LMFDB - Embedded newform 5824.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5824,2,Mod(1,5824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5824, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5824.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5824 = 2^{6} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5824.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:	$46.5048741372$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5824.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -1.58579 q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.24264 q^{11} +1.00000 q^{13} +2.24264 q^{15} +1.41421 q^{17} +7.24264 q^{19} -1.41421 q^{21} -5.82843 q^{23} -2.48528 q^{25} +5.65685 q^{27} -0.171573 q^{29} +3.24264 q^{31} +6.00000 q^{33} -1.58579 q^{35} -2.24264 q^{37} -1.41421 q^{39} +8.82843 q^{41} +5.00000 q^{43} +1.58579 q^{45} +1.58579 q^{47} +1.00000 q^{49} -2.00000 q^{51} +0.171573 q^{53} +6.72792 q^{55} -10.2426 q^{57} -0.343146 q^{59} -6.00000 q^{61} -1.00000 q^{63} -1.58579 q^{65} +14.4853 q^{67} +8.24264 q^{69} -13.0711 q^{71} -9.24264 q^{73} +3.51472 q^{75} -4.24264 q^{77} +15.4853 q^{79} -5.00000 q^{81} -13.2426 q^{83} -2.24264 q^{85} +0.242641 q^{87} +1.58579 q^{89} +1.00000 q^{91} -4.58579 q^{93} -11.4853 q^{95} +11.7279 q^{97} +4.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{13} - 4 q^{15} + 6 q^{19} - 6 q^{23} + 12 q^{25} - 6 q^{29} - 2 q^{31} + 12 q^{33} - 6 q^{35} + 4 q^{37} + 12 q^{41} + 10 q^{43} + 6 q^{45} + 6 q^{47} + 2 q^{49}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 2 * q^7 - 2 * q^9 + 2 * q^13 - 4 * q^15 + 6 * q^19 - 6 * q^23 + 12 * q^25 - 6 * q^29 - 2 * q^31 + 12 * q^33 - 6 * q^35 + 4 * q^37 + 12 * q^41 + 10 * q^43 + 6 * q^45 + 6 * q^47 + 2 * q^49 - 4 * q^51 + 6 * q^53 - 12 * q^55 - 12 * q^57 - 12 * q^59 - 12 * q^61 - 2 * q^63 - 6 * q^65 + 12 * q^67 + 8 * q^69 - 12 * q^71 - 10 * q^73 + 24 * q^75 + 14 * q^79 - 10 * q^81 - 18 * q^83 + 4 * q^85 - 8 * q^87 + 6 * q^89 + 2 * q^91 - 12 * q^93 - 6 * q^95 - 2 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.58579	−0.709185	−0.354593	−	0.935021i	$-0.615380\pi$
$5$	−1.58579	−0.709185	−0.354593	+	0.935021i	$0.615380\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.24264	0.579047
$16$	0	0
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	7.24264	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$19$	7.24264	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	−5.82843	−1.21531	−0.607656	−	0.794201i	$-0.707890\pi$
$23$	−5.82843	−1.21531	−0.607656	+	0.794201i	$0.707890\pi$
$24$	0	0
$25$	−2.48528	−0.497056
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	3.24264	0.582395	0.291198	−	0.956663i	$-0.405946\pi$
$31$	3.24264	0.582395	0.291198	+	0.956663i	$0.405946\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−1.58579	−0.268047
$36$	0	0
$37$	−2.24264	−0.368688	−0.184344	−	0.982862i	$-0.559016\pi$
$37$	−2.24264	−0.368688	−0.184344	+	0.982862i	$0.559016\pi$
$38$	0	0
$39$	−1.41421	−0.226455
$40$	0	0
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	1.58579	0.236395
$46$	0	0
$47$	1.58579	0.231311	0.115655	−	0.993289i	$-0.463103\pi$
$47$	1.58579	0.231311	0.115655	+	0.993289i	$0.463103\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	0.171573	0.0235673	0.0117837	−	0.999931i	$-0.496249\pi$
$53$	0.171573	0.0235673	0.0117837	+	0.999931i	$0.496249\pi$
$54$	0	0
$55$	6.72792	0.907193
$56$	0	0
$57$	−10.2426	−1.35667
$58$	0	0
$59$	−0.343146	−0.0446738	−0.0223369	−	0.999751i	$-0.507111\pi$
$59$	−0.343146	−0.0446738	−0.0223369	+	0.999751i	$0.507111\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.58579	−0.196693
$66$	0	0
$67$	14.4853	1.76966	0.884829	−	0.465915i	$-0.154275\pi$
$67$	14.4853	1.76966	0.884829	+	0.465915i	$0.154275\pi$
$68$	0	0
$69$	8.24264	0.992297
$70$	0	0
$71$	−13.0711	−1.55125	−0.775625	−	0.631194i	$-0.782565\pi$
$71$	−13.0711	−1.55125	−0.775625	+	0.631194i	$0.782565\pi$
$72$	0	0
$73$	−9.24264	−1.08177	−0.540885	−	0.841097i	$-0.681910\pi$
$73$	−9.24264	−1.08177	−0.540885	+	0.841097i	$0.681910\pi$
$74$	0	0
$75$	3.51472	0.405845
$76$	0	0
$77$	−4.24264	−0.483494
$78$	0	0
$79$	15.4853	1.74223	0.871115	−	0.491079i	$-0.163397\pi$
$79$	15.4853	1.74223	0.871115	+	0.491079i	$0.163397\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−13.2426	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$83$	−13.2426	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$84$	0	0
$85$	−2.24264	−0.243249
$86$	0	0
$87$	0.242641	0.0260138
$88$	0	0
$89$	1.58579	0.168093	0.0840465	−	0.996462i	$-0.473216\pi$
$89$	1.58579	0.168093	0.0840465	+	0.996462i	$0.473216\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−4.58579	−0.475524
$94$	0	0
$95$	−11.4853	−1.17837
$96$	0	0
$97$	11.7279	1.19079	0.595395	−	0.803433i	$-0.296996\pi$
$97$	11.7279	1.19079	0.595395	+	0.803433i	$0.296996\pi$
$98$	0	0
$99$	4.24264	0.426401
$100$	0	0
$101$	−10.2426	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$101$	−10.2426	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	2.24264	0.218859
$106$	0	0
$107$	20.1421	1.94721	0.973607	−	0.228232i	$-0.0732943\pi$
$107$	20.1421	1.94721	0.973607	+	0.228232i	$0.0732943\pi$
$108$	0	0
$109$	−16.7279	−1.60224	−0.801122	−	0.598501i	$-0.795763\pi$
$109$	−16.7279	−1.60224	−0.801122	+	0.598501i	$0.795763\pi$
$110$	0	0
$111$	3.17157	0.301032
$112$	0	0
$113$	2.31371	0.217655	0.108828	−	0.994061i	$-0.465290\pi$
$113$	2.31371	0.217655	0.108828	+	0.994061i	$0.465290\pi$
$114$	0	0
$115$	9.24264	0.861881
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	1.41421	0.129641
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	−12.4853	−1.12576
$124$	0	0
$125$	11.8701	1.06169
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−7.07107	−0.622573
$130$	0	0
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	7.24264	0.628017
$134$	0	0
$135$	−8.97056	−0.772063
$136$	0	0
$137$	−4.58579	−0.391790	−0.195895	−	0.980625i	$-0.562761\pi$
$137$	−4.58579	−0.391790	−0.195895	+	0.980625i	$0.562761\pi$
$138$	0	0
$139$	−6.24264	−0.529494	−0.264747	−	0.964318i	$-0.585288\pi$
$139$	−6.24264	−0.529494	−0.264747	+	0.964318i	$0.585288\pi$
$140$	0	0
$141$	−2.24264	−0.188864
$142$	0	0
$143$	−4.24264	−0.354787
$144$	0	0
$145$	0.272078	0.0225948
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	−16.2426	−1.33065	−0.665324	−	0.746554i	$-0.731707\pi$
$149$	−16.2426	−1.33065	−0.665324	+	0.746554i	$0.731707\pi$
$150$	0	0
$151$	−9.75736	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$151$	−9.75736	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$152$	0	0
$153$	−1.41421	−0.114332
$154$	0	0
$155$	−5.14214	−0.413026
$156$	0	0
$157$	3.75736	0.299870	0.149935	−	0.988696i	$-0.452094\pi$
$157$	3.75736	0.299870	0.149935	+	0.988696i	$0.452094\pi$
$158$	0	0
$159$	−0.242641	−0.0192427
$160$	0	0
$161$	−5.82843	−0.459344
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	−9.51472	−0.740720
$166$	0	0
$167$	15.3848	1.19051	0.595255	−	0.803537i	$-0.297051\pi$
$167$	15.3848	1.19051	0.595255	+	0.803537i	$0.297051\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.24264	−0.553859
$172$	0	0
$173$	−24.7279	−1.88003	−0.940015	−	0.341134i	$-0.889189\pi$
$173$	−24.7279	−1.88003	−0.940015	+	0.341134i	$0.889189\pi$
$174$	0	0
$175$	−2.48528	−0.187870
$176$	0	0
$177$	0.485281	0.0364760
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	18.7279	1.39204	0.696018	−	0.718025i	$-0.254953\pi$
$181$	18.7279	1.39204	0.696018	+	0.718025i	$0.254953\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	3.55635	0.261468
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	5.65685	0.411476
$190$	0	0
$191$	15.1716	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$191$	15.1716	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$192$	0	0
$193$	−14.4853	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$193$	−14.4853	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$194$	0	0
$195$	2.24264	0.160599
$196$	0	0
$197$	−12.3431	−0.879413	−0.439706	−	0.898142i	$-0.644917\pi$
$197$	−12.3431	−0.879413	−0.439706	+	0.898142i	$0.644917\pi$
$198$	0	0
$199$	−3.75736	−0.266352	−0.133176	−	0.991092i	$-0.542518\pi$
$199$	−3.75736	−0.266352	−0.133176	+	0.991092i	$0.542518\pi$
$200$	0	0
$201$	−20.4853	−1.44492
$202$	0	0
$203$	−0.171573	−0.0120421
$204$	0	0
$205$	−14.0000	−0.977802
$206$	0	0
$207$	5.82843	0.405104
$208$	0	0
$209$	−30.7279	−2.12549
$210$	0	0
$211$	15.9706	1.09946	0.549729	−	0.835343i	$-0.314731\pi$
$211$	15.9706	1.09946	0.549729	+	0.835343i	$0.314731\pi$
$212$	0	0
$213$	18.4853	1.26659
$214$	0	0
$215$	−7.92893	−0.540749
$216$	0	0
$217$	3.24264	0.220125
$218$	0	0
$219$	13.0711	0.883261
$220$	0	0
$221$	1.41421	0.0951303
$222$	0	0
$223$	0.757359	0.0507165	0.0253583	−	0.999678i	$-0.491927\pi$
$223$	0.757359	0.0507165	0.0253583	+	0.999678i	$0.491927\pi$
$224$	0	0
$225$	2.48528	0.165685
$226$	0	0
$227$	−26.8284	−1.78067	−0.890333	−	0.455311i	$-0.849528\pi$
$227$	−26.8284	−1.78067	−0.890333	+	0.455311i	$0.849528\pi$
$228$	0	0
$229$	−29.4558	−1.94650	−0.973248	−	0.229755i	$-0.926207\pi$
$229$	−29.4558	−1.94650	−0.973248	+	0.229755i	$0.926207\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−14.6569	−0.960202	−0.480101	−	0.877213i	$-0.659400\pi$
$233$	−14.6569	−0.960202	−0.480101	+	0.877213i	$0.659400\pi$
$234$	0	0
$235$	−2.51472	−0.164042
$236$	0	0
$237$	−21.8995	−1.42253
$238$	0	0
$239$	3.51472	0.227348	0.113674	−	0.993518i	$-0.463738\pi$
$239$	3.51472	0.227348	0.113674	+	0.993518i	$0.463738\pi$
$240$	0	0
$241$	−20.2132	−1.30205	−0.651023	−	0.759058i	$-0.725660\pi$
$241$	−20.2132	−1.30205	−0.651023	+	0.759058i	$0.725660\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−1.58579	−0.101312
$246$	0	0
$247$	7.24264	0.460838
$248$	0	0
$249$	18.7279	1.18683
$250$	0	0
$251$	−16.5858	−1.04689	−0.523443	−	0.852061i	$-0.675353\pi$
$251$	−16.5858	−1.04689	−0.523443	+	0.852061i	$0.675353\pi$
$252$	0	0
$253$	24.7279	1.55463
$254$	0	0
$255$	3.17157	0.198612
$256$	0	0
$257$	−19.4142	−1.21103	−0.605513	−	0.795836i	$-0.707032\pi$
$257$	−19.4142	−1.21103	−0.605513	+	0.795836i	$0.707032\pi$
$258$	0	0
$259$	−2.24264	−0.139351
$260$	0	0
$261$	0.171573	0.0106201
$262$	0	0
$263$	1.97056	0.121510	0.0607551	−	0.998153i	$-0.480649\pi$
$263$	1.97056	0.121510	0.0607551	+	0.998153i	$0.480649\pi$
$264$	0	0
$265$	−0.272078	−0.0167136
$266$	0	0
$267$	−2.24264	−0.137247
$268$	0	0
$269$	−9.17157	−0.559201	−0.279600	−	0.960116i	$-0.590202\pi$
$269$	−9.17157	−0.559201	−0.279600	+	0.960116i	$0.590202\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−1.41421	−0.0855921
$274$	0	0
$275$	10.5442	0.635837
$276$	0	0
$277$	7.48528	0.449747	0.224873	−	0.974388i	$-0.427803\pi$
$277$	7.48528	0.449747	0.224873	+	0.974388i	$0.427803\pi$
$278$	0	0
$279$	−3.24264	−0.194132
$280$	0	0
$281$	15.5563	0.928014	0.464007	−	0.885832i	$-0.346411\pi$
$281$	15.5563	0.928014	0.464007	+	0.885832i	$0.346411\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	16.2426	0.962131
$286$	0	0
$287$	8.82843	0.521126
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−16.5858	−0.972276
$292$	0	0
$293$	−15.3848	−0.898788	−0.449394	−	0.893334i	$-0.648360\pi$
$293$	−15.3848	−0.898788	−0.449394	+	0.893334i	$0.648360\pi$
$294$	0	0
$295$	0.544156	0.0316820
$296$	0	0
$297$	−24.0000	−1.39262
$298$	0	0
$299$	−5.82843	−0.337067
$300$	0	0
$301$	5.00000	0.288195
$302$	0	0
$303$	14.4853	0.832158
$304$	0	0
$305$	9.51472	0.544811
$306$	0	0
$307$	−13.2426	−0.755797	−0.377899	−	0.925847i	$-0.623353\pi$
$307$	−13.2426	−0.755797	−0.377899	+	0.925847i	$0.623353\pi$
$308$	0	0
$309$	11.3137	0.643614
$310$	0	0
$311$	−7.41421	−0.420421	−0.210211	−	0.977656i	$-0.567415\pi$
$311$	−7.41421	−0.420421	−0.210211	+	0.977656i	$0.567415\pi$
$312$	0	0
$313$	23.2132	1.31209	0.656044	−	0.754723i	$-0.272229\pi$
$313$	23.2132	1.31209	0.656044	+	0.754723i	$0.272229\pi$
$314$	0	0
$315$	1.58579	0.0893489
$316$	0	0
$317$	11.3137	0.635441	0.317721	−	0.948184i	$-0.397083\pi$
$317$	11.3137	0.635441	0.317721	+	0.948184i	$0.397083\pi$
$318$	0	0
$319$	0.727922	0.0407558
$320$	0	0
$321$	−28.4853	−1.58989
$322$	0	0
$323$	10.2426	0.569916
$324$	0	0
$325$	−2.48528	−0.137859
$326$	0	0
$327$	23.6569	1.30823
$328$	0	0
$329$	1.58579	0.0874272
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	0	0
$333$	2.24264	0.122896
$334$	0	0
$335$	−22.9706	−1.25502
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	0	0
$339$	−3.27208	−0.177715
$340$	0	0
$341$	−13.7574	−0.745003
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−13.0711	−0.703723
$346$	0	0
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	25.7279	1.37718	0.688592	−	0.725149i	$-0.258229\pi$
$349$	25.7279	1.37718	0.688592	+	0.725149i	$0.258229\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	−8.48528	−0.451626	−0.225813	−	0.974171i	$-0.572504\pi$
$353$	−8.48528	−0.451626	−0.225813	+	0.974171i	$0.572504\pi$
$354$	0	0
$355$	20.7279	1.10012
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−27.8995	−1.47248	−0.736240	−	0.676721i	$-0.763400\pi$
$359$	−27.8995	−1.47248	−0.736240	+	0.676721i	$0.763400\pi$
$360$	0	0
$361$	33.4558	1.76083
$362$	0	0
$363$	−9.89949	−0.519589
$364$	0	0
$365$	14.6569	0.767175
$366$	0	0
$367$	10.2426	0.534661	0.267331	−	0.963605i	$-0.413858\pi$
$367$	10.2426	0.534661	0.267331	+	0.963605i	$0.413858\pi$
$368$	0	0
$369$	−8.82843	−0.459590
$370$	0	0
$371$	0.171573	0.00890762
$372$	0	0
$373$	−8.48528	−0.439351	−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528	−0.439351	−0.219676	+	0.975573i	$0.570500\pi$
$374$	0	0
$375$	−16.7868	−0.866866
$376$	0	0
$377$	−0.171573	−0.00883645
$378$	0	0
$379$	−23.7574	−1.22033	−0.610167	−	0.792273i	$-0.708898\pi$
$379$	−23.7574	−1.22033	−0.610167	+	0.792273i	$0.708898\pi$
$380$	0	0
$381$	−2.82843	−0.144905
$382$	0	0
$383$	−20.4853	−1.04675	−0.523374	−	0.852103i	$-0.675327\pi$
$383$	−20.4853	−1.04675	−0.523374	+	0.852103i	$0.675327\pi$
$384$	0	0
$385$	6.72792	0.342887
$386$	0	0
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−29.6569	−1.50366	−0.751831	−	0.659356i	$-0.770829\pi$
$389$	−29.6569	−1.50366	−0.751831	+	0.659356i	$0.770829\pi$
$390$	0	0
$391$	−8.24264	−0.416848
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	−24.5563	−1.23556
$396$	0	0
$397$	18.2132	0.914094	0.457047	−	0.889442i	$-0.348907\pi$
$397$	18.2132	0.914094	0.457047	+	0.889442i	$0.348907\pi$
$398$	0	0
$399$	−10.2426	−0.512773
$400$	0	0
$401$	6.34315	0.316762	0.158381	−	0.987378i	$-0.449373\pi$
$401$	6.34315	0.316762	0.158381	+	0.987378i	$0.449373\pi$
$402$	0	0
$403$	3.24264	0.161527
$404$	0	0
$405$	7.92893	0.393992
$406$	0	0
$407$	9.51472	0.471627
$408$	0	0
$409$	3.24264	0.160338	0.0801691	−	0.996781i	$-0.474454\pi$
$409$	3.24264	0.160338	0.0801691	+	0.996781i	$0.474454\pi$
$410$	0	0
$411$	6.48528	0.319895
$412$	0	0
$413$	−0.343146	−0.0168851
$414$	0	0
$415$	21.0000	1.03085
$416$	0	0
$417$	8.82843	0.432330
$418$	0	0
$419$	20.8701	1.01957	0.509785	−	0.860302i	$-0.329725\pi$
$419$	20.8701	1.01957	0.509785	+	0.860302i	$0.329725\pi$
$420$	0	0
$421$	−6.72792	−0.327899	−0.163949	−	0.986469i	$-0.552423\pi$
$421$	−6.72792	−0.327899	−0.163949	+	0.986469i	$0.552423\pi$
$422$	0	0
$423$	−1.58579	−0.0771036
$424$	0	0
$425$	−3.51472	−0.170489
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−12.3431	−0.594548	−0.297274	−	0.954792i	$-0.596078\pi$
$431$	−12.3431	−0.594548	−0.297274	+	0.954792i	$0.596078\pi$
$432$	0	0
$433$	−24.9706	−1.20001	−0.600004	−	0.799997i	$-0.704834\pi$
$433$	−24.9706	−1.20001	−0.600004	+	0.799997i	$0.704834\pi$
$434$	0	0
$435$	−0.384776	−0.0184486
$436$	0	0
$437$	−42.2132	−2.01933
$438$	0	0
$439$	−34.4853	−1.64589	−0.822946	−	0.568119i	$-0.807671\pi$
$439$	−34.4853	−1.64589	−0.822946	+	0.568119i	$0.807671\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	3.68629	0.175141	0.0875705	−	0.996158i	$-0.472090\pi$
$443$	3.68629	0.175141	0.0875705	+	0.996158i	$0.472090\pi$
$444$	0	0
$445$	−2.51472	−0.119209
$446$	0	0
$447$	22.9706	1.08647
$448$	0	0
$449$	21.1716	0.999148	0.499574	−	0.866271i	$-0.333490\pi$
$449$	21.1716	0.999148	0.499574	+	0.866271i	$0.333490\pi$
$450$	0	0
$451$	−37.4558	−1.76373
$452$	0	0
$453$	13.7990	0.648333
$454$	0	0
$455$	−1.58579	−0.0743428
$456$	0	0
$457$	−7.21320	−0.337419	−0.168710	−	0.985666i	$-0.553960\pi$
$457$	−7.21320	−0.337419	−0.168710	+	0.985666i	$0.553960\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−8.82843	−0.411181	−0.205590	−	0.978638i	$-0.565911\pi$
$461$	−8.82843	−0.411181	−0.205590	+	0.978638i	$0.565911\pi$
$462$	0	0
$463$	−4.24264	−0.197172	−0.0985861	−	0.995129i	$-0.531432\pi$
$463$	−4.24264	−0.197172	−0.0985861	+	0.995129i	$0.531432\pi$
$464$	0	0
$465$	7.27208	0.337235
$466$	0	0
$467$	3.89949	0.180447	0.0902236	−	0.995922i	$-0.471242\pi$
$467$	3.89949	0.180447	0.0902236	+	0.995922i	$0.471242\pi$
$468$	0	0
$469$	14.4853	0.668868
$470$	0	0
$471$	−5.31371	−0.244843
$472$	0	0
$473$	−21.2132	−0.975384
$474$	0	0
$475$	−18.0000	−0.825897
$476$	0	0
$477$	−0.171573	−0.00785578
$478$	0	0
$479$	−6.21320	−0.283889	−0.141944	−	0.989875i	$-0.545335\pi$
$479$	−6.21320	−0.283889	−0.141944	+	0.989875i	$0.545335\pi$
$480$	0	0
$481$	−2.24264	−0.102256
$482$	0	0
$483$	8.24264	0.375053
$484$	0	0
$485$	−18.5980	−0.844491
$486$	0	0
$487$	−11.4558	−0.519114	−0.259557	−	0.965728i	$-0.583577\pi$
$487$	−11.4558	−0.519114	−0.259557	+	0.965728i	$0.583577\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−16.6274	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$491$	−16.6274	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$492$	0	0
$493$	−0.242641	−0.0109280
$494$	0	0
$495$	−6.72792	−0.302398
$496$	0	0
$497$	−13.0711	−0.586318
$498$	0	0
$499$	13.2721	0.594140	0.297070	−	0.954856i	$-0.403991\pi$
$499$	13.2721	0.594140	0.297070	+	0.954856i	$0.403991\pi$
$500$	0	0
$501$	−21.7574	−0.972047
$502$	0	0
$503$	−28.6274	−1.27643	−0.638217	−	0.769857i	$-0.720328\pi$
$503$	−28.6274	−1.27643	−0.638217	+	0.769857i	$0.720328\pi$
$504$	0	0
$505$	16.2426	0.722788
$506$	0	0
$507$	−1.41421	−0.0628074
$508$	0	0
$509$	5.10051	0.226076	0.113038	−	0.993591i	$-0.463942\pi$
$509$	5.10051	0.226076	0.113038	+	0.993591i	$0.463942\pi$
$510$	0	0
$511$	−9.24264	−0.408870
$512$	0	0
$513$	40.9706	1.80889
$514$	0	0
$515$	12.6863	0.559025
$516$	0	0
$517$	−6.72792	−0.295894
$518$	0	0
$519$	34.9706	1.53504
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	−30.9706	−1.35425	−0.677124	−	0.735869i	$-0.736774\pi$
$523$	−30.9706	−1.35425	−0.677124	+	0.735869i	$0.736774\pi$
$524$	0	0
$525$	3.51472	0.153395
$526$	0	0
$527$	4.58579	0.199760
$528$	0	0
$529$	10.9706	0.476981
$530$	0	0
$531$	0.343146	0.0148913
$532$	0	0
$533$	8.82843	0.382402
$534$	0	0
$535$	−31.9411	−1.38094
$536$	0	0
$537$	−12.7279	−0.549250
$538$	0	0
$539$	−4.24264	−0.182743
$540$	0	0
$541$	7.21320	0.310120	0.155060	−	0.987905i	$-0.450443\pi$
$541$	7.21320	0.310120	0.155060	+	0.987905i	$0.450443\pi$
$542$	0	0
$543$	−26.4853	−1.13659
$544$	0	0
$545$	26.5269	1.13629
$546$	0	0
$547$	−35.4853	−1.51724	−0.758621	−	0.651533i	$-0.774126\pi$
$547$	−35.4853	−1.51724	−0.758621	+	0.651533i	$0.774126\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−1.24264	−0.0529383
$552$	0	0
$553$	15.4853	0.658501
$554$	0	0
$555$	−5.02944	−0.213488
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	8.48528	0.358249
$562$	0	0
$563$	6.34315	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$563$	6.34315	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$564$	0	0
$565$	−3.66905	−0.154358
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	5.14214	0.215570	0.107785	−	0.994174i	$-0.465624\pi$
$569$	5.14214	0.215570	0.107785	+	0.994174i	$0.465624\pi$
$570$	0	0
$571$	−42.4558	−1.77672	−0.888361	−	0.459146i	$-0.848156\pi$
$571$	−42.4558	−1.77672	−0.888361	+	0.459146i	$0.848156\pi$
$572$	0	0
$573$	−21.4558	−0.896331
$574$	0	0
$575$	14.4853	0.604078
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	0	0
$579$	20.4853	0.851339
$580$	0	0
$581$	−13.2426	−0.549397
$582$	0	0
$583$	−0.727922	−0.0301475
$584$	0	0
$585$	1.58579	0.0655642
$586$	0	0
$587$	6.55635	0.270609	0.135305	−	0.990804i	$-0.456799\pi$
$587$	6.55635	0.270609	0.135305	+	0.990804i	$0.456799\pi$
$588$	0	0
$589$	23.4853	0.967694
$590$	0	0
$591$	17.4558	0.718037
$592$	0	0
$593$	−0.556349	−0.0228465	−0.0114233	−	0.999935i	$-0.503636\pi$
$593$	−0.556349	−0.0228465	−0.0114233	+	0.999935i	$0.503636\pi$
$594$	0	0
$595$	−2.24264	−0.0919393
$596$	0	0
$597$	5.31371	0.217476
$598$	0	0
$599$	−28.7990	−1.17669	−0.588347	−	0.808608i	$-0.700221\pi$
$599$	−28.7990	−1.17669	−0.588347	+	0.808608i	$0.700221\pi$
$600$	0	0
$601$	−38.9706	−1.58964	−0.794821	−	0.606844i	$-0.792435\pi$
$601$	−38.9706	−1.58964	−0.794821	+	0.606844i	$0.792435\pi$
$602$	0	0
$603$	−14.4853	−0.589886
$604$	0	0
$605$	−11.1005	−0.451300
$606$	0	0
$607$	−26.7279	−1.08485	−0.542426	−	0.840103i	$-0.682494\pi$
$607$	−26.7279	−1.08485	−0.542426	+	0.840103i	$0.682494\pi$
$608$	0	0
$609$	0.242641	0.00983230
$610$	0	0
$611$	1.58579	0.0641541
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	19.7990	0.798372
$616$	0	0
$617$	12.3431	0.496916	0.248458	−	0.968643i	$-0.420076\pi$
$617$	12.3431	0.496916	0.248458	+	0.968643i	$0.420076\pi$
$618$	0	0
$619$	−0.970563	−0.0390102	−0.0195051	−	0.999810i	$-0.506209\pi$
$619$	−0.970563	−0.0390102	−0.0195051	+	0.999810i	$0.506209\pi$
$620$	0	0
$621$	−32.9706	−1.32306
$622$	0	0
$623$	1.58579	0.0635332
$624$	0	0
$625$	−6.39697	−0.255879
$626$	0	0
$627$	43.4558	1.73546
$628$	0	0
$629$	−3.17157	−0.126459
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	0	0
$633$	−22.5858	−0.897704
$634$	0	0
$635$	−3.17157	−0.125860
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	13.0711	0.517083
$640$	0	0
$641$	15.3431	0.606018	0.303009	−	0.952988i	$-0.402009\pi$
$641$	15.3431	0.606018	0.303009	+	0.952988i	$0.402009\pi$
$642$	0	0
$643$	12.4853	0.492371	0.246186	−	0.969223i	$-0.420823\pi$
$643$	12.4853	0.492371	0.246186	+	0.969223i	$0.420823\pi$
$644$	0	0
$645$	11.2132	0.441519
$646$	0	0
$647$	14.5269	0.571112	0.285556	−	0.958362i	$-0.407822\pi$
$647$	14.5269	0.571112	0.285556	+	0.958362i	$0.407822\pi$
$648$	0	0
$649$	1.45584	0.0571469
$650$	0	0
$651$	−4.58579	−0.179731
$652$	0	0
$653$	−17.3137	−0.677538	−0.338769	−	0.940870i	$-0.610010\pi$
$653$	−17.3137	−0.677538	−0.338769	+	0.940870i	$0.610010\pi$
$654$	0	0
$655$	−4.48528	−0.175254
$656$	0	0
$657$	9.24264	0.360590
$658$	0	0
$659$	15.3431	0.597684	0.298842	−	0.954303i	$-0.403400\pi$
$659$	15.3431	0.597684	0.298842	+	0.954303i	$0.403400\pi$
$660$	0	0
$661$	10.7574	0.418413	0.209206	−	0.977872i	$-0.432912\pi$
$661$	10.7574	0.418413	0.209206	+	0.977872i	$0.432912\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−11.4853	−0.445380
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−1.07107	−0.0414099
$670$	0	0
$671$	25.4558	0.982712
$672$	0	0
$673$	−26.9411	−1.03850	−0.519252	−	0.854621i	$-0.673789\pi$
$673$	−26.9411	−1.03850	−0.519252	+	0.854621i	$0.673789\pi$
$674$	0	0
$675$	−14.0589	−0.541126
$676$	0	0
$677$	41.3553	1.58941	0.794707	−	0.606993i	$-0.207624\pi$
$677$	41.3553	1.58941	0.794707	+	0.606993i	$0.207624\pi$
$678$	0	0
$679$	11.7279	0.450076
$680$	0	0
$681$	37.9411	1.45391
$682$	0	0
$683$	21.1716	0.810108	0.405054	−	0.914293i	$-0.367253\pi$
$683$	21.1716	0.810108	0.405054	+	0.914293i	$0.367253\pi$
$684$	0	0
$685$	7.27208	0.277852
$686$	0	0
$687$	41.6569	1.58931
$688$	0	0
$689$	0.171573	0.00653641
$690$	0	0
$691$	−28.6985	−1.09174	−0.545871	−	0.837869i	$-0.683801\pi$
$691$	−28.6985	−1.09174	−0.545871	+	0.837869i	$0.683801\pi$
$692$	0	0
$693$	4.24264	0.161165
$694$	0	0
$695$	9.89949	0.375509
$696$	0	0
$697$	12.4853	0.472914
$698$	0	0
$699$	20.7279	0.784002
$700$	0	0
$701$	10.7990	0.407872	0.203936	−	0.978984i	$-0.434627\pi$
$701$	10.7990	0.407872	0.203936	+	0.978984i	$0.434627\pi$
$702$	0	0
$703$	−16.2426	−0.612603
$704$	0	0
$705$	3.55635	0.133940
$706$	0	0
$707$	−10.2426	−0.385214
$708$	0	0
$709$	−0.727922	−0.0273377	−0.0136688	−	0.999907i	$-0.504351\pi$
$709$	−0.727922	−0.0273377	−0.0136688	+	0.999907i	$0.504351\pi$
$710$	0	0
$711$	−15.4853	−0.580743
$712$	0	0
$713$	−18.8995	−0.707792
$714$	0	0
$715$	6.72792	0.251610
$716$	0	0
$717$	−4.97056	−0.185629
$718$	0	0
$719$	−1.75736	−0.0655384	−0.0327692	−	0.999463i	$-0.510433\pi$
$719$	−1.75736	−0.0655384	−0.0327692	+	0.999463i	$0.510433\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	28.5858	1.06312
$724$	0	0
$725$	0.426407	0.0158364
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	7.07107	0.261533
$732$	0	0
$733$	16.6985	0.616773	0.308386	−	0.951261i	$-0.400211\pi$
$733$	16.6985	0.616773	0.308386	+	0.951261i	$0.400211\pi$
$734$	0	0
$735$	2.24264	0.0827210
$736$	0	0
$737$	−61.4558	−2.26376
$738$	0	0
$739$	17.6985	0.651049	0.325525	−	0.945534i	$-0.394459\pi$
$739$	17.6985	0.651049	0.325525	+	0.945534i	$0.394459\pi$
$740$	0	0
$741$	−10.2426	−0.376273
$742$	0	0
$743$	−29.6569	−1.08800	−0.544002	−	0.839084i	$-0.683092\pi$
$743$	−29.6569	−1.08800	−0.544002	+	0.839084i	$0.683092\pi$
$744$	0	0
$745$	25.7574	0.943677
$746$	0	0
$747$	13.2426	0.484523
$748$	0	0
$749$	20.1421	0.735978
$750$	0	0
$751$	15.4853	0.565066	0.282533	−	0.959258i	$-0.408825\pi$
$751$	15.4853	0.565066	0.282533	+	0.959258i	$0.408825\pi$
$752$	0	0
$753$	23.4558	0.854778
$754$	0	0
$755$	15.4731	0.563123
$756$	0	0
$757$	−21.4853	−0.780896	−0.390448	−	0.920625i	$-0.627680\pi$
$757$	−21.4853	−0.780896	−0.390448	+	0.920625i	$0.627680\pi$
$758$	0	0
$759$	−34.9706	−1.26935
$760$	0	0
$761$	34.7574	1.25995	0.629977	−	0.776614i	$-0.283064\pi$
$761$	34.7574	1.25995	0.629977	+	0.776614i	$0.283064\pi$
$762$	0	0
$763$	−16.7279	−0.605591
$764$	0	0
$765$	2.24264	0.0810828
$766$	0	0
$767$	−0.343146	−0.0123903
$768$	0	0
$769$	52.2132	1.88286	0.941428	−	0.337214i	$-0.109484\pi$
$769$	52.2132	1.88286	0.941428	+	0.337214i	$0.109484\pi$
$770$	0	0
$771$	27.4558	0.988798
$772$	0	0
$773$	32.8284	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$773$	32.8284	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$774$	0	0
$775$	−8.05887	−0.289483
$776$	0	0
$777$	3.17157	0.113780
$778$	0	0
$779$	63.9411	2.29093
$780$	0	0
$781$	55.4558	1.98437
$782$	0	0
$783$	−0.970563	−0.0346851
$784$	0	0
$785$	−5.95837	−0.212663
$786$	0	0
$787$	24.7574	0.882505	0.441252	−	0.897383i	$-0.354534\pi$
$787$	24.7574	0.882505	0.441252	+	0.897383i	$0.354534\pi$
$788$	0	0
$789$	−2.78680	−0.0992126
$790$	0	0
$791$	2.31371	0.0822660
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	0.384776	0.0136466
$796$	0	0
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	0	0
$799$	2.24264	0.0793389
$800$	0	0
$801$	−1.58579	−0.0560310
$802$	0	0
$803$	39.2132	1.38380
$804$	0	0
$805$	9.24264	0.325760
$806$	0	0
$807$	12.9706	0.456585
$808$	0	0
$809$	17.4853	0.614750	0.307375	−	0.951589i	$-0.400549\pi$
$809$	17.4853	0.614750	0.307375	+	0.951589i	$0.400549\pi$
$810$	0	0
$811$	21.9411	0.770457	0.385229	−	0.922821i	$-0.374123\pi$
$811$	21.9411	0.770457	0.385229	+	0.922821i	$0.374123\pi$
$812$	0	0
$813$	−28.2843	−0.991973
$814$	0	0
$815$	13.4558	0.471338
$816$	0	0
$817$	36.2132	1.26694
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	20.1421	0.702965	0.351483	−	0.936194i	$-0.385678\pi$
$821$	20.1421	0.702965	0.351483	+	0.936194i	$0.385678\pi$
$822$	0	0
$823$	10.4853	0.365494	0.182747	−	0.983160i	$-0.441501\pi$
$823$	10.4853	0.365494	0.182747	+	0.983160i	$0.441501\pi$
$824$	0	0
$825$	−14.9117	−0.519158
$826$	0	0
$827$	49.4558	1.71975	0.859874	−	0.510506i	$-0.170542\pi$
$827$	49.4558	1.71975	0.859874	+	0.510506i	$0.170542\pi$
$828$	0	0
$829$	−50.7279	−1.76185	−0.880927	−	0.473253i	$-0.843080\pi$
$829$	−50.7279	−1.76185	−0.880927	+	0.473253i	$0.843080\pi$
$830$	0	0
$831$	−10.5858	−0.367217
$832$	0	0
$833$	1.41421	0.0489996
$834$	0	0
$835$	−24.3970	−0.844292
$836$	0	0
$837$	18.3431	0.634032
$838$	0	0
$839$	−33.1716	−1.14521	−0.572605	−	0.819831i	$-0.694067\pi$
$839$	−33.1716	−1.14521	−0.572605	+	0.819831i	$0.694067\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−1.58579	−0.0545527
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	13.0711	0.448070
$852$	0	0
$853$	−39.7279	−1.36026	−0.680129	−	0.733092i	$-0.738076\pi$
$853$	−39.7279	−1.36026	−0.680129	+	0.733092i	$0.738076\pi$
$854$	0	0
$855$	11.4853	0.392788
$856$	0	0
$857$	−54.7696	−1.87089	−0.935446	−	0.353469i	$-0.885002\pi$
$857$	−54.7696	−1.87089	−0.935446	+	0.353469i	$0.885002\pi$
$858$	0	0
$859$	−30.9706	−1.05670	−0.528351	−	0.849026i	$-0.677189\pi$
$859$	−30.9706	−1.05670	−0.528351	+	0.849026i	$0.677189\pi$
$860$	0	0
$861$	−12.4853	−0.425497
$862$	0	0
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	39.2132	1.33329
$866$	0	0
$867$	21.2132	0.720438
$868$	0	0
$869$	−65.6985	−2.22867
$870$	0	0
$871$	14.4853	0.490815
$872$	0	0
$873$	−11.7279	−0.396930
$874$	0	0
$875$	11.8701	0.401281
$876$	0	0
$877$	10.2426	0.345869	0.172935	−	0.984933i	$-0.444675\pi$
$877$	10.2426	0.345869	0.172935	+	0.984933i	$0.444675\pi$
$878$	0	0
$879$	21.7574	0.733858
$880$	0	0
$881$	13.1127	0.441778	0.220889	−	0.975299i	$-0.429104\pi$
$881$	13.1127	0.441778	0.220889	+	0.975299i	$0.429104\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	−0.769553	−0.0258682
$886$	0	0
$887$	19.1127	0.641742	0.320871	−	0.947123i	$-0.396024\pi$
$887$	19.1127	0.641742	0.320871	+	0.947123i	$0.396024\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	21.2132	0.710669
$892$	0	0
$893$	11.4853	0.384340
$894$	0	0
$895$	−14.2721	−0.477063
$896$	0	0
$897$	8.24264	0.275214
$898$	0	0
$899$	−0.556349	−0.0185553
$900$	0	0
$901$	0.242641	0.00808353
$902$	0	0
$903$	−7.07107	−0.235310
$904$	0	0
$905$	−29.6985	−0.987211
$906$	0	0
$907$	−1.97056	−0.0654315	−0.0327157	−	0.999465i	$-0.510416\pi$
$907$	−1.97056	−0.0654315	−0.0327157	+	0.999465i	$0.510416\pi$
$908$	0	0
$909$	10.2426	0.339727
$910$	0	0
$911$	−43.9706	−1.45681	−0.728405	−	0.685147i	$-0.759738\pi$
$911$	−43.9706	−1.45681	−0.728405	+	0.685147i	$0.759738\pi$
$912$	0	0
$913$	56.1838	1.85941
$914$	0	0
$915$	−13.4558	−0.444836
$916$	0	0
$917$	2.82843	0.0934029
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	18.7279	0.617106
$922$	0	0
$923$	−13.0711	−0.430239
$924$	0	0
$925$	5.57359	0.183259
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	36.8995	1.21063	0.605317	−	0.795985i	$-0.293047\pi$
$929$	36.8995	1.21063	0.605317	+	0.795985i	$0.293047\pi$
$930$	0	0
$931$	7.24264	0.237368
$932$	0	0
$933$	10.4853	0.343273
$934$	0	0
$935$	9.51472	0.311165
$936$	0	0
$937$	45.2132	1.47705	0.738525	−	0.674226i	$-0.235522\pi$
$937$	45.2132	1.47705	0.738525	+	0.674226i	$0.235522\pi$
$938$	0	0
$939$	−32.8284	−1.07132
$940$	0	0
$941$	−40.0711	−1.30628	−0.653140	−	0.757237i	$-0.726549\pi$
$941$	−40.0711	−1.30628	−0.653140	+	0.757237i	$0.726549\pi$
$942$	0	0
$943$	−51.4558	−1.67563
$944$	0	0
$945$	−8.97056	−0.291812
$946$	0	0
$947$	−15.1716	−0.493010	−0.246505	−	0.969142i	$-0.579282\pi$
$947$	−15.1716	−0.493010	−0.246505	+	0.969142i	$0.579282\pi$
$948$	0	0
$949$	−9.24264	−0.300029
$950$	0	0
$951$	−16.0000	−0.518836
$952$	0	0
$953$	11.1421	0.360929	0.180465	−	0.983581i	$-0.442240\pi$
$953$	11.1421	0.360929	0.180465	+	0.983581i	$0.442240\pi$
$954$	0	0
$955$	−24.0589	−0.778527
$956$	0	0
$957$	−1.02944	−0.0332770
$958$	0	0
$959$	−4.58579	−0.148083
$960$	0	0
$961$	−20.4853	−0.660816
$962$	0	0
$963$	−20.1421	−0.649071
$964$	0	0
$965$	22.9706	0.739449
$966$	0	0
$967$	17.6985	0.569145	0.284572	−	0.958655i	$-0.408148\pi$
$967$	17.6985	0.569145	0.284572	+	0.958655i	$0.408148\pi$
$968$	0	0
$969$	−14.4853	−0.465334
$970$	0	0
$971$	−43.4558	−1.39456	−0.697282	−	0.716797i	$-0.745608\pi$
$971$	−43.4558	−1.39456	−0.697282	+	0.716797i	$0.745608\pi$
$972$	0	0
$973$	−6.24264	−0.200130
$974$	0	0
$975$	3.51472	0.112561
$976$	0	0
$977$	24.0416	0.769160	0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416	0.769160	0.384580	+	0.923092i	$0.374346\pi$
$978$	0	0
$979$	−6.72792	−0.215025
$980$	0	0
$981$	16.7279	0.534081
$982$	0	0
$983$	−15.0416	−0.479754	−0.239877	−	0.970803i	$-0.577107\pi$
$983$	−15.0416	−0.479754	−0.239877	+	0.970803i	$0.577107\pi$
$984$	0	0
$985$	19.5736	0.623667
$986$	0	0
$987$	−2.24264	−0.0713840
$988$	0	0
$989$	−29.1421	−0.926666
$990$	0	0
$991$	1.02944	0.0327012	0.0163506	−	0.999866i	$-0.494795\pi$
$991$	1.02944	0.0327012	0.0163506	+	0.999866i	$0.494795\pi$
$992$	0	0
$993$	−25.4558	−0.807817
$994$	0	0
$995$	5.95837	0.188893
$996$	0	0
$997$	11.5147	0.364675	0.182337	−	0.983236i	$-0.441634\pi$
$997$	11.5147	0.364675	0.182337	+	0.983236i	$0.441634\pi$
$998$	0	0
$999$	−12.6863	−0.401377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5824.2.a.bl.1.1		2
4.3	odd	2		5824.2.a.bk.1.2		2
8.3	odd	2		1456.2.a.q.1.1		2
8.5	even	2		91.2.a.c.1.1	✓	2
24.5	odd	2		819.2.a.h.1.2		2
40.29	even	2		2275.2.a.j.1.2		2
56.5	odd	6		637.2.e.g.508.2		4
56.13	odd	2		637.2.a.g.1.1		2
56.37	even	6		637.2.e.f.508.2		4
56.45	odd	6		637.2.e.g.79.2		4
56.53	even	6		637.2.e.f.79.2		4
104.5	odd	4		1183.2.c.d.337.4		4
104.21	odd	4		1183.2.c.d.337.2		4
104.77	even	2		1183.2.a.d.1.2		2
168.125	even	2		5733.2.a.s.1.2		2
728.181	odd	2		8281.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.c.1.1	✓	2	8.5	even	2
637.2.a.g.1.1		2	56.13	odd	2
637.2.e.f.79.2		4	56.53	even	6
637.2.e.f.508.2		4	56.37	even	6
637.2.e.g.79.2		4	56.45	odd	6
637.2.e.g.508.2		4	56.5	odd	6
819.2.a.h.1.2		2	24.5	odd	2
1183.2.a.d.1.2		2	104.77	even	2
1183.2.c.d.337.2		4	104.21	odd	4
1183.2.c.d.337.4		4	104.5	odd	4
1456.2.a.q.1.1		2	8.3	odd	2
2275.2.a.j.1.2		2	40.29	even	2
5733.2.a.s.1.2		2	168.125	even	2
5824.2.a.bk.1.2		2	4.3	odd	2
5824.2.a.bl.1.1		2	1.1	even	1	trivial
8281.2.a.v.1.2		2	728.181	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5824.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -1.58579 q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.24264 q^{11} +1.00000 q^{13} +2.24264 q^{15} +1.41421 q^{17} +7.24264 q^{19} -1.41421 q^{21} -5.82843 q^{23} -2.48528 q^{25} +5.65685 q^{27} -0.171573 q^{29} +3.24264 q^{31} +6.00000 q^{33} -1.58579 q^{35} -2.24264 q^{37} -1.41421 q^{39} +8.82843 q^{41} +5.00000 q^{43} +1.58579 q^{45} +1.58579 q^{47} +1.00000 q^{49} -2.00000 q^{51} +0.171573 q^{53} +6.72792 q^{55} -10.2426 q^{57} -0.343146 q^{59} -6.00000 q^{61} -1.00000 q^{63} -1.58579 q^{65} +14.4853 q^{67} +8.24264 q^{69} -13.0711 q^{71} -9.24264 q^{73} +3.51472 q^{75} -4.24264 q^{77} +15.4853 q^{79} -5.00000 q^{81} -13.2426 q^{83} -2.24264 q^{85} +0.242641 q^{87} +1.58579 q^{89} +1.00000 q^{91} -4.58579 q^{93} -11.4853 q^{95} +11.7279 q^{97} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{13} - 4 q^{15} + 6 q^{19} - 6 q^{23} + 12 q^{25} - 6 q^{29} - 2 q^{31} + 12 q^{33} - 6 q^{35} + 4 q^{37} + 12 q^{41} + 10 q^{43} + 6 q^{45} + 6 q^{47} + 2 q^{49}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 2 * q^7 - 2 * q^9 + 2 * q^13 - 4 * q^15 + 6 * q^19 - 6 * q^23 + 12 * q^25 - 6 * q^29 - 2 * q^31 + 12 * q^33 - 6 * q^35 + 4 * q^37 + 12 * q^41 + 10 * q^43 + 6 * q^45 + 6 * q^47 + 2 * q^49 - 4 * q^51 + 6 * q^53 - 12 * q^55 - 12 * q^57 - 12 * q^59 - 12 * q^61 - 2 * q^63 - 6 * q^65 + 12 * q^67 + 8 * q^69 - 12 * q^71 - 10 * q^73 + 24 * q^75 + 14 * q^79 - 10 * q^81 - 18 * q^83 + 4 * q^85 - 8 * q^87 + 6 * q^89 + 2 * q^91 - 12 * q^93 - 6 * q^95 - 2 * q^97

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