LMFDB - Embedded newform 5776.2.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5776,2,Mod(1,5776)] 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("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-5,0,2,0,6,0,2,0,5,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.1215922075$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -3.61803 q^{5} +3.23607 q^{7} -1.47214 q^{9} +3.23607 q^{11} +1.38197 q^{13} -4.47214 q^{15} -3.38197 q^{17} +4.00000 q^{21} -5.23607 q^{23} +8.09017 q^{25} -5.52786 q^{27} -9.09017 q^{29} +1.23607 q^{31} +4.00000 q^{33} -11.7082 q^{35} +8.38197 q^{37} +1.70820 q^{39} +0.854102 q^{41} +9.23607 q^{43} +5.32624 q^{45} -4.47214 q^{47} +3.47214 q^{49} -4.18034 q^{51} -6.09017 q^{53} -11.7082 q^{55} +0.472136 q^{59} +1.38197 q^{61} -4.76393 q^{63} -5.00000 q^{65} -11.7082 q^{67} -6.47214 q^{69} -2.94427 q^{71} -5.61803 q^{73} +10.0000 q^{75} +10.4721 q^{77} +2.76393 q^{79} -2.41641 q^{81} +0.472136 q^{83} +12.2361 q^{85} -11.2361 q^{87} +8.85410 q^{89} +4.47214 q^{91} +1.52786 q^{93} +8.61803 q^{97} -4.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} + 2 q^{11} + 5 q^{13} - 9 q^{17} + 8 q^{21} - 6 q^{23} + 5 q^{25} - 20 q^{27} - 7 q^{29} - 2 q^{31} + 8 q^{33} - 10 q^{35} + 19 q^{37} - 10 q^{39} - 5 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 + 2 * q^11 + 5 * q^13 - 9 * q^17 + 8 * q^21 - 6 * q^23 + 5 * q^25 - 20 * q^27 - 7 * q^29 - 2 * q^31 + 8 * q^33 - 10 * q^35 + 19 * q^37 - 10 * q^39 - 5 * q^41 + 14 * q^43 - 5 * q^45 - 2 * q^49 + 14 * q^51 - q^53 - 10 * q^55 - 8 * q^59 + 5 * q^61 - 14 * q^63 - 10 * q^65 - 10 * q^67 - 4 * q^69 + 12 * q^71 - 9 * q^73 + 20 * q^75 + 12 * q^77 + 10 * q^79 + 22 * q^81 - 8 * q^83 + 20 * q^85 - 18 * q^87 + 11 * q^89 + 12 * q^93 + 15 * q^97 - 14 * q^99

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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	1.38197	0.383288	0.191644	−	0.981464i	$-0.438618\pi$
$13$	1.38197	0.383288	0.191644	+	0.981464i	$0.438618\pi$
$14$	0	0
$15$	−4.47214	−1.15470
$16$	0	0
$17$	−3.38197	−0.820247	−0.410124	−	0.912030i	$-0.634514\pi$
$17$	−3.38197	−0.820247	−0.410124	+	0.912030i	$0.634514\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−9.09017	−1.68800	−0.844001	−	0.536341i	$-0.819806\pi$
$29$	−9.09017	−1.68800	−0.844001	+	0.536341i	$0.819806\pi$
$30$	0	0
$31$	1.23607	0.222004	0.111002	−	0.993820i	$-0.464594\pi$
$31$	1.23607	0.222004	0.111002	+	0.993820i	$0.464594\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−11.7082	−1.97905
$36$	0	0
$37$	8.38197	1.37799	0.688993	−	0.724768i	$-0.258053\pi$
$37$	8.38197	1.37799	0.688993	+	0.724768i	$0.258053\pi$
$38$	0	0
$39$	1.70820	0.273532
$40$	0	0
$41$	0.854102	0.133388	0.0666942	−	0.997773i	$-0.478755\pi$
$41$	0.854102	0.133388	0.0666942	+	0.997773i	$0.478755\pi$
$42$	0	0
$43$	9.23607	1.40849	0.704244	−	0.709958i	$-0.251286\pi$
$43$	9.23607	1.40849	0.704244	+	0.709958i	$0.251286\pi$
$44$	0	0
$45$	5.32624	0.793989
$46$	0	0
$47$	−4.47214	−0.652328	−0.326164	−	0.945313i	$-0.605756\pi$
$47$	−4.47214	−0.652328	−0.326164	+	0.945313i	$0.605756\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	−4.18034	−0.585365
$52$	0	0
$53$	−6.09017	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$53$	−6.09017	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$54$	0	0
$55$	−11.7082	−1.57873
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	1.38197	0.176943	0.0884713	−	0.996079i	$-0.471802\pi$
$61$	1.38197	0.176943	0.0884713	+	0.996079i	$0.471802\pi$
$62$	0	0
$63$	−4.76393	−0.600199
$64$	0	0
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−11.7082	−1.43038	−0.715192	−	0.698928i	$-0.753661\pi$
$67$	−11.7082	−1.43038	−0.715192	+	0.698928i	$0.753661\pi$
$68$	0	0
$69$	−6.47214	−0.779154
$70$	0	0
$71$	−2.94427	−0.349421	−0.174710	−	0.984620i	$-0.555899\pi$
$71$	−2.94427	−0.349421	−0.174710	+	0.984620i	$0.555899\pi$
$72$	0	0
$73$	−5.61803	−0.657541	−0.328771	−	0.944410i	$-0.606634\pi$
$73$	−5.61803	−0.657541	−0.328771	+	0.944410i	$0.606634\pi$
$74$	0	0
$75$	10.0000	1.15470
$76$	0	0
$77$	10.4721	1.19341
$78$	0	0
$79$	2.76393	0.310967	0.155483	−	0.987839i	$-0.450307\pi$
$79$	2.76393	0.310967	0.155483	+	0.987839i	$0.450307\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	0.472136	0.0518237	0.0259118	−	0.999664i	$-0.491751\pi$
$83$	0.472136	0.0518237	0.0259118	+	0.999664i	$0.491751\pi$
$84$	0	0
$85$	12.2361	1.32719
$86$	0	0
$87$	−11.2361	−1.20463
$88$	0	0
$89$	8.85410	0.938533	0.469266	−	0.883057i	$-0.344518\pi$
$89$	8.85410	0.938533	0.469266	+	0.883057i	$0.344518\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	1.52786	0.158432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.61803	0.875029	0.437514	−	0.899211i	$-0.355859\pi$
$97$	8.61803	0.875029	0.437514	+	0.899211i	$0.355859\pi$
$98$	0	0
$99$	−4.76393	−0.478793
$100$	0	0
$101$	−15.6180	−1.55405	−0.777026	−	0.629468i	$-0.783273\pi$
$101$	−15.6180	−1.55405	−0.777026	+	0.629468i	$0.783273\pi$
$102$	0	0
$103$	−14.6525	−1.44375	−0.721876	−	0.692023i	$-0.756720\pi$
$103$	−14.6525	−1.44375	−0.721876	+	0.692023i	$0.756720\pi$
$104$	0	0
$105$	−14.4721	−1.41234
$106$	0	0
$107$	−9.41641	−0.910319	−0.455159	−	0.890410i	$-0.650418\pi$
$107$	−9.41641	−0.910319	−0.455159	+	0.890410i	$0.650418\pi$
$108$	0	0
$109$	−11.0344	−1.05691	−0.528454	−	0.848962i	$-0.677228\pi$
$109$	−11.0344	−1.05691	−0.528454	+	0.848962i	$0.677228\pi$
$110$	0	0
$111$	10.3607	0.983392
$112$	0	0
$113$	−9.85410	−0.926996	−0.463498	−	0.886098i	$-0.653406\pi$
$113$	−9.85410	−0.926996	−0.463498	+	0.886098i	$0.653406\pi$
$114$	0	0
$115$	18.9443	1.76656
$116$	0	0
$117$	−2.03444	−0.188084
$118$	0	0
$119$	−10.9443	−1.00326
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	1.05573	0.0951918
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	11.4164	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$127$	11.4164	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$128$	0	0
$129$	11.4164	1.00516
$130$	0	0
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	20.0000	1.72133
$136$	0	0
$137$	2.38197	0.203505	0.101753	−	0.994810i	$-0.467555\pi$
$137$	2.38197	0.203505	0.101753	+	0.994810i	$0.467555\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−5.52786	−0.465530
$142$	0	0
$143$	4.47214	0.373979
$144$	0	0
$145$	32.8885	2.73124
$146$	0	0
$147$	4.29180	0.353981
$148$	0	0
$149$	−13.0902	−1.07239	−0.536194	−	0.844095i	$-0.680139\pi$
$149$	−13.0902	−1.07239	−0.536194	+	0.844095i	$0.680139\pi$
$150$	0	0
$151$	−14.4721	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$151$	−14.4721	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$152$	0	0
$153$	4.97871	0.402505
$154$	0	0
$155$	−4.47214	−0.359211
$156$	0	0
$157$	9.38197	0.748762	0.374381	−	0.927275i	$-0.377855\pi$
$157$	9.38197	0.748762	0.374381	+	0.927275i	$0.377855\pi$
$158$	0	0
$159$	−7.52786	−0.596998
$160$	0	0
$161$	−16.9443	−1.33540
$162$	0	0
$163$	−10.6525	−0.834366	−0.417183	−	0.908822i	$-0.636983\pi$
$163$	−10.6525	−0.834366	−0.417183	+	0.908822i	$0.636983\pi$
$164$	0	0
$165$	−14.4721	−1.12665
$166$	0	0
$167$	12.4721	0.965123	0.482561	−	0.875862i	$-0.339707\pi$
$167$	12.4721	0.965123	0.482561	+	0.875862i	$0.339707\pi$
$168$	0	0
$169$	−11.0902	−0.853090
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.32624	0.100832	0.0504160	−	0.998728i	$-0.483945\pi$
$173$	1.32624	0.100832	0.0504160	+	0.998728i	$0.483945\pi$
$174$	0	0
$175$	26.1803	1.97905
$176$	0	0
$177$	0.583592	0.0438655
$178$	0	0
$179$	23.4164	1.75022	0.875112	−	0.483920i	$-0.160787\pi$
$179$	23.4164	1.75022	0.875112	+	0.483920i	$0.160787\pi$
$180$	0	0
$181$	5.41641	0.402598	0.201299	−	0.979530i	$-0.435484\pi$
$181$	5.41641	0.402598	0.201299	+	0.979530i	$0.435484\pi$
$182$	0	0
$183$	1.70820	0.126274
$184$	0	0
$185$	−30.3262	−2.22963
$186$	0	0
$187$	−10.9443	−0.800324
$188$	0	0
$189$	−17.8885	−1.30120
$190$	0	0
$191$	−3.52786	−0.255267	−0.127634	−	0.991821i	$-0.540738\pi$
$191$	−3.52786	−0.255267	−0.127634	+	0.991821i	$0.540738\pi$
$192$	0	0
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	0	0
$195$	−6.18034	−0.442583
$196$	0	0
$197$	−4.14590	−0.295383	−0.147692	−	0.989033i	$-0.547184\pi$
$197$	−4.14590	−0.295383	−0.147692	+	0.989033i	$0.547184\pi$
$198$	0	0
$199$	−10.4721	−0.742350	−0.371175	−	0.928563i	$-0.621045\pi$
$199$	−10.4721	−0.742350	−0.371175	+	0.928563i	$0.621045\pi$
$200$	0	0
$201$	−14.4721	−1.02079
$202$	0	0
$203$	−29.4164	−2.06463
$204$	0	0
$205$	−3.09017	−0.215827
$206$	0	0
$207$	7.70820	0.535757
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.23607	0.498151	0.249076	−	0.968484i	$-0.419873\pi$
$211$	7.23607	0.498151	0.249076	+	0.968484i	$0.419873\pi$
$212$	0	0
$213$	−3.63932	−0.249362
$214$	0	0
$215$	−33.4164	−2.27898
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−6.94427	−0.469250
$220$	0	0
$221$	−4.67376	−0.314391
$222$	0	0
$223$	−27.7082	−1.85548	−0.927739	−	0.373229i	$-0.878251\pi$
$223$	−27.7082	−1.85548	−0.927739	+	0.373229i	$0.878251\pi$
$224$	0	0
$225$	−11.9098	−0.793989
$226$	0	0
$227$	8.94427	0.593652	0.296826	−	0.954932i	$-0.404072\pi$
$227$	8.94427	0.593652	0.296826	+	0.954932i	$0.404072\pi$
$228$	0	0
$229$	4.38197	0.289568	0.144784	−	0.989463i	$-0.453751\pi$
$229$	4.38197	0.289568	0.144784	+	0.989463i	$0.453751\pi$
$230$	0	0
$231$	12.9443	0.851671
$232$	0	0
$233$	1.56231	0.102350	0.0511750	−	0.998690i	$-0.483703\pi$
$233$	1.56231	0.102350	0.0511750	+	0.998690i	$0.483703\pi$
$234$	0	0
$235$	16.1803	1.05549
$236$	0	0
$237$	3.41641	0.221920
$238$	0	0
$239$	−30.1803	−1.95220	−0.976102	−	0.217313i	$-0.930271\pi$
$239$	−30.1803	−1.95220	−0.976102	+	0.217313i	$0.930271\pi$
$240$	0	0
$241$	−25.4164	−1.63721	−0.818607	−	0.574354i	$-0.805253\pi$
$241$	−25.4164	−1.63721	−0.818607	+	0.574354i	$0.805253\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	−12.5623	−0.802576
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.583592	0.0369836
$250$	0	0
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	0	0
$253$	−16.9443	−1.06528
$254$	0	0
$255$	15.1246	0.947140
$256$	0	0
$257$	18.3820	1.14664	0.573318	−	0.819333i	$-0.305656\pi$
$257$	18.3820	1.14664	0.573318	+	0.819333i	$0.305656\pi$
$258$	0	0
$259$	27.1246	1.68544
$260$	0	0
$261$	13.3820	0.828323
$262$	0	0
$263$	−2.94427	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$263$	−2.94427	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$264$	0	0
$265$	22.0344	1.35357
$266$	0	0
$267$	10.9443	0.669779
$268$	0	0
$269$	14.3262	0.873486	0.436743	−	0.899586i	$-0.356132\pi$
$269$	14.3262	0.873486	0.436743	+	0.899586i	$0.356132\pi$
$270$	0	0
$271$	−26.0000	−1.57939	−0.789694	−	0.613501i	$-0.789761\pi$
$271$	−26.0000	−1.57939	−0.789694	+	0.613501i	$0.789761\pi$
$272$	0	0
$273$	5.52786	0.334562
$274$	0	0
$275$	26.1803	1.57873
$276$	0	0
$277$	−10.0902	−0.606260	−0.303130	−	0.952949i	$-0.598032\pi$
$277$	−10.0902	−0.606260	−0.303130	+	0.952949i	$0.598032\pi$
$278$	0	0
$279$	−1.81966	−0.108940
$280$	0	0
$281$	20.9787	1.25149	0.625743	−	0.780030i	$-0.284796\pi$
$281$	20.9787	1.25149	0.625743	+	0.780030i	$0.284796\pi$
$282$	0	0
$283$	−22.4721	−1.33583	−0.667915	−	0.744238i	$-0.732813\pi$
$283$	−22.4721	−1.33583	−0.667915	+	0.744238i	$0.732813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.76393	0.163150
$288$	0	0
$289$	−5.56231	−0.327194
$290$	0	0
$291$	10.6525	0.624459
$292$	0	0
$293$	22.9443	1.34042	0.670209	−	0.742172i	$-0.266204\pi$
$293$	22.9443	1.34042	0.670209	+	0.742172i	$0.266204\pi$
$294$	0	0
$295$	−1.70820	−0.0994555
$296$	0	0
$297$	−17.8885	−1.03800
$298$	0	0
$299$	−7.23607	−0.418473
$300$	0	0
$301$	29.8885	1.72275
$302$	0	0
$303$	−19.3050	−1.10904
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	−11.7082	−0.668222	−0.334111	−	0.942534i	$-0.608436\pi$
$307$	−11.7082	−0.668222	−0.334111	+	0.942534i	$0.608436\pi$
$308$	0	0
$309$	−18.1115	−1.03032
$310$	0	0
$311$	−31.2361	−1.77123	−0.885617	−	0.464415i	$-0.846264\pi$
$311$	−31.2361	−1.77123	−0.885617	+	0.464415i	$0.846264\pi$
$312$	0	0
$313$	−15.5066	−0.876484	−0.438242	−	0.898857i	$-0.644399\pi$
$313$	−15.5066	−0.876484	−0.438242	+	0.898857i	$0.644399\pi$
$314$	0	0
$315$	17.2361	0.971142
$316$	0	0
$317$	1.32624	0.0744889	0.0372445	−	0.999306i	$-0.488142\pi$
$317$	1.32624	0.0744889	0.0372445	+	0.999306i	$0.488142\pi$
$318$	0	0
$319$	−29.4164	−1.64700
$320$	0	0
$321$	−11.6393	−0.649644
$322$	0	0
$323$	0	0
$324$	0	0
$325$	11.1803	0.620174
$326$	0	0
$327$	−13.6393	−0.754256
$328$	0	0
$329$	−14.4721	−0.797875
$330$	0	0
$331$	−25.7082	−1.41305	−0.706525	−	0.707688i	$-0.749738\pi$
$331$	−25.7082	−1.41305	−0.706525	+	0.707688i	$0.749738\pi$
$332$	0	0
$333$	−12.3394	−0.676195
$334$	0	0
$335$	42.3607	2.31441
$336$	0	0
$337$	2.38197	0.129754	0.0648770	−	0.997893i	$-0.479335\pi$
$337$	2.38197	0.129754	0.0648770	+	0.997893i	$0.479335\pi$
$338$	0	0
$339$	−12.1803	−0.661545
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	23.4164	1.26070
$346$	0	0
$347$	2.65248	0.142392	0.0711962	−	0.997462i	$-0.477318\pi$
$347$	2.65248	0.142392	0.0711962	+	0.997462i	$0.477318\pi$
$348$	0	0
$349$	−33.0902	−1.77128	−0.885638	−	0.464376i	$-0.846279\pi$
$349$	−33.0902	−1.77128	−0.885638	+	0.464376i	$0.846279\pi$
$350$	0	0
$351$	−7.63932	−0.407757
$352$	0	0
$353$	−23.7426	−1.26369	−0.631847	−	0.775093i	$-0.717703\pi$
$353$	−23.7426	−1.26369	−0.631847	+	0.775093i	$0.717703\pi$
$354$	0	0
$355$	10.6525	0.565375
$356$	0	0
$357$	−13.5279	−0.715970
$358$	0	0
$359$	−11.5279	−0.608417	−0.304209	−	0.952605i	$-0.598392\pi$
$359$	−11.5279	−0.608417	−0.304209	+	0.952605i	$0.598392\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−0.652476	−0.0342461
$364$	0	0
$365$	20.3262	1.06392
$366$	0	0
$367$	−18.8328	−0.983065	−0.491532	−	0.870859i	$-0.663563\pi$
$367$	−18.8328	−0.983065	−0.491532	+	0.870859i	$0.663563\pi$
$368$	0	0
$369$	−1.25735	−0.0654552
$370$	0	0
$371$	−19.7082	−1.02320
$372$	0	0
$373$	18.3262	0.948897	0.474448	−	0.880283i	$-0.342648\pi$
$373$	18.3262	0.948897	0.474448	+	0.880283i	$0.342648\pi$
$374$	0	0
$375$	−13.8197	−0.713644
$376$	0	0
$377$	−12.5623	−0.646992
$378$	0	0
$379$	35.4164	1.81922	0.909609	−	0.415465i	$-0.136381\pi$
$379$	35.4164	1.81922	0.909609	+	0.415465i	$0.136381\pi$
$380$	0	0
$381$	14.1115	0.722952
$382$	0	0
$383$	−20.7639	−1.06099	−0.530494	−	0.847689i	$-0.677993\pi$
$383$	−20.7639	−1.06099	−0.530494	+	0.847689i	$0.677993\pi$
$384$	0	0
$385$	−37.8885	−1.93098
$386$	0	0
$387$	−13.5967	−0.691162
$388$	0	0
$389$	12.7984	0.648903	0.324452	−	0.945902i	$-0.394820\pi$
$389$	12.7984	0.648903	0.324452	+	0.945902i	$0.394820\pi$
$390$	0	0
$391$	17.7082	0.895542
$392$	0	0
$393$	−5.52786	−0.278844
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	−22.3607	−1.12225	−0.561125	−	0.827731i	$-0.689631\pi$
$397$	−22.3607	−1.12225	−0.561125	+	0.827731i	$0.689631\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.9443	0.946032	0.473016	−	0.881054i	$-0.343165\pi$
$401$	18.9443	0.946032	0.473016	+	0.881054i	$0.343165\pi$
$402$	0	0
$403$	1.70820	0.0850917
$404$	0	0
$405$	8.74265	0.434426
$406$	0	0
$407$	27.1246	1.34452
$408$	0	0
$409$	16.2705	0.804525	0.402262	−	0.915524i	$-0.368224\pi$
$409$	16.2705	0.804525	0.402262	+	0.915524i	$0.368224\pi$
$410$	0	0
$411$	2.94427	0.145230
$412$	0	0
$413$	1.52786	0.0751813
$414$	0	0
$415$	−1.70820	−0.0838524
$416$	0	0
$417$	17.3050	0.847427
$418$	0	0
$419$	6.47214	0.316185	0.158092	−	0.987424i	$-0.449466\pi$
$419$	6.47214	0.316185	0.158092	+	0.987424i	$0.449466\pi$
$420$	0	0
$421$	13.9098	0.677924	0.338962	−	0.940800i	$-0.389924\pi$
$421$	13.9098	0.677924	0.338962	+	0.940800i	$0.389924\pi$
$422$	0	0
$423$	6.58359	0.320105
$424$	0	0
$425$	−27.3607	−1.32719
$426$	0	0
$427$	4.47214	0.216422
$428$	0	0
$429$	5.52786	0.266888
$430$	0	0
$431$	−6.65248	−0.320438	−0.160219	−	0.987081i	$-0.551220\pi$
$431$	−6.65248	−0.320438	−0.160219	+	0.987081i	$0.551220\pi$
$432$	0	0
$433$	−25.0344	−1.20308	−0.601539	−	0.798843i	$-0.705446\pi$
$433$	−25.0344	−1.20308	−0.601539	+	0.798843i	$0.705446\pi$
$434$	0	0
$435$	40.6525	1.94914
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−29.2361	−1.39536	−0.697681	−	0.716409i	$-0.745785\pi$
$439$	−29.2361	−1.39536	−0.697681	+	0.716409i	$0.745785\pi$
$440$	0	0
$441$	−5.11146	−0.243403
$442$	0	0
$443$	−9.23607	−0.438819	−0.219409	−	0.975633i	$-0.570413\pi$
$443$	−9.23607	−0.438819	−0.219409	+	0.975633i	$0.570413\pi$
$444$	0	0
$445$	−32.0344	−1.51858
$446$	0	0
$447$	−16.1803	−0.765304
$448$	0	0
$449$	−22.7984	−1.07592	−0.537961	−	0.842970i	$-0.680805\pi$
$449$	−22.7984	−1.07592	−0.537961	+	0.842970i	$0.680805\pi$
$450$	0	0
$451$	2.76393	0.130148
$452$	0	0
$453$	−17.8885	−0.840477
$454$	0	0
$455$	−16.1803	−0.758546
$456$	0	0
$457$	25.2148	1.17950	0.589749	−	0.807587i	$-0.299227\pi$
$457$	25.2148	1.17950	0.589749	+	0.807587i	$0.299227\pi$
$458$	0	0
$459$	18.6950	0.872610
$460$	0	0
$461$	21.4164	0.997462	0.498731	−	0.866757i	$-0.333800\pi$
$461$	21.4164	0.997462	0.498731	+	0.866757i	$0.333800\pi$
$462$	0	0
$463$	5.05573	0.234960	0.117480	−	0.993075i	$-0.462518\pi$
$463$	5.05573	0.234960	0.117480	+	0.993075i	$0.462518\pi$
$464$	0	0
$465$	−5.52786	−0.256349
$466$	0	0
$467$	33.3050	1.54117	0.770585	−	0.637338i	$-0.219964\pi$
$467$	33.3050	1.54117	0.770585	+	0.637338i	$0.219964\pi$
$468$	0	0
$469$	−37.8885	−1.74953
$470$	0	0
$471$	11.5967	0.534350
$472$	0	0
$473$	29.8885	1.37428
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.96556	0.410505
$478$	0	0
$479$	−7.70820	−0.352197	−0.176098	−	0.984373i	$-0.556348\pi$
$479$	−7.70820	−0.352197	−0.176098	+	0.984373i	$0.556348\pi$
$480$	0	0
$481$	11.5836	0.528166
$482$	0	0
$483$	−20.9443	−0.952997
$484$	0	0
$485$	−31.1803	−1.41583
$486$	0	0
$487$	−19.2361	−0.871669	−0.435835	−	0.900027i	$-0.643547\pi$
$487$	−19.2361	−0.871669	−0.435835	+	0.900027i	$0.643547\pi$
$488$	0	0
$489$	−13.1672	−0.595441
$490$	0	0
$491$	1.52786	0.0689515	0.0344758	−	0.999406i	$-0.489024\pi$
$491$	1.52786	0.0689515	0.0344758	+	0.999406i	$0.489024\pi$
$492$	0	0
$493$	30.7426	1.38458
$494$	0	0
$495$	17.2361	0.774704
$496$	0	0
$497$	−9.52786	−0.427383
$498$	0	0
$499$	35.3050	1.58047	0.790233	−	0.612806i	$-0.209959\pi$
$499$	35.3050	1.58047	0.790233	+	0.612806i	$0.209959\pi$
$500$	0	0
$501$	15.4164	0.688754
$502$	0	0
$503$	−6.76393	−0.301589	−0.150794	−	0.988565i	$-0.548183\pi$
$503$	−6.76393	−0.301589	−0.150794	+	0.988565i	$0.548183\pi$
$504$	0	0
$505$	56.5066	2.51451
$506$	0	0
$507$	−13.7082	−0.608803
$508$	0	0
$509$	42.6869	1.89206	0.946032	−	0.324073i	$-0.105052\pi$
$509$	42.6869	1.89206	0.946032	+	0.324073i	$0.105052\pi$
$510$	0	0
$511$	−18.1803	−0.804251
$512$	0	0
$513$	0	0
$514$	0	0
$515$	53.0132	2.33604
$516$	0	0
$517$	−14.4721	−0.636484
$518$	0	0
$519$	1.63932	0.0719582
$520$	0	0
$521$	−5.27051	−0.230905	−0.115453	−	0.993313i	$-0.536832\pi$
$521$	−5.27051	−0.230905	−0.115453	+	0.993313i	$0.536832\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	32.3607	1.41234
$526$	0	0
$527$	−4.18034	−0.182098
$528$	0	0
$529$	4.41641	0.192018
$530$	0	0
$531$	−0.695048	−0.0301625
$532$	0	0
$533$	1.18034	0.0511262
$534$	0	0
$535$	34.0689	1.47293
$536$	0	0
$537$	28.9443	1.24904
$538$	0	0
$539$	11.2361	0.483972
$540$	0	0
$541$	−26.6180	−1.14440	−0.572199	−	0.820115i	$-0.693910\pi$
$541$	−26.6180	−1.14440	−0.572199	+	0.820115i	$0.693910\pi$
$542$	0	0
$543$	6.69505	0.287312
$544$	0	0
$545$	39.9230	1.71011
$546$	0	0
$547$	−5.05573	−0.216167	−0.108084	−	0.994142i	$-0.534471\pi$
$547$	−5.05573	−0.216167	−0.108084	+	0.994142i	$0.534471\pi$
$548$	0	0
$549$	−2.03444	−0.0868279
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.94427	0.380349
$554$	0	0
$555$	−37.4853	−1.59116
$556$	0	0
$557$	5.05573	0.214218	0.107109	−	0.994247i	$-0.465841\pi$
$557$	5.05573	0.214218	0.107109	+	0.994247i	$0.465841\pi$
$558$	0	0
$559$	12.7639	0.539857
$560$	0	0
$561$	−13.5279	−0.571147
$562$	0	0
$563$	−26.2918	−1.10807	−0.554034	−	0.832494i	$-0.686912\pi$
$563$	−26.2918	−1.10807	−0.554034	+	0.832494i	$0.686912\pi$
$564$	0	0
$565$	35.6525	1.49991
$566$	0	0
$567$	−7.81966	−0.328395
$568$	0	0
$569$	7.90983	0.331597	0.165799	−	0.986160i	$-0.446980\pi$
$569$	7.90983	0.331597	0.165799	+	0.986160i	$0.446980\pi$
$570$	0	0
$571$	32.5410	1.36180	0.680900	−	0.732377i	$-0.261589\pi$
$571$	32.5410	1.36180	0.680900	+	0.732377i	$0.261589\pi$
$572$	0	0
$573$	−4.36068	−0.182170
$574$	0	0
$575$	−42.3607	−1.76656
$576$	0	0
$577$	45.9230	1.91180	0.955899	−	0.293694i	$-0.0948847\pi$
$577$	45.9230	1.91180	0.955899	+	0.293694i	$0.0948847\pi$
$578$	0	0
$579$	−14.2492	−0.592178
$580$	0	0
$581$	1.52786	0.0633865
$582$	0	0
$583$	−19.7082	−0.816230
$584$	0	0
$585$	7.36068	0.304327
$586$	0	0
$587$	−2.87539	−0.118680	−0.0593400	−	0.998238i	$-0.518900\pi$
$587$	−2.87539	−0.118680	−0.0593400	+	0.998238i	$0.518900\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−5.12461	−0.210798
$592$	0	0
$593$	−34.9230	−1.43412	−0.717058	−	0.697014i	$-0.754512\pi$
$593$	−34.9230	−1.43412	−0.717058	+	0.697014i	$0.754512\pi$
$594$	0	0
$595$	39.5967	1.62331
$596$	0	0
$597$	−12.9443	−0.529774
$598$	0	0
$599$	39.1246	1.59859	0.799294	−	0.600940i	$-0.205207\pi$
$599$	39.1246	1.59859	0.799294	+	0.600940i	$0.205207\pi$
$600$	0	0
$601$	16.4721	0.671912	0.335956	−	0.941878i	$-0.390941\pi$
$601$	16.4721	0.671912	0.335956	+	0.941878i	$0.390941\pi$
$602$	0	0
$603$	17.2361	0.701907
$604$	0	0
$605$	1.90983	0.0776456
$606$	0	0
$607$	47.7771	1.93921	0.969606	−	0.244671i	$-0.0786801\pi$
$607$	47.7771	1.93921	0.969606	+	0.244671i	$0.0786801\pi$
$608$	0	0
$609$	−36.3607	−1.47341
$610$	0	0
$611$	−6.18034	−0.250030
$612$	0	0
$613$	8.38197	0.338544	0.169272	−	0.985569i	$-0.445858\pi$
$613$	8.38197	0.338544	0.169272	+	0.985569i	$0.445858\pi$
$614$	0	0
$615$	−3.81966	−0.154024
$616$	0	0
$617$	42.9443	1.72887	0.864436	−	0.502743i	$-0.167676\pi$
$617$	42.9443	1.72887	0.864436	+	0.502743i	$0.167676\pi$
$618$	0	0
$619$	−22.3607	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$619$	−22.3607	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$620$	0	0
$621$	28.9443	1.16149
$622$	0	0
$623$	28.6525	1.14794
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.3475	−1.13029
$630$	0	0
$631$	31.1246	1.23905	0.619526	−	0.784976i	$-0.287325\pi$
$631$	31.1246	1.23905	0.619526	+	0.784976i	$0.287325\pi$
$632$	0	0
$633$	8.94427	0.355503
$634$	0	0
$635$	−41.3050	−1.63914
$636$	0	0
$637$	4.79837	0.190118
$638$	0	0
$639$	4.33437	0.171465
$640$	0	0
$641$	−13.2705	−0.524154	−0.262077	−	0.965047i	$-0.584407\pi$
$641$	−13.2705	−0.524154	−0.262077	+	0.965047i	$0.584407\pi$
$642$	0	0
$643$	38.8328	1.53142	0.765708	−	0.643188i	$-0.222389\pi$
$643$	38.8328	1.53142	0.765708	+	0.643188i	$0.222389\pi$
$644$	0	0
$645$	−41.3050	−1.62638
$646$	0	0
$647$	−26.0689	−1.02487	−0.512437	−	0.858725i	$-0.671257\pi$
$647$	−26.0689	−1.02487	−0.512437	+	0.858725i	$0.671257\pi$
$648$	0	0
$649$	1.52786	0.0599739
$650$	0	0
$651$	4.94427	0.193781
$652$	0	0
$653$	−35.3951	−1.38512	−0.692559	−	0.721361i	$-0.743517\pi$
$653$	−35.3951	−1.38512	−0.692559	+	0.721361i	$0.743517\pi$
$654$	0	0
$655$	16.1803	0.632218
$656$	0	0
$657$	8.27051	0.322663
$658$	0	0
$659$	48.5410	1.89089	0.945445	−	0.325782i	$-0.105628\pi$
$659$	48.5410	1.89089	0.945445	+	0.325782i	$0.105628\pi$
$660$	0	0
$661$	−21.4164	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$661$	−21.4164	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$662$	0	0
$663$	−5.77709	−0.224363
$664$	0	0
$665$	0	0
$666$	0	0
$667$	47.5967	1.84295
$668$	0	0
$669$	−34.2492	−1.32415
$670$	0	0
$671$	4.47214	0.172645
$672$	0	0
$673$	30.3820	1.17114	0.585569	−	0.810622i	$-0.300871\pi$
$673$	30.3820	1.17114	0.585569	+	0.810622i	$0.300871\pi$
$674$	0	0
$675$	−44.7214	−1.72133
$676$	0	0
$677$	−6.61803	−0.254352	−0.127176	−	0.991880i	$-0.540591\pi$
$677$	−6.61803	−0.254352	−0.127176	+	0.991880i	$0.540591\pi$
$678$	0	0
$679$	27.8885	1.07026
$680$	0	0
$681$	11.0557	0.423656
$682$	0	0
$683$	20.1803	0.772179	0.386090	−	0.922461i	$-0.373826\pi$
$683$	20.1803	0.772179	0.386090	+	0.922461i	$0.373826\pi$
$684$	0	0
$685$	−8.61803	−0.329278
$686$	0	0
$687$	5.41641	0.206649
$688$	0	0
$689$	−8.41641	−0.320640
$690$	0	0
$691$	22.3607	0.850640	0.425320	−	0.905043i	$-0.360161\pi$
$691$	22.3607	0.850640	0.425320	+	0.905043i	$0.360161\pi$
$692$	0	0
$693$	−15.4164	−0.585621
$694$	0	0
$695$	−50.6525	−1.92136
$696$	0	0
$697$	−2.88854	−0.109411
$698$	0	0
$699$	1.93112	0.0730415
$700$	0	0
$701$	38.6312	1.45908	0.729540	−	0.683938i	$-0.239734\pi$
$701$	38.6312	1.45908	0.729540	+	0.683938i	$0.239734\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	20.0000	0.753244
$706$	0	0
$707$	−50.5410	−1.90079
$708$	0	0
$709$	6.90983	0.259504	0.129752	−	0.991546i	$-0.458582\pi$
$709$	6.90983	0.259504	0.129752	+	0.991546i	$0.458582\pi$
$710$	0	0
$711$	−4.06888	−0.152595
$712$	0	0
$713$	−6.47214	−0.242383
$714$	0	0
$715$	−16.1803	−0.605110
$716$	0	0
$717$	−37.3050	−1.39318
$718$	0	0
$719$	−12.2918	−0.458407	−0.229203	−	0.973379i	$-0.573612\pi$
$719$	−12.2918	−0.458407	−0.229203	+	0.973379i	$0.573612\pi$
$720$	0	0
$721$	−47.4164	−1.76588
$722$	0	0
$723$	−31.4164	−1.16839
$724$	0	0
$725$	−73.5410	−2.73124
$726$	0	0
$727$	30.9443	1.14766	0.573830	−	0.818975i	$-0.305457\pi$
$727$	30.9443	1.14766	0.573830	+	0.818975i	$0.305457\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−31.2361	−1.15531
$732$	0	0
$733$	−1.09017	−0.0402663	−0.0201332	−	0.999797i	$-0.506409\pi$
$733$	−1.09017	−0.0402663	−0.0201332	+	0.999797i	$0.506409\pi$
$734$	0	0
$735$	−15.5279	−0.572754
$736$	0	0
$737$	−37.8885	−1.39564
$738$	0	0
$739$	−9.12461	−0.335654	−0.167827	−	0.985816i	$-0.553675\pi$
$739$	−9.12461	−0.335654	−0.167827	+	0.985816i	$0.553675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.11146	−0.224208	−0.112104	−	0.993697i	$-0.535759\pi$
$743$	−6.11146	−0.224208	−0.112104	+	0.993697i	$0.535759\pi$
$744$	0	0
$745$	47.3607	1.73516
$746$	0	0
$747$	−0.695048	−0.0254305
$748$	0	0
$749$	−30.4721	−1.11343
$750$	0	0
$751$	24.0689	0.878286	0.439143	−	0.898417i	$-0.355282\pi$
$751$	24.0689	0.878286	0.439143	+	0.898417i	$0.355282\pi$
$752$	0	0
$753$	2.83282	0.103234
$754$	0	0
$755$	52.3607	1.90560
$756$	0	0
$757$	4.43769	0.161291	0.0806454	−	0.996743i	$-0.474302\pi$
$757$	4.43769	0.161291	0.0806454	+	0.996743i	$0.474302\pi$
$758$	0	0
$759$	−20.9443	−0.760229
$760$	0	0
$761$	26.3607	0.955574	0.477787	−	0.878476i	$-0.341439\pi$
$761$	26.3607	0.955574	0.477787	+	0.878476i	$0.341439\pi$
$762$	0	0
$763$	−35.7082	−1.29272
$764$	0	0
$765$	−18.0132	−0.651267
$766$	0	0
$767$	0.652476	0.0235595
$768$	0	0
$769$	38.7426	1.39710	0.698548	−	0.715563i	$-0.253830\pi$
$769$	38.7426	1.39710	0.698548	+	0.715563i	$0.253830\pi$
$770$	0	0
$771$	22.7214	0.818290
$772$	0	0
$773$	−8.14590	−0.292988	−0.146494	−	0.989212i	$-0.546799\pi$
$773$	−8.14590	−0.292988	−0.146494	+	0.989212i	$0.546799\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	0	0
$777$	33.5279	1.20281
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−9.52786	−0.340934
$782$	0	0
$783$	50.2492	1.79576
$784$	0	0
$785$	−33.9443	−1.21152
$786$	0	0
$787$	−28.4721	−1.01492	−0.507461	−	0.861675i	$-0.669416\pi$
$787$	−28.4721	−1.01492	−0.507461	+	0.861675i	$0.669416\pi$
$788$	0	0
$789$	−3.63932	−0.129563
$790$	0	0
$791$	−31.8885	−1.13383
$792$	0	0
$793$	1.90983	0.0678201
$794$	0	0
$795$	27.2361	0.965964
$796$	0	0
$797$	28.4377	1.00731	0.503657	−	0.863903i	$-0.331987\pi$
$797$	28.4377	1.00731	0.503657	+	0.863903i	$0.331987\pi$
$798$	0	0
$799$	15.1246	0.535070
$800$	0	0
$801$	−13.0344	−0.460549
$802$	0	0
$803$	−18.1803	−0.641570
$804$	0	0
$805$	61.3050	2.16072
$806$	0	0
$807$	17.7082	0.623358
$808$	0	0
$809$	2.74265	0.0964263	0.0482131	−	0.998837i	$-0.484647\pi$
$809$	2.74265	0.0964263	0.0482131	+	0.998837i	$0.484647\pi$
$810$	0	0
$811$	−5.59675	−0.196528	−0.0982642	−	0.995160i	$-0.531329\pi$
$811$	−5.59675	−0.196528	−0.0982642	+	0.995160i	$0.531329\pi$
$812$	0	0
$813$	−32.1378	−1.12712
$814$	0	0
$815$	38.5410	1.35003
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6.58359	−0.230049
$820$	0	0
$821$	19.2705	0.672545	0.336273	−	0.941765i	$-0.390834\pi$
$821$	19.2705	0.672545	0.336273	+	0.941765i	$0.390834\pi$
$822$	0	0
$823$	20.8328	0.726186	0.363093	−	0.931753i	$-0.381721\pi$
$823$	20.8328	0.726186	0.363093	+	0.931753i	$0.381721\pi$
$824$	0	0
$825$	32.3607	1.12665
$826$	0	0
$827$	30.5410	1.06202	0.531008	−	0.847367i	$-0.321814\pi$
$827$	30.5410	1.06202	0.531008	+	0.847367i	$0.321814\pi$
$828$	0	0
$829$	−9.72949	−0.337919	−0.168960	−	0.985623i	$-0.554041\pi$
$829$	−9.72949	−0.337919	−0.168960	+	0.985623i	$0.554041\pi$
$830$	0	0
$831$	−12.4721	−0.432654
$832$	0	0
$833$	−11.7426	−0.406859
$834$	0	0
$835$	−45.1246	−1.56160
$836$	0	0
$837$	−6.83282	−0.236177
$838$	0	0
$839$	−22.0000	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$840$	0	0
$841$	53.6312	1.84935
$842$	0	0
$843$	25.9311	0.893115
$844$	0	0
$845$	40.1246	1.38033
$846$	0	0
$847$	−1.70820	−0.0586946
$848$	0	0
$849$	−27.7771	−0.953307
$850$	0	0
$851$	−43.8885	−1.50448
$852$	0	0
$853$	−29.0902	−0.996028	−0.498014	−	0.867169i	$-0.665937\pi$
$853$	−29.0902	−0.996028	−0.498014	+	0.867169i	$0.665937\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.56231	0.190005	0.0950024	−	0.995477i	$-0.469714\pi$
$857$	5.56231	0.190005	0.0950024	+	0.995477i	$0.469714\pi$
$858$	0	0
$859$	−28.0689	−0.957698	−0.478849	−	0.877897i	$-0.658946\pi$
$859$	−28.0689	−0.957698	−0.478849	+	0.877897i	$0.658946\pi$
$860$	0	0
$861$	3.41641	0.116431
$862$	0	0
$863$	25.8885	0.881256	0.440628	−	0.897690i	$-0.354756\pi$
$863$	25.8885	0.881256	0.440628	+	0.897690i	$0.354756\pi$
$864$	0	0
$865$	−4.79837	−0.163150
$866$	0	0
$867$	−6.87539	−0.233500
$868$	0	0
$869$	8.94427	0.303414
$870$	0	0
$871$	−16.1803	−0.548250
$872$	0	0
$873$	−12.6869	−0.429387
$874$	0	0
$875$	−36.1803	−1.22312
$876$	0	0
$877$	1.74265	0.0588450	0.0294225	−	0.999567i	$-0.490633\pi$
$877$	1.74265	0.0588450	0.0294225	+	0.999567i	$0.490633\pi$
$878$	0	0
$879$	28.3607	0.956582
$880$	0	0
$881$	5.21478	0.175690	0.0878452	−	0.996134i	$-0.472002\pi$
$881$	5.21478	0.175690	0.0878452	+	0.996134i	$0.472002\pi$
$882$	0	0
$883$	17.8197	0.599679	0.299840	−	0.953990i	$-0.403067\pi$
$883$	17.8197	0.599679	0.299840	+	0.953990i	$0.403067\pi$
$884$	0	0
$885$	−2.11146	−0.0709758
$886$	0	0
$887$	−6.94427	−0.233166	−0.116583	−	0.993181i	$-0.537194\pi$
$887$	−6.94427	−0.233166	−0.116583	+	0.993181i	$0.537194\pi$
$888$	0	0
$889$	36.9443	1.23907
$890$	0	0
$891$	−7.81966	−0.261968
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−84.7214	−2.83192
$896$	0	0
$897$	−8.94427	−0.298641
$898$	0	0
$899$	−11.2361	−0.374744
$900$	0	0
$901$	20.5967	0.686177
$902$	0	0
$903$	36.9443	1.22943
$904$	0	0
$905$	−19.5967	−0.651418
$906$	0	0
$907$	0.291796	0.00968893	0.00484446	−	0.999988i	$-0.498458\pi$
$907$	0.291796	0.00968893	0.00484446	+	0.999988i	$0.498458\pi$
$908$	0	0
$909$	22.9919	0.762592
$910$	0	0
$911$	−12.0689	−0.399860	−0.199930	−	0.979810i	$-0.564071\pi$
$911$	−12.0689	−0.399860	−0.199930	+	0.979810i	$0.564071\pi$
$912$	0	0
$913$	1.52786	0.0505649
$914$	0	0
$915$	−6.18034	−0.204316
$916$	0	0
$917$	−14.4721	−0.477912
$918$	0	0
$919$	−45.8885	−1.51372	−0.756862	−	0.653575i	$-0.773268\pi$
$919$	−45.8885	−1.51372	−0.756862	+	0.653575i	$0.773268\pi$
$920$	0	0
$921$	−14.4721	−0.476873
$922$	0	0
$923$	−4.06888	−0.133929
$924$	0	0
$925$	67.8115	2.22963
$926$	0	0
$927$	21.5704	0.708466
$928$	0	0
$929$	−59.1591	−1.94095	−0.970473	−	0.241211i	$-0.922456\pi$
$929$	−59.1591	−1.94095	−0.970473	+	0.241211i	$0.922456\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−38.6099	−1.26403
$934$	0	0
$935$	39.5967	1.29495
$936$	0	0
$937$	15.8885	0.519056	0.259528	−	0.965736i	$-0.416433\pi$
$937$	15.8885	0.519056	0.259528	+	0.965736i	$0.416433\pi$
$938$	0	0
$939$	−19.1672	−0.625497
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	−4.47214	−0.145633
$944$	0	0
$945$	64.7214	2.10539
$946$	0	0
$947$	−8.18034	−0.265825	−0.132913	−	0.991128i	$-0.542433\pi$
$947$	−8.18034	−0.265825	−0.132913	+	0.991128i	$0.542433\pi$
$948$	0	0
$949$	−7.76393	−0.252028
$950$	0	0
$951$	1.63932	0.0531586
$952$	0	0
$953$	47.8115	1.54877	0.774384	−	0.632716i	$-0.218060\pi$
$953$	47.8115	1.54877	0.774384	+	0.632716i	$0.218060\pi$
$954$	0	0
$955$	12.7639	0.413031
$956$	0	0
$957$	−36.3607	−1.17537
$958$	0	0
$959$	7.70820	0.248911
$960$	0	0
$961$	−29.4721	−0.950714
$962$	0	0
$963$	13.8622	0.446704
$964$	0	0
$965$	41.7082	1.34263
$966$	0	0
$967$	−8.83282	−0.284044	−0.142022	−	0.989863i	$-0.545360\pi$
$967$	−8.83282	−0.284044	−0.142022	+	0.989863i	$0.545360\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.5410	−1.94285	−0.971427	−	0.237339i	$-0.923725\pi$
$971$	−60.5410	−1.94285	−0.971427	+	0.237339i	$0.923725\pi$
$972$	0	0
$973$	45.3050	1.45241
$974$	0	0
$975$	13.8197	0.442583
$976$	0	0
$977$	−9.03444	−0.289037	−0.144519	−	0.989502i	$-0.546163\pi$
$977$	−9.03444	−0.289037	−0.144519	+	0.989502i	$0.546163\pi$
$978$	0	0
$979$	28.6525	0.915737
$980$	0	0
$981$	16.2442	0.518637
$982$	0	0
$983$	−19.5967	−0.625039	−0.312520	−	0.949911i	$-0.601173\pi$
$983$	−19.5967	−0.625039	−0.312520	+	0.949911i	$0.601173\pi$
$984$	0	0
$985$	15.0000	0.477940
$986$	0	0
$987$	−17.8885	−0.569399
$988$	0	0
$989$	−48.3607	−1.53778
$990$	0	0
$991$	30.6525	0.973708	0.486854	−	0.873483i	$-0.338144\pi$
$991$	30.6525	0.973708	0.486854	+	0.873483i	$0.338144\pi$
$992$	0	0
$993$	−31.7771	−1.00842
$994$	0	0
$995$	37.8885	1.20115
$996$	0	0
$997$	−13.5066	−0.427758	−0.213879	−	0.976860i	$-0.568610\pi$
$997$	−13.5066	−0.427758	−0.213879	+	0.976860i	$0.568610\pi$
$998$	0	0
$999$	−46.3344	−1.46595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.t.1.2		2
4.3	odd	2		722.2.a.h.1.1	✓	2
12.11	even	2		6498.2.a.bk.1.2		2
19.18	odd	2		5776.2.a.be.1.1		2
76.3	even	18		722.2.e.p.389.2		12
76.7	odd	6		722.2.c.i.429.2		4
76.11	odd	6		722.2.c.i.653.2		4
76.15	even	18		722.2.e.p.415.1		12
76.23	odd	18		722.2.e.q.415.2		12
76.27	even	6		722.2.c.h.653.1		4
76.31	even	6		722.2.c.h.429.1		4
76.35	odd	18		722.2.e.q.389.1		12
76.43	odd	18		722.2.e.q.595.2		12
76.47	odd	18		722.2.e.q.423.1		12
76.51	even	18		722.2.e.p.245.2		12
76.55	odd	18		722.2.e.q.99.1		12
76.59	even	18		722.2.e.p.99.2		12
76.63	odd	18		722.2.e.q.245.1		12
76.67	even	18		722.2.e.p.423.2		12
76.71	even	18		722.2.e.p.595.1		12
76.75	even	2		722.2.a.i.1.2	yes	2
228.227	odd	2		6498.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.h.1.1	✓	2	4.3	odd	2
722.2.a.i.1.2	yes	2	76.75	even	2
722.2.c.h.429.1		4	76.31	even	6
722.2.c.h.653.1		4	76.27	even	6
722.2.c.i.429.2		4	76.7	odd	6
722.2.c.i.653.2		4	76.11	odd	6
722.2.e.p.99.2		12	76.59	even	18
722.2.e.p.245.2		12	76.51	even	18
722.2.e.p.389.2		12	76.3	even	18
722.2.e.p.415.1		12	76.15	even	18
722.2.e.p.423.2		12	76.67	even	18
722.2.e.p.595.1		12	76.71	even	18
722.2.e.q.99.1		12	76.55	odd	18
722.2.e.q.245.1		12	76.63	odd	18
722.2.e.q.389.1		12	76.35	odd	18
722.2.e.q.415.2		12	76.23	odd	18
722.2.e.q.423.1		12	76.47	odd	18
722.2.e.q.595.2		12	76.43	odd	18
5776.2.a.t.1.2		2	1.1	even	1	trivial
5776.2.a.be.1.1		2	19.18	odd	2
6498.2.a.be.1.2		2	228.227	odd	2
6498.2.a.bk.1.2		2	12.11	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -3.61803 q^{5} +3.23607 q^{7} -1.47214 q^{9} +3.23607 q^{11} +1.38197 q^{13} -4.47214 q^{15} -3.38197 q^{17} +4.00000 q^{21} -5.23607 q^{23} +8.09017 q^{25} -5.52786 q^{27} -9.09017 q^{29} +1.23607 q^{31} +4.00000 q^{33} -11.7082 q^{35} +8.38197 q^{37} +1.70820 q^{39} +0.854102 q^{41} +9.23607 q^{43} +5.32624 q^{45} -4.47214 q^{47} +3.47214 q^{49} -4.18034 q^{51} -6.09017 q^{53} -11.7082 q^{55} +0.472136 q^{59} +1.38197 q^{61} -4.76393 q^{63} -5.00000 q^{65} -11.7082 q^{67} -6.47214 q^{69} -2.94427 q^{71} -5.61803 q^{73} +10.0000 q^{75} +10.4721 q^{77} +2.76393 q^{79} -2.41641 q^{81} +0.472136 q^{83} +12.2361 q^{85} -11.2361 q^{87} +8.85410 q^{89} +4.47214 q^{91} +1.52786 q^{93} +8.61803 q^{97} -4.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} + 2 q^{11} + 5 q^{13} - 9 q^{17} + 8 q^{21} - 6 q^{23} + 5 q^{25} - 20 q^{27} - 7 q^{29} - 2 q^{31} + 8 q^{33} - 10 q^{35} + 19 q^{37} - 10 q^{39} - 5 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 + 2 * q^11 + 5 * q^13 - 9 * q^17 + 8 * q^21 - 6 * q^23 + 5 * q^25 - 20 * q^27 - 7 * q^29 - 2 * q^31 + 8 * q^33 - 10 * q^35 + 19 * q^37 - 10 * q^39 - 5 * q^41 + 14 * q^43 - 5 * q^45 - 2 * q^49 + 14 * q^51 - q^53 - 10 * q^55 - 8 * q^59 + 5 * q^61 - 14 * q^63 - 10 * q^65 - 10 * q^67 - 4 * q^69 + 12 * q^71 - 9 * q^73 + 20 * q^75 + 12 * q^77 + 10 * q^79 + 22 * q^81 - 8 * q^83 + 20 * q^85 - 18 * q^87 + 11 * q^89 + 12 * q^93 + 15 * q^97 - 14 * q^99

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