LMFDB - Embedded newform 384.3.h.g.65.3

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.75782 + 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 - 6.51323i) q^{9} +O(q^{10})$	\(q+(-2.75782 + 1.18087i) q^{3} -3.68481 q^{5} +5.43855 q^{7} +(6.21110 - 6.51323i) q^{9} -1.16436 q^{11} -14.4222i q^{13} +(10.1620 - 4.35127i) q^{15} -1.71276i q^{17} +28.8394i q^{19} +(-14.9985 + 6.42221i) q^{21} +35.4225i q^{23} -11.4222 q^{25} +(-9.43781 + 25.2968i) q^{27} +33.6818 q^{29} +55.5336 q^{31} +(3.21110 - 1.37496i) q^{33} -20.0400 q^{35} +30.4222i q^{37} +(17.0307 + 39.7738i) q^{39} +63.4196i q^{41} +43.0098i q^{43} +(-22.8867 + 24.0000i) q^{45} -61.5301i q^{47} -19.4222 q^{49} +(2.02254 + 4.72347i) q^{51} +56.3093 q^{53} +4.29046 q^{55} +(-34.0555 - 79.5338i) q^{57} -49.6407 q^{59} -4.73338i q^{61} +(33.7794 - 35.4225i) q^{63} +53.1430i q^{65} -7.08521i q^{67} +(-41.8293 - 97.6888i) q^{69} +96.9527i q^{71} +68.5332 q^{73} +(31.5003 - 13.4881i) q^{75} -6.33245 q^{77} -9.72900 q^{79} +(-3.84441 - 80.9087i) q^{81} +38.9156 q^{83} +6.31118i q^{85} +(-92.8883 + 39.7738i) q^{87} -0.675595i q^{89} -78.4358i q^{91} +(-153.151 + 65.5778i) q^{93} -106.268i q^{95} +86.1110 q^{97} +(-7.23199 + 7.58378i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.812833\pi$
$13$		−	14.4222i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$	10.1620	−	4.35127i	0.677468	−	0.290085i
$16$		0			0
$17$		−	1.71276i		−	0.100750i	−0.998730	−	0.0503752i	$-0.983958\pi$
$17$		−	1.71276i		−	0.100750i	0.998730	−	0.0503752i	$-0.0160417\pi$
$18$		0			0
$19$			28.8394i			1.51786i	0.651171	+	0.758931i	$0.274278\pi$
$19$			28.8394i			1.51786i	−0.651171	+	0.758931i	$0.725722\pi$
$20$		0			0
$21$	−14.9985	+	6.42221i	−0.714215	+	0.305819i
$22$		0			0
$23$			35.4225i			1.54011i	0.637978	+	0.770055i	$0.279771\pi$
$23$			35.4225i			1.54011i	−0.637978	+	0.770055i	$0.720229\pi$
$24$		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	$0.134859\pi$
$37$			30.4222i			0.822222i	−0.911585	+	0.411111i	$0.865141\pi$
$38$		0			0
$39$	17.0307	+	39.7738i	0.436685	+	1.01984i
$40$		0			0
$41$			63.4196i			1.54682i	0.633907	+	0.773409i	$0.281450\pi$
$41$			63.4196i			1.54682i	−0.633907	+	0.773409i	$0.718550\pi$
$42$		0			0
$43$			43.0098i			1.00023i	0.865960	+	0.500114i	$0.166709\pi$
$43$			43.0098i			1.00023i	−0.865960	+	0.500114i	$0.833291\pi$
$44$		0			0
$45$	−22.8867	+	24.0000i	−0.508594	+	0.533333i
$46$		0			0
$47$		−	61.5301i		−	1.30915i	−0.755996	−	0.654576i	$-0.772847\pi$
$47$		−	61.5301i		−	1.30915i	0.755996	−	0.654576i	$-0.227153\pi$
$48$		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	$-0.987647\pi$
$61$		−	4.73338i		−	0.0775965i	0.999247	−	0.0387982i	$-0.0123530\pi$
$62$		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	$-0.983162\pi$
$67$		−	7.08521i		−	0.105749i	0.998601	−	0.0528747i	$-0.0168384\pi$
$68$		0			0
$69$	−41.8293	−	97.6888i	−0.606222	−	1.41578i
$70$		0			0
$71$			96.9527i			1.36553i	0.730638	+	0.682765i	$0.239223\pi$
$71$			96.9527i			1.36553i	−0.730638	+	0.682765i	$0.760777\pi$
$72$		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	$-0.998792\pi$
$89$		−	0.675595i		−	0.00759096i	0.999993	−	0.00379548i	$-0.00120814\pi$
$90$		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	$-0.897898\pi$
$109$		−	68.7334i		−	0.630582i	0.948995	−	0.315291i	$-0.102102\pi$
$110$		0			0
$111$	−35.9246	−	83.8989i	−0.323645	−	0.755846i
$112$		0			0
$113$			71.3078i			0.631042i	0.948919	+	0.315521i	$0.102179\pi$
$113$			71.3078i			0.631042i	−0.948919	+	0.315521i	$0.897821\pi$
$114$		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	$0.185099\pi$
$137$			150.504i			1.09857i	−0.835636	+	0.549284i	$0.814901\pi$
$138$		0			0
$139$		−	51.5907i		−	0.371156i	−0.982630	−	0.185578i	$-0.940584\pi$
$139$		−	51.5907i		−	0.371156i	0.982630	−	0.185578i	$-0.0594158\pi$
$140$		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	$-0.704345\pi$
$157$		−	251.489i		−	1.60184i	0.598772	−	0.800919i	$-0.295655\pi$
$158$		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	$-0.901075\pi$
$163$		−	99.6915i		−	0.611604i	0.952095	−	0.305802i	$-0.0989246\pi$
$164$		0			0
$165$	−11.8323	+	5.06646i	−0.0717108	+	0.0307058i
$166$		0			0
$167$		−	158.483i		−	0.948999i	−0.880256	−	0.474499i	$-0.842629\pi$
$167$		−	158.483i		−	0.948999i	0.880256	−	0.474499i	$-0.157371\pi$
$168$		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	$0.295460\pi$
$181$			289.800i			1.60110i	−0.599263	+	0.800552i	$0.704540\pi$
$182$		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	$0.129360\pi$
$191$			151.005i			0.790602i	−0.918552	+	0.395301i	$0.870640\pi$
$192$		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	$0.246900\pi$
$211$			295.479i			1.40038i	−0.713959	+	0.700188i	$0.753100\pi$
$212$		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	$-0.956325\pi$
$229$		−	62.6443i		−	0.273556i	0.990602	−	0.136778i	$-0.0436747\pi$
$230$		0			0
$231$	17.4637	−	7.47779i	0.0756006	−	0.0323714i
$232$		0			0
$233$		−	270.130i		−	1.15936i	−0.814845	−	0.579679i	$-0.803178\pi$
$233$		−	270.130i		−	1.15936i	0.814845	−	0.579679i	$-0.196822\pi$
$234$		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	$0.256579\pi$
$239$			344.910i			1.44314i	−0.692342	+	0.721570i	$0.743421\pi$
$240$		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	$-0.658186\pi$
$257$		−	451.825i		−	1.75807i	0.476753	−	0.879037i	$-0.341814\pi$
$258$		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	$-0.896651\pi$
$263$		−	167.798i		−	0.638014i	0.947752	−	0.319007i	$-0.103349\pi$
$264$		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	$0.127557\pi$
$277$			216.111i			0.780184i	−0.920776	+	0.390092i	$0.872443\pi$
$278$		0			0
$279$	344.925	−	361.703i	1.23629	−	1.29643i
$280$		0			0
$281$			499.107i			1.77618i	0.459669	+	0.888090i	$0.347968\pi$
$281$			499.107i			1.77618i	−0.459669	+	0.888090i	$0.652032\pi$
$282$		0			0
$283$		−	206.468i		−	0.729569i	−0.931092	−	0.364785i	$-0.881143\pi$
$283$		−	206.468i		−	0.729569i	0.931092	−	0.364785i	$-0.118857\pi$
$284$		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	$0.158634\pi$
$307$			293.485i			0.955977i	−0.878366	+	0.477988i	$0.841366\pi$
$308$		0			0
$309$	441.289	−	188.955i	1.42812	−	0.611506i
$310$		0			0
$311$			44.7374i			0.143850i	0.997410	+	0.0719251i	$0.0229143\pi$
$311$			44.7374i			0.143850i	−0.997410	+	0.0719251i	$0.977086\pi$
$312$		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	$0.250279\pi$
$331$			468.516i			1.41545i	−0.706486	+	0.707727i	$0.749721\pi$
$332$		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	$-0.987664\pi$
$349$		−	27.0446i		−	0.0774916i	0.999249	−	0.0387458i	$-0.0123363\pi$
$350$		0			0
$351$	364.836	+	136.114i	1.03942	+	0.387789i
$352$		0			0
$353$		−	47.9572i		−	0.135856i	−0.997690	−	0.0679281i	$-0.978361\pi$
$353$		−	47.9572i		−	0.135856i	0.997690	−	0.0679281i	$-0.0216388\pi$
$354$		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	$-0.988423\pi$
$359$		−	26.1076i		−	0.0727232i	0.999339	−	0.0363616i	$-0.0115768\pi$
$360$		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	$-0.874463\pi$
$373$		−	286.644i		−	0.768483i	0.923233	−	0.384242i	$-0.125537\pi$
$374$		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	$-0.889644\pi$
$379$		−	257.560i		−	0.679579i	0.940502	−	0.339789i	$-0.110356\pi$
$380$		0			0
$381$	243.142	−	104.111i	0.638169	−	0.273257i
$382$		0			0
$383$		−	311.325i		−	0.812859i	−0.913682	−	0.406429i	$-0.866774\pi$
$383$		−	311.325i		−	0.812859i	0.913682	−	0.406429i	$-0.133226\pi$
$384$		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	$-0.784074\pi$
$397$		−	498.244i		−	1.25502i	0.778607	−	0.627511i	$-0.215926\pi$
$398$		0			0
$399$	−185.213	−	432.548i	−0.464192	−	1.08408i
$400$		0			0
$401$		−	523.495i		−	1.30547i	−0.757585	−	0.652736i	$-0.773621\pi$
$401$		−	523.495i		−	1.30547i	0.757585	−	0.652736i	$-0.226379\pi$
$402$		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	$0.0272943\pi$
$421$			72.1110i			0.171285i	−0.996326	+	0.0856425i	$0.972706\pi$
$422$		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	$-0.689245\pi$
$431$		−	714.091i		−	1.65682i	0.560119	−	0.828412i	$-0.310755\pi$
$432$		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	$-0.888989\pi$
$449$		−	306.869i		−	0.683450i	0.939800	−	0.341725i	$-0.111011\pi$
$450$		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	$-0.897911\pi$
$479$		−	302.010i		−	0.630501i	0.949008	−	0.315251i	$-0.102089\pi$
$480$		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	$0.114793\pi$
$499$			352.161i			0.705733i	−0.935674	+	0.352867i	$0.885207\pi$
$500$		0			0
$501$	187.147	+	437.066i	0.373547	+	0.872388i
$502$		0			0
$503$			130.538i			0.259519i	0.991545	+	0.129760i	$0.0414205\pi$
$503$			130.538i			0.259519i	−0.991545	+	0.129760i	$0.958579\pi$
$504$		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	$-0.901139\pi$
$521$		−	318.449i		−	0.611227i	0.952156	−	0.305613i	$-0.0988615\pi$
$522$		0			0
$523$		−	817.186i		−	1.56250i	−0.624220	−	0.781249i	$-0.714583\pi$
$523$		−	817.186i		−	1.56250i	0.624220	−	0.781249i	$-0.285417\pi$
$524$		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	$0.0961919\pi$
$541$			322.022i			0.595235i	−0.954685	+	0.297617i	$0.903808\pi$
$542$		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	$0.0813975\pi$
$547$			276.716i			0.505880i	−0.967482	+	0.252940i	$0.918602\pi$
$548$		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	$0.0646041\pi$
$569$			229.386i			0.403138i	−0.979474	+	0.201569i	$0.935396\pi$
$570$		0			0
$571$			451.354i			0.790462i	0.918582	+	0.395231i	$0.129335\pi$
$571$			451.354i			0.790462i	−0.918582	+	0.395231i	$0.870665\pi$
$572$		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	$0.0396611\pi$
$593$			147.392i			0.248554i	−0.992248	+	0.124277i	$0.960339\pi$
$594$		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	$0.114935\pi$
$599$			423.233i			0.706566i	−0.935516	+	0.353283i	$0.885065\pi$
$600$		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	$-0.935853\pi$
$613$		−	245.400i		−	0.400326i	0.979763	−	0.200163i	$-0.0641471\pi$
$614$		0			0
$615$	275.956	+	644.471i	0.448708	+	1.04792i
$616$		0			0
$617$			87.0366i			0.141064i	0.997510	+	0.0705321i	$0.0224697\pi$
$617$			87.0366i			0.141064i	−0.997510	+	0.0705321i	$0.977530\pi$
$618$		0			0
$619$			203.477i			0.328718i	0.986401	+	0.164359i	$0.0525556\pi$
$619$			203.477i			0.328718i	−0.986401	+	0.164359i	$0.947444\pi$
$620$		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	$0.213812\pi$
$641$			797.821i			1.24465i	−0.782759	+	0.622325i	$0.786188\pi$
$642$		0			0
$643$			94.7058i			0.147287i	0.997285	+	0.0736437i	$0.0234628\pi$
$643$			94.7058i			0.147287i	−0.997285	+	0.0736437i	$0.976537\pi$
$644$		0			0
$645$	187.147	+	437.066i	0.290151	+	0.677622i
$646$		0			0
$647$		−	367.344i		−	0.567765i	−0.958859	−	0.283882i	$-0.908378\pi$
$647$		−	367.344i		−	0.567765i	0.958859	−	0.283882i	$-0.0916225\pi$
$648$		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	$-0.677569\pi$
$661$		−	1121.58i		−	1.69679i	0.529364	−	0.848395i	$-0.322431\pi$
$662$		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	$0.214588\pi$
$691$			862.689i			1.24846i	−0.781239	+	0.624232i	$0.785412\pi$
$692$		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	$-0.744259\pi$
$709$		−	1020.60i		−	1.43949i	0.694238	−	0.719745i	$-0.255741\pi$
$710$		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	$0.107693\pi$
$719$			477.286i			0.663819i	−0.943311	+	0.331909i	$0.892307\pi$
$720$		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	$0.225069\pi$
$733$			952.333i			1.29923i	−0.760265	+	0.649613i	$0.774931\pi$
$734$		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	$0.00543698\pi$
$739$			25.2442i			0.0341599i	−0.999854	+	0.0170800i	$0.994563\pi$
$740$		0			0
$741$	−1147.05	+	491.156i	−1.54798	+	0.662828i
$742$		0			0
$743$		−	706.613i		−	0.951027i	−0.879708	−	0.475514i	$-0.842262\pi$
$743$		−	706.613i		−	0.951027i	0.879708	−	0.475514i	$-0.157738\pi$
$744$		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	$0.0654633\pi$
$757$			309.177i			0.408425i	−0.978927	+	0.204212i	$0.934537\pi$
$758$		0			0
$759$	48.7046	+	113.745i	0.0641694	+	0.149862i
$760$		0			0
$761$		−	400.757i		−	0.526618i	−0.964712	−	0.263309i	$-0.915186\pi$
$761$		−	400.757i		−	0.526618i	0.964712	−	0.263309i	$-0.0848139\pi$
$762$		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	$-0.881116\pi$
$787$		−	574.295i		−	0.729727i	0.931061	−	0.364863i	$-0.118884\pi$
$788$		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	$0.117340\pi$
$809$			583.032i			0.720682i	−0.932821	+	0.360341i	$0.882660\pi$
$810$		0			0
$811$			911.183i			1.12353i	0.827297	+	0.561765i	$0.189878\pi$
$811$			911.183i			1.12353i	−0.827297	+	0.561765i	$0.810122\pi$
$812$		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	$-0.928980\pi$
$829$		−	366.866i		−	0.442541i	0.975212	−	0.221270i	$-0.0710203\pi$
$830$		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	$-0.776162\pi$
$839$		−	1085.11i		−	1.29334i	0.762771	−	0.646668i	$-0.223838\pi$
$840$		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	$0.208617\pi$
$853$			1039.75i			1.21894i	−0.792810	+	0.609469i	$0.791383\pi$
$854$		0			0
$855$	−692.145	−	660.039i	−0.809527	−	0.771975i
$856$		0			0
$857$			1216.69i			1.41971i	0.704346	+	0.709857i	$0.251240\pi$
$857$			1216.69i			1.41971i	−0.704346	+	0.709857i	$0.748760\pi$
$858$		0			0
$859$		−	318.230i		−	0.370466i	−0.982695	−	0.185233i	$-0.940696\pi$
$859$		−	318.230i		−	0.370466i	0.982695	−	0.185233i	$-0.0593040\pi$
$860$		0			0
$861$	−407.294	−	951.199i	−0.473047	−	1.10476i
$862$		0			0
$863$			464.296i			0.538003i	0.963140	+	0.269001i	$0.0866936\pi$
$863$			464.296i			0.538003i	−0.963140	+	0.269001i	$0.913306\pi$
$864$		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	$0.249903\pi$
$877$			1239.89i			1.41378i	−0.707321	+	0.706892i	$0.750097\pi$
$878$		0			0
$879$	205.950	−	88.1855i	0.234300	−	0.100325i
$880$		0			0
$881$		−	49.3084i		−	0.0559687i	−0.999608	−	0.0279843i	$-0.991091\pi$
$881$		−	49.3084i		−	0.0559687i	0.999608	−	0.0279843i	$-0.00890885\pi$
$882$		0			0
$883$		−	1359.44i		−	1.53957i	−0.638303	−	0.769785i	$-0.720363\pi$
$883$		−	1359.44i		−	1.53957i	0.638303	−	0.769785i	$-0.279637\pi$
$884$		0			0
$885$	−504.450	+	216.000i	−0.570000	+	0.244068i
$886$		0			0
$887$		−	1353.53i		−	1.52597i	−0.646417	−	0.762984i	$-0.723733\pi$
$887$		−	1353.53i		−	1.52597i	0.646417	−	0.762984i	$-0.276267\pi$
$888$		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	$-0.707312\pi$
$907$		−	1442.68i		−	1.59061i	0.606213	−	0.795303i	$-0.292688\pi$
$908$		0			0
$909$	679.819	−	712.888i	0.747876	−	0.784255i
$910$		0			0
$911$		−	128.701i		−	0.141274i	−0.997502	−	0.0706372i	$-0.977497\pi$
$911$		−	128.701i		−	0.141274i	0.997502	−	0.0706372i	$-0.0225033\pi$
$912$		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	$0.117725\pi$
$929$			671.610i			0.722939i	−0.932384	+	0.361469i	$0.882275\pi$
$930$		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	$0.188046\pi$
$953$			1061.63i			1.11399i	−0.830516	+	0.556995i	$0.811954\pi$
$954$		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	$-0.972872\pi$
$977$		−	166.328i		−	0.170243i	0.996371	−	0.0851216i	$-0.0271279\pi$
$978$		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	$-0.715477\pi$
$983$		−	1532.48i		−	1.55899i	0.626411	−	0.779493i	$-0.284523\pi$
$984$		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	$0.101765\pi$
$997$			626.688i			0.628574i	−0.949328	+	0.314287i	$0.898235\pi$
$998$		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.65.3	yes	16
3.2	odd	2	inner	384.3.h.g.65.16	yes	16
4.3	odd	2	inner	384.3.h.g.65.13	yes	16
8.3	odd	2	inner	384.3.h.g.65.4	yes	16
8.5	even	2	inner	384.3.h.g.65.14	yes	16
12.11	even	2	inner	384.3.h.g.65.2	yes	16
16.3	odd	4		768.3.e.o.257.6		8
16.5	even	4		768.3.e.p.257.6		8
16.11	odd	4		768.3.e.p.257.3		8
16.13	even	4		768.3.e.o.257.3		8
24.5	odd	2	inner	384.3.h.g.65.1	✓	16
24.11	even	2	inner	384.3.h.g.65.15	yes	16
48.5	odd	4		768.3.e.p.257.5		8
48.11	even	4		768.3.e.p.257.4		8
48.29	odd	4		768.3.e.o.257.4		8
48.35	even	4		768.3.e.o.257.5		8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.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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