LMFDB - Embedded newform 9680.2.a.df.1.8

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 = [8,0,-1,0,8,0,6,0,19,0,0,0,12,0,-1,0,2] 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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
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			1.8
Root			$-2.90500$ 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)

$f(q)$	$=$	$q+2.90500 q^{3} +1.00000 q^{5} +2.18309 q^{7} +5.43902 q^{9} +6.16092 q^{13} +2.90500 q^{15} +7.12818 q^{17} -6.35983 q^{19} +6.34188 q^{21} -2.17735 q^{23} +1.00000 q^{25} +7.08534 q^{27} -3.19299 q^{29} -4.39424 q^{31} +2.18309 q^{35} +5.29453 q^{37} +17.8975 q^{39} +3.28629 q^{41} +6.83874 q^{43} +5.43902 q^{45} +4.30445 q^{47} -2.23411 q^{49} +20.7074 q^{51} -10.6160 q^{53} -18.4753 q^{57} -13.7906 q^{59} +7.35286 q^{61} +11.8739 q^{63} +6.16092 q^{65} +7.67154 q^{67} -6.32521 q^{69} -5.74133 q^{71} -4.89575 q^{73} +2.90500 q^{75} +8.11407 q^{79} +4.26586 q^{81} -4.22384 q^{83} +7.12818 q^{85} -9.27563 q^{87} +16.8037 q^{89} +13.4499 q^{91} -12.7653 q^{93} -6.35983 q^{95} -3.27563 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} + 12 q^{13} - q^{15} + 2 q^{17} + 6 q^{19} + 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} + 8 q^{29} - 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) $ 8 * q - q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 + 12 * q^13 - q^15 + 2 * q^17 + 6 * q^19 + 6 * q^21 - 10 * q^23 + 8 * q^25 - 13 * q^27 + 8 * q^29 - 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 - 3 * q^41 + 8 * q^43 + 19 * q^45 - 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 + 25 * q^57 - 25 * q^59 + 10 * q^61 + 64 * q^63 + 12 * q^65 + 2 * q^67 + 18 * q^69 - 25 * q^71 + 38 * q^73 - q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 + 2 * q^85 - 15 * q^87 + 12 * q^89 - 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * 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
$3$	2.90500	1.67720	0.838601	−	0.544746i	$-0.183374\pi$
$3$	2.90500	1.67720	0.838601	+	0.544746i	$0.183374\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.18309	0.825131	0.412566	−	0.910928i	$-0.364633\pi$
$7$	2.18309	0.825131	0.412566	+	0.910928i	$0.364633\pi$
$8$	0	0
$9$	5.43902	1.81301
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.16092	1.70873	0.854366	−	0.519671i	$-0.173945\pi$
$13$	6.16092	1.70873	0.854366	+	0.519671i	$0.173945\pi$
$14$	0	0
$15$	2.90500	0.750067
$16$	0	0
$17$	7.12818	1.72884	0.864419	−	0.502772i	$-0.167686\pi$
$17$	7.12818	1.72884	0.864419	+	0.502772i	$0.167686\pi$
$18$	0	0
$19$	−6.35983	−1.45905	−0.729523	−	0.683956i	$-0.760258\pi$
$19$	−6.35983	−1.45905	−0.729523	+	0.683956i	$0.760258\pi$
$20$	0	0
$21$	6.34188	1.38391
$22$	0	0
$23$	−2.17735	−0.454010	−0.227005	−	0.973894i	$-0.572893\pi$
$23$	−2.17735	−0.454010	−0.227005	+	0.973894i	$0.572893\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.08534	1.36357
$28$	0	0
$29$	−3.19299	−0.592924	−0.296462	−	0.955045i	$-0.595807\pi$
$29$	−3.19299	−0.592924	−0.296462	+	0.955045i	$0.595807\pi$
$30$	0	0
$31$	−4.39424	−0.789229	−0.394614	−	0.918847i	$-0.629122\pi$
$31$	−4.39424	−0.789229	−0.394614	+	0.918847i	$0.629122\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.18309	0.369010
$36$	0	0
$37$	5.29453	0.870415	0.435207	−	0.900330i	$-0.356675\pi$
$37$	5.29453	0.870415	0.435207	+	0.900330i	$0.356675\pi$
$38$	0	0
$39$	17.8975	2.86589
$40$	0	0
$41$	3.28629	0.513232	0.256616	−	0.966513i	$-0.417392\pi$
$41$	3.28629	0.513232	0.256616	+	0.966513i	$0.417392\pi$
$42$	0	0
$43$	6.83874	1.04290	0.521449	−	0.853282i	$-0.325392\pi$
$43$	6.83874	1.04290	0.521449	+	0.853282i	$0.325392\pi$
$44$	0	0
$45$	5.43902	0.810801
$46$	0	0
$47$	4.30445	0.627868	0.313934	−	0.949445i	$-0.398353\pi$
$47$	4.30445	0.627868	0.313934	+	0.949445i	$0.398353\pi$
$48$	0	0
$49$	−2.23411	−0.319159
$50$	0	0
$51$	20.7074	2.89961
$52$	0	0
$53$	−10.6160	−1.45822	−0.729110	−	0.684397i	$-0.760066\pi$
$53$	−10.6160	−1.45822	−0.729110	+	0.684397i	$0.760066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−18.4753	−2.44711
$58$	0	0
$59$	−13.7906	−1.79538	−0.897689	−	0.440630i	$-0.854755\pi$
$59$	−13.7906	−1.79538	−0.897689	+	0.440630i	$0.854755\pi$
$60$	0	0
$61$	7.35286	0.941437	0.470719	−	0.882283i	$-0.343995\pi$
$61$	7.35286	0.941437	0.470719	+	0.882283i	$0.343995\pi$
$62$	0	0
$63$	11.8739	1.49597
$64$	0	0
$65$	6.16092	0.764169
$66$	0	0
$67$	7.67154	0.937228	0.468614	−	0.883403i	$-0.344754\pi$
$67$	7.67154	0.937228	0.468614	+	0.883403i	$0.344754\pi$
$68$	0	0
$69$	−6.32521	−0.761466
$70$	0	0
$71$	−5.74133	−0.681370	−0.340685	−	0.940177i	$-0.610659\pi$
$71$	−5.74133	−0.681370	−0.340685	+	0.940177i	$0.610659\pi$
$72$	0	0
$73$	−4.89575	−0.573004	−0.286502	−	0.958080i	$-0.592493\pi$
$73$	−4.89575	−0.573004	−0.286502	+	0.958080i	$0.592493\pi$
$74$	0	0
$75$	2.90500	0.335440
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.11407	0.912904	0.456452	−	0.889748i	$-0.349120\pi$
$79$	8.11407	0.912904	0.456452	+	0.889748i	$0.349120\pi$
$80$	0	0
$81$	4.26586	0.473984
$82$	0	0
$83$	−4.22384	−0.463626	−0.231813	−	0.972760i	$-0.574466\pi$
$83$	−4.22384	−0.463626	−0.231813	+	0.972760i	$0.574466\pi$
$84$	0	0
$85$	7.12818	0.773160
$86$	0	0
$87$	−9.27563	−0.994452
$88$	0	0
$89$	16.8037	1.78119	0.890594	−	0.454800i	$-0.150289\pi$
$89$	16.8037	1.78119	0.890594	+	0.454800i	$0.150289\pi$
$90$	0	0
$91$	13.4499	1.40993
$92$	0	0
$93$	−12.7653	−1.32370
$94$	0	0
$95$	−6.35983	−0.652505
$96$	0	0
$97$	−3.27563	−0.332589	−0.166295	−	0.986076i	$-0.553180\pi$
$97$	−3.27563	−0.332589	−0.166295	+	0.986076i	$0.553180\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.91048	−0.687618	−0.343809	−	0.939040i	$-0.611717\pi$
$101$	−6.91048	−0.687618	−0.343809	+	0.939040i	$0.611717\pi$
$102$	0	0
$103$	−7.58591	−0.747462	−0.373731	−	0.927537i	$-0.621922\pi$
$103$	−7.58591	−0.747462	−0.373731	+	0.927537i	$0.621922\pi$
$104$	0	0
$105$	6.34188	0.618904
$106$	0	0
$107$	−2.80495	−0.271164	−0.135582	−	0.990766i	$-0.543290\pi$
$107$	−2.80495	−0.271164	−0.135582	+	0.990766i	$0.543290\pi$
$108$	0	0
$109$	8.02896	0.769035	0.384517	−	0.923118i	$-0.374368\pi$
$109$	8.02896	0.769035	0.384517	+	0.923118i	$0.374368\pi$
$110$	0	0
$111$	15.3806	1.45986
$112$	0	0
$113$	6.95944	0.654689	0.327344	−	0.944905i	$-0.393846\pi$
$113$	6.95944	0.654689	0.327344	+	0.944905i	$0.393846\pi$
$114$	0	0
$115$	−2.17735	−0.203039
$116$	0	0
$117$	33.5094	3.09794
$118$	0	0
$119$	15.5615	1.42652
$120$	0	0
$121$	0	0
$122$	0	0
$123$	9.54667	0.860794
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.3872	0.921716	0.460858	−	0.887474i	$-0.347542\pi$
$127$	10.3872	0.921716	0.460858	+	0.887474i	$0.347542\pi$
$128$	0	0
$129$	19.8665	1.74915
$130$	0	0
$131$	−3.48613	−0.304585	−0.152292	−	0.988335i	$-0.548666\pi$
$131$	−3.48613	−0.304585	−0.152292	+	0.988335i	$0.548666\pi$
$132$	0	0
$133$	−13.8841	−1.20390
$134$	0	0
$135$	7.08534	0.609809
$136$	0	0
$137$	−14.5471	−1.24284	−0.621422	−	0.783476i	$-0.713445\pi$
$137$	−14.5471	−1.24284	−0.621422	+	0.783476i	$0.713445\pi$
$138$	0	0
$139$	0.501893	0.0425700	0.0212850	−	0.999773i	$-0.493224\pi$
$139$	0.501893	0.0425700	0.0212850	+	0.999773i	$0.493224\pi$
$140$	0	0
$141$	12.5044	1.05306
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.19299	−0.265163
$146$	0	0
$147$	−6.49009	−0.535294
$148$	0	0
$149$	0.747620	0.0612474	0.0306237	−	0.999531i	$-0.490251\pi$
$149$	0.747620	0.0612474	0.0306237	+	0.999531i	$0.490251\pi$
$150$	0	0
$151$	3.27337	0.266383	0.133192	−	0.991090i	$-0.457477\pi$
$151$	3.27337	0.266383	0.133192	+	0.991090i	$0.457477\pi$
$152$	0	0
$153$	38.7703	3.13439
$154$	0	0
$155$	−4.39424	−0.352954
$156$	0	0
$157$	−15.0296	−1.19949	−0.599747	−	0.800190i	$-0.704732\pi$
$157$	−15.0296	−1.19949	−0.599747	+	0.800190i	$0.704732\pi$
$158$	0	0
$159$	−30.8395	−2.44573
$160$	0	0
$161$	−4.75336	−0.374617
$162$	0	0
$163$	−14.4813	−1.13426	−0.567132	−	0.823627i	$-0.691947\pi$
$163$	−14.4813	−1.13426	−0.567132	+	0.823627i	$0.691947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.304676	0.0235766	0.0117883	−	0.999931i	$-0.496248\pi$
$167$	0.304676	0.0235766	0.0117883	+	0.999931i	$0.496248\pi$
$168$	0	0
$169$	24.9570	1.91977
$170$	0	0
$171$	−34.5912	−2.64526
$172$	0	0
$173$	−9.71345	−0.738500	−0.369250	−	0.929330i	$-0.620385\pi$
$173$	−9.71345	−0.738500	−0.369250	+	0.929330i	$0.620385\pi$
$174$	0	0
$175$	2.18309	0.165026
$176$	0	0
$177$	−40.0615	−3.01121
$178$	0	0
$179$	−9.40166	−0.702713	−0.351356	−	0.936242i	$-0.614279\pi$
$179$	−9.40166	−0.702713	−0.351356	+	0.936242i	$0.614279\pi$
$180$	0	0
$181$	−13.1767	−0.979420	−0.489710	−	0.871885i	$-0.662897\pi$
$181$	−13.1767	−0.979420	−0.489710	+	0.871885i	$0.662897\pi$
$182$	0	0
$183$	21.3600	1.57898
$184$	0	0
$185$	5.29453	0.389261
$186$	0	0
$187$	0	0
$188$	0	0
$189$	15.4679	1.12513
$190$	0	0
$191$	7.35752	0.532372	0.266186	−	0.963922i	$-0.414236\pi$
$191$	7.35752	0.532372	0.266186	+	0.963922i	$0.414236\pi$
$192$	0	0
$193$	−18.4749	−1.32985	−0.664927	−	0.746908i	$-0.731537\pi$
$193$	−18.4749	−1.32985	−0.664927	+	0.746908i	$0.731537\pi$
$194$	0	0
$195$	17.8975	1.28166
$196$	0	0
$197$	9.58209	0.682696	0.341348	−	0.939937i	$-0.389117\pi$
$197$	9.58209	0.682696	0.341348	+	0.939937i	$0.389117\pi$
$198$	0	0
$199$	−13.3287	−0.944844	−0.472422	−	0.881372i	$-0.656620\pi$
$199$	−13.3287	−0.944844	−0.472422	+	0.881372i	$0.656620\pi$
$200$	0	0
$201$	22.2858	1.57192
$202$	0	0
$203$	−6.97059	−0.489240
$204$	0	0
$205$	3.28629	0.229524
$206$	0	0
$207$	−11.8427	−0.823122
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.0841	−0.694221	−0.347111	−	0.937824i	$-0.612837\pi$
$211$	−10.0841	−0.694221	−0.347111	+	0.937824i	$0.612837\pi$
$212$	0	0
$213$	−16.6785	−1.14279
$214$	0	0
$215$	6.83874	0.466398
$216$	0	0
$217$	−9.59303	−0.651217
$218$	0	0
$219$	−14.2221	−0.961044
$220$	0	0
$221$	43.9162	2.95412
$222$	0	0
$223$	−10.3500	−0.693084	−0.346542	−	0.938034i	$-0.612644\pi$
$223$	−10.3500	−0.693084	−0.346542	+	0.938034i	$0.612644\pi$
$224$	0	0
$225$	5.43902	0.362601
$226$	0	0
$227$	−10.9354	−0.725805	−0.362902	−	0.931827i	$-0.618214\pi$
$227$	−10.9354	−0.725805	−0.362902	+	0.931827i	$0.618214\pi$
$228$	0	0
$229$	22.3845	1.47921	0.739606	−	0.673040i	$-0.235012\pi$
$229$	22.3845	1.47921	0.739606	+	0.673040i	$0.235012\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.5657	−1.01974	−0.509871	−	0.860251i	$-0.670307\pi$
$233$	−15.5657	−1.01974	−0.509871	+	0.860251i	$0.670307\pi$
$234$	0	0
$235$	4.30445	0.280791
$236$	0	0
$237$	23.5714	1.53112
$238$	0	0
$239$	24.4294	1.58021	0.790103	−	0.612974i	$-0.210027\pi$
$239$	24.4294	1.58021	0.790103	+	0.612974i	$0.210027\pi$
$240$	0	0
$241$	−5.75661	−0.370816	−0.185408	−	0.982662i	$-0.559361\pi$
$241$	−5.75661	−0.370816	−0.185408	+	0.982662i	$0.559361\pi$
$242$	0	0
$243$	−8.86372	−0.568608
$244$	0	0
$245$	−2.23411	−0.142732
$246$	0	0
$247$	−39.1825	−2.49312
$248$	0	0
$249$	−12.2702	−0.777595
$250$	0	0
$251$	−12.0240	−0.758950	−0.379475	−	0.925202i	$-0.623895\pi$
$251$	−12.0240	−0.758950	−0.379475	+	0.925202i	$0.623895\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	20.7074	1.29675
$256$	0	0
$257$	−19.5364	−1.21865	−0.609324	−	0.792922i	$-0.708559\pi$
$257$	−19.5364	−1.21865	−0.609324	+	0.792922i	$0.708559\pi$
$258$	0	0
$259$	11.5584	0.718206
$260$	0	0
$261$	−17.3667	−1.07497
$262$	0	0
$263$	24.6213	1.51821	0.759106	−	0.650967i	$-0.225636\pi$
$263$	24.6213	1.51821	0.759106	+	0.650967i	$0.225636\pi$
$264$	0	0
$265$	−10.6160	−0.652135
$266$	0	0
$267$	48.8147	2.98741
$268$	0	0
$269$	1.54495	0.0941974	0.0470987	−	0.998890i	$-0.485002\pi$
$269$	1.54495	0.0941974	0.0470987	+	0.998890i	$0.485002\pi$
$270$	0	0
$271$	−22.5647	−1.37071	−0.685355	−	0.728209i	$-0.740353\pi$
$271$	−22.5647	−1.37071	−0.685355	+	0.728209i	$0.740353\pi$
$272$	0	0
$273$	39.0718	2.36473
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.8189	−0.830298	−0.415149	−	0.909754i	$-0.636270\pi$
$277$	−13.8189	−0.830298	−0.415149	+	0.909754i	$0.636270\pi$
$278$	0	0
$279$	−23.9003	−1.43088
$280$	0	0
$281$	−8.70719	−0.519427	−0.259714	−	0.965686i	$-0.583628\pi$
$281$	−8.70719	−0.519427	−0.259714	+	0.965686i	$0.583628\pi$
$282$	0	0
$283$	−1.52467	−0.0906325	−0.0453162	−	0.998973i	$-0.514430\pi$
$283$	−1.52467	−0.0906325	−0.0453162	+	0.998973i	$0.514430\pi$
$284$	0	0
$285$	−18.4753	−1.09438
$286$	0	0
$287$	7.17427	0.423484
$288$	0	0
$289$	33.8110	1.98888
$290$	0	0
$291$	−9.51569	−0.557820
$292$	0	0
$293$	23.7371	1.38674	0.693368	−	0.720584i	$-0.256126\pi$
$293$	23.7371	1.38674	0.693368	+	0.720584i	$0.256126\pi$
$294$	0	0
$295$	−13.7906	−0.802917
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.4145	−0.775781
$300$	0	0
$301$	14.9296	0.860527
$302$	0	0
$303$	−20.0749	−1.15327
$304$	0	0
$305$	7.35286	0.421024
$306$	0	0
$307$	17.7127	1.01092	0.505458	−	0.862851i	$-0.331324\pi$
$307$	17.7127	1.01092	0.505458	+	0.862851i	$0.331324\pi$
$308$	0	0
$309$	−22.0371	−1.25364
$310$	0	0
$311$	−6.78493	−0.384738	−0.192369	−	0.981323i	$-0.561617\pi$
$311$	−6.78493	−0.384738	−0.192369	+	0.981323i	$0.561617\pi$
$312$	0	0
$313$	23.8056	1.34557	0.672786	−	0.739838i	$-0.265098\pi$
$313$	23.8056	1.34557	0.672786	+	0.739838i	$0.265098\pi$
$314$	0	0
$315$	11.8739	0.669017
$316$	0	0
$317$	−1.84797	−0.103793	−0.0518963	−	0.998652i	$-0.516527\pi$
$317$	−1.84797	−0.103793	−0.0518963	+	0.998652i	$0.516527\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8.14836	−0.454797
$322$	0	0
$323$	−45.3341	−2.52245
$324$	0	0
$325$	6.16092	0.341747
$326$	0	0
$327$	23.3241	1.28983
$328$	0	0
$329$	9.39700	0.518073
$330$	0	0
$331$	−17.5654	−0.965484	−0.482742	−	0.875763i	$-0.660359\pi$
$331$	−17.5654	−0.965484	−0.482742	+	0.875763i	$0.660359\pi$
$332$	0	0
$333$	28.7970	1.57807
$334$	0	0
$335$	7.67154	0.419141
$336$	0	0
$337$	21.3539	1.16322	0.581610	−	0.813468i	$-0.302423\pi$
$337$	21.3539	1.16322	0.581610	+	0.813468i	$0.302423\pi$
$338$	0	0
$339$	20.2172	1.09805
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.1589	−1.08848
$344$	0	0
$345$	−6.32521	−0.340538
$346$	0	0
$347$	−5.36134	−0.287812	−0.143906	−	0.989591i	$-0.545966\pi$
$347$	−5.36134	−0.287812	−0.143906	+	0.989591i	$0.545966\pi$
$348$	0	0
$349$	−11.5164	−0.616456	−0.308228	−	0.951312i	$-0.599736\pi$
$349$	−11.5164	−0.616456	−0.308228	+	0.951312i	$0.599736\pi$
$350$	0	0
$351$	43.6523	2.32998
$352$	0	0
$353$	9.70208	0.516389	0.258195	−	0.966093i	$-0.416872\pi$
$353$	9.70208	0.516389	0.258195	+	0.966093i	$0.416872\pi$
$354$	0	0
$355$	−5.74133	−0.304718
$356$	0	0
$357$	45.2061	2.39256
$358$	0	0
$359$	32.5559	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$359$	32.5559	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$360$	0	0
$361$	21.4475	1.12882
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.89575	−0.256255
$366$	0	0
$367$	26.5031	1.38345	0.691725	−	0.722161i	$-0.256851\pi$
$367$	26.5031	1.38345	0.691725	+	0.722161i	$0.256851\pi$
$368$	0	0
$369$	17.8742	0.930493
$370$	0	0
$371$	−23.1757	−1.20322
$372$	0	0
$373$	25.2185	1.30577	0.652883	−	0.757459i	$-0.273559\pi$
$373$	25.2185	1.30577	0.652883	+	0.757459i	$0.273559\pi$
$374$	0	0
$375$	2.90500	0.150013
$376$	0	0
$377$	−19.6718	−1.01315
$378$	0	0
$379$	17.5353	0.900729	0.450364	−	0.892845i	$-0.351294\pi$
$379$	17.5353	0.900729	0.450364	+	0.892845i	$0.351294\pi$
$380$	0	0
$381$	30.1748	1.54590
$382$	0	0
$383$	13.2063	0.674811	0.337406	−	0.941359i	$-0.390451\pi$
$383$	13.2063	0.674811	0.337406	+	0.941359i	$0.390451\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	37.1960	1.89078
$388$	0	0
$389$	32.8461	1.66536	0.832681	−	0.553753i	$-0.186805\pi$
$389$	32.8461	1.66536	0.832681	+	0.553753i	$0.186805\pi$
$390$	0	0
$391$	−15.5206	−0.784909
$392$	0	0
$393$	−10.1272	−0.510850
$394$	0	0
$395$	8.11407	0.408263
$396$	0	0
$397$	25.2290	1.26621	0.633104	−	0.774066i	$-0.281780\pi$
$397$	25.2290	1.26621	0.633104	+	0.774066i	$0.281780\pi$
$398$	0	0
$399$	−40.3333	−2.01919
$400$	0	0
$401$	17.0748	0.852676	0.426338	−	0.904564i	$-0.359803\pi$
$401$	17.0748	0.852676	0.426338	+	0.904564i	$0.359803\pi$
$402$	0	0
$403$	−27.0726	−1.34858
$404$	0	0
$405$	4.26586	0.211972
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−0.798538	−0.0394851	−0.0197426	−	0.999805i	$-0.506285\pi$
$409$	−0.798538	−0.0394851	−0.0197426	+	0.999805i	$0.506285\pi$
$410$	0	0
$411$	−42.2593	−2.08450
$412$	0	0
$413$	−30.1060	−1.48142
$414$	0	0
$415$	−4.22384	−0.207340
$416$	0	0
$417$	1.45800	0.0713984
$418$	0	0
$419$	3.76575	0.183969	0.0919844	−	0.995760i	$-0.470679\pi$
$419$	3.76575	0.183969	0.0919844	+	0.995760i	$0.470679\pi$
$420$	0	0
$421$	−16.9537	−0.826272	−0.413136	−	0.910669i	$-0.635567\pi$
$421$	−16.9537	−0.826272	−0.413136	+	0.910669i	$0.635567\pi$
$422$	0	0
$423$	23.4120	1.13833
$424$	0	0
$425$	7.12818	0.345768
$426$	0	0
$427$	16.0520	0.776809
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.0432	−1.25446	−0.627228	−	0.778836i	$-0.715810\pi$
$431$	−26.0432	−1.25446	−0.627228	+	0.778836i	$0.715810\pi$
$432$	0	0
$433$	−22.2935	−1.07136	−0.535678	−	0.844422i	$-0.679944\pi$
$433$	−22.2935	−1.07136	−0.535678	+	0.844422i	$0.679944\pi$
$434$	0	0
$435$	−9.27563	−0.444733
$436$	0	0
$437$	13.8476	0.662421
$438$	0	0
$439$	−13.8298	−0.660060	−0.330030	−	0.943970i	$-0.607059\pi$
$439$	−13.8298	−0.660060	−0.330030	+	0.943970i	$0.607059\pi$
$440$	0	0
$441$	−12.1514	−0.578637
$442$	0	0
$443$	−15.0923	−0.717057	−0.358528	−	0.933519i	$-0.616721\pi$
$443$	−15.0923	−0.717057	−0.358528	+	0.933519i	$0.616721\pi$
$444$	0	0
$445$	16.8037	0.796571
$446$	0	0
$447$	2.17183	0.102724
$448$	0	0
$449$	17.6708	0.833936	0.416968	−	0.908921i	$-0.363093\pi$
$449$	17.6708	0.833936	0.416968	+	0.908921i	$0.363093\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9.50914	0.446778
$454$	0	0
$455$	13.4499	0.630539
$456$	0	0
$457$	−34.2329	−1.60135	−0.800674	−	0.599100i	$-0.795525\pi$
$457$	−34.2329	−1.60135	−0.800674	+	0.599100i	$0.795525\pi$
$458$	0	0
$459$	50.5056	2.35740
$460$	0	0
$461$	−10.1143	−0.471069	−0.235534	−	0.971866i	$-0.575684\pi$
$461$	−10.1143	−0.471069	−0.235534	+	0.971866i	$0.575684\pi$
$462$	0	0
$463$	11.2505	0.522857	0.261429	−	0.965223i	$-0.415806\pi$
$463$	11.2505	0.522857	0.261429	+	0.965223i	$0.415806\pi$
$464$	0	0
$465$	−12.7653	−0.591975
$466$	0	0
$467$	−34.3894	−1.59135	−0.795676	−	0.605722i	$-0.792884\pi$
$467$	−34.3894	−1.59135	−0.795676	+	0.605722i	$0.792884\pi$
$468$	0	0
$469$	16.7477	0.773336
$470$	0	0
$471$	−43.6610	−2.01179
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.35983	−0.291809
$476$	0	0
$477$	−57.7406	−2.64376
$478$	0	0
$479$	25.6708	1.17293	0.586465	−	0.809974i	$-0.300519\pi$
$479$	25.6708	1.17293	0.586465	+	0.809974i	$0.300519\pi$
$480$	0	0
$481$	32.6192	1.48731
$482$	0	0
$483$	−13.8085	−0.628309
$484$	0	0
$485$	−3.27563	−0.148739
$486$	0	0
$487$	19.3029	0.874698	0.437349	−	0.899292i	$-0.355917\pi$
$487$	19.3029	0.874698	0.437349	+	0.899292i	$0.355917\pi$
$488$	0	0
$489$	−42.0682	−1.90239
$490$	0	0
$491$	17.1628	0.774544	0.387272	−	0.921965i	$-0.373417\pi$
$491$	17.1628	0.774544	0.387272	+	0.921965i	$0.373417\pi$
$492$	0	0
$493$	−22.7602	−1.02507
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.5338	−0.562220
$498$	0	0
$499$	41.2377	1.84605	0.923027	−	0.384734i	$-0.125707\pi$
$499$	41.2377	1.84605	0.923027	+	0.384734i	$0.125707\pi$
$500$	0	0
$501$	0.885084	0.0395427
$502$	0	0
$503$	−26.7780	−1.19397	−0.596986	−	0.802252i	$-0.703635\pi$
$503$	−26.7780	−1.19397	−0.596986	+	0.802252i	$0.703635\pi$
$504$	0	0
$505$	−6.91048	−0.307512
$506$	0	0
$507$	72.5000	3.21984
$508$	0	0
$509$	−25.0638	−1.11093	−0.555467	−	0.831539i	$-0.687460\pi$
$509$	−25.0638	−1.11093	−0.555467	+	0.831539i	$0.687460\pi$
$510$	0	0
$511$	−10.6879	−0.472804
$512$	0	0
$513$	−45.0616	−1.98952
$514$	0	0
$515$	−7.58591	−0.334275
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−28.2176	−1.23861
$520$	0	0
$521$	0.959798	0.0420495	0.0210248	−	0.999779i	$-0.493307\pi$
$521$	0.959798	0.0420495	0.0210248	+	0.999779i	$0.493307\pi$
$522$	0	0
$523$	2.81911	0.123271	0.0616354	−	0.998099i	$-0.480368\pi$
$523$	2.81911	0.123271	0.0616354	+	0.998099i	$0.480368\pi$
$524$	0	0
$525$	6.34188	0.276782
$526$	0	0
$527$	−31.3229	−1.36445
$528$	0	0
$529$	−18.2591	−0.793875
$530$	0	0
$531$	−75.0071	−3.25503
$532$	0	0
$533$	20.2466	0.876977
$534$	0	0
$535$	−2.80495	−0.121268
$536$	0	0
$537$	−27.3118	−1.17859
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24.3158	−1.04542	−0.522709	−	0.852511i	$-0.675079\pi$
$541$	−24.3158	−1.04542	−0.522709	+	0.852511i	$0.675079\pi$
$542$	0	0
$543$	−38.2784	−1.64268
$544$	0	0
$545$	8.02896	0.343923
$546$	0	0
$547$	43.2096	1.84751	0.923754	−	0.382985i	$-0.125104\pi$
$547$	43.2096	1.84751	0.923754	+	0.382985i	$0.125104\pi$
$548$	0	0
$549$	39.9923	1.70683
$550$	0	0
$551$	20.3069	0.865103
$552$	0	0
$553$	17.7137	0.753265
$554$	0	0
$555$	15.3806	0.652870
$556$	0	0
$557$	8.39276	0.355613	0.177806	−	0.984065i	$-0.443100\pi$
$557$	8.39276	0.355613	0.177806	+	0.984065i	$0.443100\pi$
$558$	0	0
$559$	42.1330	1.78203
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.15829	0.259541	0.129771	−	0.991544i	$-0.458576\pi$
$563$	6.15829	0.259541	0.129771	+	0.991544i	$0.458576\pi$
$564$	0	0
$565$	6.95944	0.292786
$566$	0	0
$567$	9.31275	0.391099
$568$	0	0
$569$	4.08468	0.171239	0.0856194	−	0.996328i	$-0.472713\pi$
$569$	4.08468	0.171239	0.0856194	+	0.996328i	$0.472713\pi$
$570$	0	0
$571$	−30.8820	−1.29237	−0.646185	−	0.763181i	$-0.723637\pi$
$571$	−30.8820	−1.29237	−0.646185	+	0.763181i	$0.723637\pi$
$572$	0	0
$573$	21.3736	0.892895
$574$	0	0
$575$	−2.17735	−0.0908019
$576$	0	0
$577$	−14.4553	−0.601783	−0.300891	−	0.953658i	$-0.597284\pi$
$577$	−14.4553	−0.601783	−0.300891	+	0.953658i	$0.597284\pi$
$578$	0	0
$579$	−53.6696	−2.23043
$580$	0	0
$581$	−9.22102	−0.382552
$582$	0	0
$583$	0	0
$584$	0	0
$585$	33.5094	1.38544
$586$	0	0
$587$	−30.8993	−1.27535	−0.637676	−	0.770305i	$-0.720104\pi$
$587$	−30.8993	−1.27535	−0.637676	+	0.770305i	$0.720104\pi$
$588$	0	0
$589$	27.9466	1.15152
$590$	0	0
$591$	27.8360	1.14502
$592$	0	0
$593$	−29.6191	−1.21631	−0.608156	−	0.793818i	$-0.708090\pi$
$593$	−29.6191	−1.21631	−0.608156	+	0.793818i	$0.708090\pi$
$594$	0	0
$595$	15.5615	0.637958
$596$	0	0
$597$	−38.7198	−1.58469
$598$	0	0
$599$	6.78157	0.277087	0.138544	−	0.990356i	$-0.455758\pi$
$599$	6.78157	0.277087	0.138544	+	0.990356i	$0.455758\pi$
$600$	0	0
$601$	−17.7765	−0.725120	−0.362560	−	0.931961i	$-0.618097\pi$
$601$	−17.7765	−0.725120	−0.362560	+	0.931961i	$0.618097\pi$
$602$	0	0
$603$	41.7256	1.69920
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.8955	1.57872	0.789360	−	0.613931i	$-0.210413\pi$
$607$	38.8955	1.57872	0.789360	+	0.613931i	$0.210413\pi$
$608$	0	0
$609$	−20.2496	−0.820554
$610$	0	0
$611$	26.5194	1.07286
$612$	0	0
$613$	30.7554	1.24220	0.621100	−	0.783731i	$-0.286686\pi$
$613$	30.7554	1.24220	0.621100	+	0.783731i	$0.286686\pi$
$614$	0	0
$615$	9.54667	0.384959
$616$	0	0
$617$	23.3088	0.938377	0.469188	−	0.883098i	$-0.344547\pi$
$617$	23.3088	0.938377	0.469188	+	0.883098i	$0.344547\pi$
$618$	0	0
$619$	−34.6072	−1.39098	−0.695490	−	0.718535i	$-0.744813\pi$
$619$	−34.6072	−1.39098	−0.695490	+	0.718535i	$0.744813\pi$
$620$	0	0
$621$	−15.4273	−0.619076
$622$	0	0
$623$	36.6840	1.46971
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.7404	1.50481
$630$	0	0
$631$	−11.1871	−0.445352	−0.222676	−	0.974893i	$-0.571479\pi$
$631$	−11.1871	−0.445352	−0.222676	+	0.974893i	$0.571479\pi$
$632$	0	0
$633$	−29.2944	−1.16435
$634$	0	0
$635$	10.3872	0.412204
$636$	0	0
$637$	−13.7642	−0.545357
$638$	0	0
$639$	−31.2272	−1.23533
$640$	0	0
$641$	34.7930	1.37424	0.687121	−	0.726543i	$-0.258874\pi$
$641$	34.7930	1.37424	0.687121	+	0.726543i	$0.258874\pi$
$642$	0	0
$643$	−24.4540	−0.964370	−0.482185	−	0.876069i	$-0.660157\pi$
$643$	−24.4540	−0.964370	−0.482185	+	0.876069i	$0.660157\pi$
$644$	0	0
$645$	19.8665	0.782244
$646$	0	0
$647$	−18.7808	−0.738349	−0.369175	−	0.929360i	$-0.620360\pi$
$647$	−18.7808	−0.738349	−0.369175	+	0.929360i	$0.620360\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−27.8677	−1.09222
$652$	0	0
$653$	39.9031	1.56153	0.780765	−	0.624825i	$-0.214830\pi$
$653$	39.9031	1.56153	0.780765	+	0.624825i	$0.214830\pi$
$654$	0	0
$655$	−3.48613	−0.136214
$656$	0	0
$657$	−26.6281	−1.03886
$658$	0	0
$659$	−17.6187	−0.686326	−0.343163	−	0.939276i	$-0.611498\pi$
$659$	−17.6187	−0.686326	−0.343163	+	0.939276i	$0.611498\pi$
$660$	0	0
$661$	42.4404	1.65074	0.825371	−	0.564591i	$-0.190966\pi$
$661$	42.4404	1.65074	0.825371	+	0.564591i	$0.190966\pi$
$662$	0	0
$663$	127.576	4.95466
$664$	0	0
$665$	−13.8841	−0.538402
$666$	0	0
$667$	6.95227	0.269193
$668$	0	0
$669$	−30.0666	−1.16244
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.08633	0.234611	0.117305	−	0.993096i	$-0.462574\pi$
$673$	6.08633	0.234611	0.117305	+	0.993096i	$0.462574\pi$
$674$	0	0
$675$	7.08534	0.272715
$676$	0	0
$677$	−10.4432	−0.401366	−0.200683	−	0.979656i	$-0.564316\pi$
$677$	−10.4432	−0.401366	−0.200683	+	0.979656i	$0.564316\pi$
$678$	0	0
$679$	−7.15099	−0.274430
$680$	0	0
$681$	−31.7672	−1.21732
$682$	0	0
$683$	−28.7312	−1.09937	−0.549685	−	0.835372i	$-0.685252\pi$
$683$	−28.7312	−1.09937	−0.549685	+	0.835372i	$0.685252\pi$
$684$	0	0
$685$	−14.5471	−0.555816
$686$	0	0
$687$	65.0271	2.48094
$688$	0	0
$689$	−65.4043	−2.49171
$690$	0	0
$691$	28.0822	1.06830	0.534149	−	0.845390i	$-0.320632\pi$
$691$	28.0822	1.06830	0.534149	+	0.845390i	$0.320632\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.501893	0.0190379
$696$	0	0
$697$	23.4253	0.887296
$698$	0	0
$699$	−45.2183	−1.71031
$700$	0	0
$701$	12.0683	0.455815	0.227908	−	0.973683i	$-0.426812\pi$
$701$	12.0683	0.455815	0.227908	+	0.973683i	$0.426812\pi$
$702$	0	0
$703$	−33.6723	−1.26998
$704$	0	0
$705$	12.5044	0.470943
$706$	0	0
$707$	−15.0862	−0.567375
$708$	0	0
$709$	31.9154	1.19861	0.599304	−	0.800521i	$-0.295444\pi$
$709$	31.9154	1.19861	0.599304	+	0.800521i	$0.295444\pi$
$710$	0	0
$711$	44.1325	1.65510
$712$	0	0
$713$	9.56781	0.358317
$714$	0	0
$715$	0	0
$716$	0	0
$717$	70.9674	2.65032
$718$	0	0
$719$	2.65067	0.0988531	0.0494266	−	0.998778i	$-0.484261\pi$
$719$	2.65067	0.0988531	0.0494266	+	0.998778i	$0.484261\pi$
$720$	0	0
$721$	−16.5607	−0.616754
$722$	0	0
$723$	−16.7229	−0.621933
$724$	0	0
$725$	−3.19299	−0.118585
$726$	0	0
$727$	1.73626	0.0643944	0.0321972	−	0.999482i	$-0.489750\pi$
$727$	1.73626	0.0643944	0.0321972	+	0.999482i	$0.489750\pi$
$728$	0	0
$729$	−38.5467	−1.42765
$730$	0	0
$731$	48.7478	1.80300
$732$	0	0
$733$	33.3955	1.23349	0.616745	−	0.787163i	$-0.288451\pi$
$733$	33.3955	1.23349	0.616745	+	0.787163i	$0.288451\pi$
$734$	0	0
$735$	−6.49009	−0.239391
$736$	0	0
$737$	0	0
$738$	0	0
$739$	41.0369	1.50957	0.754784	−	0.655973i	$-0.227742\pi$
$739$	41.0369	1.50957	0.754784	+	0.655973i	$0.227742\pi$
$740$	0	0
$741$	−113.825	−4.18147
$742$	0	0
$743$	−41.8776	−1.53634	−0.768170	−	0.640246i	$-0.778833\pi$
$743$	−41.8776	−1.53634	−0.768170	+	0.640246i	$0.778833\pi$
$744$	0	0
$745$	0.747620	0.0273907
$746$	0	0
$747$	−22.9735	−0.840557
$748$	0	0
$749$	−6.12345	−0.223746
$750$	0	0
$751$	18.6398	0.680177	0.340088	−	0.940394i	$-0.389543\pi$
$751$	18.6398	0.680177	0.340088	+	0.940394i	$0.389543\pi$
$752$	0	0
$753$	−34.9298	−1.27291
$754$	0	0
$755$	3.27337	0.119130
$756$	0	0
$757$	−14.2930	−0.519488	−0.259744	−	0.965678i	$-0.583638\pi$
$757$	−14.2930	−0.519488	−0.259744	+	0.965678i	$0.583638\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.7696	−0.571649	−0.285825	−	0.958282i	$-0.592267\pi$
$761$	−15.7696	−0.571649	−0.285825	+	0.958282i	$0.592267\pi$
$762$	0	0
$763$	17.5279	0.634554
$764$	0	0
$765$	38.7703	1.40174
$766$	0	0
$767$	−84.9626	−3.06782
$768$	0	0
$769$	−16.5268	−0.595971	−0.297986	−	0.954570i	$-0.596315\pi$
$769$	−16.5268	−0.595971	−0.297986	+	0.954570i	$0.596315\pi$
$770$	0	0
$771$	−56.7532	−2.04392
$772$	0	0
$773$	39.0636	1.40502	0.702511	−	0.711673i	$-0.252062\pi$
$773$	39.0636	1.40502	0.702511	+	0.711673i	$0.252062\pi$
$774$	0	0
$775$	−4.39424	−0.157846
$776$	0	0
$777$	33.5772	1.20458
$778$	0	0
$779$	−20.9003	−0.748830
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−22.6234	−0.808496
$784$	0	0
$785$	−15.0296	−0.536430
$786$	0	0
$787$	−4.93078	−0.175763	−0.0878816	−	0.996131i	$-0.528010\pi$
$787$	−4.93078	−0.175763	−0.0878816	+	0.996131i	$0.528010\pi$
$788$	0	0
$789$	71.5247	2.54635
$790$	0	0
$791$	15.1931	0.540204
$792$	0	0
$793$	45.3004	1.60866
$794$	0	0
$795$	−30.8395	−1.09376
$796$	0	0
$797$	−50.4039	−1.78540	−0.892700	−	0.450652i	$-0.851191\pi$
$797$	−50.4039	−1.78540	−0.892700	+	0.450652i	$0.851191\pi$
$798$	0	0
$799$	30.6829	1.08548
$800$	0	0
$801$	91.3955	3.22930
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.75336	−0.167534
$806$	0	0
$807$	4.48808	0.157988
$808$	0	0
$809$	−54.5875	−1.91920	−0.959598	−	0.281375i	$-0.909210\pi$
$809$	−54.5875	−1.91920	−0.959598	+	0.281375i	$0.909210\pi$
$810$	0	0
$811$	−13.6473	−0.479223	−0.239611	−	0.970869i	$-0.577020\pi$
$811$	−13.6473	−0.479223	−0.239611	+	0.970869i	$0.577020\pi$
$812$	0	0
$813$	−65.5505	−2.29896
$814$	0	0
$815$	−14.4813	−0.507258
$816$	0	0
$817$	−43.4932	−1.52164
$818$	0	0
$819$	73.1540	2.55621
$820$	0	0
$821$	−37.3692	−1.30419	−0.652097	−	0.758136i	$-0.726111\pi$
$821$	−37.3692	−1.30419	−0.652097	+	0.758136i	$0.726111\pi$
$822$	0	0
$823$	27.5161	0.959150	0.479575	−	0.877501i	$-0.340791\pi$
$823$	27.5161	0.959150	0.479575	+	0.877501i	$0.340791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.9552	1.35460	0.677302	−	0.735705i	$-0.263149\pi$
$827$	38.9552	1.35460	0.677302	+	0.735705i	$0.263149\pi$
$828$	0	0
$829$	8.01799	0.278476	0.139238	−	0.990259i	$-0.455535\pi$
$829$	8.01799	0.278476	0.139238	+	0.990259i	$0.455535\pi$
$830$	0	0
$831$	−40.1439	−1.39258
$832$	0	0
$833$	−15.9252	−0.551774
$834$	0	0
$835$	0.304676	0.0105438
$836$	0	0
$837$	−31.1347	−1.07617
$838$	0	0
$839$	19.8626	0.685733	0.342866	−	0.939384i	$-0.388602\pi$
$839$	19.8626	0.685733	0.342866	+	0.939384i	$0.388602\pi$
$840$	0	0
$841$	−18.8048	−0.648442
$842$	0	0
$843$	−25.2944	−0.871185
$844$	0	0
$845$	24.9570	0.858546
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.42918	−0.152009
$850$	0	0
$851$	−11.5281	−0.395177
$852$	0	0
$853$	24.5518	0.840637	0.420318	−	0.907377i	$-0.361918\pi$
$853$	24.5518	0.840637	0.420318	+	0.907377i	$0.361918\pi$
$854$	0	0
$855$	−34.5912	−1.18300
$856$	0	0
$857$	9.17203	0.313311	0.156655	−	0.987653i	$-0.449929\pi$
$857$	9.17203	0.313311	0.156655	+	0.987653i	$0.449929\pi$
$858$	0	0
$859$	−2.83353	−0.0966786	−0.0483393	−	0.998831i	$-0.515393\pi$
$859$	−2.83353	−0.0966786	−0.0483393	+	0.998831i	$0.515393\pi$
$860$	0	0
$861$	20.8413	0.710268
$862$	0	0
$863$	−40.7452	−1.38698	−0.693491	−	0.720466i	$-0.743928\pi$
$863$	−40.7452	−1.38698	−0.693491	+	0.720466i	$0.743928\pi$
$864$	0	0
$865$	−9.71345	−0.330267
$866$	0	0
$867$	98.2209	3.33576
$868$	0	0
$869$	0	0
$870$	0	0
$871$	47.2638	1.60147
$872$	0	0
$873$	−17.8162	−0.602987
$874$	0	0
$875$	2.18309	0.0738020
$876$	0	0
$877$	−2.84489	−0.0960651	−0.0480325	−	0.998846i	$-0.515295\pi$
$877$	−2.84489	−0.0960651	−0.0480325	+	0.998846i	$0.515295\pi$
$878$	0	0
$879$	68.9562	2.32584
$880$	0	0
$881$	−25.7884	−0.868834	−0.434417	−	0.900712i	$-0.643046\pi$
$881$	−25.7884	−0.868834	−0.434417	+	0.900712i	$0.643046\pi$
$882$	0	0
$883$	−8.49346	−0.285828	−0.142914	−	0.989735i	$-0.545647\pi$
$883$	−8.49346	−0.285828	−0.142914	+	0.989735i	$0.545647\pi$
$884$	0	0
$885$	−40.0615	−1.34665
$886$	0	0
$887$	−25.6413	−0.860951	−0.430475	−	0.902602i	$-0.641654\pi$
$887$	−25.6413	−0.860951	−0.430475	+	0.902602i	$0.641654\pi$
$888$	0	0
$889$	22.6762	0.760537
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.3756	−0.916088
$894$	0	0
$895$	−9.40166	−0.314263
$896$	0	0
$897$	−38.9691	−1.30114
$898$	0	0
$899$	14.0308	0.467952
$900$	0	0
$901$	−75.6728	−2.52102
$902$	0	0
$903$	43.3704	1.44328
$904$	0	0
$905$	−13.1767	−0.438010
$906$	0	0
$907$	23.6343	0.784763	0.392382	−	0.919802i	$-0.371651\pi$
$907$	23.6343	0.784763	0.392382	+	0.919802i	$0.371651\pi$
$908$	0	0
$909$	−37.5862	−1.24666
$910$	0	0
$911$	47.3905	1.57012	0.785058	−	0.619422i	$-0.212633\pi$
$911$	47.3905	1.57012	0.785058	+	0.619422i	$0.212633\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	21.3600	0.706141
$916$	0	0
$917$	−7.61055	−0.251322
$918$	0	0
$919$	39.9139	1.31664	0.658319	−	0.752739i	$-0.271268\pi$
$919$	39.9139	1.31664	0.658319	+	0.752739i	$0.271268\pi$
$920$	0	0
$921$	51.4553	1.69551
$922$	0	0
$923$	−35.3719	−1.16428
$924$	0	0
$925$	5.29453	0.174083
$926$	0	0
$927$	−41.2599	−1.35515
$928$	0	0
$929$	−10.7852	−0.353851	−0.176926	−	0.984224i	$-0.556615\pi$
$929$	−10.7852	−0.353851	−0.176926	+	0.984224i	$0.556615\pi$
$930$	0	0
$931$	14.2086	0.465667
$932$	0	0
$933$	−19.7102	−0.645284
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.8842	−1.23762	−0.618812	−	0.785539i	$-0.712386\pi$
$937$	−37.8842	−1.23762	−0.618812	+	0.785539i	$0.712386\pi$
$938$	0	0
$939$	69.1552	2.25679
$940$	0	0
$941$	1.18812	0.0387317	0.0193659	−	0.999812i	$-0.493835\pi$
$941$	1.18812	0.0387317	0.0193659	+	0.999812i	$0.493835\pi$
$942$	0	0
$943$	−7.15542	−0.233012
$944$	0	0
$945$	15.4679	0.503172
$946$	0	0
$947$	−4.75475	−0.154508	−0.0772542	−	0.997011i	$-0.524615\pi$
$947$	−4.75475	−0.154508	−0.0772542	+	0.997011i	$0.524615\pi$
$948$	0	0
$949$	−30.1623	−0.979111
$950$	0	0
$951$	−5.36836	−0.174081
$952$	0	0
$953$	1.92362	0.0623120	0.0311560	−	0.999515i	$-0.490081\pi$
$953$	1.92362	0.0623120	0.0311560	+	0.999515i	$0.490081\pi$
$954$	0	0
$955$	7.35752	0.238084
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.7577	−1.02551
$960$	0	0
$961$	−11.6907	−0.377118
$962$	0	0
$963$	−15.2561	−0.491622
$964$	0	0
$965$	−18.4749	−0.594729
$966$	0	0
$967$	14.4688	0.465284	0.232642	−	0.972562i	$-0.425263\pi$
$967$	14.4688	0.465284	0.232642	+	0.972562i	$0.425263\pi$
$968$	0	0
$969$	−131.695	−4.23067
$970$	0	0
$971$	13.7317	0.440673	0.220336	−	0.975424i	$-0.429285\pi$
$971$	13.7317	0.440673	0.220336	+	0.975424i	$0.429285\pi$
$972$	0	0
$973$	1.09568	0.0351258
$974$	0	0
$975$	17.8975	0.573178
$976$	0	0
$977$	−43.0472	−1.37720	−0.688602	−	0.725140i	$-0.741775\pi$
$977$	−43.0472	−1.37720	−0.688602	+	0.725140i	$0.741775\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	43.6696	1.39426
$982$	0	0
$983$	−20.8242	−0.664190	−0.332095	−	0.943246i	$-0.607755\pi$
$983$	−20.8242	−0.664190	−0.332095	+	0.943246i	$0.607755\pi$
$984$	0	0
$985$	9.58209	0.305311
$986$	0	0
$987$	27.2983	0.868914
$988$	0	0
$989$	−14.8904	−0.473486
$990$	0	0
$991$	−42.9441	−1.36417	−0.682083	−	0.731275i	$-0.738926\pi$
$991$	−42.9441	−1.36417	−0.682083	+	0.731275i	$0.738926\pi$
$992$	0	0
$993$	−51.0276	−1.61931
$994$	0	0
$995$	−13.3287	−0.422547
$996$	0	0
$997$	−29.8397	−0.945032	−0.472516	−	0.881322i	$-0.656654\pi$
$997$	−29.8397	−0.945032	−0.472516	+	0.881322i	$0.656654\pi$
$998$	0	0
$999$	37.5135	1.18688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.df.1.8		8
4.3	odd	2		4840.2.a.bg.1.1		8
11.3	even	5		880.2.bo.k.801.1		16
11.4	even	5		880.2.bo.k.401.1		16
11.10	odd	2		9680.2.a.de.1.8		8
44.3	odd	10		440.2.y.d.361.4	✓	16
44.15	odd	10		440.2.y.d.401.4	yes	16
44.43	even	2		4840.2.a.bh.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.361.4	✓	16	44.3	odd	10
440.2.y.d.401.4	yes	16	44.15	odd	10
880.2.bo.k.401.1		16	11.4	even	5
880.2.bo.k.801.1		16	11.3	even	5
4840.2.a.bg.1.1		8	4.3	odd	2
4840.2.a.bh.1.1		8	44.43	even	2
9680.2.a.de.1.8		8	11.10	odd	2
9680.2.a.df.1.8		8	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)

\(f(q)\)	\(=\)	\(q+2.90500 q^{3} +1.00000 q^{5} +2.18309 q^{7} +5.43902 q^{9} +6.16092 q^{13} +2.90500 q^{15} +7.12818 q^{17} -6.35983 q^{19} +6.34188 q^{21} -2.17735 q^{23} +1.00000 q^{25} +7.08534 q^{27} -3.19299 q^{29} -4.39424 q^{31} +2.18309 q^{35} +5.29453 q^{37} +17.8975 q^{39} +3.28629 q^{41} +6.83874 q^{43} +5.43902 q^{45} +4.30445 q^{47} -2.23411 q^{49} +20.7074 q^{51} -10.6160 q^{53} -18.4753 q^{57} -13.7906 q^{59} +7.35286 q^{61} +11.8739 q^{63} +6.16092 q^{65} +7.67154 q^{67} -6.32521 q^{69} -5.74133 q^{71} -4.89575 q^{73} +2.90500 q^{75} +8.11407 q^{79} +4.26586 q^{81} -4.22384 q^{83} +7.12818 q^{85} -9.27563 q^{87} +16.8037 q^{89} +13.4499 q^{91} -12.7653 q^{93} -6.35983 q^{95} -3.27563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} + 12 q^{13} - q^{15} + 2 q^{17} + 6 q^{19} + 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} + 8 q^{29} - 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 8 * q - q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 + 12 * q^13 - q^15 + 2 * q^17 + 6 * q^19 + 6 * q^21 - 10 * q^23 + 8 * q^25 - 13 * q^27 + 8 * q^29 - 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 - 3 * q^41 + 8 * q^43 + 19 * q^45 - 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 + 25 * q^57 - 25 * q^59 + 10 * q^61 + 64 * q^63 + 12 * q^65 + 2 * q^67 + 18 * q^69 - 25 * q^71 + 38 * q^73 - q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 + 2 * q^85 - 15 * q^87 + 12 * q^89 - 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more