LMFDB - Embedded newform 6724.2.a.j.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.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:	$53.6914103191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 $ x^16 - 39x^14 + 594x^12 - 4428x^10 + 16529x^8 - 28236x^6 + 17856x^4 - 4032*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$3.19728$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.19728 q^{3} +2.17635 q^{5} +4.47386 q^{7} +7.22259 q^{9} +O(q^{10})$	$q+3.19728 q^{3} +2.17635 q^{5} +4.47386 q^{7} +7.22259 q^{9} +3.68489 q^{11} -4.66711 q^{13} +6.95840 q^{15} +1.23274 q^{17} -4.80904 q^{19} +14.3042 q^{21} -3.31921 q^{23} -0.263495 q^{25} +13.5008 q^{27} -0.307096 q^{29} +3.02352 q^{31} +11.7816 q^{33} +9.73669 q^{35} -3.01048 q^{37} -14.9220 q^{39} +2.66200 q^{43} +15.7189 q^{45} -6.21902 q^{47} +13.0154 q^{49} +3.94140 q^{51} -1.17121 q^{53} +8.01961 q^{55} -15.3758 q^{57} -6.26510 q^{59} -9.33487 q^{61} +32.3128 q^{63} -10.1573 q^{65} +8.77498 q^{67} -10.6124 q^{69} -5.61644 q^{71} -3.37549 q^{73} -0.842468 q^{75} +16.4857 q^{77} -3.07096 q^{79} +21.4980 q^{81} -5.15408 q^{83} +2.68287 q^{85} -0.981871 q^{87} +3.65976 q^{89} -20.8800 q^{91} +9.66704 q^{93} -10.4662 q^{95} +9.47799 q^{97} +26.6144 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) $ 16 * q + 6 * q^5 + 30 * q^9	$ 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) $ 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Display 100 coefficients 10 coefficients

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
$3$	3.19728	1.84595	0.922975	−	0.384861i	$-0.125751\pi$
$3$	3.19728	1.84595	0.922975	+	0.384861i	$0.125751\pi$
$4$	0	0
$5$	2.17635	0.973294	0.486647	−	0.873599i	$-0.338220\pi$
$5$	2.17635	0.973294	0.486647	+	0.873599i	$0.338220\pi$
$6$	0	0
$7$	4.47386	1.69096	0.845480	−	0.534007i	$-0.179315\pi$
$7$	4.47386	1.69096	0.845480	+	0.534007i	$0.179315\pi$
$8$	0	0
$9$	7.22259	2.40753
$10$	0	0
$11$	3.68489	1.11104	0.555518	−	0.831505i	$-0.312520\pi$
$11$	3.68489	1.11104	0.555518	+	0.831505i	$0.312520\pi$
$12$	0	0
$13$	−4.66711	−1.29442	−0.647212	−	0.762310i	$-0.724065\pi$
$13$	−4.66711	−1.29442	−0.647212	+	0.762310i	$0.724065\pi$
$14$	0	0
$15$	6.95840	1.79665
$16$	0	0
$17$	1.23274	0.298983	0.149491	−	0.988763i	$-0.452236\pi$
$17$	1.23274	0.298983	0.149491	+	0.988763i	$0.452236\pi$
$18$	0	0
$19$	−4.80904	−1.10327	−0.551635	−	0.834086i	$-0.685996\pi$
$19$	−4.80904	−1.10327	−0.551635	+	0.834086i	$0.685996\pi$
$20$	0	0
$21$	14.3042	3.12143
$22$	0	0
$23$	−3.31921	−0.692103	−0.346052	−	0.938215i	$-0.612478\pi$
$23$	−3.31921	−0.692103	−0.346052	+	0.938215i	$0.612478\pi$
$24$	0	0
$25$	−0.263495	−0.0526991
$26$	0	0
$27$	13.5008	2.59823
$28$	0	0
$29$	−0.307096	−0.0570263	−0.0285131	−	0.999593i	$-0.509077\pi$
$29$	−0.307096	−0.0570263	−0.0285131	+	0.999593i	$0.509077\pi$
$30$	0	0
$31$	3.02352	0.543041	0.271520	−	0.962433i	$-0.412474\pi$
$31$	3.02352	0.543041	0.271520	+	0.962433i	$0.412474\pi$
$32$	0	0
$33$	11.7816	2.05092
$34$	0	0
$35$	9.73669	1.64580
$36$	0	0
$37$	−3.01048	−0.494919	−0.247460	−	0.968898i	$-0.579596\pi$
$37$	−3.01048	−0.494919	−0.247460	+	0.968898i	$0.579596\pi$
$38$	0	0
$39$	−14.9220	−2.38944
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	2.66200	0.405951	0.202976	−	0.979184i	$-0.434939\pi$
$43$	2.66200	0.405951	0.202976	+	0.979184i	$0.434939\pi$
$44$	0	0
$45$	15.7189	2.34323
$46$	0	0
$47$	−6.21902	−0.907138	−0.453569	−	0.891221i	$-0.649849\pi$
$47$	−6.21902	−0.907138	−0.453569	+	0.891221i	$0.649849\pi$
$48$	0	0
$49$	13.0154	1.85934
$50$	0	0
$51$	3.94140	0.551907
$52$	0	0
$53$	−1.17121	−0.160878	−0.0804389	−	0.996760i	$-0.525632\pi$
$53$	−1.17121	−0.160878	−0.0804389	+	0.996760i	$0.525632\pi$
$54$	0	0
$55$	8.01961	1.08136
$56$	0	0
$57$	−15.3758	−2.03658
$58$	0	0
$59$	−6.26510	−0.815646	−0.407823	−	0.913061i	$-0.633712\pi$
$59$	−6.26510	−0.815646	−0.407823	+	0.913061i	$0.633712\pi$
$60$	0	0
$61$	−9.33487	−1.19521	−0.597604	−	0.801791i	$-0.703880\pi$
$61$	−9.33487	−1.19521	−0.597604	+	0.801791i	$0.703880\pi$
$62$	0	0
$63$	32.3128	4.07103
$64$	0	0
$65$	−10.1573	−1.25985
$66$	0	0
$67$	8.77498	1.07203	0.536017	−	0.844207i	$-0.319928\pi$
$67$	8.77498	1.07203	0.536017	+	0.844207i	$0.319928\pi$
$68$	0	0
$69$	−10.6124	−1.27759
$70$	0	0
$71$	−5.61644	−0.666549	−0.333274	−	0.942830i	$-0.608154\pi$
$71$	−5.61644	−0.666549	−0.333274	+	0.942830i	$0.608154\pi$
$72$	0	0
$73$	−3.37549	−0.395071	−0.197536	−	0.980296i	$-0.563294\pi$
$73$	−3.37549	−0.395071	−0.197536	+	0.980296i	$0.563294\pi$
$74$	0	0
$75$	−0.842468	−0.0972798
$76$	0	0
$77$	16.4857	1.87872
$78$	0	0
$79$	−3.07096	−0.345510	−0.172755	−	0.984965i	$-0.555267\pi$
$79$	−3.07096	−0.345510	−0.172755	+	0.984965i	$0.555267\pi$
$80$	0	0
$81$	21.4980	2.38867
$82$	0	0
$83$	−5.15408	−0.565734	−0.282867	−	0.959159i	$-0.591285\pi$
$83$	−5.15408	−0.565734	−0.282867	+	0.959159i	$0.591285\pi$
$84$	0	0
$85$	2.68287	0.290998
$86$	0	0
$87$	−0.981871	−0.105268
$88$	0	0
$89$	3.65976	0.387933	0.193967	−	0.981008i	$-0.437865\pi$
$89$	3.65976	0.387933	0.193967	+	0.981008i	$0.437865\pi$
$90$	0	0
$91$	−20.8800	−2.18882
$92$	0	0
$93$	9.66704	1.00243
$94$	0	0
$95$	−10.4662	−1.07381
$96$	0	0
$97$	9.47799	0.962344	0.481172	−	0.876626i	$-0.340211\pi$
$97$	9.47799	0.962344	0.481172	+	0.876626i	$0.340211\pi$
$98$	0	0
$99$	26.6144	2.67485
$100$	0	0
$101$	12.4243	1.23626	0.618131	−	0.786075i	$-0.287890\pi$
$101$	12.4243	1.23626	0.618131	+	0.786075i	$0.287890\pi$
$102$	0	0
$103$	8.08160	0.796304	0.398152	−	0.917320i	$-0.369652\pi$
$103$	8.08160	0.796304	0.398152	+	0.917320i	$0.369652\pi$
$104$	0	0
$105$	31.1309	3.03806
$106$	0	0
$107$	16.5550	1.60043	0.800214	−	0.599714i	$-0.204719\pi$
$107$	16.5550	1.60043	0.800214	+	0.599714i	$0.204719\pi$
$108$	0	0
$109$	0.246074	0.0235696	0.0117848	−	0.999931i	$-0.496249\pi$
$109$	0.246074	0.0235696	0.0117848	+	0.999931i	$0.496249\pi$
$110$	0	0
$111$	−9.62533	−0.913596
$112$	0	0
$113$	−11.4227	−1.07456	−0.537280	−	0.843404i	$-0.680548\pi$
$113$	−11.4227	−1.07456	−0.537280	+	0.843404i	$0.680548\pi$
$114$	0	0
$115$	−7.22377	−0.673620
$116$	0	0
$117$	−33.7086	−3.11636
$118$	0	0
$119$	5.51509	0.505568
$120$	0	0
$121$	2.57840	0.234400
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.4552	−1.02459
$126$	0	0
$127$	2.53377	0.224836	0.112418	−	0.993661i	$-0.464140\pi$
$127$	2.53377	0.224836	0.112418	+	0.993661i	$0.464140\pi$
$128$	0	0
$129$	8.51116	0.749366
$130$	0	0
$131$	21.9176	1.91495	0.957474	−	0.288519i	$-0.0931628\pi$
$131$	21.9176	1.91495	0.957474	+	0.288519i	$0.0931628\pi$
$132$	0	0
$133$	−21.5150	−1.86558
$134$	0	0
$135$	29.3824	2.52884
$136$	0	0
$137$	−9.67817	−0.826862	−0.413431	−	0.910535i	$-0.635670\pi$
$137$	−9.67817	−0.826862	−0.413431	+	0.910535i	$0.635670\pi$
$138$	0	0
$139$	1.62578	0.137897	0.0689486	−	0.997620i	$-0.478036\pi$
$139$	1.62578	0.137897	0.0689486	+	0.997620i	$0.478036\pi$
$140$	0	0
$141$	−19.8839	−1.67453
$142$	0	0
$143$	−17.1978	−1.43815
$144$	0	0
$145$	−0.668349	−0.0555033
$146$	0	0
$147$	41.6139	3.43226
$148$	0	0
$149$	−20.0018	−1.63861	−0.819304	−	0.573359i	$-0.805640\pi$
$149$	−20.0018	−1.63861	−0.819304	+	0.573359i	$0.805640\pi$
$150$	0	0
$151$	−6.14831	−0.500342	−0.250171	−	0.968202i	$-0.580487\pi$
$151$	−6.14831	−0.500342	−0.250171	+	0.968202i	$0.580487\pi$
$152$	0	0
$153$	8.90355	0.719809
$154$	0	0
$155$	6.58025	0.528538
$156$	0	0
$157$	16.8752	1.34679	0.673394	−	0.739284i	$-0.264836\pi$
$157$	16.8752	1.34679	0.673394	+	0.739284i	$0.264836\pi$
$158$	0	0
$159$	−3.74468	−0.296972
$160$	0	0
$161$	−14.8497	−1.17032
$162$	0	0
$163$	−19.0855	−1.49489	−0.747445	−	0.664324i	$-0.768720\pi$
$163$	−19.0855	−1.49489	−0.747445	+	0.664324i	$0.768720\pi$
$164$	0	0
$165$	25.6409	1.99614
$166$	0	0
$167$	20.8700	1.61497	0.807486	−	0.589886i	$-0.200827\pi$
$167$	20.8700	1.61497	0.807486	+	0.589886i	$0.200827\pi$
$168$	0	0
$169$	8.78191	0.675532
$170$	0	0
$171$	−34.7337	−2.65615
$172$	0	0
$173$	−14.9749	−1.13852	−0.569262	−	0.822156i	$-0.692771\pi$
$173$	−14.9749	−1.13852	−0.569262	+	0.822156i	$0.692771\pi$
$174$	0	0
$175$	−1.17884	−0.0891120
$176$	0	0
$177$	−20.0313	−1.50564
$178$	0	0
$179$	−18.4369	−1.37804	−0.689018	−	0.724745i	$-0.741958\pi$
$179$	−18.4369	−1.37804	−0.689018	+	0.724745i	$0.741958\pi$
$180$	0	0
$181$	−8.72739	−0.648702	−0.324351	−	0.945937i	$-0.605146\pi$
$181$	−8.72739	−0.648702	−0.324351	+	0.945937i	$0.605146\pi$
$182$	0	0
$183$	−29.8462	−2.20629
$184$	0	0
$185$	−6.55185	−0.481702
$186$	0	0
$187$	4.54250	0.332180
$188$	0	0
$189$	60.4006	4.39350
$190$	0	0
$191$	0.576164	0.0416898	0.0208449	−	0.999783i	$-0.493364\pi$
$191$	0.576164	0.0416898	0.0208449	+	0.999783i	$0.493364\pi$
$192$	0	0
$193$	2.36056	0.169917	0.0849585	−	0.996384i	$-0.472924\pi$
$193$	2.36056	0.169917	0.0849585	+	0.996384i	$0.472924\pi$
$194$	0	0
$195$	−32.4756	−2.32563
$196$	0	0
$197$	13.0635	0.930736	0.465368	−	0.885117i	$-0.345922\pi$
$197$	13.0635	0.930736	0.465368	+	0.885117i	$0.345922\pi$
$198$	0	0
$199$	6.97440	0.494402	0.247201	−	0.968964i	$-0.420489\pi$
$199$	6.97440	0.494402	0.247201	+	0.968964i	$0.420489\pi$
$200$	0	0
$201$	28.0560	1.97892
$202$	0	0
$203$	−1.37390	−0.0964291
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−23.9733	−1.66626
$208$	0	0
$209$	−17.7208	−1.22577
$210$	0	0
$211$	22.4332	1.54437	0.772183	−	0.635400i	$-0.219165\pi$
$211$	22.4332	1.54437	0.772183	+	0.635400i	$0.219165\pi$
$212$	0	0
$213$	−17.9573	−1.23041
$214$	0	0
$215$	5.79345	0.395110
$216$	0	0
$217$	13.5268	0.918260
$218$	0	0
$219$	−10.7924	−0.729282
$220$	0	0
$221$	−5.75332	−0.387010
$222$	0	0
$223$	25.0271	1.67594	0.837969	−	0.545718i	$-0.183743\pi$
$223$	25.0271	1.67594	0.837969	+	0.545718i	$0.183743\pi$
$224$	0	0
$225$	−1.90312	−0.126875
$226$	0	0
$227$	−6.96241	−0.462111	−0.231056	−	0.972941i	$-0.574218\pi$
$227$	−6.96241	−0.462111	−0.231056	+	0.972941i	$0.574218\pi$
$228$	0	0
$229$	−29.2862	−1.93529	−0.967644	−	0.252320i	$-0.918806\pi$
$229$	−29.2862	−1.93529	−0.967644	+	0.252320i	$0.918806\pi$
$230$	0	0
$231$	52.7093	3.46802
$232$	0	0
$233$	3.60070	0.235890	0.117945	−	0.993020i	$-0.462369\pi$
$233$	3.60070	0.235890	0.117945	+	0.993020i	$0.462369\pi$
$234$	0	0
$235$	−13.5348	−0.882912
$236$	0	0
$237$	−9.81870	−0.637793
$238$	0	0
$239$	−26.1620	−1.69228	−0.846141	−	0.532960i	$-0.821080\pi$
$239$	−26.1620	−1.69228	−0.846141	+	0.532960i	$0.821080\pi$
$240$	0	0
$241$	−8.94372	−0.576116	−0.288058	−	0.957613i	$-0.593010\pi$
$241$	−8.94372	−0.576116	−0.288058	+	0.957613i	$0.593010\pi$
$242$	0	0
$243$	28.2327	1.81113
$244$	0	0
$245$	28.3261	1.80969
$246$	0	0
$247$	22.4443	1.42810
$248$	0	0
$249$	−16.4790	−1.04432
$250$	0	0
$251$	−22.3223	−1.40897	−0.704487	−	0.709717i	$-0.748823\pi$
$251$	−22.3223	−1.40897	−0.704487	+	0.709717i	$0.748823\pi$
$252$	0	0
$253$	−12.2309	−0.768951
$254$	0	0
$255$	8.57788	0.537168
$256$	0	0
$257$	−17.0835	−1.06564	−0.532820	−	0.846228i	$-0.678868\pi$
$257$	−17.0835	−1.06564	−0.532820	+	0.846228i	$0.678868\pi$
$258$	0	0
$259$	−13.4684	−0.836888
$260$	0	0
$261$	−2.21803	−0.137292
$262$	0	0
$263$	−4.11370	−0.253662	−0.126831	−	0.991924i	$-0.540481\pi$
$263$	−4.11370	−0.253662	−0.126831	+	0.991924i	$0.540481\pi$
$264$	0	0
$265$	−2.54896	−0.156581
$266$	0	0
$267$	11.7013	0.716105
$268$	0	0
$269$	−19.7532	−1.20437	−0.602187	−	0.798355i	$-0.705704\pi$
$269$	−19.7532	−1.20437	−0.602187	+	0.798355i	$0.705704\pi$
$270$	0	0
$271$	−18.1612	−1.10322	−0.551609	−	0.834103i	$-0.685986\pi$
$271$	−18.1612	−1.10322	−0.551609	+	0.834103i	$0.685986\pi$
$272$	0	0
$273$	−66.7591	−4.04045
$274$	0	0
$275$	−0.970951	−0.0585506
$276$	0	0
$277$	0.544261	0.0327015	0.0163507	−	0.999866i	$-0.494795\pi$
$277$	0.544261	0.0327015	0.0163507	+	0.999866i	$0.494795\pi$
$278$	0	0
$279$	21.8377	1.30739
$280$	0	0
$281$	−12.2940	−0.733398	−0.366699	−	0.930340i	$-0.619512\pi$
$281$	−12.2940	−0.733398	−0.366699	+	0.930340i	$0.619512\pi$
$282$	0	0
$283$	1.90761	0.113396	0.0566978	−	0.998391i	$-0.481943\pi$
$283$	1.90761	0.113396	0.0566978	+	0.998391i	$0.481943\pi$
$284$	0	0
$285$	−33.4632	−1.98219
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.4804	−0.910609
$290$	0	0
$291$	30.3038	1.77644
$292$	0	0
$293$	0.0710168	0.00414885	0.00207442	−	0.999998i	$-0.499340\pi$
$293$	0.0710168	0.00414885	0.00207442	+	0.999998i	$0.499340\pi$
$294$	0	0
$295$	−13.6351	−0.793864
$296$	0	0
$297$	49.7489	2.88672
$298$	0	0
$299$	15.4911	0.895875
$300$	0	0
$301$	11.9094	0.686448
$302$	0	0
$303$	39.7239	2.28208
$304$	0	0
$305$	−20.3160	−1.16329
$306$	0	0
$307$	−19.1637	−1.09373	−0.546866	−	0.837220i	$-0.684179\pi$
$307$	−19.1637	−1.09373	−0.546866	+	0.837220i	$0.684179\pi$
$308$	0	0
$309$	25.8391	1.46994
$310$	0	0
$311$	−23.2312	−1.31732	−0.658660	−	0.752441i	$-0.728876\pi$
$311$	−23.2312	−1.31732	−0.658660	+	0.752441i	$0.728876\pi$
$312$	0	0
$313$	−14.5896	−0.824655	−0.412327	−	0.911036i	$-0.635284\pi$
$313$	−14.5896	−0.824655	−0.412327	+	0.911036i	$0.635284\pi$
$314$	0	0
$315$	70.3241	3.96231
$316$	0	0
$317$	3.51593	0.197475	0.0987373	−	0.995114i	$-0.468520\pi$
$317$	3.51593	0.197475	0.0987373	+	0.995114i	$0.468520\pi$
$318$	0	0
$319$	−1.13161	−0.0633582
$320$	0	0
$321$	52.9308	2.95431
$322$	0	0
$323$	−5.92828	−0.329859
$324$	0	0
$325$	1.22976	0.0682149
$326$	0	0
$327$	0.786766	0.0435083
$328$	0	0
$329$	−27.8230	−1.53393
$330$	0	0
$331$	10.6504	0.585398	0.292699	−	0.956205i	$-0.405447\pi$
$331$	10.6504	0.585398	0.292699	+	0.956205i	$0.405447\pi$
$332$	0	0
$333$	−21.7434	−1.19153
$334$	0	0
$335$	19.0974	1.04340
$336$	0	0
$337$	21.3616	1.16364	0.581819	−	0.813318i	$-0.302341\pi$
$337$	21.3616	1.16364	0.581819	+	0.813318i	$0.302341\pi$
$338$	0	0
$339$	−36.5216	−1.98358
$340$	0	0
$341$	11.1413	0.603338
$342$	0	0
$343$	26.9121	1.45312
$344$	0	0
$345$	−23.0964	−1.24347
$346$	0	0
$347$	30.7757	1.65213	0.826063	−	0.563578i	$-0.190576\pi$
$347$	30.7757	1.65213	0.826063	+	0.563578i	$0.190576\pi$
$348$	0	0
$349$	24.5175	1.31239	0.656196	−	0.754591i	$-0.272165\pi$
$349$	24.5175	1.31239	0.656196	+	0.754591i	$0.272165\pi$
$350$	0	0
$351$	−63.0096	−3.36321
$352$	0	0
$353$	−17.5527	−0.934238	−0.467119	−	0.884194i	$-0.654708\pi$
$353$	−17.5527	−0.934238	−0.467119	+	0.884194i	$0.654708\pi$
$354$	0	0
$355$	−12.2233	−0.648748
$356$	0	0
$357$	17.6333	0.933252
$358$	0	0
$359$	10.3829	0.547989	0.273994	−	0.961731i	$-0.411655\pi$
$359$	10.3829	0.547989	0.273994	+	0.961731i	$0.411655\pi$
$360$	0	0
$361$	4.12688	0.217204
$362$	0	0
$363$	8.24387	0.432691
$364$	0	0
$365$	−7.34625	−0.384520
$366$	0	0
$367$	23.5627	1.22996	0.614981	−	0.788542i	$-0.289164\pi$
$367$	23.5627	1.22996	0.614981	+	0.788542i	$0.289164\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.23982	−0.272038
$372$	0	0
$373$	12.9677	0.671443	0.335721	−	0.941961i	$-0.391020\pi$
$373$	12.9677	0.671443	0.335721	+	0.941961i	$0.391020\pi$
$374$	0	0
$375$	−36.6255	−1.89133
$376$	0	0
$377$	1.43325	0.0738161
$378$	0	0
$379$	35.1556	1.80582	0.902911	−	0.429827i	$-0.141425\pi$
$379$	35.1556	1.80582	0.902911	+	0.429827i	$0.141425\pi$
$380$	0	0
$381$	8.10117	0.415036
$382$	0	0
$383$	0.528580	0.0270092	0.0135046	−	0.999909i	$-0.495701\pi$
$383$	0.528580	0.0270092	0.0135046	+	0.999909i	$0.495701\pi$
$384$	0	0
$385$	35.8786	1.82854
$386$	0	0
$387$	19.2265	0.977340
$388$	0	0
$389$	16.4650	0.834808	0.417404	−	0.908721i	$-0.362940\pi$
$389$	16.4650	0.834808	0.417404	+	0.908721i	$0.362940\pi$
$390$	0	0
$391$	−4.09171	−0.206927
$392$	0	0
$393$	70.0766	3.53490
$394$	0	0
$395$	−6.68348	−0.336282
$396$	0	0
$397$	−31.2971	−1.57075	−0.785377	−	0.619018i	$-0.787531\pi$
$397$	−31.2971	−1.57075	−0.785377	+	0.619018i	$0.787531\pi$
$398$	0	0
$399$	−68.7894	−3.44378
$400$	0	0
$401$	36.5919	1.82731	0.913657	−	0.406486i	$-0.133246\pi$
$401$	36.5919	1.82731	0.913657	+	0.406486i	$0.133246\pi$
$402$	0	0
$403$	−14.1111	−0.702925
$404$	0	0
$405$	46.7872	2.32487
$406$	0	0
$407$	−11.0933	−0.549873
$408$	0	0
$409$	20.7919	1.02809	0.514046	−	0.857763i	$-0.328146\pi$
$409$	20.7919	1.02809	0.514046	+	0.857763i	$0.328146\pi$
$410$	0	0
$411$	−30.9438	−1.52635
$412$	0	0
$413$	−28.0292	−1.37923
$414$	0	0
$415$	−11.2171	−0.550625
$416$	0	0
$417$	5.19808	0.254551
$418$	0	0
$419$	32.2436	1.57520	0.787602	−	0.616184i	$-0.211322\pi$
$419$	32.2436	1.57520	0.787602	+	0.616184i	$0.211322\pi$
$420$	0	0
$421$	−5.44188	−0.265221	−0.132610	−	0.991168i	$-0.542336\pi$
$421$	−5.44188	−0.265221	−0.132610	+	0.991168i	$0.542336\pi$
$422$	0	0
$423$	−44.9174	−2.18396
$424$	0	0
$425$	−0.324821	−0.0157561
$426$	0	0
$427$	−41.7629	−2.02105
$428$	0	0
$429$	−54.9861	−2.65475
$430$	0	0
$431$	−6.05283	−0.291555	−0.145777	−	0.989317i	$-0.546568\pi$
$431$	−6.05283	−0.291555	−0.145777	+	0.989317i	$0.546568\pi$
$432$	0	0
$433$	2.15852	0.103732	0.0518658	−	0.998654i	$-0.483483\pi$
$433$	2.15852	0.103732	0.0518658	+	0.998654i	$0.483483\pi$
$434$	0	0
$435$	−2.13690	−0.102456
$436$	0	0
$437$	15.9622	0.763577
$438$	0	0
$439$	−15.4062	−0.735300	−0.367650	−	0.929964i	$-0.619838\pi$
$439$	−15.4062	−0.735300	−0.367650	+	0.929964i	$0.619838\pi$
$440$	0	0
$441$	94.0049	4.47643
$442$	0	0
$443$	32.6082	1.54926	0.774631	−	0.632414i	$-0.217936\pi$
$443$	32.6082	1.54926	0.774631	+	0.632414i	$0.217936\pi$
$444$	0	0
$445$	7.96492	0.377573
$446$	0	0
$447$	−63.9512	−3.02479
$448$	0	0
$449$	36.6965	1.73181	0.865906	−	0.500206i	$-0.166742\pi$
$449$	36.6965	1.73181	0.865906	+	0.500206i	$0.166742\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−19.6578	−0.923606
$454$	0	0
$455$	−45.4422	−2.13036
$456$	0	0
$457$	−8.53253	−0.399135	−0.199567	−	0.979884i	$-0.563954\pi$
$457$	−8.53253	−0.399135	−0.199567	+	0.979884i	$0.563954\pi$
$458$	0	0
$459$	16.6429	0.776825
$460$	0	0
$461$	27.8241	1.29590	0.647950	−	0.761683i	$-0.275627\pi$
$461$	27.8241	1.29590	0.647950	+	0.761683i	$0.275627\pi$
$462$	0	0
$463$	−15.0115	−0.697646	−0.348823	−	0.937189i	$-0.613419\pi$
$463$	−15.0115	−0.697646	−0.348823	+	0.937189i	$0.613419\pi$
$464$	0	0
$465$	21.0389	0.975655
$466$	0	0
$467$	32.4898	1.50345	0.751725	−	0.659477i	$-0.229222\pi$
$467$	32.4898	1.50345	0.751725	+	0.659477i	$0.229222\pi$
$468$	0	0
$469$	39.2580	1.81277
$470$	0	0
$471$	53.9547	2.48610
$472$	0	0
$473$	9.80918	0.451027
$474$	0	0
$475$	1.26716	0.0581413
$476$	0	0
$477$	−8.45915	−0.387318
$478$	0	0
$479$	32.4342	1.48195	0.740977	−	0.671530i	$-0.234363\pi$
$479$	32.4342	1.48195	0.740977	+	0.671530i	$0.234363\pi$
$480$	0	0
$481$	14.0502	0.640635
$482$	0	0
$483$	−47.4785	−2.16035
$484$	0	0
$485$	20.6274	0.936643
$486$	0	0
$487$	−34.5678	−1.56642	−0.783208	−	0.621760i	$-0.786418\pi$
$487$	−34.5678	−1.56642	−0.783208	+	0.621760i	$0.786418\pi$
$488$	0	0
$489$	−61.0216	−2.75949
$490$	0	0
$491$	31.0750	1.40240	0.701198	−	0.712967i	$-0.252649\pi$
$491$	31.0750	1.40240	0.701198	+	0.712967i	$0.252649\pi$
$492$	0	0
$493$	−0.378569	−0.0170499
$494$	0	0
$495$	57.9223	2.60342
$496$	0	0
$497$	−25.1271	−1.12711
$498$	0	0
$499$	−34.9297	−1.56367	−0.781834	−	0.623487i	$-0.785716\pi$
$499$	−34.9297	−1.56367	−0.781834	+	0.623487i	$0.785716\pi$
$500$	0	0
$501$	66.7273	2.98116
$502$	0	0
$503$	10.3718	0.462457	0.231228	−	0.972900i	$-0.425726\pi$
$503$	10.3718	0.462457	0.231228	+	0.972900i	$0.425726\pi$
$504$	0	0
$505$	27.0396	1.20325
$506$	0	0
$507$	28.0782	1.24700
$508$	0	0
$509$	−7.66867	−0.339908	−0.169954	−	0.985452i	$-0.554362\pi$
$509$	−7.66867	−0.339908	−0.169954	+	0.985452i	$0.554362\pi$
$510$	0	0
$511$	−15.1015	−0.668050
$512$	0	0
$513$	−64.9258	−2.86655
$514$	0	0
$515$	17.5884	0.775037
$516$	0	0
$517$	−22.9164	−1.00786
$518$	0	0
$519$	−47.8791	−2.10166
$520$	0	0
$521$	−9.96219	−0.436451	−0.218226	−	0.975898i	$-0.570027\pi$
$521$	−9.96219	−0.436451	−0.218226	+	0.975898i	$0.570027\pi$
$522$	0	0
$523$	−3.72806	−0.163016	−0.0815082	−	0.996673i	$-0.525974\pi$
$523$	−3.72806	−0.163016	−0.0815082	+	0.996673i	$0.525974\pi$
$524$	0	0
$525$	−3.76908	−0.164496
$526$	0	0
$527$	3.72721	0.162360
$528$	0	0
$529$	−11.9828	−0.520993
$530$	0	0
$531$	−45.2502	−1.96369
$532$	0	0
$533$	0	0
$534$	0	0
$535$	36.0294	1.55769
$536$	0	0
$537$	−58.9477	−2.54378
$538$	0	0
$539$	47.9603	2.06580
$540$	0	0
$541$	16.1285	0.693419	0.346710	−	0.937973i	$-0.387299\pi$
$541$	16.1285	0.693419	0.346710	+	0.937973i	$0.387299\pi$
$542$	0	0
$543$	−27.9039	−1.19747
$544$	0	0
$545$	0.535543	0.0229401
$546$	0	0
$547$	−26.9859	−1.15383	−0.576916	−	0.816803i	$-0.695744\pi$
$547$	−26.9859	−1.15383	−0.576916	+	0.816803i	$0.695744\pi$
$548$	0	0
$549$	−67.4219	−2.87750
$550$	0	0
$551$	1.47684	0.0629154
$552$	0	0
$553$	−13.7390	−0.584243
$554$	0	0
$555$	−20.9481	−0.889197
$556$	0	0
$557$	7.19108	0.304696	0.152348	−	0.988327i	$-0.451317\pi$
$557$	7.19108	0.304696	0.152348	+	0.988327i	$0.451317\pi$
$558$	0	0
$559$	−12.4239	−0.525473
$560$	0	0
$561$	14.5236	0.613188
$562$	0	0
$563$	−21.3398	−0.899363	−0.449682	−	0.893189i	$-0.648463\pi$
$563$	−21.3398	−0.899363	−0.449682	+	0.893189i	$0.648463\pi$
$564$	0	0
$565$	−24.8599	−1.04586
$566$	0	0
$567$	96.1790	4.03914
$568$	0	0
$569$	−12.3474	−0.517628	−0.258814	−	0.965927i	$-0.583332\pi$
$569$	−12.3474	−0.517628	−0.258814	+	0.965927i	$0.583332\pi$
$570$	0	0
$571$	10.6183	0.444362	0.222181	−	0.975005i	$-0.428682\pi$
$571$	10.6183	0.444362	0.222181	+	0.975005i	$0.428682\pi$
$572$	0	0
$573$	1.84216	0.0769573
$574$	0	0
$575$	0.874597	0.0364732
$576$	0	0
$577$	−45.9034	−1.91098	−0.955492	−	0.295018i	$-0.904674\pi$
$577$	−45.9034	−1.91098	−0.955492	+	0.295018i	$0.904674\pi$
$578$	0	0
$579$	7.54737	0.313658
$580$	0	0
$581$	−23.0586	−0.956633
$582$	0	0
$583$	−4.31577	−0.178741
$584$	0	0
$585$	−73.3618	−3.03314
$586$	0	0
$587$	8.90893	0.367711	0.183855	−	0.982953i	$-0.441142\pi$
$587$	8.90893	0.367711	0.183855	+	0.982953i	$0.441142\pi$
$588$	0	0
$589$	−14.5402	−0.599120
$590$	0	0
$591$	41.7676	1.71809
$592$	0	0
$593$	28.6777	1.17765	0.588826	−	0.808260i	$-0.299590\pi$
$593$	28.6777	1.17765	0.588826	+	0.808260i	$0.299590\pi$
$594$	0	0
$595$	12.0028	0.492066
$596$	0	0
$597$	22.2991	0.912642
$598$	0	0
$599$	−7.10718	−0.290392	−0.145196	−	0.989403i	$-0.546381\pi$
$599$	−7.10718	−0.290392	−0.145196	+	0.989403i	$0.546381\pi$
$600$	0	0
$601$	4.51613	0.184217	0.0921083	−	0.995749i	$-0.470639\pi$
$601$	4.51613	0.184217	0.0921083	+	0.995749i	$0.470639\pi$
$602$	0	0
$603$	63.3780	2.58095
$604$	0	0
$605$	5.61151	0.228140
$606$	0	0
$607$	−4.02899	−0.163532	−0.0817659	−	0.996652i	$-0.526056\pi$
$607$	−4.02899	−0.163532	−0.0817659	+	0.996652i	$0.526056\pi$
$608$	0	0
$609$	−4.39275	−0.178003
$610$	0	0
$611$	29.0249	1.17422
$612$	0	0
$613$	9.44854	0.381623	0.190811	−	0.981627i	$-0.438888\pi$
$613$	9.44854	0.381623	0.190811	+	0.981627i	$0.438888\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.0969	−1.73501	−0.867507	−	0.497425i	$-0.834279\pi$
$617$	−43.0969	−1.73501	−0.867507	+	0.497425i	$0.834279\pi$
$618$	0	0
$619$	34.1506	1.37263	0.686315	−	0.727304i	$-0.259227\pi$
$619$	34.1506	1.37263	0.686315	+	0.727304i	$0.259227\pi$
$620$	0	0
$621$	−44.8119	−1.79824
$622$	0	0
$623$	16.3732	0.655980
$624$	0	0
$625$	−23.6131	−0.944524
$626$	0	0
$627$	−56.6583	−2.26271
$628$	0	0
$629$	−3.71112	−0.147972
$630$	0	0
$631$	−4.67522	−0.186118	−0.0930589	−	0.995661i	$-0.529664\pi$
$631$	−4.67522	−0.186118	−0.0930589	+	0.995661i	$0.529664\pi$
$632$	0	0
$633$	71.7252	2.85082
$634$	0	0
$635$	5.51438	0.218831
$636$	0	0
$637$	−60.7444	−2.40678
$638$	0	0
$639$	−40.5652	−1.60473
$640$	0	0
$641$	18.6708	0.737452	0.368726	−	0.929538i	$-0.379794\pi$
$641$	18.6708	0.737452	0.368726	+	0.929538i	$0.379794\pi$
$642$	0	0
$643$	28.8247	1.13673	0.568367	−	0.822775i	$-0.307576\pi$
$643$	28.8247	1.13673	0.568367	+	0.822775i	$0.307576\pi$
$644$	0	0
$645$	18.5233	0.729353
$646$	0	0
$647$	18.4996	0.727295	0.363648	−	0.931537i	$-0.381531\pi$
$647$	18.4996	0.727295	0.363648	+	0.931537i	$0.381531\pi$
$648$	0	0
$649$	−23.0862	−0.906212
$650$	0	0
$651$	43.2490	1.69506
$652$	0	0
$653$	7.03084	0.275138	0.137569	−	0.990492i	$-0.456071\pi$
$653$	7.03084	0.275138	0.137569	+	0.990492i	$0.456071\pi$
$654$	0	0
$655$	47.7004	1.86381
$656$	0	0
$657$	−24.3798	−0.951146
$658$	0	0
$659$	−3.89876	−0.151874	−0.0759371	−	0.997113i	$-0.524195\pi$
$659$	−3.89876	−0.151874	−0.0759371	+	0.997113i	$0.524195\pi$
$660$	0	0
$661$	24.4767	0.952033	0.476017	−	0.879436i	$-0.342080\pi$
$661$	24.4767	0.952033	0.476017	+	0.879436i	$0.342080\pi$
$662$	0	0
$663$	−18.3950	−0.714401
$664$	0	0
$665$	−46.8241	−1.81576
$666$	0	0
$667$	1.01932	0.0394681
$668$	0	0
$669$	80.0185	3.09370
$670$	0	0
$671$	−34.3980	−1.32792
$672$	0	0
$673$	9.55932	0.368485	0.184242	−	0.982881i	$-0.441017\pi$
$673$	9.55932	0.368485	0.184242	+	0.982881i	$0.441017\pi$
$674$	0	0
$675$	−3.55739	−0.136924
$676$	0	0
$677$	19.9775	0.767798	0.383899	−	0.923375i	$-0.374581\pi$
$677$	19.9775	0.767798	0.383899	+	0.923375i	$0.374581\pi$
$678$	0	0
$679$	42.4032	1.62728
$680$	0	0
$681$	−22.2608	−0.853034
$682$	0	0
$683$	12.9699	0.496278	0.248139	−	0.968724i	$-0.420181\pi$
$683$	12.9699	0.496278	0.248139	+	0.968724i	$0.420181\pi$
$684$	0	0
$685$	−21.0631	−0.804780
$686$	0	0
$687$	−93.6362	−3.57244
$688$	0	0
$689$	5.46615	0.208244
$690$	0	0
$691$	0.637202	0.0242403	0.0121202	−	0.999927i	$-0.496142\pi$
$691$	0.637202	0.0242403	0.0121202	+	0.999927i	$0.496142\pi$
$692$	0	0
$693$	119.069	4.52306
$694$	0	0
$695$	3.53828	0.134214
$696$	0	0
$697$	0	0
$698$	0	0
$699$	11.5125	0.435441
$700$	0	0
$701$	−7.98163	−0.301462	−0.150731	−	0.988575i	$-0.548163\pi$
$701$	−7.98163	−0.301462	−0.150731	+	0.988575i	$0.548163\pi$
$702$	0	0
$703$	14.4775	0.546029
$704$	0	0
$705$	−43.2744	−1.62981
$706$	0	0
$707$	55.5845	2.09047
$708$	0	0
$709$	28.8178	1.08228	0.541138	−	0.840934i	$-0.317994\pi$
$709$	28.8178	1.08228	0.541138	+	0.840934i	$0.317994\pi$
$710$	0	0
$711$	−22.1803	−0.831825
$712$	0	0
$713$	−10.0357	−0.375840
$714$	0	0
$715$	−37.4284	−1.39974
$716$	0	0
$717$	−83.6473	−3.12387
$718$	0	0
$719$	−17.7951	−0.663647	−0.331823	−	0.943342i	$-0.607664\pi$
$719$	−17.7951	−0.663647	−0.331823	+	0.943342i	$0.607664\pi$
$720$	0	0
$721$	36.1559	1.34652
$722$	0	0
$723$	−28.5956	−1.06348
$724$	0	0
$725$	0.0809184	0.00300523
$726$	0	0
$727$	−36.5204	−1.35447	−0.677233	−	0.735768i	$-0.736821\pi$
$727$	−36.5204	−1.35447	−0.677233	+	0.735768i	$0.736821\pi$
$728$	0	0
$729$	25.7739	0.954588
$730$	0	0
$731$	3.28155	0.121372
$732$	0	0
$733$	−46.3510	−1.71201	−0.856006	−	0.516965i	$-0.827062\pi$
$733$	−46.3510	−1.71201	−0.856006	+	0.516965i	$0.827062\pi$
$734$	0	0
$735$	90.5665	3.34059
$736$	0	0
$737$	32.3348	1.19107
$738$	0	0
$739$	−20.5215	−0.754894	−0.377447	−	0.926031i	$-0.623198\pi$
$739$	−20.5215	−0.754894	−0.377447	+	0.926031i	$0.623198\pi$
$740$	0	0
$741$	71.7608	2.63620
$742$	0	0
$743$	23.7096	0.869822	0.434911	−	0.900474i	$-0.356780\pi$
$743$	23.7096	0.869822	0.434911	+	0.900474i	$0.356780\pi$
$744$	0	0
$745$	−43.5309	−1.59485
$746$	0	0
$747$	−37.2258	−1.36202
$748$	0	0
$749$	74.0646	2.70626
$750$	0	0
$751$	−17.2146	−0.628170	−0.314085	−	0.949395i	$-0.601698\pi$
$751$	−17.2146	−0.628170	−0.314085	+	0.949395i	$0.601698\pi$
$752$	0	0
$753$	−71.3707	−2.60089
$754$	0	0
$755$	−13.3809	−0.486980
$756$	0	0
$757$	19.1926	0.697568	0.348784	−	0.937203i	$-0.386595\pi$
$757$	19.1926	0.697568	0.348784	+	0.937203i	$0.386595\pi$
$758$	0	0
$759$	−39.1056	−1.41945
$760$	0	0
$761$	5.93643	0.215195	0.107598	−	0.994195i	$-0.465684\pi$
$761$	5.93643	0.215195	0.107598	+	0.994195i	$0.465684\pi$
$762$	0	0
$763$	1.10090	0.0398552
$764$	0	0
$765$	19.3773	0.700586
$766$	0	0
$767$	29.2399	1.05579
$768$	0	0
$769$	7.25710	0.261698	0.130849	−	0.991402i	$-0.458230\pi$
$769$	7.25710	0.261698	0.130849	+	0.991402i	$0.458230\pi$
$770$	0	0
$771$	−54.6207	−1.96712
$772$	0	0
$773$	−27.2696	−0.980819	−0.490410	−	0.871492i	$-0.663153\pi$
$773$	−27.2696	−0.980819	−0.490410	+	0.871492i	$0.663153\pi$
$774$	0	0
$775$	−0.796684	−0.0286177
$776$	0	0
$777$	−43.0624	−1.54485
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−20.6959	−0.740559
$782$	0	0
$783$	−4.14604	−0.148167
$784$	0	0
$785$	36.7264	1.31082
$786$	0	0
$787$	15.6290	0.557114	0.278557	−	0.960420i	$-0.410144\pi$
$787$	15.6290	0.557114	0.278557	+	0.960420i	$0.410144\pi$
$788$	0	0
$789$	−13.1527	−0.468247
$790$	0	0
$791$	−51.1037	−1.81704
$792$	0	0
$793$	43.5669	1.54711
$794$	0	0
$795$	−8.14973	−0.289041
$796$	0	0
$797$	−2.61006	−0.0924531	−0.0462266	−	0.998931i	$-0.514720\pi$
$797$	−2.61006	−0.0924531	−0.0462266	+	0.998931i	$0.514720\pi$
$798$	0	0
$799$	−7.66642	−0.271218
$800$	0	0
$801$	26.4329	0.933961
$802$	0	0
$803$	−12.4383	−0.438938
$804$	0	0
$805$	−32.3181	−1.13906
$806$	0	0
$807$	−63.1564	−2.22321
$808$	0	0
$809$	−37.6341	−1.32315	−0.661573	−	0.749881i	$-0.730111\pi$
$809$	−37.6341	−1.32315	−0.661573	+	0.749881i	$0.730111\pi$
$810$	0	0
$811$	3.09962	0.108842	0.0544212	−	0.998518i	$-0.482669\pi$
$811$	3.09962	0.108842	0.0544212	+	0.998518i	$0.482669\pi$
$812$	0	0
$813$	−58.0666	−2.03648
$814$	0	0
$815$	−41.5367	−1.45497
$816$	0	0
$817$	−12.8017	−0.447874
$818$	0	0
$819$	−150.808	−5.26964
$820$	0	0
$821$	31.0631	1.08411	0.542055	−	0.840343i	$-0.317646\pi$
$821$	31.0631	1.08411	0.542055	+	0.840343i	$0.317646\pi$
$822$	0	0
$823$	7.36918	0.256873	0.128437	−	0.991718i	$-0.459004\pi$
$823$	7.36918	0.256873	0.128437	+	0.991718i	$0.459004\pi$
$824$	0	0
$825$	−3.10440	−0.108081
$826$	0	0
$827$	−2.28195	−0.0793513	−0.0396756	−	0.999213i	$-0.512632\pi$
$827$	−2.28195	−0.0793513	−0.0396756	+	0.999213i	$0.512632\pi$
$828$	0	0
$829$	−21.2989	−0.739742	−0.369871	−	0.929083i	$-0.620598\pi$
$829$	−21.2989	−0.739742	−0.369871	+	0.929083i	$0.620598\pi$
$830$	0	0
$831$	1.74015	0.0603652
$832$	0	0
$833$	16.0446	0.555912
$834$	0	0
$835$	45.4205	1.57184
$836$	0	0
$837$	40.8199	1.41094
$838$	0	0
$839$	8.15153	0.281422	0.140711	−	0.990051i	$-0.455061\pi$
$839$	8.15153	0.281422	0.140711	+	0.990051i	$0.455061\pi$
$840$	0	0
$841$	−28.9057	−0.996748
$842$	0	0
$843$	−39.3073	−1.35382
$844$	0	0
$845$	19.1125	0.657491
$846$	0	0
$847$	11.5354	0.396361
$848$	0	0
$849$	6.09916	0.209323
$850$	0	0
$851$	9.99240	0.342535
$852$	0	0
$853$	4.02174	0.137702	0.0688509	−	0.997627i	$-0.478067\pi$
$853$	4.02174	0.137702	0.0688509	+	0.997627i	$0.478067\pi$
$854$	0	0
$855$	−75.5928	−2.58522
$856$	0	0
$857$	−33.5999	−1.14775	−0.573876	−	0.818942i	$-0.694561\pi$
$857$	−33.5999	−1.14775	−0.573876	+	0.818942i	$0.694561\pi$
$858$	0	0
$859$	40.8172	1.39266	0.696332	−	0.717720i	$-0.254814\pi$
$859$	40.8172	1.39266	0.696332	+	0.717720i	$0.254814\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.15525	0.175487	0.0877433	−	0.996143i	$-0.472034\pi$
$863$	5.15525	0.175487	0.0877433	+	0.996143i	$0.472034\pi$
$864$	0	0
$865$	−32.5907	−1.10812
$866$	0	0
$867$	−49.4950	−1.68094
$868$	0	0
$869$	−11.3161	−0.383874
$870$	0	0
$871$	−40.9538	−1.38767
$872$	0	0
$873$	68.4556	2.31687
$874$	0	0
$875$	−51.2490	−1.73253
$876$	0	0
$877$	26.6085	0.898506	0.449253	−	0.893405i	$-0.351690\pi$
$877$	26.6085	0.898506	0.449253	+	0.893405i	$0.351690\pi$
$878$	0	0
$879$	0.227061	0.00765856
$880$	0	0
$881$	46.3365	1.56112	0.780558	−	0.625084i	$-0.214935\pi$
$881$	46.3365	1.56112	0.780558	+	0.625084i	$0.214935\pi$
$882$	0	0
$883$	21.7341	0.731412	0.365706	−	0.930730i	$-0.380828\pi$
$883$	21.7341	0.731412	0.365706	+	0.930730i	$0.380828\pi$
$884$	0	0
$885$	−43.5951	−1.46543
$886$	0	0
$887$	38.9725	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$887$	38.9725	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$888$	0	0
$889$	11.3357	0.380188
$890$	0	0
$891$	79.2177	2.65389
$892$	0	0
$893$	29.9075	1.00082
$894$	0	0
$895$	−40.1251	−1.34123
$896$	0	0
$897$	49.5294	1.65374
$898$	0	0
$899$	−0.928511	−0.0309676
$900$	0	0
$901$	−1.44379	−0.0480996
$902$	0	0
$903$	38.0777	1.26715
$904$	0	0
$905$	−18.9939	−0.631377
$906$	0	0
$907$	4.41946	0.146746	0.0733728	−	0.997305i	$-0.476624\pi$
$907$	4.41946	0.146746	0.0733728	+	0.997305i	$0.476624\pi$
$908$	0	0
$909$	89.7355	2.97634
$910$	0	0
$911$	−25.3728	−0.840640	−0.420320	−	0.907376i	$-0.638082\pi$
$911$	−25.3728	−0.840640	−0.420320	+	0.907376i	$0.638082\pi$
$912$	0	0
$913$	−18.9922	−0.628550
$914$	0	0
$915$	−64.9558	−2.14737
$916$	0	0
$917$	98.0562	3.23810
$918$	0	0
$919$	25.6571	0.846349	0.423175	−	0.906048i	$-0.360916\pi$
$919$	25.6571	0.846349	0.423175	+	0.906048i	$0.360916\pi$
$920$	0	0
$921$	−61.2718	−2.01897
$922$	0	0
$923$	26.2125	0.862796
$924$	0	0
$925$	0.793247	0.0260818
$926$	0	0
$927$	58.3700	1.91712
$928$	0	0
$929$	4.94794	0.162337	0.0811683	−	0.996700i	$-0.474135\pi$
$929$	4.94794	0.162337	0.0811683	+	0.996700i	$0.474135\pi$
$930$	0	0
$931$	−62.5917	−2.05136
$932$	0	0
$933$	−74.2765	−2.43170
$934$	0	0
$935$	9.88607	0.323309
$936$	0	0
$937$	58.0738	1.89719	0.948594	−	0.316497i	$-0.102507\pi$
$937$	58.0738	1.89719	0.948594	+	0.316497i	$0.102507\pi$
$938$	0	0
$939$	−46.6471	−1.52227
$940$	0	0
$941$	36.8175	1.20022	0.600108	−	0.799919i	$-0.295124\pi$
$941$	36.8175	1.20022	0.600108	+	0.799919i	$0.295124\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	131.453	4.27616
$946$	0	0
$947$	4.91234	0.159630	0.0798149	−	0.996810i	$-0.474567\pi$
$947$	4.91234	0.159630	0.0798149	+	0.996810i	$0.474567\pi$
$948$	0	0
$949$	15.7538	0.511390
$950$	0	0
$951$	11.2414	0.364528
$952$	0	0
$953$	−33.0755	−1.07142	−0.535711	−	0.844401i	$-0.679956\pi$
$953$	−33.0755	−1.07142	−0.535711	+	0.844401i	$0.679956\pi$
$954$	0	0
$955$	1.25394	0.0405764
$956$	0	0
$957$	−3.61809	−0.116956
$958$	0	0
$959$	−43.2988	−1.39819
$960$	0	0
$961$	−21.8583	−0.705107
$962$	0	0
$963$	119.570	3.85308
$964$	0	0
$965$	5.13741	0.165379
$966$	0	0
$967$	19.2628	0.619450	0.309725	−	0.950826i	$-0.399763\pi$
$967$	19.2628	0.619450	0.309725	+	0.950826i	$0.399763\pi$
$968$	0	0
$969$	−18.9544	−0.608902
$970$	0	0
$971$	−2.33133	−0.0748158	−0.0374079	−	0.999300i	$-0.511910\pi$
$971$	−2.33133	−0.0748158	−0.0374079	+	0.999300i	$0.511910\pi$
$972$	0	0
$973$	7.27353	0.233179
$974$	0	0
$975$	3.93189	0.125921
$976$	0	0
$977$	−29.7827	−0.952833	−0.476417	−	0.879220i	$-0.658065\pi$
$977$	−29.7827	−0.952833	−0.476417	+	0.879220i	$0.658065\pi$
$978$	0	0
$979$	13.4858	0.431008
$980$	0	0
$981$	1.77729	0.0567445
$982$	0	0
$983$	−35.8341	−1.14293	−0.571465	−	0.820626i	$-0.693625\pi$
$983$	−35.8341	−1.14293	−0.571465	+	0.820626i	$0.693625\pi$
$984$	0	0
$985$	28.4308	0.905879
$986$	0	0
$987$	−88.9580	−2.83156
$988$	0	0
$989$	−8.83574	−0.280960
$990$	0	0
$991$	19.5167	0.619968	0.309984	−	0.950742i	$-0.399676\pi$
$991$	19.5167	0.619968	0.309984	+	0.950742i	$0.399676\pi$
$992$	0	0
$993$	34.0523	1.08062
$994$	0	0
$995$	15.1788	0.481199
$996$	0	0
$997$	−45.9652	−1.45573	−0.727867	−	0.685718i	$-0.759488\pi$
$997$	−45.9652	−1.45573	−0.727867	+	0.685718i	$0.759488\pi$
$998$	0	0
$999$	−40.6438	−1.28591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.j.1.16		16
41.20	even	20		164.2.k.a.113.1	yes	16
41.39	even	20		164.2.k.a.45.4	✓	16
41.40	even	2	inner	6724.2.a.j.1.1		16
123.20	odd	20		1476.2.bb.b.1261.3		16
123.80	odd	20		1476.2.bb.b.865.3		16
164.39	odd	20		656.2.be.e.209.1		16
164.143	odd	20		656.2.be.e.113.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.k.a.45.4	✓	16	41.39	even	20
164.2.k.a.113.1	yes	16	41.20	even	20
656.2.be.e.113.4		16	164.143	odd	20
656.2.be.e.209.1		16	164.39	odd	20
1476.2.bb.b.865.3		16	123.80	odd	20
1476.2.bb.b.1261.3		16	123.20	odd	20
6724.2.a.j.1.1		16	41.40	even	2	inner
6724.2.a.j.1.16		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q+3.19728 q^{3} +2.17635 q^{5} +4.47386 q^{7} +7.22259 q^{9} +O(q^{10})\)	\(q+3.19728 q^{3} +2.17635 q^{5} +4.47386 q^{7} +7.22259 q^{9} +3.68489 q^{11} -4.66711 q^{13} +6.95840 q^{15} +1.23274 q^{17} -4.80904 q^{19} +14.3042 q^{21} -3.31921 q^{23} -0.263495 q^{25} +13.5008 q^{27} -0.307096 q^{29} +3.02352 q^{31} +11.7816 q^{33} +9.73669 q^{35} -3.01048 q^{37} -14.9220 q^{39} +2.66200 q^{43} +15.7189 q^{45} -6.21902 q^{47} +13.0154 q^{49} +3.94140 q^{51} -1.17121 q^{53} +8.01961 q^{55} -15.3758 q^{57} -6.26510 q^{59} -9.33487 q^{61} +32.3128 q^{63} -10.1573 q^{65} +8.77498 q^{67} -10.6124 q^{69} -5.61644 q^{71} -3.37549 q^{73} -0.842468 q^{75} +16.4857 q^{77} -3.07096 q^{79} +21.4980 q^{81} -5.15408 q^{83} +2.68287 q^{85} -0.981871 q^{87} +3.65976 q^{89} -20.8800 q^{91} +9.66704 q^{93} -10.4662 q^{95} +9.47799 q^{97} +26.6144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) \) 16 * q + 6 * q^5 + 30 * q^9	\( 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) \) 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Introduction

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Modular forms

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