LMFDB - Embedded newform 384.3.b.b.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(319,384)]

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([1, 1, 0]))

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

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.3.b.b.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} -6.92820 q^{11} -13.8564i q^{13} -12.0000i q^{15} +14.0000 q^{17} -34.6410 q^{19} -20.7846i q^{21} -24.0000i q^{23} -23.0000 q^{25} -5.19615 q^{27} +34.6410i q^{29} -12.0000i q^{31} +12.0000 q^{33} -83.1384 q^{35} +27.7128i q^{37} +24.0000i q^{39} -14.0000 q^{41} -6.92820 q^{43} +20.7846i q^{45} +72.0000i q^{47} -95.0000 q^{49} -24.2487 q^{51} -62.3538i q^{53} -48.0000i q^{55} +60.0000 q^{57} +48.4974 q^{59} -55.4256i q^{61} +36.0000i q^{63} +96.0000 q^{65} -90.0666 q^{67} +41.5692i q^{69} +24.0000i q^{71} -50.0000 q^{73} +39.8372 q^{75} -83.1384i q^{77} -12.0000i q^{79} +9.00000 q^{81} +20.7846 q^{83} +96.9948i q^{85} -60.0000i q^{87} +62.0000 q^{89} +166.277 q^{91} +20.7846i q^{93} -240.000i q^{95} -146.000 q^{97} -20.7846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 56 q^{17} - 92 q^{25} + 48 q^{33} - 56 q^{41} - 380 q^{49} + 240 q^{57} + 384 q^{65} - 200 q^{73} + 36 q^{81} + 248 q^{89} - 584 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 56 * q^17 - 92 * q^25 + 48 * q^33 - 56 * q^41 - 380 * q^49 + 240 * q^57 + 384 * q^65 - 200 * q^73 + 36 * q^81 + 248 * q^89 - 584 * 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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	6.92820i	1.38564i	0.721110	+	0.692820i	$0.243632\pi$
$5$	6.92820i	1.38564i	−0.721110	+	0.692820i	$0.756368\pi$
$6$	0	0
$7$	12.0000i	1.71429i	0.515079	+	0.857143i	$0.327763\pi$
$7$	12.0000i	1.71429i	−0.515079	+	0.857143i	$0.672237\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−6.92820	−0.629837	−0.314918	−	0.949119i	$-0.601977\pi$
$11$	−6.92820	−0.629837	−0.314918	+	0.949119i	$0.601977\pi$
$12$	0	0
$13$	− 13.8564i	− 1.06588i	−0.846154	−	0.532939i	$-0.821088\pi$
$13$	− 13.8564i	− 1.06588i	0.846154	−	0.532939i	$-0.178912\pi$
$14$	0	0
$15$	− 12.0000i	− 0.800000i
$16$	0	0
$17$	14.0000	0.823529	0.411765	−	0.911290i	$-0.364913\pi$
$17$	14.0000	0.823529	0.411765	+	0.911290i	$0.364913\pi$
$18$	0	0
$19$	−34.6410	−1.82321	−0.911606	−	0.411066i	$-0.865157\pi$
$19$	−34.6410	−1.82321	−0.911606	+	0.411066i	$0.865157\pi$
$20$	0	0
$21$	− 20.7846i	− 0.989743i
$22$	0	0
$23$	− 24.0000i	− 1.04348i	−0.853105	−	0.521739i	$-0.825283\pi$
$23$	− 24.0000i	− 1.04348i	0.853105	−	0.521739i	$-0.174717\pi$
$24$	0	0
$25$	−23.0000	−0.920000
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	34.6410i	1.19452i	0.802049	+	0.597259i	$0.203744\pi$
$29$	34.6410i	1.19452i	−0.802049	+	0.597259i	$0.796256\pi$
$30$	0	0
$31$	− 12.0000i	− 0.387097i	−0.981091	−	0.193548i	$-0.938000\pi$
$31$	− 12.0000i	− 0.387097i	0.981091	−	0.193548i	$-0.0619996\pi$
$32$	0	0
$33$	12.0000	0.363636
$34$	0	0
$35$	−83.1384	−2.37538
$36$	0	0
$37$	27.7128i	0.748995i	0.927228	+	0.374497i	$0.122185\pi$
$37$	27.7128i	0.748995i	−0.927228	+	0.374497i	$0.877815\pi$
$38$	0	0
$39$	24.0000i	0.615385i
$40$	0	0
$41$	−14.0000	−0.341463	−0.170732	−	0.985318i	$-0.554613\pi$
$41$	−14.0000	−0.341463	−0.170732	+	0.985318i	$0.554613\pi$
$42$	0	0
$43$	−6.92820	−0.161121	−0.0805605	−	0.996750i	$-0.525671\pi$
$43$	−6.92820	−0.161121	−0.0805605	+	0.996750i	$0.525671\pi$
$44$	0	0
$45$	20.7846i	0.461880i
$46$	0	0
$47$	72.0000i	1.53191i	0.642891	+	0.765957i	$0.277735\pi$
$47$	72.0000i	1.53191i	−0.642891	+	0.765957i	$0.722265\pi$
$48$	0	0
$49$	−95.0000	−1.93878
$50$	0	0
$51$	−24.2487	−0.475465
$52$	0	0
$53$	− 62.3538i	− 1.17649i	−0.808684	−	0.588244i	$-0.799820\pi$
$53$	− 62.3538i	− 1.17649i	0.808684	−	0.588244i	$-0.200180\pi$
$54$	0	0
$55$	− 48.0000i	− 0.872727i
$56$	0	0
$57$	60.0000	1.05263
$58$	0	0
$59$	48.4974	0.821990	0.410995	−	0.911638i	$-0.365181\pi$
$59$	48.4974	0.821990	0.410995	+	0.911638i	$0.365181\pi$
$60$	0	0
$61$	− 55.4256i	− 0.908617i	−0.890844	−	0.454308i	$-0.849886\pi$
$61$	− 55.4256i	− 0.908617i	0.890844	−	0.454308i	$-0.150114\pi$
$62$	0	0
$63$	36.0000i	0.571429i
$64$	0	0
$65$	96.0000	1.47692
$66$	0	0
$67$	−90.0666	−1.34428	−0.672139	−	0.740425i	$-0.734624\pi$
$67$	−90.0666	−1.34428	−0.672139	+	0.740425i	$0.734624\pi$
$68$	0	0
$69$	41.5692i	0.602452i
$70$	0	0
$71$	24.0000i	0.338028i	0.985614	+	0.169014i	$0.0540583\pi$
$71$	24.0000i	0.338028i	−0.985614	+	0.169014i	$0.945942\pi$
$72$	0	0
$73$	−50.0000	−0.684932	−0.342466	−	0.939530i	$-0.611262\pi$
$73$	−50.0000	−0.684932	−0.342466	+	0.939530i	$0.611262\pi$
$74$	0	0
$75$	39.8372	0.531162
$76$	0	0
$77$	− 83.1384i	− 1.07972i
$78$	0	0
$79$	− 12.0000i	− 0.151899i	−0.997112	−	0.0759494i	$-0.975801\pi$
$79$	− 12.0000i	− 0.151899i	0.997112	−	0.0759494i	$-0.0241987\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	20.7846	0.250417	0.125208	−	0.992130i	$-0.460040\pi$
$83$	20.7846	0.250417	0.125208	+	0.992130i	$0.460040\pi$
$84$	0	0
$85$	96.9948i	1.14112i
$86$	0	0
$87$	− 60.0000i	− 0.689655i
$88$	0	0
$89$	62.0000	0.696629	0.348315	−	0.937378i	$-0.386754\pi$
$89$	62.0000	0.696629	0.348315	+	0.937378i	$0.386754\pi$
$90$	0	0
$91$	166.277	1.82722
$92$	0	0
$93$	20.7846i	0.223490i
$94$	0	0
$95$	− 240.000i	− 2.52632i
$96$	0	0
$97$	−146.000	−1.50515	−0.752577	−	0.658504i	$-0.771190\pi$
$97$	−146.000	−1.50515	−0.752577	+	0.658504i	$0.771190\pi$
$98$	0	0
$99$	−20.7846	−0.209946
$100$	0	0
$101$	20.7846i	0.205788i	0.994692	+	0.102894i	$0.0328103\pi$
$101$	20.7846i	0.205788i	−0.994692	+	0.102894i	$0.967190\pi$
$102$	0	0
$103$	84.0000i	0.815534i	0.913086	+	0.407767i	$0.133692\pi$
$103$	84.0000i	0.815534i	−0.913086	+	0.407767i	$0.866308\pi$
$104$	0	0
$105$	144.000	1.37143
$106$	0	0
$107$	131.636	1.23024	0.615121	−	0.788433i	$-0.289107\pi$
$107$	131.636	1.23024	0.615121	+	0.788433i	$0.289107\pi$
$108$	0	0
$109$	180.133i	1.65260i	0.563231	+	0.826299i	$0.309558\pi$
$109$	180.133i	1.65260i	−0.563231	+	0.826299i	$0.690442\pi$
$110$	0	0
$111$	− 48.0000i	− 0.432432i
$112$	0	0
$113$	130.000	1.15044	0.575221	−	0.817998i	$-0.304916\pi$
$113$	130.000	1.15044	0.575221	+	0.817998i	$0.304916\pi$
$114$	0	0
$115$	166.277	1.44589
$116$	0	0
$117$	− 41.5692i	− 0.355292i
$118$	0	0
$119$	168.000i	1.41176i
$120$	0	0
$121$	−73.0000	−0.603306
$122$	0	0
$123$	24.2487	0.197144
$124$	0	0
$125$	13.8564i	0.110851i
$126$	0	0
$127$	204.000i	1.60630i	0.595777	+	0.803150i	$0.296844\pi$
$127$	204.000i	1.60630i	−0.595777	+	0.803150i	$0.703156\pi$
$128$	0	0
$129$	12.0000	0.0930233
$130$	0	0
$131$	20.7846	0.158661	0.0793306	−	0.996848i	$-0.474722\pi$
$131$	20.7846	0.158661	0.0793306	+	0.996848i	$0.474722\pi$
$132$	0	0
$133$	− 415.692i	− 3.12551i
$134$	0	0
$135$	− 36.0000i	− 0.266667i
$136$	0	0
$137$	−206.000	−1.50365	−0.751825	−	0.659363i	$-0.770826\pi$
$137$	−206.000	−1.50365	−0.751825	+	0.659363i	$0.770826\pi$
$138$	0	0
$139$	48.4974	0.348902	0.174451	−	0.984666i	$-0.444185\pi$
$139$	48.4974	0.348902	0.174451	+	0.984666i	$0.444185\pi$
$140$	0	0
$141$	− 124.708i	− 0.884451i
$142$	0	0
$143$	96.0000i	0.671329i
$144$	0	0
$145$	−240.000	−1.65517
$146$	0	0
$147$	164.545	1.11935
$148$	0	0
$149$	200.918i	1.34844i	0.738530	+	0.674221i	$0.235521\pi$
$149$	200.918i	1.34844i	−0.738530	+	0.674221i	$0.764479\pi$
$150$	0	0
$151$	− 36.0000i	− 0.238411i	−0.992870	−	0.119205i	$-0.961965\pi$
$151$	− 36.0000i	− 0.238411i	0.992870	−	0.119205i	$-0.0380347\pi$
$152$	0	0
$153$	42.0000	0.274510
$154$	0	0
$155$	83.1384	0.536377
$156$	0	0
$157$	− 27.7128i	− 0.176515i	−0.996098	−	0.0882574i	$-0.971870\pi$
$157$	− 27.7128i	− 0.176515i	0.996098	−	0.0882574i	$-0.0281298\pi$
$158$	0	0
$159$	108.000i	0.679245i
$160$	0	0
$161$	288.000	1.78882
$162$	0	0
$163$	−62.3538	−0.382539	−0.191269	−	0.981538i	$-0.561260\pi$
$163$	−62.3538	−0.382539	−0.191269	+	0.981538i	$0.561260\pi$
$164$	0	0
$165$	83.1384i	0.503869i
$166$	0	0
$167$	− 240.000i	− 1.43713i	−0.695462	−	0.718563i	$-0.744800\pi$
$167$	− 240.000i	− 1.43713i	0.695462	−	0.718563i	$-0.255200\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	−103.923	−0.607737
$172$	0	0
$173$	131.636i	0.760901i	0.924801	+	0.380450i	$0.124231\pi$
$173$	131.636i	0.760901i	−0.924801	+	0.380450i	$0.875769\pi$
$174$	0	0
$175$	− 276.000i	− 1.57714i
$176$	0	0
$177$	−84.0000	−0.474576
$178$	0	0
$179$	270.200	1.50950	0.754748	−	0.656014i	$-0.227759\pi$
$179$	270.200	1.50950	0.754748	+	0.656014i	$0.227759\pi$
$180$	0	0
$181$	207.846i	1.14832i	0.818743	+	0.574160i	$0.194671\pi$
$181$	207.846i	1.14832i	−0.818743	+	0.574160i	$0.805329\pi$
$182$	0	0
$183$	96.0000i	0.524590i
$184$	0	0
$185$	−192.000	−1.03784
$186$	0	0
$187$	−96.9948	−0.518689
$188$	0	0
$189$	− 62.3538i	− 0.329914i
$190$	0	0
$191$	240.000i	1.25654i	0.777994	+	0.628272i	$0.216238\pi$
$191$	240.000i	1.25654i	−0.777994	+	0.628272i	$0.783762\pi$
$192$	0	0
$193$	−2.00000	−0.0103627	−0.00518135	−	0.999987i	$-0.501649\pi$
$193$	−2.00000	−0.0103627	−0.00518135	+	0.999987i	$0.501649\pi$
$194$	0	0
$195$	−166.277	−0.852702
$196$	0	0
$197$	131.636i	0.668202i	0.942537	+	0.334101i	$0.108433\pi$
$197$	131.636i	0.668202i	−0.942537	+	0.334101i	$0.891567\pi$
$198$	0	0
$199$	− 300.000i	− 1.50754i	−0.657140	−	0.753769i	$-0.728234\pi$
$199$	− 300.000i	− 1.50754i	0.657140	−	0.753769i	$-0.271766\pi$
$200$	0	0
$201$	156.000	0.776119
$202$	0	0
$203$	−415.692	−2.04774
$204$	0	0
$205$	− 96.9948i	− 0.473146i
$206$	0	0
$207$	− 72.0000i	− 0.347826i
$208$	0	0
$209$	240.000	1.14833
$210$	0	0
$211$	270.200	1.28057	0.640284	−	0.768138i	$-0.278817\pi$
$211$	270.200	1.28057	0.640284	+	0.768138i	$0.278817\pi$
$212$	0	0
$213$	− 41.5692i	− 0.195161i
$214$	0	0
$215$	− 48.0000i	− 0.223256i
$216$	0	0
$217$	144.000	0.663594
$218$	0	0
$219$	86.6025	0.395445
$220$	0	0
$221$	− 193.990i	− 0.877781i
$222$	0	0
$223$	− 132.000i	− 0.591928i	−0.955199	−	0.295964i	$-0.904359\pi$
$223$	− 132.000i	− 0.591928i	0.955199	−	0.295964i	$-0.0956409\pi$
$224$	0	0
$225$	−69.0000	−0.306667
$226$	0	0
$227$	−200.918	−0.885101	−0.442550	−	0.896744i	$-0.645926\pi$
$227$	−200.918	−0.885101	−0.442550	+	0.896744i	$0.645926\pi$
$228$	0	0
$229$	69.2820i	0.302542i	0.988492	+	0.151271i	$0.0483366\pi$
$229$	69.2820i	0.302542i	−0.988492	+	0.151271i	$0.951663\pi$
$230$	0	0
$231$	144.000i	0.623377i
$232$	0	0
$233$	−130.000	−0.557940	−0.278970	−	0.960300i	$-0.589993\pi$
$233$	−130.000	−0.557940	−0.278970	+	0.960300i	$0.589993\pi$
$234$	0	0
$235$	−498.831	−2.12268
$236$	0	0
$237$	20.7846i	0.0876988i
$238$	0	0
$239$	96.0000i	0.401674i	0.979625	+	0.200837i	$0.0643661\pi$
$239$	96.0000i	0.401674i	−0.979625	+	0.200837i	$0.935634\pi$
$240$	0	0
$241$	190.000	0.788382	0.394191	−	0.919029i	$-0.371025\pi$
$241$	190.000	0.788382	0.394191	+	0.919029i	$0.371025\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 658.179i	− 2.68645i
$246$	0	0
$247$	480.000i	1.94332i
$248$	0	0
$249$	−36.0000	−0.144578
$250$	0	0
$251$	−256.344	−1.02129	−0.510644	−	0.859792i	$-0.670593\pi$
$251$	−256.344	−1.02129	−0.510644	+	0.859792i	$0.670593\pi$
$252$	0	0
$253$	166.277i	0.657221i
$254$	0	0
$255$	− 168.000i	− 0.658824i
$256$	0	0
$257$	−254.000	−0.988327	−0.494163	−	0.869369i	$-0.664526\pi$
$257$	−254.000	−0.988327	−0.494163	+	0.869369i	$0.664526\pi$
$258$	0	0
$259$	−332.554	−1.28399
$260$	0	0
$261$	103.923i	0.398173i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	432.000	1.63019
$266$	0	0
$267$	−107.387	−0.402199
$268$	0	0
$269$	6.92820i	0.0257554i	0.999917	+	0.0128777i	$0.00409921\pi$
$269$	6.92820i	0.0257554i	−0.999917	+	0.0128777i	$0.995901\pi$
$270$	0	0
$271$	348.000i	1.28413i	0.766649	+	0.642066i	$0.221923\pi$
$271$	348.000i	1.28413i	−0.766649	+	0.642066i	$0.778077\pi$
$272$	0	0
$273$	−288.000	−1.05495
$274$	0	0
$275$	159.349	0.579450
$276$	0	0
$277$	41.5692i	0.150069i	0.997181	+	0.0750347i	$0.0239068\pi$
$277$	41.5692i	0.150069i	−0.997181	+	0.0750347i	$0.976093\pi$
$278$	0	0
$279$	− 36.0000i	− 0.129032i
$280$	0	0
$281$	−34.0000	−0.120996	−0.0604982	−	0.998168i	$-0.519269\pi$
$281$	−34.0000	−0.120996	−0.0604982	+	0.998168i	$0.519269\pi$
$282$	0	0
$283$	−311.769	−1.10166	−0.550829	−	0.834618i	$-0.685688\pi$
$283$	−311.769	−1.10166	−0.550829	+	0.834618i	$0.685688\pi$
$284$	0	0
$285$	415.692i	1.45857i
$286$	0	0
$287$	− 168.000i	− 0.585366i
$288$	0	0
$289$	−93.0000	−0.321799
$290$	0	0
$291$	252.879	0.869001
$292$	0	0
$293$	159.349i	0.543852i	0.962318	+	0.271926i	$0.0876606\pi$
$293$	159.349i	0.543852i	−0.962318	+	0.271926i	$0.912339\pi$
$294$	0	0
$295$	336.000i	1.13898i
$296$	0	0
$297$	36.0000	0.121212
$298$	0	0
$299$	−332.554	−1.11222
$300$	0	0
$301$	− 83.1384i	− 0.276207i
$302$	0	0
$303$	− 36.0000i	− 0.118812i
$304$	0	0
$305$	384.000	1.25902
$306$	0	0
$307$	408.764	1.33148	0.665739	−	0.746184i	$-0.268116\pi$
$307$	408.764	1.33148	0.665739	+	0.746184i	$0.268116\pi$
$308$	0	0
$309$	− 145.492i	− 0.470849i
$310$	0	0
$311$	− 48.0000i	− 0.154341i	−0.997018	−	0.0771704i	$-0.975411\pi$
$311$	− 48.0000i	− 0.154341i	0.997018	−	0.0771704i	$-0.0245886\pi$
$312$	0	0
$313$	98.0000	0.313099	0.156550	−	0.987670i	$-0.449963\pi$
$313$	98.0000	0.313099	0.156550	+	0.987670i	$0.449963\pi$
$314$	0	0
$315$	−249.415	−0.791795
$316$	0	0
$317$	173.205i	0.546388i	0.961959	+	0.273194i	$0.0880801\pi$
$317$	173.205i	0.546388i	−0.961959	+	0.273194i	$0.911920\pi$
$318$	0	0
$319$	− 240.000i	− 0.752351i
$320$	0	0
$321$	−228.000	−0.710280
$322$	0	0
$323$	−484.974	−1.50147
$324$	0	0
$325$	318.697i	0.980607i
$326$	0	0
$327$	− 312.000i	− 0.954128i
$328$	0	0
$329$	−864.000	−2.62614
$330$	0	0
$331$	20.7846	0.0627934	0.0313967	−	0.999507i	$-0.490004\pi$
$331$	20.7846	0.0627934	0.0313967	+	0.999507i	$0.490004\pi$
$332$	0	0
$333$	83.1384i	0.249665i
$334$	0	0
$335$	− 624.000i	− 1.86269i
$336$	0	0
$337$	50.0000	0.148368	0.0741840	−	0.997245i	$-0.476365\pi$
$337$	50.0000	0.148368	0.0741840	+	0.997245i	$0.476365\pi$
$338$	0	0
$339$	−225.167	−0.664208
$340$	0	0
$341$	83.1384i	0.243808i
$342$	0	0
$343$	− 552.000i	− 1.60933i
$344$	0	0
$345$	−288.000	−0.834783
$346$	0	0
$347$	408.764	1.17799	0.588997	−	0.808135i	$-0.299523\pi$
$347$	408.764	1.17799	0.588997	+	0.808135i	$0.299523\pi$
$348$	0	0
$349$	− 498.831i	− 1.42931i	−0.699475	−	0.714657i	$-0.746583\pi$
$349$	− 498.831i	− 1.42931i	0.699475	−	0.714657i	$-0.253417\pi$
$350$	0	0
$351$	72.0000i	0.205128i
$352$	0	0
$353$	−542.000	−1.53541	−0.767705	−	0.640803i	$-0.778602\pi$
$353$	−542.000	−1.53541	−0.767705	+	0.640803i	$0.778602\pi$
$354$	0	0
$355$	−166.277	−0.468386
$356$	0	0
$357$	− 290.985i	− 0.815083i
$358$	0	0
$359$	312.000i	0.869081i	0.900652	+	0.434540i	$0.143089\pi$
$359$	312.000i	0.869081i	−0.900652	+	0.434540i	$0.856911\pi$
$360$	0	0
$361$	839.000	2.32410
$362$	0	0
$363$	126.440	0.348319
$364$	0	0
$365$	− 346.410i	− 0.949069i
$366$	0	0
$367$	276.000i	0.752044i	0.926611	+	0.376022i	$0.122708\pi$
$367$	276.000i	0.752044i	−0.926611	+	0.376022i	$0.877292\pi$
$368$	0	0
$369$	−42.0000	−0.113821
$370$	0	0
$371$	748.246	2.01684
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 24.0000i	− 0.0640000i
$376$	0	0
$377$	480.000	1.27321
$378$	0	0
$379$	325.626	0.859170	0.429585	−	0.903026i	$-0.358660\pi$
$379$	325.626	0.859170	0.429585	+	0.903026i	$0.358660\pi$
$380$	0	0
$381$	− 353.338i	− 0.927397i
$382$	0	0
$383$	720.000i	1.87990i	0.341318	+	0.939948i	$0.389127\pi$
$383$	720.000i	1.87990i	−0.341318	+	0.939948i	$0.610873\pi$
$384$	0	0
$385$	576.000	1.49610
$386$	0	0
$387$	−20.7846	−0.0537070
$388$	0	0
$389$	− 20.7846i	− 0.0534309i	−0.999643	−	0.0267154i	$-0.991495\pi$
$389$	− 20.7846i	− 0.0534309i	0.999643	−	0.0267154i	$-0.00850480\pi$
$390$	0	0
$391$	− 336.000i	− 0.859335i
$392$	0	0
$393$	−36.0000	−0.0916031
$394$	0	0
$395$	83.1384	0.210477
$396$	0	0
$397$	− 221.703i	− 0.558445i	−0.960226	−	0.279222i	$-0.909923\pi$
$397$	− 221.703i	− 0.558445i	0.960226	−	0.279222i	$-0.0900766\pi$
$398$	0	0
$399$	720.000i	1.80451i
$400$	0	0
$401$	−178.000	−0.443890	−0.221945	−	0.975059i	$-0.571241\pi$
$401$	−178.000	−0.443890	−0.221945	+	0.975059i	$0.571241\pi$
$402$	0	0
$403$	−166.277	−0.412598
$404$	0	0
$405$	62.3538i	0.153960i
$406$	0	0
$407$	− 192.000i	− 0.471744i
$408$	0	0
$409$	−142.000	−0.347188	−0.173594	−	0.984817i	$-0.555538\pi$
$409$	−142.000	−0.347188	−0.173594	+	0.984817i	$0.555538\pi$
$410$	0	0
$411$	356.802	0.868133
$412$	0	0
$413$	581.969i	1.40913i
$414$	0	0
$415$	144.000i	0.346988i
$416$	0	0
$417$	−84.0000	−0.201439
$418$	0	0
$419$	103.923	0.248026	0.124013	−	0.992281i	$-0.460423\pi$
$419$	103.923	0.248026	0.124013	+	0.992281i	$0.460423\pi$
$420$	0	0
$421$	263.272i	0.625349i	0.949860	+	0.312674i	$0.101225\pi$
$421$	263.272i	0.625349i	−0.949860	+	0.312674i	$0.898775\pi$
$422$	0	0
$423$	216.000i	0.510638i
$424$	0	0
$425$	−322.000	−0.757647
$426$	0	0
$427$	665.108	1.55763
$428$	0	0
$429$	− 166.277i	− 0.387592i
$430$	0	0
$431$	− 168.000i	− 0.389791i	−0.980824	−	0.194896i	$-0.937563\pi$
$431$	− 168.000i	− 0.389791i	0.980824	−	0.194896i	$-0.0624368\pi$
$432$	0	0
$433$	526.000	1.21478	0.607390	−	0.794404i	$-0.292216\pi$
$433$	526.000	1.21478	0.607390	+	0.794404i	$0.292216\pi$
$434$	0	0
$435$	415.692	0.955614
$436$	0	0
$437$	831.384i	1.90248i
$438$	0	0
$439$	− 444.000i	− 1.01139i	−0.862712	−	0.505695i	$-0.831236\pi$
$439$	− 444.000i	− 1.01139i	0.862712	−	0.505695i	$-0.168764\pi$
$440$	0	0
$441$	−285.000	−0.646259
$442$	0	0
$443$	214.774	0.484818	0.242409	−	0.970174i	$-0.422062\pi$
$443$	214.774	0.484818	0.242409	+	0.970174i	$0.422062\pi$
$444$	0	0
$445$	429.549i	0.965278i
$446$	0	0
$447$	− 348.000i	− 0.778523i
$448$	0	0
$449$	494.000	1.10022	0.550111	−	0.835091i	$-0.314585\pi$
$449$	494.000	1.10022	0.550111	+	0.835091i	$0.314585\pi$
$450$	0	0
$451$	96.9948	0.215066
$452$	0	0
$453$	62.3538i	0.137646i
$454$	0	0
$455$	1152.00i	2.53187i
$456$	0	0
$457$	−46.0000	−0.100656	−0.0503282	−	0.998733i	$-0.516027\pi$
$457$	−46.0000	−0.100656	−0.0503282	+	0.998733i	$0.516027\pi$
$458$	0	0
$459$	−72.7461	−0.158488
$460$	0	0
$461$	769.031i	1.66818i	0.551629	+	0.834090i	$0.314006\pi$
$461$	769.031i	1.66818i	−0.551629	+	0.834090i	$0.685994\pi$
$462$	0	0
$463$	− 132.000i	− 0.285097i	−0.989788	−	0.142549i	$-0.954470\pi$
$463$	− 132.000i	− 0.285097i	0.989788	−	0.142549i	$-0.0455297\pi$
$464$	0	0
$465$	−144.000	−0.309677
$466$	0	0
$467$	−117.779	−0.252204	−0.126102	−	0.992017i	$-0.540247\pi$
$467$	−117.779	−0.252204	−0.126102	+	0.992017i	$0.540247\pi$
$468$	0	0
$469$	− 1080.80i	− 2.30448i
$470$	0	0
$471$	48.0000i	0.101911i
$472$	0	0
$473$	48.0000	0.101480
$474$	0	0
$475$	796.743	1.67735
$476$	0	0
$477$	− 187.061i	− 0.392162i
$478$	0	0
$479$	− 408.000i	− 0.851775i	−0.904776	−	0.425887i	$-0.859962\pi$
$479$	− 408.000i	− 0.851775i	0.904776	−	0.425887i	$-0.140038\pi$
$480$	0	0
$481$	384.000	0.798337
$482$	0	0
$483$	−498.831	−1.03278
$484$	0	0
$485$	− 1011.52i	− 2.08560i
$486$	0	0
$487$	444.000i	0.911704i	0.890056	+	0.455852i	$0.150665\pi$
$487$	444.000i	0.911704i	−0.890056	+	0.455852i	$0.849335\pi$
$488$	0	0
$489$	108.000	0.220859
$490$	0	0
$491$	824.456	1.67914	0.839568	−	0.543254i	$-0.182808\pi$
$491$	824.456	1.67914	0.839568	+	0.543254i	$0.182808\pi$
$492$	0	0
$493$	484.974i	0.983721i
$494$	0	0
$495$	− 144.000i	− 0.290909i
$496$	0	0
$497$	−288.000	−0.579477
$498$	0	0
$499$	381.051	0.763630	0.381815	−	0.924239i	$-0.375299\pi$
$499$	381.051	0.763630	0.381815	+	0.924239i	$0.375299\pi$
$500$	0	0
$501$	415.692i	0.829725i
$502$	0	0
$503$	− 744.000i	− 1.47913i	−0.673088	−	0.739563i	$-0.735032\pi$
$503$	− 744.000i	− 1.47913i	0.673088	−	0.739563i	$-0.264968\pi$
$504$	0	0
$505$	−144.000	−0.285149
$506$	0	0
$507$	39.8372	0.0785743
$508$	0	0
$509$	− 852.169i	− 1.67420i	−0.547048	−	0.837101i	$-0.684249\pi$
$509$	− 852.169i	− 1.67420i	0.547048	−	0.837101i	$-0.315751\pi$
$510$	0	0
$511$	− 600.000i	− 1.17417i
$512$	0	0
$513$	180.000	0.350877
$514$	0	0
$515$	−581.969	−1.13004
$516$	0	0
$517$	− 498.831i	− 0.964856i
$518$	0	0
$519$	− 228.000i	− 0.439306i
$520$	0	0
$521$	82.0000	0.157390	0.0786948	−	0.996899i	$-0.474925\pi$
$521$	82.0000	0.157390	0.0786948	+	0.996899i	$0.474925\pi$
$522$	0	0
$523$	−311.769	−0.596117	−0.298058	−	0.954548i	$-0.596339\pi$
$523$	−311.769	−0.596117	−0.298058	+	0.954548i	$0.596339\pi$
$524$	0	0
$525$	478.046i	0.910564i
$526$	0	0
$527$	− 168.000i	− 0.318786i
$528$	0	0
$529$	−47.0000	−0.0888469
$530$	0	0
$531$	145.492	0.273997
$532$	0	0
$533$	193.990i	0.363958i
$534$	0	0
$535$	912.000i	1.70467i
$536$	0	0
$537$	−468.000	−0.871508
$538$	0	0
$539$	658.179	1.22111
$540$	0	0
$541$	− 96.9948i	− 0.179288i	−0.995974	−	0.0896440i	$-0.971427\pi$
$541$	− 96.9948i	− 0.179288i	0.995974	−	0.0896440i	$-0.0285729\pi$
$542$	0	0
$543$	− 360.000i	− 0.662983i
$544$	0	0
$545$	−1248.00	−2.28991
$546$	0	0
$547$	−34.6410	−0.0633291	−0.0316645	−	0.999499i	$-0.510081\pi$
$547$	−34.6410	−0.0633291	−0.0316645	+	0.999499i	$0.510081\pi$
$548$	0	0
$549$	− 166.277i	− 0.302872i
$550$	0	0
$551$	− 1200.00i	− 2.17786i
$552$	0	0
$553$	144.000	0.260398
$554$	0	0
$555$	332.554	0.599196
$556$	0	0
$557$	− 949.164i	− 1.70406i	−0.523489	−	0.852032i	$-0.675370\pi$
$557$	− 949.164i	− 1.70406i	0.523489	−	0.852032i	$-0.324630\pi$
$558$	0	0
$559$	96.0000i	0.171735i
$560$	0	0
$561$	168.000	0.299465
$562$	0	0
$563$	−949.164	−1.68590	−0.842952	−	0.537989i	$-0.819184\pi$
$563$	−949.164	−1.68590	−0.842952	+	0.537989i	$0.819184\pi$
$564$	0	0
$565$	900.666i	1.59410i
$566$	0	0
$567$	108.000i	0.190476i
$568$	0	0
$569$	658.000	1.15641	0.578207	−	0.815890i	$-0.303752\pi$
$569$	658.000	1.15641	0.578207	+	0.815890i	$0.303752\pi$
$570$	0	0
$571$	−256.344	−0.448938	−0.224469	−	0.974481i	$-0.572065\pi$
$571$	−256.344	−0.448938	−0.224469	+	0.974481i	$0.572065\pi$
$572$	0	0
$573$	− 415.692i	− 0.725466i
$574$	0	0
$575$	552.000i	0.960000i
$576$	0	0
$577$	−526.000	−0.911612	−0.455806	−	0.890079i	$-0.650649\pi$
$577$	−526.000	−0.911612	−0.455806	+	0.890079i	$0.650649\pi$
$578$	0	0
$579$	3.46410	0.00598290
$580$	0	0
$581$	249.415i	0.429286i
$582$	0	0
$583$	432.000i	0.740995i
$584$	0	0
$585$	288.000	0.492308
$586$	0	0
$587$	464.190	0.790783	0.395391	−	0.918513i	$-0.370609\pi$
$587$	464.190	0.790783	0.395391	+	0.918513i	$0.370609\pi$
$588$	0	0
$589$	415.692i	0.705759i
$590$	0	0
$591$	− 228.000i	− 0.385787i
$592$	0	0
$593$	514.000	0.866779	0.433390	−	0.901207i	$-0.357317\pi$
$593$	514.000	0.866779	0.433390	+	0.901207i	$0.357317\pi$
$594$	0	0
$595$	−1163.94	−1.95620
$596$	0	0
$597$	519.615i	0.870377i
$598$	0	0
$599$	408.000i	0.681135i	0.940220	+	0.340568i	$0.110619\pi$
$599$	408.000i	0.681135i	−0.940220	+	0.340568i	$0.889381\pi$
$600$	0	0
$601$	−818.000	−1.36106	−0.680532	−	0.732718i	$-0.738251\pi$
$601$	−818.000	−1.36106	−0.680532	+	0.732718i	$0.738251\pi$
$602$	0	0
$603$	−270.200	−0.448093
$604$	0	0
$605$	− 505.759i	− 0.835965i
$606$	0	0
$607$	684.000i	1.12685i	0.826166	+	0.563427i	$0.190517\pi$
$607$	684.000i	1.12685i	−0.826166	+	0.563427i	$0.809483\pi$
$608$	0	0
$609$	720.000	1.18227
$610$	0	0
$611$	997.661	1.63283
$612$	0	0
$613$	498.831i	0.813753i	0.913483	+	0.406877i	$0.133382\pi$
$613$	498.831i	0.813753i	−0.913483	+	0.406877i	$0.866618\pi$
$614$	0	0
$615$	168.000i	0.273171i
$616$	0	0
$617$	−322.000	−0.521880	−0.260940	−	0.965355i	$-0.584032\pi$
$617$	−322.000	−0.521880	−0.260940	+	0.965355i	$0.584032\pi$
$618$	0	0
$619$	−90.0666	−0.145503	−0.0727517	−	0.997350i	$-0.523178\pi$
$619$	−90.0666	−0.145503	−0.0727517	+	0.997350i	$0.523178\pi$
$620$	0	0
$621$	124.708i	0.200817i
$622$	0	0
$623$	744.000i	1.19422i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	−415.692	−0.662986
$628$	0	0
$629$	387.979i	0.616819i
$630$	0	0
$631$	252.000i	0.399366i	0.979861	+	0.199683i	$0.0639912\pi$
$631$	252.000i	0.399366i	−0.979861	+	0.199683i	$0.936009\pi$
$632$	0	0
$633$	−468.000	−0.739336
$634$	0	0
$635$	−1413.35	−2.22575
$636$	0	0
$637$	1316.36i	2.06650i
$638$	0	0
$639$	72.0000i	0.112676i
$640$	0	0
$641$	−370.000	−0.577223	−0.288612	−	0.957446i	$-0.593194\pi$
$641$	−370.000	−0.577223	−0.288612	+	0.957446i	$0.593194\pi$
$642$	0	0
$643$	−1032.30	−1.60545	−0.802723	−	0.596352i	$-0.796616\pi$
$643$	−1032.30	−1.60545	−0.802723	+	0.596352i	$0.796616\pi$
$644$	0	0
$645$	83.1384i	0.128897i
$646$	0	0
$647$	− 216.000i	− 0.333849i	−0.985970	−	0.166924i	$-0.946616\pi$
$647$	− 216.000i	− 0.333849i	0.985970	−	0.166924i	$-0.0533835\pi$
$648$	0	0
$649$	−336.000	−0.517720
$650$	0	0
$651$	−249.415	−0.383126
$652$	0	0
$653$	− 145.492i	− 0.222806i	−0.993775	−	0.111403i	$-0.964466\pi$
$653$	− 145.492i	− 0.222806i	0.993775	−	0.111403i	$-0.0355344\pi$
$654$	0	0
$655$	144.000i	0.219847i
$656$	0	0
$657$	−150.000	−0.228311
$658$	0	0
$659$	−1060.02	−1.60852	−0.804260	−	0.594277i	$-0.797438\pi$
$659$	−1060.02	−1.60852	−0.804260	+	0.594277i	$0.797438\pi$
$660$	0	0
$661$	− 1302.50i	− 1.97050i	−0.171114	−	0.985251i	$-0.554737\pi$
$661$	− 1302.50i	− 1.97050i	0.171114	−	0.985251i	$-0.445263\pi$
$662$	0	0
$663$	336.000i	0.506787i
$664$	0	0
$665$	2880.00	4.33083
$666$	0	0
$667$	831.384	1.24645
$668$	0	0
$669$	228.631i	0.341750i
$670$	0	0
$671$	384.000i	0.572280i
$672$	0	0
$673$	1006.00	1.49480	0.747400	−	0.664375i	$-0.231302\pi$
$673$	1006.00	1.49480	0.747400	+	0.664375i	$0.231302\pi$
$674$	0	0
$675$	119.512	0.177054
$676$	0	0
$677$	− 270.200i	− 0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$677$	− 270.200i	− 0.399114i	0.979886	−	0.199557i	$-0.0639502\pi$
$678$	0	0
$679$	− 1752.00i	− 2.58027i
$680$	0	0
$681$	348.000	0.511013
$682$	0	0
$683$	−1198.58	−1.75487	−0.877437	−	0.479692i	$-0.840749\pi$
$683$	−1198.58	−1.75487	−0.877437	+	0.479692i	$0.840749\pi$
$684$	0	0
$685$	− 1427.21i	− 2.08352i
$686$	0	0
$687$	− 120.000i	− 0.174672i
$688$	0	0
$689$	−864.000	−1.25399
$690$	0	0
$691$	436.477	0.631660	0.315830	−	0.948816i	$-0.397717\pi$
$691$	436.477	0.631660	0.315830	+	0.948816i	$0.397717\pi$
$692$	0	0
$693$	− 249.415i	− 0.359907i
$694$	0	0
$695$	336.000i	0.483453i
$696$	0	0
$697$	−196.000	−0.281205
$698$	0	0
$699$	225.167	0.322127
$700$	0	0
$701$	588.897i	0.840082i	0.907505	+	0.420041i	$0.137984\pi$
$701$	588.897i	0.840082i	−0.907505	+	0.420041i	$0.862016\pi$
$702$	0	0
$703$	− 960.000i	− 1.36558i
$704$	0	0
$705$	864.000	1.22553
$706$	0	0
$707$	−249.415	−0.352780
$708$	0	0
$709$	568.113i	0.801287i	0.916234	+	0.400644i	$0.131213\pi$
$709$	568.113i	0.801287i	−0.916234	+	0.400644i	$0.868787\pi$
$710$	0	0
$711$	− 36.0000i	− 0.0506329i
$712$	0	0
$713$	−288.000	−0.403927
$714$	0	0
$715$	−665.108	−0.930220
$716$	0	0
$717$	− 166.277i	− 0.231906i
$718$	0	0
$719$	− 360.000i	− 0.500695i	−0.968156	−	0.250348i	$-0.919455\pi$
$719$	− 360.000i	− 0.500695i	0.968156	−	0.250348i	$-0.0805449\pi$
$720$	0	0
$721$	−1008.00	−1.39806
$722$	0	0
$723$	−329.090	−0.455172
$724$	0	0
$725$	− 796.743i	− 1.09896i
$726$	0	0
$727$	− 660.000i	− 0.907840i	−0.891042	−	0.453920i	$-0.850025\pi$
$727$	− 660.000i	− 0.907840i	0.891042	−	0.453920i	$-0.149975\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−96.9948	−0.132688
$732$	0	0
$733$	− 956.092i	− 1.30435i	−0.758066	−	0.652177i	$-0.773856\pi$
$733$	− 956.092i	− 1.30435i	0.758066	−	0.652177i	$-0.226144\pi$
$734$	0	0
$735$	1140.00i	1.55102i
$736$	0	0
$737$	624.000	0.846676
$738$	0	0
$739$	547.328	0.740633	0.370317	−	0.928906i	$-0.379249\pi$
$739$	547.328	0.740633	0.370317	+	0.928906i	$0.379249\pi$
$740$	0	0
$741$	− 831.384i	− 1.12198i
$742$	0	0
$743$	− 864.000i	− 1.16285i	−0.813599	−	0.581427i	$-0.802495\pi$
$743$	− 864.000i	− 1.16285i	0.813599	−	0.581427i	$-0.197505\pi$
$744$	0	0
$745$	−1392.00	−1.86846
$746$	0	0
$747$	62.3538	0.0834723
$748$	0	0
$749$	1579.63i	2.10899i
$750$	0	0
$751$	− 444.000i	− 0.591212i	−0.955310	−	0.295606i	$-0.904478\pi$
$751$	− 444.000i	− 0.591212i	0.955310	−	0.295606i	$-0.0955215\pi$
$752$	0	0
$753$	444.000	0.589641
$754$	0	0
$755$	249.415	0.330351
$756$	0	0
$757$	− 872.954i	− 1.15318i	−0.817035	−	0.576588i	$-0.804384\pi$
$757$	− 872.954i	− 1.15318i	0.817035	−	0.576588i	$-0.195616\pi$
$758$	0	0
$759$	− 288.000i	− 0.379447i
$760$	0	0
$761$	−14.0000	−0.0183968	−0.00919842	−	0.999958i	$-0.502928\pi$
$761$	−14.0000	−0.0183968	−0.00919842	+	0.999958i	$0.502928\pi$
$762$	0	0
$763$	−2161.60	−2.83303
$764$	0	0
$765$	290.985i	0.380372i
$766$	0	0
$767$	− 672.000i	− 0.876141i
$768$	0	0
$769$	−194.000	−0.252276	−0.126138	−	0.992013i	$-0.540258\pi$
$769$	−194.000	−0.252276	−0.126138	+	0.992013i	$0.540258\pi$
$770$	0	0
$771$	439.941	0.570611
$772$	0	0
$773$	713.605i	0.923163i	0.887098	+	0.461581i	$0.152718\pi$
$773$	713.605i	0.923163i	−0.887098	+	0.461581i	$0.847282\pi$
$774$	0	0
$775$	276.000i	0.356129i
$776$	0	0
$777$	576.000	0.741313
$778$	0	0
$779$	484.974	0.622560
$780$	0	0
$781$	− 166.277i	− 0.212903i
$782$	0	0
$783$	− 180.000i	− 0.229885i
$784$	0	0
$785$	192.000	0.244586
$786$	0	0
$787$	−1060.02	−1.34691	−0.673453	−	0.739230i	$-0.735190\pi$
$787$	−1060.02	−1.34691	−0.673453	+	0.739230i	$0.735190\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1560.00i	1.97219i
$792$	0	0
$793$	−768.000	−0.968474
$794$	0	0
$795$	−748.246	−0.941190
$796$	0	0
$797$	1461.85i	1.83419i	0.398667	+	0.917096i	$0.369473\pi$
$797$	1461.85i	1.83419i	−0.398667	+	0.917096i	$0.630527\pi$
$798$	0	0
$799$	1008.00i	1.26158i
$800$	0	0
$801$	186.000	0.232210
$802$	0	0
$803$	346.410	0.431395
$804$	0	0
$805$	1995.32i	2.47866i
$806$	0	0
$807$	− 12.0000i	− 0.0148699i
$808$	0	0
$809$	1138.00	1.40667	0.703337	−	0.710856i	$-0.251692\pi$
$809$	1138.00	1.40667	0.703337	+	0.710856i	$0.251692\pi$
$810$	0	0
$811$	630.466	0.777394	0.388697	−	0.921366i	$-0.372925\pi$
$811$	630.466	0.777394	0.388697	+	0.921366i	$0.372925\pi$
$812$	0	0
$813$	− 602.754i	− 0.741394i
$814$	0	0
$815$	− 432.000i	− 0.530061i
$816$	0	0
$817$	240.000	0.293758
$818$	0	0
$819$	498.831	0.609073
$820$	0	0
$821$	− 256.344i	− 0.312233i	−0.987739	−	0.156117i	$-0.950102\pi$
$821$	− 256.344i	− 0.312233i	0.987739	−	0.156117i	$-0.0498976\pi$
$822$	0	0
$823$	396.000i	0.481166i	0.970629	+	0.240583i	$0.0773387\pi$
$823$	396.000i	0.481166i	−0.970629	+	0.240583i	$0.922661\pi$
$824$	0	0
$825$	−276.000	−0.334545
$826$	0	0
$827$	−145.492	−0.175928	−0.0879639	−	0.996124i	$-0.528036\pi$
$827$	−145.492	−0.175928	−0.0879639	+	0.996124i	$0.528036\pi$
$828$	0	0
$829$	− 41.5692i	− 0.0501438i	−0.999686	−	0.0250719i	$-0.992019\pi$
$829$	− 41.5692i	− 0.0501438i	0.999686	−	0.0250719i	$-0.00798147\pi$
$830$	0	0
$831$	− 72.0000i	− 0.0866426i
$832$	0	0
$833$	−1330.00	−1.59664
$834$	0	0
$835$	1662.77	1.99134
$836$	0	0
$837$	62.3538i	0.0744968i
$838$	0	0
$839$	1512.00i	1.80215i	0.433668	+	0.901073i	$0.357219\pi$
$839$	1512.00i	1.80215i	−0.433668	+	0.901073i	$0.642781\pi$
$840$	0	0
$841$	−359.000	−0.426873
$842$	0	0
$843$	58.8897	0.0698573
$844$	0	0
$845$	− 159.349i	− 0.188578i
$846$	0	0
$847$	− 876.000i	− 1.03424i
$848$	0	0
$849$	540.000	0.636042
$850$	0	0
$851$	665.108	0.781560
$852$	0	0
$853$	748.246i	0.877193i	0.898684	+	0.438597i	$0.144524\pi$
$853$	748.246i	0.877193i	−0.898684	+	0.438597i	$0.855476\pi$
$854$	0	0
$855$	− 720.000i	− 0.842105i
$856$	0	0
$857$	1042.00	1.21587	0.607935	−	0.793987i	$-0.291998\pi$
$857$	1042.00	1.21587	0.607935	+	0.793987i	$0.291998\pi$
$858$	0	0
$859$	159.349	0.185505	0.0927524	−	0.995689i	$-0.470433\pi$
$859$	159.349	0.185505	0.0927524	+	0.995689i	$0.470433\pi$
$860$	0	0
$861$	290.985i	0.337961i
$862$	0	0
$863$	− 1104.00i	− 1.27926i	−0.768684	−	0.639629i	$-0.779088\pi$
$863$	− 1104.00i	− 1.27926i	0.768684	−	0.639629i	$-0.220912\pi$
$864$	0	0
$865$	−912.000	−1.05434
$866$	0	0
$867$	161.081	0.185791
$868$	0	0
$869$	83.1384i	0.0956714i
$870$	0	0
$871$	1248.00i	1.43284i
$872$	0	0
$873$	−438.000	−0.501718
$874$	0	0
$875$	−166.277	−0.190031
$876$	0	0
$877$	− 554.256i	− 0.631991i	−0.948761	−	0.315996i	$-0.897662\pi$
$877$	− 554.256i	− 0.631991i	0.948761	−	0.315996i	$-0.102338\pi$
$878$	0	0
$879$	− 276.000i	− 0.313993i
$880$	0	0
$881$	610.000	0.692395	0.346198	−	0.938162i	$-0.387473\pi$
$881$	610.000	0.692395	0.346198	+	0.938162i	$0.387473\pi$
$882$	0	0
$883$	76.2102	0.0863083	0.0431542	−	0.999068i	$-0.486259\pi$
$883$	76.2102	0.0863083	0.0431542	+	0.999068i	$0.486259\pi$
$884$	0	0
$885$	− 581.969i	− 0.657592i
$886$	0	0
$887$	− 384.000i	− 0.432920i	−0.976291	−	0.216460i	$-0.930549\pi$
$887$	− 384.000i	− 0.432920i	0.976291	−	0.216460i	$-0.0694511\pi$
$888$	0	0
$889$	−2448.00	−2.75366
$890$	0	0
$891$	−62.3538	−0.0699819
$892$	0	0
$893$	− 2494.15i	− 2.79300i
$894$	0	0
$895$	1872.00i	2.09162i
$896$	0	0
$897$	576.000	0.642140
$898$	0	0
$899$	415.692	0.462394
$900$	0	0
$901$	− 872.954i	− 0.968872i
$902$	0	0
$903$	144.000i	0.159468i
$904$	0	0
$905$	−1440.00	−1.59116
$906$	0	0
$907$	713.605	0.786775	0.393388	−	0.919373i	$-0.371303\pi$
$907$	713.605	0.786775	0.393388	+	0.919373i	$0.371303\pi$
$908$	0	0
$909$	62.3538i	0.0685961i
$910$	0	0
$911$	864.000i	0.948408i	0.880415	+	0.474204i	$0.157264\pi$
$911$	864.000i	0.948408i	−0.880415	+	0.474204i	$0.842736\pi$
$912$	0	0
$913$	−144.000	−0.157722
$914$	0	0
$915$	−665.108	−0.726893
$916$	0	0
$917$	249.415i	0.271991i
$918$	0	0
$919$	1068.00i	1.16213i	0.813856	+	0.581066i	$0.197364\pi$
$919$	1068.00i	1.16213i	−0.813856	+	0.581066i	$0.802636\pi$
$920$	0	0
$921$	−708.000	−0.768730
$922$	0	0
$923$	332.554	0.360297
$924$	0	0
$925$	− 637.395i	− 0.689075i
$926$	0	0
$927$	252.000i	0.271845i
$928$	0	0
$929$	110.000	0.118407	0.0592034	−	0.998246i	$-0.481144\pi$
$929$	110.000	0.118407	0.0592034	+	0.998246i	$0.481144\pi$
$930$	0	0
$931$	3290.90	3.53480
$932$	0	0
$933$	83.1384i	0.0891087i
$934$	0	0
$935$	− 672.000i	− 0.718717i
$936$	0	0
$937$	1630.00	1.73959	0.869797	−	0.493409i	$-0.164250\pi$
$937$	1630.00	1.73959	0.869797	+	0.493409i	$0.164250\pi$
$938$	0	0
$939$	−169.741	−0.180768
$940$	0	0
$941$	48.4974i	0.0515382i	0.999668	+	0.0257691i	$0.00820346\pi$
$941$	48.4974i	0.0515382i	−0.999668	+	0.0257691i	$0.991797\pi$
$942$	0	0
$943$	336.000i	0.356310i
$944$	0	0
$945$	432.000	0.457143
$946$	0	0
$947$	1738.98	1.83630	0.918152	−	0.396229i	$-0.129682\pi$
$947$	1738.98	1.83630	0.918152	+	0.396229i	$0.129682\pi$
$948$	0	0
$949$	692.820i	0.730053i
$950$	0	0
$951$	− 300.000i	− 0.315457i
$952$	0	0
$953$	1714.00	1.79853	0.899265	−	0.437403i	$-0.144102\pi$
$953$	1714.00	1.79853	0.899265	+	0.437403i	$0.144102\pi$
$954$	0	0
$955$	−1662.77	−1.74112
$956$	0	0
$957$	415.692i	0.434370i
$958$	0	0
$959$	− 2472.00i	− 2.57769i
$960$	0	0
$961$	817.000	0.850156
$962$	0	0
$963$	394.908	0.410081
$964$	0	0
$965$	− 13.8564i	− 0.0143590i
$966$	0	0
$967$	− 1212.00i	− 1.25336i	−0.779276	−	0.626680i	$-0.784413\pi$
$967$	− 1212.00i	− 1.25336i	0.779276	−	0.626680i	$-0.215587\pi$
$968$	0	0
$969$	840.000	0.866873
$970$	0	0
$971$	−311.769	−0.321080	−0.160540	−	0.987029i	$-0.551324\pi$
$971$	−311.769	−0.321080	−0.160540	+	0.987029i	$0.551324\pi$
$972$	0	0
$973$	581.969i	0.598118i
$974$	0	0
$975$	− 552.000i	− 0.566154i
$976$	0	0
$977$	14.0000	0.0143296	0.00716479	−	0.999974i	$-0.497719\pi$
$977$	14.0000	0.0143296	0.00716479	+	0.999974i	$0.497719\pi$
$978$	0	0
$979$	−429.549	−0.438763
$980$	0	0
$981$	540.400i	0.550866i
$982$	0	0
$983$	− 384.000i	− 0.390641i	−0.980739	−	0.195320i	$-0.937425\pi$
$983$	− 384.000i	− 0.390641i	0.980739	−	0.195320i	$-0.0625747\pi$
$984$	0	0
$985$	−912.000	−0.925888
$986$	0	0
$987$	1496.49	1.51620
$988$	0	0
$989$	166.277i	0.168126i
$990$	0	0
$991$	− 636.000i	− 0.641776i	−0.947117	−	0.320888i	$-0.896019\pi$
$991$	− 636.000i	− 0.641776i	0.947117	−	0.320888i	$-0.103981\pi$
$992$	0	0
$993$	−36.0000	−0.0362538
$994$	0	0
$995$	2078.46	2.08891
$996$	0	0
$997$	− 1441.07i	− 1.44540i	−0.691161	−	0.722701i	$-0.742900\pi$
$997$	− 1441.07i	− 1.44540i	0.691161	−	0.722701i	$-0.257100\pi$
$998$	0	0
$999$	− 144.000i	− 0.144144i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.b.b.319.2	yes	4
3.2	odd	2		1152.3.b.e.703.2		4
4.3	odd	2	inner	384.3.b.b.319.4	yes	4
8.3	odd	2	inner	384.3.b.b.319.1	✓	4
8.5	even	2	inner	384.3.b.b.319.3	yes	4
12.11	even	2		1152.3.b.e.703.1		4
16.3	odd	4		768.3.g.e.511.4		4
16.5	even	4		768.3.g.e.511.3		4
16.11	odd	4		768.3.g.e.511.1		4
16.13	even	4		768.3.g.e.511.2		4
24.5	odd	2		1152.3.b.e.703.4		4
24.11	even	2		1152.3.b.e.703.3		4
48.5	odd	4		2304.3.g.r.1279.3		4
48.11	even	4		2304.3.g.r.1279.4		4
48.29	odd	4		2304.3.g.r.1279.1		4
48.35	even	4		2304.3.g.r.1279.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.b.b.319.1	✓	4	8.3	odd	2	inner
384.3.b.b.319.2	yes	4	1.1	even	1	trivial
384.3.b.b.319.3	yes	4	8.5	even	2	inner
384.3.b.b.319.4	yes	4	4.3	odd	2	inner
768.3.g.e.511.1		4	16.11	odd	4
768.3.g.e.511.2		4	16.13	even	4
768.3.g.e.511.3		4	16.5	even	4
768.3.g.e.511.4		4	16.3	odd	4
1152.3.b.e.703.1		4	12.11	even	2
1152.3.b.e.703.2		4	3.2	odd	2
1152.3.b.e.703.3		4	24.11	even	2
1152.3.b.e.703.4		4	24.5	odd	2
2304.3.g.r.1279.1		4	48.29	odd	4
2304.3.g.r.1279.2		4	48.35	even	4
2304.3.g.r.1279.3		4	48.5	odd	4
2304.3.g.r.1279.4		4	48.11	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} -6.92820 q^{11} -13.8564i q^{13} -12.0000i q^{15} +14.0000 q^{17} -34.6410 q^{19} -20.7846i q^{21} -24.0000i q^{23} -23.0000 q^{25} -5.19615 q^{27} +34.6410i q^{29} -12.0000i q^{31} +12.0000 q^{33} -83.1384 q^{35} +27.7128i q^{37} +24.0000i q^{39} -14.0000 q^{41} -6.92820 q^{43} +20.7846i q^{45} +72.0000i q^{47} -95.0000 q^{49} -24.2487 q^{51} -62.3538i q^{53} -48.0000i q^{55} +60.0000 q^{57} +48.4974 q^{59} -55.4256i q^{61} +36.0000i q^{63} +96.0000 q^{65} -90.0666 q^{67} +41.5692i q^{69} +24.0000i q^{71} -50.0000 q^{73} +39.8372 q^{75} -83.1384i q^{77} -12.0000i q^{79} +9.00000 q^{81} +20.7846 q^{83} +96.9948i q^{85} -60.0000i q^{87} +62.0000 q^{89} +166.277 q^{91} +20.7846i q^{93} -240.000i q^{95} -146.000 q^{97} -20.7846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 56 q^{17} - 92 q^{25} + 48 q^{33} - 56 q^{41} - 380 q^{49} + 240 q^{57} + 384 q^{65} - 200 q^{73} + 36 q^{81} + 248 q^{89} - 584 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 56 * q^17 - 92 * q^25 + 48 * q^33 - 56 * q^41 - 380 * q^49 + 240 * q^57 + 384 * q^65 - 200 * q^73 + 36 * q^81 + 248 * q^89 - 584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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