LMFDB - Embedded newform 9680.2.a.cv.1.3

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 = [4,0,2,0,4,0,11,0,0,0,0,0,-7,0,2,0,-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:	$4$
Coefficient field:	4.4.2525.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77748$ 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)

$f(q)$	$=$	$q+1.77748 q^{3} +1.00000 q^{5} +4.25800 q^{7} +0.159450 q^{9} -1.36004 q^{13} +1.77748 q^{15} -2.09855 q^{17} +0.604482 q^{19} +7.56852 q^{21} +4.39768 q^{23} +1.00000 q^{25} -5.04903 q^{27} +6.63159 q^{29} +2.19493 q^{31} +4.25800 q^{35} +6.16951 q^{37} -2.41745 q^{39} -7.40557 q^{41} +12.6671 q^{43} +0.159450 q^{45} +3.07446 q^{47} +11.1305 q^{49} -3.73013 q^{51} -6.65351 q^{53} +1.07446 q^{57} -12.0372 q^{59} +5.68899 q^{61} +0.678939 q^{63} -1.36004 q^{65} -9.86416 q^{67} +7.81681 q^{69} +5.23879 q^{71} +0.722795 q^{73} +1.77748 q^{75} +5.67056 q^{79} -9.45293 q^{81} -0.952648 q^{83} -2.09855 q^{85} +11.7875 q^{87} +1.24095 q^{89} -5.79104 q^{91} +3.90145 q^{93} +0.604482 q^{95} +11.5431 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 11 q^{7} - 7 q^{13} + 2 q^{15} - 3 q^{17} + 12 q^{19} + 6 q^{21} + 9 q^{23} + 4 q^{25} + 5 q^{27} + 8 q^{29} - 3 q^{31} + 11 q^{35} - 3 q^{37} - 3 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 11 * q^7 - 7 * q^13 + 2 * q^15 - 3 * q^17 + 12 * q^19 + 6 * q^21 + 9 * q^23 + 4 * q^25 + 5 * q^27 + 8 * q^29 - 3 * q^31 + 11 * q^35 - 3 * q^37 - 3 * q^39 + 7 * q^41 + 21 * q^43 + 3 * q^47 + 15 * q^49 + 9 * q^51 - 11 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 3 * q^63 - 7 * q^65 + q^67 - 28 * q^69 + 15 * q^71 + 9 * q^73 + 2 * q^75 + 6 * q^79 - 20 * q^81 + 15 * q^83 - 3 * q^85 + 15 * q^87 - 4 * q^91 + 21 * q^93 + 12 * q^95 + 6 * 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
$3$	1.77748	1.02623	0.513116	−	0.858320i	$-0.328491\pi$
$3$	1.77748	1.02623	0.513116	+	0.858320i	$0.328491\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.25800	1.60937	0.804686	−	0.593701i	$-0.202334\pi$
$7$	4.25800	1.60937	0.804686	+	0.593701i	$0.202334\pi$
$8$	0	0
$9$	0.159450	0.0531501
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.36004	−0.377207	−0.188603	−	0.982053i	$-0.560396\pi$
$13$	−1.36004	−0.377207	−0.188603	+	0.982053i	$0.560396\pi$
$14$	0	0
$15$	1.77748	0.458944
$16$	0	0
$17$	−2.09855	−0.508972	−0.254486	−	0.967076i	$-0.581906\pi$
$17$	−2.09855	−0.508972	−0.254486	+	0.967076i	$0.581906\pi$
$18$	0	0
$19$	0.604482	0.138678	0.0693388	−	0.997593i	$-0.477911\pi$
$19$	0.604482	0.138678	0.0693388	+	0.997593i	$0.477911\pi$
$20$	0	0
$21$	7.56852	1.65159
$22$	0	0
$23$	4.39768	0.916979	0.458490	−	0.888700i	$-0.348391\pi$
$23$	4.39768	0.916979	0.458490	+	0.888700i	$0.348391\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.04903	−0.971687
$28$	0	0
$29$	6.63159	1.23145	0.615727	−	0.787959i	$-0.288862\pi$
$29$	6.63159	1.23145	0.615727	+	0.787959i	$0.288862\pi$
$30$	0	0
$31$	2.19493	0.394221	0.197111	−	0.980381i	$-0.436844\pi$
$31$	2.19493	0.394221	0.197111	+	0.980381i	$0.436844\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.25800	0.719733
$36$	0	0
$37$	6.16951	1.01426	0.507130	−	0.861869i	$-0.330706\pi$
$37$	6.16951	1.01426	0.507130	+	0.861869i	$0.330706\pi$
$38$	0	0
$39$	−2.41745	−0.387101
$40$	0	0
$41$	−7.40557	−1.15656	−0.578278	−	0.815840i	$-0.696275\pi$
$41$	−7.40557	−1.15656	−0.578278	+	0.815840i	$0.696275\pi$
$42$	0	0
$43$	12.6671	1.93171	0.965855	−	0.259084i	$-0.0834207\pi$
$43$	12.6671	1.93171	0.965855	+	0.259084i	$0.0834207\pi$
$44$	0	0
$45$	0.159450	0.0237694
$46$	0	0
$47$	3.07446	0.448456	0.224228	−	0.974537i	$-0.428014\pi$
$47$	3.07446	0.448456	0.224228	+	0.974537i	$0.428014\pi$
$48$	0	0
$49$	11.1305	1.59008
$50$	0	0
$51$	−3.73013	−0.522323
$52$	0	0
$53$	−6.65351	−0.913930	−0.456965	−	0.889485i	$-0.651064\pi$
$53$	−6.65351	−0.913930	−0.456965	+	0.889485i	$0.651064\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.07446	0.142315
$58$	0	0
$59$	−12.0372	−1.56710	−0.783552	−	0.621326i	$-0.786594\pi$
$59$	−12.0372	−1.56710	−0.783552	+	0.621326i	$0.786594\pi$
$60$	0	0
$61$	5.68899	0.728401	0.364201	−	0.931321i	$-0.381342\pi$
$61$	5.68899	0.728401	0.364201	+	0.931321i	$0.381342\pi$
$62$	0	0
$63$	0.678939	0.0855382
$64$	0	0
$65$	−1.36004	−0.168692
$66$	0	0
$67$	−9.86416	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$67$	−9.86416	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$68$	0	0
$69$	7.81681	0.941033
$70$	0	0
$71$	5.23879	0.621730	0.310865	−	0.950454i	$-0.399381\pi$
$71$	5.23879	0.621730	0.310865	+	0.950454i	$0.399381\pi$
$72$	0	0
$73$	0.722795	0.0845967	0.0422983	−	0.999105i	$-0.486532\pi$
$73$	0.722795	0.0845967	0.0422983	+	0.999105i	$0.486532\pi$
$74$	0	0
$75$	1.77748	0.205246
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.67056	0.637988	0.318994	−	0.947757i	$-0.396655\pi$
$79$	5.67056	0.637988	0.318994	+	0.947757i	$0.396655\pi$
$80$	0	0
$81$	−9.45293	−1.05033
$82$	0	0
$83$	−0.952648	−0.104567	−0.0522833	−	0.998632i	$-0.516650\pi$
$83$	−0.952648	−0.104567	−0.0522833	+	0.998632i	$0.516650\pi$
$84$	0	0
$85$	−2.09855	−0.227619
$86$	0	0
$87$	11.7875	1.26376
$88$	0	0
$89$	1.24095	0.131540	0.0657701	−	0.997835i	$-0.479050\pi$
$89$	1.24095	0.131540	0.0657701	+	0.997835i	$0.479050\pi$
$90$	0	0
$91$	−5.79104	−0.607066
$92$	0	0
$93$	3.90145	0.404562
$94$	0	0
$95$	0.604482	0.0620185
$96$	0	0
$97$	11.5431	1.17202	0.586012	−	0.810302i	$-0.300697\pi$
$97$	11.5431	1.17202	0.586012	+	0.810302i	$0.300697\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.80675	−0.279282	−0.139641	−	0.990202i	$-0.544595\pi$
$101$	−2.80675	−0.279282	−0.139641	+	0.990202i	$0.544595\pi$
$102$	0	0
$103$	0.406055	0.0400098	0.0200049	−	0.999800i	$-0.493632\pi$
$103$	0.406055	0.0400098	0.0200049	+	0.999800i	$0.493632\pi$
$104$	0	0
$105$	7.56852	0.738612
$106$	0	0
$107$	15.3985	1.48863	0.744316	−	0.667827i	$-0.232776\pi$
$107$	15.3985	1.48863	0.744316	+	0.667827i	$0.232776\pi$
$108$	0	0
$109$	3.07312	0.294352	0.147176	−	0.989110i	$-0.452982\pi$
$109$	3.07312	0.294352	0.147176	+	0.989110i	$0.452982\pi$
$110$	0	0
$111$	10.9662	1.04087
$112$	0	0
$113$	−11.2646	−1.05968	−0.529840	−	0.848098i	$-0.677748\pi$
$113$	−11.2646	−1.05968	−0.529840	+	0.848098i	$0.677748\pi$
$114$	0	0
$115$	4.39768	0.410086
$116$	0	0
$117$	−0.216858	−0.0200486
$118$	0	0
$119$	−8.93560	−0.819125
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−13.1633	−1.18689
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.113136	0.0100392	0.00501961	−	0.999987i	$-0.498402\pi$
$127$	0.113136	0.0100392	0.00501961	+	0.999987i	$0.498402\pi$
$128$	0	0
$129$	22.5155	1.98238
$130$	0	0
$131$	0.474297	0.0414395	0.0207198	−	0.999785i	$-0.493404\pi$
$131$	0.474297	0.0414395	0.0207198	+	0.999785i	$0.493404\pi$
$132$	0	0
$133$	2.57388	0.223184
$134$	0	0
$135$	−5.04903	−0.434552
$136$	0	0
$137$	−0.868693	−0.0742174	−0.0371087	−	0.999311i	$-0.511815\pi$
$137$	−0.868693	−0.0742174	−0.0371087	+	0.999311i	$0.511815\pi$
$138$	0	0
$139$	19.3894	1.64459	0.822293	−	0.569065i	$-0.192695\pi$
$139$	19.3894	1.64459	0.822293	+	0.569065i	$0.192695\pi$
$140$	0	0
$141$	5.46480	0.460219
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.63159	0.550723
$146$	0	0
$147$	19.7843	1.63178
$148$	0	0
$149$	1.11258	0.0911460	0.0455730	−	0.998961i	$-0.485489\pi$
$149$	1.11258	0.0911460	0.0455730	+	0.998961i	$0.485489\pi$
$150$	0	0
$151$	5.69465	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$151$	5.69465	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$152$	0	0
$153$	−0.334614	−0.0270519
$154$	0	0
$155$	2.19493	0.176301
$156$	0	0
$157$	1.04114	0.0830918	0.0415459	−	0.999137i	$-0.486772\pi$
$157$	1.04114	0.0830918	0.0415459	+	0.999137i	$0.486772\pi$
$158$	0	0
$159$	−11.8265	−0.937904
$160$	0	0
$161$	18.7253	1.47576
$162$	0	0
$163$	12.9115	1.01131	0.505654	−	0.862737i	$-0.331251\pi$
$163$	12.9115	1.01131	0.505654	+	0.862737i	$0.331251\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.22601	0.327019	0.163509	−	0.986542i	$-0.447719\pi$
$167$	4.22601	0.327019	0.163509	+	0.986542i	$0.447719\pi$
$168$	0	0
$169$	−11.1503	−0.857715
$170$	0	0
$171$	0.0963848	0.00737073
$172$	0	0
$173$	−1.34571	−0.102312	−0.0511561	−	0.998691i	$-0.516291\pi$
$173$	−1.34571	−0.102312	−0.0511561	+	0.998691i	$0.516291\pi$
$174$	0	0
$175$	4.25800	0.321874
$176$	0	0
$177$	−21.3959	−1.60821
$178$	0	0
$179$	−12.4288	−0.928975	−0.464487	−	0.885580i	$-0.653761\pi$
$179$	−12.4288	−0.928975	−0.464487	+	0.885580i	$0.653761\pi$
$180$	0	0
$181$	17.3092	1.28658	0.643291	−	0.765622i	$-0.277569\pi$
$181$	17.3092	1.28658	0.643291	+	0.765622i	$0.277569\pi$
$182$	0	0
$183$	10.1121	0.747508
$184$	0	0
$185$	6.16951	0.453591
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−21.4988	−1.56380
$190$	0	0
$191$	19.6503	1.42185	0.710923	−	0.703269i	$-0.248277\pi$
$191$	19.6503	1.42185	0.710923	+	0.703269i	$0.248277\pi$
$192$	0	0
$193$	−10.1616	−0.731449	−0.365724	−	0.930723i	$-0.619179\pi$
$193$	−10.1616	−0.731449	−0.365724	+	0.930723i	$0.619179\pi$
$194$	0	0
$195$	−2.41745	−0.173117
$196$	0	0
$197$	8.45375	0.602305	0.301152	−	0.953576i	$-0.402629\pi$
$197$	8.45375	0.602305	0.301152	+	0.953576i	$0.402629\pi$
$198$	0	0
$199$	6.51033	0.461505	0.230753	−	0.973012i	$-0.425881\pi$
$199$	6.51033	0.461505	0.230753	+	0.973012i	$0.425881\pi$
$200$	0	0
$201$	−17.5334	−1.23671
$202$	0	0
$203$	28.2373	1.98187
$204$	0	0
$205$	−7.40557	−0.517228
$206$	0	0
$207$	0.701211	0.0487375
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.01403	0.276337	0.138169	−	0.990409i	$-0.455878\pi$
$211$	4.01403	0.276337	0.138169	+	0.990409i	$0.455878\pi$
$212$	0	0
$213$	9.31186	0.638038
$214$	0	0
$215$	12.6671	0.863887
$216$	0	0
$217$	9.34600	0.634448
$218$	0	0
$219$	1.28476	0.0868158
$220$	0	0
$221$	2.85410	0.191988
$222$	0	0
$223$	−15.3070	−1.02503	−0.512517	−	0.858677i	$-0.671287\pi$
$223$	−15.3070	−1.02503	−0.512517	+	0.858677i	$0.671287\pi$
$224$	0	0
$225$	0.159450	0.0106300
$226$	0	0
$227$	11.5019	0.763407	0.381703	−	0.924285i	$-0.375338\pi$
$227$	11.5019	0.763407	0.381703	+	0.924285i	$0.375338\pi$
$228$	0	0
$229$	−27.5272	−1.81905	−0.909523	−	0.415653i	$-0.863553\pi$
$229$	−27.5272	−1.81905	−0.909523	+	0.415653i	$0.863553\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.20227	−0.602861	−0.301430	−	0.953488i	$-0.597464\pi$
$233$	−9.20227	−0.602861	−0.301430	+	0.953488i	$0.597464\pi$
$234$	0	0
$235$	3.07446	0.200555
$236$	0	0
$237$	10.0793	0.654723
$238$	0	0
$239$	12.6560	0.818647	0.409323	−	0.912389i	$-0.365765\pi$
$239$	12.6560	0.818647	0.409323	+	0.912389i	$0.365765\pi$
$240$	0	0
$241$	−7.33167	−0.472275	−0.236137	−	0.971720i	$-0.575882\pi$
$241$	−7.33167	−0.472275	−0.236137	+	0.971720i	$0.575882\pi$
$242$	0	0
$243$	−1.65533	−0.106190
$244$	0	0
$245$	11.1305	0.711103
$246$	0	0
$247$	−0.822118	−0.0523101
$248$	0	0
$249$	−1.69332	−0.107310
$250$	0	0
$251$	−31.2579	−1.97298	−0.986489	−	0.163826i	$-0.947617\pi$
$251$	−31.2579	−1.97298	−0.986489	+	0.163826i	$0.947617\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.73013	−0.233590
$256$	0	0
$257$	−8.32028	−0.519005	−0.259503	−	0.965742i	$-0.583559\pi$
$257$	−8.32028	−0.519005	−0.259503	+	0.965742i	$0.583559\pi$
$258$	0	0
$259$	26.2697	1.63232
$260$	0	0
$261$	1.05741	0.0654519
$262$	0	0
$263$	−2.87171	−0.177077	−0.0885386	−	0.996073i	$-0.528220\pi$
$263$	−2.87171	−0.177077	−0.0885386	+	0.996073i	$0.528220\pi$
$264$	0	0
$265$	−6.65351	−0.408722
$266$	0	0
$267$	2.20576	0.134991
$268$	0	0
$269$	−0.123970	−0.00755859	−0.00377929	−	0.999993i	$-0.501203\pi$
$269$	−0.123970	−0.00755859	−0.00377929	+	0.999993i	$0.501203\pi$
$270$	0	0
$271$	−17.0328	−1.03467	−0.517336	−	0.855783i	$-0.673076\pi$
$271$	−17.0328	−1.03467	−0.517336	+	0.855783i	$0.673076\pi$
$272$	0	0
$273$	−10.2935	−0.622990
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.7411	−1.36638	−0.683190	−	0.730241i	$-0.739408\pi$
$277$	−22.7411	−1.36638	−0.683190	+	0.730241i	$0.739408\pi$
$278$	0	0
$279$	0.349982	0.0209529
$280$	0	0
$281$	12.9499	0.772528	0.386264	−	0.922388i	$-0.373765\pi$
$281$	12.9499	0.772528	0.386264	+	0.922388i	$0.373765\pi$
$282$	0	0
$283$	15.8178	0.940273	0.470137	−	0.882594i	$-0.344205\pi$
$283$	15.8178	0.940273	0.470137	+	0.882594i	$0.344205\pi$
$284$	0	0
$285$	1.07446	0.0636453
$286$	0	0
$287$	−31.5329	−1.86133
$288$	0	0
$289$	−12.5961	−0.740947
$290$	0	0
$291$	20.5177	1.20277
$292$	0	0
$293$	24.9306	1.45646	0.728230	−	0.685333i	$-0.240343\pi$
$293$	24.9306	1.45646	0.728230	+	0.685333i	$0.240343\pi$
$294$	0	0
$295$	−12.0372	−0.700831
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.98101	−0.345891
$300$	0	0
$301$	53.9363	3.10884
$302$	0	0
$303$	−4.98895	−0.286608
$304$	0	0
$305$	5.68899	0.325751
$306$	0	0
$307$	−4.95566	−0.282835	−0.141417	−	0.989950i	$-0.545166\pi$
$307$	−4.95566	−0.282835	−0.141417	+	0.989950i	$0.545166\pi$
$308$	0	0
$309$	0.721756	0.0410593
$310$	0	0
$311$	9.59477	0.544070	0.272035	−	0.962287i	$-0.412303\pi$
$311$	9.59477	0.544070	0.272035	+	0.962287i	$0.412303\pi$
$312$	0	0
$313$	−26.9858	−1.52533	−0.762665	−	0.646794i	$-0.776109\pi$
$313$	−26.9858	−1.52533	−0.762665	+	0.646794i	$0.776109\pi$
$314$	0	0
$315$	0.678939	0.0382539
$316$	0	0
$317$	−14.2207	−0.798714	−0.399357	−	0.916795i	$-0.630767\pi$
$317$	−14.2207	−0.798714	−0.399357	+	0.916795i	$0.630767\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	27.3707	1.52768
$322$	0	0
$323$	−1.26853	−0.0705830
$324$	0	0
$325$	−1.36004	−0.0754413
$326$	0	0
$327$	5.46242	0.302073
$328$	0	0
$329$	13.0910	0.721732
$330$	0	0
$331$	8.55985	0.470492	0.235246	−	0.971936i	$-0.424410\pi$
$331$	8.55985	0.470492	0.235246	+	0.971936i	$0.424410\pi$
$332$	0	0
$333$	0.983729	0.0539080
$334$	0	0
$335$	−9.86416	−0.538937
$336$	0	0
$337$	15.6841	0.854368	0.427184	−	0.904165i	$-0.359506\pi$
$337$	15.6841	0.854368	0.427184	+	0.904165i	$0.359506\pi$
$338$	0	0
$339$	−20.0226	−1.08748
$340$	0	0
$341$	0	0
$342$	0	0
$343$	17.5878	0.949651
$344$	0	0
$345$	7.81681	0.420843
$346$	0	0
$347$	7.45509	0.400210	0.200105	−	0.979774i	$-0.435872\pi$
$347$	7.45509	0.400210	0.200105	+	0.979774i	$0.435872\pi$
$348$	0	0
$349$	30.4733	1.63120	0.815600	−	0.578616i	$-0.196407\pi$
$349$	30.4733	1.63120	0.815600	+	0.578616i	$0.196407\pi$
$350$	0	0
$351$	6.86688	0.366527
$352$	0	0
$353$	−17.5379	−0.933449	−0.466725	−	0.884403i	$-0.654566\pi$
$353$	−17.5379	−0.933449	−0.466725	+	0.884403i	$0.654566\pi$
$354$	0	0
$355$	5.23879	0.278046
$356$	0	0
$357$	−15.8829	−0.840611
$358$	0	0
$359$	−3.00432	−0.158562	−0.0792810	−	0.996852i	$-0.525262\pi$
$359$	−3.00432	−0.158562	−0.0792810	+	0.996852i	$0.525262\pi$
$360$	0	0
$361$	−18.6346	−0.980769
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.722795	0.0378328
$366$	0	0
$367$	−6.18319	−0.322760	−0.161380	−	0.986892i	$-0.551594\pi$
$367$	−6.18319	−0.322760	−0.161380	+	0.986892i	$0.551594\pi$
$368$	0	0
$369$	−1.18082	−0.0614711
$370$	0	0
$371$	−28.3306	−1.47085
$372$	0	0
$373$	−20.8707	−1.08064	−0.540321	−	0.841459i	$-0.681697\pi$
$373$	−20.8707	−1.08064	−0.540321	+	0.841459i	$0.681697\pi$
$374$	0	0
$375$	1.77748	0.0917889
$376$	0	0
$377$	−9.01921	−0.464513
$378$	0	0
$379$	2.46424	0.126580	0.0632898	−	0.997995i	$-0.479841\pi$
$379$	2.46424	0.126580	0.0632898	+	0.997995i	$0.479841\pi$
$380$	0	0
$381$	0.201098	0.0103026
$382$	0	0
$383$	−17.2036	−0.879063	−0.439532	−	0.898227i	$-0.644856\pi$
$383$	−17.2036	−0.879063	−0.439532	+	0.898227i	$0.644856\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.01977	0.102671
$388$	0	0
$389$	−18.0548	−0.915413	−0.457706	−	0.889103i	$-0.651329\pi$
$389$	−18.0548	−0.915413	−0.457706	+	0.889103i	$0.651329\pi$
$390$	0	0
$391$	−9.22873	−0.466717
$392$	0	0
$393$	0.843055	0.0425265
$394$	0	0
$395$	5.67056	0.285317
$396$	0	0
$397$	−16.7432	−0.840317	−0.420159	−	0.907451i	$-0.638026\pi$
$397$	−16.7432	−0.840317	−0.420159	+	0.907451i	$0.638026\pi$
$398$	0	0
$399$	4.57503	0.229038
$400$	0	0
$401$	28.9395	1.44517	0.722584	−	0.691283i	$-0.242954\pi$
$401$	28.9395	1.44517	0.722584	+	0.691283i	$0.242954\pi$
$402$	0	0
$403$	−2.98519	−0.148703
$404$	0	0
$405$	−9.45293	−0.469720
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.8733	1.47714	0.738569	−	0.674178i	$-0.235502\pi$
$409$	29.8733	1.47714	0.738569	+	0.674178i	$0.235502\pi$
$410$	0	0
$411$	−1.54409	−0.0761642
$412$	0	0
$413$	−51.2542	−2.52205
$414$	0	0
$415$	−0.952648	−0.0467636
$416$	0	0
$417$	34.4643	1.68772
$418$	0	0
$419$	8.29831	0.405399	0.202699	−	0.979241i	$-0.435029\pi$
$419$	8.29831	0.405399	0.202699	+	0.979241i	$0.435029\pi$
$420$	0	0
$421$	−22.8672	−1.11448	−0.557239	−	0.830352i	$-0.688139\pi$
$421$	−22.8672	−1.11448	−0.557239	+	0.830352i	$0.688139\pi$
$422$	0	0
$423$	0.490223	0.0238355
$424$	0	0
$425$	−2.09855	−0.101794
$426$	0	0
$427$	24.2237	1.17227
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.3617	−0.980789	−0.490395	−	0.871500i	$-0.663147\pi$
$431$	−20.3617	−0.980789	−0.490395	+	0.871500i	$0.663147\pi$
$432$	0	0
$433$	21.3889	1.02788	0.513942	−	0.857825i	$-0.328185\pi$
$433$	21.3889	1.02788	0.513942	+	0.857825i	$0.328185\pi$
$434$	0	0
$435$	11.7875	0.565169
$436$	0	0
$437$	2.65832	0.127165
$438$	0	0
$439$	−35.2311	−1.68149	−0.840744	−	0.541432i	$-0.817882\pi$
$439$	−35.2311	−1.68149	−0.840744	+	0.541432i	$0.817882\pi$
$440$	0	0
$441$	1.77477	0.0845127
$442$	0	0
$443$	−23.2623	−1.10523	−0.552613	−	0.833438i	$-0.686369\pi$
$443$	−23.2623	−1.10523	−0.552613	+	0.833438i	$0.686369\pi$
$444$	0	0
$445$	1.24095	0.0588265
$446$	0	0
$447$	1.97759	0.0935369
$448$	0	0
$449$	24.8518	1.17283	0.586415	−	0.810011i	$-0.300539\pi$
$449$	24.8518	1.17283	0.586415	+	0.810011i	$0.300539\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.1222	0.475580
$454$	0	0
$455$	−5.79104	−0.271488
$456$	0	0
$457$	−5.90831	−0.276379	−0.138190	−	0.990406i	$-0.544128\pi$
$457$	−5.90831	−0.276379	−0.138190	+	0.990406i	$0.544128\pi$
$458$	0	0
$459$	10.5956	0.494561
$460$	0	0
$461$	17.3587	0.808475	0.404238	−	0.914654i	$-0.367537\pi$
$461$	17.3587	0.808475	0.404238	+	0.914654i	$0.367537\pi$
$462$	0	0
$463$	−34.6937	−1.61235	−0.806176	−	0.591675i	$-0.798467\pi$
$463$	−34.6937	−1.61235	−0.806176	+	0.591675i	$0.798467\pi$
$464$	0	0
$465$	3.90145	0.180926
$466$	0	0
$467$	−9.16601	−0.424152	−0.212076	−	0.977253i	$-0.568022\pi$
$467$	−9.16601	−0.424152	−0.212076	+	0.977253i	$0.568022\pi$
$468$	0	0
$469$	−42.0015	−1.93945
$470$	0	0
$471$	1.85061	0.0852714
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.604482	0.0277355
$476$	0	0
$477$	−1.06090	−0.0485755
$478$	0	0
$479$	−40.8320	−1.86566	−0.932831	−	0.360315i	$-0.882669\pi$
$479$	−40.8320	−1.86566	−0.932831	+	0.360315i	$0.882669\pi$
$480$	0	0
$481$	−8.39076	−0.382586
$482$	0	0
$483$	33.2839	1.51447
$484$	0	0
$485$	11.5431	0.524145
$486$	0	0
$487$	26.1598	1.18541	0.592707	−	0.805418i	$-0.298059\pi$
$487$	26.1598	1.18541	0.592707	+	0.805418i	$0.298059\pi$
$488$	0	0
$489$	22.9500	1.03784
$490$	0	0
$491$	23.0012	1.03803	0.519015	−	0.854765i	$-0.326299\pi$
$491$	23.0012	1.03803	0.519015	+	0.854765i	$0.326299\pi$
$492$	0	0
$493$	−13.9167	−0.626776
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.3067	1.00059
$498$	0	0
$499$	26.2140	1.17350	0.586750	−	0.809768i	$-0.300407\pi$
$499$	26.2140	1.17350	0.586750	+	0.809768i	$0.300407\pi$
$500$	0	0
$501$	7.51167	0.335597
$502$	0	0
$503$	1.68460	0.0751124	0.0375562	−	0.999295i	$-0.488043\pi$
$503$	1.68460	0.0751124	0.0375562	+	0.999295i	$0.488043\pi$
$504$	0	0
$505$	−2.80675	−0.124899
$506$	0	0
$507$	−19.8195	−0.880214
$508$	0	0
$509$	17.8483	0.791112	0.395556	−	0.918442i	$-0.370552\pi$
$509$	17.8483	0.791112	0.395556	+	0.918442i	$0.370552\pi$
$510$	0	0
$511$	3.07766	0.136147
$512$	0	0
$513$	−3.05205	−0.134751
$514$	0	0
$515$	0.406055	0.0178929
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.39197	−0.104996
$520$	0	0
$521$	−9.54339	−0.418104	−0.209052	−	0.977905i	$-0.567038\pi$
$521$	−9.54339	−0.418104	−0.209052	+	0.977905i	$0.567038\pi$
$522$	0	0
$523$	−21.7454	−0.950862	−0.475431	−	0.879753i	$-0.657708\pi$
$523$	−21.7454	−0.950862	−0.475431	+	0.879753i	$0.657708\pi$
$524$	0	0
$525$	7.56852	0.330317
$526$	0	0
$527$	−4.60616	−0.200648
$528$	0	0
$529$	−3.66042	−0.159149
$530$	0	0
$531$	−1.91933	−0.0832918
$532$	0	0
$533$	10.0719	0.436261
$534$	0	0
$535$	15.3985	0.665737
$536$	0	0
$537$	−22.0921	−0.953343
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13.4876	−0.579876	−0.289938	−	0.957045i	$-0.593635\pi$
$541$	−13.4876	−0.579876	−0.289938	+	0.957045i	$0.593635\pi$
$542$	0	0
$543$	30.7668	1.32033
$544$	0	0
$545$	3.07312	0.131638
$546$	0	0
$547$	−24.4217	−1.04419	−0.522097	−	0.852886i	$-0.674850\pi$
$547$	−24.4217	−1.04419	−0.522097	+	0.852886i	$0.674850\pi$
$548$	0	0
$549$	0.907112	0.0387146
$550$	0	0
$551$	4.00867	0.170775
$552$	0	0
$553$	24.1452	1.02676
$554$	0	0
$555$	10.9662	0.465489
$556$	0	0
$557$	25.7940	1.09293	0.546464	−	0.837483i	$-0.315974\pi$
$557$	25.7940	1.09293	0.546464	+	0.837483i	$0.315974\pi$
$558$	0	0
$559$	−17.2277	−0.728654
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−40.3703	−1.70140	−0.850702	−	0.525649i	$-0.823823\pi$
$563$	−40.3703	−1.70140	−0.850702	+	0.525649i	$0.823823\pi$
$564$	0	0
$565$	−11.2646	−0.473903
$566$	0	0
$567$	−40.2505	−1.69036
$568$	0	0
$569$	10.5541	0.442453	0.221226	−	0.975222i	$-0.428994\pi$
$569$	10.5541	0.442453	0.221226	+	0.975222i	$0.428994\pi$
$570$	0	0
$571$	−9.77700	−0.409155	−0.204577	−	0.978850i	$-0.565582\pi$
$571$	−9.77700	−0.409155	−0.204577	+	0.978850i	$0.565582\pi$
$572$	0	0
$573$	34.9281	1.45914
$574$	0	0
$575$	4.39768	0.183396
$576$	0	0
$577$	−16.2960	−0.678411	−0.339205	−	0.940712i	$-0.610158\pi$
$577$	−16.2960	−0.678411	−0.339205	+	0.940712i	$0.610158\pi$
$578$	0	0
$579$	−18.0621	−0.750635
$580$	0	0
$581$	−4.05637	−0.168287
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.216858	−0.00896599
$586$	0	0
$587$	−33.6827	−1.39023	−0.695117	−	0.718896i	$-0.744648\pi$
$587$	−33.6827	−1.39023	−0.695117	+	0.718896i	$0.744648\pi$
$588$	0	0
$589$	1.32680	0.0546697
$590$	0	0
$591$	15.0264	0.618104
$592$	0	0
$593$	21.3355	0.876143	0.438071	−	0.898940i	$-0.355662\pi$
$593$	21.3355	0.876143	0.438071	+	0.898940i	$0.355662\pi$
$594$	0	0
$595$	−8.93560	−0.366324
$596$	0	0
$597$	11.5720	0.473611
$598$	0	0
$599$	16.4096	0.670476	0.335238	−	0.942133i	$-0.391183\pi$
$599$	16.4096	0.670476	0.335238	+	0.942133i	$0.391183\pi$
$600$	0	0
$601$	13.7954	0.562726	0.281363	−	0.959601i	$-0.409213\pi$
$601$	13.7954	0.562726	0.281363	+	0.959601i	$0.409213\pi$
$602$	0	0
$603$	−1.57284	−0.0640511
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−5.64351	−0.229063	−0.114531	−	0.993420i	$-0.536537\pi$
$607$	−5.64351	−0.229063	−0.114531	+	0.993420i	$0.536537\pi$
$608$	0	0
$609$	50.1913	2.03385
$610$	0	0
$611$	−4.18138	−0.169160
$612$	0	0
$613$	3.33104	0.134539	0.0672697	−	0.997735i	$-0.478571\pi$
$613$	3.33104	0.134539	0.0672697	+	0.997735i	$0.478571\pi$
$614$	0	0
$615$	−13.1633	−0.530795
$616$	0	0
$617$	−9.45854	−0.380786	−0.190393	−	0.981708i	$-0.560976\pi$
$617$	−9.45854	−0.380786	−0.190393	+	0.981708i	$0.560976\pi$
$618$	0	0
$619$	27.2070	1.09354	0.546771	−	0.837282i	$-0.315857\pi$
$619$	27.2070	1.09354	0.546771	+	0.837282i	$0.315857\pi$
$620$	0	0
$621$	−22.2040	−0.891017
$622$	0	0
$623$	5.28395	0.211697
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.9470	−0.516230
$630$	0	0
$631$	30.3287	1.20737	0.603683	−	0.797224i	$-0.293699\pi$
$631$	30.3287	1.20737	0.603683	+	0.797224i	$0.293699\pi$
$632$	0	0
$633$	7.13488	0.283586
$634$	0	0
$635$	0.113136	0.00448968
$636$	0	0
$637$	−15.1379	−0.599787
$638$	0	0
$639$	0.835326	0.0330450
$640$	0	0
$641$	39.8313	1.57324	0.786620	−	0.617437i	$-0.211829\pi$
$641$	39.8313	1.57324	0.786620	+	0.617437i	$0.211829\pi$
$642$	0	0
$643$	27.3517	1.07864	0.539322	−	0.842100i	$-0.318681\pi$
$643$	27.3517	1.07864	0.539322	+	0.842100i	$0.318681\pi$
$644$	0	0
$645$	22.5155	0.886547
$646$	0	0
$647$	−22.2738	−0.875675	−0.437837	−	0.899054i	$-0.644255\pi$
$647$	−22.2738	−0.875675	−0.437837	+	0.899054i	$0.644255\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.6124	0.651090
$652$	0	0
$653$	29.3913	1.15017	0.575085	−	0.818094i	$-0.304969\pi$
$653$	29.3913	1.15017	0.575085	+	0.818094i	$0.304969\pi$
$654$	0	0
$655$	0.474297	0.0185323
$656$	0	0
$657$	0.115250	0.00449632
$658$	0	0
$659$	22.8429	0.889832	0.444916	−	0.895572i	$-0.353234\pi$
$659$	22.8429	0.889832	0.444916	+	0.895572i	$0.353234\pi$
$660$	0	0
$661$	32.0302	1.24583	0.622916	−	0.782289i	$-0.285948\pi$
$661$	32.0302	1.24583	0.622916	+	0.782289i	$0.285948\pi$
$662$	0	0
$663$	5.07312	0.197024
$664$	0	0
$665$	2.57388	0.0998108
$666$	0	0
$667$	29.1636	1.12922
$668$	0	0
$669$	−27.2080	−1.05192
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−13.5740	−0.523238	−0.261619	−	0.965171i	$-0.584256\pi$
$673$	−13.5740	−0.523238	−0.261619	+	0.965171i	$0.584256\pi$
$674$	0	0
$675$	−5.04903	−0.194337
$676$	0	0
$677$	−1.58263	−0.0608254	−0.0304127	−	0.999537i	$-0.509682\pi$
$677$	−1.58263	−0.0608254	−0.0304127	+	0.999537i	$0.509682\pi$
$678$	0	0
$679$	49.1505	1.88622
$680$	0	0
$681$	20.4444	0.783432
$682$	0	0
$683$	−33.8348	−1.29465	−0.647325	−	0.762214i	$-0.724113\pi$
$683$	−33.8348	−1.29465	−0.647325	+	0.762214i	$0.724113\pi$
$684$	0	0
$685$	−0.868693	−0.0331910
$686$	0	0
$687$	−48.9291	−1.86676
$688$	0	0
$689$	9.04903	0.344741
$690$	0	0
$691$	33.7446	1.28371	0.641853	−	0.766828i	$-0.278166\pi$
$691$	33.7446	1.28371	0.641853	+	0.766828i	$0.278166\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.3894	0.735481
$696$	0	0
$697$	15.5409	0.588655
$698$	0	0
$699$	−16.3569	−0.618674
$700$	0	0
$701$	−1.88737	−0.0712851	−0.0356426	−	0.999365i	$-0.511348\pi$
$701$	−1.88737	−0.0712851	−0.0356426	+	0.999365i	$0.511348\pi$
$702$	0	0
$703$	3.72935	0.140655
$704$	0	0
$705$	5.46480	0.205816
$706$	0	0
$707$	−11.9511	−0.449468
$708$	0	0
$709$	−7.03820	−0.264325	−0.132163	−	0.991228i	$-0.542192\pi$
$709$	−7.03820	−0.264325	−0.132163	+	0.991228i	$0.542192\pi$
$710$	0	0
$711$	0.904173	0.0339091
$712$	0	0
$713$	9.65260	0.361493
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.4958	0.840121
$718$	0	0
$719$	−8.02849	−0.299412	−0.149706	−	0.988731i	$-0.547833\pi$
$719$	−8.02849	−0.299412	−0.149706	+	0.988731i	$0.547833\pi$
$720$	0	0
$721$	1.72898	0.0643906
$722$	0	0
$723$	−13.0319	−0.484663
$724$	0	0
$725$	6.63159	0.246291
$726$	0	0
$727$	29.8123	1.10568	0.552838	−	0.833289i	$-0.313545\pi$
$727$	29.8123	1.10568	0.552838	+	0.833289i	$0.313545\pi$
$728$	0	0
$729$	25.4165	0.941350
$730$	0	0
$731$	−26.5824	−0.983186
$732$	0	0
$733$	−40.9187	−1.51137	−0.755683	−	0.654938i	$-0.772695\pi$
$733$	−40.9187	−1.51137	−0.755683	+	0.654938i	$0.772695\pi$
$734$	0	0
$735$	19.7843	0.729756
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.36478	−0.160561	−0.0802805	−	0.996772i	$-0.525582\pi$
$739$	−4.36478	−0.160561	−0.0802805	+	0.996772i	$0.525582\pi$
$740$	0	0
$741$	−1.46130	−0.0536823
$742$	0	0
$743$	19.7055	0.722926	0.361463	−	0.932386i	$-0.382277\pi$
$743$	19.7055	0.722926	0.361463	+	0.932386i	$0.382277\pi$
$744$	0	0
$745$	1.11258	0.0407617
$746$	0	0
$747$	−0.151900	−0.00555773
$748$	0	0
$749$	65.5669	2.39576
$750$	0	0
$751$	38.9543	1.42146	0.710731	−	0.703464i	$-0.248364\pi$
$751$	38.9543	1.42146	0.710731	+	0.703464i	$0.248364\pi$
$752$	0	0
$753$	−55.5604	−2.02473
$754$	0	0
$755$	5.69465	0.207250
$756$	0	0
$757$	53.1675	1.93241	0.966203	−	0.257783i	$-0.0829918\pi$
$757$	53.1675	1.93241	0.966203	+	0.257783i	$0.0829918\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−51.2783	−1.85884	−0.929418	−	0.369029i	$-0.879690\pi$
$761$	−51.2783	−1.85884	−0.929418	+	0.369029i	$0.879690\pi$
$762$	0	0
$763$	13.0853	0.473721
$764$	0	0
$765$	−0.334614	−0.0120980
$766$	0	0
$767$	16.3710	0.591122
$768$	0	0
$769$	−45.0332	−1.62394	−0.811970	−	0.583699i	$-0.801605\pi$
$769$	−45.0332	−1.62394	−0.811970	+	0.583699i	$0.801605\pi$
$770$	0	0
$771$	−14.7892	−0.532619
$772$	0	0
$773$	−49.9436	−1.79635	−0.898173	−	0.439641i	$-0.855106\pi$
$773$	−49.9436	−1.79635	−0.898173	+	0.439641i	$0.855106\pi$
$774$	0	0
$775$	2.19493	0.0788442
$776$	0	0
$777$	46.6940	1.67514
$778$	0	0
$779$	−4.47653	−0.160388
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−33.4831	−1.19659
$784$	0	0
$785$	1.04114	0.0371598
$786$	0	0
$787$	9.78526	0.348807	0.174403	−	0.984674i	$-0.444200\pi$
$787$	9.78526	0.348807	0.174403	+	0.984674i	$0.444200\pi$
$788$	0	0
$789$	−5.10442	−0.181722
$790$	0	0
$791$	−47.9644	−1.70542
$792$	0	0
$793$	−7.73725	−0.274758
$794$	0	0
$795$	−11.8265	−0.419443
$796$	0	0
$797$	36.1647	1.28102	0.640509	−	0.767950i	$-0.278723\pi$
$797$	36.1647	1.28102	0.640509	+	0.767950i	$0.278723\pi$
$798$	0	0
$799$	−6.45189	−0.228251
$800$	0	0
$801$	0.197869	0.00699137
$802$	0	0
$803$	0	0
$804$	0	0
$805$	18.7253	0.659980
$806$	0	0
$807$	−0.220355	−0.00775686
$808$	0	0
$809$	−16.3357	−0.574332	−0.287166	−	0.957881i	$-0.592713\pi$
$809$	−16.3357	−0.574332	−0.287166	+	0.957881i	$0.592713\pi$
$810$	0	0
$811$	51.6959	1.81529	0.907643	−	0.419742i	$-0.137879\pi$
$811$	51.6959	1.81529	0.907643	+	0.419742i	$0.137879\pi$
$812$	0	0
$813$	−30.2756	−1.06181
$814$	0	0
$815$	12.9115	0.452270
$816$	0	0
$817$	7.65701	0.267885
$818$	0	0
$819$	−0.923382	−0.0322656
$820$	0	0
$821$	−38.6713	−1.34964	−0.674819	−	0.737983i	$-0.735778\pi$
$821$	−38.6713	−1.34964	−0.674819	+	0.737983i	$0.735778\pi$
$822$	0	0
$823$	5.30457	0.184906	0.0924528	−	0.995717i	$-0.470529\pi$
$823$	5.30457	0.184906	0.0924528	+	0.995717i	$0.470529\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.8642	−0.412558	−0.206279	−	0.978493i	$-0.566135\pi$
$827$	−11.8642	−0.412558	−0.206279	+	0.978493i	$0.566135\pi$
$828$	0	0
$829$	−49.3457	−1.71385	−0.856923	−	0.515445i	$-0.827627\pi$
$829$	−49.3457	−1.71385	−0.856923	+	0.515445i	$0.827627\pi$
$830$	0	0
$831$	−40.4219	−1.40222
$832$	0	0
$833$	−23.3579	−0.809304
$834$	0	0
$835$	4.22601	0.146247
$836$	0	0
$837$	−11.0823	−0.383059
$838$	0	0
$839$	21.4081	0.739089	0.369544	−	0.929213i	$-0.379514\pi$
$839$	21.4081	0.739089	0.369544	+	0.929213i	$0.379514\pi$
$840$	0	0
$841$	14.9779	0.516481
$842$	0	0
$843$	23.0183	0.792792
$844$	0	0
$845$	−11.1503	−0.383582
$846$	0	0
$847$	0	0
$848$	0	0
$849$	28.1160	0.964938
$850$	0	0
$851$	27.1315	0.930056
$852$	0	0
$853$	−31.0811	−1.06420	−0.532098	−	0.846683i	$-0.678596\pi$
$853$	−31.0811	−1.06420	−0.532098	+	0.846683i	$0.678596\pi$
$854$	0	0
$855$	0.0963848	0.00329629
$856$	0	0
$857$	−33.2665	−1.13636	−0.568181	−	0.822904i	$-0.692353\pi$
$857$	−33.2665	−1.13636	−0.568181	+	0.822904i	$0.692353\pi$
$858$	0	0
$859$	18.9045	0.645012	0.322506	−	0.946567i	$-0.395475\pi$
$859$	18.9045	0.645012	0.322506	+	0.946567i	$0.395475\pi$
$860$	0	0
$861$	−56.0492	−1.91015
$862$	0	0
$863$	−13.8324	−0.470862	−0.235431	−	0.971891i	$-0.575650\pi$
$863$	−13.8324	−0.470862	−0.235431	+	0.971891i	$0.575650\pi$
$864$	0	0
$865$	−1.34571	−0.0457554
$866$	0	0
$867$	−22.3894	−0.760383
$868$	0	0
$869$	0	0
$870$	0	0
$871$	13.4156	0.454571
$872$	0	0
$873$	1.84055	0.0622932
$874$	0	0
$875$	4.25800	0.143947
$876$	0	0
$877$	48.3403	1.63233	0.816167	−	0.577816i	$-0.196095\pi$
$877$	48.3403	1.63233	0.816167	+	0.577816i	$0.196095\pi$
$878$	0	0
$879$	44.3137	1.49466
$880$	0	0
$881$	−33.4457	−1.12682	−0.563408	−	0.826179i	$-0.690510\pi$
$881$	−33.4457	−1.12682	−0.563408	+	0.826179i	$0.690510\pi$
$882$	0	0
$883$	−18.2657	−0.614689	−0.307345	−	0.951598i	$-0.599440\pi$
$883$	−18.2657	−0.614689	−0.307345	+	0.951598i	$0.599440\pi$
$884$	0	0
$885$	−21.3959	−0.719214
$886$	0	0
$887$	−15.6871	−0.526721	−0.263361	−	0.964697i	$-0.584831\pi$
$887$	−15.6871	−0.526721	−0.263361	+	0.964697i	$0.584831\pi$
$888$	0	0
$889$	0.481734	0.0161568
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.85845	0.0621908
$894$	0	0
$895$	−12.4288	−0.415450
$896$	0	0
$897$	−10.6312	−0.354964
$898$	0	0
$899$	14.5559	0.485465
$900$	0	0
$901$	13.9627	0.465165
$902$	0	0
$903$	95.8710	3.19039
$904$	0	0
$905$	17.3092	0.575377
$906$	0	0
$907$	25.8952	0.859835	0.429918	−	0.902868i	$-0.358543\pi$
$907$	25.8952	0.859835	0.429918	+	0.902868i	$0.358543\pi$
$908$	0	0
$909$	−0.447537	−0.0148439
$910$	0	0
$911$	−15.4526	−0.511967	−0.255983	−	0.966681i	$-0.582399\pi$
$911$	−15.4526	−0.511967	−0.255983	+	0.966681i	$0.582399\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	10.1121	0.334296
$916$	0	0
$917$	2.01955	0.0666916
$918$	0	0
$919$	47.7134	1.57392	0.786959	−	0.617005i	$-0.211654\pi$
$919$	47.7134	1.57392	0.786959	+	0.617005i	$0.211654\pi$
$920$	0	0
$921$	−8.80861	−0.290254
$922$	0	0
$923$	−7.12495	−0.234521
$924$	0	0
$925$	6.16951	0.202852
$926$	0	0
$927$	0.0647456	0.00212652
$928$	0	0
$929$	−37.7533	−1.23865	−0.619323	−	0.785136i	$-0.712593\pi$
$929$	−37.7533	−1.23865	−0.619323	+	0.785136i	$0.712593\pi$
$930$	0	0
$931$	6.72820	0.220508
$932$	0	0
$933$	17.0546	0.558341
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.6382	1.62161	0.810805	−	0.585316i	$-0.199030\pi$
$937$	49.6382	1.62161	0.810805	+	0.585316i	$0.199030\pi$
$938$	0	0
$939$	−47.9669	−1.56534
$940$	0	0
$941$	−33.3188	−1.08616	−0.543081	−	0.839680i	$-0.682742\pi$
$941$	−33.3188	−1.08616	−0.543081	+	0.839680i	$0.682742\pi$
$942$	0	0
$943$	−32.5673	−1.06054
$944$	0	0
$945$	−21.4988	−0.699355
$946$	0	0
$947$	22.6463	0.735907	0.367954	−	0.929844i	$-0.380059\pi$
$947$	22.6463	0.735907	0.367954	+	0.929844i	$0.380059\pi$
$948$	0	0
$949$	−0.983028	−0.0319104
$950$	0	0
$951$	−25.2771	−0.819665
$952$	0	0
$953$	−48.5378	−1.57229	−0.786147	−	0.618040i	$-0.787927\pi$
$953$	−48.5378	−1.57229	−0.786147	+	0.618040i	$0.787927\pi$
$954$	0	0
$955$	19.6503	0.635869
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.69889	−0.119443
$960$	0	0
$961$	−26.1823	−0.844590
$962$	0	0
$963$	2.45530	0.0791209
$964$	0	0
$965$	−10.1616	−0.327114
$966$	0	0
$967$	−47.2983	−1.52101	−0.760505	−	0.649332i	$-0.775048\pi$
$967$	−47.2983	−1.52101	−0.760505	+	0.649332i	$0.775048\pi$
$968$	0	0
$969$	−2.25480	−0.0724345
$970$	0	0
$971$	−19.2817	−0.618780	−0.309390	−	0.950935i	$-0.600125\pi$
$971$	−19.2817	−0.618780	−0.309390	+	0.950935i	$0.600125\pi$
$972$	0	0
$973$	82.5599	2.64675
$974$	0	0
$975$	−2.41745	−0.0774202
$976$	0	0
$977$	31.8772	1.01984	0.509920	−	0.860222i	$-0.329675\pi$
$977$	31.8772	1.01984	0.509920	+	0.860222i	$0.329675\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.490010	0.0156448
$982$	0	0
$983$	−1.60184	−0.0510908	−0.0255454	−	0.999674i	$-0.508132\pi$
$983$	−1.60184	−0.0510908	−0.0255454	+	0.999674i	$0.508132\pi$
$984$	0	0
$985$	8.45375	0.269359
$986$	0	0
$987$	23.2691	0.740663
$988$	0	0
$989$	55.7057	1.77134
$990$	0	0
$991$	−48.3005	−1.53432	−0.767158	−	0.641458i	$-0.778330\pi$
$991$	−48.3005	−1.53432	−0.767158	+	0.641458i	$0.778330\pi$
$992$	0	0
$993$	15.2150	0.482833
$994$	0	0
$995$	6.51033	0.206391
$996$	0	0
$997$	22.8677	0.724228	0.362114	−	0.932134i	$-0.382055\pi$
$997$	22.8677	0.724228	0.362114	+	0.932134i	$0.382055\pi$
$998$	0	0
$999$	−31.1500	−0.985544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cv.1.3		4
4.3	odd	2		605.2.a.i.1.4		4
11.7	odd	10		880.2.bo.e.401.1		8
11.8	odd	10		880.2.bo.e.801.1		8
11.10	odd	2		9680.2.a.cs.1.3		4
12.11	even	2		5445.2.a.bu.1.1		4
20.19	odd	2		3025.2.a.be.1.1		4
44.3	odd	10		605.2.g.n.251.2		8
44.7	even	10		55.2.g.a.16.1	✓	8
44.15	odd	10		605.2.g.n.511.2		8
44.19	even	10		55.2.g.a.31.1	yes	8
44.27	odd	10		605.2.g.g.366.1		8
44.31	odd	10		605.2.g.g.81.1		8
44.35	even	10		605.2.g.j.81.2		8
44.39	even	10		605.2.g.j.366.2		8
44.43	even	2		605.2.a.l.1.1		4
132.95	odd	10		495.2.n.f.181.2		8
132.107	odd	10		495.2.n.f.361.2		8
132.131	odd	2		5445.2.a.bg.1.4		4
220.7	odd	20		275.2.z.b.49.3		16
220.19	even	10		275.2.h.b.251.2		8
220.63	odd	20		275.2.z.b.174.3		16
220.107	odd	20		275.2.z.b.174.2		16
220.139	even	10		275.2.h.b.126.2		8
220.183	odd	20		275.2.z.b.49.2		16
220.219	even	2		3025.2.a.v.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.a.16.1	✓	8	44.7	even	10
55.2.g.a.31.1	yes	8	44.19	even	10
275.2.h.b.126.2		8	220.139	even	10
275.2.h.b.251.2		8	220.19	even	10
275.2.z.b.49.2		16	220.183	odd	20
275.2.z.b.49.3		16	220.7	odd	20
275.2.z.b.174.2		16	220.107	odd	20
275.2.z.b.174.3		16	220.63	odd	20
495.2.n.f.181.2		8	132.95	odd	10
495.2.n.f.361.2		8	132.107	odd	10
605.2.a.i.1.4		4	4.3	odd	2
605.2.a.l.1.1		4	44.43	even	2
605.2.g.g.81.1		8	44.31	odd	10
605.2.g.g.366.1		8	44.27	odd	10
605.2.g.j.81.2		8	44.35	even	10
605.2.g.j.366.2		8	44.39	even	10
605.2.g.n.251.2		8	44.3	odd	10
605.2.g.n.511.2		8	44.15	odd	10
880.2.bo.e.401.1		8	11.7	odd	10
880.2.bo.e.801.1		8	11.8	odd	10
3025.2.a.v.1.4		4	220.219	even	2
3025.2.a.be.1.1		4	20.19	odd	2
5445.2.a.bg.1.4		4	132.131	odd	2
5445.2.a.bu.1.1		4	12.11	even	2
9680.2.a.cs.1.3		4	11.10	odd	2
9680.2.a.cv.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.77748 q^{3} +1.00000 q^{5} +4.25800 q^{7} +0.159450 q^{9} -1.36004 q^{13} +1.77748 q^{15} -2.09855 q^{17} +0.604482 q^{19} +7.56852 q^{21} +4.39768 q^{23} +1.00000 q^{25} -5.04903 q^{27} +6.63159 q^{29} +2.19493 q^{31} +4.25800 q^{35} +6.16951 q^{37} -2.41745 q^{39} -7.40557 q^{41} +12.6671 q^{43} +0.159450 q^{45} +3.07446 q^{47} +11.1305 q^{49} -3.73013 q^{51} -6.65351 q^{53} +1.07446 q^{57} -12.0372 q^{59} +5.68899 q^{61} +0.678939 q^{63} -1.36004 q^{65} -9.86416 q^{67} +7.81681 q^{69} +5.23879 q^{71} +0.722795 q^{73} +1.77748 q^{75} +5.67056 q^{79} -9.45293 q^{81} -0.952648 q^{83} -2.09855 q^{85} +11.7875 q^{87} +1.24095 q^{89} -5.79104 q^{91} +3.90145 q^{93} +0.604482 q^{95} +11.5431 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 11 q^{7} - 7 q^{13} + 2 q^{15} - 3 q^{17} + 12 q^{19} + 6 q^{21} + 9 q^{23} + 4 q^{25} + 5 q^{27} + 8 q^{29} - 3 q^{31} + 11 q^{35} - 3 q^{37} - 3 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 11 * q^7 - 7 * q^13 + 2 * q^15 - 3 * q^17 + 12 * q^19 + 6 * q^21 + 9 * q^23 + 4 * q^25 + 5 * q^27 + 8 * q^29 - 3 * q^31 + 11 * q^35 - 3 * q^37 - 3 * q^39 + 7 * q^41 + 21 * q^43 + 3 * q^47 + 15 * q^49 + 9 * q^51 - 11 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 3 * q^63 - 7 * q^65 + q^67 - 28 * q^69 + 15 * q^71 + 9 * q^73 + 2 * q^75 + 6 * q^79 - 20 * q^81 + 15 * q^83 - 3 * q^85 + 15 * q^87 - 4 * q^91 + 21 * q^93 + 12 * q^95 + 6 * q^97

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