LMFDB - Embedded newform 3969.2.a.u.1.4

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:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.456850$ 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)

$f(q)$	$=$	$q+2.18890 q^{2} +2.79129 q^{4} -0.913701 q^{5} +1.73205 q^{8} +O(q^{10})$	$q+2.18890 q^{2} +2.79129 q^{4} -0.913701 q^{5} +1.73205 q^{8} -2.00000 q^{10} +2.64575 q^{11} -4.00000 q^{13} -1.79129 q^{16} -3.46410 q^{17} -5.58258 q^{19} -2.55040 q^{20} +5.79129 q^{22} -3.46410 q^{23} -4.16515 q^{25} -8.75560 q^{26} -8.75560 q^{29} +9.16515 q^{31} -7.38505 q^{32} -7.58258 q^{34} +3.00000 q^{37} -12.2197 q^{38} -1.58258 q^{40} +0.913701 q^{41} +0.582576 q^{43} +7.38505 q^{44} -7.58258 q^{46} +13.1334 q^{47} -9.11710 q^{50} -11.1652 q^{52} -8.66025 q^{53} -2.41742 q^{55} -19.1652 q^{58} -3.46410 q^{59} -11.5826 q^{61} +20.0616 q^{62} -12.5826 q^{64} +3.65480 q^{65} +8.58258 q^{67} -9.66930 q^{68} -4.47315 q^{71} -15.1652 q^{73} +6.56670 q^{74} -15.5826 q^{76} +0.582576 q^{79} +1.63670 q^{80} +2.00000 q^{82} +9.66930 q^{83} +3.16515 q^{85} +1.27520 q^{86} +4.58258 q^{88} +1.82740 q^{89} -9.66930 q^{92} +28.7477 q^{94} +5.10080 q^{95} +1.58258 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64} + 16 q^{67} - 24 q^{73} - 44 q^{76} - 16 q^{79} + 8 q^{82} - 24 q^{85} + 60 q^{94} - 12 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^10 - 16 * q^13 + 2 * q^16 - 4 * q^19 + 14 * q^22 + 20 * q^25 - 12 * q^34 + 12 * q^37 + 12 * q^40 - 16 * q^43 - 12 * q^46 - 8 * q^52 - 28 * q^55 - 40 * q^58 - 28 * q^61 - 32 * q^64 + 16 * q^67 - 24 * q^73 - 44 * q^76 - 16 * q^79 + 8 * q^82 - 24 * q^85 + 60 * q^94 - 12 * 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 $

$2$	2.18890	1.54779	0.773893	−	0.633316i	$-0.218307\pi$
$2$	2.18890	1.54779	0.773893	+	0.633316i	$0.218307\pi$
$3$	0	0
$3$	0	0
$4$	2.79129	1.39564
$5$	−0.913701	−0.408619	−0.204310	−	0.978906i	$-0.565495\pi$
$5$	−0.913701	−0.408619	−0.204310	+	0.978906i	$0.565495\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.73205	0.612372
$9$	0	0
$10$	−2.00000	−0.632456
$11$	2.64575	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$11$	2.64575	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	−1.79129	−0.447822
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−5.58258	−1.28073	−0.640365	−	0.768070i	$-0.721217\pi$
$19$	−5.58258	−1.28073	−0.640365	+	0.768070i	$0.721217\pi$
$20$	−2.55040	−0.570287
$21$	0	0
$22$	5.79129	1.23471
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−4.16515	−0.833030
$26$	−8.75560	−1.71712
$27$	0	0
$28$	0	0
$29$	−8.75560	−1.62587	−0.812937	−	0.582351i	$-0.802133\pi$
$29$	−8.75560	−1.62587	−0.812937	+	0.582351i	$0.802133\pi$
$30$	0	0
$31$	9.16515	1.64611	0.823055	−	0.567962i	$-0.192268\pi$
$31$	9.16515	1.64611	0.823055	+	0.567962i	$0.192268\pi$
$32$	−7.38505	−1.30551
$33$	0	0
$34$	−7.58258	−1.30040
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−12.2197	−1.98230
$39$	0	0
$40$	−1.58258	−0.250227
$41$	0.913701	0.142696	0.0713480	−	0.997451i	$-0.477270\pi$
$41$	0.913701	0.142696	0.0713480	+	0.997451i	$0.477270\pi$
$42$	0	0
$43$	0.582576	0.0888420	0.0444210	−	0.999013i	$-0.485856\pi$
$43$	0.582576	0.0888420	0.0444210	+	0.999013i	$0.485856\pi$
$44$	7.38505	1.11334
$45$	0	0
$46$	−7.58258	−1.11799
$47$	13.1334	1.91570	0.957852	−	0.287262i	$-0.0927450\pi$
$47$	13.1334	1.91570	0.957852	+	0.287262i	$0.0927450\pi$
$48$	0	0
$49$	0	0
$50$	−9.11710	−1.28935
$51$	0	0
$52$	−11.1652	−1.54833
$53$	−8.66025	−1.18958	−0.594789	−	0.803882i	$-0.702764\pi$
$53$	−8.66025	−1.18958	−0.594789	+	0.803882i	$0.702764\pi$
$54$	0	0
$55$	−2.41742	−0.325965
$56$	0	0
$57$	0	0
$58$	−19.1652	−2.51651
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	−11.5826	−1.48300	−0.741498	−	0.670955i	$-0.765885\pi$
$61$	−11.5826	−1.48300	−0.741498	+	0.670955i	$0.765885\pi$
$62$	20.0616	2.54783
$63$	0	0
$64$	−12.5826	−1.57282
$65$	3.65480	0.453322
$66$	0	0
$67$	8.58258	1.04853	0.524264	−	0.851556i	$-0.324340\pi$
$67$	8.58258	1.04853	0.524264	+	0.851556i	$0.324340\pi$
$68$	−9.66930	−1.17258
$69$	0	0
$70$	0	0
$71$	−4.47315	−0.530866	−0.265433	−	0.964129i	$-0.585515\pi$
$71$	−4.47315	−0.530866	−0.265433	+	0.964129i	$0.585515\pi$
$72$	0	0
$73$	−15.1652	−1.77495	−0.887473	−	0.460859i	$-0.847541\pi$
$73$	−15.1652	−1.77495	−0.887473	+	0.460859i	$0.847541\pi$
$74$	6.56670	0.763364
$75$	0	0
$76$	−15.5826	−1.78744
$77$	0	0
$78$	0	0
$79$	0.582576	0.0655449	0.0327724	−	0.999463i	$-0.489566\pi$
$79$	0.582576	0.0655449	0.0327724	+	0.999463i	$0.489566\pi$
$80$	1.63670	0.182989
$81$	0	0
$82$	2.00000	0.220863
$83$	9.66930	1.06134	0.530672	−	0.847577i	$-0.321940\pi$
$83$	9.66930	1.06134	0.530672	+	0.847577i	$0.321940\pi$
$84$	0	0
$85$	3.16515	0.343309
$86$	1.27520	0.137508
$87$	0	0
$88$	4.58258	0.488504
$89$	1.82740	0.193704	0.0968521	−	0.995299i	$-0.469123\pi$
$89$	1.82740	0.193704	0.0968521	+	0.995299i	$0.469123\pi$
$90$	0	0
$91$	0	0
$92$	−9.66930	−1.00809
$93$	0	0
$94$	28.7477	2.96510
$95$	5.10080	0.523331
$96$	0	0
$97$	1.58258	0.160686	0.0803431	−	0.996767i	$-0.474398\pi$
$97$	1.58258	0.160686	0.0803431	+	0.996767i	$0.474398\pi$
$98$	0	0
$99$	0	0
$100$	−11.6261	−1.16261
$101$	−8.75560	−0.871215	−0.435608	−	0.900137i	$-0.643466\pi$
$101$	−8.75560	−0.871215	−0.435608	+	0.900137i	$0.643466\pi$
$102$	0	0
$103$	5.58258	0.550068	0.275034	−	0.961435i	$-0.411311\pi$
$103$	5.58258	0.550068	0.275034	+	0.961435i	$0.411311\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	−18.9564	−1.84121
$107$	16.5022	1.59532	0.797662	−	0.603105i	$-0.206070\pi$
$107$	16.5022	1.59532	0.797662	+	0.603105i	$0.206070\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−5.29150	−0.504525
$111$	0	0
$112$	0	0
$113$	3.36875	0.316905	0.158453	−	0.987367i	$-0.449349\pi$
$113$	3.36875	0.316905	0.158453	+	0.987367i	$0.449349\pi$
$114$	0	0
$115$	3.16515	0.295152
$116$	−24.4394	−2.26914
$117$	0	0
$118$	−7.58258	−0.698033
$119$	0	0
$120$	0	0
$121$	−4.00000	−0.363636
$122$	−25.3531	−2.29536
$123$	0	0
$124$	25.5826	2.29738
$125$	8.37420	0.749012
$126$	0	0
$127$	−16.5826	−1.47147	−0.735733	−	0.677272i	$-0.763162\pi$
$127$	−16.5826	−1.47147	−0.735733	+	0.677272i	$0.763162\pi$
$128$	−12.7719	−1.12889
$129$	0	0
$130$	8.00000	0.701646
$131$	8.03260	0.701812	0.350906	−	0.936411i	$-0.385874\pi$
$131$	8.03260	0.701812	0.350906	+	0.936411i	$0.385874\pi$
$132$	0	0
$133$	0	0
$134$	18.7864	1.62290
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−14.1425	−1.20827	−0.604136	−	0.796881i	$-0.706482\pi$
$137$	−14.1425	−1.20827	−0.604136	+	0.796881i	$0.706482\pi$
$138$	0	0
$139$	−1.58258	−0.134232	−0.0671162	−	0.997745i	$-0.521380\pi$
$139$	−1.58258	−0.134232	−0.0671162	+	0.997745i	$0.521380\pi$
$140$	0	0
$141$	0	0
$142$	−9.79129	−0.821667
$143$	−10.5830	−0.884995
$144$	0	0
$145$	8.00000	0.664364
$146$	−33.1950	−2.74724
$147$	0	0
$148$	8.37386	0.688327
$149$	21.0707	1.72618	0.863088	−	0.505054i	$-0.168527\pi$
$149$	21.0707	1.72618	0.863088	+	0.505054i	$0.168527\pi$
$150$	0	0
$151$	−11.4174	−0.929137	−0.464568	−	0.885537i	$-0.653791\pi$
$151$	−11.4174	−0.929137	−0.464568	+	0.885537i	$0.653791\pi$
$152$	−9.66930	−0.784284
$153$	0	0
$154$	0	0
$155$	−8.37420	−0.672632
$156$	0	0
$157$	−19.1652	−1.52955	−0.764773	−	0.644300i	$-0.777149\pi$
$157$	−19.1652	−1.52955	−0.764773	+	0.644300i	$0.777149\pi$
$158$	1.27520	0.101450
$159$	0	0
$160$	6.74773	0.533455
$161$	0	0
$162$	0	0
$163$	5.41742	0.424325	0.212163	−	0.977234i	$-0.431949\pi$
$163$	5.41742	0.424325	0.212163	+	0.977234i	$0.431949\pi$
$164$	2.55040	0.199153
$165$	0	0
$166$	21.1652	1.64273
$167$	−7.84190	−0.606825	−0.303412	−	0.952859i	$-0.598126\pi$
$167$	−7.84190	−0.606825	−0.303412	+	0.952859i	$0.598126\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	6.92820	0.531369
$171$	0	0
$172$	1.62614	0.123992
$173$	−14.9608	−1.13745	−0.568725	−	0.822528i	$-0.692563\pi$
$173$	−14.9608	−1.13745	−0.568725	+	0.822528i	$0.692563\pi$
$174$	0	0
$175$	0	0
$176$	−4.73930	−0.357238
$177$	0	0
$178$	4.00000	0.299813
$179$	12.2197	0.913344	0.456672	−	0.889635i	$-0.349041\pi$
$179$	12.2197	0.913344	0.456672	+	0.889635i	$0.349041\pi$
$180$	0	0
$181$	5.16515	0.383923	0.191961	−	0.981402i	$-0.438515\pi$
$181$	5.16515	0.383923	0.191961	+	0.981402i	$0.438515\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−2.74110	−0.201530
$186$	0	0
$187$	−9.16515	−0.670222
$188$	36.6591	2.67364
$189$	0	0
$190$	11.1652	0.810005
$191$	−0.818350	−0.0592137	−0.0296069	−	0.999562i	$-0.509426\pi$
$191$	−0.818350	−0.0592137	−0.0296069	+	0.999562i	$0.509426\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	3.46410	0.248708
$195$	0	0
$196$	0	0
$197$	−10.4877	−0.747214	−0.373607	−	0.927587i	$-0.621879\pi$
$197$	−10.4877	−0.747214	−0.373607	+	0.927587i	$0.621879\pi$
$198$	0	0
$199$	−10.7477	−0.761886	−0.380943	−	0.924598i	$-0.624401\pi$
$199$	−10.7477	−0.761886	−0.380943	+	0.924598i	$0.624401\pi$
$200$	−7.21425	−0.510125
$201$	0	0
$202$	−19.1652	−1.34846
$203$	0	0
$204$	0	0
$205$	−0.834849	−0.0583084
$206$	12.2197	0.851387
$207$	0	0
$208$	7.16515	0.496814
$209$	−14.7701	−1.02167
$210$	0	0
$211$	−15.7477	−1.08412	−0.542059	−	0.840340i	$-0.682355\pi$
$211$	−15.7477	−1.08412	−0.542059	+	0.840340i	$0.682355\pi$
$212$	−24.1733	−1.66023
$213$	0	0
$214$	36.1216	2.46922
$215$	−0.532300	−0.0363025
$216$	0	0
$217$	0	0
$218$	13.1334	0.889507
$219$	0	0
$220$	−6.74773	−0.454932
$221$	13.8564	0.932083
$222$	0	0
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	0	0
$225$	0	0
$226$	7.37386	0.490502
$227$	2.74110	0.181933	0.0909666	−	0.995854i	$-0.471004\pi$
$227$	2.74110	0.181933	0.0909666	+	0.995854i	$0.471004\pi$
$228$	0	0
$229$	23.9129	1.58021	0.790104	−	0.612973i	$-0.210027\pi$
$229$	23.9129	1.58021	0.790104	+	0.612973i	$0.210027\pi$
$230$	6.92820	0.456832
$231$	0	0
$232$	−15.1652	−0.995641
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−9.66930	−0.629418
$237$	0	0
$238$	0	0
$239$	−9.38325	−0.606952	−0.303476	−	0.952839i	$-0.598147\pi$
$239$	−9.38325	−0.606952	−0.303476	+	0.952839i	$0.598147\pi$
$240$	0	0
$241$	−27.5826	−1.77675	−0.888375	−	0.459119i	$-0.848165\pi$
$241$	−27.5826	−1.77675	−0.888375	+	0.459119i	$0.848165\pi$
$242$	−8.75560	−0.562832
$243$	0	0
$244$	−32.3303	−2.06974
$245$	0	0
$246$	0	0
$247$	22.3303	1.42084
$248$	15.8745	1.00803
$249$	0	0
$250$	18.3303	1.15931
$251$	2.55040	0.160980	0.0804899	−	0.996755i	$-0.474352\pi$
$251$	2.55040	0.160980	0.0804899	+	0.996755i	$0.474352\pi$
$252$	0	0
$253$	−9.16515	−0.576208
$254$	−36.2976	−2.27752
$255$	0	0
$256$	−2.79129	−0.174455
$257$	31.3676	1.95666	0.978329	−	0.207056i	$-0.0663883\pi$
$257$	31.3676	1.95666	0.978329	+	0.207056i	$0.0663883\pi$
$258$	0	0
$259$	0	0
$260$	10.2016	0.632677
$261$	0	0
$262$	17.5826	1.08626
$263$	9.57395	0.590355	0.295178	−	0.955442i	$-0.404621\pi$
$263$	9.57395	0.590355	0.295178	+	0.955442i	$0.404621\pi$
$264$	0	0
$265$	7.91288	0.486084
$266$	0	0
$267$	0	0
$268$	23.9564	1.46337
$269$	−6.92820	−0.422420	−0.211210	−	0.977441i	$-0.567740\pi$
$269$	−6.92820	−0.422420	−0.211210	+	0.977441i	$0.567740\pi$
$270$	0	0
$271$	18.7477	1.13884	0.569422	−	0.822046i	$-0.307167\pi$
$271$	18.7477	1.13884	0.569422	+	0.822046i	$0.307167\pi$
$272$	6.20520	0.376246
$273$	0	0
$274$	−30.9564	−1.87015
$275$	−11.0200	−0.664528
$276$	0	0
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	−3.46410	−0.207763
$279$	0	0
$280$	0	0
$281$	8.46955	0.505251	0.252626	−	0.967564i	$-0.418706\pi$
$281$	8.46955	0.505251	0.252626	+	0.967564i	$0.418706\pi$
$282$	0	0
$283$	−8.41742	−0.500364	−0.250182	−	0.968199i	$-0.580490\pi$
$283$	−8.41742	−0.500364	−0.250182	+	0.968199i	$0.580490\pi$
$284$	−12.4859	−0.740899
$285$	0	0
$286$	−23.1652	−1.36978
$287$	0	0
$288$	0	0
$289$	−5.00000	−0.294118
$290$	17.5112	1.02829
$291$	0	0
$292$	−42.3303	−2.47719
$293$	−16.4068	−0.958496	−0.479248	−	0.877680i	$-0.659091\pi$
$293$	−16.4068	−0.958496	−0.479248	+	0.877680i	$0.659091\pi$
$294$	0	0
$295$	3.16515	0.184282
$296$	5.19615	0.302020
$297$	0	0
$298$	46.1216	2.67175
$299$	13.8564	0.801337
$300$	0	0
$301$	0	0
$302$	−24.9916	−1.43811
$303$	0	0
$304$	10.0000	0.573539
$305$	10.5830	0.605981
$306$	0	0
$307$	20.3303	1.16031	0.580156	−	0.814505i	$-0.302992\pi$
$307$	20.3303	1.16031	0.580156	+	0.814505i	$0.302992\pi$
$308$	0	0
$309$	0	0
$310$	−18.3303	−1.04109
$311$	33.9180	1.92331	0.961657	−	0.274255i	$-0.0884311\pi$
$311$	33.9180	1.92331	0.961657	+	0.274255i	$0.0884311\pi$
$312$	0	0
$313$	−17.1652	−0.970232	−0.485116	−	0.874450i	$-0.661223\pi$
$313$	−17.1652	−0.970232	−0.485116	+	0.874450i	$0.661223\pi$
$314$	−41.9506	−2.36741
$315$	0	0
$316$	1.62614	0.0914773
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−23.1652	−1.29700
$320$	11.4967	0.642685
$321$	0	0
$322$	0	0
$323$	19.3386	1.07603
$324$	0	0
$325$	16.6606	0.924164
$326$	11.8582	0.656765
$327$	0	0
$328$	1.58258	0.0873831
$329$	0	0
$330$	0	0
$331$	−22.3303	−1.22738	−0.613692	−	0.789545i	$-0.710316\pi$
$331$	−22.3303	−1.22738	−0.613692	+	0.789545i	$0.710316\pi$
$332$	26.9898	1.48126
$333$	0	0
$334$	−17.1652	−0.939235
$335$	−7.84190	−0.428449
$336$	0	0
$337$	−1.83485	−0.0999506	−0.0499753	−	0.998750i	$-0.515914\pi$
$337$	−1.83485	−0.0999506	−0.0499753	+	0.998750i	$0.515914\pi$
$338$	6.56670	0.357182
$339$	0	0
$340$	8.83485	0.479137
$341$	24.2487	1.31314
$342$	0	0
$343$	0	0
$344$	1.00905	0.0544044
$345$	0	0
$346$	−32.7477	−1.76053
$347$	22.1751	1.19042	0.595210	−	0.803570i	$-0.297069\pi$
$347$	22.1751	1.19042	0.595210	+	0.803570i	$0.297069\pi$
$348$	0	0
$349$	−2.74773	−0.147082	−0.0735412	−	0.997292i	$-0.523430\pi$
$349$	−2.74773	−0.147082	−0.0735412	+	0.997292i	$0.523430\pi$
$350$	0	0
$351$	0	0
$352$	−19.5390	−1.04143
$353$	−9.28790	−0.494345	−0.247173	−	0.968971i	$-0.579501\pi$
$353$	−9.28790	−0.494345	−0.247173	+	0.968971i	$0.579501\pi$
$354$	0	0
$355$	4.08712	0.216922
$356$	5.10080	0.270342
$357$	0	0
$358$	26.7477	1.41366
$359$	25.0671	1.32299	0.661494	−	0.749950i	$-0.269923\pi$
$359$	25.0671	1.32299	0.661494	+	0.749950i	$0.269923\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	11.3060	0.594230
$363$	0	0
$364$	0	0
$365$	13.8564	0.725277
$366$	0	0
$367$	27.1652	1.41801	0.709005	−	0.705204i	$-0.249144\pi$
$367$	27.1652	1.41801	0.709005	+	0.705204i	$0.249144\pi$
$368$	6.20520	0.323469
$369$	0	0
$370$	−6.00000	−0.311925
$371$	0	0
$372$	0	0
$373$	29.3303	1.51867	0.759333	−	0.650703i	$-0.225525\pi$
$373$	29.3303	1.51867	0.759333	+	0.650703i	$0.225525\pi$
$374$	−20.0616	−1.03736
$375$	0	0
$376$	22.7477	1.17312
$377$	35.0224	1.80375
$378$	0	0
$379$	16.9129	0.868756	0.434378	−	0.900731i	$-0.356968\pi$
$379$	16.9129	0.868756	0.434378	+	0.900731i	$0.356968\pi$
$380$	14.2378	0.730384
$381$	0	0
$382$	−1.79129	−0.0916503
$383$	−12.4104	−0.634142	−0.317071	−	0.948402i	$-0.602699\pi$
$383$	−12.4104	−0.634142	−0.317071	+	0.948402i	$0.602699\pi$
$384$	0	0
$385$	0	0
$386$	30.6446	1.55977
$387$	0	0
$388$	4.41742	0.224261
$389$	31.7490	1.60974	0.804869	−	0.593452i	$-0.202235\pi$
$389$	31.7490	1.60974	0.804869	+	0.593452i	$0.202235\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	−22.9564	−1.15653
$395$	−0.532300	−0.0267829
$396$	0	0
$397$	−7.58258	−0.380559	−0.190279	−	0.981730i	$-0.560939\pi$
$397$	−7.58258	−0.380559	−0.190279	+	0.981730i	$0.560939\pi$
$398$	−23.5257	−1.17924
$399$	0	0
$400$	7.46099	0.373049
$401$	−17.6066	−0.879230	−0.439615	−	0.898186i	$-0.644885\pi$
$401$	−17.6066	−0.879230	−0.439615	+	0.898186i	$0.644885\pi$
$402$	0	0
$403$	−36.6606	−1.82619
$404$	−24.4394	−1.21591
$405$	0	0
$406$	0	0
$407$	7.93725	0.393435
$408$	0	0
$409$	−36.7477	−1.81706	−0.908529	−	0.417822i	$-0.862794\pi$
$409$	−36.7477	−1.81706	−0.908529	+	0.417822i	$0.862794\pi$
$410$	−1.82740	−0.0902489
$411$	0	0
$412$	15.5826	0.767698
$413$	0	0
$414$	0	0
$415$	−8.83485	−0.433686
$416$	29.5402	1.44833
$417$	0	0
$418$	−32.3303	−1.58133
$419$	29.0079	1.41713	0.708565	−	0.705646i	$-0.249343\pi$
$419$	29.0079	1.41713	0.708565	+	0.705646i	$0.249343\pi$
$420$	0	0
$421$	2.16515	0.105523	0.0527615	−	0.998607i	$-0.483198\pi$
$421$	2.16515	0.105523	0.0527615	+	0.998607i	$0.483198\pi$
$422$	−34.4702	−1.67798
$423$	0	0
$424$	−15.0000	−0.728464
$425$	14.4285	0.699885
$426$	0	0
$427$	0	0
$428$	46.0623	2.22650
$429$	0	0
$430$	−1.16515	−0.0561886
$431$	−4.91010	−0.236511	−0.118256	−	0.992983i	$-0.537730\pi$
$431$	−4.91010	−0.236511	−0.118256	+	0.992983i	$0.537730\pi$
$432$	0	0
$433$	−23.9129	−1.14918	−0.574590	−	0.818442i	$-0.694838\pi$
$433$	−23.9129	−1.14918	−0.574590	+	0.818442i	$0.694838\pi$
$434$	0	0
$435$	0	0
$436$	16.7477	0.802071
$437$	19.3386	0.925091
$438$	0	0
$439$	−16.4174	−0.783561	−0.391780	−	0.920059i	$-0.628141\pi$
$439$	−16.4174	−0.783561	−0.391780	+	0.920059i	$0.628141\pi$
$440$	−4.18710	−0.199612
$441$	0	0
$442$	30.3303	1.44267
$443$	17.3205	0.822922	0.411461	−	0.911427i	$-0.365019\pi$
$443$	17.3205	0.822922	0.411461	+	0.911427i	$0.365019\pi$
$444$	0	0
$445$	−1.66970	−0.0791512
$446$	−48.1558	−2.28024
$447$	0	0
$448$	0	0
$449$	−36.9452	−1.74355	−0.871775	−	0.489906i	$-0.837031\pi$
$449$	−36.9452	−1.74355	−0.871775	+	0.489906i	$0.837031\pi$
$450$	0	0
$451$	2.41742	0.113832
$452$	9.40315	0.442287
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	−5.33030	−0.249341	−0.124671	−	0.992198i	$-0.539787\pi$
$457$	−5.33030	−0.249341	−0.124671	+	0.992198i	$0.539787\pi$
$458$	52.3429	2.44582
$459$	0	0
$460$	8.83485	0.411927
$461$	23.7164	1.10458	0.552292	−	0.833651i	$-0.313753\pi$
$461$	23.7164	1.10458	0.552292	+	0.833651i	$0.313753\pi$
$462$	0	0
$463$	−24.9129	−1.15780	−0.578900	−	0.815399i	$-0.696518\pi$
$463$	−24.9129	−1.15780	−0.578900	+	0.815399i	$0.696518\pi$
$464$	15.6838	0.728102
$465$	0	0
$466$	0	0
$467$	−9.66930	−0.447442	−0.223721	−	0.974653i	$-0.571820\pi$
$467$	−9.66930	−0.447442	−0.223721	+	0.974653i	$0.571820\pi$
$468$	0	0
$469$	0	0
$470$	−26.2668	−1.21160
$471$	0	0
$472$	−6.00000	−0.276172
$473$	1.54135	0.0708714
$474$	0	0
$475$	23.2523	1.06689
$476$	0	0
$477$	0	0
$478$	−20.5390	−0.939433
$479$	−17.7019	−0.808821	−0.404410	−	0.914578i	$-0.632523\pi$
$479$	−17.7019	−0.808821	−0.404410	+	0.914578i	$0.632523\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−60.3755	−2.75003
$483$	0	0
$484$	−11.1652	−0.507507
$485$	−1.44600	−0.0656595
$486$	0	0
$487$	2.58258	0.117028	0.0585138	−	0.998287i	$-0.481364\pi$
$487$	2.58258	0.117028	0.0585138	+	0.998287i	$0.481364\pi$
$488$	−20.0616	−0.908146
$489$	0	0
$490$	0	0
$491$	33.0043	1.48946	0.744732	−	0.667364i	$-0.232577\pi$
$491$	33.0043	1.48946	0.744732	+	0.667364i	$0.232577\pi$
$492$	0	0
$493$	30.3303	1.36601
$494$	48.8788	2.19916
$495$	0	0
$496$	−16.4174	−0.737164
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	23.3748	1.04535
$501$	0	0
$502$	5.58258	0.249163
$503$	−26.2668	−1.17118	−0.585590	−	0.810608i	$-0.699137\pi$
$503$	−26.2668	−1.17118	−0.585590	+	0.810608i	$0.699137\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	−20.0616	−0.891847
$507$	0	0
$508$	−46.2867	−2.05364
$509$	8.56490	0.379633	0.189816	−	0.981820i	$-0.439211\pi$
$509$	8.56490	0.379633	0.189816	+	0.981820i	$0.439211\pi$
$510$	0	0
$511$	0	0
$512$	19.4340	0.858868
$513$	0	0
$514$	68.6606	3.02849
$515$	−5.10080	−0.224768
$516$	0	0
$517$	34.7477	1.52820
$518$	0	0
$519$	0	0
$520$	6.33030	0.277602
$521$	−3.46410	−0.151765	−0.0758825	−	0.997117i	$-0.524177\pi$
$521$	−3.46410	−0.151765	−0.0758825	+	0.997117i	$0.524177\pi$
$522$	0	0
$523$	15.5826	0.681378	0.340689	−	0.940176i	$-0.389340\pi$
$523$	15.5826	0.681378	0.340689	+	0.940176i	$0.389340\pi$
$524$	22.4213	0.979479
$525$	0	0
$526$	20.9564	0.913744
$527$	−31.7490	−1.38301
$528$	0	0
$529$	−11.0000	−0.478261
$530$	17.3205	0.752355
$531$	0	0
$532$	0	0
$533$	−3.65480	−0.158307
$534$	0	0
$535$	−15.0780	−0.651880
$536$	14.8655	0.642090
$537$	0	0
$538$	−15.1652	−0.653816
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	41.0369	1.76269
$543$	0	0
$544$	25.5826	1.09684
$545$	−5.48220	−0.234832
$546$	0	0
$547$	32.9129	1.40725	0.703627	−	0.710570i	$-0.251563\pi$
$547$	32.9129	1.40725	0.703627	+	0.710570i	$0.251563\pi$
$548$	−39.4757	−1.68632
$549$	0	0
$550$	−24.1216	−1.02855
$551$	48.8788	2.08231
$552$	0	0
$553$	0	0
$554$	15.3223	0.650982
$555$	0	0
$556$	−4.41742	−0.187341
$557$	−29.8263	−1.26378	−0.631890	−	0.775058i	$-0.717720\pi$
$557$	−29.8263	−1.26378	−0.631890	+	0.775058i	$0.717720\pi$
$558$	0	0
$559$	−2.33030	−0.0985613
$560$	0	0
$561$	0	0
$562$	18.5390	0.782021
$563$	20.5939	0.867930	0.433965	−	0.900930i	$-0.357114\pi$
$563$	20.5939	0.867930	0.433965	+	0.900930i	$0.357114\pi$
$564$	0	0
$565$	−3.07803	−0.129494
$566$	−18.4249	−0.774457
$567$	0	0
$568$	−7.74773	−0.325087
$569$	33.0997	1.38761	0.693805	−	0.720163i	$-0.255933\pi$
$569$	33.0997	1.38761	0.693805	+	0.720163i	$0.255933\pi$
$570$	0	0
$571$	−19.1652	−0.802037	−0.401018	−	0.916070i	$-0.631344\pi$
$571$	−19.1652	−0.802037	−0.401018	+	0.916070i	$0.631344\pi$
$572$	−29.5402	−1.23514
$573$	0	0
$574$	0	0
$575$	14.4285	0.601710
$576$	0	0
$577$	−5.25227	−0.218655	−0.109327	−	0.994006i	$-0.534870\pi$
$577$	−5.25227	−0.218655	−0.109327	+	0.994006i	$0.534870\pi$
$578$	−10.9445	−0.455231
$579$	0	0
$580$	22.3303	0.927215
$581$	0	0
$582$	0	0
$583$	−22.9129	−0.948954
$584$	−26.2668	−1.08693
$585$	0	0
$586$	−35.9129	−1.48355
$587$	19.1479	0.790319	0.395159	−	0.918613i	$-0.370689\pi$
$587$	19.1479	0.790319	0.395159	+	0.918613i	$0.370689\pi$
$588$	0	0
$589$	−51.1652	−2.10822
$590$	6.92820	0.285230
$591$	0	0
$592$	−5.37386	−0.220864
$593$	11.8383	0.486141	0.243070	−	0.970009i	$-0.421845\pi$
$593$	11.8383	0.486141	0.243070	+	0.970009i	$0.421845\pi$
$594$	0	0
$595$	0	0
$596$	58.8143	2.40913
$597$	0	0
$598$	30.3303	1.24030
$599$	−11.4014	−0.465847	−0.232923	−	0.972495i	$-0.574829\pi$
$599$	−11.4014	−0.465847	−0.232923	+	0.972495i	$0.574829\pi$
$600$	0	0
$601$	−13.5826	−0.554045	−0.277022	−	0.960863i	$-0.589348\pi$
$601$	−13.5826	−0.554045	−0.277022	+	0.960863i	$0.589348\pi$
$602$	0	0
$603$	0	0
$604$	−31.8693	−1.29674
$605$	3.65480	0.148589
$606$	0	0
$607$	−12.4174	−0.504008	−0.252004	−	0.967726i	$-0.581090\pi$
$607$	−12.4174	−0.504008	−0.252004	+	0.967726i	$0.581090\pi$
$608$	41.2276	1.67200
$609$	0	0
$610$	23.1652	0.937930
$611$	−52.5336	−2.12528
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	44.5010	1.79592
$615$	0	0
$616$	0	0
$617$	−38.6772	−1.55709	−0.778543	−	0.627591i	$-0.784041\pi$
$617$	−38.6772	−1.55709	−0.778543	+	0.627591i	$0.784041\pi$
$618$	0	0
$619$	−17.9129	−0.719979	−0.359990	−	0.932956i	$-0.617220\pi$
$619$	−17.9129	−0.719979	−0.359990	+	0.932956i	$0.617220\pi$
$620$	−23.3748	−0.938755
$621$	0	0
$622$	74.2432	2.97688
$623$	0	0
$624$	0	0
$625$	13.1742	0.526970
$626$	−37.5728	−1.50171
$627$	0	0
$628$	−53.4955	−2.13470
$629$	−10.3923	−0.414368
$630$	0	0
$631$	−12.8348	−0.510947	−0.255474	−	0.966816i	$-0.582231\pi$
$631$	−12.8348	−0.510947	−0.255474	+	0.966816i	$0.582231\pi$
$632$	1.00905	0.0401379
$633$	0	0
$634$	0	0
$635$	15.1515	0.601269
$636$	0	0
$637$	0	0
$638$	−50.7062	−2.00748
$639$	0	0
$640$	11.6697	0.461285
$641$	12.1244	0.478883	0.239442	−	0.970911i	$-0.423036\pi$
$641$	12.1244	0.478883	0.239442	+	0.970911i	$0.423036\pi$
$642$	0	0
$643$	−2.33030	−0.0918982	−0.0459491	−	0.998944i	$-0.514631\pi$
$643$	−2.33030	−0.0918982	−0.0459491	+	0.998944i	$0.514631\pi$
$644$	0	0
$645$	0	0
$646$	42.3303	1.66546
$647$	−6.01450	−0.236455	−0.118227	−	0.992987i	$-0.537721\pi$
$647$	−6.01450	−0.236455	−0.118227	+	0.992987i	$0.537721\pi$
$648$	0	0
$649$	−9.16515	−0.359764
$650$	36.4684	1.43041
$651$	0	0
$652$	15.1216	0.592207
$653$	12.1244	0.474463	0.237231	−	0.971453i	$-0.423760\pi$
$653$	12.1244	0.474463	0.237231	+	0.971453i	$0.423760\pi$
$654$	0	0
$655$	−7.33939	−0.286774
$656$	−1.63670	−0.0639024
$657$	0	0
$658$	0	0
$659$	−41.1323	−1.60229	−0.801143	−	0.598473i	$-0.795774\pi$
$659$	−41.1323	−1.60229	−0.801143	+	0.598473i	$0.795774\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	−48.8788	−1.89973
$663$	0	0
$664$	16.7477	0.649938
$665$	0	0
$666$	0	0
$667$	30.3303	1.17439
$668$	−21.8890	−0.846911
$669$	0	0
$670$	−17.1652	−0.663148
$671$	−30.6446	−1.18302
$672$	0	0
$673$	−26.4955	−1.02132	−0.510662	−	0.859781i	$-0.670600\pi$
$673$	−26.4955	−1.02132	−0.510662	+	0.859781i	$0.670600\pi$
$674$	−4.01630	−0.154702
$675$	0	0
$676$	8.37386	0.322072
$677$	−43.5873	−1.67520	−0.837598	−	0.546286i	$-0.816041\pi$
$677$	−43.5873	−1.67520	−0.837598	+	0.546286i	$0.816041\pi$
$678$	0	0
$679$	0	0
$680$	5.48220	0.210233
$681$	0	0
$682$	53.0780	2.03246
$683$	−11.4014	−0.436261	−0.218130	−	0.975920i	$-0.569996\pi$
$683$	−11.4014	−0.436261	−0.218130	+	0.975920i	$0.569996\pi$
$684$	0	0
$685$	12.9220	0.493723
$686$	0	0
$687$	0	0
$688$	−1.04356	−0.0397854
$689$	34.6410	1.31972
$690$	0	0
$691$	−29.1652	−1.10949	−0.554747	−	0.832019i	$-0.687185\pi$
$691$	−29.1652	−1.10949	−0.554747	+	0.832019i	$0.687185\pi$
$692$	−41.7599	−1.58747
$693$	0	0
$694$	48.5390	1.84252
$695$	1.44600	0.0548499
$696$	0	0
$697$	−3.16515	−0.119889
$698$	−6.01450	−0.227652
$699$	0	0
$700$	0	0
$701$	−29.4449	−1.11212	−0.556059	−	0.831143i	$-0.687687\pi$
$701$	−29.4449	−1.11212	−0.556059	+	0.831143i	$0.687687\pi$
$702$	0	0
$703$	−16.7477	−0.631652
$704$	−33.2904	−1.25468
$705$	0	0
$706$	−20.3303	−0.765141
$707$	0	0
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	8.94630	0.335749
$711$	0	0
$712$	3.16515	0.118619
$713$	−31.7490	−1.18901
$714$	0	0
$715$	9.66970	0.361626
$716$	34.1087	1.27470
$717$	0	0
$718$	54.8693	2.04770
$719$	14.4285	0.538093	0.269046	−	0.963127i	$-0.413292\pi$
$719$	14.4285	0.538093	0.269046	+	0.963127i	$0.413292\pi$
$720$	0	0
$721$	0	0
$722$	26.6283	0.991003
$723$	0	0
$724$	14.4174	0.535819
$725$	36.4684	1.35440
$726$	0	0
$727$	36.2432	1.34419	0.672093	−	0.740467i	$-0.265396\pi$
$727$	36.2432	1.34419	0.672093	+	0.740467i	$0.265396\pi$
$728$	0	0
$729$	0	0
$730$	30.3303	1.12257
$731$	−2.01810	−0.0746422
$732$	0	0
$733$	16.7477	0.618591	0.309296	−	0.950966i	$-0.399907\pi$
$733$	16.7477	0.618591	0.309296	+	0.950966i	$0.399907\pi$
$734$	59.4618	2.19478
$735$	0	0
$736$	25.5826	0.942986
$737$	22.7074	0.836436
$738$	0	0
$739$	−28.5826	−1.05143	−0.525714	−	0.850662i	$-0.676202\pi$
$739$	−28.5826	−1.05143	−0.525714	+	0.850662i	$0.676202\pi$
$740$	−7.65120	−0.281264
$741$	0	0
$742$	0	0
$743$	34.2041	1.25483	0.627413	−	0.778687i	$-0.284114\pi$
$743$	34.2041	1.25483	0.627413	+	0.778687i	$0.284114\pi$
$744$	0	0
$745$	−19.2523	−0.705349
$746$	64.2011	2.35057
$747$	0	0
$748$	−25.5826	−0.935392
$749$	0	0
$750$	0	0
$751$	−18.5826	−0.678088	−0.339044	−	0.940771i	$-0.610104\pi$
$751$	−18.5826	−0.678088	−0.339044	+	0.940771i	$0.610104\pi$
$752$	−23.5257	−0.857894
$753$	0	0
$754$	76.6606	2.79181
$755$	10.4321	0.379663
$756$	0	0
$757$	−25.1652	−0.914643	−0.457321	−	0.889301i	$-0.651191\pi$
$757$	−25.1652	−0.914643	−0.457321	+	0.889301i	$0.651191\pi$
$758$	37.0206	1.34465
$759$	0	0
$760$	8.83485	0.320474
$761$	14.5794	0.528503	0.264252	−	0.964454i	$-0.414875\pi$
$761$	14.5794	0.528503	0.264252	+	0.964454i	$0.414875\pi$
$762$	0	0
$763$	0	0
$764$	−2.28425	−0.0826413
$765$	0	0
$766$	−27.1652	−0.981517
$767$	13.8564	0.500326
$768$	0	0
$769$	−19.0780	−0.687971	−0.343986	−	0.938975i	$-0.611777\pi$
$769$	−19.0780	−0.687971	−0.343986	+	0.938975i	$0.611777\pi$
$770$	0	0
$771$	0	0
$772$	39.0780	1.40645
$773$	−10.7737	−0.387503	−0.193752	−	0.981051i	$-0.562066\pi$
$773$	−10.7737	−0.387503	−0.193752	+	0.981051i	$0.562066\pi$
$774$	0	0
$775$	−38.1742	−1.37126
$776$	2.74110	0.0983998
$777$	0	0
$778$	69.4955	2.49153
$779$	−5.10080	−0.182755
$780$	0	0
$781$	−11.8348	−0.423484
$782$	26.2668	0.939299
$783$	0	0
$784$	0	0
$785$	17.5112	0.625002
$786$	0	0
$787$	−8.08712	−0.288275	−0.144137	−	0.989558i	$-0.546041\pi$
$787$	−8.08712	−0.288275	−0.144137	+	0.989558i	$0.546041\pi$
$788$	−29.2741	−1.04285
$789$	0	0
$790$	−1.16515	−0.0414542
$791$	0	0
$792$	0	0
$793$	46.3303	1.64524
$794$	−16.5975	−0.589024
$795$	0	0
$796$	−30.0000	−1.06332
$797$	−2.20880	−0.0782398	−0.0391199	−	0.999235i	$-0.512455\pi$
$797$	−2.20880	−0.0782398	−0.0391199	+	0.999235i	$0.512455\pi$
$798$	0	0
$799$	−45.4955	−1.60951
$800$	30.7599	1.08753
$801$	0	0
$802$	−38.5390	−1.36086
$803$	−40.1232	−1.41592
$804$	0	0
$805$	0	0
$806$	−80.2464	−2.82656
$807$	0	0
$808$	−15.1652	−0.533508
$809$	41.8553	1.47155	0.735776	−	0.677224i	$-0.236817\pi$
$809$	41.8553	1.47155	0.735776	+	0.677224i	$0.236817\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	17.3739	0.608954
$815$	−4.94990	−0.173388
$816$	0	0
$817$	−3.25227	−0.113783
$818$	−80.4371	−2.81242
$819$	0	0
$820$	−2.33030	−0.0813777
$821$	−50.9923	−1.77964	−0.889821	−	0.456309i	$-0.849171\pi$
$821$	−50.9923	−1.77964	−0.889821	+	0.456309i	$0.849171\pi$
$822$	0	0
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	9.66930	0.336846
$825$	0	0
$826$	0	0
$827$	−9.38325	−0.326288	−0.163144	−	0.986602i	$-0.552163\pi$
$827$	−9.38325	−0.326288	−0.163144	+	0.986602i	$0.552163\pi$
$828$	0	0
$829$	−26.3303	−0.914489	−0.457245	−	0.889341i	$-0.651164\pi$
$829$	−26.3303	−0.914489	−0.457245	+	0.889341i	$0.651164\pi$
$830$	−19.3386	−0.671253
$831$	0	0
$832$	50.3303	1.74489
$833$	0	0
$834$	0	0
$835$	7.16515	0.247960
$836$	−41.2276	−1.42589
$837$	0	0
$838$	63.4955	2.19341
$839$	33.7273	1.16440	0.582198	−	0.813047i	$-0.302193\pi$
$839$	33.7273	1.16440	0.582198	+	0.813047i	$0.302193\pi$
$840$	0	0
$841$	47.6606	1.64347
$842$	4.73930	0.163327
$843$	0	0
$844$	−43.9564	−1.51304
$845$	−2.74110	−0.0942968
$846$	0	0
$847$	0	0
$848$	15.5130	0.532719
$849$	0	0
$850$	31.5826	1.08327
$851$	−10.3923	−0.356244
$852$	0	0
$853$	14.7477	0.504953	0.252476	−	0.967603i	$-0.418755\pi$
$853$	14.7477	0.504953	0.252476	+	0.967603i	$0.418755\pi$
$854$	0	0
$855$	0	0
$856$	28.5826	0.976932
$857$	−12.6011	−0.430446	−0.215223	−	0.976565i	$-0.569048\pi$
$857$	−12.6011	−0.430446	−0.215223	+	0.976565i	$0.569048\pi$
$858$	0	0
$859$	10.3303	0.352465	0.176233	−	0.984349i	$-0.443609\pi$
$859$	10.3303	0.352465	0.176233	+	0.984349i	$0.443609\pi$
$860$	−1.48580	−0.0506654
$861$	0	0
$862$	−10.7477	−0.366069
$863$	18.3296	0.623945	0.311973	−	0.950091i	$-0.399010\pi$
$863$	18.3296	0.623945	0.311973	+	0.950091i	$0.399010\pi$
$864$	0	0
$865$	13.6697	0.464784
$866$	−52.3429	−1.77868
$867$	0	0
$868$	0	0
$869$	1.54135	0.0522867
$870$	0	0
$871$	−34.3303	−1.16324
$872$	10.3923	0.351928
$873$	0	0
$874$	42.3303	1.43184
$875$	0	0
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	−35.9361	−1.21279
$879$	0	0
$880$	4.33030	0.145974
$881$	−37.3821	−1.25944	−0.629718	−	0.776824i	$-0.716829\pi$
$881$	−37.3821	−1.25944	−0.629718	+	0.776824i	$0.716829\pi$
$882$	0	0
$883$	3.08712	0.103890	0.0519450	−	0.998650i	$-0.483458\pi$
$883$	3.08712	0.103890	0.0519450	+	0.998650i	$0.483458\pi$
$884$	38.6772	1.30086
$885$	0	0
$886$	37.9129	1.27371
$887$	29.9216	1.00467	0.502335	−	0.864673i	$-0.332475\pi$
$887$	29.9216	1.00467	0.502335	+	0.864673i	$0.332475\pi$
$888$	0	0
$889$	0	0
$890$	−3.65480	−0.122509
$891$	0	0
$892$	−61.4083	−2.05610
$893$	−73.3182	−2.45350
$894$	0	0
$895$	−11.1652	−0.373210
$896$	0	0
$897$	0	0
$898$	−80.8693	−2.69864
$899$	−80.2464	−2.67637
$900$	0	0
$901$	30.0000	0.999445
$902$	5.29150	0.176188
$903$	0	0
$904$	5.83485	0.194064
$905$	−4.71940	−0.156878
$906$	0	0
$907$	9.41742	0.312700	0.156350	−	0.987702i	$-0.450027\pi$
$907$	9.41742	0.312700	0.156350	+	0.987702i	$0.450027\pi$
$908$	7.65120	0.253914
$909$	0	0
$910$	0	0
$911$	−22.8027	−0.755488	−0.377744	−	0.925910i	$-0.623300\pi$
$911$	−22.8027	−0.755488	−0.377744	+	0.925910i	$0.623300\pi$
$912$	0	0
$913$	25.5826	0.846660
$914$	−11.6675	−0.385927
$915$	0	0
$916$	66.7477	2.20541
$917$	0	0
$918$	0	0
$919$	−48.0780	−1.58595	−0.792974	−	0.609256i	$-0.791468\pi$
$919$	−48.0780	−1.58595	−0.792974	+	0.609256i	$0.791468\pi$
$920$	5.48220	0.180743
$921$	0	0
$922$	51.9129	1.70966
$923$	17.8926	0.588942
$924$	0	0
$925$	−12.4955	−0.410848
$926$	−54.5318	−1.79203
$927$	0	0
$928$	64.6606	2.12259
$929$	−28.8172	−0.945462	−0.472731	−	0.881207i	$-0.656732\pi$
$929$	−28.8172	−0.945462	−0.472731	+	0.881207i	$0.656732\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−21.1652	−0.692545
$935$	8.37420	0.273866
$936$	0	0
$937$	21.4955	0.702226	0.351113	−	0.936333i	$-0.385803\pi$
$937$	21.4955	0.702226	0.351113	+	0.936333i	$0.385803\pi$
$938$	0	0
$939$	0	0
$940$	−33.4955	−1.09250
$941$	−43.3966	−1.41469	−0.707345	−	0.706869i	$-0.750107\pi$
$941$	−43.3966	−1.41469	−0.707345	+	0.706869i	$0.750107\pi$
$942$	0	0
$943$	−3.16515	−0.103072
$944$	6.20520	0.201962
$945$	0	0
$946$	3.37386	0.109694
$947$	20.5939	0.669212	0.334606	−	0.942358i	$-0.391397\pi$
$947$	20.5939	0.669212	0.334606	+	0.942358i	$0.391397\pi$
$948$	0	0
$949$	60.6606	1.96913
$950$	50.8969	1.65131
$951$	0	0
$952$	0	0
$953$	−17.5112	−0.567244	−0.283622	−	0.958936i	$-0.591536\pi$
$953$	−17.5112	−0.567244	−0.283622	+	0.958936i	$0.591536\pi$
$954$	0	0
$955$	0.747727	0.0241959
$956$	−26.1914	−0.847089
$957$	0	0
$958$	−38.7477	−1.25188
$959$	0	0
$960$	0	0
$961$	53.0000	1.70968
$962$	−26.2668	−0.846876
$963$	0	0
$964$	−76.9909	−2.47971
$965$	−12.7918	−0.411783
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−6.92820	−0.222681
$969$	0	0
$970$	−3.16515	−0.101627
$971$	−28.2849	−0.907706	−0.453853	−	0.891077i	$-0.649951\pi$
$971$	−28.2849	−0.907706	−0.453853	+	0.891077i	$0.649951\pi$
$972$	0	0
$973$	0	0
$974$	5.65300	0.181134
$975$	0	0
$976$	20.7477	0.664119
$977$	38.6772	1.23739	0.618697	−	0.785630i	$-0.287661\pi$
$977$	38.6772	1.23739	0.618697	+	0.785630i	$0.287661\pi$
$978$	0	0
$979$	4.83485	0.154522
$980$	0	0
$981$	0	0
$982$	72.2432	2.30537
$983$	−28.4358	−0.906962	−0.453481	−	0.891266i	$-0.649818\pi$
$983$	−28.4358	−0.906962	−0.453481	+	0.891266i	$0.649818\pi$
$984$	0	0
$985$	9.58258	0.305326
$986$	66.3900	2.11429
$987$	0	0
$988$	62.3303	1.98299
$989$	−2.01810	−0.0641719
$990$	0	0
$991$	−8.25227	−0.262142	−0.131071	−	0.991373i	$-0.541842\pi$
$991$	−8.25227	−0.262142	−0.131071	+	0.991373i	$0.541842\pi$
$992$	−67.6851	−2.14901
$993$	0	0
$994$	0	0
$995$	9.82020	0.311321
$996$	0	0
$997$	−25.1652	−0.796988	−0.398494	−	0.917171i	$-0.630467\pi$
$997$	−25.1652	−0.796988	−0.398494	+	0.917171i	$0.630467\pi$
$998$	−43.7780	−1.38577
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.u.1.4		4
3.2	odd	2	inner	3969.2.a.u.1.1		4
7.6	odd	2		567.2.a.i.1.4	yes	4
21.20	even	2		567.2.a.i.1.1	✓	4
28.27	even	2		9072.2.a.ci.1.3		4
63.13	odd	6		567.2.f.n.379.1		8
63.20	even	6		567.2.f.n.190.4		8
63.34	odd	6		567.2.f.n.190.1		8
63.41	even	6		567.2.f.n.379.4		8
84.83	odd	2		9072.2.a.ci.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.i.1.1	✓	4	21.20	even	2
567.2.a.i.1.4	yes	4	7.6	odd	2
567.2.f.n.190.1		8	63.34	odd	6
567.2.f.n.190.4		8	63.20	even	6
567.2.f.n.379.1		8	63.13	odd	6
567.2.f.n.379.4		8	63.41	even	6
3969.2.a.u.1.1		4	3.2	odd	2	inner
3969.2.a.u.1.4		4	1.1	even	1	trivial
9072.2.a.ci.1.2		4	84.83	odd	2
9072.2.a.ci.1.3		4	28.27	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18890 q^{2} +2.79129 q^{4} -0.913701 q^{5} +1.73205 q^{8} +O(q^{10})\)	\(q+2.18890 q^{2} +2.79129 q^{4} -0.913701 q^{5} +1.73205 q^{8} -2.00000 q^{10} +2.64575 q^{11} -4.00000 q^{13} -1.79129 q^{16} -3.46410 q^{17} -5.58258 q^{19} -2.55040 q^{20} +5.79129 q^{22} -3.46410 q^{23} -4.16515 q^{25} -8.75560 q^{26} -8.75560 q^{29} +9.16515 q^{31} -7.38505 q^{32} -7.58258 q^{34} +3.00000 q^{37} -12.2197 q^{38} -1.58258 q^{40} +0.913701 q^{41} +0.582576 q^{43} +7.38505 q^{44} -7.58258 q^{46} +13.1334 q^{47} -9.11710 q^{50} -11.1652 q^{52} -8.66025 q^{53} -2.41742 q^{55} -19.1652 q^{58} -3.46410 q^{59} -11.5826 q^{61} +20.0616 q^{62} -12.5826 q^{64} +3.65480 q^{65} +8.58258 q^{67} -9.66930 q^{68} -4.47315 q^{71} -15.1652 q^{73} +6.56670 q^{74} -15.5826 q^{76} +0.582576 q^{79} +1.63670 q^{80} +2.00000 q^{82} +9.66930 q^{83} +3.16515 q^{85} +1.27520 q^{86} +4.58258 q^{88} +1.82740 q^{89} -9.66930 q^{92} +28.7477 q^{94} +5.10080 q^{95} +1.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} - 8 q^{10} - 16 q^{13} + 2 q^{16} - 4 q^{19} + 14 q^{22} + 20 q^{25} - 12 q^{34} + 12 q^{37} + 12 q^{40} - 16 q^{43} - 12 q^{46} - 8 q^{52} - 28 q^{55} - 40 q^{58} - 28 q^{61} - 32 q^{64} + 16 q^{67} - 24 q^{73} - 44 q^{76} - 16 q^{79} + 8 q^{82} - 24 q^{85} + 60 q^{94} - 12 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^10 - 16 * q^13 + 2 * q^16 - 4 * q^19 + 14 * q^22 + 20 * q^25 - 12 * q^34 + 12 * q^37 + 12 * q^40 - 16 * q^43 - 12 * q^46 - 8 * q^52 - 28 * q^55 - 40 * q^58 - 28 * q^61 - 32 * q^64 + 16 * q^67 - 24 * q^73 - 44 * q^76 - 16 * q^79 + 8 * q^82 - 24 * q^85 + 60 * q^94 - 12 * q^97

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