LMFDB - Embedded newform 3328.2.a.bm.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,-2,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.27743$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27743 q^{3} -3.27743 q^{5} -1.61023 q^{7} +7.74153 q^{9} -1.66719 q^{11} +1.00000 q^{13} -10.7415 q^{15} -3.52106 q^{17} -3.79691 q^{19} -5.27743 q^{21} -8.79849 q^{23} +5.74153 q^{25} +15.5400 q^{27} +4.24363 q^{29} +7.68615 q^{31} -5.46410 q^{33} +5.27743 q^{35} -3.94304 q^{37} +3.27743 q^{39} -10.7985 q^{41} +2.49790 q^{43} -25.3723 q^{45} -8.94462 q^{47} -4.40714 q^{49} -11.5400 q^{51} +2.12972 q^{53} +5.46410 q^{55} -12.4441 q^{57} -4.88766 q^{59} -11.3533 q^{61} -12.4657 q^{63} -3.27743 q^{65} +0.0891755 q^{67} -28.8364 q^{69} -3.05538 q^{71} +10.1329 q^{73} +18.8174 q^{75} +2.68457 q^{77} -0.292266 q^{79} +27.7067 q^{81} -4.08918 q^{83} +11.5400 q^{85} +13.9082 q^{87} -1.22047 q^{89} -1.61023 q^{91} +25.1908 q^{93} +12.4441 q^{95} -14.0190 q^{97} -12.9066 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 4 q^{13} - 18 q^{15} - 2 q^{17} - 6 q^{19} - 10 q^{21} - 12 q^{23} - 2 q^{25} + 14 q^{27} + 16 q^{29} - 10 q^{31} - 8 q^{33} + 10 q^{35} - 14 q^{37} + 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 4 * q^13 - 18 * q^15 - 2 * q^17 - 6 * q^19 - 10 * q^21 - 12 * q^23 - 2 * q^25 + 14 * q^27 + 16 * q^29 - 10 * q^31 - 8 * q^33 + 10 * q^35 - 14 * q^37 + 2 * q^39 - 20 * q^41 - 14 * q^43 - 20 * q^45 - 20 * q^47 - 2 * q^49 + 2 * q^51 + 4 * q^53 + 8 * q^55 + 4 * q^57 - 2 * q^59 - 22 * q^63 - 2 * q^65 + 6 * q^67 - 20 * q^69 - 28 * q^71 + 8 * q^73 + 16 * q^75 - 16 * q^77 - 16 * q^79 + 20 * q^81 - 22 * q^83 - 2 * q^85 - 12 * q^87 + 8 * q^89 + 36 * q^93 - 4 * q^95 - 20 * q^97 + 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.27743	1.89222	0.946112	−	0.323840i	$-0.104974\pi$
$3$	3.27743	1.89222	0.946112	+	0.323840i	$0.104974\pi$
$4$	0	0
$5$	−3.27743	−1.46571	−0.732855	−	0.680385i	$-0.761813\pi$
$5$	−3.27743	−1.46571	−0.732855	+	0.680385i	$0.761813\pi$
$6$	0	0
$7$	−1.61023	−0.608612	−0.304306	−	0.952574i	$-0.598424\pi$
$7$	−1.61023	−0.608612	−0.304306	+	0.952574i	$0.598424\pi$
$8$	0	0
$9$	7.74153	2.58051
$10$	0	0
$11$	−1.66719	−0.502677	−0.251339	−	0.967899i	$-0.580871\pi$
$11$	−1.66719	−0.502677	−0.251339	+	0.967899i	$0.580871\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−10.7415	−2.77345
$16$	0	0
$17$	−3.52106	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$17$	−3.52106	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$18$	0	0
$19$	−3.79691	−0.871071	−0.435535	−	0.900172i	$-0.643441\pi$
$19$	−3.79691	−0.871071	−0.435535	+	0.900172i	$0.643441\pi$
$20$	0	0
$21$	−5.27743	−1.15163
$22$	0	0
$23$	−8.79849	−1.83461	−0.917306	−	0.398184i	$-0.869641\pi$
$23$	−8.79849	−1.83461	−0.917306	+	0.398184i	$0.869641\pi$
$24$	0	0
$25$	5.74153	1.14831
$26$	0	0
$27$	15.5400	2.99068
$28$	0	0
$29$	4.24363	0.788023	0.394011	−	0.919106i	$-0.371087\pi$
$29$	4.24363	0.788023	0.394011	+	0.919106i	$0.371087\pi$
$30$	0	0
$31$	7.68615	1.38047	0.690236	−	0.723584i	$-0.257506\pi$
$31$	7.68615	1.38047	0.690236	+	0.723584i	$0.257506\pi$
$32$	0	0
$33$	−5.46410	−0.951178
$34$	0	0
$35$	5.27743	0.892048
$36$	0	0
$37$	−3.94304	−0.648232	−0.324116	−	0.946017i	$-0.605067\pi$
$37$	−3.94304	−0.648232	−0.324116	+	0.946017i	$0.605067\pi$
$38$	0	0
$39$	3.27743	0.524808
$40$	0	0
$41$	−10.7985	−1.68644	−0.843220	−	0.537568i	$-0.819343\pi$
$41$	−10.7985	−1.68644	−0.843220	+	0.537568i	$0.819343\pi$
$42$	0	0
$43$	2.49790	0.380926	0.190463	−	0.981694i	$-0.439001\pi$
$43$	2.49790	0.380926	0.190463	+	0.981694i	$0.439001\pi$
$44$	0	0
$45$	−25.3723	−3.78228
$46$	0	0
$47$	−8.94462	−1.30471	−0.652353	−	0.757915i	$-0.726218\pi$
$47$	−8.94462	−1.30471	−0.652353	+	0.757915i	$0.726218\pi$
$48$	0	0
$49$	−4.40714	−0.629592
$50$	0	0
$51$	−11.5400	−1.61593
$52$	0	0
$53$	2.12972	0.292539	0.146270	−	0.989245i	$-0.453273\pi$
$53$	2.12972	0.292539	0.146270	+	0.989245i	$0.453273\pi$
$54$	0	0
$55$	5.46410	0.736779
$56$	0	0
$57$	−12.4441	−1.64826
$58$	0	0
$59$	−4.88766	−0.636319	−0.318160	−	0.948037i	$-0.603065\pi$
$59$	−4.88766	−0.636319	−0.318160	+	0.948037i	$0.603065\pi$
$60$	0	0
$61$	−11.3533	−1.45365	−0.726823	−	0.686825i	$-0.759004\pi$
$61$	−11.3533	−1.45365	−0.726823	+	0.686825i	$0.759004\pi$
$62$	0	0
$63$	−12.4657	−1.57053
$64$	0	0
$65$	−3.27743	−0.406515
$66$	0	0
$67$	0.0891755	0.0108945	0.00544726	−	0.999985i	$-0.498266\pi$
$67$	0.0891755	0.0108945	0.00544726	+	0.999985i	$0.498266\pi$
$68$	0	0
$69$	−28.8364	−3.47149
$70$	0	0
$71$	−3.05538	−0.362607	−0.181303	−	0.983427i	$-0.558032\pi$
$71$	−3.05538	−0.362607	−0.181303	+	0.983427i	$0.558032\pi$
$72$	0	0
$73$	10.1329	1.18596	0.592981	−	0.805216i	$-0.297951\pi$
$73$	10.1329	1.18596	0.592981	+	0.805216i	$0.297951\pi$
$74$	0	0
$75$	18.8174	2.17285
$76$	0	0
$77$	2.68457	0.305935
$78$	0	0
$79$	−0.292266	−0.0328825	−0.0164413	−	0.999865i	$-0.505234\pi$
$79$	−0.292266	−0.0328825	−0.0164413	+	0.999865i	$0.505234\pi$
$80$	0	0
$81$	27.7067	3.07852
$82$	0	0
$83$	−4.08918	−0.448845	−0.224423	−	0.974492i	$-0.572050\pi$
$83$	−4.08918	−0.448845	−0.224423	+	0.974492i	$0.572050\pi$
$84$	0	0
$85$	11.5400	1.25169
$86$	0	0
$87$	13.9082	1.49112
$88$	0	0
$89$	−1.22047	−0.129370	−0.0646848	−	0.997906i	$-0.520604\pi$
$89$	−1.22047	−0.129370	−0.0646848	+	0.997906i	$0.520604\pi$
$90$	0	0
$91$	−1.61023	−0.168798
$92$	0	0
$93$	25.1908	2.61216
$94$	0	0
$95$	12.4441	1.27674
$96$	0	0
$97$	−14.0190	−1.42341	−0.711705	−	0.702479i	$-0.752077\pi$
$97$	−14.0190	−1.42341	−0.711705	+	0.702479i	$0.752077\pi$
$98$	0	0
$99$	−12.9066	−1.29716
$100$	0	0
$101$	1.33123	0.132462	0.0662312	−	0.997804i	$-0.478903\pi$
$101$	1.33123	0.132462	0.0662312	+	0.997804i	$0.478903\pi$
$102$	0	0
$103$	−3.62665	−0.357345	−0.178672	−	0.983909i	$-0.557180\pi$
$103$	−3.62665	−0.357345	−0.178672	+	0.983909i	$0.557180\pi$
$104$	0	0
$105$	17.2964	1.68795
$106$	0	0
$107$	−14.8174	−1.43246	−0.716228	−	0.697866i	$-0.754133\pi$
$107$	−14.8174	−1.43246	−0.716228	+	0.697866i	$0.754133\pi$
$108$	0	0
$109$	2.72573	0.261077	0.130539	−	0.991443i	$-0.458329\pi$
$109$	2.72573	0.261077	0.130539	+	0.991443i	$0.458329\pi$
$110$	0	0
$111$	−12.9230	−1.22660
$112$	0	0
$113$	1.03896	0.0977375	0.0488688	−	0.998805i	$-0.484438\pi$
$113$	1.03896	0.0977375	0.0488688	+	0.998805i	$0.484438\pi$
$114$	0	0
$115$	28.8364	2.68901
$116$	0	0
$117$	7.74153	0.715705
$118$	0	0
$119$	5.66973	0.519743
$120$	0	0
$121$	−8.22047	−0.747315
$122$	0	0
$123$	−35.3913	−3.19112
$124$	0	0
$125$	−2.43031	−0.217373
$126$	0	0
$127$	13.9082	1.23415	0.617076	−	0.786903i	$-0.288317\pi$
$127$	13.9082	1.23415	0.617076	+	0.786903i	$0.288317\pi$
$128$	0	0
$129$	8.18667	0.720796
$130$	0	0
$131$	10.5790	0.924290	0.462145	−	0.886804i	$-0.347080\pi$
$131$	10.5790	0.924290	0.462145	+	0.886804i	$0.347080\pi$
$132$	0	0
$133$	6.11392	0.530144
$134$	0	0
$135$	−50.9313	−4.38347
$136$	0	0
$137$	5.93787	0.507307	0.253653	−	0.967295i	$-0.418368\pi$
$137$	5.93787	0.507307	0.253653	+	0.967295i	$0.418368\pi$
$138$	0	0
$139$	11.6508	0.988206	0.494103	−	0.869403i	$-0.335497\pi$
$139$	11.6508	0.988206	0.494103	+	0.869403i	$0.335497\pi$
$140$	0	0
$141$	−29.3153	−2.46880
$142$	0	0
$143$	−1.66719	−0.139418
$144$	0	0
$145$	−13.9082	−1.15501
$146$	0	0
$147$	−14.4441	−1.19133
$148$	0	0
$149$	0.668770	0.0547877	0.0273939	−	0.999625i	$-0.491279\pi$
$149$	0.668770	0.0547877	0.0273939	+	0.999625i	$0.491279\pi$
$150$	0	0
$151$	−9.98358	−0.812453	−0.406226	−	0.913773i	$-0.633156\pi$
$151$	−9.98358	−0.812453	−0.406226	+	0.913773i	$0.633156\pi$
$152$	0	0
$153$	−27.2584	−2.20371
$154$	0	0
$155$	−25.1908	−2.02337
$156$	0	0
$157$	23.4209	1.86919	0.934597	−	0.355709i	$-0.115760\pi$
$157$	23.4209	1.86919	0.934597	+	0.355709i	$0.115760\pi$
$158$	0	0
$159$	6.97999	0.553549
$160$	0	0
$161$	14.1676	1.11657
$162$	0	0
$163$	−6.99842	−0.548159	−0.274079	−	0.961707i	$-0.588373\pi$
$163$	−6.99842	−0.548159	−0.274079	+	0.961707i	$0.588373\pi$
$164$	0	0
$165$	17.9082	1.39415
$166$	0	0
$167$	−10.0892	−0.780724	−0.390362	−	0.920661i	$-0.627650\pi$
$167$	−10.0892	−0.780724	−0.390362	+	0.920661i	$0.627650\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−29.3939	−2.24781
$172$	0	0
$173$	10.5928	0.805353	0.402677	−	0.915342i	$-0.368080\pi$
$173$	10.5928	0.805353	0.402677	+	0.915342i	$0.368080\pi$
$174$	0	0
$175$	−9.24521	−0.698872
$176$	0	0
$177$	−16.0190	−1.20406
$178$	0	0
$179$	0.316391	0.0236482	0.0118241	−	0.999930i	$-0.496236\pi$
$179$	0.316391	0.0236482	0.0118241	+	0.999930i	$0.496236\pi$
$180$	0	0
$181$	−9.41547	−0.699846	−0.349923	−	0.936778i	$-0.613792\pi$
$181$	−9.41547	−0.699846	−0.349923	+	0.936778i	$0.613792\pi$
$182$	0	0
$183$	−37.2097	−2.75062
$184$	0	0
$185$	12.9230	0.950120
$186$	0	0
$187$	5.87028	0.429278
$188$	0	0
$189$	−25.0231	−1.82016
$190$	0	0
$191$	−19.0611	−1.37921	−0.689605	−	0.724185i	$-0.742216\pi$
$191$	−19.0611	−1.37921	−0.689605	+	0.724185i	$0.742216\pi$
$192$	0	0
$193$	−24.8850	−1.79126	−0.895632	−	0.444796i	$-0.853276\pi$
$193$	−24.8850	−1.79126	−0.895632	+	0.444796i	$0.853276\pi$
$194$	0	0
$195$	−10.7415	−0.769217
$196$	0	0
$197$	24.7933	1.76645	0.883225	−	0.468949i	$-0.155367\pi$
$197$	24.7933	1.76645	0.883225	+	0.468949i	$0.155367\pi$
$198$	0	0
$199$	6.81429	0.483052	0.241526	−	0.970394i	$-0.422352\pi$
$199$	6.81429	0.483052	0.241526	+	0.970394i	$0.422352\pi$
$200$	0	0
$201$	0.292266	0.0206149
$202$	0	0
$203$	−6.83324	−0.479600
$204$	0	0
$205$	35.3913	2.47183
$206$	0	0
$207$	−68.1137	−4.73423
$208$	0	0
$209$	6.33018	0.437868
$210$	0	0
$211$	4.72257	0.325115	0.162558	−	0.986699i	$-0.448026\pi$
$211$	4.72257	0.325115	0.162558	+	0.986699i	$0.448026\pi$
$212$	0	0
$213$	−10.0138	−0.686133
$214$	0	0
$215$	−8.18667	−0.558327
$216$	0	0
$217$	−12.3765	−0.840172
$218$	0	0
$219$	33.2097	2.24411
$220$	0	0
$221$	−3.52106	−0.236852
$222$	0	0
$223$	−8.24617	−0.552204	−0.276102	−	0.961128i	$-0.589043\pi$
$223$	−8.24617	−0.552204	−0.276102	+	0.961128i	$0.589043\pi$
$224$	0	0
$225$	44.4482	2.96321
$226$	0	0
$227$	7.81587	0.518757	0.259379	−	0.965776i	$-0.416482\pi$
$227$	7.81587	0.518757	0.259379	+	0.965776i	$0.416482\pi$
$228$	0	0
$229$	16.6466	1.10004	0.550018	−	0.835153i	$-0.314621\pi$
$229$	16.6466	1.10004	0.550018	+	0.835153i	$0.314621\pi$
$230$	0	0
$231$	8.79849	0.578898
$232$	0	0
$233$	−6.07276	−0.397840	−0.198920	−	0.980016i	$-0.563743\pi$
$233$	−6.07276	−0.397840	−0.198920	+	0.980016i	$0.563743\pi$
$234$	0	0
$235$	29.3153	1.91232
$236$	0	0
$237$	−0.957882	−0.0622211
$238$	0	0
$239$	18.5088	1.19723	0.598616	−	0.801036i	$-0.295718\pi$
$239$	18.5088	1.19723	0.598616	+	0.801036i	$0.295718\pi$
$240$	0	0
$241$	30.3681	1.95618	0.978090	−	0.208181i	$-0.0667543\pi$
$241$	30.3681	1.95618	0.978090	+	0.208181i	$0.0667543\pi$
$242$	0	0
$243$	44.1866	2.83457
$244$	0	0
$245$	14.4441	0.922799
$246$	0	0
$247$	−3.79691	−0.241592
$248$	0	0
$249$	−13.4020	−0.849316
$250$	0	0
$251$	3.22362	0.203473	0.101737	−	0.994811i	$-0.467560\pi$
$251$	3.22362	0.203473	0.101737	+	0.994811i	$0.467560\pi$
$252$	0	0
$253$	14.6688	0.922218
$254$	0	0
$255$	37.8216	2.36848
$256$	0	0
$257$	5.11488	0.319057	0.159529	−	0.987193i	$-0.449003\pi$
$257$	5.11488	0.319057	0.159529	+	0.987193i	$0.449003\pi$
$258$	0	0
$259$	6.34922	0.394522
$260$	0	0
$261$	32.8522	2.03350
$262$	0	0
$263$	−18.9092	−1.16599	−0.582997	−	0.812474i	$-0.698120\pi$
$263$	−18.9092	−1.16599	−0.582997	+	0.812474i	$0.698120\pi$
$264$	0	0
$265$	−6.97999	−0.428777
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	9.23943	0.563338	0.281669	−	0.959512i	$-0.409112\pi$
$269$	9.23943	0.563338	0.281669	+	0.959512i	$0.409112\pi$
$270$	0	0
$271$	−16.0080	−0.972417	−0.486208	−	0.873843i	$-0.661620\pi$
$271$	−16.0080	−0.972417	−0.486208	+	0.873843i	$0.661620\pi$
$272$	0	0
$273$	−5.27743	−0.319404
$274$	0	0
$275$	−9.57223	−0.577227
$276$	0	0
$277$	−9.65910	−0.580359	−0.290180	−	0.956972i	$-0.593715\pi$
$277$	−9.65910	−0.580359	−0.290180	+	0.956972i	$0.593715\pi$
$278$	0	0
$279$	59.5025	3.56232
$280$	0	0
$281$	−17.9735	−1.07221	−0.536104	−	0.844152i	$-0.680104\pi$
$281$	−17.9735	−1.07221	−0.536104	+	0.844152i	$0.680104\pi$
$282$	0	0
$283$	−22.6391	−1.34575	−0.672877	−	0.739754i	$-0.734942\pi$
$283$	−22.6391	−1.34575	−0.672877	+	0.739754i	$0.734942\pi$
$284$	0	0
$285$	40.7846	2.41587
$286$	0	0
$287$	17.3881	1.02639
$288$	0	0
$289$	−4.60214	−0.270714
$290$	0	0
$291$	−45.9461	−2.69341
$292$	0	0
$293$	17.4261	1.01804	0.509022	−	0.860754i	$-0.330007\pi$
$293$	17.4261	1.01804	0.509022	+	0.860754i	$0.330007\pi$
$294$	0	0
$295$	16.0190	0.932660
$296$	0	0
$297$	−25.9082	−1.50335
$298$	0	0
$299$	−8.79849	−0.508830
$300$	0	0
$301$	−4.02220	−0.231836
$302$	0	0
$303$	4.36301	0.250648
$304$	0	0
$305$	37.2097	2.13062
$306$	0	0
$307$	−8.96874	−0.511873	−0.255937	−	0.966694i	$-0.582384\pi$
$307$	−8.96874	−0.511873	−0.255937	+	0.966694i	$0.582384\pi$
$308$	0	0
$309$	−11.8861	−0.676176
$310$	0	0
$311$	−16.3630	−0.927861	−0.463931	−	0.885872i	$-0.653561\pi$
$311$	−16.3630	−0.927861	−0.463931	+	0.885872i	$0.653561\pi$
$312$	0	0
$313$	13.7026	0.774515	0.387257	−	0.921972i	$-0.373422\pi$
$313$	13.7026	0.774515	0.387257	+	0.921972i	$0.373422\pi$
$314$	0	0
$315$	40.8554	2.30194
$316$	0	0
$317$	−15.4502	−0.867771	−0.433886	−	0.900968i	$-0.642858\pi$
$317$	−15.4502	−0.867771	−0.433886	+	0.900968i	$0.642858\pi$
$318$	0	0
$319$	−7.07495	−0.396121
$320$	0	0
$321$	−48.5631	−2.71053
$322$	0	0
$323$	13.3691	0.743879
$324$	0	0
$325$	5.74153	0.318483
$326$	0	0
$327$	8.93337	0.494017
$328$	0	0
$329$	14.4029	0.794060
$330$	0	0
$331$	22.6175	1.24317	0.621585	−	0.783347i	$-0.286489\pi$
$331$	22.6175	1.24317	0.621585	+	0.783347i	$0.286489\pi$
$332$	0	0
$333$	−30.5252	−1.67277
$334$	0	0
$335$	−0.292266	−0.0159682
$336$	0	0
$337$	17.7026	0.964320	0.482160	−	0.876083i	$-0.339852\pi$
$337$	17.7026	0.964320	0.482160	+	0.876083i	$0.339852\pi$
$338$	0	0
$339$	3.40513	0.184941
$340$	0	0
$341$	−12.8143	−0.693933
$342$	0	0
$343$	18.3682	0.991788
$344$	0	0
$345$	94.5092	5.08820
$346$	0	0
$347$	13.8323	0.742556	0.371278	−	0.928522i	$-0.378920\pi$
$347$	13.8323	0.742556	0.371278	+	0.928522i	$0.378920\pi$
$348$	0	0
$349$	1.79437	0.0960504	0.0480252	−	0.998846i	$-0.484707\pi$
$349$	1.79437	0.0960504	0.0480252	+	0.998846i	$0.484707\pi$
$350$	0	0
$351$	15.5400	0.829465
$352$	0	0
$353$	7.81849	0.416136	0.208068	−	0.978114i	$-0.433282\pi$
$353$	7.81849	0.416136	0.208068	+	0.978114i	$0.433282\pi$
$354$	0	0
$355$	10.0138	0.531477
$356$	0	0
$357$	18.5821	0.983471
$358$	0	0
$359$	6.08918	0.321374	0.160687	−	0.987005i	$-0.448629\pi$
$359$	6.08918	0.321374	0.160687	+	0.987005i	$0.448629\pi$
$360$	0	0
$361$	−4.58348	−0.241236
$362$	0	0
$363$	−26.9420	−1.41409
$364$	0	0
$365$	−33.2097	−1.73828
$366$	0	0
$367$	6.73321	0.351470	0.175735	−	0.984437i	$-0.443770\pi$
$367$	6.73321	0.351470	0.175735	+	0.984437i	$0.443770\pi$
$368$	0	0
$369$	−83.5968	−4.35188
$370$	0	0
$371$	−3.42934	−0.178043
$372$	0	0
$373$	−12.5549	−0.650066	−0.325033	−	0.945703i	$-0.605375\pi$
$373$	−12.5549	−0.650066	−0.325033	+	0.945703i	$0.605375\pi$
$374$	0	0
$375$	−7.96515	−0.411319
$376$	0	0
$377$	4.24363	0.218558
$378$	0	0
$379$	19.5615	1.00481	0.502404	−	0.864633i	$-0.332449\pi$
$379$	19.5615	1.00481	0.502404	+	0.864633i	$0.332449\pi$
$380$	0	0
$381$	45.5831	2.33529
$382$	0	0
$383$	10.9478	0.559405	0.279703	−	0.960087i	$-0.409764\pi$
$383$	10.9478	0.559405	0.279703	+	0.960087i	$0.409764\pi$
$384$	0	0
$385$	−8.79849	−0.448412
$386$	0	0
$387$	19.3375	0.982982
$388$	0	0
$389$	−19.0958	−0.968197	−0.484099	−	0.875013i	$-0.660852\pi$
$389$	−19.0958	−0.968197	−0.484099	+	0.875013i	$0.660852\pi$
$390$	0	0
$391$	30.9800	1.56673
$392$	0	0
$393$	34.6718	1.74896
$394$	0	0
$395$	0.957882	0.0481963
$396$	0	0
$397$	−1.59382	−0.0799915	−0.0399957	−	0.999200i	$-0.512734\pi$
$397$	−1.59382	−0.0799915	−0.0399957	+	0.999200i	$0.512734\pi$
$398$	0	0
$399$	20.0379	1.00315
$400$	0	0
$401$	−18.6414	−0.930907	−0.465454	−	0.885072i	$-0.654109\pi$
$401$	−18.6414	−0.930907	−0.465454	+	0.885072i	$0.654109\pi$
$402$	0	0
$403$	7.68615	0.382874
$404$	0	0
$405$	−90.8066	−4.51222
$406$	0	0
$407$	6.57381	0.325852
$408$	0	0
$409$	25.8785	1.27961	0.639805	−	0.768537i	$-0.279015\pi$
$409$	25.8785	1.27961	0.639805	+	0.768537i	$0.279015\pi$
$410$	0	0
$411$	19.4609	0.959938
$412$	0	0
$413$	7.87028	0.387271
$414$	0	0
$415$	13.4020	0.657877
$416$	0	0
$417$	38.1846	1.86991
$418$	0	0
$419$	3.23952	0.158261	0.0791303	−	0.996864i	$-0.474786\pi$
$419$	3.23952	0.158261	0.0791303	+	0.996864i	$0.474786\pi$
$420$	0	0
$421$	13.2477	0.645656	0.322828	−	0.946458i	$-0.395367\pi$
$421$	13.2477	0.645656	0.322828	+	0.946458i	$0.395367\pi$
$422$	0	0
$423$	−69.2450	−3.36681
$424$	0	0
$425$	−20.2163	−0.980633
$426$	0	0
$427$	18.2815	0.884706
$428$	0	0
$429$	−5.46410	−0.263809
$430$	0	0
$431$	−18.3948	−0.886048	−0.443024	−	0.896510i	$-0.646094\pi$
$431$	−18.3948	−0.886048	−0.443024	+	0.896510i	$0.646094\pi$
$432$	0	0
$433$	1.55389	0.0746753	0.0373376	−	0.999303i	$-0.488112\pi$
$433$	1.55389	0.0746753	0.0373376	+	0.999303i	$0.488112\pi$
$434$	0	0
$435$	−45.5831	−2.18554
$436$	0	0
$437$	33.4071	1.59808
$438$	0	0
$439$	−13.7943	−0.658365	−0.329183	−	0.944266i	$-0.606773\pi$
$439$	−13.7943	−0.658365	−0.329183	+	0.944266i	$0.606773\pi$
$440$	0	0
$441$	−34.1180	−1.62467
$442$	0	0
$443$	−14.0980	−0.669817	−0.334909	−	0.942251i	$-0.608705\pi$
$443$	−14.0980	−0.669817	−0.334909	+	0.942251i	$0.608705\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	2.19184	0.103671
$448$	0	0
$449$	22.8692	1.07927	0.539633	−	0.841900i	$-0.318563\pi$
$449$	22.8692	1.07927	0.539633	+	0.841900i	$0.318563\pi$
$450$	0	0
$451$	18.0032	0.847735
$452$	0	0
$453$	−32.7205	−1.53734
$454$	0	0
$455$	5.27743	0.247410
$456$	0	0
$457$	−17.6267	−0.824540	−0.412270	−	0.911062i	$-0.635264\pi$
$457$	−17.6267	−0.824540	−0.412270	+	0.911062i	$0.635264\pi$
$458$	0	0
$459$	−54.7173	−2.55399
$460$	0	0
$461$	19.2509	0.896604	0.448302	−	0.893882i	$-0.352029\pi$
$461$	19.2509	0.896604	0.448302	+	0.893882i	$0.352029\pi$
$462$	0	0
$463$	−22.3549	−1.03892	−0.519461	−	0.854494i	$-0.673867\pi$
$463$	−22.3549	−1.03892	−0.519461	+	0.854494i	$0.673867\pi$
$464$	0	0
$465$	−82.5610	−3.82867
$466$	0	0
$467$	1.49147	0.0690170	0.0345085	−	0.999404i	$-0.489013\pi$
$467$	1.49147	0.0690170	0.0345085	+	0.999404i	$0.489013\pi$
$468$	0	0
$469$	−0.143594	−0.00663053
$470$	0	0
$471$	76.7604	3.53693
$472$	0	0
$473$	−4.16447	−0.191483
$474$	0	0
$475$	−21.8001	−1.00026
$476$	0	0
$477$	16.4873	0.754900
$478$	0	0
$479$	−16.7663	−0.766070	−0.383035	−	0.923734i	$-0.625121\pi$
$479$	−16.7663	−0.766070	−0.383035	+	0.923734i	$0.625121\pi$
$480$	0	0
$481$	−3.94304	−0.179787
$482$	0	0
$483$	46.4334	2.11279
$484$	0	0
$485$	45.9461	2.08631
$486$	0	0
$487$	−19.5889	−0.887657	−0.443828	−	0.896112i	$-0.646380\pi$
$487$	−19.5889	−0.887657	−0.443828	+	0.896112i	$0.646380\pi$
$488$	0	0
$489$	−22.9368	−1.03724
$490$	0	0
$491$	35.2509	1.59085	0.795425	−	0.606051i	$-0.207247\pi$
$491$	35.2509	1.59085	0.795425	+	0.606051i	$0.207247\pi$
$492$	0	0
$493$	−14.9421	−0.672957
$494$	0	0
$495$	42.3005	1.90127
$496$	0	0
$497$	4.91988	0.220687
$498$	0	0
$499$	−7.01507	−0.314038	−0.157019	−	0.987596i	$-0.550188\pi$
$499$	−7.01507	−0.314038	−0.157019	+	0.987596i	$0.550188\pi$
$500$	0	0
$501$	−33.0665	−1.47730
$502$	0	0
$503$	−11.3344	−0.505375	−0.252688	−	0.967548i	$-0.581314\pi$
$503$	−11.3344	−0.505375	−0.252688	+	0.967548i	$0.581314\pi$
$504$	0	0
$505$	−4.36301	−0.194151
$506$	0	0
$507$	3.27743	0.145556
$508$	0	0
$509$	−16.1383	−0.715319	−0.357660	−	0.933852i	$-0.616425\pi$
$509$	−16.1383	−0.715319	−0.357660	+	0.933852i	$0.616425\pi$
$510$	0	0
$511$	−16.3163	−0.721791
$512$	0	0
$513$	−59.0040	−2.60509
$514$	0	0
$515$	11.8861	0.523763
$516$	0	0
$517$	14.9124	0.655847
$518$	0	0
$519$	34.7170	1.52391
$520$	0	0
$521$	12.1056	0.530356	0.265178	−	0.964200i	$-0.414569\pi$
$521$	12.1056	0.530356	0.265178	+	0.964200i	$0.414569\pi$
$522$	0	0
$523$	−1.22047	−0.0533674	−0.0266837	−	0.999644i	$-0.508495\pi$
$523$	−1.22047	−0.0533674	−0.0266837	+	0.999644i	$0.508495\pi$
$524$	0	0
$525$	−30.3005	−1.32242
$526$	0	0
$527$	−27.0634	−1.17890
$528$	0	0
$529$	54.4134	2.36580
$530$	0	0
$531$	−37.8380	−1.64203
$532$	0	0
$533$	−10.7985	−0.467734
$534$	0	0
$535$	48.5631	2.09957
$536$	0	0
$537$	1.03695	0.0447477
$538$	0	0
$539$	7.34756	0.316482
$540$	0	0
$541$	27.9430	1.20137	0.600683	−	0.799488i	$-0.294896\pi$
$541$	27.9430	1.20137	0.600683	+	0.799488i	$0.294896\pi$
$542$	0	0
$543$	−30.8585	−1.32427
$544$	0	0
$545$	−8.93337	−0.382664
$546$	0	0
$547$	−23.1698	−0.990670	−0.495335	−	0.868702i	$-0.664955\pi$
$547$	−23.1698	−0.990670	−0.495335	+	0.868702i	$0.664955\pi$
$548$	0	0
$549$	−87.8922	−3.75115
$550$	0	0
$551$	−16.1127	−0.686423
$552$	0	0
$553$	0.470617	0.0200127
$554$	0	0
$555$	42.3543	1.79784
$556$	0	0
$557$	32.5061	1.37733	0.688664	−	0.725080i	$-0.258197\pi$
$557$	32.5061	1.37733	0.688664	+	0.725080i	$0.258197\pi$
$558$	0	0
$559$	2.49790	0.105650
$560$	0	0
$561$	19.2394	0.812289
$562$	0	0
$563$	−4.52757	−0.190815	−0.0954073	−	0.995438i	$-0.530415\pi$
$563$	−4.52757	−0.190815	−0.0954073	+	0.995438i	$0.530415\pi$
$564$	0	0
$565$	−3.40513	−0.143255
$566$	0	0
$567$	−44.6143	−1.87362
$568$	0	0
$569$	−21.7057	−0.909951	−0.454976	−	0.890504i	$-0.650352\pi$
$569$	−21.7057	−0.909951	−0.454976	+	0.890504i	$0.650352\pi$
$570$	0	0
$571$	37.8058	1.58212	0.791061	−	0.611737i	$-0.209529\pi$
$571$	37.8058	1.58212	0.791061	+	0.611737i	$0.209529\pi$
$572$	0	0
$573$	−62.4713	−2.60978
$574$	0	0
$575$	−50.5168	−2.10669
$576$	0	0
$577$	−1.56452	−0.0651320	−0.0325660	−	0.999470i	$-0.510368\pi$
$577$	−1.56452	−0.0651320	−0.0325660	+	0.999470i	$0.510368\pi$
$578$	0	0
$579$	−81.5589	−3.38947
$580$	0	0
$581$	6.58453	0.273172
$582$	0	0
$583$	−3.55065	−0.147053
$584$	0	0
$585$	−25.3723	−1.04902
$586$	0	0
$587$	−44.3244	−1.82946	−0.914732	−	0.404062i	$-0.867598\pi$
$587$	−44.3244	−1.82946	−0.914732	+	0.404062i	$0.867598\pi$
$588$	0	0
$589$	−29.1836	−1.20249
$590$	0	0
$591$	81.2583	3.34252
$592$	0	0
$593$	23.6970	0.973120	0.486560	−	0.873647i	$-0.338252\pi$
$593$	23.6970	0.973120	0.486560	+	0.873647i	$0.338252\pi$
$594$	0	0
$595$	−18.5821	−0.761793
$596$	0	0
$597$	22.3333	0.914042
$598$	0	0
$599$	−36.8036	−1.50375	−0.751876	−	0.659304i	$-0.770851\pi$
$599$	−36.8036	−1.50375	−0.751876	+	0.659304i	$0.770851\pi$
$600$	0	0
$601$	28.6520	1.16874	0.584370	−	0.811487i	$-0.301342\pi$
$601$	28.6520	1.16874	0.584370	+	0.811487i	$0.301342\pi$
$602$	0	0
$603$	0.690355	0.0281134
$604$	0	0
$605$	26.9420	1.09535
$606$	0	0
$607$	42.9535	1.74343	0.871714	−	0.490015i	$-0.163009\pi$
$607$	42.9535	1.74343	0.871714	+	0.490015i	$0.163009\pi$
$608$	0	0
$609$	−22.3955	−0.907510
$610$	0	0
$611$	−8.94462	−0.361861
$612$	0	0
$613$	−31.2719	−1.26306	−0.631530	−	0.775352i	$-0.717573\pi$
$613$	−31.2719	−1.26306	−0.631530	+	0.775352i	$0.717573\pi$
$614$	0	0
$615$	115.992	4.67726
$616$	0	0
$617$	−8.99895	−0.362284	−0.181142	−	0.983457i	$-0.557979\pi$
$617$	−8.99895	−0.362284	−0.181142	+	0.983457i	$0.557979\pi$
$618$	0	0
$619$	4.36249	0.175343	0.0876716	−	0.996149i	$-0.472057\pi$
$619$	4.36249	0.175343	0.0876716	+	0.996149i	$0.472057\pi$
$620$	0	0
$621$	−136.729	−5.48673
$622$	0	0
$623$	1.96524	0.0787358
$624$	0	0
$625$	−20.7425	−0.829700
$626$	0	0
$627$	20.7467	0.828543
$628$	0	0
$629$	13.8837	0.553579
$630$	0	0
$631$	−10.8718	−0.432798	−0.216399	−	0.976305i	$-0.569431\pi$
$631$	−10.8718	−0.432798	−0.216399	+	0.976305i	$0.569431\pi$
$632$	0	0
$633$	15.4779	0.615191
$634$	0	0
$635$	−45.5831	−1.80891
$636$	0	0
$637$	−4.40714	−0.174617
$638$	0	0
$639$	−23.6533	−0.935711
$640$	0	0
$641$	34.4460	1.36054	0.680268	−	0.732964i	$-0.261863\pi$
$641$	34.4460	1.36054	0.680268	+	0.732964i	$0.261863\pi$
$642$	0	0
$643$	−39.1589	−1.54427	−0.772137	−	0.635455i	$-0.780812\pi$
$643$	−39.1589	−1.54427	−0.772137	+	0.635455i	$0.780812\pi$
$644$	0	0
$645$	−26.8312	−1.05648
$646$	0	0
$647$	−14.7309	−0.579131	−0.289566	−	0.957158i	$-0.593511\pi$
$647$	−14.7309	−0.579131	−0.289566	+	0.957158i	$0.593511\pi$
$648$	0	0
$649$	8.14867	0.319863
$650$	0	0
$651$	−40.5631	−1.58979
$652$	0	0
$653$	−39.2615	−1.53642	−0.768211	−	0.640196i	$-0.778853\pi$
$653$	−39.2615	−1.53642	−0.768211	+	0.640196i	$0.778853\pi$
$654$	0	0
$655$	−34.6718	−1.35474
$656$	0	0
$657$	78.4439	3.06039
$658$	0	0
$659$	−41.0748	−1.60005	−0.800023	−	0.599969i	$-0.795179\pi$
$659$	−41.0748	−1.60005	−0.800023	+	0.599969i	$0.795179\pi$
$660$	0	0
$661$	35.8132	1.39297	0.696487	−	0.717570i	$-0.254746\pi$
$661$	35.8132	1.39297	0.696487	+	0.717570i	$0.254746\pi$
$662$	0	0
$663$	−11.5400	−0.448177
$664$	0	0
$665$	−20.0379	−0.777037
$666$	0	0
$667$	−37.3375	−1.44572
$668$	0	0
$669$	−27.0262	−1.04489
$670$	0	0
$671$	18.9282	0.730715
$672$	0	0
$673$	28.6790	1.10549	0.552747	−	0.833349i	$-0.313579\pi$
$673$	28.6790	1.10549	0.552747	+	0.833349i	$0.313579\pi$
$674$	0	0
$675$	89.2234	3.43421
$676$	0	0
$677$	−20.9958	−0.806934	−0.403467	−	0.914994i	$-0.632195\pi$
$677$	−20.9958	−0.806934	−0.403467	+	0.914994i	$0.632195\pi$
$678$	0	0
$679$	22.5738	0.866303
$680$	0	0
$681$	25.6159	0.981605
$682$	0	0
$683$	38.5743	1.47601	0.738003	−	0.674797i	$-0.235769\pi$
$683$	38.5743	1.47601	0.738003	+	0.674797i	$0.235769\pi$
$684$	0	0
$685$	−19.4609	−0.743565
$686$	0	0
$687$	54.5579	2.08151
$688$	0	0
$689$	2.12972	0.0811357
$690$	0	0
$691$	−29.2696	−1.11347	−0.556735	−	0.830690i	$-0.687946\pi$
$691$	−29.2696	−1.11347	−0.556735	+	0.830690i	$0.687946\pi$
$692$	0	0
$693$	20.7827	0.789469
$694$	0	0
$695$	−38.1846	−1.44842
$696$	0	0
$697$	38.0221	1.44019
$698$	0	0
$699$	−19.9030	−0.752802
$700$	0	0
$701$	8.28154	0.312790	0.156395	−	0.987695i	$-0.450013\pi$
$701$	8.28154	0.312790	0.156395	+	0.987695i	$0.450013\pi$
$702$	0	0
$703$	14.9714	0.564656
$704$	0	0
$705$	96.0789	3.61854
$706$	0	0
$707$	−2.14359	−0.0806181
$708$	0	0
$709$	36.3005	1.36329	0.681647	−	0.731681i	$-0.261264\pi$
$709$	36.3005	1.36329	0.681647	+	0.731681i	$0.261264\pi$
$710$	0	0
$711$	−2.26259	−0.0848537
$712$	0	0
$713$	−67.6265	−2.53263
$714$	0	0
$715$	5.46410	0.204346
$716$	0	0
$717$	60.6611	2.26543
$718$	0	0
$719$	−1.93872	−0.0723020	−0.0361510	−	0.999346i	$-0.511510\pi$
$719$	−1.93872	−0.0723020	−0.0361510	+	0.999346i	$0.511510\pi$
$720$	0	0
$721$	5.83976	0.217484
$722$	0	0
$723$	99.5292	3.70153
$724$	0	0
$725$	24.3649	0.904891
$726$	0	0
$727$	−33.0642	−1.22628	−0.613142	−	0.789973i	$-0.710095\pi$
$727$	−33.0642	−1.22628	−0.613142	+	0.789973i	$0.710095\pi$
$728$	0	0
$729$	61.6983	2.28512
$730$	0	0
$731$	−8.79524	−0.325304
$732$	0	0
$733$	−20.5390	−0.758624	−0.379312	−	0.925269i	$-0.623839\pi$
$733$	−20.5390	−0.758624	−0.379312	+	0.925269i	$0.623839\pi$
$734$	0	0
$735$	47.3395	1.74614
$736$	0	0
$737$	−0.148673	−0.00547643
$738$	0	0
$739$	−32.6977	−1.20281	−0.601403	−	0.798946i	$-0.705391\pi$
$739$	−32.6977	−1.20281	−0.601403	+	0.798946i	$0.705391\pi$
$740$	0	0
$741$	−12.4441	−0.457145
$742$	0	0
$743$	14.6871	0.538818	0.269409	−	0.963026i	$-0.413172\pi$
$743$	14.6871	0.538818	0.269409	+	0.963026i	$0.413172\pi$
$744$	0	0
$745$	−2.19184	−0.0803029
$746$	0	0
$747$	−31.6565	−1.15825
$748$	0	0
$749$	23.8596	0.871809
$750$	0	0
$751$	43.9758	1.60470	0.802350	−	0.596854i	$-0.203583\pi$
$751$	43.9758	1.60470	0.802350	+	0.596854i	$0.203583\pi$
$752$	0	0
$753$	10.5652	0.385017
$754$	0	0
$755$	32.7205	1.19082
$756$	0	0
$757$	11.6024	0.421698	0.210849	−	0.977519i	$-0.432377\pi$
$757$	11.6024	0.421698	0.210849	+	0.977519i	$0.432377\pi$
$758$	0	0
$759$	48.0758	1.74504
$760$	0	0
$761$	−19.8046	−0.717917	−0.358958	−	0.933354i	$-0.616868\pi$
$761$	−19.8046	−0.717917	−0.358958	+	0.933354i	$0.616868\pi$
$762$	0	0
$763$	−4.38906	−0.158895
$764$	0	0
$765$	89.3374	3.23000
$766$	0	0
$767$	−4.88766	−0.176483
$768$	0	0
$769$	−32.8090	−1.18312	−0.591562	−	0.806260i	$-0.701488\pi$
$769$	−32.8090	−1.18312	−0.591562	+	0.806260i	$0.701488\pi$
$770$	0	0
$771$	16.7636	0.603728
$772$	0	0
$773$	−46.5472	−1.67419	−0.837093	−	0.547060i	$-0.815747\pi$
$773$	−46.5472	−1.67419	−0.837093	+	0.547060i	$0.815747\pi$
$774$	0	0
$775$	44.1302	1.58521
$776$	0	0
$777$	20.8091	0.746523
$778$	0	0
$779$	41.0009	1.46901
$780$	0	0
$781$	5.09391	0.182274
$782$	0	0
$783$	65.9461	2.35672
$784$	0	0
$785$	−76.7604	−2.73970
$786$	0	0
$787$	30.5605	1.08936	0.544681	−	0.838643i	$-0.316651\pi$
$787$	30.5605	1.08936	0.544681	+	0.838643i	$0.316651\pi$
$788$	0	0
$789$	−61.9737	−2.20632
$790$	0	0
$791$	−1.67298	−0.0594842
$792$	0	0
$793$	−11.3533	−0.403169
$794$	0	0
$795$	−22.8764	−0.811343
$796$	0	0
$797$	17.3988	0.616298	0.308149	−	0.951338i	$-0.400290\pi$
$797$	17.3988	0.616298	0.308149	+	0.951338i	$0.400290\pi$
$798$	0	0
$799$	31.4945	1.11420
$800$	0	0
$801$	−9.44830	−0.333839
$802$	0	0
$803$	−16.8934	−0.596157
$804$	0	0
$805$	−46.4334	−1.63656
$806$	0	0
$807$	30.2815	1.06596
$808$	0	0
$809$	15.0717	0.529893	0.264946	−	0.964263i	$-0.414646\pi$
$809$	15.0717	0.529893	0.264946	+	0.964263i	$0.414646\pi$
$810$	0	0
$811$	−6.69351	−0.235041	−0.117520	−	0.993070i	$-0.537495\pi$
$811$	−6.69351	−0.235041	−0.117520	+	0.993070i	$0.537495\pi$
$812$	0	0
$813$	−52.4651	−1.84003
$814$	0	0
$815$	22.9368	0.803442
$816$	0	0
$817$	−9.48429	−0.331813
$818$	0	0
$819$	−12.4657	−0.435586
$820$	0	0
$821$	−8.15931	−0.284762	−0.142381	−	0.989812i	$-0.545476\pi$
$821$	−8.15931	−0.284762	−0.142381	+	0.989812i	$0.545476\pi$
$822$	0	0
$823$	−24.8198	−0.865162	−0.432581	−	0.901595i	$-0.642397\pi$
$823$	−24.8198	−0.865162	−0.432581	+	0.901595i	$0.642397\pi$
$824$	0	0
$825$	−31.3723	−1.09224
$826$	0	0
$827$	22.6872	0.788911	0.394456	−	0.918915i	$-0.370933\pi$
$827$	22.6872	0.788911	0.394456	+	0.918915i	$0.370933\pi$
$828$	0	0
$829$	3.88608	0.134969	0.0674847	−	0.997720i	$-0.478503\pi$
$829$	3.88608	0.134969	0.0674847	+	0.997720i	$0.478503\pi$
$830$	0	0
$831$	−31.6570	−1.09817
$832$	0	0
$833$	15.5178	0.537660
$834$	0	0
$835$	33.0665	1.14431
$836$	0	0
$837$	119.443	4.12855
$838$	0	0
$839$	44.2113	1.52634	0.763172	−	0.646195i	$-0.223641\pi$
$839$	44.2113	1.52634	0.763172	+	0.646195i	$0.223641\pi$
$840$	0	0
$841$	−10.9916	−0.379020
$842$	0	0
$843$	−58.9068	−2.02886
$844$	0	0
$845$	−3.27743	−0.112747
$846$	0	0
$847$	13.2369	0.454825
$848$	0	0
$849$	−74.1980	−2.54647
$850$	0	0
$851$	34.6928	1.18925
$852$	0	0
$853$	42.9586	1.47088	0.735438	−	0.677592i	$-0.236977\pi$
$853$	42.9586	1.47088	0.735438	+	0.677592i	$0.236977\pi$
$854$	0	0
$855$	96.3363	3.29463
$856$	0	0
$857$	−0.662461	−0.0226292	−0.0113146	−	0.999936i	$-0.503602\pi$
$857$	−0.662461	−0.0226292	−0.0113146	+	0.999936i	$0.503602\pi$
$858$	0	0
$859$	−6.32387	−0.215768	−0.107884	−	0.994164i	$-0.534407\pi$
$859$	−6.32387	−0.215768	−0.107884	+	0.994164i	$0.534407\pi$
$860$	0	0
$861$	56.9882	1.94215
$862$	0	0
$863$	−29.6165	−1.00816	−0.504079	−	0.863657i	$-0.668168\pi$
$863$	−29.6165	−1.00816	−0.504079	+	0.863657i	$0.668168\pi$
$864$	0	0
$865$	−34.7170	−1.18041
$866$	0	0
$867$	−15.0832	−0.512252
$868$	0	0
$869$	0.487264	0.0165293
$870$	0	0
$871$	0.0891755	0.00302160
$872$	0	0
$873$	−108.528	−3.67312
$874$	0	0
$875$	3.91336	0.132296
$876$	0	0
$877$	46.1572	1.55862	0.779309	−	0.626640i	$-0.215570\pi$
$877$	46.1572	1.55862	0.779309	+	0.626640i	$0.215570\pi$
$878$	0	0
$879$	57.1128	1.92637
$880$	0	0
$881$	−34.2459	−1.15377	−0.576886	−	0.816825i	$-0.695732\pi$
$881$	−34.2459	−1.15377	−0.576886	+	0.816825i	$0.695732\pi$
$882$	0	0
$883$	−16.4579	−0.553852	−0.276926	−	0.960891i	$-0.589316\pi$
$883$	−16.4579	−0.553852	−0.276926	+	0.960891i	$0.589316\pi$
$884$	0	0
$885$	52.5010	1.76480
$886$	0	0
$887$	2.55255	0.0857061	0.0428530	−	0.999081i	$-0.486355\pi$
$887$	2.55255	0.0857061	0.0428530	+	0.999081i	$0.486355\pi$
$888$	0	0
$889$	−22.3955	−0.751120
$890$	0	0
$891$	−46.1924	−1.54750
$892$	0	0
$893$	33.9619	1.13649
$894$	0	0
$895$	−1.03695	−0.0346614
$896$	0	0
$897$	−28.8364	−0.962819
$898$	0	0
$899$	32.6172	1.08784
$900$	0	0
$901$	−7.49886	−0.249823
$902$	0	0
$903$	−13.1825	−0.438685
$904$	0	0
$905$	30.8585	1.02577
$906$	0	0
$907$	−44.9934	−1.49398	−0.746991	−	0.664835i	$-0.768502\pi$
$907$	−44.9934	−1.49398	−0.746991	+	0.664835i	$0.768502\pi$
$908$	0	0
$909$	10.3058	0.341820
$910$	0	0
$911$	−46.3523	−1.53572	−0.767860	−	0.640618i	$-0.778678\pi$
$911$	−46.3523	−1.53572	−0.767860	+	0.640618i	$0.778678\pi$
$912$	0	0
$913$	6.81744	0.225624
$914$	0	0
$915$	121.952	4.03162
$916$	0	0
$917$	−17.0346	−0.562533
$918$	0	0
$919$	−44.5578	−1.46983	−0.734914	−	0.678161i	$-0.762777\pi$
$919$	−44.5578	−1.46983	−0.734914	+	0.678161i	$0.762777\pi$
$920$	0	0
$921$	−29.3944	−0.968579
$922$	0	0
$923$	−3.05538	−0.100569
$924$	0	0
$925$	−22.6391	−0.744369
$926$	0	0
$927$	−28.0758	−0.922131
$928$	0	0
$929$	27.6677	0.907748	0.453874	−	0.891066i	$-0.350042\pi$
$929$	27.6677	0.907748	0.453874	+	0.891066i	$0.350042\pi$
$930$	0	0
$931$	16.7335	0.548419
$932$	0	0
$933$	−53.6286	−1.75572
$934$	0	0
$935$	−19.2394	−0.629196
$936$	0	0
$937$	34.9261	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$937$	34.9261	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$938$	0	0
$939$	44.9092	1.46555
$940$	0	0
$941$	−47.8192	−1.55886	−0.779431	−	0.626487i	$-0.784492\pi$
$941$	−47.8192	−1.55886	−0.779431	+	0.626487i	$0.784492\pi$
$942$	0	0
$943$	95.0103	3.09396
$944$	0	0
$945$	82.0113	2.66783
$946$	0	0
$947$	−35.9993	−1.16982	−0.584910	−	0.811098i	$-0.698870\pi$
$947$	−35.9993	−1.16982	−0.584910	+	0.811098i	$0.698870\pi$
$948$	0	0
$949$	10.1329	0.328927
$950$	0	0
$951$	−50.6370	−1.64202
$952$	0	0
$953$	34.3385	1.11233	0.556167	−	0.831071i	$-0.312272\pi$
$953$	34.3385	1.11233	0.556167	+	0.831071i	$0.312272\pi$
$954$	0	0
$955$	62.4713	2.02152
$956$	0	0
$957$	−23.1876	−0.749550
$958$	0	0
$959$	−9.56137	−0.308753
$960$	0	0
$961$	28.0769	0.905706
$962$	0	0
$963$	−114.710	−3.69647
$964$	0	0
$965$	81.5589	2.62547
$966$	0	0
$967$	3.38556	0.108872	0.0544361	−	0.998517i	$-0.482664\pi$
$967$	3.38556	0.108872	0.0544361	+	0.998517i	$0.482664\pi$
$968$	0	0
$969$	43.8164	1.40759
$970$	0	0
$971$	−49.1884	−1.57853	−0.789265	−	0.614052i	$-0.789538\pi$
$971$	−49.1884	−1.57853	−0.789265	+	0.614052i	$0.789538\pi$
$972$	0	0
$973$	−18.7605	−0.601434
$974$	0	0
$975$	18.8174	0.602640
$976$	0	0
$977$	−17.9884	−0.575501	−0.287750	−	0.957705i	$-0.592907\pi$
$977$	−17.9884	−0.575501	−0.287750	+	0.957705i	$0.592907\pi$
$978$	0	0
$979$	2.03476	0.0650311
$980$	0	0
$981$	21.1013	0.673713
$982$	0	0
$983$	14.8718	0.474336	0.237168	−	0.971469i	$-0.423781\pi$
$983$	14.8718	0.474336	0.237168	+	0.971469i	$0.423781\pi$
$984$	0	0
$985$	−81.2583	−2.58910
$986$	0	0
$987$	47.2046	1.50254
$988$	0	0
$989$	−21.9777	−0.698851
$990$	0	0
$991$	−4.70353	−0.149412	−0.0747062	−	0.997206i	$-0.523802\pi$
$991$	−4.70353	−0.149412	−0.0747062	+	0.997206i	$0.523802\pi$
$992$	0	0
$993$	74.1272	2.35236
$994$	0	0
$995$	−22.3333	−0.708014
$996$	0	0
$997$	12.9337	0.409613	0.204807	−	0.978802i	$-0.434343\pi$
$997$	12.9337	0.409613	0.204807	+	0.978802i	$0.434343\pi$
$998$	0	0
$999$	−61.2749	−1.93865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bm.1.4		4
4.3	odd	2		3328.2.a.bi.1.1		4
8.3	odd	2		3328.2.a.bn.1.4		4
8.5	even	2		3328.2.a.bj.1.1		4
16.3	odd	4		832.2.b.d.417.1	yes	8
16.5	even	4		832.2.b.c.417.1	✓	8
16.11	odd	4		832.2.b.d.417.8	yes	8
16.13	even	4		832.2.b.c.417.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.1	✓	8	16.5	even	4
832.2.b.c.417.8	yes	8	16.13	even	4
832.2.b.d.417.1	yes	8	16.3	odd	4
832.2.b.d.417.8	yes	8	16.11	odd	4
3328.2.a.bi.1.1		4	4.3	odd	2
3328.2.a.bj.1.1		4	8.5	even	2
3328.2.a.bm.1.4		4	1.1	even	1	trivial
3328.2.a.bn.1.4		4	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27743 q^{3} -3.27743 q^{5} -1.61023 q^{7} +7.74153 q^{9} -1.66719 q^{11} +1.00000 q^{13} -10.7415 q^{15} -3.52106 q^{17} -3.79691 q^{19} -5.27743 q^{21} -8.79849 q^{23} +5.74153 q^{25} +15.5400 q^{27} +4.24363 q^{29} +7.68615 q^{31} -5.46410 q^{33} +5.27743 q^{35} -3.94304 q^{37} +3.27743 q^{39} -10.7985 q^{41} +2.49790 q^{43} -25.3723 q^{45} -8.94462 q^{47} -4.40714 q^{49} -11.5400 q^{51} +2.12972 q^{53} +5.46410 q^{55} -12.4441 q^{57} -4.88766 q^{59} -11.3533 q^{61} -12.4657 q^{63} -3.27743 q^{65} +0.0891755 q^{67} -28.8364 q^{69} -3.05538 q^{71} +10.1329 q^{73} +18.8174 q^{75} +2.68457 q^{77} -0.292266 q^{79} +27.7067 q^{81} -4.08918 q^{83} +11.5400 q^{85} +13.9082 q^{87} -1.22047 q^{89} -1.61023 q^{91} +25.1908 q^{93} +12.4441 q^{95} -14.0190 q^{97} -12.9066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 4 q^{13} - 18 q^{15} - 2 q^{17} - 6 q^{19} - 10 q^{21} - 12 q^{23} - 2 q^{25} + 14 q^{27} + 16 q^{29} - 10 q^{31} - 8 q^{33} + 10 q^{35} - 14 q^{37} + 2 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 4 * q^13 - 18 * q^15 - 2 * q^17 - 6 * q^19 - 10 * q^21 - 12 * q^23 - 2 * q^25 + 14 * q^27 + 16 * q^29 - 10 * q^31 - 8 * q^33 + 10 * q^35 - 14 * q^37 + 2 * q^39 - 20 * q^41 - 14 * q^43 - 20 * q^45 - 20 * q^47 - 2 * q^49 + 2 * q^51 + 4 * q^53 + 8 * q^55 + 4 * q^57 - 2 * q^59 - 22 * q^63 - 2 * q^65 + 6 * q^67 - 20 * q^69 - 28 * q^71 + 8 * q^73 + 16 * q^75 - 16 * q^77 - 16 * q^79 + 20 * q^81 - 22 * q^83 - 2 * q^85 - 12 * q^87 + 8 * q^89 + 36 * q^93 - 4 * q^95 - 20 * q^97 + 2 * q^99

Introduction

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