LMFDB - Embedded newform 3969.2.a.bd.1.1

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:	$6$
Coefficient field:	6.6.59351616.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 21x^{2} - 9 $ x^6 - 12x^4 + 21x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.820103$ 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)

$f(q)$	$=$	$q-2.46050 q^{2} +4.05408 q^{4} -3.65808 q^{5} -5.05408 q^{8} +O(q^{10})$	$q-2.46050 q^{2} +4.05408 q^{4} -3.65808 q^{5} -5.05408 q^{8} +9.00071 q^{10} -0.406421 q^{11} +0.486796 q^{13} +4.32743 q^{16} +4.85584 q^{17} +1.97351 q^{19} -14.8301 q^{20} +1.00000 q^{22} -4.64766 q^{23} +8.38151 q^{25} -1.19777 q^{26} -7.64766 q^{29} -7.02720 q^{31} -0.539495 q^{32} -11.9478 q^{34} +2.32743 q^{37} -4.85584 q^{38} +18.4882 q^{40} +7.51399 q^{41} -2.32743 q^{43} -1.64766 q^{44} +11.4356 q^{46} -6.31623 q^{47} -20.6228 q^{50} +1.97351 q^{52} +3.56867 q^{53} +1.48672 q^{55} +18.8171 q^{58} +6.11839 q^{59} +8.02712 q^{61} +17.2905 q^{62} -7.32743 q^{64} -1.78074 q^{65} +3.60078 q^{67} +19.6860 q^{68} -8.46050 q^{71} -1.97351 q^{73} -5.72665 q^{74} +8.00079 q^{76} +8.16225 q^{79} -15.8301 q^{80} -18.4882 q^{82} +12.1720 q^{83} -17.7630 q^{85} +5.72665 q^{86} +2.05408 q^{88} +14.8301 q^{89} -18.8420 q^{92} +15.5411 q^{94} -7.21926 q^{95} +9.48751 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 6 * q^4 - 12 * q^8	$ 6 q - 2 q^{2} + 6 q^{4} - 12 q^{8} - 8 q^{11} + 6 q^{16} + 6 q^{22} - 4 q^{23} + 12 q^{25} - 22 q^{29} - 16 q^{32} - 6 q^{37} + 6 q^{43} + 14 q^{44} + 12 q^{46} - 56 q^{50} - 28 q^{53} + 18 q^{58} - 24 q^{64} + 6 q^{65} - 38 q^{71} - 36 q^{74} - 6 q^{79} - 30 q^{85} + 36 q^{86} - 6 q^{88} - 62 q^{92} - 60 q^{95}+O(q^{100}) $ 6 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 - 8 * q^11 + 6 * q^16 + 6 * q^22 - 4 * q^23 + 12 * q^25 - 22 * q^29 - 16 * q^32 - 6 * q^37 + 6 * q^43 + 14 * q^44 + 12 * q^46 - 56 * q^50 - 28 * q^53 + 18 * q^58 - 24 * q^64 + 6 * q^65 - 38 * q^71 - 36 * q^74 - 6 * q^79 - 30 * q^85 + 36 * q^86 - 6 * q^88 - 62 * q^92 - 60 * q^95

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 $

$2$	−2.46050	−1.73984	−0.869920	−	0.493193i	$-0.835830\pi$
$2$	−2.46050	−1.73984	−0.869920	+	0.493193i	$0.835830\pi$
$3$	0	0
$3$	0	0
$4$	4.05408	2.02704
$5$	−3.65808	−1.63594	−0.817970	−	0.575260i	$-0.804901\pi$
$5$	−3.65808	−1.63594	−0.817970	+	0.575260i	$0.804901\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−5.05408	−1.78689
$9$	0	0
$10$	9.00071	2.84628
$11$	−0.406421	−0.122540	−0.0612702	−	0.998121i	$-0.519515\pi$
$11$	−0.406421	−0.122540	−0.0612702	+	0.998121i	$0.519515\pi$
$12$	0	0
$13$	0.486796	0.135013	0.0675065	−	0.997719i	$-0.478496\pi$
$13$	0.486796	0.135013	0.0675065	+	0.997719i	$0.478496\pi$
$14$	0	0
$15$	0	0
$16$	4.32743	1.08186
$17$	4.85584	1.17771	0.588857	−	0.808237i	$-0.299578\pi$
$17$	4.85584	1.17771	0.588857	+	0.808237i	$0.299578\pi$
$18$	0	0
$19$	1.97351	0.452755	0.226378	−	0.974040i	$-0.427312\pi$
$19$	1.97351	0.452755	0.226378	+	0.974040i	$0.427312\pi$
$20$	−14.8301	−3.31612
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.64766	−0.969105	−0.484552	−	0.874762i	$-0.661018\pi$
$23$	−4.64766	−0.969105	−0.484552	+	0.874762i	$0.661018\pi$
$24$	0	0
$25$	8.38151	1.67630
$26$	−1.19777	−0.234901
$27$	0	0
$28$	0	0
$29$	−7.64766	−1.42014	−0.710068	−	0.704133i	$-0.751336\pi$
$29$	−7.64766	−1.42014	−0.710068	+	0.704133i	$0.751336\pi$
$30$	0	0
$31$	−7.02720	−1.26212	−0.631061	−	0.775733i	$-0.717380\pi$
$31$	−7.02720	−1.26212	−0.631061	+	0.775733i	$0.717380\pi$
$32$	−0.539495	−0.0953702
$33$	0	0
$34$	−11.9478	−2.04903
$35$	0	0
$36$	0	0
$37$	2.32743	0.382627	0.191314	−	0.981529i	$-0.438725\pi$
$37$	2.32743	0.382627	0.191314	+	0.981529i	$0.438725\pi$
$38$	−4.85584	−0.787721
$39$	0	0
$40$	18.4882	2.92324
$41$	7.51399	1.17349	0.586744	−	0.809772i	$-0.300409\pi$
$41$	7.51399	1.17349	0.586744	+	0.809772i	$0.300409\pi$
$42$	0	0
$43$	−2.32743	−0.354930	−0.177465	−	0.984127i	$-0.556790\pi$
$43$	−2.32743	−0.354930	−0.177465	+	0.984127i	$0.556790\pi$
$44$	−1.64766	−0.248395
$45$	0	0
$46$	11.4356	1.68609
$47$	−6.31623	−0.921317	−0.460658	−	0.887578i	$-0.652387\pi$
$47$	−6.31623	−0.921317	−0.460658	+	0.887578i	$0.652387\pi$
$48$	0	0
$49$	0	0
$50$	−20.6228	−2.91650
$51$	0	0
$52$	1.97351	0.273677
$53$	3.56867	0.490195	0.245097	−	0.969498i	$-0.421180\pi$
$53$	3.56867	0.490195	0.245097	+	0.969498i	$0.421180\pi$
$54$	0	0
$55$	1.48672	0.200469
$56$	0	0
$57$	0	0
$58$	18.8171	2.47081
$59$	6.11839	0.796546	0.398273	−	0.917267i	$-0.369610\pi$
$59$	6.11839	0.796546	0.398273	+	0.917267i	$0.369610\pi$
$60$	0	0
$61$	8.02712	1.02777	0.513884	−	0.857860i	$-0.328206\pi$
$61$	8.02712	1.02777	0.513884	+	0.857860i	$0.328206\pi$
$62$	17.2905	2.19589
$63$	0	0
$64$	−7.32743	−0.915929
$65$	−1.78074	−0.220873
$66$	0	0
$67$	3.60078	0.439905	0.219952	−	0.975511i	$-0.429410\pi$
$67$	3.60078	0.439905	0.219952	+	0.975511i	$0.429410\pi$
$68$	19.6860	2.38728
$69$	0	0
$70$	0	0
$71$	−8.46050	−1.00408	−0.502039	−	0.864845i	$-0.667416\pi$
$71$	−8.46050	−1.00408	−0.502039	+	0.864845i	$0.667416\pi$
$72$	0	0
$73$	−1.97351	−0.230982	−0.115491	−	0.993309i	$-0.536844\pi$
$73$	−1.97351	−0.230982	−0.115491	+	0.993309i	$0.536844\pi$
$74$	−5.72665	−0.665710
$75$	0	0
$76$	8.00079	0.917754
$77$	0	0
$78$	0	0
$79$	8.16225	0.918325	0.459163	−	0.888352i	$-0.348150\pi$
$79$	8.16225	0.918325	0.459163	+	0.888352i	$0.348150\pi$
$80$	−15.8301	−1.76986
$81$	0	0
$82$	−18.4882	−2.04168
$83$	12.1720	1.33605	0.668025	−	0.744139i	$-0.267140\pi$
$83$	12.1720	1.33605	0.668025	+	0.744139i	$0.267140\pi$
$84$	0	0
$85$	−17.7630	−1.92667
$86$	5.72665	0.617521
$87$	0	0
$88$	2.05408	0.218966
$89$	14.8301	1.57199	0.785996	−	0.618231i	$-0.212151\pi$
$89$	14.8301	1.57199	0.785996	+	0.618231i	$0.212151\pi$
$90$	0	0
$91$	0	0
$92$	−18.8420	−1.96442
$93$	0	0
$94$	15.5411	1.60294
$95$	−7.21926	−0.740681
$96$	0	0
$97$	9.48751	0.963311	0.481655	−	0.876361i	$-0.340036\pi$
$97$	9.48751	0.963311	0.481655	+	0.876361i	$0.340036\pi$
$98$	0	0
$99$	0	0
$100$	33.9794	3.39794
$101$	8.71176	0.866852	0.433426	−	0.901189i	$-0.357304\pi$
$101$	8.71176	0.866852	0.433426	+	0.901189i	$0.357304\pi$
$102$	0	0
$103$	−8.02712	−0.790936	−0.395468	−	0.918480i	$-0.629418\pi$
$103$	−8.02712	−0.790936	−0.395468	+	0.918480i	$0.629418\pi$
$104$	−2.46031	−0.241253
$105$	0	0
$106$	−8.78074	−0.852861
$107$	−12.8420	−1.24148	−0.620742	−	0.784015i	$-0.713169\pi$
$107$	−12.8420	−1.24148	−0.620742	+	0.784015i	$0.713169\pi$
$108$	0	0
$109$	2.60078	0.249109	0.124555	−	0.992213i	$-0.460250\pi$
$109$	2.60078	0.249109	0.124555	+	0.992213i	$0.460250\pi$
$110$	−3.65808	−0.348784
$111$	0	0
$112$	0	0
$113$	13.9502	1.31232	0.656162	−	0.754620i	$-0.272179\pi$
$113$	13.9502	1.31232	0.656162	+	0.754620i	$0.272179\pi$
$114$	0	0
$115$	17.0015	1.58540
$116$	−31.0043	−2.87867
$117$	0	0
$118$	−15.0543	−1.38586
$119$	0	0
$120$	0	0
$121$	−10.8348	−0.984984
$122$	−19.7508	−1.78815
$123$	0	0
$124$	−28.4889	−2.55837
$125$	−12.3698	−1.10639
$126$	0	0
$127$	−15.5438	−1.37929	−0.689643	−	0.724149i	$-0.742233\pi$
$127$	−15.5438	−1.37929	−0.689643	+	0.724149i	$0.742233\pi$
$128$	19.1082	1.68894
$129$	0	0
$130$	4.38151	0.384284
$131$	−8.51392	−0.743864	−0.371932	−	0.928260i	$-0.621305\pi$
$131$	−8.51392	−0.743864	−0.371932	+	0.928260i	$0.621305\pi$
$132$	0	0
$133$	0	0
$134$	−8.85973	−0.765364
$135$	0	0
$136$	−24.5418	−2.10444
$137$	−0.377242	−0.0322300	−0.0161150	−	0.999870i	$-0.505130\pi$
$137$	−0.377242	−0.0322300	−0.0161150	+	0.999870i	$0.505130\pi$
$138$	0	0
$139$	19.0013	1.61167	0.805837	−	0.592138i	$-0.201716\pi$
$139$	19.0013	1.61167	0.805837	+	0.592138i	$0.201716\pi$
$140$	0	0
$141$	0	0
$142$	20.8171	1.74693
$143$	−0.197844	−0.0165446
$144$	0	0
$145$	27.9757	2.32326
$146$	4.85584	0.401872
$147$	0	0
$148$	9.43560	0.775601
$149$	9.70175	0.794798	0.397399	−	0.917646i	$-0.369913\pi$
$149$	9.70175	0.794798	0.397399	+	0.917646i	$0.369913\pi$
$150$	0	0
$151$	−12.8348	−1.04448	−0.522242	−	0.852798i	$-0.674904\pi$
$151$	−12.8348	−1.04448	−0.522242	+	0.852798i	$0.674904\pi$
$152$	−9.97430	−0.809023
$153$	0	0
$154$	0	0
$155$	25.7060	2.06476
$156$	0	0
$157$	−20.9485	−1.67187	−0.835937	−	0.548825i	$-0.815075\pi$
$157$	−20.9485	−1.67187	−0.835937	+	0.548825i	$0.815075\pi$
$158$	−20.0833	−1.59774
$159$	0	0
$160$	1.97351	0.156020
$161$	0	0
$162$	0	0
$163$	−11.1623	−0.874295	−0.437148	−	0.899390i	$-0.644011\pi$
$163$	−11.1623	−0.874295	−0.437148	+	0.899390i	$0.644011\pi$
$164$	30.4624	2.37871
$165$	0	0
$166$	−29.9492	−2.32451
$167$	3.46023	0.267761	0.133880	−	0.990998i	$-0.457256\pi$
$167$	3.46023	0.267761	0.133880	+	0.990998i	$0.457256\pi$
$168$	0	0
$169$	−12.7630	−0.981771
$170$	43.7060	3.35210
$171$	0	0
$172$	−9.43560	−0.719458
$173$	−6.05361	−0.460247	−0.230124	−	0.973161i	$-0.573913\pi$
$173$	−6.05361	−0.460247	−0.230124	+	0.973161i	$0.573913\pi$
$174$	0	0
$175$	0	0
$176$	−1.75876	−0.132571
$177$	0	0
$178$	−36.4896	−2.73501
$179$	−9.13307	−0.682638	−0.341319	−	0.939948i	$-0.610874\pi$
$179$	−9.13307	−0.682638	−0.341319	+	0.939948i	$0.610874\pi$
$180$	0	0
$181$	−11.9478	−0.888074	−0.444037	−	0.896008i	$-0.646454\pi$
$181$	−11.9478	−0.888074	−0.444037	+	0.896008i	$0.646454\pi$
$182$	0	0
$183$	0	0
$184$	23.4897	1.73168
$185$	−8.51392	−0.625956
$186$	0	0
$187$	−1.97351	−0.144318
$188$	−25.6065	−1.86755
$189$	0	0
$190$	17.7630	1.28867
$191$	−9.14027	−0.661367	−0.330683	−	0.943742i	$-0.607279\pi$
$191$	−9.14027	−0.661367	−0.330683	+	0.943742i	$0.607279\pi$
$192$	0	0
$193$	16.9430	1.21958	0.609792	−	0.792562i	$-0.291253\pi$
$193$	16.9430	1.21958	0.609792	+	0.792562i	$0.291253\pi$
$194$	−23.3441	−1.67601
$195$	0	0
$196$	0	0
$197$	21.3173	1.51880	0.759398	−	0.650627i	$-0.225494\pi$
$197$	21.3173	1.51880	0.759398	+	0.650627i	$0.225494\pi$
$198$	0	0
$199$	−9.97430	−0.707060	−0.353530	−	0.935423i	$-0.615019\pi$
$199$	−9.97430	−0.707060	−0.353530	+	0.935423i	$0.615019\pi$
$200$	−42.3609	−2.99537
$201$	0	0
$202$	−21.4353	−1.50818
$203$	0	0
$204$	0	0
$205$	−27.4868	−1.91976
$206$	19.7508	1.37610
$207$	0	0
$208$	2.10658	0.146065
$209$	−0.802077	−0.0554808
$210$	0	0
$211$	4.89183	0.336768	0.168384	−	0.985722i	$-0.446145\pi$
$211$	4.89183	0.336768	0.168384	+	0.985722i	$0.446145\pi$
$212$	14.4677	0.993646
$213$	0	0
$214$	31.5979	2.15998
$215$	8.51392	0.580644
$216$	0	0
$217$	0	0
$218$	−6.39922	−0.433410
$219$	0	0
$220$	6.02728	0.406359
$221$	2.36381	0.159007
$222$	0	0
$223$	23.4088	1.56757	0.783786	−	0.621031i	$-0.213286\pi$
$223$	23.4088	1.56757	0.783786	+	0.621031i	$0.213286\pi$
$224$	0	0
$225$	0	0
$226$	−34.3245	−2.28323
$227$	−6.11839	−0.406092	−0.203046	−	0.979169i	$-0.565084\pi$
$227$	−6.11839	−0.406092	−0.203046	+	0.979169i	$0.565084\pi$
$228$	0	0
$229$	−1.46039	−0.0965052	−0.0482526	−	0.998835i	$-0.515365\pi$
$229$	−1.46039	−0.0965052	−0.0482526	+	0.998835i	$0.515365\pi$
$230$	−41.8323	−2.75834
$231$	0	0
$232$	38.6519	2.53762
$233$	13.2484	0.867934	0.433967	−	0.900929i	$-0.357113\pi$
$233$	13.2484	0.867934	0.433967	+	0.900929i	$0.357113\pi$
$234$	0	0
$235$	23.1052	1.50722
$236$	24.8045	1.61463
$237$	0	0
$238$	0	0
$239$	−19.3887	−1.25415	−0.627076	−	0.778958i	$-0.715748\pi$
$239$	−19.3887	−1.25415	−0.627076	+	0.778958i	$0.715748\pi$
$240$	0	0
$241$	5.05368	0.325536	0.162768	−	0.986664i	$-0.447958\pi$
$241$	5.05368	0.325536	0.162768	+	0.986664i	$0.447958\pi$
$242$	26.6591	1.71371
$243$	0	0
$244$	32.5426	2.08333
$245$	0	0
$246$	0	0
$247$	0.960699	0.0611278
$248$	35.5161	2.25527
$249$	0	0
$250$	30.4360	1.92494
$251$	−15.0928	−0.952647	−0.476324	−	0.879270i	$-0.658031\pi$
$251$	−15.0928	−0.952647	−0.476324	+	0.879270i	$0.658031\pi$
$252$	0	0
$253$	1.88891	0.118755
$254$	38.2455	2.39974
$255$	0	0
$256$	−32.3609	−2.02256
$257$	−7.71184	−0.481051	−0.240526	−	0.970643i	$-0.577320\pi$
$257$	−7.71184	−0.481051	−0.240526	+	0.970643i	$0.577320\pi$
$258$	0	0
$259$	0	0
$260$	−7.21926	−0.447720
$261$	0	0
$262$	20.9485	1.29420
$263$	−4.21206	−0.259727	−0.129864	−	0.991532i	$-0.541454\pi$
$263$	−4.21206	−0.259727	−0.129864	+	0.991532i	$0.541454\pi$
$264$	0	0
$265$	−13.0545	−0.801930
$266$	0	0
$267$	0	0
$268$	14.5979	0.891706
$269$	−20.7507	−1.26519	−0.632596	−	0.774482i	$-0.718011\pi$
$269$	−20.7507	−1.26519	−0.632596	+	0.774482i	$0.718011\pi$
$270$	0	0
$271$	−28.4889	−1.73057	−0.865287	−	0.501276i	$-0.832864\pi$
$271$	−28.4889	−1.73057	−0.865287	+	0.501276i	$0.832864\pi$
$272$	21.0133	1.27412
$273$	0	0
$274$	0.928207	0.0560750
$275$	−3.40642	−0.205415
$276$	0	0
$277$	17.1623	1.03118	0.515590	−	0.856835i	$-0.327573\pi$
$277$	17.1623	1.03118	0.515590	+	0.856835i	$0.327573\pi$
$278$	−46.7529	−2.80405
$279$	0	0
$280$	0	0
$281$	−9.44280	−0.563310	−0.281655	−	0.959516i	$-0.590883\pi$
$281$	−9.44280	−0.563310	−0.281655	+	0.959516i	$0.590883\pi$
$282$	0	0
$283$	16.8684	1.00272	0.501362	−	0.865237i	$-0.332832\pi$
$283$	16.8684	1.00272	0.501362	+	0.865237i	$0.332832\pi$
$284$	−34.2996	−2.03531
$285$	0	0
$286$	0.486796	0.0287849
$287$	0	0
$288$	0	0
$289$	6.57918	0.387011
$290$	−68.8344	−4.04210
$291$	0	0
$292$	−8.00079	−0.468211
$293$	3.72286	0.217492	0.108746	−	0.994070i	$-0.465317\pi$
$293$	3.72286	0.217492	0.108746	+	0.994070i	$0.465317\pi$
$294$	0	0
$295$	−22.3815	−1.30310
$296$	−11.7630	−0.683712
$297$	0	0
$298$	−23.8712	−1.38282
$299$	−2.26247	−0.130842
$300$	0	0
$301$	0	0
$302$	31.5801	1.81723
$303$	0	0
$304$	8.54024	0.489817
$305$	−29.3638	−1.68137
$306$	0	0
$307$	30.5691	1.74467	0.872335	−	0.488908i	$-0.162605\pi$
$307$	30.5691	1.74467	0.872335	+	0.488908i	$0.162605\pi$
$308$	0	0
$309$	0	0
$310$	−63.2498	−3.59235
$311$	10.4348	0.591702	0.295851	−	0.955234i	$-0.404397\pi$
$311$	10.4348	0.591702	0.295851	+	0.955234i	$0.404397\pi$
$312$	0	0
$313$	−0.619860	−0.0350366	−0.0175183	−	0.999847i	$-0.505577\pi$
$313$	−0.619860	−0.0350366	−0.0175183	+	0.999847i	$0.505577\pi$
$314$	51.5440	2.90879
$315$	0	0
$316$	33.0905	1.86148
$317$	−10.2484	−0.575610	−0.287805	−	0.957689i	$-0.592925\pi$
$317$	−10.2484	−0.575610	−0.287805	+	0.957689i	$0.592925\pi$
$318$	0	0
$319$	3.10817	0.174024
$320$	26.8043	1.49841
$321$	0	0
$322$	0	0
$323$	9.58307	0.533216
$324$	0	0
$325$	4.08009	0.226323
$326$	27.4648	1.52113
$327$	0	0
$328$	−37.9764	−2.09689
$329$	0	0
$330$	0	0
$331$	−20.3638	−1.11930	−0.559648	−	0.828730i	$-0.689064\pi$
$331$	−20.3638	−1.11930	−0.559648	+	0.828730i	$0.689064\pi$
$332$	49.3463	2.70823
$333$	0	0
$334$	−8.51392	−0.465861
$335$	−13.1719	−0.719658
$336$	0	0
$337$	−5.71187	−0.311145	−0.155573	−	0.987824i	$-0.549722\pi$
$337$	−5.71187	−0.311145	−0.155573	+	0.987824i	$0.549722\pi$
$338$	31.4035	1.70812
$339$	0	0
$340$	−72.0128	−3.90544
$341$	2.85600	0.154661
$342$	0	0
$343$	0	0
$344$	11.7630	0.634220
$345$	0	0
$346$	14.8949	0.800756
$347$	−8.88132	−0.476774	−0.238387	−	0.971170i	$-0.576619\pi$
$347$	−8.88132	−0.476774	−0.238387	+	0.971170i	$0.576619\pi$
$348$	0	0
$349$	−20.9749	−1.12276	−0.561379	−	0.827559i	$-0.689729\pi$
$349$	−20.9749	−1.12276	−0.561379	+	0.827559i	$0.689729\pi$
$350$	0	0
$351$	0	0
$352$	0.219262	0.0116867
$353$	−14.7654	−0.785881	−0.392941	−	0.919564i	$-0.628542\pi$
$353$	−14.7654	−0.785881	−0.392941	+	0.919564i	$0.628542\pi$
$354$	0	0
$355$	30.9492	1.64261
$356$	60.1227	3.18649
$357$	0	0
$358$	22.4720	1.18768
$359$	−7.21206	−0.380638	−0.190319	−	0.981722i	$-0.560952\pi$
$359$	−7.21206	−0.380638	−0.190319	+	0.981722i	$0.560952\pi$
$360$	0	0
$361$	−15.1052	−0.795013
$362$	29.3977	1.54511
$363$	0	0
$364$	0	0
$365$	7.21926	0.377873
$366$	0	0
$367$	10.9742	0.572850	0.286425	−	0.958103i	$-0.407533\pi$
$367$	10.9742	0.572850	0.286425	+	0.958103i	$0.407533\pi$
$368$	−20.1124	−1.04843
$369$	0	0
$370$	20.9485	1.08906
$371$	0	0
$372$	0	0
$373$	−0.543767	−0.0281552	−0.0140776	−	0.999901i	$-0.504481\pi$
$373$	−0.543767	−0.0281552	−0.0140776	+	0.999901i	$0.504481\pi$
$374$	4.85584	0.251090
$375$	0	0
$376$	31.9228	1.64629
$377$	−3.72286	−0.191737
$378$	0	0
$379$	−22.6912	−1.16557	−0.582785	−	0.812626i	$-0.698037\pi$
$379$	−22.6912	−1.16557	−0.582785	+	0.812626i	$0.698037\pi$
$380$	−29.2675	−1.50139
$381$	0	0
$382$	22.4897	1.15067
$383$	35.7139	1.82489	0.912447	−	0.409194i	$-0.134190\pi$
$383$	35.7139	1.82489	0.912447	+	0.409194i	$0.134190\pi$
$384$	0	0
$385$	0	0
$386$	−41.6883	−2.12188
$387$	0	0
$388$	38.4632	1.95267
$389$	−38.6591	−1.96010	−0.980048	−	0.198761i	$-0.936308\pi$
$389$	−38.6591	−1.96010	−0.980048	+	0.198761i	$0.936308\pi$
$390$	0	0
$391$	−22.5683	−1.14133
$392$	0	0
$393$	0	0
$394$	−52.4513	−2.64246
$395$	−29.8581	−1.50233
$396$	0	0
$397$	11.9478	0.599644	0.299822	−	0.953995i	$-0.403073\pi$
$397$	11.9478	0.599644	0.299822	+	0.953995i	$0.403073\pi$
$398$	24.5418	1.23017
$399$	0	0
$400$	36.2704	1.81352
$401$	−32.3566	−1.61581	−0.807906	−	0.589311i	$-0.799399\pi$
$401$	−32.3566	−1.61581	−0.807906	+	0.589311i	$0.799399\pi$
$402$	0	0
$403$	−3.42082	−0.170403
$404$	35.3182	1.75715
$405$	0	0
$406$	0	0
$407$	−0.945916	−0.0468873
$408$	0	0
$409$	−18.9750	−0.938254	−0.469127	−	0.883131i	$-0.655431\pi$
$409$	−18.9750	−0.938254	−0.469127	+	0.883131i	$0.655431\pi$
$410$	67.6313	3.34007
$411$	0	0
$412$	−32.5426	−1.60326
$413$	0	0
$414$	0	0
$415$	−44.5261	−2.18570
$416$	−0.262624	−0.0128762
$417$	0	0
$418$	1.97351	0.0965277
$419$	−17.2905	−0.844694	−0.422347	−	0.906434i	$-0.638794\pi$
$419$	−17.2905	−0.844694	−0.422347	+	0.906434i	$0.638794\pi$
$420$	0	0
$421$	18.6008	0.906546	0.453273	−	0.891372i	$-0.350256\pi$
$421$	18.6008	0.906546	0.453273	+	0.891372i	$0.350256\pi$
$422$	−12.0364	−0.585922
$423$	0	0
$424$	−18.0364	−0.875924
$425$	40.6993	1.97421
$426$	0	0
$427$	0	0
$428$	−52.0626	−2.51654
$429$	0	0
$430$	−20.9485	−1.01023
$431$	15.8784	0.764835	0.382418	−	0.923990i	$-0.375092\pi$
$431$	15.8784	0.764835	0.382418	+	0.923990i	$0.375092\pi$
$432$	0	0
$433$	−40.4367	−1.94326	−0.971631	−	0.236501i	$-0.923999\pi$
$433$	−40.4367	−1.94326	−0.971631	+	0.236501i	$0.923999\pi$
$434$	0	0
$435$	0	0
$436$	10.5438	0.504955
$437$	−9.17223	−0.438767
$438$	0	0
$439$	−12.4609	−0.594728	−0.297364	−	0.954764i	$-0.596108\pi$
$439$	−12.4609	−0.594728	−0.297364	+	0.954764i	$0.596108\pi$
$440$	−7.51399	−0.358216
$441$	0	0
$442$	−5.81616	−0.276646
$443$	8.23073	0.391054	0.195527	−	0.980698i	$-0.437358\pi$
$443$	8.23073	0.391054	0.195527	+	0.980698i	$0.437358\pi$
$444$	0	0
$445$	−54.2498	−2.57169
$446$	−57.5976	−2.72732
$447$	0	0
$448$	0	0
$449$	−5.64474	−0.266392	−0.133196	−	0.991090i	$-0.542524\pi$
$449$	−5.64474	−0.266392	−0.133196	+	0.991090i	$0.542524\pi$
$450$	0	0
$451$	−3.05384	−0.143800
$452$	56.5552	2.66013
$453$	0	0
$454$	15.0543	0.706534
$455$	0	0
$456$	0	0
$457$	5.06887	0.237112	0.118556	−	0.992947i	$-0.462174\pi$
$457$	5.06887	0.237112	0.118556	+	0.992947i	$0.462174\pi$
$458$	3.59330	0.167904
$459$	0	0
$460$	68.9255	3.21367
$461$	−7.77662	−0.362193	−0.181097	−	0.983465i	$-0.557965\pi$
$461$	−7.77662	−0.362193	−0.181097	+	0.983465i	$0.557965\pi$
$462$	0	0
$463$	−9.17996	−0.426629	−0.213314	−	0.976984i	$-0.568426\pi$
$463$	−9.17996	−0.426629	−0.213314	+	0.976984i	$0.568426\pi$
$464$	−33.0947	−1.53638
$465$	0	0
$466$	−32.5979	−1.51007
$467$	−13.7654	−0.636989	−0.318494	−	0.947925i	$-0.603177\pi$
$467$	−13.7654	−0.636989	−0.318494	+	0.947925i	$0.603177\pi$
$468$	0	0
$469$	0	0
$470$	−56.8506	−2.62232
$471$	0	0
$472$	−30.9228	−1.42334
$473$	0.945916	0.0434933
$474$	0	0
$475$	16.5410	0.758955
$476$	0	0
$477$	0	0
$478$	47.7060	2.18202
$479$	8.71176	0.398050	0.199025	−	0.979994i	$-0.436222\pi$
$479$	8.71176	0.398050	0.199025	+	0.979994i	$0.436222\pi$
$480$	0	0
$481$	1.13298	0.0516597
$482$	−12.4346	−0.566381
$483$	0	0
$484$	−43.9253	−1.99660
$485$	−34.7060	−1.57592
$486$	0	0
$487$	−18.0364	−0.817306	−0.408653	−	0.912690i	$-0.634001\pi$
$487$	−18.0364	−0.817306	−0.408653	+	0.912690i	$0.634001\pi$
$488$	−40.5697	−1.83651
$489$	0	0
$490$	0	0
$491$	−2.04689	−0.0923747	−0.0461874	−	0.998933i	$-0.514707\pi$
$491$	−2.04689	−0.0923747	−0.0461874	+	0.998933i	$0.514707\pi$
$492$	0	0
$493$	−37.1358	−1.67251
$494$	−2.36381	−0.106353
$495$	0	0
$496$	−30.4097	−1.36544
$497$	0	0
$498$	0	0
$499$	−39.0875	−1.74980	−0.874899	−	0.484305i	$-0.839072\pi$
$499$	−39.0875	−1.74980	−0.874899	+	0.484305i	$0.839072\pi$
$500$	−50.1484	−2.24270
$501$	0	0
$502$	37.1358	1.65745
$503$	5.11846	0.228221	0.114111	−	0.993468i	$-0.463598\pi$
$503$	5.11846	0.228221	0.114111	+	0.993468i	$0.463598\pi$
$504$	0	0
$505$	−31.8683	−1.41812
$506$	−4.64766	−0.206614
$507$	0	0
$508$	−63.0157	−2.79587
$509$	−29.5272	−1.30877	−0.654386	−	0.756161i	$-0.727073\pi$
$509$	−29.5272	−1.30877	−0.654386	+	0.756161i	$0.727073\pi$
$510$	0	0
$511$	0	0
$512$	41.4078	1.82998
$513$	0	0
$514$	18.9750	0.836952
$515$	29.3638	1.29392
$516$	0	0
$517$	2.56705	0.112899
$518$	0	0
$519$	0	0
$520$	9.00000	0.394676
$521$	1.06470	0.0466454	0.0233227	−	0.999728i	$-0.492575\pi$
$521$	1.06470	0.0466454	0.0233227	+	0.999728i	$0.492575\pi$
$522$	0	0
$523$	−13.3819	−0.585149	−0.292574	−	0.956243i	$-0.594512\pi$
$523$	−13.3819	−0.585149	−0.292574	+	0.956243i	$0.594512\pi$
$524$	−34.5161	−1.50784
$525$	0	0
$526$	10.3638	0.451883
$527$	−34.1230	−1.48642
$528$	0	0
$529$	−1.39922	−0.0608358
$530$	32.1206	1.39523
$531$	0	0
$532$	0	0
$533$	3.65779	0.158436
$534$	0	0
$535$	46.9771	2.03100
$536$	−18.1986	−0.786061
$537$	0	0
$538$	51.0572	2.20123
$539$	0	0
$540$	0	0
$541$	−34.0875	−1.46554	−0.732769	−	0.680478i	$-0.761772\pi$
$541$	−34.0875	−1.46554	−0.732769	+	0.680478i	$0.761772\pi$
$542$	70.0970	3.01092
$543$	0	0
$544$	−2.61970	−0.112319
$545$	−9.51384	−0.407528
$546$	0	0
$547$	−5.94299	−0.254104	−0.127052	−	0.991896i	$-0.540551\pi$
$547$	−5.94299	−0.254104	−0.127052	+	0.991896i	$0.540551\pi$
$548$	−1.52937	−0.0653316
$549$	0	0
$550$	8.38151	0.357389
$551$	−15.0928	−0.642974
$552$	0	0
$553$	0	0
$554$	−42.2278	−1.79409
$555$	0	0
$556$	77.0331	3.26693
$557$	30.0803	1.27454	0.637272	−	0.770639i	$-0.280063\pi$
$557$	30.0803	1.27454	0.637272	+	0.770639i	$0.280063\pi$
$558$	0	0
$559$	−1.13298	−0.0479202
$560$	0	0
$561$	0	0
$562$	23.2340	0.980069
$563$	19.6212	0.826935	0.413468	−	0.910519i	$-0.364317\pi$
$563$	19.6212	0.826935	0.413468	+	0.910519i	$0.364317\pi$
$564$	0	0
$565$	−51.0308	−2.14688
$566$	−41.5049	−1.74458
$567$	0	0
$568$	42.7601	1.79417
$569$	1.37432	0.0576144	0.0288072	−	0.999585i	$-0.490829\pi$
$569$	1.37432	0.0576144	0.0288072	+	0.999585i	$0.490829\pi$
$570$	0	0
$571$	17.3815	0.727394	0.363697	−	0.931517i	$-0.381514\pi$
$571$	17.3815	0.727394	0.363697	+	0.931517i	$0.381514\pi$
$572$	−0.802077	−0.0335365
$573$	0	0
$574$	0	0
$575$	−38.9545	−1.62451
$576$	0	0
$577$	−27.0548	−1.12631	−0.563153	−	0.826353i	$-0.690412\pi$
$577$	−27.0548	−1.12631	−0.563153	+	0.826353i	$0.690412\pi$
$578$	−16.1881	−0.673337
$579$	0	0
$580$	113.416	4.70934
$581$	0	0
$582$	0	0
$583$	−1.45038	−0.0600687
$584$	9.97430	0.412740
$585$	0	0
$586$	−9.16010	−0.378400
$587$	−7.51399	−0.310136	−0.155068	−	0.987904i	$-0.549560\pi$
$587$	−7.51399	−0.310136	−0.155068	+	0.987904i	$0.549560\pi$
$588$	0	0
$589$	−13.8683	−0.571432
$590$	55.0698	2.26719
$591$	0	0
$592$	10.0718	0.413948
$593$	−35.5808	−1.46113	−0.730565	−	0.682843i	$-0.760743\pi$
$593$	−35.5808	−1.46113	−0.730565	+	0.682843i	$0.760743\pi$
$594$	0	0
$595$	0	0
$596$	39.3317	1.61109
$597$	0	0
$598$	5.56681	0.227644
$599$	11.4821	0.469146	0.234573	−	0.972099i	$-0.424631\pi$
$599$	11.4821	0.469146	0.234573	+	0.972099i	$0.424631\pi$
$600$	0	0
$601$	−0.380061	−0.0155030	−0.00775150	−	0.999970i	$-0.502467\pi$
$601$	−0.380061	−0.0155030	−0.00775150	+	0.999970i	$0.502467\pi$
$602$	0	0
$603$	0	0
$604$	−52.0335	−2.11721
$605$	39.6346	1.61138
$606$	0	0
$607$	−18.5409	−0.752551	−0.376275	−	0.926508i	$-0.622795\pi$
$607$	−18.5409	−0.752551	−0.376275	+	0.926508i	$0.622795\pi$
$608$	−1.06470	−0.0431793
$609$	0	0
$610$	72.2498	2.92531
$611$	−3.07472	−0.124390
$612$	0	0
$613$	7.32451	0.295834	0.147917	−	0.989000i	$-0.452743\pi$
$613$	7.32451	0.295834	0.147917	+	0.989000i	$0.452743\pi$
$614$	−75.2154	−3.03545
$615$	0	0
$616$	0	0
$617$	25.4854	1.02600	0.513002	−	0.858387i	$-0.328533\pi$
$617$	25.4854	1.02600	0.513002	+	0.858387i	$0.328533\pi$
$618$	0	0
$619$	32.8963	1.32222	0.661108	−	0.750291i	$-0.270087\pi$
$619$	32.8963	1.32222	0.661108	+	0.750291i	$0.270087\pi$
$620$	104.214	4.18535
$621$	0	0
$622$	−25.6748	−1.02947
$623$	0	0
$624$	0	0
$625$	3.34221	0.133689
$626$	1.52517	0.0609580
$627$	0	0
$628$	−84.9271	−3.38896
$629$	11.3016	0.450626
$630$	0	0
$631$	29.8683	1.18904	0.594519	−	0.804082i	$-0.297343\pi$
$631$	29.8683	1.18904	0.594519	+	0.804082i	$0.297343\pi$
$632$	−41.2527	−1.64094
$633$	0	0
$634$	25.2163	1.00147
$635$	56.8603	2.25643
$636$	0	0
$637$	0	0
$638$	−7.64766	−0.302774
$639$	0	0
$640$	−69.8991	−2.76301
$641$	−11.4605	−0.452663	−0.226331	−	0.974050i	$-0.572673\pi$
$641$	−11.4605	−0.452663	−0.226331	+	0.974050i	$0.572673\pi$
$642$	0	0
$643$	−17.3816	−0.685462	−0.342731	−	0.939434i	$-0.611352\pi$
$643$	−17.3816	−0.685462	−0.342731	+	0.939434i	$0.611352\pi$
$644$	0	0
$645$	0	0
$646$	−23.5792	−0.927711
$647$	−25.3439	−0.996372	−0.498186	−	0.867070i	$-0.666000\pi$
$647$	−25.3439	−0.996372	−0.498186	+	0.867070i	$0.666000\pi$
$648$	0	0
$649$	−2.48664	−0.0976091
$650$	−10.0391	−0.393765
$651$	0	0
$652$	−45.2527	−1.77223
$653$	−14.0833	−0.551121	−0.275560	−	0.961284i	$-0.588863\pi$
$653$	−14.0833	−0.551121	−0.275560	+	0.961284i	$0.588863\pi$
$654$	0	0
$655$	31.1445	1.21692
$656$	32.5163	1.26955
$657$	0	0
$658$	0	0
$659$	−38.1708	−1.48692	−0.743462	−	0.668779i	$-0.766817\pi$
$659$	−38.1708	−1.48692	−0.743462	+	0.668779i	$0.766817\pi$
$660$	0	0
$661$	−0.353732	−0.0137586	−0.00687930	−	0.999976i	$-0.502190\pi$
$661$	−0.353732	−0.0137586	−0.00687930	+	0.999976i	$0.502190\pi$
$662$	50.1052	1.94740
$663$	0	0
$664$	−61.5183	−2.38737
$665$	0	0
$666$	0	0
$667$	35.5438	1.37626
$668$	14.0281	0.542762
$669$	0	0
$670$	32.4096	1.25209
$671$	−3.26239	−0.125943
$672$	0	0
$673$	−21.1111	−0.813773	−0.406886	−	0.913479i	$-0.633386\pi$
$673$	−21.1111	−0.813773	−0.406886	+	0.913479i	$0.633386\pi$
$674$	14.0541	0.541343
$675$	0	0
$676$	−51.7424	−1.99009
$677$	−21.1464	−0.812721	−0.406361	−	0.913713i	$-0.633202\pi$
$677$	−21.1464	−0.812721	−0.406361	+	0.913713i	$0.633202\pi$
$678$	0	0
$679$	0	0
$680$	89.7758	3.44275
$681$	0	0
$682$	−7.02720	−0.269085
$683$	−34.7716	−1.33050	−0.665249	−	0.746622i	$-0.731674\pi$
$683$	−34.7716	−1.33050	−0.665249	+	0.746622i	$0.731674\pi$
$684$	0	0
$685$	1.37998	0.0527264
$686$	0	0
$687$	0	0
$688$	−10.0718	−0.383984
$689$	1.73722	0.0661827
$690$	0	0
$691$	34.6492	1.31812	0.659059	−	0.752091i	$-0.270955\pi$
$691$	34.6492	1.31812	0.659059	+	0.752091i	$0.270955\pi$
$692$	−24.5418	−0.932940
$693$	0	0
$694$	21.8525	0.829511
$695$	−69.5083	−2.63660
$696$	0	0
$697$	36.4868	1.38203
$698$	51.6087	1.95342
$699$	0	0
$700$	0	0
$701$	48.6050	1.83579	0.917894	−	0.396826i	$-0.129889\pi$
$701$	48.6050	1.83579	0.917894	+	0.396826i	$0.129889\pi$
$702$	0	0
$703$	4.59322	0.173236
$704$	2.97802	0.112238
$705$	0	0
$706$	36.3303	1.36731
$707$	0	0
$708$	0	0
$709$	4.10817	0.154286	0.0771428	−	0.997020i	$-0.475420\pi$
$709$	4.10817	0.154286	0.0771428	+	0.997020i	$0.475420\pi$
$710$	−76.1506	−2.85788
$711$	0	0
$712$	−74.9528	−2.80898
$713$	32.6601	1.22313
$714$	0	0
$715$	0.723729	0.0270659
$716$	−37.0263	−1.38374
$717$	0	0
$718$	17.7453	0.662249
$719$	−48.2816	−1.80060	−0.900299	−	0.435271i	$-0.856653\pi$
$719$	−48.2816	−1.80060	−0.900299	+	0.435271i	$0.856653\pi$
$720$	0	0
$721$	0	0
$722$	37.1665	1.38319
$723$	0	0
$724$	−48.4375	−1.80016
$725$	−64.0990	−2.38058
$726$	0	0
$727$	−41.0302	−1.52173	−0.760863	−	0.648913i	$-0.775224\pi$
$727$	−41.0302	−1.52173	−0.760863	+	0.648913i	$0.775224\pi$
$728$	0	0
$729$	0	0
$730$	−17.7630	−0.657439
$731$	−11.3016	−0.418006
$732$	0	0
$733$	30.5428	1.12812	0.564062	−	0.825733i	$-0.309238\pi$
$733$	30.5428	1.12812	0.564062	+	0.825733i	$0.309238\pi$
$734$	−27.0021	−0.996667
$735$	0	0
$736$	2.50739	0.0924237
$737$	−1.46343	−0.0539061
$738$	0	0
$739$	23.8200	0.876234	0.438117	−	0.898918i	$-0.355646\pi$
$739$	23.8200	0.876234	0.438117	+	0.898918i	$0.355646\pi$
$740$	−34.5161	−1.26884
$741$	0	0
$742$	0	0
$743$	−10.5218	−0.386007	−0.193003	−	0.981198i	$-0.561823\pi$
$743$	−10.5218	−0.386007	−0.193003	+	0.981198i	$0.561823\pi$
$744$	0	0
$745$	−35.4897	−1.30024
$746$	1.33794	0.0489855
$747$	0	0
$748$	−8.00079	−0.292538
$749$	0	0
$750$	0	0
$751$	−10.2704	−0.374773	−0.187386	−	0.982286i	$-0.560002\pi$
$751$	−10.2704	−0.374773	−0.187386	+	0.982286i	$0.560002\pi$
$752$	−27.3330	−0.996734
$753$	0	0
$754$	9.16010	0.333591
$755$	46.9507	1.70871
$756$	0	0
$757$	8.03930	0.292193	0.146097	−	0.989270i	$-0.453329\pi$
$757$	8.03930	0.292193	0.146097	+	0.989270i	$0.453329\pi$
$758$	55.8319	2.02791
$759$	0	0
$760$	36.4868	1.32351
$761$	−27.6604	−1.00269	−0.501345	−	0.865247i	$-0.667161\pi$
$761$	−27.6604	−1.00269	−0.501345	+	0.865247i	$0.667161\pi$
$762$	0	0
$763$	0	0
$764$	−37.0554	−1.34062
$765$	0	0
$766$	−87.8742	−3.17502
$767$	2.97841	0.107544
$768$	0	0
$769$	−33.9226	−1.22328	−0.611640	−	0.791136i	$-0.709490\pi$
$769$	−33.9226	−1.22328	−0.611640	+	0.791136i	$0.709490\pi$
$770$	0	0
$771$	0	0
$772$	68.6883	2.47215
$773$	28.5956	1.02851	0.514256	−	0.857637i	$-0.328068\pi$
$773$	28.5956	1.02851	0.514256	+	0.857637i	$0.328068\pi$
$774$	0	0
$775$	−58.8986	−2.11570
$776$	−47.9507	−1.72133
$777$	0	0
$778$	95.1210	3.41025
$779$	14.8290	0.531303
$780$	0	0
$781$	3.43852	0.123040
$782$	55.5294	1.98573
$783$	0	0
$784$	0	0
$785$	76.6313	2.73509
$786$	0	0
$787$	46.0035	1.63985	0.819923	−	0.572473i	$-0.194016\pi$
$787$	46.0035	1.63985	0.819923	+	0.572473i	$0.194016\pi$
$788$	86.4222	3.07866
$789$	0	0
$790$	73.4661	2.61381
$791$	0	0
$792$	0	0
$793$	3.90757	0.138762
$794$	−29.3977	−1.04328
$795$	0	0
$796$	−40.4367	−1.43324
$797$	36.9117	1.30748	0.653739	−	0.756720i	$-0.273199\pi$
$797$	36.9117	1.30748	0.653739	+	0.756720i	$0.273199\pi$
$798$	0	0
$799$	−30.6706	−1.08505
$800$	−4.52179	−0.159869
$801$	0	0
$802$	79.6136	2.81125
$803$	0.802077	0.0283047
$804$	0	0
$805$	0	0
$806$	8.41693	0.296474
$807$	0	0
$808$	−44.0300	−1.54897
$809$	−47.6887	−1.67665	−0.838323	−	0.545174i	$-0.816463\pi$
$809$	−47.6887	−1.67665	−0.838323	+	0.545174i	$0.816463\pi$
$810$	0	0
$811$	−6.02728	−0.211646	−0.105823	−	0.994385i	$-0.533748\pi$
$811$	−6.02728	−0.211646	−0.105823	+	0.994385i	$0.533748\pi$
$812$	0	0
$813$	0	0
$814$	2.32743	0.0815764
$815$	40.8324	1.43030
$816$	0	0
$817$	−4.59322	−0.160696
$818$	46.6881	1.63241
$819$	0	0
$820$	−111.434	−3.89143
$821$	39.2642	1.37033	0.685165	−	0.728388i	$-0.259730\pi$
$821$	39.2642	1.37033	0.685165	+	0.728388i	$0.259730\pi$
$822$	0	0
$823$	22.7630	0.793469	0.396735	−	0.917933i	$-0.370143\pi$
$823$	22.7630	0.793469	0.396735	+	0.917933i	$0.370143\pi$
$824$	40.5697	1.41331
$825$	0	0
$826$	0	0
$827$	28.9286	1.00595	0.502973	−	0.864302i	$-0.332240\pi$
$827$	28.9286	1.00595	0.502973	+	0.864302i	$0.332240\pi$
$828$	0	0
$829$	33.9767	1.18006	0.590029	−	0.807382i	$-0.299116\pi$
$829$	33.9767	1.18006	0.590029	+	0.807382i	$0.299116\pi$
$830$	109.557	3.80276
$831$	0	0
$832$	−3.56697	−0.123662
$833$	0	0
$834$	0	0
$835$	−12.6578	−0.438041
$836$	−3.25169	−0.112462
$837$	0	0
$838$	42.5432	1.46963
$839$	38.0411	1.31333	0.656663	−	0.754184i	$-0.271967\pi$
$839$	38.0411	1.31333	0.656663	+	0.754184i	$0.271967\pi$
$840$	0	0
$841$	29.4868	1.01678
$842$	−45.7673	−1.57725
$843$	0	0
$844$	19.8319	0.682642
$845$	46.6881	1.60612
$846$	0	0
$847$	0	0
$848$	15.4432	0.530321
$849$	0	0
$850$	−100.141	−3.43480
$851$	−10.8171	−0.370806
$852$	0	0
$853$	−9.81491	−0.336056	−0.168028	−	0.985782i	$-0.553740\pi$
$853$	−9.81491	−0.336056	−0.168028	+	0.985782i	$0.553740\pi$
$854$	0	0
$855$	0	0
$856$	64.9046	2.21840
$857$	26.7360	0.913285	0.456642	−	0.889650i	$-0.349052\pi$
$857$	26.7360	0.913285	0.456642	+	0.889650i	$0.349052\pi$
$858$	0	0
$859$	18.4882	0.630810	0.315405	−	0.948957i	$-0.397860\pi$
$859$	18.4882	0.630810	0.315405	+	0.948957i	$0.397860\pi$
$860$	34.5161	1.17699
$861$	0	0
$862$	−39.0689	−1.33069
$863$	9.14786	0.311397	0.155698	−	0.987805i	$-0.450237\pi$
$863$	9.14786	0.311397	0.155698	+	0.987805i	$0.450237\pi$
$864$	0	0
$865$	22.1445	0.752937
$866$	99.4946	3.38097
$867$	0	0
$868$	0	0
$869$	−3.31731	−0.112532
$870$	0	0
$871$	1.75285	0.0593929
$872$	−13.1445	−0.445130
$873$	0	0
$874$	22.5683	0.763385
$875$	0	0
$876$	0	0
$877$	22.6883	0.766130	0.383065	−	0.923721i	$-0.374869\pi$
$877$	22.6883	0.766130	0.383065	+	0.923721i	$0.374869\pi$
$878$	30.6602	1.03473
$879$	0	0
$880$	6.43367	0.216879
$881$	15.3696	0.517815	0.258907	−	0.965902i	$-0.416638\pi$
$881$	15.3696	0.517815	0.258907	+	0.965902i	$0.416638\pi$
$882$	0	0
$883$	29.6372	0.997370	0.498685	−	0.866783i	$-0.333817\pi$
$883$	29.6372	0.997370	0.498685	+	0.866783i	$0.333817\pi$
$884$	9.58307	0.322313
$885$	0	0
$886$	−20.2518	−0.680371
$887$	−18.7650	−0.630069	−0.315034	−	0.949080i	$-0.602016\pi$
$887$	−18.7650	−0.630069	−0.315034	+	0.949080i	$0.602016\pi$
$888$	0	0
$889$	0	0
$890$	133.482	4.47432
$891$	0	0
$892$	94.9014	3.17753
$893$	−12.4652	−0.417131
$894$	0	0
$895$	33.4095	1.11676
$896$	0	0
$897$	0	0
$898$	13.8889	0.463479
$899$	53.7416	1.79238
$900$	0	0
$901$	17.3289	0.577310
$902$	7.51399	0.250189
$903$	0	0
$904$	−70.5054	−2.34498
$905$	43.7060	1.45284
$906$	0	0
$907$	−31.1082	−1.03293	−0.516465	−	0.856308i	$-0.672752\pi$
$907$	−31.1082	−1.03293	−0.516465	+	0.856308i	$0.672752\pi$
$908$	−24.8045	−0.823165
$909$	0	0
$910$	0	0
$911$	−17.4753	−0.578982	−0.289491	−	0.957181i	$-0.593486\pi$
$911$	−17.4753	−0.578982	−0.289491	+	0.957181i	$0.593486\pi$
$912$	0	0
$913$	−4.94695	−0.163720
$914$	−12.4720	−0.412536
$915$	0	0
$916$	−5.92054	−0.195620
$917$	0	0
$918$	0	0
$919$	42.1986	1.39200	0.696002	−	0.718040i	$-0.254960\pi$
$919$	42.1986	1.39200	0.696002	+	0.718040i	$0.254960\pi$
$920$	−85.9270	−2.83293
$921$	0	0
$922$	19.1344	0.630158
$923$	−4.11854	−0.135564
$924$	0	0
$925$	19.5074	0.641399
$926$	22.5873	0.742266
$927$	0	0
$928$	4.12588	0.135439
$929$	48.4111	1.58832	0.794159	−	0.607710i	$-0.207912\pi$
$929$	48.4111	1.58832	0.794159	+	0.607710i	$0.207912\pi$
$930$	0	0
$931$	0	0
$932$	53.7103	1.75934
$933$	0	0
$934$	33.8699	1.10826
$935$	7.21926	0.236095
$936$	0	0
$937$	22.6750	0.740762	0.370381	−	0.928880i	$-0.379227\pi$
$937$	22.6750	0.740762	0.370381	+	0.928880i	$0.379227\pi$
$938$	0	0
$939$	0	0
$940$	93.6706	3.05520
$941$	−49.9505	−1.62834	−0.814170	−	0.580627i	$-0.802808\pi$
$941$	−49.9505	−1.62834	−0.814170	+	0.580627i	$0.802808\pi$
$942$	0	0
$943$	−34.9225	−1.13723
$944$	26.4769	0.861749
$945$	0	0
$946$	−2.32743	−0.0756713
$947$	−17.7123	−0.575571	−0.287786	−	0.957695i	$-0.592919\pi$
$947$	−17.7123	−0.575571	−0.287786	+	0.957695i	$0.592919\pi$
$948$	0	0
$949$	−0.960699	−0.0311856
$950$	−40.6993	−1.32046
$951$	0	0
$952$	0	0
$953$	−38.0229	−1.23168	−0.615842	−	0.787870i	$-0.711184\pi$
$953$	−38.0229	−1.23168	−0.615842	+	0.787870i	$0.711184\pi$
$954$	0	0
$955$	33.4358	1.08196
$956$	−78.6035	−2.54222
$957$	0	0
$958$	−21.4353	−0.692544
$959$	0	0
$960$	0	0
$961$	18.3815	0.592952
$962$	−2.78771	−0.0898795
$963$	0	0
$964$	20.4881	0.659876
$965$	−61.9787	−1.99517
$966$	0	0
$967$	−31.0128	−0.997305	−0.498652	−	0.866802i	$-0.666172\pi$
$967$	−31.0128	−0.997305	−0.498652	+	0.866802i	$0.666172\pi$
$968$	54.7601	1.76006
$969$	0	0
$970$	85.3943	2.74185
$971$	−14.5675	−0.467494	−0.233747	−	0.972297i	$-0.575099\pi$
$971$	−14.5675	−0.467494	−0.233747	+	0.972297i	$0.575099\pi$
$972$	0	0
$973$	0	0
$974$	44.3786	1.42198
$975$	0	0
$976$	34.7368	1.11190
$977$	47.8797	1.53181	0.765904	−	0.642955i	$-0.222292\pi$
$977$	47.8797	1.53181	0.765904	+	0.642955i	$0.222292\pi$
$978$	0	0
$979$	−6.02728	−0.192633
$980$	0	0
$981$	0	0
$982$	5.03638	0.160717
$983$	−44.4257	−1.41696	−0.708479	−	0.705732i	$-0.750618\pi$
$983$	−44.4257	−1.41696	−0.708479	+	0.705732i	$0.750618\pi$
$984$	0	0
$985$	−77.9803	−2.48466
$986$	91.3729	2.90991
$987$	0	0
$988$	3.89476	0.123909
$989$	10.8171	0.343964
$990$	0	0
$991$	−41.6156	−1.32196	−0.660981	−	0.750403i	$-0.729860\pi$
$991$	−41.6156	−1.32196	−0.660981	+	0.750403i	$0.729860\pi$
$992$	3.79114	0.120369
$993$	0	0
$994$	0	0
$995$	36.4868	1.15671
$996$	0	0
$997$	−13.0281	−0.412606	−0.206303	−	0.978488i	$-0.566143\pi$
$997$	−13.0281	−0.412606	−0.206303	+	0.978488i	$0.566143\pi$
$998$	96.1751	3.04437
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bd.1.1		6
3.2	odd	2		3969.2.a.be.1.6		6
7.6	odd	2	inner	3969.2.a.bd.1.2		6
9.2	odd	6		441.2.f.g.148.1	✓	12
9.4	even	3		1323.2.f.g.883.6		12
9.5	odd	6		441.2.f.g.295.1	yes	12
9.7	even	3		1323.2.f.g.442.6		12
21.20	even	2		3969.2.a.be.1.5		6
63.2	odd	6		441.2.g.g.67.2		12
63.4	even	3		1323.2.g.g.667.5		12
63.5	even	6		441.2.h.g.214.5		12
63.11	odd	6		441.2.h.g.373.6		12
63.13	odd	6		1323.2.f.g.883.5		12
63.16	even	3		1323.2.g.g.361.5		12
63.20	even	6		441.2.f.g.148.2	yes	12
63.23	odd	6		441.2.h.g.214.6		12
63.25	even	3		1323.2.h.g.226.2		12
63.31	odd	6		1323.2.g.g.667.6		12
63.32	odd	6		441.2.g.g.79.2		12
63.34	odd	6		1323.2.f.g.442.5		12
63.38	even	6		441.2.h.g.373.5		12
63.40	odd	6		1323.2.h.g.802.1		12
63.41	even	6		441.2.f.g.295.2	yes	12
63.47	even	6		441.2.g.g.67.1		12
63.52	odd	6		1323.2.h.g.226.1		12
63.58	even	3		1323.2.h.g.802.2		12
63.59	even	6		441.2.g.g.79.1		12
63.61	odd	6		1323.2.g.g.361.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.g.148.1	✓	12	9.2	odd	6
441.2.f.g.148.2	yes	12	63.20	even	6
441.2.f.g.295.1	yes	12	9.5	odd	6
441.2.f.g.295.2	yes	12	63.41	even	6
441.2.g.g.67.1		12	63.47	even	6
441.2.g.g.67.2		12	63.2	odd	6
441.2.g.g.79.1		12	63.59	even	6
441.2.g.g.79.2		12	63.32	odd	6
441.2.h.g.214.5		12	63.5	even	6
441.2.h.g.214.6		12	63.23	odd	6
441.2.h.g.373.5		12	63.38	even	6
441.2.h.g.373.6		12	63.11	odd	6
1323.2.f.g.442.5		12	63.34	odd	6
1323.2.f.g.442.6		12	9.7	even	3
1323.2.f.g.883.5		12	63.13	odd	6
1323.2.f.g.883.6		12	9.4	even	3
1323.2.g.g.361.5		12	63.16	even	3
1323.2.g.g.361.6		12	63.61	odd	6
1323.2.g.g.667.5		12	63.4	even	3
1323.2.g.g.667.6		12	63.31	odd	6
1323.2.h.g.226.1		12	63.52	odd	6
1323.2.h.g.226.2		12	63.25	even	3
1323.2.h.g.802.1		12	63.40	odd	6
1323.2.h.g.802.2		12	63.58	even	3
3969.2.a.bd.1.1		6	1.1	even	1	trivial
3969.2.a.bd.1.2		6	7.6	odd	2	inner
3969.2.a.be.1.5		6	21.20	even	2
3969.2.a.be.1.6		6	3.2	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)

\(f(q)\)	\(=\)	\(q-2.46050 q^{2} +4.05408 q^{4} -3.65808 q^{5} -5.05408 q^{8} +O(q^{10})\)	\(q-2.46050 q^{2} +4.05408 q^{4} -3.65808 q^{5} -5.05408 q^{8} +9.00071 q^{10} -0.406421 q^{11} +0.486796 q^{13} +4.32743 q^{16} +4.85584 q^{17} +1.97351 q^{19} -14.8301 q^{20} +1.00000 q^{22} -4.64766 q^{23} +8.38151 q^{25} -1.19777 q^{26} -7.64766 q^{29} -7.02720 q^{31} -0.539495 q^{32} -11.9478 q^{34} +2.32743 q^{37} -4.85584 q^{38} +18.4882 q^{40} +7.51399 q^{41} -2.32743 q^{43} -1.64766 q^{44} +11.4356 q^{46} -6.31623 q^{47} -20.6228 q^{50} +1.97351 q^{52} +3.56867 q^{53} +1.48672 q^{55} +18.8171 q^{58} +6.11839 q^{59} +8.02712 q^{61} +17.2905 q^{62} -7.32743 q^{64} -1.78074 q^{65} +3.60078 q^{67} +19.6860 q^{68} -8.46050 q^{71} -1.97351 q^{73} -5.72665 q^{74} +8.00079 q^{76} +8.16225 q^{79} -15.8301 q^{80} -18.4882 q^{82} +12.1720 q^{83} -17.7630 q^{85} +5.72665 q^{86} +2.05408 q^{88} +14.8301 q^{89} -18.8420 q^{92} +15.5411 q^{94} -7.21926 q^{95} +9.48751 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 6 * q^4 - 12 * q^8	\( 6 q - 2 q^{2} + 6 q^{4} - 12 q^{8} - 8 q^{11} + 6 q^{16} + 6 q^{22} - 4 q^{23} + 12 q^{25} - 22 q^{29} - 16 q^{32} - 6 q^{37} + 6 q^{43} + 14 q^{44} + 12 q^{46} - 56 q^{50} - 28 q^{53} + 18 q^{58} - 24 q^{64} + 6 q^{65} - 38 q^{71} - 36 q^{74} - 6 q^{79} - 30 q^{85} + 36 q^{86} - 6 q^{88} - 62 q^{92} - 60 q^{95}+O(q^{100}) \) 6 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 - 8 * q^11 + 6 * q^16 + 6 * q^22 - 4 * q^23 + 12 * q^25 - 22 * q^29 - 16 * q^32 - 6 * q^37 + 6 * q^43 + 14 * q^44 + 12 * q^46 - 56 * q^50 - 28 * q^53 + 18 * q^58 - 24 * q^64 + 6 * q^65 - 38 * q^71 - 36 * q^74 - 6 * q^79 - 30 * q^85 + 36 * q^86 - 6 * q^88 - 62 * q^92 - 60 * q^95

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