LMFDB - Embedded newform 5808.2.a.bx.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5808,2,Mod(1,5808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5808 = 2^{4} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,4,0,0,0,2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.3771134940$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000 q^{5} +4.47214 q^{7} +1.00000 q^{9} -2.00000 q^{15} +4.47214 q^{17} +4.47214 q^{19} -4.47214 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +4.47214 q^{29} +8.94427 q^{35} +2.00000 q^{37} -4.47214 q^{41} -4.47214 q^{43} +2.00000 q^{45} -8.00000 q^{47} +13.0000 q^{49} -4.47214 q^{51} +6.00000 q^{53} -4.47214 q^{57} +8.94427 q^{61} +4.47214 q^{63} +12.0000 q^{67} -4.00000 q^{69} +8.00000 q^{71} -8.94427 q^{73} +1.00000 q^{75} -13.4164 q^{79} +1.00000 q^{81} +8.94427 q^{83} +8.94427 q^{85} -4.47214 q^{87} -14.0000 q^{89} +8.94427 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{15} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{37} + 4 q^{45} - 16 q^{47} + 26 q^{49} + 12 q^{53} + 24 q^{67} - 8 q^{69} + 16 q^{71} + 2 q^{75} + 2 q^{81} - 28 q^{89}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 - 4 * q^15 + 8 * q^23 - 2 * q^25 - 2 * q^27 + 4 * q^37 + 4 * q^45 - 16 * q^47 + 26 * q^49 + 12 * q^53 + 24 * q^67 - 8 * q^69 + 16 * q^71 + 2 * q^75 + 2 * q^81 - 28 * 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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	−4.47214	−0.975900
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.94427	1.51186
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.47214	−0.592349
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.94427	1.14520	0.572598	−	0.819836i	$-0.305935\pi$
$61$	8.94427	1.14520	0.572598	+	0.819836i	$0.305935\pi$
$62$	0	0
$63$	4.47214	0.563436
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−8.94427	−1.04685	−0.523424	−	0.852072i	$-0.675346\pi$
$73$	−8.94427	−1.04685	−0.523424	+	0.852072i	$0.675346\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.94427	0.981761	0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427	0.981761	0.490881	+	0.871227i	$0.336675\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	0	0
$87$	−4.47214	−0.479463
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.94427	0.917663
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−8.94427	−0.872872
$106$	0	0
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.0000	1.83340
$120$	0	0
$121$	0	0
$122$	0	0
$123$	4.47214	0.403239
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−13.4164	−1.19051	−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164	−1.19051	−0.595257	+	0.803535i	$0.702950\pi$
$128$	0	0
$129$	4.47214	0.393750
$130$	0	0
$131$	17.8885	1.56293	0.781465	−	0.623949i	$-0.214473\pi$
$131$	17.8885	1.56293	0.781465	+	0.623949i	$0.214473\pi$
$132$	0	0
$133$	20.0000	1.73422
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.94427	0.742781
$146$	0	0
$147$	−13.0000	−1.07222
$148$	0	0
$149$	−22.3607	−1.83186	−0.915929	−	0.401340i	$-0.868545\pi$
$149$	−22.3607	−1.83186	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	−13.4164	−1.09181	−0.545906	−	0.837846i	$-0.683814\pi$
$151$	−13.4164	−1.09181	−0.545906	+	0.837846i	$0.683814\pi$
$152$	0	0
$153$	4.47214	0.361551
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	17.8885	1.40981
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	4.47214	0.341993
$172$	0	0
$173$	−13.4164	−1.02003	−0.510015	−	0.860165i	$-0.670360\pi$
$173$	−13.4164	−1.02003	−0.510015	+	0.860165i	$0.670360\pi$
$174$	0	0
$175$	−4.47214	−0.338062
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−8.94427	−0.661180
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.47214	−0.325300
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−17.8885	−1.28765	−0.643823	−	0.765175i	$-0.722653\pi$
$193$	−17.8885	−1.28765	−0.643823	+	0.765175i	$0.722653\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.3607	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$197$	22.3607	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	20.0000	1.40372
$204$	0	0
$205$	−8.94427	−0.624695
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.47214	0.307875	0.153937	−	0.988081i	$-0.450805\pi$
$211$	4.47214	0.307875	0.153937	+	0.988081i	$0.450805\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−8.94427	−0.609994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.94427	0.604398
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−8.94427	−0.593652	−0.296826	−	0.954932i	$-0.595928\pi$
$227$	−8.94427	−0.593652	−0.296826	+	0.954932i	$0.595928\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.47214	−0.292979	−0.146490	−	0.989212i	$-0.546798\pi$
$233$	−4.47214	−0.292979	−0.146490	+	0.989212i	$0.546798\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	13.4164	0.871489
$238$	0	0
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	8.94427	0.576151	0.288076	−	0.957608i	$-0.406985\pi$
$241$	8.94427	0.576151	0.288076	+	0.957608i	$0.406985\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	26.0000	1.66108
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.94427	−0.566820
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−8.94427	−0.560112
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	4.47214	0.276818
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	13.4164	0.814989	0.407494	−	0.913208i	$-0.366403\pi$
$271$	13.4164	0.814989	0.407494	+	0.913208i	$0.366403\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.3050	1.86750	0.933748	−	0.357930i	$-0.116517\pi$
$281$	31.3050	1.86750	0.933748	+	0.357930i	$0.116517\pi$
$282$	0	0
$283$	13.4164	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$283$	13.4164	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$284$	0	0
$285$	−8.94427	−0.529813
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−22.3607	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$293$	−22.3607	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	−4.47214	−0.256917
$304$	0	0
$305$	17.8885	1.02430
$306$	0	0
$307$	4.47214	0.255238	0.127619	−	0.991823i	$-0.459266\pi$
$307$	4.47214	0.255238	0.127619	+	0.991823i	$0.459266\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	8.94427	0.503953
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.94427	0.499221
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−35.7771	−1.97245
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	8.94427	0.487226	0.243613	−	0.969873i	$-0.421667\pi$
$337$	8.94427	0.487226	0.243613	+	0.969873i	$0.421667\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	26.8328	1.44884
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−8.94427	−0.480154	−0.240077	−	0.970754i	$-0.577173\pi$
$347$	−8.94427	−0.480154	−0.240077	+	0.970754i	$0.577173\pi$
$348$	0	0
$349$	−26.8328	−1.43633	−0.718164	−	0.695874i	$-0.755017\pi$
$349$	−26.8328	−1.43633	−0.718164	+	0.695874i	$0.755017\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	−20.0000	−1.05851
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.8885	−0.936329
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−4.47214	−0.232810
$370$	0	0
$371$	26.8328	1.39309
$372$	0	0
$373$	−26.8328	−1.38935	−0.694675	−	0.719323i	$-0.744452\pi$
$373$	−26.8328	−1.38935	−0.694675	+	0.719323i	$0.744452\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	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
$381$	13.4164	0.687343
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.47214	−0.227331
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	−17.8885	−0.902358
$394$	0	0
$395$	−26.8328	−1.35011
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	−20.0000	−1.00125
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−26.8328	−1.32680	−0.663399	−	0.748266i	$-0.730887\pi$
$409$	−26.8328	−1.32680	−0.663399	+	0.748266i	$0.730887\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	0	0
$414$	0	0
$415$	17.8885	0.878114
$416$	0	0
$417$	13.4164	0.657004
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.94427	0.430830	0.215415	−	0.976523i	$-0.430890\pi$
$431$	8.94427	0.430830	0.215415	+	0.976523i	$0.430890\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−8.94427	−0.428845
$436$	0	0
$437$	17.8885	0.855725
$438$	0	0
$439$	13.4164	0.640330	0.320165	−	0.947362i	$-0.396262\pi$
$439$	13.4164	0.640330	0.320165	+	0.947362i	$0.396262\pi$
$440$	0	0
$441$	13.0000	0.619048
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−28.0000	−1.32733
$446$	0	0
$447$	22.3607	1.05762
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.4164	0.630358
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.8328	1.25519	0.627593	−	0.778542i	$-0.284040\pi$
$457$	26.8328	1.25519	0.627593	+	0.778542i	$0.284040\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	13.4164	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$461$	13.4164	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	53.6656	2.47805
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.47214	−0.205196
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	8.94427	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$479$	8.94427	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−17.8885	−0.813957
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	26.8328	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$491$	26.8328	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.7771	1.60482
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	8.94427	0.399601
$502$	0	0
$503$	26.8328	1.19642	0.598208	−	0.801341i	$-0.295880\pi$
$503$	26.8328	1.19642	0.598208	+	0.801341i	$0.295880\pi$
$504$	0	0
$505$	8.94427	0.398015
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	0	0
$513$	−4.47214	−0.197450
$514$	0	0
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	0	0
$519$	13.4164	0.588915
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−4.47214	−0.195553	−0.0977764	−	0.995208i	$-0.531173\pi$
$523$	−4.47214	−0.195553	−0.0977764	+	0.995208i	$0.531173\pi$
$524$	0	0
$525$	4.47214	0.195180
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−17.8885	−0.773389
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−40.2492	−1.72093	−0.860466	−	0.509507i	$-0.829828\pi$
$547$	−40.2492	−1.72093	−0.860466	+	0.509507i	$0.829828\pi$
$548$	0	0
$549$	8.94427	0.381732
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	−60.0000	−2.55146
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−40.2492	−1.70541	−0.852707	−	0.522389i	$-0.825041\pi$
$557$	−40.2492	−1.70541	−0.852707	+	0.522389i	$0.825041\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.8885	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$563$	−17.8885	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	4.47214	0.187812
$568$	0	0
$569$	4.47214	0.187482	0.0937408	−	0.995597i	$-0.470117\pi$
$569$	4.47214	0.187482	0.0937408	+	0.995597i	$0.470117\pi$
$570$	0	0
$571$	13.4164	0.561459	0.280730	−	0.959787i	$-0.409424\pi$
$571$	13.4164	0.561459	0.280730	+	0.959787i	$0.409424\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	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
$579$	17.8885	0.743423
$580$	0	0
$581$	40.0000	1.65948
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−22.3607	−0.919795
$592$	0	0
$593$	−22.3607	−0.918243	−0.459122	−	0.888373i	$-0.651836\pi$
$593$	−22.3607	−0.918243	−0.459122	+	0.888373i	$0.651836\pi$
$594$	0	0
$595$	40.0000	1.63984
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.47214	0.181518	0.0907592	−	0.995873i	$-0.471071\pi$
$607$	4.47214	0.181518	0.0907592	+	0.995873i	$0.471071\pi$
$608$	0	0
$609$	−20.0000	−0.810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	44.7214	1.80628	0.903139	−	0.429348i	$-0.141256\pi$
$613$	44.7214	1.80628	0.903139	+	0.429348i	$0.141256\pi$
$614$	0	0
$615$	8.94427	0.360668
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−62.6099	−2.50841
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.94427	0.356631
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−4.47214	−0.177751
$634$	0	0
$635$	−26.8328	−1.06483
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	8.94427	0.352180
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	35.7771	1.39793
$656$	0	0
$657$	−8.94427	−0.348949
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	40.0000	1.55113
$666$	0	0
$667$	17.8885	0.692647
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−17.8885	−0.689553	−0.344776	−	0.938685i	$-0.612045\pi$
$673$	−17.8885	−0.689553	−0.344776	+	0.938685i	$0.612045\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−31.3050	−1.20315	−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050	−1.20315	−0.601574	+	0.798817i	$0.705459\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	0	0
$681$	8.94427	0.342745
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	44.0000	1.68115
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−26.8328	−1.01783
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	4.47214	0.169152
$700$	0	0
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	8.94427	0.337340
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	20.0000	0.752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−13.4164	−0.503155
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.94427	0.334030
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−71.5542	−2.66482
$722$	0	0
$723$	−8.94427	−0.332641
$724$	0	0
$725$	−4.47214	−0.166091
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	−17.8885	−0.660728	−0.330364	−	0.943854i	$-0.607172\pi$
$733$	−17.8885	−0.660728	−0.330364	+	0.943854i	$0.607172\pi$
$734$	0	0
$735$	−26.0000	−0.959024
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.47214	−0.164510	−0.0822551	−	0.996611i	$-0.526212\pi$
$739$	−4.47214	−0.164510	−0.0822551	+	0.996611i	$0.526212\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−44.7214	−1.63846
$746$	0	0
$747$	8.94427	0.327254
$748$	0	0
$749$	−40.0000	−1.46157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−26.8328	−0.976546
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.3050	−1.13480	−0.567402	−	0.823441i	$-0.692051\pi$
$761$	−31.3050	−1.13480	−0.567402	+	0.823441i	$0.692051\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.94427	0.323381
$766$	0	0
$767$	0	0
$768$	0	0
$769$	35.7771	1.29015	0.645077	−	0.764117i	$-0.276825\pi$
$769$	35.7771	1.29015	0.645077	+	0.764117i	$0.276825\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.94427	−0.320874
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.47214	−0.159821
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	−40.2492	−1.43473	−0.717365	−	0.696698i	$-0.754652\pi$
$787$	−40.2492	−1.43473	−0.717365	+	0.696698i	$0.754652\pi$
$788$	0	0
$789$	8.94427	0.318425
$790$	0	0
$791$	26.8328	0.954065
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−35.7771	−1.26570
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	35.7771	1.26098
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−13.4164	−0.471696	−0.235848	−	0.971790i	$-0.575787\pi$
$809$	−13.4164	−0.471696	−0.235848	+	0.971790i	$0.575787\pi$
$810$	0	0
$811$	4.47214	0.157038	0.0785190	−	0.996913i	$-0.474981\pi$
$811$	4.47214	0.157038	0.0785190	+	0.996913i	$0.474981\pi$
$812$	0	0
$813$	−13.4164	−0.470534
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.3607	0.780393	0.390197	−	0.920732i	$-0.372407\pi$
$821$	22.3607	0.780393	0.390197	+	0.920732i	$0.372407\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.7214	−1.55511	−0.777557	−	0.628812i	$-0.783541\pi$
$827$	−44.7214	−1.55511	−0.777557	+	0.628812i	$0.783541\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	58.1378	2.01435
$834$	0	0
$835$	−17.8885	−0.619059
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	−31.3050	−1.07820
$844$	0	0
$845$	−26.0000	−0.894427
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−13.4164	−0.460450
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	8.94427	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$853$	8.94427	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$854$	0	0
$855$	8.94427	0.305888
$856$	0	0
$857$	40.2492	1.37489	0.687444	−	0.726238i	$-0.258733\pi$
$857$	40.2492	1.37489	0.687444	+	0.726238i	$0.258733\pi$
$858$	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
$861$	20.0000	0.681598
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	−26.8328	−0.912343
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−53.6656	−1.81423
$876$	0	0
$877$	−8.94427	−0.302027	−0.151013	−	0.988532i	$-0.548254\pi$
$877$	−8.94427	−0.302027	−0.151013	+	0.988532i	$0.548254\pi$
$878$	0	0
$879$	22.3607	0.754207
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.6656	1.80192	0.900958	−	0.433907i	$-0.142865\pi$
$887$	53.6656	1.80192	0.900958	+	0.433907i	$0.142865\pi$
$888$	0	0
$889$	−60.0000	−2.01234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−35.7771	−1.19723
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	26.8328	0.893931
$902$	0	0
$903$	20.0000	0.665558
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	4.47214	0.148331
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−17.8885	−0.591377
$916$	0	0
$917$	80.0000	2.64183
$918$	0	0
$919$	−22.3607	−0.737611	−0.368805	−	0.929507i	$-0.620233\pi$
$919$	−22.3607	−0.737611	−0.368805	+	0.929507i	$0.620233\pi$
$920$	0	0
$921$	−4.47214	−0.147362
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	58.1378	1.90539
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.6656	1.75318	0.876590	−	0.481238i	$-0.159813\pi$
$937$	53.6656	1.75318	0.876590	+	0.481238i	$0.159813\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	22.3607	0.728937	0.364469	−	0.931216i	$-0.381251\pi$
$941$	22.3607	0.728937	0.364469	+	0.931216i	$0.381251\pi$
$942$	0	0
$943$	−17.8885	−0.582531
$944$	0	0
$945$	−8.94427	−0.290957
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	22.3607	0.724333	0.362167	−	0.932113i	$-0.382037\pi$
$953$	22.3607	0.724333	0.362167	+	0.932113i	$0.382037\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	98.3870	3.17708
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−8.94427	−0.288225
$964$	0	0
$965$	−35.7771	−1.15171
$966$	0	0
$967$	−13.4164	−0.431443	−0.215721	−	0.976455i	$-0.569210\pi$
$967$	−13.4164	−0.431443	−0.215721	+	0.976455i	$0.569210\pi$
$968$	0	0
$969$	−20.0000	−0.642493
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−60.0000	−1.92351
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	44.7214	1.42494
$986$	0	0
$987$	35.7771	1.13880
$988$	0	0
$989$	−17.8885	−0.568823
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.8328	−0.849804	−0.424902	−	0.905239i	$-0.639691\pi$
$997$	−26.8328	−0.849804	−0.424902	+	0.905239i	$0.639691\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5808.2.a.bx.1.2		2
4.3	odd	2		363.2.a.g.1.2	yes	2
11.10	odd	2	inner	5808.2.a.bx.1.1		2
12.11	even	2		1089.2.a.p.1.1		2
20.19	odd	2		9075.2.a.bi.1.1		2
44.3	odd	10		363.2.e.l.130.1		4
44.7	even	10		363.2.e.a.148.1		4
44.15	odd	10		363.2.e.l.148.1		4
44.19	even	10		363.2.e.a.130.1		4
44.27	odd	10		363.2.e.a.124.1		4
44.31	odd	10		363.2.e.a.202.1		4
44.35	even	10		363.2.e.l.202.1		4
44.39	even	10		363.2.e.l.124.1		4
44.43	even	2		363.2.a.g.1.1	✓	2
132.131	odd	2		1089.2.a.p.1.2		2
220.219	even	2		9075.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.g.1.1	✓	2	44.43	even	2
363.2.a.g.1.2	yes	2	4.3	odd	2
363.2.e.a.124.1		4	44.27	odd	10
363.2.e.a.130.1		4	44.19	even	10
363.2.e.a.148.1		4	44.7	even	10
363.2.e.a.202.1		4	44.31	odd	10
363.2.e.l.124.1		4	44.39	even	10
363.2.e.l.130.1		4	44.3	odd	10
363.2.e.l.148.1		4	44.15	odd	10
363.2.e.l.202.1		4	44.35	even	10
1089.2.a.p.1.1		2	12.11	even	2
1089.2.a.p.1.2		2	132.131	odd	2
5808.2.a.bx.1.1		2	11.10	odd	2	inner
5808.2.a.bx.1.2		2	1.1	even	1	trivial
9075.2.a.bi.1.1		2	20.19	odd	2
9075.2.a.bi.1.2		2	220.219	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5808.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000 q^{5} +4.47214 q^{7} +1.00000 q^{9} -2.00000 q^{15} +4.47214 q^{17} +4.47214 q^{19} -4.47214 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +4.47214 q^{29} +8.94427 q^{35} +2.00000 q^{37} -4.47214 q^{41} -4.47214 q^{43} +2.00000 q^{45} -8.00000 q^{47} +13.0000 q^{49} -4.47214 q^{51} +6.00000 q^{53} -4.47214 q^{57} +8.94427 q^{61} +4.47214 q^{63} +12.0000 q^{67} -4.00000 q^{69} +8.00000 q^{71} -8.94427 q^{73} +1.00000 q^{75} -13.4164 q^{79} +1.00000 q^{81} +8.94427 q^{83} +8.94427 q^{85} -4.47214 q^{87} -14.0000 q^{89} +8.94427 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{15} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{37} + 4 q^{45} - 16 q^{47} + 26 q^{49} + 12 q^{53} + 24 q^{67} - 8 q^{69} + 16 q^{71} + 2 q^{75} + 2 q^{81} - 28 q^{89}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 - 4 * q^15 + 8 * q^23 - 2 * q^25 - 2 * q^27 + 4 * q^37 + 4 * q^45 - 16 * q^47 + 26 * q^49 + 12 * q^53 + 24 * q^67 - 8 * q^69 + 16 * q^71 + 2 * q^75 + 2 * q^81 - 28 * q^89 + 4 * q^97

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