LMFDB - Embedded newform 9680.2.a.db.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.90131$ 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)

$f(q)$	$=$	$q-0.614975 q^{3} +1.00000 q^{5} +2.24598 q^{7} -2.62181 q^{9} +5.19447 q^{13} -0.614975 q^{15} -3.22995 q^{17} +7.19447 q^{19} -1.38122 q^{21} -5.11377 q^{23} +1.00000 q^{25} +3.45727 q^{27} +9.61890 q^{29} -0.491468 q^{31} +2.24598 q^{35} +0.350828 q^{37} -3.19447 q^{39} -4.42090 q^{41} +10.9537 q^{43} -2.62181 q^{45} +1.03420 q^{47} -1.95557 q^{49} +1.98634 q^{51} +3.75348 q^{53} -4.42442 q^{57} -12.7082 q^{59} -6.38983 q^{61} -5.88852 q^{63} +5.19447 q^{65} +8.44020 q^{67} +3.14484 q^{69} +12.4475 q^{71} -12.0273 q^{73} -0.614975 q^{75} -2.13034 q^{79} +5.73928 q^{81} +10.7270 q^{83} -3.22995 q^{85} -5.91538 q^{87} +9.90118 q^{89} +11.6667 q^{91} +0.302241 q^{93} +7.19447 q^{95} -15.9253 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} + 2 q^{15} - 8 q^{17} + 12 q^{19} + 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} - 16 q^{29} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} - 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 + 2 * q^15 - 8 * q^17 + 12 * q^19 + 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 - 16 * q^29 + 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 - 32 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 + 16 * q^57 - 4 * q^59 - 16 * q^61 + 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 - 16 * q^73 + 2 * q^75 - 10 * q^81 + 12 * q^83 - 8 * q^85 - 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 + 12 * q^95

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
$3$	−0.614975	−0.355056	−0.177528	−	0.984116i	$-0.556810\pi$
$3$	−0.614975	−0.355056	−0.177528	+	0.984116i	$0.556810\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.24598	0.848901	0.424450	−	0.905451i	$-0.360467\pi$
$7$	2.24598	0.848901	0.424450	+	0.905451i	$0.360467\pi$
$8$	0	0
$9$	−2.62181	−0.873935
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.19447	1.44069	0.720344	−	0.693617i	$-0.243984\pi$
$13$	5.19447	1.44069	0.720344	+	0.693617i	$0.243984\pi$
$14$	0	0
$15$	−0.614975	−0.158786
$16$	0	0
$17$	−3.22995	−0.783378	−0.391689	−	0.920098i	$-0.628109\pi$
$17$	−3.22995	−0.783378	−0.391689	+	0.920098i	$0.628109\pi$
$18$	0	0
$19$	7.19447	1.65053	0.825263	−	0.564749i	$-0.191027\pi$
$19$	7.19447	1.65053	0.825263	+	0.564749i	$0.191027\pi$
$20$	0	0
$21$	−1.38122	−0.301408
$22$	0	0
$23$	−5.11377	−1.06629	−0.533147	−	0.846023i	$-0.678991\pi$
$23$	−5.11377	−1.06629	−0.533147	+	0.846023i	$0.678991\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.45727	0.665352
$28$	0	0
$29$	9.61890	1.78618	0.893092	−	0.449874i	$-0.148531\pi$
$29$	9.61890	1.78618	0.893092	+	0.449874i	$0.148531\pi$
$30$	0	0
$31$	−0.491468	−0.0882703	−0.0441352	−	0.999026i	$-0.514053\pi$
$31$	−0.491468	−0.0882703	−0.0441352	+	0.999026i	$0.514053\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.24598	0.379640
$36$	0	0
$37$	0.350828	0.0576758	0.0288379	−	0.999584i	$-0.490819\pi$
$37$	0.350828	0.0576758	0.0288379	+	0.999584i	$0.490819\pi$
$38$	0	0
$39$	−3.19447	−0.511525
$40$	0	0
$41$	−4.42090	−0.690429	−0.345214	−	0.938524i	$-0.612194\pi$
$41$	−4.42090	−0.690429	−0.345214	+	0.938524i	$0.612194\pi$
$42$	0	0
$43$	10.9537	1.67042	0.835211	−	0.549929i	$-0.185345\pi$
$43$	10.9537	1.67042	0.835211	+	0.549929i	$0.185345\pi$
$44$	0	0
$45$	−2.62181	−0.390836
$46$	0	0
$47$	1.03420	0.150853	0.0754265	−	0.997151i	$-0.475968\pi$
$47$	1.03420	0.150853	0.0754265	+	0.997151i	$0.475968\pi$
$48$	0	0
$49$	−1.95557	−0.279367
$50$	0	0
$51$	1.98634	0.278143
$52$	0	0
$53$	3.75348	0.515580	0.257790	−	0.966201i	$-0.417006\pi$
$53$	3.75348	0.515580	0.257790	+	0.966201i	$0.417006\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.42442	−0.586029
$58$	0	0
$59$	−12.7082	−1.65447	−0.827234	−	0.561858i	$-0.810087\pi$
$59$	−12.7082	−1.65447	−0.827234	+	0.561858i	$0.810087\pi$
$60$	0	0
$61$	−6.38983	−0.818134	−0.409067	−	0.912504i	$-0.634146\pi$
$61$	−6.38983	−0.818134	−0.409067	+	0.912504i	$0.634146\pi$
$62$	0	0
$63$	−5.88852	−0.741884
$64$	0	0
$65$	5.19447	0.644295
$66$	0	0
$67$	8.44020	1.03114	0.515568	−	0.856849i	$-0.327581\pi$
$67$	8.44020	1.03114	0.515568	+	0.856849i	$0.327581\pi$
$68$	0	0
$69$	3.14484	0.378594
$70$	0	0
$71$	12.4475	1.47725	0.738625	−	0.674116i	$-0.235475\pi$
$71$	12.4475	1.47725	0.738625	+	0.674116i	$0.235475\pi$
$72$	0	0
$73$	−12.0273	−1.40769	−0.703846	−	0.710353i	$-0.748535\pi$
$73$	−12.0273	−1.40769	−0.703846	+	0.710353i	$0.748535\pi$
$74$	0	0
$75$	−0.614975	−0.0710112
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.13034	−0.239682	−0.119841	−	0.992793i	$-0.538238\pi$
$79$	−2.13034	−0.239682	−0.119841	+	0.992793i	$0.538238\pi$
$80$	0	0
$81$	5.73928	0.637698
$82$	0	0
$83$	10.7270	1.17744	0.588719	−	0.808338i	$-0.299633\pi$
$83$	10.7270	1.17744	0.588719	+	0.808338i	$0.299633\pi$
$84$	0	0
$85$	−3.22995	−0.350337
$86$	0	0
$87$	−5.91538	−0.634196
$88$	0	0
$89$	9.90118	1.04952	0.524762	−	0.851249i	$-0.324154\pi$
$89$	9.90118	1.04952	0.524762	+	0.851249i	$0.324154\pi$
$90$	0	0
$91$	11.6667	1.22300
$92$	0	0
$93$	0.302241	0.0313409
$94$	0	0
$95$	7.19447	0.738137
$96$	0	0
$97$	−15.9253	−1.61697	−0.808484	−	0.588518i	$-0.799712\pi$
$97$	−15.9253	−1.61697	−0.808484	+	0.588518i	$0.799712\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.8726	−1.67889	−0.839445	−	0.543444i	$-0.817120\pi$
$101$	−16.8726	−1.67889	−0.839445	+	0.543444i	$0.817120\pi$
$102$	0	0
$103$	−7.05805	−0.695450	−0.347725	−	0.937597i	$-0.613046\pi$
$103$	−7.05805	−0.695450	−0.347725	+	0.937597i	$0.613046\pi$
$104$	0	0
$105$	−1.38122	−0.134794
$106$	0	0
$107$	7.64175	0.738756	0.369378	−	0.929279i	$-0.379571\pi$
$107$	7.64175	0.738756	0.369378	+	0.929279i	$0.379571\pi$
$108$	0	0
$109$	11.2590	1.07841	0.539206	−	0.842174i	$-0.318724\pi$
$109$	11.2590	1.07841	0.539206	+	0.842174i	$0.318724\pi$
$110$	0	0
$111$	−0.215751	−0.0204781
$112$	0	0
$113$	−0.783709	−0.0737252	−0.0368626	−	0.999320i	$-0.511736\pi$
$113$	−0.783709	−0.0737252	−0.0368626	+	0.999320i	$0.511736\pi$
$114$	0	0
$115$	−5.11377	−0.476861
$116$	0	0
$117$	−13.6189	−1.25907
$118$	0	0
$119$	−7.25441	−0.665010
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.71875	0.245141
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.20546	0.728117	0.364059	−	0.931376i	$-0.381391\pi$
$127$	8.20546	0.728117	0.364059	+	0.931376i	$0.381391\pi$
$128$	0	0
$129$	−6.73625	−0.593094
$130$	0	0
$131$	6.35603	0.555329	0.277664	−	0.960678i	$-0.410440\pi$
$131$	6.35603	0.555329	0.277664	+	0.960678i	$0.410440\pi$
$132$	0	0
$133$	16.1586	1.40113
$134$	0	0
$135$	3.45727	0.297555
$136$	0	0
$137$	−8.27612	−0.707077	−0.353538	−	0.935420i	$-0.615022\pi$
$137$	−8.27612	−0.707077	−0.353538	+	0.935420i	$0.615022\pi$
$138$	0	0
$139$	11.0778	0.939606	0.469803	−	0.882771i	$-0.344325\pi$
$139$	11.0778	0.939606	0.469803	+	0.882771i	$0.344325\pi$
$140$	0	0
$141$	−0.636005	−0.0535613
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.61890	0.798806
$146$	0	0
$147$	1.20263	0.0991910
$148$	0	0
$149$	−8.46662	−0.693613	−0.346806	−	0.937937i	$-0.612734\pi$
$149$	−8.46662	−0.693613	−0.346806	+	0.937937i	$0.612734\pi$
$150$	0	0
$151$	6.39661	0.520548	0.260274	−	0.965535i	$-0.416187\pi$
$151$	6.39661	0.520548	0.260274	+	0.965535i	$0.416187\pi$
$152$	0	0
$153$	8.46830	0.684622
$154$	0	0
$155$	−0.491468	−0.0394757
$156$	0	0
$157$	16.0273	1.27912	0.639560	−	0.768741i	$-0.279117\pi$
$157$	16.0273	1.27912	0.639560	+	0.768741i	$0.279117\pi$
$158$	0	0
$159$	−2.30830	−0.183060
$160$	0	0
$161$	−11.4854	−0.905178
$162$	0	0
$163$	10.1970	0.798694	0.399347	−	0.916800i	$-0.369237\pi$
$163$	10.1970	0.798694	0.399347	+	0.916800i	$0.369237\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.3145	−1.03031	−0.515154	−	0.857097i	$-0.672265\pi$
$167$	−13.3145	−1.03031	−0.515154	+	0.857097i	$0.672265\pi$
$168$	0	0
$169$	13.9825	1.07558
$170$	0	0
$171$	−18.8625	−1.44245
$172$	0	0
$173$	−18.7177	−1.42308	−0.711538	−	0.702647i	$-0.752001\pi$
$173$	−18.7177	−1.42308	−0.711538	+	0.702647i	$0.752001\pi$
$174$	0	0
$175$	2.24598	0.169780
$176$	0	0
$177$	7.81523	0.587429
$178$	0	0
$179$	19.5287	1.45965	0.729823	−	0.683636i	$-0.239602\pi$
$179$	19.5287	1.45965	0.729823	+	0.683636i	$0.239602\pi$
$180$	0	0
$181$	6.39136	0.475066	0.237533	−	0.971379i	$-0.423661\pi$
$181$	6.39136	0.475066	0.237533	+	0.971379i	$0.423661\pi$
$182$	0	0
$183$	3.92959	0.290483
$184$	0	0
$185$	0.350828	0.0257934
$186$	0	0
$187$	0	0
$188$	0	0
$189$	7.76497	0.564818
$190$	0	0
$191$	6.58718	0.476632	0.238316	−	0.971188i	$-0.423405\pi$
$191$	6.58718	0.476632	0.238316	+	0.971188i	$0.423405\pi$
$192$	0	0
$193$	7.20967	0.518964	0.259482	−	0.965748i	$-0.416448\pi$
$193$	7.20967	0.518964	0.259482	+	0.965748i	$0.416448\pi$
$194$	0	0
$195$	−3.19447	−0.228761
$196$	0	0
$197$	1.87253	0.133412	0.0667060	−	0.997773i	$-0.478751\pi$
$197$	1.87253	0.133412	0.0667060	+	0.997773i	$0.478751\pi$
$198$	0	0
$199$	−0.607258	−0.0430474	−0.0215237	−	0.999768i	$-0.506852\pi$
$199$	−0.607258	−0.0430474	−0.0215237	+	0.999768i	$0.506852\pi$
$200$	0	0
$201$	−5.19052	−0.366111
$202$	0	0
$203$	21.6039	1.51629
$204$	0	0
$205$	−4.42090	−0.308769
$206$	0	0
$207$	13.4073	0.931872
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.1496	0.836412	0.418206	−	0.908352i	$-0.362659\pi$
$211$	12.1496	0.836412	0.418206	+	0.908352i	$0.362659\pi$
$212$	0	0
$213$	−7.65492	−0.524507
$214$	0	0
$215$	10.9537	0.747036
$216$	0	0
$217$	−1.10383	−0.0749328
$218$	0	0
$219$	7.39651	0.499809
$220$	0	0
$221$	−16.7779	−1.12860
$222$	0	0
$223$	−0.918013	−0.0614747	−0.0307374	−	0.999527i	$-0.509786\pi$
$223$	−0.918013	−0.0614747	−0.0307374	+	0.999527i	$0.509786\pi$
$224$	0	0
$225$	−2.62181	−0.174787
$226$	0	0
$227$	−21.9513	−1.45696	−0.728479	−	0.685068i	$-0.759772\pi$
$227$	−21.9513	−1.45696	−0.728479	+	0.685068i	$0.759772\pi$
$228$	0	0
$229$	−21.9249	−1.44884	−0.724421	−	0.689358i	$-0.757893\pi$
$229$	−21.9249	−1.44884	−0.724421	+	0.689358i	$0.757893\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.2543	−0.737297	−0.368648	−	0.929569i	$-0.620179\pi$
$233$	−11.2543	−0.737297	−0.368648	+	0.929569i	$0.620179\pi$
$234$	0	0
$235$	1.03420	0.0674636
$236$	0	0
$237$	1.31010	0.0851004
$238$	0	0
$239$	−15.5108	−1.00331	−0.501654	−	0.865068i	$-0.667275\pi$
$239$	−15.5108	−1.00331	−0.501654	+	0.865068i	$0.667275\pi$
$240$	0	0
$241$	18.7929	1.21056	0.605280	−	0.796013i	$-0.293061\pi$
$241$	18.7929	1.21056	0.605280	+	0.796013i	$0.293061\pi$
$242$	0	0
$243$	−13.9013	−0.891771
$244$	0	0
$245$	−1.95557	−0.124937
$246$	0	0
$247$	37.3715	2.37789
$248$	0	0
$249$	−6.59682	−0.418057
$250$	0	0
$251$	4.84499	0.305813	0.152907	−	0.988241i	$-0.451137\pi$
$251$	4.84499	0.305813	0.152907	+	0.988241i	$0.451137\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.98634	0.124389
$256$	0	0
$257$	21.6015	1.34746	0.673732	−	0.738976i	$-0.264690\pi$
$257$	21.6015	1.34746	0.673732	+	0.738976i	$0.264690\pi$
$258$	0	0
$259$	0.787953	0.0489610
$260$	0	0
$261$	−25.2189	−1.56101
$262$	0	0
$263$	−16.2554	−1.00235	−0.501176	−	0.865346i	$-0.667099\pi$
$263$	−16.2554	−1.00235	−0.501176	+	0.865346i	$0.667099\pi$
$264$	0	0
$265$	3.75348	0.230574
$266$	0	0
$267$	−6.08898	−0.372640
$268$	0	0
$269$	−20.1959	−1.23137	−0.615684	−	0.787993i	$-0.711120\pi$
$269$	−20.1959	−1.23137	−0.615684	+	0.787993i	$0.711120\pi$
$270$	0	0
$271$	23.5368	1.42976	0.714881	−	0.699246i	$-0.246481\pi$
$271$	23.5368	1.42976	0.714881	+	0.699246i	$0.246481\pi$
$272$	0	0
$273$	−7.17472	−0.434234
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.75817	−0.165723	−0.0828613	−	0.996561i	$-0.526406\pi$
$277$	−2.75817	−0.165723	−0.0828613	+	0.996561i	$0.526406\pi$
$278$	0	0
$279$	1.28853	0.0771425
$280$	0	0
$281$	9.55050	0.569735	0.284868	−	0.958567i	$-0.408050\pi$
$281$	9.55050	0.569735	0.284868	+	0.958567i	$0.408050\pi$
$282$	0	0
$283$	2.48859	0.147931	0.0739657	−	0.997261i	$-0.476434\pi$
$283$	2.48859	0.147931	0.0739657	+	0.997261i	$0.476434\pi$
$284$	0	0
$285$	−4.42442	−0.262080
$286$	0	0
$287$	−9.92926	−0.586106
$288$	0	0
$289$	−6.56742	−0.386319
$290$	0	0
$291$	9.79366	0.574115
$292$	0	0
$293$	−7.01604	−0.409882	−0.204941	−	0.978774i	$-0.565700\pi$
$293$	−7.01604	−0.409882	−0.204941	+	0.978774i	$0.565700\pi$
$294$	0	0
$295$	−12.7082	−0.739900
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−26.5633	−1.53620
$300$	0	0
$301$	24.6018	1.41802
$302$	0	0
$303$	10.3763	0.596101
$304$	0	0
$305$	−6.38983	−0.365880
$306$	0	0
$307$	15.7618	0.899576	0.449788	−	0.893135i	$-0.351500\pi$
$307$	15.7618	0.899576	0.449788	+	0.893135i	$0.351500\pi$
$308$	0	0
$309$	4.34052	0.246924
$310$	0	0
$311$	−2.37143	−0.134472	−0.0672358	−	0.997737i	$-0.521418\pi$
$311$	−2.37143	−0.134472	−0.0672358	+	0.997737i	$0.521418\pi$
$312$	0	0
$313$	22.8728	1.29285	0.646424	−	0.762979i	$-0.276264\pi$
$313$	22.8728	1.29285	0.646424	+	0.762979i	$0.276264\pi$
$314$	0	0
$315$	−5.88852	−0.331781
$316$	0	0
$317$	16.1104	0.904848	0.452424	−	0.891803i	$-0.350559\pi$
$317$	16.1104	0.904848	0.452424	+	0.891803i	$0.350559\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.69949	−0.262300
$322$	0	0
$323$	−23.2378	−1.29299
$324$	0	0
$325$	5.19447	0.288137
$326$	0	0
$327$	−6.92398	−0.382897
$328$	0	0
$329$	2.32279	0.128059
$330$	0	0
$331$	−26.8231	−1.47433	−0.737166	−	0.675712i	$-0.763836\pi$
$331$	−26.8231	−1.47433	−0.737166	+	0.675712i	$0.763836\pi$
$332$	0	0
$333$	−0.919803	−0.0504049
$334$	0	0
$335$	8.44020	0.461138
$336$	0	0
$337$	26.6484	1.45163	0.725816	−	0.687889i	$-0.241463\pi$
$337$	26.6484	1.45163	0.725816	+	0.687889i	$0.241463\pi$
$338$	0	0
$339$	0.481962	0.0261766
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.1140	−1.08606
$344$	0	0
$345$	3.14484	0.169313
$346$	0	0
$347$	17.2459	0.925810	0.462905	−	0.886408i	$-0.346807\pi$
$347$	17.2459	0.925810	0.462905	+	0.886408i	$0.346807\pi$
$348$	0	0
$349$	7.51222	0.402120	0.201060	−	0.979579i	$-0.435561\pi$
$349$	7.51222	0.402120	0.201060	+	0.979579i	$0.435561\pi$
$350$	0	0
$351$	17.9587	0.958565
$352$	0	0
$353$	12.1232	0.645254	0.322627	−	0.946526i	$-0.395434\pi$
$353$	12.1232	0.645254	0.322627	+	0.946526i	$0.395434\pi$
$354$	0	0
$355$	12.4475	0.660646
$356$	0	0
$357$	4.46128	0.236116
$358$	0	0
$359$	15.8372	0.835853	0.417926	−	0.908481i	$-0.362757\pi$
$359$	15.8372	0.835853	0.417926	+	0.908481i	$0.362757\pi$
$360$	0	0
$361$	32.7604	1.72423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.0273	−0.629539
$366$	0	0
$367$	−23.4664	−1.22494	−0.612468	−	0.790495i	$-0.709823\pi$
$367$	−23.4664	−1.22494	−0.612468	+	0.790495i	$0.709823\pi$
$368$	0	0
$369$	11.5907	0.603390
$370$	0	0
$371$	8.43024	0.437676
$372$	0	0
$373$	17.3749	0.899639	0.449819	−	0.893120i	$-0.351488\pi$
$373$	17.3749	0.899639	0.449819	+	0.893120i	$0.351488\pi$
$374$	0	0
$375$	−0.614975	−0.0317572
$376$	0	0
$377$	49.9651	2.57333
$378$	0	0
$379$	35.7198	1.83480	0.917400	−	0.397966i	$-0.130284\pi$
$379$	35.7198	1.83480	0.917400	+	0.397966i	$0.130284\pi$
$380$	0	0
$381$	−5.04616	−0.258522
$382$	0	0
$383$	−14.8706	−0.759852	−0.379926	−	0.925017i	$-0.624051\pi$
$383$	−14.8706	−0.759852	−0.379926	+	0.925017i	$0.624051\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.7185	−1.45984
$388$	0	0
$389$	−15.2397	−0.772684	−0.386342	−	0.922356i	$-0.626262\pi$
$389$	−15.2397	−0.772684	−0.386342	+	0.922356i	$0.626262\pi$
$390$	0	0
$391$	16.5172	0.835311
$392$	0	0
$393$	−3.90880	−0.197173
$394$	0	0
$395$	−2.13034	−0.107189
$396$	0	0
$397$	17.8780	0.897271	0.448636	−	0.893715i	$-0.351910\pi$
$397$	17.8780	0.897271	0.448636	+	0.893715i	$0.351910\pi$
$398$	0	0
$399$	−9.93717	−0.497481
$400$	0	0
$401$	24.6977	1.23334	0.616672	−	0.787220i	$-0.288481\pi$
$401$	24.6977	1.23334	0.616672	+	0.787220i	$0.288481\pi$
$402$	0	0
$403$	−2.55292	−0.127170
$404$	0	0
$405$	5.73928	0.285187
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−37.4699	−1.85277	−0.926383	−	0.376583i	$-0.877099\pi$
$409$	−37.4699	−1.85277	−0.926383	+	0.376583i	$0.877099\pi$
$410$	0	0
$411$	5.08961	0.251052
$412$	0	0
$413$	−28.5424	−1.40448
$414$	0	0
$415$	10.7270	0.526566
$416$	0	0
$417$	−6.81257	−0.333613
$418$	0	0
$419$	6.28017	0.306807	0.153403	−	0.988164i	$-0.450977\pi$
$419$	6.28017	0.306807	0.153403	+	0.988164i	$0.450977\pi$
$420$	0	0
$421$	−24.2587	−1.18229	−0.591147	−	0.806563i	$-0.701325\pi$
$421$	−24.2587	−1.18229	−0.591147	+	0.806563i	$0.701325\pi$
$422$	0	0
$423$	−2.71146	−0.131836
$424$	0	0
$425$	−3.22995	−0.156676
$426$	0	0
$427$	−14.3514	−0.694514
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.0321	0.675901	0.337950	−	0.941164i	$-0.390266\pi$
$431$	14.0321	0.675901	0.337950	+	0.941164i	$0.390266\pi$
$432$	0	0
$433$	−24.3312	−1.16928	−0.584640	−	0.811293i	$-0.698764\pi$
$433$	−24.3312	−1.16928	−0.584640	+	0.811293i	$0.698764\pi$
$434$	0	0
$435$	−5.91538	−0.283621
$436$	0	0
$437$	−36.7909	−1.75995
$438$	0	0
$439$	−6.52928	−0.311626	−0.155813	−	0.987787i	$-0.549800\pi$
$439$	−6.52928	−0.311626	−0.155813	+	0.987787i	$0.549800\pi$
$440$	0	0
$441$	5.12712	0.244149
$442$	0	0
$443$	−0.922170	−0.0438136	−0.0219068	−	0.999760i	$-0.506974\pi$
$443$	−0.922170	−0.0438136	−0.0219068	+	0.999760i	$0.506974\pi$
$444$	0	0
$445$	9.90118	0.469361
$446$	0	0
$447$	5.20676	0.246271
$448$	0	0
$449$	−8.48757	−0.400553	−0.200277	−	0.979739i	$-0.564184\pi$
$449$	−8.48757	−0.400553	−0.200277	+	0.979739i	$0.564184\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.93376	−0.184824
$454$	0	0
$455$	11.6667	0.546943
$456$	0	0
$457$	15.5158	0.725799	0.362900	−	0.931828i	$-0.381787\pi$
$457$	15.5158	0.725799	0.362900	+	0.931828i	$0.381787\pi$
$458$	0	0
$459$	−11.1668	−0.521222
$460$	0	0
$461$	26.9388	1.25466	0.627332	−	0.778752i	$-0.284147\pi$
$461$	26.9388	1.25466	0.627332	+	0.778752i	$0.284147\pi$
$462$	0	0
$463$	33.9970	1.57997	0.789986	−	0.613125i	$-0.210088\pi$
$463$	33.9970	1.57997	0.789986	+	0.613125i	$0.210088\pi$
$464$	0	0
$465$	0.302241	0.0140161
$466$	0	0
$467$	−4.26684	−0.197446	−0.0987230	−	0.995115i	$-0.531476\pi$
$467$	−4.26684	−0.197446	−0.0987230	+	0.995115i	$0.531476\pi$
$468$	0	0
$469$	18.9565	0.875332
$470$	0	0
$471$	−9.85641	−0.454159
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.19447	0.330105
$476$	0	0
$477$	−9.84089	−0.450584
$478$	0	0
$479$	2.84891	0.130170	0.0650850	−	0.997880i	$-0.479268\pi$
$479$	2.84891	0.130170	0.0650850	+	0.997880i	$0.479268\pi$
$480$	0	0
$481$	1.82237	0.0830927
$482$	0	0
$483$	7.06325	0.321389
$484$	0	0
$485$	−15.9253	−0.723130
$486$	0	0
$487$	39.5054	1.79016	0.895080	−	0.445905i	$-0.147118\pi$
$487$	39.5054	1.79016	0.895080	+	0.445905i	$0.147118\pi$
$488$	0	0
$489$	−6.27093	−0.283581
$490$	0	0
$491$	38.7458	1.74857	0.874287	−	0.485409i	$-0.161329\pi$
$491$	38.7458	1.74857	0.874287	+	0.485409i	$0.161329\pi$
$492$	0	0
$493$	−31.0686	−1.39926
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.9569	1.25404
$498$	0	0
$499$	−12.0103	−0.537654	−0.268827	−	0.963189i	$-0.586636\pi$
$499$	−12.0103	−0.537654	−0.268827	+	0.963189i	$0.586636\pi$
$500$	0	0
$501$	8.18810	0.365817
$502$	0	0
$503$	−26.7888	−1.19445	−0.597226	−	0.802073i	$-0.703731\pi$
$503$	−26.7888	−1.19445	−0.597226	+	0.802073i	$0.703731\pi$
$504$	0	0
$505$	−16.8726	−0.750823
$506$	0	0
$507$	−8.59892	−0.381891
$508$	0	0
$509$	42.4779	1.88280	0.941400	−	0.337294i	$-0.109511\pi$
$509$	42.4779	1.88280	0.941400	+	0.337294i	$0.109511\pi$
$510$	0	0
$511$	−27.0131	−1.19499
$512$	0	0
$513$	24.8732	1.09818
$514$	0	0
$515$	−7.05805	−0.311015
$516$	0	0
$517$	0	0
$518$	0	0
$519$	11.5109	0.505272
$520$	0	0
$521$	−17.0296	−0.746080	−0.373040	−	0.927815i	$-0.621685\pi$
$521$	−17.0296	−0.746080	−0.373040	+	0.927815i	$0.621685\pi$
$522$	0	0
$523$	−44.4082	−1.94184	−0.970918	−	0.239413i	$-0.923045\pi$
$523$	−44.4082	−1.94184	−0.970918	+	0.239413i	$0.923045\pi$
$524$	0	0
$525$	−1.38122	−0.0602815
$526$	0	0
$527$	1.58742	0.0691490
$528$	0	0
$529$	3.15061	0.136983
$530$	0	0
$531$	33.3184	1.44590
$532$	0	0
$533$	−22.9643	−0.994692
$534$	0	0
$535$	7.64175	0.330382
$536$	0	0
$537$	−12.0097	−0.518257
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.25636	−0.182995	−0.0914975	−	0.995805i	$-0.529165\pi$
$541$	−4.25636	−0.182995	−0.0914975	+	0.995805i	$0.529165\pi$
$542$	0	0
$543$	−3.93053	−0.168675
$544$	0	0
$545$	11.2590	0.482280
$546$	0	0
$547$	15.7213	0.672194	0.336097	−	0.941827i	$-0.390893\pi$
$547$	15.7213	0.672194	0.336097	+	0.941827i	$0.390893\pi$
$548$	0	0
$549$	16.7529	0.714996
$550$	0	0
$551$	69.2029	2.94814
$552$	0	0
$553$	−4.78470	−0.203466
$554$	0	0
$555$	−0.215751	−0.00915810
$556$	0	0
$557$	−4.21547	−0.178615	−0.0893076	−	0.996004i	$-0.528465\pi$
$557$	−4.21547	−0.178615	−0.0893076	+	0.996004i	$0.528465\pi$
$558$	0	0
$559$	56.8987	2.40656
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.7102	−0.535672	−0.267836	−	0.963464i	$-0.586309\pi$
$563$	−12.7102	−0.535672	−0.267836	+	0.963464i	$0.586309\pi$
$564$	0	0
$565$	−0.783709	−0.0329709
$566$	0	0
$567$	12.8903	0.541342
$568$	0	0
$569$	−23.4920	−0.984837	−0.492418	−	0.870359i	$-0.663887\pi$
$569$	−23.4920	−0.984837	−0.492418	+	0.870359i	$0.663887\pi$
$570$	0	0
$571$	21.4826	0.899019	0.449510	−	0.893276i	$-0.351599\pi$
$571$	21.4826	0.899019	0.449510	+	0.893276i	$0.351599\pi$
$572$	0	0
$573$	−4.05095	−0.169231
$574$	0	0
$575$	−5.11377	−0.213259
$576$	0	0
$577$	−5.57447	−0.232068	−0.116034	−	0.993245i	$-0.537018\pi$
$577$	−5.57447	−0.232068	−0.116034	+	0.993245i	$0.537018\pi$
$578$	0	0
$579$	−4.43377	−0.184261
$580$	0	0
$581$	24.0926	0.999528
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−13.6189	−0.563072
$586$	0	0
$587$	39.6780	1.63769	0.818843	−	0.574017i	$-0.194616\pi$
$587$	39.6780	1.63769	0.818843	+	0.574017i	$0.194616\pi$
$588$	0	0
$589$	−3.53586	−0.145692
$590$	0	0
$591$	−1.15156	−0.0473687
$592$	0	0
$593$	−6.42867	−0.263994	−0.131997	−	0.991250i	$-0.542139\pi$
$593$	−6.42867	−0.263994	−0.131997	+	0.991250i	$0.542139\pi$
$594$	0	0
$595$	−7.25441	−0.297402
$596$	0	0
$597$	0.373449	0.0152842
$598$	0	0
$599$	−25.7321	−1.05139	−0.525693	−	0.850674i	$-0.676194\pi$
$599$	−25.7321	−1.05139	−0.525693	+	0.850674i	$0.676194\pi$
$600$	0	0
$601$	−21.9690	−0.896136	−0.448068	−	0.894000i	$-0.647888\pi$
$601$	−21.9690	−0.896136	−0.448068	+	0.894000i	$0.647888\pi$
$602$	0	0
$603$	−22.1286	−0.901145
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.29521	−0.0525709	−0.0262855	−	0.999654i	$-0.508368\pi$
$607$	−1.29521	−0.0525709	−0.0262855	+	0.999654i	$0.508368\pi$
$608$	0	0
$609$	−13.2858	−0.538369
$610$	0	0
$611$	5.37211	0.217332
$612$	0	0
$613$	−1.08300	−0.0437418	−0.0218709	−	0.999761i	$-0.506962\pi$
$613$	−1.08300	−0.0437418	−0.0218709	+	0.999761i	$0.506962\pi$
$614$	0	0
$615$	2.71875	0.109630
$616$	0	0
$617$	−12.3102	−0.495592	−0.247796	−	0.968812i	$-0.579706\pi$
$617$	−12.3102	−0.495592	−0.247796	+	0.968812i	$0.579706\pi$
$618$	0	0
$619$	−31.9610	−1.28462	−0.642311	−	0.766444i	$-0.722024\pi$
$619$	−31.9610	−1.28462	−0.642311	+	0.766444i	$0.722024\pi$
$620$	0	0
$621$	−17.6797	−0.709461
$622$	0	0
$623$	22.2379	0.890941
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.13316	−0.0451819
$630$	0	0
$631$	1.80177	0.0717272	0.0358636	−	0.999357i	$-0.488582\pi$
$631$	1.80177	0.0717272	0.0358636	+	0.999357i	$0.488582\pi$
$632$	0	0
$633$	−7.47170	−0.296973
$634$	0	0
$635$	8.20546	0.325624
$636$	0	0
$637$	−10.1582	−0.402481
$638$	0	0
$639$	−32.6350	−1.29102
$640$	0	0
$641$	−25.5940	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$641$	−25.5940	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$642$	0	0
$643$	16.8113	0.662974	0.331487	−	0.943460i	$-0.392450\pi$
$643$	16.8113	0.662974	0.331487	+	0.943460i	$0.392450\pi$
$644$	0	0
$645$	−6.73625	−0.265240
$646$	0	0
$647$	33.9970	1.33656	0.668279	−	0.743911i	$-0.267031\pi$
$647$	33.9970	1.33656	0.668279	+	0.743911i	$0.267031\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.678827	0.0266053
$652$	0	0
$653$	42.9973	1.68262	0.841308	−	0.540557i	$-0.181786\pi$
$653$	42.9973	1.68262	0.841308	+	0.540557i	$0.181786\pi$
$654$	0	0
$655$	6.35603	0.248351
$656$	0	0
$657$	31.5333	1.23023
$658$	0	0
$659$	29.3748	1.14428	0.572141	−	0.820155i	$-0.306113\pi$
$659$	29.3748	1.14428	0.572141	+	0.820155i	$0.306113\pi$
$660$	0	0
$661$	23.7553	0.923975	0.461988	−	0.886886i	$-0.347136\pi$
$661$	23.7553	0.923975	0.461988	+	0.886886i	$0.347136\pi$
$662$	0	0
$663$	10.3180	0.400717
$664$	0	0
$665$	16.1586	0.626605
$666$	0	0
$667$	−49.1888	−1.90460
$668$	0	0
$669$	0.564555	0.0218270
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16.5964	0.639745	0.319873	−	0.947461i	$-0.396360\pi$
$673$	16.5964	0.639745	0.319873	+	0.947461i	$0.396360\pi$
$674$	0	0
$675$	3.45727	0.133070
$676$	0	0
$677$	−14.6950	−0.564773	−0.282387	−	0.959301i	$-0.591126\pi$
$677$	−14.6950	−0.564773	−0.282387	+	0.959301i	$0.591126\pi$
$678$	0	0
$679$	−35.7679	−1.37265
$680$	0	0
$681$	13.4995	0.517302
$682$	0	0
$683$	42.2066	1.61499	0.807495	−	0.589874i	$-0.200823\pi$
$683$	42.2066	1.61499	0.807495	+	0.589874i	$0.200823\pi$
$684$	0	0
$685$	−8.27612	−0.316214
$686$	0	0
$687$	13.4833	0.514420
$688$	0	0
$689$	19.4973	0.742790
$690$	0	0
$691$	−48.6795	−1.85186	−0.925928	−	0.377700i	$-0.876715\pi$
$691$	−48.6795	−1.85186	−0.925928	+	0.377700i	$0.876715\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.0778	0.420205
$696$	0	0
$697$	14.2793	0.540867
$698$	0	0
$699$	6.92115	0.261782
$700$	0	0
$701$	−0.580332	−0.0219188	−0.0109594	−	0.999940i	$-0.503489\pi$
$701$	−0.580332	−0.0219188	−0.0109594	+	0.999940i	$0.503489\pi$
$702$	0	0
$703$	2.52402	0.0951953
$704$	0	0
$705$	−0.636005	−0.0239534
$706$	0	0
$707$	−37.8956	−1.42521
$708$	0	0
$709$	40.9372	1.53743	0.768714	−	0.639593i	$-0.220897\pi$
$709$	40.9372	1.53743	0.768714	+	0.639593i	$0.220897\pi$
$710$	0	0
$711$	5.58533	0.209466
$712$	0	0
$713$	2.51325	0.0941221
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.53875	0.356231
$718$	0	0
$719$	10.0377	0.374344	0.187172	−	0.982327i	$-0.440068\pi$
$719$	10.0377	0.374344	0.187172	+	0.982327i	$0.440068\pi$
$720$	0	0
$721$	−15.8522	−0.590368
$722$	0	0
$723$	−11.5572	−0.429816
$724$	0	0
$725$	9.61890	0.357237
$726$	0	0
$727$	22.1551	0.821687	0.410844	−	0.911706i	$-0.365234\pi$
$727$	22.1551	0.821687	0.410844	+	0.911706i	$0.365234\pi$
$728$	0	0
$729$	−8.66886	−0.321069
$730$	0	0
$731$	−35.3799	−1.30857
$732$	0	0
$733$	43.5669	1.60918	0.804589	−	0.593832i	$-0.202385\pi$
$733$	43.5669	1.60918	0.804589	+	0.593832i	$0.202385\pi$
$734$	0	0
$735$	1.20263	0.0443596
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−46.7110	−1.71829	−0.859146	−	0.511731i	$-0.829005\pi$
$739$	−46.7110	−1.71829	−0.859146	+	0.511731i	$0.829005\pi$
$740$	0	0
$741$	−22.9825	−0.844285
$742$	0	0
$743$	−33.3576	−1.22377	−0.611887	−	0.790945i	$-0.709589\pi$
$743$	−33.3576	−1.22377	−0.611887	+	0.790945i	$0.709589\pi$
$744$	0	0
$745$	−8.46662	−0.310193
$746$	0	0
$747$	−28.1240	−1.02900
$748$	0	0
$749$	17.1632	0.627130
$750$	0	0
$751$	24.0038	0.875910	0.437955	−	0.898997i	$-0.355703\pi$
$751$	24.0038	0.875910	0.437955	+	0.898997i	$0.355703\pi$
$752$	0	0
$753$	−2.97955	−0.108581
$754$	0	0
$755$	6.39661	0.232796
$756$	0	0
$757$	−23.0230	−0.836786	−0.418393	−	0.908266i	$-0.637407\pi$
$757$	−23.0230	−0.836786	−0.418393	+	0.908266i	$0.637407\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.87356	−0.249166	−0.124583	−	0.992209i	$-0.539759\pi$
$761$	−6.87356	−0.249166	−0.124583	+	0.992209i	$0.539759\pi$
$762$	0	0
$763$	25.2874	0.915465
$764$	0	0
$765$	8.46830	0.306172
$766$	0	0
$767$	−66.0124	−2.38357
$768$	0	0
$769$	24.6838	0.890120	0.445060	−	0.895501i	$-0.353182\pi$
$769$	24.6838	0.890120	0.445060	+	0.895501i	$0.353182\pi$
$770$	0	0
$771$	−13.2844	−0.478425
$772$	0	0
$773$	15.2591	0.548830	0.274415	−	0.961611i	$-0.411516\pi$
$773$	15.2591	0.548830	0.274415	+	0.961611i	$0.411516\pi$
$774$	0	0
$775$	−0.491468	−0.0176541
$776$	0	0
$777$	−0.484572	−0.0173839
$778$	0	0
$779$	−31.8061	−1.13957
$780$	0	0
$781$	0	0
$782$	0	0
$783$	33.2551	1.18844
$784$	0	0
$785$	16.0273	0.572040
$786$	0	0
$787$	25.9701	0.925735	0.462868	−	0.886427i	$-0.346820\pi$
$787$	25.9701	0.925735	0.462868	+	0.886427i	$0.346820\pi$
$788$	0	0
$789$	9.99668	0.355891
$790$	0	0
$791$	−1.76020	−0.0625854
$792$	0	0
$793$	−33.1918	−1.17867
$794$	0	0
$795$	−2.30830	−0.0818669
$796$	0	0
$797$	19.1403	0.677984	0.338992	−	0.940789i	$-0.389914\pi$
$797$	19.1403	0.677984	0.338992	+	0.940789i	$0.389914\pi$
$798$	0	0
$799$	−3.34040	−0.118175
$800$	0	0
$801$	−25.9590	−0.917215
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−11.4854	−0.404808
$806$	0	0
$807$	12.4200	0.437205
$808$	0	0
$809$	−19.9636	−0.701883	−0.350941	−	0.936397i	$-0.614138\pi$
$809$	−19.9636	−0.701883	−0.350941	+	0.936397i	$0.614138\pi$
$810$	0	0
$811$	−33.4667	−1.17518	−0.587588	−	0.809160i	$-0.699922\pi$
$811$	−33.4667	−1.17518	−0.587588	+	0.809160i	$0.699922\pi$
$812$	0	0
$813$	−14.4746	−0.507646
$814$	0	0
$815$	10.1970	0.357187
$816$	0	0
$817$	78.8060	2.75707
$818$	0	0
$819$	−30.5878	−1.06882
$820$	0	0
$821$	13.6778	0.477358	0.238679	−	0.971099i	$-0.423286\pi$
$821$	13.6778	0.477358	0.238679	+	0.971099i	$0.423286\pi$
$822$	0	0
$823$	47.5149	1.65627	0.828133	−	0.560532i	$-0.189403\pi$
$823$	47.5149	1.65627	0.828133	+	0.560532i	$0.189403\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.58722	−0.124740	−0.0623699	−	0.998053i	$-0.519866\pi$
$827$	−3.58722	−0.124740	−0.0623699	+	0.998053i	$0.519866\pi$
$828$	0	0
$829$	−37.2896	−1.29512	−0.647561	−	0.762014i	$-0.724211\pi$
$829$	−37.2896	−1.29512	−0.647561	+	0.762014i	$0.724211\pi$
$830$	0	0
$831$	1.69621	0.0588408
$832$	0	0
$833$	6.31639	0.218850
$834$	0	0
$835$	−13.3145	−0.460768
$836$	0	0
$837$	−1.69914	−0.0587308
$838$	0	0
$839$	−37.2210	−1.28501	−0.642507	−	0.766280i	$-0.722105\pi$
$839$	−37.2210	−1.28501	−0.642507	+	0.766280i	$0.722105\pi$
$840$	0	0
$841$	63.5231	2.19045
$842$	0	0
$843$	−5.87332	−0.202288
$844$	0	0
$845$	13.9825	0.481014
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.53042	−0.0525239
$850$	0	0
$851$	−1.79405	−0.0614993
$852$	0	0
$853$	19.3642	0.663017	0.331508	−	0.943452i	$-0.392442\pi$
$853$	19.3642	0.663017	0.331508	+	0.943452i	$0.392442\pi$
$854$	0	0
$855$	−18.8625	−0.645084
$856$	0	0
$857$	−0.844366	−0.0288430	−0.0144215	−	0.999896i	$-0.504591\pi$
$857$	−0.844366	−0.0288430	−0.0144215	+	0.999896i	$0.504591\pi$
$858$	0	0
$859$	−21.8675	−0.746110	−0.373055	−	0.927809i	$-0.621690\pi$
$859$	−21.8675	−0.746110	−0.373055	+	0.927809i	$0.621690\pi$
$860$	0	0
$861$	6.10625	0.208100
$862$	0	0
$863$	4.99125	0.169904	0.0849520	−	0.996385i	$-0.472926\pi$
$863$	4.99125	0.169904	0.0849520	+	0.996385i	$0.472926\pi$
$864$	0	0
$865$	−18.7177	−0.636419
$866$	0	0
$867$	4.03880	0.137165
$868$	0	0
$869$	0	0
$870$	0	0
$871$	43.8424	1.48554
$872$	0	0
$873$	41.7530	1.41313
$874$	0	0
$875$	2.24598	0.0759280
$876$	0	0
$877$	54.7015	1.84714	0.923570	−	0.383430i	$-0.125257\pi$
$877$	54.7015	1.84714	0.923570	+	0.383430i	$0.125257\pi$
$878$	0	0
$879$	4.31469	0.145531
$880$	0	0
$881$	−13.7463	−0.463125	−0.231562	−	0.972820i	$-0.574384\pi$
$881$	−13.7463	−0.463125	−0.231562	+	0.972820i	$0.574384\pi$
$882$	0	0
$883$	23.6026	0.794290	0.397145	−	0.917756i	$-0.370001\pi$
$883$	23.6026	0.794290	0.397145	+	0.917756i	$0.370001\pi$
$884$	0	0
$885$	7.81523	0.262706
$886$	0	0
$887$	−35.4265	−1.18950	−0.594752	−	0.803909i	$-0.702750\pi$
$887$	−35.4265	−1.18950	−0.594752	+	0.803909i	$0.702750\pi$
$888$	0	0
$889$	18.4293	0.618099
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.44050	0.248987
$894$	0	0
$895$	19.5287	0.652774
$896$	0	0
$897$	16.3358	0.545436
$898$	0	0
$899$	−4.72738	−0.157667
$900$	0	0
$901$	−12.1236	−0.403894
$902$	0	0
$903$	−15.1295	−0.503478
$904$	0	0
$905$	6.39136	0.212456
$906$	0	0
$907$	13.6049	0.451742	0.225871	−	0.974157i	$-0.427477\pi$
$907$	13.6049	0.451742	0.225871	+	0.974157i	$0.427477\pi$
$908$	0	0
$909$	44.2368	1.46724
$910$	0	0
$911$	−35.4052	−1.17303	−0.586514	−	0.809939i	$-0.699500\pi$
$911$	−35.4052	−1.17303	−0.586514	+	0.809939i	$0.699500\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.92959	0.129908
$916$	0	0
$917$	14.2755	0.471419
$918$	0	0
$919$	−33.9654	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$919$	−33.9654	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$920$	0	0
$921$	−9.69315	−0.319400
$922$	0	0
$923$	64.6584	2.12826
$924$	0	0
$925$	0.350828	0.0115352
$926$	0	0
$927$	18.5048	0.607778
$928$	0	0
$929$	−32.5351	−1.06744	−0.533722	−	0.845660i	$-0.679207\pi$
$929$	−32.5351	−1.06744	−0.533722	+	0.845660i	$0.679207\pi$
$930$	0	0
$931$	−14.0693	−0.461102
$932$	0	0
$933$	1.45837	0.0477449
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.0027	1.50284	0.751421	−	0.659823i	$-0.229369\pi$
$937$	46.0027	1.50284	0.751421	+	0.659823i	$0.229369\pi$
$938$	0	0
$939$	−14.0662	−0.459034
$940$	0	0
$941$	20.3541	0.663524	0.331762	−	0.943363i	$-0.392357\pi$
$941$	20.3541	0.663524	0.331762	+	0.943363i	$0.392357\pi$
$942$	0	0
$943$	22.6075	0.736200
$944$	0	0
$945$	7.76497	0.252594
$946$	0	0
$947$	−15.6423	−0.508308	−0.254154	−	0.967164i	$-0.581797\pi$
$947$	−15.6423	−0.508308	−0.254154	+	0.967164i	$0.581797\pi$
$948$	0	0
$949$	−62.4756	−2.02804
$950$	0	0
$951$	−9.90747	−0.321272
$952$	0	0
$953$	34.4420	1.11569	0.557843	−	0.829946i	$-0.311629\pi$
$953$	34.4420	1.11569	0.557843	+	0.829946i	$0.311629\pi$
$954$	0	0
$955$	6.58718	0.213156
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.5880	−0.600238
$960$	0	0
$961$	−30.7585	−0.992208
$962$	0	0
$963$	−20.0352	−0.645624
$964$	0	0
$965$	7.20967	0.232088
$966$	0	0
$967$	−20.3210	−0.653479	−0.326739	−	0.945114i	$-0.605950\pi$
$967$	−20.3210	−0.653479	−0.326739	+	0.945114i	$0.605950\pi$
$968$	0	0
$969$	14.2907	0.459082
$970$	0	0
$971$	60.6592	1.94665	0.973323	−	0.229440i	$-0.0736894\pi$
$971$	60.6592	1.94665	0.973323	+	0.229440i	$0.0736894\pi$
$972$	0	0
$973$	24.8805	0.797633
$974$	0	0
$975$	−3.19447	−0.102305
$976$	0	0
$977$	−27.9407	−0.893903	−0.446952	−	0.894558i	$-0.647490\pi$
$977$	−27.9407	−0.893903	−0.446952	+	0.894558i	$0.647490\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−29.5188	−0.942462
$982$	0	0
$983$	14.8724	0.474356	0.237178	−	0.971466i	$-0.423778\pi$
$983$	14.8724	0.474356	0.237178	+	0.971466i	$0.423778\pi$
$984$	0	0
$985$	1.87253	0.0596636
$986$	0	0
$987$	−1.42846	−0.0454683
$988$	0	0
$989$	−56.0146	−1.78116
$990$	0	0
$991$	54.1090	1.71883	0.859414	−	0.511280i	$-0.170828\pi$
$991$	54.1090	1.71883	0.859414	+	0.511280i	$0.170828\pi$
$992$	0	0
$993$	16.4956	0.523471
$994$	0	0
$995$	−0.607258	−0.0192514
$996$	0	0
$997$	25.8227	0.817814	0.408907	−	0.912576i	$-0.365910\pi$
$997$	25.8227	0.817814	0.408907	+	0.912576i	$0.365910\pi$
$998$	0	0
$999$	1.21291	0.0383747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.db.1.3		6
4.3	odd	2		4840.2.a.bc.1.4	✓	6
11.10	odd	2		9680.2.a.da.1.3		6
44.43	even	2		4840.2.a.bd.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.4	✓	6	4.3	odd	2
4840.2.a.bd.1.4	yes	6	44.43	even	2
9680.2.a.da.1.3		6	11.10	odd	2
9680.2.a.db.1.3		6	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.614975 q^{3} +1.00000 q^{5} +2.24598 q^{7} -2.62181 q^{9} +5.19447 q^{13} -0.614975 q^{15} -3.22995 q^{17} +7.19447 q^{19} -1.38122 q^{21} -5.11377 q^{23} +1.00000 q^{25} +3.45727 q^{27} +9.61890 q^{29} -0.491468 q^{31} +2.24598 q^{35} +0.350828 q^{37} -3.19447 q^{39} -4.42090 q^{41} +10.9537 q^{43} -2.62181 q^{45} +1.03420 q^{47} -1.95557 q^{49} +1.98634 q^{51} +3.75348 q^{53} -4.42442 q^{57} -12.7082 q^{59} -6.38983 q^{61} -5.88852 q^{63} +5.19447 q^{65} +8.44020 q^{67} +3.14484 q^{69} +12.4475 q^{71} -12.0273 q^{73} -0.614975 q^{75} -2.13034 q^{79} +5.73928 q^{81} +10.7270 q^{83} -3.22995 q^{85} -5.91538 q^{87} +9.90118 q^{89} +11.6667 q^{91} +0.302241 q^{93} +7.19447 q^{95} -15.9253 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} + 2 q^{15} - 8 q^{17} + 12 q^{19} + 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} - 16 q^{29} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} - 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 + 2 * q^15 - 8 * q^17 + 12 * q^19 + 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 - 16 * q^29 + 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 - 32 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 + 16 * q^57 - 4 * q^59 - 16 * q^61 + 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 - 16 * q^73 + 2 * q^75 - 10 * q^81 + 12 * q^83 - 8 * q^85 - 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 + 12 * q^95

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