LMFDB - Embedded newform 3344.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.302776 q^{3} -2.30278 q^{5} -1.30278 q^{7} -2.90833 q^{9} +O(q^{10})$	$q-0.302776 q^{3} -2.30278 q^{5} -1.30278 q^{7} -2.90833 q^{9} -1.00000 q^{11} -3.30278 q^{13} +0.697224 q^{15} -4.60555 q^{17} -1.00000 q^{19} +0.394449 q^{21} +1.39445 q^{23} +0.302776 q^{25} +1.78890 q^{27} -8.30278 q^{29} +3.30278 q^{31} +0.302776 q^{33} +3.00000 q^{35} +5.21110 q^{37} +1.00000 q^{39} +3.90833 q^{41} -1.09167 q^{43} +6.69722 q^{45} +6.00000 q^{47} -5.30278 q^{49} +1.39445 q^{51} -10.6056 q^{53} +2.30278 q^{55} +0.302776 q^{57} +11.2111 q^{61} +3.78890 q^{63} +7.60555 q^{65} +6.51388 q^{67} -0.422205 q^{69} +9.69722 q^{71} -2.60555 q^{73} -0.0916731 q^{75} +1.30278 q^{77} -15.8167 q^{79} +8.18335 q^{81} -14.3028 q^{83} +10.6056 q^{85} +2.51388 q^{87} +4.60555 q^{89} +4.30278 q^{91} -1.00000 q^{93} +2.30278 q^{95} +8.00000 q^{97} +2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9	$ 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9} - 2 q^{11} - 3 q^{13} + 5 q^{15} - 2 q^{17} - 2 q^{19} + 8 q^{21} + 10 q^{23} - 3 q^{25} + 18 q^{27} - 13 q^{29} + 3 q^{31} - 3 q^{33} + 6 q^{35} - 4 q^{37} + 2 q^{39} - 3 q^{41} - 13 q^{43} + 17 q^{45} + 12 q^{47} - 7 q^{49} + 10 q^{51} - 14 q^{53} + q^{55} - 3 q^{57} + 8 q^{61} + 22 q^{63} + 8 q^{65} - 5 q^{67} + 28 q^{69} + 23 q^{71} + 2 q^{73} - 11 q^{75} - q^{77} - 10 q^{79} + 38 q^{81} - 25 q^{83} + 14 q^{85} - 13 q^{87} + 2 q^{89} + 5 q^{91} - 2 q^{93} + q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9 - 2 * q^11 - 3 * q^13 + 5 * q^15 - 2 * q^17 - 2 * q^19 + 8 * q^21 + 10 * q^23 - 3 * q^25 + 18 * q^27 - 13 * q^29 + 3 * q^31 - 3 * q^33 + 6 * q^35 - 4 * q^37 + 2 * q^39 - 3 * q^41 - 13 * q^43 + 17 * q^45 + 12 * q^47 - 7 * q^49 + 10 * q^51 - 14 * q^53 + q^55 - 3 * q^57 + 8 * q^61 + 22 * q^63 + 8 * q^65 - 5 * q^67 + 28 * q^69 + 23 * q^71 + 2 * q^73 - 11 * q^75 - q^77 - 10 * q^79 + 38 * q^81 - 25 * q^83 + 14 * q^85 - 13 * q^87 + 2 * q^89 + 5 * q^91 - 2 * q^93 + q^95 + 16 * q^97 - 5 * q^99

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
$3$	−0.302776	−0.174808	−0.0874038	−	0.996173i	$-0.527857\pi$
$3$	−0.302776	−0.174808	−0.0874038	+	0.996173i	$0.527857\pi$
$4$	0	0
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	0	0
$7$	−1.30278	−0.492403	−0.246201	−	0.969219i	$-0.579182\pi$
$7$	−1.30278	−0.492403	−0.246201	+	0.969219i	$0.579182\pi$
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.30278	−0.916025	−0.458013	−	0.888946i	$-0.651439\pi$
$13$	−3.30278	−0.916025	−0.458013	+	0.888946i	$0.651439\pi$
$14$	0	0
$15$	0.697224	0.180023
$16$	0	0
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.394449	0.0860758
$22$	0	0
$23$	1.39445	0.290763	0.145381	−	0.989376i	$-0.453559\pi$
$23$	1.39445	0.290763	0.145381	+	0.989376i	$0.453559\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	1.78890	0.344273
$28$	0	0
$29$	−8.30278	−1.54179	−0.770893	−	0.636964i	$-0.780190\pi$
$29$	−8.30278	−1.54179	−0.770893	+	0.636964i	$0.780190\pi$
$30$	0	0
$31$	3.30278	0.593196	0.296598	−	0.955002i	$-0.404148\pi$
$31$	3.30278	0.593196	0.296598	+	0.955002i	$0.404148\pi$
$32$	0	0
$33$	0.302776	0.0527065
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	5.21110	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$37$	5.21110	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.90833	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$41$	3.90833	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$42$	0	0
$43$	−1.09167	−0.166479	−0.0832393	−	0.996530i	$-0.526527\pi$
$43$	−1.09167	−0.166479	−0.0832393	+	0.996530i	$0.526527\pi$
$44$	0	0
$45$	6.69722	0.998363
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−5.30278	−0.757539
$50$	0	0
$51$	1.39445	0.195262
$52$	0	0
$53$	−10.6056	−1.45678	−0.728392	−	0.685160i	$-0.759732\pi$
$53$	−10.6056	−1.45678	−0.728392	+	0.685160i	$0.759732\pi$
$54$	0	0
$55$	2.30278	0.310506
$56$	0	0
$57$	0.302776	0.0401036
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	11.2111	1.43543	0.717717	−	0.696335i	$-0.245187\pi$
$61$	11.2111	1.43543	0.717717	+	0.696335i	$0.245187\pi$
$62$	0	0
$63$	3.78890	0.477356
$64$	0	0
$65$	7.60555	0.943353
$66$	0	0
$67$	6.51388	0.795797	0.397898	−	0.917429i	$-0.369740\pi$
$67$	6.51388	0.795797	0.397898	+	0.917429i	$0.369740\pi$
$68$	0	0
$69$	−0.422205	−0.0508275
$70$	0	0
$71$	9.69722	1.15085	0.575424	−	0.817855i	$-0.304837\pi$
$71$	9.69722	1.15085	0.575424	+	0.817855i	$0.304837\pi$
$72$	0	0
$73$	−2.60555	−0.304957	−0.152478	−	0.988307i	$-0.548725\pi$
$73$	−2.60555	−0.304957	−0.152478	+	0.988307i	$0.548725\pi$
$74$	0	0
$75$	−0.0916731	−0.0105855
$76$	0	0
$77$	1.30278	0.148465
$78$	0	0
$79$	−15.8167	−1.77951	−0.889756	−	0.456436i	$-0.849126\pi$
$79$	−15.8167	−1.77951	−0.889756	+	0.456436i	$0.849126\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	−14.3028	−1.56993	−0.784967	−	0.619538i	$-0.787320\pi$
$83$	−14.3028	−1.56993	−0.784967	+	0.619538i	$0.787320\pi$
$84$	0	0
$85$	10.6056	1.15033
$86$	0	0
$87$	2.51388	0.269516
$88$	0	0
$89$	4.60555	0.488187	0.244094	−	0.969752i	$-0.421510\pi$
$89$	4.60555	0.488187	0.244094	+	0.969752i	$0.421510\pi$
$90$	0	0
$91$	4.30278	0.451053
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	2.30278	0.236260
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	2.90833	0.292298
$100$	0	0
$101$	7.81665	0.777786	0.388893	−	0.921283i	$-0.372858\pi$
$101$	7.81665	0.777786	0.388893	+	0.921283i	$0.372858\pi$
$102$	0	0
$103$	1.69722	0.167232	0.0836162	−	0.996498i	$-0.473353\pi$
$103$	1.69722	0.167232	0.0836162	+	0.996498i	$0.473353\pi$
$104$	0	0
$105$	−0.908327	−0.0886436
$106$	0	0
$107$	−3.21110	−0.310429	−0.155215	−	0.987881i	$-0.549607\pi$
$107$	−3.21110	−0.310429	−0.155215	+	0.987881i	$0.549607\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−1.57779	−0.149758
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−3.21110	−0.299437
$116$	0	0
$117$	9.60555	0.888034
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.18335	−0.106699
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	5.39445	0.478680	0.239340	−	0.970936i	$-0.423069\pi$
$127$	5.39445	0.478680	0.239340	+	0.970936i	$0.423069\pi$
$128$	0	0
$129$	0.330532	0.0291017
$130$	0	0
$131$	−19.1194	−1.67047	−0.835236	−	0.549891i	$-0.814669\pi$
$131$	−19.1194	−1.67047	−0.835236	+	0.549891i	$0.814669\pi$
$132$	0	0
$133$	1.30278	0.112965
$134$	0	0
$135$	−4.11943	−0.354544
$136$	0	0
$137$	−3.48612	−0.297839	−0.148920	−	0.988849i	$-0.547580\pi$
$137$	−3.48612	−0.297839	−0.148920	+	0.988849i	$0.547580\pi$
$138$	0	0
$139$	6.30278	0.534594	0.267297	−	0.963614i	$-0.413869\pi$
$139$	6.30278	0.534594	0.267297	+	0.963614i	$0.413869\pi$
$140$	0	0
$141$	−1.81665	−0.152990
$142$	0	0
$143$	3.30278	0.276192
$144$	0	0
$145$	19.1194	1.58778
$146$	0	0
$147$	1.60555	0.132424
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	16.4222	1.33642	0.668210	−	0.743973i	$-0.267061\pi$
$151$	16.4222	1.33642	0.668210	+	0.743973i	$0.267061\pi$
$152$	0	0
$153$	13.3944	1.08288
$154$	0	0
$155$	−7.60555	−0.610893
$156$	0	0
$157$	1.09167	0.0871250	0.0435625	−	0.999051i	$-0.486129\pi$
$157$	1.09167	0.0871250	0.0435625	+	0.999051i	$0.486129\pi$
$158$	0	0
$159$	3.21110	0.254657
$160$	0	0
$161$	−1.81665	−0.143172
$162$	0	0
$163$	11.3944	0.892482	0.446241	−	0.894913i	$-0.352762\pi$
$163$	11.3944	0.892482	0.446241	+	0.894913i	$0.352762\pi$
$164$	0	0
$165$	−0.697224	−0.0542788
$166$	0	0
$167$	12.4222	0.961259	0.480630	−	0.876924i	$-0.340408\pi$
$167$	12.4222	0.961259	0.480630	+	0.876924i	$0.340408\pi$
$168$	0	0
$169$	−2.09167	−0.160898
$170$	0	0
$171$	2.90833	0.222405
$172$	0	0
$173$	−19.3305	−1.46967	−0.734837	−	0.678244i	$-0.762741\pi$
$173$	−19.3305	−1.46967	−0.734837	+	0.678244i	$0.762741\pi$
$174$	0	0
$175$	−0.394449	−0.0298175
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.1194	0.980592	0.490296	−	0.871556i	$-0.336889\pi$
$179$	13.1194	0.980592	0.490296	+	0.871556i	$0.336889\pi$
$180$	0	0
$181$	−15.0278	−1.11700	−0.558502	−	0.829503i	$-0.688624\pi$
$181$	−15.0278	−1.11700	−0.558502	+	0.829503i	$0.688624\pi$
$182$	0	0
$183$	−3.39445	−0.250925
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	4.60555	0.336791
$188$	0	0
$189$	−2.33053	−0.169521
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−24.7250	−1.77974	−0.889872	−	0.456211i	$-0.849206\pi$
$193$	−24.7250	−1.77974	−0.889872	+	0.456211i	$0.849206\pi$
$194$	0	0
$195$	−2.30278	−0.164905
$196$	0	0
$197$	21.6333	1.54131	0.770655	−	0.637253i	$-0.219929\pi$
$197$	21.6333	1.54131	0.770655	+	0.637253i	$0.219929\pi$
$198$	0	0
$199$	2.18335	0.154773	0.0773867	−	0.997001i	$-0.475342\pi$
$199$	2.18335	0.154773	0.0773867	+	0.997001i	$0.475342\pi$
$200$	0	0
$201$	−1.97224	−0.139111
$202$	0	0
$203$	10.8167	0.759180
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−4.05551	−0.281878
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−24.6056	−1.69392	−0.846958	−	0.531660i	$-0.821569\pi$
$211$	−24.6056	−1.69392	−0.846958	+	0.531660i	$0.821569\pi$
$212$	0	0
$213$	−2.93608	−0.201177
$214$	0	0
$215$	2.51388	0.171445
$216$	0	0
$217$	−4.30278	−0.292091
$218$	0	0
$219$	0.788897	0.0533087
$220$	0	0
$221$	15.2111	1.02321
$222$	0	0
$223$	25.2111	1.68826	0.844130	−	0.536138i	$-0.180117\pi$
$223$	25.2111	1.68826	0.844130	+	0.536138i	$0.180117\pi$
$224$	0	0
$225$	−0.880571	−0.0587047
$226$	0	0
$227$	−1.81665	−0.120576	−0.0602878	−	0.998181i	$-0.519202\pi$
$227$	−1.81665	−0.120576	−0.0602878	+	0.998181i	$0.519202\pi$
$228$	0	0
$229$	11.9083	0.786924	0.393462	−	0.919341i	$-0.371277\pi$
$229$	11.9083	0.786924	0.393462	+	0.919341i	$0.371277\pi$
$230$	0	0
$231$	−0.394449	−0.0259528
$232$	0	0
$233$	9.21110	0.603439	0.301720	−	0.953397i	$-0.402439\pi$
$233$	9.21110	0.603439	0.301720	+	0.953397i	$0.402439\pi$
$234$	0	0
$235$	−13.8167	−0.901299
$236$	0	0
$237$	4.78890	0.311072
$238$	0	0
$239$	23.9361	1.54830	0.774148	−	0.633004i	$-0.218178\pi$
$239$	23.9361	1.54830	0.774148	+	0.633004i	$0.218178\pi$
$240$	0	0
$241$	13.0917	0.843309	0.421654	−	0.906757i	$-0.361450\pi$
$241$	13.0917	0.843309	0.421654	+	0.906757i	$0.361450\pi$
$242$	0	0
$243$	−7.84441	−0.503219
$244$	0	0
$245$	12.2111	0.780139
$246$	0	0
$247$	3.30278	0.210151
$248$	0	0
$249$	4.33053	0.274436
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−1.39445	−0.0876682
$254$	0	0
$255$	−3.21110	−0.201087
$256$	0	0
$257$	−4.60555	−0.287286	−0.143643	−	0.989630i	$-0.545882\pi$
$257$	−4.60555	−0.287286	−0.143643	+	0.989630i	$0.545882\pi$
$258$	0	0
$259$	−6.78890	−0.421842
$260$	0	0
$261$	24.1472	1.49467
$262$	0	0
$263$	20.3028	1.25192	0.625961	−	0.779854i	$-0.284707\pi$
$263$	20.3028	1.25192	0.625961	+	0.779854i	$0.284707\pi$
$264$	0	0
$265$	24.4222	1.50024
$266$	0	0
$267$	−1.39445	−0.0853389
$268$	0	0
$269$	−19.8167	−1.20824	−0.604121	−	0.796892i	$-0.706476\pi$
$269$	−19.8167	−1.20824	−0.604121	+	0.796892i	$0.706476\pi$
$270$	0	0
$271$	6.51388	0.395690	0.197845	−	0.980233i	$-0.436606\pi$
$271$	6.51388	0.395690	0.197845	+	0.980233i	$0.436606\pi$
$272$	0	0
$273$	−1.30278	−0.0788476
$274$	0	0
$275$	−0.302776	−0.0182581
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−9.60555	−0.575069
$280$	0	0
$281$	−0.486122	−0.0289996	−0.0144998	−	0.999895i	$-0.504616\pi$
$281$	−0.486122	−0.0289996	−0.0144998	+	0.999895i	$0.504616\pi$
$282$	0	0
$283$	18.9361	1.12563	0.562817	−	0.826582i	$-0.309718\pi$
$283$	18.9361	1.12563	0.562817	+	0.826582i	$0.309718\pi$
$284$	0	0
$285$	−0.697224	−0.0413000
$286$	0	0
$287$	−5.09167	−0.300552
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	−2.42221	−0.141992
$292$	0	0
$293$	−25.5416	−1.49216	−0.746079	−	0.665857i	$-0.768066\pi$
$293$	−25.5416	−1.49216	−0.746079	+	0.665857i	$0.768066\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.78890	−0.103802
$298$	0	0
$299$	−4.60555	−0.266346
$300$	0	0
$301$	1.42221	0.0819745
$302$	0	0
$303$	−2.36669	−0.135963
$304$	0	0
$305$	−25.8167	−1.47826
$306$	0	0
$307$	2.18335	0.124610	0.0623051	−	0.998057i	$-0.480155\pi$
$307$	2.18335	0.124610	0.0623051	+	0.998057i	$0.480155\pi$
$308$	0	0
$309$	−0.513878	−0.0292335
$310$	0	0
$311$	−25.8167	−1.46393	−0.731964	−	0.681343i	$-0.761396\pi$
$311$	−25.8167	−1.46393	−0.731964	+	0.681343i	$0.761396\pi$
$312$	0	0
$313$	18.1194	1.02417	0.512085	−	0.858935i	$-0.328873\pi$
$313$	18.1194	1.02417	0.512085	+	0.858935i	$0.328873\pi$
$314$	0	0
$315$	−8.72498	−0.491597
$316$	0	0
$317$	−13.8167	−0.776021	−0.388010	−	0.921655i	$-0.626838\pi$
$317$	−13.8167	−0.776021	−0.388010	+	0.921655i	$0.626838\pi$
$318$	0	0
$319$	8.30278	0.464866
$320$	0	0
$321$	0.972244	0.0542653
$322$	0	0
$323$	4.60555	0.256260
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−0.605551	−0.0334871
$328$	0	0
$329$	−7.81665	−0.430946
$330$	0	0
$331$	−22.7250	−1.24908	−0.624539	−	0.780994i	$-0.714713\pi$
$331$	−22.7250	−1.24908	−0.624539	+	0.780994i	$0.714713\pi$
$332$	0	0
$333$	−15.1556	−0.830521
$334$	0	0
$335$	−15.0000	−0.819538
$336$	0	0
$337$	−9.30278	−0.506754	−0.253377	−	0.967368i	$-0.581541\pi$
$337$	−9.30278	−0.506754	−0.253377	+	0.967368i	$0.581541\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.30278	−0.178855
$342$	0	0
$343$	16.0278	0.865417
$344$	0	0
$345$	0.972244	0.0523438
$346$	0	0
$347$	−2.78890	−0.149716	−0.0748579	−	0.997194i	$-0.523850\pi$
$347$	−2.78890	−0.149716	−0.0748579	+	0.997194i	$0.523850\pi$
$348$	0	0
$349$	32.4222	1.73552	0.867760	−	0.496983i	$-0.165559\pi$
$349$	32.4222	1.73552	0.867760	+	0.496983i	$0.165559\pi$
$350$	0	0
$351$	−5.90833	−0.315363
$352$	0	0
$353$	−15.2111	−0.809605	−0.404803	−	0.914404i	$-0.632660\pi$
$353$	−15.2111	−0.809605	−0.404803	+	0.914404i	$0.632660\pi$
$354$	0	0
$355$	−22.3305	−1.18518
$356$	0	0
$357$	−1.81665	−0.0961475
$358$	0	0
$359$	9.48612	0.500658	0.250329	−	0.968161i	$-0.419461\pi$
$359$	9.48612	0.500658	0.250329	+	0.968161i	$0.419461\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.302776	−0.0158916
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	7.21110	0.376416	0.188208	−	0.982129i	$-0.439732\pi$
$367$	7.21110	0.376416	0.188208	+	0.982129i	$0.439732\pi$
$368$	0	0
$369$	−11.3667	−0.591726
$370$	0	0
$371$	13.8167	0.717325
$372$	0	0
$373$	7.09167	0.367193	0.183596	−	0.983002i	$-0.441226\pi$
$373$	7.09167	0.367193	0.183596	+	0.983002i	$0.441226\pi$
$374$	0	0
$375$	−3.27502	−0.169121
$376$	0	0
$377$	27.4222	1.41232
$378$	0	0
$379$	13.4861	0.692736	0.346368	−	0.938099i	$-0.387415\pi$
$379$	13.4861	0.692736	0.346368	+	0.938099i	$0.387415\pi$
$380$	0	0
$381$	−1.63331	−0.0836769
$382$	0	0
$383$	−3.48612	−0.178133	−0.0890663	−	0.996026i	$-0.528388\pi$
$383$	−3.48612	−0.178133	−0.0890663	+	0.996026i	$0.528388\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	3.17494	0.161391
$388$	0	0
$389$	34.5416	1.75133	0.875665	−	0.482919i	$-0.160423\pi$
$389$	34.5416	1.75133	0.875665	+	0.482919i	$0.160423\pi$
$390$	0	0
$391$	−6.42221	−0.324785
$392$	0	0
$393$	5.78890	0.292011
$394$	0	0
$395$	36.4222	1.83260
$396$	0	0
$397$	−32.7527	−1.64381	−0.821906	−	0.569623i	$-0.807089\pi$
$397$	−32.7527	−1.64381	−0.821906	+	0.569623i	$0.807089\pi$
$398$	0	0
$399$	−0.394449	−0.0197471
$400$	0	0
$401$	22.6056	1.12887	0.564434	−	0.825478i	$-0.309095\pi$
$401$	22.6056	1.12887	0.564434	+	0.825478i	$0.309095\pi$
$402$	0	0
$403$	−10.9083	−0.543382
$404$	0	0
$405$	−18.8444	−0.936386
$406$	0	0
$407$	−5.21110	−0.258305
$408$	0	0
$409$	23.1472	1.14455	0.572277	−	0.820060i	$-0.306060\pi$
$409$	23.1472	1.14455	0.572277	+	0.820060i	$0.306060\pi$
$410$	0	0
$411$	1.05551	0.0520646
$412$	0	0
$413$	0	0
$414$	0	0
$415$	32.9361	1.61677
$416$	0	0
$417$	−1.90833	−0.0934512
$418$	0	0
$419$	−8.78890	−0.429366	−0.214683	−	0.976684i	$-0.568872\pi$
$419$	−8.78890	−0.429366	−0.214683	+	0.976684i	$0.568872\pi$
$420$	0	0
$421$	−30.2389	−1.47375	−0.736876	−	0.676028i	$-0.763700\pi$
$421$	−30.2389	−1.47375	−0.736876	+	0.676028i	$0.763700\pi$
$422$	0	0
$423$	−17.4500	−0.848446
$424$	0	0
$425$	−1.39445	−0.0676407
$426$	0	0
$427$	−14.6056	−0.706812
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−2.78890	−0.134336	−0.0671682	−	0.997742i	$-0.521396\pi$
$431$	−2.78890	−0.134336	−0.0671682	+	0.997742i	$0.521396\pi$
$432$	0	0
$433$	−24.2389	−1.16485	−0.582423	−	0.812886i	$-0.697895\pi$
$433$	−24.2389	−1.16485	−0.582423	+	0.812886i	$0.697895\pi$
$434$	0	0
$435$	−5.78890	−0.277556
$436$	0	0
$437$	−1.39445	−0.0667055
$438$	0	0
$439$	25.2111	1.20326	0.601630	−	0.798775i	$-0.294518\pi$
$439$	25.2111	1.20326	0.601630	+	0.798775i	$0.294518\pi$
$440$	0	0
$441$	15.4222	0.734391
$442$	0	0
$443$	−39.6333	−1.88304	−0.941518	−	0.336964i	$-0.890600\pi$
$443$	−39.6333	−1.88304	−0.941518	+	0.336964i	$0.890600\pi$
$444$	0	0
$445$	−10.6056	−0.502751
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−31.8167	−1.50152	−0.750760	−	0.660575i	$-0.770313\pi$
$449$	−31.8167	−1.50152	−0.750760	+	0.660575i	$0.770313\pi$
$450$	0	0
$451$	−3.90833	−0.184036
$452$	0	0
$453$	−4.97224	−0.233616
$454$	0	0
$455$	−9.90833	−0.464510
$456$	0	0
$457$	21.8167	1.02054	0.510270	−	0.860014i	$-0.329545\pi$
$457$	21.8167	1.02054	0.510270	+	0.860014i	$0.329545\pi$
$458$	0	0
$459$	−8.23886	−0.384557
$460$	0	0
$461$	6.42221	0.299112	0.149556	−	0.988753i	$-0.452216\pi$
$461$	6.42221	0.299112	0.149556	+	0.988753i	$0.452216\pi$
$462$	0	0
$463$	1.21110	0.0562847	0.0281424	−	0.999604i	$-0.491041\pi$
$463$	1.21110	0.0562847	0.0281424	+	0.999604i	$0.491041\pi$
$464$	0	0
$465$	2.30278	0.106789
$466$	0	0
$467$	22.6056	1.04606	0.523030	−	0.852314i	$-0.324802\pi$
$467$	22.6056	1.04606	0.523030	+	0.852314i	$0.324802\pi$
$468$	0	0
$469$	−8.48612	−0.391853
$470$	0	0
$471$	−0.330532	−0.0152301
$472$	0	0
$473$	1.09167	0.0501952
$474$	0	0
$475$	−0.302776	−0.0138923
$476$	0	0
$477$	30.8444	1.41227
$478$	0	0
$479$	−35.7250	−1.63232	−0.816158	−	0.577829i	$-0.803900\pi$
$479$	−35.7250	−1.63232	−0.816158	+	0.577829i	$0.803900\pi$
$480$	0	0
$481$	−17.2111	−0.784759
$482$	0	0
$483$	0.550039	0.0250276
$484$	0	0
$485$	−18.4222	−0.836509
$486$	0	0
$487$	29.3305	1.32909	0.664547	−	0.747247i	$-0.268625\pi$
$487$	29.3305	1.32909	0.664547	+	0.747247i	$0.268625\pi$
$488$	0	0
$489$	−3.44996	−0.156013
$490$	0	0
$491$	14.9361	0.674056	0.337028	−	0.941495i	$-0.390578\pi$
$491$	14.9361	0.674056	0.337028	+	0.941495i	$0.390578\pi$
$492$	0	0
$493$	38.2389	1.72219
$494$	0	0
$495$	−6.69722	−0.301018
$496$	0	0
$497$	−12.6333	−0.566681
$498$	0	0
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−3.76114	−0.168035
$502$	0	0
$503$	−22.1194	−0.986257	−0.493128	−	0.869957i	$-0.664147\pi$
$503$	−22.1194	−0.986257	−0.493128	+	0.869957i	$0.664147\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	0.633308	0.0281262
$508$	0	0
$509$	−18.4222	−0.816550	−0.408275	−	0.912859i	$-0.633870\pi$
$509$	−18.4222	−0.816550	−0.408275	+	0.912859i	$0.633870\pi$
$510$	0	0
$511$	3.39445	0.150162
$512$	0	0
$513$	−1.78890	−0.0789818
$514$	0	0
$515$	−3.90833	−0.172221
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	5.85281	0.256910
$520$	0	0
$521$	36.4222	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$521$	36.4222	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$522$	0	0
$523$	−19.0278	−0.832026	−0.416013	−	0.909359i	$-0.636573\pi$
$523$	−19.0278	−0.832026	−0.416013	+	0.909359i	$0.636573\pi$
$524$	0	0
$525$	0.119429	0.00521233
$526$	0	0
$527$	−15.2111	−0.662606
$528$	0	0
$529$	−21.0555	−0.915457
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.9083	−0.559122
$534$	0	0
$535$	7.39445	0.319690
$536$	0	0
$537$	−3.97224	−0.171415
$538$	0	0
$539$	5.30278	0.228407
$540$	0	0
$541$	3.81665	0.164091	0.0820454	−	0.996629i	$-0.473855\pi$
$541$	3.81665	0.164091	0.0820454	+	0.996629i	$0.473855\pi$
$542$	0	0
$543$	4.55004	0.195261
$544$	0	0
$545$	−4.60555	−0.197280
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−32.6056	−1.39157
$550$	0	0
$551$	8.30278	0.353710
$552$	0	0
$553$	20.6056	0.876237
$554$	0	0
$555$	3.63331	0.154225
$556$	0	0
$557$	−27.6333	−1.17086	−0.585430	−	0.810723i	$-0.699074\pi$
$557$	−27.6333	−1.17086	−0.585430	+	0.810723i	$0.699074\pi$
$558$	0	0
$559$	3.60555	0.152499
$560$	0	0
$561$	−1.39445	−0.0588737
$562$	0	0
$563$	−21.6333	−0.911735	−0.455868	−	0.890048i	$-0.650671\pi$
$563$	−21.6333	−0.911735	−0.455868	+	0.890048i	$0.650671\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.6611	−0.447723
$568$	0	0
$569$	12.2750	0.514596	0.257298	−	0.966332i	$-0.417168\pi$
$569$	12.2750	0.514596	0.257298	+	0.966332i	$0.417168\pi$
$570$	0	0
$571$	6.93608	0.290266	0.145133	−	0.989412i	$-0.453639\pi$
$571$	6.93608	0.290266	0.145133	+	0.989412i	$0.453639\pi$
$572$	0	0
$573$	−1.81665	−0.0758918
$574$	0	0
$575$	0.422205	0.0176072
$576$	0	0
$577$	−18.0917	−0.753166	−0.376583	−	0.926383i	$-0.622901\pi$
$577$	−18.0917	−0.753166	−0.376583	+	0.926383i	$0.622901\pi$
$578$	0	0
$579$	7.48612	0.311113
$580$	0	0
$581$	18.6333	0.773040
$582$	0	0
$583$	10.6056	0.439237
$584$	0	0
$585$	−22.1194	−0.914526
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−3.30278	−0.136088
$590$	0	0
$591$	−6.55004	−0.269433
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	−13.8167	−0.566428
$596$	0	0
$597$	−0.661064	−0.0270555
$598$	0	0
$599$	19.8806	0.812298	0.406149	−	0.913807i	$-0.366871\pi$
$599$	19.8806	0.812298	0.406149	+	0.913807i	$0.366871\pi$
$600$	0	0
$601$	4.09167	0.166903	0.0834514	−	0.996512i	$-0.473406\pi$
$601$	4.09167	0.166903	0.0834514	+	0.996512i	$0.473406\pi$
$602$	0	0
$603$	−18.9445	−0.771479
$604$	0	0
$605$	−2.30278	−0.0936211
$606$	0	0
$607$	−3.81665	−0.154913	−0.0774566	−	0.996996i	$-0.524680\pi$
$607$	−3.81665	−0.154913	−0.0774566	+	0.996996i	$0.524680\pi$
$608$	0	0
$609$	−3.27502	−0.132710
$610$	0	0
$611$	−19.8167	−0.801696
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	2.72498	0.109882
$616$	0	0
$617$	2.09167	0.0842076	0.0421038	−	0.999113i	$-0.486594\pi$
$617$	2.09167	0.0842076	0.0421038	+	0.999113i	$0.486594\pi$
$618$	0	0
$619$	34.8444	1.40052	0.700258	−	0.713890i	$-0.253068\pi$
$619$	34.8444	1.40052	0.700258	+	0.713890i	$0.253068\pi$
$620$	0	0
$621$	2.49453	0.100102
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	−0.302776	−0.0120917
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	45.0278	1.79253	0.896263	−	0.443522i	$-0.146271\pi$
$631$	45.0278	1.79253	0.896263	+	0.443522i	$0.146271\pi$
$632$	0	0
$633$	7.44996	0.296109
$634$	0	0
$635$	−12.4222	−0.492960
$636$	0	0
$637$	17.5139	0.693925
$638$	0	0
$639$	−28.2027	−1.11568
$640$	0	0
$641$	20.2389	0.799387	0.399693	−	0.916649i	$-0.369117\pi$
$641$	20.2389	0.799387	0.399693	+	0.916649i	$0.369117\pi$
$642$	0	0
$643$	−26.4222	−1.04199	−0.520995	−	0.853560i	$-0.674439\pi$
$643$	−26.4222	−1.04199	−0.520995	+	0.853560i	$0.674439\pi$
$644$	0	0
$645$	−0.761141	−0.0299699
$646$	0	0
$647$	−25.8167	−1.01496	−0.507479	−	0.861664i	$-0.669422\pi$
$647$	−25.8167	−1.01496	−0.507479	+	0.861664i	$0.669422\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.30278	0.0510598
$652$	0	0
$653$	22.3305	0.873861	0.436931	−	0.899495i	$-0.356066\pi$
$653$	22.3305	0.873861	0.436931	+	0.899495i	$0.356066\pi$
$654$	0	0
$655$	44.0278	1.72031
$656$	0	0
$657$	7.57779	0.295638
$658$	0	0
$659$	−29.0278	−1.13076	−0.565380	−	0.824830i	$-0.691271\pi$
$659$	−29.0278	−1.13076	−0.565380	+	0.824830i	$0.691271\pi$
$660$	0	0
$661$	23.6333	0.919229	0.459615	−	0.888118i	$-0.347988\pi$
$661$	23.6333	0.919229	0.459615	+	0.888118i	$0.347988\pi$
$662$	0	0
$663$	−4.60555	−0.178865
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−11.5778	−0.448294
$668$	0	0
$669$	−7.63331	−0.295121
$670$	0	0
$671$	−11.2111	−0.432800
$672$	0	0
$673$	1.30278	0.0502183	0.0251092	−	0.999685i	$-0.492007\pi$
$673$	1.30278	0.0502183	0.0251092	+	0.999685i	$0.492007\pi$
$674$	0	0
$675$	0.541635	0.0208475
$676$	0	0
$677$	−8.30278	−0.319102	−0.159551	−	0.987190i	$-0.551005\pi$
$677$	−8.30278	−0.319102	−0.159551	+	0.987190i	$0.551005\pi$
$678$	0	0
$679$	−10.4222	−0.399968
$680$	0	0
$681$	0.550039	0.0210775
$682$	0	0
$683$	17.5778	0.672596	0.336298	−	0.941756i	$-0.390825\pi$
$683$	17.5778	0.672596	0.336298	+	0.941756i	$0.390825\pi$
$684$	0	0
$685$	8.02776	0.306725
$686$	0	0
$687$	−3.60555	−0.137560
$688$	0	0
$689$	35.0278	1.33445
$690$	0	0
$691$	−32.4222	−1.23340	−0.616699	−	0.787199i	$-0.711531\pi$
$691$	−32.4222	−1.23340	−0.616699	+	0.787199i	$0.711531\pi$
$692$	0	0
$693$	−3.78890	−0.143928
$694$	0	0
$695$	−14.5139	−0.550543
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	−2.78890	−0.105486
$700$	0	0
$701$	−27.6333	−1.04370	−0.521848	−	0.853039i	$-0.674757\pi$
$701$	−27.6333	−1.04370	−0.521848	+	0.853039i	$0.674757\pi$
$702$	0	0
$703$	−5.21110	−0.196540
$704$	0	0
$705$	4.18335	0.157554
$706$	0	0
$707$	−10.1833	−0.382984
$708$	0	0
$709$	14.6972	0.551966	0.275983	−	0.961163i	$-0.410997\pi$
$709$	14.6972	0.551966	0.275983	+	0.961163i	$0.410997\pi$
$710$	0	0
$711$	46.0000	1.72513
$712$	0	0
$713$	4.60555	0.172479
$714$	0	0
$715$	−7.60555	−0.284431
$716$	0	0
$717$	−7.24726	−0.270654
$718$	0	0
$719$	25.8167	0.962799	0.481399	−	0.876501i	$-0.340129\pi$
$719$	25.8167	0.962799	0.481399	+	0.876501i	$0.340129\pi$
$720$	0	0
$721$	−2.21110	−0.0823458
$722$	0	0
$723$	−3.96384	−0.147417
$724$	0	0
$725$	−2.51388	−0.0933631
$726$	0	0
$727$	0.366692	0.0135999	0.00679993	−	0.999977i	$-0.497835\pi$
$727$	0.366692	0.0135999	0.00679993	+	0.999977i	$0.497835\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	5.02776	0.185958
$732$	0	0
$733$	−7.21110	−0.266348	−0.133174	−	0.991093i	$-0.542517\pi$
$733$	−7.21110	−0.266348	−0.133174	+	0.991093i	$0.542517\pi$
$734$	0	0
$735$	−3.69722	−0.136374
$736$	0	0
$737$	−6.51388	−0.239942
$738$	0	0
$739$	40.3583	1.48460	0.742302	−	0.670066i	$-0.233734\pi$
$739$	40.3583	1.48460	0.742302	+	0.670066i	$0.233734\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	−2.36669	−0.0868255	−0.0434128	−	0.999057i	$-0.513823\pi$
$743$	−2.36669	−0.0868255	−0.0434128	+	0.999057i	$0.513823\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	41.5971	1.52196
$748$	0	0
$749$	4.18335	0.152856
$750$	0	0
$751$	40.8444	1.49043	0.745217	−	0.666822i	$-0.232346\pi$
$751$	40.8444	1.49043	0.745217	+	0.666822i	$0.232346\pi$
$752$	0	0
$753$	7.26662	0.264810
$754$	0	0
$755$	−37.8167	−1.37629
$756$	0	0
$757$	30.5416	1.11005	0.555027	−	0.831832i	$-0.312708\pi$
$757$	30.5416	1.11005	0.555027	+	0.831832i	$0.312708\pi$
$758$	0	0
$759$	0.422205	0.0153251
$760$	0	0
$761$	20.2389	0.733658	0.366829	−	0.930288i	$-0.380443\pi$
$761$	20.2389	0.733658	0.366829	+	0.930288i	$0.380443\pi$
$762$	0	0
$763$	−2.60555	−0.0943273
$764$	0	0
$765$	−30.8444	−1.11518
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−37.6333	−1.35709	−0.678546	−	0.734558i	$-0.737390\pi$
$769$	−37.6333	−1.35709	−0.678546	+	0.734558i	$0.737390\pi$
$770$	0	0
$771$	1.39445	0.0502198
$772$	0	0
$773$	48.4222	1.74163	0.870813	−	0.491615i	$-0.163593\pi$
$773$	48.4222	1.74163	0.870813	+	0.491615i	$0.163593\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	2.05551	0.0737411
$778$	0	0
$779$	−3.90833	−0.140030
$780$	0	0
$781$	−9.69722	−0.346994
$782$	0	0
$783$	−14.8528	−0.530796
$784$	0	0
$785$	−2.51388	−0.0897242
$786$	0	0
$787$	15.0278	0.535682	0.267841	−	0.963463i	$-0.413690\pi$
$787$	15.0278	0.535682	0.267841	+	0.963463i	$0.413690\pi$
$788$	0	0
$789$	−6.14719	−0.218846
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−37.0278	−1.31489
$794$	0	0
$795$	−7.39445	−0.262254
$796$	0	0
$797$	16.6056	0.588199	0.294099	−	0.955775i	$-0.404980\pi$
$797$	16.6056	0.588199	0.294099	+	0.955775i	$0.404980\pi$
$798$	0	0
$799$	−27.6333	−0.977596
$800$	0	0
$801$	−13.3944	−0.473270
$802$	0	0
$803$	2.60555	0.0919479
$804$	0	0
$805$	4.18335	0.147444
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	33.2111	1.16764	0.583820	−	0.811883i	$-0.301557\pi$
$809$	33.2111	1.16764	0.583820	+	0.811883i	$0.301557\pi$
$810$	0	0
$811$	−29.6333	−1.04057	−0.520283	−	0.853994i	$-0.674174\pi$
$811$	−29.6333	−1.04057	−0.520283	+	0.853994i	$0.674174\pi$
$812$	0	0
$813$	−1.97224	−0.0691696
$814$	0	0
$815$	−26.2389	−0.919107
$816$	0	0
$817$	1.09167	0.0381928
$818$	0	0
$819$	−12.5139	−0.437270
$820$	0	0
$821$	−34.6056	−1.20774	−0.603871	−	0.797082i	$-0.706376\pi$
$821$	−34.6056	−1.20774	−0.603871	+	0.797082i	$0.706376\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0.0916731	0.00319165
$826$	0	0
$827$	−22.0555	−0.766945	−0.383473	−	0.923552i	$-0.625272\pi$
$827$	−22.0555	−0.766945	−0.383473	+	0.923552i	$0.625272\pi$
$828$	0	0
$829$	−27.5778	−0.957816	−0.478908	−	0.877865i	$-0.658967\pi$
$829$	−27.5778	−0.957816	−0.478908	+	0.877865i	$0.658967\pi$
$830$	0	0
$831$	3.02776	0.105032
$832$	0	0
$833$	24.4222	0.846179
$834$	0	0
$835$	−28.6056	−0.989936
$836$	0	0
$837$	5.90833	0.204222
$838$	0	0
$839$	−19.8806	−0.686354	−0.343177	−	0.939271i	$-0.611503\pi$
$839$	−19.8806	−0.686354	−0.343177	+	0.939271i	$0.611503\pi$
$840$	0	0
$841$	39.9361	1.37711
$842$	0	0
$843$	0.147186	0.00506935
$844$	0	0
$845$	4.81665	0.165698
$846$	0	0
$847$	−1.30278	−0.0447639
$848$	0	0
$849$	−5.73338	−0.196769
$850$	0	0
$851$	7.26662	0.249096
$852$	0	0
$853$	−46.4222	−1.58947	−0.794733	−	0.606959i	$-0.792389\pi$
$853$	−46.4222	−1.58947	−0.794733	+	0.606959i	$0.792389\pi$
$854$	0	0
$855$	−6.69722	−0.229040
$856$	0	0
$857$	−43.5416	−1.48735	−0.743677	−	0.668539i	$-0.766920\pi$
$857$	−43.5416	−1.48735	−0.743677	+	0.668539i	$0.766920\pi$
$858$	0	0
$859$	−8.84441	−0.301767	−0.150884	−	0.988552i	$-0.548212\pi$
$859$	−8.84441	−0.301767	−0.150884	+	0.988552i	$0.548212\pi$
$860$	0	0
$861$	1.54163	0.0525388
$862$	0	0
$863$	17.7250	0.603365	0.301683	−	0.953408i	$-0.402452\pi$
$863$	17.7250	0.603365	0.301683	+	0.953408i	$0.402452\pi$
$864$	0	0
$865$	44.5139	1.51352
$866$	0	0
$867$	−1.27502	−0.0433019
$868$	0	0
$869$	15.8167	0.536543
$870$	0	0
$871$	−21.5139	−0.728970
$872$	0	0
$873$	−23.2666	−0.787456
$874$	0	0
$875$	−14.0917	−0.476385
$876$	0	0
$877$	47.5694	1.60630	0.803152	−	0.595774i	$-0.203155\pi$
$877$	47.5694	1.60630	0.803152	+	0.595774i	$0.203155\pi$
$878$	0	0
$879$	7.73338	0.260841
$880$	0	0
$881$	−33.9083	−1.14240	−0.571200	−	0.820811i	$-0.693522\pi$
$881$	−33.9083	−1.14240	−0.571200	+	0.820811i	$0.693522\pi$
$882$	0	0
$883$	39.0278	1.31339	0.656694	−	0.754157i	$-0.271954\pi$
$883$	39.0278	1.31339	0.656694	+	0.754157i	$0.271954\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.8444	−0.834194	−0.417097	−	0.908862i	$-0.636952\pi$
$887$	−24.8444	−0.834194	−0.417097	+	0.908862i	$0.636952\pi$
$888$	0	0
$889$	−7.02776	−0.235703
$890$	0	0
$891$	−8.18335	−0.274152
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−30.2111	−1.00985
$896$	0	0
$897$	1.39445	0.0465593
$898$	0	0
$899$	−27.4222	−0.914582
$900$	0	0
$901$	48.8444	1.62724
$902$	0	0
$903$	−0.430609	−0.0143298
$904$	0	0
$905$	34.6056	1.15033
$906$	0	0
$907$	4.84441	0.160856	0.0804280	−	0.996760i	$-0.474371\pi$
$907$	4.84441	0.160856	0.0804280	+	0.996760i	$0.474371\pi$
$908$	0	0
$909$	−22.7334	−0.754019
$910$	0	0
$911$	−27.6333	−0.915532	−0.457766	−	0.889073i	$-0.651350\pi$
$911$	−27.6333	−0.915532	−0.457766	+	0.889073i	$0.651350\pi$
$912$	0	0
$913$	14.3028	0.473353
$914$	0	0
$915$	7.81665	0.258411
$916$	0	0
$917$	24.9083	0.822545
$918$	0	0
$919$	−6.11943	−0.201861	−0.100931	−	0.994893i	$-0.532182\pi$
$919$	−6.11943	−0.201861	−0.100931	+	0.994893i	$0.532182\pi$
$920$	0	0
$921$	−0.661064	−0.0217828
$922$	0	0
$923$	−32.0278	−1.05421
$924$	0	0
$925$	1.57779	0.0518776
$926$	0	0
$927$	−4.93608	−0.162122
$928$	0	0
$929$	−8.51388	−0.279331	−0.139666	−	0.990199i	$-0.544603\pi$
$929$	−8.51388	−0.279331	−0.139666	+	0.990199i	$0.544603\pi$
$930$	0	0
$931$	5.30278	0.173791
$932$	0	0
$933$	7.81665	0.255906
$934$	0	0
$935$	−10.6056	−0.346839
$936$	0	0
$937$	1.15559	0.0377515	0.0188757	−	0.999822i	$-0.493991\pi$
$937$	1.15559	0.0377515	0.0188757	+	0.999822i	$0.493991\pi$
$938$	0	0
$939$	−5.48612	−0.179033
$940$	0	0
$941$	−57.6333	−1.87879	−0.939396	−	0.342834i	$-0.888613\pi$
$941$	−57.6333	−1.87879	−0.939396	+	0.342834i	$0.888613\pi$
$942$	0	0
$943$	5.44996	0.177475
$944$	0	0
$945$	5.36669	0.174579
$946$	0	0
$947$	−25.8167	−0.838929	−0.419464	−	0.907772i	$-0.637782\pi$
$947$	−25.8167	−0.838929	−0.419464	+	0.907772i	$0.637782\pi$
$948$	0	0
$949$	8.60555	0.279348
$950$	0	0
$951$	4.18335	0.135654
$952$	0	0
$953$	−21.6333	−0.700772	−0.350386	−	0.936605i	$-0.613950\pi$
$953$	−21.6333	−0.700772	−0.350386	+	0.936605i	$0.613950\pi$
$954$	0	0
$955$	−13.8167	−0.447096
$956$	0	0
$957$	−2.51388	−0.0812621
$958$	0	0
$959$	4.54163	0.146657
$960$	0	0
$961$	−20.0917	−0.648118
$962$	0	0
$963$	9.33894	0.300943
$964$	0	0
$965$	56.9361	1.83284
$966$	0	0
$967$	−23.6333	−0.759996	−0.379998	−	0.924987i	$-0.624075\pi$
$967$	−23.6333	−0.759996	−0.379998	+	0.924987i	$0.624075\pi$
$968$	0	0
$969$	−1.39445	−0.0447961
$970$	0	0
$971$	23.9361	0.768145	0.384073	−	0.923303i	$-0.374521\pi$
$971$	23.9361	0.768145	0.384073	+	0.923303i	$0.374521\pi$
$972$	0	0
$973$	−8.21110	−0.263236
$974$	0	0
$975$	0.302776	0.00969658
$976$	0	0
$977$	27.6333	0.884068	0.442034	−	0.896998i	$-0.354257\pi$
$977$	27.6333	0.884068	0.442034	+	0.896998i	$0.354257\pi$
$978$	0	0
$979$	−4.60555	−0.147194
$980$	0	0
$981$	−5.81665	−0.185711
$982$	0	0
$983$	−23.3028	−0.743243	−0.371622	−	0.928384i	$-0.621198\pi$
$983$	−23.3028	−0.743243	−0.371622	+	0.928384i	$0.621198\pi$
$984$	0	0
$985$	−49.8167	−1.58729
$986$	0	0
$987$	2.36669	0.0753326
$988$	0	0
$989$	−1.52228	−0.0484058
$990$	0	0
$991$	25.9083	0.823005	0.411503	−	0.911409i	$-0.365004\pi$
$991$	25.9083	0.823005	0.411503	+	0.911409i	$0.365004\pi$
$992$	0	0
$993$	6.88057	0.218348
$994$	0	0
$995$	−5.02776	−0.159391
$996$	0	0
$997$	0.183346	0.00580663	0.00290332	−	0.999996i	$-0.499076\pi$
$997$	0.183346	0.00580663	0.00290332	+	0.999996i	$0.499076\pi$
$998$	0	0
$999$	9.32213	0.294939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.o.1.1		2
4.3	odd	2		418.2.a.d.1.2	✓	2
12.11	even	2		3762.2.a.bb.1.2		2
44.43	even	2		4598.2.a.bc.1.2		2
76.75	even	2		7942.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.d.1.2	✓	2	4.3	odd	2
3344.2.a.o.1.1		2	1.1	even	1	trivial
3762.2.a.bb.1.2		2	12.11	even	2
4598.2.a.bc.1.2		2	44.43	even	2
7942.2.a.bb.1.1		2	76.75	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-0.302776 q^{3} -2.30278 q^{5} -1.30278 q^{7} -2.90833 q^{9} +O(q^{10})\)	\(q-0.302776 q^{3} -2.30278 q^{5} -1.30278 q^{7} -2.90833 q^{9} -1.00000 q^{11} -3.30278 q^{13} +0.697224 q^{15} -4.60555 q^{17} -1.00000 q^{19} +0.394449 q^{21} +1.39445 q^{23} +0.302776 q^{25} +1.78890 q^{27} -8.30278 q^{29} +3.30278 q^{31} +0.302776 q^{33} +3.00000 q^{35} +5.21110 q^{37} +1.00000 q^{39} +3.90833 q^{41} -1.09167 q^{43} +6.69722 q^{45} +6.00000 q^{47} -5.30278 q^{49} +1.39445 q^{51} -10.6056 q^{53} +2.30278 q^{55} +0.302776 q^{57} +11.2111 q^{61} +3.78890 q^{63} +7.60555 q^{65} +6.51388 q^{67} -0.422205 q^{69} +9.69722 q^{71} -2.60555 q^{73} -0.0916731 q^{75} +1.30278 q^{77} -15.8167 q^{79} +8.18335 q^{81} -14.3028 q^{83} +10.6056 q^{85} +2.51388 q^{87} +4.60555 q^{89} +4.30278 q^{91} -1.00000 q^{93} +2.30278 q^{95} +8.00000 q^{97} +2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9	\( 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9} - 2 q^{11} - 3 q^{13} + 5 q^{15} - 2 q^{17} - 2 q^{19} + 8 q^{21} + 10 q^{23} - 3 q^{25} + 18 q^{27} - 13 q^{29} + 3 q^{31} - 3 q^{33} + 6 q^{35} - 4 q^{37} + 2 q^{39} - 3 q^{41} - 13 q^{43} + 17 q^{45} + 12 q^{47} - 7 q^{49} + 10 q^{51} - 14 q^{53} + q^{55} - 3 q^{57} + 8 q^{61} + 22 q^{63} + 8 q^{65} - 5 q^{67} + 28 q^{69} + 23 q^{71} + 2 q^{73} - 11 q^{75} - q^{77} - 10 q^{79} + 38 q^{81} - 25 q^{83} + 14 q^{85} - 13 q^{87} + 2 q^{89} + 5 q^{91} - 2 q^{93} + q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9 - 2 * q^11 - 3 * q^13 + 5 * q^15 - 2 * q^17 - 2 * q^19 + 8 * q^21 + 10 * q^23 - 3 * q^25 + 18 * q^27 - 13 * q^29 + 3 * q^31 - 3 * q^33 + 6 * q^35 - 4 * q^37 + 2 * q^39 - 3 * q^41 - 13 * q^43 + 17 * q^45 + 12 * q^47 - 7 * q^49 + 10 * q^51 - 14 * q^53 + q^55 - 3 * q^57 + 8 * q^61 + 22 * q^63 + 8 * q^65 - 5 * q^67 + 28 * q^69 + 23 * q^71 + 2 * q^73 - 11 * q^75 - q^77 - 10 * q^79 + 38 * q^81 - 25 * q^83 + 14 * q^85 - 13 * q^87 + 2 * q^89 + 5 * q^91 - 2 * q^93 + q^95 + 16 * q^97 - 5 * q^99

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