LMFDB - Embedded newform 9680.2.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-2,0,5,0,3,0,0,0,-6,0,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} -7.12311 q^{13} -1.56155 q^{15} -4.68466 q^{17} +5.56155 q^{19} +0.684658 q^{21} +7.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} -4.43845 q^{29} +5.56155 q^{31} -0.438447 q^{35} +11.5616 q^{37} -11.1231 q^{39} -4.24621 q^{41} +5.12311 q^{43} +0.561553 q^{45} -13.3693 q^{47} -6.80776 q^{49} -7.31534 q^{51} -2.68466 q^{53} +8.68466 q^{57} +7.12311 q^{59} +8.43845 q^{61} -0.246211 q^{63} +7.12311 q^{65} +11.1231 q^{69} +8.68466 q^{71} +7.12311 q^{73} +1.56155 q^{75} +13.3693 q^{79} -7.00000 q^{81} -6.00000 q^{83} +4.68466 q^{85} -6.93087 q^{87} -2.68466 q^{89} -3.12311 q^{91} +8.68466 q^{93} -5.56155 q^{95} -13.1231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 6 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} - 11 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} - 14 q^{39} + 8 q^{41} + 2 q^{43}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 6 * q^13 + q^15 + 3 * q^17 + 7 * q^19 - 11 * q^21 + 6 * q^23 + 2 * q^25 - 7 * q^27 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 - 14 * q^39 + 8 * q^41 + 2 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 - 27 * q^51 + 7 * q^53 + 5 * q^57 + 6 * q^59 + 21 * q^61 + 16 * q^63 + 6 * q^65 + 14 * q^69 + 5 * q^71 + 6 * q^73 - q^75 + 2 * q^79 - 14 * q^81 - 12 * q^83 - 3 * q^85 + 15 * q^87 + 7 * q^89 + 2 * q^91 + 5 * q^93 - 7 * q^95 - 18 * q^97

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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−7.12311	−1.97559	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−7.12311	−1.97559	−0.987797	+	0.155747i	$0.950222\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	−4.68466	−1.13620	−0.568098	−	0.822961i	$-0.692321\pi$
$17$	−4.68466	−1.13620	−0.568098	+	0.822961i	$0.692321\pi$
$18$	0	0
$19$	5.56155	1.27591	0.637954	−	0.770075i	$-0.279781\pi$
$19$	5.56155	1.27591	0.637954	+	0.770075i	$0.279781\pi$
$20$	0	0
$21$	0.684658	0.149405
$22$	0	0
$23$	7.12311	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$23$	7.12311	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−4.43845	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$29$	−4.43845	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.438447	−0.0741111
$36$	0	0
$37$	11.5616	1.90071	0.950354	−	0.311171i	$-0.100721\pi$
$37$	11.5616	1.90071	0.950354	+	0.311171i	$0.100721\pi$
$38$	0	0
$39$	−11.1231	−1.78112
$40$	0	0
$41$	−4.24621	−0.663147	−0.331573	−	0.943429i	$-0.607579\pi$
$41$	−4.24621	−0.663147	−0.331573	+	0.943429i	$0.607579\pi$
$42$	0	0
$43$	5.12311	0.781266	0.390633	−	0.920546i	$-0.372256\pi$
$43$	5.12311	0.781266	0.390633	+	0.920546i	$0.372256\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	−13.3693	−1.95012	−0.975058	−	0.221952i	$-0.928757\pi$
$47$	−13.3693	−1.95012	−0.975058	+	0.221952i	$0.928757\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	−7.31534	−1.02435
$52$	0	0
$53$	−2.68466	−0.368766	−0.184383	−	0.982854i	$-0.559029\pi$
$53$	−2.68466	−0.368766	−0.184383	+	0.982854i	$0.559029\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.68466	1.15031
$58$	0	0
$59$	7.12311	0.927349	0.463675	−	0.886006i	$-0.346531\pi$
$59$	7.12311	0.927349	0.463675	+	0.886006i	$0.346531\pi$
$60$	0	0
$61$	8.43845	1.08043	0.540216	−	0.841526i	$-0.318342\pi$
$61$	8.43845	1.08043	0.540216	+	0.841526i	$0.318342\pi$
$62$	0	0
$63$	−0.246211	−0.0310197
$64$	0	0
$65$	7.12311	0.883513
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	11.1231	1.33906
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	7.12311	0.833696	0.416848	−	0.908976i	$-0.363135\pi$
$73$	7.12311	0.833696	0.416848	+	0.908976i	$0.363135\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.3693	1.50417	0.752083	−	0.659069i	$-0.229049\pi$
$79$	13.3693	1.50417	0.752083	+	0.659069i	$0.229049\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	4.68466	0.508123
$86$	0	0
$87$	−6.93087	−0.743067
$88$	0	0
$89$	−2.68466	−0.284573	−0.142287	−	0.989825i	$-0.545445\pi$
$89$	−2.68466	−0.284573	−0.142287	+	0.989825i	$0.545445\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	8.68466	0.900557
$94$	0	0
$95$	−5.56155	−0.570603
$96$	0	0
$97$	−13.1231	−1.33245	−0.666225	−	0.745751i	$-0.732091\pi$
$97$	−13.1231	−1.33245	−0.666225	+	0.745751i	$0.732091\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−0.684658	−0.0668158
$106$	0	0
$107$	−1.12311	−0.108575	−0.0542874	−	0.998525i	$-0.517289\pi$
$107$	−1.12311	−0.108575	−0.0542874	+	0.998525i	$0.517289\pi$
$108$	0	0
$109$	−12.2462	−1.17297	−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462	−1.17297	−0.586487	+	0.809959i	$0.699490\pi$
$110$	0	0
$111$	18.0540	1.71361
$112$	0	0
$113$	9.12311	0.858230	0.429115	−	0.903250i	$-0.358826\pi$
$113$	9.12311	0.858230	0.429115	+	0.903250i	$0.358826\pi$
$114$	0	0
$115$	−7.12311	−0.664233
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−2.05398	−0.188288
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−6.63068	−0.597869
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.1231	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	3.80776	0.332686	0.166343	−	0.986068i	$-0.446804\pi$
$131$	3.80776	0.332686	0.166343	+	0.986068i	$0.446804\pi$
$132$	0	0
$133$	2.43845	0.211440
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	1.12311	0.0959534	0.0479767	−	0.998848i	$-0.484723\pi$
$137$	1.12311	0.0959534	0.0479767	+	0.998848i	$0.484723\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−20.8769	−1.75815
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.43845	0.368593
$146$	0	0
$147$	−10.6307	−0.876804
$148$	0	0
$149$	10.6847	0.875321	0.437661	−	0.899140i	$-0.355807\pi$
$149$	10.6847	0.875321	0.437661	+	0.899140i	$0.355807\pi$
$150$	0	0
$151$	15.1231	1.23070	0.615350	−	0.788254i	$-0.289015\pi$
$151$	15.1231	1.23070	0.615350	+	0.788254i	$0.289015\pi$
$152$	0	0
$153$	2.63068	0.212678
$154$	0	0
$155$	−5.56155	−0.446715
$156$	0	0
$157$	9.31534	0.743445	0.371723	−	0.928344i	$-0.378767\pi$
$157$	9.31534	0.743445	0.371723	+	0.928344i	$0.378767\pi$
$158$	0	0
$159$	−4.19224	−0.332466
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	−0.192236	−0.0150571	−0.00752854	−	0.999972i	$-0.502396\pi$
$163$	−0.192236	−0.0150571	−0.00752854	+	0.999972i	$0.502396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.6847	0.826804	0.413402	−	0.910549i	$-0.364340\pi$
$167$	10.6847	0.826804	0.413402	+	0.910549i	$0.364340\pi$
$168$	0	0
$169$	37.7386	2.90297
$170$	0	0
$171$	−3.12311	−0.238830
$172$	0	0
$173$	−19.1231	−1.45390	−0.726951	−	0.686689i	$-0.759063\pi$
$173$	−19.1231	−1.45390	−0.726951	+	0.686689i	$0.759063\pi$
$174$	0	0
$175$	0.438447	0.0331435
$176$	0	0
$177$	11.1231	0.836064
$178$	0	0
$179$	2.24621	0.167890	0.0839449	−	0.996470i	$-0.473248\pi$
$179$	2.24621	0.167890	0.0839449	+	0.996470i	$0.473248\pi$
$180$	0	0
$181$	8.24621	0.612936	0.306468	−	0.951881i	$-0.400853\pi$
$181$	8.24621	0.612936	0.306468	+	0.951881i	$0.400853\pi$
$182$	0	0
$183$	13.1771	0.974078
$184$	0	0
$185$	−11.5616	−0.850022
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.43845	−0.177371
$190$	0	0
$191$	6.24621	0.451960	0.225980	−	0.974132i	$-0.427442\pi$
$191$	6.24621	0.451960	0.225980	+	0.974132i	$0.427442\pi$
$192$	0	0
$193$	25.5616	1.83996	0.919980	−	0.391964i	$-0.128204\pi$
$193$	25.5616	1.83996	0.919980	+	0.391964i	$0.128204\pi$
$194$	0	0
$195$	11.1231	0.796542
$196$	0	0
$197$	26.7386	1.90505	0.952524	−	0.304462i	$-0.0984768\pi$
$197$	26.7386	1.90505	0.952524	+	0.304462i	$0.0984768\pi$
$198$	0	0
$199$	16.6847	1.18274	0.591372	−	0.806399i	$-0.298586\pi$
$199$	16.6847	1.18274	0.591372	+	0.806399i	$0.298586\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.94602	−0.136584
$204$	0	0
$205$	4.24621	0.296568
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.31534	0.503609	0.251804	−	0.967778i	$-0.418976\pi$
$211$	7.31534	0.503609	0.251804	+	0.967778i	$0.418976\pi$
$212$	0	0
$213$	13.5616	0.929222
$214$	0	0
$215$	−5.12311	−0.349393
$216$	0	0
$217$	2.43845	0.165533
$218$	0	0
$219$	11.1231	0.751630
$220$	0	0
$221$	33.3693	2.24466
$222$	0	0
$223$	−16.8769	−1.13016	−0.565080	−	0.825036i	$-0.691155\pi$
$223$	−16.8769	−1.13016	−0.565080	+	0.825036i	$0.691155\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−8.24621	−0.547320	−0.273660	−	0.961826i	$-0.588234\pi$
$227$	−8.24621	−0.547320	−0.273660	+	0.961826i	$0.588234\pi$
$228$	0	0
$229$	−0.246211	−0.0162701	−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211	−0.0162701	−0.00813505	+	0.999967i	$0.502589\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0540	0.658658	0.329329	−	0.944215i	$-0.393178\pi$
$233$	10.0540	0.658658	0.329329	+	0.944215i	$0.393178\pi$
$234$	0	0
$235$	13.3693	0.872118
$236$	0	0
$237$	20.8769	1.35610
$238$	0	0
$239$	19.6155	1.26882	0.634412	−	0.772995i	$-0.281242\pi$
$239$	19.6155	1.26882	0.634412	+	0.772995i	$0.281242\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	6.80776	0.434932
$246$	0	0
$247$	−39.6155	−2.52068
$248$	0	0
$249$	−9.36932	−0.593756
$250$	0	0
$251$	2.63068	0.166047	0.0830236	−	0.996548i	$-0.473542\pi$
$251$	2.63068	0.166047	0.0830236	+	0.996548i	$0.473542\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7.31534	0.458104
$256$	0	0
$257$	−26.4924	−1.65255	−0.826276	−	0.563266i	$-0.809545\pi$
$257$	−26.4924	−1.65255	−0.826276	+	0.563266i	$0.809545\pi$
$258$	0	0
$259$	5.06913	0.314980
$260$	0	0
$261$	2.49242	0.154277
$262$	0	0
$263$	8.05398	0.496629	0.248315	−	0.968679i	$-0.420123\pi$
$263$	8.05398	0.496629	0.248315	+	0.968679i	$0.420123\pi$
$264$	0	0
$265$	2.68466	0.164917
$266$	0	0
$267$	−4.19224	−0.256561
$268$	0	0
$269$	9.12311	0.556246	0.278123	−	0.960546i	$-0.410288\pi$
$269$	9.12311	0.556246	0.278123	+	0.960546i	$0.410288\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	−4.87689	−0.295163
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.75379	0.345712	0.172856	−	0.984947i	$-0.444701\pi$
$277$	5.75379	0.345712	0.172856	+	0.984947i	$0.444701\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	8.24621	0.491928	0.245964	−	0.969279i	$-0.420896\pi$
$281$	8.24621	0.491928	0.245964	+	0.969279i	$0.420896\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	−8.68466	−0.514435
$286$	0	0
$287$	−1.86174	−0.109895
$288$	0	0
$289$	4.94602	0.290943
$290$	0	0
$291$	−20.4924	−1.20129
$292$	0	0
$293$	−8.87689	−0.518594	−0.259297	−	0.965798i	$-0.583491\pi$
$293$	−8.87689	−0.518594	−0.259297	+	0.965798i	$0.583491\pi$
$294$	0	0
$295$	−7.12311	−0.414723
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−50.7386	−2.93429
$300$	0	0
$301$	2.24621	0.129469
$302$	0	0
$303$	3.12311	0.179418
$304$	0	0
$305$	−8.43845	−0.483184
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.5616	0.769005	0.384503	−	0.923124i	$-0.374373\pi$
$311$	13.5616	0.769005	0.384503	+	0.923124i	$0.374373\pi$
$312$	0	0
$313$	−24.7386	−1.39831	−0.699155	−	0.714970i	$-0.746440\pi$
$313$	−24.7386	−1.39831	−0.699155	+	0.714970i	$0.746440\pi$
$314$	0	0
$315$	0.246211	0.0138724
$316$	0	0
$317$	18.6847	1.04943	0.524717	−	0.851276i	$-0.324171\pi$
$317$	18.6847	1.04943	0.524717	+	0.851276i	$0.324171\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.75379	−0.0978869
$322$	0	0
$323$	−26.0540	−1.44968
$324$	0	0
$325$	−7.12311	−0.395119
$326$	0	0
$327$	−19.1231	−1.05751
$328$	0	0
$329$	−5.86174	−0.323168
$330$	0	0
$331$	−30.7386	−1.68955	−0.844774	−	0.535123i	$-0.820265\pi$
$331$	−30.7386	−1.68955	−0.844774	+	0.535123i	$0.820265\pi$
$332$	0	0
$333$	−6.49242	−0.355783
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.8078	−0.861104	−0.430552	−	0.902566i	$-0.641681\pi$
$337$	−15.8078	−0.861104	−0.430552	+	0.902566i	$0.641681\pi$
$338$	0	0
$339$	14.2462	0.773748
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	−11.1231	−0.598848
$346$	0	0
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	18.4924	0.989877	0.494938	−	0.868928i	$-0.335191\pi$
$349$	18.4924	0.989877	0.494938	+	0.868928i	$0.335191\pi$
$350$	0	0
$351$	39.6155	2.11452
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−8.68466	−0.460934
$356$	0	0
$357$	−3.20739	−0.169753
$358$	0	0
$359$	−30.7386	−1.62232	−0.811162	−	0.584822i	$-0.801164\pi$
$359$	−30.7386	−1.62232	−0.811162	+	0.584822i	$0.801164\pi$
$360$	0	0
$361$	11.9309	0.627941
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.12311	−0.372840
$366$	0	0
$367$	16.4924	0.860897	0.430449	−	0.902615i	$-0.358355\pi$
$367$	16.4924	0.860897	0.430449	+	0.902615i	$0.358355\pi$
$368$	0	0
$369$	2.38447	0.124131
$370$	0	0
$371$	−1.17708	−0.0611110
$372$	0	0
$373$	1.36932	0.0709005	0.0354503	−	0.999371i	$-0.488713\pi$
$373$	1.36932	0.0709005	0.0354503	+	0.999371i	$0.488713\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	31.6155	1.62828
$378$	0	0
$379$	23.1231	1.18775	0.593877	−	0.804556i	$-0.297597\pi$
$379$	23.1231	1.18775	0.593877	+	0.804556i	$0.297597\pi$
$380$	0	0
$381$	20.4924	1.04986
$382$	0	0
$383$	25.3693	1.29631	0.648156	−	0.761508i	$-0.275541\pi$
$383$	25.3693	1.29631	0.648156	+	0.761508i	$0.275541\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.87689	−0.146241
$388$	0	0
$389$	18.4924	0.937603	0.468802	−	0.883304i	$-0.344686\pi$
$389$	18.4924	0.937603	0.468802	+	0.883304i	$0.344686\pi$
$390$	0	0
$391$	−33.3693	−1.68756
$392$	0	0
$393$	5.94602	0.299937
$394$	0	0
$395$	−13.3693	−0.672683
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	3.80776	0.190627
$400$	0	0
$401$	10.1922	0.508976	0.254488	−	0.967076i	$-0.418093\pi$
$401$	10.1922	0.508976	0.254488	+	0.967076i	$0.418093\pi$
$402$	0	0
$403$	−39.6155	−1.97339
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	0	0
$408$	0	0
$409$	7.36932	0.364389	0.182195	−	0.983262i	$-0.441680\pi$
$409$	7.36932	0.364389	0.182195	+	0.983262i	$0.441680\pi$
$410$	0	0
$411$	1.75379	0.0865080
$412$	0	0
$413$	3.12311	0.153678
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	−6.24621	−0.305878
$418$	0	0
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	−13.6155	−0.663580	−0.331790	−	0.943353i	$-0.607653\pi$
$421$	−13.6155	−0.663580	−0.331790	+	0.943353i	$0.607653\pi$
$422$	0	0
$423$	7.50758	0.365031
$424$	0	0
$425$	−4.68466	−0.227239
$426$	0	0
$427$	3.69981	0.179047
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−13.1231	−0.630656	−0.315328	−	0.948983i	$-0.602115\pi$
$433$	−13.1231	−0.630656	−0.315328	+	0.948983i	$0.602115\pi$
$434$	0	0
$435$	6.93087	0.332310
$436$	0	0
$437$	39.6155	1.89507
$438$	0	0
$439$	−6.24621	−0.298115	−0.149058	−	0.988828i	$-0.547624\pi$
$439$	−6.24621	−0.298115	−0.149058	+	0.988828i	$0.547624\pi$
$440$	0	0
$441$	3.82292	0.182044
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	2.68466	0.127265
$446$	0	0
$447$	16.6847	0.789157
$448$	0	0
$449$	−16.2462	−0.766706	−0.383353	−	0.923602i	$-0.625231\pi$
$449$	−16.2462	−0.766706	−0.383353	+	0.923602i	$0.625231\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	23.6155	1.10955
$454$	0	0
$455$	3.12311	0.146413
$456$	0	0
$457$	8.68466	0.406251	0.203126	−	0.979153i	$-0.434890\pi$
$457$	8.68466	0.406251	0.203126	+	0.979153i	$0.434890\pi$
$458$	0	0
$459$	26.0540	1.21610
$460$	0	0
$461$	−16.0540	−0.747708	−0.373854	−	0.927488i	$-0.621964\pi$
$461$	−16.0540	−0.747708	−0.373854	+	0.927488i	$0.621964\pi$
$462$	0	0
$463$	2.63068	0.122258	0.0611291	−	0.998130i	$-0.480530\pi$
$463$	2.63068	0.122258	0.0611291	+	0.998130i	$0.480530\pi$
$464$	0	0
$465$	−8.68466	−0.402741
$466$	0	0
$467$	−12.1922	−0.564189	−0.282095	−	0.959387i	$-0.591029\pi$
$467$	−12.1922	−0.564189	−0.282095	+	0.959387i	$0.591029\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.5464	0.670263
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.56155	0.255182
$476$	0	0
$477$	1.50758	0.0690272
$478$	0	0
$479$	−2.24621	−0.102632	−0.0513160	−	0.998682i	$-0.516342\pi$
$479$	−2.24621	−0.102632	−0.0513160	+	0.998682i	$0.516342\pi$
$480$	0	0
$481$	−82.3542	−3.75503
$482$	0	0
$483$	4.87689	0.221906
$484$	0	0
$485$	13.1231	0.595890
$486$	0	0
$487$	13.7538	0.623244	0.311622	−	0.950206i	$-0.399128\pi$
$487$	13.7538	0.623244	0.311622	+	0.950206i	$0.399128\pi$
$488$	0	0
$489$	−0.300187	−0.0135749
$490$	0	0
$491$	−28.6847	−1.29452	−0.647260	−	0.762269i	$-0.724085\pi$
$491$	−28.6847	−1.29452	−0.647260	+	0.762269i	$0.724085\pi$
$492$	0	0
$493$	20.7926	0.936452
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.80776	0.170802
$498$	0	0
$499$	−30.7386	−1.37605	−0.688025	−	0.725687i	$-0.741522\pi$
$499$	−30.7386	−1.37605	−0.688025	+	0.725687i	$0.741522\pi$
$500$	0	0
$501$	16.6847	0.745416
$502$	0	0
$503$	5.12311	0.228428	0.114214	−	0.993456i	$-0.463565\pi$
$503$	5.12311	0.228428	0.114214	+	0.993456i	$0.463565\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	58.9309	2.61721
$508$	0	0
$509$	−41.6155	−1.84458	−0.922288	−	0.386504i	$-0.873683\pi$
$509$	−41.6155	−1.84458	−0.922288	+	0.386504i	$0.873683\pi$
$510$	0	0
$511$	3.12311	0.138158
$512$	0	0
$513$	−30.9309	−1.36563
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−29.8617	−1.31078
$520$	0	0
$521$	2.49242	0.109195	0.0545975	−	0.998508i	$-0.482612\pi$
$521$	2.49242	0.109195	0.0545975	+	0.998508i	$0.482612\pi$
$522$	0	0
$523$	33.1231	1.44837	0.724186	−	0.689605i	$-0.242216\pi$
$523$	33.1231	1.44837	0.724186	+	0.689605i	$0.242216\pi$
$524$	0	0
$525$	0.684658	0.0298809
$526$	0	0
$527$	−26.0540	−1.13493
$528$	0	0
$529$	27.7386	1.20603
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	30.2462	1.31011
$534$	0	0
$535$	1.12311	0.0485561
$536$	0	0
$537$	3.50758	0.151363
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.9309	−0.899888	−0.449944	−	0.893057i	$-0.648556\pi$
$541$	−20.9309	−0.899888	−0.449944	+	0.893057i	$0.648556\pi$
$542$	0	0
$543$	12.8769	0.552600
$544$	0	0
$545$	12.2462	0.524570
$546$	0	0
$547$	−35.3693	−1.51228	−0.756141	−	0.654408i	$-0.772918\pi$
$547$	−35.3693	−1.51228	−0.756141	+	0.654408i	$0.772918\pi$
$548$	0	0
$549$	−4.73863	−0.202240
$550$	0	0
$551$	−24.6847	−1.05160
$552$	0	0
$553$	5.86174	0.249267
$554$	0	0
$555$	−18.0540	−0.766349
$556$	0	0
$557$	−41.3693	−1.75287	−0.876437	−	0.481516i	$-0.840086\pi$
$557$	−41.3693	−1.75287	−0.876437	+	0.481516i	$0.840086\pi$
$558$	0	0
$559$	−36.4924	−1.54347
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.9848	0.968696	0.484348	−	0.874876i	$-0.339057\pi$
$563$	22.9848	0.968696	0.484348	+	0.874876i	$0.339057\pi$
$564$	0	0
$565$	−9.12311	−0.383812
$566$	0	0
$567$	−3.06913	−0.128891
$568$	0	0
$569$	37.6155	1.57692	0.788462	−	0.615083i	$-0.210877\pi$
$569$	37.6155	1.57692	0.788462	+	0.615083i	$0.210877\pi$
$570$	0	0
$571$	−28.3002	−1.18433	−0.592163	−	0.805818i	$-0.701726\pi$
$571$	−28.3002	−1.18433	−0.592163	+	0.805818i	$0.701726\pi$
$572$	0	0
$573$	9.75379	0.407470
$574$	0	0
$575$	7.12311	0.297054
$576$	0	0
$577$	−32.2462	−1.34243	−0.671214	−	0.741264i	$-0.734227\pi$
$577$	−32.2462	−1.34243	−0.671214	+	0.741264i	$0.734227\pi$
$578$	0	0
$579$	39.9157	1.65884
$580$	0	0
$581$	−2.63068	−0.109139
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−34.0540	−1.40556	−0.702779	−	0.711408i	$-0.748058\pi$
$587$	−34.0540	−1.40556	−0.702779	+	0.711408i	$0.748058\pi$
$588$	0	0
$589$	30.9309	1.27448
$590$	0	0
$591$	41.7538	1.71752
$592$	0	0
$593$	19.6155	0.805513	0.402757	−	0.915307i	$-0.368052\pi$
$593$	19.6155	0.805513	0.402757	+	0.915307i	$0.368052\pi$
$594$	0	0
$595$	2.05398	0.0842048
$596$	0	0
$597$	26.0540	1.06632
$598$	0	0
$599$	1.06913	0.0436835	0.0218417	−	0.999761i	$-0.493047\pi$
$599$	1.06913	0.0436835	0.0218417	+	0.999761i	$0.493047\pi$
$600$	0	0
$601$	−38.9848	−1.59022	−0.795112	−	0.606462i	$-0.792588\pi$
$601$	−38.9848	−1.59022	−0.795112	+	0.606462i	$0.792588\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.80776	0.235730	0.117865	−	0.993030i	$-0.462395\pi$
$607$	5.80776	0.235730	0.117865	+	0.993030i	$0.462395\pi$
$608$	0	0
$609$	−3.03882	−0.123139
$610$	0	0
$611$	95.2311	3.85264
$612$	0	0
$613$	11.1231	0.449258	0.224629	−	0.974444i	$-0.427883\pi$
$613$	11.1231	0.449258	0.224629	+	0.974444i	$0.427883\pi$
$614$	0	0
$615$	6.63068	0.267375
$616$	0	0
$617$	13.6155	0.548141	0.274070	−	0.961710i	$-0.411630\pi$
$617$	13.6155	0.548141	0.274070	+	0.961710i	$0.411630\pi$
$618$	0	0
$619$	−14.7386	−0.592396	−0.296198	−	0.955127i	$-0.595719\pi$
$619$	−14.7386	−0.592396	−0.296198	+	0.955127i	$0.595719\pi$
$620$	0	0
$621$	−39.6155	−1.58972
$622$	0	0
$623$	−1.17708	−0.0471588
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−54.1619	−2.15958
$630$	0	0
$631$	12.1922	0.485365	0.242683	−	0.970106i	$-0.421973\pi$
$631$	12.1922	0.485365	0.242683	+	0.970106i	$0.421973\pi$
$632$	0	0
$633$	11.4233	0.454035
$634$	0	0
$635$	−13.1231	−0.520775
$636$	0	0
$637$	48.4924	1.92134
$638$	0	0
$639$	−4.87689	−0.192927
$640$	0	0
$641$	16.4384	0.649280	0.324640	−	0.945838i	$-0.394757\pi$
$641$	16.4384	0.649280	0.324640	+	0.945838i	$0.394757\pi$
$642$	0	0
$643$	28.3002	1.11605	0.558025	−	0.829824i	$-0.311559\pi$
$643$	28.3002	1.11605	0.558025	+	0.829824i	$0.311559\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−36.8769	−1.44978	−0.724890	−	0.688864i	$-0.758110\pi$
$647$	−36.8769	−1.44978	−0.724890	+	0.688864i	$0.758110\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.80776	0.149238
$652$	0	0
$653$	24.4384	0.956350	0.478175	−	0.878264i	$-0.341298\pi$
$653$	24.4384	0.956350	0.478175	+	0.878264i	$0.341298\pi$
$654$	0	0
$655$	−3.80776	−0.148782
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−5.06913	−0.197465	−0.0987326	−	0.995114i	$-0.531479\pi$
$659$	−5.06913	−0.197465	−0.0987326	+	0.995114i	$0.531479\pi$
$660$	0	0
$661$	−30.4924	−1.18602	−0.593009	−	0.805196i	$-0.702060\pi$
$661$	−30.4924	−1.18602	−0.593009	+	0.805196i	$0.702060\pi$
$662$	0	0
$663$	52.1080	2.02371
$664$	0	0
$665$	−2.43845	−0.0945589
$666$	0	0
$667$	−31.6155	−1.22416
$668$	0	0
$669$	−26.3542	−1.01891
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.0691	0.503778	0.251889	−	0.967756i	$-0.418948\pi$
$673$	13.0691	0.503778	0.251889	+	0.967756i	$0.418948\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	−29.7538	−1.14353	−0.571765	−	0.820417i	$-0.693741\pi$
$677$	−29.7538	−1.14353	−0.571765	+	0.820417i	$0.693741\pi$
$678$	0	0
$679$	−5.75379	−0.220810
$680$	0	0
$681$	−12.8769	−0.493444
$682$	0	0
$683$	5.17708	0.198095	0.0990477	−	0.995083i	$-0.468420\pi$
$683$	5.17708	0.198095	0.0990477	+	0.995083i	$0.468420\pi$
$684$	0	0
$685$	−1.12311	−0.0429117
$686$	0	0
$687$	−0.384472	−0.0146685
$688$	0	0
$689$	19.1231	0.728532
$690$	0	0
$691$	35.6155	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$691$	35.6155	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	19.8920	0.753465
$698$	0	0
$699$	15.6998	0.593821
$700$	0	0
$701$	16.4384	0.620872	0.310436	−	0.950594i	$-0.399525\pi$
$701$	16.4384	0.620872	0.310436	+	0.950594i	$0.399525\pi$
$702$	0	0
$703$	64.3002	2.42513
$704$	0	0
$705$	20.8769	0.786269
$706$	0	0
$707$	0.876894	0.0329790
$708$	0	0
$709$	−15.7538	−0.591646	−0.295823	−	0.955243i	$-0.595594\pi$
$709$	−15.7538	−0.591646	−0.295823	+	0.955243i	$0.595594\pi$
$710$	0	0
$711$	−7.50758	−0.281556
$712$	0	0
$713$	39.6155	1.48361
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.6307	1.14392
$718$	0	0
$719$	2.82292	0.105277	0.0526386	−	0.998614i	$-0.483237\pi$
$719$	2.82292	0.105277	0.0526386	+	0.998614i	$0.483237\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.6307	0.841644
$724$	0	0
$725$	−4.43845	−0.164840
$726$	0	0
$727$	19.1231	0.709237	0.354618	−	0.935011i	$-0.384611\pi$
$727$	19.1231	0.709237	0.354618	+	0.935011i	$0.384611\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	30.2462	1.11717	0.558585	−	0.829448i	$-0.311345\pi$
$733$	30.2462	1.11717	0.558585	+	0.829448i	$0.311345\pi$
$734$	0	0
$735$	10.6307	0.392119
$736$	0	0
$737$	0	0
$738$	0	0
$739$	5.75379	0.211657	0.105828	−	0.994384i	$-0.466251\pi$
$739$	5.75379	0.211657	0.105828	+	0.994384i	$0.466251\pi$
$740$	0	0
$741$	−61.8617	−2.27255
$742$	0	0
$743$	16.4384	0.603068	0.301534	−	0.953455i	$-0.402501\pi$
$743$	16.4384	0.603068	0.301534	+	0.953455i	$0.402501\pi$
$744$	0	0
$745$	−10.6847	−0.391456
$746$	0	0
$747$	3.36932	0.123277
$748$	0	0
$749$	−0.492423	−0.0179927
$750$	0	0
$751$	51.8078	1.89049	0.945246	−	0.326358i	$-0.105822\pi$
$751$	51.8078	1.89049	0.945246	+	0.326358i	$0.105822\pi$
$752$	0	0
$753$	4.10795	0.149702
$754$	0	0
$755$	−15.1231	−0.550386
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.2462	−0.878924	−0.439462	−	0.898261i	$-0.644831\pi$
$761$	−24.2462	−0.878924	−0.439462	+	0.898261i	$0.644831\pi$
$762$	0	0
$763$	−5.36932	−0.194382
$764$	0	0
$765$	−2.63068	−0.0951125
$766$	0	0
$767$	−50.7386	−1.83207
$768$	0	0
$769$	−20.2462	−0.730097	−0.365049	−	0.930988i	$-0.618948\pi$
$769$	−20.2462	−0.730097	−0.365049	+	0.930988i	$0.618948\pi$
$770$	0	0
$771$	−41.3693	−1.48988
$772$	0	0
$773$	15.0691	0.541999	0.270999	−	0.962579i	$-0.412646\pi$
$773$	15.0691	0.541999	0.270999	+	0.962579i	$0.412646\pi$
$774$	0	0
$775$	5.56155	0.199777
$776$	0	0
$777$	7.91571	0.283975
$778$	0	0
$779$	−23.6155	−0.846114
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.6847	0.882158
$784$	0	0
$785$	−9.31534	−0.332479
$786$	0	0
$787$	39.8617	1.42092	0.710459	−	0.703739i	$-0.248487\pi$
$787$	39.8617	1.42092	0.710459	+	0.703739i	$0.248487\pi$
$788$	0	0
$789$	12.5767	0.447743
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	−60.1080	−2.13450
$794$	0	0
$795$	4.19224	0.148683
$796$	0	0
$797$	3.75379	0.132966	0.0664830	−	0.997788i	$-0.478822\pi$
$797$	3.75379	0.132966	0.0664830	+	0.997788i	$0.478822\pi$
$798$	0	0
$799$	62.6307	2.21571
$800$	0	0
$801$	1.50758	0.0532676
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.12311	−0.110075
$806$	0	0
$807$	14.2462	0.501490
$808$	0	0
$809$	−8.73863	−0.307234	−0.153617	−	0.988130i	$-0.549092\pi$
$809$	−8.73863	−0.307234	−0.153617	+	0.988130i	$0.549092\pi$
$810$	0	0
$811$	26.9309	0.945671	0.472835	−	0.881151i	$-0.343231\pi$
$811$	26.9309	0.945671	0.472835	+	0.881151i	$0.343231\pi$
$812$	0	0
$813$	−6.24621	−0.219064
$814$	0	0
$815$	0.192236	0.00673373
$816$	0	0
$817$	28.4924	0.996824
$818$	0	0
$819$	1.75379	0.0612823
$820$	0	0
$821$	18.4924	0.645390	0.322695	−	0.946503i	$-0.395411\pi$
$821$	18.4924	0.645390	0.322695	+	0.946503i	$0.395411\pi$
$822$	0	0
$823$	20.4924	0.714321	0.357160	−	0.934043i	$-0.383745\pi$
$823$	20.4924	0.714321	0.357160	+	0.934043i	$0.383745\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.8617	−0.968848	−0.484424	−	0.874833i	$-0.660971\pi$
$827$	−27.8617	−0.968848	−0.484424	+	0.874833i	$0.660971\pi$
$828$	0	0
$829$	−41.1231	−1.42826	−0.714132	−	0.700011i	$-0.753178\pi$
$829$	−41.1231	−1.42826	−0.714132	+	0.700011i	$0.753178\pi$
$830$	0	0
$831$	8.98485	0.311681
$832$	0	0
$833$	31.8920	1.10499
$834$	0	0
$835$	−10.6847	−0.369758
$836$	0	0
$837$	−30.9309	−1.06913
$838$	0	0
$839$	−52.4924	−1.81224	−0.906120	−	0.423021i	$-0.860970\pi$
$839$	−52.4924	−1.81224	−0.906120	+	0.423021i	$0.860970\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	12.8769	0.443504
$844$	0	0
$845$	−37.7386	−1.29825
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.36932	−0.321554
$850$	0	0
$851$	82.3542	2.82306
$852$	0	0
$853$	28.8769	0.988726	0.494363	−	0.869256i	$-0.335401\pi$
$853$	28.8769	0.988726	0.494363	+	0.869256i	$0.335401\pi$
$854$	0	0
$855$	3.12311	0.106808
$856$	0	0
$857$	7.80776	0.266708	0.133354	−	0.991068i	$-0.457425\pi$
$857$	7.80776	0.266708	0.133354	+	0.991068i	$0.457425\pi$
$858$	0	0
$859$	33.8617	1.15535	0.577674	−	0.816268i	$-0.303961\pi$
$859$	33.8617	1.15535	0.577674	+	0.816268i	$0.303961\pi$
$860$	0	0
$861$	−2.90720	−0.0990773
$862$	0	0
$863$	−20.8769	−0.710658	−0.355329	−	0.934741i	$-0.615631\pi$
$863$	−20.8769	−0.710658	−0.355329	+	0.934741i	$0.615631\pi$
$864$	0	0
$865$	19.1231	0.650205
$866$	0	0
$867$	7.72348	0.262303
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.36932	0.249414
$874$	0	0
$875$	−0.438447	−0.0148222
$876$	0	0
$877$	−15.6155	−0.527299	−0.263649	−	0.964619i	$-0.584926\pi$
$877$	−15.6155	−0.527299	−0.263649	+	0.964619i	$0.584926\pi$
$878$	0	0
$879$	−13.8617	−0.467545
$880$	0	0
$881$	38.4924	1.29684	0.648421	−	0.761282i	$-0.275430\pi$
$881$	38.4924	1.29684	0.648421	+	0.761282i	$0.275430\pi$
$882$	0	0
$883$	−42.5464	−1.43180	−0.715900	−	0.698203i	$-0.753983\pi$
$883$	−42.5464	−1.43180	−0.715900	+	0.698203i	$0.753983\pi$
$884$	0	0
$885$	−11.1231	−0.373899
$886$	0	0
$887$	−18.8769	−0.633824	−0.316912	−	0.948455i	$-0.602646\pi$
$887$	−18.8769	−0.633824	−0.316912	+	0.948455i	$0.602646\pi$
$888$	0	0
$889$	5.75379	0.192976
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−74.3542	−2.48817
$894$	0	0
$895$	−2.24621	−0.0750826
$896$	0	0
$897$	−79.2311	−2.64545
$898$	0	0
$899$	−24.6847	−0.823279
$900$	0	0
$901$	12.5767	0.418991
$902$	0	0
$903$	3.50758	0.116725
$904$	0	0
$905$	−8.24621	−0.274113
$906$	0	0
$907$	31.3153	1.03981	0.519904	−	0.854224i	$-0.325968\pi$
$907$	31.3153	1.03981	0.519904	+	0.854224i	$0.325968\pi$
$908$	0	0
$909$	−1.12311	−0.0372511
$910$	0	0
$911$	−45.5616	−1.50952	−0.754761	−	0.656000i	$-0.772247\pi$
$911$	−45.5616	−1.50952	−0.754761	+	0.656000i	$0.772247\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−13.1771	−0.435621
$916$	0	0
$917$	1.66950	0.0551319
$918$	0	0
$919$	47.2311	1.55801	0.779004	−	0.627018i	$-0.215725\pi$
$919$	47.2311	1.55801	0.779004	+	0.627018i	$0.215725\pi$
$920$	0	0
$921$	28.1080	0.926188
$922$	0	0
$923$	−61.8617	−2.03620
$924$	0	0
$925$	11.5616	0.380142
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.5464	−1.19905	−0.599524	−	0.800357i	$-0.704643\pi$
$929$	−36.5464	−1.19905	−0.599524	+	0.800357i	$0.704643\pi$
$930$	0	0
$931$	−37.8617	−1.24087
$932$	0	0
$933$	21.1771	0.693307
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.38447	0.273909	0.136954	−	0.990577i	$-0.456269\pi$
$937$	8.38447	0.273909	0.136954	+	0.990577i	$0.456269\pi$
$938$	0	0
$939$	−38.6307	−1.26066
$940$	0	0
$941$	28.0540	0.914533	0.457267	−	0.889330i	$-0.348828\pi$
$941$	28.0540	0.914533	0.457267	+	0.889330i	$0.348828\pi$
$942$	0	0
$943$	−30.2462	−0.984952
$944$	0	0
$945$	2.43845	0.0793227
$946$	0	0
$947$	−50.0540	−1.62654	−0.813268	−	0.581890i	$-0.802314\pi$
$947$	−50.0540	−1.62654	−0.813268	+	0.581890i	$0.802314\pi$
$948$	0	0
$949$	−50.7386	−1.64705
$950$	0	0
$951$	29.1771	0.946132
$952$	0	0
$953$	2.43845	0.0789891	0.0394945	−	0.999220i	$-0.487425\pi$
$953$	2.43845	0.0789891	0.0394945	+	0.999220i	$0.487425\pi$
$954$	0	0
$955$	−6.24621	−0.202123
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.492423	0.0159012
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	0.630683	0.0203235
$964$	0	0
$965$	−25.5616	−0.822855
$966$	0	0
$967$	−8.43845	−0.271362	−0.135681	−	0.990753i	$-0.543322\pi$
$967$	−8.43845	−0.271362	−0.135681	+	0.990753i	$0.543322\pi$
$968$	0	0
$969$	−40.6847	−1.30698
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−1.75379	−0.0562239
$974$	0	0
$975$	−11.1231	−0.356224
$976$	0	0
$977$	35.8617	1.14732	0.573659	−	0.819094i	$-0.305523\pi$
$977$	35.8617	1.14732	0.573659	+	0.819094i	$0.305523\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6.87689	0.219562
$982$	0	0
$983$	41.3693	1.31948	0.659738	−	0.751496i	$-0.270667\pi$
$983$	41.3693	1.31948	0.659738	+	0.751496i	$0.270667\pi$
$984$	0	0
$985$	−26.7386	−0.851964
$986$	0	0
$987$	−9.15342	−0.291356
$988$	0	0
$989$	36.4924	1.16039
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−48.0000	−1.52323
$994$	0	0
$995$	−16.6847	−0.528939
$996$	0	0
$997$	−50.7386	−1.60691	−0.803454	−	0.595366i	$-0.797007\pi$
$997$	−50.7386	−1.60691	−0.803454	+	0.595366i	$0.797007\pi$
$998$	0	0
$999$	−64.3002	−2.03437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bm.1.2		2
4.3	odd	2		4840.2.a.m.1.1		2
11.10	odd	2		880.2.a.k.1.2		2
33.32	even	2		7920.2.a.by.1.2		2
44.43	even	2		440.2.a.g.1.1	✓	2
55.32	even	4		4400.2.b.w.4049.2		4
55.43	even	4		4400.2.b.w.4049.3		4
55.54	odd	2		4400.2.a.bt.1.1		2
88.21	odd	2		3520.2.a.br.1.1		2
88.43	even	2		3520.2.a.bm.1.2		2
132.131	odd	2		3960.2.a.bf.1.1		2
220.43	odd	4		2200.2.b.f.1849.2		4
220.87	odd	4		2200.2.b.f.1849.3		4
220.219	even	2		2200.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.g.1.1	✓	2	44.43	even	2
880.2.a.k.1.2		2	11.10	odd	2
2200.2.a.l.1.2		2	220.219	even	2
2200.2.b.f.1849.2		4	220.43	odd	4
2200.2.b.f.1849.3		4	220.87	odd	4
3520.2.a.bm.1.2		2	88.43	even	2
3520.2.a.br.1.1		2	88.21	odd	2
3960.2.a.bf.1.1		2	132.131	odd	2
4400.2.a.bt.1.1		2	55.54	odd	2
4400.2.b.w.4049.2		4	55.32	even	4
4400.2.b.w.4049.3		4	55.43	even	4
4840.2.a.m.1.1		2	4.3	odd	2
7920.2.a.by.1.2		2	33.32	even	2
9680.2.a.bm.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.00000 q^{5} +0.438447 q^{7} -0.561553 q^{9} -7.12311 q^{13} -1.56155 q^{15} -4.68466 q^{17} +5.56155 q^{19} +0.684658 q^{21} +7.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} -4.43845 q^{29} +5.56155 q^{31} -0.438447 q^{35} +11.5616 q^{37} -11.1231 q^{39} -4.24621 q^{41} +5.12311 q^{43} +0.561553 q^{45} -13.3693 q^{47} -6.80776 q^{49} -7.31534 q^{51} -2.68466 q^{53} +8.68466 q^{57} +7.12311 q^{59} +8.43845 q^{61} -0.246211 q^{63} +7.12311 q^{65} +11.1231 q^{69} +8.68466 q^{71} +7.12311 q^{73} +1.56155 q^{75} +13.3693 q^{79} -7.00000 q^{81} -6.00000 q^{83} +4.68466 q^{85} -6.93087 q^{87} -2.68466 q^{89} -3.12311 q^{91} +8.68466 q^{93} -5.56155 q^{95} -13.1231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 6 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} - 11 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} - 14 q^{39} + 8 q^{41} + 2 q^{43}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 6 * q^13 + q^15 + 3 * q^17 + 7 * q^19 - 11 * q^21 + 6 * q^23 + 2 * q^25 - 7 * q^27 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 - 14 * q^39 + 8 * q^41 + 2 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 - 27 * q^51 + 7 * q^53 + 5 * q^57 + 6 * q^59 + 21 * q^61 + 16 * q^63 + 6 * q^65 + 14 * q^69 + 5 * q^71 + 6 * q^73 - q^75 + 2 * q^79 - 14 * q^81 - 12 * q^83 - 3 * q^85 + 15 * q^87 + 7 * q^89 + 2 * q^91 + 5 * q^93 - 7 * q^95 - 18 * q^97

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