LMFDB - Embedded newform 768.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

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:	yes
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{15} +6.00000 q^{17} -4.00000 q^{19} -3.46410 q^{21} -6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -3.46410 q^{29} +3.46410 q^{31} +12.0000 q^{35} -6.92820 q^{37} +6.00000 q^{41} -4.00000 q^{43} +3.46410 q^{45} +6.92820 q^{47} +5.00000 q^{49} -6.00000 q^{51} -3.46410 q^{53} +4.00000 q^{57} +12.0000 q^{59} +6.92820 q^{61} +3.46410 q^{63} +4.00000 q^{67} +6.92820 q^{69} +6.92820 q^{71} -2.00000 q^{73} -7.00000 q^{75} -10.3923 q^{79} +1.00000 q^{81} +20.7846 q^{85} +3.46410 q^{87} -6.00000 q^{89} -3.46410 q^{93} -13.8564 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{19} + 14 q^{25} - 2 q^{27} + 24 q^{35} + 12 q^{41} - 8 q^{43} + 10 q^{49} - 12 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{67} - 4 q^{73} - 14 q^{75} + 2 q^{81} - 12 q^{89}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^19 + 14 * q^25 - 2 * q^27 + 24 * q^35 + 12 * q^41 - 8 * q^43 + 10 * q^49 - 12 * q^51 + 8 * q^57 + 24 * q^59 + 8 * q^67 - 4 * q^73 - 14 * q^75 + 2 * q^81 - 12 * q^89 - 4 * q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.0000	2.02837
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	3.46410	0.516398
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.92820	0.887066	0.443533	−	0.896258i	$-0.353725\pi$
$61$	6.92820	0.887066	0.443533	+	0.896258i	$0.353725\pi$
$62$	0	0
$63$	3.46410	0.436436
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	6.92820	0.834058
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−7.00000	−0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.3923	−1.16923	−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923	−1.16923	−0.584613	+	0.811312i	$0.698754\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	20.7846	2.25441
$86$	0	0
$87$	3.46410	0.371391
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.46410	−0.359211
$94$	0	0
$95$	−13.8564	−1.42164
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.46410	−0.344691	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410	−0.344691	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	−17.3205	−1.70664	−0.853320	−	0.521387i	$-0.825415\pi$
$103$	−17.3205	−1.70664	−0.853320	+	0.521387i	$0.825415\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−24.0000	−2.23801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.7846	1.90532
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−3.46410	−0.307389	−0.153695	−	0.988118i	$-0.549117\pi$
$127$	−3.46410	−0.307389	−0.153695	+	0.988118i	$0.549117\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−13.8564	−1.20150
$134$	0	0
$135$	−3.46410	−0.298142
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−6.92820	−0.583460
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	−5.00000	−0.412393
$148$	0	0
$149$	−10.3923	−0.851371	−0.425685	−	0.904871i	$-0.639967\pi$
$149$	−10.3923	−0.851371	−0.425685	+	0.904871i	$0.639967\pi$
$150$	0	0
$151$	3.46410	0.281905	0.140952	−	0.990016i	$-0.454984\pi$
$151$	3.46410	0.281905	0.140952	+	0.990016i	$0.454984\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−6.92820	−0.552931	−0.276465	−	0.961024i	$-0.589163\pi$
$157$	−6.92820	−0.552931	−0.276465	+	0.961024i	$0.589163\pi$
$158$	0	0
$159$	3.46410	0.274721
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	17.3205	1.31685	0.658427	−	0.752645i	$-0.271222\pi$
$173$	17.3205	1.31685	0.658427	+	0.752645i	$0.271222\pi$
$174$	0	0
$175$	24.2487	1.83303
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	0	0
$183$	−6.92820	−0.512148
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.46410	−0.251976
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3923	0.740421	0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923	0.740421	0.370211	+	0.928948i	$0.379286\pi$
$198$	0	0
$199$	10.3923	0.736691	0.368345	−	0.929689i	$-0.379924\pi$
$199$	10.3923	0.736691	0.368345	+	0.929689i	$0.379924\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	20.7846	1.45166
$206$	0	0
$207$	−6.92820	−0.481543
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−6.92820	−0.474713
$214$	0	0
$215$	−13.8564	−0.944999
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.46410	−0.231973	−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410	−0.231973	−0.115987	+	0.993251i	$0.537003\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	27.7128	1.83131	0.915657	−	0.401960i	$-0.131671\pi$
$229$	27.7128	1.83131	0.915657	+	0.401960i	$0.131671\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	10.3923	0.675053
$238$	0	0
$239$	−27.7128	−1.79259	−0.896296	−	0.443455i	$-0.853752\pi$
$239$	−27.7128	−1.79259	−0.896296	+	0.443455i	$0.853752\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	17.3205	1.10657
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−20.7846	−1.30158
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	−3.46410	−0.214423
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−31.1769	−1.90089	−0.950445	−	0.310893i	$-0.899372\pi$
$269$	−31.1769	−1.90089	−0.950445	+	0.310893i	$0.899372\pi$
$270$	0	0
$271$	−3.46410	−0.210429	−0.105215	−	0.994450i	$-0.533553\pi$
$271$	−3.46410	−0.210429	−0.105215	+	0.994450i	$0.533553\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.8564	−0.832551	−0.416275	−	0.909239i	$-0.636665\pi$
$277$	−13.8564	−0.832551	−0.416275	+	0.909239i	$0.636665\pi$
$278$	0	0
$279$	3.46410	0.207390
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	13.8564	0.820783
$286$	0	0
$287$	20.7846	1.22688
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−3.46410	−0.202375	−0.101187	−	0.994867i	$-0.532264\pi$
$293$	−3.46410	−0.202375	−0.101187	+	0.994867i	$0.532264\pi$
$294$	0	0
$295$	41.5692	2.42025
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−13.8564	−0.798670
$302$	0	0
$303$	3.46410	0.199007
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	17.3205	0.985329
$310$	0	0
$311$	−13.8564	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$311$	−13.8564	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	0	0
$317$	−31.1769	−1.75107	−0.875535	−	0.483155i	$-0.839491\pi$
$317$	−31.1769	−1.75107	−0.875535	+	0.483155i	$0.839491\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.8564	0.766261
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−6.92820	−0.379663
$334$	0	0
$335$	13.8564	0.757056
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	24.0000	1.29212
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	6.92820	0.370858	0.185429	−	0.982658i	$-0.440632\pi$
$349$	6.92820	0.370858	0.185429	+	0.982658i	$0.440632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	−20.7846	−1.10004
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−6.92820	−0.362639
$366$	0	0
$367$	−24.2487	−1.26577	−0.632886	−	0.774245i	$-0.718130\pi$
$367$	−24.2487	−1.26577	−0.632886	+	0.774245i	$0.718130\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−20.7846	−1.07619	−0.538093	−	0.842885i	$-0.680855\pi$
$373$	−20.7846	−1.07619	−0.538093	+	0.842885i	$0.680855\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	3.46410	0.177471
$382$	0	0
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−36.0000	−1.81136
$396$	0	0
$397$	34.6410	1.73858	0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410	1.73858	0.869291	+	0.494300i	$0.164576\pi$
$398$	0	0
$399$	13.8564	0.693688
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	41.5692	2.04549
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	13.8564	0.675320	0.337660	−	0.941268i	$-0.390365\pi$
$421$	13.8564	0.675320	0.337660	+	0.941268i	$0.390365\pi$
$422$	0	0
$423$	6.92820	0.336861
$424$	0	0
$425$	42.0000	2.03730
$426$	0	0
$427$	24.0000	1.16144
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	0	0
$437$	27.7128	1.32568
$438$	0	0
$439$	−31.1769	−1.48799	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	−31.1769	−1.48799	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−20.7846	−0.985285
$446$	0	0
$447$	10.3923	0.491539
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.46410	−0.162758
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	31.1769	1.45205	0.726027	−	0.687666i	$-0.241365\pi$
$461$	31.1769	1.45205	0.726027	+	0.687666i	$0.241365\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	13.8564	0.639829
$470$	0	0
$471$	6.92820	0.319235
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−28.0000	−1.28473
$476$	0	0
$477$	−3.46410	−0.158610
$478$	0	0
$479$	−20.7846	−0.949673	−0.474837	−	0.880074i	$-0.657493\pi$
$479$	−20.7846	−0.949673	−0.474837	+	0.880074i	$0.657493\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	24.0000	1.09204
$484$	0	0
$485$	−6.92820	−0.314594
$486$	0	0
$487$	−10.3923	−0.470920	−0.235460	−	0.971884i	$-0.575660\pi$
$487$	−10.3923	−0.470920	−0.235460	+	0.971884i	$0.575660\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	−20.7846	−0.936092
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	13.8564	0.619059
$502$	0	0
$503$	34.6410	1.54457	0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410	1.54457	0.772283	+	0.635278i	$0.219115\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	24.2487	1.07481	0.537403	−	0.843326i	$-0.319406\pi$
$509$	24.2487	1.07481	0.537403	+	0.843326i	$0.319406\pi$
$510$	0	0
$511$	−6.92820	−0.306486
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−60.0000	−2.64392
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−17.3205	−0.760286
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−24.2487	−1.05830
$526$	0	0
$527$	20.7846	0.905392
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	41.5692	1.79719
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13.8564	−0.595733	−0.297867	−	0.954607i	$-0.596275\pi$
$541$	−13.8564	−0.595733	−0.297867	+	0.954607i	$0.596275\pi$
$542$	0	0
$543$	13.8564	0.594635
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	6.92820	0.295689
$550$	0	0
$551$	13.8564	0.590303
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	−10.3923	−0.440336	−0.220168	−	0.975462i	$-0.570661\pi$
$557$	−10.3923	−0.440336	−0.220168	+	0.975462i	$0.570661\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−20.7846	−0.874415
$566$	0	0
$567$	3.46410	0.145479
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−13.8564	−0.578860
$574$	0	0
$575$	−48.4974	−2.02248
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−13.8564	−0.570943
$590$	0	0
$591$	−10.3923	−0.427482
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	72.0000	2.95171
$596$	0	0
$597$	−10.3923	−0.425329
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−38.1051	−1.54919
$606$	0	0
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	34.6410	1.39914	0.699569	−	0.714565i	$-0.253375\pi$
$613$	34.6410	1.39914	0.699569	+	0.714565i	$0.253375\pi$
$614$	0	0
$615$	−20.7846	−0.838116
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	6.92820	0.278019
$622$	0	0
$623$	−20.7846	−0.832718
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−41.5692	−1.65747
$630$	0	0
$631$	17.3205	0.689519	0.344759	−	0.938691i	$-0.387961\pi$
$631$	17.3205	0.689519	0.344759	+	0.938691i	$0.387961\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.92820	0.274075
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	13.8564	0.545595
$646$	0	0
$647$	−34.6410	−1.36188	−0.680939	−	0.732340i	$-0.738428\pi$
$647$	−34.6410	−1.36188	−0.680939	+	0.732340i	$0.738428\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	−10.3923	−0.406682	−0.203341	−	0.979108i	$-0.565180\pi$
$653$	−10.3923	−0.406682	−0.203341	+	0.979108i	$0.565180\pi$
$654$	0	0
$655$	41.5692	1.62424
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−6.92820	−0.269476	−0.134738	−	0.990881i	$-0.543019\pi$
$661$	−6.92820	−0.269476	−0.134738	+	0.990881i	$0.543019\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−48.0000	−1.86136
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	3.46410	0.133930
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	−7.00000	−0.269430
$676$	0	0
$677$	3.46410	0.133136	0.0665681	−	0.997782i	$-0.478795\pi$
$677$	3.46410	0.133136	0.0665681	+	0.997782i	$0.478795\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	20.7846	0.794139
$686$	0	0
$687$	−27.7128	−1.05731
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−69.2820	−2.62802
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	10.3923	0.392512	0.196256	−	0.980553i	$-0.437122\pi$
$701$	10.3923	0.392512	0.196256	+	0.980553i	$0.437122\pi$
$702$	0	0
$703$	27.7128	1.04521
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	−10.3923	−0.389742
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.7128	1.03495
$718$	0	0
$719$	20.7846	0.775135	0.387568	−	0.921841i	$-0.373315\pi$
$719$	20.7846	0.775135	0.387568	+	0.921841i	$0.373315\pi$
$720$	0	0
$721$	−60.0000	−2.23452
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	−24.2487	−0.900575
$726$	0	0
$727$	−10.3923	−0.385429	−0.192715	−	0.981255i	$-0.561729\pi$
$727$	−10.3923	−0.385429	−0.192715	+	0.981255i	$0.561729\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	27.7128	1.02360	0.511798	−	0.859106i	$-0.328980\pi$
$733$	27.7128	1.02360	0.511798	+	0.859106i	$0.328980\pi$
$734$	0	0
$735$	−17.3205	−0.638877
$736$	0	0
$737$	0	0
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	0	0
$748$	0	0
$749$	41.5692	1.51891
$750$	0	0
$751$	−24.2487	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$751$	−24.2487	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	27.7128	1.00724	0.503620	−	0.863925i	$-0.332001\pi$
$757$	27.7128	1.00724	0.503620	+	0.863925i	$0.332001\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	20.7846	0.751469
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	38.1051	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$773$	38.1051	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$774$	0	0
$775$	24.2487	0.871039
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.46410	0.123797
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.7846	−0.739016
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	−24.2487	−0.858933	−0.429467	−	0.903083i	$-0.641298\pi$
$797$	−24.2487	−0.858933	−0.429467	+	0.903083i	$0.641298\pi$
$798$	0	0
$799$	41.5692	1.47061
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−83.1384	−2.93024
$806$	0	0
$807$	31.1769	1.09748
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	3.46410	0.121491
$814$	0	0
$815$	69.2820	2.42684
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.2487	0.846286	0.423143	−	0.906063i	$-0.360927\pi$
$821$	24.2487	0.846286	0.423143	+	0.906063i	$0.360927\pi$
$822$	0	0
$823$	31.1769	1.08676	0.543379	−	0.839487i	$-0.317144\pi$
$823$	31.1769	1.08676	0.543379	+	0.839487i	$0.317144\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	41.5692	1.44376	0.721879	−	0.692019i	$-0.243279\pi$
$829$	41.5692	1.44376	0.721879	+	0.692019i	$0.243279\pi$
$830$	0	0
$831$	13.8564	0.480673
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	−3.46410	−0.119737
$838$	0	0
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	−45.0333	−1.54919
$846$	0	0
$847$	−38.1051	−1.30931
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	48.4974	1.66052	0.830260	−	0.557376i	$-0.188192\pi$
$853$	48.4974	1.66052	0.830260	+	0.557376i	$0.188192\pi$
$854$	0	0
$855$	−13.8564	−0.473879
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−20.7846	−0.708338
$862$	0	0
$863$	13.8564	0.471678	0.235839	−	0.971792i	$-0.424216\pi$
$863$	13.8564	0.471678	0.235839	+	0.971792i	$0.424216\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	6.92820	0.233949	0.116974	−	0.993135i	$-0.462680\pi$
$877$	6.92820	0.233949	0.116974	+	0.993135i	$0.462680\pi$
$878$	0	0
$879$	3.46410	0.116841
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−41.5692	−1.39733
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.7128	−0.927374
$894$	0	0
$895$	−41.5692	−1.38951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−20.7846	−0.692436
$902$	0	0
$903$	13.8564	0.461112
$904$	0	0
$905$	−48.0000	−1.59557
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	−3.46410	−0.114897
$910$	0	0
$911$	−55.4256	−1.83633	−0.918166	−	0.396195i	$-0.870330\pi$
$911$	−55.4256	−1.83633	−0.918166	+	0.396195i	$0.870330\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	41.5692	1.37274
$918$	0	0
$919$	−51.9615	−1.71405	−0.857026	−	0.515273i	$-0.827691\pi$
$919$	−51.9615	−1.71405	−0.857026	+	0.515273i	$0.827691\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−48.4974	−1.59459
$926$	0	0
$927$	−17.3205	−0.568880
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	13.8564	0.453638
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−24.2487	−0.790485	−0.395243	−	0.918577i	$-0.629340\pi$
$941$	−24.2487	−0.790485	−0.395243	+	0.918577i	$0.629340\pi$
$942$	0	0
$943$	−41.5692	−1.35368
$944$	0	0
$945$	−12.0000	−0.390360
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	31.1769	1.01098
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.7846	0.671170
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	6.92820	0.223027
$966$	0	0
$967$	38.1051	1.22538	0.612689	−	0.790324i	$-0.290088\pi$
$967$	38.1051	1.22538	0.612689	+	0.790324i	$0.290088\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	−69.2820	−2.22108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−13.8564	−0.442401
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	27.7128	0.881216
$990$	0	0
$991$	−24.2487	−0.770286	−0.385143	−	0.922857i	$-0.625848\pi$
$991$	−24.2487	−0.770286	−0.385143	+	0.922857i	$0.625848\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	36.0000	1.14128
$996$	0	0
$997$	−20.7846	−0.658255	−0.329128	−	0.944285i	$-0.606755\pi$
$997$	−20.7846	−0.658255	−0.329128	+	0.944285i	$0.606755\pi$
$998$	0	0
$999$	6.92820	0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.a.j.1.2		2
3.2	odd	2		2304.2.a.s.1.1		2
4.3	odd	2		768.2.a.k.1.2		2
8.3	odd	2	inner	768.2.a.j.1.1		2
8.5	even	2		768.2.a.k.1.1		2
12.11	even	2		2304.2.a.u.1.1		2
16.3	odd	4		192.2.d.a.97.3	yes	4
16.5	even	4		192.2.d.a.97.4	yes	4
16.11	odd	4		192.2.d.a.97.2	yes	4
16.13	even	4		192.2.d.a.97.1	✓	4
24.5	odd	2		2304.2.a.u.1.2		2
24.11	even	2		2304.2.a.s.1.2		2
48.5	odd	4		576.2.d.b.289.1		4
48.11	even	4		576.2.d.b.289.2		4
48.29	odd	4		576.2.d.b.289.3		4
48.35	even	4		576.2.d.b.289.4		4
80.3	even	4		4800.2.d.j.1249.1		4
80.13	odd	4		4800.2.d.o.1249.3		4
80.19	odd	4		4800.2.k.j.2401.1		4
80.27	even	4		4800.2.d.j.1249.4		4
80.29	even	4		4800.2.k.j.2401.4		4
80.37	odd	4		4800.2.d.o.1249.2		4
80.43	even	4		4800.2.d.o.1249.1		4
80.53	odd	4		4800.2.d.j.1249.3		4
80.59	odd	4		4800.2.k.j.2401.3		4
80.67	even	4		4800.2.d.o.1249.4		4
80.69	even	4		4800.2.k.j.2401.2		4
80.77	odd	4		4800.2.d.j.1249.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.d.a.97.1	✓	4	16.13	even	4
192.2.d.a.97.2	yes	4	16.11	odd	4
192.2.d.a.97.3	yes	4	16.3	odd	4
192.2.d.a.97.4	yes	4	16.5	even	4
576.2.d.b.289.1		4	48.5	odd	4
576.2.d.b.289.2		4	48.11	even	4
576.2.d.b.289.3		4	48.29	odd	4
576.2.d.b.289.4		4	48.35	even	4
768.2.a.j.1.1		2	8.3	odd	2	inner
768.2.a.j.1.2		2	1.1	even	1	trivial
768.2.a.k.1.1		2	8.5	even	2
768.2.a.k.1.2		2	4.3	odd	2
2304.2.a.s.1.1		2	3.2	odd	2
2304.2.a.s.1.2		2	24.11	even	2
2304.2.a.u.1.1		2	12.11	even	2
2304.2.a.u.1.2		2	24.5	odd	2
4800.2.d.j.1249.1		4	80.3	even	4
4800.2.d.j.1249.2		4	80.77	odd	4
4800.2.d.j.1249.3		4	80.53	odd	4
4800.2.d.j.1249.4		4	80.27	even	4
4800.2.d.o.1249.1		4	80.43	even	4
4800.2.d.o.1249.2		4	80.37	odd	4
4800.2.d.o.1249.3		4	80.13	odd	4
4800.2.d.o.1249.4		4	80.67	even	4
4800.2.k.j.2401.1		4	80.19	odd	4
4800.2.k.j.2401.2		4	80.69	even	4
4800.2.k.j.2401.3		4	80.59	odd	4
4800.2.k.j.2401.4		4	80.29	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{15} +6.00000 q^{17} -4.00000 q^{19} -3.46410 q^{21} -6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -3.46410 q^{29} +3.46410 q^{31} +12.0000 q^{35} -6.92820 q^{37} +6.00000 q^{41} -4.00000 q^{43} +3.46410 q^{45} +6.92820 q^{47} +5.00000 q^{49} -6.00000 q^{51} -3.46410 q^{53} +4.00000 q^{57} +12.0000 q^{59} +6.92820 q^{61} +3.46410 q^{63} +4.00000 q^{67} +6.92820 q^{69} +6.92820 q^{71} -2.00000 q^{73} -7.00000 q^{75} -10.3923 q^{79} +1.00000 q^{81} +20.7846 q^{85} +3.46410 q^{87} -6.00000 q^{89} -3.46410 q^{93} -13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{19} + 14 q^{25} - 2 q^{27} + 24 q^{35} + 12 q^{41} - 8 q^{43} + 10 q^{49} - 12 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{67} - 4 q^{73} - 14 q^{75} + 2 q^{81} - 12 q^{89}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^19 + 14 * q^25 - 2 * q^27 + 24 * q^35 + 12 * q^41 - 8 * q^43 + 10 * q^49 - 12 * q^51 + 8 * q^57 + 24 * q^59 + 8 * q^67 - 4 * q^73 - 14 * q^75 + 2 * q^81 - 12 * q^89 - 4 * q^97

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