LMFDB - Embedded newform 2304.3.g.r.1279.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (of order $2$, degree $1$, not 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:	$62.7794529086$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.r.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.92820 q^{5} +12.0000i q^{7} +O(q^{10})$	$q+6.92820 q^{5} +12.0000i q^{7} -6.92820i q^{11} -13.8564 q^{13} -14.0000 q^{17} -34.6410i q^{19} +24.0000i q^{23} +23.0000 q^{25} -34.6410 q^{29} +12.0000i q^{31} +83.1384i q^{35} -27.7128 q^{37} -14.0000 q^{41} +6.92820i q^{43} +72.0000i q^{47} -95.0000 q^{49} -62.3538 q^{53} -48.0000i q^{55} +48.4974i q^{59} -55.4256 q^{61} -96.0000 q^{65} -90.0666i q^{67} -24.0000i q^{71} +50.0000 q^{73} +83.1384 q^{77} +12.0000i q^{79} -20.7846i q^{83} -96.9948 q^{85} +62.0000 q^{89} -166.277i q^{91} -240.000i q^{95} -146.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 56 q^{17} + 92 q^{25} - 56 q^{41} - 380 q^{49} - 384 q^{65} + 200 q^{73} + 248 q^{89} - 584 q^{97}+O(q^{100}) $ 4 * q - 56 * q^17 + 92 * q^25 - 56 * q^41 - 380 * q^49 - 384 * q^65 + 200 * q^73 + 248 * q^89 - 584 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.92820	1.38564	0.692820	−	0.721110i	$-0.256368\pi$
$5$	6.92820	1.38564	0.692820	+	0.721110i	$0.256368\pi$
$6$	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
$9$	0	0
$10$	0	0
$11$	− 6.92820i	− 0.629837i	−0.949119	−	0.314918i	$-0.898023\pi$
$11$	− 6.92820i	− 0.629837i	0.949119	−	0.314918i	$-0.101977\pi$
$12$	0	0
$13$	−13.8564	−1.06588	−0.532939	−	0.846154i	$-0.678912\pi$
$13$	−13.8564	−1.06588	−0.532939	+	0.846154i	$0.678912\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−14.0000	−0.823529	−0.411765	−	0.911290i	$-0.635087\pi$
$17$	−14.0000	−0.823529	−0.411765	+	0.911290i	$0.635087\pi$
$18$	0	0
$19$	− 34.6410i	− 1.82321i	−0.411066	−	0.911606i	$-0.634843\pi$
$19$	− 34.6410i	− 1.82321i	0.411066	−	0.911606i	$-0.365157\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	24.0000i	1.04348i	0.853105	+	0.521739i	$0.174717\pi$
$23$	24.0000i	1.04348i	−0.853105	+	0.521739i	$0.825283\pi$
$24$	0	0
$25$	23.0000	0.920000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−34.6410	−1.19452	−0.597259	−	0.802049i	$-0.703744\pi$
$29$	−34.6410	−1.19452	−0.597259	+	0.802049i	$0.703744\pi$
$30$	0	0
$31$	12.0000i	0.387097i	0.981091	+	0.193548i	$0.0619996\pi$
$31$	12.0000i	0.387097i	−0.981091	+	0.193548i	$0.938000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	83.1384i	2.37538i
$36$	0	0
$37$	−27.7128	−0.748995	−0.374497	−	0.927228i	$-0.622185\pi$
$37$	−27.7128	−0.748995	−0.374497	+	0.927228i	$0.622185\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	6.92820i	0.161121i	0.996750	+	0.0805605i	$0.0256710\pi$
$43$	6.92820i	0.161121i	−0.996750	+	0.0805605i	$0.974329\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	−62.3538	−1.17649	−0.588244	−	0.808684i	$-0.700180\pi$
$53$	−62.3538	−1.17649	−0.588244	+	0.808684i	$0.700180\pi$
$54$	0	0
$55$	− 48.0000i	− 0.872727i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	48.4974i	0.821990i	0.911638	+	0.410995i	$0.134819\pi$
$59$	48.4974i	0.821990i	−0.911638	+	0.410995i	$0.865181\pi$
$60$	0	0
$61$	−55.4256	−0.908617	−0.454308	−	0.890844i	$-0.650114\pi$
$61$	−55.4256	−0.908617	−0.454308	+	0.890844i	$0.650114\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−96.0000	−1.47692
$66$	0	0
$67$	− 90.0666i	− 1.34428i	−0.740425	−	0.672139i	$-0.765376\pi$
$67$	− 90.0666i	− 1.34428i	0.740425	−	0.672139i	$-0.234624\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.0000i	− 0.338028i	−0.985614	−	0.169014i	$-0.945942\pi$
$71$	− 24.0000i	− 0.338028i	0.985614	−	0.169014i	$-0.0540583\pi$
$72$	0	0
$73$	50.0000	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$73$	50.0000	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	83.1384	1.07972
$78$	0	0
$79$	12.0000i	0.151899i	0.997112	+	0.0759494i	$0.0241987\pi$
$79$	12.0000i	0.151899i	−0.997112	+	0.0759494i	$0.975801\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 20.7846i	− 0.250417i	−0.992130	−	0.125208i	$-0.960040\pi$
$83$	− 20.7846i	− 0.250417i	0.992130	−	0.125208i	$-0.0399600\pi$
$84$	0	0
$85$	−96.9948	−1.14112
$86$	0	0
$87$	0	0
$88$	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
$91$	− 166.277i	− 1.82722i
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	20.7846	0.205788	0.102894	−	0.994692i	$-0.467190\pi$
$101$	20.7846	0.205788	0.102894	+	0.994692i	$0.467190\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	131.636i	1.23024i	0.788433	+	0.615121i	$0.210893\pi$
$107$	131.636i	1.23024i	−0.788433	+	0.615121i	$0.789107\pi$
$108$	0	0
$109$	180.133	1.65260	0.826299	−	0.563231i	$-0.190442\pi$
$109$	180.133	1.65260	0.826299	+	0.563231i	$0.190442\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−130.000	−1.15044	−0.575221	−	0.817998i	$-0.695084\pi$
$113$	−130.000	−1.15044	−0.575221	+	0.817998i	$0.695084\pi$
$114$	0	0
$115$	166.277i	1.44589i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 168.000i	− 1.41176i
$120$	0	0
$121$	73.0000	0.603306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−13.8564	−0.110851
$126$	0	0
$127$	− 204.000i	− 1.60630i	−0.595777	−	0.803150i	$-0.703156\pi$
$127$	− 204.000i	− 1.60630i	0.595777	−	0.803150i	$-0.296844\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.7846i	− 0.158661i	−0.996848	−	0.0793306i	$-0.974722\pi$
$131$	− 20.7846i	− 0.158661i	0.996848	−	0.0793306i	$-0.0252783\pi$
$132$	0	0
$133$	415.692	3.12551
$134$	0	0
$135$	0	0
$136$	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
$139$	− 48.4974i	− 0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$	− 48.4974i	− 0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	96.0000i	0.671329i
$144$	0	0
$145$	−240.000	−1.65517
$146$	0	0
$147$	0	0
$148$	0	0
$149$	200.918	1.34844	0.674221	−	0.738530i	$-0.264479\pi$
$149$	200.918	1.34844	0.674221	+	0.738530i	$0.264479\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	83.1384i	0.536377i
$156$	0	0
$157$	−27.7128	−0.176515	−0.0882574	−	0.996098i	$-0.528130\pi$
$157$	−27.7128	−0.176515	−0.0882574	+	0.996098i	$0.528130\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−288.000	−1.78882
$162$	0	0
$163$	− 62.3538i	− 0.382539i	−0.981538	−	0.191269i	$-0.938740\pi$
$163$	− 62.3538i	− 0.382539i	0.981538	−	0.191269i	$-0.0612604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	240.000i	1.43713i	0.695462	+	0.718563i	$0.255200\pi$
$167$	240.000i	1.43713i	−0.695462	+	0.718563i	$0.744800\pi$
$168$	0	0
$169$	23.0000	0.136095
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−131.636	−0.760901	−0.380450	−	0.924801i	$-0.624231\pi$
$173$	−131.636	−0.760901	−0.380450	+	0.924801i	$0.624231\pi$
$174$	0	0
$175$	276.000i	1.57714i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 270.200i	− 1.50950i	−0.656014	−	0.754748i	$-0.727759\pi$
$179$	− 270.200i	− 1.50950i	0.656014	−	0.754748i	$-0.272241\pi$
$180$	0	0
$181$	−207.846	−1.14832	−0.574160	−	0.818743i	$-0.694671\pi$
$181$	−207.846	−1.14832	−0.574160	+	0.818743i	$0.694671\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−192.000	−1.03784
$186$	0	0
$187$	96.9948i	0.518689i
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	131.636	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$197$	131.636	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	− 415.692i	− 2.04774i
$204$	0	0
$205$	−96.9948	−0.473146
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−240.000	−1.14833
$210$	0	0
$211$	270.200i	1.28057i	0.768138	+	0.640284i	$0.221183\pi$
$211$	270.200i	1.28057i	−0.768138	+	0.640284i	$0.778817\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	48.0000i	0.223256i
$216$	0	0
$217$	−144.000	−0.663594
$218$	0	0
$219$	0	0
$220$	0	0
$221$	193.990	0.877781
$222$	0	0
$223$	132.000i	0.591928i	0.955199	+	0.295964i	$0.0956409\pi$
$223$	132.000i	0.591928i	−0.955199	+	0.295964i	$0.904359\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	200.918i	0.885101i	0.896744	+	0.442550i	$0.145926\pi$
$227$	200.918i	0.885101i	−0.896744	+	0.442550i	$0.854074\pi$
$228$	0	0
$229$	−69.2820	−0.302542	−0.151271	−	0.988492i	$-0.548337\pi$
$229$	−69.2820	−0.302542	−0.151271	+	0.988492i	$0.548337\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	498.831i	2.12268i
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	−658.179	−2.68645
$246$	0	0
$247$	480.000i	1.94332i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 256.344i	− 1.02129i	−0.859792	−	0.510644i	$-0.829407\pi$
$251$	− 256.344i	− 1.02129i	0.859792	−	0.510644i	$-0.170593\pi$
$252$	0	0
$253$	166.277	0.657221
$254$	0	0
$255$	0	0
$256$	0	0
$257$	254.000	0.988327	0.494163	−	0.869369i	$-0.335474\pi$
$257$	254.000	0.988327	0.494163	+	0.869369i	$0.335474\pi$
$258$	0	0
$259$	− 332.554i	− 1.28399i
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	−6.92820	−0.0257554	−0.0128777	−	0.999917i	$-0.504099\pi$
$269$	−6.92820	−0.0257554	−0.0128777	+	0.999917i	$0.504099\pi$
$270$	0	0
$271$	− 348.000i	− 1.28413i	−0.766649	−	0.642066i	$-0.778077\pi$
$271$	− 348.000i	− 1.28413i	0.766649	−	0.642066i	$-0.221923\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 159.349i	− 0.579450i
$276$	0	0
$277$	−41.5692	−0.150069	−0.0750347	−	0.997181i	$-0.523907\pi$
$277$	−41.5692	−0.150069	−0.0750347	+	0.997181i	$0.523907\pi$
$278$	0	0
$279$	0	0
$280$	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
$283$	311.769i	1.10166i	0.834618	+	0.550829i	$0.185688\pi$
$283$	311.769i	1.10166i	−0.834618	+	0.550829i	$0.814312\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 168.000i	− 0.585366i
$288$	0	0
$289$	−93.0000	−0.321799
$290$	0	0
$291$	0	0
$292$	0	0
$293$	159.349	0.543852	0.271926	−	0.962318i	$-0.412339\pi$
$293$	159.349	0.543852	0.271926	+	0.962318i	$0.412339\pi$
$294$	0	0
$295$	336.000i	1.13898i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 332.554i	− 1.11222i
$300$	0	0
$301$	−83.1384	−0.276207
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−384.000	−1.25902
$306$	0	0
$307$	408.764i	1.33148i	0.746184	+	0.665739i	$0.231884\pi$
$307$	408.764i	1.33148i	−0.746184	+	0.665739i	$0.768116\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	48.0000i	0.154341i	0.997018	+	0.0771704i	$0.0245886\pi$
$311$	48.0000i	0.154341i	−0.997018	+	0.0771704i	$0.975411\pi$
$312$	0	0
$313$	−98.0000	−0.313099	−0.156550	−	0.987670i	$-0.550037\pi$
$313$	−98.0000	−0.313099	−0.156550	+	0.987670i	$0.550037\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−173.205	−0.546388	−0.273194	−	0.961959i	$-0.588080\pi$
$317$	−173.205	−0.546388	−0.273194	+	0.961959i	$0.588080\pi$
$318$	0	0
$319$	240.000i	0.752351i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	484.974i	1.50147i
$324$	0	0
$325$	−318.697	−0.980607
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−864.000	−2.62614
$330$	0	0
$331$	− 20.7846i	− 0.0627934i	−0.999507	−	0.0313967i	$-0.990004\pi$
$331$	− 20.7846i	− 0.0627934i	0.999507	−	0.0313967i	$-0.00999552\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	83.1384	0.243808
$342$	0	0
$343$	− 552.000i	− 1.60933i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	408.764i	1.17799i	0.808135	+	0.588997i	$0.200477\pi$
$347$	408.764i	1.17799i	−0.808135	+	0.588997i	$0.799523\pi$
$348$	0	0
$349$	−498.831	−1.42931	−0.714657	−	0.699475i	$-0.753417\pi$
$349$	−498.831	−1.42931	−0.714657	+	0.699475i	$0.753417\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	542.000	1.53541	0.767705	−	0.640803i	$-0.221398\pi$
$353$	542.000	1.53541	0.767705	+	0.640803i	$0.221398\pi$
$354$	0	0
$355$	− 166.277i	− 0.468386i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 312.000i	− 0.869081i	−0.900652	−	0.434540i	$-0.856911\pi$
$359$	− 312.000i	− 0.869081i	0.900652	−	0.434540i	$-0.143089\pi$
$360$	0	0
$361$	−839.000	−2.32410
$362$	0	0
$363$	0	0
$364$	0	0
$365$	346.410	0.949069
$366$	0	0
$367$	− 276.000i	− 0.752044i	−0.926611	−	0.376022i	$-0.877292\pi$
$367$	− 276.000i	− 0.752044i	0.926611	−	0.376022i	$-0.122708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 748.246i	− 2.01684i
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	480.000	1.27321
$378$	0	0
$379$	− 325.626i	− 0.859170i	−0.903026	−	0.429585i	$-0.858660\pi$
$379$	− 325.626i	− 0.859170i	0.903026	−	0.429585i	$-0.141340\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	−20.7846	−0.0534309	−0.0267154	−	0.999643i	$-0.508505\pi$
$389$	−20.7846	−0.0534309	−0.0267154	+	0.999643i	$0.508505\pi$
$390$	0	0
$391$	− 336.000i	− 0.859335i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	83.1384i	0.210477i
$396$	0	0
$397$	−221.703	−0.558445	−0.279222	−	0.960226i	$-0.590077\pi$
$397$	−221.703	−0.558445	−0.279222	+	0.960226i	$0.590077\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	178.000	0.443890	0.221945	−	0.975059i	$-0.428759\pi$
$401$	178.000	0.443890	0.221945	+	0.975059i	$0.428759\pi$
$402$	0	0
$403$	− 166.277i	− 0.412598i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	192.000i	0.471744i
$408$	0	0
$409$	142.000	0.347188	0.173594	−	0.984817i	$-0.444462\pi$
$409$	142.000	0.347188	0.173594	+	0.984817i	$0.444462\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−581.969	−1.40913
$414$	0	0
$415$	− 144.000i	− 0.346988i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 103.923i	− 0.248026i	−0.992281	−	0.124013i	$-0.960423\pi$
$419$	− 103.923i	− 0.248026i	0.992281	−	0.124013i	$-0.0395765\pi$
$420$	0	0
$421$	−263.272	−0.625349	−0.312674	−	0.949860i	$-0.601225\pi$
$421$	−263.272	−0.625349	−0.312674	+	0.949860i	$0.601225\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−322.000	−0.757647
$426$	0	0
$427$	− 665.108i	− 1.55763i
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	831.384	1.90248
$438$	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
$441$	0	0
$442$	0	0
$443$	214.774i	0.484818i	0.970174	+	0.242409i	$0.0779376\pi$
$443$	214.774i	0.484818i	−0.970174	+	0.242409i	$0.922062\pi$
$444$	0	0
$445$	429.549	0.965278
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−494.000	−1.10022	−0.550111	−	0.835091i	$-0.685415\pi$
$449$	−494.000	−1.10022	−0.550111	+	0.835091i	$0.685415\pi$
$450$	0	0
$451$	96.9948i	0.215066i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1152.00i	− 2.53187i
$456$	0	0
$457$	46.0000	0.100656	0.0503282	−	0.998733i	$-0.483973\pi$
$457$	46.0000	0.100656	0.0503282	+	0.998733i	$0.483973\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−769.031	−1.66818	−0.834090	−	0.551629i	$-0.814006\pi$
$461$	−769.031	−1.66818	−0.834090	+	0.551629i	$0.814006\pi$
$462$	0	0
$463$	132.000i	0.285097i	0.989788	+	0.142549i	$0.0455297\pi$
$463$	132.000i	0.285097i	−0.989788	+	0.142549i	$0.954470\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	117.779i	0.252204i	0.992017	+	0.126102i	$0.0402467\pi$
$467$	117.779i	0.252204i	−0.992017	+	0.126102i	$0.959753\pi$
$468$	0	0
$469$	1080.80	2.30448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.0000	0.101480
$474$	0	0
$475$	− 796.743i	− 1.67735i
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−1011.52	−2.08560
$486$	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
$489$	0	0
$490$	0	0
$491$	824.456i	1.67914i	0.543254	+	0.839568i	$0.317192\pi$
$491$	824.456i	1.67914i	−0.543254	+	0.839568i	$0.682808\pi$
$492$	0	0
$493$	484.974	0.983721
$494$	0	0
$495$	0	0
$496$	0	0
$497$	288.000	0.579477
$498$	0	0
$499$	381.051i	0.763630i	0.924239	+	0.381815i	$0.124701\pi$
$499$	381.051i	0.763630i	−0.924239	+	0.381815i	$0.875299\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	744.000i	1.47913i	0.673088	+	0.739563i	$0.264968\pi$
$503$	744.000i	1.47913i	−0.673088	+	0.739563i	$0.735032\pi$
$504$	0	0
$505$	144.000	0.285149
$506$	0	0
$507$	0	0
$508$	0	0
$509$	852.169	1.67420	0.837101	−	0.547048i	$-0.184249\pi$
$509$	852.169	1.67420	0.837101	+	0.547048i	$0.184249\pi$
$510$	0	0
$511$	600.000i	1.17417i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	581.969i	1.13004i
$516$	0	0
$517$	498.831	0.964856
$518$	0	0
$519$	0	0
$520$	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
$523$	311.769i	0.596117i	0.954548	+	0.298058i	$0.0963390\pi$
$523$	311.769i	0.596117i	−0.954548	+	0.298058i	$0.903661\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 168.000i	− 0.318786i
$528$	0	0
$529$	−47.0000	−0.0888469
$530$	0	0
$531$	0	0
$532$	0	0
$533$	193.990	0.363958
$534$	0	0
$535$	912.000i	1.70467i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	658.179i	1.22111i
$540$	0	0
$541$	−96.9948	−0.179288	−0.0896440	−	0.995974i	$-0.528573\pi$
$541$	−96.9948	−0.179288	−0.0896440	+	0.995974i	$0.528573\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1248.00	2.28991
$546$	0	0
$547$	− 34.6410i	− 0.0633291i	−0.999499	−	0.0316645i	$-0.989919\pi$
$547$	− 34.6410i	− 0.0633291i	0.999499	−	0.0316645i	$-0.0100808\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1200.00i	2.17786i
$552$	0	0
$553$	−144.000	−0.260398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	949.164	1.70406	0.852032	−	0.523489i	$-0.175370\pi$
$557$	949.164	1.70406	0.852032	+	0.523489i	$0.175370\pi$
$558$	0	0
$559$	− 96.0000i	− 0.171735i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	949.164i	1.68590i	0.537989	+	0.842952i	$0.319184\pi$
$563$	949.164i	1.68590i	−0.537989	+	0.842952i	$0.680816\pi$
$564$	0	0
$565$	−900.666	−1.59410
$566$	0	0
$567$	0	0
$568$	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
$571$	256.344i	0.448938i	0.974481	+	0.224469i	$0.0720647\pi$
$571$	256.344i	0.448938i	−0.974481	+	0.224469i	$0.927935\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	249.415	0.429286
$582$	0	0
$583$	432.000i	0.740995i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	464.190i	0.790783i	0.918513	+	0.395391i	$0.129391\pi$
$587$	464.190i	0.790783i	−0.918513	+	0.395391i	$0.870609\pi$
$588$	0	0
$589$	415.692	0.705759
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−514.000	−0.866779	−0.433390	−	0.901207i	$-0.642683\pi$
$593$	−514.000	−0.866779	−0.433390	+	0.901207i	$0.642683\pi$
$594$	0	0
$595$	− 1163.94i	− 1.95620i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 408.000i	− 0.681135i	−0.940220	−	0.340568i	$-0.889381\pi$
$599$	− 408.000i	− 0.681135i	0.940220	−	0.340568i	$-0.110619\pi$
$600$	0	0
$601$	818.000	1.36106	0.680532	−	0.732718i	$-0.261749\pi$
$601$	818.000	1.36106	0.680532	+	0.732718i	$0.261749\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	505.759	0.835965
$606$	0	0
$607$	− 684.000i	− 1.12685i	−0.826166	−	0.563427i	$-0.809483\pi$
$607$	− 684.000i	− 1.12685i	0.826166	−	0.563427i	$-0.190517\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 997.661i	− 1.63283i
$612$	0	0
$613$	−498.831	−0.813753	−0.406877	−	0.913483i	$-0.633382\pi$
$613$	−498.831	−0.813753	−0.406877	+	0.913483i	$0.633382\pi$
$614$	0	0
$615$	0	0
$616$	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
$619$	90.0666i	0.145503i	0.997350	+	0.0727517i	$0.0231781\pi$
$619$	90.0666i	0.145503i	−0.997350	+	0.0727517i	$0.976822\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	744.000i	1.19422i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	387.979	0.616819
$630$	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
$633$	0	0
$634$	0	0
$635$	− 1413.35i	− 2.22575i
$636$	0	0
$637$	1316.36	2.06650
$638$	0	0
$639$	0	0
$640$	0	0
$641$	370.000	0.577223	0.288612	−	0.957446i	$-0.406806\pi$
$641$	370.000	0.577223	0.288612	+	0.957446i	$0.406806\pi$
$642$	0	0
$643$	− 1032.30i	− 1.60545i	−0.596352	−	0.802723i	$-0.703384\pi$
$643$	− 1032.30i	− 1.60545i	0.596352	−	0.802723i	$-0.296616\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	216.000i	0.333849i	0.985970	+	0.166924i	$0.0533835\pi$
$647$	216.000i	0.333849i	−0.985970	+	0.166924i	$0.946616\pi$
$648$	0	0
$649$	336.000	0.517720
$650$	0	0
$651$	0	0
$652$	0	0
$653$	145.492	0.222806	0.111403	−	0.993775i	$-0.464466\pi$
$653$	145.492	0.222806	0.111403	+	0.993775i	$0.464466\pi$
$654$	0	0
$655$	− 144.000i	− 0.219847i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1060.02i	1.60852i	0.594277	+	0.804260i	$0.297438\pi$
$659$	1060.02i	1.60852i	−0.594277	+	0.804260i	$0.702562\pi$
$660$	0	0
$661$	1302.50	1.97050	0.985251	−	0.171114i	$-0.0547365\pi$
$661$	1302.50	1.97050	0.985251	+	0.171114i	$0.0547365\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2880.00	4.33083
$666$	0	0
$667$	− 831.384i	− 1.24645i
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−270.200	−0.399114	−0.199557	−	0.979886i	$-0.563950\pi$
$677$	−270.200	−0.399114	−0.199557	+	0.979886i	$0.563950\pi$
$678$	0	0
$679$	− 1752.00i	− 2.58027i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1198.58i	− 1.75487i	−0.479692	−	0.877437i	$-0.659251\pi$
$683$	− 1198.58i	− 1.75487i	0.479692	−	0.877437i	$-0.340749\pi$
$684$	0	0
$685$	−1427.21	−2.08352
$686$	0	0
$687$	0	0
$688$	0	0
$689$	864.000	1.25399
$690$	0	0
$691$	436.477i	0.631660i	0.948816	+	0.315830i	$0.102283\pi$
$691$	436.477i	0.631660i	−0.948816	+	0.315830i	$0.897717\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 336.000i	− 0.483453i
$696$	0	0
$697$	196.000	0.281205
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−588.897	−0.840082	−0.420041	−	0.907505i	$-0.637984\pi$
$701$	−588.897	−0.840082	−0.420041	+	0.907505i	$0.637984\pi$
$702$	0	0
$703$	960.000i	1.36558i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	249.415i	0.352780i
$708$	0	0
$709$	−568.113	−0.801287	−0.400644	−	0.916234i	$-0.631213\pi$
$709$	−568.113	−0.801287	−0.400644	+	0.916234i	$0.631213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−288.000	−0.403927
$714$	0	0
$715$	665.108i	0.930220i
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	−796.743	−1.09896
$726$	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
$729$	0	0
$730$	0	0
$731$	− 96.9948i	− 0.132688i
$732$	0	0
$733$	−956.092	−1.30435	−0.652177	−	0.758066i	$-0.726144\pi$
$733$	−956.092	−1.30435	−0.652177	+	0.758066i	$0.726144\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−624.000	−0.846676
$738$	0	0
$739$	547.328i	0.740633i	0.928906	+	0.370317i	$0.120751\pi$
$739$	547.328i	0.740633i	−0.928906	+	0.370317i	$0.879249\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	864.000i	1.16285i	0.813599	+	0.581427i	$0.197505\pi$
$743$	864.000i	1.16285i	−0.813599	+	0.581427i	$0.802495\pi$
$744$	0	0
$745$	1392.00	1.86846
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1579.63	−2.10899
$750$	0	0
$751$	444.000i	0.591212i	0.955310	+	0.295606i	$0.0955215\pi$
$751$	444.000i	0.591212i	−0.955310	+	0.295606i	$0.904478\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 249.415i	− 0.330351i
$756$	0	0
$757$	872.954	1.15318	0.576588	−	0.817035i	$-0.304384\pi$
$757$	872.954	1.15318	0.576588	+	0.817035i	$0.304384\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	2161.60i	2.83303i
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	713.605	0.923163	0.461581	−	0.887098i	$-0.347282\pi$
$773$	713.605	0.923163	0.461581	+	0.887098i	$0.347282\pi$
$774$	0	0
$775$	276.000i	0.356129i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	484.974i	0.622560i
$780$	0	0
$781$	−166.277	−0.212903
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−192.000	−0.244586
$786$	0	0
$787$	− 1060.02i	− 1.34691i	−0.739230	−	0.673453i	$-0.764810\pi$
$787$	− 1060.02i	− 1.34691i	0.739230	−	0.673453i	$-0.235190\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1560.00i	− 1.97219i
$792$	0	0
$793$	768.000	0.968474
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1461.85	−1.83419	−0.917096	−	0.398667i	$-0.869473\pi$
$797$	−1461.85	−1.83419	−0.917096	+	0.398667i	$0.869473\pi$
$798$	0	0
$799$	− 1008.00i	− 1.26158i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 346.410i	− 0.431395i
$804$	0	0
$805$	−1995.32	−2.47866
$806$	0	0
$807$	0	0
$808$	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
$811$	− 630.466i	− 0.777394i	−0.921366	−	0.388697i	$-0.872925\pi$
$811$	− 630.466i	− 0.777394i	0.921366	−	0.388697i	$-0.127075\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 432.000i	− 0.530061i
$816$	0	0
$817$	240.000	0.293758
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−256.344	−0.312233	−0.156117	−	0.987739i	$-0.549898\pi$
$821$	−256.344	−0.312233	−0.156117	+	0.987739i	$0.549898\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	− 145.492i	− 0.175928i	−0.996124	−	0.0879639i	$-0.971964\pi$
$827$	− 145.492i	− 0.175928i	0.996124	−	0.0879639i	$-0.0280360\pi$
$828$	0	0
$829$	−41.5692	−0.0501438	−0.0250719	−	0.999686i	$-0.507981\pi$
$829$	−41.5692	−0.0501438	−0.0250719	+	0.999686i	$0.507981\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1330.00	1.59664
$834$	0	0
$835$	1662.77i	1.99134i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1512.00i	− 1.80215i	−0.433668	−	0.901073i	$-0.642781\pi$
$839$	− 1512.00i	− 1.80215i	0.433668	−	0.901073i	$-0.357219\pi$
$840$	0	0
$841$	359.000	0.426873
$842$	0	0
$843$	0	0
$844$	0	0
$845$	159.349	0.188578
$846$	0	0
$847$	876.000i	1.03424i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 665.108i	− 0.781560i
$852$	0	0
$853$	−748.246	−0.877193	−0.438597	−	0.898684i	$-0.644524\pi$
$853$	−748.246	−0.877193	−0.438597	+	0.898684i	$0.644524\pi$
$854$	0	0
$855$	0	0
$856$	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
$859$	− 159.349i	− 0.185505i	−0.995689	−	0.0927524i	$-0.970433\pi$
$859$	− 159.349i	− 0.185505i	0.995689	−	0.0927524i	$-0.0295665\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	83.1384	0.0956714
$870$	0	0
$871$	1248.00i	1.43284i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 166.277i	− 0.190031i
$876$	0	0
$877$	−554.256	−0.631991	−0.315996	−	0.948761i	$-0.602338\pi$
$877$	−554.256	−0.631991	−0.315996	+	0.948761i	$0.602338\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−610.000	−0.692395	−0.346198	−	0.938162i	$-0.612527\pi$
$881$	−610.000	−0.692395	−0.346198	+	0.938162i	$0.612527\pi$
$882$	0	0
$883$	76.2102i	0.0863083i	0.999068	+	0.0431542i	$0.0137407\pi$
$883$	76.2102i	0.0863083i	−0.999068	+	0.0431542i	$0.986259\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	384.000i	0.432920i	0.976291	+	0.216460i	$0.0694511\pi$
$887$	384.000i	0.432920i	−0.976291	+	0.216460i	$0.930549\pi$
$888$	0	0
$889$	2448.00	2.75366
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2494.15	2.79300
$894$	0	0
$895$	− 1872.00i	− 2.09162i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 415.692i	− 0.462394i
$900$	0	0
$901$	872.954	0.968872
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1440.00	−1.59116
$906$	0	0
$907$	− 713.605i	− 0.786775i	−0.919373	−	0.393388i	$-0.871303\pi$
$907$	− 713.605i	− 0.786775i	0.919373	−	0.393388i	$-0.128697\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	249.415	0.271991
$918$	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
$921$	0	0
$922$	0	0
$923$	332.554i	0.360297i
$924$	0	0
$925$	−637.395	−0.689075
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−110.000	−0.118407	−0.0592034	−	0.998246i	$-0.518856\pi$
$929$	−110.000	−0.118407	−0.0592034	+	0.998246i	$0.518856\pi$
$930$	0	0
$931$	3290.90i	3.53480i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	672.000i	0.718717i
$936$	0	0
$937$	−1630.00	−1.73959	−0.869797	−	0.493409i	$-0.835750\pi$
$937$	−1630.00	−1.73959	−0.869797	+	0.493409i	$0.835750\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48.4974	−0.0515382	−0.0257691	−	0.999668i	$-0.508203\pi$
$941$	−48.4974	−0.0515382	−0.0257691	+	0.999668i	$0.508203\pi$
$942$	0	0
$943$	− 336.000i	− 0.356310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1738.98i	− 1.83630i	−0.396229	−	0.918152i	$-0.629682\pi$
$947$	− 1738.98i	− 1.83630i	0.396229	−	0.918152i	$-0.370318\pi$
$948$	0	0
$949$	−692.820	−0.730053
$950$	0	0
$951$	0	0
$952$	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
$955$	1662.77i	1.74112i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2472.00i	− 2.57769i
$960$	0	0
$961$	817.000	0.850156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.8564	−0.0143590
$966$	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
$969$	0	0
$970$	0	0
$971$	− 311.769i	− 0.321080i	−0.987029	−	0.160540i	$-0.948676\pi$
$971$	− 311.769i	− 0.321080i	0.987029	−	0.160540i	$-0.0513236\pi$
$972$	0	0
$973$	581.969	0.598118
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.0000	−0.0143296	−0.00716479	−	0.999974i	$-0.502281\pi$
$977$	−14.0000	−0.0143296	−0.00716479	+	0.999974i	$0.502281\pi$
$978$	0	0
$979$	− 429.549i	− 0.438763i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	384.000i	0.390641i	0.980739	+	0.195320i	$0.0625747\pi$
$983$	384.000i	0.390641i	−0.980739	+	0.195320i	$0.937425\pi$
$984$	0	0
$985$	912.000	0.925888
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−166.277	−0.168126
$990$	0	0
$991$	636.000i	0.641776i	0.947117	+	0.320888i	$0.103981\pi$
$991$	636.000i	0.641776i	−0.947117	+	0.320888i	$0.896019\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2078.46i	− 2.08891i
$996$	0	0
$997$	1441.07	1.44540	0.722701	−	0.691161i	$-0.242900\pi$
$997$	1441.07	1.44540	0.722701	+	0.691161i	$0.242900\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.92820 q^{5} +12.0000i q^{7} +O(q^{10})\)	\(q+6.92820 q^{5} +12.0000i q^{7} -6.92820i q^{11} -13.8564 q^{13} -14.0000 q^{17} -34.6410i q^{19} +24.0000i q^{23} +23.0000 q^{25} -34.6410 q^{29} +12.0000i q^{31} +83.1384i q^{35} -27.7128 q^{37} -14.0000 q^{41} +6.92820i q^{43} +72.0000i q^{47} -95.0000 q^{49} -62.3538 q^{53} -48.0000i q^{55} +48.4974i q^{59} -55.4256 q^{61} -96.0000 q^{65} -90.0666i q^{67} -24.0000i q^{71} +50.0000 q^{73} +83.1384 q^{77} +12.0000i q^{79} -20.7846i q^{83} -96.9948 q^{85} +62.0000 q^{89} -166.277i q^{91} -240.000i q^{95} -146.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 56 q^{17} + 92 q^{25} - 56 q^{41} - 380 q^{49} - 384 q^{65} + 200 q^{73} + 248 q^{89} - 584 q^{97}+O(q^{100}) \) 4 * q - 56 * q^17 + 92 * q^25 - 56 * q^41 - 380 * q^49 - 384 * q^65 + 200 * q^73 + 248 * q^89 - 584 * q^97

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