$		0			0
$603$	−46.1476	+	44.0069i	−0.0765300	+	0.0729800i
$604$		0			0
$605$	440.866			0.728704
$606$		0			0
$607$	503.791			0.829968			0.414984	−	0.909829i	$-0.363787\pi$
$607$	503.791			0.829968			0.414984	+	0.909829i	$0.363787\pi$
$608$		0			0
$609$	−505.177	−	216.312i	−0.829520	−	0.355192i
$610$		0			0
$611$	−887.400			−1.45237
$612$		0			0
$613$			245.400i			0.400326i	0.979763	+	0.200163i	$0.0641471\pi$
$613$			245.400i			0.400326i	−0.979763	+	0.200163i	$0.935853\pi$
$614$		0			0
$615$	275.956	−	644.471i	0.448708	−	1.04792i
$616$		0			0
$617$		−	87.0366i		−	0.141064i	−0.997510	−	0.0705321i	$-0.977530\pi$
$617$		−	87.0366i		−	0.141064i	0.997510	−	0.0705321i	$-0.0224697\pi$
$618$		0			0
$619$		−	203.477i		−	0.328718i	−0.986401	−	0.164359i	$-0.947444\pi$
$619$		−	203.477i		−	0.328718i	0.986401	−	0.164359i	$-0.0525556\pi$
$620$		0			0
$621$	−896.076	+	334.311i	−1.44296	+	0.538343i
$622$		0			0
$623$			3.67426i			0.00589768i
$624$		0			0
$625$	−208.978			−0.334365
$626$		0			0
$627$	39.6530	−	92.6063i	0.0632425	−	0.147697i
$628$		0			0
$629$	52.1059			0.0828392
$630$		0			0
$631$	975.796			1.54643			0.773214	−	0.634145i	$-0.218648\pi$
$631$	975.796			1.54643			0.773214	+	0.634145i	$0.218648\pi$
$632$		0			0
$633$	−348.922	+	814.877i	−0.551219	+	1.28733i
$634$		0			0
$635$	324.870			0.511607
$636$		0			0
$637$		−	280.111i		−	0.439735i
$638$		0			0
$639$	631.475	−	602.183i	0.988224	−	0.942383i
$640$		0			0
$641$		−	797.821i		−	1.24465i	−0.782759	−	0.622325i	$-0.786188\pi$
$641$		−	797.821i		−	1.24465i	0.782759	−	0.622325i	$-0.213812\pi$
$642$		0			0
$643$		−	94.7058i		−	0.147287i	−0.997285	−	0.0736437i	$-0.976537\pi$
$643$		−	94.7058i		−	0.147287i	0.997285	−	0.0736437i	$-0.0234628\pi$
$644$		0			0
$645$	187.147	−	437.066i	0.290151	−	0.677622i
$646$		0			0
$647$			367.344i			0.567765i	0.958859	+	0.283882i	$0.0916225\pi$
$647$			367.344i			0.567765i	−0.958859	+	0.283882i	$0.908378\pi$
$648$		0			0
$649$	57.7998			0.0890599
$650$		0			0
$651$	−832.921	−	356.648i	−1.27945	−	0.547846i
$652$		0			0
$653$	−680.024			−1.04138			−0.520692	−	0.853745i	$-0.674326\pi$
$653$	−680.024			−1.04138			−0.520692	+	0.853745i	$0.674326\pi$
$654$		0			0
$655$	35.2294			0.0537853
$656$		0			0
$657$	425.667	+	446.373i	0.647895	+	0.679411i
$658$		0			0
$659$	−687.246			−1.04286			−0.521431	−	0.853293i	$-0.674602\pi$
$659$	−687.246			−1.04286			−0.521431	+	0.853293i	$0.674602\pi$
$660$		0			0
$661$			1121.58i			1.69679i	0.529364	+	0.848395i	$0.322431\pi$
$661$			1121.58i			1.69679i	−0.529364	+	0.848395i	$0.677569\pi$
$662$		0			0
$663$	68.1229	+	29.1695i	0.102749	+	0.0439962i
$664$		0			0
$665$			577.941i			0.869085i
$666$		0			0
$667$		−	1193.10i		−	1.78875i
$668$		0			0
$669$	−20.4984	−	8.77718i	−0.0306403	−	0.0131198i
$670$		0			0
$671$		−	5.51138i		−	0.00821369i
$672$		0			0
$673$	563.177			0.836816			0.418408	−	0.908259i	$-0.362588\pi$
$673$	563.177			0.836816			0.418408	+	0.908259i	$0.362588\pi$
$674$		0			0
$675$	107.801	+	288.945i	0.159705	+	0.428067i
$676$		0			0
$677$	998.773			1.47529			0.737646	−	0.675188i	$-0.235937\pi$
$677$	998.773			1.47529			0.737646	+	0.675188i	$0.235937\pi$
$678$		0			0
$679$	468.319			0.689719
$680$		0			0
$681$	843.877	+	361.339i	1.23917	+	0.530601i
$682$		0			0
$683$	147.020			0.215257			0.107628	−	0.994191i	$-0.465674\pi$
$683$	147.020			0.215257			0.107628	+	0.994191i	$0.465674\pi$
$684$		0			0
$685$			554.577i			0.809601i
$686$		0			0
$687$	73.9746	−	172.761i	0.107678	−	0.251472i
$688$		0			0
$689$			812.104i			1.17867i
$690$		0			0
$691$		−	862.689i		−	1.24846i	−0.781239	−	0.624232i	$-0.785412\pi$
$691$		−	862.689i		−	1.24846i	0.781239	−	0.624232i	$-0.214588\pi$
$692$		0			0
$693$	−39.3315	−	41.2447i	−0.0567554	−	0.0595162i
$694$		0			0
$695$		−	190.102i		−	0.273528i
$696$		0			0
$697$	108.622			0.155843
$698$		0			0
$699$	318.988	−	744.970i	0.456349	−	1.06576i
$700$		0			0
$701$	−468.489			−0.668315			−0.334158	−	0.942517i	$-0.608452\pi$
$701$	−468.489			−0.668315			−0.334158	+	0.942517i	$0.608452\pi$
$702$		0			0
$703$	−877.358			−1.24802
$704$		0			0
$705$	−267.734	+	625.270i	−0.379765	+	0.886908i
$706$		0			0
$707$	595.261			0.841954
$708$		0			0
$709$			1020.60i			1.43949i	0.694238	+	0.719745i	$0.255741\pi$
$709$			1020.60i			1.43949i	−0.694238	+	0.719745i	$0.744259\pi$
$710$		0			0
$711$	−60.4278	−	63.3673i	−0.0849899	−	0.0891241i
$712$		0			0
$713$		−	1967.14i		−	2.75896i
$714$		0			0
$715$			61.8778i			0.0865424i
$716$		0			0
$717$	−407.294	+	951.199i	−0.568052	+	1.32664i
$718$		0			0
$719$		−	477.286i		−	0.663819i	−0.943311	−	0.331909i	$-0.892307\pi$
$719$		−	477.286i		−	0.663819i	0.943311	−	0.331909i	$-0.107693\pi$
$720$		0			0
$721$	−870.244			−1.20700
$722$		0			0
$723$	−65.9418	−	28.2356i	−0.0912058	−	0.0390534i
$724$		0			0
$725$	−384.721			−0.530649
$726$		0			0
$727$	1042.27			1.43365			0.716827	−	0.697251i	$-0.245594\pi$
$727$	1042.27			1.43365			0.716827	+	0.697251i	$0.245594\pi$
$728$		0			0
$729$	−550.855	+	477.493i	−0.755631	+	0.654997i
$730$		0			0
$731$	73.6654			0.100773
$732$		0			0
$733$		−	952.333i		−	1.29923i	−0.760265	−	0.649613i	$-0.774931\pi$
$733$		−	952.333i		−	1.29923i	0.760265	−	0.649613i	$-0.225069\pi$
$734$		0			0
$735$	−197.369	−	84.5112i	−0.268529	−	0.114981i
$736$		0			0
$737$		−	8.24976i		−	0.0111937i
$738$		0			0
$739$		−	25.2442i		−	0.0341599i	−0.999854	−	0.0170800i	$-0.994563\pi$
$739$		−	25.2442i		−	0.0341599i	0.999854	−	0.0170800i	$-0.00543698\pi$
$740$		0			0
$741$	−1147.05	−	491.156i	−1.54798	−	0.662828i
$742$		0			0
$743$			706.613i			0.951027i	0.879708	+	0.475514i	$0.157738\pi$
$743$			706.613i			0.951027i	−0.879708	+	0.475514i	$0.842262\pi$
$744$		0			0
$745$	−95.0446			−0.127577
$746$		0			0
$747$	241.709	+	253.466i	0.323573	+	0.339313i
$748$		0			0
$749$	1002.57			1.33854
$750$		0			0
$751$	1414.08			1.88293			0.941466	−	0.337108i	$-0.109449\pi$
$751$	1414.08			1.88293			0.941466	+	0.337108i	$0.109449\pi$
$752$		0			0
$753$	1090.63	+	466.997i	1.44838	+	0.620182i
$754$		0			0
$755$	60.1200			0.0796291
$756$		0			0
$757$		−	309.177i		−	0.408425i	−0.978927	−	0.204212i	$-0.934537\pi$
$757$		−	309.177i		−	0.408425i	0.978927	−	0.204212i	$-0.0654633\pi$
$758$		0			0
$759$	48.7046	−	113.745i	0.0641694	−	0.149862i
$760$		0			0
$761$			400.757i			0.526618i	0.964712	+	0.263309i	$0.0848139\pi$
$761$			400.757i			0.526618i	−0.964712	+	0.263309i	$0.915186\pi$
$762$		0			0
$763$			373.810i			0.489921i
$764$		0			0
$765$	41.1062	−	39.1994i	0.0537336	−	0.0512410i
$766$		0			0
$767$		−	715.928i		−	0.933414i
$768$		0			0
$769$	804.533			1.04621			0.523104	−	0.852269i	$-0.324774\pi$
$769$	804.533			1.04621			0.523104	+	0.852269i	$0.324774\pi$
$770$		0			0
$771$	533.546	−	1246.05i	0.692018	−	1.61615i
$772$		0			0
$773$	−122.227			−0.158120			−0.0790599	−	0.996870i	$-0.525192\pi$
$773$	−122.227			−0.158120			−0.0790599	+	0.996870i	$0.525192\pi$
$774$		0			0
$775$	−634.316			−0.818472
$776$		0			0
$777$	−195.378	+	456.288i	−0.251451	+	0.587243i
$778$		0			0
$779$	−1828.98			−2.34786
$780$		0			0
$781$			112.888i			0.144543i
$782$		0			0
$783$	−317.883	−	852.042i	−0.405981	−	1.08818i
$784$		0			0
$785$		−	926.687i		−	1.18049i
$786$		0			0
$787$			574.295i			0.729727i	0.931061	+	0.364863i	$0.118884\pi$
$787$			574.295i			0.729727i	−0.931061	+	0.364863i	$0.881116\pi$
$788$		0			0
$789$	198.147	−	462.755i	0.251137	−	0.586509i
$790$		0			0
$791$		−	387.811i		−	0.490279i
$792$		0			0
$793$	−68.2658			−0.0860856
$794$		0			0
$795$	572.216	+	245.017i	0.719768	+	0.308197i
$796$		0			0
$797$	1000.03			1.25474			0.627371	−	0.778721i	$-0.284131\pi$
$797$	1000.03			1.25474			0.627371	+	0.778721i	$0.284131\pi$
$798$		0			0
$799$	−105.386			−0.131898
$800$		0			0
$801$	−4.40031	+	4.19619i	−0.00549352	+	0.00523869i
$802$		0			0
$803$	−79.7976			−0.0993744
$804$		0			0
$805$			709.867i			0.881822i
$806$		0			0
$807$	1246.77	+	533.852i	1.54494	+	0.661527i
$808$		0			0
$809$		−	583.032i		−	0.720682i	−0.932821	−	0.360341i	$-0.882660\pi$
$809$		−	583.032i		−	0.720682i	0.932821	−	0.360341i	$-0.117340\pi$
$810$		0			0
$811$		−	911.183i		−	1.12353i	−0.827297	−	0.561765i	$-0.810122\pi$
$811$		−	911.183i		−	1.12353i	0.827297	−	0.561765i	$-0.189878\pi$
$812$		0			0
$813$	−573.110	−	245.400i	−0.704932	−	0.301844i
$814$		0			0
$815$		−	367.344i		−	0.450728i
$816$		0			0
$817$	−1240.38			−1.51821
$818$		0			0
$819$	−510.871	+	487.173i	−0.623774	+	0.594839i
$820$		0			0
$821$	914.486			1.11387			0.556934	−	0.830557i	$-0.311978\pi$
$821$	914.486			1.11387			0.556934	+	0.830557i	$0.311978\pi$
$822$		0			0
$823$	−686.707			−0.834395			−0.417197	−	0.908816i	$-0.636988\pi$
$823$	−686.707			−0.834395			−0.417197	+	0.908816i	$0.636988\pi$
$824$		0			0
$825$	−36.6779	−	15.7051i	−0.0444580	−	0.0190365i
$826$		0			0
$827$	1107.30			1.33893			0.669466	−	0.742843i	$-0.266523\pi$
$827$	1107.30			1.33893			0.669466	+	0.742843i	$0.266523\pi$
$828$		0			0
$829$			366.866i			0.442541i	0.975212	+	0.221270i	$0.0710203\pi$
$829$			366.866i			0.442541i	−0.975212	+	0.221270i	$0.928980\pi$
$830$		0			0
$831$	−255.199	+	595.995i	−0.307098	+	0.717202i
$832$		0			0
$833$		−	33.2655i		−	0.0399346i
$834$		0			0
$835$		−	583.978i		−	0.699375i
$836$		0			0
$837$	−524.115	−	1404.82i	−0.626183	−	1.67840i
$838$		0			0
$839$			1085.11i			1.29334i	0.762771	+	0.646668i	$0.223838\pi$
$839$			1085.11i			1.29334i	−0.762771	+	0.646668i	$0.776162\pi$
$840$		0			0
$841$	293.466			0.348949
$842$		0			0
$843$	−589.379	+	1376.44i	−0.699145	+	1.63279i
$844$		0			0
$845$	143.707			0.170068
$846$		0			0
$847$	−650.691			−0.768230
$848$		0			0
$849$	243.812	−	569.401i	0.287175	−	0.670673i
$850$		0			0
$851$	−1077.63			−1.26631
$852$		0			0
$853$		−	1039.75i		−	1.21894i	−0.792810	−	0.609469i	$-0.791383\pi$
$853$		−	1039.75i		−	1.21894i	0.792810	−	0.609469i	$-0.208617\pi$
$854$		0			0
$855$	−692.145	+	660.039i	−0.809527	+	0.771975i
$856$		0			0
$857$		−	1216.69i		−	1.41971i	−0.704346	−	0.709857i	$-0.748760\pi$
$857$		−	1216.69i		−	1.41971i	0.704346	−	0.709857i	$-0.251240\pi$
$858$		0			0
$859$			318.230i			0.370466i	0.982695	+	0.185233i	$0.0593040\pi$
$859$			318.230i			0.370466i	−0.982695	+	0.185233i	$0.940696\pi$
$860$		0			0
$861$	−407.294	+	951.199i	−0.473047	+	1.10476i
$862$		0			0
$863$		−	464.296i		−	0.538003i	−0.963140	−	0.269001i	$-0.913306\pi$
$863$		−	464.296i		−	0.538003i	0.963140	−	0.269001i	$-0.0866936\pi$
$864$		0			0
$865$	−432.377			−0.499858
$866$		0			0
$867$	−788.919	−	337.807i	−0.909941	−	0.389627i
$868$		0			0
$869$	11.3281			0.0130358
$870$		0			0
$871$	−102.184			−0.117318
$872$		0			0
$873$	534.844	+	560.861i	0.612651	+	0.642453i
$874$		0			0
$875$	729.901			0.834172
$876$		0			0
$877$		−	1239.89i		−	1.41378i	−0.707321	−	0.706892i	$-0.750097\pi$
$877$		−	1239.89i		−	1.41378i	0.707321	−	0.706892i	$-0.249903\pi$
$878$		0			0
$879$	205.950	+	88.1855i	0.234300	+	0.100325i
$880$		0			0
$881$			49.3084i			0.0559687i	0.999608	+	0.0279843i	$0.00890885\pi$
$881$			49.3084i			0.0559687i	−0.999608	+	0.0279843i	$0.991091\pi$
$882$		0			0
$883$			1359.44i			1.53957i	0.638303	+	0.769785i	$0.279637\pi$
$883$			1359.44i			1.53957i	−0.638303	+	0.769785i	$0.720363\pi$
$884$		0			0
$885$	−504.450	−	216.000i	−0.570000	−	0.244068i
$886$		0			0
$887$			1353.53i			1.52597i	0.646417	+	0.762984i	$0.276267\pi$
$887$			1353.53i			1.52597i	−0.646417	+	0.762984i	$0.723733\pi$
$888$		0			0
$889$	−479.489			−0.539357
$890$		0			0
$891$	4.47629	−	94.2072i	0.00502390	−	0.105732i
$892$		0			0
$893$	1774.49			1.98711
$894$		0			0
$895$	−531.587			−0.593952
$896$		0			0
$897$	−1408.89	−	603.271i	−1.57067	−	0.672543i
$898$		0			0
$899$	1870.47			2.08061
$900$		0			0
$901$			96.4441i			0.107041i
$902$		0			0
$903$	−276.218	+	645.083i	−0.305889	+	0.714378i
$904$		0			0
$905$			1067.86i			1.17995i
$906$		0			0
$907$			1442.68i			1.59061i	0.606213	+	0.795303i	$0.292688\pi$
$907$			1442.68i			1.59061i	−0.606213	+	0.795303i	$0.707312\pi$
$908$		0			0
$909$	679.819	+	712.888i	0.747876	+	0.784255i
$910$		0			0
$911$			128.701i			0.141274i	0.997502	+	0.0706372i	$0.0225033\pi$
$911$			128.701i			0.141274i	−0.997502	+	0.0706372i	$0.977497\pi$
$912$		0			0
$913$	−45.3120			−0.0496297
$914$		0			0
$915$	−20.5962	+	48.1007i	−0.0225095	+	0.0525691i
$916$		0			0
$917$	−51.9964			−0.0567027
$918$		0			0
$919$	837.294			0.911092			0.455546	−	0.890212i	$-0.349444\pi$
$919$	837.294			0.911092			0.455546	+	0.890212i	$0.349444\pi$
$920$		0			0
$921$	−346.567	+	809.377i	−0.376294	+	0.878803i
$922$		0			0
$923$	1398.27			1.51492
$924$		0			0
$925$			347.489i			0.375663i
$926$		0			0
$927$	−993.864	−	1042.21i	−1.07213	−	1.12428i
$928$		0			0
$929$		−	671.610i		−	0.722939i	−0.932384	−	0.361469i	$-0.882275\pi$
$929$		−	671.610i		−	0.722939i	0.932384	−	0.361469i	$-0.117725\pi$
$930$		0			0
$931$			560.125i			0.601638i
$932$		0			0
$933$	−52.8290	+	123.378i	−0.0566227	+	0.132238i
$934$		0			0
$935$			7.34851i			0.00785937i
$936$		0			0
$937$	−598.511			−0.638753			−0.319376	−	0.947628i	$-0.603473\pi$
$937$	−598.511			−0.638753			−0.319376	+	0.947628i	$0.603473\pi$
$938$		0			0
$939$	−191.025	−	81.7947i	−0.203434	−	0.0871083i
$940$		0			0
$941$	−308.432			−0.327770			−0.163885	−	0.986479i	$-0.552403\pi$
$941$	−308.432			−0.327770			−0.163885	+	0.986479i	$0.552403\pi$
$942$		0			0
$943$	−2246.48			−2.38227
$944$		0			0
$945$	189.134	+	506.947i	0.200141	+	0.536452i
$946$		0			0
$947$	119.140			0.125808			0.0629040	−	0.998020i	$-0.479964\pi$
$947$	119.140			0.125808			0.0629040	+	0.998020i	$0.479964\pi$
$948$		0			0
$949$			988.400i			1.04152i
$950$		0			0
$951$	−33.3464	−	14.2786i	−0.0350645	−	0.0150143i
$952$		0			0
$953$		−	1061.63i		−	1.11399i	−0.830516	−	0.556995i	$-0.811954\pi$
$953$		−	1061.63i		−	1.11399i	0.830516	−	0.556995i	$-0.188046\pi$
$954$		0			0
$955$			556.424i			0.582643i
$956$		0			0
$957$	108.156	+	46.3112i	0.113015	+	0.0483920i
$958$		0			0
$959$		−	818.522i		−	0.853516i
$960$		0			0
$961$	2122.98			2.20913
$962$		0			0
$963$	1144.98	+	1200.68i	1.18898	+	1.24681i
$964$		0			0
$965$	8.18783			0.00848479
$966$		0			0
$967$	167.506			0.173223			0.0866114	−	0.996242i	$-0.472396\pi$
$967$	167.506			0.173223			0.0866114	+	0.996242i	$0.472396\pi$
$968$		0			0
$969$	−136.222	−	58.3288i	−0.140580	−	0.0601949i
$970$		0			0
$971$	−1724.86			−1.77638			−0.888189	−	0.459479i	$-0.848036\pi$
$971$	−1724.86			−1.77638			−0.888189	+	0.459479i	$0.848036\pi$
$972$		0			0
$973$			280.579i			0.288364i
$974$		0			0
$975$	−194.528	+	454.304i	−0.199516	+	0.465953i
$976$		0			0
$977$			166.328i			0.170243i	0.996371	+	0.0851216i	$0.0271279\pi$
$977$			166.328i			0.170243i	−0.996371	+	0.0851216i	$0.972872\pi$
$978$		0			0
$979$		−	0.786639i		−	0.000803512i
$980$		0			0
$981$	−447.677	+	426.910i	−0.456347	+	0.435178i
$982$		0			0
$983$			1532.48i			1.55899i	0.626411	+	0.779493i	$0.284523\pi$
$983$			1532.48i			1.55899i	−0.626411	+	0.779493i	$0.715477\pi$
$984$		0			0
$985$	643.889			0.653694
$986$		0			0
$987$	395.159	−	922.861i	0.400364	−	0.935016i
$988$		0			0
$989$	−1523.52			−1.54046
$990$		0			0
$991$	152.339			0.153722			0.0768612	−	0.997042i	$-0.475510\pi$
$991$	152.339			0.153722			0.0768612	+	0.997042i	$0.475510\pi$
$992$		0			0
$993$	−553.255	+	1292.08i	−0.557155	+	1.30119i
$994$		0			0
$995$	−517.922			−0.520524
$996$		0			0
$997$		−	626.688i		−	0.628574i	−0.949328	−	0.314287i	$-0.898235\pi$
$997$		−	626.688i		−	0.628574i	0.949328	−	0.314287i	$-0.101765\pi$
$998$		0			0
$999$	−769.584	+	287.119i	−0.770355	+	0.287406i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.h.g.65.1	✓	16
3.2	odd	2	inner	384.3.h.g.65.14	yes	16
4.3	odd	2	inner	384.3.h.g.65.15	yes	16
8.3	odd	2	inner	384.3.h.g.65.2	yes	16
8.5	even	2	inner	384.3.h.g.65.16	yes	16
12.11	even	2	inner	384.3.h.g.65.4	yes	16
16.3	odd	4		768.3.e.p.257.4		8
16.5	even	4		768.3.e.o.257.4		8
16.11	odd	4		768.3.e.o.257.5		8
16.13	even	4		768.3.e.p.257.5		8
24.5	odd	2	inner	384.3.h.g.65.3	yes	16
24.11	even	2	inner	384.3.h.g.65.13	yes	16
48.5	odd	4		768.3.e.o.257.3		8
48.11	even	4		768.3.e.o.257.6		8
48.29	odd	4		768.3.e.p.257.6		8
48.35	even	4		768.3.e.p.257.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.h.g.65.1	✓	16	1.1	even	1	trivial
384.3.h.g.65.2	yes	16	8.3	odd	2	inner
384.3.h.g.65.3	yes	16	24.5	odd	2	inner
384.3.h.g.65.4	yes	16	12.11	even	2	inner
384.3.h.g.65.13	yes	16	24.11	even	2	inner
384.3.h.g.65.14	yes	16	3.2	odd	2	inner
384.3.h.g.65.15	yes	16	4.3	odd	2	inner
384.3.h.g.65.16	yes	16	8.5	even	2	inner
768.3.e.o.257.3		8	48.5	odd	4
768.3.e.o.257.4		8	16.5	even	4
768.3.e.o.257.5		8	16.11	odd	4
768.3.e.o.257.6		8	48.11	even	4
768.3.e.p.257.3		8	48.35	even	4
768.3.e.p.257.4		8	16.3	odd	4
768.3.e.p.257.5		8	16.13	even	4
768.3.e.p.257.6		8	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)

\(f(q)\)	\(=\)	\(q+(-2.75782 - 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} +O(q^{10})\)	\(q+(-2.75782 - 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} -1.16436 q^{11} +14.4222i q^{13} +(10.1620 + 4.35127i) q^{15} +1.71276i q^{17} -28.8394i q^{19} +(-14.9985 - 6.42221i) q^{21} -35.4225i q^{23} -11.4222 q^{25} +(-9.43781 - 25.2968i) q^{27} +33.6818 q^{29} +55.5336 q^{31} +(3.21110 + 1.37496i) q^{33} -20.0400 q^{35} -30.4222i q^{37} +(17.0307 - 39.7738i) q^{39} -63.4196i q^{41} -43.0098i q^{43} +(-22.8867 - 24.0000i) q^{45} +61.5301i q^{47} -19.4222 q^{49} +(2.02254 - 4.72347i) q^{51} +56.3093 q^{53} +4.29046 q^{55} +(-34.0555 + 79.5338i) q^{57} -49.6407 q^{59} +4.73338i q^{61} +(33.7794 + 35.4225i) q^{63} -53.1430i q^{65} +7.08521i q^{67} +(-41.8293 + 97.6888i) q^{69} -96.9527i q^{71} +68.5332 q^{73} +(31.5003 + 13.4881i) q^{75} -6.33245 q^{77} -9.72900 q^{79} +(-3.84441 + 80.9087i) q^{81} +38.9156 q^{83} -6.31118i q^{85} +(-92.8883 - 39.7738i) q^{87} +0.675595i q^{89} +78.4358i q^{91} +(-153.151 - 65.5778i) q^{93} +106.268i q^{95} +86.1110 q^{97} +(-7.23199 - 7.58378i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) \) 16 * q - 16 * q^9 + 48 * q^25 - 64 * q^33 - 80 * q^49 + 32 * q^57 - 288 * q^73 + 400 * q^81 + 224 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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