LMFDB - Embedded newform 3969.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34730 q^{2} -0.184793 q^{4} +2.53209 q^{5} +2.94356 q^{8} +O(q^{10})$	$q-1.34730 q^{2} -0.184793 q^{4} +2.53209 q^{5} +2.94356 q^{8} -3.41147 q^{10} -0.467911 q^{11} -5.82295 q^{13} -3.59627 q^{16} +3.87939 q^{17} +2.18479 q^{19} -0.467911 q^{20} +0.630415 q^{22} +0.106067 q^{23} +1.41147 q^{25} +7.84524 q^{26} -8.78106 q^{29} +7.68004 q^{31} -1.04189 q^{32} -5.22668 q^{34} -7.68004 q^{37} -2.94356 q^{38} +7.45336 q^{40} -2.22668 q^{41} +1.22668 q^{43} +0.0864665 q^{44} -0.142903 q^{46} -5.33275 q^{47} -1.90167 q^{50} +1.07604 q^{52} +0.716881 q^{53} -1.18479 q^{55} +11.8307 q^{58} +0.736482 q^{59} -0.958111 q^{61} -10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} -0.716881 q^{68} -13.2344 q^{71} +10.2686 q^{73} +10.3473 q^{74} -0.403733 q^{76} -12.6382 q^{79} -9.10607 q^{80} +3.00000 q^{82} -2.73143 q^{83} +9.82295 q^{85} -1.65270 q^{86} -1.37733 q^{88} -8.11381 q^{89} -0.0196004 q^{92} +7.18479 q^{94} +5.53209 q^{95} +13.6040 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38} + 9 q^{40} - 3 q^{43} - 15 q^{44} + 3 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} - 9 q^{58} - 3 q^{59} - 6 q^{61} - 30 q^{62} + 12 q^{64} - 15 q^{65} - 12 q^{67} + 6 q^{68} - 9 q^{71} + 21 q^{73} + 30 q^{74} - 15 q^{76} - 21 q^{79} - 15 q^{80} + 9 q^{82} - 18 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 12 q^{89} - 3 q^{92} + 18 q^{94} + 12 q^{95} + 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^11 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 6 * q^20 + 9 * q^22 - 12 * q^23 - 6 * q^25 - 3 * q^26 - 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 + 6 * q^38 + 9 * q^40 - 3 * q^43 - 15 * q^44 + 3 * q^47 + 6 * q^50 + 21 * q^52 - 6 * q^53 - 9 * q^58 - 3 * q^59 - 6 * q^61 - 30 * q^62 + 12 * q^64 - 15 * q^65 - 12 * q^67 + 6 * q^68 - 9 * q^71 + 21 * q^73 + 30 * q^74 - 15 * q^76 - 21 * q^79 - 15 * q^80 + 9 * q^82 - 18 * q^83 + 9 * q^85 - 6 * q^86 + 27 * q^88 + 12 * q^89 - 3 * q^92 + 18 * q^94 + 12 * q^95 + 3 * q^97

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 $

$2$	−1.34730	−0.952682	−0.476341	−	0.879261i	$-0.658037\pi$
$2$	−1.34730	−0.952682	−0.476341	+	0.879261i	$0.658037\pi$
$3$	0	0
$3$	0	0
$4$	−0.184793	−0.0923963
$5$	2.53209	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$5$	2.53209	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.94356	1.04071
$9$	0	0
$10$	−3.41147	−1.07880
$11$	−0.467911	−0.141081	−0.0705403	−	0.997509i	$-0.522472\pi$
$11$	−0.467911	−0.141081	−0.0705403	+	0.997509i	$0.522472\pi$
$12$	0	0
$13$	−5.82295	−1.61500	−0.807498	−	0.589871i	$-0.799179\pi$
$13$	−5.82295	−1.61500	−0.807498	+	0.589871i	$0.799179\pi$
$14$	0	0
$15$	0	0
$16$	−3.59627	−0.899067
$17$	3.87939	0.940889	0.470445	−	0.882430i	$-0.344094\pi$
$17$	3.87939	0.940889	0.470445	+	0.882430i	$0.344094\pi$
$18$	0	0
$19$	2.18479	0.501226	0.250613	−	0.968087i	$-0.419368\pi$
$19$	2.18479	0.501226	0.250613	+	0.968087i	$0.419368\pi$
$20$	−0.467911	−0.104628
$21$	0	0
$22$	0.630415	0.134405
$23$	0.106067	0.0221165	0.0110582	−	0.999939i	$-0.496480\pi$
$23$	0.106067	0.0221165	0.0110582	+	0.999939i	$0.496480\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	7.84524	1.53858
$27$	0	0
$28$	0	0
$29$	−8.78106	−1.63060	−0.815301	−	0.579038i	$-0.803428\pi$
$29$	−8.78106	−1.63060	−0.815301	+	0.579038i	$0.803428\pi$
$30$	0	0
$31$	7.68004	1.37938	0.689688	−	0.724106i	$-0.257748\pi$
$31$	7.68004	1.37938	0.689688	+	0.724106i	$0.257748\pi$
$32$	−1.04189	−0.184182
$33$	0	0
$34$	−5.22668	−0.896368
$35$	0	0
$36$	0	0
$37$	−7.68004	−1.26259	−0.631296	−	0.775542i	$-0.717477\pi$
$37$	−7.68004	−1.26259	−0.631296	+	0.775542i	$0.717477\pi$
$38$	−2.94356	−0.477509
$39$	0	0
$40$	7.45336	1.17848
$41$	−2.22668	−0.347749	−0.173875	−	0.984768i	$-0.555629\pi$
$41$	−2.22668	−0.347749	−0.173875	+	0.984768i	$0.555629\pi$
$42$	0	0
$43$	1.22668	0.187067	0.0935336	−	0.995616i	$-0.470184\pi$
$43$	1.22668	0.187067	0.0935336	+	0.995616i	$0.470184\pi$
$44$	0.0864665	0.0130353
$45$	0	0
$46$	−0.142903	−0.0210700
$47$	−5.33275	−0.777861	−0.388931	−	0.921267i	$-0.627155\pi$
$47$	−5.33275	−0.777861	−0.388931	+	0.921267i	$0.627155\pi$
$48$	0	0
$49$	0	0
$50$	−1.90167	−0.268937
$51$	0	0
$52$	1.07604	0.149220
$53$	0.716881	0.0984712	0.0492356	−	0.998787i	$-0.484321\pi$
$53$	0.716881	0.0984712	0.0492356	+	0.998787i	$0.484321\pi$
$54$	0	0
$55$	−1.18479	−0.159757
$56$	0	0
$57$	0	0
$58$	11.8307	1.55345
$59$	0.736482	0.0958818	0.0479409	−	0.998850i	$-0.484734\pi$
$59$	0.736482	0.0958818	0.0479409	+	0.998850i	$0.484734\pi$
$60$	0	0
$61$	−0.958111	−0.122674	−0.0613368	−	0.998117i	$-0.519536\pi$
$61$	−0.958111	−0.122674	−0.0613368	+	0.998117i	$0.519536\pi$
$62$	−10.3473	−1.31411
$63$	0	0
$64$	8.59627	1.07453
$65$	−14.7442	−1.82880
$66$	0	0
$67$	−9.63816	−1.17749	−0.588744	−	0.808320i	$-0.700377\pi$
$67$	−9.63816	−1.17749	−0.588744	+	0.808320i	$0.700377\pi$
$68$	−0.716881	−0.0869346
$69$	0	0
$70$	0	0
$71$	−13.2344	−1.57064	−0.785318	−	0.619092i	$-0.787501\pi$
$71$	−13.2344	−1.57064	−0.785318	+	0.619092i	$0.787501\pi$
$72$	0	0
$73$	10.2686	1.20185	0.600923	−	0.799307i	$-0.294800\pi$
$73$	10.2686	1.20185	0.600923	+	0.799307i	$0.294800\pi$
$74$	10.3473	1.20285
$75$	0	0
$76$	−0.403733	−0.0463114
$77$	0	0
$78$	0	0
$79$	−12.6382	−1.42190	−0.710952	−	0.703241i	$-0.751736\pi$
$79$	−12.6382	−1.42190	−0.710952	+	0.703241i	$0.751736\pi$
$80$	−9.10607	−1.01809
$81$	0	0
$82$	3.00000	0.331295
$83$	−2.73143	−0.299813	−0.149907	−	0.988700i	$-0.547897\pi$
$83$	−2.73143	−0.299813	−0.149907	+	0.988700i	$0.547897\pi$
$84$	0	0
$85$	9.82295	1.06545
$86$	−1.65270	−0.178216
$87$	0	0
$88$	−1.37733	−0.146823
$89$	−8.11381	−0.860062	−0.430031	−	0.902814i	$-0.641497\pi$
$89$	−8.11381	−0.860062	−0.430031	+	0.902814i	$0.641497\pi$
$90$	0	0
$91$	0	0
$92$	−0.0196004	−0.00204348
$93$	0	0
$94$	7.18479	0.741055
$95$	5.53209	0.567580
$96$	0	0
$97$	13.6040	1.38128	0.690639	−	0.723200i	$-0.257329\pi$
$97$	13.6040	1.38128	0.690639	+	0.723200i	$0.257329\pi$
$98$	0	0
$99$	0	0
$100$	−0.260830	−0.0260830
$101$	−9.57398	−0.952646	−0.476323	−	0.879270i	$-0.658031\pi$
$101$	−9.57398	−0.952646	−0.476323	+	0.879270i	$0.658031\pi$
$102$	0	0
$103$	−3.04189	−0.299726	−0.149863	−	0.988707i	$-0.547883\pi$
$103$	−3.04189	−0.299726	−0.149863	+	0.988707i	$0.547883\pi$
$104$	−17.1402	−1.68074
$105$	0	0
$106$	−0.965852	−0.0938118
$107$	6.51754	0.630074	0.315037	−	0.949079i	$-0.397983\pi$
$107$	6.51754	0.630074	0.315037	+	0.949079i	$0.397983\pi$
$108$	0	0
$109$	10.6382	1.01895	0.509475	−	0.860485i	$-0.329840\pi$
$109$	10.6382	1.01895	0.509475	+	0.860485i	$0.329840\pi$
$110$	1.59627	0.152198
$111$	0	0
$112$	0	0
$113$	−5.17705	−0.487016	−0.243508	−	0.969899i	$-0.578298\pi$
$113$	−5.17705	−0.487016	−0.243508	+	0.969899i	$0.578298\pi$
$114$	0	0
$115$	0.268571	0.0250443
$116$	1.62267	0.150662
$117$	0	0
$118$	−0.992259	−0.0913449
$119$	0	0
$120$	0	0
$121$	−10.7811	−0.980096
$122$	1.29086	0.116869
$123$	0	0
$124$	−1.41921	−0.127449
$125$	−9.08647	−0.812718
$126$	0	0
$127$	−8.88207	−0.788157	−0.394078	−	0.919077i	$-0.628936\pi$
$127$	−8.88207	−0.788157	−0.394078	+	0.919077i	$0.628936\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	19.8648	1.74226
$131$	11.3628	0.992771	0.496385	−	0.868102i	$-0.334660\pi$
$131$	11.3628	0.992771	0.496385	+	0.868102i	$0.334660\pi$
$132$	0	0
$133$	0	0
$134$	12.9855	1.12177
$135$	0	0
$136$	11.4192	0.979190
$137$	5.72462	0.489087	0.244544	−	0.969638i	$-0.421362\pi$
$137$	5.72462	0.489087	0.244544	+	0.969638i	$0.421362\pi$
$138$	0	0
$139$	0.923963	0.0783695	0.0391847	−	0.999232i	$-0.487524\pi$
$139$	0.923963	0.0783695	0.0391847	+	0.999232i	$0.487524\pi$
$140$	0	0
$141$	0	0
$142$	17.8307	1.49632
$143$	2.72462	0.227844
$144$	0	0
$145$	−22.2344	−1.84647
$146$	−13.8348	−1.14498
$147$	0	0
$148$	1.41921	0.116659
$149$	−8.72462	−0.714749	−0.357374	−	0.933961i	$-0.616328\pi$
$149$	−8.72462	−0.714749	−0.357374	+	0.933961i	$0.616328\pi$
$150$	0	0
$151$	18.4270	1.49956	0.749782	−	0.661685i	$-0.230158\pi$
$151$	18.4270	1.49956	0.749782	+	0.661685i	$0.230158\pi$
$152$	6.43107	0.521629
$153$	0	0
$154$	0	0
$155$	19.4466	1.56198
$156$	0	0
$157$	−4.92396	−0.392975	−0.196488	−	0.980506i	$-0.562954\pi$
$157$	−4.92396	−0.392975	−0.196488	+	0.980506i	$0.562954\pi$
$158$	17.0273	1.35462
$159$	0	0
$160$	−2.63816	−0.208565
$161$	0	0
$162$	0	0
$163$	7.63816	0.598267	0.299133	−	0.954211i	$-0.403302\pi$
$163$	7.63816	0.598267	0.299133	+	0.954211i	$0.403302\pi$
$164$	0.411474	0.0321307
$165$	0	0
$166$	3.68004	0.285627
$167$	−5.65539	−0.437627	−0.218814	−	0.975767i	$-0.570219\pi$
$167$	−5.65539	−0.437627	−0.218814	+	0.975767i	$0.570219\pi$
$168$	0	0
$169$	20.9067	1.60821
$170$	−13.2344	−1.01503
$171$	0	0
$172$	−0.226682	−0.0172843
$173$	21.0692	1.60186	0.800932	−	0.598755i	$-0.204338\pi$
$173$	21.0692	1.60186	0.800932	+	0.598755i	$0.204338\pi$
$174$	0	0
$175$	0	0
$176$	1.68273	0.126841
$177$	0	0
$178$	10.9317	0.819366
$179$	5.12061	0.382733	0.191366	−	0.981519i	$-0.438708\pi$
$179$	5.12061	0.382733	0.191366	+	0.981519i	$0.438708\pi$
$180$	0	0
$181$	0.319955	0.0237821	0.0118910	−	0.999929i	$-0.496215\pi$
$181$	0.319955	0.0237821	0.0118910	+	0.999929i	$0.496215\pi$
$182$	0	0
$183$	0	0
$184$	0.312214	0.0230168
$185$	−19.4466	−1.42974
$186$	0	0
$187$	−1.81521	−0.132741
$188$	0.985452	0.0718715
$189$	0	0
$190$	−7.45336	−0.540724
$191$	15.5672	1.12640	0.563200	−	0.826320i	$-0.309570\pi$
$191$	15.5672	1.12640	0.563200	+	0.826320i	$0.309570\pi$
$192$	0	0
$193$	6.04189	0.434905	0.217452	−	0.976071i	$-0.430225\pi$
$193$	6.04189	0.434905	0.217452	+	0.976071i	$0.430225\pi$
$194$	−18.3286	−1.31592
$195$	0	0
$196$	0	0
$197$	−25.2344	−1.79788	−0.898939	−	0.438074i	$-0.855661\pi$
$197$	−25.2344	−1.79788	−0.898939	+	0.438074i	$0.855661\pi$
$198$	0	0
$199$	−3.04189	−0.215634	−0.107817	−	0.994171i	$-0.534386\pi$
$199$	−3.04189	−0.215634	−0.107817	+	0.994171i	$0.534386\pi$
$200$	4.15476	0.293786
$201$	0	0
$202$	12.8990	0.907569
$203$	0	0
$204$	0	0
$205$	−5.63816	−0.393786
$206$	4.09833	0.285544
$207$	0	0
$208$	20.9409	1.45199
$209$	−1.02229	−0.0707132
$210$	0	0
$211$	−5.45336	−0.375425	−0.187713	−	0.982224i	$-0.560107\pi$
$211$	−5.45336	−0.375425	−0.187713	+	0.982224i	$0.560107\pi$
$212$	−0.132474	−0.00909837
$213$	0	0
$214$	−8.78106	−0.600261
$215$	3.10607	0.211832
$216$	0	0
$217$	0	0
$218$	−14.3327	−0.970736
$219$	0	0
$220$	0.218941	0.0147610
$221$	−22.5895	−1.51953
$222$	0	0
$223$	−14.1925	−0.950402	−0.475201	−	0.879877i	$-0.657625\pi$
$223$	−14.1925	−0.950402	−0.475201	+	0.879877i	$0.657625\pi$
$224$	0	0
$225$	0	0
$226$	6.97502	0.463972
$227$	−2.89393	−0.192077	−0.0960385	−	0.995378i	$-0.530617\pi$
$227$	−2.89393	−0.192077	−0.0960385	+	0.995378i	$0.530617\pi$
$228$	0	0
$229$	−9.16756	−0.605809	−0.302905	−	0.953021i	$-0.597956\pi$
$229$	−9.16756	−0.605809	−0.302905	+	0.953021i	$0.597956\pi$
$230$	−0.361844	−0.0238593
$231$	0	0
$232$	−25.8476	−1.69698
$233$	−13.2713	−0.869429	−0.434715	−	0.900568i	$-0.643151\pi$
$233$	−13.2713	−0.869429	−0.434715	+	0.900568i	$0.643151\pi$
$234$	0	0
$235$	−13.5030	−0.880838
$236$	−0.136096	−0.00885912
$237$	0	0
$238$	0	0
$239$	−9.53714	−0.616906	−0.308453	−	0.951240i	$-0.599811\pi$
$239$	−9.53714	−0.616906	−0.308453	+	0.951240i	$0.599811\pi$
$240$	0	0
$241$	8.95811	0.577043	0.288521	−	0.957473i	$-0.406836\pi$
$241$	8.95811	0.577043	0.288521	+	0.957473i	$0.406836\pi$
$242$	14.5253	0.933720
$243$	0	0
$244$	0.177052	0.0113346
$245$	0	0
$246$	0	0
$247$	−12.7219	−0.809477
$248$	22.6067	1.43553
$249$	0	0
$250$	12.2422	0.774262
$251$	−24.9982	−1.57788	−0.788938	−	0.614473i	$-0.789369\pi$
$251$	−24.9982	−1.57788	−0.788938	+	0.614473i	$0.789369\pi$
$252$	0	0
$253$	−0.0496299	−0.00312020
$254$	11.9668	0.750863
$255$	0	0
$256$	−4.39599	−0.274750
$257$	10.8520	0.676932	0.338466	−	0.940979i	$-0.390092\pi$
$257$	10.8520	0.676932	0.338466	+	0.940979i	$0.390092\pi$
$258$	0	0
$259$	0	0
$260$	2.72462	0.168974
$261$	0	0
$262$	−15.3090	−0.945795
$263$	−26.0874	−1.60862	−0.804309	−	0.594211i	$-0.797464\pi$
$263$	−26.0874	−1.60862	−0.804309	+	0.594211i	$0.797464\pi$
$264$	0	0
$265$	1.81521	0.111507
$266$	0	0
$267$	0	0
$268$	1.78106	0.108796
$269$	−7.63310	−0.465399	−0.232699	−	0.972549i	$-0.574756\pi$
$269$	−7.63310	−0.465399	−0.232699	+	0.972549i	$0.574756\pi$
$270$	0	0
$271$	−3.40373	−0.206762	−0.103381	−	0.994642i	$-0.532966\pi$
$271$	−3.40373	−0.206762	−0.103381	+	0.994642i	$0.532966\pi$
$272$	−13.9513	−0.845922
$273$	0	0
$274$	−7.71276	−0.465945
$275$	−0.660444	−0.0398263
$276$	0	0
$277$	−5.72193	−0.343798	−0.171899	−	0.985115i	$-0.554990\pi$
$277$	−5.72193	−0.343798	−0.171899	+	0.985115i	$0.554990\pi$
$278$	−1.24485	−0.0746612
$279$	0	0
$280$	0	0
$281$	−28.3773	−1.69285	−0.846425	−	0.532508i	$-0.821249\pi$
$281$	−28.3773	−1.69285	−0.846425	+	0.532508i	$0.821249\pi$
$282$	0	0
$283$	−4.57129	−0.271735	−0.135867	−	0.990727i	$-0.543382\pi$
$283$	−4.57129	−0.271735	−0.135867	+	0.990727i	$0.543382\pi$
$284$	2.44562	0.145121
$285$	0	0
$286$	−3.67087	−0.217063
$287$	0	0
$288$	0	0
$289$	−1.95037	−0.114728
$290$	29.9564	1.75910
$291$	0	0
$292$	−1.89756	−0.111046
$293$	4.32770	0.252827	0.126413	−	0.991978i	$-0.459653\pi$
$293$	4.32770	0.252827	0.126413	+	0.991978i	$0.459653\pi$
$294$	0	0
$295$	1.86484	0.108575
$296$	−22.6067	−1.31399
$297$	0	0
$298$	11.7547	0.680929
$299$	−0.617622	−0.0357180
$300$	0	0
$301$	0	0
$302$	−24.8266	−1.42861
$303$	0	0
$304$	−7.85710	−0.450635
$305$	−2.42602	−0.138914
$306$	0	0
$307$	−12.3773	−0.706411	−0.353206	−	0.935546i	$-0.614908\pi$
$307$	−12.3773	−0.706411	−0.353206	+	0.935546i	$0.614908\pi$
$308$	0	0
$309$	0	0
$310$	−26.2003	−1.48808
$311$	−21.9855	−1.24668	−0.623340	−	0.781951i	$-0.714225\pi$
$311$	−21.9855	−1.24668	−0.623340	+	0.781951i	$0.714225\pi$
$312$	0	0
$313$	13.8898	0.785099	0.392549	−	0.919731i	$-0.371593\pi$
$313$	13.8898	0.785099	0.392549	+	0.919731i	$0.371593\pi$
$314$	6.63404	0.374380
$315$	0	0
$316$	2.33544	0.131379
$317$	6.18210	0.347222	0.173611	−	0.984814i	$-0.444457\pi$
$317$	6.18210	0.347222	0.173611	+	0.984814i	$0.444457\pi$
$318$	0	0
$319$	4.10876	0.230046
$320$	21.7665	1.21678
$321$	0	0
$322$	0	0
$323$	8.47565	0.471598
$324$	0	0
$325$	−8.21894	−0.455905
$326$	−10.2909	−0.569958
$327$	0	0
$328$	−6.55438	−0.361905
$329$	0	0
$330$	0	0
$331$	10.7314	0.589853	0.294926	−	0.955520i	$-0.404705\pi$
$331$	10.7314	0.589853	0.294926	+	0.955520i	$0.404705\pi$
$332$	0.504748	0.0277016
$333$	0	0
$334$	7.61949	0.416920
$335$	−24.4047	−1.33337
$336$	0	0
$337$	−18.5945	−1.01291	−0.506454	−	0.862267i	$-0.669044\pi$
$337$	−18.5945	−1.01291	−0.506454	+	0.862267i	$0.669044\pi$
$338$	−28.1676	−1.53211
$339$	0	0
$340$	−1.81521	−0.0984434
$341$	−3.59358	−0.194603
$342$	0	0
$343$	0	0
$344$	3.61081	0.194682
$345$	0	0
$346$	−28.3865	−1.52607
$347$	20.4124	1.09580	0.547898	−	0.836545i	$-0.315428\pi$
$347$	20.4124	1.09580	0.547898	+	0.836545i	$0.315428\pi$
$348$	0	0
$349$	3.56212	0.190676	0.0953379	−	0.995445i	$-0.469607\pi$
$349$	3.56212	0.190676	0.0953379	+	0.995445i	$0.469607\pi$
$350$	0	0
$351$	0	0
$352$	0.487511	0.0259844
$353$	10.0223	0.533433	0.266716	−	0.963775i	$-0.414061\pi$
$353$	10.0223	0.533433	0.266716	+	0.963775i	$0.414061\pi$
$354$	0	0
$355$	−33.5107	−1.77857
$356$	1.49937	0.0794665
$357$	0	0
$358$	−6.89899	−0.364623
$359$	−9.48070	−0.500372	−0.250186	−	0.968198i	$-0.580492\pi$
$359$	−9.48070	−0.500372	−0.250186	+	0.968198i	$0.580492\pi$
$360$	0	0
$361$	−14.2267	−0.748773
$362$	−0.431074	−0.0226568
$363$	0	0
$364$	0	0
$365$	26.0009	1.36095
$366$	0	0
$367$	−16.1334	−0.842157	−0.421079	−	0.907024i	$-0.638348\pi$
$367$	−16.1334	−0.842157	−0.421079	+	0.907024i	$0.638348\pi$
$368$	−0.381445	−0.0198842
$369$	0	0
$370$	26.2003	1.36209
$371$	0	0
$372$	0	0
$373$	14.0496	0.727462	0.363731	−	0.931504i	$-0.381503\pi$
$373$	14.0496	0.727462	0.363731	+	0.931504i	$0.381503\pi$
$374$	2.44562	0.126460
$375$	0	0
$376$	−15.6973	−0.809525
$377$	51.1317	2.63341
$378$	0	0
$379$	16.0574	0.824812	0.412406	−	0.911000i	$-0.364689\pi$
$379$	16.0574	0.824812	0.412406	+	0.911000i	$0.364689\pi$
$380$	−1.02229	−0.0524423
$381$	0	0
$382$	−20.9736	−1.07310
$383$	−32.0205	−1.63617	−0.818086	−	0.575095i	$-0.804965\pi$
$383$	−32.0205	−1.63617	−0.818086	+	0.575095i	$0.804965\pi$
$384$	0	0
$385$	0	0
$386$	−8.14022	−0.414326
$387$	0	0
$388$	−2.51392	−0.127625
$389$	30.0428	1.52323	0.761616	−	0.648029i	$-0.224406\pi$
$389$	30.0428	1.52323	0.761616	+	0.648029i	$0.224406\pi$
$390$	0	0
$391$	0.411474	0.0208091
$392$	0	0
$393$	0	0
$394$	33.9982	1.71281
$395$	−32.0009	−1.61014
$396$	0	0
$397$	12.3200	0.618321	0.309160	−	0.951010i	$-0.399952\pi$
$397$	12.3200	0.618321	0.309160	+	0.951010i	$0.399952\pi$
$398$	4.09833	0.205431
$399$	0	0
$400$	−5.07604	−0.253802
$401$	−20.9760	−1.04749	−0.523745	−	0.851875i	$-0.675465\pi$
$401$	−20.9760	−1.04749	−0.523745	+	0.851875i	$0.675465\pi$
$402$	0	0
$403$	−44.7205	−2.22769
$404$	1.76920	0.0880210
$405$	0	0
$406$	0	0
$407$	3.59358	0.178127
$408$	0	0
$409$	−25.6614	−1.26887	−0.634437	−	0.772975i	$-0.718768\pi$
$409$	−25.6614	−1.26887	−0.634437	+	0.772975i	$0.718768\pi$
$410$	7.59627	0.375153
$411$	0	0
$412$	0.562118	0.0276936
$413$	0	0
$414$	0	0
$415$	−6.91622	−0.339504
$416$	6.06687	0.297453
$417$	0	0
$418$	1.37733	0.0673672
$419$	−1.47977	−0.0722915	−0.0361458	−	0.999347i	$-0.511508\pi$
$419$	−1.47977	−0.0722915	−0.0361458	+	0.999347i	$0.511508\pi$
$420$	0	0
$421$	13.1070	0.638796	0.319398	−	0.947621i	$-0.396519\pi$
$421$	13.1070	0.638796	0.319398	+	0.947621i	$0.396519\pi$
$422$	7.34730	0.357661
$423$	0	0
$424$	2.11019	0.102480
$425$	5.47565	0.265608
$426$	0	0
$427$	0	0
$428$	−1.20439	−0.0582165
$429$	0	0
$430$	−4.18479	−0.201809
$431$	−17.7270	−0.853879	−0.426939	−	0.904280i	$-0.640408\pi$
$431$	−17.7270	−0.853879	−0.426939	+	0.904280i	$0.640408\pi$
$432$	0	0
$433$	5.83843	0.280577	0.140289	−	0.990111i	$-0.455197\pi$
$433$	5.83843	0.280577	0.140289	+	0.990111i	$0.455197\pi$
$434$	0	0
$435$	0	0
$436$	−1.96585	−0.0941472
$437$	0.231734	0.0110853
$438$	0	0
$439$	−29.8553	−1.42492	−0.712459	−	0.701714i	$-0.752418\pi$
$439$	−29.8553	−1.42492	−0.712459	+	0.701714i	$0.752418\pi$
$440$	−3.48751	−0.166261
$441$	0	0
$442$	30.4347	1.44763
$443$	−10.6655	−0.506733	−0.253367	−	0.967370i	$-0.581538\pi$
$443$	−10.6655	−0.506733	−0.253367	+	0.967370i	$0.581538\pi$
$444$	0	0
$445$	−20.5449	−0.973921
$446$	19.1215	0.905432
$447$	0	0
$448$	0	0
$449$	−3.55438	−0.167741	−0.0838707	−	0.996477i	$-0.526728\pi$
$449$	−3.55438	−0.167741	−0.0838707	+	0.996477i	$0.526728\pi$
$450$	0	0
$451$	1.04189	0.0490606
$452$	0.956680	0.0449985
$453$	0	0
$454$	3.89899	0.182988
$455$	0	0
$456$	0	0
$457$	5.02322	0.234976	0.117488	−	0.993074i	$-0.462516\pi$
$457$	5.02322	0.234976	0.117488	+	0.993074i	$0.462516\pi$
$458$	12.3514	0.577144
$459$	0	0
$460$	−0.0496299	−0.00231400
$461$	18.4611	0.859819	0.429910	−	0.902872i	$-0.358545\pi$
$461$	18.4611	0.859819	0.429910	+	0.902872i	$0.358545\pi$
$462$	0	0
$463$	−14.2344	−0.661530	−0.330765	−	0.943713i	$-0.607307\pi$
$463$	−14.2344	−0.661530	−0.330765	+	0.943713i	$0.607307\pi$
$464$	31.5790	1.46602
$465$	0	0
$466$	17.8803	0.828290
$467$	−3.36865	−0.155883	−0.0779413	−	0.996958i	$-0.524835\pi$
$467$	−3.36865	−0.155883	−0.0779413	+	0.996958i	$0.524835\pi$
$468$	0	0
$469$	0	0
$470$	18.1925	0.839159
$471$	0	0
$472$	2.16788	0.0997848
$473$	−0.573978	−0.0263915
$474$	0	0
$475$	3.08378	0.141493
$476$	0	0
$477$	0	0
$478$	12.8494	0.587716
$479$	−36.7665	−1.67990	−0.839952	−	0.542660i	$-0.817417\pi$
$479$	−36.7665	−1.67990	−0.839952	+	0.542660i	$0.817417\pi$
$480$	0	0
$481$	44.7205	2.03908
$482$	−12.0692	−0.549738
$483$	0	0
$484$	1.99226	0.0905572
$485$	34.4466	1.56414
$486$	0	0
$487$	−37.4175	−1.69555	−0.847773	−	0.530358i	$-0.822057\pi$
$487$	−37.4175	−1.69555	−0.847773	+	0.530358i	$0.822057\pi$
$488$	−2.82026	−0.127667
$489$	0	0
$490$	0	0
$491$	26.6705	1.20363	0.601813	−	0.798637i	$-0.294445\pi$
$491$	26.6705	1.20363	0.601813	+	0.798637i	$0.294445\pi$
$492$	0	0
$493$	−34.0651	−1.53422
$494$	17.1402	0.771175
$495$	0	0
$496$	−27.6195	−1.24015
$497$	0	0
$498$	0	0
$499$	33.7452	1.51064	0.755320	−	0.655356i	$-0.227481\pi$
$499$	33.7452	1.51064	0.755320	+	0.655356i	$0.227481\pi$
$500$	1.67911	0.0750921
$501$	0	0
$502$	33.6800	1.50321
$503$	−32.0401	−1.42860	−0.714299	−	0.699840i	$-0.753255\pi$
$503$	−32.0401	−1.42860	−0.714299	+	0.699840i	$0.753255\pi$
$504$	0	0
$505$	−24.2422	−1.07876
$506$	0.0668661	0.00297256
$507$	0	0
$508$	1.64134	0.0728227
$509$	−7.93851	−0.351868	−0.175934	−	0.984402i	$-0.556295\pi$
$509$	−7.93851	−0.351868	−0.175934	+	0.984402i	$0.556295\pi$
$510$	0	0
$511$	0	0
$512$	24.9186	1.10126
$513$	0	0
$514$	−14.6209	−0.644901
$515$	−7.70233	−0.339405
$516$	0	0
$517$	2.49525	0.109741
$518$	0	0
$519$	0	0
$520$	−43.4005	−1.90324
$521$	−14.6750	−0.642923	−0.321462	−	0.946923i	$-0.604174\pi$
$521$	−14.6750	−0.642923	−0.321462	+	0.946923i	$0.604174\pi$
$522$	0	0
$523$	−28.3432	−1.23936	−0.619680	−	0.784854i	$-0.712738\pi$
$523$	−28.3432	−1.23936	−0.619680	+	0.784854i	$0.712738\pi$
$524$	−2.09976	−0.0917283
$525$	0	0
$526$	35.1475	1.53250
$527$	29.7939	1.29784
$528$	0	0
$529$	−22.9887	−0.999511
$530$	−2.44562	−0.106231
$531$	0	0
$532$	0	0
$533$	12.9659	0.561613
$534$	0	0
$535$	16.5030	0.713487
$536$	−28.3705	−1.22542
$537$	0	0
$538$	10.2841	0.443377
$539$	0	0
$540$	0	0
$541$	11.2858	0.485215	0.242607	−	0.970125i	$-0.421997\pi$
$541$	11.2858	0.485215	0.242607	+	0.970125i	$0.421997\pi$
$542$	4.58584	0.196979
$543$	0	0
$544$	−4.04189	−0.173295
$545$	26.9368	1.15384
$546$	0	0
$547$	−29.2404	−1.25023	−0.625115	−	0.780533i	$-0.714948\pi$
$547$	−29.2404	−1.25023	−0.625115	+	0.780533i	$0.714948\pi$
$548$	−1.05787	−0.0451899
$549$	0	0
$550$	0.889814	0.0379418
$551$	−19.1848	−0.817300
$552$	0	0
$553$	0	0
$554$	7.70914	0.327530
$555$	0	0
$556$	−0.170741	−0.00724105
$557$	0.775682	0.0328667	0.0164334	−	0.999865i	$-0.494769\pi$
$557$	0.775682	0.0328667	0.0164334	+	0.999865i	$0.494769\pi$
$558$	0	0
$559$	−7.14290	−0.302113
$560$	0	0
$561$	0	0
$562$	38.2327	1.61275
$563$	24.9522	1.05161	0.525806	−	0.850605i	$-0.323764\pi$
$563$	24.9522	1.05161	0.525806	+	0.850605i	$0.323764\pi$
$564$	0	0
$565$	−13.1088	−0.551489
$566$	6.15888	0.258877
$567$	0	0
$568$	−38.9564	−1.63457
$569$	24.8033	1.03981	0.519905	−	0.854224i	$-0.325967\pi$
$569$	24.8033	1.03981	0.519905	+	0.854224i	$0.325967\pi$
$570$	0	0
$571$	8.79654	0.368124	0.184062	−	0.982915i	$-0.441075\pi$
$571$	8.79654	0.368124	0.184062	+	0.982915i	$0.441075\pi$
$572$	−0.503490	−0.0210520
$573$	0	0
$574$	0	0
$575$	0.149711	0.00624336
$576$	0	0
$577$	12.8743	0.535965	0.267983	−	0.963424i	$-0.413643\pi$
$577$	12.8743	0.535965	0.267983	+	0.963424i	$0.413643\pi$
$578$	2.62773	0.109299
$579$	0	0
$580$	4.10876	0.170607
$581$	0	0
$582$	0	0
$583$	−0.335437	−0.0138924
$584$	30.2262	1.25077
$585$	0	0
$586$	−5.83069	−0.240864
$587$	44.8631	1.85170	0.925849	−	0.377894i	$-0.123352\pi$
$587$	44.8631	1.85170	0.925849	+	0.377894i	$0.123352\pi$
$588$	0	0
$589$	16.7793	0.691379
$590$	−2.51249	−0.103438
$591$	0	0
$592$	27.6195	1.13515
$593$	3.76053	0.154426	0.0772131	−	0.997015i	$-0.475398\pi$
$593$	3.76053	0.154426	0.0772131	+	0.997015i	$0.475398\pi$
$594$	0	0
$595$	0	0
$596$	1.61225	0.0660401
$597$	0	0
$598$	0.832119	0.0340279
$599$	3.69047	0.150789	0.0753943	−	0.997154i	$-0.475978\pi$
$599$	3.69047	0.150789	0.0753943	+	0.997154i	$0.475978\pi$
$600$	0	0
$601$	21.8571	0.891570	0.445785	−	0.895140i	$-0.352925\pi$
$601$	21.8571	0.891570	0.445785	+	0.895140i	$0.352925\pi$
$602$	0	0
$603$	0	0
$604$	−3.40516	−0.138554
$605$	−27.2986	−1.10985
$606$	0	0
$607$	−24.3946	−0.990145	−0.495072	−	0.868852i	$-0.664858\pi$
$607$	−24.3946	−0.990145	−0.495072	+	0.868852i	$0.664858\pi$
$608$	−2.27631	−0.0923166
$609$	0	0
$610$	3.26857	0.132341
$611$	31.0523	1.25624
$612$	0	0
$613$	42.0215	1.69723	0.848616	−	0.529010i	$-0.177437\pi$
$613$	42.0215	1.69723	0.848616	+	0.529010i	$0.177437\pi$
$614$	16.6759	0.672986
$615$	0	0
$616$	0	0
$617$	−46.4097	−1.86838	−0.934192	−	0.356769i	$-0.883878\pi$
$617$	−46.4097	−1.86838	−0.934192	+	0.356769i	$0.883878\pi$
$618$	0	0
$619$	27.2094	1.09364	0.546820	−	0.837250i	$-0.315838\pi$
$619$	27.2094	1.09364	0.546820	+	0.837250i	$0.315838\pi$
$620$	−3.59358	−0.144322
$621$	0	0
$622$	29.6209	1.18769
$623$	0	0
$624$	0	0
$625$	−30.0651	−1.20260
$626$	−18.7137	−0.747950
$627$	0	0
$628$	0.909912	0.0363094
$629$	−29.7939	−1.18796
$630$	0	0
$631$	−29.6023	−1.17845	−0.589224	−	0.807970i	$-0.700566\pi$
$631$	−29.6023	−1.17845	−0.589224	+	0.807970i	$0.700566\pi$
$632$	−37.2012	−1.47978
$633$	0	0
$634$	−8.32913	−0.330792
$635$	−22.4902	−0.892496
$636$	0	0
$637$	0	0
$638$	−5.53571	−0.219161
$639$	0	0
$640$	−24.0496	−0.950645
$641$	0.279000	0.0110198	0.00550991	−	0.999985i	$-0.498246\pi$
$641$	0.279000	0.0110198	0.00550991	+	0.999985i	$0.498246\pi$
$642$	0	0
$643$	18.2439	0.719470	0.359735	−	0.933055i	$-0.382867\pi$
$643$	18.2439	0.719470	0.359735	+	0.933055i	$0.382867\pi$
$644$	0	0
$645$	0	0
$646$	−11.4192	−0.449283
$647$	22.4570	0.882875	0.441438	−	0.897292i	$-0.354469\pi$
$647$	22.4570	0.882875	0.441438	+	0.897292i	$0.354469\pi$
$648$	0	0
$649$	−0.344608	−0.0135270
$650$	11.0733	0.434332
$651$	0	0
$652$	−1.41147	−0.0552776
$653$	50.5313	1.97744	0.988721	−	0.149771i	$-0.0478538\pi$
$653$	50.5313	1.97744	0.988721	+	0.149771i	$0.0478538\pi$
$654$	0	0
$655$	28.7716	1.12420
$656$	8.00774	0.312650
$657$	0	0
$658$	0	0
$659$	2.67263	0.104111	0.0520554	−	0.998644i	$-0.483423\pi$
$659$	2.67263	0.104111	0.0520554	+	0.998644i	$0.483423\pi$
$660$	0	0
$661$	34.6100	1.34617	0.673086	−	0.739564i	$-0.264968\pi$
$661$	34.6100	1.34617	0.673086	+	0.739564i	$0.264968\pi$
$662$	−14.4584	−0.561942
$663$	0	0
$664$	−8.04013	−0.312018
$665$	0	0
$666$	0	0
$667$	−0.931379	−0.0360631
$668$	1.04507	0.0404351
$669$	0	0
$670$	32.8803	1.27028
$671$	0.448311	0.0173068
$672$	0	0
$673$	16.5125	0.636510	0.318255	−	0.948005i	$-0.396903\pi$
$673$	16.5125	0.636510	0.318255	+	0.948005i	$0.396903\pi$
$674$	25.0523	0.964979
$675$	0	0
$676$	−3.86341	−0.148593
$677$	−43.7579	−1.68175	−0.840877	−	0.541226i	$-0.817960\pi$
$677$	−43.7579	−1.68175	−0.840877	+	0.541226i	$0.817960\pi$
$678$	0	0
$679$	0	0
$680$	28.9145	1.10882
$681$	0	0
$682$	4.84161	0.185395
$683$	−28.2412	−1.08062	−0.540310	−	0.841466i	$-0.681693\pi$
$683$	−28.2412	−1.08062	−0.540310	+	0.841466i	$0.681693\pi$
$684$	0	0
$685$	14.4953	0.553835
$686$	0	0
$687$	0	0
$688$	−4.41147	−0.168186
$689$	−4.17436	−0.159031
$690$	0	0
$691$	29.0651	1.10569	0.552844	−	0.833284i	$-0.313542\pi$
$691$	29.0651	1.10569	0.552844	+	0.833284i	$0.313542\pi$
$692$	−3.89344	−0.148006
$693$	0	0
$694$	−27.5016	−1.04395
$695$	2.33956	0.0887444
$696$	0	0
$697$	−8.63816	−0.327193
$698$	−4.79923	−0.181654
$699$	0	0
$700$	0	0
$701$	1.10876	0.0418771	0.0209386	−	0.999781i	$-0.493335\pi$
$701$	1.10876	0.0418771	0.0209386	+	0.999781i	$0.493335\pi$
$702$	0	0
$703$	−16.7793	−0.632843
$704$	−4.02229	−0.151596
$705$	0	0
$706$	−13.5030	−0.508192
$707$	0	0
$708$	0	0
$709$	−18.4688	−0.693612	−0.346806	−	0.937937i	$-0.612734\pi$
$709$	−18.4688	−0.693612	−0.346806	+	0.937937i	$0.612734\pi$
$710$	45.1489	1.69441
$711$	0	0
$712$	−23.8835	−0.895072
$713$	0.814598	0.0305069
$714$	0	0
$715$	6.89899	0.258007
$716$	−0.946251	−0.0353631
$717$	0	0
$718$	12.7733	0.476696
$719$	−33.7769	−1.25967	−0.629834	−	0.776730i	$-0.716877\pi$
$719$	−33.7769	−1.25967	−0.629834	+	0.776730i	$0.716877\pi$
$720$	0	0
$721$	0	0
$722$	19.1676	0.713343
$723$	0	0
$724$	−0.0591253	−0.00219738
$725$	−12.3942	−0.460310
$726$	0	0
$727$	−16.8043	−0.623236	−0.311618	−	0.950207i	$-0.600871\pi$
$727$	−16.8043	−0.623236	−0.311618	+	0.950207i	$0.600871\pi$
$728$	0	0
$729$	0	0
$730$	−35.0310	−1.29655
$731$	4.75877	0.176009
$732$	0	0
$733$	13.6364	0.503672	0.251836	−	0.967770i	$-0.418966\pi$
$733$	13.6364	0.503672	0.251836	+	0.967770i	$0.418966\pi$
$734$	21.7365	0.802308
$735$	0	0
$736$	−0.110510	−0.00407345
$737$	4.50980	0.166121
$738$	0	0
$739$	−32.0419	−1.17868	−0.589340	−	0.807885i	$-0.700612\pi$
$739$	−32.0419	−1.17868	−0.589340	+	0.807885i	$0.700612\pi$
$740$	3.59358	0.132103
$741$	0	0
$742$	0	0
$743$	−33.7529	−1.23827	−0.619137	−	0.785283i	$-0.712517\pi$
$743$	−33.7529	−1.23827	−0.619137	+	0.785283i	$0.712517\pi$
$744$	0	0
$745$	−22.0915	−0.809371
$746$	−18.9290	−0.693040
$747$	0	0
$748$	0.335437	0.0122648
$749$	0	0
$750$	0	0
$751$	26.1165	0.953004	0.476502	−	0.879173i	$-0.341904\pi$
$751$	26.1165	0.953004	0.476502	+	0.879173i	$0.341904\pi$
$752$	19.1780	0.699349
$753$	0	0
$754$	−68.8895	−2.50881
$755$	46.6587	1.69808
$756$	0	0
$757$	35.6536	1.29585	0.647927	−	0.761703i	$-0.275636\pi$
$757$	35.6536	1.29585	0.647927	+	0.761703i	$0.275636\pi$
$758$	−21.6340	−0.785784
$759$	0	0
$760$	16.2841	0.590685
$761$	40.7648	1.47772	0.738861	−	0.673858i	$-0.235364\pi$
$761$	40.7648	1.47772	0.738861	+	0.673858i	$0.235364\pi$
$762$	0	0
$763$	0	0
$764$	−2.87670	−0.104075
$765$	0	0
$766$	43.1411	1.55875
$767$	−4.28850	−0.154849
$768$	0	0
$769$	−39.4270	−1.42177	−0.710886	−	0.703307i	$-0.751706\pi$
$769$	−39.4270	−1.42177	−0.710886	+	0.703307i	$0.751706\pi$
$770$	0	0
$771$	0	0
$772$	−1.11650	−0.0401836
$773$	24.9026	0.895685	0.447842	−	0.894113i	$-0.352193\pi$
$773$	24.9026	0.895685	0.447842	+	0.894113i	$0.352193\pi$
$774$	0	0
$775$	10.8402	0.389391
$776$	40.0443	1.43750
$777$	0	0
$778$	−40.4766	−1.45116
$779$	−4.86484	−0.174301
$780$	0	0
$781$	6.19253	0.221586
$782$	−0.554378	−0.0198245
$783$	0	0
$784$	0	0
$785$	−12.4679	−0.444999
$786$	0	0
$787$	30.7050	1.09452	0.547258	−	0.836964i	$-0.315672\pi$
$787$	30.7050	1.09452	0.547258	+	0.836964i	$0.315672\pi$
$788$	4.66313	0.166117
$789$	0	0
$790$	43.1147	1.53395
$791$	0	0
$792$	0	0
$793$	5.57903	0.198117
$794$	−16.5986	−0.589063
$795$	0	0
$796$	0.562118	0.0199238
$797$	−11.0137	−0.390126	−0.195063	−	0.980791i	$-0.562491\pi$
$797$	−11.0137	−0.390126	−0.195063	+	0.980791i	$0.562491\pi$
$798$	0	0
$799$	−20.6878	−0.731881
$800$	−1.47060	−0.0519935
$801$	0	0
$802$	28.2608	0.997925
$803$	−4.80478	−0.169557
$804$	0	0
$805$	0	0
$806$	60.2518	2.12228
$807$	0	0
$808$	−28.1816	−0.991425
$809$	−16.9881	−0.597271	−0.298636	−	0.954367i	$-0.596532\pi$
$809$	−16.9881	−0.597271	−0.298636	+	0.954367i	$0.596532\pi$
$810$	0	0
$811$	−37.9796	−1.33364	−0.666822	−	0.745217i	$-0.732346\pi$
$811$	−37.9796	−1.33364	−0.666822	+	0.745217i	$0.732346\pi$
$812$	0	0
$813$	0	0
$814$	−4.84161	−0.169699
$815$	19.3405	0.677468
$816$	0	0
$817$	2.68004	0.0937629
$818$	34.5735	1.20883
$819$	0	0
$820$	1.04189	0.0363843
$821$	−8.27868	−0.288928	−0.144464	−	0.989510i	$-0.546146\pi$
$821$	−8.27868	−0.288928	−0.144464	+	0.989510i	$0.546146\pi$
$822$	0	0
$823$	54.5526	1.90158	0.950792	−	0.309829i	$-0.100272\pi$
$823$	54.5526	1.90158	0.950792	+	0.309829i	$0.100272\pi$
$824$	−8.95399	−0.311927
$825$	0	0
$826$	0	0
$827$	31.8708	1.10826	0.554129	−	0.832431i	$-0.313052\pi$
$827$	31.8708	1.10826	0.554129	+	0.832431i	$0.313052\pi$
$828$	0	0
$829$	0.352349	0.0122376	0.00611879	−	0.999981i	$-0.498052\pi$
$829$	0.352349	0.0122376	0.00611879	+	0.999981i	$0.498052\pi$
$830$	9.31820	0.323439
$831$	0	0
$832$	−50.0556	−1.73537
$833$	0	0
$834$	0	0
$835$	−14.3200	−0.495562
$836$	0.188911	0.00653363
$837$	0	0
$838$	1.99369	0.0688709
$839$	25.0155	0.863630	0.431815	−	0.901962i	$-0.357873\pi$
$839$	25.0155	0.863630	0.431815	+	0.901962i	$0.357873\pi$
$840$	0	0
$841$	48.1070	1.65886
$842$	−17.6590	−0.608570
$843$	0	0
$844$	1.00774	0.0346879
$845$	52.9377	1.82111
$846$	0	0
$847$	0	0
$848$	−2.57810	−0.0885322
$849$	0	0
$850$	−7.37733	−0.253040
$851$	−0.814598	−0.0279241
$852$	0	0
$853$	−39.1908	−1.34187	−0.670933	−	0.741518i	$-0.734106\pi$
$853$	−39.1908	−1.34187	−0.670933	+	0.741518i	$0.734106\pi$
$854$	0	0
$855$	0	0
$856$	19.1848	0.655723
$857$	16.4074	0.560465	0.280232	−	0.959932i	$-0.409589\pi$
$857$	16.4074	0.560465	0.280232	+	0.959932i	$0.409589\pi$
$858$	0	0
$859$	26.8324	0.915511	0.457756	−	0.889078i	$-0.348653\pi$
$859$	26.8324	0.915511	0.457756	+	0.889078i	$0.348653\pi$
$860$	−0.573978	−0.0195725
$861$	0	0
$862$	23.8835	0.813475
$863$	−14.5057	−0.493779	−0.246890	−	0.969044i	$-0.579408\pi$
$863$	−14.5057	−0.493779	−0.246890	+	0.969044i	$0.579408\pi$
$864$	0	0
$865$	53.3492	1.81393
$866$	−7.86610	−0.267301
$867$	0	0
$868$	0	0
$869$	5.91353	0.200603
$870$	0	0
$871$	56.1225	1.90164
$872$	31.3141	1.06043
$873$	0	0
$874$	−0.312214	−0.0105608
$875$	0	0
$876$	0	0
$877$	18.9145	0.638696	0.319348	−	0.947637i	$-0.396536\pi$
$877$	18.9145	0.638696	0.319348	+	0.947637i	$0.396536\pi$
$878$	40.2240	1.35749
$879$	0	0
$880$	4.26083	0.143633
$881$	53.8976	1.81585	0.907927	−	0.419128i	$-0.137664\pi$
$881$	53.8976	1.81585	0.907927	+	0.419128i	$0.137664\pi$
$882$	0	0
$883$	43.4252	1.46137	0.730687	−	0.682712i	$-0.239200\pi$
$883$	43.4252	1.46137	0.730687	+	0.682712i	$0.239200\pi$
$884$	4.17436	0.140399
$885$	0	0
$886$	14.3696	0.482756
$887$	−38.9600	−1.30815	−0.654074	−	0.756431i	$-0.726942\pi$
$887$	−38.9600	−1.30815	−0.654074	+	0.756431i	$0.726942\pi$
$888$	0	0
$889$	0	0
$890$	27.6800	0.927837
$891$	0	0
$892$	2.62267	0.0878136
$893$	−11.6509	−0.389884
$894$	0	0
$895$	12.9659	0.433401
$896$	0	0
$897$	0	0
$898$	4.78880	0.159804
$899$	−67.4389	−2.24921
$900$	0	0
$901$	2.78106	0.0926505
$902$	−1.40373	−0.0467392
$903$	0	0
$904$	−15.2390	−0.506841
$905$	0.810155	0.0269305
$906$	0	0
$907$	34.5276	1.14647	0.573236	−	0.819390i	$-0.305688\pi$
$907$	34.5276	1.14647	0.573236	+	0.819390i	$0.305688\pi$
$908$	0.534777	0.0177472
$909$	0	0
$910$	0	0
$911$	−46.5262	−1.54148	−0.770741	−	0.637148i	$-0.780114\pi$
$911$	−46.5262	−1.54148	−0.770741	+	0.637148i	$0.780114\pi$
$912$	0	0
$913$	1.27807	0.0422978
$914$	−6.76777	−0.223858
$915$	0	0
$916$	1.69410	0.0559745
$917$	0	0
$918$	0	0
$919$	−9.95636	−0.328430	−0.164215	−	0.986425i	$-0.552509\pi$
$919$	−9.95636	−0.328430	−0.164215	+	0.986425i	$0.552509\pi$
$920$	0.790555	0.0260638
$921$	0	0
$922$	−24.8726	−0.819135
$923$	77.0634	2.53657
$924$	0	0
$925$	−10.8402	−0.356423
$926$	19.1780	0.630228
$927$	0	0
$928$	9.14889	0.300327
$929$	9.04601	0.296790	0.148395	−	0.988928i	$-0.452589\pi$
$929$	9.04601	0.296790	0.148395	+	0.988928i	$0.452589\pi$
$930$	0	0
$931$	0	0
$932$	2.45243	0.0803320
$933$	0	0
$934$	4.53857	0.148507
$935$	−4.59627	−0.150314
$936$	0	0
$937$	24.3928	0.796878	0.398439	−	0.917195i	$-0.369552\pi$
$937$	24.3928	0.796878	0.398439	+	0.917195i	$0.369552\pi$
$938$	0	0
$939$	0	0
$940$	2.49525	0.0813862
$941$	−59.5381	−1.94089	−0.970443	−	0.241331i	$-0.922416\pi$
$941$	−59.5381	−1.94089	−0.970443	+	0.241331i	$0.922416\pi$
$942$	0	0
$943$	−0.236177	−0.00769098
$944$	−2.64858	−0.0862041
$945$	0	0
$946$	0.773318	0.0251428
$947$	−8.64858	−0.281041	−0.140521	−	0.990078i	$-0.544878\pi$
$947$	−8.64858	−0.281041	−0.140521	+	0.990078i	$0.544878\pi$
$948$	0	0
$949$	−59.7934	−1.94097
$950$	−4.15476	−0.134798
$951$	0	0
$952$	0	0
$953$	−3.78249	−0.122527	−0.0612634	−	0.998122i	$-0.519513\pi$
$953$	−3.78249	−0.122527	−0.0612634	+	0.998122i	$0.519513\pi$
$954$	0	0
$955$	39.4175	1.27552
$956$	1.76239	0.0569998
$957$	0	0
$958$	49.5354	1.60042
$959$	0	0
$960$	0	0
$961$	27.9831	0.902680
$962$	−60.2518	−1.94260
$963$	0	0
$964$	−1.65539	−0.0533166
$965$	15.2986	0.492479
$966$	0	0
$967$	−32.9489	−1.05957	−0.529783	−	0.848133i	$-0.677727\pi$
$967$	−32.9489	−1.05957	−0.529783	+	0.848133i	$0.677727\pi$
$968$	−31.7347	−1.01999
$969$	0	0
$970$	−46.4097	−1.49013
$971$	−55.4570	−1.77970	−0.889850	−	0.456254i	$-0.849191\pi$
$971$	−55.4570	−1.77970	−0.889850	+	0.456254i	$0.849191\pi$
$972$	0	0
$973$	0	0
$974$	50.4124	1.61532
$975$	0	0
$976$	3.44562	0.110292
$977$	−56.5485	−1.80915	−0.904573	−	0.426318i	$-0.859811\pi$
$977$	−56.5485	−1.80915	−0.904573	+	0.426318i	$0.859811\pi$
$978$	0	0
$979$	3.79654	0.121338
$980$	0	0
$981$	0	0
$982$	−35.9331	−1.14667
$983$	28.9973	0.924871	0.462435	−	0.886653i	$-0.346976\pi$
$983$	28.9973	0.924871	0.462435	+	0.886653i	$0.346976\pi$
$984$	0	0
$985$	−63.8958	−2.03589
$986$	45.8958	1.46162
$987$	0	0
$988$	2.35092	0.0747927
$989$	0.130110	0.00413726
$990$	0	0
$991$	6.80922	0.216302	0.108151	−	0.994134i	$-0.465507\pi$
$991$	6.80922	0.216302	0.108151	+	0.994134i	$0.465507\pi$
$992$	−8.00175	−0.254056
$993$	0	0
$994$	0	0
$995$	−7.70233	−0.244180
$996$	0	0
$997$	38.9377	1.23317	0.616585	−	0.787289i	$-0.288516\pi$
$997$	38.9377	1.23317	0.616585	+	0.787289i	$0.288516\pi$
$998$	−45.4647	−1.43916
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.l.1.2		3
3.2	odd	2		3969.2.a.q.1.2		3
7.6	odd	2		567.2.a.c.1.2		3
9.2	odd	6		441.2.f.c.148.2		6
9.4	even	3		1323.2.f.d.883.2		6
9.5	odd	6		441.2.f.c.295.2		6
9.7	even	3		1323.2.f.d.442.2		6
21.20	even	2		567.2.a.h.1.2		3
28.27	even	2		9072.2.a.bs.1.1		3
63.2	odd	6		441.2.g.b.67.2		6
63.4	even	3		1323.2.g.e.667.2		6
63.5	even	6		441.2.h.d.214.2		6
63.11	odd	6		441.2.h.e.373.2		6
63.13	odd	6		189.2.f.b.127.2		6
63.16	even	3		1323.2.g.e.361.2		6
63.20	even	6		63.2.f.a.22.2	✓	6
63.23	odd	6		441.2.h.e.214.2		6
63.25	even	3		1323.2.h.b.226.2		6
63.31	odd	6		1323.2.g.d.667.2		6
63.32	odd	6		441.2.g.b.79.2		6
63.34	odd	6		189.2.f.b.64.2		6
63.38	even	6		441.2.h.d.373.2		6
63.40	odd	6		1323.2.h.c.802.2		6
63.41	even	6		63.2.f.a.43.2	yes	6
63.47	even	6		441.2.g.c.67.2		6
63.52	odd	6		1323.2.h.c.226.2		6
63.58	even	3		1323.2.h.b.802.2		6
63.59	even	6		441.2.g.c.79.2		6
63.61	odd	6		1323.2.g.d.361.2		6
84.83	odd	2		9072.2.a.ca.1.3		3
252.83	odd	6		1008.2.r.h.337.1		6
252.139	even	6		3024.2.r.k.2017.3		6
252.167	odd	6		1008.2.r.h.673.1		6
252.223	even	6		3024.2.r.k.1009.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.a.22.2	✓	6	63.20	even	6
63.2.f.a.43.2	yes	6	63.41	even	6
189.2.f.b.64.2		6	63.34	odd	6
189.2.f.b.127.2		6	63.13	odd	6
441.2.f.c.148.2		6	9.2	odd	6
441.2.f.c.295.2		6	9.5	odd	6
441.2.g.b.67.2		6	63.2	odd	6
441.2.g.b.79.2		6	63.32	odd	6
441.2.g.c.67.2		6	63.47	even	6
441.2.g.c.79.2		6	63.59	even	6
441.2.h.d.214.2		6	63.5	even	6
441.2.h.d.373.2		6	63.38	even	6
441.2.h.e.214.2		6	63.23	odd	6
441.2.h.e.373.2		6	63.11	odd	6
567.2.a.c.1.2		3	7.6	odd	2
567.2.a.h.1.2		3	21.20	even	2
1008.2.r.h.337.1		6	252.83	odd	6
1008.2.r.h.673.1		6	252.167	odd	6
1323.2.f.d.442.2		6	9.7	even	3
1323.2.f.d.883.2		6	9.4	even	3
1323.2.g.d.361.2		6	63.61	odd	6
1323.2.g.d.667.2		6	63.31	odd	6
1323.2.g.e.361.2		6	63.16	even	3
1323.2.g.e.667.2		6	63.4	even	3
1323.2.h.b.226.2		6	63.25	even	3
1323.2.h.b.802.2		6	63.58	even	3
1323.2.h.c.226.2		6	63.52	odd	6
1323.2.h.c.802.2		6	63.40	odd	6
3024.2.r.k.1009.3		6	252.223	even	6
3024.2.r.k.2017.3		6	252.139	even	6
3969.2.a.l.1.2		3	1.1	even	1	trivial
3969.2.a.q.1.2		3	3.2	odd	2
9072.2.a.bs.1.1		3	28.27	even	2
9072.2.a.ca.1.3		3	84.83	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34730 q^{2} -0.184793 q^{4} +2.53209 q^{5} +2.94356 q^{8} +O(q^{10})\)	\(q-1.34730 q^{2} -0.184793 q^{4} +2.53209 q^{5} +2.94356 q^{8} -3.41147 q^{10} -0.467911 q^{11} -5.82295 q^{13} -3.59627 q^{16} +3.87939 q^{17} +2.18479 q^{19} -0.467911 q^{20} +0.630415 q^{22} +0.106067 q^{23} +1.41147 q^{25} +7.84524 q^{26} -8.78106 q^{29} +7.68004 q^{31} -1.04189 q^{32} -5.22668 q^{34} -7.68004 q^{37} -2.94356 q^{38} +7.45336 q^{40} -2.22668 q^{41} +1.22668 q^{43} +0.0864665 q^{44} -0.142903 q^{46} -5.33275 q^{47} -1.90167 q^{50} +1.07604 q^{52} +0.716881 q^{53} -1.18479 q^{55} +11.8307 q^{58} +0.736482 q^{59} -0.958111 q^{61} -10.3473 q^{62} +8.59627 q^{64} -14.7442 q^{65} -9.63816 q^{67} -0.716881 q^{68} -13.2344 q^{71} +10.2686 q^{73} +10.3473 q^{74} -0.403733 q^{76} -12.6382 q^{79} -9.10607 q^{80} +3.00000 q^{82} -2.73143 q^{83} +9.82295 q^{85} -1.65270 q^{86} -1.37733 q^{88} -8.11381 q^{89} -0.0196004 q^{92} +7.18479 q^{94} +5.53209 q^{95} +13.6040 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38} + 9 q^{40} - 3 q^{43} - 15 q^{44} + 3 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} - 9 q^{58} - 3 q^{59} - 6 q^{61} - 30 q^{62} + 12 q^{64} - 15 q^{65} - 12 q^{67} + 6 q^{68} - 9 q^{71} + 21 q^{73} + 30 q^{74} - 15 q^{76} - 21 q^{79} - 15 q^{80} + 9 q^{82} - 18 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 12 q^{89} - 3 q^{92} + 18 q^{94} + 12 q^{95} + 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^11 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 6 * q^20 + 9 * q^22 - 12 * q^23 - 6 * q^25 - 3 * q^26 - 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 + 6 * q^38 + 9 * q^40 - 3 * q^43 - 15 * q^44 + 3 * q^47 + 6 * q^50 + 21 * q^52 - 6 * q^53 - 9 * q^58 - 3 * q^59 - 6 * q^61 - 30 * q^62 + 12 * q^64 - 15 * q^65 - 12 * q^67 + 6 * q^68 - 9 * q^71 + 21 * q^73 + 30 * q^74 - 15 * q^76 - 21 * q^79 - 15 * q^80 + 9 * q^82 - 18 * q^83 + 9 * q^85 - 6 * q^86 + 27 * q^88 + 12 * q^89 - 3 * q^92 + 18 * q^94 + 12 * q^95 + 3 * q^97

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